Automatic Control

International Review of

Automatic Control (IREACO) Theory and Applications

Contents Primary-Side Control for Series-Parallel Loosely Coupling Chargers by Yuan-Hsin Chao, Jenn-Jong Shieh

728

Voltage and Power Regulation for a Sample Power System Using Heuristics Population Search Based PID Design by Ahmed Bensenouci, A. H. Besheer

737

Transient Stability Enhancement Using SMES-Based Fault Current Limiter by Mehdi Jafari, Ali Vafamehr, Mohammad Reza Alizadeh Pahlavani

749

Dynamic and Steady-State Operational Performance of Induction and Synchronous Reluctance Motors Powered by PV Generator with MPPT by Mohammad S. Widyan, Ghassan S. Marji, Anas I. Al Tarabsheh

757

Performance Analysis of New Three Phase Seven Level Asymmetrical Inverter with Hybrid Carrier and Sine 60 Degree Reference by Johnson Uthayakumar R., Natarajan S. P., Bensraj R.

769

Parity Space Approach Based DC Motor’s Fault Detection and Isolation by A. Adouni, M. Ben Hamed, L. Sbita

776

Novel Adaptive Control Scheme for Suppressing Input Current Harmonics in Three-Phase AC Choppers by T. Suresh Padmanabhan, M. Sudhakaran, S. Jeevananthan

783

Decentralized Adaptive Sliding Mode Exciter Control of Power Systems by S. Benahdouga, D. Boukhetala, F. Boudjema, R. Khenfer, M. Meddad

790

A Novel Control Strategy for Performance Enhancement of Unified Power Quality Conditioner by Nikita Hari, K. Vijayakumar, S. S. Dash

798

Induction Machine Speed and Flux Control, Using Vector-Sliding Mode Control, with Rotor Resistance Adaptation by M. Moutchou, A. Abbou, H. Mahmoudi

804

Comparison of PWM Control Techniques for Cascaded Multilevel Inverter by R. Nagarajan, M. Saravanan

815

Overview Control Strategies of Series, Shunt, Series/Shunt FACTS Devices for Three Stability Functions of Power System by S. F. Taghizadeh, Younis M. A. A., B. Nikouei, Nadia M. L. Tan, Mekhilef S.

829

Controller Synthesis Using the Novel Fuzzy Petri Net by Mostafa Bayati, Abbas Dideban

839

(continued)

Stability Enhancement of HVDC System Using PI Based STATCOM by M. Ramesh, A. Jaya Laxmi

844

Parameter Identification of Permanent Magnet Linear Synchronous Motor Using Hartley Modulating Functions by B. Arundhati, K. Alice Mary, M. Suryakalavathi

854

Robust Design and Efficiency in Case of Parameters Uncertainties, Disturbances and Noise by K. M. Yanev, S. Masupe

860

Numerical Integration Method for Singularly Perturbed Differential-Difference Equations by Gemechis File, Y. N. Reddy

868

Single Parametric Control of Cascade Brushless DC Motor Drive by Ibrahim Al-Abbas, Mohammad Al Kawaldah, Mohammad Al-Khedher, Rateb Issa

877

Parameter Identification of a DC Motor Via Distribution-Based Approach by Dorin Sendrescu

882

Differential Equations of Synchronous Generators Dynamics for Online Assessment within Energy Control Centers by Lucian Lupsa-Tataru

892

Composite Sliding Mode Control of Induction Motors Using Singular Perturbation Theory by A. Mezouar, T. Terras, M. K. Fellah, S. Hadjeri

901

Robust Control Design for a Semi-Batch Reactor by František Gazdoš

911

Effective Detection, Identification and Measurement Strategies of Market Power and their Comparison by R. Esmaeilzadeh, H. Eskandari, M. Amjadi, M. Farrokhifar

921

Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control (I.RE.A.CO.), Vol. 5, N. 6 ISSN 1974-6059 November 2012

Primary-Side Control for Series-Parallel Loosely Coupling Chargers Yuan-Hsin Chao1, Jenn-Jong Shieh2

Abstract – This paper proposes a primary-side control for a series-parallel loosely coupling charger (SPLCC). First, the optimal values of the compensated capacitors for the series-parallel loosely systems are determined by considering the maximum output power criterion. . Furthermore, in order to impose the operation constraint of minimum volt-ampere (VA) rating of the adapted full bridge inverter, the analytical solutions for the multiple operating frequencies as well as the corresponding feasible conditions are then derived.For enhancing the practical applications, additionally, not only six operation modes but also an integral-type phase luck loop (IPLL) to control secondary output power using primary-side control are further proposed. Finally, two 24V 4.5 Ah lead-acid batteries with the proposed SPLCC is employed with some simulation and experimental results to verify the feasibility of the proposed theory. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Primary-Side Control, Inductive Power Transfer Systems, Maximum Output Power, Minimum Volt-Ampere Rating, Charger, Phase Luck Loop

I.

Nomenclature L f , L1 , L2

Inductors

Cin , C f , C1 , C2

Capacitors

Rac , Rdc

Resistor Diodes Switches Input DC voltage Output DC current Maximum output power Operation frequency Coupling coefficient of separable transformer Input impedance Mutual inductance The input voltage and current of the transformer The output voltage and current of the transformer The zero crossing point of voltage The output voltage of the S/H The output voltage of the integrator The output voltage of the error amplifier Switching frequency

D1 , D2 , D3 , D4 S1 , S 2 , S3 , S 4 Vin IO PO

ωL , ωO , ωH k Z in M vp , ip vs , is vtri vip

vir vif fs

Introduction

Loosely coupled inductive power systems have been widely used for power transmission between separate items of equipment without power lines [1]-[4]. The advantage of this system is that it avoids the risk of sparking or electrical shocks. Furthermore, without contact loss, the life time of the equipments can be further extended. Because of the above-mentioned advantages, it can be used to supply power in airtight instruments [5] or can be the power supply system for humid environment situations, mining, oil drilling or electrical vehicle which is prone to cause electrical shocks [6]. A typical SPLCC combines a separable high frequency transformer with a resonant inverter and a controlled rectifier to deliver power from the source to the load, as shown in Fig. 1. In this configuration, there is some switching loss for the resonant inverter. In addition, the leakage inductance of the loosely coupled contactless transformer is very high. As a result, the power transfer efficiency of the LCIPS is usually very low. To solve this problem, the soft-switched technique was proposed [7]-[9]. By using zero-voltage switching (ZVS) or zero-current switching (ZCS) technology to reduce the switching loss of the resonant inverter switch, the total efficiency of the LCIPTS system is enhanced. However, the enhancement of system efficiency is quite limited due to poor coupling of the separable transformer. On the other hand, the impedance-matching approach to transfer the energy stored in the leakage inductance has been studied and proved to be more effective [10]-[13]. This approach includes a series or parallel capacitor at the primary and/or the secondary side of the transformer

Manuscript received and revised October 2012, accepted November 2012

728


Yuan-Hsin Chao, Jenn-Jong Shieh

to directly solve the poor coupling problem of the separable transformer. Furthermore, in order to get the maximum output power with the minimum input volt-ampere (VA) rating, and to reduce the cost of the resonant inverter, the input side of the transformer must operate at the zero-phase-angle (ZPA) frequency [10]. Unfortunately, as the system load changes, the corresponding ZPA frequency will also be changed and there may be multiple solutions. Also, the bifurcation frequencies are related to the system load and the magnetic coupling coefficient and the corresponding output power of each bifurcation frequency will be different [10]. Therefore, to design and control the system well will be quite difficult. Up to the present time, a common way for controlling the system is to use a phase luck loop (PLL) [14]-[15] in the resonant inverter to achieve ZPA frequency, and to employ pulse width modulation (PWM) technique in the controlled rectifier of the output side to control the output voltage or current [7]. However, application of two controllers on both input and output sides will not only increase the cost but also render the system less reliable. ip + Vin −

is

+

+

vp

vs

−

−

For better understanding and clearer explanation of the operation analysis of the SPLCC, we assume all elements in the system are ideal and the input voltage and current are sinusoidal. A circuit diagram of the SPLCC is shown in Fig. 2, where the Rac denotes the battery equivalent resistance load as seen by the controlled rectifier in Fig. 1; L1 , L2 and M are the primary side self inductance, the secondary side self inductance, and the mutual inductance of the separable transformer respectively. JJG IP

+ JJG VP

1 jω C 1

jω L1

jω L 2

1 jω C 2

−

+ JJG VS −

Rac

Z in

Fig. 2. A circuit diagram of the SPLCC

First, from Fig. 2, the input impedance Z in can be derived as:

IO + VO −

Z in =

(ω M )2 R2 + 2 R2 2 + (ω L2 − X 2 )

⎡ (ω M )2 (ω L2 − X 2 ) ⎤⎥ 1 − + j ⎢ω L1 − ωC1 R2 2 + (ω L2 − X 2 )2 ⎥ ⎢⎣ ⎦ Rin + jX in

Fig. 1. A typical block diagram of the SPLCC

In view of the above, a primary-side control for a series-parallel loosely coupling charger (SPLCC) is proposed to impose the operation constraint of the minimum VA rating of the inverter. Six operation modes are developed which enable the implementation of the proposed primary-side control of the secondary output power and also an integral-type phase luck loop is proposed to minimize the input VA rating of the inverter. Furthermore, we have presented the use of the SPLCC in some simulations, and experimental results verify the effectiveness of the proposed analytical derivations.

II.

JJG IS

jω M

(1)

where: Rac

R2 =

1 + ( RacωC2 )

X2 =

(2)

2

Rac 2ωC2 1 + ( RacωC2 )

(3)

2

Three operation frequencies ωL , ωO and ωH is operated at minimum VA rating with optimum output power [11]: 1 ωO = (4) L2C2

Optimal Compensation and Operation Constraints

Generally, having a capacitor in series or in parallel with the primary side and/or the secondary side improve power transfer capability quite effectively. However, while the system input is a voltage source, a capacitor in series can enhance the voltage capacity, and, while the system output is a current source, a capacitor in parallel can enhance the current capacity. The system architecture with a capacitor in series with the primary side and a capacitor in parallel with the secondary side will be even more suitable for chargers with a voltage source as the system input.

ωL =

ωH =


α1 ⋅

(

β 2 − β1

(

)

2 ⋅ L2 ⋅ C2 ⋅ Rac ⋅ 1 − k 2

α1 ⋅

(

β 2 + β1

(

)

2 ⋅ L2 ⋅ C2 ⋅ Rac ⋅ 1 − k 2

)

(5)

)

(6)

International Review of Automatic Control, Vol. 5, N. 6

729


where:

(

α1 L2 ⋅ 1 − k

2

)≥0

From the above results, it is seen that under the condition of the optimized parameters in the SPLCC, there are three operation frequencies ωL , ωO and ωH , which can be operated to achieve the minimum VA rating. ωL and ωH are dependent on the resistance load Rac and the magnetic coupling coefficient k , but ωO is a constant. From all descriptions above, as shown in Fig. 3, it should be noticed that the output power at ωO is a local minimum as the system input is a voltage source, and output power at ωO will become a local maximum if the input is a current source. Also, from Fig. 3, one can observe that there are three solutions, namely ω1 , ωO and ω2 if the input is a voltage source. Although ωL and ωH are not local maxima under these conditions from the point of view of practical considerations, the unit power factor frequencies would be more attractive [11].

(7)

(

)

β1 L2 ⋅ k 2 − L2 + C2 ⋅ Rac 2 ⋅ k 2 ≥ 0

(

(8)

)

β 2 L2 ⋅ k 2 − L2 + 4 ⋅ C2 ⋅ Rac 2 − 3 ⋅ C2 ⋅ Rac 2 ⋅ k 2 ≥ 0 (9) M

k

(10)

L1 ⋅ L2

From (5) and (6), one can see that the conditions for the

ωL and ωH having positive real solutions are α1 ≥ 0 and β 2 ≥ β1 ≥ 0 . Hence, one can directly use (7)-(10) to obtain the following operation range of the resistance load:

(

L2 ⋅ 1 − k 2

Rac ≥

)

Ps

C2 ⋅ k 2

θ

Ps,I

(11) ω2

ω1

Ps,v θ

Additionally, the output powers based on the corresponding different operation frequencies ωL , ωO and ωH can be obtained by following equations:

PS ,V (ωL ) =

PS ,I (ωL ) =

JJJG 2 VP ⋅ C2 ⋅ Rac

(

L1 ⋅ 1 − k 2

JJG 2 I P ⋅ L1 ⋅ 1 − k 2

(

)

C2 ⋅ Rac

PS ,V (ωO ) =

PS ,I (ωO ) =

(12a)

)

JJJG 2 VP ⋅ L2 k 2 Rac ⋅ L1

JJG 2 I P ⋅ k 2 Rac ⋅ L1 L2

PS (ωH ) = PS (ωL )

ωL

(12b)

ω

We assume the parameter values in the system are optimized, so the circuit parameters L1 , L2 , C1 , C2 and k are constant, and only Rac is a variable that can be applied in general engineering application. One can know that as the system input is a constant voltage source (CVS), the characteristic of system output at operation frequency ωL and ωH is a constant current source (CCS), and at operation frequency ωO is a CVS. On the other hand, as the system input is a CCS, the characteristic of system output at operation frequency ωL and ωH is a CVS, and at operation frequency ωO is a CCS. For completeness, the circuit characteristics and mathematic relations of the six operation modes are summarized in Table I. According to the discussion in this section, when the system operates at ωL , ωO and ωH , the minimum input VA rating can be satisfied. While operated at above-mentioned operation frequencies and with a CVS/CCS input, the characteristic of system output will be a CVS or CCS that is independent of output resistor Rac .

(13a)

(13b)

(14)

(15)

On the other hand, when the system input is a current source one has: PS ,I (ωO ) ≥ PS ,I (ωH ) = PS ,I (ωL )

ωH

Fig. 3. Output power PS vs. frequency ω

Substituting (11) into (12)-(14), when the system input is a voltage source one has the following consequence: PS ,V (ωH ) = PS ,V (ωL ) ≥ PS ,V (ωO )

ωo

(16)



730


TABLE I SIX OPERATION MODES Operation Operation Input Output Mathematic relations frequency mode Source characteristic I

CCS

CVS

ωL

JG JG

II

CVS

CCS

CCS

JG

V

CVS

CCS

CVS

ωH CVS

CCS

L1 ⋅ k

JG

JG

)

JJG JJG JJG JJG PP = VP ⋅ I P = VS ⋅ I S =

2

L1 ⋅ k

(

2⋅ 2

VO ⋅ I O = Ps

(17)

JJG

2

L1 ⋅ 1 − k

2

of the primary side and the output current I S of the

)

secondary side are constants, it is operated in the II or the VI operation mode, as shown in Table I, thus the current JJG I P of the primary side would be in proportion to the JJG voltage VS of the secondary side. Therefore, as long as JJG the current I P of the primary side of the contactless JJG transformer is known, the voltage VS of the output of the

C2 C2

L1 ⋅

π

JJG From (17), one can see that when the input voltage VP

L2

JG

I S = VP ⋅

2

L2

JG

VS = I P ⋅

JG

VI

(

L1 ⋅ 1 − k

VS = VP ⋅

CVS

)

C2

JG

JG

2

C2

IS = IP ⋅

CCS

ωO IV

L1 ⋅ 1 − k

I S = VP ⋅

JG

III

(

JG

VS = I P ⋅

the secondary side caused by separable core winding, but also the proposed SPLCC can easily control the output power directly from the primary side. In other words, assuming the power transfer system is lossless, according to the principle of conservation of energy, the input power of the primary side is equal to the output power of the secondary side of the contactless transformer, i.e.:

(1 − k ) 2

III. Proposed SPLCC

secondary side of the contactless transformer can be derived. Based on this operation principle, the battery voltage VO of the output of the secondary side of the transformer can be evaluated by the input of the primary side, so the battery charging condition can be monitored.

III.1. Loop Controller Design A circuit diagram of the proposed SPLCC is shown in Fig. 4, where the Rdc represents the equivalent instantaneous resistance of the battery, which would be varied with time during battery charging, and a LC low pass filter with a full bridge rectifier is effectively adopted to reduce the output voltage ripple and to decrease the distortion of the output current on the secondary side of the transformer. As mentioned above, in order to keep the primary side of the transformer with the minimum VA of inverter, a new IPLL, which will be described in detail later, is proposed to make the system phase locked. Two PWM control loops, a PWM voltage loop and a PWM current control loop, are then used to control the output of the inverter as a CVS or CCS to charge a battery with constant current or voltage respectively. As the battery voltage rises to a reference level, the mode selector starts to operate to control the output of the inverter as a CVS to charge the battery with constant voltage until the battery is fully charged. Otherwise, the inverter is operated at CCS mode. As long as the primary side of the transformer is controlled, the output of the proposed SPLCC charger can act as either a CCS or a CVS. Hence, as the load in the output changes, the proposed IPLL can force system input as ZPA frequency and use PWM technique to control system input as CVS/CCS, thus controls of system output as a CVS or CCS can be realized. As a result, not only the control circuit can be placed at the primary side of the loosely coupled transformer to overcome the main problem of difficult output control at

III.2. The Proposed IPLL Controller In order to achieve minimum VA requirement on the primary side of the transformer, an IPLL controller, as shown in Fig. 5, is proposed to track and make sure the system is operated at ZPA frequency. In other words, when there is an out of phase conduction between the output voltage and current of the inverter, the ZPA frequency of the inverter could be achieved by using the proposed IPLL controller to increase or reduce the switching frequency of the inverter automatically. The proposed IPLL controller is constituted by five sub-circuits which include a error amplifier (EA), a integrator, a voltage to frequency converter (V/F), a sample and hold (S/H), and a zero crossing detector (ZCD) respectively. The ZCD is used to detect the zero crossing point of voltage. The S/H is used to detect the phase difference between voltage and current. The integrator is used to integrate the phase difference. The EA is used to amplify the difference between a reference signal and the feedback signal. The V/F is used to linearly convert the input voltage into output frequency. As shown in the Fig. 5, vtri takes the zero crossing point of v p by ZCD and samples the current signal i p



731


simultaneously, and then outputs the sample current vip .

IV.

An integrator is adapted to integrate current vip to

Simulation and Experimental Results

In this section, a SPLCC has been made based on the proposed theory. It employed two PE12V4.5Ah lead-acid batteries in series and charged by the constant current 1C (4.5 A). A set of design rules has been proposed to decide the parameters of the circuit so the system can be designed easily. The basic specifications of the SPLCC are given as follows: - Input DC voltage: Vin = 48VDC - Output DC current: I O = 4.5A - Maximum output power: PO = 90W - Operation frequency: f S = 20kHz (= f L ) - Coupling coefficient of contactless transformer: k = 0.6 . According to the design rule [11], one can obtain:

obtain the vir . The vif error between vir and the reference signal vc( i ) inputs the V/F and then converts vif into a new frequency signal. Figs. 6 show the exemplary operation principle of the proposed IPLL for different loads, in which Rac( i ) denotes the initial load resistor and can be obtained by Eqs. (5) and (6). Obviously, Rac is nonlinear and related to ωL and ωH . However, due to the characteristics of the negative feedback system of the proposed IPPL controller, as the frequency signal corrects, the error between voltage and current integrated by the integrator would converge since the phase difference decreases. From (1), one can see when vip ≥ 0 , the current leads

Rac = 5.48 Ω , L2 = 20.9µ H , L1 = 301.7 µ H ,

the voltage (i.e. capacitance load), and this can be corrected by raising the frequency. Conversely, if vip ≤ 0 ,

C1 = 0.21µ F and C2 = 1.94 µ F

the current lags behind the voltage (i.e. inductance load), and this can be corrected by lowering the frequency. As a result, the output current i p approaches v p until output

The above circuit parameters were used for simulation by the circuitry simulation software IsSpice. The simulation results are compared with the measured results derived from the SPLCC implementation. The implementation used the DC electrical load to make verification before charging the lead-acid batteries.

voltage and output current is in-phase. At steady-state, the switching frequency f s of the inverter is stable and Z in is resistance load now.

IO S3

S1 + Vin −

ip

M

C1

is

+

Cin

vp −

L1

L2

D1 +

Lf

+

Cf

C2 v s

Rdc

VO

−

D4

S2

S4

D3

D2

−

v p∗ ∑

vp

dv S1 S2 S3 S4

fs

di

i p∗

ip ∑

Fig. 4. Overall circuit of the proposed SP-LCIPTS charger



732


Rac ( i )

vc = f ( Rac )

vc ( i )

vif

∑

fs

V

F

vir

vip

S

ip H

vtri

vp

Fig. 5. Block diagram of the proposed IPLL

Rac''

Rac'

Rac ( i ) v

vir tH

vip

t0

tH

tH

t2

t1

t4

t3

t5

t

(a)

ip

tH

ip

vip

ip

tH

vip

vip t

t

t

tH

vp

vp vtri

vp

vtri

vtri

t (b) capacitance load

t (c) resistance load

t (d) inductance load

Figs. 6. Exemplary operation principle of the proposed IPLL controller for different loads

Figs. 7 and 8 show the waveforms of the input voltage and current on the primary side of the transformer derived from simulation and experimental results respectively. Obviously, the minimum VA can indeed be satisfied on the input of the primary side of the transformer. Figs. 9 and 10show the voltage and current waveforms on the output of the secondary side of the transformer, derived from simulation and implementation respectively. From Fig. 8 and Fig. 10, one can see that the input power on the primary side of the contactless transformer is 99.6W and the output power on the primary side of the contactless transformer is 90.9W, so that the efficiency of the transformer can achieve about 91%. In the electrical load dynamic mode, we set the resistance to alternate on 4Ω and 7Ω , and then measure the transient variation. Fig. 11 shows the waveforms of the output voltage and current transient response of the SPLCC between the alternations of the two resistances.

The total transient response time is 50 ms. When the steady state is stable, the output current difference between the two resistances is 0.16 A. Finally, the batteries are charged by the proposed SPLCC where two series PE12V4.5 (12V-4.5Ah) lead-acid batteries are discharged to 22.6V, then are charged with 1C (4.5 A). When the battery voltage reaches the designed charging voltage 28.8V, the mode selected circuit starts to operate and change the charging mode from constant current mode into constant voltage mode. In the constant voltage mode, the batteries can be fully charged without risk of overcharging. Fig. 12 shows the battery voltage and the charging current during charging. From Fig. 12, one can see that, when the charging time is between 35 minutes to 40 minutes, the vs,rms reaches the set mode-change-reference voltage (27.8V). At that time, the mode selection circuit starts to operate and change the constant current charge mode into the constant voltage



733


charge mode. Figs. 13 and 14 show the waveforms of the input voltage and current on the primary side of the transformer when the batteries are charged after 10 minutes and 30 minutes respectively.

From Figs. 12-14, one can see the proposed charger not only is operated at CCS but also the output current i p and output voltage v p is always in-phase sine the proposed IPLL that is not influent by the batteries are charged. Fig. 15 shows the photo of the experimental setup of the proposed SPLCC. From the above results one can see that the lead-acid battery is charged with constant current by the SPLCC at the beginning. When the battery voltage reaches the set reference voltage, the mode selected circuit switches the mode from constant current charge mode to constant voltage charge mode. During the charging period, the PWM control circuit is able to keep the charger output as CCS/CVS in different input modes (CVS/CCS). The IPLL controller can keep the voltage and the current on the primary side of the transformer in phase during the load variation as expected.

vp ip

vs is

Fig. 7. The input voltage and current simulation waveforms of the transformer

vp

vp

ip

ip

Fig. 10. The output voltage and current experimented waveforms of the transformer

Rdc = 7Ω

Rdc = 4Ω

Fig. 8. The input voltage and current experimented waveforms of the transformer

VO

vs is

IO

VO

IO

( 2 0 V / d iv )

0 .1 s / d iv

( 5 A / d iv )

Fig. 11. The waveforms of the output voltage and current transient response time during the load changed

Fig. 9. The output voltage and current simulation waveforms of the transformer



734


V 50

6 A

45

5

40

4

35

3

30

2

V1 V2 P , rms I1 I2 S , rms

v

v iP , rms

iS , rms 27.8V

25

1

20

0 5

10

15

20

25

30

35

40

45

50

55

60

t (min)

Fig. 12. The plot of charging current and battery voltage during charging

Vv p1 Fig. 15. Photo of the experimental setup of the proposed SPLCC

As a result, not only the intrinsic inefficiency of the separable transformer can be avoided, but also the high cost and low reliability problem in the wireless feedback of the secondary output for the transformer can be solved. Some simulated and experimental results verify the proposed theory is rather suitable for the separate battery chargers.

Ii p1

Acknowledgements The authors would like to thank the National Science Council for supporting this work under Grant NSC 99-2632-E-233-001-MY3.

Fig. 13. The experimented input voltage and current waveforms of the transformer after 10 minutes

v p1 V

References [1]

Ii p1 [2]

[3]

[4] Fig. 14. The experimented input voltage and current waveforms of the transformer after 30 minutes

V.

[5]

Conclusion

[6]

Loosely coupled inductive power systems have been widely applied in many power transmission systems. Under the optimal parameter constraints, in order to operate with the minimum VA input of the full bridge inverter used in the front stage of the SPLCC with primary side control, an IPLL controller is then proposed. Furthermore, different operation modes of the proposed SPLCC with various applications are also considered in the design of the controller.

[7]

[8]

[9]


H. Hao, G. Covic, M. Kissin and J. Boys, “A parallel topology for inductive power transfer power supplies,” 2011 IEEE Applied Power Electronics Conference and Exposition (APEC), pp. 2027-2034, 2011. J. Huh, S. Lee, C. Park, G. H. Cho and C. T. Rim, “High performance inductive power transfer system with narrow rail width for On-Line Electric Vehicles,” 2010 IEEE Energy Conversion Congress and Exposition (ECCE), pp. 647-651, 2010. Z. Pantic, B. Sanzhong and S. M. Lukic, “ZCS-Compensated Resonant Inverter for Inductive-Power-Transfer Application,” IEEE Transactions on Industrial Electronics, pp. 3500-3510, 2011. D. Yafeng, W. Changsong, W. Hui, S. Pengfei and L. Hongxia, “Study on compensation of novel planar inductive power transfer system,” 2010 2nd International Asia Conference on Informatics in Control, Automation and Robotics (CAR), pp. 388-391, 2010. A. Ghahary and B. H. Cho, “Design of transcutaneous energy transmission system using a series resonant converter,” IEEE Transactions on power electronics, vol. 7, pp. 261-269, 1992. D. A. G. Pedder, A. D. Brown and J. A. Skinner, “A contactless electrical energy transmission system,” IEEE Transactions on Industrial Electronics, vol. 46, pp. 23-30, 1999. Y. Jang and M. M. Jovanovic, “A contactless electrical energy transmission system for portable-telephone battery chargers,” IEEE Transactions on Industrial Electronics, vol. 50, 2003, pp. 520-527, 2003. C. G. Kim, D. H. Seo, J. S. You, J. H. Park, and B. H. Cho, “Design of a contactless battery charger for cellular phone,” IEEE APEC’00, vol. 2, pp. 769-773, 2000. Z. Pantic, B. Sanzhong, S. M. Lukic, “ZCS LCC-Compensated Resonant Inverter for Inductive-Power-Transfer Application,” IEEE Transactions on Industrial Electronics, pp. 3500-3510, 2011.


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[10] C. S. Wang, G. A. Covic and O. H. Stielau, “Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems,” IEEE Transactions on Industrial Electronics, vol. 51, 2004, pp. 148-157, 2004. [11] Y. H. Chao, J. J. Shieh, C. T. Pan and W. C. Shen, “A Closed-form Oriented Compensator Analysis for Series-parallel Loosely Coupled Inductive Power Transfer Systems,” Taiwan Patent No. 327402, July, 2010. [12] H. Miura, S. Arai, F. Sato, H. Matsuki and T. Sato, “A synchronous rectification using a digital PLL technique for contactless power supplies,” IEEE Transactions on Magnetics, vol. 41, 2005, pp. 3997-3999, 2005. [13] H. Abe, H. Sakamoto and K. Harada, “A noncontact charger using a resonant converter with parallel capacitor of the secondary coil,” IEEE Transactions on Industry Applications, vol. 36, pp. 444-451, 2000. [14] G. C. Hsieh and J. C. Hung, “Phase-locked loop techniques-A survey,” IEEE Trans. Ind. Electron., vol. 43, no. 6, pp. 609-615, Dec. 1996. [15] M. Helaimi, M. Benghanem, B. Belmadani, “Robust PI-Like Fuzzy Logic Controllers for High Frequency Inverter for Induction Heating Application,” International Review of Automatic Control Theory and Applications (IREACO), vol. 2, no. 5, pp. 561-567, Sep.2009.

Authors’ information 1,2

Department of Electrical and Electronic Engineering, Ta Hwa University of Science and Technology, Hsinchu, 30740, Taiwan.

Yuan-Hsin Chao was born in Jhanghua, Taiwan, in 1964. He received the B.S. degree in electrical engineering from Tamkang University, Tamsui, Taiwan, and the M.S. degrees from National Taiwan University, Taipei, Taiwan, in 1989, and 1991, respectively, all in electrical engineering. Since 1992, he has been a lecturer in the Department of Electrical Engineering, Ta Hwa Institute of Technology, Chung-Lin, Hsinchu, Taiwan. His fields of interest are power electronics and ac servo motor drives. Jenn-Jong.Shieh was born in Chi-Yi, Taiwan, in 1966. He received the B.S. degree from National Taiwan University of Science and Technology, Taipei, Taiwan, and the M.S. and Ph.D. degrees from National Tsing Hua University, Hsinchu, Taiwan, in 1990, 1992, and 1997, respectively, all in electrical engineering. Since 1998, he has been with the Department of Electrical Engineering, Ta Hwa Institute of Technology, Hsinchu, Taiwan. From 2000 to 2005 and 2006-2012, he was the Chairman of the Department of Electrical Engineering and he also was the dean of office of research of & development of the Ta Hwa Institute of Technology, respectively. Prof. Shieh has been the recipient of several outstanding special project awards. His research interests are in the areas of power electronics, motor control, and intelligent property analysis.



736


Voltage and Power Regulation for a Sample Power System Using Heuristics Population Search Based PID Design Ahmed Bensenouci1, A. H. Besheer2

Abstract – In this paper, a solution for driving a single machine infinite bus power system in a robust manner is presented by blending simple feedback linear control theory with heuristic population-based search algorithms. The proposed turbo-generator system is equipped with two decoupled control-loops, namely, the speed/power (governor) and voltage (exciter) controllers. The regulation of the voltage and power is achieved using normal structure of the Automatic Voltage Regulator (AVR) that is assisted by a conventional Power System Stabilizer (PSS). Moreover, rejection for various kinds of electromechanical modes of oscillations in the power grid is guaranteed. The design of power and voltage tunable PID controllers with the conventional PSS is formulated as an optimization problem. The goal of the design is to minimize the demand side variation effect while improving the transient performance of the terminal voltage and power. Using diverse simulation tests - where different kind of variations in load voltage, parameters change and symmetrical three phase short circuit are applied - the feasibility and effectiveness of the proposed controller are illustrated. Moreover, the superiority of the proposed design is further emphasized through a comparative study with Generalized Ziegler Nichols (GZN) based PID designs. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Bacterial Foraging Algorithm, PID Control, A.C. Turbo-Generator System, PSS

Kp, KI, KD Proportional, integral and derivative gains C1 & C2 Weighting factors for the relative controlled quality Power system stabilizer voltage Upss Kpci Critical gains Period of oscillations Tc t Simulation time in seconds final simulation time in seconds tf xcl Closed loop state variable J Objective function

Nomenclature Ud, Uq Ut ψfd xad xfd Ufd rfd e Ue δ Te/Tm Ps H ω ωo Kd Td Pt, Pb τe τg τb Ug v Kv x u

Stator voltage in d and q axis circuit Terminal voltage Filed flux linkage Stator-rotor mutual reactance Self reactance of filed winding Field voltage, Efd = (Ufd)(xad)/rfd Field resistance Busbar voltage resistance Input to exciter Rotor angle, rad Electrical / mechanical torque Steam power Inertia constant Angular frequency of rotor Angular frequency of infinite busbar Mechanical damping torque coefficient Damping torque coefficient due to damper windings Real power output at terminals and busbar Exciter time constant Governor valve time constant Turbine time constant Input to governor Governor valve position Valve constant State variables vector Control input vector

I.

Introduction

Unceasing balance between electrical generation and a varying load demand is considered as a mandatory request for any modern controlled power system. Reference has been made to the difficulties encountered when designing and implementing an online optimal control law for an ac turbo-generator [1]. One of the most important aspects in such kind of power system operation is the stability of its terminal voltage, frequency and power. Generally, the main objectives for any power system that uses large generator units with low specific inertia, and also of longer transmission lines at higher voltages are: • To drive the sample power system to cope with prescribed desired values for its terminal voltage and power.


737


Ahmed Bensenouci, A. H. Besheer

• To improve the sample power system performance. • To overcome reductions in stability limits. A variety of control techniques and approaches have been proposed in the literature for studying a sample power system that comprises a synchronous generator connected to a large network (infinite bus bar) via a stepup transformer and a transmission line. These techniques ranged from a constant-linear discrete output feedback optimal-control system requiring only the measurement of readily available signals from the power system such as [1]-[2], and on-line scheduled multiple model/controller approach to nonlinear identification and control. The resultant nonlinear multiple-controller automatic voltage regulator was achieved by online blending of multiple PID controllers, each designed for a linear sub-model [3]. In [4], based on Lyapunov theory of stability, a decentralized control of a non-linear turbogenerator system is addressed, where the complex nonlinear model of the system to be controlled is decomposed into two subsystems synchronous generator and turbine. Two PID controllers, one for the active output power and the other for the terminal voltage, are decentralized with a simplified coordinator. However, the gains of that controllers are normally fixed and usually tuned manually or using trial-and-error approach or by conventional control methods (frequency response). Therefore, it is incapable of obtaining good dynamical performance that capture all design objectives and specifications for a wide range of operating conditions and disturbances [5]-[6]. An attempt to incur the above mentioned limitations for single machine – infinite bus power system is offered in [7], using generalized ZieglerNichols (GZN) method to tune the PID controllers for ac turbo generator power system. The area of auto-tuning of PID controller using artificial intelligence such as fuzzy systems has attracted many authors [8]-[9]. The drawbacks of the fuzzy systems lies in the difficulty and the long time taken to drive the fuzzy rules as well as relying on expert knowledge. Some other control approaches have also been applied to achieve the above mentioned main objectives for such sample power system by using automatic voltage regulators (AVRs) assisted by power system stabilizer (PSS). At first, the installation of AVR on power generating units becomes a common practice since late 1950s. However the success of achieving prescribed terminal voltage level using these regulators, they are considered as a source of the well known problem of alternator rotor angle oscillation. The unavoidable low frequency oscillations in such power systems admits small amplitude and may last long periods of time which may cause the AVR to overreact and bring the oscillation to rotor angle of the synchronous machine that may result in serious consequences such as tripping the generator from the grid. Moreover, low-frequency oscillations present limitations on the power-transfer capability.

Hence, AVR alone is considered to be inadequate to reject these oscillations. To enhance system damping, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems. PSS increase the power system stability limit and extend the power-transfer capability by enhancing the system damping of lowfrequency oscillations associated with the electromechanical modes [2]. As a result, PSS contributes in maintaining reliable performance of the power system stability. Despite the potential of modern control techniques with different structures, power system utilities still prefer the conventional lead-lag power system stabilizer (CPSS) structure [10]-[11]. The reasons behind that might be the high efficiency, the ease of implementation and the online tuning capabilities. Kundur et al. [12] have presented a comprehensive analysis of the effects of the different CPSS parameters on the overall dynamic performance of the power system where the appropriate selection of CPSS parameters results in a satisfactory performance during system upsets. In addition, Gibbard [13] demonstrated that the CPSS provides satisfactory damping performance over a wide range of system loading conditions. The robustness nature of the CPSS is due to the fact that the torque-reference voltage transfer function remains approximately invariant over a wide range of operating conditions. A gradient procedure for optimization of PSS parameters at different operating conditions is presented in [14]. Unfortunately, the optimization process requires computations of sensitivity factors and eigenvectors at each iteration. This gives rise to heavy computational burden and slow convergence. Recently, intelligent optimization techniques like genetic algorithms (GA) [15]–[18], Tabu search [19], simulated annealing [20], and evolutionary programming [21] have been applied for PSS parameter optimization. These evolutionary algorithms are heuristic populationbased search procedures that incorporate random variation and selection operators. Although, these methods seem to be good methods for the solution of PSS parameter optimization problem, However, when the number of parameters to be optimized is large and those parameters are highly correlated, then the efficiency to obtain global optimum solution is degraded and also the simulation computing time is high [22]. In this paper, an evolutionary approach-based controller auto-tuning method is chosen to be applied due to its no model-based approach, simplicity, high computational efficiency, easy implementation, and stable convergence. Bacterial Foraging Algorithm (BFA) [23]-[24], a new evolutionary computation technique, has been proposed in the literature and explored in here in controllers design. The foraging (locating, handling, and ingesting food) behavior of Escherichia coli bacteria, present in our intestines, is mimicked. They undergo different stages such as chemotaxis, swarming, reproduction, elimination and dispersal. In the



738


chemotaxis stage, each bacterium can tumble or run (swim). In swarming, each bacterium signals other bacteria via attractants to swarm (group) together. In reproduction, the least healthy bacteria die and each of the healthiest splits into two bacteria that are then placed in the same location. Further, any bacterium is eliminated from the total set of bacteria just by dispersing it to a random location (elimination and dispersal). This paper provides simple and straightforward heuristic population-based procedures for designing two decentralized PID controllers for controlling ac turbogenerator power system. The first PID is assisted by PSS forming the ordinary AVR while the second PID is designed to control the output power. The procedure is based on bacterial foraging optimization method of multi-input multi-output systems. BFA is employed to design the PID controller gains. The objective of bacterial foraging optimization in this paper is to improve both the design efficiency of PID control systems and its performance to get optimal PID parameters. The paper presents design procedures for a model free bacterial foraging based PID controller. Moreover, a lead-lag type power system stabilizer is designed using the same technique. The optimal gains of the PID controllers and the PSS are obtained using integral time square errors for both terminal voltage and output power signals respectively as the performance indexes. A comparative study with GZN based PID control reflects the appealing of the suggested control technique over the traditional control methods. The main contributions of this paper are two folded. Firstly, two optimized PID controllers plus lead lag PSS for regulating two control-loops, namely, the speed/power (governor) and voltage (exciter) and damping different oscillation modes in ac turbo generator system using bacterial foraging algorithm have been designed. Secondly, a satisfactory sample power system closed loop performance – that reflects the effectiveness of the proposed design - is achieved even under diverse simulation tests namely, regulation in the controlled variables and variation in the system parameters. Furthermore, a symmetrical three-phase short-circuit is applied at the infinite bus level to test the controllers during heavy disturbances. The results are promising for further investigations and application to a multi-machine power system. This paper is organized as follows: section II presents the single machine power system configurations. BFA based PID controller is presented in section III. The single machine infinite bus simulation using the proposed technique is applied in section IV. Finally, a conclusion is set in section V.

II.

Fig. 1. Power system block diagram

In this configuration an electric alternator of synchronous type is driven by steam turbine. The alternator terminal voltage is stepped up via transformer and then connected to an infinite bus through a transmission line. Two feedback signals namely real output power Pt and terminal voltage Vt are fed to the controller who in turn produces the manipulating actions to the generator-exciter and governor-valve [1]. II.1.

Non-Linear Model

The dynamics of the used alternator is assumed to be 3rd. order nonlinear equations while the dynamics of the governor valve, steam turbine and exciter will be 1st. order model [1]. Nonlinearities of valve movement and field excitation voltage are also included. The model equations are given by Eq. (1), where the system parameters coefficients and list of symbols can be found in [3]: ⎧ x1 = x2 ⎪ ⎛ x6 − K1 x3 sin x1 − K 2 sin x1 cos x1 + ⎞ ⎪ ⎟ ⎪ x2 = ⎜ ω0 ⎜ ⎟⎟ − + K T x ( ) ⎪ ⎜ 2 d d ⎝ ⎠ 2H ⎪ ⎪ ω0 rfd ⎪ x3 = x4 + K3 x3 − K 2 sin x1 cos x1 xad ⎪ ⎨ −x ⎪ x4 = 4 + 1 U e τe τe ⎪ ⎪ − x5 Kv ⎪ ⎪ x5 = τ + τ U g g g ⎪ ⎪ − x6 x5 + ⎪ x6 = τb τb ⎩

(1)

The output y1, y2 may be expressed in terms of these state variables by:

Single Machine Infinite Bus Power System Modeling

⎧ y1 = Pt = K1 x3 sin x1 + K 2 sin x1 cos x1 ⎪ ⎨ 2 2 1/ 2 ⎪⎩ y2 = Vt = Vd + Vq

(

The configuration of the proposed sample power system is depicted in Fig. 1. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

)

(2)


739


where:

With the inclusion of both PID controllers, the system becomes:

⎧⎪Vd = K5 sin x1 ⎨ ⎪⎩Vq = K 6 x3 + K 7 cos x1

II.2.

(3) ⎧ x1 = x2 ⎪ ⎪ x = ⎛ x6 − K1 x3 sin x1 − K 2 sin x1 cos x1 + ⎞ ω0 ⎟⎟ ⎪ 2 ⎜⎜ − ( K + T ) x 2 d d ⎝ ⎠ 2H ⎪ ⎪ ω0 r fd x4 + K 3 x3 − K 2 sin x1 cos x1 ⎪ x3 = xad ⎪ ⎪ K −x ⎪ x4 = 4 + e U e ⎪ τ τe e ⎨ ⎪ − x5 K v + U ⎪ x5 = τg τg g ⎪ ⎪ −x x ⎪ x6 = 6 + 5 τb τb ⎪ ⎪ ⎪ x7 = K I 1 Vref − Vt ⎪ ⎪⎩ x8 = K I 2 Pref − Pt

Linear Model

Taylor linearization around a pre-specified operating point for Eqs. (1)-(3) leads to the linear model of the proposed power system in (4): ⎧ x = Ax + Bu ⎨ ⎩ y = Cx + Du

(4)

The matrices A, B, C and D have the form: 1 ⎡0 ⎢ − ( K d + Td ) ω0 ⎢K ⎢ 8 2H ⎢ ⎢K 0 ⎢ 10 ⎢ A=⎢ 0 ⎢0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢1 B=⎢ τe ⎢ ⎢ ⎢0 ⎢ ⎢⎣ 0

0

0

K9 K3

0

0

0

ω0 r fd

0

xad −1

0

τe

0

0

0

0

⎤ ⎥ ⎥ ⎥ ⎥ K 0 ⎥ , C = ⎡⎢ 11 ⎥ ⎣ K13 ⎥ Kg ⎥ τ g ⎥⎥ 0 ⎥⎦

⎤ ⎥ ω0 ⎥ 2H ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥ ⎥ −1 ⎥ ⎥ τb ⎦ 0

0

−1

τe −1

τb

( (

Pt = K1 x3 sin x1 + K 2 sin x1 cos x1

a1

Ps

K 6 ( K 6 x3 + K 7 cos x1 )

Vt b1 = K1 x3 cos x1 + K 2 cos 2 x1 b2 = K1 sin x1 0 K12 0 K14

ω0 rfd ⎞ ⎛ Vt = − a2 K 2 cos x1 sin x1 + a1 x2 + a2 ⎜ K3 x3 + x4 ⎟ xad ⎝ ⎠ U = K V − V + x + K V − V

0 0 0⎤ 0 0 0 ⎥⎦

e

P1

(

ref

)

t

7

D1

(

ref

t

)

⎛ ω0 r fd ⎞ Pt = b1 x2 + b2 ⎜ x4 + K3 x3 − K 2 cos x1 sin x1 ⎟ ⎝ xad ⎠ U =K P −P +x +K P − P p

T

Tm ⎤⎦ : State var. vector

P2

(

ref

t

xcl = ⎡⎣δ

δ ψ fd

w = ⎡⎣Vref

Pref ⎤⎦

)

8

E fd

D2

Ps

(

Tm

ref

t

x7

)

x8 ⎤⎦

t

t

and KPi, KIi and KDi (i=1,2) represent the PID gains. The inclusion of a lead-lag PSS whose transfer function is given by (6) and it will change the system model to be (7):

T

u = ⎡⎣U e U g ⎤⎦ : Control input vector y = [ Pt

Vt

a2 =

where: E fd

[ K5 sin x1 ]2 + [ K6 x3 + K7 cos x1 ]2 K 2 sin x1 cos x1 − ( K 6 x3 + K 7 cos x1 ) K 7 sin x1 = 5

Vt =

0 0 0

δ ψ fd

) )

where:

⎡0 0 ⎤ D=⎢ ⎥ ⎣0 0 ⎦

x = ⎡⎣δ

(5)

Vt ] : Output measurement vector T

Pt = K11 x1 + K12 x3 : Output power Vt = K13 x1 + K14 x3 : Terminal voltage δ = ω : Generator speed

VPSS

ω


=

sT sT1 + 1 ⋅ sT + 1 sT2 + 1

(6)


740


⎧ x1 = x2 ⎪ ⎪ x = ⎛ x6 − K1 x3 sin x1 − K 2 sin x1 cos x1 + ⎞ ω0 ⎟⎟ ⎪ 2 ⎜⎜ − ( K + T ) x d d 2 ⎝ ⎠ 2H ⎪ ⎪ ω0 rfd x4 + K3 x3 − K 2 sin x1 cos x1 ⎪ x3 = xad ⎪ ⎪ K −x ⎪ x4 = 4 + e U e τe τe ⎪ ⎪ ⎪ x5 = − x5 + K v U g ⎪ τg τg ⎨ ⎪ x −x ⎪ x6 = 6 + 5 τb τb ⎪ ⎪ ⎪ x7 = K I 1 Vref − Vt + VPSS ⎪ ⎪ x8 = K I 2 Pref − Pt ⎪ ⎪ x = − x9 + x2 ⎪ 9 T T ⎪ T −T ⎪ x = − x10 + 2 1 ( x − x ) 2 9 10 ⎪ T2 T22 ⎩

( (

where: G11 ( s ) = G21 ( s ) =

(7)

6.

8.

T1 ( x2 − x9 ) + x10 T2

(

)

(

U e = K P1 Vref − Vt + VPSS + x7 + K D1 Vref − Vt + VPSS

D (s)

G12 ( s ) =

,

G22 ( s ) =

K P 2G22 ( 0 )

7.

T VPSS = 1 ( x2 − x9 ) + x10 T2

h21 ( s )

,

K P1G11 ( 0 )

)

)

D (s)

h12 ( s ) D (s)

h22 ( s ) D (s)

3. Determine the steady state gains Gii(0), (i=1,2). 4. Introduce local proportional controllers with gain Kpi for each input 5. Choose an initial value of Kp1 of the first input (first loop) and calculate the corresponding gains of the second loop based on the following relation:

with: VPSS =

h11 ( s )

)

9.

This section provides conventional design procedures for PID controllers for a.c. turbo-generator power system. The procedure is based on GZN method of multi-input multi-output systems [7]. The GZN method is reformulated to suit direct application to the considered power system comprising two control loops. Normally, Ziegler-Nichols represents a method to get the optimum PID gains. Let PID transfer function be written as: K G ( s ) = KP + I + KD s (8) s

=

C1 C2

where, C1 and C2 are weighting factors for the relative controlled quality. Apply a small disturbance to the simulated system and record the outputs. See if the system makes oscillations. Change Kp1 (and hence Kp2) by a small a amount keeping the same relation (6) between the gains. Then perform step 5 above. Continuous changing Kp1 until obtaining continuous oscillations with constant amplitude. The gains corresponding to this case are called critical gains Kpci, (i=1,2) and the period of oscillations is denoted T c. Determine the optimum adjustment of the required controller parameters in each control loop from GZN Table I [7] given below: TABLE I PID GAINS USING GZN METHOD Optimum Parameters Controller Kp Ti Td P 0.5Kci ∞ 0 PI 0.45Kci 0.8Tc 0 0.5Tc 0.12Tc PID 0.6Kci

where K I =

Kp

and K D = K p Td . Ti 10. Finally, insert the appropriate controller with the selected gains in each control loop and test the overall system performance.

where Kp, KI, KD are the proportional, integral and derivative gains, respectively. GZN is summarized as follows [7]: 1. Obtain a mathematical linearized state equation model of the multi-input multi-output system as given by equation (4). 2. Determine the transfer matrix using (4) in the following form:

III. BFA Based PID Controller III.1. Bacterial Foraging Algorithm One of the effective tools for finding a solution for the combinatorial optimization problem is BFA. For a successful reproduction process in Escherichia coli bacteria foraging, strategy of eliminating poor foraging ones while keep spreading genes of those with flourishing foraging strategies is adopted.

⎡∆Vt ( s ) ⎤ ⎡ G11 ( s ) G12 ( s ) ⎤ ⎡ ∆u g ( s ) ⎤ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎣ ∆Pt ( s ) ⎦ ⎣G21 ( s ) G22 ( s ) ⎦ ⎣⎢ ∆ue ( s ) ⎦⎥ Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


741


subject to:

Poor foraging strategies are either completely eliminated or transformed into good ones after many generations are produced [23]. Chemotaxis, swarming, reproduction and elimination-dispersal are the main operations in bacterial foraging process. Moreover, tumbling or swimming are two different operations, a bacterium can perform during foraging. For example, the orientation of the bacterium is modified by the tumble action while it will move in its incumbent direction during swimming. This Chemotactic displacement is last till the direction of positive-nutrient gradient is figured out by the bacterium and the process of reproduction for half of the population is started meanwhile, the rest of the population is eliminated. In order to escape local optima, an elimination-dispersion event is carried out where some bacteria are liquidated at random with a very small probability and the new replacements are initialized at random locations of the search space [24]. The simple, straight forward and general bacterial foraging optimization algorithm can be summarized as follows: Step 1: Initialization (Assign values for the BFA parameters). Step 2: Evaluation Step 3: Chemotaxis Tumble/run Step 4: Start reproduction process (just after completing step 3). Step 5: Start elimination dispersal (just after completing step 4). Step 6: Check the optimized variables (e.g. Kp, Ki & Kd). Step 7: Go to step 1 if optimized variables are not obtained, otherwise, stop.

Ki,min < Ki < Ki,max Ti,min < Ti < Ti,max i = 1,2

where t is the simulation time in seconds, tf is the final simulation time, Ki (i=1-6) consists of the parameters of the PIDs (KP1, KI1, KD1, KP2, KI2, KD2,), and Ti (i=1-2) consists of the parameters of the PSS (T, T1, T2) when it is implemented. The proposed approach employs BFA to solve this optimization problem and search for the optimal set of controllers’ parameters. Flow chart for BFA is given below.

IV.

In this section, the proposed technique namely Bacterial Foraging based PID (BF-PID) control is applied to a single machine infinite bus power system that comprises a steam turbine driving a synchronous generator which is connected to an infinite bus via a step-up transformer and a transmission line. The technique aims at not only automatically control the output power for the speed/power governor closed loop as well as regulate the terminal voltage of synchronous generator using AVR that is equipped with a Conventional Power System Stabilizer, CPSS, in the voltage exciter closed loop but also enhance the overall dynamic performance of the system even under diverse tests conditions on power demand, a wide-range variation in system parameters and short circuit fault conditions. These control objectives are achieved by simultaneously changing the governor valve position and the exciter input to firstly allow an appropriate quantity of steam to drive the turbine and adapting the output power of the generator and secondly to change the exciter input to adjust the terminal voltage respectively. Comparative studies offered with respect to generalized Ziegler-Nichols not only reflect better dynamic behavior but also indicate the robustness of the proposed technique to different load perturbations and parameters variations.

III.2. Bacterial Foraging based PID Optimization Problem Usually, the optimization process usually consists of obtaining the controller gains such that to minimize or maximize a given objective function of the closed loop system comprising BFA based PID controller and an unknown plant. The optimization of step response of the system under control by minimizing a suitable performance criterion is the aim of this work. The effectiveness of the proposed ant based PID is quantified by the following performance criteria. The performance index (cost/objective function) summarizes the desired performance that the controlled system should fulfill. Here, to achieve this, the Integral of Time Squared Error (ITSE) is considered using the errors in Vt, and Pt. ITSE is a better criterion which keeps account of errors at the beginning but also emphasizes the steady state [4]. Therefore, the design problem can be formulated as the following optimization problem: t =t f

Minimize J =

⎡ ∫ t ⎣⎢(Vref − Vt ( t ) ) + ( Pref − Pt ) 2

2⎤

t =0

⎦⎥

Single Machine Infinite Bus Simulation Using the Proposed Technique

IV.1. Closed Loop Control The stator output power and terminal voltage are chosen as the feedback variables. In the first closed loop, the coordination between PID1 and the lead/lag PSS (or CPSS) results in the control action signal Ue for the exciter. The output power signal is compared with the required power reference signal. The power error signal is fed to PID2 controller that outputs another control action Ug signal for the governor valve. A linear multi-input multi-output model of the turbogenerator system, in (4), is used to design the PID controllers. Proper tuning of the PID controllers and

(9)



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CPSS is required in order to give satisfactory dynamic performance of the proposed sample power system. The idea is to start the proposed algorithm with certain PID gains then enhance the dynamic response of the closed loop system according to the performance index in (9) via the steps of the algorithm. Finally, the algorithm will reach to the optimal parameters of the PID controllers as well as CPSS that can be used in controlling the output power and the terminal voltage of the sample power system maintaining reliable dynamic performance of the system stability. (a) Terminal voltage

IV.2. Closed Loop Control Simulation Results Simulation experiments, using MATLAB/SIMULINK environment, are performed in this section to study the dynamic response of a single machine infinite bus power system with bacterial foraging algorithm based PID closed loop control. To demonstrate the effectiveness of the proposed design, tests such as terminal voltage and output power regulation, attenuation of exogenous disturbance represented by symmetrical three phase short circuit and parameters variation for the studied system are carried out. The design is based on the nonlinear model of the plant. Due to the presence of dynamic oscillations in the output power, terminal voltage, rotor angle and rotor speed under such tests, lead/lag type of PSS is designed and its output signal is summed up with PID1 controller to form the ordinary AVR structure to offer better damping for different electromechanically modes of oscillation. The BFA design parameters and the obtained PID controller gains from the proposed optimization algorithm are found in the Appendix.

(b) Output power

IV.2.1. Effect of Voltage Load Demand Variation without PSS In this test, the nonlinear model of single machine infinite bus power system is subjected to is subjected to a 10% step increase and decrease in the load voltage demand (Vref) while load power demand kept constant. Figs. 2 show the system time response of the system terminal voltage Vt, output power Pt, with the system driven by the BF-PID and GZN-PID controllers. It can be noticed that BF-PID controllers gives oscillatory response but stable. High frequency dynamic oscillations in the terminal output power signal as well as over and under shoot in the terminal voltage are shown. The need for PSS is clearly reflected.

(c) Rotor angle

IV.2.2. Three-Phase Short-Circuit Disturbance without PSS This test is provided to show the feasibility of the designed BF-PID in attenuating heavy exogenous disturbance, represented by symmetrical three-phase short-circuit that is applied at the infinite bus and lasting for 100 millisecond.

(d) Rotor speed Figs. 2. System response due to load demand step variation without PSS



743


At the beginning, the nonlinear model of a single machine infinite bus power system reference values are set to Vref=1.2 pu and Pref=0.8 pu. Once the system has settled down to its steady-state, a 100-ms balanced threephase short-circuit is applied (t=20s) at the terminal of the machine. Figs. 3 show the system time response of the system terminal voltage Vt, output power Pt, rotor angle δ, and the rotor speed ω, with the system driven by the BF-PID and GZN-PID controllers. As shown in Figs. 3, oscillations are present in Pt, δ, and ω signals. The system regains its stability after few seconds. It is clear that BFA shows improved performance summarized in faster response (5s for BFA and 15s for GZN) with fewer oscillations than GZN.

(a) Terminal voltage

IV.2.3. Fault Condition and System Parameters Variation without PSS In this test, severe conditions are applied to the proposed power system equipped with optimized PID controllers BF-PID to show the robustness and effectiveness of BF-PID controllers to the system parameters variation while subjected to a 100-ms symmetrical three-phase fault at the machine terminal bus voltage. The test is carried out with 20% decrease in the inertia constant H, 20% increase in the rotor resistance rfd, 10% decrease in Td, and a 20% decrease in xt, with the system being subjected to a 100-ms threephase short-circuit at the machine bus. The time response of Vt, Pt, δ, and ω are shown in Figs. 4. Heavy oscillations are shown in Pt, δ, and ω signals. It is clear that BFA shows improved performance summarized in faster response with fewer oscillations than GZN. Remark: a) However, the two BF-PID controllers of the single machine infinite bus power system ensure the stability for the closed governor and exciter loops under different severe test conditions. The obtained dynamic responses contain heavy oscillations that need to be damped. b) Introduction of PSS to assist the exciter loop will offer fast damping for electromechanical modes in the proposed system as can be seen in the remaining tests.

(b) Output power

(c) Rotor angle

IV.2.4. Effect of Voltage Load Demand Variation with PSS In this test, coordination between the AVR represented by PID1 and lead/lag conventional type PSS is performed. The AVR (PID1) assisted by PSS and the PID2 are optimized using BFA. The designed PSS is also incorporated with GZN for comparison purposes. A simultaneous increase in Vref from 1.2 to 1.5 pu and Pref from 0.8 to 1 pu is applied at t=50s. Figs. 5 show the system time response of Vt, Pt , δ, and ω.

(d) Rotor speed Figs. 3. System response following a three-phase short-circuit at the infinite bus without PSS



744




(b) Output power

(b) Output power

(c) Rotor angle

(c) Rotor angle

(d) Rotor speed

(d) Rotor speed

Figs. 4. System response following a three-phase short-circuit and parameters change without PSS

Figs. 5. System response due to demand side step variation with the presence of PSS



745


proposed sample power system is demonstrated in Fig. 6 under a heavy disturbance, a 100-ms three-phase shortcircuit, and parameters change. It is clear that the system equipped with BF-PID and PSS exhibits better performance (only one oscillation with fast damping) compared to GZN-PID with PSS, as depicted in the Figs. 6. Comparison between system dynamic responses in Figs. 4 and Figs. 6 show the need for PSS to well damp different modes of oscillations in system parameters.

The optimized parameters for the PSS can be found in the appendix. It is clear that the oscillations that were found previously have been damped efficiently by a proper design of the selected type of PSS. Less over/undershoots are also noticed. IV.2.5. Fault Condition and System Parameters Variation with PSS The effectiveness of the proposed optimization technique for the AVR assisted by PSS in the exciter loop as well as for the PID in the governor loop for the

(b) Output power


(d) Rotor speed

(c) Rotor angle

Figs. 6. System response due to a three-phase fault and parameters change with the presence of PSS

V.

technique. Its effect is very clear in damping the oscillations and improving the system performance. The proposed control method outperforms the traditional control method. As an extension to this work, application to a multi-machine power system will be targeted.

Conclusion

The problem of driving both the terminal voltage and power of a single machine infinite bus power system to a certain desired values is addressed. Using BFA, two PID controllers for the governor and exciter closed loops are designed. A certain class of heuristic population based search algorithms that mimic the search of food for the bacteria is tailored to match the tuning problem in the PID control scheme. A stable but oscillatory dynamic performance for the sample power system during different diverse simulation tests, namely regulation, fault and parameters change is obtained. For this purpose, the AVR is assisted by conventional PSS and designed using the same BFA

Appendix • BFA parameters selection: p=6 Dimension of the search space S=50 Number of bacteria in the population Nc=100 Number of chemotactic steps per bacteria lifetime Ns=4 Limits the length of a swim when it is on a



746


[2]

gradient Nre=4 Number of reproduction steps Sr=S/2 Number of bacteria reproductions (splits) per generation (this choice keeps the number of bacteria constant) Ned=2 Number of elimination-dispersal events ped=0.25Probabilty that each bacteria will be eliminated / dispersed C=0.1 Basic run length step dattractant=0.1 Magnitude of secretion of attractant by a cell wattractant=0.2 How the chemical cohesion signal diffuses (smaller makes it diffuse more) hrepellant=0.1 Sets repellant (tendency to avoid nearby cell) wrepellant=10 Makes small area where cell is relative to diffusion of chemical signal

[3]

[4]

[5] [6]

[7]

[8]

• The resulted PID controller gains from BF are: Vt-control: KP1 = 3.713, KI1 = 6.4036, KD1 = 0.93337 Pt-control: KP2 = 0.11646, KI2 = 1.0483, KD2 = 0.01

[9]

• The resulted PID controller gains from GZN are: Vt-control: KP1 = 0.72583, KI1 = 1.3801, KD1 = 0.091617 Pt-control: KP2 = 0.4608, KI2 = 0.87616, KD2 = 0.058164

[10]

[11]

• The resulted PSS controller gains from BF are: Suitable values found using BF are T1=0.6248 s and T2= 0.0250s. Also, T= 10.

[12]

[13]

• System Parameters [14]

MVA 37.5 MW 30 p.f. 0.8 lag kV 11.8 r/min 3000 xd 2 pu xq 1.86 pu xad 1.86 pu xfd 2 pu Rfd 0.00107 pu H 5.3 MWs/MVA Td 0.05 s

xt 0.345 pu xl 0.125 pu e 1 pu τe 0.1 s τg 0.1 s τb 0.5 s Kv 1.889 Ke 0.01 Vd 0.5586 Vq 1.1076 Vt 1.2405 K11.2564

K2 -0.9218 K3 -0.5609 K4 0.4224 K5 0.7983 K6 0.5905 K7 0.3650 K8 -39.559 K9 -27.427 K10 -0.2955 K11 1.268 K12 0.8791 K13 0.0287 K14 0.52726

[15]

[16]

[17]

[18]

[19]

Acknowledgements [20]

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah.

[21]

[22]

References [1]

P.A.W. Walker, M.I.E.E., and O.H. Abdalla, Discrete control of an a.c. turbogenerator by output feedback, PROC. IEE, Control & Science, Vol. 125, No. 9, OCTOBER 1978.

[23]


A. R. Daniels, and H. Lu, Nonlinear single-variable optimisation studies of a.c. turbogenerator performance, PROC. IEE, Control & Science, Vol. 126, No. 5, MAY 1979. Lixin Ren, George W. Irwin, and Damian Flynn, Nonlinear Identification and Control of a Turbogenerator,An On-Line Scheduled Multiple Model/Controller Approach, IEEE Transactions On Energy Conversion, VOL. 20 (Issue 1): 237245, MARCH 2005. Jan Murgas, Andrej Dobrovic, Eva Miklovicova and Jozef Dubravsky, Two Level Turbogenerator Control System, Journal of Electrical Engineering, VOL. 55, NO. 3-4: 83-88, 2004.. K. Ogata, Modern Control Engineering (Prentice- Hall, 1990). K.J. Astrom, T. Hagglund, C.C. Zhang, and W.K. Ho, Automatic tuning and Adaptation for PID controllers, IFAC Journal Control Eng. Practice , Vol. 1, No. 4: 699-714,1993. A.M. Abdel Ghany and Ahmed Bensenouci, PID Controllers for a.c Turbo-generator Connected to Infinite-Bus Using Generalized Ziegler-Nichols Method, Proc. 4th Saudi Technical Conference & Exhibition, STCEX’2006, Riyadh, KSA, Dec. 2-6, 2006. A. Visioli, Evaluation of Analogue PID, Digital PID and Fuzzy Controllers for a Servo System, IEE Proc. Control Theory and Applications, Vol. 148, No. 1: 1-8, 2001. Mohammad Yaghoubi Bejomeh, Behnam Ganji and Abbas Z. Kouzani, Evaluation of Analogue PID, Digital PID and Fuzzy Controllers for a Servo System, International Review of Automatic Control (IREACO), Vol. 5. No. 2: 255-261, March 2012. E. Larsen and D. Swann, Applying power system stabilizers, IEEE Trans. Power App. Systems, Vol. PAS-100(Issue6):30173046, 1981. G. T. Tse and S. K. Tso, Refinement of conventional PSS design in multimachine system by modal analysis, IEEE Trans. on Power Systems, Vol. 8(Issue 2):598-605, 1993. P. Kundur, M. Klein, G. J. Rogers, & M. S. Zywno, Application of power system stabilizers for enhancement of overall system stability, IEEE Trans. on Power Systems, Vol.4(Issue 2):614-626, 1989. M. J. Gibbard, Robust design of fixed-parameter power system stabilizers over a wide range of operating conditions, IEEE Trans. On Power Systems, Vol. 6(Issue 2): 794-800, 1991. V. A. Maslennikov, S. M. Ustinov, Method and software for coordinated tuning of power system regulators, IEEE Trans. On Power Systems, Vol. 12(Issue 4): 1419-1424, 1997. Abdel-Magid YL, Abido MA, AI-Baiyat S, Mantawy AH. Simultaneous stabilization of multimachine power systems via genetic algorithms. IEEE Trans Power Syst, Vol.14, (Issue 4): 1428–1439, 1999. Abido MA, Abdel-Magid YL. Hybridizing rule-based power system stabilizers with genetic algorithms. IEEE Trans Power Syst Vol. 14, No 2: 600–607, 1999. Rouzbeh Jahani, Heidar Ali Shayanfar, Omid Khayat, GAPSOBased Power System Stabilizer for Minimizing the Maximum Overshoot and Setting Time. International Review of Automatic Control (IREACO); Vol. 3(Issue 3): 270-278, May 2010. Abdel-Magid YL, Abido MA. Optimal multiobjective design of robust power system stabilizers using genetic algorithms. IEEE Trans Power Syst; Vol.18(Issue 3): 1125–1132, 2003. Kaouther Laabidi, Mekki Ksouri. Genetic Algorithm and Tabu Search For Nonlinear Constrained Generalized Predictive Control. International Review of Automatic Control (IREACO), Vol. 2. No. 1: 27-33, Jan. 2009. Abido MA. Robust design of multimachine power system stabilizers using simulated annealing. IEEE Trans Energy Convers; Vol. 15(Issue 3): 297–304, 2003. Abido MA, Abdel-Magid YL. Optimal design of power system stabilizers using evolutionary programming. IEEE Trans Energy Convers; Vol. 17(Issue 4): 429–436,2002. M. Zamani, M. Karimi-Ghartemani, N. Sadati, and M. Parniani, “Design of a fractional order PID controller for an AVR using particle swarm optimization,” Control Engineering Practice, vol. 17, no. 12: 1380–1387, 2009. T. Deepa, P. Lakshmi, Coordinated Controller Tuning of Boiler Turbine Unit using Bacteria Foraging based Particle Swarm


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Optimization, European Journal of Scientific Research, Vol.64, No.3: 446-455, 2011. [24] V. Rajinikanth and K. Latha, I-PD Controller Tuning for Unstable System Using Bacterial Foraging Algorithm: A Study Based on Various Error Criterion, Applied Computational Intelligence and Soft Computing, Vol. 2012, Article ID 329389, 10 pages.

Ahmed Bensenouci (MIEEE’96-SMIEEE'03) obtained his Ph.D. in Electrical Engineering in 1988 from Purdue University Indiana, USA; and his Master of Engineering in Electric Power Engineering in 1983 from Rensselaer Polytechnique Institute, NY, USA. He worked as a staff member in University of 7th April, Libya; University of Malaya, KL, Malaysia; College of Technology at Al-Baha, KSA; and Qassim University, AlBuraydah, KSA. Actually, he is a Professor at the Electrical and Computer Engineering department, King Abdulaziz University, Jeddah, KSA. His research areas include analysis and control of power systems, and the application of modern/robust techniques using LMI. He published more than 90 papers in divers Conferences and Technical Journals.

Authors’ information 1

Electrical & Computer Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, KSA. Fax: 00966 2 6952 686 E-mail: [email protected]

A. H. Besheer (Corresponding author) got his PhD in Electrical Power & Machines in 2006 from Cairo University. He is currently holding a position of Assistant Professor in Electrical Engineering department, University of Tabuk, Saudi Arabia. His research interest includes; recent optimization technique, fuzzy system, renewable energy. Dr. Besheer is IEEE member in Industrial Electronics, Computational Intelligence Societies & CIS Social Media Subcommittee 2012 under Member Activities.

2

Electrical Engineering Department, Faculty of Engineering, University of Tabuk, KSA. On leave from Environmental Studies & Research Institute (ESRI), Menoufia University, Egypt. P.O. Box 7031, Unit No. 1, Tabuk 47315-3470 KSA Fax: 00966 4 4250 965 E-mail: [email protected]



748


Transient Stability Enhancement Using SMES-Based Fault Current Limiter Mehdi Jafari, Ali Vafamehr, Mohammad Reza Alizadeh Pahlavani

Abstract – In this paper, transient stability improvement of power system using fault current limiter (FCL) based on superconducting magnetic energy storage (SMES) is proposed. SMES technology is capable to make dc reactors with large values because of its superconducting property. Using such dc reactors in FCL structure can effectively limit the fault current during short circuits in power lines. To show the additional merits of such FCL structure, it is applied to transient stability of single machine infinite bus (SMIB) system with a double circuit transmission line. Analytical analysis including transient stability study is presented. In addition, simulation results using EMTDC/PSCAD software are included to confirm the analytical analysis accuracy. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Fault Current Limiter, Superconducting Magnetic Energy Storage, Transient Stability, Synchronous Generator

I.

Superconducting magnetic energy storage (SMES) technology firstly introduced by Ferrier in 1969 for storing electrical energy aims. But, more studies on this field illustrated that the SMES has better performance in power system operating applications than the energy storage application [14]-[18]. One of the applications of SMES technology is using its superconducting reactor in FCL structure [19]. SMES technology can produce dc reactor with large value which is needed in solid state FCLs. Such reactor will have large value and low losses due to its superconducting characteristic, while its costs will be high. However, developments on superconductors’ technology in recent years make its costs reasonable. Recently, some studies are performed on transient stability enhancement of power system using fault current limiters. These studies focus on superconducting FCLs (SFCLs) which operate by quenching of superconductor. Base of all of these structures is to limit the fault current and restore the bus voltage and so, help to proper power flow of parallel transmission lines during the fault. This action can help the generator to maintain its stability. Therefore, transient stability of power system will be improved in short circuit conditions [20]-[23]. However, these FCL structures have two main problems: firstly, they make the superconductor to change from superconducting state to normal state and vise versa which leads to power losses; secondly, they have recovery time due to quenching phenomenon [24]. In this paper, a SMES-based FCL is applied to enhancement of SMIB system’s transient stability. It is observed that using SMES technology can help the limiting characteristics of FCL.

Introduction

Power systems have become more expanded and complicated because of the growth of electric power demand. To reach the more reliability for power supply and overcome increasing power demand, the electric power systems are interconnected each other and the power generation systems are incremented. So, the available fault current levels may exceed the maximum short circuit rating of power system equipments. Under these conditions, limiting the fault currents is an important subject [1]-[4]. Traditionally, to moderate the cost of switchgear and bus replacements, the most common ways to limit highlevel fault currents are: splitting the power grid and introducing higher voltage connections, using currentlimiting fuses or series reactors or high-impedance transformers, and using complex strategies like sequential network tripping. A better idea to limit the fault currents and prevent high costs is usage of fault current limiters (FCLs) [5]-[8]. Different topologies of FCLs are introduced in literature, such as solid state, superconducting and resonance type FCLs [9]-[13]. Conventionally, solid state FCLs use a dc reactor to limit the increasing rate of current during the short circuit faults. To place a dc reactor in ac power system, thyristor or diode bridge is employed which changes the ac current of power line to dc current. The dc reactor used in such FCL structures should have some characteristics as follows: large value for better limitation of fault current, low power losses for high efficiency of structure and low cost for economic reasons.


749


Mehdi Jafari, Ali Vafamehr, Mohammad Reza Alizadeh Pahlavani

Such structure has not recovery time and has low power losses in comparison with the other structures which are used for this purpose. The analytical analysis for FCL operation and transient stability are presented in detail. The simulation study on SMIB system with double circuit transmission line including the proposed FCL is established using EMTDC/PSCAD software. Results are discussed carefully to show the performance of this FCL on transient stability.

II.

not be any recovery time and so, just after fault removal, FCL retreats from the system. A B

Isolation Transformer

Secondary Side

D2

D5

D4

D1

Power Circuit Topology

The three phase power circuit topology of the proposed FCL is shown in Fig. 1. This structure is composed of three main parts which are described as follows: 1) The three phase transformer in series with the system that is named “isolation transformer”. 2) The three phase diode rectifier bridge. 3) A SMES magnet as dc reactor with large value. As a conventional method, isolation transformer is needed to direct the line current to current limiting part. Three phase diode bridge is ac/dc converting tool for FCL and SMES magnet is the main part of FCL which has the current limiting task during the short circuit fault. II.2.

Primary Side

C

Power Circuit Topology and Principles of Operation II.1.

i L(t)

Diode Bridge

D3

D6

Vdc idc (t)

Lsmes SMES Magnet

Cryostat

Fig. 1. Three phase power circuit topology of the proposed FCL

III. Analytical Analysis of the Proposed FCL

Principles of Operation

This section deals with the analytical analysis of the proposed FCL’s performance in the current limiting during the fault. To calculate the equation of line current and dc current, two modes are considered as follows: A. Pre-fault condition B. Fault duration

In normal operation of power system, diode bridge rectifies line current to dc current and this dc current charges SMES magnet. When the dc current reaches to the peak of line current, SMES magnet behaves as short circuit because of its superconducting characteristic and so, voltage drop on it becomes almost zero. Very small voltage drop on SMES magnet is due to dc current ripple. By this way, total voltage drop on FCL will be related to voltage drop on diodes and isolation transformer which is negligible in comparison with the feeder’s nominal voltage. Therefore, FCL has not considerable effect on the normal operation of power system. As fault occurs, line current starts to increase. But, the SMES magnet limits its increasing rate and prevents fault current rapid increment. In this case, fault energy will be stored in SMES magnet. Since the value of SMES magnet is large, this current limitation is in acceptable range which will be shown in simulation results. By this manner, the voltage of connected bus does not experience considerable sag in comparison with the case of no FCL. So, power flow in system will not be affected by the fault and transient stability of system will be improved. By removal of the fault, system returns to its normal state and SMES magnet starts to discharge. Since the SMES magnet is in its superconducting state during the fault and has not quenching phenomenon, there will

III.1. Pre-Fault Condition In pre-fault condition, line current, iL ( t ) , and dc current, idc ( t ) have two modes: Charging mode and Discharging mode. Dc current and diodes’ currents are shown in Fig. 2. Enlarged view of these currents is shown in Fig. 3. Charging mode begins at t0 and ends at t1. In this mode, one diode from each phase is ON. So, the following equation can be written: Vm sin (ωt ) = ri ( t ) + L

di ( t ) dt

+ VD

(1)

where: i = idc = iL , r = rs + rFCL + rL , L = Ls + Lsmes + LL

rs and Ls: source side’s resistor and inductor, respectively;



750


rL and LL: load side’s resistor and inductor, respectively; rFCL : equivalent resistance of FCL’s elements; VD : voltage drop on each diode. Solving Eq. (1) leads to: Vm VD ⎛ ⎜ I 0 − z sin (ωt0 − ϕ ) + r ⎝ V V + m sin (ωt − ϕ ) − D z r

i (t ) = e

L(

−r

t − t0 )

⎞ ⎟+ ⎠

I 0 = ⎛⎜ 1 − ⎝

2

(

and ϕ = tg −1 Lω

r

By using Eq. (9), formula of Idc can be written as follow: ⎛ Tr ⎞ 2TVD I dc = I max ⎜ 1 − D ⎟ − ⎝ Lsmes ⎠ Lsmes

(3)

idc ( t ) = e

rD

Lsmes (

t −t1 )

⎛ 2VD ⎞ 2VD ⎜ I max + ⎟− rD ⎠ rD ⎝

III.2. Fault Duration Considering Fig. 4, fault occurs at tf. The fault condition has two modes: M1 and M2. Mode M1 is in t3 to t4 time interval and in this mode, diodes D4 and D5 are ON, while D1, D2, D3 and D6 are OFF. So, B phase current is equal to negative value of A phase current and C phase current is zero (Fig. 5). The zero sections which are appeared in the line current are due to the commutation of diodes. In the normal operation, commutation of diodes was based on their current, while, in the fault condition, there will be voltage commutation on diodes. In such condition, formula of A phase current can be written as follow:

(4)

Considering charging and discharging modes of current, it is possible to calculate the average value (Idc) and ripple (ir) of dc current. As it is obvious, dc current’s ripple leads to voltage drop on SMES magnet and therefore on FCL. For Idc and ir, we can write: I dc = I max −

ir

2

1 ir = ( I max − I 0 ) 2

(5)

Vm sin (ωt ) = ri A ( t ) + L

(6)

−

rD

Lsmes (

t2 − t1 )

⎛ 2VD ⎞ 2VD ⎜ I max + ⎟− rD ⎠ rD ⎝

By considering that e− x ≈ (1 − x ) and t2 − t0 = T

di A ( t ) dt

+ VD

Vm VD ⎛ ⎜ I 3 − z sin (ωt3 − ϕ ) + r ⎝ V V + m sin (ωt − ϕ ) − D z r

iA ( t ) = e

where, Imax is the peak of line current. I0 in Eq. (6) can be calculated from Eq. (4) in t2 instant as follow: I0 = e

(11)

It is obvious that the average of SMES magnet’s current is very close to the peak of line current. This value can be used for designing the SMES magnet.

where, rD is the diode’s resistance. Note that its value is very small. By solving Eq. (3), dc current formula will be as follow: −

(9)

It is important to note that its value is very small due to very small value of rD/Lsmes. Considering Eq. (9), the voltage drop on SMES magnet can be calculated as Eq. (10): r ⎛ 2V ⎞ Vsmes = 24 D ⎜ I max + D ⎟ (10) Lsmes ⎝ rD ⎠

)

didc + rD idc + 2VD = 0 dt

(8)

⎛ 2TrD ⎞ ⎛ 2VD ⎞ ir = ⎜ ⎟ ⎜ I max + ⎟ rD ⎠ ⎝ Lsmes ⎠ ⎝

(2)

Discharging mode starts at t1 and continues to t2. In this mode, both diodes of each phase are ON and therefore the line current is sinusoidal waveform and FCL behaves as a series transformer with short-circuited secondary. In such condition, dc current differential equation can be derived as follow: Lsmes

2V ⎞ 2V ⎞⎛ I + D − D 9 Lsmes ⎟⎠ ⎜ max r ⎟ r ⎝ D ⎠ D

Therefore, ir can be derived from combination of Eq. (6) and (8):

where: I 0 = i ( t0 ) , z = r 2 + ( Lω )

rDT

(7)

−r

L(

t −t3 )

(12) ⎞ ⎟+ ⎠ (13)

where: r = rs + rFCL , L = Ls + Lsmes , z = r 2 + ( Lω )

,

6 where T is the time period of power system, Eq. (7) can be simplified to Eq. (8):

(

2

)

ϕ = tg −1 Lω r and I 3 = iA ( t3 )



751


Id1

Id2

Id3

Id4

Id5

Similar to the previous mode, dc current follows the A phase current relation. The B phase and C phase currents can be driven from Eq. (14) with corresponding phase shift. This manner of variation will be repetitive for next steps in the fault condition.

Id6

Current

Idc

Time

IV.

Fig. 2. Dc current and diodes’ currents in the normal operation of power system Id1

Id2

Id3

Charging Mode

In this section, it will be shown that the proposed FCL can improve the transient stability of power system by absorbing generator energy during the fault and restoring the voltage of connected bus. Transient stability analysis is established on single line diagram of Fig. 6 which includes a power system with the proposed FCL at the beginning of one of the parallel lines. Before the fault, the system is operating in steady state. So, the transferred power can be expressed by:

Idc

Discharging Mode

Current

Imax I0

t0

t1

t2

Time

Fig. 3. Enlarged view of Fig. 2 Idc

Id1

Id2

Id3

Id4

Id5

Id6

Pg = ( EV X ) sin δ 0

M2

Current

M1

tf

Time

t3

t4 t5

B phase

C phase

Current

A phase

tf

t3

t4

t5

Time

Fig. 5. Line current and dc current in the fault condition

Note that the dc current relation is same as A phase current in this mode. For B phase and C phase, as pointed previously: iB ( t ) = −i A ( t ) and iC (t ) = 0 . In mode M2, Diodes D4 and D1 are in commutation, D5 is ON and other diodes are OFF. So, one diode from each phase is ON and therefore, sum of phases’ current is zero (Fig. 5). The line current in this time interval follows the Eq. (14): Vm VD ⎞ ⎛ ⎜ I 4 − z sin (ωt4 − ϕ ) + r ⎟ + ⎝ ⎠ (14) Vm VD sin (ωt − ϕ ) − + z r

iA ( t ) = e

−r

L(

(15)

where: - E : RMS line to line voltage of synchronous generator; - V : RMS line to line voltage of infinite bus; - X : Total reactance ( X t = X d + X t + X L 2 ); - X d : Unsaturated reactance of generator; - X t : Transformer reactance; - X L : Line reactance; - δ 0 : Load angle. A three phase fault is considered at the power line L2 which its location is determined by the factor of k. In this condition, without using the proposed FCL, the transferred power to the infinite bus will be reduced. So, it is possible that the synchronous generator be unstable. Using the proposed FCL at the beginning of the parallel lines can ensure stability of the synchronous generator. Since the FCL absorbs the generator’s active power and stores it in SMES magnet during the fault, it can be modeled by an element which consumes the generator’s power. It is possible to model the FCL by a resistor during the fault, but it should be considered that this resistor is not an ordinary resistor and should be calculated. For this calculation, we used the relation of energy stored in SMES magnet during the fault as follow:

Fig. 4. Dc current and diodes’ currents in the fault condition Idc

Transient Stability Analysis Using the Proposed FCL

t − t4 )

w=

1 2 Lsmes I smes 2

(16)

where, w is the energy stored in SMES magnet during the fault and Ismes is the SMES magnet’s current during the fault and can be written as Eq. (17), approximately.

where, I 4 = i A ( t4 ) .



752


I smes = I max +

Vdc t Lsmes

(17) R fcl

where, Vdc is the dc side voltage rectified by the diode bridge. So, the instantaneous power of SMES magnet can be concluded as follow: Pdc =

V2 dw = Vdc I max + dc t dt Lsmes

(18)

Vdc2 tf 2 Lsmes

Vdc =

⎧ Z a = b + c + ( bc a ) ⎪ ⎪ Z b = a + c + ( ac b ) ⎪ ⎛ j R fcl + jkX L X L ⎪ ⎪a = Z F + ⎜ ⎜ R fcl + j ( 2 X L + kX L ) ⎪⎪ ⎝ ⎨ 2 ⎪b = − X L R fcl + j ( 2 X L + kX L ) ⎪ ⎛ j R fcl + jkX L X L ⎪ ⎜ ′ = + c jX ⎪ ⎜ R fcl + j ( 2 X L + kX L ) ⎪ ⎝ ⎪ ′ ⎪⎩ X ′ = X d + X t

(19)

( (

π

sin( )Vm π 3

(20)

(

Rfcl (the model of FCL during the fault) can be concluded as Eq. (21): R fcl =

π Vm

Xd

Xt

L2 Rfcl

kX L

)

(

)

) ) )

⎞ ⎟ ⎟ ⎠

(24)

⎞ ⎟ ⎟ ⎠

where, Z F and X d ′ are fault impedance and unsaturated transient reactance of the generator, respectively. Considering three phase fault Z F is equal to zero, approximately. So, during the fault, output power of the synchronous generator can be expressed by:

Infinite Bus

XL

L1

(

(21)

⎛ ⎞ 3 3Vm 2 3 ⎜ I max + tf ⎟ ⎜ 2π Lsmes ⎟⎠ ⎝ PCC

Generator

(22)

I g = ( E ∠δ Z b ∠α 2 ) + ( ( E ∠δ − V ∠0 ) Z a ∠α1 ) (23)

Finally, considering this fact that the ac side and dc side active powers are equal and Vdc is equal to: 6

Lsmes → ∞

In Fig. 6, Rfcl is the proposed FCL’s model that will appear in current path during the fault. Fig. 7 shows the equivalent circuit during the fault with the proposed FCL after applying star to delta transformation in Fig. 6. To compute the output power of generator, firstly, we calculate output current of the synchronous generator ( I g ) in Fig. 7. So we have:

Considering Eq. (18) and fault duration equal to tf, average of active power absorbed by SMES magnet will be as Eq. (19): Pdc − ave = Vdc I max +

Lsmes = 0

⎧0 ⎪ → ⎨ π Vm ⎪ 2 3I max ⎩

(

) (

)

Pf = real I g ∗ E ∠δ = E 2 Z a cos α1 +

(1-k)X L

(

+ E

Fault

Fig. 6. Single line diagram of power system with the proposed FCL

2

)

Zb cos α 2 + ( EV Z a ) sin (δ + α1 − π 2 )

(25)

Considering Eq. (21), (22) and (25), larger value of Lsmes will lead to larger value of Rfcl and consequently better limitation of fault current and better enhancement of transient stability. However, large value of Lsmes will increase the design and construction difficulties and costs. Therefore, the value of Lsmes should be selected considering the maximum acceptable fault current. In other words, the maximum line current and minimum time in which the circuit breakers can open the line should be determined and then the value of Lsmes should be calculated by using Eq. (17).

Fig. 7. Equivalent circuit of power system during the fault with the proposed FCL

V. Simulation using performed on Fig. 6.

It is important to note that this resistor has minimum and maximum limits as follows:


Simulations EMTDC/PSCAD

software

is


753


Table I shows the parameters of simulation. To study the performance of proposed FCL on current limitation and transient stability improvement, a threephase fault considered at t = 20s with the duration of 0.2s (10 cycles of power frequency). Two sets of simulations are performed:

Voltage (V)

800 400 0 -400 -800 19

Rotor speed (rpm)

X t = 0.00145 H

23

25

27

29

A phase current

dc current

200

LsmesA = 1.04H , LsmesB = 0.52H

Current (A)

Power system parameters FCL data

21

By using the proposed FCL and considering the SMES magnet’s value equal to 1.04H (Case A) current of generator is limited to 3Imax as shown in Fig. 12. This makes the generator stable considering its current (Fig. 13) and its terminal voltage (Fig. 14). Rotor speed variation in such condition is shown in Fig. 15. According to this figure, generator’s speed experiences some oscillations and then becomes stable by damping these oscillations. In this case, maximum deviation of rotor speed from its base value (1500rpm) is about 7rpm.

VDF = 5V ,rD = 0.001Ω

100 0 19.9

20

20.1

20.2

-100 -200

Time (s)

1000

Fig. 12. Current of faulted line by using the proposed FCL (Case A) 150

500

100

0 19.9

19.95

20

20.05

20.1

20.15

Current (A)

C urrent (A )

1500

Fig. 11. Rotor speed response of generator without using the FCL

50Hz , 10kVA , a = 1

Isolation transformer parameters

25

Time (s)

X d = 1.227p.u. , X ′ = 0.394p.u. d 380/380 V , 50kVA ,

X L = 0.0064 H

24

3000

19

S b = 40kVA , Pm = 0.7p.u .

Transmission lines dc side parameters

23

0

TABLE I SIMULATION PARAMETERS Generator 4 poles, 380V, L-L RMS , 50Hz,

400V, L-L RMS

22

Fig. 10. Generator terminal voltage without using the FCL

In first case, value of Lsmes is calculated 1.04H and in second case it is calculated 0.52H considering Eq. (17) in section IV. Fig. 8 shows the current of faulted line without using the proposed FCL. Considering this figure, line current is increased extremely during the fault. Fig. 9 shows the generator current in such condition. This figure shows that the synchronous generator becomes unstable. Terminal voltage and rotor speed response of the generator are shown in Figs. 10 and 11, respectively. Unstability of the generator is obvious in these figures.

Infinite bus

21

Time (s)

Case (A): for current limitation to 3Imax; Case (B): for current limitation to 5Imax.

Transformer data

20

20.2

50 0 -50 -100

-500

-150

Time (s)

19

20

21

22

23

24

25

Time (s)

Fig. 8. Current of faulted line without using the FCL Fig. 13. Generator current with the proposed FCL (Case A) 800

250

Voltage (V)

Current (A)

500

0 -250

400 0 -400

-500 19

20

21

22

23

24

25

-800 19

Time (s)

20

21

22

23

24

25

Time (s)

Fig. 9. Generator current without using the FCL

Fig. 14. Generator terminal voltage with the proposed FCL (Case A)



754


Comparison of Figs. 15 and 19 shows that the peak of rotor speed’s oscillation in the case A is less than the case B. This result proves the discussions of section IV. As mentioned previously, larger value of SMES magnet will lead to larger value of Rfcl and consequently better enhancement of transient stability.

Rotor speed (rpm)

1515

1505

1495

1485 19

21

23

25

27

29

Time (s)

VI.

Fig. 15. Rotor speed response of generator wit proposed FCL (Case A)

In this paper, enhancement of power system’s transient stability by using a SMES-based FCL is proposed. The proposed SMES-based FCL can improve the transient stability in two ways: firstly by preventing the voltage sag in connected bus during the fault and consequently help to proper power flow in parallel transmission line; and secondly by absorbing the acceleration power of generator in the fault interval and storing it in SMES magnet. By this manner, the synchronous generator can keep its stability after the short circuit fault occurrence. In general, analytical analysis and simulation results using EMTDC/PSCAD software show that the proposed FCL has acceptable performance in transient stability enhancement in addition to fault current limiting.

In case B, value of the SMES magnet is set 0.52H. In this case, current of faulted line is limited to 5Imax as shown in Fig. 16. This value of SMES magnet has led to the stability of synchronous generator, too. The current and terminal voltage of generator (Figs. 17 and 18, respectively) proves this fact obviously. Also, Fig. 19 shows the rotor speed response of the synchronous generator in this case. Considering this figure, rotor speed is returned to its normal condition after some oscillations. Note that the deviation of rotor speed in this case is about 12rpm. A phase current

dc current

Current (A)

200 100 0 19.9

20

20.1

20.2

References

-100 -200

[1]

M. Jafari, S. B. Naderi, M. Tarafdar Hagh, M. Abapour, S. H. Hosseini, “Voltage Sag Compensation of Point of Common Coupling (PCC) Using Fault Current Limiter,” IEEE Trans. Power Del., vol. 26, no. 4, pp. 2638-2646, Oct. 2011. [2] S. B. Naderi, M. Jafari, M. Tarafdar Hagh, “Parallel Resonance Type Fault Current Limiter,“ IEEE Trans. Ind. Electron. Early Access, 2012. [3] S. P. Valsan and K. Sh. Swarup, “High-speed fault classification in power lines: theory and FPGA-based implementation,” IEEE Trans., Ind. Electron., vol. 56, no. 5, pp. 1793−1800, May 2009. [4] P. Rodriguez, A. V. Timbus, R. Teodorescu, M. Liserre and F. Blaabjerg, “Flexible active power control of distributed power generation systems during grid faults,” IEEE Trans., Ind. Electron., vol. 54, no. 5, pp. 2583−2592, Oct. 2007. [5] M. Fazli R. Jahani, Ali Fazli, J. Olamaei, H. A. Shayanfar, “New Method to Connect Wind Turbines Equipped With DFIGs to the Power Grid Using FCL and STATCOM,” International Review on Modelling and Simulations (IREMOS), vol. 3, no. 4, pp. 598603, Aug. 2010. [6] Lin Ye, LiangZhen Lin, and Klaus-Peter Juengst, “Application Studies of Superconducting Fault Current Limiters in Electric Power Systems,” IEEE Trans. Appl. Supercond., vol. 12, no. 1, Mar. 2002. [7] T. Madiba, M. W. Siti, A. A. Jimoh, “Protection of Three phase Power Transformers feeding a Load by using Current Limiting Technologies Devices in Power Grids,” International Review on Modelling and Simulations (IREMOS), vol. 3, no. 5, Oct. 2010(part A). [8] Mehrdad Tarafdar Hagh, Mehdi Abapour, “Nonsuperconducting Fault Current Limiter With Controlling the Magnitudes of Fault Currents,” IEEE Trans. Power Electron., vol. 24, no. 3, Mar. 2009. [9] H. Ohsaki, M. Sekino and S. Nonaka, “Characteristics of resistive fault current limiting elements using YBCO superconducting thin film with meander-shaped metal layer,” IEEE Trans., Appl. Supercond., vol. 19, no. 3, pp. 1818−1822, Jun. 2009. [10] V. Sokolovsky, V. Meerovich, I. Vajda and V. Beilin, “Superconducting FCL: design and application,” IEEE Trans.,

Time (s)

Fig. 16. Current of faulted line by using the proposed FCL (Case B) 150

Current (A)

100 50 0 -50 -100 -150 19

20

21

22

23

24

25

Time (s)

Fig. 17. Generator current with the proposed FCL (Case B)

Voltage (V)

800 400 0 -400 -800 19

20

21

22

23

24

25

Time (s)

Fig. 18. Generator terminal voltage with the proposed FCL (Case B) 1515

Rotor speed (rpm)

Conclusion

1505

1495

1485 19

21

23

25

27

29

Time (s)

Fig. 19. Rotor speed response of generator with proposed FCL (Case B)



755


Appl. Supercond., vol. 14, no. 3, pp. 1990−2000, Sept. 2004. [11] S. H. Lim, H. S. Choi, D. Ch. Chung, Y. H. Jeong, Y. H. Han, T. H. Sung and B. S. Han, “Fault current limiting characteristics of resistive type SFCL using a transformer,” IEEE Trans., Appl. Supercond., vol. 15, no. 2, pp. 2055−2058, Jun. 2005. [12] T. Hoshino, K. M. Salim, M. Nishikawa, I. Muta and T. Nakamura, “DC reactor effect on bridge type superconducting fault current limiter during load increasing,” IEEE Trans., Appl. Supercond., vol. 11, no. 1, pp. 1944−1947, Mar. 2001. [13] M. T. Hagh and M. Abapour, “Non-superconducting fault current limiters,” Euro. Trans. Electr. Power, vol. 19, no. 5, pp. 669−682, Jul. 2009. [14] M. Ferrier, “Stockage d’energie dans un enroulement supraconducteur,” in Low Temperature and Electric Power, Pergamon Press, pp. 425-432, 1970. [15] J.D. Rogers et al., “30-MJ SMES system for electric utility transmission stabilization,” Proc. IEEE, 71, pp. 1009-1107, 1983. [16] R.J. Loyd, S.M. Schoenung, T. Nakamura, W. Hassenzahl, J.D. Rogers, J.R. hrcell, D.W. Lieurance, and M.A. Hilal. “Design Advances in Superconducting Magnetic Energy Storage for Electric Utility Load Leveling,” IEEE Trans. Mag., vol. 23, pp. 1323-1330, 1987. [17] M. R. I. Sheikh, S. M. Muyeen, R. Takahashi, T. Murata, J. Tamura, “Wind Generator Stabilization by PWM Voltage Source Converter and Chopper Controlled SMES,” International Review of Automatic Control (IREACO), vol. 1, no. 3, pp. 311-320, Sept. 2008. [18] Roozbeh Kamali and Hooman Akbarzadeh, “Design of a Superconducting Magnetic Energy Storage Controller for Power System Stability Improvement,” International Review of Automatic Control (IREACO), vol. 4, no. 6, pp. 890-896, Nov. 2011. [19] Eung Ro Lee, Seungje Lee, Chanjoo Lee, Ho-Jun Suh, Duck Kweon Bae, Ho Min Kim, Yong-Soo Yoon, and Tae Kuk Ko, “Test of DC Reactor Type Fault Current Limiter Using SMES Magnet for Optimal Design,” IEEE Trans. Appl. Supercond., vol. 12, no. 2, pp. 850-853, Mar. 2002. [20] Y. Shirai, K. Furushiba, Y. Shouno, M. Shiotsu, and T. Nitta, “Improvement of power system stability by use of superconducting fault current limiter with ZnO device and resistor in parallel,” IEEE Trans., Appl. Supercond., vol. 18, no. 2, pp. 680−683, Jun. 2008. [21] K. Furushiba, T. Yoshii, Y. Shirai, K. Fushiki, J. Baba and T. Nitta, “Power system characteristics of the SCFCL in parallel with a resistor in series with a ZnO device,” IEEE Trans., Appl. Supercond., vol. 17, no. 2, pp. 1915−1918, Jun. 2007. [22] B. Ch. Sung, D. K. Park, J. W. Park and T. K. Ko, “Study on a series resistive SFCL to improve power system transient stability: modeling, simulation and experimental verification,” IEEE Trans., Ind. Electron., vol. 56, no. 7, pp. 2412−2419, Jul. 2009. [23] B. Ch. Sung and J. W. Park, “Optimal parameter selection of resistive SFCL applied to a power system using eigenvalue analysis,” IEEE Trans., Appl. Supercond., vol. 20, no. 3, pp. 1147−1150, Jun. 2010. [24] M. T. Hagh, M. Jafari and S. B. Naderi, “Transient stability improvement using non-superconducting Fault Current Limiter,” in Proc. Power Electronic & Drive Systems & Technologies Conference (PEDSTC), 2010, pp. 367−370.

Authors’ information Mehdi Jafari (S’10) was born in Ahar, Iran. He received the B.S. and M.Sc. degrees in power engineering from the University of Tabriz, Tabriz, Iran, in 2008 and 20011, respectively. His current research interests include fault current limiters, power quality and power system transient. He is currently with the Islamic Azad University, Sarab Branch. E-mail: [email protected] Ali Vafamehr was born in Urmia, Iran. He received the B.S. degree in Electronic Engineering from the University of Tabriz, Tabriz, Iran in 2005. He is currently M.S. student in Power Electronics at the Urmia University, Urmia, Iran. His research interests include power electronics, Multi level Converters and Fault Current Limiters. Faculty of Electrical Engineering, University of Urmia, Urmia, Iran. E-mail: [email protected] Mohammad Reza Alizadeh Pahlavani received his degree in electrical engineering from Iran University of Science and Technology (IUST) in 2009. He is the author of more than 110 ISI, Transactions, Journals, International, and National Conference papers in field of electromagnetic systems, electrical machines, power electronic, FACTS devices, and pulsed power. Address: Department of Electrical Engineering, Malek-Ashtar University of Technology (MUT), Tehran, Iran. E-mail: [email protected]



756


Dynamic and Steady-State Operational Performance of Induction and Synchronous Reluctance Motors Powered by PV Generator with MPPT Mohammad S. Widyan1, Ghassan S. Marji2, Anas I. Al Tarabsheh1 Abstract – This paper presents comprehensive dynamic and steady-state performance characteristics of Induction Motor (IM) and Synchronous Reluctance Motor (SRM) powered by Photovoltaic (PV) cells. The PV generator is interfaced to the motors via DC-DC buck-boost switch mode converter, DC-AC switch mode inverter and filter. The duty cycle of the DC-DC converter is controlled such that to track the Maximum Power Point (MPP) of the output characteristics of the PV generator. The DC-AC inverter has fixed modulating and frequency indices. More sinusoidal input voltage to motors is accomplished by placing LC filter across the output terminals of the inverter. The dynamical study comprises successive step changes on the static load coupled to the motors at various solar irradiances. The PV cells are designed such that their maximum power point, at full solar illumination, is at the rated conditions of each motor. The results are compared to the case of supplying the motors by fixed terminal voltage. Furthermore, the dynamical responses of the motors after successive step changes in the solar intensities are obtained. Steady-state output characteristics of PV-powered motors at different solar illuminations are also outlined and compared with the case of supplying them by fixed terminal voltage. The system under investigation is fully modeled including the nonlinearity of the solar-cell generator and all numerical simulations are carried out based on the nonlinear dynamical mathematical model of the complete system. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Induction Motor, Synchronous Reluctance Motor, PV Array Design and Characteristics, DC-DC Buck-Boost Switch Mode Converter, Maximum Power Point Tracker

Nomenclature Induction Motor (IM)

ird , irq isd , isq

J Llr Lm Lrr Lss P Rr

d-axis and q-axis rotor current components, respectively d-axis and q-axis stator current components, respectively Rotor moment of inertia Rotor winding leakage reactance

V2d , V2q

d-axis and q-axis motor terminal voltage

ωr ωs δr

Rotor rotational speed Synchronous radians frequency Rotor angle of the motor

Synchronous Reluctance Motor (SRM)

Bm

Rotor mechanical friction coefficient

Mutual inductance between stator and rotor windings Self inductance of the rotor winding

isd , isq

Self inductance of the stator winding

J Ld , Lq

d-axis and q-axis stator current components, respectively d-axis and q-axis damper windings currents, respectively rotor moment of inertia d-axis and q-axis stator inductances, respectively d-axis and q-axis damper windings self inductances d-axis and q-axis mutual indutances, repectively Number of poles d-axis and q-axis damper windings resistances, respectively

ikd , ikq

Number of poles Rotor winding resistance

Rs

Stator winding resistance

TL Vrd , Vrq

Load torque

Lkkd , Lkkq Lmd , Lmq

d-axis and q-axis rotor voltage components, respectively. Vrd = Vrq = 0 in case of

P Rkd , Rkq

squirrel cage rotor


757


Mohammad S. Widyan, Ghassan S. Marji, Anas I. Al Tarabsheh

Rs

Stator winding resistance

TL

Load torque

V2 d , V2q

d-axis and q-axis motor terminal voltage

ωr ωs δr

Rotor rotational speed

Oriented field control is used for the induction motor control and PWM control strategy is used for the converters. The dynamic and steady-state characteristics of PVpowered DC and permanent-magnet DC motors at different solar intensities, different loading conditions and different system controllers & configurations have been proposed [6]-[15]. Similar studies for AC machines are presented [4] and [16]-[21]. Normally, systems incorporated with PV generators are forced to adjust their operating point at the location where the maximum power is extracted. A survey of the algorithm for seeking the Maximum Power Point (MPP) is proposed [22]. They are categorized into direct and indirect methods. Indirect methods are featured by the fact that the MPP is estimated based on the measurements of the voltage and/or current, the illumination or based on some mathematical manipulations for certain given PV modules and therefore none of them is able to obtain the MPP exactly. On the other hand, direct methods offer the advantage that they obtain the actual MPP and therefore they are suitable for all solar illuminations and working temperatures. All algorithms, direct and indirect, can be included in the DC-DC or DC-AC converter controllers [23]. In this study, the dynamic and steady-state characteristics of three-phase IM and SRM at different solar irradiances and various loading conditions as compared to the case of supplying them by fixed terminal voltage are presented. The PV generators and the two AC motors are interfaced together via DC-DC buck-boost switch mode converter, DC-AC switch mode inverter and LC filter. The duty cycle of the DC-DC converter is automatically adjusted such that to keep the PV generator supplying its maximum power. The MPP algorithm used in this study is the open-circuit voltage PV generator method which is one of the indirect MPPT discussed in [22]. In this method the open-circuit voltage of the PV generator is measured periodically while the DC-DC converter switch is open. It is reported that the MPP of the PV generator is located at the point where the voltage is at 0.73 to 0.8 of the open-circuit voltage. In this study, it is taken as 0.75. The modulating and frequency indices of the DC-AC inverter are kept constant during this study. The LC filter is used to suppress the unwanted inherent harmonics incorporated in the output voltage of the DC-AC inverter. The PV generator is designed to provide its maximum power at the rated conditions of the motors when the PV array is fully illuminated. The nonlinearities of the output characteristics of the PV system are included. The dynamical study comprises the responses after successive step changes on the load coupled to the motor at various solar irradiances. Additionally, the responses of the systems after successive step changes on the solar intensities at certain loading condition are discussed. The rest of the paper is structured as follows: systems description and main design characteristics of the photovoltaic cells are

Synchronus radians frequency Rotor angle of the motor

DC-DC Buck-Boost Switch Mode Converter Controller D TD

Duty ratio or duty cycle of the controller Controller time constant

VPV

Input voltage to the DC-DC converter, which is the output voltage from the PV generator Voltage set value of the controller, which is the output PV voltage corresponding to the MPP resulting from the measurement process

Vset

I.

Introduction

Three-phase induction motors are asynchronous machines, running below the synchronous speed. They are considered the workhorses for most industrial systems. The rotor of induction machine comes in two forms; squirrel cage or slip-ring. They are characterized by their lower prices, robust construction and very little maintenance. Synchronous Reluctance motor, on the other hand, combines the best features of the synchronous and induction motors. Similar to induction motor, synchronous reluctance motor is self starting via getting the benefits of the rotor damper windings. Similar to synchronous machine, it runs at constant steady-state rotational speed and has no DC rotor circuit. Basically, synchronous reluctance motor is brushless and has wide industrial applications. Due to the progressive growth of the PV-market, continuous reduction in the prices of photovoltaic technology and the ongoing improvements of their efficiency, their applications as static pollution-free power supplies for DC and AC stand-alone electrical systems is considered a promising area [1] and [2]. A system composed of PV generator, DC-DC converter, a storage battery and vector controlled three-phase induction motor with MPPT is investigated [3]. It is concluded that this approach is feasible and effective. Optimization for performance of a photovoltaic threephase induction motor driving pumping system is studied [4]. The optimization of the motor efficiency is described. Simulation results show that the proposed system permits to combine the performances of the system with constant efficiency. A bond graph model to enable testing the PV-powered water pumping system consisting of DC-DC and DC-AC converters and three-phase induction motor including numerical simulations for MPPT is investigated [5].



758


presented in Section II. The dynamical mathematical model of the PV generator, DC-DC converter and its controller, DC-AC inverter, LC filter, IM and SRM are all outlined in Section III. Section IV discusses the numerical simulation results. Finally, conclusions are drawn in Section V.

II.

polynomial curve approximation in addition to the corresponding P/I characteristics. Clearly, the 10th order polynomial curve fitting is practically accurate enough to characterize the output of the PV generator. As for the SRM, its rated conditions are 600W, 400V and 3A. Using Eq. (1) and with ma = 0.8 and rated motor power factor of 0.8, the voltage and current of the photovoltaic cells at MPP at full solar irradiance should be about 815V and 2.03A. N s and N p

Systems Description and Main Design Characteristics of Photovoltaic Cells

for SRM should be as given in Table I.

A brief summary for photovoltaic cells design and output characteristics along with the descriptions for the elements of the systems under study are presented in this section. Fig. 1 shows a PV generator with N s series and

TABLE I TECHNICAL SPECIFICATIONS OF THE DESIGNED PV ARRAY AT FULL ILLUMINATION

N p parallel modules feeding three-phase AC motor via

IM SRM

DC-DC switch mode converter, DC-AC switch mode inverter and LC filter. PV generator series modules are used to increase the output voltage of the array and parallel modules are used to increase the current capability of the generator. The duty cycle of the DC-DC converter is automatically adjusted to track the MPP of the PV generator at certain solar irradiance. The LC filter is placed as an intermediate stage between the inverter and the AC motor to reduce the ripple of the output voltage and therefore to provide the AC motor with a higher quality power. The filter is designed such that to have its cut off frequency at almost 13 times the fundamental frequency of the system which are normally the most dominant harmonics in the line-line output voltage for low values of frequency modulating index which should be odd and multiple of three [24]. The rated conditions of the induction motor are 1hp, 200V and 4.27A. The rms value of the fundamental component of the line-line voltage output from the switch mode inverter VLL is:

VLL =

3 2 2

maVo

VOC

I SC

VMPP

I MPP

Ns

NP

554 1100

3.41 2.37

409 815

2.82 2.03

60 119

46 32

III. System Dynamical Mathematical Model This section presents the detailed mathematical model of the system which includes the PV generator, the DCDC converter and its controller, the DC-AC inverter, the LC filter and the two motors. III.1. PV Generator The output DC voltage of the PV generator as function of the output DC current is highly nonlinear. In this study, it has been approximated by the 10th order polynomial for different solar illuminations considered as: 11

−n VPV = ∑ α n I 11 PV

(2)

n =1

where α1 , α 2 … α11 are the polynomial coefficients.

(1)

III.2. DC-DC Switch Mode Converter and its Controller

where Vo input DC voltage to the inverter (output

DC-DC converters are devices used to change the input DC voltage to another level. Buck (step-down) converters have an output average voltage equal to or lower than the input average voltage. The output voltage of boost (step-up) converters is equal to or greater than the input average voltage. In buck-boost converters, the average value of the output voltage is higher, equal to or lower than the input voltage. They are used in regulated power supplies, motor drive systems and renewable energy systems an intermediate stage for controlling purposes. The output voltage is controlled by the duty ratio D which is the ratio between the on-state duration of the switch ton and the switching period Ts . The duty ratio is usually controlled by Pulse Width Modulation (PWM) technique. The circuit of commonly used buckboost switch mode converter is given in Fig. 3 [24].

voltage from the DC-DC converter), ma amplitude modulating index of the Pulse Width Modulation (PWM) technique of the inverter. With ma = 0.8, the output voltage of the PV array at full illumination will be about 409V. With the rated motor power factor of 0.8, the current output from the PV array at full illumination will be about 2.82A. To have this voltage and current at the maximum power point of the V/I characteristic of the PV array at full illumination, N s and N p should be as given in Table I, [13] & [14]. The V/I characteristics of the designed PV array at full illumination, 0.75 of full illumination and 0.5 of full illumination are given in Figs. 2 including their 10th order



759


Fig. 1. PV array consisting of N s series- and N p parallel-connected modules feeding three-phase AC motor via DC-DC Converter, DC-AC Inverter and LC Filter 600

Full Illumination

Voltage (V)

500 400

0.75 of Full Illumination

300

Solid Line: Actual Charateristics Dashed Line: 10th Order Polynomial Approximation

200 100 0

0

0.5

1


1.5

2

2.5

3

3.5

(a) 1200 1000

Power (W)

Full Illumination 800 600


400


200 0

0

0.5

1

1.5

2

2.5

3

3.5

Current (A) (b) Figs. 2. (a) V/I characteristics at full solar illuminations, 0.75 and 0.5 of full illumination for a PV array consisting of 60 series and 46 parallel-connected modules and the 10th order polynomial fitted curve, (b) the corresponding P/I characteristics

The output voltage of the converter Vo , which is the input voltage to the DC-AC switch mode inverter is given by: D (3) Vo = VPV 1− D where VPV is the input voltage to the converter, which is the output voltage of the photovoltaic cells and D is the duty ratio. Clearly, the output voltage of the converter is controllable via the duty ratio D . In this study, the output voltage of the DC-DC converter is adjusted such that to have it at the Maximum Power Point (MPP) of the PV generator output characteristic at the given solar illumination. The MPPT technique used is the open-circuit approach.

Fig. 3. Buck-boost switch mode converter circuit



760


The open-circuit voltage of the PV generator is measured when the switch of the DC-DC converter is open. The MPP of the PV generator is normally in the range of 0.73 to 0.8 of its open-circuit voltage [22]. In this study, it is taken as 0.75. The output of this process is taken as the set value for the duty ratio controller. The block diagram of the MPPT controller system is given in Fig. 4. The output is the duty cycle which will be, for a given operating solar illumination, adjusted to provide the output voltage of the PV generator corresponding to the MPP.

the frequency of the triangle signal to the frequency of the control signal is called the frequency modulating index m f . The output of the three-phase inverter is three-phase waveform with an rms value for the line voltage VLL as given in Eq. (1) at a fundamental frequency equal to the same frequency of the control signals. The magnitude of the output phase voltage Vφ is:

Vφ =

1 2 2

maVo

(7)

The output line voltage has some harmonics depending on the values of ma and m f . These unwelcome harmonic components can be greatly reduced using LC filter. Fig. 4. Block diagram of the MPPT controller

Mathematically, the measurement process of the opencircuit voltage of the PV generator can be expressed as: Tm

dVset = 0.75VPV −oc − Vset dt

(4)

and the equation governing the duty cycle D of the DCDC converter is: Fig. 5. Circuit of three-phase inverter

TD

Vset dD = −D dt VPV + Vset

(5)

III.4. LC Filter The LC filter is used to suppress the inherent unwanted harmonic components in the output voltage of the inverter. The circuit of the LC filter is shown in Fig. 6. The input voltage V1 represents the output phase

Substituting for VPV from Eq. (2) gives: TD

dD = dt

Vset 11

∑

−n α n I 11 PV

−D

(6)

voltage from the DC-AC inverter and V2 is the filtered output voltage which is the input to the three-phase motor.

+ Vset

n =1

At steady-state where the time derivative terms are zero, Vset is equal to 0.75VPV −oc and the operating point of the PV generator will always be at 0.75 of its opencircuit voltage which is the point corresponding to the maximum power output from the PV generator. III.3. DC-AC Switch Mode Inverter The circuit of three-phase inverter is shown in Fig. 5. It consists of six switches and six anti-parallel diodes. The input is a DC voltage which is the output of the DC-DC converter. The switches are controlled by the Pulse Width Modulation (PWM) technique where three control signals of same amplitude phase shifted by 120° in time crossing a triangle signal at a certain amplitude. The ratio between the peak value of the control signal to the peak value of the triangular signal is called the amplitude modulating index ma and the ratio between

Fig. 6. Circuit of LC filter

The equation of the voltage across the inductor L in phase notation is:

⎡ L 0 0 ⎤ ⎡iLa ⎤ ⎡V1a − V2 a ⎤ ⎢ 0 L 0 ⎥ d ⎢i ⎥ = ⎢V − V ⎥ ⎢ ⎥ dt ⎢ Lb ⎥ ⎢ 1b 2b ⎥ ⎢⎣ 0 0 L ⎥⎦ ⎢⎣iLc ⎥⎦ ⎢⎣V1c − V2c ⎥⎦


(8)


761


Applying Park’s transformation provides the following in d-q stationary reference frame (assuming balanced conditions) [25]: L

L

diLd = V1d − V2 d + ω LiLq dt

diLq dt

= V1q − V2q − ω LiLd

Lm

(9) Lm

C

dV2q dt

= iCq − ωCV2d

dt

J

d ωr 3 P = Lm ird isq − Lm irq isd − TL dt 4

(

C

dV2q dt

disd di + Lmd kd = V2 d − Rs isd + dt dt + ωr Lq isq + ωr Lmq ikq

Ld

(12)

Lq

(13)

disq dt

Lmd

= iLq − isq − ωCV2d

(19)

(20)

(21)

and for the three-phase SRM in d-q stationary reference frame is [27]:

(12) and (13) will be: dV2 d = iLd − isd + ωCV2 q dt

)

d δ r ωr − ω s = ωs dt

but iCd = iLd − isd and iCq = iLq − isq and therefore Eqs.

C

(18)

dirq

= Vrq − (ωs − ωr ) Lrr ird + dt − (ωs − ωr ) Lm isd − Rr irq

+ Lrr

(11)

Applying Park’s transformation provides the following in d-q stationary reference frame (assuming balanced conditions) [25]: dV C 2 d = iCd + ωCV2 q dt

disq

(10)

The equation of the current through the capacitor C in phase notation is: ⎡C 0 0 ⎤ ⎡V2 a ⎤ ⎡iCa ⎤ ⎢ 0 C 0 ⎥ d ⎢V ⎥ = ⎢i ⎥ ⎢ ⎥ dt ⎢ 2b ⎥ ⎢ Cb ⎥ ⎢⎣ 0 0 C ⎥⎦ ⎢⎣V2c ⎥⎦ ⎢⎣iCc ⎥⎦

disd di + Lrr rd = Vrd + (ωs − ωr ) Lrr irq + dt dt + (ωs − ωr ) Lm isq − Rr ird

(14)

Lmq (15) J

In this study, ma is 0.8 and m f is chosen as 15. For

+ Lmq

dikq

= V2 q − Rs isq + dt − ωr Ld isd − ωr Lmd ikd

disd di + Lkkd kd = − Rkd ikd dt dt

disq dt

(22)

+ Lkkq

dikq dt

= − Rkq ikq

⎛ Ld isd isq + Lmd ikd isq + ⎞ d ωr = 0.75 P ⎜ + ⎜ − Lq isq isd − Lmq ikq isd ⎟⎟ dt ⎝ ⎠

(23)

(24)

(25)

(26)

− TL − Bmωr

these values, the lowest frequency of the most dominant harmonic components is 13 times the fundamental frequency of the line voltages.

dδ r = ω s − ωr dt

(27)

III.5. IM and SRM

V1d appearing in Eq. (9) can be written as V1 cos (δ r )

The nonlinear dynamical mathematical model of the three-phase IM in d-q stationary reference frame can be summarized as [26]:

and V1q appearing in Eq. (10) can be expressed as

Lss

Lss

disd di + Lm rd = V2 d + ωs Lss isq + dt dt + ωs Lm irq − Rs isd disq dt

(16)

dirq

= V2 q − ωs Lss isd + dt − ωs Lm ird − Rs isq

+ Lm

V1 sin (δ r ) where V1 is the phase output voltage from the DC-AC inverter. Eqs. (4), (6), (9), (10), (14), (15), (16)(21) represent the nonlinear dynamical mathematical model of PV-powered three-phase IM and Eqs. (4), (6), (9), (10), (14), (15), (22)-(27) represent the nonlinear dynamical mathematical model of PV-powered threephase SRM. Both are fed via controlled DC-DC converter, DC-AC inverter and LC filter with MPPT including the dynamics of the PV generator. The numerical parameters of the system are presented in Appendix.

(17)



762


IV.

voltage increases to 3.93A and to 3.75A in case of PV cells with a motor phase terminal voltage of about 81V. The rotational speed of the motor in case of fixed terminal voltage is now 1733rpm and 1713rpm in case of PV cells. The output mechanical power from the motor is about 574W. After the new steady-state, the load coupled to the motor jumps to 0Nm. The drawn current from the PV generator becomes 3.17A at a motor phase terminal voltage of 101V. The dynamics of the system after successive step changes on the solar irradiances are presented. Fig. 9 shows the IM stator current, rotational speed and duty cycle of the DC-DC converter after step change of the solar intensity from full illumination to 0.75 of full illumination followed by step increase to 0.9 of full illumination at a fixed load torque of 3.6Nm. With fully illuminated PV array, the motor draws a current of 4.05A at rotational speed of 1722rpm with an operating duty cycle of 0.48. All of a sudden, the solar irradiance reduces to 0.75 of full illumination. The stator current becomes about 3.92A, the rotational speed becomes about 1699rpm and the duty cycle increases to about 0.64. As the solar intensity increases again to 0.9 of full illumination, the stator current becomes about 3.98A, the rotational speed increases to about 1713rpm and the duty cycle returns back to about 0.49. The steady-state characteristics of the PV-powered induction motor which is the rotational speed as function of the load torque with full illumination, 0.9 of full illumination and 0.75 of full illumination as compared to the case of supplying it by fixed terminal voltage are presented in Fig. 10. Apparently, as the load increases the rotational speed decreases. The characteristic in case of fully illuminated PV cells is slightly higher as the voltage supplied is slightly higher. At the rated conditions they are almost identical as the PV cells are designed to provide their maximum power point at the rated conditions of the motor. With 0.75 of full illumination, the characteristic is lower with a maximum output mechanical power of about 514W ( ≈ 0.69hp). For higher loads, the partially illuminated PV cells could not develop the power required.

Numerical Simulation Results

Numerical simulation results carried out at PVpowered three-phase IM and SRM are presented in this section at different solar illuminations. In all of the study, the results in case of feeding the motors by PV cells are compared with the case of supplying them by fixed terminal voltage. IV.1. Induction Motor

Fig. 7 shows the stator current of the IM, its rotational speed, the PV generator and motor terminal voltages after a step change in the load torque from 2Nm to 4.2Nm followed by step change to no-load (0Nm) in case of fully illuminated PV cells and fixed terminal voltage. Initially at 2Nm, the drawn current from the fixed terminal voltage source is about 3.66A at a rotational speed of 1760rpm and from the fully illuminated PV array is about 3.63A at a rotational speed of 1759rpm. The terminal phase voltage of the induction motor in case of PV array at this loading condition is about 115V at an output mechanical power of 368W. All of a sudden, the load torque jumps to 4.2Nm. Interestingly, the steady-state stator current in case of PV cells and fixed terminal voltage increases to the rated value of 4.27A as the PV array is designed to provide the rated voltage of the motor at the rated current. The steady-state rotational speed is about 1708rpm which is almost identical in both cases. The output power from the motor is about 750W ( ≈ 1hp). When the static load coupled to the motor becomes 0Nm (no-load) the steady-state value of the stator current in case of fixed terminal voltage becomes about 3.57A and 3.53A in case of fully illuminated PV array. The corresponding PV generator terminal voltage becomes 445V. In all loading conditions the motor terminal voltage is about 115V. This constant running voltage comes as a result of the DC-DC buck-boost switch mode converter where its duty cycle is changed to keep the constant voltage corresponding to the MPP. Numerical simulations for partially illuminated PV cells are carried out. Fig. 8 shows the stator current of the three-phase IM, the rotational speed, the PV terminal voltage and the motor terminal voltage after a step change in the load torque from 2Nm to 3.2Nm followed by step reduction to 0Nm (no-load) at 0.75 of full solar illumination as compared with the case of supplying the motor by fixed terminal voltage. Initially at 2Nm, the motor draws 3.66A from the fixed terminal voltage supply and 3.34A from the PV array. The rotational speed in case of PV cells is about 1749rpm and in case of fixed terminal voltage is about 1760rpm. The terminal phase voltage of the motor in case of PV cells at this operating condition is about 100V and the terminal voltage of the PV cells is about 353V with an output motor mechanical power of about 366W. All of a sudden, the load torque jumps to 3.2Nm. The steady-state value of the stator current in case of fixed terminal

IV.2. Synchronous Reluctance Motor

Fig. 11 shows the stator current and motor rotational speed of the SRM after a step change in the load torque from 2Nm to the rated torque of 3.6Nm followed by step reduction to no-load (0Nm) for the case of supplying the motor by PV cells at full illumination as compared to the case of supplying it by fixed terminal voltage. The corresponding PV cells and the motor terminal voltages are also presented. Initially, the SRM draws the same current of about 2.79A in both cases of fully illuminated PV cells and fixed terminal voltage as the DC-DC converter controller is designed to adjust the voltage across the terminals of the motor in case of PV cells to the rated value.



763


Stator Current (A)

4.5

Solid Line: PV at Full Intensity Dashed Line: Fixed Terminal Voltage

4

3.5

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

7

8

9

10

Rotational Speed (rpm)

1850

Solid Line: PV at Full Intensity Dashed Line: Fixed Terminal Voltage

1800 1750

PV Generator and Motor Terminal Voltages (V)

1700

0

1

2

3

4

5

500 400

Solid Line: PV Generator Terminal Voltage Dashed Line: Motor Terminal Voltage

300 200 100

0

1

2

3

4

5

6

Time (s)

Fig. 7. IM stator current, its rotational speed and the corresponding PV generator and motor terminal voltages after successive step changes of the load torque from 2Nm to 4.2Nm to 0Nm (no-load) with PV cells at full illumination as compared to the case of supplying the motor by fixed terminal voltage

Stator Current (A)

4

Solid Line: Fixed Terminal Voltage Dashed Line: PV at 0.75 of Full Intensity

3.5

3

0

1

2

3

4

5 Time (s)

6

7

8

9

10

7

8

9

10

8

9

10


1850

Solid Line: Fixed Terminal Voltage Dashed Line: PV at 0.75 of Full Intensity

1800

1750


1700

0

1

2

3

4

400

5 Time (s)

6

Solid Line: PV Generator Terminal Voltage Dashed Line: Motor Terminal Voltage

300 200 100 0

0

1

2

3

4

5 Time (s)

6

7

Fig. 8. IM stator current, its rotational speed and the corresponding PV generator and motor terminal voltages after successive step changes of the load torque from 2Nm to 3.2Nm to 0Nm (no-load) with PV cells at 0.75 of full illumination as compared to the case of supplying the motor by fixed terminal voltage Stator Current (A)

6 5 4 3 2

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30


1740 1720 1700

Duty Cycle of the DC-DC Converter

1680

0.8 0.7 0.6 0.5 0.4

Time (s)

Fig. 9. IM stator current, rotational speed and duty cycle of the DC-DC converter after successive step changes in the solar illumination from full illumination to 0.75 to 0.9 of full illumination for a fixed load torque of 3.6Nm



764


1800

1780

Full Solar Illumination 1760

Rotational Speed (Nm)

Fixed Terminal Voltage 1740

0.75 of Full Solar Illumination

1720

1700

0.9 of Full Solar Illumination 1680

1660

1640

1620

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Developed Torque (Nm) Fig. 10. Steady-state characteristics of three-phase induction motor when fed by fixed terminal voltage, fully illuminated PV generator and partially illuminated PV generator at 0.9 and 0.75 of full intensity

power output from the motor is about 314W. Upon step increase of the load torque to 2.6Nm, the steady-state value of the stator current increases to 2.77A in case of feeding the motor by the partially illuminated PV generator and 2.88A when the motor is powered by fixed terminal voltage. As a result the steady-state value of the terminal voltage of the PV cells decreases to about 183V. Despite this, the steady-state value of the motor terminal voltage remains about 217V. Upon reducing the load coupled to the motor to 0Nm, the current drawn by the motor becomes 2.54A in case of powering it by the PV generator and becomes 2.70A in case of supplying it by fixed terminal voltage. The corresponding terminal voltage of the PV generator is now about 700V. Clearly, in all operating conditions of this case, the motor rotational speed and the motor terminal voltage are both kept constant. Concerning the steady-state analysis of the synchronous reluctance motor, it is well known that it has a constant rotational speed. Fig. 13 shows the drawn current as function of the load torque coupled to the SRM for the cases of feeding the motor by fixed terminal voltage, feeding the motor by PV generator with full illumination and 0.8 of full illumination. It can be concluded that the characteristics in cases of fully illuminated PV generator and fixed terminal voltage are almost identical. With partially illuminated PV generator of 0.8 of full illumination, the characteristic is lower with a maximum output power of about 400W, which is almost 67% of full load.

The fully illuminated PV cells voltage is about 861V and the motor phase terminal voltage is about 230V. The output mechanical power from the motor is about 314W. Upon the step increase in the load torque to 3.6Nm, the stator current increases in both cases to a steady-state rated value of 3A. Despite the reduction of the PV cells terminal voltage to the steady-state value of about 803V, the steady-state terminal voltage of the motor remains constant. The output mechanical power from the motor is about 565W. As the static load coupled to the motor jumps to 0Nm, the stator current becomes about 2.70A and the PV generator terminal voltage increases to 880V. Again the motor terminal voltage has remain constant as a result of the automatic adjusing of the duty cycle of the DC-DC switch mode converter. In all operating conditions, the synchronous reluctance motor has never lost synchronism with steady-state rotational speed of 1500rpm which the synchronous speed of the motor. Numerical simulations at 0.8 of full illumination have also been executed. Fig. 12 shows the stator current of the SRM, the rotational speed, the PV generator and motor terminal voltages after a step change in the load torque from 2Nm to 2.6Nm to 0Nm for the case of supplying the motor by PV cells at 0.8 of full illumination as compared with the case of supplying the motor by fixed terminal voltage. Initially, the SRM draws 2.65A in case of feeding the motor by PV generator while the current drawn from the fixed terminal voltage is about 2.8A. The PV cells terminal voltage is about 580V and the motor terminal voltage is about 217V. The mechanical Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


765


Stator Current (A)

3.4

Solid Line: Fixed Terminal Voltage Dashed Line: PV at Full Intensity

3.2 3 2.8 2.6 0

0.5

1

1.5

0.5

1

1.5


1540

1500 1480



1520

0

1000 800

Solid Line: Motor Terminal Voltage Dashed Line: PV Generator Terminal Voltage

600 400 200

0

0.5

1

1.5

Time (s)

Fig. 11. SRM stator current, rotational speed, PV generator voltage and motor terminal voltage after successive step changes in the load torque from 2 to 3.6 to 0Nm when the motor is fed by PV cells at full illumination as compared to the case of supplying the motor by fixed rated terminal voltage

Stator Current (A)

2.9 2.8 2.7


2.6 2.5

0

0.5

1

1.5

1

1.5


1530 1520

1500 1490



1510

0

0.5

800

Solid Line: Motor Terminal Voltage Dashed Line: PV Generator Terminal Voltage

600 400 200 0

0

0.5

1

1.5

Time (s)

Fig. 12. SRM stator current, rotational speed, PV generator and motor terminal voltages after successive step changes in the load torque from 2 to 2.6 to 0Nm (no-load) when the motor is fed by PV cells at 0.8 of full illumination as compared to the case of supplying the motor by fixed rated terminal voltage

V.

generator at the MPP. The MPPT system adopted in this study is the open-circuit approach where the point corresponding to the maximum power is at 0.75 of the open circuit voltage at certain given illumination. The dynamical study is carried out at different loading conditions and various realistic solar illuminations. The nonlinearity of the output characteristics of the PV generator is taken into account. The steady-state output characteristics of both motors at different solar irradiances are obtained and compared with the case of feeding it by fixed terminal voltage. It can be concluded that feeding the IM and SRM from PV generator with MPPT is feasible and effective way as far as the power generated by the PV generator is practically realistic.

Conclusion

The dynamic and steady-state characteristics of threephase induction and synchronous reluctance motors powered by PV cells have been investigated. Numerical simulations with fully and partially illuminated PV array for the two motors as compared to the case of feeding them by fixed terminal voltage are outlined. The PV arrays feeding the motors are designed such that to provide their maximum power point at the rated conditions of the motors. The two motors are interfaced to the PV generator via DC-DC buck-boost converter, DC-AC switch mode inverter and LC filter. The duty cycle of the DC-DC converter is automatically controlled such that to keep the operating point of the PV



766


3.2

3.1

Stator Current (A)

3

Fixed Terminal Voltage 2.9

PV Generator at Full Illumination

2.8

2.7

PV Generator at 0.8 of Full Illumination

2.6

2.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Load Torque (Nm)

Fig. 13. Stator current of SRM as function of load torque coupled to the motor when fed by fixed terminal voltage, fed by PV generator at full illumination and 0.8 of full illumination

Appendix

[6]

The following are the numerical parameters of the motors, the DC-DC converter and the LC filter:

[7]

Induction Motor Lss = 0.1707 H ,

[8]

Lm = 0.16373 H , ωs = 188.5rad/s, Rs = 3.35 Ω , Vt = 200V, Llr = 0.00694 H , Rr = 1.99 Ω , J = 0.1 kgm2, P = 4.

[9]

Synchronous Reluctance Motor Ld = 0.54H, Lq = 0.21H, Lmd = 0.154H, Lmq = 0.088H,

[10]

Rs = 7.8 Ω , Rkd = 1 Ω , Rkq = 1 Ω , P = 4 , J = 0.038

[11]

kgm2, Lkkd = 61.25mH, Lkkq = 73.59mH, Bm = 0.0029 Nms, ωs = 157.08rad/s.

[12]

DC-DC Buck-Boost Switch Mode Converter Controller TD 0.1ms.

[13]

References

[14]

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[2]

[3]

[4]

[5]

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Mohammadd S. Widyan, Ghassan G S. Ma arji, Anas I. Al A Tarabsheh

[20] L. D. Parttain, Solar cells and a their applicaations, (John Wiley & Sons, Inc.,, 1995). [21] Br. Khiarri, A. Sellami, and a R. Andoulssi, A Novel Strrategy Control off Photovoltaic Soolar Pumping Sysstem Based on Sliding Mode Coontrol, Internatioonal Review off Automatic Coontrol (IREACO)), Vol. 5(2): 113-1117, 2012. [22] V. Salas, E. Olis, A. Barrrado and A. Lazaro, Review of o the m Power Point Tracking T Algrithhms for Stand-A Alone Maximum Photovoltaaic Systems, Solaar Energy Materrials and Solar Cells, Vol. 90, 15555-1578, 2006. [23] Tamer T. N. Khatib, Azaah Mohamed, Marwan M Mahmoudd and P Point Traacking Nowshad Amin, An Efficcient Maximum Power Controllerr for a Standalonne Photovoltaic System, Internaational Review onn Modelling and Simulations S (IRE EMOS), Vol. 3(2):: 129139,2010 [24] N. Mohann, T. M. Undelannd W. P. Robbinns, Power Electrronics, Converterss, Applications and a Design, (Johnn Wiley & Sons, Inc., 2003). m Stability and Control, (New--York, [25] P. Kunduur, Power System McGraw Hill, H Inc. 1993). [26] Chee-Munn Ong, Dynamicc Simulation off Electric Machhinery, (Prentice Hall H PTR, Upper Saddle River, Neew Jersy 07458, 1998). 1 [27] T. Lubin,, H. Razik and A. Rezzoug, On-Line Efficciency Optimizatiion of a Synchronnous Reluctance Motor, Electric Power P Systems Reesearch, Vol. 77: 484-493, 2007.

Anas Al Tarrabsheh was bo orn on 30 June,, 1978, in Irbidd, Jordan. He recceived his B.Sc.. and M.Sc. degrees in Electronics E andd n University off Communicatioons from Jordan Science and T Technology in 2000 2 and 2002,, respectively. From 15 Junee 2002, to 244 a a Lecturer inn February 20033, he worked as Electrical Enngineering Depaartment at Thee Hash hemite Universityy, Jordan. The Phh.D. was awardeed from Stuttgartt Univ versity, Germany,, on 29 January 22007. Currently Dr. D Al Tarabshehh is an n Assistant Professor in Electricall Engineering Deepartment at Thee Hash hemite Universitty. His fields of interest are Semiconductorr Mateerials and Charactterization of Phottovoltaic Cells.

Authors'' informatioon 1

Electrical Enginneering Dept, Thhe Hashemite University, 13115 Zaarqa Jordan. 2

Electrical Enggineering Dept.,Alhuson Univerrsity College, Balqa Aplied Universiity, 21510 Huson, Jordan. mad Widyan waas born on 5 Deceember Mohamm 1976 in Irbid, Jordan. He H received his B.Sc. wer Engineering from degree inn Electrical Pow Yarmoukk University,Jorddan in 2000 annd his M.Sc. degree in Control and a Power Engineeering from Joordan Universitty of Science and Technoloogy in 2002. Froom 2002 to 20003 he worked as a a lecturer in Electrical E Engineeering Department at The Hashemite University, Jordan. He receiveed his Ph.D. degree froom Berlin Univerrsity of Technoloogy, Germany in 2006. Currently, Dr. Widyan is ann Associate Professor in Elecctrical Engineering Deepartment at Thee Hashemite Uniiversity. His fiellds of interest are Power P System and Electrical Machine Dynaamics, Bifurcation Thheory (Modern Nonlinear Thheory) and Coontrol, Permanent-Maggnet and Convenntional Electrical Machine Designn and Finite Element Technique. T n Marji was bornn on 22 Februaryy 1964 Ghassan is a full f time instrructor of Elecctrical Engineering. He receivedd his M.Sc in Power P J and Conttrol Engineering in 2002 from Jordan Universitty of Science andd Technology. He H has been apppointed as Leecturer in Elecctrical Engineering Department at Alhuson Univversity Al-Balqa Applieed University since College/A 2002. His reseaarch interests aree Power Optimizzation, Power Syystem Stability and Coontrol.

Copyright © 20012 Praise Worthyy Prize S.r.l. - Alll rights reserved

Internnational Review oof Automatic Con ntrol, Vol. 5, N. 6

768


Performance Analysis of New Three Phase Seven Level Asymmetrical Inverter with Hybrid Carrier and Sine 60 Degree Reference Johnson Uthayakumar R., Natarajan S. P., Bensraj R.

Abstract – Carrier based Pulse Width Modulation (PWM) techniques have been used widely for switching of multilevel inverters due to their simplicity, flexibility and reduced computational requirements compared to Space Vector Modulation (SVM). A novel carrier based PWM technique for three phase Asymmetrical Multi Level Inverter (AMLI) with sine sixty degree reference is proposed in this paper. The technique is based on the combination of the Control Freedom Degrees (CFD). The combination of inverted sine carrier and triangular carrier is used as hybrid carrier in this work to produce pulses for the power switches used in the proposed three phase seven level AMLI. This paper investigates the potentials of hybrid carrier based cascaded multilevel inverter in the development of medium power AC power supplies with specific emphasis on Power Conditioning Systems (PCS) for alternate sources of energy. The performance of chosen inverter is evaluated through MATLAB-SIMULINK simulation. The performance indices used are Total Harmonic Distortion (THD), RMS value of output voltage and DC bus utilization. It is observed that POD and COPWM relatively provide better DC bus utilization and PDPWM & PODPWM techniques create relatively less distortion for modulation index ma=0.7-1. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Sine 60, PD, POD, APOD, CO, VF, THD, DF

I.

conventional four level inverter and found that it exhibits lesser switching losses and lesser harmonics. Seyezhai and Mathur [9],[10] have done a comparative evaluation between hybrid modulation strategy and the conventional Phase Disposition PWM method in terms of output voltage quality, power circuitry complexity, Distortion Factor (DF) and THD. Wang ShuZheng et al [11] have proposed a three phase cascaded multilevel inverter for grid controled photo voltaic system. Lau et al [12] have found an analytical solutions for determining the spectral characteristics of multicarrier based multilevel PWM pulses for the inverters. Josh et al [13] have done a comparative analysis of multicarrier control techniques for Sine PWM (SPWM) controlled cascaded H-bridge multilevel inverter. A Carrier-based PWM modulation for THD and losses reduction on multilevel inverters was studied by Barreto et al [14]. Hachemi et al [15] have studied the vector control of three phase inverter applied to Permanent Magnet Synchronous Motor. Boulkhrachef and Berkouk [16] have found the implementation and control using Fuzzy Systems to Control DC-Buses of Five-Level NPC VSI. Satish Kumar et al [17] have done an Analytical SpaceVector PWM Method for Multi-Level Inverter Based on Two-Level Inverter. A seven level output voltage is achieved with two

Introduction

The topologies of MLIs are classified into three types: the flying capacitor inverter, the diode clamped inverter and the modular H-bridge inverter. In literature it is found that these topologies have been used by several researchers. Chiasson et al [1] has proposed in his work that the lower cascaded bridge can be removed and replaced by capacitors. Tolbert et al [2] have demonstrated several modulation strategies such as Sub Harmonic PWM (SH-PWM) for the six level Diode Clamped Multilevel Inverter (DCMLI). Zhong et al [3] have proposed a Cascaded Multi Level Inverter (CMLI) with a single DC source and alsoinverter with three DC sources instead of two DC sources. Ayob and Salam [4] have proposed multiple trapezoidal reference signal for CMLI. The novel Phase Disposition PWM (PDPWM) stratergy is proposed by Zhao et al [5] which produces lesser THD. A survey of topologies, controls and applications of MLIs has been carried out by Rodriguez et al [6]. Sun [7] has presented a new asymmetrical multilevel inverter topology. The new topology can improve the number of output voltage levels greatly using a bidirectional auxiliary switch. Further he has proposed multicarrier PWM method for the asymmetrical inverter [7]. Nami et al [8] have optimized asymmetrical arrangement compared with a


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Johnson Uthayakumar R., Natarajan S. P., Bensraj R.

bridges in asymmetrical inverter whereas only five level output voltage will be achieved with three bridges in case of conventional cascaded MLI. In AMLI with lesser number switches more voltage levels can be achieved. Fig. 1 shows the chosen asymmetrical three phase inverter. Each cell has two pairs of complementary switches Sa and Sd and Sb and Sd. There are six cells used in the three phase inverter each leg containing two cells each. Since the carrier based methods have good CFD, this paper focusses on the hybrid carrier arrangement using triangular carrier in the positive side and inverted sine carrier in the negative side with Phase Disposition (PD), Phase Opposition Disposition (POD), Alternate Phase Opposition Disposition (APOD), Carrier Overlapping (CO) and Variable frequency (VF) PWM strategies. In some application with different DC input sources such as electric vehicles, a modular H-bridge asymmetrical inverter can be used to drive a traction motor from a set of solar cells or fuel cells. Fig. 2 shows a sample SIMULINK model developed for PWM strategy of a three phase AMLI.

II.

The sixty degree curve is obtained by just cutting the sine wave at sixty degree. In this strategy, the modulation index ma can be increased beyond ma=1.0 without moving into over modulation. The frequency modulation index: mf = fc/fm The amplitude modulation index: ma = 2Am/ (m-1) Ac where: fc – Frequency of the carrier signal fm – Frequency of the reference signal Am –Amplitude of the reference signal Ac – Amplitude of the carrier signal ma =Am / (m / 4) · Ac (COPWM) II.1.

PDPWM Strategy

In this strategy carriers have same frequency, amplitude and phase, but they are just different in DC offset to occupy contiguous bands as shown in Fig. 3. For this technique, significant harmonic energy is concentrated at the carrier frequency fc but because it is a co-phase component, it does not appear in the line voltage. It should be noted that the other harmonic components are centered on the carrier frequency as sidebands. This technique employs (m-l) carriers which are all in phase for a m level inverter. In seven level converter all the six carrier waves are in phase with each other across all the bands as described in Fig. 3 for a phase leg of a seven level AMLI structure with ma = 0.8. The pulse pattern for AMLI with hybrid carriers Sine 60o PDPWM strategy is shown in Fig. 4.

Hybrid Carrier Based Bipolar Modulation Schemes With Sine 60° Reference

The maximum modulation index of a three phase inverter can be increased by sine 60O reference waveform of each phase leg.

II.2.

PODPWM Strategy

This technique employs (m-1) carriers which are all in phase above and below the zero reference. In seven level converters all the three carrier waves above zero reference are phase shifted by 180 degrees with the ones below zero reference. The PODPWM is explained in the Fig. 5 in which all the carriers above the zero reference are in phase and carriers below the zero reference are also in phase but are phase shifted by 180 degree with respect to that above zero reference. Fig. 5 illustrates the POD PWM hybrid carrier and sine 60o reference arrangements for a phase leg of a three phase seven level AMLI structure with ma = 0.8. The pulse pattern for same AMLI with hybrid carrier sine 60o PODPWM strategy is shown in Fig. 6.

Fig. 1. A three phase asymmetrical cascaded seven level inverter

II.3.

PODPWM Strategy

This technique requires each of the m-l carrier waveforms for an m-level phase waveform to be phase displaced from each other by 180 degrees alternatively.

Fig. 2. A sample SIMULINK model developed for PWM strategy of a three phase inverter



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II.4.

COPWM Strategy

For an m-level inverter using carrier overlapping technique, (m-1) carriers with the same frequency (fc) and same peak-to-peak amplitude (Ac) are disposed such that the bands they occupy overlap each other; the overlapping vertical distance between each carrier is 0.5Ac. The reference waveform has amplitude of Am and frequency of fm and it is centered in the middle of the carrier signals. Fig. 9 shows the hybrid carrier and sine 60o reference arrangements for COPWM of a phase leg of a three phase seven level AMLI structure with ma =0.8 and corresponding pulsse pattern is in Fig. 10.

Fig. 3. Carrier arrangements for hybrid carrier strategy PDPWM with sine 60o reference

Fig. 4. Pulse pattern for AMLI with hybrid carrier Sine 60 PDPWM strategy

Fig. 7. Carrier arrangements for hybrid carrier APODPWM strategy with Sine 60o as reference

Fig. 5. Carrier arrangements for hybrid carrier strategy PODPWM with Sine 60o as reference Fig. 8. Pulse pattern for chosen AMLI with hybrid carrier APODPWM strategy with Sine 60o reference

II.5.

VFPWM Strategy

Variable frequency PWM strategy is used in order to equalize the number of switching’s for all the switches, as illustrated in Fig. 11. The carrier frequency of the intermediate switches is properly increased to balance the numbers of switching for all the switches. The hybrid carrier and sine 60o reference arrangements of VFPWM a phase leg for a three phase seven level AMLIS structure with ma=0.8 are illustrated in Fig. 11 and the corresponding pulse pattern are displayed in Fig. 12.

Fig. 6. Pulse pattern for chosen AMLI with hybrid carrier PODPWM strategy with Sine 60o reference

The APOD hybrid carrier and sine 60 degree reference arrangements for a phase leg of a three phase seven level cascaded structure with ma=0.8 are illustrated in Fig. 7 and their pulse pattern for AMLI with hybrid carrier Sine 60-APODPWM strategy are displayed in Fig. 8.

III. Simulation Results Simulation results have been obtained by using MATLAB-SIMULINK power system toolbox software. The input DC sources are asymmetrical i.e. one of the cascaded bridges is fed with Vdc/2 and the other by Vdc.



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Figs. 13-17 illustrate the output voltages of three phase asymmetrical cascaded seven level inverter for ma=0.8 only. THD and RMS value, form Factor (FF) and Crest Factor (CF) of output voltage are evaluated using appropriate formule and for various hybrid carrier modulation schemes and for various ma (0.7-1) as in Tables I-IV. Figs. 18-22 display the frequency spectra and %THD for chosen strategies. It is found that the DC bus utilisation of the three phase seven level cascaded inverter is relatively higher with COPWM and PODPWM as in Table II. PODPWM and PDPWM techniques create relatively less distortion(Table I). It is seen that 3rd,7th, 37th and 39th harmonics are dominant in PDPWM and 3rd,7thharmonics are dominant in POD PWM where as 3rd,7th,27th,29th,31st and 39th harmonics are dominant in APOD PWM. COPWM creates 3rd,37th,39th harmonic dominant energy where as in VFPWM 3rdand 7th harmonics are dominant. The fundamental RMS output voltage is higher for modulation indices (1- 0.85) in PODPWM and in case of COPWM the RMS values are higher with ma = 0.8-0.7 as shown in Table II. FF is a measure of the shape of the output voltage given by FF=VRMS/Vavg where VRMS is the RMS value of output voltage and Vavg is the DC content in the output voltage and is calculated and displayed in Table III. CF is the measure of peak current rating of the device and components and is calculated and displayed in Table IV. The following parameter values are used for simulation: Vdc =100V, Vdc/2 =50V Ac=1, mf=40 and R(load) = 100 ohms for each phase.

Fig. 9. Carrier arrangements for hybrid carrier COPWM strategy with Sine 60o as reference

Fig. 10. Pulse pattern for chosen AMLI with hybrid carrier COPWM strategy with Sine 60o reference

Fig. 11. Carrier arrangements for hybrid carrier VFPWM strategy with Sine 60o reference

Fig. 13. Simulated output voltage of hybrid carrier PDPWM strategy with sine 60o reference

Fig. 12. Pulse pattern for chosen AMLI with hybrid carrier VFPWM strategy Sine 60o reference

Asymmetrical topology reduces voltage stress on switches and results in better power quality in comparison to two level PWM. lesser THD and hence smaller Inductor-Capacitor-Inductor (LCL) output filter

Fig. 14. Simulated output voltage of hybrid carrier PODPWM strategy with sine 60o reference



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Fig. 15. Simulated output voltage of hybrid carrier APODPWM strategy with sine 60o reference

Fig. 19. FFT spectrum for hybrid carrier PODPWM strategy for ma=0.8, mf =40 with sine 60o reference

Fig. 16. Simulated output voltage of hybrid carrier COPWM strategy with sine 60o reference

Fig. 20. FFT spectrum for hybrid carrier APODPWM strategy for ma=0.8, mf =40 with sine 60o reference

Fig. 17. Simulated output voltage of hybrid carrier VFPWM strategy with sine 60o reference

Fig. 21. FFT spectrum for hybrid carrier COPWM strategy for ma=0.8, mf =40 with sine 60o reference

Fig. 18. FFT spectrum for hybrid carrier PDPWM strategy for ma=0.8, mf =40 with sine 60o reference

Fig. 22. FFT spectrum for hybrid carrier VFPWM strategy for ma=0.8, mf =40 with sine 60o reference



773


TABLE IV CF OF OUTPUT VOLTAGE OF AMLI FOR DIFFERENT PWM STRATEGIES AND VARIOUS VALUES OF ma ma PD POD APOD CO VF 1 1.414 1.414 1.415 1.414 1.415 0.95 1.414 1.414 1.414 1.414 1.432 0.9 1.414 1.415 1.414 1.414 1.413 0.85 1.415 1.414 1.413 1.414 1.414 0.8 1.414 1.414 1.414 1.415 1.414 0.75 1.414 1.414 1.414 1.414 1.414 0.7 1.414 1.415 1.414 1.414 1.415

IV. Fig. 23. %THD vs ma for all strategies

Conclusion

The new hybrid modulation strategy proposed for chosen three phase AMLI is studied for various ma=0.71. It is observed that COPWM and PODPWM provide better DC bus utilization and less harmonic distortion excists in PDPWM and PODPWM for various ma=0.7-1.

References [1]

[2]

[3]

Fig. 24. RMS output voltage vs ma for all strategies TABLE I %THD OF OUTPUT VOLTAGE of AMLI FOR VARIOUS ma ma 1 0.95 0.9 0.85 0.8 0.75 0.7

PD 25 27 28.3 29 28.89 27.3 23.5

POD 23.7 26.16 27.49 28.47 29.02 28.49 26.34

APOD 25.71 27.64 28.78 29.18 29.43 28.16 25.43

CO 27.11 28.66 29.68 30.48 30.52 30.6 29.98

[4]

VF 25.06 27.22 28.26 28.82 28.39 26.86 23.9

[5]

[6]

TABLE II RMS (FUNDAMENTAL) VALUE OF OUTPUT VOLTAGE OF AMLI FOR DIFFERENT PWM STRATEGIES AND VARIOUS ma ma 1 0.95 0.9 0.85 0.8 0.75 0.7

PD 117 111 105 100 94.5 89.3 84.2

POD 123 117.6 112.4 107.2 101.2 95.45 87.09

APOD 117.9 111.8 105.9 100.7 94.86 90.1 85.38

CO 119.1 114.4 110.2 105.8 102.1 97.74 93.58

[7]

[8]

VF 117 110.9 106 99.43 94.71 89.74 84.76

[9]

[10]

TABLE III FF OF OUTPUT VOLTAGE OF AMLI FOR DIFFERENT PWM STRATEGIES AND VARIOUS VALUES OF Ma ma 1 0.95 0.9 0.85 0.8 0.75 0.7

PD 3.05E+01 2.68E+01 2.93E+01 2.75E+01 3.00E+01 3.84E+01 5.81E+01

POD 4.78E+01 3.89E+01 2.69E+01 2.84E+01 2.51E+01 2.08E+01 2.83E+01

APOD 4.53E+01 3.92E+01 4.32E+01 3.60E+01 4.52E+01 8.79E+01 1.26E+09

CO 1.58E+01 1.45E+01 1.30E+01 1.39E+01 1.30E+01 1.29E+01 1.33E+01

[11]

VF 3.06E+01 2.84E+01 3.34E+01 2.71E+01 3.35E+01 4.08E+01 6.65E+01

[12]

[13]


J. N. Chiasson, L. M. Tolbert, and O. Ridge, “A Five-Level Three-Phase Hybrid Cascade Multilevel Inverter Using a Single DC Source for a PM Synchronous Motor Drive”, Applied Power Electronics Conference, APEC 2007 pp.1504-1507, L. M. Tolbert and F. Z. Peng, “Multilevel PWM Methods at Low Modulation Indices”, IEEE Transactions on Power Electronics, vol.15,Jul 2000, no.4, pp.719-725. Zhong Du, L. M. Tolbert, J. N. Chiasson, and B. Özpineci, “A Cascade Multilevel Inverter Using a Single DC Source”, Applied Power Electronics Conference and Exposition, 2006. APEC '06. Twenty-First Annual IEEE, pp. 5 pp., 19-23. S.M Ayob, Z.Salam, “A New PWM Scheme For Cascaded Multilevel Inverter Using Multiple Trapezoidal Modulation Signals,”University Teknoloai Malavsia. Malavsia. Reproduced 2004.pp. 242-246, J. Zhao, X. He, S. Member, and R. Zhao, “A Novel PWM Control Method for Hybrid-Clamped Multilevel inverter”, IEEE Transactions on Industrial Electronics, July 2010. vol.57, no.7, pp.2365-2373, J.Rodriguez, J.S.Lai, and F.Z.Peng,“Multilevel Inverter: A Survey of Topologies, Controls and Applications”, IEEE Trans. on Industrial Electronics, Vol. 49 (2000), No. 4 pp. 724-738 . X. Sun, “Hybrid Control Strategy for A Novel Asymmetrical Multilevel Inverter”, IEEE Transactions on Industrial Electronics 2010.pp. 5-8, A. Nami et al., “Comparison between Symmetrical and Asymmetrical Single Phase Multilevel Inverter with DiodeClamped Topology”, IEEE Transactions on Power Electronics, 2008 pp. 2921-2926, R.Seyezhai, "Investigation of Performance Parameters For Asymmetric Multilevel Inverter Using Hybrid Modulation Technique”, International Journal of Engineering Science and Technology (IJEST),vol.3,2012.pp.8430-8443. R.Seyezhai, B.L.Mathur “Hybrid Multilevel Inverter using ISPWM Technique for Fuel Cell Applications”, International Journal of Computer Applications (0975 – 8887) volume 9,No.1, November 2010.ppWang Shu Zheng; Zhao Jian Feng and Shi Chao "Research on a Three-Phase Cascaded Inverter For Grid-Connected Photovoltaic Systems", Conference on Advanced Power System Automation and Protection (APAP), 2011 International, 16-20 Oct2011 vol.1, no., pp.543-548. W.H Lau, Bin Zhou and H.S.H Chung, "Compact analytical solutions for determining the spectral characteristics of multicarrier-based multilevel PWM", IEEE Transactions on Circuits and Systems, Aug.2004,vol.51, no.8, pp. 1577- 1585. F.T Josh, J.Jerome and A. Wilson, "The comparative analysis of


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[14]

[15]

[16]

[17]

multicarrier control techniques for SPWM controlled cascaded Hbridge multilevel inverter", International Conference on Emerging Trends in Electrical and Computer Technology (ICETECT),2011,vol.,no.,pp.459-464,23-24. L. H. S. C. Barreto, G. A. L. Henn, P. P. Praca, R. N. A. L. Silva, D. S. Oliveira, and E. R. C. da Silva, “Carrier-based PWM modulation for THD and losses reduction on multilevel inverters”, in Applied Power Electronics Conference and Exposition (APEC), 2012 Twenty-Seventh Annual IEEE, 2012, pp.2436-2441. K.Hachemi, B. Mazari, Y. Miloud, M. Laouer, “Fuzzy Controller with Vector Control Applied to Permanent Magnet Synchronous Motor”, International Review of Automatic Control (IREACO), January 2009,vol.2.n.1,pp69-74. S. Boulkhrachef, E. M. Berkouk “Multilevel PWM-CSR Using Fuzzy Systems to Control DC-Buses of Five-Level NPC VSI”, International Review of Automatic Control (IREACO), January 2009,vol.2.n.1,pp75-82. P. Satish Kumar, J. Amarnath, S. V. L. Narasimham,” An Analytical Space-Vector PWM Method for Multi-Level Inverter Based on Two-Level Inverter” International Review on Modelling and Simulations (IREMOS), February 2010, Vol. 3. n. 1 pp. 1-9.

R. Bensraj was born in 1973 in Marthandam, India. He is currently working as a Assistant Professor in the Department of Electrical Engineering, Annamalai University. He has publications in 12 international journals and two national journals. His fields of interest include multilevel inverters, power quality and power electronics in power systems. Contact number+91- 9942429311. E-mail: [email protected]

Authors’ information Department of Electronics and Instrumentation Engineering, Annamalai University, India. R. Johnson Uthayakumar obtained his B.E Electrical and Electronics Engineering from National Institute of Technology, Trichirapalli formerly named Regional Engineering College, Trichirapalli.Tamil Nadu, India and his post graduate degree in Shanmugha Engineering College presently a deemed University SASTRA since the year 2000. He has been in the field of education for engineers for about 12 years. He is at present persuing his Ph.D. degree in the field of power electronics from the Department of Electronics and Instrumentation Engineering, Annamalai University . He is interested in the field of inverters and DC-DC converters. S. P. Natarajan was born in 1955 in Chidambaram. He has obtained B.E (Electrical and Electronics) and M.E (Power Systems) in 1978 and 1984 respectively from Annamalai University securing distinction and then Ph.D in Power Electronics from Anna University, Chennai in 2003. He is currently Professor and Head in Instrumentation Engineering Department at Annamalai University where he has put in 31 years of service. He produced nine Ph.Ds and presently guiding nine Ph.D Scholars and also so far guided eignty M.E students. His research papers (60) have been presented in various / IEEE international / national conferences in Mexico, Virginia, Hong Kong, Malaysia, India, Singapore and Korea. He has 25 publications in national journals and 40 in international Journals. His research interest are in modeling and control of DC-DC converters and multiple connected power electronic converters, control of permanent magnet brushless DC motor, embedded control for multilevel inverters and matrix converters etc. He is a life member of Instrument Society of India and Indian Society for Technical Education. Tel: +91-9443185211 E-mail: [email protected]



775


Parity Space Approach Based DC Motor’s Fault Detection and Isolation A. Adouni, M. Ben Hamed, L. Sbita

Abstract – The main object of this paper is the detection and isolation of sensors and actuator’s DC motor faults. To perform fast diagnosis of these defects, discrete-time parity equations with structural residuals approach has been used. It allows, from a model description, to generate signals that deviate from zeros in case of abnormal operation. These signals are, then, compared to the theoretical residuals in order to make the decision. All the faults are successfully detected and isolated. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Analytical Redundancy, DC Motor, Fault Detection, Isolation, Linear System, Parity Space Method, Residual Generation, Diagnosis, Sensor Fault, Actuator Fault

I.

In reference [17], the author discussed the relationship between the PSM and the H2 approach to linear discrete time fault detection. The results were discussed and proven upon a numerical example. In reference [18], parity residuals and parameter estimates are used for only DC motor fault detection. The DC machine is frequently used in industrial applications due to its robustness, it’s actually moderate cost and its simple monitoring. Its behavior could be described by an analytical model which is sufficiently accurate. Therefore, in this paper, the DC motor sensors and actuators installation equipment defect is detected and isolated using the PSM. Compared to the work published in [19], the main contribution in this article is the detection and isolation of sensors (speed and current sensors) and actuator faults. The remaining parts of this paper are organized as follows. Section 2, briefly reviews the parity space method with structural residuals and its application for the DC motor case. Section 3, illustrates the simulation results for FDI sensor and actuator. The final section concludes the paper research work.

Introduction

It is necessary that an industrial installation should be controlled at each time in order to detect incipient faults and to locate the deteriorated components. The early diagnosis allows one to plan the required maintenance actions and to decrease the number of emergency shutdowns of any operating process. The hardware redundancy was the first solution to detect the faults. It means that three or four sensors are used to measure the same quantity. A voting scheme evaluates its most likely value. However, this requires additional space; as a result the equipment becomes quite costly. To avoid these drawbacks, many approaches were developed. Generally, they could be classified into two main categories: model-free methods and model-based methods. The first does not require a mathematical representation of the process such as spectrum analysis of plant measurements [1]. The second category named also analytical redundancy or quantitative, which is based on the plant model. This last one is called parity space method (PSM). It is already used because it is the best stability compared to other model-based FDI techniques [2]-[5]. Historically, the static model types were used and after the dynamic models were concerned. The first efforts begun with Willsky in (1976) [6], Isermann in (1984) [5]-[6]; Patton, et al in (1989) [7], Frank in (1990) [7], Gertler in (1991) [8] and Patton in (19940) [9] etc. The PSM was discussed in many papers both in the linear system [10]-[12] and nonlinear system [13]-[16]. In reference [2], the (PSM) has been generalized to the continuous-time case. It was shown that as in the discrete-time the parity equations (PE) are essentially a deadbeat observation problem.

II.

Dynamic Parity Space II.1.

Parity Equation

A known linear state model can often represent a dynamic system with unknown nonlinearity. Known part can be obtained from the physical construction of the system, while, the unknown non-linear part includes the uncertainty and the unknown inputs such as modeling errors and external disturbances. It can be described by [14]: ⎪⎧ x ( k + 1) = Ax ( k ) + Bu ( k ) ⎨ ⎪⎩ y ( k ) = Cx ( k )


776

(1)


A. Adouni, M. Ben Hamed, L. Sbita

where x ( k ) is the system’s state vector, is the system’s

Otherwise, r ( k ,s ) is nonzero. Therefore, faults can

input, y ( k ) is the system’s sensors outputs and A , B

be detected from r ( k ,s ) as follows:

and C are the system’s matrix with appropriate dimension. Let us design with s the order of the parity space. For s 0 , the measurement equation is given by [20]-[33] with: Y ( k ,s ) = H ( s ) x ( k ) + G ( s ) U ( k ,s )

Y ( k ,s ) = ⎡⎣ y ( k ) ,..., y ( k + s ) ⎤⎦

T

U ( k ,s ) = ⎣⎡u ( k ) ,...,u ( k + s ) ⎦⎤ H ( s ) = ⎡⎣C CA ... CAs ⎤⎦

T

r ( k ) ≥ rk r ( k ) = rk = 0

where, r ( k ) is the threshold obtained from experience

(3)

based on the safety and reliability requirement for the system. As it is shown by (8), r ( k ,s ) is function of all actuator and sensor faults. Therefore, r ( k ,s ) can be used to detect and isolate faults. However, the obtained equations are not necessary independent, specially, where the order s is high. The auto redundancy and inter redundancy concepts are evolved to resolve this problem.

(5)

0 0 … 0⎤ ⎡ C ⎢ CA ⎥ 0 ⎢ ⎥ ⎥ CB G ( s ) = ⎢ CAB ⎢ ⎥ CB ⎢ ⎥ ⎢CAs −1 B CAs − 2 B … CB 0 ⎥ ⎣ ⎦

II.2.

(6)

with:

Y j ( k ,s ) = ⎡⎣ y j ( k ) ... y j ( k + s ) ⎤⎦ H j ( s ) = ⎡⎣C j

(7)

r ( k ,s ) = ΩZ ( k ,s ) − G ( s ) U c ( k ,s )

(8)

Z ( k ,s ) = ⎡⎣ z ( k ) ... z ( k + s ) ⎤⎦

(9)

U c ( k ,s ) = ⎡⎣uc ( k ) ... uc ( k + s ) ⎤⎦

T

C j A ... C j As ⎤⎦

0 ⎡ Cj ⎢ 0 ⎢ CjB ⎢ C AB CjB Gj (s) = ⎢ j ⎢ ⎢ ⎢⎣C j As −1 B C j As − 2 B

Definition 2: The parity equation at a time k is given by:

T

Auto Redundancy Equations

They are obtained by writing the relation (2) for each sensor: Y j ( k ,s ) = H j ( s ) x ( k ) + G j ( s ) U ( k ,s ) (13)

Definition1: Ω is a parity matrix which verify the following equation [12]-[14]: ΩH ( s ) = 0

(12)

normal

(2)

(4)

T

(11)

faulty

T

(14)

T

(15)

0⎤ ⎥ ⎥ ⎥ ⎥ (16) ⎥ ⎥ C j B 0 ⎥⎦ …

0

The term C j represents the jth line vector of the matrix. Let us design with nj the maximum rank of

(

(10)

)

H j ( s ) matrix. The null space of the matrix H j n j − 1 th

where, r ( k ) is the residual, U c ( k ,s ) is the normal input

is known as the unobservable subspace of the j sensor. In these conditions, it is possible to write the nj components of state vector with nj outputs y j ( k )

contain faults. From equation (8), any actuator or sensor fault is included in r ( k ,s ) . Therefore, for a normal actuator we

to y j k + n j − 1 . Equation (13) can be written as:

to the actuator and Z ( k ,s ) is the sensor output that may

(

)

(

)

( + G j ( s ) U ( k ,n j − 1)

)

G j ( s ) = Y k ,n j − 1 = H j n j − 1 x ( k ) +

have U c ( k ,s ) = U ( k ,s ) and for a normal sensor we get Z ( k ,s ) = Y ( k ,s ) . Based on Eqn. (8), r ( k ,s ) is decoupled from the

(17)

To obtain the redundancy, a supplementary line is

(

system state if the system is normal and it is identically zero if the unknown non linear terms are identically zero.

)

added to a H j n j − 1 matrix. As a result, (7) is given



777


by:

where:

( )

Ω j = Ω j / Ω jH j nj = 0

II.3.

x ( k ) = ⎡⎣ia ( k ) ω ( k ) ⎤⎦

(18)

Inter Redundancy Equations

The inter redundancy equations are obtained by using all sensors measurement together. They are obtained by considering the (j=1 to q) independent relations obtained with (8). The parity equations are a vector. The last obtained by elimination of the unknown state. This fact consists of looking for a matrix satisfying the relation (11):

{

) }

(

Ω = Ω / ΩH k ,n j − 1 = 0

)

(

) (

(19)

)

ia −

f J

⎡1 0 ⎤ C=⎢ ⎥ ⎣0 1 ⎦

(27)

+ω ( k ) − 15.2061ua ( k − 1)

As it is shown with Table I, it is impossible to isolate faults. Now, considering the auto redundancy approach, the parity vectors are described with (18) and (20):

(21)

r1 ( k ) = 0.9372 ia ( k − 2 ) − 1.9369 ia ( k − 1) + ia ( k ) +

(22)

r2 ( k ) = 0.9372 ω ( k − 2 ) − 1.9369 ω ( k − 1) + ω ( k ) − 0.035ua ( k − 2 )

where ia is the armature current, ω is the shaft speed, ua is the input voltage, Ra and La are the armature resistance and inductance respectively, K e and Kt are the speed and torque proportionality constants, is J the moment of the inertia and f is the viscous damping constant. The discrete state space equation derived from (21) and (22) is: ⎪⎧ x ( k + 1) = Ax ( k ) + Bu ( k ) ⎨ ⎪⎩ y ( k ) = Cx ( k )

(29)

TABLE I THEORETICAL SYMPTOMS MATRIX FOR INTER REDUNDANCY Current sensor Speed sensor Actuator r1 1 1 1 r2 1 1 1

+ 0.0913ua ( k − 2 ) − 0.0914 ua ( k − 1) ω

(28)

Based on these equations, the theoretical symptoms matrix is established (Table I).

Fault diagnosis of DC motors has attracted considerable interest as they are often used in practical control systems. Self-excited DC motor can be described by the ordinary differential equations:

Ka J

(26)

r2 ( k ) = −156.0983 ia ( k − 1) + 166.37751ia ( k )

III. Application to DC Motor

=

⎡ 0.9380 − 0.0060 ⎤ ⎡ 0.0078⎤ ⎥ A = ⎢⎢ ⎥, B = ⎢ 0 ⎥ ⎣ ⎦ ⎢⎣ 0.0387 0.9989 ⎥⎦

+ω ( k − 1) − 15.227 ua ( k − 1)

To isolate the occurred faults, an isolation method based on the comparison of the theoretical symptoms matrix to an experimental one. The theoretical and experimental symptoms matrixes are obtained as follows: the residual vector components values are zero when the later are insensitive to the considered faults and a value one is setup and affected to the residual when it is sensitive to the considered faults.

dω dt

(25)

r1 ( k ) = −156.2298 ia ( k − 1) + 166.5591ia ( k )

Faults Isolation

dia R K u = − a ia − e ω + a dt La La La

u ( k ) = ua ( k )

From (23), there are one actuator and two sensors. The order of the parity space is identified to 2. The parity vectors using the inter redundancy are obtained using (20) and they are defined by the two recurrent equations:

r ( k ) = Ω ⎡Y k ,n j − 1 − G n j − 1 U k ,n j − 1 ⎤ (20) ⎣ ⎦

II.4.

(24)

and:

Finally:

(

T

(30)

(31)

Using (30) and (31), the theoretical symptoms matrix is established. As it is shown with Table II, the residual vectors are decoupled from each other. As a result, each fault can be detected. TABLE II THEORETICAL SYMPTOMS MATRIX FOR AUTO REDUNDANCY Current sensor Speed sensor Actuator r1 1 0 1 r2 0 1 1

(23)



778


6

500

400

4 speed (pr.m)

Armature current (A)

5

3 2

200

100

1 0

300

0

1

2

3 Time (s)

4

5

0

6

0

1

(a) armature current

2

3 Time (s)

4

5

6

(e) shaft speed

48.5

5

Control volatge (V)

Armature voltage (V)

48

47.5

47

46.5

46

0

1

2

3

4

5

4.5

4

3.5

3

6

0

1

2

Time (s)

(b) armature voltage 2

x 10

6

Speed residual, r2

Current residual, r1

5

6

x 10

4

5

6

-14

4

0 -1 -2 -3

2 0 -2 -4

0

1

2

3 Time (s)

4

5

-6

6

0

1

(c) current residual 1

0.5

0.5

0

-0.5

-1

2

3 Time (s)

(g) speed residual

1

Speed decision, d2

Current decision, d1

4

(f) control voltage

-15

1

-4

3

Time (s)

0

-0.5

0

1

2

3 Time (s)

4

5

-1

6

0

1

(d) current decision

2

3 Time (s)

4

5

6

(h) speed decision Figs. 1. Simulation results in case of absence fault



779


-3

8

x 10

4 3 current residual, r1

Armarure current (A)

6

4

2

2 1 0 -1

0 -2

-2

0

1

2

3 Time (s)

4

5

-3

6

0

1

0.1

1500

0.05

Speed residual, r2

shaft speed (pr.m)

3 Time (s)

4

5

6

(e) current residual

(a)armature current 2000

1000

500

0

2

0

-0.05

0

1

2

3 Time (s)

4

5

0

6

1

2

3 Time (s)

4

5

4

5

6

(f) speed residual

(b) shaft speed 50

1

current decision, d1

armature voltage (V)

45 40 35 30

0.8 0.6 0.4 0.2

25

0 20

0

1

2

3 Time (s)

4

5

0

6

1

2

3 Time (s)

6

(g) current residual

(c) armature voltage 5

0.8

4.5

Speed decision, d2

Control volatge (V)

1

4

0.6 0.4 0.2

3.5

0 3

0

1

2

3

4

5

6

0

1

Time (s)

2

3 Time (s)

4

5

6

(h) speed residual

(d) control voltage Figs. 2. Simulation results with sensor and actuator faults



780


IV.

Simulation Results

V.

To highlight the performance of the used method extensive simulation results are conducted using matlab/simulink. In first case, we discuss the behavior of different outputs, residuals and decisions in normal function, then in presence of the faults. The faults studied in this work are additive faults for sensors faults as to for the actuator the fault consists in disconnection of the voltage ( ua ) .

In this paper, we have studied the problem of fault detection and isolation in a DC motor described by linear models. The parity space approach has been performed to provide the FDI process. The considered methods allow one to provide a reliable and early detection and isolation of the sensors and actuator faults. In this paper, linear model has been considered. The later is often deduced from physical laws. For certain processes, physical descriptions are not always available for that reason the parity space approach could not be used. To solve this problem we suggest the artificial intelligent techniques for fault detection and isolation systems. To improve the performances of the used algorithms, a robust model is to be considered. This will be the subject of author’s future work.

IV.1. Absence of Faults (Normal Function) The Figs. 1(a), (b) and (c) illustrate respectively the evolution of the speed, the current and the voltage in case of normal function. It is clear that the residual r1 and r2 (Figs. 1(e) and (f)) are unchanged. They are approximately equal to zero. As well as the decision d1 and d2 (Figs. 1(g) and (h)) are equal to the zeros.

References [1]

[2]

IV.2. Sensor and Actuator Faults In the case of defects, the behavior of residuals is different to that in normal operation. Indeed, we distinguish three scenarios. The first is during [1.3 1.45]s when the fault influences only on the evolution of the current while the speed keeps its state. Consequent to that, only the first residual deviates. They indicate the appearance of a defect. The vector [ d1

[3]

[4]

[5]

d 2 ] becomes equal to [1 0] . T

T

[6]

By referring to the second table and to the evolution of decision, we deduce that the element affected by the defect is the sensor current. The second scenario is totally the opposite of the first. During the interval [2 2.4]s only the second residue deviates from zero when the speed deviates from the expected value gained. As for the decisions we observe that d2 equal to the unit value during the same interval of the application of the defect while d1 remains unchanged. The vector

[ d1

[7] [8]

[9]

[10]

d 2 ] becomes equal to [ 0 1] . By referring to the

[11]

decision model we deduce that it is a defect sensor speed. Now, the actuator fault which consists in a disconnection of the voltage ( ua ) feeding the inverter is

[12]

T

T

[13]

applicated during the interval [4.3 4.5]s. This defect has led to a decline at the level of the acquired speed as well as the current which deviated to that delivered by DC motor. This is clearly reflected in the behavior of residuals and decisions, which deviate from the no value. The vector

[ d1

d2 ]

T

becomes equal to

Conclusion

[14]

[15]

[1 1]

T

[16]

which asserts that the actuator is marked by the defect. [17]


J.Gertler, Survey of model-based failure detection and isolation in complex plants., IEEE Control Systems Magazine, vol 8, 1988, pp 3–11. A. Medevedev, Fault detection and isolation by a continuous parity space method, Elsevier Science. Automatic, vol.31. No. 7, 1995, pp. 1039-1044. Y. Menasria, N. Debbache, A Robust Actuator Fault Detection and Isolation Approach for Nonlinear Dynamic Systems, IREACO, Vol. 1. n. 2 : 169-176, July 2008. Ahmed Hafaifa, Ferhat Laaouad, Kouider Laroussi, Centrifugal Compressor Surge Detection and Isolation with Fuzzy Logic Controller, IREACO, Vol. 2. n. 1: 108-114, January 2009. M. N. Saadi, H. Kherfane, B. Bensaker, An Observer-Based Approach for Induction Machine Faults Diagnosis, International Review of Automatic Control (IREACO), Vol. 2. n. 3: 314-319, May 2009. M. Kinnaert, Fault diagnosis based on analytical models for linear and nonlinear systems a tutorial, free Bruxelles University. A.S. Willsky, A survey of design methods for failure detection in dynamic systems, Automatica, vol. 12, 1976, pp.601-611. R. Isermann,Process fault detection based on modelling and estimation methods: A survey. Automatica, vol.20 , 1984, pp. 387-404. R. Isermann, Integration of fault detection and diagnosis methods, the 8th Conference . IFAC-Symposium safe process , Espoo, pp. 597-612, 1994. R.J. Patton, P.M. Frank and R.N. Clark , Fault diagnosis in dynamic systems - Theory and applications, Prentice Hall. Control Engineering Series, vol. 12, 1989, pp. 601-611. P.M. Frank, Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy, Automatica, vol.26, 1990 , pp.459-474. J. Gertler,Analytical redundancy methods in failure detection and isolation, Conference, IFAC-Symposium SAFEPROCESS , BadenBaden, 1991 ,pp.9-21. R.J. Patton, Robust model-based fault diagnosis: The state of the art, the Conf. IFAC Symposium SAFEPROCESS, Espoo, 1994 , pp.1-24. PM. Frank, Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy – a survey and some new results,. Automatica vol.26, 1990, pp.459–74. J. Gertler, Residual generation in model-based fault diagnosis, Control Theory Adv Technol, vol.9, 1993, pp.259–85. J. Gertler and R. Monajemy R, Generating directional residuals with dynamic parity relation, Automatica, vol.31, pp.627–635, 1995. C. Guernez, JP. Cassar, M. Staroswiecki, Extension of parity space to nonlinear polynomial dynamic”,


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[18] V. Krishnaswami and G. Rissoni, Nonlinear parity equation residual generation for fault detection and isolation. the IFAC Symposium SAFEPROCESS. Espoo, vol. 1, 1994,pp. 317–322. [19] LA. Mironovsky, Functional diagnosis of nonlinear discrete-time processes. Autom Remote Control, vol.6, 1989, pp.150–157. [20] A. Shumsky, Parity relation method and its application to fault detection in nonlinear dynamic systems, Autom Remote Control, vol.9, 1998 , pp.155–165. [21] P. Zhanga, H.Yeb, S.X. Dinga, G.Z.Wangb and D.H. Zhoub, On the relationship between parity space and H2 approaches to fault detection , Systems & Control Letters , vol.55, 2006, pp 94 – 100. [22] T. Hiifling and R. Isermann, Fault detection based on adaptive parity equations and single parameter tarcking,, Control Eng. Practice, vol. 4, 1996 , pp. 1361-1369. [23] J. Gertler, and D. Singer, A new structural framework for parity equation based failure detection and isolation, Automatica, vol. 26, 1986, pp.381–388. [24] F. Gustafsson, Stochastic fault diagnosis stability in parity spaces, the 15th Triennial World Congress, Barelona. [25] T. Kailath , Linear systems, Prentice-Hall, 1980. [26] B. Liu, and J. Si, Fault detection and isolation for linear time invariant systems, th 33rd Conference on Decision and Control, Lake Buena Vista, 1994 , pp. 3048–3053. [27] M. Massoumnia, and W. E. Vander Velde, Generating parity relations for detecting and identifying control system component failures, Journal of Guidance, Control and Dynamics, no. 1 1987, pp. 60–65. [28] M. Massoumnia, G. C. Verghese, and A. S. Willsky, Failure detection and identification, IEEE Trans. Autom. Control, no. 3, 1989, pp. 316–321. [29] P.M. Frank, Analytical and qualitative model-based fault diagnosis - a survey and some new results, European Journal of Control, no. 1, 1994, pp. 6–28. [30] J. J. Gertler and M. M. Keunwer., Optimal residual decoupling for robust fault diagnosis, International Journal of Control, no. 2, 1995, pp. 395–421. [31] D. Henry and A. Zeolghadri , Design and analysis of robust residual generators for systems under feedback control, Automatica, no. 2, 2005, pp. 251–264. [32] R. Isermann and P. Ballé, Trends in the application of modelbased fault detection and diagnosis of technical process, Control Engineering Practice, no. 5, 1997, pp. 709–719. [33] H. Niemann, Performance based fault diagnosis, the American Control Conference, Anchorage, USA, 2002, pp. 3943 – 3948.

Authors’ information Adouni Amel obtained the master degree on July 23 in 2011 in automatic and intelligent techniques from National engineering School of Gabes. She is actually a Doctorate in electrical engineering. Her fields of interest include power electronics, machine drives, automatic control, fault detection, Isolation and renewable energies.

Mouna Ben Hamed obtained the master degree on September 22 in 2006 and PhD degree on Marsh 04 in 2009 in automatic and intelligent techniques from National engineering School of Gabes. She works actually as an associate professor at the High Institute of Industrial System of Gabes (ISSIG)-Tunisia. Her fields of interest include power electronics, machine drives, automatic control, modeling, observation, identification, fault detection, localization and Isolation and renewable energies. Lassaad Sbita obtained the doctorate thesis on July 1997 in Electrical engineering from ESSTT of Tunis, Tunisia. He works as Professor at the electrical and automatic department of the National Engineering School of Gabes, Tunisia. He received the “Habilitation de Diriger des Recherches” (HDR) from the National Engineering School of Sfax, Tunisia in Marsh 08 2008. He is Director of research unit entitled “Photovoltaic, Wind and Geothermal Systems”. His fields of interest include power electronics, machine drives, automatic control, modeling, observation and identification real time control and renewable energies.



782


Novel Adaptive Control Scheme for Suppressing Input Current Harmonics in Three-Phase AC Choppers T. Suresh Padmanabhan1, M. Sudhakaran2, S. Jeevananthan3 Abstract – Selective harmonic elimination pulse width modulation (SHEPWM) techniques are special kind of optimal PWM techniques. They can generate high quality output waveform through elimination of selected lower order harmonics with a low number of switching transitions. In this paper, a new adaptive current harmonic selective elimination algorithm is proposed for performing harmonic elimination in the supply current of three-phase power converters. The algorithm uses Least Mean Square (LMS) method, which can eliminate any numbers of selected harmonics just by knowing their frequencies and it can also improve the total harmonic distortion (THD) of the line current. This proposed adaptive method is effective in predicting fundamental and harmonic current information from the line current through the self-adjustment of weights of unit sine and cosine components. The predicted harmonic currents are used as the reference signals for selective elimination. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: AC Voltage Controller, Least Mean Square (LMS) Algorithm, Selective Harmonic Elimination (SHE)

Harmonics can also cause harmonic torques, which have little effect on mean torque, however can produce significant torque pulsations and noise. The switching devices can be operated more effectively at low switching frequency with improved voltage and current harmonic distortion requirement and the switching scheme is called as selective harmonic elimination(SHE)[9]. This results in significant reduction of electromagnetic interference (EMI) emission and filtering effort. Power loss of the main circuit can be reduced, because of the lower switching voltage [10]. With the growth of power electronic and digital electronic technologies, switched mode ac chopper based system has been studied by few researchers and control strategies and simulations have been developed. The bygone techniques of controlling the ac voltage controller are phase angle control (PAC) and On-Off control. To improve the current waveform, passive or active filters are usually used to cancel existing current harmonics to meet specific power quality related standards [11]. S. Sangeetha et al suggested an adaptive filtering scheme based on least mean square (LMS) algorithm to eliminate selective current harmonics in three-phase inverters. Harmonic elimination in this innovative approach is achieved by adding weighted sine and cosine components of respective selected frequencies to match their amplitudes and phase angles present in the line current and then subtracting this sum from the line current [12].

Nomenclature d, q ∇k µ Xk Dk Yk Wk α-β

εk

Uc PLL

Direct and quadrature axes quantity Error gradient Adaptation gain constant Reference input Input signal Output signal Weight updating Two phase stationary coordinate system Error estimation Control input Phase locked loop

I.

Introduction

Rapid developments in the field of power electronics and miniaturization/mass production of control electronics have reached such a stage that ac voltage controller drives are becoming increasingly popular in today’s motor drives [1]-[5]. A switching mode ac to ac converter is one type of the modern power electronic systems [6]-[8].The function of a switching mode ac voltage controller is to produce an ac voltage/current, with controllable magnitude and frequency. These devices generate significant harmonic currents and hence affect the performance of rotating machines. Harmonic currents increase electromechanical power losses in rotating machines, which will heat the machine and cause reduction in the service life of the machine. Manuscript received and revised October 2012, accepted November 2012

783


T. Suresh Padmanabhan, M. Sudhakaran, S. Jeevananthan

The adaptive selective harmonic elimination in output currents of ac voltage controller has been successfully implemented using LMS method by Suresh Padmanabhan.T et al [13]. This method eliminates the dominant harmonics in output current and it requires only the knowledge of the frequency of the particular harmonic to be eliminated. The main objective of this paper is developing an algorithm suitable for the most general case of a threephase ac voltage controller circuit to eliminate the selected lower order current harmonics and improves the total harmonic distortion (THD). The adaptive gains of the proposed Least Mean Square algorithm based method can be chosen relatively large to obtain faster convergence. The stability of the proposed method is guaranteed. This adaptive harmonic selective algorithm involves abc to d-q transformation and its inverse transformation. MATLAB/SIMULINK is used for simulation study and results such as THD, magnitude of fundamental component and lower order harmonics are taken for analysis.

II.

LMS algorithm uses a gradient-based method of steepest decent. It uses the estimates of the gradient vector from the available data. LMS incorporates an iterative procedure that makes successive corrections to the weight vector in the direction of the negative of the gradient vector, which eventually leads to the minimum mean square error. The algorithm is commonly explained by considering a pot like Mean Square Error (MSE) structure as indicated in Fig. 1. Initial assumption is that weights are at the point indicated by the arrow mark. Though the LMS algorithm tries to reach the steepest gradient, the final weights and the corresponding error will reach the bottom point of the pot. II.2.

A structure suitable for performing single frequency adaptive selective harmonic elimination (ASHE) is illustrated in Fig. 2. The LMS algorithm is implemented with regular plant control is shown in the figure. The algorithm changes (adapts) the weight values so that the error is minimum. Combiner combines the weighted reference signals and the output of the combiner Yk is given to the plant transfer function, which is connected to the load. In a closed loop, the iterations of LMS algorithm will stop only if the estimation of the mean square error is minimum. Though the adaptation process is slow, the LMS algorithm neither interferes with system dynamics nor alters the transfer function of the plant and related control.

Adaptive Selective Harmonic Elimination II.1.

LMS Algorithm in SHE

Least Mean Square Algorithm

The Least Mean Square (LMS) algorithm is most commonly used adaptive algorithm because of its simplicity and a reasonable performance. Since it is an iterative algorithm it can be used in a highly time varying signal environment. It has a stable and robust performance against different signal conditions [14]. The LMS algorithm, introduced by Widrow and Hoff in 1959 is an adaptive algorithm, which incorporates an iterative procedure that makes successive corrections to the weight vector in the direction of the negative of the gradient vector which eventually leads to the minimum mean square error. Compared to other algorithms LMS algorithm is relatively simple; it does not require correlation function calculation nor does it require matrix inversions [15].

Fig. 2. Single frequency ASHE canceling filter structure

The error estimation is given by:

ε k = Dk − Yk = Dk − X kT Wk

(1)

The reference and weight vectors are defined by: X kT = [ X a X b ]

Fig. 1. MSE surface


(2)


784


WkT = [Wa Wb ]

harmonic components still present) is introduced to the error inputs of the blocks 5 and 7. The primary input Dk=Uc+Udis. Where, ‘Uc’ is the control input and ‘Udis’ is the undesirable harmonic component of the converter input.

(3)

The gradient ∇ k is estimated as the following: ⎡ δε k2 ⎤ ⎢ 1⎥ δ w ⎥ = 2ε ∇k = ⎢ k ⎢ δε 2 ⎥ ⎢ k2 ⎥ ⎢⎣ δ w ⎥⎦

⎡ δε k ⎤ ⎢ 1⎥ ⎢ δ w ⎥ = −2ε k xk ⎢ δε k ⎥ ⎢ 2⎥ ⎣δ w ⎦

(4)

The weight updating formula is given by: Wk +1 = Wk − µ∇ k = Wk + 2 µε k X k

(5)

where µ is the adaptation gain constant. The adaptive process attenuates substantial amount of noise in the gradient estimate. The detailed algorithmic steps involved in proposed ASHE is diagrammed in Fig. 3.

Fig. 4. ASHE for multiple harmonics

The proposed ASHE algorithm has advantages such as: (i) (ii) (iii) (iv) (v) (vi)

Easy tuning of adaptive gains for harmonic reference fixing Better convergence properties, which makes the harmonic selective elimination better Dynamic and steady state performance Solution of stability problems associated with adaptive detection algorithms Fast frequency tracking ability when power system has big frequency change Works well when system currents have dc offset.

Fig. 3. Flowchart for LMS algorithm

II.3.

III. ASHE in Three-Phase AC Voltage Controller

Multiple Harmonics Elimination

The adaptive selective elimination scheme for multiple harmonics is illustrated in Fig. 4. Harmonics, 5th and 7th are eliminated from the plant input current. The expansion to eliminate the other harmonic components can be done by adding blocks like fifth and seventh. Additional blocks will have the same error input ε, frequency of the reference signal will be equal to the input harmonic component to be eliminated and the output will be added to the outputs of the previous blocks (5 and 7). Each ASHE block has a complete system as indicated in Fig. 2. The first harmonic (fundamental component) is taken out of the primary input, first. Then filtered first harmonic output at the block 1 (but fifth and seventh

The application of ASHE in the three-phase ac voltage controller is illustrated in Fig. 5. It has PI controller U_reg for dc bus voltage control and two PI regulators Iq and Id implemented in synchronous reference frame for current control. PLL block is used for generating the sine and cosine reference angles with frequency of fundamental component, and frequencies of fifth and seventh harmonic components. Selected frequency components can then be separated from other frequency components in harmonic line currents. This process is explained clearly in the Fig. 4. For harmonic reference generation, consider the 5th harmonic extraction. Source currents ia, ib and ic are



785


measured and sent to the 3 phase-to-2phase transformation (abc/dq transformation). The d-q transformation is a transformation of coordinates from the three-phase stationary coordinate system to the d-q rotating coordinate system [16]-[17]. This transformation is made of two steps: (i) Transformation from the three phase stationary coordinate system to the two phase α-β stationary coordinate system; (ii) Transformation from the α-β stationary coordinate system to the d-q rotating coordinate system. The two currents in rotating frame are dc components now, which prove that the three phase ac quantities can be transformed to dc components in a rotating reference frame with the speed of the ac quantities. This transformation can also be applied to currents.

An inverse transformation can transfer the dc components back to ac components. If the synchronous signals sinθ5 and cos θ5 are well provided through a PLL, the 5th harmonic currents become dc components in this d-q rotating frame while other frequency components still keep their AC form but with different frequencies. The same method can be applied to any frequency component. Different Phase Locked Loops (PLLs) are required to synchronize each rotating reference frame frequency. Sinθ1 and cosθ1 through sinθn and cosθn are required for the transformations; where, ‘n’ indicates the order of harmonics to be eliminated. The direct component and quadrature component will have a phase difference of 90 degrees for an ac system. Figs. 6, 7, 8 show the relevant transformations. Fig. 9 indicates the output of MF_ASHE block.

Fig. 5. ASHE support three phase ac voltage controller

Fig. 6. abc to dq- stationary transformation

Fig. 7. dq- stationary to dq- rotationary transformation



786


Fig. 11. Updating weights of fundamental component

Fig. 8. dq- rotationary to dq- stationary transformation

(a)

Fig. 9. Output of MF SHE block

(b)

(a) (c) Figs. 12. Updating weights (a) 5th harmonics, (b) 7th harmonics and (c) 11th harmonics

IV.

Results

Three-phase ac voltage controller is simulated without and with ASHE algorithms, separately. Figs. 10 show the wave forms of input line currents and harmonic spectrum without ASHE algorithms. The weight updating process is indicated in Fig. 11 and Figs. 12. The improved wave forms of input line currents and harmonic spectrum after implementation of ASHE are shown in Figs. 13. Table I provides the comparison of THD, magnitude of fundamental component and harmonics for the system with and without ASHE algorithm. About 40% reduction

(b) Figs. 10. (a) Input current waveform and (b) Harmonic Spectrum without ASHE filter



787


scheme by eliminating 5th, 7th and 11th harmonics and also improving the total harmonic distortion. The proposed system has suitability of implementation with ac drives like induction motor drive, synchronous drive etc.

in THD is evidenced from the results. The percentage reductions in 5th, 7th and 11th harmonics are 79, 66 and 64 respectively.

References [1]

[2]

[3]

[4]

(a)

[5]

[6]

[7]

[8] (b) [9]

Figs. 13. (a) Input Current waveform and (b) harmonic spectrum with ASHE filter

[10]

TABLE I HARMONIC SUPPRESSION EFFECT OF ASHE ALGORITHM Without ASHE With ASHE LMS Harmonic Order (% of fundamental) (% of fundamental) 1 100 100 5 38.03 7.87 7 21.44 7.22 11 14.12 5.06 THD 99.60 60.54

V.

[11] [12]

[13]

Conclusion

Generation of current harmonics is very common problem associated with adjustable speed drives. Performing selective current harmonic elimination is the best option to guarantee the higher quality torque in any drive system. The existing selective harmonic elimination solution suffer by any one intricacy such as dependency on starting values guess, involvement of high order polynomials, probabilistic nature, slow speed of convergence, multiple sets of solutions, local optima etc. The Least Mean Square algorithm based adaptive scheme developed in this paper is triumph in performing current harmonic elimination three-phase system. The simulation results demonstrated the effectiveness of the

[14]

[15] [16]

[17]


Blaabjerg. F, Consoli. A, Ferreira. J. A, and Van Wyk.J.D, “The Future of Power Processing and Conversion,” IEEE Transactions on Industry Applications, vol.41, No.1, 2005, pp.3-8. Wang.S, Lembeye.Y, and Ferrieux J.P, “Design and Implementation of a High Switching Frequency Digital Controlled SMPS”, Proceedings of the Conference, IEEEPESC’2007, Orlando, June 17-21, 2007, pp.219-223. Kimmo Rauma, “FPGA Based Control Design for Power Electronic Applications”, Ph.D. thesis, Acta Universitatis, Lappeenrantaensis, 2006. Foley.R, Kavanagh.R, Marnane.W, and Egan.M, “Multiphase Digital Pulse Width Modulator”, IEEE Transactions on Power Electronics,vol.21,No.3,2006, pp.842-846. Cichowski.A, and Nieznanski.J,“Self-Tuning Dead-Time Compensation Method for Voltage-Source Inverters”, IEEE Power Electronics Letters, Vol.3, No.2, 2005, pp.72-75. Alesina A., and M. G. B. Venturini,”Analysis and Design of Optimum Amplitude Nine-Switch Direct Ac-Ac Converters,”IEEE Transactions on Power Electronics, vol. 4, No.1, January 1989, pp.101-112. Casadei. D, Serra.G,Tani.A, and Zarri. L, ‘‘Matrix Converter Modulation Strategies: A New General Approach Based on Space-Vector Representation of the Switch State”, IEEE Transactions on Industrial Electronics, Vol. 49, No. 2, April 2002, pp 370-381. Patrick W. Wheeler, José Rodríguez, Clare.J.C, Empringham.L, and Weinstein.A,‘‘Matrix Converters: A Technology Review”, IEEE Transactions on Industrial Electronics, vol. 49,No. 2, April 2002, pp. 276-288. Y. S. Lai, and S. R. Bowes, “A Novel Harmonic Elimination Pulse-Width Modulation Technique for Static Converter and Drives,” Proceedings of IEEE APEC’98, Vol.1, 1998,Pp.108–115. F. Mihalic and D. Kos, “Reduced Conductive EMI in SwitchedMode Dc-Dc Power Converters Without EMI Filters: PWM versus Randomized PWM,” IEEE Transactions on Power Electronics, vol. 21, no. 6, Nov. 2006, pp. 1783–1794. Alexander Kusko, and Marc T. Thompson, Power Quality in Electrical Systems (McGraw Hill, New York, 2007). S. Sangeetha, CH. Venkatesh and S. Jeevananthan, “Selective Current Harmonic Elimination in a Current Controlled DC-AC Inverter Drive System using LMS Algorithm”, Proceedings of International conference on Computer Application in Electrical Engineering Recent Advances (CERA-2009), paper no.333, Indian Institute of Technology, ROORKEE, 19th to 21st Feb. 2010. T. Suresh Padmanabhan, M. Sudhakaran and S. Jeevananthan,” Selective Current Harmonic Elimination in an AC Voltage Controller Drive System using LMS Algorithm”,Proceedings of 46th International Universities’ Power Engineering Conference (UPEC 2011), Paper-169, 5th-8th September 2011, South Westphalia University of Applied Sciences, Soest, Germany. Vladamir Blasko, “A Novel Method for Selective Harmonic Elimination in Power Electronic Equipment,” IEEE Transactions on Power Electronics, vol.22, No.1, January 2007, pp 223-228. S. Haykin, Adaptive Filter Theory (4th Edition, Prentice-Hall, 2002). L. Qian, D. Cartes and Q. Zhang, "Three-Phase Harmonic Selective Active Filter Using Multiple Adaptive Feed Forward Cancellation Method”,IEEE International Power Electronics and Motion Control Conference (IPEMC 2006), Volume 2, Aug. 2006, pp.1-5. H. R. Hafezinasab, H. Feshki Farahani,”A Quadratic Model for Total Harmonic Distortion in a Dual- Tap Chopping Stabilizer


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with Four-Quadrant Switches”, International Review on Modelling and Simulations (IREMOS), August 2011 (Part B), Vol. 4 N. 4, pp. 1857-1863. [18] S. V. Heidari, M. Sedighizadeh, A. Rezazadeh, M. Ahmadzadeh,” Lyapunov Based Self-tuning Control of Wind Energy Conversion System” International Review on Modelling and Simulations (IREMOS), October 2010 (part A), Vol. 3. n. 5, pp.864-869. [19] Vincent A. Akpan, George D. Hassapis,” Training Dynamic Feedforward Neural Networks for Online Nonlinear Model Identification and Control Applications”, International Review of Automatic Control (IREACO), May2011,Vol. 4 N. 3, pp. 335350. [20] R. Krishnan, Electric Motor Drives: Modelling, Analysis and Control (Prentice Hall Inc., 2001).

Authors’ information T. Suresh Padmanabhan was born in Nagercoil, India, in June 1980.He received the B.E degree in Electrical and Electronics Engineering from Noorul Islam College of Engineering, in 2001.He obtained his M.E degree in Power Electronics and Industrial Drives from Sathyabama University, Chennai, India, in 2005.Also, he is pursuing the Ph.D. degree in Electrical and Electronics Engineering from Pondicherry University, Pondicherry, India. Since August 2008, he has been with Bharathiyar College of Engineering and Technology, karaikal, India. He is working as Associate Professor in Electrical and Electronics Engineering department. He is having ten years of teaching experience. In addition, he has authored/co-authored in several international journals and conferences. His current research interests include power electronic converters, artificial intelligence (AI) applications in power electronic systems, etc. Dr. M. Sudhakaran was born in Madurai, India.He received the B.E. degree in Electrical and Electronics Engineering from Manonmaniam Sundaranar University, Tirunelveli, India, in 1997, and the M.E. degree from Thiyagarajar College of Engineering, Madurai, India, in 1998. He completed his Ph.D. degree from Madurai Kamaraj University in 2004. Since 2001, he has been with the Department of Electrical and Electronics Engineering, Pondicherry Engineering College, Pondicherry, India, where he is an Associate professor. He has made a significant contribution to the power systems through his publications. He has published many papers in international and national conference proceedings and professional journals. He has received many honors and awards in academics. He is an active member of the professional societies, IE (India) and ISTE. Dr. S. Jeevananthan was born in Nagercoil, India on May 25, 1977. He received the B.E. degree in Electrical and Electronics Engineering from MEPCO SCHLENK Engineering College, Sivakasi, India, in 1998, and the M.E. degree from PSG College of Technology, Coimbatore, India, in 2000. He completed his Ph.D. degree from Pondicherry University in 2007. Since 2001, he has been with the Department of Electrical and Electronics Engineering, Pondicherry Engineering College, Pondicherry, India, where he is an Associate professor. He has made a significant contribution to the PWM theory through his publications and has developed close ties with the international research community in the area. He has authored more than 50 papers published in international and national conference proceedings and professional journals. Dr. S. Jeevananthan regularly reviews papers for all major IEEE TRANSACTIONS in his area and AMSE periodicals (France). He is an active member of the professional societies, IE (India), MISTE, SEMCE, and SSI.



789


Decentralized Adaptive Sliding Mode Exciter Control of Power Systems S. Benahdouga, D. Boukhetala, F. Boudjema, R. Khenfer, M. Meddad

Abstract – The robustness of parameter variations and external disturbances is the major property of sliding mode control system. The aim of this paper is to present a decentralized adaptive sliding mode controller (DASMC) with a nonlinear sliding surface for nonlinear multimachine power systems model. The combination of adaptive approach and sliding mode controller is used to achieve the decentralization of the control and to exploit the advantage of the sliding mode control. The feedback linearization technique is used in order to handle the nonlinearities of power systems model. Each machine is modeled as an independent uncertain dynamic subsystem where the uncertainty is a disturbance that represents the effects of the rest of the system on that particular machine. This disturbance is expressed as a polynomial function of transient EMF deviation and its parameters are estimated by an adaptive control. A local adaptive sliding mode stabilizer is designed to regulate the machine angle and to stabilize the terminal voltage of each generating unit in the global system under high level external disturbances. This method is illustrated with a three machines-infinite bus power systems. Simulations show that the proposed (DASMEC) provides high-performance in dynamic characteristics and its robustness with regard to external disturbance. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Power System Stability, Feedback Linearization Approach, Sliding Mode Control, Adaptive Sliding Mode Controller

in late 1960s in the former USSR that was led by Prof. V. I. Utkin and Prof. S. V. Emelyanov [1] and has been widely applied to robust control of nonlinear systems in recent years such as power electronics, power electrical systems, and robot manipulators [2]. The sliding mode control employs a discontinuous control law to drive the state trajectory towards a specified sliding surface and maintains its motion along the sliding surface in the state space. Due to the switching and time delays in system dynamics, it is difficult for the system trajectory to reach the ideal sliding mode and therefore a high frequency motion described as chattering occurs[3], and it is the main drawback of (SMC). Centralized control as well as decentralized one and (hierarchical) combination of these schemes have been used to design Electrical Power Systems (EPSs) controllers. Since the EPSs with interconnected generators through transmission lines are modeled as complex large-scale systems, the implementation of the centralized controller could be difficult while the decentralized control allows simplifying the power system controller design procedure, rejecting interactions between subsystems and only local measurements are required [4]. The idea of using a feedback linearization approach is used to linearize the system by considering the term of

Nomenclature Id , Iq

d and q are axis stator circuit currents

Ed' ,Eq'

d and q are components of the transient EMF

Vd ,Vq

d and q are components of the terminal voltage

Vt

xd ,xq

Is the terminal voltage Are rotor speed and rated rotor speed of the generator Are machine angle and the reference machine angle d and q are axis synchronous reactance

x'd , x'q

d and q are axis transient reactance

r H D E fd

Is the armature resistance Is the inertia constant Is the damping constant Is the exciter voltage

Pm Pe

Is the mechanical power Is the electrical power

ω , ω0 δ , δ ref

I.

Introduction

The Sliding mode control (SMC), based on the theory of variable structure systems (VSS), was discussed first Manuscript received and revised October 2012, accepted November 2012

790


S. Benahdouga, D. Boukhetala, F. Boudjema, R. Khenfer, M. Meddad

interconnection as disturbance and is expressed as a polynomial function of transient EMF deviation [4]. The Adaptive control is used to estimate the coupling term parameters in order to decentralize the control law. The proposed system consists of three generating units. Each unit is considered as a subsystem which is connected to the rest of the system through voltage transmission lines. Each subsystem is modeled by a nonlinear dynamic model. The objectives of the design control are to regulate the machine angle of each synchronous generator and to decentralize the controller. The paper is organized as follows: The nonlinear mathematical model of the electric power system is described in the first section [5]-[8]. In the second section, the feedback linearization approach is applied to each subsystem model in order to handle the nonlinearities and to develop models of interconnection terms [9]-[13]. In third section, the adaptive control is presented in order to permit the decentralization of the global system by the approximation of coupling term and to use, only the local measurements of each controller [9]-[11]. The adaptive sliding mode exciter control method is applied to each subsystem by using the linearized model that is investigated in section 2. Finally, the performances of the (DASMEC) according to robustness tests are shown through nonlinear simulations of the power system responses.

II.

)

(

∑ ( Bij ⋅ cos (δ ij )

j =1 n

j =1

' Eqi

( ))

(6)

+ Gij ⋅ sin δ ij

⎧ x1i = x2i ⎪ ⎪⎪ x2i = ω0 Pmi − x3i I qi − Di (ωi − ω0 ) 2Hi 2Hi ⎨ ⎪ 1 ⎪ x3i = ' E fd i − x3i − ∆xdi I di Tdoi ⎪⎩

(

)

(

(7)

)

(1) when: xi = [ x1i x2i x3i ]

) )

(

I qi =

n

This section is devoted to the feedback linearization approach where the nonlinear control law transforms the nonlinear system into an equivalent linear system [12][13]. The decentralized and linearized controllers for the multi-machine power systems can be used in practice. For this purpose, the decentralized one is obtained by this technique so the control law must cancel the inherent system nonlinearities and the obtained state transformation leads to the equivalence of the feedback linear system where the decentralized information of the system has to be used. Thus, the following problem is very important in practice: how to design decentralized controllers for multi-machine power systems to improve the transient stability [12]-[14]. The methodology is first to transform the nonlinear system into a controllable linear system and then to apply the Decentralized Adaptive Sliding Mode Exciter Control (DASMEC). We start this section by calculating the relative degree of each subsystem and then we transform the models into the canonicals form to obtain the linearized and decoupled control laws [12]-[13]. The system studied is composed of three generators infinite bus, each generator is considered as a subsystem of the global system. The subsystem is presented by the following states equations [5], [9]-[11]:

In this section, the mathematical formulation of a large-scale power system has been developed for a power system with interconnected n-generating units through a transmission network. The mechanical and electrical models of ith synchronous machine are given by the following equations [5], [9]-[11]:

(

∑ Eqi' ( Bij ⋅ sin (δ ij ) − Gij ⋅ cos (δij ) )

III. Nonlinear Feedback Linearization

Electric Power System Model

⎧ dδ ⎪ i = ωi − ω0 ⎪ dt ⎪⎪ d ω D ω i = 0 Pmi − Pei − i (ωi − ω0 ) ⎨ 2H 2H ⎪ dt ' ⎪ dEqi 1 ' ⎪ = + xdi − x'di I di E fdi − Eqi Tdoi ⎪⎩ dt

I di =

T

is the stat vector, for ith subsystem where: ' Pei = Eqi I qi

(2)

Vdi = xqi I qi

(3)

' Vqi = Eqi − x'di I di

(4)

Vti =

(

Vdi2 + Vqi2

i = 1, 2 , 3

III.1. Relative Degree

)

Consider a dynamic system of the form: x = g ( t, x ) + b ( t,x ) u, y = h ( t ,x )

(8)

(5)



791


⎡ Z1i ⎤ ⎡ 0 1 0 ⎤ ⎡ Z1i ⎤ ⎡ 0 ⎤ ⎡0⎤ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Z 2i ⎥ = ⎢ 0 0 1 ⎥ ⋅ ⎢ Z 2i ⎥ + ⎢ 0 ⎥ v1i + ⎢ 0 ⎥ di ⎢ ⎥ ⎢⎣1 ⎥⎦ ⎣ Z 3i ⎦ ⎢⎣ 0 0 0 ⎥⎦ ⎣⎢ Z 3i ⎦⎥ ⎢⎣1 ⎥⎦

where x ∈ R n , u ∈ R is control, y is only measured output g, b and y are smooth functions. In our case the output variable coincide with the sliding surface so:

∂s( ) r s ( ) = g ( t,x ) + b ( t,x ) u , =b≠0 ∂u r

The linearized and decoupled control law is given as follow:

where the relative degree r is defined by the conditions [12]-[13]: L f h = L f h = ... = L f 2

r −2

h=0

,Lg Lrf−1h

Linearization Control Law

III.3.

(9)

(12)

E fd i = −

≠0

1 Lg L2f hi1

⎡ − L3f h1i ( x ) + vi + di ⎤ ⎦

( x) ⎣

(13)

where: and as our objective is to control the rotor angle of each synchronous machine, we choose it as the first output for calculating the relative degree. For this, we pose:

Lg L2f hi1 ( x ) = L3f hi1 ( x ) =

hi = x1i − xd1i = δ1ι − δ d1i

d 2t d 3 hi

=

ω0 d 2 H i dt

=−

(

where: Pmi − x3i I qi

αi =

) (10)

ω0 ⎡ 1 ⎛ E fdi − x3i + ⎞

⎢ ⎜ 2 H i ⎢⎣ Td' 0i ⎜⎝ −∆xdi I di d 3t . ⎤ ∆x ⋅ ω ⎛ . + x3i ⋅ I qi ⎥ − di 0 ⎜ I qi ⋅ I di 2Hi ⎝ ⎦

So the obtained relative degree is

. ⎞ + I qi ⋅ I di ⎟ ⎠

di =

( r = n = 3) where

The canonical form of the system is given by the differential Eqs. (10) where it is calculated by the following nonlinear transformation: Z1i = h1i ( x ) Z 3i = L2f h1i ( x ) =

ω0 2Hi

2 HTd' 0i

Eq' I qi and βi =

−ω0 2 HTd' 0i

I qi

ω0 ⎡

. . ⎛. ⎞⎤ ⎢ x3i ⋅ I qi + ∆xdi ⎜ I qi ⋅ I di + I di ⋅ I qi ⎟ ⎥ 2H ⎣ ⎝ ⎠⎦

(15)

In the literature, many methods are used for decentralized the control law, L. Fan which is based on taking into account the maximum measuring of the di and also A. Karimi applied an adaptive backstepping to approximate the coupling term and H. Huerta applied a nonlinear observer for estimating the rotor fluxes and mechanical torque. Where the coupling term di is given by (15), this term includes local and remote information. In this paper, it is expressed as an uncertain polynomial function of transient EMF deviation, i.e. with parameters that will be estimated and are local information [15]:

Canonical Form

Z 2i = L f h1i ( x ) = x2i

ω0

(14)

So, the interconnection terms sleepers di is given by fellow Eqn. (15):

⎟⎟ I qi + ⎠

the model of subsystem is completely or exactly linearized. III.2.

)

vi = α i + β i E fdi

dhi dx1i = = x2i dt dt d hi

(

w0 ' Eqi − ∆xdi I qi 2 H iTd 0i

The virtual control vi as defined by:

Then, the first total derivatives of hi are calculated by follows [9]-[13]:

2

2 H iTd' 0i ω0 I qi

(11)

( Pmi − x3i I qi ) − 2DHi (ωi − ω0 )

' '2 di ≈ θ1i ⋅ Eqi + θ 2i ⋅ Eqi

(16)

i

where θ1i and θ 2i are the uncertain values that are not known in priori which need to be estimated through adaptation laws. In generic terms, the equations set (12) for ith generator, are given by:

Z i = [ Z1i , Z 2i , Z 3i ] , is the new state variable vector.

Then the linearized state space model is written into the canonical form as follow:



792


Z1i = Z 2i Z 2i = Z3i ' '2 + θ 2i Eqi Z3i = vi + θ1i Eqi

IV.1. Equivalent Control The main steps for sliding mode controller design, is to calculate the equivalent control Veqi. and it’s designed to keep the states moving on sliding manifold. The equivalent control concept can be summarized as follows: - Selecting switching surface Si ( x,t ) = 0 that provides

(17)

The quadratic polynomial estimate of the disturbance has given adequate results.

IV.

Sliding Mode Control Design -

The first goal in this paper is to characterize the class of manifold on which the control objective is achieved. We recall that the sliding mode control objective consists of: first, designing a suitable manifold Ψ ( x,t ) ∈ R defined by ψ = { x ∈ R / S ( x ) = 0} to restrict

the desired asymptotic behaviour in steady state where x, is the system’s vector. Obtaining the equivalent control veqi . veqi is

Si = 0 ⇔ K i Zi − K1i Z di = 0

the state trajectories of the plan to this manifold to result in the desired behavior such as tracking, regulation, and stability; then, determining a switching control law v ( x,t ) which is able to drive the state trajectory to this

−1

−1

⇒ veqi = − ( k1/k 3) ⋅ Z 2i − ( k 2 /k 3) ⋅ Z 3i

(21)

IV.2. States Feedback Vector

manifold Ψ ( x,t ) ∈ R is made attractive and invariant

Replace veqi in Equation (14):

[1]-[2],[16]-[18]. Consider the following dynamical linearized system:

Y1i = Ci Z1i

(20)

⇒ veqi = − [ Ki Bi ] Ki Ai Z i − [ Ki Bi ] Ki B p Z di

manifold and maintain it on this manifold for all the time. That is, v ( x,t ) is determined such that the selected

Zi = Ai ⋅ Z i + Bi ⋅ Vi + B pi di

calculated by:

.

Z eqi ⎡ Ai − Bi [ Ki Bi ] Ki Ai ⎤ Z i + ⎣ ⎦

(18)

−1

−1 + ⎡ I − Bi [ Ki Bi ] Ki ⎤ B p di ⎣ ⎦

where i = 1, 2, 3 and Z i = [ Z1i Z 2i Z 3i ] is the new state

−1 Aeqi = ⎡ Ai − Bi [ Ki Bi ] Ki Ai ⎤ ⎣ ⎦

T

vector. The vectors Bi , B pi , Ci and the matrix Ai are given by:

(22)

(23)

where Aeqi is a dynamic matrix of the equivalent system (18):

⎡0 ⎤ ⎡0 ⎤ ⎡0 1 0⎤ ⎢ ⎥ ⎢ ⎥ Bi = ⎢0 ⎥ B pi = ⎢0 ⎥ Ai = ⎢⎢ 0 0 1 ⎥⎥ ⎢⎣1 ⎦⎥ ⎣⎢1 ⎦⎥ ⎣⎢ 0 0 0 ⎦⎥ Ci = [1 0 0]

1 ⎡0 ⎢ Aeq = ⎢ 0 0 ⎣⎢ 0 − K1i /K3i

⎤ 1 ⎥⎥ − K 2i /K3i ⎦⎥

0

(24)

The characteristic polynomial equation given by: At first, we develop a sliding manifold and we search the properties that must be fulfilled in order to achieve our control objective. A pole placement approach is used to design a state feedback controller. The switching surfaces are given by: Si ( z ) = [ K1i

K 2i

⎡ Z1i ⎤ K3i ] ⋅ ⎢⎢ Z 2i ⎥⎥ = 0 ⎢⎣ Z 3i ⎥⎦

Pi ( λ ) = λi ⎡⎣λi2 + ( K 2i / K3i ) λi + K1i / K3i ⎤⎦

(25)

Pole-placement method is used to design sliding manifold. According to system performance indices, we choose − ρι + jρι and − ρι − jρι as dominant poles. So, the constant gains are directly determined according to the choice of these poles:

(19)

K3i = Cont

After this, we calculate the feedback gain vector Ki by using the Pole-placement method. Z di is the reference = constant.

K1i = 2 ρi2 K3i K 2i = 2 ρi K3i



793


⎛ k1i ⋅ Z 2i + k2i ⋅ Z3i + Vi si ,θi = si ⋅ ⎜ ' ⎜ + k3i ⋅ v1i + θˆ1i ⋅ Eqi + θˆ2i ⋅ Eqi2' ⎝

Finally the feedback vector Ki is given by:

(

Ki = ⎡⎣ 2 ⋅ ρi 2Cont 2 ⋅ ρ Cont Cont ⎤⎦

(26)

(27)

where gi , is positive constant. The following step is to calculate the sliding mode control law. Consider the whole control signal given by: vi ( Z ) = veqi − gi sign ( si )

2i

E fd i

(28)

V.

k1i k w ⋅ w2i − 2i ⋅ 0 ( Pmi − Pei ) + k3i k3i 2 H ' '2 − θˆ1i Eqi − θˆ2i Eqi − gi ⋅ Sat ( si )

vi = −

A standard Lyapunov stability analysis performed on the following function [15], [19]-[20]:

)

T 1 1 Vi si ,θi = si 2 + ⎡⎣θ1i ⋅ θ2i ⎤⎦ Γ −1 ⎡⎣θ1i ⋅ θ2i ⎤⎦ = 2 2 1 2 1⎡ θ1i − θˆ1i ⋅ θ2i − θˆ 2i ⎤ Γ −1 = (30) = si + ⎦ 2 2⎣

(

)(

)(

)

VI.

)

= ⎡ θ1i − θˆ1i ⋅ θ2i − θˆ 2i ⎤ ⎣ ⎦

Γ = diag [ Γ1 , Γ 2 ] , is an adaptation gain matrix.

Whose derivative is given by:

)

Vi si ,θi = si ⋅ si − θ1i ⋅ Γ1−1θˆ1i − θ2i ⋅ Γ 2 −1θˆ 2i = ⎛ ⎛ v1i + θˆ1i ⋅ E' qi ⎞ ⎞ ⎟⎟ + = si ⋅ ⎜ k1i ⋅ Z 2i + k2i ⋅ Z 3i + k3i ⋅ ⎜ ⎜ ⎜ + θˆ 2i ⋅ E 2' qi ⎟ ⎟ (31) ⎝ ⎠⎠ ⎝ ' 2' + s ⋅ k ⋅θ ⋅ E + s ⋅ k ⋅θ ⋅ E + i

3i

1i

qi

i

3i

2i

(34)

(35)

Simulation Results

The simulation results for the centralized sliding mode controllers compared with decentralized adaptive sliding mode controllers are presented in the below. In this section, we apply the proposed (DASMEC) method to the global interconnected system which has been linearized via the feedback linearization approach. The design objectives are to decentralize the controls lows by using adaptive approach firstly, to regulate the rotor angle and to stabilize the terminal voltage of each synchronous generator in the power system Fig. 1, secondly. Some robustness tests are made in order to visualize the performance of the proposed control method. Figs. 2-5 show the performance and the robustness of the proposed centralized (solid line) and decentralized controllers (dash-dotted line) responses under a threephase short-circuit fault occurring at t=0.1 s with a duration of 0.05s. Figs. 6-7 show the coupling terms estimate and the responses of the transient EMF in the two cases: centralized and decentralized controllers. The references angles, the nominal parameters of the system and the initial conditions are given in Appendix. The performance of the proposed controller is compared with centralized control.

T

where θi = θi − θˆi , θˆ1i and θˆ 2i are estimate of θ1i , θ2i .

(

i

Now for obtained the reels excitations field voltage Efd, we replace the (34) in (13). The Chattering phenomena is most problem in sliding mode control, for this, we are removed it by using saturation (Sat) function.

)

(

i

k1i k w ⋅ w2i − 2i ⋅ 0 ( Pmi − Pei ) + k3i k3i 2 H ' '2 ˆ ˆ − θ1i Eqi − θ 2i Eqi − gi ⋅ sign ( si )

Design Decentralized Adaptive Sliding Mode Exciter Control

(

qi

vi = −

⎡ − L3f h1i ( x ) + ⎤ ⎢ ⎥ (29) =− Lg L2f hi1 ( x ) ⎢ + veqi − gi sign ( si ) − di ⎥ ⎣ ⎦

(

)

Finally, these controls laws are written in terms of the original state variables as:

Now the reels controllers’ lows using in the global system are given as follows: 1

(

⎞ ⎟ (32) ⎟ ⎠

The controllers vi are then designed to make Vi ≤ 0 is given: k k ' vi = − 1i ⋅ Z 2i − 2i ⋅ Z 3i − θˆ1i Eqi + k3i k3i (33) '2 ˆ − θ E − k ⋅ sign ( s )

The corrective control term vci can be given as: vici = − gi sign ( si )

)

qi

− θ1i ⋅ Γ1−1θˆ1i − θ2i ⋅ Γ 2 −1θˆ 2i

Choose adaptation laws θˆ1i and θˆ 2i in (31) as: θˆ1i = Γ1 ⋅ k3i ⋅ si ⋅ E' qi and θˆ 2i = Γ 2 ⋅ k3i ⋅ si ⋅ Eqi2'

(31) become:



794


4

8 5 G1 1 A 6

2

7

3

G

G B

Time(s)

C

Fig. 5. Response of Excitation Control under fault

Fig. 1. A three machine infinite bus power system

δ1‐4 δ2‐4

δ3‐4 Time(s)

Time(s)

Fig. 6. Response of Transient EMF under fault

Fig. 2. Response of Machine Angle under fault

Time(s)

Time(s)

Fig. 7. Coupling terms estimated dˆ 1 , dˆ 2 and dˆ 3 under

Fig. 3. Response of Speed Deviation under fault

fault with decentralized controllers

The proposed controller is a decentralized control which uses only local information. More importantly, the interface is modeled easily by simple polynomial function of transients EMF. The simulation results given in above figures prove the robustness of the proposed control, in order to estimate the coupling terms, regulating the rotor angle and stabilizing the terminal voltage around the operating points. It means that the decentralized adaptive sliding mode exciter control has successfully addressed the system requirements.

Time(s) Fig. 4. Response of Terminal Voltage under fault



795


VII.

Conclusion

References [1]

In this paper, two controllers are applied to enhance the transient stability, to regulate the machine angle and also to stabilize the terminal voltage around a nominal value of a nonlinear multi-machine power system trough excitation control. Each generator is modeled as an uncertain dynamic subsystem. The first controller was the direct sliding mode controller which can be directly used to control a system. However, the feedback linearization was used in order to handle the nonlinearity and to decentralize the control law by using the second controller, adaptive sliding mode controller. The uncertainty represents the effects of the rest of the system on a particular generator and it is expressed as a polynomial of transient EMF. The results showed that the (DASMEC) can guarantee the overall stability under extreme conditions (three phases shortcircuit) and illustrate its effectiveness.

[2]

[3]

[4]

[5] [6]

[7] [8]

Appendix

[9]

Switching surfaces gains:

ρ = 2.5

[10]

k13 = 10 , k11 = 2 ⋅ ρ 2 ⋅ k13 and k12 = 2 ⋅ ρ ⋅ k13 k23 = 10 , k21 = 2 ⋅ ρ 2 ⋅ k23 and k22 = 2 ⋅ ρ ⋅ k23

[11]

k33 = 12, k31 = 2 ⋅ ρ 2 ⋅ k33 and k32 = 2 ⋅ ρ ⋅ k33 [12]

The correctives gains: g1 = 50 , g 2 = 50 and g3 = 70

[13]

The references rotor angles are around the initials conditions:

[14]

δ1ref = 37.920 , δ 2ref = 320 and δ 3ref = 20.80

[15]

[16]

Loads Parameters: All data are given in per unit: A=0.4257-j2.038 p.u B=0.1121-j1.176 p.u C=0.4218-j1.475 p.u

[17]

[18]

TABLE I MACHINES PARAMETERS Machine 1 2 3

xd

( p ⋅u) 1.68 0.88 1.02

'

xd

( p ⋅u)

[19]

Td 0 ( p ⋅ u )

H (s)

D

4.0 8.0 7.76

2.31 3.40 4.63

0 0 0

'

0.32 0.33 0.20

[20]

TABLE II INITIALS CONDITIONS Machine

δ ( deg )

Pm ( p ⋅ u )

1 2 3

37.93 32.07 20.88

0.8005 0.6863 0.5004

Ef

( p ⋅ u)

0.3770 0.4513 0.6077

Vt ( p ⋅ u )

Q. P. Ha, Q. H. Nguyen, D.C. Rye, H.F. Durrant-Whyte, Fuzzy Sliding-Mode Controllers with Applications, IEEE Transactions on Industrial Electronics, vol. 48, n. 1, February 2001. A. Boubakir, F. Boudjema, S. Labiod, A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface Applied to the Coupled Tanks System,” International Journal of Automation and Computing., vol. 06, n. 1, pp72-80, February 2009. J.Y. Hung, W. Gao, J.C. Hung, Variable Structure Control: A Survey, IEEE Transactions on Industrial Electronics, vol.40, n.1, pp2-22, 1993. H. Huerta, G.L. Alexander, M.C. Jose, Decentralized sliding mode lock control of multi-machine power systems, Electrical Power and Energy Systems, vol.32, pp1–11, 2010. P.M. Anderson, A.A. fouad, Power System Control And Stability, (IOWA state university press, Ames, IOWA,1977). J.W. Chapman, M.D. Lic, C.A. King, L. Eng, H. Kaufman, Stabilizing a Multi-machine Power System via Décentralized Feedback Linearizing Excitation control, IEEE Transaction on Power system, vol.8, n.3, August 1993. H. Nijmeijer, A.J. van, S. der, Nonlinear Dynamical Control Systems, (Springer-Verlag, New York Inc,1990). J. Machowski, J.W. Bialek, J.R. Bumby, Power System Dynamics: Stability and Control, (Second edition, John Wiley & Sons, Ltd, 2008). Laurence D. Colvara, Luiz Flávio X. Sá, A Mechanical Analogous Model for the Power System with Automatic Voltage Regulator, International Review on Modelling and Simulations (IREMOS), Vol. 3. n. 1 pp. 57-63, February 2010. M. Rahi, A. Feliachi, H∞ Robust Decentralized Controller for Nonlinear Power Systems, IEEE Transactions on Power Systems; vol.13, n4, pp1401–1406, 1998. J. Machowski, S. Rboak, J.W. Bialek, J.K. Bumby, N. Abi-Samra, Decentralized Stability Enhancing Control of synchronous generator, IEEE Transactions on power systems, Vol 15, n 4, November 2000. D. Cheng, T.J. Tarn, A. Isidori, Global external linearization of nonlinear systems via feedback, IEEE Transaction Automatic Control, AC-30, August 1985. C.I. Byrnes, A. Isidori, New results and examples in nonlinear feedback stabilization, Syst. Control Lett, pp437–442, December 1989. Z. Xi, G. Feng, D. Cheng, Q. Lu, Nonlinear Decentralized Saturated Controller Design for Power Systems, IEEE Transa on Cont sys techno, vol.11, no. 4, july2003. A. Karimi, A. Feliachi, Decentralized adaptive backstepping control of electric power systems, Electric Power Systems Research, vol.78, pp484–493, 2008. D. Boukhetala, F. Boudjema, T. Madani, M.S. Boucherit, N.K. M’Sirdi, “A New Decentralized Variable Structure Control for Robot Manipulators,” International Journal of Robotics and Automation, vol. 8, n.1, pp28–40, 2003. S. Benahdouga, D. Boukhetala, F. Boudjema Decentralized high order sliding mode control of multimachine power systems, Electrical Power and Energy Systems, Vol 43 issue1, pp1081– 1086, 2012. H. Bühler, Réglage par mode de glissement, ( Lausanne, Switzerland, Polytechniques Romandes, 1986). C. Ghorbel, A. Abdelkrim, M. Benrejeb, An Adaptive Stabilization of Nonlinear Systems using Polyquadratic Lyapunov Function, International Review of Automatic Control (IREACO), Vol. 1. n. 4, pp. 407-412, November 2008. E. Dorsaf, D. Tarak, Adaptive Sliding Mode Observer for Interconnected Fractional Nonlinear Systems, International Review of Automatic Control (IREACO), Vol. 4 N. 2, pp. 170179, March 2011.

Authors’ information

0.9999 1.0200 1.0399

Seddik Benahdouga was born in Algiers, Bordj Bou Arreridj, on 1979.



796


He received the Engineer degree in Automatic Engineering and Master degree in Automatic from the National Polytechnic School, Algiers, Algeria in 2003 and 2006, respectively. In 2008, he joined the department of Electromechanical Engineering at the Bordj Bou Arreridj, Algiers, as an Assistant Professor. His research interests include application of sliding mode control, decentralized control, adaptive control, power systems and solar energy. Djamel Boukhetala is a professor in Automatic Control in Departement of Electrical Engineering of the National Polytechnic School of Algiers. From 1996 to 1999. He was the head of the Automatic Control Department and currently, he is the director of the laboratoire de Commande des Processus. His research interests are decentralized control, nonlineare control, fuzzy control and artificial neural networks control applied to robotics, aerospace systems, active suspension and power systems. Farès Boudjema was born in Algiers, Algeria, on March 28, 1962. He received the Engineer degree in Electrical Engineering from the Ecole Nationale Polytechnique, Algiers, Algeria in 1985, the DEA degree, and the Doctorat degree in Automatic Control from the University of Paul Sabatier, Toulouse, France, in 1987, and 1991, respectively. In 1991, he joined the department of Electrical Engineering at the Ecole Nationale Polytechnique, Algiers, as an Assistant Professor. He was promoted to Associate Professor in 1994, and Professor in 2000. He was the head of the control process laboratory from 2000 through 2005. Since March 2010, he has been the head of the automatic control department. His research interests include application of sliding mode control, artificial neural network control, fuzzy control, and decentralized control in the field of the Electrical machines, power systems, robotics and plus house energy. Riadh Khenfer was born in setif in Algeria, on November 07, 1975. He received his B.Sc. andM.Sc.in electrical engineering in 2002 and 2005 from the ferhat abbes University of Technology (UFAS),Setif, Algeria. Now, he is the Ph.D. student in the ferhat abbes University of Technology (UFAS),and he is a teacher researcher in Mohamed Bachir El Ibrahimi University of Technology (UBBA),of Bordj Bou-Arréridj, Algeria. His research interests include diagnosis of systems, renewable energy, control and power electronics and Industrial automation systems. Mounir Meddad is an associate professor of electrical engineering from 2006. He received his engineering degree in electrical engineering from the University of Setif, Ageria in 1992, and his magister degree in electrical engineering from the University of Setif in 1999. His current field of interest focuses on vibration damping, energy harvesting and structural health monitoring using piezoelectric, pyroelectric or electrostrictive devices, as well as autonomous, self-powered wireless systems and power systems.



797


A Novel Control Strategy for Performance Enhancement of Unified Power Quality Conditioner Nikita Hari, K. Vijayakumar, S. S. Dash Abstract – The spiraling growth of power electronics based equipments has produced a significant impact on the quality of electric power supply. Conventional power quality mitigation equipment is proving to be inadequate for an increasing number of applications, and this fact has attracted the attention of power engineers to develop dynamic and novel solutions to Power Quality (PQ) problems. This has led to the development of Custom Power Devices (CPD).One modern and very promising CPD that deals with both load current and supply voltage imperfections is the Unified Power Quality Conditioner (UPQC). This paper investigated the development of UPQC control schemes and algorithms for power quality improvement and implementation of a novel control strategy to enhance the performance of UPQC. The proposed control scheme gives better steady-state and dynamic response. The validity of the proposed control method is verified by means of MATLAB/SIMULINK. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: PQ, CPD, UPQC, Power Quality

I.

The presence of harmonic currents also cause additional losses and voltage waveform distortions leading to poor power quality. Also, the number of sensitive loads that require ideal sinusoidal supply voltages for their proper operation has increased manifold. The increasing use of electronic equipment sensitive to power variations drives the interest in power conditioning technologies. The power quality at the point of common coupling (PCC) with the utility grid is governed by the various standards and the IEEE-519 standard is widely accepted. So, in order to keep the power quality within limits proposed by standards, it is necessary to include versatile compensation devices satisfying both utilities and customers [4]. To solve these problems, a new set of devices called Custom Power Devices(CPDs)emerged.[6] One modern solution that deals with both load current and supply voltage imperfections is the Unified Power Quality Conditioner (UPQC) , which was first presented in 1995 by Hirofumi Akagi. Such a solution can compensate for almost all the power quality problems such as: sags, swells, voltage imbalance, flicker, harmonics and reactive currents [5]. Control strategy plays a major role in enhancing the performance of the compensating device. In [11]-[17] various control schemes for CPDs are reported. All these schemes are specifically designed to mitigate a particular PQ problem or certain common PQ issues. This paper presents a versatile and simple control strategy for UPQC which provides a cluster of functions such as load balancing, voltage regulation, harmonic filtering and voltage flicker reductions.

Introduction

Reliability of supply and quality of power are the two most important attributes of any power delivery system today [1]. Not so long ago, the main concern of consumers of electricity was the continuity of supply. However nowadays, consumers want not only continuity of supply, but the quality of power is also equally important to them. Though the power quality (PQ) problems in distribution power systems are not new, customer awareness of these issues has recently increased to a great extent. Utilities and researchers all over the world have for decades worked on the improvement of power quality. There are few sets of conventional solutions to the power quality problems, which have existed for a long time. However these conventional solutions use passive elements and do not always respond instantly and accurately as the nature of the power system conditions change [2]. This led to the advent of active power filters which proved to be a better power compensator. The increased power capabilities, ease of control, and reduced costs of modern semiconductor devices have made active power filters affordable in a large number of applications [3]. But nowadays equipments made with semiconductor devices appears to be highly sensitive and polluting causing a lot of power quality disturbances. Non-linear devices, such as power electronics converters, increase the overall reactive power demanded by the equivalent load, and inject harmonic currents into the distribution system. This reactive power demand causes a drop in the feeder voltage and thereby increases the losses.


798


Nikita Hari, K. Vijayakumar, S. S. Dash

The output voltages of the series VSI do not have the shape of the desired signals, but contain switching harmonics, which are filtered out by the series low pass filter. The amplitude, phase shift, frequency and harmonic content of injected voltages are controllable [8]. The design of UPQC control circuit includes the selection of the following three main factors: • signal conditioning; • derivation of compensating signals; • generation of gating signals to control converters.

The paper is organized as follows. The structure of the UPQC is presented in Section II. Then, in Section III, the control principles are described in detail. The simulation model is presented in Section IV. Simulation results in this section demonstrate the efficacy and versatility of the proposed design technique. Finally, Section V gives the conclusion.

II.

Structure of UPQC

The UPQC is a custom power device that integrates the series and shunt active filters, connected back-to-back on the dc side and sharing a common DC capacitor, as shown in Fig. 1. It employs two voltage source inverters (VSIs) that are connected to a common DC energy storage capacitor. One of these two VSIs is connected in series with the feeder and the other is connected in parallel to the same feeder [12].

III. Control Strategy The effectiveness of any power filter depends on the design characteristics of the current/voltage controller, the method implemented to generate the required reference signal, the modulation technique used and its ability to follow the generated reference signal to compensate the distorted current or voltage with a minimum error and time delay. A. Control of the series active filter The series component of UPQC is controlled to inject the appropriate voltage between the point of common coupling (PCC) and load, such that the load voltages become balanced, distortion free and have the desired magnitude. Two UPQC terms are defined in depending on the angle of the injected voltage: UPQC-Q and UPQC-P. In the first case (UPQC-Q) the injected voltage is maintained 90 degrees in advance with respect to the supply current, so that the series compensator consumes no active power in steady state. In second case (UPQC-P) the injected voltage is in phase with both the supply voltage and current, so that the series compensator consumes only the active power, which is delivered by the shunt compensator through the dc link. In the case of quadrature voltage injection (UPQC-Q) the series compensator requires additional capacity, while the shunt compensator VA rating is reduced as the active power consumption of the series compensator is minimized and it also compensates for a part of the load reactive power demand [9], [18]. In UPQC-P case the series compensator does not compensate for any part of the reactive power demand of the load, and it has to be entirely compensated by the shunt compensator. Also the shunt compensator must provide the active power injected by the series compensator. Thus, in this case the VA rating of the shunt compensator increases, but that of the series compensator decreases. In the case when the UPQC-P control strategy is applied, the injected voltage is in phase with the supply voltage; hence the load voltage is in phase with the supply voltage and there is no need for calculating the angle of the reference load voltage. Thus, the reference load voltage is determined by multiplying the reference magnitude (which is constant) with the sinusoidal template phase-locked to the supply

Fig. 1. Power circuit diagram of a three-phase UPQC

The shunt active filter is responsible for power factor correction and compensation of load current harmonics and unbalances. Also, it maintains constant average voltage across the DC storage capacitor. The shunt part of the UPQC consists of a VSI (voltage source inverter) connected to the common DC storage capacitor on the dc side and on the ac side it is connected in parallel with the load through the shunt interface inductor and shunt coupling transformer. The shunt interface inductor, together with the shunt filter capacitor are used to filter out the switching frequency harmonics produced by the shunt VSI. The shunt coupling transformer is used for matching the network and VSI voltages [7]. The series active filter compensation objectives are achieved by injecting voltages in series with the supply voltages such that the load voltages are balanced and undistorted, and their magnitudes are maintained at the desired level. This voltage injection is provided by the dc storage capacitor and the series VSI. Based on measured supply and/or load voltages the control scheme generates the appropriate switching signals for the series VSI switches.



799


Voltage waveform is also distorted. Fig. 4 shows the results after UPQC is connected to the system. The waveforms are balanced and THD is greatly reduced. This confirms that UPQC compensates harmonics to a great extent.

voltage. Then, the reference series filter voltage is obtained. Comparing the techniques for calculating the reference voltage of the series compensator, presented above, it can be concluded that the UPQC-P algorithm has the simplest implementation (it involves very little computation). In the UPQC-P case the voltage rating of the series compensator is considerably reduced. Also, the UPQC-Q compensation technique does not work in the case when the load is purely resistive. Therefore, the UPQC-P control strategy has been used in the UPQC simulation model. PI controller has been used for dc link voltage control in the UPQC simulation model [10], [18].

Fig. 3. Steady state source voltage and load current waveforms (without UPQC) THD of source voltage:25.9% ; THD of load current:63%

B. Control of the shunt active filter The control scheme of a shunt active power filter must calculate the current reference waveform for each phase of the inverter, maintain the dc voltage constant, and generate the inverter gating signals. The hysteresis control appears to be the most preferable for shunt active filter applications. Therefore, in the UPQC simulation model (presented in Fig. 2), a hysteresis controller has been used. The hysteresis control method has simpler implementation, enhanced system stability, increased reliability and response speed [11].

IV.

Fig. 4. Steady state source voltage and load current waveforms (with UPQC) THD of source voltage:0.98% ;THD of load current:1.03%

Fig. 5. Steady State DC link voltage (500V)

Simulations and Results

Figs. 6-8 show the simulation results when a three phase fault is introduced, the series active filter (DVR) injects the compensating voltage so that the source voltage is maintained constant. This shows that voltage imperfections are compensated by the series part of UPQC.

A UPQC simulation model (Fig. 2) has been created in MATLAB/Simulink so as to investigate UPQC circuit waveforms, the dynamic and steady-state performance, and voltage and current ratings [18].

Fig. 6. Source voltage when a three phase fault is introduced from 0.3 to 0.4 s. (without UPQC) (THD:27%) Fig. 2. UPQC simulation system

The following typical case studies have been simulated and the results are presented: 1. Short duration three phase fault conditions. 2. Long duration three phase fault conditions. 3. Dynamic load and three phase fault conditions. 4. Harmonic compensation 5. DC link voltage regulation for the above conditions is also verified. Fig. 3 shows the simulation results for the case where the system is in steady state. Due to power electronic load the current waveform is distorted and is unbalanced.

Fig. 7. Compensating voltage injected by series active filter

Fig. 8. Source voltage when a three phase fault is introduced from 0.3 to 0.4 s (with UPQC) (THD: 0.50%)



800


Fig. 15 validates that UPQC can mitigate long duration sags effectively. Fig. 16-18 shows that long duration current faults are mitigated by D-STATCOM. Fig. 19 shows the transient case where both fault and dynamic load is introduced at different time instants. DVR compensates for the sag by injecting more voltage when sag occurs in the system. Thus the source voltage is regulated and maintained constant by the UPQC.

Figs. 9-11 show that when there is current distortions, the shunt part (D-STATCOM) maintains balance and filters out harmonics. The DC link voltage is maintained constant by the shunt compensator.

Fig. 9. Load current when a three phase fault is introduced from 0.3 to 0.4 s (without UPQC)

Fig. 16. Load current when a three phase fault is introduced from 0.3 to 0.7 s (without UPQC) Fig. 10. Compensating current injected by shunt active filter

Fig. 11. Load current when a three phase fault is introduced from 0.3 to 0.4 s (with UPQC)

Fig. 17. Compensating current injected by shunt active filter

Fig. 12. DC link voltage when a three phase fault is introduced from 0.3 to 0.4 s (with UPQC)

Fig. 18. Load current when a three phase fault is introduced from 0.3 to 0.7 s. (with UPQC)

Fig. 13 shows the case when the fault duration is increased to 0.4s. Fig. 14 shows that DVR injects the required voltage during fault.

Fig. 19. Source voltage when a three phase fault is introduced from 0.1 to 0.2 s and an RLC load from 0.25 to 0.35 s (without UPQC) Fig 13. Source voltage when a three phase fault is introduced from 0.3 to 0.7 s (without UPQC)

Fig. 20. Compensating voltage injected by series active filter Fig 14. Compensating voltage injected by series active filter

Fig. 21. Source voltage when a three phase fault is introduced from 0.1 to 0.2 s and an RLC load from 0.25 to 0.35 s (with UPQC) Fig. 15. Source voltage when a three phase fault is introduced from 0.3 to 0.7 s (with UPQC)



801


current tracking for shunt active filter and PWM controlled series active filter. The performance of the UPQC is compared with DVR and DSTATCOM. The objectives laid down have been successfully realized through software implementation in MATLAB/SIMULINK. Simulation results show that, when the UPQC applying such control strategy is used for the compensation of the nonlinear/unbalance load conditions in three-phase three-wire system, the harmonic reduction is better; unbalance/distortion of load current and source voltage are compensated well and dc voltage gets regulated all of which verifies the effectiveness of applying such a flexible control strategy in UPQC.

Fig. 22 shows the case when load current is distorted and unbalanced. D-STATCOM injects current waveforms of opposite polarity and mitigates the swell in current.

Fig. 22. Load current when a three phase fault is introduced from 0.1 to 0.2 s and an RLC load from 0.25 to 0.35 s (without UPQC)

References [1] Fig. 23. Compensating current injected by shunt active filter [2] [3] [4] [5] Fig. 24. Load current when a three phase fault is introduced from 0.1 to 0.2 s and an RLC load from 0.25 to 0.35 s (with UPQC)

[6]

[7]

[8] Fig. 25. DC link voltage when a three phase fault is introduced from 0.1 to 0.2 s and an RLC load from 0.25 to 0.3s (with UPQC)

[9]

DC link voltage dips slightly during fault condition but is maintained constant by UPQC. Simulation results show that UPQC mitigates deeper sags, harmonic compensation is better, does better load regulation and balancing for dynamic loads and can tolerate long duration fault conditions effectively. Thus it gives enhanced performance when compared to DSTATCOM and DVR. Results also show that it gives good steady state and transient performance. The proposed control scheme is feasible and simple to implement although further work is needed to optimize the parameters of the UPQC.

V.

[10]

[11]

[12]

[13]

Conclusion

[14]

The main objective of this work was to develop a novel UPQC control scheme for power quality improvement. This paper investigated the development of UPQC control schemes and algorithms for power quality improvement and implementation of a flexible control strategy to enhance the performance of UPQC. The enhanced steady state and dynamic performance of UPQC is due to this versatile control strategy using average dc voltage regulation, hysteresis controller based

[15]

[16]


C. Sankaran, Power quality(Boca Raton, Fla.: CRC Press LCC, 2002). Math H.J. Bollen, Understanding power quality problems: Voltage sags and Interruptions( New York: IEEE Press, 2000). Hirofumi Akagi, Active harmonic filters, Proceeding of the IEEE, vol. 93, no. 12, Dec 2005 pp. 2128-2141. N.G. Hingorani, Introducing custom power, IEEE Spectrum, vol. 32, no. 6, June 1995, pp. 41-48. Arindam Ghosh and Gerard Ledwich, Power quality enhancement using custom power devices( Boston, Kluwer Academic Publishers, 2002). Hirofumi Akagi, New trends in active filters for power conditioning, IEEE Transactions on Industry Applications, vol. 32, no. 6,Dec 1996,pp. 1312-1322. N.belhaouchet, Rahmani, Begag, Three phas shunt active power filter with high performance operation, International Review of Automatic Control (IREACO), vol 1.n.3 Sept 2008, pp 294-302 Arindam Ghosh, Compensation of Distribution System Voltage Using DVR, IEEE Transaction on Power Delivery, vol. 17, no. 4,Oct 2002, pp 1030 – 1036. Chris Fitzer, Atputharajah Arulampalam, Mike Barnes, and Rainer Zurowski, Mitigation of Saturation in Dynamic Voltage Restorer Connection Transformers, IEEE Transactions on Power Electronics, vol. 17, no. 6, Nov. 2002, pp. 1058 – 1066. A.D Falehi, M.Rostmani, Verification of a novel proposed control system for DVR to correct the voltage sag , International Review of Automatic Control (IREACO), vol.3, no.6, Dec 2010 pg.14611467 S. Sajedi, F. Khalifeh, T. Karimi, Z. Khalifeh, Modeling and Application of a D-STATCOM in Voltage Flicker Mitigation, International Review of Automatic Control (IREACO), vol.4 , no.4, August 2011, pp. 1812-1816. Hideaki Fujita, Hirofumi Akagi, The Unified Power Quality Conditioner: The Integration of Series- and Shunt-Active Filters, IEEE Transactions on Power Electronics, vol. 13, no. 2, March 1998 ,pp. 315-322. S.W. Middlekauff and E.R. Collins, System and Customer Impact: Considerations for Series Custom Power Devices, IEEE Transactions on Power Delivery, vol. 13, no. 1,Jan 1998, pp. 278282. B. Han, B. Bae, S. Baek, and G. Jang, New Configuration of UPQC for Medium Voltage Application, IEEE Trans. on Power Delivery, vol. 21, no. 3, July 2006, pp. 1438 -1444. Shyh-Jier Huang and Jinn-Chang Wu, A control algorithm for three-phase three wired active power filter under non ideal mains voltages, IEEE Transactions on Power Electronics, vol. 14, no. 4, July 1999, pp. 753 – 760. Iurie Axente, Malabika Basu, Michael F. Conlon and Kevin Gaughan, A Study of Shunt Active Filter Generating the DC Biased Current, Proceedings of the 41st International Universities Power Engineering Conference (UPEC) , 6th –8th September 2006 Northumbria University, Newcastle upon Tyne, UK.


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Nikita Hari, K. K Vijayakumar, S. S. Dashh

[17] Arindam Ghosh, Gerarrd Ledwich, Load Compensating OM in Weak AC A Systems, IEE EE Trans. on Power P DSTATCO Delivery, vol. v 18, no. 4, Oct 2003,pp. 1302-1309. [18] M. Moham mmadi, R. Ebrahiimi, M. Najafi, Voltage V Sag Mitiggation with Dynaamic Voltage Reestorer (DVR) in Distribution Sysstems, Internationnal Review of Automatic Conntrol (IREACO), vol. 4,no.2, Appril 2011, pp. 824-828. [18] Iurie Axeente ,Some Invesstigations on Unified U Power Quality Q Conditioneer, PhD Thesis, Dublin D Institute of o Technology, Ireeland, May 2009. B Michael F. F Conlon and Kevin K [19] Iurie Axente, Malabika Basu, D L Laboratory Prottotype Gaughan, A 12-kVA DSP-Controlled UPQC Caapable of Mitigatting Unbalance in Source Voltagge and Load Currrent, IEEE Transs. on Power Delivery, vol. 25, no. n 6, June 2012, pp. 1302-1309.

Authors’’ information Nikita Hari H received herr B.Tech degree from Cochin University U Of Sccience & Technoology, Cochin, India, in 20077 in Electronicss and M Instrumenntation engineerring, and the M.Tech degree from f SRM Insttitute of Technoology, Chennai, India, in 2011, 2 in Elecctrical Engineering. Her researrch interests innclude G Grid Integgration of Renew wable Smart Grids, wer Devices. Energy Sources and Custom Pow K. Vijayyakumar receivved his Bachelor of Engineering degree froom the Facultty of Engineering and Techhnology, Annaamalai Universitty,India and Master of Engineeering from the same universityy both in the arrea of electricall engineering. His areea of researchh interests inccludes Computaational Intelligennce applicationns in wer System Operaation and Controll. Power Systems, FACTS and Pow Dr. S. S. S Dash receivedd his M.E. degrree in Power Syystems from U. C. E, Orissa, India in the year 1996 and receiveed his Ph.D. deggree in Electricall Engineering frrom Anna Univeersity, Chennai, India in the yearr 2006. He holds more than tenn years of ressearch and teaaching experiencce. His reseaarch interests incclude Power Sysstems, FACTS and Pow wer Quality.

Copyright © 20012 Praise Worthyy Prize S.r.l. - Alll rights reserved

Internnational Review of Automatic Con ntrol, Vol. 5, N. 6

803


Induction Machine Speed and Flux Control, Using Vector-Sliding Mode Control, with Rotor Resistance Adaptation M. Moutchou, A. Abbou, H. Mahmoudi

Abstract – This paper presents a vector control of induction machine improved by using a sliding mode control and by using Flux sliding mode observer and a MRAS estimator of rotor resistance, this techniques combination allows to have a good performance in terms of trajectory tracking and robustness towards the parametric uncertainties or parametric variations. The effectiveness and the good performance of this control technique have been proved by simulation results. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Vector Control, Sliding Mode Control, MRAS Observer, Lyapunov Stability, Induction Motor Drive

and generally less expensive than other machines. However, its model is complicated for various reasons. The dynamic behavior of the motor is described by a fifth-order highly coupled and nonlinear dynamical system, the rotor electric variables (fluxes and currents) are practically not measured; and some of its physical parameters may vary significantly while operating the motor (stator and mainly rotor resistance, due to heating, magnetizing induction due to saturation ); These features, together with the broad use of this kind of machines in industry, have motivated a great research effort in the last few years. Different control strategies like Field Oriented Control (FOC) proposed by Blaschke [1], this control technique has led to a radical change in control of the asynchronous machine, thanks to the quality of dynamic performance that it brings. In vector control the torque and flux are decoupled by a suitable decoupling network. Then the flux component and the torque are controlled independently and respectively by stator direct-axis current and stator quadratic-axis current to control the induction motor (IM) as a separately excited DC motor [2], [22]. Achieving decoupling is the main aim of vector control. The ideal decoupling will not be obtained, due to plant uncertainties. Consequently the efficiency and the performance of the motor drive are degraded. Several means were used to reduce effects of parameters variations, such as on-line tuning techniques have been reported [3], [4], [5]. In this work our choice fell on the on-line adaptation of parameters to achieve decoupling, associated with the sliding mode control which is a special case of the theory of variable structure systems. Based primarily on solving differential equations in discontinuous second member, initiated in 1960 by Filippov [6], it will be used since the publication of books Emelyanov [7], of Itkis [8] and Utkin [9].

Nomenclature ˆx , x

x or

Are respectively the estimation and estimation error of the x dx dt

Φr

Time-derivative of the x

Φ rd ,Φ rq

Rotor flux Flux component in the coordinate ( d,q )

isd ,isq

Stator currents in the coordinate ( d,q )

vsd ,vsq

Stator voltages in the coordinate ( d,q )

Φ rα , Φ r β

Flux component in the stationary (α , β )

isα , isβ

axis Stator currents in the stationary (α , β )

vsα ,vsβ

axis Stator voltages in the stationary (α , β )

Rr Rs Lr , Ls Lm Ω p J f TL

axis Rotor resistance Stator resistance Are respectively, rotor and stator cyclic inductance Mutual cyclic inductance Rotor speed Number of pole pairs Rotor inertia Moment Viscous friction coefficient Load torque

I.

Introduction

Induction motors are widely used in industry especially in variable speed applications. An induction motor is simple in operation, rugged, maintenance free


804


M. Moutchou, A. Abbou, H. Mahmoudi

Automaticienne community quickly realized the value of this [10], [11], [12]. The principle of sliding mode control is based on the definition of a surface called sliding surface depending on system states so that it is attractive. The synthesized global control consists of two terms: the first allows the stat trajectory to approach this surface and the second maintaining and sliding along it towards the origin of the phase plane. The global control ensures, in addition to good tracking performance, a fast response time and robust control. However, this control law represents some drawbacks which can be summarized in tree points. The first is that the performance expected by using this control, cause generally a high level of control variable. The second is the need to have accurate information on the evolution of the system in state space and the upper bounds of uncertainties and disturbances. However, the uncertain nature of nonlinear systems makes it difficult if not impossible to have an analytical description of the system dynamics. The third drawback is the use of the sign function in the control law to ensure the passage of the approach phase to the sliding phase. This gives rise to the phenomenon of chatter which consists of spikes and fast control signal, which can excite the high frequency of the process and cause damage. To overcome these problems, several approaches have been presented in the literature [13], [14], [15], [16], [17]. In this research, we chose to use a kind of sign function which ensures a smooth transition between the two extreme values of saturation (positive and negative). These control techniques cannot guarantee good performances without the use of suitable state observer. Among the observation technique used, there are the sliding mode techniques used in this work to estimate flux, and MRAS techniques used for rotor resistance adaptation. MRAS technique is used, in particular, in Sensorless IM drives at the first time by Schauder(1992). MRAS is interesting since it leads to relatively easy to implement system with high speed adaptation [18], [19], [20], [21].

II.

In order to transform the initial representation (treephase) in the coordinate system (d,q), we have chosen to use the transformation of Concordia which conserves power value, followed by a transformation of Park. This transformation allows as to have the necessary (d,q) coordinate rotation, in order to orientate the rotor flux on the direct axis (d), so we get Φ r = Φ rd and Φ rq = 0 . Then the model obtained is as follows: vsd k ⎧ disd ⎪ dt = −γ ⋅ isd + ωs ⋅ isq + T ⋅ Φ r + σ L r s ⎪ ⎪ disq vsq = −ωs ⋅ isd − γ ⋅ isq − k ⋅ p ⋅ Ω ⋅ Φ r + ⎪ σ Ls ⎪ dt ⎪ ⎨Tr ⋅ Φ r + Φ r = Lm ⋅ isd ⎪ i ⎪ω = pΩ + Lm ⋅ sq ⎪ s Tr Φ r ⎪ L T Ω d f ⎪ = p m Φ r ⋅ isq − ⋅ Ω − L ⎪ dt J ⋅ Lr J J ⎩

(

(1)

)

where: Lm 2 L L ; k = m ; Tr = r ; Lr ⋅ Ls σLs Lr Rr

σ = 1−

γ=

1 σLs

⎛ R ⋅L 2 ⎜⎜ Rs + r 2m Lr ⎝

⎞ ⎟⎟ ⎠

In order to have a decoupled control of flux and torque, we conducted a decoupling using compensation. The equivalent model obtained, consists of two decoupled sub-systems which operate in parallel, respectively representing the Flux control through the axis "d", and torque control through the axis "q": ⎧ di u sd = −γ ⋅ isd + sd ⎪ σ Ls ⎪ dt ⎪ di usq sq ⎪ = −γ ⋅ isq + ⎪ dt σ Ls ⎪⎪ + Φ = L ⋅i ⎨ Tr ⋅ Φ r r m sd ⎪ i ⎪ ω = pΩ + Lm ⋅ sq s ⎪ Tr Φ r ⎪ Lm TL ⎪ dΩ f ⎪ dt = p J ⋅ L Φ r ⋅ isq − J ⋅ Ω − J ⎪⎩ r

Model Description of the Induction Machine

The mathematical model of the induction machine is derived from assumptions that the electromagnetic structure satisfies the following conditions: • the gap is assumed to be constant, • electromagnetic induction space distribution is sinusoidal, • the magnetic materials of the stator and the rotor have a linear characteristic of magnetization B=f(H) (no saturation in the magnetic circuit) We have chosen in this work to represents the model of the induction machine in a rotating coordinate (d,q), that we'll oriented in the direction of the rotor flux so as to cancel the Flux quadratic component.

(

(2)

)

III. Design of Induction Machine Sliding Mode Control (SMC) In this section we will design the flux and speed sliding mode control, which will provide the references

(

of stator current isd ,isq


)ref .


805


ˆ − ( a + ∆a ) ⋅ i + ( b + ∆ b ) ⋅ Φ e1 = Φ ref sd r

Using PI controller, these references will be used to obtain the control voltages that supply the induction machine, as showing in the general structure represented Fig. 1. We take two step approaches to design the SMC: 1. Define the sliding mode. This is a surface that is invariant of the controlled dynamics, where the controlled dynamics are exponentially stable, and where the system tracks the desired set-point. 2. Define the control that drives the state to the sliding mode in finite time.

Ω ref

isd ref

e1

SMC

e2 SMC

i sq

ref

u sd

z1 z 2 P.I

u sq

Ω

This later equation can be expressed as below: e1 ( t ) = −b ⋅ e1 ( t ) + Γ1 ( t ) + ∆1 ( t )

with:

ˆ Φ r

The sliding surface S1 that we considered is as below: MRAS Rr Adaptation

t

S1 ( t ) = e1 ( t ) − ( k1 − b ) ⋅ ∫ e1 (τ ) ⋅ dτ

when the sliding mode is reached: S1 ( t ) = S1 ( t ) = 0

Fig. 1. Control structure

III.1. Flux Sliding Mode Control

and we obtain:

We start designing Flux sliding mode control, so we use the following equation, given by model equations (2): = Lm ⋅ i − 1 Φ Φ (3) r sd r Tr Tr

= a ⋅i − b⋅Φ Φ r sd r

(4)

V ( S ) ≤ −λ ⋅ S ⇒ V ( S ) = 0 ⇒ S = 0

with: b=

The control, that we will design, will consist of two parts. The first part Γ equ ( t ) will bring the system to the

1 Tr

sliding surface, while the second part Γ n ( t ) maintains

In order to design a control that deal with parameters variations and parameters uncertainties, we considered the parameters uncertainties ∆a and ∆b corresponding respectively to parameters a and b , by expressing Eq. (4) as follows: ˆ = ( a + ∆a ) ⋅ i − ( b + ∆b ) ⋅ Φ ˆ Φ r sd r

e1 = ( k1 − b ) ⋅ e1

If we choose k1 < b the flux well converge asymptotically to its reference. Now, we determine the necessary control to force the system to that surface, then the behaviour of the system slides to the desired equilibrium point, by maintaining the existence of sliding condition:

Equation (3) can be rewritten as below:

Lm ; Tr

(8)

0

(v sd , v sq )

a=

(7)

reach zero, and ∆1 ( t ) represent the uncertainties.

v sq

(i sd , i sq )

Flux SM

Observer

(6)

where Γ1 ( t ) represent the control that will bring e1 ( t ) to

MAS + Inverter

(i sd , i sq ) Rˆ r

− a ⋅i + b⋅Φ Γ1 ( t ) = Φ ref sd ref

ˆ ∆1 ( t ) = −∆a ⋅ isd + ∆b ⋅ Φ r

v sd

Decoupling

Φ ref

ˆ − ai − ∆a ⋅ i + b ⋅ Φ − b ⋅ e + ∆b ⋅ Φ e1 = Φ ref sd sd ref 1 r

the system trajectory on this surface, and it’s used to eliminate parameters uncertainties and parameters variations. So the control form is as below: Γ1 ( t ) = Γ equ ( t ) + Γ n ( t )

(5)

The control function will satisfy reaching sliding surface and maintaining sliding conditions is as follow:

We define the flux error as:

Γ1 ( t ) = k1 ⋅ e1 ( t ) − d1 ⋅ sign ( S1 )

ˆ e1 = Φ ref − Φ r

(9)

where sign ( ⋅) is the proposed saturation function. It can

The dynamic of flux error is as below:

be defined as below:



806


⎧−1 if x < 0 ⎪ sign ( x ) = ⎨ 0 if x = 0 ⎪ 1 if x > 0 ⎩

We can write equation above as below: e2 = −c ⋅ e2 + Γ 2 ( t ) + ∆ 2 ( t )

with:

But the sliding mode method using this sign function produces chattering effect caused by its discontinuous differential equation. In this research we choose to use tan inverse function that can delete or smooth chattering effect:

+ c ⋅Ω −α ⋅i + β Γ2 (t ) = Ω ref ref sq ˆ − ∆α ⋅ i + ∆β ∆ 2 ( t ) = ∆c ⋅ Ω sq

(12)

(13)

To control the induction machine speed, we define the corresponding sliding surface as below:

sign ( x ) = arctg ( x )

t

S2 ( t ) = e2 ( t ) − ( k2 − c ) ⋅ ∫ e2 (τ ) ⋅ dτ

From (6) and (7) we define the following direct current reference equation: isd _ ref =

+ b ⋅ Φ − k ⋅ e (t ) + ⎞ 1 ⎛Φ ref ref 1 1 ⎜ ⎟ ⎟ a ⎜⎝ + d1 ⋅ sign ( S1 ) ⎠

0

when the sliding mode will be reached, the switching surface will verify:

(10)

S2 ( t ) = S2 ( t ) = 0

III.2. Speed Sliding Mode Control Then we obtain:

We consider the following equation, representing mechanical dynamic, taken from model Eq. (2): L T dΩ f = p m ⋅ Φ r ⋅ isq − ⋅ Ω − L dt J ⋅ Lr J J

The speed error well converge asymptotically to zero, If we choose k2 < c . The sliding mode control of speed is designed as below, in order to drive the speed error to the sliding surface and keep the system motion on this surface:

That we can express as below: dΩ = α .i − β − c.Ω sq dt

Γ 2 ( t ) = k2 ⋅ e2 ( t ) − d 2 ⋅ sign ( S2 )

T β= L; J

f c= J

isq _ ref =

Now we consider the parameters uncertainties of α , β and c : ˆ dΩ ˆ = (α + ∆α ) ⋅ isq − ( β + ∆β ) − ( c + ∆c ) ⋅ Ω dt

(14)

From (12) and (13) the quadratic current reference equation is the following:

with: L ⋅Φ α=p m r; J ⋅ Lr

e2 = ( k2 − c ) ⋅ e2

+ c ⋅ Ω + β − k ⋅ e (t ) + ⎞ 1 ⎛Ω ref ref 2 2 ⎜ ⎟ ⎟ α ⎜⎝ + d 2 ⋅ sign ( S2 ) ⎠

(15)

III.3. Stability Analysis

(11)

Let us check the tracking error stability of speed and flux sliding mode control, by choosing the Candidate Lyapunov Function (CLF) below:

We define the speed tracking error as below: ˆ e2 = Ω ref − Ω

V=

1 2 1 ⋅ S1 + ⋅ S2 2 2 2

(16)

The corresponding dynamic is as follow:

The derivative of (6) gives:

ˆ − (α + ∆α ) ⋅ i + ( β + ∆β ) + ( c + ∆c ) ⋅ Ω e2 = Ω ref sq

V = S1 ⋅ S1 + S2 ⋅ S2 ⇒ V = S1 ( e1 − ( k1 − b ) ⋅ e1 ) + S2 ( e2 − ( k2 − c ) ⋅ e2 )

− α ⋅ i − ∆α ⋅ i + β + ∆β + e2 = Ω ref sq sq

ˆ + c ⋅ Ω ref − c ⋅ e2 + ∆c ⋅ Ω

By using (6) and (12) we can write V as below:



807


V = S1 ( −b ⋅ e1 + Γ1 ( t ) + ∆1 ( t ) − ( k1 − b ) ⋅ e1 ) +

with:

+ S2 ( −c ⋅ e2 + Γ 2 ( t ) + ∆ 2 ( t ) − ( k2 − c ) ⋅ e2 ) ⇒

T

T

I s = ⎡⎣isα isβ ⎤⎦ ; Φ r = ⎡⎣Φ rα Φ r β ⎤⎦ ; Vs = ⎡⎣ vsα vsβ ⎤⎦

V = S1 ( − k1 ⋅ e1 + Γ1 ( t ) + ∆1 ( t ) ) +

⎡ λ ω = p⋅Ω ; A = ⎢ r ⎣ −ω

+ S2 ( −k2 ⋅ e2 + Γ 2 ( t ) + ∆ 2 ( t ) )

T

ω⎤ 1 1 ; λr = ; β = ⎥ λr ⎦ Tr σ ⋅ Ls

The choosing control (9) and (14) above, gives:

IV.2. Rotor Flux S.M Observer Model

V = S1 ( −d1 ⋅ sign ( S1 ) + ∆1 ( t ) ) +

The equations representing the observer model are given below:

+ S2 ( −d 2 ⋅ sign ( S2 ) + ∆ 2 ( t ) ) ⇒ V = − d1 ⋅ S1 + S1 ⋅ ∆1 ( t ) − d 2 ⋅ S2 + S2 ⋅ ∆ 2 ( t )

(

)

(

⎧ Iˆ = −γ ⋅ Iˆ + k ⋅ A ⋅ Φ ˆ + β ⋅V + D ⋅ u ⎪ s s r s i s ⎨ ˆ = L ⋅ λ ⋅ Iˆ − A ⋅ Φ ˆ + D ⋅u ⎪⎩Φ ϕ r m r s r s

)

⇒ V ≤ − d1 − ∆1 ( t ) ⋅ S1 − d 2 − ∆ 2 ( t ) ⋅ S2

If we consider that ∆1max and ∆ 2 max are the upper bounds of uncertainty ∆ ( t ) and ∆ ( t ) , V can be 1

(18)

The dynamics of the estimation error is expressed by the following equations:

2

forced to be negative definite, by choosing:

⎧ I = −γ ⋅ I + k ⋅ A ⋅ Φ − D ⋅u ⎪ s s r i s ⎨ = L ⋅ λ ⋅ I − A ⋅ Φ − D ⋅u ⎪⎩Φ r m r s r s ϕ

d1 > ∆1max and d 2 > ∆ 2 max

III.4. Stator Current Control of I.M

(19)

where: Is = I s − Iˆ s : is the stator current estimation error. ˆ : is the rotor flux estimation error. = Φ −Φ Φ

The decoupling used to linearize the electrical part of the system (stator voltage - stator current), allows us to represent this part by the following transfer function:

r

r

r

us = ⎡⎣ sign ( S1 ) sign ( S2 ) ⎤⎦ T S = [ s s ] = η ⋅ I : is the sliding mode surface. T

1 isd ( s ) = ⋅u (s) σ ⋅ Ls ( s + γ ) sd

1

2

( s)

Dϕ , Di , η : are the matrix (2x2) that we will determine 1 isq ( s ) = ⋅u (s) σ ⋅ Ls ( s + γ ) sq

later.

IV.3. Design of the Rotor Flux SM Observer In order to ensure that the current tracks there reference given by sliding mode controller designing above. We will use two identical PI controllers, one for each, who’s the transfer function is the following:

R (s) = kp +

IV.

We consider the following Lyapunov candidate function: 1 V = ⋅ ST ⋅ S 2 Its time derivative is the following:

ki s

V = S T ⋅ S

Rotor Flux Sliding Mode Observer

We suppose that

IV.1. Induction Machine Model (α , β )

V = S T ⋅η ⋅ Is

In this section we use a model of asynchronous machine formulated in a stationary stator reference frame (α , β ) , as showing below: ⎪⎧ Is = −γ ⋅ I s + k ⋅ A ⋅ Φ r + β ⋅ Vs ⎨ ⎪⎩Φ r = Lm ⋅ λr ⋅ I s − A ⋅ Φ r

dη = 0 , thus we obtain: dt

(

)

− S T ⋅η ⋅ D ⋅ u V = S T ⋅η −γ ⋅ Is + k ⋅ A ⋅ Φ r i s

In order to have V negative definite and satisfy the condition of attractiveness, we must have:

(17)



808


(

S T ⋅η −γ ⋅ Is + k ⋅ A ⋅ Φ r

) < S T ⋅η ⋅ Di ⋅ us

To complete the design of the observer, it now suffices to choose properly the observer parameters. The ( p1 , p2 ) parameters must be chosen to determine the

If we put: ⎡µ 0 ⎤ η ⋅ Di = ⎢ 1 ⎥ ⎣ 0 µ2 ⎦

dynamics of observer convergence and the

( µ1 , µ2 )

parameters to satisfy the condition of attractiveness and stability of the observer.

Then we obtain the condition below: µ1 ⋅ S1 + µ2 ⋅ S2 > S T ⋅η ( −γ ⋅ Is + k ⋅ A ⋅ Φ r

V.

)

By considering that the load torque is constant, we can reconstitute the electromagnetic torque by using the estimated rotor flux, as showing by the following expression:

When the sliding mode will be reached, the switching surface will verify: I = I = 0 s

s

(

L ˆ ˆ Tˆe = p m Φ rα ⋅ is β − Φ r β ⋅ isα Lr

Therefore we obtain: us = Di −1 ⋅ k ⋅ A ⋅ Φ r

VI.

By putting this equation in (18) we get this equation:

(

)

We put: = −Ρ ⋅ Φ A + Dϕ ⋅ Di −1 ⋅ k ⋅ A = Ρ ⇒ Φ r r

In order to have an exponential convergence we choose P under the following form: ⎡ p1 0 ⎤ Ρ=⎢ ⎥ ⎣0 p2 ⎦

Rotor Resistance MRAS Observer

and by using measurements of the stator currents and voltages, we build two estimators of rotor flux, as showing below.

where p1 and p2 are a positive constants. Then we obtain de following equation: Dϕ = ( Ρ − A ) ⋅ A ⋅ k

)

The rotor resistance is estimated by using the model reference adaptive system approach (MRAS). This method consists of using two models. The first one is the reference model and the second is an adjustable one in which two components of the rotor flux, are estimated by using the measurement of the currents and stator voltages. The output of the reference model is compared with an adjustable observer. The error is fed into an adaptation mechanism which is designed to ensure the stability of the MRAS. By using the dynamic model of the asynchronous machine, formulated in a stator reference frame (α , β ) ,

= − A + D ⋅ D −1 ⋅ k ⋅ A ⋅ Φ Φ r ϕ i r

−1

Electromagnetic Torque Estimation

VI.1. Reference Model −1

⋅ Di

The reference model is chosen to be independent of estimated parameter (rotor resistance). It is represented by equations system below:

Now if we put:

η = A−1 ⋅ k −1

⎧ d Φ rα Lr ⎡ di ⎤ vsα − Rs ⋅ isα − σ Ls ⋅ sα ⎥ = ⎪ ⎢ Lm ⎣ dt ⎦ ⎪ dt ⎪ ⎨ ⎪ ⎪ d Φ r β = Lr ⎡ v − R ⋅ i − σ L ⋅ disβ ⎤ ⎢ sβ s sβ s dt ⎥ ⎪ dt Lm ⎣ ⎦ ⎩

So we finally find the following equations: ⎡ µ1 0 ⎤ Di = k ⋅ A ⎢ ⎥ ⎣0 µ2 ⎦ ⎡ µ1 0 ⎤ Dϕ = ( Ρ − A ) ⎢ ⎥ ⎣0 µ2 ⎦

(20)

VI.2. Adjustable Model

Consider the two last equation of induction machine model:

Finally the condition of attractiveness becomes: µ1 ⋅ S1 + µ2 ⋅ S2 > S T ⋅ Φ r



809


⎧ d Φ rα ⎪⎪ dt = λr ( Lm ⋅ isα − Φ rα ) − ω ⋅ Φ r β ⎨ ⎪ d Φrβ = λ L ⋅ i − Φ r m sβ r β + ω ⋅ Φ rα ⎪⎩ dt

(

( (

(21)

)

In order to determine the observer stability condition, and then determine the adaptation mechanism that gives us the rotor resistance estimation, let us consider the following Lyapunov Candidate Function (LCF):

We can see that Eq. (20) doesn't involve the parameter Rr, this estimator is considered as reference model, and Eq. (21), which contain Rr, is regarded as an adjustable model. The following figure represents the observation technique structure. [Vs]

Reference Model

[Is]

V=

[Φ r ] +

[Φ~ r ]

+ Φ + 1 ⋅ λ ⋅ λ ⋅Φ ⋅Φ V = Φ rα rα rβ rα r r Γ

[Φˆ r ]

(

)

(

)

ˆ ⎡ Lm ⋅ isα − Φ ⎤ rα ⋅ Φ rα + ⎢ ⎥ + λr ⎢ ⎥ λ r ⎥ ˆ ⎢ + Lm ⋅ isβ − Φ rβ ⋅ Φrβ + ⎣ Γ⎦

(

Fig. 2. Rotor Resistance MRAS Observer structure

We define the error flux between the states of two models as below:

)

In order to make V to be negative definite, we can for example force the second term to be null, then we can write:

ˆ = Φ −Φ ⎧⎪Φ rα rα rα ⎨ ˆ Φ = Φ − Φ ⎪⎩ r β rβ rβ

ˆ ˆ + L ⋅i − Φ + λr = 0 Lm .isα − Φ Φ ⋅ Φ rα rα m sβ rβ rβ Γ 2 2 +Φ ⇒ V = −λ Φ 0 , Fi ≥ Ei + Gi and Ei ≤ Gi

h ⎛ ε ⎞⎛ ⎞ h⎛ δ ⎞ Gi = 1 − ⎜1 − ⎟ ⎜ pi +1 − pi′+1 ⎟ − ⎜ 1 − ⎟ qi +1 + 2 ⎝ h ⎠⎝ ⎠ 2⎝ h ⎠ (18) h h⎛ η ⎞ − ri +1 − ⎜ 1 + ⎟ si +1 6 2⎝ h⎠ Hi =

⎛ ⎞ Gi Wi = ⎜ ⎟ ⎝ Fi − EiWi −1 ⎠

To solve these recurrence relations for i = 1, 2, 3,…, N-1, we need the initial conditions for W0 and T0 . If we choose W0 = 0 , then we get T0 = φ0 . With these initial values, we compute Wi and Ti for i = 1, 2, 3,…., N-1 from Eqs. (24) and (25) in forward process, and then obtain yi in the backward process from (21). The conditions for the Discrete Invariant Imbedding Algorithm to be stable are, see ([12], [11] and [13]):

where: Ei =

(23)

By comparing (23) and (21), we get the recurrence relations for Wi and Ti :

y ( xi +1 + η ) = y ( xi +1 ) + η y ′ ( xi +1 ) =

η ⎛ η⎞ = ⎜1 + ⎟ yi +1 − yi h ⎝ h⎠

(22)

(26)

In our method, one can easily show that if the and assumptions α ( x) + β ( x) + ω ( x) < 0

β ( x )η − α ( x ) δ > 0 hold, then the above conditions (26) hold and thus the discrete invariant imbedding algorithm is stable.

(19) II.3.

This gives us the tridiagonal system which can be solved easily by Thomas Algorithm described in the next section.

Numerical Examples with Left End Boundary Layer

To demonstrate the applicability of the method we have applied it to three boundary value problems of the



870

Gemechis File, Y. N. Reddy

II.4.

type given by equations (1)-(3) with left-end boundary layer. The exact solution of such boundary value problems having constant coefficients (i.e. a ( x ) = a,

α ( x) = α ,

β ( x) = β ,

ω ( x) = ω,

We now consider (1)–(3) and assume that a ( x ) + β ( x )η − α ( x ) δ ≤ M < 0 throughout the interval

f ( x) = f ,

[0,1] where M is constant. This assumption merely implies that the boundary layer will be in the neighborhood of x = 1. By using Taylor series expansion in the neighborhood of the point x, we have:

φ ( x ) = φ and γ ( x ) = γ are constants) is given by: y ( x ) = c1 exp ( m1 x ) + c2 exp ( m2 x ) + where: c1 =

c2 =

m1 =

m2 =

f c

(27)

− f + γ c + exp ( m2 )( f − φ c )

( exp ( m1 ) − exp ( m2 ) ) c

− ( a − αδ + βη ) +

+ r ( x) y ( x) + s ( x) y ( x +η ) + t ( x)

2ε

(30)

(28b) where:

( a − αδ + βη )2 − 4ε c

( a − αδ + βη )2 − 4ε c

(29)

and consequently (1) is replaced by the following first order differential difference equation of the form: y′ ( x ) = p ( x ) y′ ( x + ε ) + q ( x ) y ( x − δ ) +

p ( x) =

(28c)

2ε

− ( a − αδ + βη ) −

y ( x + ε ) ≈ y ( x ) + ε y′ ( x )

(28a)

( exp ( m1 ) − exp ( m2 ) ) c

f − γ c + exp ( m1 )( − f + φ c )

Right End Boundary Layer Problems

s ( x) =

α ( x) ω ( x) 1 , q ( x) = , r ( x) = , 1− a ( x) 1− a ( x) 1− a ( x) β ( x)

1− a ( x)

, t ( x) =

− f ( x)

(31)

1− a ( x)

(28d)

c = α + β +ω

Now we divide the interval [0, 1] into N equal parts h. Let with constant mesh length 0 = x0 , x1 , x2 , ..., xN = 1 be the mesh points. Then we have xi = ih, i = 0 , 1, 2 , ..., N . Integrating Eqn. (30) with respect to x from xi −1 to xi , we get:

(28e)

Example 1: Consider the model boundary value problem given by Eqs. (1)-(3) with:

yi − yi −1 = pi y ( xi + ε ) − pi −1 y ( xi −1 + ε ) +

a ( x ) = 1, α ( x ) = 2 , β ( x ) = 0 , ω ( x ) = −3, f ( x ) = 0 ,

φ ( x ) = 1, γ ( x ) = 1.

xi

−

∫ p′ ( x ) y ( x + ε ) dx +

(32)

xi −1

The exact solution of the problem is given by (27)(28). The numerical results are given in Tables I, II for ε=0.001 and 0.0001 respectively. Example 2: Consider the model boundary value problem given by equations (1)-(3) with:

⎡q ( x ) y ( x − δ ) + r ( x ) y ( x ) + ⎤ ⎥ dx ⎥⎦ xi −1 xi

+

∫ ⎢⎢⎣+ s ( x ) y ( x + η ) + t ( x )

where:

a ( x ) = 1, α ( x ) = 0, β ( x ) = 2, ω ( x ) = −3,

y ( xi ) = yi , p ( xi ) = pi , q ( xi ) = qi , r ( xi ) = ri ,

f ( x ) = 0, φ ( x ) = 1, γ ( x ) = 1

s ( xi ) = si , t ( xi ) = ti , f ( xi ) = fi

The exact solution of the problem is given by (27)(28). The numerical results are given in Tables III, IV for ε=0.001 and 0.0001 respectively. Example 3: Consider the model boundary value problem given by Eqs. (1)-(3) with:

By using Trapezoidal rule to evaluate the integral in (32), we get the following Eq. (33): h ⎞ ⎛ yi − yi −1 = ⎜ pi − pi′ ⎟ y ( xi + ε ) + 2 ⎠ ⎝

a ( x ) = 1, α ( x ) = −2, β ( x ) = 1, ω ( x ) = −5, f ( x ) = 0,

φ ( x ) = 1, γ ( x ) = 1

h h ⎡ qi −1 y ( xi −1 − δ ) + ⎤ ⎛ ⎞ − ⎜ pi −1 + pi′−1 ⎟ y ( xi −1 + ε ) + ⎢ ⎥+ 2 2 ⎢⎣ + qi y ( xi − δ ) ⎝ ⎠ ⎥⎦

The exact solution of the problem is given by (27)(28). The numerical results are given in tables 5, 6 for ε= 0.001 and 0.0001 respectively.

+


h h ⎡s y ( x +η ) +⎤ h [ ri −1 yi −1 + ri yi ] + ⎢ i −1 i −1 ⎥ + [ti −1 + ti ] 2 2 ⎢⎣ + si y ( xi + η ) ⎥⎦ 2 International Review of Automatic Control, Vol. 5, N. 6

871


II.5.

Again, by means of Taylor series expansion and then by approximating y ′ ( x ) by linear interpolation, we get:

To demonstrate the applicability of the method we have applied it to three boundary value problems of the type given by equations (1)-(3) with right-end boundary layer. The exact solution of such boundary value problems having constant coefficients (i.e. a ( x ) = a,

δ ⎛ δ⎞ y ( xi − δ ) = y ( xi ) − δ y ′ ( xi ) = ⎜ 1 + ⎟ yi − yi +1 (34) h ⎝ h⎠ y ( xi −1 − δ ) = y ( xi −1 ) − δ y ′ ( xi −1 ) =

δ ⎛ δ⎞ = ⎜ 1 + ⎟ yi −1 − yi h ⎝ h⎠

α ( x) = α , β ( x) = β , ω ( x) = ω, f ( x) = f , φ ( x) = φ

(35)

and γ ( x ) = γ are constants) is given by (27)-(28).

Example 4: Consider the model boundary value problem given by Eqs. (1)-(3) with:

ε ⎛ ε⎞ y ( xi + ε ) = y ( xi ) + ε y ′ ( xi ) = ⎜1 − ⎟ yi + yi +1 (36) h ⎝ h⎠ ε⎞

a ( x ) = −1, α ( x ) = −2, β ( x ) = 0, ω ( x ) = 1,

ε

⎛ y ( xi −1 + ε ) = y ( xi −1 ) + ε y ′ ( xi −1 ) = ⎜ 1 − ⎟ yi −1 + yi (37) h ⎝ h⎠

f ( x ) = 0, φ ( x ) = 1, γ ( x ) = −1 The exact solution of the problem is given by (27)(28). The numerical results are given in Tables VII, VIII for ε=0.01 and 0.005 respectively. Example 5: Consider the model boundary value problem given by Eqs. (1)-(3) with:

η ⎛ η⎞ y ( xi + η ) = y ( xi ) + η y ′ ( xi ) = ⎜ 1 − ⎟ yi + yi +1 (38) h ⎝ h⎠ y ( xi −1 + η ) = y ( xi −1 ) + η y ′ ( xi −1 ) =

η ⎛ η⎞ = ⎜1 − ⎟ yi −1 + yi h ⎝ h⎠

(39)

a ( x ) = −1, α ( x ) = 0, β ( x ) = −2, ω ( x ) = 1, f ( x ) = 0, φ ( x ) = 1, γ ( x ) = −1

By making Eqs. (34)-(39) in (33) and rearranging we get the three term recurrence relation:

The exact solution of the problem is given by (27)(28). The numerical results are given in Tables IX, X for ε=0.01 and 0.005 respectively. Example 6: Consider the model boundary value problem given by Eqs. (1)-(3) with:

Ei yi −1 − Fi yi + Gi yi +1 = H i ; i = 1, 2 , ..., N − 1 (40)

where:

a ( x ) = −1, α ( x ) = −2, β ( x ) = −2, ω ( x ) = 1,

h ⎛ ε ⎞⎛ ⎞ h⎛ δ ⎞ Ei = −1 + ⎜ 1 − ⎟ ⎜ pi −1 + pi′−1 ⎟ − ⎜ 1 + ⎟ qi −1 + 2 h ⎝ ⎠⎝ ⎠ 2⎝ h ⎠ (41) h h⎛ η ⎞ − ri −1 − ⎜ 1 − ⎟ si −1 6 2⎝ h⎠

f ( x ) = 0, φ ( x ) = 1, γ ( x ) = −1 The exact solution of the problem is given by (27)(28). The numerical results are given in Tables XI, XII for ε=0.01 and 0.005 respectively.

h ⎞ ε⎛ h ⎛ ε ⎞⎛ ⎞ Fi = −1 + ⎜ 1 − ⎟ ⎜ pi − pi′ ⎟ − ⎜ pi −1 + pi′−1 ⎟ + 2 ⎠ h⎝ 2 ⎝ h ⎠⎝ ⎠ h⎛ δ ⎞ h δ η − qi −1 + ⎜ 1 + ⎟ qi + ri + si −1 + (42) 2 2⎝ h⎠ 2 2

III. Discussion and Conclusions We have presented an integration method to solve singularly perturbed differential difference equations with the delay and advance parameters. To demonstrate the efficiency of the method, we considered three examples with left end boundary layer and three with right end boundary layer for different values of δ, η and ε. The approximate solution is compared with the exact solution. From the computational results, it is observed that the proposed method approximates the exact solution very well (see Tables I-XII). Thus, we have devised an alternative technique of solving boundary value problems for singularly perturbed differential difference equations, which is easily implemented on computer and is also practical.

h⎛ η ⎞ + ⎜ 1 − ⎟ si 2⎝ h⎠

ε⎛ h ⎞ δ η Gi = − ⎜ pi − pi′ ⎟ + qi − si h⎝ 2 ⎠ 2 2 Hi =

Numerical Examples with Right End Boundary Layer

h ( ti −1 + ti ) 2

(43)

(44)

We solve the above tridiagonal system by using the method of Discrete Invariant Imbedding Algorithm described in section II.2. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


872


TABLE I NUMERICAL RESULTS OF EXAMPLE 1 FOR ε=0.001, N=100 x 0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90 1.00

δ=0.0009, η=0.00 Numerical Solution 1.0000000 0.3802869 0.3826408 0.3903325 0.3982238 0.4490395 0.5485464 0.6701040 0.8185989 0.9047645 1.0000000

δ=0.0005=η

Exact Solution 1.0000000 0.3750170 0.3825990 0.3903342 0.3982259 0.4490415 0.5485483 0.6701056 0.8185998 0.9047651 1.0000000

Numerical Solution 1.0000000 0.3805332 0.3829340 0.3906261 0.3985170 0.4493270 0.5488098 0.6703185 0.8187299 0.9048370 1.0000000

δ=0.00, η= 0.0005 Exact Solution 1.0000000 0.3753111 0.3828929 0.3906278 0.3985190 0.4493290 0.5488116 0.6703200 0.8187307 0.9048374 1.0000000


Exact Solution 1.0000000 0.3756783 0.3832599 0.3909945 0.3988851 0.4496878 0.5491403 0.6705877 0.8188941 0.9049277 1.0000000

TABLE II NUMERICAL RESULTS OF EXAMPLE 1 FOR ε=0.0001, N=100 x 0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90 1.00


δ=0.00005=η

Exact Solution 1.0000000 0.3752817 0.3828635 0.3905984 0.3984897 0.4493002 0.5487853 0.6702986 0.8187176 0.9048302 1.0000000


δ=0.00, η= 0.00005

Exact Solution 1.0000000 0.3753111 0.3828929 0.3906278 0.3985190 0.4493290 0.5488116 0.6703200 0.8187307 0.9048374 1.0000000


Exact Solution 1.0000000 0.3753479 0.3829297 0.3906645 0.3985557 0.4493649 0.5488446 0.6703469 0.8187471 0.9048465 1.0000000

TABLE III NUMERICAL RESULTS OF EXAMPLE 2 FOR ε=0.001, N=100 x 0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90 1.00

δ=0.0005,η=0.00 Numerical Solution 1.0000000 0.3808371 0.3832961 0.3909888 0.3988791 0.4496820 0.5491350 0.6705834 0.8188916 0.9049262 1.0000000

δ=0.0005=η

Exact Solution 1.0000000 0.3756783 0.3832599 0.3909945 0.3988851 0.4496878 0.5491403 0.6705877 0.8188941 0.9049277 1.0000000


δ=0.00, η= 0.0009 Exact Solution 1.0000000 0.3760452 0.3836265 0.3913607 0.3992508 0.4500463 0.5494686 0.6708549 0.8190573 0.9050178 1.0000000


Exact Solution 1.0000000 0.3763385 0.3839196 0.3916534 0.3995431 0.4503328 0.5497310 0.6710684 0.8191876 0.9050899 1.0000000

TABLE IV NUMERICAL RESULTS OF EXAMPLE 2 FOR ε=0.0001, N=100 x 0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90 1.00


Exact Solution 1.0000000 0.3753479 0.3829297 0.3906645 0.3985557 0.4493649 0.5488446 0.6703469 0.8187471 0.9048465 1.0000000

δ=0.00005=η Numerical Solution 1.0000000 0.3754407 0.3829617 0.3906966 0.3985877 0.4493963 0.5488734 0.6703703 0.8187615 0.9048544 1.0000000


δ=0.00, η= 0.00009

Exact Solution 1.0000000 0.3753847 0.3829664 0.3907013 0.3985924 0.4494008 0.5488775 0.6703736 0.8187635 0.9048555 1.0000000


Exact Solution 1.0000000 0.3754140 0.3829958 0.3907306 0.3986217 0.4494296 0.5489038 0.6703951 0.8187765 0.9048627 1.0000000


873


TABLE V NUMERICAL RESULTS OF EXAMPLE 3 FOR ε=0.001, N=100 x 0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90 1.00


δ=0.0005=η

Exact Solution 1.0000000 0.0029107 0.0032791 0.0036941 0.0041616 0.0085073 0.0280120 0.0922351 0.3037022 0.5510918 1.0000000


δ=0.00, η= 0.0005

Exact Solution 1.0000000 0.0029191 0.0032883 0.0037043 0.0041729 0.0085273 0.0280614 0.0923436 0.3038808 0.5512539 1.0000000


Exact Solution 1.0000000 0.0029023 0.0032698 0.0036839 0.0041504 0.0084873 0.0279626 0.0921265 0.3035235 0.5509297 1.0000000

TABLE VI NUMERICAL RESULTS OF EXAMPLE 3 FOR ε=0.0001, N=100 x

δ=0.00005, η=0.00

δ=0.00005=η

0.00 0.02 0.04 0.06 0.08 0.20 0.40 0.60 0.80 0.90

Numerical Solution 1.0000000 0.0028922 0.0031580 0.0035605 0.0040143 0.0082448 0.0273612 0.0908009 0.3013319 0.5489371

Exact Solution 1.0000000 0.0028063 0.0031638 0.0035669 0.0040213 0.0082574 0.0273926 0.0908703 0.3014469 0.5490418

1.00

1.0000000

1.0000000

Numerical Solution 1.0000000 0.0028926 0.0031589 0.0035615 0.0040153 0.0082467 0.0273659 0.0908113 0.3013491 0.5489527 1.00000 00

δ=0.00, η= 0.00005

Exact Solution 1.0000000 0.0028071 0.0031647 0.0035679 0.0040224 0.0082594 0.0273975 0.0908811 0.3014649 0.5490583

Numerical Solution 1.0000000 0.0028919 0.0031571 0.0035595 0.0040132 0.0082428 0.0273563 0.0907901 0.3013140 0.5489207

Exact Solution 1.0000000 0.0028055 0.0031629 0.0035659 0.0040202 0.0082554 0.0273877 0.0908594 0.3014289 0.5490254

1.0000000

1.0000000

1.0000000

TABLE VII NUMERICAL RESULTS OF EXAMPLE 4 FOR ε=0.01, N=100 x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00

δ=0.009, η=0.00 Numerical Solution 1.0000000 0.9041162 0.8174261 0.6681849 0.5461911 0.4464689 0.4022469 0.3900150 0.3656129 0.2926230 -1.0000000

δ=0.005=η

Exact Solution 1.0000000 0.9041153 0.8174245 0.6681829 0.5461891 0.4464684 0.4035918 0.3951149 0.3841594 0.3541659 -1.0000000

Numerical Solution 1.0000000 0.9048357 0.8187277 0.6703150 0.5488054 0.4493208 0.4052268 0.3931687 0.3692478 0.2973938 -1.0000000

δ=0.00, η= 0.005 Exact Solution 1.0000000 0.9048374 0.8187308 0.6703200 0.5488117 0.4493290 0.4065081 0.3980641 0.3872665 0.3580557 -1.0000000


Exact Solution 1.0000000 0.9057251 0.8203378 0.6729542 0.5520498 0.4528673 0.4101149 0.4016902 0.3909290 0.3615241 -1.0000000

TABLE VIII NUMERICAL RESULTS OF EXAMPLE 4 FOR ε=0.005, N=100

x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00

δ=0.0009, η=0.00 Numerical Solution Exact Solution 1.0000000 1.0000000 0.9051209 0.9051247 0.8192438 0.8192506 0.6711604 0.6711717 0.5498440 0.5498578 0.4504563 0.4504714 0.4076952 0.4077327 0.3994674 0.3996845 0.3899488 0.3917950 0.3674278 0.3840614 -1.0000000 -1.0000000

δ=0.0005=η Numerical Solution Exact Solution 1.0000000 1.0000000 0.9051933 0.9051962 0.8193749 0.8193802 0.6713750 0.6713839 0.5501078 0.5501186 0.4507443 0.4507563 0.4079887 0.4080229 0.3997629 0.3999752 0.3902543 0.3920862 0.3677891 0.3843529 -1.0000000 -1.0000000


δ=0.00, η= 0.0005 Numerical Solution Exact Solution 1.0000000 1.0000000 0.9052814 0.9052855 0.8195345 0.8195418 0.6716367 0.6716488 0.5504295 0.5504442 0.4510960 0.4511121 0.4083471 0.4083852 0.4001239 0.4003383 0.3906277 0.3924499 0.3682322 0.3847170 -1.0000000 -1.0000000


874


TABLE IX NUMERICAL RESULTS OF EXAMPLE 5 FOR ε=0.01, N=100 x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00


δ=0.005=η

Exact Solution 1.0000000 0.9057251 0.8203378 0.6729542 0.5520498 0.4528673 0.4101149 0.4016902 0.3909290 0.3615241 -1.0000000




Exact Solution 1.0000000 0.9072822 0.8231609 0.6775939 0.5577688 0.4591334 0.4165221 0.4082114 0.3981311 0.3731723 -1.0000000

TABLE X NUMERICAL RESULTS OF EXAMPLE 5 FOR ε=0.005, N=100 x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00


δ=0.0005=η

Exact Solution 1.0000000 0.9052855 0.8195418 0.6716488 0.5504442 0.4511121 0.4083852 0.4003383 0.3924499 0.3847170 -1.0000000


δ=0.00, η= 0.0009

Exact Solution 1.0000000 0.9053746 0.8197032 0.6719133 0.5507694 0.4514674 0.4087471 0.4007010 0.3928132 0.3850807 -1.0000000


Exact Solution 1.0000000 0.9054458 0.8198320 0.6721246 0.5510292 0.4517514 0.4090364 0.4009908 0.3931035 0.3853713 -1.0000000

TABLE XI NUMERICAL RESULTS OF EXAMPLE 6 FOR ε=0.01, N=100 x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00


δ=0.005=η

Exact Solution 1.0000000 0.7450659 0.5551232 0.3081618 0.1710678 0.0949637 0.0707149 0.0664081 0.0605775 0.0414836 -1.0000000




Exact Solution 1.0000000 0.7491820 0.5612738 0.3150282 0.1768171 0.0992428 0.0742943 0.0697266 0.0626316 0.0337035 -1.0000000

TABLE XII NUMERICAL RESULTS OF EXAMPLE 6 FOR ε=0.005, N=100 x 0.00 0.10 0.20 0.40 0.60 0.80 0.90 0.92 0.94 0.96 1.00


Exact Solution 1.0000000 0.7438487 0.5533110 0.3061530 0.1693978 0.0937297 0.0697207 0.0657141 0.0619378 0.0583784 -1.0000000

δ=0.0005=η Numerical Solution 1.0000000 0.7439999 0.5535358 0.3064019 0.1696044 0.0938821 0.0698336 0.0656999 0.0607726 0.0464807 -1.0000000




Exact Solution 1.0000000 0.7442760 0.5539469 0.3068571 0.1699825 0.0941613 0.0700820 0.0660622 0.0622730 0.0587012 -1.0000000


875


References [1] [2]

[3] [4]

[5] [6]

[7]

[8]

[9]

[10]

[11]

[12] [13]

Authors’ information 1,2

Y. Kuang, Delay Differential equations with applications in population dynamics (Academic Press, 1993). L. E. El'sgol'ts, Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs 12, (American mathematical society, 1964). R. B. Stein, Some Models of Neuronal Variability, Biophysical Journal, vol. 7, 37-68, January 1967. H. C. Tuckwell, and D. K. Cope, Accuracy of Neuronal Interspike Times Calculated from a Diffusion Approximation, Journal of Theoretical Biology, vol. 83, 377-387, April 1980. R. Bellman and K. L. Cooke, Differential-Difference Equations (Academic Press, 1963). C. G. Lange, and R. M. Miura, Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. V. Small Shifts with Layer Behavior, SIAM J. Appl. Math. vol. 54, 249-272, February 1994. C. G. Lange, and R. M. Miura, Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Layer Behavior, SIAM J. Appl. Math. vol. 54, 273-283, February 1994. M. K. Kadalbajoo, and A. S. Yadaw, An ε-uniform Ritz-Galerkin finite element method for numerical solution of singularly perturbed delay differential equations, Int. Journal of Pure and Appl. Math. vol. 55, 265-286, 2009. G. M. Amiraliyev, and E. Cimen, Numerical Method for a Singularly Perturbed Convection Diffusion Problem with Delay, Applied Mathematics and Computation, vol. 216: 2351-2359, June 2010. M. K. Kadalbajoo, and K. K. Sharma, Numerical treatment of a mathematical model arising from a model of neural variability, Journal of Math. Anal. Appl. Vol. 307, 606-627, July 2005. L. E. Elsgolt's, and S. B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments (Academic Press, 1973). E. Angel, and R. Bellman, Dynamic Programming and Partial differential equations (Academic Press, 1972). M. K. Kadalbajoo, and Y. N. Reddy, A non asymptotic method for general linear singular perturbation problems, Journal of Optimization Theory and Applications, vol. 55 256-269, November 1987.

Department of Mathematics, National Institute of Technology, WARANGAL -506004 (A. P), INDIA. E-mails: [email protected] or [email protected] [email protected] Mr. Gemechis File Duressa (Ethiopia, 21st June 1979) M. Sc (Mathematics) in 2006 from Addis Ababa University, Ethiopia. Gemechis has published so far 4 research papers in international journals in the area of Numerical Methods. Mr. Gemechis is life member of MSE.

Prof. Y. N. Reddy (India, 9th October 1958) PhD (Mathematics) in 1986 from IIT Kanpur, India. Reddy has published so far 64 research papers in international journals in the area of Numerical Methods. Prof. Reddy is life member of ISTE, ISTAM, and APSMS.



876


Single Parametric Control of Cascade Brushless DC Motor Drive Ibrahim Al-Abbas1, Mohammad Al Kawaldah2, Mohammad Al-Khedher3, Rateb Issa4 Abstract – The proportional plus integral (PI) control is the most widely used algorithm to regulate the armature current and speed of Brushless DC motors in cascade motor drive control systems. However, even when effective tuning methods are employed to satisfy the desired performance, the output overshoot is of higher values. In this paper the PI current controller is replaced with a proportional-integral-derivative (PID) controller to eliminate the overshoot in current loop and then the overshoot in speed loop. Methods of computing PID current controller parameters are derived using Internal Model Control as a function of motor parameters. The transfer function of overall closed loop current is used to determine PI speed controller parameters. Simulation results show the robustness and the effectiveness of the proposed method. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Cascade Control, Current Control, Dc Drive, PID Controller

In this type of control there are two control loops, inner one for controlling current and outer one for speed control. In cascade control, proportional-integral (PI) type controllers are used, which removes the delay and provides fast control [1]-[3]. The PI controller parameters are determined by neglecting the internal back emf of the motor [4], [5] or by calculating the reaction curve parameters using mathematical expressions [6]. In this type of design the time response of the current loop is fast, but the maximum overshoot is high [7] and this overshoot will be higher when the speed loop is added to the system [8]. In our proposed technique, to reduce or eliminate the output overshoot a PID current controller is used and Internal Model Control (IMC) system strategy is used to determine the current controller type and parameters [9]. The IMC approach has the advantages that it considers the model uncertainty, allows for easier tradeoffs between control system performance and robustness, and eliminates the steady state error and the current overshoot. The drive system is simulated using MATLAB/Simulink and its time response is plotted and investigated.

Nomenclature ω (t )

Shaft speed

i(t) v(t) TL(t) J b Kt Ls Rs Lsi M Ke f λ G p ( s )

Motor current Supply voltage Load torque Mechanical inertia of motor shaft Viscous friction coefficient of motor shaft Torque constant Stator inductance Stator resistance per phase Self inductance per phase Mutual inductance per phase Back electromagnetic-force constant Low pass filter with steady gain of unity Realizable factor Process model

G*c(s)

IMC controller model

I.

Introduction

Brushless Direct Current (BLDC) motors are one of the motor types rapidly gaining popularity. BLDC motors are used in industries such as appliances, automotive, aerospace, consumer, medical, industrial automation equipment and instrumentation. BLDC motors have many advantages over brushed DC motors and induction motors. A few of these are: Better speed versus torque characteristics, high dynamic response, high efficiency, long operating life, noiseless operation and higher speed ranges. The speed of BLDC motor can be controlled below the rated speed decreasing the armature voltage of the dc motor to achieve required speed using cascade control.

II.

Mathematical Model of Brushless Dc Motor Drive

The BLDC motor dynamics can be well approximated by the following linear time-invariant (LTI) state equation [9], [10]:


877

Ls di ( t ) / dt = − Ri ( t ) − K eω ( t ) + va ( t )

(1)

Jd ω ( t ) / dt = −bω ( t ) − TL ( t ) + Kt i ( t )

(2)


Ibrahim Al-Abbas, Mohammad Al Kawaldah, Mohammad Al-Khedher, Rateb Issa

where, ω ( t ) is the shaft speed, i(t) is the motor current,

III. Current Controller Design

v(t) is the supply voltage, TL(t) is the load torque, J and b are the overall mechanical inertia and the viscous friction coefficient at the motor shaft, respectively, Kt is the torque constant, Ls=2(Lsi-M) and R=2Rs, Ls is the stator inductance and Rs stator resistance per phase, Lsi is the self inductance per phase, M is the mutual inductance per phase and Ke is the back electromagnetic-force constant and Ke = Kt. Taking Laplace transform of the two forgoing equations assume zero initial conditions for all variables, the DC motor can be represented in block diagram form as shown in Fig. 1.

To avoid the oscillation of the armature current and to limit the starting current, the PID controller is implemented and designed using internal model control based on an approximated process model. The controller settings are related to the model parameters in a straightforward manner. This will simplify the consideration of model uncertanity and the tradoff between control system performances and robustness. Fig. 3(a) shows the classical feedback control strategy while Fig. 3(b) shows a simplified block diagram of internal model control [12] where G p ( s ) is the process model and G*c(s) is the IMC controller model. The IMC controller design involves two steps: Step 1. The process control model is factored as:

G p ( s ) = G p + ( s ) G p − ( s )

where G p + ( s ) contains any time delay and unstable

Fig. 1. BLDC motor block diagram

zeros and it is specified so that its steady state gain is unity. Step2. The IMC controller is specified as:

To decouple the current loop from the machine – inherent induced emf loop, it is necessary to split the transfer function between the speed and voltage into two cascade transfer functions, first function is between speed and armature current and the other function is between armature current and input voltage [11]. These transfer functions are derived from BLDC motor block diagram and have the following form:

ω (s)

Va ( s )

where: G1 ( s ) =

Ia ( s )

Va ( s )

ω ( s ) Ia ( s )

=

=

G2 ( s ) =

Ia ( s )

ω (s)

=

K e2

Ke Js + b

(4)

Vc ( s )

=

K ch Tch s + 1

= G1 ( s ) = ⋅

(5)

(8)

Js + b ⋅ s

(

(9)

1

)

RJ + Ls b + Rb + K e2 / s + Ls Js

Comparing the second term of Eq. (9) with the transfer function of PID controller: Gc ( s ) = K p + Ki / s + K d s

The transfer function of transistor chopper is given by: V (s)

1 f Gp− ( s )

where f is a low pass filter with steady gain of unity. In this work the single phase chopper is considered as a filter. The transfer function of the current open loop of the BLDC motor can be written as:

Va ( s )

Js + b

Ia ( s )

Gc* ( s ) =

(3)

I a ( s ) Va ( s )2

( Ls s + R ) ( Js + b ) +

(7)

(6)

(10)

The IMC-PID tuning parametrs are obtained as: K pc = ( RJ + Ls b ) / λ

where Kch and Tch are the gain and the time constant of chopper circuit. The block diagram of the cascade system of BLDC motor drive is shown in Fig. 2.

(

(11a)

Kic = Rb + K e2 / λ

)

(11b)

K d c = Ls J / λ

(11c)

Fig. 2. Block diagram of cascade SLDC motor drive system

where λ is the realizable factor and a good rule of thumb is to choose λ to be twice as fast as the open loop response.



878


(12)

following parameter values are chosen for the test runs [4]: Base current, I b = 17.35 A Base torque, Tb = 0.89Nm, Maximum phase current, I max = 2 I b = 34.7 A, Maximum torque, Tmax = 2 Tb = 1.78 Nm, Gain of the chopper, K ch = 16 V/V, Time constant of the chopper, Tch = 50 µs, Phase resistance, R = 1.4 Ohm, Phase inductance, Ls = 2.44 mH, Phase time constant, Ts = Ls R = 1.743 ms, emf constant, K e = 0.051297 Vs, Total friction coefficient, Bt = 0.002125 Nm/rad/s,

(13)

Inertia, J = 0.0002 kg m 2 , Motor and load mechanical time constant, Tm = J Bt = 94.1 ms, Current feedback gain Ki = 0.288 V/A, Speed feedback gain, Kω = 0.02387 Vs/rad, Speed.

Figs. 3. Feedback control strategies: (a) Classical feedback control, (b) Internal model control

IV.

Speed Controller Design

Using the block diagram of the cascade control system shown on Fig. 3(a), the current closed loop transfer function can be written as: I (s)

I ref ( s )

where: s1,2 = −

=

K ch ( Js + b ) / λTch

( s + s1 )( s + s2 )

( λ + JKch Ki ) ± r

2λTch

2

⎛ λ + JK ch Ki ⎞ K ch Ki b ⎜ ⎟ − λ λTch 2 T ch ⎝ ⎠

The components of the cascade system connected together are shown on Fig. 3(b); where the current loop contains the armature circuit (electrical part), the chopper and PID current controller. While the speed loop contains the mechanical part and PI speed controller.

The speed open loop transfer function relating th feedback speed signal and the speed error is:

ω f (s)

ωref ( s ) − ω f ( s )

=

K n K ch K e Kω / λTch s ( s + s1 )

(14)

V.1.

The dynamic response of the closed loop armature current is shown in Fig. 4 when PI and PID current controllers are employed. In PI control, the overshoot of the starting current is 43% which is not acceptable for the armature winding. In traditional cascade control a limiter with saturation is connected after the controller to limit this current, this result in slower response and introduces nonlinearity in the control system. This overshoot in the staring current is eliminated by PID controller as shown. Both time response curves approximately have the same settling time.

If the time constant of speed PI controller is selected to cancle the largest time constant of the closed current loop T2=1/s2, then the speed closed loop transfer function can be simpified as:

ω f (s)

ωref ( s )

=

K n K ch K e Kω / λTch s + s1s + K n K ch K e Kω / λTch 2

(15)

The speed closed loop system is a standard second order system and the gain of the speed PI controller is determined acording to the desired damping ratio and natural undamped frequency:

ωn = K n K ch K e Kω / λTch

(16)

ζ = s1 / 2ωn

(17)

Armature Current Time Response

For a given damping ratio, the proportional gain of PI speed controller can be determined using equations (16) and (17).

V.

Simulation and Results

Fig. 4. Start up step response of stator current

A brushless DC motor with name plate ratings of 373 W, 60V, 4000 rev/min is used in all simulation runs; the

To simplify tuning procedure of PID current controller, the realizable factor λ is split from controller parameters.



879


To tune the hall drive system first calculate the current PID controller parameters without the realizable factor, and set these parameters, then vary the factor λ to accomplish the desired motor current time response. Before that set the time constant of PI current controller to compensate the largest time constant of the closed current loop and then tune the gain of PI speed controller so that the maximum starting current and speed of rotation is of acceptable values and type.

To evaluate the effect of changing the value of the realizable factor on the current response, the current response for different realizable factor values were obtained. The results are shown in Fig. 5. It is clear that reducing the realizable factor reduces the overshoot of the current.

VI.

In the present study, mathematical expressions for PID current controller design using internal model conrol to avoid the oscillatoon of the armature current and to limit the starting current were developed. These expressions directly relate the controller setting to the motor parameters in a staightforward manner. The response of the current loop is achieved only varying the realizable factor. For speed control loop, selecting the time constant of speed PI controller to compensate the higher time constant of the current closed loop, the closed loop speed control system is of a second order, and the parameters of the controller is simply obtained using the natural frequency and the damping ratio of this system . The type of speed time response curve is adjusted by varying the gain of speed PI controller. The cascade system is simulated using MATLAB/Simulink and the time response for current loop and speed loop is plotted and investigated. The proposed method for determination of the parameters of the time response parameters and the controller parameters can be used in DC drive systems, AC drive systems and also in process control system with a single closed loop where PI controller is replaced by PID controller.

Fig. 5. Closed loop current response for different realizable factor values of: 1- λ = Tch ; 2- λ =0.1 Tch ; 3- λ =0.01 Tch

V.2.

Conclusion

Angular Speed Closed Loop Response

The start up time response of the speed closed loop system for different proportional gain values of PI speed controller is shown on Fig. 6 this plot indicates that higher the proportional gain the time response is faster and vice versa, in addition the response is free overshoot .The starting current for the same values of proportional gain are depicted on Fig. 7. When the gain Kp=2, the maximum starting current is twice its reference value and is acceptable. For this gain the response of the current loop is very fast.

References [1]

[2] Fig. 6. Start up speed step response of closed loop system: 1-Kn=3;2-Kn=2;3-Kn=1 [3]

[4]

[5] Fig. 7. Start up current step response of closed loop system: 1-Kn=3;2-Kn=2;3-Kn=1


P. Crnosija, R. Krishnan and T. Bjazic, Optimization of PM brushless DC motor drive, Proceedings of the IEEE International Symposium Industrial Electronics, pp. 566-569, Rio de Janeiro, Brazil, June 2003. M.V.Ramesh, J.Amarnath, S.Kamakshaiah, Srinivasa Rao.Gorantla, M.Balakrishna, Direct Torque Control Of Brushless DC Motor With Trapezoidal Back EMF, International Review of Automatic Control (IREACO), Vol. 5 (Issue 2): 202-208, 2012. M. Shafiei, M. Bahrami Kouhshahi, M. B. B. Sharifian, M. R. Feyzi, Position Sensorless for Controlling Brushless DC Motor Drives Based on Sliding Mode and RLS Estimators Using NSGAII Algorithm Optimization, International Review on Modelling and Simulations (IREMOS), Vol. 4 (Issue 3): 1121-1130, 2011. P. Crnosija, R. Krishnan and T. Bjazic, Optimization of PM Brushless DC Motor Drive Speed Controller Using Modification of Ziegler- Nichols Methods Based on Bode Plots, Power Electronics and Motion Control Conference, pp. 343-348, Portorož, Slovenia, August 2006. H. Boubertakh, M. Tadjine, P.-Y. Glorennec, S. Labiod, Comparison between Fuzzy PI, PD and PID Controllers and Classical PI, PD and PID Controllers, International Review of Automatic Control (IREACO), Vol. 1 (Issue 4): 413-421, 2008.


880


[6]

I. Al-Abbas, Methodical Tuning of Proportional plus Integral Controllers for separately Excited DC, American Journal of Applied Sciences, Vol. 9 (Issue 11): 1891-1898, 2012. [7] L. Ching-Hung, A survey of PID controller design based on gain and phase margin, International Journal of Computational Cognition, Vol. 2 (Issue 3): 63-100, September 2004. [8] S. Sahoo, A. Sahoo and R. Sultana, LabVIEW Based Speed Control of DC Motor using Modulus Hugging Approach, European Journal of Scientific Research, Vol. 68 (Issue 3): 367376, 2012. [9] A. Pisano, A. Davila, L. Fridman, and E. Usai, Cascade Control of PM DC Drives Via Second-Order Sliding-Mode Technique, IEEE Transactions on Power Electronics, Vol. 55 (Issue 11): 268 – 273, November, 2008. [10] P. Pillay and R. Krishnan, Modeling, simulation and analysis of Permanent-magnet motor drives, part-II: the brushless DC motor drives, IEEE Trans. on Industry Applications, Vol. 25: 274-279, March/April 1989. [11] K. J. Astrom, and T. Hagglund, Revisiting the Ziegler–Nichols step response method for PID control, Journal of Process Control, Vol. 14: 635–650, 2004. [12] K. Ogata, Modern Control Engineering (Englewood Cliffs, NJ: Prentice Hall, 2001).

Authors’ information 1,2,3,4

Al-Balqa Applied University, Mechatronics Department, P.O. 15008 Amman 11134 Jordan. E-mail: [email protected] Tel: (+962 777)527538 Fax: (+962 6) 4790350

Engineering

Ibrahim Al-Abbas was born in Mafraq, Jordan, in 1953. He received the master degree in 1978 and the Ph.D degree in 1983 in electric drive and automation from Moscow Power Engineering Institute, Moscow, USSR. His research areas are speed control and dynamic analysis of engineering systems. Dr. Al-Abbas is a member of Jordanian Engineering Association. Mohammad Al Khawaldah received his B.S. in electronic engineering in 1999 from University of Technology in Iraq, and M.S. degree in Mechatronics engineering from AlBalqa Applied University in Jordan in 2005. He received his PhD (Exploration and Map-building by cooperating mobile robots) from University of Hertfordshire/UK in 2010. He is currently an assistant professor in Mechatronics engineering department in Al-Balqa Applied University. His research interests include control theory, mobile robot exploration and robot simulation. M. Al-Khedher (Corresponding author) received his BSc in Mechatronics Engineering from JUST, Jordan, MSc in Mechatronics Engineering from American university of Sharjah, UAE and PhD degrees in Mechanical Engineering from Washington State University, USA. He worked as an assistant professor at AlBalqa Applied University since 2007 at Department of Mechatronics Engineering. His research interests are: Artificial intelligence, Robotics, Real-time control and MEMS. Rateb Isaa, an associate professor at Al-balqa applied university was born in Yarqa, Jordan, in 1952. He received the master degree in 1978 and the Ph.D degree in 1987 in Electromechanical and electric drive respectively from Moscow Power Engineering Institute, Moscow, USSR. His research areas are: Energy Saving in Electric Drive Systems, Optimal Control Systems, and Adaptive Control Systems.



881


Parameter Identification of a DC Motor Via Distribution-Based Approach Dorin Sendrescu

Abstract – In this paper one presents an algorithm for a DC motor parameters identification from sample data using the distribution approach. While most of the latest methods used in identification utilize a discrete-time model, the distribution method is an alternative approach to directly identify a continuous-time model from discrete-time data. The relation between the state variables is represented by functionals using techniques from distribution theory. Based on these relations, an algorithm for off-line parameter identification is developed. The method is applied to identify the parameters of a real experimental platform. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Parameter Identification, DC Motor, Distributions

In the last period there has been an increasing interest in continuous-time approaches for system identification from sampled data [1], [2]. Identification of continuoustime models is indeed a problem of significant importance in various disciplines. A simplistic way of estimating the parameters of continuous-time models by an indirect approach is to use the sampled data to estimate a discrete-time model and then transform it into an equivalent continuous-time model [3]. The second step, i.e. obtaining an equivalent continuous-time model from the estimated discrete-time model, is not always easy [4]. Difficulties are encountered whenever the sampling time is either too large or too small [5]. Whereas a large sampling interval may lead to loss of information, a small sampling period may create numerical problems because the poles are constrained to lie in a small area of the z-plane close to the unit circle. Some conversion methods use the matrix logarithm which may produce complex arithmetic when the matrix has negative eigenvalues. Moreover, the zeros of the discrete-time model are not as easily transformable to continuous-time equivalents as the poles are. In every tuning algorithm, the most difficult phase is the identification one, the whole control design depending on it [6]. We can underline two approaches of identification algorithms: on-line identification algorithms and off-line identification algorithms. In online identification approach, the result is obtained in the same moment with a new observation data acquisition. The on-line identification deals with parametric methods (deterministic or stochastic), which identify the parameters of a mathematical model with a structure apriori known. The main on-line methods can be found in [7], [8]. In off-line identification approach it is possible to identify both the structure of linear time invariant

Nomenclature T E i

La Ra u Ke J Kt B Ωn ℜ Fq

Motor torque Back EMF Armature current (A) Rotor speed Equivalent inductance of armature circuit (H) Equivalent resistance of armature circuit ( Ω ) Terminal voltage of armature circuit (V) Voltage coefficient of DC motor (V·s/rad) Inertia moment of the rotor (kg·m2) Torque coefficient of DC motor (N·m/A) Viscous friction coefficient (N·m·s/rad) Fundamental space from the distribution theory Set of real numbers Distribution (generalized function)

ϕ (t )

Test function

u(t) y(t) Hcc(s) ai, bi

Input signal Output signal D.c. Motor transfer function D.c. Motor transfer function coefficients Column vector of unknown parameters Vector of estimated parametrs Digital filter Sampling period Matrix of distributions

ω

θ θˆ

H(z) Ts Fw

I.

Introduction

Since the development of digital computers and the availability of data provided by the acquisition boards, most system identification algorithms usually aim at identifying the parameters of discrete-time models based on sampled input-output data.


882


Dorin Sendrescu

systems and the parameters of the mathematical model using observations over a larger time interval, including the steady state [9]. The distribution method used in this paper is an off-line integral method. In this approach the set of linear differential equations describing the state evolution (or input-output evolution) is mapped into a set of linear algebraic equations respect to the model parameters. DC motors have long been widely used in many industrial applications. A DC motor can be considered as a single input, single output system having torque-speed characteristics compatible with most mechanical loads. This makes a DC motor controllable over a wide range of speeds by proper adjustments of its terminal voltage. Mathematical modeling is one of the most important and often the most difficult step towards understanding a physical system [10], [11]. In modeling a dc motor, the aim is to find the governing differential equations that relate the applied voltage with the produced speed of the rotor and to determine the parameters of the model. System identification of dc motors is a topic of great practical importance, because for almost every servo control design a mathematical model is needed. This paper is structured as follows. Section 2, describes the dynamic of the separately excited dc motor. Section 3 presents the problem statement of continuous time systems identification based on distribution approach. Section 4 is dedicated to identification algorithm analysis. In section 5, the identification algorithms are applied to the parameter identification of a dc motor. Finally, conclusions of the paper are summarized in section 6.

II.

motor parameters identification is not accurate and leads to poor controlling [12]. A DC motor is designed to run on DC electric power. By far the most common DC motor types are the brushed and brushless types, which use internal and external commutation respectively to create an oscillating AC current from the DC source so they are not purely DC machines in a strict sense. The classic DC motor design generates an oscillating current in a wound rotor, or armature, with a split ring commutator, and either a wound or permanent magnet stator. A rotor consists of one or more coils of wire wound around a core on a shaft; an electrical power source is connected to the rotor coil through the commutator and its brushes, causing current to flow in it, producing electromagnetism. The commutator causes the current in the coils to be switched as the rotor turns, keeping the magnetic poles of the rotor from ever fully aligning with the magnetic poles of the stator field, so that the rotor never stops but rather keeps rotating indefinitely (as long as power is applied and is sufficient for the motor to overcome the shaft torque load and internal losses due to friction, etc.). A DC motor can be considered as a single input, single output (SISO) system having torque-speed characteristics compatible with most mechanical loads. A DC motor consists of two sub-processes: electrical and mechanical. The electrical sub-process consists of armature inductance, armature resistance and the magnetic flux of the stator. A second sub-process in the motor is a mechanical one. The traditional model of DC motor is a 2-order linear one, which ignores the dead nonlinear zone of the motor. The DC motor equivalent circuit under rating excitation is shown in Fig. 1. The motor torque, T, is related to the armature current, ia, by a constant factor Kt. The back emf, E, is related to the rotational velocity by the constant factor Ke. The voltage balance equation of DC motor armature circuit is expressed as:

DC Motor Linear Model

A mathematical model for a physical device must often reflect a compromise. It must not attempt to mirror the real device in such great detail that the model becomes cumbersome; on the other hand it should not be so simplified that predictions and explanations based on it are either trivial or far from reality. In this work one used the second order linear model over other models due to its simplicity. The main difficulty with the nonlinear models is the requirement of numerical solution and the use of this model in those applications of adaptive and optimal control which require a digital computer. The second-order linear model assumes the following: 1. The static friction is negligible and the frictional torque can be considered directly proportional to angular velocity. 2. The brush voltage drop is negligible. 3. Armature reaction can be neglected. 4. The resistance and the inductance of the armature can be regarded as constant. There is a variation of the inductance of the armature with armature current, so conventional methods for dc

u = Ra ⋅ i + K e ⋅ ω + La

di dt

(1)

where, i is armature current (A); La is equivalent inductance of armature circuit (H); Ra is equivalent resistance of armature circuit ( Ω ); u is terminal voltage of armature circuit (V); Ke is voltage coefficient of DC motor (V·s/rad). The torque balance equation of DC motor is expressed as:

Kt ⋅ i − B ⋅ ω = J

dω dt

(2)

where, J is the inertia moment of the rotor (kg·m2); Kt is the torque coefficient of DC motor (N·m/A); B is viscous friction coefficient (N·m·s/rad). In the state-space form, the equations above can be expressed by choosing the rotational speed and electric current as the state variables



883

Dorin Sendrescu

and the voltage as an input. The output is chosen to be the rotational speed, so by representing (1) and (2) in a model of state space form provides: ⎧ dω ⎪⎪ J dt = − B ⋅ ω + Kt ⋅ i ⎨ ⎪ L di = − K ⋅ ω − R ⋅ i + u e a ⎪⎩ a dt y =ω

•

Choosing the model parameters to fit the model as well as possible to the measurements: selection of a ”goodness of fit” criterion. • Validating the selected model. Estimation approaches can be divided into two categories: offline estimation and online estimation. Offline techniques use specific test inputs, measure the corresponding output signals and then try to establish the relation between them. Online techniques use, for example, observers and Kalman filters to recursively estimate parameters. Distribution based technique is an off-line estimation method. A choice should be made within all the possible mathematical models that can be used to represent the system. Again a wide variety of possibilities exist, such as: - Parametric versus nonparametric models: in a parametric model, the system is described using a limited number of characteristic quantities called the parameters of the model, whereas in a nonparametric model the system is characterized by measurements of a system function at a large number of points [16]. Examples of parametric models are the transfer function of a filter described by its poles and zeros and the motion equations of a piston. An example of a nonparametric model is the description of a filter by its impulse response at a large number of points. Usually it is simpler to create a nonparametric model than a parametric one because the modeler needs less knowledge about the system itself in the former case. However, physical insight and concentration of information are more substantial for parametric models than for nonparametric ones. - White box models versus black box models: in the construction of a model, physical laws whose availability and applicability depend on the insight and skills of the experimenter can be used (Kirchhoff’s laws, Newton’s laws, etc.). Specialized knowledge related to different scientific fields may be brought into this phase of the identification process [17]. The modeling of a loudspeaker, for example, requires extensive understanding of mechanical, electrical, and acoustical phenomena [18]. The result may be a physical model, based on comprehensive knowledge of the internal functioning of the system. Such a model is called a white box model. Another approach is to extract a black box model from the data. Instead of making a detailed study and developing a model based upon physical insight and knowledge, a mathematical model is proposed that allows sufficient description of any observed input and output measurements. This reduces the modeling effort significantly. The choice between the different methods depends on the aim of the study: the white box approach is better for gaining insight into the working principles of a system, but a black box model may be sufficient if the model will be used only for prediction of the output. - Linear models versus nonlinear models: in real life, almost every system is nonlinear. Because the theory of

(3)

Eliminating state variable i from this system of equation one obtains the input – output differential equation: d2y dy a2 2 + a1 + a0 y = b0 u (4) dt dt where: a2 = La ⋅ J a1 = La ⋅ B + Ra ⋅ J a0 = K e ⋅ Kt + Ra ⋅ B

(5)

b0 = Kt

Fig. 1. DC motor equivalent circuit

III. Distribution Based Identification of Linear Systems Accurate mathematical models and their parameters are essential when designing controllers because they allow the designer to predict the closed loop behavior of the plant [13]. Errors in parameter values can lead to poor control and instability. The conventional way of characterizing a dc motor is to perform a separate test for each parameter, but this is not only time consuming, but can yield misleading results if the parameters are measured under static or no load conditions [14]. Therefore, estimation techniques must be used to estimate the unknown or inaccurate parameters values with precision. Each identification session consists of a series of basic steps. Some of them may be hidden or selected without the user being aware of his choice. This can result in poor or suboptimal results [15]. In each session the following actions should be taken: • Collecting information about the system. • Selecting a model structure to represent the system.



884

Dorin Sendrescu

k new distribution Fq( ) ∈ Ω n uniquely defined by the

nonlinear systems is very involved, these are mostly approximated by linear models, assuming that in the operation region the behavior can be linearized [19]. This kind of approximation makes it possible to use simple models without jeopardizing properties that are of importance to the modeler. This choice depends strongly on the intended use of the model. For example, a nonlinear model is needed to describe the distortion of an amplifier, but a linear model will be sufficient to represent its transfer characteristics if the linear behavior is dominant and is the only interest. One important direction in continuous-time system identification is to transform the system of differential equations to an algebraic system that reveals the unknown parameters [20]. By using some measures, the direct computation of the input-output data derivatives can be completely avoided. For linear system identification, several methods are reported on this direction: identification based on the Laplace transformation and then use the Laguerre filter or transforming the continuous-time system to the frequency domain. The idea of utilizing test functions in system identification was proposed by Pearson and Lee [21] in terms of modulating functions in order to ameliorate the noise handling for deterministic least-squares identification based on time limited data. In this approach the set of linear differential equations describing the state evolution is mapped into a set of linear algebraic equations respect to the model parameters. Using techniques utilized in distribution approach, the measurable functions and their derivatives are represented by functionals on the fundamental space of testing functions [22]. The main advantages of this method are that a set of algebraic equations with real coefficients results and the formulations are free from boundary conditions [23]. Let us denote by Ω n the fundamental space from the distribution theory of the real functions ϕ : ℜ → ℜ , t → ϕ ( t ) with compact support T, having

relations:

( )

k k k Fq( ) (ϕ ) = ( −1) Fq ϕ ( ) , ∀ϕ ∈ Ω n

ϕ → Fq( k ) (ϕ ) = ( −1)

(k ) ∫ q ( t ) ϕ ( t ) dt ∈ ℜ

(9)

ℜ

where:

ϕ ( k ) : ℜ → ℜ, t → ϕ ( k ) ( t ) =

d kϕ (t ) dt k

(10)

is the k-order time derivative of the test function. When q ∈ C k ( ℜ ) , then: k k k k Fq( ) (ϕ ) = q( ) ( t ) ϕ ( t ) dt = ( −1) q ( t ) ϕ ( ) ( t ) dt (11)

∫

∫

ℜ

ℜ

that means the k-order derivative of a distribution generated by a function q ∈ C k ( ℜ ) equals to the distribution generated by the k-order time derivative of the function q. So, in place of the states and their time derivatives of a system one utilize the corresponding distributions and, in some particular cases, it is possible to obtain a system of equations linear in parameters. If the system is compatible the model parameters are structurally identifiable. In our study have been utilized three types of test functions characterized by a bounded support T = ( ta tb ) , ta < tb , all of these accomplishing the condition:

ϕ ( t ) = 0; ∀t ∈ ( −∞ ,ta ] ∪ [tb , +∞ )

(12)

The nonzero restriction is one of the following three types: 1. Exponential:

continuous derivatives at least up to the order n. Let q : ℜ → ℜ , t → q ( t ) be a function which admits a

⎛

⎞ ⎟⎟ ⎝ ( t − ta )( t − tb ) ⎠ ta tb

ϕ ( t ) = exp ⎜⎜

Riemann integral on any compact interval T from ℜ . Using this function, a unique distribution (or generalized function): Fq : Ω n → ℜ , ϕ → Fq (ϕ ) ∈ ℜ

k

(8)

(13)

2. Sinusoidal: (6)

ϕ ( t ) = sin p ( ( t − tb ) π / ( tb − ta ) )

(14)

can be built by the relation:

Fq (ϕ ) = ∫ q ( t ) ϕ ( t ) dt,∀ϕ ∈ Ω n

3. Polynomial: (7)

ϕ ( t ) = ( t − ta )

ℜ

In distribution theory, the notion of k-order derivative is introduced. If Fq ∈ Ω n , then its k-order derivative is a

p

( t − tb ) p

(15)

where p ≥ 2 is an integer. Figs. 1 and 2 present the



885

Dorin Sendrescu

k k Fu( ) (ϕ ) = Fu ( k ) (ϕ ) = u ( ) ( t ) ϕ ( t ) dt =

∫

exponential type test function and its first-order derivative for T = [0.1 0.9]. One can note that these functions and their derivatives vanish on the ends of the interval T.

R

= ( −1)

k

(19)

(k ) ∫ u ( t )ϕ ( t ) dt R

0.7

The unknown parameters are grouped in a column vector:

0.6

θ = [bm ,...,bk ,...,b0 ,an ,...,ak ,...,a1 ] = ⎡⎣θ1 ,...,θ p ⎤⎦ (20)

0.5

t

t

0.4

Because θ has p components it is necessary to use a finite number N ≥ p of test functions ϕi ,i = 1 : N to get a linear system of algebraic equations in the unknown parameters:

0.3 0.2 0.1 0

0

0.1

0.2

0.4

0.6

0.8

0.9

Fwθ =Fv

1

(21)

where Fw is a real matrix ( N × p ) :

Fig. 2. Exponential type test function 5

Fw = ⎡⎣ Fwt (ϕ1 ) ,...,Fwt (ϕi ) ,...,Fwt (ϕ N ) ⎤⎦

t

(22)

2.5

If the matrix rank is r=rank(Fw)=p then the system has a unique solution: 0

(

θˆ = Fwt Fw -2.5

-5

0

0.1

0.2

0.4

0.6

0.8

0.9

1

For a linear system consider the input – output differential equation: n

m

k =0

k =0

(16)

IV.

integrating over ℜ one get the following algebraic equation: n

where: k k Fy( ) (ϕ ) = Fy ( k ) (ϕ ) = y ( ) ( t ) ϕ ( t ) dt =

∫ R

= ( −1)

∫

k y ( t ) ϕ ( ) ( t ) dt

Analysis of the Algorithm Properties

Identifiability is a necessary prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input–output data, under ideal conditions of noise-free observations and error-free model structure. A remarkable feature of distributionbased identification procedure is that it provides a linear reparameterization of the input–output relation of the nonlinear system. This reparametrization of the system involves a very simple identifiability condition to be accomplished, that is the existence of the matrix

(17)

k =0

k

(23)

IV.1. Identifiability

m

k =0

Fwt Fv = θ *

In the following one presents some consistency and numerical aspects related to the presented algorithm.

Multiplying both sides with a test function ϕ ( t ) and

∑ ak Fy( k ) ( t ) = ∑ bk Fu( k ) , m ≤ n,an ≠ 0

−1

Remark 1. The consistency of estimates is influenced by the sampling period. The consistency analysis of estimates using integral filters is presented in section 4. Remark 2. This procedure can also be applied in the case of state space equations (that are first order linear differential equations) for identification of state space matrices. Obviously, the states must be measureable. In order to illustrate this, in section 5 are presented the numerical results obtained by simulation.

Fig. 3. First-order derivative of exponential type test function

∑ ak y( k ) ( t ) = ∑ bk u( k ) ( t ), m ≤ n,an ≠ 0

)

(18)

R



886

Dorin Sendrescu

(F

T w

⋅ Fw

)

−1

Ri = −

or, equivalently, Fw is of full rank.

Obviously, consistency of the estimates is directly influenced by the precision of numerical integration used to compute the value of the distributions. There are available numerous integration methods (often called numerical quadrature) with various degree of precision. One presents shortly the techniques used in the simulations to the approximate calculation of a

b

∫

f ( x ) dx =

a

b

RT = −

a

where coefficients {ci} are

i =0

derived from interpolating polynomial fitted for points {xi, fi}. A Newton–Cotes formula of any degree n can be constructed. One of the simplest integration methods is the trapezoidal rule. The trapezoidal rule is based on linear interpolation of f(x) at x1=a and x2=b, i.e. f(x) is approximated by:

b−a

a

(24)

h

∫ f ( x ) dx = 3 ( f0 + 4S1 + 2S2 + fn ) + RT

(30)

S1 = f1 + f3 +· · ·+fn−1, S2 = f2 + f4 +· · ·+fn−2

(31)

are sums over odd and even indices, respectively. The remainder is: RT =

h4 ( b − a ) f IV (ς ) ,ς ∈ [ a,b ] 180

(32)

This shows that one have gained two orders of accuracy compared to the trapezoidal rule, without using more function evaluations. Let’s study properties of Newton-Cotes formulas in frequency domain (or spectral properties). Newton-Cotes rules are symmetric (hence linear phase) digital filters with finite impulse response. One of the most important characteristic of digital filter is magnitude/frequency response – function which shows how much filter damps or amplifies magnitude of particular frequency contained in input data. So, for trapezoidal rule and Simpson’s rule one obtains:

(25)

To increase the accuracy one subdivides the interval [a,b] and assume that fi = f(xi) is known on a grid of equidistant points x0 = a, xi = x0 + ih, xn = b, where h=(b−a)/n is the step length. The trapezoidal approximation for the ith subinterval is:

H (z) =

xi +1

h ∫ f ( x ) dx = 2 ( fi + fi +1 ) + Ri

(29)

where:

hence: b−a ( f ( a ) + f (b )) 2

∑

a

1 ( f ( a ) + f (b)) 2

∫ f ( x ) dx =

h3 i = n −1 h2 f " (ς i ) = − ( b − a ) f " (ς ) , 12 i = 0 12

b

The integral of p(x) equals the of trapezoid with base (b-a) times the average height:

b

(28)

This shows that by choosing h small enough we can make the truncation error arbitrarily small. In other words, we have asymptotic convergence when h → 0 . In the composite Simpson’s rule one divides the interval [a, b] into an even number n = 2m steps of length h and uses the formula

N −1

f (b) − f ( a )

∑

ς ∈ [ a,b ]

function and [a, b] a finite interval. Interpolatory quadrature formulas, where the nodes are constrained to be equally spaced, are called Newton–Cotes formulas. These are especially suited for integrating a tabulated function (such is our case). Newton-Cotes numerical integration rule If it is a weighted sum of function

p ( x) = f (a) + ( x − a)

i = n −1 h fi + RT ( f0 + f n ) + h 2 i=2

The global truncation error is:

definite integral I f = ∫ f ( x ) dx where f(x) is a given

∑ ci ⋅ f ( xi )

(27)

Summing the contributions for each subinterval [xi,,xi+1], i = 0:n, gives:

IV.2. Consistency

values: I f =

h3 f " (ς i ) ,ς i ∈ [ xi ,xi +1 ] 12

(26)

xi

h 1 + z −1 h 1 + 4 z −1 + z − 2 ⋅ H z = ⋅ and ( ) 2 1 − z −1 3 1 − z −1

respectively. These transfer functions have the amplitude-frequency Bode characteristics from Figs. 4.

where:



887

Dorin Sendrescu

Ke=0.00767; J=3.87e-7; B=1.5e-3; La=180e-6; The system was simulated on a time interval of 40 seconds using a fourth order Runge-Kutta integration method in three cases: Case 1: Ts = 40 ms, noise free. Case 2: Ts = 40 ms, noisy measurements (SNR=40dB). Case 3: Ts = 100 ms, noise free. As input signal a sum of sinusoids of different amplitudes and frequencies was used that constitute a persistently exciting signal for the identification. Part of the input signal and measured signals in the noise free case are presented in Figs. 5 and 6.

As one can see classical Newton-Cotes formulas suppress high frequencies (noise) in the input data. In that sense they can be considered low-pass filters. From Figs. 4 one observes that trapezoidal rule offers a better suppression of noise than the Simpson’s rule, so in the algorithm implementation one used the trapezoidal rule for numerical integration. Bode Diagram 20

Magnitude (dB)

0

-20

-40

-60

16 15

-80

14 13 0

1

10

2

10

Amplitude [V]

-100 -1 10

10

Frequency (rad/s)

(a) Bode Diagram 40

12 11 10 9 8

20

7 6

0

2

4

6

8

Magnitude (dB)

10

12

14

16

18

20

8

9

10

Time [sec]

0

Fig. 5. Input signal (voltage)

-20

16 -40

14 -60

12

-80

-100 -1 10

0

1

10

10

Amplitude

10

2

10

Frequency (rad/s)

8

6

(b) 4

Figs. 4. Amplitude-frequency bode characteristics: (a) trapezoidal rule (b) Simpson’s rule

2

0

V.

2

3

4

5

6

7

Time [sec]

Experimental Results

Fig. 6. System response (time evolution of state variables: speed [rad/s] – continuous line and current [A] – dotted line)

The performance of the proposed identification algorithm was tested by numerical simulations for the state space model and on a real plant (using an experimental platform) for the input – output case.

V.1.

1

TABLE I CASE 1: REAL AND ESTIMATED DC MOTOR PARAMETERS (Ts=40ms, NOISE FREE) Parameter Ra Kt Ke J B La real 2.6 0.00767 0.00767 3.87e-7 1.5e-3 180e-6 estimated 2.6067 0.00687 0.00687 3.56e-7 0.00148 0.000178 Ts=40 ms

Identification of the State Space Model

If both current and speed of the load gear (state variables) are available for measurements one can identify all the motor parameters. The system described by state space equations in section 2 was simulated using the following parameter values: Ra=2.6; Kt=0.00767;

TABLE II CASE 2: REAL AND ESTIMATED DC MOTOR PARAMETERS (Ts=40 ms, NOISY MEASUREMENTS, SNR=40dB) Parameter Ra Kt Ke J B La real 2.6 0.00767 0.00767 3.87e-7 1.5e-3 180e-6 estimated 2.5344 0.00502 0.00502 2.29e-8 0.00348 0.000195 SNR=40dB



888

Dorin Sendrescu

TABLE III CASE 3: REAL AND ESTIMATED DC MOTOR PARAMETERS (Ts=100 ms, NOISE FREE) Parameter Ra Kt Ke J B La real 2.6 0.00767 0.00767 3.87e-7 1.5e-3 180e-6 estimated 1.8432 0.00232 0.00232 1.47e-7 0.000511 0.000049 Ts=100ms

the Fig. 8. A high quality DC servo motor is mounted in a solid aluminum frame. The motor drives a built-in 14:1 gearbox whose output drives an external gear. The motor gear drives a gear attached to an independent output shaft that rotates in a precisely machined aluminum ball bearing block. The output shaft is equipped with an encoder. This second gear on the output shaft drives an anti-backlash gear connected to a precision potentiometer. The potentiometer is used to measure the output angle. The external gear ratio can be changed from 1:1 to 5:1 using various gears. Two inertial loads are supplied with the system in order to examine the effect of changing inertia on closed loop performance. In the high gear ratio configuration, rotary motion modules attach to the output shaft using two 8-32 thumbscrews. The square frame allows for installations resulting in rotations about a vertical or a horizontal axis. The system is interfaced by means of a data acquisition card and driven by Wincon 5.0 based real time software. WinCon™ is a real-time Windows application. It allows you to run code generated from a Matlab/Simulink diagram in real-time on the same PC (also known as local PC) or on a remote PC. Data from the real-time running code may be plotted on-line in WinCon Scopes and model parameters may be changed on the fly through WinCon Control Panels as well as Simulink. The automatically generated real-time code constitutes a stand-alone controller (i.e. independent from Simulink) and can be saved in WinCon Projects together with its corresponding user-configured scopes and control panels. WinCon software actually consists of two distinct parts: WinCon Client and WinCon Server. They communicate using the TCP/IP protocol. WinCon Client runs in hard real-time while WinCon Server is a separate graphical interface, running in user mode. The measured input–output data are transferred to the computer by a data acquisition card (Quanser Q4, 33 MHz PCI bus interface, 12 bit high speed A/D converter [24]). The data acquisition card permits use of user defined programs interfaced with Matlab. The output speed is obtained from the tacho-generator. One obtains the following transfer function:

As test functions for signals processing three functions of exponential type (and their derivatives) were used. The corresponding results for the analyzed cases are presented in Tables I – III. The simulation results reveal good noise rejection properties of the estimation algorithm. This fact is due to the filtering properties of the integration operation. The estimates are more sensitive to the sampling period that influences the truncation error in the integration stage.

V.2.

Identification of Input – Output Model

If we want to build a model for a system, we should get information about it. This can be done by just watching the natural fluctuations, but most often it is more efficient to set up dedicated experiments that actively excite the system. In the latter case, the user has to select an excitation that optimizes his own goal (for example, minimum cost, minimum time, or minimum power consumption for a given measurement accuracy) within the operator constraints. The quality of the final result can depend heavily on the choices that are made.

Input

Physical system

Output

PC with Data Acquisition board Fig. 7. System identification experimental setup

Fig. 7 shows the experimental setup requirement prior to the parameter identification. This is the recording phase: 1. Deploy a data acquisition system, which can record the input and output at the required sampling frequency (according to the system dynamics). 2. Feed the system with rich inputs. (Inputs must change with time). 3. Record the inputs and corresponding outputs simultaneously using this data acquisition system. To illustrate the performance of the proposed identification algorithm, one identifies a real Quanser experiment using a DC servomotor with built in gearbox. The “rotational series” that we have is the SRV-02ET (E-encoder, T-tachometer), and the DC servo is shown in

H cc ( s ) =

b0 a2 s + a1s + a0 2

where: b0 = 0.00753 a2 = 2.852e − 005 a1 = 0.00011116 a0 = 0.000901

The input signal and system response are presented in Fig. 9. The validation of the model is realized by the



889

Dorin Sendrescu

comparison between the output of the identified model and the real plant at the same input (that was a sum of sinusoids). The result is presented in Fig. 10.

shown to be versatile when applied to parameter estimation, without requiring a detailed mathematical representation of the identification problem. This procedure is a functional type method, which transforms a differential system of equations to an algebraic system in unknown parameters. The relation between the state variables of the system is represented by functionals using techniques from distribution theory based on testing function from a finite dimensional fundamental space. The identification algorithm allows obtaining a linear algebraic system of equations in the unknown parameters. The coefficients of this algebraic system are functionals depending on the input and state variables and are evaluated through some testing functions from distribution theory. The effectiveness of system identification using the distribution algorithm was researched and a satisfactory performance was obtained. The simulation results show that the proposed method achieved a minimum tracking error and estimated the parameter values with high accuracy. The method was also applied to estimate the parameters of a real DC motor commonly used in industry. The influence of the sampling period, initial conditions, test functions type, input type and noise on the parameters estimates was empirically analyzed. The algorithm provides very good results even the measurements are noise contaminated because the evaluation of states derivatives is completely avoided.

Fig. 8. Quanser SRV02ET DC Motor experiment 35 input signal [V] output signal [rad/sec]

30 25

Amplitude

20 15 10 5

Acknowledgements

0 -5

This work was supported by the strategic grant POSDRU/89/1.5/S/61968, Project ID61968 (2009), cofinanced by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007-2013.

-10 -15

0

5

10

15

20

25

30

35

40

45

50

Time [sec]

Fig. 9. The input signal and system response (dotted line) 35

References

30

Speed [rad/sec]

25

[1]

20

Estimated model response

15

[2]

10 5

[3] 0

Real system respons

-5 -10

0

5

10

[4]

15

Time [sec]

Fig. 10. Real system (dashed line) and estimated model (continuous line) responses to the same input signal [5]

VI.

[6]

Conclusion

In this paper, a novel parameter estimation method for linear system identification based on distributions was developed. The distribution based algorithm has been

[7]


R. Johansson, Identification of continuous-time models, IEEE Transactions on Signal Processing, Vol. 42 (Issue 4): 887-896, April 1994. N. K. Sinh, G. P. Rao, Identification of continuous–time systems (Dordreht: Kluwer Academic Press, 1991). H. Unbehauen, G.P. Rao, Continuous-time approaches to system identification – a survey. Automatica, Vol. 26 (Issue 1): 23-35, January 1990. A. C. Megherbi, H. Megherbi, K. Benmahamed, A. G. Aissaoui, A. Tahour, Parameters Identification of a Nonlinear System Based on Genetic Algorithms with an Optimized Cost Function, International Review of Automatic Control (IREACO), Vol. 1. No. 1, pp. 8-14, May 2008. R. H. Middleton, G. C. Goodwin, Digital Control and Estimation, a Unified Approach (Prentice-Hall, Englewood Cliffs, 1990). A. Ben Amor, S. Hajri, M. Gasmi ,Optimization of Stepping Motor Characteristics by Genetic Algorithm, International Review on Modelling and Simulations (IREMOS), Vol. 3. no. 2,pp. 168177, April 2010. T. Soderstrom, P. Stoica, System identification. (Cambridge, UK: Prentice Hall, 1989).


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Dorin Sendrescu

[8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18] [19]

[20]

[21]

[22] [23]

[24]

P. Eykhoff, System Identification (J. Wiley, London, 1974). T. Bastogne, H. Garnier, P. Sibille & M. Mensler, PMF-based subspace method for continuous-time model identification deterministic study, Proceedings of the 11th IFAC symposium on system identification (SYSID’97), Fukuoka, pp. 1665 – 1670, July 1996. A. Kaddouri, M. Ghribi, J. Ghouili, Input-Output Linearization Based Adaptive Nonlinear Control of a Permanent-Magnet Synchronous Motor, International Review of Automatic Control (IREACO), Vol. 1. no. 2, pp. 192-199, July 2008 Laurence D. Colvara, Luiz Flávio X. Sá, A Mechanical Analogous Model for the Power System with Automatic Voltage Regulator, International Review on Modelling and Simulations (IREMOS), ,Vol. 3. n. 1,pp. 57-63, February 2010. N. Sinha, C. Dicenzo, and B. Szabados, Modelling of DC motors for control applications, IEEE Trans. Industrial Electronics And Control Instrumentation, vol. IECI-21, pp. 84-88, 1984. J. Wang, J. Pu, and P. Moore, A practical control strategy for servo-pneumatic actuator systems, Control Engineering Practice Vol. 7:1483–1488, 1999. Y. G. Jung, K. Cho, Y. Lim, J. Park, and Y. H. Change, Time domain identification of brushless DC motor parameters, Proceedings of the IEEE International Symposium on Industrial Electronics, vol. 2, 593-597, 1992. S. Kamoun,Parametric Estimation of Non-Linear Stochastic Systems Described by Input-Output Mathematical Models, International Review of Automatic Control (IREACO), Vol. 1. no. 4, pp. 422-434, November 2008. A. V. Bos, Parameter Estimation for Scientists and Engineers (First Edition, Wiley, Inc, 2007). Saber Tlili, Hassen Mibar, A New Approach of Approximation and Stability for a Class of Discrete Nonlinear Systems, International Review of Automatic Control (IREACO), Vol. 5 no. 1, pp. 40-48, January 2012. P. Diniz, Adaptive Filtering: Algorithms and Practical Implementation (Springer Publishers, Second edition, 2002). R. K. Pearson, and M. Pottmann, Grey-box identification of block-oriented nonlinear models. Journal of Process Control, Vol. 10 (Issue 4): 301–315, April 2000. C. Marin, System identification based on distribution theory, Proceedings of IASTED Int. Conf. Applied Simulation and Modelling (ASM), Crete, pp. 456-462, 2002. A. Pearson, F. Lee, On the identification of Polynomial inputoutput differential systems. IEEE Transaction on Automatic Control, Vol. 30 (Issue 8): 778-782, August 1985. L. Schawarz, Théorie des distribution (Paris, 1965). A. Ohsumi, K. Kameiama, Subspace identification for continuoustime systems via distribution-based approach, Automatica, Vol. 38 (Issue 1): 63-79, January 2002. *** Quanser Consulting Inc., Quanser SRV02-ET Experiment. User manual, 1998.

Authors’ information Department of Automatic Control, University of Craiova, Romania. E-mail: [email protected] Dorin Sendrescu received his B.S. and M. Sc. degrees (1998) in automatic control and the Ph. D. degree in control systems (2007) from University of Craiova, Romania. In 2005 he was a Marie Curie Fellow at Université de Technologie de Compiègne, France. Since 1998 he is with the University of Craiova where he is currently Associate Professor in the Department of Automatic Control. He has been and currently is involved in national and international research projects in the field of automatic control. His present research interests are on identification of linear and nonlinear systems. He published more than 60 journal and conference papers and he is author or co-author of 4 books. Dr. Sendrescu is member of IEEE, SRAIT and ARR.



891


Differential Equations of Synchronous Generators Dynamics for Online Assessment within Energy Control Centers Lucian Lupsa-Tataru

Abstract – The aim of present investigation is the formulation of a new system of differential equations, in normal form, to describe the saturated synchronous generators dynamics by advancing the d-q axis magnetizing inductances as expressions in terms of d-q axis winding flux linkages only. It will be proven that the representation of magnetizing inductances in this original manner makes it possible to improve the structure of synchronous generators winding flux linkage state-space model with the purpose of online assessment. The selection of the winding flux linkage state-space model is practically justified having in view the model well-known structural simplicity, what makes it suitable for straight implementation within energy control centers. The procedure of representing the d-q axis magnetizing inductances as functions of only winding flux linkages will be carried out in a deductive manner that is without distorting the structural equations of the generalized d-q axis mathematical model of synchronous generators. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Differential Equations, Mathematical Modelling, Nonlinear Systems, Energy Control Center, Synchronous Generators, Magnetizing Inductance

Q m

Nomenclature Symbols u , i,ψ R, L

ω im imd , imq

λdq Lmd , Lmq

α , β ,γ

ωd ij , ωqij ⎧ δ ,δ ⎪⎪ d 0 d 1 ⎨ δ q 0 , δ q1 ⎪ ⎪⎩ L pd , L pq

σ

Voltage, current and flux linkage, respectively Resistance and inductance, respectively Angular frequency Total magnetizing current i.e. magnetizing current space-phasor modulus Magnetizing current d-q axis components

T

Associated with q-axis damper circuit Associated with main flux path Suffix to denote leakage inductances Transposed matrix (vector)

I.

Introduction

Due to the increasing complexity and decreasing stability of power systems, the formulation of suitable adaptive models has become a topic of growing significance, in connection with the development of microprocessors architecture. These models are necessary for online dynamic security analysis, so that the operation of each power system component can be predicted and eventually improved [1]-[3]. In order to solve the contingency cases within an energy control center by extending the analysis to an online dynamic one, the time interval appointed for the computation is of utmost importance [4]-[6]. However, to mend the computational capabilities, like most of electric power components, synchronous generators are usually delineated by means of linear, reduced-order dynamic dq axis models that encompass exclusively constant parameters. This is due to the conveniently accomplishing of a contingency analysis when implementing linear models of power system dynamics on parallel or array processors. Because of its internal structural simplicity, the wellestablished winding flux linkage state-space model of synchronous generators, derived by selecting all d-q axis

Quantity varying with d-q axis winding flux linkages only d-q axis magnetizing inductances Parameters within representations of d-q axis magnetizing inductances Flux linkages coefficients in the structure of winding flux linkage state-space model Constant quantities

Subscripts/Superscripts Designates the d-axis (“direct” axis) d components q Designates the q-axis (“quadrature” axis) components Associated with field winding f Associated with d-axis damper circuit D


892


Lucian Lupsa-Tataru

that angular frequency ω , interfering in stator voltage Eqs. (1) and (2), is maintained at the rated (synchronous) value. It is to be noticed that the well-established structure (1)-(4) provides the time-related derivatives of winding flux linkages in terms of both stator d-q axis winding flux linkages and all winding currents; (ii) The flux equations as algebraic correlations between each winding flux linkage and the corresponding winding currents i.e.:

winding flux linkages as state variables, is being intensively exploited with the purpose of dynamic security assessment. However, with the increasing emphasis on economy, the power generators are now operated in the proximity of security limits, thus undergoing large variations of magnetic stress. To account for these variations, the approach in the present paper is geared toward advancing the synchronous generators magnetizing inductances as expressions in terms of d-q axis winding flux linkages only. Thus, the connection with synchronous generators winding flux linkage state-space model becomes immediate. The approach in [7], materialized in formulation of an auxiliary algebraic equation to allow the magnetizing current computation when only the values of winding flux linkages are available, becomes in this case the most suitable one. The treatment in this paper further develops the idea of main flux saturation modelling validated in [7]. Unlike [7], wherein the methodology is aimed at magnetizing current computation, this contribution is redirected at disclosing the d-q axis magnetizing inductances as continuous functions of the state variables of winding flux linkage state-space model. The proof sequence employs the well-formed formulae that depict the synchronous generators generalized d-q axis model.

II.

ψ d = Lσ id + Lmd imd = Ld id + Lmd ⋅ ( i f + iD )

(5)

ψ q = Lσ iq + Lmq imq = Lq iq + Lmq iQ

(6)

ψ f = L f σ i f + Lmd imd = L f i f + Lmd ⋅ ( id + iD )

(7)

ψ D = LDσ iD + Lmd imd = LD iD + Lmd ⋅ ( id + i f

(8)

ψ Q = LQσ iQ + Lmq imq = LQ iQ + Lmq iq where in

and

Lmd

)

(9)

stand for magnetizing

Lmq

inductances whilst: imd = id + i f + iD , imq = iq + iQ

The Generalized d-q Axis Mathematical Model of Synchronous Generators

(10)

denote the d-q axis components of magnetizing current space-phasor [12]. The magnetizing current space-phasor modulus designates the so-called total magnetizing current:

In the d-q reference frame fixed to the rotor, and with rotor quantities referred to stator, the commonly accepted picture of synchronous generators is described by means of two distinctive sets of structural equations [7]-[11]: (i) Associating the voltage and current positive signs for the case of generator operating mode, the generalized voltage equations as a set of ordinary differential equations i.e.: d ψ d = ω ψ q − Rid − ud dt

(1)

d ψ q = − ω ψ d − Riq − uq dt

(2)

im =

2 imd

2 + imq

⎡ ⎢ id + i f + iD ⎢ 2 ⎢⎣ + iq + iQ

(

=

(

)

)

2

+⎤ ⎥ ⎥ ⎥⎦

(11)

Relationships (5)-(9) yield the expressions of the stator and rotor self inductances in terms of the corresponding magnetizing inductance: Ld = Ld ( Lmd ) = Lσ + Lmd

(

)

Lq = Lq Lmq = Lσ + Lmq L f = L f ( Lmd ) = L f σ + Lmd

(12)

LD = LD ( Lmd ) = LDσ + Lmd d ψ f = − Rf if + u f dt

(3)

d d ψ D = − RD iD , ψ Q = − RQ iQ dt dt

(4)

(

)

LQ = LQ Lmq = LQσ + Lmq

Since they are provided by means of algebraic correlations, the flux Eqs. (5)-(9) permit the selection of the d-q axis state variables (currents and/or flux linkages) in different variants. The selection of the vector of state variables has to be in accordance with the problem on hand that could be represented by synthesis of the

Having in view that the motion equation is irrelevant here, to improve legibility in presentation, we assume



893

Lucian Lupsa-Tataru

vector-controlled system, offline or online assessment, respectively. The approach in the present investigation, directed toward online assessment, assumes all winding flux linkages as state variables.

the winding flux linkage state-space model in the following split matrix form: ⎡ψ d ⎤ ⎡ψ d ⎤ ⎢ψ ⎥ ⎡ ud q d ⎢ ⎥ ⎢ ψ f ⎥ = Ω d × ⎢⎢ ⎥⎥ − ⎢ −u f ⎢ ψ dt ⎢ ⎥ ⎢ f ⎥ ⎢⎣ 0 ⎣ψ D ⎦ ⎣⎢ψ D ⎦⎥

III. The Winding Flux Linkage State-Space Model of Synchronous Generators The winding flux linkage state-space model of synchronous machines is derived by selecting all d-q axis winding flux linkages as state variables. The model possesses a simple structure due to the fact that the set of voltage Eqs. (1)-(4) of the generalized d-q axis mathematical model already incorporates explicitly the time-related derivatives of all d-q axis winding flux linkages. In order to develop the flux linkage state-space model, it is necessary to solve system (5)-(9) in relation to the d-q axis winding currents. By solving the algebraic system (5), (7), (8) in relation to d-axis winding currents i.e. id , i f , iD and the

d dt

− Lmd

⎡ Lmd ⎢ ⎢ LDσ ⎢L ⎣ fσ

LDσ Lmd Lσ

⎡ iq ⎤ 1 ⎡ LQ ⎢ ⎥= ⎢ ⎢⎣iQ ⎥⎦ ∆ q ⎢⎣ − Lmq

0 Ld LD 0

0 ⎤ ⎥ 0 ⎥+ Ld L f ⎥⎦ ⎞ ⎡ψ d ⎤ ⎟ ⎢ ⎥ ⎟ × ⎢ψ f ⎥ ⎟ ⎢ ⎥ ⎠ ⎣ψ D ⎦

Lfσ ⎤ ⎥ Lσ ⎥ Lmd ⎥⎦

− Lmq ⎤ ⎡ψ q ⎤ ⎥×⎢ ⎥ Lq ⎥⎦ ⎢⎣ψ Q ⎥⎦

) (

)

+ Lσ LQσ = δ q1 Lmq + δ q 0

(18)

T

The flux linkages coefficients ωd ij in (17) and ωqij in (18), respectively, depend on d-axis and q-axis magnetizing inductance as follows:

ωd 11 ( Lmd ) = − R

(13)

( L f σ + LDσ ) Lmd + L f σ LDσ ∆ d ( Lmd )

=

(19a)

( L f σ + LDσ ) Lmd + L f σ LDσ = −R δ d 1 Lmd + δ d 0

ωd 13 ( Lmd ) = R

(14)

ωd 14 ( Lmd ) = R

⎛ Lσ L f σ + Lσ LDσ + ⎞ ∆ d ≡ ∆ d ( Lmd ) = ⎜ ⎟⎟ Lmd + ⎜ + L f σ LDσ (15) ⎝ ⎠ + Lσ L f σ LDσ = δ d 1 Lmd + δ d 0

(

⎡ψ ⎤ ⎡ψ q ⎤ ⎢ d ⎥ ⎡uq ⎤ ⎢ ⎥ = Ω q × ⎢ψ q ⎥ − ⎢ ⎥ ⎢⎣ψ Q ⎥⎦ ⎢ ⎥ ⎣0⎦ ⎣⎢ψ Q ⎦⎥

T ⎡ωd 11 ωd 21 ωd 31 ⎤ ⎡ −ω 0 ⎤ ⎢ ω ⎥ ⎢ ⎥ 0 0 ⎥ , Ω q = ⎢ωq12 ωq 22 ⎥ Ωd = ⎢ ⎢ωd 13 ωd 23 ωd 33 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ωq13 ωq 23 ⎥⎦ ω ω ω d 24 d 34 ⎦ ⎣ d 14

with the system determinants depending on the value of d-axis and q-axis magnetizing inductance, respectively:

∆ q ≡ ∆ q Lmq = Lσ + LQσ Lmq +

(17)

where in:

algebraic system (6), (9) in relation to q-axis winding currents i.e. iq , iQ , we receive the correlations [7]: ⎛ ⎡ L f LD ⎡ id ⎤ ⎢ ⎥ 1 ⎜ ⎢ ⎢i f ⎥ = ∆ ⎜ ⎢ 0 d ⎜ ⎢ ⎢i ⎥ 0 ⎣ D⎦ ⎝ ⎣

⎤ ⎥ ⎥ ⎥ ⎦

ωd 21 ( Lmd ) = R f

LDσ Lmd

∆ d ( Lmd ) L f σ Lmd ∆ d ( Lmd )

LDσ Lmd

∆ d ( Lmd )

ωd 23 ( Lmd ) = − R f (16) = − Rf

=R

=R

= Rf

LDσ Lmd

δ d 1 Lmd + δ d 0 L f σ Lmd

δ d 1 Lmd + δ d 0 LDσ Lmd

δ d 1 Lmd + δ d 0

( Lσ

+ LDσ ) Lmd + Lσ LDσ

( Lσ

+ LDσ ) Lmd + Lσ LDσ

∆ d ( Lmd )

(19b)

(19c)

(19d)

=

(19e)

δ d 1 Lmd + δ d 0

where in:

ωd 24 ( Lmd ) = R f

δ d 0 = Lσ L f σ LDσ , δ q 0 = Lσ LQσ , δ d 1 = Lσ L f σ + Lσ LDσ + L f σ LDσ , δ q1 = Lσ + LQσ

ωd 31 ( Lmd ) = RD

By replacing the d-q axis winding currents in (1)-(4) with the expressions yielded by (13) and (14), we receive Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

Lσ Lmd

∆ d ( Lmd ) L f σ Lmd ∆ d ( Lmd )

= Rf

= RD

Lσ Lmd

δ d 1 Lmd + δ d 0 L f σ Lmd

δ d 1 Lmd + δ d 0

(19f)

(19g)


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Lucian Lupsa-Tataru

ωd 33 ( Lmd ) = RD

Lσ Lmd

∆ d ( Lmd )

ωd 34 ( Lmd ) = − RD = − RD

Lσ Lmd

= RD

δ d 1 Lmd + δ d 0

( Lσ + L f σ ) Lmd + Lσ L f σ ∆ d ( Lmd )

Lmq = Lmq ( im ) =

(19h)

⎛ = Lmq ⎜ ⎝

=

δ d 1 Lmd + δ d 0

and:

ωq12 ( Lmq ) = − R

Lmq + LQσ

(

ωq13 ( Lmq ) = R

ωq 22 ( Lmq ) = RQ

ωq 23 ( Lmq ) = − RQ

)

∆ q Lmq Lmq

(

∆ q Lmq Lmq

(

∆ q Lmq

)

Lmq + Lσ

(

∆ q Lmq

)

Lmq + LQσ

=−R

)

=R

= RQ

(20a)

δ q1 Lmq + δ q 0 Lmq

δ q1 Lmq + δ q 0

(20b) Lmd ( im ) =

Lmq

= − RQ

Lmq + Lσ

δ q1 Lmq + δ q 0

Lmq ( im ) =

(20c)

δ q1 Lmq + δ q 0

(20d)

L pq

Lmd =

Lmd = Lmd ( im ) =

( id + i f + iD )

=

(

+ iq + iQ

)

2

⎞ ⎟ ⎠

⎞ (22) ⎟ ⎠

2

β + im γ − L pq im

(23)

β + im

( = ( Lσ

−1

+

LQ−σ1

)

)

−1

(24)

−1

One will perceive that constant quantities (24) are deliberately incorporated in (23) with the purpose of making feasible the expressing of magnetizing current in terms of d-q axis winding flux linkages. The main flux saturation modelling by means of (23) holds the possibility of overspreading a range having great extent. For low values associated with magnetizing current (11), the magnetizing inductances (23) approach to the unsaturated values equal to parameters ratio α β and γ β , respectively. Still, it has to be pointed out that nonlinearities (23) yield the magnetizing inductances as dependencies on total magnetizing current (11), i.e. as functions of the d-q axis currents:

It is assumed that leakage flux saturation and main flux saturation can be treated independently. Since only main flux path saturation is discussed here, leakage inductances (denoted with subscript σ ) are constants. The effects of main flux saturation (including the cross-coupling effect) in synchronous machines are accurately taken into account by changing the d-q axis magnetizing inductances into dependencies upon the total magnetizing current (11) as follows [9]-[14]:

2

)

α − L pd im

L pd = Lσ−1 + L f−σ1 + LD−σ1

Introducing Magnetizing Inductances as Dependencies on Magnetizing Current

⎛ = Lmd ⎜ ⎝

(

+ iq + iQ

where leakage inductances interfere here through quantities:

Assuming constant i.e. saturation-independent magnetizing inductances, the flux linkages coefficients in (17) and (18), respectively, provided by expressions (19) and (20), are preserved at constant values and, implicitly, structure (17), (18), with voltages as input quantities, will describe a linear system of differential equations in relation to d-q axis winding flux linkages.

IV.

2

with a view to online dynamic security assessment, in order to make the running of a contingency analysis fast enough, the system analysts should avoid the employment of transcendental functions for depicting nonlinearities (21) and (22). Thus, an agreement between accuracy and the simplicity of expressing has to be considered. Having in view the nonlinearities of type (21), (22), in order to account here for main flux saturation, we will employ rational fraction approximations in the form originally adopted in [15] for the specific case of induction machine. Hence, we will fix on dependencies:

(19i)

( Lσ + L f σ ) Lmd + Lσ L f σ

( id + i f + iD )

α − L pd

2 2 + imq β + imd

α − L pd β+

(21)


2 2 imd + imq

=

( id + i f + iD ) ( id + i f + iD )

2

2

(

+ iq + iQ

(

+ iq + iQ

)

)

2

(25)

2


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Lucian Lupsa-Tataru

Lmq =

=

γ − L pq

2 2 imd + imq

2 2 + imq β + imd

γ − L pq β+

replace the magnetizing current d-q axis components in (11). We successively receive:

=

( id + i f + iD ) ( id + i f + iD )

2

2

(

+ iq + iQ

(

+ iq + iQ

)

)

2

( imq = ∆ q−1 ⋅ ( LQσ ψ q + Lσ ψ Q )

imd = ∆ d−1 ⋅ L f σ LDσ ψ d + Lσ LDσ ψ f + Lσ L f σ ψ D

(26)

2

and, eventually:

Particularly, to select winding currents as state variables, in the generalized voltage equations (1)-(4), the winding flux linkages have to be taken as currentbased expressions by employing correlations (5)-(9). Nevertheless, with magnetizing inductances expressions (25), (26), the inherent time-related differentiation of winding flux linkages, provided by (5)-(9), would lead in this situation to a complex systematization due to the necessity of differentiating the expressions of magnetizing inductances. However, as already pointed, to allow of an immediate conjunction with the generator winding flux linkage state-space model, the derivation here is geared toward representing the magnetizing inductances in terms of winding flux linkages only.

V.

The Nonlinear Winding Flux Linkage State-Space Model

V.1.

Representing Magnetizing Inductances in Terms of Winding Flux Linkages

)

im =

⎛ λd ⎜ ⎜ Lmd + L pd ⎝

2

⎞ ⎛ λq ⎟ +⎜ ⎟ ⎜ Lmq + L pq ⎠ ⎝

⎞ ⎟ ⎟ ⎠

2

(27)

with λd and λq as flux-dependent quantities:

λd = λd ( ψ d ,ψ f ,ψ D ) = =

L f σ LDσ ψ d + Lσ LDσ ψ f + Lσ L f σ ψ D

(28)

L f σ LDσ + Lσ LDσ + Lσ L f σ

λq = λq ( ψ q ,ψ Q ) =

LQσ ψ q + Lσ ψ Q

(29)

LQσ + Lσ

Substituting the d-q axis magnetizing inductances in (27) with the adopted expressions (23), we obtain the flux-based expression of magnetizing current: im = β

It comes out that the d-axis magnetizing inductance Lmd interferes within the flux linkages coefficients in the set of differential Eqs. (17) whilst the q-axis magnetizing inductance Lmq is contained in the flux

λdq

(30)

1 − λdq

with λdq interfering as quantity varying with d-q axis winding flux linkages only. More precisely, we have:

linkages coefficients of differential equations (18). Considering any nonlinearities of type (21), (22), which traditionally will yield the magnetizing inductances in terms of magnetizing current that is in terms of d-q axis currents, the flux linkages coefficients in differential equations (17) and (18), respectively, given by (19) and (20), will come forth as functions of currents, which are not state variables. Hence, in order to preserve structure (17), (18) as a system of differential equations, we have to represent the magnetizing inductances in terms of selected state variables i.e. just the winding flux linkages. Moreover, it has to be emphasized that besides the fact that the d-q axis magnetizing inductances do interfere in expressions (19), (20) of the flux linkages coefficients in differential Eqs. (17), (18), they also decide the d-q axis winding currents in terms of winding flux linkages by means of correlations (13), (14). Thus, the advancing of magnetizing inductances as functions of state variables appears to be the crucial point in the proof sequence here. Having in view (10) and considering the currents flux linkages relationships (13) and (14), we proceed to

λdq =

⎛ λd ⎜ ⎜ α + β L pd ⎝

2

⎞ ⎛ λq ⎟ + ⎜ ⎟ ⎜ γ + β L pq ⎠ ⎝

⎞ ⎟ ⎟ ⎠

2

(31)

Replacing the magnetizing current variable in (23) with flux-based expression (30), the magnetizing inductances change to linear functions of the fluxdependent quantity (31): Lmd λdq =

( )

⎞ α ⎛α − ⎜ + L pd ⎟ λdq β ⎝β ⎠

(32)

( )

⎞ γ ⎛ γ − ⎜ + L pq ⎟ λdq β ⎝β ⎠

(33)

Lmq λdq =

Hence, by means of quantity (31), relations (32) and (33) disclose the d-q axis magnetizing inductances as expressions in terms of d-q axis winding flux linkages only:



896

Lucian Lupsa-Tataru

Lmd =

⎞ α ⎛α − ⎜ + L pd ⎟ × β ⎝β ⎠

×

Lmq =

×

(

⎡ λd ψ d ,ψ f ,ψ D ⎢ α + β L pd ⎢ ⎣

) ⎤⎥

2

) ⎤⎥

2

(

⎡ λq ψ q ,ψ Q + ⎢ ⎥ ⎢ γ + β L pq ⎦ ⎣

) ⎤⎥

2

) ⎤⎥

2

implementation within energy control centers. The flux linkages coefficients in differential equations (17) and (18), respectively, come now to be saturation-dependent but independent on currents, varying only with the new flux-dependent quantity (31).

⎥ ⎦

⎞ γ ⎛ γ − ⎜ + L pq ⎟ × β ⎝β ⎠

(

⎡ λd ψ d ,ψ f ,ψ D ⎢ α + β L pd ⎢ ⎣

(

⎡ λq ψ q ,ψ Q + ⎢ ⎥ ⎢ γ + β L pq ⎦ ⎣

⎥ ⎦

Thus, the new variable (31), given in terms of state variables only, acts here as a substitute for magnetizing current variable. It appears to be of interest the expressing of the flux-dependent quantity (31) in terms of magnetizing current. One observes that (30) is equivalent to the following change of variable:

λ dq =

im β + im

Fig. 1. Magnetizing inductances of a salient-pole synchronous generator together with the new quantity (31), provided as functions of magnetizing current variable (11)

(34)

Moreover, employing (34) in (32) and (33), respectively, one receives the following dependencies: Lmd ( im ) = Lmq ( im ) =

im ⎞ im α⎛ ⎜1 − ⎟ − L pd β ⎝ β + im ⎠ β + im

γ β

⎛ i ⎜1 − m ⎜ β +i m ⎝

⎞ i ⎟ − L pq m , ⎟ β + im ⎠

Fig. 2. Magnetizing inductances, provided as functions of the fluxdependent quantity (31) by means of relations (32) and (33)

which are identical to nonlinearities (23). Thus, representations (32), (33) are symbolically validated. For a 440 kVA salient-pole synchronous generator (Appendix), nonlinearities (23) along with dependency (34), which yields the new (flux-dependent) variable (31) in relation to magnetizing current variable (11), are depicted in Fig. 1. Taking into account the p.u. values of machine leakage inductances (Appendix) and of fitting parameters in (23) i.e.:

Related to the set of differential equations (17), we receive the coefficients that depend on d-axis magnetizing inductance Lmd , now provided by (32). We have:

( L f σ + LDσ ) ⋅ Lmd ( λdq ) + L f σ LDσ δ d 1 Lmd ( λdq ) + δ d 0 = ωd 11 ( λdq )

ωd 11 = − R

α = 3.135 , β = 1.43 , γ = 1.695 linearities (32) and (33) are established, being plotted in Fig. 2.

V.2.

The Flux Linkages Coefficients in the Structure of Winding Flux Linkage State-Space Model

With representations (32) and (33), structure (17), (18), explicitly incorporating the magnetizing inductances, is preserved as a system of differential equations in relation to selected state variables i.e. the winding flux linkages, thus being suitable for


=

(35a)

ωd 13 = R

( ) =ω λ d 13 ( dq ) δ d 1 Lmd ( λdq ) + δ d 0

(35b)

ωd 14 = R


(35c)

ωd 21 = R f


(35d)

LDσ Lmd λdq

L f σ Lmd λdq

LDσ Lmd λdq


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Lucian Lupsa-Tataru

ωd 23 = − R f

( Lσ

( ) δ d 1 Lmd ( λdq ) + δ d 0

+ LDσ ) ⋅ Lmd λdq + Lσ LDσ

( )

ωq13 = R

( ) =ω λ q13 ( dq ) δ q1 Lmq ( λdq ) + δ q 0

(36b)

ωq 22 = RQ

( ) =ω λ q 22 ( dq ) δ q1 Lmq ( λdq ) + δ q 0

(36c)

=

(35e)

Lmq λdq

= ωd 23 λdq

ωd 24 = R f

ωd 31 = RD

ωd 33 = RD

( ) =ω λ d 24 ( dq ) δ d 1 Lmd ( λdq ) + δ d 0 Lσ Lmd λdq

(35f)

ωq 23 = − RQ


(35g)


(35h)

L f σ Lmd λdq

Lσ Lmd λdq

( Lσ + L f σ ) ⋅ Lmd ( λdq ) + Lσ L f σ δ d 1 Lmd ( λdq ) + δ d 0 = ωd 34 ( λdq )

ωd 34 = − RD

(

=

(35i)

id =

)

( ) ( ) ( ) δ d 1 Lmd ( λdq ) + δ d 0

T ⎡ L f σ + LDσ ⋅ Lmd λdq + L f σ LDσ ⎤ ⎡ψ d ⎤ ⎢ ⎥ ⎢ ⎥ × ⎢ψ ⎥ − LDσ Lmd λdq ⎢ f⎥ ⎢ ⎥ ⎢ψ ⎥ ⎢ ⎥ ⎣ D⎦ − L f σ Lmd λdq ⎣ ⎦

iq =

(33) in terms of state variables. We have:

( ) = ωq12 ( λdq ) δ q1 Lmq ( λdq ) + δ q 0 Lmq λdq + LQσ

( ) = ωq 23 ( λdq ) (36d) δ q1 Lmq ( λdq ) + δ q 0 Lmq λdq + Lσ

On the other hand, with representations (32) and (33), the stator d-q axis winding currents result out of (13) and (14) just in terms of winding flux linkages:

Related to the set of differential Eqs. (18), we obtain the flux linkages coefficients depending on the q-axis magnetizing inductance Lmq , which is now yielded by

ωq12 = − R

Lmq λdq

,

T ( ) ( ) δ q1 Lmq ( λdq ) + δ q 0

⎡ Lmq λdq + LQσ ⎤ ⎡ψ ⎤ ⎢ ⎥ ×⎢ q⎥ ⎢ −L ⎥ ⎣⎢ψ Q ⎦⎥ mq λdq ⎣ ⎦

(36a)

TABLE I SAMPLES OF RECORDED FLUX LINKAGES COEFFICIENTS DEPENDING ON DIRECT AXIS MAGNETIZING INDUCTANCE

λdq

ωd 11

ωd 13

ωd 14

ωd 21

ωd 23

ωd 24

ωd 31

ωd 33

ωd 34

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

-0.114375 -0.114465 -0.114559 -0.114656 -0.114758 -0.114865 -0.114976 -0.115093 -0.115215 -0.115343 -0.115477 -0.115618 -0.115767 -0.115924 -0.116089 -0.116264 -0.116449 -0.116646 -0.116854 -0.117076 -0.117313 -0.117567 -0.117838 -0.118129 -0.118443 -0.118782

0.060213 0.060182 0.060151 0.060117 0.060082 0.060046 0.060008 0.059968 0.059927 0.059883 0.059837 0.059789 0.059738 0.059685 0.059629 0.059569 0.059506 0.059439 0.059368 0.059292 0.059211 0.059125 0.059032 0.058933 0.058826 0.058710

0.047076 0.047052 0.047027 0.047001 0.046974 0.046945 0.046915 0.046884 0.046852 0.046818 0.046782 0.046744 0.046705 0.046663 0.046619 0.046572 0.046523 0.046470 0.046415 0.046355 0.046292 0.046225 0.046152 0.046075 0.045991 0.045900

0.007527 0.007523 0.007519 0.007515 0.007510 0.007506 0.007501 0.007496 0.007491 0.007485 0.007480 0.007474 0.007467 0.007461 0.007454 0.007446 0.007438 0.007430 0.007421 0.007411 0.007401 0.007391 0.007379 0.007367 0.007353 0.007339

-0.009836 -0.009837 -0.009839 -0.009840 -0.009841 -0.009843 -0.009845 -0.009846 -0.009848 -0.009850 -0.009852 -0.009854 -0.009856 -0.009858 -0.009861 -0.009863 -0.009866 -0.009869 -0.009872 -0.009875 -0.009879 -0.009882 -0.009886 -0.009890 -0.009895 -0.009900

0.002007 0.002006 0.002005 0.002004 0.002003 0.002002 0.002000 0.001999 0.001998 0.001996 0.001995 0.001993 0.001991 0.001989 0.001988 0.001986 0.001984 0.001981 0.001979 0.001976 0.001974 0.001971 0.001968 0.001964 0.001961 0.001957

0.161823 0.161740 0.161655 0.161565 0.161472 0.161374 0.161272 0.161165 0.161053 0.160936 0.160813 0.160683 0.160547 0.160403 0.160252 0.160091 0.159922 0.159742 0.159550 0.159347 0.159129 0.158897 0.158649 0.158381 0.158094 0.157783

0.055195 0.055167 0.055138 0.055107 0.055076 0.055042 0.055007 0.054971 0.054933 0.054893 0.054851 0.054807 0.054760 0.054711 0.054660 0.054605 0.054547 0.054486 0.054420 0.054351 0.054277 0.054198 0.054113 0.054022 0.053923 0.053817

-0.223514 -0.223536 -0.223559 -0.223583 -0.223608 -0.223634 -0.223661 -0.223689 -0.223719 -0.223750 -0.223783 -0.223818 -0.223854 -0.223892 -0.223933 -0.223976 -0.224021 -0.224069 -0.224120 -0.224174 -0.224232 -0.224294 -0.224360 -0.224432 -0.224508 -0.224591



898

Lucian Lupsa-Tataru

TABLE II SAMPLES OF RECORDED FLUX LINKAGES COEFFICIENTS DEPENDING ON QUADRATURE AXIS MAGNETIZING INDUCTANCE

λdq

ωq12

ωq13

ωq 22

ωq 23

λdq

ωq12

ωq13

ωq 22

ωq 23

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

-0.170078 -0.170157 -0.170239 -0.170324 -0.170413 -0.170506 -0.170603 -0.170704 -0.170811 -0.170922 -0.171039 -0.171163 -0.171292

0.161108 0.161003 0.160894 0.160780 0.160662 0.160538 0.160409 0.160273 0.160131 0.159982 0.159826 0.159662 0.159489

0.226558 0.226411 0.226257 0.226098 0.225931 0.225757 0.225575 0.225384 0.225185 0.224975 0.224756 0.224524 0.224281

-0.243378 -0.243574 -0.243778 -0.243991 -0.244213 -0.244446 -0.244688 -0.244942 -0.245208 -0.245488 -0.245781 -0.246089 -0.246413

0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

-0.171429 -0.171573 -0.171726 -0.171888 -0.172059 -0.172241 -0.172435 -0.172641 -0.172862 -0.173099 -0.173353 -0.173627 -0.173923

0.159307 0.159114 0.158911 0.158695 0.158467 0.158224 0.157966 0.157690 0.157396 0.157080 0.156741 0.156376 0.155982

0.224025 0.223754 0.223468 0.223165 0.222844 0.222503 0.222140 0.221752 0.221338 0.220894 0.220417 0.219904 0.219349

-0.246755 -0.247115 -0.247497 -0.247901 -0.248329 -0.248784 -0.249268 -0.249785 -0.250338 -0.250929 -0.251565 -0.252250 -0.252989

Having at hand dependencies (32) and (33), the computer storage of flux linkages coefficients (35), (36) becomes feasible. In our case, having in view linearities (32), (33), plotted in Fig. 2 for the generator adopted in this paper (Appendix), samples of recorded coefficients (35) and (36) are presented in Table I and Table II, respectively.

VI.

of time-related differentiation of magnetizing inductances. Consequently, the enhanced structure, although encompassing saturation-dependent flux linkages coefficients, possesses not only the simplicity of the original structure of winding flux linkage state-space model but also the accuracy of the commonly accepted picture of synchronous generators, delineated by the well-formed formulae of generalized d-q axis mathematical model. Since the differentiation of magnetizing inductances is not a requisite, further developments could employ nonlinearities (23) to construct continuous piecewise representations with the mention that, in this case, linearities (32) and (33) will describe segments within appropriate piecewise linear representations.

Conclusion

The present paper introduces a simple and accurate method of extending the winding flux linkage state-space model of synchronous generators by including the main flux saturation effects with the purpose of implementing within energy control centers. The enhanced structure of winding flux linkage starespace model is linked by deductive reasoning with the generalized mathematical model of saturated synchronous generators. The crucial point in the course of derivation procedure here is represented by the replacement of the d-q axis currents in the expression that defines the synchronous machines magnetizing current (magnetizing current space-phasor modulus) by considering appropriate fluxbased expressions. This symbolic manipulation, distinguishing the present paper from the contributions presented hitherto, allowed us to advance both magnetizing inductances in terms of winding flux linkages only. Implicitly, such kind of representation enables an immediate connection with the synchronous generators winding flux linkage state-space model, well-known for its structural simplicity, suitable for online assessment. The derivation of the d-q axis magnetizing inductances as expressions in terms of d-q axis winding flux linkages i.e. in terms of state variables of the winding flux linkage state-space model has been carried out without altering the sets of equations of the synchronous generators generalized mathematical model. On the other hand, following the selection of winding flux linkages as state variables, the resulted differential equations do not include additional parameters (differential inductances), which could arise as an effect

Appendix: Synchronous Generator Rated Parameters (in per unit) [7] -

Stator winding resistance: R = 0.0256

-

The rotor (field and damper) windings resistances: R f = 0.0032 , RD = 0.088 , RQ = 0.036

-

Stator winding leakage inductance: Lσ = 0.088

-

The rotor windings leakage inductances: L f σ = 0.258 , LDσ = 0.33 , LQσ = 0.066

-

Rated operating point: im, n = 0.9246 , Lmd , n = 1.31 , Lmq, n = 0.705

References [1]


J. Arrillaga, N.R. Watson, Computer Modelling of Electrical Power Systems (John Wiley & Sons, 2001).


899

Lucian Lupsa-Tataru

[2]

[3]

[4]

[5]

[6] [7]

[8] [9]

[10] [11] [12]

[13]

[14]

[15]

D. Barrie, D.S. Hill, and A.H. Yuen, Computer Configuration for Ontario Hydro’s New Energy Management System, IEEE Transactions on Power Systems, Vol. 4(Issue 3):927-934, August 1989. F.A. Rahimi, N.J. Balu, and M.G. Lauby, Assessing Online Transient Stability in Energy Management Systems, IEEE Computer Applications in Power, Vol. 4(Issue 3):44-49, July 1991. M. Nagata, Fast Computation and Assessment Methods in Power System Analysis: Fast Power System Analysis Techniques for Online Dynamic Security Assessment, IEEJ Transactions on Power and Energy B, Vol. 128(Issue 5):705-708, May 2008. K. Radha Rani, J. Amarnath, and S. Kamakshaiah, Transient Stability and Contingency Analysis of Power System in Deregulated Environment, International Review on Modelling and Simulations (IREMOS), Vol. 4(Issue 3):1257-1265, June 2011. M.E. El-Hawary, Electrical Energy Systems (CRC Press, 2000, Ch. 8: “The Energy Control Center”). L. Lupsa-Tataru, An Extension of Flux Linkage State-Space Model of Synchronous Generators with a View to Dynamic Simulation, WSEAS Transactions on Power Systems, Vol. 1(Issue 12):2017-2022, December 2006. A.A. Gorev, Transient Processes of Synchronous Machine (Nauka, 1985). I. Boldea, The Electric Generators Handbook: Synchronous Generators (CRC Press, 2006, Ch. 5: “Synchronous Generators: Modeling for Transients”). I. Boldea, Electric Machine Parameters: Identification, Estimation and Validation (Academia Romana, 1991). P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach (Clarendon, 1992). J. Stepina, Space-Phasor Theory? - What is that for? European Transactions on Electrical Power Engineering, Vol. 5(Issue 6):409-412, November-December 1995. J.E. Brown, K.P. Kovacs, and P. Vas, A Method of Including the Effects of Main Flux Path Saturation in the Generalized Equations of AC Machines, IEEE Transactions on Power Apparatus and Systems, Vol. 102(Issue 1):96-103, January 1983. I. Boldea, S.A. Nasar, A Unified Analysis of Magnetic Saturation in Orthogonal Axis Models of Electric Machines, Electric Machines and Power Systems, Vol. 12(Issue 3):195-204, January 1987. L. Lupsa-Tataru, A Flux-Based Expression of Induction Machine Magnetizing Inductance, IEEE Transactions on Energy Conversion, Vol. 25(Issue 1):268-270, March 2010.

Authors’ information Department of Electrical Engineering and Applied Physics, Faculty of Electrical Engineering and Computer Science, Transilvania University of Brasov, Brasov 500036, Romania. E-mails: [email protected] [email protected] Lucian Lupsa-Tataru was born on March 28, 1969. He received the Diploma degree and the Doctor degree at Transilvania University of Brasov, Romania, in 1993 and 2001, respectively, all in the field of Electrical Engineering. He joined the Electrical Engineering Department of Transilvania University in 1993. His major field of study is algorithm engineering. He is currently involved in research activities in the areas of mathematical modelling of electrical systems, microcomputer programming, and energy conversion.



900


Composite Sliding Mode Control of Induction Motors Using Singular Perturbation Theory A. Mezouar1, T. Terras1, M. K. Fellah2, S. Hadjeri2

Abstract – In this paper, a composite sliding mode control-observer approach is presented for the induction motor drive. This approach, based on the singular perturbation theory, decomposes the original system into separate slow and fast subsystems and permits that separate slow and fast control and observer can be designed for each subsystem and then combined into a composite control and observer for the original system. The controller design uses the sliding mode technique and is divided in two phases: slow control and fast control so that a final composite control is obtained. In addition and assuming that only the fast states are available; a two time scale sliding mode observer design is proposed for which a stability analysis is easily made. The simulations results validate the performance of the proposed approach. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Induction Motor, Singular Perturbation, Sliding Mode Control, Observer, Stability

(VSC) strategy using the sliding mode concept has been widely studied and developed for control and state estimation problems since the works of Utkin [1]-[3]. For induction motor drive, this control technique has many good properties to offer such as insensitivity to parameter variations, external disturbance rejection and fast dynamic response [4]-[7]. Furthermore, sliding mode observers have been used for estimating the states of the control system. Sliding mode observers, also, have the same robust features as the sliding mode control [8]-[12]. In other hand, singular perturbation theory provides the mean to decompose two time scale systems into slow and fast subsystems of lower order described in separate time scales, which greatly simplify their structural analysis and any subsequent control design[13]-[15]. Then, the control (and/or observer) design may be done for each lower order subsystem, and the combined results yield to a composite control (and/or observer) for the original system. So, the idea of combining singular perturbation theory and sliding mode technique constitutes a good possibility to achieve classical control objectives for systems having unmodeled or parasitic dynamics and parametric uncertainties [16]-[20]. Such a structure needs to have information about all state variables of the process. So, it is necessary to estimate the inaccessible states of the process by using a state observer [21]-[24]. Recently, singular perturbation theory has been widely used in observers for sensorless control drives, because it greatly simplifies the observer design [25][27]. This paper is organized as follows: In section II, we briefly review the two time scale approach based on the singular perturbation theory.

Nomenclature I.M. ω , ω* vsd , vsq isd , isq

φsd , φsq

ωs , ωsl

Induction motor Electrical rotor speed and reference rotor speed Stator voltages in the synchronous rotating frame Stator currents in the synchronous rotating frame Stator fluxes in the synchronous rotating frame

τ

Synchronous and slip frequencies Stator and rotor inductances Stator and rotor resistances Stator and rotor time constants Mutual inductance and leakage factor Moment of rotor inertia Coefficient of viscous friction Number of pole pairs Electromagnetic and load torques Slow time scale (real time) Fast time scale τ = ( t − t0 ) / ε

Sc

Sliding mode control surface

Sc,s

Slow sliding mode control surface

Sc, f

Fast sliding mode control surface Sliding mode observer surface Estimated and reference value of x

Ls , Lr Rs , Rr Ts , Tr M ,σ J f p Tem , TL t

S ˆx , x*

I.

Introduction

In the past two decades, the variable structure control


901


A. Mezouar, T. Terras, M. K. Fellah, S. Hadjeri

The general design of a two time scale sliding mode observer is presented in section III. In section IV, the two time scale sliding mode control of induction motors is briefly reviewed. In section V, we present the design of the proposed two time scale sliding mode observer for the induction motors. In that section, a study of stability analysis of this observer is made via singular perturbation method and Lyapunov stability theory. In section VI, and through simulation, the studied observer is associated to the sliding mode composite control of the induction motor where stator fluxes are replaced by those delivered by the observer. Finally, in section VII, we give some comments and conclusions.

II.

II.2.

In the limiting case, as ε → 0 in (2), the asymptotically stable fast transient decays ‘instantaneously’ leaving the reduced order model in the t time scale defined by the quasi steady states xs ( t ) and zs ( t ) :

d xs = f s ( xs ) + g s ( xs ) us , dt

xs ( t0 ) = x0

zs = h ( xs ) = − F2−1 ( xs ) ⎣⎡ f 2 ( xs ) + g 2 ( xs ) u s ⎦⎤

(3)

(4)

where xs , zs and us denote the slow components of the original variables x , z and u , respectively and:

Two Time Scale Approach Review

The two time scale approach, based on the singular perturbation theory, can be applied to systems where the state variables can be split into two sets, one having “fast” dynamics, the other having “slow” dynamics. The difference between the two sets of dynamics can be distinguished by the use of a small multiplying scalar ε . Generally, the scalar parameter ε is the speed ratio of the slow versus fast phenomena. If the slow states are expressed in the t time scale, then, the fast ones will be in the τ time scale defined by:

τ = ( t − t0 ) / ε

Slow Reduced System

f s ( xs ) = f1 ( xs ) − F1 ( xs ) F2−1 ( xs ) f 2 ( xs ) g s ( xs ) = g1 ( xs ) − F1 ( xs ) F2−1 ( xs ) g 2 ( xs )

The slow invariant manifold can be defined as:

{

}

M ε = z ∈ Dz ⊂ ℜm : z = h ( xs )

where Dz is closed, bounded and centered in z = 0 , and the so-called manifold condition:

(1)

ε

where t0 is the initial time. The reader is referred to [13], [14] and [15] for the general theory on singular perturbation.

∂h ⎡ f1 ( x ) + F1 ( x ) z + g1 ( x ) u ⎤⎦ = ∂x ⎣ = f 2 ( x ) + F2 ( x ) z + g 2 ( x ) u

must be satisfied for M ε to an invariant manifold [15]. II.1.

Singularly Perturbed Systems

In this paper, we consider the following class of nonlinear singularly perturbed systems described by the so-called standard singularly perturbed form: d x = f1 ( x ) + F1 ( x ) z + g1 ( x ) u, x ( t0 ) = x0 dt d ε z = f 2 ( x ) + F2 ( x ) z + g 2 ( x ) u, z ( t0 ) = z0 dt

II.3.

The fast dynamic (also known as boundary layer system) is obtained by transforming the slow time scale t to the fast time scale τ = ( t − t0 ) / ε . We rewrite (2) in the fast time scale τ and introducing the derivation of z from M ε , i.e., z f = z − h ( xs ) , so:

(2)

dx = ε ⎡⎣ f1 ( x ) + F1 ( x ) ⎡⎣ z f + zs ⎤⎦ + g1 ( x ) u ⎤⎦ dτ

where x ∈ ℜn is the slow state, z ∈ ℜm is the fast state, u ∈ ℜ p is the control input, ε is a small positive parameter such that ε ∈ [ 0 , 1] . The matrices f1 , f 2 ,

dz f dτ

F1 ,F2 ,g1 and g 2 are assumed to be bounded with their components and analytic real vector fields. It is also assumed that the matrix F2 is nonsingular for all x .

(5)

= F2 ( x ) z f + g 2 ( x ) ( u − us ) + f 2 ( x ) + + F2 ( x ) h + g 2 ( x ) us −

An additional assumption is that f1 ( 0 ) = f 2 ( 0 ) = 0

and, for u = 0 , the origin

Fast Reduced System

∂h dx ∂x dτ

(6)

where z f ( 0 ) = z0 − h ( x0 ) , x ( 0 ) = x0 and u f = u − us is

( x,z ) = ( 0,0 ) is an isolated

the fast control, and again examine the limit as ε → 0 .

equilibrium state. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


902


Then dx / dτ = 0 , that is x = constant in the fast time scale. So, the only fast variations are the variation of z from its quasi steady state zs . Making ε = 0 in (5) and

where Γ s = sign ( S ( y, ˆy ) ) is the switching function.

(6), we obtain an O ( ε ) approximation of the fast

(m × m)

subsystem:

The observer sliding surface S can be chosen as a linear function of ( y − ˆy ) as given in [11] and [27], so:

dz f dτ

Gx and Gz are the observer gains with ( n × m ) and

dimensions respectively, to be determined.

= F2 ( x ) z f + g 2 ( x ) u f , z f ( 0 ) = z0 − h ( x0 ) (7)

II.4.

S ( y, ˆy ) = Λ ( y − ˆy )

Composite Control

where and Λ

The fast and slow control laws can be combined into a composite control structure: u ( x,z ) = us ( x ) + u f ( z − h ( x ) )

(11)

( y − ˆy )T = ⎡⎣( y1 − ˆy1 ) ( y2 − ˆy2 ) ... ( ym − ˆym )⎤⎦ is ( n × m ) gain matrix to be specified.

The error dynamics is calculated by subtracting (10) from (9):

(8)

where us and u f denotes the slow and fast components of the control law, respectively.

⎧⎪ex = f x ( x,z,u,ε ) − f x ( ˆx,z,u,ε ) − Gx Γ s ⎨ ⎪⎩ε ez = f z ( x,z,u,ε ) − f z ( ˆx,z,u,ε ) − Gz Γ s

(12)

⎧ex = ∆f x − Gx Γ s ⎨ ⎩ε ez = ∆f z − Gz Γ s

(13)

or:

III. Two Time Scale Sliding Mode Observer Now, consider the above continuous nonlinear singularly perturbed system of (2) which can be expressed as follows: ⎧⎪ x = f x ( x,z,u,ε ) ⎨ ⎪⎩ε z = f z ( x,z,u,ε )

where: ex = x − ˆx , ez = z − ˆz ∆f x = f x ( x,z,u,ε ) − f x ( ˆx,z,u,ε )

(9)

∆f z = f z ( x,z,u,ε ) − f z ( ˆx,z,u,ε )

where f x and f z are assumed to be bounded and analytic real vector fields, and consider a vector of measurement that is linearly related to the fast state vector as:

Since (13) is a singularly perturbed system, the observer design can be based on sequential application of resulted subsystems of (13) by applying singular perturbation methodology.

y = z, y ∈ ℜ m

III.2. Stability Analysis in the Fast Time Scale

It is also assumed that the above system is controllable and observable [23]. Consequently, the observer design may be considered for the state observation of the slow variables from the measurement of the fast variables.

For fast error dynamic subsystem, the associated time scale is defined by τ = ( t − t0 ) / ε , then (13) can be transformed into: ⎧ dex ⎪⎪ dτ = ε ( ∆f x − Gx Γ s ) ⎨ ⎪ dez = ∆f − G Γ z z s ⎪⎩ dτ

III.1. Sliding Mode Observer Design

By structure, observer based on sliding mode technique is very similar to the standard full order observer with replacement of the linear corrective terms by a discontinuous function [11], [23] and [27]. The corresponding sliding mode observer for the system of (9) can be written as a replica of the system with an additional nonlinear auxiliary input term as follows: ⎧⎪ ˆx = f x ( ˆx,z,u,ε ) + Gx Γ s ⎨ ⎪⎩ε ˆz = f z ( ˆx,z,u,ε ) + Gz Γ s

(14)

Setting ε = 0 in (14), it yields: dez = ∆f z − G z Γ s dτ

(15)

In this time scale, the stability analysis consists of determining Gz so that in this time scale (τ ) , the surface

(10)



903


S (τ ) = 0 is attractive.

to [16]-[20] for more information on the singularly perturbed sliding mode control design.

It can be shown that when sliding mode occurs on S (τ ) , the equivalent value of the discontinuous observer

IV.1. Induction Motor Model

maxillary input is found by solving the Eq. (15) for Gz Γ s after insuring zero for dez dτ : Gz Γ s = ∆f z

The classical state space model of the induction motor expressed in the ( d ,q ) axis rotating reference frame

( isd ,isq ,φsd ,φsq ,ω ) as state variables ( vsd ,vsq ,ωsl ) as control variables is [28]:

(16)

with

and the equivalent switching vector is obtained as: Γ s = Gz−1∆f z

(17)

⎧ ⎛1 1⎞ d 1 ⎪σ Ls isd = − Ls ⎜ + ⎟ isd + φsd + ωφsq + dt T T T r ⎠ r ⎝ s ⎪ ⎪ + σ Ls isqωsl + vsd ⎪ ⎪ ⎛1 1⎞ d 1 ⎪σ Ls isq = − Ls ⎜ + ⎟ isq − ωφsd + φsq + dt Tr ⎪ ⎝ Ts Tr ⎠ ⎪ − σ Ls isd ωsl + vsq (20) ⎨ ⎪ ⎪ d φ = − R i + ωφ + φ ω + v s sd sq sq sl sd ⎪ dt sd ⎪d ⎪ φsq = − Rs isq − ωφsd − φsd ωsl + vsq ⎪ dt ⎪ dω p f = (Tem − TL ) − ω ⎪ J J ⎩ dt

III.3. Stability Analysis in the Slow Time Scale

Slow error dynamic subsystem can be found by making ε = 0 in (16), so: dex = ∆f x − G x Γ s dt

(18)

0 = ∆f z − Gz Γ s

(19)

and

From (19), the equivalent switching vector can be found: Γ s = Gz−1∆f z

where

ωsl

is

the

slip

frequency

ωsl = ωs − ω ,

Ts = Ls / Rs , Tr = Lr / Rr and σ = 1 − M / ( Ls Lr ) . 2

Therefore, by appropriate choice of Gx , the desired rate of convergence ex → 0 can be obtained.

IV.

The electromagnetic torque expressed in terms of the state variables is:

(

Tem = p φsd isq − φsq isd

Two Time Scale Sliding Mode Control of induction motors

The design of the classical sliding mode control consists generally of two stages: design of the switching surface Sc and design of the sliding mode controller [2], [3]. The control law used is of the type:

)

(21)

IV.2. Singularly Perturbed Induction Motor Model

Based on the well known of the induction machine model dynamics [21], [25]-[27], the slow variables are

(ω ,φsd ,φsq ) and the fast variables are ( isd ,isq ) .

u = ueq + u N

Therefore, the corresponding standard singularly perturbed

where ueq is the equivalent control which acts when the

(

form

z = Ls isd ,Ls isq

system is restricted to Sc = 0 , while u N is the discontinuous part of the control u acting when Sc ≠ 0 . The sliding mode control should be chosen such that the candidate Lyapunov function satisfies the Lyapunov stability criteria. The two time scale sliding mode control for the system (2) is designed in two steps [14]-[17]. First, the sliding mode controllers for each reduced subsystem are designed separately. Then, they are combined to obtain a composite control for the complete system. In this work, the composite control with sliding mode is considered as a case study. The reader is referred

)

T

ε =σ ,

with

(

and u = vsd ,vsq ,ωsl

(

x = ω ,φsd ,φsq

)

T

)

T

,

is:

f p ⎧ ⎪ x1 = k ( x2 z2 − x3 z1 ) − J x1 − J TL ⎪ ⎪ x2 = −α z1 + x1 x3 + x3u3 + u1 ⎪ (22) ⎨ x3 = −α z2 − x1 x2 − x2 u3 + u2 ⎪ε z = − α + β z + β x + x x + ε z u + u ( )1 2 1 3 2 3 1 ⎪ 1 ⎪ε z = − (α + β ) z + β x − x x − ε z u + u 2 3 1 2 1 3 2 ⎪ 2 ⎩

with:



904


α = Rs / Ls ; β = Rr / Lr ; k = p 2 / ( JLs )

subsystem. For the design of the slow control, we use again the sliding mode concept. For this control, another sliding mode control surface must be proposed. We have used the sliding surface given by:

IV.3. Fast Reduced Subsystem

⎡ S1s ⎤ ⎡ k1s ( x1s − x1d ) ⎤ ⎥ ⎢ ⎥ ⎢ Sc,s ( xx ) = ⎢ S2 s ⎥ = ⎢ k2 s ( x2 s − x2 d ) ⎥ ⎢⎣ S3s ⎥⎦ ⎢ k ( x − x ) ⎥ ⎣ 3 s 3 s 3d ⎦

Following the methodology presented in Section II, the O ( ε ) approximation of the exact fast subsystem is given by: d dτ

⎡ z1 f ⎢ ⎢⎣ z2 f

⎤ ⎡α + β ⎥ = −⎢ ⎣ 0 ⎥⎦

0 ⎤ ⎡ z1 f ⎢ α + β ⎥⎦ ⎢⎣ z2 f

⎤ ⎡ u1 f ⎥+⎢ ⎥⎦ ⎢⎣u2 f

⎤ ⎥ ⎥⎦

(23)

where x1d ,x2 d and x3d are the reference angular speed, and d-q reference fluxes. k1s , k2s and k3s are positive constants that allow to ensure proper stability performances of the closed loop system. For both fast and slow control laws, the same demonstration used in [21] can be applied to carry out its stability analysis. Finally, the composite control can be synthesized as: (30) u = u f + us

or:

d z f = − Af z f + u f dτ

(24)

IV.4. Slow Reduced Subsystem For the slow reduced subsystem, we can obtain:

(

or:

)

p ⎤ ⎡ f − x − λ x22s + x32s − TL ⎥ ⎡ x1s ⎤ ⎢ J 1s J ⎥ ⎢ ⎥ ⎢ = − δ α x x x x ( ) ⎢ ⎥+ 1s 3 s 2s ⎢ 2s ⎥ ⎥ ⎢⎣ x3s ⎥⎦ ⎢⎣ −δ ( x1s x2 s + α x3s ) ⎦ 0 ⎤ ⎡u1s ⎤ ⎡ −λ x3s λ x2 s ⎢ ⎥ + ⎢⎢ δ 0 x3s ⎥⎥ ⎢u2 s ⎥ − x2 s ⎦⎥ ⎢⎣u3s ⎥⎦ δ ⎣⎢ 0

(25)

xs = f s ( xs ) + g s ( xs ) us

(26)

V.

β k , λ= α +β α +β

and: ⎡ z1s ⎤ 1 ⎧⎪ ⎡ β ⎨⎢ ⎢ ⎥= z ⎣ 2 s ⎦ α + β ⎪⎩ ⎣ − x1s

x1s ⎤ ⎡ x2 s ⎤ ⎡ u1s ⎤ ⎪⎫ + ⎬ β ⎥⎦ ⎢⎣ x3s ⎥⎦ ⎢⎣u2 s ⎥⎦ ⎭⎪

V.1.

The composite sliding mode control for the induction motor is made on the basis of the decomposed subsystems (23) and (25). Following the procedure described in [14] and [16], the fast and slow control laws can be easily formulated. For the design of the fast control u f , the proposed sliding mode control surface

f p ⎧ ⎪ ˆx1 = k ( ˆx2 z2 − ˆx3 z1 ) − J x1 − J TL + Gx1Γ s + ⎪ ⎪ + qx1 ( x1 − ˆx1 ) ⎪ ⎪ ˆx2 = −α z1 + x1 ˆx3 + ˆx3u3 + u1 + Gx 2 Γ s ⎪ ˆx = −α z − x ˆx − ˆx u + u + G Γ 2 1 2 2 3 2 x3 s ⎨ 3 ⎪ε ˆz = − (α + β ) z + β ˆx + x ˆx + ε z u + 1 2 1 3 2 3 ⎪ 1 ⎪ + u1 + Gz1Γ s ⎪ ⎪ε ˆz2 = − (α + β ) z2 + β ˆx3 − x1 ˆx2 − ε z1u3 + ⎪ + u 2 + Gz 2 Γ s ⎩

was:

( )

Sc, f z f

⎤ ⎡ k1 f z1 f ⎥=⎢ ⎥⎦ ⎢⎣ k2 f z2 f

⎤ ⎥ ⎥⎦

Singularly Perturbed Observer

With reference to the above singularly perturbed induction motor model of (22), and considering the measured stator currents as the system outputs, the corresponding sliding mode observer can be constructed as follows:

(27)

IV.5. Composite Control

⎡ S1 f =⎢ ⎢⎣ S2 f

Two Time Scale Sliding Mode Observer Design

For induction motor, rotor speed and stator currents are easily measured but stator fluxes are rather difficult to measure. In fact, different observer structures have been proposed to estimate those fluxes from rotor speed and state currents [21] and [27]. In this paper, we use a sequential methodology for designing a sliding mode observer for induction motor drive using singular perturbation theory. The two time scale decomposition of the original system of the observer error dynamics into separate slow and fast subsystems permits a simple design and sequential determination of the observer gains.

with:

δ=

(29)

(28)

where k1 f and k2 f are positive constants chosen in

(31)

order to assure proper stability proprieties of the fast Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


905


where ˆxi and ˆz j are the estimation of xi and z j for

System of Eqs. (35) gives:

i ∈ {1, 2,3} and j ∈ {1, 2} . Gx1 , Gx 2 , Gx 3 , Gz1 , Gz 2

⎧ dex1 ⎪ ⎪ dτ ⎪ dex2 ⎪ ⎪ dτ ⎪⎪ dex3 ⎨ ⎪ dτ ⎪ dez1 ⎪ ⎪ dτ ⎪ dez2 ⎪ ⎪⎩ dτ

and qx1 are the observer gains. The switching vector Γ s is: ⎡ sign ( s1 ) ⎤ Γs = ⎢ (32) ⎥ ⎢⎣ sign ( s2 ) ⎥⎦ with: ⎡ s1 ⎤ ⎡ z1 − ˆz1 ⎤ S = ⎢ ⎥ = Λ −1 ⎢ (33) ⎥ ⎣ s2 ⎦ ⎣ z2 − ˆz2 ⎦ and: ⎡ β Λ=⎢ ⎣ − x1

x1 ⎤ β ⎥⎦

(34)

i ∈ {1, 2 ,3} and j ∈ {1, 2} , the estimation error dynamics

are:

(

=ε

( ( x1 + u3 ) ex

2

)

( ez ,ez ) 1

2

(

2

(35)

)

)

(36)

= β ex2 + x1ex3 − Gz1Γ s = β ex3 − x1ex2 − Gz 2 Γ s

⎧ dez1 = β ex2 + x1ex3 − Gz1Γ s ⎪ ⎪ dτ ⎨ ⎪ dez2 = β e − x e − G Γ x3 z2 s 1 x2 ⎪⎩ dτ

(37)

⎧ ⎡ ex ⎤ ⎫ d [ez ] = Λ ⎪⎨ ⎢ e 2 ⎥ − Λ −1 [Gz ] Γ s ⎪⎬ dτ ⎩⎪ ⎣⎢ x3 ⎦⎥ ⎭⎪

(38)

By appropriate choice of the observer gain terms Gz1 and Gz 2 , sliding mode occurs in (37) along the manifold S = ez = 0 . Proposition (1): Assume that ex2 and ex3 are bounded in this time, and consider system (37) with the following observer gain matrices: ⎡ Gz1 ⎤ ⎢G ⎥ = ΛΦ ⎣ z2 ⎦

are fast variables and

3

where:

of the above system consists of determining Gz1 and Gz 2 to ensure the attractiveness of the sliding surface S = 0 in the fast time scale. Thereafter Gx1 , Gx 2 and G x 3 are determined, such that the reduced order system obtained when S = S = 0

(39)

0⎤ ⎡ϕ Φ=⎢ 1 ⎥ and ϕ1 , ϕ 2 > 0 ⎣ 0 ϕ2 ⎦

The attractivity condition of sliding surface S (τ ) = 0 is given by:

is locally stable. V.2.

)

or:

( ex ,ex ,ex ) are slow variables. So, the stability analysis 1

− Gx 2 Γ s

= ε − ( x1 + u3 ) ex3 − Gx 3Γ s

Exploiting the time properties of multi time scales systems of (35),

)

Making ε = 0 in the above system, it yields:

The choice of the discontinuous function Γ s is made to get a simple observer gain synthesis as we will see after. Setting exi = xi − ˆxi and ez j = z j − ˆz j for

⎧ dex1 = k z2 ex2 − z1ex3 − Gx1Γ s − qx1ex1 ⎪ ⎪ dt ⎪ dex2 = ( x1 + u3 ) ex2 − Gx 2 Γ s ⎪ ⎪ dt ⎪⎪ dex3 = − ( x1 + u3 ) ex3 − Gx 3Γ s ⎨ ⎪ dt ⎪ dez1 = β ex2 + x1ex3 − Gz1Γ s ⎪ε ⎪ dt ⎪ dez2 = β ex3 − x1ex2 − Gz 2 Γ s ⎪ε ⎪⎩ dt

((

= ε k z2 ex2 − z1ex3 − Gx1Γ s − qx1ex1

⎛ dS ⎞ ST ⎜ ⎟ < 0 ⎝ dτ ⎠

Fast Reduced Order Error Dynamics

From singular perturbation theory, the fast reduced order system of the observation errors can be obtained by introducing the fast time scale τ :

In this time scale

dxi = 0 for i = 1, 2 ,3 so: dτ

⎛ dS ⎞ ST ⎜ ⎟ = ST ⎝ dτ ⎠

τ = ( t − t0 ) / ε


(40)

⎧⎪ ⎡ex2 ⎤ ⎫⎪ ⎨ ⎢ ⎥ − ΦΓ s ⎬ e ⎩⎪ ⎢⎣ x3 ⎥⎦ ⎭⎪

(41)


906


System (44) can be written as the following system:

Thus, (40) is verified with the set defined by the following inequalities: ⎧ϕ1 > ex 2 max ⎪ ⎨ ⎪ϕ2 > ex3 max ⎩

⎡ ex ⎤ ⎡ −q ⎢ 1⎥ ⎢ 1 ⎢ ex2 ⎥ = ⎢ 0 ⎢ ⎥ ⎢ 0 ⎢⎣ ex3 ⎥⎦ ⎣

(42)

( Gz Γ s )eq ,

which can be

VI.

calculated from the subsystem (37) assuming ez = 0 , e = 0 . If S (τ ) = 0 the equivalent switching vector Γ z

is obtained as:

V.3.

(43)

Slow Reduced Order Error Dynamics

For slow error dynamics, we use the system (36) and setting ε = 0 . So, we can write: ⎡ ex ⎤ ⎡ − q −kz1 ⎤ ⎡ ex1 ⎤ kz2 ⎢ 1⎥ ⎢ 1 ⎥⎢ ⎥ 0 ⎢ ex2 ⎥ = ⎢ 0 ( x1 + u3 ) ⎥ ⎢ ex2 ⎥ + ⎢ ⎥ ⎢ 0 − ( x1 + u3 ) ⎥⎦ ⎢⎢ ex ⎥⎥ 0 ⎢⎣ ex3 ⎥⎦ ⎣ ⎣ 3⎦ G ⎡ x1 ⎤ − ⎢⎢Gx 2 ⎥⎥ Γ s ⎢⎣ Gx3 ⎥⎦

k2 f = k1s = k2s = k3s =1.

Following the design considerations of section (V), ϕ1 = ϕ2 = 500 and the observer gains are q1 = q2 = q3 = 10 . The problem of chattering is remedied by replacing the switching function by a continuous one in the sliding surface neighborhood. Fig. 1 shows the rotor speed response of the motor; a very good speed regulation is obtained. In Fig. 2 is shown the slip frequency. The corresponding composite controls are shown in Fig. 3. Fig. 4 shows the stator fluxes responses; after a short initial time, they converge to their desired values. Stator currents are shown in Fig. 5. These results show that the composite sliding mode control with the proposed observer can track the reference command accurately and quickly.

(44)

with: ⎡ β ⎢− x ⎣ 1

x1 ⎤ ⎡ ex2 ⎤ ⎡Gz1 ⎤ Γ =0 ⎢ ⎥− β ⎥⎦ ⎣⎢ ex3 ⎦⎥ ⎢⎣Gz 2 ⎥⎦ s

(45)

ω (rad/s)

From Eq. (45), we can get the equivalent switching vector Γ as:

300.00

s

⎡ ex2 ⎤ Γ s = Φ −1 ⎢ ⎥ ⎢⎣ ex3 ⎥⎦

Simulations Results

The performances of the proposed control observer scheme for the induction motor model in closed loop system developed in the previous sections were studied through simulations. Some simulations were carried out when the motor is started without torque load. The controller should smoothly regulate the angular speed at 300 rad/s, keep the stator d-component flux φsd at its rated value 1.0 Wb and align the stator flux with the d-axis (i.e. constrain φqs to 0). The control law gains were chosen as k1 f =

s

⎡ ex2 ⎤ Γ s = Φ −1 ⎢ ⎥ ⎣⎢ ex3 ⎦⎥

(49)

which is stable for q1 ,q2 ,q3 > 0 .

According to the equivalent control method, the system in sliding mode behaves as if Gz Γ s is replaced by its equivalent values

0 ⎤ ⎡ ex1 ⎤ ⎢ ⎥ 0 ⎥⎥ ⎢ ex2 ⎥ − q3 ⎥⎦ ⎢⎢ ex ⎥⎥ ⎣ 3⎦

0 − q2 0

(46) 200.00

In this time scale and according to the equivalent control method, we can replace Γ s by Γ s , and with: Gx1 = k [ z2 − z1 ] Φ

100.00

(47)

and: q1 ⎡ Gx 2 ⎤ ⎡ ⎢G ⎥ = ⎢− x + u ⎣ x3 ⎦ ⎣ ( 1 3 )

( x1 + u3 )⎤ q2

⎥Φ ⎦

0.00 0.0

(48)

t(s)

0.2

0.4

0.6

0.8

1.0

Fig. 1. Angular speed



907


ω sl (rad/s) 500.00 400.00 300.00 200.00 100.00 0.00 0.0

t(s)

0.2

0.4

0.6

0.8

1.0

Fig. 2. Composite slip frequency input

v sd (V)

v sq (V)

50.00

350.00

0.00

300.00

-50.00

250.00

-100.00

200.00

-150.00

150.00

-200.00 0.0

0.2

0.4

0.6

0.8

1.0

100.00 0.0

t(s)

0.2

0.4

0.6

0.8

1.0

0.8

1.0

t(s)

Fig. 3. Composite control voltage inputs

φsq (Wb) 0.80

φ sd (Wb) 1.20

0.60

0.80

0.40

0.40 0.20

0.00

-0.40 0.0

0.00

0.2

0.4

0.6

0.8

1.0

-0.20 0.0

t(s)

t(s)

0.2

0.4

0.6

Fig. 4. d-q stator fluxes



908


isd (A)

isq (A)

8.00

16.00

4.00

12.00

0.00

8.00

-4.00

4.00

-8.00 0.0

0.00 0.0

t(s)

0.2

0.4

0.6

0.8

1.0

t(s)

0.2

0.4

0.6

0.8

1.0

Fig. 5. d-q stator currents

VII.

[2]

Conclusion

[3]

In this paper, we have shown singular perturbation theory to be an effective tool in the analysis of induction motors control-observer problems. Using the assumption of separate time scales, a full order observer has been easily designed in order to estimate the slow variables (stator fluxes) under the assumption that only the fast variables (stator currents) and rotor speed are available for measurement. It has been shown by the simulation results that this controller-observer scheme is may be useful in controlling induction motors with rotor speed and motor fluxes in order to obtain high dynamic performance. Sensitivity of the control-observer structure to torque disturbances and uncertainties in the electrical and mechanical parameters are under investigation.

[4]

[5]

[6]

[7]

[8]

Appendix [9] TABLE A1 INDUCTION MOTOR NOMINAL PARAMETERS

1.5 kW

220/380 V

3.68 / 6.31 A

N = 1420 rpm

Rs = 4.85 Ω

Rr = 4.805 Ω

M = 0.258

Ls = 0.274 H

p=2

J = 0.031 kg ⋅ m

[10]

Lr = 0.274 H 2

[11]

f = 0.00114 N×m×s/rd

[12]

Acknowledgments The authors would like to acknowledge the financial support of the Algeria's Ministry of Higher Education and Scientific Research, under CNEPRU project: J02036 2010/0005.

[13] [14]

References [1]

[15]

V.I. Utkin, Sliding Mode in Control and Optimization (SpringerVerlag, Berlin, 1992).


C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications (Taylor & Francis, 1998). W. Perruquetti, J.P. Barbot, Sliding Mode Control in Engineering (Taylor & Francis, 2002). B. Castillo-Toledo, S. Di-Gennaro, A.G. Loukianov, J. Rivera, Discrete Time Sliding Mode Control with Application to Induction Motors, Automatica, Vol. 44 (Issue 12): 3036-3045, December 2008. K.B. Mohanty, Sliding Mode Controllers for Sensorless Induction Motor Drive With Variable Inertia Load, International Review of Automatic Control (IREACO), Vol. 1 (Issue 4): 466-473, November 2008. B. Veselic, B. Perunicic-Drazenovic, C. Milosavljevic, HighPerformance Position Control of Induction Motor using DiscreteTime Sliding-Mode Control, IEEE Transactions on Industrial Electronics, Vol. 55 , (Issue 11): 3809 – 3817, November 2008. M.M. Krishan, Fuzzy Sliding Mode Control with MRAC Technique Applied to an Induction Motor Drives, International Review of Automatic Control (IREACO), Vol. 1 (Issue 1): 42-48, May 2008. N. Inanc, A Robust Sliding Mode Flux and Speed Observer for Speed Sensorless Control of an Indirect Field Oriented Induction Motor Drives, Electric Power Systems Research, Vol. 77 ( Issue 12): 1681-1688, October 2007. N. Inanc, A New Sliding Mode Flux and Current Observer for Direct Field Oriented Induction Motor Drives. Electric Power Systems Research, Vol. 63 (Issue 2): 113–118, September 2002. S. J. Salehi, M. Manoochehri, A Nonlinear Sliding Mode Control of Induction Motor Based on Second Order Speed and Flux Sliding Mode Observer, International Review on Modelling and Simulations (IREMOS), Vol. 4 (Issue 3): 1057-1065, June 2011. A. Benchaib, A. Rachid, E. Audrezet, M. Tadjine, Real-Time Sliding-Mode Observer and Control of an Induction Motor, IEEE Transaction on Industrial Electronics, Vol. 46 (Issue 1): 128-138, February 1999. M. Jafarifar, R. Kianinezhad, S.Gh. Seifossadat, S.S. Mortazavi, Sliding Mode and Disturbance Observer: Two Viable Schemes for Sensorless Control of Induction Machines, International Review of Automatic Control (IREACO), Vol. 3 (Issue 1): 75-82, January 2010. E.M. De Jager, J. Furu, The Theory of Singular Perturbations (North Holland, 1996). D.S. Naidu, Singular perturbations and time scales in control theory and applications: An overview, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, Vol. 9 (Issue 2): 233-278, 2002. R.S. Johnson, Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering


909


(Springer, 2005). [16] B.S. Heck, Sliding Mode Control for Singularly Perturbed Systems, International Journal of Control, Vol. 53(Issue 4): 9851001, 1991. [17] M. Innocenti, L. Greco, L. Pollini, Sliding Mode Control for TwoTime Scale Systems: Stability Issues, Automatica, Vol. 39 (Issue 2): 273-280, February 2003. [18] J. Wang, K.M. Tsang, Second-Order Sliding Mode Controllers for Nonlinear Singular Perturbation Systems, International Society of Automation Transactions, Vol. 44 (Issue 1): 117-129, January 2005. [19] T.-H.S. Li, Kuo-Jung Lin, Composite Fuzzy Control of Nonlinear Singularly Perturbed Systems, IEEE Transactions on Fuzzy Systems, Vol. 15 (Issue 2): 176 – 187, April 2007. [20] X. Han, E. Fridman, S.K. Spurgeon, Sliding Mode Control in the Presence of Input Delay: A Singular Perturbation Approach, Automatica, Vol. 48 (Issue 8): 1904-1912, August 2012. [21] J. De-Leon, J. Alvares, R. Castro, Sliding Mode Control and State Estimation for Non-Linear Singularly Perturbed Systems: Application to An Induction Electric Machine, 34th IEEE Conference on Control and Applications, pp. 998-1003, New York, USA, September 1995. [22] J. De-Leon, R. Castro, J. Alvares, Two-Time Sliding Mode Control and State Estimation for Non-Linear Systems, 13th Triennial World Congress, IFAC, pp.265-270, San Francisco, USA, July 1996. [23] R. Castro-Linares, J. Alvarez-Gallegos, V. Vásquez-López, Sliding Mode Control and State Estimation for a Class of Nonlinear Singularly Perturbed Systems, Dynamics and Control, Vol. 11 (Issue 1) 25-46, January 2001. [24] J. Alvarez-Gallegos, G Silva-Navarro, Two time scale sliding mode control for a class of nonlinear systems, International Journal of Robust and Nonlinear Control, Vol. 7 (Issue 9): 865 – 879, September 1997. [25] H. Hofmann, S. R. Sanders, Speed Sensorless Vector Control of Induction Machines using Two-Time-Scale Approach, IEEE Transaction on Industrial Application, Vol. 34 (Issue 1): 169-177, January/February 1998. [26] T. Song, M.F. Rahman, K.W. Lim, M.A. Rahman, A Singular Perturbation Approach to Sensorless Control of a Permanent Magnet Synchronous Motor Drive, IEEE Transactions on Energy Conversion, Vol. 14 (Issue 4): 1359 – 1365, December 1999. [27] A. Mezouar, M.K. Fellah, S. Hadjeri, Adaptive Sliding Mode Observer For Induction Motor Using Two-Time-Scale Approach, Electric Power Systems Research, Vol. 77 (Issue 5-6): 604-618, April 2007. [28] W. Leonhard, Control of Electrical Drive. (Springer-Verlag, Berlin, 1996).

Tahar Terras was born in Saida, Algeria, in 1966. He received the B.S. degree in Electrical Engineering from Djillali Liabes University, Sidi Bel Abbès, Algeria, in 1991 and the M.S. degree from High National Schools of Technical Studies in 2002. He is currently a member of the Laboratory of Electrical Engineering, Taher Moulay University, Saida, Algeria, where he prepare his Ph.D. degree. His research interests include electrical machines and drives, observers and sensorless methods. Mohammed-Karim Fellah was born in Oran, Algeria, in 1963. He received the Eng. degree in Electrical Engineering from University of Sciences and Technology, Oran, Algeria, in 1986, and The Ph.D degree from National Polytechnic Institute of Lorraine, Nancy, France, in 1991. Since 1992, he is Professor at the Department of Electrical Engineering, University of Sidi-Bel-Abbes, Algeria. His current research interests include Power Electronics, HVDC links, and Electrical Drives. Samir Hadjeri was born in 1961. He received the Eng. degree in Electrical Engineering from University of Sciences and Technology, Oran, Algeria, in 1986, and the Master's degrees in Electrical Engineering from the University of Laval, Quebec, Canada, in 1990. The PhD degrees from the University of Sidi Bel-Abbes, Algeria, in 2003. He is currently a professor at the Department of Electrical Engineering, Faculty of Science Engineering, Djillali Liabès University, Sidi-Bel-Abbes, Algeria. His research there focused on power system analysis, HVDC links and FACTS.


Laboratory of Electrical Engineering, Taher Moulay University, (20 000) Saida, Algeria.

2

Intelligent Control and Electrical Power Systems Laboratory, Djillali Liabès University, (22 000) Sidi Bel Abbès, Algeria.

Abdelkader Mezouar was born in Mascara, Algeria, in 1974. He received the B.S. and M.S. degrees in Electrical Engineering from National Polytechnic School, Algiers, Algeria, in 1997 and 1999 respectively. Subsequently, he received the Ph.D. degree and the university habilitation from Djillali Liabes University, Sidi Bel Abbès, Algeria, in 2006 and 2008 respectively. He is currently a member of the Laboratory of Electrical Engineering, Taher Moulay University, Saida, Algeria. His research interests include electrical machines and drives, sliding mode control, observer techniques and renewable energy.



910


Robust Control Design for a Semi-Batch Reactor František Gazdoš

Abstract – The paper presents control system design for an exothermic semi-batch reactor used for tanning waste recovery. It demonstrates that this highly nonlinear system can be successfully controlled by a relatively simple fixed controller designed robustly. For this purpose the systematic algebraic approach with some useful tools from the robust control theory are fruitfully exploited. The first part presents a simplified mathematical model of the process together with all its parameters and limits. Further in the paper, a complete procedure of control system design including system identification, controller design, robustness analysis and simulation verification are described in detail. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Algebraic Approach, Modelling, Robust Control, Semi-Batch Reactor, Simulation

Nomenclature

SISO

Reactor variables Value [Unit] Variable Meaning Pre-exponential factor 219.6 [s-1] A Chromium sludge mass [-] aFK concentration Chromium sludge specific 4400 [J·kg-1·K-1] cFK heat capacity Specific heat capacity of 4500 [J·kg-1·K-1] cR the reactor content Coolant specific heat 4118 [J·kg-1·K-1] cv capacity Activation energy 29968 [J·mol-1] E Reaction heat 1392350 [J·kg-1] ∆H r Reaction rate constant [s-1] k Conduction coefficient 200 [J·m-2· K-1·s-1] K m Total mass in the reactor [kg] Initial filling 1810 [kg] mP Coolant mass flow 1 [kg·s-1] m v Chromium sludge mass 0÷3 [kg·s-1] m FK flow Coolant mass 220 [kg] mvR Gas constant 8.314 [J·mol-1·K-1] R S Heat transfer surface 7.36 [m2] Temperature in the reactor [K] T Coolant temperature [K] Tv Chromium sludge 293.15[K]=20[°C] TFK temperature Input coolant temperature 288.15[K]=15[°C] Tvp

t.f. b, a

General notation Abbreviation/Symbol deg MIMO

Single Input – Single Output Transfer function (Process model) t.f. polynomials (Controller) transfer function (Loop) characteristic polynomial Control error and its Laplace transform (Controlled process) transfer function Process gain Controller t.f. polynomials Complex Laplace variable Time-constants Control (manipulated) input Disturbances and their Laplace transforms Reference (set-point) signal and its Laplace transform Controlled signal and its Laplace transform Controller tuning parameter

C

d

e, E G ks q, p s τ , T1 , T2 u vu , v y , Vu , V y

w, W y, Y

α

I.

Introduction

Chemical reactors are essential parts of many industrial processes. Therefore their analysis, optimal design and control are of special interest, e.g. [1]-[4]. As it is often simpler, cheaper, safer and less time consuming than real-time experiments the modelling and simulation tools play an important role in this field nowadays. They can be used for e.g. analyses of system

Meaning (Polynomial) degree Multi Input – Multi Output


911


František Gazdoš

behaviour in the pre-production and production phase, examination of different (often dangerous) states, operators training or optimal control system design. The basics of process modelling and simulation can be found in e.g. books [5]-[8]. In this paper modelling and simulation tools are fruitfully utilised to design a suitable control system for a semi-batch reactor used for the tanning waste recovery [9]. The waste comes from the tannery industry where the process of leather-to-hide conversion takes place. It contains chromium chemicals with negative impact on the environment problematic to recycle. One approach to deal with this tanning waste is the enzymatic hydrolysis which separates the chrome from protein in the form of the chromium filter cake [10]. Control oriented simulation analysis of the semi-batch reactor used for this process can be found in e.g. [11]. This reactor is successfully controlled by means of predictive and adaptive control in the works [12], [13]. In this contribution it is shown that the process can be controlled by a relatively simple fixed-parameters controller designed in a robust way. The controller is designed using the polynomial approach, e.g. [14]-[17] and some results form the robust control theory [18][21]. The paper starts with the reactor description and modelling, followed by a clear presentation of the complete procedure of control system design including process identification, controller design and robust setting of the loop. The contribution continues by simulation of both open-loop and closed-loop responses showing the performance of the designed control system. The final section discusses the achieved results in the context of the previous works and suggests possible extensions.

II.

Nomenclature. The reactor constants were obtained analytically, experimentally, estimated or taken from the literature; for details see [9]. During the reaction a considerable quantity of heat is developing so that reaction control is necessary. In the beginning the reactor contains initial filling m p [kg] given by the solution of chemicals without the chromium sludge (filter cake). This is fed into the reactor by m FK [kg/s] to control the developing heat since the temperature has to stay under a certain critical level ( T ( t ) < 100°C ) otherwise the reactor could be destroyed. The reaction is cooled by the water flow inside the reactor jacket. The goal is to utilise the maximum capacity of the reactor to process the maximum amount of waste in the shortest possible time (higher temperature is desirable). Therefore an optimal control strategy has to find a reasonable trade-off between these opposite requirements on the operation temperature.

II.1.

Mathematical Model

A simplified mathematical model of the reactor was suggested in [9]. It was further refined and analysed for control purposes in e.g. [11]. The model is described by the following four nonlinear ordinary differential equations: d m (t ) dt d m FK = k m ( t ) aFK ( t ) + ⎣⎡ m ( t ) aFK ( t ) ⎦⎤ dt m FK cFK TFK + ∆H r k m ( t ) aFK ( t ) = m FK =

d ⎡ m ( t ) cR T ( t ) ⎤⎦ dt ⎣ m v cv Tvp + K S ⎡⎣T ( t ) − Tv ( t ) ⎤⎦ =

= K S ⎡⎣T ( t ) − Tv ( t ) ⎤⎦ +

Reactor Description

The chromium sludge from the tannery industry is processed in a semi-batch chemical reactor sketched in Fig. 1 by an exothermic chemical reaction with chrome sulphate acid.

= m v cvTv ( t ) + mvR cv

(1)

d Tv ( t ) dt

where all the used variables and symbols are clearly defined in Nomenclature. The first equation expresses the total mass balance of the chemical solution in the reactor. The second equation represents the chromium sludge mass balance where the expression k m ( t ) aFK ( t ) defines the chromium sludge extinction by the chemical reaction. Here, k is the reaction rate constant expressed by the Arrhenius equation (2):

k = Ae

Fig. 1. Chemical reactor scheme

−

E RT ( t )

(2)

The third equation describes the enthalpy balance. The input heat entering the reactor in the form of the

All the variables and symbols appearing in this figure and further in the paper are clearly defined in the Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


912

František Gazdoš

chromium sludge is expressed by the term m FK cFK TFK , the heat arising from the chemical reaction is given by the expression ∆H r k m ( t ) aFK ( t ) and the heat

Next sections describe the procedure of control system design step-by-step starting with the adopted approximate linear model and its identification.

transmission through the reactor wall is expressed by the formula K S ⎡⎣T ( t ) − Tv ( t ) ⎤⎦ .

III.1. Approximate Linear Model

The last equation describes coolant heat balance. The input heat is given by m v cvTvp , the heat entering the

Based on the works [22], [11] it is possible to approximate the nonlinear process by the following linear transfer function:

coolant by the reactor wall is expressed by K S ⎣⎡T ( t ) − Tv ( t ) ⎦⎤ , the heat going out with the coolant

G (s) =

is described as m v cvTv ( t ) and the heat accumulated in the double wall describes the last term mvR cvTv′ ( t ) .

(T1s + 1)(T2 s + 1)

=

b1s + b0 s + a1s + a0 2

(3)

where k s is the process gain, τ , T1 , T2 are timeconstants of the numerator and denominator and “ s ” is the complex Laplace variable. Then among the transfer function coefficients the following relations hold:

Variables m FK ,m v ,TFK ,Tvp are manipulated signals, however, practically only m FK and m v are usable. The temperature change of TFK or Tvp is inconvenient due to the economic reasons (great energy demands).

II.2.

k s (τ s + 1)

b1 = k sτ (T1T2 ) ; b0 = k s (T1T2 ) ; a1 = (T1 + T2 ) (T1T2 ) ; a0 = 1 (T1T2 ) ;

Starting Conditions and Limitations

Starting conditions of the reactor are defined as: m ( 0 ) = m p = 1810 kg (initial reactor filling), aFK ( 0 ) = 0

(4)

This model arises from the linearization of the nonlinear model (1) in a general operating point and neglecting the minor terms. Parameters k s , τ , T1 , T2

(initial mass concentration of the chromium sludge in the reactor), T ( 0 ) = 323.15 K = 50°C (initial temperature of

and consequently also coefficients

the reactor filling) and initial coolant temperature Tv ( 0 ) = 293.15 K = 20°C.

{bi ;ai }i =0 ,1 of

this

model are changing in time and are functions of the reactor operating conditions. In the time-domain, the approximate model (3) can be expressed using the step-function as:

Maximum filling of the reactor is limited by its volume to the value of m < 2450 kg approximately. Then the process of feeding by the chromium sludge m FK has to be stopped. The feeding can be practically

ks h (t ) = T1 − T2

realized in the range m FK ∈ 0; 3 kg·s-1. As stated in the process description, the temperature cannot exceed the limit T ( t ) < 100o C which holds also for the coolant

−t −t ⎤ ⎡ ⎢T1 − T2 + (τ − T1 ) e T1 + (T2 − τ ) e T2 ⎥ (5) ⎢ ⎥ ⎣ ⎦

III.2. Identification

(water).

Using the simulation means, as a response to different operating conditions of the reactor (changes in the manipulated variable m FK ) the following possible intervals of the model parameters were obtained:

III. Control System Design Present control strategy of the introduced reactor uses only the chromium sludge mass flow m FK as the main manipulated variable to control the process dynamics. As explained in the process description section, the goal is to process the maximum amount of the waste in the shortest possible time while maintaining the process quantities within the defined limits. Main critical variable is the temperature in the reactor which has to stay under the limit of 100°C during the whole process. Therefore the controlled variable is the temperature inside the reactor T ( t ) and the manipulated variable is

k s ∈ 82; 540 [°C×s kg ] ; τ ∈ −650; 8293 [s ] ;

T1 ∈ 202; 9536 [s ] ; T2 ∈ 734; 6601 [s ] ;

(6)

or for the coefficients {bi ;ai }i = 0 ,1 : b1 ∈ −0.1715; 0.1219 b0 ∈ 1.069 × 10−5 ; 2.637 × 10−4

(7)

the chromium sludge mass flow m FK . Consequently, from the control theory point of view the process can be seen as a single input – single output (SISO) system.

a1 ∈ 2.937 × 10−4 ; 5.10 × 10−3



a0 ∈ 1.980 × 10−8 ; 7.499 × 10−7

913

František Gazdoš

These intervals were obtained using the identification of the step-responses of the nonlinear model (1) to different changes of the input signal m FK . For the identification the approximate linear model described by (3), (5) was employed and standard MATLAB functions for nonlinear regression were applied. It is necessary to say that all the simulated stepresponses were well-fitted by this approximate model. Several records of the simulated step-responses of the nonlinear reactor model (1) are presented in Fig. 2 to show complex dynamics of the process. Here the simulations were performed only until the limits of the reactor were reached (maximum capacity or temperature, see section II.2 for details).

presented further in this work, namely the Kharitonov’s theorem and related tools.

III.3. Controller Design – Theoretical Framework For the control system design the classical control setup of Fig. 3 is considered where G denotes the controlled process, C stands for the controller and the signals w , e , u , y describe the reference (set-point), control error, control input and controlled variable respectively. Signals vu and v y represent disturbances.

Fig. 3. Control system set-up

The process can be approximated by the transfer function (3) with the nominal values (8) as described in the previous section: G (s) = =

Fig. 2. Limited step-responses of the reactor model

From the results it is obvious that the system is highly nonlinear changing its dynamics significantly with gain from tens to hundreds and time-constants from several minutes to several hours. In addition the process can behave as a non-minimum phase system in some conditions (time-constant τ and consequently the coefficient b1 can become negative). These properties class the process generally as difficult to control by conventional fixed parameters controllers (e.g. the widespread PI or PID regulators). A nominal process model used for the further control system design is based on the middle values of the uncertainty intervals of its parameters (6)-(7). Therefore it takes the form: G (s) = =

b1s + b0 s + a1s + a0 2

a (s)

=

b1s + b0 s + a1s + a0 2

=

−2.479 × 10−2 s + 1.372 × 10−4

(9)

s 2 + 2.698 × 10−3 s + 3.849 × 10−7

Further, the controller C can be also described by a transfer function (10) with q ( s ) , p ( s ) coprime polynomials satisfying (11): C (s) =

q (s)

p (s)

deg p ( s ) ≥ deg q ( s )

(10)

(11)

Requirements for the control system are formulated as stability, asymptotic tracking of the reference signal, disturbances attenuation and inner properness. Besides these the system has to be robust in order to cope with the real nonlinear plant (not only with the adopted nominal linear model) and possible disturbances. From the scheme of Fig. 3 and assuming (9), (10) it is easy to derive following relationships between the controlled variable y ( Y ( s ) in the complex domain)

=

−2.479 × 10−2 s + 1.372 × 10−4

b(s)

(8)

s 2 + 2.698 × 10−3 s + 3.849 × 10−7

and input signals w , vu and v y ( W ( s ) , Vu ( s ) and

This model is used for the further controller design, however, the resultant control system has to work properly for the whole range of the parameters uncertainty intervals (6)-(7). This is ensured by useful tools from the robust control theory, e.g. [18], [19],

V y ( s ) similarly); the argument “ s ” is in these formulas

omitted somewhere to keep them more compact and readable):



914

František Gazdoš

and, further suppose that disturbances vu ( t ) , v y ( t ) can

G ⋅C G ⎧ ⎪Y ( s ) = 1 + G ⋅ C ⋅ W ( s ) + 1 + G ⋅ C ⋅ Vu ( s ) + ⎪ 1 ⎪ + ⋅Vy ( s ) ⎪ 1+ G ⋅C ⎪ b⋅q b⋅ p ⎪Y ( s ) = ⋅W ( s ) + ⋅ Vu ( s ) + ⎪ a⋅ p +b⋅q a⋅ p +b⋅q (12) ⎨ ⎪ a⋅ p + ⋅Vy ( s ) ⎪ a⋅ p +b⋅q ⎪ ⎪ b⋅q b⋅ p a⋅ p ⋅W ( s ) + ⋅ Vu ( s ) + ⋅ Vy ( s ) ⎪Y ( s ) = d d d ⎪ ⎪Y ( s ) = T ⋅ W ( s ) + Su ⋅ Vu ( s ) + S ⋅ V y ( s ) ⎩

be also approximated by step-functions as in (16): Vu ( s ) =

E (s) =

p ⎡ a ⋅ W ( s ) − b ⋅ Vu ( s ) − a ⋅ Vy ( s ) ⎤⎦ d⎣

vy0 ⎞ v p ⎛ w0 − b ⋅ u0 − a ⋅ ⎜a⋅ ⎟ d⎝ s s s ⎠

(17)

which shows that in order to guarantee zero-control error in the steady-state, the denominator polynomial of the controller p ( s ) needs to be divisible by the “ s ”-term. This will be fulfilled for this polynomial in the form: p ( s ) = s ⋅ p ( s )

(13)

Symbols S , T , Su denote important transfer functions of the loop known as the sensitivity function, complementary sensitivity function, and input sensitivity function S respectively. The sensitivity function further helps to make the designed control system robust. Similarly, it is straightforward to derive the formula (14) for the control error:

E (s) =

(16)

Then substituting (15)-(16) into (14) yields:

here, the symbol d defines the characteristic polynomial of the closed-loop given generally as: a⋅ p +b⋅q = d

vy0 vu 0 , Vy ( s ) = s s

(18)

Then the controller (10) can be written as: C (s) =

q (s) s ⋅ p ( s )

(19)

and the Diophantine equation (13) defining stability will be: a ⋅ s ⋅ p + b ⋅ q = d (20)

(14)

III.3.3. Control System Inner Properness Inner properness of the control system is satisfied if all its parts (transfer functions) are proper. With regard to the proper approximate transfer function of the process (9), condition (11) and taking into account solvability of (13) it is possible to derive following formulae for degrees of the unknown polynomials q , p and d :

III.3.1. Control System Stability From (12) it is clear that the control system of Fig. 3 will be stable if the characteristic polynomial d ( s ) given by (13) is stable. This Diophantine equation, after a proper choice of the stable polynomial d ( s ) , is used to compute unknown controller polynomials q ( s ) , p ( s ) .

deg q ( s ) = deg a ( s )

Sometimes it is useful to require also so called strong stability which guarantees also stability of the designed controller, i.e. stability of the polynomial p ( s ) in (10).

deg p ( s ) ≥ deg a ( s ) − 1 deg d ( s ) ≥ 2 ⋅ deg a ( s )

As the controlled process is nonlinear with possible non-minimum phase behaviour and the suggested design methodology relies on the approximate linear model only, the strong stability condition is also considered in this work for safety reasons.

Equalities are chosen in the formulas above in order to obtain the simplest controller structure satisfying the given requirements.

III.3.4. Robust Setting of the Loop

III.3.2. Asymptotic Tracking of the Reference Signal and Disturbances Attenuation

In this work the control system design is based on the nominal linear model of the system in the form (8) with uncertainty intervals of its coefficients (7). These uncertainty intervals describe nonlinearities of the original nonlinear process model (1). In order to fulfill the requirements introduced in the previous sections not only for the nominal model but for the

Let us assume that the reference signal w ( t ) is a step function, defined in the complex domain as: W (s) =

w0 s

(21)

(15)



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František Gazdoš

whole family of models given by the uncertainty intervals the robust control approach is fruitfully utilized, e.g. [18]-[21]. A good measure of the control loop robustness is the peak gain of the sensitivity function frequency response [23]. The sensitivity function S , see (12), is defined as: S (s) = =

Y (s)

Vy ( s )

=

1 = 1+ G (s) ⋅C (s)

a (s) ⋅ p (s)

a (s) ⋅ p (s) + b (s) ⋅ q (s)

=

a (s) ⋅ p (s)

Having the prescribed behaviour of the loop given by the characteristic polynomial d ( s ) (24), from (20) it is possible to derive the matrix relation (25) between the unknown controller coefficients, known nominal model coefficients and the tuning parameter α : ⎡1 ⎢a ⎢ 1 ⎢ a0 ⎢ ⎢0 ⎢⎣ 0

(22)

d (s)

the process output y ; moreover, it gives the relative sensitivity of the closed-loop transfer function T ( s ) to the relative plant model error. Therefore it can be utilized to make the control system robust. In this work this is done via tuning some of the closed loop poles – roots of the characteristic polynomial d ( s ) in (13). In order to ensure that the control system will be stable not only for the nominal model (8) but also for the original nonlinear model (1), i.e. the family of linear models given by uncertainty intervals (7) the concept of robust stabilization is employed. The Kharitonov’s theorem (and its derivations, e.g. [18]) is a useful tool for this task. It enables to check stability of interval polynomials relatively simply. Therefore it is used further to check whether the controller stabilizes the whole family of models given by the uncertainty intervals.

(23)

hence, it is a real (filtered) PID controller. Its coefficients are obtained by a solution of the polynomial equation (13) for some stable characteristic polynomial d ( s ) . Therefore, the next task is to choose this polynomial which must be, according to (21), of the 4th order. Here it is suggested to have it in this simple form: 4

1 a1

b1 b0

0 b1

a0

0

b0

0

0

0

0 ⎤ ⎡ p1 ⎤ ⎡ 1 ⎤ ⎢ ⎥ 0 ⎥⎥ ⎢⎢ p 0 ⎥⎥ ⎢ 4α ⎥ 2 0 ⎥ ⎢ q2 ⎥ = ⎢ 6α ⎥ ⎢ ⎥ ⎥⎢ ⎥ b1 ⎥ ⎢ q1 ⎥ ⎢ 4α 3 ⎥ ⎢ ⎥ b0 ⎥⎦ ⎢⎣ q0 ⎥⎦ ⎢ α 4 ⎥ ⎣ ⎦

(25)

which robust stabilization is ensured. In this interval, stabilization of the whole family of models given by the uncertainty intervals (7) is guaranteed, i.e. the designed controller will stabilize not only the nominal model (8) but all the models represented by the uncertainty intervals of its coefficients (7). Consequently it can be supposed that the controller will stabilize also the nonlinear process model (1) and the real plant. When checking the strong stability condition, i.e. also the controller stability, see section III.3, one has to ensure that the controller coefficient p 0 in (23) is positive (since from (25) p1 = 1 ). Further computations show that this condition is fulfilled for α ≥ 0.0006 . As a result, a “safe” interval for the tuning parameter is:

Given the nominal model of the process (8) it is easy to derive a suitable controller structure using the formulas (21). The resultant controller has the form:

d ( s ) = ( s+α )

0

those close to the zero value, result in very sensitive control loop. Values higher than the minimum lead to relatively robust control loop, however, higher values of the parameter give rise to higher control action of the controller. Therefore one has to find a reasonable trade-off between the robustness of the loop and limitations on the control input. Further computations with the help of the Kharitonov’s tools provide this interval α ∈ 0.0005; 0.0017 for the tuning parameter α in

III.4. Controller Design – Implementation

q (s) q ⋅ s 2 + q1 ⋅ s + q0 = 2 s ⋅ p ( s ) s ⋅ ( p1 ⋅ s + p 0 )

0

This matrix equation is used to compute the unknown controller coefficients for a chosen parameter α > 0 . Dependence of the infinity norm H ∞ (peak value) of the sensitivity function (22) upon the tuning parameter α is depicted in Fig. 4. It is obvious that smaller values of α close to zero result in very sensitive control system. The detail in Fig. 5 reveals a minimum for α 0.0008 leading to the most robust control loop. Values in the left interval α ∈ ( 0; 0.0008 ) , especially

and it describes the impact of output disturbance v y on

C (s) =

0

(24)

α ∈ 0.0006; 0.0017

Although this simple choice limits possible behaviour of the designed loop it enables to tune the control loop simply using one parameter α > 0 .

(26)

where both, robust stabilization and controller stability is ensured.



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Fig. 6. Total mass response Fig. 4. Sensitivity function infinity-norm with α

Fig. 7. Chromium sludge mass concentration response Fig. 5. Sensitivity function detail

IV.

The response of Fig. 8 shows the temperature increase inside the reactor – it can be seen how the temperature rises as a result of the chemical reaction. The faster input flow rate of the chromium sludge, the faster reaction and temperature increase. Then, the next increase is limited by the restriction on the maximum possible mass in the reactor followed by gradual temperature fall. From the graph it is also clear that for the simulated range of m FK the temperature goes beyond the allowed limit T ( t ) < 100 [°C] for higher values of m FK , therefore the

Simulated Experiments

Simulation experiments with the nonlinear model of the reactor (1) were performed with the help of the MATLAB/Simulink system which offers both powerful computations and user-friendly simulation interface.

IV.1. Open-loop Responses

process needs to be controlled properly. The record of the coolant temperature presented in Fig. 9. reveals that the media temperature for the whole range of m FK is not critical since water is used for the cooling (for its defined flow rate m v = 1 [kg/s]).

Open-loop (without control) step-responses of the reactor are presented in the next figures, Figs. 6-9 [11]. They reveal complex dynamics of the process for different changes of the manipulated variable – m FK (chromium sludge mass flow) in the admissible interval m FK = [ 0.05 0.1 0.5 1 3] ⎡⎣ kg ⋅ s-1 ⎤⎦ . The first figure shows increase of the total mass in the reactor for various input flow rates of the chromium sludge. The simulation reveals integrating, astatic behaviour limited to the defined reactor max. capacity. Next response presented in Fig. 7 reveals derivative behaviour of the chromium sludge mass concentration aFK ( t ) for various values of m FK .

IV.2. Closed-Loop Responses – Simulation of Control For the simulation of the reactor control, the reference value of the temperature was set as: w ( t ) = 98 [°C] for safety reasons – the temperature cannot exceed the limit of 100 [°C] as described in III.2, however higher temperature is desirable in order to process the maximum amount of waste in the shortest possible time-interval. A resultant controller for a chosen tuning parameter α = 0.0014 from the suggested range of robust stability



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and controller stability (26) has the form of (27) and provides the control response presented in Figs. 10-12: C (s) =

0.0265s 2 + 7.51× 10−5 s + 2.8 × 10−8 s ( s + 0.00356 )

(27)

As can be seen from the first graph the control is stable with only a minor overshoot reaching and tracking the desired value relatively quickly.

Fig. 10. Robust control response for α = 0.0014

Fig. 8. Temperature-in-the-reactor response

Fig. 11. Control input response for α = 0.0014

Fig. 9. Coolant temperature response

Next figure shows that the manipulated variable m FK is within the allowed limit 0÷3 [kg·s-1] during the whole process of control. The last graph, Fig. 12 displays increase of the reactor total mass before reaching the maximum capacity m < 2450 kg . Then the feeding by the chromium sludge m FK is stopped and after cooling the reactor is emptied and the process continues with a next batch. Next two graphs, Figs. 13-14 show the influence of the tuning parameter α on the control process. It can be seen that its higher values speed-up the control process but result in higher overshoots and wider range of the manipulated variable m FK .

Fig. 12. Total mass response for α = 0.0014

Therefore, in the real application one has to find a reasonable trade-off between the speed of the control response and particular process limitations. Last three graphs show non-robust setting of the tuning parameter – out of the suggested range of the robust and controller stability (26). The first picture reveals poor control response with unacceptable overshoots and tracking. Next figure reveals that limits of the manipulated variable are reached very soon and therefore the control process does not work properly.



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Fig. 13. Robust control response – influence of the tuning parameter α

Fig. 15. Non-robust control response

Fig. 14. Control input response – influence of the tuning parameter α

Fig. 16. Non-robust control input response

Last figure shows the total mass increase in the reactor – in the first case ( α = 0.0003 ) the reactor is soon filled up without the expected results of proper control. In the second case ( α = 0.0030 ) it is not limited so early, however, as a result of control input limitation the control process also does not work properly.

IV.3. Discussion of the Results Presented results confirm the need for robust control of the reactor when the control system design is based on the linear nominal model only together with a fixed controller. Another approach is adopted in the works [12], [13] where predictive and adaptive control were applied to this reactor successfully, however, these control design methodologies are more complex and computationally demanding. The aim of this contribution was to show that this process can be also successfully controlled by a relatively simple (PID) fixed-parameters controller designed in a robust way. It can be expected that more complex choice of the characteristic polynomial (24) will provide even better responses and more robust control loops, however, for the cost of optimising not only one, but more, up to four poles of the characteristic polynomial.

Fig. 17. Non-robust control – total mass response

In the limited space of this contribution it can be just added that two optimised parameters provided more robust control loop, however, the achieved responses were very slow and consequently not suitable for the defined goals of this reactor control application. In this work the classical control set-up of Fig. 3 with one feedback controller was employed. Different control configurations, e.g. with also the feedforward part of the controller filtering the reference signal could help to reduce overshoots of the controlled variable and decrease the control action in order to stay in its defined limits.



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V.

[10] K. Kolomazník, M. Mládek, F. Langmaier, M. Taylor, E. Diefendorf, E. Brown, W. Marmer, and L. Tribula, Process for Preparing a Hydrolyzate of Proteinaceous Waste of Animal Origin, CZ Patent 280 655, Czech Republic, 1996. [11] F. Gazdoš, L. Macků, Control-Oriented Simulation Analysis of a Semi-Batch Reactor, International Review of Automatic Control (IREACO), Vol. 2(Issue 5):584-591, September 2009. [12] D. Sámek, L. Macků, Semi-Batch Reactor Predictive Control Using Artificial Neural Network, Mediterranean Conference on Control and Automation, pp. 1532–1537, Ajjacio, Corsica, France, June 2008. [13] D. Novosad, L. Macků, Pole Placement Controller with Compensator Adapted to Semi-batch Reactor Process, Journal of Mathematical Models and Methods in Applied Sciences, Vol. 5(Issue 7):1265-1272, February 2011. [14] V. Kučera, Diophantine Equations in Control – A Survey, Automatica, Vol. 29(Issue 6):1361-1375, 1993. [15] K.J. Hunt, Polynomial Methods in Optimal Control and Filtering (Institution of Engineering and Technology, 1993). [16] M.J. Grimble, V. Kučera, Polynomial Methods for Control Systems Design (Springer, 1996). [17] M.J. Grimble, Robust Industrial Control Systems: Optimal Design Approach for Polynomial Systems (Wiley, 2006). [18] B.R. Barmish, New Tools for Robustness of Linear Systems (Macmillan, 1994). [19] S.P. Bhattacharyya, H. Chapellat, and L.H. Keel, Robust Control The Parametric Approach (Prentice-Hall, 1995). [20] M. Morari, E. Zafirou, Robust Process Control (Prentice-Hall, 1989). [21] K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control (Prentice-Hall, 1995). [22] F. Gazdoš, L. Macků, Analysis of a Semi-batch Reactor for Control Purposes, European Conference on Modelling and Simulation, pp. 512–518, Nicosia, Cyprus, June 2008. [23] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis and Design (Wiley, 2005).

Conclusion

In this contribution a simulation study of control system design for a semi-batch reactor used for tanning waste recovery has been presented. The complete procedure of control system design is shown in detail, including process modelling, identification, controller design and simulation verification. It is shown that this highly non-linear complex process can be successfully controlled by a relatively simple fixed-parameters controller designed in a robust way. For this purpose the polynomial approach and some useful tools from robust control theory were exploited. In this work, the process was controlled as a single input – single output (SISO) system only – the temperature in the reactor was controlled by means of the chromium sludge mass flow m FK . It would be interesting and practically possible to include also the coolant mass flow m v as the manipulated variable and treat this system as multi input – multi output (MIMO). It can be supposed that the achieved results could be even better. In conclusion it must be added that the analysis and synthesis performed in this contribution relies upon the adopted nonlinear model of the process and that verification on the real system of the reactor should follow. Next weak point of the analysis may be in the identification of the approximated linear model of the process where only a limited number of simulated responses were used to compute the uncertainty intervals of its coefficients. Therefore the resultant interval linear model may not handle all possible states of the adopted nonlinear process model/real process.

Authors’ information Tomas Bata University in Zlín, Faculty of Applied Informatics, nam. T.G. Masaryka 5555, 760 01 Zlín, Czech Republic. Tel: +420-576-035-199 Fax: +420-576-032-719 E-mail: [email protected]

Acknowledgements A previous version of this paper was presented at the International Workshop on Applied Modeling and Simulation, Rome, Italy, 24-27 September 2012.

František Gazdoš (Corresponding author) was born in Zlín, Czech Republic in 1976, and graduated from the Brno University of Technology in 1999 with MSc. degree in Automation. He then followed studies of Technical Cybernetics at Tomas Bata University in Zlín, obtaining Ph.D. degree in 2004. He became Associate Professor for Machine and Process Control in 2012 and now works in the Department of Process Control, Faculty of Applied Informatics of Tomas Bata University in Zlín, Czech Republic. He is author or co-author of more than 70 journal contributions and conference papers giving lectures at foreign universities, such as Politecnico di Milano, University of Strathclyde Glasgow, Universidade Técnica de Lisboa and others. His research activities cover the area of modelling, simulation and control of technological processes.

References [1] [2] [3]

[4] [5] [6] [7] [8] [9]

G.F. Froment, K.B. Bischoff, and J. de Wilde, Digital Chemical Reactor Analysis and Design (Wiley, 2010). A.C. Dimian, Integrated Design and Simulation of Chemical Processes (Elsevier Science, 2003). J. Ingham, I.J. Dunn, E. Heinzle, and J.E. Prenosil, Chemical Engineering Dynamics: Modelling with PC Simulation (Wiley, 2000). W.L. Luyben, Chemical Reactor Design and Control (Wiley, 2007). P.E. Wellstead, Introduction to Physical System modelling (Academic Press, 1979). L. Ljung, G. Torkel, Modeling of Dynamic Systems (PrenticeHall, 1994). B.A. Ogunnaike, W.H. Ray, Process Dynamics, Modeling, and Control (Oxford University Press, 1994). F.L. Severance, System Modeling and Simulation: An Introduction (Wiley, 2001). L. Macků, Control design for the preparation of regenerate for tanning, Ph.D. dissertation, Faculty of Technology, Tomas Bata University in Zlín, Czech Republic, 2004.



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Effective Detection, Identification and Measurement Strategies of Market Power and their Comparison R. Esmaeilzadeh1,2, H. Eskandari1, M. Amjadi1, M. Farrokhifar2 Abstract – To cure a market which is infected by market power, or to vaccinate the market against it, first of all market power should be detected then, measured and identified via appropriate tools and indices. Therefore more powerful methods and indices to search created market power or market power potential will result the most appropriate solution to decrease it. In this paper, different methods of analysis, identification and detection are studied and then drawbacks and advantages of each one encountered to different conditions of market power applying are proved. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Market Power, Power Market, Detection Methods, Identification Method

I.

Totally, market power indices could be divided into three categories of structural indices, behavioral indices and simulation analysis. Although structural indices are simple, they generally do not contemplate reactionary capability of the load. Behavioral indices will be suitable for operational market power search not for potential one. Of course behavioral indices are being used to search market power. In this approach the price should be estimated and this estimation has its own problems. The most appropriate methods are simulation analyses which estimate market power after market simulation. In simulation models reactionary capability of the load and even problems related to transmission congestion can be considered. Finally, in table indices, advantages and disadvantages are summarized [3], [4].

Introduction

As stated in prior papers, one of the most important duties of supervision unit is the development of the one and market power analysis tools [1]. An ideal index of market power is an index which provides a simple number depicting ability of market power. A measurement of appropriateness for this index is its ability to predict market power applying or to state correlation rate of participants in increment of price up to a level higher than competitive one. Therefore some measurements applied to other markets are not suitable for electricity market and better measurements are required [2], [3], [6]. The other aim of this article is to describe the best effective indexes and analysis methods of market power applying in different markets to measure market power and to demonstrate advantages and disadvantages of them each. Also applications of each index are stated in different cases. In this paper first different categorization of analysis and detection methods such as structural indexes, behavioral indexes and simulation models are explained. Then the measurement is engaged [1]. Experiences in different worldwide electricity markets show that assumption of competitive market without supervisory unit is an inexcusable assumption. Therefore existence of supervisory unit is necessary to save and develop competition and to prevent market power. Supervision organization should be independent of all market participants to have a rightful supervision and should be under direct supervision of market directorate. Market supervision unit must measure arisen market power and potential of market power appearance to vaccinate market with suitable retrenchment methods against the market power. Market power is being measured by different indices which have their own advantages and drawbacks.

II.

Detection Methods

Market power detection is not so simple and is being performed in all markets without exception. Of course some certain properties in electricity market have caused differences in market power appearance respect to other markets. For instance in electricity market, participants suggest their output rate based on the price range while in other markets just market settlement price is suggested. Other differentiation characteristic of electricity market is that technologic data such as thermal production rate and producers’ capacity are accessible. Consequently, cost and price estimation in electricity market is quite accurate in respect to other markets. Other apparent characteristic of electricity industry is that great share of costs variations in short term periods are due to fuel cost which has accessible price. In categorizing different methods of market power detection, the most important differentiation is the


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R. Esmaeilzadeh, H. Eskandari, M. Amjadi, M. Farrokhifar

products consists of energy production, (excess energy reserve), short term capacity or long term capacity) and then geographical range of market must be considered (which must be considered as rival) to calculate the value. Having production type, market size and market share calculation, some market power criterion must be defined in order to determine what amount of market share could be assumed as market power possibility. In federal energy regulation commission 20% is introduced as the criterion for market power. It means that if a company has market share more than 20%, this shows market power. Market share index is a general tool which is utilized both academically and as a market supervision tool. This index can be calculated easily and used in long term studies although most of the people who use this index are aware of its vast drawbacks.

differentiation between methods which are measuring or searching market power potential (Ex-Ant) and methods detecting imposed market power (Ex-Post). Second effective differentiation is that some methods just perform detection and some do identification too. Also indices could be categorized based on time horizon meaning that some methods are being used in long term time horizon which are applicable in the analysis of combination of two companies or market design investigations, and some are being used in short term time interval. Of course some other categorization approaches are possible for market power detection methods. Factors which should be considered through evaluation of competition in electricity market are: i) Market Share ii) Market concentration iii) Load traction iv) Rate and distribution of excess capacities v) Contracts vi) Price determination process vii) Simplicity of entering to market viii) Transmission grid II.1.

II.1.2. Herfindal-Hirshman Index One of the critics on market share index is that the ability of one company with 20% share to impose power market is different in the states that the mentioned company is the greatest player in non-concentrated market or is second or third player in a market with high concentration. One suitable index to consider this characteristic is Herfindal-Hirshman index. Herfindal-Hirshman index is obtained via summation of squares of market participant shares in the market:

Structural Indices

Generally the first step in market competition evaluation is investigation of market structure such as participants market share as long as market structure is effective on market power exposure. Market structure must be in a way that all participants are provided with competitive environment. It means that the market should not be concentrated and market participants have equal share and power in order to have a fair competition. The most famous structural index which has applications in all markets is Herfindal-Hirshman index (HHI) which measures market concentration. Physical rules governing electricity market distribution network have lead to more complete and general indices which include both network and variations as long as electricity market is dynamic and production and consumption rates are equal. The target in this chapter is to introduce structural indices and state their applications.

HHI = S12 + " + Sn2

(1)

Si is market share percentage of company i. For example, when 10 companies with equal shares of 10% are in the market, we will have: HHI = 10 × 102 = 1000

Herfindal-Hirshman evaluation can be divided into three portions: None-concentrated Semi- concentrated Highly- concentrated

II.1.1. Market Share Index Concentration index introduces market producers’ concentration and is stated with one scalar number. The idea behind this index is that if the market has more concentration, participants of that market will impose market power with more probability. Two famous concentration indices which are widely used are the index of market share and Herfindal-Hirshman index. The ratio of market share concentration is market share percentage of the greatest company in the market. In order to calculate this index, products related to the market should be identified (in electricity market, these

HHI ∑ GC j − Dt

if

j =1

if

Dt − ∑ GC j > 0 j ≠i

if

(6)

Dt − ∑ GC j ≤ 0 j ≠i

Dt : Total load of the market; GC j : Maximum capacity of jth generator.

2

(3)

Pivotal supplier index for time interval of T is calculated via summation of pivotal supplier index for hours: 1 T PSI i = ∑ PSI it (7) T t =1

HHI G : Herfindal index of the group MWH i : Production of unit i in group G TotalG : Total production in group G

iii) Herfindal Index of the system: Herfindal index of the system is defined as square root of weighted mean square for HHI 2 on the bases of group share percentage. If the power system is composed of n groups, system Herfindal index at a specific hour will be calculated as below:

Supply margin assessment is another name for pivotal supplier index which is chosen by federal energy regulation commission in 2001 as depicter of power market replacing 20% market share index [1]. II.1.4. Residual Supply Index

HHI s =

n

∑

j =1

⎡ MWH i 2 ⎤ × HHI Gj ⎢ ⎥ ⎣ Totals ⎦

Residual supply index is similar to pivotal supplier index but the difference is in its continuous calculation. One critic on pivotal supplier index is that a company may impose market power when it is very near to effective state but it is not effective supplier company. Residual supply index for company i is the residual production capacity percent in the market after subtracting capacity of company i in production:

(4)

HHI s : Herfindal index of the system

HHI Gj : Herfindal index of jth group MWH i : Total production in group j Totals : Total production in the system

(Total Capacity − Company i s Capacity ) ,

RSI i =

II.1.3. Pivotal Supplier Index Pivotal Supplier Index tries to add market conditions into production conditions.

Total Demand

(8)

Total Capacity: Total region production capacity plus maximum external input of the grid.



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total demand curve (in a real time). Companies are not aware of demand curve accurately. Of course one of priorities of electricity market is that the necessary information in order to form residual demand curve is available in the market. In a competitive market, companies will ran into elastic residual demand curve and will not be able to increase price more than competitive level by withholding its capacity.

Company i’s Relevant Capacity: Capacity of company i minus the capacity of company sold in contracts and obligations: Total Demand: Total measured load When residual supply index is more than 100%, other producers except i are able to supply the load and the company i will have little effect in the determination of market settlement price. On the other hand if residual supply index is less than 100%, company i will be needed to supply the load and the company i will be a very important and effective player in the market. As long as residual supply index is calculated for each company in the market, it is possible to calculate it for total market too. Normally, residual supply index for total market is defined as the minimum residual supply index existing in the market. Sheffrin discussed in 2002 using residual supply index in order to market supervision that market power will not exist if : Residual supply index is less than 5% of total hours in the year less than 110% or residual supply index is more than 95% or total hours in the year more than 110%. Superiority of residual supply index respect to pivotal supplier index is that its threshold is variable but threshold for pivotal supplier index is 100%. Residual supply index threshold is regulated experimentally. Fig. 1 shows distribution of residual supply index in CAISO [3].

II.2.

Behavioral Indices

While structural indices look for potential of imposing market power, behavioral indices investigate behavior and real actions of participants to detect market power. Therefore, the most important difference between behavioral and structural indices is that structural indices calculate market power not ignoring action and implemented strategy by market participants, while behavioral indices investigate market output (market settlement price) and players’ behavior not considering market structure to find imposed market power. It means that behavioral indices investigate and analyze price suggestion and quantity of each player. Of course high prices and low level of suggested price are not reasons for power market by themselves. Consequently, there is an obvious requirement for conceptual and applicable indices to separate high prices and power market. II.2.1. Bid-Cost Margin In a competitive market, companies must suggest price in their marginal cost. Therefore, the comparison between their suggested and marginal price is the most important measurement in order to highlight market power imposed by a company in the market. There are two types of electricity sellers in the electricity market: Price taker and price maker. For price takers, suggested price is the same as their marginal cost. But the story is not the same for price makers; actually market price is regulated by price makers. Therefore the comparison between producers’ suggestion and their marginal cost is the most important measurement to signify market power existence in electricity market. The result for this comparison is stated by Lerner index (LI) or Price Cost-Margin Index (PCMI): P − MC LI = (9) P

Fig. 1. Distribution of residual supply index in CAISO

II.1.5. Residual Demands Analysis

PCMI =

Residual demands analysis is a better index to calculate companies’ tendency to impose market power. This calculation is obtained via analyzing residual demand cure with which company i faces. Residual demand curve is obtained by subtracting all curves suggested by producing companies in the market from

P − MC MC

(10)

MC: Marginal Cost P: Market Settlement Price LI: Lerner Index PCMI: Price Marginal Cost Index



924


1. Economical withhold which reduces the output (of production), because it suggests a higher price than the competitive price of the market. 2. Physical withholds which does not suggest its output to the market. Economical withhold is described in this section. Economical withhold is measured with output gap estimation. Output gap is defined as the difference between the capacity of units which have economical production based on market price and real production value of units:

In order to estimate marginal price, fuel price and thermal efficiency are utilized which is one of the biggest problems. Other drawbacks of this index are: 1. There are other variable costs which are really hard to be valued such as increasing cost of machinery depreciation. 2. Data for variable costs are confidential and therefore hard to be obtained. 3. Even in completely competitive markets, market price can excess marginal price of producers (if the production is limited). This price increment is not meant as market power in some cases. In some markets that market price equals the last accepted bid; increase in the suggestion of marginal generators has great effect on market price increase respect to its competitive level.

Piecon − Pi prod

(11)

Piecon : Economical value of production and competitive suggestion for unit i based on market price prod Pi : Real production of unit i This analysis is based on variable cost estimation the same as the suggestion marginal cost price. Also in the calculation of Pi prod , transmission constraints, obligatory exit and other effective factors on production must be considered. Positive value for output gap estimation means economical withhold. If this gap is small (less than 1% of capacity), this withhold will not lead to severe problems.

II.2.2. Net Revenue Benchmark Analysis [1] Other type of analysis which is being performed on cost data is net revenue benchmark analysis. A high level of net revenue does not mean market power (as high price does not mean either) but pure revenue is monitored as a beneficial tool. Also one of the ongoing activities is try to estimate generators’ pure revenue. In long term, return obtained from energy, capacity and market of ancillary services must cover costs for new power plants consisting capital return. If the revenue is less than this level, it will discourage participants from entering market and finally the pressure will thrust upward to the price. On the other hands, if the revenue is more than this value, it will result in entering new players in the market and the pressure will be toward low prices. The difference between income resulted by market by power plants and variable costs, cooperates in covering constant costs such as non-variable costs of operations, maintenance costs and investments. This difference could be foreseen having novel producing units’ variable costs, hourly certain energy price and estimation of capacity income and ancillary services.

II.2.2.2. Physical Withhold Producers do not suggest the market by physical withhold. They withhold their capacity from market by not operating their units in the nominal state, for example reducing high operation limit. Normally there are two methods for production units to operate in non-nominal states. 1. Unit exit 2. Equating high operation limit (HOL) with a positive value less than maximum production capacity. In nominal state analysis, usually scheduled and obligatory exit are not considered as they are very similar to physical withhold strategy and hide physical withhold by saying that the exit was obligate. In nominal state, lack of operation is not as a sign of market power by itself. Simple statistical methods are used in nominal states for pattern evaluation as a sign of physical withhold. To do so, it is possible to estimate capacity of units by comparing unit exit with reliability indices such as obligatory exit rate. Actually the problem is that estimation of reliability indices for each unit is dependant to former use of unit and maintenance performed on it.

II.2.2.1. Economical Withhold [1] If one generator benefits from selling electricity of additional units but does not sell, this producer has imposed market power. Consequently with this point of view, in order to evaluate market power in electricity market we must not focus on the price; but we have to consider the output. It means that the production capacity must be searched which is beneficiary based on the market price to be produced but is not being produced. The target from withhold analysis is to determine production capacity which is beneficiary to produce but refuses too. There are two types of withhold:

II.3.

Market Power Analysis by Simulation

There are two approaches in simulation method; one of them is the price behavior estimation with simulation analysis and the other one is multi-pole equilibrium analysis. Comparison with competitive price is possible



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SFE model sounds much more suitable respect to others for electricity market as it is similar to players’ decisions in electricity markets such as California auction market. In Bertrand model it is assumed that the producer that offers less price respect to others and wins is able to posses the whole market and increase its output (production) up to load junction. As the production cost increase with increasing output and production capacity limitation is a very important constraint, this model is not applicable for electricity market. Consequently, Bertrand model is not able to provide useful information and only Karnat and SFE models are appropriate for electricity markets’ analysis.

through price behavior estimation and market power will be signified using marginal price-cost index. In the following, these two methods are illustrated [7].

II.3.1. Estimation of Price Behavior Implementing Simulation Analysis There are three general methods for price estimation. First of them is to study strategies of optimal suggestions of power suppliers’ and estimate market settlement price and utilize them to detect market power. Participants’ behavior is dependent on several factors such as bilateral contracts, estimated load, required product reserve and former obtained experiments. Second method is to accept simulation models such as production simulation model in order to simulate electricity market. One good simulation model can improve market power analysis accuracy at least via two methods which will be represented at follows: 1. Considering exchange between producers and loads in different parts of transmission grid. 2. Considering operational constraints and cost characteristic of production unit and transmission grid. Of course existing production simulation models cannot be used for electricity markets directly as bilateral contracts are not assumed in them for example. Therefore some corrections are necessary [4]. 3. Third method is analysis based on former data. In this method, market powers abuse are investigated implementing historical data of the market and statistical techniques. It means that it is possible to demonstrate whether electricity price is far from its marginal costs or not. Actually in this method competitive price of the market or marginal cost of generators must be estimated.

III. Numerical Example In this example a market with IEEE grid of 30 buses (Fig. 2) and two production companies is simulated. Company A is the owner of unit 3 and company B is the owner of units 5 and 6. Units 2, 1 and 4 are competitive also the government owns them. This means that the suggestion of them is always their marginal costs. The information about the grid is shown in [5]. By the way, line capacities are 0.8 of nominal capacity and maximum production of power of generators 1, 2 and 4 is assumed 55MW.

II.3.2. Multi-Pole Equilibrium Analysis Several models of multi-pole restrict equilibriums are available. The most famous of them are Karnat model, Bertrand model and Supply Function Equilibrium (SFE). Actually these models are models for players in electricity market with incomplete competition and show strategic behavior of producers. In Karnat model, producers choose their production level and believe that their rivals in the market do not change their production and the price changes only with production level strategy of producers and load curve. In the Bertrand model, producers choose the price. This model states that when each participant is proposed to announce just one price and there is no limitation on production capacity, then complete competition will occur and market price will equate marginal cost. In SFE model, a relation is proposed between quantity and price, actually participants suggest the price based on production level.

Fig. 2. IEEE 30 bus grid

Load in different buses changes in 24 hours as a factor of the load mentioned in 30 buss IEEE grid. These coefficients are actually normalized values of total market load in a specific day. Fig. 3 shows total market load for 24 hours. It is assumed that companies A and B are suggesting a factor of their marginal costs. Marginal costs of generators are as shown in Fig. 4 approximately. Simulation results which are actually the value of optimal k (for units 3, 5 and 6), market price, revenue of production companies A and B and their produced power are shown in Fig. 5 to Fig. 8 respectively.



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Fig. 3. Total demand load during 24 hours Fig. 6. Market price

Fig. 4. Boundary costs of generators Fig. 7. Benefits of companies

Fig. 5. K (scalar) for optimal bid Fig. 8. Generated powers of units 3, 5 and 6

The most suitable index in power market measurement is price-marginal cost index. In order to obtain this we have to calculate competitive price first. To find this it is assumed that all generators offer in their marginal costs; then market price will be competitive price. Competitive price in 24 hours is shown in Fig. 9.

HHI, RSI and PCMI indices could be calculated to measure power market. But based on what discussed before, the most suitable one is price-marginal cost index. The values for these indices for 24 hours are shown in Fig. 10 to Fig. 12 respectively.



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obtain difference of market price that the company I offers in its marginal cost and the state that offers strategic suggestions.

Fig. 9. Market competitive price

Fig. 12. PCMI index

For instance, in order to calculate power market for company A, we have to calculate market price assuming that company A offers in marginal cost (other companies offer the suggestion in which their return is maximized). Market price in this state is shown in Fig. 13.

Fig. 10. HHI index

Fig. 13. Market price while price of company A is on boundary cost

The difference of this price (the price obtained assuming that company A is competitive) and market price in the state that company A offers strategic suggestions demonstrates the ability of company A to impose market power. This difference in percentage is actually market power of company A. Fig. 14 depicts market power of company A. As it can be seen, in hours 8, 9, 10, 18 to 22 companies A has the ability to impose market power. Power market for company B can be calculated in the same way. Fig. 15 and Fig. 16 show market price for the cases that company B is and is not competitive respectively.

Fig. 11. RSI index

The comparison between these indices illustrates that HHI index is not able to detect market power. PCMI index shows that in hours 8, 9, 10, 18 to 22 which the value for this index is more than 50% there is potential possibility of market power. In order to identify which company has the most possibility to impose market power, it is necessary to



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Fig. 16. Market power of company A

Fig. 14. Market power of company A

IV.

Market power is measured using different methods which have their own benefits and drawbacks. All market power indices could be categorized into three major groups of structural indices, behavioral indices and simulation analysis. Although structural indices are simple, they generally do not consider the possibility of the load elasticity. Behavioral indices are suitable to search imposed market power rather than potential market power. Of course behavioral indices could be used for market power search in which the price must be estimated which has its own problems. The most suitable method is simulation analysis which estimates market power after market simulation. In simulation models, load elasticity and even the problems related to transmission congestion are considered. Table I represents indices advantages and disadvantages of indices briefly.

Fig. 15. Market price while price of company B is in bound boundary cost

Categories

Structural indices and analysis

Behavioral indices and analysis

Simulation models

Index Type

Conclusion

TABLE I COMPARISON OF DIFFERENT INDICES Application Advantages Easiness of analysis understanding just by reliable assumptions (It doesn’t depend on auction data)

Market share and Herfindan (HHI) index

Ex-Ante

Pivotal supplier index and residual supply index

Ex-Ante & Ex-Post

Residual demands analysis

Ex-Post

Lerner index (LI) and price-cost margin index (PCMI)

Ex-Ante & Ex-Post

Easy understanding and independency on geographical region definition

Net revenue benchmark analysis

Ex-Post

It includes outage and entrance to market by guaranteeing investments

Behavior estimation by simulation analysis

Ex-Post

It identifies an index for market efficiency

Multi-pole monopoly models

Ex-Ante

It gathers some market power factors in frame

It is available to track market dynamical variation in power market, it is feasible and practical in real market It includes load and market elasticity


Disadvantages Low practical and empirical justification, ignoring the load, strategic incentives and problems of line congestion, improperness for dynamic market conditions, difficulty in specifying the geographical regions of market Difficulty in determining the proper geographical regions, ignoring the elasticity of changeable factors of market Requirement to data on selling bid few empirical doing is done with it Difficulty in evaluating the cost and the competitive reference level of factors other than market price are also affects the marginal price Difficulty in specifying the costs, the factors other than competition also affect this index Difficulty to determine the cost and competitive level, it is not possible to determine that with generator applies the competition There are many suppositions which lower the precision of results


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Majid Amjadi was born in Tabriz, Iran, 1978. He received the B.Sc. degree in electrical engineering from Sahand University of Technology and the M.Sc. degree in electrical engineering from University of Tabriz in 1999 and 2003 respectively. Currently, he is in charge of power market department of Azerbaijan regional electric company, Tabriz, Iran. He is consultant in power market and operation office. His research interests are in the application of intelligent theories and optimization algorithms on restructured space, dynamic load modeling and energy management.

References [1]

[2]

[3] [4]

[5]

[6]

[7]

P. Twomey, R. Green, K. Neuhoff and D. Newbery, A Review of the Monitoring of Market Power, Cambridge Working Papers in Economics CWPE 0504, CMI Working paper 71, 2005. J. Yang and G. Jodan, System Dynamic Index for Market Power Mitigation in the Restructuring Electricity Industry, IEEE Power Engineering Society Summer Meeting, pp. 2217–2222, July 2000. A. Sheffrn, Predecting Market Power Using the Residual Supply Index, URL: http://www.caiso.com, December 2002. A. K. David and F. Wen, Market Power in Electricity Supply, IEEE Trans. on Energy Conversion, Vol. 16, No. 4, pp. 352-360, December 2001,. O. Alsac and B. Stot, Optimal Load Flow with Steady State Security, IEEE Transactions on power Systems, Vol. PAS 93, No. 3, pp. 745-751, 1974. A. Shishebori, M. S. Javadi and F. Taki, Generation Simulation in Energy and Reserve Market and their Economic Analysis in Iran, International Review on Modelling and Simulations (IREMOS), Vol. 4, No. 2, Part B, pp. 843-850, April 2011. A.M Kimiagari, M. Fattahi and V. Nourbakhsh, Using Simulator with Learning Capability to Determine Bidding Strategy in a Multi-Agent Wholesale Power Market, International Review on Modelling and Simulations (IREMOS), Vol. 4, No. 2, Part B, pp. 858-864, April 2011.

Meysam Farrokhifar was born in Tabriz, Iran, in 1981. He received the B.Sc. and M.Sc. degrees in electrical power engineering from University of Tabriz, Iran, in 2004 and 2007, respectively. Currently, he is studying Ph.D. in electrical power engineering at the Politecnico di Milano, Italy. He is also a member of IEEE industrial electronics society and Iranian national electro technical committee (INEC). He has published more than 17 technical papers. His research interests are power system optimization, energy management, electrical machines, transformers and intelligent methods for optimization.


Azarbaijan Regional Electric Company, Tabriz, Iran.

2 Electrical Engineering Department, Politecnico di Milano, Milan, Italy.

Rasoul Esmaeilzadeh was born in Tabriz, Iran, 1978. He received the B.Sc. and M.Sc. degrees in electrical engineering in 1999 and 2003 from University of Tabriz. Currently, he is studying Ph.D. in electrical power engineering at the Politecnico di Milano. His research interests are in the application of estimation theories and optimization algorithms to power system control design, dynamic load modeling and energy bidding. Hamideh Eskandari was born in Tabriz, Iran. She received the B.Sc. degree in electrical engineering from University of Tabriz and the M.Sc. degrees in industrial system planning in from Tarbiat Modares University. She was head of wireless measurements group in Azerbaijan dispatching center until 2004. After that she has been the manager of power market office in Azerbaijan regional electric company of Iran. Her research interests are around intelligent dispatching methods and economic energy management in power grids.



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