Fault Detection of Nonlinear Systems Based on ...

International Review of

Automatic Control (IREACO) Theory and Applications

Contents A Review of the State of the Art of Modulation Techniques and Control Strategies for Matrix Converters by C. G. Richards, P. J. Ehlers, D. V. Nicolae, E. Monacelli, Y. Hamam

298

Fault Detection of Nonlinear Systems Based on Takagi-Sugeno Fuzzy Models by Parity Relations by Majid Ghaniee Zarch, Mahdi Aliyari Shoorehdeli, Mohammad Teshnehlab

309

On the Developments and Performances of Synchronous Reluctance Machine by A. S. O. Ogunjuyigbe, A. A. Jimoh

316

Microgrid Stabilization by Superconducting Magnetic Energy Storage with Optimal Energy Capacity by Genetic Algorithm by Issarachai Ngamroo

327

Reduction of Excessively High Voltages in Transmission System by Milodrag P. Košarac, Čedomir B. Vujović

333

Enhanced Control Technique for Stabilization of Power Systems Using Controllable Series Capacitor by R. Latha, Atchyutarao Golla, J. Kanakaraj

338

UPS Mode Operation of UPQC with Mixed Mode Control by Srinath S., Selvan M. P.

342

Optimal Power Factor Control of Three-Phase Induction Motor Drives Using PIC-Microcontroller by Hussein Sarhan, Rateb Issa, Mohammad Alia

349

Stability Study of Power Plants and their Integration Into the Electric Power Grid by M. Bouchahdane, A. Bouzid, I. Bouchareb

354

Computer Analysis of Ionization as a Cause of Grounding Impedance Nonlinearity by Rasim Gačanović, Hamid Zildžo

359

Optimal Nonlinear Control of Electrostatic Micro Actuator System by Mohammad Nasser-Moghadasi, Farbod Setoudeh, Seyed A. Olamaei

364

A Systematic Approach to a Time Series Neural Model Development for River Flow Forecasting by Petar Matić, Ozren Bego, Ranko Goić

367

The Effect of Load Variations on Operation of a Micro-Turbine Generation System in Grid Connected Mode by Mohammad Mahdi Mahmoodi, Seyed Morteza Alizadeh, Fatemeh Nadimi, Sedigheh Babaei Sedaghat

373

(continued)

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

New Method to Determine the Memory of Volterra Model by Safa Chouchane, Kais Bouzrara, Hassani Messaoud

379

Multivariable Generalized Predictive Control Based on CARIMA Model with non Identity Disturbance Matrix by Badreddine Louhichi, Ahmed Toumi

386

Control and Operation of Wind Farm Connected to Grid Using Multi-Terminal VSC-HVDC by Reza Noroozian, Mohammad Reza Safari Tirtashi

395

Review and Comparing of Conventional Buck, Boost and Buck-Boost Dc to Dc Converters with their Interleaved Topology Via Equations, Simulation and Experimental Results by M. Jahanmahin, A. Hajihosseinlu, E. Afjei

405

Dissolved Gas Analysis and Fault Diagnosis Interface for Oil-Immersed Transformer Using Fuzzy Logic by Yunus Biçen, Faruk Aras, Melih İnal, Hasbi İsmailoğlu

414

Comparative Analysis of P-I, Self Tuned Fuzzy and a Hybrid Controller for Indirect Vector Controlled Induction Generator for Wind Energy Application by Kanungo Barada Mohanty, Swagat Pati, Madhu Singh

421


International Review of Automatic Control (I.RE.A.CO.), Vol. 5, N. 3 ISSN 1974-6059 May 2012

A Review of the State of the Art of Modulation Techniques and Control Strategies for Matrix Converters C. G. Richards1, P. J. Ehlers1, D. V. Nicolae1, E. Monacelli2, Y. Hamam3

Abstract – The reliability and stability of the Matrix Converter has improved during the last years due to the enhanced control algorithms. The traditional direct transfer function control mode has been replaced by more complex – digitally implemented control methodologies. These methodologies allow for real time calculation of the optimal switching interval of each individual switch of the matrix converter. These new switching algorithms allow optimal performances, ensuring sinusoidal outputs at any desired power factor. This paper will first revise the underlying theory of matrix converters, then review the various control limitations and finally review the current control algorithms. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved Keywords: Matrix Converter, Modulation, Control

The paper mathematically defined the output amplitude limitations of the matrix converter. It was shown that, depending on the modulation technique, the output voltage can be either 50% or 86.6% of the maximum input voltage. This control method was later referred to as the direct transfer function approach [4]. Although [4] is a review, the discussion is limited to brief summaries of the various aspects of matrix converters. This paper places an emphasis on present modulation and control techniques.

Nomenclature vi va ( t ) vb ( t ) vc ( t )

Input voltage matrix Input voltage system

vo v A ( t ) vB ( t ) vC ( t )

Output voltage matrix Output voltage system

Skj S ii ia ( t ) ib ( t ) ic ( t )

Bilateral switch Switching matrix Input currents matrix Input current system

io iA ( t ) iB ( t ) iC ( t )

Output currents matrix Output currents system

fs, Ts

Switching frequency, period

I.

Skj

Introduction

A 3x3 matrix converter is a direct AC-to-AC forced commutated cyclo-converter (see Figure 1). The topology of a 3x3 matrix converter does not have an intermediate energy storing dc-link capacitor. This specific topology provides for both voltage and/or infinite frequency modification, through a single stage converter, directly connecting the input source to the output load. The specific switching elements of the matrix converter consist of a controlled bi-directional four quadrant switch. A matrix converter can provide an output of varying frequency and amplitude. Matrix converters are growing in popularity from an interesting power electronic converter to a viable industrialised direct AC-AC converter. The term ‘matrix converter’ was coined by Venturini and Alensia when they presented their work on direct AC-AC converters in [1] and [2]. The authors developed and formalized the analysis and design of a 3x3 matrix converter in [3].

Fig. 1. Matrix Converter

The theory and modulation of a 3x3 matrix converter was further researched. A novel control method was introduced by Rodriguez [5]. This method has been termed the indirect transfer function. Rodriguez proposed that the switching pattern should resemble that of a Voltage Source Inverter (VSI): the output line/load is switched between the most positive and most negative rail of the input using a pulse width modulation technique. This method implies the use of a virtual DClink capacitor in the control methodology. Research

Manuscript received and revised April 2012, accepted May 2012


298

C. G. Richards, P. J. Ehlers, D. V. Nicolae, E. Monacelli, Y. Hamam

where va,b,c (t) is the respective output voltage and vA,B,C (t) is the respective input voltage. The relationship between the input and output voltage may be expressed as the instantaneous transfer matrix:

topics progressed further with the focus shifting from a theoretical approach to more practical applications for variable speed drives for electrical motors [6], [7], [8], [9], [10] and [11]. The direct frequency conversion functionality of the matrix converter is practically useful as a variable frequency drive; due to the smaller form factor and lack of bulky energy storage capacitors. This is evidenced and documented in [12], [13] and [14]. Naturally for development of industrial applications, other operational aspects of the matrix converter were covered. Protection, commutation and operation under abnormal conditions are discussed in [15], [16], [17] and [18]. Figure 2 shows how the various subsystems are integrated for a matrix converter.

vo = S ⋅ v i

(5)

⎡ S Aa ( t ) S Ba ( t ) SCa ( t ) ⎤ ⎢ ⎥ S = ⎢ S Ab ( t ) S Bb ( t ) S Bc ( t ) ⎥ ⎢⎣ S Ac ( t ) S Bc ( t ) SCc ( t ) ⎥⎦

(6)

where:

In the same fashion the input and output current may be expressed as: io = ⎡⎣i A ( t ) iB ( t ) iC ( t ) ⎤⎦

T

(7)

and: ii = ⎡⎣ia ( t ) ib ( t ) ic ( t ) ⎦⎤

T

(8)

where ia,b,c (t) is the respective input currents and iA,B,C (t) is the respective output current.The relationship between the input and output current may be expressed as the transpose of the instantaneous transfer matrix: ii = S T ⋅ io

Fig. 2. Subsystems of a Matrix Converter

II. II.1.

An additional equation for the relationship between the phase voltage values and the line voltage values may be written by inspection:

Commutation

Fundamental Definitions for Matrix Converter

⎡ v AB ⎤ ⎢v ⎥ = ⎢ BC ⎥ ⎣⎢ vCA ⎦⎥

The fundamentals of matrix converters described in [1], [2] and [4] are summarised below. The switching function of the bi-directional switch may be defined as [19]: ⎪⎧1 switch S Kj closed S Kj = ⎨ ⎪⎩0 switch S Kj open

K = { A,B,C}

j = {a,b,c}

⎡ S Aa − S Ab = ⎢⎢ S Ab − S Ac ⎢⎣ S Ac − S Aa

(1)

subject to the constraint:

II.2.

S Aj + S Bj + SCj = 1

T

(3)

and: vi = ⎡⎣v A ( t ) vB ( t ) vC ( t ) ⎤⎦

T

S Ba − S Bb S Bb − S Bc S Bc − S Ba

SCa − SCb ⎤ ⎡ va ⎤ SCb − SCc ⎥⎥ ⋅ ⎢⎢ vb ⎥⎥ SCc − SCa ⎥⎦ ⎢⎣ vc ⎥⎦

(10)

Bilateral Switch Configuration

The matrix converter requires a bi-directional switch capable of blocking voltage and current in both directions. Successful attempts at manufacturing a functional bidirectional have been reported in [20] and [21]. The device characterised of a practical monolithic bidirectional switch is described in [22]. A semiconductor manufacturer has made a monolithic bidirectional switch for matrix converters commercially available [23]. The bi-directional switch may also be constructed from discrete components. The configuration may be Common Collector back-to-back, Common Emitter back-to-back or a diode-switch configuration.

(2)

The input and output voltages, expressed as vectors, are referenced the neutral of the supply: vo = ⎡⎣va ( t ) vb ( t ) vc ( t ) ⎤⎦

(9)

(4)


International Review of Automatic Control, Vol. 5, N. 3

299


Both the common collector (Fig. 3(a)) and the common emitter (Fig. 3(b)) configurations consist of two IGBT’s with two anti-parallel diodes. The diode-switch configuration (Fig. 3(c)) requires four diodes and a single IGBT. A fourth switch configuration (Fig. 3(d)) uses a reverse blocking IGBT. The conduction losses of the reverse blocking IGBT compares favourably with standard IGBT’s, however the switching losses are high resulting in a slower switching frequency for the converter [24]. All the above configurations allow for independent control of the direction of the current.

commutation strategy; however additional protective devices (Figure 4) are required to prevent damage to the converter. Overlap current commutation requires that the incoming switch is triggered before the out-going switching element is turned off. This should cause a short circuit between phases but extra line inductance slows the rise in current so that reliable commutation is achieved. This is not a desired method as the additional inductors would be large. Additionally the switching time for each current commutation is increased causing possible control instability. II.3.

(a)

A clamping circuit may be used to protect the matrix converter (see Figure 4). If the supply for the input is unbalanced and distorted, circulating currents may circulate in the clamping circuit and possibly affect the output current profile. The energy stored in the capacitor of the clamping circuit could be used to drive the auxiliary circuits via a switched mode power supply. An alternative passive protection method using varistors is presented in [24] when the matrix converter is connected to an induction machine.

(b)

(c)

Bilateral Switch Configuration

(d)

Figs. 3. Discrete Switch layout

The common emitter configuration allows both IGBT switches to be controlled from one isolated gate drive power supply. However, in practice, the number of auxiliary power supplies required to drive the IGBT’s is dependent on the method of commutation and the topology of the bi-lateral switch. This number can be between 9 and 18. The common collector configuration is preferred as only six isolated power supplies are needed to supply the gate drive signals. Commutation is a process of switching an individual switch off and another switch on. This current transfer is called commutation. Several authors have described this problem and presented possible solutions [4], [25], [26] and [27]. Several current commutation techniques have been proposed and implemented on matrix converters. Commutation in a matrix converter must be actively controlled, since there is no natural freewheeling path, as in Voltage Source Inverters. Two fundamental constraints must be complied with, (as expressed in (2)), in order to maintain operational stability: • No two bi-directional switches must be switched on at the same time as this would result in a potential short circuit between phases, resulting in excessively high currents, likely damaging the converter. • The bi-directional switches for each output phase should not all be turned off at any instant as this would result in an excessively high overvoltage since there would be no path for the inductive load current. These rules are in conflict for practical solid-state electronic switches since no device can switch on or off instantaneously owing to propagation delays and finite switching times. The various commutation techniques may be summarised from [1], [16], [25], [28], [29] and [30]. The two rules are transgressed in the basic current

Fig. 4. Voltage Clamp Circuit

II.4.

Commutation Overview

The current direction based commutation is more reliable than current overlap commutation [27]. This strategy complies with the rules and uses a four step commutation process in which the direction of the current flow through the commutating switches can be controlled. The difficulty with this method is that the bidirectional switch must be designed in such a way so that the direction of the current flow can be actively controlled in each switch. A dynamic switching pattern is determined from the relative magnitudes of the input voltages to control the current commutation. This method is called the relative voltage magnitude based commutation and is relatively simple to implement, [31] as only the relative amplitude of the input voltage needs to be measured to calculate the space-vector. Resonant switching techniques are well known in general power electronic applications to reduce switching losses. In matrix converters, resonant switching techniques have the additional benefit of solving the current commutation issue. However the additional circuitry increases the component count, increasing conduction losses and



300


requires modification of the control algorithm to operate under all possible conditions. A matrix converter does not have automatic commutation or free-wheeling paths. This is a fundamental issue; matrix converters do not have any natural free-wheeling paths for load current.

1

1

4

3 B

4

1

4

3

A 1

2

II.5.

Commutation Sequences

1

Commutation in a matrix converter must allow for switching to take place without violating the operational stability as described above. In [26], all 12 possible commutation steps are shown for a transition from one switch to another switch. The diagram shows a 4-step commutation technique. In sector A VAB>0 and iLOAD>0, sector B VAB0, Sector C VABEC), a process called ionization appears around the electrode, [2]-[8]. The ionization phenomenon is also affecting the impulse grounding impedance. The goal of this paper is to analyze ionization as a cause of grounding impedance nonlinearity. Soil ionization is a very complex nonlinear phenomenon. Current impulse intensity, configuration and frequency, electrode geometry, soil structure and quality, electrical parameters (resistivity - ρ, permeability - µ, permittivity - ε), chemical parameters (salinity, acidity, alkalinity, ...), heating parameters, humidity, content and arrangement of air pockets, gravity, (non)homogeneity are factors that affect ionization. Due to the ionization model efficiency, certain simplifications have been done. The model focuses on key ionization origination factors – geometry and electric parameters, homogeneity, and uniformity of the observed ground as a medium that surrounds the analyzed electrode, [9].

Nomenclature E EC ρ

εr

J I l i(t) U(t) t1 t2 Zj R, L, C and G EMTP ION.GR EdF IHT RWTH EP B&H -

Electric field in the soil Criticalel ectric field strength in the soil Soil resistance Relative dielectric constant Current density in the soil Magnitude of current impulse Lengthof ground electrode Current impulse Electrode potential Rise time of transient current i(t) Decline time of transient current i(t) Impulse impedance of ground electrode Resistance, inductivity, capacity and conductance – Concentrated Π parameters Electromagnetic transients program Ionization calculation program Electricite de France Institute of High Voltage IHT RWTH Aachen, Germany ELEKTROPRENOS Bosnia and Herzegovina (Company for the Transmission of Electric Power in Bosnia and Herzegovina)

I.

II.

Introduction

The ION.GR ionization simulation model, [8], [9], is based on the fact that the grounding electrode is divided into segments of equal length. The electrode segment is modeled as a transmission line, and as such is represented with Π concentrated parameters R, L, C and

Spreading around the grounding electrode, the currentis generating an electric field in the ground, [1]: E = Jρ

Model

(1)



359

Rasim Gačanović, Hamid Zildžo

edf - the results related to research EdF (Electricite de France); IHT - the results related to research IHT RWTH Aachen, Germany; EP B&H - the results concerning the study made by the Faculty of Electrical Engineering Sarajevo on behalf of Elektroprenos B&H; ION.GR results that are related to research in this paper. Impulse currents applied in the experiment and related research is given in Fig. 2.

G circuit, [8]-[12]. The model computes ionization radius ri for each electrode segment in every time step, calculating the potential on the grounding electrode.

III. Analyzed Configuration As stated earlier, the goal of this paper is to analyze ionization as a cause of grounding impedance nonlinearity. For easier result comparison and verification of this study, the selected configuration was based on references from experimental measurements, or similar computer-based models, Fig. 1. The analyzed configuration is shown on Fig. 1, and it’s parameters are: the grounding electrode is a horizontally laid round-shaped copper wire with a radius of r=6.078mm (S=116mm2), the electrode length is 15m, and is trenched at 0.6m in a soil, that has soil resistivity equal to ρ=65Ωm, and relative dielectric constant εr=15. The experimental research results from the French Power Utility - Electricite de France (EdF), and computer-based simulation model from IHT RWTH Aachen, Germany and results from a study by ELEKTROPRENOS Sarajevo which was done by the Faculty of Electrical Engineering Sarajevo are available for this grounding configuration. The french power utility Electricite de France (EdF) has published extensive experimental research of grounding impulse characteristics 1977/78 and 1985 in EdF laboratoriesLes Renardieres and St-Brieuc, [13]-[15]. In the mid 90-ties The IHT RWTH Instute from Aachen, Germany developed a frequency dependent simulation model for grounding grid, using frequency dependent lines, which can be found in computer software EMTP (Electro Magnetic Tranisents Program). Using the EdF experimental results, IHT RWTH Aachen made a review of their computer based simulation model, [16]-[18]. In 2004, the Faculty of Electrical Engineering Sarajevo, has made calculations for Elektroprenos B&H, and the results were published in a study entitled Modeling frequency dependent grounding parameters, [19], [20]. The model is based on a decomposition of the power system on a two subsystems: subsystem – substation (equipment and lightning protect system) and subsystem – Grounding. Interconnection of these subsystems is done in each time step using Thevenin's theorem. Frequency dependence of impedance grounding is represented by the parallel combination of resistance-inductance, which allows the simulation in time domain. The EdF, IHT RWTH Aachen and Elektroprenos B&H studies, do not take into account soil ionization around the grounding electrode, whereas this study use a computer model ION.GR which takes into account ionization, [9].

IV.

A Length l = 15m

0.6 m

2

i(t)

AIR SOIL

Electrode: Cu, 116 mm

ρ=65Ωm, ε

A ELECTRODE RADIUS r DEPTH h

CONSTANT CONDUCTIVITY NONCONSTANT COND.

ρ, ε IONISATION FILAMENTARY SPACE ELECTRODE

View A - A

i[A]

Fig. 1. Analyzed configuration 40 35 30 25 20 15 10 5 0

edf ION.GR

0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 time [µs] Fig. 2. Current Impulses i(t)

In ION.GR, the generated current shape is 0.6/5µs (t1/t2 - ”rise time of transient current i(t)”/”decline time of transient current i(t)”), with a simulation time of 0.7µs, which shows quite good agreement with experiments. The grounding electrode potential as a function of time, obtained by different methods is given in Fig. 3. Zj–impulse impedance of ground is defined as the ratio of current voltage and current impulse values at the beginning of the grounding:

Calculation

On the diagrams below, the following labeling is used: Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


360


Z(t)=U(t)/i(t)

Fig. 6 shows the variability of the impulse impedance in time for the current impulse i(t)...0.6/5µs intensity 35.46A, 1kA and 50kA. Nonlinearity Z(t) is significant in the time interval 0.0-0.2 µs, after which it subsequently becomes a linearized function.

(2)

where: U(t)- potential of electrode, i(t) - current impulse. ION.GR

EP B&H

IHT

edf

35.46A

2000

50kA

150 Z [Ohm]

U [V]

1500 1000 500

100 50 0

0,7

0,6

0,65

0,55

0,5

0,4

0,45

0,3

0,35

0,2

0,25

0,15

0,1

0

0,05

0

0

time [µs]

EP B&H

IHT

edf

i [A] 0,7

0,6

0,65

0,55

0,5

0,45

0,4

0,35

0,3

0,25

0,2

0,15

0,3

0,4

Calculation and comparison of results was made for the ground configuration from Fig. 1, excited by current impulse from Fig. 7 (I = 32.77A). The program ION.GR the generated current shape is 0.8/8µs, with a duration of 3µs.

60 50 40 30 20 10 0 0,1

0,2

Fig. 6. Impulse impedance Z as function in time for different values of the intensity of current pulses (Current impulse i(t)…0.6/5µs )

Fig. 4 shows the results of the impulse impedance Z(t) for the selected configuration of the ground from Fig. 1, excited by a current impulse from Fig. 2 (I = 35.46A). ION.GR

0,1

time [µs]

Fig. 3. Grounding electrode potential

Z [Ohm]

1kA

200

time [µs] Fig. 4. Impulse impedance as a function of time

40 35 30 25 20 15 10 5 0

ION.GR EP B&H

0

This study analyzed the impact of the nonlinearity ionization impulse impedance through a change of intensity electric pulses (35.46A, 1kA, 2kA, 5ka, 10kA, 20kA and 50kA), at time points 0.1, 0.3, 0.5 and 0.7 microseconds - grounding excited by impulse i(t)...0.6/5µs, duration 0.7µs, Fig. 5.

1

2

3

time [µs] Fig. 7. Current impulses i(t)

ION.GR

EP B&H

edf

60 50

at 0.3mic.s. at 0.7mic.s.

Z [Ohm]

at 0.1mic.s. at 0.5mic.s. Z [Ohm]

40

40 30 20 10

20

0 0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9 2,1 2,3 2,5 2,7 2,9

0

time [µs] 0

10

20 30 Current [kA]

40

50 Fig. 8. Impulse grounding impedance as a function of time

Fig. 5. Change of impulse grounding impedance Z, depending on the intensity of currents at various points in time (Current impulsei(t)…0.6/5µs )

Fig. 8 depicts the results of the impulse impedance Z(t) for the selected configuration of the ground from Fig. 1, excited by current impulses from Fig. 7 (I =



361


This paper also analyses the possible impact of the change in impedance of the pulse depending on the form of excitation functions of the current at the beginning of grounding electrodes – the current impulse. As it can be seen from Fig. 11, Fig. 6 and Fig. 10, the impulse impedance function Z(t) for the current impulses 0.6/5µs and 0.8/8µs has a tendency to significant diverge at the beginning of the injection impulse (the time interval 0-0.1 µs). The slight difference exists in the time interval up to 0.1-0.2 µs, and then it functions almost coincide.

32.77A). Fig. 9 presents the influence of the nonlinearity ionization impulse impedance by changing the intensity of electrical impulses (32.77A, 1kA, 2kA, 5kA, 10kA, 20kA and 50kA), in time points 0.1, 0.3, 0.8, 1.5 i 3µs – the grounding excited by current impulse i(t)...0.8/8µs, duration 3µs. Fig. 10 shows the variability (nonlinearity) of impulse impedance in time for the current i(t)...0.8/8µs with intensity 32.77A, 1kA and 50kA. at 0.1mic.s.

at 0.3mic.s.

at 1.5mic.s.

at 3.0mic.s.

at 0.8mic.s.

V.

50

Research results and analyses conducted in this paper show that there is a significant functional connection between the ionization phenomena and nonlinear impulse grounding impedance. This functional relationis especially pronounced in the change (increase) of intensity current impulse from 1-20 kA, when ionization is intense. When generating impulse impedance nonlinearities, other factors are important, such as the form of excitation impulses of current in grounding, observation periodof Z(t), pulse duration, grounding configuration, etc. For sure, it would be good to continue research, including other factors that influence the ionization phenomena, such as heat, humidity.

Z [Ohm]

40 30 20 10 0 0

10

20 30 Current [kA]

40

50

Fig. 9. Change of impulse impedance grounding Z depending on the intensity of current at various points in time (Current impulse i(t)…0.8/8µs)

32.77 A

50

1kA

50kA

40 Z [Ohm]

Conclusion

References

30

[1]

20 [2]

10 0 0

0,2

0,4 0,6 time [µs]

0,8

[3]

1

Fig. 10. The function of the pulse impedance Z in time for different values of the intensity of current (Current impulse i(t)…0.8/8µs) 0.6/5microsec.

[4]

0.8/8microsec.

80

[5]

60 Z [ohm]

[6]

40 20

[7]

0 0

0,2

0,4 time [µs]

0,6

0,8 [8]

Fig. 11.Change of impulse impedance in time for the current impulses 0.6/5µs and 0.8/8µs [9]


E. D. Sunde, Earth Conductors Effects in Transmission Systems, D. Van Nostrand Company, Inc., Toronto, New York, London 1949. Mousa A. M., “The soil ionization gradient associated with discharge of high currents into concentrated electrodes,” IEEE Trans. on Power Delivery, Vol. 9, 1669–1677, Jul. 1994. D’Angola, A., M. Capitelli, G. Colonna, C. Gorse, “Transport properties of high temperature air in local thermodynamic equilibrium,” The European Physical Journal D, Atomic, Molecular, Optical and Plasma Physics, Vol. 11, No. 2, 279–289, Jul. 2000. Capitelli, M., G. Colonna, A. D’Angola, “Thermodynamic properties and transport coefficients of high temperature air,” Pulsed Power Plasma Science. PPPS-2001. Digest of Technical Papers, Vol. 1, No. 17–22, 694–697, Jun. 2001. G. Ala, M. L. D. Silvestre, F. Viola, Soil Ionization due to high pulse Transient currents leaked by earth electrodes, IEEE, Progress In Electromagnetics Research B, Vol. 14, 1–21, 2009. R. Gačanović, H. Zildžo, H. Matoruga, Computer Model Analysis of Variations of Parameters of the Short-Horizontal Grounding Electrode for Ionization, International Review on Modelling and Simulations (IREMOS),Vol. 3, N. 4, pp. 634-638, August 2010. R. Gačanović, H. Zildžo, H. Matoruga, Effect of Changes in the Shapeof Input Wave to Ionization of the Horizontal Grounding – Computer Model Analysis, International Review on Modelling and Simulations (IREMOS), Vol. 3, N. 6, pp. 1256-1260, December 2010. R. Gačanović, I. Hadžizulfić, S. Sadović, M. Raščić, R. Mahmutćehajić, Principles and Methods of Modelling Soil ionization on Grounding Installations, VII Symposium BHK CIGRE, Neum 2005. I. Hadžizulfić, Modeling of ionization on grounding system, Master thesis - Mentor R. Gacanovic, ETF Sarajevo, 2006.


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[10] Geri A., Veca G. M., Garbagnati E., Sartorio G., Non-linear behaviour of ground electrodes under lightning surge currents: computer modelling and comparison with experimental results, IEEE Transactions on Magnetics, Vol. 28, Issue 2, pp. 1442-1445, 1992. [11] Golde R. H, Lightning currents and related parameters, Vol.1, Physics of lightning, Academic Press, pp. 309-350, 1977. [12] Y. Liu, Transient Response of Grounding Systems Caused by Lightning: Modeling and Experiments, Ph.d. Thesis, Acta Universitates Upsaliensis, Upsala 2004. [13] P. Kouteynikoff: 'Modele pour le calcul de l'impedamce d'une preise de terre longue aux basses et hautes frequeces' DER-EDF HM/72-04670, January 1982. [14] F. Villefranque: 'Reponse de differentes prises de terre parcourues par un courant impulsionnel' DER-EDF HM/72-3915 FV/PS, April 1977. [15] F. Villefranque: 'Comportement de differentes prises de terre reelles parcourues par des courant impulsionnels ayant differetes durees de front' DER-EDF HM/72-3915 FV/PS, February 1978. [16] H. Rochereau: 'Comportement des prises de terre localisees parcourues par des courants a front raide' DER-EDF HM/73-322 HR/TPLD, January 1988. [17] P. Anzel: 'Comportement en regime impulsionnel des prises de terre localisess: Nouveauh resultats experimentaux et synthese de connaissances actuelles' DER-EDF HM/73-110 PA/Ar/BF, January 1976. [18] B. Merheim: 'Modellierung von Hochspannungs-Erdersystemen und Vergleich mit Messungen ihres dynamischen Verhatens' IHT RWTH Aachen , Diplomarbeit , Oktober 1992. [19] H. Zildžo, S. Sadović, R. Gačanović, Z. Haznadar, R. Mahmutćehajić, F. Imamović, Modeling of Frequency dependence of the Grounding Parameters – Study, Transmission of Electric Power Co., Sarajevo, Bosnia&Herzegovina. Faculty of Electrical Engineering, University of Sarajevo, Sarajevo, 2004. [20] S. Sadović, R. Gačanović, Impulse Characteristics of Grounding, Textbook, Faculty of Electrical Engineering - Universiti of Sarajevo (B&H), Sarajevo 2008.

Hamid Zildzo was born in Sarajevo (Bosnia and Herzegovina) on 05th of June, 1960. He received a B.S. degree in Electrical Engineering, M.S. degree and Ph.D. degree from University of Sarajevo (Bosnia and Herzegovina), in 1983, 1986 and 1995, respectively. His topics of interest are numerical calculation electromagnetic field. From 1997 he has been working at the Faculty of Electrical Engineering, University of Sarajevo

Authors’ information Rasim Gacanovic was born in Sarajevo, Republic of Bosnia-Herzegovina in 1950. He received the Diploma Engineer and Master of Science degree from the Sarajevo University (B&H) and Doctor of Science degree from Zagreb University (Croatia) in 1974, 1980, and 1991 respectively. Dr Gacanovic has been with the Transport Institute IPS-a Sarajevo (Research Fellow) and with Energoinvest Electric Power Research Institute – IRCE (Research Scientific). In 1991 he was with the Osijek University (Croatia), were he was the Scientific Associate. Since 1993 till now Dr Gacanovic is with Sarajevo University (Professor in Electric Power Engineering). He is Member of IEEE. From 1998 – 2000 Prof.dr.sci.Rasim Gacanovic was Mayor of Sarajevo City. Areas of Interesting: Traveling waves in electrical Engineering; Electromagnetic Compatibility; Frequency dependence modeling in Power system transients simulations; Power analysis; Grounding in Power system and Telematic tehnology.



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Optimal Nonlinear Control of Electrostatic Micro Actuator System Mohammad Nasser-Moghadasi1, Farbod Setoudeh2, Seyed A. Olamaei3 Abstract – This paper propose the control of electrostatic micro actuator, the feedback linearization method has been used for linearization of system and then linear quadratic regulator (LQR) has been used for stabilizing and controlling of electrostatic micro actuators which can be represented as a nonlinear system. The simulation results are very promising. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Electrostatic Micro Actuator, Feedback Linearization, Linear Quadratic Regulator

I.

The proposed control schemes guarantee the stability of the closed loop system [8], Sabzehmeidani et al. designed Hybrid Fuzzy-based Robust Controller for Pneumatically Actuated Micro Robot [9], Fakhrabadi et al. worked on Static and Dynamic Behaviors of an Electrostatically Actuated Micro Beam [10], Kuzu et al. designed High-order Sliding Mode Controllerfor MEM Optical Switches [11]. In this paper the optimal control of electrostatic micro actuator has been proposed, section 2 proposed the reviewelectrostatic micro actuator and nonlinear control of electrostatic micro actuator proposed in section 3.

Introduction

Micro electromechanical systems (MEMS) (also written as micro-electro-mechanical, Micro Electromechanical or microelectronic and micro electromechanical systems) is the technology of very small mechanical devices driven by electricity; it merges at the nano-scale into nano electromechanical systems (NEMS) and nanotechnology. MEMS are also referred to as micro machines (in Japan), or Micro Systems Technology - MST (in Europe). MEMS are separate and redistrict from the hypothetical vision of molecular nanotechnology or molecular electronics [1]. In the recent years many researchers has been worked on this electrostatic micro-actuators (E µ As) MEMS electrostatic actuators, also termed as micro-actuators, are the key devices allowing MEMS to perform physical movements [2]. Some examples of the applications OF electrostatic actuators are: micro-mirrors, optical gratings, variable capacitors, and micro-accelerometers [3]. The control of micro-scale devices has gained increasing attention in recent decades, such as Vagia et al. designed a robust switching PID controller coupled to a feed forward compensator for set-point regulation of an electrostatic micro-actuator [4], Tee et al. presented adaptive output feedback control for a class of 1DOF electrostatic microactuator systems, such that the movable plate is able to track asymptotically a reference trajectory within the air gap without knowledge of the plant parameters, and without any contact between the movable plate and the electrodes [5]. Maithripala et al. proposed Control of an electrostatic micro electromechanical system using static and dynamic output feedback [6]. Edwards described the mathematical modeling and closed-loop voltage control of a MEMS electrostatic actuator. The control goal is to extend the travel range of the actuator beyond the open-loop pull-in limit of one third of the initial gap [7]. Zribi et al. designed a feedback linearization controller, a static sliding mode controller and a dynamic sliding mode controller for the micro electromechanical system.

II.

A Review on Electrostatic Actuated MEMS Resonator

Fig. 1 presents the electro statically actuated structure [4], the dynamic equation of system is:

mη + bη + kη = Fx

(1)

where m is the plate’s mass, b is the damping caused mainly by the motion in the air, k the spring’s stiffness, η is the displacement of the plates from the relaxed position, η max is the distance between the plates when the spring is relaxed, A is the area of the plate, e is the dielectric constant, m is the plate’s mass, b is the damping caused mainly by the motion in the air, k the spring’s stiffness, U is the applied voltage between the capacitor’s plates, and Fx is the actuation force or electrically induced force. The net actuation force Fx can be expressed as: Fx =

ε AU 2 2 (ηmax − η )

2

(2)

where A is the area of the plate, ε is the dielectric constant, U is the applied voltage between the



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Mohammad Nasser-Moghadasi, Farbod Setoudeh, Seyed A. Olamaei

capacitor’s plates and η max is the distance between the plates when the spring is relaxed [4]. If we consider η = x1 ⋅η = x2 then state equation of system can be write as following equations: 1

Using lemma 1 we can see that there exist a region D, such that the vector containing the origin, in is linear independent in D and the vector , is involutive in D such that . Using Feedback linearization Method the control input has been considered as:

(3)

1

(5)

That is the control signal. Then, the linearized system yield as bellows:

1

(6)

Next, we design a linear state feedback controller: Fig. 1. Electrostatic actuated MEMS structure

III. Nonlinear Control of Electrostatic Micro-Actuator If we consider (3) can be written as:

To stabilize the Chua's circuit after feedback linearizationlinear quadratic regulator method has been used. For a practical case the parameters for (6) are: .7 10 , 9 10 . .8, 1.8 10 , In order to illustrate the effectiveness of the proposed controller, in this paper we simulate system (4) in initial 0.1 condition . Figure 2 shows simulation result for 0 uncontrolled and controlled systems in initial 0.1 condition . 0

as input control the equation (4)

which:

1 and: 0 2 Following lemma shows sufficient condition for feedback linearizability.

Lemma 1 [9]: The nonlinear system:

with , smooth vector fields and 0 0 is feedback linearizable if and only if there exists a region D, containing the origin, in in which the following conditions are satisfied: ,…, are (a) The vector fields , linearly independent. (b) The set , …, is involutive in D, that is, all the Lie brackets of each pair of the vector fields of the set have to be a linear combination of them.

Fig. 2. Simulation of close loop controlled and open loop without control of electrostatic actuated MEMS structure in initial condition equal to (0.1,0)



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IV.

Mohammad Naser-Moghadasi was born in Saveh, Iran, In 1959. He received the B.Sc. degree in Communication Eng. in 1985 from the Leeds Metropolitan University (formerly Leeds polytechnic), UK. Between 1985 and 1987 he worked as an RF design engineer for the Gigatech company in Newcastle Upon Tyne, UK. From 1987 to 1989, he was awarded a full scholarship by the Leeds educational authority to pursue an M.Phil. Studying in CAD of Microwave circuits. He received his Ph.D. in 1993, from the University of Bradford, UK. He was offered then a two years Post Doc. To pursue research on Microwave cooking of materials at the University of Nottingham, UK. From 1995, Dr. Naser-Moghadasi joined Islamic Azad University, Science & Research Branch, Iran, where he currently is head of postgraduate studies and also member of Central Commission for Scientific Literacy & Art Societies. His main areas of interest in research are Microstrip antenna, Microwave passive and active circuits, RF MEMS. Dr. Naser-Moghadasi is member of the Institution of Engineering and Technology, MIET and the Institute of Electronics, Information and Communication Engineers (IEICE). He has so far published over 130 papers in different journals and conferences.

Conclusion

Nonlinear controllers are proposed for a electrostatic micro actuator. The feedback linearization method has been used for linearization of system and then linear quadratic regulator (LQR) has been used for controlling of electrostatic micro actuators which can be represented as a nonlinear system. The simulation results are very promising.

References [1]

W. Jean-Baptiste, Nano computers and Swarm Intelligence, London: ISTE John Wiley & Sons. [2] Y. Nemirovsky, A methodology and model for the pull-in parameters of electrostatic actuators, Journal of Micro electromechanical Systems, Vol. 10, No. 4, pp. 601-615, Dec. 2001. [3] J. Seeger, Charge control of parallel-plate, electrostatic actuators and the tip-in instability, Journal of Microelectromechanical Systems,Vol. 12, No. 5, pp. 656-671, Oct. 2003. [4] M. Vagia, G. Nikolakopoulos, A. Tzes, Design of a robust PIDcontrol switching scheme for an electrostatic micro actuator, Journal of Control Engineering Practice, Volume 16, Issue 11, November 2008, pp. 1321–1328. [5] Keng-Peng Tee, Shuzhi Sam Ge and Eng Hock Tay, OutputFeedback Adaptive Control of Electrostatic Micro actuators, American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12, pp. 4215-4220, 2009. [6] Jason M. Edwards, Modeling and feedback control of a mems electrostatic actuator, Master of Science Thesis in Electrical Engineering at the Cleveland State University, December 2008. [7] M. Zribiy, M. Karkoub, Nonlinear Control of an Electrostatic Micro electromechanical System, Common Nonlinear Sci Number Simulat 2010. [8] K. Chih, J. Chih, and C. Liang, Input-state linearization of a rotary inverted pendulum, Asian Journal of Control, March 2004, Vol. 6, No. 1, pp. 130-135. [9] Y. Sabzehmeidani, M. Mailah, M. Hussein, E. Gatavi, M. Z. MdZain, A Hybrid Fuzzy-based Robust Controller for Pneumatically Actuated Micro Robot, International Review on Modelling and Simulations (IREMOS), Vol. 3 N. 6, pp. 13081316, December 2010. [10] M. M. S. Fakhrabadi, B. Dadashzadeh, V. Norouzifard, M. Dadashzadeh, Static and Dynamic Behaviors of an Electrostatically Actuated Micro Beam, International Review on Modelling and Simulations (IREMOS),, Vol. 4 N. 2, pp. 710-717, April 2011. [11] Ahmet Kuzu, Seta Bogosyan, Metin Gokasan, High-order Sliding Mode Approach for the Control of MEM Optical Switches, International Review of Automatic Control (IREACO), Vol. 4 N. 3, pp. 431-438, May 2011.

Farbod Setoudeh received B.S. in Electrical Engineering from Islamic Azad University, Arak, Iran and M.S. degrees in Electronics from Islamic Azad University, Arak, Iran, in 2006 and 2008 and he is now Ph.D. candidate in Electrical Engineering (Electronic) from Science and Research Branch, Islamic Azad University, Tehran, Iran. He is teaching now in the Department of Electrical Engineering at IUST University and Arak University, His research interests are linear and non-linear control system and systems identification, Neural Network, Modeling, stability, circuits design, and Earthquake Prediction. Seyed A. Olamaei, Received his M.S. degree in Electronics & Communications Engineering from the Islamic Azad University – south Tehran Branch, IRAN in 1996 and graduated in PhD degree in Electronics & Communications Engineering at Science and Research Branch Islamic Azad University, Iran in 2003. He is currently an Asistant Professor in the Department of Electrical Engineering, at Islamic Azad University – south Tehran Branch. From 2003 to 2006 He was a member of technical staff of the wireless Network Department of TCI, IRAN and also a member of ITU-T Study Group 5 - Environment and climate change. He has been involved in various projects in Research Institute for ICT, IRAN. His current research interests include Rx. & Tx. in wireless Network, Electromagnetic waves absorption in cellular Network, protection against Electromagnetic waves quality of service in wireless Network, LAN/MAN Architectures and protocols, Cellular Radio Telecommunication Services, etc.

Authors’ information 1

Department of Electrical and Computer Eng., Science and Research Branch, Islamic Azad University, Tehran Iran. E-mail: [email protected] 2

Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran. 3 Department of Electrical Engineering, Islamic Azad University-South Tehran Branch Tehran, Iran.



366


A Systematic Approach to a Time Series Neural Model Development for River Flow Forecasting Petar Matić1, Ozren Bego2, Ranko Goić2

Abstract – This paper aims to apply a systematic approach to a time series neural model development procedure. The model is developed for flow forecasting of river Cetina, technoeconomically the most important basin in Croatia according to the annual energy production. Multi-Layer Perceptron was used to model hydrological time series of a measured daily river flow. The best model was determined through an experiment based on a values comparison of different error measures (SEE, RMSE, MAE, and CE). In order to determine the best model, 780 MLP neural networks were trained using Levenberg-Marquardt training algorithm. Simulation results indicate high accuracy of flow forecast for one-step-ahead and therefore provide encouragement for further research. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Artificial Neural Networks, Forecasting, Modeling, River Flow, Time Series

However, the classical time series models assume a linear relationship between input and output variables, and since the hydrological systems are highly nonlinear [2], the error resulting from use of these models is inevitable. Artificial neural networks have proven to be a powerful tool for modelling different sorts of nonlinearities, as shown in [3], [4] and [5]. An extensive report from [6] confirmed good forecasting abilities of diverse neural models, but also emphasized the lack of a systematic approach to the neural model development procedure. A systematic approach can be described as a set of well defined steps in a model development procedure. It is usually advised to start from the simplest solutions, by applying commonly used, proven methods for every step of the procedure to form a “Benchmark model” [7]. Furthermore, there are many meteorological and hydrological variables influencing the river flow, but variables for flow modelling must be carefully selected [7]. Therefore, it is advised to start only with the flow variable, which means modelling a time series. In this paper, systematic approach is applied to develop a neural model for daily flow forecasting of river Cetina. A basic idea of the systematic approach is presented in a third chapter of this paper, and model development process is presented in the fourth chapter. Artificial neural networks are shortly described in second chapter, and conclusion remarks from this research as well as suggestions for the further work are stated in the fifth chapter.

Nomenclature ϕ x o k E f η w Q ∆ NH NI NS ∂ v t m

Artificial neuron's activation function Artificial neuron's input Artificial neuron's output (response) Calculation step Error Function Intensity of adjustment ("speed of learning") Interconnection strength between artificial neurons (weights) Mean daily value of the river flow Modification (adjustment) Number of artificial neurons of the hidden layer Number of artificial neurons of the input layer Number of training samples (input-output pairs) Partial-derivative symbol Sum of the "weighted" artificial neuron's inputs Time step (discrete) Total number of the artificial neuron's inputs

I.

Introduction

Natural water resources are limited in terms of quantity and appearance and therefore should be used rationally regardless of its purpose. In order to optimize the use of water, hydrological models are used to obtain hydrological forecasts. In the past few decades several types of hydrological models have been developed. These models can generally be divided into statistical and physical models. Based on the set of measured values it is possible to form a statistical time series model that can predict values of the observed variable with a certain degree of accuracy, as in [1].

II.

Artificial Neural Networks

Artificial Neural Network (ANN) can be described as



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P. Matić, O. Bego, R. Goić

a simplified mathematical representation of the process performed by a biological neural network. It consists of a set of interconnected artificial neurons whose final shape is formed after the learning process is over. Fig. 1 shows a general representation of the most commonly used network architecture, a Multi-Layer Perceptron (MLP).

Neural network training can be defined as an iterative process of adjusting network parameters, which is performed according to a training algorithm that aims to determine the values of network parameters for which the error would be minimal. Training algorithms can be generally divided into local and global algorithms. First-order local algorithms are based on the linear approximation of error function (4). Step adjustment (∆wk) is calculated by the algorithm in order to reduce the error, where η defines the intensity of adjustment: ∆wk = −η

∂Ek ∂wk

(4)

The calculated adjustment of network parameters is then added to the current value as expressed by (5): wk +1 = wk + ∆wk

Error Back-Propagation algorithm (BP) is a first-order local algorithm, basic and most well known method for MLP network parameter optimization. Its advantage is reflected in a relatively simple determination of partial derivatives in (4). A various improvements of the original BP algorithm have been developed over the years, but “the problem of local minimum jamming” remained. Second-order algorithms are based on the quadratic approximation of the error function, which enables a more direct search for the error function minimum. Furthermore, if the objective function is known it is possible to develop more efficient, specialized algorithm to adjust the network parameters [8]. This is the case with Levenberg-Marquardt (LM) algorithm, developed particularly to train MLP networks. LM algorithm is also a second-order algorithms and it has proven to be the most appropriate algorithm for training the network of the modest structure containing up to several hundreds of adjustable parameters [3]. LM algorithm is used in this study and its detailed description can be found in [8]. Detailed description of ANNs in general can be found in [9].

Fig. 1. Multi-Layer Perceptron

MLP is a static, feed-forward neural network (therefore, also referred to as FFNN) that consists of at least three layers of neurons: input, hidden and output layer. In general case, every neuron of one layer is connected to every neuron of the next layer. An artificial neuron in the k-th step calculates the response ok (1), where vk denotes the sum of m-inputs as defined by the expression (2): ok = ϕ ( vk )

vk =

(1)

m

∑ ( wk ⋅ xk )n

(2)

n =0

Expressions (1) and (2) represent the mathematical model of artificial neuron whose inputs are marked with the xk, and the values of interconnection strength (weights) are denoted by wk. The properties of the network depend on the properties of the neuron, i.e. the type of activation function (φ). The most commonly used activation functions are linear, linear limited, unipolar and bipolar sigmoid function. During the training process the network is provided with a set of input and output pairs in order to “set an example” for the network. Then, a training algorithm is used to adjust network parameters (w) in order to make network produce the desired output for the associated input. Initially excited network would certainly produce response different from the desired output, i.e. it would produce an error because of unadjusted network parameters. Therefore, the error can be defined as function of network parameters (3): E = f ( w)

(5)

III. Systematic Approach to River Flow Forecasting with ANN Despite a significant amount of research activity on the use of ANNs for prediction and forecasting of water resources variables in river systems, little of this is focused on methodological issues. Consequently, there is still a need for the development of robust ANN model development approach [6]. Based on a proposed scheme found in [7] and [6], a systematic approach to the neural model development for the flow forecasting can be shortly described as a sevenstep-procedure.

(3)



368


III.1. Step 1: Goal Determination

Besides MLP, Radial Basis Function (RBF) neural networks are often used, as well as different temporal neural network types, such as: RNN (Recurrent Neural Networks), TDNN (Time Delayed Neural Networks), TLNN (Time Lagged Neural Networks) or IDNN (Input Delayed Neural Networks).

To determine a model’s objective or goal means to determine the output variable. Therefore, if the goal is to forecast a daily flow, than the model output variable is a current flow. III.2. Step 2: Selection of the Input Variables

III.6. Step 6: Neural Model Development Procedure

The input variables of the model are chosen according to the type of model that is being developed. For example, in the case of a time series (TS) modelling, input variables are members of a single time series variable (e.g. flow), and in case of the rainfall-runoff (RR) modelling, input variables are members of the two time series variables (e.g. rain and flow), as in [10]. Since the flow forecasting is based on time series, it is necessary to determine the appropriate number of consecutive samples of the input variables that have a significant impact on the current values of output variables. Both linear (e.g. autocorrelation function) and nonlinear methods (e.g. mutual information) are often used to determine the appropriate lags.

Neural model development procedure can be divided into three stages: structure selection, model calibration and model evaluation. These three stages are sequentially repeated until the best model is found, according to one or more quality measures. In case of MLP architecture, determining the final network structure is reduced to determining the number of hidden layer neurons, since the number of input neurons is determined by the dimension of the input vector, and the number of output neurons is determined by the output vector dimension: ⎛ N ⎞ N H ≤ min ⎜ 2 N I + 1, S ⎟ NI +1 ⎠ ⎝

III.3. Step 3: Data Pre-Processing

(6)

Although there is no widely accepted way to determine the optimal number of hidden neurons, it is proposed taking into account the minimum of NH from (6), where NI denotes the number of inputs, and S denotes the number of training samples [11]. Calibration is the process of adjusting model parameters, i.e. network training. It is important to emphasize the importance of the network parameters initialization that determines the final outcome of the network training. The most commonly used algorithms for network training, local algorithms, use the set of randomly chosen initial values of network parameters as a starting point. Therefore, in such cases it is advised to repeat training for different sets of initial values [3]. Model evaluation, as a part of model development procedure, is usually performed by a single error measure and Mean Squared Error (MSE) is commonly used.

Processing of data in this case means the transformation of data into a form suitable for neural network operation. The simplest and most commonly used method of data processing to form the neural model is a linear transformation [11], which allows data (inputs) to be rescaled within the desired limits, or to be standardized to a certain value. Generally, it is recommended either to scale data between -1 and 1, or standardizing them to a mean of zero and standard deviation of one. III.4. Step 4: Data Division First, all available data are divided into training and validation data, because the data with which model is validated should not be presented to the network during training. Second, a set of training data is divided into sub-sets of data for calibration, testing and evaluation. Such division is essential if the early stopping method is applied as a measure against overtraining, i.e. overfitting [3]. Since the network can only predict values within the range encountered during training [12], all sets of data must contain the same statistical properties [13].

III.7. Step 7: Model Validation Different quality (error) measures are used to evaluate different aspects of model accuracy, i.e. to validate the model. According to [14], Legates and McCabe (1999) suggested that these include at least one relative error measure, such as the coefficient of efficiency (CE), and at least one absolute error measure, such as Root Mean Squared Error (RMSE), or Mean Absolute Error (MAE). Also, to evaluate the accuracy of predictions a Standard Error of Estimate (SEE) can be used.

III.5. Step 5: Selection of Network Architecture The selection of network architecture represents the selection of the type of network that will be used to form the neural model. MLP neural network with universal approximation property [3] is commonly used for hydrological modeling. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


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IV.

encountered during training, [13], [12], data from the year 2009 were selected for validation because it does not contain the extreme minimum nor maximum values of the entire set. Data used for validation were not presented to the network during training.

A Day-Ahead Flow Forecast for River Cetina

A neural model for predicting daily flow of the river Cetina (i.e. an inflow into the reservoir Peruća) for oneday-ahead is developed based on the flow data measured at station Vinalić. A number of 1461 mean daily values of the river flow for the period of 2007 to 2010 were provided by the Croatian Meteorological and Hydrological Service.

IV.3. Neural Model Development Procedure To develop a flow forecasting neural model of the river Cetina an MLP with one hidden layer was used, as recommended in [11]. Since the number of output neurons is determined by dimension of the target (output) vector, all created models have one output neuron. Number of input neurons is determined by the dimension of input vector, so that the first group of models (ANN1) has one input neuron, the second group (ANN2) has two input neurons, etc. Number of hidden layer neurons is determined through an experiment. First, ten neural networks with one neuron in the hidden layer (ANN1-1-1) were trained and validated with the use of four different error measures. The procedure was then repeated with sequentially increasing the number of hidden layer neurons for all of the neural models from Table I. Increasing the number of hidden layer neurons was stopped at thirteen because it was the least strict criterion obtained from expression (6), and it was applied to all of the model groups in order to train all models in the same conditions. Consecutive training of each network was applied in order to reduce the effect of random outcome, and number of 10 was determined as optimal according to [6]. Therefore, for the purposes of determining the best model a total of 780 neural networks were trained. All networks were trained with LM algorithm using MSE as the objective function and early stopping method to avoid overtraining the network. The data was scaled between -1 and 1 during training. The error measure mean values (from 10 consecutive trained networks) were calculated for all obtained models and the mean values were compared. The results of this comparison are shown graphically in Figs. 3 and numerically in Table II. The results clearly define the neural model with three inputs and six hidden layer neurons (ANN3-6-1) as the best model, since it provides the lowest SEE, RMSE and MAE and the highest CE (see Table II). The trend lines drawn on graphs from Figs. 3 indicate that the extreme values are probably the points of global extremes which means that further increase of the network parameter number (i.e. number of input and hidden neurons) would not increase the model quality.

IV.1. Determination of the Input and Output Vectors Time series of daily flow are input variables, and current flow (Qt) is the output variable. The autocorrelation function (ACF) was used to determine time series lags.

Fig. 2. ACF of the modeled time series data

The results indicate the influence of 77 consecutive members of the time series to the current flow value (see Fig. 2). However, what number of previous samples used as inputs would produce the best model cannot be concluded from this analysis. In order to choose the best model, six groups of models with different members of the input vector were formed (see Table I). A sequence of six consecutive samples was chosen as the highest, since the correlation coefficient falls below 0,75 afterwards. Therefore, it is assumed that influence of the remaining members of the time series is not of great significance. TABLE I INPUT VECTORS FOR DIFFERENT GROUP MODELS Group Input vector members

ANN1 ANN2 ANN3 ANN4 ANN5 ANN6

Qt-1 Qt-1 , Qt-2 Qt-1 , Qt-2 , Qt-3 Qt-1 , Qt-2 , Qt-3 , Qt-4 Qt-1 , Qt-2 , Qt-3 , Qt-4 , Qt-5 Qt-1 , Qt-2 , Qt-3 , Qt-4 , Qt-5 , Qt-6

TABLE II NUMERICAL ERROR MEASURE COMPARISON SEE RMSE MAE ANN 1-2-1 4,3183 4,4039 1,9344 ANN 2-3-1 3,8217 3,8678 1,7328 ANN 3-6-1 3,6485 3,6794 1,6521 ANN 4-5-1 3,7426 3,7738 1,7075 ANN 5-3-1 3,6755 3,7032 1,8241 ANN 6-3-1 3,7325 3,8494 2,0164

IV.2. Data Division The available data set is divided into two subsets, network training and validation data sets. For network training a 75% of collected samples were used, and 25% were used for model validation. Since the neural network can make predictions only within the set of values that it


CE 0,8741 0,9012 0,9086 0,8999 0,8981 0,8817


370


(a)

Fig. 5. The Error of the best model (ANN3-6-1)

V.

Conclusion

The lack of systematic approach present in hydrological systems modeling was reported more than a decay ago [13], nevertheless, adequate progress hasn’t yet been made, according to the report from [6]. There are still some concerns about arbitrary determination of model inputs, data subsets for network training and the final model structure determination which are all commonly done in an ad-hoc fashion. There is also a problem of inconsistency in model validation process, where inappropriate error measures are often used. This paper investigated the possibility of applying a systematic approach to hydrological time series modeling for forecasting daily flow of the river Cetina with the use of artificial neural networks. Inputs were determined based on ACF, and final structure of a neural model was determined through an experiment. The model was validated based on different error measures, as recommended from [14]. The best model is one that takes into account previous three members of the time series, i.e. has three input neurons and six hidden neurons (ANN3-6-1). It can be also noted that model ANN3-6-1 satisfies the constraint defined by the expression (6), NH ≤ 7. Since the experiment produced encouraging results for one day flow forecast, the next step would be to determine the prediction horizon, i.e. to examine how many days ahead prediction could be made with a satisfactory rate of accuracy. Further work should also be focused on determination of appropriate variables that should be included in flow-forecasting model in order to predict sudden flow changes.

(b)

(c)

(d) Figs. 3. Graphical error measure comparison: (a) SEE, (b) RMSE, (c) MAE, (d) CE

Comparison of the best model response and measured values of the flow for year 2009 are shown graphically in Fig. 4, and the error produced by the best model is shown in Fig. 5.

Acknowledgements The author of this paper would like to express gratitude to the employers of the Croatian Meteorological and Hydrological Service for providing the data that enabled this work.

References [1]

Fig. 4. Comparison of the best model response (ANN3-6-1) and measured values


Sh. Khorshidi, M. Towhidi, M. Karimi, F. Babazadeh, Optimal Predictive Order Selection Criterion for the Autoregressive (AR)


371


[2]

[3] [4]

[5]

[6]

[7]

[8]

[9] [10]

[11]

[12]

[13]

[14]

Model, International Review of Automatic Control (IREACO), Vol. 2. n. 6 614-620, November 2009. B. Sivakumar, Nonlinear dynamics and chaos in hydrologic systems: latest developments and a look forward, Stochastic Environmental Research and Risk Assessment, Volume 23 N. 7, 1027-1036, 2009. H. B. Demuth, M. H. Beale, Neural Network Toolbox™ User’s Guide (The MathWorks, Inc, 2004). N. M. Bellaaj, L. Bouzidi, M. El Euch, ANN Based Prediction of Wind and Wind Energy, International Review of Automatic Control (IREACO), Vol. 2. n. 6, pp. 700-707, September 2010. A. Errachdi, I. Saad, M. Benrejeb, On-line Identification Method Based on Dynamic Neural Network, International Review of Automatic Control (IREACO), Vol. 3. n. 5, pp. 474-479, September 2010. H. R. Maier, A. Jain, G. C. Dandy, K. P. Sudheer, Methods used for the development of neural networks for the prediction of water resource variables in river systems: Current status and future directions, Environmental Modeling & Software, Vol. 25, pp. 891–909, 2010. R. J. Abrahart, P. E. Kneale, L. M. See, Neural Networks for Hydrological Modelling (A. A. Balkema Publishers, a member of Taylor & Francis Group, 2005). M. Hagan, M. B. Menhaj, Training Feedforward Networks with the Marquardt Algorithm, IEEE Transactions on Neural Networks, Vol. 5 No. 6, pp. 989-993, November 1994. S. Haykin, Neural Networks: A Comprehensive Foundation (Pearson Education Inc., 1999). N. J. de Vos, T. H. M. Rientjes, Constraints of artificial neural networks for rainfall-runoff modeling: trade-offs in hydrological state representation and model evaluation, Hydrology and Earth System Sciences, Vol. 9, pp.111–126, 2005. G. B. Kingston, Bayesian Artificial Neural Networks in Water Resources Engineering, Ph.D. dissertation, Faculty of Eng., Comp. and Math. Sci., Univ. of Adelaide, Australia, 2006. A. W. Minns, M. J. Hall, Artificial neural networks as rainfallrunoff models, Hydrological Sciences Journal, Vol. 41 No. 3, pp. 399-417, June 1996. H. R. Maier, G. C. Dandy, Neural networks for the prediction and forecasting of water resources variables: a review of modeling issues and applications, Environmental Modeling & Software, Vol. 15, pp. 101–124, 2000. D. R. Legates, G. J. McCabe Jr., Evaluating the use of “goodnessof-fit” measures in hydrologic and hydro-climatic model validation, Water Resources Research, Vol. 35 No. 1, pp. 233241, January 1999.

Ozren Bego was born in Split, Croatia in 1966. He received the B.S. degree in electrical engineering from the University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Croatia. He received M.S. and Ph. D. degree from the Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia. Currently he is working as assistant professor at Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split. His research interests include digital control systems, embedded computer systems, industrial process automation. Ranko Goić was born in Supetar, Croatia in 1969. He obtained his Ph.D. degree in electrical engineering, field of power system engineering, from the University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Croatia. Currently he is working as associate professor at same Faculty, as associate professor on Power System Engineering Department. His research interests include power system analyses, as well as power system planning, optimization and economy. Dr. Goic is a member of the CIGRE and the IEEE.

Authors’ information 1 University of Split, Faculty of Maritime Frankopanska 38, 21000 Split, Croatia.

Studies,

Zrinsko-

2 University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Ruđera Boškovića bb, 21000 Split, Croatia.

Petar Matić was born in Split, Croatia in 1981. He received the Dipl.ing. degree in Electrical Engineering from University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Croatia in 2006. Since 2007, he has been employed as an assistant at University of Split, Faculty of Maritime Studies. He is a doctoral student at University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture. His interests and main field of research include prediction of hydrological variables with the use of artificial neural networks.



372


The Effect of Load Variations on Operation of a Micro-Turbine Generation System in Grid Connected Mode Mohammad Mahdi Mahmoodi, Seyed Morteza Alizadeh, Fatemeh Nadimi, Sedigheh Babaei Sedaghat Abstract – In this paper, we study the effect of load variations on the operation of a microturbine generation system (MTGS) in grid connected mode. The model is developd with the consideration of the main parts including: compressor-turbine, permanent magnet (PM) generator, three phase bridge rectifier and inverter. We show the variations of active and reactive powers and current according to the variations of load in the proposed system. The model is developed in Matlab / Simulink. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Distributed Generation, Inverter, Microturbine, PM Generator, Rectifier

The integration of the increasing portion of DG within the existing infrastructure requires a full understanding of its impact on the distribution feeders and its interaction with the loads. Some of the operational aspects which require full understanding are voltage control, stability, system protection etc. Such studies require accurate modeling of Distributed Generation (DG) sources including distribution system [23]. There are essentially two types of microturbine designs. One is a split shaft design that uses a power turbine rotating at 3600 rpm and a conventional generator (usually induction generator) connected via a gearbox. The power inverters are not needed in this design. Another is a highspeed single-shaft design with the compressor and turbine mounted on the same shaft as the permanent magnet synchronous generator [2]. Although different works have been undertaken on modeling of microturbine, it is essential to develop more models with higher precision. The load following performance and modeling of split shaft microturbine is developed in [7]. Ref [2] discuss about the operation of MTGS in stand alone mode base on the variation of load. Ref [3] presents detailed models of the components and controls forming the thermo-mechanical and electric subsystems of a microturbine power plant. The modeled thermomechanical subsystem includes different control loops: a speed controller for primary frequency control (droop control), an acceleration control loop, which limits the rotor acceleration in case of sudden loss of load or in case of start-up, and a controller to limit the temperature of the exhaust gases below the maximum admissible temperature. In [16]-[20] a dynamic model for combustion gas turbine has been discussed. In these references, the model was used to represent the gas turbine dynamics, including speed, temperature, acceleration and fuel controls. However, these works

Nomenclature V DC ,ref V DC V AC XC I DC PMSG MTGS

α

GA PSO P ref Q ref J E T tmax xi vi

Desired DC voltage The error between desired Dc voltage and the voltage of DC bus RMS phase to phase voltage level on the AC side of the rectifier Reactance of the elements on the AC side of the rectifier DC link Current Permanent Magnet Synchronous Generator Microturbine generation system Firing angle for the rectifier thyristor Genetic Algorithm Particle Swarm Optimization Desired active power Desired reactive power Inertia coefficient The error between desired and real active power of phase a Current iteration number in PSO Maximum iteration number in PSO Position vector of PSO particles Velocity vector of PSO particles

I.

Introduction

The fundamental concepts for the penetration of DG technologies are the high efficiency of the energy conversion process and the limited emission of pollutants as compared to conventional power plants [1]-[23]. Besides offering a higher flexibility and load management, they provide a number of significant local benefits.



373

M. M. Mahmoodi, S. M. Alizadeh, F. Nadimi, S. Babaei Sedaghat

deal with heavy-duty gas turbine. A non-linear model of the micro-turbine implemented in NETOMAC software and a linear modelling of grid connected MTG system are reported in [21] and [22], respectively. Ref [4] presents the analyzed results of a permanent-magnet synchronous generator-based microturbine generator (MTG) system connected to a distribution system through an AC-to-DC converter and a DC-to-AC inverter. An alternative filter and control approach for the design of a microturbine's utility interface is proposed in [5]. Ref [6] presents modeling, simulation, and analysis of load following behavior of a microturbine (MT) as a distributed energy resource (DER). This study is mainly organized into six sections. Following the introduction, the microturbine design and analyses are given in Section II. In Section III, the design considerations of PM generator are discussed in details. Section IV shows the role of converter in MTG system. Section V provides simulation results in none load and part load condition. Finally, some conclusions are drawn in Section VI.

II.

The valve positioner transfer function is according to (1): KV E1 = Fd (1) TV s + c and the fuel system actuator transfer function is as in (2): Wf =

Kf Tf s + s

(2)

E1

In (1) and (2), KV , ( K f ) is the valve positioner (fuel system actuator) gain, TV , T f are the valve positioner and fuel system actuator time constants, c is a constant, Fd and E1 are the input and output, respectively, of the valve positioned and W f is the fuel demand signal in pu.

ω f is the per unit turbine speed. The per unit value for Ve corresponds directly to the per-unit value of the mechanical power from the turbine in steady state [2]. The fuel flow control as a function of Ve is shown in Fig. 2 [9].

Microturbine Model 0.23

The dynamic model used for microturbine has three main parts: speed control, fuel system and compressorturbine. In the following, each of these parts is discussed.

Kv

Kf

Tv.s+C

Tf.s+C

Transfer Fcn1

Transfer Fcn2

Constant1 Add

1 -K-

From LVG Product

2

1 Wf- Fuel Flow

Gain1

N

II.1.

Speed Control

Fig. 2. Fuel control system

In this paper a lead-lag transfer function is used for modeling the speed governer, as shown in Fig. 1 [2]. However a PID controller can also be used as a model for speed governer. The input of the model is rotor speed in per unit and the output is turbine torque. The output of the governor goes to a low value select to produce a value for Vce, the least amount of fuel needed for that particular operating point and is an input to the fuel system. X.s+1

1

25

Y.s+Z Ref. Speed

Transfer Fcn

Gain

II.3.

Compressor-Turbine

The block diagram of the compressor-turbine package is shown in Fig. 3. The torque characteristics of the single-shaft gas turbine is essentially linear with respect to fuel flow and turbine speed and are given by the following (3):

(

)

Torque = K HHV + W f − 0.23 + 0.5 (1 − N ) ( N m ) (3)

1

where K HHV is a coefficient which depends on the enthalpy or higher heating value of the gas stream in the combustion chamber. The K HHV and the constant 0.23 in the torque expression cater for the typical power/fuel rate characteristic, which rises linearly from zero power at 23% fuel rate to the rated output at 100% fuel rate.

To LVG

1 pu rotor speed

Fig. 1. Speed controller block 1

1.3

II.2.

2

0.2s+1

Fuel System

Gain2 1

The fuel system consists of the fuel valve and actuator. The fuel flow from the fuel system results from the inertia of the fuel system actuator and of the valve positioner, whose equations are given below [8].

Torque

0.23

Wf

Transfer Fcn3

-KAdd1

Constant2

1

Gain4 0.5

Constant3

Gain3

1 pu rotor speed

Fig. 3. Copressor-Turbine Model



374


The input to this subsystem is the fuel demand signal W f and output is the turbine torque [2]. II.4.

IV.

The electrical output frequency of a turbo alternator is typically from 1000 to 3000 Hz and must, in most cases, be converted to a 50 Hz or 60 Hz useable output. To achieve this goal, first, a thyristor rectified the high frequency AC voltage to 800 V DC. The control scheme for the thyristor situated in machine side is shown in Fig. 6 [2]. According to Fig. 6, the rectifier control block is based on a PI regulator, acting on the VDC error ( VDC ,ref − VDC ), that controls the firing angle α for the

Subsystem of Microturbine Developed with Simulink

According to the models developed above for each parts, the dynamic model considered for microturbine is shown in Fig. 4 [9]. 0.23 Constant1 Add X.s+1

speed governer

Gain

Saturation

Product

1 0.4s+1

T ransfer Fcn1

-K-

K

Y.s+Z

1 0.05s+1

Transfer Fcn2 Transport Delay

Gain1

rectifier thyristor, obtained from (4):

1

1.3

Power Conditioning

0.2s+1 Gain2 1

0.23

Transfer Fcn3

1 Torque

Add1

Constant

0.5 Gain3

1

VDC =

Constant2 1 Constant3

3 2

π

VAC cos α −

3X C

π

(4)

I DC

pu rotor speed

VAC is the RMS phase to phase voltage level on the

Fig. 4. Dynamic model of micro-turbine

AC side of the rectifier, X C is the reactance of the elements on the AC side of the rectifier; I DC is the DClink current [2].

III. High Speed Generator A conventional, small, gas turbine generator set consists of a high-speed turbine, operating at say 50000 r min , coupled to a low-speed electrical machine (alternator), typically operating at 3000 r min , through a reduction gearbox. However, the power output and size of an electrical machine is proportional to speed. Consequently, if the electrical machine is run at turbine speed it is so small that it can be integrated into the engine on the same shaft as the turbine machinery forming a compact high-speed alternator, referred to as a permanent magnet synchronous generator or turbo alternator. The gearbox is no longer required and the alternator also acts as a starter motor to further reduce the size of the generator set. Fig. 5 illustrates a 50 kW high-speed alternator rotor mounted with the gas turbine compressor and turbine wheels and in the foreground an 110 kW high-speed alternator rotor.

v

PI

1

+ -

Vdc Vdc ref

alpha degree

Discrete PI C ontroller

800

Fig. 6. Rectifier control block

The DC voltage is then inverted back to 60 or 50 Hz AC, and, then filtered to reduce harmonic distortion [10]. The MTG system line side converter can operate both in grid connected control mode and stand alone control mode [9]. So depending on the status of the power plant, there are two different control strategies for the inverter: • PQ control strategy, when MTG system is in grid connected operation; • V-f control strategy, for operating in stand-alone mode. In this work, we study the grid connected operation. In this strategy, The inverter controls the active and reactive power injected to the grid, which should follow the set-points Pref and Qref. These set-points can be chosen by customer or by a remote control (PoMS) [2]. The initial value of Pref is 9kW, Qref is 3kVAr, then Pref steps up to 18 (kW) and Qref steps up to 5250 (VAR) in 0.5 s. The inverter control system model is shown in Fig. 7. The inverter is current controlled, and two PI controllers are used to regulate the grid current components, i.e. id and iq. In section V, the values of the coefficients related to each of these two PI controllers, i.e. kp1 and ki1 for first PI controller and kp2 and ki2 for second one, are determined using GA and PSO.

Fig. 5. Turbo alternator

The generator is equipped with a 2-poles permanent magnet rotor. The generated terminal voltage has a very high frequency which is not appropriate for load. In the following, we describe how this high frequency voltage convert to an appropriate voltage for the load.



375


1 vabc

dq0 sin_cos Selector1

abc_to_dq0 Transformation 2

abc dq0

iabc

-K-

sin_cos

Gain1

Selector 3

abc_to_dq0 Transformation1 Add

sin-cos

dq0

4

abc

Pref

Divide

sin_cos

Add1

Product

kp1

5 Qref

At no -load operation, the variations of the active power is very disorganized and the its value is between 15 kW and -30 kW. By contrast, the active power follows a certain plan as it operate at part load condition. As can be seen from Fig. 8, the amount of active power is about 9kW after coming the first load, i.e. 50kW at t = 0.45. The active power steps up to 18kW to follow the Pref at t = 0.5 s. It, then, decreases to 17.5 kW and

abc

dq0_to_abc Transformation

Out1

Divide1

1 vabc-inv

17kW as the second and third load come at t = 0.575 s and t = 0.8 s, respectively. When it turns to the reactive power, the level of the variations is more clear. Similar to the active power plan, the reactive power varies drastically at no-load operation from -20kVAr to more than 35 kVAr. The amount of the reactive power steps up to -3 kW as the first load (50 kW) comes into the system. To follow the Qref , the reactive power steps

ki1

PI_1 6 kp1

Add2

7

Product1

kp2

Add3

Out

ki1 ki2

8 PI_2

kp2 9

-K-

ki2

0 Constant

Gain

Fig. 7. Inverter control loop

V.

down to -5.5kW at t = 0.5 s. Fig. 9 shows that the amount of Q decreases to -8kW at t= 0.575 s and -9.6kW at t = 0.8 s as the loads of 150 kW and 300 kW come into the system, respectively. Fig. 10 shows the active power and reactive power variations in one frame.

Simulation Results

The test system used in this study has a nominal speed of 66000 rpm / min and a rated capacity of 55kW. A load of 50 kW with the frequency of 50 Hz is applied on the MTG system. Initially the system is operating at no-load. At t = 0.45 s a load of 50kW is applied on the MTG system, and at t = 0.575 s, the load is increased to 150 kW. Finally the load is increased to 300 kW at t = 0.8 s. Figs. 8 and 9 show the variations of active power and reactive powers according to the load variations.

3

x 10

4

Active Power ( W ) Reactive Power ( VAR )

Active and Reactive Power

2 1 0 -1 -2 -3 -4 -5 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time ( s )

Fig. 10. The effect of load variations on active and reactive power

Fig. 11 shows how the current of phase a changes during the period. It varies from -900 (A) to 900 (A) when the system is operating at no-load. Fig. 12 shows the variation details of the current at no-load over small duration.

Fig. 8. The effect of load variations on active power

Fig. 11. The effect of load variations on current

According to the Fig. 11, the amplitude of the current is 50 (A) as the system feeds the first load. Changing the

Fig. 9. The effect of load variations on reactive power



376


[3]

value of reference active and reactive powers causes that the magnitude of current increase to about 100 (A), at t = 0.5 s. At t = 0.575 s, when the second load comes into the system, the initial value of the current is -500 (A) and then the magnitude of current leaps to 150 (A). By adding the load of 300 kW the initial value of current is not noticeable and the amplitude of current increases to 250 (A).

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Fig. 12. Variation details of the current at no-load

VI.

Conclusion

[11]

The main goals of this study was: 1. Presenting a dynamic model for a microturbine generation system in grid connected operation; 2. Studying the effect of load variations on the operation of the proposed model. According to the results, the system experienced a disorder period at no-load, which is opposed to the time when the system feeded the loads. Increasing the loads leaded to increase the injected active power and received reactive power. It also increased the amplitude of current. Having a dynamic model of a micro-turbine generation system provides an useful tool for studying the various operational aspects of micro turbines. However there are some constraints to simulate such a system considering which software is used. Here we used Matlab software. Because our test system contained nonlinear elements, the ode23tb variable-step stiff solver with relative tolerance set to 1e-4 should be applied in order to get best accuracy and simulation performance. In addition to, it is recommended that the "Solver reset method" parameter be set to "Robust".

[12]

References

[21]

[1]

[2]

[13] [14]

[15]

[16]

[17]

[18]

[19]

[20]

M. Sedighizadeh, M. Fallahnejad, M. R. Alemi, M. Omidvaran, D. Arzaghi-haris, " Optimal Placement of Distributed Generation Using Combination of PSO and Clonal Algorithm, 2010 IEEE International Conference on Power and Energy (PECon2010), Nov 29 – Dec 1, Kuala Lampur, Malaysia. Zhou Yunhai, Stenzel J., Simulation of a microturbine generation system for grid connected and islanding operation, Power and Energy Engineering Conferences ~APPEEC 2009~, Asia-Pacific, 27 – 31 march 2009, pp. 1 – 5.

[22]

[23]


Grillo S., Massucco S., Morini A., Pitto A., Silvestro F., “Microturbine Control Modeling to Investigate the Effects of Distributed Generation in Electric Energy Networks" Systems Journal, IEEE, vol. 4, September 2010. Li. Wang Guang-Zhe Zheng, Nat. Cheng Kung " Analysis of a Microturbine Generator System Connected to a Distribution System Through Power-Electronics Converters" Sustainable Energy, IEEE Transactions on, Vol.2, April 2011. Fu-Sheng Pai, " An Improved Utility Interface for Microturbine Generation System With Stand-Alone Operation Capabilities" Industrial Electronics, IEEE Transactions on, Vol. 53, October 2006. Saha A. K. Chowdhury S., Chowdhury S. P., Crossley P. A., "Modeling and Performance Analysis of a Microturbine as a Distributed Energy Resource, Energy Conversion, IEEE Transactions on, Vol. 24, June 2009. Zhu, K. Tomsovic, Development of models for analyzing the load-following performance of microturbines and fuel cells, Journal of Electric Power Systems Research, vol. 62, pp. 1-11, 2002. Sreedhar R. Guda, C. Wang, M. H. Nehrir, “A Simulink-based microturbine model for distributed generation studies”, Power Symposium, 2005. Proceedings of the 37th Annual North American, 23- 25 Oct. 2005, pp:269 – 274 Gaonkar D. N., Patel R. N., Pillai G. N., Dynamic Model of Microturbine Generation System for Grid Connected/Islanding Operation, Industrial Technology, IEEE International Conference~ICIT 2006~, on 15-17 Dec. 2006, pp:305 – 310. D. N. Gaonkar, R. N. Patel, "Modeling and simulation of microturbine based distributed generation system", in Proc IEEE Power India conference, New-Delhi, India, April 2006, pp. 256260. J. Kennedy, R. Eberhart, Particle swarm optimization. IEEE Int Conf Neural Networks 1995;4:1942–8. R. Eberhart, Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the congress on evolutionary computing, vol. 1. 2000, p. 84–8. J. Kennedy, R. Eberhart, Swarm Intelligence, Academic press, San Diego, CA, 2001. Y. Shi, R. Eberhart, A modified particle swarm optimizer, Proceedings of IEEE International Conference on Evolutionary Computation, Anchorage, May 1998. Kahouli A., Guesmi T., Hadj Abdallah, Ouali A.," A genetic algorithm PSS and AVR controller for electrical power system stability", System, Signal and Devices, 2009. SSD'09. 6th International Multi-Conference on, pp. 1-6, 23-26 March 2009. W. I. Rowen, "Simplified mathematical representations of heavy duty gas turbine," Journal of Engineering for power, Trans. ASME, vol.105, no. 4, pp. 865-869, October 1983. Hannet, Afzal Khan, .Combustion turbine dynamic model validation from tests,. IEEE Trans. on Power Systems, vol.8, no.1, pp.152-158, February 1993. Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance Studies, dynamic models for combined cycle plants in power system studies, IEEE Trans. Power System, vol. 9, no.3, pp. 1698-1708, August 1994. L. N. Hannett, G. Jee, B. Fardanesh, .A governor/turbine model for a twin-shaft combustion turbine, IEEE Trans. on Power System, vol.10, no. 1, pp. 133-140, February 1995. L. M. Hajagos, G. R. Berube, .utility experience with gas turbine testing and modeling,. in Proceedings, IEEE PES Winter Meeting, vol.2, pp. 671-677 January /February 2001, Columbus, OH. H. Nikkhajoei, M. R. Iravani, Modeling and analysis of amicroturbine generation system, IEEE Power Engineering SocietySummer Meeting, vol. 1, pp. 167 -169, 2002. R. Lasseter, Dynamic models for micro-turbines and fuel cells, In Proc. IEEE PES Summer Meeting, vol. 2, Vancouver, Canada, 2001, pp.761-766. W. G. Scott, “Micro-Turbine Generators for Distribution Systems,” IEEE Industry Applications Magazine, May/June 1998.


377


Fatemeh Nadimi was born in Tehran, Iran in 1981. She is graduated in electronic engineering from Shahid Beheshti University. She works in electrical laboratory of Sazeh Gostar Saipa Company (S.G.S Co) since 2004. Her research interest is Renewable energy.

Authors’ information Islamic Azad University, Saveh Branch, Iran.

Sedighe Babaei Sedaghat was born in Lahijan, Iran, in1985. She received the B.Sc. (with the highest distinction) degree in electrical engineering from Islamic Azad University in lahijan and her M.S. degree in electrical engineering from Razi University, Iran, Kermansha, in 2008 and 2011 respectively. Her current research interests are in high-frequency integrated circuits design and analog integrated circuits design in CMOS technology.

Mohammad Mahdi Mahmoodi was born in Tehran, Iran in 1980. He received the B.Sc. degree from Power and Water University of Technology (PWUT) in 2004, Tehran, Iran and M.Sc. degree from Islamic Azad University, Science and Research branch in 2007, Tehran, Iran, both in power electrical engineering. Since 2008, he is a member of faculty of Islamic Azad University, Saveh branch. His research interests are Micro grids, Smart grids and Renewable energies. E-mail: [email protected] Seyed Morteza Alizadeh was born in Rasht, Iran, in 1985. He received the B.Sc. degree in electronic engineering from Islamic Azad University in 2008 and the M.Sc. degree in power electronic engineering from Islamic Azad University in 2010. He has several papers in the field of microturbine, coverters, modelling and simulation, hybrid systems, artificial intelligence controller and faults . His principle research interests in electronic drives, converters, magnetic machines and artificial intelligence. Mr. Alizadeh is one of the members of faculty of Lahijan Islamic Azad University and one of the members of Engineering Organization, Gilan, Iran. E-mail: [email protected]



378


New Method to Determine the Memory of Volterra Model Safa Chouchane, Kais Bouzrara, Hassani Messaoud

Abstract – This paper proposes a new method to determine, from input / output measurements, the memory of Volterra model describing a non linear system when its order is known. The proposed algorithm is based on calculating the determinant of a specific matrix defined for an increasing value of the memory and for which we prove that it becomes singular once this value exceeds the minimal value. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Nonlinear System, Volterra Model, Memory, Determinant Ratio

matrix containing the lagged inputs and outputs and proving that this matrix goes to singularity when this value exceeds the minimal value of the memory required to reproduce the system behavior. Therefore, we consider this value as the minimal memory value. This paper is organized as follows. In section 2, we remind the Volterra model and we define its structure parameters and kernels. Section 3 is devoted to memory estimation. The proposed algorithm is illustrated by simulation results and the yielded Volterra model is validated.

Nomenclature u(k) y(k) hi ( m1 ,… ,mi )

System input System output Volterra kernel of order i

np

Parameters number of Volterra model

,M P E[ ]

Memories of Volterra model Order of Volterra model Mathematical expectation

N B

ϕ xy (τ )

Observation number Matrix Cross-correlation function

DR (

Determinant ratio

)

det(X) fi ( k − j )

Determinant of the matrix x Defined functions

SNR NMSE

Signal to noise ratio Normalized mean square error

II.

Volterra Model

A discrete-time Volterra model with memory M and order P is described by the following input-output relation: y (k ) =

P

M

M

∑∑ ∑

i =1 m1 =1 m2 = m1

I.

Introduction

M

∑

The Volterra series [1] are a very powerful mathematical tool for developing models governing the behavior of certain classes of non linear systems as the provided models are linear with respect to their parameter which enables to use the available results in linear parameter identification, it is an alternative to classical non linear models such as NARMAX [7] and Neural Network [8], [9]. However this linearity is penalized by the increase of the model parameter number which may bereaves these models to be engaged in a real time control strategy. This number depends on model structure parameters such as the non linearity degree P and the system memory M which are generally arbitrarily chosen high to cope with the system features. This paper proposes a new algorithm to estimate the minimal value of the memory when the non linearity degree is known. This algorithm consists on building, for an increasing value of the memory, a given

(1) i

mi = mi −1

hi ( m1 ,… ,mi ) ∏ u ( k − me ) e =1

where u(k) and y(k) are the system input and output respectively and hi ( m1 ,… ,mi ) is the Volterra kernel of order i. The parameters number of model (1) is: np =

P

( M −1 + i )!

∑ ( M − 1) !i!

(2)

i =1

This number depends mainly on the order P and the memory M. Hence any increase of the latter causes an increase in the Volterra model coefficient number and consequently in its complexity.



379

S. Chouchane, K. Bouzrara, H. Messaoud

with:

III. Memory Estimation

⎧ A = v1 ( M + 1) v1T ( M + 1) ⎪ ⎪ A ∈ ℜ( 2 M + 2 )×( 2 M + 2 ) ⎪ ⎪ B = v1 ( M + 1) v2T ( M + 1) ⎪⎪ ⎛ ( M +1)( M + 2 ) ⎞ ( 2 M + 2 )×⎜ ⎟ ⎨ 2 ⎝ ⎠ ⎪B ∈ ℜ ⎪ T ⎪C = v2 ( M + 1) v2 ( M + 1) ⎪ ⎛ ( M +1)( M + 2 ) ⎞ ⎛ ( M +1)( M + 2 ) ⎞ ⎜ ⎟×⎜ ⎟ ⎪ 2 2 ⎠ ⎝ ⎠ ⎪⎩C ∈ ℜ⎝

In this section, we propose a new method for the memory determination. We start with the case of P = 2 then we generalize the method for any value of P. For P = 2 and a memory , the model (1) becomes:

∑ h1 ( m1 ) u ( k − m1 ) +

y (k ) =

m 1 =1

+

∑ ∑

m 1 =1 m 2 = m 1

(3) h2 ( m1 ,m2 ) u ( k − m1 ) u ( k − m2 )

Since N is supposed very large we can assume the ergodicity of signals u and y. Therefore the matrix V ( M + 1) converges in probability to the matrix

where h1 ( m1 ) and h2 ( m1 ,m2 ) are the Volterra kernels of order 1 and 2 respectively. Let’s define the vectors v1 ( ) , v2 ( ) and v ( ) as: v1 ( v2 (

⎡u ( k − 1) y ( k − 1) )=⎢ ⎢⎣ u ( k − ) y ( k −

α=

(4)

E [ B ]⎤ ⎥ E [ C ]⎥ ⎦

(10)

where E [ x ] denotes the mathematical expectation of x. From (8) - (10) we conclude that V ' ( M + 1) is a square

T

matrix with dimension γ = 2M + 2 +

2

( )⎤ β ⎥ ∈ℜ , β = 2 +α ⎢⎣ v2 ( ) ⎥⎦ ⎡v

)=⎢ 1

Using the vector v (

),

(6)

⎡ϕuu ( 0 ) ϕuy ( 0 ) ⎢ ⎢ϕuy ( 0 ) ϕ yy ( 0 ) ⎢ ⎢ϕuu (1) ϕuy (1) ⎢ E = [ A] ⎢ϕuy (1) ϕ yy (1) ⎢ ⎢ ⎢ϕuu ( M ) ϕuy ( M ) ⎢ ϕ ( M ) ϕ yy ( M ) ⎣⎢ uy ϕuy (1) ϕuu ( M )

we define the square matrix

): 1 N

V(

)=

V(

)∈ℜ

+N

∑ v ( ) v T ( ),

k = +1

(7)

β ×β

where N is the observation number assumed to be very high. In the following, we show that the matrix V ( )

ϕ yy (1) ϕuy ( 0 )

tends to singularity when the model memory exceeds the system memory M. In fact, for = M + 1 the expression (7) becomes:

ϕ yy ( 0 )

M +1+ N

⎡A B ⎤ 1 V ( M + 1) = ⎢ ⎥ N k = M + 2 ⎣ BT C ⎦

∑

( M + 1)( M + 2 )

2 and its ith column is composed of the ith column of the matrix E[A] and the ith column of the matrix E[BT] if i = 1, …, 2M+2 and the ith column of the matrix E[B] and the ith column of the matrix E[C] if i = 2M+3, …, γ . Since the input/output observations are real, using the autocorrelation and cross-correlation functions the matrix E [A] can be written as:

(5)

) ∈ℜα ( + 1)

v(

V(

⎡ E [ A] V ' ( M + 1) = ⎢ T ⎢⎣ E ⎡⎣ B ⎤⎦

)=

⎡u 2 ( k − 1) u ( k − 1) u ( k − 2 ) ⎤ ⎢ ⎥ 2 = ⎢u ( k − 1) u ( k − ) u ( k − 2 ) u ( k − 2 ) u ( k − 3) ⎥ ⎢ ⎥ ⎢⎣ u ( k − 2 ) u ( k − ) u 2 ( k − ) ⎥⎦ v2 (

V ' ( M + 1) given by:

T

⎤ 2 ⎥ ∈ℜ )⎥⎦

(9)

(8)


ϕuy ( M )

ϕuu (1) ϕuy (1) ϕuu ( 0 ) ϕuy ( 0 ) ϕuu ( M − 1) ϕuy ( M − 1) ϕuy ( M ) ⎤

(11)

⎥

ϕ yy ( M ) ⎥ ⎥

ϕuu ( M − 1) ϕuy ( M − 1) ⎥ ϕuy ( M − 1) ϕ yy ( M − 1) ⎥⎥

ϕuy ( M − 1)

ϕuu ( 0 )

ϕuy ( 0 )

ϕ yy ( M − 1)

ϕuy ( 0 )

ϕ yy ( 0 )

⎥ ⎥ ⎥ ⎥ ⎥⎦


380


with:

ϕ xy (τ ) = E ⎡⎣ x ( k + τ ) y ( k ) ⎤⎦

( ( ( (

(12)

For the matrix B, we define the functions: fi ( k − j ) = u ( k − j ) u ( k − i − j + 1)

(13)

i, j = 1,… ,M + 1

( (

The vector v2 ( M + 1) can be written as: v2 ( M + 1) = ⎡ f1 ( k − 1) f 2 ( k − 1) f M +1 ( k − 1) ⎢ ⎢ f1 ( k − 2 ) f 2 ( k − 2 ) f M ( k − 2 ) =⎢ ⎢ f1 ( k − 3) f 2 ( k − 3) f M −1 ( k − 3) ⎢ f ( k − M − 1) ⎣ 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

B2

B3

BM

Bi =

Bi2

…

(21)

with:

(14)

⎧i = 1,… , M + 1 ⎪ ⎪ j = 1,… , M + 2 − i ⎪k = 1,… , M + 1 ⎨ ⎪ 1 si i < k ⎪α k = ⎨⎧ ⎪⎩ ⎩2 si i > k

BM +1 ]

(15) The matrix C is written as:

where the dimension of Bi is ( 2 M + 2 ) × ( M + 2 − i ) : ⎡ Bi1 ⎣

) )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

T

and the matrix B can be partitioned as: B = [ B1

) ) ) )

α1 ⎡ ϕ u f j ( i − 1)( −1) ⎢ ⎢ α1 ⎢ ϕ y f j ( i − 1)( −1) ⎢ α2 ⎢ ϕ u f j ( i − 2 )( −1) ⎢ α2 E ⎡⎣ Bi j ⎤⎦ = ⎢ ϕ ⎢ y f j ( i − 2 )( −1) ⎢ ⎢ ⎢ α M +1 ⎢ϕu f j ( i − M − 1)( −1) ⎢ α ⎢ϕ i − M − 1)( −1) M +1 ⎢⎣ y f j (

BiM + 2 −i ⎤⎦

C = [C1

( )

CM +1 ]

(22)

with: Ci = ⎡⎣Ci1 Ci2 … CiM + 2 −i ⎤⎦

(23)

⎛ ( M + 1)( M + 2 ) ⎞ dim ( Ci ) = ⎜ ⎟(M + 2 − i) 2 ⎝ ⎠

(24)

⎡ f1 ( k − 1) f j ( k − i ) ⎤ ⎢ ⎥ ⎢ f 2 ( k − 1) f j ( k − i ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ f M +1 ( k − 1) f j ( k − i ) ⎥ ⎢ ⎥ ⎢ f1 ( k − 2 ) f j ( k − i ) ⎥ i = 1,… ,M + 1, j (25) Ci = ⎢ ⎥ ⎢ f 2 ( k − 2 ) f j ( k − i ) ⎥ j = 1,… ,M + 2 − i ⎢ ⎥ ⎢ ⎥ ⎢ fM ( k − 2) f j ( k − i ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ f ( k − M − 1) f ( k − i ) ⎥ 1 j ⎣ ⎦

(18)

from (15), the mathematical expectation of the matrix B is: ⎡ E [ B1 ] E [ B2 ] E [ B3 ]⎤ (19) E [ B] = ⎢ ⎥ ⎢⎣ E [ BM ] E [ BM +1 ] ⎥⎦ with: ⎡ E ⎡ Bi1 ⎤ E ⎡ Bi2 ⎤ ⎤ ⎣ ⎦ ⎣ ⎦⎥ E [ Bi ] = ⎢ ⎢… E ⎡ B M + 2−i ⎤ ⎥ ⎣ i ⎦ ⎦⎥ ⎣⎢

CM

(16)

⎡ u ( k − 1) f j ( k − i ) ⎤ ⎢ ⎥ ⎢ y ( k − 1) f j ( k − i ) ⎥ ⎢ ⎥ ⎢ u ( k − 2 ) f j ( k − i ) ⎥ i = 1,… ,M + 1 (17) Bi j = ⎢ y ( k − 2 ) f j ( k − i ) ⎥ ⎢ ⎥ j = 1,… ,M + 2 − i ⎢ ⎥ ⎢ ⎥ ⎢ u ( k − M − 1) f j ( k − i ) ⎥ ⎢ y ( k − M − 1) f ( k − i ) ⎥ j ⎣ ⎦ dim Bi j = 2 ( M + 1)

C2

so: (20)

E [C ] = ⎡⎣ E [C1 ] E [C2 ]

E [CM ] E [CM +1 ]⎤⎦ (26)

with: j

From (17) and (12), the expectation of the matrix Bi ,

E [Ci ] = ⎡ E ⎡⎣Ci1 ⎤⎦ ⎣

E ⎡⎣ Bi j ⎤⎦ is:


E ⎡⎣Ci2 ⎤⎦ … E ⎡⎣CiM + 2−i ⎤⎦ ⎤ (27) ⎦


381


ϕ yy ( 0 ) = E ⎡⎣ y ( k − 1) y ( k − 1)⎤⎦ =

and: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ E ⎡⎣Ci j ⎤⎦ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ϕ ⎣

f1 f j

α1

ϕ ϕ ϕ ϕ ϕ

((i − 1)( −1) ) ((i − 1)( −1) ) α1

ϕ

f2 f j

f2 f j

((i − 1)( −1) ) ((i − 2)( −1) ) ((i − 2)( −1) )

fM f j

((i − 2)( −1) )

α1

f M +1 f j

α2

f1 f j

α2

α2

((i − M − 1)( −1) ) α M +1

f1 f j

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

= h1 (1) ϕuy (1) + h1 ( 2 ) ϕuy ( 2 ) + +

+ h1 ( M ) ϕuy ( M ) +

+ h2 (1,1) ϕ y f1 (1) + h2 (1, 2 ) ϕ y f 2 (1) + +

+ h2 (1,M ) ϕ y f M (1) +

+ h2 ( 2 , 2 ) ϕ y f1 ( 2 ) + h2 ( 2,3) ϕ y f 2 ( 2 ) + +

(28)

+

+ h2 ( 2,M ) ϕ y f M −1 ( 2 ) + + h2 ( M ,M ) ϕ y f1 ( M )

ϕuy (1) = E ⎡⎣u ( k − 2 ) y ( k − 1) ⎤⎦ = = h1 (1) ϕuu ( 0 ) + h1 ( 2 ) ϕuu (1) + +

+ h1 ( M ) ϕuu ( M − 1) +

+ h2 (1,1) ϕu f1 ( 0 ) + h2 (1, 2 ) ϕu f 2 ( 0 ) + +

+ h2 (1,M ) ϕu f M ( 0 ) +

+ +

+ h2 ( 2,M ) ϕu f M −1 (1) +

+ h2 ( M ,M ) ϕu f1 ( M − 1)

ϕ yy (1) = E ⎡⎣ y ( k − 2 ) y ( k − 1) ⎤⎦ = = h1 (1) ϕuy ( 0 ) + h1 ( 2 ) ϕuy (1) + +

+ h1 ( M ) ϕuy ( M − 1) +

E [ A] and the second row of the matrix E [ B ] .

+ h2 (1,1) ϕ y f1 ( 0 ) + h2 (1, 2 ) ϕ y f 2 ( 0 ) +

In the following we rewrite all the components of the second column of E [ A] given by (11), where we

+ + +

ϕuy ( 0 ) = E ⎡⎣u ( k − 1) y ( k − 1) ⎤⎦ =

+ h2 ( 2 , 2 ) f1 ( k − 3) + h2 ( 2,3) f 2 ( k − 3) + +

+ h2 ( M ,M ) f1 ( k − M − 1)

]=

+ +

(29)

+

+ h1 ( M ) ϕuu ( M ) +

+

(33)

+ h2 ( 2,M ) ϕu f M −1 ( M − 2 ) + + h2 ( M ,M ) ϕu f1 ( 0 )

= h1 (1) ϕuy ( M − 1) +

+ h2 (1,M ) ϕu f M (1) + h2 ( 2 , 2 ) ϕu f1 ( 2 ) +

+ h2 ( 2 ,3) ϕu f 2 ( 2 ) +

+ h2 (1,M ) ϕu f M ( M − 1) +

ϕ yy ( M ) = E ⎡⎣ y ( k − M − 1) y ( k − 1) ⎤⎦ =

+ h2 (1,1) ϕu f1 (1) + h2 (1, 2 ) ϕu f 2 (1) + +

+ h1 ( M ) ϕuu ( 0 ) +

+ h2 ( 2 , 2 ) ϕu f1 ( M − 2 ) +

= h1 (1) ϕuu (1) + h1 ( 2 ) ϕuu ( 2 ) + +

+ h2 ( M ,M ) ϕ y f1 ( M − 1)

+ h2 (1,1) ϕu f1 ( M − 1) +

+ h2 (1,M ) f M ( k − 2 ) +

+ h2 ( 2 ,M ) f M −1 ( k − 3) +

+ h2 ( 2 ,M ) ϕ y f M −1 (1) +

= h1 (1) ϕuu ( M − 1) +

+ h1 ( M ) u ( k − M − 1) + h2 (1,1) f1 ( k − 2 ) +

+

(32)

ϕuy ( M ) = E ⎡⎣u ( k − M − 1) y ( k − 1) ⎤⎦ =

= E ⎡⎣u ( k − 1) ( h1 (1) u ( k − 2 ) + h1 ( 2 ) u ( k − 3) + + h2 (1, 2 ) f 2 ( k − 2 ) +

+ h2 (1,M ) ϕ y f M ( 0 ) +

+ h2 ( 2 , 2 ) ϕ y f1 (1) + h2 ( 2,3) ϕ y f 2 (1) +

y ( k − i ) , i = 1,… ,M + 1 , its expression

obtained from model (1) for P = 2:

+

(31)

+ h2 ( 2 , 2 ) ϕu f1 (1) + h2 ( 2 ,3) ϕu f 2 (1) +

⎧i = 1,… , M + 1 ⎪ j = 1,… , M + 2 − i ⎪ ⎪ with: ⎨k = 1,… , M + 1 . ⎪ 1 si i < k ⎪α k = ⎨⎧ ⎪⎩ ⎩2 si i > k From (10), the second column of the matrix V ' ( M + 1) is given by the second column of the matrix

substitute to

(30)

+ h1 ( M ) ϕuy ( 0 ) +

+ h2 (1,1) ϕ y f1 ( M − 1) +

+ h2 ( 2,M ) ϕu f M −1 ( 2 ) +

+ h2 (1,M ) ϕ y f M ×

× ( M − 1) + h2 ( 2 , 2 ) ϕ y f1 ( M − 2 ) +

+ h2 ( M ,M ) ϕu f1 ( M )

× ( 2 ,M ) ϕ y f M −1 ( M − 2 ) +


(34)

+ + h2 ×

+ h2 ( M ,M ) ϕ y f1 ( 0 )


382


From (29) - (34), we note that the second column of the matrix E [ A] is a linear combination of odd columns

ϕ y f1 ( M ) = E ⎡⎣ f1 ( k − M − 1) y ( k − 1) ⎤⎦ =

3,5,… , 2M − 1 of the same matrix, the columns 1, 2,… ,M of the matrix E [ B2 ] , the columns

= h1 (1) ϕu f1 ( M − 1) +

+ h1 ( M ) ϕu f1 ( 0 ) + h2 (1,1) ϕ f1 f1 ( M − 1) +

1, 2,… ,M − 1 of the matrix E [ B3 ] ... the columns 1, 2 of

+

the matrix E [ BM ] and the column 1 of the matrix

+

ϕ y f1 ( 0 ) = E ⎡⎣ f1 ( k − 1) y ( k − 1) ⎤⎦ = + h1 ( M ) ϕu f1 ( M ) + h2 (1,1) ϕ f1 f1 (1) +

(35)

+

+ h2 ( 2,M ) ϕ f1 f M −1 ( 2 ) +

1, 2,… ,M − 1 of the matrix E [C3 ] ,…, the columns 1, 2

+ h2 ( M ,M ) ϕ f1 f1 ( M )

of the matrix E [CM ] and the column 1 of the matrix E [CM +1 ] .

ϕ y f M +1 ( 0 ) = E ⎡⎣ f M +1 ( k − 1) y ( k − 1) ⎤⎦

So the second column of the matrix V ' ( M + 1) is a

= h1 (1) ϕu f M +1 (1) + h1 ( 2 ) ϕu f M +1 ( 2 ) +

+

linear combination of other columns of the same matrix and hence the matrix V ' ( M + 1) is singular. Thus the

+ h1 ( M ) ϕu f M +1 ( M ) +

+ h2 (1,1) ϕ f M +1 f1 (1) + h2 (1, 2 ) ϕ f M +1 f 2 (1) + +

+ h2 (1,M ) ϕ f M +1 f M (1)

matrix V ( M + 1) tends to singularity. This result is quite (36)

important for the memory’s determination of the Volterra model. We define for each value of the determinant ratio DR ( ) as:

+ h2 ( 2 , 2 ) ϕ f M +1 f1 ( 2 ) + h2 ( 2,3) ϕ f M +1 f 2 ( 2 ) + + +

+ h2 ( 2,M ) ϕ f M +1 f M −1 ( 2 )

+ h2 ( M ,M ) ϕ f M +1 f1 ( M )

DR (

ϕ y f1 (1) = E ⎡⎣ f1 ( k − 2 ) y ( k − 1) ⎤⎦ = = h1 (1) ϕu f1 ( 0 ) + h1 ( 2 ) ϕu f1 (1) +

+

+ h2 (1,M ) ϕ f1 f M ( 0 ) +

+

ϕ y f M (1) = E ⎡⎣ f M ( k − 2 ) y ( k − 1) ⎤⎦ + +

M

M

1

M

i = 0,… , − 1 . By observing the relation (29), ϕuy ( 0 ) ,which is present in all columns of the matrix

(38)

E ( A ) , contains ϕuu ( M ) that doesn’t exist in any

2

column of the matrix E ( A ) for all

M −1

M

M

(40)

those that contain elements of the form ϕuy ( i ) where

f1

M

2

max

is to look for columns that can be a linear combination of other columns. For the matrix E ( A ) , these columns are

(0) + + h2 (1, 2 ) ϕ f f ( 0 ) + + h2 (1,M ) ϕ f f ( 0 ) + + h2 ( 2 , 2 ) ϕ f f (1) + h2 ( 2 ,3) ϕ f f (1) + + + h2 ( 2 ,M ) ϕ f f (1) + + + h2 ( M ,M ) ϕ f f ( M − 1) M

; = 1, 2 ,… ,

will be taken as the system memory. To prove that this value corresponds to the minimal one, we show that the matrix V ' ( + 1) cannot be singular for ≤ M . The idea

+ h2 ( M ,M ) ϕ f1 f1 ( M − 1)

+ h1 ( M ) ϕu f M ( M − 1) + h2 (1,1) ϕ f M

)) det (V ( + 1) )

the determinant ratio for successive values of , the value for which the quantity DR ( ) increases rapidly

+ h2 ( 2 ,M ) ϕ f1 f M −1 (1) +

= h1 (1) ϕu f M ( 0 ) + h1 ( 2 ) ϕu f M (1) +

det (V (

singularity of the matrix V ( + 1) . Thus by calculating

(37)

+ h2 ( 2 , 2 ) ϕ f1 f1 (1) + h2 ( 2 ,3) ϕ f1 f 2 (1) + +

)=

with det(X) is the determinant of the matrix X. For = M , DR ( ) increases rapidly because of the

+ h1 ( M ) ϕu f1 ( M − 1) + h2 (1,1) ϕ f1 f1 ( 0 ) + + h2 (1, 2 ) ϕ f1 f 2 ( 0 ) +

+ h2 ( M ,M ) ϕ f1 f1 ( 0 )

3,5,… , 2 M + 1 of the same matrix, the columns 1, 2,… ,M of the matrix E [C2 ] , the columns

+ h2 ( 2 , 2 ) ϕ f1 f1 ( 2 ) + h2 ( 2,3) ϕ f1 f 2 ( 2 ) + +

+

As above, it raises that the second column of the matrix E ⎡⎣ B T ⎤⎦ , which is the second row of the matrix E [ B ] , is a linear combination of odd columns

= h1 (1) ϕu f1 (1) + h1 ( 2 ) ϕu f1 ( 2 ) +

+ h2 (1,M ) ϕ f1 f M (1) +

(39)

+ h2 ( 2 ,M ) ϕ f1 f M −1 ( M − 2 ) +

of the matrix E [ B ] are written as:

+ h2 (1, 2 ) ϕ f1 f 2 (1) +

+ h2 (1,M ) ϕ f1 f M ( M − 1) +

+ h2 ( 2 , 2 ) ϕ f1 f1 ( M − 2 ) +

E [ BM +1 ] . Similarly, the components of the second row

+

+

≤M .

1



383


y ( k ) = 0.5 y ( k − 1) + 0.3 y ( k − 1) u ( k − 1) +

For the matrix E ( B ) , columns that can be a linear

+0.2u ( k − 1) − 0.5 y 3 ( k − 1) + 0.7u 2 ( k − 1) + ε ( k )

combination of the other are those that contain the quantities ϕ y f j ( i ) where i = 0,… , − 1 and j = 1,… , .

where ε ( k ) is a measurement noise chosen so that the

By observing the relation (35), ϕ y f1 ( 0 ) , which is

signal to noise ratio SNR = 20, 10 and 5. The observation number is N = 800 . We propose to determine the minimal value of the memory for a fixed order of the Volterra model describing this system. In Table I we summarize the value of the determinant, the determinant ratio and the corresponding normalized mean square error (NMSE) for different values of the Volterra model memory and for a fixed order P = 2. It resorts that the determinant ratio jumps rapidly when the model memory reaches the value 4. Therefore the minimal value so that the model fits the system behavior is 4. This assumption is emphasized by the value of the NMSE which reaches a small value that still unchangeable for bigger values assigned to model memory. Fig. 1 plots the decimal logarithm of the determinant ratio for different values of the model order and for the three values of the signal to noise ratio SNR = 20, 10 and 5. This figure illustrates clearly the jump at = 4 for the different SNR values. Moreover this ratio still approximately constant or bigger for high values of . In Fig. 2, we draw simultaneously the process output and the output of the model with P = 2 and M = 4. This figure raises a perfect concordance between both outputs.

present in all columns of the matrix E ( B ) , contains

ϕu f1 ( M ) that doesn’t exist in any column of the matrix E ( B ) for all

≤ M . Finally, the matrix V ' ( + 1) and

not be singular for all ≤ M and also the matrix V ( + 1) . Then we assume that the matrix V ( + 1) becomes singular only when the model memory exceeds the system memory M. To generalize this idea for any order P > 2 , let’s write the vector v ( ) as:

vP (

⎡ v1 ( ⎢ v ( ) = ⎢⎢ 2 ⎢ ⎣⎢vP (

)⎤ ) ⎥⎥ ⎥ ⎥ )⎦⎥

(41)

u (k −

) y ( k − ) ]T (42)

with: v1 (

) = ⎡⎣u ( k − 1) y ( k − 1) vi (

) = ⎡⎣u ( k − m1 ) u ( k − m2 )… u ( k − mi )⎤⎦

T

(43) TABLE I THE VALUE OF THE DETERMINANT, THE DETERMINANT RATIO AND THE CORRESPONDING NORMALIZED MEAN SQUARE ERROR (NMSE) FOR DIFFERENT VALUES OF THE VOLTERRA MODEL MEMORY AND FOR A FIXED ORDER P = 2 1 2 3 4 5 6 SNR -5 -10 det [V( )] 14 000 11900 302.2 0.555 6.1 10 5 10 5 1.16 39.51 546.26 8740 1.2 10 DR( ) 20 NMSE 8.37 % 6.82 % 5.78 % 5.01 % 4.96% 4.92 % -4 7 10-9 det [V( )] 14600 17500 746.23 2.397 4.9 10 0.83 23.49 311.32 4910 70050 DR( ) 10 NMSE 8.89 % 7.02 % 6.32 % 5.38 % 5.27 % 5.21 % 44.93 0.02 8.5 10-7 det [V( )] 17300 47400 5330 104 0.36 8.89 118.64 1920 2.74 DR( ) 5 NMSE 9.03 % 8.72 % 7.56 % 6.42 % 6.36 % 6.24%

m1 = 1,… , m2 = m1 ,… , mi = mi −1 ,… ,

Algorithm: We fix the order P. 1. For = 1,… ,n (n is an integer):

1.1 COMPUTE vi ( 1.2 Compute V

1.3 Compute

P

)

(

i = 1,… ,P 1 )= N

AND

vP (

∑ v ( ) ( v ( )) P

P

,

k = +1

)) DR ( ) = P det (V ( + 1) )

2. The value of

), T

+N

(44)

(

det V P (

6

.

for which the quantity DR (

log ( DR (

))

SNR = 20 SNR = 10 SNR = 5

5 4

)

3

increases rapidly is considered as the system’s memory.

2 1

IV.

Simulation Results 0

Consider a SISO non linear system described as follows:

-1 1

2

3

Fig. 1. Evolution of DR (


4

5

)


384


y (k ) ym ( k )

References

1

[1]

0.8

[2] [3]

0.6 0.4

[4] 0.2 0

[5] 20

0

40

60

80 100 120 itérations

140

160

180

200

[6] Fig. 2. Validation phase: P = 2, M = 4 and SNR = 20

When we modify the SNR, the structure parameters of Volterra model don’t change. For P = 2 and M = 4 the NMSE remains large. In order to improve the performances of the system, the order is increased. TABLE II RESULTS FOR P = 3 AND SNR = 20 1 2 3

A det (V ( A ) )

312.17

Dr ( A )

3.41 10

NMSE

2.32%

9.13 10 8

−7

5.23 10

[7]

[8]

[9] 4 −10

3.45 10

−13

1745.69

1515.94

--

2.27%

2.26%

2.24%

M. Schetzen, "The Volterra and Wiener Theories of nonlinear systems", John Wiley and Sons, New York, 1980. N. Wiener, "Nonlinear problems in random theory", New York:Wiley, 1958. A. Khouaja, "Modélisation et identification de systèmes non linéaires à l’aide de modèle de Volterra à complexité réduite", Thèse de doctorat à Université de Nice Sofia Antipolis, France ( Mars 2005). Alain Y. Kibangou, G. Favier, M. Hassani, "A growing approach for selecting generalized orthonormal basis functions in the context of system modelling", IEEE-EURASIP Workshop on Non linear Signal and Imag Processing, NSIP’03, Grado, Italy, 2003. S. Sagara, H. Gotanda, K. Wada (1982) Dimensionally recursive order determination of linear discrete system. Int J, control, Vol 35, Nov, 637-651. D. Dembele, "Identification structurelle des systèmes linéaires à temps discret stochastiques et monovariables", Diplôme d’études approfondies, l’école normale supérieure de l’enseignement technique, Tunis (1989). S. Rejeb, F. Ben Hmida, A. Chaari, "Unbiased Iterative Identification of Hammerstein ARMAX Model", International Review of Automatic Control (IREACO), Vol. 3. n. 2, pp. 125133, March 2010. G A. Errachdi, I. Saad, M. Benrejeb "On-line Identification Method Based on Dynamic Neural Network", International Review of Automatic Control (IREACO), Vol. 3. n. 5, pp. 474-479, September 2010. Alireza Rezazadeh, Alireza Askarzadeh, Mostafa Sedighizadeh "ANN-Based PEMFC Modeling by a New Learning Algorithm", International Review of Automatic Control (IREACO), Vol. 3. n. 2, pp. 168-177, April 2010.

Authors’ information Laboratory of Automatic Control, Signal and Image Processing (ATSI), National School of Engineers of Monastir (ENIM), 5000, Monastir, Tunisia.

y (k ) ym ( k ) 1 0.8

Safa Chouchane was born in 1981. She received her Electrical Engineering diploma in 2005 and the master degree in 2006 both from the National School of Engineers of Monastir (ENIM), Tunisia. She is currently preparing the PhD degree in Electrical Engineering in the laboratory of Automatic Control, signal and Image Processing (ATSI). She is majoring on the modelling and identification of nonlinear system using Volterra model.

0.6 0.4 0.2

Kais Bouzrara was born in 1974. He received his Electrical Engineering diploma in 1998 from National School of Engineers of Monastir (ENIM), his master degree from Faculty of Sciences of Monastir and the Ph.D degree in Electrical Engineering from the National School of Engineers of Tunis (ENIT), Tunisia in 2007. He is presently an assistant Professor at (ENIM). His research is related to modeling, identification and control of linear and nonlinear system.

0 0

20

40

60

80 100 120 itérations

140

160

180

200

Fig. 3. Validation phase: P = 3, M = 1 and SNR = 20

V.

Hassani Messaoud was born in 1959. He received the Master degree in Electrical Engineering and the Ph.D degree in Automatic Control from the High Normal School of Techniques Education of Tunisia in 1985. In 2001, he received the ability degree from the National School of Engineers of Tunisia. He is presently Professor in the National School of Engineers of Monastir (ENIM) and Director of the laboratory ATSI. He has been the supervisor of several PhD theses and he is author or coauthor of several Journal articles.

Conclusion

In this paper, we have addressed the problem of structure identification of Volterra models. We have proposed a new algorithm for estimating the order and the memory simultaneously. The proposed algorithm has been tested in simulation and the model with estimated structure parameters has been validated with the real process.



385


Multivariable Generalized Predictive Control Based on CARIMA Model with Non Identity Disturbance Matrix Badreddine Louhichi1, Ahmed Toumi2

Abstract – In this paper, Multivariable Generalized Predictive Control (MGPC) is presented. If a physically realizable multivariable process can be described by a Controlled Auto-Regressive Integrated Moving Average (CARIMA) model with non identity disturbance polynomial matrix :

( )

C q −1 ≠ I m , the way to get MGPC controller coefficients can be simplified. Since, there exist

direct expressions describing the relationship between the open-loop model parameters and the MGPC controller coefficients according to a certain set of tuning parameters. The control moves are just the product of the process known information and the MGPC controller coefficients. This technique is based on the Generalized Predictive Control (GPC) approach. The adopted strategy consists, first of all, in the formulation of the output’s predictor of the model. Secondly, the elaboration of the control law for the linear system is then envisaged. This stage is conceived by means of the transformation of the quadratic criterion to minimize. Lastly, the main characteristic of the algorithm is to take into account the coloring polynomials of the noise. The proposed control algorithm is evaluated on experiment results. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Generalized Predictive Control, Multivariable System, Non Identity Disturbance Polynomial Matrix

Nomenclature CARIMA GPC MGPC MIMO SISO m

m,1 Y (k ) ∈ ( )

Controlled Auto-Regressive Integrated Moving Average Generalized Predictive Control Multivariable Generalized Predictive Control Multi Input Multi Output Single Input Single Output Number of inputs / outputs The output vector

Yc ( k ) ∈

( m,1)

U (k ) ∈

( m,1)

A white noise vector

q −1 d nA

The shift operator The delay

nC

−1

The degree of polynomial

( )

F j q −1

nK


( )

K j q −1

nL


( )

L j q −1

hi hp hc R∈

The free response vector of the predictor vector The output reference vector sequence The input vector

m,1 V (k ) ∈ ( )

The degree of polynomial B q −1

nF

m,1 Yˆ ( k + j / k ) ∈ ( ) The output predictor vector

m,1 Yo ( k + j ) ∈ ( )

( ) The degree of polynomial C ( q )

nB

Q∈

The initial horizon The prediction horizon Ithe control horizon A symmetric positive definite matrix A symmetric positive definite matrix of level-headedness of the control The quadratic criterion

m*m

m*m

J

I.

( )

The degree of polynomial A q

−1

Introduction

The studies concerning the predictive control really began in the sixties with the predictor from Smith [1]. Afterward, Aström and Wittenmark [2] proposed the control with minimal variance. Two other methods were



386

B. Louhichi, A. Toumi

then presented: the pole placement [4] and the concept of prediction at wide horizon proposed by Richalet [3]. For the purpose of self-tuning and stable control, Generalized Predictive Control (GPC) algorithm was proposed by Clarke et al. [5] in 1987. Since GPC uses long-range predictive control ideas to adapt to process dead time, time-delay and parameter variations, it is widely used for industrial and dynamic processes. GPC is one of the long-range prediction control methods which have good robustness properties for using the future behavior of the system. The control law is a result of minimization of the prediction output error and the incremental control values over a finite horizon [11]. In the literature, numbers of works have treated the nonlinear generalized predictive control. For example, Meziane et al. have implemented in [19] the NGPC algorithm and the Kalman Filter algorithm to estimate and control the rotor speed and the rotor flux norm of the motor in fixed stator. Bourebia et al. have proposed in [20] the GPC algorithm with the Takagi-Sugeno-Kang (TSK) fuzzy modeling approach. Laabidi et al. have used in [21] the Constrained Nonlinear Generalized Predictive Control CNGPC for nonlinear dynamic systems based on Artficial Neural Networks (ANN), Tabu Search (TS) and Genetic Algorithms (GAs). Multivariable systems (MIMO systems) having multiple delays are often difficult to control using conventional SISO controllers. Because of this way, several works have proposed different strategies that give a better solution than those obtained with classical controllers. In practice, a majority of systems have many variables that have to be controlled (outputs) and many manipulated variables or variables used to control the plant (inputs). In this case, the plant must be considered to be a process with multiple inputs and outputs (MIMO). The control of MIMO processes has been extensively treated in literature; perhaps the design philosophy behind GPC has been extended to the MIMO case in [6] using the Tfiltering design and in [8] using the Youla parameterization. However, there is a drawback to the application of the GPC algorithm in industrial practice: the robustness limits for stability decrease very quickly when the dead time of the system increases, that is, GPC can be easily unstabilised by small errors in the estimation of the dead time of the process [9]. The basic idea is to use a multivariable CARIMA model where

( ) is a non identity matrix : C ( q ) ≠ I

matrix C q

−1

−1

m;

The paper is organized as follows: • During the section 2, the problem is formulated. • Section 3 is devoted to the development of the Multivariable Generalized Predictive Control (MGPC). In the section 4, we develop the control law of GPC. • In the section 5, we estimate these algorithms on multivariable processes and we comment the obtained results. • Some concluding remarks given in section 6 end of this paper.

II.

Problem Formulation

The process model plays a crucial role to determine the control law performance. Indeed, it must reflect, with sufficient accuracy, the dynamics of the system to future moments, while using the output past value, the control and the control future increments. This representation must be adequate for industrial plants or measurable disturbances. In practice the most frequent disturbances are a step signal and random occurrences, hence it is adequate to use CARIMA model. Let us consider a multivariable input-output dynamic system with m inputs and m outputs. The MIMO CARIMA model is given by the following equation:

( )

( )

A q −1 Y ( k ) = q − d B q −1 U ( k ) +

( )V (k ) ∆ (q )

C q −1 −1

(1)

with:

( )

A q −1 = I m + A1q −1 + A2 q −2 + ... + AnA q − nA , Ai ∈

m*m

( )

B q −1 = B0 + B1q −1 + B2 q −2 + ... + BnB q − nB , Bi ∈

m*m

( )

C q −1 = I m + C1q −1 + ... + CnC q − nC , Ci ∈

( )

m*m

∆ q −1 = I m − I m q − 1

it

is a disturbance polynomial matrix representing the coloring polynomial of the noise to compute the predictions of the output of the plant and to calculate a sequence of future control signals to minimize a multistage cost function defined over a control horizon, as in the GPC proposed by Clarke et al. [5]. On the other hand, as shown subsequently, the design of filters in the GPC is simple because it does not modify the primary controller and the robustness index is defined with the correct choice of the filter.

is the difference operator.

•

m,1 Y ( k ) ∈ ( ) is the output vector,

•

m,1 U ( k ) ∈ ( ) is the input vector,

•

m,1 V ( k ) ∈ ( ) is a white noise vector,

•

q −1 is the shift operator,

• d is the delay. •

( )

nA, nB and nC are degrees of polynomials A q −1 ,

( )

( )

B q −1 and C q −1 respectively.

The CARIMA model (1) can be rewritten as follows:



387


( )

( )

( )

α q −1 Y ( k ) = q − d B q −1 ∆U ( k ) + C q −1 V ( k ) (2)

model by introducing an integrator. However, the choice

( )

of the model disturbance C q −1 = I m is not sufficiently to represent a majority of practice systems. The object of the following work is to take a CARIMA model process that we choose an explicit model of

with:

( ) ( ) ( )

α q −1 = ∆ q − 1 A q − 1 =

( )

disturbance where C q −1 ≠ I m in order to better

− n +1 = I m + α1q −1 + α 2 q −2 + ... + α nA +1q ( A )

αi ∈

m* m

describe a practice plant. To derive a j-step ahead predictor of Y(k+j) based on (3), let consider the polynomial identity of BEZOUT :

as:

( )

α i = Ai − Ai −1 for i = 2...n A α1 = A1 − I m α nA +1 = − AnA

( )

Multiplying equation (2) by α q −1

Y ( k ) = q−d

−1

(5)

( )

− j −1 E j q −1 = I m + E1q −1 + E2 q −2 + ... + E j −1q ( )

, we get:

( )

F j q −1 = F0j + F1 j q −1 + F2j q −2 + ... + Fnj q − nF

C q −1

−1

( )

with:

( ) ∆U ( k ) + ( ) V ( k ) α (q ) α (q ) B q −1

( ) ( )

C q −1 = E j q −1 α q −1 + q − j F j q −1

F

(3)

−1

Ei ∈

m* m

, Fi j ∈

m*m

III. Generalized Predictive Control

and nF = max ( nC − j,n A ) . The previous polynomial

We wish to calculate for the process, a sequence of command which minimizes the quadratic criterion J given by [10], [12] – [15]:

and the prediction horizon j. Using the equations (3) and (5), we shall have:

( ) ( )

matrix is unique solution defined by α q −1 , C q −1 , ∆

J ( k ,hi,hp,hc ) =

Y (k + j) =

⎧ hp ⎡Y ( k + j ) + ⎤T ⎡Y ( k + j ) + ⎤ ⎪ =ζ ⎨ ⎢ ⎥ R⎢ ⎥+ ⎪⎩ j = hi ⎣⎢ −Yc ( k + j ) ⎦⎥ ⎣⎢ −Yc ( k + j ) ⎦⎥

∑

(4)

( )V (k ) + E q V (k + j) + ( ) α (q )

⎫⎪ + ∆U T ( k + j − 1) Q∆U ( k + j − 1) ⎬ j =1 ⎪⎭

the

output

V (k ) =

reference

−1

From equation (3), V(k) is written as:

where: • hi is the initial horizon, • hp is the prediction horizon, • hc is the control horizon, is

j

(6)

−1

∑

m,1 Yc ( k ) ∈ ( )

−1

F j q −1

hc

•

( ) ∆U ( k + j − d ) + α (q ) B q −1

vector

( ) ⎢Y ( k ) − B ( q ) ∆U ( k − d )⎤⎥ ⎥ C (q ) ⎢ α (q ) ⎣ ⎦

α q −1 ⎡

−1

−1

−1

(7)

Substituting the above expression of V(k) in equation (6), we can write:

sequence, • R ∈ m*m is a symmetric positive definite matrix chosen to settle the dynamics of the error of the output. • Q ∈ m* m is a symmetric positive definite matrix of level-headedness of the control. More of works in the literature have assumed that

Y (k + j) =

( ) ( ) ⎤⎥ ∆U ( k + j − d ) + ( ) ( ) ⎥⎦ F (q ) Y ( k ) + E ( q )V ( k + j ) + C (q ) =

( )

B q −1 ⎡ F j q −1 ⎢ I − q− j m C q −1 α q −1 ⎢ ⎣ j

C q −1 = I m in the CARIMA model [7], [16] – [18],

(8)

−1

j

−1

−1

[22], [23]. By proceeding in this way, we try to cancel any static error towards a constant disturbance in the CARIMA

So:



388


( ) ( ) ∆U ( k + j − d ) + C (q )

Yˆ ( k + j / k ) =

B q −1 E j q −1

Y (k + j) =

( ) ( ) ( ) + ( K ( q ) + H ( q ) F ( q ))Y ( k )

= H j q −1 B q −1 E j q −1 ∆U ( k + j − d ) +

−1

(9)

( ) Y (k ) + E q V (k + j ) + ( ) C (q ) F j q −1

−1

j

−1

j

−1

( )

As E j q −1

( )

( ) ( ) ( )

decomposed into two parts: the one depends on the future control and the other one of the past by using the identity of BEZOUT:

E j q −1 V ( k + j ) are all in the future, so that the

( ) ( ) ( ) ) = G (q ) + q ( L (q ) H j q −1 B q −1 E j q −1 =

optimal predictor, given measured output data up to time k and any given U(k+i) for i > 1, is clearly:

−1

( ) L (q ) = L

−1

j

(11)

+M

To derive a j-step ahead predictor of Y(k+j) based on (11), let consider the polynomial identity of BEZOUT: Im = H

(q )C (q ) + q −1

−1

−j

K

j

(q ) −1

where: M

(12)

( )

− j −1 H j q −1 = H 0 + H1q −1 + H 2 q −2 + ... + H j −1q ( )

( )

m*m

multiplying by H

(I

j

−1

j

j

−1

j

(16)

−1

(q ) = K (q ) + H (q ) F (q ) −1

j

−1

j

−1

−1

j

j

( q ) Y ( k ) (17) −1

( )

( q ) , we shall have :

−1

−1

−1

Yˆ ( k + j / k ) = G j q −1 ∆U ( k + j − d ) + Yo ( k + j ) (18)

All these outputs predictors for j going of 1 to hp, is given by:

− q − j K j q −1 Yˆ ( k + j / k ) = j

j

j

−1

( )) = H ( q ) B ( q ) E ( q ) ∆U ( k + j − d ) + +H ( q ) F ( q )Y ( k ) m

( ) ( q ) Y ( k ) + L ( q ) ∆U ( k − 1)

The predictor is:

and nK = nC − 1 . Using the equations (11) and (12), then j

∈

m*m

Yo ( k + j ) = L j q −1 ∆U ( k − 1) + M

K

, Ki j ∈

,

( )

K j q −1 = K 0j + K1j q −1 + K 2j q −2 + ... + K nj q − nK m* m

Lij

The first term of the prediction in (16) corresponds to the forced response due to future control increments, while the last two terms correspond to the free response and are generated by past input increments and past output. Let take:

with:

Hi ∈

L

m* m

Yˆ ( k + j / k ) = G j q −1 ∆U ( k + j − d ) +

−1

j

+ L1j q −1 + L2j q −2 + ... + Lnj q − nL

and nL = nB + j + d − 3 . Substituting the equation (15) in (14), we obtained the following predictor:

( ) = B ( q ) E ( q ) ∆U ( k + j − d ) + +F (q )Y ( k ) j

j 0

Gi ∈

Yˆ ( k + j / k ) =

−1

−1

j

with: C q

(15)

−1

− j −d G j q −1 = G0 + G1q −1 + G2 q −2 + ... + G j − d q ( )

−1

−1

j

with:

−1

(q ) Y (k ) + C (q ) F

( ) ( ) ∆U ( k + j − d ) + C (q ) (10)

B q −1 E j q −1

− j − d +1

−1

j

j

−1

j

The term H j q −1 B q −1 E j q −1 ∆U ( k + j − d ) is

is of degree j-1, the noise components

Yˆ ( k + j / k ) =

−1

j

(14)

( ) Yˆ ( k + 2 / k ) = G ( q ) ∆U ( k + 2 − d ) + Y ( k + 2 ) Yˆ ( k + 1 / k ) = G1 q −1 ∆U ( k + 1 − d ) + Yo ( k + 1)

(13)

−1

2

−1

o

(

Using equation (13), we get:

)

( ) (

)

(

h Yˆ k + h p / k = G p q −1 ∆U k + h p − d + Yo k + h p


)


389


By replacing Yˆhp

This group of hp equations can be written in the key vector form: Yˆhp = G ∆U hp + Yhp ,o (19) ⎡ G0 ⎤ ⎢ G G0 0 ⎥⎥ 1 ∈ with : G = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢⎣Ghp −1 … G1 G0 ⎥⎦

(

= ⎡ ∆U ⎣

T

IV.

(

(

) ( T

)

⎤ R G ∆U hp + Yhp ,c − Yhp ,o ⎥ ⎦

)

min J ( k ,hi,hp,hc ) =

∆U h p

) ∆U T ( k + h p − d ) ⎤ ⎦

(

)

⎡ ∆U T GT RG + Q ∆U + hp ⎢ hp ⎢ T = min ⎢ −2 Yhp ,c − Yhp RG ∆U hp + ∆U h p ⎢ T ⎢ ⎢⎣ + Yhp ,c − Yhp R Yhp ,c − Yhp

(

)

(

Control Law

) (

(

The term Yhp ,c − Yhp

)

) R (Y T

h p ,c

− Yhp

)

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(23)

is independent

of ∆U hp , so the solution of the equation (23) is

min J k ,hi ,h p ,hc =

equivalent to the following criterion:

⎧ hp ⎡Yˆ k + j + ⎤T ⎡Yˆ k + j + ⎤ ) ) ⎪ hp ( hp ( = min ⎨ ⎢ ⎥ R⎢ ⎥ + (20) ∆U h p ⎢⎣ −Yc ( k + j ) ⎥⎦ ⎪⎩ j = hi ⎢⎣ −Yc ( k + j ) ⎥⎦

min J ( k ,hi,hp,hc ) =

∑

∆U h p

(

)

⎫ ( k + j − 1) Q∆U hp ( k + j − 1)⎪⎬ ⎪⎭

⎡ ∆U hT GT RG + Q ∆U h + ⎤ p p ⎢ ⎥ = min ⎢ T ⎥ ∆U h p ⎢⎣ −2 Yhp ,c − Yhp RG ∆U hp ⎥⎦

Minimize the criterion (20) means writing:

Minimize the criterion (24) means writing:

+

(22)

After development of the equation (22), we obtain:

Using the optimal predictor (18), the minimization of the quadratic criterion (4) can thus rewrite: ∆U h p

)

+ G ∆U hp + Yhp ,c − Yhp ,o

m.hp *m.hp

YoT k + h p ⎤ ⎦

(k +1− d )

(

min J k ,hi ,h p ,hc = min ⎡ ∆U hTp Q ∆U hp + ∆U h p ⎣

Yˆ T k + h p / k ⎤ ⎦

YhTp ,0 = ⎡YoT ( k + 1) ⎣ ∆U hTp

problem (21) can be written again: ∆U h p

(

YˆhTp = ⎡Yˆ T ( k + 1 / k ) ⎣

by its value, the minimization

hc

∑

j =1

T ∆U hp

(

so :

We distinguish, according to the choice of hi and hc , two cases: • hi = 1 and hc = 1 Using the vector form (19), the problem (20) is written as:

) R (Yˆ

hp

∆U ( k ) = [ I m 0m

)

− Yhp ,c +

(

(21)

⋅GT R Yhp ,c − Yhp ,o

+∆U hp T Q∆U hp ⎤ ⎦

where: Yhp ,cT = ⎡YcT ( k + 1) ⎣

(

m.hp*m.hp

Q = diag ( Q ) ∈

m.hp*m.hp

−1

(

GT R Yhp ,c − Yhp ,o

)

(25)

(

0m ] GT RG + Q

)

)

−1

⋅

(26)

• hi = 1 and 1 < hc < hp To reduce the dimension and the calculation time, we impose on the input to be constant after the control horizon hc ( ∆U ( k + j − d ) = 0 for j > hc ) . It is in this

)

YcT k + h p ⎤ ⎦

R = diag ( R ) ∈

)

We consider only the increment vector of control at the moment k, we obtain [15]:

min J ( k ,hi,hp,hc ) = T

(

∆U hp = GT RG + Q

∆U h p

(

)

∂J =0 ∂∆U k

∂J =0 ∂∆U k

⎡ = min ⎢ Yˆhp − Yhp ,c ∆U h p ⎣

(24)

level that lives the power of the GPC. In other words, it means imposing, after any some times, an infinite levelheadedness on the control actions. In this case, the



390


⎡ 0.0119 −0.0061⎤ B q −1 = ⎢ ⎥+ ⎣ −0.0007 −0.0369 ⎦ ⎡ −0.0521 0.0325 ⎤ −1 +⎢ ⎥q + ⎣ 0.0625 −0.1091⎦

( )

problem is written as follows: min J ( k ,hi,hp,hc ) =

∆U h p ,hc

(

)

⎡ ∆U hT ,h GhT RGh + Qh ∆U h ,h p c c c c p c = min ⎢ T ⎢ ∆U h p ,hc ⎢⎣ −2 (Yc − Y ) RGhc ∆U hp ,hc

+⎤ ⎥ ⎥ ⎦⎥

(27)

⎡ −0.0358 0.1018 ⎤ −2 +⎢ ⎥q ⎣ 0.2451 0.0517 ⎦

where:

0 ⎤ −1 ⎡1 0 ⎤ ⎡ −0.85 C q −1 = ⎢ +⎢ q ⎥ −0.85⎥⎦ ⎣0 1 ⎦ ⎣ 0

( )

Ghc = ⎡ G0 ⎢ G ⎢ 1 ⎢ ⎢ =⎢ ⎢Ghp − hc ⎢ ⎢ ⎢ ⎣⎢ Ghp −1

⎤ ⎥ G0 ⎥ ⎥ ⎥ G0 ⎥ ∈ G1 ⎥ … … … … ⎥ ⎥ ⎥ … … … … Ghp − hc ⎦⎥

Qhc = diag ( Q ) ∈

The parameters of the control low synthesis are: h p = 5, hc = 2, Q = = [ 0.099 0; 0 0.1] I 2 , R = I 2 , d = 1

m⋅hp*m⋅hc

The simulation results for 240 seconds are shown in Figures 1 to 4. From t = 60 seconds to the end, a 0.2 step disturbance through first-order dynamics is applied to the three outputs. From t = 100 seconds to the end, a white noise sequence, with zero mean and variance σ 2 = 0.03 , is applied to the both outputs. The reference trajectories are supposed to be equal to:

m.hc*m.hc

(

T ∆U hp ,hc = Ghc RGhc + Qhc

)

−1

(

)

T Ghc R Yhp ,c − Yhp ,o (28)

The optimal control is obtained in a similar way in the equation (26). The introduction of the control horizon hc offers advantages. Indeed, when the matrix GT RG is not full rank, it is then taken, even for a minimum phase system, to choose a weighting matrix Q positive definite of levelheadedness of the control, in order to calculate the control law. In fact, there is some connections between the choice of control horizon hc and the choice weighting matrix on the control vector.

V.

• Ref. 1 :

• Ref. 2 :

Simulation

⎧0 ⎪-25 ⎪ y2 c ( k ) = ⎨ ⎪0 ⎪⎩ y2c ( k − T )

for 0 ≤ k ≤ 20 for 20 ≤ k ≤ 45 for 45 ≤ k ≤ 120 for 120 ≤ k ≤ 240 for 0 ≤ k ≤ 75 for 75 ≤ k ≤ 100 for 100 ≤ k ≤ 175 for 175 ≤ k ≤ 240

Figures 1 and 2 represent respectively the evolution of the disturbances and the noises. Figures 3 and 4 show respectively the evolution of the control law and the outputs of the plant. It is shown that the proposed GPC controller accomplishes the fast and stable responses in the Figure 4. Also, the proposed GPC controller has lower overshoot in dynamic responses and smaller fluctuation amplitudes in steady state tracking. This behavior is considered to result from the output prediction characteristics of GPC controller at the control process. This behavior is considered to result from the mismatching of the CARIMA predictive model. This disadvantage of the proposed GPC controller is suggested to be overcome by using an on-line parameter identification algorithm to adjust the parameter values of CARIMA predictive model.

In this section, we present the simulation results of the MGPC algorithm. We consider a process described by a multivariable CARIMA model having two inputs and two outputs. Its model is given by the system of equation (1) with: ⎡1 0 ⎤ A q −1 = ⎢ ⎥+ ⎣0 1 ⎦

( )

⎡ −1.0908 +⎢ ⎣ −0.1549 ⎡0.2504 +⎢ ⎣0.2119

⎧0 ⎪25 ⎪ y1c ( k ) = ⎨ ⎪0 ⎪⎩ y1c ( k − T )

−0.1882 ⎤ −1 q + −1.1585⎥⎦ 0.2898⎤ −2 q 0.1661⎥⎦

⎡ −0.0052 −0.0705⎤ −3 +⎢ ⎥q ⎣ −0.0621 0.1356 ⎦



391


Control law u1(k)

Disturbance e1(k)

0.2

20

0.15

10

u1(k)

e1(k)

30

0.1

0

0.05

-10

0

-20

-0.05

-30

0

50

100

150

200

0

50

100

150

200

250

200

250

time k

time k

Control law u2(k)

Disturbance e2(k)

0.2

30

0.15

20

u2(k)

e2(k)

40

0.1

10

0.05

0

0

-10

-0.05

-20

0

50

100

150

200

0

50

100

Fig. 3. The GPC control law evolution

Fig. 1. Evolution of the disturbances

Output y1(k), Refernce yc1(k) and Predictive Output yp1(k)

Noise br1(k)

30

0.2

25

0.1

20

0

15

-0.1

10

-0.2

5

-0.3

0

br1(k)

0.3

-0.4

0

50

100

150

150 time k

time k

200

-5

250

Refernce yc1(k) Output y1(k) Predictive Output yp1(k)

0

50

100

time k

150

200

250

time k

Noise br2(k)

Output y2(k), Refernce yc2(k) and Predictive Output yp2(k) 5

0.2

0

0.1

-5

0

-10

-0.1

-15

-0.2

-20

-0.3

-25

-0.4

-30

br2(k)

0.3

0

50

100

150

200

250

time k

0

Refernce yc2(k) Output y2(k) Predictive Output yp2(k) 50 100

150

200

250

time k

Fig. 2. Evolution of the noises Fig. 4. The GPC Outputs evolution



392


( )

We can analyze the role of C q −1 in two ways. In

[6]

the first part and in the case where the polynomial system

( )

C q −1 is known, it is used to make a prediction of the

[7]

optimal output to minimize the variance. In this case,

[8]

( )

C q −1

plays the role of a filter to minimize the

( )

[9]

prediction error. In the second part, we can use C q −1

[10]

as a system of parameters used to make robust control.

( )

C q −1

can be seen, then, as an observer or a

[11]

prefiltering system. In this case, we lose optimality in the prediction, but increase the robustness. The performances obtained from the generalized predictive control algorithm with disturbance

[12]

( )

( C q −1 ≠ I m ) deduct of the adequate choice of both [13]

matrices of level-headedness R and Q which were fixed after a certain number of tests to lead finally to the values attributed for these two matrices.

[14]

VI.

Conclusion

In this work, we bent in the study of the multivariable control and its stake in work over industrial processes. We were interested in fact with multivariable generalized

[15]

( )

predictive control with disturbance ( C q −1 ≠ I m ).

[16]

The centralized GPC is not appropriate for large scale systems. Complete decentralized GPC does not have a good performance. In this paper, a method is proposed to improve the performance of the GPC. The GPC in the case of measurable disturbance is described and then used for the proposed method. This method is simulated in a 2-input 2-output system and the results are presented. Simulation results have shown effectiveness and have great benefit for implementing advanced process control strategies used by practice plant. Future works can include improving the stability of this method.

[17]

[18]

[19]

[20]

Acknowledgements We thank the Ministry of Higher Education and Scientific Research of Tunisia for funding this work.

[21]

References

[22]

[1] [2] [3]

[4]

[5]

O. J. M. Smith, "A controller to overcome dead-time", Instrument Society of America Journal, vol. 6, n° 2, (1959) pp. 28-33. K. J. Aström, B. Wittenmark, "On self-tuning regulators", Automatica, vol. 21, (1973) pp. 185-199. J. Richalet, A. Rault, J. L. Testud, J. Papon, "Model predictive heuristic control : applications to industrial processes", Journal of Automatica, vol. 14, (1978) pp. 413- 428. K. J. Aström, B. Wittenmark, "On self-tuning controllers based on pole-zero placement" Proc. I.E.E, vol. 127, n° 3, (1980) pp. 120130. D. W. Clarke, C. Mothadi, P. Tuffs, "Generalized predictive control. Part I : the basic algorithm. Part II : extensions and interpretations", Automatica, vol. 23, n° 2, (1987) pp. 137- 160.

[23]

S. L. Shah, C. Mothadi, D. W. Clarke, "Multivariable adaptive control without prior knowledge of the delay matrix", System Control Lett., vol 9, (1987) pp. 295-306. M. Ait Lafkih, "Commande Prédictive et Adaptative des Procédés Multivariables", Thèse de Docteur de 3ème cycle, Université Cadi Ayyad, Marrakech (Maroc), (1993). B. Kouvaritakis, J. A. Rossiter, "Multivariable stable generalized predictive control", IEEE Proc. D. Control theory Appl., 140 (5), (1993) pp. 364-372. E.F. Camacho, C. Bordons, "Model predictive control", Springer Verlag (1999). S. M. Moon, L. R. Clark, D. G. Cole, "The recursive generalized predictive feedback control: theory and experiments", Journal of Sound and Vibration 279, (2005) pp. 171-199. Z. Wang, Z. Chen, Q. Sun, Z. Yuan, "Multivariable decoupling predictive control based on QFT theory and application in CSTR chemical process", Chinese J. Chem. Eng., 14 (6) (2006) pp. 765769. C. Ogab, S. Hasseine, S. Laribi, A. Ben Diabdelallah, "Commande Prédictive Généralisée d'un Moteur Synchrone à Aimants Permanents", International Conference STA′2006, Tunisia, (2006). L. Xiuxia, W. Xiaoye, L. Zhiwei, C. Zengqiang, Y. Zhuzhi, "Study on Algorithm of Constrained Multivariable Generalized Predictive Control", Machine Learning and Cybernetics, 2006 International Conference on Volume (2006) pp. 336-339. L. Xiuxia, Q. Yanhua, Z. Yan, C. Zengqiang, Y. Zhuzhi, "Study on Algorithm of Constraint Multivariable Generalized Predictive Control Based on Multi-objective Programming", Industrial Electronics and Applications ICIEA’2007. 2nd IEEE Conference on Volume, (2007) pp. 432-435. B. Louhichi, A. Toumi, "Commande Prédictive Généralisée Adaptative des Procédés Multivariabes", Nouvelles Tendances Technologiques en Génie Electrique et Informatique GEI'2008, Tunisia, (2008). B. Louhichi, A. Toumi, "Constrained Multivariable Generalized Predictive Control", 9th International conference on Sciences and Techniques of Automatic control & computer engineering STA'2008, Tunisia, (2008). L. X. Niu, X. J. Liu, "Multivariable generalized predictive scheme for gas turbine control in combined cycle power plant", Cybernetics and Intelligent Systems, IEEE Conference on Volume, (2008) pp. 791 – 796. M. Sedraoui, "Apport des principales méthodes d'optimisation dans la CPG multivariable sous contraintes". Thèse de doctorat, Université Mentouri-Constantine, Faculté des Sciences de l’Ingénieur, Département d’électronique, Algérie (2008). S. Meziane, R. Toufouti, H. Benalla, "Generalized Nonlinear Predictive Control of Induction Motor", International Review of Automatic Control (IREACO), Vol. 1, n. 1, (2008) pp. 65-71. O. Bourebia, K. Belarbi, "Fuzzy Generalized Predictive Control for Nonlinear Systems with Coordination Technique", International Review of Automatic Control (IREACO), Vol. 1, n. 2, (2008) pp. 169-176. K. Laabidi, M. Ksouri, "Genetic Algorithm and Tabu Search For Nonlinear Constrained Generalized Predictive Control". International Review of Automatic Control (IREACO), Vol. 2, n. 1, (2009) pp. 27-33. V. A. Akpan, G. D. Hassapis, “Nonlinear model identification and adaptive model predictive control using neural networks”, ISA Transactions, vol. 50, no. 2, pp. 177-194, (2011). V. A. Akpan, “Development of new model adaptive predictive control algorithms and their implementation on real-time embedded systems”, Ph.D., Aristotle University of Thessaloniki, 2011.

Authors’ information 1,2

Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA).



393


1 Faculty of the Sciences of Sfax (FSS), Tunisia. Tel. : +216 74 85 16 29 GSM : +216 98 95 17 28 +216 22 35 80 04 E-mails: [email protected] [email protected] 2 National School of Engineering of Sfax (ENIS), Tunisia. Tel.: + 216 74 27 40 88 Fax: + 216 74 27 55 95 E-mails: [email protected] [email protected]

Badreddine Louhichi was born in Sfax (Tunisia) on August 1976. He received the Electrical Engineering Diploma from Sfax Engineering National School (ENIS), University of Sfax, Tunisia, in 2001. He prepared in collaboration with the Polytechnique School, University of Nantes, France, his diploma of DEA. He received the DEA (Master) in automatic control from ENIS in 2002. He is currently preparing the Ph.D. in the field of automatic control. He is a member of the Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA) of the University of Sfax. He is currently an Associate Professor of electric engineering in the Faculty of the Sciences of Sfax, Tunisia. Ahmed Toumi was born in Sfax (Tunisia) on November 1952, he is Professor in the Sfax Engineering National School (ENIS). He received the Electrical Engineering Diploma from (ENIS/Tunisia), the DEA (Master) in Instrumentation and Measurement from University of Bordeaux-1/France in 1981 and the Doctoral Thesis from the University of Tunis in 1985. He joined the Sfax Engineering National School (ENIS), as an Associate Professor of Electric Engineering, since October 1981. He obtained the Doctorat thesis (PhD) in 1985, form the Faculty of the Sciences of Tunis. In 2000, he obtained the University Habilitation (HDR) from the Sfax Engineering School (ENIS). He is at present, Professor on Automatic Control, and the Director of the Electrical Engineering Department in ENIS. The main research area concerns the modeling, the stability of electric machines, and the fuzzy logic control. He is a member of the Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA) of the University of Sfax. He is the Head of the Research Team of Industrial Processes Control (CPI) of the Sfax University. Since 2002, he is the President of the international conference on Sciences and Techniques of Automatic control (STA) which have taken place every year in a number of tourist cities of Tunisia.



394


Control and Operation of Wind Farm Connected to Grid Using Multi-Terminal VSC-HVDC Reza Noroozian, Mohammad Reza Safari Tirtashi Abstract – Wind energy is one of the most important sources for renewable energy that has been utilizing worldwide. Interconnecting the wind farm to the grid is noteworthy issue in modern power system. In this paper, considered wind farm composed of variable speed wind turbines driving squirrel cage induction generators. HVDC connection based on Voltage Source Converter (VSC) has been selected for interconnecting the wind farm to the AC grid. The proposed wind farm is 50MW, consists of 10 induction generators based 5MW generators with two VSC, while the turbines are clustered in two groups and the space between turbines in each group is 300m. Perturbation and observation method has been used to obtain the maximum power from wind turbines. Investigation on transient and normal operation of the system based PSCAD/EMTDC software has been done. Simulation results show fine operation of the system even when the severe fault has been applied. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: VSC-HVDC, Wind Farm, Induction Generator

I.

has been done. In [9], modelling and simulation of offshore wind power plants connected to the onshore power system grid by VSC based HVDC transmission has been taken into consideration. In [10], the investigation of subsynchronous resonance with VSC-based HVDC has been carried out based on linear analysis and nonlinear transient simulation. In the other hand, to obtain the maximum power and reducing the loud noises of wind turbines, using the variable speed wind turbine has attracted a lot of interest from the power industry. For maximum power tracking, there are many approaches such as duty cycle ratio control, using a lookup table, and etc. These methods are based on characteristics of the wind turbine either before or during the execution. Many of previous publications such as [1], [11] used these methods to reach the maximum power point. In this paper the perturbation and observation method that eliminates the turbine characteristics measurement [12] has been used to obtain the maximum power point tracking (MPPT). Turbines of wind farm have been divided into two groups and multi-terminal VSC-HVDC technology has been used to interconnect the considered wind farm to the grid. Principle of flux oriented vector control has been applied in each wind farm side converters to control the speed of induction generators. Moreover, the operation and control of proposed system both in normal and transient modes have been investigated. To evaluate the operation of proposed system in transient mode, both single and three phase faults have been applied as a network disturbance and the simulation

Introduction

There is currently significant interest in offshore wind farm development because of a range of technoeconomic and environmental benefits [1]. Since the most of planning offshore wind farms are 100 km away from onshore, HVDC transmission solutions will be feasible and more competitive than the AC solutions [2], [3]. Multi-terminal DC wind farm topologies are attracting increasing research effort. For grid connection of wind farms, the topology uses high-voltage direct current transmission using voltage-source converters (VSCHVDC) [4], [5]. The introduction of VSC in high-voltage high-power applications such as DC transmission, reactive power compensation devices and active power filtering devices, offers several features such as [6]: • Ability to provide voltage/reactive power support to the AC network. • Decoupling of the AC systems which results in improved fault ride-through capability. • Since power electronic converters are current control devices, they do not change significantly the fault level at the point of connection. • Facilitates connection of weak systems such as wind farm, independently of the effective short circuit ratio (ESCR). • Black start capability, this eliminates the need for the start-up generator [7]. The use of VSC-HVDC for offshore wind farm has been mentioned in some previous literature. In [8], integration of large offshore doubly fed induction generator-based wind farms with a common collection bus controlled by a STATCOM into the main onshore grid using line-commutated high-voltage dc connection Manuscript received and revised April 2012, accepted May 2012


395

Reza Noroozian, Mohammad Reza Safari Tirtashi

results have been shown.

where, VMean is base wind velocity, VRamp is ramp wind component, VGust is Gust wind component, VStep is step

II.

Description of the Case Study

wind component and VTur is Turbulence wind component. Base wind velocity ( VMean ) is a constant value that shows the wind speed initial average. Other components are defined as follows:

The overall scheme of the case study is shown in Fig. 1. As shown in this figure, the wind farm includes two groups of generator. The proposed wind farm has been constructed based on variable speed wind turbines with induction generators. Each group of generators includes five wind turbine induction generators and the rated power for wind farm is 50MW. All machine parameters are given in Appendix A. The 115 kV , 100 MVA grid with equivalent impedance assuming X/R=5.67 has been perceived. The VSC-HVDC has been used for interconnecting the wind farm to the AC grid while the length of DC line is 100km and, the reference DC bus voltage, Vdcref is

t ≤ Tsr ⎧0 ⎪ ⎛ t − Tsr ⎞ ⎪ Vramp ( t ) = ⎨ Awr ⎜ ⎟ Tsr < t < Ter ⎝ Ter − Tsr ⎠ ⎪ ⎪0 Ter ≤ t ⎩

where, Awr is the amplitude of ramp, Tsr is the beginning time and Ter is the end time: Vgust ( t ) =

118 kV for grid side converter and 123 kV for wind farm side converter. The DC link capacitance in Fig. 1 can be calculated by the following equation:

C=

2τ Srated

⎧0 ⎪ ⎪⎪ Awg =⎨ ⎪ 2 ⎪ ⎪⎩0

(1)

V dcref 2

(3)

where, τ is the time constant and usually 5ms< τ 0

ωref = ωref − ∆ω

P (k ) − P (k − 1) = 0

Yes

P (k ) − P (k − 1) > 0

No

Yes

ωref = ωref + ∆ω

ω (k ) − ω (k − 1) > 0

ωref = ωref − ∆ω

No

ωref = ωref + ∆ω

Return Fig. 5. Perturbation and Observation algorithm



398


leads to lower power coefficient, consequently the obtained power of turbines reduces.

VI.1. Wind Farm Side Converters

Wind farm side converters controller is depicted in Fig. 6. As shown in this figure, to obtain the maximum power and reducing the loud noises of wind turbines, the variable speed wind turbine induction generators have been used to build the wind farm. Flux oriented vector control is suitable method for controlling the generators speed [1]. As shown in Fig. 6, MPPT algorithm based perturbation and observation method [12], determines the rotor speed reference signal to get the maximum power. Corresponding to Fig. 6, Pt and Wt are the power and speed of turbine, W1 until W5 are the speed of generators, V/w is the rated flux, Vac is the terminal voltage of generator, I dc is the DC current of each VSC and Vabc is the offshore grid voltage. When converters are parallel, then DC currents in each converter determine the active power flow [1]. Based on vector control principle, the active and reactive power of generators have controlled independently by controlling M m and M θ . Phase locked loop (PLL) synchronizes the control signals with the offshore grid voltage. As shown in Fig. 6, when the reference signals are prepared, they compared with triangular carrier to build gate signals.

W4 W3 + +

W5

+

W2 + W1 + Wt Pt

1/5 + − I dc

PI

MPPT Wg-ref

V/w Vac +

+

−

−

PI

PI

Vabc Mθ

PLL

Mm

Mm & to M ref θ Mθ Comparator

VI.2. Grid Side Converter

Carrier Signal

The grid side converter controls the DC bus voltage. Consequently, the power flow between converters will be controlled. For this purpose, as shown in Fig. 7, the DC reference voltage is compared to DC bus voltage and finally constructs the angle of grid side VSC. Comparison between ac voltage reference and ac bus voltage determines the modulation index for grid side converter. After that, similar to previous section, the reference signals for PWM are built while the PLL synchronizes the control signals with grid voltage. Corresponding to Fig. 7, Vabc is the onshore grid voltage. The controller gains are determined using trial and error method.

M ref Gate Signals

Fig. 6. Wind farm side converter controller

Vdc

−

+

G 1 + sT +

1 sT

Vdcref

Vacref V ac + −

+

VI.3. Blade Angle Controller

PI

For maintaining the machine power/speed within rated value at high winds, each of the turbines is equipped with blade angle controller. In this paper, the used controller is based on PSCAD/EMTDC blade angle controller which is shown in Fig. 8. In this figure ωm is mechanical speed,

Vabc Mθ

PLL

ωref is reference speed, Pref is power demand, Pg is

PI

Mm

Mm & to M ref θ Mθ

power output of the machine, K s is the gain, K p is the

Comparator

proportional gain, Ki is integral gain, Gm is gain multiplier and K 4 is blade actuator integral gain. When the wind speed increases, to maintain the machine power/ speed in rated values, the blade angle controller increases the angle of blade. As shown in Fig. 3, increase of the β

Carrier Signal

M ref Gate Signals

Fig. 7. Grid side converter controller



399


Terminal voltage and speed of typical generator in group 1 are depicted in Figs. 10(c) and (d) respectively. Despite the random operation of wind, based on these figures, it is self-evident that induction generators have a good voltage profile and the rotor speed is controlled.

Pg

+

Wm

Pref

Kp

+

Wref

−

KI s

20

Ks

Limiter

wind speed(m/s)

−

Gm

+

−

−1

15 10 5 0

Filter

2

4

6

+

12

14

(a)

−

200 DC bus voltage(KV)

Rate Limiter

K4 s

β Fig. 8. Blade Angle Controller

VII.

8 10 time(s)

sending

100 50 0

Simulation Results

recieving

150

2

4

6

To evaluate the control of grid connected wind farm by using multi-terminal VSC-HVDC, simulation studies have considered for various operating conditions.

8 10 time(s)

12

14

(b)

VII.1.

bus voltage(PU)

sending

Dynamic Characteristics Analysis

In this section the stability of the system is investigated. The simulations have been done in PSCAD/EMTDC but the figures are plotted in MATLAB. The wind speed data is shown in Fig. 9(a), while the rated speed for wind is 12m/s. The MPPT algorithm determines the optimum rotor speed for obtaining the maximum power. The controllers can maintain the DC bus voltage for both sending and receiving side converters at rated values as shown in Fig. 9(b). Corresponding to Fig. 9(c), it is clear that, voltages of grid side and wind farm side converters are adjusted in rated values. Active and reactive power for sending and receiving converters are shown in Figs. 10(a) and (b) respectively. As shown in these figures, the groups of generators produce maximum power and send it to the grid. Clearly, the reactive power is flowed from wind farm side converters towards wind farm groups to compensate the needing of induction generators.

recieving

1.1 1 0.9 0.8

2

4

6

8 10 time(s)

12

14

(c) Figs. 9. (a) Wind speed data, (b) wind farm side and grid side DC bus voltages and (c) wind farm and grid side ac bus voltages

VII.2.

Transient Characteristics Analysis

For more investigation, the transient operation of the system under both single and three phase faults has been considered. In these cases, the wind speed has been kept constant and is equal to the rated speed for the turbines.



400


This is because during the duration of fault, wind speed does not change tremendously. VII.3.

VII.4.

Three phase fault is one of the most severe faults that rarely happen at the grid. Similar to single phase fault, the fault has been applied at t=4s, and the CBs on the faulted line are opened at t=4.1s, and at t=4.9s the CBs are reclosed. As mentioned in the previous sections, control of DC bus voltages is crucial to stabilize the system during the faults. Great performance of the system to control the DC bus voltages is self-evident based on Fig. 12(a). Effect of three phase fault on the other important characteristics of the system has been shown in the Figs. 12(b), (c) and (d). It is priceless to mention that, network side fault has minute effect on wind farm side variables except for DC bus voltage. Moreover, for typical generator in group 1, the terminal voltage is shown in Fig. 13. In short, the simulation results show that this wind farm has a good fault ride-through and the controllers can defeat severe faults.

Single Phase Fault

The fault has been applied at t=4s as a grid disturbance, and the circuit breakers (CBs) on the faulted line are opened at t=4.1s, and at t=4.9s the CBs are reclosed. The transient response of the system for DC bus voltages are shown in Fig. 11(a). When the fault applies to the grid side converter, the DC bus voltages disrupt for a short time, then controllers stabilize the system and the grid side and wind farm side DC bus voltages return to their original states, quickly. This happens for other variables as demonstrated in the Figs. 11(b), (c) and (d). These figures show that the proposed system has great ability to deal with single phase fault.

40 sending

reactive power(Mvar)

active power(MW)

100 recieving

50

0

-50

Three Phase Fault

2

4

6

8 10 time(s)

12

sending

0 -20 -40

14

recieving

20

2

4

6

(a)

14

12

14

1.05 generator speed(PU)

terminal voltage(KV)

12

(b)

6

4

2

0

8 10 time(s)

1

0.95

2

4

6

8 10 time(s)

12

14

2

4

(c)

6

8 10 time(s) (d)

Figs. 10. (a) sending and receiving active powers, (b) sending and receiving reactive powers, (c) terminal voltage of induction generator and (d) Rotor speed of induction generator



401


140

80 active power(MW)

DC voltage(KV)

sending recieving

130 120 110 100 3.9

4

4.1 4.2 time(s)

4.3

sending recieving

60 40 20 0 3.9

4.4

4

sending

bus voltage(PU)


50 recieving

0

4

4.3

4.4

(b)

(a)

-50 3.9

4.1 4.2 time(s)

4.1 4.2 time(s)

4.3

1.2

sending

1.1

recieving

1 0.9 0.8 0.7 3.9

4.4

4

4.1 4.2 time(s)

4.3

4.4

(d)

(c)

Figs. 11. (a) wind farm and grid side DC bus voltages, (b) wind farm side and grid side active powers, (c) wind farm side and grid side reactive powers and (d) wind farm and grid side ac bus voltages

150 sending

active power(MW)

DC bus voltage(KV)

200 recieving

150

100

50

4

4.2 time(s)

4.4

sending

100 50 0 -50 -100

4.6

recieving

4

4.2 time(s)

4.4

4.6

(b)

(a)

sending

bus voltage(PU)


100 recieving

50 0 -50 -100

4.2 time(s)

4.4

4.6

(c)

recieving

1 0.8 0.6 0.4 0.2

4

sending

1.2

4

4.2 time(s)

4.4

4.6

(d)

Figs. 12. (a) wind farm and grid side DC bus voltages, (b) wind farm side and grid side active powers, (c) Wind farm side and grid side reactive powers and (d) wind farm and grid side ac bus voltages



402


terminal voltage(KV)

4.4 [3]

4.2 4 [4]

3.8 4

4.2 time(s)

4.4

[5]

4.6

Fig. 13. Terminal voltage of induction generator [6]

VIII. Conclusion

[7]

In this paper, investigation on VSC-HVDC technology for interconnecting the wind farm to the grid is done. The considered wind farm composed of 10 wind turbine induction generators which clustered in two groups. For controlling each of the groups, one VSC is used based on flux oriented vector control method. To obtain the maximum power, the perturbation and observation algorithm has been applied in wind farm side converter controllers. Blade pitch angle controller is applied to maintain the machine power/speed in rated values at high wind speed. Grid side converter undertook the controlling of DC bus voltage which is the most important factor for good operation of multi-terminal VSC-HVDC. To investigate the transient operation of the system, single and three phase fault have been applied. Simulation studies have been done with PSCAD/EMTDC software. Results showed that the system has a good operation both in normal and transient modes. Moreover proposed VSC-HVDC can transfer the wind farm power to the grid satisfactorily.

[8]

[9]

[10]

[11]

[12]

[13]

[14]

Appendix A

[15]

TABLE A1 PARAMETERS OF INDUCTION GENERATOR AND TURBINE (IN PER UNIT UNLESS INDICATED SPECIALLY) Parameter Value Parameter Value Unsaturated Rated power 5MW 3.86 magnetizing reactance Rotor unsaturated Rated voltage 2.4kV 0.122 mutual reactance Second cage Rated frequency 60Hz 0.105 unsaturated reactance Polar moment of Pole number 6 3.5s inertia Stator resistance 0.066 Mechanical damping 0.008 0.298 Rated wind speed 12m/s First cage resistance Second cage 0.018 Gearbox ratio 70 resistance Stator unsaturated 0.046 Turbine radius 60m leakage reactance

[16]

Authors’ information E. E. Department, University of Zanjan. Reza Noroozian was born in Iran. He is an assistant professor with the department of Power Engineering, University of Zanjan, Iran. He received his B.Sc. from Tabriz University, Tabriz, Iran, in 2000. He received his M.Sc. and also Ph.D degrees in electrical engineering from Amirkabir University of Technology (AUT), Iran, in 2003 and 2008, respectively. His areas of interest include: Power Electronic, Power System, Power Quality, Integration and Control of Renewable Generation Units, Custom Power, Micro-grid Operation, Distributed Generation Modeling, Operation and Interface Control. Dr Noroozian is a member of the institute of electrical and electronics engineers (IEEE). E-mail: [email protected]

References [1]

[2]

HVDC grid, in Proc. Nordic Wind Power Conference, Bornholm, Denmark, sept. 10-11, 2009. B. Liu, J. Xu, R. E. Torres, T. Undeland, Faults Mitigation Control Design for Grid Integration of Offshore Wind Farms and Oil & Gas Installations Using VSC HVDC, International Symposium on Power Electronics Electrical Drives Automation and Motion, Trondheim, Norway, 14-16 June 2010, pp. 792-797. P. Bresesti, W. L. Kling, R. L. Hendriks, R. Vailati, HVDC connection of off shore wind farms to the transmission system, IEEE Transaction Energy Conversion, vol. 22, n.1, pp. 37–43, Mar. 2007. J. Yang, J. E. Fletcher, J. Reilly, Muiltiterminal DC Wind Farm Collection Grid Internal Fault Analysis and Protection Design, IEEE Transaction on power Delivery, vol. 25, n. 4, 2010, pp. 2308-2318. G. Li, G. Li, L. Haifeng, Y. Ming, Research on Hybrid HVDC, International Conference on Power System Technology, 2004. PowerCon 2004, Nov. 21-24, 2004, Baoding, China. G. O. A, G .P. Adam, O. Anaya-Lara, K. L. Lo, Grid Integration of Offshore Wind Farms Using Multi-terminal DC Transmission System (MTDC), 5th IET International Conference on Power Electronics, Machines and Drives (PEMD 2010), pp. 1-6. S. Bozhko, G. Asher, R. Li, J. Clare, L. Yao, Large Offshore DFIG-Based Wind Farm With Line-Commutated HVDC Connection to the Main Grid: Engineering Studies, IEEE Transaction Energy Conversion, vol. 23, n. 1, 2008, pp. 119-127. S. K. Chaudhary, R. Teodorescu, P. Rodriguez, P. C. Kjær, P. W. Christensen, Modelling and Simulation of VSC-HVDC Connection for Wind Power Plants, in Proc. 5th Nordic Wind Power Conference, Bornholm, Denmark, sept. 10-11, 2009. N. Prabhu, K. R. Padiyar, Investigation of Subsynchronous Resonance With VSC-Based HVDC Transmission Systems, IEEE Transaction on power Delivery, vol. 24, n. 1, 2009, pp. 433-440. Y. Higuchi, N. Yamamura, M. Ishida, T. Hori, An Improvement of Performance for Small-Scaled Wind Power Generating System with Permanent Magnet Type Synchronous Generator, 26th Annual Conference of the IEEE Industrial Electronics Society IECON, vol. 2, 2000, pp. 1037-1043. G. Esmaili, Application of Advanced Power Electronics in Renewable Energy Sources and Hybrid Generating Systems, Ph.D. dissertation, The Ohio State University, USA, 2006. G. H. Riahi, P. Freere, Dynamic Controller to Operate a Wind Turbine in Stall, Proceedings of Solar97, Canberra, Australia, 1997. S. Benjanirat, Computational studies of the horizontal axis wind turbines in high wind speed condition using advanced turbulence models, Ph.D. dissertation, Georgia Institute of Technology, 2006. J. G. Slootweg, Wind Power, Modeling and Impact on Power System Dynamics, Ph.D. dissertation, Delft, ISBN 90-9017239-4, 2003. S. M. Muyeen, R. Takahashi, J. Tamura, Operation and Control of HVDC-Connected Offshore Wind Farm, IEEE Transaction on Sustainable Energy, vol. 1, n. 1, 2010, pp. 30-37.

D. Jjovcic, N. Strachan, Offshore wind farm with centralised power Conversion and DC interconnection, IET Generation. Transmission. Distribution, vol. 3, n. 6, pp. 586-595, 2009. E. Koldby, M. Hyttinen, Challenges on the road to an offshore



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M. R. Safari Tirtashi was born in behshahr, Iran in 1986. He received B.Sc. in Electrical and Electronic Engineering from Mazandaran University, Iran, in 2008. He is M.Sc. student at Electrical Engineering department of Zanjan University, now. His research interests control and stabilization of power systems, power electronic and application of intelligent methods in power systems. E-mail: [email protected]



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Review and Comparing of Conventional Buck, Boost and Buck-Boost Dc to Dc Converters with their Interleaved Topology Via Equations, Simulation and Experimental Results M. Jahanmahin, A. Hajihosseinlu, E. Afjei

Abstract – A Dc to Dc converter is used to change the input voltage level to a desired output voltage level less/more than the input voltage magnitude by utilizing pulse width modulation. Interleaved configuration is well-known topology that is used to increasing output power by employing two storage elements as well as reducing the output ripple voltage. This paper compare characterizes of three kinds of Dc to Dc conventional converter against them interleaved one via equations, simulation and experimental results. In interleaved technique, two inductors are used for feeding the load by two independent switches. One inductor charges up by the source voltage for up to 50% of the total switching period while the other inductor is discharging its energy into the load during this step. The output power production is almost doubled while the ripple voltage is reduced by a factor of two when compared to a conventional Dc to Dc converter. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Dc-Dc Converter, Conventional Converter, Interleaved Converter, Voltage Ripple

For example, in a Buck converter with one switch, reduce voltage ripple reduction is accomplished by select a bigger capacitor at the output but the transient time, as well as cost will increase. In order to decrease power losses in the power switches, one can decrease the switching frequency but this process will increase ripple voltage and can adverse effect on the output voltage waveform. In this paper, the interleaved configurations of Buck, Boost and Buck-boost converter are analyzed which operates under current continuous mode of operation (CCM) [16]. In this configuration by utilizing two storage devices will have less damaging effects on circuit parameters. these interleaved converters has two inductors two switches which can improve several factors over conventional converters by considering a delay time between these two switches which will be explained in the next section. These factors are namely: output voltage ripple, maximum input current, transient time and maximum of transferable power. These interleaved converters can be used in BLDC [17] for reduce torque ripple, and also in inverter [18]-[19] for prepare the dc input voltage with lower voltage ripple.

Nomenclature D T ILmax ILmin Vout

Duty cycle Period of signal Maximum of inductor current Minimum of inductor current Averaged of output voltage

I.

Introduction

In several power conversion applications, it is required to convert a constant dc voltage source to a variable dc voltage source. The dc-dc converters are devices which convert a specific dc voltage to another specific level [1]-[3]. In fact, the dc-dc converter of dc grid is similar to transformer in the ac systems. These kinds of converters are used in some applications such as regulated DC power supplies, renewable energy systems, electrical vehicles, cranes, distributed generation systems, and power factor Correction process [4]-[8]. There are several studies that have been done about reducing the output voltage ripple of dc-dc converters [9]-[15]. The changes in the structure of the converter using new topology can improve the operation of the converter. A good design requires special attentions to many circuit parameters such as voltage ripple, maximum current of each element, power losses, voltage stress, and etc. These parameters usually have tradeoff in such way that improving of one parameter might have a big effect on another parameter.

II.

Operational Principle of the Converters

Figs. 1, 2, and 3 show the topologies of the interleaved Buck, Boost and Buck-boost converters respectively. Each topology composed of two inductors (i.e. L1, L2), two diodes (i.e. D1, D2) and two switches



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M. Jahanmahin, A. Hajihosseinlu, E. Afjei

(i.e.S1, S2). In experimental cases these ideal switches can be replaced by power transistors. These switches have the same switching frequency (f) and duty cycle (D) in PWM applications [1]. The only difference between these switches is that one switch has a delay time; this delay time is half of the switching period (T/2) as shown in Fig. 4 when D is smaller than 0.5 and is shown in Fig. 5 when D is larger than 0.5.

Fig. 5. The gate pulses of S1 and S2 when D > 0.5

III. Analysis of the Interleaved Converters Fig. 1. The interleaved Buck Converter

The interleaved Buck, Boost and Buck-boost converters undergo four different topological states as shown in Figs. 6, 7, and 8 respectively. Every state for each converter is explained in following manner: State 1: In this state, S1, S2 are turned on. Both inductors (L1, L2) are charging up. D1, D2 are in reversed bias mode. This condition only can happen when D > 0.5. State 2: In this state, S1 is switched off and L1 is discharging into the load by forward biasing of diode D1. S2 is turned on, L2 is charging up and D2 is in reversed bias mode. State 3: In this state, S1 is turned on; L1 is charging up and D1 is in reversed bias mode. S2 is switched off and L2 is discharging in the path that contains L2, load and D2. State 4: In this state, S1, S2 are turned off. Both inductors (L1, L2) are discharging into the load by forward biasing of two diodes D1 and D2. This condition can only occur when D < 0.5. All these states can be summarized in Tables I, II, and III respectively for interleaved Buck, Boost and Buck-boost converters.

Fig. 2. The interleaved Boost Converter

Fig. 3. The interleaved Buck-Boost Converter

Fig. 6. Four variety conditions of the interleaved Buck Converter

Fig. 4. The gate pulses of S1 and S2 when D < 0.5



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TABLE III DIFFERENT STATES FOR INTERLEAVED BUCK-BOOST CONVERTER Components

State1

State 2

State 3

State 4

S1 S2 D1 D2 L1 L2

On On Off Off Charging Charging

Off On On Off Discharging Charging

On Off Off On Charging Discharging

Off Off On On Discharging Discharging

Elements which feeding the Load

C

L1 & C

L2 & C

L1 & L2 &C

As mentioned in the above, at any moment, the load is being fed by two independent paths. Hence, for a given voltage level it can transfer more power in comparison with conventional converters.

Fig. 7. Four variety conditions of the interleaved Boost Converter

III.1. The Interleaved Buck Converter While S1 is switched on, the voltage drops on L1 which is Vin-Vout will take DT seconds. When S1 is switched off, the voltage drops on L1 during this time is -Vout and it will take (1-D)T seconds. So: (Vin – Vout)DT + Vout (1 – D)T = 0

(1)

Therefore the expression for the input voltage (Vin) and the average of output voltage (Vout) can be summarized as: (2) Vout = DVin while L1 is charging up, its current increases from IL1,min to IL1,max . Thus current variation for L1 can be expressed as: (3) ∆IL1 = IL1,max - IL1,min

Fig. 8. Four variety conditions of the interleaved Buck-boost Converter TABLE I DIFFERENT STATES FOR INTERLEAVED BUCK CONVERTER Components

State1

State 2

State 3

State 4

S1 S2 D1 D2 L1 L2 Elements which feeding the Load





L1 & L2 &C

L1 & L2 &C

L1 & L2 &C

L1 & L2 &C

The duration of this condition is: Vin – Vout = L1 (∆IL1 / DT) Similarly, for L2:

State1

State 2

State 3

State 4

S1 S2 D1 D2 L1 L2 Elements which feeding the Load





C

L1 & C

L2 & C

L1 & L2 &C

∆IL2 = IL2,max - IL2,min

(5)

Vin – Vout = L2 (∆IL2 / DT)

(6)

and:

TABLE II DIFFERENT STATES FOR INTERLEAVED BOOST CONVERTER Components

(4)

It is assumed that this interleaved Buck converter is operating in continuous current mode and L1=L2=L, therefore the equations for minimum and maximum inductors current with the load R are:


IL1,max = (DVin / 2RL)( L + (1.5-D)RT)

(7)

IL1,min = (DVin / 2RL) L + (D-0.5)RT)

(8)

IL2,max = (DVin / 2RL)(L + (0.5-D)RT)

(9)

IL2,min = (DVin / 2RL)(L + (D-1.5)RT)

(10)


407



III.2. Analysis of the Interleaved Boost Converter While S1 is switched on, the voltage drops on L1 is Vin and it takes DT seconds. When S1 is switched off, the voltage drops on L1 is Vin-Vout and it takes (1-D)T seconds. So: Vin DT + (Vin – Vout)(1 – D)T = 0

The duration for this condition is:

(12)


(25)

Vin = L2 (∆IL2 / DT)

(26)

It is assumed that this interleaved Buck-boost converter operating in continuous current mode of operation and L1=L2=L, therefore the equations for minimum and maximum of inductors current with R as a load are:

(13)

IL1,max = (DVin / [4RL(1-D)2] )( 2L – RT(2D-3)(1-D) ) (27)

The duration for this condition is: Vin = L1 (∆IL1 / DT)

(24)

and:

While L1 is charging up, its current increases from IL1,min to IL1,max . Thus current variation for L1 can be expressed as: ∆IL1 = IL1,max - IL1,min

Vin = L1 (∆IL1 / DT) Similarly, for L2:

(11)

Therefore the expression for the input voltage (Vin) and the average of output voltage (Vout) can be summarized as: Vout = Vin / (1 – D)

(23)

IL1,min = (DVin / [4RL(1-D)2] )( 2L – RT(1-2D)(1-D) ) (28)

(14)

IL2,max = (DVin / [4RL(1-D)2] )( 2L – RT(2D-1)(1-D) ) (29)

Similarly, for L2: ∆IL2 = IL2,max - IL2,min

IL2,min =(DVin / [4RL(1-D)2] )( 2L – RT(3-2D)(1-D) ) (30)

(15)

This circuit simulated by MATLAB. The inductors currents for the each interleaved topology are shown in Figs. 9, 10 and 11.

and: Vin = L2 (∆IL2 / DT)

(16)

It is assumed that this interleaved Boost converter is operating in continuous current mode and also L1=L2=L. The equations for minimum and maximum of inductors current with R as the load are: IL1,max = (Vin / [4RL(1-D)2])(2L + RTD(1-D)2)

(17)

IL1,min = (Vin / [4RL(1-D)2])(2L - RTD(1-D)2)

(18)

IL2,max = IL1,max

(19)

IL2,min = IL1,min

(20) Fig. 9. Inductors Current for the interleaved Buck Converter

III.3. Analysis of Interleaved Buck-boost Converter While S1 is switched on, the voltage drops on L1 is Vin and it takes DT seconds. When S1 is switched off, the voltage drops on L1 is Vout and it takes (1-D)T seconds. So: Vin DT + Vout (1 – D)T = 0

(21)

Therefore the expression for the input voltage (Vin) and the average of output voltage (Vout) can be summarized as: Vout = – DVin / (1 – D)

(22)

while L1 is charging up, its current increases from IL1,min to IL1,max. Thus current variation for L1 can be expressed as:

Fig. 10. Inductors Current for the interleaved Boost Converter



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(31)

IL,min = (DVin / 2RL)(2L - (1-D) RT)

(32)

IL,max = (Vin / [2RL(1-D)2])(2L + RTD(1-D)2)

(33)

IL,min = (Vin / [2RL(1-D)2])(2L - RTD(1-D)2)

(34)

IL,max =(DVin / [2RL(1-D)2])(2L + R(1-D)2)

(35)

IL,min =(DVin / [2RL(1-D)2])(2L + R(1-D)2)

(36)

The first advantage of these interleaved converters is that their maximum ripples voltage which is almost half as much as the conventional converters. This is summarized in Table IV, and shown in Figs. 15, 16 and 17 respectively for Buck, Boost and Buck-boost converters. Interleaved Buck converter has a special trait. The ripple voltage of this converter is almost zero when duty cycle is 0.5. This is driven from Table II and is obtained at Fig. 18.

Fig. 11. Inductors Current for the interleaved Buck-boost Converter

IV.

IL,max = (DVin / 2RL)(2L + (1-D) RT)

Advantages and Simulation Results

In this section some of the benefits that can be obtained from the interleaved configurations in comparison to the conventional converters are presented. The conventional Buck, Boost and Buck-boost are shown in Figs. 12, 13 and 14 respectively. The maximum and minimum of inductor current equations are also presented in (31) and (32) for conventional Buck converter, in (33) and (34) for conventional Boost converter and in (35) and (36) for conventional Buck-boost converter, respectively.

TABLE IV VOLTAGE RIPPLE OF CONVERTER Type of Converter

Voltage ripple of conventional converter

Voltage ripple of interleaved Converter

Buck Boost Buck-boost

(RTVin / L)D(1-D) Vin / (1-D)2 Vin D / (1-D)2

(RTVin / L)D(1-2D) Vin / [2(1-D)2] Vin D / [2(1-D)2]

Fig. 12. The Conventional Buck Converter

Fig. 15. Voltage Ripple of Buck Converter Fig. 13. The Conventional Boost Converter

Fig. 14. The Conventional Buck-boost Converter

Fig. 16. Voltage Ripple of Boost Converter



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Fig. 19. Input Currents of Buck Converter

Fig. 17. Voltage Ripple of Buck-boost Converter

Fig. 18. Ripple Voltage for Buck Converter in D = 0.5

Fig. 20. Input Currents of Boost Converter

The second advantage is the ratio of the maximum inductor current in the interleaved converters presented by equations (7), (17), and (27) respectively to the maximum of inductor current equations in conventional converters presented in (31), (33) and (35) respectively which show higher attainable current magnitude almost doubled for the interleaved converter circuits. Therefore in the given voltage level, the maximum of transferable power increases by a factor of two. The third advantage is the maximum current out of voltage source in the interleaved converters when D < 0.5, is almost half of the maximum input current in a conventional converters. Of course only interleaved Buck and interleaved Buck-boost converters have this advantage; in the interleaved Boost converter, the input current is continues. This issue is simulated by MATLAB and is shown in Figs. 19, 20 and 21. The fourth advantage is related to power losses regarding the power switches. In these interleaved converters the power losses are almost half of the power losses in conventional converters. This issue has been achieved for two reasons. First is the maximum of inductors current in interleaved converter is almost half of the maximum of inductors current in conventional converters and the second one is the power losses in each switch is proportion to square of current, I, flowing through them where I is the switch current. Finally, the last advantage obtained is related to the transient response time. It is smaller than the response time of the conventional Dc to Dc converters. This fact is shown in Figs. 22, 23 and 24 for these interleaved converters.

Fig. 21. Input Currents of Buck-boost Converter

Fig. 22. The response time of Buck Converter



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Fig. 27. IL1 and IL2 Currents in time division=5us for Boost Converter

Fig. 23. The response time of Boost Converter

Fig. 28. IL1 and IL2 Currents in time division=5us for Buck-boost Converter

Figs. 29, 30 and 31 show the comparison between the output voltages of converters with a conventional one having the same load value. Figs. 32, 33 and 34 show the comparison between the input currents of the interleaved converters with conventional ones. In these figures the input current of conventional converters is placed on top while the input current of interleaved converters is shown in the bottom part of these figures.

Fig. 24. The response time of Buck-boost Converter

V.

Experimental Results

These interleaved converters are built in the laboratory and their circuits are shown in Fig. 25.

Fig. 25. The Circuit Structures

The currents of the two inductors for the interleaved converters are obtained shown as Figs. 26, 27 and 28.

Fig. 29. Output Voltage of: A) Conventional Buck B) The interleaved Buck

Fig. 26. IL1 and IL2 Currents in time division=5us for Buck Converter

Fig. 30. Output Voltage of: A) Conventional Boost B) The interleaved Boost



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All the figures have been obtained by laboratory experiment which shows the same results as the ones obtained through simulations.

VI.

Conclusion

This paper presents the analysis, simulation and experimental results of interleaved configurations of three kind Dc to Dc converters. It then continues with the comparison with conventional converters. The interleaved converters show higher output power (almost twice as much) as well as lower ripple factor (half) when compared to the conventional converters.

Fig. 31. Output Voltage of: A) Conventional Buck-boost B) interleaved Buck-boost

References [1] [2]

[3] [4]

[5]

Fig. 32. Input Current of: A) Conventional Buck B) The interleaved Buck

[6]

[7]

[8]

Fig. 33. Input Current of: A) Conventional Boost B) The interleaved Boost

[9]

[10] [11]

[12]

[13]

[14]

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Fig. 34. Input Current of: A) Conventional Buck-boost B) interleaved Buck-boost



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[15] J. Xu, M. Qin, “Multi-pulse train control technique for Buck converter in discontinuous conduction mode,” IET Power Electron., vol. 3, no. 3, pp. 391-399, May 2010. [16] A. Reatti, M. K. Kazimierczuk “Small-Signal Model of PWM Converters for Discontinuous Conduction Mode and Its Application for Boost Converter,” IEEE Transactions of Circuits and Systems-I: Fundamental Theory and Applications, vol. 50, no. 1, pp. 65-73, January 2003. [17] F. Montazeri and D. Arab Khaburi, “A Modified Direct Torque Control for BLDC Motor Drives,” International Review on Modelling and Simulations (IREMOS), Vol. 3. n. 4, pp. 553-559, August 2010. [18] H. Chahkandi Nejad, R. Jahani, J. Olamaei, H. Shayanfar, “Using an Exhaustive-Based Search Algorithm for Selective Harmonic Reduction of Multilevel Inverters,” International Review on Modelling and Simulations (IREMOS), Vol. 3 N. 6, pp. 12291233, December 2010. [19] B. Husodo, M., Shahrin M., Taufik, “Analysis and Simulation of a New Three Phase LLCC Resonant Inverter for Fuel Cell Applications,” International Review on Modelling and Simulations (IREMOS), Vol. 4 N. 1, pp. 1-9, February 2011.

Authors’ information Mohamad Jahanmahin was born in jahrom, Iran, in 1985. He received the B.S degree in electrical engineering from Najafabad Islamic University, Esfahan, Iran, at 2008. He is currently pursuing the M.S degree in electrical engineering at Shahid Beheshti University, Tehran, Iran and is assistance of full Professor Ebrahim Afjei. He has been working in power electronics laboratories. His research interests are power electronics and it’s applications in power system and drives. E-mail: [email protected] Amin Hajihosseinlu was born in Bandarabbas, Iran, in 1990. He is currently pursuing the B.S degree in electrical engineering at Shahid Beheshti University, Tehran, Iran. He has been research assistant of full professor Ebrahim Afjei and has been working in power electronics laboratories. His research interests are power electronics and it’s applications in power system, hybrid electric vehicles, digital control applied to Power Electronics, renewable-energy systems and electrical machines. E-mail: [email protected] Ebrahim S. Afjei received the B.S. degree in electrical engineering from the University of Texas in 1984, the M.S. degree in electrical engineering from the University of Texas in 1986, and the Ph.D. degree from New Mexico State University, Las Cruces, in 1991. He is currently a Professor in the Department of Electrical Engineering, Shahid Beheshti University, Tehran, Iran. His research interest is in switched reluctance motor drives and power electronics. E-mail: [email protected]



413


Dissolved Gas Analysis and Fault Diagnosis Interface for Oil-Immersed Transformer Using Fuzzy Logic Yunus Biçen, Faruk Aras, Melih İnal, Hasbi İsmailoğlu

Abstract – In this study, the dissolved gases in oil due to the incipient failures in the oilimmersed transformer are analyzed and diagnosed by fuzzy logic approach considering the framework of IEC 60599 standard. In addition to the incipient failures of transformer, more precise results are obtained about the condition of cellulosic material by using CO2/CO rate. Fuzzy logic membership functions and rules designed to ensure compliance with the standards. For efficient use of this approach, a user friendly interface is created by using LabVIEW software package. The results obtained are compared with the results of diagnostic in the literature. The fault diagnosis success of over 95% is achieved. Copyright © 2012 Praise Worthy Prize S.r.l. All rights reserved.

Keywords: Dissolved Gas Analysis, Fault Diagnosis, Fuzzy Logic, Transformer

I.

When the use of these methods attains a more effective level, this will prevent large-scale failures, shorten their maintenance period, decrease energy cuts and costs in total to a significant degree. In literature, several new approaches which take dissolved gas analysis methods/techniques and standards as reference have been suggested in recent years. The dissolved gas data obtained from power transformers have been examined using fuzzy logic approach based on Duval triangle method considering IEC 60599 standard [13]. In the study, two fuzzy logic units were used, the first was used to define the dissolved gas analysis results and the outputs obtained from this unit were applied to the second unit as inputs. Thus, the situation of paper-oil insulation was assessed [13]. In another study, the failure detection chart was developed based on neuro-fuzzy and fuzzy logic membership functions such as triangular, trapezoidal and gaussian, in terms of degree of effectiveness were also examined by Duraisamy et al. [14]. Algorithms which assess dissolved gas analysis results by using only artificial neural networks and detect failures were developed. In a study executed in this context, it was shown that the network structure trained by taking Roger gas ratios method as basis also produced output for possibilities remaining outside the limits of this method and was more effective in detecting the real failures [15]. There are also other studies available in literature, which have carried out failure detection by using other artificial intelligence methods, again in line with the standards besides the methods of fuzzy logic and/or artificial neural network. [16]-[21]. In this study, the fuzzy logic approach is used to diagnose the incipient failures of the transformer by evaluating the key gases.

Introduction

It is an issue of great importance in electric industry in today’s world that power transformers, which are one of the most significant parts of energy transmission and distribution system, complete their economic life without any failure or shutting down. In literature, although the failure rates of power transformers are less than other electrical equipments on yearly basis, their out of service periods are quite high[1], [2]. In case of a large-scale failure, this period can extend further. This situation gives rise to hitches in energy transmission and distribution. Production, installation and maintenance costs of power transformers are quite high. For this reason, today, studies on monitoring operation performance of power transformers and detecting the failures in the initial phase before a large-scale failure occurs continue [3]-[6]. The most effective ones among these studies are dissolved gas analyses conducted on the oil sample taken from inside of transformer tank [7], [8]. Through the dissolved gas analysis, the nine different gases in the oil (hydrogen, methane, acetylene, ethane, ethylene, carbon monoxide, carbon dioxide, nitrogen, and oxygen) are identified according to their ppm values and a incipient failure can be detected with the help of ratios obtained [8]-[12]. These analyses are usually executed in laboratory environment. Some measurement instruments which enable online-monitoring of nine different gases have been developed in recent years. However, as their costs are not at reasonable levels, they are used for systems at critical points which have great economic value.



414

Yunus Biçen, Faruk Aras, Melih İnal, Hasbi İsmailoğlu

TABLE I FAULT TYPES AND CRITICAL RATE OF GASES Gases Faults

For this purpose, the standards are reviewed and the standard IEC 60599 is taken as the basis due to its success in the literature in the rate of identifying the transformer failures. The fuzzy logic membership functions and the rule base are designed as to provide the most accurate output considering the standard using by user friendly interface. The obtained results are analyzed as well.

II.

H2

Dissolved Gas Analysis

As it is known, there are basically two different insulators, oil and cellulosic structure in the transformer. In the event of a fault (discharge, excessive heating, cellulosic deformation,…) oil and paper material react chemically and form some gas compounds. The compound structures of these emerging gases that are defined as key gas are shown in Fig. 1.

OİL

Ethane

Partial discharges

Ethylene

Circulation currents

TRANSFORMER TANK

Carbon‐ Mon/Di‐ oxide

C2H4

C2H2

CO

CO2

9

9 200

TABLE II GAS ANALYSIS METHODS AND TECHNIQUES Reference Standards IEEE Tools IEEE C57, IEC 60599 PC57, 104-1991 / 1999 104 D11d Individual & TDCG 9 9 Doernenburg Ratios 9 Rogers Ratios 9 9 Basic Gas Ratios 9 Key Gas Ratios 9 9 TCG Procedure 9 TDCG Procedure 9 9 Duval Triangle 9

Overheating CELLULOSE

C2H6

There are many techniques in literature that have been developed to determine the fault by using these rates of increase of these gases. Generally these techniques take three standards as basis as shown in Table II. The most successful one of the fault diagnosis techniques that have been developed taking the standards shown in Table II is the Duval Triangle technique, developed by M. Duval [8], [27]. The Duval Triangle takes the IEC 60599 standard as basis [28].

Hydrogen Methane

Arcing

CH4

Degradation of 9 cellulose Low-energy 9 9 9 discharges in oil High-energy 9 9 9 9 9 discharges in oil Over-heating for oil 9 9 9 9 9 and paper According to IEC 60599 5 2 2 2 0,1 50 (ml/day) H2: Hydrogen, CH4: Methane, C2H2: Acetylene, C2H4: Ethylene, C2H6: Ethane, CO: Carbon-monoxide, CO2: Carbon-dioxide

Acetylene

Fig. 1. The key gases caused by the reaction

Basically, each or some of the different nine gases that can be found in the transformer oil indicate specific faults, only the changes at the nitrogen and oxygen gases are not related to a fault in the transformer [10], [22]. All the other gases provide information on the faults that occur or have occurred in the transformer tank. However, the types of oil used in the transformer systems may vary. Especially in the recent years the usage rates of the natural oils are increasing due to their operation advantages and environmental characteristics [23], [24]. The studies that are carried out demonstrate that similar dissolved gas analysis techniques can be used for the gases that emerge as a consequence of faults in the transformers, in which natural oil is used as insulating material [25]. Table I demonstrates specific faults and the gases that change related to these faults. In order for the transformer to operate safely, the daily increasing amounts of these gases should not exceed the limit values given in Table I [26]. The exceeded values mean a fault prognostication in the transformer. The rates of increase of these gases change according to the type of the fault.

As it is mostly preferred in literature and due to its high sensitivity in fault diagnosis, the IEC 60599 standard is taken as the basis in this study and fuzzy logic based fault diagnosis is carried out. As a difference from other studies, in order to be able to interpret the status of the paper insulator, the CO2/CO gas rate is evaluated using the fuzzy logic approach.

III. Fuzzy Logic Approach Fuzzy logic is used to mathematically model the behavior of the system within uncertain boundaries expressed by an expert, by using linguistic expressions in the decision-making process. In other words, it produces significant results by operating the rule based relation between input and output, without any necessity for complicated mathematical equations in the systems that cannot be expressed with definite boundaries, but can be stated with approximately knowable band ranges and boundary changes [29], [30].



415

Yunus Biçen, Faruk Aras, Melih İnal, Hasbi İsmailoğlu

TABLE III GAS RATES AND THE CODING TABLE ACCORDING TO THE IEC 60599 STANDARD Coding according to the limit Characteristic gas ratios (b) Coding according to the limit values values (IEC 60599)

Characteristic gas ratios (a) R(K1) = C2H2/ C2H4 R(K2) = CH4/ H2 K1 K2* R(K3) = C2H4/ C2H6 R < 0.1 0: low 1: low 0.1 < R < 1 1: med 0: med 17 3

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