Automatic Control

International Review of

Automatic Control (IREACO) Theory and Applications

Contents A New Sensitivity Function Loop Shaping Design Based on Extended Observers by L. Saidi, S. Benacer, M. Boulemden

594

Fuzzy Logic and Causal Reasoning for FDI of Bond Graph Uncertain Parameters Systems by W. Bouallegue, S. Bouslama Bouabdallah, M. Tagina

602

Fault Detection and Isolation for Nonlinear Dynamic Power System by A. Thabet, M. Boutayeb, G. Didier, S. Chniba, M. N. Abdelkrim

610

Sensor Fault Detection and Localization Methodology Using Principal Component Analysis by Mohamed Guerfel, Anissa Ben Aicha, Kamel Benothman

620

An Optimal Fuzzy Sliding Mode Control Based on Direct Thrust Force Control for Permanent Magnet Linear Synchronous Motors by M. Abroshan, J. Milimonfared, M. B. Menhaj, S. Tarafdar

630

Nonlinear MPPT and PFC Control of SCIG Wind Farm by M. Benchagra, Y. Errami, M. Hilal, M. Ouassaid, M. Maaroufi

640

Sliding Mode Control of Buck Converter for Low Voltage Applications by S. Chander, P. Agarwal, I. Gupta

649

Field Oriented Control of Salient Pole Wound Field Synchronous Machine in Stator Flux Coordinate Based Dampers Windings Effect by Seyed Jafar Salehi

659

Enhancement of One-Comparator Counter-Based PWM Control via Sawtoothed Wave Injection by K. I. Hwu, Y. T. Yau

669

A Wide Input-Output Voltage Range AC-DC Converter with a Fuzzy PI+D Controller by Mohammad Bagher Akbari Haghighat, Farhad Ghadaki, Mohammad-Ali Shamsi-Nejad

678

Brushless DC Motor Power Factor Correction by Converters BOOST and SEPIC by H. Amiri, E. Afjei, M. Jahanmahin, V. Mirzaee

688

Design and Implementation of Current-Controlled Voltage Source Converter in Wind Turbine Application by M. S. Majid, S. M. Hussin, H. Abdul Rahman, M. Y. Hassan

694

(continued)

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Dynamic Response Improvement of DFIG Driven by Wind Turbines Using a Novel Algorithm by Mostafa Eidiani, Natan Asghari, Hossein Zeynal

700

Controlling Chaos in the Voltage Transformer by a Time Delay Feedback Controller by A. Abbasi, S. H. Fathi, A. Gholami, H. R. Abbasi

707

Chaos Control in the Voltage Transformer with Nonlinear Core Loss Model by H. R. Abbasi, A. Gholami, M. Rostami, A. Abbasi

719

Optimal Placement of FACTS Devices Considering Power System Loadability and Cost of Installation by S. Gh. Seifossadat, M. Heidari Orejloo, R. Kianinezhad, D. Mirabbasi

733

Fuzzy Logic Controller Design Based SVC for Improving Power System Damping by H. Hasanvand, B. Bakhshideh Zad, B. Mozafari, H. Maskani

740

A New Adaptive Neuro Fuzzy Based Algorithm to Identify Power Swing by A. Esmaeilian

749

Regional Gradient Stabilisation for Linear Distributed Systems by E. Zerrik, Y. Benslimane, A. El Jai

755

Design and Analysis of SVC Complementary Controller to Improve Power System Stability Using RCGA-Optimization Technique by A. D. Falehi

766

Determining the Contribution of Harmonic Distortion Generated by Utility and Customer in a Radial Distribution System by Farzaneh Bagheri, Ali Ajami

773

A New Equivalent Linear Mixed-Integer Expression Model of the Unit Commitment Problem with Minimum Operating Units by N. Zendehdel, A. Karimpour, M. Oloomi

782

Optimal Mixed Variable-Pitch-Angle Control & Variable-Speed Control of Wind Turbines by S. Sh. Alaviani, G. H. Riahy

791

An IMM Algorithm for Tracking a Maneuvering Target by Ahmadreza Amirzadeh, Naser Pariz, Mahdi Ghadiri

798

A Review of Singularity Avoidance in the Inverse Kinematics of Redundant Robot Manipulators by Samer Yahya, M. Moghavvemi, Haider A. F. Mohamed

807

Optimisation of Defects in Composite Materials Using an Improved Wavelet Analysis Basic Algorithm by Benhamou Amina, Benyoucef Boumedienne

815

Development of a Screening Tool for Cervical Cancer of the Uterus by Artificial Intelligence Tools Using the Uterine Cervico – Smears by Guesmi Lamia, Nabli Lotfi, Bedoui Mohamed Hédi

821


International Review of Automatic Control (I.RE.A.CO.), Vol. 4, N. 5 September 2011

A New Sensitivity Function Loop Shaping Design Based on Extended Observers L. Saidi, S. Benacer, M. Boulemden Abstract – A new approach for active noise and disturbance control based on sensitivity function loop shaping design is presented. The proposed method combines pole placement with sensitivity function loop shaping in the frequency domain using extended observers. The approach is non adaptive and the frequencies are chosen according to the desired shape of the sensitivity function in the corresponding bandwidth. Damped sine wave models are introduced in the observer for disturbance tuning reduction, active damping control, and stability robustness achievement. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Loop Shaping, Sensitivity Function, Disturbance Estimator, Damping Factor

Nomenclature S (s)

Sensityvity function

T (s)

Complementary sensitivity function

WB km Φm sm

ω ξi Ai

ωd i

ϕi G J

H ∞ loop shaping is a popular design method to form the open loop by introduction of weights, in order to fulfill certain aims as disturbance rejection, reference tracking, etc. However, weighting function selection is not an easy task and the order of the final controller, which is designed by this technique, is usually high [3][6]. The two-degree-of-freedom structure has been also shown to be very effective for rejecting disturbances and loop shaping design. The RST configuration (where R, S and T are polynomials to determine) is a widely used method in the design of controllers using classical pole placement [7]-[9]. However, it requires solving the equation of Bezout on one hand and an increased number of memory locations on the other. The Bezout equation may be solved by expanding the various polynomials as powers of the indeterminate variable and equate coefficients of like powers. This leads to a set of linear equations in the coefficients of the unknown polynomials, which are known as the Sylvester equations [8]. In [10], [11] a Virtual Reference Feedback Tuning (VRFT) method is used for the shaping of the sensitivity function. The VRFT is a data based method in the design of feedback controllers for a linear plant whose transfer function is unknown. It is a solution of the one degree of freedom model reference control problem. The design is based on a set of I/O data. The choice of the filter and a weighting factor used in this approach are not determined analytically but chosen on the basis of the operator's experience. In [16] and [17] both performance weight and controller structure are simultaneously determined by genetic algorithms. The performance and robust stability conditions of the designed system satisfying the H∞ loop shaping are formulated as the objective function in the optimization problem. More recently, In [18], the authors propose a solution of the loop shaping in quantitative

Bandwith Gain margin Phase margin Modulus margin Angular frequency Damping factor Sinusoidal disturbance amplitude Sinusoidal disturbance pulsation Sinusoidal disturbance phase Gain vector A gain providing zero state error

I.

Introduction

During the recent years, the reduction of noises, vibrations and loop shaping in various plants has become a main line of research and is the object of numerous publications. Solutions to the problem have been actively studied in recent years in the literature and many approaches are proposed [1]-[15]. For example, repetitive structure has been shown to be very effective for rejecting repetitive disturbances [1]. However, this method requires an increased number of memory locations. Furthermore, plant uncertainty and bad knowledge of the disturbance frequency make it difficult to design a procedure providing good tracking performance. Recently a robust repetitive control was introduced in [2] to solve such problem.

Manuscript received and revised August 2011, accepted September 2011

594


L. Saidi, S. Benacer, M. Boulemden

design: good setpoint tracking and disturbance rejection ( S ≈ 0 ,T ≈ 1 ) has to be traded off against suppression of measurement noise ( S ≈ 1,T ≈ 0 ) [19], [20]. It is small for low frequencies and approaches the value 1 at high frequencies. Values greater than 1 and peaking are to be avoided. Peaking easily happens near the point where the curve crosses over the level 1 (the 0 dB line). The desired shape for the sensitivity function S implies a matching shape for the magnitude of the complementary sensitivity function T = 1 − S . When S is as shown in Fig. 1, T is close to 1 at low frequencies and decreases to 0 at high frequencies. It may be necessary to impose further requirements on the shape of the sensitivity function if the disturbances have a distinct frequency profile.

feedback theory (QFT) using fractional compensators, which give singular properties to automatically shape the open loop gain function. Disturbance observer has also been known to be very effective to compensate disturbances. However, it is not very efficient when the disturbance frequency is poorly known or varying with the rotational speed. This paper proposes a new methodology based on multi-models of the disturbances where damped sine wave models are introduced. The damping factor gives robustness to disturbance angular frequency variations. We consider the problem of rejecting sinusoidal disturbances whose magnitude and phase are unknown. The frequency is time-varying around a specific value. The proposed method combines setpoint tracking with sensitivity function loop shaping. The sensitivity function provides important information on the disturbances rejection and it constitutes a good indicator for the robustness of the controller.

II.

Magnitude

S Frequency

The Output Sensitivity Function T

The transfer function between the disturbance P and the output y is called the output sensitivity function. It is an indicator for assessing both disturbances rejection performances and robustness [19]. In particular the inverse of its modulus is equal to the modulus margin [19]. The smaller S ( jω ) is, with ω ∈ ℜ , the more the

Fig. 1. Typical form of sensitivity S and complementary sensitivity T

disturbances are attenuated at the angular frequency ω . |S| is small if the magnitude of the loop gain is large. Hence, for disturbance attenuation it is necessary to shape the loop gain so it is large over those frequencies where disturbance attenuation is needed. Making the loop gain large over a large frequency band easily results in error signals e and resulting plant inputs u that are larger than the plant can absorb. Therefore, the loop gain can only be made large over a limited frequency band. This is usually a low-pass band, that is, a band that ranges from frequency zero up to a maximal angular frequency WB (bandwidth of the feedback loop). Effective disturbance attenuation is only achieved up to the angular frequency WB . The larger the “capacity” of the plant is, the greater the inputs the plant can handle before it saturates are. For plants whose transfer functions have zeros with nonnegative real parts, the maximally achievable bandwidth is limited by the location of the right-half plane zero closest to the origin [20]. Fig. 1 shows a typical shape of the magnitude of the sensitivity function and the complementary sensitivity function T. The complementary sensitivity function T ( s ) derives

To provide total disturbances rejection at certain frequencies, the sensitivity function value is zero at the desired frequency. We have to define a template of acceptable sensitivity function which depends on the plant to control on one hand and on the control specifications on the other. In general, the shape is based on the requirements of strong disturbances attenuation at low frequencies, and eventually a total disturbance rejection in steady state with regards to a minimum acceptable modulus margin which defines an upper boundary for the modulus of the sensitivity function. It is desirable to make S(s) as "small" as possible. Thus S ( s ) can be made small only over a finite frequency range. The bandwidth WB can serve as a simple closed-loop performance measure. It is related to S ( jω ) by [19]:

S ( jω ) < 1 / 2

(2)

III. Stability Robustness The closed-loop system remains stable under perturbations of the loop gain as long as the Nyquist plot of the perturbed loop gain does not encircle the point −1. Naturally, this may be accomplished by “keeping the Nyquist plot of the nominal feedback system away from the point −1”.

its name from the equality: S (s) + T (s) = 1

∀ω < WB

(1)

This illustrates one of the basic trade-offs in feedback


International Review of Automatic Control, Vol. 4, N. 5

595


equation:

The classic gain margin and phase margin are wellknown indicators for how closely the Nyquist plot approaches the point −1. In the classical feedback system design, robustness is often specified by establishing minimum values for the gain and phase margin. Practical requirements are km >2 for the gain margin and 30°< Φ m 0.5 . Adequate margin of this type is not only needed for robustness, but also to achieve a satisfactory time response of the closed-loop system. The gain margin km and the phase margin Φ m are related to the modulus margin sm by the inequalities: ⎛s ⎞ Φ m = 2 arcsin ⎜ m ⎟ ⎝ 2 ⎠

1 km ≥ , 1 − sm

xd = Ad xd

and the output equation: d = Cd xd

(7)

The matrices Ad and Cd are given as: 0 ⎤ ⎡ Ad1 ⎢ ⎥ % Ad = ⎢ ⎥ ⎢ 0 ⎥ A dN ⎦ ⎣

(8)

Cd = ⎡⎣Cd1

(9)

" Cd N ⎤⎦

The individual block entries in these block matrices follow from the state space of a damped sine wave as: ⎡ −ξiωdi Adi = ⎢ ⎢⎣(ξi + 1) ωdi

(3)

(ξi − 1) ωd −ξiωdi

Cdi = [ 0 1]

This means that if sm ≥ 0 ,5 then km ≥ 2 and Φ m ≥ 28,96 [19]. The converse is not true in general.

IV.

(6)

i

⎤ ⎥ ⎥⎦

(10)

(11)

This choice of a damped sinusoidal model is guided by the fact that this model allows to introduce two complex conjugate roots:

Extended Disturbance Observer si = −ξiωdi ± jωdi 1 − ξi 2 ( ξi is the damping factor)

For instance, let us assume that the state space plant model is given by the four-tuple ( A,B,C , 0 ) completely

The damping factor of these zeros has to be scaled in accordance to the desired reduction of the shape and/or the peak of the sensitivity function. The aim of the two added complex double root is to decrease the sensitivity function magnitude in the high frequency range and to shift its maximum to a lower frequency. As a result, resonant zeros will be introduced in the sensitivity function. With the N disturbances acting upon the plant output, an overall model of the plant with the extended observer follows as:

controllable and observable of order n: x = Ax + Bu y = Cx + d

(4)

where d is a sinusoidal disturbance acting upon the plant output. To reduce the influence of the disturbance d on the output y, the approach is to generate an estimate of this disturbance and use this estimate as a control signal. A disturbance observer is used to generate this estimate (Fig. 2). A typical spectrum of the disturbances consists of harmonics that are multiple of the fundamental frequency. For control design purposes, it is therefore assumed that the disturbance signal d is a sum of N sine signals with amplitude Ai , pulsation ωdi and phase ϕi ,

ˆxd = Ad ˆxd + Ae ˆx = Axˆ + Bu + Le ˆy = Cxˆ + dˆ dˆ = C ˆx

(12)

d d

i.e.: d=

T where dˆ is the reconstructed disturbance and [ L A ] is

N

∑ Ai sin (ωd i t + ϕi )

(5)

the matrix gain of the observer. This observer design can be carried out by pole placement or by designing an optimal stationary Kalman filter. The new state vector consists of the system and the

i =1

For this, the disturbance is modelled as an output of an autonomous state space model with the state transition Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


596


states of the N disturbances model: X = ⎡⎣ x

is the observer which is assigned to adapt to the shape of the disturbance.

" xd N ⎤⎦

xd1

T

(13)

Control Signal

Reference

The control law is therefore given by:

Signal

u = − K Xˆ + J r

(14) State Estimate

where X = ⎡⎣ x xd1 " xd N ⎤⎦ and K = [ K G ] is the extended feedback gain. G is a gain vector which permits to eliminate the disturbance effect and J is a gain which provides zero steady state error: T

V. 1 − K ( A) B −C ( A ) B

So as to reject the sinusoidal disturbance, the sensitivity function has to be forced to zero. In this case, from equations ((4)-(6)), we can write:

{−C ( sI − A + BK )

−1

BGX di + Cdi X di

}

s = si

= 0 (16)

The disturbance state can be determined using equations ((6)-(7)) and ((10)-(11)):

X di

( (

) )

⎡γ 1 s + ξiωd + (ξi − 1) ωd γ 2 ⎤ i i i i ⎥ =⎢ ⎢γ s + ξ ω + (ξ + 1) ω γ ⎥ i di i di 1i ⎦ ⎣ 2i

minimum acceptable modulus gain). Fig. 3 shows an example of a desired sensitivity function shape. s1 corresponds to the frequency weighting function used in H∞ and s2 the maximum peak value of the sensitivity function.

(17)

where γ 1i and γ 2i are disturbance initial conditions, i.e. X di ( 0 ) = ⎡⎣γ 1i

Definition of a Template

We have to determine a template of acceptable output sensitivity function. This template depends on the plant to control and on the control specifications, but we propose a general shape method which ensures good robustness and good disturbance rejection. It is based on the following requirements: • Disturbance reduction at low frequencies and total rejection in steady state. • For specified frequencies disturbance rejection, it is necessary to shape the sensitivity so that it is large over those frequencies. • Maximum for peak value of the magnitude of the sensitivity function S ( jω ) (which defines a

(15)

−1

Extended Observer

Fig. 2. Extended disturbance observer

−1

J=

Output Plant

Controller

T

γ 2i ⎤⎦ . Fixing:

|S|

−1

−C ( si I − A + BK ) B =

(

)

(

= α ξi ,ωdi + j β ξi ,ωdi

)

s2

(18)

s1

Frequency

Then equation (16) gives: G1i = −

G2i

(

) (

β ξi ,ωdi 1 + ξi 2 1 − ξi α ξi ,ωd + β 2 ξi ,ωd i i

(

(

)

) = 2 2 α (ξi ,ωd ) + β (ξi ,ωd )

)

α ξi ,ωdi i

(19) Fig. 3. Template of sensitivity function

i

VI.

Note that the components of the vector gain G are independent of the initial conditions of the state vector X di . This result is foreseeable because these coefficients

Loop Shaping Procedure

The design of a robust controller by shaping the sensitivity function can be obtained using the following procedure: − Choose the dynamic of the state feedback and the observer following the desired dynamics,

are calculated to compensate the disturbance dynamics. It



597


− Design the regulator and check the shape of the sensitivity function. Generally, make a judicious choice of the poles of the state feedback and those of the observer for a compromise robustness / performance allows having a satisfactory modulus margin. The obtained sensitivity function will be considered as a reference concerning the estimation of the influence of the disturbances on the plant output, − If the values of the disturbance attenuation and the bandwidth are not acceptable, introduce a damped sine wave model in the observer. This model will introduce a pair of complex combined poles whose effect is to move the maximum of the sensitivity function towards high frequencies and to increase the disturbances rejection interval, − If again, the values of the disturbance attenuation and the bandwidth are not acceptable or if the maximum of the sensitivity function is too strong, a solution consists of introducing other resonant roots generated by a second sinusoidal disturbance model. The effects of these additional zeros in the controller are in general a reduction of the sensitivity function around the resonant frequency of the zeros added and an increase of S ( jω ) magnitude in high frequency. If necessary, increase the value of the auxiliary poles and decrease the dominant poles.

VII.

eccentricity, track irregularities, mechanical vibrations and shocks. The position of the pick-up is controlled by two cooperative actuators; a fine actuator and a coarse actuator, which are briefly depicted in Fig. 4. As shown in [24], we will consider only the fine actuator because it is corrupted by the disturbance whose frequency is proportional to the disc rotation speed. The CD-ROM drive (for LG 52X) can be represented by the following model [24]: P (s) =

1.022775 × 109 s 2 + 64.73s + 166800

(20)

This transfer function describes a voice coil motor actuator from voltage input to position output. It takes into consideration the sensor gain which converts the position displacement into voltage. The optical disc drive measures the position of the pick-up by a relative position error between the desired track and the actual position of the optical spot. The spindle motor frequency is assumed equal to 63.5 Hz. Spindle-Motor

Disc

Fine Actuator

Application to Track Following Problem

For many multimedia applications, it is desired to achieve a high speed in increasing data rate and reducing access time. Track following problem, for optical disk drive such as CD-ROM, is to control the position of the optical spot in such away that it follows the desired track (within 0,1 µm) of optical disk media which is usually deviated from the concentric circles due to the disk eccentricity. The displacement error caused by this last one amounts to 280 µm in the worst case. The optical disk drive measures the position of the spot by a relative position error between the desired track and the actual position of the spot. Therefore, the disc eccentricity affects this measure as a sinusoidal disturbance whose frequency is the one of the disk spindle motor [22], [23]. The basic problem for the compact disc mechanism control is the huge variety of sources for disturbances and model errors which pose rather conflicting constraints on the control system in terms of bandwidth, precision, mechanical vibrations, and shocks …etc. A disc drive mechanism has many control loops, most of which are rather slow and of less significance. Track following problem is to control the position of optical spot so that it follows the desired track of the optical disc media. However, this goal is usually deviated due to the internal and external disturbances. The most important disturbances present in optical disc drive are rotation

Pick-up

Coarse Actuator

Fig. 4. Optical disc drive scheme

VIII. Simulation Results To demonstrate the effectiveness of the proposed technique, several simulation examples are applied to the CD-ROM model described previously. This latter is subject to periodic disturbances with time varying fundamental frequency Fig. 5 shows the shape of the sensitivity function for one disturbance model. Disturbance is supposed by pulsation 2 rd/s. Fig. 6 shows the effect of the increase of the bandwidth of disturbance rejection (theorem of Bode) by moving the pulsation of the model of the disturbance towards great frequencies. The choice of the first pulsation (of weak value) comes from the fact that we are interested in rejecting sinusoidal disturbances to the neighbourhood of this frequency as well as disturbances of the constant type. The second pulsation, superior to the first, is introduced to move the maximum of the sensitivity function towards superior frequencies. It is scaled on the spindle motor frequency.



598


Fig. 8 shows the effect of the damping factor on the shape of the sensitivity function. This additional ratio constitutes an additional parameter of very interesting regulation. Table I shows that the damping factor constitutes a good parameter for disturbance tuning reduction and active damping control to achieve stability robustness ( ωd1 = 5 rd/s, ξ1 = 0.5 , ωd 2 = 60 rd/s, ξ 2 = 0.3 ).

10 0 -10

Magnitude (db)

-20 -30 -40 -50 -60

Figs. 9 and 10 show the controller adaptability to the disturbance angular frequency variations.

-70 -80

0

-90 -1 10

0

10

1

2

10 pulsation (rd/s)

3

10

10

-5

-10

Fig. 5. Sensitivity function shape for ω d =2 rd/s, ξ = 0.1 Magnitude (dB)

-15

10

0

-20

-25

-30

-10

-35

Magnitude (dB)

-20

-40

-45

-30

-50 -40

3

10

398.9823 pulsation (rd/s)

-50

Fig. 8. Influence of the damping factor for ωd =398.9823 rd/s

-60

-70 0 10

3

(----- ξ 3 =0.12 and ____ ξ 3 =0.4) 1

2

10

3

10

4

10

10

5

10

pulsation (rd/s)

TABLE I DAMPING FACTOR VERSUS MODULUS GAIN AND SENSITIVITY FUNCTION PEAK ( ωd =398.9823 rd/S)

Fig. 6. Sensitivity function shape for ωd =20 rd/s, ξ1 = 0.1 , 1

3

ωd =398.9823 rd/s, ξ 2 = 0.2

Damping Factor 0.1 0.2 0.3 0.4 0.5 0.6

2

We can introduce an additional model of pulsation ωd3 in the observer with a certain damping factor (Fig. 7). This model will shape the sensitivity function around medium frequencies. The Price to be paid is a regulator of more and more raised order.

Modulus Gain 0.5466 0.5526 0.5683 0.5750 0.5820 0.5929

Sensitivity Function Peak 5.2474 5.1510 4.9081 4.8067 4.7016 4.5402

-3

6

x 10

10

5 0

4

Tracking error (V)

Magnitude (dB)

-10

-20

3

2

-30

1 -40

0 -50

-1 0 -60 0 10

1

10

2

3

10

10

4

10

5

0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

1

10

pulsation (rd/s)

Fig. 7. Sensitivity function shape for ωd =5 rd/s, ξ1 = 0.5 , ωd =60

Fig. 9. Sinusoidal disturbance rejection. Disturbance model: ξ=0.01, ωd =5 rd/s

rd/s, ξ 2 = 0.3 ; ωd =398.9823 rd/s, ξ 3 = 0.12

Disturbance pulsation: ω d + 10%ω d

1

2

3



599


-3

7

x 10

[11] 6

5

[12]

Tracking error (V)

4

3

[13]

2

1

[14] 0

-1

0

0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

1

[15]

Fig. 10. Sinusoidal disturbance rejection. Disturbance model: ξ=0.01, ω d =5 rd/s

[16]

Disturbance pulsation: ωd - 10%ωd [17]

IX.

Conclusion

In this paper a new methodology of design of robust controllers for SISO plants is presented. It combines pole placement and sensitivity function loop shaping. The approach is progressive and allows obtaining a loop shaping following wished frequency specifications. Thus, we showed that the fact of introducing a damping factor allows better shape of the sensitivity function. The results obtained in simulation show that the strategy provides a robust performance when disturbances and parameter uncertainties are present. Such a strategy has been derived using multi-models disturbance observer and extended state feedback.

[18]

[19]

[20]

[21]

[22] [23] [24]

References [1]

J.H. Moon, M.N. Lee, M.J. Chung, Repetitive control for trackfollowing servo system of an optical disk drive, IEEE Transactions on Control Systems Technology, vol. 6 n. 5, 1998, pp. 663-670. [2] M. Steinbuch, Repetitive control for systems with uncertain period-time, Automatica, vol. 38, n. 12, 2002, pp. 2103-2109. [3] P. Blue, L. Güvenc, D. Odenthal, Large Envelope Flight Control Satisfying H∞ Robustness and Performance Specifications, American Control Conference, June 25-27, 2001, pp. 1351-1356, Arlington, USA. [4] D. McFarlane, K. Glover, A Loop Shaping Design Procedure Using H∞ Synthesis, IEEE Transactions on Automatic Control, vol. 37, n. 6, 1992, pp. 759-769. [5] G. Vinnicombe, Uncertainty and Feedback: H∞ Loop-shaping and the µ-gap Metric (Imperial College Press, London, 2000). [6] H. Panagopoulos, K.J. Aström, PID Control Design and H∞ Loop Shaping, International Journal of Robust Nonlinear Control, vol. 10, n. 15, 2000, pp. 1249-1261. [7] K.J. Aström, B. Wittenmark, Computer Controlled Systems, Theory and Design (Prentice-Hall International Editions, 1990). [8] P. De Larminat, Automatique, Commande des Systèmes Linéaires (Hermès Editions, Paris, 1993). [9] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory (Mac Millan Inc., N.Y., 1992). [10] A. Lecchini, M.C. Campi, S.M. Savaresi, Sensitivity Shaping Via Virtual Reference Feedback Tuning. The 40th IEEE Conference

on Decision and Control, December, 2001, pp. 750-755, Orlando, Florida, USA. A. Lecchini, Virtual Reference Feedback Tuning: a New Direct Data-based Method for the Design of Feedback Controllers, Ph.D. dissertation, Brescia University, Italy, 2001. Z. Ding, Asymptotic Rejection of Disturbances Generated by Nonlinear Exosystems in Linear Dynamic Systems, International Review of Automatic Control (IREACO), Vol. 1, n. 1, 2008, pp. 17. M. Hedayati, S. M. Bashi, N. Mariun, H. Hizam, A Short Review of Different Optimal H∞ Robust FACTS Controller Designs, International Review of Automatic Control (IREACO), Vol. 2, n. 1, 2009, pp. 8-16. A. Tahar, M. N. Abdelkrim, Multimodel H∞ Loop Shaping Control of Uncertain Weakly Coupled Systems, International Review of Automatic Control (IREACO), Vol. 4, N. 3, 2011, pp. 351-361. Z. Souar, L. Mostefai, F. Olivie, New Nonlinear Controller Based on Active Disturbance Rejection for Friction Compensation, International Review of Automatic Control (IREACO), Vol. 1, n. 4, 2008, pp. 500-506. S. Kaitwanidvilai, P. Olranthichachat, I. Ngamroo, Weight Optimization and Structure specified Robust H∞ Loop shaping Control of a Pneumatic Servo System Using Genetic Algorithm, International Journal of Robotics and Automation, vol. 25, 2010. P. Olranthichachat, S. Kaitwanidvilai, Enhancing the Perfor– mance of Fixed-Structure Robust Loop Shaping Control using Genetic Algorithm Approach, International Multi-Conference of Engineers and Computer Scientists, March 16-18, 2011, Hong Kong. J. Cervera, A. Baños, Automatic Loop Shaping in QFT Using CRONE Structures, Journal of Vibration and Control, vol. 14, 2008. I.D. Landau, Placement de pôles avec calibrage des fonctions de sensibilité, In La robustesse, Analyse et Synthèse de Commandes Robustes (Hermès Ed., 1994, 160-171) J.S. Freudenberg, D.P. Looze, Right Half Plane Poles and Zeros and Design Trade-offs in Feedback Systems, IEEE Transactions on Automatic Control, vol. 30, n. 6, 1985, pp. 555-565. H. K. Sung, S. Hara, Properties of Complementary Sensitivity Function in SISO Digital Systems, International Journal of Control, vol. 50, n. 4, 1989, pp. 1283-1295. K.C. Pohlmann, The Compact Disc Handbook (A-R Editions, Inc., 1992). J. Watkinson, The Art of Data Recording (Focal Press, 1994). H. Shim, H. Kim, C.C. Chung, Design and experiment of add-on track following controller for optical disc drives based on robust output regulations, International Conference on Control, Automation and Systems, June 3-July 2, 2004, pp. 1829-1835, Boston, Massachusetts, USA.

Authors’ information Department of Electrical Engineering, University of Batna, 05000 Batna, Algeria. Tel/fax : +213-33-80-54-94 E-mail: [email protected] Lamir Saidi received his Engineering Master degree from University of Constantine, Algeria, in 1991 and the Ph.D. degree from Savoie University, France, in 1996. Currently, he is Associate Professor at the Electrical Engineering department, University of Batna, Algeria. His interests include Digital Motion Control, Fuzzy control, Robust control Mechatronics, and Digital Signal Processing. He is member of CISE and RST.



600


Saddok Benacer received his Engineering Master degree and his magister in Electronics from the University of Batna, Algeria. Currently, he is a PhD student. His research interests concern robotics, renewable energy and system optimization.

Prof. Mohammed Boulemden, is a graduate from the National Polytechnic School of Algiers (ENPA). He obtained his Ph.D degree in electronics from Nottingham University, UK. He carried out research work in fields related to information theory, signal processing, robotics and sensors and was an expert member in various scientific committees. Currently, he is director of research at the university of Batna where he launched a laboratory in Signals and Intelligents Systems.



601


Fuzzy Logic and Causal Reasoning for FDI of Bond Graph Uncertain Parameters Systems W. Bouallegue, S. Bouslama Bouabdallah, M. Tagina

Abstract – In this paper, a method for on-line fault detection and isolation (FDI) of non linear uncertain parameters systems modelled by bond graphs (BG) is proposed. Residuals are generated from the Diagnostic Bond Graph (DBG) which is a direct residuals generator. Detection is based on fuzzy logic approach and its different parameters are determined off line. For isolation, three methods based on causal properties of the BG model, that we deduced, are used: Fault Signature Matrix (FSM), exoneration and cover of a causal graph. A real simulation example is provided to show the efficiency of the proposed detection method and a comparison between the three isolation approaches is made. Copyright © 2011 Praise Worthy Prize S.r.l. All rights reserved.

Keywords: Fault Detection and Isolation (FDI), Bond Graph (BG), Parameters Uncertainties, Fuzzy Logic, Causal Reasoning

I.

process and the association of observations (symptoms) to failures using qualitative operators. In process history based methods [9]-[19]-[23], there is no need to any kind of models. Only the availability of large amount of historical process data is needed [19]. There are different ways in which this data can be transformed and presented as a priori knowledge to a diagnostic system. This is known as feature extraction. In case of uncertain parameters systems, robust diagnosis algorithms are proposed in [10]-[11]-[18]-[24]. For FDI, the choice of a modelling formalism is an important step. Because of its behavioral, structural and causal properties, the Bond Graph (BG) tool is used in complex processes modelling and FDI [1]-[6]-[7]-[8]. BG gives mathematical and graphical representations that make easy the task of monitoring and the design of supervision systems. Its causal properties are used to determine the fault origins; BG models are exploited in both qualitative and quantitative diagnosis methods [1]. The main contribution of this paper is to use the bond graph model directly in the task of robust FDI in case of uncertain parameters systems. The detection module is based on a fuzzy logic system. For isolation, we proposed three methods exploiting causal properties of the BG model. This paper is organized as follows: Section 2 details the notion of DBG used to generate directly the residuals in the BG model. Section 3 describes the proposed fuzzy detection method. Section 4 gives the details of three used isolation methods: Fault Signature Matrix, exoneration and covering causal graphs. Section 5 presents the model of the hydraulic benchmark composed of interconnected three tanks. Finally,

Introduction

The increasing complexity of modern automated processes and the increasing demands for quality, reliability, availability, safety and cost efficiency require better safety management and supervision. Generally, the function of a supervisory control system is to detect and isolate faults in the system [1]-[23]. FDI methods can be divided into three categories: quantitative methods, qualitative methods and process history based methods [16]-[17]-[18]. The most frequently quantitative diagnosis approaches are based on Analytical Redundancy Relations (ARRs), state observers and parameters estimation [17]. The ARRs are relations comparing informations given by the real process to those generated by the theorical model. The parity space relations transform the input-output or model state space to rearranged and usually transformed variants; the method check the parity (consistency) of the plant models with sensor outputs (measurements) and known process inputs to detect faults. Observers design a state estimator with minimum estimation error to be compared to the measured state from the real system to generate residuals [22]. And the parameters estimation supposes the existence of an accurate dynamic model of the process with exact parameters; it requires on-line parameter estimation methods such as least squares, instrumental variables method. ARRs are generated by direct comparison of the real parameters to the estimated ones. The qualitative methods are based on qualitative models such as causal graphs, fault trees or abstraction hierarchies [8]-[18]. These models are obtained by analyzing the cause and effect relationships in the


602


W. Bouallegue, S. Bouslama Bouabdallah, M. Tagina

different results and their interpretations are given in section 6.

II.

decision on the presence or absence of faults in a monitored system In ideal conditions, the residuals’ values are equals to zero in fault free context. In practice, due to the uncertainty and the measurement noise, residuals are different from zero. Thresholds are used to deduce whether systems are in normal functioning mode or faulty mode. Unfortunately, thresholds near to zero can cause false alarms problems because of noise variation, and assigning larger thresholds reduce the fault detection sensitivity [21]. Fuzzy logic is the most common solution to overcome the uncertainty problem. Many works used this approach in residuals processing in order to know the system’s state [3]-[13]-[14]. The principle of residuals evaluation using fuzzy logic can be summarized in a three-step process (Fig. 2). Firstly, the residuals have to be fuzzified, then they have to be evaluated by an inference mechanism using fuzzy IF-THEN rules, and finally they have to be defuzzified [20].

Residuals Generation from the DBG

Fuzzification

Inference

Defuzzification

Decision

Residuals

The generation of Analytical Redundancy Relations (ARRs) from the bond graph model is a recursive procedure and a hard computational task especially in case of non linear systems [1]. It uses the structural relations given by the conservation law in all 0 and 1 junctions of the model and aims to express the unknown variables by those known (inputs and sensors). This method cannot deal with algebraic loops, so, unknown variables cannot be eliminated and the structural independence of the different residuals has to be checked with existing residuals. In [4], a direct method for ARR generation from BG model is proposed. The causality inversion of detectors (which are considered as sources) has been proposed as a unified approach to generate residuals [4]. When we assign preferred differential causality to BG model and invert sensors causalities, if necessary, the following five compositions are possible [4]: • Inverted causality in effort sensor (De), • Inverted causality in flow sensor (Df), • Non-inverted causality in effort sensor (De), • Non-inverted causality in flow sensor (Df), • Inversion of signal sensor, Ds, to signal source, Ss (for controllers). Let us consider the case of inverted causality in the effort sensor, De, (Figs. 1). This sensor will be equivalent to an effort source (measurements from real process), so, expression for the source loading flow variable is equated to zero [4]. In fact, this expression is a residual (it does not involve any states, since all storage elements are in differential causality) which’s measured by a virtual flow sensor [4].

Fig. 2. Scheme of fuzzy residuals evaluation

In this work, we propose the use of residuals generated from the DBG in the task of fault detection in case of parameters uncertainty. From residuals values, two features can be extracted: • The absolute value of the residuals: r • Residuals’ variation over a sliding time window d given by: d=

1 t + N −1 ⋅ | r ( i ) − r ( i − 1) | N i =t

∑

(1)

The descriptor sets associated with each feature fuzzy partition are: r= {“small”, “large”} d= {“small”, “large”}

(a)

(b)

These two variables are fuzzified using two trapezoidal membership functions (Fig. 3). So four parameters have to be determined for each function.

(c)

Figs. 1. (a) sensor e in behavioral model, (b) inverted causality in e and (c) substituted representation for inverted causality in e resulting in the residual sensor f

The bond graph of a system with all substitutions using preferred derivative causality is called the Diagnostic Bond Graph (DBG) [2].

III. Detection Using Fuzzy Logic Approach The problem of fault detection consists in making the

Fig. 3. Fuzzy sets of residuals



603


Many procedures issued from FDI and Artificial Intelligence communities (DX) were proposed in the task of Fault Isolation. Exploiting the causal properties of the BG tool we can use: Fault Signature Matrix (FSM), the exoneration algorithm and the causal graph.

The trapezoidal boundaries of the set small are given by: r: µsmall=[0, 0, r-max, Rmin] d: µsmall=[0, 0, d-max, Dmin] and for the set large, they are given by:

IV.1. Fault Signature Matrix

r: µLarge=[r-max, Rmin, Rmax, Rmax] d: µLarge=[d-max, Dmin, Dmax, Dmax]

In FDI terminology, the Fault Signature Matrix (FSM) crosses ARRs in rows and faults in column [4] [5]. Fault isolation uses structural properties of the ARR expressed in terms of a binary fault signature matrix S, which describes the participation of various components (physical devices, sensors, actuators and controllers) in each residual and forms a structure that links the discrepancies in components to changes in the residuals [4]. Let us consider that Fj is a fault affecting component Cj then in the binary fault signature matrix S:

where: r-max respectively d-max is the maximum value of the residual respectively residual variation in normal operating mode: Rmin = K* r-max Dmin= K*d-max Rmax respectively Dmax: is the maximum value respectively variation of the concerned residual. Detection decision is summarized in Table I.

⎧0 , Sij = ⎨ ⎩1,

TABLE I RULES BASE OF THE FUZZY SYSTEM Large Small r d Small Large

Normal Fault

if the occurence of fault Fj does not affect ARRi if the fault Fj will violate ARRi

From the DBG model, the analysis of the causal paths to each residual is used to generate these signatures [2]. In fact, every component causally linked to the residual detector can affect its value. Let us consider the DBG of Fig. 5.

Fault Fault

From this table, the next following rules base are deduced: • If (residual is Large) or (residual’s variation is Large) then (fault) • If (residual is Small) and (residual’s variation is Small) then (Normal) Rules are obtained using MIN-MAX inference method, the MIN operator represents the logic function AND, and the MAX operator for the logic function OR. The output of this fuzzy system is a fault index indicating whether the concerned residual is in normal operating mode or faulty mode. The defuzzification step of each residual is given by (Fig. 4).

Fig. 5. Example of DBG

If we consider residual detector r1, the next causal paths can be found: CÆf4Æf2Æf1Ær1 RÆf5Æf2Æf1Ær1 PÆe6Æe2Æe4ÆCÆf4Æf2Æf1Ær1 Fig. 4. Residuals defuzzification

IV.

Then, any variation in components C, R and sensor P can affect the value of residual r1. In the same way, and using all residuals we can deduce the FSM. In this work, fuzzy detection module output is exploited in isolation task. So, a fuzzy fault signature matrix is defined as follows:

Isolation Methods

The fault isolation is performed only after the deliverance of a fault detection alarm. Once a change is detected from the model parameter, it may be necessary to isolate which or which set parameters have changed. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


604


⎧ small , if the occurence of fault Fj does not affect ARRi Sij = ⎨ ⎩large, if the fault Fj will violate ARRi

causal links between the residuals detectors and the different components.

P

R

IV.2. Exoneration

C

IV.2.1. Principle of Exoneration Exoneration principle is a fundamental concept that is often used implicitly in diagnosis [4] [19]. It uses consistency of tests (residuals) to check if its support can be faulty or not. The supports of a residual are variables than can affect it and change its value. The exoneration algorithm manages two lists, a list of components whose state is normal, LN, and a list of suspect components LS. LS is made by the union of the inconsistent test supports that are not exonerated by the consistent tests [19]. The steps of the algorithm are the followings [19]: • Initialize LN and LS to the empty list LN = LS = {∅} .

r Fig. 6. Causal graph

To isolate the faulty components, backward/forward procedures are used. When the fuzzy detection module indicates a faulty residual, the backward search bounds the fault space by eliminating the normal measurements causally upstream. Then each possible primary deviation generates a hypothesis, which is forward tested using the states of the variables and the functions of the arcs [21].

• At each sampling time and for each test Ti: ¾ IF Ti result is consistent, THEN Ti support Ci is considered normal thus added to LN, LN={CiULN} and deleted from LS, LS=LS\Ci ¾ IF Ti result is inconsistent, Ti support Ci is suspected of being faulty and its components that are not in LN are added to LS, LS={CiULS}\LN • Finally, after the analysis of the tests, the components in LS represent the final diagnosis.

V. V.1.

Application Example Presentation of the Test Bench

Let us consider the following hydraulic system (Fig. 7).

IV.2.2. Exoneration Improved by Fuzzy Logic The first algorithm is based on binary logic to check the consistency of the residuals. To use the fuzzy detection module results, we make the following improvements in the precedent algorithm. So we propose the following algorithm: • Initialize LN and LS to the empty list LN = LS = {∅} • At each sampling time and for each test Ti: ¾ IF Ti result is NORMAL, then Ti support Ci is considered normal thus added to LN, LN={CiULN} and deleted from LS, LS=LS\Ci. ¾ IF Ti result is FAULT, then Ti support Ci is suspected of being faulty and its components that are not in LN are added to LS, LS={CiULS}\LN. • Finally, after the analysis of the tests, the components in LS represent the final diagnosis.

Fig. 7. Real hydraulic system

A representation of this system is given in Fig. 8.

IV.3. Causal Graph The causal graph is obtained by analyzing the cause (faults) and effect (symptoms) relationships in the process. Since the BG model express the causality explicitly, we can use it to infer a causal graph [21]. The DBG described below (see Fig. 5), connects directly the faults’ origins i.e components, sensors or controllers to the symptoms which are the fictive residuals detectors. Since that, we can deduce the causal graph of Fig. 6 from DBG of Fig. 5 by checking the

Fig. 8. A representation of the system

It’s composed of three tanks T1, T2 and T3 respectively of diameters D1, D2 and D3; water level in tanks, H1, H2 and H3 (proportional respectively to the pressures P1, P2 and P3) is measured by level sensors. This system is fed by two pumps which deliver flows Sf1



605


(flow of entrance on T1) and Sf2 (flow of entrance on T3). Tanks T1 and T2 communicate through valve V12 and tanks T2 and T3 communicate through valve V23 of diameter Sv. Each tank has a draining valve noted Vi (i=1 to 3). Flow going out from valves V1 and V2 is measured by flow sensors f1 and f2. In Table II, various physical parameters used in the system’s BG model as well as the associated uncertainties are presented. TABLE II NOMINAL PARAMETERS AND UNCERTAINTIES VALUES Parameters Value Uncertainties Value 9.81 g (kg m-1) 103 ρ (kg/m3) µ1 1 ∆µ1 ±5% µ12 1 ∆µ12 ±5% µ2 1 ∆µ2 ±5% µ23 1 ∆µ23 ±5% µ3 1 ∆µ3 ±5% D1 0.25 D2 0.28 D3 0.25 Sv 0.009 ∆Sf1, ∆Sf2 0.00025 ±5% Sf1, Sf2 (m3/s)

V.2.

Fig. 11. DBG of the three tanks system TABLE III FAULT SIGNATURE MATRIX OF THE THREE TANKS SYSTEM r1

r2

r3

r4

r5

Sf1 Sf2

1 0

0 0

0 1

0 0

0 0

V1

1

0

0

1

0

V12

1

1

0

0

0

V2

0

1

0

0

0

V23

0

1

1

0

0

V3

0

0

1

0

1

System Bond Graph Modelling

A procedure described in [12] allows elaborating the BG model of the system in integral causality (Fig. 9) as well as the corresponding model in derivative causality (Fig. 10). Fig. 12. Causal graph of the three tanks system

The supports of each residual deduced from the DBG that will be used in the exoneration algorithm are given below: C1:{Sf1, V1, V12} C2:{V12, V2, V23} C3:{Sf2, V23, V3} C4:{V1} C5:{V3}

Fig. 9. BG model in preferred integral causality

VI.

Experimental Results

VI.1. Case of Normal Operating Mode In Fig. 13, different residuals in normal operating mode are presented. We notice that residuals have low values around zero; the variations represented in Fig. 13 are due to parameters uncertainties. In ideal case, these values should be equal to zero. From Fig. 13, we deduced different numeric values of the boundaries of the trapezoidal memberships functions in the fuzzy detection module (r-max, Rmin, Rmax, d-max, Dmin and Dmax) for each residual.

Fig. 10. BG model in preferred differential causality

The coupling between the two precedent models produces the DBG of Fig. 11. From Fig. 11, the following FSM can be determined (Table III), and the causal graph of Fig. 12 is determined.



606


Residuals variation

-5

1

r1

0 -5

0 -5 x 10

1000

0 -5 x 10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000 5

0 5000

6000

7000

8000

9000

10000

r3

5

0 -5 x 10

1000

2000

3000

4000

0 -5 x 10

1000

2000

3000

4000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000 1 0 -1

1 0 -1

0 -1 1

5000

6000

7000

8000

9000

10000

5000

6000

7000

8000

9000

10000

0 -1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 -5 x 10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 -5 x 10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 -5 x 10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 -5

r4

1

r4

4000

0 -5

r5

3000

r5

r3

5

2000

0 -4 x 10

0 -5

-1

Residuals evolution

-4

x 10

5

0 -5

r2

x 10

r2

r1

5

Time(s)

Time (s)

Fig. 15. Residuals evolution in case of V12 fault

Fig. 13. Residuals evolution in normal operating mode

Different isolation methods were applied with the determined boundaries. For isolation, we generated fault indexes showing the state of the monitored components. If the fault index of a variable is equal to 1, then the corresponding component is faulty, otherwise, the concerned component is in normal operating mode. Isolation results are given in Fig. 14; we notice that all fault indexes of different methods are equals to zero indicating that there is no faulty component.

Signature matrix 1

defSF1 0 -1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1

defSF2 0 -1 1

defR1 0 -1 1

defR2 0 -1 1

defR3 0 -1 1

defR12 0.5

0 1

defR23 0 -1

Time(s)

Fault indexes 1

defSF1 0 -1 1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Fig. 16. Fault indexes with fault signature matrix

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

We remark then that the fault’s origin is perfectly identified. In fact, fault index DefR12 passes to 1 at instant 5000s. The results of isolation by exoneration algorithm are shown in Fig. 17.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

defSF2 0 -1 1

defR1 0 -1 1

defR2 0 -1 1

defR3 0 -1 1

defR12 0 -1 1

defR23 0 -1

Exoneration algorithm 1

Time(s)

defSF1 0.5 0 0 1

Fig. 14. Fault indexes with fault signature, exoneration and causal graph

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

defSF2 0 -1 0 1

defR1 0 -1 1

VI.2. Case of Faulty Operating Mode

defR2 0.5

In case of faulty operating mode, some residuals deviate from their normal values and the isolation methods are then used to identify the faulty component.

defR3 0 -1 1

defR12 0.5

defR23 0 -1

0

Time (s)

VI.2.1. First Case: Fault Affecting Valve V12

Fig. 17. Fault indexes with exoneration algorithm

We totally close valve V12 at instant 5000s; Fig. 15 illustrates the evolution of the different residuals. We notice, at the occurrence of the fault, that residuals sensors r1 and r2 make deviations; all the other residuals do not show any abnormal behaviour. Once the fault is detected by the fuzzy logic mechanism, isolation methods will be used to identify the faulty component. In Fig. 16, fault indexes generated by the FSM method are presented.

In this case, the fault indexes of components SF1, V2 and V12 passed to 1 at time 5000s. In fact, residuals r1 and r2 are faulty, so their supports are added to LS However, other LS = {Sf 1, V 1, V 12 , V 2 , V 23} . residuals are normal, so their supports are deleted from LS and added to LN. Then, we obtain finally 3 candidates components: {SF1, V2, V12}.



607


The same results of isolation are obtained by causal graph localization method (Fig. 18).

Signature matrix 1

defSF1 0 -1 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

defSF2 0

Causal graph

-1 0 1

1

defR1 0

defSF1 0.5 0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

-1 0 1

10000

defSF2 0

defR2 0

-1 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

-1

10000

1

defR1 0 -1

defR3 0.5 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1

defR12 0

defR2 0.5 0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

-1 0 1

10000

defR3 0

defR23 0

-1

-1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1

Time (s)

defR12 0.5 0 0 1

1000

2000

3000

4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

5000

6000

7000

8000

9000

10000

defR23 0 -1

Fig. 20. Fault indexes with fault signature matrix 0

Time (s)

Exoneration algorithm 1

defSF1 0

Fig. 18. Fault indexes from causal graph

-1

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1

defSF2 0.5

From the residuals detectors r1 and r2, the backward covering of the causal graph of Fig. 12 gives the initial suspect set composed of {SF1, V1, V2, V12, V23}. As the residual detectors r3 and r4 do not show any abnormal behavior, the forward covering eliminates default on component V1 and V23. Finally, we obtain the set of faulty component {SF1, V2, V12}.

defR1 0 -1 0 1

defR2 0 -1 1

defR3 0.5

defR12 0 -1 0 1

defR23 0 -1

0

Time (s)

VI.2.2. Second Case: Fault Affection Valve V3 We suppose that valve V3 is closed at instant 5000s, Fig. 19 presents the evolution of different residuals in this case.

Fig. 21. Fault indexes with exoneration algorithm Causal graph 1

defSF1 0 -1

r1

5

r2

1

r3

5

r4

2

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

5

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0 1

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

-1 0 1

defR2 0 -1 0 -4 x 10

1000

0 -5 x 10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1

defR3 0.5

2000

3000

4000

5000

6000

7000

8000

9000

defR12 0

10000

-1 0 1

0 -2

r5

0 -5 x 10

0 -5

2000

defR1 0

0 -1

1000

defSF2 0.5

0 -5

0

1

Residuals evolution

-5

x 10

defR23 0 -1 0 -4 x 10

1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

Time (s)

0 -5

2000

3000

4000

5000 Time (s)

6000

7000

8000

9000

10000

Fig. 22. Fault indexes from causal graph

From the causal graph, the backward/forward propagation algorithm localizes also the fault in components SF2 and V3.

Fig. 19. Residuals evolution in case of V3 fault

We notice that residuals r3 and r5 are affected by this fault and make a distinctive variation from their normal values. In Fig. 20, fault indexes obtained by signature matrix method are shown. This method localizes perfectly the faulty component (V3). Localization indexes given by exoneration procedure are given in Fig. 21. The used algorithm identify two candidates to that fault: SF2 and V3. Causal graph based isolation method result is given by Fig. 22.

VII.

Conclusion

In this paper, fuzzy logic approach and causal properties of the BG model are exploited for FDI. The residuals generated from the DBG are processed in the fuzzy detection module. Firstly, each residual is fuzzified. Then, an inference mechanism is used. Finally, the defuzzification of the outputs gives a fault index indicating whether the concerned residual is faulty or not.



608


Causal properties of the BG model allow using 3 isolation methods: FSM, exoneration algorithm and backward/forward covering of the causal graph. The different isotion methods have shown diverse performances. Our principle reason to judge those performances is the false alarm criterion. The FSM has proved to be the most performant technique to localize fault in case of single fault hypothesis. The 2 others methods give a conflict set composed by more than one element in most cases.

[17]

[18]

[19]

[20]

References [1] [2]

[3]

[4]

[5]

[6] [7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

A.K. Samantaray, B. Ould Bouamama, Model-based Process Supervision: A Bond Graph Approach (Springer, 2008). A.K. Samantaray, K. Medjaher, B. Ould Bouamama, M. Staroswiecki and G. Dauphin-Tanguy, Diagnostic bond graphs for online fault detection and isolation, Simulation Modelling Practice and Theory, Volume 14, Issue 3, April 2006, pp 237-262. A. Evsukoff, S. Gentil, J. Montmain, Fuzzy reasoning in cooperative supervision systems, Control Engineering Practice, Volume 8, Issue 4, 2000, pp. 389-407. Cordier, M.O., Dague, P., Dumas, M., Levy, F., Montmain, J., Staroswiecki, M., and Trave-Massuyes, L. Conflicts versus Analytical Redundancy Relations: a comparative analysis of the Model-based Diagnosis approach from the Artificial Intelligence and Automatic Control perspectives, IEEE Trans. on Systems, Man, and Cybernetics (Part B), 2004, pp. 2163-2177. G. Biswas, X. Koutsoukos, A. Bregon and B. Pulido, Analytic Redundancy, Possible Conflicts, and TCG-based Fault Signature Diagnosis applied to Nonlinear Dynamic Systems , Proceedings of the IFAC-Safeprocess, Barcelona, Spain, 2009. G. Dauphin-Tanguy, Les bond graphs (Hermès, Paris 2000). G. Dauphin-Tanguy, M. Tagina, La méthodologie bond graph: Principes et applications (Centre de publications universitaires, Tunis 2003). J. Montmain, S. Gentil, Causal Modeling for Supervision, Proceedings of the IEEE International Symposium In Intelligent Control/Intelligent Systems and Semiotics, 1999. J. Ben Slimane Dhifallah, K. Laabidi, M. Ksouri Lahmari, Support Vector Machines for Failures Diagnosis, International Review of Automatic Control (IREACO), vol 2. n. 5, September 2009, pp 505-511. M.A. Djeziri, B. Ould Bouamama and R. Merzouki, Modelling and robust FDI of steam generator using uncertain bond graph model, Journal of process control 19, 2009, pp 149–162. M. H. Moulahi, F. Ben Hmida, M. Gossa, Robust Fault Detection for Stochastic Linear Systems in Presence the Unknown Disturbance: Using Adaptive Thresholds, International Review of Automatic Control (IREACO), vol. 3. n. 1, January 2010, pp 1123. M. Tagina, L'application de la modélisation bond graph à la surveillance des systèmes complexes, Ph.D. University of Lille1, France 1995. S. Bouslama Bouabdallah, M. Tagina, A fuzzy approach for fault detection and isolation of uncertain parameter systems and comparison to binary logic, IFAC, IEEE 3rd International Conference on Informatics in Control, Automation and Robotics, ICINCO 2006. Portugal, pp 98-1066. S. Bouslama-Bouabdallah, M. Tagina, A Fault Detection and Isolation Fuzzy System Optimized by Genetic Algorithms and Simulated Annealing, International Review on Modelling and Simulations (IREMOS), Vol. 3. n. 2, pp. 212-218. V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, K. Yin, A review of process fault detection and diagnosis Part I: Quantitative model-based methods, Computers and Chemical Engineering 27, 2003, pp 293-311. V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, K. Yin, A review of process fault detection and diagnosis Part II: Qualitative

[21]

[22]

[23]

models and search strategies, Computers and Chemical Engineering, 2003, pp 313-326. V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, K. Yin, A review of process fault detection and diagnosis Part III: Process history based methods, Computers and Chemical Engineering, 2003, pp 327-346. W. Bouallègue, S. Bouslama Bouabdallah, M. Tagina, Diagnosis of Bond Graph modeled uncertain parameters systems using residuals sensitivity, IEEE International conference on Systems, Man, and Cybernetics, Turkey, 2010, pp 593-600. I. Fagarasan, S. Ploix, S. Gentil, Causal Fault Detection and Isolation Based on a Set-Membership Approach, Automatica, vol 12, 2004, pp 2099-2110. P. M. Frank, B. Köppen-Seliger, Fuzzy logic and neural network applications to fault diagnosis, International Journal of Approximate Reasoning, Volume 16, Issue 1, January 1997, pp 67-88. S. Gentil, J. Montmain, C. Combastel, Combining FDI and AI approaches within causal model based diagnosis, IEEE Transactions on Systems, Man, and Cybernetics, vol 34, 2004, pp 2207-2221. E. Khadri, M. Tagina, Robust Diagnosis of the Hybrid Systems by Proportional Integral Observer, International Review of Automatic Control (IREACO), vol. 2. n. 6, November 2009, pp 628-637. A. Braham, H. Keskes, Z. Lachiri, Multiclass Support Vector Machines for Diagnosis of Broken Rotor Bar Faults Using Advanced Spectral Descriptors, International Review of Electrical Engineering (IREE), vol. 5. n.5, 2010, pp 2095-2105.

Authors’ information W. Bouallegue obtained the Engineering Diploma from National Engineering School of Tunis, Tunisia, in 2007, received the M.S. degree in Automatic and Signal Processing in 2008. He is currently studying for a Ph.D. degree in Electrical Engineering. His research interests include bond graph modelling, system monitoring and Fault Detection and Isolation. S. Bouslama_Bouabdallah was born in Tunis, Tunisia, on July 25, 1973. She obtained the Engineering Diploma from National Engineering School of Tunis, Tunisia, in 1997; she obtained her PhD thesis in 2008. Since September 2009 she has been teaching in Nabeul Preparatory Engineering Institute (Tunisia). Her research interests include bond graph modelling, system monitoring, fault diagnosis and Artificial intelligence. M. Tagina was born in Sfax, Tunisia, on July 6, 1968, and went to the Ecole Centrale de Lille (France), where he studied Engineering and obtained his degrees and Master of Automatic in 1992 and PhD thesis in 1995. He teached one year at Ecole Centrale de Lille moving in 1996 to University of Science of Monastir. He obtained his "University Habilitation" from National Engineering School of Tunis in 2002. Since 2003 he has been Professor at the National School of Computer Sciences (Tunisia). His research interests include bond graph modelling, Systems monitoring, robotics and Artificial intelligence.



609


Fault Detection and Isolation for Nonlinear Dynamic Power System A. Thabet1, M. Boutayeb2, G. Didier3, S. Chniba4, M. N. Abdelkrim5

Abstract – In this paper, the Fault Detection and Isolation (FDI) problem for nonlinear dynamic power systems based on observers is treated. The nonlinear dynamic model based on differential algebraic equation (DAE) is transformed to ordinary differential equation (ODE). By combining a relevant version of the Extended Kalman Filter with moving horizon (E.K.F-MH) and the version with Unknown Inputs (U.I.E.K.F), we propose a robust FDI. Simulation of IEEE 3 buses test system demonstrates the robustness and effectiveness of the proposed FDI for nonlinear dynamic power systems. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: FDI, Dynamic Power System, U.I.E.K.F, E.K.F-MH, Convergence Analysis

Fault detection and isolation (FDI) is an active area of research due to increasing complexity of process plants and growing demand for safety and reliability. Many fault detection and isolation techniques have been proposed in literature. These techniques can, in general, be classified as hardware redundancy, logic-based or information flow graphs, data-driven approaches, model-based approaches, knowledge-based systems and analytical redundancy techniques. An impressive number of works has been devoted to fault detection for nonlinear systems [2], [3]. The robustness of FDI Observer-based approaches is an important research topic, in which the unknown input observer (UIO) scheme is very famous [4]. The basic idea is to design a fault diagnosis observer decoupled from the unknown disturbances. For nonlinear systems the extended Kalman filter (E.K.F) is very famous and has been widely used as an estimator [5]. The convergence of the E.K.F used as an observer for nonlinear deterministic discrete-time systems was discussed by [6] and [7] which is based on selection of Qk and Rk and the convergence analysis are also discussed. In this paper, the basic idea of the adopted approach is to combine a new proposed version of E.K.F with moving horizon (E.K.F-MH) to fault detection with a group of Unknown Inputs E.K.F (U.I.E.K.F) to fault isolation after the development of a nonlinear dynamic model of power system. The local convergence of the E.K.F-MH and E.K.FMH for deterministic systems is studied based on the choice of covariance matrices of the system noises and measurements with inserting some numerical approximations for the calculation of Jacobean matrix. In the last section, numerical simulations of IEEE 3 buses test system show the relevance and efficiency of the proposed FDI.

Nomenclature DAE ODE LMI M D δ ω ωs PM Pj, Qj Pc,d Ybus Gij + jBij N ng nl PGi θi ,Vi

Differential-Algebraic Equation Ordinary Differential Equation Linear Matrix Inequality Inertia constant of the generator Damping constant of the generator Mechanical rotor angle of the rotating machine Mechanical angular velocity Electrical angular velocity Mechanical power input Nodal active and reactive power Transit power Nodal admittance matrix Real and imaginary terms of bus admittance matrix corresponding to ith row and jth column Total number of system buses Number of generator buses Number of load buses Electrical power supplied by the generator Phase and voltage at bus i

I.

Introduction

Safety has become a major concern in power systems in order to avoid accidents which have sometimes tragic consequences. In fact an accident can be costly in human lives, harmful to the environment and also to the economy. It is therefore important to know the operating conditions and the design for which a power system can safely operate. Process development and continuous request for productivity lead to an increasing complexity of industrial units. In power systems, it is absolutely necessary to control the process and any drift or anomaly must be detected as soon as possible in order to prevent risks and accidents [1]. Manuscript received and revised August 2011, accepted September 2011

610


A. Thabet, M. Boutayeb, G. Didier, S. Chniba, M. N. Abdelkrim

II.

same value. The mechanical rotor angle is the same as the electrical phase angle of the voltage therefore δ now refers to the electrical angle. To further simplify the notation, the transient reactance is incorporated into the system Ybus, resulting in θi as the generator terminal phase and Vi as the terminal voltage. If we take node 1 as reference, the set of equation of this network is given by [8]:

Dynamic Power System Model

-

The dynamics of a power system can be modeled with a combination of nonlinear differential equations and nonlinear algebraic equations [15]. These sets of equations are often solved separately in different analysis techniques. The solution is accomplished in an iterative way, with the important feature that all the desired system characteristics are included. The general form of the DAE model is given as: ⎧x d ( t ) = Fd ( x d ( t ) , x a ( t ) ,u ( t ) ) ⎪⎪ ⎨0 = g ( x d ( t ) , x a ( t ) ) ⎪ ⎪⎩y ( t ) = h ( x d ( t ) , x a ( t ) )

⎧ f I i : δi = ωi − ωs ⎪ ⎞ ⎪ II ωs ⎛ PMi + ⎜ ⎟ ⎪ f i : ωi = 2M ⎜⎝ − PGi δ i ,θ j ,V j − Dωi ⎟⎠ ⎪⎪ ⎨ I ⎪ g j : Pj − Pj δ i ,θ j ,V j = 0 ⎪ II ⎪ g j : Q j − Q j δ i ,θ j ,V j = 0 ⎪ ⎪⎩ yq = Pc,d δ i ,θ j ,V j

(

(1)

(

with: x d ( t ) ∈ \ nd and x a ( t ) ∈ \ na are respectively dynamic and algebraic states, Fd ( t ) ∈ \ the

nonlinear

differential

nd

(

)

(

(2)

)

)

a function representing equations,

with: i=1...ng-1; j=(ng+1)...(ng+nl); q=1...m; c,d=1...N and ⎡Gij cos δ i − θ j + ⎤ N ⎥. PGi = ∑ Vi V j ⎢ ⎢ + B sin δ − θ ⎥ j =1 ij i j ⎣ ⎦ Therefore, the model (2) can be rewritten under this form: ⎪⎧ F ( x , x ,β ) = u ⎨ ⎪⎩y = h ( x ,β )

g ( ⋅) ∈ \ na

(

represents the nonlinear algebraic constraints (equations), u ( t ) ∈ \ p the control and y ( t ) ∈ \ m the

)

(

output system. The problem with the system (1) is that x a ( t ) does not appear explicitly. II.1.

)

Problem Formulation

To put out, in details, the physical dynamic power model, we will treat the case of the IEEE 3 buses test system given in Fig. 1 (with ng=2 and nl=1).

)

with:

x = ⎡⎣δ i

T

ωi θ j V j ⎤⎦ ,

F ( ⋅) = ⎡⎣ fi ,g j ⎤⎦

T

u=

PMi , M

β = {Ybus } ,

and y = Pc,d where u et y will be

respectively the control and the output of the system. Thus for this network, the state vector and the system equations are given by (3) and (4): x = [ x1

x2

x3

x4 ] = T

= [δ 3 ω3 θ 2 V2 ]

T

Fig. 1. IEEE 3 buses test system

-

-

-

⎧ f I : x1 = x2 − ωs ⎪ ⎞ ωs ⎛ PM 3 + ⎪ II ⎜ ⎟ ⎪ f : x2 = ⎜ 2 M ⎝ − PG 3 ( x1 ,x3 ,x4 ) − Dx2 ⎟⎠ ⎪ ⎨ I ⎪ g : P2 − P2 ( x1 ,x3 ,x4 ) = 0 ⎪ II ⎪ g : Q2 − Q2 ( x1 ,x3 ,x4 ) = 0 ⎪ y = P ( x ,x ,x ) 3 ,2 1 3 4 ⎩

In this study, some assumptions are made [8]: The internal field currents are constant, providing the representation of the machine as a constant voltage behind the direct axis transient reactance. The mechanical power provided by the prime mover is constant and all dynamics of the prime mover are neglected. All generators are rotating at synchronous speed (steady state) and are round rotors. All generators in the system are identical, and therefore the inertia constant (Mi) along with the damping constant (Di) of each generator have the

(3)

(4)

with x1 and x2 are the dynamic variables, x3 and x4 are the algebraic variables, where:



611


⎧ ⎡ ⎛ G2 ,1 cos ( x3 ) + ⎞ ⎪ P2 ( x1 ,x3 ,x4 ) = x4 ⎢U1imp ⎜ ⎟+ ⎜ + B sin ( x ) ⎟ ⎪ ⎢⎣ 3 ⎠ ⎝ 2 ,1 ⎪ ⎪ ⎛ G2 ,3 cos ( x3 − x1 ) + ⎞ ⎤ ⎪+ ( x4G2 ,2 ) + U 3imp ⎜ ⎟⎥ ⎜ + B sin ( x − x ) ⎟ ⎥ ⎪ 3 1 ⎠⎦ ⎝ 2 ,3 ⎪ ⎡ imp ⎛ G2 ,1 sin ( x3 ) + ⎞ ⎪ ⎟+ ⎪⎪Q2 ( x1 ,x3 ,x4 ) = − x4 ⎢U1 ⎜⎜ ⎟ ⎢⎣ ⎨ ⎝ − B2 ,1 cos ( x3 ) ⎠ ⎪ ⎛ G2 ,3 sin ( x3 − x1 ) + ⎞ ⎤ ⎪ imp ⎟⎥ ⎪− ( x4 B2 ,2 ) + U 3 ⎜⎜ ⎟ ⎪ ⎝ − B2 ,3 cos ( x3 − x1 ) ⎠ ⎦⎥ ⎪ imp 2 ⎪ P3,2 ( x1 ,x3 ,x4 ) = U 3 G3,2 + ⎪ ⎪ imp ⎛ G3,2 cos ( x1 − x3 ) + ⎞ ⎟ ⎪−U 3 x4 ⎜⎜ ⎟ ⎪⎩ ⎝ + B3,2 sin ( x1 − x3 ) ⎠

II.2.

with J is the Jacobean matrix used in the Load Flow calculation excepted for generators terms, which allows us to verify that this det ( g ( x d , x a ) ) ≠ 0 and g is

solvable for any x a (the elements of this matrix are the components of the diagonal Jacobean matrix used in load flow). Finally, the complete model in form ODE is according to: ⎧ ⎛ x d ⎞ ⎪x = ⎜ ⎟ = f ( x d , x a ,u ) = ⎝ x a ⎠ ⎪ ⎪ ⎛ ⎞ Fd ( x d , x a ,u ) ⎪= ⎜ ⎟ ⎨ ⎜ −1 ⎟ ⎪ ⎝ −g xa ( x d , x a ) g xd ( x d , x a ) Fd ( x d , x a ,u ) ⎠ ⎪ ⎪y = ⎛ 0 ⎞ = h ( x , x ) = ⎛ g ( xd ,xa ) ⎞ ⎜⎜ ⎟⎟ ⎜ ⎟ d a ⎪ ⎝y⎠ ⎝ h ( xd , xa ) ⎠ ⎩

In the expression of h ( x d , x a ) the purpose of adding

Semi-explicit DAE of Index 1

the algebraic constraint g ( x d , x a ) is to check it. It

If at an equilibrium point, the system (1) is called semi-explicit [9], index-1 property requires that

(

should be noted that the assumptions and the propositions given can be generalized for the other forms of dynamic power system models.

)

g ( x d , x a ) is solvable for x a and det g xa ( x d , x a ) ≠ 0

(to simplify x d ( t ) = x d , x a ( t ) = x a ):

III. Extended Kalman Filter

⎧⎪0 = g xd ( x d , x a ) x d + g xa ( x d , x a ) x a ⎨ ⎪⎩0 = g xd ( x d , x a ) Fd ( x d , x a ,u ) + g xa ( x d , x a ) x a

where: g xa ( x d , x a ) =

and: g xd ( x d , x a ) =

The main problem in fault diagnosis of dynamic power system is that few methods are applicable. Effectively, the numerous and strong nonlinearities in presence lead generally to the use of Extended Kalman Filter to resolve the problem of diagnosis. The advantages of the EKF are its simplicity, the fact that it is a recursive algorithm and so it’s modest computational load. The EKF is suitable for real-time industrial-scale applications with the development of the Digital Signal Processor devices. We propose here a relevant FDI scheme based on the E.K.F-MH to fault detection (the choice is based on a comparative study given by [8] and shows that the E.K.F-MH generates the best residual signal that E.K.F in term of bounded variation and false alarm) and a group of U.I.E.K.F to fault Isolation (based on the application given by [4] for three-tank system). A study of local convergence will be presented.

(5)

∂g ( x d , x a ) ∂x a ∂g ( x d , x a ) ∂x d

In other words, the differentiation index is 1, if, by differentiation of the algebraic equations with respect to time, an implicit ODE system results:

⎧⎪x d = Fd ( x d , x a ,u ) ⎨ −1 ⎪⎩x a = −g xa ( x d , x a ) g xd ( x d , x a ) Fd ( x d , x a ,u )

(6)

III.1. where

g −xa1

( xd , xa ) ∈ \

na ×na

and g xd ( x d , x a ) ∈ \

na ×nd

.

∂g ( x d , x a ) ∂x a

⎛ g xa 1 =⎜ ⎜ gx 3 ⎝ a

g xa 2 ⎞ ⎟ [J] g xa 4 ⎟⎠

Extended Kalman Filter with Moving Horizon

We propose here the use of an E.K.F which takes into account a moving horizon of measurements [8], based on the filter with delay [11], to improve the precision as well as the robustness of estimation. We present in this section the synthesis of the estimator. We consider the following system (we used Euler method with step size Te, x k +1 = x k + Te f ( x k ,u k ) to

A study of nature and stability of DAE system is given by [10]. It should be noted that: g xa ( x d , x a ) =

(8)

(7)

discretize the continuous model (1) :



612


⎪⎧x k +1 = f ( x k ,u k ) + v k ⎨ ⎪⎩ y k = h ( x k ,u k ) + w k

f ( x k ) − f ( ˆx k ) = Fk x k h ( x k ) − h ( ˆx k ) = H k x k

(9) where:

where v k and w k are the process and observation noises which are both assumed to be zero mean multivariate Gaussian noises with covariance Q k and R k respectively. The proposed observer is given by:

ˆx k +1

⎛ ⎞ y k − h ( ˆx k ) ⎜ ⎟ y k −1 − h ( ˆx k −1 ) ⎟ = f ( ˆx k ) + K k ⎜ ⎜ ⎟ # ⎜⎜ ⎟⎟ ⎝ y k − M +1 − h ( ˆx k − M +1 ) ⎠

⎧ ∂ ( x k + Te f ( x k ,u k ) ) ⎪Fk = F ( ˆx k ,u k ) = ∂xk ⎪ x k = ˆx k ⎪ ⎪⎪ ⎛ ∂g ( x k ) ⎞ (10d) ⎨ ⎜ ⎟ ⎪ ∂h ( x k ,u k ) ⎜ ∂x k ⎟ =⎜ ⎪H k = H ( ˆx k ,u k ) = ∂x k ∂h ( x k ) ⎟ ⎪ ⎜ ⎟ ⎜ ∂x ⎟ ⎪ k ⎠ xk = ˆxk ⎝ ⎪⎩

(10a)

There are some attempts to apply Kalman Filter on linearized DEA system [12], but our proposition is to apply it in the classic nonlinear general form with some numerical approximations that we propose for the Jacobean matrix calculation. Initially, it should be noted that due to the difficulty of finding Fk (following the transformation of the algebraic variables in ODE model), we will make the following numerical approximation (11):

with M is a size of moving horizon. In what follows we calculate the various parameters of the filter. We have:

(

Pkk = E x k x Tk

)

(10b)

x k +1 = x k +1 − ˆx k +1 We consider the following approximations:

(

))

(

⎧ ∂ x dk + Te Fd x d k , x ak ,u k ⎪ ⎪ ∂ x d k , x ak ⎪ Fk = F ( ˆx k ,u k ) = ⎨ ⎪ ∂ x ak + Te −g −xa1 x d k , x ak g xd x dk , x ak Fd x d k , x ak ,u k ⎪ ⎪ ∂ x d k , x ak ⎩

(

(

(10c)

( ) ) ( ( )

(

) (

)))

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(11)

The numerical approximation is used on the second term of Fk since it is very difficult to determine, it is calculated as follows:

(

) ( ) ( ))) ≈ ∂ ( xd ,xa ) ⎛ ∂Fd ( x d , x a ,u k ) ⎞ ⎞ ⎟⎟ + Te ⎜ −g −x 1 ( x d , x a ) g x ( x d , x a ) ⎜ ∂ ( xd , xa ) ⎟ ⎟ ⎝ ⎠⎠

(

(

∂ x ak + Te −g −xa1 x d k , x ak g xd x dk , x ak ⋅ Fd x dk , x ak ,u k k

⎛ ≈ ⎜ I na ⎜ ⎝

k

k

a

k

k

d

k

(12)

k

k

k

k

for x d k = ˆx d k and x ak = ˆx ak . The terms g −xa1 and g xd are calculated numerically. We develop Pkk++11 to obtain: Pkk++11 = Fk Pkk FkT + K k Ck Pk CTk K Tk − Fk ⎡⎣ Pkk Pkk −1 … Pkk − M +1 ⎤⎦ CTk K Tk + ⎛ Pkk ⎞ ⎜ ⎟ ⎜ Pkk−1 ⎟ T T −K k Ck ⎜ ⎟ Fk + K k R k K k + Q k # ⎜ ⎟ ⎜ Pk ⎟ ⎝ k − M +1 ⎠ Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

(13)


613


where Ck = diag ⎡⎣ H k ( ˆx k ) … H k ( ˆx k − M +1 ) ⎤⎦ .

by inserting the E.K.F for the nonlinear case based on [4]. Consider the following nonlinear discrete-time system with unknown disturbance:

In the expression (13), a total estimation error covariance matrix Pk intervenes. This matrix is calculated as follows:

Pk +1

⎛ Pkk++11 ⎜ ⎜ P k +1 =⎜ k ⎜ # ⎜ P k +1 ⎝ k −M +2

Pkk+1 … Pkk+−1M + 2 ⎞ ⎟ ⎟ . . . ⎟ . . % ⎟ . " Pkk−−MM++22 ⎟⎠

⎧⎪x k +1 = f ( x k ,u k ) + E ( x k ) d k + w k ⎨ ⎪⎩y k +1 = h ( x k +1 ,u k +1 ) + v k +1

(14)

where: f, h et E are assumed to be smooth and known and dk ∈ \q . Then like the E.K.F, we extend the U.I.K.F to system (20) as follows:

with in each iteration we must calculate the first component of Pk with the relation (13). The other elements are calculated by the following expression:

(

Pkk+−1i = E x k +1x Tk −i

Pkk+−1i = Fk Pkk −i

)

(15)

∂K k

(16)

)

−1

(18)

(24)

L k +1 = K k +1 + ηk +1Π k +1

(25)

K k +1 = Pk +1\k HTk +1Vk−+11

(26)

ˆ ηk +1 = ( I − K k +1H k +1 ) E k

(27)

Π k +1 =

(

ˆ = ⎡ H k +1E k ⎢⎣

The fact of using a moving horizon to the measures introduces a matrix Pk . The calculation of K k , then, takes into account preceding measures which differs from classical E.K.F. The initialization of the E.K.F-MH is given by the E.K.F in its classical formulation: .F E.K .F ⎤ P0 = diag ⎡⎣ PME.K .F PME.K −1 … P0 ⎦

)

T

(

−1

) ( H k +1Eˆ k )

ˆ ⎤ Vk−+11 H k +1E k ⎥ ⎦

T

Vk−+11

(28)

Vk +1 = H k +1Pk +1\k HTk +1 + R k +1

(29)

∂f |ˆx ,u ∂x k\k k

(30)

∂h |ˆx ,u ∂x k +1\k k +1

(31)

where: Fk =

(19)

where PkE.K .E is an estimation error covariance matrix of E.K.F. In the same way, in the EKE the scalar residual is generated, with the possibility of choosing: = yk − M +1 −

(23)

with:

thus, we obtain Kk which satisfies (17):

rkE.K .F − MH

(22)

Pk +1\k +1 = ( I − K k +1H k +1 ) Pk +1\k +

(17)

(

Pk +1\k = Fk Pk\k FkT + Q k

+ ηk +1Π k +1Vk +1ΠTk +1ηTk +1

) =0

K k = Fk ⎡⎣ Pkk Pkk −1 … Pkk − M +1 ⎤⎦ CTk Ck Pk CTk + R k

(21)

+ L k +1 ( y k +1 − h ( ˆx k +1\k ,u k +1 ) )

We calculate then Kk in order to minimize the trace of error covariance matrix ( Pkk++11 ):

(

ˆx k +1\k = f ( ˆx k\k ,u k )

ˆx k +1\k +1 = f ( ˆx k +1\k ,u k ) +

⎛ Pkk −i ⎞ ⎜ ⎟ ⎜ Pkk−−1i ⎟ − K k Ck ⎜ ⎟ ⎜ # ⎟ ⎜ P k −i ⎟ ⎝ k − M +1 ⎠

∂trace Pkk++11

(20)

H k +1 =

ˆ = E ( ˆx ) E k k\k

yke − M +1

(32)

Compared to the linear case, the U.I.E.K.F algorithm is mainly different from the U.I.K.F in the calculation of the matrices Fk and H k +1 (as that in E.K.F-MH) and

III.2. Unknown Input Extended Kalman Filter In what follows, we present an extension of U.I.K.F Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


614


rank ( O ( k − A + 1,k ) ) =

ˆ . E ( x k ) is substituted by its estimation E k

H k − A+1 ⎡ ⎤ ⎢H ⎥ F = ⎢ k − A+ 2 k − A+1 ⎥ = ( nd + na ) ⎢ ⎥ .... ⎢ ⎥ H F F ... ⎣ k k −1 k − A+1 ⎦

III.3. Convergence Analysis

In this section, we present an extension of the principal theorem for convergence analysis of E.K.E and E.K.F-MH based on the method of [13], [6] and [14]. This method includes an unknown diagonal matrix to model linearization errors and a Lyapunov function which leads us to the resolution of a LMI which depends on the choice on R k and Q k . We present some guiding steps for the case of E.K.FMH (Briefly, many steps and demonstrations are omitted). Initially, the error vector x k and a candidate Lyapunov function Vk +1 are defined by:

in practice, we use a numerical rank test on O ( k − A + 1,k ) . ii. Fk,, Hk are uniformly bounded matrices and Fk−1 exist. iii. The matrices R k and Q k are chosen such as: a. For E.K.F-MH: Q k = σ e f Tk e f k I nd + na + χ I nd + na

x k = x k − ˆx k

(

T Vk +1 = x k +1 Pkk++11

)

−1

(35)

R k = Ck Pk CTk + ρ I M

x k +1

(36)

where: and: ⎧x k +1 = β k Fk x k ⎪α e = H x k +1 k +1 ⎪⎪ k +1 k +1 ⎨β = diag β ,..., β 1k ( nd + na ) k ⎪ k ⎪ ⎪⎩α k +1 = diag α1k +1 ,...,α mk +1

(

(

)

efk

)

and: σ and have to be chosen small and positive and χ and ρ a positive scalar. b. For U.I.E.K.F:

We have then:

(

Vk +1 = α k F k x k T = x k F kT α k

) ( P ) (α F x ) = ( F P F + Q ) α F x k +1 −1 k +1

T

−1

T k k k

A decreasing sequence

k k k

k

{Vk }k =1,...

Q k = γ eTk ek I nd + na + λ I nd + na

(33)

R k = ς H k +1Pk +1/ k HTk +1 + τ I m

k k k

Vk +1 − (1 − ξ ) Vk ≤ 0

therefore, which gives us this LMI:

(

− (1 − ξ )

( )

−1 Pkk

)

−1

α k F k +

(37)

where γ have to be chosen sufficiently large and positive, λ positive scalar which is small enough, ς and τ positive scalars fixed by the user, while verifying that the following expression must be negative: γ k′T+1N k +1γ k′ +1 < 0

means that there

exists a positive scalar 0 < ξ < 1 so that:

F kT α k F k Pkk FkT + Q k

⎛ ⎞ y k − h ( ˆx k ) ⎜ ⎟ ˆ h − y x ( k −1 ) ⎟ k −1 =⎜ ⎜ ⎟ # ⎜⎜ ⎟⎟ ⎝ y k − M +1 − h ( ˆx k − M +1 ) ⎠

where: ⎧⎪N k +1 = Vk−+11 ⎡ Π k +1 − I n + n ⎤ d a ⎦ ⎣ ⎨ ⎪⎩γ k′ +1 = α k +1H k +1Fk xk\k

(34)

≤0

With the same reasoning used in [13], [8] and to ensure local convergence, we must verify these conditions : i. The system (9) is A-locally uniformly rank observable, there exists k ≥ A − 1 where the observability matrix:

and rank ( H k +1Ek ) =rank ( Ek ) =q .

IV.

Fault Detection and Isolation

We present in this section a Robust FDI strategy of nonlinear dynamic power systems using a combination



615


rank ( H k +1Tk ) =rank ( Tk ) =q and formed to locate faults

of E.K.F-MH and a group of U.I.E.K.F. We proceed with the same logic used by [4], [16]. In fact, both unknown disturbances and faults can be described by E ( x k ) d k , where d k is the magnitude of

in a specific node and not a parameter. The isolation task can be performed using this simple test: • If ei > e j then fault occurs in node i.

unknown disturbances or parameters of fault, and E ( ⋅)

• If ei < e j then fault occurs in node j.

represents the distribution matrix of unknown disturbances. We assume that all unknown disturbances in the system are E ( x k ) d k and the fault is described by T ( x k ) φk , where

T ( x k ) = ⎡⎣T

1

( xk )"T ( xk )⎤⎦ ∈ \ s

• If ei = e r

then fault occurs between the nodes i

and r (where r represent the nodes connected to the reference node i).

n× s

and φk = ⎡⎣φk1 "φks ⎤⎦ . To detect fault, we need to design an E.K.F-MH. The

V.

Simulation Results

Simulations studies are carried out on the IEEE 3 buses test system to evaluate the performance of the proposed FDI scheme. For the discretization of the dynamic model (8), we used Euler method with a step size Te equal to 10-3s. The measurement values are generated by generated by the Toolbox SimPowerSystems of MATLAB® and by adding low variance noise (±5% of real value) to the calculated measurements (transit power P3,2) . Fig. 2 shows the evolution of the rank of the observability matrix (numerical calculation with A=4) to verify the observability.

residual signal used for detection is e f k = rkE.K .F − MH . Fault will be detected by comparing e f k with 0. For fault isolation, a group of U.I.E.K.F is needed to design structured residuals. The most commonly used scheme in designing the residual set is to make each residual sensitive to one fault. Design N U.I.E.K.F (called U.I.E.K.F{i}, with estimated state ˆxik\k ,i = 1,… N ) which is sensitive to the i th fault, respectively, besides disturbances. Then U.I.E.K.F{i} is designed as: ⎧ ⎡d k ⎤ i ⎪x k +1 = f ( x k ,u k ) + ⎡⎣ E ( x k ) T ( x k ) ⎤⎦ ⎢ i ⎥ + w k (38) ⎨ ⎣⎢ φk ⎦⎥ ⎪ ⎩y k +1 = h ( x k +1 ,u k +1 ) + v k +1

6

rank of Observability matrix

5

The distribution matrices can be treated as constant matrices, where φk is the magnitude of the default. Since each node in power system is represented by two variables, consequently we generalize to a network of N nodes whose state vector is as follows:

4

3

2

1

x = [δ i ,ωi ,θi ,Vi ] . Then to locate faults at node i, T is T

0

0

500

1000

1500

2000

chosen such that:

ωi θi Vi ] T

(

ωi θi Vi ]

(39)

T

5000

k − 4 ,k

)

T1 = [1 1 0 0]

T

used for localization faults at

node 3.

T

T i = [ 0 0 1 1]

4500

number of estimated states, the observability of the system is verified. We focus our attention on faults detection and isolation. There are, therefore, two stages U.I.E.K.F and we define:

ii. For loads buses: The states representing each load node are Vi and θi : x = [δ i

4000

As the value of rank O3( buses ) = 4 which equal to the

T

T i = [1 1 0 0]

3500

Fig. 2. Evolution of the rank of Observability matrix

i. For generators buses: The states representing each generator node are δ i and ωi : x = [δ i

2500 3000 Iterations (k)

T2 = [ 0 0 1 1]

T

used for localization faults at

node 2. We consider two types of faults in this simulation: - Decrease in the power generator: φk1 = − PG3 ( k )

(40)

applied in the expression of x2 ( k ) .

It should be noted that this choice of T i is imposed by the verification of this condition Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


616


Break-Line between nodes 3 and 2: φk2 is formed by admitting that the real and imaginary terms of bus admittance between nodes 3 and 2 are equal to zero ( G3,2 = B3,2 = G2 ,3 = B2 ,3 = 0 in the expression of

-

in generator node 3) gives a clear idea that the node 3 remains in default by verifying the first condition of the proposed FDI scheme eT1 ( k + 1) eT2 ( k + 1) . To validate the proposed FDI scheme, we inject the second defect φk2 between iterations 2900 and 3000. We present, in the same way, the evolution of residuals signals in Fig. 5 and Fig. 6.

x2 ( k ) ,x3 ( k ) and x4 ( k ) ).

For simplicity, we don't consider any other unknown disturbance here ( d k = 0 and we replace Ek in equations (27), (28) and (32) by T i ). Then we can design an E.K.F-MH for fault detection and two U.I.E.K.Fs (U.I.E.K.F1 consider T1 and U.I.E.K.F2 with T2) for fault isolation, each of which is sensitive to faults in the corresponding node respectively. Fault isolation logic is according to the proposed scheme. Now, we consider the first defect φk1 between iterations 1450 and 2600. We present in Fig. 3 and Fig. 4 the evolution of the residuals signals of these three observers.

300

250

residu

200

150

100

50

35

0

0

500

1000

1500

2000


3500

4000

4500

5000

30

Fig. 5. Evolution of residual signal with E.K.F-MH 25 300 T1

residu

20

T2

250 300

10

200 200 norm(e)

15

5

0

0

500

1000

1500

2000


3500

4000

4500

150 100

0 2885

100

5000

2890

2895

2900

2905

2910

50

Fig. 3. Evolution of residual signal with E.K.F-MH 0

600

0

500

1000

1500

2000

T2


3500

4000

4500

5000

T1 500

Fig. 6. Evolution of the residual signal with U.I.E.K.F 500

400

Fig. 5 shows that the defect is detected by the variation of the residue generated by E.K.F-MH. Result given in Fig. 6 shows, clearly, that

norm(e)

400 300

300

eT1 ( k + 1) = eT2 ( k + 1)

200 200 100 100

0

0 1500

0

500

1000

1500

2000

1550

1600


1650

3500

1700

4000

1750

verifies

the

third

condition given in the proposed logic test. We are interested in the convergence of two versions of the extended Kalman filter (proposed E.K.F-MH and U.I.E.K.F). The measurement values are generated by adding high variance noise to the measurements ±15% of real value). We consider: • The classical values of Q k and R k given by (Standard versions: S-E.K.F-MH / S-U.I.E.K.F):

1800

4500

which

5000

Fig. 4. Evolution of the residual signal with U.I.E.K.F

From Fig. 3 it is clear that after the fault occurs, the residual of E.K.F-MH increases immediately ( e f ( k ) ), then the fault is detected. On the other hand, the residuals of U.I.E.K.Fs (1 and 2) will depart from zero as well as the E.K.F-MH does, which are illustrated in Fig. 4 indicating that the residuals signal generated by the first U.I.E.K.F (with T1 and used to indicate a fault is located

QUk .I .E .K .F = Q kE.K .F − MH = 10−5 I 4 RUk .I .E .K .E = 10−3 R kE.K .F − MH = 10−3 I 4



617


• Modified proposed values given by (36) and (37) (Modified versions based on the proposed conditions: M-E.K.F-MH / M-U.I.E.K.F):

References [1]

⎧QUk .I .E.K .F = 100eTk ek I 4 + 0.5I 4 ⎪ U .I .E.K .F = 0.01H k Pk HTk + 0.05 ⎪⎪R k ⎨ E.K .F − MH = 0.001e f Tk e f k I 4 + 0.005I 4 ⎪Q k ⎪ E.K .F − MH = 0.001Ck Pk CTk + 0.005I 4 ⎪⎩R k

[2]

[3]

[4]

We consider 100 simulations while varying the initial values in a random way (variation of ±20% with respect to the actual initial values) and we present in Table I the rate of convergence.

[5]

TABLE I RATE OF CONVERGENCE (%)

[7]

Observers S-E.K.F-MH S-U.I.E.K.F M-E.K.F-MH M-U.I.E.K.F

[6]

Rate of Convergence 51 % 49 % 96 % 94 %

[8]

[9]

In the general case, the studied algorithms converge to the good values only when they are initialized near to their actual values (the voltages are selected close to the values of the generators voltages and the phases equal to 0). The values obtain in Table I show that the appropriate choice of matrices Q k and R k given by (36) and (37) insure the convergence and increases the estimation quality.

VI.

[10]

[11] [12]

[13]

[14]

Conclusion

In this paper, an observer based approach for robust fault detection and isolation was presented. A new filter design based on fundamental problem of residual generation concepts was elaborated for nonlinear dynamic power system. A relevant E.K.F-MH has been described and investigated based on moving horizon to generate a perfect residual signal to fault detection and a group of U.I.E.K.F is used for geographical isolation of the network. We also used some numerical approximations for the calculation of Jacobean matrix which was preceded by a convergence analysis. Numerical results demonstrated the robustness of the proposed FDI scheme with the occurrence of the majority of real types of defect. The remaining open questions are the experimental test of the proposed method and its application to large scale power test systems (IEEE 118 bus test system for example). These two issues will be investigated in the near future.

[15]

[16]

Chetouani, Y, “Using the kalman filtering for the fault detection and isolation (FDI) in the nonlinear dynamic processes” Int .J. of Chem. Reactor Eng., vol 6, 2008, pp.1-20. D.F. Leite, M.B. Hell and P.J.F. Gomide, “Real-time fault diagnosis of nonlinear systems” Int. Multidisciplinary J. on Nonlinear Analysis, vol 71, 2009, pp-2665–2673. Zhang, X., T. Parisini and M. Polycarpou, “Sensor bias fault isolation in a class of nonlinear systems” IEEE Trans. On Autom. Control, vol 50, 2005, pp370–376. Lingali ,L.,Z. Donghua, W. Youqing and S. Dehui, “Unknow input ex-tended kalman filter and applications in nonlinear fault diagnosis” Chin. J. Chem. Eng., vol 13, 2005, pp783–790. Chen, J. and R.J. Patton , Robust Model-Based Faults Diagnosis for dynamic systems, Dordrecht: Kulwer Academic Press, 2edition, 1999) Boutayeb, M., “Identification of nonlinear systems in the presence of unknown but bounded disturbances” IEEE Trans. on Autom. Control, vol 45, 2000, pp.1503–1507. Guo, L. and Q. Zhu, “A fast convergent extended kalman observer for non linear discrete-time systems” Int. J. of Syst. Sci., vol 33, 2002, pp.1051–1058. Thabet, A. , M. Boutayeb, G. Didier, S. Chniba and M.N. Abdelkrim “Fault diagnosis for dynamic power system” 8th International Multi-Conference on Syst., Signals and Devices, Conference on Systems, Analysis and Automatic Control, Sousse, TUNISIA, 2011, pp.1–7. Gordon, B. W. “Dynamic sliding manifolds for realization of high index differential-algebraic systems” Asian J. of Control, vol 5, 2003, pp.454–466. Tarraf, D. C. and H. H. Asada, “On the nature and stability of differential-algebraic systems”, Proc. American Control Conference, Anchorage, Al USA, 2002, pp.3546–3551. Boutayeb, M., “Observers design for linear time-delay systems” Syst. and control Lett., vol 44, 2001, pp.103–109. Becerra, V. M., P. D. Roberts and G. W. Griffiths, “Applying the extended kalman filter to systems described by nonlinear differential-algebraic equations” Control Eng. Practice, vol 9, 2001, pp.267–281. Boutayeb, M. and C. Aubry, “A strong tracking extended kalman observer for nonlinear discrete-time systems” IEEE Trans. on Autom. Control, vol 44, 1999, pp.1550–1556. Y. Song and J. Grizzle, “The extended kalman filter as a local asymptotic observer for nonlinear discrete-time systems” J. of Mathematical Syst. Estimation and Control, vol 5, 1995, pp.59– 78. Rahmat, S. Jovanovic, K.L. Lo, Reliability and Availability Modelling of Uninterruptible Power Supply Systems Using Monte-Carlo Simulation, International Review of Electrical Engineering (IREE), vol. 1 n. 3, August 2006, pp. 374 – 380. H. Mokhlis, H. Mohamad, A. H. A Bakar and H. Y. Li, “Evaluation of Fault Location based on Voltage Sags Profiles: a Study on the Influence of Voltage Sags Patterns”, International Review of Electrical Engineering (IREE), vol. 6 n.2, April 2011, pp. 874-880.

Authors’ information 1

Unit Modeling, Analysis and Control of Systems (MACS), University of Gabes –TUNISIA. Tel.: +21696960475 E-mail: [email protected] 2 Research center of automatic of Nancy (CRAN), CRAN–LONGWY -IUT Henri Poincaré, 186 Street of Lorraine -54400 Cosnes Et Romain, France. E-mail: [email protected] 3 Group of Research in Electrical engineering and Electronics of Nancy (GREEN),



618


Faculty of Science and Technology, B.P. 239-54506 Vandoeuvre lès Nancy, France. E-mail: [email protected] 4 Energetic and environment Unit (ENENV), University of Gabes –Tunisia. E-mail: [email protected] 5 Unit Modeling, Analysis and Control of Systems (MACS), University of Gabes –Tunisia. E-mail: naceur.abdelkrim@ enig.rnu.tn)

Thabet Assem, Gabès, 24/09/1981. Master of Automatic and Intelligent technologies, National School of Engineer of Gabès, Gabès, TUNISIA, 2008. Diploma of engineer in electric-automatic, National School of Engineer of Gabès, Gabès, TUNISIA, 2006. Diploma of Superior technician in industrial electricity, High Institute of Technological Studies of Gabès , Gabès, TUNISIA, 2003. He is an electric engineer at company GTSP (2007-2008) and now assistant with the higher institute of industrial systems of Gabès.



619


Sensor Fault Detection and Localization Methodology Using Principal Component Analysis Mohamed Guerfel1,2, Anissa Ben Aicha3, Kamel Benothman1

Abstract – This work proposes a new methodology for sensor fault detection and localization using principal component analysis (PCA). Several fault detection indices in the literature are analyzed and unified. A proposed index is adopted in order to detect simple and multiple faults affecting the dependent and independent process variables. A new iterative selection method of principal component number is presented. This method determines a model allowing the detection of faults without a priori knowledge of their natures. The fault localization is carried out using hierarchical contribution plots applied to the adopted detection index. The performance of this approach becomes poor for a bad partitioning of the variables into blocs. A new partitioning method is proposed to identify correctly all the faults affecting the process. The whole proposed results were applied to a non linear noisy system subjected to simple and multiple faults. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: PCA, Number of Principal Components, Sensor Fault, Detection, Localization, Contribution Plots

Nomenclature m

Matrix formed by the eigenvectors associated to the variables of the bth bloc Eigenvalues matrix of ∑ qth eigenvalue of Λ

Pb

x (k )

Number of measured process variables Number of retained principal components Process data vector

λq

x0 ( k )

Non noisy process data vector

ˆ Λ

x* ( k )

Non faulty process data vector

Λ

υ (k )

Measurements noise vector

ˆx ( k )

Estimated data vector obtained from the PCA model Vector containing the measured variables of the bth bloc ith component of xb ( k )

ˆ C,C SPE(k) Di(k) SWE(k) Gi(k) T2(k) Ti 2 ( k )

Squared prediction error Partial squared prediction error Squared weighted error Proposed index Hotelling statistics Partial Hotelling statistics

D (k) ϕ (k )

Mahalanobis distance Combined index

d(k)

Proposed unified form for detection indices Unified vector form Computed index Thresholds of the computed indices Chi –square distribution with i liberty degrees and α as a confidence limit Identity matrix ∈ m×m jth column vector of Im

xb ( k ) xb,i ( k ) e (k ) t (k )

Data error vector obtained from the PCA model Process data matrix Principal component vector

tq ( k )

qth component of t ( k )

ˆt ( k )

Vector of significant process variations

t (k )

Noise data vector

∑ P pq

Process correlation matrix Eigenvectors matrix of ∑ qth principal vector

Pˆ

Matrix formed by the first principal vectors Matrix formed by the m − last principal vectors

XN

P

Λ

aQ Q

δ χ i2,α

Im ξj cont j ( k )


620

Diagonal matrix formed by the first eigenvalues Diagonal matrix formed by the m − last eigenvalues PCA model matrices

Contribution of the jth variable to the Gi (k) index


M. Guerfel, A. Ben Aicha, K. Benothman

con*j

The mean of the jth variable contribution

contb ( k )

Contribution of the bth bloc of variables to the Gi (k) index Contribution of the sth variable to the bth bloc Threshold of detection for the cont j ( k )

contb,s ( k )

τ 2j ,α Z

determines the blocs according to the variables contributions in the nominal operating mode. The application of this method permits to identify correctly the simple and multiple faults affecting the dependent and independent variables. The paper is organized as follows. Section 2 is a brief recall of the PCA principle. Section 3 analyzes several detection indices and gives a unified form for all of them. The proposed criterion for the PCA model fixation is presented in the section 4. Section 5 presents the localization via the classical and hierarchical contribution plots in the case of the adopted index. A new method permitting the partitioning of the system variables into blocs is also presented. The last section illustrates the application of the previous sections results on a noisy non linear system affected with simple and multiple faults.

Vector containing the mean values of the variables contribution

I.

Introduction

With the development of automation and sophisticated plants, fault diagnosis methods become very important in process monitoring [1]-[3]. Those methods can be broadly classified into two categories: non-statistical methods and statistical methods. Principal Component Analysis (PCA) belongs to the second category of methods. PCA was successfully used as a tool for the diagnosis of sensor faults [4]-[8]. PCA transforms the initial variables in limited number called principal components, which are linear combinations of the original variables while preserving the data variance measured on a system. Thus PCA defines a representation subspace containing significant data variations and a residual subspace containing noise measurements and carried by the redundancy relations. The fault detection is achieved by a detection index in one of these subspaces or in the total space [9]. Most of the indices are insensitive to the independent variables faults [10]. In order to mitigate this disadvantage, a new index [11] allowing the detection of faults affecting the dependent and independent variables is adopted in this work. The PCA based fault detection stage depends closely on the retained number of principal components which defines the dimension of the representation subspace. Most of the methods permitting the choice of this number [12] do not take into account the fault effect on the computed PCA model [7]. Inspired from the work of [5], a new iterative method is proposed for the determination of the principal component number to retain in the PCA model. This method uses conjointly nominal operating data to identify the PCA model and faulty data in order to fix its structure. The PCA based fault localization can be carried out using many methods [13]. The contribution plots is a widely used localization method [9], [14]-[17]. However, it can give wrong localizations for the simple and multiple fault case [19], [13]. In order to minimize the wrong localizations, a pertinent approach called hierarchical contribution plots was proposed [19]. This approach consists in dividing the process variables into blocs. The computation of the contributions permits the localization of the faulty bloc(s). Those blocs are analyzed in order to identify the faulty variable(s). The performance of this approach becomes poor for a bad partitioning of the variables into blocs. To avoid this risk, a new method of variable partitioning is proposed. It

II. Let

Principal Component Analysis x ( k ) = ⎣⎡ x1 ( k ) … xm ( k ) ⎦⎤ , T

be

a

vector

containing the m centered and reduced observed inputs and outputs in the instant k. The data matrix XN formed by staking the data vector x(k) over a period of observation equal to N is written: X N = ⎡⎣ x ( k ) … x ( k + N − 1) ⎤⎦

T

(1)

Modeling a process via static PCA consists in seeking an optimal linear transformation (with respect to a variance criterion) of the vector x(k) into a new one called t(k) and defined as follows : t ( k ) = PT x ( k ) = ⎣⎡t1 ( k ) … tm ( k ) ⎦⎤

T

(2)

The variables tq ( k ) , q ∈ {1, . . . , m}, called principal components are uncorrelated and arranged in decreasing variance order. The column vectors pq of matrix P represent the eigenvectors corresponding to eigenvalues λq obtained from the diagonalization of correlation matrix Σ of XN:

∑=

1 X NT X N = PΛPT ; PT P = PPT = I m N −1

the the the the

(3)

The notation Λ = diag (λ1 . . . λm) designates the diagonal matrix of eigenvalues arranged in the decreasing magnitude order λ1 ≥ λ2 ≥ . . . ≥ λm. With the triple partitioning: ˆ 0⎤ ⎡Λ Λ=⎢ ⎥ ; P = ⎡⎣ Pˆ | P ⎤⎦ ; T = ⎡⎣Tˆ | T ⎤⎦ 0 Λ ⎣ ⎦

(4)

from equation (4), the principal components vector t can



621


x ( k ) = x0 ( k ) +ν ( k )

be written in every sample k as follows: t ( k ) = ⎡⎣ˆt ( k ) | t ( k ) ⎤⎦

T

The components of the vector ˆt ( k ) ∈

where x0 ( k ) represents the vector of the process true

(5)

with ˆt ( k ) = Pˆ T x ( k ) and t ( k ) = PT x ( k )

values and ν ( k ) represents the vector of the measurements noise with null mean and which components are identically distributed. From equations (7) and (5) the residual vector t ( k ) is

carry the

significant process variations. The components of the vector t ( k ) ∈ m − represent the quasi null relations

written: =0

between the process variables xq ( k ) .

t (k ) = P x

T 0

For every sample k, the vector x ( k ) can be written: x ( k ) = ˆx ( k ) + e ( k )

(7)

( k ) + P Tν ( k )

(8)

The first term of the vector t ( k ) is null in absence of

(6)

faults. The second term of t ( k ) has a small magnitude

The vectors ˆx ( k ) and e ( k ) represent, respectively,

and the bounds of its norm is small and can be computed under certain statistical hypothesis [22]. Thus, if the first term of t ( k ) is nonzero, the norm of the vector t ( k )

the estimates vector and the error vector from the PCA model. ˆ ˆ t and C = I − Cˆ form the The matrices Cˆ = PP m static PCA model of the process [20]. The first ℓ eigenvectors forming the matrix Pˆ ∈ m× constitute the representation subspace whereas the last

will be over its bounds in nominal operating mode and a fault can be declared. Variation due to a fault occurrence can be highlighted using this vector. The detection indices SPE [22] and SWE [23] are built from the residual subspace. They use the statistical proprieties of the t ( k ) vector. The application of those indices in

ˆ ( k ) and e ( k ) = Cx ( k ) . with ˆx ( k ) = Cx

( ) m-ℓ eigenvectors forming the matrix P ∈ constitute the residual subspace. The identification of the static PCA model thus consists in estimating its parameters by an eigenvalue /eigenvector decomposition of the matrix Σ and determining its structural parameter which is the number of principal components ℓ to retain. Thus, this model divides the data space into two orthogonal subspaces: the principal subspace of representation formed by the ℓ first eigenvectors and the residual subspace formed by the m-ℓ last eigenvectors. An incorrect choice (too large or too small) of ℓ could mask the changes occurring in the modeled process or gives false alarms which affect the change detection procedure [21]. Many methods were proposed to choose ℓ for PCA modeling. The reader can find more details in [12].

different subspaces of the residual subspace permits the generation of the indices Di (for SPE [5]) and Gi (for SWE [24]). From equations (7) and (5), the vector ˆt ( k ) is written:

III. Fault Detection via PCA

may stay in its nominal bounds because of its large range variation. Indices using the statistical properties of ˆt ( k )

m× m −

≠0

ˆt ( k ) = Pˆ T x 0 ( k ) + Pˆ Tν ( k )

(9)

The first term of ˆt ( k ) has significant norm and regroups the variations of all process variables. The bounds of the norm of the first term of ˆt ( k ) can be computed under certain statistical hypothesis [25]. The norm of the second term of ˆt ( k ) is insignificant compared to the norm of the first term. After the occurrence of a small magnitude fault, the norm of ˆt ( k )

The detection stage is based on the analysis of the residuals also called indicators or detection indices. In order to compute the detection indices, it is necessary to build in a first time the matrices of eigenvalues and eigenvectors (Λ and P) of the matrix Σ which is obtained from data in nominal operating mode. The number ℓ of retained principal component is chosen so that the model represents all the redundancy relations of the process. In nominal operating mode and at every sample k, the measurements vector x(k) is written:

may not detect small magnitude faults but they detect jumps in the mean of the process variables. The indices exploiting the statistical proprieties of ˆt ( k ) are: the Hotelling statistics T2 [25] and the statistics Ti 2 [26] which are obtained from the application of T² in different subspaces of the principal subspace. Owing to the complementary nature of the indices using the principal and the residual subspaces [9], indices using the whole space of representation were proposed. They use all the principal components in their computation. Those indices



622


The quantity β designates the number of nonzero

are: the Mahalanobis distance D [27] and the combined index ϕ [28]. All the indices mentioned above can be put, in every sample k, in a unified form:

elements of aQ , α designates the used confidence limit and g and h are given as follows:

m

d ( k ) = ∑ ahQ th2 ( k )

g=

(10)

h =1

The quantities ahQ , Q designates the computed index,

for Q ∈

}

(11)

{SPE,Di ,SWE,Gi ,

⎡ ⎤ ⎢ 0 … 01 … 1⎥ ⎢ ⎥ ⎣ ⎦

for

SPE

⎡ m −i ⎤ ⎢ 0 … 01 … 1⎥ ⎢ ⎥ ⎣ ⎦

for

Di

⎡ ⎤ ⎢ 0 … 0 λ −+11 … λm−1 ⎥ ⎢ ⎥ ⎣ ⎦

(12a) for SWE

⎡ m −i ⎤ ⎢ 0 … 0 λm−1−i +1 … λm−1 ⎥ for ⎢ ⎥ ⎣ ⎦

( )

trace Ψ 2

(14)

for SPE ( k ) for Di ( k ) for

(15)

ϕ (k )

The index chosen in this work for detection and localization purposes is Gi [24]. This index is based on the sum of the squares of the i last principal components weighted by the inverse of their variances. The index Gi is robust regarding the choice of i and permits the detection of weak magnitude faults affecting the process [29]. In the case where i is equal to m, the proposed index corresponds to a Mahalanobis distance D [27]. This distance allows the detection of faults affecting independent variable [10]. Thus, the significant feature of the proposed index resides in the fact that it can be extended to cover different subspaces in the data space. Consequently, this index has the aptitude to detect any type of fault affecting the process. In some cases, it is not necessary to exploit all the data space because the augmentation of the detection threshold can prevent the detection of weak magnitude faults. Thus, an adequate selection of i adapted to the type of fault to detect can solve this compromise.

T 2 ,Ti 2 ,D,ϕ are: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q a =⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

; h=

⎧ Λ ⎪ Ψ = ⎪ diag ( λi … λm ) ⎨ ˆ ˆ −1 ˆ T −1 T ⎪ PΛ P + PΛ P ⎪ δ2 χ 2,α ⎩ SPE

are components of a vector aQ :

The values of aQ

trace ( Ψ )

( traceΨ )2

According to the computed index, the matrix Ψ is written:

The index d(k) can be either SPE(k), Di(k), SWE(k), Gi(k), T²(k), Ti 2 ( k ) , ϕ ( k ) or D ( k ) .

a Q = ⎡⎣ a1Q … amQ ⎤⎦

( )

trace Ψ 2

Gi

⎧ ⎡ λ1−1 … λ −1 0 … 0 ⎤ for T 2 ⎪⎣ ⎦ (12b) a =⎨ −1 −1 ⎡ 0 … 0 ⎤⎦ for Ti 2 ⎪ ⎣ λ1 … λi ⎩ Q

(

)

IV.

( ) ) ( )

−1 ⎧ ⎡ ⎤ 2 −1 … λ χ2 ⎪ ⎢ λ1 χ ⎥ for ϕ ⎪⎪ ⎢ −1 −1 ⎥ Q 2 2 a =⎨ ⎢ (12c) δ SPE … δ SPE ⎥ ⎦ ⎪ ⎣ ⎪ ⎡ λ −1 … λ −1 λ −1 … λ −1 ⎤ for D +1 m ⎦ ⎩⎪ ⎣ 1

(

The number of the retained principal component ℓ has a significant impact on the fault detection stage [26]. Several criteria were proposed for the selection of this number [12], [30]. Valle et al. have demonstrated in [12] that the VNR criterion [31] is the most interesting because it takes into account the redundancies between the process variables. However, this criterion does not take into account the influence of the faults on the choice of ℓ and gives an average number ℓ for all kind of faults affecting the process. To take into account the influence of faults on the choice of ℓ, the authors in [7] propose to determine this number according to the fault directions. However, the generalization of this method to multiple faults is unrealizable due to the great number of cases to take into account. All the methods cited previously use nominal operating data and determine off-line the

A fault is declared if the index d(k) is higher than a threshold δ 2 . This latter is computed in the following manner: ⎧ ⎪⎪ χ β2 ,α 2 δ =⎨ ⎪ 2 ⎪⎩ g χ h,α

for for

SWE ( k ) ,Gi ( k ) ,T 2 ( k ) , Ti 2 ( k ) , D ( k )

Proposed Method for the Fixation of the Model Structure

(13)

SPE ( k ) ,Di ( k ) ,ϕ ( k )



623


number ℓ. Inspired from the work of [5], one proposes an iterative selection method. It uses conjointly nominal operating data to identify the PCA model and faulty data to fix ℓ in an on-line mode. The principle of this method, illustrated on Fig. 1, is explained as follows:

V.

Localization via Contribution Plots

Three approaches can be used for fault localization via PCA. The first is based on the residual structuration [32]. The second uses a bench of models sensitive to a particular subset of faults [33]-[35]. The third approach, treated in this work, is based on the computation of the contribution of different variables to the detection index [9], [16], [19]. The variables with the greatest contributions are suspected to be faulty. This method presents several drawbacks. On one hand, most of the works are based on the definition of the approximate contributions of the variables to the statistics SPE and T2 [13]-[15]. On the other hand, those contributions are sensitive to the variables amplitudes [18]. The variables with the higher values have great probability to be suspected as faulty. To mitigate the disadvantages mentioned above, a pertinent approach called hierarchical contribution plots was proposed [14], [19]. This approach divides the variables into multiple blocs based on the knowledge of the process. The hierarchical contribution plots gives better localization results compared to the classical approach [9]. This section defines the classical and hierarchical contribution plots in the case of the proposed index Gi.

1. Initialization i = 1. 2. Acquire nominal operation data. Compute XN, Σ, Λ and P. 3. Compute Gi(k) from data in failure mode and χ i,2α . 4. If Gi(k) > χ i,2α then go to 6. Else go to 5. 5. If i < m then i = i + 1, go to 3. Else fault non detectable, go to 7. 6. Localization via contribution plots using Gi(k). 7. End algorithm. The proposed method can be applied in the simple and multiple fault case. This method determines the smallest number i which enables the fault detection for the first time without a priori knowledge on the fault(s), their direction(s) and their type(s). Its disadvantage lies in the use of faulty process data to ensure the choice of the PCA model structure. In order to suppress false alarms, the process is considered in failure mode (Gi(k) > χi2,α ), if Gi(k) has

Classical Contribution Plots

V.1.

shown six succeeding values larger than χi2,α . The value

The index Gi i ∈ ; {1, . . ., m} can also be written as follows:

”six” is determined in an empirical way and must be adjusted according to the treated application.

Gi ( k ) =

xT ( k ) Hx ( k ) = H 1/ 2 x ( k )

where H = PΛ −1 PT ∈

m× m

2

(16)

i is the Frobenius

and

norm. Using the definition in [17], the equation (16) can be written: Gi ( k ) =

∑( m

j =1

ξ Tj H 1/ 2 x ( k )

) ∑ cont 2

=

m

j =1

j

(k )

(17)

The classical contribution of the jth variable to the index Gi is [29]:

(

cont j ( k ) = ξ Tj H 1/ 2 x ( k )

)

2

(18)

where ξ j designates a column vector having 1 in the jth position and 0 elsewhere. The jth variable is considered faulty, if: cont j ( k )

τ 2j ,α

>1

(19)

The quantity τ 2j ,α represents the threshold of contj(k)

Fig. 1. Algorithm of the proposed sensor fault detection and isolation method



624


(

computed for a confidence level equal to α. The results of [18] permit the determination of its coefficient rj and its liberty degree bj:

τ 2j ,α ≡ r j χb2j ,α

th

If the b bloc is declared to be in failure, a contribution plots computation of its different variables is necessary in order to determine the origin of the fault. The contribution of the sth variable to the bth bloc is:

(20)

with: rj =

ξ Tj H ∑ H ξ j

; b j =1

ξ Tj H ξ j

)

and ∑b = cov x*b ( k ) .

1/ 2 contb,s ( k ) = H b,s xb,s ( k )

(21)

m

∑

h = m −i +1

=

V.2.

Hierarchical Contribution Plots

The hierarchical contribution plots divides the m variables in n blocs containing mb variables, b ∈{1 , . . . , n}. The Gi(k) index can be written [29]: Gi ( k ) =

∑ b =1

H b1/ 2 xb

contb,s ( k )

(k )

2 τ b,s

(22) T

λh

>1

(29)

(23)

notation

contb ( k ) = H b1/ 2 xb ( k )

2

=

V.3.

The choice of the blocs and their variables has a major impact on the performance of the localization results. Qin [9] subdivides the different variables according to the physical knowledge of the process. In this work, one proposes a new partitioning method of the variables into blocs. Let Z be a vector defined as follows:

2

mb

1/ 2 xb,s ( k ) ∑ H b,s

Proposed Method for the Choice of Blocks

(24)

s =1

Z=

where xb,s(k) and Hb,s represent respectively the sth component of xb(k) and the sth line of Hb. The bth bloc is considered faulty, if: contb ( k )

τ b2

>1

{

2

trace {∑b H b }

}

; bb

∑

where y ( k ) = H 1/ 2 x* ( k ) ∈

(25)

m

(30)

T conm* ⎤⎦

.

The vector x ( k ) designates the process data vector *

in nominal operating conditions. The notation M ( k ) = diag ( y1 ( k ) … ym ( k ) ) designates a matrix containing the vector y(k) in its diagonal and zeros elsewhere. The quantity con*j represents the mean computed on l

with: trace ( ∑b H b )

1 l M (k ) y (k ) = l k =1

= ⎡⎣ con1* …

here τ b2 designates the threshold of contb(k), it is given as follows: τ b2 ≡ rb χb2b ,α (26)

rb =

(28)

Remark The thresholds computed in a theoretical manner can be inadequate in the case of the classical and hierarchical contribution plots. For this reason, they can be adapted by training on nominal data.

i×mb PbT = ⎡⎣ pb,1 … pb,mb ⎤⎦ ∈ design- nates the matrix formed via the juxtaposition of the eigenvectors pb,s s ∈{1 , . . . , mb} associated to the bth bloc. The expression of the contribution of the bth bloc is [29]:

The

2 2 pb,hs xb,s (k )

2 The threshold τ b,s is computed similarly to τ b2 .

mb where xb ( k ) = ⎡⎣ xb,1 ( k ) … xb,mb ( k ) ⎤⎦ ∈ is a th vector containing the variables of the b bloc. The matrix Hb ∈ mb ×mb is given as follows:

H b = Pb ΛPbT

=

where pb,hs designates the hth component of the vector pb,s. The sth variable of the bth bloc is faulty, if:

2

n

2

( trace {∑b H b }) =

{

trace ( ∑b H b )

2

samples of the contribution of the jth variable in nominal operating mode. The partitioning of the m variables into n blocs is determined in a heuristic way according to the magnitude of the vector Z components (30). The variables xj

2

}

(27)



625


associated to the great coefficients con*j are gathered

X N via (1). The diagonalization of the correlation

into a bloc, the variables corresponding to the average coefficients are gathered into another bloc and so on. This partitioning method brings back the average contributions of variables of every bloc to the same order of magnitude in nominal operation. In the presence of faults in the bth bloc, the contributions of all its variables increase with respect to their nominal values. The most significant augmentation is that of the faulty variables. This augmentation can be masked by the variables having the greatest contributions in nominal operation in the case where this partitioning is not respected. Thus, this partitioning method minimizes the probability of erroneous localizations.

matrix of X N permits the identification of its eigenvalues and its eigenvectors.

VI.

VI.1. Simple Fault Case The system (31) is simulated a second time on 500 samples and a fault is added to the dependent variable x6b since the sample 300 till the end of this simulation. This fault is represented by a constant bias of amplitude equal to 6% of the variation domain of x6b . The application of the proposed method for PCA model selection gives i = 1 (the fault is detected via G1). The evolution of this index is illustrated on the Fig. 3. In order to identify the faulty variable, one applies the classical contribution plots on the index G1. The left hand side of Fig. 4 shows the classical contribution of variables weighted by the inverse of their thresholds at the sample 400. This figure shows that the variables x1, x4, x6 and x7 are faulty but only x6 is really responsible of the fault. The analysis of the coefficients of the matrix Z (30), computed with l = 20 samples in nominal operation mode, permits the division of the system variables into three blocs. A bloc A containing the variables x1, x2 and x3, a bloc B gathering the variables x4, x5 and x6 and a bloc C formed by the variables x7 and x8. The right hand side of Fig. 4 illustrates the contribution of these three blocs weighted by the inverse of their thresholds at the sample 400. Only the contribution of the bloc B exceeds 1 which indicates that this bloc is responsible of the fault.

Application

In order to test the efficiency of the new methodology for the detection and localization of simple and multiple faults affecting the dependent and independent variables, one considers the system described by the following equations: ⎧ z b ( k ) = sin ( k / 4 ) + υ ( k )2 + 1 ⎪ 1 ⎪ z2b ( k ) = cos ( k / 4 )3 exp ( − k / N ) ⎪ ⎪ x1b ( k ) = z1b ( k ) + ε1 ( k ) ⎪ b b ⎪ x2 ( k ) = z2 ( k ) + ε 2 ( k ) ⎪ b 3 b ⎪ x3 ( k ) = z2 ( k ) + ε 3 ( k ) ⎨ b b b ⎪ x4 ( k ) = z1 ( k ) + z2 ( k ) + ε 4 ( k ) ⎪ b b b ⎪ x5 ( k ) = z2 ( k ) − z1 ( k ) + ε 5 ( k ) ⎪ b b b ⎪ x6 ( k ) = 2 z1 ( k ) + z2 ( k ) + ε 6 ( k ) ⎪ b 3 b b ⎪ x7 ( k ) = z1 ( k ) + z2 ( k ) + ε 7 ( k ) ⎪ b ⎩ x8 ( k ) = υ1 ( k ) + ε 8 ( k )

(31)

where xbj ( k ) ; j ∈ {1, . . . , 8} designate the system measurements (inputs or outputs). z1b ( k ) and z2b ( k ) designate the real system inputs. The quantities υ ( k ) and

υ1 ( k ) designate random variables following a reduced centered normal law. The variables ε j ( k ) represent the measurements noise. They are obtained via realizations of random variables following centered normal law with steady deviation equal to 0.095. The system (31) presents linear and non linear redundancy relations as well as an independent variable ( x8b ( k ) ). The system is simulated a first time for N = 500 observations. The evolution of its measurements is illustrated on the Figure 2. After the centering and the reduction of the measurements, they are used to build the data matrix

Fig. 2. Evolution of the variables xbj ( k )



626


The Fig. 5 presents the values of the normalized contribution of the different variables of the bloc B at the sample 400. This figure reveals that only x6 is responsible of the fault. Thus, the hierarchical contribution plots permits to identify correctly the faulty variable contrary to the classical contribution plots.

VI.2. Multiple Fault Case A third simulation is carried out on 500 samples and three bias type faults affecting the dependent variables x1b , x4b (of amplitude equal to 10% of their variation domain) and the independent variable x8b (of amplitude equal to 150% of its variation domain) are simultaneously introduced from the sample 300 till the end. The application of the proposed method for PCA model selection gives i = 1 (the fault is detected via G1). The evolution of this index is illustrated on the Fig. 6. The left hand side of Fig. 7 illustrates the value of the normalized classical contributions obtained at the sample 400. This figure shows that the variables x4, x5 and x6 are faulty. The bloc partitioning used in this case for the hierarchical contribution is the same as the one used in the second simulation. The values of the normalized bloc contributions at the sample 400 are shown on the right hand side of the Fig. 7. This figure shows that the blocs A, B and C contain faulty variables. The Fig. 8 illustrates the normalized contributions of each bloc variables. From this figure, one can conclude that the faulty variables are x4, x6 and x8.

Fig. 5. Contribution of the variables of the bloc B at the sample 400 (second simulation case)

Fig. 6. Evolution of G1 in the third simulation case

Fig. 3. Evolution of G1 in the second simulation case

Fig. 7. Classical and hierarchical contribution plots at the sample 400 (third simulation case)

The performance of the hierarchical contribution plots can be degraded for a bad partitioning of the system variables into blocs. One considers the case where the system variables are splitted into three blocs as follows: a bloc A1 containing the variables x1, x5 and x7, a bloc B1 formed with x2, x3 and x8 and a bloc C1 gathering x4 and x6. The value of the normalized contribution of these blocs, in the third simulation case, at the sample 400 is

Fig. 4. Classical and hierarchical contribution plots at the sample 400 (second simulation case)



627


shown on the Fig. 9. This figure shows that the blocs A1 and C1 are not responsible of the fault whereas A1 contains the faulty variable x1 and C1 contains the faulty variable x4.

[2]

[3]

[4]

[5]

[6]

[7]

Fig. 8. Contribution of the variables of the blocs A, B and C (third simulation case)

[8]

[9] [10]

[11] Fig. 9. Contribution of the variables of the blocs A1, B1 and C1 (third simulation case) [12]

VII.

Conclusion

Indices using the residual subspace are the best in the detection of weak magnitude sensor faults. All the detection indices can be put in a unified form. A new iterative method is proposed to determine the structure of the PCA model. Unlike existing methods, this one determines the principal components number in an online mode. This modeling method allows the detection of the faults without a priori knowledge on their natures. The localization method, adopted in this work, uses the contribution plots applied to the proposed index. The hierarchical contribution plots are more efficient than the classical one in fault localization. However, the performance of the hierarchical contribution plots depends closely on the partitioning of the blocs and its variables. A partitioning, computed from the suggested method, permits the correct localization of simple and multiple faults affecting the dependent and independent variables.

[13]

[14]

[15]

[16]

[17] [18]

[19]

[20]

References [1]

[21]

Y. Menasria, N. Debbache, A Robust Actuator Fault Detection and Isolation Approach for Nonlinear Dynamic Systems,


International Review of Automatic Control (IREACO), vol. 1. n. 2, July 2008. Kang, Ning, Liao, Yuan, New Fault Location Technique for Series Compensated Transmission Lines, International Review of Automatic Control (IREACO), vol. 4. n. 6, 2009. M. Yazdani-Asrami, E. Samadaei, S. Darvishi, M. Taghipour, A Robust Actuator Fault Detection and Isolation Approach for Nonlinear Dynamic Systems, International Review of Automatic Control (IREACO), vol. 5. n. 4, 2010. L. Nabli, K. Ouni, The indirect supervision of a system of production by the Principal Components Analysis and the Average Dynamics of the Metrics, International Review of Automatic Control (IREACO), vol. 1. n. 4, November 2008. M. F. Harkat, G. Mourot, J. Ragot, An improved PCA scheme for sensor. FDI: application to an air quality monitoring network, Journal of Process Control, vol. 16, 2006, pp. 625-634. Y. Tharrault, G. Mourot, J. Ragot, D. Maquin, Fault detection and isolation with robust principal component analysis. International Journal of Applied Mathematics and Computer Science AMCS, vol. 18, 2008, pp. 429-442. M. Tamura, S. Tsujita, A study on the number of principal components and sensitivity of fault detection using PCA, Journal of Computers and chemical Engineering, vol. 31, 2007, pp. 10351046. M. Guerfel, A. Ben Aicha, K. Ben Othman, M. Benrejeb, An improved principal component analysis scheme for sensor fault detection and isolation : application to a three tanks system, 17th Mediterranean Conference on Control and Automation ~MED’09~ June 24-26, 2009, Makedonia Palace, Thessaloniki, Greece. S. J. Qin, Statistical process monitoring: basis and beyond, Journal of Chemometrics, vol. 17, 2003, pp. 480-502. Y. Tharrault, Diagnostic de fonctionnement par analyse en composantes principales: Application à une station de traitement des eaux usées, Ph.D. dissertation, National Polytechnic Institute of Lorraine, Nancy, France, 2008. M. Guerfel, S. Ghachem, K. Ben Othman, M. Benrejeb, A new principal component analysis methodology for sensor fault detection and isolation, International Review on Modeling and Simulations (IREMOS), vol. 4. n.5, 2011. S. Valle, L. Weihua, S. J. Qin, Selection of the number of principal components: The variance of the reconstruction error criterion with a comparison to other methods, Industrial and Engineering Chemistry Research, vol. 38, 1999, pp. 4389-4401. M. F Harkat, G. Mourot, J. Ragot. Différentes méthodes de localisation de défauts basées sur les dernières composantes Conférence Internationale Francophone principales. d’Automatique CIFA, 2002, Nantes, France. S. J. Qin, S. Valle, M. Piovoso. On unifying multiblock analysis with applications to decentralized process monitoring. Journal of Chemometrics, vol. 15, 2001, pp. 715-742. T. Kourti, J. F. MacGregor, Recent Developments in Multivariate SPC Methods for Monitoring and Diagnosing Process and Product Performance, Journal of Quality Technology, vol. 6, 1996, pp. 409–428. J. A. Westerhuis, S. P. Gurden, A. K. Smilde, Generalized contribution plots in multivariate statistical process monitoring. Chemometrics and Intelligent Laboratory Systems, vol. 51, 2000, pp. 95–114. C. F. Alcala, S. J. Qin, Reconstruction based contribution for process monitoring, Automatica, vol. 45, 2009, pp. 1593-1600. H. Yue, S. J. Qin. Reconstruction based fault identification using a combined index. Industrial and Engineering Chemistry Research, vol. 40, 2001, pp. 4403-4414. J.F. MacGregor, C. Jaeckle, C. Kiparissides, M. Koutoudi. Process monitoring and diagnosis by multiblock PLS methods. AIChE Journal, vol. 40, 1994, pp. 826-828. I. T. Jolliffe. Principal component analysis. Springer-Verlag, New York, 2002. M. Kano, K. Nagao, S. Hasebe, I. Hashimoto, H. Ohno, Strauss R, Comparison of multivariate statistical process control monitoring methods with applications to the Eastman challenge problem.


628


[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32] [33]

[34]

[35]

Computers and Chemical Engineering, vol. 26, 2002, pp. 161– 174. G. E. P. Box, Some theorems on quadratique forms applied in the study of analysis of variance problems: Effect of inequality of variance in one-way classification, The Annals of Mathematical Statistics, vol. 25, 1954, pp. 290-302. P. Oxby, S. L. Shah. A critique of the use of PCA for fault detection and diagnosis, Technical report, 1998, Matrikon, Edmonton, Alberta, Canada. M. Guerfel, G. Mourot, J. Ragot, M. Benrejeb, K. Benothman, Comparaison des indices de détection de changements des modes de fonctionnement par ACP. Cas des indices basés sur l’estimation d’état. 1st International Workshop on Systems Engineering Design & Applications ~ SENDA’08~ October 24-26, 2008, Monastir, Tunisia. H. Hotelling, Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, vol. 24, 1933, pp. 417-441. M. Kano, H. Ohno, S. Hasebe, I. Hashimoto. New multivariate statistical process monitoring method using principal component analysis. Computers and Chemical Engineering, vol. 25, 2001, pp. 1103-1113. K.V. Mardia. Mahalanobis distances and angles. Multivariate Analysis-IV, Krishnaiah PR (ed.). North-Holland : Amsterdam, pp. 495-511. 1977. H. H. Yue, S. J. Qin. Reconstruction-based fault identification using a combined index. Industrial & Engineering Chemistry Research, vol. 40, pp. 4403-4414, 2001. A. Benaicha, M. Guerfel, N. Bouguila, K. Benothman, New PCAbased methodology for sensor fault detection and localization, 8th International Conference of Modeling and Simulation ~ MOSIM’10 ~ May 10-12, 2010, Hammamet, Tunisia. S. J. Qin, R. Dunia, Determining the number of principal components for best reconstruction, Journal of Process Control, vol. 10, 2000, pp. 245-250. R. Dunia, S. J. Qin, Joint diagnosis of process and sensor faults using principal component analysis, Control Engineering Practice, vol. 6, 1998, pp. 457-469. J. Gertler, J. Cao, PCA based fault diagnosis in the presence of control and dynamics, AIChE Journal, vol. 50, 2004, pp. 388-402. Y. Huang, J. Gertler, T. McAvoy, Sensor and actuator fault isolation by structured partial PCA with nonlinear extensions, Journal of Process Control, vol. 10, 2000, pp. 444-459. C. L. Stork, D. J. Veltkamp, B. R. Kowalski, Identification of multiple sensor disturbances during process monitoring. Analytical Chemistry, vol. 69, 1997, pp. 5031-5036. R. Dunia, S. J. Qin, Subspace approach to multidimentional identification and reconstruction, AIChE Journal, vol. 44, 1998, pp. 1813–1831.

Mohamed Guerfel was born in Tunisia in May 1979. He received the engineering degree in electro mechanics from the National Engineering School of Sfax in Tunisia in 2003 then he obtained the Msc degree in automatics and industrial maintenance from the National Engineering School of Monastir in Tunisia in 2005; he is currently a student in the PhD program in Electrical Engineering of the National Engineering School of Tunis in Tunisia. He is holding the position of assistant professor at the High Institute of applied mathematics and informatics of Kairouan in Tunisia. His interests include fault diagnosis, multi statistical process control, linear and non linear modeling and interval arithmetic. Mr. Guerfel is a member of the IEEE group for interval arithmetic standardization. Anissa Ben Aicha was born in Tunisia in July 1983. She received the engineering degree in electric from the National Engineering School of Monastir in Tunisia in 2007 then he obtained the Msc degree in automatics and industrial maintenance from the National Engineering School of Monastir in Tunisia in 2008; she is currently a student in the PhD program in Electrical Engineering of the National Engineering School of Monastir in Tunisia. Her interests include fault diagnosis, multi statistical process control, and interval arithmetic. Kamel Benothman was born in Tunisia in July 1958. He received the license in mechanical and energetic engineering in 1980 from the University of Valencienne in France. He obtained the engineering diploma in mechanics and energetic, the Msc degree in automatics and signal processing in 1981 and the PhD in automatics and signal processing in 1984 of the same University. He received the DSc degree from the National Engineering School of Tunis in Tunisia in 2008. He is currently a professor at the High Institute of sciences and energy technologies of Gafsa in Tunisia. His interests include reliability, fault diagnosis, multi statistical process control, interval arithmetic and fuzzy systems. Mr Benothman is a member of the Association of the Electrician Specialists in Tunisia ASET.


Research unit LARA–Automatique, National engineering school of Tunis, BP 37, Le Belvédère, 1002 Tunis, Tunisia. 2

Institut Supérieur des Mathématiques Appliquées et d’Informatique de Kairouan, Avenue Assad Ibn Fourat, Kairouan, Tunisia. 3 Research unit ATSI, National engineering school of Monastir, Avenue Ibn Aljazzar, 5000 Monastir, Tunisia.



629


An Optimal Fuzzy Sliding Mode Control Based on Direct Thrust Force Control for Permanent Magnet Linear Synchronous Motors M. Abroshan1, J. Milimonfared2, M. B. Menhaj2, S. Tarafdar2

Abstract – This paper presents a fuzzy sliding mode position control based on direct thrust force control for permanent magnet linear synchronous motors and field weakening strategy for utilizing maximum capacity of these motors. A Particle Swarm Optimization (PSO) based optimum control scheme is fully derived. The fuzzy inference mechanism is used to reduction of chattering phenomena in sliding mode control. The proposed scheme incorporates maximum thrust force per ampere and field weakening strategies to keep operate the drive within the voltage and current limits of the motor/inverter. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Direct Thrust Force Control, Field Weakening Strategy, Fuzzy Logic Controller, Maximum Force Per Ampere, Permanent Magnet Linear Synchronous Motor, Sliding Mode Control

tools, semi-conductor manufacture equipment, robotics, factory automation, and numerical control systems. The direct drive of mechanical application based on permanent magnet linear synchronous motor benefits from simple structure, less loss, less friction, faster response and high precision resulting into a higher reliability compared to rotary motors [1]. Since PMLSM is driven directly in mechanical system then its servo mechanism is affected by the uncertainties such as force ripples, parameter variations, load disturbances and cogging force. Therefore, for compensation of these equivalent forces and to achieve an accurate tracking in high performance position control systems, a sophisticated control strategy is often required [2]. There are some papers have recently published on the PMLSM servo drive [3]-[8]. With no exception, in all of them, implementation of force control was established using currents. However, the direct thrust force control (DTFC) strategy has a very fast response to flux and force changes and it is robust against motor parameters’ variations and perturbations as well, which is a suitable candidate for the substitution of field oriented control strategy [3]. The basic principle of DTFC ِis to directly select stator voltage vectors according to the differences between the references of force and stator flux linkage and their actual values [4]. Current vector controllers followed by Pulse Width Modulation PWM) or hysteresis comparators and coordinate transformations to and from one of the reference frames are not used in DTFC systems. Consequently, the mover position sensor is no longer mandatory [4]. Variable structure systems (VSS) and Sliding-mode controllers (SMC) because of good control performance

Nomenclature

d ref

Friction factor Mover position Position command

Fcogging

Cogging force

FL

Load force Electrical thrust force d- and q-axis stator currents

B dm

FT id ,iq

Ld ,Lq

Maximum stator current d- and q-axis stator inductances

M P RS ud ,uq

Mover mass Number of pole pairs Phase winding resistance d- and q-axis stator voltages

U am vm

δ τ ϕd ,ϕq

Maximum stator voltage Motor base velocity Mover velocity Load angle Pole pitch Permanent magnet flux linkage

ψf

Permanent magnet flux linkage

ψS

Stator flux linkage

I am

vbase

I.

Introduction

Permanent magnet linear synchronous motors (PMLSM) are used extensively in precision machine


630


M. Abroshan, J. Milimonfared, M. B. Menhaj, S. Tarafdar

for nonlinear systems and robustness to the parameter variations are an effective controller for servo control drives and robotics [9]. There have many studies investigating for reducing the chattering phenomena in sliding mode controllers, Yager and Filev [10] determined the fuzzy rules according to the sliding mode condition. F.J. Lin and S.L. Chiu [9] combined the best features of fuzzy control and sliding mode control to achieve rapid and accurate tracking control of a vector controlled ac drive. Using Maximum Thrust force per Ampere (MTPA) strategy below the base velocity leads to copper loss minimization and increase the maximum thrust force of motor instead of id = 0 strategy, and by using field weakening strategy, maximum capacity of motor is utilized in constant power region [3]. In linear motors because of the relatively low speeds, the core losses are mostly small compared to the copper losses, and therefore its effect is negligible [2]. In this study, combination of fuzzy control and sliding-mode control for DTFC drive system is proposed, which combines the merits of the sliding-mode control, the fuzzy inference mechanism and the DTFC system. Sliding mode controller is used as robust state feedback control system and fuzzy inference mechanism is used for reducing chattering phenomena in the sliding-mode controller. The MTPA strategy is utilized for achieving greater maximum force and the FW strategies one used for obtain response to position command. Finally all the control parameters are optimized by Particle Swarm Optimization (PSO) algorithm.

using finite element method (FEM) the cogging force is extracted and is shown in Fig. 2. The thrust force of the PM linear synchronous motor consists of three components: main synchronous force, reluctance force and cogging force. The cogging force is a function of position and is independent of the load angle. Due to the slotted nature of the primary core, the cogging force is periodic and repeats itself over every slot pitch [12]. By using Fourier series the cogging force can described as: Fcogging ( d m ) = ⎛πd ⎞ ⎛ πd = 2.13 + 0.29 cos ⎜ m ⎟ − 3.61cos ⎜ 2 m ⎝ τ ⎠ ⎝ τ ⎛ πd ⎞ ⎛ πd ⎞ − 1.69 cos ⎜ 3 m ⎟ + 2.24 cos ⎜ 4 m ⎟ + τ ⎝ ⎠ ⎝ τ ⎠ ⎛πd ⎞ ⎛ πd ⎞ + 0.55 sin ⎜ m ⎟ + 6.48 sin ⎜ 2 m ⎟ + ⎝ τ ⎠ ⎝ τ ⎠ ⎛ πd ⎞ ⎛ πd ⎞ − 0.49 sin ⎜ 3 m ⎟ + 1.21 sin ⎜ 4 m ⎟ τ ⎝ ⎠ ⎝ τ ⎠

⎞ ⎟+ ⎠

(4)

Stator

Y

Z Move r

Stator

(a) Stator

Y

II.

Modeling of PMLSM X

The machine model of a PMLSM can be described in rotor reference frame as follows [3]: d ⎡ R + Ld ⎡ ud ⎤ ⎢ s dt ⎢ ⎥=⎢ u π q ⎢ ⎣ ⎦ P vm Ld ⎣⎢ τ FT =

−P

π ⎤ vm Lq ⎥ i ⎡ d ⎤ ⎡⎢ τ +

0 ⎥⎢ ⎥ π ⎢P v ψ d i Rs + Lq ⎥ ⎣ q ⎦ ⎣⎢ τ m f dt ⎦⎥

3 π 1 P ψ S ⎡⎣ 2ψ f Lq sin δ + 2 τ Ld Lq

(

)

⎤ ⎥ (1) ⎥ ⎦⎥

(b) Figs. 1. Machine structure of the PMLSM 10

(2) Cogging force, N

− ψ S Lq − Ld sin 2δ ⎤ ⎦

dv FT + Fcogging ( d m ) = M m + Bvm + FL dt

mover MOVER

(3)

5

0

-5

-10

The motor that used in this study is moving secondary and has surface mount permanent magnet (PM) as shown in Figs. 1 which are based on a real PMLSM [11]. The parameters of the PMLSM are tabulated in Table A.1. By

0

5

10 Axial position, mm

15

20

Fig. 2. Cogging force of the PMLSM



631


U a = ud2 + uq2 ≤ U am

ν 1 > ν base > ν 2 ν 1 ν base ν 2

T hrus t forc e, N

300

v max

Further one current limit which is independent of saturation becomes:

200

I a = id2 + iq2 ≤ I am

(6)

100

0

0

0.005 0.01 0.015 0.02 0.025 Amplitude of stator flux linkage, Wb

The maximum current, I am , is either continuous armature current rating in a continuous operation mode or a maximum available current of the inverter obtained in limited short-time operation. The maximum available force can be produced by the MTPA strategy below the base velocity. In order to achieve the MTPA for a given force demand, the line

0.03

Fig. 3. Control trajectories and motor limitations in the force- ψ s plane

current

III. Direct Thrust Force Control

( ) ≈ 0 ) then the terminal voltage, U

dLq iq dt

a

amplitude,

or

id2 + iq2 ,

equivalently

is

minimized to achieve the maximum force within the current and voltage constraints of (5) and (6). So:

The basic principle of DTFC ِis to directly select stator voltage vectors according to the differences between the references of force and stator flux linkage and their actual values. Switching states of the inverter, the dc-link voltage (sensed) of the inverter and two of the motor currents (sensed) can be used to obtain the stator voltage and current vectors of the motor in the stator DQ frame. Force and stator flux linkage are controlled using two hysteresis comparators which operate independently. The outputs of these controllers generate appropriate voltage vectors via the inverter in a way to force the two variables to track some predefined trajectories [3], [4]. The block diagram of the conventional DTFC for an IPMLSM position servo drive is shown in Fig. 4. By using MTPA/FW and limitation block the position tracking can be done by utilizing one PID position controller. Fig. 5 demonstrated the block diagram of the DTFC based drive position control system with one PID controller. The maximum voltage, U am , is the maximum available output voltage of the inverter depending on the dc-link voltage. If magnetic saturation is neglected (

(5 )

∂FT ∂iq

=0

(7)

Vm

If the magnetic saturation is neglected, the relation between the id and iq currents for unsaturated IPMLSM becomes [13]: id =

(

ψf

2 Lq − Ld

−

)

ψ 2f

(

4 Lq − Ld

)

2

+ iq2

(8)

Using the maximum thrust force per ampere strategy in a direct thrust force control based drive mode of the permanent magnet linear synchronous motor drives leads to achieve minimum copper losses which in turn leads to a better utilization of motor capacity. When the mover velocity is increased above the base velocity, the area within the voltage limit contour, which is allowable operation area, will be decreased. As a result, the stator flux linkage for FW operation must be reduced appropriately according to (6) or (10).

is bounded

as: ∆d

d ref dm

∆v m

v ref

vm

Fref*

∆F

Fref

vm

ψS

ref

∆ ψS

θn

V

n

V

V , Vdc n

V

ψS

Q

V

dc

D

iQ

F

iD

dm

Xm

Fig. 4. Block diagram of the conventional DTFC-based drive for PMLSM



632


disturbances:

In this paper, for the sake of simplification, the voltage limits are considered in the steady state by applying (9) instead of (6):

(ψ q ) + (ψ d ) 2

2

τ U om 2 ≤ Pπ vm

Fref = k1 x1 + k2 x2 + F0

by combination equations (3), (13), (14) and (16), this equation can be represented as:

(9 )

x2 = α Fe − β x2 + α FL

where: U om = U am − I am Rs

(10)

α=

2

(

+ (α + ∆α ) FL + FCogging

2 ⎛ π ⎞ Since the left side of (11) is equal to ⎜ p vm ⎟ ψ s , τ ⎝ ⎠ this equation becomes:

x2 = α Fe − β x2 − E

The stator flux linkage and the rated stator flux linkage, which is corresponded to point A in Fig. 3. The voltage limit trajectory for each velocity is indicated as a vertical line, as defined by (12). When the mover velocity is below the base velocity, the intersection of the MTPA and current limit trajectories is within the voltage limit contour, and therefore the voltage limit is always satisfied with MTPA trajectory control.

IV.

(

−

dt

− ( β + ∆β ) Vref

lim SS < 0

(22)

S →0

(13) by considering eqn, (22) , (20) and (15), the control law becomes: Fref = −ka Sign ( S ⋅ x1 ) ⋅ x1 − kb Sign ( S ⋅ x2 ) ⋅ x2 +

⎧⎪ x2 = d m − d ref ⎨ ⎪⎩ x2 = vm − d ref

− f 0 Sign ( S )

(14)

S = x2 + kC x1 = 0

ka > 0 , kb > B − kC M and f 0 > E

(15)

and

(24)

according to Lyapunov`s theory, the S=0 surface is globally asymptotically stable and the states x1 and x2 will slide into the origin.

Fref defined in (16) is a control signal which is

velocity

(23)

where ka, kb and f0 are some constants satisfying the following:

The switching surface for the sliding mode position controller can be written as below, [6], [9] and [14]

error,

dVref

(21)

For finding the switching inputs and parameters limitation, Lyapunov function is selected to guarantees the existence condition of the sliding mode as follows [15]:

From (1), (3) and (13) the state variable can be written

position

)

− (α + ∆α ) FLoad + Fcogging +

as:

to

(20)

E = α∆Fe + ∆α Fe + ∆α∆Fe − ∆β x2 +

Sliding Mode Position Control

⎧⎪ x1 = d m − d ref ⎨ ⎪⎩ x2 = x1

(19)

where E is called the lumped uncertainty and is defined by:

In this section, the proposed sliding mode position controlled DTFC-based drive for PMLSM shown in Fig. 5. So define the state variables as:

proportion

)

where ∆α , ∆β are denoted as the uncertainties introduced by system parameters. After some algebras manipulate, eqn. (19) is simplified:

(12)

base

(18)

x2 = (α + ∆α ) Fe − ( β + ∆β ) x2 +

2

π π vm ψ s = p vbase ψ s τ τ

1 B , β= M M

Considering eqn. (17) with uncertainties:

π ⎛ π ⎞ ⎛ π ⎞ 2 ⎜ P vm Ld id + P vmψ f ⎟ + ⎜ P vm Lq iq ⎟ ≤ U om (11) τ ⎝ τ ⎠ ⎝ τ ⎠ 2

(17)

where:

with combination (9),(10) and (1) the voltage limits can be written as:

U om = p

(16)

load



633


d ref

+_

∆d

Fref

PID

MTPA / FW & Limitations

dm

Fref*

+

ψS

vm

∆F

1

-

0

+ ref

Switching Table

Inverter

1

-

∆ ψS

0

V

θn

n

V

V , Vdc n

V

ψS

V

Q

PMLSM dc

D

iQ

Flux & Force Computation F

2

iD

3

d/dt

Encoder dm

Xm

Filter

Fig. 5. Block diagram of the DTFC-based drive system with one PID position controller for PMLSM

they will slide to origin, then by considering equation (23), one component ( f 0 Sign ( S ) ) is generating

For reducing the chattering phenomena in sliding mode control the Sign function can be replaced by Saturation function as below [14]: Sat ( x ) =

x x +λ

chattering. In this study, fuzzy inference mechanism is used for tuning the f0 parameter by considering the dynamic of the SMC states. The control block diagram of the fuzzy sliding-mode controller is shown in Figs. 7. The fuzzy logic controller (FLC) has two inputs and one output, its inputs are the SMC Surface and its derivative proportion to time and its output is the f0 parameter. The membership functions for the fuzzy sets corresponding to the switching surface S, S and f 0 are defined in Fig. 8 and the fuzzy rule-based are given in Table I. Fuzzy output f 0 is calculated by the centre of area (COA) defuzzification method.

(25)

where λ > 0 is a smooth factor. For satisfying the current and voltages constraint, it is necessary that the SMC output be modified, therefore the stator flux linkage ( ϕ Sref ) and reference force ( Fref ) tune by the MTPA/FW and limitation block. For achieving minimum position tracking deviation, the SMC control parameters (ka, kb, kc and f0) are optimized by PSO algorithm which is described in continue.

V.

Fuzzy Sliding Mode Position Control

TABLE I FUZZY INFERENCE RULES N Z

S_dot

P

S

However using the Saturation function in sliding mode control leads to decreasing the chattering phenomena, but it decrease the quickness responsibility of the system to the variations. When the state variables (x1, x2) trapped on the switching surface, because of the stability of the system, ∆d

d ref

Fref*

Fref

N

L

M

S

Z

Z

Z

Z

P

S

M

L

∆F

∆ ψS

ψS

dm vm

ref

vm

θn

V

n

V

n V

F

Q

V

, V dc V

dc

D

iQ

ψS

iD

dm

Xm

Fig. 6. Block diagram of the Sliding mode position controlled DTFC-based drive for PMLSM



634


Z

Degree of membership

1

S

M

optimized by PSO for achieving minimum chattering and minimum position tracking deviation.

L

0.8

VI.

0.6

PSO is a population-based optimization technique proposed firstly by Kennedy and Eberhart in [16] for the unconstrained minimization problem. Particle swarm optimization (PSO) is inspired from studies of various animal groups and has been proven to be a powerful competitor to other evolutionary algorithms such as genetic algorithms [17]-[19]. The particle swarm optimization is based on changing the velocity and position of each particle toward its pbest and gbest locations according to the equations (26) and (27), respectively, at each time step:

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

f0

(a) N


1

Z

Particle Swarm Optimization

P

0.8 0.6

vid ( t + 1) = ω vid ( t ) + c1r1 ( pid − xid ( t ) ) +

0.4

(

+ c2 r2 pgd − xid ( t )

0.2

)

(26)

0 -1

-0.5

0 s

0.5

xid ( t + 1) = xid ( t ) + vid ( t + 1)

1

(b) N


1

Z

(27)

where ω is the inertia weight, c1 and c2 are acceleration constants, r1 and r2 are random variables in

P

the range [0,1], xid ∈ [ xmin ,xmax ] and vid ∈ [ −Vmax ,Vmax ] ,

0.8

where xmin and xmax denote minimum and maximum of particle position, respectively. And Vmax specify maximum of velocity. Further details about the algorithm are given in [20]. In this study four systems is optimized: 1-DTFC based system with one PID position controller (PID – DTFC). [Fig. 4] 2-DTFC based system with two PID position and velocity controller (PID2 – DTFC). [Fig. 5] 3-DTFC based system with sliding mode position controller (SMC – DTFC). [Fig. 6] 4-DTFC based system with fuzzy sliding mode position controller (Fuzzy SMC –DTFC). [Fig. 8]

0.6 0.4 0.2 0 -1

-0.5

0 sdot

0.5

1

(c) Figs. 7. Optimized Term sets of membership functions: (a) f0; (b) S; (c) S

The fuzzy-SMC controller parameters contain ka, kb, kc and the membership functions of the FLC are S

f0

S

∆d

d ref

Fref*

Fref

∆ ψS

ψS dm

∆F

vm

ref

vm

θn

V

n

V

n V

F

Q

V

, V dc V

dc

D

iQ

ψS

iD

dm

Xm

Fig. 8. Block diagram of the Fuzzy Sliding mode position controlled DTFC-based drive for PMLSM



635


⎧⎪0 d ref = ⎨ − t / 0.03 ⎪⎩0.15 1 − e

The limitations of equation 24 must be established for optimizing the SMC parameters. In these optimizations, number of population and the number of iterations are set to 50 and 100 respectively. In order to simultaneously achieve zero steady-state error, minimum position tracking deviation of the DTFC drive, the following fitness function is selected which can cover all the above purposes:

(

t < 0s

)

(31)

t ≥ 0s

The position, force, and velocity responses for the drive systems are shown in Figs. 10 to Fig. 13. 0.2

fitness =

∫0 e dt

Mover position, m

0.15

t

(28)

e = ∆d = d ref − d m

(29)

0.1

0.05

0

where e is position tracking error. Convergence curves of the PSO algorithm, for the four systems, are depicted in Fig. 9 and the optimized parameters for each system are tabulated in Table II.

Mover position Position command -0.05

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

0.3

0.4

0.5

0.3

0.4

0.5

Time, S

(a) 2

0.02

1.5

0.016

Velocity, m/s

PID – DTFC PID2 – DTFC SMC – DTFC Fuzzy SMC –DTFC

0.018

Fitness

0

0.014 0.012

1

v base

0.5 0 -0.5

0.01

-1

0

0.1

0.2 Time, S

0.008

10

20

30

40 50 60 70 Number of iterations

80

90

(b)

100 200 150

Fig. 9. Convergence of the optimization algorithm for the four systems

100

System

Optimized Parameters

PID-DTFC

k I = 127.2 ,k P = 8.2 ,k D = 2.4

PID2-DTFC

⎧ k I 1 = 140 ,k P1 = 47 ,k D1 = 2 ⎨ ⎩ k I 2 = 14,k P 2 = 0.01,k D 2 = 0.01

Thrust force, N

TABLE II OPTIMIZED PARAMETERS FOR THE FOUR SYSTEMS

50 0 -50 -100 -150

Fuzzy-SMCDTFC

0.1

0.2

(c)

kC = 59.5,ka = 92.4 ,kb = 79.28

5

f 0 = 183 kC = 59.5,ka = 128.4 ,kb = 184.3

Optimized membership function of the fuzzy controller is shown in Fig. 7.

VII.

0

Time, S

Phase a stator current, A

SMC-DTFC

-200

Simulation

To establish the effectiveness of the proposed drive system and, also, to examine the proper strategies, four “MATLAB” package simulations are designed. For testing systems, the following conditions have been considered: t < 0.25s ⎧0 (30) FLoad = ⎨ t > 0.25s ⎩35N

0

-5

0

0.1

0.2 Time, S

(d) Figs. 10. Closed loop responses for PID-DTFC system: (a) position response, (b) velocity response, (c) force response, (d) the phase current of stator winding



636


0.2

0.2

Mover position, m

Mover position, m

0.15

0.1

0.05

0.15

0.1

0.05

0

Mover position Position command

Mover position Position command -0.05

0

0.1

0.2

0.3

0.4

0 0.5

0

0.1

Time, S

(a)

0.4

0.5

0.4

0.5

0.4

0.5

0.4

0.5

(a) 2

2

1.5 Velocity, m/s

1.5 Velocity, m/s

0.2 0.3 Time, S

v base

1

0.5

v base

1 0.5 0

0

-0.5 -0.5

0

0.1

0.2

0.3

0.4

0

0.1

0.5

0.2 0.3 Time, S

Time, S

(b)

(b) 200

200 150

Thrust force, N

Thrust force, N

100 50 0 -50

100

0

-100

-100 -150 -200

0

0.1

0.2

0.3

0.4

-200

0.5

0

0.1

Time, S

(c)

0.2 0.3 Time, S

(c) 1

5

Surface of SMC

Phase a stator current, A

0

0

-1 -2 -3 -4 -5

-5

0

0.1

0.2

0.3

0.4

-6 0

0.5

Time, S

0.1

0.2 0.3 Time, S

(d)

(d)

Figs. 11. Closed loop responses for PID2-DTFC system: (a) position response, (b) velocity response, and (c) force response, (d) the phase current of stator winding

Figs. 12. Closed loop responses for SMC-DTFC system: (a) position response, (b) velocity response, (c) force response, (d) the surface of sliding mode



637


The velocity responses of the all systems indicate the proper performance of the MTPA-based drive systems in transition from the constant force region into the field weakening region. As shown in these figures, the chattering phenomena and the force ripple are much reduced for the Fuzzy SMC-DTFC system. The simulation results of SMCDTFC and Fuzzy SMC-DTFC systems indicate that these systems have good responses for position command tracking with load disturbances.

Mover position, m

0.2 0.15 0.1

Error=3.3648

0.05 0

Mover position Position command

0

0.1

0.2 0.3 Time, S (a)

0.4

0.5

VIII. Conclusion This study presents an optimum DTFC based drive system with fuzzy sliding-mode position controller which is robust with regard to plant uncertainties, load disturbances and cogging force. The dynamics of the proposed drive’s responses fully satisfied the desired position and velocity of a mover in the position control mode, in addition, chattering phenomena and the force ripple are much reduced for fuzzy sliding-mode controllers.

2

Velocity, m/s

1.5

v base

1 0.5 0 -0.5 0

0.1

0.2 0.3 Time, S

0.4

0.5

Appendix

(b)

TABLE A.1 DATA OF THE PMLSM USED IN SIMULATION

Thrust force, N

200 100 0 -100 -200

0

0.1

0.2 0.3 Time, S (c)

0.4

Ia m

4.8 A

Maximum voltage

Uam

24 V

Base speed

vbase

1.2 m/s

Maximum speed

vmax

1.92 m/s

Number of pole pairs

Pn

4

d-axis inductance

Ld

1.5 mH

q-axis inductance

Lq

1.5 mH

R

Stator resistance

0.5

Magnetic flux-linkage Mover mass

150

f0, m/s

Maximum current

100

0.5 Ω

ψf

0.017 Wb

M

10 Kg

Pole pitch

τ

10.5 mm

Friction coefficient

Bm

0.001 N.s/m

References [1]

50

[2] 0 0

0.1

0.2 0.3 Time, S (d)

0.4

0.5

[3]

Figs. 13. Closed loop responses for Fuzzy SMC-DTFC system: (a) position response, (b) velocity response, and (c) force response, (d) the fuzzy output [4]


J. G. Gieras and Z. J. Piech, Linear Synchronous Motors. Boca Raton, FL. CRC, 2000. J. Rivas-Conde, G. González-Palomino, E. Laniado-Jacome and J. Montoya-Larrahondo, "New Method of Vibration Analysis of Signal Force of Skew Permanent Magnet in Permanent Magnet Linear Synchronous Motors", International Review of Electrical Engineering (IREE), Vol. 5, Issue 5, Part A, pp. 1994-1999, September-October 2010. M. Abroshan, J. Milimonfared, K. Malekian and B. Abdi, "An Optimal Direct Thrust Force Control for Interior Permanent Magnet Linear Synchronous Motors Incorporating Field Weakening", International Symposium on Power Electronics, Electrical Drives, Automation and Motion, SPEEDAM 2008, Italy, pp. 130-135, 2008. M. F. Rahman, L. Zhong and K. W. Lim, "A Direct Torque-


638


[5]

[6]

[7]

[8]

[9]

[10] [11]

[12]

[13]

[14] [15]

[16]

[17]

[18]

[19]

[20]

Controlled Interior Permanent Magnet Synchronous Motor Drive Incorporating Field Wakening", IEEE Transactions on Energy Conversion, Vol. 18, Issue 1, pp. 17-22, 2003. M. Hajji, M. Khoidja, M. A. Nasr and B. Ben Salah, "Direct Thrust Control of a Linear Induction Motor with End Effects", International Review of Electrical Engineering (IREE), Vol. 4, Issue 4, pp. 532-538, 2009. F. J. Lin, K. Shyu and C. Lin, "Incremental Motion Control of Linear Synchronous Motor", IEEE Transactions on Aerospace and Electronic Systems, Vol. 38, Issue 3, pp. 1011-1022, 2002. F.J. Lin, J.C. Hwang, P.H. Chou, Y.C. Hung, "FPGA-Based Intelligent-Complementary Sliding-Mode Control for PMLSM Servo-Drive System", IEEE Transactions on Power Electronics, Vol. 25, Issue 10, pp. 2573-2587, 2010. F. J. Lin and P. Shen, "Linear Synchronous Motor Servo Drive Based on Adaptive Wavelet Neural Network", Proceedings. 2005 IEEE International Symposium on Computational Intelligence in Robotics and Automation, CIRA 2005, pp. 673-678, 2005. F.J. Lin, S.L. Chiu, "Adaptive fuzzy sliding-mode control for PM synchronous servo motor drives", IEE Proceedings - Control Theory and Applications, Vol. 145, Issue 1, pp. 63-72, 1998. R.R. Yager and D.P. Filev, "Essential of fuzzy modeling and control", Wiley, New York , 1994. S.T. Boroujeni, J. Milimonfared, M. Ashabani, "Design, Prototyping, and Analysis of a Novel Tubular Permanent-Magnet Linear Machine", IEEE Transactions on Magnetics, Vol. 45, Issue 12, pp. 5405-5413, 2009. Y.W. Zhu and Y.H. Cho, "Thrust Ripples Suppression of Permanent Magnet Linear Synchronous Motor", IEEE Transactions on Magnetics, Vol. 43, Issue 6, pp. 2537-2539, 2007. M. Abroshan, J. Milimonfared, K. Malekian and A. Rahnamaee, "An optimal control for saturated interior permanent magnet linear synchronous motors incorporating field weakening", 13th Power Electronics and Motion Control Conference, EPE-PEMC 2008, Poznan, pp. 1117-1122, 2008. W. Perruquetti and J. P. Barbot, "Sliding Mode Control In Engineering", Marcel Dekker, 2002 O. Kaynak, K. Erbatur and M. Ertugrul, "The Fusion of Computationally Intelligent Methodologies and Sliding-Mode Control—A Survey", IEEE Transactions on Industrial Electronics, Vol. 48, Issue 1, pp. 4-17, 2001. J. Kennedy, and R. Eberhart, “Particle Swarm Optimization”, IEEE International Conference on Neural Networks, pp. 19421948, 1995. R.C. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: Proceedings of the Sixth International Symposium on Micro Machine and Human Science (MHS’95), Nagoya, Japan, 1995, pp. 39–43. H. Boukef, M. Benrejeb and P. Borne, "Genetic Algorithm and Based Particle Swarm Optimization Comparison for Solving a Flow-Shop Multiobjective Scheduling Problem in Pharmaceutical Industries", International Review of Automatic Control (IREACO), Vol. 2, no. 2, pp. 223-228, March 2009. R. Jahani, H.A. Shayanfar and O. Khayat, "GAPSO-Based Power System Stabilizer for Minimizing the Maximum Overshoot and Setting Time", International Review of Automatic Control (IREACO), Vol. 3. n. 3, pp. 270-278, May 2010. P. J. Angeline, “Evolutionary optimization versus particle swarm optimization: Philosophy and performance differences,” in Proc. 7th Annu. Conf. Evol. Programming VII, 1998, pp. 601–610.

Mohammad Abroshan was born in Tehran, Iran, on September 21, 1983. He received the B.Sc. degree in electrical engineering from Tabriz University of Tarbiat Moallem, Tabriz, Iran, in 2006 and the M.Sc. degree from the Amirkabir University of Technology (Tehran Polytechnic Uni.), Tehran, Iran, in 2009. His research interests cover many aspects of Power Electronics, Renewable energies, Variable speed Drives, Wind Energy, Optimization algorithms, Intelligent Controls, as well as Design and Analysis of Electrical Machines. Jafar Milimonfared was born in Tehran, Iran in 1953. He received the B.Sc. degree in electrical engineering from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 1978 and the M.Sc. and Ph.D. degrees in electrical engineering from Paris VI University, Paris, France in 1981 and 1984, respectively. Dr. Milimonfared joined the Amirkabir University of Technology as an assistant professor in 1984 where he is now a professor of electrical engineering. His research interests include electrical machines analysis and design, power electronics and variable speed drives. He is with Amirkabir University of Technology as a Professor of Electrical Engineering. Mohammad Bagher Menhaj received his PhD from the School of Electrical and Computer Engineering at OSU in 1992. After completing one year with OSU as postdoctoral fellow in 1993, he joined Amirkabir University of Technology, Tehran, Iran. From December 2000 to August 2003, he was with School of Electrical and Computer Engineering and Department of Computer Science at OSU as a visiting faculty member and research scholar. His main research interests are the theory of computational intelligence, learning automata, adaptive filtering and its applications in control, power systems, image processing, pattern recognition, communications, rough set theory, and knowledge discovery. Soroush Tarafdar was born in Tehran, Iran, on 1982. He received his B.Sc. degree in Electerical Engineering from K.N.Toosi University of Technology (KNTU), Tehran, Iran in 2006 and Master of Science degree in Electrical Engineering in field of energy management from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 2010. His research interest includes Renewable energies, Power Electronics, DGs and their connection to the grid, Power Quality Analysis, Reliability and Cost Assessment and Energy storage systems.


Electrical Engineering Department, Damavand Branch- Islamic Azad University, Damavand, Iran. 2

Electrical Engineering Department, Amirkabir Uni. of Tech., Center of Excellence in Power Systems, Tehran, Iran.



639


Nonlinear MPPT and PFC Control of SCIG Wind Farm M. Benchagra1, Y. Errami2, M. Hilal3, M. Ouassaid4, M. Maaroufi5 Abstract – This paper presents a nonlinear control of 900-kW wind farm, to track the maximum power point (MPPT) and control Power Factor Correction (PFC) of wind farm using three Squirrel Cage Induction Generators (SCIGs) driven by 300-kW wind turbines. The wind farm delivers an active and reactive power to grid via common DC-bus and Voltage Source Inverter (VSI), The proposed control provides perfect tracking performances of the DC-bus voltage and the active and reactive powers to their references trajectories. The proposed control laws are derived from the Lyapunov approach using backstepping controllers. All theoretical and simulations results verified the accuracy and effectiveness of the proposed MPPT and PFC nonlinear control laws. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Wind Farm, Scigs, MPPT, PFC, VSC, VSI, Nonlinear Control, Buckstepping Controller, Lyapunov Approach

Jt Jg Ps Udc is ig C vg123 ig123 vi123 Rt

Nomenclature Pv Pm Cp δ β ρ A v R G Pm-opt Tem Tm ωm ωt ωs ωr vds vqs vdr vqr λds λqs λdr λqr ids iqs idr iqr Rs Rr Ls Lr Lm J

Wind turbine input power Mechanical power of the wind turbine Power coefficient Tip speed ratio Blade pitch angle Air density Rotor blades area Wind speed Blade radius Gear ratio Optimal mechanical power Electromagnetic torque Mecanical torque Generator speed Turbine speed Stator electrical speed Rotor electrical speed Direct stator voltage Quadrature stator voltage Direct rotor voltage Quadrature rotor voltage Direct stator flux Quadrature stator flux Direct rotor flux Quadrature rotor flux Direct stator current Quadrature stator current Direct rotor current Quadrature rotor current Resistance per phase of a stator winding Resistance per phase of a rotor winding Inductance per phase of a stator winding Inductance per phase of a rotor winding Mutual inductance Total inertia

Lt Pg Qg iqg-ref vdqg idqg vdqi V σ f ei ki

Turbine inertia Generator inertia SCIG active power DC-bus voltage SCIG side current Grid side current DC-link capacitor Grid voltages Output wind farm currents Output VSI voltages Equivalent resistance of the grid-side transmission line Equivalent inductance of the grid-side transmission line Output active power of the wind farm Output reactive power of the wind farm Reactive current command on the VSI Direct and quadrate grid voltages Direct and quadrate distribution line currents Direct and quadrate output VSI voltages Lyapunov function Dispersion coefficient Friction coefficient Error function Constant positive

I.

Introduction

Wind power is one of the most-effective systems available today to generate electricity from renewable sources. Therefore, with the rapid development of wind power and power electronic technology, wind turbine of constant speed constant frequency has been gradually replaced by variable speed constant frequency (VSCF) wind power technology [1]. For power electronic, the back to back voltage source converter is usually used to


640


M. Benchagra, Y. Errami, M. Hilal, M. Ouassaid, M. Maaroufi

connect the generator to the grid, the Voltage Source Converter (VSC) can increase the robustness and MPPT tracking control [2], the DC-link capacitor provides the decoupling between the generator and grid. The VSI can control the magnitude, frequency and phase angle of the output voltage as well as the power factor correction (PFC) [3]. In order to control VSI connected-grid, several control schemes have been proposed, in most of these schemes, the reference PWM voltages are generated by regulating currents using proportional integral (PI) controllers. This consists of two control loop i.e. outer voltage loop and inner current loop. It can realize decoupling control of active current and reactive current. Thus the active power and reactive power can be regulated independently [4]-[6]. Wind farm have so many uncertainties due to erratic nature of wind-based systems. Therefore, the controller should accommodate the effects of uncertainties and keep the system stable. The conventional PI-based controllers cannot fully satisfy stability and performance requirements. Nonlinear control methods can be used to effectively solve this problem [7]-[10]. Especially in these years, there has been tremendous amount of activity on a special control schemes known as “Backstepping” approach [11] and [12], the backstepping approach is a nonlinear method which is perfectly suited for SCIG-wind power. Through studying the characteristics of wind turbine and nonlinear control of SCIG for MPPT control, which are detailed in [2], the focus of this paper is show that the nonlinear backstepping MPPT, DC-bus voltage and powers control schemes combined with three nonlinear wind turbines model assure a good control even during a wind speed conditions, and that a good tracking of reference trajectories can be achieved, so a perfect control of wind farm.

II.

(4)

In order to make full use of wind power, in low wind speed β should be equal to zero. Fig. 2 illustrates the wind turbine power curve when β is equal to zero. From the Fig. 2 we can see there is one specific angular frequency at which the output power of wind turbine is maximum. Connected all the maximum power point of each power curve, the optimal power curve (Pm-opt curve) is gotten. When the wind turbine is in the Pm-opt curve, the turbine will get the maximum power Pmax. In this case, the maximum value of Cp is CPmax=0.47, is achieved for β=0° and δopt=8.1, Fig. 3. II.2.

SCIGs Models

The SCIG mathematical model can be expressed in a reference frame rotating with arbitrary angular speed by the following equations:

(5)

The stator and rotor flux can be expressed as:

(6)

Modeling of Wind Turbine and SCIG II.1.

Wind Turbine Characteristics

The wind turbine input power usually is [3]:

The electromagnetic torque can be calculated as: (1)

(7)

The output mechanical power of wind turbine is:

where p, is the pair-pole number. The voltage and flux equation can be supplemented by the mechanical equation for the drive train (gearbox) by (8) to complete the model of generator used in this paper:

(2) where δ is defined as the ratio of the tip speed of the turbine blades to wind speed:

(8)

(3)

where Tm is the mechanical torque. J is the total inertia, it can be calculated as:

We consider a generic equation to model a power coefficient Cp, based on the modeling turbine characteristics described by [13]:

(9)



641


Wind turbine 1 ia1

va1

ia

va

VSC

is1

SCIG 1

Wind Gear box

v

Wind turbine

Ps

SCIG Gear box

Wind

abc/dq

PWM

v isdq

v 1

MPPT

Nonlinear MPPT PWM Control

ωm-ref

abc/dq isdq

1

ωm-ref

va2

ia2

MPPT SCIG 2

is

is2 Nonlinear Backstepping Controller

v

ia

va

VSI

ig

id c

vi1

Grid

C

VSC

Ps

Ps

Ps

SCIG

abc/dq

PWM

idc

MPPT

ωm-ref

abc/dq

abc/dq

isdq

Wind turbine 3

va3

ωm-refia3

v

abc/dq

Nonlinear PFC PWM Control

Nonlinear MPPT PWM Control

Ps Grid

PWM

C

isdq

v

vg1

MPPT SCIG 3

is3 Nonlinear Backstepping Controller

Nonlinear Backstepping Controller

Gear box

Wind turbine

va

ia

Ps

SCIG Gear box

abc/dq isdq

v

MPPT

ωm-ref

abc/dq

PWM Nonlinear MPPT PWM Control

isdq

Fig. 1. Schematic diagram of the studied wind farm connected-grid

The above model (5) to (9) can be presented as differential equations for the stator currents and rotor flux vector components under the following form [2]:

where:

;

(10) II.3. (11)

;

;

Nonlinear MPPT Control Strategy

The basic idea of the backstepping design is the use of the “virtual control” quantities to decompose systematically a complex non-linear control structure problem into simpler and smaller ones. Backstepping design is divided into two various design steps. In each one we deal with an easier and single-input-single-output design problem and each step provide a reference for the next design step [9].

(12) (13) (14)



642


ids and iqs are considered to be the virtual control input. Thus, there references are obtained as: (19)

(20) kω and kλ are positive design constants that determine the closed loop dynamics. Equations (19) and (20) indicate that the virtual controls should be in order to satisfy the control objectives. So they provide the references for the next step and try to make the signals ids and iqs behave as desired. So, we define again the error signals involving the desired variables in (19) and (20):

Fig. 2. Wind turbine power curve

(21) (22) Finally, we extended the Lyapunov function in (17) to include the states variables eids , eiqs as: (23) Control laws are derived by differentiating the Lyapunov function with respect to time:

Fig. 3. Characteristics Cp vs δ, for various values of pitch angle β

In order to realize the MPPT control, the control law should be based on the optimal wind turbine power curve Pm-opt and the wind speed to calculate the mechanical speed reference ωm-ref. A nonlinear MPPT backstepping law control using rotor oriented field was proposed in [2] for wind turbine power capture optimization. Fixing the d-axis of the rotating reference frame on the rotor flux vector, we have λdr=λr=cte and λqr=0. In this case, two state variables have been proposed for describing the SCIG model in order to extracting maximum power from turbine, mechanical speed and rotor flux, as follows:

(24)

(25)

The Lyapunov function derivative is given by: (26)

(15) The controller of MPPT is implemented by (24) and (25) which is simulated by using Matlab/simulink.

(16) Therfore, the error is defined using the rotational speed and rotor flux and the first Lyapunov function is defined as:

III. Nonlinear PFC Control Strategy Considering the grid and the output from a three phase voltages sourced dc/ac inverter as ideal voltage sources. According to Fig. 1, the relationship between the grid, inverter voltages, and line currents is given as:

(17) The time derivative of V is given by:

(27) (18)



643


Transforming the voltage equation (27) using dq transformation in the rotating reference frame at the grid frequency gives:

(37)

(38) (28) (39) (29) where b1=Rt/Lt ; b2=ω ; b3=1/Lt Now we turn to the backstepping design steps. First, for the DC-bus voltage regulation tracking objective, define the tracking error as:

The controller assumes that the voltage is balanced and hence the vog component is not present in equations. The instantaneous active and reactive power outputs, seen from the inverter side, can be defined as [14]:

(40) (30)

Then, the error dynamical equation is:

(31)

(41)

Now the initial angle of the d-q reference frame is set to π/2 and the initial angle of the phase 1 is set to 0, this causes the vqg component to be zero and the vdg component to be equal to vg1 . In this reference frame the above P and Q equations will become:

Since our objective requires that the error converge to zero, we could satisfy the objective by viewing idg as virtual control variable in the above equation and use it to control the error . We use de Lyapunov function defined as:

(32)

(42)

(33)

The derivative of

along the error equation is:

=

Thus the active and reactive power flows are controlled by the idg and iqg respectively. The equation for the voltage across the DC-bus is given by:

=

=

=-

(43)

(34) This yields to:

is positive design constant that determine the closed loop dynamics. Thus the tracking objective will be satisfied if we choose:

(35) Eq. (35) can be rewritten as:

(44) (36)

Since the choice of (41) give: (45)

From Eq. (36), it is clear that the DC-bus voltage controller provides the active grid power reference value (or direct grid current reference) and Ps acts as a disturbance. The above model (28) to (36) can be presented as differential equations for the grid currents and DC-bus voltage under the following form:

Eq. (44) indicates that the virtual control should be in order to satisfy the control objective of active power. So it provides the reference for the next step and try to make the signal idg behave as desired. So, we define again the error signals involving the desired variables idg and iqg (iqg-ref = ) in order to control active and reactive powers respectively:



644


(46) (47) Then the error equation in (41) can be expressed as:

= =

(48)

(53)

Also, the dynamical equation for the error signals eidg , eiqg can be computed as: (49) (50)

From the above we can obtain the control law as:

Now we extended the Lyapunov function in (42) to include the states variables eidg , eiqg as:

(54)

(51) (55)

We use Veg to derive the control algorithm. To this end, we compute again the derivative of Veg along with the error equations (41), (49) and (50):

This leads to:

(52)

(56)

So:

The controller of PFC is implemented by (54) and (55) which is simulated by Matlab/simulink.

+ + vdg -

Equation (54)

vdi

Equation ()

vds

Equation (55)

vqi

Equation ()

vqs

vqg ω idg ids

iqg iqg-ref iqs

+ i*dg

+

Equation (44)

+

Equation ()

+

Fig. 4. Detailed diagram of the nonlinear Backstepping PFC control



645


IV.

In order to evaluate nonlinear backstepping PFC control strategy proposed in this paper, Matlab/simulink is used to carry out the simulation.

Simulations Results

Figs. 5 illustrate simulation of 900-kW wind farm using SCIGs driven by wind turbines. The parameters of wind turbine and SCIG are given in appendix. The PWM VSC and PWM VSI are operated at 10 kHz.

(e) Output DC-bus voltage response

(a) mechanical speeds responses

(f) Output currents in d-q frame

(b) Power coefficients responses

(g) One phase output current and voltage

(c) SCIG1 stator voltage and current

(h) Zoom in one phase output current and voltage

(d) Zoom in SCIG1 stator current curve

Figs. 5. Simulations results of wind farm using backstepping MPPT and PFC control



646


TABLE A.2 PARAMETERS OF THE SCIG

The control law is designed as followed: initially, wind speed is set at 10 m/s, then starting at t=0.5 s for SCIG1, the same gust of wind is applied to SCIG2 and SCIG3, respectively with 1.5 seconds and 2.5 seconds delays. Figs. 5(a) and (b), illustrate the mechanical speed and power coefficient for each SCIG of wind farm designed in Fig. 1. According to the wind turbine characteristics, the optimal mechanical speed of SCIG is calcult as:

Parameters Rated power No. of poles Rated speed Stator resistance Stator inductance Mutual inductance Rotor resistance Rotor inductance Generator inertia

(57)

300 kW 2 158.7 rad/s 0.0063 Ω 0.0118 H 0.0116 H 0.0048 Ω 0.0116 H 10 kg/m2

References

So, SCIG mechanical speed is: ωm-opt=14v

[1]

(58)

[2]

According to (*), when wind speed v is 10 m/s, the optimal mechanical speed is 140 rad /s, it is seen that in MPPT control, the generator speed has to change to keep maximum power coefficient (Cp=0.47) Fig. 5(c) shows the SCIG1 stator voltage and current. Fig. 5(e) presents the output DC-bus voltage of the wind farm, as desired, the DC-bus voltage is perfectly and quickly tracked to their input reference (Udc-ref =1160 V). Fig. 5(f) give the simulation results of the proposed backstepping PFC controller, it is seen clearly that under the action of controller (44) and the controllers (54) and (55) the system outputs (idg and iqg) follow the desired references signals well. Figs. 5(g) and (h) shows the perfect PFC control. All results verified the effectiveness of the proposed nonlinear MPPT and PFC control laws for wind farm.

[3]

[4]

[5]

[6]

[7]

V.

Values

Conclusion [8]

In this paper, 900-kW wind farm based on three 300kW wind turbines is connected to grid. Wind turbine use squirrel cage induction generators. Variable speed wind turbine driving SCIG with current controlled voltage source inverter has been proposed. Detailed modeling and nonlinear control strategies of the wind farm has been developed. It is found that under the proposed backstepping control strategy the system run smoothly under quickly wind condition. Finally, it is concluded that Backstepping PFC control works very effectively to operate the Wind farm in both dynamic and transient conditions.

[9]

[10]

[11]

Appendix

[12]

TABLE A.1 PARAMETERS OF THE TURBINE Parameters

Values

Density of Air Area swept by blades, A Speed-up gear ratio, G Base wind speed Turbine inertia

1.22 kg/m3 615.8 m2 23 12 m/s 50 kg m2

[13]

[14]


X. Zheng, L. Li, D. Xu, Sliding mode MPPT control of variable speed wind power system, Power and Energy Engineering Conference, APPEEC 2009, pp. 1-4, Asia-Pacific, 2009. M. Benchagra, Y. Errami, M. Hilal, M. Ouassaid, M. Maaroufi, Nonlinear MPPT Control of SCIG Wind Power Generation, Journal of Theoretical and Applied Information Technology, Submitted for publication. M. Pahlevaninezhad, A. Safaee, Z. Eren, A. Bakhshai, P. Jain, Adaptative nonlinear maximum power point tracker for a WECS based on permanent magnet synchronous generator fed by a matrix converter, Energy Conversion Congress and Exposition, ECCE 2009 IEEE, pp. 2578-2583, 2009 H. Zhang, Y, Zhao, Vector decoupling controlled PWM rectifier for wind power grid-connected inverter, 2009 International Conference on Energy and Environment Technology, IEEE, 2009. S.M. Muyeen, A. Al-Durra, J. Tamura, Variable Speed Wind Turbine Generator System with Current Controlled Voltage Source Inverter, Energy Conversion and Management, vol. 52 , pp. 2688-2694, February 2011. M. Benchagra, M. Maaroufi, M. Ouassaid, Modeling and Control of SCIG based Variable-Speed with Power Factor Control, International Review of Modelling and Simulations (IREMOS), vol. 4, n. 3, pp. 1007-1014, Jun 2011. B. Boukhezzar, H. Siguerdidjane, Nonlinear Control with Estimation of a DFIG Variable Speed Wind Turbine for Power Capture Optimization, Energy Conversion and Management, vol. 50, pp. 885-892, January 2009. J. Yu, B. Chen, H. Yu, Position Tracking Control of Induction Motor via Adaptive Fuzzy Backstepping, Energy Conversion and Management, vol. 51, pp. 2345-2352, April 2010. A.L. Nemmour, F. Mehazzem, A. Khezzar, M. Hacil, L. Louze, R. Abdessemed, Advanced Backstepping Controller for Induction Generator Using Multi-scalar Machine Model for Wind Power Purposes, Journal of Renewable Energy, vol. 35, n 1, pp. 23752380, February 2010. J. Hu, L. Shang, Y. He, Z.Q. Zhu, Direct Active and Reactive Power Regulation of Grid-Connected DC/AC Converters Using Sliding Mode Control Approach, IEEE Transactions on Power Electronics, vol. 26, n. 1, pp. 210-222. January 2011. A.H. Hamida, A. Allag, M.Y. Hammoudi, S.M. Mimoune, S. Zarouali, M.Y. Ayad, M. Becherif, E. Miliani, A. Miraoui, A nonlinear adaptive backstepping approach applied to a three phase PWM AC-DC converter feeding induction heating, Communication in Nonlinear Science and Numerical Simulation 14, pp. 1515-1525, 2009. M. Ouassaid, M. Maaroufi, M. Cherkaoui, Decentralized Nonlinear Adaptive Control and Stability Analysis of Multimachine Power System, International Review of Electrical Engineering (IREE), vol. 5 n. 6, pp. 2754-2763, December 2010. R. Pena, J.C. Clare and G.M. Asher, Doubly fed induction generator using back-to-back PWM converters and its application to variable speed wind-energy generation, Electric Power Applications, IEE Proceeding, pp. 231-241, 1996. S. Sarkar, P. Vijayan, D.C. Aliprantis, V. Ajjarapu, Effect of grid voltage unbalance on operation of a bi-directional converter,


647


Power Symposium, 2008. NAPS '08. 40th North American, sept 28-30, pp. 1-7, 2008, Calgary, AB.

Mohamed Hilal was born in Assoul-Errachidia, Morocco. He received this diploma in Electrical Engineering from high school of education of technology, in specialty electrical and power electronics engineering in Rabat, Morocco in 1993, and his higher education depth DESA in Electrical Engineering from Mohamed V University - Mohammadia school of engineers, Rabat, Morocco, in 2008. He is currently Professor of engineering at high School of technologies, Salé, Morocco. His research interests are electric drives, power electronics, power systems and renewable energy.

Authors’ information 1,2,3,5

Department of electrical Engineering, Ecole Mohammadia d’ingénieurs, Mohammed V University, Rabat, Morocco. E-mails: [email protected] [email protected] [email protected] [email protected]

Mohammed Ouassaid was born in Rabat, Morocco, in 1970. He received the « Diplôme d’agrégation » in Electrical Engineering from Ecole Normal Supérieur de l’Enseignement Technique, Rabat, in 1999, and the M.Sc.A. and Ph. D. degrees, in Electrical Engineering from Ecole Mohammadia d’Ingénieur, Université Mohamed V , Rabat, Morocco, in 2002 and 2006, respectively. He is currently an Assistant Professor at National School of Application Sciences (ENSA-Safi) Cadi Aayad University, Morocco. His research interests are electric drives, power electronics, power systems and renewable energy. Dr. Ouassaid is a member of the IEEE.

4

Department of industrial Engineering, National School of Application Sciences - Safi, Cadi Aayad University, Morocco. E-mails: [email protected] [email protected] Mohamed Benchagra was born in Beni-Mellal, Morocco, in 20/03/1982. He received the diplôme d’ingénieur d’application degree from the Faculty of Technical Sciences, Béni-Mellal, Morocco, in 2004 and the DESA in 2006 in electrical and power electronics engineering from Mohammadia School of Engineering (Mohamed V University) Rabat-Morocco, where he pursues his doctoral program. His research is interested in the modeling and control of wind farm based on inductions generators, power systems and renewable energy. He is also interested in all electrical energy research. M. Benchagra is a graduate student member of the IEEE.

Mohammed Maaroufi was born in Marrakech, Morocco, in 1955. He received the diplôme d’ingénieur d’état degree from the Ecole Mohammadia, Rabat, Morocco, in 1979 and the Ph.D. degrees from the Liége University, Belgium, in 1990, in Electrical Engineering. In 1990, he joined the Department of Electrical Engineering, Ecole Mohammadia, Rabat, Morocco, where is currently a Professor and University Research Professor. His current research interests include electrical network, renewable energy, motor drives and power systems.

Youssef Errami received the Agregation in electrical engineering from Ecole Normale Supérieure de Rabat, Morocco in 2001 and the DESS from laboratory of physical, Chouab Doukkali University,Eljadida,Morocco, in 2005. In 2009, he joined electric machines Laboratory at the Department of Electrical Engineering of Ecole Mohammadia d’Ingénieur, Rabat, Morocco .He is pursuing Ph.D His research interests are in the areas of Power Electronics Systems, electric drives, power systems and Wind Power Energy. Youssef Errami is a graduate student member of the IEEE



648


Sliding Mode Control of Buck Converter for Low Voltage Applications S. Chander1, P. Agarwal2, I. Gupta3

Abstract – The switch-mode power supplies have been controlled using different control algorithms like PID control, current mode control etc. Since, Switch-mode power supplies represent a particular class of variable structure systems (VSS). Thus, they can take advantage of non-linear control techniques developed for the variable structure systems. In this paper the sliding mode control is analyzed and developed with preference to buck converter. The simulation model of buck converter with its control circuit was build up in MATLAB/SimulinkTM and simulation results are obtained. Then the dynamic response of buck converter controlled by SMC is studied. Sliding mode control extends the properties of hysteresis control to multi-variable environments, resulting in stability even for large supply and load variations. It presents a good dynamic response and simple implementation. The performance of SM control is compared with that of conventional PID control. It has been shown that the use of SM control can lead to an improved robustness in providing consistent transient responses over a wide range of operating conditions. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: DC-DC Converter, Non-Linear Controller, Sliding-Mode (SM) Control

Nomenclature CCM EMI HC PID PWM SMC VSS vi Vc IR, V0 VSC S(x) fs,Ts fc u ±δ Tmin RL IL,max Vc,max fmax Vrpp Kdpwm ∆t1 ∆t2

I.

Continuous Conduction Mode Electromagnetic Interference Hysteresis Control Proportional-Integral-Derivative Pulse Width Modulator Sliding Mode Control Variable structure systems Input voltage Capacitor voltage Output current, voltage Variable Structure Control Sliding function Switching frequency, time Corner frequency Switching function Hysterises Band Minimum time period of one cycle Load resistance Maximum inductor current Maximum capacitor voltage Maximum switching frequency Peak-to-peak ripple voltage DPWM gain Time taken by state variable to move from position X to Y Time taken by state variable to move from position Y to Z

Introduction

The control of DC-DC converters has been widely investigated in the past. Many linear control techniques have been proposed and analyzed. Among them, the most popular are Voltage Mode Control (VMC) and Current Mode Control and its derivations like Peak Current Control and Average Current Control etc. [l],[2]. Controllers based on these techniques are simple to implement and easy to design, but their parameters generally depend on the operating point. Efforts to achieve large signal stability with the controller result in reduction of the useful bandwidth and adversely affecting performances of converter. Moreover, application of these techniques to high-order DC-DC converters makes selection of control parameters and response stabilization very critical. Different control schemes have been proposed for the design of controllers for switching converters [2]-[13]. But a great deal of work is still required for improvement of line and load regulation, especially in applications where any uncompensated small disturbance in the switching converters is intolerable, and it may seriously affect the performance of the system employing this converter [3]. Sometimes in DC-DC Converters, considerable modifications in parameters values take place during operation [3]. The fixed controller with large modifications of parameters causes slow transition and insufficient damping. In such cases, the dynamic properties of the controlled system over wide operating range can be improved by the use of the Variable


649


S. Chander, P. Agarwal, I. Gupta

Structure Control (VSC) and are characterized as variable–structured systems. The control techniques derived from the variable structure systems (VSS) theory like sliding-mode (SM) control can also be applied to the DC-DC converters [4]. Accordingly, SM control suits to the non-linear nature of these converters. SLIDING-MODE (SM) controller is a nonlinear controller It was introduced for controlling the variable structure systems (VSSs) [1],[4],[5],[13], and[15].The SM control guaranteed stability and the robustness against parameter, line, and load variations [15]. SM controller has a high degree of flexibility in choice of parameters, during design and is relatively easy to implement as compared with other types of nonlinear controllers [15]. Such properties make it highly suitable for control applications in nonlinear systems, especially in application to high-order converters, yielding improved performances as compared to classical control techniques. The SM control has several advantages [15]: • SM control offers large signal handling capability. • It provides stability even for large supply and load variations. Thus, it provides better regulation over a wide operating range. • Robustness are excellent, as for other hysteretic control against load, supply and parameter variations • It provides fast dynamic response. Since all control loops act concurrently • The system response depends only slightly on actual converter parameters • Simple implementation as compare to other nonlinear controllers. In addition to above advantages, conventional SM control has some drawbacks also [15]: • The switching frequency is not constant as in case of PWM controller. The switching frequency varies depending on the working point, due to its hysteretic nature. • The steady-state errors can affect the controlled variables e.g output voltage. • All state variables must be sensed. • The sliding Mode control theory is too complex. Due to this complexity, the selection of control parameters may be difficult Design of SM controller is more heuristic in absence of any systematically defined procedure. Moreover, the SM controllers are not available in Integrated-circuit (IC) forms for power-electronic applications, unlike the PWM controller [15]. Above all, there is a strong reluctance to the use of SM controllers in DC-DC converters, because of their inherently high and variable switching frequency. The high and variable switching frequency causes excessive power losses and electromagnetic–interference (EMI) [15], which further, complicates the design of the filter. Discussions regarding the usefulness and advantages of SM controllers have been going on and thus these are rarely used in practical DC-DC [15].

II.

SM Control for DC-DC Converter

One of the most important features of the sliding mode regime, in the variable structure system (VSS), is the ability to achieve response that is independent of the system parameters. The DC-DC converter is particularly suitable for the application of the SMC because of its controllable state, since every state variable can be affected by the input signal. The output voltage V0 and inductor iL are both continuous and accessible for measurement. Most of the DC-DC converters used in practice, the rate of change of the current is much faster than the rate of change of output voltage [6]. The control problem can be solved by using cascaded control structure with two control loops, an inner current control loop and an outer voltage control loop. The combined loops represent the SMC. The voltage control is usually realized with standard linear control technique, and the current control is implemented using either PWM or hystereses control (HC), which is the standard practice [6]. Sliding Mode control scheme of buck converter is shown in Fig. 1. Vi and V0 are the input and output voltages, respectively. The inductor current (iL) and capacitor voltage (Vc) are the internal state variables of the converter. The buck converter operating in the continuous conduction mode (CCM), have measurable continuous controllable states [7]. Switch ‘SW’ accounts for the system non-linearity and indicates that the converter may assume only two linear sub-topologies as shown in Figs. 2, each corresponding to each of the state of switch.

Fig. 1. Basic structure of the Sliding Mode controlled buck converter

The above condition also implies that the mathematical approach presented here is valid only for CCM operation. According to SM control theory, the two state variables of the converter are sensed and the corresponding errors X1 and X2 are generated. The errors are obtained by taking difference of the steady state values of the two state variables, respectively. These error values are then multiplied by the proper gains K1 and K2 and added together to form the sliding function S(x) as in (1) and (2). The hysteretic block maintains the function S(x) to zero, such that for N-order system (No. of state variables):



650


s ( x) =

II.2.

N

∑ Ki X i = 0

(1)

Buck converter to be considered for sliding mode control design, requires selection of gain parameters K1 and K2. The selection must be done in order to ensure the following three constraints: • The hitting condition-requires that the system trajectories across the sliding line, irrespective of their starting point • The existence condition-requires that the system trajectories near the sliding line in both the regions are directed towards the line itself. Mathematically, this condition can be express in (3) and (4) as:

1

For 2nd order system like buck converter, the sliding function is given (2): s ( x ) = K1 X 1 + K 2 X 2 = 0

Conditions for Sliding Motions

(2)

A hyperplane constitutes the sliding surface whose equation in the state space is expressed by the linear combination of state variable errors as expressed in (1). Equation (1) represents a hyper plane in the state error space passing through the origin. Each of the two regions, separated by this plane, is associated by HC block to one converter sub-structure. If the existence condition of the SM are satisfied i.e the state trajectories near the surface, in both the regions are directed towards the sliding plane, system state can be enforced to remain near (lie on) the sliding plane by proper operation of the converter switch.

lim+

ds 0 dt

(4)

S →0

S →0

• The stability condition of the system motion on the sliding line, i.e the motion must be towards the equilibrium point II.3.

Control Design

In practice, controller design for buck converter only, requires the proper selection of the sliding surface (2) i.e of coefficients K1 and K2 so as to ensure that the existence condition, hitting condition and the stability of the system trajectories on the sliding plane are complied. In order to ensure that the control system operates properly, the existence condition and stability must be verified.

III. Theoretical Derivations

Figs. 2. DC-DC buck converter (a) sub-topology during switch ON time (b) sub-topology during switch OFF time

II.1.

This section covers the theoretical aspects of SM control buck converter. The capacitor voltage (Vc) and inductor current (iL) are taken as state variables of the converter. The complete mathematical derivations for the designs are presented in this section.

Principles of Sliding Mode Control

The basic principle of SM control is to employ a certain sliding surface as a reference path such that the controlled state variables’ trajectory can be directed toward the desired equilibrium. Theoretically, such objective of the SM control can be fully achieved only if certain conditions, namely, the hitting condition, the existence condition, the stability condition, and the condition that the system operates at an infinite switching frequency are absolutely complied. This however requires, an idealized controlled system, wherein no external disturbances or system’s uncertainties can affect the ideal control performance i.e zero regulation error and very fast dynamic response. Hence, in a certain sense, the SM controller is actually a type of ideal controller for the class of VSSs.

III.1. Mathematical Modeling of Buck Converter Figs. 2 shows the basic structure of the power stage of buck converter. Fig. 2(a) and 2(b) show the subtopologies corresponding to switch state when switch is ON and OFF respectively. The state space description of the buck converter under sliding mode control, where the control parameters are the capacitor voltage Vc and the inductor current iL (in phase canonical form) is first discussed. When switch SW is ON, then according to Fig. 2(a). The rate of change of inductor current and capacitor voltage is given by:



651


diL (Vi − Vc ) = dt L

(5)

dVc iL V = − c dt C CRL

(6)

Also: B u = V − [ A]V − C

In SMPS, it is better to sense state variable errors like capacitor voltage and inductor current errors, given by (14) and (15), respectively. The sliding function S(x) given by (2), can be rewritten as (16):

When switch SW is OFF, then according to Fig. 2(b).The rate of change of inductor current and capacitor voltage is given by: diL −Vc = dt L

(7)


(8)

(9)


(10)

)

(

X = AX + B u + D

)

(16)

(17)

where, X is the vector of state variable errors ∗⎞ ∗ ⎛ X = ⎜ V − V ⎟ and V is the vector of state variable ⎝ ⎠ ∗ ⎡ iLref ⎤ references V = ⎢ ⎥. ⎢⎣Vcref ⎥⎦ From (17), D can be given as: D = X − AX − Bu

(18)

Substituting the values of X , X and Bu (eq. (13)) and after simplification: (11) *

D = AV + C

(19)

* ⎡0 ⎤ D = AV , as C = ⎢ ⎥ ⎣0 ⎦

(20)

(12) ⎡ ⎢0 [ D] = ⎢ 1 ⎢ ⎢C ⎣

where: ⎡ ⎡ diL ⎤ ⎢0 ⎢ dt ⎥ V =⎢ ⎥ , [ A] = ⎢ ⎢1 ⎢ dVc ⎥ ⎢C ⎢⎣ dt ⎥⎦ ⎣

(15)

According to variable structure system theory, the state space model of the system can be derived in following form [16]:

Equation (11) can be written in canonical form as: V = [ A]V + B u + C

X 2 = Vc − Vcref

(

In standard matrix form, the (9) and (10) are written −1 ⎤ ⎡V ⎤ ⎡0 ⎤ L ⎥ ⎡ iL ⎤ ⎢ i ⎥ ⎥⎢ ⎥+ L u+⎢ ⎥ 1 ⎥ ⎣Vc ⎦ ⎢ ⎥ ⎣0 ⎦ − ⎣⎢ 0 ⎦⎥ RL C ⎥⎦

(14)

K1 iL − iLref + K 2 Vc − Vcref

as: ⎡ diL ⎤ ⎡ 0 ⎢ dt ⎥ ⎢ ⎢ ⎥=⎢ ⎢ dVc ⎥ ⎢ 1 ⎢⎣ dt ⎥⎦ ⎢⎣ C

X 1 = iL − iLref

S ( x ) = K1 X 1 + K 2 X 2 =

where, L, C and RL are the inductance, capacitor and load resistance respectively. Vi and Vc is the input and capacitor voltage respectively. Let u be the switching function which represents the switching state of switch SW. It has the value of 1 or 0 for ON and OFF position of the switch SW. By combining above expression for both states of the SW, the expressions are given by: V diL Vi = u− c dt L L

(13)

−1 ⎤ ⎡Vi ⎤ L ⎥ ⎥ , [ B] = ⎢ L ⎥ , 1 ⎥ ⎢ ⎥ − ⎣⎢ 0 ⎦⎥ RL C ⎥⎦

−1 ⎤ L ⎥ ⎡ iLref ⎤ ⎥⎢ ⎥ 1 ⎥ ⎢Vcref ⎥ ⎣ ⎦ − RL C ⎥⎦

⎡ −Vcref ⎤ ⎢ ⎥ L ⎢ ⎥ D = [ ] ⎢i Vcref ⎥ Lref − ⎢ ⎥ RL C ⎦ ⎣ C

⎡i ⎤

⎡0⎤ and [C ] = ⎢ ⎥ ⎣0⎦ ⎣ c⎦

[V ] = ⎢VL ⎥


(21)

(22)


652


III.2. Design of Sliding Mode (SM) Controller

The control can enforce the system state to remain near the sliding plane by proper operation of the converter switch SW. To make the system states to move towards the sliding surface, necessary and sufficient, condition (28) is which is same as (27) [9]:

In SM control, the controller employs a switching function u to decide its input states to the system [8]. The switching state u for SM controller, can be determined from the control parameters X1 and X2 and using the switching function of (2): S ( x ) = K1 X1 + K 2 X 2 = K T X

S ( x ) < 0 , S ( x ) > 0,

(23)

where K1 and K2 are control parameters. The switching function and its derivative are given as (24) and (25): S ( X ) = [ K1

⎡X ⎤ K2 ] ⎢ 1 ⎥ ⎣X2 ⎦

S ( X ) = K T X

(24)

lim S ⋅ S < 0

S ( X ) = K T ( AX + Bu + D )

(29)

S ( X ) = K T AX + K T Bu + K T D

(30)

S ( X ) = K T AX + K T B + K T D > 0, if S ( X ) < 0 (31) S ( X ) = K T AX + K T D < 0 , if S ( X ) > 0

(32)

From practical point of view assuming the error variable Xi smaller than reference V*, the KTAX can be neglected. Accordingly, above equations can be modified into following: S ( X ) = K T B + K T D > 0 , if S ( X ) < 0

(33)

S ( X ) = K T D < 0, if S ( X ) > 0

(34)

Substituting B, D and K in (33) and (34), the final expressions are:

(26)

S ( X ) = [ K1

Equation (26), simply conveys that when switch is OFF, the function S(X) must decrease, while when it is ON, S(X) must increase. The control law of (26) only provides the general requirement that the trajectories will be driven towards the sliding line and there is no assurance that the trajectory can be maintained on this line. This local reachability i.e to ensure that the trajectory is maintained on the sliding line, the existence condition, which is derived from Lyapunov’s second method [8] must be satisfied: S →0

(28)

(25)

⎡X ⎤ K 2 ] and X = ⎢ 1 ⎥ . ⎣X2 ⎦ By enforcing S(X)=0, a sliding line can be obtained. The purpose of this line is to serve as a boundary to split the phase plane into two regions. Each of the regions is specified with a switching state to direct the phase trajectory towards the sliding line. When the phase trajectory reaches the sliding line and tracks it towards the origin, the system is considered to be the stable i.e X1=X2=0. When the trajectory is within a small vicinity of the sliding surface, it is said to be in SM operation. The SM controller will give a series of control actions through switching, to maintain the trajectory within a small vicinity of the sliding surface. When the trajectory is at any position, above the sliding line i.e S(X)=0, the switching function u=0 must be employed so that the trajectory is directed towards the sliding line. On the other hand, when the phase trajectory is at any position below the sliding line, u=1 must be employed for the trajectory to be directed towards the sliding line. This forms the basis for the control law and is given as:

when S ( X ) > 0 when S ( X ) < 0

if S ( X ) < 0

SMC is obtained by means of control strategy of (25), which relates to the status of the switch u with value of S(x). The existence condition stated in (28) can be expressed in the following form, taking consideration of the value of switching function u. Using (17), (25) can be written as:

where, K T = [ K1

⎪⎧0 = OFF u=⎨ ⎪⎩1 = ON

if S ( X ) > 0

+ [ K1

S ( X ) = [ K1

⎡ −Vcref ⎤ ⎢ ⎥ L T ⎥+ K2 ] ⎢ ⎢ iLref Vcref ⎥ − ⎢ ⎥ RL C ⎦ ⎣ C ⎡ Vi ⎤ T ⎢ ⎥ K2 ] L ⎢ ⎥ ⎢⎣ 0 ⎥⎦

K2 ]

T

⎡ −Vcref ⎤ ⎢ ⎥ L ⎢ ⎥ ⎢ iLref Vcref ⎥ − ⎢ ⎥ RL C ⎦ ⎣ C

(35)

(36)

(27)



653


⎡ K1Vi K1Vcref − + ⎢ L L ⎢ S(X ) = ⎢ K2 ⎢ + R C RL iLref − Vcref ⎣ L

(

)

⎤ ⎥ ⎥ > 0 , S ( X ) < 0 (37) ⎥ ⎥ ⎦

⎪⎧0 = OFF u=⎨ ⎪⎩ 1 = ON

when when

S(X ) >δ

S ( X ) < −δ

(39)

where, δ is an arbitrarily small value. A hysteresis band S(X)= δ and S(X)=-δ with the boundary conditions is introduced to overcome the chattering effect. This is to provide a form of control to the switching frequency of the converter [8]. With this modification, the operation of converter is altered such that if the parameter of the state variables are such that S(X)>δ, the switch SW of the converter is turn off and it will turn on when S(X 0 (38) S ( X ) = ⎢ ⎢ ⎛ iLref Vcref ⎞ ⎥ − ⎢+ K2 ⎜ ⎟⎥ RL C ⎠ ⎦⎥ ⎝ C ⎣⎢

The above inequalities give the conditions for existence and, therefore provide the range of employable sliding coefficients K1 and K2 that will ensure that the converter stay in SM operation when its state trajectory is near the sliding surface. No other information relating the sliding coefficients to the converter performance can be drawn. If the SM is to exist in the system defined by (11), the sufficient condition is that K1 and K2 need to be nonnegative. III.3. Introduction of Hysteresis Band Ideally, a converter will switch at infinite frequency with its phase trajectory moving on the sliding line when it enters SM operation (Fig. 3(a)). But, practically, this is not possible due to switching imperfections such as switching delay, switching time constant etc [8]. The discontinuity in the feedback control will produce a particular dynamic behavior in the vicinity of surface trajectory known as Chattering (Fig. 3(b)) [8],[13],[14]. The converter will become oscillatory, if the chattering is left uncontrolled [8], particularly at high switching frequency corresponding to chattering dynamics. This is undesirable as high switching frequency will result in excessive switching loss, inductor and transformer core losses and EMI noise [8].

III.4. Calculation of Maximum Switching Frequency Control of the switching frequency of the converter, requires that the relationship between the hysteresis band 2δ and the switching frequency must be known. Fig. 4 shows the view of the phase trajectory when it is operating in SM.

Fig. 4. Phase trajectory of Practical SM operation

The vectors of state variable velocity for u=0 and u=1 are shown. ∆t1 and ∆t2 are the time taken by state variable velocity vector to move from position x to y, and from position y to z as given by (46) and (47) respectively. T denotes the switching period i.e is the time period for one cycle. Since switching period is inversely propositional to the, switching frequency fs, then:

Figs. 3. Phase trajectory for (a) ideal SM operation and (b) practical SM operation with chattering

Furthermore, since chattering is introduced by the imperfection of controller ICs, gate driver, and power switches, it is difficult to predict the exact switching frequency [8]. Hence, the design of the converter and the selection of the components will be difficult. Control law of (26) to solve these problems, is redefined as:

T = ∆t1 + ∆t2

(40)

and: fs =


1 ( ∆t1 + ∆t2 )

(41)


654


From Fig. 4:

S ( X ) = S ( X ) ∆t1

(42)

2δ = S ( X ) ∆t1

(43)

or:

∆t1 =

2δ S(X )

(44)

∆t2 =

−2δ S ( X )

(45)

f max =

IV.

⎡ K1Vi K1Vcref ⎤ K − + 2 RL iLref − Vcref ⎥ ⎢ L RL C ⎣ L ⎦

∆t2 =

(

)

−2δ ⎡ K1Vcref ⎤ K2 + RL iLref − Vcref ⎥ ⎢− L RL C ⎣ ⎦

(

(46)

(47)

)

1) Selection of Vcmax Vcref ( max ) = Vc max = V0 +

( ∆t1 + ∆t2 ) = 2δ

=

⎡ K1Vi K1Vcref ⎤ K − + 2 RL iLref − Vcref ⎥ ⎢ L L R C L ⎣ ⎦ −2δ + ⎡ K1Vcref ⎤ K2 + RL iLref − Vcref ⎥ ⎢− L RL C ⎣ ⎦

(

(

)

Standard Design of SM Controller for Buck Converter

The SM parameters using previously deduced equations for the buck converter with the following specifications, are as under: - Output voltage V0 = 2.5 V - Voltage ripple Vrpp =2% of V0 = 50 mV - Steady state output current I0 =1.25 A - Current ripple ILpp = 20% of I0 = 0.25 A - Maximum switching frequency fmax = 200 kHz. - Inductor value L = 10uH (ESL=15 mΩ) - Capacitor C = 47 uF (ESR= 2mΩ) - Hysteresis band ± δ = ±0.3 - Load resistance RL=2 ohms

Using (37) and (38) respectively, the ∆t1 and ∆t2 are given by: 2δ

+

I Lref ( max ) = I Lref = I 0 +

f max =

Vc max K1 ⎡Vi − Vc max ⎤ ⎢ ⎥ 2δ L ⎣ Vi ⎦

= 2.525 V

(52)

I Lpp 2

= 1.357 A

(53)

K1 ratio L For specified value of fmax, δ, Vcmax Vcref ,Vi and using K (51). The 1 ratio is given by : L K1 = 96009 L

(54)

For specified value of L, K1 is given by: K1 = 0.96

voltage reference crossing its maximum value Vc max . The max. value of frequency is obtained as:

Vcref

2

3) Determination of

Determine maximum frequency, requires determining Tmin the minimum time period for one cycle. The minimum time period for one cycle for which the phase trajectory moves from position x to z is given by (49). The expression for maximum value of frequency is obtained with the assumption that converter is operating 1 in no load condition ( iL = 0 and = 0 ) and the output RL

2δ Vi ⎛ K1Vi K1Vcref ⎞ − ⎜ ⎟ L ⎠ ⎝ L

Vrpp

2) Selection of ILref

(48)

)

Tmin = ( ∆t1 + ∆t2 ) =

(51)

Since, the cycle is repeated (cyclic) through out the SM operation, the maximum frequency of converter when it operating in SM can be expressed as (51).

Similarly:

∆t1 =

Vc max K1 ⎡ Vc max ⎤ ⎢1 − ⎥ Vi ⎦ 2δ L ⎣

K2 ratio C Taking ILref (max) =ILmax and using equations (37) and K2 is given by (55), and the (38), the range of C corresponding range for K2 for specified value of C, is given by (56):

4) Determination of

(49)

(50)



655


−2091078
0,∀λ ∈ σ ( A ) ,∃φ ∈ N ( A − λ I )

is r.w.g.s (r.s.g.s, r.e.g.s) on ω. From the previous definition, we have the following points: • If the system (1) is regionally gradient stabilizable on ω ⊂ Ω , then it is regionally gradient stabilizable on ω ' using the same control, for any ω' ω. • Stabilizing regionally a gradient of system is cheaper than: 1. Stabilizing it on the whole domain. 2. Stabilizing regionally the state in the same region. Indeed if we consider the following cost functional: q (v) =

Characterizations

(7)

Re λ t = e ( ) ∇ω φ

n

≥ ∇ω φ

2 n

2 n

>0

where:

Hence the system (2) isn't r.w.g.s on ω.

{

v ∈ Uad (ω ) = v ∈ L2 ( 0 , +∞ ;U ) ;v stabilizes

2) For every z0 ∈ H 1 ( Ω ) , we have:

the gradient of (1) on ω and q ( v ) < ∞}

then we have:

∇ω S ( t ) z0 = min q ( v ) ≥ min q ( v )

Uad (ω )

Uad ( Ω )

∑

eλn t

n≥0

rn

∑

z0 ,φnk ∇ωφnk

k =1

where rn is the multiplicity of the eigenvalue λn , then:

The second point is immediate since every control which stabilizes regionally the state, stabilizes also regionally the gradient: • Regional gradient stabilization problem may be seen as a special case of an output stabilization problem with partial observation y = ∇ω z .

∇ω S ( t ) z0

n

≤ e−α t z0

So the system (2) is r.e.g.s on ω. As illustration of the above result, we consider the example 2.3.

( )

We have ∀i, j ∈ ` ,∇ωϕij ≠ 0 ⇒ Re λij ≤ −π 2 then



757

E. Zerrik, Y. Benslimane, A. El Jai

1 − e−2 µt ∇ω S ( t ) z 2µ

the system (5) is r.e.g.s. The following result gives conditions ensuring the regional gradient exponential stability and a bounded state on the whole domain. Proposition 2.6 Assume that

(

M ω ∈ L 0, +∞; \ 2

t

≤ ∫e

−2 µ ( t − s )

t

= ∫e

2

n

−2 µ ( t − s )

∇ω S ( t ) z n ds 2

0

‖ ∇ω S ( t − s ) z ‖n2 M ω2 ( s ) ds

0

there

+

exists

) such that:

a

function

≤ C32 z

2

for some C3 > 0

Thus: ∇ω S ( t + s ) ≤ M ω ( t ) ∇ ω S ( s )

∀t ,s ≥ 0

(8)

∇ω S ( t ) z

and ∇ω S ( mt ) ≤ ∇ω S ( t )

m

∀t ≥ 0 ∀m ∈ ` *

≤ C z ∀t ≥ 0,z ∈ H 1 ( Ω )

n

(11)

for some C>0. We will show that:

(9)

w0 := inf

Then the system (2) is r.e.g.s if:

ln ||| ∇ω S ( t ) |||

t >0

ln ||| ∇ω S ( t ) |||

= lim

t →+∞

t

t

0

2

Using the uniform boundedness principle we obtain for all m ∈ I N * ||| Rm |||≤ ξ for some ξ independent of m. In addition there exist C1 > 0 and µ > 0 such that

implies

that

t∈[ 0 ,t1 ]

there exists

t≥0

which

implies

( ∇ω S ( t ) z )

n t ≤t1

≤

ln ||| ∇ω S ( mt1 ) ||| ln S ( t − mt1 ) + t t

With (9) we have: ln ||| ∇ω S ( t ) ||| t

≤

mt1 ln ||| ∇ω S ( t1 ) ||| ln N 2 + t t1 t

that

it follows that:

≤ C2 e µt z for some C2 > 0 .

The operators

which

t

N 2 = sup S ( t ) ,

and

ln ||| ∇ω S ( t ) ||| t

n

N1 z

m ∈ I N such that mt1 ≤ t < ( m + 1) t1 for each t ≥ t1 ,

0

∇ω S ( t ) z

≤

n

2

N12 z , for some N1 > 0

then:

m

for

2

2

w0 < 0 ∀t ≥ t0 for some t0 > 0.

and

am = ∫ ∇ω S ( t ) z n dt

S ( t ) z ≤ C1e µt z

∫ M ω ( s ) ∇ω S ( t − s ) z n ds 0

n L ⎛⎜ [ 0 , +∞[ ; L2 (ω ) ⎞⎟ , where: ⎝ ⎠ 2

(12)

are bounded for some

lim sup t →∞

t1 > 0 . For t > t1 , we calculate:

ln ||| ∇ω S ( t ) ||| t

≤ inf

ln ||| ∇ω S ( t ) |||

t >0

≤ lim inf t →∞


t ln ||| ∇ω S ( t ) ||| t


758


+∞

then (12) is satisfied. Hence for all α ∈ ]0 , −ω0 [ , there exists N3 such that:

By (2)-(14),

∫ || ∇ω S ( s ) z0 ||n ds < ∞ and from (15), 2

0

we have:

|| ∇ω S ( t ) z ||n ≤ N3e−α t z , ∀z ∈ H 1 ( Ω ) , t ≥ 0

∂ || ∇ω S ( t ) z0 ||n2 ≤ 0 ∂t

So the system is r.e.g.s. The converse is immediate. Then :

Corollary 2.7 Suppose that (8) and (9) are verified. If in addition ||| ∇ω S ( t0 ) |||< 1 for some t0 > 0 , then

t

t || ∇ω S ( t ) z0 ||n2 = || ∇ω S ( t ) z0 ||n2 ds ≤

∫

(2) is r.e.g.s on ω .

0 t

≤ || ∇ω S ( s ) z0 ||n2 ds

∫

Proof From the proof of proposition 2.6, we have ln ||| ∇ω S ( t ) ||| then w0 < 0. Thus the system w0 = lim t →+∞ t (2) is r.e.g.s. The following result gives sufficient condition for regional gradient strong stability.

0

We deduce: || ∇ω S ( t ) z0 ||n2 ≤

Proposition 2.8 Suppose there exists a self adjoint positive operator

(

0

t. Then lim ∇ω S ( t ) z0 t →+∞

< Az,Pz > + < Pz, Az > + < Rz,z >= 0 , z ∈ D ( A ) (13)

n

= 0.

Since D ( A ) is dense in H 1 ( Ω ) , and the function z0 6 f ( z0 ,t ) then (16) holds for all z0 ∈ H 1 ( Ω ) . Thus

)

where R ∈ L H 1 ( Ω ) is a self adjoint operator such that:

(2) is r.s.g.s.

(14)

III. Regional Gradient Stabilizability

for some α > 0 . Moreover if the following condition holds: Re ( < Gω Az,z > ) ≤ 0 , z ∈ D ( A )

, t > 0 , z0 ∈ D ( A ) (16)

t

)

〈 Rz,z 〉 ≥ α ‖ ∇ω z ‖2n ∀z ∈ H 1 ( Ω )

t

with f ( z0 ,t ) = ∫ || ∇ω S ( s ) z0 ||n2 ds which is bounded in

P ∈ L H 1 ( Ω ) such that:

(

f ( z0 ,t )

The aim of this section is to find a control which stabilizes regionally the gradient of system (1). With the same assumption on A and B, let denote by S k ( t ) the (15)

strongly continuous semigroup generated by the operator A + BK on Z = H 1 ( Ω ) .

then (2) is r.s.g.s on ω . Proof Let V ( t,z ) = < Pz,z > for every z ∈ H 1 ( Ω ) and

III.1. Decomposition Approach In the following we show that stabilizing the gradient of system (1) on a subregion ω turns up to stabilizing the gradient of a finite dimensional system. Let δ > 0 be fixed and consider the subsets of the spectrum σ ( A) of A, defined as

t ≥ 0 . For z0 ∈ D ( A ) , we have:

d V ( t ,S ( t ) z0 ) =< PAS ( t ) z0 ,S ( t ) z0 > + dt + < PS ( t ) z0 , AS ( t ) z0 >=

σ u ( A ) = {λ ; Re ( λ ) ≥ −δ } , σ s ( A ) = {λ ; Re ( λ ) < −δ } . If the set σ u ( A ) is bounded and is separated from the

= − < RS ( t ) z0 ,S ( t ) z0 >

set σ s ( A ) in such a way that a rectifiable simple closed

which gives:

curve may be drawn so as to enclose an open set containing σ s ( A ) in its interior and σ u ( A ) in its

+∞

∫ ds ≤ V ( 0,z0 )

exterior, (for example A is self adjoint with compact resolvent, there are at most finitely many nonnegative eigenvalues of A, each with finite dimensional

0



759


eigenspace), then the state space Z may be decomposed (see [8]) according to: Z = Zu + Z s

(1) is regionally exponentially (resp strongly) gradient stabilizable on ω using the feedback operator K = ( Ku ,0 )

(17)

Proof We give the proof for the strongly case. In view of the above decomposition, we have sup Re (σ ( As ) ) ≤ −δ . Hence if As satisfies (21) then for

with Z u = PZ , Z s = ( I − P ) Z , and P ∈ L ( Z ) is the 1 ( λ I − A)−1d λ , ∫ c 2π i where C is a curve surrounding σ s ( A ) . projection operator giving by P =

some M 1 and β ∈ ]0 ,δ [ , we have:

Then the system (1) may be decomposed into: ⎧ ∂zu ( t ) = Au zu ( t ) + PBv ( t ) ⎪ ⎪ ∂t ⎨ z = Pz ⎪ u ⎪⎩ z0u = Pz0

∇ω S s ( t ) ≤ M1e− β t , t ≥ 0 (18)

It follows that the system (19) is regionally exponentially gradient stabilizable taking the null control. We have zu ( t ) = e Fu t z0u , with

⎧ ∂zs ( t ) = As zs ( t ) + ( I − P ) Bv ( t ) ⎪ ⎪ ∂t ⎨z = I − P z ) 0 ⎪ 0s ( ⎪ zs = ( I − P ) z ⎩

Fu = Au + PBK u ∇ω ∈ L ( Z u ) , and:

(19)

lim ∇ω zu ( t )

t →+∞

t > t0 , v ( t ) U
0 ∃t0 > 0 such that for:

where As and Au are the restrictions of A on Z s and

lim

(22)

∫

(21)

=0:

t0

− β t −τ − β t −τ e ( ) v (τ ) U = e ( ) v (τ ) U dτ +

∫

0

0 t

1) If the system (18) is regionally exponentially (resp strongly) gradient stabilizable on Ω by a feedback n control v = K u ∇ω zu , with Ku ∈ L ⎛⎜ L2 (ω ) ,U ⎞⎟ , then ⎝ ⎠ the system (1) is regionally exponentially (resp strongly) gradient stabilizable on ω using the control v = ( v, 0 ) .

(

n

− β t −τ + e ( ) v (τ ) U

∫

)

t0

ε − β t −t ≤ De ( 0 ) + n 2 t

∫e

2) If the system (18) is regionally exponentially (resp strongly) gradient stabilizable on Ω by the feedback control v = K u zu , with K u ∈ L ( Z u ,U ) , then the system

− β ( t −τ )

v (τ ) U dτ → 0 as t → +∞

0



760


< Gω z,z >≥ dRe < ( A + BK ) z,z > ,z ∈ D ( A) ,

1

t ⎞2 2 1 ⎛0 ⎜ v ( t ) dt ⎟ U ⎟ 2β ⎜⎝ 0 ⎠

Hence lim ∇ω zs ( t ) t →+∞

n

then the state of system (1) remains bounded in Ω \ ω .

= 0.

Finally the gradient of system (1) excited by v ( t ) ∇ω z ( t )

satisfies

n

(27)

for some d > 0

∫

where D =

≤ ∇ω zu ( t ) n + ∇ω zs ( t )

n

Proof Applying the results of proposition 2.8 to system (6), we obtain the first and the second results of the proposition. 3. For z0 ∈ D( A ), we have:

which

shows that system (1) is regional strongly gradient stabilizable on ω . 3) It follows from similar above techniques.

Re ( 〈( A + BK ) z ( t ) ,z ( t )〉 ) =

Corollary 3.2 Let A satisfy the spectrum decomposition assumption and suppose (21) verified, if in addition 1. Z u is a finite dimensional space, 2. the system (18) is controllable on Z u then system (1) is regionally exponentially gradient stabilizable on ω .

∫ ∇ω z ( s ) n ds ≥ 2 ( z ( t ) t

and from (27)

1 ∂ z (t ) 2 ∂t

d

2

2

− z0

2

2

).

0

Since the system is regionally exponentially gradient stabilizable on ω then

+∞

∫

∇ω z ( t ) dt < +∞ , so there 2

0

Proof The system (18) is controllable in finite dimension space Z u then it's stabilizable, then it's regionally stabilizable on ω , hence it's regionally gradient stabilizable on ω .

exists C > 0 such that z ( t ) ≤ C, for all t ≥ 0 and the conclusion by the density of D ( A ) in Z. Remarks 3.4 1. If the operator ∇Ω

III.1.1. Riccati Approach

ω

is continuous then the third

point of the proposition implies that the gradient also remains bounded on the residual part Ω ω 2. If the operator ∇ Ω ω isn't continuous we can

Let us consider the system (1) with the same assumptions on A and B. Let R ∈ L ( Z ) be a self adjoint operator which satisfy

replace the condition (27) by the condition:

(14) and consider the steady state Riccati equation: < Gω z,z >≥ d

< Az,Pz > + < Pz, Az > + + < Rz,z > − < B* Pz,B* Pz >= 0,

(t )

∫ ‖ ∇ω z ( s ) ‖ ds ≥ d ( ∇Ω t

2

verifies the conditions (8) and (9) then

< Gω ( A + BK ) z,z > ≤ 0, z ∈ D ( A )

ωz

(t )

2

− ∇Ω

0

system (1) is regionally exponentially gradient stabilizable on ω using the control v ( t ) = Kz ( t ) . 2. If:

> ,z ∈ D ( A )

(28)

to obtain that the gradient remains bounded on the residual part. Indeed: For z0 ∈ D ( A ) , from (28) we have:

Proposition 3.3 Suppose there exists a self adjoint positive operator P ∈ L ( Z ) satisfying equation (25) and let K = − B* P

1. If S

ω z,z

for some d > 0

(25)

z ∈ D ( A)

K

d < GΩ dt

ω z0

2

)

From the regional exponential gradient stabilizability on ω we have ∇Ω ω z ( t ) ≤ C, for some C > 0 , t ≥ 0

(26)

and the conclusion by the density of D ( A ) in H 1 ( Ω ) . Now, we will give an application of the above result to conservative systems. For this, let U = H 1 ( Ω ) and

then (1) is regionally strongly gradient stabilizable on ω. 3. Suppose that system (1) is regionally exponentially gradient stabilizable on ω , if in addition the feedback operator K verifies:

R = BB* , so the operator P = I is a solution of the following equation:

〈 Az,Pz 〉 + 〈 Pz, Az〉 = 0 , ∀z ∈ D ( A )


(29)


761


Suppose that ‖ B* z ‖≥ d ‖ ∇ω z ‖ for some d > 0, then we have the following result:

where: q (v) =

Corollary 3.5 If Re〈Gω ( A + BK ) z,z〉 ≤ 0 , z ∈ D ( A ) is verified,

+∞

0

0

2 ∫ 〈 Rz ( t ) ,z ( t )〉 dt + ∫ v ( t ) dt

{

then the feedback v ( t ) = − B* z ( t ) stabilizes the system

(32)

}

and Uad = v ∈ L2 ( 0, +∞;U ) ;q ( v ) < +∞ .

gradient (1) strongly on ω .

Let R be a linear bounded operator mapping H 1 ( Ω ) into itself. It is known (see [4]) that if U ad ≠ ∅ for each initial

Example 3.6 Let Ω = ]0 ,1[ , we consider the following system

state z0 , and R is coercive ( 〈 Ry, y〉 ≥ α y , ∀y ∈ H 2

equation: ⎧ ∂z ( t ) = ∆z ( t ) + Gω v ( t ) ⎪i ⎪⎪ ∂t ⎨ ∂z ( 0,t ) = ∂z (1,t ) ⎪ ∂x ∂x ⎪ 1 ⎪⎩ z ( x, 0 ) ∈ H ( Ω )

( x,t ) ∈ ]0, +∞[ × Ω

for some α > 0 ) then there exists a unique control v* solution of (31) and stabilizes exponentially the system (1), this control is given by:

(30)

t ∈ ]0, +∞[

v* ( t ) = − B* Pz ( t )

where i = −1, and ω = ]0,a[ with 0< a 0 , subregion ω , D and N is the dimension of the projection space(large enough) Step 2: giving a time sequence (ti )i ≥1 , ti +1 = ti + δ , with δ > 0 small enough for numerical consideration Step 3: solving (32) using Newton-Kleinman algorithm gives PN

Π N : H1 (Ω) → Z N N

z 6 ∑ 〈 z,ϕi 〉 H 1 ( Ω ) ϕi ( x ) i =0

Thus: for z ∈ H 1 ( Ω ) , lim Π N ( z ) − z = 0

Step 4:Until & ∇ω z N (ti ) &< ε repeat ||solving (33) which gives z N (ti ) .

N →∞

then lim PN Π N z − Pz = 0 for each z ∈ H 1 ( Ω ) , that is N →∞

PN Π N converges to P strongly in H 1 ( Ω ) , (see [10]).

IV.3. Stimulation

The projection of equation (25) on Z N gives: PN AN + A*N PN − PN BN B*N PN + RN = 0

Example 4.2 Let Ω = ]0 ,1[ , we consider the system equation:

(36)

⎧ ∂z ( t,x ) = ∆z ( t,x ) + z ( x,t ) + χ D v ( t,x ) ⎪ ⎪ ∂t ⎪ ∂z t, 0 ⎨ ( ) = ∂z ( t ,1) = 0 ⎪ ∂x ∂x ⎪ 2 3 ⎪⎩ z ( x, 0 ) = x (1 − x )

where AN , BN , PN and RN are respectively the projections of A, B, P and R on Z N . N

Now z N ( t ) = ∑ α i ( t ) ϕi with z N ( t ) = Π N ( z ( t ) ) , i =0

which implies that

∂z N ( t ) ∂t

=

AN z N

D is an open set of Ω and, v ( t ) ∈ H 1 ( Ω ) ∀t ≥ 0 .

i =0

N

Let us consider the functional cost (32) with Rω , is

N

∑ αi ( t ) ∑ 〈 Aϕi ,ϕ j 〉ϕ j = i =0

=

N

= ∑ αi ( t ) ϕi , and:

N

the operator mapping H 1 ( Ω ) into itself, defined by:

j =0

⎧ z ( x ) if x ∈ω Rω ( z ) = ⎨ 0 else ⎩

N

∑∑ αi ( t ) 〈 Aϕi ,ϕ j 〉ϕ j j =0 i =0

BN v ( t ) = − BB* PN z N ( t ) = =

N

N

N

∑ αi ( t ) ∑∑ ∑ 〈 Bϕk ,ϕ j 〉〈 B*ϕl ,ϕk 〉〈 PN ϕi ,ϕl 〉ϕ j i =0 N

=

N

and satisfying (14). Let Z N be the subspace of H 1 ( Ω ) spanned by

{ϕi / i = 0,..., N }

l =0 k =0 j =0

N

N

N

βi =

∑∑∑∑ αi ( t ) 〈 Bϕk ,ϕ j 〉〈 B*ϕl ,ϕk 〉〈 PN ϕi ,ϕl 〉ϕ j j =0 i =0 l =0 k =0

2 1 + ( iπ )

2

where

ϕi ( x ) = βi cos ( iπ x )

with

.

Applying the previous algorithm, taking ω = ]0.3, 0.8[ , and D = ]0, 0.3[ , we have the following results.

Then solve the system (1) turns up solve the differential system:



763


Let's fixed ω = ]0, 0.3[ the following results establish a link between control area and the cost of gradient stabilization on ω (Table I). TABLE I CONTROL AREA-STABILIZATION COST D ]0.3,0.4[ ]0.3,0.5[ ]0.3,0.6[ ]0.3,0.7[ Cost 118.02×10-2 118.0×10-2 118×10-2 117.98×10-2 D Cost

]0.3,0.8[ 117.94×10-2

]0.3,1[ 117.92×10-2

The gradient stabilizing cost increases as the control area support decreases. There is a relation between the gradient stabilization cost, stabilization error, and the area of the target region ω as shown in the following tabular. Let's fixed D = ]0 .8 ,1[ (Table II).

Fig. 1. Flux evolution at t=0

TABLE II REGION AREA-STABILIZATION ERROR AND STABILIZATION COST ]0,0.3[ ]0,0.4 ]0,0.6[ ]0,0.7[ ]0,0.9[ ]0,1[ ω Error (×10-9)

6.6039

6.7428

7.6575

8.5953

9.1983

9.3011

Cost (×10-2)

117.83

135.17

155.13

159.52

162.03

165.18

This shows that more the area of subregion increases more the error and the cost increase. Example 4.3 Let Ω = ]0 ,1[ , we consider the state space system

equation:

Fig. 2. Flux evolution

⎧ ∂z ( t,x ) = ∆z ( t,x ) + χ D v ( t,x ) ⎪ ⎪ ∂t ⎪ ∂z t, 0 ⎨ ( ) = ∂z ( t ,1) = 0 ⎪ ∂x ∂x ⎪ 2 2 ⎪⎩ z ( x, 0 ) = x ( x − 1) cos ( 0.5π x )

Let's take ω = ]0 , 0.45[ , D = ]0, 0.8[ and applying the same process we have the following Fig. 4. The cost of the regional gradient stabilization is 4.53 ×10−2 and the error is 9.9659 ×10−6 . There is a relation between the gradient stabilization cost, error of this stabilization, and the area of the target region. Also there is relation between the cost of the gradient stabilization and the control support, we summarize these results in the following Table III. This shows that the area of subregion increases as the error and the cost increase. Let's fixed ω = ]0 ,0.45[ .

Fig. 3. Flux evolution

It is clear that the gradient is stabilized on ω = ]0.3, 0.8[ with error 3.7933 × 10−9 and cost equal 14.99 × 10−2 .



764


[3]

R. F.Curtain,A. J. Pritchard, Infinite Dimensional Linear Systems Theory (Springer-Verlag Berlin 1978). [4] R. F. Curtain, H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory (Springer-Verlag Berli , 1995). [5] A. J. Prichard, J. Zabczyk, Stability and Stabilizability of Infinite Dimensional Systems, Siam Review, Vol. 23(Issue 1): 25-51, 1981. [6] R. Triggiani, On the Stabilizability Problem in Banach Space, J. Math. Anal. Appl, Vol. 52(Issue 3): 383-403, December 1975. [7] E. Zerrik, M. Ouzahra, Regional stabilization for infinitedimensional systems, Int.J.Control, Vol.76(Issue 1): 73-81, 2003. [8] T. Kato, Perturbation theory for linear operators (SpringerVerlag Berlin 1980). [9] E. Zerrik, Y. Benslimane, An output stabilization problem of distributed linear systems Approaches and simulations, submitted in International Journal of Dynamical and Control System. [10] H. Banks, K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. control Optim, Vol. 22 684-696, 1984. Fig. 4. Flux evolution

Authors’ information 1,2 Moulay Ismail University, Faculty of sciences.

TABLE III REGION AREA-STABILIZATION ERROR AND STABILIZATION COST ]0,0.17[ ]0,0.33[ ]0,0.36[ ]0,0.45[ ]0,0.78[ ω Error

9.9530

9.9534

9.9609

9.9659

3

Yassine Benslimane was born in Meknes, Morocco, in 1982. he is a researcher student at "Mathematics for analysis and control of systems" (Macs), preparing his PHD in systems theory at Moulay Ismail University of Meknes in Morocco. His research is focused on analysis and stability and stabilization of distributed systems. E-mail: [email protected]

(×10−6 )

Cost

3.68

4.26

4.3

4.53

6.4

(×10−2 )

D Cost

TABLE IV CONTROL AREA-STABILIZATION COST ]0,0.75[ ]0,0.7[ ]0,0.65[ ]0,0.6[ ]0,55[ 4.74 5 5.37 5.87 6.38

Perpignan University.

9.9729

]0,0.5[ 6.76

(×10−2 )

More the control area support more the gradient stabilizing cost increases decreases.

V.

Conclusion

In this work we characterize controls that stabilize gradient of a distributed system on a subregion of the system domain. Also we give examples and simulations that illustrate different established results. Various questions remain open. This is the case where the subregion is a part of the boundary of the system domain. Also the case of bilinear system is of a great interest. The work is under consideration.

Acknowledgements The work presented here was carried out within the help of the Academy Hassan II of Sciences and Technology.

References [1]

[2]

A. V. Balakrishnan, Strong Stability and the Steady State Riccati Equation, Applied Mathematics and Optimization. Vol. 7, (Issue 1):335-345. March 1981. C. D. Benchimol, Feedback Stabilizability in Hilbert Spaces', Appl. Math. Optim., Vol. 4(Issue.1):209-223.



765


Design and Analysis of SVC Complementary Controller to Improve Power System Stability Using RCGA-Optimization Technique A. D. Falehi

Abstract – In this paper a complementary damping controller is proposed for static VAR compensator (SVC). Power system stability improvement using this damping controller has been perused. The Real Coded Generic Algorithm (RCGA) which has been known to be immediately perceptive, well performing, and capable to impressively solve highly non-linear objective, has been applied to the proposed controller based on an optimization problem to achieve optimal controller parameter of SVC. In this paper both local and remote signals with associated time delays are considered and these signals have been compared together. The SVC-based damping controller is evaluated under severe disturbances for single-machine infinite-bus & multi-machine (two-machine) power system. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: SVC-Based Damping Controller, Transient Stability, Remote Signal, Local Signal, RCGA Optimization Technique, Single-Machine Infinite-Bus Power System, MultiMachine Power System

ω1 and ω2

Nomenclature SVC TCR TSC RCGA HTG FCT TS1 , TS 2 ,

Static Var Compensator Thyristor Controlled Reactor Thyristor Switch Capacitor Real Coded Generic Algorithm Hydraulic Turbine and Governor Fault Clearing Time SVC time constants

tsim LS RS

I.

N n ( 0,1)

SVC gain SVC washout block SVC voltage signal Number of variables Variable n Highest number in the variable range Lowest number in the variable range Normalized value of variable Mother chromosome Father chromosome nth variable in the mother chromosome nth variable in the father chromosome Parameter where crossover occurs Mixing value for continuous variable crossover Standard deviation of the normal distribution Standard normal distribution

∆ω

Speed deviation of rotor

TWS VS N var Pn Phi Pli Pnorm m d pmn

Pdn

α β σ

Introduction

Recent advances in the field of Power Electronics prepare an appropriate bed in order to wide utilization of Flexible AC Transmission System (FACTS) devices in power system [1]. FACTS devices have high ability to control network status in unexpected & rapid events and this particular feature increased the power system transient stability. Shunt FACTS devices with compensation of reactive power, plays an important role in controlling active power in power grid, beside of enhancement of voltage fluctuation and increasing the transient stability in it [2], [3]. SVC is a shunt FACTS device that can absorb/or inject reactive power to the network. Such devices increase both steady state capacity and dynamic performance of system, now been used in power system widely [2]. Different techniques and controllers have been presented to improve transient stability and reducing oscillations by FACTS devices [4]-[6], [8]. FACTS also been introduced as a key device for damping power system swings [9], [10]. Optimum location of FACTS devices is essential to increase the power system stability and decrease the losses [11]-[14]. A number of conventional techniques have been applied to design power system stabilizers, for example:

TS 3 and TS 4 KS

Speed of generator G1 and G2, respectively Time range of the simulation Local Signal Remote Signal


766


A. D. Falehi

mathematical programming, gradient process for optimization and modern control theory. Unluckily, the conventional techniques necessitate heavy computation along with slow convergence. In addition, the search process is likely to be trapped in local minima and the result may not be optimum. The progressive methods build up a technique to search for the optimum solutions via some of directed random search processes. An advantage of the evolutionary methods is to reach the result without prior problem perception. In recent years, computation techniques based on Genetic Algorithms (GAs) have been applied as optimization implements in various probe areas, including the design of controllers [15], [16] and solution of optimum location of FACTS device [17]-[22]. The GAs has been used for coordinated design of PSS and FACTSs [23], [24] and the design of FACTS controllers [25], [26] to enhance the power system stability. For design of a drastic damping controller a proper input signal initially must selected. When a disturbance occurs in the power system by means of input signal, power system stability improvement can be determined. To design of damping controller, studies are based on either local signal or remote signal [6], [7], [27]. Line active power as local signal and speed deviation as remote signal are considered to be input signals for complementary proposed damping controller of SVC. Also relation of signal transmission delays in power system stability improvement thoroughly investigates. By using of the RCGA optimization technique optimal controller parameter of SVC by reduction of cost function or minimizing power system swings will be obtained. Efficiency of proposed damping controller under severe disturbance in single-machine infinite-bus power system and multi-machine power system will be assessed. In addition, obtained results from local and remote signals have been compared and analyzed.

II. II.1.

control, increasing system loading capability, and enhancing power system stability.

Fig. 1. Single-machine infinite-bus power system with SVC-based damping controller

II.2.

Multi-Machine Power System with SVC

To evaluate proposed SVC based damping controller and optimization of controller parameters, two-machine power system considered which is presented in Figure. 2. It is almost similar to the power system used in Ref [11]. This system consists of two generators, connected via 500km transmission line. The generator is equipped with HTG and excitation system. All of the other relevant parameters are given in Appendix B.

System Model Fig. 2. Multi-machine power system with SVC-based damping controller

SMIB Power System with SVC

The model of the single-machine infinite-bus system under study has been created using MATLAB/SIMULINK environment that is shown in Figure 1. It is almost similar to the power system used in Ref [28], [29]. The system contains a synchronous generator through a transformer and a SVC and a double circuit transmission line connected to an infinite-bus. The generator is equipped with HTG and excitation system [30]. All of the other relevant parameters are given in Appendix A. Generally SVC is made from the TCR and TSC bank that is connected in parallel to networks to be treated [3]. SVC actually acts as a variable reactance to maintain or control specific power system variables; typically the bus voltage [6]. It is used extensively to provide fast reactive power and voltage regulation support. The main reasons for installing a SVC are to improve dynamic voltage

III. Proposed Approach III.1. Structure of SVC-Based Damping Controller A lead-lag structure is presented as complementary damping controller for SVC to regulate the VS, which is exhibited in Figure 3. This structure comprised of a delay block, a gain block with gain KS, a signal washout block and two-stage phase compensation blocks. The phasecompensation blocks provide the proper phase-lead characteristics to compensate the phase lag between input and the output signals. Time delay is applied due to delay block which is depended on the type of input signal. For local input signals only the sensor block and for remote signals both sensor block and the signal transmission delays are involved. The signal washout



767

A. D. Falehi

block with the time constant TWS is used as a high-pass filter which allows associated oscillator signal in input signal to pass unaltered. Commonly the value of TWS has predefined in literatures [27], [30].

J2 =

t =tsim

∫t =0

ω1 − ω2 ⋅ t ⋅ dt

For multi machine

(3)

The time-domain simulation of the nonlinear system model has been performed. It is aimed to minimize this objective function in order to improve the system response, in terms of the settling time, undershoot and overshoots. The problem constraints are the SVC controller parameter bounds. Therefore, it can be formulated as the following optimization problem:

Fig. 3. Structure of proposed complementary damping controller

Minimize J III.2. Problem Formulation

subject to:

The value of TWS 10s is considered in this study and parameters of the controller gain (KS) and the time constants (T1S, T2S, T3S and T4S) are to be determined. During steady state conditions ∆VS and VS-ref are constant. During dynamic conditions, the VS is adjusted to damp system oscillations. The value of VS in dynamic conditions is presented by: VS = VS -ref + ∆VS

K Smin ≤ K S ≤ K Smax max T1min S ≤ T1S ≤ T1S max T2min S ≤ T2 S ≤ T2 S

T3min S T4min S

J1 =

∫t =0

∆ω ⋅ t ⋅ dt

For single machine

(5)

≤ T3S ≤ T3max S ≤ T4 S ≤ T4max S

(1) To solve this problem and search RCGA optimization technique has been employed for optimal set of SVCbased damping controller parameters.

Selection of the suitable input signal for design of a robust damping controller is an important issue. When a disturbance occurs in the power system input signal is determinant the correction of control actions. Both local and remote signals are applied as input control. To improve the system reliability, input signal should be measured locally. Although, a local control signal is easy to take, but not included the favorite oscillation modes. Thus, they are not as highly controllable and observable, compared to wide-area signals. Because of the recent promotions in optical fiber communication and GPS, the wide-area measurement system can comprehend phasor measurement synchronously and transfer it to the control center almost in real time, which makes the wide-area signal a good option for control input [27], [31]. In worst condition communication channels should not have more than 50ms delay for the transmission of measured signals. Usually active and reactive power, line current magnitude and bus voltage magnitudes are measured as local signals for input signals of the FACTS damping controller and also generator rotor angle and speed deviation are considered as remote signals. Finally, for local input signals: active power and for remote input signals: speed deviations preferred in this study. For local signals 15 ms sensor time constant and for remote signals a 50 ms signal transmission delay plus 15 ms sensor time constant are considered. In the present study, Integral Time Absolute Error (ITAE) of the speed signals deviations is considered as the objective function J: t =t sim

(4)

IV.

Description of the Implemented Real Coded Genetic Algorithm Technique

Genetic Algorithm is a kind of random search optimization technique based on the mechanism of natural generation selection. Due to GAs are usually more flexible and robust than other methods, they have been successfully used in power system planning. GA maintains and controls a population of solutions and enhances performance of fitness function in their search for better solutions. Reproducing the generation and keeping the best individuals for next generation, the best gens will be obtained. The RCGA optimization process can be described as below. IV.1. Initialization To commence the RCGA optimization process, initial population shall be specified. An initial population can randomly be generated or obtain from other methods [32]. The length limitation of variables should determine for optimization problem: p = ( phi − plo ) pnorm + plo

(6)

IV.2. Objective Function Each individual represents a possible solution to optimize the fitness function. The fitness for each individual in the population is evaluated by taking

(2)



768

A. D. Falehi

objective function. Eliminating the worst individuals, a new population is created, while the most highly fit members in a population are selected to pass information to the next generation: chromosome ( variables ) = [ P1 ,P2 ,...,PN var ]

(7)

cost = f ( chromosome ) = f ( P1 ,P2 ,...,PN var )

(8)

chromosome as before: offspring _ 1 = [ pm1 pm 2 ...pnew1 ...pdN var ]

(14)

ofspring _ 2 = [ pd 1 pd 2 ...pnew2 ...pmN var ]

(15)

IV.4.2. Mutation The mutation process is used to avoid missing significant information at a special situation in the decisions. Mutation is usually considered as an auxiliary operator to extend the search space and cause release from a local optimum when used cautiously with the selection and crossover systems. With added a normally distributed random number to the variable, uniform mutation will be obtained:

IV.3. Selection Function The selection function attempts to implement pressure on the population like natural biological systems. The selection function decides which of the individuals can survive and transfer genetic characteristic to the next generation. The selection function specifies which individuals are selected for crossover. Several methods exist that parents are chosen according to efficiency of their fitness. In this paper, roulette wheel selection method is considered and is described in details in [33].

p'n = pn + σ N n ( 0 ,1)

(16)

IV.5. Stopping Criterion The stopping scale can be considered as: the maximum number of generation, population convergence criteria, lack of improvement in the best solution over a specified number of generations or target value for the objective function. With ending of generation the best individuals will be obtained. Flowchart of the RCGA optimization technique process is presented in Figure. 4.

IV.4. Genetic Operator There are two main operators in GA optimization process which are basic search mechanism of the GA techniques: crossover and mutation. They are used to create new population based on acquirement the best solution. IV.4.1. Crossover Crossover is the core of genetic operation, which helps to achieve the new regions in the search space. Conceptually, pairs of individuals are chosen randomly from the population and fit of each pair is allowed to mate. Thus, parameter where crossover occurs expressed as: α = roundup {random* N var} (9) Each pair of mates creates a child bearing some mix of the two parents: parent 1 = [ pm1 pm 2 ...pmα ...pmN var ]

(10)

parent 2 = [ pd 1 pd 2 ...pdα ...pdN var ]

(11)

Then the selected variables are combined to form new variables that will appear in the children: pnew 1 = pmα − β [ pmα − pdα ]

Fig. 4. Flowchart of the RCGA optimization technique

(12)

V. pnew 2 = pdα + β [ pmα − pdα ]

(13)

V.1.

Simulation Results

Single-Machine Infinite-Bus Power System

To assess the efficiency and robustness of the proposed controller, a severe disturbance (3-phase fault) position considered at the middle of one transmission

where, β is also a random value between 0 and 1. The final step is to complete the crossover with the rest of the



769

A. D. Falehi

line connecting bus-2 and bus-3 in Fig. 1, at t = 0.1 s and FCT is considered 0.246 s. Heavy loading equal to 0.95pu electric power with initial rotor angle of 53 (deg) has been considered for this study. The speed deviation signal is taken into account as the remote signal to the proposed SVC-based controller. By using RCGA optimization technique best of damping controller parameter set for SVC have obtained which are presented in Table I. Also the convergence of objective function J with the number of generations is shown in Fig. 5. Usually after 50 generation best value of objection function will be obtained. According to Figures 7, 8, it’s clear that power system stability is significantly increased by employing SVCbased damping controller, while time delay is assumed 50ms. Choosing different values of time delay for input remote signal, sensitivity of time delay in transient stability improvement is determined and is shown in Figure 9. V.2.

Fig. 7. Variation of rotor angle under 3-ph fault in transmission line

Multi-Machine Power System

To approve the transient performance of the proposed controller, a three phase fault occurs at sending end bus in t = 0.1 s and FCT is considered to be 0.138 s. Initial power outputs of the generators are P1 = 0.8 pu and P2 = 0.4 pu.

Fig. 8. Variation of speed deviation under 3-ph fault in transmission line

TABLE I SVC-BASED DAMPING CONTROLLER PARAMETERS FOR INPUT REMOTE SIGNAL WITH TIME DELAY 50ms KS T1S T2S T3S T4S 145.2974 0.9827 0.5056 0.9004 0.5768

Fig. 9. Variation of rotor angle with different time delay under 3-ph fault in transmission line

The speed deviation signal as a remote signal and line active power at sending end as a local signal are considered to evaluate the proposed SVC-based controller performance. In this situation, 50ms time delay is considered for a remote signal (Figures 10-12). By using RCGA optimization technique the best of damping controller parameters of SVC for both local and remote input signal have been obtained and these parameters are presented in Table II. Considering the Figures 10-12, it's clear that remote signal is more effective than the local signal to use as an input signal of SVC based damping controller. Also considering the different value of time delay for input remote signal, effect of time delay on transient stability improvement is determined and is shown in Figure 13.

Fig. 5. Convergence of objective function

Fig. 6. Variation of SVC voltage signal under 3-ph fault in transmission line



770

A. D. Falehi

TABLE II SVC-BASED DAMPING CONTROLLER PARAMETERS FOR REMOTE SIGNAL & LOCAL SIGNAL Input signals KS T1S T2S T3S T4S Remote 274.0127 0.9706 0.7749 0.3922 0.8235 Local 0.0096 0.1570 0.3668 0.0236 0.3788

VI.

Conclusion

In this study, a static VAR compensator (SVC)-based damping controller is presented to enhance the power system transient stability. Finding suitable controller parameters were formulated as an optimization problem followed by using RCGA optimization technique. The proposed controller is evaluated under severe disturbance for single-machine infinite-bus power system and twomachine power system, which approves SVC performance and abilities to stabilize power system effectively. Both local and remote signals (as an input signals) were associated with time delays to do sensitivity analysis. Results show that remote signal significantly acts better than local signal to damp the power system oscillations while a disturbance occur in power system. In addition to signal transmission delay, sorely violate performance of SVC-based damping controller.

Fig. 10. Variation of SVC voltage signal under 3-ph fault at sending end

Appendix A. Single-machine infinite-bus power system Generator: SB=2100MVA, H=3.7s, VB=13.8kV, RS=2.8544e-3, f=60 Hz, X d =1.305 p.u., X d′ = 0.296 p.u, X d′′ = 0.252 p.u., X q =0.474p.u., X q′ =0.243 p.u., X q′′ =0.18 p.u., ′′ =0.1 s. Td =1.01 s, Td′ =0.053 s, Tqo

Load at Bus-2: 250 MW. Transformer: 2100MVA, 13.8/500kV, 60Hz, R1=R2=0.002 p.u, L1=0, L2=0.12p.u., D1/Ygconnection, Rm=500p.u., Lm=500p.u. Transmission line: 3-Ph, 60Hz, Length=300km each, R1=0.02546 Ω/km, R0=0.3864 Ω/km, L1=0.9337e-3 H/km, L0=4.1264e-3 H/km, C1=12.74e-9 F/km, C0=7.751e-9 F/km. Hydraulic Turbine and Governor: Ka=3.33, Ta= 0.07, Gmin=0.01, Gmax=0.97518, Vgmin=-0.1p.u./s, Td=0.01s, β=0 Tw=2.67s Vgmax=0.1p.u./s, Rp=0.05, Kp=1.163, Ki= 0.105 B. Multi-machine power system Generators parameters: M1=1400MVA, M2=700MVA, V=13.8kKV, f=60Hz, X d =1.305pu, X d′ =0.296pu, X d′′ =0.255pu, X q =0.474pu, X q′ =0.243, X q′′ =0.18pu

Fig. 11. Variation of rotor angle under 3-ph fault at sending end

Fig. 12. Variation of speed deviation under 3-ph fault at sending end

Transformer parameters: T1=1500 MVA, T2=800 MVA, 13.8/500 kKV, R2 = 0.002 pu, L2 = 0.12 pu, Rm = 500 pu, Xm = 500 pu. Transmission line: R1=0.1755 Ω/km, R0 = 0.2758 Ω/km, L1 = 0.8737 e-3 H/km, L0 = 3.22 e-3 H/km, C1 = 13.33 e-9 F/km, C0 = 8.297 e-9 F/km.

Fig. 13. Variation of rotor angle for remote signal with different time delay under 3-ph at sending end



771

A. D. Falehi

[20] G. Stephane, C. Rachid and J.G. Alain, Optimal location of multitype FACTS devices in a power system by means of genetic algorithms, IEEE Transactions on Power systems, Vol. 16, No. 3, August 2001, pp. 537-544. [21] G. I. Rashed, H.I. Shaheen and S.J. Cheng, Optimal location and parameter setting of TCSC by both genetic algorithm and particle swarm optimization, IEEE Conference on Industrial Electronics and Applications, Harbin, China, 2007, pp. 1141-1147. [22] M. Rashidinejad, H. Farahmand, M. Fotuhi-Firuzabad, A.A. Gharaveisi., “ATC enhancement using TCSC via artificial intelligent techniques” Electric Power Systems Research. Vol. 78, 2008, pp. 11–20. [23] Sukumar Mishra, Neural-Network-Based Adaptive UPFC for Improving Transient Stability Performance of Power System, IEEE Transactions on Neural, Vol. 17, No. 2, March 2006, pp. 461-470. [24] Karim Sebaa, Mohamed Boudour., Power System Dynamic Stability Enhancement via Coordinated Design of PSSs and SVCbased Controllers using Hierarchical Real Coded NSGA-II, IEEE conference, 2008. [25] Sidhartha Panda, Differential evolutionary algorithm for TCSCbased controller design, Simulation Modeling Practice and Theory, Vol. 17, No. 10, November 2009, pp. 1618-1634. [26] M.A. Abido, Analysis and assessment of STATCOM-based damping stabilizers for power system stability enhancement, Electric Power Systems Research, Vol. 73, No. 2, February 2005, pp. 177-185. [27] Sidhartha Panda and Narayana Prasad Padhy, Optimal location and controller design of STATCOM for power system stability improvement using PSO, Journal of the Franklin Institute, Vol. 345, No. 2, March 2008, pp. 166-181. [28] A. D. Falehi, M. Rostami, A Robust Approach Based on RCGAOptimization Technique to Enhance Power System Stability by Coordinated Design of PSS and AVR, International Review of Electrical Engineering (IREE), Vol. 6, No. 1, February 2011, pp. 371-378. [29] A. D. Falehi, M. Rostami, Design and Analysis of a Novel Dualinput PSS for damping of power system oscillations Employing RCGA-Optimization Technique, International Review of Electrical Engineering (IREE), Vol. 6, No. 2, April 2011. [30] S. Panda, S.C. Swain, P.K. Rautray, R.K. Malik and G. Panda., Design and analysis of SSSC-based supplementary damping controller, Simulation Modeling Practice and Theory, Vol. 18, No. 9, October 2010, pp. 1199-1213. [31] Y. Chang, Z. Xu, A novel SVC supplementary controller based on wide area signals, Elect. Power Syst. Res. Vol. 77, 2007, pp. 1569–1574. [32] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, New York: Wiley, 2004. [33] Luonan Chen, Hideya Tanaka, Kazuo Katou, Yoshiyuki Nakamura., Stability analysis for digital controls of power systems, Electric Power System Research, Vol. 55, No .2, August 2000, pp.79-86.

References [1]

[2]

[3] [4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

A. D. Falehi, A. Dankoob, S. Amirkhan, H. Mehrjardi, Coordinated Design of STATCOM-Based Damping Controller and Dual-Input PSS to Improve Transient Stability of Power System, International Review of Electrical Engineering (IREE), Vol. 6, No. 3, June 2011. HINGORANI, N. G.GYUGYI, L., Understanding FACTS: Concepts and Technology of Flexible AC Transmission System, IEEE Press, New York, 2000. Enrique Acha, Claudio, Fuerte-Esquiv and Hogo Ambriz, modeling and simulation in power network, Wiley, England, 2004. L.Angquist, B.Lundin, J.Samuelsson, Power oscillation damping using controlled reactive power compensation and comparison between series and shunt approaches, IEEE Transaction on Power System, Vol. 8, No. 2, May 1993, pp. 687-695. M. Nooroozian and G. Andersson, Damping of power system oscillations by use of controllable components, IEEE Transaction on Power System, Vol. 9, No. 4, October 1994, pp. 2046-2054. Nadarajah Mithulananthan, Claudio A. Canizares, John Reeve,G. J. Rogers, Comparison of PSS, SVC, and STATCOM Controllers for Damping Power System Oscillations, IEEE Transaction on Power System, Vol. 18, No. 2, May 2003, pp. 786-792. M. Ghandhari, G. Andersson, M. Pavella, and D. Ernst, A control strategy for controllable series capacitor in electric power systems, Automatica, Vol. 37, No. 10, October 2001, pp. 1575–1583. N. Senthil kurnar, R. Serinivasan and M. Abdullah khan, Damping Improvement by FACTS Device A comparison between STATCOM, SSSC, UPFC, International Journal of Electrical and Power Engineering, Vol. 2, No. 3, 2008, pp. 171-178. D. Povh, Advantages of power electronic equipment in AC system in International Colloquium on HVDC and FACTS Systems, Wellington, New Zealand, Sept. 29, 1993. G. D. Galanos et al., Advanced static compensator for flexible AC transmission, IEEE Trans. Power Systems, Vol. 8, No. 1, 1993, pp. 113–121. Sidhartha Panda and Ramnarayan N. Patel, Improving power system transient stability with an off-center location of shunt FACTS devices, Journal of Electrical Engineering, Vol. 57, No. 6, 2006, pp. 365-368. M. H. Haque, Optimal location of shunt FACTS devices in long transmission lines, IEE. Pros-Gener. Transm Distrib, Vol. 147, No. 4, July 2000, pp. 218-22. Dheeman Chatterjee and Arindam Ghosh, Transient Stability Assessment of Power Systems Containing Series and Shunt Compensators, IEEE Transaction on Power System, Vol. 22, No. 3, August 2007, pp. 1210-1220. Zuwei Yu and D. Lusan, Optimal placement of FACTs devices in deregulated systems considering line losses, Electrical Power and Energy Systems, Vol. 26, No. 10, December 2004, pp. 813-819. WANG G., ZHANG M., XU X., JIANG C., Optimization of controller parameters based on the improved genetic algorithms, Proc. 6th World Congress on Intelligence Control and Automation, Dalian, China, 21–23 June 2006. Serhat Duman, Ali Öztürk, Robust Design of PID Controller for Power System Stabilization by using Real Coded Genetic Algorithm, International Review of Electrical Engineering (IREE), Vol. 5, N. 5, October 2010, pp. 2199-2208. P.P. Narayana, M.A. Abdel-Moamen and B.J. Praveen-Kumar, Optimal location and initial parameter settings of multiple TCSCs for reactive power planning using genetic algorithm, IEEE Power Engineering Society General Meeting, Denver, CO. , pp. 11101114, 10-10 June, 2004. L. Ippolito and P. Siano, Selection of optimal number and location of thyristor-controlled phase shifters using genetic based algorithms, IEE. Proc. Gener. Trasm. Distrib. Vol. 151, No. 5, 2004, pp. 630-637. D. Radu, Y. Besanger., A multi-objective genetic algorithm approach to optimal allocation of multi-type FACTS devices for power system security, IEEE Power Engineering Society General Meeting, 2006, pp. 8.

Authors’ information Department of Electrical Engineering, Izeh Branch, Islamic Azad University, Izeh, Iran. E-mail: [email protected] Ali Darvish Falehi was born in Izeh, Khouzestan, in 1985. He received the M.Sc degree from the Shahed University, Tehran, Iran, in 2010. Ali has been published 15 papers in International and domestic journals and conferences like IREE, TUBITAK, IREMOS and ENG. He is currently IEEE member in region 8. His interests include FACTS, DFACTS, PSS, power system stability, power quality, optimization techniques and designing controllers.



772


Determining the Contribution of Harmonic Distortion Generated by Utility and Customer in a Radial Distribution System Farzaneh Bagheri1, Ali Ajami2

Abstract – This paper presents an improved method for determining the contribution of harmonic distortion generated by utility and customer at the Point of Common Coupling (PCC) in radial distribution systems. For this purpose, first the magnitude and phase of voltage and current at the PCC in each frequency are estimated by adaptive Kalman filter. Then the parameters of Thevenin equivalent circuits of load and utility sides are estimated using the recursive least squares technique based on singular value decomposition (SVD). Finally, the contribution of utility and customer in harmonic distortion of the 3-phase voltage waveforms has been calculated by three approaches. A case study has been made to verify the accuracy of the proposed method. Also, the presented method has been used in a 13-bus IEEE standard distribution system. Presented simulation results show that the proposed method can accurately determine the harmonic contributions of utility and customer for measurements made at the PCC. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Contribution of Harmonic Distortion, Adaptive Kalman Filter, Recursive LeastSquares Based Singular Value Decomposition

by the consumer and utility. Before taking the necessary harmonic control measures, it is important to know who is responsible for the cause of harmonic distortion. Several methods have been proposed to identify the location of harmonic sources so as to determine whether the source is from the utility or customer side. In [3] the classification of industrial loads into “conforming” and “non-conforming” types has been presented. Another method for harmonic source localization is based on the real power flow direction [4]. However, the accuracy of the real power flow direction method is less than 50% and therefore the reliability of this method is questionable. This method is impractical because it requires knowledge of actual impedances of the system for its calculation [5]-[6]. Other methods for harmonic source localization are such as the critical impedance method [7] and voltage magnitude comparison method [8] which requires implementation of switching tests for obtaining the harmonic impedance. Hence, the switching tests do not allow its application in practical power systems. A recent method for harmonic source localization which is called as the harmonic vector method (HVM) [9] uses resistance as the reference impedance for modeling the customer side and uses the equations in [5]-[6] for determining the harmonic contribution of utility and customer. However, modeling the customer side by an equivalent resistance may introduce inaccuracy in calculating the harmonic contribution factors especially in cases where loads contain inductive elements such as motors. In [10], the total harmonic distortion (THD) is used for finding the

Nomenclature FACTS HVDC HVM THD HSE ICA PCC SVD PCC CI

Flexible AC Transmission System High Voltage Direct Current Harmonic Vector Method Total Harmonic Distortion Harmonic State Estimation Independent Component Analysis Point of Common Coupling Singular Value Decomposition Point of Common Coupling Critical Impedance

I.

Introduction

Use of nonlinear loads, such as thyristor controlled inductors for FACTs devices, converters for HVDC [1] transmission and large adjustable speed motor drives [2], is expected to grow rapidly. All of these loads inject harmonic currents and reactive power into the power system. These harmonics distort fundamental voltage and current waveforms and have many negative effects on power systems. It may cause resonance problems, overheating in capacitor banks and transformers, wrong operation of protection devices and reduction of power quality which eventually increases the maintenance costs of the system. Power distribution companies are now considering the application of penalties in the energy tariff in order to decrease the waveform distortion. This has led to the need for estimating the respective contributions to voltage waveform distortion at the PCC Manuscript received and revised August 2011, accepted September 2011

773


F. Bagheri, A. Ajami

share of harmonic distortion from utility and customer sides. The disadvantage of this method is that the THD value cannot show the variation of contributions caused by changes in phase angle of harmonic sources. In [11][14], several multiple harmonic sources localization methods were developed based on harmonic state estimation (HSE) and independent component analysis (ICA). In HSE based method, a complete knowledge about system parameters at different harmonic frequencies is necessary but these parameters are usually unknown. In addition, the method requires various types of harmonic measurements such as voltage, active and reactive power measurements, which are costly for large systems. The ICA based method, however, requires historical load data and harmonic impedance matrix of the system to eliminate indeterminacies caused by the ICA algorithm [15]-[16]. All the techniques mentioned above require the impedance of the consumer’s load .In this paper, a technique is proposed for estimating the utility’s and consumer’s contribution to voltage waveform distortion at the PCC. The input data required are the voltage and current waveforms at the PCC. By applying parameter estimation techniques, the equivalent circuit of the consumer’s load and utility has been determined. (In contrast, the techniques presented in the literature require this impedance as input data). Subsequently, an analysis of the utility’s equivalent circuit is carried out and the relative contributions of the utility and consumer to waveform distortion are calculated. Finally, waveform measurements have been made in a 13-bus standard IEEE distribution system and sources of harmonic distortion have been estimated. This paper addresses the problem of tuning Kalman filters so that they can properly track harmonic fluctuations. A method for self-tuning of the model error covariance is presented and tested, showing fast adaptive capability under sudden changes of the input signal. To eliminate the problems of the least square algorithm particularly reducing the computational requirement and using in the on-line monitoring a least square algorithm based on singular value decomposition(SVD) is used. The SVD is a powerful and computationally stable mathematical tool for solving rectangular matrices which eliminates the matrix inversion [21]-[22].

bus voltage and current at fundamental and harmonic frequencies by Adaptive Kalman filter [17]-[19]. The Kalman filter is an optimal estimator that takes into account the presence of white noise in the measurements. At the end of this step, samples of the voltage phasor at the PCC (Vix,ω+jViy,ω) and the load current phasor (Iix,ω+jIiy,ω) are both available at each angular frequency ω. In the second step of the procedure the identification of the Thevenin equivalent circuit parameters at fundamental and harmonic frequencies is performed by recursive least squares method using the estimates of the voltage and current phasors given by the adaptive Kalman filters in the first step of the procedure. In the third step the utility’s and consumer’s contribution to harmonic distortion will be estimated by the approaches which will be explained. PCC

Electric Distribution System

Load

i v

Measurement System

Fig. 1. Measurement of waveforms at the PCC

III. Load Side and Utility Side Parameter Estimation III.1. Parameter Estimation in Single Phase Circuits The proposed technique represents the load & the utility by its equivalent at each frequency, as shown in Fig. 2. The current phasor (Iix,ω+jIiy,ω) and the voltage phasor (Vix,ω+jViy,ω) are both available from the Adaptive Kalman filter. Superscript i denote the sample number and subscript ω indicates the angular frequency under consideration. I xi ,ω + jI iy ,ω

Rωi

jX ωi

PCC +

Vxi,ω + jV yi ,ω

Vxi0,ω + jV yi 0,ω

-

II.

On-line Identification Procedure Fig. 2. Load & utility side model at angular frequency ω

The proposed technique is based on measurement of the waveforms of the 3-phase voltages and currents at the PCC, as shown in Fig. 1. Typically, 20000 samples of each waveform are acquired over a 1-sec window and stored in a file. The process of estimating the utility’s and consumer’s contribution to voltage waveform distortion at the PCC will be done in three steps. In the first step, the sampled measurements of voltage and current at the PCC are used to estimate the phasors of

Applying Kirchhoff’s voltage law to the equivalent circuit of the load: Vx,i ω + jVyi ,ω = Vxi0 ,ω + jV yi 0 ,ω +

(

)(

i i + Rωi + jX ωi ⋅ I x, ω + jI y ,ω


)

(1)


774


⎡ Ipix,ω + jIpiy ,ω ⎤ ⎢ ⎥ ⎢ Inix,ω + jIniy ,ω ⎥ = ⎢ ⎥ ⎢ Iz ix,ω + jIz iy ,ω ⎥ ⎣ ⎦

Equations (2) may be put in the form of a matrix equation:

i ⎡ I x, ⎣ ω

I iy ,ω

⎡ R ⎢ 1 1⎤⎦ ⋅ ⎢ X 1 ⎢ i ⎣⎢Vx 0 ,ω

X2 ⎤ ⎥ R2 ⎥ = ⎡⎣Vx,i ω ⎥ V yi 0 ,ω ⎦⎥

V yi ,ω ⎤⎦ (2)

where R1 = R2 = Riω and X2 = -X1 = Xiω. Considering several successive samples, the recursive least-squares estimate of the parameters R1, X1, R2, X2, Vix0,ω, Viy0,ω are obtained subject to the constraints R1 = R2 and X1 = -X2. The recursive least-squares procedure for estimating the parameters values is fully described in [19]. The procedure explained before will be repeated for the utility side. Fig. 3 shows the equivalent circuit of the load and the utility system at the PCC. I xi ,ω + jI iy ,ω

RT

jωLT

PCC

Rωi

where a=ej120. After that the parameters of the equivalent circuit in positive, negative and zero sequences will be estimated by using recursive least square method. III.3. Estimating the Contribution to Voltage Distortion The procedure for estimating the contribution to voltage distortion will be applied to each sequence, at each frequency. For example the pair (Vpix,ω+jVpiy,ω), (Ipix,ω+jIpiy,ω) are used to estimate the positive-sequence voltages at the PCC due the utility and the consumer. In a similar manner, the zero-sequence and negativesequence contributions at the PCC are calculated. In order to estimate these contributions three approaches will be used: 1) Superposition principle 2) Critical impedance 3) Voltage rate. With applying the superposition principle in Fig. 3, the contribution to the voltage at the PCC in frequency, ω, is given by:

jX ωi

+

E xi 0,ω + jE iy 0,ω Vxi,ω + jV yi ,ω Vxi0,ω + jV yi 0,ω

Utility

Consumer

Fig. 3. Equivalent circuit of the load and utility sides at the PCC

III.2. Parameter Estimation in 3-phase circuits At the point of common coupling between a 3-phase distribution system and an industrial load, the variables to be measured are the 3-phase voltages (Vai, Vbi, Vci) and the currents (Iai, Ibi, Ici). The frequencies in each of the waveforms are estimated and the Kalman filter provides estimates of the voltage and current phasors for each phase, at each frequency. Let the voltage phasors be (Vaix,ω+jVaiy,ω), (Vbix,ω+jVbiy,ω), (Vcix,ω+jVciy,ω) and the current phasors be (Iaix,ω+jIaiy,ω), (Ibix,ω+jIbiy,ω), (Icix,ω+jIciy,ω). Then the voltage and current phasors are resolved into the positive, negative and zero sequence components, at each frequency, using the symmetrical components transformation: ⎡Vpix,ω + jVp iy ,ω ⎤ ⎢ ⎥ ⎢Vnix,ω + jVniy ,ω ⎥ = ⎢ ⎥ ⎢ Vz ix,ω + jVz iy ,ω ⎥ ⎣ ⎦ ⎡1 a 1 ⎢ = ⋅ ⎢1 a 2 3 ⎢ ⎢⎣1 1

i i a ⎡Vax,ω + jVa y ,ω ⎤ ⎥ ⎥ ⎢ i i ⎥ a ⎥ ⋅ ⎢Vbx, ω + jVby ,ω ⎥ ⎥ ⎢ 1 ⎥ ⎢Vci + jVci ⎥ y ,ω ⎦ ⎦ ⎣ x,ω 2⎤

(4)

i i a 2 ⎤ ⎡ Iax,ω + jIa y ,ω ⎤ ⎥ ⎥ ⎢ i a ⎥ ⋅ ⎢ Ibx,ω + jIbyi ,ω ⎥ ⎥ ⎥ ⎢ 1 ⎥ ⎢ Ici + jIci ⎥ y ,ω ⎦ ⎦ ⎣ x,ω

⎡1 a 1 ⎢ = ⋅ ⎢1 a 2 3 ⎢ ⎢⎣1 1

VUtility ,ω = =

Rωi + jX ωi

( Rω + jX ω ) + ( R i

i

T

+ jω LT )

(

)

(5)

(

)

(6)

⋅ E xi 0 ,ω + jE iy 0 ,ω

VConsumer ,ω = =

(

RT + jω LT

)

Rω + jX ωi + ( RT + jω LT ) i

⋅ Vxi0 ,ω + jVyi 0 ,ω

In critical impedance approach [7] a quantitative index will be defined by the name of critical impedance: CI = 2

Q I2

(7 )

The Q is reactive power which is generated by the load and “I” is the current in the Fig. 3. The Algorithm of critical impedance method for determining the contribution harmonic distortion generated by utility and customer is:

(3)



775


If CI > 0 then the load is dominant in producing distortions. If CI < 0 then there will be three conditions: 1- If CI > X max , then the utility side is the main harmonic contributor ( X max is the maximum of all possible X values) 2- If CI < X min then load side is the main harmonic contributor ( X min is the minimum of all possible X values) 3- If X min < CI < X max then no definite conclusion can be drawn. In voltage rate approach [8] the following rate has been proposed:

θv = Z + Zc / Z − Zu

Fig. 4. Three-phase circuit

Then, these waveforms were used as the input data for the parameter estimation algorithm and the electrical circuit of the load and supply were estimated. The estimated parameters and their exact values are shown in Table I. The agreement between them is satisfactory.

(8 )

With applying the voltage rate in Fig. 3, the contribution to the voltage at the PCC in frequency ω is given by: Vu = ( Ec ) / θ v (1 + Zu / Z c ) (9 ) Vc = θ v ( Eu ) / (1 + Zu / Z c )

IV.1.1. Determination of Contribution of Utility and Load After estimating the parameters of load and utility, the three methods that had been introduced in section III.3 are used to estimate the contribution of load and utility to harmonic distortion. The results are shown in Tables II, III and IV.

(10)

The voltage rate method uses the following decision criteria for localizing dominant harmonic sources: ⎧ Dc ⎪ D = ⎨ Dn ⎪D ⎩ u

if if if

θv < 1 θv = 1 θv > 1

IV.2. Simulation Results of IEEE 13-bus Standard Test System

(11)

The IEEE-13 bus test distribution power system shown in Fig. 5 is used as a test case for above described approach. In this case study it was assumed that the source of the harmonic is a Thyristor based 6-pulse drive which is connected to the node 680 and there are not other harmonic sources in the system and the point PCC is considered node 680. The firing angle of the Thyristor is considered 45 degree from 0 to 1 seconds, 75 degree from 1 to 2 seconds and 30 degree from 2 to 3 seconds. Three-phase bus voltage and current waveforms at PCC were observed during three second. All of signals were sampled with 20000 Hz (128 samples/cycle). 3-Phase voltage and current waveforms of bus 680 at PCC is shown in Fig. 6. It is obvious that the voltage and current waveforms are severely distorted because of the harmonics. The fundamental, 5th, 7th and 11th harmonic components of utility and load voltage are shown in Figures 7, 8, 9, and 10. From these figures, it is clear that the fundamental and 7th harmonic components are predominantly positive-sequence and that the 5th and 11th harmonics are mainly negative-sequence. Figures 11, 12, 13 and 14 show the estimation of the load parameters at the fundamental, 5th, 7th and 11th harmonics.

where, Dc, Dn and Du, are the decisions as bellow: ‘Customer is dominant’, neutral decision and ‘utility is dominant, respectively.

IV.

Simulation Results

IV.1. Simulation for a Hypothetical System The procedure described above for estimating the parameters of the load & supply circuit was verified for the Three-phase circuit of Fig. 4. In this circuit, the impedances and voltage sources behind the utility’s impedances have been set to the values shown in Table I. The voltage source behind the consumer’s impedance and utility system impedance comprises the positivesequence fundamental frequency component, as well as the Negative-sequence 5th, positive-sequence 7th and Negative-sequence 11th harmonic components. The impedance of the load comprises the series combination of 130 Ω resistor and 66.3mH inductor. The utility system impedance is 2.5 Ω resistor and 39.8mH inductor. The circuit was analyzed and the voltage and current waveforms at the PCC were extracted.



776


f (Hz)

Z Utility (Exact)

Z Utility (Estimated)

60 300 420 660

2.5+j15 2.5+j73 2.5+j105 2.5+j165

2.5+j15 2.5+j73 2.5+j105 2.5+j165

f (Hz)

ZConsumer (Exact)

ZConsumer (Estimated)

60 300 420 660

130+j25 130+j125 130+j175 130+j275

129.97+j25 130+j125 130+j175 130+j275

TABLE I PARAMETER ESTIMATION FOR THE HYPOTHETICAL SYSTEM PositivePositive-sequence Negative Negativesequence -sequence sequence

Zerosequence

Zerosequence

Utility Source

Utility Source

Utility Source

Utility Source

|V|∠θ (Exact) 100∠0.00º 0∠0.00º 0.12∠34.95º 0∠0.00º

|V|∠θ (Estimated) 99.8277∠-1.1 0∠0.00º 0.12∠27.52 0∠0.00º

|V|∠θ (Estimated) 0∠0.00º 0∠0.00º 0∠0.00º 0∠0.00º

Consumer Source

|V|∠θ (Exact) 0.36∠85.94º 0∠0.00º 6.95∠30.37º 0∠0.00º

|V|∠θ (Estimated)

Consumer Source

0.37∠86.3372 0∠0.00º 6.5667∠24.8817 0∠0.00º

Utility Source

|V|∠θ (Estimated)

Utility Source

0∠0.00º 0.94∠32.09º 0∠0.00º 0.09∠12.61º

0∠0.00º 0.9417∠30 0∠0.00º 0.09∠11

|V|∠θ (Exact) 0∠0.00º 0∠0.00º 0∠0.00º 0∠0.00º

Consumer Source

Consumer Source

Consumer Source

Consumer Source

|V|∠θ (Exact) 0∠0.00º 11.409∠37.82º 0∠0.00º 3.08∠35.52º

|V|∠θ (Estimated) 0∠0.00º 10.6951∠34.11 0∠0.00º 2.9054∠25.17

|V|∠θ (Exact) 0∠0.00º 0∠0.00º 0∠0.00º 0∠0.00º

|V|∠θ (Estimated) 0∠0.00º 0∠0.00º 0∠0.00º 0∠0.00º

|V|∠θ (Exact)

TABLE II THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH SUPERPOSITION PRINCIPLE |V|∠θ Utility |V|∠θ Utility |V|∠θ Consumer Contribution |V|∠θ Consumer Contribution Dominant f (Hz) Contribution Contribution source (Exact) (Estimated) (Exact) (Estimated) utility 60 95.65∠ -5.73º 95.4852∠-7º 0.04∠119.75º 0.0407∠146 consumer 300 0.71∠19.48º 0.7096∠14º 3.57∠69.33º 3.5685∠64º consumer 420 0.09∠23.49º 0.0892∠17.2º 2.36∠54.43º 2.3528∠46.7º consumer 660 0.06∠4.01º 0.0618∠7.51º 1.12∠51.57º 1.1039∠39º TABLE VII THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH VOLTAGE RATE (0-1SECONDS) |V|∠θ Consumer Dominant f (Hz) θv |V|∠θ Utility Contribution Contribution source

TABLE III THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH CRITICAL IMPEDANCE Dominant f (Hz) Q(var) CI source -20.6258 -87 utility 60 0.0414 40 consumer 300 0.0024 9.7796 consumer 420 0.6050 25 consumer 660

60 300 420 660

0.1332 31.52 15.65 1.12

Utility consumer consumer consumer

3000.77∠64.9º 7.1314∠0º 14.8132∠-117º 0.18∠-43º

396.3831∠21.1 149.24∠156º 970.35∠-9º 1.80∠-146.34º

TABLE IV THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH VOLTAGE RATE |V|∠θ Utility |V|∠θ Consumer Dominant f (Hz) θv Contribution Contribution source (Estimated) (Estimated) 0.0036 utility 60 95.64∠80º 0.34∠-5 12.14 consumer 300 0.71∠25.4º 8.63∠19.7º 57.92 consumer 420 0.08∠19º 4.89∠23º 34.22 consumer 660 0.06∠26º 2.03∠4º TABLE V THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH SUPERPOSITION PRINCIPLE (0-1SECONDS) |V|∠θ Consumer |V|∠θ Utility Contribution Dominant source f (Hz) Contribution (Estimated) (Estimated) utility 60 3000.18∠ 21º 423.6∠85.5 consumer 300 9.1∠150º 285.89∠-121º consumer 420 17.3∠-4.3º 270.55∠-38.5º consumer 660 3.2∠-43º 3.56∠173.34º TABLE VI THE CONTRIBUTION OF LOAD & UTILITY FOR THE HYPOTHETICAL SYSTEM WITH CRITICAL IMPEDANCE (0-1SECONDS) Dominant source f (Hz) Q(VAr) CI -4e4 -53.86 Utility 60 -25 -0.62 |CI|n), the vast majority of research has involved resolution through the use of the pseudoinverse J+ of the Jacobian matrix J [14]: θ = J + x (4) This solution minimizes

(6)

⎡ ∂ϕ ∂ϕ ⎤ ∇ϕ = ⎢ " ⎥ ∂ ∂ θ θn ⎦ ⎣ 1

(7)

The value of α allows us to realize a trade-off between the minimization of and the optimization of φ(θ). As it was mentioned earlier that secondary performance criteria can be optimized, φ(θ) is used for minimizing the norm of the joint velocities, avoiding obstacles, avoiding singular configurations, avoiding joints limits, or minimizing driving joint torques. Because this work deals with the singularity avoidance in redundant manipulators, only the measurements used for singularity avoidance is reviewed here. Clearly, the effect of a singularity is experienced not only at the singular configuration itself but also at its neighborhood. For this reason, it is important to be able to characterize the distance from singularities through suitable measures; these can then be exploited to counteract undesirable effects. During the past three decades, many researchers have suggested several methods to fully utilize the redundancy of robots. Reference [15] proposes a pseudoinverse method that minimizes the sum of squares of joint velocities to avoid kinematically singular points. But this method cannot generate trajectories to automatically avoid kinematic singularities and guarantee repeatability. Reference [19] suggests the Resolved Motion method using a general solution. This method includes terms that minimize joint velocity norms and self-motion that does not affect endeffector motion. Self-motion is attributed to maximized performance index. The concept of task priority is illustrated in Reference [20]. The manipulator task is divided into an ordered sequence of sub-tasks. If it is not possible to perform all of the sub-tasks in their entirety because of the shortage of DOF, this method carries out the most significant sub-task before any of the others and

By differentiating this equation with respect to time and defining J=df/dθ, the following equation is obtained:

θ = J −1 x

)

θ = J + x + α I n − J + J ∇ϕ with:

(1)

x = J θ

(5)

where z is an arbitrary (n*1) vector in the θ space. The MATLAB Simulink of this equation is shown in Fig. 1. The second term on the right-hand side of the equation belongs to the null space of J. It corresponds to a self-motion of the joints that does not move the endeffector. This term, which is called homogeneous solution or optimization term, can be used to optimize a desired function φ(θ) [16]. In fact, taking where is the gradient of this function with respect to θ minimizes the function φ(θ) when α < 0 and maximizes when α > 0. Equation (5) is rewritten as:

Inverse Kinematics of Redundant Manipulator

x = f (θ )

)

θ = J + x + I n − J + J z

2

θ . Because of this

minimizing property, it is hoped [15] that singularities will automatically be avoided [16]. It is shown that without modification, this approach does not avoid singularity [17]. Moreover, reference [18] points out that it does not produce cyclic behavior, which is a serious practical problem. For these reasons, another component belonging to the null space of the Jacobian has to be added to the pseudoinverse solution to realize the secondary objective function. The basic redundancy resolution scheme is the Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


808

Samer Yahya, M. Moghavvemi, Haider A. F. Mohamed

leaves the less important sub-tasks until later using the remaining DOF. In reference [21], it is proved that the pseudoinverse method of [15], without modification, will not avoid singular configurations. Therefore, the extended Jacobian method is proposed [21]. This method specifies an additional number of equality constraints equal to the degree of arm redundancy. This means that the Jacobian matrix is extended until it becomes a square 1-to-1 mapping between the extended end-effector velocity space and joint velocity space. It is easy to solve the inverse Jacobian matrix using a square matrix, but it is difficult to obtain new sufficient conditions and additional algorithmic singularities as well as kinematic singularities. This problem means that the extended Jacobian method is singular even when the robot manipulator is not in a singular configuration. Various performance indices have been proposed to utilize redundancy. Yoshikawa uses the manipulability measure [22],[23]. The manipulability, which is basically the determinant of the Jacobian matrix (product of singular values), faces two problems when it is used to find design solutions [24],[25]: scale dependency and order dependency. These two problems create difficulties for design, since we need to compare manipulators with

different sizes. The scale dependency prevents a fair comparison between a longer manipulator and a shorter manipulator, and the order dependency makes it impossible to derive the physical meaning of the manipulability. The measure ψ of a symmetric matrix JJT is presented in reference [26]. It is defined as the trace of JJT divided by the order m. This measure represents the arithmetic mean of eigenvalues of JJT. This is independent of the order, but is dependent on the scale. A critical drawback of this measure is that it cannot provide any information of proximity to singular configurations, which prevents the use of this measure. The manipulability M was defined as the geometric mean of the eigenvalues of JJT and the measure ψ as their arithmetic mean. Thus, ψ is always greater than M, and equal to M when all eigenvalues are the same. The equality of all eigenvalues implies isotropy of the mdimensional ellipsoid (that is, m-dimensional sphere). Kim defines a new measure of isotropy of the manipulability ellipsoid by the ratio of these two means [26], which is independent of scale since both M and ψ have the dimension of length. The new measure has an upper bound of one. A larger value implies a more isotropic ellipsoid.

Fig. 1. The MATLAB Simulink of Eq. (5)

Reference [27] proposes a condition number that indicates the uniformity of the Jacobian transformation with respect to direction. Klein suggests a minimum singular value in [28]. This could be the distance from the singular point, but be short of a physical meaning. This measure is suggested because the minimum eigenvalue changes more radically near singularities than they do near other eigenvalues [21]. To indicate the uniformity of the Jacobian transformation, the condition

number may be used [28]. Reference [29] uses J-minor index ΦG based on the concept of aspect. This measure tends to give more balanced minors so that it preserves an aspect. This approach keeps all the J-minors at nonzero. Therefore, the repeatability, singularity and discontinuity problems in motion due to switching of aspects can be prevented. Nevertheless, the analytic form of the gradient of ΦG will also be very complicated and hard to obtain. In addition, if at least one of the J-minors



809


is zero initially, the gradient of ΦG becomes infinite, and this method will fail [30]. Reference [31] shows that a key factor for the failure of the manipulability measure and condition number is that they do not include the information about the task or of the direction where the manipulator should move. This reference proposes a direction manipulability measure that gives information about dexterity of the manipulator in the direction of the path. Since each singularity is associated with a rank loss of J, one conceptually simple possibility in the case of a square Jacobian matrix (m=n) is to compute its determinant. A generalization of this idea that also works for a non-square Jacobian matrix is the manipulability measure [23]. It is one of the earliest proposed and most recognized Jacobian based manipulator performance metric:

( )

H1 = det JJ T

The condition number has values ranging from 1, at configurations in which all the singular values are equal to ∞, at singular configurations. Note that when H2=1 all the singular values are equal and thus the end-effector has the same motion capability in all task-space directions – i.e., the arm is at an isotropic configuration – whereas at a singularity it loses mobility in some taskspace directions [12]. An even more direct measure of the distance from singular configurations is the smallest singular value of the Jacobian matrix [28], i.e. defined as follows: H 3 = σ min

It should be noted that the manipulability measure may remain constant even in the presence of significant variations of either the condition number or the smallest singular value of J. On the other hand, since the smallest singular value changes more radically near singularities than the other singular values [35], it dominates the behavior of the determinant and the condition number of the Jacobian matrix; therefore, the most effective measure of distance from singular configurations is the smallest singular value of the Jacobian matrix [12]. Reference [21] presents a coordinate-free version of the manipulability index. This index is defined as follows:

(8)

This metric can be expressed in terms of minors as follows [29],[32]: p

H1 =

∑∆i2

(9)

i =1

with ∆i‘s, i=1,2,…,p, with p=nCm, are minors of rank m of the matrix J. It can be recognized that the manipulability measure (H1) is equal to the product of the singular values of Jacobian matrix J [12], i.e.:

( )

H 4 = tr JJ T

(10)

i =1

where σ is the singular value of J and thus its zeros coincide with the singularities. The manipulability index has proven effective when measuring and optimizing motion smoothness and avoiding kinematic singularities. Unfortunately, since the manipulability index is derived using only the manipulator Jacobian matrix, it suffers from scale, order and dimensional homogeneity dependencies that prevent an accurate comparison of performance between two or more competing morphologies [33],[34]. Another disadvantage of the manipulability measure is that theoretically, there is nothing to stop one or more of the minors in the manipulability measure equation becoming zero whilst still maintaining an optimal value; therefore, it cannot prevent the switching of aspects [32]. Another possible measure of distance from a singular configuration is the condition number of the Jacobian matrix [28], defined as: H2 =

σ max σ min

−1

(13)

As long as this measure has the same properties of (H1) and (H3) [36], the smaller the value this index has the farther the manipulator is from the singularity configuration. Another measure using the trace of JJT is presented in [26] and shown in the following equation:

m

H1 = Πσ i

(12)

H5 =

( )

tr JJ T m

(14)

This measure can be rewritten as: H5 =

λ1 + λ2 + ⋅⋅⋅ + λm m

(15)

This measure represents the arithmetic mean of eigenvalues of JJT. This is independent of the order, but is dependent on the scale. A critical drawback of this measure is that it cannot provide any information of proximity to singular configurations, which prevents the use of this measure [26]. Both the geometric mean and arithmetic mean of the eigenvalues of JJT have been used to present a new dexterity measure of isotropy of the manipulability ellipsoid [33],[34]. This measure is:

(11)



810


( ) ( )

⎛ m det JJ T ⎞ ⎜ ⎟ ⎠ H6 = ⎝ tr JJ T

y = 20 sin ( t + 5 ) + 30

(16)

The inverse kinematics of the manipulator is calculated using the well-known method of manipulability measure (H1) which is currently used as one of the most reliable method for singularity avoidance. Figure 2 shows the values of the joint angles using this method.

m

which has an upper bound of 1. A larger value of the measure of isotropy implies a more isotropic ellipsoid. The advantage of this measure is that it can be expressed analytically as a function of joint angles, which is beneficial for real-time control of a manipulator [26]. In the case of hyper redundant manipulators with high number of degrees of freedom, the computational burden of pseudo inverse Jacobian becomes prohibitive, despite proposed improvements. Furthermore, most of the proposed schemes handle the inverse kinematic problem at the velocity level only [37]. Therefore many approaches were presented, and they did not require calculation of the pseudo inverse of the Jacobian. The more recent techniques for inverse kinematic solution of hyper-redundant manipulators are the Virtual Link/Displacement Distribution Scheme [38]-[40]. The method is singularity free, and provides a robust solution and it is based on defining virtual layers, and dividing them into virtual/real three-link or four-link sub-robots. It starts by solving the inverse kinematic problem for the sub-robot located in the lowest virtual layer, which is then used to solve the inverse kinematic equations for the sub-robots located in the upper virtual layers. A new geometrical method for the planar and specific type of spatial redundant manipulators is presented in reference [41]. The proposed method is a very effective way to control redundant manipulators far from their singularity configurations. The main aim for the proposed method is to set the angles between the adjacent links so that they are equal and the links do not line up while the end-effector is moving, i.e. it guarantees to find the inverse kinematics of redundant manipulators for any end-effector target point in the manipulator workspace far from singularity configurations.

6 5 theta1 4

joint angles (rad)

3 2 1

theta4 theta3

0 -1

theta2

-2 -3

theta5 0

5

10

15

time (s)

Fig. 2. Values of the joints angles using the manipulability index method (H1)

Case Two: A five degrees of freedom manipulator is examined in this case. The lengths of the links are as follows, l=[22,20,18,16,14]T and the lengths are in units length. In this case it is desired to move the end-effector on the path defined as: x = 14 cos ( t + 3) + 15

(19)

y = 14 sin ( t + 3) − 25

(20)

The inverse kinematics of the manipulator is calculated using the condition number index method (H2). Figure 3 shows the values of the joint angles using this method. 2.5

III. Simulation Results

theta1

theta5 2

Because the simulation of all the mentioned methods have been examined and built using the MATLAB Simulink, this section shows the result of using these methods.

1.5 1

theta3

joint angle (rad)

0.5

Case One: A five-links redundant manipulator is simulated in this case. The lengths of the links are as follows, l=[19,18,17,16,15]T and the lengths are in units length. In this case it is desired to move the end-effector on the path defined as: x = 20 sin ( t + 5 ) − 25

(18)

0 -0.5 theta2 -1 -1.5 theta4 -2 -2.5

0

5

10

15

20

25

time (s)

(17)

Fig. 3. Values of the joints angles using the condition number index method (H2)



811


5

Case Three: The lengths of the links of the manipulator of this case are, l=[22,20,18,16,14]T and the lengths are in units length. In this case it is desired to move the end-effector on the path defined as:

3 2 joint angle (rad)

x = 14 cos ( t + 3) + 30

theta 1

4

(21)

theta 3

1 0 theta 5

theta 4

-1

y = 14 sin ( t + 3) − 25

(22)

-2 theta 2 -3

The inverse kinematics of the manipulator is calculated using the smallest singular value index method (H3). Figure 4 shows the values of the joint angles using this method.

-4

0

5

10

15

20

25

30

time (s)

Fig. 5. Values of the joints angles using the index method (H4)

3

Figure 6 shows the values of the joint angles using the method of index (H5).

theta 1

2

1

6

joint angle (rad)

theta 4 theta 3

0

5

theta 2

4

theta 1

-1

joint angels (rad)

3 -2

theta 5

-3

2 1 theta 4 0

-4 0

5

10

15

20

theta 3

25

time (s)

theta 2

-1

theta 5 -2

Fig. 4. Values of the joints angles using the smallest singular value index method (H3)

-3

0

5

10

15

20

25

30

time (s)

Case Four: In this case, a manipulator is simulated in which the lengths of the links are as follows, l=[19,18,17,16,15]T. The lengths are in unit length. The inverse kinematics of the manipulator is calculated using the method of index (H4). It is desired to move the end-effecor of the manipulator on the path defined:

Fig. 6. Values of the joints angles using the index method (H5)

Case Six: In this case, a manipulator is simulated in which the lengths of the links are as follows, l=[22,20,18,16,14]T. The lengths are in unit’s length. The inverse kinematics of the manipulator is calculated using the method of isotropy measure (H6). It is desired to move the endeffector on the path defined as:

x = 15 sin ( t + 5 ) − 25

(23)

y = 15 cos ( t + 5 ) + 30

(24)

x = 12 cos ( t + 3.5 ) − 25

(27)

Figure 5 shows the values of the joint angles using the method of index (H4).

y = 12 sin ( t + 3.5 ) + 30

(28)

Case Five: The same manipulator used in the previous case is examined in this case. It is desired effect to move the end-effector on the path defined as:

Figure 7 shows the value of the joint angles of the manipulator using the method of the isotropy index (H6).

x = 9.5 sin ( t + 5 ) − 25

(25)

y = 9.5 cos ( t + 5 ) + 30

(26)

Case Seven: The same manipulator of case one with the same desired path of end-effector of that case, this is examined using the method of reference [41]. Figure 8 shows the value of the joint angles of the manipulator using the method [41].



812


[3]

2.5 theta 1

2

theta 5

1.5

[4]

joint angles (rad)

1 theta 3

0.5 0

[5]

-0.5

theta 2

-1

[6]

-1.5 theta 4

-2 -2.5

[7] 0

5

10

15

20

25

time (s)

[8]

Fig. 7. Values of the joints angles using the isotropy index method (H6) 1.2

[9]

theta 1

1.1 1

[10]

joint angles (rad)

0.9 0.8 0.7

[11]

0.6

theta 2,3,4,5

[12]

0.5

[13]

0.4 0.3 0.2

[14] 0

5

10

15

time (s)

[15]

Fig. 8. Values of the joints angles using the method of [41] [16]

IV.

Conclusion

[17]

Because redundant manipulators have infinite solutions for its inverse kinematics, secondary performance criteria can be optimized. Avoiding singular configurations has been taken as secondary performance for the methods reviewed in this paper. The paper reviews all the well-known methods used to find the inverse kinematics of redundant manipulators that guarantee moving the manipulators far from their singularity configurations. The advantages and disadvantages of these methods have been explained as well. The simulation of all the mentioned methods in this paper has been built and examined using the MATLAB Simulink.

[18]

[19]

[20] [21]

[22]

References [1]

[2]

[23]

Fernando B. M. Duarte, Tenreiro Machado J. A. Pseudoinverse Trajectory Control of Redundant Manipulators: A Fractional Calculus Perspective. IEEE International Conference on Robotics and Automation (ICRA '02). 2002. Vol. 3, 2406 - 2411. Graca, R.A., and You-Liang Gu. Application of the Fuzzy Learning Algorithm to Kinematic Control of a Redundant Manipulator with Subtask. IEEE World Congress on Computational Intelligence, Proceedings of the Third IEEE Conference on Fuzzy Systems. 1994. Vol. 2, 843 – 848.

[24] [25] [26]


Antonelli, G. Stability Analysis for Prioritized Closed-Loop Inverse Kinematic Algorithms for Redundant Robotic Systems. IEEE International Conference on Robotics and Automation (ICRA 2008). 2008. 1993 - 1998. Omrčen, D., Žlajpah, L., and Nemec, B. Compensation of velocity and/or acceleration joint saturation applied to redundant manipulator. Robotics and Autonomous Systems, 2007: vol. 55, no. 4, 337-344. Chirikjian, G. S., and Burdick, J. W. A model Approach to HyperRedundant Manipulator Kinematics. IEEE Transaction on Robotics and Automation, 1994: vol. 10, no. 3, 343-354. Chirikjian, G. S., General Methods for Computing HyperRedundant Manipulator Inverse Kinematics. IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 93). 1993. vol. 2, 1067 – 1073. Braganza, D., Dawson, D. M., Walker, I. D., and Nath, N. Neural Network Grasping Controller for Continuum Robots. IEEE Conference on Decision and Control. 2006. 6445 – 6449. Tesar, Chetan Kapoor and Delbert. Kinematics abstractions for general manipulator control. DETC99: ASME Design Engineering Technical Conferences. 1999. 1-12. Das, H., Slotine, J-J. E., and Sheridan, T. B. Inverse kinematic algorithms for redundant systems. IEEE International Conference on Robotics and Automation. 1988. vol. 1, 43 – 48. Wang, Youshen Xia and Jun. A Dual Neural Network for Kinematic Control of Redundant Robot Manipulators. IEEE Transactions on Systems, Man, and Cybernetics, 2001: Vol. 31, 147 – 154. Duffy, J. Analysis of Mechanisms and Robot Manipulators. New York: Wiley, 1980. Bruno Siciliano, Oussama Khatib. Springer Handbook of Robotics. Springer-Verlag Berlin Heidelberg, 2008. Ramdane-Cherif, A., Daachi, B., Benallegue, A., and Levy, N. Kinematic inversion. IEEE/RSJ International Conference on Intelligent Robots and Systems. 2002. vol. 2, 1904 - 1909. Albert, A. Regression and the Moore-Penrose Pseudo-inverse. New York: Academic Press, 1972. Whitney, D. E. Resolved motion rate control of manipulators and human prostheses. IEEE Transactions on Man Machine Systems, 1969: vol. MMS-10, no. 2, 47 – 53. Khalil, W., and Dombre, E. Modeling, Identification and Control of Robots. Hermes Penton Ltd, 2002. Baillieul, J. Kinematic programming alternatives for redundant manipulators. IEEE International Conference on Robotics and Automation. 1985. 722 – 728. Klein, C. A., and Huang, C. Review of Pseudo Inverse Control for use with Kinematically Redundant Manipulators. IEEE Transactions on Systems, Man, and Cybernetics, 1983: vol. SMC13, no. 2, 245–250. Liegeois, A. Automatic Supervisory Control of the Configuration and Behaviour of Multibody Mechanisms. IEEE Transactions on Systems, Man, and Cybernetics, 1977: vol. SMC-17, no. 1, 868– 878. Nakamura. Advanced Robotics: Redundancy and Optimization. Addison-Wesley, 1991. Baillieul, J., A constrained Oriented Approach to Inverse Problems for Kinematically redundant manipulators. IEEE International Conference on Robotics and Automation. 1987. 1827 – 1833. Yoshikawa, T., Analysis and Control of Robot Manipulators with redundancy. Robotics Research: The First International Symposium. Cambridge: M. Brady and R. Paul, Eds., MIT Press, 1984. 735-747. Yoshikawa, T. Manipulability of Robotic Mechanisms. The International Journal of Robotics Research, 1985: vol. 4, no. 2, MIT Press, Cambridge, MA. Strang, G. Linear Algebra and its Application, 3rd ed., San Diego: Harcourt Brace Jovanovich Publishers, 1988. Leon, S. J. Linear Algebra with Applications. Macmillan, 1986. Kim, J. O., and Khosla, P. K. Dexterity Measures for Design and Control of Manipulators. IEEE/RSJ International Workshop on Intelligent and Systems. 1991. 758-763.


813


[27] Salisbury, J. K., and Craig, J. J. Articulated Hands: Force Control and Kinematic Issue. The International Journal of Robotics Research, 1982: vol. 1, no. 1, 4-7. [28] Klein, C. A., and Blaho, B. E. Dexterity Measures for the Design and Control of Kinematically Redundant Manipulators. The International Journal of Robotics Research, 1987: vol. 6, no. 2, 72-83. [29] Chang, P. H., A Dexterity Measure for Kinematic Control of Redundant Manipulators. American Control Conference. 1989. 496-502. [30] Cheng, F. T., Chen, J. S., and F. C., Kung. Study and Resolution of Singularities for a 7-DOF Redundant Manipulator. IEEE Transactions on Industrial Electronics, 1998: vol. 45, no. 3, 469480. [31] Zlajpah, L. Dexterity Measures for Optimal Path Control of Redundant Manipulators. International workshop on Robotics in Alpe-Adria-Danube Region. 1996. 85-90. [32] Stevenson, R., Shirinzadeh, B., and Alici, G. Singularity Avoidance and Aspect Maintenance in Redundant Manipulators. International Conference Control, Automation, Robotics and Vision. 2002. 857-862. [33] Hammond, F. L., and Shimada, K. Improvement of redundant manipulator task agility using multiobjective weighted isotropybased placement optimization. IEEE International Conference on Robotics and Biomimetics. 2009. 645-652. [34] Hammond, F. L., and Shimada, K., Morphological Design Optimization of Kinematically Redundant Manipulators using Weighted Isotropy Measures. IEEE International Conference on Robotics and Automaton. 2009. 2931-2938. [35] Choi, B. W., and Chung, M. J. Performance Evaluation of Dexterity Measures Using Measure Constraint Locus. IEEE/RSJ International Conference on Intelligent Robots and Systems. 1992. 1943-1950. [36] Choi, B. W., and Chung, M. J., Evaluation of Dexterity Measures for a 3-Links Planar Redundant Manipulator Using Measure Constraint Locus. IEEE Transactions on Robotics and Automation, 1995: 282-285. [37] Haider A. F. Mohamed, Samer Yahya, M. Moghavvemi, and Yang S. S. A New Inverse Kinematics Method for Three Dimensional Redundant Manipulators. ICROS-SICE International Joint Conference. 2009. 1557-1562. [38] Chung W. J., Chung W. K., and Youm Y. Kinematic Control of Planar Redundant Manipulators Extended Motion Distribution Scheme. Robotica, 1992: Vol.10, 255-262. [39] Chung W. J., Chung W. K., and Youm Y., Inverse Kinematics of Planar Redundant Manipulators Using Virtual Link and Displacement Distribution Schemes. IEEE International Conference on Robotics and Automation. 1991. 926-932. [40] Maghami Asl, F., Ashrafiuon, H., and Nataraj C. A general solution for the position, velocity, and acceleration of hyperredundant planar manipulators. Journal of Robotic Systems, 2002: vol. 19, no. 1, 1–12. [41] Samer Yahya, M. Moghavvemi, Haider A.F. Mohamed. Geometrical approach of planar hyper-redundant manipulators: Inverse kinematics, path planning and workspace. Simulation Modelling Practice and Theory, Available online 10 August 2010.

Mr. Samer Yahya was born in 1980 in Iraq. He received his M.Sc. in Control and Systems Engineering from the University of Technology, Iraq in 2006. He worked as a lecturer for two years. Currently, he is pursuing his Ph.D. in Robotics and Automation at the Department of Electrical Engineering in University of Malaya, Malaysia. His most important research concern is modeling, controlling and path planning of hyper redundant robots. Dr. Haider A. F. Mohamed received his PhD in Electrical Engineering from the University of Malaya, Malaysia in 2006. He worked as a computer engineer for two years and as a researcher for four years before he became a lecturer in the Department of Electrical Engineering in University of Malaya, Malaysia, in 2000. In 2008, Haider joined the School of Electrical and Electronics Engineering in the University of Nottingham Malaysia Campus as an associate Professor. His main research fields are identification and nonlinear intelligent control of various systems such as robot arms, automated guided vehicles, and electric drives. Haider is active in participating in many competitions and conferences where he won some awards and medals. He received SICE International scholarship award in 2002 and 2008. Prof. Dr. M. Moghavvemi obtained his BSc in Electrical Engineering from the State University of New York, MSc from University of Bridgeport and PhD from the University of Malaya. He joined University of Malaya in 1991 and currently holds the chair of electrical engineering. Prof. Moghavvemi is the Director of Centre for Research in Applied Electronics (CRAE). His current research interests are; Electronic circuit design; application toward sensory interface electronics in industrial, commercial, scientific, transportation, and biomedical systems.

Authors’ information Center of Research in Applied Electronics (CRAE), University of Malaya, 50603 Kuala Lumpur, Malaysia. Tel.: 0060172841560 E-mails: [email protected] [email protected] *Corresponding author. 1 Department of Electrical & Electronic Engineering, The University of Nottingham Malaysia Campus Jalan Broga, 43500 Semenyih, Selangor, Malaysia.



814


Optimisation of Defects in Composite Materials Using an Improved Wavelet Analysis Basic Algorithm Benhamou Amina1, Benyoucef Boumedienne2 Abstract – The present work carries on the use of a method based on the wavelet transform to detect the internal flaws of composite materials. The objective of this work consists in working out a data processing sequence of an ultrasonic signal identifying nearly flaws in composite laminate materials and estimating their position. The use of a numerical signal processing technique, based on the Fast Wavelet Transforms is applied. The method is implanted and optimized for detection and classification of delamination and porosity flaws in manufactured materials. Since the information about the signal requires a large amount of computation time and resources, a technique is used to reduce the dimensions of the sampling signals. In Non-destructive evaluation of stratified composite materials, the identification of some defect features requires more recent and advanced methods than classical techniques. Notably, in thin composite materials, the reflected NDE ultrasonic signals are overlapping. As a result, the flaws evaluation is becoming unfeasible. Many works dedicated to advanced signal processing based on time-frequency analysis has been widely used in non-destructive evaluation (NDE) applications. To evaluate the nearly flaw detection of delamination and porosity enclosed in composite multilayer plate, the wavelet analysis is applied to ultrasound waveforms acquired by immersion pulse-echo technique. The obtained results offer some defect features relating their nature and position. The applied wavelet analysis provided excellent results for the investigated materials containing artificial delamination and porosity flaws. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Ultrasonic Signal, Wavelet Transforms, Composite Materials, Nearly Flaws, Algorithm

I.

This problem raises a notorious difficulty in the applications of non destructive ultrasonic evaluating methods. As a result, it needs the use of different approaches of signal processing to analyse the signal of the propagating wave in the probed material. Some explicit methods based on the Discrete Wavelet Transform (DWT) have been proposed to solve this kind of problems. The basic algorithm used is based on the DWT to analyse the measurement errors of the size and to locate the position of the artificial defects in the tested composite materials. The principal advantage of those approaches is that they allow a fine description of the local regularity of signals. The Fourier analysis did not make possible the connection of the local regularity of a function with the behaviour of its coefficients. The wavelet analysis allows a local description of the signals and obtaining new characterizations of the regularity. The wavelet decomposition of the majority of the signals reveals significant coefficients with only one small number of positions in the time-scale plan, i.e., that the energy is preferentially localised at some frequencies and positions. The received ultrasonic signals are composed of the signal components and an additive Gaussian noise. So, the wavelet coefficients represent both the signal and

Introduction

The design and analysis of composite materials being more complex and more expensive than the conventional materials, such as the steel and the aluminum, the anisotropic behaviour of these materials makes their structural analysis difficult, so the use of more sophisticated methods for their analysis is required [1], [2]. Developing the applications of composites in the mechanical structures requires the use of the non destructive evaluation of the mechanical behaviour of these materials. The field of the multilayer materials evaluation aims to strengthen the diagnosis in matter of researching defects especially the fibres basis composites of carbon or glass and epoxy (delamination, porosity, overlapping, bridging and inclusions) in particular, the distinction between micro-cavity and enriched areas with resin as well as the position of the echo signal resulting from the cavity near the first reflected echo signal. Let’s note that the existence of defects in a composite can lead to a significant alteration of its mechanical characteristics. For the evaluation of reinforced fibre composites, emerges the problem of a great acoustic attenuation of the measuring signals.


815


Benhamou Amina, Benyoucef Boumedienne

the analysed signal and the family of wavelets translated in time and dilated of the scale parameter a as follow. They are obtained by calculating the scalar product of the signal to be analyzed and the family of wavelets shifted in time and dilated or compressed by using the parameter a and b:

the noise. To eliminate the noise, the best tool to use is the wavelet processing technique [3], [4]. Concerning the position of the problem, this study consists in working out a data processing sequence of an ultrasonic signal for which would consist in the following steps: a. Detection of a flaw in a material, b. Estimation of the exact position of the flaw in order to operate at the good place, c. Simple and general modelling of the signals allowing a better analysis, d. Identification of a signal by a well-known software, such as MATLAB, e. Application of the wavelet transform for the timefrequency analysis to measure the thickness of a flaw [5].

II.

S ( b,a ) =

1

∞

s (t ) ⋅ψ a∫ 0

*

⎛t −b⎞ ⎜ ⎟ dt ⎝ a ⎠

(3)

where s(t) is the signal to be analysed and ψ * the daughter wavelet.

Methods of Applications of the Wavelet Transform

The methods of wavelets anticipate powerful tools to analyze, quantify, compress, rebuild, and model signals and images. They are useful to locally capture, identify, and analyze processes, with multi-scales, non-stationary signals, to make it possible to explore aspects of data that other analysis techniques omit. A family of elementary functions is constructed by translation and dilation (or contraction) starting from a basic function, the mother or analyzing wavelet. There exist an infinite number of wavelets. The analyzing wavelet most often used, called Morlet wavelet or Mexican hat (Figure 1), is due to Morlet [6], [7] and is given by: ⎛ t2 ⎝ 2

ψ ( t ) = exp ( ict ) ⋅ exp ⎜⎜ −

⎞ ⎟⎟ ⎠

Fig. 1. Morlet wavelet

III. Fast Wavelet Transform and Applications III.1. Experimental Results The detection of the flaws implies many factors, which influence the transmitted ultrasonic signal in the materials under investigation [8], [9]. The acoustic theory of propagation in the materials shows that the parameters of the transmitting ultrasonic signal depend on many factors of which the most important are as follows: - Frequency and bandwidth of the ultrasonic signal; - Inspection of the trajectory and distance; - Position of flaws and their dimension; - Materials Properties; - The ultrasonic signals processing methods among which, the wavelet transform. The realization of the whole measurements is done by an ultrasonic bench including ultrasonic transducers with longitudinal waves, excited by a generator of ultrasounds transmitter-receiver. According to Figure 2, control is done in a tank filled with water or any other medium ensuring a satisfactory coupling between the transducer and the part to be controlled, and mobility with 3 directions of these transducers, so the possibility of examination of the complex parts by using a remotecontrolled manipulator arm. The signals are visualized on a numerical oscilloscope allowing their samplings. The unit is controlled by a micro-computer via a GPIB interface. Then the wavelet transform of the ultrasonic signal was analysed and the preliminary obtained results were presented. The training set was comprised of scanned signals from both damaged and undamaged

(1)

We use a family of wavelets of functions ψ a,b indexed by two labels (a and b):

ψ a,b ( t ) =

⎛ t −b ⎞ ⋅ψ ⎜ ⎟ a ⎝ a ⎠

1

(2)

To remind, a and b are the translation and dilation or contraction parameters. The functions ψ a,b are called "wavelets"; the function ψ is called "mother wavelet". Note that ψ is implicitly assumed to be real, and if not, complex conjugate has to be introduced. As a changes, equation (2) with b=0 covers different frequency ranges (large values of a, correspond to small frequencies, while small values of a, correspond to high frequencies). This wavelet transform allows us to lead to a multi-resolution analysis. Changing the parameter b as well allows moving the time localization centre: each t is localized around t=b. It follows that the wavelet transform provides a time-frequency description. The wavelet coefficients are then computed using the inner product of



816


•

samples using stratified composite materials. One dimensional wavelet transform is used to test our algorithm. This allowed us to study and identify the changes introduced in an ultrasound waveform by the presence of flaws. Scale-time plots were studied. Some flaws (delamination, porosity) in some composite materials were investigated for initial time-position of flaw peaks. Notice that when an ultrasound signal propagates through a material, the frequency components are modified. The composite materials themselves modify the frequencies: they degrade them like seen in the following Figures 4 and 5.

For intermediate wavelength, reflection /transmission coefficients decrease/increase linearly.

(a)

Fig. 2. Control by immersion in echo mode (b)

(a)

(c) Figs. 4. (a) Measured Signal without Defects, (b) its decomposition with White Noise, (c) elimination of the white noise

(b) Figs. 3. (a) Measured signal without defects and (b) its decomposition

So the defects behave as frequency filters [9], [10]. The above problems confirm that testing composite materials require a fine frequency selection and a good signal interpretation. The filtering action can be formulated as a change in transmission and reflection coefficients as follows [11]: • For low wavelength, transmission and reflection coefficients stay unchanged; • For high wavelength, the wave is completely transmitted, that is doesn’t interact with the defect;

Fig. 5. Flowchart of the basic algorithm



817


III.2. Fast Wavelet Transform Flowchart The use of a straightforward Algorithm using a prohibitive number of points in a sample leads to a prohibitive computation time and a non-comprehensible decomposition. The first purpose of the proposed algorithm (Figure 5) is a more effective computation procedure to compress the data sets needed for the calculation of the wavelet coefficients by simplifying the multidimensional [9] data sets to lower dimensions. This allowed us to achieve very high processing rate whose consequence is a better decomposition of a signal and a better interpretation of its graph. The wavelet transform of S(t) is given by equation (3) where ψ* denotes the complex conjugate of ψ, a is tied to the frequency and b to the time (Figures 4). To compute the coefficients, the real part of the Morlet wavelet is used [12].

IV.

(a)

Analysis and Results Discussion

The time-frequency analysis allows observing the nature of the signal whose parameters are unknown (rate of amplitude, frequency modulation, etc.). It also allows a global and a local analysis of the signal, i.e., a panoramic and a detailed analysis. The scalogram, by its positive character can assume the energy density function, which is not the case for a number of timefrequency representations (i.e. Wigner-Ville representation which is a pseudo- representation of the energy since it can be negative). A basic algorithm is used to compute the wavelet coefficients by means of numerical methods for obtaining results whose time and space complexities are more reduced. The results are exact and do not admit any approximations due to the truncation of the wavelet since the wavelet is taken on the same number of samplings. The decomposed ultrasound signals for undamaged stratified composite material are shown in Figures 3(a), 3(b) and 3(c). While in Figures 6(a) and 6(b), decompositions of damaged stratified composite materials are presented. For both materials, the front surface echo can easily be detected. For the undamaged material, the noise could be eliminated by suppressing some wavelet coefficients not corresponding to any current frequency. So, the composite materials modify the frequency (they act like filters) and degrade the frequency. When the material is undamaged, the oscillogram presents only oscillations due to the entering side (a) and those due to the bottom of the piece of material (b) (Figures 3). On the other hand, if there is a defect near the surface of the damaged material, the oscillogram presents two overlapped oscillations: echo of the surface of the investigated material and the one due to the defect (Figures 7(a) and 7(b)). Comparing with non-defect (Figures 3) and defect signals (Figures 7(a) and 7(b)) using the wavelet transform, the results of the decomposition show that specific signal characteristics appear only at defect regions.

(b) Figs. 6. (a) Ultrasound Measured Signal with Delamination Defects, (b) its Wavelet transform

(a)

(b) Figs. 7. (a) Ultrasound Measured Signal with Porosity Defect, (b) Its Wavelet transform

This provides that the amplitude of the waveform and the peak magnitude of frequency are parameters of defects (delamination, porosity) [18]. For the analysis of the artificial defect relating to the delamination, the default echo represented in the ultrasound signal, the experimental s1(t) in Figure 8 has been detected at t2 while t1 is the entering echo. The time separating the entering echo and the defect echo denotes the time of flight.



818


Compared to the referenced values, the error represents 4.2% in the first case and 6.21% in the second case. Those are quite good results.

V.

The main objective of this work was to detect manufacturing defects and calculate their thicknesses using ultrasonic immersion technique. The use of a numerical signal processing technique, based on the FWT was applied. The method was optimized and implanted for detection and classification of delamination and porosity defects in composite materials. Since the information about the signal requires a large amount of computation time and resources, a technique to reduce the dimensions of the sampling signals was used. The implementation presented improves the time complexity of the basic algorithm by an exponential factor while compressing the data sample. The scalogram characterizes the distribution of the signal energy in the time-frequency plane. In echograms, the wavelet analysis is useful for locating defects in some composite materials. A signal is emitted then received, and the analysis of the perturbed area of the received signal allows to localize exactly the area of the defect, since having the velocity of the beam and the time the perturbation appears, the distance is just velocity times time. It would be interesting to establish in a practical way, in each domain a semantic better adapted to simplify interpretations of the results. The treatment of the signals by the wavelet transform makes it possible on one hand, to better understand the physical phenomena brought into action and on the other hand, to improve the ratio signal noise of the impact echo signals. Starting from a visual inspection of ultrasonic echoes, delamination and porosity defectiveness can be illustrated. In case of delamination, the decomposed signal presents an initial peak initiated by the reflection of the ultrasonic signal on the medium surface and a second peak initiated by the reflection on the defect. While for porosity, the first peak is due to the reflection of the ultrasonic signal on the medium surface while the other peaks are generated by the reflection on the defect which, in this case is represented by multiple peaks. The peak amplitude in the case of porosity flaw is strongly lower than that of the delamination case.

Fig. 8. Delamination defects for the experimental signal

For s2(t), in Figures 9, τ=0.757µs. The depth of the artificial defect is computed by: d = V ⋅τ / 2 τ = t2 − t1

(4)

where V is the velocity and τ represents the time of flight. Taking V= 3.916×106 mm/s, d= 1.482mm for s1 (t). For theoretical validation, we presented an analytical signal (s2(t)) that simulates an overlapping resulting from two signals: an interface signal and a reflected signal caused by an obstacle (bottom echo). Performing the same calculations on s2(t), τ= 0.4455 µs and d=0.872mm. The comparative results are shown in Table I.

(a)

References

(b) [1] Figs. 9. (a) An analytical signal without noise and (b) its decomposition TABLE I COMPARISON OF THEORETICAL MEASURES Measured Performed Reference Defect Measurement Measurement using the method Experimental s1(t) τ = 0.757 µs τ = 0.766 µs Simulated s2(t) τ = 0.4455 µs τ = 0.475 µs

Conclusion

Performed depth (mm)

[2]

d=1.482 d=0.872

[3]


Med. O. Si-Chaib, N. Rezig, A. Yahiaoui, M.S. Bouamrane, Y. Chevalier, A. Nour, “Nearly Detection and localisation by ultrasounds in thin composite plates by using the wavelet coefficients analysis.”, 3éme Congrés International, “Conception et Modélisation des Systèmes Mécaniques”, CMSM 2009 A. Yahiaoui, Med. O. Si-Chaib, M. Benantar, “Contribution of the Wavelet Transforms to the Analysis of the Composites”, Advanced Materials for Applications in Acoustics and Vibration, Cairo, Egypt, 4-6, Jan 2009. Bilgutay N.M., R. Murthy, U. Bencharit, and J. Saniie, Spatial Processing for Coherent Noise Reduction in Ultrasonic Imaging," Journal of the American Statistical Association, Feb 1990, Vol.


819


[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17] [18]

87, No. 2, pp 728-736. Guilherme Cardoso, Jafar Saniie, Data Compression and Noise Suppression of Ultrasonic NDE Signals Using Wavelets, Ultrasonics, 2003 IEEE Symposium on , vol.1, no., pp. 250- 253 Vol.1, 5-8 Oct. 2003. Ramazan Demirli, Jafar Saniie, Model-Based Estimation of Ultrasonic Echoes Part II: Nondestructive Evaluation Applications, Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on , vol.48, no.3, pp.803-811, May 2001. Soman, A.K., and P.P. Vaidyanathan, "On Orthonormal Wavelets and Para unitary Filter Banks," IEEE Transactions on Acoustics, Speech and Signal Processing, Mar 1993, Vol.41, pp 1171183. Li, X., N.M. Bilgutay and R. Murthy, "Spectral Histogram Using the Minimization Algorithm: Theory and Applications to Flaw Detection," IEEE Transactions on Ultrasonic Ferroelectrics and Frequency Control, Mar 1992, Vol. 39, No. 2, pp 279-284. R. Kazys, D. Pagodinos, O. Tumsys, Application of the HilbertHuang Signal Processing to Ultrasonic Nondestructive Testing of Composite Materials, ISSN 1392-2114 ULTRAGARSAS, Nr. 1 (50). 2004. R. Kazys, O. Tumsys, D. Pagodinos, A new ultrasonic technique for detection and location of defects in three-layer plastic pipes with a reinforced internal layer, ISSN 1392-2114 ULTRAGARSAS (ULTRASOUND), Vol. 63, No. 3, 2008. Xin, J.Q., Detection and Resolution of Multiple Targets Using Time-Frequency and Deconvolution Techniques, PhD Thesis, 1994, Drexel University. Murthy, R., "Statistical Characterization of Frequency Diverse Signals," PhD Thesis, Drexel University, 1992. Rioul, O. and M. Vetterli, "Wavelets and Signal Processing," IEEE Signal Processing Magazine, Oct 1991, pp 14-38. Daubechies, I., Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992 IEEE Transactions on Information Theory, Special Issue, Sep 1990, Vol. 36, No. 5. Mallat, S., "A Theory for Multiresolution Signal Processing: The Wavelet Representation," IEEE Transactions on Pattern Analysis and Machine Intelligence, Jul 1989, Vol. 11, pp 674-693. Bilgutay, N.M., J. Saniie, E.S. Furgason, V.L. and Newhouse, "Flaw-Torain Echo Enhancement", Proceedings of Ultrasonics International, May 1979, pp 152-157. Cavaccini G., Agresti M., Borzacchiello G., An evaluation Approach to NDT Ultrasound Process by Wavalet Transform , 15th World Conference on NDT - 2000 - Rome (Italy). Saniie, J., K.D. Donohue, and N.M. Bilgutay, "Order Statistics Filters as Post- detection Processors," IEEE Transactions on Acoustics, Speech and Signal Processing, Oct 1990,Vol. 38, No. 10, pp 1722-32. Kaya, O., "Ultrasonic Target Detection Using Wavelet Decomposition”, MS Thesis, 1994, Drexel University. S. Lee, Y. Ha, J. Lee, J. Byun, ‘Experimental Evaluation of Delamination in CFRP using laser-based Ultrasound, IVth NDT in PROGRESS. November 05–07, 2007, Prague, Czech Republic.

Amina Benhamou, born in Tlemcen , Algeria, June.08th ; 1974. Organization: Boumerdes University, Algeria. Doctor; maitre de conference. Chief of researcher groups; in motor’s dynamics and vibro acoustic Laboratory. Education: Doctor es sciences physiques , in materials and energetic (university of Tlemcen – Algeria 2011 ; Master of energetic (University of Mostaganem, Algeria, 1998), Engineer in energetic (University of Mostaganem, Algeria, 2001), Experience: Large experience in research, CDER (2001-2003) Asistant Professor (2003-2010). Boumerdes University Main range of scientific interests: Renewable energy, Solar Drying,Thermodynamic properties of Food, Food storage Heat transfer; electricity and wavelet, signal treatment. Publications: 7 papers in international scientific journals, 35 communications in international meetings. Boumediene Benyoucef, born in Tlemcen, Algeria, Sep. 24, 1950. Did his Ph.D. degree in New Energies at Paris VI University, in 1986 He is the Director of “Research Unit of Materials and Renewable Energies” Abou Bekr Belkaid University of Tlemcen, Algeria. He has authored more than 400 papers published in international/national conference proceedings and technical journals in the area as well as many patents. He supervised more 120 Ph.D. & M.Sc. theses. He is a Scientific Expert in Materials and Renewable Energies and expert of “Ministère de l’Enseignement Superieur et de la Recherche Scientifique”, Algeria. He chaired a number of international congresses. Prof. Benyoucef is an active member of the Renewable Energies International Society.


Motor’s Dynamic and Vibroacoustic Laboratory, University of Boumerdes, Algeria. E-mail: [email protected] 2 Materials and Renewable Energies Laboratory, University of Tlemcen, Algeria.



820


Development of a Screening Tool for Cervical Cancer of the Uterus by Artificial Intelligence Tools Using the Uterine Cervico – Smears Guesmi Lamia1,3, Nabli Lotfi1,2, Bedoui Mohamed Hédi3 Abstract – Research in the field of cyto-pathology was able to develop artificial intelligence systems for the diagnosis based on the development of new analytical technologies and segmentation of the image cell. These tools are intended to facilitate the task of the expert without pretending to replace him. The objective of this work is to present a means for detecting cancerous and precancerous lesions using the technique of human and automatic intelligent supervisor. Three types of supervisors were tested; the first based on a neural network and has a success rate of 43.3%, the second by the technique of fuzzy logic with a success rate equal to 56.7% and a third by a neuro-fuzzy approach with a success rate of around 94%. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Cancer of the Cervix, Uterine Cervico - Smears (UCSs), Artificial Intelligence, Supervisor, Fuzzy Logic, Neural Networks

I.

-

Abnormalities in the structure of nuclei: regular and fine-grained, granular with masses of chromatin, Opaque, Nucleolus voluminous, hyper chromatic. The analysis of these parameters can give a classification of precancerous lesions and cancer of the cervix.

Introduction

Pathological Anatomy and Cytology, we distinguish two types of reviews: Histology is the observation that the cutting of tissue and cytology, the examination of a spreading cell. In the latter, samples are spread on a slide and then fixed and stained to recognize the different cells present. The smears are then examined under a microscope by a cytotechnologist to identify cells of interest. This step of reading the blade is a visual assessment of these cells on a slide for cytology. The purpose of this step is either the detection of abnormal or suspicious cells or quantification of cells. This is therefore a key concern of the cyto-pathologist must establish a reliable and valid diagnosis especially in the case of classification UCSs for cancer screening of the cervix. In order to assist the cyto-pathologist, we tested the artificial intelligence tools to facilitate this task with a very high success rate based on the technical supervisor of human and automatic after having illustrated a priori information used to recognize the cells as size, shape, texture and color above. • Background on the achievement of Uterine Cervico-Smear (UCS) The smear is a cytological examination, representing a range of existing cells in the ectocervix and endocervix uteri. Before being considered by the cyto-pathologist, these cells go through two stages: the smears and staining using the Papanicolaou (Pap smear) [1], [2], [3]. We note that the cell undergoes structural deformations carcinogen [4], [5] as follows: - Abnormalities of cell shape: Polygon, Oval, Round, ellipsoid, irregular, elongated. - Abnormalities in the shape of nuclei: Oval, Round, Irregular, Multi nuclear.

II. II.1.

Segmentation of UCSs Elimination of Inflammatory Cells

In order to get rid UCSs the inflammatory cells which obscure the image to be processed (Fig. 1), these cells recognize the same shades of color than the nuclei of cells to be tested, we spent the gray level of the image to properly locate and illustrate these cells (Fig. 2) applications using the programming software MATLAB. After the cleaning step, we turn to the segmentation of the UCS. II.2.

Image Segmentation in the Experimental Phase

Segmentation plays a very important role in pattern recognition. It was after this stage that we can extract the parameters of quality that will be defined later. It may happen that we have more than one cell in the image examined. We then separated the whole cell from other cells that may appear on the edges of the image, that is to say, incomplete cells. Here we enumerate the steps required to segment such an image: Playing the picture; Detecting internal and external contours of a cell; Linearization of the outlines of objects bulging contours of objects filling internal objects; removing objects exceeding the image border,


821


Guesmi Lamia, Nabli Lotfi, Bedoui Mohamed Hédi

smallest circle included the nucleus; SN2: Surface of the largest circumscribed circle to the nucleus; SC1: size of the smallest circle including the cytoplasm and SC2: Surface of the largest circumscribed circle to the cytoplasm.

and the softening of the object and segmentation (outer contour only). We reached an UCS segmented as follows (Fig. 3).

IV.

Classification Methodologies by Supervisor of UCS

The implementation of the supervisor in systems including human operators requires the realization of an intelligent interface [6], [7]. This interface also adapts to changes in the human operator. It is necessary to take into account: its experience operating system, physical condition and mental state especially for the cytopathologist. All these criteria correspond to imperfect information modeled by fuzzy sets based on fuzzy logic [6], [7], [8]. The supervisor makes a number of models that simulate the relationship between man as an operator and the automation as a tool. These models are: Model Rasmussen [9] Model CMO [10] (Optimal Control Model), Model of regulation of human activity [11] models the technical system [10], and model decision making human [10]. For the structure of cooperation between the automatic and the operator man, two structures are used: vertical cooperation and horizontal cooperation and dynamic task allocation [11]. We selected as a model of the supervisor of human decision making is characterized by a human solver's main problem (if the doctor). And we have chosen as the structure of mixed vertical and horizontal cooperation which it has the operator as responsible for all process variables and it may appeal, if necessary, the tool helps the decision will provide advice.

Fig. 1. UCS: the background is inflammatory and contains blue parasites. The squamous cells showed clear perinuclear halo (Papanicolaou staining)

Fig. 2. UCS: grayscale contrast: the background is dark black inflammatory

V.

Supervisor Application Using the Tools of Artificial Intelligence

In our case, we focus on the diagnosis-based models and database of a hybrid system because our support, equal to 120 UCSs of distributed equally into four classes of cancer (Cancer (C), High Grade Dysplasia (HGD), Lower Grade Dysplasia (LGD) and Normal (N)). These smears will be processed and exploited to extract quantitative information. The latter will form the basis of data for the development and testing of the tool to develop diagnostic techniques based on Artificial Intelligence (AI).

Fig. 3. UCS segmented: we can determine the overall shape of the cell carcinogenic and size of its nucleus and the cytoplasm

III. Shape Parameters After segmentation of the image we had in carcinogenic cell sorting parameters to locate the most appropriate forms, each grade cancer (class) is described by the parameters of quality based on the shape of the nucleus and cytoplasm more significant. They represent the best indicators used for the diagnosis and achieve the classification of cancer. Their values are measured manually to reach the extract. We chose eight shape parameters using the physician are: E1 = (DFMax) kernel / (DFMax) cytoplasm; E2 = (DFMin) kernel / (DFMin) cytoplasm; DF1 = (DFMin / DFMax) cytoplasm; DF2 = (DFMin / DFMax) kernel ; U1 = SN1/SC1, U2= SN2/SC2; SC1/SC2 and SN1/SN2, with: DFMax: Maximum Feret diameter; DFMin: Minimal Feret diameters; SN1: Surface of the

V.1. 9

Supervisor Achieved by the Technique of Neural Networks (Multilayer Perceptron(MLP))

Size a multilayer perceptron MLP

Size a multilayer perceptron returns to the determination ofnumber of neurons in each of two layers that constitute it. For the input layer that amounts to the number of existing information at the outset to achieve the expected result; so for every information he



822


associated a neuron that carries its characteristics. For the output layer, depending on the nature of the outcome is expected. If you need "n" numbers out, must be "n" neurons in the output layer, as for the perceptron monolayer. For the hidden layer, things are much simpler. There is no law, no rules, no theorem that would determine the number of neurons in the hidden layer set for a neural network optimal. The method of approach it is to try "random" number of numbers of neurons in hidden layer to be of the most convincing possible after training. However, there are results for certain types of functions we wish to approach. Unfortunately, none of these findings is yet generic enough to be exploited in normal cases [12], [13]. 9

-

9

certain number. The choice of the initial architecture of the network remains a difficult problem. This choice can be made by experience.Methods called "self-constructive" exist: it is to add neurons during learning so that learning is done well. Choice of weight

Despite this limitation, the threshold perceptrons have an interesting property that occurs there is a (actually several) algorithm (s) that allow (s) to a perceptron to adjust its weight to a set of examples so to obtain the classification for this set expected. Thus, if the set of examples is large enough (the examples are quite varied). We can obtain a perceptron that will work for appropriate examples not met.

Learning Multilayer Perceptron

The learning phase of a MLP as monolayer perceptron is based on an algorithm to adjust the weights against a set of examples. This algorithm is called back propagation gradient (see Appendix A) [13], [14]. This algorithm uses the same rule of weight change ("delta rule") that the Widrow-Hoff algorithm. The algorithm will be given in its most general, i.e. with several hidden layers. The Widrow-Hoff algorithm or "delta rule" seeks to reap the errors on each such test to reach the correct weight. This method developed by Widrow and Hoff is to modify the weights after each example, and not after all have examples show. This will therefore minimize the error precisely, and that on each instance. Instinctively, we see that the neural network will improve much better and much faster will tend to be classified perfectly (almost) each of the examples, although more efficient methods still exist (see Appendix B). It should be noted that the application several times this algorithm (Widrow-Hoff) help to refine the error correction and to obtain a neural network more efficient. Be careful, because applying too many times lead to what is called "overfitting" (over-learning), meaning that your network is very efficient on the examples used for the learning, but little or no fails to generalize for any information [13]. Returning to the algorithm for back propagation gradient characterized by a set of properties, listed as follows: - The algorithm back propagation gradient is an extension of the algorithm of Widrow-Hoff. Indeed, in both cases, the weights are updated each presentation as an example and so it tends to minimize the error calculated for each example and not the overall error. - The method gives good practical results. In most cases, we encounter few problems that accrue to local minima, but there are a. However, it is less efficient than other algorithms, error propagation: it tends to weight more slowly, more or less optimal. - There is no stop condition to repeat it. It was at the computer to set the standard. We can repeat this example until the error on each sample drops below a

9

Our selected case

Neural networks have several types; the most used is the MLP which is characterized by its learning algorithm and the back propagation of errors [15]. The MLP is an artificial neural network-oriented organized in layers, where information flows in one direction, from input layer to output layer. The input layer is always a virtual layer containing the inputs to the system [16]. The following layers are the hidden layers that admit "m" number of layers according to the need for resolution of the system. It ends with the output layer representing the results of classification system [17]. In the case suggested, the MLP consists of an input layer which is formed of eight shape parameters that are specified above, a hidden layer which consists of 26 neurons and an output layer which is composed of four neurons formants the top four grades of cancer cervix. This type of perceptron is chosen after several tests in which varied in the number of neurons in the hidden layer, the learning period and the vector normalization. We used activity as a function of the "Log-Sigmoid" with the normalization vector as V = [0.01, 0.99], because the database is formed by values between 0 and 1. The simulation results The results are summarized in Table I below: Success rate of HGD = 17/30 = 56.7%

(1)

Success rate of LDG = 12/30 = 40%

(2)

Success rate of C = 10/30 = 33.3%

(3)

Success rate of N = 13/30 = 43.3%

(4)

The overall success rate = = (17 +12 +10 +13) / 120 = 43.3%

(5)

We note that the overall success rate of neural networks is low. We used another technique of artificial



823


•

intelligence to improve it: it is the technique of fuzzy logic.

Membership function of "E2"

TABLE I EVALUATION OF THE TECHNIQUE OF NEURAL NETWORKS HGD

LGD

C

N

HGD

17

03

05

05

LDG

03

12

07

08

C

13

04

10

03

N

08

04

05

13

V.2.

Supervisor Achieved by the Technique of Fuzzy Logic [18]-[20]

Fig. 5. Membership function of “E2” E2min= 0 ; E2Topt= 0,1859; E2opt=0,3388; E2moyop=0,4288; E2moypes= 0,6096 ; E2pes=0,6115 ; E2Tpes=0,9983; E2max=1.0000

The construction of a model based on fuzzy logic for diagnosis and classification is to move mainly by three main stages: fuzzification, inferences and déffuzzification. To achieve these three steps, we must determine the membership functions of input variables and output ones.

•

9 Definition of membership functions of input variables and output: The membership functions are most commonly used form: Singleton, triangular, trapezoidal, Gaussian. We retained the trapezoidal shapes for ease of coding and manipulation, so they cover the entire range of variation of shape parameters. The number of trapezoidal shape in the representation of the membership function will be kept to a minimum four to describe the scope of a variable. In this case, they will be associated with the terms: low - medium - large-very important [21]. We associate a variable (X) the following values: minimum (Xmin), very optimistic (XTopt), optimistic (Xopt) average optimistic (XmoyO) average pessimistic (XmoyP), pessimistic (Xpes), very pessimistic (Xtpes ) and maximum (Xmax).

Membership function of « DF1 »

Fig. 6. Membership function of “DF1” DF1min= 0 ; DF1Topt= 0,3834; DF1opt=0,3908; DF1moyop=0,4846; DF1moypes= 0,8331; DF1pes=0,936; DF1Tpes=0,9601; DF1max=1.0000

•

Membership function of « DF2 »

• Membership function of "E1"

Fig. 7. Membership function of “DF2” DF2min= 0 ; DF2Topt= 0,4208; DF2opt=0,4934; DF2moyop=0,5191; DF2moypes= 0,9032; DF2pes=0,9777; DF2Tpes=0,9831; DF2max=1.0000

Fig. 4. Membership function of “E1” E1min = 0; E1Topt = 0.1344; E1opt = 0.265; E1moyop=0,4076 ; E1moypes=0,5172 ; E1pes=0,6681 ; E1Tpes=0,881; E1max=1.0000



824


•

•

Membership function of « U1 »

Fig. 8. Membership function of “U1” U1min= 0 ; U1Topt= 0,0187; U1opt=0,081; U1moyop=0,1741; U1moypes= 0,1763; U1pes=0,6224; U1Tpes=0,9627; U1max=1.0000

•

Fig. 11. Membership function of “SN1/SN2” SN1/SN2min= 0 ; SN1/SN2Topt= 0,2087; SN1/SN2opt=0,2102; SN1/SN2moyop=0,2425; SN1/SN2moypes= 0,8837; SN1/SN2pes=0,9145; SN1/SN2Tpes=0,9909; SN1/SN2max=1.0000

•

Membership function of « U2 »

Fig. 9. Membership function of “U2” U2min= 0 ; U2Topt= 0,0152; U2opt=0,0807; U2moyop=0,1654; U2moypes= 0,2899; U2pes=0,4771; U2Tpes=0,808; U2max=1.0000

•

Membership function of « SN1/SN2 »

Defining the membership function of output variable "S"

Fig. 12. Membership function of “SN1/SN2” Smin = 0, STopt = 0.15, Sopt = 0.17, SmoyO = 0.25, SmoyP = 0.27, Spes = 0.4, STpes = 0.5 , Smax = 0.7

9 Fuzzification: After defining the membership functions according to the expert, we had to convert our input parameters of numerical values of physical quantities in translating linguistic variables. The fuzzy logic algorithm requires accurate numerical variables (1, 2, 3, 4, 5, ...) to be exploited later by the algorithm, then the eight input parameters will admit four lexical classes: "LOW" "MIDDLE" "IMPORTANT" and "VERY IMPORTANT" where we have allocated there, this lexical fields, respectively, the values '1 ', '2', '3 'and '4'. The output vector allows four lexical classes "HGD", "LGD", "C" and "N" where we've combined their respective values '1 ', '2', '3 'and '4' [21].

Membership function of « SC1/SC2 »

9 Rules of inference or inference: This block is composed by all the fuzzy rules that exist between input variables and output variables, expressed both in linguistic forms. We have chosen as a method of inference method "MAX-MIN". It is based on the use of two logical operations: the "AND" logic associated with

Fig. 10. Membership function of “SC1/SC2” SC1/SC2min= 0 ; SC1/SC2Topt= 0,1537; SC1/SC2opt=0,1742; SC1/SC2moyop=0,2079; SC1/SC2moypes= 0,754; SC1/SC2pes=0,8024; SC1/SC2Tpes=0,9132; SC1/SC2max=1.0000



825


the minimum and the "OR" logic associated to the maximum. These rules are of the form: IF (AND ... ....) THEN (decision) OR .... [21]-[25]. We have 185 possible rules of inference in the form of the maximum number of rules.

The simulation results We reached these results to evaluate the following approach (Table III):

9 Defuzzification: The inference methods provide a membership function resulting "µ" for the output variable "X". It is therefore fuzzy information that must be transformed into physical quantity. This is the inverse of the conversion phase "Fuzzification" [22]. The simulation results We have summarized the results of this technique in Table II below: (6)

Success rate of LDG = 20/30 = 66.7%

(7)


(8)

Success Rate of N = 18/30 = 60%

(9)

The overall success rate= = (13 +20 +17 +18) / 120 = 56.7%

(10)

C

N

HGD

13

02

15

00

LDG

05

20

00

05

C

11

02

17

00

N

00

11

01

18

Cervico graphy

LDG

The overall success rate of the technique of fuzzy logic is better than the neural network. To improve it, we have combined the two approaches: the hybrid technique neuro - fuzzy. Supervisor Directed by the Hybrid Approach : Neuro-Fuzzy

Speculoscopy

V.3.

Success rate of LDG = 29/30 = 96.7%

(12)


(13)

Success rate of N = 28/30 = 93%

(14)

The overall success rate = = (30 +29 +26 +28) / 120 = 94%

(15)

TABLE IV EVALUATION OF THE HYBRID APPROACH: NEURO- FUZZY (PART I)

TABLE II EVALUATION OF THE TECHNIQUE OF FUZZY LOGIC HGD

(11)

TABLE III EVALUATION OF THE HYBRID APPROACH: NEURO- FUZZY HGD LDG C N 30 0 0 0 HGD 0 29 0 1 LDG 26 0 4 0 C 0 2 0 28 N

Methods

Success rate of HGD =13/30 = 43.3%

Success rate of HGD = 30/30 = 100%

Polaprobe

Such a hybrid system is characterized by the following features: learning is performed by an algorithm derived from a neural network, the general architecture of this system is represented by a recurrent network, and this system is interpreted in terms of rules the form "if ... then ...." Learning is done from the semantics of the underlying fuzzy model, thus preserving the linguistic interpretability of the model. This model performs a function approximation. Therefore, the use of neuro-fuzzy approach overcomes the drawbacks inherent in each of the previous approaches (neural networks and fuzzy logic), while retaining their respective advantages [25], [26], [27].


Procedure

Benefits

Disadvantages

NONCYTOLOGICAL METHODS [29] Photograph of the * Detection * Low sensitivity cervix acid of lesions for lesions application cervical minors. high grade. acetic acid. *Rate Review 4% defective 28%. * Reconvening of patients. * Low sensitivity for lesions high grade. *Rate Review 4% defective 28%. * Reconvening of patients. Based on the * The * To the principle association UCS association a reflection of the increases the UCS decreases fluorescent light sensitivity specificity of 99% proportional to 31% to 83%. to 87%, and the intensity value keratinization of positive predictive cells 90% to 47%. epithelium of the cervix. *Increased rate of reconvening 2.6% of patients to 14%. OptoSafety. Ease of * Relevance electrical technolo work. non diagnostic gy Fast results. confirmed. based on the use Low cost. combined pulse Useful in screening low electrical mass. voltage and pulse light frequency variables.


826


RAD

Cryotherapy

Methods

TABLE V EVALUATION OF THE HYBRID APPROACH: NEURO- FUZZY (PART II)

Procedure

Benefits

VII.

The supervisor directed by the hybrid approach neuro - fuzzy won the record classification UCSs a success rate Total of about 94% that we can improve if we increase our database by other cases of precancerous lesions and cancerous encountered.

Disadvantages

CYTOLOGICAL METHODS [29] *Cryotherapy is a Cryotherapy is a * Possible side method of outpatient effects: loss aqueo relatively simple treatment, quick and method that us easy (no more destroysprecancer than 15 minutes), ous which does not cells by cryoprese require anesthesia. It rvation of the can be cervix, using performed safely acompressed and effectively by gas coolant such both general as dry ice practitioners than (CO2) or liquid nitrogen(N2O). by non-physician * Efficiency: 8695% * Anaesthesia: Not required * Relative cost: Low Also known ARD has the double * Possible side as excision of the advantage, first, to effects: bleeding cove off be a simple surgical * Anesthesia: Loc the transformation procedure and, al zone secondly, to provide anesthesia require (LLETZ for Large a sample of excised d Loop Excision of tissuethat can * Sample tissue fo thebe sent to r histopathology: Processing Zone), the histopathology yes LEEPuses a thin * Efficiency: 91* Relative Cost: electric 98% * High wire, shaped hand le, to remove the regionabnormal ce rvical. It is usually performed under colposcopic control and under local anesthesia, in centers of secondary or tertiary care

VI.

Conclusion

Acknowledgements I especially thank my supervisor Mr. Mohamed Hadi Bedoui and my co- encadror Mr. Lotfi Nabli for their help in setting up my database and the moral support throughout my project.

References [1]

[2]

[3]

[4]

[5] [6]

[7] [8]

Discussion

[9]

The success rate achieved through the three approaches to artificial intelligence is very important especially if the neuro-fuzzy can be improved if we increase our database used. Neuro-fuzzy method has areas of confusion between two classes that are close to the point of view as a cell carcinogen, in the case of couples (HGD, C) and (LDG, N). As a remedy, we can add another shape parameter that expresses a specific quality indicator or as we mentioned earlier we grow our database. Despite these false positive cases, this method is the refuge of screening for cancer cervix now. In Tables IV and V, other means of diagnosis of Cancer Cervix are reported.

[10]

[11]

[12]

[13]

[14]


Telmoudi A. J., Chetouani Y., Nabli L., M’hiri R., Réseau de neurones doublement récurrent à base de fonctions radiales : RR2FR Application au pronostic prévisionnel, Journal Européen des Systèmes Automatisés, JESA, VOL 43/7-9, pp.871-887, 2009. Baillet. Cancérologie. DCEM3. Service de radiothérapie. Faculté de médecine ; Pierre et Marie Curie. Université. Paris-VI. pp.105. 2002 – 2003. Mise à jour: 6 janvier 2004. GRAISyHM GT5 : Méthodologie de Conception des Systèmes de Supervision. Développement d’une plate – forme expérimentale dédiée à la supervision. Thème B12 : distribution des traitements entre automatismes et opérateurs humains. (Permanent: LAMIH : B.Riera ; Doctorants : LAMIH : G.Martel, E.Chérifi, M.Lambert) .08/08/2007. http://www.univ-valenciennes.fr/graisyhm/rapgt59798. Romaine. R ; Etude pilote de dépistage du cancer du col de l’utérus dans une région rurale camerounaise ; Thèse de doctorat en médecine ; Université de Genève ; Faculté de médecine ; Département de gynécologie-obstétrique ; pp.34-5; 2002. Body. G; PH. Descamps; ML. Jourdan; et al. Néoplasie intra épithéliale du col. EMC 597-A-10 60-200-A-10. Renard-Oldrini. S ; Implémentation de la radiothérapie conformationnelle avec modulation d’intensité dans les cancers du col utérin au centre Alexis Vautrin ; Thèse pour le doctorat en médecine ; Université Henri Poincaré, Nancy 1 ; Faculté de Médecine Nancy ; pp.24-20; 11 Juin 2010. P. Millot. Supervision des procédés automatisés et ergonomie .Edition HERMES. Paris. Pp.5; Décembre 1988. Anastadiadou. M ; Imagerie Polarimétrique : Développements Instrumentaux et Applications Biomédicales ; Thèse de doctorat de l’école polytechnique ; CNRS ;pp. 135-131; 11 Décembre 2007. Rouidi .B, Larabi. M. S. ; Diagnostic Neuro-Flou : Application à la machine asynchrone ; Thèse de doctorat ; Faculté des sciences et sciences de l’ingénieur ; pp.34-17;Juin 2005. M. Hoc J. Supervision et contrôle de processus, La cognition en situation dynamique. Presses Universitaires de Grenoble, pp.12; 1996. Argent. D. Cancer du col de l’utérus, épidémiologie anatomie pathologique diagnostic évolution principes du traitement, dépistage. pp.49: 1933-1923; Rev Prat (paris) 1999. Hafiane.M.L. ; Conception d’un capteur de pression intelligent ; Mastère en micro électrique ; Option IC Design ; Université de Batna; faculté des sciences de l’ingénieur ; pp18 ; 28/12/2005. Mahul. A. ; Apprentissage de la qualité de service dans les réseaux multiservices : applications au routage optimal sous contraintes ; Thèse doctorat ; Université BLAISE PSCAL – CLERMONT-FERRAND II ; Spécialité Informatique ; pp37, 3433 ; 23 Novembre 2005. Locatelli .L. ; Direct search for higgs boson in LHCB and contribution to the development of the vertex detector ; Thèse doctorat ; Ecole polytechnique fédérale de LAUSANNE ; pp8786 ;26 Octobre 2007.


827


[15] M. Da Silveira. La distribution avec redondance partielle de modèles à événements discrets pour la supervision de procédés industriels. Thèse de l’Université Paul Sabatier de Toulouse. Laboratoire d’analyse et d’Architecture des Systèmes du CNRS. Version Provisoire ; pp.73-54; 16/07/2003. [16] Racoceanu.D; Contribution à la surveillance des Systèmes de Production en utilisant les Techniques de l’Intelligence Artificielle ; Habilitation à diriger des recherches ; Université de Franche – Comté de Besançon ; pp.197-189; 19 Janvier 2006. [17] Telmoudi A. J., Chetouani Y., Nabli L., M’hiri R., Réseau de neurones doublement récurrent à base de fonctions radiales : RR2FR Application au pronostic prévisionnel, Journal Européen des Systèmes Automatisés, JESA, VOL 43/7-9, pp.871-887, 2009. [18] Cimuca. G. - O.; Système inertiel de stockage d’énergie associe à des générateurs éoliens ; Thèse doctorat. Spécialité Electrique ; Ecole nationale supérieure d’arts et métiers centre de Lille ; pp162 ; 2005. [19] Guesmi. L ; Contribution au développement d’un outil d’aide au diagnostic des lésions pré cancéreuses et cancéreuses par FCV ; Mastère : Automatique et Maintenance Industrielle ; Présenté à l’Ecole Nationale d’Ingénieurs de Monastir (ENIM) ; pp37-36 ; 16 Juin 2006. [20] Baghli.L. ; Contribution à la commande de la machine asynchrone, utilisation de la logique floue, des réseaux de neurones et des algorithmes génétiques ; Thèse doctorat ; UFR Sciences et Techniques : STMIA ; Spécialité : InformatiqueAutomatique-Electrotechnique-Electronique-Mathématique ; Université Henri Poincaré, Nancy-I ; pp31, 20-19 ; 14 Janvier 1999. [21] Nabli L., Ouni K., The indirect supervision of a system of production by the Principal Components Analysis and the Average Dynamics of the Metrics, International Review of Automatic Control (IREACO), ISSN: 2070-3961, vol.1 n° 4. pp 148-153, November 2008. [22] El Zoghbi. M. Analyse électromagnétique et outils de modélisation couplés. Application à la conception hybride de composants et modules hyperfréquences. Thèse doctorat. Université de Limoges. Discipline: Electronique des Hautes fréquences et Optoélectronique. Spécialité: "Communications Optiques et Microondes. pp.37; 14 Octobre 2008. [23] Nabli.L. Contribution à la Conduite des Systèmes de Production par L’utilisation des Techniques de l’Intelligence Artificielle. Habilitation Universitaire. Discipline: Génie Electrique. Université de Monastir Tunisie, 01 février 2010. [24] Coquelle. L ; Simulation de comportements individuels instinctifs d'animaux dans leur environnement, De la description éthologique à l'exécution de comportements réactifs ; Thèse de doctorat ; Université de Bretagne Occidentale; pp. 134-131; 01/12/2005 [25] Michaud. F ; Nouvelle architecture unifiée de contrôle intelligent par sélection intentionnelle de comportements ; Thèse de doctorat ; Spécialité: génie électrique et génie informatique ; Université de Sherbrooke ; Faculté des sciences appliquées ;pp. 72-59; 1995. [26] Baghli.L.; Contribution à la commande de la machine asynchrone, utilisation de la logique floue, des réseaux de neurones et des algorithmes génétiques. Thèse doctorat. UFR Sciences et Techniques: STMIA. Université Henri Poincaré, Nancy-I. pp.16; 14 Janvier 1999. [27] Främling.K.; Les réseaux de neurones comme outils d'aide à la décision floue. Rapport de D.E.A. Spécialité : informatique. Equipe Ingénierie de l'Environnement .Ecole Nationale Supérieure des Mines de Saint-Etienne. pp.12-11; Juillet 1992. [28] OMS; La lutte contre le cancer du col de l’utérus : guide des pratiques Essentielle ; p.287; 2007. [29] Manuel à l’usage des organisateurs ; Planification et mise en œuvre des programmes de prévention ; Alliance pour la Prévention du Cancer du Col 2006 ; Centre international de Recherche sur le Cancer (CIRC), JHPIEGO, Organisation panaméricaine de la Santé (PAHO), Program for Appropriate Technology in Health (PATH). Mention à utiliser en cas de citation: Alliance pour la Prévention du Cancer du Col (APCCP) ; Lyon (France) ;pp10-12; 2006.

[30] P. Millot. Systèmes Homme – Machine et Automatique. Université de Valenciennes et du Hainaut – Cambrésis. Laboratoire : LAMIH CNRS. Journées Doctorales d'Automatique JDA'99, Conférence Plénière, Nancy, pp.23-21; Septembre 1999.


National School of Engineers of Monastir, Road PO Box 5000 Kairouan, Monastir, Tunisia. 2 Higher Institute of Applied Science and Technology of Sousse Box, 4000 Sousse, Tunisia. 3 Laboratory Technology and Medical Imaging (TIM), Faculty of Medicine, Monastir 5019, Tunisia.

E-mails:

[email protected] [email protected] [email protected]

Guesmi Lamia was born in 01/07/1978 in Kef (Tunisia). In 2000 she was admitted in national competition for entry into engineering schools. In 2000-2003, she had graduated from Cycle Specialty in electrical engineering and admission to the National Engineering School of Monastir (ENIM), the graduation project is entitled "Improvement of transport of power, " With reference to Excellent. In 2003-2005, she admitted the Master "Industrial Maintenance and Automation" option "Industrial Maintenance" in stating ENIM obtained in the defense of the memory is fine. The thesis is entitled "Contribution to the development of a tool for the diagnosis of precancerous and cancerous lesions by FCV" and supervised by Mr. Lotfi Nabli (ENIM) and Dr. Sihem BEN HAJ SALAH HMISSA (CHU Farhat Hached Sousse - Tunisia). My research laboratory is named "Laboratory of Anatomy and Cytology Pathology CHU Farhat Hached Sousse - Tunisia" and chaired by Professor Sadok Korbi. She is preparing her doctoral thesis (fifth year) is entitled "Joint Monitoring Approach for the evaluation of the integration of human and automatic supervisors" in the laboratory (TIM) Technology and Medical Imaging, UR 08 - 27 , Laboratory of Biophysics Faculty of Medicine of Monastir – Tunisia seminars are: "The regulation and thermal energy in new buildings in Tunisia," in Hotel Abou Nawas Tunisia "Practical Methods for Energy," The Office Training: Office Chaar Ridha (CCR) - Elite International Training, Instructor: Dr. HSAIRI. And training on the UNIX server AIX to drive in the Laboratory of Anatomy and Cytology, CHU Farhat Hached, Sousse - Tunisia. With the trainer Mr Abderrazek Jemai. Nabli Lotfi is the Assistant Professor of higher education at the National School of Engineers of Monastir, Department of Electrical Engineering, a researcher at the Research Unit ATSI. In June 1989, he was the Master of Science and Technology of the Normal School Superior Technical Education Tunis option Electrical Construction In June 1991 DEA in Automatic Control, High normal School of Technical Education of Tunis, with distinction In 2000, he had the PhD in Automatic and Computer Engineering at the University of Science and Technology of Lille, supported at the Central School of Lille, Honours. It is entitled "Forward-looking Indirect Conditional Preventive Monitoring: Application to Textile spinning unit," which is its scientific director Professor E. CRAYE (Central School of Lille), currently director of the Central School of Lille. AndCo-Scientific Director is Professor TOGUEYNI AKA (Central School of Lille), currently director of the lens to IG2I.



828


Mohamed Hedi Bedoui received his PhD degree from Lille University in 1993. He currently teaches with the position of Professor of biophysics in the Faculty of Medicine of Monastir (FMM), Tunisia. He is a member of Medical Technology and image processing team (TIM), UR 08-27. His research interests are real-time and embedded systems, image & signal processing and hardware/software design in medical field, electronic applications in biomedical instrumentation. He is the president of the Tunisian Association of Promotion of Applied Research.



829

1974-6067(201109)4:5;1-S Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Automatic Control

Automatic Control

Suggest Documents