Adaptive Fuzzy Sliding Mode Power System Stabilizer Using Nussbaum Gain

Transcription

1 International Journal of Automation an Computing 1(4, August 213, DOI: 1.17/s Aaptive Fuzzy Sliing Moe Power System Stabilizer Using Nussbaum Gain Emira Nechai 1 Mohame Naguib Harmas 1 Najib Essounbouli 2 Abelaziz Hamzaoui 2 1 Electrical Engineering Department, Ferhat Abbas University of Sétif 1, Algeria 2 Centre of Research for Science an Information Technology an Communication Laboratory, Champagne Arennes University, France Abstract: Power system stability is enhance through a novel stabilizer evelope aroun an aaptive fuzzy sliing moe approach which applies the Nussbaum gain to a nonlinear moel of a single-machine infinite-bus (SMIB an multi-machine power system stabilizer subjecte to a three phase fault. The Nussbaum gain is use to avoi the positive sign constraint an the problem of controllability of the system. A comparative simulation stuy is presente to evaluate the achieve performance. Keywors: Multi-machine power system stabilizer, aaptive fuzzy, sliing moe, Nussbaum gain, Lyapounov stability. 1 Introuction Power systems are complex nonlinear systems that often exhibit low frequency oscillations ue to insufficient amping cause by averse operating conitions which can lea to a evastating loss of synchronism [1]. Power system stabilizers are use to suppress these oscillations an improve the overall stability [1 4]. The computation of the fixe parameters of these stabilizers is usually base on the linearize moel of the power system aroun a nominal operating point [5 7]. Operating conitions often change as a result of loa variations an/or major isturbances. These changes affect power system ynamic behavior which requires ajustment of stabilizer parameters. Keeping the latter at fixe values will greatly egrae power system performance [7]. Conventional stabilizers, using lea lag compensators have been base on linearize power system moel to amp oscillations. Disturbances, varying loaing conitions an therefore frequently changing operating point were not taken into consieration [8 11]. However, a lot of researches about the esign of power system stabilizers have been conucte, using a wie range of strategies, such as sliing controller [12], aaptive controller [13 15], an aaptive fuzzy controllers [16, 17]. A comparison of some approaches to esigning power system stabilizers was presente in [18]. One of these possible methos is the application of aaptive fuzzy sliing controller. Remarkable research effort has been one in the last ecae to put forwar intelligent fuzzy logic base power system stabilizer (PSS as well as optimality in aapting to changing operating conitions as in [19 21]. However, these linear moel base control strategies often fail to provie satisfactory results over a wie range of operating conitions. Moreover, uring severe isturbances, PSS action may actually cause the generator uner its control to lose synchronism in an attempt to control its excitation fiel. For the last few years, optimization techniques for a conventional [22 23] an ual PSS [24 26] have also been applie using ifferent algorithms such as particle swarm optimization (PSO, genetic algorithms (GA, chaotic opti- mization algorithm (COA an neuro-fuzzy system (NFS. Contribution mae in this paper consists in a new aaptive fuzzy sliing moe power system stabilizer using a Nussbaum gain. Stability of the overall system is guarantee via Lyapunov synthesis. In the following sections of this paper, a nonlinear power system moel is presente first, followe by the evelopment of an aaptive fuzzy sliing moe controller using Nussbaum gain an the aresse stability issue. In Section 3, the stuy of simulation results for ifferent operating conitions on single-machine an multi-machine power system is escribe. 2 Power system moel The power system moel consiere in this paper is a nonlinear moel representing a synchronous machine connecte to an infinite bus via a ouble circuit transmission line. Fig. 1 shows the power system schematic iagram incluing turbine, transformer, automatic voltage regulator an PSS [4]. Fig. 1 Single-machine infinite-bus (SMIB power system A nonlinear representation of the power system uring a transient perio after a major isturbance has occurre in the system [17, 27, 28] : { ẋ 1 = x 2 (1 ẋ 2 = f (x + g (x u [ ] T [ where x = x 1 x 2 = ω P M ] T R 2 is the Manuscript receive December 5, 212; revise April 16, 213

2 282 International Journal of Automation an Computing 1(4, August 213 state vector, ω is the spee eviation, P = P m P e is the accelerating power, M is inertia moment coefficient of the synchronous machine, u R is the input, f (x an g (x are nonlinear functions, an g (x in the controllability region (see the appenix. The block iagram of a conventional lea-lag power system stabilizer is shown in Fig. 2, in which a conventional single-input is presente. Parameters an etails can be foun in [22]. The two inputs of an IEEE power system stabilizer, unlike the conventional single-input PSS, is shown in Fig. 3, in which a ual-input PSS3B is presente. Parameters an etails can be foun in [24]. Fig. 2 Fig. 3 Conventional power system stabilizer Dual-input power system stabilizer 3 Aaptive fuzzy sliing moe power system stabilizer Consier a single-input-single-output (SISO nonlinear system escribe by (1, with the sliing surface as S = x 2 + βx 1 (2 where β is a positive constant. The time erivative of the sliing surface (2 is given by Ṡ = f (x + βx 2 + g (x u. (3 Assuring the existing conition (4 Hence, the control law is SṠ <. (4 u = g 1 (x ( f (x βx 2 ksgn (S. (5 Control law (5 allows ensuring the system stabilization an robustness, but it has the rawback in the computation of k, which is not a straightforwar task. In a more realistic case where f (x an g (x are unknown, they are replace by their fuzzy estimations. In previous researches on the inirect aaptive fuzzy metho, the controller with ĝ 1 (x can be singular because it cannot be guarantee that ĝ (x is not equal to zero at any moment, where ĝ (x enotes the approximation of g (x. In this paper, a Nussbaum gain will be incorporate into the control law in orer to estimate function ĝ 1 (x [29]. Definition 1 [3]. A function is calle a Nussbaum-type function if it has the following properties: lim y sup 1 y lim y inf 1 y y y N (ς ς = + (6 N (ς ς =. (7 Throughout this paper, the even Nussbaum function: ( π N (ς = cos ςe ς2 (8 2 is employe, an ς is a variable to be etermine later. Lemma 1 [3]. Let V ( an ς ( be smooth functions efine on [, t f, with V (t, t [, t f, an N ( be an even Nussbaum-type function. If the following inequality hols: V (t c + e c 1t e c 1t g (x (τn (ς ςe c 1τ τ+ ςe c 1τ τ, t [, t f (9 where c represents some suitable constant, c 1 is a positive constant, an g (x (τ is a time-varying parameter which takes values in the unknown close interval I = [ l, l +], with / I,then V (t, ς (t an g (x (τn (ς ςτ must be boune on [, t f. The proof of this lemma can be foun in [3]. Base on the universal approximation theorem, the unknown function f (x an constant k can be approximate by (1 an (11, respectively: ˆf (x, θ f = ξ T (x θ f (1 ˆk (x, θ k = ξ T (x θ k (11 where θ = [θ 1, θ 2,, θ m] T is the parameter vector an ξ = [ξ 1, ξ 2,, ξ m] T is the vector of fuzzy basis functions. The approximation error is given by δ f = f (x ξ T (x θ f (12 δ k = k ξ T (x θ k (13 in which θf an θk are the optimal parameter values. We efine θ f = θ f θf (14 θ k = θ k θ k. (15 Theorem 1. For the nonlinear system (1, if we choose the following control law ( u = N (ς ˆf (x βx 2 ˆksgn (S (16 with ς = S (θf T ξ (x + βx 2 an the following aaptation laws: (17 θ f = γ 1Sξ (x γ 1θ f (18 θ k = γ 2g (x N (ς S ξ (x γ 2θ k (19

3 E. Nechai et al. / Aaptive Fuzzy Sliing Moe Power System Stabilizer 283 such that δ f S ε, γ 1, γ 2 are positive constants an g (x = g (x, then the stability of the close loop system can be guarantee. Proof. Choose the Lyapunov function caniate to be Therefore, V = 1 2 ST S + 1 2γ 1 θt f θf + 1 2γ 2 θt k θk. (2 V = S T Ṡ + 1 γ 1 θt f θf + 1 γ 2 θt k θk = S T (f (x + g (x u + βx θt f θf + 1 θt k θk = ( γ 1 γ 2 S T ( g (x N (ς+1 ( θf T ξ (x+βx 2 θ f T ξ (x+δ f ˆkg (x N (ς sgn (S + 1 θt f θf + 1 θt k θk = γ 1 γ 2 [ g (x N (ς+1 ] ς θ f T Sξ (x+δ f S k g (x N (ς S T S S θ T k g (x N (ς S ξ (x + 1 γ 1 θt f θf + 1 γ 2 θt k θk. (21 Using (18 an (19, we can obtain V = k g (x N (ς S T S θf T S θ f θ T k θ k + δ f S + ( g (x N (ς + 1 ς. (22 The following inequalities are vali θ T f θ f 1 2 θ T f θ f θ f 2 (23 θ T k θ k 1 2 θ T k θ k θ k 2. (24 An we can rewrite V as V k g (x N (ς S T S 1 S 2 θ f T θ f 1 2 θ k T θ k + 1 θ f θ k 2 + ε + ( g (x N (ς + 1 ς. (25 { ( } Let α = min λ 2k g(xn(ς min, λ s min (γ 1, λ min (γ 2 an β = 1 2 θ f θ k 2 + ε. Then, (25 becomes V αv + β + ( g (x N (ς + 1 ς. (26 Multiplying both sies of (26 by e αt, we can obtain ( V (t e αt βe αt + e αt [ g (x N (ς + 1 ] ς. (27 t After integrating (27 over [, t f ], it follows that V (t β [ α + V ( β ] e αt + α e αt [ g (x N (ς + 1 ] e ατ ςτ. (28 Noting that < e αt < 1 an β α e αt >, we have [ V ( β α ] e αt V (. Then the above equality becomes V (t η + e αt [ g (x N (ς + 1 ] e ατ ςτ (29 where η = β + V (. α Accoring to Lemma 1, it can be conclue from (32 that V (t an g (x N (ς ςτ are boune, i.e., S, θ f, θ k, ς (t an [g (x N (ς + 1] e ατ ςτ are also boune. Fig. 4 Spee eviations in nominal, heavy loaing an light loaing cases

4 284 International Journal of Automation an Computing 1(4, August Simulation The sounness of the propose PSS was teste, an the performance as well as robustness tests were conucte an compare with a conventional stabilizer an a ual-input power system stabilizer through simulations. Goo transient behavior with the propose control uner severe operating conitions were illustrate by the following case stuies. The spee variation an accelerating power are chosen as the power system control variables. Five fuzzy sets for each input are sufficient for the PSS to be esigne. 4.1 Simulation cases for an SMIB To assess the performance of the propose controller, simulations were carrie out for ifferent operating conitions. A three-phase fault test is applie to an infinite bus as in Fig. 1, lasting 6 ms before being cleare. When the power system is strongly perturbe, the propose stabilizer reacts rapily an prevents an eventual loss of synchronism. Therefore, it enables the system to reach a stable operating point very quickly. The simulation results for nominal loa, heavy loa an light loa are shown in Fig. 4. It is clear that the propose PSS (PPSS exhibits superior performance to the conventional PSS (CPSS an ual-input PSS (DPSS power system stabilizers. The simulation results shown in Fig. 4 inicates a goo transient behavior of the propose PSS. 4.2 Simulation of multi-machine power system To evaluate the performance of the propose control, we performe simulation for multi-machine power system as in Fig. 5 with the aim to compare the performance of the propose PSS with the conventional PSS an ual-input PSSs. In orer to valiate stability enhancement ue to the propose stabilizer, a three-phase fault test is applie to bus 7 (Fig. 5 with a uration of 6 ms before it is cleare. Three operating conitions were investigate: Cases of nominal loa, light loa an heavy loa. The fault is cleare an the controller helps the system to reach a stable operating point very quickly. As shown in Fig. 6, the propose approach shows better control performance than the conventional PSS an ual PSSs in terms of settling time an amping effect. Fig. 5 Multi-machine power system To further evaluate the robustness of the propose stabilizer, the power system is subjecte to heavy loa operation (Fig. 7 an light loa operation (Fig. 8. The results show again the clear oscillations amping of the propose controller in multi-machine power system, as compare with its conventional counterpart, but we have a small eterioration in performance. Fig. 6 Spee eviations in nominal case

5 E. Nechai et al. / Aaptive Fuzzy Sliing Moe Power System Stabilizer 285 Fig. 7 Spee eviations in heavy loaing of the system Fig. 8 Spee eviations in light loaing of the system In this stuy, we use ifferent operating points to emonstrate the effectiveness of the propose control in oscillations amping after the occurrence of large isturbance on a power system by proviing better transient response an stronger robustness than other stabilizers. The aaptive fuzzy sliing moe using Nussbaum gain scheme permits to take account of severe loa variation an varying operating conitions. 5 Conclusions Base on the aaptive fuzzy sliing moe controller an the Nussbaum gain, we introuce a new power system stabilizer that enhances amping an improves transient ynamics of a single-machine infinite-bus an multi-machine power system stabilizers. Different loa conitions as well as severe perturbations were use to evaluate the propose

6 286 International Journal of Automation an Computing 1(4, August 213 power system stabilizer effectiveness in rapily reucing oscillations that coul lea to loss of synchronism if not treate. Simulation results exhibite its superior performance over classical PSSs. Real power system remains to be thoroughly investigate uner the propose stabilizer. Appenix Parameters of single (SMIB operating conitions: Nominal loa: P =.9/unit, Q =.3/unit Heavy loa: P = 1.6/unit, Q =.6/unit Light loa: P =.45/unit, Q =.35/unit Multi-machine power system parameters: G1:Nominal loa: P =.72/unit, Q =.27/unit. Heavy loa: P = 2.21/unit, Q = 1.9/unit. Light loa: P =.36/unit, Q =.16/unit. G2:Nominal loa: P = 1.63/unit, Q =.7/unit. Heavy loa: P = 1.92/unit, Q =.56/unit. Light loa: P =.8/unit, Q =.11/unit. G3:Nominal loa: P =.85/unit, Q =.11/unit. Heavy loa: P = 1.28/unit, Q =.36/unit. Lightloa: P =.45/unit, Q =.2/unit. Nonlinear functions: x e + x f (x = T (xe + x ( P Pm + V 2 2T (xe + x ( 2 (xe + x (x xq + (x x (x e + x sin (2δ + q ( ( Q + V 2 sin2 (δ + cos2 (δ + (x xq cos (2δ x e + x q x e + x (x e + x q (x e + x KAV sin (δ ω sm ω + T (xe + (Vref Vt x KAV sin (δ g (x = T (xe + x. References [1] P. Kunur. Power System Stability an Control, New York, USA: McGraw-Hill Inc, [2] F. P. DeMello, C. A. Concoria. Concepts of synchronous machine stability as affecte by excitation control. IEEE Transactions on Power Apparatus an Systems, vol. PAS- 88, no. 4, pp , [3] P. M. Anerson, A. A. Foua. Power System Control an Stability, Lowa State, USA: IEEE Press, [4] E. V. Larsen, D. A. Swann. Applying power system stabilizers, Part I, II, III. IEEE Transactions on Power Apparatus an Systems, vol. PAS-1, no. 6, pp , [5] M. L. Kothari, J. Nana, K. Bhattacharya. Design of variable structure power system stabilisers with esire eigenvalues in the sliing moe. IEE Proceeings C of Generation, Transmission an Distribution, vol. 14, no. 4, pp , [6] K. Bhattacharya, M. L Kothari, J. Nana. Design of iscrete-moe variable structure power system stabilizers. International Journal of Electrical Power & Energy Systems, vol. 17, no. 6, pp , [7] Y. M. Park, W. Kim. Discrete-time aaptive sliing moe power system stabilizer with only input/output measurements. International Journal of Electrical Power & Energy Systems, vol. 18, no. 8, pp , [8] Z. H. Jiang. Design of power system stabilizers using synergetic control theory. In Proceeings of Power Engineering Society General Meeting, IEEE, Tampa, FL, pp. 1 8, 27. [9] A. L. Elshafei, K. A. El-Metwally, A. A. Shaltout. A variable structure aaptive fuzzy logic stabilizer for single an multi-machine power systems. Control Engineering Practice, vol. 13, no. 4, pp , 25. [1] P. Hoang, K. Tomsovic. Design an analysis of an aaptive fuzzy power system stabilizer. IEEE Transactions on Energy Conversion, vol. 11, no. 2, pp , [11] G. J. Li, T. T. Lie, C. B. Soh, G. H. Yang. Design of statefeeback ecentralize nonlinear H controllers in power systems. International Journal of Electrical Power & Energy Systems, vol. 24, no. 8, pp , 22. [12] A. Y. Sivaramakrishnan, M. V. Hariharan, M. C. Srisailam. Design of variable-structure loa-frequency controller using pole assignment technique. International Journal of Control, vol. 4, no. 3, pp , [13] A. Ghosh, G. Lewich, O. P. Malik, G. S. Hope. Power system stabilizer base on aaptive control techniques. IEEE Transactions on Power Apparatus an Systems, vol. PAS- 13, no. 8, pp , [14] S. J. Cheng, Y. S. Chow, O. P. Malik, G. S. Hope. An aaptive synchronous machine stabilizer. IEEE Transactions on Power Systems, vol. 1, no. 3, pp , [15] D. A. Pierre. A perspective on aaptive control of power systems. IEEE Transactions on Power Systems, vol. 2, no. 2, pp , [16] N. Hossein-Zaeh, A. Kalam. A irect aaptive fuzzy power system stabilizer. IEEE Transactions on Energy Conversion, vol. 14, no. 4, pp , [17] N. Hossein-Zaeh, A. Kalam. An inirect aaptive fuzzylogic power system stabiliser. International Journal of Electrical Power & Energy Systems, vol. 24, no. 1, pp , 22. [18] A. L. Elshafei, K. El-Metwally. Power system stabilization via aaptive fuzzy-logic control. In Proceeings of the 12th IEEE International Symposium on Intelligent Control, IEEE, Istanbul, Turkey, pp , [19] S. S. Lee, J. K. Park. Design of power system stabilizer using observer/sliing moe, observer/sliing moe-moel following an H /sliing moe controllers for small-signal stability stuy. International Journal of Electrical Power & Energy Systems, vol. 2, no. 8, pp , [2] T. Hussein, M. S. Saa, A. L. Elshafei, A. Bahgat. Damping inter-area moes of oscillation using an aaptive fuzzy power system stabilizer. Electric Power Systems Research, vol. 8, no. 12, pp , 21. [21] G. H. Hwang, D. W. Kim, J. H. Lee, Y. J. An. Design of fuzzy power system stabilizer using aaptive evolutionary algorithm. Engineering Applications of Artificial Intelligence, vol. 21, no. 1, pp , 28.

7 E. Nechai et al. / Aaptive Fuzzy Sliing Moe Power System Stabilizer 287 [22] M. Soliman, A. L. Elshafei, F. Benary, W. Mansour. LMI static output-feeback esign of fuzzy power system stabilizers. Expert Systems with Applications, vol. 36, no. 3, pp , 29. [23] H. Shayeghi, H. A. Shayanfar, A. Safari, R. Aghmasheh. A robust PSSs esign using PSO in a multi-machine environment. Energy Conversion an Management, vol. 51, no. 4, pp , 21. [24] H. Shayeghi, H. A. Shayanfar, S. Jalilzaeh, A. Safari. Multi-machine power system stabilizers esign using chaotic optimization algorithm. Energy Conversion an Management, vol. 51, no. 7, pp , 21. [25] A. Chatterjee, S. P. Ghoshal, V. Mukherjee. Chaotic ant swarm optimization for fuzzy-base tuning of power system stabilizer. International Journal of Electrical Power an Energy Systems, vol 33, no. 3, pp , 211. [26] A. Sharma, M. L. Kothari. Intelligent ual input power system stabilizer. Electric Power Systems Research, vol. 64, no. 3, pp , 23. [27] S. P. Ghoshal, A. Chatterjee, V. Mukherjee. Bio-inspire fuzzy logic base tuning of power system stabilizer. Expert Systems with Applications, vol. 36, no. 5, pp , 29. [28] M. A. M. Hassan, O. P. Malik, G. S. Hope. A fuzzy logic base stabilizer for a synchronous machine. IEEE Transactions on Energy Conversion, vol. 6, no. 3, pp , [29] A. Boulkroune, M. Tajine, M. MSaa, M. Farza. Fuzzy aaptive controller for MIMO nonlinear systems with known an unknown control irection. Fuzzy Sets an Systems, vol. 161, no. 6, pp , 21. [3] S. S. Ge, H. Fan, T. H. Lee. Aaptive neural control of nonlinear time-elay systems with unknown virtual control coefficients. IEEE Transactions on Systems, Man, an Cybernetics Part B: Cybernetics, vol. 34, no. 1, pp , 24. Emira Nechai receive her bachelor egree in control systems, master egree in control systems, an Ph. D. egree in control systems, all from the University of Sétif, Algeria. in 22, 24, an 213, respectively. Her research interests inclue sliing moe control, aaptive fuzzy control, fuzzy systems, synergetic control an power systems. nechaiamira@yahoo.fr (Corresponing author Mohame Naguib Harmas receive the Ph. D. egree in control systems from Ferhat Abbas University of Sétif 1, an M. Sc. egree in electrical engineering an electronics from California State University of Sacramento, Sacramento, USA. He is currently a professor at Sétif University, Algeria. His research interests inclue robust nonlinear control, power system an power electronics control, an fuzzy synergetic esign. mharmas@yahoo.fr Najib Essounbouli receive his bachelor egree in electrical from the University of Sciences an Technology of Marrakech (FSTG Morocco, an DEA. an Ph. D. egrees both in electrical engineering from Reims University of Champagne Arennes, in 2 an 24 respectively. From 25 to 21, he was an assistant professor with University Institute of Troyes, Reims Champagne Arennes University. Since 21, he has been a professor with the same institute. His research interests inclue fuzzy logic control, robust aaptive control, renewable energy an control rive. najib.essounbouli@univ-reims.fr Abelaziz Hamzaoui receive his bachelor egree in electrical engineering from the Polytechnic School of Algiers (ENPA, Algeria, in 1982, an DEA. an Ph. D. egrees both in electrical engineering from Reims University of Champagne Arennes, in 1989 an 1992, respectively. He is currently a professor an the irector of the Technology Institute of Troyes, Reims Champagne Arennes University. His research interests inclue intelligent control, fuzzy control, an robust aaptive control. abelaziz.hamzaoui@univ-reims.fr