Research Article Sliding Mode Control for the Synchronous Generator

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1 ISRN Applied Mathematics Volume 014, Article ID 56504, 7 pages Research Article Sliding Mode Control for the Synchronous Generator Yaote Chang and Chih-Chin Wen Department of Electrical Engineering, Kao Yuan University, Kaohsiung 81, Taiwan Correspondence should be addressed to Yaote Chang; t005@cc.kyu.edu.tw Received 3 November 013; Accepted 19 February 014; Published 0 March 014 Academic Editors: J. Shen and K.-V. Yuen Copyright 014 Y. Chang and C.-C. Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the Lyapunov stability theorem and sliding mode control technique, a design of the nonlinear controller is proposed for the dual-excited and steam-valving control of the synchronous generators with matched and mismatched perturbations in this paper. By using some constant gains designed in the sliding surface function, the perturbations in the power system can be suppressed, and the property of asymptotical stability of the rotor angle and the voltage can be achieved at the same time. 1. Introduction To achieve a high degree of reliability in the power systems, many works [1 11] have studied the stability of generators. In general, there are two ways to stabilize the generators: the excited control [1 3] andthesteam-valvingcontrol[4 7]. Using the excited control, Xie et al. [1] designed a linear matrix inequality (LMI) controller for a class of multimachine power systems with uncertain parameters to achieve the property of asymptotical stability. Galaz et al. [] proposed a passivity-based controller and discussed the domain of attraction of the equilibria in power systems. Huang et al. [3] utilized a physical exact linearization method to design a controller for a dual-excited synchronous generators. For the steam-valving control, Zhang and Sun [4], Fu [5], Li et al. [6], and Li et al. [7] designed the adaptive backstepping controller for single machine infinite bus system in the presence of internal and external disturbances to achieve the property of asymptotical stability. As for the systems with both steam turbine dynamics and the excited generator, Xi et al. [8] andmaetal.[9] presented a novel nonlinear controller based on Hamiltonian energy theory steam for the turbine dynamics and single excited generator to achieve the property of asymptotical stability. The dual-excitation means the system has d-axis and q-axis field winding simultaneously. Each field voltage can be adjusted separately and hence the control objectives canbeachievedmoreflexibly.basedonthepassivelemma, Wang and Lin [10] designed the bounded passivity controller for the synchronous generators to achieve the property of asymptotical stability. Using the coordinated passivation technique, Chen et al. [11] designed backstepping controller for steam-valving and dual-excited synchronous generators to achieve the property of asymptotical stability. However, the perturbations were not considered in the works [10, 11]. Sliding mode control () is well known to possess several advantages, for example, fast response, good transient performance, robustness of stability, and insensitivity to matched parameter variations and external disturbances [1, 13]. However, the property of asymptotical stability is in general hard to achieve by using the traditional technique if the mismatched perturbations are presented in the systems [14]. A lot of researchers applied techniques to solve the tracking problems with mismatched perturbations [15 17]. For example, Shieh and Shyu [15], Chen and Dunnigan [16], and Kwan [17] employedtechniques foraninductionmachinewithanuncertainloadtorque; however, the mismatched perturbations considered in these works [15 17] belong to the unknown constants. In this paper, we have proposed the nonlinear sliding mode controller for the dual-excited and steam-valving control of the synchronous generators with matched and

2 ISRN Applied Mathematics mismatched perturbations to achieve the property of asymptotical stability. Our proposed control scheme can be thought of as the extension work of [10, 11], where no perturbations are considered in the works [10, 11]. Furthermore, the mismatched perturbations considered in this paper can be time varying.. System Model Consider a machine power system with dynamic equations [11] and model uncertainties given by δ=ω ω 0, modeling errors, uncertainties, and disturbance in the control system. Then, (1) canbewrittenas x 1 =x +d 1, (3a) x =f (x 1,x )+z x 3 z 3 x 4 sin (x 1 +δ 0 ) z 4 x 5 sin (x 1 +δ 0 )+d, x 3 =f 3 (x 3 )+( C +η T 1 )μ+d 3 (x), H x 4 =f 4 (x 1,x 4 )+( 1 Td0 +η )E fd +d 4 (x), (3b) (3c) (3d) E q = E d = where X d T d0 X d X q T q0 X q ω= D H (ω ω 0)+ ω 0 H (P m P e ), P m = P m0 P m +( C +η T H T 1 )μ, H E q + X d X d Td0 V s cos δ+( 1 X T +η )E fd, d d0 E d + X q X q Tq0 V s sin δ+( 1 X T +η 3 )E fq, q q0 (1) x 5 =f 5 (x 1,x 5 ) + ( 1 Tq0 +η 3 ) E fq +d 5 (x), where P m0 =P e (0) [11], f (x 1,x ) D H x +P m0 P e (x 1), P e (x 1)=z 3 E q0 sin (x 1 +δ 0 )+z 4 E d0 cos (x 1 +δ 0 ) +z 5 sin (x 1 +δ 0 ), f 3 (x 3 ) 1 T H x 3, f 4 (x 1,x 4 ) z 6 (x 4 +E q0 )+z 7 cos (x 1 +δ 0 ), f 5 (x 1,x 5 ) z 8 (x 5 +E d0 ) z 9 sin (x 1 +δ 0 ), (3e) (4) P e E q X d V s sin δ E d X q V s cos δ X d X q X q X d V s cos δ, δ, ω, P m, P e, D, H, V s, T H,andCare the power angle, relative speed, mechanical input power, electromagnetic power, perunit damping constant, inertia constant, infinite bus voltage, steam-valving control time constant, and power coefficient, respectively. μ, E fd,ande fq are steam-valving controller, daxis field voltage, and q-axis field voltage, respectively. T d0 and T q0 are the d-axis and q-axis transient short-circuit time constants, respectively. X d =X d +X T +X L, X d =X d + X T +X L, X q =X q +X T +X L, X q =X q +X T +X L,where X q, X q, X d, X d, X L,andX T are the q-axis transient reactance, q-axis reactance, d-axis transient reactance, d-axis reactance, reactance of transmission line, and reactance of transformer, respectively. E q and E d are the q-axis internal transient voltage and q-axis internal transient voltage, respectively. Let (δ 0, ω 0, P m0, E q0, E d0 ) be an operation point, and define the state variable by x 1 = δ δ 0, x = ω ω 0, x 3 = P m P m0, x 4 = E q E q0, x 5 = E d E d0,andx = [x 1 x x 3 x 4 x 5 ] T. We further consider that the model perturbations d i, 1 i 5,maybeappliedinthepower system (1) because the perturbation may come from the () z = ω 0 /H, z 3 = ω 0 V s /HX q, z 4 = ω 0 V s /HX d, z 5 = ((X d X q )/HX q X d )ω 0V s, z 6 =X d /T d0 X d, z 7 = ((X d X d )/T d0 X d )V s, z 8 = X q /T q0 X q,andz 9 = ((X q X q )/T q0 X q )V s. Remark 1. The assumptions of the mismatched perturbations d 1 (x 1 ) and d (x 1,x,x 4,x 5 ), not in the range of any control effort (μ, E fd,e fq ),canbeseeninsomeliteratures[18, 19]. However, the stability analysis is not proposed in these works [18, 19]. d i, 3 i 5,arethematchedperturbations. Assumption. The upper bounds of the following vanished perturbations [0] are assumed as d 1 (x 1 ) a 1 x 1, d (x 1,x,x 4,x 5 ) b 1 x 1 +b x 1 +a x +a 3 x 4 +a 4 x 5, where a i, 1 i 4,areknownpositiveconstants.When x 4 =x 5 =0,theupperboundof d can be computed as d b 1 x 1 +b x 1 +a x, (6) wherethesystemhasbeenintheslidingmode.ontheother hand, if the information of this upper bound is unknown, (5)

3 ISRN Applied Mathematics 3 the adaptive mechanism can be used to estimate these parameters. 3. Design of the Sliding Surface Fortacklingthe perturbationin (3a) (3e), the sliding surface σ [σ 1 σ σ 3 ] T R 3 can be designed as σ 1 =x z [(ε +ε 1 )φ+x 1 ε a x +f ], σ =x 4, σ 3 =x 5, where φ=x α, α= ε a x 1, ε a =3ε 1 +a 1,andε 1 is a designed positive constant. ε =a +a 3 /4ε 1 +b x 1 /4ε 1,wherea 6 is the known parameter defined in Appendix A. (7) (rad) Figure 1: Responses of the power angle δ. Theorem 3. Consider the perturbed power system (3a) (3e).If theslidingsurfacefunctionisdesignedas(7),thetrajectoryof state x =[x 1 x x 3 x 4 x 5 ] T will reach zero asymptotically when the system is in the sliding mode. Proof. Please see Appendix A. 4. Design of Controllers According to (3a) (3e), the robust controller can be designed as ε 4 = ε u ( u n + B 1 ε 3)+ε 1, 1 ε u B 1 C 0 0 T HΣ 1 B = 0 Td0 0, [ 1 ] 0 0 [ Tq0 ] (9) where μ n = T H C μ μ n +μ s u = [ E fd ] = [ E fdn +E fds ], (8) [ E fq ] [ E fqn +E fqs ] {f { ε z n φ+(ε +ε 1 ) [f +z x 3 z 3 x 4 sin (x 1 +δ 0 ) z 4 x 5 sin (x 1 +δ 0 )+ε a x ]+x ε a [f +z x 3 z 3 x 4 sin (x 1 +δ 0 ) z 4 x 5 sin (x 1 +δ 0 )]} + df n }, E fdn = T d0 f 4, E fqn = T q0 f 5, u n =[μ n E fdn E fqn ] T, μ s u s = [ E fds ] = { (ε 3 +ε 4 ) B 1 σ, if σ { σ =0, [ E fqs ] { 0, if σ =0, where ε 3 and ε u < 1 are known positive constants. df /dt and ε can be divided into df /dt df n +df d and ε = ε n + ε d, respectively, where df n and ε n are the nominal parts of df /dt and ε,respectively.df d and ε d are the parts of df /dt and ε which contain perturbations. Theorem 4. Consider the system (3a) (3e). Ifthecontrollers are designed as (8), the sliding variable σ will approach zero in afinitetime. Proof. Please see Appendix B. 5. Simulation The system parameters are given in [11]. In the simulation, we assume that the desired reference signals are (δ 0,ω 0,P m0,e q0,e d0 ) = (0.5, 3150, 0.6, 1.1, 1.1) and the initial conditions of the tracking errors are x = [15.3 0, 15, 0.4, 0.8, 0.8] T. We also assume that the disturbances d 1 = 0.x 1,d = 0.1x x x 0.3x x 5,d 3 = x 1 x 4,d 4 = x 1 0.x 1 x 4,andd 5 = cos 3x 4 are suddenly applied from 10 sec onwards. Figures 1,, 3, 4,and5 show that the responses of δ, ω, P m, E q, E d of the proposed control scheme () can achieve the robust performance with a short transient time even if perturbations d i,1 i 5, exist, whereas the coordinated passivation

4 4 ISRN Applied Mathematics (deg) PU Figure : Responses of the relative speed ω. Figure 5: Responses of the d-axis internal transient voltage E d. PU Figure 3: Responses of the mechanical power P m σ 1 σ σ 3 Figure 6: The sliding variable σ. 1.4 PU Figure 4: Responses of the q-axis internal transient voltage E q. control () may lose the asymptotical stability. Figure 6 demonstrates that the sliding variable σ will approach zero in afinitetime. 6. Conclusions In this paper, a sliding mode controller has been successfully designedforthedual-excitedandsteam-valvingcontrolof the synchronous generators with perturbations. Even though the dynamics of the controlled systems are affected by the nonlinear perturbations, some constant gains designed in the sliding surface can effectively overcome these perturbations and achieve asymptotical stability. The proposed control scheme also demonstrates the robustness against the perturbations in the simulation.

5 ISRN Applied Mathematics 5 Appendices A. The Dynamic of the System in the Sliding Mode According to (3a)and(7), one can obtain the dynamics of x 1 as x 1 =x +d 1 =α+d 1 +x α= ε a x 1 +d 1 +φ. (A.1) Choose the first Lyapunov function candidate as V 1 = (1/)x 1. Using Assumption, the time derivative of V 1 along the trajectory of (A.1)canbegivenby V 1 =x 1 x 1 = ε a x 1 +x 1d 1 +x 1 φ 3ε 1 x 1 a 1x 1 + x 1 d 1 +x 1φ = 3ε 1 x 1 +x 1φ. (A.) When the system is in the sliding mode, σ = 0, from(8), it canbeseenthatx 4 =x 5 =0and x 3 1 z [(ε +ε 1 )φ+x 1 ε a x +f ]. (A.3) Using (3b)and(A.3), the closed-loop reduced dynamics of x canberewrittenas where ε 1 x 1 + φ (a 3 x 1 +a φ ) = ε 1 ( x 1 a 3 ε φ ) +(a + a 3 1 4ε1 )φ (a + a 3 4ε1 )φ, ε 1 x 1 +b φ x 1 = ε 1 x 1 (1 b x 1 b x 1 4ε1 φ. ε 1 φ ) + b x 1 4ε1 φ (A.7) Equation (A.6) impliesthatφ and x 1 will approach zero as t.from(7), x =(φ+α)alsoreaches zero as t because α = ( ε a x 1 ) approaches zero as t.using(3a) (3e), it is also noted that f (0, 0) (D/H) 0+P m0 P e (0) = 0 as t.similarly,x 3 =( 1/z [(ε +ε 1 )φ+x 1 ε a x +f ]) also reaches zero as t because f and ε a x approach zero as t in accordance with (A.3). Since x 4 =x 5 =0 and x 1,x,x 3 0 as t, the state trajectory of the state variable x = [x 1 x x 3 x 4 x 5 ] T will approach zero asymptotically when the system is in the sliding mode. x =f +z x 3 z 3 x 4 sin (x 1 +δ 0 ) z 4 x 5 sin (x 1 +δ 0 )+d (ε +ε 1 )φ x 1 +ε a x +d. Let d d =d ε a d 1.Itcanbeseenthat (A.4) B. The Proof of the Reaching Mode From (7) and(8),one can obtain the time derivative of the sliding variable σ as σ = ζ + Bu s, (B.1) d d d + ε ad 1 b 1 x 1 +b x 1 +a x +ε aa 1 x 1 a 3 x 1 +b x 1 +a φ, (A.5) where a 3 b 1 +ε a (a 1 +a ) in accordance with Assumption, (6), and (7). To show that the state trajectory of the state variable x will approach zero asymptotically, one can select the nd Lyapunov function candidate as V =V 1 + (1/)φ. From Assumption,(A.), and (A.5), we can obtain the time derivative of V along the trajectory of (A.4)as V = V 1 +φ( x α) 3ε 1 x 1 ε 1φ ε φ +φd d ε 1 (x 1 +φ ) ε φ +(a + a 3 4ε 1 = ε 1 (x 1 +φ ), + b x 1 4ε1 )φ (A.6) where σ [ σ 1 σ σ 3 ] T, σ 1 = x [ ε z φ+(ε +ε 1 ) φ+ x 1 ε a x + f ] =f 3 + C T H (μ n +μ s )+η 1 μ+d {( ε z n + ε d )φ+(ε +ε 1 ) =ζ 1 + [f +z x 3 z 3 x 4 sin (x 1 +δ 0 ) z 4 x 5 sin (x 1 +δ 0 )+d ε a (x +d 1 )] +x +d 1 ε a [f +z x 3 z 3 x 4 sin (x 1 +δ 0 ) C T H μ s, z 4 x 5 sin (x 1 +δ 0 )+d ]+df n +df d }

6 6 ISRN Applied Mathematics σ = x 4 =f Td0 (E fdn +E fds )+η E fd +d 4 =ζ + 1 Td0 E fds, σ 3 = x 5 =f Tq0 (E fqn +E fqs )+η 3 E fq +d 5 =ζ Tq0 E fqs and ζ [ζ 1 ζ ζ 3 ] T, ζ 1 η 1 μ+d [ ε z d φ+(ε +ε 1 )(d ε a d 1 ) ζ η E fd +d 4, ζ 3 η 3 E fq +d 5. +d 1 ε a d +df d ], (B.) (B.3) The lumped perturbations ζ can be assumed to satisfy the constraints ζ ε 3 +ε u u. (B.4) See [1]. To prove that the sliding variable σ will approach zero in a finite time, we define a Lyapunov function candidate as V σ = (1/)σ T σ.byusing(8) and(b.1), one can obtain the time derivative of V σ as V σ = σ T σ = σ T (ζ + Bu s ) σ (ε 3 +ε u u ) (ε 4 +ε 3 ) σ =( ε 4 +ε u u ) σ { ε 4 +ε u [ u n + B 1 (ε 3 +ε 4 )]} σ =[ ε 4 (1 ε u B 1 )+ε u ( u n + B 1 ε 3)] σ = ε 1 σ <0. (B.5) The preceding equation indicates that the values of σ will approach zero in a finite time. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] S. Xie, L. Xie, Y. Wang, and G. Guo, Decentralised control of multimachine power systems with guaranteed performance, IEE Proceedings Control Theory and Applications, vol.147,no. 3, pp , 000. [] M. Galaz, R. Ortega, A. S. Bazanella, and A. M. Stankovic, An energy-shaping approach to the design of excitation control of synchronous generators, Automatica,vol.39,no.1,pp , 003. [3]J.Huang,G.Y.Tu,D.S.Chen,andT.S.Chung, Improved nonlinear excitation control of dual-excited synchronous generators, in Proceedings of the 4th International Conference on Advances in Power System Control, Operation and Management, pp , Hong Kong, November [4] J. Zhang and Y. Sun, Backstepping design of nonlinear optimal control, in Proceedings of the IEEE/PES Transmission and Distribution Conference and Exhibition: Asia and Pacific,Dalian, China, August 005. [5] J. Fu, Extended backstepping approach for a class of non-linear systems in generalised output feedback canonical form, IET ControlTheoryandApplications,vol.3,no.8,pp , 009. [6] W.-L. Li, Y.-W. Jing, and X.-P. Liu, Adaptive robust backstepping design for nonlinear steam valve controller, Proceedings of the Chinese Society of Electrical Engineering, vol.3,no.1,pp , 003. [7] W. Li, S. Liu, G. Jiang, and G. M. Dimirovski, Adaptive robust backstepping design for turbine valve controller, in Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA 06), pp , Dalian, China, June 006. [8] Z.Xi,G.Feng,D.Cheng,andQ.Lu, Nonlineardecentralized saturated controller design for power systems, IEEE Transactions on Control Systems Technology, vol.11,no.4,pp , 003. [9] J. Ma, Z.-R. Xi, S.-W. Mei, and Q. Lu, Nonlinear stabilizing controller design for the steam-valve and excitation system basedonhamiltonianenergytheory, Proceedings of the Chinese Society of Electrical Engineering,vol.,no.5,pp.88 93,00. [10] B. Wang and W. Lin, Bounded control of dual-excited synchronous generator by using a passivity-based approach, in Proceedings of the World Congress on Intelligent Control and Automation (WCICA 11), pp , Taipei, Taiwan, June 011. [11] H. Chen, H.-B. Ji, B. Wang, and H.-S. Xi, Coordinated passivation techniques for the dual-excited and steam-valving control of synchronous generators, IEE Proceedings Control Theory and Applications,vol.153,no.1,pp.69 73,006. [1] J.Y.Hung,W.Gao,andJ.C.Hung, Variablestructurecontrol. A survey, IEEE Transactions on Industrial Electronics, vol. 40, no. 1, pp., [13] R. A. DeCarlo, S. H. Zak, and G. P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proceedings of the IEEE, vol. 76, no. 3, pp. 1 3, [14] S. K. Spurgeon and R. Davies, A nonlinear control strategy for robust sliding mode performance in the presence of unmatched uncertainty, International Control, vol.57,no.5,pp , [15] H.-J. Shieh and K.-K. Shyu, Nonlinear sliding-mode torque control with adaptive backstepping approach for induction motor drive, IEEE Transactions on Industrial Electronics, vol. 46,no.,pp ,1999. [16] F. Chen and M. W. Dunnigan, Sliding-mode torque and flux control of an induction machine, IEE Proceedings: Electric Power Applications,vol.150,no.,pp.7 36,003. [17] C.-M. Kwan, Robust adaptive control of induction motors, International Control,vol.67,no.4,pp ,1997.

7 ISRN Applied Mathematics 7 [18]X.-G.Yan,C.Edwards,S.K.Spurgeon,andJ.A.M.Bleijs, Decentralised sliding-mode control for multimachine power systems using only output information, IEE Proceedings Control Theory and Applications, vol.151,no.5,pp , 004. [19] Q.Lu,S.Mei,W.Hu,F.F.Wu,Y.Ni,andT.Shen, Nonlinear decentralized disturbance attenuation excitation control via new recursive design for multi-machine power systems, IEEE Transactions on Power Systems, vol. 16, no. 4, pp , 001. [0] H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, USA, 00.

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