Adaptive sliding mode reliable excitation control design for power systems

Transcription

1 Acta Technca 6, No. 3B/17, c 17 Insttute of Thermomechancs CAS, v.v.. Adaptve sldng mode relable exctaton control desgn for power systems Xuetng Lu 1, 3, Yanchao Yan Abstract. In ths paper, the problem of adaptve sldng mode relable exctaton control desgn procedure was consdered for mult-machne power systems wth possble actuator faults, whch the upper bound s not exactly known, usng lnear matrx nequalty technque and Lyapunov stablty theory. The control objectve s to nvestgate the relable desgn on both passve and adaptve sldng mode exctaton controllers. Fnally, the numercal smulaton results based -area 4-machne power system valdated the effectveness of the proposed method n terms of mprovement of the system stablty and relablty effectvely. Key words. relable. Sldng mode exctaton control, adaptve, electrc power system, stablty, 1. Introducton Wth the development of scence and technology, the power ndustry s an mportant pllar of the natonal economy n every country n the world. In recent years, the contnuous development of power system and safe and stable operaton to the world each country s natonal economc and socal development, the tremendous power and effcency. So how to ensure the stable and effcent operaton of the power system and the relable qualty of the power supply for the users s the basc problem to be consdered. However, hstorcal experence shows that once the mult-machne power system of natural and man-made fault, f not promptly and effectvely control and lose stablty, and even the collapse of the power grd, wll lead to blackouts, wll brng serous nfluence to the socety and the consequences. Wth the rapd growth of the power demand of each country n the world, the scale of the power grd and the degree of nterconnecton are ncreasng. The safe 1 College of Electrcal and Electronc Engneerng, Shandong Unversty of Technology, Zbo 5549, Chna School of Scence, Shandong Unversty of Technology, Zbo 5549, Chna 3 Correspondng author; e-mal: lhk868@163.com

2 594 XUETING LIU, YANCHAO YAN and relable operaton of power system s becomng more and more serous. How to evaluate the grd durng operaton, especally the relablty level n the fault state and specal operaton mode of the power system has become an urgent problem n the world, the majorty of electrcty workers concerned [1 6]. The so-called relable control refers to the stablty of the closed-loop system and the other performance n the acceptable range of control when the actuator or sensor falure occurs. Relable control research has always been an actve research area [7 14]. In ths paper, the problem of adaptve sldng mode relable exctaton control desgn procedure was consdered for mult-machne power systems whch allows the actuator falure usng lnear matrx nequalty technque and Lyapunov stablty theory. The control objectve s to nvestgate the relable desgn on both passve and adaptve sldng mode exctaton controllers. Fnally, the numercal smulaton results based -area 4-machne power system valdated the effectveness of the proposed method n terms of mprovement of the system stablty and relablty effectvely.. Mathematcal model In ths paper, a mult-machne power system wth possble actuator faults, external dsturbances and uncertan parameters, s consdered, assumng that the nput mechancal power s constant. Neglectng the equvalent dampng wndngs (E d =, X q = X q and transent salent pole effect of q axs (X d = X q. Then the state equaton of the N-machne power system can be descrbed as δ = ω ω, ω = ξ T ω ωiq E q + ω T P m + ξ T ω + d 1 (t, T (1 Ė q = 1 T E d q + I d(x d x d T 1 d T E d f + d (t, where = 1,,, N, d 1 (t and d (t represent the errors and external dsturbances exstng n the model, but the upper bound s not exactly known; ξ represents the dampng coeffcent and s an uncertan parameter, δ represents the power angle, ω represents the rotor angular velocty, ω represents the rated angular velocty where ω = πf. Symbol T represents the nerta tme constant, P m s the mechancal power that s assumed constant, E f represents the exctaton voltage, that s, the desgn of the control, I d and I q represent the d-axs and the q-axs components of the armature current, and can be regarded as the mplct functons of the system state. The COI sgnal s defned as follows M =1 δ COI = T M δ =1 M =1 T, ω COI = T ω M =1 T.

3 ADAPTIVE SLIDING MODE 595 Take the followng coordnate transformaton for (1: x 1 = (δ δ (δ COI δ COI, x = ω ω COI, where The formula (1 can be transformed to A = where x 3 = ω T ( Pm E q I q ωcoi. ẋ (t = Ax (t + ξ f (x (t + B 1 u F (t + B d 1 + B 3 d 3, ( 1 1, B 1 = 1, B = 1, f (x (t = ω ω Suppose that system ( may be actuator falure, and falure modes satsfy. u F (t = (I Λ (t u (t, (3 Λ (t = dag( δ 1 (t, δ (t, δ 3 (t, δ j δ j (t δ j < 1, j= 1,, 3, and I s three-order unt matrx. If the falure (3 occurs, the system ( can be changed as ẋ (t = Ax (t + ξ f (x (t + B 1 u F (t + B d 1 + B 1 d 3. (4 For system (3 the objectve of ths paper s to desgn the adaptve sldng mode relable exctaton control law, whch makes the state trajectory tend to the specfed sldng surface, and the sldng mode moton along the specfed sldng surface s asymptotcally stable even f the actuator falure occurs. The sldng surface of the sldng mode s s (t = G x (t =, (5 where G = B1 T P 1 and P >. It s easy to see that G B 1 s nonsngular. The am of ths paper s that the sldng mode control law can make the state trajectory tend to the specfed sldng surface, and asymptotcally tends to zero along the sldng surface. Because of the possble falure of the system, ths paper studes the adaptve sldng mode control

4 596 XUETING LIU, YANCHAO YAN 3. Controller desgn In order to acheve the purpose of ths paper, the adaptve sldng mode relable exctaton controller s desgned as follows u (t = (I Λ (t 1 (K + G Ax (t ρ (t sgn (s (t, (6 ρ (t = where 1 [ µ + B T ξ + 1 δ max The update rates are defned as follows ] (G B 1 1 G Ax (t + (K + G A x (t. = l 1 x T P 1 B 1 f (x (t, ε1 = l x T P 1 B d 1, ε = l 3 x T P 1 B 1 d, (8 δ j (t = P r {U j } = [δ j,δ j], δj (t = δ j and U j, or δ j (t = δ j and U j, U j, other cases, U j = η j x T (P 1 B 1 j (1 δ j (t 1(K + G A j x (t, and l 1, l, l 3, η j are adaptve gan coeffcents. Symbol Λ (t = dag { δ1 (t, δ (t,, δ } m (t, M j and M j are row j and column j of the matrx M respectvely. Pr{ } represents the projecton operator, whose functon s to project the estmate value δ j (t onto the nterval [δ j, δ j ]. Theorem 3.1: For system (4, the adaptve sldng mode controller s desgned as (6, (7, (8. If there exst matrces L R m n, P >, such that the followng lnear matrx nequalty s establshed ℵ B 1 B1 T P A T P <, (9 P where ℵ = AP + P A T B 1 K P P K TBT = AP + P A T B 1 L P L T, K = L P 1, then even f the actuator (3 shows the fault, the closed-loop system s asymptotcally stable. Proof: Substtutng (6 nto (4 ẋ (t = Ax (t + ξ f (x (t B 1 (I Λ [(I Λ (t 1 (K + G A x (t +ρ (t sgn (s (t] + B d 1 + B 1 d 3 (7

5 ADAPTIVE SLIDING MODE 597 = Ax (t + ξ f (x (t B 1 Λ [(I Λ (t 1 (K + G A x (t B 1 (K + G A x (t B 1 (I Λ ρ (t sgn (s (t] + B d 1 + B 1 d 3, (1 where Λ (t = dag { δ1 (t, δ (t,, δ } m (t, δj (t = δ j (t δ j (t beng the estmaton error. Construct now the Lyapunov functon V 1 = x T P 1 x + 1 η j ( δj δ j l 1 ( θ θ + 1 l ( ε 1 ε l 3 ( ε ε. (11 Takng the dervatve on both sdes of (11, V 1 = x T P 1 (Ax + ξ f (x B 1 (I Λ [(I Λ 1 (K + G A x +ρ (t sgn (s (t] + B d 1 + B 1 d 3 + δj ( δj δ j η j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε l 1 l l 3 = x T P 1 (Ax B 1 (K + G A x x T P 1 B 1 Λ [(I Λ 1 (K + G A x x T P 1 B 1 (I Λ ρ (ρ (t sgn (s (t ξ f (x + δj ( δj δ j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε, (1 η j l 1 l l 3 where G = B T 1 P 1. It s easy to deduce that x T P 1 B 1 G Ax x T (P 1 B 1 B1 T P 1 (B 1 B1 T P 1 x + x T A T P 1 Ax, (13 x T P 1 B 1 (I Λ ρ (t sgn (s (t = ρ (t (1 δ j s j (t ρ (t ( 1 δ,max s (t 1 ρ (t ( 1 δ,max s (t, (14 x T P 1 (ξ f (x x T P 1 B 1 Λ(I Λ 1 (K + G A x + B d 1 + B 1 d 3

6 598 XUETING LIU, YANCHAO YAN + δj ( δj δ j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε. (15 η j l 1 l l 3 By means of Schur s Lemma and (9, Ψ = P 1 A + A T P 1 + (P 1 P 1 B 1 K K T B1 T P 1 + B 1 B1 T P 1 (B 1 B1 T P 1 + A T P 1 A <. (16 Substtutng (13, (14, (15 and (16 nto (1, we have V 1 <. (17 Thus system (1 s asymptotcally stable, the estmaton errors are bounded. The followng concluson proves that the controller s able to make the system (3 the state trajectory tend to be specfed by the sldng surface Theorem 3. For system (, the adaptve sldng mode controller s desgned as (6,(7,(8. If there exst matrces L R m n, P >, such that the lnear matrx nequalty (9 s establshed, then even f the actuator (3 shows the fault, the desgned adaptve sldng mode controller s stll able to guarantee the sldng mode surface s (t =. Proof: Constructng Lyapunov functon V = s T (G B s + ( δj δ j ( θ θ ( ε 1 ε ( ε ε. η j l 1 l l 3 Takng the dervatve on both sdes of (18, (18 V = s T (G B 1 1 G (Ax + ξ f (x B 1 (I Λ [(I Λ 1 (K + G A x +ρ (t sgn (s (t] + B d 1 + B 1 d 3 + δj ( δj δ j η j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε l 1 l l 3 = s T (G B 1 1 G (Ax B 1 (K + G A x x T P 1 B 1 Λ [(I Λ 1 (K + G A x s T (G B 1 1 G B 1 (I Λ (ρ (t T (s (t ξ f (x

7 ADAPTIVE SLIDING MODE δj ( δj δ j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε, (19 η j l 1 l l 3 where G = B T 1 P 1. It s easy to deduce that s T (G B 1 1 G B 1 (I Λ ρ (t sgn (s (t = ρ (t (1 δ j s j (t ρ (t ( 1 δ,max s (t 1 ρ (t ( 1 δ,max s (t, ( s T P 1 B 1 Λ(I Λ 1 (ξ f (x + B d 1 + B 1 d 3 + δj ( δj δ j η j + ( θ θ + ε1 ( ε 1 ε 1 + ε ( ε ε. (1 l 1 l l 3 Substtutng (, (1 nto (19, and by means of Schur s Lemma and nequalty (9, V s T (G B 1 1 G Ax (t + (K + G A x (t ρ (t ( 1 δ,max s (t µ s (t. 4. Smulaton results In ths secton, the numercal smulaton results based -area 4-machne power system valdated the effectveness of the proposed method n terms of mprovement of the system stablty and relablty effectvely. Fgure 1 shows that the trackng error s acceptable and tends to. Obvously, the smulaton results verfy the theoretcal analyss. 5. Concluson Ths paper studes the problem of adaptve sldng mode relable exctaton control desgn procedure for mult-machne power systems whch allows the actuator falure usng lnear matrx nequalty technque and Lyapunov stablty theory. The control objectve s to nvestgate the relable desgn on both passve and adaptve sldng mode exctaton controllers, and the reachablty of the specfed sldng surface was analyzed.

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9 ADAPTIVE SLIDING MODE 61 [13] Z. Xang, Q. Chen: Robust relable control for uncertan swtched nonlnear systems wth tme delay under asynchronous swtchng. Appled Mathematcs and Computaton 16 (1, No. 3, [14] J. Dong, Y. G. Nu: Relable control of uncertan systems va adaptve sldng mode technque. Journal of East Chna Unversty of Scence and Technology (Natural Scence Edton (14, No. 6, Receved Aprl 3, 17

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