Adaptive Controller Design for a Synchronous Generator with Unknown Perturbation in Mechanical Power

Transcription

1 308 International Journal of Control Xiaohong Autoation Jiao Yuanzhang and Systes Sun vol and 3 Tielong no (special Shen edition) pp June 005 Adaptive Controller Design for a Synchronous Generator with Unknown Perturbation in Mechanical Power Xiaohong Jiao Yuanzhang Sun and Tielong Shen Abstract: Transient stabilization with voltage regulation is investigated for a synchronous generator when the echanical power is perturbed to any unknown value within its physical bounds so that the operating point of the syste shifts to an unknown point An adaptive excitation controller is designed based on the backstepping ethod with tuning functions It will be shown that the adaptive control law proposed can achieve the convergence of the syste states to the new equilibriu point in correspondence with the real value of the unknown echanical power and the regulation of the terinal voltage to the required value Siulation results are given to deonstrate the effectiveness of the proposed controller for the transient stabilization and voltage regulation Keywords: Adaptive control echanical power synchronous generator transient stabilization voltage regulation INTRODUCTION Transient stability of power systes is a classical dynaic control systes proble The application of nonlinear control ethods to design the excitation control in order to enhance transient stability has been given uch attention throughout various literatures since the late 980s Since the uncertainty in practical power systes represented by sudden echanical (load shedding and generation tripping) and electrical (short circuits with changes in the power network structure) perturbations ay destabilize the operating conditions the research of robustness issues is ore challenging and widely applicable Therefore the transient stabilization proble consists in the design of an excitation feedback control for the power systes that will keep the generator at synchronous speed and the terinal voltage at the prescribed value when perturbation occurs In the last decade nonlinear control theory has been exploited to solve the transient stabilization proble Feedback linearization based excitation schees have Manuscript received Septeber ; accepted Deceber 004 Recoended by Guest Editor Youyi Wang Xiaohong Jiao is with the Departent of Electrical Engineering Tsinghua University Beijing China and the Departent of Autoatic Control Yanshan University Qinhuangdao China (e-ail: jiaoxh@tsinghuaeducn) Yuanzhang Sun is with the Departent of Electrical Engineering Tsinghua University Beijing China (eail: yzsun@tsinghuaeducn) Tielong Shen is with the Departent of Mechanical Engineering Sophia University Tokyo Japan (eail: tetu-sin@sophiaacjp) been proposed by [-3] Adaptive versions of the feedback linearizing controls are then developed in [45] Robust nonlinear state feedback controls have also been investigated in [67] In [89] robust adaptive nonlinear excitation controls with L disturbance attenuation are developed for power systes with additive disturbances and unknown electrical paraeters Very recently the adaptive control proble has been addressed in [0] for power systes involving unknown echanical power When the echanical power is perturbed to a new constant value the equilibriu of the syste will shift to a new corresponding point oreover the new point will be unknown if the echanical power is unknown In [0] an estiation function is introduced for the unknown echanical power and the angular speed and with the estiation the trajectories for the angle the angular speed and the active power are given such that the excitation control drives the states of the syste to track the trajectories and the trajectories to converge to the equilibriu point In this paper we consider the transient stabilization with voltage regulation proble for synchronous generators with unknown perturbation in the echanical power Motivated by [0] we also introduce an estiation function in the proposed state feedback excitation control law However we will show that with soe tuning functions such a feedback controller can be directly constructed so that the syste is stabilized at the new equilibriu corresponding to the unknown echanical power The reaining portion of the paper is organized as follows The proble is forulated in Section In Section 3 based on the adaptive back-stepping design technique

2 Adaptive Controller Design for a Synchronous Generator with Unknown Perturbation in Mechanical Power 309 an adaptive state feedback controller achieving the transient stabilization and voltage regulation is designed Siulation results will be given in Section 4 to support the theoretical clais Concluding rearks follow in the final Section PROBLEM FORMULATION We consider the classical third order dynaic odel of a synchronous generator connected to an infinity-bus which is given as follows []: δ = ω ωs D ωs ω = ( ω ωs) + { P Pe( t)} H H Xd X d E q = E cos q + V s δ + uf Td 0 where Pe () t is the active power here s q V q s d δ dσ qσ dσ () VE e sin X X P t = X X X sin δ () δ () t denotes the rotor angle Eq () t is the q -axis transient potential and V s is the voltage at the infinite bus ω ( t) is the angular speed of the generator andω s is the synchronous rotating speed P is the echanical power D is the daping constant H is the inertia constant = Xd + XT + XL is the total reactance that takes into account X d the generator direct axis reactance X T the reactance of the transforer and the transission line reactance X L d d T L X Σ = X + X + X with X d denotes the generator direct axis transient reactance XqΣ = Xq + XT + XL with X q denotes the generator q -axis transient reactance T d 0 is tie constant of the field winding u f is the control input of the SCR aplifier of the generator Suppose that the echanical power P is constant Then it is well known that for given noinal values of the angle δ s the angular speed ω s and the transient potential E if we set the excitation control u f as qs Xd Xd f cos qs s δs u = E V δs and E qs should satisfy s qs V q s d sinδ sin s δ s Pes XqΣ VE X X = = Then the closed loop syste is asyptotically stable at the point of equilibriu s s Eq Eqs δ = δ ω = ω = () π where δs (0 ) Correspondingly the generator terinal voltage will be regulated at V tr which is deterined by the following well known relation [] V t = + + V XsPe XdVs XsXd s sin δ X dσ Pecotδ (3) with the stable equilibriu and Xs = XT + XL However for syste () there exists another unstable equilibriu ( δu ωs Eqs) with sinδu = sinδs which ay be close to the stable one This fact signifies that the stable equilibriu ay have a very sall stability region so that a disturbance for instance the perturbation in the echanical power P ay drive the generator to be unstable or propt loss of synchronousness and inability to achieve voltage regulation Therefore under unknown echanical power the design proble of the adaptive controller which stabilizes the generator at an appropriate equilibriu corresponding to the echanical power and guarantees terinal voltage to be regulated to its prescribed value V tr is an iportant issue in the power syste The design proble considered in this paper is as follows Suppose the echanical power P is an unknown constant and let the corresponding unknown equilibriu be ( δs ω s Eqs) Find an adaptive state feedback controller of the for u= α( δ ω E q ) (4) P = ( β δ ω Eq ) such that the resulting closed-loop syste is stable at the unknown equilibriu and δ δs ω ωs E q qs E as t where P is the estiate of P Furtherore the terinal voltage V t is regulated to the prescribed value V tr 3 CONTROLLER DESIGN For the sake of siplicity we define the state variable by x = δ x = ω ωs x = E the 3 q

3 30 Xiaohong Jiao Yuanzhang Sun and Tielong Shen control law qs u = u and the unknown equilibriu f ( δ s0 E ) = ( xe0 x3e) = xe then the dynaics of the syste can be represented by the following state space odel: x = x x = ax + a P ax sin x + a sin x x 3 = cx3 + ccos x+ c3u where the paraeters are defined by D ωs ωsvs ωsv Xq X s d = = 3 = 4 = H H H HX dσ X qσx dσ (5) a a a a Xd Xd = = s 3 = Td0 X T d d0 X T Σ dσ d0 c c V c In the following we will seek a solution to the design proble by recursively constructing a Lyapunov function In order to achieve the voltage regulation perforance δs with the real value of the echanical power P and the set point V tr ust satisfy (3) ie b P xe = arccot b + Vtr P b (6) Vs XdVs where b = and b = X s X d Σ Motivated by this equation we can use the following estiation for the equilibriu point δ s corresponding the estiate P : b P xe = arccot b + V tr b e (7) Clearly x x e when P We start the recursive design process with this estiation First we define a positive definite function W( x) = x (8) with x = x x e The tie derivative of W along the trajectory of subsyste x is obtained x W ( x) = xx x (9) P e Define x = x α( x) and choose the virtual control law α( x ) as α ( x ) = ( x x ) (0) e then we obtain the following equality x W x x x x x P () ( ) = + e Let z3: = ax 3 3sinx a4sinx and consider the subsyste ( x x ) x = x x = ax + a z3 () Fro here we construct a positive definite function W( x x) = W( x) + x (3) Then we can obtain the tie derivative of W as: W = x ax + a z3 + x x x ~ x e x e x ~ x P ~ x P (4) P x e P Define z 3 = z 3 α( x x P ) and choose the virtualcontrol law α () as α = ax + x x + x + ap ν x where ν( x x P ) is a sooth function deterinedlater so that the tie derivative of W satisfies W = x x xz 3 + xa( ) ~ x e x e x P ~ + x P ν (6) x e P P For the whole syste constructing the positive definite function W3( x x z3) = W( x x) + z 3 (7) we have the tie derivative of W 3 along the trajectories of (5) W = z { x + a ( c x + c cos x + c u)sin x } z 3 axx 3 3 cos x ax 4 cos x x x z 3 ( ax + ap ax 3 3sinx+ a4sin x) x

4 Adaptive Controller Design for a Synchronous Generator with Unknown Perturbation in Mechanical Power 3 ~ x e ~ z ~ ( 3 z3 + xa ) (8) x e P ~ x e ~ x ~ ~ x P e + ν x x x P P x e The control law chosen as u = { 3 3 cos 4 cos 3 3sin x a x x x + ac x a x x + ( ax + a ax 3 3sinx+ a4sin x) (9) x + x + ν z 3} + ( cx3 ccos x) x c3 where ν ( x x x3 P ) is a sooth function deterined later renders x e x e W ~ x ~ x ~ z ~ x P ~ x P 3 = 3 + ν x e P P ~ α x e z P α P 3 ν (0) x e P P + x z 3 a( ) x Furtherore we choose a positive definite function r V( x x z3 ) = W3( x x z 3) + P () where P = and r is a given positive constant Then we calculate the tie derivative of V as follows: x e x e V = ~ x ~ x ~ z ~ x P + ~ α x P 3 ν P x e P ~ α x e z P α P 3 ν () x e P P + P x a z 3 a r x Let P = x a z 3 a (3) r x x e ~ x e ν ~ = a x + a x (4) r P r x e P x e ~ ~ ν = + 3 x a z a r x e P x x e ~ x e a ~ x a x r x e P x r P x (5) we obtain the following equality V = x x z (6) 3 With this equality we can reach the following conclusion: Proposition : For syste (5) if the state feedback controller is chosen as (9) and the paraeter adaptation law as (3)~(5) then for any constant echanical power P that guarantees the existence of the operating point xe( x e < π ) the closed loop syste is locally Lyapunov stable at the point of equilibriu and P P x xe x 0 x3 x3e Vt Vtr as t Proof: For the closed loop syste coordinated by ( x x z 3 P ) the Lyapunov function V that satisfies the equality (6) ensures stability at the origin Furtherore with the equality we can conclude fro Barbalats Lea that x 0 x 0 and z 3 0 as t ie x x e x 0 and z3 = ax 3 3 sin x a4 sin x α(00 P ) as t To coplete the proof we will show that ( x 0 e x3e ) will converge to the unknown equilibriu ( x e 0 x3e P ) Since x = x +α ( x ) = 0 at the origin x = 0 x = 0 we obtain fro the defini-tion of z 3 that z 3 α (00 P ) Note that the tuning function ν (00 P ) = 0 Therefore we have fro (5) that z3 = a3x3sinx a4 sinx a x sinx a sinx 3 3e e 4 e a P Alternatively fro the second equation of the dynaics of the generator a a3x 3e sinx e a4 sinx e 0 = + This eans P as t Consequently x xe x3 x3e as t Furtherore fro (6) it follows Vt Vtr Reark : It should be noted that in both [0] and this paper the convergence to the new equilibriu point corresponding the real value of the unknown echanical power is achieved by introducing the estiation function However in [0] the estiations for both the echanical power and the angular speed are required to construct the trajectories tracked by the states of the syste In this paper the echanical power is estiated to design the adaptive stabilizing

5 3 Xiaohong Jiao Yuanzhang Sun and Tielong Shen controller with the help of the tuning functions allowing the design algorith to be siple Moreover the proof of the result is given by analyzing directly the physical character of the syste 4 SIMULATION The physical paraeters of the power syste eployed for the siulation are given as follows: ω = pu D = 0 pu H = 7sV = 0995 pu T = 8s s s d0 d = 8 d = 03 q = 76 X pu X pu X pu X T = 0 5 pu XL = 035 pu and Xs X dσ X dσ XqΣ can be calculated as X s = XT + XL = 0475 = Xd + X s = 85 = Xd + X s = 0775 XqΣ = Xq + Xs = 35 Moreover according to the definition in Section 3 the values of the paraeters in odel (5) are obtained as follows: a = 0043 a = 049 a3 = 0834 a4 = c = c = 043 c3 = 050 and when the echanical power P = 09 and the excitation control input signal is a constant u fs = 097 the syste has a stable equilibriu at x e = ( ) and the terinal voltage of the generator V t is prescribed as V tr = 0475 Now we consider the following two cases in which there exists the unknown perturbation in the echanical power: (a) Unknown disturbance occurs to the echanical power when the synchronous generator is operating at stable equilibriu and after several occurrences the echanical power is recovered ie 09 0 t < 40[sec] = t 5[sec] 09 5 < t (b) Sudden turbine failure happens and the fault is not recovered so that the echanical power abruptly changes to an unknown value ie 09 0 t < 40[sec] = t The adaptive controller is designed as shown in (9) and (3)~(5) for the transient stabilization with voltage regulation of the generator We first give the siulation research for case (a) When the syste (5) is not forced by any feedback excitation control law but only the constant u fs the response of the syste under case (a) (in siulation = 03 ) is shown in Fig It can be seen that the P equilibriu can be recovered and the terinal voltage can be regulated to the prescribed value however the recovering tie is 56s When the syste is forced by the proposed adaptive control law (9) with (3)~(5) the response of the closed loop syste is presented in Fig where the control paraeter r is chosen as r = 05 It follows fro the results of Fig that the adaptive controller guarantees the syste to recover quickly the operating equilibriu and the prescribed value of the terinal voltage (the recovering tie is 5s) Fig The response of the syste without feedback control under case (a) Fig The response of the syste with feedback control under case (a)

6 Adaptive Controller Design for a Synchronous Generator with Unknown Perturbation in Mechanical Power 33 Fig 3 The response of the syste without feedback control under case (b) Fig 4 The response of the syste with feedback control under case (b) Now we consider case (b) ie the echanical power is varied fro the noral value 09 to soe unknown constant (in the siulation it is chosen as ) The response of syste (5) with the constant input u fs is shown in Fig 4 It is seen that the syste is unstable However fro the siulation result given by Fig 4 it is indicated that the proposed adaptive control law can render the closed loop syste to converge quickly to a new equilibriu point corresponding the real value of the unknown echanical power and the terinal voltage can be regulated to the prescribed value quickly 5 CONCLUSIONS In this paper we investigated the transient stabilization with the voltage regulation for the synchronous generator when the echanical power is perturbed to an unknown constant value so that the operating point of the syste is also unknown We designed a nonlinear adaptive excitation controller by way of backstepping ethod with tuning functions It was shown that the presented excitation controller can drive the syste to a stable equilibriu corresponding to the real value of the unknown echanical power and siultaneously achieve good regulation of the generator terinal voltage REFERENCES [] Q Lu and Y Z Sun Nonlinear stabilization control of ultiachine systes IEEE Trans on Power Systes vol 4 no pp [] L Gao L Chen Y Fan and H Ma A nonlinear control design for power systes Autoatica vol 8 pp [3] Y Wang D J Hill R H Middleton and L Gao Transient stability enhanceent and voltage regulation of power systes IEEE Trans on Power Systes vol 8 no pp [4] Y Tan and Y Wang Augentation of transient stability using a supperconduction coil and adaptive nonlinear control IEEE Trans on Power Systes vol 3 no pp [5] Y Wang D J Hill R H Middleton and L Gao Transient stabilization of power systes with an adaptive control law Autoatica vol 30 pp [6] Y Wang and D J Hill Robust nonlinear coordinated control of power systes Autoatica vol 3 pp [7] M Galaz R Ortega A S Bazanella and A M Stankovic An energy-shaping approach to the design of excitation control of synchronous generators Autoatica vol 39 no pp [8] T Shen R Ortega Q Lu S Mei and K Taura Adaptive L disturbance attenuation of hailtonian systes with paraetric perturbation and application to power systes Asian Journal of Control vol 5 no pp [9] T Shen S Mei Q Lu W Hu and K Taura Adaptive nonlinear excitation control with L disturbance attenuation for power systes Autoatica vol 39 no pp [0] G Da R Marino and L L Francoise Adaptive nonlinear output feedback for transient stabilization and voltage regulation of power generators with unknown paraeters International Journal of Robust and Nonlinear Control vol 4 pp [] Q Lu Y Z Sun and S Mei Nonlinear Control Systes and Power Syste Dynaics Kluwer Acadeic Publishers Boston 000

7 34 Xiaohong Jiao Yuanzhang Sun and Tielong Shen Xiaohong Jiao received his BEng and MSc degrees in Autoatic Control fro Northeast Heavy Machinery Institute China in 988 and 99 respectively and his PhD degree in Mechanical Engineer-ing fro Sophia University Tokyo Japan 004 Currently she is working towards her PhD in the Departent of Electrical Engineering Tsinghua University China Siultaneously she is eployed as a Professor in the Departent of Autoatic Control Yanshan University China Her ain areas of interest are robust control and adaptive control of nonlinear systes and tie-delay systes Yuanzhang Sun received his BS degree fro Wuhan University of Hydro and Electrical Engineering China in 978 his MS degree fro the Electric Power Research Institute (EPRI) China in 98 and his PhD degree fro Tsinghua Univ Beijing in 988 all in Electrical Engineering Fro 98 to 985 he was with the EPRI China working in the Power Syste Research Departent He is currently a Professor at Tsinghua University and the Head of the Dynaic Siulation Power Syste Lab He was appointed as a ChangJiang Scholar in 999 Tielong Shen received his BEng and MSc degrees in Autoatic Control fro Northeast Heavy Machinery Institute China in 98 and 986 respectively and his PhD degree in Mechanical Engineer-ing fro Sophia University TokyoJapan 99 Since April 99 hehas been an Assistant Professor in the Departent of Mechanical Engineering at Sophia University He is the author/co-author of three textbooks His current research interests include H control theory and robust control of linear and nonlinear systes and its applications in echanical systes