ADAPTIVE CONTROL DESIGN FOR A SYNCHRONOUS GENERATOR

Transcription

1 ADAPTIVE CONTROL DESIGN FOR A SYNCHRONOUS GENERATOR SAEED ABAZARI MOHSEN HEIDARI NAVID REZA ABJADI Key word: Adaptive control Lyapunov tability Tranient tability Mechanical power. The operating point of a power yte change to an unknown point with an unknown change in the echanical input power. In thi paper a nonlinear adaptive controller i deigned for ecitation yte of the generator baed on the backtepping control technique in order to achieve tranient tability enhanceent in the preence of uncertaintie in echanical power. The deigned controller guarantee the convergence of yte tate to new deired value correponding to unknown echanical power. A power yte coniting of a ynchronou generator connected to an infinite bu through a double circuit traniion line i ued in control deign and the iulation tudie. Coputer iulation verifie the effectivene and the validity of the propoed control conidering faulted yte with a clearance and change in network topology.. INTRODUCTION There are variou tability proble in power yte due to all and large diturbance. Aong the tranient tability i a very iportant atter. For tranient tability analyi and control deign the power yte ut be decribed by nonlinear differential equation [ ]. To aintain a high degree of reliability the effect of diturbance uch a 3-phae fault and echanical power udden change hould be conidered. Large diturbance on the traniion yte caue difficultie in active power traniion epecially during electrical fault. Thi lead to an acceleration of the rotor of the ynchronou generator due to unequal echanical and electrical torque acting on the haft. Rapid detection of the fault and fat acting circuit breaker are norally ued to iolate the faulted ection of the traniion yte. In addition to the rapid iolation of faulted ection of the traniion equipent it i generally recognized that the auiliary control action would be neceary to enhance the pot fault region to aure the tability of a power yte. The firt power yte tabilizer ued in power yte operate with linear ethod. Linear power yte tabilizer (PSS) [3] are often ued to provide uppleentary daping through ecitation control to iprove the dynaic tability liit. An approiate linearized odel a a deign odel and a linear controller are ued for a pecific operating point. Therefore the uual tabilizer deign baed on approiate linearization odel i not adequate for large 3 Sahrekord Univerity Faculty of Engineering Shahrekord Iran E-ail: aeedabazari@yahoo.co h39.63@gail.co navidabjadi@yahoo.co Rev. Rou. Sci. Techn. Électrotechn. et Énerg p. 4 4 Bucaret 4

2 4 Saeed Abazari Mohen Heidari Navid Reza Abjadi diturbance. In other word with large diturbance the operating point of power yte change and the linear deigned controller ay be failed. The application of nonlinear control in thi field ha been dicued in reearch paper ince 98. Since there are alway uncertaintie in power yte due udden change in input echanical power and electrical fault the overall yte ay becoe even untable. To retain tability and to achieve good perforance and generator ynchronization ophiticated control ethod are needed. In recent two decade nonlinear control theory ha been applied to any practical control proble. In [4 5] feedback linearization technique i eployed for ecitation yte. By uing direct feedback linearization (DFL) technique an ecitation controller and coordinated controller are propoed in [6] to achieve both tranient tability enhanceent and voltage regulation of a power yte. However the DFL ethod cannot work when there are uncertaintie in power yte. In [7 8] adaptive feedback control i ued for the ae proble. In [9] robut nonlinear control and energy haping approach are ued for power yte control; however when a fault occur at one-tenth of the traniion line fro the generator terinal the yte uing an adaptive controller baed on thee reearch work can no longer aintain tranient tability. In [ ] a nonlinear adaptive controller i deigned for power yte with unknown echanical power but the peed of tate convergence to equilibriu point i low. One of the nonlinear control ethod which ay guarantee global tability i backtepping approach [3 6]. In [3] an adaptive backtepping control i deigned for a ingle achine with an infinite bu without conidering any change in network topology. The etiation and control of [3] are coplicated and the convergence ay be failed. In [4] adaptive backtepping i ued in a otor drive yte. An adaptive backtepping control i deigned for a power yte with unified power flow controller (UPFC). In [6] an adaptive backtepping control i deigned for a power yte with a ingle achine. The preent paper odifie the control ethod of [6]. The obtained control law reult fat repone copared to [6]. The obtained reult how the propoed control retain the tability of the overall yte even with changing the power yte topology. Tranient tability with voltage regulation of ynchronou generator under an unknown input echanical power i conidered. An adaptive controller i deigned for ecitation yte with an etiation function for input power. Uing the propoed controller the operating point of the yte i hifted to new equilibriu point and the yte will be table. The organization of the paper i a follow: in ection the power Fig. Power yte diagra.

3 3 Adaptive controller deign for a ynchronou generator 43 yte odel and control proble are preented; in ection 3 an adaptive backtepping nonlinear controller i deigned for the yte and the iulation reult are illutrated in ection 4 including the coparion of the propoed controller to conventional power yte tabilizer (PSS).. MODELING AND CONTROL PROBLEM The power yte under tudy i hown in Fig.. The paraeter of the yte are given in Appendi. To iplify the proce of controller deign a third order odel in conidered for ynchronou achine but in yte iulation the ynchronou achine i odeled with a ore accurate odel cloe to the phyical real achine. The tate equation of the yte for controller deign are a follow: δ = ω ω r ω D ω = ( P P ) ( ω ω ) () r H e H r ' E q d d ed q eq q = V in δ( ωr ω) Eq + V coδ + U f ( t) ed ed ' τ' do ed ' τ' do ed ' τ' do where δ i the power angle of generator ω r i the rotor peed of the generator ω i the ynchronou peed P i the echanical power P e i the electrical power E q i the q-ai potential q i the q-ai reactance d i the d-ai reactance ' d i the d-ai tranient reactance eq i the q-ai total reactance ed i the d-ai total reactance ' ed i the d-ai total tranient reactance T i the tranforer reactance L i the traniion reactance V i the infinite bu voltage V t i the terinal voltage and V tr i the precribed value of the terinal V. Eq voltage Pe = inδ ed' = ' d+ T + L eq = q + T + L ed = d + T + L eq = T + L. The operating point of the yte i deterined a b P V qv = arccot b + Vtr with: b = b = P b eq δ V E q eq in δ = P e = P E q P eq =. () V in δ

4 44 Saeed Abazari Mohen Heidari Navid Reza Abjadi 4 Firt auing the echanical power in each intant and during each diturbance occurrence i eaurable. One can ee that fro () (3) after a known diturbance the new operating point ( δ Eq) i obtained. The following ecitation control can tabilize the yte. U f ' τ' ' ' τ' ed d q d ed d q () t [ V inδ( ω ω ) + E V coδ]. = r q eq ed ed d ed ' τ' d When a diturbance occur in the yte and the tate depart fro the operating point thi controller return the to the operating point. The following equation give the terinal voltage cot in Pe V q V t = + + Pe δ. (4) V δ eq eq If the input echanical power i not known or it i changed to an unknown value due a diturbance in turbine the new operating point ( δ Eq ) can not be deterined. In the following an adaptive ecitation control i deigned. The control law and etiation law are conidered a follow U f = F ( δ ωr Eq P ) P = Y ( δ ωr Eq P ) (3) (5) where P i an etiation of P and the ain ai of control deign i the following convergence: δ δ ωr ω Eq Eq. According to () Eq i the ae a electrical power of generator which i ent to power network. 3. ADAPTIVE CONTROLLER DESIGN In thi ection adaptive backtepping control theory i eployed to achieve the entioned control ai. To achieve the deired terinal voltage δ and P hould atify the following relation δ b P = arccot b + Vtr P b. (6) Baed on the above equation an etiation of δ Eq correponding to echanical power i obtained a follow

5 5 Adaptive controller deign for a ynchronou generator 45 δ b P = arccot b + Vtr P b P E. (8) eq q = V in δ (7) It i evident that if P P then δ δ and E q E q ; to iplify the control deign the following new tate and notation are introduced ( δ E = δ q ) = ( e e The yte dynaic i given by 3e = ω ) r ω = e P 3 3 = E q 3e. (9) = ap + f + g () + g3. U ( ) 3 = f3 f t where ( ) ( ) f = β J3ein e + g = Jin e + g3 = L 3e f3 = P in( ) ( 3 3) co( ) + KV e + M e + + NV e + P ω a = H D β = H ω V J = H ' d q ed eq M = N = L =. ed '. τ' d ed '. τ' d ed '. τ' d Now baed on tate equation () the ecitation control i deigned in uch a way that the derivative of a poitive definite Lyapunov function around equilibriu point becoe negative definite. Thi condition hould be atified for π e < to achieve local ayptotically tability. Here backtepping control theory i ued. Conider the following priary Lyapunov function eq K = V ( ) =. () q ed d

6 46 Saeed Abazari Mohen Heidari Navid Reza Abjadi 6 Differentiating it one can obtain V = = e P. () Adding and ubtracting virtual input in the bracket V ( e = + P ). (3) If the virtual input i conidered a = k k ) then ( > V =. (4) e k + ( ) P Now conider the following etended Lyapunov function V = V + ( ). (5) Again with differentiating V = V + ( )( ) = (6) = k + ( )( + ap + f + g ) P. e 3 Suppoe there i another virtual input notated with 3 ; if 3 i deigned a 3 = ( k( ) + + f + v ) (7) g where k > and v i deterined later then V = k k( ) + ( ) g ( 3 3 ) + e + ( )( ap v ) P. P The overall yte Lyapunov function i conidered a follow (8) V3 = V + ( 3 3 ). (9) DifferentiatingV 3 one can obtain

7 7 Adaptive controller deign for a ynchronou generator 47 V = V + ( )( ) = ( f ) ( ) ( 3 3) ( ) = k k + g + f + gu + e + ( )( ap v ) P. P () The ecitation control input i appeared in the above epreion. If it i conidered a follow the derivative of the Lyapunov function becoe negative U f = [ k3( 3 3 ) + g ( ) + f3 + v ] k3 > () g 3 where v another function which i deterined later. Subtituting for one can obtain U f V = k k k P + e 3 ( ) 3( 3 3 ) + ( )( v ) + ( )( ap v ) () To etiate the input echanical power and to achieve an adaptive control the lat Lyapunov function i copleted a follow where r > and P = P P. Differentiating V 4 V 4 = V3 + rp (3) V = V rpp = k k k P + e 4 3 ( ) 3( 3 3 ) k k + β k k + β + ( 3 3 ) 3 v + ( ) ap ( ) ap + g g + ( ) a( P+ P ) v rpp = ( ) (4) = k k k P + ( ) 3( 3 3) e P

8 48 Saeed Abazari Mohen Heidari Navid Reza Abjadi 8 ( ) k k + β + ( 3 3 ) 3 v + ( ) ap + ( ) ap v + g k k + β P a( + ) a( 3 3 )( ) rp. g The lat three ter of the above epreion ay becoe poitive or negative o they hould be oitted and the following etiation law and function for v and v are obtained P = a( r k k + β ) a( 3 3 )( ) g (5) e v = ap = ap + k k P (6) k k + β v = 3 + ( ) ap (7) g or 3 = ( k ( ) ) + + v + f + g 3 g v = [ k k + + ap + k k β J co( + ) + e 3e e g k k + β + g 3 ] + ( ) ap g ( e e e ) 3 3 J co( + ) + Jco( + ) ]. (8) Uing thee epreion one can obtain V = k k ( ) k ( ) (9) P 4 e P It i clear that after an unknown diturbance in input echanical power the yte equilibriu point of the yte i changed and the ecitation input hould be changed uch that the yte converge to the new equilibriu point. The above control and etiation law guarantee that for t li V 4 = which yield a tie tend to infinite = = = k = and 3 = 3. Uing () (7) and (6) one can obtain = ap + f + g 3 = v + f + g 3 v = ap P = P.

9 9 Adaptive controller deign for a ynchronou generator 49 One ay note fro (7) (8) with convergence of etiated echanical power to it real value e e 3e 3e and terinal voltage regulation i achieved. In fact etiated equilibriu point ( e 3e ) converge to new equilibriu point ). ( e 3e 4. SIMULATION AND DISCUSSION Uing yte paraeter given in appendi conidering echanical power a P =.6 p.u and ecitation input a U f =.656 p.u the operating point of yte becoe ( ) which i table. The terinal voltage becoe V tr =.7 p.u. In thi ection through two tet the perforance of the propoed controller i invetigated and copared to PSS. The tuned controller paraeter are k = 3 k =. k3 = 4 r = 5. a) Tet a. Conidering an unknown change in input echanical power for a all duration a in Fig. a. The iulation reult of thi tet are hown in Fig. a to d. A can be een fro Fig. a and d with the change ΔP =. p.u then ΔP =.5 p.u then ΔP =.5 p.u the propoed ecitation control input with echanical power etiation and new equilibriu point etiation trie to tabilize the yte and regulate the terinal voltage. After thi change i oitted the yte tate return to their previou value. The reult with AVR+PSS are alo illutrated in Fig. a tod it can be een that the convergence and daping of the propoed controller are uch better copared to conventional AVR+PSS. The propoed controller caue a fat dynaic. b) Tet b. Short circuit in a duration tie and clearing the fault with changing the network topology: In thi tet a hort-circuit fault at t = inline L very cloe to bu i appeared. The fault i cleared by opening line at t =.5. Siulation reult are hown in Fig 3a to 3b. It i clear that the propoed controller i able to tabilize the yte very quickly with changing network topology in the new equilibriu point. (a) etiate of echanical power(b) achine angle

10 4 Saeed Abazari Mohen Heidari Navid Reza Abjadi (c) terinal voltage (d) control law Fig. Siulation reult of Tet a conidering an unknown change in input echanical power for a all duration. (a) achine angle (b) Angular velocity Fig. 3 Siulation reult with changing the network topology. 5. CONCLUSION In thi paper conidering change in input echanical power a an unknown diturbance an adaptive ecitation control i deigned for power yte baed on backtepping control. Since the deigned control i baed the original nonlinear tate equation it ha uperiority over conventional control obtained for linearized equation. By uing adaptive technique with Lyapunov direct ethod which i applied for each ubyte eparately the echanical input power i etiated. The effectivene and validity of the propoed control i illutrated through iulation reult. The tability of the power yte i guaranteed even in the preence of uncertaintie in power network. The perforance of the propoed controller i copared to conventional AVR+PSS. It can be een that the daping and dynaic of the propoed controller are uch better that conventional AVR+PSS. It i hown that the propoed controller i able to tabilize the network even in the cae of network topology changing and even if the fault point i near to terinal of ynchronou generator. The tranient tability i teted by applying a three phae fault near the ending end with conidering a clearance tie.

11 Adaptive controller deign for a ynchronou generator 4 APPENDIX Syte paraeter: P =.6 p.u V = p.u Vte =.7 p.u.757 p.u =.5845 p.u ' =.5945 p.u =.364 p.u τ' d = q d T ω d = p.u D =. p.u H = 3.54 = 6.66 L = L Received on January 4 L =.546 p.u. REFRENCES. Yao Nan Yu. Electric Power Syte Dynaic Acadeic Pre New York P. Kundur Power Syte Stability and Control Power Syte Engineering Serie McGraw-Hill New York E. V. Laren and D. A. Swann Applying power yte tabilizer IEEE Tran. Power Appr.yt. pp Q. Lu and Y. Z. Sun Nonlinear tabilization control of ultiachine yte IEEE Tran. on Power Syte 4 pp L. Gao L. Chen Y. Fan H. Ma A nonlinear control deign for power yte Autoatica 8 pp Y. Wang D. J. Hill R. H. Middleton L. Gao Tranient tability enhanceent and voltage regulation of power yte IEEE Tran. on Power Syte 8 pp Y. Tan and Y. Wang Augentation of tranient tability uing a upper conduction coil and adaptive nonlinear control IEEE Tran. on Power Syte 3 pp Y. Wang D. J. Hill R. H. Middleton L. Gao Tranient tabilization of power yte with an adaptive control law Autoatica 3 pp Y. Wang D. J. Hill Robut nonlinear coordinated control of power yte Autoatica 3 pp Deign of ecitation control of ynchronou generator Autoatica 39 pp Ide Deign of ecitation control of ynchronou generator Autoatica 39 pp G. Da R. Marino L. L. Francoie Adaptive nonlinear output feedback for tranient tabilization and voltage regulation of power generator with unknown paraeter International Journal of Robut and Nonlinear Control 4 pp Q. Lu Y. Z. Sun and S. Mei Nonlinear Control Syte and Power Syte Dynaic Kluwer Acadeic Publiher Boton. 3. R. Yan Z. Y. Dong T. K. Saha R. Majuder Power Syte Tranient Stability enhanceent with an adaptive control chee uing backtepping deign IEEE conf. June A. Tahour A. G. Aiaoui A. C. Megherbi Poition control of witched reluctance otor uing an adaptive backtepping controller Rev. Rou. Sci. Techn. Électrotechn. et Énerg pp S. Shojaeian J. Soltani Low frequency ocillation daping of a power yte including unified power flow controller baed on adaptive backtepping control Rev. Rou. Sci. Techn. Électrotechn. et Énerg. 58 pp X. Jiao Y. Sun T. Shen Adaptive Controller Deign for a Synchronou Generator with Unknown Perturbation in Mechanical Power International Journal of Control Autoation and Syte 3 pp