Stability of an Adaptive Switched Controller for Power System Oscillation Damping using Remote Synchrophasor Signals

Transcription

1 Stablty of an Adaptve Swtched Controller for Power Sytem Ocllaton Dampng ung Remote Synchrophaor Sgnal Farhad R. Pour Safae 1, Scott G. Ghocel 2, João P. Hepanha 1, and Joe H. Chow 3 Abtract Th paper concerned wth the tablty of an adaptve wtched controller for nterarea power ytem ocllaton dampng ung remote gnal. Thee gnal ntroduce varable latency nto the dampng control ytem due to phaor proceng and communcaton network delay. In prevou work, we degned an adaptve controller that dynamcally wtche among dfferent compenator dependng on the latency of the ncomng meaurement. To prevent wtchng ntablty, a long reet tme wa mplemented. We demontrated the tablty of the adaptve ytem ung mulaton. In th work, we ue the concept of average dwell tme (ADT) of wtched control ytem to develop a uffcent condton that guarantee the tablty of our adaptve wtchng algorthm. Ung th condton, we formulate a feablty problem to compute the mnmum average dwell tme for the et of degned compenator. We apply the algorthm to a thyrtor-controlled ere compenator on a two-area power ytem and how that the adaptve controller table for tme-varyng latency. I. INTRODUCTION The advent of ynchronzed phaor meaurement unt (PMU) at many locaton acro power grd enable the ue of uch remote gnal for nterarea ocllaton dampng. Unlke local gnal, remote meaurement can provde better obervablty of nterarea mode and thu more effectve control ung power-electronc-baed devce uch a a thyrtor-controller ere compenator (TCSC). However, communcaton of data from remote PMU can ntroduce data lo, corrupton, and latency. Data lo and corrupton can be partally mtgated ung data recontructon method or tate etmaton technque [1]. On the other hand, latency nherent n remote gnal and ncreaed by uch data proceng algorthm. Dfferent level of congeton n communcaton network produce varyng amount of meaurement latency to be compenated. In prevou work [2], we developed an adaptve wtched controller to damp nterarea ocllaton for a two-area power ytem ung remote PMU gnal. The controller conted of everal compenator, each degned for a pecfc amount of data latency n the PMU. Thu the adaptve cheme would Th work at RPI wa upported n part by the Engneerng Reearch Center Program of the Natonal Scence Foundaton and the Department of Energy under NSF Award Number EEC and the CURENT Indutry Partnerhp Program. Th reearch at UCSB wa upported by the Inttute for Collaboratve Botechnologe through grant W911NF from the U.S. Army Reearch Offce. 1 F. R. Pour Safae (farhad@ece.ucb.edu) and J. P. Hepanha (hepanha@ece.ucb.edu) are wth the Dept. of Electrcal and Computer Eng., Unverty of Calforna, Santa Barbara, CA. 2 S. G. Ghocel (ghocel@exponent.com) wth the Electrcal Engneerng and Computer Scence Practce, Exponent, Inc., New York, NY. 3 J. H. Chow (chowj@rp.edu) wth the Dept. of Electrcal, Computer, and Sytem Eng., Renelaer Polytechnc Inttute, Troy, NY. wtch among the compenator dependng on the meaured latency of the nput gnal. To prevent ntablty due to wtchng among the dfferent compenator, we degned the algorthm to have a uffcently long reet tme, much longer than the typcal decay tme of the ocllaton. Through mulaton, we howed that the wtchng algorthm exhbted tablty. In th work, we formally demontrate the tablty of the adaptve controller degn ung a theoretcal reult for wtched ytem. We employ the concept of average dwell tme (ADT) wtchng equence to contruct a tablty proof for our degned adaptve controller. Ung thee theoretcal reult, we compute the mnmum average dwell tme to guarantee tablty for a elected et of compenator under our adaptve control cheme. We how that the reet tme choen n our prevou work uffcent to enure table operaton. Other reearcher have alo condered tablty of networked control ytem wth tme varyng delay [3]. However, the man drawback of [3] that t more computatonally expenve compared to the algorthm ntroduced n th paper. II. PMU DATA LATENCY Phaor meaurement unt (PMU) are dtrbuted acro wde geographcal regon and the data tranmtted acro long communcaton lnk. Generally the data frt collected by a local utlte at ther phaor data concentrator (PDC), then treamed to the central PDC of the regonal ytem operator. An example of th herarchcal arrangement hown n Fgure 1. The phaor meaurement face a number of delay along the gnal path. Frt, the frequency etmaton and phaor calculaton algorthm typcally requre multple cycle of meaured data to compute the phaor quantte. Th type of fxed delay et a floor for the overall tme-varyng meaurement latency. After the phaor computed, the data tranmtted acro communcaton network to the local PDC and fnally to the central PDC. In addton to the delay acro the communcaton network nfratructure, whch can be calculated baed on the type of communcaton lnk [4], the data alo encounter mall delay at each PDC due to proceng. A an example, we lt the etmated delay for the Quebec power ytem n Table I [5]. We model the meaurement latency ung the ame approach a n our prevou work, namely a mnmum delay wth a varable component, takng the approach ued n [6] to calculate total tme delay T ld a T ld = T + T b + T p + T r

2 Phaor Meaurement Unt Hgh-voltage Subtaton A GPS Sgnal 3-phae current and voltage Hundred of km apart Local Phaor Data Concentrator Phaor Meaurement Unt Hgh-voltage Subtaton B PMU data GPS Sgnal 3-phae current and voltage Internet Local Phaor Data Concentrator Internet PMU data Central Phaor Data Concentrator Dampng Controller PMU data GPS Sgnal Control actuaton Fg. 1. PMU Data Communcaton Path 0 + u TABLE I PMU DATA LATENCY IN THE QUEBEC POWER SYSTEM PMU proceng tme 73 m Local data concentraton 16 m 2,000 km n optcal fber m Central data concentraton m Total etmated data latency 9 m TW 1+ TW Fg. 2. ( 1+ T ) 2 δ Delay d( t) Glead (, T ) d d( t) TCSC Adaptve Control Sytem Power Sytem where T = P /D r the eral delay, P the packet ze n bt, D r the tranmon rate of the lnk n bt/, and T b the delay between data packet. Moreover, T p = L/ν the propagaton delay, where L the lnk length n km and ν the propagaton peed n the lnk n km/, and T r the routng delay. III. CONTROL DESIGN The adaptve controller follow a mlar control degn a n [2] wth ome mnor change to reflect a more practcal mplementaton. For dampng control wth a thyrtorcontroller ere compenator (TCSC), we typcally ue a dervatve flter wth a mall tme contant T δ n addton to the wahout flter and thu no addtonal phae lead requred. For our controller degn, we adapt the controller from [7] and add a phae lead compenator to counteract the phae lag caued by data latency. The overall controller G c (, T d ) = K(T d ) 1+Tnum(T d) 1+T den (T d ) (1+T δ ) 2 T w 1+T w (1) and the ytem hown n Fgure 2. Note that we wtch among everal phae lead compenator whch are each degned for a pecfc level of delay T d. A the nput gnal latency d(t) vare, we ue the adaptve algorthm from [2] to elect the controller delay T d uch that T d d(t), but wth dwell tme conderaton to prevent wtchng ntablty. The algorthm can be ummarzed a follow: Adaptve Control Algorthm Prepecfy a et of T d value: 0 < T d1 < T d2 < < T dn. At tme t = t k, where t the tme at the controller, the tme delay T d ued to et the controller. for t = t k + t, where t the amplng perod of the PMU data f the next data pont already n the nput data buffer, or the ncremental tme of arrval of the next data pont f the nput data buffer empty. f the data delay larger than the T d, wtch to a controller wth the lowet latency T dj whch hgher than the data delay. elef the maxmum latency of all the data n the lat T r le than T dj, where T dj < T d, wtch n the controller wth a lower latency T dj. ele contnue wth the ame controller. end End algorthm Note that the reet tme T r ued to lmt rapd wtchng and prevent ntablty. In the next ecton, we wll derve a uffcent condton to guarantee tablty of the cloed-loop wtched ytem. IV. STABILITY OF THE SWITCHED SYSTEM It well-known [8] that a wtched ytem mght be table under uffcently low wtchng equence, uch that the tranent effect dpate after each wtch. The mplet way to pecfy low wtchng equence to retrct the cla of admble wtchng gnal to atfy a dwell tme contrant. That to ntroduce a number τ d > 0 uch that

3 the wtchng tme atfy the nequalty t k+1 t k τ d. Specfyng a dwell tme contrant may be too retrctve n the context of controlled wtchng. Thu, one can conder an enlarged famly of wtchng gnal that occaonally have conecutve dcontnute eparated by le than τ d, but for whch the average nterval between dcontnute no le than τ d. Th concept wa frt formalzed n [9] a average dwell tme. Let N σ (t, T ) denote the number of dcontnute of a wtchng gnal σ on the tme nterval (t, T ). We ay that σ ha average dwell tme τ d f there ext potve number N 0 and τ d uch that N σ (t, T ) N 0 + T t τ d T t 0. Our goal to etablh a tablty crtera for a cla of wtched delay ytem under an average dwell tme contrant. We conder a cla of wtched delay ytem of the form ẋ(t) = A σ(t) x(t) + B σ(t) x(t d t ), (2) where x R n denote the tate of the ytem, σ(t) : [0, ) S = {1, 2,..., N} the pecewe contant wtchng gnal. We propoe a uffcent condton that guarantee the tablty of the adaptve wtchng mechanm n [2]. We formulate the problem for arbtrary number of mode. Let u denote the latency of the arrvng data by d t [0, d max ]. Aume that controller ha been degned to work n the range of d t [0, d ] for S wth d +1 > d and d N = d max. We conder the followng wtchng mechanm. In mode, If the data latency larger than d, wtch to the controller wth the mallet delay d j whch greater than the data latency. If the data latency maller than d j where d j < d, wtch to the controller wth a lower delay, conderng that the average tme between wtche at leat τ d. Our goal to fnd a mnmum value of τ d for whch the wtched ytem reman table. The followng theorem provde a lower bound for τ d. We defne F := [ A 0 ] and H := [ ] 0 B. Theorem 1: Conder the delayed wtched ytem (2) wth d t [0, d max ). Aume that controller ha been degned to work n the range of d t [0, d ] for S wth d +1 > d and d N = d max. If there ext potve defnte ymmetrc matrce P, Z R n n, matrce N R 2n n and a contant µ 1 uch that wth P µp j Z µz j, j S (3) [ ] φ N Φ := N d 1 < 0 (4) Z [ φ = F P 0 ] + [ P 0 ] F [ + H P 0 ] + [ P 0 ] H + d max (F + H ) Z (F + H ) [ ] [ ] N I I I I N then the ytem table for any average dwell tme wtchng equence wth τ d > log µ α where α gven by α = mn λ mn ((φ + d N Z 1 N )) max λ max (P ) + d2 max 2 max λ max (Z ). (5) It worth notng that for a fxed value of µ 1, (3)-(4) are lnear matrx nequalty condton. By performng a lne earch on µ, one hould look for the mallet µ for whch (3)-(4) are feable. Proof: Aume that the ytem n the th mode for t [t k, t k+1 ]. We have ẋ(t) = A x(t) + B x(t d t ) + B x(t) B x(t) = (A + B )x(t) B ẋ() d. We chooe the Lyapunov functon V (t) = V 1 (t) + V 2 (t), V 1 (t) = x(t) P σ(t) x(t), V 2 (t) = 0 d max t+θ ẋ() Z σ() ẋ() d dθ. Let u compute V (t) for t [t k, t k+1 ). We aume that n th tme nterval σ(t) = V 1 (t) = x(t) ((A + B ) P + P (A + B )) x(t) 2x(t) P B ẋ() d V 2 (t) = d max ẋ(t) Z ẋ(t) ẋ() Z ẋ() d td max Therefore, V (t) x(t) ((A + B ) P + P (A + B )) x(t) 2x(t) P B + d max ẋ(t) Z ẋ(t) ẋ() d ẋ() Z ẋ() d = x(t) ((A + B ) P + P (A + B )) x(t) 2x(t) P B ẋ() d + d max (A x(t) + B x(t d t )) Z ( A x(t) + + B x(t d t ) ) ẋ() Z ẋ() d = x(t) ((A + B ) P + P (A + B )) x(t) 2x(t) P B (x(t) x(t d t )) + d max x(t) A Z A x(t) + 2d max x(t) A Z B x(t d t ) + d max x(t d t ) B Z B x(t d t ) ẋ() Z ẋ() d (6)

4 TABLE II PHASE LEAD COMPENSATORS G c() Controller # Delay (T d ) Lag (Delay) Lead Damp. K T N T D 1 m % m % m % m % m % Defnng ξ(t) := [ x(t) N, we have 2ξ N [ I I ] ξ = 2ξ N x(t d t ) ], For any et of matrce + ẋ() d d(t)ξ N Z 1 N ξ Combnng (6) and (7), we have ẋ() Z ẋ() d. (7) V (t) x(t) ((A + B ) P + P (A + B )) x(t) 2x(t) P B (x(t) x(t d t )) + d max x(t)a Z A x(t) + 2d max x(t) A Z B x(t d t ) + d max x(t d t ) B Z B x(t d t ) 2ξ [ ] N I I ξ + d ξ N Z 1 N ξ = ξ (F [ P 0 ] + [ P 0 ] F + H [ P 0 ] + [ P 0 ] H + d max (F + H ) Z (F + H ) [ ] [ ] N I I I I N + d N Z 1 N )ξ Thu, we have V (t) ξ (φ + d N Z 1 n (5), one can how that V (t) αv (t). N )ξ. Wth α gven Therefore, the Lyapunov functon decreae between wtchng. Followng (3), at a wtchng ntant t k, we have V σ(tk )(t k ) µv σ(t k )(t k ). Ung the reult of [8, Theorem 3.2], we conclude that the ytem table for any average dwell tme wtchng wth τ d > log µ α. It worth notng that the number of decon varable n (4) a lnear functon of the number of mode. However the number of decon varable n [3] grow quadratcally wth the number of mode. In partcular, f N denote the number of mode and n the dmenon of the cloed-loop ytem, the number of decon varable n (4) N(3n 2 + n) whle [3] requre N(5N + 1)n(n + 1)/2 decon varable. A. Two-Area Power Sytem V. SIMULATION RESULTS To llutrate the capablty of our degn, we conder a two-area, four-generator ytem adapted from [] and hown n Fgure 3. We ue the ame parameter from [2], uch that the ytem ha gnfcant nterarea power tranfer and prone to untable ocllaton followng a hort crcut fault. In th ytem, Generator 1 and 2 n Area 1 are coherent and Generator 11 and n Area 2 are coherent and all generator are repreented ung detaled machne model wth exctaton ytem. Gen 1 Gen 2 Area Load 4 Load 14 Fg. 3. Four-Machne, Two-Area Sytem Area Gen 11 Gen The TCSC located n ere wth one of the nterarea lne, and a hort crcut fault appled near Bu 999 to excte the ytem. We chooe the dfference between the averaged angle θ a n each area a our nput gnal θ a = 0.5(θ 1 + θ 2 ) 0.5(θ 11 + θ ) where θ the phae angle of the voltage at Bu. Th nput gnal exhbt good obervablty of the nterarea mode, and often preent n PMU deployment. Further dcuon on the choce of nput gnal can be found n [2]. B. Multple Compenator for Tme-Varyng Data Latency In degnng an adaptve controller for tme-varyng latency, we elect delay level tartng at 50 m, at ncrement of 50 m, up to 250 m. We alo nclude a delay level of m to how the nearly-deal cae. We mplement the overall controller from (1), where the tme contant of the hgh-pa wahout flter T w = and the tme contant of the dervatve flter T δ = 0.04, the ame a n [7]. The adaptve parameter K(T d ), T 1 (T d ), and T 2 (T d ) are gven n Table II. C. Average Dwell Tme Baed on Theorem 1, we formulate a lnear matrx nequalty (LMI) feablty problem and calculate the ADT for the degned controller n Table II. The LMI feablty problem become prohbtvely large a the number of wtched controller and ytem tate ncreae. To mprove the optmzaton convergence and obtan a oluton n a reaonable amount of tme, we reduce the tate-pace model

5 of the power ytem before applyng the TCSC controller. There are two part to the proce. Frt, we remove the ytem mode, whch are located at the orgn of the root-locu. Thee pole correpond to the fact that the machne angle have multple table oluton wth perodcty 2π. We perform a mple tranformaton on the angle to elmnate thee pole, whch ha the effect of lghtly hftng the frequency of the nterarea mode. The econd tep to reduce the order of the power ytem dynamc model. After removng the 2 ytem mode, we have a plant wth 35 tate. We then perform a balanced reducton to reduce the ytem to 3rd order. Becaue the nterarea mode untable and the TCSC doe not gnfcantly affect the other tate, the nterarea mode kept n the 3rd order ytem and t behavor cloely reemble the orgnal ytem. Fgure 4 compare the root-locu plot of the reduced model and orgnal ytem. TABLE III AVERAGE DWELL TIME RESULTS Controller # Delay (m) Mn. ADT () (4,5) (150,200) (3,4,5) (0,150,200) (2,3,4,5) (50,0,150,200) (1,2,3,4,5) (,50,0,150,200) mulate data latency wth varable arrval tme baed on a Poon tochatc proce, wth a mnmum latency of 90 m. The parameter of the probablty dtrbuton functon are choen uch that the data latency n the range d(t) [90, 1] m n 99% of cae. For a 1- nterval durng the dturbance, latency varablty ncreaed to repreent a bref congeton. We et the mnmum reet tme to T r =, uch that t gnfcantly larger than the mnmum average dwell tme. Imagnary Ax (econd -1 ) Orgnal Sytem Reduced Model Root Locu Angle Dff. (degree) Angle Dfference (Input) Sgnal Actual Value Delayed Sgnal Controller Input Tme () 0 Fg Real Ax (econd -1 ) Root-Locu Plot for Orgnal and Reduced Sytem Let (A p, B p, C p, 0) and (A c, Bc, Cc, Dc) be the tate pace realzaton for the reduced plant and the controller n mode, repectvely. Aume that A p R n pl n pl and A c R ncn ncn. The matrce n (2) whch correpond to the cloed-loop matrce are gven by [ ] A A = p B p Cc 0 ncn n pl A c [ ] Bp D B = cc p 0 npl n cn BcC. p 0 ncn n cn Ung Theorem 1, we can how that for the group of controller n Table III, the decrbed wtchng mechanm tablze the cloed-loop ytem. Note that a addtonal controller are combned, the mnmum average dwell tme ncreae to guarantee tablty. D. Adaptve Controller Performance Th adaptve algorthm appled to the 2-area ytem for the ame hort-crcut dturbance. The adaptve control performance hown n Fgure 5. In th 15-econd mulaton, a data buffer functon created n PST. We Fg. 5. Dampng Performance of the Adaptve Controller Fgure 5 how the angle dfference gnal θ a meaured ntantaneouly at the bue, the tme that the gnal θ a actually arrve at the controller, and the θ a waveform that ued a the dampng controller nput. A cloe-up hown n Fgure 6. Note the ntal delay for the phaor data θ a to be pcked up by the data buffer. In Fgure 7, we how the phae compenaton electon by the adaptve algorthm. The algorthm tart wth T d = 0 m compenator and wtche to the T d = 200 m Angle Dff. (degree) Fg. 6. Actual Value Delayed Sgnal Controller Input Tme () Performance wth Data Latency and Controller Delay

6 Delay (m) Angle Dff. (degree) X: Y: 200 Sgnal delay Controller delay Tme () Fg. 8. Fg. 7. Adaptve Compenator Selecton Actual Value Delayed Sgnal Controller Input Tme () Cloe-Up of Input Sgnal durng Compenator Swtchng compenator before the fault and t doe not affect the performance. At t = 1.588, a data pont arrve wth greater than 150 m latency, o the algorthm wtche to the T d = 200 m compenator. We ee that 150 m later (at t = ), the controller hold the lat data pont n nput queue for 50 m whle t wtche to the T d = 200 m compenator (Fgure 8). After wtchng n the 200 m latency compenator, the controller performance tll excellent. To damp the ocllaton, the controller aturate a t drve the effectve reactance of the TCSC branch connecton Bue 3 and 13 to a mnmum, a hown n Fgure 9. X eq (p.u.) VI. CONCLUSION In th paper, we analyzed the tablty of an adaptve control cheme for a power ytem nterarea dampng controller ung remote PMU data wth tme-varyng latency. The adaptve control cont of a controller wtchng algorthm baed on the latency of PMU data, and a phae compenaton degn of the controller for a gven et of latency. Latency requre addng phae lead compenaton, and we ue a bank of phae-lead controller governed by a latency-montorng, adaptve algorthm to wtch among them. We developed a uffcent condton to guarantee that th wtched control ytem table a long a an average dwell tme contant met. Fnally, we llutrated the control degn and demontrated the performance ung a 2-area power ytem. Future work nclude the development and analy of adaptve control algorthm for multple actuator conderng packet lo, late data arrval, and cooperatve control. Another topc for future reearch to degn a et of controller that are robut both to the nput gnal delay and to the varaton n the operatng condton of the ytem. REFERENCES [1] S. Ghocel, J. Chow, G. Stefopoulo, B. Fardaneh, D. Maragal, B. Blanchard, M. Razanouky, and D. Bertagnoll, Phaormeaurement-baed tate etmaton for ynchrophaor data qualty mprovement and power tranfer nterface montorng, Power Sytem, IEEE Tranacton on, vol. 29, no. 2, pp , March [2] J. H. Chow and S. G. Ghocel, An adaptve wde-area power ytem dampng controller ung ynchrophaor data, n Control and Optmzaton Method for Electrc Smart Grd, er. Power Electronc and Power Sytem, A. Chakrabortty and M. D. Ilc, Ed. Sprnger New York, 20, vol. 3, pp [3] B. Demrel, C. Brat, and M. Johanon, Determntc and tochatc approache to upervory control degn for networked ytem wth tme-varyng communcaton delay, CoRR, vol. ab/ , [4] J. Kuroe and K. Ro, Computer Networkng: A Top-Down Approach, 5th ed. New York: Addon-Weley, 20. [5] C. Cyr and I. Kamwa, WACS degn at Hydro-Quebec, n Proc. of IEEE PES General Meetng, Mnneapol, MN, July 20. [6] J.W. Stahlhut, T.J. Browne, G.T. Heydt, and V. Vttal, Latency vewed a a tochatc proce and t mpact on wde area power ytem control gnal, IEEE Tran. Power Syt., vol. 23, pp , Feb [7] E.V. Laren, J.J. Sanchez-Gaca, and J.H. Chow, Concept for degn of FACTS controller to damp power wng, IEEE Tran. Power Syt., vol., pp , May [8] D. Lberzon, Swtchng n Sytem and Control. Boton: Brkhäuer, [9] J. Hepanha and A. More, Stablty of wtched ytem wth average dwell-tme, n Proc. 38th IEEE Conf. on Decon and Control, 1999, pp [] M. Klen, G.J. Roger, and P. Kundur, A fundamental tudy of nterarea ocllaton n power ytem, IEEE Tran. Power Syt., vol. 6, pp , Aug Tme () Fg. 9. TCSC Control Acton