Impact of Realistic Bus Frequency Measurements on Wide-Area Power System Stabilizers

Transcription

1 Impact of Realitic Bu Frequency Meaurement on Wide-Area Power Sytem Stabilizer Georgio Tzouna, Muyang Liu, Mohammed Ahan Adib Murad, Federico Milano School of Electrical and Electronic Engineering, Univerity College Dublin, Ireland {georgio.tzouna, muyang.liu, Abtract The paper tudie the dynamic repone of a Wide- Area Power Sytem Stabilizer (WAPSS) for damping inter-area ocillation. The WAPSS utilize remote bu frequency meaurement obtained with frequency etimation. The geometric approach i adopted for the election of the mot effective ignal and input placement. Then, the uitability of Padé approximant to capture the impact of communication delay on ignal obervability i tackled. The reult of the mall-ignal tability analyi are dicued through non-linear time domain imulation with incluion of time-varying tochatic delay. Index Term Inter-area ocillation, wide-area ignal, frequency etimation, obervability/controllability, time delay, tranient tability analyi. A. Motivation I. INTRODUCTION Feedback ignal and input location election are critical deciion during a control deign and often not traightforward to be made, epecially when communication network mediate between meaurement and control. A relevant example of thi kind are Power Sytem Stabilizer (PSS) coupled with Wide-Area Meaurement Sytem (WAMS). The utilization of available ignal from the tranmiion ytem a well a the tudy of communication delay impact on control ignal election have motivated thi work. B. Literature Review Low frequency ocillation are inherent in interconnected power ytem and are caued by poorly damped mode. Power ytem how both local mode with a ingle or a group of generator ocillating within an area (typically -2 Hz) and inter-area mode with a group of generator ocillating with repect to a group of a different area (typically 0.- Hz) []. The tandard olution for uppreing the local ocillation mode i the intallation of PSS with local ignal []. On the other hand, effective damping of inter-area mode uually require the incluion of remote ignal [2]. Thi i becaue inter-area mode are often obervable from a local ignal but effectively controllable only from another location [3]. With the development of WAMS and the broad deployment of Phaor Meaurement Unit (PMU), remote ignal can be Thi work i upported by the Science Foundation Ireland, by funding Georgio Tzouna, Muyang Liu, Mohammed Ahan Adib Murad and Federico Milano, under Invetigator Programme Grant No. SFI/5/IA/3074. obtained and tranmitted to a Wide-Area Damping Controller (WADC) through a communication network [4]. The evaluation of wide-area ignal play an important role in the deign of a WADC. The comparion of two common modal obervability/controllability method, namely the reidue and the geometric approach, ha deduced that the ignal recommended by the geometric approach can alway obtain the bet robutne and performance [5]. Thu, the geometric approach i adopted in the preent work. Concerning frequency-baed WADC, the reult in [2], [6], [7] how that inter-area ocillation are better obervable through the differential rotor peed of two machine from different area. However, PMU are intalled on ytem bue and, typically, only bu frequency meaurement are available to a WADC. In thi paper, we conider both ideal and Phae-Locked Loop (PLL) etimated bu frequency meaurement. The involvement of the communication network in remote ignal tranmiion introduce time delay with incluion of noie, data diordering and dropout [8], [9]. Depite the fact that eigen-baed method have been uccefully applied to approximate the pectrum of power ytem modeled with Delay Differential Algebraic Equation (DDAE) [0], [], global modal analyi of delayed power ytem i till an open reearch topic. An approximation of the effect of delay on the obervability of wide-area ignal can be provided uing Padé polynomial [2]. We evaluate the accuracy of uch an approach with a detailed delay model that i able to emulate the phyical behavior of WAMS [3]. C. Contribution The contribution of the paper are twofold. A comparion of the dynamic behavior of WAPSS uing either ideal bu frequency ignal or PLL-baed etimation. Ideal ignal are obtained with the frequency divider propoed in [4]. Etimation are obtained with the commonly ued Synchronou Reference Frame PLL (SRF-PLL) [5]. A dicuion on the uitability of Padé approximant to capture the effect of delay on wide-area ignal obervability. Concluion are drawn baed on non-linear time domain imulation with incluion of realitic WAMS delay [3].

2 D. Organization The remainder of the paper i organized a follow. Section II recall the tate pace model of power ytem and preent the applied control cheme. Section III outline the delay model utilized in thi work. Section IV dicue the cae tudy by carrying modal and tranient tability analye on the two-area tet ytem. Concluion are drawn in Section V. II. POWER SYSTEM MODEL AND CONTROL SCHEME A. State Space Repreentation Power ytem are conventionally decribed through a et of Differential Algebraic Equation (DAE): ẋ = f(x,y,u) 0 = g(x,y,u), where f (f : R n+m+p R n ), g (g : R n+m+p R m ) are the differential and algebraic equation; x, x R n, and y, y R m, are the tate and algebraic variable, repectively; and u, u R p, are the controlled input. Differentiating () around an equilibrium point (x 0,y 0,u 0 ) yield: ẋ = f x x+f y y +f u u (2) 0 = g x x+g y y +g u u, where f x, f y, f u, g x, g y and g u are the ytem Jacobian matrice; x = x x 0, y = y y 0 and u = u u 0. The Multiple-Input Multiple-Output (MIMO) tate pace repreentation of the ytem i: ẋ = A x+b u w = C x+d u, where A = f x f y g y g x, B = f u f y g y g u and w are output meaurement. B. Modal Analyi The mode of the open-loop power ytem ẋ = A x can be obtained by olving the characteritic equation det(a λi) = 0. The correponding right and left eigenvector aociated with a mode λ k are determined by olving Aφ k = λ k φ k and ψ k A = λ k ψ k, repectively. That aid, the geometric obervability/controlability are defined a follow [6]. The geometric obervability gm oµ (k) of the mode λ k from output w µ i: () (3) gm oµ (k) = co(θ (c T µ,φ k )) = c µφ k φ k c µ, (4) where c µ i the µ th row of the output matrix C; θ i the acute angle between c µ and the right eigenvector φ k ; and are the modulu and Euclidian norm, repectively. The geometric controllability gm cν (k) of the mode λ k from input u ν i: gm cν (k) = co(θ 2 (ψ k,b ν )) = b νψ k ψ k b ν, (5) where b ν i the ν th column of the input matrix B; θ 2 i the acute angle between b ν and the left eigenvector ψ k. Baed on thee meaure, a comparion among different output and input can be carried out, o that the one that provide the maximum joint geometric obervability/controllability meaure are elected. The joint obervability/controllability meaure i defined by: gm cok (µ,ν) = gm cν (k) gm oµ (k). (6) C. Wide-Area Power Sytem Stabilizer (WAPSS) Model The WAPSS in thi tudy i a decentralized controller intalled at the j th ynchronou machine. The wide-area tabilizing ignal v i i a differential frequency of the form: v i = ω Bi ω Gj, (7) where ω Gj i the local rotor peed of the j th ynchronou machine; ω Bi i the frequency at bu i. v i K w T w T w+ T + T 2+ Fig. : WAPSS block diagram. T 3+ T 4+ v min v max The tructure of the WAPSS (ee Fig. ) can be decribed by the following et of DAE: v = (K w v i +v )/T w v 2 = (( T T 2 )(K w v i +v ) v 2 )/T 2 v 3 = (( T 3 T 4 )(v 2 + T T 2 (K w v i +v )) v 3 )/T 4 0 = v 3 + T 3 T 4 (v 2 + T T 2 (K w v i +v )) v o, where K w i the WAPSS gain, T w i the wahout time contant, T, T 2, T 3, T 4 are the 4 tabilizing block time contant, v, v 2, v 3 are the WAPSS tate variable and v o the output ignal. The output ignal v o i an additional input to the local Automatic Voltage Regulator (AVR) initial reference (v0 ref ), o that the controller damp electromechanical ocillation through excitation control. The reulting voltage reference (v ref ) i: D. Bu Frequency Etimation v o (8) v ref = v ref 0 +v o. (9) In order to obtain the wide-area ignal in (7), an etimation of the frequency ω Bi i required. To thi aim, we utilize the commonly ued SRF-PLL (ee Fig. 2) model. In Fig. 2, the contant delay implie that the bu phae angle (θ Bi ) meaurement i not intantaneou. The error between the meaured and etimated phae angle i fed to a Proportional-Integral (PI) controller, which output the etimated bu frequency deviation ( ω Bi ). The frequency at bu i i obtained a: ω Bi = ω 0 + ω Bi, (0)

3 θ + Bi ω Bi e τ K p + Ki ω 0 Fig. 2: SRF-PLL block diagram. where ω 0 i the fundamental frequency of the ytem. PLL-baed frequencie are accurate but include a compromie between noie and peed of repone. They are alo prone to problem during the numerical integration [5]. In thi work, we compare the reult obtained with the SRF-PLL with the one produced when ideal bu frequency etimation are ued. Ideal bu frequencie are etimated uing the frequency divider formula (FDF) propoed in [4]. FDF i a general expreion which allow the etimation of the frequency at the bue of a tranmiion ytem. In per unit, the FDF i formulated a: ω B = (B BB +B B0 ) B BG (ω G ), () where ω B are the etimated bu frequencie; ω G are the ynchronou machine rotor peed; B BB, B B0, B BG are ytem uceptance matrice. The accuracy, the numerical robutne and the computational efficiency of the FDF have been dicued in [4], [7] and [8]. A. DDAE and WAMS delay III. TIME DELAY MODELS Introducing time delay in () change the et of DAE into a et of DDAE. For the purpoe of thi paper, the index- Heenberg i an adequate formulation to model power ytem with delay [9]: + ˆx = ˆf(ˆx,ŷ,ˆx d,ŷ d ) 0 = ĝ(ˆx,ŷ,ˆx d ), + ω Bi θbi (2) where ˆx d and ŷ d are the delayed tate and algebraic variable, repectively. In thi tudy, the delay are included in the remote bu frequency meaurement. The reult in [2] how that timeynchronizing peed-baed local and remote feedback ignal ha a detabilizing impact, regardle of the control method. Hence, we alo conider non-time-ynchronized ignal. The delayed bu frequency ignal ω Bid i: ω Bid = ω Bi (t τ(t)), (3) where τ(t) i the WAMS delay, which i of the form [3]: where τ(t) = τ 0 +τ p (t)+τ (t), (4) τ 0 i a contant component that expree the proceing time of the meaurement unit plu the unavoidable latency impoed by the communication medium; τ p (t) i a periodic component which implie that data packet are delivered in dicrete time intant. It i τ p (t) = t t k, where t k i the time when the lat uccefull data packet delivery occured; the delivery period of a packet without dropout i T ; τ (t) i a tochatic component which account for uncertaintie and noie. It i τ (t) = Gamma(α,β,t) p, which aume a Gamma ditribution with cale factor α, hape factor β [20]; p i the probability of a dropout. The mean value of τ(t) i [3]: τ = τ 0 + T 2( p) + α p β (5) The profile of WAMS delay model i illutrated in Fig. 3. τ(t) [] Time [] Fig. 3: WAMS delay (τ 0 = 0.05, T = 0.04, p = 0.2, α = 0.02, β = 2). Modal obervability/controllability analyi of DDAE i not an eay tak, ince differentation of (2) lead to a trancendental characteritic equation, with infinitely many eigenvalue and aociated eigenvector [9]. Different technique, e.g., Chebyhev dicretization, Padé approximant [0], have been employed to overcome thi problem. In thi paper, we make ue of the Padé approximant in order to approximate the effect of communication delay on wide-area ignal obervability. B. Padé Approximant Padé approximant are baed on the well-known Taylor erie expanion of e τ in the frequency domain: e τ = τ! + (τ)2 (τ) ! 3! b 0 +b τ+...+b q (τ) q (6) a 0 +a τ+...+a p (τ) p, where a 0,a,...,a p and b 0,b,...,b q are defined o that the firt p + q coefficient are the ame a thoe of the Taylor expanion. The advantage of thi approach i that it allow to repreent a contant delay a a et of linear differential equation. Therefore, tracking the change of the obervabilitie with repect to delay i relatively imple to implement through modal analyi. On the other hand, thi technique i known to be of limited accuracy. Furthermore, it cannot

4 approximate properly crucial information of a WAMS delay, i.e., tochaticity and periodicity. IV. CASE STUDY The ytem conidered in thi tudy i hown in Fig. 4. It conit of two identical area connected through a relatively weak tie; eleven bue B,B2,...,B and four ynchronou machine G, G2, G3, G4 connected at the medium voltage level of 20 kv; the nominal value of the high voltage tranmiion line i 230 kv. Each machine i equipped with an AVR of type IEEE DC- and a turbine governor. All reult dicued in thi ection are obtained with Dome, a Python-baed oftware tool for power ytem analyi [2]. G km 0km 25km 0km 0km 25km 3 G3 We conider wide-area ignal a decribed in Section II. Two cenario are examined: ideal bu frequencie (ω Bi ) etimated uing the FDF; and bu frequencie obtained uing SRF-PLL ( ω Bi ). Table IV and V ummarize the bet ignal and their geometric meaure. The SRF-PLL ha a negative impact on the obervability of the mode by the wide-area ignal. The reult for both cenario ugget that differential frequencie B5 G3 and B G3 are the mot uitable for damping the inter-area ocillation. Hence, in the remainder of the paper we aumed that the WAPSS i intalled at G3. A expected, wide-area ignal are more effective than local rotor peed (Table III). Since B i the medium voltage ide of the tep-up tranformer, it typically belong to the generation ide, and thu, a meaurement at thi point may not be available. For thi reaon, we proceed with the frequency B5 G3. TABLE IV: Candidate with bet gm co (ideal ω Bi). G2 2 Area Area 2 4 G4 Fig. 4: Two-area four-machine tet ytem. A. Wide-Area Signal and Input Location Selection Table I how the critical mode of the two-area ytem. The mot poorly damped mode i λ,2 = 0.095±j3.5809, which i an inter-area mode with the machine of Area ocillating with repect to the one of Area 2. The controllability of λ,2 from the four AVR i preented in Table II. TABLE I: Two-area ytem critical mode Eigenvalue Frequency (Hz) Damping ratio (%) λ,2 = ±j λ 3,4 = 0.544±j λ 5,6 = ±j TABLE II: Controllability of λ,2 from j-th AVR j gm c The geometric meaure of λ,2 for local rotor peed ignal are hown in Table III. If only local meaurement are conidered, the inter-area mode i better obervable/controllable from G3. TABLE III: Obervability of λ,2 from ω Gj and joint meaure Signal gm o AVR gm c gm co ω G G ω G G ω G G ω G G Angular frequency ωg3 [pu] Signal gm o AVR gm c gm co ω B5 ω G G ω B ω G G ω B6 ω G G ω B2 ω G G ω B7 ω G G ω B5 ω G G TABLE V: Candidate with bet gm co ( ω Bi from SRF-PLL) Signal gm o AVR gm c gm co ω B ω G G ω B5 ω G G ω B2 ω G G ω B6 ω G G ω B ω G G ω B7 ω G G ω B5 ω G3 ω B5 ω G3 ω G Time [] Fig. 5: Rotor peed of G3 for local and wide-area control ignal. A time domain imulation i carried out with the WAPSS intalled at G3 and impoing the outage of the load connected at B7 at t =.0. The implicit trapezoidal formula i ued for the numerical integration of the ytem. The olution at

5 each time tep i obtained by employing the Newton-Raphon iterative method. Figure 5 how the angular frequency of G3 for the local rotor peed ω G3 and the wide-area ignal ω B5 ω G3 and ω B5 ω G3. The wide-area ignal confirm to be the mot effective to damp electromechanical ocillation. The obervation that uing the SRF-PLL to meaure the bu frequency degrade the damping effect i alo confirmed. B. Effect of Time Delay In thi ubection, the impact of time delay on the damping effect of wide-area ignal i examined. Firt, we attempt to approximate thi effect uing Padé polynomial. The joint obervability/controllability of wide-area ignal with repect to the time delay i hown in Fig. 6 and 7 for ideal and SRF- PLL etimated bu frequencie. The time delay degrade the obervability of wide-area ignal in both cae. In the region that i typical for network-induced delay, i.e., m, the wide-area ignal are till the bet option to damping interarea ocillation. For larger delay, however, i.e., τ > 400 m, reult indicate that uing the local rotor peed i more effective. Obervability/controllability of λ, ω G3 ω B2d ω G3 ω B6d ω G3 ω Bd ω G3 ω B5d ω G Time Delay [] Fig. 6: Effect of approximated contant delay on the obervability of wide-area ignal (ideal ω Bi). Obervability/controllability of λ, ω G3 ω B2d ω G3 ω B6d ω G3 ω Bd ω G3 ω B5d ω G Time Delay [] Fig. 7: Effect of approximated contant delay on the obervability of wide-area ignal ( ω Bi from SRF-PLL). Since Padé polynomial provide an approximation of the delay effect, the reult obtained in Fig. 6 and 7 are further evaluated by carrying time domain imulation with incluion of WAMS delay. Two delay, τ (t) and τ 2 (t), with different expected (mean) value are examined. The parameter of the examined delay are given in Table VI. Both delay include a contant component of 50 m, a periodic component with 40 m delivery period and a 20% probability dropout rate. The two delay have different hape and cale factor of the Gamma ditribution. Finally, the expected value are τ = 25 m and τ 2 = 450 m, repectively. TABLE VI: WAMS delay parameter Delay τ i (t) τ 0 [] T [] p α β τ i [] τ (t) τ 2 (t) Figure 8 and 9 how the angular frequency of G3 for ideal and SRF-PLL-baed bu frequencie. The delay degrade the WAPSS damping ability. For the mall delay (τ (t)), the damping ability of the wide-area ignal i better than uing the local rotor peed. Thi i in agreement with the reult obtained with Padé approximant. However, Fig. 8 and 9 ugget that the wide-area ignal i clearly preferable even for large delay (τ 2 (t)), independently whether the ideal or SRF-PLL etimated bu frequency i ued. Thi contradict Padé approximant reult. Overall, we conclude that the tudy of a ytem uing Padé approximant i reliable only if mall delay are conidered. Neverthele, in order to ae whether the approximation for a conidered delay i acceptable or not, i.e., whether a delay i mall enough to trut Padé approximant, it i neceary to evaluate the reult through detailed time domain imulation. Angular frequency ωg3 [pu] ω B5( τ2=450m) ω G3 ω B5( τ=25m) ω G3 ω B5 ω G3 ω G Time [] Fig. 8: Rotor peed of G3 (ideal ω Bi). V. CONCLUSIONS The paper dicue a decentralized WAPSS with remote bu frequency meaurement. Ideal bu frequency ignal are compared with realitic meaurement ignal from PLL that include meaurement noie and delay. Modal and tranient tability reult baed on the Kundur two-area ytem how that wide-area ignal are more effective that local ignal to damp inter-area ocillation. Reult alo indicate that the PLL have a non-negligible and negative impact on the obervability

6 Angular frequency ωg3 [pu] ω B5( τ2=450m) ω G3 ω B5( τ=25m) ω G3 ω B5 ω G3 ω G Time [] Fig. 9: Rotor peed of G3 ( ω Bi etimated with a SRF-PLL). of the inter-area mode and, hence, on the overall damping ability of the WAPSS. The impact of time delay on the obervability of widearea ignal uing Padé approximant i alo invetigated. Reult how that thi approach provide correct reult only for mall contant delay but it i not reliable for modal analyi of power ytem with incluion of WAMS delay. Further reearch will focu on exploiting eigen-baed method, uch a [0], which allow carrying a global modal analyi of DDAE. REFERENCES [] P. Kundur, Power ytem tability and control. New York: Mc-Grall Hill, 994. [2] M. E. Aboul-Ela, A. A. Sallam, J. D. McCalley, and A. A. Fouad, Damping controller deign for power ytem ocillation uing global ignal, IEEE Tran. on Power Sytem, vol., no. 2, pp , May 996. [3] J. H. Chow, J. J. Sanchez-Gaca, H. Ren, and S. Wang, Power ytem damping controller deign-uing multiple input ignal, IEEE Control Sytem Magazine, vol. 20, no. 4, pp , Aug [4] H. Wu, K. S. Takali, and G. T. Heydt, Evaluation of time delay effect to wide-area power ytem tabilizer deign, IEEE Tran. on Power Sytem, vol. 9, no. 4, pp , Nov [5] A. Heniche and I. Kamwa, Aement of two method to elect widearea ignal for power ytem damping control, IEEE Tran. on Power Sytem, vol. 23, no. 2, pp , May [] C. Li, G. Li, C. Wang, and Z. Du, Eigenvalue enitivity and eigenvalue tracing of power ytem with incluion of time delay, IEEE Tran. on Power Sytem, vol. 33, no. 4, pp , July 208. [6] W. Yao, L. Jiang, Q. H. Wu, J. Y. Wen, and S. J. Cheng, Delaydependent tability analyi of the power ytem with a wide-area damping controller embedded, IEEE Tran. on Power Sytem, vol. 26, no., pp , Feb. 20. [7] Y. Wang, P. Yemula, and A. Boe, Decentralized communication and control ytem for power ytem operation, IEEE Tran. on Smart Grid, vol. 6, no. 2, pp , Mar [8] S. Wang, X. Meng, and T. Chen, Wide-area control of power ytem through delayed network communication, IEEE Tran. on Control Sytem Technology, vol. 20, no. 2, pp , Mar [9] B. P. Padhy, S. C. Srivatana, and N. Verma, A wide-area damping controller conidering network input and output delay and packet drop, IEEE Tran. on Power Sytem, vol. 32, no., pp , Jan [0] F. Milano, Small-ignal tability analyi of large power ytem with incluion of multiple delay, IEEE Tran. on Power Sytem, vol. 3, no. 4, pp , July 206. [2] S. Ghoh, K. A. Folly, and A. Patel, Synchronized veru nonynchronized feedback for peed-baed wide-area PSS: Effect of timedelay, IEEE Tran. on Smart Grid, vol. 9, no. 5, pp , Sep [3] M. Liu, I. Daio, G. Tzouna, and F. Milano, Stability analyi of power ytem with incluion of realitic-modeling of WAMS delay, IEEE Tran. on Power Sytem, pp., 208. [4] F. Milano and Á. Ortega, Frequency divider, IEEE Tran. on Power Sytem, vol. 32, no. 2, pp , Mar [5] Á. Ortega and F. Milano, Comparion of different PLL implementation for frequency etimation and control, in Proc. of the 8th International Conference on Harmonic and Quality of Power (ICHQP), May 208, pp. 6. [6] H. Hamdan and A. Hamdan, On the coupling meaure between mode and tate variable and ubynchronou reonance, Electric Power Sytem Reearch, vol. 3, no. 3, pp. 65 7, 987. [7] Á. Ortega and F. Milano, Comparion of bu frequency etimator for power ytem tranient tability analyi, in Proc. of the IEEE International Conference on Power Sytem Technology (POWERCON), Sept 206, pp. 6. [8], Impact of frequency etimation for VSC-baed device with primary frequency control, in Proc. of the IEEE PES Innovative Smart Grid Technologie Conference Europe (ISGT-Europe), Sept 207, pp. 6. [9] F. Milano and M. Anghel, Impact of time delay on power ytem tability, IEEE Tran. on Circuit and Sytem - I: Regular Paper, vol. 59, no. 4, pp , Apr [20] O. Solomon and E. Fridman, Sytem with gamma-ditributed delay: A lyapunov-baed analyi, in Proc. of the 52nd IEEE Conference on Deciion and Control, Dec 203, pp [2] F. Milano, A Python-baed oftware tool for power ytem analyi, in Proc. of the IEEE PES General Meeting, Vancouver, BC, Jul. 203.