PMU-Base System Ientification for a Moifie Classic Generator Moel Yasser Wehbe, Lingling Fan, Senior Member, IEEE Abstract The paper proposes to use PMU measurements (voltage phasor, real an reactive powers) for system ientification. A secon-orer synchronous generator moel with real an reactive power as input an voltage phasor as output will be evelope in this paper base on the two-axis synchronous generator ynamic moel. Parameters of the moel will be ientifie base on the PMU measurements using least suare estimation techniue. Two case stuies are presente: a synchronous generator estimation an a power subsystem estimation. In both cases, the system will be represente by the propose moel with parameters ientifie from PMU measurements. Time-omain simulation results will valiate the accuracy of the propose moel an its ientification. The contribution of this paper is twofol: i) A moel suitable for PMU measurement base ientification is evelope; ii) better accuracy of system parameter estimation can be achieve using the propose moel compare to the classical moel. Inex Terms Least suare estimation, parameter estimation, synchronous machine, system ientification. I. INTRODUCTION Phasor Measurements Units (PMU) provie time stampe phasor ata such as voltage magnitue (V ) an angle (θ), active power (P e ) an reactive power (Q e ), at a reporting rate of -6 Hz. Due to such characteristics, PMU is regare as in important tool in Wie Area Measurement System (WAMS) [1. The objective of this research is to use PMU ata for system ientification, i.e., given time series ata of input an output of a system an its ynamic moel structure, the entire set of the parameters of the moel will be ientifie using least suare estimation. Due to the low sampling rate of PMU ata, PMU measurements o not reflect electromagnetic ynamics aeuately. However, the sampling rate is high enough to reflect the lowfreuency electromechanical ynamics in power gris. Compare to some of the approaches in the literature, where all kins of measurements can be obtaine, PMU measurements are limite to voltage phasors an powers. Therefore, the ynamic moel suitable for PMU ata application will focus on electromechanical ynamics. Meanwhile, the parameters ientifie will be a subset of generator parameters. Two estimation approaches are ominant: Kalman Filter (KF) base estimation an least suares base estimation. KF approach is a Bayesian recursive metho aiming at reucing a covariance matrix whereas the least suares approach uses a non Bayesian methos consiering the complete ata set in the time winow. KF approaches have been propose to Y. Wehbe is with Ventyx, Texas. L. Fan is with the Department of Electrical Engineering, University of South Floria, Tampa, FL 336, email: linglingfan@usf.eu. estimate synchronous machine electromechanical parameters using PMU measurements. In [, extene Kalman filter (EKF) base metho is propose to calibrate the parameters of a classical generator. In [3, EKF is applie to calibrate the multi-machine power system parameters. Our research in [4 has also examine EKF application for PMU ata base estimation an the Unscente Kalman Filter in [5. Least suares base system ientification has been seen plenty in the literature in variety of applications. The objective of system ientification is to compare the output of a system with the projecte output of the moel given the same input. The parameters of the moel will be optimize to reach a minimum error suares over the experimental perio. System ientification has been use in power electronic research in ientifying a power converter moel [6, [7, to moel large signal power electronics systems [8, an to estimate DC link moel parameters in VSC-HVDC system [9. It has also been use in generator parameter ientification [1, Prony analysis [11, [1, ARX-base generator moel ientification [13, [14 an in fining the state space system for multi-input multioutput moels of power systems [15. In PMU ata base estimation, least suare estimation base system ientification approach has not been examine. This approach will be investigate in this paper. A key challenge in least suare estimation base system ientification is the ifficulty to estimate some parameters since they are insensitive to the output ata. This makes the estimation of some parameters not reliable. Burth et al. [16 points out to the ifficulty in estimating synchronous machine parameters base on measurements such as terminal voltage an current. This is especially a problem in PMU measurements since these measurements are the only measurements can be obtaine an use. Burth et al. [16 suggests to apply parameters sub-set selection to fin a best set of parameters that can be estimate with reasonable amount of precision. In [16, a Hessian matrix of the objective function is applie to fin the parameters subset. Another approach for sub-set selection is the stuy of sensitivity matrix as shown by Cintron-Arias et al. [17. The two approaches are correlate an both are base on Jacobian calculation. In this research, sensitivity matrix will be use to emonstrate the choice of moel orer. The rest of this paper is organize as follows: Section II will erive a linear moel suitable for PMU ata base system ientification. Section III presents case stuies applying Matlab System Ientification Toolbox. A single machine ientification an a an area ientification are presente. Section IV presents the conclusion of this paper.
II. PROPOSED MODEL SUITABLE FOR PMU DATA BASED IDENTIFICATION To facilitate system ientification, PMU measurements will be separate into two sets: the input an the output. The active an reactive powers will be treate as the input an the voltage magnitue an phase will be consiere as the output. The purpose of this section is to buil a linearize ΔP e ΔQ e x Fig. 1. Propose state space moel Ax Bu y Cx Du ΔV Δθ state-space moel as shown in Fig. 1. In orer to ientify which set of parameters can be better represente in the PMU measurements, sensitivity analysis will be conucte. Furthermore, a reuce-orer moel will be evelope. A. Subset selection Synchronous machine two-axis moel with no governor nor exciter controls an ignoring the sub-transient ynamics [18 can be escribe by: δ t = ω ω (1) H ω ω t = P m P e () E = E (x x t )I + E f (3) T o T o E = E (x x t )I (4) The internal voltage, current an terminal voltage relationship is expresse as follows. [E +(x x )I +E = jx (I +ji )+V e j(θ δ +π/) (5) Phasor iagram in also explains the above relationship. axis axis E E jx I Vs δ γ δ I E Fig.. Phasor iagram for a classical moel from [19. [17 proposes to use sensitivity matrix to select a subset of parameters to be estimate. Sensitivity matrix can be use stuy the impact of the various parameters on the output of a system, in other wors, it tries to fin the most influential an the least influential parameter on the output. In this research, for a two-axis generator moel, the machine parameter set is efine as M= { H, x, x, x, T o, T } o where H is the inertia, x -axis transient reactance, x an x are the -axis reactances, T o an T o are the -axis opencircuit time constants. The sensitivity matrix of the system ()-(1) is the Jacobian matrix χ of the output Y wrt. the parameter set M. χ ij = Y i M j (6) where i enotes i-th output an j enotes j-th parameter. Y i is the i th measurement an has N samples. Accoringly, χ for the system will be N 6 where N is the total number of samples. The importance of the sensitivity matrix in least suares base estimations (like system ientification) comes from the objective of the least suares estimation of minimizing the output error by manipulating M aroun a value M [17 to fin its estimate ˆM: ˆM = arg min M N (Y (i) Ŷ (i M)) (7) i=1 The estimate parameter set has an expression as follows: ˆM = M + (χ T χ) 1 χ T ζ (8) where ζ is the 1 N error (noise) matrix associate with the output euation. A goo estimation of M will reuce the impact of ζ by having (χ T χ) far from singularity. Shoul (χ T χ) be close to singular, then (χ T χ) 1 will amplify the impact of ζ an istorts the estimation of M. Singular value ecomposition (SVD) of χ provies an insightful relation between each parameter singular value an the possibility of the least suares estimation to fin a goo value of the parameter. The singular value ecomposition of of χ is χ = ΥSΩ T. where Υ is an N N orthogonal matrix, Ω is a 6 6 orthogonal matrix, an S is the singular value matrix an is χ is N 6 matrix. The first 6 iagonal elements of S are the singular values of χ an the rest of the matrix euals to. The estimation of M aroun M can now be written as [17: ˆM = M + 6 i=1 o i u T i s i ζ (9) where o i, u i are the ith columns of Ω an Υ. s i is the ith iagonal value of S. Euation (9) shows the inverse proportional impact of the singular value s i associate with a parameter M i. As s i ecreases the error ζ introuces more istortion on the estimate parameter an leas to a larger eviation from the correct value. Accoringly, it is better to estimate the parameters with high singular values. In orer to fin the singular values, the matrix χ nees to be calculate.
χ can be calculate analytically [ in case the the associate ifferential euations are simple. In other cases, χ can be calculate numerically. The sensitivity matrix χ was calculate numerically in [1 using finite ifferences by perturbing each parameter M i asie by a value h an then recoring the output of the system. The recore output before (Y ) an after (Y h ) perturbation are use for the Jacobian calculation:χ = Y h Y h. Following the calculation of the χ, SVD is performe in orer to extract S. 3.9. S =.1.8.4.3 (1) It was foun that x is associate with 3.9, H with., x with.1, x with.8, T o with.4, T D o with.3. Therefore, it is obvious that x an H can be estimate reliably, while the two time constants cannot be estimate as reliably as x an H. In the following subsection, a secon-orer generator moel will be erive from the two-axis moel. By reucing the orer of the moel, the two time constants no longer nee to be estimate. B. Moifie Classical Generator Moel For the two-axis synchronous generator moel, the powers can be expresse by terminal voltage an current or terminal voltage an internal voltages. Euations (11) an (1) present the powers: { P e = I V sin(δ θ) + I V cos(δ θ) Q e = I V cos(δ θ) I V sin(δ (11) θ) P e = E V sin(δ θ) E V cos(δ θ) x Q e = E V cos(δ θ)+e V sin(δ θ) V x (1) Note that (1) has an assumption of neglecting the transient saliency. Therefore x = x. The synchronous machine classical moel is obtaine by reucing the two-axis moel when setting T o is set to zero (i.e. ignoring the uick axis ampers ynamics) an T o is extene to therefore E is a constant. Conseuently, such simplification keeps the electromechanical ynamics in (1) an () an completely ecouples from E an E ynamics. In classical moel, the electric circuit, the generator is a voltage source E behin a transient reactance x. The angle between E an V is δ. The ifference between the voltage source angle δ an the rotor angle δ is ignore in classical generator moel. In this paper, this ifference will not be ignore. Accoring to [19 an [18 the ifference γ between the rotor angle δ an the voltage source angle δ is almost constant an is negligible when stuying angle ynamics. Fig. [19 shows γ, δ, an δ. Accoringly: δ = δ + γ δ = δ + γ (13) Careful attention at γ (Fig. ) shows the following: ( ) E γ = tan 1 E E γ = E + E E + E + E (14) When E an E ynamics are ignore γ =. In this paper, the effect of ynamics will be consiere. Base on (13) an (14): δ = δ γ δ = ω + E E + E We nee to fin ( ) an generator moel, we have: E E + E E (15) ( E ). Base on the two-axis ( ) = 1 T o [ E (x x ) I + E f (16) ( ) = 1 T o [ E (x x ) I (17) Since P e an Q e are treate as the input to the moel, the erivatives ( ), ( ) shoul be expresse in terms of states ( E, E ) an powers ( P e an Q e ) not in terms of I an I. I an I can be expresse in terms of E, E, P e, an Q e using (11) an (1) (by removing V sin(δ θ) an V cos(δ θ)). Accoringly, the small signal system 1 (16) an (17) will become: [ [ [ E J E J = E E E J J E + E E [ J E Pe J E Pe E E J E Qe J E Qe [ Pe Q e + [ 1/T o E f (18) Base on the power expression in (1), the terminal voltage phase angle an magnitue can also be expresse in terms of states, an power: [ θ V [ [ [ 1 = δ JθPe J + θqe Pe + J V P J V Qe Q e [ [ JθE J θe E J V E J V E E (19) (J θpe, J θqe, J V Pe, J V Qe, J θe, J θe, J V E, an J V E ) represent the value of the Jacobian of θ an V aroun the euilibrium point. The state space system for the linearize two-axis moel can be formulate by using (18) in (15), linearizing () an aing (18), an (19) in orer to have: δ 1 J δe J δe ω = JE J E E E JE E J E E 1 The appenix shows the etaile term etaile expressions are shown in the appenix δ ω E E +
J δpe J δqe ω /H JE J Pe E Qe JE J Pe E Qe [ [ θ 1 JθE = V [ Pe Q e J θe J V E J V E [ JθPe J θqe J V P [ Pe J V Qe Q e + J δef δ ω 1/T o E E + E f () (1) J δpe, J δqe, an J δef represent the influence of the input an the electric states E an E on the electromechanical state δ. The state space system ()-(1) can be ownsize into a system with two states only δ an ω with an aitional error Err1 an Err, since the euations of these two states inclue all the parameters in M. [ [ [ δ 1 δ ω = ω [ [ [ θ 1 δ = V ω + [ J δpe J δqe ω /H [ Pe + Err1 Q e () [ [ JθPe J + θqe Pe + Err J V P J V Qe Q e (3) In the above state space moel, J δpe an J δqe are etermine by steay-state values of E, E, terminal voltage, rotor angle an time constants. These two terms, JeltaP e an JeltaQ e, will be treate as inepenent parameters an will be estimate. J θpe, J θqe, J V P an J V Qe are all relate to the transient reactance x. Hence these four terms are not inepenent parameters an will be expresse in terms of x. In the case of a classical generator moel, the ynamics of E an E are completely ignore, then J an J δpe δqe are zeros. As a summary, the system ientification will ientify these parameters for the propose moel: H, x, J an J δpe δqe, while ientifying H, x for the classical moel. The above system can be written as: Ẋ = [AX + [BU + Err1 Y = [CX + [DU + Err (4) with the state vector X = [ δ ω T, the observation (or measurement) vector Y = [ θ V T, the input vector U = [ P e Q e T, an the error vectors Err1 an Err. Note that the system matrix [A is not stable. In ifferential euation integration, this can cause numerical error propagation. Therefore, this matrix is moifie with small numerical number ɛ (ɛ < ). [ 1 A = (5) ɛ ɛ III. SIMULATION AND VALIDATION Matlab system ientification toolbox is use for the case stuies. A four-machine two-area system is use for case stuy. Four sets of measurements for P e, Q e, V, an θ are taken following a three phase line to groun fault at a point right outsie the machine or the power subsystem. Small signal uantities P e, Q e, V an θ are erive by removing the steay state value of the measurements. P e an Q e are consiere as the input an V an θ are consiere as output. Both systems representing the propose moel an the classical moel are implemente with MATLAB System Ientification Toolbox grey box. For simplification an generalization purposes, Err1 an Err are treate as zeros.the simulation ata is shown in the appenix. Once the moels are ientifie, the moels of Fig. 3 are implemente in MATLAB Simulink an fe with the input P e, Q e in aition to the estimate parameters. The output of the Simulink system an the states ( δ an ω) are compare to the simulate output an mechanical states δ an ω. Input ΔP e, ΔQ e Classic moel Propose moel Fig. 3. Valiation process of propose moel Output ΔV, Δ ΔV, Δ The purpose of this case is to represent Area 1 machines (Fig. 4) by one single machine an run the estimation algorithm to fin the euivalent machine parameters. Valiation of the estimate machine will show if the euivalent machine truly represents Area 1. A similar approach can be use to represent Area then the whole system can be scale own to two euivalent machines connecte by a raial transmission line. G1 G 1 Area 1 PMU 3 13 Loa#1 Fig. 4. Case : Four-machine two-area System Loa# Area 11 1 1 G3 G4 The simulation was carrie out in Power System Toolbox (PST) [. The simulate machines are similar an were built aroun sub-transient moel an euippe with c exciters an
Parameter H x cost function Simulate euivalent machine 13.7 Propose moel 1.3.31.5e 11 Classic moel 16.31 4.4e 1 TABLE I CASE : ESTIMATED PARAMETERS OF THE PROPOSED MODEL AND CLASSIC MODEL θ.15.1.5.5 Simulate Propose moel Classic moel governors. The simulation etails are shown in the appenix. Input an Output ata were extracte in bus were a PMU is suppose to be installe. The resulting input an output ata are shown in Fig. 5. V.1 5 1 15 5..1 Simulate Propose moel Classic moel Input.5 Pe Qe.1. 5 1 15 5 Time (s) Fig. 6. Case : Valiate output of the propose moel an classic moel -.5 5 1 15 5.15 Estimation V.1 ( =-.1).3..1 Simulate Propose moel Classic moel Output.5 -.5 -.1 5 1 15 5 Time (s) Fig. 5. Case : Input an Output The stabilizing term ɛ formulate in (5) is set to -.9 which is higher than -.5 use in case 1 (subsection??). The value -.9 for ɛ is still small compare to the other factor ( ω /H) affecting ω which is aroun -15. The estimate parameters along with the cost function 3 provie by the system ientification algorithm are shown in Table I. The simulate euivalent machine of the power subsystem has theoretically a total inertia euals the sum of the inertias of its iniviual machines (when perfectly coherent) an a transient reactance euals the Thevenin euivalent of the reactances seen from bus (i.e. H = 13 an x =.7). The valiate output of the propose moel an the classic moel are shown in 6. The impact of various values for ɛ on angle valiation is shown in Fig. 7, which clearly shows the the propose moel is better in every case. The impact on V was insignificant. ( =-.5) ( =-.9) -.1 5 1 15 5.15.1.5 -.5 Simulate Propose moel Classic moel -.1 5 1 15 5.15.1.5 -.5 Simulate Propose moel Classic moel -.1 5 1 15 5 Time (s) Fig. 7. Case : The impact of varying ɛ on the angle output of the propose an classic moels ynamics of electric variables such as the internal voltages are completely ignore, this moel inclues the impact of these ynamics. System ientification base on this moel is emonstrate in two case stuies: a single generator moel ientification an a subsystem ientification. In both case stuies, the propose moel is teste against the classical moel an shows more accurate preiction of parameters an system responses. IV. CONCLUSION This paper investigates using least suare estimation base system ientification to apply PMU ata for generator moel ientification. A secon-orer moel is propose in this paper base the two-axis generator moel. This moel has inputs from real power an reactive power an outputs as the terminal voltage phasor. Unlike the classic generator moel where the of 3 It is assume that the simulate machine, being the base moel, has a cost APPENDIX The etaile erivation of the Jacobian matrix is shown in the longer version of the paper poste at http://power.eng.usf.eu. Case 1: Simulation ata for the machine: H = 3.7 s, x =.4 pu, x =.4 pu, x = 1.81 pu, x = 1.81 pu, r s =, x =.15 pu, x =.15 pu, T = 8 s, T = 1 s, T =.3, T =.7 s. Exciter: T r = s, K a =, T a =.3 s, K e = 1, T e =
.1 s, T b = 5 s, T c =.3 s, K f =.5e 3, T f =.1 s, Ef min = 11.5, Ef max = 11.5, K p = Case : Simulation ata for the machines: H = 6.5 s, x =.3 pu, x =.3 pu, x = 1.81 = pu, x = 1.7 pu, r s =.5, x =.5 pu, x =.5 pu, T = 8 s, T =.4 s, T =.3, T =.5 s. DC Exciter: T r =.1 s, K a = 46, T a =.6 s, K e =, T e =.46 s, T b = s, T c = s, K f =.1, T f = 1 s, V r min =.9, V r max = 1 [19 J. Machowski, Power system ynamics stability an control. Chichester, U.K: Wiley, 8. [ Y. Bar, Nonlinear parameter estimation. New York: Acaemic Press, 1974. [1 J. R. S. H. Thomas Banks, Marie Daviian an K. L. Sutton, MATHE- MATICAL AND STATISTICAL ESTIMATION APPROACHES IN EPI- DEMIOLOGY. Springer, 9, ch. An Inverse Problem Statistical Methoology Summary, pp. 49 3. [ J. Chow, G. Rogers, an K. Cheung, Power System Toolbox, Tech. Rep. REFERENCES [1 A. Phake an R. e Moraes, The wie worl of wie-area measurement, Power an Energy Magazine, IEEE, vol. 6, no. 5, pp. 5 65, september-october 8. [ Z. Huang, P. Du, D. Kosterev, an B. Yang, Application of extene kalman filter techniues for ynamic moel parameter calibration, in Power Energy Society General Meeting, 9. PES 9. IEEE, july 9, pp. 1 8. [3 K. Kalsi, Y. Sun, Z. Huang, P. Du, R. Diao, K. Anerson, Y. Li, an B. Lee, Calibrating multi-machine power system parameters with the extene kalman filter, in Power an Energy Society General Meeting, 11 IEEE, july 11. [4 L. Fan an Y. Wehbe, Extene kalman filtering base real-time ynamic state an parameter estimation using pmu ata, Electric Power Systems Research, vol. 13, pp. 168 177, 13. [5 Y. Wehbe an L. Fan, UKF base estimation of synchronous generator electromechanical parameters from phasor measurements, in North American Power Symposium (NAPS), 1, sep. 1. [6 J.-Y. Choi, B. Cho, H. VanLaningham, H. soo Mok, an J.-H. Song, System ientification of power converters base on a black-box approach, Circuits an Systems I: Funamental Theory an Applications, IEEE Transactions on, vol. 45, no. 11, pp. 1148 1158, nov 1998. [7 B. Miao, R. Zane, an D. Maksimovic, System ientification of power converters with igital control through cross-correlation methos, Power Electronics, IEEE Transactions on, vol., no. 5, pp. 193 199, sept. 5. [8 K. Chau an C. Chan, Nonlinear ientification of power electronic systems, in Power Electronics an Drive Systems, 1995., Proceeings of 1995 International Conference on, feb 1995, pp. 39 334 vol.1. [9 L. Xu an L. Fan, System ientification base vsc-hvc c voltage controller esign, in North American Power Symposium (NAPS), 1, sep. 1. [1 L. Fan, Z. Miao, an Y. Wehbe, Application of ynamic state an parameter estimation techniues on real-worl ata, IEEE Transactions on Smart Gri, vol. 4, no., pp. 1133 1141, 13. [11 J. Pierre, N. Zhou, F. Tuffner, J. Hauer, D. Trunowski, an W. Mittelstat, Probing signal esign for power system ientification, Power Systems, IEEE Transactions on, vol. 5, no., pp. 835 843, may 1. [1 J. Khazaei, L. Fan, W. Jiang, an D. Manjure, Distribute prony analysis on real-worl pmu ata, submitte, IEEE Trans. Power Systems, 15. [13 B. Mogharbel, L. Fan, an Z. Miao, Least suares estimation-base synchronous generator parameter estimation using pmu ata, accepte, IEEE PESGM 15, 15. [14 L. Fan, Least suares estimation an kalman filter base ynamic state an parameter estimation, accepte, IEEE PESGM 15, 15. [15 I. Kamwa an L. Gerin-Lajoie, State-space system ientification-towar mimo moels for moal analysis an optimization of bulk power systems, Power Systems, IEEE Transactions on, vol. 15, no. 1, pp. 36 335, feb. [16 M. Burth, G. Verghese, an M. Velez-Reyes, Subset selection for improve parameter estimation in on-line ientification of a synchronous generator, Power Systems, IEEE Transactions on, vol. 14, no. 1, pp. 18 5, feb 1999. [17 A. Cintron-Arias, H. T. Banks, A. Capali, an A. L. Lloy, A sensitivity matrix base methoology for inverse problem formulation. Journal of Inverse & Ill-Pose Problems, vol. 17, no. 6, pp. 545 564, 9. [18 P. Sauer an M. Pai, Power System Dynamics an Stability. Prentice Hall, 1998.