Deriving AR Moels for Synchronous Generators Yangkun u, Stuent Member, IEEE, Zhixin Miao, Senior Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract Parameter ientification of a synchronous generator base on Phasor Measurement Unit (PMU) ata requires autoregressive exogenous (AR) moels his paper presents AR moel erivation an valiation of two synchronous generator moels he first one is a thir-orer AR moel incluing electromechanical ynamics, turbine-governor ynamics an primary frequency control Input of the moel is power while output is frequency he secon one is a fifth orer generator moel extening from the simplifie one, with aitional rotor flux ynamics an excitation ynamics Generator terminal voltage is treate as input while real power, reactive power an frequency are treate as outputs Nonlinear moel simulation ata from Power System oolbox (PS) are use as PMU ata he erive AR moels are valiate by comparing the measurement outputs with the outputs generate from the AR moels High matching egrees are observe he coefficients of the AR moels are also compare base on two approaches: (i) calibration from the known generator parameters an (ii) system ientification High matching egrees are also observe Inex erms linear analysis, non-linear simulation, AR moel I INRODUCION Accurate estimation of synchronous generator parameters an states plays an important role in power system operation Various ynamic state an parameter estimation methos have been evelope recently for Phasor Measurement Unit (PMU) ata here are two major system ientification methos he first approach is Kalman filter base estimation [] [8] he estimation can be carrie out at each time step he secon approach is least squares estimation (LSE) [9] [] A time winow of measurements an a iscrete AR moel are require to carry out the estimation For phasor measurement unit (PMU) ata base estimation, [9] shows a linear continuous moel in Laplace omain is converte to a Z-omain moel A iscrete AR moel is erive base on the z-omain moel Converting a Laplace continuous moel to a z-omain requires specific treatment for each transfer function hus, this metho (zero-orer hol) ocumente in [9] requires time consuming erivation his metho will be ifficult to be applie to a general moel he objective of this paper is to provie a more general approach to form an AR moel for LSE for PMU ata base generator parameter estimation we will start from linear continuous state-space moels, which will be further converte to iscrete linear state-space moels Discrete AR moels are then obtaine With a iscrete AR moel, a linear estimation problem can be formulate an the coefficients of the AR moel can be foun From these coefficients, we can further recover the generator parameters Y u, Z Miao an L Fan are with Dept of Electrical Engineering, University of South Floria, ampa FL 336 Email: linglingfan@usfeu In this paper, we present two AR moels with ifferent level of complexity he first one is a simplifie generator moel incluing swing ynamics, turbine-governor ynamics an primary frequency control roop he secon ones extens the first one by incluing rotor flux ynamics an exciter ynamics For the first moel, real power is treate as the input while the frequency is treate as the output For the secon moel, the terminal voltage magnitue is treate as the input while the power, reactive power an frequency are treate as the outputs For valiation, PMU ata is generate by time-omain simulation, which is carrie out for a two-area four-machine test system via PS [] he AR moels are valiate by comparing output of real ata with those generate from the AR moels he AR moel coefficients obtaine from two approaches (calibration base on known parameters an system ientification) are also compare he experiments show high matching egrees on both output ata an coefficients, which emonstrates the feasibility an accuracy of the erive AR moels he rest of this paper is organize as follows Section II escribes the PMU ata an the synchronous generator Section III presents two AR moel Section IV presents the numerical estimation results Section VI conclues the paper II SYNCHRONOUS GENERAOR AND PMU DAA his paper uses ata recore by PS (generator terminal voltage V a, machine spee ω, electrical power P e an Q e ) at the terminal of the synchronous machine after a fault within few secons he synchronous generator is assume to inclue electromechanical parameters, primary frequency control roop, turbine-governor ynamics an excitation ynamics he following parameters: inertia constant H, amping coefficient D, regulation parameter R, governor time constant g, exciter time constant e, voltage regulator gain K A, -axis open circuit transient time constant o, reactance q, an transient reactance ), are to be estimate through PMU ata A synchronous machine connecte to power gri is shown in the Figure he generator is moele by a subtransient moel in PS All system ata can be foun in Appenix he PMU ata inclue the terminal voltage V a, phase angle θ, frequency f, an real an reactive power (P an Q) In orer to carry out LSE-base system ientification using PMU ata, AR moels are require here are multiple ways to separate the PMU ata into input an outputs In this paper, two ways are examine: ) P as input, frequency as output We will show in Section III that this moel is suitable to estimate parameters relate to swing ynamics an turbine-governors: H, D, R, an g However, this moel oes not reflect the other parameters such
P e G V a, f P Q q Power gri P ref s g P m R Hs D s Fig Synchronous generator connecte to power gri as,,, etc ) V a as input, frequency, P an Q as output his moel reflects all the aforementione parameters A AR moel III AR MODELS DERIVAION he AR moel structure can be expresse as: A(z)y(k) = B(z)u(k n k ) e(k) () where u(k) is the system inputs, y(k) is the system outputs, n k is the system elay an e(t) is the white noise isturbance A(z) an B(z) are polynomials an efine by the following equations: A(z) = a z a na z na B(z) = b b z b nb z n b [n a, n b, n k ] represent the AR moel s orers an time elay In Matlab software, we can use AR comman to fin the coefficients of the polynomials for the given input/output ata an [n a, n b, n k ] B Moel : Power ( P e ) to frequency ω he swing equations are expresse as follows { ω t = H (P m P e D (ω )) δ t = ω (ω ) where δ is the rotor angle in raius, ω is the rotor spee (frequency) in pu, P m is the mechanical power (pu), an P e is the electrical power (pu) he small-signal moel is as follows { ω = H ( P m P e D ω) δ (3) = ω ω he mechanical power P m is prouce after the turbine which is controlle by a governor he ynamic of turbine governor with primary frequency control can be expresse as: () P m = g ( P ref R ω P m) (4) With (3) an (4), he block iagram of a simplifie thir orer synchronous generator is esigne as shown in Fig, where g is the time constant of the turbine-governor (secons), an R is the spee regulation constant (pu) P e is consiere as input an ω is output We name this generator moel as Moel Fig Block iagram of a simplifie synchronous generator Base on Fig, the plant moel will be built in the form of linearize state-space vector ifferential equations δ ω δ ω = D P H H ω m H P e gr g P m ω = [ ] δ ω (5) P m Discretizing the continuous state-space moel leas to the following iscrete state-space moel δ ω k h ω k = D h h H H δ k ω k h P mk h h H P gr mk g }{{}}{{} A B ω k = [ ] δ k ω k (6) }{{} P C mk where h is sampling time Convert the above moel to z-omain: y u = C (zi A ) B (7) We can fin the equivalent iscrete time AR moel as: where A(z)y(k) = B(z)u(k ) (8) A(z) = a z a z B(z) = b b z he orers an elay of Moel is [,, ] Given time series measurements of input u an output y, an overetermine problem Y = φθ can be built as follows: P ek y(k ) y(k ) y(k) u(k ) u(k) a y(i) = y(i ) y(i ) u(i ) u(i ) a b b y(n) y(n ) y(n ) u(n ) u(n ) (9) where y(k ), y(k), u(k ), u(k) are elaye input an output ata is referee to as regressors he coefficients can be solve by the normal equation of least square estimation θ = (φ φ) φ Y or Matlab AR comman
C Moel : erminal voltage ( V a ) to frequency ω, electrical active power P e an reactive power Q e Consiering the swing equations an rotor flux ynamics, there are three state variables: E a, frequency, ω, an the rotor angle δ he ynamic moel of the generator is expresse as follows δ t = ω (ω ) ω () t = E a t = o H (P m P e D (ω )) (E f E a ) where E f can be viewe as fiel voltage v F in equivalent stator terms, E a is an internal voltage whose magnitue is proportional to the rotor flux λ F, E a is the stator open circuit voltage he voltage an current phasor iagram is shown in Fig 3 Using the same way to linearize (4), P e = P e E a a P e δ δ P e V a V a = V a sin(δ ) a [ ( E a V a cos(δ ) Va ) ] q cos(δ ) δ [ ( E a sin(δ ) ) ] q V a sin(δ ) V a = k a δ p V a (5) he electrical reactive power Q e is given by: ' E a E a j I q aq Q e = E av a ( ) cos(δ) cos(δ) Va sin(δ) q (6) I aq I a I a δ V a Fig 3 Phasor iagram showing E a an E a j ' Ia E a in terms of E a, δ an V a can be expresse as: ' a j( ) I E a = E a V a cos(δ) () A a small perturbation to linearize the E a equation: a = E a E a a E a = ( a δ δ E a V a V a cos(δ ) V a ) V a sin(δ ) δ = k 3 a k 4 δ e V a () Substituting () into the ifferential equation of E a in () leas to: o a t a = = f k 3 a k 4 δ e V a k 3 o k 3s ( f k 4 δ e V a ) (3) he electrical active power P e in terms of E a, δ an V a can be expresse as: P e = E av a sin(δ) V a ( ) q sin(δ) (4) Linearizing the Q e equation: Q e = Q e E a a Q e δ δ Q e V a V a = V a cos(δ ) a [ ( E av a sin(δ ) Va [ ( E a cos(δ cos(δ ) ) ) ] sin(δ ) δ q ) V a ] V a sin(δ ) q = q a q δ q 3 V a (7) Linearizing ynamics () by substituting (5) an () into them: δ = ω ω ω = H ( P m (k a δ p V a ) D ω) ( ) a = o f ( k 3 a k 4 δ e V a ) (8) A simplifie exciter moel is use in the single machine infinite bus system In this, the transfer function is K A /( e s) he ynamic of f can be expresse as: E f = e (K A V ref K A V a f ) (9) With (8), (4) an (9), the block iagram of fifth orer generator moel is esigne as shown in Fig 4 he input is the terminal voltage V a, the outputs are frequency w, real power P e an reactive power Q e Finally, writing (8), (4) an (9) in compact form, we
ΔP ref Δ V a p s g - R ΔV ref - Δv e e ΔP M k - k 3 A s e ΔE f - k 3 o s - ΔP e k ΔE a Hs D s k 4 Fig 4 Linearize fifth orer generator moel consiering terminal voltage variation obtain δ w ω D K δ H H H H a f = k 4 o ω k 3 o o P a f e m P m R g g p H e o Va K A e ω P e = δ ω k a Q e q q p V a () f q 3 P m Discretizing (), the complete iscrete state-space moel has the following form: w h δ k ω k h H D h k h h δ k H H H H a k fk = k 4h k o 3 o h h k 3 o o ω k a e h k P e fk mk h g h P mk R g g p h H e h o V ak K Ah e ω k P ek = δ k ω k k a Q ek q q k p V ak () fk q 3 P mk hree AR moels can be formulate in terms of corresponing output he orers an elay of them are presente in able I he structure of AR moel for output w is given by: where A(z)y(k) = B(z)u(k ) () A(z) = a z a z a 3 z 3 a 4 z 4 a 5 z 5 B(z) = b b z b z b 3 z 3 b 4 z 4 he orers an elay of Moel is [5 5 ] he structure of AR moel for output P an Q are same, both can be expresse as: where A(z)y(k) = B(z)u(k) (3) A(z) = a z a z a 3 z 3 a 4 z 4 a 5 z 5 B(z) = b b z b z b 3 z 3 b 4 z 4 b 5 z 5 he orers an elay of Moel is [5 6 ] ABLE I input V a Output A(z) orer B(z) orer Delay ω 5 5 P e 5 6 Q e 5 6 IV CASE SUDIES he two-area four-machine test system is shown in Fig 5 he synchronous generators are moele by subtransient moels in PS he generators are equippe with turbinegovernors, primary frequency roop, excitation systems A three-phase fault occurs at t = s on Bus 3 to bus he fault is cleare at Bus 3 after 5s he power system ynamics stuy is carrie out by PS [] he simulation time is set to 5 sec an the sampling time is h = sec Machine spee ω, machine terminal voltage V a, electrical active power P e an reactive power Q e of generator are recore as shown in Fig 6 he recore raw ata are preprocesse he ata for AR moel estimation are taken starting from 5 sec In aition, the DC offset will be remove from recore ata he smooth ata are use to ientify the AR moels A Case stuy-i: Moel Valiation In this case stuy, P e is consiere as input an ω is output he AR moel is generate with the [n a =, n b =, n k = ] We compares the output of the AR moel with the measurement ata as shown in Fig 7 he normalize root mean square measure of the gooness of fit is 9563%
G 3 3 G 3-3 5 4 ime Response Comparison Valiation ata moel: 9563% 3 Loa# Loa# Amplitue ω G G4 - Fig 5 wo-area four-machine system - 5-3 5 5 3 35 4 45 5 ime (secons) V a -5 5 5 5 3 35 4 45 5 Fig 7 Comparison of the valiation ata vs moel Output signal ( ω) is epicte -3 5 ω -5 5 5 5 3 35 4 45 5 P e - 5 5 5 3 35 4 45 5 For P e or Q e is output, the gooness of fit are 859% an 83% respectively he coefficients of preictive AR moel an the real AR moel are presente in able III Q e - 5 5 5 3 35 4 45 5 ime (secons) Fig 6 Recore ata by PS: magnitue response to three-phase fault he coefficients of preicte AR moel an the compute coefficients base on known paramters are presente in able II ω -3 5 ime Response Comparison Valiation ata moel: 975% ABLE II Case Coefficients Moel Data by PS a - 999-995 a 9995 995 b -8736 5-769 5 b 873 5 7677 5 It can be observe that the coefficients of A(z) polynomial from measurement base ientification match very well with the compute coefficients he error of estimation of b an b is approximately % B Case stuy-ii: Moel Valiation In this case stuy, V a is input here are three outputs to be consiere For ω as output, the gooness of fit is 975% Amplitue P e Q e -5 5-5 - 5-5 5 5 3 35 4 45 5 ime (secons) Valiation ata moel: 859% Valiation ata moel: 83% Fig 8 Comparison of the valiation ata vs Moel Output signals ( ω, P e an Q e) are epicte
ABLE III Output ω Output P e Output Q e Coefficients Estimation Calibration Estimation Calibration Estimation Calibration a -5-499 -4999-499 -4995-499 a 997 999 997 998 997 a 3 - -995-999 -995-997 -995 a 4 5 497 499 497 498 497 a 5 - -99 - -99 - -99 b -86 4-493 4-94 -639-595 -3 b 74 4 974 4 469 3 7965 56 b - 4-965 4-9356 -64-59 -3 b 3 74 4 979 4 939 64 589 9 b 4-85 4-495 4-465 -35-7933 -54 b 5 976 644 585 3 he comparison shows that the AR moel output has a high matching egree with the measurement ata he preicte AR moel coefficients for A(z) polynomial are very close to the values compute base on the known parameters he coefficients for B(z) polynomial are not close to the real values his experiment inicates that if the coefficients of A(z) from system ientification can be use for parameter recovery However, the coefficients of B(z) from system ientification nee to be further examine V CONCLUSION his paper presents AR moel erivation an valiation of two synchronous generator moels he first one is a thir-orer AR moel incluing electromechanical ynamics, turbine-governor ynamics an primary frequency control Input of the moel is power while output is frequency he secon one is a fifth orer generator moel extening from the simplifie one, with aitional rotor flux ynamics an excitation ynamics Generator terminal voltage is treate as input while real power, reactive power an frequency are treate as outputs Nonlinear moel simulation ata from Power System oolbox (PS) are use as PMU ata he erive AR moels are valiate by comparing the measurement outputs with the outputs generate from the AR moels High matching egrees are observe he coefficients of the AR moels are also compare base on two approaches: (i) calibration from the known generator parameters an (ii) system ientification High matching egrees are also observe he erive AR moels are suitable for generator parameter estimation REFERENCES [] E Wan, R Van Der Merwe et al, he unscente kalman filter for nonlinear estimation, in Aaptive Systems for Signal Processing, Communications, an Control Symposium AS-SPCC he IEEE IEEE,, pp 53 58 [] G Evensen, he ensemble kalman filter: heoretical formulation an practical implementation, Ocean ynamics, vol 53, no 4, pp 343 367, 3 [3] R Van Der Merwe, E Wan et al, he square-root unscente kalman filter for state an parameter-estimation, in Acoustics, Speech, an Signal Processing, Proceeings(ICASSP ) IEEE International Conference on, vol 6 IEEE,, pp 346 3464 [4] G Evensen, Data assimilation: the ensemble Kalman filter Springer Science & Business Meia, 9 [5] P L Houtekamer an H L Mitchell, A sequential ensemble kalman filter for atmospheric ata assimilation, Monthly Weather Review, vol 9, no, pp 3 37, [6] Z Huang, K Schneier, an J Nieplocha, Feasibility stuies of applying kalman filter techniques to power system ynamic state estimation, in Power Engineering Conference, 7 IPEC 7 International IEEE, 7, pp 376 38 [7] Z Huang, P Du, D Kosterev, an B Yang, Application of extene kalman filter techniques for ynamic moel parameter calibration, in Power & Energy Society General Meeting, 9 PES 9 IEEE IEEE, 9, pp 8 [8] L Fan an Y Wehbe, Extene kalman filtering base real-time ynamic state an parameter estimation using pmu ata, Electric Power Systems Research, vol 3, pp 68 77, 3 [9] B Mogharbel, L Fan, an Z Miao, Least squares estimation-base synchronous generator parameter estimation using pmu ata, ariv preprint ariv:5354, 5 [] Soerstrom, H Fan, B Carlsson, an S Bigi, Least squares parameter estimation of continuous-time arx moels from iscrete-time ata, Automatic Control, IEEE ransactions on, vol 4, no 5, pp 659 673, 997 [] D Simon, Least squares estimation, Optimal State Estimation: Kalman, H, an Nonlinear Approaches, pp 79 5 [] J Chow an G Rogers, Power system toolbox, Cherry ree Scientific Software,[Online] Available: http://www ecse rpi eu/pst/ps html, vol 48, p 53, APPENDI Generator moel ata for the swing moel in per unit is as follows: H = 65 g = s R = 4 D = Generator moel ata for the 5th orer moel in per unit is as follows: H = 65s g = 5s o = 8s R = 4 D = 377 = 8 = 3 = 5 q = 7 q = 55 q = 5 f = 4 r f = 5 Excitation system parameters in per unit are as follows: k A = 46 e = 46s