Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation

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1 Aaptive Ajustment of Noise Covariance in Kalman Filter for Dynamic State Estimation Shahroh Ahlaghi, Stuent Member, IEEE Ning Zhou, Senior Member, IEEE Electrical an Computer Engineering Department, Binghamton University, State University of New Yor, Binghamton, NY 392, USA {sahlag, Abstract Accurate estimation of the ynamic states of a synchronous machine (e.g., rotor s angle an spee is essential in monitoring an controlling transient stability of a power system. It is well nown that the covariance matrixes of process noise (Q an measurement noise (R have a significant impact on the Kalman filter s performance in estimating ynamic states. he conventional a-hoc approaches for estimating the covariance matrixes are not aequate in achieving the best filtering performance. o aress this problem, this paper proposes an aaptive filtering approach to aaptively estimate Q an R base on innovation an resiual to improve the ynamic state estimation accuracy of the extene Kalman filter (EKF. It is shown through the simulation on the two-area moel that the propose estimation metho is more robust against the initial errors in Q an R than the conventional metho in estimating the ynamic states of a synchronous machine. Inex erms Kalman filter, ynamic state estimation (DSE, innovation/resiual-base aaptive estimation, process noise scaling, measurement noise matching. I. INRODUCION imely an accurately estimating the ynamic states of a synchronous machine (e.g., rotor angle an rotor spee is important for monitoring an controlling the transient stability of a power system over wie areas []. With the worlwie eployment of phasor measurement units (PMUs, many research efforts have been mae to estimate the ynamic states an improve the estimation accuracy using PMU ata [2] [], among which the Kalman filtering (KF techniques play an essential role. For instance, Huang et al. [2] propose an extene Kalman filtering (EKF approach to estimate the ynamic states using PMU ata. Ghahremani an Innocent [3] propose the EKF with unnown inputs to simultaneously estimate ynamic states of a synchronous machine an unnown inputs. [4]-[7] propose the unscente Kalman filtering to estimate power system ynamic states. Zhou et al. [8] propose an ensemble Kalman filter approach to simultaneously estimate the ynamic states an parameters. Ahlaghi, Zhou an Huang [9]-[] propose an aaptive interpolation approach to mitigate the impact of non-linearity Zhenyu Huang, Senior Member, IEEE Pacific Northwest National Laboratory, Richlan, WA 99352, USA zhenyu.huang@pnnl.gov in ynamic state estimation (DSE. hese stuies have lai a soli groun for estimating the ynamic states of a power system an also reveale some nees for further stuies. One important problem that nees to be aresse in using the KF is how to properly set up the covariance matrixes of process noise (i.e., Q an measurement noise (i.e., R. Note that the performance of the KF is highly affecte by Q an R [2]. Improper choice of Q an R may significantly egrae the KF s performance an even mae the filter iverge [3]. o etermine Q an R, almost all the previous DSE stuies use an a-hoc proceure, in which Q an R are assume to be constant uring the estimation an are manually ajuste by trial-an-error approaches. Note that because the noise levels may change for ifferent applications an users of DSE can have ifferent bacgrouns, it can be very challenging to use such an a-hoc approach to properly set up Q an R. o aress this challenge, this paper proposes an estimation approach to aaptively ajust Q an R at each step of the EKF to improve DSE accuracy. An innovation-base metho is use to aaptively ajust Q. A resiual-base metho is use to aaptively ajust the R. A simple example is use to evaluate the impact of Q an R on the performance of EKF. hen, performance of the propose approach is evaluate using a two-area moel []. he rest of paper is organize as follows: Section II reviews the ynamic moel of a synchronous machine use for DSE. In Section III, the aaptive EKF approach is propose. Sections IV an V present a case stuy an simulation results. Conclusions are rawn in Section VI. II. DYNAMIC SAE ESIMAION MODEL his section gives a brief review on the ynamic moel of a synchronous machine to be use by the EKF for DSE. he 4 th orer ifferential equations of a synchronous machine in a local -q reference frame is given by (. (Reaers may refer to [9]-[] for more etails: his paper is base on wor sponsore by the U.S. Department of Energy (DOE through its Avance Gri Moeling program. Pacific Northwest National Laboratory is operate by Battelle for DOE uner Contract DE- AC5-76RL83.

2 2 δ = ωδωr t Δωr = Δ t 2H e q = t e = t q ( K ω m e D r ( Ef e q ( x x i ( e ( xq xq iq (.a (.b (.c (. In (, the 4 states, δ, ω r, e an e q, are the rotor angles in raians, rotor spees in per-unit (pu an transient voltages in pu along an q axes, respectively. ω = 2π f is the synchronous spee; m an e are the mechanical an the electric air-gap torque in pu; an parameters H an K D are the inertia an amping factor, respectively; E f is the internal fiel voltage. Variables x an x q are the synchronous reactance; x an x q are the transient reactance along an q axes, respectively. i an i q are the stator currents along an q axes, respectively. an q are the open circuit time constants in the q frame. o facilitate the notation for applying the EKF to DSE of a synchronous machine, ( is transforme into a general iscrete state space moel as shown in (2 an (3 with sampling interval of Δ t using the moifie Euler metho []. x =Φ ( x, u w z = h( x, u v Φ( x, u h( x, u Φ, H [] [] x x x = x = δ Δω q m f R I = [ R I] x e e u E i i z e e (2.a (2.b Here, subscript is the time inex, which inicates the time instance at Δ t. Symbols x, u, an z are the state, input an measurement output, respectively. Functions Φ ( an h( are the state transition an measurement function, [] respectively. Φ is the Jacobian matrix of the state transition [] matrix at step -, an H is the Jacobian matrix of the measurement function at step. In (2, vectors w an v are the state process noise an measurement noise, respectively. heir mean an variance are enote by (4 []. Here, symbol E( represents the expecte value. Symbols Q an R are the covariance matrixes of process noise an measurement noise respectively at step. (, ( E w = E w w = Q (4.a (, ( Ev = Evv = R (4.b III. ADAPIVE EXENDED KALMAN FILER APPROACH his section escribes the conventional extene Kalman (3 filter (CEKF an proposes an aaptive extene Kalman filter (AEKF approach which aaptively estimates Q - an R. A. Conventional Extene Kalman Filter he CEKF consists of the following 3 steps. Reaers may refer to [] for more etails about the CEKF. Step ( Initialization: o initialize the CEKF, the mean values an covariance matrix of the states are set up at = as in (5. = E( x (5.a P ˆ ˆ = E ( x x( x x (5.b where the superscript inicates that the estimate is a posteriori, an P is the state covariance matrix. Step (I Preiction: he state an its covariance matrix at - are projecte one step forwar to obtain the a priori estimates at as in (6. ( ˆ =Φ x, u (6.a Preicte State Estimate [ ] [ ] =Φ Φ Priori Covariance Matrix P P Q (6.b Step (II - Correction: he actual measurement is compare with preicte measurement base on the a priori estimate. he ifference is use to obtain an improve a posteriori estimate as in (7. ( ˆ z h x Measurement innovation [ ] [ ] = Innovation Covariance (7.a S H P H R (7.b Kalman Gain [] [ ] K = P H S (7.c [ ] ˆ = x K (7. Posteriori State Estimate [] P = I KH P { } (7.e Posteriori Covariance Matrix Note that to run the CEKF, users nee to provie Q - in (6.b an R in (7.b. Performance of a CEKF epens on how well users can select the right Q - an R for ifferent applications. Conventionally, R is often assigne as a constant matrix base on the instrument accuracy of the measurements. Q - is assigne as a constant matrix using a trial-an-error approach, which relies on users experiences an bacgroun. As such, selection of Q - an R is a challenge for the users of the CEKF. B. Aaptive Extene Kalman Filter (AEKF o aress this challenge, this paper proposes an aaptive estimation approach to estimate Q - an R in the EKF. Mehra [4] classifie the aaptive estimation approaches into four categories: Bayesian, correlation, covariance matching an maximum lielihoo approaches. he covariance matching is

3 3 one of the well-nown aaptive estimation approaches, which tunes the covariance matrix of the innovation or resiual base on their theoretical values [5]. At the EKF s preication step, the innovation is the ifference between the actual measurement an its preicte value, an it can be calculate by (7.a. On the other han, the resiual is the ifference between actual measurement an its estimate value using the information available at step, an it can be calculate by (8. ε ( ˆ z h x (8 resiual Base on the above efinitions, the Q - an R can be estimate as the follows. Resiual Base Aaptive Estimation of R he innovation base approach estimates the covariance matrix R using (9 [2]. [ ] [ ] R = S H P H (9 Here S is the covariance matrix of the innovation. Note that theoretically speaing, R shoul be positive efinite because it is a covariance matrix. Yet, its estimation equation (9 coul not guarantee that the estimate R be a positive efinite matrix because the R is estimate by subtracting the two positive efinite matrixes. herefore, to ensure a positive efinite matrix, the resiual base aaptive approach propose by [6] is use by this paper to estimate R using (. [ ] [ ] S = E εε = E νν H P H ( [] [] R = E εε H P H o implement (, the expectation operation on ε ε is approximate by averaging ε ε over time. Instea of the using the moving winow, this paper introuces a forgetting factor < α in ( to aaptively estimate R. Note that a larger α puts more weights on previous estimates an therefore incurs less fluctuation of R, an longer time elays to catch up with changes. his paper set α =.3 for all the stuies. [ ] [ ] R = αr ( α( εε H P H ( 2 Innovation Base Aaptive estimation of Q o aaptively estimate the Q -, base on (2, the process noise can be calculate using (2. w = x Φ ( x, u (2 From (6 an (7, it can be conclue that: wˆ ˆ ˆ = x Φ( x, u = x ˆ ( ˆ x = K z h x (3 = K herefore, E wˆ wˆ = E K ( K = K E K Qˆ = KSK (4 Similar to the previous subsection, the paper uses a forgetting factor α to average estimates of Q over time as in (5. Q = αq ( α( K K (5 An implementation flowchart of the propose AEKF algorithm is summarize in Fig.. Note that similar to the CEKF, users nee to select the initial Q an R for AEKF in the initialization step. Different from the CEKF which eeps Q - an R constant, the Q - an R of the AEKF are aaptively estimate an upate uring each correction step. ˆx =Φ(, u ( ˆ z h x = K ε ( ˆ z h x P, Q, R [] [] P =Φ P Φ Q [] [] R = αr ( α( εε H P H [] [] [] K = P H H P H R [] P = { I KH } P, Q = αq ( α( K K Fig.. Implementation flowchart of the propose AEKF IV. CASE SUDY BASED ON A SIMPLE MODEL In this section, a simple linear moel escribe by (6 is use to compare the impact of the choice of R an Q - on the performance of the CEKF an the AEKF. he simple an linear moel with nown noise features is use in this stuy to eliminate the potential impacts from non-linearity. he moel in (6 is a moel of a vehicle tracing problem, which the vehicle is constraine to move in a straight line with a constant velocity. Let p an p& represent the vehicle position an velocity. he system state can be escribe by x = [ p, p& ]. It is assume that sample observations are acquire at iscrete time interval Δt. he w - an v are Gaussian white noise, whose variances are efine by (6.b. x = Ax w z = Hx v = (6.a Δt A=, [ ] H = (6.b 3 2 Δt 3 Δt 2 Qtrue = q [] 2, Rtrue = r Δt 2 Δt he scalars q, r an Δt are set to be.,. an, respectively. It is assume that the vehicle starts from rest so that x = [, ]. For testing the performance of the CEKF an AEKF, time steps of simulation are generate using (6. o evaluate the impact of Q an R on the estimation accuracy, x is set to its true values an P is set to zeros to

4 4 eliminate their impacts on the estimation. For the CEKF, Q an R are set by scaling Q true an R true. As shown in able I, the scaling factors are the multiples of. Using the same setup, the resulting mean square errors (MSEs of estimate position, i.e., x(, are summarize in able I for the CEKF an in able II for the AEKF. It can be observe in able I that the MSEs on the iagonal are same. Note that the ratio between Q an R are same for the iagonal elements. his observation inicates that it is the ratio between Q an R (instea of their iniviual values that etermines the performance of the CEKF. Also observe that the major iagonal, where Q :R = Q true :R true, have the smallest MSE (i.e..5. he observation suggests that the optimal Q/R ratio is aroun their true ratios. Also observe that the MSEs increase monotonously when the Q/R ratio increases or ecreases from its true value. ABLE I. MSES OF HE ESIMAED POSIION FROM HE CEKF MSE. Q true. Q true Q true Q true Q true. R true R true R true R true R true ABLE II. MSE OF HE ESIMAED POSIION FROM HE AEKF MSE. Q true. Q true Q true Q true Q true. R true R true R true R true R true Comparing able I an able II, one can observe that in general, the propose AEKF prouces smaller MSEs than the CEKF. he performance improvement of the AEKF is more significant when the MSEs of the CEKF are larger (at the bottom left of the table. he only exception to this improvement is at the major iagonal where the Q :R = Q true :R true, which is alreay an optimal Q/R ratio setup for the CEKF. he MSEs for the AEKF is slightly larger than the CEKF. his may be because averaging operations in ( an ( are use to approximate the expectation operation, which will incur some estimation errors in Q an R. Notice that in a real worl application, the true value of Q/R ratio an states are often not available an have to be estimate. he propose AEKF provies a systematic way of estimating the Q/R ratios an can often achieve stable an smaller MSEs than most guesse values. Similar observations are mae with the other state (i.e., x(2 spee an are not presente here to be concise. V. CASE SUDY BASED ON HE WO-AREA MODEL o evaluate the performance of the propose AEKF approach on the DSE of synchronous machines, the two-area four-machine system [] shown in Fig. 2 is use to generate the simulation ata. he simulation is performe using the Power System oolbox (PS [7]. A three-phase fault is applie to sening en of the line between buses an 2 at. s. o reuce integration errors an capture the ynamics, the simulation time step is set to be. s. G 2 G MW G3 4 wo-area Four Machine System G4 Area 2 Area Fig. 2. he two-area four-machine system []. Fig. 3. Comparison of AEKF an CEKF when the initial Q is set to be relatively less than the proper value. It is assume that all the generation buses are equippe with PMUs to measure the voltage phasors an current phasors in (3. o mimic the fiel measurements from the PMUs, the simulation ata is ecimate to 25 samples/s. An 4.% noise in total vector errors [] is ae to the current an voltage phasors to consier the noise introuce by potential transformers an current transformers. Also, 4.% noise is ae to E f an m. Similar to section IV, in the initialization step, x is set to its true values an P is set to zeros to eliminate their impacts on the estimation. Assume that the R is nown base on the accuracy of measurement evice an is set to be iag([.4,.4] 2 to match the ae measurement noise. he initial Q is ajuste to set up the following four scenarios for comparing the estimation accuracy of the AEKF an CEKF. Scenario #: Q is set to very small values (i.e. *e-8. he states estimate by the AEKF an CEKF are shown in Fig. 3 an their MSEs are summarize in able III. Fig. 3 shows that with the same setup, the CEKF iverges while the AEKF converges. able III shows that the MSEs of the AEKF is much smaller than those of the CEKF for all the estimate states. he observation inicates that when Q is set up to be too small, the AEKF is robust against the improper setup an can estimate states accurately while the CEKF iverges. Scenario #2: Q is set to very large values (i.e.. he states estimate by the AEKF an CEKF are shown in Fig. 4 an their MSEs are summarize in able III. Fig. 4 shows that both the CEKF an AEKF converges an the states estimate by the AEKF stay closer to the true states than those estimate by the CEKF. able III shows that the MSEs of the AEKF is smaller than those of the CEKF for all the estimate states. he observation inicates that when Q is set up to be too large, both the AEKF an CEKF converge an the AEKF is more accurate than the CEKF, measure by MSEs.

5 5 Scenario #3: Q is set to be close to the true values. As the true value of Q is not accurately nown, the final Q resulting from the AEKF in Scenarios #2 is use. he MSEs of the estimate states are summarize in able III. able III shows that the MSEs of the AEKF is similar to those of the CEKF for all the estimate states. he observation inicates that when Q is set up to be close to the true values, both the AEKF an CEKF converge an the AEKF has similar accuracy as the CEKF in the sense of MSEs. Scenario #4: he setups of this scenarios are same as those for scenario # except that the Monte-Carlo simulation is use to generate N = 2 instances of simulation ata with ranom noise. he estimate states are summarize in Fig. 5, which shows that observations mae uner scenarios # also apply to ifferent noise instances. ABLE III. COMPARISON OF HE MSES OF HE ESIMAED DYNAMIC SAES FROM HE CEKF AND AEKF Scenario # δ MSE Δω e e q CEKF e AEKF 7.e-5.25e-7.5e-4 3.3e-6 2 CEKF.74e e e e-5 AEKF 2.53e-5.5e e-5 3.e-6 3 CEKF 8.38e e e e-5 AEKF.57e-5.9e-7 9.2e e-6 Fig. 4. Comparison of AEKF an CEKF when the initial Q is set to be relatively greater than the proper value. From the results of this section, it can be conclue that the propose AEKF approch is robust agaist the initial errors in setting up the corvariance matrixes of process noise (i.e., Q an measurement noise (i.e., R. he reason for scenarios #2 an #3 to have goo performance is that we assume true R an the measurements are ieal (meaning they match the moel with no outliers, no losses. In this case, a large Q woul bias the EKF to believe the measurements, which woul generate goo estimate. If R is unnown an/or measurements are not ieal, a blin selection of large Q woul fail to generate goo estimates. We are testing Q only as the first step to mae it easier to show the effect of the AEKF. Scenario # is the most important case to examine. Future wor will continue the research to test unnown R an imperfect measurements. VI. CONCLUSIONS his paper proposes an AEKF approach to aaptively estimate an ajust covariance matrixes Q - an R for estimating the ynamic states of a synchronous machine. Also, it is shown through simulations using a simple moel an the two-area system that the AEKF is more robust against the improper choice of initial Q an R than the CEKF. hese simulation results suggest that the propose AEKF can aaptively estimate Q as well as R an therefore relieve users buren of choosing proper Q an R in the EKF. Fig. 5. Comparison of DSE results from the AEKF an CEKF approaches. REFERENCES [] P. Kunur, Power System Stability an Control. New Yor, NY, USA: McGraw-Hill, 994. [2] Z. Huang, K. Schneier, an J. 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