Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation Shahrokh Akhlaghi, Student Member, IEEE Ning Zhou, Senior Member, IEEE Zhenyu Huang, Senior Member, IEEE Electrical and Computer Engineering Department, Pacific Northwest National Laboratory, Binghamton University, State University of New York, Richland, WA 99352, USA Binghamton, NY 13902, USA [email protected] {sakhlag1, ningzhou}@binghamton.edu Abstract—Accurate estimation of the dynamic states of a in dynamic state estimation (DSE). These studies have laid a synchronous machine (e.g., rotor’s angle and speed) is essential in solid ground for estimating the dynamic states of a power monitoring and controlling transient stability of a power system. system and also revealed some needs for further studies. It is well known that the covariance matrixes of process noise (Q) and measurement noise (R) have a significant impact on the One important problem that needs to be addressed in using Kalman filter’s performance in estimating dynamic states. The the KF is how to properly set up the covariance matrixes of conventional ad-hoc approaches for estimating the covariance process noise (i.e., Q) and measurement noise (i.e., R). Note matrixes are not adequate in achieving the best filtering performance. To address this problem, this paper proposes an that the performance of the KF is highly affected by Q and R adaptive filtering approach to adaptively estimate Q and R based [12]. Improper choice of Q and R may significantly degrade on innovation and residual to improve the dynamic state the KF’s performance and even make the filter diverge [13]. estimation accuracy of the extended Kalman filter (EKF). It is To determine Q and R, almost all the previous DSE studies shown through the simulation on the two-area model that the used an ad-hoc procedure, in which Q and R are assumed to be proposed estimation method is more robust against the initial constant during the estimation and are manually adjusted by errors in Q and R than the conventional method in estimating the dynamic states of a synchronous machine. trial-and-error approaches. Note that because the noise levels may change for different applications and users of DSE can Index Terms— Kalman filter, dynamic state estimation (DSE), have different backgrounds, it can be very challenging to use innovation/residual-based adaptive estimation, process noise such an ad-hoc approach to properly set up Q and R. scaling, measurement noise matching. To address this challenge, this paper proposes an estimation I. INTRODUCTION approach to adaptively adjust Q and R at each step of the EKF Timely and accurately estimating the dynamic states of a to improve DSE accuracy. An innovation-based method is synchronous machine (e.g., rotor angle and rotor speed) is used to adaptively adjust Q. A residual-based method is used important for monitoring and controlling the transient stability to adaptively adjust the R. A simple example is used to of a power system over wide areas [1]. With the worldwide evaluate the impact of Q and R on the performance of EKF. deployment of phasor measurement units (PMUs), many Then, performance of the proposed approach is evaluated research efforts have been made to estimate the dynamic states using a two-area model [1]. and improve the estimation accuracy using PMU data [2]– The rest of paper is organized as follows: Section II [11], among which the Kalman filtering (KF) techniques play reviews the dynamic model of a synchronous machine used an essential role. For instance, Huang et al. [2] proposed an for DSE. In Section III, the adaptive EKF approach is extended Kalman filtering (EKF) approach to estimate the proposed. Sections IV and V present a case study and dynamic states using PMU data. Ghahremani and Innocent [3] simulation results. Conclusions are drawn in Section VI. proposed the EKF with unknown inputs to simultaneously estimate dynamic states of a synchronous machine and II. DYNAMIC STATE ESTIMATION MODEL unknown inputs. [4]-[7] proposed the unscented Kalman This section gives a brief review on the dynamic model of a filtering to estimate power system dynamic states. Zhou et al. th [8] proposed an ensemble Kalman filter approach to synchronous machine to be used by the EKF for DSE. The 4 simultaneously estimate the dynamic states and parameters. order differential equations of a synchronous machine in a Akhlaghi, Zhou and Huang [9]-[10] proposed an adaptive local d-q reference frame is given by (1). (Readers may refer interpolation approach to mitigate the impact of non-linearity to [9]-[11] for more details): This paper is based on work sponsored by the U.S. Department of Energy (DOE) through its Advanced Grid Modeling program. Pacific Northwest National Laboratory is operated by Battelle for DOE under Contract DE- AC05-76RL01830. 2 ⎧ dδ (1.a) filter (CEKF) and proposes an adaptive extended Kalman =Δωω ⎪ dt 0 r filter (AEKF) approach which adaptively estimates Qk-1 and ⎪ dΔω 1 Rk. ⎪ r =−−Δ()TTKω (1.b) ⎪ me Dr ⎪ dt2 H A. Conventional Extended Kalman Filter ⎨ de′ 1 ⎪ q =−−−()Eexxi′′() (1.c) The CEKF consists of the following 3 steps. Readers may dt T ′ fd q d d d ⎪ d 0 refer to [11] for more details about the CEKF. ⎪ de′ 1 ⎪ d =−+−()′′ exxidqqq() (1.d) ⎪ ′ Step (0) – Initialization: ⎩ dt Tq0 To initialize the CEKF, the mean values and covariance In (1), the 4 states, δω ′ and ′ , are the rotor angles in , r , ed eq matrix of the states are set up at k = 0 as in (5). radians, rotor speeds in per-unit (pu) and transient voltages in + = (5.a) xExˆ00() ω = π pu along d and q axes, respectively. 2 f is the +++T 0 0 =−−⎡ ˆˆ⎤ (5.b) PExxxx00000⎣()()⎦ synchronous speed; Tm and Te are the mechanical and the electric air-gap torque in pu; and parameters H and KD are the where the superscript “+” indicates that the estimate is a inertia and damping factor, respectively; Efd is the internal posteriori, and P is the state covariance matrix. field voltage. Variables xd and xq are the synchronous Step (I) – Prediction: reactance; x′ and x′ are the transient reactance along d and q d q The state and its covariance matrix at k-1 are projected one axes, respectively. id and iq are the stator currents along d and step forward to obtain the a priori estimates at k as in (6). ′ ′ q axes, respectively. Td 0 and Tq0 are the open circuit time −+ xxuˆˆ=Φ(,)−− (6.a) constants in the dq0 frame. 1kkk4442 411 443 Predicted State Estimate To facilitate the notation for applying the EKF to DSE of a −+=Φ[11] Φ[ ]T + PPkkkkk−−11 − 1 Q − 1 (6.b) synchronous machine, (1) is transformed into a general 14444244443 Priori Covariance Matrix discrete state space model as shown in (2) and (3) with sampling interval of Δt using the modified Euler method [11]. Step (II) - Correction: =Φ + ⎧xxuwkkkk(,)−−11 − 1 The actual measurement is compared with predicted ⎪ ⎨ (2.a) measurement based on the a priori estimate. The difference is ⎪ =+ used to obtain an improved a posteriori estimate as in (7). ⎩ zhxuvkkkk(,) =− − ∂Φ(,)xu ∂ hxu (,) dzhxkkkk⎡ ()ˆ ⎤ Φ≈[1]kk−−11 [1] ≈ kk ⎣ ⎦ (7.a) kk−1 , H (2.b) 14442 4 4 43 ∂∂xx==+− Measurement innovation xkkxxxˆˆ−1 kk [11] − [ ]T T SHPHR=+ xee=Δ⎡⎤δω′′ kkkkk (7.b) kqd⎣⎦ 144424443 Innovation Covariance T (3) uTEii=⎡⎤ − []1 T −1 kmfdRI⎣⎦ = [] KPHSkkkk (7.c) =[]T 14442 4 4 43 zeekRI Kalman Gain ˆˆ+−=+[] Here, subscript k is the time index, which indicates the xxKdkk kk (7.d) 14442 4 443 Posteriori State Estimate time instance at kΔt . Symbols xk, uk, and zk are the state, input Φ∗ +−=− []1 and measurement output, respectively. Functions () and PIKHPkkkk{ } 144424443 (7.e) h()∗ are the state transition and measurement function, Posteriori Covariance Matrix [1] Φ Note that to run the CEKF, users need to provide Qk-1 in respectively. k −1 is the Jacobian matrix of the state transition [1] (6.b) and Rk in (7.b). Performance of a CEKF depends on how matrix at step k-1, and Hk is the Jacobian matrix of the well users can select the right Qk-1 and Rk for different measurement function at step k. In (2), vectors w and v are k k applications. Conventionally, Rk is often assigned as a constant the state process noise and measurement noise, respectively. matrix based on the instrument accuracy of the measurements. Their mean and variance are denoted by (4) [11]. Here, Qk-1 is assigned as a constant matrix using a trial-and-error ∗ symbol E( ) represents the expected value. Symbols Qk and approach, which relies on users’ experiences and background. Rk are the covariance matrixes of process noise and As such, selection of Qk-1 and Rk is a challenge for the users of measurement noise respectively at step k. the CEKF. ()==T Ewkkkk0 , Eww() Q (4.a) B. Adaptive Extended Kalman Filter (AEKF) To address this challenge, this paper proposes an adaptive ()==T Evkkkk0 , Evv() R (4.b) estimation approach to estimate Qk-1 and Rk in the EKF. Mehra [14] classified the adaptive estimation approaches into four III. ADAPTIVE EXTENDED KALMAN FILTER APPROACH categories: Bayesian, correlation, covariance matching and This section describes the conventional extended Kalman maximum likelihood approaches. The covariance matching is 3 one of the well-known adaptive estimation approaches, which forgetting factor α to average estimates of Q over time as in tunes the covariance matrix of the innovation or residual (15). based on their theoretical values [15]. At the EKF’s =+−ααTT QQ− (1 )( KddK ) (15) predication step, the innovation is the difference between the k k1 kkk k actual measurement and its predicted value, and it can be An implementation flowchart of the proposed AEKF calculated by (7.a).