Source-linked AI summary
Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation
Shahrokh Akhlaghi, Ning Zhou, Zhenyu Huang
TL;DR
Accurate synchronous-machine state estimation matters for transient-stability monitoring and control, but conventional EKF performance depends on difficult manual selection of Q and R. The paper adaptively estimates these covariances from innovation and residual information, and simulations show greater robustness to initial covariance errors than the conventional EKF.
Problem
Conventional ad-hoc, fixed covariance selection for Q and R is inadequate, although these matrices strongly affect Kalman-filter performance in synchronous-machine state estimation.
Method
The paper proposes an adaptive EKF that estimates Q and R during filtering using innovation-based and residual-based methods.
Results
Simulations using simple and two-area models show that the AEKF is more robust to improper initial Q and R choices than the CEKF.
Takeaways & Limitations
The AEKF can adaptively estimate Q and R, reducing users’ burden of selecting proper covariance matrices for the EKF.
Abstract
from arXiv · showhide
Accurate estimation of the dynamic states of a synchronous machine (e.g., rotor s angle and speed) is essential in monitoring and controlling transient stability of a power system. It is well known that the covariance matrixes of process noise (Q) and measurement noise (R) have a significant impact on the Kalman filter s performance in estimating dynamic states. The conventional ad-hoc approaches for estimating the covariance matrixes are not adequate in achieving the best filtering performance. To address this problem, this paper proposes an adaptive filtering approach to adaptively estimate Q and R based on innovation and residual to improve the dynamic state estimation accuracy of the extended Kalman filter (EKF). It is shown through the simulation on the two-area model that the proposed estimation method is more robust against the initial errors in Q and R than the conventional method in estimating the dynamic states of a synchronous machine.
I. INTRODUCTION
Dynamic state estimation of synchronous machines supports monitoring and control of transient stability, but Kalman-filter performance depends critically on selecting process and measurement noise covariances Q and R. The paper addresses limitations of manual covariance tuning with an adaptive EKF approach.
- Dynamic state estimates such as rotor angle and speed are important for monitoring and controlling power-system transient stability.
- Kalman-filter performance is highly affected by the process-noise covariance Q and measurement-noise covariance R.
- Improper Q and R choices can significantly degrade filtering performance and even cause the filter to diverge.
- Conventional DSE studies typically hold Q and R constant and tune them manually through trial and error, despite application- and user-dependent noise levels.
- The proposed method adaptively adjusts Q and R at each EKF step, using innovation-based estimation for Q and residual-based estimation for R.
II. DYNAMIC STATE ESTIMATION MODEL
The paper represents synchronous-machine dynamics in a general discrete state-space form for EKF-based estimation. The model distinguishes states, inputs, measurements, transition and measurement functions, and process and measurement noise covariances.
- The four states represent rotor angle, rotor speed, and transient voltages along the d and q axes.
- A fourth-order synchronous-machine model is transformed into a discrete state-space model for EKF dynamic state estimation.
- The modified Euler method performs the discretization with sampling interval Δt.
- The discrete model uses state x_k, input u_k, and measurement output z_k with state-transition function Φ and measurement function h.
- Process noise w_k and measurement noise v_k have zero means and covariance matrices Q_k and R_k, respectively.
III. ADAPTIVE EXTENDED KALMAN FILTER APPROACH
The paper contrasts a conventional EKF with an adaptive EKF that estimates the process- and measurement-noise covariances during filtering.
- The adaptive extended Kalman filter estimates Q_{k-1} and R_k rather than treating them as fixed inputs.
- The proposed AEKF is introduced as an adaptive alternative to the conventional extended Kalman filter.
- The approach is designed to improve dynamic state estimation by adapting covariance matrices during EKF operation.
A. Conventional Extended Kalman Filter
The conventional EKF initializes state estimates, predicts the next state and covariance, and corrects them using measurements. Its performance depends on user-supplied Q and R, which are conventionally selected as fixed matrices.
- The CEKF has three stages: initialization, prediction, and correction.
- Initialization: Initialization sets the state mean and covariance matrix at k = 0.
- Prediction: Prediction projects the state and covariance at k−1 forward one step to obtain a priori estimates at k.
- Correction: Correction compares the actual measurement with its prediction and uses the difference to obtain an improved posteriori estimate.
- Users must provide Q_{k−1} and R_k; R_k is often fixed from instrument accuracy, while Q_{k−1} is tuned by trial and error.
B. Adaptive Extended Kalman Filter (AEKF)
The AEKF adaptively estimates process-noise covariance Q and measurement-noise covariance R during EKF correction steps, using innovation- and residual-based covariance matching with forgetting-factor averaging.
- The paper proposes adaptively estimating Q_k-1 and R_k in the EKF rather than keeping them fixed.
- Residual-based estimation of R_k is used because innovation-based subtraction does not guarantee a positive definite covariance matrix.
- A forgetting factor α averages R estimates over time; larger α reduces fluctuation but delays adaptation, and the studies use α = 0.3.
- Innovation-based estimation adjusts Q_k-1, while residual-based estimation adjusts R_k.
- A forgetting factor also averages Q estimates over time, and the updated Q_k-1 and R_k are applied during each AEKF correction step.
IV. CASE STUDY BASED ON A SIMPLE MODEL
A simple linear, constant-velocity vehicle-tracking model isolates how choices of Q_k−1 and R_k affect CEKF and AEKF performance. Across 100 simulated time steps, the AEKF generally achieves lower position-estimation MSEs than the CEKF, especially when CEKF errors are larger.
- Model and setup: The study uses a linear vehicle-tracking model with known noise features to isolate the effects of Q_k−1 and R_k on CEKF and AEKF performance.The vehicle moves in a straight line at constant velocity, avoiding potential impacts from nonlinearity.
- Model and setup: The vehicle state comprises position and velocity, with observations sampled at discrete intervals and Gaussian white process and measurement noise.The sampling interval is Δt, and the noise variances are defined by the model.
- Model and setup: The experiment generates 100 simulation time steps, uses x_0 at its true value and P_0=0, and evaluates position-estimation MSEs under scaled Q_k and R_k settings.For CEKF, Q_k and R_k are scaled relative to Q_true and R_true; results are summarized in Tables I and II.
- CEKF results: CEKF performance depends on the Q_k:R_k ratio rather than the individual covariance values, with the true ratio producing the smallest reported MSE of 0.051.Equal ratios yield equal diagonal MSEs, and the major diagonal corresponds to Q_true:R_true.
- AEKF results: The AEKF generally produces smaller MSEs than the CEKF, with larger improvements where CEKF MSEs are higher.At the true Q_k:R_k ratio, already optimal for CEKF, AEKF MSEs are slightly larger, possibly because of the averaging operation.
V. CASE STUDY BASED ON THE TWO-AREA MODEL
The two-area four-machine case study evaluates AEKF and CEKF dynamic-state estimation under different initial Q settings. AEKF remains more accurate when Q is substantially misinitialized, while the study notes boundaries related to known R and ideal measurements.
- Case-study setup: The study uses a two-area four-machine system to generate synchronous-machine dynamic-state estimation data.A three-phase fault is applied at 10.1 s, and the simulation step is 0.001 s.
- Case-study setup: The simulation assumes PMU measurements with decimated sampling and added noise on phasors, Efd, and Tm.The data are decimated to 25 samples/s, with 4.0% noise added to the specified measurements.
- Results by initial Q: With very small initial Q0, CEKF diverges while AEKF converges and produces much smaller MSEs for all estimated states.Scenario #1 sets Q0 to 1*e-08.
- Results by initial Q: With very large initial Q0, both filters converge, but AEKF stays closer to the true states and has smaller MSEs for all estimated states.Scenario #2 sets Q0 to 1000.
- Results by initial Q: When Q0 is close to its true value, AEKF and CEKF both converge with similar MSE accuracy.Scenario #3 uses the final Q from Scenario #2 because the true Q0 is not accurately known.
- Scope boundary: The study tests Q as a first step under known true R and ideal measurements; unknown R or imperfect measurements remain future work.The authors state that blind selection of large Q may fail when R is unknown or measurements are not ideal.
VI. CONCLUSIONS
The paper proposes AEKF to adaptively estimate Qk-1 and Rk for synchronous-machine dynamic-state estimation. Simulations indicate greater robustness than CEKF to improper initial covariance choices and suggest reduced burden in selecting Q and R.
- Contribution: The proposed AEKF adaptively estimates and adjusts Qk-1 and Rk for synchronous-machine dynamic-state estimation.The conclusion identifies adaptive covariance estimation as the central approach.
- Conclusion: Simulations using a simple model and the two-area system show AEKF is more robust than CEKF to improper initial Q and R choices.The comparison concerns dynamic-state estimation of a synchronous machine.
- Implication: The results suggest AEKF can adaptively estimate Q and R, relieving users’ burden of choosing proper covariance values in EKF.This is presented as a conclusion suggested by the simulation results.