A Review of Dynamic Phasor Estimation by Non-Linear Kalman Filters

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Phasor estimation under dynamic conditions has been under study recently by relaxing the amplitude and phase of the static phasor. This paper will review some methods to estimate dynamic phasor by nonlinear Kalman filters. Nonlinear Kalman filters have found an application in the tracking of time-varying amplitude and phase. Five nonlinear Kalman filtering methods for dynamic phasor estimation are examined in this paper. These methods are: EKF1 stands for first-order Extended Kalman filter, EKF2 stands for second-order Extended Kalman, UKF stands for Unscented Kalman filter, GHKF stands for Gauss Hermite Kalman filter, and finally CKF stands for Cubature Kalman filter. This paper describes the theoretical processes of these methods and demonstrates their effectiveness in dynamic phasor estimation by some test signal simulations in MATLAB. The simulation section shows that nonlinear Kalman filters give more accuracy than linear Kalman filters when the phasor is relaxed by modulated amplitude and phase. Moreover, comparative assessments among the performance of five nonlinear Kalman filters are done for dynamic phasor estimation, and also their performances are compared with the other methods which have already been published. According to the simulation results, EKF1 gives the highest accuracy during steady-state ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 10^{-13}$ </tex-math></inline-formula> ) because the signal model is more similar to the estimation model of EK1 during the steady-state condition. However, the other non-linear Kalman filters show better performances in dynamic conditions. When the phasor is a time-varying amplitude and phase, filters give the same accuracy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.5 \%$ </tex-math></inline-formula> ). A step-change in amplitude and phase creates different overshoot and response times, but EKF2 shows the least overshoot (3.2%) and the longest response time (7.6 ms). Computation burden and noise indices can discriminate the methods from other viewpoints. The computation burden of GHKF is drastically increased when the number of states gives rise. CKF shows an appropriate performance when the number of samples and the number of the state increased in the input signal. EKF1 is not a good solution for noise infiltration when SNR is less than 40 dB, but CKF gives the highest accuracy in high noise levels. Compared to the six already pblished methods, CK shows the best performance with a reasonable estimation error ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.1101 \%$ </tex-math></inline-formula> ) and simulation time (0.6146 ms).