Comparisons on Kalman-Filter-Based Dynamic State Estimation Algorithms of Power Systems

Abstract

The Kalman-filter-based algorithms as the mainstream algorithms of dynamic state estimation of power systems have been extensively used to provide accurate data for power system applications. However, few comparisons are made to show their advantages and disadvantages. In this paper, four Kalmanfilter-based algorithms (i.e., extended Kalman filter, unscented Kalman filter, cubature Kalman filter, and ensemble Kalman filter) are compared to show their differences from implementation complexity, estimation accuracy and calculation efficiency, the resistance to measurement errors, and the sensitivity to system scales. Finally, the simulation results on the 3-machine, 10-machine, and 48-machine power systems show their advantages and disadvantages. INDEX TERMS Cubature Kalman filter, dynamic state estimation, ensemble Kalman filter, extended Kalman filter, unscented Kalman filter. I. NOMENCLATURE a Admittance angle matrix C Equal weight cubature points vector D Damping coefficient e Unit column vector E EnKF sample set f (•) State transition function F Jacobian matrix of state transition function h(•) Measurement function H Jacobian matrix of measurement function I Unit matrix k Filtering gain p e Electromagnetic power of the generator p m Mechanical power of the generator P k Estimated error covariance P zz Innovation covariance matrix P xz The cross-covariance matrix Q System noise variance R Measurement noise variance The associate editor coordinating the review of this manuscript and approving it for publication was Wuhui Chen. S Weighted Sigma points vector T Inertia constant of the generator u Constant control vector v Measurement noise vector V Voltage amplitude w System noise vector W c Weight vectors of covariance W m Weight vectors of mean x State vector Y Reduced node admittance matrix z Measurement vector δ Generator power angle ω Generator speed ω s Generator reference speed ω δ System noise vector of generator power angle ω ω System noise vector of generator speed h Difference step size i, j The ith/jth generator k Current time n Number of generators n en Number of samples N Number of times α, β, η, λ, l Setting parameters of UKF

Application of Physical Function Model to State Estimations of Nonlinear Mechanical Systems

“…In general, as shown in [30], the nonlinear system needs to be linearized by using the Taylor series expansion around the state vector to apply EKF to it. Furthermore, some comparison studies between EKF and UKF have been conducted [31], [32]. Unfortunately, EKF induces heavy calculation load when deriving the Jacobian, which is the partial differentiation of nonlinear functions with respect to states.…”

“…In spite of an approach where multiple piece-wise linear systems are switched depending on the contact states in backlash [33], its application range is limited, and how to handle multiple nonlinearities is not considered. According to [31], EKF cannot also be applied for large-scale systems because of the complicated Jacobian matrix calculation. On the other hand, UKF, in which a probability density function is approximated by a Gaussian distribution, can be used to a wide range of nonlinear characteristics.…”

Pyramidal approximation for power flow and optimal power flow

“…The subscript j is the line index here. Equations (5) and (6) are the active and reactive power balance equations, respectively. The matrices M PQ and M l are used to represent the connection between the buses and the lines.…”

“…The PF equations (PFEs) are the most important formulas and constraints in PF and OPF formulation. The general AC PFEs in the polar coordinates, called the bus injection model (BIM) in this paper, are complex non-linear equations with sin and cos functions [4][5][6]. In most cases, the AC PFEs can be kindly solved by the Newton method.…”

A Robust INS/SRS/CNS Integrated Navigation System with the Chi-Square Test-Based Robust Kalman Filter

“…To keep the reliability and improve the accuracy of the output of an integrated navigation system, information fusion is also important. The conventional Kalman filter (KF) has been a primary algorithm for linear navigation system integration [ 15 , 16 ]. However, in order to achieve information fusion when using the traditional KF, the accuracy system model and exact noise statistics are required [ 17 ].…”