30 Power System Dynamic Model
We can represent a nonlinear dynamical power system for state estimation in a discrete form with the help of the following equations:\begin{align*}&\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_{k})+\mathbf{w}_{k}\tag{1}\\ &\mathbf{z}_{k+1}=\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{v}_{k+1}\tag{2}\\ &\mathbf{w}_{k}=N(0,\,\mathbf{Q}_{k})\tag{3}\\ &\mathbf{v}_{k+1}=N(0,\,\mathbf{R}_{k+1}), \tag{4}\end{align*} where, is the state vector at the instant; is the measurement vector at the instant; is Gaussian process noise with zero mean; is the process noise error covariance; and is the Gaussian measurement noise with zero mean. In addition, is the measurement noise error covariance, and and are the nonlinear functions for a state space of size and the measurement space of size , respectively. For the PSHDSE formulation, state vector comprises the bus voltage and the angle state subvectors. The measurement vector comprises the subvectors of voltage magnitude measurements , voltage angle measurements , real power injection measurements , reactive power injection measurements , real power flow measurements , and reactive power flow measurements , received from the RTUs and PMUs at the instant , which are given by\begin{align*}&\mathbf{x}_{k}=[\mathbf{V}_{k}\vert \theta_{k}]^{\mathbf{T}}\tag{5}\\ &\mathbf{z}_{k}=[\mathbf{V}_{mk}\vert\theta_{mk}\,\vert\,\mathbf{Pr}_{mk}\vert\mathbf{Qr}_{mk}\,\vert\, \mathbf{PF}_{mk}\,\vert\,\mathbf{QF}_{mk}]^{\mathbf{T}}, \tag{6}\end{align*} where subscript indicates the size of the corresponding measurement samples at the instant.