## 3 Hybrid State Estimator Considering PMU Measurements
### Estimator with PMU Measurements  
The state estimator based on SCADA measurements is nonlinear, while the state estimator based on PMU measurements can be linear. In this paper, a hybrid state estimator considering PMU measurements and SCADA measurements is proposed. At the sampling time of SCADA, the first state estimator is carried out with the SCADA measurements and PMU measurements at the nearest moment, and the result of state estimation \(\hat{\mathbf{x}}_{\text{s}}\) is saved. This state estimator is nonlinear. At this moment, the measurements include the voltage amplitude and phase angle of the bus, the amplitude and phase angle of branch current, the active power and reactive power of branch, the active power and reactive power injected at the bus. In this paper, the real and imaginary parts of slack bus voltage and branch current are chosen as state variables [30]. The measurement function and Jacobian matrix are obtained by the relationship between the measurements and state variables. PMU collects the amplitudes and phase angles of bus voltage and branch current, while the state variables of the estimator are the real and imaginary parts of bus voltage and branch current, so PMU measurements need to be equivalently transformed. Coordinate transformation involves the transmission of measurement errors. Taking the branch current as an example, the covariance of branch current in terms of real and imaginary parts can be calculated as Equation (10):  
\[\left[ \begin{array}{cc}\sigma_{\vec{r}}^{2}&\sigma_{\vec{n}\pm}^{2}\\ \end{array} \right]=\left[ \begin{array}{cc}\frac{\partial\sigma_{\vec{n}\pm}^ {\prime}}{\partial\vec{n}\pm}&\frac{\partial\sigma_{\vec{n}\pm}^{\prime}}{ \partial\vec{n}\pm}\\ \frac{\partial\sigma_{\vec{n}\pm}^{\prime}}{\partial\vec{n}\pm}&\frac{\partial \sigma_{\vec{n}\pm}^{\prime}}{\partial\vec{n}\pm}\\ \end{array} \right]\times\left[ \begin{array}{cc}\sigma_{\vec{l}\pm}^{2}&0\\ 0&\sigma_{\vec{l}\pm}^{2}\\ \end{array} \right]\times\left[ \begin{array}{cc}\frac{\partial\sigma_{\vec{n} \pm}^{\prime}}{\partial\vec{n}\pm}&\frac{\partial\sigma_{\vec{n}\pm}^{\prime}} {\partial\vec{n}\pm}\\ \frac{\partial\sigma_{\vec{n}\pm}^{\prime}}{\partial\vec{n}\pm}&\frac{\partial \sigma_{\vec{n}\pm}^{\prime}}{\partial\vec{n}\pm}\\ \end{array} \right]^{T}, \tag{10}\]  
where \(w\) and \(k\) are the bus number; \(\sigma_{\vec{r}}^{2}\) and \(\sigma_{\vec{r}\pm}^{2}\) are the measurement error covariances of the real and imaginary parts of branch current; \(i_{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\textThen, at the sampling time of PMU, the second linear state estimator is carried out with the last result of state estimation \(\hat{\pi}_{S}\) and PMU measurements at the current moment. The covariance matrix of the estimated state error can be calculated as Equation (11):  
\[\mathbf{R}_{S}=\left(\mathbf{H}^{T}\mathbf{W}\mathbf{H}\right)^{-1}, \tag{11}\]  
where \(\mathbf{R}_{S}\) represents the covariance matrix of the estimated state error of \(\hat{\mathbf{x}}_{S}\). Then the weight matrix of measurements of this state estimator can be expressed as Equation (12):  
\[\mathbf{W}_{P}=\left[ \begin{array}{ccc}\mathbf{R}_{S}&&\\ &\mathbf{R}_{I_{PMU}}&\\ &&\mathbf{R}_{I_{PMU}}\end{array} \right]^{-1}, \tag{12}\]  
where \(\mathbf{W}_{P}\) is the weight matrix of measurements of this state estimator; \(\mathbf{R}_{I_{PMU}}\) and \(\mathbf{R}_{I_{PMU}}\) are the covariances of branch current and bus voltage measurement errors provided by PMU at the current moment. The estimated state provided by the second linear state estimator can be defined as Equation (13):  
\[\hat{\mathbf{x}}_{P}=\left(\mathbf{H}_{P}{}^{T}\mathbf{W}\mathbf{H}_{P}\right)^{-1}\mathbf{H}_{P}{ }^{T}\mathbf{W}\mathrm{z}, \tag{13}\]  
where \(\hat{\mathbf{x}}_{P}\) is the estimated state provided by the second state estimator; \(\mathbf{H}_{P}\) is the Jacobian matrix of this state estimator, its elements are constant which can be obtained by Equation (6).