## 3 Hybrid State Estimator Considering PMU Measurements  
### The Basic Model of State Estimation  
The state estimator proposed in this paper is based on WLS. Assuming that the number of state variables is \(N\) and the number of measurements is \(M\), the measurements can be expressed as Equation (1):  
\[\mathbf{z}=\mathbf{h}(\mathbf{x})+\mathbf{v}, \tag{1}\]  
where \(\mathbf{z}=\left[z_{1},\ z_{2},\ \cdots,\ z_{M}\right]^{T}\) is the \(M\)-dimensional vector of measurements; \(\mathbf{x}=\left[x_{1},\ x_{2},\ \cdots,\ x_{N}\right]^{T}\) is the \(N\)-dimensional vector of state variables; \(\mathbf{v}=\left[v_{1},\ v_{2},\ \cdots,\ v_{M}\right]^{T}\) is the \(M\)-dimensional vector of measurement errors, whose entry \(v_{i}\) represents the error of measurement \(i\); \(\mathbf{h}\) is the vector of functions, which represents the relationship between \(\mathbf{z}\) and \(\mathbf{x}\).  
According to the measurement vector \(\mathbf{z}\) and measurement function \(\mathbf{h}\), the objective function that minimizes the sum of squares of estimation errors can be expressed as Equation (2):  
\[\min\ \ \ J(\mathbf{x})=\left(\mathbf{z}-\mathbf{h}(\mathbf{x})\right)^{T}\mathbf{W}(\mathbf{z}-\mathbf{h}( \mathbf{x})), \tag{2}\]  
where \(J(\mathbf{x})\) is the sum of squares of estimation errors; \(\mathbf{W}\) is the weight matrix of measurements, \(\text{diag}\ \mathbf{W}=\left[w_{1},\ w_{2},\ \cdots,\ w_{M}\right]^{T}\); \(w_{i}\) represents the weight of measurement \(i\), which can be described as Equation (3):  
\[\mathbf{W}=\mathbf{R}^{-1}=\left[\begin{array}{cccc}\sigma_{1}^{2}&&&\\ &\sigma_{2}^{2}&&\\ &&\ddots&\\ &&&\sigma_{M}^{2}\end{array}\right]^{-1}, \tag{3}\]  
where \(\mathbf{R}\) is the covariance matrix of the measurement error; \(\sigma_{i}\) is the variance of the measurement \(i\).  
To minimize the sum of squares of estimation errors, \(J(\mathbf{x})\) is guided by \(\mathbf{x}\), which can be expressed as Equation (4):  
\[\frac{\partial J(\mathbf{x})}{\partial\mathbf{x}}=0. \tag{4}\]  
According to Equation (2), Equation (4) can be modified to Equation (5):  
\[\mathbf{H}(\mathbf{x})\mathbf{W}(\mathbf{z}-\mathbf{h}(\mathbf{x}))=0, \tag{5}\]  
where \(\mathbf{H}(\mathbf{x})\) is named as the Jacobian matrix, which can also be defined as Equation (6):  
\[\mathbf{H}(\mathbf{x})=\frac{\partial\mathbf{h}(\mathbf{x})}{\partial\mathbf{x}}. \tag{6}\]As \(\mathbf{h}\) is the vector of nonlinear functions, the Taylor series is applied to solve Equation (5), and the high terms with orders greater than 1 are ignored to get a linear Equation (7):  
\[\mathbf{h}(\mathbf{x})=\mathbf{h}(\mathbf{x}_{0})+\mathbf{H}(\mathbf{x}_{0})\Delta\mathbf{x}, \tag{7}\]  
where \(\mathbf{x}_{0}\) represents the initial state vector.  
According to Equations (5) and (7), the iterative equations can be expressed as Equation (8):  
\[\left\{\begin{array}{c}\Delta\mathbf{x}^{(k)}=\Big{[}\mathbf{H}^{\rm T}(\mathbf{x}^{(k) })\mathbf{W}\mathbf{H}(\mathbf{x}^{(k)})\Big{]}^{-1}\mathbf{W}\Big{[}\mathbf{z}-\mathbf{h}(\mathbf{x}^{(k) })\Big{]}\\ \mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\Delta\mathbf{x}^{(k)}\end{array}\right., \tag{8}\]  
where \(\mathbf{x}^{(k+1)}\) and \(\mathbf{x}^{(k)}\) are the state variable vectors obtained by the \(k-\text{th}\) and \(\left(k+1\right)-\text{th}\) iterations respectively.  
Newton method is applied to solve the iterative Equations (8). The estimated state vector \(\mathbf{\hat{x}}\) can be obtained when the convergence criterion is satisfied. The convergence criterion can be defined as Equation (9):  
\[\text{max}\Big{|}\Delta\mathbf{x}^{(k)}\Big{|}<\varepsilon, \tag{9}\]  
where \(\varepsilon\) represents the value of convergence.