## 3 Distribution System State Estimation
\(e=[e_{1},\cdots,e_{M}]^{T}\) is the vector measurement error, which follows the Gaussian distribution \(e\sim\mathrm{N}(0,W)\), where \(W\) is the covariance matrix of the measurement errors \(W=diag\big{\{}\sigma_{z_{1}}^{2},\sigma_{z_{2}}^{2},\cdots,\sigma_{z_{m}}^{2} \big{\}}\) and the zero mean. The WLS technique is commonly applied to solve an SE problem. In this method, state vectors are obtained by minimizing the sum of the residual squares as follows. \[\hat{x}=\arg\,\min_{x}\,J(x) \tag{8}\] \[=\arg\,\min_{x}\,\big{(}(z-h(x))^{T}W^{-1}(z-h(x)).\]  
the iterative Gauss-Newton numerical method is applied to solve the optimization problem (8) with objective function \(J(x)\). \[x=[\theta_{1},\cdots,\theta_{N},V_{1},\cdots,V_{N}]. \tag{2}\]0 \[x=[\theta_{1},\cdots,\theta_{N},V_{1},\cdots,V_{N}]. \tag{2}\]1 \[x=[\theta_{1},\cdots,\theta_{N},V_{1},\cdots,V_{N}]. \tag{2}\]2  
where \(G\) is the gain matrix, and \(\Delta x^{k}\) is the updated state vector to determine a new vector \(x^{k}\). The iterative calculation proceeds until the preset convergence criterion is attained. The prefix threshold \(\varepsilon\) is compared to the largest absolute value of the updated state vector \((\Delta x^{k})\), and the state variables will be determined in the last iteration when \(\max(|\Delta x|)<\varepsilon\). \(H(x)=\partial h(x)/\partial(x)\) is the Jacobian matrix, which is derived from the function vector \(h(x)\). \[H(x)=\frac{\partial h(x)}{\partial(x)}=\begin{bmatrix}\frac{\partial h_{1}(x) }{\partial x_{1}}&\frac{\partial h_{1}(x)}{\partial x_{2}}&\cdots&\frac{ \partial h_{1}(x)}{\partial x_{n}}\\ \frac{\partial h_{2}(x)}{\partial x_{1}}&\frac{\partial h_{2}(x)}{\partial x _{1}}&\cdots&\frac{\partial h_{2}(x)}{\partial x_{n}}\\ \vdots&\vdots&\vector [73]. In (11), the Jacobian matrix is shown when traditional (\(H_{Traditional}\)) and smart (\(H_{Smart}\)) measurements are included in the measurement vector, respectively. \[H_{Traditional}=\left[ \begin{array}{ccc}\dfrac{\partial V_{SCADA}}{\partial \theta}&\dfrac{\partial V_{SCADA}}{\partial V}\\ \dfrac{\partial P_{inj}}{\partial\theta}&\dfrac{\partial P_{inj}}{\partial V} \\ \dfrac{\partial Q_{inj}}{\partial\theta}&\dfrac{\partial Q_{inj}}{\partial V} \\ \dfrac{\partial P_{flow}}{\partial\theta}&\dfrac{\partial P_{flow}}{\partial V} \\ \dfrac{\partial Q_{flow}}{\partial\theta}&\dfrac{\partial Q_{flow}}{\partial V} \end{array} \right]H_{Smart}=\left[ \begin{array}{ccc}\dfrac{\partial V_{SCADA}} {\partial\theta}&\dfrac{\partial V_{SCADA}}{\partial V}\\ \dfrac{\partial^{PAUI}_{\theta_{V}}}{\partial\theta}&\dfrac{\partial V_{PMU}} {\partial V}\\ \dfrac{\partial^{PAUI}_{\theta_{I}}}{\partial\theta}&\dfrac{\partial I_{PMU}} {\partial V}\\ \dfrac{\partial I_{imag}}{\partial\theta}&\dfrac{\partial I_{imag}}{\partial V }\\ \dfrac{\partial P_{inj}}{\partial\theta}&\dfrac{\partial P_{inj}}{\partial V} \\ \dfrac{\partial Q_{inj}}{\partial\theta}&\dfrac{\partial Q_{inj}}{\partial V} \\ \dfrac{\partial P_{flow}}{\partial\theta}&\dfrac{\partial P_{flow}}{\partial V} \\ \dfrac{\partial Q_{flow}}{\partial\theta}&\dfrac{\partial Q_{flow}}{\partial V} \end{array} \right] \tag{11}\]