# A Hybrid State Estimator Based on SCADA and PMU Measurements for Medium Voltage Distribution System
## 3 A Hybrid State Estimator Based on SCADA and PMU Measurements  
### State Variables and Measurement Equations in Robust State Estimation  
At the time of SCADA data acquisition, the available measurements including the bus voltage amplitude, the bus voltage phasor, the current amplitude, the current phasor and power measurements, and the mathematical observability is guaranteed by pseudo-measurements. This paper follows the method of Reference [19] to deal with the voltage amplitude and voltage phasor measurements. Taking the branch currents as the state variables, the root bus voltage is expanded into the state variables, and the state variables can be expressed in the form of rectangular coordinates as Equation (1):  
\[\mathbf{x}_{\phi}=\left[v^{r}_{slack\phi},v^{x}_{slack\phi},i^{r}_{1\phi} \ldots i^{r}_{N\phi},i^{x}_{1\phi}\ldots i^{x}_{N\phi}\right] \tag{1}\]  
where, \(v^{r}_{slack\phi}\) and \(v^{x}_{slack\phi}\) are the real and imaginary parts of the root bus voltage on phase \(\phi\), respectively, and \(i^{x}_{1\phi}\ldots i^{x}_{N\phi}\) and \(i^{x}_{1\phi}\ldots i^{x}_{N\phi}\) are the real and imaginary parts of branch currents on phase \(\phi\) respectively. Since the phase angle difference between the two ends of the lines in the distribution network is very small, the voltage phase angles of all buses are not much different from that of the root bus [34], so the measurement of the bus voltage amplitude on the phase \(\phi\) can be represented by the state variable as Equation (2):  
\[ \begin{array}{ll}v_{h_{\phi}}&\simeq v^{r}_{slack\phi}\cos\left(\delta^{(k-1 )}_{v_{h_{\phi}}}\right)+v^{x}_{slack\phi}\sin\left(\delta^{(k-1)}_{v_{h_{\phi }}}\right)-\sum\limits_{l\in\mathbb{T}_{h_{\phi}}}\sum\limits_{\phi=1}^{3} \left(r_{l_{l\phi\phi}}i^{r}_{l_{l\phi\phi}}-x_{l_{l\phi\phi}}i^{x}_{l_{l\phi \phi}}\right)\cos\left(\delta^{(k-1)}_{v_{h_{\phi}}}\right)\\ &-\sum\limits_{l\in\mathbb{T}_{h_{\phi}}}\sum\limits_{\phi=1}^{3}\left(r_{l_{l \phi\phi}}i^{x}_{l_{l\phi\phi}^{x}}+x_{l_{l\phi\phi}}i^{r}_{l_{l\phi\phi}} \right)\sin\left(\delta^{(k-1)}_{v_{h_{\phi}}}\right)\end{array}  \tag{2}\]  
where \(\mathbb{T}_{h_{\phi}}\) is the path between the bus \(h\) and the root bus. \(r_{l_{l\phi\phi}}\) and \(x_{l_{l\phi\phi}}\) are the resistance and reactance of branch \(l\) on the path \(\mathbb{T}_{h_{\phi}}\), respectively, and \(\delta^{(k-1)}_{v_{h_{\phi}}}\) is the voltage phase angle of the bus \(h\) at the \((k-1)\)th iteration. The Jacobian matrix elements corresponding to the bus voltage amplitude are given by Equation (3):  
\[\left\{ \begin{array}{ll}\frac{v_{h_{\phi}}}{v^{r}_{slack\phi}}=\cos\left( \delta^{(k-1)}_{v_{h_{\phi}}}\right)&\frac{v_{h_{\phi}}}{v^{r}_{slack\phi}}= \sin\left(\delta^{(k-1)}_{v_{h_{\phi}}}\right)\\ \frac{v_{h_{\phi}}}{v^{r}_{l_{l\phi\phi}}}=-r_{l_{l\phi\phi}}\cos\left(\delta^ {(k-1)}_{v_{h_{\phi}}}\right)-x_{l_{l\phi\phi}}\sin\left(\delta^{(k-1)}_{v_{h_ {\phi}}}\right)&\frac{v_{h_{\phi}}}{v^{r}_{l_{l\phi\phi}}}=x_{l_{l\phi\phi}} \cos\left(\delta^{(k-1)}_{v_{h_{\phi}}}\right)-r_{l_{l\phi\phi}}\sin\left( \delta^{(k-1)}_{v_{h_{\phi}}}\right)\end{array} \right. \tag{3}\]  
The bus voltage phasor measurements can be represented by state variables as Equation (4):  
\[\dot{v}_{h_{\phi}}=v^{r}_{h_{\phi}}+jv^{x}_{h_{\phi}}=v^{r}_{slack\phi}-\sum \limits_{l\in\mathbb{T}_{h_{\phi}}}\sum\limits_{\phi=1}^{3}\left(r_{l_{l\phi }}i^{r}_{l_{l\phi\phi}}-x_{l_{l\phi}}i^{x}_{l_{l\phi\phi}}\right)+j\left(v^{x}_ {slack\phi}-\sum\limits_{l\in\mathbb{T}_{h_{\phi}}}\sum\limits_{\phi=1}^{3} \left(r_{l_{l\phi}}i^{x}_{l_{l\phi\phi}}+x_{l_{l\phi}}i^{r}_{l_{l\phi\phi}} \right)\right) \tag{4}\]  
and the Jacobian matrix elements corresponding to the bus voltage phasor are given by Equation (5):  
\[\left\{