# 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
### The Improved Robust State Estimation with Normalized Residuals and Adaptive Thresholds  
In the traditional WLS, the measurement weight matrix contains the covariance of each measurement error. During the measuring process, due to the reliabilities of sensors or communication, the bad data with errors greater than \(\pm 3\sigma\) may appear. The traditional WLS does not distinguish normal data from bad data, which will cause a large estimation error, thereby affecting the subsequent decision-making credibility of the distribution system. However, considering the good convergence performance of WLS, this paper uses an improved robust state estimator based on WLS at the time of SCADA data acquisition. The concept of equivalent weight is used to integrate the residuals into measurement weights, and the Institute of Geodesy and Geophysics III (IGGIII) [37] robustness scheme is used to divide the measurement data into normal, suspicious, and harmful segments so that the corresponding weights are divided into the preservation zones, decentralized zones, and elimination zones. Compared with dividing the measurement data into normal measurements and harmful measurements, dividing the measurement data into three segments can make full use of measurement information. The weights are expressed as Equation (8):  
\[\overline{w}_{i}=\left\{ \begin{array}{*{20}{c}}{w_{i}},&{|r_{i}|\leq k_{0}}\\ {w_{i}}\frac{{k_{0}d_{i}^{2}}}{{|r_{i}|}},&{k_{0}<|r_{i}|\leq k_{1}}\\ {0},&{|r_{i}|>k_{1}}\\ \end{array} \right. \tag{8}\]  
where \(w_{i}\) is the \(i\)th diagonal elements of the measurement weight matrix \(\mathbf{W}\) in the traditional WLS, \(k_{0}\) and \(k_{1}\) are the robust thresholds, and \(d_{i}\) is the smooth factor, where \(d_{i}=(k_{1}-|r_{i}|)/(k_{1}-k_{0})\). When the value of \(r_{i}\) is smaller than \(k_{0}\), its weight maintains the original measurement weight \(w_{i}\). When the value of \(r_{i}\) is between \(k_{0}\) and \(k_{1}\), its weight decreases as \(r_{i}\) increases. When the value of \(r_{i}\) is larger than \(k_{1}\), its weight is \(0\). The validity of the estimation in the robust estimation depends largely on the rationality of the equivalent weight. The efficiency of robust estimation is different from statistics and robust thresholds, especially the rationality of decentralized zone weight threshold and elimination zone weight threshold. For all measurement data, if the thresholds of the robust scheme are the constant experience values, the measurements that originally belong to the normal segment may be defined as the harmful segment or suspicious segment, which will lose a certain amount of efficiency in the state estimation. For this problem, this paper refers to Reference [29] and gives a robust estimation method with variable thresholds. The key to the variable thresholds is that during the iteration process, the thresholds gradually change with the residuals, which can overcome the shortage of the traditional thresholds determined by the experience. The robust thresholds are designed in accordance with Equation (9):  
\[\left\{ \begin{array}{l}k_{0}=\min(k_{t1},k_{t2})\\ k_{1}=\max(k_{t1},k_{t2})\end{array} \right. \tag{9}\]  
where \(k_{t1}=\alpha r_{i}\) and \(k_{t2}=k_{t1}+(|r_{i}|-k_{t1})/3\). To make the robust thresholds adapt to the normal condition, \(\alpha\) is set to 1.438 [29; 38]. The residuals can be expressed as Equation (10):  
\[ \begin{array}{ll}\mathbf{r}&=\mathbf{z}-\mathbf{h}(\mathbf{x})-\left( \mathbf{H}\Delta\mathbf{x}\right)\\ &=\Delta\mathbf{z}-\mathbf{H}\left(\mathbf{H}^{\mathsf{T}}\overline{\mathbf{W} }\mathbf{H}\right)^{-1}\mathbf{H}^{\mathsf{T}}\overline{\mathbf{W}}\Delta \mathbf{z}\\ &=\left[\mathbf{I}-\mathbf{H}\left(\mathbf{H}^{\mathsf{T}}\overline{\mathbf{W} }\mathbf{H}\right)^{-1}\mathbf{H}^{\mathsf{T}}\overline{\mathbf{W}}\right] \Delta\mathbf{z}\\ &=\left(\mathbf{I}-\mathbf{J}\right)\Delta\mathbf{z}\end{array}