# A Review on Distribution System State Estimation Algorithms
## 3 DSSE Fundamentals and Main Algorithms
### DSSE Algorithms
#### 3.2.1 Conventional, Model-Based Algorithms  
The most common approach regarding the core of DSSE is the Weighted Least Square (WLS) algorithm. This is a model-based solution, denoting that the details of the distribution \begin{table}
\begin{tabular}{l l l} \hline \hline  & **NV-DSSE** & **BC-DSSE** \\ \hline Configurations & All (radial, meshed, etc.) [23] & Mostly limited to radial \\ \hline Coordinates & Polar or rectangular [17,23,24,25,26,27] & Mostly rectangular [27,28,29,30] \\ \hline Implementation & More complex & Relatively easy [9] \\ \hline Sensitivity (weight variations, etc.) [11] & Higher (might affect convergence) & Lower \\ \hline Computational time [9,11] & Higher & Lower \\ \hline Compatibility with transmission system state estimation & Yes [22] & No \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between NV-DSSE and BC-DSSE [9,11,12,13,22,23,24,25,26,27,28,29,30]. network need to be known to the operator beforehand. The purpose of WLS is to minimize the weighted residuals between the estimated and measured values, subjected to the distribution network's constraints. Provided that the residual vector \(\mathbf{r}\) is calculated with (10), where \(\mathbf{z}\) is the measurement vector, \(\mathbf{x}\) is the state vector, and \(\mathbf{h}(\mathbf{x})\) is the measurement function calculated upon \(\mathbf{x}\), the objective function of the WLS is presented in (11). In the objective function, \(\mathbf{W}\) is the weight matrix that denotes the operator's confidence in the measured data. It should be noted that the size of \(\mathbf{z}\) is (m \(\times\) 1) where m refers to the number of measurements, the size of \(\mathbf{x}\) is (n \(\times\) 1) where n refers to the number of states, and the size of \(\mathbf{W}\) is (m \(\times\) m). Obviously, m can only be lower than (or equal to) n.  
\[\mathbf{r}=\mathbf{z}\ -\ \mathbf{h}(\mathbf{x}) \tag{10}\]  
\[\text{minF}=\mathbf{r}^{\text{T}}\mathbf{W}\mathbf{r} \tag{11}\]  
The aforementioned method is characterized by its simplicity and wide use, e.g., implementation in the work of [31; 32; 33; 34; 35; 36]. However, it is known to be sensitive to bad data, noise of measured values, etc. For this purpose, variations of WLS have been developed, improving its robustness at the cost of simplicity and computational resources. For example, the Least Absolute Value (LAV) constitutes a common alternative. According to this approach, the purpose is to minimize the sum of absolute values of measurement residuals, as presented in (12). Detailed implementation can be found in the work of [37; 38]. This approach can be further enhanced by adding weights to each residual. This is also known as the Weighted Least Absolute Value (WLAV). The respective objective function is presented in (13), where \(\text{w}_{\text{i}}\) denotes the weight of the i-th residual. Research regarding WLAV can be found in [39]. \[\text{minF}=\sum_{\text{i=1}}^{\text{m}}|\mathbf{r}_{\text{i}}| \tag{12}\]  
\[\text{minF}=\sum_{\text{i=1}}^{\text{m}}\text{w}_{\text{i}}|\mathbf{r}_{\text {i}}| \tag{13}\]  
Another advanced and robust model-based approach is the Least Trimmed Squares (LTS). In this case, the squared values of the residuals are calculated and ordered from the lowest to the highest. The objective function aims to select a total number, \(\text{u}\), of lowest values and minimize their sum, as presented in (14). Related work can be found in [40]. \[\text{minF}=\sum_{\text{i=1}}^{\text{u}}\mathbf{r}_{\text{i}}^{2} \tag{14}\]  
Furthermore, the Least Median of Squares (LMS) method has been used for DSSE by the authors of [41]. In this case, the aim of the objective function (15) is to minimize the median-instead of the sum-of the squared, weighted residuals, where \(\mathbf{r}_{\mathbf{w}}^{2}\) is the matrix of squared, weighted residuals.