# A Review on Distribution System State Estimation Algorithms
## 4 Auxiliary Algorithms
### Observability  
The limited availability of real time data may challenge the observability of the distribution system. In order to provide the DSSE with enough data to converge and produce accurate results, pseudo-measurements are used [69]. Pseudo-measurements can be generated in a variety of ways. In literature, two main categories are distinguished, i.e., (i) probabilistic and statistical approaches and (ii) learning-based approaches [9; 13], as presented in Table 3.  
Following probabilistic and statistical approaches, the authors of [70; 71; 72] have used Gaussian Mixture Models (GMM) and Expectation Maximization (EM). The Gaussian mixture, \(\mathsf{f}(\mathsf{z}|\gamma)\), comprises a weighted, finite sum of Gaussian probability density functions, \(\mathsf{f}(\mathsf{z}|\mathsf{\mu}_{\mathsf{j}},\Sigma_{\mathsf{i}})\), as presented in (18), where \(\mathsf{M}_{\mathsf{c}}\) is the number of mixture components. EM is used in order to obtain the parameters of the mixture components, i.e., weights, means, and variances, as presented in [73]. \[\mathsf{f}(\mathsf{z}|\gamma)=\sum_{\mathsf{i}=1}^{\mathsf{M}_{\mathsf{c}}}w_{ \mathsf{i}}\mathsf{f}(\mathsf{z}|\mathsf{\mu}_{\mathsf{j}},\Sigma_{\mathsf{i}}) \tag{18}\]  
Another way to obtain pseudo-measurements that falls into the category of probabilistic and statistical approaches is presented in [72] and involves the calculation of correlation coefficients between measurements obtained from the main substation and non-monitored \begin{table}
\begin{tabular}{l l l} \hline \hline
**Method** & **Description** & **Advantages and Disadvantages** \\ \hline \multirow{4}{*}{Kalman filter} & FASE, providing recursive updates of the & Changes in normal operation can be tracked. \\  & FASE, providing recursive updates of the & Further, since FASE is based on forecasts by nature, \\  & estimated states with the use of Kalman filter and & missing measurements can be addressed with the \\  & dynamic models such as (17). & use of the forecasted states. However, this sort of \\  & & algorithms is more complex and not as widely \\  & & used as WLS. \\ \hline \multirow{2}{*}{EKF [49; 50; 51; 52]} & FASE, extended variation of Kalman filter with the & Widely used variation of Kalman filter for \\  & use of Taylor series for linearization of & linearization of nonlinear systems. Yet, not as \\  & nonlinear systems. & widely used as WLS. \\ \hline \multirow{2}{*}{UKF [52; 53]} & FASE, variation of Kalman filter with the use of UT & Variation of Kalman filter for linearization of \\  & for linearization of nonlinear systems. & Jacobian or Hessian matrices is necessary. Limited \\  & & use of this approach in literature. \\ \hline ANN [61; 62] & Purely data-driven DSSE approach including three & Modern, data-driven approach, where the \\  & layers, trained with data provided by the DSO. & sensitivity/accuracy regarding model parameters, \\ \hline DNN [63] & Data-driven approach, sub-category of ANN, with & initial conditions, etc. does not affect the result. \\  & more hidden layers. & However, an adequate database is required. \\ \hline \multirow{4}{*}{Physics-aware neural network [64; 65; 66]} & Data-driven approach where the neural network & Advanced, data-driven approach where the \\  & takes into account the structure of the & physics of the network are not overlooked, thus \\  & distribution network. & prohibiting the overfitting behavior of the \\  & & algorithm. Yet, in this case too, an adequate \\  & & database is required. \\ \hline \multirow{4}{*}{Hybrid [67; 68]} & Various possibilities, e.g., [67]: DNN-based DSSE, & Combination of the advantages of various sorts of \\  & supported by random forest algorithm for & algorithms at the cost of having a less simple \\ \cline{1-1}  & topology identification. WLAV is used to ensure & methodology. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._electrical quantities and the application of regression analysis.