# A Review on Distribution System State Estimation Algorithms
## 2 DSSE Special Attributes  
Distribution systems have a number of special attributes that render their detailed representation more complex than transmission systems. Firstly, distribution systems are unbalanced by nature, as presented in Figure 3. This is a result of the unbalanced consumption as well as the existence of mutual impedences between the lines. It should be noted that in some parts of the grid, phases may be missing by design, according to the requirements and geographical location of the loads, thereby enhancing the occurring unbalances. Thus, the modelling of the components, including lines, transformers, loads, etc., as well as the power flow equations, need to be adjusted. For example, the impendence matrix of a distribution line is formulated as presented in (1), where Z is the impendence [16]. Furthermore, the active and reactive power flow between two nodes is formulated as presented in (2) and (3), respectively, and the power injection is formulated as presented in (4) and (5), where P\({}_{\text{i,j}}^{\text{ph}}\) and Q\({}_{\text{i,j}}^{\text{ph}}\) are the active and reactive power, respectively, flowing in phase ph between nodes i and j, P\({}_{\text{i}}^{\text{ph}}\) and Q\({}_{\text{i}}^{\text{ph}}\) are the active and reactive power injection, respectively, in phase ph at node i, V is the voltage magnitude and \(\delta\) is the voltage angle, G and B refer to the real and imaginary parts of the admittance matrix, respectively, I is the index referring to each of the three phases of the line, and n refers to the number of neighboring nodes [17; 18]. The complexity of these formulas lies in the separate calculation of each phase's values, as the occurring unbalances cannot be modelled via single-phase equivalent approaches [17; 18]. \[\text{Z}= \begin{bmatrix}\text{Z}_{\text{aa}}&\text{Z}_{\text{ab}}&\text{Z}_{ \text{ac}}\\ \text{Z}_{\text{ba}}&\text{Z}_{\text{bb}}&\text{Z}_{\text{bc}}\\ \text{Z}_{\text{ca}}&\text{Z}_{\text{cb}}&\text{Z}_{\text{cc}}\end{bmatrix} . \tag{1}\]  
\[\begin{split}\text{P}_{\text{i,j}}^{\text{ph}}=\text{V}_{\text{i} }^{\text{ph}}\sum\limits_{\text{l=1}}^{3}\text{V}_{\text{i}}^{\text{l}}[& \text{G}_{\text{ij}}^{\text{ph,l}}\cos\left(\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text\[ \begin{split} Q_{ij}^{ph}=-V_{i}^{ph}\sum\limits_{1=1}^{3}V_{i}^{l}[G_{ij }^{ph,l}\sin\left(\mathrm{g}_{i}^{ph}-\delta_{i}^{l}\right)-B_{ij}^{ph,l}\cos \left(\mathrm{g}_{i}^{ph}-\delta_{i}^{l}\right)]\\ -V_{i}^{ph}\sum\limits_{l=1}^{3}V_{j}^{l}[G_{ij}^{ph,l}\sin\left( \mathrm{g}_{i}^{ph}-\delta_{j}^{l}\right)-B_{ij}^{ph,l}\cos\left(\mathrm{g}_{i }^{ph}-\delta_{j}^{l}\right)]\end{split}  \tag{3}\]  
\[ \begin{split} P_{i}^{ph}=V_{i}^{ph}\sum\limits_{l=1}^{3}\sum \limits_{j=1}^{n}V_{j}^{l}[G_{ij}^{ph,l}\cos\left(\mathrm{g}_{i}^{ph}-\delta_{ j}^{l}\right)+B_{ij}^{ph,l}\sin\left(\mathrm{g}_{i}^{ph}-\delta_{j}^{l} \right)]\end{split}  \tag{4}\]  
\[ \begin{split} Q_{i}^{ph}=V_{i}^{ph}\sum\limits_{l=1}^{3}\sum \limits_{j=1}^{n}V_{j}^{l}[G_{ij}^{ph,l}\sin\left(\mathrm{g}_{i}^{ph}-\delta_{ j}^{l}\right)-B_{ij}^{ph,l}\cos\left(\mathrm{g}_{i}^{ph}-\delta_{j}^{l} \right)]\end{split}  \tag{5}\]  
Another issue that the distribution systems face is the limited availability of real-time data as a result of the sparse meter placement. Therefore, the accuracy or even the convergence of the DSSE might be negatively affected. This is an issue that can be tackled either with the placement of more instruments, which provide the DSSE with more accuracy but have a certain cost, or with the use of pseudo-measurements [19]. Pseudo-measurements are used to augment the available real measurements and are usually calculated using short-term forecasts or historical data. As a result, they are not as accurate as an actual measurement, however they require no further actions or cost from the side of the DSO [20]. A more detailed analysis regarding the methods to obtain pseudo-measurements is presented in sub-Section 4.1.