# Voltage Optimization in PV-Rich Distribution Networks--A Review
## 4 Voltage Optimization in PV-Rich Distribution Networks--Formulation and Solution Methods
### Formulation and Solution Methods--Discussion  
Analytical solution methods require a strictly mathematical formulation of the OPF problem, which can then be solved by an appropriate analytical method. Since there are a few effective analytical algorithms for solving nonlinear problems (especially with integer variables), most of the papers in which analytical methods are used transform the  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Formulation**} & \multicolumn{2}{c}{**Computational Intelligence**} & \multirow{2}{*}{**Reference**} \\  & **Methods** & \\ \hline NLP & GA & [37; 57] \\ MINLP & GA & [62] \\ NLP & NDSGA II & [83] \\ NLP & PSO & [68; 82] \\ MINLP & PSO & [64; 85; 88; 114] \\ NLP & SCA & [66] \\ MINLP & C\&CG algorithm & [53] \\ NLP & SSA & [74] \\ MINLP & GWO & [81] \\ MINLP & Modified PSO, DLF algorithm & [75] \\ MINLP & MOPSO, IPM & [77] \\ MINLP & GA,PSA & [79] \\ NLP & PREA, SPEA2, NSGA-II, & [55] \\ NSGA-III, ToP, DDNN & & \\ MINLP & FP & [67] \\ MINLP & PSA & [80] \\ MINLP & NSGA-III & [89] \\ MINLP & \(\epsilon\)-constrained method and FL & [90] \\ MINLP & DDPG & [113] \\ MINLP & ANN & [114] \\ \hline \hline \end{tabular}
\end{table}
Table 5: Overview of formulation and computational intelligence methods.  
original NLP (or MINLP) into some of the more convenient forms--usually QP or MIQP. Transformations into a linear form (LP or MILP) are very rare due to the nature of the problem, i.e., the quadratic function of voltage deviation and poor performance of DC power flows in the environment of the DN (ratio \(R\)/\(X\) is not as small as in the TN).  
According to Table 5, the popularity of computational intelligence methods can be observed. According to Figure 10, a decision about which solution method would be used depends on the objective and mathematical formulation. To briefly address Table 3, multi-objective optimization problems dominate. In single-objective problems, analytical solution methods prevail. Analytical methods require that a multi-objective (usually known as Pareto optimization) problem transforms (scalarizes) into a single-objective using weighting coefficients, which is not a straightforward procedure. Computational intelligence methods are most used for multi-objective problems. A comparison of analytical and computational intelligence methods is presented in Table 6.  
Analytical methods are well-developed and applicable in systems where the requirements of modeling accuracy are low. Analytical methods are able to straightforwardly find an optimal solution but there is no guarantee that the optimum is global. If multiple local optima exist, global optima cannot be guaranteed and the analytical method can stuck in local optima. To apply the analytical method, it is necessary to perform a transformation of the original problem to a level that it can solve. This is where the problem of trade-off comes in. On the one hand, there is an accurate real-life system description and, on the other hand, there is an applicable solution method. Some shortcomings of analytical methods are solved by computational intelligence methods. These methods do not depend on mathematical formulation because they required only parameters that can be calculated separately (for example solution of the power flows). Compared to analytical methods, a hard computational effort is required and there is no guarantee of finding an optimal solution thus some expert knowledge of the system is needed. In recent years, computational intelligence methods are used in co-simulation with proven power flow tools DIgSILENT PowerFactory [41], DLF [75], OpenDSS [115], etc. This approach simplified the application of computational intelligence methods for large-scale DNs.  
Figure 10: Solution methods for different objectives and formulations. (**a**) Multi-objective problem and continuous formulation. (**b**) Multi-objective problem including integer variables. (**c**) Single-objective problem.