*4.1. Analytical Methods*

Classical analytical methods are used in multi-field optimization problems. In addition to the basic methods such as LP, QP [69], NLP, MINLP [76], and MILP, some papers deal with problems that reduce to the basic ones (see Table 4). In [65], SLP is developed to solve the optimization problem in real time. To obtain global optima, SQP has been developed in [38]. The iterative gradient projection method is implemented to specify VAR outputs for voltage optimization in [87] and additionally, active power outputs in [84]. A method for solving linear and nonlinear optimization problems was developed in [34,78]. The alternating direction method of multipliers (ADMM), which is one of the augmented Lagrangian-based methods, was developed as one of the most used methods for a network optimization problem. In ADMM, the optimization problem is decomposed into subproblems to deal with it. The subproblems are coordinated to seek the global optimal solution. The authors in [70,72,73] solve the voltage optimization problem using ADMM.

**Table 4.** Overview of formulation and analytical methods.


#### *4.2. Computational Intelligence Methods*

In recent years, computational intelligence methods have been increasingly used to solve voltage optimization problems. One of the main advantages of computational intelligence methods is that they do not require a mathematical formulation of the optimization problem. For the sake of unification and formality, the authors keep the formulation of the OPF problem in Table 5 even though it is not required for computational intelligence methods. The most commonly used methods are GA [37,57,62,83] and PSO [64,82,85,88]. Other computational intelligence methods that appear in the literature are the sine-cosine algorithm (SCA) [66], feasibility pump (FP) method [67], sparrow search algorithm (SSA) [74], pattern search algorithm [80], and grey wolf optimization (GWO) [81]. In some papers, hybrid methods are applied. In [79], a GA is applied to solve the day-ahead scheduling optimization problem in the first stage, while the pattern search algorithm (PSA) is used to solve the real-time optimization problem in the second stage. The branch-bound algorithm is combined with the IPM in [86] to solve the discrete problem. The combination of modified PSO and direct load flow (DLF) is used in [75]. DLF is used for power flow analysis and according to obtained data and PSO is used to evaluate network performance. In [77], the authors used both computational intelligence methods and analytical methods. In [55], five multi-objective evolution algorithms (MOEAs), named promising-region-based evolutionary many-objective algorithm (PREA), strength Pareto evolutionary algorithm 2 (SPEA 2), nondominated sorting genetic algorithm II (NSGA-II), nondominated sorting genetic algorithm III (NSGA-III), and two-phase framework (ToP), are used to determine the reactive power capacity of PVs and EVs. The results obtained by MOEAs are used to train a deep deconvolution neural network (DDNN) to solve the problem of voltage deviation and loss minimization. For inverter coordination, the authors in [113] use deep deterministic policy gradient (DDPG).


**Table 5.** Overview of formulation and computational intelligence methods.

#### *4.3. 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 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, multiobjective 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**) Singleobjective problem.


**Table 6.** Comparison of analytical and computational intelligence methods.

#### *4.4. Test Network Models*

To validate the efficiency of different solution methods for the voltage optimization problem, the authors use test network models that can be divided into a standard test model and a test model based on real-life examples. The most common test network models used in literature represent IEEE test network models which, depending on the number of buses, can be IEEE–13 bus, IEEE–15 bus, IEEE–33 bus, IEEE–34 bus, IEEE–37 bus, IEEE–69 bus, IEEE–123 bus, and IEEE–8500 bus. Almost all have radial topology. According to the processed problem, some authors modify standard test models. For instance, the standard test model [70] is modified according to balance. Real-life-based models represent urban residential feeders located in the US, China, Italy, Egypt, Australia, Ireland, and the UK. Unbalanced networks are mostly low voltage and belong to real-life models. In Table 7, test network models, their voltage level, and balance are summarized. According to the reviewed literature, more authors utilize standard test network models IEEE–33 bus, IEEE–69 bus, and IEEE–123 bus node due to their flexibility and robustness. One possible problem that can appear is the OPF application for unbalanced DNs. For instance, the authors in [72] reduce an unbalanced system to a balanced assuming that voltage magnitudes between phases are analogous and phase angles on nodes are not drastic. Therefore, an unbalance between phases is low, and almost balanced. For more, see [72].

**Table 7.** Overview of test network models.



**Table 7.** *Cont.*

#### **5. Conclusions**

This paper aims to systematize and categorize scientific papers that are dealing with the optimization of voltage in the DN using the reactive power management of PV inverters. Additionally, the papers are categorized according to the optimization problem formulation and applied solution methods. It can be observed that the original voltage optimization problem is nonlinear due to a quadratic objective function and nonlinear power flow equations. Additionally, some authors propose a mixed-integer nonlinear formulation due to integer variables such as the OLTC tap setting. To solve such complex optimization problems, some authors use analytical methods and some use computational ones. In this review paper, the authors tried to point out the advantages and shortcomings of both approaches without favoring one. When analytical methods are used, the compromise regarding the transformation of the original problem into a standard one is present but the analytical approach enables the straightforward method to find the optimum of a well-defined optimization problem (although special attention is required in order to determine whether a calculated optimum is local or global). On the other hand, computational intelligence methods can solve complex optimization problems without the transformation of the original formulation but they required higher computational performance as well as more computational time. One of the trends in applying computational intelligence methods is using well-known power flow calculation tools in order to feed the computational intelligence method with multiple power flow solutions. This principle is recognized as co-simulation.

The research potential of the reviewed field lies in the fact that more and more inverterbased sources are installed in distribution networks worldwide. The importance of voltage optimization is specifically stressed in microgrids where voltage supports depend mainly on the inverter-based source. Since the PV active power production depends on variable and stochastic sun irradiation, further research direction in the field of voltage optimization will strive to create an adequate probabilistic formulation of the OPF problem which is computationally more demanding since large numbers of possible scenarios need to be analyzed. Some probabilistic OPF solutions are already created for the transmission system environment but their replication in the distribution network (or microgrid) is not straightforward. It is hard to foresee which solution methodology (analytical or computational

intelligence) will show better performance in a probabilistic environment and there is still plenty of research challenges and gaps present for further research.

**Author Contributions:** Conceptualization, M.D. and K.F.; methodology, M.D. and K.F.; formal analysis, M.D., K.F. and D.T.; investigation, M.D.; writing—original draft preparation, M.D., K.F. and D.T.; writing—review and editing, D.T. and M.B.; funding acquisition, D.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by Croatian Science Foundation under the project "Prosumer-rich distribution power network" (project number: UIP-2020-02-5796).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The work of doctoral student Marina Dubravac has been fully supported/supported in part by the "Young researchers' career development project—training of doctoral students" (DOK-2021-02-1257) of the Croatian Science Foundation. This work was founded by the Croatian Science Foundation under the project "Prosumer-rich distribution power network" (project number: UIP-2020-02-5796).

**Conflicts of Interest:** The authors declare no conflict of interest.
