*3.2. Objectives and Variables—Discussion*

Figure 7 represents objectives quantitatively. Almost all objectives include VD. In multi-objective problems, VD is most combined with losses, however, many other objectives also appear.

If OPF is regarded as a part of the distribution energy management system (DEMS), the dominance of the multi-objective formulation of OPFs is logical. DSO tries to reach the optimal operation point regarding several objectives and the most commonly used ones are loss minimization together with voltage deviation minimization. Additionally, active power curtailment (APC) minimization is frequently a combined objective with the minimization of voltage deviations. Other objectives are rare and they are used only in a few papers.

The variables are presented quantitatively in Figure 8.

A similar conclusion can be made regarding optimization variables. DSO tries to utilize all the available controls such as OLTC tap settings, CB reactive power, and ESS variables. Some of the variables are continuous but some are discrete (such as OLTC tap settings), which will affect the formulation of the OPF problem (the appearance of integer variables) and largely the choice of the solution method.

**Figure 8.** Quantitative presentation of the variables.

## **4. Voltage Optimization in PV-Rich Distribution Networks—Formulation and Solution Methods**

The complexity of the OPF depends on the power flow formulation approach. If the original AC power flow equations are used for OPF formulation, an optimization problem is nonlinear and hard to solve. Thus, many researchers try to simplify OPF formulation in order to obtain a linear or quadratic optimization problem that is easier to solve. According to [60], the OPF formulation can be classified into:


In early papers, continuous NLP formulation is used. All discrete variables are approximated as continuous for simplicity. This formulation includes nonlinear objectives and constraints. The LP formulation uses the DC power flow approach, i.e., both the objectives and the constraints are linear. Due to its simplicity, robustness, speed, and welldeveloped solution methods, it is an attractive OPF formulation, especially for industry [51]. However, due to the modeling assumptions, LP is not adequate for problems such as minimizing power losses, and a global optimum cannot be guaranteed. QP is a special case of NLP with quadratic objective and linear constraints and represents an alternative to LP. The inclusion of discrete variables (transformer tap settings, shunt capacitor settings, etc.) in NLP results in MINLP being the most realistic and accurate formulation of OPF, but also the most complex and difficult to solve. However, there is a trade-off between the system description and the tractability of the problem. One way is to linearize and apply MILP. For more details, see [60,92].

The OPF formulation determines which solution method is used. Figure 9 presents the mathematical formulation and solution methods. For each solution method in Figure 9, a scientific paper in which it is applied is listed. The basic categorization of OPF solution methods is divided into analytical methods and computational intelligence methods. The most commonly used analytical methods for linear OPF are the well-developed simplex methods [93], sequential linear programming (SLP) [94], and interior point methods (IPMs) [95]. In the first period, analytical iterative methods were applied to the NLP OPF. They were Newton-based methods [96]; gradient methods—reduced gradient method (RG) [97], conjugate gradient method (CG) [98], and generalized reduced gradient method (GRG) [99]; IPMs [100]; sequential quadratic programming (SQP) [101]. Recently, computational intelligence methods have been applied to solving OPF problems. Computational intelligence methods have been developed to overcome the weak capabilities of analytical methods for solving global optimization [102]. Although computational intelligence methods do not require a precise mathematical formulation of the OPF problem, the authors include them in the group of solution methods that can solve nonlinear and integer formulations of the OPF since they can take into account nonlinearities in the original problem. Computational intelligence methods include artificial neural networks (ANNs) [103], genetic algorithms (GAs) [104], particle swarm optimization (PSO) [105,106], ant colony optimization (ACO) [107,108], bacterial foraging algorithm (BFA) [109], simulated annealing (SA) [110], tabu search (TS) [111], and fuzzy logic (FL) [112].

**Figure 9.** Mathematical formulation and solution methods used in OPF solving.
