# Voltage Optimization in PV-Rich Distribution Networks--A Review
## 3 Voltage Optimization in PV-Rich Distribution Networks--Objectives and Variables
### General Formulation--Objectives and Variables  
The OPF problem can be described as minimizing the objective function while taking equality and inequality constraints into account [48]:  
\[minF(\textbf{x},\textbf{u})=0 \tag{2}\]  
\[g(\textbf{x},\textbf{u})=0 \tag{3}\]  
\[h(\textbf{x},\textbf{u})\leq 0 \tag{4}\]  
where \(F(\textbf{x},\textbf{u})\) represents the objective function and \(g(\textbf{x},\textbf{u})\) represents nonlinear equality constraints i.e., power flow equations, \(h(\textbf{x},\textbf{u})\) represents nonlinear inequality constraints. The vectors **x** and **u** present state variables, and control variables, respectively. In [48], the generally used objectives for OPF formulation are given. It should be noted that the objectives and constraints must be modeled accurately to obtain a satisfactory solution. Scientific papers are included in this review if at least one of the objectives is voltage optimization and if one of the optimization variables is PV inverter reactive power injection. Furthermore, the voltage optimization problem is mostly described as the objective of voltage deviation (VD) minimization, i.e., maintaining voltages within boundaries determined by grid codes. The general mathematical expression for VD is:  
\[V_{dev}=\sum_{i\in N}{(V_{i}-V^{nom})^{2}} \tag{5}\]  
where:  
\(V_{dev}\)--voltage deviation;  
\(V_{i}\)--voltage at bus \(i\);  
\(V^{nom}\)--nominal voltage. Another objective that appears is related to the voltage unbalance, commonly presented as the voltage unbalance factor (VUF). The definition of VUF is given in [52] as the ratio of negative \(V^{-}_{sequence}\) and positive \(V^{+}_{sequence}\) voltage sequences and is most often expressed in percentages:  
\[VUF=\frac{V^{-}_{sequence}}{V^{+}_{sequence}}. \tag{6}\]  
In addition to voltage optimization, the following objectives also appear: (i) power loss minimization [53; 54; 55], (ii) on load tap changer (OLTC) switching operation minimization [56], (iii) PV cost minimization [38], (iv) reactive power injection/absorption minimization [57], (v) active power curtailment (APC) minimization [58], (vi) cost of purchased energy minimization [59], (vii) peak shaving minimization [59], and (viii) security margin index (SMI) minimization [59]. The mathematical expressions of the commonly used objectives are given in Table 2.  
In power systems, the conventional power flow is both nonlinear and nonconvex and commonly solved by the Newton-Raphson iterative method. In constrained OPF applications, equality constraints incorporate conventional power flow equations and other constraints to ensure balance. A detailed version of the power flow is named AC power flow [60]. AC power flow as a constraint in OPF is most often formulated in the polar form [60]:  
\[P_{i}=\sum_{k=i}^{N}|V_{i}||V_{k}||Y_{ik}|\cos(\delta_{i}-\delta_{k}-\theta_{ ik}) \tag{7}\]  
\[Q_{i}=\sum_{k=i}^{N}|V_{i}||V_{k}||Y_{ik}|\sin(\delta_{i}-\delta_{k}-\theta_{ ik}) \tag{8}\]  
where:  
\(P_{i}\)--active power at bus \(i\);  
\(Q_{i}\)--reactive power at bus \(i\);  
\(V_{i}\)--voltage magnitude at bus \(i\);  
\(V_{k}\)--voltage magnitude at bus \(k\);  
\(Y_{k}\)--\(ik\)th element of bus admittance matrix \(\mathbf{Y}_{bus}\);  
\(\delta_{k}\)--voltage phase angle at \(i\)th bus;  
\(\delta_{k}\)--phase angle at \(k\)th bus;  
\(\theta_{ik}\)--phase angle of \(ik\)th element of bus admittance matrix \(\mathbf{Y}_{bus}\). Besides AC power flow, the authors use two other formulation approaches: decoupled AC power flow [49] and DC power flow [50]. In decoupled AC power flow, active and reactive powers are decoupled as a function of voltage angle and voltage magnitude, respectively. Assumptions made for the DC power flow formulation include purely imaginary elements of Y and a small difference between two voltage angles of two adjacent busses. Various inequality constraints are given in [48; 61]: