# 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
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Objective** & **Formulation** & **Explanation** \\ \hline Power loss minimization [53; 54; 55] & \(\sum_{k=i}^{N}g_{ik}(V_{i}^{2}+V_{k}^{2}-2V_{i}V_{k}\cos\theta_{ik})\) & \(\theta_{ik}\)—phase angle \(i\)th and \(k\)th elements of conductance \(g_{ik}\) \\  & \(\sum_{(i)k\in B}r_{ik}l_{ik,t}\) and \(r_{ik}\)—square value of current and resistance of branch line from bus \(i\) to bus \(k\) \\ \hline OLTC tap operation minimization [56] & \(\sum_{m=1}^{M}s_{m,t}\) & M—number of discrete devices \\  & \(s_{m,t}\)—status of discrete device \(m\) at time \(t\) \\ \hline APC minimization [58] & \(\sum_{i=1}^{N}|P_{t,i}^{PV,curt}|\) & \(P_{t,i}^{PV,curt}\)—curtailed active power of PV at time \(t\) \\ \hline PV inverter loss minimization [38] & \(\sum_{p=1}^{3}\sum_{i\in\tau_{\tau}}(k_{i1}^{p}{S_{PVi}^{p}}^{2}+k_{i2}^{p}{S_{ PVi}^{p}}+k_{i3}^{p})\) & \(p_{i1}^{p},k_{i2}^{p},k_{i3}^{p}\)—coefficients of each inverter’s efficiency data \\ \hline Reactive power inj./abs. & \(|Q_{inj.}|/|Q_{abs.}|\) & \(Q_{inj.}\) and \(Q_{abs.}\)—injected/absorbed reactive power \\ \hline Cost of energy minimization [59] & \(\sum_{t\in T}a_{t}\)P\({}_{1,t}\Delta t\) &
\begin{tabular}{c} \(a_{t}\)—price of energy at \(t\)th time \\ \(P_{1,t}\)—active power imported from the external network at time \(t\) \\ \(\Delta t\)—duration of time intervals \\ \end{tabular} \\ \hline Security margin maximization [59] & \(\sum_{t\in T}(1-min|\frac{l_{t}-l_{t}}{l_{t}^{2}}|)\) & \(l_{t,t}\)—line current in \(l\)th line at time \(t\) \\  & \(l_{t}^{2}\)—ampacity of line current in \(l\)th line \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical expressions of the commonly used objectives in voltage optimization problems. control variables limit;
* limits for power generation (active and reactive power upper and lower limits);
* network operational limit determined in the network analysis (e.g., MVA limit). Voltage optimization needs to meet the following constraint requirements:  
1. Power flow equations given as Equations (7) and (8);
2. Voltage constraint \[V_{i}^{min}\leq V_{i}\leq V_{i}^{max}\] (9) where \(V_{i}^{min}\) and \(V_{i}^{max}\) are the lower and upper voltage limits. 3. PV active and reactive power constraint \[0\leq P_{PV,i}\leq P_{PV,av,j}\] (10) \[-\sqrt{S_{PV,i}^{2}-P_{PV,i}^{2}}\leq Q_{PV,i}\leq\sqrt{S_{PV,i}^{2}-P_{PV,i}^{ 2}}\] (11) where \(P_{PV,i}\), \(Q_{PV,i}\), and \(S_{PV,i}\) are active, reactive, and apparent powers at bus \(i\). \(P_{PV,av,j}\) is available active power at bus \(i\). 4. Line current (thermal) constraint \[I_{ik}^{min}\leq I_{ik}\leq I_{ik}^{max}\] (12) where \(I_{ik}^{min}\) and \(I_{ik}^{max}\) are the lower and upper limits of the line current between buses \(i\) and \(k\). 5. OLTC tap position constraint (if it is included) \[T_{i}^{min}\leq T_{i}\leq T_{i}^{max}\] (13) where \(T_{i}^{min}\) and \(T_{i}^{max}\) are the lower and upper positions of OLTC tap at bus \(i\). 6. Capacitor constraint (if it is included) \[Q_{Ci}^{min}\leq Q_{Ci}\leq Q_{Ci}^{max}\] (14) where \(Q_{Ci}^{min}\) and \(Q_{Ci}^{max}\) are the lower and upper limits of capacitor reactive power at bus \(i\). 7. Energy storage constraint (if it is included) \[SoC_{i,t}^{min}\leq SoC_{i,t}\leq SoC_{i,t}^{max}\] (15) where \(SoC_{i,t}^{min}\) and \(SoC_{i,t}^{max}\) are the lower and upper limits of the charge state of the storage system at time \(t\). The voltage optimization problem can be single-objective or multi-objective. OPF objectives and variables used in the review papers are categorized and summarized in Table 3.