## 3 OPDG Problem Formulation
### Objective Functions
#### 3.1.3 Maximization of Voltage Stability Index  
As shown in Figure 6, the voltage stability index (VSI) of bus \(j=2,3,\ldots,n\) of an RDS is defined by  
\[VSI_{j}={\left|{V_{i}}\right|^{4}}-4*{\left({P_{j}X_{k}-Q_{j}R_{k}}\right)^{2}} -4*{\left({P_{j}R_{k}+Q_{j}X_{k}}\right)}*{\left|{V_{i}}\right|^{2}}. \tag{16}\]  
The VSI is usually calculated to evaluate whether the system is stable or not, where for stable systems, VSI should be more than zero for all buses along the feeder so that the system can avoid the voltage collapse. Thus, this index needs to be maximized [25]. In this case, the objective function is written as  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Function** & **Value** & **GA** & **PSO** & **DE** & **FA** & **ABC** & **SCA** & **QODELFA** \\ \hline \multirow{4}{*}{\(f_{1}(x)\)} & Min. & 0.00150000 & **7.9936E-15** & 5.5078E-05 & 3.5994E-05 & 0.001499 & 3.3672E-05 & 3.2346E-06 \\  & Max. & 1.50180000 & **1.5099E-14** & 9.1151E-05 & 5.0207E-05 & 0.005632 & 0.00136427 & 1.8930E-05 \\  & Mean & 0.49401000 & **1.1546E-14** & 7.2796E-05 & 4.3948E-05 & 0.003136 & 0.00052077 & 7.6498E-06 \\  & SD & 0.650915210 & **3.7449E-15** & 1.3982E-05 & 4.1056E-06 & 0.001235 & 0.00039639 & 4.6941E-06 \\ \hline \multirow{4}{*}{\(f_{2}(x)\)} & Min. & 1.67057E-07 & 1.855-13 & 1.89310E-06 & 4.48524E-08 & 0.023929 & 3.53044E-05 & **3.01981E-14** \\  & Max. & 0.007396679 & 0.058921 & 6.47944E-05 & 0.009864731 & 0.305490 & 0.260080741 & **0.022126734** \\  & Mean & 0.00740400 & 0.026802 & 1.42411E-05 & 0.002465738 & 0.088867 & 0.079647319 & **0.007140086** \\  & SD & 0.02338778 & 0.019734 & 1.857875E-05 & 0.004206263 & 0.085039 & 0.008410261 \\ \hline \multirow{4}{*}{\(f_{3}(x)\)} & Min. & 0.994959 & 0 & 2.01E-11 & 0.994959 & 0.200500 & 0 & **0** \\  & Max. & 5.969749 & 1.989918 & 1.155-09 & 5.969754 & 1.657200 & 9.32E-11 & **3.34E-13** \\  & Mean & 2.984877 & 0.596975 & 3.80E-10 & 3.283365 & 0.768219 & 9.40E-12 & **1.19E-13** \\  & SD & 1.483196 & 0.8390203 & 5.352-10 & 1.410988 & 0.537652 & 2.94E-11 & **1.23E-13** \\ \hline \multirow{4}{*}{\(f_{4}(x)\)} & Min. & 3.0262-09 & **3.282E-19** & 6.41E-10 & 9.21E-11 & 0.000138 & 1.065931 & 1.80E-12 \\  & Max. & 0.998176 & **3.91E-18** & 2.03E-09 & 1.32E-10 & 0.001058 & 1.279176 & 4.79E-10 \\  & Mean & 0.172109 & **2.65E-18** & 1.14E-09 & 1.19E-10 & 0.000406 & 1.180877 & 9.38E-11 \\  & SD & 0.335319 & **1.52E-18** & 4.79E-10 & 1.27E-11 & 0.000265 & 0.085863 & 1.45E-10 \\ \hline \multirow{4}{*}{\(f_{5}(x)\)} & Min. & 0.000224 & 3.50E-05 & 0.003839 & 8.36E-09 & 0.001526 & 0.0204023 & **8.50E-11** \\  & Max. & 0.0051547 & 0.000233 & 1.001516 & 0.031277 & 0.025481 & 1.154009 & **1.91E-09** \\  & Mean & 0.017779 & 0.000159 & 0.031891 & 0.005996 & 0.04177 & 0.523433 & **7.76E-10** \\  & SD & 0.018239 & 7.25E-05 & 0.030668 & 0.009571 & 0.008200 & 0.299470 & **5.70E-10** \\ \hline \multirow{4}{*}{\(f_{6}(x)\)} & Min. & 1.50E-05 & **5.76E-13** & 6.34E-05 & 7.05E-09 & 0.119930 & 0.001221 & 4.21E-06 \\  & Max. & 0.000791 & **3.05E-11** & 0.000117 & 1.08E-08 & 0.221020 & 0.727696 & 9.49E-05 \\  & Mean & 0.000243 & **8.88E-12** & 9.21E-05 & 9.15E-09 & 0.169986 & 0.887205 & 3.10E-05 \\  & SD & 0.000294 & **9.99E-12** & 1.57E-05 & 1.02E-09 & 0.032625 & 1.029825 & 2.83E-05 \\ \hline \multirow{4}{*}{\(f_{7}(x)\)} & Min. & 2.91E-08 & **2.22E-17** & 1.13E-07 & 7.16E-08 & 6.58E-05 & 1.14E-06 & 1.19E-09 \\  & Max. & 0.002864 & **5.91E-15** & 2.52E-07 & 1.09E-07 & 0.000191 & 0.001364 & 6.48E-08 \\  & Mean & 0.0030366 & **1.21E-15** & 1.84E-07 & 9.21E-08 & 0.000109 & 0.000251 & 1.87E-08 \\  & SD & 0.000900 & **1.93E-15** & 3.93E-08 & 1.36E-08 & 3.71E-05 & 0.000455 & 2.12E-08 \\ \hline \multirow{4}{*}{\(f_{8}(x)\)} & Min. & 0.000104 & 3.97E-09 & 0.001110 & 6.21E-08 & 0.002016 & 0.068476 & **1.36E-11** \\  & Max.