## 4 Simulation Results
### IMGO for PV Source Penetration and Shunt Capacitors in Distribution Systems  
Optimizing the placement and sizing of PV distributed generation units and shunt capacitors in power distribution systems can effectively reduce power and energy losses. This approach enhances system efficiency, lowers operational costs, and mitigates adverse environmental effects. The PV dispersed power generation and capacitor model can be \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Best Cost** \\ \hline SSA [70] & 6059.7144371950 \\ ADEA [69] & 6059.71433504 \\ PSO-DE [69] & 6059.714335 \\ MFO [70] & 6059.7143350521 \\ Improved accelerator PSO [69] & 6059.7143 \\ Multi-level cross entropy [69] & 6059.7143 \\ Eco geography-based optimization (EBO) [70] & 6092.5671060055 \\ MGO & 5885.558392 \\ Proposed IMGO & 5885.332939 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparisons of the best cost between IMGO, MGO, and the other reported algorithms for the PVD problem. Figure 12: Heatmap of the obtained cost using MGO and IMGO for the PVD problem. optimized to lower energy losses, as demonstrated by Equation (21), and raise the system's total energy efficiency. In Equation (21), the daily losses of energy are denoted by \(Energy_{Loss}\), \(h\) denotes one hour, the number of hours per day is denoted by \(N_{h}\), and \(Loss_{h}\) represents the total amount of power lost in one hour in the system. The hourly losses are depicted in Equation (22), where \(N_{LD}\) refers to the number of the existing branches in the system, \(I_{LD}\) denotes the current flowing through each line (\(LD\)), and \(R_{LD}\) represents the resistance of each branch (\(LD\)). \[Energy_{Loss}=\sum_{h=1}^{N_{h}}Loss_{h} \tag{21}\]  
\[Loss_{h}=\sum_{LD=1}^{N_{LD}}I_{LD}^{2}\times R_{LD} \tag{22}\]  
The following two types of design variables are defined, the locations and capacities of PV sources and capacitors. The PV sources can be installed and connected at any bus within the distribution network, with their capacities restricted as specified in Equations (23) and (24). These equations establish the limitations of the PV systems, where \(PV\_{bus}j\) represents the maximum number of PV units allowed at each bus, \(N_{PV}\) denotes the total number of PV units to be integrated, \(P_{PV,limit}\) is the upper capacity limit, and \(P_{PV,j}\) signifies the candidate capacity of each PV unit under consideration. Additionally, the positioning and sizing of capacitors are subject to constraints defined in Equations (25) and (26). These equations specify the limits for capacitor placement, where \(CP_{k}\) designates each potential bus location for capacitor installation. \(N_{CP}\) represents the maximum number of capacitors that can be considered, while \(Q_{CP,k}\) and \(Q_{CP,limit}\) indicate the candidate size and the upper limit of the capacitors at each designated bus \(CP_{k}\), respectively. In addition, it is essential to regulate the voltage levels at each distribution node to ensure system stability. This requirement can be represented mathematically, as shown in Equation (27), where \(V_{Bs,min}\) and \(V_{Bs,max}\) denote the minimum and maximum allowable voltage limits for the bus nodes. Alongside voltage control, maintaining security constraints on power flow is also crucial. Equation (28) models this condition [78], where \(I_{LD,limit}\) reflects the thermal capacity of each distribution line, ensuring current flow stays within safe operational boundaries. Additionally, it is crucial to ensure that the balance constraints for active and reactive power flows are met, as outlined in Equations (29) and (30). In these equations, \(P_{Grid,h}\) and \(Q_{Grid,h}\) denote the active and reactive power provided by the grid substation on an hourly basis, respectively. Similarly, \(Pd_{k,h}\) and \(Qd_{k,h}\) represent the active and reactive power demands at each endpoint \(k\) for each hour.