# Optimal Design of a Residential Photovoltaic Renewable System in South Korea
## 4 Computational Results  
The residential PV design model is optimized with various practical data, as proposed in the above sections. Figures 5 and 6 show the total PV design cost, as specified in Equation (1), with different PV sizes (\(0\leq S^{PV}\leq 3\) kW, by 0.2 kW) and tilt angles (\(15^{\circ}\leq A^{PV}\leq 60^{\circ}\), by \(2.5^{\circ}\)). In this resolution, 639,919 KRW, with a PV size of 1.2 kW and a PV tilt angle of \(27.5^{\circ}\) is the minimal design solution for the system.  
When we narrowed down the PV size (\(0.95\leq S^{PV}\leq 1.3\) kW) and the tilt angle (\(26.4^{\circ}\leq A^{PV}\leq 28.6^{\circ}\)), and then divided them into finer intervals (0.05 kW for the PV size and 0.1\({}^{\circ}\) for the tilt angle), Figures 7 and 8 were obtained. At this resolution, we obtained a better solution (639,901 KRW with PV size of 1.2 kW and PV tilt angle of \(28.3^{\circ}\)-\(28.4^{\circ}\)) than that with coarse resolution (639,919 KRW with PV size of 1.2 kW and PV tilt angle of \(27.5^{\circ}\)).  
Figure 4: Location of the solar data source (Incheon) from Google Maps.  
Figure 5: The total PV design cost.  
Figure 6: Map of the total PV design cost.  
Figure 8: Map of the total PV design cost at a finer resolution.  
Figure 7: The total PV design cost at a finer resolution.  
In order to find a global optimal solution, we applied a genetic algorithm [10] as a global search meta-heuristic algorithm [11] to this PV design problem. When this meta-heuristic optimization algorithm was applied, we obtained an even better solution (639,824 KRW) at different solution spot (a PV size of 1.1904 kW and a PV tilt angle of 26.7013\({}^{\circ}\)) than those at the previous two resolutions. This phenomenon means that there exist local optimal solutions within the solution space.  
Here, it should be noted that this PV design problem cannot be solved by using calculus-based approaches, because the monthly electric bill, as a part of the objective function, possesses stepped piecewise linearity, as shown in Figure 1. At certain stepped points such as 100, 200, 300, 400, and 500 kW, this cost function is not differentiable. Although a previous research [6] tackled this problem with a gradient-based approach, named SQP (Sequential Quadratic Programming), it had to sacrifice the accuracy of the objective function by smoothing out this step function with polynomial curve fitting.  
In order to compare our approach by using a genetic algorithm with the old approach, using SQP, we first performed a polynomial regression based on the electrical rate data in Table 1, and obtained the following second-order polynomial function:  
\[C_{Electric}^{m}=0.5632x^{2}-76.207x+7612.3\text{with}R^{2}=0.9947 \tag{6}\]  
Then, based on Equation (6), the SQP optimization was performed, obtaining an optimal cost of 617,529 KRW, with a PV size of 1.3935 and a PV tilt angle of 29.7229\({}^{\circ}\). It appears that the solution (617,529 KRW) from SQP was better than that (639,824 KRW) of our approach. However, when we verified the SQP solution with a real cost table (Table 1), we obtained 644,252 KRW, which is worse than our solution.