## 4 Numerical Study
### Results and Discussion  
Using Algorithm (2), the following results can be calculated under a series of valley electricity selling prices \(Pr_{v}^{e}\): (1) the maximum profit of PIES; (2) the optimal peak electricity selling price; (3) the maximum profit of consumers; and (4) the optimal TR. In Figure 11, the left area of the dotted line \(l\) is the benefit-sharing zone, which means that the total profit generated by changing the peak and valley electricity selling price is shared by consumers and PIES. In the benefit-sharing zone, as the valley electricity selling price increases, the peak electricity selling price increases gradually in order to make the PIES gain the maximum profit, while the consumers profits decrease slowly. Additionally, from Figure 12, the total profit is constant in the benefit-sharing zone. The reason is that the left area of the dotted line \(l\) has no load shifted, as shown in Figure 13. In the benefit-sharing zone of Figure 12, point B is the optimal solution of the P-M model; that is, the daily profit of PIES is 337,861 yuan, the peak selling price is 0.842 yuan/kWh, the valley selling price is 0.56 yuan/kWh, and the consumers' electricity bill reduction is 63,421 yuan. In order to reflect the effectiveness of the proposed pricing model, the economic benefits are calculated without considering the impact of demand shift on the capacity of the PIES in the planning and design stage, based on the existing typical price leadership model. The results show that using the proposed model can reduce the initial investment of PIES by 10.69 million yuan and increase the total daily profit by 21,967 yuan, including PIES and consumers. Furthermore, from Figure 12, the right area of the dotted line \(l\) is the benefit-exclusive zone, which means that the total profit is taken by the PIES, and the consumers do not need to pay extra for the electricity compared with that associated with using the large power grid. The benefit-exclusive zone in Figure 12 corresponds to the load-shifted zone in Figure 13. In the benefit-exclusive zone, in order to reduce electricity bills, the consumers will adjust their energy habits (by DR) according to the time-of-use scheme formulated by the PIES. In fact, this adjustment (by DR) will neither reduce electricity bills nor increase electricity bills, as shown in Figure 12. Therefore, the adjustment of consumers in the benefit exclusive zone is meaningless for consumers, which is called the reactive demand response (RDR) in this paper. However, for the PIES, the RDR brings some profit growth, as shown in Figure 12. From the perspective of the total benefit for consumers and the PIES, the total profit in the RDR zone is lower than the total profit in the the benefit-sharing zone. In addition, if the valley price resides in the RDR zone for a long time, the consumers will find this trap based on actual experience and, thus, refuse to respond to the time-of-use scheme of PIES. Hence, the TR will drop to zero; that is, it will return to the no-load-shifted zone (benefit sharing zone), as shown in Figure 13. In summary, from the perspective of the long-term benefit, the PIES sharing its profits with consumers (the benefit sharing zone) is the ultimate choice for PIES and consumers; that is, point B is the optimal solution for the long-term situation.  
Additionally, from the perspective of P-M model optimization, point A in Figure 12 can be considered as the optimal solution for the short-term scenario. At point A, it is possible to ensure a reduction in the peak load and an increase in the valley load, simultaneously, corresponding to a PIES daily profit of 357,725 yuan, a peak selling price of 1.034 yuan/kWh, a valley selling price of 0.78 yuan/kWh, and a TR of 6.668%. The shifted load of point A is illustrated in Figure 14, where no new peak appears.  
Figure 12: Results of valley price, peak price, and profit.  
Figure 13: Results of valley price, peak price, and TR.