# Calculation Method for Electricity Price and Rebate Level in Demand Response Programs
## 5 Numerical Simulation Results
### Numerical Simulation Results for Incentive-Based DRs  
By using Equations (25) and (26), the optimal rebate levels and the acceptable conditions in the incentive-based DR were calculated as:  
\[r_{t}^{\prime}=\frac{-A\big{(}d_{t}^{\prime 2}-d_{t}^{*2}\big{)}+(p^{*}-B)(d_{t}^ {\prime}-d_{t}^{*})}{|d_{t}^{\prime}-d_{t}^{*}|}, \tag{39}\]  
\[\frac{r_{t}}{\leq r_{t}}\leq r_{t}^{\prime}. \tag{40}\]  
where \(\underline{r_{t}}=\frac{p_{t}^{\prime}(d_{t}^{\prime}-d_{t}^{*})_{t}-X_{t}\log \frac{d_{t}^{\prime}+Z_{t}}{d_{t}^{\prime}+Z_{t}}}{\big{|}d_{t}^{\prime}-d_{t} ^{*}|}\).  
Tables 6 and 7 summarize the calculation results in the case that the power suppliers set the DR target as a 1% decrease (Scenario 1) or increase (Scenario 2) of the standard electricity consumption. Figures 11 and 12 illustrate changes in the utility functions of the power suppliers and the consumers. Changes in the social welfare functions were omitted in Figures 11 and 12, because they were the same with the changes of the consumers' utility functions in the authors' proposal.  
In Table 6, the calculated rebate levels were in the range of 0.95 JPY/kWh to 16.07 JPY/kWh. Meanwhile, the rebate levels in Table 7 were in the range of \(-2.61\) JPY/kWh to 7.70 JPY/kWh. As opposed to the results of price-based DRs, the time variation in Table 6 and Figure 11 had a similar trend to the electricity consumption. As shown in Table 7 and Figure 12, the optimal rebate levels were lower than \(\underline{r_{t}}\) from 18:00 to 22:00, and thus, the consumers' utility became negative during the periods. It indicated that the power suppliers could not ensure the resources for the DR, and the DR request was impossible  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{t}\) & \(\underline{r_{t}}\) (\(\mathbf{\times}\mathbf{10^{-2}}\) JPY/kWh) & \(\mathbf{r_{t}^{{}^{\prime}}}\) (JPY/kWh) & \(\mathbf{t}\) & \(\underline{r_{t}}\) (\(\mathbf{\times}\mathbf{10^{-2}}\) JPY/kWh) & \(\mathbf{r_{t}^{{}^{\prime}}}\) (JPY/kWh) \\ \hline
1 & 10.28 & 5.90 & 13 & 9.84 & 3.41 \\
2 & 10.28 & 6.00 & 14 & 9.97 & 4.14 \\
3 & 10.34 & 6.37 & 15 & 10.03 & 4.60 \\
4 & 10.52 & 7.70 & 16 & 9.96 & 4.02 \\
5 & 10.33 & 6.27 & 17 & 9.90 & 3.62 \\
6 & 9.74 & 2.22 & 18 & 9.30 & \(-0.88\) \\
7 & 9.55 & 0.80 & 19 & 9.10 & \(-2.61\) \\
8 & 9.68 & 1.94 & 20 & 9.13 & \(-2.33\) \\
9 & 9.89 & 3.55 & 21 & 9.12 & \(-2.36\) \\
10 & 10.07 & 4.75 & 22 & 9.24 & \(-1.53\) \\
11 & 10.04 & 4.77 & 23 & 9.61 & 1.19 \\
12 & 9.95 & 4.15 & 24 & 9.86 & 3.01 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Numerical simulation results of incentive-based DRs under scenario 2.  
Figure 11: Changes in each function by the 1% DR request in the day of the highest electricity consumption.  
Figure 12: Changes in each function by the 1% DR request in the day of the lowest electricity consumption. In this scenario, the power suppliers could not request the DR cooperation from 18:00 to 22:00.  
during the periods. In Figures 11 and 12, there were no burden on the power suppliers and no decrement in the social welfare excepting the period from 18:00 to 22:00 in scenario 2.  
As shown in Figures 13 and 14, the optimal rebate levels reduced in response to the decrease (Figure 13) or increase (Figure 14) of the target values of the DR. In contrast, the lower limits of the rebate levels were gradually raised. This is because the resources for the DR were limited in association with the utility function of the power suppliers.  
These results showed that the authors' proposal functioned properly in the incentive-based DR as well.