*5.2. Equilibrium Result for Group N*

This section shows the Nash equilibrium among the residential communities in group *N*. In our proposed scenario, at the beginning of each week, the community will choose to participate in DR or not, and then, communities participating in DR have to bid for the dispatching amount in the day-ahead market. Therefore, in each week exists one equilibrium solution and there are altogether 70 equilibrium solutions, considering the DR project was conducted in 70 weeks. For the limitation of paper space, we took the equilibrium solutions of two special weeks (i.e., 1st week and 70th week) as examples to illustrate the optimal bidding strategy.

Figures 6 and 7 are the optimal bidding amounts of each community in the 1st week and 70th week, respectively. In which, communities 1–4 are the communities who were willing to participate in DR at the beginning of the DR project. From the figures, we can see that the bidding amount in the 1st week was much more than that in the 70th week in all 12 time slots. Additionally, the bidding strategies of four communities in each time slot were all different in the 1st week, while four communities had the same bidding strategies in the 70th week. The main reason was that, at the beginning of the DR project, the bidding market needed a large amount of DR resources, hence the community with more DR resources will compete for more dispatching amount. However, when the DR project was conducted for 70 weeks, the DR resource in the market was saturated and the bidding price was also saturated with the lowest value. Consequently, the bidding amount of the community was reduced to the minimal value, even for those communities with abundant DR resources. Actually, in this case study, the bidding amount of each community was only related with bidding price parameters and utility model parameters after the bidding market reached saturation. The bidding amount changed with the change of bidding price parameters. For example, we see that from Figure 7, the values of bidding price parameters were different between *t* = 1–5 and *t* = 6–10, then the bidding amount between *t* = 1–5 and *t* = 6–10 were also different. Concretely, the average bidding amount of all communities in group *N* is presented in Table 1. From the table, it shows that the maximal bidding amount was 0.739 MWh between 19:45–20:00 in the 1st week, while the maximal bidding amount was only 0.203 MWh between 19:15–20:30 in the 70th week.

**Figure 6.** Optimal bidding amount of each community in the 1st week.

**Figure 7.** Optimal bidding amount of each community in the 70th week.


**Table 1.** Average bidding amount of all communities in group *N.*

In addition, the bidding price in the 1st week and 70th week is shown in Figure 8. It depicts that the bidding price in the 1st week was much higher than the price in 70th week. For example, the highest bidding price in the 1st week was 675 dollars/MWh during 20:00–20:15, while the highest price in the 70th week was only 203 dollars/MWh. Therefore, from the aspect of the dispatching center

of the smart grid, the designed bidding price mechanism can effectively reduce the dispatching cost of the grid. Of course, the dispatching center of the smart grid can also control the DR resource amount in the market by regulating price parameters. The next section analyzes the influence of bidding price parameters on DR.

**Figure 8.** Bidding price in the 1st week and 70th week.
