*5.1. Evolution Result in Groups N and M*

Based on the above simulation parameters, the population evolution result and the Nash equilibrium was obtained by performing Algorithm 1. Note that Algorithm 1 is set to 70 iterations and each iteration means one week. Accordingly, Figure 3 is the convergence process of the population in groups *N* and *M*. It depicts that the population in each group gradually converged to the evolutionary equilibrium in 42 weeks. Specifically, in the first 20 weeks, group *N*'s population increased rapidly from four communities to 36 communities, while between the 21st week and 42nd week, group *N*'s population increased slowly from 37 communities to 44 communities. Note that, since the average number of communities in a group is deduced from Equation (45), in which the probability Pr*<sup>N</sup>* and Pr*<sup>M</sup>* are non-integral values, the group's population in Figure 2 was also a non-integral value. But the group's population was rounded as an integral value when searching the Nash equilibrium in the day-ahead bidding market.

**Figure 3.** Convergence process of population in groups *N* and *M.*

Figure 4 is the convergence process of the total bidding amount in group *N* in all dispatching time slots *T*. From the figure, one can see that, at the beginning of the DR project, the total bidding amount of four communities was only about 20 MWh, while the bidding amount reached 80 MWh after performing the DR project for about 10 weeks. The main reason for such rapid growth was due to the increase of the population in group *N*. However, the bidding amount tends to reach saturation after 10 weeks. It was because the bidding price was reduced gradually with the further increase of group *N*'s population and the bidding amount of each community was declined as well. Moreover, the comprehensive income of group *N* in all dispatching time slots *T* is shown in Figure 5. It demonstrates that the total income of group *N* increased firstly and then decreased dramatically in 10 weeks, and finally converged to the fixed value in the weeks after 10 weeks. The reason for such a trend was mainly related with the variation of economic compensation and utility. In the first five weeks, the bidding amount in the market was not very large, then the bidding price was relatively high and the economic compensation increased dramatically with the increase of group *N*'s population. Therefore, the income in the first five weeks was mainly dependent on the economic compensation. However, with the increase of the bidding amount, the utility that the community lost in DR increased gradually but the economic compensation increased slowly. Therefore, the comprehensive income in the 6th weeks and 10th weeks decreased rapidly. Since the bidding amount tends to be stable after 10 weeks, the total income of group *N* was also convergent.

From the above evolution result, it is clear that most of the residential communities will be attracted to participate in DR with the implementation of the DR project. However, not all communities in the set *I* will be involved in DR. When residential communities in the bidding market become saturated, the average economic compensation of each community will be decreased to the minimal value. Consequently, some members in group *N* will be unwilling to participate in DR and switch from group *N* to group *M* with a high probability. At last, the population of groups *N* and *M* will reach the dynamic balance.

**Figure 4.** Convergence process of total bidding amount in group *N*.

**Figure 5.** Convergence process of total comprehensive income in group *N*.
