*4.1. Transition Probability Model*

To describe the population evolution briefly, we define the state of residential community *i* ∈ *I* at week *h* as *Si*(*h*), in which *Si*(*h*) = 0 represents community *i* belonging to group *M* is unwilling to participate in DR, *Si*(*h*) = 1 represents community *i* belonging to group *N* adopts the DR project. Then, the transition probability can be calculated following four cases.

(1) Case 1: *Si*(*h*) = 0 → *Si*(*h* + 1) = 1

When the state of community *i* is *Si*(*h*) = 0 at week *h*, then it may participate in DR if some of its neighboring communities have adopted it. That is, community *i* can be infected by each neighboring community with probability β, where 0 ≤ β ≤ 1. Such probability shows the effect of social networking or mutual imitation among DR communities. Note that the probability β is a networking-related parameter. Therefore, the greater the number of neighboring communities who participate in DR, the higher the probability community *i* will adopt the DR project. Consequently, the corresponding transition probability can be expressed as follows:

$$P\left(S\_i(h+1) = 1 \middle| S\_i(h) = 0\right) = 1 - (1 - \beta)^{N\_i^{\hat{h}}} \tag{23}$$

where *N<sup>h</sup> <sup>i</sup>* is the total number of neighboring communities in group *N* that have connection to the community *i*; 1 − β represents the probability that community *i* is not infected by one neighboring community; (1 − β) *Nh <sup>i</sup>* represents the probability that community *i* is not infected by *N<sup>h</sup> i* neighboring communities.

(2) Case 2: *Si*(*h*) = 0 → *Si*(*h* + 1) = 0

Since community *i* with *Si*(*h*) = 0 can only choose *Si*(*h* + 1) = 1 or *Si*(*h* + 1) = 0, the probability that community *i* still remains in group *M* can be obtained according to Equation (23). That is:

$$P\left(\mathcal{S}\_{i}(h+1) = 0 \Big| \mathcal{S}\_{i}(h) = 0\right) = 1 - P\left(\mathcal{S}\_{i}(h+1) = 1 \Big| \mathcal{S}\_{i}(h) = 0\right) = \left(1 - \beta\right)^{N\_{i}^{0}}\tag{24}$$

(3) Case 3: *Si*(*h*) = 1 → *Si*(*h* + 1) = 0

When the state of community *i* is *Si*(*h*) = 1 at week *h*, it may switch back to *Si*(*h* + 1) = 0 at week *h* + 1. For example, a community may find the DR project is inconvenient or uneconomical and thus abandon it. In this paper, we assume that the probability from *Si*(*h*) = 1 to *Si*(*h* + 1) = 0 are correlated with economic compensation and energy consumption utility. If a community in the group *N* switches from state 1 to state 0, it will lose economic compensation, but will obtain the corresponding utility. Hence, when economic compensation decreases due to the increasing of group *N*'s population, the probability from *Si*(*h*) = 1 to *Si*(*h* + 1) = 0 will increase gradually. To quantitatively measure such probability, the average comprehensive income for all communities in group *N* is defined as:

$$\overline{w}^{\text{h}} = \frac{1}{N} \sum\_{i \in \mathcal{N}} \left( e\_i^{\text{h}} - u\_i^{\text{h}} \right) \tag{25}$$

Basically, the probability from *Si*(*h*) = 1 to *Si*(*h* + 1) = 0 is defined as:

$$
\alpha = \eta \left( 1 - \frac{\overline{w}^{l}}{\overline{w}^{l, \text{max}}} \right) \tag{26}
$$

where <sup>η</sup> is a constant parameter; *wh*,max <sup>=</sup> maximize *w*1 , *w*<sup>2</sup> , ... , *w<sup>h</sup>* represents the maximal value of group *N*'s average income in the preceding *h* weeks. Equation (26) shows that the lower income group *N* receives, the higher probability the community switches from state 1 to state 0. Specifically, when the average income in group *N* reaches the maximal value, the corresponding probability will reach the minimal value. According to Equations (25) and (26), the transition probability from state 1 to state 0 can be expressed as:

$$P\left(S\_i(h+1) = 0 \middle| S\_i(h) = 1\right) = \alpha = \eta \left(1 - \frac{\overline{w}^h}{\overline{w}^{h,\max}}\right) \tag{27}$$

(4) Case 4: *Si*(*h*) = 1 → *Si*(*h* + 1) = 1

Similarly, since community *i* with *Si*(*h*) = 1 can only choose *Si*(*h* + 1) = 1 or *Si*(*h* + 1) = 0, the probability that community *i* still remains in group *N* can be obtained according to Equation (27). That is:

$$P\left(S\_i(h+1) = 1 \Big| S\_i(h) = 1\right) = 1 - P\left(S\_i(h+1) = 0 \Big| S\_i(h) = 1\right) = 1 - a \tag{28}$$

In summary, the transition probability model from *Si*(*h*) to *Si*(*h* + 1) can be expressed as follows:

$$P\left(S\_{i}(h+1)\middle|S\_{i}(h)\right) = \begin{cases} 1 - (1-\beta)^{N\_{i}^{h}} & (S\_{i}(h), S\_{i}(h+1)) = (0,1) \\ (1-\beta)^{N\_{i}^{h}} & (S\_{i}(h), S\_{i}(h+1)) = (0,0) \\ \alpha & (S\_{i}(h), S\_{i}(h+1)) = (1,0) \\ 1-\alpha & (S\_{i}(h), S\_{i}(h+1)) = (1,1) \end{cases} \tag{29}$$

## *4.2. Markov Model for Group Population*

In reality, the state of a residential community in past weeks has no effect on the decision in future weeks, and the decision result in week *h* + 1 is only correlated with a community's state in week *h*. Therefore, this paper adopts the Markov chain to describe the decision-making process of residential community. The Markov chain mainly indicates that the future decision-making state is independent

of the past state and is only dependent on the current state [34]. In our proposed scenario, the Markov chain can be expressed as:

$$P\left(S\_i(h+1)\middle|S\_i(1),\cdots,S\_i(h)\right) = P\left(S\_i(h+1)\middle|S\_i(h)\right) \tag{30}$$

where *Si*(1), ··· , *Si*(*h*) are decision-making states from week 1 to week *<sup>h</sup>*. Assume that Pr*<sup>i</sup> <sup>N</sup>*(*h*) and Pr*i <sup>M</sup>*(*h*) are the probabilities of residential community *i* in group *N* and group *M* in week *h*. Then, the Markov state evolution can be described as:

$$\begin{split} \text{Pr}\_{N}^{i}(h+1) &= \text{Pr}\_{N}^{i}(h) \text{P} \Big( \mathbb{S}\_{i}(h+1) = 1 \Big| \mathbb{S}\_{i}(h) = 1 \Big) + \text{Pr}\_{M}^{i}(h) \text{P} \Big( \mathbb{S}\_{i}(h+1) = 1 \Big| \mathbb{S}\_{i}(h) = 0 \Big) \\ &= (1-a) \text{Pr}\_{N}^{i}(h) + \Big( 1 - (1-\beta)^{N\_{i}^{\#}} \Big) \text{Pr}\_{M}^{i}(h) \end{split} \tag{31}$$

and:

$$\begin{split} \Pr\_{M}^{i}(h+1) &= \Pr\_{M}^{i}(h)P\Big(\mathcal{S}\_{i}(h+1) = 0 \Big| \mathcal{S}\_{i}(h) = 0\Big) + \Pr\_{N}^{i}(h)P\Big(\mathcal{S}\_{i}(h+1) = 0 \Big| \mathcal{S}\_{i}(h) = 1\Big) \\ &= (1-\beta)^{N\_{i}^{b}}\Pr\_{M}^{i}(h) + a\Pr\_{N}^{i}(h) \end{split} \tag{32}$$

In Equations (31) and (32), (1 − β) *Nh <sup>i</sup>* can be rewritten as:

$$\begin{array}{rcl} \left(1-\beta\right)^{\mathsf{N}\_{i}^{\mathsf{N}}} & = \prod\_{i \in I\_{i}} \left(\Pr\_{N}^{i}(h)(1-\beta) + \Pr\_{M}^{i}(h)\right) = \prod\_{i \in I\_{i}} \left(\Pr\_{N}^{i}(h)(1-\beta) + \left(1-\Pr\_{N}^{i}(h)\right)\right) \\ & = \prod\_{i \in I\_{i}} \left(1-\beta \Pr\_{N}^{i}(h)\right) \approx 1-\beta \sum\_{i \in I\_{i}} \Pr\_{N}^{i}(h) \end{array} \tag{33}$$

where *Ii* represents the set of community *i*'s neighbors in set *I*. Therefore, Equations (31) and (32) can also be expressed as follows:

$$\mathrm{Pr}\_{N}^{i}(h+1) = (1-\alpha)\mathrm{Pr}\_{N}^{i}(h) + \left(1 - \left(1 - \beta \sum\_{i \in I\_{i}} \mathrm{Pr}\_{N}^{i}(h)\right)\right) \mathrm{Pr}\_{M}^{i}(h) \tag{34}$$

and:

$$\Pr\_{M}^{i}(h+1) = \left(1 - \beta \sum\_{i \in I\_{i}} \Pr\_{N}^{i}(h)\right) \Pr\_{M}^{i}(h) + \alpha \Pr\_{N}^{i}(h) \tag{35}$$

According to the Markov state Equations (34) and (35), the probability for each community *i* in groups *N* or *M* can be calculated. However, the decision of a single community is not our concern and the main purpose in this section is to analyze the group population. For simplification, assume that *I* residential communities have good communication and each community can be infected by any other *I*−1 communities. Then, each residential community in *I* has equal probability to participate in DR. Consequently, we have:

$$
\overline{\mathbf{Pr}}\_{N} = \mathbf{Pr}\_{N'}^{i} \,\overline{\mathbf{Pr}}\_{M} = \mathbf{Pr}\_{M'}^{i} I\_{i} = I \tag{36}
$$

where Pr*<sup>N</sup>* and Pr*<sup>M</sup>* represent the probability of any community being in group *N* or *M*. Then, Equations (34) and (35) are simplified to:

$$
\overline{\text{Pr}}\_{\text{N}}(h+1) = (1-a)\overline{\text{Pr}}\_{\text{N}}(h) + \beta I \overline{\text{Pr}}\_{\text{N}}(h)\overline{\text{Pr}}\_{\text{M}}(h) \tag{37}
$$

and:

$$
\overline{\text{Pr}}\_{\text{M}}(h+1) = \left(1 - \beta l \overline{\text{Pr}}\_{\text{N}}(h)\right) \text{Pr}\_{\text{M}}^{i}(h) + a \overline{\text{Pr}}\_{\text{N}}(h) \tag{38}
$$

Therefore, the average number of residential communities in groups *N* and *M* in week *h* can be expressed as:

$$\begin{cases} N(h) = \overline{\text{Pr}}\_{N}(h)I \\ M(h) = \overline{\text{Pr}}\_{M}(h)I \end{cases} \tag{39}$$

According to Equations (37) and (38), the average number of residential communities in groups *N* and *M* in week *h* + 1 can be calculated as:

$$\begin{cases} N(h+1) = (1-a)N(h) + \beta N(h)M(h) \\ M(h+1) = (1-\beta N(h))M(h) + aN(h) \end{cases} \tag{40}$$

Equation (40) is used to describe the population evolution in groups *N* and *M*.
