Abstract
Residential flexible resource is attracting much attention in demand response (DR) for peak load shifting. This paper proposes a scenario for multi-stage DR project to schedule energy consumption of residential communities considering the incomplete information. Communities in the scenario can decide whether to participate in DR in each stage, but the decision is the private information that is unknown to other communities. To optimize the energy consumption, a Bayesian game approach is formulated, in which the probability characteristic of the decision-making of residential communities is described with Markov chain considering human behavior of bounded rationality. Simulation results show that the proposed approach can benefit all residential communities and power grid, but the optimization effect is slightly inferior to that in complete information game approach.
THE development and implementation of new technologies and strategies for solving the energy problems are critical to meeting the increasing energy demand in all walks of life. The main concern in this field is how to alleviate the contradiction between the energy supply and demand on the premise of environmental friendliness. Many advancements have been achieved in generation side, where distributed generation is an effective way to solve the contradiction between the energy supply and demand from “source side” [
Currently, there exists abundant research on residential DR problems. For example, [
In view of this, game theory, which is excellent in solving multi-player decision problem, has been widely employed in the field of DR [
According to the deficiency of current research, this paper proposes a repeated game approach considering incomplete information to schedule the energy consumption of users, which considers the construction of probability characteristic with Markov chain. Considering that a DR project may be implemented for many years, some users may participate in the DR while some may exit the project during the period. In order to facilitate the management and improve the efficiency, in the proposed scenario, we assume that the DR project is divided into different stages. The duration time of each stage can be one week or one month. At the end of each stage, residential community can re-decide whether to participate in the DR in the next stage. Additionally, this paper adopts a kind of agent mechanism, in which residential community is deployed as an agent of native users to participate in the DR decision. In the proposed scenario, the decision of one residential community in each stage is private information that is unknown to other residential communities, and one residential community only knows the decisions of other residential communities in the previous stage. Communities with different decisions will have different energy consumption arrangements that will cause the change of real-time demand and price.
Since each residential community does not know the decisions of its opponents, it is difficult to predict the energy demand and price. Consequently, it is difficult for each residential community to schedule its own energy consumption with complete information game approach. To solve the issue, each residential community needs to speculate the decisions of its opponents according to the probability characteristic of decisions in the previous stage. And then, each residential community can schedule its energy consumption by considering the optimal scheduling strategies of other residential communities with different decisions. In summary, the contributions of this paper are as follows.
1) A scenario is proposed for DR project to schedule energy consumption of residential communities considering the incomplete information, in which the dynamic decision-making process of residential community on whether to participate in DR is considered in multi-stage DR project.
2) The probability characteristic of the decision-making of residential communities is described with Markov chain, in which the absolute rationality assumption for residential community is removed and the human behavior of bounded rationality, i.e., imitation and randomness, is considered.
3) A Bayesian game approach is proposed for the proposed scenario to reduce the daily cost with the formulated probability characteristic of the decision-making of residential communities, and the existence of Bayesian Nash equilibrium is proven mathematically.
The rest of this paper is organized as follows. The system model is introduced in Section II. In Section III, the Bayesian game approach is formulated and the existence of Bayesian Nash equilibrium is proven. Then, the probability characteristics for decision-making and Bayesian Nash equilibrium are analyzed in Section IV. The simulation results are presented in Section V. Finally, the conclusions are drawn in Section VI.
The proposed scenario for residential communities participating in DR is shown in

Fig. 1 Scenario for residential community participating in DR.
When DR project is implemented, the DR center will broadcast DR information to all communities. Based on the broadcasted DR information, each residential community who is willing to participate in DR independently executes the scheduling algorithm to obtain the optimal energy consumption. Those communities who are unwilling to participate in DR will consume energy in their own ways. After all communities finish the scheduling, they send the desired demand information to the DR center. Then, the DR center sends the desired demand information of all communities to public power grid. Since residential communities in the scenario cannot communicate with each other, there are no privacy issue and heavy network traffic. In addition, whether a residential community is willing to participate in DR in the next stage is generally related with the decision in current stage. Hence, in the scenario, we assume that the decision-making of each residential community on whether to participate in DR in each stage has Markov property.
The two-state Markov chain for residential community is shown in

Fig. 2 Two-state Markov chain for residential community.
Suppose that the set of N residential communities is denoted by and a day is divided into H time slots with . In addition, all shiftable household appliances of residential community n are denoted by set .
Assume that for any residential community in time slot , its non-shiftable load consumes energy and its shiftable load consumes energy . The shiftable load is relatively insensitive to energy consumption time, which can be shifted in a certain time interval. Suppose that any shiftable household appliance consumes energy in time slot h, and it has to satisfy (1).
(1) |
where is the shiftable time interval of appliance a; and En,a is the daily energy demand of appliance a. Therefore, the whole energy consumption of residential community n is expressed as:
(2) |
where Ln,h is the whole energy consumption of residential community n in time slot h. Accordingly, the set of the feasible energy consumption scheduling corresponding to residential community n can be expressed as:
(3) |
where is the scheduling vector of appliance a.
It is clear that the energy price mechanism plays an important role in attracting more residential communities to participate in DR. In recent years, various price mechanisms have been studied, including time-of-use price [
(4) |
(5) |
where is the function of real-time price; and are the coefficients of energy price with higher values during peak hours; and Lh is the whole energy consumption of all communities.
Based on the energy price model (4), the cost of each residential community can be calculated. Accordingly, the total daily cost of residential community n can be calculated as:
(6) |
The objective of the residential community participating in DR is to minimize the daily cost, i.e.,
(7) |
Residential communities can obtain the optimal result by solving the optimization problem (7).
In this section, the complete information game approach will be firstly formulated among residential communities, in which the game information of each residential community is known to all communities. Then, the Bayesian game approach for the energy consumption is formulated considering the incomplete information.
In the complete information game approach, the DR information of each residential community such as the initial energy consumption in each time slot and the decision on whether to participate in DR is well known to all communities. Consequently, each residential community will try to minimize its daily cost by speculating the scheduling strategies of other communities. Therefore, according to (6), the complete information game approach among residential communities can be formulated as follows.
1) Players: all communities who are willing to participate in DR.
2) Strategies: each residential community n schedules its energy consumption of shiftable load xn,a to minimize the daily cost.
3) Payoffs: the payoff of residential community n is defined as:
(8) |
where is the scheduling strategies of other communities except residential community n.
It needs to be noted that, the above formulated game approach is mainly designed for the communities participating in DR. For some communities who do not participate in DR, they just need to consume energy in their initial state and the scheduling strategy xn,a is assigned with initial energy consumption. When these communities schedule energy consumption to maximize their own payoff based on the strategies of opponents until all strategies are unchanged, such state is called Nash equilibrium. Assume that is the corresponding Nash equilibrium of the formulated complete information game approach, we can obtain:
(9) |
Once the Nash equilibrium is reached, no residential community will break such equilibrium state. Otherwise its payoff will be reduced.
In the complete information game approach, each residential community has the full information of other communities and can speculate the scheduling strategies of other communities via solving the optimization problem (8). However, a lot of information in reality is not public information. For example, in the proposed scenario, each residential community only knows the decision of opponents on whether to participate in DR in the historical stage, and it is not easy to know the decision of opponents in the future stage. That is to say, when the residential community participates in energy management, it is not easy to know if other communities participate in DR in the next stage. Consequently, one residential community cannot speculate the scheduling strategies of other communities because the payoff function is unknown completely. Hence, the modeling process of the complete information game approach is not suitable for the incomplete information game approach. In this paper, Bayesian game approach is employed to describe the competition behavior among residential communities considering the incomplete information. Different from the complete information game approach, the basic elements of Bayesian game approach have to introduce the types of players and the probability distribution of the types except for players, strategies, and payoffs.
Assume that residential communities are divided into two types according to the decision on whether to participate in DR. That is, the type space Tn of residential community n has elements and the actual type of residential community n is tn. Herein, let represent that the residential community is willing to participate in DR, otherwise . Accordingly, represents the type space combination for all communities and the actual type combination of all communities is . Since each residential community does not know the types of its opponents, it needs to speculate the types of opponents according to the probability distribution of the types. On the basis of Bayesian formula, we can obtain:
(10) |
where and are the type combination for other communities except residential community n and the actual type combination of these communities, respectively; is the conditional probability of under the condition that the type of residential community n is tn; is the probability that the type of residential community n is tn; and is the joint probability distribution for type combination t.
It is clear that, the types of all communities can be deduced from the prospect of probability with the Bayesian condition probability (10). In other words, the incomplete information game can be translated into various complete information games by dividing different type combinations of all communities and each complete information game appears with a certain probability. Therefore, the payoff of the Bayesian game is actually the expected value of all payoffs of these complete information games. When the type of residential community n is tn, it will speculate the type combination of opponents with Bayesian formula and then formulate the payoff function of Bayesian game. Accordingly, the payoff of Bayesian game of residential community n with type tn can be expressed as:
(11) |
where is the scheduling strategy of residential community n with type tn; and is the scheduling strategies of other communities with type combination .
(12) |
where is the Bayesian Nash equilibrium corresponding to type combination . It needs to point out that, this paper focuses on the multi-stage DR, hence, the Bayesian game approach in this section is formulated for any DR stage. In order to have a better expression, the identifier “s” for DR stage is omitted in the above analysis, but it will be added in the analysis of joint probability distribution in next section.
Based on the formulated Bayesian game approach, one can know that the determination of joint probability distribution is significant for the equilibrium solution. Therefore, this section mainly focuses on the analysis of joint probability distribution and Bayesian Nash equilibrium.
In the proposed scenario, the implement process of the DR project is divided into different stages. At the end of each stage, residential community can freely decide whether to participate in DR in the next stage. Since a residential community whether to participate in DR in next stage is only related with the current stage, the decision-making process has the Markov property [
(13) |
where are the actual type of residential community n from stage 1 to stage s; and is the probability of residential community n being type under the condition of being type . Before Bayesian game approach is applied, the state of each residential community has to be determined separately. In fact, the determination of can be obtained from
1) Case 1: residential community n participates in DR in stage s, i.e., , and still participates in DR in stage , i.e., . The corresponding conditional probability is expressed as:
(14) |
2) Case 2: residential community n participates in DR in stage s, i.e., , and exits DR in stage s+1, i.e., . The corresponding conditional probability is expressed as:
(15) |
3) Case 3: residential community n does not participate in DR in stage s, i.e., and participates in DR in stage , i.e., . The corresponding conditional probability is expressed as:
(16) |
4) Case 4: residential community n does not participate in DR in stage s, i.e., and does not participate in DR in stage as well, i.e., . The corresponding conditional probability is expressed as:
(17) |
Therefore, the transition probability of Markov chain for residential community n can be expressed as:
(18) |
According to the characteristic and the transition probability of Markov chain, we can obtain the probability of residential community n with type 1 in stage as:
(19) |
(20) |
Assume that is the set of all neighbors of residential community n and residential community j is any one in the set . Clearly, residential community j will participate in DR with the probability and not participate in DR with the probability . When residential community j participates in DR, the probability that residential community n is not infected is equal to ; otherwise, the corresponding probability is equal to 1. That is, the probability that residential community n is not infected by residential community j can be expressed as:
(21) |
Accordingly, the probability that residential community n is not infected by all neighbors can be rewritten as:
(22) |
Therefore, the probability of residential community n with type 1 and type 2 can be expressed as:
(23) |
(24) |
Considering residential communities are in a well-connected information network, the probability characteristic of the decisions of communities on whether to participate in DR in previous stage is the public information in the network. Therefore, the set can be considered as set \n, then the probability of residential community n with type 1 and type 2 is equal to (25) and (26), respectively.
(25) |
(26) |
Based on (25) and (26), the probability of each residential community with type 1 or 2 in stage can be easily obtained. Consequently, the joint probability distribution will be determined for all type combinations. Furthermore, Bayesian conditional probability will be obtained with (10).
According to the above analysis, it is obvious that Bayesian Nash equilibrium is closely correlated with the type of players. Hence, the residential community will have different Bayesian Nash equilibriums for different types. In this subsection, the existence and uniqueness of the Bayesian Nash equilibrium will be proven for any actual type combination t.
Proposition 1: in the formulated complete information game approach, the Bayesian Nash equilibrium is unique.
Proof: obviously, the payoff function is continuously differentiable in for the fixed . Hence, the Hessian matrix of function can be obtained as:
(27) |
Due to the coefficient of energy price , is a diagonal matrix with all negative elements. Therefore, the function is concave in . Consequently, the existence of Nash equilibrium can be proven according to Theorem 1 in [
Proposition 2: in the formulated Bayesian game approach, Bayesian Nash equilibrium is unique for any actual type combination t.
Proof: similarly, we just need to prove the concavity of payoff function . Accordingly, the corresponding Hessian matrix is expressed as:
(28) |
Since , (28) can also be rewritten as:
(29) |
Accordingly, the Bayesian Nash equilibrium is unique for any actual type combination t.
In order to search the optimal solution of the Bayesian game approach, a distributed algorithm executed by community center is proposed, which is shown in
In stage 7, given , the problem (11) has only local variable and can be solved with the mature mathematic programming algorithm. In this paper, CPLEX optimization solver is adopted, which has a high convergence and efficiency. Additionally, for the Bayesian Nash equilibrium, each residential community has to constantly readjust its strategy until the equilibrium is reached. The dynamic decision-making process of the Bayesian game can be summarized as follows.
1) Community n calculates the optimal scheduling strategies of other residential communities one by one according to its own strategy.
2) Community n updates the original strategy according to the new scheduling strategies of other residential communities.
3) Repeat 1) and 2) until the equilibrium is achieved.
Such dynamic process is realized with stages 3-12. Since the formulated game model has Nash equilibrium,
In this section, simulation results are presented to show the effectiveness of the formulated Bayesian game approach and the performance of the designed distributed algorithm.
Assume that there are 3 residential communities and each residential community contains 800 users in the case. A day is divided into 24 time slots and each time slot is 1 hour. Initial energy consumption of each residential community before DR is given with a random energy demand value between the upper limit and the lower limit that are set in

Fig. 3 Energy demand range of each residential community.
Such shiftable loads can be scheduled uniformly by community center during the permitted time interval. Herein, we assume that the permitted time slots of shiftable loads are as follows: electric vehicle can be charged from 17:00 to 24:00 and 00:00 to 06:00; washing machine can operate from 17:00 to 23:00; and the dishwasher can operate from 17:00 to 23:00. In addition, considering the difference of operation characteristics of the shiftable appliances, we assume that the electric vehicle can be charged at any time during the permitted interval, while the washing machine or dishwasher only works once a day and the operation time is 1 hour [
Since each stage has an equilibrium solution, this subsection will take the result of stage as an example. According to (25) and (26), the probability of any residential community n with type 1 is equal to and . Since 3 residential communities are considered, there are
(31) |
where , 2, 3 represents the number of residential communities with type 1 in actual type combination . Additionally, we assume that the actual type combination of 3 residential communities is in stage . Based on the above simulation parameters, the equilibrium can be obtained by executing
Tables
From the tables, it can be seen that electric vehicles in 3 residential communities are all shifted to the off-peak hours, i.e., time slots 1-6, but the operation strategies of the dishwasher and washing machine are different between the two approaches. In the complete information game approach, the operation time of dishwasher and washing machine in 3 residential communities are scheduled into 6 time slots from time slot 18 to time slot 23. That is, each time slot only has one shiftable appliance, for example, time slot 18 only has the dishwasher of residential community 3. However, in Bayesian game approach, some time slots have more than one shiftable appliance, for example, time slot 21 has the washing machine of residential community 1 and the dishwasher of residential community 2. The main reason for the difference is that, since the decision of each residential community on whether to participate in DR in stage 2 is unknown to other communities, each residential community cannot deduce the strategy of its opponents precisely. Consequently, residential community can only make a compromise strategy to maximize the expected payoff by considering all possible type combinations.
Residential community and public power grid can effectively obtain the benefits from the proposed Bayesian game approach. However, due to the lack of game information, the obtained benefits in the Bayesian game are less than those in the complete information game.

Fig. 4 Optimal energy demand of each residential community in Bayesian game. (a) Residential community 1. (b) Residential community 2. (c) Residential community 3.
The corresponding energy demand of all communities is shown in

Fig. 5 Energy demand of all communities in complete information and Bayesian game approaches. (a) Complete information game approach. (b) Bayesian game approach.
In order to quantitatively analyze the variation of energy demand, peak to average ratio (PAR) can be introduced, which is calculated as [
(32) |
(33) |
where PARg is the PAR in public power grid; and PARn is the PAR in residential community n. Basically,

Fig. 6 PAR in public power grid and each residential community.
Before DR project is implemented, no matter in public power grid or in each residential community, the PAR is very high. However, although the PAR in each residential community is high after the two game approaches have been employed, the PAR in public power grid is reduced greatly. It indicates that the two game approaches can realize the demand complementation among communities. The daily cost of each residential community is shown

Fig. 7 Daily cost of each residential community.
In the above case, for the convenience of analysis, we assume that the actual types of 3 residential communities are all type 1, but such probability is only 0.059. In addition, the above case only concerns the optimal result in a certain stage. Therefore, this subsection mainly concentrates on the evolution analysis for the probability of communities participating in DR with the execution of DR project. Specially, the different evolution results for communities with different degrees of rationality will be analyzed by regulating parameters and . Since different groups of users have various degrees of rationality in the reality, the obtained result can provide the reference in the design and implementation of DR project considering different user groups. The evolution result of the probabilities of residential communities with types 1 and 2 is shown in

Fig. 8 Evolution result of probabilities of residential communities with types 1 and 2.
It depicts that the probabilities of communities with types 1 and 2 gradually converge to fixed values after 20 stages. Finally, each residential community will participate in DR with the probability of 0.75. In fact, the evolution result of the probabilities of communities is closely related with parameters N, , and .

Fig. 9 Probability of type 1 with different parameters. (a) and . (b) .
It is clear that, with the growth of the number of communities, the probability of type 1 increases gradually. The main reason is that, each residential community will have a higher probability to be attracted into DR with the growth of the number of communities in DR. However, with the growth of the value of , the probability of type 1 decreases dramatically. The values of and are correlated with the DR project such as the DR price and DR experience. The growth of demonstrates that the DR project has a low attraction or experience, and the residential community is unwilling to participate in DR or wants to exit DR. Therefore, it is important to design a good DR mechanism for the better performance of DR in smart grid. According to the results in

Fig. 10 Daily cost of 3 residential communities with different values of and .
Additionally,
In this paper, a Bayesian game approach is formulated to schedule energy consumption of shiftable load in residential communities considering the incomplete information. In the proposed scenario, each residential community can decide whether to participate in DR in any stage of DR project. The decision of each residential community is the private information, which is unknown to other communities. Therefore, each residential community needs to evaluate such information based on the probability distribution of decision. Accordingly, the Markov model for joint probability distribution is proposed to describe the decision-making process of residential community considering human behavior of bounded rationality. Simulation results demonstrate that the proposed approach can reduce the daily cost and PAR of the overall energy demand. However, compared with the complete information game approach, it shows that the optimization effect on the Bayesian game approach is weakened due to the loss of game information.
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