Abstract
This paper proposes a scheduling strategy based on realtime pricing in smart grids. A hierarchical game is employed to analyze the decisionmaking process of generators and consumers. We prove the existence and uniqueness of Nash equilibrium and utilize a backward induction method to obtain the generation and consumption strategies. Then, we propose two dynamic algorithms for the generators and consumers to search for the equilibrium in a distributed fashion. Simulation results demonstrate that the proposed scheduling strategy can match supply with demand and shift load away from peak time.
1. Introduction
Matching supply with demand has been an active topic in power systems. Traditionally, we have to provide enough generation capacity to meet peak demand [1]. This requires substantial infrastructures to be idle for all but a few days a year. In recent years, demand response (DR) has been proposed to reduce the peak demand by managing the consumption of consumers [2]. DR can be implemented by various pricing strategies, such as time of use (TOU), critical peak pricing (CPP), extreme day CPP (EDCPP), extreme day pricing (EDP), and realtime pricing (RTP) [3]. For the RTP strategy, the electricity price is published on a rolling basis. The price for a given time period is determined before the start of the period. With the development of smart grids, RTP can be implemented within a shorter period.
Recently, game theory has been applied to the DR programs [4]. For instance, the congestion game was applied to control the electricity consumption by dynamic pricing [5], the noncooperative game was utilized to find the pricing condition for demand response [6] and minimize the electricity cost to consumers [7, 8], and the Stackelberg game was applied to residential power scheduling [9], energy exchange between electric vehicles and smart grids [10], and demand response with one energy provider [11] and multiple energy providers [12]. Nevertheless, few papers are devoted to the DR program with the integration of generation scheduling. In this study, we propose an integrated scheduling strategy for the generators and the consumers based on a hierarchical game approach and formulate the competitions among the generators into a noncooperative game at the leader level. The main contributions are as follows.(i)The generation and consumption scheduling problem is formulated as a hierarchical game, where the generators act as the leaders and the consumers act as the followers.(ii)The existence and uniqueness of Nash equilibrium are proved, and the optimal generation and consumption strategies are obtained by a backward induction method.(iii)Two dynamic algorithms are proposed to search for the optimal generation and consumption strategies in a distributed fashion.
The rest of the paper is organized as follows. The scheduling problem is formulated as a hierarchical game in Section 2. We give the game analysis by the backward induction method in Section 3 and propose two dynamic algorithms to search for the equilibrium in Section 4. Section 5 shows the numerical results. Finally, conclusions are summarized in Section 6.
2. Problem Formulation
As shown in Figure 1, we consider smart grids consisting of generators, consumers, and an Independent Electricity System Operator (IESO). The IESO is a nonprofit organization, which is responsible for operating the grids and directing the operation of the participants by pricing. Each consumer installs a smart meter to implement twoway communications with the IESO. The intended operation cycle is divided into time slots indexed by . In each time slot, the generators determine the generation, and then the consumers manage the consumption according to the realtime price published by the IESO.
2.1. Consumer Level Formulation
Let denote the set of consumers and denote the consumption profile, where is the consumption of consumer in time slot . To describe the profits of consumers, we consider the following utility functions with linear decreasing marginal benefit [13]: where represents the willingness to increase the consumption in time slot , and denotes the maximal demand for consumer . For instance, the utility functions with different are shown in Figure 2, from which we see that the utility function has the following properties:(i) is increasing with , for ;(ii) is a strictly concave function of , for ;(iii) equals zero if , for .
Then, we can define the payoff of consumer as where is the electricity price, and is a positive parameter. Each consumer manages the consumption to maximize its payoff, which can be described as the following optimization problems:
2.2. Generator Level Formulation
The generators aim to maximize the profit without too much generation cost. Thus, the payoff of generator is defined as where and are the profit and the cost to generator , respectively. As shown in Figure 3, the generation cost is defined as a quadratic function [14]: where denotes the generation of generator in time slot . The generation profile is denoted as . The parameters , , and are positive. Then, the optimal generation can be obtained from the following optimization problems: where denotes the set of generators.
2.3. IESO Pricing Model
The IESO operates the wholesale markets and sets the electricity price to balance the supply and the demand. Following the principle of locational marginal pricing (LMP), the price needs to reflect the value of electricity at a specific location at the time that it is delivered. To this end, we construct a realtime pricing function based on the difference between the demand and the supply: where is a basic price. The parameter is to implement the elastic pricing such that the supply can meet the demand.
3. Analysis of Hierarchical Game
In smart grids, the IESO can acquire information from the consumers based on an advanced metering infrastructure (AMI). Thus, the generation scheduling and the consumption scheduling are performed with information asymmetry. Specifically, the generators determine the generation first, and then the consumers manage the consumption in response. This multistage decision process can be formulated as a hierarchical game consisting of a generator level subgame and a consumer level subgame. In particular, the generators act as the leaders, and the consumers act as the followers. Next, we utilize the backward induction method to analyze the hierarchical game and omit for convenience in the following sections.
3.1. Consumer Level Subgame
We first give the definitions of the consumer level subgame and the Nash equilibrium in the game [15].
Definition 1. A consumer level subgame is defined as a triple , where is the set of active consumers participating in the game, is the set of possible strategies that consumer can take, and is the payoff function.
Definition 2. An energy consumption profile is a Nash equilibrium in the consumer level subgame if , for all , where is an energy consumption profile of all consumers except for consumer , and denotes the payoff function of consumer given the energy consumption of the other consumers.
It is noted that the Nash equilibrium is a set of strategies where no one has an incentive to change unilaterally. Differentiating with respect to , we have According to Definition 2, we know that the Nash equilibrium can be obtained from , which gives the following equations: Next, we derive the Nash equilibrium in the following theorem.
Theorem 3. The consumer level subgame has a unique Nash equilibrium if and the equilibrium is given as
Proof. The constraints (11) can be obtained directly from the pricing condition in [6], and the consumer level subgame has a unique Nash equilibrium. Next, we will give the explicit expression of the unique equilibrium. From (10), we have Adding the two sides of (13) from 1 to , we obtain Then, the optimal consumption (12) is obtained by substituting (14) into (13).
The objective of the IESO is to achieve the balance between supply and demand: Combining with (14), we calculate the pricing parameter as Substituting (15) and (16) into (7), we obtain the pricing function: with which, the balance between supply and demand is achieved.
3.2. Generator Level Subgame
In this section, we consider the profit allocation among the generators with the knowledge of pricing function in (17). Substituting (17) into (4), we obtain the payoff of generator : Then, the generator level subgame can be defined as follows.
Definition 4. A generator level subgame is defined as a triple , where is the set of active generators participating in the game, is the set of possible strategies that generator can take, and the payoff is denoted by (18).
Next, we need to study the existence and uniqueness of the Nash equilibrium for the generator level subgame , which requires the following lemma.
Lemma 5 (see [16]). A Nash equilibrium exists in the generator level subgame , if for all ,(1) is a nonempty, convex, and compact subset of some Euclidean space ;(2) is continuous in and quasiconcave in .
Theorem 6. The generator level subgame has a unique Nash equilibrium if
Proof. Let the strategy set be a nonempty, convex, and compact subset of some Euclidean space . From (18), we know that is continuous in . Taking the second order derivative of with respect to , we have
It is straightforward to see that the second order derivative of with respect to is negative, and thus is concave in . According to Lemma 5, we conclude that the generator level subgame has at least one or more Nash equilibriums.
For each , differentiating it with respect to , we obtain
and differentiating with respect to and , , respectively, we obtain the Jacobian matrix:
We construct a matrix ,Given the constraints (19), is strictly diagonally dominant, and all its diagonal elements are negative. Following Gershgorin’s theorem [17], is negative definite. Thus, the generator level subgame has a unique Nash equilibrium [18].
Remark 7. To implement RTP in smart grids, we choose the linear pricing function. The reasons are as follows. First, according to the technical report from U. S. Department of Energy [19], the electricity price is almost linearly increased with the total load. Second, a pricing condition in [6] shows that the linear pricing function is sufficient to make the demand response stable. Last but not least, the linear pricing function is easy to ensure the existence of a unique Nash equilibrium and derive the explicit expression of the Nash equilibrium in the consumer level subgame, which is necessary to the backward induction method in hierarchical model. In practice, the pricing function (7) is not necessary to be linear. For example, we can also choose the quadratic pricing function. Substituting (24) into the payoff function (2), we obtain Then, we can obtain the second derivative of the payoff function (25) with respect to : We can choose sufficient small to make the second derivative (26) negative, that is, the existence of Nash equilibrium. However, it is infeasible to find the conditions of to ensure (26) always to be negative. Further, we cannot obtain the explicit expression of the Nash equilibrium from .
Remark 8. For the LMP in wholesale electricity markets, we need to set different prices for different areas in the power system. In fact, the proposed pricing function (7) can be seen as the electricity price at a specific area in one time slot, and describes the power usage in the area. For a predefined pricing parameter , we see that the price is increased with the difference between the total load and the total supply, which consists with the roles of LMP.
Remark 9. The scheduling model proposed in this work only deals with the power balance in one area. In practice, we need to consider the scheduling with operating constraints, for example, power flow between different areas, which will introduce the voltage phase angles to the pricing function. In that case, the pricing function is defined as where and are the indices of different areas in power system, is the susceptance of bus , and and are the voltage phase angles of area and , respectively. In this study, and are neglected because we only consider the electricity scheduling in one area. For electricity scheduling in the whole power system, we also need to consider the transmission congestion on the transmission line, which requires additional pricing models. The pricing method to solve the transmission congestion is out of the scope of this study.
4. Distributed Algorithms and Implementations
4.1. Iterative Algorithms
In smart grids, the generators and the consumers need to search for the optimal generation and consumption in a distributed fashion. We propose two distributed iterative algorithms to search for the optimal generation and consumption. The generation scheduling algorithm for generator is defined as where is the price to be announced to the generators for generation scheduling:
The consumption scheduling algorithm for consumer is defined as where is the price published to the consumers for consumption scheduling: In the iterative algorithms, and are iterative steps, and and are step sizes. Algorithms (28) and (30) are implemented on different time scales. Specifically, the generators first search for the optimal generation by algorithm (28), and then the consumers search for the optimal consumption by algorithm (30). The pricing function (29) is equal to the pricing function (31) at the Nash equilibrium because of the balance between supply and demand. Since the price is known to the generators and the consumers, the iterative algorithms (28) and (30) in a distributed fashion. Next, we give the convergence conditions of the algorithms.
Proposition 10. Suppose the generator level subgame has a unique inner Nash equilibrium . The dynamic algorithm (28) converges to the unique Nash equilibrium if
Proposition 11. Suppose the consumer level subgame has a unique inner Nash equilibrium . The dynamic algorithm (30) converges to the unique Nash equilibrium if
The proofs are omitted here. Interesting readers can find the details in [6].
4.2. Implementations
In this section, we provide distributed implementations for the scheduling strategy in smart grids. The programming pseudocode of the scheduling strategy is given in Algorithm 1. Specifically, the generators compete to maximize the payoff (18) and search for the optimal generation by algorithm (28). Then, the consumers decide the optimal consumption by algorithm (30). Both of the prices (29) and (31) are announced by the IESO.

5. Numerical Results
In the simulations, the entire time cycle is divided into 24 time slots representing 24 hours in a day. The numbers of consumers and generators are set to be 100 and 5, respectively. Before giving the numerical results, we define the daily average price as To characterize the peak load shifting, we adopt the peaktoaverage ratio (PAR) defined in [7].
The balance between the total electricity consumption and the supply is shown in Figure 4 for the RTPbased scheduling strategy. In comparison, the difference between the total electricity consumption and the supply is large for the flat pricingbased scheduling strategy. From Figure 5, we see that the RTPbased scheduling strategy can implement peak load shifting by flexible pricing. Specifically, the IESO increases the price at the peak time and decreases the price at the nonpeak time. We show the convergence of algorithms (28) and (30) in Figures 6 and 7, respectively. The results indicate that the generators and the consumers can both search for the optimal strategies within 30 iterations, which demonstrate the rapid convergence speed.
We compare the average price, the PAR, and the daily load in Table 1. The results demonstrate that the RTPbased scheduling strategy lowers the average price compared with the flat pricingbased scheduling strategy and also reduces the PAR. Furthermore, the daily load under the RTPbased scheduling strategy is smaller than that under the flat pricingbased scheduling strategy, which indicates the advantage of electricity saving.
6. Conclusions
In this paper, we propose a RTPbased scheduling strategy based on the hierarchical game in smart grids, where the generators act as the leaders and the consumers act as the followers. The distributed scheduling algorithms are proposed to be implemented by the generators and the consumers with rapid convergence speed. It is shown that the RTPbased scheduling strategy outperforms the flat pricingbased scheduling strategy in terms of the load matching and load shifting and reduces the average price, the daily load, and the PAR. This can benefit both the consumers and the generators in electricity markets. In the future, we will consider the robustness of the proposed scheduling strategy with disturbances and delays.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported in part by National Natural Science Foundation of China under Grant 61074088 and Grant 61203104.