A Game Theoretical Approach to Modeling Energy Consumption with Consumer Preference

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1 A Game Theoretical Approach to Modeling Energy Consumption with Consumer Preference Lihui Bai and Guangyang Xu Department of Industrial Engineering University of Louisville Louisville, Kentucky Qipeng P. Zheng (IEEE member) Department of Industrial Engineering and Management Systems University of Central Florida Orlando, Florida Abstract We propose a new game theoretical equilibrium model to analyze residential users electricity consumption behavior in smart grid where energy usage and price data are exchanged between users and utilities via advanced communication. Consideration is given to users possible preference on convenience over cost-saving under the real-time pricing in smart grid, and each user is assumed to have a preferred time window for using a particular appliance. As a result, each user (player) in the proposed energy consumption game wishes to maximize a payoff or utility consisting of two parts: the negative of electricity cost and the convenience of using appliances during their preferred time windows. Extensive numerical tests suggest that users with less flexibility on their preferred usage times have larger impact on the system performance at equilibrium. This provide insights for utilities to design their pricing based demand response schemes. I. INTRODUCTION The department of energy estimates that residential buildings accounted for 37% of the total electricity consumption in 2008, and the percentage will grow much higher upon a broad adoption of electrical vehicles. Thus, energy consumption management in residential buildings becomes a pressing issue for our society to be sustainable. Particularly, when consumers use of electricity is mainly driven by convenience, coincident demand occurs, resulting in electric load peaks that greatly increase the generation costs. In the advent of smart grid where price and usage data can be exchanged between consumers and utilities, demand side management (DSM) or demand response (DR) presents significant opportunities for load shifting and leveling. Early approach of consumer response can date back to the 1980 s [1]. More recently, the study of strategies to flatten load curves in DSM/DR has required detailed modeling of consumers energy consumption behaviors. To this end, gametheoretic approaches have been the main stream (e.g., Saad et al. [2] and Fadlullah et al. [3]) for modeling electricity consumers behaviors. Furthermore, pricing-based approaches (e.g., Samadi et al. [4]) are proposed to incentivize consumers to shift their electrical loads from peak to off-peak hours. In addition, incentive-based demand response approaches are also effective in load shifting (e.g., Zhong et al. [5]). In both pricing and incentive based DSM/DR approaches, because all consumers respond to the price signals by adjusting their consumption behaviors, many use Stackelberg games to determine proper incentives or pricing schemes(e.g., Meng and Zeng [6] and Maharjan et al. [7]). More related to our work, Mohsenian-Rad et al. [8] propose a distributed algorithm to study consumers optimal energy consumption scheduling. Their work assumes that each consumer minimizes his/her utility payment, while we assume consumers value the convenience of using appliances at preferred times and incorporate this in the model. In addition, Chen et al. [9] and Mohsenian-Rad et al. [10] allow users to delay the starting time for using appliances in order to reduce cost. However, users experience on convenience is not directly modeled in their works as is in our approach. Furthermore, Li et al. [11] consider regulated utilities who wish to maximize social welfare as well as individual consumer behaviors (under the game theoretical framework). They show that there exist time-varying pricing schemes that would maximize individuals own benefits while simultaneously maximizing social welfare. Finally, Maharjan et al. [7] assume users will choose different utility companies through a Stackelberg game. In summary, the current paper assumes that consumers take into consideration not only their energy costs but also the convenience of their energy consumption schedule, and the latter is the focus for subsequent numerical simulations. The contribution of the paper is three fold. First, we propose a game theoretical user equilibrium model to describe users energy consumption behavior that explicitly considers users possible preference on convenience over cost-saving. In this user equilibrium model, each user maximizes his/her payoff consisting of convenience and cost. This makes the proposed model more realistic. Indeed, in a recent smart meter/appliances pilot program by General Electric (GE) and Louisville Gas & Electric Company (LG&E), participants praised the program for it allows them to override the costsaving based schedule. Second, a centralized optimization model is developed as a benchmark for the user equilibrium model. The system model is for a central controller to minimize the system-wide energy costs. Third, extensive sensitivity analysis focusing on the effects of users value for convenience on system performance at equilibrium is conducted. It provides insights on how consumers with various monetary values for convenience and with various flexibility on their energy

2 consumption schedule affect the system equilibrium differently. Numerical experiments show that system performance is affected by both consumers preferences and their value for convenience. Users who are less flexible in shifting their consumption schedule have more influence on the equilibrium. Further, consumers who value convenience higher will have a larger impact on the system s total cost. The remainder of the paper is organized as follows. Sections II and III present the system optimum and user equilibrium models respectively. Section IV discusses numerical experiments, and section V offers conclusions and future research. II. THE SYSTEM OPTIMUM MODEL Consider a local area power system with n users and a set of appliances A for each user. Assume each user i has a daily energy demand D i,a for appliance a A. We define a 24- hour daily cycle with t T = {1, 2,, 24}. Further, let E i,a be the maximum amount of energy consumed by user i on appliance a during one unit time. In addition, define a set of unacceptable time intervals Ti,a 0 T during which user i does not wish to use appliance a, and a set of preferred time intervals Ti,a 1 T (T i,a 1 T i,a 0 = ) during which user i prefers to use appliance a. Using this notation and letting decision variable x t i,a be the amount of energy consumed on appliance a by user i at time t, the system model for the energy controller can be formulated as follows: SO: min T t=1 [f t( i,a xt i,a ) i,a xt i,a ] (1) T s.t. x t i,a = D i,a, i, a (2) t=1 x t i,a E i,a, i, a, t (3) x t i,a = 0, i, a, t : t T 0 i,a (4) x t i,a 0, (5) where f t ( ) represents the unit electricity (generation) cost at time t, which is a monotone increasing function of the total electricity consumption at time t, i.e., i,a xt i,a. In the SO model, the objective in (1) is for the central controller to minimize the total electricity cost required to serve all users. Constraints (2) ensure that user i s energy demand for appliance a is met. Constraints (3) state that the total energy used by user i s appliance a during each time interval does not exceed E i,a, an upper bound due to technical specification. Finally, constraints (4) ensure that user i does not use appliance a at any time interval t Ti,a 0. The SO model assumes a centralized decision system where the central area controller in the power distribution network wishes to coordinate energy consumption for all of its subscribers. Thus, the SO model provides an energy consumption profile with the least electricity cost. On the other hand, in practice each subscriber may be more interested in their own electricity usage, but not so much in others or even the average usage for the entire system. Thus, a game theoretical user equilibrium model is suitable for describing the individual subscriber s energy consumption behavior. III. THE USER EQUILIBRIUM MODEL In modeling a user s decision on when and how much to use his/her appliances, we consider not only cost but also the convenience for the user to be able to use an appliance during his/her preferred times. Thus, the user equilibrium model assumes each user i maximizes the following payoff or utility: U i = T t=1 [f t(x i, x i ) a xt i,a ] + u i (x i ) (6) where f t ( ), as previously defined, is a function of consumption profiles for user i, x i, and those for all others x i. Thus, under the non-cooperative game framework in the sense of Nash and Cournot, electricity consumers are the players in the game who wishes to maximize their U i non-cooperatively. In (6), the first term represents the total energy cost, thus the disutility, for user i, and the second term defines the convenience experienced by user i, calculated by his/her personal utility function u i ( ) in terms of monetary value. In general, u i (x i ) can incorporate different monetary values or functions toward different appliances and time periods, e.g., t T πt i,a νt i,a (xt i,a ), where πt i,a and νt i,a are u i (x i ) = a A the monetary value of convenience and the utility function for user i to use appliance a at time t, respectively. Further, the reason, for a user to take into account the unit generation cost to calculate his/her total consumption cost, is the wide use of smart meters which enable the user-to-generator and user-touser communication in the smart grid. Hence, in a distributed manner, each user solves the following user s problem: UO i : max T t=1 [f t(x i, x i ) a xt i,a ] + u i (x i )(7) T s.t. x t i,a = D i,a, a (8) t=1 x t i,a E i,a, a, t (9) x t i,a = 0, a, t : t T 0 i,a (10) x t i,a 0, a, t (11) The UO i model is for each user i to maximize his/her payoff assuming the knowledge of others usage profile x i {x t j,a } a,j i. The decision variables in UO i only pertain to user i s energy consumption profile x t i,a. Thus, the objective (7) for user i is to maximize the total utility, i.e., the conveniencebased utility less the energy cost. Note that in calculating the energy cost, the generation cost function is rewritten as f t (x i, x i ) in order to distinguish user i s decision variable x i from the input parameter x i. Constraints (8)-(11) for the UO model are similar to (2)-(5) in the SO model. Finally, if each user solves his/her own UO i, then the system of n user problems, i.e., {UO i } i=1,,n, may reach an equilibrium defined below. Definition 1. Let X i = {x i (8) (11)} be the set of feasible consumption profiles and x {x i } i=1,,n be the usage profile for an n-user system. Then, x is at user equilibrium if and only if each user i does not have the incentive to

3 unilaterally change his/her optimal consumption profile x i, i.e., U i (x i ; x i ) U i(x i ; x i ), x i X i, i = 1,, n. (12) Under some mild convexity assumptions, the above user equilibrium exists uniquely. The next section presents numerical results on solving the systems of {UO i } i=1,,n. IV. COMPUTATIONAL RESULTS In this section, we discuss the results from our numerical simulations for evaluating the proposed system and user equilibrium models. Both SO and UO models were implemented and solved in GAMS [12], a state-of-the-art modeling language for nonlinear programs. All simulations were run on a 16-core dual Opteron CPU server with 32GB of memory running opensuse 11 Linux. Common to all numerical examples and test instances is the way we model the convenience experienced by users. To do so, we introduce the notion of preferred usage window. The preferred usage window, denoted as Ti,a 1 T, represents a set of time periods when user i prefers to use appliance a. Note that associated with user i and appliance a, there is also a time window Ti,a 0 T when the user does not wish to use appliance a. Thus, Ti,a 0 T i,a 1 = holds. Using the preferred usage window, the (second) convenience term as required in the utility function (6) is defined as ( ) 1/2 pi,a u i (x i ) = π i, (13) a D i,a where π i is the utility coefficient representing the monetary value of convenience for user i and p i,a = t T x t i,a 1 i,a is the amount of electricity from appliance a used by user i during his/her preferred usage window. Subsequently, u i is the utility value based on the proportion of total demand for user i that is fulfilled during preferred usage window. We note that this specific form of utility function is widely used in economics and decision analysis for modeling users preference (e.g., Keeney and Raiffa [13]). Without loss of generality, in this section, we simplify the presentation by using [α, β] to denote T \ Ti,a 0 and [α p, β p ] to denote Ti,a 1. Thus, 1 α α p β p β 24. A. Relationship between UO and SO consumption profiles To begin the evaluation of the proposed user equilibrium model, we illustrate the difference between the outcomes of the SO and UO models, where utilities for all users π i are set to zero in the UO model. The latter is because we would like to study without introducing the notion of convenience utility, how the proposed UO model performs against the SO model. In section IV-B, we discuss extensively the UO model with non-zero utilities π i. Consider a simple example with two users, two appliances and four time intervals. The attributes of the two users are as in Table I, where α, β, α p and β p are as defined above. For simplicity, we use the unit electricity price f t (l t ) = l t, where l t is the total load at time t. Then, solving TABLE I PARAMETER VALUES FOR EXAMPLE 1 user appliance D i,a E i,a α β α p β p the SO and UO models for Example 1 yields the optimal consumption profiles displayed in Table II, where the SO solution is denoted by x t i,a and the UO solution by ˆxt i,a. Note again the UO solution is obtained by setting utilities = π 2 = 0. Table II suggests that the optimal solutions to the SO and UO models (even when the latter has zero utilities for both users) are different. This necessitates the study of the user equilibrium model, while using the system model as a benchmark. Second, the total electricity cost for the SO solution is , less than the that for the UO solution (502.25). This is consistent with the general knowledge that in a non-cooperative game the equilibrium solution does not necessarily yield the maximal system-wide payoff. TABLE II OPTIMAL UO AND SO SOLUTIONS FOR EXAMPLE 1 user appliance time SO- x t i,a UO-ˆx t i,a aggregate cost for user aggregate cost for user aggregate cost for the system B. Impacts of various users groups on the equilibrium The data used in extended numerical studies are based on daily energy consumptions of three prototypical appliances: dishwasher, plug-in hybrid electric vehicle (PHEV) and air conditioner. The unit cost is f t (l t ) = c 0 +cl t where c 0 = 7.43 cents and c = 1.55 cents per kwh. This section reports extensive sensitivity analysis for the UO and SO models based on variations to the baseline scenario summarized in Table III. In words, there are two users, three appliances and 24 time intervals. Setting [α, β] = [1, 24] allows both users to be able to use all three appliances any time. Note that both users share the same profile with respect to demand D i,a and upper bound E i,a. Given special attention is the design of the preferred usage window for users 1 and 2. User 1 s preferred usage window

4 for using appliance 1 is [3,10], three hours shorter than that for user 2 ([3,13]). Similar observations can be made for appliances 2 and 3 between user 1 and user 2. Practically, user 1 represent those with less flexibility on their preferred time, while user 2 represents those with more flexibility. One focus of our subsequent sensitivity analysis is how these two groups may affect the equilibrium solution differently. TABLE III PARAMETER VALUES FOR BASELINE SCENARIO user appliance D i,a E i,a α β α p β p Utilities: =π 2 =5 Table IV displays the aggregate measures for the SO and UO solutions when varying the utility coefficient for user 2, i.e., π 2. Recall that π i represents the user i s monetary value for the convenience utility. The aggregate measures include the total electricity cost as well as the total disutility (the total electricity cost minus the total convenience utilities), for both the SO and UO solutions. Note that the SO is independent of utility coefficients, thus the electricity cost for the SO solution is fixed regardless values of π i. From Table IV, as user 2 s monetary value of convenience increases from 5 (the baseline value) to 100, the system-wide electricity cost for the UO solution increases from to On the other hand, the disutility of the UO solution decreases from to , which indicates the increase of the electricity cost is dominated by the increase of the convenience. Similarly, the disutility of the SO solution decreases (from to ) as well when π 2 increases. Finally, for all cases 1 through 11, the disutility of the UO solution is consistently more appealing, i.e., less, than that of the SO solution. This validates the user equilibrium model in that it provides the best utility for users, whereas the system model provides the best cost for the central controller. Similar results are obtained when varying user 1 s utility coefficient. TABLE IV AGGREGATE MEASURES VS. UTILITY FOR USER 2 SO UO Case π 2 Cost Disutility Cost Disutility More interestingly, Figure 1 indicates that user 1 and user 2 have different impact on the system cost at equilibrium. In Figure 1, the series corresponds to the system-wide electricity cost with varying, and the series with varying π 2. From the figure, the series is consistently above the series, implying that increasing the utility value for user 1, who has less flexibility on the preferred usage window, has larger effect than increasing that for user 2, who has more flexibility. To illustrate, point A in the series in Figure 1 represents the case where = 5 and π 2 = 55 and point B in the series represents the case where = 55 and π 2 = 5. Clearly, = 55 yields a higher electricity cost ( ) than does π 2 = 55 ( ), when compared to the baseline cost for the UO solution ( ). UO Cost π 2 B(π 2 =55) A( =55) π or 2 Fig. 1. UO Cost vs. Utility Finally, we examine the convenience experienced by users for the SO and UO solutions under various scenarios. Recall that the (second) convenience term in the utility function (6) is defined as u i (x i ) = π i a ( pi,a D i,a ) 1/2, where p i,a = t T x t i,a 1 i,a is the amount of electricity used from appliance a by user i during the preferred usage window. In subsequent analysis, we define the average percentage of preferred usage (APPU) to be APPU = n i=1 p i/n, where p i = p i,a a A D i,a / A. In other words, p i is user i s average percentage of preferred usage over all appliances, and APPU is the average of p i over all users. Figure 2 compares how the APPU for both users is affected by the change of and π 2. From this figure, the system APPU is higher in the situation where π 2 is fixed and increases, compared to the situation where is fixed and π 2 increases. This suggests that user 1 has more influence on the equilibrium in terms of the convenience utility, which is consistent with previous observations in terms of the system costs. We also study the effect of the changing preferred usage window, i.e., α p and β p, on the equilibrium solution. The baseline scenario in this experiment is the same as the baseline in previous sensitivity studies except for = π 2 = 50. Table V displays the SO and UO solutions for four cases. Case 1 corresponds to the baseline. Case 2 is constructed from the baseline by spreading out user 1 s preferred windows for the three appliances so that they do not overlap with each other. As a result, the electricity cost and disutiliuty for the UO solution decrease from to , and from to , respectively. This implies that if a user is willing to spread out his/her preferred usage window between various

5 APPU π or π 2 Fig. 2. Average Percentage of Preferred Usage vs. Utility appliances, the entire system is better off at the equilibrium. Compared to case 1, case 3 spreads out the preferred usage window for user 2, although these intervals still overlap but to lesser degree than in case 1. Similar to case 2, in case 3 the total cost and disutility decrease from to , and from to , respectively, when compared to the baseline case 1. Finally, case 4 represents a scenario where there is a large overlap amongst all six preferred usage windows for three appliances and two users. Consequently, the system is worse off at equilibrium, experiencing higher cost ( vs ) and disutility ( vs ). Case TABLE V UO SOLUTIONS FOR VARIOUS PREFERRED WINDOWS Appliance User 1 User 2 UO [α p, β p] [α p, β p] Cost Disutility 1 [3,10] [3,13] 2 [4,9] [4,12] [5,15] [5,18] 1 [1,8] [3,13] 2 [9,14] [4,12] [14,24] [5,18] 1 [3,10] [1,11] 2 [4,9] [16,24] [5,15] [11,24] 1 [3,10] [3,13] 2 [3,8] [3,11] [3,13] [3,16] V. CONCLUSIONS We consider residential end users in a smart grid who wish not only to minimize their electricity cost but to maximize the convenience of using appliances during their preferred times. A game theoretical user equilibrium model is developed for each user to maximize his/her utility, which equals the monetary value for convenience minus cost. As a benchmark, a centralized system model is also developed for central controllers to minimize the total system-wide electricity cost. Numerically, we demonstrate in general the solutions to the two models are different, even when the convenience utility is not considered. In terms of the total electricity cost of all users, the system model yields lower costs than does the equilibrium model. Our sensitivity analysis suggest that: 1) as users increase their values for convenience, the system s total electricity cost increase at the equilibrium; 2) users with less flexibility on their preferred time windows for various appliances have larger impact on the total systemwide measures (e.g., cost and average percentage of preferred usage) at equilibrium; 3) the lesser degree of overlap amongst all users preferred usage windows for various appliances yields a better system performance (e.g., reduced total cost) at equilibrium. These conclusions provide unique insights for utilities to properly design their demand response programs. Future research includes large scale numerical simulations enabled by efficient algorithms for computing equilibria, and possible pricing models to reduce the gap between the system and equilibrium models. ACKNOWLEDGEMENTS This research was partly supported by the Kentucky Science and Engineering Foundation grant KSEF-2808-RDE-016, the NSF grant ECCS , and the Leigh Ann Conn Graduate Fellowship at the Conn Center For Renewable Energy Research at the University of Louisville. REFERENCES [1] B. Daryanian, R. E. Bohn, and R. D. Tabors, Optimal demand-side response to electricty spot prices for storage-type customers, IEEE Transactions on Power Systems, vol. 4, no. 3, pp , August [2] W. Saad, Z. Han, H. V. Poor, and T. Basar, Game theoretic methods for the smart grid, February 2012, arxiv: v1 [cs.it]. [3] Z. Fadlullah, Y. Nozaki, A. Takeuchi, and N. Kato, A survey of game theoretic approaches in smart grid, in Wireless Communications and Signal Processing (WCSP), 2011 International Conference on, 2011, pp [4] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V. Wong, and J. 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