Impact of Communication Availability in a Demand-Side Energy Management System: Differential Game-Theoretic Approach Ryohei Arai∗ , Koji Yamamoto∗ , Takayuki Nishio∗ , and Masahiro Morikura∗ ∗ Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan E-mail:
[email protected]
Abstract—Differential games are multi-agent versions of optimal control problems, which have been used for modeling control systems of smart grids. Thus, a differential-game theoretic approach is a promising mathematical framework enabling to discuss a demand-side energy management system where there are multiple decision-making entities. This paper first indicates that the optimal demand-side management of multiple demandside actors (e.g., consumers, houses, buildings, communities) in a decentralized way is formulated as a differential game. In addition, we point out that an information structure (e.g., open-loop, feedback), that is the most important property of differential games, corresponds to availability of communications. When information shared by communications is available, it enables demand-side actors to act in an appropriate manner, i.e., demand-side actors conduct decentralized feedback control. On the other hand, when shared information is unavailable due to communication failure, demand-side actors do not necessarily give up control and they may use another type of control, such as prediction-based control. For example, demand-side actors are assumed to manage the power consumption to minimize disutility and the electricity rates allocated to the power grid, and to maintain the demand-supply balance by considering a photovoltaic power generation. Numerical analyses demonstrate that the proposed framework enables a trade-off analysis for decentralized controls considering the availability of shared information.
I. I NTRODUCTION Smart grids have attracted attention recently as a technology for managing electric power efficiently. In this system, new power sources are connected with existing power networks and controlled using information and communication technologies. There are also various demand-side actors such as houses, buildings, and communities. These sources and actors should share information, which allows them to act more efficiently because they interact with each other for the use of electric power. Similarly, the supply and demand sides participate in the energy management system, which is one of the most important technologies and known as the “demand-side management.” Many studies have investigated the demand-side management [1]–[4]. Controlling the power consumption of household electrical appliances has been the focus of many demandside management studies. For example, the optimal temperature setting for air-conditioning systems based on the dayahead electricity rate and outdoor temperature forecasts was investigated in [1] where a computer server was connected
to all devices via power-line and wireless communication. The communication technology is important for demand-side management. In [2], the scheduling strategy in the MAC layer of wireless networks comprised of smart meters was investigated, which assumed that the information used to decide the policy was collected via wireless communication. In addition, the energy storage system has been used to determine the demand-side management. In [3], the optimal demand response was considered with energy storage and the concept of disutility was introduced to control the power consumption within an appropriate range. The demand-supply balance is a significant criterion that affects the demand-side management. The demand-side management of a large population was considered based on the energy supply [4]. Optimal control is commonly used for applications of smart grids [5]–[8]. In [5], scheduling methods of power demand tasks using optimal control to minimize the grid operational cost was investigated. In [6], an aggregator for vehicle-to-grid frequency regulation is developed using an optimal control strategy. Previous studies [1]–[5] have assumed that the demand-side actors can share information through communications and they do not consider communication failure. However, it is possible that information is unavailable due to packet loss or electrical equipment failure. Even if communication fails, demand-side actors do not necessarily give up control and they can act using predicted data instead of the actual information. This paper introduces a differential game [9]–[11] to develop a comprehensive framework for decentralized control taking into account the availability of communication. As mentioned above, previous studies that apply optimal control have investigated demand-side management and there are few studies [4] applying differential games. Optimal control cannot perform optimization of multiple inputs but a single one. In smart grids, however, because there are various actors and more than one input that interact with each other, optimal control cannot treats this situation directly. By contrast, differential games regarded as extensions of optimal control are appropriate ways to analyze the situation where there are multiple inputs and they interact with each other. Thus, differential games are highly effective for demand-side management system comprised of multiple actors. In this paper, we formulate a demand-side energy management system and propose a framework for two decen-
tralized control schemes, i.e., decentralized feedback control and decentralized open-loop control. Demand-side actors in the demand-side energy management system use decentralized feedback control when information is available and decentralized open-loop control when information is unavailable. The aim of control is to schedule the power consumption of an electrical appliance used by demand-side actors considering the photovoltaic power generation. In particular, the demandside actors aim to suppress disutility [3], improve the demandsupply balance [4], and minimize the electricity rates on a power grid. In our previous study [12], the information is assumed to be always available between two houses and focused on the data that predicts the output of a photovoltaic power system. In contrast, the present study focuses on the information shared between two houses via communication and discuss the performance with and without communication. This paper is organized as follows. Section II proposes a demand-side energy management system and the relationship between the system and communication. Section III describes the differential game for decentralized control of the system. Section IV formulates the system as a differential game and derives the decentralized open-loop control and decentralized feedback control. Section V makes numerical analysis and discusses the control methods based on the communication state. Concluding remarks are provided in Section VI. II. S YSTEM M ODEL A. Demand-side Energy Management System Fig. 1 shows the model discussed in this paper. This model is a simplified version of the demand-side energy management system model introduced in [3] and [4]. This model comprises two houses and a power grid for the sake of simple analysis. Note this model is scalable in terms of the number of houses. In each house, there is an electrical appliance and the power consumption is assumed to be controllable. Let the power consumption of the electrical appliance in house i (i ∈ {1, 2}) at time t be denoted as pi (t). In each house, there is also a photovoltaic power generation system that produces electrical power ri (t). We define di (t) = pi (t) − ri (t) as insufficient power (di (t) ≥ 0) or surplus power (di (t) < 0) for house i．If di (t) ≥ 0, house i buys power from the power grid and if di (t) < 0, house i sells power to the power grid at a unit electricity price η. Thus, the electricity rates to the grid is written as ηdi (t) = η(pi (t) − ri (t)), where ηdi (t) < 0 represents the income of house i, and ηdi (t) > 0 is the expenses. We define x(t) as the total amount of insufficient power and surplus power for two houses over time t as follows: x(t) ˙ =d1 (t) + d2 (t), x(0) = 0.
(1)
Next, we discuss how the houses select their strategy. Each house aims to minimize expenses paid to the grid when buying power to compensate for insufficient power, i.e., each house attempts to minimize ηdi (t). In this case, the minimization of ηdi (t) results in the minimization of power consumption pi (t). However, if the electrical appliances are heating equipment,
The remainder of the power system
Fig. 1. System model.
insufficient heating may be uncomfortable for the occupants. Thus, we introduce a term to prevent insufficient heating. This term indicates the disutility for consumers in house i, which is defined as Di (pi (t)) = βi (Pi,target − pi (t))2 , where βi is a parameter and Pi,target represents the target power consumption value for house i. The disutility increases as the actual power consumption pi (t) deviates from the target value. Based on [4], we consider the target value of the grid power supply. We denote the target value of the power supply by S and we introduce a term αi (x(T )−S)2 to balance the demand x(t) and supply S, where αi is a parameter for house i, and T represents the end time of control. Note that x(T ) is the total power demand of two houses at the end time of control. Given these factors, the cost for house i, Ji , can be formulated as ∫ T [ ] Ji = ηdi (t) + βi (Pi,target − pi (t))2 dt 0
+ αi (x(T ) − S)2 , i = 1, 2. (2) We refer to the first term as the “integral cost,” the second term as the “terminal cost,” and Ji as the “total cost.” House i minimizes (2) by controlling pi (t). B. Communication State and Decentralized Control Two houses are assumed to control independently power consumption, pi (t), i.e., turn some electrical appliances on and off. When controlling pi (t) to minimize (2), the situation may differ greatly depending on whether both houses can obtain information with respect to x(t). Each house can control the power consumption according to x(t) if it can obtain x(t) at each time t. We refer to this control as “decentralized feedback control.” By contrast, if each house cannot obtain x(t) after communication loss, it can only be controlled based on the last x(t) that can be obtained. We refer to this control as “decentralized open-loop control.” III. D ECENTRALIZED C ONTROL Decentralized feedback control and decentralized openloop control can be analyzed using feedback and open-loop information structures in a differential game. In this section,
we describe the differential game for decentralized control. The following definitions and equations refer to [9]–[11]. A. Differential game A differential game is a framework for analyzing the actions of multiple decision-making entities known as “players” in game theory. In a dynamic and continuous-time system, players act independently to achieve their purpose. At that time, the action of each player has affects the overall state, while the state effects the action of each player. Differential games are used as extensions of optimal control problems. An optimal control problem treats the situation where there is a single player with one objective function, whereas a differential game treats the situation where multiple players interact with each other for their purpose. An N -person linear quadratic differential game is generally written as follows: minimize
Ji (y, u1 , . . . , uN ) ∀i ∈ N , N = {1, 2, . . . , N } ,
subject to
Ji (y, u1 , . . . , uN ) ∫ 1 T 2 ∑ 1 = qi y + rij u2j dt + qiT y 2 , 2 0 2 j∈N ∑ y(t) ˙ = ay(t) + bi ui (t) + c(t), y(0) = y0 ,
ui
i∈N
(3) where a, bi ∈ R, rii > 0, qi ≥ 0, and qiT ≥ 0. c(t) is a known and continuous function. [0, T ] represents time horizon of the game. y(t) denotes the state of the game at time t and y0 represents the initial state of y(t). ui (t) denotes the action of player i. Ji represents the cost function of player i. In the game (3), each player selects a strategy to minimize his cost function. B. Nash Equilibrium As in general non-cooperative games, Nash equilibria are defined in differential games. An action u?i that satisfies (3) is known as an equilibrium action. Moreover, a set of equilibrium actions (u?1 , . . . , u?N ) is known as a Nash equilibrium, which satisfies the following condition: Ji (u?1 , . . . , u?i−1 , u?i , u?i+1 , . . . , u?N ) ≤ Ji (u?1 , . . . , u?i−1 , ui , u?i+1 , . . . , u?N ), ∀i ∈ N . (4) At Nash equilibrium, the cost increases if players change their actions independently. Therefore, players have no incentive to change their actions. C. Information Structure In differential games, the strategy that each player selects and the property of Nash equilibrium vary according to information he can obtain. The framework of available information is known as an “information structure.” Typical information structures are open-loop and feedback. In an open-loop information structure, each player knows the initial value of a state in a game and determines his strategy
as a function of the initial state and time t. This is written as follows: η(t) = {y0 }, t ∈ [0, T ], where ηi (t) represents the information that a player can obtain from the game. With a feedback information structure, however, players know only the value of a state in a game at time t, so they determine their strategy by considering the state y(t): η(t) = {y(t)}, t ∈ [0, T ]. IV. R ELATIONSHIP BETWEEN THE S YSTEM M ODEL AND D ECENTRALIZED C ONTROL In this section, we formulate (1) and (2) as the differential game described in Section III. This section also derives the open-loop Nash equilibrium and the feedback Nash equilibrium. A. Translation from a System Model into a Differential Game The system model described in Section II can be formulated as a differential game, where each house is a player, the power consumption of its appliance is a strategy, and (2) is a cost function that needs to be minimized. We define additional variables as follows: y(t) = x(t) − S, ) ( η , ui (t) = pi (t) − Pi,target − 2βi η c(t) = P1,target − − r1 (t) 2β1 η + P2,target − − r2 (t), 2β2 2 Ji,given = βi Pi,target T ( )2 ] ∫ T[ η − ηri (t) + βi Pi,target − dt. 2βi 0 Using these variables, (1) and (2) are formulated as the following differential game: minimize ui
Ji ∀i ∈ {1, 2} , ∫
T
Ji = Ji,given + 0
subject to
βi ui (t)2 dt + αi y(T )2 , (5)
y(t) ˙ = u1 (t) + u2 (t) + c(t), x(0) − S = −S.
B. Open-loop Nash Equilibrium In this section, we solve a linear quadratic differential game (5) for an open-loop information structure and derive the openloop Nash equilibrium based on [10]. We assume that the following Riccati differential equation has a set of solutions Mi (t): ∑ 1 Mj (t) = 0, M˙ i (t) − Mi (t) 2βj j∈{1,2}
Mi (T ) = 2αi , i ∈ {1, 2}. (6) Thus, the differential game (5) has an open-loop Nash equilibrium solution as follows: 1 [Mi (t)y ? (t) + mi (t)], i ∈ {1, 2}, (7) u?i (t) = − 2βi
where mi (t) represents the solution of the linear differential equation: m ˙ i (t) + Mi (t)c(t) − Mi (t)
∑ j∈{1,2}
1 mj (t) = 0, 2βj
mi (T ) = 0, i ∈ {1, 2}, (8) and y ? (t) represents an equilibrium trajectory generated by ∫ t y ? (t) = Φ(t, 0)y0 + Φ(t, σ)ψ(σ)dσ, 0
˙ σ) = F (t)Φ(t, σ); Φ(σ, σ) = I, Φ(t, ∑ 1 F (t) = − Mi (t), 2βi
D. Time Consistency Time consistency is one of the most important properties in the differential game and it has two types, i.e., weak time consistency and strong time consistency [11]. Let the differential game (3) be denoted by Γ (0, y0 ). In addition, we define the subgame Γ (t1 , yt1 ) of Γ (0, y0 ) as the differential game that starts at t = t1 with the initial state yt1 , i.e., Γ (t1 , yt1 ), which is defined as follows: minimize
Ji (y, u1 , . . . , uN ) ∀i ∈ N , N = {1, 2, . . . , N } ,
subject to
Ji (y, u1 , . . . , uN ) ∫ 1 1 T 2 ∑ qi y + rij u2j dt + qiT y 2 , = 2 t1 2 j∈N ∑ y(t) ˙ = ay(t) + bi ui (t) + c(t), y(t1 ) = yt1 .
ui
(9)
i∈{1,2}
ψ(t) = c(t) −
∑
i∈{1,2}
1 mi (t). 2βi
i∈N
As shown above, the open-loop Nash equilibrium (7) is a function that depends on the initial state y0 and time t. With decentralized open-loop control, house i acts according to (7). C. Feedback Nash Equilibrium In this section, we solve a linear quadratic differential game (5) for a feedback information structure and derive the feedback Nash equilibrium based on [10]. We assume that the following Riccati differential equation has a set of solutions Zi (t): 1 Z˙ i (t) + 2Zi (t)F˜ (t) + Zi (t)2 = 0, 2βi Zi (T ) = 2αi , i ∈ {1, 2}. (10) F˜ (t) is written as follows: ∑ 1 F˜ (t) = − Zi (t). 2βi
(11)
i∈{1,2}
Thus, the differential game (5) has a feedback Nash equilibrium solution as follows: 1 u?i (t, y) = − [Zi (t)y(t) + ζi (t)], i ∈ {1, 2}, (12) 2βi where ζi (t) represents the solution of the linear differential equation: 1 ζ˙i (t) + F˜ (t)ζi (t) + Zi (t)β(t) + Zi (t)ζi (t) = 0, 2βi ζi (T ) = 0, i ∈ {1, 2}, (13) where β(t) is written as: ∑ β(t) = c(t) − i∈{1,2}
1 ζi (t). 2βi
(14)
As shown above, it is obvious that the feedback Nash equilibrium (12) is a function that depends on the game state y(t) at each time t. With decentralized feedback control, house i acts according to (12).
The concept of weak time consistency and strong time consistency are introduced using this subgame. Weak time consistency means that at all time t1 ∈ (0, T ), the equilibrium solution remains the equilibrium solution if y(t1 ) is on the equilibrium trajectory y ? (t). By contrast, strong time consistency means that the equilibrium solution remains the equilibrium solution even when all of the initial states yt1 are off the equilibrium trajectory y ? (t). Next, we discuss the time consistency of the open-loop Nash equilibrium and the feedback Nash equilibrium introduced in Sections IV-B and IV-C, respectively. The open-loop Nash equilibrium has weak time consistency whereas the feedback Nash equilibrium has strong time consistency. In the openloop Nash equilibrium, at t = 0, all equilibrium trajectories after t = 0 are determined and player i makes an equilibrium action u?i (t) according to the trajectories. This is obvious from (7). Therefore, if y(t1 ) = y ? (t1 ) at t1 ∈ (0, T ), an equilibrium action u ˜i (t) in subgame Γ (t1 , yt1 ) is equal to an equilibrium action u?i (t) in differential game Γ (0, y0 ). However, if y(t1 ) 6= y ? (t1 ), u ˜(t) 6= u? (t). Thus, the openloop Nash equilibrium has weak time consistency. By contrast, in the feedback Nash equilibrium, because (12), each player makes an action that considers the game state y(t) at each t. Therefore, if y(t1 ) 6= y ? (t1 ) at t1 ∈ (0, T ), each player acts after considering the difference, and finally, the equilibrium solution u ˜i (t) is equal to u?i (t). Thus, the feedback Nash equilibrium has strong time consistency. V. N UMERICAL A NALYSIS In this section, we evaluate the performance of decentralized open-loop control and decentralized feedback control for the model introduced in Section II based on a numerical analysis. This paper derives the equilibrium strategy by defining the time horizon as one day. With decentralized feedback control, the communication state is normal throughout the day. With decentralized open-loop control, however, the communication fails at the beginning of the day and is not restored all day. On the other hand, when the communication state is normal
TABLE I PARAMETERS .
1 JPY/(kWh)2 0.60 kW 22 JPY/kWh
βi S T
3
0.6
Pattern A Pattern B Pattern C
2.5 ri (kW)
0.7
30 JPY · h/(kWh)2 15.44 kWh 24 h pi (kW)
αi Pi,target η
0.8
2
0.5 0.4
1.5
Decentralized feedback Decentralized open-loop Target
0.3
1 0.2
0.5 0
0
4
8
12 t (h)
16
20
0
4
8
12 t (h)
24
Fig. 2. The amount of power from a photovoltaic power generator [13].
16
20
24
Fig. 3. The power consumption of an electrical appliance (predicted weather: sunny, actual weather: sunny). 0.035
throughout the day, the decentralized open-loop can also be used with the information only at time t = 0.
Decentralized feedback Decentralized open-loop
0.03
The parameters are summarized in Table I. We use Pattern A in Fig. 2 as the amount of power produce by a photovoltaic power generator. Patterns A, B, and C in Fig. 2 correspond to the data for a sunny day, a cloudy day, and a rainy day, respectively. These three values are calculated based on the output pattern of a Sakai photovoltaic power plant [13]. To be specific, they are calculated by multiplying output pattern of Sakai photovoltaic power plant (=actual output/maximum output) by the capacity of a general household photovoltaic power generation system. We assume that the capacity of a solar cell is 4.2 kW, which is based on the capacity of the solar cells installed in newly-built houses between April and September 2012, as described in [14]. B. Performance without Disturbance Fig. 3 shows the power consumption per house when controlling an electrical appliance using each control method. Table II shows the integral cost, i.e., the electricity rates and the disutility, the terminal cost, i.e., demand-supply balance, and the total cost in (2). The performances of both houses are the same because we assume that the conditions of both houses correspond. We also assume that there are no disturbances. Both control methods does not use the actual amount of power generated each day and the predicted data is used when determining the control strategy. In this section, the actual weather is assumed to correspond to the predicted data. Fig. 3 and Table II show that the cost with decentralized open-loop control is smaller than that with decentralized feedback control. This is because of the power gain with both control methods. With a lower power gain, the control is more moderate and the actions are more inefficient. The power TABLE II C OST ( PREDICTED WEATHER : SUNNY, ACTUAL WEATHER : SUNNY ).
Integral cost (JPY) Terminal cost (JPY) Total cost (JPY)
Feedback 10.93 221.10 232.03
Open-loop 10.43 161.14 171.57
gain (1/h)
0.025
A. Parameter
0.02 0.015 0.01 0.005 0
0
4
8
12
16
20
24
t (h)
Fig. 4. Power gain. −1 T gain with decentralized open-loop control is Rii Bi Mi (t) in (7); the power gain with decentralized feedback control is −1 T Rii Bi Zi (t) in (12). Fig. 4 shows both power gains, and it is obvious that the power gain with decentralized openloop control is always greater than that with decentralized feedback control. Therefore, decentralized open-loop control is more efficient than decentralized feedback control because decentralized open-loop control determines the overall equilibrium strategy at t = 0 h whereas decentralized feedback control considers a state at each time t and its control action is moderate. As mentioned above, decentralized open-loop control can control the power consumption more efficiently, irrespective of the communication state.
C. Performance with Disturbance The advantage of decentralized feedback control is that it considers a state y(t) at each time t, so it works efficiently even when there is disturbance. We treat the difference between the actual weather and the predicted data as a disturbance and evaluate the performance of both control methods with disturbance. We treat pattern A in Fig. 2 as the predicted data and pattern B as the actual weather. The power consumption of an electrical appliance is shown in Fig. 5. Fig. 5 shows that with both control methods, each house can maintain the power consumption near the target value Pi,target at t < 7 h, although they could not control the power consumption well at t > 7 h. In particular, the values
0.8
Decentralized feedback Decentralized open-loop Target
0.7
pi (kW)
0.6 0.5 0.4 0.3 0.2
0
4
8
12
16
20
24
t (h)
Fig. 5. The power consumption of an electrical appliance (predicted weather: sunny, actual weather: cloudy). TABLE III D IFFERENCE IN THE TOTAL COSTS COL − CFB (JPY).
Actual weather
Sunny Cloudy Rainy
Predicted weather Sunny Cloudy Rainy −60.46 24.51 93.40 0.33 −12.37 7.07 63.90 3.35 −1.43
with decentralized open-loop control are more distant from Pi,target . Both controls are accurate at t < 7 h because the difference between patterns A and B in Fig. 2 is low at t < 7 h, i.e., there is no sunshine before t = 7 h on a sunny day or a cloudy day. Table III shows the difference between the total cost of decentralized open-loop control COL and that of decentralized feedback control CFB , i.e., COL − CFB using a combination of the predicted weather and the actual weather. We use patterns A, B, and C in Fig. 2 as the data of sunny, cloudy, and rainy days, respectively. Table III shows that COL is lower than CFB when the predicted weather corresponds to the actual weather, whereas CFB is less when it does not correspond. This is because of the time consistency discussed in Section IV-D. Decentralized open-loop control has weak time consistency and the equilibrium strategy determined at t = 0 h remained the equilibrium strategy at t > 0 h only when the state of game y(t) is on the equilibrium trajectory y ? (t), which is determined according to the predicted weather. However, if the actual weather is different from the predicted data, there is a difference between y(t) and y ? (t). Therefore, with decentralized open-loop control, the equilibrium trajectory determined at t = 0 h is not the equilibrium trajectory at t > 7 h when the values of Patterns A and B in Fig. 2 are different. As a result, decentralized open-loop control could not work optimally and the total cost increased if the predicted weather does not correspond with the actual weather. By contrast, decentralized feedback control has strong time consistency and the equilibrium strategy at t = 0 h remains the equilibrium strategy at t > 0 h even when y(t) is off y ? (t). Therefore, by considering the disturbance, the decentralized feedback control works well and the total cost decreases. VI. C ONCLUSION This paper introduced a framework for decentralized control in smart grid based on the availability of communication.
In this framework, we introduced a differential game to formulate two decentralized control schemes. In particular, this study addressed demand-side energy management system and discussed the methods used for controlling the power consumption of electrical appliances in each house considering photovoltaic generation. Our numerical analysis showed that when information is available, decentralized feedback control can work well even if the predicted weather does not correspond to the actual weather. When information is unavailable, however, decentralized open-loop control can work efficiently if the predicted weather corresponds to the actual weather. We would like to emphasize that we have discussed the case of two houses and some fixed parameters for the sake of simplicity and the proposed framework using differential games are scalable in terms of the number of demand-side actors and time dependencies of parameters. ACKNOWLEDGEMENT This work was supported in part by JSPS KAKENHI Grant Number 24246069 and 23760333. R EFERENCES [1] Y. Y. Hong, J. K. Lin, C. P. Wu, and C. C. Chuang, “Multi-objective airconditioning control considering fuzzy parameters using immune clonal selection programming,” IEEE Trans. Smart Grid, vol. 3, no. 4, pp. 1603–1610, Dec. 2012. [2] H. Li, L. Lai, and R. C. Qiu, “Scheduling of wireless metering for power market pricing in smart grid,” IEEE Trans. Smart Grid, vol. 3, no. 4, pp. 1611–1620, Dec. 2012. [3] L. Huang, J. Walrand, and K. Ramchandran, “Optimal demand response with energy storage management,” Proc. IEEE SmartGridComm 2012, pp. 61–66, Tainan, Taiwan, Nov. 2012. [4] Q. Zhu and T. Bas¸ar, “A multi-resolution large population game framework for smart grid demand response management,” in Network Games, Control and Optimization (NetGCooP), 2011 5th International Conference on, pp. 1–8, Oct. 2011. [5] I. Koutsopoulos and L. Tassiulas, “Challenges in demand load control for the smart grid,” Network, IEEE, vol. 25, no. 5, pp. 16–21, 2011. [6] S. Han, S. Han, and K. Sezaki, “Development of an optimal vehicleto-grid aggregator for frequency regulation,” IEEE Trans. Smart Grid, vol. 1, no. 1, pp. 65–72, June 2010. [7] D. Livengood and R. Larson, “The energy box: Locally automated optimal control of residential electricity usage,” Service Science, vol. 1, no. 1, pp. 1–16, 2009. [8] S. Paudyal, C. Canizares, and K. Bhattacharya, “Optimal operation of distribution feeders in smart grids,” IEEE Trans. Industrial Electronics, vol. 58, no. 10, pp. 4495–4503, Oct. 2011. [9] Z. Han, D. Niyato, W. Saad, T. Bas¸ar, and A. Hjørungnes, Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. New York: Cambridge University Press, 2011. [10] T. Bas¸ar and G. J. Olsder, Dynamic Noncooperative Game Theory (Classics in Applied Mathematics), 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics, 1999. [11] J. Engwerda, LQ Dynamic Optimization and Differential Games. Chichester: Wiley, 2005. [12] R. Arai, K. Yamamoto, T. Nishio, and M. Morikura, “Differential gametheoretic framework for a demand-side energy management system,” Proc. IEEE SmartGridComm 2013, Vancouver, Canada, Oct. 2013. [13] Real-Time Information in Sakai Photovoltaic Power Plant. (in Japanese): The Kansai Electric Power Company. [Online]. Available: http://www1.kepco.co.jp/energy/newenergy/monitor.html [14] The Number of Houses with Delivery of Subsidies for Domestic Photovoltaic Power Generation 2012. (in Japanese): Japan Photovoltaic Expansion Center, Japan Photovoltaic Energy Association, Oct. 2012. [Online]. Available: http://www.jpec.or.jp/information/doc/pdat h24koufu 20121022.pdf
