Game Theory and Pricing Strategies for Demand ... - Semantic Scholar
Game Theory and Pricing Strategies for Demand-Side Management in the Smart Grid Firooz B. Saghezchi∗ , Fatemeh B. Saghezchi† , Alberto Nascimento∗‡ , and Jonathan Rodriguez∗ ∗ Instituto
de Telecomunicac¸o˜ es, Campus Universit´ario de Santiago, 3810-193 Aveiro, Portugal {firooz, jonathan}@av.it.pt † Departamento de Economia, Gest˜ao e Engenharia Industrial, Universidade de Aveiro, Portugal [email protected] ‡ Universidade da Madeira, Funchal, Portugal [email protected] Abstract—Demand-side management is an effective means to optimize resource utilization in the electricity grid. The smart grid can enable the utility company to shape the users consumption by adopting appropriate pricing strategies. Having received the price information, each user may independently schedule its appliances to minimize its electricity payment. In this paper, we consider a smart grid scenario with a single utility company and multiple users where the utility company adopts day-ahead pricing strategy. We formulate the problem as a binary linear programming problem. The simulation results show that users can reduce their bills by 25% using the proposed technique. Keywords—game theory, demand-side management, pricing strategy, PAR reduction, appliance scheduling, smart grid.
I.
I NTRODUCTION
Game theory is a mathematical tool to analyse potentially arising conflict of interest among independent rational agents who are intelligent and seek to maximize their own benefit when they strategically interact with each other [1]–[3]. Having classically used as a toolkit to analyse economic and political problems, game theory has recently attracted considerable amount of interest from engineers and researchers to analyse smart grid problems including the demand-side management (e.g. see [3]–[6] and references therein). Integration of information technology with the power system not only can enable protection and troubleshooting but also can enable utility company and customers to actively participate for the demand-side management (DSM). The utility company can control and shape the customers’ consumption by controlling the price of electricity to reduce peak-to-average (PAR) load ratio (e.g. by increasing the price of electricity during peak hours). On the other hand, the supporting information technology enables customers to monitor their hourly energy consumption, as well as the variations of the price of electricity in the market (e.g. either due to variations of the price of oil or due to intermittent nature of renewable resources such as wind and solar energy) to react accordingly in order to conserve energy or reduce their electricity payments by shifting their elastic1 demand from peak hours to off-peak hours–when the price of electricity is lower. Pricing is as an effective means for a utility company to control and shape electricity consumption of the users [7]. For 1 Price elasticity is a microeconomic term defined as the variation in the quantity demanded when the price increases one unit.
example, inclining block pricing (IBP) has been adopted for many years to make electricity affordable for low-income users to fulfill their basic needs such as lighting and refrigerating while charging higher rates for users how consume higher amount of energy for the purposes such as air conditioning or heating. There are also several other pricing strategies used by utility companies to control the demand response such as critical-peak pricing (CPP), time-of-use pricing (TUP), realtime pricing (RTP), and day-ahead pricing (DAP). Specifically, in DAP strategy, the utility company sets the price of energy for the next 24 hours and advertises it to the users. Non-cooperative game theory techniques can be employed to analyse the customers behavior in response to any change in the price of electricity in the market. In a non-cooperative game, every player adheres to a strategy that maximizes its own utility, without necessarily concerning about the social welfare. The term non-cooperative does not necessarily mean that there is no cooperation between players; it rather implies that there is no communication or binding agreement between them. A price-aware user may shift its unnecessary demand from peak hours, when the price of electricity is high, to off-peak hours, when the price becomes lower. A residential user has generally two types of appliances. The first type includes items such as lights and refrigerator that are price inelastic; i.e. no matter how much is the price of electricity in the market, these appliances are needed to be always on and cannot be switched off or shifted to other hours of day. In contrast, the second type of appliances includes items such as washing machine, dish washer, cloth dryer, and plug-in hybrid electric vehicle (PHEV) that are price elastic; unless their task is finished before their associated deadline they can be shifted to other hours of day during which the price of electricity is more affordable. In this paper, we use game theory and DAP strategy to optimize the demand response by motivating the consumers to schedule their price elastic demand from peak hours to off-peak hours. Specifically, we consider a scenario with ten residential users where every user has multiple elastic and nonelastic appliances. The rest of this paper is organized as follows. Section II describes the system model. Section III presents the game model for the energy consumption scheduling of home appliances, and Section IV formulates its Nash Equilibrium (NE) solution
Fig. 1.
Assumed demand side management (DSM) scenario.
as a binary linear programming problem. Section V depicts the simulation setup, and Section VI discusses the simulation results. Finally, Section VII concludes the paper. II.
S YSTEM M ODEL
We assume a DSM problem that involves a utility company and several residential consumers as illustrated by Fig. 1. The utility company first estimates the electricity demand for next hours and sets the price for different hours. Then, the utility company advertises these prices to the customers over the supporting digital communication network. Then, each user optimizes its energy consumption by adopting the best scheduling for its appliances based on this received price information from the utility company. In the following, we assume a day-ahead pricing (DAP) strategy where the utility company announces in advance the price of electricity for each hour for the next 24 hours. Next, we summarize game theory formulation for the DSM problem. Let N denote the set of users and |N | = N . For each customer n ∈ N , let lnh denote the total load at hour h ∈ H = {1, ..., H}, where H = 24. The daily load for user n is denoted by ln = [ln1 , ..., lnH ]. Based on these definitions, the total aggregate load of all users at each hour of day h ∈ H can be calculated as X Lh = lnh . (1) n∈N
Fig. 2. Scheduler embedded in smart meter to schedule household appliances.
n ∈ N at hour h ∈ H. Clearly, the total load of user n ∈ N is obtained as X lnh = xhn,a , h ∈ H. (6) a∈An
As illustrated by Fig. 2, the task of scheduler in user n’s smart meter is to determine the optimal choice of the energy consumption vector xn,a for each appliance a ∈ An to shape user n’s daily load profile. Next, we identify the feasible choices of the energy consumption scheduling vectors based on users’ energy needs. For each user n ∈ N and each appliance a ∈ An , we denote the predetermined total daily energy consumption as En,a . Note that the scheduler does not aim to change the amount of energy consumption, but instead to systematically manage and shift it, e.g., in order to reduce the PAR or minimize the energy cost. In this case, the user needs to determine the beginning αn,a ∈ H and the end βn,a ∈ H of a time interval that appliance a can be scheduled. Clearly αn,a ≤ βn,a . For example, a user may select αn,a = 6 PM and βn,a = 8 AM for its PHEV to have it ready before going to work. This imposes certain constraint on scheduling vector xn,a . Furthermore, we denote that βn,a
The daily peak and average load levels are calculated as Lpeak = max Lh h∈H
and Lavg
1 X = Lh , H
X (2)
(3)
respectively. Therefore, the PAR in load demand is Lpeak H maxh∈H Lh = P . Lavg h∈H Lh
(4)
For each user n ∈ N , let An denote the set of household appliances such as lights, washing machine, dish washer, refrigerator, PHEV, and so on. For each appliance a ∈ An , we define an energy consumption scheduling vector xn,a = [x1n,a , ..., xH n,a ]
(7)
h=αn,a
and
h∈H
P AR =
xhn,a = En,a
(5)
where scalar xhn,a denotes the corresponding one-hour energy consumption that is scheduled for appliance a ∈ An by user
xhn,a = 0, ∀h ∈ H\Hn,a
(8)
where Hn,a = {αn,a , ..., βn,a }. For each appliance, the time interval provided by the user needs to be larger than or equal to the time interval needed to finish the operation. The total energy consumed by all appliances in the system over 24 hours is equal to sum of the daily energy consumption of all loads/appliances. That is, we always have the following energy balance relationship. X X X Lh = En,a . (9) h∈H
n∈N a∈An
In general, the operation of some appliances may not be time shiftable and they may have strict energy consumption scheduling constraints. For example, a refrigerator may have
TABLE I.
N ON - SHIFTABLE A PPLIANCES
Appliance Light Refrigerator
Power (W) 300 30
Stove
1200
Oven
3500
Kettle
2000
Toaster
850
TV DVD Player Modem Hair Dryer
200 10 10 2000
Grill
1800
Deep Fryer
1600
Coffee Machine
1600
Freezer PC
20 200
Start 19:00 00:00 12:00 19:00 12:15 19:15 08:30 19:00 21:00 08:30 20:00 08:00 08:00 00:00 08:10 12:40 19:40 12:45 19:45 08:30 13:30 00:00 09:00
TABLE II.
End 24:00 24:00 13:00 20:00 13:00 20:00 08:35 19:05 21:05 08:35 20:05 24:00 24:00 24:00 08:20 13:00 20:00 13:00 20:00 08:35 13:35 24:00 24:00
and the We define the minimum standby power level max for each appliance a ∈ An and maximum power level γn,a for each user n ∈ N . We assume that
X
III.
250 200 2100 700 1800 2000 5500 5 1500 250 250 700 12
Duration 09:00 1:00 1:00 01:00 03:30 02:00 02:00 00:10 00:30 02:30 02:00 01:00 01:00 01:00 00:15 5:00
m∈N a∈Am
Un (x∗n ; x∗−n ) ≥ Un (xn ; x∗−n ), ∀n ∈ N .
h=αn,a
(11) An energy consumption scheduling vector calculated by the scheduler in user n’s smart meter is valid only if we have xn ∈ Xn .
3000
Ventilation Washing Machine Tumble Dryer Dish Washer Vacuum Cleaner Iron Water Heater Mobile Phone Charger Electric Fire Dehumidifier Towel Rail Juicer Tablet Charger
Deadline 08:00 18:00 08:00 22:00 20:00 22:00 19:00 22:00 24:00 08:00 08:00 24:00 22:00 24:00 13:00 08:00
The Nash equilibrium (NE) of this game is defined as a strategy profile that no player can benefit by unilaterally deviating from it. That is, the energy consumption scheduling vector x∗n , ∀n ∈ N form an NE if and only if
xhn,a = En,a , xhn,a = 0 ∀h ∈ H\Hn,a ,
min max γn,a ≤ xhn,a ≤ γn,a ∀h ∈ Hn,a }.
Space Heater
Start 18:00 15:00 03:00 09:00 09:00 09:00 14:00 9:00 09:00 00:00 18:00 18:00 09:00 00:00 09:00 18:00
Here, x−n = [x1 , ..., xn−1 , xn+1 , ..., xN ] denotes the energy consumption scheduling vectors of all users other than user n ∈ N , and ph (·) denotes the price of electricity during hour h ∈ H which might be a function of the aggregate energy consumption of users (including user n) during the same hour.
(10)
βn,a
Xn = {xn |
Power (W) 1100
h=1
min γn,a
For notational simplicity, we introduce vector xn for each user n ∈ N , which is formed by stacking up energy consumption scheduling vectors xn,a for all appliances a ∈ An . In this regard, we can define a feasible set for energy consumption scheduling vector for user n ∈ N as follows.
Appliance PHEV
its appliances, which is given by the following utility function. P a∈An En,a P Un (xn ; x−n ) = − P × m∈N a∈Am Em,a (12) H X X X h xm,a ) ph (
to be on all the time. In that case, αn,a = 1 and βn,a = 24. As shown in Fig. 2, the scheduler in the smart meter has no impact on the energy consumption scheduling of non-shiftable household appliances.
min max γa,n ≤ xhn,a ≤ γn,a , ∀h ∈ Hn,a .
S HIFTABLE A PPLIANCES
(13)
As long as the cost function ph (·) is a strictly convex and increasing function for each hour h ∈ H, the NE of the energy consumption scheduling game always exists and is unique [4, Theorem 1]. Moreover, this unique NE maximizes the social welfare (or equivalently minimizes the total aggregate energy cost of all users) [4, Theorem 2].
E NERGY C ONSUMPTION S CHEDULING G AME
We are now ready to define the energy consumption scheduling game. Note that we model the energy consumption scheduling problem as a strategic game since in general the strategy of each player for adopting an energy consumption scheduling vector may affect other users through influencing the price of electricity in the market. The energy consumption scheduling game is defined as follows: •
Players: Registered users in set N .
•
Strategies: Each user n ∈ N selects its energy consumption scheduling vector xn to maximize its payoff.
•
Payoffs: The payoff for user n ∈ N is defined as the negative of its daily total energy cost for all of
IV.
NASH E QUILIBRIUM OF THE E NERGY C ONSUMPTION S CHEDULING G AME
The NE of the game is the energy consumption scheduling vector xn that minimizes the electricity payment of each user n ∈ N . In the following, we formulate the problem of finding the NE as a binary linear programming problem. To do so, we first define a new auxiliary binary decision vector yn which is the same size as xn and is equal to 1 when xn 6= 0 and is equal to 0 otherwise. Considering DAP pricing strategy with price vector f = [f 1 , ..., f 24 ], which includes the price for different hours of day, the NE of the game is reduced to the solution of the following binary linear programming problem.
Demand (kWh)
0.2 0.1 0
Fig. 3.
1
8
Hour (h)
18
40 20 0
1
7
24
14 Hour (h)
20
24
20
24
20
24
20
24
(a) Aggregate demand 10 Cost (Euro)
Price (Euro/kWh)
0.3
Price of electricity during different hours of day.
5 0
min fT yn
1
7
yn
14 Hour (h)
(b) Aggregate cost
s.t. : βn,a
Fig. 4. h pn,a yn,a = En,a
Hourly demand and cost before scheduling.
(14)
h=αn,a h yn,a = 0 ∀h ∈ H\Hn,a h yn,a ∈ {0, 1} ∀h ∈ Hn,a .
where pn,a is the power consumption of appliance a ∈ An of user n ∈ N .
Demand (kWh)
X
60 40 20 0
1
7
14 Hour (h)
(a) Aggregate demand
S IMULATION S ETUP
For simulations, we consider N = 10 residential users. Each user has 10 to 15 non-shiftable appliances (e.g. refrigerator, light, and stove) as well as 10 to 15 shiftable appliances (e.g. PHEV, washing machine, and dish washer). Table I and Table II provide complete lists of these non-shiftable and shiftable appliances, respectively, as well as their power consumption values. Each table has two parts. The first part, which includes the first ten appliances, represents the basic appliances that we assume every user has. Whereas, the second part, which is separated by a double horizontal line, includes five optional devices that each user may have. In simulations, we generate two random integer numbers between 1 and 5 for each user to select its optional appliances: one for shiftable and the other for non-shiftable appliances. For instance, if the random number is 2 for non-shiftable appliances of a user, we add grill and deep fryer to its list of non-shiftable appliances. Table I also provides the start and end times of each nonshiftable appliance. For example, the lights need to be on in the evening from 19:00 to 24:00. Similarly, Table II provides the start time, duration, and the deadline for each shiftable appliance. As an example, PHEV needs 9 hours to charge that can start once the user arrives home at 18:00 and it needs to be ready by 8:00 AM the day after when the user goes to work. For the purpose of simulation, we consider the DAP pricing model illustrated by Fig. 3 As seen in the figure, the price is lower over night (from mid-night to 8:00 AM), comparing 10 cents per kWh to 20 cents per kWh during the day. Moreover, the price further increases during the peak hours (between 18:00 and 23:00). Finally, we assume appliances have no power control mechanism and are either on or off. When they are on, they consume their full power rate, while they consume min no power when they are off, i.e. γn,a = 0.
Cost (Euro)
V.
10 5 0
1
7
14 Hour (h)
(b) Aggregate cost Fig. 5.
Hourly demand and cost after scheduling.
VI.
N UMERICAL R ESULTS AND D ISCUSSION
Fig. 4(a) illustrates the aggregate hourly electricity demand before activating the schedulers embedded in the smart meters. As seen in the figure the peak demand occurs at around 20:00, while between 4:00 AM and 6:00 AM the demand is considerably low. It is also worth mentioning that the significant demand between the mid-night and 3:00 AM is mostly due to PHEV charging that starts at 6:00 PM and finishes at 3:00 AM. Correspondingly, Fig. 4(b) illustrates the aggregate hourly cost considering the assumed DAH pricing model of Fig. 3. Fig. 5(a) shows the aggregate hourly electricity demand when the users activate the schedulers, while Fig. 5(b) shows the corresponding electricity cost. As clear from Fig. 5(a), the assumed pricing strategy successfully shifts the demand from peak hours to off-peak hours. However, the synchronization of the shiftable appliances introduces new peak hours at formerly off-peak hours such as just after the mid-night or at 8:00 AM. To solve this problem, we need a pricing model where the price of electricity at any hour increases as the aggregate demand increases. For example, we can use the following convex and
Demand (kWh)
60 40 20 0
Fig. 6.
Daily payment (Euro)
w/o scheduling w scheduling
1
7
14 Hour (h)
20
24
Aggregate elastic demand before and after scheduling. w/o scheduling w scheduling
12 10
6 4 2
Fig. 7.
C ONCLUSION
In this paper, we addressed how pricing strategy and gametheoretic optimization techniques can be used to optimize the demand response in the smart grid. The utility company sets the price of electricity for different hours and communicates it to the users. When the users receive the price information from the utility company, they maximize their utility functions independently, contributing to the global optimization. Providing a scalable and distributive solution is the main advantage of applying game theory techniques to demand-side management problem. Our simulations results indicate that the users can reduce their electricity bills by 25% when they use the proposed scheduler to react to DAP pricing strategy. This study can be extended in several ways. First, appropriate mechanisms should be designed to encourage rational agents to reveal their consumption information truthfully. Second, similar to [9], DSM problem can be extended taking into account the storage and generating capabilities of the users. Finally, convex pricing models that have briefly been touched here need to be comprehensively investigated.
8
0
VII.
1
2
3
4
5 6 User (n)
7
8
9
10
ACKNOWLEDGMENT
Daily electricity bills of users before and after scheduling.
increasing price model. p(x) = αx2
(15)
where α is a constant coefficient that can be different during the day hours and over the night. Using this price model, the optimization problem of Eq. (14) becomes a convex binary linear programming problem as follows:
The research leading to these results has received funding from the FEDER through Programa Operacional Factores de Competitividade COMPETE - Fundac˜ao para a Ciˆencia e a Tecnologia and the ENIACs JU and ARTEMIS through projects E2SG (ENIAC/0002/2011 Grant Agreement no 296131) and ACCUS (ARTEMIS/0005/2012, Grant Agreement no 333020). Firooz B. Saghezchi would also like to acknowledge his PhD grant funded by the Fundac˜ao para a Ciˆencia e a Tecnologia (FCT-Portugal) with reference number SFRH/BD/79909/2011. R EFERENCES
min f (yn ) yn
[1]
s.t. : [2]
βn,a
X
h pn,a yn,a
= En,a
(16)
h=αn,a
[3]
h yn,a = 0 ∀h ∈ H\Hn,a h yn,a ∈ {0, 1} ∀h ∈ Hn,a
where f (.) is the convex nonlinear price function. This nonlinear binary programming problem can be solved using interior point method (IPM) [8] for example. We plan to study peak power shaving through incorporating this convex binary programming problem in our future research. Fig. 6 compares the elastic (i.e. shiftable) hourly demand before and after scheduling. As clear in the figure, the scheduler completely shifts the elastic demand from peak hours to other hours when the electricity is more affordable. Finally, Fig. 7 compares the daily electricity bills of users with and without employing the scheduler. Note that the fluctuation of the payment from one user to the other is due to different sets of optional appliances that each user possesses. As clear from the figure, every user pays less when it activates the scheduler. In fact, every user pays 25% less in average when they activate the scheduler.
[4]
[5]
[6]
[7]
[8] [9]
R. B. Myerson, Game Theory Analysis of Conflict, Harvard University Press, 1991. M. F´elegyh´azi and J. Hubaux, “Game Theory in Wireless Networks: A Tutorial,” Technical Report LCA- REPORT-2006-002, EPFL, 2006. W. Saad, Z. Han, V. Poor, and T. Baar, “Game-Theoretic Methods for the Smart Grid: An overview of micro-grid systems, demand-side management, and smart grid communications,” IEEE Signal Processing Magazine, vol. 29, no. 5, pp. 86-105, Sept. 2012. A. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich, R. Schober, and A. Leon-Garcia, “Autonomous Demand-Side Management Based on GameTheoretic Energy Consumption Scheduling for the Future Smart Grid,” IEEE Transactions on Smart Grid, vol. 1, no. 3, pp. 320-331, Dec. 2010. P. Yang, G. Tang, and A. Nehorai, “A Game-Theoretic Approach for Optimal Time-of-Use Electricity Pricing,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 884-892, May 2013. P. Samadi, H. Mohsenian-Rad, R. Schober, and V. W. S. Wong, “Advanced Demand Side Management for the Future Smart Grid Using Mechanism Design,” IEEE Transactions on Smart Grid, vol. 3, no. 3, pp. 1170-1180, Sept. 2012. A. Mohsenian-Rad and A. Leon-Garcia, “Optimal Residential Load Control With Price Prediction in Real-Time Electricity Pricing Environments,” IEEE Transactions on Smart Grid, vol. 1, no. 2, pp. 120-133, Sept. 2010. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. I. Atzeni, L. Ord´on˜ ez, G. Scutari, D. P. Palomar, and J. Rodr´ıguez, “Demand-Side Management via Distributed Energy Generation and Storage Optimization,” IEEE Transactions on Smart Grid, vol. 4, no. 2, pp. 866-876, June 2013.
de Telecomunicac¸o˜ es, Campus Universit´ario de Santiago, 3810-193 Aveiro, Portugal {firooz, jonathan}@av.it.pt † Departamento de Economia, Gest˜ao e Engenharia Industrial, Universidade de Aveiro, Portugal [email protected] ‡ Universidade da Madeira, Funchal, Portugal [email protected] Abstract—Demand-side management is an effective means to optimize resource utilization in the electricity grid. The smart grid can enable the utility company to shape the users consumption by adopting appropriate pricing strategies. Having received the price information, each user may independently schedule its appliances to minimize its electricity payment. In this paper, we consider a smart grid scenario with a single utility company and multiple users where the utility company adopts day-ahead pricing strategy. We formulate the problem as a binary linear programming problem. The simulation results show that users can reduce their bills by 25% using the proposed technique. Keywords—game theory, demand-side management, pricing strategy, PAR reduction, appliance scheduling, smart grid.
I.
I NTRODUCTION
Game theory is a mathematical tool to analyse potentially arising conflict of interest among independent rational agents who are intelligent and seek to maximize their own benefit when they strategically interact with each other [1]–[3]. Having classically used as a toolkit to analyse economic and political problems, game theory has recently attracted considerable amount of interest from engineers and researchers to analyse smart grid problems including the demand-side management (e.g. see [3]–[6] and references therein). Integration of information technology with the power system not only can enable protection and troubleshooting but also can enable utility company and customers to actively participate for the demand-side management (DSM). The utility company can control and shape the customers’ consumption by controlling the price of electricity to reduce peak-to-average (PAR) load ratio (e.g. by increasing the price of electricity during peak hours). On the other hand, the supporting information technology enables customers to monitor their hourly energy consumption, as well as the variations of the price of electricity in the market (e.g. either due to variations of the price of oil or due to intermittent nature of renewable resources such as wind and solar energy) to react accordingly in order to conserve energy or reduce their electricity payments by shifting their elastic1 demand from peak hours to off-peak hours–when the price of electricity is lower. Pricing is as an effective means for a utility company to control and shape electricity consumption of the users [7]. For 1 Price elasticity is a microeconomic term defined as the variation in the quantity demanded when the price increases one unit.
example, inclining block pricing (IBP) has been adopted for many years to make electricity affordable for low-income users to fulfill their basic needs such as lighting and refrigerating while charging higher rates for users how consume higher amount of energy for the purposes such as air conditioning or heating. There are also several other pricing strategies used by utility companies to control the demand response such as critical-peak pricing (CPP), time-of-use pricing (TUP), realtime pricing (RTP), and day-ahead pricing (DAP). Specifically, in DAP strategy, the utility company sets the price of energy for the next 24 hours and advertises it to the users. Non-cooperative game theory techniques can be employed to analyse the customers behavior in response to any change in the price of electricity in the market. In a non-cooperative game, every player adheres to a strategy that maximizes its own utility, without necessarily concerning about the social welfare. The term non-cooperative does not necessarily mean that there is no cooperation between players; it rather implies that there is no communication or binding agreement between them. A price-aware user may shift its unnecessary demand from peak hours, when the price of electricity is high, to off-peak hours, when the price becomes lower. A residential user has generally two types of appliances. The first type includes items such as lights and refrigerator that are price inelastic; i.e. no matter how much is the price of electricity in the market, these appliances are needed to be always on and cannot be switched off or shifted to other hours of day. In contrast, the second type of appliances includes items such as washing machine, dish washer, cloth dryer, and plug-in hybrid electric vehicle (PHEV) that are price elastic; unless their task is finished before their associated deadline they can be shifted to other hours of day during which the price of electricity is more affordable. In this paper, we use game theory and DAP strategy to optimize the demand response by motivating the consumers to schedule their price elastic demand from peak hours to off-peak hours. Specifically, we consider a scenario with ten residential users where every user has multiple elastic and nonelastic appliances. The rest of this paper is organized as follows. Section II describes the system model. Section III presents the game model for the energy consumption scheduling of home appliances, and Section IV formulates its Nash Equilibrium (NE) solution
Fig. 1.
Assumed demand side management (DSM) scenario.
as a binary linear programming problem. Section V depicts the simulation setup, and Section VI discusses the simulation results. Finally, Section VII concludes the paper. II.
S YSTEM M ODEL
We assume a DSM problem that involves a utility company and several residential consumers as illustrated by Fig. 1. The utility company first estimates the electricity demand for next hours and sets the price for different hours. Then, the utility company advertises these prices to the customers over the supporting digital communication network. Then, each user optimizes its energy consumption by adopting the best scheduling for its appliances based on this received price information from the utility company. In the following, we assume a day-ahead pricing (DAP) strategy where the utility company announces in advance the price of electricity for each hour for the next 24 hours. Next, we summarize game theory formulation for the DSM problem. Let N denote the set of users and |N | = N . For each customer n ∈ N , let lnh denote the total load at hour h ∈ H = {1, ..., H}, where H = 24. The daily load for user n is denoted by ln = [ln1 , ..., lnH ]. Based on these definitions, the total aggregate load of all users at each hour of day h ∈ H can be calculated as X Lh = lnh . (1) n∈N
Fig. 2. Scheduler embedded in smart meter to schedule household appliances.
n ∈ N at hour h ∈ H. Clearly, the total load of user n ∈ N is obtained as X lnh = xhn,a , h ∈ H. (6) a∈An
As illustrated by Fig. 2, the task of scheduler in user n’s smart meter is to determine the optimal choice of the energy consumption vector xn,a for each appliance a ∈ An to shape user n’s daily load profile. Next, we identify the feasible choices of the energy consumption scheduling vectors based on users’ energy needs. For each user n ∈ N and each appliance a ∈ An , we denote the predetermined total daily energy consumption as En,a . Note that the scheduler does not aim to change the amount of energy consumption, but instead to systematically manage and shift it, e.g., in order to reduce the PAR or minimize the energy cost. In this case, the user needs to determine the beginning αn,a ∈ H and the end βn,a ∈ H of a time interval that appliance a can be scheduled. Clearly αn,a ≤ βn,a . For example, a user may select αn,a = 6 PM and βn,a = 8 AM for its PHEV to have it ready before going to work. This imposes certain constraint on scheduling vector xn,a . Furthermore, we denote that βn,a
The daily peak and average load levels are calculated as Lpeak = max Lh h∈H
and Lavg
1 X = Lh , H
X (2)
(3)
respectively. Therefore, the PAR in load demand is Lpeak H maxh∈H Lh = P . Lavg h∈H Lh
(4)
For each user n ∈ N , let An denote the set of household appliances such as lights, washing machine, dish washer, refrigerator, PHEV, and so on. For each appliance a ∈ An , we define an energy consumption scheduling vector xn,a = [x1n,a , ..., xH n,a ]
(7)
h=αn,a
and
h∈H
P AR =
xhn,a = En,a
(5)
where scalar xhn,a denotes the corresponding one-hour energy consumption that is scheduled for appliance a ∈ An by user
xhn,a = 0, ∀h ∈ H\Hn,a
(8)
where Hn,a = {αn,a , ..., βn,a }. For each appliance, the time interval provided by the user needs to be larger than or equal to the time interval needed to finish the operation. The total energy consumed by all appliances in the system over 24 hours is equal to sum of the daily energy consumption of all loads/appliances. That is, we always have the following energy balance relationship. X X X Lh = En,a . (9) h∈H
n∈N a∈An
In general, the operation of some appliances may not be time shiftable and they may have strict energy consumption scheduling constraints. For example, a refrigerator may have
TABLE I.
N ON - SHIFTABLE A PPLIANCES
Appliance Light Refrigerator
Power (W) 300 30
Stove
1200
Oven
3500
Kettle
2000
Toaster
850
TV DVD Player Modem Hair Dryer
200 10 10 2000
Grill
1800
Deep Fryer
1600
Coffee Machine
1600
Freezer PC
20 200
Start 19:00 00:00 12:00 19:00 12:15 19:15 08:30 19:00 21:00 08:30 20:00 08:00 08:00 00:00 08:10 12:40 19:40 12:45 19:45 08:30 13:30 00:00 09:00
TABLE II.
End 24:00 24:00 13:00 20:00 13:00 20:00 08:35 19:05 21:05 08:35 20:05 24:00 24:00 24:00 08:20 13:00 20:00 13:00 20:00 08:35 13:35 24:00 24:00
and the We define the minimum standby power level max for each appliance a ∈ An and maximum power level γn,a for each user n ∈ N . We assume that
X
III.
250 200 2100 700 1800 2000 5500 5 1500 250 250 700 12
Duration 09:00 1:00 1:00 01:00 03:30 02:00 02:00 00:10 00:30 02:30 02:00 01:00 01:00 01:00 00:15 5:00
m∈N a∈Am
Un (x∗n ; x∗−n ) ≥ Un (xn ; x∗−n ), ∀n ∈ N .
h=αn,a
(11) An energy consumption scheduling vector calculated by the scheduler in user n’s smart meter is valid only if we have xn ∈ Xn .
3000
Ventilation Washing Machine Tumble Dryer Dish Washer Vacuum Cleaner Iron Water Heater Mobile Phone Charger Electric Fire Dehumidifier Towel Rail Juicer Tablet Charger
Deadline 08:00 18:00 08:00 22:00 20:00 22:00 19:00 22:00 24:00 08:00 08:00 24:00 22:00 24:00 13:00 08:00
The Nash equilibrium (NE) of this game is defined as a strategy profile that no player can benefit by unilaterally deviating from it. That is, the energy consumption scheduling vector x∗n , ∀n ∈ N form an NE if and only if
xhn,a = En,a , xhn,a = 0 ∀h ∈ H\Hn,a ,
min max γn,a ≤ xhn,a ≤ γn,a ∀h ∈ Hn,a }.
Space Heater
Start 18:00 15:00 03:00 09:00 09:00 09:00 14:00 9:00 09:00 00:00 18:00 18:00 09:00 00:00 09:00 18:00
Here, x−n = [x1 , ..., xn−1 , xn+1 , ..., xN ] denotes the energy consumption scheduling vectors of all users other than user n ∈ N , and ph (·) denotes the price of electricity during hour h ∈ H which might be a function of the aggregate energy consumption of users (including user n) during the same hour.
(10)
βn,a
Xn = {xn |
Power (W) 1100
h=1
min γn,a
For notational simplicity, we introduce vector xn for each user n ∈ N , which is formed by stacking up energy consumption scheduling vectors xn,a for all appliances a ∈ An . In this regard, we can define a feasible set for energy consumption scheduling vector for user n ∈ N as follows.
Appliance PHEV
its appliances, which is given by the following utility function. P a∈An En,a P Un (xn ; x−n ) = − P × m∈N a∈Am Em,a (12) H X X X h xm,a ) ph (
to be on all the time. In that case, αn,a = 1 and βn,a = 24. As shown in Fig. 2, the scheduler in the smart meter has no impact on the energy consumption scheduling of non-shiftable household appliances.
min max γa,n ≤ xhn,a ≤ γn,a , ∀h ∈ Hn,a .
S HIFTABLE A PPLIANCES
(13)
As long as the cost function ph (·) is a strictly convex and increasing function for each hour h ∈ H, the NE of the energy consumption scheduling game always exists and is unique [4, Theorem 1]. Moreover, this unique NE maximizes the social welfare (or equivalently minimizes the total aggregate energy cost of all users) [4, Theorem 2].
E NERGY C ONSUMPTION S CHEDULING G AME
We are now ready to define the energy consumption scheduling game. Note that we model the energy consumption scheduling problem as a strategic game since in general the strategy of each player for adopting an energy consumption scheduling vector may affect other users through influencing the price of electricity in the market. The energy consumption scheduling game is defined as follows: •
Players: Registered users in set N .
•
Strategies: Each user n ∈ N selects its energy consumption scheduling vector xn to maximize its payoff.
•
Payoffs: The payoff for user n ∈ N is defined as the negative of its daily total energy cost for all of
IV.
NASH E QUILIBRIUM OF THE E NERGY C ONSUMPTION S CHEDULING G AME
The NE of the game is the energy consumption scheduling vector xn that minimizes the electricity payment of each user n ∈ N . In the following, we formulate the problem of finding the NE as a binary linear programming problem. To do so, we first define a new auxiliary binary decision vector yn which is the same size as xn and is equal to 1 when xn 6= 0 and is equal to 0 otherwise. Considering DAP pricing strategy with price vector f = [f 1 , ..., f 24 ], which includes the price for different hours of day, the NE of the game is reduced to the solution of the following binary linear programming problem.
Demand (kWh)
0.2 0.1 0
Fig. 3.
1
8
Hour (h)
18
40 20 0
1
7
24
14 Hour (h)
20
24
20
24
20
24
20
24
(a) Aggregate demand 10 Cost (Euro)
Price (Euro/kWh)
0.3
Price of electricity during different hours of day.
5 0
min fT yn
1
7
yn
14 Hour (h)
(b) Aggregate cost
s.t. : βn,a
Fig. 4. h pn,a yn,a = En,a
Hourly demand and cost before scheduling.
(14)
h=αn,a h yn,a = 0 ∀h ∈ H\Hn,a h yn,a ∈ {0, 1} ∀h ∈ Hn,a .
where pn,a is the power consumption of appliance a ∈ An of user n ∈ N .
Demand (kWh)
X
60 40 20 0
1
7
14 Hour (h)
(a) Aggregate demand
S IMULATION S ETUP
For simulations, we consider N = 10 residential users. Each user has 10 to 15 non-shiftable appliances (e.g. refrigerator, light, and stove) as well as 10 to 15 shiftable appliances (e.g. PHEV, washing machine, and dish washer). Table I and Table II provide complete lists of these non-shiftable and shiftable appliances, respectively, as well as their power consumption values. Each table has two parts. The first part, which includes the first ten appliances, represents the basic appliances that we assume every user has. Whereas, the second part, which is separated by a double horizontal line, includes five optional devices that each user may have. In simulations, we generate two random integer numbers between 1 and 5 for each user to select its optional appliances: one for shiftable and the other for non-shiftable appliances. For instance, if the random number is 2 for non-shiftable appliances of a user, we add grill and deep fryer to its list of non-shiftable appliances. Table I also provides the start and end times of each nonshiftable appliance. For example, the lights need to be on in the evening from 19:00 to 24:00. Similarly, Table II provides the start time, duration, and the deadline for each shiftable appliance. As an example, PHEV needs 9 hours to charge that can start once the user arrives home at 18:00 and it needs to be ready by 8:00 AM the day after when the user goes to work. For the purpose of simulation, we consider the DAP pricing model illustrated by Fig. 3 As seen in the figure, the price is lower over night (from mid-night to 8:00 AM), comparing 10 cents per kWh to 20 cents per kWh during the day. Moreover, the price further increases during the peak hours (between 18:00 and 23:00). Finally, we assume appliances have no power control mechanism and are either on or off. When they are on, they consume their full power rate, while they consume min no power when they are off, i.e. γn,a = 0.
Cost (Euro)
V.
10 5 0
1
7
14 Hour (h)
(b) Aggregate cost Fig. 5.
Hourly demand and cost after scheduling.
VI.
N UMERICAL R ESULTS AND D ISCUSSION
Fig. 4(a) illustrates the aggregate hourly electricity demand before activating the schedulers embedded in the smart meters. As seen in the figure the peak demand occurs at around 20:00, while between 4:00 AM and 6:00 AM the demand is considerably low. It is also worth mentioning that the significant demand between the mid-night and 3:00 AM is mostly due to PHEV charging that starts at 6:00 PM and finishes at 3:00 AM. Correspondingly, Fig. 4(b) illustrates the aggregate hourly cost considering the assumed DAH pricing model of Fig. 3. Fig. 5(a) shows the aggregate hourly electricity demand when the users activate the schedulers, while Fig. 5(b) shows the corresponding electricity cost. As clear from Fig. 5(a), the assumed pricing strategy successfully shifts the demand from peak hours to off-peak hours. However, the synchronization of the shiftable appliances introduces new peak hours at formerly off-peak hours such as just after the mid-night or at 8:00 AM. To solve this problem, we need a pricing model where the price of electricity at any hour increases as the aggregate demand increases. For example, we can use the following convex and
Demand (kWh)
60 40 20 0
Fig. 6.
Daily payment (Euro)
w/o scheduling w scheduling
1
7
14 Hour (h)
20
24
Aggregate elastic demand before and after scheduling. w/o scheduling w scheduling
12 10
6 4 2
Fig. 7.
C ONCLUSION
In this paper, we addressed how pricing strategy and gametheoretic optimization techniques can be used to optimize the demand response in the smart grid. The utility company sets the price of electricity for different hours and communicates it to the users. When the users receive the price information from the utility company, they maximize their utility functions independently, contributing to the global optimization. Providing a scalable and distributive solution is the main advantage of applying game theory techniques to demand-side management problem. Our simulations results indicate that the users can reduce their electricity bills by 25% when they use the proposed scheduler to react to DAP pricing strategy. This study can be extended in several ways. First, appropriate mechanisms should be designed to encourage rational agents to reveal their consumption information truthfully. Second, similar to [9], DSM problem can be extended taking into account the storage and generating capabilities of the users. Finally, convex pricing models that have briefly been touched here need to be comprehensively investigated.
8
0
VII.
1
2
3
4
5 6 User (n)
7
8
9
10
ACKNOWLEDGMENT
Daily electricity bills of users before and after scheduling.
increasing price model. p(x) = αx2
(15)
where α is a constant coefficient that can be different during the day hours and over the night. Using this price model, the optimization problem of Eq. (14) becomes a convex binary linear programming problem as follows:
The research leading to these results has received funding from the FEDER through Programa Operacional Factores de Competitividade COMPETE - Fundac˜ao para a Ciˆencia e a Tecnologia and the ENIACs JU and ARTEMIS through projects E2SG (ENIAC/0002/2011 Grant Agreement no 296131) and ACCUS (ARTEMIS/0005/2012, Grant Agreement no 333020). Firooz B. Saghezchi would also like to acknowledge his PhD grant funded by the Fundac˜ao para a Ciˆencia e a Tecnologia (FCT-Portugal) with reference number SFRH/BD/79909/2011. R EFERENCES
min f (yn ) yn
[1]
s.t. : [2]
βn,a
X
h pn,a yn,a
= En,a
(16)
h=αn,a
[3]
h yn,a = 0 ∀h ∈ H\Hn,a h yn,a ∈ {0, 1} ∀h ∈ Hn,a
where f (.) is the convex nonlinear price function. This nonlinear binary programming problem can be solved using interior point method (IPM) [8] for example. We plan to study peak power shaving through incorporating this convex binary programming problem in our future research. Fig. 6 compares the elastic (i.e. shiftable) hourly demand before and after scheduling. As clear in the figure, the scheduler completely shifts the elastic demand from peak hours to other hours when the electricity is more affordable. Finally, Fig. 7 compares the daily electricity bills of users with and without employing the scheduler. Note that the fluctuation of the payment from one user to the other is due to different sets of optional appliances that each user possesses. As clear from the figure, every user pays less when it activates the scheduler. In fact, every user pays 25% less in average when they activate the scheduler.
[4]
[5]
[6]
[7]
[8] [9]
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