A Bayesian Game Theoretic Based Bidding Strategy for Demand Response Aggregators in Electricity Markets

Journal Pre-proof A Bayesian Game Theoretic Based Bidding Strategy for Demand Response Aggregators in Electricity Markets Saeid Abapour, Behnam Mohammadi-Ivatloo

PII:

S2210-6707(19)30544-X

DOI:

https://doi.org/10.1016/j.scs.2019.101787

Article Number:

101787

Reference:

SCS 101787

To appear in: Received Date:

24 February 2019

Revised Date:

8 July 2019

Accepted Date:

15 August 2019

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A Bayesian Game Theoretic Based Bidding Strategy for Demand Response Aggregators in Electricity Markets

Saeid Abapour, Behnam Mohammadi-Ivatloo*

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University of Tabriz, 29 Bahman Blvd, [email protected]

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Highlights

network operator market.

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 A competitive model is represented for DR aggregators to participate in the

problem.

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 A non-cooperative static game with incomplete information is used to solve

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 A responsive load economic model is applied to DR implementation.  The proposed DR program is based on price elasticity and customer benefit

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function.

 The Nash equilibrium idea is used to solve this game.

Abstract: In recent years, significant development in smart metering and remote sensing systems in the electricity industry, especially on the side of consumers, it has made possible to implement demand response programs in peak periods. This paper presents a game theoretical approach to the optimal bidding strategies of demand response (DR) aggregators in deregulated energy market. This model will be based on customer benefit function and price elasticity so that an economic responsive load model is applied to DR implementation. The interaction among the system operator and

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aggregators in a deregulated market is modeled in this paper, where DR aggregator provides DR service to the system operator. It is assumed which a system operator collects bids from aggregators and determine the share of each aggregator in the demand response programs by maximizing its

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revenue function and offers rewards to DR aggregators to reach this goal. On the other hand, DR

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aggregators compete together to offer their DR services to the network operator and in this way provide compensation to customers. The competition among DR aggregator participants is modeled as

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a non-cooperative game considering incomplete information. This game solved using the Nash equilibrium idea.

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Keywords: System Operator, aggregator, demand response, game theory, Nash equilibrium.

Nomenclature

i, j

Index of time

n

Index of aggregator number

k

Index of iterations

s

Index of scenario

q, r

Indicate the types for aggregators

Variables

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Indices

The selling price of electricity to end users at hour i

di

The load demand in local electricity system at hour i

ci (d i )

The cost of purchasing electricity from the wholesale market at hour i

n

Given monetary rewards by the operator to aggregator n



DR gain for network operator

d i ( )

Local electricity demand considering monetary rewards

p n,i

Financial compensation paid by aggregator n to all end users at hour i

d n,i

The cumulative load of aggregator n at hour i

APn

The net profit of aggregator n

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Pr

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i

EL (i, j)

The price elasticity of demand hour i versus hour j

Ai

Incentive paid to the customers for each kWh load reduction at hour i

B (d i )

The income of customer in hour i

SO

The benefit of customer during hour i

 n ,i

The strategic bidding of aggregator n at hour i

 Aq (sA )

The bidding strategy vector for aggregator A when its scenario is sA

Sn

The scenario space of the DR aggregator n

 Aq (r)

The conditional probability that aggregator A with type q is playing against aggregator B with type r probability which aggregator A and B are type q and type r, respectively.

EAPAq

The expected profit of aggregator A

EAPBr

The expected profit of aggregator B

 (f)

Probability distribution to the uncertainty modeling of aggregator costs.

ks

bid coefficient of aggregator at strategy s

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pr

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 q ,r

1. Introduction

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With the increase of social welfare, energy consumption increases and designed power generation and transmission networks will not be adequate for power consumption. In Iran for example, electricity

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networks may be faced with the highest possible demand at several hot summer days due to high air conditioning usage. Investing in the development and construction of new electricity generation units

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and distribution networks has heavy costs for governments. At the same time, during peak demand periods in summer days, the wholesale electricity prices in Iran electricity market are often faced with rising prices.

Advancements in communication and control technologies, the integration of electrical networks with telecommunication networks and considering weather information as temperature, humidity, wind

speed and solar irradiation measurements make it possible for demand-response (DR) programs implementation [1]. Demand response is a service used as an energy reserve, and consumers reduce their electricity consumption during peak hours or critical times [2]. DR programs have been implemented in the large and industrial consumers, successfully and the electricity prices reduction, spice prices and network load demand has been proven [3]. However, it will be faced with more challenges in small consumers for example the residential sector. In order to

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aggregate small consumers and design appropriate DR programs for them, another entity between end users and the operator will be needed. DR aggregators (DRAs) are new entities in the power market which work as intermediate between the network operator and volunteer customers.

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DRAs offer participate in DR programs to customers, so volunteer customers can contribute in

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electricity markets, indirectly. This work depends on a set of hardware components such as advanced metering devices for the control and monitoring of real-time consumption of volunteer customers and

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necessary infrastructure for supporting the implementation of demand response programs [4]. Electricity market clearing with regard to the DRAs presences is investigated in the research works [5-

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17].

In [5], a DR bidding strategy framework is proposed for participation in the day-ahead power markets. In the proposed framework, DRAs consign aggregated DR offers to the ISO and ISO optimizes final

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determination on aggregators' DR share in energy market. The strategies of DRAs for load reduction

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include shift and curtailment of load, the use of auxiliary generation and battery storage devices. In [6], a load management method is considered by the DRA in the wholesale and real time markets. Various contracts of load curtailment have been investigated and uncertainty parameters have been modeled using scenario based approach.

In [7], a load scheduling model is proposed with regard to the industrial loads curtailment. In this model, and the first stage, satisfaction and profit of costumers is considered, and at the second stage, DRA investment return is investigated with the assumption of the uncertainty of DR resources. In [8], in order to implement the model in the household sector, DRA consider the two main parameters. The flexibility in the consumption pattern changing as the customers’ behavior and their awareness of the dynamic energy pricing that is discussed in this paper. In [9], authors have

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considered a wind power producer that offer in an energy market while adjust DR contracts with a DRA to compensate its generation. In a bi-level problem, the wind power producer is the leader, which submits offers into an energy market while the DRA that sells its DR service to the wind power

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producer or the power market is modeled as follower. In [10], a new model of reward for participating

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in demand response programs based on the costumer coupon is proposed. A fairness function is defined for DRAs. Simulation is performed on a distribution network and energy losses are also

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considered in the objective function. In [11], authors have presented a framework of game theory to model competition between DRAs that sell energy stored in battery storage devices to other

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participants in an energy market. Non-cooperative game is considered to model competition between two DRAs. The optimal bidding strategy for DRA is derived by the Nash equilibrium. In [12] a new model is proposed to integration of wind power units by misshaping the electrical load in the power

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system. The DR Programs provide the required load reduction through DR resources in the power

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market. The customers in the program send their proposed packages to in the electricity market. The proposed framework determines the generation units’ commitment, the load reduction procured by the DR customers and the energy and reserve planned by units. In [13], authors have presented a stochastic model to investigate the behavior of wind power producers and responsive customers in the electricity market. The players optimize their strategies for participating in the market. Responsive customers are included residential consumers and electric vehicle owners which participate in DR

programs. Incomplete information game theory is used to model the interaction of players in variety of power markets. In [14] authors are proposed a DR scheme from the retailers’ point of view. The retailer decides how to deal DR from aggregators. DR agreements are proposed in real-time and longterm period. These agreements include spike-order options, reward-based DR, pool-order options and forward DR contracts. In [15] the competition of DRAs is modeled by game theory. The objective of DRAs is selling the

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stored energy in storage units to other aggregators in the defined market. A repeated game-theoretic framework is proposed in this study. Dynamic economic dispatch is performed after finding optimum strategies of the aggregators in each time period and the demand is updated based on the dispatch of

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generators. The proposed method optimally plans the generators in the supply side to minimize the

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fuel and operation costs. In [16], a game theory framework is proposed for the optimal scheduling of microgrid based distribution networks within a retail electric market. Three types of players are

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considered namely retailer, generator and consumer. The Nash equilibrium point achieves using the Nikaido-Isoda Relaxation Algorithm in a game framework in every hour. Competition between more

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players leads to lower prices. The uncertainty of the demand and supply are considered using appropriate stochastic models. In [17] authors are presented a competition model between thermal and hydropower generators in an electricity market with considering incentive-based DR. Generators

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compete to close bilateral contracts with consumers and network operator which provides

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transmission services. A Nash–Cournot equilibrium is obtained at peak times when the system is in normal condition. The results indicate that DR reduce the energy consumption at peak times, shifts the energy to off-peak time's period. The profit of generators also reduces due to the reduction of exchanged energy and power market prices. This paper consist a new DR bidding framework in the operators' day-ahead market which changes the energy consumption patterns and also energy supplied by the grid. In this scheme, we use non-

cooperative static game with incomplete information to model competition between DR aggregators in supplying DR resources to the network operator. In this paper, the proposed game has not been used in previous DR studies. In our model, each player has full information on its own cost while its information on other participants’ costs is imperfect. A responsive load economic model is applied to DR implementation. DR program will be based in price elasticity and customer benefit function. The network operator receives bids from aggregators and specifies the share of each aggregator from the

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DR programs. DR aggregators compete to contract DR services with the network operator and provide compensation to costumers. Thus, costumers will modify their consumption pattern. The game with incomplete information is transformed into a complete game with imperfect information. this game is

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solved by using the Nash equilibrium idea.

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The remainder of this paper is organized as follows. In section 2, the structure of the DR bidding framework and problem formulation is introduced. The DR aggregators’ game is presented in section

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3. Case study and proposed methodology to obtain Nash equilibrium is presented in section 4. The simulation results are given in section 5. Finally, conclusions are present in section 6.

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2. Problem formulation

In this paper is considered an electricity market consisting of an electric system operator, a number of

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DRAs and a set of the end users. All contributors are self-interested and rational. Each end user participates in the market through contract with a predetermined aggregator based on a contract, so

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that users do not move between aggregators. We assume a day-ahead market and in which the day is divided into 24 equal periods corresponding the set of hours. In order to supply demand, the operator has either to purchase electricity from power market with high price or purchase electricity from third parties (in here DR aggregators). If the network operator could control the user demands, directly, its objective could be represented as:

T

max

 i 1

i

 d i  c i (d i )

(1)

Since the operator does not have direct control on the load demand, then in this paper are proposed indirect methods such as dynamic pricing. In this purpose, the DRA is introduced as intermediary entities between the operator and the costumers [3]. In the proposed model, the operator troops monetary rewards   n  0, n  A  to DRAs so that they

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enforce DR on the end users. Therefore the load vector d will be dependent on . On the other hand, the operator is willing to provide a partial its DR gain,  , to the DRAs,    n . In fact, the DR gain n A

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is the decrement of the power generation cost. The values of  is assumed to be normalized. T

  c (d ( ))  [c ino DR  c i (d i ( ))]

(2)

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i 1

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where c ino DR is the power generation cost at ith time period if no DR program is used. Therefore with considering of the DRAs, the objective function of the network operator will be as follows:

   i 1

s .t .

i

 d i ( )  c i (d i ( ))    c i (d i ( ))  

0  1

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T

max

n  0 n  A

(3)

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Equation (3) captures both reward to the DRAs and power generation cost.

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Since the end users cannot contract directly with the operator, they negotiate with an aggregator to take part in DR program provided by the DR aggregator. In fact, the DRA provides DR services to the operator and also guarantees reduce electricity bills for consumers. We suppose that a DRA encourages users to modify their consumption profile by a compensation mechanism which will be discussed later.

The compensation vector p n   p n,i : n  A , i T  will be the strategy of DRA n. Let d n  d n,i : n  A , i T  express the cumulative load of each DRA n at ith time period that results from compensation pn. The gain of DRA n at ith time period,  c i , depends on its own compensation strategy pn, and also on the compensation strategy of other DRAs expressed by

p  n   p1 , . . ., pn 1 , pn 1 , . . ., pN  . The net profit of DRA n is expressed as follows: T

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max APn = n,i  c n,i (p n , P n )  p n,i  d n,i (p n,i )  i 1

(4)

The first term is the cashed reward from the network operator and the second term indicates the

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compensation paid to the end users. The third term represents the cost of energy deviations from the

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contracted energy value by the DRA. The energy deviation term denotes the corrective actions of the DRA to a possible realization in the real-time market.

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In this paper, the customers participation in DR programs is formulated and propose an economic model that presents the changes of the customer’s demand due to change of the market price [18]. The

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aggregators provide monetary compensation to motivate the costumers to move load to off peak consumption periods. In this model we consider only incentive to the participation of customers and the customers penalties is not developed here. At first we describe the price elasticity of demand as

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follows [19]:

(5)

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 0j d i EL (i, j)  0  d i  j

Equation (5) defines the price elasticity of the ith period versus jth period. Due to changes in electricity prices, the load of users divided into two categories [20]. Some loads are unable to move from one period to another such as lighting loads so that they will be only on or off. This loads type is called self-elastic and they have a negative elasticity value. In the second category, the consumption could be

transferred to outside the peak hours like as process loads. These loads are measured by cross elasticity, which their values are always positive. In order to model the responsive load economic, at first we model elastic loads for single period. Assume which the final consumer changes the demand level based on the amount that is considered as a incentive in the contract. d i  d i  d i0

(6)

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where d i0 is the initial value of load. The total incentive fee paid to customers to participate in DR programs can be defined as follows:

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P (d i )  Ai  (d i  d i0 )

(7)

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where A i $/MWh is incentive fee paid to the end user in ith hour for each kWh load reduction. In this

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paper penalty is not considered for customer who does not commit in DR program. The income and benefit of end user during ith hour due to the use of di kWh of electric energy is calculated as follows. (8)

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SO  B (d i )  i  d i  P (d i )

where B (d i ) is assumed to be the end user revenue during ith hour and SO is the customer’s benefit

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in ith hour.

The end user benefit will be maximized if,

SO should be equal to zero: d i

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P (d i ) SO B (d i )   i  0 d i d i d i

B (d i )  i  A i d i

Schweppe et al. [21] was proposed a quadratic benefit function:

(9)

(10)

B (d i )  B i0  i0  (d i  d i0 )  [1 

d i  d i0 ] 2ELi  d i0

(11)

By substituting the (11) into (10) and calculating

i0  [1 

B (d i ) , we will have: d i

d i  d i0 ]  i  A i ELi ,i  d i0

(12)

d i  [ELi ,i 

d i0



0 i

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So, changes of the users' consumption will be obtained as follows:  ( i  i0  Ai )]

(13)

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For multi period elastic loads, we have considered a 24 hours period. Then, we will have the multi period equation as follows:

 0j

j 1 j i

 (  j   0j  Ai )]

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d i0

24

d i  [ ELi , j 

(14)

d i0

i0

24

 ( i  i0  Ai )   ELi , j  j 1 j i

d i0

 0j

 (  j   0j  Ai )]

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d i  [ELi ,i 

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An economic model of the responsive load will be get by combining (13) and (14), as follows: (15)

The above equation indicates how much can be the end user’s consumption to have the most profit by participating in the DR program.

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3. DR Aggregators’ Game

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In this paper, the aggregators' competition game in a day-ahead market can be defined using the following terms.

Players: A set of aggregators. Strategies: For each aggregator, a bidding strategy vector. Payoffs: The net profit of aggregator expressed in (4). Note that the aggregator sets the prices based on the model in (15).

In order to model the nth aggregator’s strategic bidding,  n ,i , we will use an incomplete information game. The following relation needs to be met.

 n ,i  An,i  d n,i

(16)

In a complete information game, the nth DR aggregator will not only knows its own payoff function, but also have information from its opponents’ payoff functions. However, in a competitive electricity market, each DR aggregator would know its own payoffs function, but may have lack information of

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other aggregators. Then, each DRA will model the strategic biddings of its opponents into different scenario [22]. The different strategic biddings (bid high, bid low, etc.) are the scenarios of each DR

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aggregator.

Suppose Sn be the scenario space of the nth DR aggregator with sn as an element, and

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S  n  S 1  S 2  ...S n 1  S n 1  ...S N indicate the scenarios of the nth aggregator’s opponents with s-n as

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an element. We define  n (sn ) as the bidding strategy vector for the nth DR aggregator when its scenario is sn, while   n (s n ) is the bidding strategy vector of the nth aggregator’s opponents when

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their scenarios are indicated by s-n.

At first, we model a two-aggregator game so that players A and B compete together. This procedure can be expanded for a game with N-players case. Player A model the uncertainty of the player B’s

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cost (compensation vector), so that player A supposes which there are R types of player B. The same

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assumption is also made by the player B. Therefore, the superscripts q=1, ...,Q and r=l, …,R indicate the types of player A and player B, respectively. The basic probability distribution in this game will be a probability matrix  so that element  q ,r shows the probability which player A is type q and player B is type r. Also the conditional probability

 Aq (r) shows the probability which player A with type q will play against player B with type r and it is expressed as follows:

 Aq (r) 

 q ,r R

 r 1

q ,r

(17)

And similarly,

 q ,r

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 Br (q ) 

Q

 q 1

q ,r

(18)

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We define  Aq (sA ) as the bidding strategy vector for DR aggregator A when its scenario is sA with

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type q. The benefit of aggregator A depends on two factors: first factor is his strategy and second will be his opponent’s strategies:

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APAq  APAq ( Aq (sA ),  Br (sB ),q, r) . The profit APAq is referred to as conditional profit. Aggregator A

as follows [23]:

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should be aware of his opponent’s type to maximize APAq . The expected profit of aggregator A will be

EAPAq   APAq ( Aq (sA ),  Br (sB ),q, r) Aq (r) r 1

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And similarly,

EAPBr   APBr ( Br (sB ),  Aq (sA ), q, r) Br (q )

(20)

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q 1

(19)

DR aggregator A and B in (19)-(20) consider all potential types together. The incomplete game is defined now as a (Q+R)-player as complete game. In fact this game is a complete game but with imperfect information. Players do not aware to the type of opponents. In this paper the expected profit of EAPAq and EAPBr will be payoff functions. Nash equilibrium in new complete game is defined [24]

as step 6 in the next section. The amount of power reduction that aggregator bid, d n,i (pn,i ) , to operator is a function of the compensation strategy. The higher the monetary reward consequent with the higher load reduction. 4. Proposed methodology and case study In this section, the problem assumption is represented to extract numerical result. Also the annual peak demand curve of the Ghom electric distribution company (GEDC) in Iran on 21/07/2016, has

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been applied for our studies. Figure 1 indicates the load curve divided into three different periods, namely off-peak period (2:00 am–11:00 am), medium-peak period (18:00 pm–2:00 am) and peak

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period (11:00 am–18:00 pm). The price of electricity in Iran on that date is also shown in Fig. 2 [25].

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Fig. 1. GEDC peak load curve on 21/07/2016.

Fig. 2. Iranian electricity price curve on 21/07/2016.

The implementation potential of DR programs is assumed to be 15%, in the other words, the total customer contract to take part in these programs is 15% of the total load demand. In this case, the network operator can decrease the peak demand about 102 MW in the best condition and this act reduces the load shedding probability. The price elasticity of the demand is listed in Table 1. Table 1. Self and cross price elasticities. Medium-peak

Off-peak

Peak

-0.1

0.018

0.014

Medium-peak

0.018

-0.1

0.01

Off-peak

0.014

0.01

-0.1

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Peak

Step 1: Define the types of aggregator We define Q=2 and R=2 with:

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We will find Nash equilibrium in the following steps.

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[15,16.5] $ / h at peak hours p1,i   [10,11.5] $ / h at medium peak hours

In fact p1,i is monetary compensation paid to end users by DRA 1 at hour i (in this paper DRA A).

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The first and second elements corresponds to q=l and q=2, respectively. The aggregator B's cost is:

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[14,18] $ / h at peak hours p 2,i   [10,12] $ / h at medium peak hours

DRA 1 estimates that he may be act with two types: type 1 with lower compensation for residential users and type 2 with higher compensation for industry users. A probability distribution  (f) is applied to model the uncertainty on its costs that each DRA has been considered for the contracted

F

end users in their own area that   (f)  1 . In this paper, two cost scenarios are used for the DRA f 1

(low cost, high cost) with probability  (1)  0.6 and  (2)  0.4 . The first and second scenarios correspond to low and high cost, respectively. Step 2: Define the basic probability distribution for this game The expected value all cost scenarios  q ,r when DRA A is type q and DRA B is type r is obtained

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using the conditional probability. Therefore  q ,r will be as:

 q ,r  A (q)  B (r )

(21)

0.24  0.16 

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0.36  0.24

pr

In this paper will be:

Pr

The main incomplete game is transformed into a complete game with imperfect information using matrix  . In the new complete game, each DRA knows its costs and he can compute his own

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monetary compensation vector. Nevertheless, he does not aware his opponent's monetary compensation so that he has little information on his opponents. In (17) and (18), the conditional

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probability vectors for DRA A and B are:

Br  0.6 0.4 ,

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Aq  0.6 0.4 , ,

The conditional probability vectors indicate that DRA A considers a positive correlation between his type and DRA's B type. Step 3: DRAs' strategies:

Each DRA's type has three strategies that are describe by different bid coefficient which each three bids are above the marginal cost of DRA. The coefficient k s is considered to be 1.1, 1.2 and 1.3 respectively.

 n ,i  k s  pn,i In peak hours

$/MWh ,

1  B, i  15.4 16.8 18.2

 A2 ,i  18.15 19.8 21.45

$/MWh ,  B,2 i  19.8 21.6 23.4

$/MWh ,

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 A1 ,i  16.5 18 19.5

(22)

$/MWh

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Each element in the bidding strategy vectors shows a bid coefficient for the respective combination aggregator-type-strategy. For example the second element in vector  A2 ,i is the bidding strategy of

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DRA A when its type is 2 and the bid coefficient is k 2  1.2 .

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Step 4: Computing the conditional benefit of DRAs

For each combination of DRA's type-strategy, equation (23) will be solved and the benefit of each

(23)

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Δd A,i +Δd B,i =Δd

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participant is obtained using (4).

The conditional benefits DRA A with type 1 versus DRA B with type 1 are:

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328 305 325  AP (1)   426 348 368 $ / day  419 320 335 1 A

Each row in APA1 corresponds to a strategy selected by DRA A with type 1. Similarly, each column corresponds to a strategy selected by DRA B with type 1. For example, the element in row 3 &

column 2 will corresponds to the benefits of DRA A in the case which he is type 1 and choose a high bid with k 3  1.3 when against DRA B is type 1 and he is bidding at k 2  1.2 . DRA B's conditional benefits with type 1 in this case are:

379 395 412  AP (1)  523 551 550  $ / day  468 475 486 

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1 B

The remaining conditional benefits APA1 (2) , APA2 (1) , APA2 (2) , APB2 (1) , APB1 (2) and APB2 (2) are

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Step 5: Determining expected DRA profit matrices

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obtained similarly.

Using (19) and (20) and the conditional benefit matrices from Step 4, the expected DRA profit

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matrices are calculated as:

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z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33  A1 (1) 331 336 340 318 322 326 330 334 338    EAPA1   A1 (2)  411 423 429 365 377 382 377 389 394     A1 (3) 395 406 417 336 346 358 344 355 367   

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z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33  A2 (1)  205 210 215 178 184 188 181 187 191    EAPA2   A2 (2)  254 259 270 210 215 226 214 220 231    A2 (3)  256 262 275 189 195 209 193 200 213  

x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33  B1 (1) 388 394 399 398 404 408 408 414 418    EAPB1   B1 (2) 538 540 548 555 557 565 554 556 564     B1 (3)  481 484 495 485 488 500 492 495 506   

x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33  B2 (1) 114 117 119 113 116 118 117 119 121    EAPB2   B2 (2) 128 135 139 122 129 133 132 140 143    B2 (3) 111 118 123 104 110 115 112 119 124   

Each row in matrices EAPA1 and EAPA2 show the strategy of DRA A when its type is 1 and type 2, respectively. Also, each column in EAPA1 and EAPA2 shows a combination of DRA B's strategies. For

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example, column z23 shows the DRA A’ benefits when DRA B will to play strategy 2 when its type is 1 and strategy 3 when its type is 2. Similarly, each row in EAPA1 and EAPA2 shows a strategy for

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DRA B when its type is 1 and type 2, respectively. The column in EAPB1 and EAPB2 shows a

combination of DRA A's strategies. Also, the column y13 shows when DRA A will play strategy 1

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when its type is 1 and strategy 3 when its type is 2.

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Step 6: The Nash equilibrium

In this paper, the Nash equilibrium pairs of the game is obtained using EAPA1 , EAPA2 , EAPB1 and EAPB2

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; In other words, this paper look the best strategies in which each DRA will be versus the other DRA's strategy. If all players foretaste certain Nash equilibrium, then there will not be an incentive for the

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various games.

By control of EAPB1 and EAPB2 , rows  B1 (2) and  B2 (2) dominate all other rows; it means that,

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regardless of the DRA's strategy A uses, DRA B attains higher benefits using k=1.2 cost margin bidding than any other strategy. Therefore, a rational DRA B will always select medium biding strategy regardless of his type. Then the optimum strategy of DRA B is indicated by column z22 in matrices EAPA1 and EAPA2 .

Due to analysis, aggregator A notices which DRA B will bid medium bid in all cases; therefore, DRA A investigates only the z22 column in matrices EAPA1 and EAPA2 . By investigating column z22, we find which a rational DRA A bids  A1 (2) and  A2 (2) when its type is 1 and type 2, respectively. This strategy will be the best response to DRA B's decision indicated by column x22 in matrices

EAPB1 and EAPB2 . Therefore the pair of strategies z22 – x22 is the Nash equilibrium of this game. The

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optimal bids of DRAs have been extracted for this equilibrium point. These strategies maximize the conditional payoff of all over participants in this game.

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5. Discussion

In this section, we will investigate the results of the game output. It should be noted that we have

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assumed that both DRAs are rational. Any player knows that other players are rational and knows that every other player assumes which every player is rational. This refers to the concept of common

and 2 is second strategy.

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knowledge in game theory. According to the Nash equilibrium, best strategy for DRAs in both types 1

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Table 2 summarizes the obtained simulation result in different states. In order to compare and demonstrate the effect of DR programs on the system, the results for before and after DR are also

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calculated and listed in Table 2. By using Nash equilibrium, the best solution is selecting strategy 2 (k=1.2) for each states. The operator benefit has increased with applying DR programs and modifying

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electricity tariff. The maximum benefit of DRA A is in type 2 when DRA B is type 2 and conversely. The maximum benefit of operator is when DRA A and B are type 2. Table 2: Simulation results in different states Objective functions

Before DR

A (type 1)

A (type 2)

A (type 1)

A (type 2)

B (type 1)

B (type 1)

B (type 2)

B (type 2)

Operator benefit ($/day)

270138

278191

282396

285580

289741

Aggregator A benefit ($/day)

-

348

198

420

240

Aggregator B benefit ($/day)

-

551

566

117

148

Figure 3 shows the load pattern with and without presence of DRAs. It is assumed that customers located on the local network under the operator are divided equally between DRAs. As it can be seen,

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the load profile becomes more flat with considering DR programs. In Figure 3 also shows the shifted

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load values in network in different statuses.

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(a) The load of aggregator A and B at state 1, 2

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Figure 3: Load profile result in different states

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6. Conclusion

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(b) The load of aggregator A and B at state 3, 4

In this paper we model the competition among DR aggregators as a non-cooperative game. The aim of

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aggregators is bidding the optimal strategies of DR in deregulated energy market. In this model each aggregator knows its own costs but they do not aware the other aggregator costs. The Nash

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equilibrium idea is used to solve the game. An economic based DR programs model has been proposed. The result of Nash equilibrium shows that in a competitive environment, basically the best answer is not the highest bid for aggregators. It has been shown that customer’s load profile depends

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on not only to the incentive values and electricity price but also the price elasticity of demand can

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improve the demand characteristics. In this paper as an example a two-player game is used; however, the proposed method can be developed to more than two participants games.

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