Effects of Option Contracts on Electricity Markets: A Cournot Equilibrium Analysis

Zhang Shaohua, Fu Xinhua, Wang Xian Key Laboratory of Power Station Automation Technology, Department of Automation,

Shanghai University, Shanghai 200072, China E-mail: eeshzhan@126.com

Abstract—Option contracts have been widely applied in electricity markets for their advantages of flexibility and diversity. While risk management using option contracts has been extensively investigated, impacts of these option contracts on generation companies’ (Gencos’) strategic bidding behaviors and the market equilibrium outcomes have not been well understood. For this purpose, this paper addresses the electricity market equilibrium problem when Gencos sell financial call options or buy financial put options. A Cournot equilibrium model for electricity wholesale market competition is developed taking into account Gencos’ option contracts and the forced outages of generation units. The uncertainty in load demand is also considered to reflect option contracts’ advantages of flexibility. The introduction of option contracts and generation unit outages leads to a non-smooth and complicated equilibrium problem. An iterative solution method is employed to solve this equilibrium problem. Numerical examples show that call options with relatively low strike prices or put options with relatively high strike prices will help to reduce Gencos’ interests to raise market price by strategic behaviors, thus mitigate their market power abuse in electricity markets.

Key words - Electricity market, call option, put option, Cournot model, equilibrium analysis.

I. INTRODUCTION The restructuring and deregulation of electric power

industries, which aim at creating competitive and efficient electricity markets, have taken place in many countries since the 1990s. The practice of market-oriented reforms around the world has shown that the emerging electricity market structure is more akin to oligopoly than perfect market competition [1]. In such market structure, Gencos have market power to manipulate market price by strategic bidding, which will seriously affect the efficiency and reliability of electricity markets. Therefore, analysis of Gencos’ strategic behaviors and market power using oligopolistic equilibrium models has become an important research area and is of great relevance to the design and operation of electricity markets [1-4].

It is widely acknowledged that electricity market prices are bound to be volatile as a consequence of the unique physical attributes of electricity. An increasing number of market participants are recognizing the importance and necessity of risk management, at least after observing the market anomalies in California during 2000/2001 electricity crisis. Furthermore, to mitigate environmental pollution and global warming, renewable energy (e.g. wind and solar power generation) has

been widely used and rapidly developed in recent years. These energy resources have features of relatively strong randomness, volatility and intermittence. Thus large-scale renewable energy penetration in power systems will inevitably increase risks faced by electricity market participants. In order to manage these risks, financial derivatives, such as forward, future and option contracts, have been broadly employed in electricity markets [5-8].

There are mainly two types of option contract, i.e. call option and put option. The holder of a call option or a put option has the right to buy or sell a certain amount of underlying commodity at a given price, called strike price, but does not have the obligation to exercise this right. To own the right, the option holder needs to pay option premiums. Compared to forward contracts, option contracts have the advantages of flexibility and diversity. Especially, option contracts can be used to hedge both price and quantity risks [9]. In electricity markets, there are many quantity risks such as fluctuations in load demand and generation output. A common approach of dealing with quantity fluctuations by means of inventories is not possible in electricity markets where the underlying commodity cannot be effectively storable. As such, option contracts are being widely used as an effective measure of risk management in electricity markets [5-8].

It is widely recognized that as means for risk management, forward contracts can also affect Gencos’ strategic behaviors in the wholesale market competition. Theoretical research based on oligopolistic equilibrium models has shown that forward contract can reduce Gencos’ interest in raising market price by capacity withholding, thus mitigate Gencos’ market power abuse and improve the efficiency of electricity markets [10-16]. Up to now, research work related to electricity option contracts mainly concentrates on two areas. The first is to address the optimization problems for Gencos and retailers to manage market risks using option contracts [17-20]. The other is to address the capacity adequacy issue in electricity markets by means of call options (reliability options) auction [21-25]. It can be noted that to date, there is few research work to address the impacts of option contracts on Gencos’ strategic behaviors and the equilibrium outcomes in the wholesale market competition.

Given this background, this paper addresses the electricity market equilibrium problem when Gencos sell financial call options or buy financial put options. A Cournot equilibrium model for electricity wholesale market competition is developed taking into account Gencos’ option contracts and the

This work is supported by National Natural Science Foundation of China(No.70871074) and "11th Five-Year Plan" 211 Construction Project of Shanghai University.

978-1-4577-0547-2/12/$31.00 ©2012 IEEE

forced outages of generation units. The uncertainty in load demand is also considered to reflect option contracts’ advantages of flexibility. The introduction of option contracts and generation unit outages leads to a non-smooth and complicated equilibrium problem. An iterative solution method is employed to solve this equilibrium problem. Numerical examples are presented to show how the call and put option contracts affect Gencos’ strategic behaviors and market equilibrium outcomes. The impacts of option contracts’ strike prices and option volume are also discussed.

II. THEORETICAL MODEL

A. Assumptions There are N strategic Gencos in a pool-based electricity

wholesale market. The Cournot mode is assumed for the competition of these Gencos. This is, Gencos compete in the market by bidding their power outputs. The demand at a future time T (1h) is represented by a liner demand function as follows:

ePLPD −=)( (1)

where, D is the market demand at time T; P is the market price at time T; e is a constant coefficient taking values greater than 0; L is assumed to follow a known probability distribution.

Genco i (i=1,2,…,N) is assumed to own (Ni−1) generation uints. The capacity and marginal cost of the jth unit of Genco i is represented by Xij and cij, respectively. To account for Genco i’s ability to purchase electricity at a relatively high cost from an outside source, we define an artificial generation unit Ni which is always available and have infinite capacity and a relatively high marginal cost of , ii Nc , , ii N ijc c> , j=1,2,…,Ni−1.

Each Genco dispatches its generation units according to the loading order of economic merit. That is, in order to meet a Genco’s bid output, the unit with the lowest marginal cost is loaded first, followed by the unit with the next lowest marginal cost, and so on.

B. Consideration of Generation Units’ Forced Outages It has been shown from [26] that the availability of

generation units has a significant impact on Gencos’ strategic behaviors in electricity markets. Thus the forced outages of generation units are taken into account in this paper.

Each generation unit is represented by a two-state capacity model. Let qij denote the forced outage rate (FOR) of the jth unit of Genco i, its availability probability will be aij=1−qij . Let {j= 1,2, ... , Ni} denote the merit loading order for Genco i. Given that Genco i’s power output is xi, the expected cost Ci(xi) can be calculated as follows.

We define )( iij xC as the expected cost of Genco i when its units 1,2,…, j−1 are not available. The following recursive relationship can be derived [26]:

ii NiiiNi cxxC ,, )( = (2)

, 1

, 1 , 1

( ), if 0 ( ) [ ( )] ( ), if

1, 2, , ; 1, 2, ,1

ij ij i ij i j i i ij

ij i ij ij ij i j i ij ij i j i i ij

i i

a c x q C x x XC x a c X C x X q C x x X

i N j N N

+

+ +

⎧ + ≤ ≤⎪= + − + ≥⎨⎪ = = − −⎩

(3)

The calculation starts from unit Ni and end with unit 1 in the loading order. The expected cost Ci(xi) can be derived as follows:

1( ) ( )i i i iC x C x= (4)

C. Incorporation of Gencos’ Option Contracts Suppose that Gencos have bought or sold a certain amount

of option contracts for time T before they participate in the wholesale market. The option premiums and strike prices are assumed to be determined by an option contract auction market. Two cases are considered here. One is for Gencos to sell financial call options and the other is for Gencos to buy financial put options.

Consider the case for Gencos to sell financial call options with option premium Fc and strike price cλ . Genco i’s call option volume is denoted by kci. At the option delivery time T, Genco i will pay ( )ci ck P λ− to the option holder if P> cλ , but

no payment is made if P< cλ . Consequently, at the option delivery time T, Genco i’s payment for selling financial call option can be expressed as:

( ) max{ ,0}ci ci cV P k P λ= − (5) Consider the case for Gencos to buy financial put options

with option premium Fp and strike price pλ . Genco i’s put option volume is denoted by kpi. At the option delivery time T, Genco i will be paid ( )pi pk Pλ − from the other part to the

option contract if P< pλ , but no payment is made if P> pλ . Consequently, at the option delivery time T, Genco i’s revenue from buying financial put option can be written as:

( ) max{ ,0}pi pi pV P k Pλ= − (6)

D. Cournot Equilibrium Model The uncertainty in load demand is temporarily ignored in

this section. In the Cournot competition mode, Genco i chooses its power output xi to maximize its profit, provided that all other Gencos’ power output ix− will keep unchanged. In this optimization process, the following balance condition between power supply and demand should be respected.

ii xxePLPD −+=−=)( (7) Equation (7) can be rewritten as:

exxLP ii )( −−−= (8) Consider the case for Gencos to sell financial call options,

Genco i’ profit from bidding power output of xi at time T can be expressed as:

( ) ( )( ) max{ ,0}

i i i i ci

i i i ci c

x P C x V Px P C x k P

πλ

= ⋅ − −= ⋅ − − −

(9)

Consider the case for Gencos to buy financial put options, Genco i’ profit from bidding power output of xi at time T can be expressed as:

( ) ( )

( ) max{ ,0}i i i i pi

i i i pi p

x P C x V P

x P C x k P

πλ

= ⋅ − +

= ⋅ − + − (10)

Hence, the profit maximization problem for Genco i with option contracts can be formulated as follows:

. . ( )

iix

i i

Maximize

s t P L x x e

π

−= − − (11)

The Cournot equilibrium model taking into account Gencos’ option contracts can be formulated by gathering N Gencos’ optimization problems expressed by (11). Obviously each Genco’s objective function is non-smooth and complex due to the inclusion of option contracts and generation unit outages. Thus it is difficult to solve this equilibrium model using analytic methods.

E. Solution Method and Deamnd Uncertainty Treatment Substituting (8) into (9) and (10) will lead to an

unconstrained optimization problem for Genco i. Thus Genco i’s optimum power output xi can be obtained through one-dimensional search method, provided that all other Gencos’ power output ix− is fixed. As such, an iterative algorithm to solve the equilibrium model can be developed as follows.

Step 1: Set the value of the error toleranceε and the iteration number k=0; Initialize k

ix (i=1,2,…,N). Step 2: For i=1,2,…,N,

)|(maxarg1 ki

kii

Rx

ki xxx

i

−∈

+

+

= π

Step 3: If 1| |k ki i ix x x+Δ = − ≤ε for i=1,2,…,N, then Gencos’

equilibrium outputs are obtained and go to Step 4; Otherwise, k:=k+1, go to Step 2.

Step 4: Calculate the equilibrium market price using (8).

To deal with the uncertainty in load demand, the Monte Carlo simulation technique is adopted. First, enough samples of the random variable L in (1) are taken according to its probability distribution. Then, for each sample of L, the iterative algorithm is employed to obtain the corresponding equilibrium outputs for each Genco. The final equilibrium outputs of Gencos is calculated by averaging the equilibrium outputs for each sample of L, from which the final equilibrium market price can also be obtained.

III. NUMERICAL EXAMPLES Suppose that there are two symmetric Gencos in a pool-

based wholesale market. The parameters of generation units and configuration of Gencos are given in Table 1, where the last row refers to the artificial generation unit which is always available and have infinite capacity and a relatively high

TABLE I PARAMETERS OF GENERATION UNITS AND CONFIGURATION OF GENCOS

Unit Parameters Number of units owned by Gencos

Unit Capacity (MW)

Forced outage

rate

Marginal cost

($/MWh)

Genco 1 Genco 2

1 700 0.12 20.0 1 1 2 500 0.10 25.0 1 1 3 300 0.08 30.0 1 1 4 ∞ --- 100.0 1 1

marginal cost. In the market demand function, the parameter e takes a value of 20.0 (MW)2h/$, and L is assumed to follow a normal distribution with a mean of 2500MW and a standard deviation of 100MW. In the iterative algorithm, the value of error toleranceε is 0.1MW and the sample number of Monte Carlo simulation is 5000.

A. Impacts of Gencos’ selling Call Options on Market Equilibrium Suppose that an equal volume of call option is sold by the

two Gencos. Two cases, in which each Genco’s call option volume takes values of 200MWh and 300MWh, are considered. The impacts of the call option strike price on Genco 1’s equilibrium output, the market price and Genco 1’s profit are illustrated in Fig.1, Fig.2 and Fig.3, respectively.

It can be observed that Genco’s output increases with decreasing strike price, and both the market price and Genco’s profit are lowered with decreasing strike price. This is because

660

680

700

720

740

760

50 55 60 65Strike price ($/MWh)

Gen

co 1

's ou

tput

(MW

)

Option volume of 200MWhOption volume of 300MWh

Figure 1. Impact of call option on Genco 1’s output

50

55

60

65

50 55 60 65Strike price ($/MWh)

Mar

ket p

rice

($/M

Wh) Option volume of 200MWh

Option volume of 300MWh

Figure 2. Effect of call option on market price

20000

21000

22000

23000

24000

25000

50 55 60 65Strike price ($/MWh)

Gen

co 1

's pr

ofit

($/h

)

Option volume of 200MWOption volume of 300MW

Figure 3. Effect of call option on Genco 1’s profit

that with a relatively low strike price, the possibility to exercise the call option will be relatively large, which makes the call option contract act more like a forward contract. Thus call options with relatively low strike prices will be helpful to reduce Gencos’ interests to raise market price by strategic behaviors. This effect becomes more obvious when the call option volume increases. It can also be noted that for relatively high strike prices, the possibility to exercise the call option will be relatively low, which will motivate Gencos to raise the market price and their profits by output withholding.

B. Impacts of Gencos’ Buying Put Option on Market Equilibrium Suppose that an equal volume of put option is bought by

the two Gencos. Two cases, in which each Genco’s put option volume takes values of 200MWh and 300MWh, are considered. The impacts of the put option strike price on Genco 1’s equilibrium output, the market price and Genco 1’s profit are illustrated in Fig.4, Fig.5 and Fig.6, respectively.

It can be seen that Genco’s output becomes larger with increasing strike price, and both the market price and Genco’s profit are reduced with increasing strike price. This is because that with a relatively high strike price, the possibility to exercise the put option will be relatively large, which makes the put option contract act more like a forward contract. Thus put options with relatively high strike prices will help to reduce Gencos’ interests to raise market price by strategic behaviors. This effect becomes more obvious when the put option volume increases. It can also be observed that for relatively low

660

680

700

720

740

760

40 45 50 55Strike price ($/MWh)

Gen

co 1

's ou

tput

(MW

) Option volume of 200MWh

Option volume of 300MWh

Figure 4. Effect of put option on Genco 1’s output

50

55

60

40 45 50 55Strike price ($/MWh)

Mar

ket p

rice

($/M

Wh)

Option volume of 200MWhOption volume of 300MWh

Figure 5. Effect of put option on market price

22000

23000

24000

25000

40 45 50 55

Strike price ($/MWh)G

enco

1's

prof

it ($

/h)

Option volume of 200MW

Option volume of 300MW

Figure 6. Effect of put option on Genco 1’s profit

strike prices, the possibility to exercise the put option will be relatively low, which will induce Gencos to raise the market price and their profits by output withholding.

IV. CONCLUSIONS This paper addresses the electricity market equilibrium

problem when Gencos sell financial call options or buy financial put options. A Cournot equilibrium model for electricity wholesale market competition is developed taking into account Gencos’ option contracts and the forced outages of generation units. To reflect option contracts’ advantages of flexibility, the uncertainty in load demand is also considered. The introduction of option contracts and generation unit outages leads to a non-smooth and complicated equilibrium problem. To solve this equilibrium problem, an iterative solution method is employed. Numerical examples are presented to demonstrate how the call and put option contracts affect Gencos’ strategic behaviors and market equilibrium outcomes. It is shown that call options with relatively low strike prices or put options with relatively high strike prices will be helpful to reduce Gencos’ interests to raise market price by strategic behaviors, thus mitigate their market power abuse in electricity markets.

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