Abstract—Because generators are limited rationality and their bidding strategies exist risk and uncertainty, this paper proposes a generators’ bidding model which is constructed by value function and weight function based on cumulative prospect theory. Value function shows the changes of losses and gains, and weight function reflects players’ psychological risk factors. The prospect value of bidding strategy is decided by value function and weight function, and the optimal bidding strategy is the biggest one among these prospect values. Finally, an example is employed, and the numerical results tell us that this method is feasible.

I. INTRODUCTION HE Cumulative Prospect Theory has been introduced by Daniel Kahneman and Amos Tversky in 1992.The main

contribution of this theory lie in the more factors have been concerned about people’s decision than the Expected Utility Theory. It puts forward that people focus on relative value changes of wealth rather than ultimate value of wealth when they make decisions. Thus, people are risk aversion for gains but risk seeking for losses. It refers that the attitude of people facing risks is not determined by utility function alone, but determined by value function and weight function together. Finally, it establishes the whole theory including value function and weight function. This theory gives up the hypotheses about people’s rationality in the Expected Utility Theory and puts forward a more realistic behavior choice theory. It has been widely used in financial and investment fields as well as multi-objective evaluation. For example, the decision models were developed in financial and investment fields respectively in the papers [1], [2] through the Cumulative Prospect Theory. While the multi-objective and multi-path selection method was concerned in the paper [3] based on this theory and a model which is appropriated to route guidance and path replacing was established by using this theory, finally, a fast calculate algorithm was given. The bidding of generators in electric power market is actually the decision making process under the limited rationality owing to risks and uncertainties, so the issue of generators bidding is corresponded with the condition of Cumulative Prospect Submitted date: 2011-3-2

Project Supported by National Natural Science Foundation of China (70971038).

Authors：Mr. LI Chun-jie (1952—), Professor, North China Electric Power University.

Research filed: Electricity economics, Electric power industry policy.E-mail: lichunjie518@suhu.com;

Mr. WU Jun-you, Industrial Economics, Master, E-mail: junyou.no.1@163.com;

Miss. CHENG Yan-cong, Industrial Economics, Master, E-mail: chengyancong@126.com.

(Contact: CHENG Yan-cong, Economics and Management School, North China Electric Power University, Beinong Road,

Huilongguan, Changping District, Beijing, P.R. China, 102206, Tel: 15811460579, Fax: 010-51963835)

Theory. In day-ahead trading, there are several bidding strategies

for generators to choose corresponding to different profits (or losses) in each auction period. Due to the auction process day after day, generators can make estimates about the possibilities of profits (or loses) that can be simulated by Bayesian Model. Bayesian auction model was applied to analyze the price in electricity distribution market. It also used the linear combination of strategies to calculate the equilibrium solution in Bayesian game [6]. Under the assumption that competitors’ bidding obeys some known distribution in the given range of bidding rules, bidding strategies under uncertain demands were analyzed in a market consisting of two partners [7]. As we all know, Bayesian theory is based on strictly assumptions that there are no distinctiveness in people’s subjective judgments on the same event. Obviously, it does not coincide with the fact exactly.

In this paper, we use Cumulative Prospect Theory and Bayesian method focused on the process of day-ahead trading to establish generators’ bidding strategy model.

II. BIDDING ANALYSIS

A. Market Messages And Bidding Rules The messages generators get before auction are as follows:

the market clearing price and their winning capacities in the last round; the forecast of load curve, maximum and minimum prices of this round, etc. Generators bid by blocks. They offer to sell specific quantities of electricity at specific prices by trading periods. Ten contribution levels and corresponding prices for each period are offered. The offer curve is non-decreasing and the first value must be the least output of sets. For each bidding period, the power exchange (PX) sorts the offered prices from low to high and selects the relevant capacity until it meets fixed quantity demanded, thus setting a market clearing price. The market clearing price is uniformly paid to all suppliers.

B. Bidding Strategy Coefficient (BSC)

Suppose that the cost function of generator i can be described as 2( ), , ,C Q a Q b Q ci i j i i j i i j i= + + ,where ai 、 bi

and ci are the constants standing for cost features of the set;

,Qi j is the jth capacity of set i . According to the principles

of economics, the marginal cost of the set is ( ) 2, ,MC Q a Q bi i j i i j i= + . Since generators may make a

variety of bidding choices at each output point, the function of

Study on bidding strategies of Gencos based on Cumulative Prospect Theory and Bayesian Learning Model

Chunjie Li, Junyou Wu, and Yancong Cheng

T

International Conference on Information Science and Technology March 26-28, 2011 Nanjing, Jiangsu, China

978-1-4244-9442-2/11/$26.00 ©2011 IEEE 958

bidding strategy can be expressed as ( ),( ) 2, ,i mB Q h a Q bi i j i i j i= + , where ,hi m is defined as BSC. It

represents generator’s special biddings in corresponding period of time. It is clear that the more ,hi m are, the more

depicted bidding strategies are. Following from that, it is even more able to reveal the actual events. Therefore, the values are different based on various actual needs.

We use ( ),1B Qi i and ( ),10B Qi i to stand for the ceiling

price maxp and floor price minp in the same period of bidding time. In view of the limited price in trading rules,

( ),10B Qi i cannot be higher than the ceiling price maxp , so there

is max10,max10, )2()( pbQahQB iiiii ≤+= , where maxhi stands for the cap of BSC. And then there is

)2/( 10,maxmax iiii bQaph += （1） In addition, generators’ bidding will not normally be lower

than the marginal cost. So this paper sets the flooring price in the market as the lowest marginal cost of every generating set. Ibid, )( 1,ii QB cannot be lower than the floor price minip ，so

there is min1,min1, )2()( pbQahQB iiiii ≥+= ，where minih denotes as the upset of BSC. It can be calculated as follows:

)2/( 1,minmin iiii bQaph += （2）

Obviously, the BSC for each generator is between minhi

and maxhi , that is to say [ , ]maxminh h hi ii∈ . In order to simulate bidding strategies with every possible BSC, we divide the interval [ , ]maxminh hii into Mi sections by a step 0.05λ = , the terminal point of each section representing a specific BSC, and then it

gives max min( ) 1h hi iM Inti λ

−= + . The BSC which is

available for generator i is selected from the set { , , , , , }maxmin min minh h h m hii i iλ λ+ + ,

0,1, 2, , 1m Mi= ⋅ ⋅ ⋅ − .

C. Market Clearing Model We only consider demand restriction and price constraint.

The market clearing model is represented as *m in ,

* (2 ), ,

, ,m ax,m in*

m axm in

,1

G

D

nt tp Qi jitp h a Q bi m i i j i

tQ Q Qi j iis t

p p pn tQ Qi ji

C = ∑

≥ +

≤ ≤

⋅ ⋅ ≤ ≤

=∑=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(3)

Where GC denotes the cost for purchasing electricity. This model aims to minimize GC . N represents the numbers

of generators and ( ), 2 ,i mh a Q bi i j i+ is the linear offer curve of

generator i in each period of auction. ,maxQi and ,minQi are

the maximum and minimum capacities of generator i ; , [1,10], ji jQ ∈ is the capacity of generator i in this period of

auction, which is between ,minQi

and ,maxQi . The sum of

offers from suppliers equals to the total demand DQ . In actual bidding, each generator strives to achieve more

profits. The dilemma they are facing is that a higher price strategy will raise the market clearing price p ∗ , thus profiting more, whereas there is a risk of losing bid; Conversely, a lower price strategy will lower market clearing price p ∗ , but with more capacities to load. The bidding is actually a strategy selection process in which the behaviors of both other competitors and itself must be taken into account. In bidding curve ( ,(2 ), i jh a Q bi m i i+ ) of generator i , the slop is

determined by BSC( ,hi m ) and the constant of cost

feature ( )ai , and the intercept is determined by ,hi m and

bi , In consequence of different BSCs( ,hi m ) and cost

features ( , )a bi i of sets, there is no equilibrium solution in once bidding when there are many generators(n>2) in the market. Generators learn to adjust their strategies to get close to the dominant strategy during the bidding process day after day, and market equilibrium is formed at last. In this paper, we use Cumulative Prospect Theory and Bayesian Method to simulate the process of generators’ consistency strategies, that is, generators amend their strategies with market messages and subjective judgments. Then a more realistic theory to study generators’ bidding is established.

III. THE APPLICATION OF CUMULATIVE PROSPECT THEORY IN

GENERATORS’ BIDDING

A. Cumulative Prospect Theory Kahneman and Tversky hold that: 1) the carriers of value

are gains and losses, not final assets. For strategies, they are prospect values. 2) expected profits and losses of strategies are calculated by each prospect value multiplying its weight respectively rather than its probability of occurrence.

Let S be a finite set of states of nature; subsets of S are called events. Let S be a set of consequences of these events. We define prospect value ix as an uncertain event, ix X∈ . f is a function from the set of states of nature S to the set of

consequences X ,that is :f S X→ . iA denotes a partition

of S ,called event iA .When event iA occurs, the set of consequences { } klixi ,0,, −= comes into being, where

i j> , if ji xx > . We arrange each expected value in the order of increasing by degrees, that is

1 0 1... ...l l kx x x x x− − +≤ ≤ ≤ ≤ ≤ ≤ . We assume 0x as a

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referent point, denoted 0 for neutral state, 0ix > for gains and 0ix < for losses. According to Cumulative Prospect Theory,

decision makers have different considerations for gains and losses.

The Cumulative prospect theory asserts that there is a strictly increasing value function v , where 0( ) (0) 0v x v= = . By increasing the decision weight +

iπ for gains or −iπ for

losses, it can treat gains and losses differently. For f , there are:

00

( ) ( ) ( )

( ) ( )

( ) ( )

k

i ii

i ii l

V f V f V f

V f v x

V f v x

π

π

+ −

+ +

=

− −

=

⎧ = +⎪⎪ =⎪⎨⎪⎪ =⎪⎩

∑

∑ (4)

If the prospect function ( , )i if x A= is given by a probability distribution ( )i ip A p= , the decision problem can be viewed as probabilistic prospect ( , )i ix p .In that case, decision weights are defined as follows:

( )k kw pπ + += ； ( )l lw pπ − −−= ；

1( ... ) ( ... )i i k i kw p p w p pπ + + ++= + + − + +

0 1i k≤ ≤ − 1( ... ) ( ... )i l i l iw p p w p pπ − − −

− − −= + + − + + 1 0l i− ≤ ≤ Where w+ and w− are strictly increasing,

satisfying (0) (0) 0w w+ −= = (1) (1) 1w w+ −= = .

B. The Determination Of Value Function In Bidding Kahneman and Tversky hold that value function has three

characteristics: 1) gains and losses are relative to reference point.2) people are risk aversion for gains but risk seeking for losses.3) the decision maker will be more sensitive to losses than to gains. Based on the characteristics of value function, Tversky, as well as others, describes value function as power function centering on the problem of determination under risks and uncertainties, which is:

, 0( )

( ) , 0i i

ii i

x xv x

x x

α

βθ⎧ ≥⎪= ⎨− − <⎪⎩

Where α represents the sensitivity of gains zone and

β represents the sensitivity of losses zone in value function; 1α < and 1β < reflect the decline of sensitivity. θ shows

the characteristic that losses zone is steeper than gains zone, and 1θ > represents loss aversion.

According to cumulative prospect theory, decision maker refers to previous gains (or losses), and we use jix , to represents this reference point. We assume that generator i regards the break-even point of the capacities ,i jQ of each generator as reference point for the first bidding. After this round, the result of previous bidding will be considered as reference point for next round.

Value function of generator i is expressed as:

*, , , , ,

1( / ) ( )i j i m i j i j i ju Q h p Q C QF

= −∑

Where F depicts the times of bidding and *,

1i jp

F ∑ is the

mean value of market clearing prices under these special BSCs ,hi m and the distribution of each ,i jQ loaded.

So the prospect result of generator i under a specific BSC

,hi m is , , , ,( / )i j i j i m i jx u Q h x= − , that is the difference between the gains(or losses) and the reference point under a specific BSC ,hi m . And then, the corresponding value function is

, , , ,,

, , , ,

, ( / )( )

( ) , ( / )i j i j i m i j

i ji j i j i m i j

x u Q h xv x

x u Q h x

α

βθ

⎧ ≥⎪= ⎨− − <⎪⎩

（5）

Where ,( )i jv x is the prospect value corresponding to the

prospect result ,i jx . Generators have many prospect values for different bidding strategies in each round of bidding.

C. The Decision Of Weight Function In Bidding Process In weight function, the possibility is transformed to

decision weight. On account of decision-making problem under the condition of risks and uncertainties, Kahneman and Tversky define the decision weights for gains and losses as:

1

1

1 1

( ) ( )

( ) ( )

k k

i j jj i j i

i i

i j jj j

w p w p

w p w p

π

π

+ + +

= = +

−− − −

= =

⎧= −⎪

⎪⎨⎪ = −⎪⎩

∑ ∑

∑ ∑ （6）

Where the set of subscripts can be expressed as

{ }1,2,..., ,...,h k ,the scheduling of the prospect results is

1 2 ... ...h kx x x x≤ ≤ ≤ ≤ ≤ and hx is the reference point.

w+ and w− are expressed as non-linear weight functions for gains and losses respectively, that are ( )k kw pπ + += and

1 1( )w pπ − −= . There are multiple prospect values in generators’ bidding. Since there are two or more results of risk prospect, the functions are calculated by Prelec as follows[8]:

1 1

1 1

( ) exp( ( ln( )) )

( ) exp( ( ln( )) )

k k

j jj j

i i

j jj j

w p p

w p p

ϕ

ϕ

γ

γ

+ +

= =

− −

= =

⎧= − −⎪

⎪⎨⎪ = − −⎪⎩

∑ ∑

∑ ∑ （7）

Where γ + ， γ − and ϕ are the parameters of the model,

0γ + > ， 0γ − > ， 0ϕ > .

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The possibility of bidding can be obtained by Bayesian learning model under a specific BSC ,hi m and the

distribution of each ,i jQ loaded.

IV. THE BAYESIAN LEANING MODEL IN GENERATORS’

BIDDING

A. The Prior Probability Of Generators’ Bidding

In each round of bidding, generator i has ten possible results of winning capacity. Without any market message to refer to, we assume that the possibility of winning subjects to normal distribution under a specific BSC ,hi m and the

distribution of each ,i jQ loaded. The posterior probability of this round is the prior probability for next round.

B. The Conditional Probability Of Generators’ Bidding Before the next round of bidding, the initial state of

generator i is defined as 0 1, ( , ), 0,1, 2, ,i mS P n m Mi −= = ⋅ ⋅ ⋅ . 0, ( , )i mS P n= represents the distribution of times on the

condition of winning capacity ,Qi j and a specific ,hi m after

n rounds. If ,Qi j is loaded jn times and10

1j

j

n n=

=∑ , then the

probability for each ,Qi j under a given ,hi m can be described

as [ ( ), ( ), , ( )],1 ,2 ,10P p Q p Q p Qi i i= ⋅⋅⋅ , where ( ),n j

p Qi j n= .

From the initial state 0 ( , ),S P ni m = , generator i continues to

observe n rounds under a given ,hi m . When ,Qi j happens

jn times, the initial state transforms to ( , ),S P ni m = , where:

[ ( ), ( ), , ( )],1 ,2 ,10

( ),

10

1

Q Q Qi i in nj jQi j n n

n njj

P P P P

P

= ⋅ ⋅ ⋅

+=

+

=∑=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(8)

( ),Qi jP is get under a specific ,i mh ,and the conditional

probability is ,( / ),i mh Qi jP , that is ( ),Qi jP .

C. The Posterior Probability Of Bidding According to market clearing situation and conditional

probability, the posterior probability can be calculated in the light of Bayesian formula as follows:

, ,,

, ,

( / ) ( ),( / ),( / ) ( ),1

i j i ji j

i j i j

P h Q P Qi mP Q hi m nP h Q P Qi mj

=∑=

(9)

Where ( / ),,P Q hi mi j expresses new knowledge about

the distribution of the possible capacity loaded in the next round after the progress of learning.

We can put the posterior probability to formula （7）and get

weight function.

V. THE BIDDING PROCESS BACED ON CUMULATIVE PROSPECT

THEORY

A. Market Stability Generators choose the optimal bidding strategy in the light

of value function and weight function, in order to maximize profits. Value function expresses that those generators who are limited rationality, focus on the relative changes of gains (or losses) when they make decisions. Weight function reflects that generators try to find optimal strategy in bidding. Based on the posterior probability derived from Bayesian learning model, generators bid tentatively to find the optimal strategy which is suitable for itself after so many rounds of bidding. Generators are fast learning and incompletely informed. They can adjust their bidding strategies quickly according to market trends. The process of making strategies conforms to best-response dynamic mechanism of game theory [12]. Therefore, many generators can gain the stable strategy driven by market evolution after multiple rounds of bidding, that is, when all generators’ bidding strategy coefficients do not change with the increasing of bidding times, the market is in a stable state. On basis of best-response dynamic mechanism, the stability is group oriented and has the character of anti-interference. Every generator chooses the ESS (evolution stability strategy) which is the best reaction strategy responding to each other’s.

B. The Conditional Probability Of Generators’ Bidding According to the generator’s bidding strategy based on

Cumulative Prospect Theory, the concrete steps are as follows:

1. In line with market messages and the situation of itself, generator i can get the set of bidding strategy coefficients by formula (2), that is ]1[ , , , , ,,0 ,1 ,2 i

h h h hi Mi i i − , and bid

tentatively in order to find the optimal strategy. 2. Before bidding, the profit of the generator is zero, so

reference point is 0ix = .Based on Cumulative Prospect Theory, the generator is risk seeking at that time and will adopt high-price strategy, corresponding the maximum of bidding strategy coefficients. And then we can obtain each prospect value ,( )i jv x by formula (5), where 0.88α β= = , 2.25λ = , defined by Tversky and Kahneman in the experiment[9].

3. Generator i assumes that winning capacity subjects to normal distribution under a speCIAL ,hi m , which is the initial probability distribution. After the first round of bidding, the generator puts the result to formula (8) and (9), and obtains the possibility of each bidding strategy in the next round. The weight can be revised according to formula (7), where

0.8γ + = ， 0.8γ − = ， 1.0ϕ = , defined by Goda K and Hong H P in the experiment[11]. 4. Substitute the results which are get from step 2 and 3 into

the utility formula (4) and then gain the prospect values in

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various bidding strategies. Select the strategy for the next round that is the maximum prospect value in this round.

5. Each generator offers his bidding curve and carries out market cleaning, and then it submits the market-clearing price and the winning biding.

6. The market keeps stable when all the generators do not change their bidding strategies in consecutive three bidding rounds. Otherwise return to step 1 in the next round of bidding.

VI. CASE STUDY For example, there are five competitors in power supplying

market of Jilin Province. Every generator has a set for bidding .Table 1 displays the parameters of cost, the largest and the lowest active powers. The total demand DQ is 1000MW. The ceiling price and floor price are ¥102.362 per MW·h and ¥307.085 per MW·h respectively. Using function (1) and (2), the BSC of each generator can be obtained and the results are shown in table 2.

Based on the process of adjusting strategies, we take advantage of MATLAB software to do simulation calculation. Each generator does not change its BSC from the 17th round which is shown in table 3. It is also shown that the generator comes to a stable state quickly after short bidding adjustments.

It is considered that people are risk aversion for gains but risk seeking for losses on basis of Cumulative Prospect Theory. The decision maker is more sensitive to losses than to gains. Generators 4 and 5 have lower cost and can gain profits easily. Since they are risk aversion, they tend to take low-price strategy and gain more profits through more capacity loaded; while the others have higher cost and are vulnerable to suffering losses. They tend to raise cleaning price to gain more profits because they are risk seeking.

TABLE I MESSAGES ON COST AND OUTPUT

Generator ai

bi ci ,minQi ,maxQi

gen.1 0.204 104 9255 120 300

gen.2 0.215 110 9175 100 250

gen.3 0.176 92 8150 200 370

gen.4 0.165 85 8540 200 500

gen.5 0.224 113 9057 110 280

TABLEⅡ

THE BSCS AVAILABLE FOR GENERATORS

ijh

Gen.1 Gen.2 Gen.3 Gen.4 Gen.5

,0hi 1 1 1 1 1

,1hi 1.05 1.05 1.05 1.05 1.05

,2hi 1.1 1.1 1.1 1.1 1.1

,3hi 1.15 1.15 1.15 1.15 1.15

,4hi 1.2 1.2 1.2 1.2 1.2

,5hi 1.25 1.25 1.25 -- 1.25

,6hi 1.3 1.3 1.3 -- --

,7hi 1.35 1.35 1.35 -- --

,8hi -- 1.4 -- -- --

TABLEⅢ

MARKET CLEARRING RESULTS

Rounds Gen.1 Gen.2 Gen.3 Gen.4 Gen.5

1 1.35 1.4 1.35 1.2 1.25

2 1.3 1.35 1.3 1.15 1.2

3 1.25 1.3 1.25 1.1 1.15

4 1.2 1.25 1.2 1.05 1.1

5 1.15 1.2 1.15 1 1.05

6 1.1 1.15 1.1 1.2 1

7 1.05 1.1 1.05 1.15 1.25

8 1.35 1.05 1.35 1.1 1.2 9 1.3 1.4 1.3 1.05 1.15

10 1.25 1.35 1.25 1.15 1.1

11 1.2 1.3 1.2 1.1 1.05

12 1.15 1.25 1.15 1.15 1.25

13 1.1 1.2 1.35 1.1 1.2 14 1.35 1.15 1.3 1.15 1.15 15 1.3 1.35 1.35 1.1 1.2 16 1.25 1.3 1.3 1.1 1.15 17 1.35 1.35 1.3 1.1 1.2 18 1.35 1.35 1.3 1.1 1.2 19 1.35 1.35 1.3 1.1 1.2

VII. CONCLUSION A bidding strategy model for generators is introduced in

this paper based on Cumulative Prospect Theory and Bayesian method. There are improvements in three ways. First, Cumulative Prospect Theory is applied in bidding model which reflects that generators are limited rationality and risk-considering. Second, bidding strategy is improved from pure high and low price to complete bidding choices. Third, Bayesian learning model is established, in which different individuals have different reactions toward the same message, rather than performing in the same way.

By the three improvements above, we make more realistic theoretical study to reveal generators’ bidding.

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REFERENCES

[1] K .D .David,C .Charlie, S. Raymond." Prospect Theory,Analyst Forecasts, and Stock Returns" . Multinational Financial Management, 14(4/5), pp.425-442,2004 .

[2] S. K. Albert,O. Y .Hui, X. Wei. "Prospect Theory and Liquidation Decisions".Economic Theory, 2006,129 (1),pp. 273-288.

[3] Wang Zheng-wu,Luo Da-yong,Huang Zhong-xiang,Wang Jun-yi. "Multi-criteria Multi-route Choice under Uncertainty". Systems Engineering, 24(3),pp.355-359,2009 ．

[4] D. L. Torres, J .Contreras,A. J. Conejo, "Finding Multi Period Nash Equilibria in Pool-based Electricity Markets.Transactions on Power System,9(1) ,pp.643～651,2004．

[5] A. Tversky, C. R. Fox.Weighting Risk and Uncertainty.Psychological Review,102(2),pp.269-238,1995．

[6] Zhang Xin-hua, Ye Ze."Bayesian Game Model on Electric Power Bidding under Uncertain Demand".Systems Engineering, 22(2),pp.215-220, 2007．

[7] Chen Yong-quan, XIAO Xiang-ning, SONG Yong-hua．"Bidding Strategy Model for Generation Companies Considering Bidding Experiences" .in Proceedings of the CSEE,25(29) ,pp.71-77,2009．

[8] D. Prelec.Compound Invariant Weight Functions in Prospect Theory .Cambridge：Cambridge University Press，2000．

[9] A. Tversky, D. Kahneman."Advances in Prospect Theory: Cumulative Representation of Uncertainty". Riskand Uncertainty, 1992, 5(4), pp.297-323.

[10] Huang Tao."The Evolution of Subjective Probability Judgment" . Quantitative & Technical Economics，5,pp.45-50,1998．

[11] K. Goda, H .P. Hong. "Application of Cumulative Prospect Theory: Implied Seismic Design Preference". Structural Safety, 30(6) ,pp.506-516, 2008.

[12] Wang Xin, Li Yan."Best-response Dynamic Model for Gencos' Bidding Strategies in Regional Electricity Market". North China Electric Power University,33(6),pp.51-54,2006．

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