Electric Power Systems Research 81 (2011) 329–339

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Long-term Nash equilibria in electricity markets David Pozo a , Javier Contreras a,∗ , Ángel Caballero b , Antonio de Andrés b a b

Department of Applied Mechanics and Project Engineering – Universidad de Castilla – La Mancha, Avda. Camilo José Cela s/n. 13071 Ciudad Real, Spain Gas Natural Fenosa, 28033 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 27 May 2009 Received in revised form 12 May 2010 Accepted 22 September 2010 Available online 25 October 2010 Keywords: Pure Nash equilibrium Mixed Nash equilibrium Market simulation Meta-game Hermite interpolation Exponential smoothing

a b s t r a c t In competitive electricity markets, companies simultaneously offer their productions to obtain the maximum profits on a daily basis. In the long run, the strategies utilized by the electric companies lead to various long-term equilibria that can be analyzed with the appropriate tools. We present a methodology to find plausible long-term Nash equilibria in pool-based electricity markets. The methodology is based on an iterative market Nash equilibrium model in which the companies can decide upon their offer strategies. An exponential smoothing of the bids submitted by the companies is applied to facilitate the convergence of the iterative procedure. In each iteration of the model the companies face residual demand curves that are accurately modeled by Hermite interpolating polynomials. We introduce the concept of meta-game equilibrium strategies to allow companies to have a range of offer strategies where several pure and mixed meta-game Nash equilibria are possible. With our model it is also possible to model uncertainty or to generate price scenarios for financial models that assess the value of a generating unit by real options analysis. The application of the proposed methodology is illustrated with several realistic case studies. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In a restructured electric environment, electricity markets represent an effective system for the purchase and sale of electricity. Many electricity markets in the world have been established applying a pool-based auction paradigm. In an auction-based day-ahead market [1–3], the market operator processes the bid information provided by the producers and consumers and aggregates this information creating hourly offer and demand curves, respectively. Both producers and consumers bid with the target of maximizing their profits. Once the bids are submitted, a market-clearing algorithm matches the production and demand curves producing a series of hourly prices and accepted quantities. Searching for possible market equilibria is a desirable objective both for market participants and regulators. For participants, because an equilibrium shows long-term bidding strategies of their rivals; for regulators, because market power monitoring and corrective measures are possible. The knowledge of long-term equilibria represents a valuable tool for electric companies to implement their bidding strategies. In addition, electric companies need to know what their offer strategy should be against every possible offer strategy of their competitors for long time periods. To find market equilibria, it is necessary: (i) to simulate how

∗ Corresponding author. Tel.: +34 926 295464; fax: +34 926 295361. E-mail address: [email protected] (J. Contreras). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.09.008

participants generate their offers and bids, (ii) to establish a marketclearing procedure, and (iii) to identify plausible equilibria. Steps (i) and (ii) are possible by means of optimization techniques [4,5] and step (iii) relies on the concept of Nash equilibrium [6,7]. Existing literature related to equilibrium models in electricity markets mainly focuses either on Nash–Cournot models or on supply function equilibria (SFE) models. The Nash–Cournot equilibrium concept has been applied to calculate equilibria in multi-period settings either by iterative simulation, as in [8,9], or by mathematical optimization methods, as in [10,11]. SFE models have been also applied since its introduction by the seminal paper from Klemperer and Meyer [12]. One of its first applications was in the British spot market by Green and Newbery [13] and subsequent studies by Baldick [14], among others [15], where uncertainty is considered in their approach [16]. Finding Nash equilibria by simulation is also possible combining mathematical optimization and game theory. Game theory simulators are closely connected with market equilibrium models, several works have tackled the use of game theory models and/or agent-based models within electricity markets’ simulators [17,18]. In addition, other types of equilibria, such as Forchheimer or Bertrand are studied in an electricity market simulator framework [19]. One of the main problems of iterative market simulation models is the lack of convergence [8]. This issue can be interpreted considering that lower prices in an iteration result in smaller offered quantities in the next iteration and vice versa and no stable solution can be found. We have solved this problem using an exponential

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smoothing scheme of the energies bid by the companies weighting past energy bids with exponentially decreasing weights. Additionally, the calculation of the residual demand curves as a result of the subtraction of the companies’ bids from the total demand is accurately represented by means of a Hermite polynomial approximation. Therefore, to have a realistic equilibrium model of a pool-based electricity market we have developed in this paper: (i) an iterative electricity market bidding model and a market-clearing model that finds Nash equilibria strategy following a SFE approach when each of the companies selects a particular offer, (ii) the concept of offer parameter to bid in the market, (iii) a meta-game equilibrium model that finds all the equilibria for a range of strategies of the companies, (iv) a Hermite polynomial approximation of the residual demand curves faced by the companies, and (v) an exponential smoothing procedure to achieve convergence in the iterative bidding. The paper is organized as follows. Section 2 presents the iterative market equilibrium model including the notion of offer parameter. Section 3 introduces the concept of meta-game equilibrium. Section 4 presents several case studies. In Section 5, extensions to the model are described, and conclusions are outlined in Section 6.

• • • • • •

Period 1: Monday, hours 1–8. Period 2: Monday, hours 9–24. Period 3: Tuesday to Friday, hours 1–8. Period 4: Tuesday to Friday, hours 9–24. Period 5: Saturday, hours 1–24. Period 6: Sunday, hours 1–24.

2.2. Market-clearing algorithm The market-clearing algorithm used determines the power produced by each generating unit so that the social welfare is maximized and the demand is met during all the periods. Social welfare objective function: The objective function to maximize is the social welfare for the entire time horizon. It is defined as the area between the aggregate bid curve and the aggregate offer curve, both ordered by decreasing and increasing prices, respectively. Maximizing this function is equivalent to minimizing the production cost if the demand is inelastic. The mathematical expression of the social welfare is:

⎡ ⎤ NJ NB NT ND NH      G-offer ⎣ ⎦ max pD D-bid − pG  dth dth jtb jtb

pG ,pD jtb

2. Market equilibrium iterative model To model a general pool-based electricity market we consider that the agents offer and/or bid using discrete blocks. Although the demand is generally assumed inelastic in actual markets, we have modeled it using a step-wise function. Both uncertainty and network modeling are disregarded since we aim to find realistic long-term equilibria in electricity markets without serious congestion problems. Uncertainty could be included in the model for demands, prices and bids but it will be part of future research. For a more detailed economic dispatch model under uncertainty and step-wise offers see [20], or, with a generating company focus, see [21]. It is assumed that a unique price exists resulting from the matching of generation and demand exists for the entire market. As regards to the market-clearing algorithm, we only consider the upper and lower bounds of the produced power, disregarding other constraints such as ramps, start-up and shut-down times. The production costs of the generating units are modeled using linear functions. Note that the aforementioned constraints and the ones derived from the inclusion of the network produce highly complex models that can lead to exceedingly long running times for long-term simulations. Our focus is to produce an equilibrium based upon an iterative method, in which the agents are considered as rational and compete to obtain the maximum possible profits. In the proposed model each company optimizes the power to supply the day-ahead market using different offer strategies.

2.1. Demand model We assume that the demand can be elastic by using discrete demand bidding blocks. Since in long-term simulations (1 year or more), market bidding and subsequent equilibria involve 24 × 365 = 8760 periods that would mean that the level of detail may be cumbersome to deal with. A more logical approach would be to split the days into several hourly periods. For example, for 1 day, hours 1–8 may correspond to the first period, and hours 9–24 to the second. In Fig. 1 we observe that for a week with 168 h, the number of periods is 6, reducing the problem by 28. We have used the following periods in the model:

dth

t=1

d=1 h=1

(1)

j=1 b=1

Constraints: The constraints that have to be met are described as follows: - The first block offered by each generating unit must be equal to its lower production bound: pG = Pj jt1

∀t ∈ ˝T , ∀j ∈ ˝J

(2)

- The power generated by any generating unit in any given period is the summation of its corresponding production blocks. This power must be lower than or equal to the upper production bound of the unit: NB 

pG = pG ≤ Pj jtb jt

∀t ∈ ˝T , ∀j ∈ ˝J

(3)

b=1

- The power produced by each block of a generating unit is limited by the size of the block: G

∀t ∈ ˝T , ∀j ∈ ˝J , b = 2, . . . , NB

0 ≤ pG ≤ pjtb jtb

(4)

- The power consumed by any demand in any given period is the summation of its corresponding consumption blocks: NH 

∀t ∈ ˝T , ∀d ∈ ˝D

pD = pD dth dt

(5)

h=1

- The power consumed by each block of a demand is limited by the size of the block: D

0 ≤ pD ≤ pdth dth

∀t ∈ ˝T , ∀d ∈ ˝D , h = 1, . . . , NH

(6)

- The production must match the demand in every period: ND NH   d=1 h=1

pD = dth

NJ NB  

pG jtb

∀t ∈ ˝T .

(7)

j=1 b=1

2.3. Generating companies quantity optimization model To model a pool-type day-ahead market we consider thermal units, where all of them maximize their profits. Thermal units encompass both fossil-fueled and nuclear units. The latter ones can be seen as thermal units with very low production costs. The objective function of a generating company is obtained subtracting the

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Actual demand (MW)

Agregate demand (MW)

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0

24

48

72

96

120

144

168

Hours Fig. 1. Weekly aggregation of the demand in six periods.

production costs from the income resulting from the generating units that it owns, so that:

max pG ,pit jt

NT 



⎣it (pit )pit −

t=1





Cj (pG )⎦ ˛t jt

(8)

Note that we have omitted the iteration index corresponding to the exponential smoothing procedure in pit for simplicity of notation. 2.4. Generating companies offer strategy

j ∈ i

As a result, the optimal power production of each of the generating units of a company is obtained for every period, where pit is the total power produced by the generating company, it (pit ) is the residual demand function faced by the company, and ˛t is the weight associated with the number of hours of period t. The cost term Cj (pG ) is obtained by aggregating the product of the conjt stant marginal cost blocks of every unit times the optimal power produced by each of the blocks. The objective function is subject to constraints that ensure that the unit operates within its feasible operating region. - Residual demand function: For every period, a company faces a residual demand function it (pit ) which results from subtracting all the offers of the competitors from the total demand. The resulting curve is approximated by a piecewise cubic Hermite polynomial interpolation function [22]. The cubic Hermite poly-

This subsection is devoted to describe the strategy that can be carried out by the generating companies to offer their previously calculated optimized power. An aspect that needs to be considered is the residual demand function that the company faces each period. We describe the offer parameter strategy as heuristic method to generate offers to the pool-market. Each company knows its optimal power, Popt , by finding pit in (8), and also the market price, opt , resulting from the intersection between the residual demand curve and a vertical line at Popt . As seen in Fig. 2, the offer could vary from a marginal cost offer to an offer at which the intersection with the residual demand function occurs exactly at the point (Popt , opt ). The linearization of the offer curve can facilitate that a company changes its offer using a parametric factor.

120

Optimal power

k+1

k+1

, first derivative in the interval [k, k + 1], (d(it (pit )))/(d(pit )) k is continuous. The method chooses the slopes at the endpoints of any interval in such a way that the shape of the original residual demand curve is preserved. Moreover, the new curve is always monotonic on each interval. Note that the curve obtained by Hermite interpolation has more resemblance with the residual demand curve than other approximation methods, such as the spline method, because it does not distort the resulting curve (i.e., there are no jumps). - Power limits of the units: each generating unit cannot exceed its upper and lower production bounds so that: P j ≤ pG ≤ Pj jt

∀t ∈ ˝T , ∀j ∈ i

(9)

- Total power produced by a company: Must be equal to the sum of all the optimal productions of its units:

 j ∈ i

pG = pit jt

∀t ∈ ˝T .

(10)

100

Prices ($ / MWh)

, is generated between every two nomial function, it (pit ) k contiguous points, k and k + 1, of the residual demand function. The interpolating function value is equal to the residual demand curve value at the endpoints of each interval. In addition, the

80

Residual demand function Optimal price

60 100% 40 a

b

0% Maximal power Marginal cost

20

0

Offers

0

500

1000 1500 2000 2500 3000 3500 4000 4500

Power (MW) Fig. 2. Generating company offer parameter strategy.

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We define the offer parameter as the factor that relates an offer with respect to the linear offer that results from connecting point “a” (minimum power offered at marginal cost) to point “b” (optimal power intersection with the residual demand function), as seen in Fig. 2. For example, a 0% offer parameter means that the company’s offer is at marginal costs, and a 100% offer parameter means that the offer is constructed from a straight line that connects points “a” and “b”; any offer in between can be parametrized between these two extreme values. The way to calculate a parametrized offer is as follows:

a subsequent decrease in prices in the next iteration, producing a sawtooth oscillatory effect [5]. It can be observed that the oscillations present recognizable patterns repeated over time. This phenomenon is the result of all the companies simultaneously making their bids with the knowledge of the previous iteration results, not the current ones. This could be avoided if the companies made their bids using historic information from previous iterations. To do so, an exponential smoothing procedure [23] is carried out, as shown in (11):

Step 1. Calculate the optimal price–power point “b” from the quantity optimization model (8)–(10). Step 2. Calculate the linear offer connecting points “a” and “b”. Step 3. Calculate the difference between the linear offer “ab” and the step-wise marginal cost curve. Step 4. Multiply the difference obtained in step 3 by the offer parameter (from 0 to 100%) for each offer block. Step 5. Add the amount obtained in step 4 to the step-wise marginal cost offer curve for each offer block.

where pit (m) is the power produced by the ith company in period t at iteration m of the equilibrium model explained. In (11), the final quantity value is obtained as the sum of two terms: the current optimal quantity obtained from (8) at iteration m, scaled by the  factor, and the smoothed quantity from a previous iteration (m − 1), which considers the history of the bidding process, scaled by (1 − ). Therefore, (11) can be reformulated for period t and company i as a function of the optimal values calculated in previous iterations as follows:

Note that we assume a linear production cost function that produces a single marginal cost block per generating unit. However, in step 4, the number of offer blocks that a generating unit uses can be chosen. The first block represents the lower production bound, the following n-1 blocks are split into two pieces: one up to the optimal power, Popt , and the second up to the maximal power of the unit. In the first segment, a 50% offer parameter is used; in the second one, the offer prices are increased a certain percentage above the optimal price, opt . The optimal price is the result of the intersection of the offer curve of the company and the residual demand curve that the company is facing, as shown in Fig. 2.

pˆ it (m) = pit (m) + (1 − )pit (m − 1) + (1 − )2 pit (m − 2)

2.5. Iterative market equilibrium model If the optimized quantities and the offer parameters are known by all the companies, the iterative process to find the market equilibrium can start. For every company, since the residual demand cannot be obtained for the first iteration, all its power is offered at marginal costs. Later, with the first market-clearing, each company is able to build its residual demand curve. This curve allows measuring the influence that a company can have upon the final price. Once known its residual demand (since the aggregate offers of the competitors are assumed to be publicly available in this model), each company optimizes the quantity to be offered. The quantity optimization model is run per company and produces the optimal power for each unit of the company, as shown in (8). The model uses a piecewise cubic Hermite polynomial interpolation of the residual demand curve bounded above and below by the upper and lower production limits of the company, respectively. At this point, the companies can choose their offer parameters. Once all the companies have created their own offer curves and submitted them, the clearing of the market takes place. In the next iteration, the companies, knowing the aggregate offers of their competitors from the previous iteration, construct new residual demand curves and generate new offers accordingly. Note that we assume that the offer parameter of each company does not change throughout the entire iterative process. The iterative process is repeated until the difference between the power produced by each generating unit, from one iteration to the next, does not change significantly. In iterative simulation models like this one, it is common to observe dramatic variations of the results during the first iterations. In the case of market prices, they may reach high values initially, making the quantity bids by the companies increase. This leads to

pˆ it (m) = pit (m) + (1 − )ˆpit (m − 1) ∀t ∈ ˝T , ∀i ∈ ˝I

+ (1 − )3 pit (m − 3) + . . . ∀t ∈ ˝T , ∀i ∈ ˝I

(11)

(12)

3. Meta-game equilibrium model In the described market equilibrium model, utility companies can offer their energy using a particular offer parameter, or they can offer it at marginal costs, for instance. Therefore, depending on the strategies selected by all the companies in the market, different equilibria are obtained. We can define the result of the previously described iterative model as a noncooperative Nash equilibrium, since all the players maximize their own profits simultaneously. However, they only use a single offer strategy each. To account for the fact that they can have many offer strategies at hand, we define the meta-game equilibrium concept. A meta-game equilibrium is a meta-equilibrium in strategies where the companies have a range of possible strategies at hand. If the meta-game is depicted in its normal form, each of the resulting Nash equilibria payoffs is obtained from the market-clearing algorithm (1)–(7). Note that this meta-game equilibrium represents a high-level type of Nash equilibrium. 3.1. Meta-game equilibrium methods Finding a meta-game equilibrium can be a computational burden; if there are many companies and strategies then several techniques can be applied to reduce the size of the problem. We study meta-game equilibria in coupled and decoupled forms. For long periods of time, a year for example, it is possible to find long-term equilibria or to find each of the 52 weekly equilibria separately. Consequently, we can define (i) a coupled meta-game equilibrium, for which the profits are accumulated throughout the entire period of study, and (ii) a decoupled meta-game equilibrium, where the profits are calculated for each of the 52 weeks. 3.1.1. Coupled meta-game equilibrium The steps to create this model are as follows: 1. Each company selects a unique offer parameter for all the 52 weeks of the year of study.

300

60

200

40

50 C1 C2 C3 C4

40

30

20

Frecuency

Generating Marginal Cost ($/MWh)

60

333

30

100

20

Marginal Cost ($/MWh)

D. Pozo et al. / Electric Power Systems Research 81 (2011) 329–339

15 10

10

0 0

500

1000

1500

2000

2500

3000

0

3500

0 0

1000

Installed Power (MW)

1. Each company selects a unique offer parameter for all the 52 weeks of the year of study. 2. An iterative equilibrium is obtained for each week separately. 3. Each company changes its offer parameter, then go to step 2. 4. The procedure is repeated, obtaining pure and mixed strategy equilibria for the 52 weeks. 4. Case studies 4.1. Description of the generating companies We consider an oligopolistic market composed of 32 generating units, from which 4 are nuclear units and the rest are thermal units. There are 4 companies in the market that own 40%, 19%, 13% and 27% of the installed power, respectively. Fig. 3 presents the marginal costs per company. 4.2. Description of the demand The period of study is 1 year, with 365 days and 24 h per day, which results in 8760 periods. The demand is modeled with 20 blocks for the sake of generality. The maximum bidding price is set to $80/MWh. The year is divided into weekly periods and the 168 h of the week are reduced to just 6 representative periods, as shown in Fig. 1. An iterative equilibrium problem of 8760 periods is converted into 52 weekly problems of 6 periods each. Fig. 4 shows the aggregated marginal costs of the entire generation park vs. the histogram of all possible demands. If the companies offered their quantities at marginal costs the prices would oscillate between $10 and $15/MWh.

4000

5000

6000

7000

Fig. 4. Marginal cost of total installed power vs. demand histogram.

4.3. Iterative algorithm settings The stopping criterion of the algorithm described in Section 2.5 is applied to check that the difference between the power assigned by the market-clearing algorithm from one iteration to the next is less than 2% of total installed power of each company. If this situation is not reached, the maximum number of iterations is set to 40. To verify the convergence of the exponential smoothing procedure we simulate three possible ways in which the companies can use the information from the previous two iterations to send their quantities. In the first case we leave the quantities free, as obtained from the quantity optimization model; in the second case we set the weight of the last iteration to be 30%, the weight of the next-to-last to 20% and the current iteration weight to 50%. The third case uses exponential smoothing with the  parameter equal to 0.1. The first five iterations in all the three cases are just the optimized quantities (first case). Fig. 5 illustrates the quick convergence of the exponential smoothing procedure vs. the other two methods. All the simulations are done using a 4-processor AMD Opteron Dual Core at 2.0 Ghz with 8GB of RAM. MATLAB is used to implement the market-clearing algorithm and the quantity opti-

300

250

Weekly Energy (GWh)

3.1.2. Decoupled meta-game equilibrium In this case, 52 meta-games are created, one for each of the weeks of study. The procedure is similar to the one described for the coupled meta-game:

3000

Installed power and histogram of demand (MW)

Fig. 3. Thermal units marginal costs per company.

2. An iterative equilibrium is obtained for the entire year; the overall profit of every company is obtained by aggregating the profits of the 52 weeks. 3. Each company changes its offer parameter, then go to step 2. 4. Annual meta-game equilibria (based on pure or mixed strategies) are obtained.

2000

200 Free iterations Linear-weighted Exponential smoothing

Company 1 150 Company 2 100

50

0

5

10

15

20

25

30

35

40

Number of iterations Fig. 5. Exponential smoothing vs. lineal-weighted and free iterations methods.

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300

20

Energy company 1

19

250

17

200

Market clearing price

16

150

15 14

Energy company 2

100

Energy company 3

Price ($/MWh)

Energy produced (GWh)

18

13 12

50

11

Energy company 4 0 0

4

8

12

16

20

24

28

32

36

40

44

48

52

10

Weeks Fig. 6. Market-clearing prices and energy production per company for the 52 weeks.

mizations. Gambit [24] is used to find pure and mixed Nash equilibria in the meta-games. 4.4. Case study 1: iterative equilibria results The first case study shows the resulting iterative equilibria of 52 weeks when the offer strategies are fixed for the companies: 60% for company 1 and 40% for companies 2, 3 and 4. These offer parameters are valid up to the optimal production and price (intersection with the residual demand function) of each company. From this point on the offers are made using

a fixed 15% bid price increase with respect to the optimal price. Fig. 6 shows the resulting market-clearing prices plus the energy produced per company. It can be observed that company 3 has an almost flat weekly energy profile (not the daily/hourly profile) as a result of its technology costs (see Fig. 3) between $5.28 and $21.71/MWh that prevent the company to increase its production for prices that oscillate between $14 and $19/MWh. In addition, the energy profile of companies 2 and 4 is very similar since their marginal costs are comparable (see Fig. 3). The energy produced by company 1 is more closely related with the weekly

70

60 Residual demands

50 40

Last offers

Last optimal point

30

Power produced

50

Power produced

Price ($/MWh)

Price ($/MWh)

60

Marginal costs

20

Residual demands

40

Marginal costs

Last offers

30

Last optimal point

Market clearing price

20 10

10

Market clearing price

0 0

500

1000

1500

2000

2500

3000

0

3500

0

500

1000

Power (MW) 60

60 Power produced

40

Last optimal point

30

Last offers Residual demands Market clearing price

20

Power produced

50

Marginal costs

10

Price ($/MWh)

50

Price ($/MWh)

1500

Power (MW)

40

Marginal costs

Last optimal point

30

Market clearing price

20 10

0 0

500

1000

1500

Power (MW)

2000

2500

Last offers

0

Residual demands 0

100

200

300

400

500

600

700

800

900

1000

Power (MW)

Fig. 7. Residual demands, final market-clearing price and power produced in period 3 (working day low demand) for the 29th week. Left-up company 1, right-up company 2, left-down company 3, and right-down company 4.

D. Pozo et al. / Electric Power Systems Research 81 (2011) 329–339

Market clearing price ($/MWh)

30

335

with energy constrains without energy constrains

25

20

15

10 Monday low demand

Monday high demand

Working day low demand

Working day high demand

Saturday

Sunday

Fig. 8. Final market-clearing prices with and without energy withholding in the 6 representative periods of the 29th week.

120

1.8 constrained unconstrained

1.6

100 1.4

Profits (M$)

Energy Produced (GWh)

constrained unconstrained

80

60

40

1.2 1 0.8 0.6 0.4

20 0.2 0

Monday low demand

Monday high demand

Working Working day low day high Saturday Sunday demand demand

0

Monday low demand

Monday high demand

Working Working day low day high Saturday Sunday demand demand

Fig. 9. Energy produced (left) and profits (right) by company 1 per period with and without energy withholding.

prices than the other companies. This happens because company 1 is the most competitive company but, since it offers all its energy, the final prices are almost always fixed by the more expensive competitors, even if company 1 increases its offer prices. Therefore, this case study represents a reasonably competitive market scenario, because prices do not increase significantly with respect to marginal costs (see Fig. 3). This effect is reversed when company 1 restricts its quantity offers, as it will be seen in the next case study. 70 Power produced

Price ($/MWh)

60 Last optimal power

50

Market clearing price

40

Residual demands

30 20 Marginal costs

Last offers

10 0

Finally, we show Fig. 7 where the residual demands are approximated by Hermite polynomials, the optimal power resulting from the quantity optimization model and the final Nash equilibria are depicted for the 4 companies in a particular week, the 29th. It is noticeable that the slope of the residual demand curve of company 1 is higher, meaning that the company is able to exert market power by restricting the offered power to increase prices. However, in this case study, all the companies offer all their available power, but at different prices. This is why the optimal quantity for company 1 is around 500 MW and the final power produced is around 850 MW. For the other companies, the residual demands are flatter and the last optimal power obtained from the quantity optimization models coincide with the actual power produced. There is a step-like jump in the offers due to a technology change in costs that prevents these companies from increasing their respective quantity offers. Running times to find the 52 weekly iterative equilibria are 3502 s of CPU.

4.5. Case study 2: iterative equilibria results with energy withholding 0

500

1000

1500

2000

2500

3000

3500

Power (MW) Fig. 10. Residual demands, final market-clearing price and power produced for company 1 and period 3 (working day low demand) for the 29th week.

Based upon the previous case study we analyze the effect of withholding energy by company 1 in the 29th week. As seen in Fig. 8, company 1 is able to withhold energy from the market to

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Table 1 Pure and mixed equilibria of the weekly meta-games and annual coupled meta-game equilibria.

Company 2

Company 3

Company 4

Company 1

Company 2

Company 3

Company 4

week 0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40%

week 0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40%





15

1

0

0

0

1

0

0

0

0

0

1

0

0

0

1

43

1

0

0

0

1

0

0

0

0

0

1

0

0

1

0

15

1

0

0

0

0

0

0

1

0

1

0

0

1

0

0

43

1

0

0

0

0

0

0

1

1

0

0

0

0

0

1

15

1

0

0

0

0

0

0

1

0

0

0

1

0

1

0

43

0

1

0

0

0

0

1

0

1

0

0

0

0

0

1

15

0

0

0

1

0

0

0

1

0

1

0

0

0

0

1

44

1

0

0

0

1

0

0

0

0

0

0

1

0

0

1

15

0

0

0

1

0

1

0

0

0

0

1

0

0

0

1

44

1

0

0

0

0

0

0

1

0

1

0

0

0

0

1

15

0

0

0

1

0

1

0

0

0

0

0

1

1

0

0

44

0

1

0

0

0

1

0

0

1

0

0

0

0

0

1

16

1

0

0

0

0

1

0

0

1

0

0

0

0

1

0

44

0.56 0.44

0

0

0

0

1

0 0.73 0

0 0.27 0

0

1

16

1

0

0

0

0

0

0

1

0

0

1

0

0

0

1

44

0.59 0.41

0

0 0.09 0.06 0.85

0 0.77 0

0 0.23 0

0

1

16

0

1

0

0

0

0

1

0

0

0

1

0

0

0

1

16

0

1

0

0

0

0

0

1

1

0

0

0

0

0

1

17

1

0

0

0

0

1

0

0

0

0

1

0

0

1

0

18

1

0

0

0

0

1

0

0

1

0

0

0

0

1

18

0

1

0

0

0

0

1

0

0

0

0

1

0

18

0

0

0

1

0

0

1

0

0

1

0

0

… Equilibriums

Equilibriums

Company 1

50

0

0

1

0

1

0

0

0

0

1

0

0

0

1

0

50

0

0

1

0

0

0

1

0

0

0

1

0

0

0

1

0

50

0

0

0

1

0

0

1

0

0

0

1

0

0

1

0

1

0

50

0

1

0

0

0 0.54 0.46

0

0

0

1

0 0.99 0.01

0

0

0

1

50

0

0

1

0 0.91 0.09

0

0

0.35 0.65 0

0

0

0

1

… 1

0

0

0

0

1

0

0

0

0

0

1

0

0

1

29

1

0

0

0

0

0

1

0

0

1

0

0

0

0

1

29

1

0

0

0

0

0

0

1

0

1

0

0

0

1

0

29

0

1

0

0

0

1

0

0

1

0

0

0

0

0

1

29

0

1

0

0

0

0

1

0

0

0

1

0

0

1

0

29

1

0

0

0

0

0

0

0

1

0

0.02 0.98 0

0.1 0.9

increase prices and total profits, especially in periods 3 and 4 of high demand. Note that, in this case, company 1 only offers its optimal quantity, not all its production. Fig. 9 depicts the energy produced during all the periods for both the unrestricted and restricted cases and shows that periods 3 and 4 also give the highest increase in profits when subject to energy constraints. Fig. 10 depicts the residual demand curves corresponding to each of the iterations. For each residual demand there is an optimal point in terms of power. As the iteration count increases, the curves shift upwards due to the other companies’ offers. The last optimal point is on top of the curve meaning that company 1 is restricting the energy in order to obtain more profits. The running time to find the weekly iterative equilibria for the 29th week has been 69 s of CPU.

Company 1

Company 2

Company 3

Company 4

0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40% 60% 0% 20% 40% ANNUAL

29

1

0

0

0

0

0

0

1

0

0

0

1

0

0

1

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1

0

1

0

0

0

0

1

0

0

0

0

1

0

1

0

4.6. Case study 3: coupled and decoupled meta-game equilibria results To find meta-game equilibria both coupled and decoupled methods are possible. In the coupled case, the offers resulting from the equilibrium in the previous week can be used to start the iterative process for the subsequent week to reduce running times. In the decoupled case, the weekly simulations can be run separately. Each generating unit of every company uses 8 offer blocks in this simulation. In this case study, we find the meta-game equilibria for the 52 weeks of the case study allowing the companies to change their offer parameters. The resulting pure and mixed equilibria for both the decoupled case (only showing selected weeks for lack of space reasons) and the coupled case (1 year) are shown part in Table 1. It

Table 2 Profits of the annual coupled meta-game in millions of $. Equilibria that dominate in payoffs are shown in bold. Strategies Company 2

Company 1 Company 3

Company 4

0%

20%

40%

0%

20%

40%

60%

0% 20% 40% 40% 60%

0% 20% 40% 60% 60%

0% 20% 40% 0% 20% 40% 0% 20% 40% 0% 20% 40%

96.941 97.714 95.734 100.268 97.960 96.359 96.293 97.460 95.255 96.983 95.454 93.189

36.957 37.195 36.934 37.580 37.360 37.660 37.046 37.556 37.403 37.314 36.859 36.670

36.467 36.905 36.602 37.281 37.174 37.274 36.553 37.250 37.033 36.778 36.482 36.234

4.644 5.419 5.405 4.910 5.394 5.759 4.604 5.433 5.670 4.954 5.224 5.055

95.566 96.947 94.897 96.484 95.870 97.819 96.252 96.259 95.364 93.780 96.779 93.650

36.444 36.866 37.100 36.949 37.070 37.695 36.992 37.261 37.469 36.409 37.367 36.788

35.938 36.576 36.553 36.491 36.708 37.529 36.458 36.934 37.056 35.717 37.091 36.389

4.519 5.246 5.520 4.596 5.254 5.959 4.652 5.345 5.575 4.646 5.643 5.220

88.306 92.191 90.909 90.814 94.170 92.260 90.092 92.375 91.316 88.834 91.878 90.830

35.039 36.265 36.159 35.745 36.918 36.683 35.931 36.820 36.419 35.760 36.403 36.447

34.169 35.510 35.463 35.004 36.377 36.078 35.032 36.142 35.853 34.870 35.698 35.803

4.115 4.974 4.947 4.259 5.360 5.258 4.459 5.244 5.087 4.489 4.955 5.155

83.918 84.565 87.526 83.896 – 89.598 85.238 – 86.819 – – –

34.419 34.652 35.937 34.510 – 36.512 34.803 – 36.096 – – –

33.167 33.598 34.994 33.392 – 35.753 33.727 – 35.129 – – –

3.559 3.885 4.877 3.444 – 5.029 3.768 – 4.866 – – –

4.664 5.553 5.612 4.734 5.835 5.371

90.777 91.875 92.277 90.312 90.985 91.880

36.081 36.797 36.819 35.947 36.475 36.839

35.314 36.043 36.264 35.145 35.607 36.061

4.408 5.205 5.398 4.649 4.904 5.283

– – 88.189 – – –

– – 36.258 – – –

– – 35.438 – – –

– – 5.003 – – –

4.726 5.667 6.019 4.783 5.669 5.662

90.597 92.259 92.751 90.621 91.477 92.630

35.965 36.656 37.163 36.254 36.516 37.124

35.208 36.033 36.625 35.294 35.796 36.410

4.466 5.209 5.856 4.759 5.170 5.245

– 86.536 87.874 – – 88.006

– 35.875 36.713 – – 36.300

– 34.791 35.668 – – 35.405

– 4.594 5.290 – – 5.050

– – 0% 20% 40% 0% 20% 40%

97.141 94.811 93.953 94.700 97.895 94.255

37.496 37.034 37.110 36.811 37.588 37.064

36.958 36.547 36.695 36.169 37.293 36.566

4.856 5.398 5.647 4.880 5.715 5.202

95.730 95.469 93.967 94.582 98.642 95.145

36.916 37.236 37.213 36.945 37.812 37.308

36.365 36.764 36.731 36.255 37.560 36.940 – –

0% 20% 40% 0% 20% 40%

98.377 97.666 93.825 95.318 96.398 97.491

37.336 37.770 37.431 36.874 37.278 37.917

36.961 37.455 36.984 36.243 36.835 37.581

5.125 5.848 5.876 4.838 5.670 5.848

95.937 97.320 94.937 95.097 96.697 94.989

37.011 37.828 37.428 36.818 37.268 37.377

36.560 37.411 37.084 36.316 36.957 36.945

D. Pozo et al. / Electric Power Systems Research 81 (2011) 329–339

0%

20%

60%



337

338

D. Pozo et al. / Electric Power Systems Research 81 (2011) 329–339

must be noted that the equilibria that result from the simulations in Table 1 are filtered through a payoff dominance criterion [7]. That means that the Nash equilibria in which all the companies’ payoffs have lower values (with respect to the dominant Nash equilibria) are removed. In the decoupled case, the equilibria during the week with the lowest demand, week 16, are the most competitive and company 1 bids using low offer parameters. The opposite happens in the week with the highest demand, week 50, where the offer bids by company 1 are riskier. Note that the payoff dominance criterion has removed many equilibria from Table 1 whose values are very different from the ones shown. In the annual coupled meta-game, there are 12 equilibria, 5 pure and 7 mixed, but there are only 3 that dominate in payoffs, which are the ones shown in bold Table 2. The matrix values left blank mean that there is no equilibrium for these particular set of offer strategies. Company 1 tends not to take risks in its offers since higher offer parameters diminish profits; on the contrary the other companies tend to risk more in general, especially when company 1 does not. This is noticeable in Table 2, where higher offer parameters of company 1 diminish its own profits regardless of what the other companies do. To solve the coupled meta-game initialized by the previous week and the 52 decoupled meta-games, 2 days and 12 h and 2 days and 17 h of CPU are necessary, respectively. 5. Extensions In this section, extensions to stochastic Nash equilibria models and applications to price scenario generation are described. 5.1. Stochastic Nash equilibria The Nash equilibria obtained in the meta-games described in the paper are based on the deterministic model shown in Section 3. The demand, fuel costs, and physical characteristics for each generating unit are perfectly known. An extension of the meta-game model assumes stochastic demand. The demand realization tree and the scenario probabilities can be obtained from different well-known sampling methodologies derived from statistical models [25] suitable to scenario reduction techniques [26]. Each scenario is defined by its probability and a payoff matrix. In a stochastic meta-game, each strategy vector is made up of pure strategies (one per company) and the payoff is generated from the expected value that results when all companies choose the same strategy vector for all demand realizations. Thus, each company’s expected value is the sum for all scenarios of the payoffs multiplied by their probabilities. The Stochastic Nash equilibria comes from the solution of the stochastic meta-game matrix, whose elements are the aforementioned expected values. Note that, as in non-stochastic Nash equilibria, the solutions of the stochastic game can be pure or mixed, being the offer parameters of each company. In addition, the stochastic matrix generation process can be applied to coupled or decoupled meta-games, resulting in annual or weekly equilibria, respectively. 5.2. Generation of price scenarios Financial markets need market-clearing prices (MCP) inputs to assess the value of a generator in a market environment, as is the case of real options analysis [27–30]. Most financial models that evaluate generation assets use MCP scenarios as an alternative to forecasting time series or modeling electricity prices with Geometric Brownian Motion (GMB) [28], among other methods. In particular, the model presented in Section 2 is suitable to generate

MCP scenarios for each demand realization. The resulting MCP scenarios are based on demand scenarios (generally more accurate to forecast), on market rules, and on strategic bidding, as modeled in the iterative Nash equilibria algorithm. Thus, by changing the bidding and the demand patterns in our model we can produce a set of scenarios that can feed a real options model, in which assessment of generating units or long-term investment decisions can be made. 6. Conclusions We present an iterative Nash equilibrium model for electricity markets. The model is suitable for application in short- and long-term analysis of pool-based electricity markets. It uses an iterative market simulation tool, in which the offer strategies and the market-clearing algorithm are defined. Among the offer strategies, the offer parameter strategy is explained in detail. An exponential smoothing procedure is applied to achieve convergence in the market-clearing algorithm and a piecewise cubic Hermite polynomial interpolation is applied to accurately model the residual demand curves. To find the range of plausible equilibria we introduce the concept of meta-games. Meta-games are high-level games in which the companies select different bidding strategies either in short- or long-term games. The contributions of the model are: (i) the use of realistic parametrized offer strategies by the companies, (ii) the use of Hermite polynomials to accurately describe residual demand curves, (iii) the concepts of coupled (long-term) and decoupled (shortterm) meta-game equilibria, where the companies can use a range of offer strategies and (iv) the effect of exponential smoothing in the convergence of the algorithm. As a result of having all these features implemented, the model is able to find market equilibria for fixed strategies and meta-game equilibria for a range of strategies. This makes the model a valuable tool for companies and regulators to predict future market prices and energy quotas and also to find and compare pure and mixed short- and long-term equilibria in different realistic scenarios. An application of the model to generate MCP scenarios suitable to real options analysis and an stochastic Nash equilibria extension are briefly described. Acknowledgement The authors are grateful for the financial support of the Gas Natural Fenosa. Appendix A. List of symbols

A. Indexes b offer blocks of each generating unit d demands h bid blocks of each demand i generating companies in the market j thermal generating units m iteration index t time periods of the time horizon B. Constants NB number of offer blocks for each generating unit ND number of demands number of bid blocks for each demand NH NJ number of generating units number of periods of the time horizon NT D-bid price bid by the hth block of the dth demand in period t dth G-offer

jtb

price offered by the bth block of the jth generating unit in period t

D. Pozo et al. / Electric Power Systems Research 81 (2011) 329–339

Pj

pjtb

upper bound of the power output of the jth generating unit lower bound of the power output of the jth generating unit size of the bth block of the jth generating unit in period t

pdth pG jt1 

size of the hth block of the dth demand in period t size of the first block of the jth generating unit in period t weighting factor of the exponential smoothing procedure

Pj G

D

C. Variables pD power consumed by the dth demand in period t dt pD power consumed by the hth block of the dth demand in dth period t pG power produced by the jth generating unit in period t jt pG jtb

˛t

power produced by the bth block of the jth generating unit in period t power produced by the ith company in period t power produced by the ith company in period t using the exponential smoothing procedure weight associated with the number of hours of period t

D. Sets ˝D ˝I ˝J ˝T i

set of all demands set of all generating companies set of all generating units set of all the periods of the time horizon set of all generating units belonging to company i

pit pˆ it

E. Functions Cj (pG ) production cost for period t of the jth generating unit jt it (pit )

residual demand function of the ith company in period t, where it (pit ) expresses the resulting market price as a function of the power produced by the ith company in period t, pit

k+1

it (pit )

k

piecewise cubic Hermite polynomial function by the ith company in period t for the [k, k+1] interval of the residual demand curve

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Long-term Nash equilibria in electricity markets

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