On the Variance of Electricity Prices in Deregulated Markets

1. On the Variance of Electricity Prices in Deregulated Markets Ph.D. Dissertation Claudio M. Ruibal University of Pittsburgh August 30, 2006

2. Agenda • Characteristics of electricity and of its price • Object of study and uses • Electricity markets • Pricing models • Mean and variance of hourly price • Mean and variance of average price • Conclusions and contributions of this work • Recommendations for future work

3. Characteristics of electricity • Electricity is not storable. • Electricity takes the path of least resistance. • The transmission of power over the grid is subject to a complex series of interactions (e.g., Kirchhoff’s laws). • Electricity travels at the speed of light. • Electricity cannot be readily substituted. • It can only be transported along existing transmission lines which are expensive and time consuming to build.

4. Goals of competition in electricity markets through deregulation • Improving efficiency in both supply and demand side. • Providing cost-minimizing incentives • Stimulating creativity to develop new energy-saving technologies. • Making better investments. • Promoting energy conservation. • But as a consequence electricity prices show an extremely high variability.

5. Comparing prices of five days Source: PJM Interconnection, Hourly Average Locational Price

6. Comparing load of five days Source: PJM Interconnection, Hourly Load

7. Two months Source: PJM Interconnection, Hourly Average Locational Price

8. A year Source: PJM Interconnection, Hourly Average Locational Price

9. Object of study • The expected value and variance of hourly and average electricity prices with a fundamental bid-based stochastic model. • Hourly price: the price for each hour. • Average price: a weighted average of the hourly prices in a period (e.g., on-peak hours, a day, a week, a month, etc.)

10. Uses of the variancesHourly prices • Pricing: decisions on offer curves • Measuring profitability of peak units • Scheduling maintenance • Determining the type of units needed for capacity expansion.

11. Uses of the variancesAverage Prices • Prediction of prices • Budgeting cash flow • Calculating Return over Investment (ROI) • Managing risk • Valuation of derivatives • Calculation of VaR and CVaR • Computation of the expected returns -variance of returns objective function.

12. Electricity marketplace Transmission Companies Retail Companies Charge a fee for the service of transmitting electricity Charge a fee for the service of connecting, disconnecting and billing Retailing Generation Companies Distribution Companies End users Transmission process Distribution process Retail Marketplace Wholesale Marketplace

13. Electricity Market at work

14. Real time market Today's Outlook

15. Energy risk management • There is a need for the firms to hedge the risk associated with variability of prices. • Derivatives prices depend on the variance. • Value-at-Risk and Conditional Value-at-Risk (Rockafellar and Uryasev, 2000). • Expected returns – variance of returns objective function (Markowitz, 1952)

16. Value-at-Risk and Conditional Value-at-Risk mean CVaR Figure extracted from http://www.riskglossary.com/link/value_at_risk.htm

17. Markowitz’s Expected return-variance of returns Variances Attainable E,V combinations Efficient E,V combinations Expected values

18. Electricity price models

19. Electricity market modeling trends

20. The model selected • Combined imperfect-market equilibrium/ stochastic production-cost model. • Based on fundamental drivers of the price. • It considers uncertainty from two sources: • Demand • Units’ availability • It compares three equilibrium models: • Bertrand • Cournot • Supply Function Equilibrium

21. Bidding behavior

22. Supply Function Equilibrium (SFE) Klemperer and Meyer (1989) Green and Newbery (1992) Supply function equilibria for a symmetric duopoly are solutions to this differential equation: Here, p is bounded by to satisfy the non-decreasing constraint.

23. Allowable Supply Functions

24. Rudkevich, Duckworth, and Rosen (1998) Assumptions: • step-wise supply functions • n identical generating firms • Dp = 0 (which zero price-elasticity of demand) • perfect information • equal accuracy in predicting demand • taking the lowest SFE which means that the price at peak demand equals marginal cost, i.e. p(Q*) = dM The Nash Equilibrium solution to the differential equation is:

25. Rudkevich, Duckworth, and Rosen’s supply functions

26. Modeling supply The system consists of N generating units dispatched according to the offered prices (merit order). The jth unit in the loading order has cjcapacity (MW) dj marginal cost ($/MWh) pj= j/(j+j) proportion of time that it is up j failure rate j repair rate There exists the possibility of buying energy outside the system, which is modeled as a (N+1)th generating unit, with large capacity and always available.

27. Operating state of the units The operating state of each generating unit j follows a two-state continuous time Markov chain Yj(t), For i j, Yi(t) and Yj(s) are statistically independent for all values of t and s.

28. Mean and variance of the hourly price where:

29. Probability distribution of the marginal unit The following events are equivalent: and So, to know the distribution of J(t), we should evaluate the argument of the RHS for all j:

30. Auxiliary variable Excess of load Xj(t) that is not being met by the available generated power up to generating unit j, with a cumulative distribution function Gj(x:t).

31. Edgeworth expansion Where:

32. Equivalent load price It captures the uncertainty of demand and of units’ availability at the same time p(t) Missing ci quantity L(t) Equivalent L(t) This approach is useful to determine the price and the marginal unit.

33. Modeling electricity prices under Bertrand model

34. Modeling electricity prices under Cournot model

35. Approximation of the equivalent load expected value

36. Modeling electricity prices under Supply Function Equilibrium

37. Average electricity price • Daily load profile considered to be deterministic. • Joint probability distribution of marginal units at two different hours. • Expressions for the expected values and variances for the three models: Bertrand, Cournot and Rudkevich.

38. Daily load profile

39. Expected value and variance of hourly load

40. Covariance of prices

41. Joint probability distribution of marginal units at two hours

42. Bivariate Edgeworth expansion

43. Bertrand model

44. Cournot model

45. Rudkevich model

46. Numerical results • Supply model: 12 identical sets of 8 units. • Load model: data from PJM market, Spring 2002, scaled to fit the supply model. • Sensitivity analysis on: • Number of competing firms: 3 to 12 • Slope of the demand curve Dp: -100 to -300 (MWh)2/$ • Anticipated peak demand as percentage of total capacity: 60% to 100%.

47. Hourly prices (Cournot)

48. Hourly prices (Rudkevich)

49. Rudkevich supply functions (6 firms)

50. Rudkevich supply functions (12 firms)