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 variances Hourly 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 variances Average 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

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

18. Electricity price models
Game theory models
Study the agents’ strategic behavior
Production-cost models
Simulate energy production and market processes
Time series model
Perform statistical analysis on price data

19. Electricity market modeling trends
Technique
Applied to
Comments
Optimization models
Profit maximization of one firm
Well known and robust algorithms
Equilibrium models
Simplified overall markets
Cournot: tractable
SFE: more realistic
Simulation models
Complex overall markets
Capture iterative characteristic of markets

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 Type of Competition Strategies Equilibrium Bertrand
prices
Marginal cost
Cournot
quantities
Marginal cost plus a term depending on price elasticity of demand
Supply Function Equilibrium
supply functions
Marginal cost plus a term

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
cj capacity (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)

51. Rudkevich supply functions (3 firms)

52. Average hours13-16

53. Average hours 3-6

54. Stochastic model of the load
So far, hourly loads were considered normally distributed (load model 1).
The effect of temperature on the load is studied in models 2 and 3.
The remaining term, after removing the effect of temperature, is considered as:
Normally distributed (load model 2)
Time series (load model 3)
Results for a data set for Spring–Summer 1996 are shown to compare models.

55. Correlation load-temperature

56. Load model 2 where L(t) is the load at hour t
f(t) accounts for part of the load that is ascribed to the ambient temperature t.
x(t) is normally distributed, not independent

57. Load model 3 where L(t) is the load at hour t
f(t) accounts for part of the load that is ascribed to the ambient temperature t.
xt follows an ARIMA (1,120,0) process
zt is a Gaussian white noise with mean zero and standard deviation z

58. Actual and expected load (3 models)

59. Variance of hourly load (3 models)

60. Probability distribution marginal unit (load model 1)

61. Probability distribution marginal unit (load model 2)

62. Probability distribution marginal unit (load model 3)

63. Joint probability (model 1)

64. Joint probability (model 2)

65. Hourly prices (3 models)

66. Average prices (3 models, hours 13-18)

67. Conclusions
The number of firms in the market is an important factor for the mean and variance of prices.
Increasing elasticity will bring down prices and variances significantly.
Rudkevich model presents a nice trade off between excess capacity and price. Being tight to full capacity brings prices up.
An accurate forecast of temperature can reduce significantly the prediction error of prices.
A rigorous time series analysis of the load does not increase the accuracy of prediction.

68. Contributions of this work
It is new model for electricity prices combining a statistical approach and a game theory viewpoint.
The expected values and variances of hourly and average prices can be computed with closed form expressions.
The covariances of hourly prices have been calculated.

69. Recommendations for future research
Calibrating the model for a real market
Incorporating fuel cost as another source of uncertainty
Extending the model for asymmetric firms
Incorporating transmission constraints
Incorporating the unit commitment problem

70. Thank you!