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

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

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.

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.

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

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

Two months Source: PJM Interconnection, Hourly Average Locational Price

A year Source: PJM Interconnection, Hourly Average Locational Price

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.)

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.

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.

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

Electricity Market at work

Real time market Today's Outlook

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)

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

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

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

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

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

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

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.

Allowable Supply Functions

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:

Rudkevich, Duckworth, and Rosen’s supply functions

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.

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.

Mean and variance of the hourly price where:

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:

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).

Edgeworth expansion Where:

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.

Modeling electricity prices under Bertrand model

Modeling electricity prices under Cournot model

Approximation of the equivalent load expected value

Modeling electricity prices under Supply Function Equilibrium

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.

Daily load profile

Expected value and variance of hourly load

Covariance of prices

Joint probability distribution of marginal units at two hours

Bivariate Edgeworth expansion

Bertrand model

Cournot model

Rudkevich model

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%.

Hourly prices (Cournot)

Hourly prices (Rudkevich)

Rudkevich supply functions (6 firms)

Rudkevich supply functions (12 firms)

Rudkevich supply functions (3 firms)

Average hours13-16

Average hours 3-6

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.

Correlation load-temperature

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

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

Actual and expected load (3 models)

Variance of hourly load (3 models)

Probability distribution marginal unit (load model 1)

Probability distribution marginal unit (load model 2)

Probability distribution marginal unit (load model 3)

Joint probability (model 1)

Joint probability (model 2)

Hourly prices (3 models)

Average prices (3 models, hours 13-18)

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.

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.

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

Thank you!