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On the Variance of Electricity Prices in Deregulated Markets

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On the Variance of Electricity Prices in Deregulated Markets

Ph.D. Dissertation

Claudio M. Ruibal

University of Pittsburgh

August 30, 2006

- 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

- 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.

- 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.

Source: PJM Interconnection, Hourly Average Locational Price

Source: PJM Interconnection, Hourly Load

Source: PJM Interconnection, Hourly Average Locational Price

Source: PJM Interconnection, Hourly Average Locational Price

- 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.)

- Pricing: decisions on offer curves
- Measuring profitability of peak units
- Scheduling maintenance
- Determining the type of units needed for capacity expansion.

- 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.

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

Today's Outlook

- 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)

mean

CVaR

Figure extracted from http://www.riskglossary.com/link/value_at_risk.htm

Variances

Attainable

E,V combinations

Efficient

E,V combinations

Expected values

- 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

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.

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:

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.

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.

where:

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:

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).

Where:

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.

- 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.

- 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%.

- 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.

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

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.

xtfollows an ARIMA (1,120,0) process

zt is a Gaussian white noise with mean zero and standard deviation z

- 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.

- 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.

- 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!