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. Agenda. Characteristics of electricity and of its price Object of study and uses Electricity markets Pricing models

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On the variance of electricity prices in deregulated markets

On the Variance of Electricity Prices in Deregulated Markets

Ph.D. Dissertation

Claudio M. Ruibal

University of Pittsburgh

August 30, 2006


Agenda

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

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

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

Comparing prices of five days

Source: PJM Interconnection, Hourly Average Locational Price


Comparing load of five days

Comparing load of five days

Source: PJM Interconnection, Hourly Load


Two months

Two months

Source: PJM Interconnection, Hourly Average Locational Price


A year

A year

Source: PJM Interconnection, Hourly Average Locational Price


Object of study

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

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.


Uses of the variances average prices

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.


Electricity marketplace

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

Electricity Market at work


Real time market

Real time market

Today's Outlook


Energy risk management

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

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

Markowitz’s Expected return-variance of returns

Variances

Attainable

E,V combinations

Efficient

E,V combinations

Expected values


Electricity price models

Electricity price models


Electricity market modeling trends

Electricity market modeling trends


The model selected

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

Bidding behavior


Supply function equilibrium sfe

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

Allowable Supply Functions


Rudkevich duckworth and rosen 1998

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

Rudkevich, Duckworth, and Rosen’s supply functions


Modeling supply

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.


Operating state of the units

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

Mean and variance of the hourly price

where:


Probability distribution of the marginal unit

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

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

Edgeworth expansion

Where:


Equivalent load

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 Bertrand model


Modeling electricity prices under cournot model

Modeling electricity prices under Cournot model


Approximation of the equivalent load expected value

Approximation of the equivalent load expected value


Modeling electricity prices under supply function equilibrium

Modeling electricity prices under Supply Function Equilibrium


Average electricity price

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

Daily load profile


Expected value and variance of hourly load

Expected value and variance of hourly load


Covariance of prices

Covariance of prices


Joint probability distribution of marginal units at two hours

Joint probability distribution of marginal units at two hours


Bivariate edgeworth expansion

Bivariate Edgeworth expansion


Bertrand model

Bertrand model


Cournot model

Cournot model


Rudkevich model

Rudkevich model


Numerical results

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 (Cournot)


Hourly prices rudkevich

Hourly prices (Rudkevich)


Rudkevich supply functions 6 firms

Rudkevich supply functions (6 firms)


Rudkevich supply functions 12 firms

Rudkevich supply functions (12 firms)


Rudkevich supply functions 3 firms

Rudkevich supply functions (3 firms)


Average hours13 16

Average hours13-16


Average hours 3 6

Average hours 3-6


Stochastic model of the load

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

Correlation load-temperature


Load model 2

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

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.

xtfollows 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

Actual and expected load (3 models)


Variance of hourly load 3 models

Variance of hourly load (3 models)


Probability distribution marginal unit load model 1

Probability distribution marginal unit (load model 1)


Probability distribution marginal unit load model 2

Probability distribution marginal unit (load model 2)


Probability distribution marginal unit load model 3

Probability distribution marginal unit (load model 3)


Joint probability model 1

Joint probability (model 1)


Joint probability model 2

Joint probability (model 2)


Hourly prices 3 models

Hourly prices (3 models)


Average prices 3 models hours 13 18

Average prices (3 models, hours 13-18)


Conclusions

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

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

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

Thank you!


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