Capacity expansion and forward contracting in a duopolistic power sector

The surge in demand for electricity in recent years requires that power companies expand generation capacity sufficiently. Yet, at the same time, energy demand is subject to seasonal variations and peak-hour factors that cause it to be extremely volatile and unpredictable, thereby complicating the decision-making process. We investigate how power companies can optimise their capacity-expansion decisions while facing uncertainty and examine how expansion and forward contracts can be used as suitable tools for hedging against risk under market power. The problem is solved through a mixed-complementarity approach. Scenario-specific numerical results are analysed, and conclusions are drawn on how risk aversion, competition, and uncertainty interact in hedging, generation, and expansion decisions of a power company. We find that forward markets not only provide an effective means of risk hedging but also improve market efficiency with higher power output and lower prices. Power producers with higher levels of risk aversion tend to engage less in capacity expansion with the result that together with the option to sell in forward markets, very risk-averse producers generate at a level that hardly varies with scenarios.

1. According to the International Energy Outlook 2011 by the U.S. Energy Information Administration, energy consumption is expected to increase by 53 % from 2008 to 2035. Worldwide total consumption is predicted to expand by 1.6 % yearly, with OECD countries’ demand rising by 0.6 % annually, and energy consumption in non-OECD countries growing by 2.3 % a year on average.

2. As the models in this paper are relatively simple and involve few parameters and scenarios, the problem is easily tractable and takes less than a second on average to solve in GAMS. PATH is amenable for addressing large-scale systems, e.g., Gabriel et al. (2005) use it to solve a complementarity model for the North American natural gas network. Solution times for an implementation with over 4,000 variables are \(<\)100 s.

3. For the monopoly case, we use \(c^G_{2,i}=\frac{1}{2}\) in order to have the same industry-wide supply curve as in a duopoly.

\(\varOmega \) :

Scenarios

\(\mathcal I \) :

Producers

\(\mathcal N \) :

Forward blocks

\(\omega \) :

Scenario index, \(\omega \in \varOmega \)

\(i \) :

Producer index, \(i \in \mathcal I \)

\(n \) :

Forward block index, \(n \in \mathcal N \)

\(p^S_{i,\omega }\) :

Power sold in the spot by producer \(i\) in scenario \(\omega \) (MW)

\(p^F_{i,n}\) :

Power sold forward by producer \(i\) in forward block \(n\) (MW)

\(p^G_{i,\omega }\) :

Total power generated by producer \(i\) in scenario \(\omega \) (MW)

\(\lambda ^S_\omega \) :

Spot price of power in scenario \(\omega \) ($/MW)

\(\Delta _i\) :

Amount of capacity expansion done by producer \(i\) (MW)

\(R^S_{i,\omega }\) :

Spot revenue earned by producer \(i\) in scenario \(\omega \) ($)

\(\zeta _i\) :

Value-at-risk (VaR) of producer \(i\) ($)

\(\eta _{i,\omega }\) :

Auxiliary variable that varies with scenario \(\omega \) used to calculate the CVaR of producer \(i\)

\(\pi _\omega \) :

Probability of scenario \(\omega \)

\(\lambda ^{S0}_\omega \) :

Intercept of the inverse spot demand curve in scenario \(\omega \) ($/MW)

\(\gamma \) :

Slope of the inverse spot demand curve ($/MW\(^2\))

\(\lambda ^F_n\) :

Forward price of power in block \(n\) ($/MW)

\(c^G_{1,i}\) :

Linear coefficient of cost function for producer \(i\) ($/MW)

\(c^G_{2,i}\) :

Quadratic coefficient of cost function for producer \(i\) ($/MW\(^2\))

\(P^{max}_i\) :

Maximum initial generating capacity of producer \(i\) (MW)

\(Q^F_{i,n}\) :

Maximum quantity of forward sales by producer \(i\) in block \(n\) (MW)

\(\bar{\Delta }_i\) :

Maximum capacity expansion permitted for producer \(i\) (MW)

\(c^E_i\) :

Per-unit cost of expansion for producer \(i\) ($/MW)

\(\alpha \) :

Confidence level used for calculation of CVaR

\(\beta _i\) :

Measure of risk aversion of producer \(i\)

\(\phi _{i,\omega }\) :

Dual price for capacity constraint of producer \(i\) in scenario \(\omega \) ($/MW)

\(\rho _i\) :

Dual price for expansion constraint of producer \(i\) ($/MW)

\(\theta _{i,\omega }\) :

Dual price of the constraint imposed to calculate the CVaR of producer \(i\) in scenario \(\omega \)

\(\delta _{i,n}\) :

Dual price of the forward block-quantity constraint for producer \(i\) in block \(n\) ($/MW)

We are grateful to feedback received from the participants of the Computational Management Science Conference in London, UK (18–20 April 2012). Suggestions from the editor and two anonymous referees have helped to improve this work. Seminars by Ruud Egging during the Oppdal Winter School (6–13 March 2011) provided the initial idea for this paper. All remaining errors are the authors’ own.

Authors and Affiliations

1. Department of Statistical Science, University College London, London, UK

   Dorea Chin & Afzal Siddiqui

2. Department of Computer and Systems Sciences, Stockholm University, Stockholm, Sweden

   Afzal Siddiqui

Corresponding author

Correspondence to Afzal Siddiqui.

Appendix A: Tables of numerical results

See Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.

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