# Capacity expansion and forward contracting in a duopolistic power sector

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## Abstract

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.

### Keywords

Stochastic programming Complementarity modelling Energy markets Risk management Capacity expansion### Sets

- \(\varOmega \)
Scenarios

- \(\mathcal I \)
Producers

- \(\mathcal N \)
Forward blocks

### Indices

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

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

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

### Decision variables

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

### Parameters

- \(\pi _\omega \)
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 \(i\)

### Dual variables

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

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