Investments in Power Generation Under Uncertainty—a MIP Specification and Large-Scale Application for EU

Abstract

Investments in power generation constitute a typical budget allocation problem in the context of multiple objectives, while all factors influencing investor’s decisions for power plants are subject to considerable uncertainties. The paper introduces a multi-objective stochastic model designed to optimize budget allocation decisions for power generation in the context of risk aversion taking into account several sources of uncertainty, especially with regard to volatility of fossil fuel and electricity prices, technological costs, and climate policy variability. Probability distributions for uncertain factors influencing investment decisions are directly derived from the stochastic global energy model PROMETHEUS and thus they take into account complex interactions between variables in the systemic context. In order to fully incorporate stochastic characteristics of the problem, the model is specified as an optimization problem in which the probability that an objective exceeds a given threshold is maximized (risk aversion) subject to a set of deterministic and probabilistic constraints. The model is formulated as a mixed integer program providing complete flexibility on the joint distributions of rates of return of technologies competing for investments, as it can handle non-symmetric distributions and take automatically into account complex covariance patterns as emerging from comprehensive PROMETHEUS stochastic results. The analysis shows that risk is a crucial factor for power generation investments with investors not opting for technologies subject to uncertainty related to climate policies and fossil fuel prices. On the other hand, combination of options with negative covariance tends to benefit in the context of risk-hedging behavior.

Appendices

Annex A

The figure below presents various types of distributions of rates of return obtained from PROMETHEUS stochastic results (distributions are hypothetical and are used for illustrative purposes). The distribution at the bottom right-hand side can be adequately represented by normal distribution with the same mean and standard deviation. The distributions at the top of figure are typical of the more speculative/uncertain power generation technologies that are asymmetric and display high skewness, i.e., the top left figure illustrates a case with high probabilities for low profitability and relatively small probabilities for high rates of return. The multi-modal distribution defies all attempts to fit a normal approximation and it is essentially characterized by virtually discrete clusters for low or high return on investment.

Examples of the probability distributions of rates of return of technologies as emerging from PROMETHEUS analysis:

Annex B

Equations below are used to calculate revenues and upfront capital investment expenditures respectively (in each Monte Carlo experiment). Revenues are calculated on annual basis and then are aggregated over the lifetime of investment to determine annual rate of return of each technological option.

$$ {\mathsf{rev}}_{\mathit{\mathsf{j}}}=\mathsf{hours}\times {\mathsf{capfact}}_{\mathit{\mathsf{j}}}\left(\mathsf{elcprice}-{\mathsf{vom}}_{\mathit{\mathsf{j}}}-\frac{\left({\mathsf{fprice}}_{\mathit{\mathsf{j}}}+{\mathsf{CV}}^{\ast }{\mathsf{emfac}}_{\mathit{\mathsf{j}}}\right)}{{\mathsf{eff}}_{\mathit{\mathsf{j}}}}-\mathsf{cdist}\right)-{\mathsf{fc}}_{\mathit{\mathsf{j}}} $$

(16)

$$ {\mathsf{cc}}_{\mathit{\mathsf{j}}}=\mathsf{ovcc}{\mathsf{2030}}_{\mathit{\mathsf{j}}}\times \left[\frac{{\mathsf{ctime}}_{\mathit{\mathsf{j}}}\times \mathsf{ir}\times {\left(\mathsf{1}+\mathsf{ir}\right)}^{{\mathsf{ctime}}_{\mathit{\mathsf{j}}}}}{{\left(\mathsf{1}+\mathsf{ir}\right)}^{{\mathsf{ctime}}_{\mathit{\mathsf{j}}}}-\mathsf{1}}+{\mathsf{decshare}}_{\mathit{\mathsf{j}}}\times {\mathit{\mathsf{e}}}^{-\left(\mathsf{ir}+\mathsf{0.04}\right)\times {\mathsf{lft}}_{\mathit{\mathsf{j}}}}\right] $$

(17)

Parameter hours represents the required operation hours of power plants (considered as 7000 per year in our base load example), capfact _j is the availability factor of technology j (uncertain and derived from PROMETHEUS results), while elcprice represents electricity price for industrial consumers, which is also uncertain, derived from PROMETHEUS stochastic projections and is based on the evolution of long-term average power generation and distribution cost. Parameters vom _j and eff _j symbolize variable costs of Operation and maintenance (O&M) and technical efficiency of technologies, respectively, which are derived from the technology progress mechanisms of PROMETHEUS, fprice _j represents the price for fossil fuels used in electricity production (from PROMETHEUS analysis), parameter CV is the uncertain carbon price (used as a measure for climate policy intensity), emfac _j is the deterministic emissions factor per fossil fuel, cdist is the transport and distribution costs of electricity to industrial consumers, and the variable fc _j represent fixed O&M costs.

Stochastic capital outlays are calculated from Eq. (17) where ovcc2030 _j represents overnight cost for investment in 2030 (which is stochastic and derived from PROMETHEUS results), decshare _j refers to decommissioning costs of power production technologies, ctime _j is the uncertain construction period of time of technology j, lft _j is the lifetime of technology j, while parameter ir represents stochastic long term interest rate that is derived from PROMETHEUSS macro-economic module.