A REAL OPTIONS MODEL fOR Electricity CAPACITY expansion

1. A REAL OPTIONS MODEL fOR Electricity CAPACITY expansion

2. Christos Nakos, NTUA, Postgraduate Student Optimal Management of the Dynamic Systems of the Economy and the environment THALES RESEARCH WORKSHOP

3. This presentation mainly follows the developments produced in : Gahungu, Jand Y. Smeers (2011), “ A Real Options Model for Electricity Capacity Expansion“, CORE discussion paper, UniversiteCatholique de Louvain, Belgium Basic Literature

4. Why use such kind of model for the capacity expansion of the power system? • Better representation of risks (economic, regulatory) • Traditional capacity expansion models of the optimization type become intractable when extended to a stochastic setting • Provide an intuitive financial representation • It offers a complementary field of investment analysis compared to the well known evaluation method of NPV. INTRODUCTION

5. Real option capacity expansion models for the power sector should take into account the following crucial particularities: • Electricity is a differentiated product, i.e. cannot be stored • Technologies differ by both operation and investment cost, a fact inconsonant with the assumptions considered by the majority of the real options capacity expansion models • Profits accruing from new capacities are not given in a closed form but need to be computed numerically by an optimization problem Particularities of the power sector

6. Assumption 1 : A competitive electricity market is considered, where a portfolio of k different generation technologies, candidates for investment, is introduced. These technologies have different technoeconomic characteristics(investment cost, fuel and VAROM costs, FIXOM costs) Assumption 2: The annual load demand is modelled as a decreasing step function (annual load curve), meaning that a certain (fixed) level of load is activated for each load segment respectively. Description of the Model

7. In reality, the annual load curve is non-linear and generally follows the form shown in figure. However, it is a well-known technique to approach the annual load curve as a step function under the condition that the area under the line, i.e the annual energy consumed remains equal in both cases. Description of the Model

8. Assumption 3 : An activity index (e.g. GDP)is considered, which is subject through time random disturbances (e.g). This index drives the demand for electricity in the market. Real options capacity expansion models under uncertainty in perfect competitive markets can be described by an optimal control problem. In such control problems a social benevolent planner is considered to maximize the welfare by controlling the capital stock K, i.e. the new capacity going to be installed in the market. Lucas and Prescott (1971) The equivalence between perfect competition and maximization of the welfare in real options context is theoretically guaranteed. Dixit and Pindyck(1994) Description of the Model

9. Assumption 4 : The new capacity to be installed , functions as the control variable and is modelled as a stochastic process. More than that it has to be non-decreasing, since it represents the cumulative investment installed in the market throughout the horizon of the study. Description of the Model

10. The social planner’s instantaneous welfare problem The numerical approach for the computation of is based on the following program : s.t. Description of the Model

11. The social planner’s capacity expansion problem Assuming that the instantaneous welfare problem is tractable, i.e. knowing , then the social planners observes the evolution of and develops/controls its capital stock , targeting the maximization of the economic value of the power sector. Important to note that investment is considered irreversible. In mathematical terms the problem is formulating as follows: Description of the Model

12. Stochastic control problems may be transformed in a succession of optimal stopping problems • The classical Dynamic Programming Techniques fail to apply when the control variable has to be non-decreasing • The equivalence between the two formulations (SCPOSP) has been proved, only when certain conditions are met. Baldursson and Karatzas(1997) Stochastic CONTROL PROBLEM AS a succession of optimal STOPING PROBLEMS

13. The situation when the equivalence holds is often stated as optimality of myopia • The term myopia is used to reflect the fact that each agent in the market acts assuming that a new capacity addition will be the last to be made in the horizon of the study. • The sufficient and necessary conditions for the existence of myopia optimality, are : • Investments should be defined in an incremental way • The economy should be convex • The agents should be homogenous • The profit should be additively separable for each technology, or else each technology should have the same investment costs Conditions of Equivalence

14. For a stopping time τ define the function: C(Y,,τ) (1) Consider now the optimal stopping problem: arginfC(Y,,τ), (2) And the so called Risk Function: rC (3) Which by intuition can be considered: r The optimal sTopping problem –Single Technology model

15. From (2) can be derived the following FBP: + =0, (5) = I (6) =0 (7) Remember that to solve PROBLEM 2, having assumed that (4) holds, it is required to find the function F and the level such that (5),(6),(7) also hold Moving to A FREE boundary problem –single Technology

16. Assuming that the FBP has been solved, one has to integrate the risk function with respect to the state variableand compute . However for multi-technology models both the diversification of the techno-economic characteristics of the different generation technologies as well as the weakness to assign a certain contribution for each technology at the computation of , prevents the direct application of results drawn from single technology real options capacity expansion models. Issues for Multi-tEchnology models

17. It has been stated that we cannot have Ψ in a closed analytical form. A simplification adopted is to approximateΨ by a smooth analytical function using linear combinations of basis functions The capability to compute the value of Ψ under the desired combinations of (Y,K) by the optimization program gives the opportunity to make use of regression scheme Two regression schemes will be used supposing thatis: concave and additively separable, and 2. concave but not additively separable The difference of performance under the two regression schemes canbe intuitively used as a measure of validity of the optimality of myopia. Implementation insights

18. The proposed methodology offers the opportunity to find for which prices of the stochastic variables (Y,K) is it optimal to invest in a certain technology. • The so called investment triggers indicate that once Y reaches a certain level then it is optimal to invest. • Assuming one performs the regression schemes properly, then the analytical forms of the investment triggers are derived from the following equations: • Regression Scheme 1 • Regression Scheme 2 Expected Results

19. Additionally, this methodology offers also the possibility of sensitivity analysis regarding • The drift(average growth) of • The volatility of • The capacity vector • The elasticity of demand • The behavior of the investor Further Analysis

20. Expected Results Investment trigger with average growth of Y Investment trigger with volatility of Y

21. Expected Results Investment trigger for nuclear technology with capital stock vectors Investment trigger for a coal power-generation technology with capital stock vectors

22. Develop this setting also for renewable technologies Investigate robustness of the optimality of myopia under violation of additive separability condition Realize synergies with other models or methodologies that deal with the same problem Extensions and considerations

23. THANK YOU!