A REAL OPTIONS MODEL FOR ELECTRICITY CAPACITY EXPANSION

Christos Nakos,NTUA, Postgraduate Student

Optimal Management of the Dynamic Systems of the Economy and the environmentTHALES RESEARCH WORKSHOP

BASIC LITERATURE

This presentation mainly follows the developments produced in :

Gahungu, J and Y. Smeers (2011), “ A Real Options Model for Electricity

Capacity Expansion“, CORE discussion paper, Universite

Catholique de Louvain, Belgium

INTRODUCTION 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.

PARTICULARITIES OF THE POWER SECTOR 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

DESCRIPTION OF THE MODEL 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 ofload is activated for each load segment respectively.

DESCRIPTION OF THE MODEL 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.

0

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1 558 1115 1672 2229 2786 3343 3900 4457 5014 5571 6128 6685 7242 7799 8356

MW

Step wise approximation of the annual load curve

DESCRIPTION OF THE MODEL 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 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 MODELThe social planner’s instantaneous welfare problemThe numerical approach for the computation of is based on the following program :

s.t.

DESCRIPTION OF THE MODELThe 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:

STOCHASTIC CONTROL PROBLEM AS A SUCCESSION OF OPTIMAL STOPING PROBLEMS

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 (SCPOSP) has been proved, only when certain conditions are met.

Baldursson and Karatzas(1997)

CONDITIONS OF EQUIVALENCE 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

THE OPTIMAL STOPPING PROBLEM –SINGLE TECHNOLOGY MODEL

For a stopping time τ define the function: C(Y,,τ) (1)

Consider now the optimal stopping problem:

arg infC(Y,,τ), (2)And the so called Risk Function:

r C (3)Which by intuition can be considered:

r

MOVING TO A FREE BOUNDARY PROBLEM –SINGLE TECHNOLOGY

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

ISSUES FOR MULTI-TECHNOLOGY MODELS 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.

IMPLEMENTATION INSIGHTS It has been stated that we cannot have Ψ in a closed analytical

form. A simplification adopted is to approximate Ψ 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 can be intuitively used as a measure of validity of the optimality of myopia.

EXPECTED RESULTS 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

FURTHER ANALYSIS 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

EXPECTED RESULTS

Investment trigger with average growth of Y

Investment trigger with volatility of Y

EXPECTED RESULTS

Investment trigger for nuclear technology with capital stock vectors

Investment trigger for a coal power-generation technology with capital stock vectors

EXTENSIONS AND CONSIDERATIONS 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

THANK YOU!