Good-Deal Investment Valuation in stochastic Generation Capacity Expansion problems

Transcription

1 Good-Deal Investment Valuation in stochastic Generation Capacity Expansion problems Abstract Generation capacity expansion models have a long tradition in the power industry. Designed as optimization problems for the regulated monopoly industry, they can be interpreted as equilibrium models in a competitive environment. While often written as deteristic problems, they can be adapted to accommodate the wide range of uncertainties that currently assail the industry. We consider a stochastic optimization version of the capacity expansion model where risk is assessed through risk functions. In order to combine both the criteria of coherence and time consistency while at the same time sacrificing nothing in terms of computational tractability and economic interpretation, we formulate the model using the so-called good deal risk function. We show that the resulting model takes the form of a conic optimization program that can be interpreted as a multistage hedging optimization problem in an incomplete market. I. INTRODUCTION Solutions of optimization models of generation capacity expansion can be seen as equilibriums in perfectly competitive economies. The property is easily interpreted for a deteristic model that represents a risk free economy: investment and operations levels found by the optimization model are equilibrium quantities and dual variables are equilibrium prices. The property remains valid in a perfectly competitive economy in a risky world but its interpretation can be much more complex. The paradigm of perfect competition with uncertainties is an economy that trades all risks at some price. Risk trading takes place through so called Arrow Debreu securities that pay one currency unit in a given state of the world and zero otherwise. The price of risk is found as the price of these Arrow Debreu securities at equilibrium. Stochastic capacity expansion models bypass the introduction of Arrow Debreu securities by supposing that all agents of the economy are risk-neutral. The interpretation of the stochastic optimization models as a perfectly competitive economy is then similar to the one of the deteristic model. Investment and operations levels at the optimum of the stochastic capacity expansion model are equilibrium quantities in different states of the world; the dual variables of the model are prices in these states of the world and the probabilities of the model are risk neutral probabilities used by the agents to compute expectations of profits or surplus. This implies that exogenously given probabilities of the stochastic capacity expansion model are exogenously given risk neutral probabilities or alternatively, prices of those Arrow Debreu securities that are not represented explicitly in the model. A stochastic capacity expansion model can thus be interpreted as a partial equilibrium model where the price of risk is given. While partial equilibrium models are common in practice, this one embeds an internal contradiction: it deteres endogenous electricity prices but considers that the probability of occurrence of these prices is exogenously given. The easy interpretation of the model and the implicit contradiction that it contains disappear when one dispenses with risk neutral agents, as one may wish to do in the very risky environment where utilities are operating today. The risk neutral probabilities are no longer given to the models as exogenous probabilities of the stochastic capacity expansion model; in contrast their derivation should be made part of the problem. Ehrenmann and Smeers [1] address that problem that they cast in the context of stochastic discount rates (see e.g. Cochrane in [] for the concept) where risk neutral probabilities are made endogenous to the problem. They consider two different cases. One assumes a stochastic discount factor that is linear in some risk factors. This is a reinterpretation of the standard CAPM or APT methodologies of corporate finance that only consider systematic risk and suppose that the price of idiosyncratic risk is zero. The authors show that the associated model of the competitive economy is still described by a capacity expansion model that can be solved as a stochastic program. While it is common in corporate finance to assume a zero price for non systematic risk, the question arises as to what one can do when supposing that agents of the economy also take idiosyncratic risk into account, that is that they are not risk neutral with respect to idiosyncratic risk. Ehrenmann and Smeers [1] also consider this case and expand it in Ehrenmann and Smeers [3] where the authors represent the behavior of agents with respect to idiosyncratic risk through CVaRs. This extension however departs from the optimization paradigm (cost imization) and requires a full equilibrium model. Ralph and Smeers [4] consider a CVaR based capacity expansion model of the optimization type and elaborate on its interpretation in terms of equilibrium in physical and financial assets. Even though the CVaR is gaining in popularity as a risk measure that overcomes some defects of the well-established VaR, it still suffers from some difficulties. One is that it is not time consistent in the sense of Artzner et al. (1998), the other is that it may induce inappropriate hedging behaviors as pointed out in Eisenberg [5]. The purpose of this paper is

2 to propose an optimization version of the capacity expansion model where the discounting of the pay-off results from a risk function that is both time consistent and exhibits more satisfactory behavior. This work should serve as a preliary to extension to equilibrium models. This risk function is known in the literature as the Good-Deal and was first presented in Cochrane and Saa-Requejo [6].We treat the problem as follow. Section presents the simple stochastic capacity expansion model with risk neutral agents and expands on that model to consider different versions of risk-aversion. Section 3 discusses various interpretations related to the primal and dual forms of the Good-Deal risk measure. Among them it presents a reinterpretation of the capacity expansion in terms of replicating portfolios in an incomplete market. Section 4 elaborates on the relations between the Good deal and the more usual CAPM and APT theories while section 5 extends the discussion to the multi-period case. The section 6 illustrates the model with a simplified example and the conclusion section summarizes the paper. II. STOCHASTIC CAPACITY EXPANSION PROBLEM The restructuring of the electricity system provides the contextual background of this paper. Changes occured in this sector from at least two sides: the sector moved from a monopoly to a competitive regime; it also went from an almost risk-free environment to one assailed by uncertainties from all sides. We only treat the second problem and discuss optimization capacity expansion models in a risky environment. Capacity expansion in electricity is typically a multi-stage problem. For the sake of simplification, we begin with a two stage program and work on a discrete probability space (Ω,P). Let ω and p ω denote a scenario and its probability. We adopt the following standard formulation of stochastic programs (e.g. [7]) and consider a two stage framework where some investment decisions x need to be made in the first stage and other operations decisions y take place in a second stage after uncertain parameters ω have been revealed. Both x and y may be subject to stage specific constraints but the second stage operations variables y are also constrained by investment decisions x, implying a relation between x and y. Mathematically, all operations constraints on y are expressed by the linear relations T ω x + W ω y = h ω. These relations always embed constraints representing the satisfaction of the demand, the production set of the generators and relations linking the operations variables y to the level of investment x. The constraint set can also embed more general constraints as the electric grid limited capacity due to the thermal limits of electric lines. All parameters of the models (e.g. demand, fuel cost, availability of older power plant) can be taken as uncertain. This paper conducts the discussion in terms of capacity generation and does not refer to transmission. Our intent is to lay the ground for representing investments in generation in models of the restructured industry as the movement from optimization to equilibrium in capacity expansion becomes much more difficult when transmission is included 1 [10]. Still the interpretation of a planning model as a competitive market (with price taking agents) holds for a more general generation and transmission set up provided the whole model is formulated as a continuous convex optimization problem. This implies that the representation of transmission neglects both the second Kirchhoff law and project indivisibilities. This simplification was common practice in the times of integrated companies (e.g. the ORTIE model in [11]). Current studies (appendix C in [1]) dealing with drastic restructuring of the infrastructure because of wind penetration still make that same assumption as this is the only way to practically represent transmission expansion together with generation. A. An expectation model We begin with the traditional formulation of the second stage (the operations cost) in term of expectation. We invoke the Present Value (PV) and the CAPM which are the most widely used techniques for investment purposes. The CAPM proceeds by taking the expectation of second stage cash flows in the probability P and discounting it using an exogenous risk adjusted cost of capital that combines the cost of debt and equity. It is appropriate to simplify the discussion by neglecting debt and focussing on the case of a project entirely financed on equity and to assume a CAPM environment to compute its cost (e.g. chapter on the CAPM in [13]). In a two stage framework the problem is stated as: x X It x + E Q(x, ω) s.t. X := {x Ax = b, x 0} Q(x, ω) = c t y ωy T ω x + W ω y = h ω y 0 In problem (1), the I is the vector of annual investment costs of the different technologies computed with the CAPM risk adjusted cost of capital. The vector c ω represents the operations cost in scenario ω. 1 In a restructured world, transmission is a natural monopoly subject to regulation for most of its investments. To the best of our knowledge, there is today no capacity expansion model of a restructured electricity system where transmission is considered as a price taking agent (see [8] for an attempt to do so at the cost of introducing non linear prices and [9] for an alternative model). In an industrial institutional context based on the separation between generation and transmission owning companies, the development of the generation market should preferably be studied with a focus on generation under exogenous assumptions of transmissions or through interactions between generation and transmission models (e.g. [9]). Needless to say this introduces complex questions, whether in practice (defining the scenarios of transmission) or in theory (constructing complex combinations of generation and transmission models). At this stage it is fair to say that these questions are far from solved. (1)

3 B. Stochastic discount factor Problem (1) is formulated by assug a CAPM based discounting factor. The question of the appropriate discounting of E Q(x, ω) or equivalently of the computation of the annual investment cost is certainly or from a stochastic programg point of view. It is quite relevant in capacity expansion whether formulated as an optimization or equilibrium model. As a first departure from the standard analysis, we modify the objective function in (1) into It x + E ζ(ω) Q(x, ω) () x X We value every asset in terms of zero-coupon bonds 3. I is thus computed from the overnight investment cost using the risk free rate R f. The variable ζ(ω) is a stochastic vector that satisfies E [ζ(ω)] = 1. All other elements of the problem (1) remain unchanged. The stochastic vector ζ(ω) is referred to either as state price or as a stochastic discount rate. Models of type () allow one to deal with CAPM or APT formulations by choosing a stochastic discount factor compatible with these theories (Ehrenmann and Smeers [1]). As a further step in the analysis, we suppose that ζ(ω) takes its values in a convex set of probabilities U and replace the expression () by x X It x + max E ζ(ω) Q(x, ω) (3) ζ U Denoting Q(x, ω) as Z(ω), the expression introduces a new function ρ(z) of the type ρ(z) = max E [ζ(ω)z(ω)] (4) ζ U In the context of the quantification of the risk of financial positions, ρ(z) is referred to as a static risk measure 4. Risk measures were further specialized into coherent 5 risk measures, in the seal papers of Artzner, Delbaen, Eber & Heath [15], [16] and systematically embedded into optimization by Shapiro et al. in [7]. Stochastic optimization models with risk measures of the form x X It x + ρ Q(x, ω) (5) have been extensively treated in [7]. An extension of this notion to an equilibrium model of the investment capacity type can be found in Ehrenmann and Smeers [1]. 3 We choose zero coupon bond with maturity T, the final time of the capacity expansion problem. The price of a bond at time t is given by the random variable P t,t, where P T,T =1, that is, the bond is default free. As noted in [14], it is only after this choice of numéraire, that risk measure can be qualified as monetary. 4 A risk measure is a function ρ which maps the space of risky payoffs (i.e. the set of all possible real-valued functions on Ω) into the extended real line R = R {+ } { } 5 By definition, coherence means that the risk measure satisfies four axioms respectively defined as convexity, monotonicity, translation equivariance and positive homogeneity. The appendix A recalls the axioms. This paper deals with problem (5) with a particular focus on the following risk measure. ρ(z) = max {ζ} s.t. ζ(ω) 0 E ζf i 1 = f i 0 E ζ A This expression, known as the Good-Deal, has been introduced by Cochrane and Saá-Requejo in [6]. It is characterized by the set U GD of feasible ζ, i.e. U GD := {ζ ζ ω 0; E ζf i 1 = f i 0 ; E [ζ] = 1; E ζ A }. Note that the constraint E [ζ] = 1 is contained in the set by taking an f i 1 = f i 0 =1. This latter expression has the interpretation that the ζ should price the zero coupon bound. We motivate this set later in section III but first conclude this section with some mathematical properties. Problem (6) is a conic 6 program (see e.g. [17]). It admits the following strong dual formulation. ρ(z) = f t w i,η ω 0 0w + AE η 1 f t 1,ωw + η ω Z ω 0 We can reformulate the problem (5) as a full imization by using the dual formulation of the Good-Deal measure. The capacity expansion problem can be restated as the following conic two stages problem: x X,w It x + w t f 0 + E [Q(x, w,ω)] 1 Where Q(x, w,ω) is the optimal value of the second stage problem Q(x, w,ω)= s.t. A ηω y ω,η ω f1,ωw t + η ω c t ωy ω W ω y = h ω T ω y ; y 0 We discuss the interpretation of these models in the following section. III. THE GOOD-DEAL RISK MEASURE The valuation of an uncertain pay-off can be approached either through the axiomatics of risk measures or from the point of view of risk hedging. Cochrane and Saá-Requejo introduced the Good-Deal by refering to this latter context, that is, to the notion of absence of arbitrage in an incomplete market. The principles can be summarized as follows. The absence of arbitrage opportunity allows one to uniquely value uncertain pay-offs in a complete market where prices can be detered on the basis of underlying traded assets and without reference to agents risk-aversion. This pricing methodology breaks down in incomplete markets. Prices 6 Recall that E ζ = ω pωζ ω and so the inequality (A, p ωζ ω) L Ω+1, defines the Lorentz cone. (6) (7) (8) (9)

4 are no longer unambiguously detered but it is still possible to bound them. Alternatively, one can overcome the undeteracy of prices by assug market prices of risk for missing securities (which then have to be estimated from data 7 ). It is well known that price bounds obtained by imposing the sole no arbitrage constraint in an incomplete market are too wide to be helpful in practice. Cochrane and Saá-Requejo s pricing scheme is a mix of arbitrage valuation and equilibrium theory. It gives tight bounds on prices of uncertain pay-offs, that the authors call Good-Deal bounds. Cochrane and Saá-Requejo impose that the pricing kernel rule out arbitrage opportunities (i.e. the price kernel should be positive) and too high Sharpe ratio. It is common in finance to regard high Sharpe ratio as Good-Deal that are not sustainable at equilibrium. For example, the CAPM theory specifies that the market portfolio is mean-variance efficient, i.e. no asset can have a higher Sharpe ratio than the market. Cerny and Hodges [18] extend the work of Cochrane and Saá-Requejo and present a theory of generalized arbitrage pricing that is based on the absence of attractive investment opportunities at equilibrium but does not necessarily invoke a high Sharpe ratio. Price bounds can then be derived in their theory by considering super-hedging and arbitrage arguments. These notions are related to the coherent risk measures proposed in Artzner, Delbaen, Eber & Heath [15], [16]. Specifically, Jaschke and Kuchler [19] show that generalized arbitrage bounds are essentially equivalent to coherent risk measures. The Good-Deal risk measure used in this paper is the original notion introduced by Cochrane and Saá-Requejo in [6]. It is based on Sharpe ratio, it is coherent and already formulated in the representation theorem 8 of coherent risk measure. We show in section 5 that it also turns out to be time-consistent while remaining computationally attractive. The following further elaborates on the interpretation of this risk measure. A. The primal Good Deal problem Consider an incomplete market. The interpretation of the Good-Deal measure stated in problem (6) is as follows. Z is the random cost that one wants to value. ζ is a stochastic price kernel (see Cochrane [] for the implications of p = E [mx]); it is contrained to be positive, which implies that there is no arbitrage opportunity. The f i are selected traded assets with f0,f i 1 i being the prices expressed in the numéraire at t = 0 and t = 1 respectively. The scalar A is equal to (1 + h ), where h is the maximal admissible Sharpe ratio. The f i can take a natural interpretation of long-term power or 7 This estimation can only be performed when the pay-offs are traded. 8 The representation theorem states that every coherent risk-adjusted value ρ can be represented as an expectation taken with respect to the probability measure ζdp selected from some convex set according to the following formula (see [16]). fuel derivatives contracts in a generation capacity expansion problem. They can also refer to the market portfolio, which then leads to a CAPM-like interpretation (see section IV). B. The dual Good Deal problem Coherent risk measures have both primal and dual representations. We here comment the dual of Cochrane and Saá- Requejo s model. Problem (7) can be interpreted in terms of portfolio replication. The portfolio is constituted in t = 0 and composed of the assets f i with weight w. Because the market is incomplete, this portfolio cannot perfectly reproduce the random cost Z. The variables η ω in problem (7) measure the under performance in t =1of the replicating portfolio for hedging the cost Z. Indeed, the optimality conditions of (7) give : η ω = max(0,z ω f t 1,ωw) (10) The expectation E η of this under-performance is called a regret in the literature of replicating portfolio (see e.g. Dembo and Rosen [0]). When the A, the appearance of the regret in the objective function of problem (7) imposes that the replicating portfolio over-hedges the cost Z and the Good-Deal value is equal to the bound computed under the sole absence of arbitrage 9. To sum up, the Good-Deal value in t =0of an uncertain loss Z occurring in t = 1 can be interpreted as the price in t = 0 of a portfolio replicating the loss in t = 1 plus a deviation term, representing the part of the risk (i.e. the idiosyncratic risk) in Z for which there is no observable market price. IV. GOOD-DEAL AND FACTORS PRICING THEORIES The assets f i used in the Good-Deal play a role analogous to the risk factors in the standard CAPM or APT models. In order to see this, consider the following complementarity condition obtained from the KKT conditions of problem (6): 0 ζ ω ζ ω i α i f i 1,ω α z Z ω 0 (11) A strictly positive pricing kernel ζ would be linear in the risk factors, as in the two cited theories. Alternatively one would immediately obtain a linear pricing kernel by dropping the non arbitrage conditions, as in the CAPM or APT. One can actually recover the CAPM formula from the Good-deal measure by introducing some assumptions of that theory. Suppose, as in the the CAPM that the market portfolio is mean-variance efficient (i.e. no asset can have a higher Sharpe ratio) and consider only the risk-free asset (R f ) and the market portfolio (R M ) as risk factors; dropping the 9 Note that this interpretation can be directly derived from the primal problem (6).

5 positivity constraint on the pricing kernel (that is, allowing for arbitrage opportunities), the Good-deal problem (6) becomes ρ(z) = max {ζ} E [ζz] E [ζr f ]=1 ; E [ζr M ]=1 σ (ζ) (E [R M ] R f ) σ M R f (1) The optimality conditions of problem (1) give a pricing kernel 10 ζ = a b(r M E [R M ]) identical to the one of the CAPM. V. THE MULTIPERIOD PROBLEM These notions extend to the multiperiod capacity expansion problem. Let (Ω, F, P) be a probability space equipped with a filtration (F t ) t 1,...,T (i.e a sequence of increasingly refined algebras of subsets of Ω). Because investment and operations variables occur in each period, they can no longer be associated to specific stages as in the two stage model (respectively investment in stage 0 and operations in stage 1). To simplify notation, we collect the investment vector x t, the total installed capacity k t and the operations variable y t in a vector u t =(x t,k t,y t ) for each t. The values taken by u t is a function of the available information (i.e. fuels and investment cost, load level,...) up to time t. In other words u t is F t measurable. Because operations variables y t are now constrained by the total capacity k t, we modify the stage specific constraints so that they relate y t and k t. This total installed capacity is obtained by the following recursive formula k t = k t 1 +x t 1. The time t contribution to the objective function and constraint of the problem are as follows: g t (u t,ω) c t t,ωy t (ω)+i t t,ωx t (ω) k t (ω) =k t 1 (ω)+x t 1 (ω) x t X t,ω u t χ t (u t 1 (ω),ω) T t,ω k t 1 (ω)+w t,ω y t (ω) =h t,ω y t (ω) 0 (13) The expression is slightly different in stage 1, which only involves the investment variables, i.e. g 1 (u 1 )=I t x 1 as no investment has been committed before (k 1 =0). The situation is again different in the final stage, which only involves operations costs, therefore leading to g T (u T,ω)=c t ωx T (ω). Cochrane and Saá-Requejo [6] extended the Good-Deal measure to multiperiod and continuous time environments. We limit ourselves to the discrete form and consider their recursive definition that we state through a sequence (Z 1,...,Z T ) of F t 10 Where a =1/R f and b =(E [R M ] R f )/(σ M R f ) measurable losses Z t using the following risk function: ρ(z 1,...,Z T )= ζ(ω) Z 1 + E ζ (ω) Z + ζ 3 (ω) ζ T 1 (ω)(z T 1 + ζ T (ω)z T ) F 0 s.t. ζ (ω) U GD (ω),...,ζ T (ω) U GD T (ω) where the sets U GD are defined by U GD t+1 (ω) = ζ t+1 (ω) f i t,ω = E ζ t+1 (ω)f i t+1,ω F t ; E ζ t+1 (ω) F t A t,ω ; (14) ζ t+1 (ω) 0 (15) One shall note that the measures appearing in that definition are conditional: the pricing kernel ζ t is a function of the conditioning information available at time t. This construction raises the question of both the coherence and time consistency of the obtained risk function 11. We first discuss time consistency and come back later to coherence. Time consistency in a multiperiod risk function requires that judgements based on this risk measure be non contradictory over time (i.e. a project preferred to another one in all states of the world in period t remains preferred in all states of the world in period t 1). Time consistency has been formulated in different ways (see among [7], [14], [3]); we here focus on the Shapiro s definition, which has been tailored to risk optimization but also applies to pure risk evaluation: a decision policy (or evaluation) should not involve states that cannot happen in the future [4]. A multiperiod risk measure defined in a recursive way as a composition of conditional risk mappings (that satisfies the conditions of convexity, monotonocity, translation equivariance and positive homogeneity [5]) always satisfies this definition. Apply this definition to the Good Deal and consider with: ρ(z 1,...,Z T )=Z 1 + ρ F1 Z ρ T 1 FT ZT 1 + ρ T FT 1 [Z T ] (16) ρ t+1 Ft (Z t+1 (ω)) = E [ζ t+1 (ω)z t+1 (ω) F t ] ζ t+1(ω) U GD t+1 (ω) (17) Because the corresponding static good deal measure is coherent it is straightforward to show that this risk measure is coherent. Applying these formulations, the multiperiod capacity expansion problem can be stated as g 1 (u 1 )+ρ F1 inf g (u,ω)+... u 1 χ 1 u χ (u 1,ω) +ρ T FT 1 [ inf g T (u T,ω)] (18) u T χ T (u T 1 (ω),ω) This recursive derivation applies to all coherent risk functions, therefore leading to coherent, time consistent multiperiod 11 The question of the information monotonicity (as defined in [1], []) of the Good-Deal measure is still open.

6 risk functions. But the operation generally suffers from two important defects. First the recursive formulation is not directly amenable to a general-purpose solver. A restatement of the recursive formulation into a single static risk function optimization problem would be useful for computation. Second the multistage and single stage formulation are not interpretable in the same terms: the single stage risk function is lost in the recursive formulation (i.e. the CVaR of a CVaR is not a CVaR). The Good Deal offers a remedy to both defects. First the multistage model can be reformulated a single static optimization model that can be submitted to a general-purpose code. Second the multistage and single stage problem can both be interpreted in terms of optimal hedging in incomplete markets. In order to see this consider first a restatement of problem (18) obtained by recursively calling upon the dual formulation of the Good-Deal. This leads to the problem: u,w,η s.t. g 1 (u 1 )+f t 1w 1 + AE η F 1 1 u 1 χ 1, u t χ t (u t 1 (ω),ω),t=,...,t f t t,ω(w t 1 (ω) w t (ω)) + η t (ω)... g t (u t (ω),ω)+ae η t+1 (ω) 1 F t (19) The obtained problem is a single conic program that can be submitted to a general purpose code of conic programg. Recall that conic problems are the same degree of difficulty as linear programs of the same size. One can also see that the obtained model takes the form of a dynamic hedging problem in an incomplete market. VI. STYLIZED EXAMPLE Our model was developed as an approach to overcome shortcogs arising from the use of standard stochastic programg models in real industrial conditions. We argued in the introduction that multistage stochastic planning models have a natural interpretation of competitive markets. These models, while affected by usual numerical difficulties due to their size, also raise their own problems when used in this economic context. They rely on risk-adjusted discount rates that are difficult to estimate these days because of rapidly changing economic conditions. Standard stochastic programg models also reveals plants with quite different risk exposures that investors maybe reluctant to value on the sole basis of expectation. The Good Deal function remedies those two difficulties. It bypasses the question of the risk adjusted discount rate by using a pricing kernel that prices the free asset (E [ζ] = 1 in (6)). It also allows investors to directly insert a market interpretable parameter of risk aversion (the Sharpe ratio) in the evaluation of their plants. The full implementation of this type of model in an industrial context is a subject for further work 1. We rather illustrate the approach on a simple generation capacity expansion problem with two risks factors, namely 1 We believe that it overcomes two major difficulties encountered with existing stochastic programg models. demand growth and gas price. The problem is inspired by [3]. Consider a 3-stage scenario tree representing the evolution of the corresponding data process. Stage (time) t =0consists of a single root node denoted ω 0. Stage t =1comprises 5 nodes corresponding to different realizations of the demand and gas price. Each of them is connected to the root node. We have 65 different scenarios at stage t =. A generic node is denoted ω 1 at stage t =1and ω at t =. Each node ω 1 has 5 successors nodes. We denote Ω t the set of all possible nodes at stage t. Figure 1. The tree representation Because we are dealing with an investment problem, these periods are multiyear and can be interpreted as five years long. Multiyear periods imply some discount technicalities in the calculation of investment and operating costs. These are standard and not discussed here. The industry initially invests at stage t =0. Depending on the realization of data at t = 1, the industry operates those capacities and possibly reinvests in new ones. Finally, at t =, the industry operates the total capacity installed according to the realization of demand and gas prices. The set K of capacity types consists of Coal, Combined Cycle Gas Turbine (CCGT) and Open Cycle Gas Turbine (OCGT). Each equipment has both an investment cost and operating cost. The investment costs are annualized costs in thousand euro per MW that affect existing capacities in each year of a period. They are computed from overnight construction and fixed operating cost using a standard annualisation procedure (see Table V). The operating costs are derived from fuel prices and the heat rate (thermal efficiency) of the technology: c t (k, ω t )=HR(k) p fuel t (ω t ). For the sake of simplicity we assume that the price of the coal remains constant at 1 /MW th and do not consider CO regulation or emission market. Demand is price insensitive and described by a load duration curve decomposed in different time segments L as depicted in Figure 1. The shape of the load duration curve is given in a reference scenario noted d 0. Table II reports the

7 Coal CCGT OCGT I(k): Investment cost [k /MW] HR(k): Heat rate [MW th /Mw] Table I FIXED ANNUAL COST AND HEAT RATE decomposition of this reference load duration curve in time segments of duration τ() and level d 0 () d 0 : demand [GW] τ: duration [k hours] Table II REFERENCE LOAD DURATION CURVE The evolution of demand is random. We model this risk factor in stage t = 1 through 5 different scenarios detered by the growth α of the load duration curve 13 : d 1 (, ω 1 )=α(ω 1 ) d 1 (). The load duration curve at stage t =is similarly obtained using the same stochastic growth rate α and the realization of the load duration curve at t =1 : d (, ω F 1 ) = α(ω ) d 1 (, ω 1 ). Table IV represents the different scenarios and their probabilities for the demand growth rate α. α: demand growth - 5% 0% 5% 10% 15% Probability 10% 15% 30 % 5 % 0 % Table III DEMAND GROWTH Demand growth in Table IV is stated for five years periods. The corresponding annual growth rate therefore ranges from less than 1% to less than +3%. The price of the gas fuel is the second source of uncertainty. It impacts the operations cost of the CCGT and OCGT technologies. Similarly to the load, we model the gas evolution by assug a random growth rate β. The reference gas price is set at 14 /MW th. The two growth rates are stochastically independent, generating 5 scenarios at the stage t =1and 65 scenarios at t =. Again one should interpret these growth rates as referring to five year periods. β: Gas price growth 0% 1% 4% 36% 48% Probability 10% 40% 0 % 0 % 10 % Table IV GAS PRICE GROWTH 13 We assume that all demand blocks varies with the same growth rate α We introduce the following notation: x t (k, ω t ) is the capacity (in MW) of technology k K in stage t; this capacity is operated at level y t (k,, ω t ) (in MW) in time segment. The variable VOLL is the Value of Lost Load, that is, the economic value ( set to 500 /MWh) of unsatisfied electricity demand. The VOLL notion has been around since several decades but has so far escaped any precise evaluation. 500 /MWh is definitively a low value inspired by regulation concerns. It does not provide a sufficient incentive to invest. We accordingly introduce z t (, ω t ), the unsatisfied demand in demand segment (in MW) in scenario ω t and demand segment. Finally we assume a risk free rate R f equal to 1.0 and set the scalar A to 1.5 in the good-deal risk measure. The model (19) adapted to this simplified example leads to the following conic problem. u,w,η k K I(k)x 0 (k, ω 0 )+1w 0 (ω 0 )+AE η 1 (ω 1 ) F 0 1 s.t. x 0 (k, ω 0 ) 0 ; x 1 (ω 1 ) x 0 (k, ω 0 ) y t (k,, ω t ) 0 ; z t (, ω t ) 0 t =1, x t 1 (k, ω t 1 ) y t (k,, ω t ) t =1, y t (k,, ω t )+z t (, ω t ) d t (, ω t ) 0 t =1, k K (R f )(w 0 (ω 0 ) w 1 (ω 1 )) + η 1 (ω 1 )... τ() c 1 (k, ω 1 )y 1 (k,, ω 1 )+VOLL z 1 (, ω 1 ) L + k K k K I(k)x 1 (k,, ω 1 )+AE η (ω ) F 1 1 (R f ) w 1 (ω 1 )+η (ω )... τ() c (k, ω )y (k,, ω )+VOLL z (, ω ) L k K (0) Problem (0) has variables. The expressions E ηt+1 F t define a total of 6 Lorentz cones, each of dimension 5. We solve the problem using yalmip [6] and the MOSEK solver. The solver running time is approximatively 3 seconds.

8 Simulation results The following table reports the investments in the different types of plants in stage 0 (operated in stage 1 and ) and the expected investment in stage 1 (operated in stage ). Recall that there is no initial exogenous capacity in this model and hence that investment in t =0have to make up for the bulk of d 1. The industry invests principally in coal and OCGT technologies, mainly due to the peaky characteristic of the load duration curve (interpreted as resulting from a significant wind penetration). Investment [GW] Coal CCGT OCGT x 0 (k, ω 0 ) E [x 1 (k, ω 1 ) F 0 ] Table V INVESTMENT As expected the investment in stage t =1strongly depends on the realization of the demand growth and the gas price. The total level of investment is more impacted by demand growth and the optimal technology mix depends more on the gas price. This is highlighted in figure and 3 showing the investment in t =1in Coal and CCGT. These are standard stochastic programg phenomena. Figure. Investment in Coal at t =1 Less standard, table VI shows the optimal mix of technology at t =0varying with the risk aversion parameter A.The more the industry is risk averse, the more it invests in capacities and also neglects the CCGT technology. The higher investment is meant to avoid curtailments and hence VOLL costs. The second effect is intended to avoid a lower utilization of relatively high capital investment plants (CCGT) when demand is low. VII. CONCLUSION This paper cast the standard multiperiod capacity expansion planning in a risk measure context. The Good Deal Figure 3. Investment in CCGT at t =1 A=1.05 A=1. A=.3 Coal CCGT OCGT Table VI INITIAL INVESTMENT (t =0) FOR DIFFERENT SHARPE RATIO introduced by Cochrane and Saá Requejo in [6] is the chosen risk measure. We explain that this formulation has several advantages. From the point of view of industrial practice, the Good-Deal can be seen as an extension of the stochastic discount factor constructed from standard corporate finance theories such as the CAPM and the APT. In term of optimization of risk measure, the multiperiod Good-Deal benefits from the two very desirable properties of coherence and time consistency. The risk measure has also economic interpretation and computational advantages. The multi-period capacity expansion model under Good-Deal is a conic program and hence amenable to a treatment of very large size. From an economic point of view, one can interpret the dual of the capacity expansion model in terms of portfolio replication in an incomplete market. Future work will extend this optimization problem to a multi-period equilibrium context. REFERENCES [1] A. Ehrenmann and Y. Smeers, Stochastic equilibrium models for generation capacity expansion, in Handbook on Stochastic Optimization Methods in Finance and Energy, Springer, Ed. M. Bertochi, G. Consigli and M. Dempster, 010, to be published. [] J. H. Cochrane, Asset Pricing. Princeton University Press, 001. [3] A. Ehrenmann and Y. Smeers, Generation capacity expansion in a risky environment: A stochastic equilibrium analysis, Operations Research, 010, accepted for publication. [4] D. Ralph and Y. Smeers, Risk averse stochastic equilibria : a perfectly competitive two stage market in capacity with financial products, Judge Business School, University of Cambridge, Working Paper, 010. [5] L. K. Eisenberg, The marginal price of risk with a cvar constraint, New Jersey Institute of Technology, Working Paper Series, 007. [6] J. H. Cochrane and J. Saá-Requejo, Beyond arbitrage: Good deal asset price bounds in incomplete markets, Journal of Political Economy, vol. 108, pp , February 00.

9 [7] A. Shapiro, D. Dentcheva, and A. Ruszynski, Lectures on Stochastic Programg : Modeling and Theory. SIAM, 009. [8] Y. Smeers, Long term locational prices and investment incentives in the transmission of electricity, in Competitive electricity markets and sustainability, F. Lévêque, Ed. Edward Elgar Publishers, 006. [9] E. Sauma and S. Oren, Proactive transmission investment in competitive power systems, in Proceeding of the IEEE PES Annual Meeting, Montreal, Canada, 006. [10], Conflicting investment incentives in electric transmission, in Proceeding of the IEEE PES Annual Meeting, San Francisco, CA, 005. [11] J. C. Dodu and A. Merlin, Dynamic model for long-term expansion planning studies of power transmission systems: the ortie model, International Journal of Electrical Power & Energy Systems, [1] Roadmap 050 : A practical guide to a prosperous, low-carbon europe, European Climate Foundation, Tech. Rep., 010. [Online]. Available: [13] D. G. Luenberger, Investment Science. Oxford University Press, [14] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, and H. Ku, Coherent multiperiod risk adjusted values and bellmans principle, Annals of Operations Research, vol. 15, pp. 5, 007. [15] P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, Thinking coherently, Risk, no. 10, pp , [16], Coherent measures of risk, Mathematical Finance, vol. 9, no. 3, [17] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for Industrial and Applied Mathematic, 001. [18] A. Cerny and S. D. Hodges, The theory of good-deal pricing in financial markets, FORC Preprint, No. 98/90, 004. [19] S. Jaschke and U. Küchler, Coherent risk measures and good-deal bounds, Finance and Stochastics, vol. 5, no., pp , 001. [0] R. Dembo and D. Rosen, The practice of portfolio replication: A practical overview of forward and inverse problem, Annals of Operations Research, no. 85, pp , [1] G. C. Pflug, A value-of-information approach to measuring risk in multi-period economic activity, Journal of Banking & Finance, vol. 30, no., pp , February 006. [] G. C. Pflug and R. Kovacevic, Modeling, Measuring and Managing Risk. World Scientific Publishing Company, 007. [3] F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications, vol. 11, no., pp , 004. [4] A. Shapiro, On a time consistency concept in risk averse multistage stochastic programg, Operations Research Letters, vol. 37, pp , 009. [5] A. Ruszynski and A. Shapiro, Conditional risk mappings, Mathematics of Operations Research, vol. 31, no. 3, pp , August 006. [6] J. Lofberg, Yalmip : A toolbox for modeling and optimization in matlab, in Proceedings of the CACSD Conference, 004. A. Coherent risk measure APPENDIX Let (Ω, F,P) be a probability space (equipped with a sigma algebra F and a probability measure P ), Z := L p (Ω, F,P) be the set of all F -measurable functions Z such that Ω Z(ω) p dp (ω) <. A risk measure is a function ρ(z) that maps Z into R. A risk measure is coherent if it satisfies the following axioms. - Convexity : Z 1,Z Z, t [0, 1] : ρ(tz 1 +(1 t)z ) tρ(z 1 )+(1 t)ρ(z ) - Monotonicity: If Z 1,Z Z and Z 1 Z, then ρ(z 1 ) ρ(z ) - Translation equivariance: If a R and Z Z, then ρ(z + a) =ρ(z)+a - Positive homogeneity: If t>0 and Z Z, then ρ(tz) = tρ(z)