Stochastic optimization models for power generation capacity expansion with risk management Maria Teresa Vespucci, Marida Bertocchi, Stefano Zigrino
Laureano F. Escudero
Department of Management, Economics and Quantitative Methods University of Bergamo Via dei Caniana 2, Bergamo 24127, Italy email:
[email protected]
Department of Statistics and Operations Research Universidad Rey Juan Carlos c/Tulipan, 28933 Mostoles, Spain email:
[email protected]
Abstract—We propose a two-stage stochastic optimization model for maximizing the profit of a price-taker power producer who has to decide his own power generation capacity expansion plan in a long time horizon, taking into account the uncertainty of the following parameters: fuel costs; market electricity prices, as well as prices of green certificates and CO2 emission allowances; market share. The parameter uncertainty is represented by scenarios on their values along the planning horizon and the associated probability of occurrence. We first discuss the risk neutral stochastic model, that maximizes over all scenarios the net present value of the expected profit along the planning horizon. The risk neutral model does not take into account the variability of the objective function value over the scenarios and, then, the possibility of realizing in some scenarios a very low profit. Several approches have been introduced in the literature for measuring the profit risk. In this work we consider the Conditional Value at Risk, that requires a confidence level to be defined, and the Firstorder Stochastic Dominance constraints, for which a benchmark need to be assigned. By using a realistic case study, we report the main results of considering risk averse strategies under different hypotheses of the available budget, analysing the impact on the expected profit.
I. I NTRODUCTION The incremental selection of power generation capacity is of great importance for energy planners. In this paper we deal with the case of the single power producer who wants to determine the optimal mix of different technologies for electricity generation (ranging from coal, nuclear and combined cycle gas turbine to hydroelectric, wind and photovoltaic), taking into account the existing plants, the cost of investment in new plants, the maintenance cost, the purchase and sales of CO2 emission allowances and Green Certificates to satisfy regulatory requirements over a long term planning horizon (typically 30 years, or more). Uncertainty of prices (fuels, electricity, CO2 emission allowances and Green Certificates) should be taken into account, see [5]. However, in practice, producers generally use the Levelized Cost of Electricity (LCoE) to compute the convenience of investments in new power generation technology, i.e. to find the technology that gives the lowest price of electricity or, equivalently, the net
present value of the investments equals to zero. In this work, we assume that the producer is a price-taker, i.e. he cannot influence the price of electricity, and propose alternative models for finding an optimal trade-off between the expected profit and the risk of getting a negative impact on the solution’s profit, due to a not-wanted scenario to occur. So this weighted mixture of expected profit maximization and risk minimization undoubtedly may be perceived as a more general model than LCoE. See some approaches in [2], [5], [6], [8], [9], [10], [13], [15], among others. Our model is based on the approach introduced in [15], which we extend by introducing the following elements: 1) consideration of fixed and variable costs in the objective function; 2) design of a risk-averse two-stage stochastic mixedinteger optimization model, by using different risk measures, as an extension of our previous work, see [22]; 3) computational comparison of the risk measures strategies, so as to determine the reduction of expected net profit due to risk minimization; We propose a decision support model for a power producer who wants to determine the optimal planning for investment in power generation capacity in a long term horizon. The power producer operates in a liberalized electricity market, where rules are issued by the Regulatory Authorities with the aim of promoting the development of power production systems with reduced CO2 emissions. Indeed, CO2 emission allowances have to be bought by the power producer as a payment for the emitted CO2 . Moreover, the Green Certificate scheme supports power production from Renewable Energy Sources (RES) (e.g. geothermal, wind, biomass and hydro power plants) and penalizes production from conventional power plants (e.g. CCGT, coal and nuclear power plants). Indeed, every year a prescribed ratio is required between the electricity produced from RES and the total electricity produced. In case the actual ratio, attained in a given year, is less than the prescribed one, the power produces must buy Green Certificates, in order to
satisfy the related constraint. On the contrary, when the yearly attained ratio is greater than the prescribed one, the power produces can sell Green Certificates in the market.
II. T HE RISK - NEUTRAL TWO - STAGE STOCHASTIC MIXED
The power producer aims at profit maximization over the planning horizon. Revenues from sale of electricity depend both on the electricity market price and on the amount of electricity sold. The latter is bounded above by the power producer’s market share; it also depends on the number of operating hours per year characterizing each power plant in the production system. Costs greatly differ among the production technologies. Investment costs depend both on the plant rated power and on the investment costs per power unit. Typically, for thermal power plants rated powers are higher and unit investment costs are lower than for RES power plants. Variable generation costs for conventional power plants highly depend on fuel pricing. Revenues and costs associated to the Green Certificate scheme depend on the Green Certificate price, as well as on the yearly ratio between production from RES and total annual production of the producer. Finally, costs for emitting CO2 depend on the price of the emission allowances, as well as on the amount of CO2 emitted, that greatly varies among the production technologies.
In order to introduce the risk neutral stochastic mixed integer linear programming model, the following sets are defined:
The evolution of prices along the planning horizon is not known at the time when the investment decisions have to be done. The model presented in this paper takes into account the risk associated with the capacity expansion problem due to the uncertainty of prices, as well as the uncertainty of market share. Uncertainty is included in the model by means of scenarios that represent different hypotheses on the future evolution of the market share and of the prices of electricity, Green Certificates, CO2 emission allowances and fuels. The proposed decision support model determines the evolution of the production system along the planning horizon, taking into account all scenarios representing the uncertainty. Since construction time and industrial life greatly vary among power plants of different technology, the model determines the number of power plants for each technology, whose construction is to be started in every year of the planning period. Each new power plant is then available for production when its construction is completed and its industrial life is not ended. The annual electricity production of each power plant in the production system, and the corresponding Green Certificates and CO2 emission allowances, are determined for each year of the planning horizon under each scenario into consideration. The paper is organized as follows. Section II presents the risk neutral two-stage mixed-integer linear stochastic programming model. Section III describes how the model in Section II is generalized in order to introduce risk aversion. Section IV presents the case study. Section V reports the computational experience on using the risk aversion strategies considered in this work to analysing the hedging effect against negative impact on the profit in case of non-wanted scenarios in the generation capacity expansion. Section VI concludes and outlines future research work.
INTEGER MODEL FOR THE GENERATION CAPACITY EXPANSION PROBLEM
J K I Ω
set of candidate technologies, indexed by j; set of power plants owned by the producer at the beginning of the planning horizon, indexed by k; set of years in the planning horizon, indexed by i; set of scenarios, indexed by ω.
The set J is partitioned into two subsets, i.e. J = J T ∪ J R , where J T is the subset of candidate technologies for thermal power production and J R is the subset of candidate technologies for power production from RES. Analogous partition is defined for the set K of power plants owned by the producer at the beginning of the planning horizon, i.e. K = K T ∪ K R , where K T is the set of thermal power plants and K R is the set of power plants using RES. Uncertainty is taken into account by means of the following parameters related to year i under scenario ω: E πi,ω GC πi,ω CO2 πi,ω M i,ω J vj,i,ω K vk,i,ω
market electricity price [e /M W h]; Green Certificate price [e /M W h]; CO2 emission price [e /t]; market share [GW h]; variable production unit cost [ke ] of a thermal power plant of candidate technology j ∈ J T ; variable production unit cost of thermal power plant k ∈ KT .
Note that fuel cost is the main component of the variable production unit costs of thermal power plants. The stochastic mixed integer linear programmimg model determines the optimal power generation capacity expansion plan, represented by the integer decision variables wj,i , which indicates the number of power plants of technology j whose construction has to be started in year i. For every candidate technology j the total number of new power plants constructed along the planning horizon is bounded above by the number Z j of sites ready for construction of new power plants of that technology, i.e. sites for which all the necessary administrative permits have been released: X (1) wj,i ≤ Z j . i∈I
An upper bound Z j,i to wj,i could alternatively be considered. The capacity expansion plan must also satisfy constraints related to the budget availability, to the power producer’s market share and to the rules imposed by the Regulatory Authority. These constraints are expressed in terms of the number Wj,i of power plants of technology j available for production in year i, which is defined as X Wj,i = wj,l , (2) i−(Sj +LJ j )+1≤l≤i−Sj
where Sj [years] and LJj [years] represent construction time and industrial life, respectively, of a power plant of candidate technology j. The new power plants of technology j available for production in year i are those for which both construction is completed and industrial life is not ended. The annual debt repayment (i.e. the annualized investment cost) for investment in new power plants of technology j is expressed by Rj Wj,i , with Rj [ke ] given by Rj =
1000 r J
1 Lj 1 − ( 1+r )
Ij PjJ ,
(3)
where Ij [e /M W ] is the investment cost of a power plant of candidate technology j, PjJ [M W ] is the rated power of a power plant of candidate technology j and r is the interest rate. The sum of the present annual debt repayments, corresponding to the number Wj,i of new power plants of technology j available for production in year i, is required not to exceed the available budget B [M e ] X X 1 Rj Wj,i ≤ B. (4) (1 + r)i j∈J
i∈I
The electricity production in year i depends on the realization of the uncertain parameters in scenario ω. The J electricity production Ej,i,ω [GW h] of all new power plants of technology j in year i under scenario ω is nonnegative and bounded above by the number of new plants available for production in year i times the maximum annual production J E j,i of a plant of technology j, i.e. 0≤
J E j,i Wj,i
(5)
1 − νjJ J J P H , 1000 j j
(6)
J Ej,i,ω
with J
E j,i =
≤
where HjJ [h] and νjJ are the number of operating hours per year and the loss percentage, respectively, of a power plant of technology j. K Analogously, the electricity production Ek,i,ω [GW h] of power plant k in year i under scenario ω is nonnegative and K bounded above by its maximum annual production E k,i in year i, i.e. K K (7) 0 ≤ Ek,i,ω ≤ E k,i
The total electricity generated in year i cannot exceed the power producer’s market share M i,ω [GW h] under scenario ω X X J K Ej,i,ω + Ek,i,ω ≤ M i,ω . (9) j∈J
k∈K
The amount of electricity Gi,ω [GW h] for which the corresponding Green Certificates have to be purchased, if Gi,ω < 0, or are sold, if Gi,ω > 0, in year i under scenario ω is given by X X X X J K J K Ej,i,ω Ek,i,ω Gi,ω = + −βi Ej,i,ω + Ek,i,ω j∈J R
K E k,i
(
1−νkK K K 1000 Pk Hk
k∈K
(10) where βi is the ratio, required in year i, between the electricity produced from RES and the total electricity produced. The amount Qi,ω [t] of CO2 emissions the power producer must pay for in year i under scenario ω is defined as X X J K Qi,ω = θjJ Ej,i,ω + θkK Ek,i,ω (11) j∈J T
k∈K T
where θkK [t/GW h] and θjJ [t/GW h] are the CO2 emission rates of thermal power plant k ∈ K T and thermal power plant of candidate technology j ∈ J T , respectively. Among all power generation capacity expansion plans that satisfy the above constraints, one has to be chosen so as to maximize the objective function X p ω Fω , (12) ω∈Ω
where Fω is the actualized profit under scenario ω and pω is the scenario probability. The profit Fω under scenario ω is defined as X X X 1 E J K πi,ω Ej,i,ω + Ek,i,ω + Fω = (1 + r)i j∈J
i∈I
+ −
GC πi,ω Gi,ω
X
k∈K
−
CO2 πi,ω Qi,ω
J J vj,i,ω Ej,i,ω
−
j∈J T
−
X
+
X
K K vk,i,ω Ek,i,ω +
k∈K T J J vj,i Ej,i,ω
j∈J R
−
X
K K vk,i Ek,i,ω +
k∈K R
−
with
j∈J
k∈K R
X
J
Rj + fj Wj,i .
j∈J
if i ≤ LK k if i > LK k .
(8)
(13)
For power plant k, owned by the power producer at the beginning of the planning horizon, PkK [M W ] denotes the rated power, HkK [h] the number of operating hours per year, K LK k [years] the residual life and νk the loss percentage. Notice that for some technologies a lower bound to the annual electricity production could be imposed, in order to take into account technical limitations.
The hypotheses on the values of prices of electricity, Green Certificates, CO2 emission allowances and fuels in year i under scenario ω are represented by the stochastic parameters CO2 E GC J K πi,ω , πi,ω , πi,ω , vj,i,ω and vk,i,ω , respectively. The variable production unit costs of RES power plants are assumed to be known with certainty and therefore are represented by J the deterministic parameters vj,i , for a RES power plant of R K candidate technology j ∈ J , and vk,i , for RES power plant
=
0
k ∈ K R . Parameter fjJ [ke ] denotes the fixed production cost of a power plant of technology j. Summarizing, the risk-neutral two-stage stochastic mixed integer linear model for the generation capacity expansion problem determines the values of the decision variables wj,i , K J , Ek,i,ω , Gi,ω and Qi,ω , for j ∈ J, k ∈ K, i ∈ I Wj,i , Ej,i,ω and ω ∈ Ω, that maximize the objective function (12), with Fω defined by (13), subject to constraints (1), (2), (4), (5), (7), (9), (10), (11) and integrality of wj,i . III. R ISK AVERSE VERSIONS OF THE STOCHASTIC POWER GENERATION CAPACITY EXPANSION MODEL
The risk neutral approach determines the values of the decision variables that maximize, over all scenarios, the expected profit along the planning horizon. It does not take into account the variability of the objective function value over the scenarios and, then, the possibility of realizing in some scenarios a very low profit. In the literature several approches have been introduced for measuring the profit risk. In this section we present the two risk measures we have used in order to reduce the risk of having low profits in some scenarios.
the expected value of profit (12), with Fω defined by (13), subject to constraints (1), (2), (4), (5), (7), (9), (10), (11) and to constraints X pω µpω ≤ τ p ∀p ∈ P, (17) ω∈Ω
where variables µpω , by which the shortfall probability with respect to threshold φp is computed, are defined by constraints φp − Fω ≤ Mω µpω µpω ∈ {0, 1}
ω∈Ω
see [14], [19], [20] and [21]. Based on CVaR, the risk averse version of the generation capacity expansion model determines J K the values of the decision variables wj,i , Wj,i , Ej,i,ω , Ek,i,ω , Gi,ω , Qi,ω , dω and V , for j ∈ J, k ∈ K, i ∈ I and ω ∈ Ω, that maximize the objective function ( ! " !#) X 1 X max (1 − ρ) p ω Fω + ρ V − pω dω , α ω∈Ω ω∈Ω (15) with Fω defined by (13), subject to constraints (1), (2), (4), (5), (7), (9), (10), (11), integrality of wj,i and to constraints dω ≥ 0 ,
∀ p ∈ P , ω ∈ Ω.
(19)
The price to be paid is the increase of the number of constraints and variables. In this work we only analyze the results obtained by the model by plain use of a MIP solver. For large-scale instances, decomposition approaches must be used, like the exact multistage Branch-and-Fix Coordination method [10], multistage metaheuristics, like Stochastic Dynamic Programming [6], [11], for very large scale instances, and Lagrangian heuristic approaches, see [16]. IV. C ASE STUDY
Given the confidence level α, the CVaR of profit is the expected value of the profit smaller than the α-percentile, or Value at Risk (VaR), of the profit distribution and it is the defined as !# " 1 X p ω dω , (14) max V − V α
and
(18)
with
A. Conditional Value at Risk (CVaR)
dω ≥ V − Fω
∀ p∈P , ω∈Ω
∀ω ∈ Ω.
(16)
The value of V in the optimal solution is the VaR. Parameter ρ ∈ [0, 1] is the so called CVaR risk averse weighting factor. In a different context, see [1], [3], [7] and [18]. B. First-order Stochastic Dominance (FSD) In the risk aversion strategy FSD the expected profit maximization is hedged against a set of non-wanted scenarios, with the hedging represented by the requirement of forcing the scenario profit to be not smaller than a set of thresholds, with a bound on failure probability for each of them, see [4] and [17]. A benchmark is given by assigning a set P of profiles (φp , τ p ), p ∈ P , where φp is the threshold to be satisfied by the profit at each scenario and τ p is the upper bound of its failure probability. The profit risk is hedged by maximizing
The stochastic model introduced in Section II and its risk averse versions, based on either CVaR or FSD, presented in Section III, have been implemented in GAMS 23.2.1 and solved by the MIP solver CPLEX 12.1.0. Model validation has been performed on a testbed of realistic instances. Here we report the results obtained for a power producer who owns at the beginning of the planning horizon (i.e. in year 0) the medium-size generation system described in Table I, where rated power, residual life and efficiency are reported for every power plant of the system. Note that this generation system, with total capacity of 8629 M W , is biased toward the CCGT technology and is therefore highly dependent on the gas price. A planning horizon of 50 years is considered. TABLE I P OWER PLANTS OWNED BY THE POWER PRODUCER IN YEAR 0. power plant k 1 2 3 4 5 6 7 8 9
type CCGT CCGT CCGT CCGT CCGT CCGT Coal Coal Coal
rated power PkK [M W ] 840 425 1090 532 75 227 600 500 402
residual life ˆ K [years] L k 19 18 16 15 9 7 8 3 12
efficiency ηkK [−] 0.52 0.52 0.50 0.51 0.52 0.50 0.40 0.39 0.41
Since in this paper we plainly use the MIP solver, we limit the computational time by considering only fuel price uncertainty. The market electricity price is 106 e /M W h. The ratio βi ”electricity from RES / total electricity produced” is set according to the 20-20-20 European target from year 2011 to year 2020 (i.e. from year 1 to year 8 of the planning period); after 2020 a further increase of 1% is assumed. The GenCo
can satisfy the imposed βi ratio by either producing from RES or buying Green Certificates. The Green Certificate price is assumed to be 73.3 e /M W h. For the price of CO2 emission allowances a 3% increase per year is assumed. For the market share a 2% increase per year is assumed, which implies that the maximum amount of energy that can be sold to the market in 50 years is 4229 T W h. For every candidate technology the technical characteristics of a power plant are reported in Table II. The CO2 emission rates of coal and gas are 338 and 200 [t/GW h] respectively. TABLE II C HARACTERISTICS OF CANDIDATE TECHNOLOGIES .
Coal CCGT Nuclear Wind
investment cost [ e /W ] 1 0.47 3.2 1.55
rated power [M W ] 600 800 1200 100
operating hours [h/year] 7446 7446 7884 2215
construction time [years] 4 2 7 1
industrial life [years] 25 25 40 20
Tables III and IV provide six alternative values of the ratio ”estimated price over price in year 0” that are considered, together with the associated probabilities, for coal and gas, respectively. Analogously, Table V provides three alternative values and the associated probabilities for the nuclear fuel. By combining all alternatives, 108 independent scenarios of fuel prices are obtained. TABLE III A LTERNATIVE VALUES OF ” ESTIMATED PRICE OVER PRICE IN YEAR 0” RATIO ( COAL PRICE IN YEAR 0: 115 e /t (12.3 e /M W h) ratio probability
0.786 0.035
0.9 0.1
1.00 0.565
1.246 0.26
1.6571 0.035
2.0 0.005
TABLE IV A LTERNATIVE VALUES OF ” ESTIMATED PRICE OVER PRICE IN YEAR 0” RATIO ( GAS PRICE IN YEAR 0: 0.29 e /N m3 (31.3 e /M W h) ratio probability
0.786 0.035
0.9 0.1
1.00 0.565
1.246 0.26
1.6571 0.035
2.0 0.005
TABLE V A LTERNATIVE VALUES OF ” ESTIMATED PRICE OVER PRICE IN YEAR 0” RATIO ( NUCLEAR FUEL PRICE IN YEAR 0: 2100 e /kg (2.21 e /M W h) ratio probability
0.8313 0.08
1.00 0.72
1.28 0.2
V. N UMERICAL RESULTS The HW/SW platform used for the numerical experiments has the following characteristics: Intel Core i7-2670QM, 2.20 GHz, 8 GB RAM and the Microsoft Windows 7 Home Premium 64 bit. In this section we present the results obtained by the risk neutral model, its risk averse version based on CVaR and its risk averse version based on FSD. The results obtained
with other risk measures, i.e. Shortfall Probability, Expected Shortage and Second-orded Stochastic Dominance can be found in [23]. Table VI shows the model dimensions in our case study. The row headings are as follows: m, number of constraints; nc, number of continuous variables; nd, number of discrete variables; n0/1, number of 0-1 variables; nel, number of nonzero elements in the constraint matrix; and den [%], constraint matrix density. In the case study the range of the integer variables, i.e. the maximum number of new power plants of each technology available in the planning horizon are 20 CCGT plants, 30 coal plants, 15 nuclear plants and 60 wind power plants. TABLE VI P ROBLEM DIMENSIONS IN THE CASE STUDY. risk neutral CVaR FSD
m 1863 2430 4703
nc 2451 3019 5286
nd 550 550 3385
n0/1 550 550 3385
nel 16028 642564 3145868
den(%) 0.0035 0.0876 0.1265
A. Numerical results with CVaR In order to evalute the Net Present Value of the investment in a single technology, we first solved the risk neutral model with only one technology a time (and without existing plants) for a fixed budget of 3.84 Ge and different electricity market prices. We observed that if the market price is fixed at 110 e /M W h, the most convienent technology is the CCGT, followed by Geothermal, Coal, Mini-hydro, Wind, Nuclear and then the other RES technologies, which have low NPVs. As the electricity price decreases, the optimal solution changes: with a price of 100 e /M W h the first technology is the Geothermal, followed by Mini-hydro, Coal, CCGT, Wind, Nuclear and the other RES technologies. In the rest of the case study the candidate technologies are assumed to be the three thermal technologies and the wind power technology. The optimal technology mix obtained with different values of the initial budget and market electricity price are shown in Table VII, where profits are in Ge . In the case of high electricity price and low initial budget, CCGT plants are preferred as they are characterized by relatively high production per unit of investment cost. As the budget increases, the number of power plants increases until it reaches the market share, so the best decision consists of investing in technologies that are more capital intensive like Coal, Wind and Nuclear. For further budget increase the wind power plants are preferred and for very high budget, nuclear plants come to place. We can observe that nuclear plants appear only when the maximum number of wind plants (60, in this case) is reached. If the electricity market price is low, the optimal mix consists mainly of coal and wind plants due to their lower variable costs. B. Numerical results with CVaR A market electricity price of 106 e /M W h and two values for the initial budget are considered, in order to show their influence on the resulting technology mix. The confidence
TABLE VII R ISK NEUTRAL MIX DEPENDING ON BUDGET VALUE AND ELECTRICITY PRICE . Electricity price [ e /M hW ] 110
106
100
95
Technology CCGT Nuclear Coal Wind Profit CCGT Nuclear Coal Wind Profit CCGT Nuclear Coal Wind Profits CCGT Nuclear Coal Wind Profits
Budget 3.84 [Ge ] 11 20.471 11 16.617 9 1 11.013 6 3 7.049
Budget 7.68 [Ge ] 20 5 1 30.459 7 10 24.769 2 12 1 18.584 13 1 14.185
Budget 14.5 [Ge ] 14 14 60 42.668 1 16 60 36.730 14 59 30.078 10 58 25.224
Budget 30 [Ge ] 4 4 20 60 63.936 7 5 9 60 57.281 1 5 7 60 48.506 5 6 60 41.844
level α = 0.05 is used to define the CVaR. For the case corresponding to the budget B = 3.84 Ge , the profit distributions are shown in Fig. 1.
TABLE VIII T ECHNOLOGY MIX OBTAINED WITH CVaR AND DIFFERENT RISK AVERSION PARAMETERS - B UDGET: 3.84 Ge . Case ρ CCGT coal wind expected profit [Ge ] CVaR
0 0 13 0 1 22.173 -1.026
1 0.1 9 2 0 21.990 2.682
2 0.2 5 3 0 21.019 8.910
3 0.3 1 6 1 20.367 10.585
4 0.8 0 4 10 18.357 11.164
Fig. 2. Profit distribution for different risk aversion parameters using CVaR with α = 0.05, budget 14.5 Ge and market electricity price 106 e /M W h.
Fig. 3. Percentage of new capacity installed with CVaR, α = 0.05, budget 14.5 Ge and ρ = 0.5. Fig. 1. Profit distribution for different risk aversion parameters using CVaR with α = 0.05, budget 3.84 Ge and market electricity price 106 e /M W h.
In Table VIII we can observe a regular change in the technology mix as the risk aversion parameter ρ increases. Notice the difference between the case with ρ = 0 (risk neutral) and the case with ρ = 0.2 (risk averse): the espected profit decreases by only 5% (1.154 Ge ), while the CVaR has grown by 9.926 Ge . For larger values of the risk aversion parameter (ρ = 0.3 and ρ = 0.8) the corresponding optimal technology mixes have a small growth of the CVaR but have a big decrease in the expected profit. For the case with a higher budget, say 14.5 Ge , the profit distributions are shown in Fig. 2. The nuclear technology appears in the mix (see Table IX) only with high risk aversion (ρ ≥ 0.5) and it represents a low percentage of the new capacity mix (see Fig. 3).
With a budget lower than 14.5 Ge the nuclear technology is not included in the optimal mix, due to the rated power of nuclear plants. TABLE IX T ECHNOLOGY MIX OBTAINED WITH CVaR AND DIFFERENT RISK AVERSION PARAMETERS - B UDGET B = 14.5 Ge . Case ρ CCGT nuclear coal wind Expected profit CVaR
0 0 17 0 1 60 41.48 23.23
1 0.2 10 0 6 60 41.24 27.23
2 0.4 8 0 5 60 40.63 28.44
3 0.5 6 1 2 60 39.50 29.58
4 0.6 5 1 1 60 38.79 30.21
5 0.8 4 1 0 59 37.90 30.77
C. Numerical results with FSD We considered a budget of 3.84 Ge and four benchmarks, the profiles of which are shown in Table X. TABLE X B ENCHMARKS USED FOR TESTING THE FSD RISK AVERSION STRATEGY.
Benchmark 0
Benchmark 1
Benchmark 2
Benchmark 3
p 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
φp [Ge ] -4 5 16 29 0 10 21 29 8 13 16 20 10 12 15 18
τp 0.02 0.08 0.38 0.88 0.02 0.08 0.38 0.88 0.02 0.08 0.28 0.58 0.02 0.08 0.28 0.49
For Benchmark 0 the same optimal mix of technologies is obtained as in the risk neutral case. This is due to the magnitude of the first three thresholds, which have low values and high probabilities, therefore allowing large losses. The expected profit is the highest among the various tested benchmarks, see Table XI and Fig. 4. Table XI shows the optimal values and times occurred in solving the associated MIP problems related to the different benchmarks.
TABLE XI O PTIMAL TECHNOLOGY MIX OBTAINED WITH THE FSD RISK AVERSION STRATEGY USING THE BENCHMARKS DEFINED IN TABLE X. Benchmark CCGT coal wind expected profit [Ge ] VaR [Ge ] CVaR [Ge ] TM IP [s]
0 13 0 1 22.173 8.016 -1.026 274.72
1 9 2 0 21.990 8.174 2.682 402.49
2 3 5 0 21.028 11.490 8.463 4173.22
3 0 3 14 17.650 12.472 10.583 >50000
capacity expansion planning in a long time horizon with respect to changes in electricity prices and available budget with uncertain fuel prices. It shows a modification in the optimal investment portfolio towards RES plants when the budget increases. The nuclear plants appear only with very high budget (30Ge). With low electricity prices the optimal investment portfolio moves from gas plants towards coal and wind power plants, because of their lower variable costs. The introduction of the risk measures in the model shows the role of fuel price volatility in the selection of the optimal technology mix. The increase of risk aversion moves the optimal technology mix towards technologies with lower variable costs. The comparison among the CVaR and the FSD approaches shows that the FSD risk measure gives a more complete description of the investor’s adversion to risk, at the cost of higher computational time with the standard MIP solver. It is planned to use ad hoc decomposition algorithms, in order to increase the dimension of the solvable cases.
ACKNOWLEDGMENT The authors acknowledge the support from the grant by Regione Lombardia ”Metodi di integrazione delle fonti energetiche rinnovabili e monitoraggio satellitare dell’impatto ambientale”, EN-17,ID 17369.10, and University of Bergamo grants (2011, 2012) coordinated by M. Bertocchi and M.T. Vespucci. R EFERENCES Fig. 4. FSD: Profit distributions for the different benchmarks.
The profit distributions related to the four benchmarks show a decreasing risk in Fig. 4. It is clear from the figure that a comprehensive risk analysis requires to consider several benchmarks. Indeed, we can notice that for some of them the associated values of the expected profit and CVaR are more favourable than in other benchmarks. However, this analysis may require an unaffordable computing time by plain use of a MIP solver and, then, a decomposition algorithm is required, see e.g. [6], [10], [12]. VI. C ONCLUSIONS This paper analyzes the sensitivity of a two-stage riskneutral stochastic optimization model for power generation
[1] Alonso-Ayuso A., di Domenica N., Escudero L.F., Pizarro C., Structuring bilateral energy contract portfolios in competitive markets. In Bertocchi M., Consigli G., Dempster M.A.H., Stochastic Optimization Methods in Finance and Energy. International Series in Operations Research and Management Science, Springer Science + Business Media, 163(2), 203-226 (2011) [2] Bjorkvoll T., Fleten S.E., Nowak M.P., Tomasgard A., Wallace S.W., Power generation planning and risk management in a liberalized market, IEEE Porto Power Tech Proceedings, 1, 426-431 (2001) [3] Cabero J., Ventosa M.J., Cerisola S., Baillo A. Modeling risk management in oligopolistic electricity markets: a Benders decomposition approach IEEE Transactions on Power Systems, 25, 263-271 (2010) [4] Carrion M., Gotzes U., Schultz R., Risk aversion for an electricity retailer with second-order stochastic dominance constraints Computational Management Science, 6, 233-250 (2009) [5] Conejo A.J., Carrion M., Morales J.M., Decision making under uncertainty in electricity market, International Series in Operations Research and Management Science. Springer Science+Business Media, New York (2010)
[6] Cristobal M.P., Escudero L.F., Monge J.F., On Stochastic Dynamic Programming for solving large-scale tactical production planning problems, Computers and Operations Research, 36, 2418-2428 (2009) [7] Ehrenmann A., Smeers Y., Generation capacity expansion in a risky environment: a stochastic equilibrium analysis, Operations Research, 59, 1332-1346 (2011) [8] Ehrenmann A., Smeers Y., Stochastic equilibrium models for generation capacity expansion. In Bertocchi M., Consigli G., Dempster M.A.H., Stochastic Optimization Methods in Finance and Energy. International Series in Operations Research and Management Science, Springer Science + Business Media, 163(2), 203-226 (2011) [9] Eppen G.D., Martin R.K., Schrage. L., Scenario approach to capacity planning, Operations Research 34, 517-527 (1989) [10] Escudero L.F., Garin M.A., Merino M., Perez G., An algorithmic framework for solving large scale multistage stochastic mixed 0-1 problems with nonsymmetric scenario trees, Computers and Operations Research, 39: 1133-1144 (2012) [11] Escudero L.F., Monge J.F., Romero-Morales D., Wang J., Expected future value decomposition based bid price generation for largescale network revenue management. Transportation Science, doi: 10.1287/trsc.1120.0422 (2012) [12] Escudero L.F., Garin M.A., Merino M., Perez G., Risk management for mathematical optimization under uncertainty. Part II: Algorithms for solving multistage mixed 0-1 problems with risk averse stochastic dominance constraints strategies. To be submitted (2012) [13] Fabian C., Mitra G., Roman D., Zverovich V., An enhanced model for portfolio choice SSD criteria: a constructive approach, Quantitative Finance, 11, 1525-1534 (2011) [14] Gaivoronski A.A., Pflug G., Value-at-Risk in portfolio optimization: properties and computational approach, Journal of Risk, 7, 11-31 (2005) [15] Genesi C., Marannino P., Montagna M., Rossi S., Siviero I., Desiata, L., Gentile G., Risk management in long term generation planning, 6th Int. Conference on the European Energy Market, 1-6 (2009) [16] Gollmer R., Gotzes U., Schultz R., A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse, Mathematical Programming, Ser. A, 126, 179-190 (2011) [17] Gollmer R., Neise F., Schultz. R., Stochastic programs with firstorder stochastic dominance constraints induced by mixed-integer linear recourse, SIAM Journal on Optimization, 19, 552-571 (2008) [18] Kannan A., Shanbhag U.V., Kim H.M. Risk-based generalized Nash games in power markets: characterization and computation of equilibria, University of Illinois, USA, submitted for publication (2009) [19] Rockafellar R.T., Uryasev S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-41 (2000) [20] Rockafellar R.T., Uryasev S., Conditional value-at-risk for general loss distributions, Journal of Banking and finance, 26, 1443-1471 (2002) [21] Schultz R., Tiedemann S., Conditional Value-at-Risk in stochastic programs with mixed-integer recourse, Mathematical Programming, Ser. B, 105, 365-386 (2006) [22] Vespucci M.T, Bertocchi M., Innorta I., Zigrino S., Deterministic and stochastic models for investment in different technologies for electricity production in the long period, submitted to CEJOR Journal (2012) [23] Vespucci M.T, Bertocchi M., Zigrino S., Escudero, L.F. A risk-averse stochastic optimization model for power generation capacity expansion planning, Technical Report, DMEQM (2013)
A. Short bio Maria Teresa Vespucci received the Ph.D. degree in numerical optimization from the University of Hertfordshire, UK, in 1992. She is currently an Associate Professor of Operations Research at the University of Bergamo, Italy. Her research interests include linear algebra, numerical optimization, mathematical programming and stochastic programming, with applications to operations, planning and economics of electric energy systems. Marida Bertocchi received the degree in Mathematics from the University of Milano, Italy, in 1974. She is currently a full Professor of Mathematical Finance at the University of Bergamo and the scientific coordinator of the PhD program in
Economics, Applied Mathematics and Operational Research. Her research interests include stochastic programming and its applications to finance and energy systems. Stefano Zigrino received the Master degree in Mechanical Engineering from the University of Bergamo in 2010. He is currently pursuing the Ph.D. degree at the University of Bergamo. His research interests include optimization and investment evaluation applied to energy systems. Laureano F. Escudero received the Ph.D. degree from the Universidad de Deusto, Spain, in 1974. He is currently a full Professor at Universidad Rey Juan Carlos, Madrid, Spain. His research interests include planning and economics of electric energy systems, as well as mathematical optimization (linear, nonlinear, integer, continuous) under uncertainty (stochastic optimization).
