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Chapter 10Structuring Bilateral Energy Contract Portfoliosin Competitive Markets

Antonio Alonso-Ayuso, Nico di Domenica, Laureano F. Escudero,and Celeste Pizarro

Abstract A multistage complete recourse model for structuring energy contractportfolios in competitive markets is presented for price-taker operators. The mainuncertain parameters are spot price, exogenous water inflow to the hydro system andfuel-oil and gas cost. A mean-risk objective function is considered as a compositefunction of the expected trading profit and the weighted probability of reaching agiven profit target. The expected profit is given by the bilateral contract profit andthe spot market trading profit along the time horizon over the scenarios. The uncer-tainty is represented by a set of scenarios. The problem is formulated as a mixed0–1 deterministic equivalent model. Only 0–1 variables have nonzero coefficientsin the first-stage constraint system, such that the continuous variables only showup in the formulation of the later stages. A problem-solving approach based on asplitting variable mathematical representation of the scenario clusters is considered.The approach uses the twin node family concept within the algorithmic frameworkpresented in the chapter. The Kyoto protocol-based regulations for the pollutantemission are considered.

Keywords Energy trading contracts portfolio · Stochastic programming · Meanrisk ·Mixed 0–1 models · Splitting variable · Branch-and-fix coordination

10.1 Introduction

Given a power generation system and a set of committed as well as candidate energytrading contracts, the Energy Contract Portfolio Problem (ECP2) is concerned withselecting the bilateral contracts for energy purchasing and selling to be deliveredalong a given time horizon. It is one of the main problems faced today by thepower generation companies and the energy service providers. The main uncertainparameters are spot market price, exogenous water inflow to the hydro system, andfuel-oil and gas cost. The maximization of the weighted reaching probability (RP)is included as a risk measure in the objective function, together with the expected

L.F. Escudero (B)Departamento de Estadística e Investigación Operativa, Universidad Rey Juan Carlos,Móstoles (Madrid), Spaine-mail: [email protected]

M. Bertocchi et al. (eds.), Stochastic Optimization Methods in Finance and Energy,International Series in Operations Research & Management Science 163,DOI 10.1007/978-1-4419-9586-5_10, C© Springer Science+Business Media, LLC 2011

203

204 A. Alonso-Ayuso et al.

energy trading profit along the time horizon over the scenarios, and the economicvalue of satisfying the Kyoto protocol-based regulations for the pollutant emissionsby the thermal generators. The constraint system comprises the power generationconstraints for the thermal and hydro generators, the logistic constraints relatedto the energy contracts, and the power load already committed to purchasing andselling. See Tanlapco et al. (2002) for different types of financial hedging con-tracts. See also Fleten et al. (2001) for an approach to hedge electricity portfoliosvia stochastic optimization. An approach in Pritchard et al. (2005) uses dynamicprogramming for problem solving. Shrestha et al. (2005) present a medium-termhydropower planning with bilateral contracts. Sen et al. (2006) integrate the unitcommitment model with financial decision-making and spot market considerationsin a non-hydropower generation environment. Fleten and Kristoffsen (2007) presenta model for determining optimal bidding strategies taking uncertainty into accountby a price-taking hydropower producer.

Nowadays, most of the traded energy (around 75%) is done via bilateral contracts(e.g. the daily market APX in the Netherlands has only between 10 and 15% of thetotal energy of the country and the daily market EEX in Germany sells somethingless than 15% of the total energy). Moreover, there are some other countries (Spainis a good example, see Perez Arriaga (2005)), with a very small bilateral contractsdevelopment due, mainly, to disruption of market regulations.

The deterministic version of the problem can be represented as a mixed 0–1model. Given today’s state-of-the-art optimization tools, no major difficulties shouldarise for problem solving of moderate size instances, at least. However, given theuncertainty of the main parameters, the modelling and algorithmic approach to theproblem makes ECP2 an interesting application case of stochastic integer program-ming.

The aim of the model that we present in the chapter consists of helping a betterdecision making on the energy selling/purchasing (so-called first-stage) policy. Forthis purpose, the future uncertainties in the spot market and generation availabilityare considered. (Since the model assumes price-taking decision maker, no restric-tions are considered for satisfying the uncertain demand.)

Some of the stochastic approaches related to power generation only consider con-tinuous variables. Moreover, there are different schemes to address the energy powergeneration planning under uncertainty, by modelling it with 0–1 and continuousvariables; see Carpentier et al. (1996), Caroe and Schultz (1998, 1999), Dentchevaet al. (1997), Dentcheva and Römisch (1998), Gollmer et al. (1999), Hemmecke andSchultz (2001), Klein Haneveld and van der Vlerk (2001), Ni et al. (2004), Nürnbergand Römisch (2002), Schultz et al. (2005), Takriti and Birge (2000), and Triki et al.(2005), among others. Most of these approaches only consider mean (expected)objective functions and Lagrangian decomposition schemes, some of which proposebidding strategies for 24 h horizons.

Alternatively, very few approaches in general deal with the mean-risk measuresby considering semi-deviations (Ogryczak and Ruszczynski 1999), excess probabil-ities (Alonso-Ayuso et al. 2009; Schultz and Tiedemann 2004), value-at-risk (VaR),conditional VaR (Rockafellar and Uryasev 2000, Schultz and Tiedemann 2006) and

10 Structuring Bilateral Energy Contract Portfolios in Competitive Markets 205

stochastic dominance constraint strategies (Gollmer et al. (2011) and referencestherein); see also Dert (1998); Lulli and Sen (2004); Schultz (2003) and Valente(2002). These approaches are more amenable for large-scale problem solving thanthe classical mean – variance schemes, particularly, in the presence of 0–1 variables.

In this chapter we present a mixed 0–1 deterministic equivalent model (DEM)for the multistage stochastic ECP2 with complete recourse, where the parameters’uncertainty is represented by a set of scenarios. Only 0–1 variables (that representstrategic trading decisions) have nonzero coefficients in the first-stage constraintsystem, such that the continuous variables (that represent operational decisions)only show up in the formulation of the later stages. The values of the variablesdo consider all scenarios without being subordinated to any of them. The reachingprobability (RP) is considered as a risk measure in contrast with approaches whereonly mean functions appear in the objective function to optimize.

The model has no direct relationship with the enterprise-wide risk managementfunction or the bidding to each submarket function. However, the max mean-riskfunction to consider assumes this interrelation with the enterprise-wide risk man-agement function. On the other hand, the elements of the spot market in the modelassume the interrelation with the bidding function in an aggregated way. After struc-turing the portfolio, the model can be run again, in a rolling horizon approach,once new opportunities for energy selling/purchasing, restructuring and cancellationappear in the horizon.

One of the main contributions of the chapter consists of considering a couplingof a generic electricity generation planning problem and the contract managementproblem, since in our opinion the structuring of the bilateral contract portfolio musttake into account the spot market availability and the energy generation restrictionsalbeit in an aggregate way.

We present a branch-and-fix coordination (BFC) scheme to exploit the structureof the splitting variable representation of the mixed 0–1 DEM and, specifically, thestructure of the nonanticipativity constraints for the 0–1 variables. The algorithmmakes use of the twin node family (TNF) concept introduced in Alonso-Ayuso et al.(2003a). It is specially designed for coordinating and reinforcing the branching nodeand the branching variable selections at each branch-and-fix (BF) tree. The treesresult from the relaxation of the nonanticipativity constraints in the splitting variablerepresentation of the DEM for the scenario clusters.

Additionally, the proposed approach considers the DEM at each TNF integer set,i.e. each set of integer nodes, one for each BF tree, where the nonanticipativityconstraints for the 0–1 variables are satisfied. By fixing those variables to the nodes’values, the DEM has only continuous variables. The approach ensures the satisfac-tion of the nonanticipativity constraints for the continuous variables associated witheach scenario group in the scenario tree and so, provides the LP optimal solution forthe given TNF integer set.

The remainder of the chapter is organized as follows. Section 10.2 states theECP2 and introduces the mixed 0–1 DEM for the multistage stochastic model, com-pact representation. Section 10.3 introduces the mean-risk function and the splittingvariable representation of the DEM. Section 10.4 presents our approach for problem


solving. Section 10.5 concludes. The Appendix gives more algorithmic details ofthe approach presented in this chapter.

10.2 Problem Description

10.2.1 Problem Statement

Consider a set of thermal power generators, and a set of hydropower generatorsdistributed along river basins whose reservoirs can store water from one period tothe next one in a given time horizon. Consider also a set of scenarios for the uncer-tain parameters, mainly, spot market price, exogenous water inflow to the hydrosystem, and fuel-oil and gas cost. Finally, let a set of candidate energy bilateraltrading (purchasing/selling) contracts be considered. Each contract is characterizedby the energy price depending on the scenario to occur and the lower and upperbounds of the purchasing/selling load to deliver in each period of the time horizon.The load in the selling contracts will be assumed to be dependent on the scenario tooccur. It is assumed that the energy generator/trading company is a price taker whenoperating in the spot market. Note that the contract portfolio can combine contractsin different ways to reduce risk and profit less or, vice versa, increase profit andincrease risk.

The problem consists of deciding which candidate purchasing/selling contracts toaccept, such that some logistic constraints related to the combination of the contractsare satisfied, the power to generate satisfies the thermal and hydro generation con-straints, the balance equations of the water flow along the periods in the river basinsare satisfied, the Kyoto protocol-based pollutant emission constraints are satisfiedby the thermal units, and the purchasing/selling loads already committed are alsosatisfied. See also Conejo and Prieto (2001) and Escudero (2001). The objectivefunction to maximize is a composite function of the estimated profit along the timehorizon and the weighted RP of having a profit greater than or equal to a giventhreshold over the scenarios.

We present below a mixed 0–1 model for structuring the energy contract port-folio, where the uncertainty is treated such that the occurrence of the events isrepresented by a multistage scenario tree.

10.2.2 Scenario Tree

Let the tree shown in Fig. 10.1 represent the stochasticity of the problem to be dealtwith. Each node in the figure is associated with a point in time where a decisioncan be made. Once a decision is made, some contingencies can happen (in thisexample, the number of contingencies is 3 for time period t = 2), and informationrelated to them is available at the beginning of the period. Each root-to-leaf pathin the tree represents one specific scenario and corresponds to one realization of


Fig. 10.1 Scenario tree

the whole set of the uncertain parameters along a time horizon. Each node in thetree can be associated with a scenario group, such that two scenarios belong to thesame group for a given time period, provided that they have the same realizations ofthe uncertain parameters up to that period. The nonanticipativity principle stated byWets (1974) and restated in Rockafellar and Wets (1991), see Birge and Louveaux(1997), among many others, requires that both scenarios have the same values forthe related variables up to the given period.

The following notation related to the scenario tree will be used throughout thechapter:

T , set of stages (periods, in our case). T − = T − {|T |}Ω , set of (consecutively numbered) scenarios!, set of multiperiods for pollutant emission regulation. Note: The regulation is

performed at the multiperiod level (e.g. quarter, year), instead of the periodlevel (e.g. week)

Mγ ⊆ T , set of (consecutive) time periods for multiperiod γ , for γ ∈ !.τγ , last period for multiperiod γ , for γ ∈ !G, set of (consecutively numbered) scenario groupsGt , set of scenario groups in period t , for t ∈ T , such that the scenarios that

belong to the same group are identical in all realizations of the uncertainparameters up to period t (Gt ⊆ G)

t (g), period for scenario group g, for g ∈ G. Note: g ∈ Gt (g)

Ωg , set of scenarios in group g, for g ∈ G (Ωg ⊆ Ω)N g , set of scenario groups {k} such thatΩg ⊆ Ωk , for g ∈ G (N g ⊆ G), that is,

set of ancestor nodes of node gφ(g), immediate ancestor node of node g, for g ∈ G. Note: φ(g) ∈ N g


10.2.3 Mixed 0–1 DEM: Mean Function Maximization

The following is additional notation used in ECP2.

10.2.3.1 Sets

D, set of thermal-power generator types (coal, fuel-oil, gas, combined cycle, etc.)I, set of thermal-power generators in the problemd(i), type of generator i , such that d(i) ∈ D, for i ∈ I. Note: Two generators

that belong to the same type share the allowed pollutant emission regulationsJ , set of hydro generation units in the river basins under considerationU j , set of immediate up-stream hydro units to hydro unit j to represent the topol-

ogy of the river basins, for j ∈ JFcu, current energy bilateral selling contractsKcu, current energy bilateral purchasing contractsFca, candidate energy bilateral selling contractsKca, candidate energy bilateral purchasing contractsF = Fcu ⋃Fca and K = Kcu ⋃Kca

10.2.3.2 Deterministic Function and Parameters

p j

(xg

j , zgj

), power output of hydro unit j for a given water release, say, xg

j

and storage level, say, zgj at period t (g) under scenario group g, for g ∈ G,

j ∈ J . Although the formulation in terms of the variables z j would providea better representation of the influence of the water-related variables in thehydropower generation function, it would result in a significant increase inproblem complexity. As a compromise, a representative value for the storedwater, say zg

j , is used in the formulation to linearize the resulting expression.

It is represented by a concave piecewise function on xgj .

z j , upper bound of the stored water in the reservoir associated with hydro unit jat any period, for j ∈ J

xy j , upper bound of the (released and spilled) water flow downstream from thereservoir associated with hydro unit j at any time period, for j ∈ J , t ∈ T

p j , upper bound of the power to generate by hydro unit j at any time period, forj ∈ J

pi , upper bound of the power to generate by thermal unit i , for i ∈ Iθkt , θkt , lower and upper bounds for energy bilateral purchasing contract k during

period t , respectively, for k ∈ K, t ∈ T . This instrument is evidently a physicalinstrument, but it can be used as a financial one and, in particular, can be usedas a load factor contract, see its definition in Mo et al. (2001)

vd , unit pollutant emission from thermal generator type d, for d ∈ D"d , threshold for the non-penalized pollutant emission for the Kyoto protocol

regulation in the thermal generation type d, for d ∈ D


ζ , unit penalty (res., reward) for Kyoto protocol-based pollutant emission excess(res., deficit) from the allowed threshold. It is constant for the time horizon, atleast

a, unit trading fee for energy purchasing/selling from the spot market

10.2.3.3 Feasibility Spaces

Φi , operating feasible region for the production of thermal generator i at anytime period, for i ∈ I. There is ample bibliography in the open literaturedescribing the operating-constraint system to define the feasible space; seeArroyo and Conejo (2000), Dentcheva et al. (1997), Conejo et al. (2002),Dentcheva and Römisch (1998), Groewe-Kuska et al. (2001), Hobbs et al.(2001), Jimenez Redondo and Conejo (1997), Klein Haneveld and van derVlerk (2001), Schultz et al. (2005), Nürnberg and Römisch (2002), and Woodand Wollenberg (1996), among many others. However, since the problemtreated in this chapter considers a long time horizon, the set of operating con-straints has not been considered.

�, feasible region for the set of energy bilateral selling contracts to be chosen,such as maximum percentage for each contract, regional considerations, exclu-sivity constraints mainly in the financial contracts, etc.

Ψ , feasible region for the set of energy bilateral purchasing contracts to be cho-sen, such as maximum allowed budget, maximum percentage for each con-tract, regional considerations, and exclusivity constraints mainly in the finan-cial contracts.

10.2.3.4 Stochastic Function and Parameters for Period t(g) Under ScenarioGroup g, for g ∈ G

cgi (p

gi ), production cost for power output, say, pg

i of thermal generator i , fori ∈ I. It can be represented by a convex piecewise linear function, where thestochasticity is mainly due to the cost of the fuel-oil and gas along the timehorizon.

zgj , lower bound of the stored water in reservoir j , for j ∈ J . Special attentionshould be given to the lower bounds of the stored water during the last periodof the time horizon.

xygj , lower bound of the (released and spilled) water flow downstream from reser-voir j , for j ∈ J .

egj , net exogenous water inflow to reservoir j , for j ∈ J .

ιgf , energy requirement from bilateral selling contract f , for f ∈ F .

μgf , unit payment received from bilateral selling contract f , for f ∈ F .

λgk , unit payment due for bilateral purchasing contract k, for k ∈ K.ηg , unit energy price in the spot market.

Let ωg denote the weight assigned to scenario group g, for g ∈ G.


10.2.3.5 Generation and Trading Variables for Period t(g) Under ScenarioGroup g, for g ∈ G

θgk , energy requirement from bilateral purchasing contract k, for k ∈ K.

pgi , power output of thermal generator i , for i ∈ I."+d

g, "−d

g, excess and deficit pollutant emissions by the thermal generators of

type d from the allowed target for satisfying the Kyoto protocol-based regula-tions, respectively, for d ∈ D, g ∈ Gτγ , γ ∈ Γ .

xgj , released water from reservoir j , for j ∈ J .

ygj , spilled water from reservoir j , for j ∈ J .

zgj , stored water at (the end of) period t (g) in reservoir j , for j ∈ J .

sg , power output to be sold to the spot market by the generation/energy servicecompany.

bg , power to be purchased from the spot market by the generation/energy servicecompany.

10.2.3.6 First-Stage Variables: Candidate Contracts

δ f , 0–1 variable having the value 1 if the candidate energy bilateral selling con-tract f is chosen and, otherwise, its value is 0, for f ∈ Fca .

βk , 0–1 variable having the value 1 if the candidate energy bilateral purchasingcontract k is chosen and, otherwise, its value is 0, for k ∈ Kca .

The following is a so-called compact representation of the mixed 0–1 DEM for themultistage stochastic problem with complete recourse.

Objective: Determine the portfolio of the energy bilateral trading contracts to maxi-mize the mean (expected) profit from the contract portfolio exploitation and spotmarket trading along the time horizon over the scenarios, subject to constraints(10.2)–(10.14):

QE =∑g∈G

∑f ∈F cu

wgμgf ι

gf +max

∑g∈G

wg

⎡⎣ ∑

f ∈F ca

μgf ι

gf δ f + (ηg − a)sg

−⎛⎝∑

k∈Kλ

gk θ

gk +

∑i∈I

cgi (p

gi )+ (ηg + a)bg+

∑d∈D|g∈Gτγ ,γ∈!

ζ("+dg−"−d g

)

⎞⎠⎤⎦ .

(10.1)

Constraints:

∑i∈I

pgi +

∑j∈J

p j (xgj , zg

j )+∑k∈K

θgk + bg =

∑f ∈F cu

ιgf +

∑f ∈F ca

ιgf δ f + sg ∀g ∈ G,

(10.2)


0 ≤ pgi ≤ pi ∀i ∈ I, g ∈ G, (10.3)

vd

∑k∈N g |t (k)∈Mγ

∑i∈I|d(i)=d,

pki + "−d g = "d + "+d g

d ∈ D, g ∈ Gτγ , γ ∈ !,

(10.4)

p j (xgj , zg

j ) ≤ p j ∀ j ∈ J , g ∈ G, (10.5)

−zφ(g)j −∑

u∈U j

(xgu + yg

u )+ xgj + yg

j + zgj = eg

j ∀ j ∈ J , g ∈ G, (10.6)

zgj ≤ zg

j ≤ z j ∀ j ∈ J , g ∈ G, (10.7)

xygj≤ xg

j + ygj ≤ xy j ∀ j ∈ J , g ∈ G, (10.8)

θk,t (g) ≤ θ gk ≤ θk,t (g) ∀k ∈ Kcu, g ∈ G, (10.9)

θk,t (g)βk ≤ θ gk ≤ θk,t (g)βk ∀k ∈ Kca, g ∈ G, (10.10)

δ f ∈ Δ ∀ f ∈ F , (10.11)

βk ∈ Ψ ∀k ∈ K, (10.12)

xgj , yg

j , sg, bg ≥ 0 ∀i ∈ I, j ∈ J , g ∈ G, (10.13)

"+dg, "−d

g ≥ 0 ∀d ∈ D, g ∈ Gτγ , γ ∈ !. (10.14)

The model determines the selling/purchasing contract (so-called first-stage) pol-icy to be followed when restructuring the energy bilateral contracts portfolio. Therestructuring aims to maximize the expected profit given by the exploitation of theportfolio as well as the spot market trading as represented in (10.1). The balanceequations (10.2) equate the power availability (i.e., thermal and hydro generationplus purchased power already committed plus candidate power to purchase in bilat-eral trading contracts and power to purchase on the spot market) on one side andthe power needs (i.e., sold power already committed plus candidate power to sellin bilateral trading contracts and power to sell on the spot market) on the otherside. Bounds (10.3) ensure the feasibility of the thermal power to generate. Equa-tions (10.4) define the positive and negative output from the pollutant thermal sys-tem to satisfy the regulation based on the Kyoto protocol. Constraints (10.5) forceupper bounds on the hydropower generation. Balance equations (10.6) ensure thewater availability along the time horizon, given the topology of the river basinsunder the control of the generation company (see in Cervellera et al. (2006) anapproach for optimizing large-scale water reservoir network by stochastic dynamicprogramming).

Bounds (10.7) and (10.8) ensure the bounding of the stored water in the reser-voirs and the lower and upper bounds of the water flow (i.e., released and spilledwater) in the segments of the river basins. Bounds (10.9) ensure that the generationcompany/energy service provider will purchase the required energy from the currentbilateral contracts satisfying the requirements stated in the contracts. Constraints(10.10) ensure that the delivery of the energy from the candidate bilateral purchasingcontracts will satisfy the requirements of the contracts, if any. Memberships (10.11)


and (10.12) ensure that the logistic constraints are satisfied for the set of bilateralenergy selling and purchasing contracts to be chosen.

It is worth pointing out the different mechanism of the energy bilateral sellingand purchasing contracts. The model decides the energy to purchase, within a givenset of lower and upper bounds. On the other hand, the decisions on the energy tobe sold in a given period are exogenous to the model (i.e., they are made by thebuyer within committed lower and upper bounds); they are represented by uncertainenergy requirements which are modeled under the scenario groups for both types ofcontracts, current, and candidate.

Notice that the decision on the acceptance of the contracts considers the scenario-related system, but it is not subordinated to any scenario.

Remark: The bilateral contracts enter the constraint system in the power balanceequation restriction (10.2), but they also appear in the constraints that define thefeasible spaces (10.11) and (10.12) besides the objective function (10.1) (and func-tion (10.15)). It is enough for deciding the selling/purchasing (so-called first-stage)policy. Notice also that the physical contracts are the main contracts in the model;however, the financial ones are also allowed. The differences between the two typesof contracts are considered in the spaces defined by (10.11) and (10.12). It is up tothe contract manager to selling or purchasing a financial contract but (s)he shouldtake into account according to the model that there is a cost for energy purchasing(including fees) and an income for energy selling (subtracting the related fees), suchthat the solution must also satisfy constraints (10.2), (10.11), and (10.12).

10.3 Mean-Risk Function Maximization and VaR

As an alternative to maximizing the risk neutral mean function (10.2), we proposeto consider a risk averse mean-risk objective function, defined as a composite func-tion of the expected trading profit and RP, the probability of reaching a given profitthreshold, say, τ . It can be formulated as follows:

max QE + ρQP, (10.15)

where

QE =∑ω∈Ω

wωOω, (10.16)

wω gives the weight assigned to scenario ω, Oω gives the profit to obtain underscenario ω for ω ∈ Ω , ρ is a nonnegative weighting parameter, and QP is thereaching probability, RP. The profit Oω can be expressed as


Oω =∑

g∈N n

⎡⎣ ∑

f ∈F cu

μgf ι

gf +

∑f ∈F ca

μgf ι

gf δ f + (ηg − a)sg

⎤⎦

(10.17)

−∑

g∈N n

⎡⎣∑

k∈Kλ

gk θ

gk +

∑i∈I

cgi (p

gi )+(ηg + a)bg + ζ

∑d∈D|g∈Gτγ ,γ∈Γ

("+dg − "−d

g)

⎤⎦ ,

where n ∈ G|T | such that ω ∈ Ωn , and RP can be expressed as

QP = P(ω ∈ Ω : Oω ≥ τ) . (10.18)

Based on an approach for modeling RP given in Schultz and Tiedemann (2004),a more amenable expression of (10.15) for computational purposes, at least, isgiven by

max∑ω∈Ω

wω(Oω + ρνω) ,

s.t. Oω + M(1− νω) ≥ τ ∀ω ∈ Ω,νω ∈ {0, 1} ∀ω ∈ Ω,

(10.19)

where M is a parameter whose value is chosen as the smallest value which does noteliminate any feasible solution under any scenario, and νω is a 0–1 variable suchthat

νω =

⎧⎪⎨⎪⎩

1, if the trading profit Oω under scenario ω

is greater or equal than threshold τ

0, otherwise

∀ω ∈ Ω.

Note: The term M(1− νω) in (10.19) allows the benefit Oω to be negative for somescenario ω ∈ Ω .

As alternatives to the mean-risk optimization, the value at risk (VaR) and con-ditional VaR functionals to maximize for a given α-risk, where 0 ≤ α < 1, can beexpressed as

max VaR

s.t. Oω + M(1− ϑω) ≥ VaR ∀ω ∈ Ω∑ω∈Ω

wωϑω ≥ 1− α

ϑω ∈ {0, 1} ∀ω ∈ Ω,

(10.20)

for the value at risk (VaR), where ϑω is a 0–1 variable, and


max CVaR

s.t. κ + 1

1− α∑ω∈Ω

[wωOω − κ

]+ ≥ CVaR,(10.21)

for the conditional VaR, where [x]+ gives the positive value of x and κ gives theVaR value. Note: The Kyoto-based pollutant emission regulation has not been con-sidered.

Notice that the replacement of the expected trading profit (10.2) by the mean-risk system (10.19) with (10.17) does not change the structure of the model. Onthe contrary, the maximization of VaR (10.20) and the maximization of conditionalVAR (10.21) destroy the structure of the model, since the related constraints arenon-separable.

The instances of the mixed 0–1 DEM (10.1)–(10.14) can have such large dimen-sions that the plain use of state-of-the-art optimization engines becomes unafford-able.

Additionally, a splitting variable representation can be introduced by replacingthe variables with their siblings, such that θ g

k , pgi , xg

j , ygj , zg

j , sg and bg are replacedwith θωkt , pωi t , xωj t , yωj t , zωj t , sωt and bωt , respectively, for t = t (g), ω ∈ Ωg , g ∈ G;

"+dg

and "−dg

are replaced with "+dtω

and "−dtω

, respectively, for ω ∈ Ωg , g ∈ Gt ,t = τγ , γ ∈ Γ ; and δ f and βk are replaced with δωf and βωk , respectively, for ω ∈ Ω .Additionally, the nonanticipativity constraints (10.22)–(10.32) are appended to themodel, ∀ω ∈ Ωg , g ∈ Gt , t ∈ T −, where from now on ω + 1 is assumed to be anyscenario in set Ωg , but the last one, whenever ω is the last scenario in the set:

θωkt − θω+1kt = 0 ∀k ∈ K, (10.22)

pωi t − pω+1i t = 0 ∀i ∈ I, (10.23)

"+dtω − "+dt

ω+1 = 0 ∀d ∈ D, γ ∈ !, where t = τγ , (10.24)

"−dtω − "−dt

ω+1 = 0 ∀d ∈ D, γ ∈ !, where t = τγ , (10.25)

xωj t − xω+1j t = 0 ∀ j ∈ J , (10.26)

yωj t − yω+1j t = 0 ∀ j ∈ J , (10.27)

zωj t − zω+1j t = 0 ∀ j ∈ J , (10.28)

sωt − sω+1t = 0, (10.29)

bωt − bω+1t = 0, (10.30)

δωf − δω+1f = 0 ∀ f ∈ Fca, (10.31)

βωk − βω+1k = 0 ∀k ∈ Kca. (10.32)

By considering the nonanticipativity constraints (10.22)–(10.32) as the “dif-ficult” ones, Benders (1962) decomposition schemes can be used; see Andradeet al. (2006), Birge and Louveaux (1997), Escudero et al. (2007), and Laporte


and Louveaux (1993), among others. Alternatively, by dualizing these constraintsat each so-called supernode of a branch-and-bound phase, Lagrangian decompo-sition schemes can be used for solving the model given by the function (10.19)with (10.17) and the constraint system (10.2)–(10.14); see Caroe and Schultz(1999), Hemmecke and Schultz (2001), Klein Haneveld and van der Vlerk (1999,2001), Nürnberg and Römisch (2002), Römisch and Schultz (2001), Schultz (2003),Schultz et al. (2005), Schultz and Tiedemann (2004), and Takriti and Birge (2000),among others. However, some heuristics could be needed for obtaining solutions tothe problem, such that the nonanticipativity constraints are also satisfied. Indepen-dently of the constraints’ dualization, a Branch-and-Fix Coordination approach canbe used, see below.

10.4 A Branch-and-Fix Coordination (BFC) Approach

10.4.1 BFC Methodology

By slightly abusing the notation, consider the following splitting variable represen-tation of the DEM: maximize (10.19) with (10.17) subject to (10.2)–(10.14) and(10.22)–(10.32).

Z IP = max∑ω∈Ω

wω

(∑r∈R

eωr γωr +

∑t∈T

cωt vωt + ρνω

)

s.t.∑r∈R

eωr γωr +

∑t∈T

cωt vωt + M(1− νω) ≥ τ ∀ω ∈ Ω,

Aωt γω + B ′ωt vωt−1 + Bωt v

ωt = bωt ∀ω ∈ Ω, t ∈ T ,

γ ωr − γ ω+1r = 0 ∀ω ∈ Ω, r ∈ R,

vωht − vω+1ht = 0 ∀ω∈Ωg, g∈Gt, t ∈T −, h∈H,

vωht ≥ 0 ∀ω ∈ Ω, t ∈ T , h ∈ H,γ ωr , ν

ω ∈ {0, 1} ∀ω ∈ Ω, r ∈ R,(10.33)

where vωt denotes the vector of the continuous variables {vωht , ∀h ∈ H} for ω ∈ Ω,t ∈ T , representing the variables θωkt , pωi t , "

+dtω

, "−dtω

, xωj t , yωj t , zωj t , sωt and bωt inECP2; H is the set of indices of the variables in vωt ; γ ω denotes the vector of the0–1 variables {γ ωr , ∀r ∈ R} for ω ∈ Ω representing the variables δωf ∀ f ∈ F andβωk ∀k ∈ K; R is the set of indices of the variables in γ ω; cωt is the row vector ofthe objective function coefficients for the vector vωt and eωr is the objective function


coefficient for the variable γ ωr ; Aωt , B ′ωt , and Bωt are the constraint matrices for thevectors γ ω, vωt−1, and vωt , respectively; bωt is from now on the right-hand side (rhs)vector for ω ∈ Ω, t ∈ T ; and τ and ν are as defined above, all of them withconformable dimensions.

Notice that the relaxation of the nonanticipativity constraints

γ ωr − γ ω+1r = 0 ∀ω ∈ Ω, r ∈ R (10.34)

vωht − vω+1ht = 0 ∀ω ∈ Ωg, g ∈ Gt , t ∈ T −, h ∈ H (10.35)

in model (10.33) results in a set of |Ω| independent mixed 0–1 models, where(10.36) is the model for scenario ω ∈ Ω:

max∑r∈R

eωr γωr +

∑t∈T

cωt vωt + ρνω

s.t.∑r∈R

eωr γωr +

∑t∈T

cωt vωt + M(1− νω) ≥ τ

Aωt γω + B ′ωt vωt−1 + Bωt v

ωt = bωt ∀t ∈ T ,

vωht ≥ 0 ∀t ∈ T , h ∈ H,γ ωr , ν

ω ∈ {0, 1} ∀r ∈ R.

(10.36)

To improve computational efficiency, it is not necessary to relax all constraintsof the form (10.34) and (10.35). The number of constraints to relax in a givenmodel will depend on the dimensions of the scenario-related model (10.36) (i.e.,the parameters |T |, |I|, |J |, |D|, |Fca|, and |K| in model (10.2)–(10.14)). Let qdenote the number of scenario clusters that results after taking into account the sizeconsiderations mentioned above. Let Ωp denote the set of scenarios that belong tocluster p. The criterion for the choice of scenario clusters for the sets Ω1, . . . ,Ωq ,such that Ωp ∩ Ωp′ = ∅, p, p′ = 1, . . . , q : p �= p′ and Ω = ∪q

p=1Ωp couldalternatively be based on the smallest internal deviation of the uncertain parameters,the greatest deviation, etc. We favour the assignation of a higher priority to includein the same cluster those scenarios with greater number of common ancestor nodesin the associated scenario tree.

By slightly continuing to abuse the notation, the model to consider for scenariocluster p = 1, . . . , q can be expressed by the compact representation (10.37), whereω for n ∈ G|T | is such that ω is the unique scenario in Ωn and, on the other hand,G p = {g ∈ G : Ωg ⋂Ωp �= ∅}:

Z I Pp = max

∑n∈G|T |⋂Gp

wω

⎡⎣ ∑

g∈N n

(egγ p + cgvg)+ ρνω

⎤⎦


subject to

∑g∈N n

(egγ p + cgvg)+ M(1− νω) ≥ τ ∀n ∈ G|T |

⋂Gp,

Agγ p + B ′gvφ(g) + Bgvg = bg ∀g ∈ Gp,

vgh ≥ 0 ∀g ∈ Gp, h ∈ H,γ

pr , ν

ω ∈ {0, 1} ∀ω ∈ Ωp, r ∈ R,

(10.37)

where vg is the vector of the continuous variables for the constraint system related toperiod t (g) under scenario group g, γ p is the vector of the 0–1 variables, eg and cg

are the row vectors of the objective function coefficients for the γ p and vg vectors,respectively, Ag , B ′g , and Bg are the constraint matrices, bg is the rhs under scenariogroup g, and the other parameters are as above, all with conformable dimensions.Note that model (10.37) for q = 1 is the compact representation equivalent to thesplitting variable representation (10.33), and it is the model (10.36) for q = |Ω|.

The q models (10.37) are linked by the nonanticipativity constraints:

γ p − γ p′ = 0, (10.38)

vg p − vg p′ = 0, (10.39)

∀g ∈ Gp⋂

Gp′, p, p′ = 1, . . . , q : p �= p′, and γ p and γ p′ are the γ vectors andvg p

and vg p′are the v vectors for submodels (10.37) related to the clusters p and

p′, respectively.We could apply a branch-and-bound procedure to ensure the integrality condition

in models (10.37), for p = 1, . . . , q. However, instead of obtaining independentlythe optimal solution for each one, elsewhere (Alonso-Ayuso et al. 2003a, b) we pro-pose an approach, the so-called Branch-and-Fix Coordination (BFC). Our special-ization in this chapter is designed to coordinate the selection of the branching nodeand branching variable for each scenario cluster-related so-called Branch-and-Fix(BF) tree, such that the relaxed constraints (10.38) are satisfied when fixing theappropriate γ variables to either 1 or 0. The proposed approach also coordinatesand reinforces the scenario cluster-related BF node pruning, the variable fixing, andthe objective function bounding for the submodels attached to the nodes. See alsoAlonso-Ayuso et al. (2009), Escudero (2009), Escudero et al. (2007, 2009a, b, 2010)

Let Cp denote the BF tree associated with scenario cluster p and Ap the set ofactive nodes in Cp for p = 1, . . . , q. Any two nodes, say, a ∈ Ap and a′ ∈ Ap′ , aresaid to be twin nodes if either they are the root nodes or the paths from their rootnodes to each of them in their own BF trees Cp and Cp′ , respectively, have branched

on/fixed at the same values of some or all their (first-stage) variables γ pr and γ p′

r ,for p, p′ = 1, . . . , q : p �= p′. Note that in order to satisfy the nonanticipativityconstraints (10.38), the γ variables must be branched on/fixed at the same 0–1 valuefor the twin nodes. A Twin Node Family (TNF), say, Ln is a set of active nodes, such


that any one is a twin node to all the other members of the family, for n ∈ ϒ , whereϒ is the set of TNFs.

10.4.2 On Obtaining LP Optimal Solutions for TNF Integer Sets

Recall that a TNF integer set is each set of integer nodes, one for each BF tree,where the nonanticipativity constraints (10.38) for the 0–1 variables are satisfied.

Consider the following LP model, obtained after fixing in model (10.33) the γand ν variables to their 0–1 values, say, γr ∀r ∈ R and νω ∀ω ∈ Ω for a given TNFinteger set, respectively,

ZLPTNF =

∑ω∈Ω

wω(eωγ + ρνω)+max

∑ω∈Ω

∑t∈T

wωcωt vωt

subject to

∑t∈T

cωt vωt ≥ τ − eωγ − M(1− νω) ∀ω ∈ Ω,

B ′ωt vωt−1 + Bωt vωt = bωt − Aωt γ ∀ω ∈ Ω, t ∈ T ,

vωht − vω+1ht = 0 ∀ω ∈ Ωg, g ∈ Gt , t ∈ T −, h ∈ H,

vωht ≥ 0 ∀ω ∈ Ω, t ∈ T , h ∈ H.

(10.40)

Notice that the RP-related constraint for scenario ω is redundant for νω = 0,ω ∈ Ω .

10.4.3 On TNF Integer Set Bounding

Let R f denote the set of non-yet branched on/fixed at γ variables in a given TNFinteger set, such that R01 = R − R f . Let also Aωt = (A f ωt , A01ωt ) ∀ω ∈ Ω ,where A f ωt and A01ωt are the constraint submatrices related to the sets R f andR01, respectively, for t ∈ T , ω ∈ Ω . Similarly, let Ω = Ω f

⋃Ω01, such that Ω f

and Ω01 are the set of scenarios where the ν variables have not yet been branchedon/fixed at 0–1 and the set where these variables have already been branchedon/fixed at integer 0–1 values, respectively.

Model (10.41) enforces the satisfaction of the nonanticipativity constraints(10.34) and (10.35) but, on the other hand, it allows the γ and ν variables from thesets R f and Ω f, respectively, to take continuous values for the given TNF integerset.

ZLPCON =

∑ω∈Ω

∑r∈R01

wωerγ r + ρ∑ω∈Ω01

wωνω

+max∑ω∈Ω

∑r∈R f

wωerγ r +∑ω∈Ω

wωcωvω + ρ∑ω∈Ω f

wωνω


subject to

eω f γ ω f + cωvω + Mω f (1− νω) ≥τ − eω01γ ω01 − Mω01(1− νω) ∀ω ∈ Ω,A f ωt γ

ω f + B ′ωt vωt−1 + Bωt vωt = bωt − A01ωt γ

ω01 ∀ω ∈ Ω, t ∈ T ,γ ω f − γ ω f+1 = 0 ∀ω ∈ Ω,vωht − vω+1

ht = 0 ∀ω ∈ Ωg, g ∈ Gt , t ∈ T −, h∈H,vωht ≥ 0 ∀ω ∈ Ωg, g ∈ Gt , t ∈ T , h ∈ H,0 ≤ γ ω f ≤ 1 ∀ω ∈ Ω,0 ≤ νω ≤ 1, ∀ω ∈ Ω f,

(10.41)

where γ ω01 and γ ω f give the subvectors of the vector γ ω for the sets R01 andR f , respectively, similarly for the subvectors eω01 and eω f , and the subvectorsM f ω = M and M01ω = M for ω ∈ Ω f and ω ∈ Ω01, respectively, and, otherwise,they are zero.

10.4.4 BFC Algorithm

Consider the following algorithm for solving model (10.33) by using the scenariocluster-related submodels (10.37).

Step 1: Solve the LP relaxations of the q models (10.37). Each model is attachedto the root node in the trees Cp ∀p = 1, . . . , q. If the integrality constraints andthe constraints (10.38) and (10.39) are satisfied then stop, the optimal solutionto the original mixed 0–1 model (10.33) has been obtained.

Step 2: The following parameters are saved in a centralized device (CD): thevalues of the variables and the solution value (i.e., the optimal objective func-tion value) of the LP models attached to the nodes in Ap ∀p = 1, . . . , q, aswell as the appropriate information for branching on the 0–1 γ and ν variablesin the TNFs Ln ∀n ∈ Υ . A decision is made in CD for the selection of thebranching TNF and the branching variable. The decision is made available forthe execution of each scenario cluster-related BF phase.

Step 3: Optimization of the LP models attached to the newly created TNF afterbranching on the chosen 0–1 variable. Prune the TNF if its related LP modelis infeasible or its solution value (i.e., the weighted sum of the solution valuesof its node members) is not greater than the value of the incumbent solutionand, then, go to step 8.

Step 4: If the optimal solution that has been obtained in step 3 has 0–1 valuesfor all the γ and ν variables and satisfies constraints (10.38) (i.e. it is a TNFinteger set), either of the two following situations has happened:


(a) The nonanticipativity constraints (10.39) have been satisfied and, then, anew solution has been found for the original mixed 0–1 model (10.33). Therelated incumbent solution can be updated and, additionally, the updatingof the sets Ap at the trees Cp ∀p = 1, . . . , q can also be performed. In anycase, the TNF is pruned. Go to step 8.

(b) The nonanticipativity constraints (10.39) have not been satisfied. Go tostep 5.

Otherwise, go to step 2.Step 5: Optimizing the LP model (10.40) that results from fixing the γ and ν

variables in model (10.33) to the 0–1 values given in the TNF integer setwhose associated models (10.37) have been optimized in step 3. See Section10.4.2. If the LP model is feasible (and, thus, its optimal solution has beenobtained), then the updating of the incumbent solution and active node setscan be performed.

Step 6: Solve the LP model (10.41) that results from fixing the branched on/fixedat γ and ν variables in model (10.33) to their integer values and allowing theother 0–1 variables to take continuous values. See Section 10.4.3.

Step 7: If the solution value of model (10.41) for the given TNF is not greaterthan the incumbent solution value, or the LP models to be optimized in steps 5and 6 yield the same solution value, then the family is pruned.

Step 8: If the sets of active nodes are empty then stop, since the optimality of theincumbent solution has been proved, if any. Otherwise, go to step 2.

10.5 Conclusions

In this chapter we have introduced a multistage stochastic mixed 0–1 model withcomplete recourse for structuring portfolios of bilateral energy trading contracts incompetitive electricity markets. The treatment of the uncertainty plays a central rolein the approach that we have presented. Scenario treatment has proved to be a usefulmechanism to represent the uncertainty. The risk measure given by the compositefunction of the expected trading profit and the reaching probability functional isconsidered, instead of optimizing mean values alone. The modelling approach toselect candidate purchasing and selling contracts has been proved to be successfulfor maximizing the mean-risk function and, thus, balancing the net revenue fromthe contracts exploitation and the trading in the spot market under each scenariogroup along the time horizon. It is worth noting that the approach can be used forportfolio structuring by energy service providers in addition to energy generationcompanies. It can also be easily adapted to the existence of fixed costs associatedwith the contracts and to the consideration of contract cancellations. The modelconsiders the influence of the constraints imposed by the Kyoto protocol-based pol-lutant emission regulations. A specialization of the Branch-and-Fix Coordinationalgorithm has been proposed for optimizing multistage stochastic problems with0–1 variables in the first stage and continuous variables in the other stages.


Appendix

Different types of implementations can be considered within the algorithmic frame-work presented in Section 10.4. This appendix presents the version that we favour.

We use the depth-first strategy for selecting the branching TNF. The branchingcriterion followed for the energy selling contracts is to perform first a “branching onthe ones” and, afterwards, a “branching on the zeros”. However, the branching cri-terion for the energy purchasing contracts performs first a “branching on the zeros”and, afterwards, a “branching on the ones”.

Given the significance of the ν variables, they have the highest priority forbranching purposes. The priority is given according to the non-increasing scenarioweight criterion. It is branched first on the ones and then on the zeros.

Another topic of interest is the branching ordering for the energy trading con-tracts. We have considered a static ordering as follows: The highest priority is givento the selling contracts over the purchasing contracts. Within each category, thepriority is given according to the non-increasing expected income criterion for theselling contracts

∑g∈G

wgμgf ι

gf ∀ f ∈ Fca,

and the non-decreasing expected cost criterion for the purchasing contracts,

∑g∈G

wgλgk

1

2

(θk,t (g) + θk,t (g)

)∀k ∈ Kca.

Notice that a TNF can be pruned for any of the following reasons: (a) the LPrelaxation of the scenario cluster model (10.37) attached to any node member isinfeasible, (b) there is no guarantee that a better solution than the incumbent one canbe obtained from the best descendant TNF integer set (currently, it is based on theLP objective function value), (c) the LP model (10.40) attached to the TNF integerset is infeasible or its solution value is not better than the value of the incumbentsolution, in case that all γ and ν variables have already been branched on/fixed atfor the family, and (d) see below the case where some integer-valued γ or ν variablehas not yet been branched on/fixed at.

Once a TNF has been pruned, the same branching criterion allows us to performeither a “branching on the zeros (resp., “on the ones”) in the case that the TNFhas already been “branched on the ones” for the energy selling contracts (resp., “onthe zeros” for the energy purchasing contracts), or a backtracking to the previousbranched TNF if it has already been branched “on the ones” and “on the zeros”.

It is worth noting, see also Escudero et al. (2007), that if ZLPTNF (10.40) = ZLP

CON(10.41), then the related TNF can be pruned. Notice that the solution space definedby model (10.40) is included in the space defined by model (10.41) and, so, in thiscase, there is not a greater solution value than ZLP

TNF from the descendant nodes to


be obtained by branching on the non-yet branched on/fixed at γ and ν variables. Forthe same reason, the family is also pruned if ZLP

CON is not greater than the incumbentsolution value.

To gain computational efficiency, the optimization of model (10.41) should notbe performed for a small number of the γ and ν variables already branched on/fixedat in the given TNF. Let λ denote the minimum fraction of the branched on/fixed atvariables that is required for optimizing the model, where 0 ≤ λ ≤ 1.

For presenting the detailed BFC algorithm to solve model (10.33), let the follow-ing additional notation:

LPp, LP relaxation of the scenario cluster-related model (10.37) attached to anode member from the BF tree Cp in the given TNF, for p = 1, . . . , q.

ZLPp , solution value of the LP model LPp, for p = 1, . . . , q.

ZIP

, upper bound of the solution value of the original model (10.33) to beobtained from the best descendant TNF integer set for a given family. It is

computed as ZIP =∑

p=1,...,q ZLPp .

Z IP, lower bound of the solution value of the original model (10.33). It is theincumbent solution value, and it ends up with the value of the optimal solution.

By convention, ZLP∗ = −∞ for any infeasible LP∗ model.

Procedure

Step 0: Initialize Z IP := −∞.Assign σr := 1 where r represents a scenario or an energy selling contract,and σr := 0 if it is an energy purchasing contract, for r ∈ Ω⋃

R.Step 1: Solve the q independent LP models LPp, ∀p = 1, . . . , q, and com-

pute ZIP. If the γ and ν variables take integer values and the nonanticipativity

constraints (10.38) and (10.39) are satisfied, then the optimal solution to the

original model (10.33) has been found and, so, Z IP := ZIP

and stop.Step 2: If there is any γ or ν variable that takes continuous values or there is a γ

variable that takes different values for some of the q scenario clusters then goto step 3.Solve the LP model (10.40) to satisfy constraints (10.35) for the v variables inthe given TNF integer set (in this case, the set of root nodes in the BF trees).Note that the solution value is denoted by ZLP

TNF.

Update Z IP := ZLPTNF. If Z IP = Z

IPthen the optimal solution to the original

model has been found and stop.Step 3: Initialize r := 1 and go to step 5.Step 4: Reset r := r + 1.

If r = |Ω⋃R| + 1 then go to step 9.

Step 5: Branch γ pr := σr , ∀p = 1, . . . , q.

Step 6: Solve the linear models LPp, ∀p = 1, . . . , q and compute ZIP.


If ZIP ≤ Z IP then go to step 8.

If there is any γ or ν variable that takes continuous values or there is a γvariable that takes different values for some of the q scenario clusters then goto step 4.If all the v variables take the same value for all scenario clusters p = 1, . . . , q

then update Z IP := ZIP

and go to step 8.Solve the LP model (10.40) to satisfy constraints (10.35) for the v variables inthe given TNF integer set.Update Z IP := max{ZLP

TNF, Z IP}. If r = |Ω⋃R| then go to step 8.

If r < λ|Ω⋃R| then go to step 4.

Step 7: Solve the LP model (10.41), where the 0–1 continuous variables are thenon-yet been branched on/fixed at γ and ν variables in the solution of thecurrent TNF. Notice that the solution value is denoted by ZLP

CON. If ZLPTNF

< ZLPCON and ZLP

CON > Z IP then go to step 4.Step 8: Prune the branch.

If γ pr = σr , ∀p = 1, . . . , q then go to step 11.

Step 9: Reset r := r − 1.If r = 0 then stop, since the optimal solution Z IP has been found, if any.

Step 10: If γr = 1− σr , ∀p = 1, . . . , q then go to step 9.Step 11: Branch γr := 1− σr ∀p = 1, . . . , q.

Go to step 6.

Acknowledgements This research has been supported by the grants RM URJC-CM-2008-CET-3703 and RIESGOS CM from Comunidad de Madrid, and OPTIMOS2 MTM2009-14039-C06-03and PLANIN MTM2009-14087-C04-01 from Ministry of Science and Innovation, Spain.

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