Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market

European Journal of Operational Research 181 (2007) 1354–1369

www.elsevier.com/locate/ejor

Risk-averse profit-based optimal scheduling of a hydro-chainin the day-ahead electricity market

Javier Garcıa-Gonzalez a,*, Ernesto Parrilla a, Alicia Mateo b

a Instituto de Investigacion Tecnologica (IIT), E.T.S.I. (ICAI), Universidad Pontificia Comillas,

Santa Cruz de Marcenado 26, 28015 Madrid, Spainb Wind To Market, Plaza Pablo Ruiz Picasso, 1, pta. 24 28015 Madrid, Spain

Received 1 December 2004; accepted 1 November 2005Available online 12 May 2006

Abstract

This paper presents a profit-based model for short-term hydro scheduling adapted to pool-based electricity markets.The objective is to determine a feasible and realistic operation of a set of coupled hydro units belonging to a small or med-ium-size hydroelectric company in order to build the generation bids for the next 24 hourly periods. The company isassumed to be price-taker, and therefore, market prices are considered exogenous variables and modeled via scenarios gen-erated by an Input/Output Hidden Markov Model (IOHMM). In order to be protected against low prices scenarios, twodifferent risk-aversion criteria are introduced in the model: a minimum profit constraint and a minimum conditional Value-at-Risk (CVaR) requirement, which can be formulated linearly in the context of the optimization problem. In order toensure a feasible operation, the model takes into account a very detailed representation of the generating units, whichincludes forbidden discharge intervals, spatial–temporal constraints among cascaded reservoirs, etc. The non-linear rela-tionship among the electrical power, the net-head and the turbine water discharge is treated by means of an under-relaxediterative procedure where net-heads are successively update until convergence is reached. During each algorithm stage, pre-vious iterations’ information is used to build the input–output curves. This way, the hydro scheduling problem can be for-mulated as a MILP optimization problem, where unit-commitment decisions are modeled by means of binary variables.The model has been successfully applied to a real-size example case, which is also presented in this paper.� 2006 Elsevier B.V. All rights reserved.

Keywords: Hydroelectric power generation; Mixed integer linear programming; Short-term hydro scheduling; Day-ahead energy markets;Profit maximization; Risk-aversion; CVaR

1. Introduction

This paper presents an optimization model tohelp a hydro-generation company to schedule its

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2005.11.047

* Corresponding author.E-mail addresses: [email protected] (J. Garcıa-

Gonzalez), [email protected] (E. Parrilla), [email protected] (A. Mateo).

hydroelectric units in the very short-term (up to 24hours) under a competitive environment. The modelis formulated as a stochastic profit-based hydroscheduling problem and the pool is supposed to beorganized as day-ahead market.

In a day-ahead market, the market operator isthe coordinating authority who receives the offersto buy and to sell electricity and performs the auc-tion model to obtain the hourly marginal prices

.

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J. Garcıa-Gonzalez et al. / European Journal of Operational Research 181 (2007) 1354–1369 1355

(used to remunerate all the generation), and the setof accepted and rejected bids. Generation compa-nies are the power producers who try to sell electric-ity in the market. Energy service companiesrepresent the load requirement of electricity custom-ers and therefore they submit offers to buy electric-ity. Each hour, marginal price is found as theintersection between the hourly aggregated supplyand demand functions (see Fig. 1).

Some international experiences of this kind ofmarkets can be found in Spain, Scandinavia (Nord-pool), Netherlands, PJM, Ontario, New England,Australia, etc.

In this framework, a pure hydro-generation com-pany has to:

1. Elaborate a daily operation plan of its hydroresources in order to assess the available energythat could be offered in the day-ahead market,and build the hourly bids to be submitted.

2. Throughout the operating day, operate its hydro-units trying to fulfill the market clearing sche-dule, and modify the program in the intra-dayenergy markets if necessary as real-time opera-tion is getting closer.

This paper is focused on the first step, and there-fore, the real-time reservoir management related tothe second one, is out of the scope of this work.

In electricity markets it is possible to discriminatetwo kinds of selling agents depending on theircapacity to alter market prices: oligopolistic agentsand price-taker agents. In both cases, as in anyother business, the utilities’ criterion should be themaximization of their expected profit, defined asthe difference between market revenues and opera-tion costs. However, the difference between themis that oligopolistic agents cannot consider market

clearing price

π

p

[/M

Wh]

[MWh]

demand function

supply function

Fig. 1. Market clearing.

prices as exogenous variables, as their own decisionscan affect market results. Therefore, an oligopolisticagent has to estimate its competitors’ behavior andto consider explicitly its influence on clearing prices.This can be achieved by the estimation of the hourlyresidual demand functions (Garcıa-Gonzalez andBarquın, 2000; Baıllo et al., 2004).

In case of a price-taker, its schedule cannot mod-ify the hourly market clearing prices. Thus, marketuncertainty is limited to the expected prices for thenext 24 hours (Conejo et al., 2002). It is importantto note that during the first stages of the deregu-lation process, many electricity markets can beconsidered oligopolies, i.e. with a relatively smallnumber of suppliers. However, as the marketevolves, the demand grows and the system getsmature, new entrants might decide to participateas selling agents, normally as price-takers. There-fore, there exists a real need for tools adapted tothese new entrants.

The model presented here is generic and compre-hensive to any type of hydroelectric system. How-ever, it is specially indicated for systems formed bymany but small reservoirs, in which the daily policyof water releases can influence notably on the hourlynet heads1 and therefore, it can affect the generationof the whole river basin. In this context, this paperaims to help the generating utility to build thehourly bids, introducing the market uncertaintyvia prices scenarios.

The main features of the proposed model are thefollowing:

• A very detailed representation of hydro equip-ment is taken into account: start-up and shut-down decisions, initial and final reservoir levelconditions, forbidden discharge intervals, waterrights, detraction flows for water consumption,spillage management, space-temporal relation-ships among cascaded reservoirs, etc. This way,the power profile obtained with the model is con-sistent with the water releases at each dam andwith the reservoir levels along the temporalhorizon.

• The model is stochastic and considers simulta-neously a predetermined number of price sce-narios weighted with their correspondingprobabilities.

1 The net head is defined as the difference between the forebayelevation, the tailrace elevation and the penstock head loss.

1356 J. Garcıa-Gonzalez et al. / European Journal of Operational Research 181 (2007) 1354–1369

• Two risk-aversion criteria are introduced: mini-mum profit and minimum Conditional Value-at-Risk (CVaR) constraints.

Regarding the first point, a special emphasis hasbeen done in the modeling of the head-dependantunits. The output power of a hydro unit dependsnon-linearly on the product of the turbine dischargeand the net hydraulic head of the corresponding res-ervoir (Wood and Wollenberg, 1984). Depending onthe particular structure of the hydro sub-system (forexample, in large reservoirs where the net-headremains almost constant throughout the day) thisdependence might be neglected in the short-term.However, in many hydro systems, it is necessaryto take into account this dependency in order toobtain realistic results. For instance, Conejo et al.(2002) propose to simplify the problem by consider-ing a discrete family of curves (up to 3 or 5)corresponding to some prefixed net heads. InGarcıa-Gonzalez and Castro (2001), the truthfulinput–output surface is approximated by meshingand triangulation using Mixed-Integer LinearProgramming (MILP). However, this approach isonly suitable for small systems due to size limita-tions, specially, if additional binary variables areintroduced to model other particularities. Anotherapproach that can be found in the literature is toimplement an iterative procedure where each itera-tion considers a fixed head successively updated(Pereira and Pinto, 1983). However, it can lead tooscillating solutions.

In Feltenmark and Lindberg (1997), theauthors study a problem similar to the core of thispaper, taking into account a very detailed repre-sentation of the hydro units (head dependencyand varying efficiency of the turbine). The result-ing non-linear problem is solved by a specializednetwork flow algorithm, with very satisfactoryresults. In fact, the network flow approach wouldbe the natural way to model hydro systems(Nabona et al., 1992). However, its main draw-back is that it cannot consider binary variables.Real operation of a hydro chain in the short-termrequires managing binary variables in order tomodel the discrete unit-commitment decisions.For instance, when the unit is ‘‘on’’, the water dis-charge has to be higher than a minimum outflowin order to avoid cavitation damages and largevibrations in the turbine. Other forbidden dis-charge intervals could appear when the plant isformed by a set of hydro units. Besides this, Nils-

son and Sjelvgren (1997) show how the start-upcosts related to the loss of water, wear and tearof the equipment, might not be negligible. To takeinto account all these issues, it is necessary tointroduce in the optimization model binary vari-ables and to apply an alternative optimizationtechnique.

In this paper, the non-linear net-head effect istreated by an iterative procedure based on the clas-sic under-relaxed non-linear programming tech-nique (Ortega and Rheinboldt, 1970). The mainadvantages of this new approach are the followingones:

• The under-relaxed updating strategy avoids theoscillating solutions that can arise using otherupdating strategies.

• The company does not have to decide a priori thecandidate input–output curves because the modelselects iteratively the most accurate ones.

• Binary variables related to unit-commitmentdecisions, forbidden discharge intervals, start-upand shutdown decisions can be taken intoaccount.

To implement the under-relaxed algorithm, eachiteration requires solving a simplified problem. Sev-eral solution techniques can be found in the litera-ture such as dynamic programming (Arce et al.,2002), Lagrangian Relaxation (Ni et al., 1999), Lin-ear Programming (Medina et al., 1994), Mixed Inte-ger Linear Programming (MILP) (Chang et al.,2001), optimal feedback control, artificial neuralnetworks, etc. In this paper, the problem solvedeach iteration has been stated as a MILP problem(solved with the standard Branch & Bound), becauseof its modeling flexibility, modularity and quality ofthe solutions obtained.

This paper is laid out as follows. First of all, Sec-tion 2 presents the proposed model overview. Then,the mathematical formulation of the optimizationproblem solved each iteration is presented in Section3. After that, a real application is presented in Sec-tion 4. Finally, concluding remarks are given in Sec-tion 5.

2. Model overview

The problem of planning the operation of a set ofcoupled reservoirs is normally decomposed in sev-eral sub-problems, attending to different temporalscopes. Fig. 2 shows a typical hierarchy of sub-prob-


lems, where the model presented in this paper corre-sponds to the last level, i.e. the daily problem.

In the mid- and long-term planning (for instance,1 year), uncertainty plays a crucial role. Besides thestochastic nature of water inflows (Escudero et al.,1996), market participants in deregulated systemsface the price uncertainty. In Fleten et al. (1997) alinear stochastic model for the portfolio manage-ment of a Scandinavian power supplier with hydrau-lic power plants under uncertain inflow and marketprice conditions is introduced. Fleten et al. (2002)represent price uncertainty by scenarios and maxi-mize risk-adjusted profit within an asset-liabilityframework. Other stochastic programming modelsin energy can be seen in Wallace and Fleten (2003).

The link between the mid-term problem andthe weekly problem can be either the water valuecurves, or directly the reservoirs’ levels that shouldbe reached at the end of the week, as in (Nurn-berg and Romisch, 2002). The weekly problem isa multi-stage problem involving mixed-integerdecisions and subject to uncertainty (inflows andmarket price). One approach for solving suchproblem is the stochastic Lagrangian relaxationof coupling constraints (Nowak and Romisch,2000). In the case of this paper, the first-stagevariables should be the reservoirs’ levels at theend of the first day.

Finally, in the short-term (up to 24 hours), theobjective is to find the optimal 24 supply functions

mid-long termproblem

dailyproblem

weekly problem

1week 2week ...

Incremental water valuesReservoir volumes at theend of the week

Reservoir volumes , at the end of the day

24 supply functions

fiv i∀

1day 2day 3day 4day 5day 6day 7day

...

.........

1day 2day 3day 4day 5day 6day 7day

...

.........

Fig. 2. Model hierarchy.

to be submitted simultaneously to the market oper-ator. In hydro systems formed by many but smalland connected reservoirs, small variation in thehourly planned schedule can make the probleminfeasible. Thus, in order to avoid unnecessarycosts in the intra-day energy markets, or to avoidthe penalization payments when real productiondiffers from the final program, a possible strategyis to build the hourly bids (quantity–price pairs)setting a very low price to the expected generationof each hour. However, as the final energy pro-gram depends as well on the other competitors’bids, the feasibility of the cleared schedule cannotbe absolutely guaranteed. A more sophisticatedbidding strategy for just one reservoir can be foundin Pritchard et al. (2004), although net head effectis not taken into account.

In the daily temporal scope, water inflows canbe predicted with a fairly good precision in manysystems. For that reason, in this paper uncertaintyis limited to market price as the price volatilitythroughout the day can affect notably the incomesof the generating utility. The boundary conditionsfixed by the weekly problem are the reservoirlevels that must be reached at the end of theday, limiting the available energy that can be soldin the market. As the utility is assumed to beprice-taker, the optimal short-term problemcan be divided into two separate phases (seeFig. 3):

electricity prices model

profit-based hydro scheduling problem(PBS)

simplified profit-basedhydro scheduling problem

(SPBS)

NO:update net-heads

initial net-heads

24-hourprices scenarios

convergence?

YES: final optimal schedule

Fig. 3. Daily problem overview.


1. The forecasting of the day-ahead market prices.2. The optimization of its expected profit subject to

the forecasted prices, to a feasible schedule of thegenerating units, and to some risk-aversion crite-ria. This problem will be denoted hereafter as theProfit-Based hydro Scheduling problem (PBS).

Regarding the first point, Section 2.1 presents adiscussion about the most appropriate methodsavailable in the literature to forecast these prices.The PBS description, and the resolution methodproposed in this paper is presented in Section 2.2.

2.1. Electricity prices models for the generation of

price scenarios for the day-ahead spot market

Several models for electricity prices have beendeveloped in the literature for the short-term (Bunn,2004). In Mateo et al. (2005) electricity prices mod-els based on time series are reviewed and a novelclassification is proposed. First of all, authors dis-criminate between stationary and non-stationarymodels. Although electricity prices exhibit a well-known non-stationary component due to its multi-ple seasonalities (related to daily, weekly andmonthly periodicity), several authors have proposeddifferent stationary models for electricity prices ser-ies. For example, Dynamic Regression Models andLinear Transfer Function Models (Nogales et al.,2002) or ARIMA models (Contreras et al., 2003).In order to apply this kind of models the non-sta-tionarity component has to be firstly removed byapplying different techniques provided by classicalstatistics. However, as it is pointed out by Mateoet al., this could not be enough in electricity marketsas spot prices reflect a switching nature related todiscrete changes in participants’ strategies.

A more suitable alternative for modeling electric-ity prices is to apply non-stationary models. Econo-metric and financial world have given rise the mostof the non-stationary electricity prices models:Mean-Reversion models (Knittel and Roberts,2001), GARCH models (Batlle and Barquın,2002), two factor models (Schwartz and Smith,2000), or jump diffusion models (Deng, 2000).Besides this, different authors have proved that neu-ral networks can properly be used for modeling theevolution of the electricity prices series (Szkutaet al., 1999). However the switching nature of spotprices requires to apply switching models. In thisgroup, the most important econometric model isproposed in Hamilton (1990) and Fabra and Toro

(2002), where price series is modeled through a Mar-kovian switching process among autoregressiveregimes, adapting to occasional discrete shifts inthe level, variance and autoregressive dynamics ofthe series.

Finally in Mateo et al. (2005) the authors intro-duce a novel approach for modeling and forecastingelectricity prices by the Input/Output Hidden Mar-kov Model (IOHMM) originally proposed in Ben-gio and Frasconi (1996). The switching nature ofthe electricity market, related to discrete changesin competitors’ strategies, can be represented by aset of dynamic models sequenced together by aMarkov chain. In the IOHMM different marketstates are firstly identified and characterized by theirmore relevant explanatory variables. Moreover, aconditional probability transition matrix governsthe probabilities of remaining in the same state, orchanging to another. Finally, at each time step k

the IOHMM model provides the probability densityfunction (f) of the price (p) conditioned to a set ofinput variables (u), i.e. f(pkjuk). This feature is usedin this paper to generate the price scenarios. Theprocess can be briefly summarized as follows. Thefirst step is to select the number of scenarios, N.Then, in the first period, the IOHMM providesthe probability function f(p1ju1) that can be usedto randomly draw N values of p1. Each one of thesevalues will be the initial point of the resulting N

price trajectories. This way, the probability of eachgenerated scenario will be 1/N. As the price in theprevious hour is one of the input variables of theIOHMM model, the randomly drawn values of p1,together with the rest of input variables, can be usedto build the probability density functions of the nextperiod. This is done for each one of the N trajecto-ries, fn(p2ju2). At this point, these probably func-tions can be used to randomly draw one value ofp2 for each trajectory. The algorithm continues for-ward until the last hour is reached. A more detaileddescription of this process can be seen in Mateo(2005).

2.2. The PBS problem

In order to compute the global production of thehydro chain in the mentioned PBS, it is necessary toexpress accurately the hydroelectric generationfunctions of its hydro units. This can be attainedby introducing in the PBS the following equation:

pik ¼ Uiðqik; hikÞ 8i; 8k; ð1Þ


where Ui(Æ) is the non-linear function that links thepower produced by unit i in hour k (pik) with theturbine discharge (qik) and with the net-head ofthe associated reservoir (hik).

When hik is constant, pik = Ui(qik) and only onecurve is needed to characterize the generating unit.However, when the variation in the storage pondis a fairly large percentage of the overall net hydrau-lic head, it cannot be considered constant. There-fore, each hydro unit is characterized by its ownfunction Ui(Æ), which besides the efficiency curve ofthe turbine, the gravity constant and the water den-sity, it might allow to model forbidden operatingareas, multi-group effects, etc. The direct consider-ation of the whole real input–output curves inPBS could require excessive computational effortsand long execution times (Garcıa-Gonzalez andCastro, 2001) (Fig. 4).

Suppose the case where the net head values foreach reservoir along the temporal horizon wereknown, ht

ik. In that case, we could build the follow-ing time-varying functions where the net headdependence has been removed by substituting inUi(Æ) those known net head values:

/tikðqikÞ ¼ Uiðqik; h

tikÞ 8i; 8k: ð2Þ

The hydro scheduling problem can be rewrittenreplacing the equations in (1) by the following one:

pik ¼ /tikðqikÞ 8i; 8k: ð3Þ

This way, the original problem is simplified, and thenew one will be denoted as SPBS (Simplified Profit-

Based hydro Scheduling problem). The SPBS re-

net hydraulic headh [m]

elec

tric

al p

ower

p[M

W]

( , )p q h=

net hydraulic headh [m]

elec

tric

al p

ower

p[M

W]

( , )p q h= Φ

water dischargeq [m3/s]q [m /s]

Fig. 4. Input–output characteristic surface.

quires less computational effort to be solved thanthe PBS, as all the units are treated as non-headdependent. However, their solutions will remain dif-ferent unless the functions /t

ikð�Þ are the correctones. For that reason, assume that the optimal solu-tion of the PBS were known and denoted with anasterisk. In this case, the optimal power generationp�ik must satisfy

p�ik ¼ Uiðq�ik; h�ikÞ 8i; 8k: ð4Þ

Thus, functions /tikð�Þ leading to an equivalence be-

tween PBS and SPBS are the following ones:

/�ikðqikÞ ¼ Uiðqik; h�ikÞ 8i; 8k: ð5Þ

As the net heads are variables of the problem, univ-ocally related to the reservoir levels at each time per-iod, it is impossible to determine a priori theoptimal evolution of net head values ðh�ikÞ, so it isnecessary to introduce an iterative procedure toreach the optimal value ðh�ikÞ. In Appendix, there isa description of the under-relaxed iterative proce-dure implemented to solve this problem.

3. SPBS mathematical formulation

The SPBS problem is formulated as a MILP opti-mization problem, which is solved at each iterationof the mentioned under-relaxed algorithm.

The following subsections include the nomencla-ture, the mathematical formulation of the objectivefunction and the considered constraints.

3.1. Nomenclature

3.1.1. Sets and indexes

N, n set and index of price scenariosK, k set and index of hourly periodsI, i set and index of hydro unitsXi subset of units direct upstream of unit I

3.1.2. Parameters

pkn market price in hour k in scenario n (€/MW h)

wik lateral inflows of unit i in k (Hm3/h)�hik , hik water rights of unit i in k (Hm3/h)�pik, pik output power limits of unit i in k (MW)�qik, qik turbine discharge limits of unit i in k (Hm3/h)

qmxefik outflow at maximum efficiency point of unit

i in k (Hm3/h)


pmxefik output at maximum efficiency point of unit

i in k (MW)�vi; vi reservoir capacity limits of unit i (Hm3)ci start-up cost of unit i (€)a under-relaxation parameterqn probability of price scenario n (p.u)

3.1.3. Variables

Bn profit in scenario n (€)pik output power of unit i in k (MW)hik net hydraulic head of unit i in k (MW)mik reservoir level at the end of the interval

(Hm3)qik turbine discharge of unit i in k (Hm3/h)qa

ik discharge over the minimum outflow ofunit i in k (Hm3/h)

qbik discharge over the maximum efficiency flow

of unit i in k (Hm3/h)sik spillage (Hm3/h)yik start-up decision of unit i in k {0,1}uik commitment of unit i in k {0,1}zik shutdown decision of unit i in k {0,1}

3.2. Objective function

The main objective is to maximize the expectedprofit of the hydro chain in the day-ahead market,considering possible start-up costs. The objectivefunction takes into account all the price scenariosat once, weighed by their occurrence probability

Maximize:XN

n¼1qn � Bn

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{expected profit

¼XN

n¼1

XK

k¼1qn � pkn �

XI

i¼1ðpikÞ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{expected income

�XI

i¼1

XK

k¼1ci � yikð Þ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{start-up costs

; ð6Þ

where qn is the probability associated to the scenarion; pkn is the price is the scenario n at the hour k; pik

is the hourly electricity production of the hydro uniti; and ci is the start-up cost unit i.

3.3. Risk-aversion constraints

Several techniques can be found in the related lit-erature to introduce risk management into decisionmodels. The simplest criterion is to impose a mini-

mum profit constraint. This can lead to unfeasiblesolutions, and therefore, Fleten et al. (2002) proposeto minimise the negative deviations of each sce-nario’s profit respect to a pre-fixed target.

In this paper, the implemented risk-aversion cri-teria are the minimum profit constraint and the min-imum Conditional Value-at-Risk. Hereafter, theyare formulated.

3.3.1. Minimum profit

The risk-aversion of the hydro utility can bemodeled by the following constraint, where a mini-mum profit Bmin is required for every scenario n:

XK

k¼1pkn �

XI

i¼1pik

h i�XI

i¼1

XK

k¼1ci � yik

h izfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Bn

P Bmin; 8n:

ð7Þ

The minimum profit requirement is a very simpleconstraint, which allows the generator to be pro-tected against the worst scenario. However, this cri-terion might be too conservative and inappropriateto model the risk-aversion of the hydro utility whenfaces the problem of building the bids for the day-ahead market. For that reason, another risk mea-sure is proposed in the next section.

3.3.2. Minimum Conditional Value-at-Risk (CVaR)

Conditional Value-at-Risk (CVaR) is anotherapproach which has been applied successfully tofind the optimal hedging strategies in the electricitymarkets, as in Unger (2002). Another application ofthe CVaR can be found in the paper (Cabero et al.,2005), which presents an original methodology toaddress the integrated risk management problemof a hydrothermal generation company.

In this paper, the CVaR technique is introducedin the hydro scheduling model for two main rea-sons. The first one is the protection that this crite-rion provides against low probable but very severescenarios. The second one is that it can be formu-lated as linear programming for the discrete sce-nario setting.

Value-at-Risk (VaR) estimates the likelihood thata given portfolio’s losses will exceed a certainamount. In this paper, instead of losses, the VaR

is expressed in terms of profits (B). Let f be theValue-at-Risk at a confidence level of d, i.e., theminimum profit that will be reached with a proba-bility d. Given the percentile d, CVaRd representsthe mean of the profit in the worst d100% cases(see Fig. 5). Mathematically, it can be defined as

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

( )f B

[ ]BCVaRδ

nB

nB−

ζ

Fig. 6. VaR and CVaR for a d% confidence interval.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

( )f B

[ ]BζVaRδ =CVaRδ

area 1–δ=

Fig. 5. VaR and CVaR with a confidence level of d %.


CVaRd Bð Þ ¼ E BjB < fð Þ: ð8ÞThe discrete formulation is the following one:

CVaRd Bð Þ ¼P

n2N Bn<fj qn � BnPn2N Bn<fj qn

¼P

n2N Bn<fj qn � Bn

1� d:

ð9Þ

Defining B�n as an auxiliary variable which is equalto zero when the profit is higher than the Value-at-Risk f, and it is equal to the difference betweenthem in the opposite case, the minimum CVaRmin

can be introduced in the model by the followingset of linear constraints (10)–(12):

f�P

n2Nqn � B�n1� d

P CVaRmin; ð10Þ

B�n P f� Bn; 8n; ð11ÞB�n P 0; 8n: ð12Þ

Assuming that the expected profit is maximizedin the objective function, previous constraints canbe introduced in the schedule model in order toensure a minimum Conditional Value-at-Risk, andin the optimum, when constraint (10) is active, thevariable f contains the Value-at-Risk at the proba-bility level d indicated as input data. Fig. 6 showsa graphical representation of this formulation.

3.4. Technical constraints

3.4.1. Reservoir water balance

For each reservoir i at each time period k, the fol-lowing constraint establishes the water balanceequation. The reservoir level at the end of a periodmik is the reservoir level at the end of the previousperiod mi(k�1) minus the released volume (turbinedischarged qik or spilled sik) plus the volume coming

from direct upstream reservoirs (noted as the set Xi)and natural inflows wik

mik ¼ miðk�1Þ þwik � ðqik þ sikÞ þPj2Xi

ðqjk þ sjkÞ 8i; 8k;

ð13Þwhere for k = 1 it is assumed that miðk�1Þ ¼ m0

i (initiallevel).

3.4.2. Final reservoir level

The final reservoir level (mfi Þ, fixed by the weekly

problems, is stated as follows:

mik ¼ mfi 8i; k ¼ K ð14Þ

3.4.3. Reservoir capacity limits

The reservoirs management has to take intoaccount their upper �mik and lower mik dynamic capac-ity limits

mik 6 mik 6 �mik 8i; 8k ð15Þmik 6 mik 6 �mik 8i; 8k ð16Þ

3.4.4. Water rights

It is usual in hydro systems the existence of waterrights derived from ecological flows, irrigationrequirements, etc. This water discharged limits(hik; �hikÞ are applied to both the turbine dischargesand the spillages

hik 6 qik þ sik 6�hik 8i; 8k ð17Þ

3.4.5. Input–output curve modeling

In order to model the truthful input–outputcurve for a given net head, a piece-wise linearapproximation has been implemented taking intoaccount the minimum, the maximum, and the maxi-mum efficiency discharge points (see Fig. 7). Eqs.

aq

p

p

q

p

bq

Maximum efficiency discharge

mxefqq q

Linear approximation

True curve

Forbidden interval

Fig. 7. Input–output curve approximation for a given net-head.

r4

r2

u1

u4

u3

r6

r1

r5

r3

u5

u2

reservoir

lateral inflow

hydropower unit

run-of river unit


(18)–(20) state that when the unit is off (uik = 0), theoutflow is zero and when it is on, the outflow mustbe within the interval [qik; �qik]. Therefore, the inter-val [0,q

ik] is a forbidden discharge interval because

releases below qik

can damage the turbine. Note thatthese limits depend on the current net-head

qik ¼ uik � qik þ qaik þ qb

ik 8i; 8k; ð18Þ

qaik 6 uik � qmxef

ik � qik

� �8i; 8k; ð19Þ

qbik 6 uik � �qik � qmxef

ik

� �8i; 8k; ð20Þ

where qaik is the turbine discharged over the mini-

mum one (qik

) and qbik is the turbine discharged over

the maximum efficiency point qmxefik .

After defining the water discharge physic limits, itis possible to express the power generation functionpik. In this case this function has been modeled by apiece-wise linear approximation presented in Eq.(21)

pik ¼ uik � pik þ qaik �

ðpmxefik � pikÞðqmxef

ik � qikÞ

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{slope of the 1st segment

þ qbik

� ð�pik � pmxefik Þ

ð�qik � qmxefik Þ

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{slope of the 2nd segment

8i; 8k: ð21Þ

u7

r7

u6

Fig. 8. Hydro chain topology.

3.4.6. Logic constraint

This constraint ensures the coherence among thebinary variables related to the discrete commitment(uik), start up (yik) and shutdown (zik) decisions. For

instance, it does not allow to start-up a unit which isalready on

yik þ uik�1 � uik � zik ¼ 0 8i; 8k: ð22ÞThis formulation allows including additional logicconstraints, such us minimum up and down con-straints, etc.

4. Study case

The presented model has been implemented inGAMS, using the commercial solver CPLEX 7.1to solve the MILP problems of each iteration. Thissection presents its application to a fictitious butreal size hydro chain, for a temporal scope of 24hourly periods. The convergence criterion was fixedto e = 0.5% (see Appendix).

4.1. Input data

This hydro chain consists of ten cascaded reser-voirs and seven hydraulic generating units. Fig. 8shows the complete topology. The characteristicparameters of each unit are fictitious, although theyare in the range of realistic values. Table 1 summa-rizes the most relevant characteristic parameters ofeach unit.

The hydropower units start-up costs have beenestimated as a function of its nominal output power,

Table 1Characteristics parameters

�v m m0 mf �p �q w c

(Hm3) (MW) (m3/s) (€)

u1 – – – – 12 10 5 30u2 85 20 45 45 56 45 1 125u3 – – – – 30 38 – 75u4 – – – – 24 36 – 67.5u5 213 80 123.5 122 58 110 – 145u6 20 13 16.4 16.7 120 124 – 350u7 – – – – 31 120 - 85r1 40 15 25 25 – 9 – –r2 15 3 12 12 – 11 – –r3 23 6 19.4 19.1 – 31 – –r4 0.49 0.11 0.3 0.3 – 35 1 –r5 1.5 0.6 1.38 1.3 – 48 – –r6 1.3 0.1 0.72 0.67 – 56.5 1 –r7 12.55 7.1 10 9.79 – 120 12 –


ci ¼ p � 2:5 €=MW (Nilsson and Sjelvgren, 1997).Besides these technical characteristics, unit u2 hasto fulfill a maximum outflow of �hu2

¼ 38 m3=s dur-ing the whole day.

Regarding the characteristic input–output sur-faces used in the example, Fig. 9 shows the surfacefor unit u2. It is important to highlight that in theprevious table, minimum outflow limits did notappear, as this parameter depends on the net head.Therefore, depending on the net-head active eachiteration, the minimum outflow has to be obtainedfrom this surface. The same occurs for the maxi-mum efficiency discharge point.

The number of prices scenarios considered in theoptimization problem is N = 250. This number hasbeen selected arbitrarily as a compromise betweenthe uncertainty modeling and the problem size.

2030

4050

6070

80

15

20

25

30

35

40

450

10

20

30

40

50

60

2u

[MW]p

3 [m /s]q 3 [Hm ]v20

3040

5060

7080

15

20

25

30

35

40

450

10

20

30

40

50

60

3 [m /s]q 3 ]

Fig. 9. Input–output surface of unit u2.

The price scenarios have been generated applyingthe methodology proposed in Mateo et al. (2005)and Mateo (2005), where an Input/Output HiddenMarkov Model (IOHMM) was trained to fit theSpanish daily-market hourly prices. The historicdata used to train the model was the hourly pricesof a whole year. The distinction among workingdays and weekend days was introduced by theexplanatory variables such as the system demand.Other explanatory variables were the aggregatedproduction of each technology and the price in theprevious hour. The resulting price scenarios areshown in Fig. 10.

4.2. Results without risk-aversion constraints

Fig. 11 and Table 2 present the resulting hydroscheduling without risk-aversion constraints. Theexpected profit of such schedule is Bwithout ¼P

nðqn � BnÞ=n ¼ 74326:32€. The production of u1

is a good example of a run-of-river unit, which isforced to produce its own natural inflow duringthe whole day. The other units distribute their pro-duction during the higher expected prices hours.

Another interesting result is the reservoir man-agement. Fig. 12 shows the evolution of thevolume of water in the two smallest reservoirs(r4 and r6). Note that water levels decrease inthe hours corresponding to high prices as duringthese hours, water is released through the hydrochain.

In order to illustrate the effect of the under-relaxed factor value, the case without risk-aversion

Fig. 10. Prices scenarios considered in the example case.

Fig. 11. Hydro units’ generation without risk-aversion constraints.

Table 2Detailed hydro schedule without risk-aversion constraints (MW)

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19 h20 h21 h22 h23 h24

u1 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4u2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.9 41.9 36.7 0.0 0.0 0.0 0.0u3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.4 25.1 25.1 25.1 0.0 0.0 0.0 0.0u4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18.5 22.7 22.7 22.7 0.0 0.0 0.0 0.0u5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 15.7 51.0 39.2 28.4 12.1 0.0 0.0 0.0u6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 47.3 47.2 47.2 47.2 0.0 0.0 0.0 0.0 47.2 109.0 83.1 47.3 0.0 0.0 0.0 0.0u7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 30.5 30.5 30.5 7.5 7.5 7.5 7.5 7.5 30.5 30.5 30.5 7.5 7.5 20.8 0.0


constraints was solved for different values ofa. Table 3 resumes the evolution of the errorbetween consecutive iterations. Notice that fora = 1, the problem does not converge. The bestperformance result took place for a = 0.7, wherethe convergence was reached after 6 iterations(Fig. 13).

4.3. Results with risk-aversion constraints

In this example case, the minimum profit con-straint has been set to Bmin = 69,500€. When thisminimum profit constraint is introduced in themodel, the obtained schedule varies slightly (seeTable 4), and the obtained expected profit is

Reservoir 4 volume variation

80%

90%

100%

110%

120%

130%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time [h]va

riat

ion

[%]

Reservoir 6 volume variation

80%

90%

100%

110%

120%

130%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time [h]

vari

atio

n [%

]

Fig. 12. Evolution of the storage level in reservoirs 4 and 6 (% respect to the initial value).

Table 3Effect of a on the convergence

a = 1 a = 0.8 a = 0.7 a = 0.5

iter1 4.03% 4.03% 4.50% 4.03%iter2 1.16% 1.72% 1.10% 1.90%iter3 1.70% 2.31% 2.25% 1.88%iter4 1.70% 1.59% 1.69% 1.93%iter5 1.21% 1.52% 0.91% 1.24%iter6 1.18% 2.28% 0.28% 2.06%iter7 0.92% 1.86% – 0.96%iter8 2.12% 2.56% – 1.38%iter9 2.59% 2.19% – 1.82%iter10 1.00% 2.39% – 1.43%iter11 2.56% 2.30% – 0.29%iter12 2.03% 1.26% – –iter13 1.71% 1.43% – –iter14 1.95% 2.18% – –iter15 1.99% 2.33% – –iter16 2.03% 0.89% – –iter17 2.68% 0.42% – –iter18 2.34% – – –iter19 2.01% – – –iter20 1.70% – – –

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

α=1α=0.8α=0.7α=0.5

Fig. 13. Evolution of the maximum error e between consecutiveiterations, for different values of a.


BBmin ¼ 74175:61€, which is smaller than theobtained without risk-aversion constraints. Regard-

Table 4Detailed hydro schedule with minimum profit constraint (MW)

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h1

u1 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.u2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.u3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.u4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.u5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.u6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 47.3 47.2 109.4 47.2 0.u7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 12.9 30.5 30.5 30.5 0.

ing the schedule, note that units u6 and u7 ‘‘move’’some energy from the evening to the morning hours(Fig. 14).

For a minimum CVaR95% = 70,000€, theobtained results are also different. In Fig. 15 andTable 5 it can be seen how the generation duringhours in the interval [h21:h23] is increased. Thiscan be achieved by moving some energy from

3 h14 h15 h16 h17 h18 h19 h20 h21 h22 h23 h24

4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.40 0.0 0.0 0.0 0.0 41.9 41.9 36.7 0.0 0.0 0.0 0.00 0.0 0.0 0.0 20.4 25.1 25.1 25.1 0.0 0.0 0.0 0.00 0.0 0.0 0.0 18.5 22.7 22.7 22.7 0.0 0.0 0.0 0.00 0.0 0.0 0.0 12.1 45.9 44.3 28.4 15.7 0.0 0.0 0.00 0.0 0.0 0.0 0.0 108.7 65.4 47.3 0.0 0.0 0.0 0.00 0.0 0.0 0.0 7.5 30.5 30.5 30.5 15.0 7.5 30.5 0.0

Fig. 14. Hydro units’ generation with minimum profit constraint.

Fig. 15. Hydro units’ generation with minimum CVaR constraint.

Table 5Detailed hydro schedule with minimum CVaR constraint (MW)

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17 h18 h19 h20 h21 h22 h23 h24

u1 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4u2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.9 41.9 36.7 0.0 0.0 0.0 0.0u3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 25.1 25.1 24.7 6.5 6.5 7.6 0.0u4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 22.7 22.7 22.4 6.0 6.0 7.0 0.0u5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 27.6 51.0 39.2 28.4 0.0 0.0 0.0 0.0u6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 47.3 47.2 47.2 47.2 0.0 0.0 0.0 0.0 47.2 108.9 83.4 47.3 0.0 0.0 0.0 0.0u7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 30.5 30.5 30.5 7.5 7.5 7.5 7.5 7.5 30.5 30.5 30.5 7.5 7.5 20.8 0.0


peak-hours to off-peak hours. In this case, theexpected profit is BCVaR ¼ 74285:71€.

A very interesting comparison among the threecases is presented in Fig. 16. The histograms ofthe profit obtained for each scenario show that

when the minimum profit constraint is active, theleft-tail of the distribution is pushed to the right.

Besides this, the minimum CVaR formulationimplemented in the scheduling model gives satisfac-tory results. In the first case, the CVaR95% =

6.8 7 7.2 7.4 7.6 7.8 80

0.02

0.04

0.06

0.08

0.10

0.12

0

0.02

0.04

0.06

0.08

0.10

0.12

6.8 7 7.2 7.4 7.6 7.8 8 6.8 7 7.2 7.4 7.6 7.8 80

0.02

0.04

0.06

0.08

0.10

0.12

0

0.02

0.04

0.10

0.12

0

0.02

0.04

0.06

0.08

0.10

0.12

0

0.10

minBminCVaRwithout risk constraints

freq

uenc

y

x 104B x 104 x 104B x 104x 104B x 104

Fig. 16. Histograms of the obtained profits for each case.


69,885€ and in the second case, CVaR95% =69,917€. When adding the minimum CVaR formu-lation, the resulting value is CVaR95% = 70,025€,which is slightly higher than the right-hand side ofthe constraint (70,000€).

5. Conclusions

This paper presents an optimization model tohelp a hydro-generation company to schedule itshydroelectric units in a day-ahead electricity mar-ket. The objective function is the maximization ofthe expected profit, defined as the difference betweenexpected market revenues and the start-up costs.The hydro-generation company is supposed to beprice-taker, and therefore, market price is consid-ered as an exogenous variable. A discussion aboutthe most appropriate methods available in the liter-ature to forecast these prices is also presented.Regarding the hydro units modeling, the net-headeffect has been considered by means of an under-relaxed iterative procedure, which overcomes themain drawbacks of other approaches found in theliterature. Different risk-aversion constraints havebeen implemented, and the application to an exam-ple case has been satisfactory.

Acknowledgements

We thank Dr. Julian Barquın for his manyvaluable comments during the development of thisresearch, and Dr. Antonio Munoz for his contribu-tions to the price-forecasting section. Finally, wewould like to thank the anonymous referees fortheir suggestions to improve the quality of thepaper.

Appendix. Under-relaxed iterative procedure

The under relaxed method is a technique toupdate the variables involved in an iterative process.It was firstly introduced by the authors to the reso-lution of a traditional short-term hydro schedulingin Garcıa-Gonzalez et al. (2003), so further informa-tion could be found in this paper. The application ofthis method to the presented problem could be sum-marized in four steps:

Step 1. Initialize the net-heads assuming an a priorireservoir management. The net-head hi (m)of a hydro plant i measures the differencebetween the forebay elevation and thetailrace elevation. Therefore, it can beexpressed as a function of its reservoir stor-age mi (Hm3) and the immediate down-stream reservoir storage, mj (Hm3).

hi ¼ qiðmi; mjÞ: ð23Þ

In the Spanish system, the tailrace elevationcan be considered constant in most of thereservoirs. Therefore, this relationship canbe simplified

hi ¼ qiðmiÞ: ð24Þ
Step 2. Functions /t
ikðqikÞ are built by applying (2)to the current values ht

ik.Step 3. Once the hydro units have been character-

ized by their time-varying input–outputfunctions, the SPBS solution is obtainedby solving the MILP optimization problempresented in Section 3.

Step 4. The aim of this step is to check whether theconvergence has been reached or not, and inthis case, to prepare the input-data for the


next iteration t + 1. Firstly, it is necessaryto define a convergence measure which in

this case is e ¼ mjk � mtjk

��=mt

jk, where j is the

index of the reservoir in which the storedwater in period k has the highest mismatchbetween two consecutive iterations. If valueof e is smaller than a given tolerance (e.g.0.1%) the iterative process finishes, but ifnot, the last solution of the SPBS providesnew values for the reservoir levels mik, thatcould be used directly to update htþ1

ik andthe algorithm would continue in step 2

htþ1ik ¼ qi mikð Þ: ð25Þ

However, in order to avoid undesirable divergingoscillations, net heads can be updated using alsoprevious iterations information. Let define therelaxation parameter a > 0. The updated net headscan be obtained by the following equation:

htþ1ik ¼ qi mtþ1

ik

� �¼ qi mt

ik þ a � mik � mtik

� � �: ð26Þ

Note that (25) is just a particular case of (26) whena = 1. The selection of the best under-relaxation fac-tor is empiric and unfortunately, it can be case-dependent. However, a general rule can be stated:for the early stages of iterations lower values ofthe under-relaxation factor will help to avoid diver-gence, and as the iterations get closer to the con-verged state, values very close to 1 help to speedup the progress.

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Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market

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