Fleten '07

7/31/2019 Fleten '07

1/13

O.R. ApplicationsStochastic programming for optimizing bidding strategies

of a Nordic hydropower producer

Stein-Erik Fleten a , Trine Krogh Kristoffersen b, *a Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology,

NO-7491 Trondheim, Norwayb Department of Operations Research, University of Aarhus, Ny Munkegade, Bygning 1530, DK-8000 A rhus C, Denmark

Received 23 September 2005; accepted 17 August 2006Available online 13 October 2006

Abstract

From the point of view of a price-taking hydropower producer participating in the day-ahead power market, marketprices are highly uncertain. The present paper provides a model for determining optimal bidding strategies taking thisuncertainty into account. In particular, market price scenarios are generated and a stochastic mixed-integer linear pro-gramming model that involves both hydropower production and physical trading aspects is developed. The idea is toexplore the effects of including uncertainty explicitly into optimization by comparing the stochastic approach to a deter-ministic approach. The model is illustrated with data from a Norwegian hydropower producer and the Nordic power mar-

ket at Nord Pool. 2006 Elsevier B.V. All rights reserved.

Keywords: OR in energy; Electricity markets; Bidding; Stochastic programming; Scenarios

1. Introduction

The increased interest in power optimizationproblems within recent years has been stimulatedby the tendency to decentralize and deregulate the

power sector. Whereas traditional operating andplanning procedures were based on centralized opti-mization, novel approaches rest on independentoptimization of separate power plants. In anattempt to increase efficiency, markets have been

liberalized which has forced former procedures toconform to more market oriented approaches.

With the pioneering act of 1990 Norway wasamong the rst countries in the world to deregulate,and already in 1991 a Norwegian power market

was established. From 1996 to 2000 the nationalNorwegian power market, developed into a multi-national Nordic power market also encompassingthe three neighboring countries, Sweden, Finlandand Denmark. Today, the Nordic power markethas successfully adapted to the new competitiveenvironment and serves as a model for the restruc-turing of other power markets. An important com-ponent of the market is the presence of a powerexchange, that facilitates physical trading and is

0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2006.08.023

* Corresponding author. Tel.: +45 89423536.E-mail addresses: [email protected] (S.-E. Fleten),

[email protected] (T.K. Kristoffersen).

European Journal of Operational Research 181 (2007) 916928www.elsevier.com/locate/ejor
mailto:[email protected]:[email protected]:[email protected]:[email protected]

7/31/2019 Fleten '07

2/13

effective immediately. The spot market, Elspot atthe Nordic power exchange Nord Pool, takes theform of a pool-based market in which market par-ticipants exchange power contracts for physicaldelivery the following operation day and is referred

to as the day-ahead market. In 2004 a total of 167 TWh was exchanged at Elspot, representing42%1 of the overall consumption in the Nordicregion.

Inevitably, physical trading is of vital importancein the economic activity of the power sector with theso-called bidding problem being a major challengein this respect. Bidding involves the submission of sales and purchase bids to the power exchange aday ahead of physical trading. Since day-aheadmarket clearing prices are determined by the bal-ance between sales and purchase bids, bidding takesplace ahead of market clearing and, thus, with onlylimited information on day-ahead market prices. Asa result, bidding is a rather complicated task. In thefollowing, the problem of submitting bids to theday-ahead market will be referred to as the biddingproblem.

In the Nordic area, power production comprisesnuclear and thermal power in addition to hydro andother kinds of renewable power such as wind. Swe-den and Finland use nuclear, fossil-fuel and hydro-power production, Denmark makes use of

conventional thermal power plants, combined heatand power facilities and accounts for most of thewind power while almost only hydropower is usedin Norway. In general, thermal power productionis located in the south whereas hydropower produc-tion is found in the northern parts of the Nordicregion and is transmitted to the heavily populatedsouth. Approving of the comparatively low costsof hydropower production, the market prefers thisenergy source over thermal power. The number of hydropower producers participating in Nord Poolamounts to around 48 and the total hydropowergeneration in the Nordic region i s 183 TWh or54% of the total generation in 2004. 2

A crucial difference between thermal power andhydropower production is the possibility of storingenergy. If production is disposed of in the powermarket, storing allows the producer to respond tothe development in market prices. That is, whenprices are high, water is released and energy is pro-duced and sold immediately, whereas when prices

are low, the water is held back and the energy issaved for future disposal at higher prices. The exi-bility makes bidding strategies extremely relevant tohydropower producers.

As already implied, the bidding problem of a

hydropower producer involves both hydropowerproduction and day-ahead market exchange. Ineffect, the decision process is divided into stages.Bidding decisions take place with only limited infor-mation on day-ahead market prices whereas pro-duction decisions are deferred until informationhas been fully disclosed. Thus, the problem ts theframework of two-stage stochastic programmingand, more specically, the bidding problem maybe formulated as a two-stage stochastic mixed-inte-ger linear program. The objective is to maximizesales and production prots subject to a numberof bidding and operational constraints. The rststage concerns day-ahead market exchange whereasthe second stage takes in real-time hydropower pro-duction. Uncertainty of the day-ahead marketprices is represented by a known probability distri-bution. Formally speaking, the probability distribu-tion is obtained by the modeling of a stochasticprocess calibrated from historical data. For similarpower optimization models in stochastic program-ming, see the review of [27].

In practice short-term power planning and oper-

ation is often based on deterministic optimizationtools, such as the Short-term Hydro OperationPlanning (SHOP) model [8,10,11] used in Norway,whereas the setting up of bidding tables rests onskills and experience of the operating engineers.Until now, price uncertainty has not been handledexplicitly neither when planning and operating norwhen bidding.

One of the aims of the present paper is to com-pare the stochastic approach with a deterministicapproach. The idea is to explore the consequencesof including uncertainty into the optimizationmodel, in particular, the effects on the objectivefunction value, i.e. the sales and production prot,as well as on the solution, i.e. the structure of thebids. It can be shown that higher expected protsare obtained by taking the stochastic approachrather than the deterministic approach. Moreover,we expect the structure of the bids in the stochasticsolution to differ from that in the deterministic solu-tion. The deterministic model suggests the use of hourly bids only whereas the stochastic model pro-poses the combination of hourly bids and blockbids. If the day-ahead market prices are known in

1 Reference: www.nordel.org and www.nordpool.no .2 Reference: www.nordel.org .

S.-E. Fleten, T.K. Kristoffersen / European Journal of Operational Research 181 (2007) 916928 917
http://www.nordel.org/http://www.nordpool.no/http://www.nordel.org/http://www.nordel.org/http://www.nordpool.no/http://www.nordel.org/

7/31/2019 Fleten '07

3/13

advance, hourly bids dominate block bids as theyadjust hourly to price variations. On the other hand,if the day-ahead market prices are unknown at thetime of bidding, block bids protect against unex-pected price changes over time and have their

relevance.Although the paper is motivated by the needs of small Nordic hydropower producers the biddingmodel is applicable to other price-taking power pro-ducers who act in a pool-based day-ahead marketaccepting piecewise linear bids and block bids.

As regards the bidding problem, other contribu-tions within the literature include the following.The paper [5] addresses the problem from the per-spective of a price-taking thermal power producerbidding in the day-ahead market under market priceuncertainty. Based on time series analysis and fore-casts of market prices, a simple bidding strategy isdeveloped. Essentially, the expected value problemis solved, i.e. the stochastic prices are replaced byexpected prices, and the optimal solution is usedto construct piecewise constant bidding curves.For comparison, our model includes uncertaintyexplicitly by the use of price scenarios. In a spiritsimilar to that of the present paper, [9] proposes astochastic programming formulation of the biddingproblem for a price-taking power marketer whosupplies power to end users through purchases in

the day-ahead market. Solving the problem, theresult is piecewise linear bidding curves. Again, bid-ding is subjected to market price uncertainty. Unlikeour paper, uncertainty is represented by scenariosthat are generated by moment matching. Anotherstochastic programming approach related to thebidding problem appears in [21]. A model for thesimultaneous planning of production and day-ahead energy sales of a hydropower plant is givenand hence the output includes both productionand bidding schedules. As opposed to the modelhere, the plant is not a price-taker and the structureof the bidding process is incorporated directly in themodel so that uncertainty concerns the competitorsbids. Likewise, [28,19] build linear bidding curvesfor competitive suppliers in a day-ahead power mar-ket who do not have full information on the com-petitors bidding curves. The bidding curves arehandled using estimates that are based on expecta-tions and correlations. The authors suggest the useof maximum prot bidding strategies and, if not fea-sible for dispatch, minimum output bidding strate-gies. Ref. [1] studies strategies of producers whomake offers into power markets in circumstances

under which uncertainty also concerns the behaviorof other market participants in a competitive envi-ronment. Uncertainty is modeled by a so-calledmarket distribution and it is shown how to derivesmooth bidding curves solving an optimal control

problem. It should be stressed that unlike all previ-ous studies, our model allows for both piecewise lin-ear bids and block bids.

As regards pure hydropower problems in the lit-erature, we give a few references. Ref. [7] forms anoptimal control problem that relates to the short-term operation of a hydropower plant with reser-voirs in a cascade. Inows are unknown. Assuminga known probability distribution, the problem canbe reformulated as a multi-stage stochastic pro-gram. Ref. [22,26] develops deterministic short-termoptimization models for power systems that com-prises thermal and hydro generation whereas [21]presents a stochastic version of the like. Finally,[23] considers hydropower unit commitment subjectto uncertain demand in a multi-stage stochastic pro-gramming framework.

The outline of the present paper is as follows. Weexplain the composition of the day-ahead marketand model the bidding process in Section 2. In Sec-tion 3 we describe the hydropower plant and modelthe production in a relatively standard way. Section4 introduces uncertainty and the resulting stochastic

programming model whereas Section 5 is devoted toscenario generation. Section 6 illustrates with a casestudy from a Norwegian hydropower producer andthe Nordic power exchange, Nord Pool.

2. Day-ahead bidding

Elspot at Nord Pool is a spot market in whichcontracts for physical delivery the following opera-tion day are exchanged. The power exchange offersan access to the physical market at low transactioncosts as well as a possibility of settlement close toreal-time operation. The Elspot contracts are powerobligations to deliver or receive power of a durationof one hour or longer. Contracts are divided intohourly bids, block bids and exible hourly bids.All bids consist of a price and a volume. Submittinghourly bids, sales bids have to be listed in ascendingorder and purchase bids in descending order accord-ing to price. Consistent with the rules, for each hourElspot will make a linear interpolation between theprice-volume points to construct the bidding curve.The volume dispatched is determined by the pointon the bidding curve that corresponds to the market

918 S.-E. Fleten, T.K. Kristoffersen / European Journal of Operational Research 181 (2007) 916928

7/31/2019 Fleten '07

4/13

price. All transactions are settled at market price.Block bids are aggregated bids valid for a numberof consecutive hours and associated with only oneprice and volume. The so-called mean price condi-tion determines whether a block bid is either

rejected or accepted as a whole. If the price of a salesbid is at most equal to the average market price or if the price of a purchase bid is at least equal to theaverage market price for the hours of the block,the bid is accepted. All transactions are settled atthe mean price. Flexible hourly bids are hourly bidsthat are accepted in the hour with the highest priceprovided this price exceeds a certain threshold price.Such bids are omitted here as the bids are mainlyused by companies able to close down power inten-sive production. Participants post the price-differen-tiated bids for all hours of the following operationday before deadline at noon.

It is important to note that the volumes dis-patched and the prices at which transactions are set-tled are unknown until the market has been clearedand market clearing prices have been determined.Once this is done, the participants receives a noti-cation. The market price calculations are the samefor each individual hour. The bidding curves, i.e.the sales and purchase curves, are collocated to anaggregated demand curve and an aggregated supplycurve respectively. The intersection of the demand

curve and the supply curve denes a candidate of the unconstrained market price. Through an itera-tive process the bidding curves are updated accord-ing to the so-called priority rules by includingaccepted block bids and accepted exible hourlybids and a new unconstrained market price is found.The Nordic grid is divided into xed price zones.Sweden, Finland, East and West Denmark are onezone each and Norway is divided into four zones.If the contractual ow between zones does notexceed the allocated grid capacity, the uncon-strained market price simply applies to all zones.Otherwise, separate prices are established throughcounter purchases and corresponding iterations of price calculations in order to relieve grid congestion.

The following model and strategies should workas a tool of a price-taking hydropower producerfor planning how to commit in the day-ahead mar-ket and how to produce in accordance with thecommitment. It is assumed that the hydropowerproducer does not participate in bilateral exchange,as this does not affect the problem of bidding, butinstead disposes of the entire production in theday-ahead market and the real-time market.

The time horizon of 24 hours is divided intohourly time intervals and is denoted T f 1; . . . ; T g. From this the set of blocks B f b1; . . . ; b Bg is constructed. A block is a number of minimum two consecutive hours and the total num-

ber of such blocks within 24 hours isB

= 276.Examples of blocks are b140 = {1, . . . ,7}, b165 ={8, . . . , 18} and b265 = {19, . . . ,24}.

Regarding the modeling of the bidding process,the problem of selecting both bid prices and bid vol-umes is nonlinear. However, we have chosen towork with a linear model as the problem, even for-mulated as a mixed-integer linear program, is rela-tively hard to solve in terms of computing times,standard software packages are available for solvingthe problem and nally the problem is amenable tospecially designed stochastic programming algo-rithms. Hence, for computational reasons, nonlin-earities are avoided by xing prices in advance sothat only volumes have to be selected. LetI f 1; . . . ; I g index the possible bid prices anddenote these prices p i; i 2 I where p i 6 pi +1 . Thecorresponding bid volumes are represented by thevariables xit 2 R ; i 2 I ; t 2 T for hourly bidsand xib 2 R ; i 2 I ; b 2 B for block bids. Thevariables y t 2 R ; t 2 T and y b 2 R ; b 2 B arethe volumes dispatched, for hourly bids and blockbids respectively. The hourly market prices are

denoted q t ; t 2 T and average market prices forthe blocks qb; b 2 B where q b 1=jbjPt 2 bq t .Disposing of hydropower production in the day-ahead market, total sales revenues accumulate to

Xt 2 T q t y t Xb2 B q b y b:Hourly bids are handled as in [9]. For each hour,t 2 T , the bids xit ; p i; i 2 I are interpreted asprice-volume points on a bidding curve that deter-mines the relation between volumes bid and vol-umes dispatched. The curve is constructed bymaking a linear interpolation between the pointswhich results in a piecewise linear curve. Thus, interms of prices, the bidding curve can be expressedas (see Fig. 2.1)

q t

p 1 p 2 p 1

x2t x1t y t x1t if x1t 6 y t < x2t ;

.

.

.

p i1 p i p i1

xit xi1t y t xi1t if xi1t 6 y t < xit ;

.

.

.

p I 1 p It p I 1t x It x I 1t y t x I 1t if x I 1t 6 y t 6 x It

8>>>>>>>>>>>>>>>:


7/31/2019 Fleten '07

5/13

or equivalently, in terms of volumes,

y t

q t p 1 p 2 p 1

x2t p 2q t p 2 p 1 x1t if p 1 6 q t < p 2;

.

.

.

q t p i1 p i p i1 xit p iq t p i p i1 xi1t if p i1 6 q t < p i;...

q t p I 1 p I p I 1

x I 1t p I q t p I p I 1 x It if p I 1 6 q t 6 p I :

8>>>>>>>>>>>>>>>>>: 2:1

Considering sales bids, it would b e natural for thebidding curve to be increasing, i.e. 3

xit 6 xi 1t ; i 2 I n f I g; t 2 T : 2:2

For each b 2 B , the block bids are xib ; p i; i 2 Iand the relation between volumes bid and volumesdispatched is

y b X j : p j 6 qb x jb ; b 2 B ; 2:3i.e. in a given block, the volume dispatched com-prises the volumes of accepted bids. For example,consider the block b140 = {1, . . . , 7}. If two bidsare given by (x1,140 , p1) = (50,100) and(x2,140 , p2) = (100,200) and the average market priceis q140 150, then only one bid is accepted and the

volume dispatched is y140 = 50.

3. Short-term hydropower production

Modeling the hydropower production side is rel-atively standard and follows the lines of for example[23]. This section presents a simple but illustrativemodel of a small hydropower plant. Still, it is

straightforward to combine a different modeling of the hydropower plant with the modeling of the bid-ding process of the preceding section. The plantconsists of two reservoirs in a cascade, that is, a lar-ger upper reservoir and a smaller lower reservoir.

There is a time delay between the two reservoirs.The combination of time delay and size differencesrestricts the exibility of the system. This featurecontributes to understanding the importance of including uncertainty which will be discussed later.Each reservoir is connected to a power station witha single generator. Now hydropower productionworks as follows. Upstream water reaching theplant ows to the upper reservoir where it is storeduntil released for generation. When released, thewater from the upper reservoir ows to the lowerreservoir and is again stored until used for genera-tion. Electricity is generated by changing the poten-tial energy of the water into electrical energy. Waterthat is not discharged on purpose and used for gen-eration is considered spill. Leaving the plant, thewater proceeds downstream. For an illustration,see Fig. 3.1.

To model the production side, let J f1; 2gindex the reservoirs, let the variables u jt 2 f 0; 1g; j 2 J ; t 2 T represent the on/off states of the gen-erators, w jt 2 R ; j 2 J ; t 2 T the generation lev-els and v jt 2 R ; j 2 J ; t 2 T the corresponding

discharges from the reservoirs. Moreover, let thevariables l jt 2 R ; j 2 J ; t 2 T be the reservoirstorage levels and r jt 2 R ; j 2 J ; t 2 T the spill.

Direct costs of hydropower production includeonly start-up costs as operating costs are negligible.Start-up costs amount to

Xt 2 T X j2 J S ju jt 1; u jt with the cost functions beingS ju jt 1; u jt c j maxf u jt u jt 1; 0g; j 2 J ; t 2 T ;

Fig. 2.1. Bidding curve of time interval t 2 T .

Fig. 3.1. Two reservoirs in a cascade.

3 Nord Pool does not require that the bidding curve beincreasing and therefore such constraints may in principle beomitted. Omitting the constraints, however, does not affect theoptimal objective function values.


7/31/2019 Fleten '07

6/13

where c j ; j 2 J are the unit start-up costs. Notethat the cost functions are consistent with amixed-integer linear formulation. The initial condi-tions u j0 u jinit ; j 2 J must be added. Costs alsotake in opportunity costs of releasing water as the

water could be stored and saved for future disposal.Thus, such costs are measured as the value of storedwater, which is usually available from more long-term models. Here, we have used an average of prices of futures contracts and prices of forwardcontracts to estimate water values. The opportunitycosts are

X j2 J V j l j0 V j l jT with

V j l jt minh2 H f d 1hj l jt d

2hj g; j 2 J ; t 2 T ;

where d 1hj ; d 2hj ; h 2 H ; j 2 J are the coefficients of

the concave water value functions. Again, the for-mulation corresponds to a mixed-integer linearformulation.

The following bounds are imposed on the waterdischarges. The upper reservoir is either not in oper-ation or operated at maximum capacity, which leadsto the constraints

w1t u1t wmax1 ; t 2 T ; 3:1

where wmax1 is the maximum generation level. Thelower reservoir, however, can be operated anywherebetween minimum and maximum capacity.

u2t wmin2 6 w2t 6 u2t wmax2 ; t 2 T ; 3:2

where wmin2 and wmax2 are minimum and maximumgeneration levels. Similar bounds apply to the dis-charges, i.e.,

vmin j 6 v jt 6 vmax j ; j 2 J ; t 2 T : 3:3

Here,v

min j ; j 2

J andv

max j ; j 2

J are the minimumand maximum discharges. Finally, the storage levelshave to adhere to the bounds

l min j 6 l jt 6 lmax j ; j 2 J ; t 2 T ; 3:4

where l min j ; j 2 J and lmax j ; j 2 J denote minimal

and maximal storage levels.As the reservoir has to balance, reservoir inow

and storage from previous periods either appear asdischarge, storage or spill. In the case of the upperreservoir the balance equations are

l 1t l 1t 1 v1t r 1t m1t ; t 2 T ; 3:5

where m1t ; t 2 T are the inows from upstream. Theinitial storage level is l 10 = l 1init . In the case of thelower reservoir, the balance equations are

l 2t l 2t 1 v2t r 2t v1t s ; t 2 T ; 3:6

where s is the time delay between the upper and low-er reservoirs. Again, the initial storage level isl 20 = l 2init .

Generation and discharge are essentially propor-tional if it is ignored that generation efficiency is infact nonconstant and omitting effects of the reser-voir water levels on both generation and discharge.This leads to the constraints

w jt c j v jt ; j 2 J ; t 2 T : 3:7

Here, c j ; j 2 J are the generation efficiencycoefficients.

Imbalances between volumes produced and vol-umes dispatched in the day-ahead market are settledin a real-time balancing market. Nevertheless, onaverage, day-ahead market exchange should notcause imbalances as the day-ahead market shouldfunction as a de facto spot market. In particular, ahydropower producer should not be allowed savewater and postpone production for disposal in thereal-time balancing market as the day-ahead marketshould reect the physical conditions of the system.Thus, the primary focus in the short-term planning

of a hydropower plant should be the day-ahead mar-ket. Justied by the discussion, we approximate thebalancing effects of the real-time market. We imposea penalty or a reward on imbalances. The penalty ishigher than the day-ahead market price and paid if volumes dispatched exceed the volumes produced,i.e. in hours of up-regulation, and the reward islower than the day-ahead market price and is paidin hours of down-regulation. By imposing a penaltyor a reward, the producer retains exibility to rampup or down and bid this exible capacity into thebalancing market close to real-time. In this way,the modeling does not prevent the producer fromparticipating in the balancing market, but is alonean attempt to avoid planned imbalances. As penaltyand reward, we take the average real-time marketprice of hours in which the market has been up-reg-ulated and down-regulated respectively. Such penal-ties and rewards would apply if the producer isalways regulated in the same direction as the market.In reality, a producer is very rarely regulated in theopposite direction of that of the local market. Fora different modeling approach, see also [9]. Letthe variables z t ; z t 2 R ; t 2 T represent the


7/31/2019 Fleten '07

7/13

imbalances and let l t ; l t ; t 2 T denote the corre-sponding penalty and reward. Then the total penaltyand reward amount to

Xt 2 T

l t z t l

t z

t

and the balance constraints are

y t Xb2 B : t 2 b y b X j2 J w jt z t z t ; t 2 T : 3:84. Day-ahead bidding under uncertainty

The model of the preceding sections do not takeinto considerations the uncertainty of the data. Inthis context data uncertainty can arise with respectto water inow and market prices. Whereas inowforecasting is rather complicating during summer,this is not the case in winter. Moreover, since theidea is to analyze uncertainty that relates directlyto bidding, we ignore the possibility of inows tobe stochastic. Prices, on the other hand, are affectedby market conditions that may be hard to predict.To a small market participant, the market clearingprocess is governed by the behavior of other marketparticipants and generally the information on othermarket participants is not available. Thus, as marketprices are determined by clearing the market, prices

are unknown at the time of bidding. In other words,to a price-taker, the process of bidding is subjectedto uncertainty of the market clearing prices.

Uncertainty can be handled by means of stochas-tic programming. For an introduction to the subject, see [2,16,24]. A stochastic program ischaracterized by the partition of decisions intostages according to information ow in a way thatdecisions of one stage do not depend on informationof following stages. The most obvious objective cri-terion is expectation-based. Here, decisions aregrouped into two, i.e. are to be made before andafter observing uncertainty respectively. Hence, atwo-stage stochastic programming model is appro-priate. The rst stage involves the bidding processwhereas the second stage includes the productionaspects. As bids are submitted before the markethas cleared, prices are unknown at the time of rst-stage decision making. In contrast, second-stage decision making is put off until the markethas cleared and take advantage of the additionalinformation from observing prices. The aim is toobtain the optimal bidding strategies in terms of expected sales and production prot.

To incorporate uncertainty assume market pricesf q t gt 2 T form a stochastic process on some probabil-ity space. Assume the multivariate distribution isknown and in particular that it is discrete with anite number of realizations S f1; . . . ; S g

referred to as scenarios. Denote the scenario proba-bilities by p s; s 2 S and the corresponding marketprices by f q st gt 2 T ; s2 S . Whereas rst-stage decisions xit ; xib ; i 2 I ; t 2 T ; b 2 B , i.e. volumes bid,should be independent of future market prices, sec-ond-stage decisions y st ; y sb; z

; st ; z ;

st ; v s jt ; w s jt ; l

s jt 2 R ;

u s jt 2 f 0; 1g, i.e. volumes dispatched as well as pro-duction decisions, are allowed to depend on therealization of future market prices and are indexedby the scenario superscript s 2 S . The stochasticprogram then consists in maximizing the expectedsales and production prot subject to the biddingand operational constraints. The constraints (2.1) (2.3) couples rst-stage and second-stage decisionsthrough the relation between volumes bid and vol-umes dispatched whereas the constraints (3.1) (3.8) applies to second-stage decisions only andmodel hydropower production. The two-stage sto-chastic mixed-integer program formulated as itsdeterministic equivalent is

max X s2 S p s Xt 2 T q st y st Xb2 B q sb y sb Xt 2 T l t z ; st l t z ; st X j2 J V j l s j0 V j l s jT Xt 2 T X j2 J S ju s jt 1; u s jt !

s:t: 2:1 2:3; 3:1 3:8; xit ; xib 2 R ; i 2 I ; t 2 T ; b 2 B ; y st ; y

sb; z

; st ; z

; st ; v

s jt ; w

s jt ; l

s jt 2 R ; u

s jt 2 f 0; 1g;

j 2 J ; t 2 T ; b 2 B ; s 2 S :

4:1

It should be remarked that (2.1) and (2.3) can besimplied. Consider xed t 2 T and s 2 S . If theprice points i 2 I are xed in advance, the realizedmarket price q st is located between two adjacentpoints. The remaining price points are irrelevantfor determining the volume to be dispatched. Let-ting it ; s maxf i 2 I : p i 6 q st g the point of dis-patch y st ; q st is located on the line segmentbetween (x i (t ,s ) t , pi ( t ,s )) and (x i (t ,s )+1 t , pi ( t ,s )+1 ). Fromthis, (2.1) is equivalent to

y st q st p it ; s

p it ; s1 p it ; s xit ; s1t

p it ; s1 q st p it ; s1 p it ; s

xit ; st ;

t 2 T ; s 2 S :


7/31/2019 Fleten '07

8/13

Similarly, letting ib; s maxf i 2 I : p i 6 q sbg,(2.3) can be rewritten as

y sb X j6 ib; s x jb ; b 2 B ; s 2 S :We operate with two different ways of xing theprice points p i; i 2 I in advance, one way beingto x equidistant price points and the other beingto x price points such that the number of realizedmarket prices q st ; t 2 T ; s 2 S between any twopoints is always the same. The latter reects the dis-tribution of market prices and consequently theprice graduation is more crude in areas where mar-ket prices are less likely. We examined the effect onthe optimal bidding curves of varying the number of xed price points with price points xed accordingto the distribution of market prices. In general, themore crude the price graduation the more crudethe bidding curves, although small changes in thenumber of xed price points did not alter the bid-ding curve. Also, we found that xing equidistantprice points can at rst sight induce a rather nebidding curve. However, within the interval of real-ized of market prices the bidding curve derived fromequidistant price points is in fact cruder than thatderived from the other price points.

5. Scenario generation

To describe the behavior of day-ahead marketprices, the ARMA methodology may be applied,modeling an advanced stochastic process calibratedfrom historical price proles. Basically, ARMA pro-cesses are a specic class of stochastic processesadopted for the analysis of time series and date backto Box and Jenkins [3]. The ARMA method hasbeen found to be mathematical sound and at thesame time effective at predicting electricity prices,see [6] for an example. Taking the ARMA frame-

work as a starting point, we establish a statisticalmodel from which electricity price scenarios canbe generated. A similar approach is seen in [13,14]which considers electricity load scenarios. Otherrelated approaches to model electricity pricesembrace regression analysis in which electricity loadhas been included as an explanatory variable, cf.[20], and GARCH models, cf. [12]. Among the latestwork, long memory processes within the ARFIMAframework have proven very promising, cf. [15].

An ARMA model can be formulated as

U Bq t H B t ; t 2 Z ;

where U(B ) and H (B ) are polynomials of the formU B 1 Pi/ i B

i and H B 1 Pihi Bi and B

denotes the back-shift operator, i.e. B i q t = q t i .The innovations f t gt 2 Z are assumed a Gaussianwhite noise process, i.e. t ; t 2 Z are independent

normally distributed random variables with zeromean and constant variance. The development of the proposed ARMA model follows the steps:

1. Identify a statistical model of the historical data.2. Estimate the parameters of the model.3. Validate the model.4. Use simulation to generate scenarios.1. For identication an hourly price prole of a

year is given. The analysis begins with a carefulinspection of the main characteristics of the timeseries. Non-constant mean and variance as wellas calendar effects and seasonal trends corre-sponding to daily and weekly periodicity areobserved. In the creation of a trial model the datais made stationary by a transformation of the ori-ginal data. The logarithm is applied to attain astable variance and the inclusion of the factors(1 B ), (1 B 24 ) and (1 B 168 ) is used to stabi-lize the mean. The structure of the polynomials isdetermined by investigating the autocorrelationsand partial autocorrelations of the transformeddata. In successive trials, renements can be

made based on the autocorrelations and partialautocorrelations of the residuals. The nalARMA model is identied as

1 B1 B241 B168log q t

1 h1 B1 h7 B71 h23 B23 h24 B24

h25 B25 h47 B47 h48 B48 h49 B49

1 h168 B168 t ; t 2 Z :

2. On completion of the identication parameterestimates may be computed by the use of maxi-mum likelihood optimization. Estimates are dis-played in Table 1 .

3. The model is validated by testing the assump-tions of a Gaussian white noise process madeon the innovations which is achieved by studyingthe autocorrelations and partial autocorrelationsof the residuals as well as the LjungBox statis-tics. The ability of the model to forecast is illus-trated in Fig. 5.1.

4. Simulated price scenarios f q st gT t 1; s 1; . . . ; S

can be generated by withdrawing the startingvalues q t , t = 192, . . . ,0 and t , t = 223, . . . , 0


7/31/2019 Fleten '07

9/13

from the historical data and sampling from theindependent identically normally distributed ran-dom variables t , t = 1, . . . , T . Monte Carlo sam-pling has been used to simulate a large number of scenarios. Some descriptive statistics can befound in Table 2 . For computational accessibil-

ity, however, the number of scenarios has beenreduced using the scenario reduction approachin [13]. An example of some simulation scenariosis shown in Fig. 5.2.

Steps 14 are all carried out by the statisticalsoftware package SAS, version 8.2.

Although in practice the distribution of electric-ity prices is continuous, for computational reasons

it is approximated by a discrete distribution. Thequality of the approximation is directly linked tothe quality of the scenarios which makes it relevantto evaluate the scenario generation method. As con-cerns practical performance, the in-sample and theout-of-sample stability as well as the stability of the solution was tested, cf. [17,18]. Where the truedistribution was needed so-called back-testing wasmade with historical data. The stability analysisgives an indication on the number of scenarios toinclude in order to represent uncertainty. The sto-chastic program is found to fulll the stabilityrequirements in a satisfying way for as few as 10scenarios.

For illustration purposes a number of demon-stration scenarios have been generated. In these sce-narios day-ahead market prices are very volatile. Anexample of some demonstration scenarios is plottedin Fig. 5.3.

Table 1Parameter estimates of hourly day-ahead market prices

Parameter h1 h2 h3 h4 h5 h6 h7Estimates 0.1858 0.1409 0.0392 0.1292 0.0912 0.0670 0.0810Parameter h23 h24 h25 h47 h48 h49 h168Estimates 0.0921 0.7321 0.0987 0.0526 0.1527 0.0500 0.9843

Fig. 5.1. Realized vs. forecasted prices.

Table 2Descriptive statistics of hourly day-ahead market prices (NOK/MW h)

Sample size Mean Std. dev. Minimum Maximum

1000 174.59 11.66 141.48 221.63

Fig. 5.2. 10 simulation scenarios. Fig. 5.3. 10 demonstration scenarios.


7/31/2019 Fleten '07

10/13

6. Case study

The case study concerns two reservoirs of a smallNorwegian hydropower plant located near Trond-heim and run by the company TrnderEnergi. For

computational reasons the real dimensions of thereservoirs and the power stations are scaled downby a factor 10. Initial reservoir levels and reservoirinows are real data from a typical day of 2005.To generate day-ahead market prices in 2005,Elspot at Nord Pool has provided real data from2004 that applied to the Norwegian prize zone,NO2 which is the Trondheim area.

The two-stage stochastic programming problemcontains 18,300 continuous variables and 1440 con-straints in the rst stage and 590 continuous vari-ables, 48 binary variables and 740 constraints inthe second stage. Formulated as its deterministicequivalent, (4.1), the problem is a large-scalemixed-integer linear program solvable by standardsoftware. Here, the problem was solved with themixed-integer linear programming solver fromOPL Studio version 3.7 calling CPLEX 9.0 on anIntel Xeon 2.67 GHz processor with 4 GB RAM.The time of compiling and solving the problem var-ied between 1 and 3000 seconds since no specialeffort was made to make the code efficient. In thatthe problem is a stochastic mixed-integer linear pro-

gram special purpose algorithms such as progressivehedging [25] and dual decomposition [4] mightprove useful.

This section compares the stochastic approach tothe bidding problem with a deterministic approach.The aim is to explore the consequences of includinguncertainty into the optimization model, in particu-lar, the effects on the objective function value, i.e.the sales and production prot, as well as on thesolution, i.e. the structure of the bids.

To explore the structure of the bids, we solvedthe bidding problem as a stochastic problem andas a deterministic problem. The former problemconcerned the stochastic programming problemand the latter the corresponding expected valueproblem formed by replacing random prices byexpected prices. Moreover, we computed theexpected result of using the expected value solution.

Consider the deterministic bidding problems forone reservoir and for two reservoirs in a cascade.The structure of the solution is simple. In both caseshourly bids are sufficient. Indeed, if the market priceis known in advance, only two hourly bids are rele-vant. The relevant bids determine the part of the

bidding curve that passes through the point givenby the market price and the optimal productionlevel. However, the resulting bidding curve may bevery sensitive to changes in the market price. Inmore extreme scenarios the deviations from the

expected market price may enforce heavy balancingas dispatches are far from met by production. Anexample is given in Table 3 in which the differences(Imbal.) between the total volumes dispatched of hourly bids (H. disp.) and block bids (B. disp.)and the volumes produced (Prod.) are rather largein the most extreme scenarios of a given hour.

For the stochastic bidding problems the structureof the solution is more complex. In the one-reser-voir-case, the structure of the solution depends onwhether start-up costs are included or not. Withno start-up costs hourly bids are sufficient whereaswith start-up costs both hourly bids and block bidsare necessary. Hourly bids follow prices closely.These are, however, accepted independently of eachother and do not take into account intertemporali-ties due to start-ups. Block bids, on the other hand,are valid for a number of consecutive hours andthus tend to support a regular production with lessstart-ups. In the two-reservoir-case the structure of the solution also depends on whether start-up costsare included or not. Moreover, the presence of abottleneck in the system has a similar impact on

the structure of the solution. A bottleneck occurswhen the capacity of the upper reservoir exceedsthe capacity of the lower reservoir. Water is releasedfrom both the upper and the lower reservoirs inhours with a high market price. Due to the bottle-neck and the time delay between the reservoirs,however, too much water released from the upperreservoir may lead to forced releases in the lowerreservoir in hours with a low market price. Withno bottleneck hourly bids are sufficient whereaswith a bottleneck both hourly bids and block bidsare necessary. An explanation is that as hourly bidsare accepted independently they ignore intertempo-ralities generated by dependencies between the res-ervoirs. On the other hand, since block bids arevalid for a number of consecutive hours such bidscan be used as protection against major price uctu-ations over time.

If applying the deterministic approach to the bid-ding problem hourly bids will suffice whereas if applying the stochastic approach both hourly bidsand block bids are relevant. The difference in thestructure of the bids between the expected valueproblem (EVP) and the stochastic programming


7/31/2019 Fleten '07

11/13

problem (SPP) is illustrated in Table 3 in which H.disp. refers to the total volume dispatched as hourlybids and B. disp. refers to the total volume dis-patched as block bids of a given hour. It shouldbe remarked that if formulating the problems differ-ently both hourly bids and block bids might be rel-evant in a stochastic problem as well as in adeterministic problem. Still, from the present for-mulation it is clear that uncertainty is one way of justifying the use of block bids and that block bidsprovide motivation for the inclusion of uncertaintyinto the bidding problem.

Based on 10 demonstration scenarios and 10 sim-ulation scenarios respectively, bidding curves havebeen drawn in Figs. 6.1 and 6.2 . The curves areshown for both the deterministic problem (EVP),cf. the dashed lines, and the stochastic problem(SPP), cf. full-drawn lines. The vertical lines repre-sent block bids and the other curves hourly bids.Although the bidding curves appear as piece-wise

constant, the curves are in fact only piece-wise lin-ear. Nearly piece-wise constant bidding curves,however, are consistent with the bidding practiceof the current application and reects an almostprice insensitive behavior between certain price lev-els. The gures also illustrate the discussion above.Consider Fig. 6.1. For an expected market price of 261.90 NOK/MW h the dispatch is the same forthe deterministic and the stochastic problems, i.e.2.24 MW h. Nevertheless, as already stated, thedeterministic bidding curves are very sensitive tochanges in the market price. If for instance the priceturns out to be 28.00 NOK/MW h (scenario 10) or389.00 NOK/MW h (scenario 1), the dispatch mustbe very small (0.00 MW h) or very large(2.29 MW h) which results in balancing andincreased costs. The same situation applies toFig. 6.2.

To examine the effect of including uncertainty onproduction and sales prot we solved the stochasticprogramming problem and the expected value prob-

Table 3Scenario solutions of a given hour, demonstration scenarios

Sce. EEV SPP

H. disp. B. disp. Prod. Imbal. H. disp. B. disp. Prod. Imbal.

1 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.00

2 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.003 0.00 0.00 1.37 1.37 0.00 2.24 2.24 0.004 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.005 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.006 0.00 0.00 1.37 1.37 0.00 2.24 2.24 0.007 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.008 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.009 2.29 0.00 2.24 0.05 0.00 2.24 2.24 0.00

10 0.00 0.00 1.37 1.37 0.00 2.24 2.24 0.00

The columns H. disp. and B. disp. show dispatch generated by hourly bids and block bids, separately. Prod. denotes production. Thecolumn Imbal. shows imbalances between production and dispatch.

Fig. 6.1. Bidding curves of a given hour, demonstrationscenarios.

Fig. 6.2. Bidding curves of a given hour, simulation scenarios.


7/31/2019 Fleten '07

12/13

lem with both the demonstration scenarios and thesimulation scenarios, in the latter case by varyingthe number of scenarios. For the demonstration sce-narios we have xed equidistant price pointswhereas for the simulation scenarios we have xed

price points according to the distribution of marketprices. We recorded the optimal value of the sto-chastic program as well as the expected result of using the expected value solution (EEV). To com-pare the two we calculated the difference referredto as the value of the stochastic solution (VSS).The value of the stochastic solution measures theeffect of including stochastic prices explicitly intothe bidding problem rather than simply usingexpected prices. Averages of 10 different runs arereported in Tables 4 and 5 . We focus on the compu-tational results using the simulation scenarios. Itfollows that the value of the stochastic solution is7.93% on average. On a per day basis the monetarygains of solving the stochastic program rather thanthe expected value problem may seem moderate.However, the dimension of the power plant hasbeen scaled down by a factor 10. Hence, if the prob-lem is solved every day of the year, signicant prof-its can be earned by applying stochasticprogramming.

As concluded, solutions are less sensitive tochanges in the data if applying the stochastic

approach than if applying the deterministicapproach. Still, the expectation-based stochasticprogram lacks robustness in the sense that solutionstend to be unstable. Almost equal objection func-tion values are obtained by structurally differentsolutions which indicates a at objective function.Although structurally different, the solutions areequally good as long as the expectation-based objec-tive criterion is acknowledged. However, the expec-

tation-based objective criterion can be claimed toignore both risk attributes and prot distributionissues. In order to obtain a more robust stochasticprogramming model a risk measure can be includedin the objective function. The result is a so-calledmean-risk model. The inclusion of the downside riskmeasure semideviation penalizes deviations fromthe expected value and has the advantage of beingconsistent with the mixed-integer linear formula-

tion. With semideviation, the objective function of (4.1) turns into

X s2 S p s z s q X s2 S p s max X s2 S p s z s z s; 0( )with

z s Xt 2 T q st y st Xb2 B q sb y sb Xt 2 T l t z ; st l t z ; st X j2 J V j l s j0 V j l s jT Xt 2 T X j2 J S ju s jt 1; u s jt ;

where q is a weight. The mean-risk problem ensuresmore stable solutions and a more equal prot distri-bution among the scenarios. We have solved themean-risk problem for varying weights and in Table6 we report the expected value and the risk for aver-ages of 10 different runs using simulation scenarios.As the objective function is at more stable solu-tions can be obtained with only very small protreductions. Finally, note that the inclusion of thedownside risk measure semideviation does not alterthe structure of the solution in a way that contra-dicts the analysis already made.

Acknowledgments

The authors would like to thank BernhardKvaal, Gunnar Aronsen and Lars Olav Hoset atTrnderEnergi for indispensable information ontheory and practice of both hydropower productionand bidding, Matthias P. Nowak at SINTEF forfundamentals and suggestions on how to structurethe paper, Olav Bjarte Fosso and Michael M. Bels-nes at Norwegian University of Science and

Table 4Computational results for demonstration scenariosS Opt. val. EEV VSS

10 34801.51 33821.17 980.34

Table 5Computational results for simulated scenarios

S Opt. val. EEV VSS

10 33395.45 30992.49 2402.9630 33420.60 30676.07 2744.5350 33405.75 30805.37 2600.39

100 33419.60 30566.44 2853.17

Table 6Computational results for semideviation, simulation scenarios

q 0.001 1 10 100 1000

Exp.val.

33389.34 33389.34 33192.04 32922.73 32716.62

Risk 107.56 107.56 7.94 1.02 0.00


7/31/2019 Fleten '07

13/13

Technology for helpful discussions of the problemand its relevance, Nina Detlefsen and Frede Aak-mann Tgersen at Forskningscenter Foulum as wellas Preben Blsild and Anders Holst Andersen at theUniversity of Aarhus for assistance concerning the

time series analysis.

References

[1] E.J. Anderson, A.B. Philpott, Optimal offer construction inelectricity markets, Mathematics of Operations Research 27(1) (2000) 82100.

[2] J.R. Birge, F. Louveaux, Introduction to Stochastic Pro-gramming, Springer, New York, 1997.

[3] G.E.P. Box, G.M. Jenkins, Time Series Analysis, HoldenDay, 1970.

[4] C.C. Caroe, R. Schultz, Dual decomposition in stochasticinteger programming, Operations Research Letters 24 (1998)

3745.[5] A.J. Conjeno, F.J. Nogales, J.M. Arroyo, Price-takerbidding strategy under price uncertainty, IEEE Transactionson Power Systems 17 (4) (2002) 10811088.

[6] J. Contreras, R. Espinola, F.J. Nogales, A.J. Conejo,ARIMA models to predict next-day electricity prices, IEEETransactions on Power Systems 18 (3) (2003) 10141019.

[7] G.B. Dantzig, G. Infanger, Intelligent control and optimi-zation under uncertainty with application to hydro power,European Journal of Operations Research 97 (1997) 396 407.

[8] N. Flatab, A. Haugstad, B. Mo, O.B. Fosso, Short-termand medium-term generation scheduling in the Norwegianhydro system under a competitive power market structure,VIII SEPOPE02, Brasil, 2002.

[9] S.E. Fleten, E. Pettersen, Constructing bidding curves for aprice-taking retailer in the Norwegian electricity market,IEEE Transactions on Power Systems 20 (2) (2005) 701708.

[10] O.B. Fosso, M.M. Belsnes, Short-term hydro scheduling in aliberalized power system, in: Proceedings of the InternationalConference on Power System Technology, 2124 November2004, Singapore.

[11] O.B. Fosso, A. Gjelsvik, A. Haugstad, B.I. Mo, I. Wagens-teen, Generation scheduling in a deregulated system. TheNorwegian case, IEEE Transactions on Power Systems 14(1) (1999) 7581.

[12] R.C. Garcia, J. Contreras, M.V. Akkeren, J.B.C. Garcia, AGARCH forecasting model to predict day-ahead electricityprices, IEEE Transactions on Power Systems 20 (2) (2005)867874.

[13] G. Growe-Kuska, K.C. Kiwiel, M.P. Nowak, W. Ro misch,I. Wegner, Power management under uncertainty byLagrangian relaxation, in: Proceedings of the 6th Interna-tional Conference Probabilistic Methods Applied to PowerSystems PMAPS 2000 2, INESC Porto, 2000.

[14] N. Growe-Kuska, K.C. Kiwiel, M.P. Nowak, W. Ro misch,I. Wegner, Power management in a hydro-thermal systemunder uncertainty by Lagrangian relaxation, in: C. Green-gard, A. Ruszczynski (Eds.), Decision Making under Uncer-tainty: Energy and Power, IMA Volumes in Mathematicsand its Applications, vol. 128, Springer-Verlag, New York,2002, pp. 3970.

[15] N. Haldrup, M.O. Nielsen, A regime switching long memorymodel for electricity prices, Journal of Econometrics 135 (1 2) (2006) 349376.

[16] P. Kall, S.W. Wallace, Stochastic Programming, Wiley,Chichester, 1994.

[17] M. Kaut, S.W. Wallace, Stability analysis of a portfoliomanagement model based on the conditional value-at-riskmeasure, 2003 (Downloadable from website ).

[18] M. Kaut, S.W. Wallace, Evaluation of scenario-generationmethods for stochastic programming, 2003 (Downloadablefrom website ).[19] X. Ma, F. Wen, J. Liu, Towards the development of risk-constrained optimal bidding strategies for generation com-panies in electricity markets, Electric Power SystemsResearch 73 (2005) 305312.

[20] F.J. Nogales, A.J. Conejo, R. Espinola, Forecasting next-day electricity prices by time series models, IEEE Transac-tions on Power Systems 17 (2) (2002) 342349.

[21] M.P. Nowak, R. Nurnberg, W. Romisch, R. Schultz, M.Westphalen, Stochastic programming for power productionand trading under uncertainty, in: W. Ja ger, H.J. Krebs(Eds.), Mathematics Key Technology for the Future,Springer, Berlin, 2003, pp. 623636.

[22] P. Oliveira, S. McKee, C. Coles, Optimal scheduling of ahydro thermal power generation system, European Journalof Operations Research 71 (1993) 334340.

[23] A.B. Philpott, M. Craddock, H. Waterer, Hydro-electric unitcommitment subject to uncertain demand, European Journalof Operations Research 125 (2000) 410424.

[24] A. Prekopa, Stochastic Programming, Kluwer, Dordrecht,1995.

[25] R.T. Rockafellar, L.A. Wets, Scenarios and policy aggrega-tion in optimization under uncertainty, Mathematics of Operations Research 16 (1991) 119147.

[26] Z.K. Shawwash, T.K. Siu, S.O. Russel, The B.C. Hydroshort term hydro scheduling optimization model, IEEETransactions on Power Systems 15 (3) (2000) 11251131.

[27] S.W. Wallace, S.E. Fleten, Stochastic programming modelsin energy, in: A. Ruszczynski, A. Shapiro (Eds.), StochasticProgramming, Handbooks in Operations Research andManagement Science, vol. 10, Elsevier, North-Holland,2003, pp. 637677.

[28] F.S. Wen, A.K. David, Strategic bidding for electricitysupply in a day-ahead energy market, Electric PowerSystems Research 59 (2001) 197206.

http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.http://work.michalkaut.net/CV_and_study/SG_evaluation_abstract.htm.http://work.michalkaut.net/CV_and_study/SG_evaluation_abstract.htm.http://work.michalkaut.net/CV_and_study/SG_evaluation_abstract.htm.http://work.michalkaut.net/CV_and_study/SG_evaluation_abstract.htm.http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.http://work.michalkaut.net/CV_and_study/CVaR_stability_abstract.htm.

Documents

Fleten '07