Modeling the e!ects of nuclear fuel reservoir operation ina competitive electricity market.

Jean-Michel Glachant!, Pascal Gourdel†, Maria Lykidi, ‡

February 2010

Abstract

In many countries, the electricity systems are quitting the vertically integrated monopolyorganization for an operation framed by competitive markets. In such a competitiveregime one can ask what is the optimal operation of the nuclear generation set? Thereare two approaches of that issue, according to the relevant temporal horizon : the short-term and the medium-term. The short term is related to the daily demand variationswhile the medium-term takes into account the seasonal variation of the demand level be-tween winter (high demand) and summer (low demand). In this paper we focus on theseasonal dimension. It corresponds to the fact that nuclear fuel functions like a reservoirsince nuclear plants stop periodically (12/ 18 months) to reload their fuel while the actuallength of a nuclear production campaign can be shortened or expanded if economicallye!cient. The operation of the reservoir allows di"erent profiles of nuclear fuel uses duringthe two di"erent seasons: with the high and the low demand. We analyze it in a generaldeterministic dynamic model. We study the optimal management of the nuclear pro-duction as the management of a nuclear fuel reservoir in a perfectly competitive marketwith both nuclear and non-nuclear thermal generation. Then we run a very simple nu-merical model of power stations with nuclear plants being not operated strictly base-loadbut within a flexible dispatch frame (like the French nuclear set). A flexible nuclear setis operated to follow a part of the demand variations. Our simulation shows how thatnuclear set could rationally follow the seasonal variations of demand and exercise some“capacity withholding” market power if not fearing the market monitor. It suggests thatnon-nuclear thermal could stay marginal during most of the year including the months oflow demand (summer season).

Key words : Nuclear technology, thermal technology, electricity, nuclear fuel “reser-voir”, perfect competition, merit order, follow-up of load, seasonal demand.

JEL code numbers : C61, C63, D24, D41, L11.

!Director of the Florence School of Regulation, Loyola de Palacio Professor at the European UniversityInstitute in Florence - Robert Schuman Center.

†Professor in Mathematics at University of Paris 1 Pantheon - Sorbonne, Center of Economy of Sorbonne,Paris School of Economics.

‡PhD Student in Economic Sciences at University Paris - Sud 11, Department of Economics UniversityParis-Sud 11 ADIS - GRJM.

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1 Introduction

The nuclear generation technology is mainly di!erentiated from other generation technologiesby its very high fixed cost and relatively low marginal cost. Consequently, nuclear is used toserve base-load: targeting a given and constant minimum demand. In the past, in an integratedmonopoly regime, nuclear could “easily” be operated at its maximum capacity; and this didhelp to covering its fixed costs (e. g. United Kingdom, see Ref. [8]). However in numerouscountries, electricity systems are passing from monopoly, to a frame of competitive markets (e.g. European Union) which reopens -both empirically and theoretically- the question of nuclearoperation.

Economic reasoning supports that in a changing environment for the production process,the choice and operation of generation technology should also change. It therefore questionshow nuclear plants should be operated in an open market frame. What could be the optimalmanagement of a nuclear set in a competitive setting? Within this new competitive framework,we assume that we have to distinguish two time horizons of operation: the short-term and themedium-term.

The short term operation of plants is related to daily variations of demand. The core pointhere is the daily to intra-day flexibility of nuclear generation. Can the plant manager adjustdaily or intraday its power to follow the demand in order to maximize its “costs versus revenue”margin. Of course the nuclear output flexibility is constrained by the generation ramping ratethat bounds the variation of the output between two steady production periods. The shortterm horizon is therefore organized around a “hard” technological constraint: the inherentoperational flexibility of a given nuclear plant technology. Di!erent nuclear technologies havedi!erent operational flexibilities. In France that short term flexibility is quite high for a nuclearset.

However we do believe that the second time horizon - the medium - deserves at least the sameor even more attention than short term. While the short term horizon is caped by a straighttechnological constraint (the operational flexibility of nuclear output), the medium term horizonappears to be a “pure” economic strategy question. In the medium-term, the nuclear managerhas to set his seasonal variation of output according to her forecast of demand level. We assumeonly two stylized seasons: a “winter” season (with high demand) and a “summer” season (withlow demand). In this medium term horizon, a core feature is that nuclear fuel appears to be a“reservoir” of energy - partly similar to a water reservoir of energy. Thus, we will look at it as arational economic analysis of the operation of a nuclear fuel “reservoir”. The nuclear manager isallocating a limited and exhaustible amount of nuclear fuel between the di!erent seasons havingdi!erent demands and pricing features. The characteristic of nuclear as reservoir is based onthe discontinuous reloading of the nuclear reactor. Nuclear plants stop only periodically (from12 to 24 months) to reload their fuel. Then managers have to decide what is the expectedand actual length of each “campaign of production”; as the final amount and actual temporalprofile of fuel uses.

To build the corresponding modeling, we aim at establishing a microeconomic model ofoperation of nuclear power stations in a flexible market based operation framework. We ab-solutely do not claim that the French nuclear producer did or is doing what we are modeling.We only treat academically a hypothetical case while borrowing some key features from theexisting world. The French nuclear set is of course very appealing for us : because of the nu-clear importance (80% of the French electricity production being nuclear); because the Frenchnuclear set does not entirely operate as base load and has developed a unique load-followingmanagement to partly respond to the daily and seasonal variations of demand; because theparticular geographical position of France connected to six di!erent countries and the core of

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continental Europe; also, to end, because the existing economic literature on this precise topicis more than extremely reduced and close to a vacuum.

Assuming that nuclear plants and hydro storage plants have in common a few similar reser-voir characteristics, despite their strong operational di!erences, we start in section 2 with ananalysis of nuclear fuel as “reservoirs”. In section 3, we build a model to study the operationof “market based” nuclear reservoirs in a perfect competitive setting. This model can be usedlike a benchmark to trace and measure an hypothetical market power exercised by nuclearproducers (See Smeers (2007)). In section 4, we collect some basic data to feed our model. Insection 5, we run numerical tests of our model with that set of data. Section 6 concludes.

2 Medium term aspect : The characteristic of the nu-clear fuel “reservoir”.

There are few theoretical analyses of the operation of nuclear plants in a competitive marketwhile the di"culties of that modeling are numerous. It is obvious that gas or coal power stationsoperate a load follow-up, which implies a variable fuel consumption and supply. This is not thecase with nuclear power. The existing economics of nuclear estimate that nuclear plants shouldrun all the year at the maximum of their capacity to cover their extremely high fixed costs.Such nuclear plants in a competitive market should roughly be price-taker. This is why nucleartechnology is assumed to resemble to the hydraulic run-of-river because the latter does not tryto make any follow-up of load. In the French case, nuclear generation is not of that kind (seeRef. [15]) France is distinct from other countries like UK because its far higher generation ofnuclear power implies not to run nuclear plants strictly as base-load units. In the French casethe similarity between hydro and nuclear would spontaneously be that both are reservoirs.

From a technical point of view, the heart of a French nuclear reactor consists in a bunchof nuclear fuel bars controlled trough neutralizing graphite bars moving under control fromoutside. These reactors stop periodically to reload their fuel and neutralizing bars after theopening of the heart of the reactor). After this reloading a new period (named “campaign”) ofproduction starts. A campaign consists in transforming the potential energy contained in theuranium bars in electricity to be consumed (between 12 and 18 months generally). The regularlength of a campaign depends on many factors (technical specificities of the reactor, size, age,management decision to reload the reactor’s heart per third or quarter of its full capacity, typeof nuclear fuel put into the fuel bars, forecasted average rate of use of the reactor, regulatoryconstraints issued by safety inspectors...) (see Ref. [11], [19], [20])

Reloading of reactors is to be avoided when the level of demand is high (which is winter inFrance). For operational reasons, the normal duration of a campaign is determined in advanceto get a general scheduling of reloading. That action requires the intervention of many qualifiedpersons external to the nuclear operator. It also has to be consistent with the scheduling ofall the 58 reactors of the French set. As a result the manager of a nuclear plant has a givenhorizon in which to manage the fuel stock. However the exact duration of each plant campaigncan be shortened or expanded at the request of the general nuclear set manager.

Assuming that nuclear energy has to be sold on the wholesale market, we bet that it willbe sold like a stream of “energy blocks”.

Energy blocks are fixed quantity sold over a very short period of time at a price determinedby the market at each period (then a “spot price”). The French market has periods of half anhour, which means 48 prices per day, 17520 prices per year. Such spot prices are very volatilefrom day to day, during the day and along the year at the same time (see Figure 1, Source :Powernext). These 17520 prices are essentially determined by three characteristics (hour, labor

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day in opposition to the end of the week or at holidays, monthly components). There is also astrong seasonal variation.

01/07/2008 International Symposium on Forecasting2

Spot prices on the French market (Powernext)peak hours

0.00

100.00

200.00

300.00

400.00

500.00

600.00

Summer 2006

Heat wave

Winter 2007

Cold wave, strike, low wind

prod…

Spike

definition

Potential

explanations

Figure 1: Spot prices on the French market (Powernext).

Of course, the total value of the electricity produced during a campaign of nuclear fuelreservoir depends in a crucial way on the temporal profile of generation and how it can respondto the variation of demand and market price.

We can then benefit from an analogy with a hydraulic producer managing her reservoir andhaving to allocate the water of her basin between di!erent periods of generation. To analyzethe management of the reservoir of nuclear fuel, we can now draw from the important literatureon the optimal management of hydraulic reservoir (see Ref. [1], [2]).

There are however di!erences between the nuclear plants and the hydro storage stationswith respect to the characteristics of the “reservoir”. An important point of di!erentiationconcerns the timing of the reloading of the “reservoir” (nuclear / hydraulic). In the case ofnuclear the reloading of the reservoir depends on the producer who is responsible for the optimalmanagement of shut downs of the nuclear unit. While the hydraulic reservoir will be reloadedonly when rain will have enough fallen. Another di!erence is that nuclear reloading stopsproduction. Hydraulic reservoir stations do not stop during the reservoir’s reloading while theycannot choose when and how much to reload (typical “seasonal reloading”).

However this seasonality of reloading has also to be considered in the nuclear case. A“good” seasonal allocation of the shut-downs of nuclear plants consists in avoiding shut downsin winter (high demand) and concentrating them to the maximum between May and September(low demand) (see Figure 2). Thus, the producer takes into account the level of demand whenshe chooses when to reload the heart of the reactor. A fundamental point of the optimization ofthe French nuclear set is therefore the allocation of the shut downs. Their timing and frequency

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0 10 20 30 40 500

2

4

6

8

10

12

14

7

8

9

10

11

12

13

consumption 2006 (TWh)Number of shut-downed units (2006)

time (weeks)

Figure 2: Availability of nuclear units (EDF).

are determining the length of the campaign for nuclear plants. In a market based electricityindustry, the goal is here the maximization of the production’s value.

Optimality and suboptimality of the nuclear set. A numerical example. It is usuallythought that a flexible management of the generation of nuclear plants does not make sense. Itis because, the nuclear plants are deemed to cover only the base-load demand by operating ina constant way to their maximum capacity in order to recover their fixed costs on the biggestpossible amount of energy generated. In a competitive market if the marginal technology1 isnuclear all the year, the nuclear producer cannot cover his fixed costs. Indeed, the fixed costsand the variable costs will be covered let say on a yearly basis only if the nuclear set has itsoptimal size within the whole generation set (see Ref. [9]). Spector gives the following numericalcase. The optimal nuclear set for France corresponds in a duration of nuclear marginality of40%. This exactly means that nuclear plants can cover all their fixed-costs trough market-based competitive prices during the 60% of time period of marginality of the other generationtechnologies (except wind: basically coal, gas and fuel oil) on the basis of marginal costs of thelatter.

However, the nuclear set could be smaller than its optimal level. In this case, even in presenceof perfect competition, it would be remunerated above its marginal costs more than 60% oftime,... Consequently, its holders would profit from a scarcity rent, whatever the intensity ofcompetition would be on the wholesale market. A temporary scarcity rent can also occur ifa sudden modification a!ects the supply or the demand (e. g. increase of the cost of fossilenergies, increase of national consumption or foreign demand), because the nuclear set cannotadjust instantaneously.

Spector estimates that vis-a-vis the actual size of continental European market the Frenchnuclear set is “sub-dimensioned”, which makes EDF recipient of a scarcity rent (see Ref. [10]).

1According to the merit order, which is a way of ranking available technologies of electricity generation, inthe same order like their marginal costs of production, a combination of di"erent generation technologies isrealized to reach the level of the demand at a minimum cost. The price of the market is therefore determinedby the marginal cost of the “last technology” used to equilibrate supply and demand (perfect competitive case).This technology is also called marginal technology.

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However starting from the year 2007, a program of renewal/extension of the French nuclear setis under consideration. An expansion of the French nuclear set would notably be possible at thetime of the renewal of the current set, between 2017 and the decade 2030, if the current reactorswere replaced by new more powerful generation reactors (like EPRs -“European PressurizedReactors”-).

3 Model : Perfect competitive case

In this section, we describe our general deterministic model of a perfectly competitive electricitymarket where the producers manage both nuclear and non-nuclear thermal plants. We assumea perfect competition according to which firms treat price as a parameter and not as a choicevariable. Price taking firms guarantees that when firms maximize their profits (by choosingthe quantity they wish to produce and the technology of generation to produce it with) themarket price will be equal to marginal cost. The general frame is also characterized by perfectequilibrium between supply and demand and perfect information among producers. First, ourmodeling aims at determining the optimal management of the nuclear generation set in thatcompetitive regime. More precisely we want to focus on the medium term horizon which ischaracterized by the seasonal variation of the level of demand between winter and summer.Second, the constraints imposed by generation capacity and fuel storage play a central role todetermine the equilibrium outcomes in this electricity market.

3.1 Modeling of the demand

The demand, being given, is considered perfectly inelastic. It is obviously a simplification. Itcan nevertheless be motivated by two arguments. In short-term to medium-term, the demandis less sensitive to price because it is already determined by previous investments in electricaldevices and ways of life whose evolutions require several years.

Electricity is sold to consumers2 by retailing companies. There is no bilateral contractingregime between retailers and producers. The wholesale spot prices are paid by the retailers tothe producers.

3.2 Modeling the time horizon

The time horizon of the model is T= 36 months3 beginning by the month of December. Adiscretization of the time frame has been conducted by using weeks instead of months and itleads to the same conclusion. A nuclear producer has mainly two options with respect to thescheduling of reloading : (i) 1/3 of fuel reservoir that corresponds to 18 months of campaignand 396 days equivalent to full capacity for a unit of 1300 MW, (ii) 1/4 of fuel reservoir thatcorresponds to 12 months of campaign and 258 days equivalent to full capacity for a unit of 1500MW. We retain the second mode of reloading and therefore a duration of campaign equivalentto 12 months in order to have a cyclic model with a periodicity of one year. The period ofcampaign is then decomposed into 11 months being the period of production and 1 monthcorresponding to the of reloading of the fuel. We also assume that value is not discountedduring the period of 36 months.

2In reality in the French case, most of the consumers use a fixed price contract or a regulated price contractset by the government. The regulated price does not follow the wholesale market price evolution while theprivate “eligible” contract does each six or twelve months.

3The time horizon of the model is a multiplicative of twelve being expressed in months. Therefore it couldbe modified.

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3.3 Modeling the generating units

We define 12 types of generating units both for nuclear or for non-nuclear thermal generatingunits. A nuclear or a non-nuclear producer operates with several types of generating units.How do we define our twelve “types” then ? Each type indexed by j = 1, · · · , 12 correspondsto a di!erent month of reloading for the nuclear units. More precisely, let us consider that aunit which belongs to the type of unit j = 1 (respectively j = 2, · · ·, j = 12) shut-down themonth of December (respectively January, · · ·, November). We assume that the nuclear unitsonly di!er by the month of reloading, which means that all these units have the same capacityand the same cost. The level of the nuclear) production during the month t = 1, · · · , T for theunit j will be denoted by qnuc

jt . Furthermore, the maximum nuclear production realized by theunit j during a month is given by the parameter Qj,nuc

max , while the minimum nuclear productionis equal to Qj,nuc

min . The variable Sjt , which represents the quantity of fuel stored in the nuclear

reservoir and available to the unit j at the beginning of the month t, is the potential energythat can be produced by this stock.

Symmetrically, the non-nuclear units also have their own common capacity and cost. Thelevel of the non-nuclear production during the month t = 1, · · · , T for the unit j will be denotedby qth

jt ). Furthermore the maximum non-nuclear production that the unit j can realize duringa month is given by the parameter Qj,th

max, while there is no minimum non-nuclear productionQj,th

min = 0.

3.4 Modeling the production costs

The nuclear cost function is composed by the fixed part which is determined by the cost ofinvestment, the fixed cost of exploitation as well as the taxes and the variable part whichincludes the variable cost of exploitation and the fuel cost. We assume that the cost functionCnuc

j (.) of the nuclear production is linear and defined as

Cnucj (qnuc

jt ) = anuc + bnucqnucjt .

The thermal cost function is composed by the fixed part which corresponds to the cost ofinvestment, the fixed cost of exploitation as well as the taxes and the variable part which isconstituted by the variable cost of exploitation, the fuel cost, the cost of CO2 as well as thetaxes on the gas fuel. We assume that the non-nuclear production has a quadratic cost functionCth

j (.) which is the following

Cthj (qth

jt ) = ath + bthqthjt + cthqth

jt2.

The nuclear and non-nuclear cost functions are monotone increasing and convex functionsof qnuc

jt and qthjt respectively. We choose a quadratic cost function in the case of non-nuclear

thermal because of the increasing marginal cost of the non-nuclear production since it resultsfrom di!erent electricity generation technologies (e. g. coal, gas -combined cycle or not-, fueloil). Furthermore, the non-nuclear production needed a non constant function in order torecover its fixed costs. So, we assume that the marginal cost of nuclear is a constant functionof qnuc

jt while that of the thermal is an increasing function of qthjt .

3.5 Modeling the nuclear fuel stock

Let us denote Sjreload, the nuclear fuel stock of reloading available to the unit j. The evolution

of the nuclear fuel stock is determined by the following rules

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Sj1 given, Sj

t+1 =

!Sj

t " qnucjt , if no reload during month t for unit j

Sjreload, if unit j reloads during month t

(1)

The relationship 1 gives the evolution of the stock given the flow of the nuclear production.In the case that t is the month during which the producer reloads the nuclear fuel of thereactor, the stock at the beginning of the following month (beginning of the campaign) is equalto Sj

reload. Moreover, we impose

SjT+1 ! Sj

1 (2)

The constraint 2 implies that the producer must return his nuclear units at the end of thegame in the same state as the initial one. This means that each of them has to finish theperiod T at least with the same quantity of stock as the initial one. In this way the producerhas to spare his nuclear fuel during the production period. Such a constraint is implicit if theend of the period T coincides with the end of the campaign. In the case of virtual plants theconstraint 2 has to be imposed together with a system of penalty.

3.6 A “naive” modeling of the optimal production behavior

If the unit j dispose at time t the stock Sjt , then it could try to solve the following optimal

production problem

maxqnucjt ,qth

jt

pt · (qnucjt + qth

jt )" Cnucj (qnuc

jt )" Cthj (qth

jt )

subject to the constraints

!Qj,nuc

min " qnucjt " Qj,nuc

max , if no reload during month t for unit jqnucjt = 0, if unit j reloads during month t

(3)

0 " qthjt " Qt,th

max (4)

where the price pt is given (perfect competition) by the equality between supply and demand.The quantity Qt,th

max represents the maximum thermal production (coal, gas, fuel, etc...) that canbe realized during the month t by all units and corresponds to the nominal thermal capacity.

The constraint 3 shows that the nuclear production of each month is bounded by the min-imum/maximum quantity of nuclear production which can be realized during a month. Thethermal production is a non negative quantity which is also bounded by the maximum thermalproduction (constraint 4); the producer may use the thermal resources to produce electricityuntil he reaches the level of demand of the corresponding month by respecting at the same timethe constraint 4.

The solution of this problem determines the new level of stock Sjt+1. Unfortunately, such a

process does not take su"ciently into account the constraints of the stock. In particular, onemay face an insu"cient level of stock in order to produce Qj,nuc

min every month.

3.7 Alternative constraint of the nuclear fuel stock

In order to involve all the “auxiliary variables” of the nuclear fuel stock Sjt in the optimization

problem, let us first remark that there exist implicit conditions. Let us consider by examplethat the unit j at month t has 3 months of campaign that remain until the month of reloading(including the month t), then the constraints 1 and 3 imply the following condition:

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Sjt ! 3 · Qj,nuc

min

This inequality results from the comparison between the current level of stock and thequantity 3 · Qj,nuc

min which is equivalent to the total quantity of nuclear fuel that the unit j hasto reserve in order to realize the minimum nuclear production at each of the remaining monthsuntil the end of the campaign. In addition, one has to take into account the final constraint(constraint 2). Let us introduce by backward induction the variable Sj

t,min which is the quantityof nuclear fuel that the unit j has to reserve at the beginning of the month t in order to “finish”the campaign; that is to produce at least Qj,t+1,nuc

min during the month t + 1, Qj,t+2,nucmin during

the month t + 2, · · · until one reaches either the month of reloading or the end of the game.The notation Qj,t,nuc

min represents the minimum nuclear production realized by the unit jduring the period t. Let us notice that during the month of reloading the minimum nuclearproduction provided by the unit j is zero. Otherwise, Qj,t,nuc

min equals to the minimum nuclearproduction Qj,nuc

min that can be realized during a month. More precisely,

Qj,t,nucmin =

!Qj,nuc

min , if no reload during month t for unit j0, if unit j reloads during month t

We define Sjt,min as following

for t < T, Sjt!1,min =

"#

$

0, if unit j reloads during month t0, if month t is the last month of campaign for unit jSj

t + Qj,t,nucmin , in other cases

for t = T, SjT,min =

!Sj

1, if no reload during month T for unit j0, if unit j reloads during month T

It is important to mention that the maximum nuclear production that the unit j is ableto provide during the month t depends obviously on the level of stock at the beginning of themonth t. In particular, if the stock is close to zero, then the maximum nuclear production willbe close to zero too.

We define Qj,t,nucmax as the maximum nuclear production realized by the unit j during the

month t. We remark that Qj,t,nucmax is equal to the minimum between the remaining stock (quan-

tity of stock available to the unit j at the beginning of the month t minus the reserve) whichis available at the beginning of t and the maximum nuclear production that can be realizedduring a month. However, if t is the month of reloading then Qj,t,nuc

max is equal to zero.

Qj,t,nucmax (Sj

t ) =

!min(Sj

t " Sjt,min, Q

j,nucmax ), if no reload during month t for unit j

0, if unit j reloads during month t

Later, we will use the reduced notation Qj,t,nucmax for the variable Qj,t,nuc

max (Sjt ). This notation

is not ambiguous since the problem will be solved recursively; firstly one computes the solutionfor t = 1, which allows to determine the level of stock Sj

1, then we can solve the problem fort = 2, etc...

So, the first constraint of our optimization problem will take the reduced form

Qj,t,nucmin " qnuc

jt " Qj,t,nucmax , for all j, t.

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3.8 Decentralization rules

The comparison between the aggregate maximum nuclear production realized each month bythe di!erent type of units and the corresponding demand determines the perfect competitiveprice as well as the optimal levels of nuclear and thermal production. Note that the monthlydemand which is considered in this model results from the di!erence between the level ofdemand D(t) observed at the month t and the aggregate hydraulic production Qt,hyd

Tot providedduring the month t. Let us also remark that since the hydraulic technology (run-of-river) is abaseload generation technology which is never marginal, it is necessary to call up nuclear tocover the di!erent levels of demand either partially or totally.

At each date t, the price pt is determined by the equality between supply and demand :

12%

j=1

qnucjt (pt) +

12%

j=1

qthjt (pt) + Qt,hyd

Tot = D(t),

where (qnucjt (pt), qth

jt (pt)) is the solution of the optimization problem involving the parameter pt.Note that this condition is not correctly written since the solution is not necessarily unique,the nuclear production is not a function of the price but a correspondence. This is why we willdistinguish the two following rules.

3.8.1 Decentralization rule when nuclear is marginal

More precisely, if the demand at the month t, D(t) " Qt,hydTot , is inferior or equal to the corre-

sponding aggregate maximum nuclear production available at the date t, then the nuclear is the“last technology” used to equilibrate supply and demand (marginal technology) and accordingto the “merit order” the price is determined by the marginal cost of the nuclear productionwhich is constant. In this case, the thermal production is zero and the total nuclear productionis redistributed between the j units in order to respect the constraints of our optimizationproblem and a rule of “equal treatment”. This rule guaranties that the o!er is equal to thedemand, all constraints are satisfied and the ratio of use of the “mobilizable capacities” is thesame. The level of the nuclear and thermal production is determined respectively as

qnucjt = Qj,t,nuc

min + (Qj,t,nucmax "Qj,t,nuc

min )

D(t)"Qt,hydTot "

12%

j!=1

Qj!,t,nucmin

12%

j!=1

Qj!,t,nucmax "

12%

j!=1

Qj!,t,nucmin

, for all j

and

qthjt = 0, for all j,

The value of the perfect competitive price is

pt = bnuc.

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3.8.2 Decentralization rule when thermal is marginal

If the aggregate maximum nuclear production is not su"cient to cover the demand D(t)"Qt,hydTot

during the month t, the producer uses his thermal resources to generate electricity in order toreach the level of demand at t. This means that the thermal generation technology is themarginal one and according to the “merit order” the perfect competitive price is given by themarginal cost of the thermal production which is a linear increasing function of the thermalproduction qth

jt . In this case, each nuclear unit j produces to the maximum of its availablecapacity Qj,t,nuc

max and each thermal unit produces symmetrically in order to cover the “residualdemand” (demand D(t)"Qt,hyd

Tot at month t minus the aggregate maximum nuclear production atmonth t) which corresponds to it. Consequently, the quantity of nuclear and thermal productionis respectively

qnucjt = Qj,t,nuc

max , for all j

and

qthjt =

D(t)"Qt,hydTot "

12%

j!=1

Qj!,t,nucmax

12 , for all j.

The perfect competitive price is defined as

pt = bth + 2cthqthjt .

3.9 Nuclear production planning

According to the modeling of the optimal production behavior introduced in subsection 3.7,the production plan resulting from the decentralization rules is inadmissible because of theviolation of the production constraints. In particular, we observe the violation of the maximumthermal production constraint. More precisely, the thermal production is insu"cient since it isnot able to cover the demand especially during the last months of the period T .

12

0 5 10 15 20 25 30 35 40

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

Nuclear Production

Time

0 5 10 15 20 25 30 35 40

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

Thermal Production

Time

Figure 3: Nuclear/Thermal production in the first scenario

We can see in the Figure 3 that the thermal production exceeds its maximum value to coverthe demand during the last two months of the period T . Therefore, the nuclear set has to bemanaged so that the equality between supply and demand is respected.

For this reason, we provide a second scenario in which we propose a planning of the nuclearproduction in order to obtain a compatible production which satisfy the following condition :

(i) It respects the minimum/maximum production constraints as well as the constraints ofthe nuclear fuel stock. In particular, it guarantees that the maximum thermal production (coal,gas, fuel, etc...) will not be exceeded by the monthly demand. More precisely, the plannednuclear production is superior or equal to the monthly demand minus the maximum thermalproduction

&12j=1 qplan

jt ! D(t)"Qt,hydTot "Qt,th

max, for all t.

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Our algorithm determines a variable (qplanj,t ) satisfying (i). The variable qplan

j,t , which iscomputed by backward induction, tries to treat symmetrically the units (same level of activity).However, we have to take into account the disparities between the units (level of nuclear fuelstock, being or not close to the moment of reloading, etc...)

Note that there are several solutions which satisfy the condition (i).

4 Data

The data used in our numerical dynamic model concerns the level of french demand during theyear 2006 " 2007, the fixed and variable costs of nuclear, coal and gas production, the levelsof capacity of the hydraulic (run-of-river), nuclear and thermic technology and the nuclear fuelstock of reloading. More precisely, the information concerning the consumption comes from themanager of the French Transmission & System Operator (called RTE in France). We find therethe historic of the daily consumption in MWh for the month of December 2006 and the entireyear 2007. Then, we aggregate the daily consumption for each month in order to determine thetotal amount of monthly consumption. We use RTE’s data in order to determine the annualcapacity of nuclear as well as the annual capacity of gas and coal. The given values concern thefrench set of reference for the year 2009. In addition, the informations concerning the nuclearfuel stock of reloading as well as the annual capacity and production of the hydraulic generationtechnology are provided by EDF (Electricity of France).

The data related to the costs of production comes from the report “Reference Costs of Elec-tricity Production“ realized by the General Direction of Energy and Raw Materials (DIDEME)[47] in 2003. The part in which we are more interested is the second part of the report wherewe can find informations with respect to the technical characteristics, the costs, the resultsand the analysis of sensibility for the di!erent types of energy (nuclear, coal, gas, fuel). In thesection which is relative to the technical characteristics of each energy, there are numbers withrespect to the installations of reference, the life duration, the availability of the energy unitsas well as the management of the fuel in the case of nuclear. These numbers are estimatedfor the year 2007 and 2015. In addition, the section of costs gives informations for the cost ofinvestment, the variable / fixed cost of exploitation, the cost of fuel as well as the external costs(e. g. cost of CO2, cost of a major nuclear accident, etc...). The section which includes theresults provides us with the total cost of production for a base (8760h) and semi-base (3000h)operation. Furthermore, we find a decomposition of the total cost of each technology which isthe following : cost of investment, variable and fixed cost of exploitation, fuel cost, taxes, R&Dcosts for the nuclear and cost of CO2 per ton in the case of coal and gas for the same levelsof operation with these mentioned previously. These costs are estimated for the year 2007 and2015 and for di!erent levels of discount (3%, 5%, 8%, 11%) taking also into account the influ-ence of dollar on the value of the production cost. Finally, there is an analysis of sensibility ofproduction costs in base with respect to the principal parameters retained for each technology(e. g. investment cost, availability, life duration, etc...).

It is important to emphasize that the estimation of the fixed/variable costs depends a loton the rate of discount. For example, the value of the cost of investment for nuclear is 6, 4Euros/MWh using a rate of discount equal to 3% while the same cost is estimated at 16, 3Euros/MWh for a discount rate of 8%. The main reason is that some of the operations realizedin the case of nuclear have a high duration. For example, the construction of a nuclear unitneeds at least 5 years which is a long period in comparison with other technologies whose timeof construction is less important (e. g. 26 months for gas). Furthermore, the life durationof the currently thermal units is 30 years, while the new generation reactors EPR (European

14

Pressurized Reactor) are conceived for a functioning of 60 years.

Figure 4: Nuclear fuel cycle.

In addition, more than other energy resources such as coal, oil and natural gas, uraniumhas its own distinctive and very complicated fuel cycle. There are several steps in the nuclearfuel cycle - mining and milling, conversion, enrichment, and fuel fabrication (see Figure 4). It’sobvious that the duration of the nuclear fuel cycle is significantly high. Finally, one of the maincharacteristic of nuclear electricity producer units is the dismantling of nuclear plants whichexploitation duration lasts about 25 years (Source : Brite/Euram III : Projects). Therefore,the rate of discount used in each case is an important factor which can significantly a!ect thenuclear costs.

Moreover, we need to mention that the future value of the costs of thermal production isvolatile for di!erent reasons. One of them concerns the price volatility of CO2. The price too!set one ton of CO2 varies substantially. Volatile CO2 prices have been observed already inEurope, especially in the Phase I of the Emission Trading System (ETS)4 that ran from 2005to 2007. The CO2 futures prices for 2007 delivery ranged between almost 0 and 30 euros perton of CO2 during just the twelve month period May 2006 - May 2007 in Phase I (Source : TheBrattle Group, Cambridge).

Another one is the extreme movements in oil prices which have been observed in the lastfew years (see Figure 5, Source : Oilnergy).

4The European Union Emission Trading System is the largest multi-national, emissions trading scheme inthe world and is a major pillar of EU climate policy. Under the EU ETS, large emitters of carbon dioxide withinthe EU must monitor and annually report their CO2 emissions, and they are obliged every year to return anamount of emission allowances to the government that is equivalent to their CO2 emissions in that year.

15

Figure 5: Average monthly data from January 1978 through March 2009.

15

Figure 6: Evolution of electricity production costs.

In the contrary, the cost of nuclear fuel plus the exploitation costs3 are less volatile duringtime than the costs of thermal production (See Figure 6, Source : World Nuclear Association,Global Energy Decisions).

6 Numerical Illustration

7 Conclusions

8 Future research

3The above data refer to fuel plus operation and maintenance costs only, they exclude capital, since thisvaries greatly among utilities and states, as well as with the age of the plant.

Figure 6: Evolution of electricity production costs.

On the contrary, the cost of nuclear fuel plus the exploitation costs5 are less volatile duringtime than the costs of thermal production (see Figure 6, Source : World Nuclear Association,Global Energy Decisions). This is due mainly to two reasons, the first is that the NaturalUranium cost which is one of the component of the fuel cost enters only for one very weakshare the cost of nuclear kWh, the second is the stabilization of the price as a result of the hugedelay between extraction of natural Uranium and fabrication of the nuclear fuel.

For our modeling, we choose the scenario according to which one dollar is equal to one euro,

5The above data refers to fuel plus operation and maintenance costs only, they exclude capital, since thisvaries greatly among utilities and states, as well as with the age of the plant.

16

the level of discount is 8%, the cost of CO2 per ton reaches the 20 euros, the price of coal is 30dollars per ton and the price of gas is 3.3 dollars per MBtu (1 MBtu=293.1 KWh). In addition,we have to mention that the value of the coe"cient ath involved in the thermal cost functioncorresponds to the fixed cost provided by the data (investment cost, fixed exploitation cost),while the other coe"cients have been determined by interpolation in order to meet the variablecost of coal and gas provided by our data base (fuel cost, variable exploitation cost, CO2 cost,taxes). Furthermore, the level of capacity of each nuclear unit has been simulated in order toapproximate the graphic of the figure 2, which shows the availability of the nuclear units perweek. Moreover, the initialization of the nuclear fuel stock has been done by simulating thenuclear fuel stock of each unit available at the beginning of the time horizon of the model.Finally, we considerate the losses of electricity during the transportation on the network, whichare estimated by RTE.

5 Numerical Illustration

We study the nuclear and thermal production decisions as well as the storage decisions analyzedin the previous section, within a simple numerical model.

0 5 10 15 20 25 30 35 40

2.8e+07

3.0e+07

3.2e+07

3.4e+07

3.6e+07

3.8e+07

4.0e+07

4.2e+07

4.4e+07

4.6e+07

Demand

Time

Figure 7: Simulated demand

The levels of the monthly demand (in MWh) obtained for the time horizon of our model(December 2006 " November 2009) are presented in the figure 7 (we suppose an exponentialaugmentation of the demand by using a rate of 1% per year). We can see the seasonal variationof the demand’s level between winter (high demand) and summer (low demand). More precisely,we observe high levels of demand during the months of November, December, January withdemand peaks realized during the month of December while the demand falls significantlyduring almost the entire period of spring (exception is the month of March) as well as duringthe period of summer (June " September). On the contrary, we do not observe any demandpeaks during the summer period which implies that there were no important increases of thetemperature.

17

0 5 10 15 20 25 30 35 40

0

10000

20000

30000

40000

50000

60000

70000

Production

Time

Figure 8: Simulated hydraulic/nuclear/thermal production

0 5 10 15 20 25 30 35 40

25000

30000

35000

40000

45000

50000

55000

60000

65000

Nuclear production

Time

Figure 9: Simulated nuclear production

18

0 5 10 15 20 25 30 35 40

0

2000

4000

6000

8000

10000

12000

14000

16000

Thermal Production

Time

Figure 10: Simulated thermal production

0 5 10 15 20 25 30 35 40

1.2e+08

1.4e+08

1.6e+08

1.8e+08

2.0e+08

2.2e+08

2.4e+08

2.6e+08

Nuclear fuel stock

Time

Figure 11: Simulated nuclear fuel stock

19

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

Price

Time

0 5 10 15 20 25 30 35 40

!1.4e+09

!1.2e+09

!1.0e+09

!8.0e+08

!6.0e+08

!4.0e+08

!2.0e+08

0.0e+00

2.0e+08

Profit

Time

Figure 12: Simulated price/Aggregated total profit

20

General simulation resultsWe notice that the thermal generation technology is marginal during the majority of the

months of the period T in order to equilibrate supply and demand while the nuclear technologyis marginal only at the beginning of the period T. More precisely, nuclear stays marginalduring almost the entire period of winter, spring and summer of 2007 (exception is the monthof March). In addition, nuclear follows the seasonal variations of the demand by becomingdecreasing during summer and increasing during winter. Furthermore, we observe that themonthly nuclear production never reaches its maximum value6 (see Figure 8, Figure 9, Figure10). Note that the reader should not focus on the precise value of the profit since the datadepends on many type of uncertainties (euro/dollar, oil prices, CO2 cost, discounting levels,etc...) and the modeling does not take into account the importations/exportations and theproduction coming from the renewable generation technologies (including hydro) (see Figure12).

We separate the period T into three subperiods according to the evolution of both nuclearand thermal production. More precisely, according to Figure 8, we distinguish the first subpe-riod during which nuclear is mainly the marginal technology, the medium subperiod which ischaracterized by the periodical evolution of the nuclear and thermal production and finally thelast subperiod.

First subperiodThe total nuclear production approaches its maximum level during the first months of pro-

duction (December 2006, January 2007) due to the overconsumption of the nuclear fuel stock.On the contrary, the thermal production is equal to zero during the first months of the periodT (with only exception the month of March) since the demand is covered exclusively by thenuclear production (see Figure 8, Figure 9, Figure 10, Figure 11). The price7 reaches its lowestvalues at the beginning of the period T because of the marginality of the nuclear production.Only exception is the month of March 2007 during which the thermal technology becomesmarginal. However, the price during this month is significantly lower than the price during thesame month of the following years because of the importance of the nuclear production at thebeginning of the period T which leads to a less important thermal production compared to thatrealized the month of March of the next years.

Note that if nuclear is overused at the beginning of the time horizon of the model, thencolossal losses are generated (see Figure 12).

Medium subperiodWe notice that the nuclear production follows the seasonal variations of the demand (high

production during winter – low production during summer) which means high levels of nuclearfuel stock during summer and low levels of nuclear fuel stock during winter. Therefore, theperiodical evolution of the nuclear production implies a periodical evolution for the nuclear fuelstock too. Note that the trend of the stock appears significantly below the “stock of reference”8

(see Figure 9, Figure 11).Moreover, we observe that the thermal and the nuclear production increase (respectively

decrease) simultaneously during almost the entire time horizon of our model, which correspondsto the notion of comonotonicity introduced by Yaari (1987) (see Figure 8). In addition, we cansee that the thermal production is high during winter (respectively low during summer) because

6The maximum nuclear production during a month is represented by the purple dotted line and correspondsto the nominal capacity of the french nuclear set.

7The red (respectively green) dotted line indicates the value of the price in the case that nuclear (respectivelythermal) is the marginal technology.

8The “stock of reference” is represented by the blue dotted line which shows the value of stock at thebeginning that is also the value of stock at the end.

21

of the high (respectively low) levels of demand. In particular, thermal production is increasingduring winter (beginning from October) until it reaches its peak values during the month ofDecember and March. Afterwards, thermal production decreases because of the summer periodwhich is a low demand season. However, it stays marginal during summer because of the lowlevels of the nuclear production (see Figure 10). Consequently, price is high during the monthsof winter by taking its highest value during the month of December and low during summer.The aggregate profit obtained by the producer is high during winter and at the beginning ofspring of 2008, 2009 while lower profits are realized during summer (see Figure 12).

Last subperiodHowever, the total nuclear production is significantly low during the last four months of the

period T (August, September, October, November) which allows the rebuilding of the nuclearfuel stock so that the producer returns his nuclear units at the end of the game in the samestate as the initial one (see Figure 9, Figure 11). On the contrary, the thermal units increasesignificantly their production during these months in order to cover the increased levels ofdemand because of the important decrease of the nuclear production. In particularly, thermalproduction reaches its maximum value9 during the last two months of the model’s time horizon(see Figure 8, Figure 10). For this reason, the price and the production profit reach theirhighest levels during this period (see Figure 12).

It is important to notice that for this data, on this level of time slicing, we don’t observethe conclusion of Spector about the actual size of the nuclear set, which does not seem to besignificantly below the “optimal size”. Note also that if T ! 36 months, then the evolutionof the nuclear and thermal production during the first and the last subperiod as well as theperiodical evolution of the production during the medium subperiod is the same. However, themedium subperiod last longer than that resulting from the initial time horizon (T = 36). Thisis not the case for the first and the last subperiod.

6 Conclusion

In this paper, we examine the question of the optimal management of the nuclear generationset in a perfectly competitive regime. In particular, we emphasize the medium-term approachwhich takes into account the seasonal variation of the demand between winter (high demand)and summer (low demand). The innovative idea of this paper consists on the fact that nuclearfuel functions like a “reservoir”, which allows di!erent placements of the nuclear productionduring the di!erent seasons of the year. We describe the characteristic of the nuclear fuel“reservoir” and we give a numerical example provided by Spector in order to have a betterunderstanding of the notion of optimality and suboptimality of the french nuclear set. Then,we propose a deterministic multi-period model to study the perfect competitive case in anelectricity market where the producers control both the nuclear and the thermal generationtechnology. Afterwards, we propose the “decentralization rules” considering some operationalconstraints related to the level of the nuclear production and the nuclear fuel stock. The e"cientproduction levels (nuclear, thermal) as well as the price value resulting from these rules dependon which generation technology is marginal during the observation period.

We distinguish three di!erent scenarios in this paper : The first scenario supports that thenuclear technology is used to cover the base-load demand by functioning in a constant way tothe maximum of its capacity. However, this is not the case for the french nuclear set which wasused to meet not only the base-load demand (share of constant consumption throughout the

9The maximum thermal production during a month is represented by the white blue dotted line and corre-sponds to the nominal thermal capacity (including coal, gas, fuel, etc...) of the french set.

22

year), but also partly the semi-base demand (share of variable consumption) (see Ref. [16]).We also studied a second scenario resulting from an alternative attempt to model the nuclearproduction. In the two first scenarios, the minimum/maximum production constraints are notrespected, so we provide a third scenario in which a planning of the nuclear production isproposed in order to achieve a compatible nuclear production.

According to the simulation results of the third scenario, we find high levels of nuclearproduction during the months of high demand (winter) and low levels during the months oflow demand (summer), which shows that nuclear is a load-following generation technology. Aswe expected, the evolution of the nuclear fuel stock is the opposite of the evolution of thenuclear production (low levels of stock during winter – high levels of stock during summer).In addition, we noticed that the di!erent values of the nuclear fuel stock obtained during theperiod T remain significantly lower than the “reference” value of stock. Furthermore, the unitsincrease their thermal production during winter and they decrease it during summer, accordingto the corresponding demand levels and the levels of the nuclear production. However, thermaltechnology remains marginal during both seasons in the medium and last subperiod of thetime horizon T . Consequently, the price is determined by its marginal cost most of the timeexcept at the beginning of the period T . In particular, we observed that the price takes itshighest values during the period of winter and its lowest during the period of summer. Finally,producer obtains higher profits during winter and lower profits during summer.

Note that the optimal production behavior is determined by an optimization per monthproduction problem, which consists in the maximization of the production value during amonth given the production of the previous month. However, this mode of operation could bequalified myopic because it’s not based on the optimization of the production over the entireperiod of the campaign. More precisely, the intertemporal optimization will result from themaximization of the value of the production that is realized during the period of campaign (11months) and it will lead to the determination of the global optimum of the optimal productionproblem. This study will constitute the object of a future work.

23

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