A MEAN FIELD MODEL FOR THE DEVELOPMENT OF RENEWABLE CAPACITIES CLÉMENCE ALASSEUR MATTEO BASEI CHARLES BERTUCCI AND ALEKOS CECCHIN

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A MEAN FIELD MODEL FOR THE DEVELOPMENT OF RENEWABLE

CAPACITIES

CLÉMENCE ALASSEUR, MATTEO BASEI, CHARLES BERTUCCI, AND ALEKOS CECCHIN

Abstract. We propose a model based on a large number of small competitive producers of

renewable energies, to study the eﬀect of subsidies on the aggregate level of capacity, taking

into account a cannibalization eﬀect. We ﬁrst derive a model to explain how long-time equi-

librium can be reached on the market of production of renewable electricity and compare this

equilibrium to the case of monopoly. Then we consider the case in which other capacities of

production adjust to the production of renewable energies. The analysis is based on a master

equation and we get explicit formulae for the long-time equilibria. We also provide new numer-

ical methods to simulate the master equation and the evolution of the capacities. Thus we ﬁnd

the optimal subsidies to be given by a central planner to the installation and the production in

order to reach a desired equilibrium capacity.

1. Introduction

As electricity production is responsible for around a third of worldwide CO2emission, its

decarbonization is of course a huge challenge. As such, the share of electricity produced from

renewable capacities such as hydro, wind, photovoltaic is widely growing. However, to achieve

decarbonization objectives, this share must still grow in a very large proportion. In 2021, the

share of all renewable production had reached 30%1. But many new investments are required

in order to achieve a net-zero economy by 2050 as 90% of global electricity generation should

come from renewable sources at that point2.

Electricity market is characterized by the constraint that production must be equal to the

consumption at any time. In case of non respect of this constraint, the system can incur a power

outage whose consequences might be highly problematic. For example, the total economic cost

of the August 2003 blackout in the USA was estimated to be between seven and ten billion

dollars [15]. As electricity can hardly be stored, hydro storage is limited in size, and developing

a large ﬂeet of batteries is still highly costly, the power production capacity must be high enough

to cope with major peak load events, which can reach extreme levels compared to the average

load. The consequence of such constraints on the production system is that some power plants

are used rarely (only during extreme peak load) but remain necessary for the system security,

and insuring their economical viability with energy markets only is not guaranteed. This ques-

tion has already motivated a great amount of economic literature under the name of missing

money [19]. That’s also why some regulators took the lead on this matter through "strategic"

reserves (see for example [10]) or by the design of capacity remunerations (see for example [12]

for a review of theoretical studies and implementations of capacity remuneration mechanisms).

Renewable energies have low variable costs so their introduction has made electricity prices

lower, see for example [11] or[21]. This eﬀect is also sometimes referred as the cannibalisation

Date: May 20, 2023. Second version.

2020 Mathematics Subject Classiﬁcation. 91A16, 91B50, 49N80.

Key words and phrases. Energy transition, mean ﬁeld equilibrium, master equation, optimal incentives.

C. Alasseur acknowledges support from FIME and ANR ECOREES (project ANR-19-CE05-0042). C. Bertucci

acknowledges a partial support from the FDD Chair. A. Cecchin beneﬁted from the support of LABEX Louis

Bachelier Finance and Sustainable Growth (through project ANR-11-LABX-0019), ECOREES ANR Project,

FDD Chair and Joint Research Initiative FiME.

1https://www.iea.org/reports/global-energy-review-2021/renewables

2https://www.iea.org/reports/net-zero-by-2050

arXiv:2210.15023v2 [math.OC] 25 May 2023

2 CLÉMENCE ALASSEUR, MATTEO BASEI, CHARLES BERTUCCI, AND ALEKOS CECCHIN

eﬀect (see [23]). This could reinforce the lack of incentives to invest in new capacities. To which

level and at which rhythm new renewable capacity would develop in electricity markets is a

key issue to regulators. Indeed, regulators can remove barriers by modifying the market design

such as the rules of the markets, the level of competition or by providing ﬁnancial incentives to

the new capacities. The model we develop in this paper intends to provide explicit elements to

guide regulators in such issues.

We based our model on MFG (Mean Field Games) approach to represent numerous renew-

able producers who can decide to invest in new capacities. The use of MFG to analyse power

system is not new. For example, it has been used to study how consumers can optimise their

ﬂexilibities against a dynamic price, see [14] for EV (Electrical Vehicle) charging, [16], [18] or

[5] for micro-storages or [17] or [7] for TCL (Thermostatic Control Load).

MFG are dynamic games involving an inﬁnite number of small players. Such problems have

been studied for a long time in Economics and a mathematical framework has been proposed

in [20, 22]. One of the key feature of the MFG theory is that it provides a tool to analyze the

eﬀect of an aggregate shock (or common noise) on the system, by means of the so-called master

equation. When it is well posed, this equation, usually set on the set of probability measures

[13], contains all the information on the MFG. In several situations, the master equation can

be posed on a ﬁnite dimensional space [8],[9] and it is thus easier to study. In this paper, we

adopt a framework which is similar to the one of [9]. In particular we shall study a master

equation whose derivation relies on the expression of dynamical equilibria on markets, even if

no precise MFG is introduced. This type of approach is quite natural in Economics like for

example equations which result from no-arbitrage assumptions. Indeed, in those situations, it

is not necessarily the mathematical problem of the arbitrageurs which is important but rather

the equilibrium relations that it implies.

In this paper, we analyse the long-term competition between producers who invest in re-

newable assets taking into account the cannibalisation eﬀect. Moreover, we study the eﬀect of

subsidies on the aggregate level of capacity of production of renewable energies. We adopt a

master equation formulation as in [9]. For a given level of subsidies, we are able to explicitly

characterise the level of capacities which would develop within the competition framework and

compare it to the level of renewable capacities achieved in a monopoly setting. The equilibrium

we obtain is explicit but also unique and our model provides a way to analyse the impact of

the level of subvention. We prove that competition strengthens the development of renewable

achieving a larger total renewable capacities and diminishes the proﬁtability of producers com-

pared to a non-regulated monopoly setting. Numerically, we can analyse the rhythm at which

renewable develops and how fast a given renewable capacity target can be achieved. We are also

able to compute the optimal levels of subsidies for a central planner wants to achieve a target

of renewable capacity while saving the distributed subventions. In particular, we demonstrate

that from a regulator point of view, to subsidize the cost of production as a decreasing amount

of installed capacities is more eﬃcient than keeping a ﬁxed subvention over time. We provide

this study in two cases: one in which the renewable energies are the only adjustable capacities

of production on the market and one in which the other capacities of production, the reserve,

adjust to the production of renewable energies. When the strategic reserve adapts to the level

of renewable capacity, we show in the paper the explicit and unique equilibrium of competitive

and monopoly situation.

Mathematically, this problem can be named as the one of controlling a MFG equilibrium.

Even though this problem seems quite natural from the point of view of applications, it has

received only few attention for the moment. This may be due to the intrinsic diﬃculty of the

problem, that we shall not enter in in this paper. Here, we shall restrict ourselves to our par-

ticular problem and illustrate the results we obtain with numerical simulations.

A MEAN FIELD MODEL FOR THE DEVELOPMENT OF RENEWABLE CAPACITIES 3

The rest of the paper is organized as follows. In Section 2, we introduce and study a simple

model in which the capacity of electric production outside the renewable source is ﬁxed. In

this setting, we develop the case with a constant subvention to the production and also with a

decreasing subvention with the total renewable capacity. In Section 3, we then proceed to study

the more involved case in which this capacity evolves with the installed capacity of renewable.

2. Renewable capacity development with fixed reserve

2.1. A ﬁrst general model. Assume that we have a multitude of producers of a renewable

energy, that they are symmetric and non-atomic in the sense that every one of them is too small

to have an inﬂuence on the market. Assume that the total (aggregate) available capacity of

renewable energy Kt(in MW) at time t(in years) evolves according to

Kt=−δKt+Xt.(1)

The parameter δ > 0is given and represents the decay of old installations due to time (in

years−1). The quantity Xtis the variation of capacity, at time t, induced by the behaviour of

the producers. Let Ptbe the market remuneration assimilated at the spot price in e/MWh and

assume that it depends on the total (aggregate) available capacity Ktin a decreasing way:

Pt=p

Kt+ε,(2)

where εis a ﬁxed parameter in MW and pa constant in e/h. This model for the spot price is

directly inspired by structural approaches to model electricity prices, such as models developed

in [6] or in [3], which represent spot prices as the intersection of the production and demand

curves in a stylised way. Because, in our model, the new capacities Xare renewable technolo-

gies with very low cost of production, large introduction of this type of technology mechanically

decreases the spot price most of the time.

Let ctbe the cost of production, in e/MW, at time t. We introduce also a parameter h

(in hours/year) which accounts for the number of hours of production of the renewable energy,

per year. Indeed, renewable technologies have the particularity to depend on meteorological

conditions and as such do not run at full capacity all the time.

We want to characterize the value of a single unit of production at time t, which we denote

by ut. Note that, equivalently, it represents the value of the "game" for a producer detaining a

single unit of production at time t. From the previous assumptions, we can write utas the sum

of the expected payments given by a unit of production, before it defaults, which happens at

rate δ > 0. This yields (denoting by a+the positive part of a)

ut=E"Zτt

e−r(s−t)hp

Ks+ε−cs+

ds#.(3)

The random variable τtis an exponential random variable of parameter δ−1which models the

time at which the unit of production will default.

The strategic equilibrium which takes place in our model can be seen through the link be-

tween the equations (3) and (1). Indeed to compute the value of a unit of production, we need to

have anticipation on the evolution of the future total capacity installed. But on the other hand,

the evolution of the total capacity installed should depend on the value of a unit of production:

the more proﬁtable the production of renewable, the more capacity should be installed. To

make this link more apparent, we make the assumption that Xtis a function of only the time t,

the aggregate capacity of production Ktand the value utof a unit of production, and we write

Xt=F(t, Kt, ut). We shall come back on this assumption later on.

4 CLÉMENCE ALASSEUR, MATTEO BASEI, CHARLES BERTUCCI, AND ALEKOS CECCHIN

The previous assumption hints to seek utas a function of only tand Ktwhich we note

ut=U(t, Kt). This leads to a sort of dynamic programming formulation for utwhich takes the

form

U(t, Kt) = E"Zt+dt∧τt

e−r(s−t)hp

Ks+ε−cs+

+{t+dt<τt}U(t+dt, Kt+dt)#,

for dt > 0. Assuming that ct=c(t, Kt), simply computing d

dt U(t, Kt), applying the chain rule

and using (1) and the above equation, we ﬁnd that Ushould solve the PDE

∂tU(t, k)+(F(t, k, U(t, k)) −δk)∂kU(t, k)−(r+δ)U(t, k) + hp

k+ε−c(t, k)+

= 0 (4)

on the domain (t, k)∈(0,∞)×[0,∞). Note that the previous equation is backward in time,

that no boundary condition is needed at t=∞because of the presence of the discount factor

rand that no boundary condition is needed at k= 0 provided that F(t, 0, U(t, 0))) ≥0. In the

terminology of the MFG theory, this PDE is the master equation of the problem.

Let us note that the well-posedness (existence and uniqueness) of the previous PDE yields

the existence and uniqueness of an equilibrium in our model. Indeed, if we are able to compute

Uthrough (4), then we can also compute the evolution of Ktthrough (1). We stress that (4)

is not the master equation of a mean ﬁeld game, because Udoes not represent the value of

an agent in a Nash equilibrium, but the value of a unit. Instead, in our model, (4) is simply

derived by the chain rule, and we call it master equation because it has the same mathematical

structure of the mean ﬁeld game master equation (in one space dimension), so that we easily

get well-posedness results. It is in fact an equation for the value of a unit in equilibrium, but

not a game, and the sole fact we use of MFG theory is the idea that producers are non-atomic

(or small), as explained above.

Equilibrium with a large number of competitive producers. We now make two

assumptions to make the precedent model more tractable and get, in particular, an explicit

expressions for the equilibrium capacity. We assume ﬁrst that the model is stationary, that is

that no quantity depends explicitly on the time variable. This leads to the master equation

−(r+δ)U(k)+(F(k, U(k)) −δk)∂kU(k) + hp

k+ε−c(k)+

= 0 (5)

which thus reduces to a singular ODE since only one variable is remaining.

We now make the more crucial assumption that, for some constant λ > 0and function

α:R+→R,

F(k, U) = λ(U−α(k)).(6)

Remark 1. This assumption has the important property that higher the value of a unit of

production, the higher is the installation of new capacity and is justiﬁed by the following model.

We now present a simple model with Nsmall competitive producers to justify the equations

(6) and (5) when Nis large. Consider a producer whose capacity of production in (MW) at

time tis given by St. Let

•α(k)be the cost of installation, in e/MW, given the installed (aggregate) capacity k

•βNbe an adjustment cost, in eyear/MW2which represents a friction to large volume

of installation.

We shall come back later on why it is natural that βdepends on the number Nof producers.

These costs depend on the type of the production technology. Later, we will consider the

possibility for a central planner of subsidizing the cost of production cand the cost of installation

α.

We assume that the producer controls its (new) installed capacity Stthrough

St=−δSt+yt

A MEAN FIELD MODEL FOR THE DEVELOPMENT OF RENEWABLE CAPACITIES 5

where (yt)t≥0is its control, which represents the intensity of installation, in MW/year. Its

reward is then Z∞

0e−rtSthp

Kt+ε−c(Kt)+

−α(Kt)yt−βNy2

tdt. (7)

which is in inﬁnite horizon with a discount factor rand, at every time t(in years) given by the

spot price minus the cost of production, multiplied by the capacity Stand the hours per year

of production h, while it pays the installation cost, multiplied by the intensity of installation,

and an adjustment cost.

We assume, as above, that the producer neglects the impacts its production has on the

aggregate production Kt, meaning that is is small compared to the total multitude of producers

(as in mean ﬁeld models). This implies, in particular, that Ktis treated as a function of time

only in the maximization of the above reward. The maximization problem, in inﬁnite horizon,

is easily solved by means of Pontryagin’s maximum principle: the Hamiltonian is

H(S, y, v)=(−δ+S)v+Sh p

Kt+ε−c(Kt)+

−α(Kt)y−βNy2,

where vis the adjoint variable, which solves the ODE ˙v=−∂H

∂S +rv and the optimal control

maximized the Hamiltonian. Thus the adjoint equation associated to this problem is given by

˙vt=δvt−hp

Kt+ε−c(Kt)+

+rvt(8)

and the maximization leads to

yt=vt−α(Kt)

2βN

,(9)

which does not depend on St. From (8), expressing vt=V(Kt)as a function of Kt, we then

deduce (using the chain rule) that

−(r+δ)V(Kt)+( ˙

Kt)∂kV(Kt) + hp

Kt+ε−c(Kt)+

= 0.

On the other hand, we can compute the evolution of the aggregate capacity Ktas the sum of

the evolution of the capacity of all players. Let Stbe the new aggregate available capacity of the

total multitude of producers, i.e. St=PN

i=1 Si

t, with S0= 0, thus the total available capacity

Kt=St+k0e−δt,(10)

so that K0=k0is the initial capacity. This leads to

Kt=Nyt−δ

i=1

t−δk0e−δt

=NV(Kt)−α(Kt)

2βN

−δKt.

Assuming that, in the limit N→ ∞,βNscales as Nβ for some β > 0, we ﬁnally obtain that

Kt=V(Kt)−α(Kt)

2β−δKt.

Hence, setting λ= (2β)−1, we recover the required form F(k, U) = λ(U−α(k)).

Remark 2. Recall that in this Nplayers framework, βNis an adjustment cost. Hence it is

natural that this cost grows with the number of producers (or at least of active producers) in the

model. Indeed if βNis small, then the adjustment cost is also small and we can expect that a

few big producers can adapt their production optimally quite easily. However, as βNgrows, this

adjustment becomes more and more important and the slowness of the producers to adjust allow

rooms for smaller producers to install some small, non zero, capacity of production. Somehow

this is what the relation (9) yields.

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AMEANFIELDMODELFORTHEDEVELOPMENTOFRENEWABLECAPACITIESCLÉMENCEALASSEUR,MATTEOBASEI,CHARLESBERTUCCI,ANDALEKOSCECCHINAbstract.Weproposeamodelbasedonalargenumberofsmallcompetitiveproducersofrenewableenergies,tostudytheeffectofsubsidiesontheaggregatelevelofcapacity,takingintoaccountacannibalizationeffect...

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A MEAN FIELD MODEL FOR THE DEVELOPMENT OF RENEWABLE CAPACITIES CLÉMENCE ALASSEUR MATTEO BASEI CHARLES BERTUCCI AND ALEKOS CECCHIN

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