Oskar Lecuyer
email: oskar.lecuyer@vwi.unibe.ch

Adrien Vogt-Schilb
email: avogtschilb@worldbank.org

Céline Guivarch

Louis-Gaëtan Giraudet

Stéphane Halle- Gatte

Jean-Charles Hourcade

Guy Meunier

Jean-Pierre Ponssard

Antonin Pottier

Philippe Quirion

Julie Rozenberg

François Salanié

Ralph Winkler

Patrice Dumas

Optimal transition from coal to gas and renewable power under capacity constraints and adjustment costs

Keywords: climate change mitigation, path dependence, optimal timing, investment, resource extraction, dynamic efficiency, early-scrapping, JEL classification: Q54, Q58

This paper studies the optimal transition from existing coal power plants to gas and renewable power under a carbon budget. It solves a model of polluting, exhaustible resources with capacity constraints and adjustment costs (to build coal, gas, and renewable power plants). It finds that optimal investment in renewable energy may start before coal power has been phased out and even before investment in gas has started, because doing so allows for smoothing investment over time and reduces adjustment costs. Gas plants may be used to reduce short-term investment in renewable power and associated costs, but must eventually be phased out to allow room for carbon-free power. One risk for myopic agents comparing gas and renewable investment is thus to overestimate the lifetime of gas plants -e.g., when computing the levelized cost of electricity -and be biased against renewable power. These analytical results are quantified with numerical simulations of the European Commission's 2050 energy roadmap.

emissions from other sectors [START_REF] Williams | The technology path to deep greenhouse gas emissions cuts by 2050: The pivotal role of electricity[END_REF][START_REF] Kriegler | The role of technology for achieving climate policy objectives: overview of the EMF 27 study on global technology and climate policy strategies[END_REF].

Two important features of the electricity sector are that (i) it depends on long-lived capital that in general is tied to a specific fuel, and (ii) today's power production greatly relies on polluting fossil fuels such as coal (IEA, 2014). Several alternatives are thus available to abate GHG emissions from electricity production. Emissions may be reduced by replacing coal plants with new gas power plants, 1 or with more-expensive but almost-carbon-free options such as renewable power. Also, decision makers can either wait for existing plants to reach their natural lifetime, or decide to decommission them earlier in order to switch faster to cleaner energy sources.

This paper analyzes the optimal timing of investment made to transition from coal power to gas and renewable power under a carbon budget. To our knowledge, it proposes the first theoretical model capable of investigating this question, as related issues have been treated in separate strands of the analytical literature.

On the one hand, a strand of the literature studies the optimal usage of different fossil fuels through the lens of nonrenewable resources theory [START_REF] Hotelling | The economics of exhaustible resources[END_REF][START_REF] Herfindahl | Depletion and economic theory[END_REF], with little attention to the dynamics of capital accumulation [START_REF] Chakravorty | Ordering the extraction of polluting nonrenewable resources[END_REF][START_REF] Smulders | Climate policy and the optimal extraction of high-and low-carbon fossil fuels[END_REF][START_REF] Van Der Ploeg | Too much coal, too little oil[END_REF]. For instance, [START_REF] Chakravorty | Ordering the extraction of polluting nonrenewable resources[END_REF] find that a carbon price may accelerate the burning of the most polluting resource: coal would be used first, followed only later by natural gas. In this literature, renewable energy (modeled as a clean backstop) should never be used early, that is before fossil fuel production stops.

More recently, [START_REF] Amigues | Equilibrium transitions from non renewable energy to renewable energy under capacity constraints[END_REF] study the extraction of a renewable and a non-renewable energy source, taking into account that the extraction of the renewable source requires to first invest in appropriate capital (renewable power plants) and pay for adjustment costs. They prove that optimal investment in renewable is in this case independent from existing fossil resources; the best strategy is to start to build renewable power early and spread investment over time. 2 Amigues et al. (2013) let climate policy for further research (the authors do mention that climate policies is an obvious application), consider only one type of fossil fuel, and disregard capacity constraints limiting the extraction of fossil fuels.

On the other hand, there is a nascent literature on the optimal investment in abatement capital [START_REF] Fischer | Optimal investment in clean production capacity[END_REF][START_REF] Williams | Setting the initial time-profile of climate policy: The economics of environmental policy phase-ins[END_REF][START_REF] Vogt-Schilb | Optimal timing, cost and sectoral dispatch of emission reductions: abatement cost curves vs. abatement capital accumulation[END_REF]. This literature disentangles optimal emission reduction pathways from optimal investment pathways and from carbon prices. In particular, [START_REF] Vogt-Schilb | Optimal timing, cost and sectoral dispatch of emission reductions: abatement cost curves vs. abatement capital accumulation[END_REF] study optimal investment in different economic sectors, and find that the same carbon price translates in more short-term investment in sectors with larger abatement potential and sectors where abatement capital is more expensive. Also, [START_REF] Rozenberg | Transition to clean capital, irreversible investment and stranded assets[END_REF] analyze the trade-off between (i) earlyscrapping existing dirty capital to reduce emissions quickly and (ii) using dirty capital at full capacity to increase short-term production. They find that earlyscraping part of the dirty capital built before climate policies were implemented is optimal for achieving stringent climate targets. 3 These two last studies are close to our question, but they focus on investment in perfectly clean capital (e.g. renewable power), letting aside any intermediate option (e.g., switching from coal to gas).

This paper attempts to bridge the gap between these two veins of the literature. 4 We model different types of nonrenewable resources (cheap coal and more expensive gas) and an expensive renewable source, all subject to capacity constraints and adjustment costs (to make irreversible investment in coal, gas, and renewable power plants). We study the optimal transition from existing coal plants to gas and to renewable power under the constraint of a carbon budget.

The analysis brings three findings. First, we confirm and extend the result by [START_REF] Amigues | Equilibrium transitions from non renewable energy to renewable energy under capacity constraints[END_REF]: under capacity constraints and adjustment costs, different energy sources may be used simultaneously. In particular, investment in gas can start before coal is phased out, and most importantly investment in renewable can start before gas or coal are phased out. The reason is that smoothing investment over time reduces adjustment costs. In other words, the availability of appropriate resources (skilled workers, production lines) and the stringency of the carbon budget set an optimal speed at which to deploy renewable power.

We also extend the finding from [START_REF] Rozenberg | Transition to clean capital, irreversible investment and stranded assets[END_REF]: it may be optimal to build gas power plants and to decommission them before their natural end of life, even in the absence of a policy shock. Doing so allows for reducing investment in expensive renewable power plants in the short term, moving some efforts to the medium term, when gas power is under-used and replaced by renewable power.

Finally, we extend the result from [START_REF] Vogt-Schilb | Optimal timing, cost and sectoral dispatch of emission reductions: abatement cost curves vs. abatement capital accumulation[END_REF], as we find that more investment should go to the expensive zero-carbon renewable power than 3 We focus on the analytical literature. Early decommissioning of existing coal power plants has also been studied using numerical models (e.g., [START_REF] Rogelj | Emission pathways consistent with a 2℃ global temperature limit[END_REF]IEA, 2014). 4 The interaction of investment and natural resources extraction is also the subject of the theory of the mine (e.g., [START_REF] Campbell | The effect of capital intensity on the optimal rate of extraction of a mineral deposit[END_REF][START_REF] Gaudet | Optimal investment and adjustment costs in the economic theory of the mine[END_REF][START_REF] Lasserre | Exhaustible-resource extraction with capital[END_REF], in which installed capital similarly limits the extraction rate of (a single type of) minerals. After reviewing this literature, [START_REF] Cairns | The microeconomics of mineral extraction under capacity constraints[END_REF] notes that "there can be three phases in the exploitation of the mine, namely (1) a period of positive investment after time t = 0, in which production is at full capacity, then (2) a period in which investment is zero and production is at full capacity, and finally (3) a period of declining production". This paper is different as we model several resources and a climate constraint, we find however a similar trajectory for exploitation of gas resources.

In addition, [START_REF] Dasgupta | The optimal depletion of resources[END_REF], [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF] and [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF] have started a literature that studies the impact of resource finitude on growth, in green Ramsey models that feature both capital accumulation and resource extraction -van der Ploeg and Withagen (2014) is a recent example. This literature also focuses on a single type of capital and a single fossil resource, while in the present paper we study the transition from coal to gas and renewable power plants.

to the intermediate low-carbon gas power. This is not only because renewable power plants save fossil fuels and carbon taxes, but also because they stay valuable for ever, while gas plants eventually become obsolete.

These results stress the importance of correctly anticipating the effective lifetime of different types of power plants when choosing among them. Such effective lifetime is not only a technical parameter, but also an endogenous choice depending on future investment and production decisions. For instance, the levelized cost of electricity (LCOE), in theory a good metric for comparing investment in different power plants, may be biased against renewable power in practice.

Indeed, different types of power plants are usually compared using the LCOE (e.g. [START_REF] Kost | Levelized cost of electricity renewable energies[END_REF][START_REF] Eia | Levelized cost of new generation resources in the annual energy outlook[END_REF]IPCC, 2014), that is the ratio of discounted costs of building and using a power plant, including fuel and GHG emission costs, over the discounted expected electricity production. We find that the LCOE computed against future production, investment and prices should be equal to the average future electricity price. But in practice, fossil fuel, carbon and electricity prices and the resale value of power plants of any given technology may be difficult to anticipate. They depend on future investment and production from all technologies, which depend in turn on future prices and regulations. 5 We thus investigate whether LCOEs computed against current signals can provide an operational rule of thumb to guide investment decisions. 6In a numerical calibration using data from the European electricity sector, we find that such operational LCOE should be several times higher for new renewable power plants than for new gas power plants. Indeed, the operational LCOE does not account for future increases in fuel and carbon prices, and does not consider that gas plants will be under-used in the medium term once replaced by renewable power. As a result, if an agent was to assess investment in new capacity for the European electricity market by computing LCOEs from current prices, she would recommend to invest too much in gas capacity, and not enough in renewable capacity.

The remainder of the paper is structured as follows. Section 1 details the model. Section 2 derives the first-order conditions, optimal production decisions at each point in time, and the electricity price. It then studies the different phases through which the power sector may go during its transition to renewable power. Section 3 tackles optimal investment decisions and the levelized cost of electricity. Section 4 provides numerical simulations calibrated with data from the European electricity sector. Section 5 concludes.

Model

A social planner controls the supply of electricity, using and investing in three different technologies: an existing high-carbon technology (h, coal power), a fossil-fueled low-carbon technology (ℓ, gas), and an inexhaustible zero-carbon technology (z, renewable power).

At each time t, the social planner chooses the physical investment x i,t in technology i. The investment adds to the installed capacity k i,t , which depreciates at the constant rate δ (dotted variables denote temporal derivatives):

7 ∀i, ki,t = x i,t -δk i,t (1)
Physical investment is made at a positive, increasing and convex cost c i (x i,t ):

c i > 0, c ′ i > 0, c ′′ i > 0 (2)
This captures the increasing opportunity cost to use scarce resources (skilled workers and appropriate capital) in order to build and deploy capacities faster. We assume that low-carbon capacity is cheaper than zero-carbon capacity in the sense that:

∀x c ′ ℓ (x) < c ′ z (x) (3)
Without loss of generality, we assume low-carbon and zero-carbon capacities to be nil at the beginning (k ℓ,t0 = k z,t0 = 0). 8

The social planner then chooses how much output to produce from each technology. We assume production exhibits constant returns to scale: two gas plants can produce twice as much electricity as one gas plant. 9 The positive production q i,t with technology i cannot exceed the installed capacity k i,t : ∀i, 0 ≤ q i,t

q i,t ≤ k i,t (4)
7 Through this paper, capacity is to be understood as equivalent capacity, e.g., in kWh/yr, unless otherwise specified. For instance if 2 kW of windmills are required to provide as much output per year as 1 kW of coal, then 2 kW of windmills are accounted as 1 kWeq.

8 The reason why this assumption does not result in a loss of generality is that our model studies the transition from the existing situation to cleaner and to clean power. 9 We thus neglect the imperfect substitution between gas and renewable plants that comes from their respective cost structures (in terms of fixed versus variable costs), for instance making gas more suitable for peak power than coal. We are thus implicitly assuming that such considerations have more impact on the optimal electricity mix at the equilibrium than on the transition from the existing state to that equilibrium, which is the focus of the paper (see also [START_REF] Vogt-Schilb | Optimal timing, cost and sectoral dispatch of emission reductions: abatement cost curves vs. abatement capital accumulation[END_REF].

For simplicity, we assume the existing carbon-intensive capital is overabundant, such that (4) is not binding for coal; this assumption is relaxed and confirmed in the numerical application. 10Let F i be the carbon intensity (or emission factor) of technology i. The highcarbon technology is more carbon-intensive than the low-carbon technology:

F h > F ℓ > F z = 0 (5)
The social planner is subject to an exogenous carbon budget (or emission ceiling), cumulative emissions cannot exceed a given ceiling M :

∀t, m t ≤ M (6)
Where cumulative emissions m t grow with emissions F i q i,t :

ṁt = i F i q i,t (7)
Cumulative emissions have been found to be a good proxy for global warming [START_REF] Allen | Warming caused by cumulative carbon emissions towards the trillionth tonne[END_REF][START_REF] Matthews | The proportionality of global warming to cumulative carbon emissions[END_REF]. Some policy instruments, such as an emission trading scheme with unlimited banking and borrowing, set a similar constraint on firms [START_REF] Slechten | Intertemporal links in cap-and-trade schemes[END_REF].

Using fossil fuel (gas or coal) requires to extract exhaustible resource from an initial stock, such that the current stock S i,t classically satisfies:

S i,t0 given Ṡi,t = -q i,t (8)
S i,t ≥ 0

While it is convenient to use the above general notations (indexed by i), we will follow van der Ploeg and Withagen (2012) and focus on the case where the zero carbon technology is renewable and coal is overabundant (S z,t0 = S h,t0 = ∞). This is equivalent to recognize the carbon budget is more stringent than the scarcity of coal resources -as it will be discussed later, this assumption is likely to hold for gas resources as well, especially with the recent developments of non-conventional extraction methods. Consumers derive utility u (∑ i q i,t ) from electricity consumption, where u satisfies the Inada conditions and is smooth enough. The program of the social planner consists in determining the trajectories of investment x i,t and production q i,t that maximize discounted utility net from investment costs while complying with the carbon budget M and the various constraints:

max xi,t,qi,t ∞ 0 e -rt u i q i,t - i c i (x i,t ) dt (9) s.t. ki,t = x i,t -δk i,t (ν i,t ) q i,t ≤ k i,t (γ i,t ) q i,t ≥ 0 (λ i,t ) ṁt = i F i q i,t (µ t ) m t ≤ M (η t ) Ṡi,t = -q i,t (α i,t ) S i,t ≥ 0 (β i,t )
Where r is the constant discount rate and the Greek letters in parentheses are the costate variables and Lagrange multipliers (all chosen such that they are positive): among them, ν i,t is the shadow cost of new power plants; γ i,t , the social cost of the capacity constraint, can also be interpreted as the shadow rental cost of capital or the marginal productivity of capital; µ t is the shadow carbon price; and α i,t the shadow price of resource i (all notations are gathered in Table A.5).

Sequencing production from and investment in competing technologies

Optimal production decisions

We start the resolution with the first order conditions, which translate the equilibrium between production decisions and electricity prices at each point in time. Indeed, during a time interval (τ + i , τ - i ) when production and investment with technology i are strictly positive, the first-order conditions simplify to the following (see Appendix A for the complete set of equations):

∀t ∈ (τ + i , τ - i ), (δ + r) c ′ i (x i,t ) - d dt c ′ i (x i,t ) = u ′ t -α i,t -µ t F i (10)
On the right hand side of (10), u ′ t is the competitive electricity price (u ′ t stands for u ′ (∑ i q i,t )), α i,t is the fuel cost, F i the carbon intensity of fuel i, and µ t is the carbon price.

The left hand side is what Vogt-Schilb et al. (2014b) call the marginal implicit rental cost of capital (MIRCC), extending the concept of implicit rental cost of capital proposed by [START_REF] Jorgenson | The theory of investment behavior[END_REF] to the case of endogenous capacity prices. It is the rental price that ensures agents are indifferent between buying and renting capacity. More specifically, the relation means there is no profitable tradeoff between the two following strategies: (i) buy capital at t at a cost c ′ i (x i,t ), rent it out during one period dt at the rental price, then sell the depreciated (δ) capacities at t + dt at a price c

′ i (x i,t ) + d dt c ′ i (x i,t
)dt or (ii) simply lend money at the interest rate r.

Equation ( 10) can thus be seen as an application of the equimarginal principle: it provides a simple rule to arbitrate production decisions at each moment. It shows that at each time step, production decisions may be taken relying only on current prices -the rental price of capacities and the price of electricity.

Electricity price and types of phases

Equation ( 10) can also be used to determine the price of electricity, depending on which one is the (short-term) marginal power plant: Lemma 1. The output price u ′ t is given by the variable costs of the marginal technology, which includes a strictly positive rental cost only if capacities are fully used. Three types of situations are possible:

1. When coal capacities are used at less than full capacity, electricity price is given by variable costs from coal generation:

0 < q h,t < k h,t ⇒ u ′ t = α h,t + F h µ t (11)
2. When gas capacities are used at less than full capacity, electricity price is given by variable costs from gas generation:

0 < q ℓ,t < k ℓ,t ⇒ u ′ t = α ℓ,t + F ℓ µ t (12)
3. If all capacities are either used at full or not used at all, rental costs adjusts such that electricity price is set by the implicit rental cost of renewable capacity:

∀i, q i,t ∈ {k i,t , 0} ⇒ u ′ t = α i,t + F i µ t + (δ + r) c ′ i (x i,t ) - d dt c ′ i (x i,t ) (13) = (δ + r) c ′ z (x z,t ) - d dt c ′ z (x z,t ) (14)
In particular, it is not possible that at a given date gas and coal are both used at less than full capacity,11 and renewable power is never used at less than full capacity.

Proof. The results are straightforward once it is noted that γ i,t , the Lagrange multiplier associated with the capacity constraint, is zero when capacity i is underused, and that γ i,t equals the implicit rental cost of capital (Appendix A).

Lemma 1 thus defines implicitly three types of phases through which electricity production may go during the transition from coal-fired to renewable power -and implies that the price may jump when the system changes phases and capacity constraint start or stop being binding [START_REF] Wang | Monopoly extraction of a nonrenewable resource facing capacity constrained renewable competition[END_REF], offer a discussion of price jumps).

As discussed in the next subsection, the ordering of these three types of phases may vary. Fig. 1 illustrates one particular situation where investment in both renewable and gas power start at the beginning of the period -a sufficient condition is that c ′ i (0) = 0. In a first phase, coal is progressively replaced by renewable and gas, coal capacity is underused and electricity price is given by the variable cost of coal. When coal is phased out follow two phases during which renewable power progressively replaces gas. First gas is used at full capacity, and electricity price depends on the marginal implicit rental cost of renewable capacities. At one point meeting the emission constraint requires gas production to decrease faster than its capacity depreciates: gas plants are underused and the price is set by variable costs from gas. During the final phase, all production comes from renewable power. Output price during the three types of phases.

(1):

u ′ t = (δ + r) c ′ z (x z,t ) -d dt c ′ z (x z,t )

Ordering investment in low-and zero-carbon capacity

In this subsection we study the various phases through which the system may go during the transition from coal to renewable. Three particular dates play a pivotal role.

To comply with the carbon budget constraint, all production must eventually switch to renewable power, at a first date that we denote T (Fig. 2). At one point, polluting coal production is thus completely phased out, at a second date T - h ≤ T . Similarly, while gas plants may be used during the transition, at one point gas production has to decline, possibly faster than the speed at which its capacity depreciates. This may result in an underutilization of gas capacity, at a third date T γ .

In Appendix B, we systematically list the various possible transition shapes, and identify three possible sequences of investment in gas and renewable power:

Proposition 1. Depending on the parameters, investment phases may be ordered in three ways:

1. Two successive transitions. Gas first completely replaces coal, then renewable power replaces gas. In this case, investment in renewable power starts after coal is phased out, but before production from gas is phased out (Fig. 2a).

2. Gas and wind simultaneously replace coal. In this case, investment in renewable starts before production from any fossil resources is phased-out (Fig. 2b).

3. Starting with investment in renewable power. In this case, investment in renewable power starts before investment in gas power (Fig. 2c).

Proof. See Appendix B.

Assessing investment in competing technologies

In this section, we study optimal investment decisions. We first derive a general investment decision rule, showing that all future prices and the future resale value of capital are required to determine the optimal investment strategy. We then discuss the levelized cost of electricity and investigate consequences of failing to anticipate all future prices.

General investment decision equation

During a time interval (τ + i , τ - i ) when production and investment are strictly positive, the optimal marginal investment cost c ′ i (x i,t ) is the solution of the differential equation ( 10). In general, such solution reads:

12 ∀t ∈ (τ + i , τ - i ), (15)
c ′ i (x i,t ) = τ - i t e -(r+δ)(θ-t) (u ′ θ -µ θ F i -α i,θ )dθ + e -(r+δ)(τ - i -t) c ′ i (x i,τ - i )
The right hand side decomposes in two terms: (i) the present value of all future revenues from selling the output minus costs from emission and resource usage (u ′ tµ t F iα i,t ) produced by the depreciated marginal unit of capacity (e -(r+δ)(t-θ) ), plus (ii) the contribution of the marginal investment to the capacity installed at τ - i , valued at its replacement cost or resale value c ′ i (x i,τ - i ). Again, the optimal investment trajectory is given by a marginalist argument: the capacity cost is equal to all the future discounted net revenues, including a resale value. Equation ( 15) means that the optimal investment cost in one technology can be superior or inferior to the other, depending in particular on the dates when it is optimal to invest and use each technology (τ + i , τ - i ).

The intertemporal merits of investment in gas and renewable power

Proposition 1 means that it may be optimal to invest in both renewable and gas power at the same time. Equation (15) implies that in this case, more investment should go to renewable power.

Indeed, producing electricity from renewable power does not require to buy fossil fuel or pay for the carbon price. Moreover, renewable power plants keep a residual value forever (as during the final phase, all investment goes to maintain renewable power at its maximum capacity), while investment in gas power eventually stops, leaving gas plants with a lower residual value: Corollary 1. On the optimal pathway, when the social planner invests in both the zero-and the low-carbon technology, it builds zero-carbon capacity at a higher marginal investment cost than low-carbon capacity.

Proof. From (15), now denoting τ + i the date when investment in capacity i optimally starts and τ - ℓ the date when investment in gas optimally stops:

∀t ∈ [max i (τ + i ) , τ - ℓ ], c ′ z (x z,t ) -c ′ ℓ (x ℓ,t ) = τ - ℓ t e -(r+δ)(θ-t) (µ θ F ℓ + α ℓ,θ ) dθ ∆p + c ′ z (x z,τ - ℓ ) -c ′ ℓ (0) e (r+δ)(t-τ - ℓ ) ∆c ′ (16)
∆p is the discounted value of emissions and fossil fuels that the marginal renewable capacity built at time t allows saving before τ - ℓ when compared to the marginal gas capacity built at time t. ∆c ′ is the difference between the discounted values of the marginal capacities built at t which will depreciate from t to τ - ℓ . It is strictly positive, as c ′ z (x z,τ - ℓ ) > c ′ z (0) by assumption (2) and c ′ z (0) > c ′ ℓ (0) by assumption (3).

Levelized cost of electricity (LCOE)

Here we investigate whether the Levelized cost of electricity (LCOE) can be used as an operational rule of thumb to value different power plants. Let us consider an incremental unit of capacity built during an interval when investment is positive (at t ∈ (τ + i , τ - i )). Dividing its optimal investment costs (15) with its discounted production over the same time interval yields:

∀t ∈ (τ + i , τ - i ), ∫ τ - i t e -(r+δ)(θ-t) u ′ θ dθ ∫ τ - i t e -(δ+r)(θ-t) dθ = ∫ τ - i t e -(r+δ)(θ-t) (µ θ F i + α i,θ )dθ ∫ τ - i t e -(δ+r)(θ-t) dθ (17) + c ′ i (x i,t ) -e -(r+δ)(τ - i -t) c ′ i (x i,τ - i ) ∫ τ - i t e -(δ+r)(θ-t) dθ
Equation ( 17) defines the levelized cost of electricity (LCOE) produced with technology i. The first term in the right hand side is the discounted variable cost (from carbon and resource prices) over the discounted production; the second term corresponds to the investment costs of the marginal unit of capacity at t minus its residual value at τ - i , over the discounted future production (the levelized cost of the capacity net of its residual value). The left-hand side is the average output price weighted by discounted production during (t, τ - i ). Equation ( 17) states that at the optimum, the LCOE is equal to the weighted average future price of the output. The relevant weight is the discounted production from the deprecating capacity. In other terms, electricity priced at the optimal LCOE covers for the complete cost of building and operating a power plant. 13 In practice, using this relationship to assess investment requires to know in advance the future prices of additional capacity, resources, emissions and output. This may be challenging for actual agents willing to schedule or monitor investment in different types of power plants. For instance, it is recognized that governments have in general little capacity to commit, and in particular to commit to specific future carbon price levels [START_REF] Helm | Credible carbon policy[END_REF][START_REF] Golombek | Carbon taxes and innovation without commitment. The B.E[END_REF][START_REF] Ulph | Optimal climate change policies when governments cannot commit[END_REF]. 14The question we address here is whether LCOEs computed against current price signals may provide an operational rule of thumb to value different types of power plants. 15 Such an operational LCOE reads:

L i,t = (r + δ) c ′ i (x i,t ) + µ t F i + α i,t (18)
L i,t is the sum of the annualized investment cost, the carbon cost and the fuel cost.

Proposition 2. Equalizing the operational LCOE across technologies would lead to invest relatively too much in gas power and too little in renewable power

Proof. Let xz,t and xℓ,t be two investment schedules that equalize the operational LCOE accross technologies, such that:

(r + δ)c ′ z (x z,t ) + µF z + α z = (r + δ)c ′ ℓ (x ℓ,t ) + µF ℓ + α ℓ (19) ⇒ c ′ z (x z,t ) -c ′ ℓ (x ℓ,t ) = µF ℓ + α ℓ r + δ (20)
Comparing the allocation implied by ( 20) with the optimal rule stated in Corollary 1 shows that equalizing the operational LCOE across technologies misses the future increase for carbon and fossil resources prices, and misses the term correcting for future changes in the value of capacities, leading to relatively too much investment in gas power and not enough investment in renewable power.

In the next section we use numerical simulations calibrated on the European power sector and find that during the optimal transition to clean power, renewable power plants should be built at an operational LCOE several times higher than new gas power plants.

Numerical application to the European electricity sector

Functional forms, data and calibration

We calibrate the model with data from the European power sector as described in the European 2050 Energy Roadmap (EU, 2011). In this numerical application, efficient gas power plants (the low-carbon technology) and onshore wind (the zero-carbon technology) are used to phase out the existing polluting capacities represented as the average legacy thermal production mix (Table 1).

To better fit the data, we express installed capacity k i,t in peak capacity (GW), and production q i,t in GWh/yr. Production is constrained by a maximum number of operating hours H i . For instance, a given windmill will produce electricity only at the moments where it is windy, which expectedly happens a given number of hours per year.

We assume for simplicity that all technologies have the same depreciation rate δ, calibrated as δ = 1 lifetime assuming a lifetime of 30 years (IEA, 2010). We consider that Europe is price-taker on the fossil resources costs α i,t , which follows the trajectory assumed in the European 2050 Roadmap (Table 2). Note that α h corresponds to the average price of the resources of the legacy technology mix (Table 1).

To calibrate the cost functions, we assume that when investment equals the actual average annual investment flow in Europe between 2009 and 2011 (X i ), the marginal investment cost C m i is equal to the OECD median value for 2010 (as found in IEA, 2010). We write the cost function as:

c i (x i,t ) = C m i ⋅ X i ⋅ A x i,t X i + 1 -A 2 x i,t X i 2 (21) ⇒ c ′ i (X i ) = C m i (22)
A ∈ (0, 1) is a convexity parameter, equal across technologies for simplicity. When A = 1, the marginal investment cost are constant, there is no adjustment cost for production capacity, and optimal investment pathways can exhibit jumps. When A = 0, the marginal cost curves are linear, capacity accumulated at very low speed is free (lim xi,t→0;A=0 c ′ i (x i,t ) = 0), and the cost of new capacity doubles when the investment pace doubles. For intermediate value A ∈ (0, 1), new capacity is always costly, and the marginal cost of new capacity increases with the investment pace. We first assume quadratic costs (that is linear marginal costs and A = 0), and perform later a sensitivity anaysis on A.

The demand is assumed inelastic in the short-term, and follows an exogenous long-term growth. Its value in 2008 is calibrated as the reference fossil energy (from coal, oil and gas), that is D 0 = 1 940 TWh/yr (ENERDATA, 2012). The central scenario of the Roadmap forecasts a 700 TWh/yr increase of electricity consumption between 2008 and 2050, that we model as a linear growth of G = 16.5 TWh/yr 2 . Demand after time t is thus:

D t = D 0 + t ⋅ G (23)
Acording to [START_REF] Trotignon | Échanges de quotas en période d'essai du marche européen du CO 2 : ce que révèle le CITL[END_REF], emission allowances allocated to the power sector in 2008 amounted to E ref = 1.03 GtCO 2 /yr , leading to a reference emission rate of 530 tCO 2 /GWh. A linear decrease of these emissions until 2050, as planned in the Roadmap, yields a carbon budget of M = 22 GtCO 2 . A sensitivity analysis on M is performed later. We use r = 5 %/yr for the social discount rate.

Results

Fig. 3 shows production, investment, the investment cost, the electricity price and the operational LCOE obtained in the numerical application. The following lists several interesting results.

The social planner does not invest in the legacy capacity; moreover, existing legacy plants start being decommissioned as soon as the climate policy is implemented (Fig. 3a). In the central scenario, legacy fossil-fueled plants are entirely phased out in 2028. With our technology assumptions, the carbon budget is too loose to justify a complete transition by 2050, but instead phases out all fossil production by 2064. Investment in renewable starts at the beginning of the transition, at the same time as gas and before coal or gas are phased out (Fig. 3c). It grows over time, and only decreases for a short period from 2055 to 2064 when gas is phased out, as most of the power plants have already been replaced (Fig. 3a). Investment in renewable starts at 22 GW/yr in 2008, more than twice the actual average investment rate observed between 2009 and 2011, and reaches 70 GW/yr in 2060. Investment grows again after 2064 to maintain the installed renewable capacity and meet the growing demand.

Gas plants operate as a temporary bridge between coal and renewable. As seen in Fig. 3c, investment in gas starts since the beginning of the transition in 2008, lowering the need for investment in renewable. In 2035, however, investment in gas stops and gas capacities start being underused, allowing gas to be replaced faster by renewable power. The sensitivity analysis (Fig. 4d) shows that it is optimal to build gas power plants transiently for any positive carbon budget ranging up to 45 GtCO 2 : investment always starts in 2008 and, with quadratic investment costs, plants are underused immediately after investment stops. Only carbon budgets lower than 17 GtCO 2 make gas investment stop before 2020 (for comparison, the Roadmap implicitly sets a budget of 22 GtCO 2 ).

Fig. 4c shows that the optimal maximal gas capacity increases with the convexity of the investment functions (when A tends to 0). The more convex the investment costs, the more renewable investment must be smoothed and the more gas is needed. In contrast, for constant investment costs (when A tends to 1), the social planner would replace all the stock of capital with renewable power plants at once. The capacity cost is always higher for renewable than for gas -as predicted by Corollary 1. Fig. 3d displays the marginal costs for new capacities (gas and renewable) along the period. They essentially follow the investment trajectories. Gas investment costs decrease from 2015 on, as the average power plants becomes less and less carbon-intensive, making investment in low carbon capacity less and less profitable.

Electricity prices are displayed in Fig. 3b. In a first phase, before 2028, the marginal capacity is the legacy dirty technology. The endogenous carbon price is 137 $/tCO 2 , representing 73 $/MWh or 63 % of the electricity price during the phase while the legacy power plant is marginal. After the dirty technology has been phased out, gas becomes the marginal technology and is under-used from 2036 to 2064. In between and after fossil technologies have been phased out, the electricity price equals the rental cost of renewable power plants (Lemma 1).

Finally, along the simulated optimal path, the levelized cost of electricity computed against current prices and current returns (the operational LCOEs) is several times higher for renewable power than for gas. Fig. 3b shows the LCOEs and the electricity price. The LCOE is higher than the electricity price for renewable plants, and lower for gas plants. Fig. 4a and4b show the operational LCOEs of renewable and gas in 2008, along with the electricity price for a range of parameters. 16 For a wide range of carbon budget and convexity parameter values, the operational LCOE is 2 to 5 times higher for renewable power than for gas. This difference comes from the wrong anticipations one makes when considering only current signals. These are biased toward gas for two reasons: the future increase in fuel and carbon price is not accounted for, and gas is assumed to produce at full capacity for its whole lifetime. These simulations thus suggest that if an agent was to assess investment in new capacity for the European electricity market by computing LCOEs from current prices, she would recommend to invest too much in gas capacity, and not enough in renewable capacity.

Conclusion

Our analysis makes several simplifications. For instance, we do not account for knowledge accumulation and directed technical change, which have been found to play a key role in the optimal transition from fossil to renewable energy [START_REF] Tahvonen | Economic growth and transitions between renewable and nonrenewable energy resources[END_REF][START_REF] Acemoglu | The environment and directed technical change[END_REF][START_REF] André | Fueling growth when oil peaks: Directed technological change and the limits to efficiency[END_REF][START_REF] Gerlagh | The optimal time path of clean energy r&d policy when patents have finite lifetime[END_REF]. Knowledge spillovers would tend to increase the short-term gap between optimal investment in renewable and gas power [START_REF] Rosendahl | Cost-effective environmental policy: implications of induced technological change[END_REF][START_REF] Del Rio Gonzalez | Policy implications of conflicts between short-term and long-term efficiency in CO 2 emissions abatement[END_REF], adding to the effect of adjustment costs studied here. We also disregarded uncertainty, known to play a key role in optimal accumulation of capital [START_REF] Pindyck | Irreversibility, uncertainty, and investment[END_REF], optimal emission reduction pathways [START_REF] Duong | Influence of socioeconomic inertia and uncertainty on optimal CO2-emission abatement[END_REF], and optimal extraction of several energy sources [START_REF] Gaudet | The efficient use of multiple sources of a nonrenewable resource under supply cost uncertainty[END_REF]. Finally, we did not account for the possibility to retrofit existing plants (notably with carbon capture and storage), and the possibility to use cleaner fuels (in particular derived from biomass) in existing power plants.

Notwithstanding these limitations, our analysis suggests that capacity constraints and adjustment costs play an essential role in the transition from coal to gas and renewable energy. In particular, investment in renewable power should start before coal and gas resources are phased out, and should be higher than investment in gas power. The transition to carbon-free power may nonetheless benefit from temporary investment in intermediate technologies such as gas, to decrease the need for costlier renewable in the short term. The resulting plants would however need to be subsequently decommissioned before the end of their natural lifetime, to give room to more carbon-free power in the medium term.

Our results shed light on technical choices (e.g. investors can consider gas plants with shorter scheduled lifetimes or an option to retrofit) as well as policy decisions (when setting milestones for carbon-free power or managing the social consequences of decommissioning stranded fossil-fueled plants). Moreover, we warn that the levelized cost of electricity (LCOE) computed against current price signals is biased toward gas power and against renewable power. An accurate valuation of different types of power plants requires to anticipate a comprehensive long-term investment and production pathway, and in particular that gas built to replace coal in the short term may itself be replaced with renewable in the medium term.

Finally, while this paper focuses on power generation, it more broadly informs on the ordering and assessment of investment in polluting fossil-fueled and clean capital; its results could be extended to other capitalistic and energyintensive sectors, such as buildings or transportation.

≥ T, ṁt = 0 ⇒ q ℓ,t = q h,t = 0 (B.2)
When the carbon budget is binding, the date when the ceiling on atmospheric carbon is reached and the steady state necessarily coincide: T = T η .

B.2. Output production and fossil fuel deposits

Initially, all the output is produced from coal by assumption. Coal production is eventually replaced by gas or renewable power. Let T - h be the date when high-carbon production stops:

∀t ≥ T - h , q h,t = 0 (B.3)
This happens necessarily before the steady state:

T - h ≤ T (B.4)
Similarly, let T + i be the date when production of technology i ∈ {ℓ, z} starts, and T - ℓ the date when gas production stops. Note that only low-carbon production eventually stops, since zero-carbon energy is used indefinitely during the steady state.17

Let T β be the date when the low-carbon fossil fuel deposit is depleted. If it exists (i.e. if the gas deposit is eventually depleted), T β coincides with the end of production from low-carbon fuel:

∃T β ⇒ T β ℓ = T - ℓ (B.5)
Optimality then requires that fossil fuel prices follow the Hotelling rule: their current price α ℓ,t is increasing at the discounting rate before they are exhausted (A.6, A.9): ∀t < T β , α ℓ,t = α ℓ e rt > 0 (B.6)

B.3. Investment and under-utilization of capacities

For coal to be phased out, production in one of the alternative technologies must have started:

T - h ≥ min(T + ℓ , T + z ) (B.7)
Let τ + i ≤ T + i be the date when investment in capacity i starts, and τ - i the date when it stops. Note that only the investment in gas stops eventually, since renewable capacity is used indefinitely during the steady state and must be maintained as δ > 0.

If gas is used during the transition, production from gas must decline before the steady state, either because gas reserves are depleted, or to reduce GHG emissions to zero. If it declines faster than the depreciation rate of capacities, installed gas capacities may be underused. Let T γ ∈ [T + ℓ , T - ℓ ] be the date when gas production is lower than its capacity: ∀t ≥ T γ , k ℓ,t > 0 ⇒ q ℓ,t < k ℓ,t (B.8)

The social planner stops to invest in gas power plants when the marginal cost of new capacities exceeds its marginal benefit:

c ′ ℓ (0) = Tγ τ - ℓ e -(r+δ)(θ-τ - ℓ ) (u ′ θ -µ θ F i -α i,θ )dθ (B.9)
such that in general she stops investing before starting to underuse gas capacities

τ - ℓ ≤ T γ (B.10)
In particular, if c ′ ℓ (0) = 0 then τ - ℓ = T γ (this is the case in Fig. 3). We assume that coal capacity is overabundant at the beginning of the transition, meaning that its capacity is underused from the start. In other words, coal is marginal during the first phase of the transition. As two technologies cannot be marginal at the same time (Lemma 1), gas can be the marginal technology (and gas capacity gets underused) only after coal is completely phased out:

T - h ≤ T γ ≤ T - ℓ (B.11)
If gas is used, it's thus used from the moment there are some gas capacities ∃(T + ℓ , τ + ℓ ) ⇒ T + ℓ = τ + ℓ (B.12) B.11 also means that if gas is used, once gas production ends, coal production has ended, and emissions are null. The end of gas production thus coincides with the steady state:

∃T - ℓ ⇒ T - ℓ = T (B.13)
If production does not decrease with time, the moment when gas starts being underused necessarily happens after wind started to produce:

T + z ≤ T γ (B.14)
Finally, because variable costs from renewable are nil, its never optimal not to use renewable at full capacity:

T + z = τ + z (B.15)

B.4. Summary of the transition phases

This leaves us with three possible transition profiles:

1. Two successive transitions. Gas first completely replaces coal, then renewable power replaces gas. 2. Gas and wind simultaneously replace coal, starting with gas. B.6 and the following cost function:

c ′ i (x i,t ) = C m i ⋅ A ⋅ I + C m i ⋅ (1 -A) x i,t X i (B.19)

  (a) Production and capacity (b) Output price

  Figure 1:

  (a) Two successive transitions (b) Gas and wind simultaneously replace coal (c) Starting with investment in renewable power

Figure 2 :

 2 Figure 2: Numerical simulations of three possible transition profiles. Figures on the left display production and capacity, figures on the right display optimal investment. The parameters used to produce these figures are gathered in Table B.6.

( a )

 a Production and capacity (b) Operational levelized cost of electricity and electricity price (c) Investment (d) Marginal investment cost

Figure 3 :

 3 Figure 3: Numerical application to the European electricity sector

( a )Figure 4 :

 a4 Figure 4: Sensitivity analysis on the carbon budget and the convexity parameter

  Fig.2illustrates the three possible transition profiles, by using the parameters gathered in TableB.6 and the following cost function:

Table 1 :

 1 Technology sets considered in the numerical model

	Set	Abbreviation	Composition
			Average thermal production mix in
	High carbon technology	Legacy	2008: 40% gas , 50% coal , 10% oil (EN-
			ERDATA, 2012)
	Low carbon technology	Gas	Efficient gas
	Zero carbon technology	Wind	Onshore wind

Table 2 :

 2 Fuel price trajectories of the fossil technology sets in $/MWh (EU,

	2011)
		2008	2025	2035	2050
	Legacy	42.5	46	48	39
	Gas	65	76	75	60

Table 3 :

 3 Technology-specific data used in the numerical application

		Description	Unit	Legacy Gas Wind	Source
	C m i	Nominal investment costs	$/kW	1 800	1 200 2 000	IEA (2010)
		Average annual
	Xi	new capacity in	GW/yr	4.2	11	10	ENERDATA (2012)
		Europe
	Hi	Average annual operating hours	h/yr	7 500	7 500 2 000	IEA (2010)
	Fi	Carbon intensity gCO2/kWh	530	330	0	ENERDATA (2012); Trotignon and Delbosc (2008)

Table 4 :

 4 General parameter values used in the numerical application

		Description	Unit	Value	Source
	r	Discount rate	%/yr	5
	M	Carbon budget (central value)	GtCO2	22	UE (2011)
	D0 Electricity demand in 2008	TWh/y 1 940	ENERDATA (2012); EU (2011)
	G	Annual growth of demand	TWh/y	16.5	EU (2011)
	δ	Depreciation rate	%/yr	3.33	IEA (2010)
	A	Convexity parameter (central value)	⋅	0

Table A . 5 :

 A5 Variables and parameters notations used in the model.

	is stable, hence emissions and production from high-and low-carbon fossil fuels
	must be nil (4, 6, 7):
		∀t
		Description	Possible units
	i	technology index
	h	high-carbon technology (h)
	l	low-carbon technology (ℓ)
	z	zero-carbon technology (z)
	ki,t	capacity of technology i at time t	GW
	qi,t	production of technology i at time t	GW
	xi,t	physical investment in technology i at time t	GW/yr
	νi,t	shadow price of new capacity i	$/GW
	µt	cost of emissions	$/tCO2
	αi,t	cost of resource used by technology i	$/MWh
	γi,t	shadow rental cost of existing capacity i	$/(GW⋅ yr)
	u ′ t	output price	$/GWh
	ci(xi,t)	monetary investment in technology i at time t	$/yr
	mt	stock of atmospheric carbon	tCO2
	δ	depreciation rate	yr -1
	r	discount rate	yr -1
	Fi	emission rate of technology i	tCO2/GWh
	M	carbon budget	tCO2
	u (∑ i qi,t) consumer utility	$/yr

  TableB.6: Parameters used to produce Fig.2

		Fig. 2c Fig. 2b Fig. 2a		Fig. 2c Fig. 2b Fig. 2a
	δ	.0333	.0333	.0333	αz	0	0	0
	M	42	38	40	α h	.055	.055	.055
	D	1940	1940	1940	α ℓ	.1	.06	.06
	r	.05	.05	.05	C m z	75000	75000	80000
	Rz	0	0	0	C m h	18000	18000	18000
	R h	.00063	.00063	.00063	C m ℓ	35000	12000	12000
	R ℓ	.0004	.0004	.0003	Xz	.00003	.0005	.001
	Hz	7500	7500	7500	X h	.005	.005	.005
	H h	7500	7500	7500	X ℓ	.0001	.001	.01
	H ℓ	7500	7500	7500	A	.9	.9	.9
					I	12	12	2

This paper assumes gas is a low-carbon substitute for coal. The relative carbon content of gas and coal may actually depend on the type of gas and coal and on the particular processes used for extracting and transporting the fuels (e.g.,Alvarez et al.,

2012). The relative merits of coal and gas also depend on factors disregarded here, such as impact on energy security or impact on local pollution.2 A whole literature started by[START_REF] Kemp | On two folk theorems concerning the extraction of exhaustible resources[END_REF] has established that if extraction of an expensive energy source is exogenously constrained, it may be optimal to use that source simultaneously with cheaper alternatives (e.g.,[START_REF] Amigues | On the optimal order of natural resource use when the capacity of the inexhaustible substitute is limited[END_REF][START_REF] Holland | Extraction capacity and the optimal order of extraction[END_REF].[START_REF] Amigues | Equilibrium transitions from non renewable energy to renewable energy under capacity constraints[END_REF] confirm this result for endogenous constraints. Other reasons for not extracting resources according to a strict Herfindahl sequence include heterogeneity of both producers and consumers -[START_REF] Gaudet | Modelling the use of nonrenewable resources with multiple demands and multiple sources[END_REF] provides an extensive review.

One way such investment and production pathways are provided in practice is through numerical models of the electricity sector (e.g.,[START_REF] Seebregts | Energy/environmental modeling with the MARKAL family of models[END_REF][START_REF] Loulou | ETSAP-TIAM: the TIMES integrated assessment model[END_REF] or wider energy-environment-economy models (e.g.,[START_REF] Weyant | Introduction and overview to the EMF 19 study on technology and climate change policy[END_REF][START_REF] Waisman | Peak oil profiles through the lens of a general equilibrium assessment[END_REF][START_REF] Kriegler | The role of technology for achieving climate policy objectives: overview of the EMF 27 study on global technology and climate policy strategies[END_REF]. These models do not always model adjustment costs explicitly, but they take into account the limited ability to switch quickly from high-to low-carbon capital by building on maximum investment speeds[START_REF] Wilson | Future capacity growth of energy technologies: are scenarios consistent with historical evidence?[END_REF]. Several authors have stressed that in this case optimal investment pathways are in part driven by such exogenous bounds, therefore calling for explicit discussion of their calibration[START_REF] Wilson | Future capacity growth of energy technologies: are scenarios consistent with historical evidence?[END_REF][START_REF] Iyer | Diffusion of low-carbon technologies and the feasibility of long-term climate targets[END_REF][START_REF] Vogt-Schilb | Long-term mitigation strategies and marginal abatement cost curves: a case study on Brazil[END_REF].

6 Such usage of the LCOE would be consistent with the view, popular in the early literature on investment, that firms can take optimal investment decisions based on current prices and current productivity of capital only (e.g.,[START_REF] Arrow | Optimal capital policy, the cost of capital, and myopic decision rules[END_REF] Jorgenson, 1967, p. 145). It has already been signaled that the efficiency of such myopic investment decisions critically depends on investment prices being exogenous: under endogenous investment costs, optimal investment decisions require to perfectly anticipate all future prices[START_REF] Gould | Adjustment costs in the theory of investment of the firm[END_REF][START_REF] Mussa | External and internal adjustment costs and the theory of aggregate and firm investment[END_REF]. The bad performance of imperfectly computed LCOEs is thus not surprising (and is not a theoretical challenge under the rational expectation hypothesis). The novelty here is in determining the direction and magnitude of the error induced by myopic decision rules in the particular case of the transition to clean electricity.

[START_REF] Rozenberg | Transition to clean capital, irreversible investment and stranded assets[END_REF] offer a proof that carbon-intensive capital becomes overabundant at the moment when a carbon price is implemented.

We disregard the case where fuel costs compensate exactly differences in carbon intensities α ℓα h = µ (F h -F ℓ ) as it requires a very restrictive set of assumptions.

Equation 15 is the textbook solution of equation 10 (e.g.,Wikibooks contributors, 2014)

We ignore here all intermittency issues, which add a cost to renewable sources, and may be a rationale to use some gas during the final phase[START_REF] Joskow | Comparing the costs of intermittent and dispatchable electricity generating technologies[END_REF][START_REF] Ambec | Electricity provision with intermittent sources of energy[END_REF][START_REF] Bazilian | Re-considering the economics of photovoltaic power[END_REF][START_REF] Ueckerdt | System LCOE: What are the costs of variable renewables?[END_REF].

See also[START_REF] Spiro | Resource prices and planning horizons[END_REF] on the limited capacity of investors and regulators to anticipate prices over long time horizons.

As discussed in footnote 6, such operational LCOE is consistent with the idea that agents can take optimal investment decisions based on current price signals only.

Note that computing the LCOE only make sense if investment is strictly positive.

As the marginal utility tends to infinity when energy consumption tends to zero energy production is always strictly positive.

The present value Hamiltonian associated with Problem 9 reads:

The first-order conditions are:

The complementary slackness conditions are: ∀i, t, λ i,t ≥ 0, q i,t ≥ 0 and λ i,t q i,t = 0 (A.7) ∀i, t, η t ≥ 0, Mm t ≥ 0 and η t Mm t = 0 (A.8) ∀i, t, β i,t ≥ 0, S i,t ≥ 0 and β i,t S i,t = 0 (A.9) ∀i, t, γ i,t ≥ 0, k i,tq i,t ≥ 0 and γ i,t (k i,tq i,t ) = 0 (A.10)

The transversality condition is replaced by the terminal condition that at some point the atmospheric carbon reaches its ceiling (6).

B. Investment and production phases

B.1. Carbon budget constraint and steady state

Let T η be the date when the ceiling on atmospheric carbon is reached. The carbon-free atmosphere can be seen as a non renewable resource depleted by GHG emissions. In this context, the optimal current carbon price follows Hotelling's rule, i.e. grows at the discount rate r (A.5, A.8):

where µ is the carbon price at t = 0. Eq. B.1 reflects the fact that abatement realized at any time contributes equally to meet the carbon budget. We focus on the case where the carbon budget is binding, so that the carbon price µ t is strictly positive.

Let T be the date when the system reaches a steady state. During the steady state, the clean backstop produces all the output. Indeed, atmospheric carbon


Arthur Hellouin De Menibus
email: arthur.hellouin-de-menibus@cea.fr

Quentin Auzoux

Ousmane Dieye

Vincent Macdonald

Jacques Besson

Jérôme Crépin

Hydride blisters formation, characterization and effect on the fracture of Zircaloy-4 cladding tubes under reactivity initiated accident conditions

INTRODUCTION

During service exposure, Zircaloy-4 alloys used as PWR fuel claddings are oxidized by the water coolant. It results in the growth of an oxide layer (zirconia), and a fraction of the released hydrogen is absorbed and diffuses into the cladding. The zirconia is submitted to compression stresses due to oxide lower density than zirconium. It results in the development of cracks in the oxide due to micro-buckling. When the zirconia thickness reaches more than 100µm, these cracks may interconnect causing oxide spallation on a localized area (typically 1-5mm on the cladding outer diameter). Because hydrogen diffuses in thermal gradient, it could results in the development of localized high hydrogen concentration areas, named hydride blisters, on the outer diameter of highly irradiated Zircaloy-4 cladding. These zones are brittle. Blistering phenomenon was first observed in 1977 [START_REF] White | A computer model for hydride-blister growth in zirconium alloys[END_REF] and blister induced failure in 1983 [START_REF] Field | An Analysis of the Pressure Tube Rupture at Pickering NGS "A" Unit 2[END_REF], when an axial crack developed in a CANDU pressure tube following an array of hydride blisters on the external surface due to contact with the cold calendria tube. It was later shown in dedicated experiments in research reactors that blisters reduce the cladding ductility when submitted to a Reactivity Initiated Accident (RIA). One of the most critical hypothetical RIA scenarios is the Rod Ejection Accident. It corresponds to the unwanted remove a control rod from the PWR (Pressurized Water Reactor) core. It results in a fast power pulse transient that is intrinsically limited in time due to the Doppler effect of the uranium from the fuel pellets and the reduction of the moderator effect of the coolant water when the temperature increases.

The CEA Saclay has been performing studies to assess the cladding mechanical behavior for years. A specific program named PROMETRA [START_REF] Desquines | Release of the PROMETRA v2.5 material database[END_REF] is devoted to characterize the cladding under RIA conditions, by uncoupling or partially uncoupling the different parameters that may affect the cladding mechanical resistance. This study is part of this program. This paper described a thermo-diffusion setup that was designed to generate hydride blisters on PWR Zircaloy-4 cladding in thermal conditions that optimized the blister growth rate. Then, it provides extensive experimental characterization of hydride blisters, such as their morphology, the hydrides crystallographic phases, or the hydrogen concentration inside and outside the blister. A newly developed mechanical test is presented. It allows improving the representativity of standard mechanical tests to RIA conditions. Eventually, the results of these mechanical tests performed at 25°C, 350°C and 480°C are presented and compared with previous study.

FORMATION OF HYDRIDE BLISTERS

Hydride blisters were artificially obtained in laboratory by reproducing the blistering scenario that may occurs in reactor [START_REF] Domizzi | Blister growth in zirconium alloys: experimentation and modeling[END_REF]. First, cold-worked stress-relieved and unirradiated Zircaloy-4 claddings were hydrogen pre-charged at 300wppm by a gaseous charging technique. The initial hydrogen content was homogeneous on the cladding section. The cladding was then partitioned in 15, 20 or 30mm axial tubes. The samples were heated to dissolve a fraction of the precipitated hydrides, and a 1mm in diameter copper rod was simultaneously put in contact with the sample to generate a cold spot. This creates the thermal gradient necessary to trigger hydrogen thermal diffusion and to generate a blister. Because many samples were required for subsequent mechanical tests, hydrogen thermodiffusion calculations were performed to define the best thermal gradient to apply to optimize the blister growth rate. The hydrogen diffusion in thermal and concentration gradient can be modeled by the equation 1 [START_REF] Sommer | Thermal diffusion of hydrogen in nonstoichiometric zirconium-dihydride[END_REF]: [START_REF] White | A computer model for hydride-blister growth in zirconium alloys[END_REF] where J is the hydrogen flux, D the diffusion coefficient of hydrogen in zirconium, C the hydrogen in solid solution concentration, Q* the heat of transport of hydrogen in zirconium, R the perfect gas constant equal to 8,314J/mol/K and T the temperature. The equation 1 was solved in 1D with a finite difference scheme implemented in Matlab. The procedure requires taking into account the solubility limit of hydrogen in zirconium because the term C is only the hydrogen in solid solution, not the total hydrogen concentration. Moreover, the solubility limit hysteresis was also taken into account. This means the solubility limit is not unique; it is lower in dissolution (TSSD -Terminal Solid Solubility in Dissolution) than in precipitation (TSSP -Terminal Solid Solubility in Precipitation) for a given temperature. Eventually, it was supposed hydrogen does not diffuse in hydrides. The implementation validity was checked by simulating the hydrogen profile obtained by Kammenzind [START_REF] Kammenzind | Hydrogen pickup and redistribution in alpha-annealed Zircaloy-4[END_REF] in a thermodiffusion experiment. It was chosen to do not overpass 380°C for the several hours thermal treatment to obtain hydride blister to do not restaurate or recrystallized the cold worked specimens. Then, a thermodiffusion calculation was performed to identify cold spot temperature that optimizes the blister growth rate. It was taken into account the diffusion situation was reach by cooling down the specimen from a higher temperature (385°C). The D, TSSD and TSSP parameter values identified by [START_REF] Kammenzind | Hydrogen pickup and redistribution in alpha-annealed Zircaloy-4[END_REF] were used, Q* = 25kJ/mol was fixed as an average of several authors reported values ( [START_REF] Kammenzind | Hydrogen pickup and redistribution in alpha-annealed Zircaloy-4[END_REF][START_REF] Sawatzky | The diffusion and solubility of hydrogen in the alpha phase of Zircaloy-2[END_REF][START_REF] Markowitz | The thermal diffusion of hydrogen in alpha-delta Zircaloy-2[END_REF][START_REF] Sawatzky | The heat of transport of hydrogen in Zirconium Alloys[END_REF][START_REF] Morozumi | Effects of alloying elements and cold work on the redistribution of hydrogen in zirconium under a temperature gradient[END_REF][START_REF] Sugisaki | Thermal diffusion of tritium and protium in alpha phase of zirconium[END_REF][START_REF] Hashizume | Temperature Dependence of Heat of Transport of Hydrogen in Zirconium. Defect and Diffusion Forum[END_REF][START_REF] Hong | Thermotransport of hydrogen in Zircaloy-4 and modified Zircaloy-4[END_REF]). The calculation was performed assuming spherical symmetry and logarithmic temperature variation (in lines with experimental observations). It was found cold spot temperature ranging from 200°C to 330°C allowed high blister growth rate.

The blister morphology obtained consists in a protrusion on the external diameter and a growth of a lenticular shape in the cladding thickness (figure 1). The blister depth reaches 217µm after 12h and 296µm after 24h. The square root fit of the blister depth in function of the thermodiffusion time gave depth = 6.4858 t 0.3335 with t in seconds. The reason of the protrusion is the hydrides lower density than the zirconium [START_REF] Ferguson | Computed X-Ray powder diffraction patterns and densities for corundum, aluminium, zirconium, delta-UZr2, and the zirconium hydrides[END_REF]. The ratio protrusion height to blister depth ratio was measured to be equal to 22.2% by metallography analysis on 4 samples. Post mortem analysis on 19 failed samples containing blisters showed the ratio is 21.5±1.2%, therefore confirming the ratio is about 22%. This ratio was used to check that the grown blisters correspond to the expected depth prior to mechanical testing by nondestructive measurement of the protrusion height. This also confirmed the good reproductibility of the developed setup in generating blisters. The figure 1 shows many hydrides are present under the blister. The hydrogen concentration is about 450wppm on the inner side to 1000wppm at the blister frontier based on measurement with the image analysis software HYDURO developed at the CEA. While Zircaloy-4 exhibit mainly circumferential hydrides (hydrides aligned with the tube circumferential direction) under standard reactor conditions, the figure 1 show a "sunburst" of radial hydrides around the blister. These radial hydrides are a clear mark of the stress field induced by the blister growth. The blister obtained microstructure is representative of some blisters observed in irradiated Zircaloy-4 claddings (figure 2). Several characterizations such as X-Ray diffraction and micro X-ray diffraction in synchrotron, nano-hardness or micro-Elastic Recoil Detection Analysis (ERDA) were performed to measure the hydrogen content inside and outside the blister and the crystallographic phase of the hydrides in the blister. Only the ERDA measurements are detailed here for sake of brevity. ERDA is a nuclear technique well adapted for quantitative absolute measurement of hydrogen concentration [START_REF] Raepsaet | Quantification and local distribution of hydrogen within Zircaloy-4 PWR nuclear fuel cladding tubes at the nuclear microprobe of the Pierre Sue Laboratory from l-ERDA[END_REF]. Preliminary tests showed the accuracy at high content is about 2,5% on the hydride stoichiometry ZrHx. ERDA measurements showed the blister is a hydrogen concentration gradient mainly constituted of hydrides (more than 85% in volume fractionfigure 3). The concentration at the top of the blister is high enough for the transformation from zirconium to hydride to be complete. This result has been confirmed by XRD measurement in synchrotron, where peaks associated with zirconium completely disappeared in the upper area of the blister, and by nano-hardness measurements. All the hydrides were found to be "δ-hydrides".

DEVELOPPEMENT OF A NEW MECHANICAL TEST : HB-EDC

A newly developed test named High Biaxiality Expansion Due to Compression (HB-EDC) was developed to reach stress biaxiality ratio (axial to circumferential) of 0.5 while the standard EDC test biaxiality is limited to 0. The EDC test consists in compressing a soft pellet inside a tubular sample, to simulate the fuel pellet expansion (figure 4). The end of the sample are free to move axially, therefore no or very limited axial stresses are generated in the cladding. The basic idea of the HB-EDC is to block the axial displacement at the end of the sample (figure 4). Finite element computations showed this allows reaching a 0.5 biaxiality ratio that is more representative of RIA conditions where the biaxiality ratio is included between 0.5 and 1. The HB-EDC setup is easy to use, is reusable (more than 40 tests in the present study), requires only small samples (30mm long), allows to test samples with displacement controlled loading up to high strain rate, at high temperatures (up to 480°C in the present study but no part of the device limit higher temperatures), and is adaptable to standard tensile machine. The HB-EDC test has therefore all the characteristics needed to characterise the fracture of irradiated materials in shielded cells. It implementation in CEA shielded cells is planned in the near future.

APPLICATION OF EDC AND HB-EDC TEST TO STUDY THE LOADING BIAXIALITY INFLUENCE ON THE CLADDING FRACTURE WITH HYDRIDE BLISTERS

Several EDC and HB-EDC mechanical tests were performed at 0,1/s strain rate and 25, 350 and 480°C, monitored with an high speed optical camera (1000hz) to detect accurately the fracture of the cladding. The circumferential strain was measured as being the sample diameter on the last picture before cladding failure (figure 5). Zircaloy-4 cladding showed a marked influence of the loading biaxiality without hydride blisters. At 25°C, the fracture strain is divided by 2 to 3 between the EDC and HB-EDC. EDC tests at 350°C and 480°C without blisters did not allowed to produce samples failure while they failed at fracture strain NNN.6 from 20% to 30% with HB-EDC. This is illustrated by the figure 6(a) were the final aspect of EDC and HB-EDC after a 350°C test without blister can be observed. The presence of blisters clearly embrittle the cladding, as one could see on the sample post mortem aspect on the figure 6(b). Nevertheless, the loading biaxiality seems to have a limited or no effect on the fracture strain in presence of blisters (figure 5). This result is in lines with the previous observations of Glendening [START_REF] Glendening | Failure of hydrided Zircaloy-4 under equal biaxial and plane-strain tensile deformation[END_REF] who have reported no effect of biaxiality between biaxiality ratios of 0.5 and 1 in the fracture strains of Zircaloy-4 sheets.

CONCLUSION AND PERSPECTIVES

A setup was developed to generate hydride blisters on Zircaloy-4 claddings. Preliminary thermodiffusion calculations were performed to determine the best thermal conditions to optimize the blister growth rate. Hydride blister cladding half-thickness deep were obtained in 24h. Extensive characterization of hydride blister has shown blisters are representative of some blisters observed in irradiated Zircaloy-4 claddings. A blister is a hydrogen gradient, where the complete transformation of zirconium to hydride can be achieved. A new mechanical named HB-EDC was developed. It allows performing tests on tubular samples and has all the characteristics needed for use on irradiated materials in shielded cells in the near future. The results obtained comparing EDC and HB-EDC tests showed a mark effect of the loading biaxiality without blister, but a negligible effect in presence of blister. Even in presence of deep blister (>200µm), the unirradiated material can maintain a certain ductility (≈4 to 5%) at 350°C.

This work has several perspectives. The very first step is to perform numerical computations to simulate the blister induced embrittlement.

figure 1 : 4 figure 2 :

 142 figure 1: Blister morphology in the radial circumferential plane and induced protrusion on the external surface (optical microscopy).

figure 3 :

 3 figure 3: Hydrogen concentration determined by ERDA (a) global cartography in wppm, (b) scan along the horizontal black line and (c) scan along the vertical black line.

figure 4 :

 4 figure 4: Schematic view of EDC versus HBEDC tests.

figure 5 :

 5 figure 5: Circumferential deformation measured with the last picture before fracture of EDC and HB-EDC tests at 25, 350 and 480°C, without blister or for different blister depths.

figure 6 :

 6 figure 6: Post-mortem aspect of (a) EDC and HB EDC sample without blister tested at 350°C and (b) aspect of EDC test without or with a 250µm deep blister at 350°C.

AKNOWLEDGEMENTS

This work has envolved several persons from CEA and Mines ParisTech. S. Allègre, M. Azéra, J.L. Béchade, P. Berger, S. Bosonnet, J.C. Brachet, E. Foy, T. Guilbert, A. Meddour, D. Menut, C. Poussard, M. Sonnaert, J.C. Teissedre, C. Toffolon are especially acknowledged. Useful suggestions from M. Daymonds (Queens University, Canada), F. Garzarolli (ANT international, Sweden), A. Motta (Penn State University, USA) and M. Puls (MPP Consulting, Canada) are particularly acknowledged.