Optimal Transition from Coal to Gas and Renewable Power Under Capacity Constraints and Adjustment Costs

Abstract

Given a cap on carbon emissions, what is the optimal transition from coal to gas and renewable energy? To answer this question, we model the dependence of the energy sector on both (1) polluting exhaustible resources and (2) long-lived capital such as power plants. To minimise adjustment costs, optimal investments in expensive renewable energy start before phasing out fossil-fuel consumption, and may even start before investing in gas-fired plants. Investment in gas-fired plants can reduce the need for expensive renewable investment in the short term, but they eventually need to be decommissioned to make room to carbon-free power in the long term. Simulations of the European Commission’s Energy Roadmap illustrate and quantify these results.

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Fig. 1

Notes: The horizontal axis shows the installed capacity for renewable power (r), gas (g) and coal (\(k_{c,0^-}=\infty \)) at a given point in time. The vertical axis shows the marginal costs and the price of electricity. The production technologies are ranked according to their variable production costs, excluding capacity rents (in other words, accounting only for the resource costs \(\alpha _{i}\) plus the cost of emissions \(\mu _0 F_{i}\)). Time subscripts, \(_t\), are omitted. The price p and quantity q are set by the intersection of the demand curve \(u'\) and the merit-order curve. If there is such an intersection, it means that only one type of power plant is not used at full capacity: the marginal resource. All technologies used at full capacity receive a capacity rent \(\gamma _i = p -\alpha _{i}-\mu _0 F_{i}\)

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Notes

  1. 1.

    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 impacts on energy security (see Guivarch et al. 2015, for a review) and non-GHG pollution (e.g., Shindell 2015). The case of carbon capture, another option to decarbonise power generation, is not treated in this paper.

  2. 2.

    Whether a rapid phasing out of coal would be politically feasible, or even compatible with existing transmission infrastructure (since coal power plants are concentrated in a few countries in Europe) is beyond the scope of this paper.

  3. 3.

    The interaction of investment and natural resource extraction is also the focus of theoretical work on mining, in which installed capital similarly limits the extraction rate of a single type of minerals (e.g., Campbell 1980; Gaudet 1983; Lasserre 1985); see Cairns (1998) for a review. Other reasons for not extracting resources in a sequential way à la Herfindahl include time to build (?), imperfect substitution (Smulders and van der Werf. 2008), heterogeneity of producers/consumers, and transportation costs (Gérard and Salant 2014 provide a recent review).

    In addition, Dasgupta and Heal (1974), Solow (1974), and Stiglitz (1974) have started a literature that studies the impact of resource scarcity on growth, in green Ramsey models that feature both capital accumulation and resource extraction. This literature also focuses on a single type of capital and a single fossil resource (e.g., Van der Ploeg and Withagen 1991, 2014), while we model several resources and resource-specific capital.

  4. 4.

    In our paper, investment refers to the building of new power plants. In practice, when private owners of power plants sell an existing plant to another electricity production firm, the latter also undertakes investment. However, at the social level, such investment represents a transfer of property rights in existing capacities rather than the net accumulation of new capital.

  5. 5.

    A carbon-budget constraint is equivalent to a carbon-ceiling constraint if natural dilution of CO\(_2\) in the atmosphere is negligible: in such a case, the carbon budget is simply equal to the carbon ceiling target minus initial atmospheric carbon stock. This equivalence no more holds with natural dilution. With a carbon-ceiling constraint and natural dilution of CO\(_2\) in the atmosphere, the transition to renewables would be modified. As coal is assumed to be over abundant, coal would still be used in the long-run and consumed at a level such that natural dilution just balances emissions from coal. The assumption regarding the dynamics of natural dilution is also important. Were natural dilution constant, most of our results would still hold as the variable cost of fossil fuels would still increase at the interest rate. Were natural dilution proportional to the carbon stock, the carbon price would increase at the rate \(\rho +\beta \), with \(\rho \) being the social discount rate and \(\beta \) being the natural dilution rate of CO\(_2\). The most polluting resource could be used first to build up the carbon stock in order to increase natural dilution. Chakravorty et al. (2008) analyse this problem in a setting without capacity constraints and two exhaustible resources with different pollution contents.

  6. 6.

    Setting a carbon tax equal to the shadow cost of pollution would decentralise the optimum if agents have perfect foresight about the future tax path, resource owners are in perfect competition, and both coal and gas are used in the transition towards renewables. This has been demonstrated in Coulomb and Henriet (2018), who study a problem similar to Problem 9 (abundant coal, scarce gas, abundant renewables) but ignore capital constraints. Their proof can be easily amended to add capital constraints.

  7. 7.

    Assuming abundant gas reserves would not change the main results of our analysis. Were gas reserves abundant, the gas price would still tend to increase through time because of the carbon budget constraint, and gas would be phased out for economic reasons. In the numerical simulations of Sect. 4, gas reserves are found to be abundant enough to not be exhausted in the low-carbon transition.

  8. 8.

    In particular, the transversality condition ensures that \(\nu _{i,t}=\int _{t}^{\infty }{e^{-(\delta + \rho )(\theta -t)}\gamma _{i,\theta }\, d\theta }\) is definite.

  9. 9.

    It is not optimal to use fossil fuels forever. If fossil fuels were used forever, the long-run price of gas or coal would be higher than the long-term energy price, \(u'(\bar{k}_{r})\). Indeed, \(min(p_{g,t}, p_{c,t})>min((\alpha _{g,0}+F_g\mu _0)e^{\rho t}; F_c\mu _0e^{\rho t})\), thus \(\lim _{t \rightarrow T} min((\alpha _{g,0}+F_g\mu _0)e^{\rho t} ; F_c\mu _0e^{\rho t})=\infty > u'(\bar{k}_r)\) if T infinite.

  10. 10.

    Indeed, the same consumption path could be reached at a lower cost by smoothing investments in renewables and using less of the gas capacities and postponing date T.

  11. 11.

    From Eq. (15), it follows that \(\forall t,\) such that \( x_{g,t}x_{r,t}>0\), \(\nu _{r,t}- \nu _{g,t} \ge \int _{t}^{T}{ e^{-(\delta +\rho )(\theta -t)} (\alpha _{g,\theta }+F_g\mu _\theta })d\theta + \int _{T}^{\infty } { e^{-(\delta +\rho )(\theta -t)} p_\theta d\theta } \ge 0\).

  12. 12.

    Indeed, in our setting, shadow costs are all continuous through time. For examples of optimal control problems with jumps in state and/or co-state variables, see Blaquière (1985) and Seierstad and Sydsaeter (1987).

  13. 13.

    First, \(\forall k_{g,0^-} < \bar{k}_{g,0^-}\) there is a positive threshold in the value of the investment cost under which investment occurs. As a result, the f curve cannot intersect the x-axis for a value of \(k_{g,0^-}\) strictly smaller that \( \bar{k}_{g,0^-}\). Second, if one assumes that \(f(\bar{k}_{g,0^-}) >0\), there would exist cases with \(c'_g(0)>0\) and \({k}_{g,0^-}\) arbitrarily close to \(\bar{k}_{g,0^-}\) with investments in gas still being optimal. This is a contradiction, because the capacity rent of gas tends to zero when \({k}_{g,0^-}\) tends to \(\bar{k}_{g,0^-}\) due to the continuity of the solution with respect to marginal parameters changing.

  14. 14.

    The relative quantity (in GW/year) of power plants that should be built is ambiguous. Renewable plants have a higher value than fossil-fuel plants (see earlier subsections), but the latter are cheaper to build.

  15. 15.

    This is on the lower range of most studies, but these usually ignore capital stocks, focus on past price variations that are small compared to the changes considered in the present study, and ignore the fact that a substantial part of demand is still regulated in Europe (see e.g., Espey and Espey 2004 or Labandeira et al. 2012). Note that, in our framework, the long-term price elasticity of demand will typically be much higher when taking changes in the capital stock into account.

  16. 16.

    We let the adjustment costs vary from 0 to 100%; i.e., at the average annual investment flow, \(X_i\), the marginal costs vary from \(\frac{C_i^m}{2}\) to \(\frac{3C_i^m}{2}\). The carbon budget varies from 5 to 100 GtCO\(_2\), with the central carbon budget value being 22 GtCO\(_2\). To analyse the sensitivity to the pre-existing gas capacities, we let the initial gas capacity vary from 0 to 400 GW; i.e., approximately three times the installed capacity in the EU in 2012. To analyse the sensitivity to the marginal investment cost in gas, we let the nominal investment costs vary from 0 to 6 $/W, or approximately four times its central value.

  17. 17.

    In the analytical part, we assumed that pre-existing coal capacities were abundant, in which case, investing in coal capacities would never be optimal.

  18. 18.

    “GAMS user’s guide.” www.gams.com/latest/docs/UG_MAIN.html (Accessed Nov. 2017).

  19. 19.

    “CONOPT Home Page” www.gams.com/latest/docs/S_CONOPT.html (Accessed Nov. 2017).

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Correspondence to Oskar Lecuyer.

Additional information

We thank Mook Bangalore, Céline Guivarch, Louis-Gaëtan Giraudet, Stéphane Hallegatte, Jean- Charles Hourcade, Guy Meunier, Jean-Pierre Ponssard, Antonin Pottier, Philippe Quirion, Julie Rozenberg, François Salanié, Ralph Winkler, one anonymous referee from the FAERE Working Paper series and two other anonymous referees for useful comments and suggestions on earlier versions of this paper. We are responsible for all remaining errors. We are grateful to Patrice Dumas for technical support, and to Institut pour la Mobilité Durable (Renault and ParisTech), Ecole des Ponts ParisTech and EDF R&D for financial support. The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reflect the views of the Inter-American Development Bank, its executive directors, or the countries they represent.

Appendices

Appendix

Efficiency Conditions

Table 3 Notation used for variables and parameters in the model

The Hamiltonian associated with Problem 9 is (Table 3):

$$\begin{aligned} \begin{aligned} \mathcal {H}&= e^{- \rho t} \left[ u\left( \sum _i{q_{i,t}}\right) - \sum _i{c_i(x_{i,t}) } \right. - \sum _i{\nu _{i,t}\left( \delta \, k_{i,t}- x_{i,t}\right) }\\&\qquad -\mu _t\sum _i{ F_iq_{i,t}} - \eta _t\left( m_t-\bar{M}\right) - \sum _i{\left( \alpha _{i,t}\, q_{i,t}-\beta _{i,t}\, S_{i,t}\right) }\\&\qquad \left. - \sum _i{\gamma _{i,t}\left( q_{i,t} - k_{i,t} \right) } + \sum _i{ \lambda _{i,t}\, q_{i,t}} + \sum _i{ \xi _{i,t}\, x_{i,t}} \right] \end{aligned} \end{aligned}$$
(A1)

The first-order conditions are:

$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial x_{i}} = 0\iff & {} &c_i'(x_{i,t}) = \nu _{i,t}+ \xi _{i,t}\end{aligned}$$
(A2)
$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial q_i} = 0\iff & {} &\lambda _{i,t}- \mu _tF_i -\alpha _{i,t}+ u'_t= \gamma _{i,t}\end{aligned}$$
(A3)
$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial k_{i}} = \frac{-d\left( e^{- r t}\nu _{i,t}\right) }{dt}\iff & {} &(\delta +\rho )\,\nu _{i,t}- \dot{\nu }_{i,t}= \gamma _{i,t}\end{aligned}$$
(A4)
$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial m_t}= \frac{-d\left( e^{- r t}\mu _t\right) }{dt}\iff & {} &\dot{\mu }_t- \rho \,\mu _t= - \eta _t\end{aligned}$$
(A5)
$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial S_{i}} = \frac{-d\left( e^{- r t}\alpha _{i,t}\right) }{dt}\iff & {} &\dot{\alpha }_{i,t} - \rho \,\alpha _{i,t}= -\beta _{i,t}\end{aligned}$$
(A6)

The complementary slackness conditions are:

$$\begin{aligned} \forall i,t, \quad&\xi _{i,t}\ge 0,\quad x_{i,t}\ge 0&\text {and}&\xi _{i,t}\, x_{i,t}= 0 \end{aligned}$$
(A7)
$$\begin{aligned} \forall i,t, \quad&\lambda _{i,t}\ge 0,\quad q_{i,t}\ge 0&\text {and}&\lambda _{i,t}\, q_{i,t}= 0 \end{aligned}$$
(A8)
$$\begin{aligned} \forall i,t, \quad&\eta _t\ge 0,\quad \bar{M}-m_t\ge 0&\text {and}&\eta _t\,\left( \bar{M}-m_t\right) = 0 \end{aligned}$$
(A9)
$$\begin{aligned} \forall t, \quad&\beta _{g,t} \ge 0,\quad S_{g,t} \ge 0&\text {and}&\beta _{g,t} \, S_{g,t}= 0 \end{aligned}$$
(A10)
$$\begin{aligned} \forall i,t, \quad&\gamma _{i,t}\ge 0, \quad k_{i,t}- q_{i,t}\ge 0&\text {and}&\gamma _{i,t}\, \left( k_{i,t}- q_{i,t}\right) =0 \end{aligned}$$
(A11)

The transversality conditions at infinity are:

$$\begin{aligned}&\lim _{t \rightarrow \infty } \alpha _{g,t} e^{-\rho t} S_{g,t}= \alpha _{g,0} \lim _{t \rightarrow \infty } S_{g,t}= 0 \end{aligned}$$
(A12)
$$\begin{aligned}&\lim _{t \rightarrow \infty } \mu _t e^{-\rho t}(\bar{M}- m_t)= \mu _0 \lim _{t \rightarrow \infty } (\bar{M}- m_t)=0. \end{aligned}$$
(A13)
$$\begin{aligned}&\lim _{t \rightarrow \infty } \nu _{i,t} e^{-\rho t}k_{i,t}=0, \quad i \in \{g,r\}. \end{aligned}$$
(A14)

Scarcity Rent of Gas and Shadow Cost of Pollution

From (A9) and (A10), we see that while the carbon budget and the gas stock are not exhausted, their positive dual variables \(\eta _t\) and \(\beta _{g,t}\) are zero. From (A5) and (A6), we then get the classic Hotelling rule:

$$\begin{aligned}&\alpha _{g,t}= \alpha _{g,0}\, e^{\rho t} \\&\mu _t= \mu _0 e^{\rho t} \end{aligned}$$
Fig. 7
figure7

Phase diagram: renewables-only phase

Steady State

In the main text, we have shown that fossil fuels cannot be used forever. Dropping the time subscript, the long-term renewable capital level is the solution to the following system of equations:

$$\begin{aligned}&\dot{k}_r=-\delta k_r+x_r \end{aligned}$$
(A15)
$$\begin{aligned}&\dot{\nu }_r=(\rho +\delta )\nu _r-\gamma _r=(\rho +\delta )\nu _r-u'^{-1}(k_r). \end{aligned}$$
(A16)

Using a simple phase diagram (Fig. 7), we show that renewable capacities evolve smoothly from \(k_{r,T} \) to \(\bar{k}_r\) defined by:

$$\begin{aligned} c_r'(\delta \bar{k}_r)=\frac{ u'(\bar{k}_r)}{\rho +\delta }. \end{aligned}$$

This represents the steady state. Transition towards the steady state is represented by the bold curve in Fig. 7. If \(k_{r,T}<\bar{k}_r\), renewable capacities increase towards \(\bar{k}_r\), investment must be larger than \(({\rho +\delta })c_r'(\delta \bar{k}_r)\). If \(k_{r,T}>\bar{k}_r\), renewable capacities decrease towards \(\bar{k}_r\), investment must be lower than \(({\rho +\delta })c_r'(\delta \bar{k}_r)\). Below, we show that there is no incentive to deploy renewable capacities to any value larger than their long-term level \(\bar{k}_r\). As initial capacities are negligible, this result implies that \(k_{r,T}<\bar{k}_r\), and thus investments are larger than \(({\rho +\delta })c_r'(\delta \bar{k}_r)\) during the renewables-only phase: capacities increase towards \(\bar{k}_r\) and the energy price decreases. The bold curve in the grey area in Fig. 7 displays the transition to the steady state when only renewables are used if pre-existing renewable capacities are small.

Accumulation of Renewable Capacities Before Date T

We first show that \(k_{r,T} \le \bar{k}_r\), i.e. \(p_T \ge \bar{p}\). Assume that \(k_{r,T}>\bar{k}_r\), i.e. \(p_T<\bar{p}\), where \(\bar{p}\) is the long-term energy price, \(\bar{p}=c_r'(\delta \bar{k}_r)(\rho +\delta )\). We call \(t^*\) the last date at which we invest in renewable capacities to expand them to a level of at least \(\bar{k}_r\). As \(k_{r,0^-}=0<\bar{k}_r\), such a date necessarily exists. \(t^*\) is such that \(x_r(t^*) \ge \delta k_{r,t^*}\) with \(k_{r,t^*} \ge \bar{k}_r\). As \(x_r(t^*) \ge \delta \bar{k}_{r}\), \(\nu _{r,t^*}=c_r'(x_{r,t^*}) \ge c_r'(\delta \bar{k}_{r})\). For all \(t^*\le t \le T\), \(k_{r,t} \ge k_{r,T}\) and \(p_t \le p_T <\bar{p}\). From the phase diagram in Fig. 7, \(k_{r,T}>\bar{k}_r\)\(\implies \)\(\dot{p}(t)>0\) and \(\bar{p}>p_t>p_T\) for all \(t>T\). If follows that \(\nu _{r,t^*} <\frac{\bar{p}}{\rho +\delta }\). As \(\bar{p}=c_r'(\delta \bar{k}_r)(\rho +\delta )\), it comes that \(\nu _{r,t^*}<c_r'(\delta \bar{k}_r)\). Contradiction, thus \(k_{r,T} \le \bar{k}_r\). There is no incentive to expand the capacities to a value of at least \(\bar{k}_r\) before date T. Indeed, were this optimal, there would exist a date, \(t^{**}\), at which \(x_{r,t^{**}} \ge \delta \tilde{k}_r\) where \( \tilde{k}_ r \ge \bar{k}_r\) is defined as the maximum of the value of the capacities reached before T. The limit of \(\dot{x}_{r,t}\) when t tends to the right of \(t^{**}\) is negative as \(\tilde{k}_r\) is the maximum of \(k_{r,t}\) for \(t \le T\). It follows that \(\dot{\nu }_{r,t}\) is negative when t tends to the right of \(t^{**}\) as \(\dot{\nu }_{r,t}= c_r''({x}_{r,t}) \dot{x}_{r,t}\) with \(c_r''({x}_{r,t})>c_r''(0)>0\). We get that for t larger than \(t^{**}\) but close enough, \((\rho +\delta ){\nu }_{r,t}<p_t\). As some fossil fuel is used at \(t^{**}\), thus \( \tilde{k}_ r \ge \bar{k}_r\) implies \(p_{t^{**}} < \bar{p}\). As \(p_{t^{**}} < \bar{p}\), we get that \((\rho +\delta ){\nu }_{r,t}< p_{t^{**}}< \bar{p}\) for t larger than \(t^{**}\) but close enough. This implies that \({\nu }_{r,t^{**}} < c_r'(\delta \bar{k}_r)\) for t larger than \(t^{**}\) but close enough. This contradicts the fact that \(x_{r,t^{**}} \ge \delta \tilde{k}_r \ge \delta \bar{k}_r\). Thus, there is no incentive to expand the capacities to a value of at least \(\bar{k}_r\) before the exhaustion of the carbon budget.

Fossil Fuels Use in the Transition to the Renewables-Only Phase

A NECESSARY CONDITION ON THE PRICE ELASTICITY OF DEMAND TO HAVE THE ELECTRICITY PRICE INCREASE AT A RATE STRICTLY LARGER THAN THE DISCOUNT RATE—If the electricity price increases at a rate strictly larger than the interest rate, the marginal resource must be gas with full capacity use. Were coal or gas used at partial capacity, the price would increase at the discount rate (Eq. 17). Assume that at date \(t^*\), the electricity price increases at a rate higher than the discount rate. Call \(p^*\) the price at date \(t^*\). \(\dot{D}(p^*)=D'(p^*)\dot{p}^*\), with \(\dot{p}^*>\rho {p^*}\). Denoting by \(\epsilon \) the price elasticity of demand, we get: \(\dot{D}(p^*)<D'(p^*)\rho {p^*}=\epsilon \rho D(p^*)\). If \(\epsilon \le -\frac{\delta }{\rho }\), then \(\frac{\dot{D}(p^*)}{D(p^*)}<-{\delta } \). Demand decreases at a rate larger that the depreciation rate, thus capacities cannot be fully used around date \(t^*\) and the energy price increases at the discount rate at that date. Thus, a necessary condition to have the electricity price increase over an interval of time at a rate strictly larger than the interest rate is that \(\exists p \in W, \epsilon _p > -\frac{\delta }{\rho }\), where W represents the range of prices covered in the transition to renewables.

USING A RESOURCE OVER DISJOINT INTERVALS OF TIME—It is clear that renewables are not used over disjoint time intervals as their marginal costs equal zero. Below, we show that if the demand is elastic enough, fossil fuels cannot be used over disjoint intervals of time. First, recall that while a fossil fuel is used under full capacity, its price grows exponentially at the rate \(\rho \). If a fossil fuel i is used again after it has momentarily stopped being used, the electricity price must grow at rate strictly larger than the discount rate over a period in which this fossil fuel is not used, so that the electricity price “catches up” with the variable cost of the fossil fuel i. This requires that \(\exists p \in W, \epsilon _p > -\frac{\delta }{\rho }\), where W represents the range of prices covered in the transition to renewables, as shown in Sect. A.A5. If \(\epsilon \le -\frac{\delta }{\rho }\), the electricity price cannot increase at a rate faster than the social discount rate under full use of capacities and resources cannot be used over disjoint intervals of time.

PARTIAL USE OF GAS CAPACITIES THEN FULL USE OF GAS CAPACITIES—Assume that gas capacities are not fully used, the gas price increases at the rate \(\rho \). Coal is not used because otherwise the social planner would be indifferent between gas and coal, and the pre-existing gas capacities would be “abundant”. Installed capacities follow the motion law: \(\dot{k}_{r,t}+\dot{k}_{g,t}=-\delta (k_{r,t}+k_{g,t})+ x_{r,t}+x_{g,t}\). The energy demand is smaller that the supply based on full use of the gas capacities: \(D(p(t))< {k_{r,t}+k_{g,t}}\). There must exist an interval of time over which the electricity price increases at the interest rate, but the demand decreases at a lower speed than existing capacities, otherwise the gas capacities will never be a binding constraint. Call \(t^*\) a date included in this time interval and \(p^*=p_{t^*}\) the gas price at date \(t^*\). The variation through time of the demand is given by \(\dot{D}(p^*)=D'(p^*)\dot{p}^*\), with \(\dot{p}^*= \rho {p^*}\). Writing \(\epsilon \) as the price elasticity of demand, we get: \(\dot{D}(p^*)=D'(p^*)\rho {p^*}=\epsilon \rho D(p^*)\). If \(\epsilon \le -\frac{\delta }{\rho }\), then \(\frac{\dot{D}(p^*)}{D(p^*)}\le -{\delta } \). As investments are positive or nil, \(\dot{k}_r+ \dot{k}_g \ge -\delta ({k_r+k_g})\). Thus, if \(\epsilon \le -\frac{\delta }{\rho }\), \(\frac{\dot{D}(p^*)}{D(p^*)}\le \frac{\dot{k}_r+ \dot{k}_g }{{k_r+k_g}}<0.\) The demand would decrease through time faster than installed capacities depreciate and gas capacities would never be fully used in the future. \(\epsilon \le -\frac{\delta }{\rho }\) is a sufficient condition to avoid a path on which gas capacities are partially used then fully used (assuming coal and gas are interchangeable; i.e., excluding case 2 in Sect. 3.3).

Transitions Towards the Renewables-Only Phase

Coal and gas production are eventually replaced by renewable power. Let \(T^{-}_i\) be the date when the production of technology i definitely stops:

$$\begin{aligned} \forall t \ge T^{-}_i,\forall i \in \{c,g\},\quad q_{i,t}= 0 \text { and } T_c^-<T_g^-=T \end{aligned}$$
(A17)

This is always true if there is no indifference between gas and coal use. \(T_c^-<T_g^-\) simply means that gas phases out coal as indicated in Sect. 3.3. \(T_g^-=T\) means that gas is necessarily used before the carbon budget is exhausted.

Similarly, let \(T^+_i\) be the date when production of technology \(i \in \{c,g,r\}\) starts. As gas capacities pre-exist, \(T^+_g=0\). As the marginal utility tends to infinity when energy consumption tends to zero, energy production is always strictly positive: only the emitting production eventually stops.

Let \(t^X\) be the date when the gas deposit is depleted. If it exists (i.e., if the gas deposit is eventually depleted), \(t^X\) coincides with the end of production from gas fuel:

$$\begin{aligned} \exists t^X \implies t^X=T^{-}_g \end{aligned}$$
(A18)

Let \(\tau ^+_i \le T^+_i\) be the date when investment in capacity i starts, and let \(\tau ^{\text {-}}_i\) be the date when it stops. Investment in renewables is required before the carbon budget is exhausted:

$$\begin{aligned} \exists \tau _r^+, \tau _r^+ < T=T_g^- \end{aligned}$$

For each technology, the sequence of dates is:

$$\begin{aligned}&0 = T_g^+ \le \tau _g^+< \tau _g^-< T_g^- = T \\&0 \le T_c^+ \le T_c^- \le T \\&0 \le T_r^+ = \tau _r^+ <T \end{aligned}$$

We can characterise different extraction paths by ordering the first investment date of each technology, and the dates they stopped, to be used. Overlapping transition means that renewables are developed before coal is phased out. Sequential transition means that renewables are developed after coal has been phased out by gas.

  • Coal is not used (case 1):

    • Investments in gas

      $$\begin{aligned} 0 \le \tau _r^+ \le \tau _g^+< T_g^- =T\\ 0 \le \tau _g^+\le \tau _r^+ < T_g^- =T \end{aligned}$$
    • No investments in gas

      $$\begin{aligned} 0 \le \tau _r^+ < T_g^- =T \end{aligned}$$
  • Coal and gas are used

    • Abundant gas capacities (case 2): Undefined transition

      $$\begin{aligned} 0 \le \tau _r^+ < T \end{aligned}$$
    • Scarce gas capacities; no investments in gas (case 3)

      $$\begin{aligned}&0 \le \tau _r^+ \le T_c^-< T_g^- =T \textit{ Overlapping transition} \\&0< T_c^- \le \tau _r^+ < T_g^- =T \textit{ Sequential transition} \end{aligned}$$
    • Scarce gas capacities; investments in gas (case 3)

      $$\begin{aligned}&0 \le \tau _r^+ \le T_c^- \le \tau _g^+< T_g^- =T \textit{ Overlapping transition} \\&0 \le \tau _r^+ \le \tau _g^+ \le T_c^-< T_g^- =T \textit{ Overlapping transition} \\&0 \le \tau _g^+\le \tau _r^+ \le T_c^-< T_g^- =T \textit{ Overlapping transition} \\&0 \le \tau _g^+ \le T_c^- \le \tau _r^+< T_g^- =T \textit{ Sequential transition} \\&0< T_c^- \le \tau _g^+ \le \tau _r^+< T_g^- =T \textit{ Sequential transition} \\&0< T_c^- \le \tau _r^+ \le \tau _g^+ < T_g^- =T \textit{ Sequential transition} \end{aligned}$$

Web Appendix: Numerical Algorithm

A discretised version of the model (with 1000 timesteps of .2 years) was coded in the GAMS modelling languageFootnote 18 as a nonlinear problem, using the CONOPT solving algorithm.Footnote 19 The algorithm used in GAMS/CONOPT is based on the GRG algorithm first suggested by Abadie and Carpentier (1969), specifically designed for large nonlinear programming problems. The actual implementation has many modifications to make it efficient for large models and for models written in the GAMS language. Details on the algorithm can be found in Drud (1985, 1992).

The original GRG method helps achieve reliability and speed for models with a high degree of nonlinearity, i.e., difficult models, and CONOPT is often preferable for highly nonlinear models and for models for which feasibility is difficult to achieve. Extensions to the GRG method such as pre-processing, a special phase 0, linear mode iterations, and a sequential linear programming and a sequential quadratic programming component makes CONOPT efficient for easier and mildly nonlinear models as well.

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Coulomb, R., Lecuyer, O. & Vogt-Schilb, A. Optimal Transition from Coal to Gas and Renewable Power Under Capacity Constraints and Adjustment Costs. Environ Resource Econ 73, 557–590 (2019). https://doi.org/10.1007/s10640-018-0274-4

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Keywords

  • Climate change mitigation
  • Low-carbon investment
  • Renewables
  • Fossil fuels
  • Resource extraction
  • Early-scrapping

JEL Classification

  • Q54
  • Q58