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.

Appendices

Appendix

1.1 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)

1.2 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

Phase diagram: renewables-only phase

1.3 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.

1.4 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.

1.5 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).

1.6 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 language¹⁸ as a nonlinear problem, using the CONOPT solving algorithm.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.