Optimal policies to promote efficient distributed generation of electricity

Abstract

We analyze the design of policies to promote efficient distributed generation (DG) of electricity. The optimal policy varies with the set of instruments available to the regulator and with the prevailing DG production technology. DG capacity charges often play a valuable role in inducing optimal investment in DG capacity, allowing payments for DG production to induce the optimal production of electricity using non-intermittent DG technologies. Net metering can be optimal in certain settings, but often is not optimal, especially for non-intermittent DG technologies.

Appendix

Part A of this Appendix outlines the proofs of key propositions in the text.⁵⁸ Part B provides the details of the numerical analysis presented in Sect. 7.

1.1 A. Proofs of the propositions in the text

The proof of Proposition 1 follows immediately from the proof of Proposition 2. The proof of Proposition 3 follows immediately from the proof of Proposition 6 in Brown and Sappington (2017b).

Proof of Proposition 2

Let $\lambda \ge 0$ denote the Lagrange multiplier associated with constraint (5). Using (1), (2), (3), and the Envelope Theorem, the necessary conditions for a solution to [RP-$\theta e$] with respect to $R_{j}$ and $K_{G}$ reveal that $\lambda \,=\,1$ and conclusion (i) in Proposition 1 holds. Furthermore, because $\,\frac{\partial Q_{t}^{v}}{\partial X_{t}^{j}} \,=\,1\,$ and $\,\frac{\partial V_{t}^{j}(X_{t}^{j}(\cdot ,\theta _{t}),\theta _{t})}{\partial X_{t}^{j}}\,=\,r_{jt}(\theta _{t})$ for $\,j\in \{D,N\}$ and $t\in \{L,H\}$, the corresponding necessary condition with respect to $\,r_{jt}(\theta _{t})$ provides:

$$\begin{aligned} r_{jt}(\theta _{s})=\frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}(\cdot ,\theta _{s})}+\psi _{tv}(\cdot )\text { for all }\,\theta _{s}\,\in \,\left[ \,\underline{\theta }_{t},\,\overline{\theta } _{t}\,\right] . \end{aligned}$$

(9)

Because $\lambda =1$, $\frac{\partial Q_{t}^{v}}{\partial Q_{t}^{n}}\,=\,-\,1\,$, $\,\frac{\partial K_{Dy} }{\partial k_{yt}}$ is not a function of $\,\theta _{s}$, $\,k_{y}$ only affects $Q_{t}^{v}$ through $K_{Dy}$, $\,\frac{dQ_{t}^{v}(\cdot ,\theta _{s}) }{dK_{Di}}\,=\,-\,\mu _{t}\,\theta _{s}\,$, and $\frac{\partial Q_{t}^{i}(\cdot ,\theta _{s})}{\partial K_{Di}}\,=\,\mu _{t}\,\theta _{s}$, the necessary conditions with respect to $k_{i}$ and $k_{n}$ reveal:

$$\begin{aligned} k_{i}= & {} -\frac{\partial T(\cdot )}{\partial K_{Di}}+\sum \limits _{t\,\in \,\{L,H\}}\int _{\underline{\theta }_{t}}^{ \overline{\theta }_{t}}\,\bigg [\,\frac{\partial C_{t}^{G}(Q_{t}^{v}(\cdot ,\theta _{s}),K_{G})}{\partial Q_{t}^{v}}-w_{it}(\theta _{s})+\psi _{tv}(\cdot ) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad -\,\psi _{ti}(\cdot )\,\bigg ] \,\mu _{t}\,\theta _{s}\,\,dF_{t}(\theta _{s})\,\text {, and} \end{aligned}$$

(10)

$$\begin{aligned} k_{n}= & {} -\frac{\partial T(\cdot )}{\partial K_{Dn}} +\sum \limits _{t\,\in \,\{L,H\}}\int _{\underline{\theta }_{t}}^{\overline{ \theta }_{t}}\left[ \,\frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}}-w_{nt}(\theta _{s})+\psi _{tv}(\cdot )-\psi _{tn}(\cdot )\,\right] \nonumber \\&\qquad \quad \qquad \qquad \qquad \qquad \quad \quad \; \times \,\, \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial K_{Dn}}\,dF_{t}(\theta _{s}). \end{aligned}$$

(11)

Because $\lambda =1$, $\frac{\partial Q_{t}^{i}(\cdot ,\theta _{s})}{\partial K_{Di}} \,=\,\,\mu _{t}\,\theta _{s}$, $\,\frac{\partial Q_{t}^{i}(\cdot ,\theta _{s})}{\partial w_{it}(\theta _{s})}\,=\,0$, and $\,\frac{\partial Q_{t}^{v}(\cdot ,\theta _{s})}{\partial K_{Di}}\,=\,-\,\mu _{t}\,\theta _{s}$ , the necessary condition with respect to $w_{it}(\cdot )$ implies that for all $\theta _{s}\,\in \,[\,\underline{\theta }_{t}, \overline{\theta }_{t}\,]$:

$$\begin{aligned} k_{i}= & {} -\frac{\partial T(\cdot )}{\partial K_{Di}}+\left[ \, \frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}}-w_{it}(\theta _{s})+\psi _{tv}(\cdot )-\psi _{ti}(\cdot )\,\right] \nonumber \\&\qquad \quad \qquad \qquad \times \,\mu _{t}\,\theta _{s}\,\,f_{t}(\theta _{s}). \end{aligned}$$

(12)

(10) and (12) imply:

$$\begin{aligned} \,w_{it}(\theta _{s})\,=\,\frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}}+\psi _{tv}(\cdot )-\psi _{ti}(\cdot )\, \, \text {for all }\,\theta _{s}\,\in \,\left[ \,\underline{\theta }_{t}\,\, ,\overline{\theta }_{t}\,\right] .\nonumber \\ \end{aligned}$$

(13)

(12) implies that $k_{i}\,$ $=$ $-\,\frac{\partial T(\cdot )}{\partial K_{Di}}$ when (13) holds.

Because $\lambda =1$ and $\frac{\partial Q_{t}^{v}}{\partial Q_{t}^{n}}\,=\,-\,1$, the necessary condition with respect to $w_{nt}(\cdot )$ implies that for each $\theta _{s}\,\in \,[\,\underline{\theta }_{t},\,\overline{\theta }_{t}\,]$:

$$\begin{aligned}&\left[ \,\left( \,w_{nt}(\theta _{s})-\frac{\partial C^{G}(\cdot )}{\partial Q_{t}^{v}}+\psi _{tn}(\cdot )-\psi _{tv}(\cdot )\,\right) \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial K_{Dn}}\,f_{t}(\theta _{s})+k_{n}+ \frac{\partial T(\cdot )}{\partial K_{Dn}}\,\right] \nonumber \\&\quad \times \frac{\partial K_{Dn}}{ \partial w_{nt}(\theta _{s})} +\,\left[ \,w_{nt}(\theta _{s})\,-\frac{\partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}+\psi _{tn}(\cdot )-\psi _{tv}(\cdot )\,\right] \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial w_{nt}(\theta _{s})}\,f_{t}(\theta _{s})=0.\nonumber \\ \end{aligned}$$

(14)

Since $\frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial K_{Dn}} \,>\,0$, $\frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial w_{n}(\theta _{s})}\,>\,0$, and $\frac{\partial K_{Dn}}{\partial w_{nt}(\theta _{s})}\,>\,0\,$ for all $\theta _{s}\,\in \,[\, \underline{\theta }_{t},\,\overline{\theta }_{t}\,]$, (14) implies that if $\,w_{nt}(\theta _{s})\,>\,\frac{\partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}+\psi _{tn}(\cdot )-\psi _{tv}(\cdot )$ , then $\,k_{n}+\frac{\partial T(\cdot )}{\partial K_{Dn}} \,<\,0$. (14) also implies that if $\,w_{nt}(\theta _{s})\,<\,\frac{\partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}+\psi _{tn}(\cdot )-\psi _{tv}(\cdot )$, then $\,k_{n}+\frac{\partial T(\cdot )}{ \partial K_{Dn}}\,>\,0$. Therefore, because $\,k_{n}+\frac{ \partial T(\cdot )}{\partial K_{Dn}}$ does not vary with $\theta _{s}$, it must be the case that:

$$\begin{aligned} \,w_{nt}(\theta _{s})=\frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}}+\psi _{tv}(\cdot )-\psi _{tn}(\cdot )\,\,\text { for all }\theta _{s}\,\in \,\left[ \,\underline{\theta }_{t},\,\overline{\theta } _{t}\,\right] . \qquad \end{aligned}$$

(15)

(11) and (15) imply $k_{n}$ $=\,\,-\,\frac{ \partial T(\cdot )}{\partial K_{Dn}}\,$. $\square$

Proof of Proposition 4

Let $\lambda$ denote the Lagrange multiplier associated with constraint (5). As in the proof of Proposition 2, it is readily shown that $\lambda =1$ at the solution to [RP]. Furthermore, because $\,\frac{\partial Q_{t}^{v}}{\partial X_{t}^{j}} \,=\,1$, $\frac{dQ_{t}^{v}}{dK_{G}}\,=\,0\,$, and $\frac{\partial V_{t}^{j}(X_{t}^{j}(r,\theta _{s}),\theta _{s})}{\partial X_{t}^{j}}=r$ for $j\in \{D,N\}$ and $t\in \{L,H\}$, the necessary conditions for $K_{G}$ and r reveal that conclusion (i) of the proposition and conclusion (i) in Proposition 3 hold.

Because $\lambda =1$, $\,\frac{\partial K_{Dn} }{\partial k_{n}}$ is not a function of $\,\theta _{s}$, and $k_{n}$ only affects $Q_{t}^{v}$ through $K_{Dn}$ and the corresponding impact on $Q_{t}^{n}$, the necessary condition with respect to $k_{n}$ reveals:

$$\begin{aligned}&-\bigg [\,\sum _{t\,\in \,\{\,\,L,H\,\,\}}\int _{ \underline{\theta }_{t}}^{\overline{\theta }_{t}}\left( \,w_{n}+\frac{ \partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}\,\frac{\partial Q_{t}^{v}(\cdot ,\theta _{s})}{\partial Q_{t}^{n}}\,\right) \frac{\partial Q_{t}^{n}}{\partial K_{Dn}}\,dF_{t}(\theta _{s})+k_{n}+\frac{\partial T(\cdot )}{\partial K_{Dn}}\,\,\bigg ] \\&\quad \times \frac{\partial K_{Dn}}{\partial k_{n}}=0. \end{aligned}$$

Therefore, since $\frac{\partial Q_{t}^{v}}{\partial Q_{t}^{n}}\,=\,-\,1$, conclusion (iii) in the proposition holds for $y=n$. The proof of conclusion (iii) for $y=i$ is analogous.

Because $\lambda =1$, $\frac{\partial K_{Dn}}{\partial w_{n}}\,$ is not a function of $\theta _{t}$ , and $\,\frac{\partial Q_{t}^{v}}{\partial Q_{t}^{n}}\,=\,-\,1$, the necessary condition with respect to $w_{n}$ provides:

$$\begin{aligned}&-\bigg [\,\sum _{t\,\in \,\{\,\,L,H\,\,\}}\,\int _{ \underline{\theta }_{t}}^{\overline{\theta }_{t}}\left( \,w_{n}-\frac{ \partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}\,\right) \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial K_{Dn}}\,dF_{t}(\theta _{s})+\,k_{n}+ \frac{\partial T(\cdot )}{\partial K_{Dn}}\,\bigg ]\,\frac{\partial K_{Dn}}{ \partial w_{n}} \nonumber \\&\quad -\sum _{t\,\in \,\{\,\,L,H\,\, \}}\,\int _{\underline{\theta }_{t}}^{\overline{\theta }_{t}}\left( \,w_{n}- \frac{\partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}\,\right) \frac{ \partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial w_{n}}\,dF_{t}(\theta _{s}) =0 \nonumber \\&\quad \Rightarrow \,\,\sum _{t\,\in \,\{\,,L,H \,\,\}}\,\int _{\underline{\theta }_{t}}^{\overline{\theta }_{t}} \left[ \,w_{n}-\frac{\partial C_{t}^{G}(Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}) }{\partial Q_{t}^{v}}\,\right] \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{ \partial w_{n}}\,dF_{t}(\theta _{s})=0.\nonumber \\ \end{aligned}$$

(16)

(16) reflects conclusion (iii) in the proposition for $y=n$. Since $\, \frac{\partial Q_{t}^{n}}{\partial w_{n}}$ is not a function of $\theta _{s}$, (16) implies that conclusion (ii) of the proposition holds. $\square$

Proof of Proposition 5

Let $\lambda \ge 0$ denote the Lagrange multiplier associated with constraint (5). As in the proof of Proposition 2, it is readily shown that $\lambda =1$ at the solution to [RP-$\theta k$]. Furthermore, because $\,\frac{ \partial Q_{t}^{v}}{\partial Q_{t}^{n}}\,=\,-\,1\,$, the necessary condition with respect to $w_{nt}(\cdot )$ can be written as:

$$\begin{aligned}&\left[ \,\left( \,w_{nt}(\theta _{s})-\frac{\partial C^{G}(\cdot )}{\partial Q^{v}}\,\right) \frac{\partial Q_{t}^{n}(\cdot ,\theta _{s})}{\partial K_{Dn} }\,f_{t}(\theta _{s})+\frac{\partial T(\cdot )}{\partial K_{Dn}}\,\right] \frac{\partial K_{Dn}}{\partial w_{nt}(\theta _{s})} \\&\quad +\,\left[ \,w_{nt}(\theta _{s})\,-\frac{\partial C_{t}^{G}(\cdot )}{\partial Q_{t}^{v}}\,\right] \frac{ \partial Q_{t}^{n}(\cdot )}{\partial w_{nt}(\theta _{s})}\,f_{t}(\theta _{s}) =0\,\text { for each }\theta _{s}\,\in \,\left[ \, \underline{\theta }_{t},\,\overline{\theta }_{t}\,\right] . \end{aligned}$$

Therefore, conclusion (ii) of the proposition for $y=n$ holds for each $\theta _{s}\,\in \,[\,\underline{\theta }_{t},\,\overline{ \theta }_{t}\,]$. The corresponding proof for $y=i$ is analogous.

$\frac{\partial V_{t}^{j}(X_{t}^{j}(r,\theta _{s}),\theta _{s})}{\partial X_{t}^{j}}=r_{jt}(\theta _{s})$ for each $j\in \{D,N\}$ and $t\in \{L,H\}$. Also $\,\frac{\partial Q_{t}^{v} }{\partial X_{t}^{j}}=1$. Therefore, because $\lambda \,=\,1$ , the necessary condition with respect to $r_{jt}(\cdot )$ implies that for each $\theta _{s}\,\in \,[\,\underline{\theta } _{t},\,\overline{\theta }_{t}\,]\,$ and for $t,\,t^{\prime }\in \{L,H\}$ ($t^{\prime }\ne t$):

$$\begin{aligned}&\left[ \,r_{jt}(\theta _{s})-\frac{\partial C_{t}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s}),K_{G}\right) }{\partial Q_{t}^{v}(\cdot ,\theta _{s})}\,\right] \frac{ \partial X_{t}^{j}(\cdot ,\theta _{s})}{\partial r_{jt}(\theta _{s})} \,\,f_{t}(\theta _{s}) \nonumber \\&\quad +\,\,\int _{\underline{ \theta }_{t^{\prime }}}^{\overline{\theta }_{t^{\prime }}}\left[ \,r_{jt^{\prime }}(\theta _{s^{\prime }})\,-\frac{\partial C_{t^{\prime }}^{G}\left( Q_{t}^{v}(\cdot ,\theta _{s^{\prime }}),K_{G}\right) }{\partial Q_{t^{\prime }}^{v}(\cdot ,\theta _{s^{\prime }})}\,\right] \frac{\partial X_{t^{\prime }}^{j}(\cdot ,\theta _{s^{\prime }})}{\partial r_{jt}(\theta _{s})} \,dF_{t^{\prime }}(\theta _{s^{\prime }})=0.\nonumber \\ \end{aligned}$$

(17)

(17) is satisfied if conclusion (i) in the proposition holds. $\square$

1.2 B. Details of the numerical analysis in Sect. 7

The baseline setting is designed to reflect settings where S primarily employs non-coal resources to serve a relatively large number of customers. Consumer $j\in \left\{ D,N\right\}$’s demand for electricity in period $t\in \{L,H\}$ given state $\theta _{t}$ and unit price r is assumed to be:

$$\begin{aligned} X_{t}^{j}(r,\theta _{t})=m_{jt}\left[ \,1+\theta _{t}^{\beta _{jt}}\,\right] -\alpha _{jt}\,r. \end{aligned}$$

(18)

Period H is the period of peak demand between 11:00 am and 6:00 pm. Period L consists of the other (off-peak) hours of the day.

The $\beta _{jt}$ parameters in (18) are set to ensure that the elasticity of demand with respect to solar intensity ($\theta _{t}$) is 0.1 in each period at the relevant expected values of demand and $\theta$. The $\alpha _{jt}$ parameters in (18) are set to ensure the price elasticity of demand is $-0.25$ at $\overline{r} =143.8165$ ($/MWh), the average unit retail price of electricity in California in 2014 (California PUC 2015), and at $\overline{X}_{t}^{j}$, the average baseline level of demand for consumer j in period t. This average baseline demand is the product of: (i) $\overline{X}_{t}$, the average hourly total electricity consumption in period t in California in 2014 ($\overline{X}_{H}\,=31,328$ and $\overline{X}_{L}=16,149$);⁵⁹ and (ii) $\eta _{j}$, the fraction of total consumption accounted for by consumers of type j. We assume $\eta _{D}=0.1$ (and $\eta _{N}=0.9$) to reflect the potential deployment of PV panels in the U.S. in the near future.⁶⁰

Recall that the distribution of the solar intensity variable ($\theta$) reflects variation in the production of electricity from installed DG capacity. To specify this distribution, we first plot the ratio of the MW’s of electricity produced by photo-voltaic (PV) panels to the year-end installed generating capacity ($\overline{K}_{D}=3254$ MW) of PV panels in California for each of the 8760 h in 2014.⁶¹ We then employ maximum likelihood estimation to fit a density function to the observations that are strictly positive in each of the relevant periods. Standard tests reveal that the beta density with parameters (2.834641, 1.712483) fits the data well in period H and the beta density with parameters (0.8267931, 1.434153) fits the data well in period L. These functions are used as the densities for $\theta$ in the two periods.

S’s capacity costs are taken to be $C^{K}(K_{G})=a_{K}\,K_{G}+b_{K}\left( K_{G}\right) ^{2}$. Estimates of the cost of the generation capacity required to produce a MWh of electricity range from $\$16.1$/MWh for a conventional combined cycle natural gas unit to $\$81.9$/MWh for a nuclear facility (EIA 2015a). We set $a_{K}$ at the lower bound of this range (16.1). We also set $b_{K}=0.00045$ to ensure that the marginal cost of capacity required to generate a MWh of electricity is $\$81.9$ at the observed level of centralized non-renewable generation capacity in California in 2014 ($\,\overline{K}_{G}=$ 72,926 MW) (California Energy Commission 2015).

S’s TDM costs are assumed to be $T(K_{G},K_{Di},K_{Dn})= a_{T}^{G}\,K_{G}+a_{T}^{Di}\,K_{Di}+a_{T}^{Dn}\,K_{Dn}$. Estimates of these costs vary widely. EIA (2015a) estimates utility transmission capacity costs associated with generating a MWh of electricity are between $\$1.2$ and $\$3.5$ for centralized, non-renewable generation, and between $\$4.1$ and $\$6.0$ for PV generation. Reflecting the midpoints of these cost estimates, we initially assume $a_{T}^{G} =a_{T}^{Dn}=2.35$ and $a_{T}^{Di}=5.05$.

We assume that S’s cost of generating $Q^{v}$ units of electricity when it has $K_{G}$ units of capacity is $C^{G}(Q^{v},K_{G})=\left[ \,a_{v}+\frac{c_{v}}{K_{G}}\,\right] Q^{v}+b_{v}\left( Q^{v}\right) ^{2}$, where $a_{v}$, $b_{v}$, and $c_{v}$ are positive constants. We set $b_{v}=0.0015$ and $a_{v}+\frac{c_{v}}{K_{G}}=28.53$, reflecting Bushnell (2007)’s estimates.⁶² $c_{v}$ is chosen to equate the observed marginal benefit ($\frac{c_{v}\,\overline{Q}^{v}}{\left( \overline{K} _{G}\right) ^{2}}$) and marginal cost ($a_{K}+2\,b_{K}\,\overline{K} _{G}+a_{T}^{G}\,$) of S’s capacity.⁶³

The cost of installing $K_{Di}$ units of intermittent DG capacity and $K_{Dn}$ units of non-intermittent DG capacity is assumed to be $\,C_{D}^{K}(K_{Di},K_{Dn})=a_{Di} \,K_{Di}+b_{Di}\left( K_{Di}\right) ^{2}+a_{Dn}\,K_{Dn}+b_{Dn}\left( K_{Dn}\right) ^{2}$. Estimates of the unsubsidized cost of residential PV capacity vary between $\$100$ and $\$400$/MWh (Branker et al. 2011; EIA 2015a). When the 30 percent federal income tax credit (ITC) is applied, these estimates decline to between $\$70$ and $\$280$/MWh. State subsidies further reduce these estimates to between $\$45$ and $\$255$/MWh (NCCETC 2015a, b, c). We initially set $a_{Di}=150$, the midpoint of this lattermost range. We also set $\,b_{D}=0.0038$ to ensure ensure that the marginal cost of DG capacity when $K_{Di}=\overline{K}_{D}=3254$ (i.e., $a_{D}+2\,b_{D}\, \overline{K}_{D}$) is 175, the midpoint of the range of estimated costs after applying the ITC.

Estimates of the cost of DG capacity using natural gas reciprocating engine and gas turbines range from $\$34.09$ to $\$87.44$/MWh (PacifiCorp 2013, Table 6.8). We set $a_{Dn}$ equal to the lower bound of this range (34.09) and select $b_{Dn}$ to ensure the marginal cost of capacity at the level of combined heat and power (CHP) DG in California, $\overline{K}_{Dn}=8518$ MW (ICF 2012), reflects the upper bound of this range, so $a_{Dn}+2b_{Dn}\overline{K}_{Dn}= 87.44$.

To capture the social losses from environmental externalities ($\psi (\cdot )$), we introduce the parameters $e_{c}=37.231$, $e_{g}=21.029$, and $e_{o}=0$, where $\,e_{j}$ is the estimated unit loss from environmental externalities for production technology $j\in \{c,g,o\}$, where c denotes coal, g denotes natural gas, and o denotes other. The $e_{j}$ estimate is the product of $\$38$, the estimated social cost of a metric ton of CO$_{2}$ emissions (EPA 2013), and the metric tons of $\hbox {CO}_{2}$ emissions that arise when technology j is employed to produce a MWh of electricity.⁶⁴

We assume $\,\psi _{t}(Q_{t}^{v},\,Q_{t}^{i},\,Q_{t}^{n})\,=e_{v}\,Q_{t}^{v}+e_{o} \,Q_{t}^{i}+e_{g}\,Q_{t}^{n}$ where $e_{v}=\phi _{c}\,e_{c}+\phi _{g}\,e_{g}+\phi _{o}\,e_{o}$.⁶⁵ $\,\phi _{c}$, $\phi _{g}$, and $\phi _{o}$ denote the fraction of S’s electricity production that is generated by coal, natural gas, and other units, respectively.⁶⁶ In the baseline setting, we assume $\phi _{c}=0.064$ and $\phi _{g}=0.445$, reflecting the fraction of electricity generated by California utilities in 2014 using coal and natural gas generating units, respectively (California Energy Commission 2015). Consequently, $e_{v}=11.746$.