The prosumers and the grid

Abstract

Prosumers are households that are both producers and consumers of electricity. A prosumer has a grid-connected decentralized production unit and makes two types of exchanges with the grid: energy imports when the local production is insufficient to match the local consumption and energy exports when local production exceeds it. There exists two systems to measure the exchanges: a net metering system that uses a single meter to measure the balance between exports and imports and a net purchasing system that uses two meters to measure separately power exports and imports. Both systems are currently used for residential consumption. We build a model to compare the two metering systems. Under net metering, the price of exports paid to prosumers is implicitly set at the price of the electricity that they import. We show that net metering leads to (1) too many prosumers, (2) a decrease in the bills of prosumers, compensated via a higher bill for traditional consumers, and (3) a lack of incentives to synchronize local production and consumption.

Appendix

Let $L=C\left( F\left( \hat{z}_{\varphi }\right) \right) +F\left( \hat{z}_{\varphi }\right) \frac{\left( \hat{\varphi }-\bar{\varphi }\right) ^{2}}{2}+\lambda \pi ^{D}$, the Lagrangian function of the problem with $\lambda \ge 0$ substituting (10) and (9). The Khun and Tucker FOC write, for $i=m,x:$

$$\begin{aligned}&\frac{\partial L}{\partial r_{i}}=0\Rightarrow f\left( \hat{z}_{\varphi }\right) \left[ C^{\prime }\left( F\left( \hat{z}_{\varphi }\right) \right) +\frac{\left( \hat{\varphi }-\bar{\varphi }\right) ^{2}}{2 }\right] \frac{d\hat{z}_{\varphi }}{dr_{i}}+\left[ \frac{C\left( F\left( \hat{z} _{\varphi }\right) \right) }{\partial \hat{\varphi }}+F\left( \hat{z}_{\varphi }\right) \left( \hat{\varphi }- \bar{\varphi }\right) \right] k \\&\quad -\lambda \left\{ \frac{1}{k}\frac{\partial \pi ^{D}}{\partial r_{i}}+f\left( \hat{z}_{\varphi }\right) k\left[ -\left( r_{m}-\theta \right) \hat{\varphi } +r_{x}(1-\hat{\varphi })-\frac{K_{l}}{k}\right] \frac{d\hat{z}_{\varphi }}{ dr_{i}}+\left( \theta -r_{m}-r_{x}\right) F\left( \hat{z}_{\varphi }\right) k^{2}\right\} =0,\\&\quad \lambda ~\pi ^{D}=0 \end{aligned}$$

As $\frac{d\hat{z}_{\varphi }}{dr_{m}}=\hat{\varphi }$ ; $\frac{d\hat{z} _{\varphi }}{dr_{x}}=-\left( 1-\hat{\varphi }\right)$, $\frac{\partial \pi ^{D}}{\partial r_{m}}=q-F\left( \hat{z}_{\varphi }\right) \hat{\varphi }k$ and $\frac{\partial \pi ^{D}}{\partial r_{m}}=(1-\hat{\varphi })F\left( \hat{z }_{\varphi }\right) k$, after substitutions and some manipulations this leads to

$$\begin{aligned} \frac{\partial L}{\partial r_{m}}= & {} 0\Rightarrow \left( 1+\lambda \right) \left\{ \hat{\varphi }\left( \hat{z}_{\varphi }-z_{\varphi }^{*}\right) + \frac{F\left( \hat{z}_{\varphi }\right) }{f\left( \hat{z}_{\varphi }\right) k }\left( \hat{\varphi }-\varphi ^{*}\right) \right\} -\lambda \frac{ q-F\left( \hat{z}_{\varphi }\right) \hat{\varphi }k}{f\left( \hat{z}_{\varphi }\right) k}=0, \\ \frac{\partial L}{\partial r_{x}}= & {} 0\Rightarrow \left( 1+\lambda \right) \left\{ -\left( 1-\hat{\varphi }\right) \left( \hat{z}_{\varphi }-z_{\varphi }^{*}\right) +\frac{F\left( \hat{z}_{\varphi }\right) }{f\left( \hat{z} _{\varphi }\right) k}\left( \hat{\varphi }-\varphi ^{*}\right) \right\} -\lambda \frac{F\left( \hat{z}_{\varphi }\right) }{f\left( \hat{z}_{\varphi }\right) }(1-\hat{\varphi })=0, \\ \lambda ~\pi ^{D}= & {} 0. \end{aligned}$$

We see that $\lambda ^{*}=0$ implies $\hat{\varphi }=\varphi ^{*}$ and $\hat{z}_{\varphi }=z_{\varphi }^{*}$ with a grid tariff structure $\left( \hat{r}_{m},\hat{r}_{x}\right) =(\theta -\frac{K_{l}}{k},\frac{K_{l}}{k})$ but with such tariffs $\pi ^{D}=-\frac{q}{k}K_{l}<0$: a contradiction. So $\lambda ^{*}>0$ and the joint first best cannot be implemented and the break-even constraint is necessarily binding. Then solving the FOC with respect to $\hat{z}_{\varphi }$ and $\hat{\varphi }$ leads to

$$\begin{aligned} \hat{\varphi }-\varphi ^{*}= & {} \left( 1-\hat{\varphi }\right) \frac{ \lambda ^{*}}{1+\lambda ^{*}}\frac{q}{F\left( \hat{z}_{\varphi }\right) }\Rightarrow \hat{\varphi }>\varphi ^{*}, \\ \hat{z}_{\varphi }-z_{\varphi }^{*}= & {} \frac{\lambda ^{*}}{1+\lambda ^{*}}\left\{ \frac{q-F\left( \hat{z}_{\varphi }\right) k}{f\left( \hat{z} _{\varphi }\right) k}\right\} \Rightarrow \hat{z}_{\varphi }>z_{\varphi }^{*}, \\ \lambda ^{*}= & {} \frac{qf\left( \hat{z}_{\varphi }\right) }{q-F\left( \hat{z}_{\varphi }\right) k}\left( r_{m}-\theta \right) >0. \end{aligned}$$

which in turns implies in order to verify the break-even constraint that:

$$\begin{aligned} \hat{r}_{m}>\theta \quad \text {and}\quad \hat{r}_{x}<\frac{K_{l}}{(1-\varphi )k}. \end{aligned}$$

We verify that $\lambda ^{*}=\frac{qf\left( \hat{z}_{\varphi }\right) }{ q-F\left( \hat{z}_{\varphi }\right) k}\left( \hat{r}_{m}-\theta \right) >0$.