Firming renewable power with demand response: an end-to-end aggregator business model

Abstract

Environmental concerns have spurred greater reliance on variable renewable energy resources (VERs) in electric generation. Under current incentive schemes, the uncertainty and intermittency of these resources impose costs on the grid, which are typically socialized across the whole system, rather than born by their creators. We consider an institutional framework in which VERs face market imbalance prices, giving them an incentive to produce higher-value energy subject to less adverse uncertainty. In this setting, we consider an “aggregator” that owns the production rights to a VER’s output, and also signs contracts with a population of demand response (DR) participants for the right to curtail them in real time, according to a contractually specified probability distribution. The aggregator bids a day ahead offer into the wholesale market, and is able to offset imbalances between the cleared day-ahead bid and the realized VER production by curtailing DR participants’ consumption according to the signed contracts. We consider the optimization of the aggregator’s end-to-end problem: designing the menu of DR service contracts using contract theory, bidding into the wholesale market, and dispatching DR consistently with the contractual agreements. We do this in a setting in which wholesale market prices, VER output, and participant demand are all stochastic, and possibly correlated.

Appendix: Proofs

Proof of Lemma 2.8

Assumption 1 ensures the conditions sufficient for this claim. Note that $L(\cdot )$ is absolutely continuous, and differentiable except at two points. Further, recall that the support of $F(\cdot \vert \tau , \omega )$ is constant. Therefore we can apply integration by parts, and the antiderivative term drops out:

$$\begin{aligned} U(\tilde{\tau } \vert \tau ) = \int _{\varOmega _\text {RT}} \int _\Theta - L'(\theta ) F(\theta | \tau , \omega ) \mathrm{d}\theta \, k(\tilde{\tau },\omega ) \mathrm{d}P_\text {RT}(\omega ) \, . \end{aligned}$$

The partial derivative in question is therefore

$$\begin{aligned} \frac{\partial }{\partial \tau } U(\tilde{\tau } \vert \tau )&= \int \int - L'(\theta ) \frac{\partial F(\theta | \tau , \omega )}{\partial \tau }k(\tilde{\tau },\omega ) \mathrm{d}P_\text {RT}(\omega ) \le 0 \, . \end{aligned}$$

Each term in the integrand is uniformly bounded, which guarantees that this partial derivative is uniformly bounded, so that the function $U(\tilde{\tau } \vert \cdot )$ is Lipschitz continuous, and thus absolutely continuous.

Proof of Proposition 2.10

By rearrangement, the payment is the equilibrium utility minus the net option value:

$$\begin{aligned} t(\tau )&= \ u(\tau ) - U(\tau ) \, . \end{aligned}$$

(31)

First we simplify the expression for $u(\tau )$ from line (8). Under merit order curtailment, $k(\tau ,\omega ) = \mathbbm {1}_{\{ \tau \le \hat{\tau }(\omega )\}}$. Exchanging the order of integration so that we integrate first with respect to $\tau$, we get that:

$$\begin{aligned} u(\tau ) = u({\overline{\tau }}) - \int _{\varOmega _\text {RT}} \int _\Theta L(\theta ) \bigl (f(\theta \vert \hat{\tau }(\omega ),\omega ) - f(\theta \vert \tau , \omega ) \bigr ) k(\tau ,\omega ) \mathrm{d}P_\text {RT}(\omega ). \end{aligned}$$

(32)

Subtracting from this the net option value $U(\tau )$, we arrive at the desired result.

Proof of Proposition 2.12

Consider the difference in utility between the case where an increment of ex ante type $\tau$ reports truthfully, versus mis-reporting as $\tilde{\tau }$.

$$\begin{aligned}&u(\tau ) - \bigl (U(\tilde{\tau } \vert \tau )+t(\tilde{\tau })\bigr ) \\&\quad = \, \int _\varOmega \int _\Theta L(\theta ) f(\theta \vert \tau , \omega ) \mathrm{d}\theta \bigl (k(\tau ,\omega ) - k(\tilde{\tau },\omega ) \bigr ) \mathrm{d}P(\omega ) + t(\tau ) - t(\tilde{\tau }) \\&\quad =\, \int \int L(\theta ) \bigl ( f(\theta \vert \tau , \omega )-f(\theta \vert \hat{\tau }(\omega ), \omega ) \bigr ) \mathrm{d}\theta \bigl (k(\tau ,\omega ) - k(\tilde{\tau },\omega ) \bigr ) \mathrm{d}P(\omega )\\&\quad =\, \int \int \underbrace{- L'(\theta )}_{\ge 0}\bigl ( F(\theta \vert \tau , \omega )-F(\theta \vert \hat{\tau }(\omega ), \omega ) \bigr ) \mathrm{d}\theta \bigl (k(\tau ,\omega ) - k(\tilde{\tau },\omega ) \bigr ) \mathrm{d}P(\omega ) \, . \end{aligned}$$

Suppose $\tau > \tilde{\tau }$. Merit order curtailment implies that $k(\tau ,\omega ) - k(\tilde{\tau },\omega ) = - \mathbbm {1}_{\{\tilde{\tau } \le \hat{\tau }(\omega ) < \tau \}} \le 0$. On this set, FOSD implies that $F(\theta \vert \tau , \omega ) \le F(\theta \vert \hat{\tau }(\omega ), \omega )$. This implies that the integral’s value is nonnegative. Similarly, if $\tau < \tilde{\tau }$, then $k(\tau ,\omega ) - k(\tilde{\tau },\omega ) = \mathbbm {1}_{\{\tau \le \hat{\tau }(\omega ) < \tilde{\tau } \}} \ge 0.$ On this set, $F(\theta \vert \tau ,\omega ) \ge F(\theta \vert \hat{\tau }(\omega ), \omega )$. Again, the integral’s value is nonnegative. $\square$

Proof of Proposition 2.14

The substance of the Proposition is that (ignoring the penalty receipts term)

$$\begin{aligned}&\int _{{\underline{\tau }}}^{{\overline{\tau }}} t(\tau ) g(\tau ) \mathrm{d}\tau \nonumber \\&\quad = - \int _{\varOmega _\text {RT}} \int _{{\underline{\tau }}}^{\hat{\tau }(\omega _\text {RT})}\Biggl ( \frac{G(\tau )}{g(\tau )}\frac{\partial }{\partial \tau } z(\tau , \omega _\text {RT}) + z(\tau ,\omega _\text {RT}) \Biggr )g(\tau ) \mathrm{d}\tau \mathrm{d}P(\omega _\text {RT})\, . \end{aligned}$$

(33)

Integrating Eq. (9) under merit order curtailment, we get that

$$\begin{aligned} \int _{{\underline{\tau }}}^{{\overline{\tau }}} t(\tau ) g(\tau ) \mathrm{d}\tau&= \int _{\varOmega _\text {RT}} \int _{{\underline{\tau }}}^{\hat{\tau }(\omega _\text {RT})} z(\hat{\tau }(\omega _\text {RT}))g(\tau )\mathrm{d}\tau \mathrm{d}P(\omega _\text {RT}) \end{aligned}$$

(34)

$$\begin{aligned}&= \int _{\varOmega _\text {RT}} z(\hat{\tau }(\omega _\text {RT}))G(\hat{\tau }(\omega _\text {RT})) \mathrm{d}\tau \mathrm{d}P(\omega _\text {RT}) \,. \end{aligned}$$

(35)

To obtain the desired formula, we first apply the fundamental rule of calculus to express the integrand as $\int _{{\underline{\tau }}}^{\hat{\tau }(\omega _\text {RT})} z(\tau )G(\tau ) \mathrm{d}\tau$. in (35) differentiate the integrand in (35) using the product rule, factor out a “$g(\tau )$,” and re-integrate.

Proof of Proposition 3.8

(FONC for $q^*$ ) In order to differentiate this objective with respect to q, we decompose RT outcomes into three sets: overproduction = $\{ DR + s > q\}$, shortfall = $\{ DR + s < q \}$, and no imbalance = $\{DR + s = q\}$.²⁹ From expression (15), we see that

$$\begin{aligned} \frac{\partial }{\partial q} \hat{\tau }^*(\omega _\text {RT} \vert q)&= \frac{1}{ DR '( DR ^{-1}(q-s))}\mathbbm {1}_{\{DR(\hat{\tau }^*) = q-s\}} \\&= \frac{1}{ DR '(\hat{\tau }^*)}\mathbbm {1}_{\{DR(\hat{\tau }^*) = q-s\}} = \frac{1}{y(\hat{\tau }^*)g(\hat{\tau }^*)}\mathbbm {1}_{\{DR(\hat{\tau }^*) = q-s\}}. \end{aligned}$$

Here we ignore certain “edge” events, which we assume have probability zero: abusing notation,

$$\begin{aligned} s_{\text {inf}}&= \inf \{s : DR (\hat{\tau }^*(s)) + s = q\} \\ s_{\text {sup}}&= \sup \{s : DR (\hat{\tau }^*(s)) + s = q\}. \end{aligned}$$

Also, note that the contributions to the expected derivative from the overproduction and shortfall terms are both zero when $DR + s = q$, because by assumption, the shortfall quantity is constantly zero on this set. When overproduction and underproduction are strict, then $\frac{\partial }{\partial q} \hat{\tau }^*(\omega _\text {RT} \vert q) =0$.

This gives us that

$$\begin{aligned} \frac{\partial }{\partial q} J(q \vert \omega _\text {DA}) =\,&p - \mathbb {E}[a \vert DR (\hat{\tau }^*) + s> q] \Pr \{ DR (\hat{\tau }^*) + s > q\} \\&-\, \mathbb {E}[b \vert DR (\hat{\tau }^*) + s< q] \Pr \{ DR (\hat{\tau }^*) + s < q\} \\&- \, \mathbb {E}[c(\hat{\tau }^*) \vert DR (\hat{\tau }^*) + s = q] \Pr \{ DR ^* + s = q\}. \end{aligned}$$

The first order condition follows. $\square$