Coupling a Power Dispatch Model with a Wardrop or Mean-Field-Game Equilibrium Model

Abstract

In this paper, we propose an approach for coupling a power network dispatch model, which is part of a long-term multi-energy model, with Wardrop or Mean-Field-Game (MFG) equilibrium models that represent the demand response of a large population of small “prosumers” connected at the various nodes of the electricity network. In a deterministic setting, the problem is akin to an optimization problem with equilibrium constraints taking the form of variational inequalities or nonlinear complementarity conditions. In a stochastic setting, the problem is formulated as a robust optimization with uncertainty sets informed by the probability distributions resulting from an MFG equilibrium solution. Preliminary numerical experimentations, using heuristics mimicking standard price adjustment techniques, are presented for both the deterministic and stochastic cases.

Appendix

1.1 Proof of Propositions  1 and 2

Propositions 1 and 2 are special cases of a more general theorem in RO, by taking into account that $\Delta _a(\tau ) \le \Delta _\ell (\tau )$ and $\Delta _u(\tau ) \le \Delta _a(\tau )$, respectively. For the interested readers, we state now the general theorem and prove Proposition 1 as a corollary. In order to show the derivation, we shall use the concise notation

$$\begin{aligned} z_0+ \sum _\tau z_\tau \xi _\tau \le 0 \end{aligned}$$

(72)

to represent the inequality (53). The coefficients $\hat{z}$ are easily identified as

$$\begin{aligned} z_0= & {} \sum _{\tau =0}^{T-1} \Delta _\ell (\tau ) \\ z_\tau= & {} \Delta _a(\tau ) - \Delta _\ell (\tau ). \end{aligned}$$

The proof is similar for Proposition 2 considering

$$\begin{aligned} z_0= & {} \sum _{\tau =0}^{T-1} \Delta _u(\tau ) \\ z_\tau= & {} -(\Delta _u(\tau ) - \Delta _a(\tau )). \end{aligned}$$

The main result can be formulated as

Theorem 2

Let $\eta _\tau$ be T independent random variables with common support $[-1,1]$ and known means $E(\eta _\tau ) = \nu _\tau$. The probabilistic inequality $\hat{z}_0+ \sum _\tau \hat{z}_\tau \eta _\tau \le 0$ is satisfied with probability at least $(1-\epsilon )$ if there exists a vector $w\in \mathbb R^\tau$ such that the deterministic inequality

$$\begin{aligned} \hat{z}_0 + \sum _\tau \hat{z}_\tau \nu _\tau + \sum _\tau ( |w_k | - w_{\tau }\nu _{\tau }) + \sqrt{2T \ln \frac{1}{\epsilon }}\max _\tau |\hat{z}_\tau - w_\tau | \le 0 \end{aligned}$$

(73)

is satisfied.

Note that the range of the random factors is now $[-1,1]$. The above theorem is the formal statement of the theory for inequalities with random factors having known means $\nu _{\tau }$ and common range $[-1,1]$ as discussed in [9, example 2.4.9, p. 55].

We show now how to prove Proposition 1 as a corollary of Theorem 2.

Proof

(Proposition 1)

Let us start with (72) and define the variables $\eta _{\tau } = 2\xi _{\tau } - 1$. In view of Assumption  4 the range of $\eta _{\tau }$ is $[-1,1]$ and $E(\eta _{\tau }) = \nu _{\tau } = 2\mu _{\tau } - 1 \le 0$. Inequality (72) becomes

$$\begin{aligned} z_{0} +\frac{1}{2} \sum _{\tau }z_{\tau } + \frac{1}{2} \sum _{\tau }z_{\tau }\eta _{\tau } \le 0. \end{aligned}$$

Let $\hat{z}_{0} = z_{0} +\frac{1}{2}\sum _{\tau }z_{\tau }$ and $\hat{z}_{\tau } = z_{\tau }/2$. The hypotheses of Theorem 2 for the inequality $\hat{z}_{0} + \sum _{k}\hat{z}_{k}\eta _{k} \le 0$ are verified. Hence,

$$\begin{aligned} \hat{z}_0 + \sum _\tau \hat{z}_\tau \nu _\tau + \sum _\tau ( |w_\tau | -w_{\tau }\nu _{\tau }) + \sqrt{2T \ln \frac{1}{\epsilon }}\max _\tau |\hat{z}_\tau - w_\tau | \le 0 \end{aligned}$$

is a sufficient condition to ensure the constraint satisfaction with probability at least $(1 - \epsilon )$. If we substitute $\nu _{\tau }$, $\hat{z}_{0}$ and $\hat{z}_{\tau }$ by their values, we obtain the condition

$$\begin{aligned} z_{0} + \sum _{\tau }\mu _{\tau } z_{\tau } + \sum _\tau ( |w_\tau | +w_{\tau } -2\mu _{\tau } w_{\tau }) + \sqrt{\frac{T}{2} \ln \frac{1}{\epsilon }}\max _\tau | z_\tau - 2 w_\tau | \le 0. \end{aligned}$$

(74)

Recall that $z_{\tau } = \Delta _a(\tau ) - \Delta _\ell (\tau ) \ge 0$. We claim that only positive values $w\ge 0$ need to be considered. Indeed, the theorem does not specify the value it should take. In particular, we can choose w so as to have to minimize the right-most component $\sum _\tau ( |w_\tau | +w_{\tau } -2\mu _{\tau } w_{\tau }) + \sqrt{\frac{T}{2} \ln \frac{1}{\epsilon }}\max _k | z_\tau - 2 w_\tau |$. If for some $\tau '$, $w_{\tau '} <0$, then $|w_{\tau '}| + w_{\tau '} = 0$ and the contribution of term $\tau '$ in the summation is $-2\mu _{\tau '}w_{\tau '} + \max \{ z_{\tau '} - 2 w_{\tau '}, \max _{\tau \ne \tau '}|z_{\tau } - w_{\tau }|\}$. Clearly this term can be made smaller by taking $w_{\tau '}= 0$. Hence, we can assume $w_{\tau '} \ge 0$.

With $w\ge 0$ inequality (74) becomes

$$\begin{aligned} z_{0} + \sum _{\tau }\mu _{\tau } z_{\tau } + \sum _\tau (1 - \mu _{\tau })2 w_{\tau } + \sqrt{\frac{T}{2} \ln \frac{1}{\epsilon }}\max _\tau | z_\tau - 2 w_\tau | \le 0. \end{aligned}$$

Writing u for 2w in the above inequality, we obtain the condition

$$\begin{aligned} z_{0} + \sum _{\tau }\mu _{\tau } z_{\tau } + \sum _\tau (1 - \mu _{\tau }) u_{\tau } + \sqrt{\frac{T}{2} \ln \frac{1}{\epsilon }}\max _\tau | z_\tau - u_\tau | \le 0. \end{aligned}$$

Using the same argument as before, we easily prove that we can restrict our choice of u to $u \le z$. Hence, $| z_\tau - u_\tau | = z_\tau - u_\tau \ge 0$ and using the additional scalar variable $v\ge 0$ we can transform our inequality into

$$\begin{aligned} z_{0} + \sum _{\tau }\mu _{\tau } z_{\tau } + \sum _\tau (1 - \mu _{\tau }) u_{\tau } + \sqrt{\frac{T}{2} \ln \frac{1}{\epsilon }}\cdot v\le & {} 0 \\ z_\tau - u_\tau\le & {} v,\; \tau =0, \ldots , T-1 \\ u \ge 0,v\ge 0. \end{aligned}$$

This concludes the proof of the proposition.