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

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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.

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  1. Electric Vehicles and Plug-in Hybrid Electric Vehicles. In the rest of the paper, we shall refer only to PHEVs.

  2. Most of these activities take place at distribution level see [1, 2] .

  3. ETEM-SG for Energy Technology Environment Model with Smart-Grids is a technology-rich capacity expansion model used to assess long-term energy policies.

  4. Energy-Technology-Environment-Model with Smart-Grid

  5. It is currently used to assess the possible transition to 100% renewable energy in non-interconnected regions (typically islands and remote territories) in France.

  6. Usually the swing bus is numbered 1 for the load flow studies. This bus sets the angular reference for all the other buses. Since it is the angle difference between two voltage sources that dictates the real and reactive power flow between them, the particular angle of the swing bus is not important.

  7. In electrical engineering, susceptance (\(\mathbf {B}\)) is the imaginary part of admittance. The inverse of admittance is impedance and the real part of admittance is conductance. In SI units, susceptance is measured in siemens.

  8. This power flow model corresponds to the approximate DC flow where power flow obeys Kirchoff’s voltage law, reactive power is ignored and phase angle differences are small and per unit voltages are set to 1.

  9. Note that, to obtain time continuous versions of the nodal price and target charging demand at a generic node, we consider spline approximations along the time slices.


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This research is supported by the Canadian IVADO programme (VORTEX Project). First and third authors gratefully acknowledge the support provided by Qatar National Research Fund under Grant Agreement No. NPRP10-0212-170447. The first author also acknowledges support provided by FONDECYT 1190325 and by ANILLO ACT192094, Chile.

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This article is part of the topical collection “Dynamic Games for Smart Energy Systems”.



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

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

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

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

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.


(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

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

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.

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Babonneau, F., Foguen, R.T., Haurie, A. et al. Coupling a Power Dispatch Model with a Wardrop or Mean-Field-Game Equilibrium Model. Dyn Games Appl 11, 217–241 (2021).

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