Reliable Frequency Regulation through Vehicle-to-Grid: From EU Legislation to Robust Optimization

Daniel Kuhn, Dirk Lauinger, François Richard Vuille
2020
Article

Résumé

Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to the electricity grid. We formulate a robust optimization problem in continuous time that maximizes the expected profit from selling primary frequency regulation to the grid and guarantees that vehicle owners can meet their market commitments for all frequency deviation trajectories in an uncertainty set that encodes applicable EU legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem non-convex. By exploiting a total unimodularity property of the proposed uncertainty sets and an exact linear decision rule reformulation, we prove that this non-convex robust optimization problem in continuous time is equivalent to a tractable linear program. Through extensive numerical experiments based on real-world data we investigate how the value of vehicle-to-grid depends on the penalties for non-delivery of promised regulation power, the delivery guarantees, and the vehicle's battery, charger, and yearly mileage.

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https://infoscience.epfl.ch/record/277540?ln=fr

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Future low-carbon societies will need to store vast amounts of electricity to stabilize electricity grids and to power electric vehicles. Vehicle-to-grid allows vehicle owners and grid operators to share the costs of electricity storage by making the batteries of electric vehicles available to the grid. In practice, vehicle owners decide when to reserve their cars for driving and when to make them available for grid services. Vehicle aggregators then decide how to commit the vehicles to grid services. For vehicle-to-grid to succeed, both vehicle owners and grid operators must be able to trust aggregators, i.e., vehicles should be available for driving and for grid services when the aggregators promise they will be. In this thesis, we solve a decision-making problem that ensures reliable commitments by vehicle aggregators for a particular grid service known as primary frequency regulation, considered one of the most profitable applications of vehicle-to-grid. Mathematically, we first formulate a robust optimization problem with functional uncertainties that maximizes the expected profit from selling primary frequency regulation to grid operators and guarantees that vehicle owners can meet their market commitments for all frequency deviation trajectories in an uncertainty set that encodes applicable European Union regulations. Functional uncertainties ensure that vehicle owners and grid operators can trust the decisions of aggregators at all times. Faithfully modeling the energy conversion losses during battery charging and discharging renders the optimization problem nonconvex. By exploiting a total unimodularity property of the proposed uncertainty sets and an exact linear decision rule reformulation, we prove that the nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Somewhat counterintuitively, the underlying deterministic problem for a known frequency deviation trajectory does not reduce to a linear program but results in a large-scale mixed-integer linear program, even if time is discretized. We believe that we have thus discovered the first practically interesting class of optimization problems that become dramatically easier through robustification. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. In particular, we find that the prevailing penalties for non-delivery of promised regulation power are too low to incentivize aggregators to honor their promises toward grid operators. For general electricity storage devices, we then solve a simplified version of the decision-making problem analytically to understand how the optimal frequency regulation commitments depend on the roundtrip efficiency of the storage device, the dispersion of the frequency deviations, and the EU delivery guarantee. We show how the marginal cost of frequency regulation decreases with roundtrip efficiency and increases with frequency deviation dispersion. For energy-constrained storage devices, we find that the profits from frequency regulation are inversely proportional to the length of time for which storage operators commit regulation power. Establishing an intra-day market for frequency regulation would thus make electricity storage devices more competitive with other regulation providers, such as thermal power plants.

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Many decision problems in science, engineering, and economics are affected by uncertainty, which is typically modeled by a random variable governed by an unknown probability distribution. For many practical applications, the probability distribution is only observable through a set of training samples. In data-driven decision-making, the goal is to find a decision from the training samples that will perform equally well on unseen test samples. In this thesis, we leverage techniques from distributionally robust optimization to address decision-making problems in statistical learning, behavioral economics and estimation problems. In particular, Wasserstein distributionally robust optimization is studied where the decision-maker learns decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In the first part of the thesis we study regression and classification methods in supervised learning from the distributionally robust perspective. In the classical setting the goal is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce training data, overfitting is typically mitigated by adding regularization terms to the objective that penalize hypothesis complexity. We introduce new regularization techniques using ideas from distributionally robust optimization, and we give new probabilistic interpretations to existing techniques. In the second part of the thesis we consider data-driven inverse optimization problems where an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the predicted decision (i.e., the decision implied by a particular candidate objective) differs from the agent's actual response to a random signal. In the final part of the thesis we study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the nonconvex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.

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Distributional Robustness in Mechanism Design

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