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We develop a Nash equilibrium problem representing a perfectly competitive market wherein all players are subject to the same source of uncertainty with an unknown probability distribution. Each player—depending on her individual access to and confidence over empirical data—builds an ambiguity set containing a family of potential probability distributions describing the uncertain event. The ambiguity set of different players is not necessarily identical, yielding a market with potentially heterogeneous ambiguity aversion. Built upon recent developments in the field of Wasserstein distributionally robust chance-constrained optimization, each ambiguity-averse player maximizes her own expected payoff under the worst-case probability distribution within her ambiguity set. Using an affine policy and a conditional value-at-risk approximation of chance constraints, we define a tractable Nash game. We prove that under certain conditions a unique Nash equilibrium point exists, which coincides with the solution of a single optimization problem. Numerical results indicate that players with comparatively lower consumption utility are highly exposed to rival ambiguity aversion.

Convex optimization finds many real-life applications, where - optimized on real data - optimization results may expose private data attributes (e.g., individual health records, commercial information, etc.), thus leading to privacy breaches. To avoid these breaches and formally guarantee privacy to optimization data owners, we develop a new privacy-preserving perturbation strategy for convex optimization programs by combining stochastic (chance-constrained) programming and differential privacy. Unlike standard noise-additive strategies, which perturb either optimization data or optimization results, we express the optimization variables as functions of the random perturbation using linear decision rules; we then optimize these rules to accommodate the perturbation within the problem's feasible region by enforcing chance constraints. This way, the perturbation is feasible and makes different, yet adjacent in the sense of a given distance function, optimization datasets statistically similar in randomized optimization results, thereby enabling probabilistic differential privacy guarantees. The chance-constrained optimization additionally internalizes the conditional value-at-risk measure to model the tolerance towards the worst-case realizations of the optimality loss with respect to the non-private solution. We demonstrate the privacy properties of our perturbation strategy analytically and through optimization and machine learning applications.

Although distribution grid customers are obliged to share their consumption data with distribution system operators (DSOs), a possible leakage of this data is often disregarded in operational routines of DSOs. This paper introduces a privacy-preserving optimal power flow (OPF) mechanism for distribution grids that secures customer privacy from unauthorised access to OPF solutions, e.g., current and voltage measurements. The mechanism is based on the framework of differential privacy that allows to control the participation risks of individuals in a dataset by applying a carefully calibrated noise to the output of a computation. Unlike existing private mechanisms, this mechanism does not apply the noise to the optimization parameters or its result. Instead, it optimizes OPF variables as affine functions of the random noise, which weakens the correlation between the grid loads and OPF variables. To ensure feasibility of the randomized OPF solution, the mechanism makes use of chance constraints enforced on the grid limits. The mechanism is further extended to control the optimality loss induced by the random noise, as well as the variance of OPF variables. The paper shows that the differentially private OPF solution does not leak customer loads up to specified parameters.

We address the market design issue for a local energy community, comprising prosumers, consumers, photovoltaic, and energy storage systems, all connected as a community to a distribution grid. Our work explores different market design options based on cooperative and non-cooperative game-theoretic models that enable an economic access to the benefits of energy storage for prosumers without a direct ownership of a storage system. We compare market outcomes in terms of the community cost as well as the individual cost. We pay special attention to potential uncertainties, and investigate financial instruments that allow storage systems to be utilized by multiple prosumers. In particular, we explore a case where a prosumer that owns a storage system provides rights, either physical or financial, rather than participating in the local market as an arbitrageur. Moreover, we consider a cooperative market design where energy community members agree on the Shapley value or the nucleolus as a community cost allocation rule. Our results show that an access economy for energy storage systems enhances energy communities by reducing the cost volatility for most prosumers, while the expected operational cost of the community as a whole remains unchanged.