Distributed Nash Equilibrium Seeking for Aggregative Games With Quantization Constraints

The key elements in an indirect V2G system architecture are aggregators. They act as an interface between the grid and a group of plug-in hybrid electric vehicles (PHEVs). In this paper, we design an optimal vehicle to grid (V2G) aggregator to control the charging and frequency regulation processes of a group of PHEVs. We consider a problem that an aggregator has to minimize the overall cost of PHEV fleet in a multiple time slot horizon and meet the required battery level when PHEVs plug out. We adopt summation of PHEVs' expenditure in a finite number of time slots as our objective function, which is a quadratic optimization problem. A model predictive control based (MPC-based) PHEV charging and regulation algorithm is proposed to schedule the charging and regulation processes. Through the numerical experiments, we obtain the optimal charging and frequency regulation sequences for each PHEV, the effect of price prediction error on PHEV's cost as well as the impact of penalty factor to plug-out State of Charge (SOC). It is also shown that by taking the optimal control sequences, the PHEV owner can reduce his cost and depart with desired SOC.

This article studies a distributed Nash equilibrium (NE) seeking problem for multiple agents of aggregative games. A novel distributed algorithm is presented to assure both fast convergence and nonovershoot performance by the combination of an accelerated method and a gradient-based NE seeking algorithm. To be specific, an accelerated method (i.e., the heavy-ball method or Nesterov’s accelerated method) is introduced in the designed algorithm for the fast convergence of agents’ strategies to the NE. Moreover, a suitable switching mechanism is proposed to improve the transient performance by switching the distributed algorithm based on the accelerated method to the gradient-based algorithm. As a result, the presented distributed algorithm is modeled by a hybrid dynamical system (HDS). The semi-globally practical convergence is established by analyzing the stability of a parameterized HDS. An example of distributed energy resources is taken to verify the presented algorithm.

We address the Nash equilibrium problem in a partial-decision information scenario, where each agent can only observe the actions of some neighbors, while its cost possibly depends on the strategies of other agents. Our main contribution is the design of a fully-distributed, single-layer, fixed-step algorithm, based on a proximal best-response augmented with consensus terms. To derive our algorithm, we follow an operator-theoretic approach. First, we recast the Nash equilibrium problem as that of finding a zero of a monotone operator. Then, we demonstrate that the resulting inclusion can be solved in a fully-distributed way via a proximal-point method, thanks to the use of a novel preconditioning matrix. Under strong monotonicity and Lipschitz continuity of the game mapping, we prove linear convergence of our algorithm to a Nash equilibrium. Furthermore, we show that our method outperforms the fastest known gradient-based schemes, both in terms of guaranteed convergence rate, via theoretical analysis, and in practice, via numerical simulations.

Distributed Nash equilibrium seeking for aggregative games with second-order nonlinear players

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

This article considers the (two-player) composite Nash equilibrium (CNE) problem with a separable nonsmooth part, which is known to include the composite saddle-point (CSP) problem as a special case. Due to its two-block structure, this problem can be solved by any algorithm belonging to the block-decomposition hybrid proximal-extragradient (BD-HPE) framework proposed in [R. D. C. Monteiro and B. F. Svaiter, SIAM J. Optim., 23 (2013), pp. 475--507]. The framework consists of a family of inexact proximal point methods for solving a more general two-block structured monotone inclusion problem which, at every iteration, solves two prox subinclusions according to a certain relative error criterion. By exploiting the fact that the two prox subinclusions in the context of the CNE problem are equivalent to two composite convex programs, this article proposes a new instance of the BD-HPE framework that approximately solves them using an accelerated gradient method. It is shown that the new instance is able to take significantly larger prox stepsizes than other instances from this framework that perform single composite gradient steps for solving the subinclusions. As a result, it is shown that the first instance has better iteration-complexity than the latter ones. Finally, it is also shown that the new accelerated BD-HPE instance computationally outperforms several state-of-the-art algorithms on many relevant classes of CSP and CNE instances.

We consider the Nash equilibrium problem with vector payoffs in a topological vector space. By employing the recent concept of relative (pseudo) monotonicity, we establish several existence results for vector Nash equilibria and vector equilibria. The results strengthen in a major way existence results for vector equilibrium problems which were based on the usual (generalized) monotonicity concepts.

We consider the problem of finding Nash equilibrium for two-player turn-based zero-sum games. Inspired by the AlphaGo Zero (AGZ) algorithm, we develop a Reinforcement Learning based approach. Specifically, we propose Explore-Improve-Supervise (EIS) method that combines exploration, improvement and supervised learning to find the value function and policy associated with Nash equilibrium. We identify sufficient conditions for convergence and correctness for such an approach. For a concrete instance of EIS where random policy is used for exploration, Monte-Carlo Tree Search is used for improvement and Nearest Neighbors is used for supervised learning, we establish that this method finds an $\varepsilon$-approximate value function of Nash equilibrium in $\widetildeO(\varepsilon^-(d+4))$ steps when the underlying state-space of the game is continuous and d-dimensional. This is nearly optimal as we establish a lower bound of $\widetildeOmega (\varepsilon^-(d+2) )$ for any policy.

In this paper, we formulate the random bimatrix game as a chance-constrained game using chance constraint. We show that a Nash equilibrium problem, corresponding to independent normally distributed payoffs, is equivalent to a nonlinear complementarity problem. Further if the payoffs are also identically distributed, a strategy pair where each player’s strategy is the uniform distribution over his action set, is a Nash equilibrium. We show that a Nash equilibrium problem corresponding to independent Cauchy distributed payoffs, is equivalent to a linear complementarity problem.

The use of electric vehicles (EVs) are increasing in the transportation world. In the electricity market, if a large number of EVs are connected in the power system, but the EVs charging power demand is scheduled appropriately, then the EV owners, as well as the grid, will make the profit. This work focuses on customer-oriented EV charging in the day-ahead market with the objective of reducing the charging cost for EV owners while also considering the impact on spot prices of EV demand in the day-ahead market. An aggregative game has been proposed for satisfying the EVs charging demand in the dayahead electricity market. In the day-ahead aggregative game approach, we formulate the energy charging problem where each EV chooses its own demand strategy to maximize its own benefit selfishly while taking into account the strategies of other EVs. The Nash equilibrium(NE) for the proposed game model has been proved.

Distributed convergence to Nash equilibria in network and average aggregative games

This article considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players' actions, such as their neighbors' actions. Thus, the players are supposed to communicate with each other to estimate other players' actions. To solve the problem, a leader-following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player' local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players' actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations.

We introduce a linearized alternating direction method of multipliers (ADMM)-like Nash equilibrium seeking algorithm (LANA) for a class of non-cooperative games over generally connected networks. This model differs from conventional settings because the communication graph is not necessarily the same as the players' objective dependency network and thus players have to deal with incomplete information issues. To solve this game theoretic problem, the introduced algorithm involves every player performing gradient (projection) play to minimize his own objective selfishly while sharing, retrieving, and combining information locally among his network neighborhood. Convergence guarantees are provided for the algorithm. We further extend the introduced algorithm to asynchronous updates and find it works well. Numerical experiments verify the viability of the algorithms.

Global stability of unique Nash equilibrium in Cournot oligopoly and rent-seeking game

With the advancement of power-to-gas (P2G) technology and local energy markets, new opportunities emerge for small-scale gas distributors (GDs) to unlock additional value. However, research on multiple GDs within the same power network participating in local energy markets remains unexplored, potentially leading to suboptimal collaborative efficiency and market imbalance. To bridge this gap, this paper proposes a heterogeneous game framework that integrates mixed-integer Nash equilibrium problem (MI-NEP) and generalized Nash equilibrium problem (GNEP). The framework assumes rational GDs simultaneously engage in non-convex GNEP for optimal power flow (OPF) coordination in distribution systems while addressing MI-NEP governing pipeline flow control in local gas market transactions. We develop a collaborative computing solution combining the best response (BR) method with a Korpelevich-inspired algorithm to obtain Nash equilibrium (NE) solutions ensuring fairness and operational stability. Comprehensive case studies on integrated 10-node power distribution networks with 7-node and 20-node gas networks demonstrate that the algorithm converges within 10 iterations with a residual error below [Formula: see text]. The proposed framework effectively ensures market fairness by reducing the cost disparity between isomorphic agents to zero.