Game Design and Analysis for Price-Based Demand Response: An Aggregate Game Approach

Distributed Edge-Based Nash Equilibrium Seeking With Event-Triggered Quantized Communication

This article investigates a noncooperative game of multiagent systems in incomplete information scenarios. To cooperatively seek the Nash equilibrium (NE), each agent aims to minimize its own cost function by interacting with its neighbors over undirected communication networks. While existing distributed NE seeking methods alleviate the computational burden, they also entail higher communication costs. To reduce communication frequency and bandwidth, we propose a class of distributed edge-based NE seeking methods by leveraging the advantages of event-triggered mechanisms and quantization techniques. In the proposed framework, a buffer is equipped on every communication channel, thereby reducing the workload of both agents at either end. It is shown that the convergence error can be made arbitrarily small by tuning a constant threshold, and it can asymptotically converge to zero by setting an exponentially decaying threshold or a dynamic threshold. Moreover, in the case of unawareness of any global information, we further provide a fully distributed event-triggered quantized algorithm, by which the convergence error is ultimately uniformly bounded. Finally, two numerical examples are utilized to illustrate the effectiveness of the proposed algorithms.

Distributed Nash Equilibrium Seeking for Multicluster Aggregative Game of Euler-Lagrange Systems With Coupled Constraints

This article considers the distributed Nash equilibrium seeking problem of a multicluster aggregative game subject to local set constraints, consensus constraints in the same cluster, and coupled linear equality and nonlinear inequality constraints among all clusters. In the considered game, each cluster is composed of a group of players formulated by uncertain Euler-Lagrange (EL) dynamics, and its objective is to minimize its own cost function, which is the sum of the local functions of all players in the cluster. The local cost function of each player depends on its own decision and an aggregate of the decisions of all the players. An adaptive continuous-time distributed strategy is developed for uncertain EL systems to reach the generalized Nash equilibrium (GNE) of multicluster aggregative game. In particular, a new auxiliary system is synthesized using a projection operator, gradient descent, and dynamic average consensus to estimate the GNE. Based on the outputs of the auxiliary system, an adaptive tracking algorithm is developed for an EL system with uncertain parameters. Using the Lyapunov stability theory, it is shown that the developed distributed strategy achieves accurate convergence to the GNE. Finally, a numerical example is presented to demonstrate the theoretical results.

Operator-as-a-Consumer: A Novel Energy Storage Sharing Approach Under Demand Charge

Energy storage systems (ESSs)-based demand response (DR) is an appealing way to save electricity bills for consumers under demand charge and time-of-use (TOU) price. In order to counteract the high investment cost of ESS, a novel operator-enabled ESS sharing scheme, namely, the "operator-as-a-consumer (OaaC)," is proposed and investigated in this article. In this scheme, the users and the operator form a Stackelberg game. The users send ESS orders to the operator and apply their own ESS dispatching strategies for their own purposes. Meanwhile, the operator maximizes its profit through optimal ESS sizing and scheduling, as well as pricing for the users' ESS orders. The feasibility and economic performance of OaaC are further analyzed by solving a bilevel joint optimization problem of ESS pricing, sizing, and scheduling. To make the analysis tractable, the bilevel model is first transformed into its single-level mathematical program with equilibrium constraints (MPEC) formulation and is then linearized into a mixed-integer linear programming (MILP) problem using multiple linearization methods. Case studies with actual data are utilized to demonstrate the profitability for the operator and simultaneously the ability of bill saving for the users under the proposed OaaC scheme.

Distributed Algorithm Design for Aggregative Games of Euler-Lagrange Systems and Its Application to Smart Grids

The aggregative games are addressed in this article, in which there are coupling constraints among decisions and the players have Euler-Lagrange (EL) dynamics. On the strength of gradient descent, state feedback, and dynamic average consensus, two distributed algorithms are developed to seek the variational generalized Nash equilibrium (GNE) of the game. This article analyzes the convergence of two algorithms by utilizing singular perturbation analysis and variational analysis. The two algorithms exponentially and asymptotically converge to the variational GNE of the game, respectively. Moreover, the results are applied to the electricity market games of smart grids. By the algorithms, turbine-generator systems can seek the variational GNE of electricity markets autonomously. Finally, simulation examples verify the methods.

Nonsmooth Continuous-Time Distributed Algorithms for Seeking Generalized Nash Equilibria of Noncooperative Games via Digraphs

In this article, the problem of distributed generalized Nash equilibrium (GNE) seeking in noncooperative games is investigated via multiagent networks, where each player aims to minimize his or her own cost function with a nonsmooth term. Each player's cost function and feasible action set in the noncooperative game are both determined by actions of others who may not be neighbors, as well as his/her own action. Particularly, feasible action sets are constrained by private convex inequalities and shared linear equations. Each player can only have access to his or her own cost function, private constraint, and a local block of shared constraints, and can only communicate with his or her neighbours via a digraph. To address this problem, a novel continuous-time distributed primal-dual algorithm involving Clarke's generalized gradient is proposed based on consensus algorithms and the primal-dual algorithm. Under mild assumptions on cost functions and graph, we prove that players' actions asymptotically converge to a GNE. Finally, a simulation is presented to demonstrate the effectiveness of our theoretical results.

Distributed Nash Equilibrium Seeking Over Markovian Switching Communication Networks

We aim to address the Nash equilibrium (NE) seeking problem for multiple players over Markovian switching communication networks in this article, where a new type of distributed synchronous discrete-time algorithm is proposed and utilized. Specifically, each player in the present game model is assumed to employ a gradient-like projection algorithm to choose its action based upon the estimated ones for all the others. Under the mild condition that the union network of all communication network candidates is connected, we show that the players' actions could converge to an arbitrarily small neighborhood of the NE in the mean-square sense by adjusting the algorithm parameters. It is further found that the unique NE is mean-square stable when it is not restricted by any constraint set. In addition, we show that the proposed distributed discrete-time NE seeking algorithm can be utilized to deal with the energy trading problem in microgrids where each microgrid is modeled as a rational player using a purchase price as its action to buy energy from other microgrids with surplus supplies. The energy market allocates the excess energy according to the principle of proportional distribution. Some numerical simulations are finally presented to verify the validity of the present discrete-time NE seeking algorithm in solving the energy trading problem.

DisEHPPC: Enabling Heterogeneous Privacy-Preserving Consensus-Based Scheme for Economic Dispatch in Smart Grids

These days, the increasing incremental cost consensus-based algorithms are designed to tackle the economic dispatch (ED) problem in smart grids (SGs). However, one principal obstruction lies in privacy disclosure for generators and consumers in electricity activities between supply and demand sides, which may bring great losses to them. Hence, it is extraordinarily essential to design effective privacy-preserving approaches for ED problems. In this article, we propose a two-phase distributed and effective heterogeneous privacy-preserving consensus-based (DisEHPPC) ED scheme, where a demand response (DR)-based framework is constructed, including a DR server, data manager, and a set of local controllers. The first phase is that Kullback-Leibler (KL) privacy is guaranteed for the privacy of consumers' demand by the differential privacy method. The second phase is that (ε,δ) -privacy is, respectively, achieved for the generation energy of generators and the sensitivity of electricity consumption to electricity price by designing the privacy-preserving incremental cost consensus-based (PPICC) algorithm. Meanwhile, the proposed PPICC algorithm tackles the formulated ED problem. Subsequently, we further carry out the detailed theoretical analysis on its convergence, optimality of final solution, and privacy degree. It is found that the optimal solution for the ED problem and the privacy preservation of both supply and demand sides can be guaranteed simultaneously. By evaluation of a numerical experiment, the correctness and effectiveness of the DisEHPPC scheme are confirmed.

Distributed Generalized Nash Equilibrium Seeking and Its Application to Femtocell Networks

In this article, distributed algorithms are developed to search the generalized Nash equilibrium (NE) with global constraints. Relations between the variational inequality and the NE are investigated via the Karush-Kuhn-Tucker (KKT) optimal conditions, which provide the underlying principle for developing the distributed algorithms. Two time-varying consensus schemes are proposed for each agent to estimate the actions of others, by which a distributed framework is established. The algorithm with fixed-gains is designed with certain system knowledge, while the adaptive algorithm is proposed to address the problem when the system parameters are not available. The asymptotic convergence to the NE is established through the Lyapunov theory and the consensus theory. The power control problem in a femtocell network is formulated as a Nash game and is solved by the proposed algorithms. The simulation results are provided to verify the effectiveness of theoretical development.

Distributed Robust Seeking of Nash Equilibrium for Networked Games: An Extended State Observer-Based Approach

This article aims to accommodate networked games in which the players' dynamics are subjected to unmodeled and disturbance terms. The unmodeled and disturbance terms are regarded as extended states for which observers are designed to estimate them. Compensating the players' dynamics with the observed values, the control laws are designed to achieve the robust seeking of the Nash equilibrium for networked games. First, we consider the case in which the players' dynamics are subject to time-varying disturbances only. In this case, the seeking strategy is developed by employing a smooth observer based on the proportional-integral (PI) control. By utilizing the designed strategy, we show that the players' actions would converge to a small neighborhood of the Nash equilibrium. Moreover, the ultimate bound can be adjusted to be arbitrarily small by tuning the control gains. Then, we further consider the case in which both an unmodeled term and a disturbance term coexist in the players' dynamics. In this case, we adapt the idea from the robust integral of the sign of the error (RISE) method in the strategy design to achieve the asymptotic seeking of the Nash equilibrium. Both strategies are analytically investigated via the Lyapunov stability analysis. The applications of the proposed methods for a network of velocity-actuated vehicles are discussed. Finally, the effectiveness of the proposed methods is verified via conducting numerical simulations.

On Distributed Nash Equilibrium Computation: Hybrid Games and a Novel Consensus-Tracking Perspective

With the incentive to solve Nash equilibrium computation problems for networked games, this article tries to find answers for the following two problems: 1) how to accommodate hybrid games, which contain both continuous-time players and discrete-time players? and 2) are there any other potential perspectives for solving continuous-time networked games except for the consensus-based gradient-like algorithm established in our previous works? With these two problems in mind, the study of this article leads to the following results: 1) a hybrid gradient search algorithm and a consensus-based hybrid gradient-like algorithm are proposed for hybrid games with their convergence results analytically investigated. In the proposed hybrid strategies, continuous-time players adopt continuous-time algorithms for action updating, while discrete-time players update their actions at each sampling time instant and 2) based on the idea of consensus tracking, the Nash equilibrium learning problem for continuous-time games is reformulated and two new computation strategies are subsequently established. Finally, the proposed strategies are numerically validated.

Mechanism Design for Demand Management in Energy Communities

We consider a demand management problem in an energy community, in which several users obtain energy from an external organization such as an energy company and pay for the energy according to pre-specified prices that consist of a time-dependent price per unit of energy as well as a separate price for peak demand. Since users’ utilities are their private information, which they may not be willing to share, a mediator, known as the planner, is introduced to help optimize the overall satisfaction of the community (total utility minus total payments) by mechanism design. A mechanism consists of a message space, a tax/subsidy, and an allocation function for each user. Each user reports a message chosen from her own message space, then receives some amount of energy determined by the allocation function, and pays the tax specified by the tax function. A desirable mechanism induces a game, the Nash equilibria (NE), of which results in an allocation that coincides with the optimal allocation for the community. As a starting point, we design a mechanism for the energy community with desirable properties such as full implementation, strong budget balance and individual rationality for both users and the planner. We then modify this baseline mechanism for communities where message exchanges are allowed only within neighborhoods, and consequently, the tax/subsidy and allocation functions of each user are only determined by the messages from their neighbors. All of the desirable properties of the baseline mechanism are preserved in the distributed mechanism. Finally, we present a learning algorithm for the baseline mechanism, based on projected gradient descent, that is guaranteed to converge to the NE of the induced game.

Distributed Nash Equilibrium Seeking for Aggregative Games With Nonlinear Dynamics Under External Disturbances

In this paper, we study the distributed Nash equilibrium (NE) seeking problem for a class of aggregative games with players described by uncertain perturbed nonlinear dynamics. To seek the NE, each player needs to construct a distributed algorithm based on information of its cost function and the exchanging information obtained from its neighbors. By combining the internal model principle and the average consensus technique, we propose a distributed gradient-based algorithm for the players. This paper not only assures the NE seeking of aggregative games but also achieves the disturbance rejection of external disturbances.

Asynchronous Distributed Algorithms for Seeking Generalized Nash Equilibria Under Full and Partial-Decision Information

This paper investigates asynchronous algorithms for distributedly seeking generalized Nash equilibria with delayed information in multiagent networks. In the game model, a shared affine constraint couples all players' local decisions. Each player is assumed to only access its private objective function, private feasible set, and a local block matrix of the affine constraint. We first give an algorithm for the case when each agent is able to fully access all other players' decisions. By using auxiliary variables related to communication links and the edge Laplacian matrix, each player can carry on its iteration asynchronously with only private data and possibly delayed information from its neighbors. Then, we consider the case when agents cannot know all other players' decisions, called a partial-decision information case. We introduce a local estimation of the overall agents' decisions and incorporate consensus dynamics on these local estimations. The two algorithms do not need any centralized clock coordination, fully exploit the local computation resource, and remove the idle time due to waiting for the "slowest" agent. Both algorithms are developed by preconditioned forward-backward operator splitting, and their convergence is shown by relating them to asynchronous fixed-point iterations, under proper assumptions and fixed and nondiminishing step-size choices. Numerical studies verify the algorithms' convergence and efficiency.

Average Quasi-Consensus Algorithm for Distributed Constrained Optimization: Impulsive Communication Framework

This paper presents the impulsive average quasi-consensus algorithm for distributed constrained convex optimization. First, the constrained optimization problem can be transformed into an unconstrained problem using the interior point method, and then a distributed algorithm is modeled by means of impulsive differential equation. In the framework of the continuous-time gradient method and algebraic graph theory, each agent can deal with one local objective function with local constraints. At the impulsive instants, each agent can communicate with its neighboring agents over the network. Under certain conditions, the impulsive average quasi-consensus is achieved. It is shown that the state of average quasi-consensus is the optimal solution of the aforementioned unconstrained optimization problem, and the state of each agent can also reach the neighborhood of the optimal solution. Finally, two numerical examples show the effectiveness of the proposed impulsive average quasi-consensus algorithm. Moreover, the feasibility of the approach is verified by an application to one sensor network localization problem.

Distributed Nash Equilibrium Seeking for Quadratic Games with Security

Considering a game with quadratic cost functions, this paper presents a distributed algorithm with security, whereby each player updates its strategy variable without using its private data and still achieves the Nash equilibrium. By using the theory of differential inclusions, Lyapunov function and invariance principle, the algorithm is proved to be convergent. Our algorithm can be used when it is required to seek the Nash equilibrium without disclosure of private data.

Distributed Algorithms for Searching Generalized Nash Equilibrium of Noncooperative Games

In this paper, the distributed Nash equilibrium (NE) searching problem is investigated, where the feasible action sets are constrained by nonlinear inequalities and linear equations. Different from most of the existing investigations on distributed NE searching problems, we consider the case where both cost functions and feasible action sets depend on actions of all players, and each player can only have access to the information of its neighbors. To address this problem, a continuous-time distributed gradient-based projected algorithm is proposed, where a leader-following consensus algorithm is employed for each player to estimate actions of others. Under mild assumptions on cost functions and graphs, it is shown that players' actions asymptotically converge to a generalized NE. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.

Distributed Generalized Nash Equilibrium Seeking Algorithm Design for Aggregative Games Over Weight-Balanced Digraphs

In this paper, two aggregative games over weight-balanced digraphs are studied, where the cost functions of all players depend on not only their own decisions but also the aggregate of all decisions. In the first problem, the cost functions of players are differentiable with Lipschitz gradients, and the decisions of all players are coupled by linear coupling constraints. In the second problem, the cost functions are nonsmooth, and the decisions of all players are constrained by local feasibility constraints as well as linear coupling constraints. In order to seek the variational generalized Nash equilibrium (GNE) of the differentiable aggregative games, a continuous-time distributed algorithm is developed via gradient descent and dynamic average consensus, and its exponential convergence to the variational GNE is proven with the help of Lyapunov stability theory. Then, another continuous-time distributed projection-based algorithm is proposed for the nonsmooth aggregative games based on differential inclusions and differentiated projection operations. Moreover, the convergence of the algorithm to the variational GNE is analyzed by utilizing singular perturbation analysis. Finally, simulation examples are presented to illustrate the effectiveness of our methods.

Pricing Mechanism With Noncooperative Game and Revenue Sharing Contract in Electricity Market

In this paper, a pricing mechanism is proposed for the electricity supply chain, which is consisting of one generation company (GC), multiple consumers, and competing utility companies (UCs). The UC participates in electricity supply chain management by a revenue sharing contract (RSC). In the electricity supply chain, the electricity real-time balance has an important role in the stable operation of the power system. Therefore, we introduce the demand response into the electricity supply chain to match supply with demand under forecast errors. Hence, we formulate a noncooperative game to characterize the interactions among the multiple competing UCs, which set the retail prices to maximize their profits. Besides, the UCs select their preferred contractual terms offered by the GC to maximize its profits and coordinate the electricity supply chain simultaneously. The existence and uniqueness of the Nash equilibrium (NE) are examined, and an iterative algorithm is developed to obtain the NE. Furthermore, we analyze the RSC that can coordinate the electricity supply chain and align the NE with the cooperative optimum under the RSC. Finally, numerical results demonstrate the superiority of the proposed model and the influence of market demand disruptions on the profits of the UCs, GC, and supply chain.

Distributed Nash Equilibrium Seeking in Multiagent Games Under Switching Communication Topologies

This paper investigates distributed Nash equilibrium seeking in multiagent games under switching communication topologies. To be specific, the communication topology is supposed to be switching among a set of strongly connected digraphs, which might suffer from occasional loss of communication due to sensor failure, packet loss, etc. The synthesis of the leader-following consensus protocol and the gradient play is exploited to achieve the distributed Nash equilibrium seeking under the switching communication topologies. Switching topology without loss of communication is firstly considered, followed by switching topology subject to missing communication within some time slots. For both situations, nonquadratic and quadratic games are addressed separately. Local convergence results are presented for nonquadratic games and nonlocal convergence results are provided for quadratic games. The theoretical results are verified by numerical examples.

Game-Theory Modeling for Social Welfare Maximization in Smart Grids

In this paper, we study the Stackelberg game-based evolutionary game with two players, generators and energy users (EUs), for monetary profit maximization in real-time price (RTP) demand response (DR) systems. We propose two energy strategies, generator’s best-pricing and power-generation strategy and demand’s best electricity-usage strategy, which maximize the profit of generators and EUs, respectively, rather than maximizing the conventional unified profit of the generator and EUs. As a win–win strategy to reach the social-welfare maximization, the generators acquire the optimal power consumption calculated by the EUs, and the EUs obtain the optimal electricity price calculated by the generators to update their own energy parameters to achieve profit maximization over time, whenever the generators and the EUs execute their energy strategy in the proposed Stackelberg game structure. In the problem formulation, we newly formulate a generator profit function containing the additional parameter of the electricity usage of EUs to reflect the influence by the parameter. The simulation results show that the proposed energy strategies can effectively improve the profit of the generators to 45% compared to the beseline scheme, and reduce the electricity charge of the EUs by 15.6% on average. Furthermore, we confirmed the proposed algorithm can contribute to stabilization of power generation and peak-to-average ratio (PAR) reduction, which is one of the goals of DR.

Distributed Coordination for Optimal Energy Generation and Distribution in Cyber-Physical Energy Networks

This paper proposes three coordination laws for optimal energy generation and distribution in energy network, which is composed of physical flow layer and cyber communication layer. The physical energy flows through the physical layer; but all the energies are coordinated to generate and flow by distributed coordination algorithms on the basis of communication information. First, distributed energy generation and energy distribution laws are proposed in a decoupled manner without considering the interactive characteristics between the energy generation and energy distribution. Second, a joint coordination law to treat the energy generation and energy distribution in a coupled manner taking account of the interactive characteristics is designed. Third, to handle over- or less-energy generation cases, an energy distribution law for networks with batteries is designed. The coordination laws proposed in this paper are fully distributed in the sense that they are decided optimally only using relative information among neighboring nodes. Through numerical simulations, the validity of the proposed distributed coordination laws is illustrated.

Distributed Secure Coordinated Control for Multiagent Systems Under Strategic Attacks

This paper studies a distributed secure consensus tracking control problem for multiagent systems subject to strategic cyber attacks modeled by a random Markov process. A hybrid stochastic secure control framework is established for designing a distributed secure control law such that mean-square exponential consensus tracking is achieved. A connectivity restoration mechanism is considered and the properties on attack frequency and attack length rate are investigated, respectively. Based on the solutions of an algebraic Riccati equation and an algebraic Riccati inequality, a procedure to select the control gains is provided and stability analysis is studied by using Lyapunov's method.. The effect of strategic attacks on discrete-time systems is also investigated. Finally, numerical examples are provided to illustrate the effectiveness of theoretical analysis.

Coalition Formation and Spectrum Sharing of Cooperative Spectrum Sensing Participants

In cognitive radio networks, self-interested secondary users (SUs) desire to maximize their own throughput. They compete with each other for transmit time once the absence of primary users (PUs) is detected. To satisfy the requirement of PU protection, on the other hand, they have to form some coalitions and cooperate to conduct spectrum sensing. Such dilemma of SUs between competition and cooperation motivates us to study two interesting issues: 1) how to appropriately form some coalitions for cooperative spectrum sensing (CSS) and 2) how to share transmit time among SUs. We jointly consider these two issues, and propose a noncooperative game model with 2-D strategies. The first dimension determines coalition formation, and the second indicates transmit time allocation. Considering the complexity of solving this game, we decompose the game into two more tractable ones: one deals with the formation of CSS coalitions, and the other focuses on the allocation of transmit time. We characterize the Nash equilibria (NEs) of both games, and show that the combination of these two NEs corresponds to the NE of the original game. We also develop a distributed algorithm to achieve a desirable NE of the original game. When this NE is achieved, the SUs obtain a Dhp-stable coalition structure and a fair transmit time allocation. Numerical results verify our analyses, and demonstrate the effectiveness of our algorithm.