This paper considers multiple park-level integrated energy systems (PIESs) connected to electricity, heat, and gas networks. Cooperative alliances form among PIESs due to lower internal trading prices for surplus electricity and heat. Production costs may be overstated by industrial PIES to raise energy prices to residential PIES, while demand may be underreported by residential PIES to lower energy prices from industrial PIES. Fraudulent behaviors due to asymmetric information in large-scale heterogeneous PIESs are often overlooked. Moreover, heat grade differences in PIESs heat trading, unlike electricity trading, have not been considered, potentially increasing entropy and reducing energy efficiency. Due to privacy concerns, existing centralized optimization methods cannot be applied in large-scale PIES energy trading. To address these challenges, a distributed optimization strategy for multi-grade heat cascading in PIESs, considering equilibrium fraud and privacy preservation, is proposed. The fraud factor is updated by adjusting the relaxation coefficient using a quasi-gradient descent method. Heterogeneous PIESs are graded by temperature ranges, and waste heat recovery devices are used to construct multi-grade heat equilibrium functions. An adaptive step-size alternating direction multiplier method (ADMM) is introduced for efficient optimization. Simulation results show a 17.41% improvement in energy efficiency with multi-grade heat, and the total benefit of cooperative alliances increases by 25% when fraud is considered.

This study investigates the problem of event-based privacy-preserving platooning control for connected automated vehicles subject to limited communication resources and data falsification attacks. First, a novel dynamic event-triggered mechanism (DETM)-based encryption and decryption strategy is proposed to regulate the frequency of data encryption and transmission in response to real-time network bandwidth occupancy and vehicular states, thereby safeguarding data privacy while improving communication resource utilization. Then, a unified and general attack modeling method is established to characterize various forms of data falsification attacks on sensor measurements and driving commands. Building upon this, a secure observer-based platooning control protocol is derived, where the controller gains, observer parameters, and communication topology are co-designed to ensure the desired secure platooning performance while ensuring vehicular data privacy and efficient utilization of communication resources. Finally, several comparative case studies with quantified metrics are conducted to demonstrate the efficacy and merits of the proposed methods from the perspectives of platoon stability, security, communication efficiency and data privacy.

Dear Editor, This letter deals with incentive design problem for noncooperative dynamical systems to achieve social welfare maximization with uncertain types of mixed dynamics. The agents are allowed to liberally choose either the pseudo-gradient or best-response dynamics for continuous-time decision-making. Sufficient conditions are derived for the incentive mechanism to ensure the diagonal stabilization around the social optimum of the social welfare function in the mixed noncooperative dynamical systems without using the knowledge of the types. A numerical example is presented to illustrate the efficacy of our results.

This paper considers the problem of steering the states of agents in non-cooperative games to the social optimum. To this end, we design an incentive mechanism based on the radial basis function neural network without requiring the detailed payoff functions from agents. To construct phased incentive mechanism, we propose an updating rule based on event-triggered framework for the coefficients of the incentive functions. We analytically prove the convergence of proposed method. A simulation example is given to demonstrate the performance of the designed method.

Self-interested behaviors in noncooperative games often lead to social inefficiency. In many practical scenarios, individual cost functions are private and unavailable to a central regulator, which makes the design of effective incentive mechanisms challenging. In this paper, we consider pricing mechanism design for uncertain noncooperative games with pseudo-gradient-based agents, aiming at achieving social cost minimization without requiring knowledge of individual cost functions. In this pricing mechanism, the central regulator designs pricing signals to influence agent dynamics through a regulated cost structure. To address the unknown pseudo-gradient information, radial basis function neural networks are employed to approximate the agents’ dynamics. To further reduce communication and computational burdens, an asynchronous event-triggered updating scheme is developed, which allows intermittent price updates with or without knowledge of the optimal target state. In addition, a robust term is included to mitigate neural network approximation errors and ensure stability. Sufficient conditions are derived to guarantee boundedness and asymptotic convergence of the closedloop system to the social optimum. The presented methods are applied to a demand management problem in energy retail systems, demonstrating the satisfactory performance of the designed methods with substantially reduced computational burdens.

This article focuses on prescribed-time distributed robust Nash equilibrium seeking for monotone games impacted by unknown and time-varying disturbances. First, a regularization term with a prescribed-time decaying parameter is introduced to compensate for the absence of strong monotonicity in the merely monotone game. Based on the regularization technique, a new prescribed-time signum-based distributed Nash equilibrium seeking algorithm incorporating an integral sliding mode method, a leader-following consensus protocol, and a gradient algorithm is presented for monotone games with unknown but bounded disturbances. Then, to dispose of the unknown bounds of disturbances, a prescribed-time distributed adaptive integral sliding mode based Nash equilibrium seeking strategy is devised. On the basis of the proposed strategies, some sufficient conditions are obtained to guarantee that the players’ actions are capable of converging to the least-norm Nash equilibrium of the monotone games in a prescribed time. In the end, numerical simulations on least-distance formation control of a network of players testify to the performance of the proposed seeking strategies.

A distributed optimal double-layer formation control problem for multi-cluster systems, which has rarely been reported, is addressed in this paper. Specifically, the agents in each cluster are required to form a desired shape, while the centroids of the clusters must also adhere to a specified geometric pattern. Beyond this double layer formation task, the agents are faced with a global optimization problem, which should be resolved concurrently. The paper mainly faces the following challenges: (1) how to deal with both inter-cluster and intra-cluster formation problems; (2) how to simultaneously achieve the double-layer formation control and global optimization; (3) how to estimate the averaged state of each cluster, which is unknown to all agents. To achieve optimal double-layer formation control, a multi-cluster aggregative double-layer formation (MCADF) game is first formulated. Following the MCADF game formulation, an efficient broadcasting method is developed to estimate and disseminate the averaged state of each cluster. Moreover, distributed Nash equilibrium seeking strategies for multi-cluster aggregative games are established by incorporating regularization techniques, gradient algorithms, dynamic average consensus protocols and “improved” consensus tracking protocols. It is formally proven by Lyapunov stability analysis that the proposed strategies can steer agents to achieve optimal double-layer formation. Finally, a numerical simulation example on a multiple target enclosing problem is given to verify the effectiveness of the proposed method.

This paper considers a multi-Unmanned Surface Vehicle (multi-USV) escort problem, in which a group of slave USVs need to achieve formation and interception of intruders so as to protect the master vessel against the intruders. The mission scope includes maintaining a protective formation around the master vessel, selecting appropriate interceptors upon detect intruders while retaining followers to preserve coverage, and adapting online to uncertain intruder maneuvers through real-time decision making. An Adaptive Model Predictive Control (AMPC) escort model is established for the considered problem, which takes the future maneuvers of the intruders into consideration. An offline-learned, online-corrected maneuver predictor based on a Mixture Density Network (MDN) and a Kalman Filter (KF) is developed: the MDN captures multi-modal uncertainty in intruder strategies, while the KF corrects predictions when strategies shift, thereby providing robustness even with limited historical data. To enhance solution efficiency, the Crested Porcupine Optimizer is improved via chaotic initialization, random restarts and enhanced cyclic population reduction (CPR) mechanism as desensitization strategies, jointly accelerating convergence and strengthening exploration. Numerical simulations show improvements in interception performance, prediction accuracy, generalization to unknown intruder strategies, and computational efficiency.

This paper studies the problem of aggregative optimization in open multi-agent systems (OMAS), where agents are allowed to join and leave the system in a free manner during the decision-making process. A multi-aggregator communication mechanism is proposed to facilitate information exchange among agents, in which the aggregators are responsible for collecting information from agents and exchanging them with neighboring aggregators. Based on the multi-aggregator communication mechanism, a novel semi-distributed algorithm is designed. Dynamic regret is taken as a metric to evaluate the performance of the proposed algorithm. It is analytically shown that the dynamic regret can be bounded by the sum of agents' individual regrets, each of which grows sub-linearly with its active period length under locally diminishing step-sizes and a slowly changing environment. Moreover, to quantify the impact of agents' joining and leaving on algorithm performance, the regret is analyzed in the case of agent replacement, wherein some of the agents are replaced by new ones though the number of agents remains constant over time. Additionally, the algorithm is simplified in some special cases, resulting in a tighter regret bound. Finally, two numerical examples on target surrounding problems and price based energy management are given to verify the effectiveness of the proposed methods.

This paper proposes a novel secure frequency control approach for microgrids with online attack detection-based data scheduling and remedy mechanisms. First, taking into account the dynamics of photovoltaic (PV) units, energy storage systems (ESSs) and electric vehicle (EV) aggregators with state of charge (SOC)-dependent charging/discharging characteristics, a generic frequency control system model of PV-ESS-EV microgrids is established. Second, a two-step state estimation method involving a one-step-ahead state predictor and a measurement-update-based state estimator is proposed to estimate the microgrid system state, generating optimal prediction and estimation ellipsoids enclosing the true states during the whole operation time. Third, an attack detection scheme is provided for online detection of potential attacks by calculating the intersection over union (IOU) of the two generated ellipsoids. Upon successful attack detection, novel active data scheduling and two-stage data remedy mechanisms are derived for responsively reducing the negative effects of adversarial attacks. Then, a co-design approach, dependent on time-varying IOU signals, to the secure frequency controller and the scheduling mechanism parameters is presented. Finally, the efficacy and merits of the proposed frequency control strategy with attack detection-based data scheduling and remedy are verified through several case studies.

This paper formulates a new distributed Nash equilibrium seeking problem with a dynamic set of players in which players are allowed to join and leave the network in a free manner during the decision-making process. To accommodate the dynamic joining and leaving behaviors of the players, a status estimation mechanism, which is capable of estimating in a finite time whether the players are active or inactive, is introduced. Based on the status estimation mechanism, a gradient play based algorithm is developed for distributed Nash equilibrium seeking in the dynamic environment. It is shown that under strongly connected communication graphs, players' actions are convergent to a small neighborhood of the new Nash equilibrium linearly every time the player set changes. Moreover, the convergence accuracy and convergence rate can be adjusted by suitably tuning the step-size. To cover more general communication scenarios, strongly connected graphs are further relaxed to be B-jointly connected graphs, under which the convergence properties of the proposed algorithm are analytically studied. Furthermore, the upper bound of the average tracking error is quantified to evaluate the dynamic performance of the proposed algorithm. In the last, a simulation study on energy consumption games is given to verify the effectiveness of the proposed algorithm. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper considers differentially private distributed Nash equilibrium seeking for aggregative games under an event-triggered mechanism. By incorporating an eventtriggered scheme with a privacy protected aggregate estimation mechanism, a distributed Nash equilibrium seeking strategy is proposed. In the proposed strategy, the privacy protection is achieved by using Laplace noises to mask the information exchanged among the players, in which the transmission instants are determined by an event-triggered mechanism. By gradually weakening inter-player interactions, the proposed strategy ensures that players' actions can be driven exactly to the Nash equilibrium while ensuring rigorous $\epsilon$-differential privacy with reduced communication costs. The effectiveness of the proposed strategy is verified using a numerical example.

In games involving nonlinear coupled-dynamic systems, players' states are influenced by each others. In such problems, the state-based objective function's partial gradient with respect to input for each player is challenging to acquire explicitly, and existing equilibrium-seeking algorithms fail to achieve (i) zeroth-order partial gradient estimation and (ii) fully distributed implementation. To address this challenge, this paper proposes a novel distributed two-stage decoupled equilibrium-seeking algorithm framework. Each player estimates local partial gradients by communicating with neighbors and constructing local optimization models. Theoretical analysis of performance of proposed algorithm is provided. Experimental results in electricity markets validate its practical feasibility and effectiveness. (c) 2026 Published by Elsevier Ltd.

In this paper, we consider the Nash equilibrium (NE) seeking problem for a class of high-order uncertain nonlinear multi-agent systems (MASs) under event-triggered communication (ETC). To address this problem, a hierarchical control method is proposed, which includes a fully distributed NE seeking algorithm layer, a seeking variable performance improvement layer, and a decentralized tracking control layer. Specifically, an event-triggered fully distributed NE seeking algorithm is proposed to achieve the NE seeking objective in the fully distributed NE seeking algorithm layer, which composes a gradient term and a consensus term to drive the variable to its N. To enable this approach to accommodate high-order nonlinear MASs, an improved adaptive NE variable generator with the form of lower triangular is proposed in the seeking variable performance improvement layer. Driven by the newly designed variable, an adaptive controller is designed using the backstepping technique in the decentralized tracking control layer. Compared with the existing NE methods, the developed method is applicable to a wider class of high-order uncertain nonlinear MASs under distributed ETC. Finally, a simulation example is given to demonstrate the efficacy of our developed method. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper investigates distributed platooning control of connected and autonomous vehicles (CAVs) under practical spacing policies with vehicular data privacy protection. A practical spacing policy is first designed to adaptively regulate inter-vehicle distances by incorporating real-time traffic flow variations, road friction conditions, and safety distance requirements. The resulting spacing-policy-induced uncertainties are then addressed through an augmented system-based estimation approach. To safeguard vehicular privacy, a differential privacy-enhanced encoding-decoding mechanism is developed, enabling protection of both initial vehicular state information and broadcast vehicular data against eavesdropping attacks. Upon that, a co-design strategy is further proposed to jointly synthesize the privacy protection scheme, estimator, and distributed controller, ensuring reliable platoon tracking performance with data confidentiality. Finally, comparative case studies are conducted to validate the enhanced platoon tracking stability, driving safety, and privacy preservation in realistic traffic scenarios.

This paper studies a distributed online optimization problem over partially Free-In and Free-Out (FIFO) networks, in which a set of unfixed agents cooperate to minimize the sum of a group of time-varying functions over a time horizon. To be specific, the agents are divided into static agents and dynamic agents. The static agents are those who remain in the network during the whole time horizon, while the dynamic agents are allowed to join and leave the network freely. Based on the dual averaging technique, two novel distributed algorithms are developed to address the distributed optimization problem in such a dynamic environment. In the case where agents can distinguish whether their out-neighbors are dynamic agents or static agents, a weighting matrix based algorithm is developed. In the case where the identities of out-neighbors are unavailable, a gradient-storage based algorithm is developed, which has higher communication and local storage requirements than the weighting matrix based algorithm. By assuming that each dynamic agent has at least one in-neighbor and one out-neighbor who are static agents, the running average regret is shown to be upper bounded by $\mathcal {O}(\frac{1}{\sqrt{\mathit {T}}})$ under suitable step-sizes, where $\mathit {T}$ is the time horizon. A simulation study on binary classification is given to verify the effectiveness of the developed algorithms. In addition, a numerical example is used to highlight the advantages of the proposed methods over the distributed sub-gradient decent method.

A privacy-preserving incentive mechanism to maximize the weighted social welfare for pseudo-gradient-based noncooperative dynamical systems is developed. Sensitive information that could reveal the payoff functions is referred to as the privacy of agents. In the proposed approach, at least half of the agents (named as nonessential agents) can provide some nonsensitive information in addition to their state values, enabling the system manager to influence agents' dynamical decisions by imposing taxes or offering subsidies with a sustainable budget constraint. To protect the privacy of all agents, our approach shifts the stationary points of the nonessential agents and hence indirectly moves the Nash equilibrium of the entire system. By constructing the additional nonsensitive information from nonessential agents as the timewise information of stationary-point mappings related to certain secret matrices, sufficient conditions are derived where the Nash equilibrium is shifted toward the target state achieving Pareto efficiency under the incentive mechanism. We also provide some stability conditions to ensure that agents' state converges to the target state under a sustainable budget for the given initial state. Several numerical examples are used to illustrate the efficacy of our results.

This paper considers a network of distributed energy resources (DERs) that intend to optimize their power generation in an open price-based energy management system. Specifically, in the considered problem, the cost function of each DER, depends on not only its own decision variable but also an aggregate of all DERs’ energy decision variables. The DERs are cooperative to minimize their total cost of power generations, thus forming a distributed aggregative optimization problem. In addition, it is considered that the DERs are allowed to frequently arrive in and leave the system, which brings great difficulties in the algorithm design and analysis. To address this problem, a novel distributed optimization algorithm is proposed, by integrating gradient descent algorithms with a multi-level storage-based consensus mechanism. In addition, the performance of the proposed algorithm is evaluated by the dynamic regret, which quantifies the sum of all DERs’ loss between the actual cost and real time optimal cost during its active period length. It is shown that the upper bound of the all active DERs’ dynamic regret over active period length grows sublinearly under diminishing stepsizes. Finally, a numerical simulation is provided to verify the effectiveness of the proposed method.

This article is concerned with game-based distributed least-distance formation tracking with a moving target. More specifically, the agents aim to form a desired shape while enclosing a moving target with a least overall distance. Different from most of the existing results on formation tracking, the key challenge for least-distance formation tracking under consideration is that the distances between the agents and the target are not required to be predefined but are autonomously adjusted by solving a distance optimization problem. In particular, with the target being moving, the distance optimization problem is time-varying. To address this issue, a game-based formulation of distributed least-distance formation tracking is first presented, where first- and second-order integrator-type agents are taken into account, respectively. Then, under the scenario in which all the agents know the position of the moving target, a new distributed Nash equilibrium seeking strategy that can be utilized to achieve distributed least-distance formation is constructed based on gradient search algorithms, regularization techniques and consensus protocols. Moreover, when it is further considered that only a subset of the agents can access the information of the moving target, a leader-following tracking protocol is introduced to estimate unknown information distributively. It is theoretically shown that the proposed strategies can globally steer the agents to form a desired shape while minimizing the overall distance to the moving target. Finally, the effectiveness of the established strategies is verified by, respectively, conducting simulation studies for a network of vehicles and quadrotors.

This article proposes a new aggregative game model over free-in and free-out networks, in which each player can freely join and leave the network at its timing and aims to minimize its total cost during its active period. To enable the players to make self-beneficial decisions in such a dynamic environment, it is assumed that the players can locally store and exchange the historical action information. Based on the stored information, a distributed strategy is established, in which each player updates its action by a dual averaging method. In order to prevent excessive storage requirements, a storage mechanism is designed so that only the information generated from a certain historical time horizon is retained. The performance of the proposed strategy is evaluated by the static regret, which quantifies player's loss between the actual cost and the cost with the fixed best response during its active period. It is shown that the upper bound of each player's static regret grows sublinearly under local diminishing step sizes, indicating that the proposed strategy performs as well as choosing the best stationary action in hindsight for the long-term players. Finally, a simulation study on energy consumption games is given to verify the effectiveness of the developed methods.