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Aggregate load modeling is of fundamental importance for systematic analysis and design of various demand response strategies. Instead of keeping track of the trajectories of individual loads, the aggregate modeling problem focuses on characterizing the density evolution of the load population. Most existing models are only applicable to thermostatically controlled loads (TCLs) with first-order linear dynamics. This paper develops a unified aggregate modeling approach that can be used for general TCLs as well as deferrable loads. We propose a deterministic hybrid system model to describe individual load dynamics under demand response rules, and develop a general stochastic hybrid system (SHS) model to capture the population dynamics. We also derive a set of partial differential equations (PDEs) that govern the probability density evolution of the SHS. Our results cannot be obtained using the existing SHS tools in the literature as the proposed SHS model involves both random and deterministic switchings with general switching surfaces in multidimensional domains. The derived PDE model includes many existing aggregate load modeling results as special cases and can be used in many other realistic modeling scenarios that have not been studied in the literature. [ABSTRACT FROM AUTHOR]

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively. [ABSTRACT FROM AUTHOR]

Within the realm of smart buildings and smart cities, dynamic response management is playing an ever-increasing role, thus attracting the attention of scientists from different disciplines. Dynamic demand response management involves a set of operations aiming at decentralizing the control of loads in large and complex power networks. Each single appliance is fully responsive and readjusts its energy demand to the overall network load. A main issue is related to mains frequency oscillations resulting from an unbalance between supply and demand. In a nutshell, this paper contributes to the topic by equipping each consumer with strategic insight. In particular, we highlight three main contributions and a few other minor contributions. First, we design a mean-field game for a population of thermostatically controlled loads, study the mean-field equilibrium for the deterministic mean-field game, and investigate on asymptotic stability for the microscopic dynamics. Second, we extend the analysis and design to uncertain models, which involve both stochastic or deterministic disturbances. This leads to robust mean-field equilibrium strategies guaranteeing stochastic and worst-case stability, respectively. Minor contributions involve the use of stochastic control strategies rather than deterministic and some numerical studies illustrating the efficacy of the proposed strategies. [ABSTRACT FROM PUBLISHER]

This paper considers two classes of large population stochastic differential games connected to optimal and robust decentralized control of large-scale multiagent systems. The first problem (P1) is one where each agent minimizes an exponentiated cost function, capturing risk-sensitive behavior, whereas in the second problem (P2 ) each agent minimizes a worst-case risk-neutral cost function, where the “worst case” stems from the presence of an adversary entering each agent’s dynamics characterized by a stochastic differential equation. In both problems, the individual agents are coupled through the mean field term included in each agent’s cost function, which captures the average or mass behavior of the agents. We solve both P1 and P2 via mean field game theory. Specifically, we first solve a generic risk-sensitive optimal control problem and a generic stochastic zero-sum differential game, where the corresponding optimal controllers are applied by each agent to construct the mean field systems of P1 and P2. We then characterize an approximated mass behavior effect on an individual agent via a fixed-point analysis of the mean field system. For each problem, P1 and P2, we show that the approximated mass behavior is in fact the best estimate of the actual mass behavior in various senses as the population size, $N$, goes to infinity. Moreover, we show that for finite $N$, there exist $\epsilon$- Nash equilibria for both P1 and P2, where the corresponding individual Nash strategies are decentralized in terms of local state information and the approximated mass behavior. We also show that $\epsilon$ can be taken to be arbitrarily small when $N$ is sufficiently large. We show that the $\epsilon$- Nash equilibria of P1 and P2 are partially equivalent in the sense that the individual Nash strategies share identical control laws, but the approximated mass behaviors for P1 and P2 are different, since in P2, the mass behavior is also affected by the associated worst-case disturbance. Finally, we prove that the Nash equilibria for P1 and P2 both feature robustness, and as the parameter characterizing this robustness becomes infinite, the two Nash equilibria become identical and equivalent to that of the risk-neutral case, as in the one-agent risk-sensitive and robust control theory. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton –Jacobi– Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean–Vlasov equations, Fokker–Planck–Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean–Vlasov dynamics. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to synthesize initial mean consensus behavior of a set of agents from the fundamental optimization principles of i) stochastic dynamic games, and ii) optimal control. In the stochastic dynamic game model each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents. In this formulation we derive the limiting infinite population mean field equation system and explicitly compute its unique solution. The resulting mean field (MF) control strategies drive each agent to track the overall population's initial state distribution mean, and by applying these decentralized strategies, any finite population system reaches mean consensus asymptotically as time goes to infinity. Furthermore, these control laws possess an \varepsilonN-Nash equilibrium property where \varepsilonN goes to zero as the population size N goes to infinity. Finally, the analysis is extended to the case of random mean field couplings. In the social cooperative formulation the basic objective is to minimize a social cost as the sum of the individual cost functions containing mean field coupling. In this formulation we show that for any individual agent the decentralized mean field social (MF Social) control strategy is the same as the mean field Nash (MF Nash) equilibrium strategy. Hence $MF-Nash\ Controls\ UNash^N=\rm MF-Social\ Controls\ USoc^N.$On the other hand, the solution to the centralized LQR optimal control formulation yields the Standard Consensus (SC) algorithm whenever the graph representing the corresponding topology of the network is Completely Connected (CC). Hence $Cent.\ LQR\ Controls\ UCent^N=\rm SC-CC\ Controls\ USC^N.$Moreover, a system with centralized control laws reaches consensus on the initial state distribution mean as time and population size N go to infinity. Hence, asymptotically in time $\eqalignno&MF-Nash\ Controls\ UNash^N\!=\!\,\rm MF-Social\ Controls\ USoc^N\cr&\!=\!\,\rm Cent.\ \rm LQR\ Controls\ UCent^\infty \!=\! \,\rm SC-CC\ Controls\ USC^\infty.$Finally, the analysis is extended to the long time average (LTA) cost functions case. [ABSTRACT FROM AUTHOR]