Foresighted Demand Side Management

We consider a smart grid with an independent system operator (ISO), and distributed aggregators who have energy storage and purchase energy from the ISO to serve its customers. All the entities in the system are foresighted: each aggregator seeks to minimize its own long-term payments for energy purchase and operational costs of energy storage by deciding how much energy to buy from the ISO, and the ISO seeks to minimize the long-term total cost of the system (e.g. energy generation costs and the aggregators' costs) by dispatching the energy production among the generators. The decision making of the entities is complicated for two reasons. First, the information is decentralized: the ISO does not know the aggregators' states (i.e. their energy consumption requests from customers and the amount of energy in their storage), and each aggregator does not know the other aggregators' states or the ISO's state (i.e. the energy generation costs and the status of the transmission lines). Second, the coupling among the aggregators is unknown to them. Specifically, each aggregator's energy purchase affects the price, and hence the payments of the other aggregators. However, none of them knows how its decision influences the price because the price is determined by the ISO based on its state. We propose a design framework in which the ISO provides each aggregator with a conjectured future price, and each aggregator distributively minimizes its own long-term cost based on its conjectured price as well as its local information. The proposed framework can achieve the social optimum despite being decentralized and involving complex coupling among the various entities

Global stability of day-to-day dynamics for schedule-based Markovian transit assignment with boarding queues

Schedule-based transit assignment describes congestion in public transport services by modeling the interactions of passenger behavior in a time-space network built directly on a transit schedule. This study investigates the theoretical properties of scheduled-based Markovian transit assignment with boarding queues. When queues exist at a station, passenger boarding flows are loaded according to the residual vehicle capacity, which depends on the flows of passengers already on board with priority. An equilibrium problem is formulated under this nonseparable link cost structure as well as explicit capacity constraints. The network generalized extreme value (NGEV) model, a general class of additive random utility models with closed-form expression, is used to describe the path choice behavior of passengers. A set of formulations for the equilibrium problem is presented, including variational inequality and fixed-point problems, from which the day-to-day dynamics of passenger flows and costs are derived. It is shown that Lyapunov functions associated with the dynamics can be obtained and guarantee the desirable solution properties of existence, uniqueness, and global stability of the equilibria. In terms of dealing with stochastic equilibrium with explicit capacity constraints and non-separable link cost functions, the present theoretical analysis is a generalization of the existing day-to-day dynamics in the context of general traffic assignment.Comment: 26 pages, 3 figure

A route-swapping dynamical system and Lyapunov function for stochastic user equilibrium

An analysis of the continuous-time dynamics of a route-swap adjustment process is presented, which is a natural adaptation of that which was presented in Smith (1984) for deterministic choice problems, for a case in which drivers are assumed to make perceptual errors in their evaluations of travel cost, according to a Random Utility Model. We show that stationary points of this system are stochastic user equilibria. A Lyapnuov function is developed for this system under the assumption of monotone, continuously differentiable and bounded cost-flow functions and a logit-based decision rule, establishing convergence and stability of trajectories of such a dynamical system with respect to a stochastic user equilibrium solution

Stability of Intelligent Transportation Network Dynamics: A Daily Path Flow Adjustment considering Travel Time Differentiation

A theoretic formulation on how traffic time information distributed by ITS operations influences the trajectory of network flows is presented in this paper. The interactions between users and ITS operator are decomposed into three parts: (i) travel time induced path flow dynamics (PFDTT); (ii) demand induced path flow dynamics (PFDD); and (iii) predicted travel time dynamics for an origin-destination (OD) pair (PTTDOD). PFDTT describes the collective results of user’s daily route selection by pairwise comparison of path travel time provided by ITS services. The other two components, PTTDOD and PFDD, are concentrated on the evolutions of system variables which are predicted and observed, respectively, by ITS operators to act as a benchmark in guiding the target system towards an expected status faster. In addition to the delivered modelings, the stability theorem of the equilibrium solution in the sense of Lyapunov stability is also provided. A Lyapunov function is developed and employed to the proof of stability theorem to show the asymptotic behavior of the aimed system. The information of network flow dynamics plays a key role in traffic control policy-making. The evaluation of ITS-based strategies will not be reasonable without a well-established modeling of network flow evolutions

Game-theoretical control with continuous action sets

Motivated by the recent applications of game-theoretical learning techniques to the design of distributed control systems, we study a class of control problems that can be formulated as potential games with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems. The method employed is to analyse the learning process under study through a mean-field dynamical system that evolves in an infinite-dimensional function space (the space of probability distributions over the players' continuous controls). To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge to equilibrium in the case of potential games. These results combine to give a provably-convergent learning algorithm in which players do not need to keep track of the controls selected by the other agents.Comment: 19 page

A splitting rate model of traffic re-routeing and traffic control

Non peer reviewedPublisher PD

Route choice and traffic signal control: a study of the stability and instability of a new dynamical model of route choice and traffic signal control

This paper presents a novel idealised dynamical model of day to day traffic re-routeing (as traffic seeks cheaper routes) and proves a stability result for this dynamical model. (The dynamical model is based on swapping flow between paired alternative segments (these were introduced by Bar Gera (2010)) rather than between routes.) It is shown that under certain conditions the dynamical system enters a given connected set of approximate equilibria in a finite number of days or steps. This proof allows for saturation flows which act as potentially active flow constraints. The dynamical system involving paired alternative segment swaps is then combined with a novel green-time-swapping rule; this rule swaps green-time toward more pressurised signal stages. It is shown that if (i) the delay formulae have a simple form and (ii) the “pressure” formula fits the special control policy P0 (see Smith, 1979a, b), then the combined flow-swapping / green-time-swapping dynamical model also enters a given connected set of approximate consistent equilibria in a finite number of steps. Computational results confirm, in a simple network, the positive P0 result and also show, on the other hand, that such good behaviour may not arise if the equi-saturation control policy is utilized. The dynamical models described here do not represent blocking back effects

The Day-to-Day Dynamics of Route Choice

This paper reviews methods proposed for modelling the day-to-day dynamics of route choice, on an individual driver level. Extensions to within-day dynamics and choice of departure time are also discussed. A new variation on the approaches reviewed is also described. Simulation tests on a simple two-link network are used to illustrate the approach, and to investigate probabilistic counterparts of equilibrium uniqueness and stability. The long-term plan is for such a day-to-day varying demand-side model to be combined with a suitable microscopic supply-side model, thereby producing a new generation network model. The need for such a model - particularly in the context of assessing real-time transport strategies - has been identified in previous working papers

Fluctuations, response, and resonances in a simple atmospheric model

We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation–dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics