Dynamic programming for a Markov-switching jump–diffusion

We consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a Markov-switching jump–diffusion stochastic differential equation. Our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems. More specifically, we provide a detailed proof of Bellman’s optimality principle (or dynamic programming principle) and obtain the corresponding Hamilton–Jacobi–Belman equation, which turns out to be a partial integro-differential equation due to the extra terms arising from the Lévy process and the Markov process. As an application of our results, we study a finite horizon consumption– investment problem for a jump–diffusion financial market consisting of one risk-free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state Markov process. We provide a detailed study of the optimal strategies for this problem, for the economically relevant families of power utilities and logarithmic utilities.

Distributed Energy Resources Management using Plug-in Hybrid Electric Vehicles as a Fuel-Shifting Demand Response Resource

In the smart grids context, distributed energy resources management plays an important role in the power systems’ operation. Battery electric vehicles and plug-in hybrid electric vehicles should be important resources in the future distribution networks operation. Therefore, it is important to develop adequate methodologies to schedule the electric vehicles’ charge and discharge processes, avoiding network congestions and providing ancillary services. This paper proposes the participation of plug-in hybrid electric vehicles in fuel shifting demand response programs. Two services are proposed, namely the fuel shifting and the fuel discharging. The fuel shifting program consists in replacing the electric energy by fossil fuels in plug-in hybrid electric vehicles daily trips, and the fuel discharge program consists in use of their internal combustion engine to generate electricity injecting into the network. These programs are included in an energy resources management algorithm which integrates the management of other resources. The paper presents a case study considering a 37-bus distribution network with 25 distributed generators, 1908 consumers, and 2430 plug-in vehicles. Two scenarios are tested, namely a scenario with high photovoltaic generation, and a scenario without photovoltaic generation. A sensitivity analyses is performed in order to evaluate when each energy resource is required.