Analytic Solutions to the Dynamic Programming Subproblem in Hybrid Vehicle Energy Management

Abstract

The computationally demanding Dynamic Programming (DP) algorithm is frequently used in academic research to solve the energy management problem of an Hybrid Electric Vehicle (HEV). This paper is focused exclusively on how the computational demand of such a computation can be reduced. The main idea is to use a local approximation of the gridded cost-to-go and derive an analytic solution for the optimal torque split decision at each point in the time and state grid. Thereby it is not necessary to quantize the torque split and identify the optimal decision by interpolating in the cost-to-go. Two different approximations of the cost-to-go are considered in the paper: i) a local linear approximation, and ii) a quadratic spline approximation. The results indicate that computation time can be reduced by orders of magnitude with only a slight degradation in simulated fuel economy. Furthermore, with a spline approximated cost-to-go it is also possible to significantly reduce the memory storage requirements. A parallel Plug-in HEV is considered in the paper but the method is also applicable to an HEV. I. INTRODUCTIONDuring the last fifteen years significant attention has been given to the topic of optimal energy management for Hybrid Electric Vehicles (HEVs) and Plug-in HEVs (PHEVs). The task is non-trivial to solve since a priori information regarding the future driving conditions is needed. Furthermore, the plant model is generally non-linear with both continuous and integer decisions, i.e. torque split between engine/motor, choice of gear and engine on/off. Many different methods have been proposed for solving the energy management problem. Some examples are: rule based methods, Dynamic Programming (DP), convex optimization, and the Equivalent Consumption Minimization Strategy (ECMS) that is derived from the Pontryagin maximum principle. Refer to [1]-[3] for a review of different methods. This paper will focus exclusively on DP, which has been used in numerous studies [4]- [15]. The main advantage with DP is that it is a very versatile algorithm that can handle a wide range of problem formulations. It provides the global optimal solution, which cannot be guaranteed by a rule based method. In contrast to a convex optimization formulation, DP can handle integer decision variables without any need for approximations or iterative methods. Moreover, an important advantage compared to an ECMS strategy is that state constraints can be treated in a more formal way.

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“…The fuel consumption is identical for both (3.54 l/100km). The dynamic programming (DP) algorithm has also been used to compute a solution [30]. It is denoted as "DP" in the Fig.…”

“…In the sequel, a piecewise expression is considered each time the coefficients Equation (28) can be rewritten to show that the Hamiltonian is a piecewise affine function of the single variable P , according to (29) where P S is a so-called switching function defined by (30).…”

A Novel Highly Integrated Hybrid Energy Storage System for Electric Propulsion and Smart Grid Applications

“…For this reason, another popular approach is determining BP and UM reference power profiles by minimizing suitable cost functions over a given time horizon. Hence, different optimal solving techniques can be used, such as model predictive control, mixed-integer/linear programming, nonlinear programming, and dynamic programming [52][53][54][55]. However, such solving techniques are generally complex to implement and quite time-demanding.…”

GRAB-ECO for Minimal Fuel Consumption Estimation of Parallel Hybrid Electric Vehicles

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