Optimal motion control for energy-aware electric vehicles

Abstract

We study two problems of optimally controlling how to accelerate and decelerate a non-ideal energy-aware electric vehicle so as to (a) maximize its cruising range and (b) minimize the traveling time to a specified destination under a limited battery constraint. Modeling an electric vehicle as a dynamic system, we adopt an electric vehicle power consumption model (EVPCM) and formulate two respective optimal motion control problems. Although the full solutions can only be obtained numerically, we propose approximate controller structures such that the original optimal control problems are transformed into nonlinear parametric optimization problems, which are much easier to solve. Numerical examples illustrate the solution structures and support their accuracy.

Introduction

The emergence of plug-in hybrid electric vehicles (HEV) and fully electric vehicles (EV) is motivated by the goals of reduced oil dependency and greenhouse gas emissions. However, both HEVs and EVs heavily rely on limited battery power, thus raising such issues as vehicle cruising ranges and accessibility to charging sources.

In the HEV literature, work has been focused on designing optimal strategies for power distribution between the electric motor and the combustion engine in order to minimize fuel consumption (Sciarretta et al., 2004, Salmasi, 2007). Moreover, efforts have been dedicated to establishing control-oriented models of the traction dynamics of the vehicles, which are the vital ingredients for active controllers to achieve desired accuracy and energy-efficiency. For example, a dynamic model of the power-train of HEVs is proposed in Powell, Bailey, and Cikanek (2002); Hori, Toyoda, and Tsuruoka (2002) address the traction control of an EV; and speed and acceleration controllers are studied for an energy-aware two-wheeled EV in Dardanelli et al. (2011).

On the other hand, the expanding number of HEV and EV fleets brings out new research issues related to insufficient power supplied to vehicles, allocation of limited charging stations and power balance of electric grids. From the vehicle side, a circuit-based battery is commonly used to represent the electricity source when considering the supervisory control of an HEV Guzzella and Sciarretta (2007). Sundstrom and Binding (2011) employed this model to optimally plan the charging behaviors of EV fleets in terms of grid power balancing. In order to minimize the waiting time for EV charging, a scheduling problem in a network of EVs and charging stations was studied in Qin and Zhang (2011). From the grid side, an optimization methodology of allocating the recharging infrastructure for EVs in an urban environment was developed in Gallego and Larrodeg (2011). More recently, a decentralized protocol for negotiating day-ahead charging schedules for EVs via pricing control was proposed in Gan, Topcu, and Low (2011) to fill the overnight electricity demand valley. However, despite the variety of research on HEV/EV energy-aware systems, there is little work investigating HEV/EV motion control from a power management perspective, which is mainly because the relationship between vehicle dynamics and power consumption is complicated. In Kuriyama, Yamamoto, and Miyatake (2010) an “Eco-Driving” technique based on dynamic programming is developed for EVs seeking an optimal speed profile to minimize the amount of total energy consumption under fixed origin and destination, running time and track condition. Assuming a complete knowledge of the upcoming traffic light timing via infrastructure to vehicle (I2V) communication, De Nunzio, Canudas de Wit, Moulin, and Domenico (2013) address the problem of energy-optimal velocity profiles in urban traffic network. Formulating this as an optimal control problem, an algorithm is proposed which provides a fast sub-optimal solution for the energy minimization problem. In Dib, Serroa, and Sciarretta (2011), a dynamic programming algorithm was developed to minimize energy consumption for an EV during a short trip by optimizing the speed trajectory. Two alternative formulations were proposed which are suitable for off-line and on-line implementation.

Closely related work is presented in Petit and Sciarretta (2011) and Dib, Chasse, and Sciarretta (2012) which aim to minimize energy consumption for an EV over a time and distance horizon. Formulating this as an optimal control problem (OCP) and doing Hamiltonian analysis, a two point boundary value problem (TPBVP) was obtained in Petit and Sciarretta (2011). Then, using the inversion-based methodology, a closed-form solution for a simple case when the aerodynamic friction and the road slope are ignored was proposed. In Dib et al. (2012) a similar problem is revisited in an urban environment where a trip is broken into several segments and the optimal control is formulated for each segment. Similar to Petit and Sciarretta (2011), a TPBVP was obtained by Hamiltonian analysis and a closed form solution for the optimal trajectories was derived when the road slope is constant and the aerodynamic friction is ignored. In the current paper, we discuss two optimization problems for an EV traveling through a highway: (a) cruising range maximization and (b) traveling time minimization. In both, we start by formulating the problem as an OCP with a free terminal time. Note that for the cruising range maximization problem, which is closer to the problems discussed in Petit and Sciarretta (2011) and Dib et al. (2012), there is no terminal condition for amount to travel. Next, we solve them numerically and derive the structure of the optimal controllers based on which we propose simpler nonlinear parametric optimization problems giving approximate solutions. Using this method we can obtain approximate solutions for more general cases relative to Petit and Sciarretta (2011) and Dib et al. (2012) where we can include aerodynamic friction and road slope.

An analytical power consumption model for an EV was proposed in Tanaka, Ashida, and Minami (2008), which presents a comprehensive relationship between velocity, acceleration and power consumption rate. Motivated by the model in Tanaka et al. (2008), in this paper we formulate two optimal motion control problems, i.e., a cruising range maximization problem and a vehicular traveling time minimization. Although the intricate state dynamics of the problems require resorting to numerical solutions, they do serve to formulate approximate solution structures so as to transform the original optimal control problems into simpler nonlinear parametric optimization problems. The approximate solutions possess a much simpler structure while preserving accuracy. This allows us to apply optimal motion planning to various interesting issues in EV-based systems, such as vehicular routing, charging station deployment and EV-to-smart-grid (V2G) charging scheduling, thus opening up a wide spectrum of research directions. In the area of optimal routing for electric vehicles, we have addressed several problems for a single EV with energy constraints as well as multi-vehicle routing problems considering traffic congestion effects on both traveling time and energy consumption on their paths (Wang et al., 2014, Pourazarm and Cassandras, 2014).

Our results are illustrated by examples based on data available for certain EV classes. We point out, however, that with EV technology rapidly evolving, the model data used for specific examples are likely to become obsolete. Thus, the main contribution of the paper is the formulation of the optimal control problems and the derivation of controller structures which remain valid regardless of EV model used.

The structure of the paper is as follows. In Section 2, we introduce the analytical power consumption model for EVs. In Section 3, we formulate an EV cruising range maximization problem based on a power consumption model with a prescribed initial energy. The explicit numerical solution, as well as an approximate one, is presented and the accuracy of the latter is verified. Based on the approximate solution structure, the optimal control problem is reduced to a nonlinear parametric optimization problem, which is easier to solve. Section 4 explores an EV traveling time minimization problem with the same procedure as Section 3. Finally, conclusions and further research directions are described in Section 5.