Finding Ultimate Limits of Performance for Hybrid Electric Vehicles

Hybrid electric vehicles are seen as a solution to improving fuel economy and reducing pollution emissions from automobiles. By recovering kinetic energy during braking and optimizing the engine operation to reduce fuel consumption and emissions, a hybrid vehicle can outperform a traditional vehicle. In designing a hybrid vehicle, the task of finding optimal component sizes and an appropriate control strategy is key to achieving maximum fuel economy. In this paper we introduce the application of convex optimization to hybrid vehicle optimization. This technique allows analysis of the propulsion systems capabilities independent of any specific control law. To illustrate this, we pose the problem of finding optimal engine operation in a pure series hybrid vehicle over a fixed drive cycle subject to a number of practical constraints including: • nonlinear fuel/power maps • min and max battery charge • battery efficiency • nonlinear vehicle dynamics and losses • drive train efficiency • engine slew rate limits We formulate the problem of optimizing fuel efficiency as a nonlinear convex optimization problem. This convex problem is then accurately approximated as a large linear program. As a result, we compute the globally minimum fuel consumption over the given drive cycle. This optimal solution is the lower limit of fuel consumption that any control law can achieve for the given drive cycle and vehicle. In fact, this result provides a means to evaluate a realizable control law's performance. We carry out a practical example using a spark ignition engine with lead acid (PbA) batteries. We close by discussing a number of extensions that can be done to improve the accuracy and versatility of these methods. Among these extensions are improvements in accuracy, optimization of emissions and extensions to other hybrid vehicle architectures. INTRODUCTION Two areas of significant importance in automotive engineering are improvement in fuel economy and reduction of emissions. Hybrid electric vehicles are seen as a means to accomplish these goals. The majority of vehicles in production today consist of an engine coupled to the road through a torque converter and a transmission with several fixed gear ratios. The transmission is controlled to select an optimal gear for the given load conditions. During braking, velocity is reduced by converting kinetic energy into heat. For the purposes of this introduction, it is convenient to consider two propulsion architectures: pure parallel and pure series hybrid vehicles. A parallel hybrid vehicle couples an engine to the road through a transmission. However, there is an electric motor that can be used to change the RPM and/or torque seen by the engine. In addition to modifying the RPM and/or torque, this motor can recover kinetic energy during braking and store it in a battery. By changing engine operating points and recovering kinetic energy, fuel economy and emissions can be improved. A series hybrid vehicle electrically couples the engine to the road. The propulsion system consists of an engine, a battery and an electric motor. The engine is a power source that is used to provide electrical power. The electrical power is used to recharge a battery or drive a motor. The motor propels the vehicle. This motor can also be used to recover kinetic energy during braking. For a given type of hybrid vehicle, there are three questions of central importance: • What are the important engine, battery and motor requirements? • When integrated into a vehicle, what is the best performance that can be achieved? • How closely does a control law approach this best performance? Answers to these questions can be found by solving three separate problems: • Solving for the maximum fuel economy that can be obtained for a fixed vehicle configuration on a fixed drive cycle independent of a control law. • Given a method to find maximum fuel economy, vary the vehicle component characteristics to find the optimal fuel economy. • Apply the selected control law to the system and determine the fuel consumption. Calculate the ratio between this control laws fuel consumption and the optimal value to give a metric for how close the control law comes to operating the vehicle at its maximum performance. There are many hybrid vehicle architectures[1]. For the sake of simplicity, a pure series hybrid was chosen for this study. However, the methods used for series hybrid vehicles can be extended to apply to other hybrid vehicle architectures. This study was restricted to minimizing fuel economy. This method can be extended to include emissions. DISCUSSION: FINDING THE MAXIMUM FUEL ECONOMY FOR A GIVEN VEHICLE There are many approaches that can be used to determine the maximum fuel economy that can be obtained by a particular vehicle over a particular drive cycle. One common approach is to select a control law and then optimize that control law for the system. Other techniques search through control law architectures and control parameters simultaneously. Since these techniques select a control law before beginning the optimization, the minimum fuel economy found is always a function of the control law. This leaves open the question of whether selection of a better control law could have resulted in better fuel economy. The approach presented here finds the minimal fuel consumption of the vehicle independent of any control law. Because a control law is not part of the optimization, the fuel economy found is the best possible. It is noncausal in that it finds the minimum fuel consumption using knowledge of future power demands and past power demands. Therefore it represents a limit of performance of a causal control law. Furthermore, since the problem is formulated as a convex problem and then a linear program, the minimum fuel consumption calculated is guaranteed to be the global minimum solution. The discussion that follows details: 1. The formulation of the fuel economy minimization problem as a convex problem. 2. The reduction of this convex problem to a linear program. 3. Solution of the linear program to find the minimum fuel consumption. DESCRIBING THE PROBLEM To solve for maximum fuel economy, a model of the series hybrid vehicle is used. To simplify the model, the following assumptions are made: • The voltage on the electrical bus is constant. Voltage droop and ripple can be ignored. • The relationship between power output from the engine and fuel consumption can be assumed to be a fixed relationship that is not affected by transients. • The batterys storage efficiency is constant. It does not change with state of charge or power levels. These simplifications are used to reduce the complexity of the resulting linear program and to maintain a problem description which is convex. These simplifications illustrate one of the challenges that arises in the application of convex analysis to engineering problems  finding a description of the problem which is convex.