#### Abstract

Due to the complexity of the hybrid powertrain, the control is highly involved to improve the collaborations of the different components. For the specific powertrain, the components' sizing just gives the possibility to propel the vehicle and the control will realize the function of the propulsion. Definitely the components' sizing also gives the constraints to the control design, which cause a close coupling between the sizing and control strategy design. This paper presents a parametric study focused on sizing of the powertrain components and optimization of the power split between the engine and electric motor for minimizing the fuel consumption. A framework is put forward to accomplish the optimal sizing and control design for a heavy parallel pre-AMT hybrid truck under the natural driving schedule. The iterative plant-controller combined optimization methodology is adopted to optimize the key parameters of the plant and control strategy simultaneously. A scalable powertrain model based on a bilevel optimization framework is built. Dynamic programming is applied to find the optimal control in the inner loop with a prescribed cycle. The parameters are optimized in the outer loop. The results are analysed and the optimal sizing and control strategy are achieved simultaneously.

#### 1. Introduction

The parallel power-train is one of the most effective configurations for hybrid electric vehicles (HEVs). The benefits of the parallel power-train result from its ability to drive with the engine or the electric motor only, or with both. How to minimize the fuel consumption of this type of HEV is presently quite hot in the academic community. Several energy management strategies have been studied or implemented in the literatures [1–5]. Sciarretta and Guzzella [6] suggested that HEV energy control strategy can be mainly categorized into four groups—the numerical optimization method, the analytic optimization method, the equivalent consumption minimization strategy, and the heuristic strategy. Dynamic programming (DP) is a numerical method for solving multi-stage decision-making problems and has been widely applied to explore the possible maximum fuel saving for the parallel HEVs [7, 8]. However, an optimal control strategy with the inappropriate component sizing could not guarantee the best fuel economy. It means that component sizing should be studied along with power management strategy to acquire the optimal performance. Hence, the combined optimization problem of the power management and component sizing for HEV is important. The combined plant/controller optimization problem has been researched a lot. For example, [9] discussed several implementations for the combined optimization strategy: the sequential, iterative, bilevel, and simultaneous manners, in which bilevel form was most commonly used [10]. Wu et al. [11, 12] optimized the components’ sizes and rule-based control strategy parameters for a hybrid electric vehicle. The highly accurate models were considered in the bilevel framework in [13]. A parameterized powertrain model and the near-optimal controller constituted a combined optimization problem for a fuel cell hybrid vehicle [14]. However, due to the near-optimal controller, the vehicle fuel saving was a bit unsatisfactory. Delphine et al. built a scalable powertrain model to form an integrated optimization problem, in which the outer loop chose the battery capacity, maximum torque of engine and motor as the variables, while dynamic programming was applied to find the optimal control strategy in the inner loop. Each simulation adjusted merely one parameter while keeping the remainder fixed [15]. Therefore, the coupling effects among component parameters were neglected.

In this study, the combined power management and sizing optimization problem for a heavy parallel hybrid electric truck is formulated and solved in a bilevel manner. The paper starts from the power train modelling, including the engine, the motor, the battery, and the transmission. Through the bilevel framework, a scalable vehicle model is developed and integrated in the optimal design process. DP is applied for the power management in the inner loop and the main parameters of the components are optimized simultaneously in the outer loop. The coupling among the component parameters is studied and the considerable fuel economy improvements are achieved.

#### 2. Vehicle Model

##### 2.1. Vehicle Configuration

The baseline vehicle is shown in Figure 1. The hybrid electric truck is a pre-transmission parallel HEV. The engine is connected to an automatic clutch, and then to the transmission. The parameters of this vehicle are given in Table 1.

##### 2.2. Model Simplification

It is highly desirable to perform the extensive simulations for HEVs with the different component configurations at the preliminary system design and optimization. It also means that the scalable model is in great demand at that stage. To avoid the dependence on the specific efficiency maps, a universal representation of the internal combustion engine based on the Willans line concept has been adopted [16]. Considering the complexity of the combined optimization, a simplified scalable motor model is also built later. Those models only consider the dynamic effects related to the low frequency power flows. The transient phenomena, such as chemical reactions in the battery and electric dynamics in the motor, are ignored. Due to the fact that the computation cost increases exponentially as the number of state increases, only the gear number and SOC are chosen to be the system states.

*(1) Engine Modeling*

The mean effective pressure and the mean piston speed are used to describe the engine power and the operating condition. The following three normalizations are used to define the engine efficiency by avoiding the quantities which depends on the engine size [17]:
where is the engine’s displaced volume, is the stoke, is the fuel mass flow rate, and is the fuel low heating value. is the engine effective torque, is the engine angular speed, and can be interpreted as an available mean pressure. When the energy converting efficiency is considered, the following exist:
where is the engine efficiency, is the thermodynamic efficiency, and is the available torque that would be generated by engine if all the chemical energy were converted into mechanical form. is the inner loss. Associating (2.1) and (2.2), a dimensionless definition of the engine efficiency can be acquired:
The two parameters and are the functions of the engine speed and load. The following parameterizations have been experimentally validated on the different engines [18]:
The coefficients, ,and , remain unchanged for the different engines in the same family and are obtained through the bench test and parameter identification. Hence the actual engine behavior from the same family is defined by the two size parameters, the swept volume and the piston stroke . Figure 2 compares the engine model with the actual data collected from the bench experiments for a prototype 7.0 L compression-ignition engine.

*(2) Motor Modeling*

The motor is modeled based on the experimental data. The motor efficiency is considered as a constant because of its high average efficiency in its feasible working area. Due to the battery power and the motor torque limits, the final motor torque becomes
where is the requested motor torque. and are the maximum motor torques in the motoring and charging modes, respectively. and are the torque bounds due to the battery current limits in the discharging and charging modes.

*(3) Battery Modeling*

The thermal-temperature effects and transients are ignored. SOC is calculated by
where the internal resistance and the open circuit voltage are functions of the battery SOC, obtained through the bench test. is the maximum battery charge, is the terminal resistance, and is the efficiency of the motor.

*(4) Driveline Modeling*

The driveline is defined as the system from the transmission input shaft to wheels. Assuming perfect clutches and gear shifting, the following equations describe the transmission and final drive gear models:
where is the transmission gear ratio, is the final drive gear ratio, and are the transmission and final drive efficiency, respectively. and are the transmission input torque and output torque, respectively. is the transmission viscous-loss coefficient, is the transmission input speed, and is the wheel speed.

The gear-shifting sequence of the AMT is modeled as a discrete dynamic system:
where the state is the gear number, and the control to the transmission is constrained to take on the values of −1, 0, and 1, representing down shifting, sustaining, and upshifting, respectively.

*(5) Vehicle Dynamics*

It is a common practice that only the vehicle longitudinal dynamics is considered. The longitudinal vehicle dynamics is modeled as a point-mass:
where is the friction brake torque, and are the rolling resistance force, and the aerodynamic drag force, and is the dynamic tire radius. is the effective mass of the vehicle, and is the equivalent inertia of the rotating components in the vehicle.

#### 3. Combined Optimization Problem Formulation

##### 3.1. Combined Optimization Framework

Given the particular system parameters, DP can be used to find the optimal control theoretically subject to some constraints under a specific driving schedule. When the system parameters vary in the feasible scope, DP is iteratively applied. The optimal combination of the parameters and control will be identified simultaneously. The bilevel combined plant/controller optimization is adopted, consisting of two nested optimization loops. The outer loop evaluates the system parameters. The inner loop generates the optimal control strategy for the parameters selected by the outer loop. These two loops form the integrated plant/controller optimization, which guarantees the global optimal design for the system parameters and control strategy. The combined optimization problem is complicated, due to the interaction between system parameters and control optimization, and computationally expensive due to the bilevel iterative search process. In order to improve the computational efficiency, once the constraints in the inner loop are violated, the current search stops, and the current cost will be set to a huge infeasible value. The flow chart of the bilevel combined optimization process is shown in Figure 3.

##### 3.2. The Scaled Model and Optimization Problem Formulation

The scaled models are needed to parameterize the system conveniently in the optimization. The scope of the motor torque, the motor speed, the motor power, the engine volume, the cylinder stoke, the battery numbers, and the capacity of battery are scaled by mot_tor, mot_spd, motorp, , , bat_num, and bat_ah,respectively. The final drive ratio , varying within a certain range without a scale enlarging, is one of the design parameters. The component parameters are described as follows: where and are the baseline battery capacity and the baseline number of the battery pack, is the baseline open circuit voltage of battery pack as a function of SOC. and are the baseline internal resistance and terminal resistance. , , and are the baseline parameters of the motor, while and are those of the engine. The baseline parameters are listed in Table 1. The variables in the left hand of the equations are the scaled parameters that need to be transferred to the inner loop.

The degree of hybridization (DOH) is often adopted to measure the relative contributions of the primary and second power sources. As to the parallel hybrid electric vehicles, the engine is often the primary power source and the battery the secondary power source. The DOH is constrained to be within [0, 0.4] and calculated by where is the maximum power that the motor offers, and is the maximum power that the engine provides. The combined optimal problem is formulated with all the constraints by subject to where represents the dynamics (2.1)–(2.9). The dynamic performance should be limited in the constraints when both the engine and motor propel the car. The constraints on the scaled parameters constitute the design space of the component sizing optimization.

#### 4. Algorithms and Methods

Design of experiments (DOE) technique is first applied to explore the response map in all the feasible design space based on Optimal Latin Hypercube sampling. Then the Nonlinear Programming by Quadratic Lagrangian (NLPQL) algorithm is applied to obtain the global optimal solution [19]. The group of parameters derived from DOE is optimal among the randomly selected points and will be the initial design point for NLPQL algorithm which can build a quadratic approximation. The quadratic programming problem is iteratively solved to find an improved solution until the final convergence to the optimum design.

Dynamic Programming (DP) is a powerful tool for solving optimization problems due to its guaranteed global optimality even for nonlinear dynamic systems with constraints. For a given driving cycle, DP can obtain the optimal operating strategy minimizing fuel consumption.

For maximizing the fuel saving of HEV, the cost function to be minimized has the following form: where is the instantaneous cost of the fuel use. The vehicle drivability is constrained by to avoid excessive shifting, in which is a positive weighting factor. A terminal constraint on SOC, represented by , is imposed on the cost function due to the charge-sustaining strategy. During the optimization, it is necessary to enforce the following inequality constraints to ensure safe and smooth operation for the engine, the battery, and the motor: where is the engine speed, SOC is the battery state of charge. and are selected to be 0.4 and 0.8, respectively. is the engine torque, and is the motor torque. A generic DP algorithm is implemented in MATLAB and applied to solve the above optimal control problem [20, 21].

#### 5. Simulations and Results

The heavy-duty vehicle driving schedule used to evaluate the fuel economy of the hybrid electric truck is shown in Figure 4.

The Pareto figure indicating the influence of the various factors on the fuel consumption is shown in Figure 5. It is determined by ordering the scaled and normalized coefficients of a standard least-squares second-order polynomial fit to the contribution to the fuel consumption from the different parameters. It is evident that motorp, , and individually have a significant effect on the fuel consumption. These three parameters represent the motor’s maximum power, the engine’s maximum power, and the final drive ratio, respectively. However, the interaction between the maximum motor power and the engine volume has the largest impact on the fuel consumption. The effects of the battery capacity on the fuel consumption are not as significant as other parameters; the percentage is less than 3%. Therefore, the battery supplying enough power for the motor can be chosen based on the cost effectiveness.

The specific influences on the fuel economy from the power sizing of the engine and the motor are shown in Figure 6. Note that alteration of the engine volume brings the change of the engine maximum power. It can also be concluded that the fuel consumption does not decrease as the engine size reduces or the motor size increases. Both of them should be chosen within a specific range in order to obtain the impressive fuel economy.

The parameters of the three typical groups with the different components’ sizes are listed in Table 2. The second and third group only differs in the final drive ratio and the third one has the same final ratio with the first one. The three groups of parameters are marked in Figure 7. It may allow the conclusion that the final drive ratio should be selected within a limited range, roughly between 3 and 4, slightly smaller than the initial value, to keep the good fuel economy, regardless of the DOH. The improper selection of the final drive ratio can lead to the increasing fuel consumption despite the optimal control strategy.

The engine working area and the gear shifting of the first and second group is shown in Figures 8, 9, 10, and 11. The second group with a smaller engine has a fundamentally different gear shifting from the first one. It is easy to extract the shifting rule from Figure 9, whereas difficult to obtain a shifting line for the second group because there is no apparent boundary between neighboring gears in Figure 11. The improper selection of the final drive ratio will result in low efficiency working area for the engine more possibly and could not be compensated by optimizing gear shifting and power distribution.

It is clear that the component parameters can affect HEV fuel economy directly. Sometimes a slight parameter discrepancy may lead to the considerable change of the fuel consumption. It emphasizes that the component sizing of HEV should be designed with a great cautiousness.

The optimal and initial parameters are listed in Table 3. The battery capacity decreases to 30 Ah from the original value, 60 Ah, although its voltage increases a bit. The final ratio decreases to 3.3 from the original value 4.769. Although the motor power is decreased, the motor max torque is found to increase by 63% to meet the performance constraints. Around 9% improvement is observed in the fuel economy through the combined optimization. The feasibility of the components in the actual engineering applications, however, needs more investigation in the view of the reliability and cost effectiveness.

#### 6. Conclusion

A bilevel optimization problem for the combined component sizing and power distribution of a heavy hybrid electric truck is formulated and solved. DOE and NPQRL algorithms are applied to find the optimal component parameters in the outer loop, while DP is used to find the optimal energy strategy in the inner loop. Simulation results show that the complex relationships between the component sizes and fuel consumption can be efficiently analyzed by solving the combined optimization problem. The law extracted from the optimization results can provide the suggestions for the actual hybrid vehicle system optimization and control. The results also indicate that the comprehensive bilevel optimization framework can facilitate the enhancement of HEV fuel economy, and the components sizing is as important as the control strategy.

#### Acknowledgments

This research is financially supported by China Natural Science Funding Project (50905015), China 863 High Technology Project (2011AA11A223), and China University Discipline Talent Introduction Program (B12022).