Performance Evaluation and Control Strategy Comparison of Supercapacitors for a Hybrid Electric Vehicle

Electrification of powertrain system is a great technical progress of traditional vehicle, leading to a significant reduction of fuel consumption and emission pollution. Energy storage system (ESS) normally consisting of batteries is a key component of an electric vehicle or hybrid electric vehicle. An ESS can recover braking energy during the regenerative braking process. Currently, lithium-ion batteries are the main energy storage device due to their high energy density. However, sometimes, a sudden large increase of operation current is required during acceleration or regenerative braking processes, which will jeopardize the operation life of batteries. A supercapacitor takes advantage of high power density and can tolerate large current in a short time. Application of supercapacitor in an ESS can reduce the peak current of batteries effectively, and the life time of batteries can be extended. Meanwhile, the braking energy can also be recovered sufficiently. Supercapacitors can be used solely in some hybrid electric vehicles. In this chapter, the application of supercapacitors in electric vehicles or hybrid electric vehicles is reviewed briefly. Then, the performance of a series hybrid transit bus, which uses a compressed natural gas engine and supercapacitors as power sources, is analyzed.


Introduction
A transit bus is a prime commuting tool for city residents, which consumes lots of fuel every year and produces a huge amount of poisonous emissions [1]. Electrification of transit bus is a good solution for these problems. Normally, the energy storage device consists of batteries [2]. The supercapacitor is a newly developed high-power electrochemical energy storage component [3,4]. Farkas and Bonert have predicted that a supercapacitor is a good solution for hybrid vehicles [5]. Recently, many researches were performed to study the characteristics of supercapacitors as the energy storage system (ESS) of a hybrid electric vehicle (HEV) [6,7]. velocity and has a huge computation load. Therefore, it is infeasible for a practical real-time vehicle system.
In this chapter, a series hybrid transit bus powered by a compressed natural gas (CNG) engine and supercapacitors is studied. The energy conversion characteristics of the designed series hybrid powertrain are analyzed using a mathematical model. First, two rule-based control strategies-the thermostatic control and the power follower control-are designed and compared. Later, the maximum potential of energy savings is estimated using an optimal control strategy based on dynamic programming. Finally, the operation characteristics of the series hybrid powertrain using the different control strategies are discussed. The results of this study provide a demonstration on how to design an ESS for an electric vehicle or hybrid electric vehicle from the systematic level.

System description
The designed series hybrid powertrain for a transit bus is described as follows. A CNG engine is directly connected with a permanent magnetic synchronous generator (PMSG). A high-voltage power line is connected to the generator and the ESS as well as a permanent magnetic synchronous motor (PMSM). The PMSM connected to the final drive is a specially designed low-speed and high-power PMSM. The engine is a Yuchai 6.5 L CNG engine whose rated power is 140 kW. Both the PMSG and the PMSM are developed by Jing-Jin Electric Technologies (Beijing) Co., Ltd. The PMSG is a low-speed and high-efficiency electric machine whose rated power is 135 kW. The rated power of the PMSM is slightly greater than that of the PMSG. These two high-power electric machines have already been applied to several different types of hybrid vehicles successfully. The ESS includes three parallel groups of supercapacitors, and each group consists of 13 units of Maxwell 48 V module connected in series. The total energy capacity of the ESS is 2.115 kWh, and the rated voltage is 624 V. To keep the supercapacitors from overdischarging, the minimum operation voltage of the ESS is set to 300 V.
The working principle of the designed series hybrid powertrain can be explained by an energy flow diagram shown in Figure 1. When the series hybrid transit bus is running, four different operation modes are defined according to the working conditions. Two modes are used for the driving conditions, and the other two modes are for the regenerative braking conditions. For the driving mode A, the CNG engine is running, and the APU together with the ESS supplies electric power to the motor. This driving mode is activated if the required driving power is high or the SOC is low. When the SOC is high and the required driving power is less than a certain value, the ESS provides electric energy to the motor alone as the driving mode B shows. If the series hybrid bus is braking, the regenerative energy output from the motor is supplied to the ESS. Meanwhile, the APU can be activated or deactivated denoted by the braking modes A and B, respectively.

Mathematical model
A mathematical model is established according to the working principle of the designed series hybrid powertrain. The corresponding parameters of the hybrid transit bus are listed in Table 1. The tractive force acting on a rear-wheel-driven two-axle vehicle can be determined according to the corresponding longitudinal dynamic equation expressed as The wheel and axle model first calculates the front axle load and the rear axle load according to the technical parameters of the series hybrid transit bus. Then, the tractive force coefficient of the rear tires can be determined. The slip of the tires can be modeled as a function of the tractive force coefficient. Finally, the angular speed of the rear tire can be obtained [29].
If the transit bus is operating at the regenerative braking process, the required braking force F b can be determined according to the deceleration of the vehicle. Then the regenerative braking force of the rear axle F br is obtained based on the following force distribution equation: The model of the final drive takes into account the friction loss T l0 , and the inertia of rotating parts J 0 and is expressed as where T w is the output torque of the final drive and T m and ω m are the input torque and speed of the final drive, respectively. The electric motor model determines the requested torque T mr according to the motor inertia J m and the output torque T m . Then the input power P m is calculated based on a two-dimensional (2D) lookup table measured from a motor test bench.
The mathematical model of the generator is similar to the motor. If the detailed physical mechanism of supercapacitors is studied, an electrochemical model or a high-order equivalent circuit model must be adopted [30]. In this study, we only consider the systematic performance of the ESS. Therefore, the RC equivalent circuit model is built using Advisor. The internal series resistance R s and capacitance C can be obtained according to the capacitor test procedure. The relation between the current i and the voltage U c is expressed as With regard to the energy conversion processes during charging or discharging, the current is obtained by where P L is the output power of the ESS. During discharging, the energy efficiency η dch is calculated as For charging process, the energy efficiency η ch is expressed as The CNG engine model computes the requested torque T er according to the engine output torque T e and the engine speed ω e determined by the control strategy block: Subsequently, the instantaneous fuel consumption m e is determined from a 2D map measured by an engine test bench.
m e ¼ f ω e ; T er ð Þ: Then, the equivalent fuel consumption Q e can be obtained according to the integral values of the fuel consumption and the driving distance using the density of diesel fuel ρ f : where t f is the final time of the driving cycle.

Rule-based control strategy
To evaluate the performance of the designed series hybrid powertrain, a simulation program is developed according to the established mathematical model with MATLAB/Simulink and Advisor. Advisor is a modeling and simulation tool for hybrid electric vehicle based on MATLAB and Simulink developed by the US National Renewable Energy Laboratory [31]. The designed program is shown in Figure 2. Using a backward-facing method, the performance and fuel consumption of the series hybrid transit bus can be calculated. First, the driving force is determined based on the longitudinal dynamic equation. The rotating speed and driving torque of the rear axle are computed according to the tire model. Subsequently, the input torque and speed are determined for the final drive, and the input power of the motor is obtained. Then, the control strategy decides the power distribution between the ESS and the APU. As a result, both the output powers of the ESS and the APU are specified by the power bus. The supercapacitor model calculates the current and energy loss of the ESS. Moreover, the control strategy determines the operation torque and speed of the CNG engine according to the output power of the APU. Then, the fuel consumption is determined based on the performance maps of the CNG engine and the PMSG.
Rule-based control strategy takes advantage of a small computation load, which is very suitable for real-time applications. Therefore, two rule-based control strategies-the thermostatic control and the power follower control-are set up, respectively. The thermostatic control is adopted first for hybrid electric vehicles due to its simple logic. When the HEV is running, if the SOC value drops to a lower bound, the engine starts and operates at a fixed point until the SOC value reaches to an upper bound. The operation condition of the engine is set to the point with the highest effective thermal efficiency.
Large current variation in the ESS may occur for the thermostatic control strategy which results in high energy loss. Therefore, a more sophisticated power follower control is developed. According to this strategy, if the engine state is on, the output power of the APU follows the power demand of the transit bus along an optimal operation line (OOL). Hence, the output power of the APU is decreased, and the operation current of the ESS can be alleviated. The designed power follower control strategy is shown in Figure 3. The decision algorithm for the engine state is designed using Stateflow.
Since the power follower control needs to define an OOL, the energy efficiency characteristics of the APU must be studied first. The performance map of the CNG engine is shown in Figure 4a. The blue contour denotes the engine power in kW. The black contour is the brake-specific fuel consumption (bsfc) in g/kWh. It can be seen that the minimum bsfc of the CNG engine is 196 g/kWh, which is better than that of a diesel engine. The effective thermal efficiency map of the CNG engine shown in Figure 4b is obtained based on the performance map of Figure 4a. The maximum engine efficiency achieves 36.8%, and in most of the operation regions, the effective thermal efficiency of the CNG engine is greater than 30%. The efficiency map of the PMSG is given in Figure 4c, where the highest energy efficiency is 94.5%. In most of the operation regions, the generator efficiency is greater than 89%. The efficiency decreases obviously if the generator speed is less than 500 r/min. According to the results of Figure 4b and c, the energy efficiency map of the APU is obtained as the product of the efficiencies of the CNG engine and the generator. The engine speed ranges from 900 to 2500 r/min, and the maximum engine torque is 650 Nm, which can be covered completely by the generator's operation domain. The results are given in Figure 4d. In this figure, the x-axis is the engine speed, and the y-axis is the engine torque. The blue contour represents the APU output power in kW. The black contours denote the energy efficiency of APU, which decreases with the engine torque and is greater than 30% over the regions of middle and high engine torques. The maximum efficiency is 34.06% located very close to the point with a maximum of engine torque.
Subsequently, the OOL is determined according to the efficiency map of the APU. The efficiency for each point of the OOL is the maximum at each power contour. The result is shown as the green line in Figure 4d. The OOL is the same with the external profile when the engine speed is greater than 1600 r/min. Meanwhile, the engine speed of the OOL remains at 900 r/min if the APU power is less than 50 kW. The OOL appears to have a U shape when the APU power is between 50 and 100 kW. Finally, the parameters of the rule-based control strategies must be optimized. The maximum energy efficiency of the APU is 34.06%. This point is denoted as point P in Figure 8. The corresponding engine speed and torque are 1543 r/min and 650 Nm, respectively, which are specified as the operation point of the thermostatic control strategy. The lower and upper bounds of the SOC is set to 0.58 and 0.99. The other parameters are also optimized for the power follower control strategy.

Optimal control using dynamic programming
In order to evaluate the maximum potential of energy savings, the theoretical minimum fuel consumption of the designed series hybrid powertrain is calculated using an optimal control algorithm based on dynamic programming. Dynamic programming is a static backward-facing optimal algorithm according to Bellman's principle of optimality. The boundary conditions of the optimal algorithm are the same with the thermostatic control.
The optimization target is the total fuel consumption based on the Chinese Transit Bus City Driving Cycle (CTBCDC). The SOC of the ESS is used as the state variable, and the output power of the APU is used as the input variable. If the output power of the APU keeps constant, the fuel consumption achieves the minimum when the CNG engine operates along the OOL. Therefore, the corresponding engine output torque and speed are determined. The target function calculates the instantaneous fuel consumption at time t and is denoted by m(x(t), u(t), t). Function F(x(t), u(t), t) is the state equation determining the SOC for the next time. The initial and final states of the SOC are set to the same values as a constraint. Moreover, the SOC and the output power of the APU must be limited within the allowable ranges all the time. Neither the target function nor the state equation of the designed series hybrid powertrain can be expressed as an explicit equation. Therefore, the analytic solution of this optimal problem cannot be obtained. However, a numerical optimal solution can be determined by a discrete optimal model translated from the aforementioned designed program: subject to Based on the established model of the series hybrid powertrain, the numerical solution of this discrete optimization problem can be determined. The solution approximates a theoretical minimum of the continuous model if the discrete computational grids for the state and input variables are fine enough. In this study, the discretization steps for the SOC and the APU output power are set to 0.005 and 1 kW, respectively. Since the total driving time of the CTBCDC is 1305 s, a time step of 1 s is used.
The optimization problem for the designed series hybrid powertrain contains the final state constraint, which can be translated to a problem without constraint via a penalty function. Sundstrom et al. have successfully applied this method to optimize the energy management problem of a parallel hybrid electric vehicle [32]. In this research, the penalty function for the state variable x i at time stage k is defined as Accordingly, the cost-to-go function is defined by A program is developed in MATLAB based on the designed optimal algorithm, and its working principle can be explained by Figure 5. The entire driving cycle is discretized along the time horizon from stage 1 to stage N shown as the yellow dashed lines. For time stage k, the state variable is discretized from SOC min to SOC max in a step of 0.005 and is expressed by x i (k), i = 1, 2, …, 79. At each state variable x i (k), the value of the cost-to-go function is denoted by J i (k), and the corresponding penalty function is Φ i (k). The input variable is discretized from 0 to P apu,max in a step of 1 kW and is denoted by u j (k), j = 1, 2, …, 136. The fuel consumption from stage k to stage k + 1 at the state x i (k) is denoted by the function m i (k). The initial and final states are constrained to the two red points S 0 shown in Figure 5.
The algorithm first calculates the penalty function J N at the last stage for the state vector x i : Then taking into account each state variable of stage N À 1, a state vector at the next time stage is computed corresponding to the input variable vector based on the state equation. This process is described by a group of lines from one state point of stage N À 1 to different positions of stage N. Because the calculated state vector for stage N may not locate exactly at the computational grid points, a linear interpolation is used to determine the values of the corresponding cost-to-go function. Meanwhile, the fuel consumption function for each state variable is calculated. As a result, the values of cost-to-go function at time stage N À 1 is obtained according to Eq. (18). The optimal path for each state variable is represented by a blue line in Figure 5. The above recurrence calculation process is repeated along the time horizon one by one until to the first time stage. Finally, an optimal map of the costto-go function for all the time stages and the state variables is obtained.
In order to determine the optimal policy, a forward calculation process is performed based on the optimal map. The following equation is used to compute the minimum fuel consumption from the initial state S 0 to the same final state, and the corresponding input variable is recorded: where x op (k) is the optimal state variable at stage k. The optimal state value for the next time stage is determined by where U 0 is the rated voltage of the ESS. Repeating the calculation process until the last time stage, an optimal policy is obtained shown as the green line in Figure 5. The algorithm uses a weighting factor θ for the penalty function whose value specifies the importance of the SOC deviation relative to the fuel consumption. In order to make the final state of the optimal path converge to S 0 , various values of θ are tried. Finally, a value of 120 is specified, and the relative error of the SOC at the last state is 0.44%.

Comparison of rule-based control strategies
The system performance and fuel economy of the series hybrid transit bus are evaluated using the CTBCDC driving cycle. The performances of the thermostatic control and the power follower control are compared. The results of the thermostatic control strategy are given in Figure 6. Figure 7 shows the results of the power follower strategy.
The target vehicle speed and the achievable vehicle speed are given in Figures 6a and 7a, which are denoted by the blue and magenta lines, respectively. The achievable vehicle speeds for both of the rule-based strategies can trace the target one perfectly. Therefore, both can satisfy the requirements of drivability. Figure 6b shows the input power of the PMSM as the blue lines and the output power of the CNG engine by the red lines. Figure 7b shows the results of the power follower strategy. The engine operation time of the thermostatic control is less than that of the power follower strategy, which is 141 s for the thermostatic control while 151 s for the power follower control. The engine power keeps constant for the thermostatic control. However, the engine power of the power follower control varies along the OOL within a small range. The results of the engine state for these two control strategies are shown in Figures 6c and 7c, where the engine state ON is represented by 1, and the engine state OFF is denoted by 0. The engine demonstrates a regular alternative start and stop for the thermostatic control, whereas the engine starts more frequently for the power follower control, which will worsen the engine emissions. The engine speed and torque are shown in Figure 6d and e for the thermostatic control. Compared to the results of the power follower control given in Figure 7d and e, the engine can operate more stably for the thermostatic control, which will be beneficial for the engine working life. The output power of the ESS for the thermostatic control is shown in Figure 6f, where the positive values mean discharging and the negative values denote charging (this expression is used for the following figures). Figure 7f shows the results of the power follower control. Both output powers vary all the time except for the stopping conditions. Furthermore, the variation magnitude of the thermostatic control is obviously greater than that of the power follower control. The maximum discharging power for the thermostatic control is 126.8 kW, while this value reduces to 95.68 kW for the power follower control. By contrast, the maximum charging power of the ESS for the thermostatic control is 202.8 kW, while this value reduces significantly to 106.6 kW for the power follower control. The profiles of the SOC are given in Figures 6g and 7g, respectively. In terms of the thermostatic control, the SOC shows an alternative variation process that first decreases slowly then increases rapidly. However, the SOC of the power follower control shows a relative slow augmentation process, which is in favor of the life span of the ESS.
The energy efficiencies of the ESS are obtained according to Eqs. (10) and (11). The results are given in Figures 6h and 7h. The average discharging and charging efficiencies of the power follower control are 99.1 and 98.5%. As a contrast, these two values are 99.1 and 98.4% for the thermostatic control. The results indicate that the energy efficiency of the power follower control is slightly higher than the thermostatic control. The energy efficiency of supercapacitor will decrease obviously if the operation temperature is too high. Therefore, a temperature control system for the ESS is required in practice. The voltage profiles are given in Figures 6i and 7i. Both of them operate within the constraint range. The average voltage for the power follower control is 450 V, while the average voltage is 505 V for the thermostatic control. Enhancement of the operation voltage is helpful to improve the energy efficiency of the electric motor. Because the SOC value has a linear relation with the operation voltage, the variation tendency of the SOC is consistent with that of the voltage. The current profiles are given in Figures 6j and 7j. The maximum discharging and charging currents for the power follower control are 247 and 269 A. However, these two values are increased significantly to 300 and 471 A for the thermostatic control. Although supercapacitors can work with a high power rate, a lower current will be in favor of their life span. Therefore, it seems that the power follower control strategy is better than the thermostatic control. Table 2 gives the equivalent fuel consumptions of these strategies. The equivalent fuel consumption is 17.32 L/100 km for the power follower strategy, while it equals 17.51 L/100 km for the thermostatic control strategy. In contrast to a conventional vehicle powered solely by the same CNG engine, the fuel consumptions of the two rule-based strategies are decreased by 52%. Figure 8 is used to explain the reason from a viewpoint of energy efficiency. The OOL is displayed as the green line. Point P is the operation point of the thermostatic control whose energy efficiency is 34.06%. Because most of the APU output power of the thermostatic control first charges to the ESS and then outputs to the power line, the overall energy efficiency of the series hybrid powertrain decreases slightly to 33.2%, which is denoted by a blue contour L in Figure 8. The operation points of the power follower control are described by the cyan points. Most of the APU output power is delivered directly to the power line for the power follower strategy. The energy efficiencies for the thermostatic control approximate the line L, while the energy efficiencies of the power follower control remain at the cyan points. Therefore, the fuel economy of the power follower control is a little higher than that of the thermostatic control.

Results of optimal control
The optimal performance of the series hybrid powertrain is shown in Figure 9. Figure 9a is the velocity profile of the CTBCDC cycle. The corresponding input power required by the PMSM is shown in Figure 9b, where the positive values are used for the driving mode and the negative values are used for the regenerative braking mode. These two profiles are used as input parameters for the optimal algorithm. The optimal results of the output power of the APU are given in Figure 9c, which demonstrates a series of short impulse when the transit bus undergoes an acceleration process. Furthermore, the number of the impulse increases as the power demand of the motor rises. The APU stops if the power demand of the motor is negative. The corresponding engine output torque and speed are shown in Figure 9d and e, respectively. The optimal engine speed and torque remain around 1500 r/min and 600 Nm. The reason for such an optimal trajectory can be explained as follows. The operation points of the CNG engine for the optimal control are described by the red points in Figure 8. These red points are very close to the point P where the energy efficiency is the highest. Therefore, an overall minimum of the fuel consumption is realized. On the other hand, the Relative to the conventional bus with a CNG engine. Table 2.
Results of fuel consumption. internal series resistance of the ESS consumes part of the energy during the charging or discharging process, especially for large current conditions. To avoid too much energy loss of the ESS, the optimal policy will try to use the APU power to satisfy the power demand of the PMSM exactly. In other words, the optimal policy is obtained if the APU operates at the maximum energy efficiency point and the amount of the output power equals to the power demand of the PMSM, leading to the energy supply in the form of a series of impulse. The output power of the supercapacitors is given in Figure 9f. Generally, the variation tendency of the output power is similar to that of the power follower control shown in Figure 7f. However, a series of impulses occur under the driving modes due to the same reason for the output power of the APU. The SOC profile is given in Figure 9g. The optimal SOC remains within a small interval around the initial valleys. Only two valleys occur at the high-velocity condition with rapid braking. The energy loss of the ESS is decreased if the current of the ESS is limited within a small range. Thus, the fuel consumption can be reduced. The calculated energy efficiencies of the ESS are shown in Figure 9h. The average energy efficiency is very close to the results of the rule-based strategies. The output voltage and current are given in Figures 9i and j, respectively. The output voltage is proportional to the SOC. The optimal current demonstrates a similar tendency as that of the power follower control. However, more spikes occur at the driving conditions.
The optimal equivalent fuel consumption is 15.72 L/100 km listed in Table 2. Compared to the conventional transit bus, the optimal fuel consumption of the hybrid bus can be decreased by 57% if the quantity of fuel consumed during the starting processes is ignored. In practice, the fuel consumption including starting process will be increased slightly. Taking the optimal result as a reference, the fuel consumptions of the rule-based control strategies are increased by approximate 1.7 L/100 km.

Conclusions
In this chapter, the energy efficiency of a series hybrid transit bus powered by a CNG engine and supercapacitors was evaluated. A mathematical model was established, and three different control strategies were designed. The performance characteristics of two rule-based control strategies were compared to the result of the optimal control using dynamic programming. Based on our analysis, the following can be concluded: 1. The performance of the designed series hybrid transit bus powered by a CNG engine and supercapacitors can fulfill the requirements of the vehicle. Because supercapacitors take advantage of low energy loss and can recover the vehicle's kinetic energy efficiently during regenerative braking, the fuel consumption of the designed series hybrid powertrain is decreased significantly by approximately 52% compared to a conventional transit bus under the CTBCDC driving cycle. The results indicate that supercapacitors are a good solution for the ESS of the hybrid transit bus.
2. The energy efficiency of the designed series hybrid transit bus using the power follower control strategy is slightly greater than that of the thermostatic control strategy because the operation current of the ESS remains at a low level. The thermostatic control has a relatively simple logic and less starting times compared with the power follower control. In practice, the most suitable control strategy should be selected according to the requirements of system performance including efficiency, cost, size, reliability, emissions, and so on.
3. The maximum energy efficiency of the designed series hybrid powertrain is estimated using an optimal algorithm based on dynamic programming. The optimal fuel consumption can be decreased by 57% compared to the results of the conventional transit bus. Although the fuel consumption from the rulebased control strategies is slightly higher than that of the optimal control, taking into account the operation requirements of the engine, the rule-based control strategies are more practical for real-time application. However, the result of the optimal control provides some useful insights for real-time control strategy design.