Optimal Hybridization of Conventional ICE Vehicles

: Most commercially available hybrid electric vehicle (HEV) drivetrains are made of small internal combustion (IC) engines and large electric drives to improve fuel economy. They usually have higher cost than the conventional IC-engine-based vehicles because of the high costs of the electric drives. This paper proposes a hybridized powertrain composed of the original full-size engine of the vehicle and a universally optimum size parallel electric drive. The dynamic programming (DP) algorithm was used to obtain the sensitivity of the maximum miles per gallon (MPG) values versus the power rating of the electric drive. This sensitivity was then analyzed to determine the optimal window of the electric drive power ratings. This was proven to be universal for all passenger cars of various masses and engine powers. The fuel economy and vehicle performance of this HEV was compared with those of the 2019 Toyota Corolla, a conventional IC-engine-based vehicle, and the 2019 Toyota Prius, a commercially available HEV. The results showed that the proposed universally optimized HEV powertrain achieved better fuel economy and vehicle performance than both the original ICE and HEV vehicles, at low additional vehicle cost.


Introduction
Conventional vehicles, powered by internal combustion (IC) engines, are a major source of carbon dioxide emission, causing global warming [1]. They also pollute the air with significant emissions of toxic gases such as nitrogen oxides (NOx), carbon monoxide (CO), and unburned hydrocarbons [1]. Another drawback of the IC engine is its low efficiency. Its typical efficiency is around 20% which is much lower than an electric motor whose efficiency is around 85% [2]. This low efficiency leads to the poor fuel economy of IC engine based vehicles, especially in urban driving cycles.
It is now known that electric vehicles (EV) have certain advantages over IC engine based vehicles, such as higher efficiency, no tailpipe emissions, smoother operation and less noise [1]. However, they also have several disadvantages, such as short travel range, long battery recharging time, and high comparative costs. For example, the 2020 Chevrolet Bolt, a commercially available EV, can have an additional travel range of only 90 miles after its battery is recharged for 30 min at a Level 3 charging station [3]. This time is much longer than the time needed for filling a gasoline tank [2]. In addition, the travel range of EV can be even shorter under lower ambient temperatures. For instance, it can be shown that the travel distance of the Mitsubishi i-MiEV, a commercial EV, decreases at a rate of 2.5 km per 1 • C temperature drop in the ambient temperature range of +20 • C to −15 • C [4]. Furthermore, the electric drive and battery in the EV have a significantly higher total cost than an IC engine and its gasoline tank, leading to a higher cost for the EV.
To combine the advantages of the IC engine based vehicle and the EV, the hybrid electric vehicle (HEV) typically contains an IC engine to deliver the average tractive power and an electric motor to supply the peak power. In this way, the power rating of the IC engine can be reduced to less than half of that in an equivalent conventional vehicle with the same mass and dynamic specifications [1,5]. However, the HEV must use this The drivetrain configuration of a torque-coupling parallel HEV with a full-size engine is shown in Figure 1. It contains two power plants, an IC engine and an electric motor. The engine is connected to a torque-coupler via a multi-gear transmission, and the motor is connected to the same torque-coupler via a single-gear transmission. The torque-coupler enables the torque on the driving shaft to be equal to the sum of the torques on the engine shaft and the motor shaft, whereas the speed of the driving shaft is proportional to the speeds of both the engine shaft and the motor shaft. A final drive is used to distribute the driving shaft torque to each driving wheel through a differential. The load power on the driving wheels, P l , has a relation with powers of engine and motor expressed as where P e_w and P m_w are the engine power and the motor power transmitted onto the driving wheels, respectively.
Eng 2021, 2, FOR PEER REVIEW 3 have different curb masses and IC engine power ratings, as shown in Table 1, which can represent compact, medium-sized and large-sized passenger cars on the market. The drivetrain configuration of a torque-coupling parallel HEV with a full-size engine is shown in Figure 1. It contains two power plants, an IC engine and an electric motor. The engine is connected to a torque-coupler via a multi-gear transmission, and the motor is connected to the same torque-coupler via a single-gear transmission. The torque-coupler enables the torque on the driving shaft to be equal to the sum of the torques on the engine shaft and the motor shaft, whereas the speed of the driving shaft is proportional to the speeds of both the engine shaft and the motor shaft. A final drive is used to distribute the driving shaft torque to each driving wheel through a differential. The load power on the driving wheels, Pl, has a relation with powers of engine and motor expressed as where Pe_w and Pm_w are the engine power and the motor power transmitted onto the driving wheels, respectively. The parameters of the three IC engine vehicles and of the designed motors used in this study are listed in Tables 1 and 2, where δ1 and δ2 are used to calculate the rotational inertia factor which converts the inertia of all rotating components to the equivalent mass and can be expressed as where, and are the gear ratios of the transmissions connected to the engine and the motor, respectively, and is the gear ratio of the final drive. For an eight-gear transmission, can be any one of the eight gear ratios of the engine transmission listed in Table 1, depending on which gear is used at a time. To demonstrate the universality associated with the power rating of the electric drive (discussed later in Section 4), the motor parameters in Table 2 are designed to be the same for all the three IC engine vehicles,  except which is designed to keep the product of and the same for all the three IC engine vehicles. The parameters of the three IC engine vehicles and of the designed motors used in this study are listed in Tables 1 and 2, where δ 1 and δ 2 are used to calculate the rotational inertia factor which converts the inertia of all rotating components to the equivalent mass and can be expressed as where, i ge and i gm are the gear ratios of the transmissions connected to the engine and the motor, respectively, and i 0 is the gear ratio of the final drive. For an eight-gear transmission, i ge can be any one of the eight gear ratios of the engine transmission listed in Table 1, depending on which gear is used at a time. To demonstrate the universality associated with the power rating of the electric drive (discussed later in Section 4), the motor parameters in Table 2 are designed to be the same for all the three IC engine vehicles, except i gm which is designed to keep the product of i gm and i 0 the same for all the three IC engine vehicles. The mechanical resistances opposing the vehicle movement consist of rolling resistance, aerodynamic drag, and grading resistance. Taking the vehicle acceleration into consideration, we can express the load power P l as where, V is the vehicle speed, and i is the road grade. The electric drive consists of the motor, power electronic converters, and a battery pack. The maximum torque of the motor is constant when the motor speed is below the base speed, whereas the maximum power of the motor is constant when the motor speed is above the base speed. That is, the motor has nearly ideal torque-speed and power-speed profiles. Therefore, only a single-gear transmission is needed for the motor. The widely used battery packs for HEVs and EVs are nickel metal hydride (NiMH) batteries and Li-ion batteries [15]. Research in [16][17][18][19] shows that the energy density and efficiency of Li-ion batteries are significantly higher than those of NiMH batteries. Hence, the Li-ion battery pack is used here to deliver electrical power to the motor. The total mass of the electric drive should be considered during the optimization process of the electric drive power rating because it has an adverse effect on the fuel economy of the HEV. We assume that motors with power ratings from 3 kW to 40 kW have a constant mass per kW factor which is 2.5 in this paper. We also assume that the specific power of the battery pack is 1.28 kW/kg, and Eng 2021, 2 596 the energy/power ratio of the battery pack is 0.03 h. Based on these assumptions, the masses of electric drives with various power ratings from 3 kW to 45 kW are estimated and listed in Table 3.

Dynamic Programming Algorithm in HEV
The dynamic programming (DP) algorithm optimizes a cost function and finds the optimal sequences of control variables in the whole process by making control decisions, step-by-step, in time domain [20,21]. The cost function in HEV drivetrain control is the total fuel consumption of the engine during a certain driving cycle.
Research in [5,22] uses the DP algorithm whose control variable is the output power or torque of the engine. This algorithm will either lead to the depletion of the battery or select the pure electric mode for only a short distance and switch to the engine-alone mode after the battery SOC drops to its lower bound, which is not an optimal control for an HEV. Although this situation can be avoided by adding electric energy usage penalties (i.e., electric penalties) to the cost function, the weights of the gasoline usage cost and the electric penalties will highly affect the control sequences, leading to the non-optimality of the obtained solution.
To overcome the disadvantages of algorithms applied in previous research, the DP algorithm used in this paper selects the battery SOC as well as the number of gears of the engine transmission as the control variables. This selection can avoid battery depletion without adding the electric penalties, and thus the optimality of the obtained solution is guaranteed. In this algorithm, the output powers of the engine and the motor are calculated according to the battery SOCs, SOC(k) and SOC(k + 1), at two adjacent time steps, k and k + 1, in a driving cycle. If SOC(k) > SOC(k + 1), the battery is discharged, and the motor power on the driving wheels is expressed as where Q n is the battery energy capacity, η b_w is the power efficiency from the battery to the driving wheels, and ∆t is the time interval between time steps k and k + 1. Otherwise, the battery is charged if SOC(k) < SOC(k + 1), and the motor power on the driving wheels is expressed as where η w_b is the power efficiency from the driving wheels to the battery. If SOC(k) = SOC(k + 1), there is no energy delivered from or to the battery, and thus P m_w = 0. The battery energy capacity, Q n , is related to the output power rating of the electric drive, P m_rated , which is expressed as where, R e/p is the energy/power ratio of the battery and is assumed to be 0.03 h in this paper.

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Using (1), the engine power on the driving wheels, P e_w , can also be calculated, with the load power, P l , obtained from the driving cycle as expressed in (3). The output power, P e , and speed, n e , of the engine on its own shaft are calculated as where, i ge is determined by the gear number of the engine transmission, as shown in Table 1.
The fuel consumption, F c , in the time interval ∆t is calculated as where ρ f is the mass density of gasoline, which is assumed to be 0.75 kg/L in this paper, and g e is the brake specific fuel consumption (BSFC) of the engine, which is determined by P e and n e in the engine fuel consumption map. Suppose the driving cycle has N + 1 discrete time steps from time step 0 to time step N. Because the engine delivers the average tractive power and the motor only supplies the peak power, all the net energy used during the entire driving cycle is only delivered by the engine. Therefore, the battery SOC must be the same value, SOC init , at both time steps 0 and N. At each of other time steps, the battery SOC can be any value within its full usable range, SOC min ≤ SOC ≤ SOC max . Suppose the range is discretized into i m values, that is, Suppose the value of the battery SOC is SOC i at time step k and SOC r at time step k + 1, where SOC i and SOC r are within the SOC range, 1 ≤ i, r ≤ i m , and 1 ≤ k ≤ N−2. Suppose the jth gear of the engine transmission, G j , is selected at time step k, where 1 ≤ j ≤ j m , and j m = 6 for a six-gear transmission. For the time interval ∆t k from time step k to time step k + 1, we have a recursive relation of fuel consumption expressed as where J * SOC i (k) and J * SOC r (k + 1) are the minimum total fuel consumptions from time step 0 to time step k with SOC i at time step k and from time step 0 to time step k + 1 with SOC r at time step k + 1, respectively, and F c (SOC i,r (k), G j (k)) is the fuel consumption in the time interval ∆t k with SOC i at time step k, SOC r at time step k + 1, and the jth gear selected at time step k. Using (11), we can obtain the minimum total fuel consumption from time step 0 to time step k + 1 for every SOC r within SOC min ≤ SOC r ≤ SOC max at time step k + 1 and every k within 1 ≤ k ≤ N−2.
In the initial time interval ∆t 0 from time step 0 to time step 1 and the terminal time interval ∆t N−1 from time step N−1 to time step N, the recursive relations of fuel consumption are expressed as where, F c (SOC init,r (0), G j (0)) is the fuel consumption in the time interval ∆t 0 with SOC r at time step 1 and the jth gear selected at time step 0, F c (SOC i,init (N − 1), G j (N − 1)) is the fuel consumption in the time interval ∆t N−1 with SOC i at time step N−1 and the jth gear selected at time step N−1, and J * SOC init (N) is the minimum total fuel consumption during the entire driving cycle. The initial fuel consumption at time 0, J * SOC init (0), is assumed to be zero. Using (12), the minimum total fuel consumption from time step 0 to time step 1, J * SOC r (1), is obtained for every SOC r within SOC min ≤ SOC r ≤ SOC max at time step 1, whereas J * SOC init (N) obtained in (13) is a unique value solely for the single initial and terminal SOC value, SOC init , which is also within the SOC usable range.
Using (1)- (3) and (4)- (9), we can obtain P e_w , P m_w and F c in every time interval ∆t k for every set of SOC i (k), SOC r (k + 1) and G j (k) values. However, not all of the discrete values within the SOC usable range can be assigned to SOC i and SOC r in practical conditions because P e_w and P m_w must satisfy the following constraint conditions expressed as P e_min (n e (k)) ≤ P e_w (k) ≤ P e_max (n e (k)).
where the upper and lower bounds of P e_w and P m_w are functions of the engine speed, n e (k), and the motor speed, n m (k), respectively. The functions are determined by the power ratings and the power-speed characteristics of the engine and the motor. Every set of SOC i (k), SOC r (k + 1) and G j (k) values which cause P e_w or P m_w to go beyond the corresponding constraint condition in (14) or (15) should be excluded from the optimization processes expressed in (11)-(13).
Besides the final minimum value of the cost function, J * SOC init (N), we can also obtain the optimal values of SOC i (k) and G j (k) that lead to J * SOC init (N) with k = N−1 from (13). Then, with k = N−2, we let SOC r (k + 1) be equal to the optimal value of SOC i (N−1) and use (11) to find the optimal values of SOC i (k) and G j (k). Repeat this with k = N−3, N−4, . . . , 1. Finally, substituting the optimal value of SOC r (1) = SOC i (1) into (14), we obtain the optimal value of G j (0). Letting G j (N) = G j (0) and SOC i (N) = SOC i (0) = SOC init , we obtain the optimal sequences of the control variables, SOC i (k) and G j (k) with k = 0, 1, . . . , N. Using (1)- (3) and (4)-(8), we can also obtain the optimal sequences of P e_w , P m_w , P e and n e from those of the control variables to see how the total tractive power is distributed among the engine and the motor and to obtain the optimal engine operating points analyzed in the next section.

Optimization of Electric Drive Power Rating
To optimize the power rating of the electric drive, P m_rated , in a full-size engine HEV, the DP algorithm discussed in the previous section is used to calculate the minimum total fuel consumption, which is equivalent to the maximum MPG, during four typical driving cycles (FTP75 Urban, FTP75 Highway, LA92, and SC03) under various values of P m_rated . Then, the sensitivity of the maximum MPG to P m_rated is analyzed.
Research in [23][24][25][26][27] has proposed an optimization methodology which fixes either the total power rating of the power plants or the acceleration performance of the vehicle and obtains the trend of maximum MPG by increasing the electric drive power rating. Under this methodology, the engine power rating has to be reduced when the electric drive power rating increases. This reduction will potentially worsen the vehicle performances because the smaller engine is the only available power plant to propel the vehicle if the battery SOC drops to its lower bound.
In this section, to ensure that the vehicle performance is not degraded in the optimization process, the power rating of the full-size engine in each of the three IC engine vehicles is kept constant. The performance and fuel economy are further improved with the addition of the electric drive. Therefore, the optimal window of P m_rated , obtained from the sensitivity analysis of the maximum MPG trend with the increasing P m_rated , will be practical for the full-size engine HEV.
The parameters of the DP algorithm used in the optimization of this section are listed in Table 4. The discrete SOC values expressed in (10) are uniformly distributed within the SOC usable range, and thus the difference between any two adjacent SOC values, SOC ∆ , is kept constant at 5 × 10 −4 . The time interval between two adjacent time points, ∆t k , is one second in all of the four driving cycles, whereas the total time duration, N, is different in different driving cycles.
For hybridization of each IC engine vehicle with each of the electric drive power rating values listed in Table 3, the maximum MPG during each driving cycle and the average maximum MPG of the four driving cycles are calculated and plotted in Figure 2. The maximum MPG of each IC engine vehicle without the electric drive, i.e., P m_rated = 0 kW, is also calculated and plotted for comparison and analysis. sensitivity analysis of the maximum MPG trend with the increasing Pm_rated, will be practical for the full-size engine HEV. The parameters of the DP algorithm used in the optimization of this section are listed in Table 4. The discrete SOC values expressed in (10) are uniformly distributed within the SOC usable range, and thus the difference between any two adjacent SOC values, SOCΔ, is kept constant at 5 × 10 −4 . The time interval between two adjacent time points, Δtk, is one second in all of the four driving cycles, whereas the total time duration, N, is different in different driving cycles.
For hybridization of each IC engine vehicle with each of the electric drive power rating values listed in Table 3, the maximum MPG during each driving cycle and the average maximum MPG of the four driving cycles are calculated and plotted in Figure 2. The maximum MPG of each IC engine vehicle without the electric drive, i.e., Pm_rated = 0 kW, is also calculated and plotted for comparison and analysis.  It is shown in Figure 2 that a vehicle with large mass and high engine power rating (e.g., Chrysler 300 2016) has lower maximum MPG than that with small mass and low engine power rating (e.g., Toyota Corolla 2019) after hybridized with electric drives of the same power rating. However, Figure 2 shows that all the three vehicles during all the four driving cycles have the similar trend of sensitivity of MPG to the electric drive power rating Pm_rated. Specifically, the maximum MPG starts to increase rapidly as Pm_rated increases from 0 kW to 10 kW, which means the fuel economy is significantly improved with the help of the electric drive. As Pm_rated keeps on increasing from 10 kW to 30 kW, the rate of It is shown in Figure 2 that a vehicle with large mass and high engine power rating (e.g., Chrysler 300 2016) has lower maximum MPG than that with small mass and low engine power rating (e.g., Toyota Corolla 2019) after hybridized with electric drives of the same power rating. However, Figure 2 shows that all the three vehicles during all Eng 2021, 2 600 the four driving cycles have the similar trend of sensitivity of MPG to the electric drive power rating P m_rated . Specifically, the maximum MPG starts to increase rapidly as P m_rated increases from 0 kW to 10 kW, which means the fuel economy is significantly improved with the help of the electric drive. As P m_rated keeps on increasing from 10 kW to 30 kW, the rate of increase in the maximum MPG is still positive and significant but starts to decline because the effect of fuel economy improvement is partially offset by the increasing resistance brought by the additional mass of the electric drive. After P m_rated exceeds 30 kW, the maximum MPG shows negligible improvement or even starts to decrease as P m_rated increases, which indicates that there is little room for further improvement on the fuel economy, whereas the significant mass increase of the electric drive starts to dominate the MPG trend. Although for different vehicles and during different driving cycles, there are minor differences in the P m_rated value where the maximum MPG reaches its peak and starts to decline, the differences do not play a significant role in the optimization of P m_rated . This is because the additional cost brought by the increase of P m_rated overwhelms the little, if any, improvement on the fuel economy after P m_rated exceeds 30 kW. Therefore, the upper bound of the optimal window of P m_rated is determined to be 30 kW, and the lower bound of this window is determined to be 10 kW because 10 kW is the upper limit of P m_rated with the largest rate of increase in MPG. This optimal window of P m_rated between 10 kW and 30 kW is universal regardless of vehicle mass, engine power rating and driving cycle, i.e., it is applicable to all the vehicles with different masses and engine power ratings during all the driving cycles, as illustrated in Figure 2. Therefore, it is possible to produce the same motor-battery package for all different sizes of passenger cars. For example, if the 30 kW electric drive package is produced for the purpose of optimizing fuel economy, it will be suitable for hybridizing all passenger cars, which will significantly decrease the production cost of the electric drive.
The MPG trends shown in Figure 2 can be also explained from the perspective of the distribution of operating points of the engine in the fuel consumption map. Here, each operating point of the engine refers to a set of the engine speed n e and the engine output power P e at a certain time point in the driving cycle. For example, for the hybridization of Toyota Corolla 2019 during the FTP75 Highway driving cycle, the engine operating points, overlapping the fuel consumption map with various P m_rated values, are shown in Figure 3. The operating points in Figure 3 are classified into seven operating modes (hybrid traction, engine-alone traction, charging battery from engine, motor-alone traction, regenerative braking alone, hybrid braking, and mechanical braking alone), which determine the flow direction of motor power P m and load power P l at each operating point, as shown in Table 5. For load power P l , positive power means traction, and negative power means braking. For motor power P m , positive power means the battery is discharged so that the motor provides propulsion force to the vehicle, and negative power means the battery is charged so that the motor provides regenerative braking force to the vehicle. For engine power P e , positive power means the engine provides propulsion force to the vehicle, and zero power means the engine is shut down. P e cannot be negative because the engine cannot be used for regenerative braking purpose. Thus, mechanical braking power P b is needed for braking if the motor cannot supply sufficient braking power to meet the braking demand of the vehicle load. charged so that the motor provides propulsion force to the vehicle, and negative power means the battery is charged so that the motor provides regenerative braking force to the vehicle. For engine power Pe, positive power means the engine provides propulsion force to the vehicle, and zero power means the engine is shut down. Pe cannot be negative because the engine cannot be used for regenerative braking purpose. Thus, mechanical braking power Pb is needed for braking if the motor cannot supply sufficient braking power to meet the braking demand of the vehicle load.

Operating Modes Polarities of Powers
Hybrid traction P e > 0, P m > 0, P l > 0 Engine-alone traction P e > 0, P m = 0, P l > 0 Charging battery from engine P e > 0, P m < 0, P l > 0 Motor-alone traction P e = 0, P m > 0, P l > 0 Regenerative braking alone P e = 0, P m < 0, P l < 0, P b = 0 Hybrid braking P e = 0, P m < 0, P l < 0, P b < 0 Mechanical braking alone P e = 0, P m = 0, P l < 0, P b < 0 In Figure 3a with P m_rated = 0 kW, most operating points of the engine lie in the region with significant engine power and high BSFC. According to (9), the fuel consumption F c is large, leading to poor fuel economy. The reason for this undesirable point distribution is that the absence of the electric drive forces the engine power to follow the load power which usually corresponds to points in the fuel consuming region of the map.
To improve the fuel economy, the engine operating points should move away from this fuel consuming region. According to (9), this can be accomplished by either lowering the points towards the region with negligible engine power (Region A), or lifting the points towards the region with low BSFC (Region B). In Region A, most of the operating points are under the motor-alone traction mode or the regenerative braking mode because the negligible engine power is insufficient to meet the load power requirement. In Region B, the operating points are under the charging battery from engine mode because the engine power exceeds the load power requirement. Thus, the electric drive is needed in both desirable regions to let the motor power offset the power gap between the engine and the load. The larger rated power P m_rated the electric drive has, the more motor power it can deliver to offset the power gap, and thus the more operating points it can move towards the two desirable regions, leading to better fuel economy. This principle is shown in Figure 3b,c.
With P m_rated = 10 kW in Figure 3b, all the lowered operating points lie in Region A, leading to significant MPG improvement. Whereas, most of the lifted operating points are only closer to but do not lie in Region B, and a small portion of the operating points still stay in the fuel consuming region, indicating that there is still some room for further improvement on MPG. With P m_rated = 30 kW in Figure 3c, almost all the operating points are moved into either Region A or Region B, indicating that the MPG is close to its peak value shown in Figure 2, and there is little room for further improvement on MPG if P m_rated keeps on increasing. Similar trends in the distribution of engine operating points can also be seen for the other driving cycles. Therefore, we can conclude that the optimal window of the electric drive power rating is from 10 kW to 30 kW.
The independence of the optimal window of the electric drive power rating from vehicle mass and engine power rating can be explained from the distribution of operating points of the vehicle load with respect to the power-speed characteristics of the motor. Here, each operating point of the vehicle load refers to a set of the vehicle speed V and the load power P l on the driving wheels at a certain time point in a driving cycle. For example, for each of the three IC engine vehicles without the electric drive, the distribution of the load operating points during the FTP75 Urban driving cycle with respect to the power-speed characteristics of the 10 kW rated motor and of the 30-kW-rated motor are shown in Figure 4. To make the comparisons between the load power and the motor power more convenient, the motor power-speed characteristics in Figure 4 are also transmitted onto the driving wheels.
Note that the engine traction mode in Figure 4 is divided into two submodes, one with full-linkage of the engine to the transmission and another with semi-linkage of the engine to the transmission. The engine usually operates in the full-linkage submode, where almost all of the engine power can be transmitted onto the driving wheels with high efficiency. However, the engine must switch to the semi-linkage submode if the speed of the engine operating in the full-linkage submode falls below the minimum permissible speed of the engine. This usually happens when the vehicle speed is very low. The semi-linkage submode will lead to low efficiency of the engine power transmission because there is a significant friction between the engine shaft and the transmission gearbox if they are only partially linked. Figure 4a shows that the 30-kW-rated motor has the capacity to supply all the load powers of a compact vehicle like Toyota Corolla 2019. As the vehicle mass increases, the load powers also increase, as shown in Figure 4b,c for a medium-sized vehicle like Toyota Camry 2016 and for a large-sized vehicle like Chrysler 300 2016. However, the 30-kWrated motor can still meet most of the load power demands for both heavier vehicles. Furthermore, for all the three vehicles, the 30-kW-rated motor can supply the powers of all the loads at which the engine has to operate in the semi-linkage submode. This means the 30-kW-rated motor alone can be used for propulsion to prevent the engine from inefficiently operating in the semi-linkage submode, which can significantly help improve the fuel efficiency. On the other hand, there are few or no load operating points lying outside the power range of the 30-kW-rated motor, indicating that there will be little or no further improvement on meeting load power demands as the motor power rating keeps on increasing beyond 30 kW. This explains why 30 kW is the universal upper bound of the optimal window of the electric drive power rating. high efficiency. However, the engine must switch to the semi-linkage submode if the speed of the engine operating in the full-linkage submode falls below the minimum permissible speed of the engine. This usually happens when the vehicle speed is very low. The semi-linkage submode will lead to low efficiency of the engine power transmission because there is a significant friction between the engine shaft and the transmission gearbox if they are only partially linked.   Figure 4a shows that the 30-kW-rated motor has the capacity to supply all the load powers of a compact vehicle like Toyota Corolla 2019. As the vehicle mass increases, the load powers also increase, as shown in Figure 4b,c for a medium-sized vehicle like Toyota Camry 2016 and for a large-sized vehicle like Chrysler 300 2016. However, the 30-kWrated motor can still meet most of the load power demands for both heavier vehicles. Furthermore, for all the three vehicles, the 30-kW-rated motor can supply the powers of all the loads at which the engine has to operate in the semi-linkage submode. This means the 30-kW-rated motor alone can be used for propulsion to prevent the engine from inefficiently operating in the semi-linkage submode, which can significantly help improve the fuel efficiency. On the other hand, there are few or no load operating points lying outside the power range of the 30-kW-rated motor, indicating that there will be little or no further improvement on meeting load power demands as the motor power rating keeps on It is also shown in Figure 4 that for all the three vehicles, the density of load operating points within the power range of the 10 kW rated motor is significantly higher than that of the points beyond 10 kW rated motor power range. Therefore, the capability of meeting load power demands can be rapidly enhanced as the motor power rating increases from 0 to 10 kW, whereas the rate of this enhancement slows down as the motor power rating goes beyond 10 kW. This explains why 10 kW is the universal lower bound of the optimal window of the electric drive power rating.

Comparison with Commercially Available Vehicles
To demonstrate the advantages of the full-size engine HEVs over IC engine vehicles and commercially available HEVs, the fuel economy, acceleration performance, cost of power plants and cost payback mileage of the full-size engine HEVs are compared with those of Toyota Corolla 2019, an IC engine vehicle without hybridization, and with those of 2019 Toyota Prius, a commercially available HEV consisting of a large electric drive and a relatively small IC engine.
A model of Corolla-based full-size engine HEV is constructed by adding a relatively small electric drive in parallel to the full-size (98 kW) engine of Corolla 2019. Because 10 kW and 30 kW are the lower and upper bounds of the optimal window of the electric drive power rating P m_rated , we make two models of the Corolla-based full-size engine HEVs, one with P m_rated = 10 kW and another with P m_rated = 30 kW.
Using the DP algorithm introduced in this paper, we calculate the maximum MPGs of the two HEV models during FTP 75 Urban and FTP 75 Highway driving cycles, and compare them with those of the Corolla 2019. To obtain the overall fuel economy during the two driving cycles, a combined MPG taking 55% of the urban MPG and 45% of the highway MPG is also calculated and compared. To make fair comparisons under the same conditions, the maximum MPGs of the Corolla 2019 are also calculated using the same DP algorithm, rather than directly copied from its commercial MPG data. The maximum MPGs of the three vehicles are listed in Table 6, which shows that both full-size engine HEVs have significantly better fuel economy than the Corolla 2019. The acceleration performance is specified in this study as the acceleration time for the vehicle speed from 0 to 60 mph (26.8 m/s) on a flat road, which can be calculated as where V 1 = 0, V 2 = 26.8 m/s, T e_w and T m_w are the engine torque and the motor torque transmitted onto the driving wheels, respectively. Using (15), we can calculate the acceleration time from 0 to 60 mph for the three vehicles, as listed in Table 7, which shows that both full-size engine HEVs have better acceleration performance than the Corolla 2019. Table 7. Acceleration time from 0 to 60 mph for Corolla-based vehicles.

Vehicles
Acceleration Time (s) Corolla 2019 7.53 HEV (P m_rated = 10 kW) 7.14 HEV (P m_rated = 30 kW) 6.58 The Toyota Prius 2019 is a series-parallel HEV whose maximum MPGs cannot be calculated using the DP algorithm introduced in this paper because this DP algorithm can only be used to calculate the maximum MPGs of parallel HEVs. To estimate the fuel economy of the Prius 2019 under optimal control, an equivalent parallel HEV model (Modified Prius) is made. The electric drive power rating P m_rated of the Modified Prius is designed to be the same as the Prius 2019, which is 53 kW, so that the Modified Prius will have roughly the same weight as Prius 2019, and its initial cost will also be close to Prius 2019. The reason for this design is that the fuel economy is affected by the vehicle weight, and the initial cost is largely determined by P m_rated because the price per kW of the motor is much higher than that of the engine. The engine power rating of the Modified Prius is determined to be 31 kW, which ensures that the Modified Prius has the same acceleration performance (acceleration time from 0 to 60 mph) as the Prius 2019.
To construct full-size engine HEV models based on the Prius 2019, an IC engine vehicle model (Engine-only Prius) is made first. The engine power rating of the Engineonly Prius is determined to be 73 kW, which ensures that the Engine-only Prius has the same performance (acceleration time from 0 to 60 mph) as the Prius 2019. Then, a relatively small electric drive is added in parallel to the full-size engine of the Engine-only Prius, Eng 2021, 2 605 making a Prius-based full-size engine HEV. Similar to the Corolla-based HEVs, two models of the Prius-based full-size engine HEVs are made, one with P m_rated = 10 kW and another with P m_rated = 30 kW.
The maximum MPGs of all the Prius-based vehicles are listed in Table 8. They are all calculated using the DP algorithm introduced in this paper, except for those of the Prius 2019, which is estimated to be lower than those of the Modified Prius. The reason for such estimation is that the Prius 2019 has a much larger engine (71 kW) than the Modified Prius (31 kW). To compare the costs of power plants for all the Prius-based vehicles, we assume that the average prices of the induction motor, the IC engine, and the battery pack are $110 per kW, $35 per kW, and $137 per kWh, respectively [2,28,29]. To calculate the cost payback mileages, we assume that the price of gasoline is $4 per gallon [2]. The costs of power plants and payback mileages for the Prius-based vehicles are listed in Table 9.  Table 8 shows both the full-size engine HEVs have significantly better fuel economy than the Engine-only Prius, and the fuel economy of the full-size engine HEV with P m_rated = 30 kW is close to that of the Modified Prius and may be better than that of the Prius 2019. In the meantime, the costs of power plants in both full-size engine HEVs are significantly lower than those in the Modified Prius and the Prius 2019, as shown in Table 9. Therefore, the cost payback mileages of both full-size engine HEVs are significantly shorter than those of the Modified Prius and the Prius 2019, as shown in Table 9.
The acceleration time from 0 to 60 mph for all the Prius-based vehicles are listed in Table 10, which shows that both full-size engine HEVs have better acceleration performance than the Engine-only Prius, the Modified Prius, and the Prius 2019. Note that the acceleration time for the Modified Prius and the Engine-only Prius is by design the same as that for the Prius 2019, which is found on [30]. Table 10. Acceleration time from 0 to 60 mph for Prius-based vehicles.

Conclusions
This paper proposed a universally optimized parallel HEV drivetrain, which uses the full-size IC engine as the primary power plant to ensure that the vehicle performance and the travel range of the hybridized vehicle are at least the same as the original IC engine vehicle. To improve the fuel economy and performance, an optimal size electric drive, including an electric motor, a battery bank, and power electronic converters, was added in parallel with this full-size engine.
Three commercially available IC engine vehicles with different curb masses and engine power ratings were chosen as representative passenger cars on the market. The original power ratings of the engines were kept for this hybridization.
To find the optimal power rating of the electric drive, we developed a DP algorithm to calculate the maximum MPG and find the optimal sequence of the engine operating points during the driving cycle. The optimality of the solution obtained from the algorithm was guaranteed by selecting the battery SOC and the number of gears in the engine transmission as the control variables of the algorithm.
The optimal window of electric drive power rating was found to be between 10 kW and 30 kW. This was shown to be independent of IC engine power rating and vehicle mass by analyzing the sensitivity of the maximum MPG to the electric drive power rating during each test driving cycle. Therefore, all passenger cars, regardless of their masses and engine sizes, can be universally optimally hybridized with one size of electric drive packages, further reducing the hybridization cost. The sensitivity of the maximum MPG to the electric drive power rating and the universality of the optimal window were further explained in terms of the distribution of the optimal engine operating points in the engine fuel consumption map and the distribution of the load operating points with respect to the power-speed profiles of the motor.
To show the advantages of the full-size engine HEVs proposed in this paper over the conventional IC-engine based vehicles and commercially available HEVs, comparisons were made with two types of popular passenger cars for fuel economy, acceleration performance, cost of power plants and cost payback mileage. The proposed full-size engine HEV has significantly better fuel economy as well as better performance than the equivalent IC engine vehicle. This HEV was also shown to have better performance and shorter cost payback than conventional HEVs.
Further research may be done to find a real-time control strategy for the full-size engine HEV. This will be independent of the driving cycle.