# Sizing of the Propulsion System for a Heavy-Duty Fuel Cell Commercial Vehicle

Authors: Weiguang Zheng; Weiwei Xin; Enyong Xu; Shuilong He; Jirong Qin; Heng Wang
Journal: Scientific Programming
 (2021)

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## Abstract

This paper presents a methodology for the sizing of a heavy-duty fuel cell commercial vehicle. The parameters scanning model and the long-term stochastic drive cycle are adopted for this proposed sizing framework. The dynamic programming algorithm is employed as the energy management strategy to assess the performance of sizing. The efficacy of this framework is evaluated, and a detailed analysis for the hydrogen consumption is given in the results. In addition, a prediction analysis based on the price performance of the next decade is also given in this work.


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## Body

## 1. Introduction

The energy and environment are two limitations for the transportation industry [1, 2]. The heavy-duty commercial vehicle is imperative to turn to new energy for its high fuel consumption and emission. As a fuel-renewable and environment-friendly on-board propulsion system, hydrogen fuel cells are of great significance for alleviating the current energy and environment dilemma.Restricted by the dynamic properties of fuel cell, fuel cell vehicles (FCVs) are always equipped with at least two energy sources: a fuel cell stack and a battery pack or a supercapacitor, which are also known as fuel cell hybrid electric vehicles (FCHEVs) [3]. Therefore, FCHEV is generally treated as a particular electric vehicle which equipped with two or three power sources [4]. Many approaches are presented in the literature to optimize the vehicle operating cost, which can be divided into two levels: the optimization level and the design level. The optimal operation aims to find the optimal operating points for fuel cell to achieve the minimum hydrogen cost, which is determined by the energy management strategy (EMS). The optimal design aims to determine the optimal sizes of vehicle components [5, 6].The key role of EMS for a FCHEV is to decide the power splitting for energy sources at each instant, while at the same time, the drivability constraints of vehicle must be satisfied [7]. There have been many researches on EMS to improve the fuel economy. In general, rule-based EMS and optimization-based EMS are two approaches widely studied [8, 9]. Under current technology, rule-based EMS has been widely used in real vehicles’ controller for its advantages in real time, simplicity, and cost. However, subjectivity and uncertainty of this strategy may lead the control rules far to the optimal rules [8]. On the other hand, thanks to the increasing computing power of on-board processors, more complex EMS has the potential to be applied in real vehicles; optimization-based EMS has become the focus of current academic research. Optimization-based EMS is often represented as optimization problems for constrained systems in finite-time domain; the common optimization algorithms include dynamic programming (DP) [10], Pontryagin’s minimum principle (PMP) [11], and genetic algorithm (GA) [12]. In addition, two optimization frameworks (equivalent consumption minimization strategy (ECMS) [13] and model prediction control (MPC) [14–16]) are also applied to this strategy to improve the control effect.In addition to the EMS, many researchers focused their attention on the sizing of energy sources to improve vehicle performance. The main purpose of sizing of energy sources is to optimize the dynamic and fuel economy with fewer cost; reasonable parameters can maximize the potential of a vehicle’s EMS [17]. With the development of electric vehicle, numbers of algorithms have been adopted to solve this problem. As early as 2004, Lukic and Emadi [18] verified the influence of drivetrain hybridization on energy consumption and dynamic performance. In their work, the hybridization factor was defined, and the EMS was implemented as charge sustaining. Kim and Peng [19] proposed a combined optimization method of EMS and parameters sizing for FCHEV. Different subsystem-scaling models were adopted to predict the characteristics of components on different sizes, and a near-optimal EMS was designed with the inspiration of stochastic dynamic programming results. In [20], the scholars constructed a stochastic drive cycle to simulate the randomness of real-world driving conditions, and ECMS was used as an online EMS; the equivalent factors of ECMS were adjusted with the proposed sizing methodology. In [21], a parameters’ sizing methodology based on ordinal optimization and dynamic programming was proposed. Hu et al. [22] gave a compared analysis for two optimization-based EMS, and the influence of downsizing the battery was also studied. In their further study, the influence of driving pattern on sizing was studied in [23]; the comparison of three different electrochemical energy storage systems and the sizing framework for hybrid electric vehicle are also given [24, 25]. In addition, Karaoğlan et al. [26] studied the influence of gear ratio on fuel economy and emissions.For the EMS, optimization-based strategy has been the focus of current research; and for the parameters sizing, systematic algorithms are used to find the optimal/near-optimal sizes of components for FCHEV; rule-based EMS or optimization-based EMS are adopted as an associated problem for these algorithms. Although existing research has proposed a number of approaches for parameters sizing, these approaches are based on the few drive cycles or only one standard drive cycle, and parameters’ sizing for hybrid heavy-duty commercial vehicles is little studied.In this paper, a sizing methodology for a fuel-cell/battery commercial vehicle is presented. It is based on the DP-based EMS and parameters scanning model, and a long-term stochastic drive cycle is adopted for the sizing framework. The main contributions are as follows: (a) A long-term stochastic drive cycle that is based on the cluster and stochastic procedure is adopted, and the processes of generate stochastic drive cycle are also available. (b) A detailed analysis on oversizing the fuel cell stack and battery is given. (c) A predictive analysis is also given with considering the price performance in the next decade.The remainder of this paper is organized as follows. In Section2, the model of a heavy-duty fuel cell commercial vehicle is established. In Section 3, the design process of the long-term stochastic drive cycle is proposed. In Section 4, the DP-based EMS is introduced. In Section 5, the sizing methodology of propulsion system is presented. Section 6, provides results and discussion, and in Section 7, the conclusion of this paper is given.


## 2. Mathematical Model of an FCHEV

In general, for an FCHEV, fuel cell stack is employed as the main power source, and battery is adopted as an energy storage system, which is also known as the auxiliary source. The structure of selected FCHEV is shown in Figure1. The electric motor is the only conversion equipment of electric energy and kinetic energy, fuel cell stack and battery pack composed of the propulsion system to provide power to permanent magnet synchronous motor and the motor’s inverter (DC-AC), and a DC-DC converter is used to boost the voltage from fuel cell stack to DC bus. In this work, the powertrain system is divided into a combined propulsion subsystem and a transmission subsystem. The parameters sizing of propulsion system is the main work of this paper.Figure 1
Block diagram of FCHEV. The blue dotted lines, red thin solid lines, and black thick solid lines represent the hydrogen gas flow channel, the electrical connections and mechanical connections of FCHEV, respectively.


### 2.1. Vehicle and Drivetrain

The mathematical model building of FCHEV is reversed with the target of tracking drive cycle, and the impacts of lateral dynamics and rotating mass are ignored. At every moment of vehicle’s operation, the torqueTw and speed ωw of wheels are given by(1)Tw=rmdvdt+12ρCdAfv2+mgfcosα+mgsinα,ωw=vr,where r is the rolling radius of the wheel, m the FCHEV’s mass with full load; v the vehicle’s velocity at the current moment; ρ the air density; Af the frontal area; Cd the aerodynamic drag’s coefficient; f the rolling resistance coefficient; α the road’s inclination angle. The motor’s torque Tm, speed ωm, and power Pm are formulated as follows:(2)Tm=Twηfdrfd,Tw≥0,Twηfdrfd,Tw<0,ωm=ωwrfd,Pm=Tmωmηm,Tm≥0,Tmωmηm,Tm<0,where ηfd and rfd are the efficiency and gear ratio of the final drive, respectively.For this work, the motor power is defined as positive during traction while negative during breaking. The propulsion system consists of fuel cell stack and battery pack outputs the power demand of the motor, as shown in the following formula:(3)Pm=Pstack+Ppack,where Pstack and Ppack are the power of fuel cell stack and battery pack. Parameters of the heavy-duty fuel cell commercial vehicle are listed in Table 1.Table 1
Parameters of the FCHEV.
ParametersValueVehicle total mass (kg)9000Air density (kg/m3)1.2Aerodynamic drag coefficient0.563Vehicle frontal area (m2)6.6454Wheel radius (m)0.413Gear ratio6.67Efficiency of transmission (%)98Rolling resistance coefficient0.015Efficiency of converter (%)98Efficiency of inverter (%)98Efficiency of charge (%)90Efficiency of discharge (%)100


### 2.2. Fuel Cell Stack

Fuel cell is an electrical device that converts chemical energy into electricity. Proton exchange membrane fuel cell (PEMFC) is the most common on-board propulsion system for fuel cell vehicles. A complete on-board PEMFC system includes fuel cell stack, hydrogen storage system, hydrogen circuit, air circuit, water circuit, and coolant circuit [3]. The models for these systems are very complicated. This work is mainly concerned with the power-split of the bus but not the detailed conversion process of fuel cells. Therefore, a simple efficiency graph model is used for the simulation.In this selected model, hydrogen consumption and efficiency of the fuel cell stack are both formulated as the function of power demand; these datasets are obtained through testing, which have been provided in ADVISOR (FC_ANL50H2) [27]. Scaling models are adopted to facilitate parameters’ sizing, as shown in Figure 2.Figure 2
Scaling models of fuel cell stack (on the basis of a fuel cell stack with 50 kW peak power on ADVISOR). (a) Hydrogen consumption at different stack power. (b) Efficiency at different stack power.
(a)(b)


### 2.3. Battery Pack

Benefiting from the high power density and high reliability, lithium-ion batteries have become the most commonly used energy storage system for FCHEV. The purpose of adding a battery pack into the propulsion system is to improve the drivability and efficiency performance of the vehicle.For the model of battery pack, the circuit model is used for simulation. In this model, the battery is equivalent to a circuit with an ideal voltage source and an internal resistance. When the battery pack is operating, its current and power are formulated as follows:(4)Ibat=Voc−Voc2−4RintPpack2Rint,where Voc is the open circuit voltage, Ibat the current, Rint the internal resistance, and Ppack the power of battery pack. These parameters can be obtained from equations (5) to (7).(5)Ppack=Pbatηbat,(6)Voc=Vocnsnp,(7)Rint=rintnsnp,where ηbat is the conversion efficiency for charge/discharge, ns the series number of cells, and np the parallel number of cells. The state of charge (SOC) of battery at time step k is defined as(8)sock=sock−1−IbatQbat,where Qbat is the capacity of battery pack.Rint and Ppack are the functions of SOC, as shown in Figure 3.Figure 3
Circuit voltage and internal resistance of the cell. (a) Resistances of cell for different SOC in charge and discharge state. (b) Open circuit voltage of cell for different SOC.
(a)(b)It is worth mentioning that the battery pack is composed of strings connected in parallel, with each string containing the same number of cells in series. More detailed descriptions about configuration and modelling of battery pack are available in [23].


### 2.4. Drive Motor

For an FCHEV, the drive motor’s function mainly includes the following two parts: (a) converting the electrical energy from bus to mechanical energy to drive the vehicle and (b) recycling the vehicle’s braking power to charge the battery. A permanent magnet synchronous motor (PMSM) is adopted for its compact and high efficiency. The model of drive motor is based on its efficiency map, as shown in Figure4.Figure 4
Efficiency map of PMSM.


## 2.1. Vehicle and Drivetrain

The mathematical model building of FCHEV is reversed with the target of tracking drive cycle, and the impacts of lateral dynamics and rotating mass are ignored. At every moment of vehicle’s operation, the torqueTw and speed ωw of wheels are given by(1)Tw=rmdvdt+12ρCdAfv2+mgfcosα+mgsinα,ωw=vr,where r is the rolling radius of the wheel, m the FCHEV’s mass with full load; v the vehicle’s velocity at the current moment; ρ the air density; Af the frontal area; Cd the aerodynamic drag’s coefficient; f the rolling resistance coefficient; α the road’s inclination angle. The motor’s torque Tm, speed ωm, and power Pm are formulated as follows:(2)Tm=Twηfdrfd,Tw≥0,Twηfdrfd,Tw<0,ωm=ωwrfd,Pm=Tmωmηm,Tm≥0,Tmωmηm,Tm<0,where ηfd and rfd are the efficiency and gear ratio of the final drive, respectively.For this work, the motor power is defined as positive during traction while negative during breaking. The propulsion system consists of fuel cell stack and battery pack outputs the power demand of the motor, as shown in the following formula:(3)Pm=Pstack+Ppack,where Pstack and Ppack are the power of fuel cell stack and battery pack. Parameters of the heavy-duty fuel cell commercial vehicle are listed in Table 1.Table 1
Parameters of the FCHEV.
ParametersValueVehicle total mass (kg)9000Air density (kg/m3)1.2Aerodynamic drag coefficient0.563Vehicle frontal area (m2)6.6454Wheel radius (m)0.413Gear ratio6.67Efficiency of transmission (%)98Rolling resistance coefficient0.015Efficiency of converter (%)98Efficiency of inverter (%)98Efficiency of charge (%)90Efficiency of discharge (%)100


## 2.2. Fuel Cell Stack

Fuel cell is an electrical device that converts chemical energy into electricity. Proton exchange membrane fuel cell (PEMFC) is the most common on-board propulsion system for fuel cell vehicles. A complete on-board PEMFC system includes fuel cell stack, hydrogen storage system, hydrogen circuit, air circuit, water circuit, and coolant circuit [3]. The models for these systems are very complicated. This work is mainly concerned with the power-split of the bus but not the detailed conversion process of fuel cells. Therefore, a simple efficiency graph model is used for the simulation.In this selected model, hydrogen consumption and efficiency of the fuel cell stack are both formulated as the function of power demand; these datasets are obtained through testing, which have been provided in ADVISOR (FC_ANL50H2) [27]. Scaling models are adopted to facilitate parameters’ sizing, as shown in Figure 2.Figure 2
Scaling models of fuel cell stack (on the basis of a fuel cell stack with 50 kW peak power on ADVISOR). (a) Hydrogen consumption at different stack power. (b) Efficiency at different stack power.
(a)(b)


## 2.3. Battery Pack

Benefiting from the high power density and high reliability, lithium-ion batteries have become the most commonly used energy storage system for FCHEV. The purpose of adding a battery pack into the propulsion system is to improve the drivability and efficiency performance of the vehicle.For the model of battery pack, the circuit model is used for simulation. In this model, the battery is equivalent to a circuit with an ideal voltage source and an internal resistance. When the battery pack is operating, its current and power are formulated as follows:(4)Ibat=Voc−Voc2−4RintPpack2Rint,where Voc is the open circuit voltage, Ibat the current, Rint the internal resistance, and Ppack the power of battery pack. These parameters can be obtained from equations (5) to (7).(5)Ppack=Pbatηbat,(6)Voc=Vocnsnp,(7)Rint=rintnsnp,where ηbat is the conversion efficiency for charge/discharge, ns the series number of cells, and np the parallel number of cells. The state of charge (SOC) of battery at time step k is defined as(8)sock=sock−1−IbatQbat,where Qbat is the capacity of battery pack.Rint and Ppack are the functions of SOC, as shown in Figure 3.Figure 3
Circuit voltage and internal resistance of the cell. (a) Resistances of cell for different SOC in charge and discharge state. (b) Open circuit voltage of cell for different SOC.
(a)(b)It is worth mentioning that the battery pack is composed of strings connected in parallel, with each string containing the same number of cells in series. More detailed descriptions about configuration and modelling of battery pack are available in [23].


## 2.4. Drive Motor

For an FCHEV, the drive motor’s function mainly includes the following two parts: (a) converting the electrical energy from bus to mechanical energy to drive the vehicle and (b) recycling the vehicle’s braking power to charge the battery. A permanent magnet synchronous motor (PMSM) is adopted for its compact and high efficiency. The model of drive motor is based on its efficiency map, as shown in Figure4.Figure 4
Efficiency map of PMSM.


## 3. Long-Term Stochastic Drive Cycle

The drive cycle refers to the driving characteristics of a certain type of vehicle (passenger car, bus, and cargo vehicle, etc.) in a specific condition (urban, suburban, highway, etc.); it is generally a set of points representing speed versus time. There have been many systems of typical driving cycles around the world.Although the representativeness of cycles will be considered in the design process of typical drive cycle, the optimization results obtained under few drive cycles cannot satisfy the randomness of real driving conditions [20]. Therefore, a long-term stochastic drive cycle is built to overcome this drawback. The constructed drive cycle is based on the typical drive cycles of heavy commercial vehicles, and cluster analysis and random process are adopted. The flowchart of this procedure is shown in Figure 5.Figure 5
Flowchart of long-term stochastic drive cycle generation. DC: drive cycle.In this work, eight typical drive cycles for commercial vehicles are introduced as the DC library, which are numbered and shown in Figure6. Twelve characteristic parameters are used for clustering, and the corresponding descriptions of these parameters are listed in Table 2. The characteristic parameters for each drive cycle are calculated and listed in Table 3. These parameters will be the raw data of clustering.Figure 6
The serial number of the selected drive cycles.Table 2
The characteristic parameters used for clustering.
ParameterDenotationAverage velocity (m/s)v¯Average acceleration (m/s2)a¯Average deceleration (m/s2)d¯Standard deviation of velocityσvMaximum velocity (m/s)vmaxMaximum acceleration (m/s2)amaxMaximum deceleration (m/s2)dmaxThe percentage of time in speed interval 0–20 km/h (%)θ0−20The percentage of time in speed interval 20–40 km/h (%)θ20−40The percentage of time in speed interval 40–60 km/h (%)θ40−60The percentage of time in speed >60 km/h (%)θ60−maxThe percentage of time in idle state (%)θ0Table 3
The characteristic parameters that are calculated for each drive cycle.
DenotationDC1DC2DC3DC4DC5DC6DC7DC8v¯11.253.7815.227.198.426.073.929.62a¯0.230.300.200.330.480.390.470.22d¯0.280.390.210.420.580.460.540.29σv7.814.589.776.568.875.824.237.61vmax24.2516.0127.1520.0325.9319.5816.0924.58amax0.821.141.421.291.961.162.061.22dmax1.172.241.862.162.071.791.951.25θ0−2021.9341.2615.8524.8616.4931.5633.9822.56θ20−4023.1017.199.2113.5115.8319.4325.6322.67θ40−6021.6511.2911.8328.5312.0624.767.2820.61θ60−max26.49053.847.9322.342.64019.89θ06.3830.269.2725.1733.2721.6233.1114.28Hierarchical clustering and Euclidean distance are adopted to obtain the similar degree of the different drive cycles. The clustering results are shown in Figure7. The library is divided into 5 classes with the benchmark of similarity to 1: the first category includes drive cycles 1, 4, 6, and 8, and drive cycles 2, 7, 3, and 5 are classified as the second to fifth categories, respectively. The drive cycles 8 (CHTC_HT), 2 (WVUCITY), 7 (NYCOMP), 3 (WVUINTER), and 5 (UDDSSHCV) are selected as the representatives of each category. These five drive cycles will be regarded as the originals in the long-term stochastic drive cycle.Figure 7
Results of hierarchical clustering.The next step for this work is to build the combined drive cycle, which is generated from the original drive cycles. After that, the combined drive cycle will be divided into microtrips and microidles. A microtrip is a segment of the drive cycle where the velocity is not equal to zero, and a microidle is the segment where the velocity is equal to zero. The relationship between microtrips and microidles is depicted in Figure8. Then, the divided segments are added to the stochastic drive cycle randomly. In this process, the microtrips and microidles are scaled with a stochastic procedure. The scaling processes for microtrips are for their velocity, and for microidles, it is for their length of time, the scaling factors are both defined as a stochastic number from 0.8 to 1.2. To follow the objective laws for drive cycles, the microtrip and microidle should be added in turn, which means after adding a microtrip, a microidle must be added as its follow-up. In addition, the long-term stochastic drive cycle can more effectively reflect the randomness of the drive cycle, the total length of stochastic drive cycle is set as 10 times that in the combined drive cycle in this work.Figure 8
Relationship of the microtrips and microidles. The MI(k) and MT(k) represent the kth microidle and microtrip, respectively.Another noteworthy point for this process is that after the scaling of a microtrip, three indexes should be detected: the maximum speed, maximum acceleration, and maximum deceleration. If any index is higher than the combined drive cycle, the scaling process should be repeated until satisfied.After completing the construction of long-term stochastic drive cycle, evaluation indexes are introduced to evaluate the rationality. Figure9 illustrates the distribution of two drive cycles; Table 4 depicts the comparison of characteristics of combined drive cycle and long-term stochastic drive cycle. The above charts show that the long-term stochastic drive can accurately reflect the characteristics of original cycles.Figure 9
The distribution of velocity and acceleration for two drive cycles. (a) The percentage of speed for combined drive cycle. (b) The percentage of speed for long-term stochastic drive cycle. (c) The percentage of acceleration for combined drive cycle. (d) The percentage of acceleration for long-term stochastic drive cycle.
(a)(b)(c)(d)Table 4
The characteristics of two drive cycles.
CharacteristicsCombined DCStochastic DCDuration (s)693969393Idle time (%)22.0324.38Maximum velocity (m/s)27.1527.08Average velocity (m/s)8.736.85Standard deviation of velocity8.667.06Maximum acceleration (m/s2)2.072.06Maximum deceleration (m/s2)3.243.17Average acceleration (m/s2)0.290.28Average deceleration (m/s2)0.350.34


## 4. DP-Based EMS

In general, the EMS for FCHEV has a great impact on efficiency. In this work, an optimal EMS based on dynamic programming algorithm is adopted to assess the economic potential of sizing.Dynamic programming is a numerical method for solving multistage decision-making problems. It can provide the optimal solution for problems of any complexity level in the limits of computational capabilities [7].For a controlled multistage decision-making problem, the state transfer function can be described as(9)sk+1=fksk,uk,where sk is the state variable at time k and uk is the control variable. For a control policy u=u0,u1,…uN−1, the cost in time step N is defined as(10)Js0,u=LNsN+∑k=1N−1Lksk,uk,where Lk is the instantaneous cost function. The optimal cost is(11)J∗s0=minJs0,u,u.The optimal control policyu∗=u1∗,u2∗,…,uN−1∗.For the DP-based EMS for FCHEV, the SOC of battery is selected as the state variable, and the output power of fuel cell stack is adopted as the control variable. The state transfer function is defined as follows:(12)SOCk+1=SOCk+hkPstack,wherehkPstack is the change rate of state in time k, which can be obtained with equations (7) to (11).For the structure of non-plug-in FCHEV, a constraint of terminal SOC is required to sustain the energy storage system. In principle, the constraint of terminal SOC can be taken into account in two different ways, namely, as a soft constraint or a hard constraint. In this study, a hard constraint is adopted by requiring that the energy storage stored at the end of the mission equal to the value at the start of the mission, as shown in equation (13). A more detailed explanation will be found in [28–30]. Other parameters under constraints are listed in equation (14). In particular, the SOCmin and SOCmax are set to 0.4 and 0.8, respectively.(13)SOCend=SOC0,(14)Tm_min≤Tm≤Tm_max,ωm_min≤ωm≤ωm_max,Ibat_min≤Ibat≤Ibat_max,Pbat_min≤Pbat≤Pbat_max,Pstack_min≤Pstack≤Pstack_max,SOCmin≤SOC≤SOCmax.


## 5. Structure of Parameters’ Sizing

The objective of sizing is to obtain the near-optimal sizes of maximum power of fuel cell stackPstack_max and the number of battery cell nbat. The sizes of propulsion should satisfy the following objectives: (a) meet the drivability and (b) minimize the operating cost of the vehicle during its life cycle.For drivability, passenger cars typically have higher velocity and acceleration targets, while for commercial vehicles, equipped with large load mass, the vehicle’s climbing performance always receives more attention. Therefore, the sizing method is tested with three subobjects: (a) The peak power of fuel cell stack alone must be able to sustain the maximum velocity; (b) the peak power with fuel cell stack and battery pack together must meets the maximum climbing slope index at a given velocity; (c) the power with fuel cell stack and battery pack together should fulfil the power demand of long-term stochastic drive cycle. The first subobject is used to obtain the low limiting value ofPstack_max, and the others are used to determine nbat. It should be noted that the for the subobject (c), the verification results are related to the selected EMS, DP-based strategy is employed for this work, which has been introduced in Section 4.This sizing methodology is based on the generated long-term stochastic drive cycle, a parameters’ scanning model is added for sizing, and the dynamic programming algorithm is adopted for the EMS. The flowchart of this proposed methodology is shown in Figure10.Figure 10
Flowchart of the sizing methodology.In this sizing methodology, the range of fuel cell stack’s peak power is divided intoimax independent nodes; the range of battery cell’s number is divided into jmax independent notes; thses two groups of nodes constitute a crossover model with imax×jmax crossover points, and for each crossover point, there is an FCHEV model corresponding to it. The optimal cost for each size is obtained by DP-based EMS. In addition, the low limiting value of Pstack_max should meet the power demand of the vehicle at a maximum velocity (equation (15)), and the peak-powers provided by the battery and fuel cell should meet the requirements of the vehicle’s climbing performance (equation (16)). The peak power of the battery pack can be expressed as the product of its capacity and the maximum discharge rate equation (17):(15)Pstack_maxηcon≥Pvη¯mηinv,(16)Pstack_maxηcon+Pbat_maxηbat≥Pcη¯mηinv,(17)Pbat_max=QbatCmax,where Pv is the power demand for maximum velocity, Pc is the power demand for climbing, η¯m is the average efficiency of motor, ηbat is the charge/discharge efficiency of battery, and ηinv and ηcon are the efficiency of DC-AC inverter and DC-DC converter, respectively.


## 6. Results and Discussion

This simulation is carried out with MATLAB; the interval is set to 10 kW and 25 for the fuel cell stack’s peak power and the battery cells’ number, respectively.Under the current technical, lithium-ion batteries can maintain a discharge time of 5 s at the peak discharge rate of 30 C [31]. However, high rate discharge will seriously affect the life of the battery. In this paper, the maximum discharge rate of the battery is set as 20 C. In addition, the capacity of the battery cell used in the simulation is 3 Ah. Therefore, on the basis of equations (18) to (20), the minimum constraints of Pstack_max and nbat will be estimated, as shown in Figure 11.Figure 11
The minimum constraints of thePstack_max and nbat.Figure12 shows the hydrogen consumption of vehicles with different sizes under long-term stochastic drive cycle. For a fair comparison, all the hydrogen costs are computed with the SOC deviation. Two conclusions can be obtained according to this figure:(1)
Starting fromPstack_max=70 kW, the hydrogen consumption decreases with the increasing of the peak power of fuel cell stack, but it increases when the peak power exceeds 120 kW(2)
For a determinedPstack_max, the hydrogen consumption decreases with the increasing of the number of battery cells, but the slope of decline gradually decreasesFigure 12
Hydrogen consumption in different sizes (Ps: Pstack_max).To understand the causes of the changing of hydrogen consumption, we consider three relevant performance indicators: the global propulsion efficiency, the braking-recovery-energy to negative-load-power ratio, and the global fuel cell stack efficiency.The global propulsion efficiency is formulated as follows:(18)ηg,pro=∑i=0NPveh+iΔtLHVH2∑i=0NmH2iΔt×100,where Pveh+ is the positive vehicle power, N is the length of stochastic drive cycle, mH2 is the hydrogen consumption, and Δt is the simulation step.The braking-recovery-energy to negative-load-power ratio is defined as follows:(19)γre=∑i=0NPreiΔt∑i=0NPveh−kΔt×100,where Pveh− is the negative vehicle power and Pre is the braking energy recovered by battery pack, which is formulated as follows:(20)Pre=Pbat,ifPveh<0,0,otherwise.The global fuel cell stack efficiency is defined as follows:(21)ηg,FC=∑i=0NPstackiΔtLHVH2∑i=0NmH2iΔt×100.To make the figure clearer, four types of fuel cell stack peak power are selected for this work:Pstack_max=70kW,Pstack_max=100kW,Pstack_max=120kW, andPstack_max=170kW. The rest results of other types are visible in Figure 13.Figure 13
Comparative results of optimal costs.
(a)(b)(c)Figure14 shows the effect of Pstack_max and nbat on defined relevant performance indicators. Figure 14(a) shows the maximum global propulsion efficiency is approximately 55.56%, which is reached when Pstack_max=120 kW and nbat>1000. Increasing or decreasing the type of fuel cell will worsen this indicator. On the other hand, increasing the number of battery cells can improve the global efficiency, but its growth trend gradually slows down and eventually to level. Figure 14(b) shows the size of fuel cell stack has less affected in braking-recovery-energy, while the number of battery cells has a great impact on this indicator: when nbat<400, limited by the physical constraints (e.g., the maximum SOC or maximum voltage), the battery pack is not enough to recover all the braking energy. This means that friction brakes will dissipate power from fuel cell stack, which is not friendly to the global efficiency. Figure 14(c) shows that the global fuel cell stack efficiency performs best when Pstack_max=120 kW, and the increase in nbat will benefit this indicator. Similarly, the growth trend gradually slows down with the increase of nbat.Figure 14
The test results of performance indicators with increasing cell number in different fuel cell sizes. (a) The global propulsion efficiency. (b) The braking-recovery-energy to negative-load-power ratio. (c) The global fuel cell stack efficiency.
(a)(b)(c)The results in Figure10 clarifies that the reduction of hydrogen can be achieved with an appropriate Pstack_max, and the increase of nbat will also benefits the fuel economy. It is beneficial to reduce the running cost, but the sharp increase in vehicle’s price led by the increase of fuel cell and battery is unacceptable for both manufactures and users. Vehicle’s price should also be considered as a further indicator.To balance the cost of hydrogen consumption and component costs, a cost function which is defined as the cost per 100 km of the propulsion system is proposed, as shown in equation (18).(22)J100=Jstack100+Jbat100+JH2100.In this function, the vehicle’s costJ100 is divided into fuel cell stack cost Jstack100, battery cost Jbat100, and hydrogen cost JH2100. The fuel cell stack cost and the battery cost are cycle-normalized cost, as expressed in equations (19) and (20) [28, 29]. In addition, the cost of hydrogen is the value of hydrogen consumed when the vehicle’s total driving distance reaches 100 km in the constructed long-term stochastic drive cycle, as shown in equation (21).(23)Jstack100=dsyv1+Pcyv+12jstackPstack_max,Jbat100=dsyv1+Pcyv+12jbatQbat,JH2100=dXdcjH2mH2,where d is the length of cycle, Xdc is the distance under a stochastic drive cycle, s is the average travelled distance of the vehicle per year, yv is the vehicle lifetime, and Pc is the yearly interest rate. Here, it is assumed that there are no components replaced during three-year operation, and the corresponding travelled distance is fifty thousand kilometres per year. These parameters above are given in Table 5. jstack is the fuel cell stack price per kilowatt, jbat is the battery price per kilowatt hour, and jH2 is the hydrogen price per gram. Three price conditions on the years 2020, 2025, and 2030 [32] are chosen for this study, as shown in Table 6.Table 5
The parameters of simulation.
DescriptionParametersValueLength of cycle (km)d100Yearly travelled distance (km)s5e4Vehicle lifetime (years)yv3Yearly interest rate (%)Pc5Stochastic cycle distance (km)Xdc269Table 6
The price trends of components cost.
YearFuel cell price ($/kW)Battery price ($/kW)Hydrogen price ($/kg)202070.412565.32202537.001735.05203029.841175.03Figure15 shows the price impact on vehicle’s total cost. Figure 14(a) shows the vehicle total cost with the price in 2020. In this case, the unit prices of fuel cells and battery cell are higher; the optimal cost is 25.07 US dollars per 100 km with the corresponding size to Pstack_max  = 70 kW and nbat  = 696. The cost previews of 2025 and 2030 are shown in Figure 14(b) and 14(c) respectively. On the preview of 2025, benefiting from the production increases and technology improves, the unit price of fuel cell and battery will decrease rapidly, and the optimal cost will drop to 21.46 US dollars per 100 km with the corresponding size to Pstack_max  = 80 kW and nbat  = 640. In 2030, with the further decrease of component prices, the optimal cost will drop to 20.41 US dollars per 100 km. The optimal sizes and costs for different price conditions are summarized in Table 7.Figure 15
Performance indicators with increasing cell number in different fuel cell sizes. (a) The global propulsion efficiency. (b) The braking-recovery-energy to negative-load-power ratio. (c) The global fuel cell stack efficiency.
(a)(b)(c)Table 7
The optimal sizes under different price conditions.
202020252030Optimal cost ($/100 km)25.0721.4620.41Fuel cell size (kW)708080Cell number696640640String number444Bus nominal voltage626576576Cell number per string174160160Figure16 gives the comparative results of optimal costs under three price conditions. Based on the price trends in future, the optimal cost gradually decreases from 2020 to 2030. Compared to 2020, the decreases have reached 14.4% and 18.59% in 2025 and 2030, respectively. The main reason is the sharp decrease of fuel cell and battery price. At the same time, benefiting from the decrease of fuel cell and battery price, the proportion of lifetime cost decreases rapidly. For example, based on the price performance in 2020, the proportion of lifetime cost is 25.56%, and it decreases to 18.26% and 14.35% in 2025 and 2030, respectively. Correspondingly, the hydrogen costs are 74.44%, 81.74%, and 85.65% for the price performance in 2020, 2025, and 2030, respectively. Along with the decreasing of hydrogen and components price, the hydrogen cost will be the more important performance index for vehicle cost in the near future.Figure 16
Vehicle costs for 100 km in different price conditions. (a) In 2020. (b) In 2025. (c) In 2030.


## 7. Conclusion

This paper presents a sizing methodology for a heavy-duty fuel cell commercial vehicle on the basis of long-term stochastic drive cycle, parameters’ scanning model, and DP-based EMS. The causes of hydrogen consumption under different sizes are studied in detail, and a discussion of the vehicle cost in the next decade is also given, on the basis of propulsion system and hydrogen price performance.The main findings are as follows:(1)
The hydrogen consumption will be minimized when the peak power of fuel cell stack reaches 120 kW, and the global propulsion efficiency will be improving with the battery cell number increase.(2)
When the number of battery cells is small, the global propulsion efficiency is affected by the braking-recovery-energy and the fuel cell stack efficiency, while as the number of battery cells continues to increase, the main cause for that is the fuel cell stack efficiency. Limited by the maximum efficiency of fuel cell stack, the effect of increasing the battery cells number on the global propulsion efficiency will gradually decrease.(3)
The size ofPstack_max  = 70 kW and nbat  = 696 will be a better choice for the vehicle total cost, under the consideration of the cost performance in 2020. The predictive analysis shows that the larger fuel cell stack will be suitable for the FCHEV with the decrease of fuel cell and battery price.



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