Multi-objective component sizing based on optimal energy management strategy of fuel cell electric vehicles

Applied Energy xxx (2015) xxx–xxx

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Multi-objective component sizing based on optimal energy management strategy of fuel cell electric vehicles q Liangfei Xu a,c,1, Clemens David Mueller a,b,1, Jianqiu Li a,c,1, Minggao Ouyang a,⇑, Zunyan Hu a,1 a

State Key Lab of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 100084, PR China Institute of Aeronautics and Astronautics, RWTH, Aachen 52062, Germany c Collaborative Innovation Center of Electric Vehicles in Beijing, PR China b

h i g h l i g h t s  A non-linear model regarding fuel economy and system durability of FCEV.  A two-step algorithm for a quasi-optimal solution to a multi-objective problem.  Optimal parameters for DP algorithm considering accuracy and calculating time.  Influences of FC power and battery capacity on system performance.

a r t i c l e

i n f o

Article history: Received 22 August 2014 Received in revised form 1 February 2015 Accepted 5 February 2015 Available online xxxx Keywords: Electrolyte membrane fuel cell electric vehicle Parameter sizing Energy management Dynamic programming Fuel economy Durability

a b s t r a c t A typical topology of a proton electrolyte membrane (PEM) fuel cell electric vehicle contains at least two power sources, a fuel cell system (FCS) and a lithium battery package. The FCS provides stationary power, and the battery delivers dynamic power. In this paper, we report on the multi-objective optimization problem of powertrain parameters for a pre-defined driving cycle regarding fuel economy and system durability. We introduce the dynamic model for the FCEV. We take into consideration equations not only for fuel economy but also for system durability. In addition, we define a multi-objective optimization problem, and find a quasi-optimal solution using a two-loop framework. In the inside loop, for each group of powertrain parameters, a global optimal energy management strategy based on dynamic programming (DP) is exploited. We optimize coefficients for the DP algorithm to reduce calculating time as well as to maintain accuracy. For the outside loop, we compare the results of all the groups with each other, and choose the Pareto optimal solution based on a compromise of fuel economy and system durability. Simulation results show that for a ‘‘China city bus typical cycle,’’ a battery capacity of 150 Ah and an FCS maximal net output power of 40 kW are optimal for the fuel economy and system durability of a fuel cell city bus. Ó 2015 Published by Elsevier Ltd.

1. Introduction Most cars and buses are driven by internal combustion engines (ICEs) running either on gasoline, diesel, or natural gas [1]. The combustion process emits pollutants such as CO2 or NOX, causing much damage to the environment and human health [2]. Furthermore, the reserves and resources of crude oil are limited and it is likely that the current supply will not be able to satisfy

q This paper is included in the Special Issue of Clean Transport edited by Prof. Anthony Roskilly, Dr. Roberto Palacin and Prof. Yan. ⇑ Corresponding author. Tel.: +86 10 62773437; fax: +86 10 62785708. E-mail addresses: [email protected] (L. Xu), clemens.mueller@ rwth-aachen.de (C.D. Mueller), [email protected] (J. Li), ouymg@tsinghua. edu.cn (M. Ouyang), [email protected] (Z. Hu). 1 Tel.: +86 10 62785706, +86 10 62773437; fax: +86 10 62785708.

increased demands within the next decades. A solution must be developed that changes the energy system for vehicles. Currently, there are several candidates: plug-in hybrid, battery electric, and fuel cell electric vehicles (FCEVs). FCEVs have attracted much attention in the past few years because they run on hydrogen and the only output is pure water. They are highly energy-efficient, have zero emission, and are very quiet; FCEVs are a very promising way to provide sustainable transportation [3,4]. A typical topology of an FCEV contains at least two power sources, a fuel cell system (FCS) and an energy storage system (ESS), e.g. a lithium battery system or a super capacitor (SC). The FCS provides stationary power, and the ESS delivers dynamic power. For a pre-defined driving cycle, two factors determine performances in fuel economy and system durability of an FCEV: parameter sizing and energy management strategy (EMS). EMS is widely studied in several fields besides new energy vehicle

http://dx.doi.org/10.1016/j.apenergy.2015.02.017 0306-2619/Ó 2015 Published by Elsevier Ltd.

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powertrains, e.g. wind power system, microgrid systems or green buildings [5–12]. There is a sole optimized EMS for a pre-defined driving cycle and cost function corresponding to one group of powertrain parameters. These two problems always couple with each other. When we solve a parameter sizing problem, theoretically, we cannot avoid the optimized EMS problem. There have been many papers on parameter sizing in literature. We can separate these papers into several categories, depending on the consideration of optimal EMS when choosing powertrain parameters. Some authors tend to ignore the EMS and focus on the optimal sizing problem to reduce the computational load [13]. Wu et al. [14] proposed a methodology for optimal component sizing using a parallel chaos optimization algorithm (PCOA). Using the PCOA, the authors defined a cost minimization problem and regarded the requirements on vehicle performance as constraints. Cai et al. [15] presented a sizing-design methodology based on a flow diagram, and applied it to an unmanned underwater vehicle. Parameters for the ESS, and the fuel cell system were optimized. Eren et al. [16] studied the multi-objective optimum sizing of hybrid electric vehicles. The EMS was simply treated based on power flow analysis. Using this method, the authors defined a multi-objective problem and solved it using the mixed integer linear programming (MILP) method. Some authors try to include a simple EMS, e.g. rule-based or thermostat strategy. Doucette et al. [17] compared the cost and fuel economy of integrating high-speed flywheels, batteries, or ultracapacitors with an FCS into an FCEV. Rule-based strategies were developed for different powertrain parameters. Ribau et al. [18] proposed the parameter sizing method for fuel cell city buses based on efficiency, cost, and life cycle CO2. Analysis was conducted in ADVISOR with the default control strategy of thermostat. Ravey et al. [19] gave a new methodology based on energy flow analysis for a statistical description of driving cycles. A fuzzy logic-based EMS was developed to verify the effectiveness of the design. Sorrentino et al. [20] presented an integrated tool for component sizing of FCEVs. A thermostatic EMS was included in the model, and several parameters of the strategy were optimized. Kim et al. [21] set up an FCEV model with fuzzy control EMS and optimized powertrain parameters by comparing simulation results with different parameters. Cipollone et al. [22] presented a modelbased design and an optimization method. Other authors tried to optimize the powertrain parameters and EMS simultaneously. Murgovski et al. [23] solved the component sizing and energy management problem via convex optimization. Compared to dynamic programming (DP), convex optimization has similar results, and no curse of dimensionality. Similar results were also achieved by Hu et al. [24], who proposed the optimal sizing and EMS for an FCEV. These authors proposed a convex optimization framework and used a CVX tool for parameter optimization. Jain et al. [25] studied an optimal component sizing of an FCEV using a multi-objective generic algorithm regarding fuel economy and vehicle performance. Several Pareto optimal solutions were found for the vehicle. However, this study did not incorporate system durability. Vasallo et al. [26] studied the problem of optimal sizing for UPS systems based on battery and FCS. An optimal EMS was involved in the whole framework, but not explained in detail. Masoud et al. [27] proposed the problem for optimum sizing and EMS for battery life improvement. A DP algorithm was adopted as the optimal EMS. Hung et al. [28] studied a combined optimal sizing and EMS for in-wheel motors of EVs. They used a global search method (GSM) to solve the problem. Kim et al. [29] suggested a comprehensive and systematic framework to optimize EMS and component sizing simultaneously for FCEVs. A near-optimal EMS based on statistic dynamic programming (SDP) was chosen in simulation.

For EMS research, recent papers mostly focused on an optimized algorithm with single- or multi-objectives. Hemi et al. [30] presented an EMS for an FCEV based on Pontryagin’s Minimal Principle (PMP) and the Markov chain. The Markov chain was adopted for power prediction in the algorithm. Chen et al. [31] gave a rule-based multi-mode algorithm for a range-extended electric vehicle. The rules-based strategy was developed according to a two-point boundary DP algorithm. Driving pattern recognition technology was also developed based on this strategy. Zheng et al. [32] studied an optimal control EMS based on PMP for an FCEV. Results show that a constant co-state can be used instead of a PMP algorithm because the open circuit voltage and resistance of the battery is kept almost constant during the operation. Segura et al. [33] proposed an EMS based on sliding control theory for the DC converter, which joins constant and variable frequency control technologies. Trovao et al. [34] presented a multi-level EMS for a multi-source electric vehicle. The strategy is separated into energy level and power level, and is mainly developed by using a rule-based algorithm. From previous analyses, we derive the following viewpoints. (1) EMS is always coupled with component sizing for a hybrid electric vehicle. It can be simplified using some rule-based strategies or near-optimal strategies, or treated as a lowlevel optimization problem in the whole component sizing optimization loop. (2) Some algorithms, such as convex optimization, can solve the combined optimized problem for component sizing and EMS simultaneously. This paper proposes a multi-objective optimization method for parameter sizing of an FCEV. An optimized EMS based on DP is included in the component sizing problem. Section 2 describes the powertrain topology of a fuel cell city bus, and presents the dynamic model regarding fuel economy and system durability. Section 3 defines the problem of parameter sizing, and introduces a two-loop framework based on Pareto optimization. The inside loop is built on a DP-based optimal EMS, and the outside loop is set up on comparison of different powertrain parameters for fuel economy and system durability. Section 4 gives the simulation results, and Section 5 is the conclusion.

2. Powertrain description and dynamic model 2.1. Powertrain structure There are several different powertrain topologies in FCEV development history. At the beginning, a PEM FCS was the sole power source. However, it could not meet the quick dynamic requirement of an electric vehicle. Later, an ESS, e.g. a battery system or a super capacitor, was installed to compensate for the dynamic load. The FCS and an ESS can be connected with each other through different modes. They can be connected directly (upper part of Fig. 1(a)), or via a DC converter (middle and bottom parts of Fig. 1(a)). The directly connected system is simple, but the two power sources are uncontrollable. Since the FCS prefers to provide stationary power and the battery can provide dynamic power, a DC converter on the FCS side is favored (middle part of Fig. 1(a)). This paper focuses on the powertrain system illustrated as in the middle part of Fig. 1(a). The PEM FCS provides stationary power, which equals to the average or part of the average power of the electric vehicle. A lithium battery system is installed as the ESS of the whole vehicle. A boost or bulk DC converter regulates the output power of the FCS. An EMS is designed (1) to fulfill the power requirement of the powertrain, (2) to minimize hydrogen consumption,

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FCS

ESS

d.c. converter

FCS

Motor controller

Electric motor

ESS

FCS

Bi-directional d.c. converter

ESS

(a) '&&'&'&FRQYHUWHU %06

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(b) Fig. 1. Fuel cell electric vehicles powertrain (a) three typical structures and (b) layout of the key components of a fuel cell city bus.

and (3) to prolong the working lifetime of the FCS [35–40,13]. The basic parameters of a fuel cell city bus and its powertrain can be found in [38–40]. This vehicle is 12 m long and weighs 11 tons. The maximum velocity needs reach 60 km h1, and an accelerate time from zero to 50 km h1 should be less than 25 s. Normally an electric motor with a rated power of 100 kW is installed. Fig. 1(b) illustrates the layout of the key components in a fuel cell city bus. Several battery packages are installed in the middle bottom of the vehicle. The electric motor and its controller (electric motor controller, MCU), the DC converter, and the fuel cell system are at the rear part of the vehicle. A distributed controller area network (CAN) is designed for the vehicle controller unit (VCU), battery management system (BMS), MCU, DC converter controller (DCC), and fuel cell system controller (FCC). It is shown as a red dashed line. The VCU is the supervisory controller unit to coordinate the power sources. There are two electric voltage systems: the high voltage system (around 100 V) for power distribution and the low voltage system (5–25 V) for signal exchange. Fig. 1(a) and (b) in [38] show a description of the control system.

2.2. Fuel economy and system durability modeling The basic non-linear dynamic model of the fuel cell city bus contains several parts: longitudinal dynamics of vehicles, efficiency models of the FCS, DC converter, lithium battery, and electric motor. Our previously published papers [38–40] describe the entire model. For the study in this paper, we pay special attention to the fuel economy and system durability of the PEM FCS and the battery. 2.2.1. PEM fuel cell system modeling In a PEM fuel cell stack, hydrogen is fed into the anode, and humidified air is fed into the cathode. Electricity is generated out of the electro-chemical reaction [41]. Hydrogen flows along the anode when the fuel cell is in operation mode. A proportion pressure control valve decreases the hydrogen pressure from highpressed tanks. A maintaining valve keeps the hydrogen pressure into the anode channels remaining constant. The air along the cathode is compressed first and humidified before going into the

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_ H2 can be cathode channels. The hydrogen fuel mass flow rate m calculated as follows:

_ H2 ¼ m

NIfc MH2 uf ne F

ð1Þ C fc ¼

where N is number of serially connected cells in the stack, Ifc is fuel cell stack current, MH2 is molar mass of hydrogen gas, uf is fuel utilization rate, ne is number of transferred electrons, and F is the _ air can be calculated as follows: Faraday’s constant. The air flow rate m

_ air ¼ m

k0 Lst NIfc Moxy 2ne F

ð2Þ

where Lst is air coefficient, k0 is excess air ratio, and Moxy is the molar mass of oxygen gas. An air compressor, which consumes part of the stack power, supplies the air stream with a certain flow rate. We can calculate power consumption by the air compressor Pcp as follows:

Pcp ¼

_ air cp;air Tatm m



j1

pcpj  1



gcp

ð3Þ

where cp;air is heat capacity of the air, T atm is temperature of surrounding air, pcp is pressure ratio, j is polytropic exponent, and gcp is the efficiency of the air compressor. The fuel net efficiency gfc can be obtained by applying the following equation:

gfc ¼

Pfc  Pcp  PAux _ H2 LHV m

ð4Þ

where Pfc is fuel cell stack power, Paux is power consumption of auxiliary components, and LHV is the lower heating value of hydrogen (120 MJ kg1). The net output power of the FCS Pfcnet is the difference of the fuel cell stack power and all the auxiliary power.

Pfcnet ¼ Pfc  Pcp  PAux

ð5Þ

The FCS net power is controlled by the DC converter:

Pfcnet ¼ Pdc =gdc

ð6Þ

where gdc is the average static efficiency of the DC converter. The converter efficiency is affected by input/output voltages and powers. The dynamic properties are very important for real-time control. In the component sizing problem, however, we focus on low frequency properties, so the average static efficiency is adopted for system optimization. 2.2.2. Lithium battery modeling Within simulation models for hybrid powertrains, batteries can be modeled according to a Rint model [42]. This model was originally developed for a lead acid battery. It is also widely adopted for other kinds of batteries in powertrain optimization and control problems, e.g. for lithium batteries [21,24,29,31,32]. The battery’s output current Ibat can be obtained by using its output power Pbat.

Ibat ¼

V ocv 

parts, the actual hydrogen consumption by the FCS and the equivalent hydrogen consumption by the battery. Hydrogen consumption by the FCS is calculated as follows:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2ocv  4Rbat P bat 2Rbat

ð7Þ

V ocv is open circuit voltage and Rbat is battery resistance. They are all functions of the battery State of Charge (SoC) and temperature. The state function of the battery is as follows:

_ ¼  Ibat gcolm SOC Q bat

ð8Þ

where gcolm is coulombic efficiency; Q bat is battery capacity. 2.2.3. Fuel economy and system durability of the powertrain Fuel economy is evaluated by the consumed hydrogen mass per 100 km driving mileage of a certain driving cycle. It includes two

Z

T

_ H2 dt m

0

ð9Þ

where Cfc is the actual hydrogen gas consumption in a driving cycle and T is the duration of the driving cycle. During operation, the battery should be kept in charge sustaining. We use an equivalent hydrogen consumption model [43] to calculate the equivalent _ H2equ , hydrogen consumption rate of the battery system m

_ H2equ ¼ m

8 <

_ H2avg Pbat m

gdis gchgavg Pdcavg ; P bat

: Pbat gdisavg gchg m_ H2avg ; P P dcavg

P0

bat

ð10Þ <0

where gdis , gchg are battery discharge and charge efficiencies, gdisavg , _ H2avg gchgavg are average battery discharge and charge efficiencies, m is average hydrogen flow rate of the FCS, and Pdcavg is the average output power of the DC converter. Based on this information, we can calculate the equivalent hydrogen consumption by the battery as follows:

C bat ¼

Z

T

_ H2equ dt m

ð11Þ

0

where Cbat is the equivalent hydrogen consumption by the battery. The equivalent hydrogen consumption of the powertrain Cpw is the summary of the two:

C pw ¼ C fc þ C bat

ð12Þ

We evaluate system durability of the whole powertrain by performance degradations of the PEM FCS and the lithium battery system. The performance degradation of a PEM FCS D/FCdeg (%) is defined by the percentage of power reduction at a constant stack current compared to the original value. From an operation viewpoint, the unsuitable electric loads, e.g. long time idles, frequent start-stops, high loads, and fast dynamic loads are the major reasons for power degradation of an FCS. These unsuitable loads lead to water flooding or membrane drying, and finally cause irreversible damage to the MEA, the gas diffusion layer (GDL), or the channels. We can express the influences of four typical loads on performance degradation of a PEM FCS as follows [44]:

D/FCdeg ¼ ðk1 t1 þ k2 t2 þ k3 t 3 þ k4 nÞkp

ð13Þ

where D/FCdeg is power degradation in %, and t1, t2, t3 and n, are idle time, dynamic load time, high load time, and start-stop times in one cycle, respectively. k1, k2, k3 and k4 are performance degradation coefficients for idling, dynamic load, high load, and start-stop, respectively. These four coefficients are measured by accelerating tests on benches, the coefficient kp is a modified coefficient for on-road systems. Table 1 presents the values for each coefficient, and the conditions for four typical loads. The performance degradation of the lithium battery system is defined as the percentage of capacity loss compared to the original value after operating for a certain period of time. It depends on the cell activation energy, cycling times, operation temperature, and calendar time. The performance degradation rate n (%.s1) of a lithium battery can be expressed as follows [45]: Ea

n ¼ AeRT n

ð14Þ

where A is the degradation coefficient, R is the ideal gas constant, T is operation temperature, Ea is cell activation energy, n is the times of fully charge–discharge cycling. This paper discusses powertrain performances in one driving cycle, and the battery is kept almost

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where the maximal net power of the FCS changes between 10 kW and 80 kW, and the battery capacity varies from 50 Ah to 250 Ah. There are two constraints on battery SoC in Eq. (16). The first is for limitation of the battery SoC during the driving cycle. The second is to limit the end values of SoC. It should not exceed a range of [SoC0  DSoC, SoC0 + DSoC], DSoC = 0.1. Bus voltage Vbus affects the driving efficiency and ability of the electric motor. Typically, there are two levels of bus voltages in the automobile industry, a rated voltage of 300–400 V or a rated voltage of 600– 700 V. When we design a fuel cell city bus, the bus voltage is pre-defined as between 600 V and 700 V. The maximum power of the DC converter Pdcmax, is changed with maximum net power of the FCS Pfcnetmax, as in Eq. (6). According to our previously published paper [13], the relationship between battery capacity Qbat, charging/discharging resistances Rbat,chg, Rbat,dis, and open circuit voltage Vocv are:

Table 1 Coefficients for performance degradation model. Coefficient

Value (unit)

Condition

k1

0.00356 (h1) 0.00196 (cycle1) 0.00126 (h1) 0.00147 (h1) 1.47

Output power less than 10% of rated power

k2 k3 k4 kp

One fully start-stop Absolute value of power change rate is larger than 10% of rated power per seconds Higher than 90% of maximal power –

charge sustaining. Eq. (14) corresponds to a full charge–discharge cycle, where the absolute change of battery SoC is 200%. Therefore, when the battery SoC changes DSoC, the performance degradation will be Ea

Dn ¼ 0:5AeRT jDSoCj

ð15Þ

3. Problem definition and two-loop framework for Pareto optimization



Rbat;i ¼ Di V ocv =Q bat Di ¼ Rcell;i Q cell =V ocvcell

where i = ‘‘chg’’ or ‘‘dis’’ corresponds to charge or discharge. Rcell,i, Qcell and Vocvcell are charge/discharge resistance, capacity, and open circuit voltage of a single battery cell. Therefore, the minimal and maximal powers of battery are also linked with battery capacity and open circuit voltage.

8 < Pbatmin ¼ V busmax ðVR ocv V busmax Þ batchg

3.1. Problem definition

ð18Þ

: Pbatmax ¼ V busmin ðV ocv V busmin Þ R

ð19Þ

batdis

The performance of a PEM fuel cell city bus is determined by three aspects: driving cycle, powertrain parameters, and EMS. In this study, we try to find the optimal powertrain parameters for the ‘‘China city bus typical cycle’’ [46]. It is a standard cycle for city bus design in China, especially for new energy vehicles. The maximal velocity is 60 km h1, the average velocity is 16.1 km h1, and the driving distance is 5.84 km. In such a pre-defined cycle, component sizing and EMS determine the performance. A multi-objective optimization problem combining component sizing and EMS is defined as follows:

minJ ¼ ½J FE ; J DE  ~ x 8 ~ x ¼ ðPfcnetmax ; Q bat ÞT 2 U > > > > > v ¼ v cycle ðtÞ; t 2 ½0; T > > > h > > > < SoC 2 ½SoCL ; SoCH  s:t: jSoC0  SoCend j 6 DSoC > > > > Pbat 2 ½Pbatmin ; Pbatmax  > > > > > Pdc 2 ½0; Pdcmax  > > : V bus 2 ½V busmin ; V busmax 

ð16Þ

where J is the multi-objective optimization function, JFE is the fuel economy index in the unit of kg (100 km)1, and JDE is the system x is the optimized performance degradation index in the unit of %. ~ vector, which contains the maximal net power of a PEM FCS Pfcnetmax, and the capacity of the battery Qbat. Vh is vehicle velocity, Vcycle is cycle velocity, SoCL and SoCH are the lower and upper limits of battery SoC during driving, SoC0 is the initial SoC value, SoCend is the end SoC value, DSoC is the maximal SoC change in the cycle, Pbatmin and Pbatmax are the minimal and maximal power of the battery, Pdcmax is the maximal power of the DC converter, Vbusmin and Vbusmax are the minimal and maximal voltages of the bus, and U is the optimization space,

 U¼

ðPfcnetmax ; Q bat ÞjPfcnetmax 2 f10; 20; 30; 40; 50; 60; 70; 80gkW

3.2. Two-loop Pareto optimization framework with a DP-based EMS The problem defined in Eq. (16) is a typical multi-objective optimization problem that combines component sizing and EMS. It contains two indexes, fuel economy, and system durability. These two indexes have different physical meanings, and are difficult to evaluate in one equation. We can find a Pareto optimal as a near-optimal solution. According to paper [47], vector ~ x can be regarded as a Pareto optimal solution if there does not exist x 2 U, such that JFE(~ x) 6 JFE(~ x ) and JDE(~ x) 6 JDE(~ x ) another vector, ~ x is a Pareto optimal simultaneously, which means that Vector ~ solution if it is at least the optimal solution for minJFE or minJDE. Fig. 2(a) illustrates the basic idea of Pareto front and Pareto x 2 U. In the JFE–JDE optimal points for one parameter vector ~ space, each point (JDE, JFE) corresponds to one EMS. The left boundary in the red dashed line of all the possible points is the Pareto front, which consists of different Pareto optimal points, e.g. A, B, C, D, and E. In order to reduce computational load and keep the accuracy of the simulation, we find a Pareto optimal solution based on DP. Fig. 2(b) shows a two-loop Pareto optimal framework with a DP-based EMS for the multi-objective problem defined in Eq. (16).  Loop 1: to find the minimal fuel economy JFE corresponding to x using a DP algorithm, and to calculate the each vector ~ corresponding fuel economy and system durability indexes (JDE, JFE).  Loop 2: to compare the system performance index (JDE, JFE) of all x with an the specified vectors, and to choose a suitable vector ~ acceptable fuel economy and system durability as the Pareto optimal solution.



; Q bat 2 ½50; 75; 100; 125; 150; 175; 200; 225; 250 Ah ð17Þ

The DP algorithm is the core of the whole framework. It is a discrete global optimization problem with constraints as follows:

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J FE

Pareto front

A

B C D E

(a)

JDE

(c) (b) Fig. 2. Basic ideas for the two-loop optimization framework (a) Pareto front (dashed curve) and Pareto optimization points (A–E), (b) structure of the two-loop optimization framework and (c) process of the dynamic programming.

xÞ ¼ C fc þ C bat minJ FE ð~ Pdc 8 ~ x ¼ ðPfcnetmax ; Q bat ÞT 2 U > > > > > v ¼ v cycle ðkÞ; k 2 ½0; N  1 > > > h > > > < SoC 2 ½SoCL ; SoCH  s:t: jSoC0  SoCend j 6 DSoC > > > > Pbat 2 ½Pbatmin ; Pbatmax  > > > > > Pdc 2 ½0; Pdcmax  > > : V bus 2 ½V busmin ; V busmax 

k 2 ½0; k1 Þ, we define a bilateral penalty function to constrain SoC within [SoCL, SoCH] as follows:

ð20Þ

where N is the number of sample points for the discrete driving cycle. For an EMS problem, normally, we choose dt = 1 s. Thus, N = 1304 for the China city bus typical cycle. We can fulfill the constraint on Pdc and Pbat by defining the optimization space of Pdc. We can fulfill constraints on SoC and Vbus by limiting the variation of battery SoC. Therefore, the problem defined in Eq. (20) is equivalent to following:

minJ 0FE ð~ xÞ ¼ C fc þ C bat þ C SoC P dc

ð21Þ

where J 0FE is the equivalent optimized function without constraints, and CSoC is the penalty function regarding battery SoC. Note that because there are two constraints on battery SoC, we separate the driving cycle into two stages: k 2 ½0; k1 Þ is normal operation, and k 2 ½k1 ; N is final operation. For the normal operation stage

C SoC

8 2 > < 1 þ aðSoC  SoCH þ 1Þ ; SoC 2 ½SoCH  e; 1 ¼ 0; SoC 2 ðSoCL  e; SoCH  eÞ > : 1 þ aðSoC  SoCL þ 1Þ2 ; SoC 2 ½0; SoCL þ e

ð22Þ

where a > 0 is a positive coefficient that is large enough, e.g. 105. e > 0 is a positive coefficient that is smaller than the discrete step of SoC. For the final operation stage k 2 ½k1 ; N, we define a unilateral penalty function to constrain SoC within a small range around SoCend,

( C SoC ¼

0; SoC 2 ½SoCend  e; 1 1 þ aðSoC  SoCend þ 1Þ2 ; SoC 2 ½0; SoCend  eÞ

ð23Þ

where a > 0 and e > 0 is the same value as defined in Eq. (22). The problem defined in Eqs. (21)–(23) can be solved using a DP algorithm, where Pdc is the decision variable, SoC is the state variable, and Eqs. (7) and (8) are state transfer equations. A detailed description for the process of solving the DP algorithm is shown in our previous paper [40,48] and Fig. 2(b). Fig. 2(c) illustrates the process in the battery SoC space. The DP algorithm can be solved step by step from the end to the start of the driving cycle. As a result, the optimal DC power Pdc,opt (k, SoC) is a function of time and battery SoC. We calculate it using an inverse iteration process, as shown in Fig. 2(b) in Loop 1. In

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system simulation, the optimal DC power is calculated using a Lookup Table with time and SoC as input signals. We can obtain a battery SoC trajectory as the optimal curve using a DP algorithm, shown as the red dashed curve in Fig. 2(c). 3.3. Optimization of parameters for the DP algorithm We can find the optimal EMS for each powertrain parameter x by using a DP algorithm. Considering Eq. (17), there are vector ~ x, meaning that we need to run the DP totally 72 candidates of ~ algorithm 72 times. Thus, it is necessary to reduce the calculating time for each DP. Generally, there is a trade-off between the calculating time and result accuracy, which are determined by discrete steps of input cycles (dt), decision variable (dPdc), and state variable (dSoC). For an EMS problem, normally, we choose dt = 1 s. The steps for decision and state variables are chosen from following sets.



dPdc 2 f1; 2; 4; 6; 8; 10gkW dSoC 2 f0:01; 0:02; 0:04; 0:06; 0:08; 0:09; 0:10; 0:11; 0:12g ð24Þ

Equivalent hydrogen consumption per 100km [kg/(100km)]

Fig. 3(a) and (b) present the simulation results of the DP algorithm with different steps (dPdc, dSoC). As in Fig. 3(a), the equivalent hydrogen consumption Cpw per 100 km varies with dSoC, and is almost independent with dPdc when dPdc 6 6 kW. When dSoC e [0.01, 0.08], the equivalent hydrogen consumption converges to 4.6 and 4.7 kg (100 km)1. The accurate value is 4.6 kg (100 km)1. When dSoC increases, the equivalent hydrogen consumption fluctuates around this value. The related errors are 2.1%, 2.0%, 3.7%, and 9.1%, corresponding to dSoC = 0.09, 0.1, 0.11, and 0.12. Fig. 3(b) illustrates the end value of battery SoC in the DP algorithm. Similarly, SoCend varies with dSoC, and is independent with dPdc when dPdc 6 5 kW. When dSoC e {0.01, 0.02, 0.04, 0.08}, SoCend converges to SoC0 = 0.6. When dSoC equals other values, SoCstep = 0.01

5.2

SoCstep = 0.02

5.0

SoCstep = 0.04 SoCstep = 0.06

4.8

SoCstep = 0.08 SoCstep = 0.09

4.6 4.4

SoCstep = 0.10 SoCstep = 0.11

0

5

10

15

SoCstep = 0.12

dPdc

(a)

SOCend [%]

63

SoCstep = 0.01

62

SoCstep = 0.02

61

SoCstep = 0.04

60

SoCstep = 0.06

59

SoCstep = 0.08

58

SoCstep = 0.09

57

SoCstep = 0.10

56

SoCstep = 0.11

0

5

10

15

SoCstep = 0.12

dPdc

(b) Fig. 3. Simulation results for different discrete steps of DP algorithm, SoC0 = 0.6 (a) powertrain equivalent hydrogen consumption and (b) SOCend.

SoCend fluctuates around this value. The related errors are 2.5%, 3.0%, 1.3%, 1.7%, and 5.0%, corresponding to dSoC = 0.06, 0.9, 0.1, 0.11, and 0.12. Large discrete steps mean short calculating time. Considering the results accuracy and calculating time, we choose dSoC = 0.8 and dPdc = 5 kW as the optimal discrete steps in simulation. Table 2 shows all the parameters for the DP algorithm. 4. Optimal parameters based on simulation x are carried out with DP Simulations for each parameter vector ~ algorithms. In the following paragraphs, we analyze and compare the fuel economy and system durability for 72 parameter vectors. 4.1. Fuel economy Figs. 4(a) and (b) show the relationship of the actual hydrogen consumption Cfc per 100 km and the battery capacity Qbat and maximal FC net power Pfcnetmax. Generally, it decreases with increasing of Qbat when Qbat < 150 Ah. For battery capacities that are larger than 150 Ah, the consumption increases for some of the graphs because while operating the vehicle, the FCS charges the battery sometimes to prevent it from exceeding SOC limits. When we calculate the equivalent hydrogen consumption, the hydrogen for charging the battery is removed from the completely consumed fuel. The hydrogen consumption as a function of Pfcnetmax, which is presented in Fig. 4(b), shows basically rising graphs that level off or experience a slight decrease for Pfcnetmax values larger than 40 kW. When the FCS is too small, the battery delivers a major part of the whole driving power, resulting in a reduction in battery SoC. Otherwise, the FCS charges the battery, causing an increase in battery SoC. Because the battery is not strictly kept charge sustaining in the simulation, the actual hydrogen consumption Cfc cannot reflect the fuel economy of the FCEV. The powertrain equivalent hydrogen consumption Cpw = Cfc + Cbat is more suitable to evaluate fuel economy, taking the equivalent hydrogen consumption of the battery into consideration. Fig. 5(a) and (b) present the relationship between powertrain equivalent hydrogen consumption Cpw per 100 km, Pfcnetmax, and Qbat. With an increased capacity, a battery can store more energy that is recovered during braking. A larger amount of the kinetic energy can be reused and therefore, the hydrogen consumption decreases with an increasing Qbat. Fig. 5(a) shows this effect. When Pfcnetmax becomes larger, Cpw becomes less sensitive with Qbat. When the battery capacity Qbat is larger than 150 Ah, we can ignore the influence of Qbat on Cpw. For a fixed Qbat, Cpw decreases with growing Pfcnetmax, as Fig. 5(b) shows. The largest Pfcnetmax of 80 kW demands the lowest Cpw. The influence of Pfcnetmax on Cpw is slightly affected by Qbat. _ H2 and Fig. 6(a) and (b) give a close look at hydrogen flow rate m stack power Pfc with different configurations. Fig. 6(a) shows curves with Pfcnetmax = 10 kW, Qbat = 175 Ah. With this configuration, the FC starts working in the first seconds of the driving cycle. The average output power of the FCS cannot fulfill the

Table 2 Parameters for the DP algorithm. Coefficient

Value (unit)

dSoC dPdc

0.8 5 (kW) 105 0.2 1304 1204

a e N k1

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L. Xu et al. / Applied Energy xxx (2015) xxx–xxx

Hydrogen consumption [kg/(100km)]

8

Pfce_max = 80 kW

6.0

Pfce_max = 70 kW

5.5

Pfce_max = 60 kW

5.0 Pfce_max = 50 kW

4.5

Pfce_max = 40 kW

4.0

Pfce_max = 30 kW

3.5 3.0

Pfce_max = 20 kW

0

50

100

150

200

250

300

Pfce_max = 10 kW

Q bat [Ah]

Hydrogen consumption [kg/(100km)]

(a) 6.0

Qbat = 250 Ah

5.5

Qbat = 225 Ah

5.0

Qbat = 200 Ah Qbat = 175 Ah

4.5

Qbat = 150 Ah

4.0

Qbat = 125 Ah

3.5

Qbat = 100 Ah

3.0

Qbat = 75 Ah

0

20

40

60

Pfcnetmax [kW]

80

Qbat = 50 Ah

(b)

Equivalent hydrogen consumption [kg/(100km)]

Fig. 4. Relationship between actual hydrogen consumption Cfc per 100 km, Pfcnetmax and Qbat (a) hydrogen consumption v.s. Qbat and (b) hydrogen consumption v.s. Pfcnetmax.

10

Pfce_max = 80 kW

9

Pfce_max = 70 kW

8

Pfce_max = 60 kW

7

Pfce_max = 50 kW

6

Pfce_max = 40 kW

5

Pfce_max = 30 kW

4

0

100

200

300

Qbat [Ah]

Pfce_max = 20 kW Pfce_max = 10 kW

Equivalent hydrogen consumption [kg/(100km)]

(a) 10

Qbat = 250 Ah

9

Qbat = 225 Ah

8

Qbat = 200 Ah

7

Qbat = 175 Ah Qbat = 150 Ah

6

Qbat = 125 Ah

5

Qbat = 100 Ah

4

0

20

40

60

Pfcnetmax [kW]

80

Qbat = 75 Ah Qbat = 50 Ah

(b) Fig. 5. Relationship between powertrain equivalent hydrogen consumption Cpw per 100 km, Pfcnetmax and Qbat (a) equivalent hydrogen consumption v.s. Qbat and (b) equivalent hydrogen consumption v.s. Pfcnetmax.

powertrain requirement, and the battery needs to provide part of the energy. This part of energy needs to be added to the equivalent amount of hydrogen consumption. The behavior of the FCS is

Fig. 6. Hydrogen flow rate and stack power with different configurations (a) Pfcnetmax = 10 kW, Qbat = 175 Ah and (b) Pfcnetmax = 80 kW, Qbat = 175 Ah.

different for Pfcnetmax = 80 kW, Qbat = 175 Ah. As Fig. 6(b) indicates, the FC does not contribute energy to the powertrain in the first 200 s. The average output power of the FCS is more than the average powertrain requirement, and the battery is charged. This part of the energy needs to be removed to the equivalent hydrogen consumption. The difference in the actual/equivalent hydrogen consumption for varying Pfcnetmax values is caused by the DP-based energy management system. It regulates which power source activates and where the power comes from for different driving situations, and tends to activate the FCS earlier for a smaller FCS than for a larger one. An FCS with smaller power cannot fulfill the requirement of an FCEV. Its output energy is less than an FCS with larger power, and the battery discharges. The actual hydrogen consumption Cfc increases with Pfcnetmax when it is less than 40 kW. When Pfcnetmax becomes larger than 40 kW, it can fulfill the power requirement and part of the hydrogen energy is charged into the battery. The actual hydrogen consumption Cfc almost keeps constant in this range, shown as in Fig. 4(b). The battery charging/discharging energy converts into the equivalent hydrogen consumption when calculating the equivalent powertrain hydrogen consumption Cpw, which is primarily determined by the average efficiency of the FCS. As illustrated in [38,39], the fuel efficiency reaches its maximum level at about 15% power ratio. The average power requirement of the powertrain is about 15 kW. Note that Pfcnetmax e [10, 80]kW, the power ratio e [0.19,1.5], where the average efficiency decreases with increments of power ratio. Thus, the equivalent powertrain hydrogen consumption Cpw decreases with Pfcnetmax, as Fig. 5(b) shows. Additionally, the simulating values of the hydrogen consumption are lower than the values obtained by a real testing procedure [36,38–40]. There are two main influences leading to a reduced consumption in this simulation. (1) We assume the regeneration ratio to be 1 in simulation, meaning that all the braking energy is recycled by the electric motor. (2) Power consumption of auxiliary components Paux is set to 0 in simulation. Both assumptions contribute to reduced hydrogen consumption and explain the difference in the values from the testing procedure. 4.2. System durability Fig. 7(a) and (b) show the power degradation of the FCS in one driving cycle. The loss in the output power of the FCS depends on

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L. Xu et al. / Applied Energy xxx (2015) xxx–xxx Pfce_max = 80 kW

FC Degradation [e-3] [%]

80

Pfce_max = 70 kW

70

Pfce_max = 60 kW

60

Pfce_max = 50 kW

50 40

Pfce_max = 40 kW

30

Pfce_max = 30 kW Pfce_max = 20 kW

20 0

50

100

150

200

250

300 Pfce_max = 10 kW

Q bat [Ah]

(a) Qbat = 250 Ah

FC Degradation [e-3] [%]

80 Qbat = 225 Ah

70

Qbat = 200 Ah

60

Qbat = 175 Ah

50

Qbat = 150 Ah

40

Qbat = 125 Ah Qbat = 100 Ah

30

Qbat = 75 Ah

20 0

20

40

60

80

Qbat = 50 Ah

Pfcnetmax [kw]

(b)

Battery Degradation [e-3] [%]

Fig. 7. FCS power degradation as a function of (a) Qbat and (b) Pfcnetmax.

60

Pfce_max = 80 kW

50

Pfce_max = 70 kW

40

Pfce_max = 60 kW

30

Pfce_max = 50 kW

20

Pfce_max = 40 kW

10

Pfce_max = 30 kW Pfce_max = 20 kW

0 0

50

100

150

200

250

300 Pfce_max = 10 kW

Qbat [Ah]

Battery Degradation [e-3] [%]

(a) 60

Qbat = 250 Ah

50

Qbat = 225 Ah Qbat = 200 Ah

40

Qbat = 175 Ah

30

Qbat = 150 Ah

20 Qbat = 125 Ah

10

Qbat = 100 Ah

0 0

20

40

60

Pfcnetmax [kW]

80

100

9

and 3.7 ⁄ 102%. When the FCS becomes large, the influence of Qbat on D/FCdeg becomes small. However, D/FCdeg decreases with the increment of Pfcnetmax when Pfcnetmax < 40 kW. It almost keeps constant when Pfcnetmax > 40 kW. The value of Qbat slightly affects the relationship between D/FCdeg and Pfcnetmax. The results for the capacity degradation of the lithium battery system in Fig. 8(a) and (b) show that the degradation is clearly reduced for larger capacities. The lowest degradation value occurs for capacities of 250 Ah. Smaller capacities show larger degradation values. When the capacity is reduced beginning at 250 Ah, the degradation does not grow linearly, but with an exponential gradient. The influence of the FC output power is negligible. The performance degradation of a lithium battery is primarily determined by charge/discharge ratio and working temperature. A large charge/discharge ratio causes quick degradation. With a large capacity of a lithium battery system, the charge/discharge ratio is reduced, resulting in a reduction in performance degradation. 4.3. Optimal parameter selection Finally, considering the fuel economy and system degradation, 150 Ah is the best choice for the capacity. The actual hydrogen consumption reaches the best results between 125 Ah and 175 Ah (Fig. 4(a)). In addition, the equivalent hydrogen consumption achieves good and stable values for Qbat larger than or equal to 150 Ah (Fig. 5(a)). The result for the FC degradation is acceptable for these Qbat values. Moreover, with a degradation of less than 0.015% (Fig. 7(a)), the battery degradation shows good predictions for this parameter choice (Fig. 8(a)). For maximum FC net output power Pfcnetmax, a dimension of 40 kW is recommended. The analysis of the FCS degradation indicates very good results for 40 kW and 50 kW (Fig. 7(b)). Output powers of 10 kW, 20 kW, and 30 kW indicate higher degradation values. For Pfcnetmax values larger than 50 kW, the degradation of the FCS is almost constant (Fig. 7(b)), but the price and weight increase. As the battery degradation is rather independent from Pfcnetmax, 40 kW is also a good choice. Considering the equivalent hydrogen consumption, 40 kW leads to acceptable values that are in the middle of the range of the results. Focusing on a Pfcnetmax value of 40 kW, a capacity of 125–175 Ah achieves the lowest hydrogen consumption (Fig. 4(a)). The simulating results in fuel economy are smaller than what we have tested in fuel cell city buses previously [43], around 4.5 kg (100 km)1 vs. around 8.0 kg (100 km)1. There are several reasons. (1) The vehicle mass in simulation is much less than on paper [43]. (2) All the braking energy is recovered in this study. (3) The efficiency of the fuel cell system, the electric motor, the battery, and the DC converter are higher than the bus in real testing. For system durability, there are similar problems. However, such differences will not affect the parameter sizing problem because the optimal parameters are chosen based on the relative values of system performance, but the absolute values are not.

Qbat = 75 Ah Qbat = 50 Ah

(b) Fig. 8. Battery capacity degradation as a function of (a) Qbat and (b) Pfcnetmax.

Pfcnetmax as well as on the battery’s capacity Qbat. The power degradation D/FCdeg moves within the range of 2.46 ⁄ 102% and 7.01 ⁄ 102%. D/FCdeg decreases with the increment of Qbat. When Qbat > 150 Ah, D/FCdeg keeps almost constant between 2.5 ⁄ 102%

5. Conclusion This paper studies the multi-objective optimization problem of powertrain parameters for a pre-defined driving cycle regarding fuel economy and system durability. We propose a two-loop framework based on a DP algorithm and the Pareto optimal principle. We also analyze the influence of the discrete steps of the DC output power dPdc and of the SoC dSoC on the accuracy of the DP. Big discrete steps lead to low accuracy and short calculating time. An optimal (dSoC, dPdc) is chosen as (0.08, 5 kW), because

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with these two values, both the accuracy and calculating time are acceptable. Step values larger than these two will either lower the accuracy or make the results unstable. Applying this algorithm to the model of a fuel cell city bus gives the opportunity to choose the best values for battery capacity Cbat and the maximal net output power of the FCS Pfcnetmax. The optimal value for Qbat of 150 Ah is chosen based on the following reasons: The actual hydrogen consumption Cfc and powertrain equivalent hydrogen consumption Cpw show the lowest results. The optimal value for Pfcnetmax of 40 kW is chosen based on the following reasons: The FCS power degradation reaches its lowest level, and the capacity degradation of the battery system is acceptable. In the future, several aspects need to be studied. First, it is necessary to improve and evaluate the non-linear system model, especially the durability submodels for the fuel cell and battery. Second, we need to find some other algorithms that can shorten the calculating time while keeping accuracy compared to the DP algorithm, e.g. the Pontryagin’s Minimum Principle (PMP) algorithm or convex optimization. Third, we only find a quasi-optimal solution to the multi-objective problem based on Pareto optimal principles. It will be more helpful to find the Pareto front, which is the assembly of all the Pareto solutions. Fourth, we plan to place all the models and algorithms into one software, to make it easier for the potential users. In addition, we will verify our methodology by cooperating with our industrial partners to build a prototype fuel cell city bus and test it on the road. Acknowledgments The work was support by NSFC (National Natural Science Foundation of China) under Grand No. 61004075, by MOST (Ministry of Science and Technology of China) under Grand Nos. 2013BAG16B01 and 2014DFG71590, by BMSTC (Beijing Municipal Science and Technology Commission) under Grand No. Z131110001513101, and by the State Key Laboratory of Automotive Safety and Energy under Grand No. ZZ2014-034. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apenergy.2015. 02.017. References [1] Wagner DV, An F, Wang C. Structure and impacts of fuel economy standards for passenger cars in China. Energy Policy 2009;37(10):3803–11. [2] Shi H, Wang Y, Huisingh D, Wang J. On moving towards an ecologically sound society: with special focus on preventing future smog crises in China and globally. J Clean Prod 2014;64:9–12. [3] Li J, Fang C, Xu L. Current status and trends of the research and development for fuel cell vehicles. J Automot Safety and Energy 2014;5:17–29 (in Chinese). [4] Chan CC. The state of the art of electric, hybrid, and fuel cell vehicles. Proc IEEE 2007;95(4):704–18. [5] Li Y. Nejabatkhah F. Overview of control, integration and energy management of microgrids, Journal of Modern Power Systems and Clean Energy 2014;2:212–22. [6] Pavan YV. Kumar, R. Bhimasingu, Renewable energy based microgrid system sizing and energy management for green buildings, Journal of Modern Power Systems and Clean, Energy 2015;3:1–13. [7] Chen Z. Wind power in modern power systems. Journal of Modern Power Systems and Clean Energy 2013;1:2–13. [8] Xu G, Xu L, Yao L. Wind turbines output power smoothing using embedded energy storage systems. Journal of Modern Power Systems and Clean Energy 2013;1:49–57. [9] Smith R, Meng K, Dong Z, Simpson R. Demand response: a strategy to address residential air-conditioning peak load in Australia. Journal of Modern Power Systems and Clean Energy 2013;1:223–30. [10] Zhang Y, yao F, Iu HHC, Fernando T, Wong KP. Sequential quadratic programming particle swarm optimization for wind power system operations considering emissions, Journal of Modern Power Systems and Clean. Energy 2013;1:231–40.

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Please cite this article in press as: Xu L et al. Multi-objective component sizing based on optimal energy management strategy of fuel cell electric vehicles. Appl Energy (2015), http://dx.doi.org/10.1016/j.apenergy.2015.02.017