Optimal Design of Hybrid Fuel Cell Vehicles - Optimal Design Laboratory

Proceedings of FUELCELL2006 The 4th International Conference on FUEL CELL SCIENCE, ENGINEERING and TECHNOLOGY June 19-21, 2006, Irvine, CA, USA

FUELCELL2006-97161

OPTIMAL DESIGN OF HYBRID FUEL CELL VEHICLES

Jeongwoo Han∗, Michael Kokkolaras, Panos Papalambros {jwhan,mk,pyp}@umich.edu Department of Mechanical Engineering, University of Michigan G.G. Brown Bldg., Ann Arbor, Michigan 48109

ABSTRACT Fuel cells are being considered increasingly as a viable alternative energy source for automobiles because of their clean and efficient power generation. Numerous technological concepts have been developed and compared in terms of safety, robust operation, fuel economy, and vehicle performance. However, several issues still exist and must be addressed to improve the viability of this emerging technology. Despite the relatively large number of models and prototypes, a model-based vehicle design capability with sufficient fidelity and efficiency is not yet available in the literature. In this article we present an analysis and design optimization model for fuel cell vehicles that can be applied to both hybrid and non-hybrid vehicles by integrating a fuel cell vehicle simulator with a physics-based fuel cell model. The integration is achieved via quasi-steady fuel cell performance maps, and provides the ability to modify the characteristics of fuel cell systems with sufficient accuracy (less than 5% error) and efficiency (98% computational time reduction on average). Thus, a vehicle can be optimized subject to constraints that include various performance metrics and design specifications so that the overall efficiency of the hybrid fuel cell vehicle can be improved by 14% without violating any constraints. The obtained optimal fuel cell system is also compared to other, not vehicle-related, fuel cell systems optimized for maximum power density or maximum efficiency. A tradeoff between power density and efficiency can be observed depending on the size of compressors. Typically, a larger compressor results in higher fuel cell power density at the cost of fuel cell efficiency because it operates in a wider current region. When optimizing the fuel cell

∗ Corresponding

author, Phone/Fax: (734) 647-8402/8403

system for maximum power density, we observe that the optimal compressor operates efficiently. When optimizing the fuel cell system to be used as a power source in a vehicle, the optimal compressor is smaller and less efficient than the one of the fuel cell system optimized for maximum power density. In spite of this compressor inefficiency, the fuel cell system is 9% more efficient on average. In addition, vehicle performance can be improved significantly because the fuel cell system is designed both for maximum power density and efficiency. For a more comprehensive understanding of the overall design tradeoffs, several constraints dealing with cost, weight, and packaging issues must be considered.

1

Introduction Currently, PEM fuel cells are agreed upon as the most suitable technology for vehicular applications because of their mobility and high power density [1]. Nevertheless, several issues still exist that must be addressed in order to assess and improve viability of fuel cell vehicles, including high vs. low pressure fuel cell vehicles and hybrid vs. non-hybrid fuel cell vehicles. Several fuel cell vehicle concepts and fuel cell system designs have been proposed and studied in terms of safety, robust operation, fuel economy, and vehicle performance. To the best of our knowledge, despite the relatively large number of models and prototypes, an integrated model-based vehicle design methodology with sufficient fidelity and efficiency is not yet available in the literature. In order to analyze the behavior of fuel cell vehicles, models must be developed for subsystems, such as fuel cell, bat1

c 2006 by ASME Copyright

tery, and motor. The essence of fuel cell system modeling is the membrane-electrode assembly (MEA) model, which describes mathematically the entire physical environment of the electrochemical reactions; the transport phenomena of gases (hydrogen, oxygen, vapor, etc.), water, protons, and current; and the relationships among fuel cell voltage, current, temperature, material (electrode, catalyst and membrane) properties, and transport parameters. MEA modeling has been accomplished by analyzing physical effects of reactant gases [2, 3], performing experiments on actual stacks [4], or integrating the physical and experimental models [5]. Many publications on MEA models have concentrated on analyzing the water transport as well as the gas diffusion [6–8]. Heat transfer and thermodynamics were included to predict the temperature and humidity profiles in both transient and steady-state conditions [9, 10]. Computational fluid dynamics have been used extensively to analyze air and water transport behavior of fuel cell systems [11]. Unlike the relatively wide availability of MEA models, only a few publications are available on fuel cell system modeling. Pukrushpan et al. [12] applied reactant flow dynamics in order to estimate the net power output as a function of reactant partial pressures and the power losses in flow devices. Using MEA and fuel cell system models, optimization studies have been conducted to minimize the weighted sum of the inverse of functional performance and product cost [13] and to maximize power density by adjusting proper operating conditions [14]. However, objectives in these papers do not reflect the requirements of the “supersystems” in which the designed fuel cell is used. Usually, requirements cannot be expressed using a single attribute; a combination of several performance and cost metrics is required. For example, design objectives of a fuel cell system to be used as a power source in a vehicle, include maximum power, power density, product cost, and reliability, and are possibly conflicting with design objectives of other vehicle systems. In this paper, the fuel cell system is designed to be suitable for a certain type of vehicle by solving an integrated vehicle and fuel cell design problem. The considered fuel cell system and fuel cell vehicle models are presented in Section 2. The process of integrating these models is also introduced. In order to ensure fidelity and demonstrate efficiency, the vehicle simulation results obtained quasi-steady and full dynamic fuel cell models are compared. In Section 3, the design optimization problem is solved using derivative-free optimization algorithms. Concluding remarks are provided in Section 4.

For example, most of the control-oriented fuel cell system models require computations that are prohibitively expensive for optimization purposes. On the other hand, many vehicle simulation models cannot handle significant changes in the design of fuel cell system components because of the low fidelity and limitations of the models. Therefore, the purpose of the model used in this paper is to capture the effects of variations in design accurately with an efficiency that enables optimization studies. 2.1

Quasi-static Fuel Cell System Model The quasi-static fuel cell system model is based on the transient fuel cell model developed by Pukrushpan et al. [12]. This model generates a static performance map that represents the maximum power for a certain range of fuel consumption with given control constraints. 35

water injected

Hydrogen Tank humidi fier

cooler

compre ssor

Pressure Adjusted

Figure 1.

Humidity Temperature Adjusted Adjusted

Figure 3.3: Reactant supply subsystems

Reactant supply subsystems (modified from [12])

In this study, we assume that the properties of the inlet reactant flow except for the partial pressures can be perfectly controlled to make the problem simple. Additionally, the

The power output from a fuel cell system P can be determined as the difference between the power generated from a fuel With assumptions, output, expressed by in equation 3.20,components can be reduced to cellthose stack Pfc andthe thevoltage power consumed auxiliary Pcon : a function of current density and the oxygen partial pressure as netas a function of pca . pressure of the anode, pan is also assumed to be instantly regulated

i) v − Pcon , Pnet = Pfc − vPfcon =c (pnOfc,caI,fc c = vf fc 2

(3.21) (1)

where pO2 ,ca is the partial pressure of oxygen. Since the oxygen partial pressure, in turn,

where nfc is the number of cells, Ifc is stack current, and vfc is

iscell controlled by the If output of the compressor, the performance the fuel cell is voltage. the pressure composition and structure of the ofcells is de-

termined, the cell voltage is a function of stack current density and reactant flow properties, including partial pressures, humidbyity, a feedforward control (FigureThe 3.4). Thus, the designed output ofby the reactant fuel cell can and temperature. properties arepower governed suppliers consisting of four flow systems: (i) hydrogen supply be obtained by applying a proper feed-forward control on vcm [25]. system, (ii) air supply system, (iii) cooling system, and (iv) huSince there system are many (Figure sources of1). disturbance due tofocuses the transient of flows midifying This study on irregularity high pressure fuel cell systems with a compressor because most of the protoand environmental variation, a feedback controller is typically used to operate the fuel cell types are developed using high pressure fuel cells due to their more consistently. this study, however, is interest in the optimization based on the higher powerBecause density. governed by the compressor input, which is determined as a function of the stack current

2

Fuel Cell Hybrid Vehicle Model In addition to the fuel cell model mentioned in the previous section, several vehicle simulation models have been developed using simple fuel cell performance maps obtained from experiments. Each model has a different purpose and perspective, and so each model has several limitations for optimization studies. 2


Compressor

We assume that the properties of the inlet reactant flow, except for the partial pressures, can be controlled perfectly without transient irregularity. Additionally, the pressure at the anode is also assumed to be regulated instantly as a function of cathode pressure. Ambient air is assumed to be constant. The assumed properties are given in Table 1. Under these assumptions, the

WN ,in 2

WO ,in 2

Cathode Side

Cathode

Thermodynamical parameters used in the model

Value

Ambient Temperature Tamb (Kelvin)

298

Stack Temperature Tst (Kelvin)

353

Ambient pressure pamb (bar)

1

Ambient Relative Humidity

0.5

Relative Humidity of Cathode Inlet Flow

0.8

Anode Relative Humidity

1

WO ,out WN2,out Wvap,out

WH ,an 2

Wvap,gen

Wvap,mem Membrane

WH ,rea 2

Anode

2

Figure 2.

Return Manifold

Diagram of reactant flows in a PEM fuel cell

n I WO2 ,rea = MO2 fc fc , 4F

(4)

where WH2 ,rea and WO2 ,rea , MH2 and MO2 , and F are the rates of reacted hydrogen and oxygen, the molar masses of hydrogen and oxygen, and the Faraday constant (= 96485 C/mol), respectively. At steady state, since the oxygen is directly supplied from the ambient and transient manifold filling effect is ignored, the rate of oxygen supplied to the cathode equals the rate of oxygen from the ambient. Therefore, the total mass flow rate of the inlet air can be represented as a function of stack current and oxygen excess ratio:

(2)

where E is the fuel cell open circuit voltage, and vact , vohm , and vconc are overvoltages due to the activation loss, ohmic loss, and concentration loss, respectively. The detailed explanations and formulas are presented in [12]. The overvoltage due to the fuel crossover and internal currents is neglected because the loss is relatively small in PEM fuel cells. Since the oxygen partial pressure is controlled by the output pressure of the compressor, the performance of the fuel cell is governed by the compressor input, which is determined as a function of the stack current by feedforward control. Thus, the designed power output of the fuel cell can be obtained by applying a proper feed-forward control on the compressor command voltage, vcm [12]. In this study, the feedforward controller is designed to meet the target values of oxygen excess ratio λ(Ifc ). Thus, the stack power can be simplified as a function of stack current and oxygen excess ratio. Given ambient air properties, air pressure and mass flow rate of the compressor outlet can be calculated from the mass conservation principle and thermodynamic and psychrometric gas properties under a quasi-static assumption. Figure 2 illustrates reactant flows under steady-state conditions. As the stack current Ifc is drawn from the fuel cell, the rates of hydrogen and oxygen consumed in the reaction can be calculated as n I WH2 ,rea = MH2 fc fc 2F

Wvap,an

Wvap,sat

cell voltage can be reduced to a function of current density and oxygen partial pressure pO2 ,ca : vfc = vfc (pO2 ,ca , Ifc ) = E − vact − vohm − vconc ,

Hydrogen Tank

Humidifier

2

Parameter

Anode Side

Wvap,inj

WO ,rea

Table 1.

Cooler

Wvap,in

Win = WO2 ,in +WN2 +Wvap,in WO2 ,in = λWO2 ,rea WN2 =

(1 − MFO2 )MN2 MFO2 · MO2 Mvap pvap,amb

WO2 ,in

Wvap,in = (W +WN2 ), Mair pair,amb O2 ,in

(5) (6) (7) (8)

where WO2 ,in , WN2 , and Wvap,in are the inlet mass flow rates of oxygen, nitrogen, and vapor to the cathode side, MN2 , Mvap , and Mair are the molar masses of nitrogen, vapor, and dry air, MFO2 is the oxygen mass fraction in dry air (= 0.21), and pvap,amb and pair,amb the partial pressures of vapor and dry air in the ambient, respectively. The flow rates of the other flows can be obtained similarly. Once the mass flow rate of each reactant gas is obtained, the pressure of each component can be calculated readily by balancing them. Taking into account the pressure drops in flow channels, the required pressure raise of the compressor can be expressed as a function of the inlet air flow rate

(3)

pcp = pcp (Win ) = pcp (Ifc , λ). 3

(9)


In order to control the properties of reactant gases, auxiliary components consume significant amount of energy. Since the compressor consumes more than 80% of all auxiliary energy consumption in high pressure PEM fuel cells, other energy losses are commonly neglected when calculating system net power loss. Assuming a constant mechanical static motor efficiency of 0.9, the compressor power consumption is   γ−1 γ pcp T − 1 , Pcon = C pWin amb  0.9ηcp pamb

tional Renewable Energy Laboratory [17]. The VESIM and ADVISOR models have been validated and proven to be powerful tools to study different vehicle aspects. Since both of them are MATLAB/SIMULINK-based models, they provide ease of reconfiguration and subsystem coupling. The model used here represents the powertrain system of FC hybrid vehicles, consisting of seven main modules: fuel cell stack, batteries, electrical motor, driver, vehicle controller, drivetrain, and vehicle dynamics (Figure 3). The engine model in

(10) req. Power by Motor/ gen. Power by Reg. Brake Motor Command

where C p and γ are specific heat capacity (1004J/(kg · K)) and ratio of specific heats (1.4) of air, respectively, Win is the mass flow rate of system inlet flow, ηcp is compressor efficiency, and pcp and pamb are the pressures of the compressor outlet flow and the ambient, respectively. The compressor is assumed to be static, driven by a static motor. Thus, a static compressor map is used to determine the efficiency corresponding to the required pressure ratio and the air mass flow rate. The performance of various compressors needs to be investigated. Because of lack of data, the compressor in this study is scaled geometrically from the Allied signal compressor given in [15]. Using the similarity principle, the map of a geometrically scaled compressor can be found readily since there is no difference between the flow characteristics of the original and the scaled compressor at a given point in the map. The flow and power is predicted to vary with pump size as pcp η0cp = ηcp (Win , ) pamb 0 = α2cpWin Win 0 0 Pcon = α2cp Pcon ,

Fuel Cell Power vehicle speed accel

DrvingCycle

speed set

decel

Electric Motor/Generator

Fuel Cell

T motor

T_shaft

w_wheel

w_shaft

Battery DRIVETRAIN

Vehicle Dynamics

Battery Power Batt. PowerLimit FC PowerLimit

Figure 3. Simulink model of hybrid electric fuel cell vehicles with a regenerative brake module

VESIM is replaced by a newly developed FC system module that is simplified as a first-order system with the performance map in order to make computation affordable. The performance map is generated by the high-fidelity FC model, described in the previous section. The motor/generator used in this study is selected from the library of motors published in ADVISOR (MC-AC187). In this motor/generator we assume that the motor/generator loss during braking equals the loss during acceleration. The battery and fuel cell control rules are selected from ADVISOR. Since the regenerative braking is employed to enhance the fuel economy, the battery is charged by the generator during braking. The mass of the vehicle mveh is affected by any changes in these components according to

(11) (12) (13) 0

where αcp is the geometric scaling factor, defined as αcp = DD , Win and Pcon are the inlet mass flow rate and power consumption 0 and P0 of the unscaled compressors, and Win con are the inlet mass flow rate and power consumption of a newly scaled compressors, respectively. The pressure ratio is invariant. Using the above relations, the power consumed by the compressor motor and the net power output from the fuel cell system can be expressed as a function of stack current and oxygen excess ratio.

mveh = mbody + nfc · Afc · ρfc + mcp · α3cp + maux +mbatt · nbatt · αbatt + mmot · αmot ,

(14)

where mveh , mbody , mcp , maux , mbatt , and mmot are the mass of whole vehicle, vehicle body, initial (unscaled) compressor, fuel cell auxiliary components, battery, and motor, respectively, Afc and ρfc are the cell density and area of fuel cells, nbatt is the number of battery modules, and αbatt and αmot are the scale factor of the battery capacity and motor torque, respectively. Table 2 lists all model parameters and their values.

2.2

Fuel Cell Vehicle Model The Fuel Cell Vehicle model for OPTimization (FCVOPT) is a version of the Vehicle Engine SIMulation (VESIM) model, developed at the University of Michigan [16], with battery, motor, and controller modules from ADVISOR, developed at Na4


Vehicle parameters used in the model 61

Inner Loop

0.7

Vehicle Frontal Area

3.58m2

Vehicle Auxiliary Power Load (Fxed)

1kW

Vehicle Body Mass (VESIM Initial), mbody

4272kg

Fuel Cell Density, ρfc

3.77kg/(m2 · cell)

Fuel Cell Area, Afc

0.038m2

Compressor Initial Mass, mcp

15kg

Fuel Cell Performance Model : Optimize the Fuel Cell Fuel Cell Optimal !(Ist)?Performance Model Reactant Control No : Optimize the Fuel Cell Yes Yes Yes Reactant Control No Fuel Cell No Fuel Cell Variables Variables No Changed? Changed? NewNew MapMap -./(0

Yes

$%!&'()&*

-./(0

$%!&'()&*+,-./(0

Drag Coefficient

61 Steady State Fuel Cell Model : Simulate the fuel cell reactant system

!"#

$%!&'()&*+,-./(0

Value

$%!&'()&*+,-./(0

Parameter

$%!&'()&*

-./(0

$%!&'()&*

-./(0

Map New MapOldOld Map !"#

!"#

$%!&'()&*+,-./(0

Table 2.

Fuel Cell Variables Changed?

$%!&'()&*

FCVOPT FCVOPT Old Map : :Simulate Simulate the the designed design vehiclefor forfuel fueleconomy economy FCVOPT vehicle andacceleration acceleration: Simulate the design and performancevehicle for fuel economy performance and acceleration performance No No Optimum? Optimum? !"#

Fuel Cell Auxiliary Mass,maux

55kg

Battery Mass, mbatt

11kg /module

Motor Initial Mass, mmot

91kg

Lower Limit of SOC

0.8

Upper Limit of SOC

0.7

Yes

Yes

No

Optimum? Yes

Figure4.3.10:Process Process Integration Figure integration Figure 3.10: Process Integration

2.3

Integration of Fuel Cell Simulation One of the objectives of this study is to develop a highfidelity fuel cell model that is capable of predicting the effects of design changes in fuel cell systems, and to integrate it with a vehicle simulator for design optimization of fuel cell vehicles. ¿From the quasi-static FC model, developed in Section 2.1, a FC performance map is generated for a given FC design and passed to the FCVOPT model to estimate hydrogen consumption. The optimizer determines whether the design is optimal or not based on given criteria. In order to enhance computational efficiency, ten previous designs for fuel cells are stored, and if the same fuel cell design is requested, the corresponding map is reused (Figure 4). The performance map indicates the hydrogen consumption or energy consumption to produce maximum power at a given stack current. For a given stack current, the hydrogen consumption is determined by Equation (3). Therefore, the maximum efficiency can be obtained when the maximum power is achieved by providing the optimal compressor command. In other words, at a given stack current, maximum power or maximum efficiency can be achieved by adjusting the oxygen excess ratio. Therefore, a nested optimization is used to find optimal oxygen excess ratios for a given fuel cell design. The compressor, used in the FC model, is associated with two constraints: upper and lower bounds for mass flow rate. The mass flow rate can not be larger than the specified maximum flow rate. In this study, we assume that the maximum flow rate is also scaled using Equation (12). On the other hand, for centrifugal and axial compressors, a minimum amount of mass flow rate is required at any given pressure ratio. Otherwise, because of an

max

Pnet = Pst − Pcp

increased pressure across the compressor, a temporary reversal Pnet = Pst − Pcp to fuelmax cell modeling equations flow occurs and subject the compressor becomes unstable. This(3.79) unsta(3.79) p fuel cell modeling equations subject to ble phenomenon is calledWcompressor ) surge. Compressor surge, in ≥ Wmin ( p p or compressor stall, can lead to lossWof power ) serious comWmin ( p and in ≥ where the Wmin is the minimum mass flow rate determined by the surge control line. The pressor damage [18]. In a compressor map, a surge line connects where the Wmin is the minimum mass flow rate determined by the surge control line. The continuesflow to determine the optimal ratio, λ, for possible stack theoptimizer minimum rate points foroxygen the excess entire operating pressure optimizer continues to determine the optimal oxygen excess for possible stack range. compressor operates above the surge line,ratio, theλ,comcurrents If untila the power output reaches the maximum. pressor will surge. Thus, it is standard to employ a surge control currents until the power output reaches the maximum. line positioned to the right of the surge line and set the operating point below the control line (Figure 5). Since the maximum efficiency occurs near the surge line, a bigger margin between the surge line and the control line leads to a reduction of power output. In addition, several sophisticated anti-surge controllers, or surge avoidance controllers, can extend the stable region of compressors. The controllers, however, also have their own limitations due to the control logic and controlling resources. Thus, the control line needs to be set properly in order to prevent performance reduction and excessive controller commands. In this study, the control line is set to be close to the surge line. The inner-loop problem, including the compressor constraints, can be formulated as ca

amb

ca

amb

max Pnet = Pst − Pcon with respect to λ(Ifc ) subject to fuel cell modeling equations p Win ≥ α2cpWmin ( p ca ) amb Win ≤ α2cpWmax , 5

(15)


3.5

60 with Quasi−steady FC Model with Full Dynamic FC Model 76

50

3

72

rge

40

78

Velocity(mph)

Pressure Ratio

Su 2.5

Re gio

n 64

72

2

control line

78

20

56

48

surge line

1.5

30

40

10

76

1

0 0

Figure 5.

0.01

0.02

0.03

0.04 0.05 Flow(kg/sec)

0.06

0.07

0.08

0.09

0

Figure 6.

Surge and control lines on a compressor map

200

400

600 800 Time(sec)

1000

1200

1400

Comparison of results of fuel cell vehicle simulator with a full

dynamic fuel cell model and a quasi-steady fuel cell model

Table 3. Comparison between full dynamic model and quasi-steady model in terms of fuel economy and simulation time

where Wmin is the minimum mass flow rate determined by the surge control line and Wmax is the maximum mass flow rate of the unscaled compressor. Note that the vehicle (outer-loop) design optimization problem is presented in Section 3.1. The optimizer continues to determine the optimal oxygen excess ratio λ for possible stack currents until the power output reaches the maximum. For this nested (inner-loop) optimization, the Sequential Quadratic Programming (SQP) algorithm, a popular and effective gradient-based algorithm, is used because of its computational efficiency.

Driving Cycle

Full Dynamics

Quasi-steady

Error Time Reduction

EPA HWFET

EPAUDDS

US06

Fuel Cell VESim Simulation with Dynamic and Static FC models In order to validate the fidelity of the quasi-steady fuel cell model, two vehicle models, one with a quasi-steady model and one with a full dynamic model, are compared in terms of fuel economy and tracking performance for several testing cycles. For a full dynamic model, the model developed for the FORD P2000 fuel cell [12] is scaled to make the fuel cell generate the same amount of power as the quasi-steady model. Figure 6 shows the simulated vehicle speeds of both models for the EPA Urban Dynamometer Driving Schedule (EPA UDDS). Based on the simulation, agreement is satisfactory. Table 3 shows the differences in fuel consumption between the two models depending on the driving cycle. The maximum difference is smaller than 5%. Typically, the simulation using the quasi-static model is two orders of magnitude faster than the simulation using the full dynamic model. Therefore, it can be concluded that the quasi-static model is appropriate for investigating overall vehicle performance. The quasi-static model, however, cannot detect several transient problems, including oxygen starvation and surge. Based

fuel economy

16.26mpg

16.73mpg

-2.9%

sim. time

500 sec

12 sec

97.6%

fuel economy

15.81mpg

15.07mpg

+4.7%

sim.time

2310 sec

35 sec

98.5 %

fuel economy

10.26 mpg

10.46 mpg

-1.9 %

sim.time

260 sec

5.3 sec

98.0%

2.4

on the simulation results, oxygen starvation problem is not a serious issue because of the power demand limiter in the fuel cell controller. Since oxygen starvation occurs when a sharp increase in the power demand happens, the power demand limiter reduces the possibility of oxygen starvation (Figure 7(a)). Figure 7(b) shows the transient response on the compressor map. The compressor surges excessively when the power demand decreases sharply. Therefore, an additional controller that detects and prevents surge is required when short-time stability is important. Since this paper focuses on the overall vehicle performance during a relatively long cycle, transient stability is assumed to be satisfied.

3

System Design Optimization and Parametric Study The FC vehicle model with battery hybridization and regenerative braking is optimized with respect to design variables including six component size variables and two fuel cell control limits. Design variables and their bounds are listed in Table 4. One of the most significant benefits of hybrid electric vehicles is 6


200

400

600 800 time(sec)

1000

1200

1400

Figure 3.12: Oxygen excess ratio during the cycle 4

64

43

Table 4.

2.4

Design variables and their lower and upper bounds of optimiza-

tion for hybrid electric FCVs with a regenerative brake 2.2

2.5 3.5

Nominal Operation

2 Pressure Ratio

32

1.5 2.5

21

0

200

400

600 800 time(sec)

1000

1200

1400

1.5

g in rg on Su egi R

Oxygen Excess Ratio Oxygen Excess Ratio

3.5

Design Variables

Lower

Upper

x1 : number of cells, nfc

800

2000

x2 : compressor scale, αcp

0.5

1.7

x3 : motor torque scale, αmot

0.5

1.5

1

60

0.5

2.5

x6 : lower limit of FC power command

0.01

0.4

(b) on the compressor map Figure 3.13: Transient x7 :response upper limit of FC power command

0.6

0.99

1

5

1.8

1.6

1.4

Transient Variation

1.2

Figure 3.12: Oxygen excess ratio during the cycle 1

1

0

200

400

600 800 time(sec)

1000

1200

1400

(a) Figure 3.12: Oxygen excess ratio during the cycle

0

x4 : number of battery modules,nbatt

0.01

x : battery capacity scale, αbatt 0.03 0.04 0.05 0.06

5 0.02

Flow(kg/s)

2.4

x8 : final gear ratio 2.2

Nominal Operation

Pressure Ratio

1200

1400

1.8 2.2 1.6 2 1.4 1.8 1.2 1.6 1.41 0

regenerative braking benefits. Acceleration criteria (g1 - g3 ) are defined to be the same as those of the baseline diesel vehicle (HMMWV 4.5L International). Maximum discrepancy between driving cycle-prescribed speed vcyc (t) and actual vehicle speed, vveh (t) is limited to less than 2 mph in order to ensure tracking performance during both acceleration and deceleration (g4 ). The difference between final and initial battery SOC (g5 ) is also restrained to less than 0.5% in order to reduce the effect of the initial SOC on fuel economy.

Nominal Operation

g in rg on Su egi R

Pressure Ratio

2 2.4

g in rg on Su egi R

1000

0

Transient Variation 0.01

0.02

0.03 Flow(kg/s)

0.04

0.05

0.06

Transient 1.2 Variation

(b) on the compressor map Figure 3.13: Transient response

ring the cycle

1

0

0.01

0.02

0.03

0.04

0.05

3.2

Optimization Algorithms Based on a design-of-experiments study, the objective and constraint functions are found to exhibit considerable numerical noise. Noisy responses pose difficulties and challenges to gradient-based optimization algorithms. Therefore, two derivative-free optimization codes are used in this study for the outer optimization loop: DIRECT (DIvided RECTangles), and NOMADm. The DIRECT algorithm was developed by Donald Jones [19]. The code in this study is a Matlab implementation of DIRECT, written by Kenneth Holmstrom for a previously public domain optimization toolbox called TOMLAB [20]. An advantage of DIRECT is that it does not require any parameter tuning. The most significant disadvantage of DIRECT is the lack of a convergence criterion. The algorithm iterates until the userspecified maximum number of function evaluations or iterations is exceeded. There is a rule of thumb that requires at least 200d function evaluations, where d is the number of variables, but this rule is subject to the curse of dimensionality. DIRECT is neither effective nor efficient when the number of variables is larger than, say, 10. NOMADm is the Matlab implementation of the Generalized Pattern Search (GPS) and Mesh Adaptive Direct Search (MADS)

0.06

Flow(kg/s) Figure 7. Oxygen excess ratio during the cycle and transient response on the compressor map Figure 3.13: Transient response on the compressor map

the presence of a regenerative brake, which can transform decelerating torque into electricity and charge the batteries. Utilizing power that would be otherwise wasted, a regenerative brake improves the efficiency of hybrid vehicles significantly.

Nominal Operation

3.1

0.05

0.06

compressor map

Problem Formulation The vehicle design optimization problem is formulated as maximize f (x) = (fuel economy) subject to g1 ≡ (max velocity) ≥ 80mph g2 ≡ (0-60 mph time) ≤ 23sec g3 ≡ (30-50 mph time) ≤ 11sec g4 ≡ max{|vveh (t) − vcyc (t)|} ≤ 2mph g5 ≡ |SOCfinal − SOCinitial | ≤ 0.5%

(16)

Since fuel economy is relative to the used driving cycle, EPA UDDS, an urban driving cycle, is chosen in this study to assess 7


Table 5. Initial Design

Optimization Results DIRECT

NOMADm

limit of the FC power command, the optimizer can improve efficiency. This power-splitting trend can be seen clearly in Figure 8. During the cycle, the fuel cell turns off frequently, and the depen-

NOMADm w/o Reg. Brk.

1823

1936

1969

1.1

1.36

1.27

1.27

x3

1

0.83

0.83

0.88

x4

12

30

26

24

x5

1

0.86

1

1

x6

0.1

0.27

0.13

0.05

x7

0.95

0.98

0.79

0.74

x8

3

2.53

2.875

2.875

g1

-0.03

-0.09

-0.07

-0.00

g2

0.17

-0.15

-0.13

-0.15

g3

-0.02

-0.01

-0.00

-0.09

g4

-0.46

-0.57

-0.57

-0.67

g5

9.76

-0.46

-0.06

-0.43

f

21.9

Number of Func. Eval.

24.9

24.8

21.0

1800

491

452

150 Fuel Cell Power (kW)

1400

x2

100

50

0

0

200

400

600

800

1000

1200

1400

200

400

600 800 Time (sec)

1000

1200

1400

100 Battery Power (kW)

x1

50 0 !50 !100 0

a) Power drawn from fuel cell and battery in vehicles with regenerative braking

family of derivative-free algorithms for mixed variable programming [21]. This family of algorithms have proven convergence properties [22, 23]. The NOMADm package offers a variety of options to choose from (including the flexibility of performing a user-defined search and the ability of using derivatives even if they are only partially available) to increase the effectiveness of the algorithm for a given problem. However, it does not require expertise, and the default options can be used with high rate of success for most problems. Convergence is based on final mesh size. In our experience, NOMADm will find at least as good solutions as DIRECT in far fewer function evaluations for most problems.

Fuel Cell Power (kW)

150

100

50

0

0

200

400

600

800

1000

1200

1400

200

400

600 800 Time (sec)

1000

1200

1400

Battery Power (kW)

100

50

0

!50 0

3.3

Results The results are presented in Table 5. It can be seen that the overall efficiency of the hybrid fuel cell vehicle can be improved by 14% after optimizing component sizes and control parameters. Even though the power management is based on simple rule-based control, the improvement is remarkable. Additionally, the same vehicle is optimized without accounting for regenerative braking to verify associated benefits. As expected, a regenerative brake can improve fuel economy by 18%. Interestingly, there are no significant differences between the optimal vehicle designs with and without the regenerative brake. On the other hands, control limitations are affected a lot. For the vehicle with a regenerative brake, the optimizer tends to reduce the operating range of the fuel cell. Since the regenerative brake frequently charges the batteries, the latter take more power load than batteries in a vehicle without a regenerative brake, especially for low power load cycles, where battery efficiency is considerably higher than fuel cell efficiency. Thus, by increasing the lower

b) Power drawn from fuel cell and battery in vehicles without regenerative braking

Figure 8. Simulation of a hybrid electric fuel cell vehicle model with/without regenerative braking

dency on batteries is greater. Moreover, the fuel cell power load becomes more stable because the batteries buffer the high frequency power demand even though the dynamic stability of fuel cell systems is not taken into account in this problem formulation. This smooth power load is even more desirable for fuel cell system control because it reduces the probability of compressor surge and oxygen starvation. Note that the number of cells is large. If we assume that the average thickness of each cell is about 1 mm, the length of the fuel cell stack will be more than 1.8 m, which is longer than the 8


Table 6.

0.8

0.8

width of a small vehicle. The large number of cells is due to the low power density, which is because the fuel cell voltage model is based on the FORD P2000 prototype that was manufactured in 1999. The fuel cell stack in this prototype has 381 cells and generates 50 kW, equivalent to 131 W/cell. During the last five years, a tremendous amount of research has been conducted on fuel cell technology, and several new membrane and cell designs have improved the power output per cell significantly. A recently reported fuel cell stack can generate 66.5 kW with 450 cells (148 W/cell) [24].

Vehicle Optimum

Vehicle Optimum

0.7

0.7

Max Power Density Max Efficiency

System Efficiency (%)

System Efficiency (%)

0.5

0.4

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0

Max Power Density Max Efficiency

0.6

0.6

0

0

0.1

0.2

0.3

0.4 0.5 0.6 Power Ratio (P(Ist)/Pmax)

0.7

a) x axis: P(Ist)/Pmax

Figure 10.

0.8

0.9

1

0

0.2

0.4

0.6

0.8

1

1.2

Current Density (A/cm2)

b) x axis: current density

Efficiency versus power ratio and stack current of the fuel cell

systems of Table 6

Comparison of compressor scales for fuel cell systems that are

operates most efficiently. In spite of the compressor inefficiency, a smaller compressor makes a fuel cell system more efficient because it consumes less power and supplies a smaller mass flow of air that restrict the operation of fuel cell system within only a smaller current density region where the efficiency of fuel cells is significantly higher (Figure 10). Since the objective of the vehicle is equivalent to the overall efficiency, the fuel cell system also tends to be more efficient, which in turn requires a smaller compressor and a large number of cells. The compressor, however, cannot be smaller than the optimal values because the acceleration and maximum velocity constraints require the fuel cell system to have sufficient amount of power. Since the vehicle model in this paper represents heavy-duty small trucks weighing more than 4000 kg, the vehicle performance is less sensitive to the increased fuel cell system mass compared to passenger car cases. Therefore, at the optimal design, the fuel cell system becomes more similar to one optimized for maximum efficiency. It is expected that if cost, weight, and packaging constraints are taken into account, the optimizer will yield a smaller number of cells and a larger compressor size so that fuel cell power density is increased.

optimized for vehicle objectives, maximum power density, and maximum efficiency (number of cells = 1936) Optimized for

Vehicle

Max. Power Density

Max. Effi. 0.99

Compressor Scale

1.27

2.25

Power Densitya (kW/kg)

0.49

1.03

0.33

Max. Effi.

55%

46%

59%

Ave. Effi.

66%

57%

72%

a

Power density is calculated as the maximum power divided by the mass of the fuel cell stack and compressor, excluding other auxiliary units.

3.5

Vehicle Optimum Max Power Density Max Efficiency

Pressure Ratio

3

2.5

2

1.5

4 1

0

Figure 9.

0.01

0.02

0.03

0.04 0.05 Flow(kg/sec)

0.06

0.07

0.08

Conclusion A fuel cell vehicle simulation model was developed to achieve sufficient fidelity and efficiency for design optimization. The model integrates two submodels: a fuel cell system performance model and a fuel cell vehicle model. In the fuel cell system performance model, the stoichiometry is optimized to produce the maximum power at given currents for each fuel cell system design. By integrating the two models, the efficiency of the new model was improved significantly (average 98% reduction in simulation time) without sacrificing accuracy (less than 5% error). The fuel cell vehicle model was used to maximize fuel economy with respect to six component size variables and two control limits. Two derivative-free optimization algorithms, DIRECT and NOMADm, were used. Fuel economy was improved by 14% by balancing the power loads on fuel cell stack and battery

0.09

Operating points of the fuel cell systems of Table 6

In addition to the old membrane model, the relatively small compressor size also decreases power density. Table 6 includes compressor scales for fuel cell systems that are optimized for vehicle objective, maximum power density, and maximum efficiency. A tradeoff between power density and efficiency can be observed depending on compressor size. Looking at the operating points on the compressor maps in Figure 9, the compressor of the fuell cell system optimized for maximum power density 9


modules and the tradeoff between power density and efficiency of the fuel cell system. Regenerative braking was found to be significantly beneficial even though it has almost no impact on design. Since the vehicle in this paper is less sensitive to component mass changes, several constraints dealing with cost, weight, and packaging must be considered for a more comprehensive understanding of the overall design tradeoffs.

[11] Wang, Z., Wang, C., and Chen, K., 2001. “Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells”. Journal of Power Sources, 94, pp. 40–50. [12] Pukrushpan, J. T., Stefanopoulou, A. G., and Peng, H., 2004. Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design. Springer. [13] Xue, D., and Dong, Z., 1998. “Optimal fuel cell system design considering functional performance and production costs”. Journal of Power Sources, 76(1), pp. 69–80. [14] Mawardi, A., Yang, F., and Pitchumani, R., 2005. “Optimization of the operating parameters of a proton exchange membrane fuel cell for maximum power density”. Journal of Fuel Cell Science and Technology, 2(2), pp. 121–135. [15] Cunningham, J., Hoffman, M., Moore, R., and Friedman, D., 1999. “Requirements for a flexible and realistic air supply model for incorporation into a fuel cell vehicle (fcv) system simulation”. Future Transportation Technology Conference and Exposition, Costa Mesa, California, August 17-19. 1999-01-2912. [16] Assanis, D., Filipi, Z., Gravante, S., Grohnke, D., Louca, L., Rideout, G., Stein, J., and Wang, Y., 2000. “Validation and use of simulink integrated, high fidelity, engine-invehicle simulation of the international class VI truck”. SAE Paper. 2000-01-0288. [17] Markel, T., B. A. H. T. J. V. K. K. K. B. O. M. S. S., and Wipke, K., 2002. ADVISOR: a system analysis tool for advanced vehicle modeling. [18] Murphy, K., Kalata, P., Fischl, R., and Marchio, D., 1995. “On modeling surge avoidance controllers (sac) in compressors: Design procedure”. Proceedings of the American Control Conference, Seatle, Washington, June. TP2-3:50. [19] Jones, D., 1999. DIRECT, Encyclopedia of Optimization. Kluwer Academic Publishers. [20] Holmstrom, K., 1989. gclsolve.m: A standalone version of direct, software documentation, revision 2.00. HKH MatrisAnalys AB, Sweden. [21] Abramson, M., 2005. Nomadm version 3.31 user’s guide. http://en.afit.edu/ENC/Faculty/MAbramson/NOMADm.html, April, 2005. [22] Audet, C., and J. E. Dennis, J. “Mesh adaptive direct search algorithms for constrained optimization”. to appear in SIAM Journal on Optimization. [23] Abramson, M. A., Audet, C., and Dennis Jr., J., 2004. “Generalized pattern searches with derivative information”. Mathematical Programming, 100(1), pp. 3 – 25. [24] Cunningham, J., Moore, R., and Ramaswamy, S., 2003. “A comparison of energy use for a direct-hydrogen hybrid versus a direct-hydrogen load-following fuel cell vehicle”. 2003 SAE World Congress, Detroit, Michigan, March 3-6. 2003-01-0416.

ACKNOWLEDGMENT This work was partially supported by a Korea Science and Engineering Foundation Fellowship, and by the Automotive Research Center, a US Army Center of Excellence at the University of Michigan. Any opinions expressed in this publication are only those of the authors.

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