Proceedings of FuelCell2008
Sixth International Fuel Cell Science, Engineering and Technology Conference
June 16-18, 2008, Denver, Colorado, USA
FUELCELL2008-65111
OPTIMAL DESIGN OF HYBRID ELECTRIC FUEL CELL VEHICLES UNDER
UNCERTAINTY AND ENTERPRISE CONSIDERATIONS
Jeongwoo Han∗, Panos Papalambros
{jwhan, pyp}@umich.edu
Department of Mechanical Engineering, University of Michigan
G.G. Brown Bldg., Ann Arbor, Michigan 48109
ABSTRACT
System research on Hybrid Electric Fuel Cell Vehicles
(HEFCV) explores the tradeoffs among safety, fuel economy,
acceleration, and other vehicle attributes. In addition to engineering considerations, inclusion of business aspects is important in a preliminary vehicle design optimization study. For
a new technology, such as fuel cells, it is also important to
include uncertainties stemming from manufacturing variability
to market response to fuel price fluctuations. This paper applies a decomposition-based multidisciplinary design optimization strategy to an HEFCV. Uncertainty propagated throughout
the system is accounted for in a computationally efficient manner. The latter is achieved with a new coordination strategy
based on sequential linearizations. The hierarchically partitioned
HEFCV design model includes enterprise, powertrain, fuel cell,
and battery subsystem models. In addition to engineering uncertainties, the model takes into account uncertain behavior by
consumers, and the expected maximum profit is calculated using
probabilistic consumer preferences while satisfying engineering
feasibility constraints.
vehicle performance [1–5]. As a result, safety and vehicle performance have been significantly improved to a practical level.
System research on hybrid electric fuel cell vehicles (HEFCV)
has aimed at exploring the tradeoffs among safety, fuel economy,
acceleration, and other vehicle attributes [6, 7]. A model-based
vehicle design methodology using a quasi-static fuel cell model,
which could be used to design both the vehicle and its fuel cell
system, was recently presented [7]. The model offered sufficient
fidelity and efficiency for engineering design studies. Such studies, however, can be more valuable for preliminary design if business aspects are included in the optimization study. Inclusion of
business aspects makes the design problem more complex, requiring multidisciplinary analyses with significant interactions.
Additionally, a new technology, such as fuel cells, is subject to
increased uncertainty stemming from manufacturing variability
to market response to fuel price fluctuations. Such uncertainty
should be accounted for to the extent possible when studying the
feasibility of HEFCV designs.
Due to the increased complexity and uncertainty, an AllIn-One (AIO) method where all subsystems are considered all
together in a single problem may not be practical or reliable.
In such cases, it is advisable to use decomposition strategies
in a Multidisciplinary Design Optimization (MDO) framework,
where the system is broken down into several manageable subsystems whose solution is coordinated to produce the overall
system solution. Decomposition strategies often use two levels:
subproblems, typically representing different aspects (or disciplinary analyses), are optimized concurrently, while a systemlevel problem coordinates the interactions between the subprob-
1
Introduction
Automotive use of fuel cells has received increased attention as a viable alternative energy source for automobiles due to
clean and efficient power generation. Several fuel cell vehicle
concepts and fuel cell system designs have been proposed and
studied in terms of safety, robust operation, fuel economy, and
∗ Corresponding
author, Phone/Fax: (734) 647-8402/8403
1
c 2008 by ASME
Copyright Inputs
Level index i
Element index j
j=1
i=1
j=2
i=2
Tij
j=4
X̄ij = {Xij , T(i+1)k }
local objective
fij
(a)
maximize Profit
w.r.t. {enterprise decisions}
s.t. market constraints
vehicle
targets
local constraints Pr[gij,m ≤ 0] ≥ αij,m
j=3
j=5
Outputs
design variable
responses
i=3
Rij
j=6
Rij = aij (X̄ij )
R(i+1)k
ptpt
Fuel
Fueleconomy
economy( fefe )
pt)
Powertrain
Powertraincost
cost( C
Cpt
meet vehicle targets
w.r.t. {vehicle variables}
s.t. performance constraints
T(i+1)k
(b)
fuel cell
targets
Figure 1. Example of index notation in ATC and information flow for an
element Oi j
fc
# of cells ( nfc )
fc
Mass ((m
Weight
Map
Wfc),),Map
fc
Specific Cost ( SCfc)
meet fuel cell targets
w.r.t. {fuel cell variables}
s.t. fuel cell constraints
lems [8–11]. Analytical Target Cascading (ATC) is an optimization method for multilevel hierarchical systems typically partitioned into physical subsystems or objects (see Figure 1(a)) [12].
Each block in the hierarchical structure, referred to as an element,
is an optimization subproblem. An element can be coupled with
only one parent element but with multiple children elements. The
linking variables between a parent and children are design targets
and analysis responses. Targets are set by parents and propagated
to their children; the children are optimized to obtain responses
that are as close to the targets as possible. Thus, targets and responses are updated and coordinated iteratively to achieve consistent values for the overall system.
Compared to the deterministic formulation, only a few publications are available that solve hierarchical system design optimization problems under uncertainty due to the difficulty in
incorporating uncertainty into linking variables. Using random
variables to represent uncertainty, the so-called Probabilistic Analytical Target Cascading (PATC) has been formulated from the
deterministic ATC by Kokkolaras et al. [13], and generalized
with more general probabilistic characteristics by Liu et al. [14].
As pointed out in [14], the choice of random variable representation is an important issue in MDO under uncertainty. A popular
way to define uncertainty is using random variables, assuming
that their probability density functions (PDFs) can be inferred.
These distributions are assumed as inputs to the optimization
problem. Solving the optimization problem requires estimating
propagation of these uncertainties throughout the system, which
can be a computationally expensive process for nonlinear systems. In order to overcome this numerical difficulty, new coordination strategies using sequential linearizations were developed
to solve hierarchically decomposed design problems [15, 16].
These strategies take advantage of the simplicity and ease of uncertainty propagation for linear systems by solving a hierarchy
of linearized problems successively.
In this study, a hierarchically decomposed HEFCV design
model that includes enterprise, powertrain, fuel cell and battery
models is developed (see Figure 2) and solved using the aforementioned decomposition strategies. In addition to engineering
Figure 2.
battery
targets
Mass ((m
Weight
Map
Wbtbt),),Map
bt
Capacity ( Cpbt)
meet battery targets
w.r.t. {battery variables}
s.t. battery constraints
Hybrid electric fuel cell vehicles design problem
uncertainties, the model takes into account uncertain behavior by
consumers, and the expected maximum profit is calculated using
probabilistic consumer preferences while satisfying engineering
feasibility constraints.
The article is organized as follows. In Section 2, a Sequential Linear Programming (SLP) coordination strategy for PATC
is briefly introduced while Section 3 explains the development of
the comprehensive HEFCV design model. Optimization results
and discussion are presented in Section 4, followed by conclusions in Section 5.
2
SLP coordination for PATC
In this section, a formulation of PATC is explained briefly
and the computational advantage of sequential linearization is
presented. Also, the subproblem formulation of the SLP coordination strategy for PATC is provided. In PATC, we consider
a Probabilistic All-In-One (PAIO) system design problem expressed as follows:
min E[ f (X)]
X
subject to Pr[gm (X) ≤ 0] ≥ αm ,
(1)
m = 1, ..., Mc ,
where Mc is the number of constraints. In Eq. (1), f and X are
the system objective function and the vector of all random design
variables, respectively. Design constraints are expressed in an
probabilistic feasibility formulation, in which the probability of
satisfying gm (X) ≤ 0 is greater than the required reliability level
αm . For target matching problems, f (X) is expressed as ||T −
R(X)|| p where T and R are the system’s targets and responses
1
and || · || p is a p-norm, mathematically expressed as (∑ | · | p ) p .
Assuming that the system objective and constraints are separable, Eq. (1) is decomposed hierarchically into N elements at
M levels. Quantities with indices i j are related to element j at
2
c 2008 by ASME
Copyright level i. As shown in Figure 1(b), Xi j , Ti j and Ri j denote local
design variables, targets and responses to the element Oi j while
fi j , gi j and ai j are local objective, constraint and response functions. Consistency between elements is relaxed and coordinated
through penalty functions. Then the generalized PATC formulation for an element Oi j with a quadratic penalty function is
expressed as follows:
Given Ti j , R(i+1)k ,
min E[ fi j (X̄i j )] + ||wi j ◦ (Ti j − Ri j )||22
Figure 3.
Benefit of sequential linearization in decomposition strategies
X̄i j
+ ∑ ||w(i+1)k ◦ (T(i+1)k − R(i+1)k )||22
k∈Ci j
a considerable amount of computational cost, depending on the
accuracy of estimation. On the other hand, once the system is approximated linearly and the random variables are normally distributed, the linking variables also have normal distributions. In
other words, no estimators are needed, as shown in Figure 3 (b).
In [15], we presented SLP coordination strategies for PATC
by extending the SLP algorithm to hierarchically decomposed
design optimization problems under uncertainty. In these strategies, probabilistic constraints are approximated by equivalent deterministic linear constraints using either the first or second order reliability method (FORM/SORM). The linking variables are
represented only with means and standard deviations. Among
them, the means of linking variables are treated as optimization
variables, while their standard deviations are estimated at every
iteration. Therefore, consistency of random variables does not
require significant computation in estimating and matching distributions. Moreover, weighted L∞ norms are used in the penalty
functions in order to maintain linearity of subproblems. The resulting formulation for an element Oi j with L∞ penalty function
can be expressed as follows :
(2)
subject to Pr[gi j,m (X̄i j ) ≤ 0] ≥ αi j,m , m = 1, ..., Mc,i j
where Ri j = ai j (X̄i j ), X̄i j = [Xi j , T(i+1)k ],
∀k ∈ Ci j , ∀ j ∈ Ei , i = 1, ..., N,
where Ci j is the set of the children of element j at level i and
Ei is the set of elements at level i while wi j is the linking
variable deviation weighting coefcient vector for element Oi j
that is updated successively. In Eq. (2), the ◦ operation indicates the component-wise multiplication of two vectors such that
{a1 , ..., ak }T ◦ {b1 , ..., bk }T = {a1 b1 , ..., ak bk }T . For detailed discussion on PATC formulations, readers are referred to [13, 14].
In Eq. (2), random variables are often used to define uncertainty and assumed to have independent normal distributions. As mentioned earlier, estimating the propagated uncertainty throughout the system is required but can be a computationally expensive task for nonlinear functions even with a simple univariate function. On the other hand, since the output of
independent normal distributions through a linear function is normally distributed, the uncertainty propagation for a linear system
with normally distributed inputs can be obtained efficiently.
In order to take advantage of the simplicity and ease in
estimating uncertainty propagation for linear systems, Chan et
al. [17] proposed the use of SLP to solve reliability-based design optimization problems for a single system, with the goal of
achieving an appropriate balance between accuracy, efficiency
and convergence behavior. Thus, assuming that random design
variables or parameters are normally distributed, the algorithm
can solve the optimization problem under uncertainty with sufficient accuracy and efficiency by linearizing and solving a problem successively.
The benefit of sequential linearization can be more significant for decomposed systems because of the system consistency
with random linking variables. Since it is not practical to match
two distributions exactly, the first few moments, such as means
and variances, were used in the previous literature to maintain
consistency in linking variables. In order to obtain the first few
moments, however, additional estimators between subsystems
were used for linking variables when their PDFs are unknown,
as illustrated in Figure 3 (a). The estimators typically require
Given µTi j , dµT , µR(i+1)k , dµR
ij
min
∇ fi j (µx̄i j )T d̄i j + εi j +
(i+1)k
,
∑ ε(i+1)k
k∈Ci j
with respect to d̄i j , εi j , ε(i+1)k
subject to ∇gi j,m (x̄Mi j,m )T d̄i j + gi j,m (x̄Mi j,m ) ≤ 0,
wi j ◦ (µTi j + dµT − µRi j − dµR ) ≤ ±εi j ,
ij
(3)
ij
{w ◦ (µT + µdT − µR − µdR )}(i+1)k ≤ ±ε(i+1)k ,
where µRi j = ai j (µx̄i j ) + ∇ai j (µx̄i j )T d̄i j
d̄i j = [dµX , dµT
], ||d̄i j ||∞ ≤ ρ,
ij
(i+1)k
∀k ∈ Ci j , ∀ j ∈ Ei , i = 1, ..., N, m = 1, ..., Mc,i j ,
where µx are the mean values of variables x where the linear
approximations are made, while dx is the solution vector of x
at the current iteration. In Eq. (3), x̄Mi j,m is the most probable
point (MPP) obtained by FORM/SORM while ρ is a trust region
radius updated at every iteration. The maximum consistency error of linking variables for element Oi j , εi j , is used to maintain
the system consistency because the L∞ penalty function cannot
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c 2008 by ASME
Copyright air
H2
water
injected
comp
ressor
pressure
adjusted
Figure 5.
Figure 4.
cal parts
humidifier
hydrogen
Tank
humidity
adjusted
cooler
stack
temperature
adjusted
Reactant supply subsystems (modified from [18])
Decoupling of a hybrid powertrain into mechanical and electriTable 1.
Fuel cell system operating conditions and geometries
Parameter
ambient pressure
ambient temp.
stack temp.
active area (Afc )
be incorporated with the objective function [15]. The convergence, accuracy and effectiveness of the strategy were discussed
in [15, 16].
Value
1bar
298K
353K
769cm2
Parameter
ambient relative humidity
cathode relative humidity
anode relative humidity
Value
0.5
0.8
1.0
termined as the difference between the power generated from a
fuel cell stack, Pstfc , and the power consumed by auxiliary comfc , expressed as:
ponents, Pcon
3
Hybrid Electric Fuel Cell Vehicle Design Model
In this section, a comprehensive HEFCV design model,
which takes into account profit, cost and market demand issues,
is developed and decomposed hierarchically into four elements at
three levels as illustrated in Figure 2. Blocks in the figure represent subsystems in the problem while the variables between them
denote the linking variables. Some design variables are chosen as
random variables to investigate the effect of uncertainties in engineering design and customer behavior on the overall enterprise
decisions.
The HEFCV under consideration is a light truck (or small
sports utility vehicle (SUV)) whose curb weight is about 2480kg
including hydrogen storage. Figure 4 illustrates its powertrain
configuration. The proton exchange membrane (PEM) fuel cells
and lithium ion batteries are used as primary and secondary
power sources in the powertrain, respectively. The study focuses
on high-pressure fuel cell systems with a compressor because
most vehicular application prototypes are developed using highpressure fuel cells due to their higher power density.
fc
fc
fc
Pnet
= Pstfc − Pcon
= nfc Istfc vfc
cl − Pcon ,
(4)
where nfc , Istfc and vfc
cl are the number of cells, stack current and
cell voltage of a fuel cell system, respectively. If the composition
and structure of the cells are determined, then the cell voltage
is a function of stack current density and reactant flow properties, including partial pressures, humidity, and temperature. As
shown in Figure 5, these properties are governed by reactant suppliers consisting of four flow subsystems: a hydrogen tank and
compressor determine hydrogen and air pressure throughout the
system, and a humidifier and cooler adjust the humidity and temperature of reactant gases to the fuel cell stack. Since the map
generated in the model is quasi-static, the transient irregularity
in the properties of the inlet reactant flow is ignored. It is also
assumed that the pressure at the anode depends on the cathode
pressure. Table 1 summarizes the system operating conditions
and active area. Additionally, power losses other than the comfc in Eq. (4),
pressor one are ignored from the calculation of Pcon
because the compressor consumes more than 80% of all auxiliary power consumption in high-pressure PEM fuel cells. Under
these assumptions, the net power output, cell voltage, vfc
cl , and
stack voltage, vfc
st , can be reduced to a function of the stack current, and the typical relation between them is presented in Figure 6.
Effectiveness of decomposition strategies depends on the
number of linking variables. Since matching two maps accurately requires significant computational cost due to the large
number of linking variables related to the maps, a simple yet accurate representation of performance maps needs to be defined.
As shown in Figure 6 (a), the net power output can be approxifc = afc I fc 2 + bfc I fc , where afc and bfc are the coeffimated as Pnet
st
st
3.1
Fuel Cell System Model
The study employs the quasi-static PEM fuel cell model
in [7] developed for design optimization from a dynamic fuel
cell model by Pukrushpan et al. [18]. The fuel cell model combines fluid dynamics with static Membrane-Electrode-Assembly
(MEA) and compressor efficiency models obtained from experimental data. Since the MEA properties are normalized by a unit
area, the MEA model can be scaled by multiplying the active
area while the compressor and flow channels can be scaled by
the similarity principle. Thus, the quasi-static model can generate a static performance map that represents the maximum power
for a certain range of fuel consumption with given control constraints and design variables.
fc , can be deThe power output from a fuel cell system, Pnet
4
c 2008 by ASME
Copyright 4
9
(ICEs), the 2010 U.S. Department of Energy (DOE) targets for
fuel cell stacks are used here [20]. Due to lack of data, it is assumed that the power density per unit area in year 2010 is identical to that in year 2005 and the stack cost and mass here are set
to $130/m2 and 2.7kg/m2 , respectively. Also, assuming that the
baseline auxiliary components is for a 100kW system and satisfies the 2010 U.S. DOE targets ($20/kW), the cost of baseline
auxiliary components is set to $2000. If the cost of auxiliaries
is assumed to increase linearly with the compressor volume, we
can define the fuel cell system cost Cfc as follows:
x 10
Data
Fitted
8
Net Power (W)
7
6
5
4
3
2
1
0
0
40
80
120 160 200 240 280 320 360 400
Stack Current (A)
Cfc = 130Afc nfc + 2000α3cp .
(a) Net power vs. stack current
Current Density (A/cm2 )
0.2
0.3
0.4
System Voltage(Data)
System Voltage(Fitted)
Cell Voltage
400
System Voltage (V)
350
0.8
0.6
250
0.5
200
0.4
150
mfc = 2.7Afc nfc + 15α3cp + 20αch + 50.
0.2
fc
Imin
40
fc
Imax
80
0.1
120 160 200 240 280 320 360 400
Stack Current (A)
(b) System voltage vs. stack current
Cell voltage vs. current density
Figure 6.
Typical fuel cell system performance maps
cients of the approximation that depend on fuel cell design variables, namely, the number of fuel cells in a stack (nfc ) and the
geometric scaling factors of the compressor and reactant channel
fc = vfc I fc , then
in length (αcp and αch , respectively). Since Pnet
net st
fc
fc
fc
fc
vnet = a Ist + b . As shown in Figure 6 (b), the net voltage has
a peak at low current. Let the current with the peak voltage be
fc . Then, the linear approximation of the net voltage is valid
Imin
fc and the maximum current I fc . Thus,
for the range between Imin
max
assuming that the designed fuel cell system is operated only in
fc , and I fc and I fc = I fc /10, the fuel cell
the range between Imin
max
max
min
map can be represented as follows:
2
fc = afc I fc + bfc I fc ,
Pnet
st
st
fc ≤ I fc ≤ I fc .
Imin
st
max
(7)
Due to lack of data, the parameters used for the auxiliaries in
Eq. (7) are assigned arbitrarily. Based on some parametric studies, however, fuel economy and acceleration are less sensitive to
the parameters than to the efficiency and maximum power of the
fuel cell system due to the mass of vehicle if the total mass of
auxiliaries is between 70kg and 120kg.
While current costs for fuel cell stack and fuel cell system
are $67/kW and $108/kW, respectively, a fuel cell stack should
cost less than $50/kW in mass production to be competitive in
the automotive market [21]. Therefore, assuming the ratio between the costs of a fuel cell stack and a fuel cell system remains
similar, we can consider market acceptability as follows:
0.3
100
0
0
0.9
0.7
300
50
Also, assuming the mass of auxiliaries increases linearly with the
compressor volume and flow channel radius, the fuel cell system
mass can be expressed as follows:
0.5
Cell Voltage (V)
0.1
450
(6)
SCfc = Cfc /(rated power) ≤ $80/kW.
(8)
Then, the fuel cell subproblem in deterministic ATC formulation
can be expressed as follows:
Given tfc = {tnfc ,tmfc ,tSCfc ,tafc ,tbfc ,tImax
fc }
min π(tfc − rfc )
with respect to xfc = {nfc , αcp , αch }
subject to gfc = SCfc − $80/kW ≤ 0,
{200, 0.8, 0.8} ≤ xfc ≤ {1000, 1.1, 1.2}
where rfc = afc (xfc ).
(5)
(9)
The compressor and channel scaling factors are assumed to
be normally distributed with σαcp = σαch = 0.02. On the other
hand, the number of cells is considered deterministic and large
enough to be relaxed. Further, we assume that the linking variables related to the map representation are deterministic and the
remaining, tmfc ,tSCfc , are random. Moreover, the local constraint,
gfc , is treated as a probabilistic constraint with p f = 0.13%.
As shown in Figure 2, the other linking variables between
the fuel cell and powertrain are mass, mfc , specific cost, SCfc ,
and number of cells, nfc . According to [19], a fuel cell stack surveyed in year 2005 costs and weighs $360/m2 and 3.9kg/m2 ,
respectively, including membranes, electrodes, gas diffusion layers, bipolar plates and seals. Since fuel cells are too expensive and heavy compared to current internal combustion engines
5
c 2008 by ASME
Copyright separator
negative
electrode
positive
electrode
hn
hp
hs
1.35
Area Specific Resistance (ohm m2 )
current
collector
A bt
current
collector
Figure 7. Li-ion cell sandwich consisting of composite negative and positive electrode and separator (adapted from [23])
x 10
-3
1.3
charge
apprx. charge
discharge
apprx. discharge
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.55
0.6
0.65
0.7
0.75
SOC (A)
0.8
0.85
0.9
Figure 8. Discharging and charging resistances of a Li-ion battery showing agreement between estimated resistances and quadratic approxima-
3.2
Battery System Model
In a hybrid powertrain, a secondary power source (such as a
rechargeable battery) stores energy from a primary power source
(such as an ICE or a fuel cell) and provides the stored energy
under conditions where the primary power source operates inefficiently. Among various secondary power sources, Li-ion batteries have gained significant attention due to their high energy
density, high open circuit voltage, no memory effect and a slow
loss of charge when not in use. A Li-ion battery cell consists of
layers of a negative electrode, a positive electrode and a separator
sandwiched by current collectors from both ends, as illustrated in
Figure 7. The electrodes are made of two different insertion compounds that determine cell properties including an open circuit
voltage and load resistances. More detailed explanation on Liion battery cells and the insertion reaction can be found in [22].
tions
electrolyte of LiPF6 in EC:DMC.
The theoretical Coulomb capacity of an electrode can be determined by the amount of Li content based on its stoichiometry
ranging from 0 to 1. Since the actual range of the stoichiometry
should be narrower than the theoretical range, we assumed that
the actual capacity is 80% of the theoretical capacity. Also, in a
balanced cell, the actual capacities of each electrode are equal.
Therefore, the ratio between the electrode thicknesses is set to a
constant (hp = 0.585hn ). In addition, we assume that the separator thickness, hs , is set to 25µm.
The model requires a relatively high computational cost for
design optimization, and so a simple internal resistance battery
model is developed by characterizing battery cells, as described
in the Partnership for Next Generation of Vehicles (PNGV) battery test manual [25]. In the battery model, the estimated voltage,
vbt
net , can be expressed as follows:
The rate of the insertion reaction depends not only on cell
properties (such as diffusion coefficients of lithium-ions) but also
on cell geometries (such as cell thicknesses or active areas). With
given cell properties, a wider active area, Abt , is desirable because it typically results in a lower resistance and higher energy
content. Moreover, because a typical discharge voltage of Li-ion
cell is less than 4.0V, a number of cells need to be connected in
series to produce sufficiently high voltage for automotive applications. In this study, all cells are assumed to be connected in
series and the number of battery cells in series connection, nbt , is
considered as a design variable in powertrain design.
In order to simulate the behavior of a given Li-ion battery
cell for a load cycle, a 1-D full cell model of Li-ion batteries has
been developed in [23, 24], assuming the cell is uniform in the
directions parallel to the current collectors. Since most output
quantities are normalized by a unit area, the resulting output can
be easily scaled by multiplying the active area. The cell temperature can vary with time if temperature-dependent material
properties are provided. Due to lack of data, however, we assume that the temperature of the system is uniform and constant
at 25◦ C. Also we take the cell thicknesses, hbt
i , and cell area,
Abt , as the design variables for the battery, assuming the other
properties and geometries, such as insertion materials, porosities
and number of windings, are fixed. The negative and positive
insertion materials are graphite and CoO2 , respectively, with an
bt
bt
vbt
net = E − Rl Il ,
(10)
where E bt , Rl and Ilbt are an open circuit voltage, a internal resistance and a load current, respectively. Since the open circuit
voltage does not depend on cell geometries but on materials, the
open circuit voltage can be easily obtained from the 1-D battery
cell model, expressed as E bt = −1.92x3 + 3.78x2 − 1.48x + 4.16,
where x is the State Of Charge (SOC) divided by 100. Resistances depend on cell geometries, such as cell thicknesses.
Thus, in order to measure the resistance, the 1-D battery cell
model is simulated for a load cycle, corresponding to the hybrid pulse power characterization (HPPC) tests described in [25].
The HPPC tests are performed for the SOC range of 55-85% because the range is wide enough for batteries to run a cycle and
the internal resistances can be approximated accurately by a second order polynomial function over the range. Then, the HPPC
test results are used to estimate the resistance assuming the dual
mode operation [25]. Note that two different internal resistances
are estimated, namely, charging and discharging resistances.
6
c 2008 by ASME
Copyright Figure 8 presents the estimated discharging and charging resistances and their quadratic approximations. As shown in the
figure, the quadratic approximations agree with the estimated resistances. Note that the resistances show similar curvatures to
each other over the SOC range. Thus, in order to reduce the
number of linking variables, the cell resistances are modeled as
follows:
bt
bt
Discharging : Rdis = (abt x2 + bbt
dis x + cdis )A
bt
2
bt
Charging : Rchr = (a x + bchr x + cchr )Abt .
The vehicle includes two motors; one for each wheel on the rear
axis. Using the cited motor model, a motor map is generated as
a function of motor variables, namely, rotor radius, number of
turns per stator coil and rotor resistance. Here, the rotor radius,
rm, is assumed to be the only designable geometry. Since motors can cover wider speed and torque ranges more efficiently
than conventional ICEs, the conventional gearbox is removed
and each motor is connected to each wheel through a belt and
pulley system. Thus, the final drive ratio is determined by the
pulley speed ratio, pr. With a given rotor radius and pulley speed
ratio, the mechanical part model estimates the required power,
pt
Preq (t; rm, pr). The masses of the fuel cell, battery and motors
are also taken into account.
(11)
The error resulted from the use of the same abt for Rdis and Rchr
is smaller than 5% over the design space defined in Eq. (12). The
resistances of a pack can be calculated by multiplying Rdis and
Rchr with nbt .
For powertrain simulation, the mass, mbt , and the capacity
of batteries, Cpbt , are estimated from the densities and capacity
of the materials provided in the library of the 1-D model. Then,
the battery subproblem in a deterministic ATC formulation can
be expressed as follows:
In the electrical part, Pbt (t) and Pfc (t) are estimated from
pt
Preq (t) based on the power-management strategy and used to calculate SOC and hydrogen consumption, respectively. In order to
estimate fuel economy from the hydrogen consumption, the final
SOC (SOC f ) should be sustained within a small range from the
pt
initial SOC (SOCi ) or gSOC = |SOC f − SOCi | ≤ 0.001.
Provision of sufficient power, speed and torque is assured by
imposing the design constraints:
Given tbt = {tmbt ,tCpbt ,tabt ,tbbt ,tcbt ,tbbt ,tcbt },
dis
dis
chr
chr
min π(tbt − rbt )
bt
(12)
with respect to xbt = {hbt
n ,A }
bt
bt
bt
subject to {50µm, 0.5A0 } ≤ x ≤ {200µm, 3A0 }
2
where rbt = abt (xbt ), Abt
0 = 0.528m .
pt
pt
pt
gpower = max{Preq (t) − Pavl (t)} ≤ 0,
pt
gspeed = max{wmax − wmt (t)} ≤ 0,
(13)
pt
mt
mt
mt
mt
gtorque = max{τmax (w ) − τ (t), τ (t) − τmin (w )} ≤ 0,
Both local variables are assumed to have normal distributions with σhbt = 2µm, σAbt = 0.02Abt
0 . Also, the mass and capacity are considered random while the linking variables related
to the resistance maps are deterministic.
where τmax and τmin are the maximum and minimum torques,
while wmax is the maximum angular velocity. For acceleration
performance, the 0-60 mph time, t0-60 , is measured and should
pt
be less than 8 sec, expressed as g0-60 = t0-60 − 8 ≤ 0.
The cost of the powertain, Cpt , estimated for enterprise decisions, can be expressed as follows:
3.3
Powertrain Model
In the powertrain illustrated in Figure 2, the power bus splits
the power demand from the mechanical part into power demands
to the fuel cell and the battery, and combines the powers supplied from the two sources to drive the motors based on a powermanagement strategy. A poorly designed power-management
strategy may result in worse fuel economy than that of conventional vehicles. Among the variety of power-management strategies, such as a rule-based control [7], dynamic programming
(DP) [26], stochastic dynamic programming (SDP) [27] and
equivalent consumption minimization strategy (ECMS) [2, 28],
ECMS is employed here because it provides robust power management compared to other strategies according to [29].
Note that the powertrain is decoupled into the electrical and
mechanical parts because the mechanical part does not depend
on the power-management strategy. Models for the mechanical parts are developed in [30], including a motor design model.
Cpt = Cfc +Cbt +Cmt −Cic ,
(14)
where Cfc ,Cbt and Cmt are the costs of fuel cell, battery and motor, while Cic is the cost of a target ICE whose max power is
200kW. Since the specific cost of ICEs is not readily available,
we employ the same assumption as in [31], i.e., (ICE specific
fc .
cost) = 19$/kW. Thus, Cic = $ 3800. Also, Cfc = SCfc Pmax
For the battery and motor, cost models presented in [32] are
used. Since the battery model is developed for NiMH batteries,
the reference manufacturing cost and the reference specific energy are modified for Li-ion batteries. Then, the powertrain subproblem in a deterministic ATC formulation can be expressed as
7
c 2008 by ASME
Copyright Table 2.
follows:
Historical product price and demand data points and demand
values adjusted for expected new product penetration [32]
Given tpt = {tfept ,tCpt }, rfc , rbt ,
min π({tpt − rpt , tfc − rfc , tbt − rbt })
with respect to xpt = {rm, pr, nbt }, tfc , tbt
pt
pt
pt
subject to gpower ≤ 0, gspeed ≤ 0, gtorque ≤ 0, (15)
pt
pt
gSOC ≤ 0, g0-60 = t0-60 − 8 ≤ 0,
{0.2, 1, 25} ≤ xpt ≤ {0.3, 3, 100},
where rpt = apt (xpt , tfc , tbt ).
=
927.83
872.14
Lifecycle Mileage of a light truck for the first twelve years [36]
Age
Miles
Age
Miles
Age
Miles
Age
Miles
1
2
3
28,951
26,479
24,226
4
5
6
22,173
20,301
18,593
7
8
9
17,035
15,613
14,314
10
11
12
13,128
12,043
11,052
(17)
¯ 01|02 is the average of 2001 and 2002 market prices
where Pent
of the current conventional light truck design, which is set to
$24,109. Because the value of V ent is not verified in this study,
we will treat it as a parameter in the optimization. Its value is
determined after the following discussion on the fuel cost saving.
In order to estimate the fuel cost saving, miles traveled, the
rate of fuel consumption and fuel price need to be known. Lifecycle mileage of light trucks for the first twelve years of vehicle life is presented by Environmental Protection Agency [36]
(see Table 3); the rate of fuel consumption is the inverse of fuel
economy obtained from the powertrain model, assuming that the
initial fuel economy is maintained for the period. On the other
hand, the fuel price is uncertain because it fluctuates across time.
In [34], the fuel price is assumed to follow the mean-reverting
process, expressed as,
∆Ddsl = αent
(Ddsl − D̄dsl )∆t + σdsl ∆z,
dsl √
ent
∆z = η
∆t, ηent ∼ N(0, 1),
ent
ent ent
λ
∆Pent
θ − ∆P
q + λ Sent
Sent ,
∆qent
∆qent
Pent
Adjusted quantity (k)
9278.3
8721.4
ent
Sent − (Pent − P̄01|02
) ≥ V ent ,
Enterprise Decision Model
The objective of enterprise decisions is to maximize profit
subject to marketing constraints. Here we consider a simple
gross profit πent , calculated as the total revenue minus the cost of
obtaining the revenue. Revenue equals price, Pent , times quantity, qent , considering the sale of the designed vehicle the only
economic activity. Also, this study considers only the manufacturing cost of the vehicle, ignoring operational expenses, such as
marketing and sales [34].
Under standard microeconomic assumptions, a negative linear relationship between price and quantity demanded of conventional light class trucks can be drawn from the two pairs of price
and annual sales data in 2001 and 2002, shown in Table 2 [32].
We assume that the enterprise has decided to allocate 10% of its
existing capacity for the production of the new product. Moreover, following the argument in [34], demand is assumed to be
shifted by the fuel cost saving, Sent . The resulting demand curve
can be expressed as,
Pent
Quantity (k)
the ratio λSent is unknown and treated as a random variable with
σλent /λent = 0.02.
P
S
To determine the demand curve, the mean of the consumer
behavior, µλent /λent , is realized through consumer’s aversion toP
S
ward the new technology that can be modeled by a net utility
threshold V ent [35]. Then, for market acceptability, the difference
between fuel saving from a hybrid fuel cell vehicle and change
in price should be greater than the threshold [34], expressed as,
3.4
ent
Price
$23,632
($24,585)98
Table 3.
The rotor radius and the number of battery cells are deterministic while the pulley ratio is normally distributed with
σ pr = 0.002. In this subproblem, the local constraints except
pt
for g0-60 are assumed deterministic. Due to the nested optimizapt
tion and ECMS, gSOC is not violated unless the power sources
are too small for the vehicle. For fuel economy estimation, Simplified Federal Urban Driving Schedule (SFUDS) is used here,
which has the same average speed and maximum acceleration
and braking values as the federal urban drive schedule (FUDS)
used in U.S. urban fuel economy estimates, but runs for only 360
seconds while FUDS runs for 1500 seconds [33].
∆q
ent + ∆q Sent ⇒
qent = θ − ∆P
ent P
∆Sent
Year
2001
2002
(16)
(18)
where α is the speed of reversion, D̄dsl is the normal level of Ddsl
and σdsl is the volatility of diesel fuel price, estimated from historical monthly diesel fuel prices from March 1994 to October
2007 [37]. The mean-reverting process can be used for predicting the diesel fuel price for an ICE vehicle. At present there is
no such commodity market for hydrogen, and data for hydrogen
where λSent = ∆qent /∆Sent and λPent = ∆qent /∆Pent . Here λSent
can be interpreted as the fuel cost saving elasticity of demand,
meaning the responsiveness of the quantity demanded of a good
to a change in the expected fuel cost saving. Due to lack of
knowledge about consumer behavior toward the new technology,
8
c 2008 by ASME
Copyright prices are not rich enough for the mean-reverting process to be
applied. The Department of Energy set the 2005 target for the
end-user cost of hydrogen to 2.00 - 3.00$/kg [20]. Therefore,
this study assumes that the hydrogen price is $3/kg currently and
increases at a static inflation rate, rent , that is assumed to be 3%.
The model for hydrogen prices, therefore, is not suitable for a
long-term prediction. Instead, we can assume that both price
models are valid in the short-run, such as 2 years. For diesel
price, we can generate a random walk for the period based on
Eq. (18). Discounting back with the static inflation rate, rent , the
diesel fuel expense can be calculated in:
Cdsl =
ent
Z 2yr
Ddsl Mt e−r t
0
fedsl
dt,
mulation can be expressed as follows:
Given rpt
min π({−πent = (Pent −Cpt −CPent )qent , tpt − rpt })
λent
with respect to xent = { λSent , qent }, tpt
subject to
≤0
{0.1, 60}
gent
1
P
ent
ent
gent
2 ≤ 0 g3 = q − 1200
ent
≤ x ≤ {0.9, 1200}.
(22)
≤ 0,
The lack of understanding of market behavior is taken into
account as uncertainties in local variables with σλent /λent = 0.02,
P
S
σqent = 12. Also, the local constraints are treated as probabilistic
constraints with p f = 0.13%.
(19)
4
Results
Noting that Eq. (9), (12), (15) and (22) correspond to the
blocks in Figure 2, the hierarchically decomposed PATC problem is solved by the SLP coordination strategy, starting from the
solution obtained first from solving the deterministic ATC problem. Results are shown in Table 4. The numbers in parentheses
indicate those from the initial point, i.e., the deterministic optimal design. As shown in the table, the initial point is superior to
the probabilistic solution based on nominal values with gent
1 and
pt
g0-60 active. Because the initial point is located at the boundary of
the constraints, they are violated severely when uncertainty is introduced. On the other hand, the solution satisfies the constraints
under uncertainty but with reduced profit. For example, in the
pt
powertrain subproblem, g0-60 is satisfied at the solution with a
larger pr that increases the acceleration of the vehicle. Also,
the design changes in the fuel cells are more substantial than in
the batteries, which could mean that batteries are less sensitive
to uncertainty than fuel cells. At the solution, αcp is reduced
fc is considerably sensitive to the uncersignificantly because Pmax
tainty in αcp with the given uncertainty. More specifically, σPmax
fc
is 0.94 at the solution while it is 8.2 at the initial point where
gfc is violated. Since a smaller compressor typically results in
smaller net power per cell, more cells are used to compensate
the power shortage. Thus, all changes to make the fuel cell less
sensitive to uncertainty (or more reliable) result in a heavier and
more expensive fuel cell system, which causes the smaller and
less powerful motors to maintain the vehicle mass. The smaller
motor and larger pr decreases the fuel economy, and the reduced
fuel economy with the increased fuel cell cost decreases the price
and profit of HEFCV at the solution.
Figure 9(a) presents the power from the fuel cell (solid line)
and the battery (dashed line) during SFUDS. The power demand
during this schedule is not aggressive compared to the maximum
power available from the fuel cell and battery. Moreover, Figure 9(b) shows the SOC history during the cycle. As shown in
both figures, ECMS splits the power demands properly so that
where Mt denotes miles traveled while fedsl is the fuel economy
of a conventional light truck whose average value is reported to
be 22.3 mpg in [38]. In order to consider multiple future scenarios, the process is repeated 100,000 times and the mean of the
fuel expenses is used for the rest of model. On the other hand, because hydrogen price increases at rent , the hydrogen fuel expense
and fuel cost saving can be expressed as,
CH2 =
3(28951+26479)
fept
and
Sent = Cdsl −CH2 .
(20)
Returning to consumer preference, we assume that consumers want their return on investment after 2 years to be larger
than half of the cost of the investment. Additionally, for a longterm prediction, five times the fuel cost saving should be larger
than the price difference by V ent =$ 10,000. Both constraints can
be expressed mathematically as follows:
ent
ent
ent ≤ 0,
gent
1 = (P − P̄01|02 ) − 2S
ent
ent
ent
gent
2 = (P − P̄01|02 ) − 5S + 10000 ≤ 0.
(21)
Modeling a more sophisticated customer preference is possible
but beyond the scope of this demonstration.
The manufacturing cost includes the production cost Cpent
and the powertrain cost Cpt . While Cpt is estimated from the powertrain model, the production cost remains to be defined. Due to
lack of data, the regression model in [34] is scaled down by the
ratio between the prices of light and medium trucks, expressed
2
as CPent = 3.05 × 104 − 44.5qent + 0.0443qent . Then, assuming
that the enterprise has allocated the maximum monthly capacity
to 1200, the enterprise subproblem in a deterministic ATC for9
c 2008 by ASME
Copyright Table 4.
Summary of results (design variables indicated by boldfaces)
120
400
data
fit
100
FUEL CELL
nfc
αcp
αch
mfc
Cfc
fc
Pmax
SCfc
Fuel Cell
Battery
0
SOC
Power (W)
150
300
0
350
range of approximation
0
50
100
150
200
250
Stack Current (A)
300
350
(b) Net voltage vs. stack current
!1
0.7
0.699
!2
0.6985
300
350
(a) Power generated by fuel cell and battery
0.698
0
50
100
150
200
time (sec)
250
300
Fuel cell system performance map
Parameters used in parametric study
Original
PS:FC
PS:BT
PS:H2
130
50
3
190
50
3
130
100
3
130
50
4
price (denoted by PS:FC, PS:BT and PS:H2, respectively). The
parameters are summarized in Table 5, and Figure 11 shows the
parametric study results. Due to the increased costs, all profits drop to negative. It is important to note that changes in the
cost analyses and the fuel price prediction affect not only enterprise decisions but also engineering decisions because of the
strong coupling between the two domains. When the fuel cell
or battery cost is high, the enterprise sets the fuel economy and
powertrain cost targets by balancing them, but in different ways
for each case. In the case when the fuel cell stack cost is high,
the fuel economy decreases significantly to enable cost to be as
low as possible. This is because the fuel cell cost in the original
solution comprises a significant portion of the powertrain cost.
Thus, to make the powertrain more inexpensive, a smaller fuel
cell is favorable and power can be drawn from the more powerful battery resulting in slightly increased battery cost. On the
other hand, in the case when the battery manufacturing cost is
high, fuel economy increases despite the increased battery and
powertrain costs, because the battery in the original solution is
at least three times less expensive than the fuel cell. Also, the
battery cost in this study is more dependent on its capacity than
its power resulting in a significant drop in the optimal battery
capacity. Based on profit-loss comparisons between these two
cases, the cost reduction of fuel cells is more important than that
of batteries.
Another important factor for market feasibility is hydrogen
price. Similar to other types of vehicles, as the fuel price rises,
fuel economy becomes more important. Since higher fuel economy requires more efficient but expensive powertrains, vehicle
cost also increases significantly. That is, decisions on HEFCV
require accurate fuel price models in addition to reliable engineering models. While the mean-reverting process is applied to
diesel fuel price, the hydrogen price is assumed to increase by the
0.6995
Figure 9.
150
200
250
Stack Current (A)
Fuel cell stack cost
Battery manu. cost $/kg
Hydrogen price $/kg
0.7015
250
100
$/m2
0
150
200
time (sec)
50
Table 5.
1
100
0
Figure 10.
0.701
50
200
50
(a) Net power vs. stack current
0.7005
0
250
100
192 (192)µm
112 (112)µm
0.883 (0.861)m2
2.04 (1.99)kAh
36.8 (35.9)kg
$2,268 (2,221)
88.1 (88.1)kW
38 (38)
hn
hp
Abt
Cpbt
mbt
Cbt
bt
Pmax
nbt
2
!3
40
range of approximation
x 10
3
60
20
4
4
80
BATTERY
0.228 (0.234)m
1.86 (1.34)
84.1 (106)mpg
7.96 (8.00)sec
3,197 (3,217)kg
488 (514)kg
$1,952 (2,004)
191 (207)kW
rm
pr
fept
t0-60
mveh
mmt
Cmt
mt
Pmax
300
527 (489)
1.04 (1.08)
0.862 (0.822)
193 (188)kg
$7,517 (7,402)
101 (117)kW
74.1 (51.0)$/kW
POWERTRAIN
350
Net Voltage (V)
$2.20 (3.39) ×106
0.81 (0.899)
647 (592)
$31,555 (33,210)
$28,155 (27,480)
$3,844 (4,551)
$3,400(5,730)
πent
λS /λP
qent
Pent
Cpt +Cpent
Sent
ent
π /qent
Net Power (kW)
ENTERPRISE
350
(b) History of SOC
Simulation of a hybrid electric fuel cell vehicle
the final SOC is maintained close to the initial SOC and satisfies the SOC constraint. Note that the maximum deviation from
the initial SOC is only 1.3% and the fuel economy is quite high
due to the short and mild duty cycle (SFUDS) where the battery
does not need to be charged by the fuel cell. If a longer and
more aggressive cycle, such as the Urban Dynamometer Driving
Schedule (UDDS) is used, the fuel economy plummets to 43.7
mpg with around 10% of the maximum SOC deviation.
Figure 10 shows the performance maps of the fuel cell at
the solution . Even though the approximated net voltage output
from the fuel cell for 37A to 105A is not as accurate as that for
the other net current, the net power approximation shows good
agreement with the actual output because the excess in the net
voltage results in power loss below 1kW (or 5%).
The practical feasibility of the solution depends highly on
parameters in the cost models, which have not yet been validated.
In order to investigate the importance of these parameters, a parametric study is conducted on the fuel cell stack cost per area, the
battery reference manufacturing cost per mass and the hydrogen
10
c 2008 by ASME
Copyright Normalized Difference (%)
Normalized Difference (%)
validated.
For a more comprehensive understanding of the overall design tradeoffs, several constraints with packaging and safety issues must be considered, and these constraints require multidisciplinary analyses and decisions. For example, many safety issues
of fuel cell vehicles are related to hydrogen carriers and storage
that affect packaging and vehicle performance. Safety depends
also on battery materials. In this study, a metal oxide-based cathode material (LiCoO2 ), commonly used for electronics, is used
due to availability of material properties. Since the load profiles of automotive applications are considerably more aggressive
than those of power electronics, a novel group of olivine-based
cathode materials, such as phospho-olivine LiFePO4 , can improve safety significantly and be suitable for hybrid vehicles [40].
Since these constraints require a multidisciplinary approach, use
of the optimization strategies employed here could be advantageous in investigating system tradeoffs.
200
100
PT Cost
Profit
0
1
2
3
4
FC Cost
-100
BT Cost
No Profit
-200
PS:FC
PS:BT
PS:H2
PS:FC
PS:BT
PS:H2
-300
20
Fuel Economy
10
0
FC Power
1
2
BT Power
3
4
-10
BT Capacity
-20
Figure 11.
Parametric study on cost and fuel price models
inflation rate due to lack of historical price data. Since the hydrogen price is expected to have low volatility, the fuel cost saving
would be larger, and demand and profit might be improved if a
hydrogen price model for a longer time horizon is provided.
ACKNOWLEDGMENT
This work was partially supported by the Automotive Research Center, a US Army Center of Excellence in Modeling
and Simulation of Ground Vehicle Systems at the University of
Michigan, and NSF Grant DMI-0503737. This support is gratefully acknowledged. The results and opinions expressed here are
solely those of the authors.
5
Conclusion
A hierarchically decomposed HEFCV design model was
developed, including enterprise decisions, powertrain, fuel cell
and battery models. A PATC problem was formulated considering uncertainties in engineering design and marketing decisions. Customer preference and demand were assumed to be
random variables. Since the linking variables between the powertrain model and its children contain performance maps, the maps
were approximated in order to reduce the number of linking variables. The approximation of the performance maps at the solution agreed with the actual maps with less than 5% error. The
problem was solved by the SLP coordination strategy presented
in [15, 16] that takes advantage of the simplicity and ease in estimating propagated uncertainties through linear functions.
Among the nine constraints, a customer preference conpt
straint, gent
1 , and an acceleration constraint, g0-60 , were active,
and would be violated severely if the deterministic optimal design were to be chosen. Given the assumptions on costs and hydrogen price predictions, the resulting HEFCV was expected to
achieve a profit of $2.20×106 for the particular light truck market segment. Moreover, the cost analyses and price prediction are
as critical as engineering models for market feasibility based on
the parametric study. Clearly, these are results based on the assumed parameter values and models. It is important to note that
many models and parameters used in this study are not representing the current state of technology due to limited availability to
the models and parameters: the fuel cell models are based on the
Ford P2000 prototype in 1999 [39] while most cost and weight
models are based on the 2010 U.S. DOE targets and need to be
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