4

Battery Energy Storage

Energy, as available in nature for conversion and usage by the society, is

known as primary energy. The sources of primary energy are in the wells or
in other natural environments such as in the wind or in the sun. Examples
of primary energy sources are fossil fuels, wind, solar, hydro, wave, nuclear,
geothermal, and biomass. Energy is extracted from the primary source either
by burning it or by transforming the source energy through an energy con-
version system. When primary energy is transformed by one or more energy
conversion processes and/or devices, it is known as secondary energy; addi-
tional energy conversion devices use the secondary energy to deliver useful
work. Several stages of energy conversion are necessary, first, for processing
of the primary energy, and then for delivering it to the end user. Inefficiencies
at varying degrees are associated in each of these conversion stages.
Primary sources have energy stored in them in chemical, heat, kinetic, or
some other alternative form. For example, energy is stored in fossil fuels in
chemical form; energy is extracted from it in an internal combustion (IC)
engine vehicle by burning it in a heat engine. Wind energy is available in
kinetic form, which can be converted to electrical energy using a wind tur-
bine. Electrical energy is an example of secondary energy that can be con-
verted to mechanical work by an electric machine. Electrical energy can be
obtained from the primary source by burning fossil fuels in thermal power
plants, or from renewable primary sources of water, wind, solar, or other pri-
mary sources. Electrochemical devices also produce electrical energy from
chemical energy.
Energy derived from sources other than the burning of fossil fuels is
known as alternative energy. Ideally, converters used for the processing of
alternative energy must avoid the usage of fossil fuels during any stage of
the energy conversion process. For example, the ideal solution for recharg-
ing batteries in electric vehicles is to use electricity derived from renewable
energy sources such as wind or solar. If electricity from coal-fired power
plants is used for recharging, then the environmental problems are only
being shifted from one location to another.
The energy content of various raw fuels or materials refers to the energy
that can be extracted from it for useful work. The parameter for energy
content evaluation is specific energy or energy density. Specific energy is the
energy per unit mass of the energy source, and its unit is W h/kg. For fossil
fuels, the energy content refers to the calorific or thermal energy that can be
extracted from it by burning. Energy content for other materials is similarly

99
100 Electric and Hybrid Vehicles: Design Fundamentals

Table 4.1
Nominal Energy Densities of Sources
Nominal Specific Energy
Energy Source (W h/kg)
Gasoline 12,500
Diesel 12,000
Biodiesel 10,900
Natural gas 9,350
Ethanol 8,300
Methanol 6,050
Hydrogen 33,000
Coal (bituminous) 8,200
Lead-acid battery 35
Li-polymer battery 200
Flywheel (carbon-fiber) 200

evaluated in terms of specific energy for a level comparison. The specific

energies of several energy sources are given in Table 4.1. These energies are
shown without taking containment into consideration. The specific energies
of hydrogen and natural gas would be significantly lower than that of gaso-
line when containment is considered.
One of the challenges of energy is to store it in a convenient form so that it
can be utilized when needed. In terms of storage, fossil fuels have the biggest
advantage since these can be conveniently stored in a container. On the other
hand, electrical energy is very useful in delivering work on demand using a
highly efficient electromechanical device; however, energy storage in electri-
cal form is not simple.
The mechanism for energy delivery must also be considered for the sys-
tem under consideration. The energy transportation system has to be secure,
efficient, and environmentally friendly. Energy transportation over long
distances using electrical transmission and distribution systems is highly
efficient. Once an infrastructure is in place, energy converted to electrical
form at one end (which is the generation side) can be delivered to industrial
and residential units through the transmission and distribution system. On
the other hand, fossil fuels are transported long distances using pipelines,
ocean liners, and land transportation before being finally dispersed at the
gas stations.
The electrochemical devices are the most promising alternative technolo-
gies to the conventional fossil fuel–burning power plants, both in vehicle
and utility power stations. The electrochemical energy conversion processes
have the advantages of high conversion efficiency, large enough power out-
put, and the availability of a wide selection of fuels. Batteries and fuel cells
are energy storage and power generating devices that are suitable for both
portable and stationary applications. The topics covered in this chapter
Battery Energy Storage 101

include the fundamentals of battery structure and operation, and battery

applications in electric and hybrid vehicles. The presentations will help a
design engineer size the energy storage system, and then select the appro-
priate battery technology for an application. Additional alternative energy
storage devices are covered in the next chapter.

4.1 Batteries in Electric and Hybrid Vehicles

The basic requirement for purely electric vehicles is a portable supply of elec-
trical energy, which is converted to mechanical energy in the electric motor
for vehicle propulsion. The electrical energy is typically obtained through
the conversion of chemical energy stored in devices such as batteries and
fuel cells. The portable electrical energy storage presents the biggest obstacle
in the commercialization of electric vehicles. One solution for minimizing
the environmental pollution problem due to the lack of a suitable, high-
energy density energy storage device for electric vehicles is the hybrid elec-
tric vehicles that combine the propulsion efforts from gasoline engines and
electric motors.
Among the available choices of portable energy sources, batteries have
been the most popular choice of energy source for electric vehicles since the
beginning of research and development programs in these vehicles. The
electric vehicles and hybrid electric vehicles that are available commercially
today use batteries as the electrical energy source. The desirable features of
batteries for electric and hybrid electric vehicle applications are high specific
power, high specific energy, high charge acceptance rate for both recharging
and regenerative braking, and long calendar and cycle lifes. Additional tech-
nical issues include methods and designs to balance the battery segments
or packs electrically and thermally, accurate techniques to determine a bat-
tery’s state of charge, and recycling facilities for battery components. Above
all, the cost of batteries must be reasonable for electric and hybrid vehicles to
be commercially viable.
There are two basic types of batteries: a primary battery and a secondary
battery. Batteries that cannot be recharged and are designed for a single
discharge are known as primary batteries. Examples of these are the lith-
ium batteries used in watches, calculators, cameras, etc., and the manga-
nese dioxide batteries used to power toys, radios, flashlights, etc., Batteries
that can be recharged by flowing current in the direction opposite to that
during discharge are known as secondary batteries. The chemical reac-
tion process during cell charge operation when electrical energy is con-
verted into chemical energy is the reverse of that during discharge. The
batteries needed and used for electric and hybrid vehicles are all second-
ary batteries, since they all are recharged either during a mode of vehicle
102 Electric and Hybrid Vehicles: Design Fundamentals

operation or during the recharging cycle in the stopped condition using a

charger. All the batteries discussed in this book are examples of secondary
batteries.
The lead-acid type of battery has the longest development history of all
battery technology, particularly for their need and heavy use in the indus-
trial electric vehicles, such as golf cars in sports, passenger cars in airports,
forklifts in storage facilities and supermarkets. The power and energy den-
sities of lead-acid battery are lower compared to several other battery tech-
nologies. The research and development for alternative batteries picked
up momentum following the resurgence of interest in electric vehicles
and hybrid vehicles in the late 1960s and early 1970s. The sodium-sulfur
batteries showed great promise in the 1980s with high energy and power
densities, but safety and manufacturing difficulties led to the abandoning
of the technology. The development of battery technology for low power
applications, such as cell phones and calculators opened the possibilities
of scaling the energy and power of nickel-cadmium and lithium-ion-type
batteries for electric and hybrid vehicle applications. The major types of
rechargeable batteries used or being considered for electric and hybrid
vehicle applications are

• Nickel-metal-hydride (NiMH)
• Lithium-ion (Li-ion)
• Lithium-polymer (Li-poly)
• Sodium-sulfur

The Li-ion battery technology is the most promising among the four battery
chemistries mentioned above. It must also be stated that there are several
different types of Li-ion battery technologies that are being developed for
electric and hybrid vehicles.
The development of batteries is directed toward overcoming the signifi-
cant practical and manufacturing difficulties. The theoretical predictions are
difficult to match in manufactured products due to practical limitations. The
theoretical and practical specific energies of several batteries are given in
Table 4.2 for comparison.
The battery technology has gone through extensive research and develop-
ment efforts over the past 30 years, yet there is currently no battery that can
deliver an acceptable combination of power, energy, and life cycle at a reason-
able cost for high-volume production urban usage electric vehicles. However,
the extensive research and interest in alternative vehicles have resulted in
several promising battery technologies. Pure electric vehicles are available
commercially, but the cost is prohibitive since these require large capacity
batteries. Hybrid electric vehicles minimize the battery capacity through
using the combination of IC engine and electric machines. Even though
Battery Energy Storage 103

TABLE 4.2
Specific Energies of Batteries
Specific Energy (W h/kg)
Battery Theoretical Practical
Lead-acid 108 50
Nickel-cadmium 20–30
Nickel-zinc 90
Nickel-iron 60
Zinc-chlorine 90
Zinc-bromide 70
Silver-zinc 500 100
Sodium-sulfur 770 150–300
Aluminum-air 300
Nickel metal 70
Hydride 150
Li-ion

the hybrid technology is much more complex than either a conventional or

electric vehicle alone, the advantages gained in fuel economy through the
addition of a smaller electric powertrain component are significant. Hybrid
electric vehicles are commercially available for more than 10 years. Plug-in
hybrid vehicles are the next step forward where larger capacity batteries are
required. Several automotive manufacturers have announced the produc-
tion plans of plug-in hybrid vehicles in the coming years. These steps along
with societal interest and governmental incentives promote innovations that
will ultimately drive the cost down. The next step toward the use of battery
electric vehicles for urban transportation will be well within reach as trac-
tion battery cost comes down and technology improves. The move toward a
battery electric vehicle becomes easier if consumers can accept vehicles with
a range of 50 mi instead of 300 mi. This is certainly possible since the average
daily commute is 32.8 mi/day in the United States, assuming that an average
vehicle is driven 12,000 mi/year. The average number of miles driven per day
is much less in other countries of the world.
The fundamentals of a battery are in a unit cell, and include the electrodes
and electrolytes used, the chemical reactions involved, and the cell potential
developed. In the following section, cell structure and chemical reactions are
discussed with examples of some of the common cell chemistries. This will
enable us to define the parameters of a battery, which are needed for the mac-
roscopic point of view. We will then dig deeper into the theoretical aspects
of a battery cell for analyzing, evaluating, and modeling different types
of batteries. The chapter will then present cell chemistries of some of the
promising battery technologies for electric and hybrid vehicle applications.
104 Electric and Hybrid Vehicles: Design Fundamentals

Finally in this chapter, battery-pack design for electric and hybrid vehicles’
applications will be addressed.

4.2 Battery Basics

The batteries are made of unit cells containing the stored chemical energy
that can be converted to electrical energy. One or more of these electrochemi-
cal cells are connected in series to form one battery. The grouped cells are
enclosed in a casing to form a battery module. A battery pack is a collection
of these individual battery modules connected in a series and/or parallel
combination to deliver the desired voltage and energy to the power elec-
tronic drive system.

4.2.1 Battery Cell Structure

The energy stored in a battery is the difference in free energy between chem-
ical components in the charged and discharged states. This available chemi-
cal energy in a cell is converted into electrical energy only on demand using
the basic components of a unit cell; these components are the positive and
negative electrodes, the separators, and the electrolytes. The electrochemi-
cally active ingredient of the positive or negative electrode is called the active
material. Chemical reactions take place at the two electrodes, one of which
releases electrons while the other consumes those. The electrodes must be
electronically conducting and are located at different sites separated from
each other by a separator, as shown in Figure 4.1. The connection points
between the electrodes and the external circuit are called the battery ter-
minals. The external circuit ensures that stored chemical energy is released
only on demand and utilized as electrical energy.

Vcell
+ – Negative
electrode
Separator

Positive Cell
electrode container
Negative electrode
(a) (b) Electrolyte

FIGURE 4.1
Components of a battery cell. (a) Cell circuit symbol; (b) cell cross-section.
Battery Energy Storage 105

The components of the battery cell are described as follows:

1. Positive electrode: The positive electrode is an oxide or sulfide or

some other compound that is capable of being reduced during cell
discharge. This electrode consumes electrons from the external
circuit during cell discharge. Examples of positive electrodes are
lead oxide (PbO2) and nickel oxyhydroxide (NiOOH). The electrode
materials are in the solid state.
2. Negative electrode: The negative electrode is a metal or an alloy that is
capable of being oxidized during cell discharge. This electrode releases
electrons to the external circuit during cell discharge. The examples of
negative electrodes are lead (Pb) and cadmium (Cd). The negative elec-
trode materials are in the solid state within the battery cell.
3. Electrolyte: The electrolyte is the medium that permits ionic conduc-
tion between positive and negative electrodes of a cell. The electro-
lyte must have high and selective conductivity for the ions that take
part in electrode reactions, but must be a nonconductor for electrons
in order to avoid self-discharge of batteries. The electrolyte may be
liquid, gel, or solid material. Also, the electrolyte can be acidic or
alkaline, depending on the type of battery. Traditional batteries such
as lead-acid and nickel-cadmium use liquid electrolytes. In lead-
acid batteries, the electrolyte is the aqueous solution of sulfuric acid
(H2SO4(aq)). Advanced batteries for electric vehicles, such as sealed
lead-acid, NiMH, and lithium-ion batteries use an electrolyte that is
gel, paste, or resin. Lithium-polymer batteries use a solid electrolyte.
4. Separator: The separator is the electrically insulating layer of mate-
rial, which physically separates electrodes of opposite polarity.
Separators must be permeable to the ions of the electrolyte and may
also have the function of storing or immobilizing the electrolyte.
Present day separators are made from synthetic polymers.

4.2.2 Chemical Reactions

During battery operation, chemical reactions at each of the electrodes are
sustainable only if electrons generated at the electrodes are able to flow
through an external electrical circuit that connects the two electrodes. When
a passive electrical circuit element is connected to the electrode terminals of
a battery, electrons are released from the negative electrode and consumed
at the positive electrode, resulting in current flow into the external circuit. In
this process, the battery gets discharged. The supply of electrons is due to the
chemical reactions at the electrode surfaces inside the battery cell, which are
collectively known as reduction and oxidation (redox) reactions. During bat-
tery discharge, the positive electrode gets chemically reduced as it absorbs
electrons from the external circuit; the negative electrode gets oxidized as it
106 Electric and Hybrid Vehicles: Design Fundamentals

releases electrons to the external circuit. For battery charging, a source with
voltage higher than the battery terminal voltage has to be applied so that
current can flow into the battery in the opposite direction. During charging,
electrons are released at the positive electrode and consumed at the negative
electrode; consequently, the positive electrode is oxidized and negative elec-
trode is reduced.
Regardless of the battery cell chemistry, redox reactions take place at the
electrodes during both cell charging and discharging for the release and
absorption of electrons at the terminals. The generalized redox reactions are
given by [1]

Charge

aA ←  → cC + nE+ + ne − (4.1)
Discharge

for the positive electrode, and

Charge

bB + nE + + ne − ←  → dD (4.2)
Discharge

for the negative electrode. The combined chemical reaction is

Charge

aA + bB ←  → cC + dD (4.3)
Discharge

Chemical reactions 4.1 and 4.2 illustrate that electrons are released and
absorbed during any redox reaction. The positive electrode reaction 4.1
shows that during cell charging, species A within the electrode is oxidized
and becomes energized species C, releasing electron(s) into the external cir-
cuit and positive ion(s) into the electrolyte. Similarly, the negative electrode
reaction 4.2 shows that species B at the electrode combines with positive
ion(s) from the electrolyte and electron(s) from the external circuit to form
energized species D. The converse is true at the two electrodes during cell
discharging. The coefficients a, b, c, and d represent the numbers of moles
associated with the species in the reactions; the coefficient n represents the
number of electrons and ions involved in the redox reactions.
In electric traction applications, battery cell operation is that of cell dis-
charging when the energy is supplied from the battery to the electric motor
for propulsion power and of cell charging when energy is supplied from
an external source to store energy in the battery. In conventional vehicles,
battery cells supply power to electrical accessories while discharging, and
accept energy from an external device to replenish the stored energy during
charging. We will next review the redox reactions during cell charging and
discharging in a few battery chemistries, starting with the lead-acid battery
Battery Energy Storage 107

RL

Electron flow

+ –
e– e– e– e–

H+
PbO2(s) H+
H+ SO42– Pb(s)
H+
SO42–
Water
PbSO4(s) PbSO4(s)
2H2O

FIGURE 4.2
Lead-acid battery: cell discharge operation.

cell. Lead-acid is still the battery choice for powering electrical accessories in
conventional, electric, and hybrid electric vehicles.
Figure 4.2 shows the cell discharge operation of a lead-acid battery cell into
a passive resistive element. The positive electrode made of lead-oxide (PbO2)
is reduced by consuming electrons and ions. The electron supply is through
the external circuit which originates at the negative electrode. The current
flow is therefore out of the positive electrode into the electrical load with the
battery acting as the source. The positive electrode reaction is given by

PbO 2 (s) + 4H + (aq) + SO 24− (aq) + 2e − → PbSO 4 + 2H 2O(l)

A highly porous structure is used for the positive electrode to increase

PbO2(s)/electrolyte contact area. A porous electrode structure results in higher
current densities since PbO2 is converted to PbSO4(s) during cell discharge.
As discharge proceeds, the internal resistance of the cell rises due to PbSO4
formation and decreases the electrolyte conductivity as H2SO4 is consumed.
PbSO4(s) deposited on either electrode in a dense, fine-grain form can lead to
sulfation. The discharge reaction is largely inhibited by the buildup of PbSO4,
which reduces cell capacity significantly from the theoretical capacity.
The negative electrode is made of solid lead (Pb); during cell discharge
lead is oxidized, releasing electrons into the external circuit. The negative
electrode reaction during cell discharge is

Pb(s) + SO 24− (aq) → PbSO 4 + 2e −

The production of PbSO4(s) can degrade battery performance by making the

negative electrode more passive.
108 Electric and Hybrid Vehicles: Design Fundamentals

Electric current

Electron flow
+

e– e– e– e–
H2O
PbO2(s) PbSO4(s) H2O
PbSO4(s) Pb(s)
SO42–
H+
H+
H+
SO42–

FIGURE 4.3
Lead-acid battery: cell charge operation.

The overall cell discharge chemical reaction is

Pb(s) + PbO 2 (s) + 2H 2SO 4 (aq) → 2PbSO 4 + 2H 2O(l)

The cell charge operation is the reverse of the cell discharge operation. An
external electrical source supplies current into the battery to reverse the
chemical reactions as shown in Figure 4.3. During cell charging, the lead
sulfate is converted back to the reactant states of lead and lead oxide. The
electrons are consumed from the external source at the negative electrode,
while the positive electrode produces the electrons. The current flows into
the positive electrode from the external source, thereby delivering electrical
energy into the cell where it gets converted into chemical energy. The posi-
tive electrode is oxidized, releasing electrons during cell charging as follows:

PbSO 4 (s) + 2H 2O(L) → PbO 2 (s) + 4H + (aq) + SO 24− (aq) + 2e −

The negative electrode is reduced during cell charging absorbing electrons,

and the chemical reaction is

PbSO 4 (s) + 2e − → Pb(S) + SO 24− (aq)

The overall chemical reaction during cell charging is

2PbSO 4 (s) + 2H 2O(l) → Pb(s) + PbO 2 (s) + 2H 2SO 4 (aq)

The nickel-cadmium (NiCd) and NiMH batteries are examples of alkaline

batteries where electrical energy is derived from the chemical reaction of a
Battery Energy Storage 109

metal with oxygen in an alkaline electrolyte medium. The specific energy of

alkaline batteries is lowered due to the mass addition of the carrier metal.
The NiCd battery employs a nickel-oxide positive electrode and a metallic-
cadmium negative electrode. The reactions take place in the presence of potas-
sium hydroxide (KOH) electrolyte. The positive electrode chemical reaction is

Discharge

NiOOH + H 2O + e − ←  → Ni ( OH ) + OH −
Charge 2

The negative electrode chemical reaction is

Discharge

Cd + 2OH − ←  → Cd(OH)2 + 2e −
Charge

The overall chemical reaction is

Discharge

Cd + 2NiOOH + 2H 2O ←  → 2Ni(OH)2 + Cd(OH)2
Charge

In NiMH batteries, the positive electrode is a nickel oxide similar to that used
in a NiCd battery, while the negative electrode has been replaced by a metal
hydride which stores hydrogen atoms. The concept of NiMH batteries is
based on the fact that fine particles of certain metallic alloys, when exposed to
hydrogen at certain pressures and temperatures, absorb large quantities of the
gas to form the metal hydride compounds. Furthermore, the metal hydrides
are able to absorb and release hydrogen many times without deterioration.
The two electrode chemical reactions in a NiMH battery are as follows:
At the positive electrode,

Discharge

NiOOH + H 2O + e − ←  → Ni(OH)2 + OH −
Charge

At the negative electrode,

Discharge

MH x + OH − ←  → MH x −1 + H 2O + e −
Charge

where
M stands for metallic alloy, which takes up hydrogen at ambient tempera-
ture to form the metal hydride MHx
x is the number of hydrogen atoms absorbed
110 Electric and Hybrid Vehicles: Design Fundamentals

The overall chemical reaction is

Discharge

NiOOH + MH x ←  → Ni(OH)2 + MH x −1
Charge

4.3 Battery Parameters

In this section, the various battery parameters including capacity and state
of charge (SoC) are defined. The parameters mostly relate to the terminal
characteristics of the battery that have the practical implications for use in
an application.

4.3.1 Battery Capacity
The amount of charge released by the energized material at an electrode
associated with complete discharge of a battery is called the battery capacity.
The capacity is measured in A h (1 A h = 3600 C or Coulomb, where 1 C is the
charge transferred in 1 s by 1 A current in the SI unit of charge).
The theoretical capacity of a battery can be obtained by Faraday’s law of
electrolysis, which states that the mass of the elemental material altered at
an electrode is directly proportional to the element’s equivalent weight for
a given quantity of electrical charge. The equivalent weight of the elemen-
tal material is given by the molar mass divided by the number of electrons
transferred per ion for the reaction undergone by the material. This number
is known as the valency number of ions for the substance. Mathematically,
Faraday’s law can be written as

Q Mm
mR = (4.4)
F n

where
mR is the mass of the limiting reactant material altered at an electrode
Q is the total amount of electric charge passing through the material
F is the Faraday number or Faraday constant
Mm is the molar mass
n is the number of electrons per ion produced at an electrode

Mm/n is the equivalent weight of the reactant substance. The Faraday num-
ber is given by the amount of electric charge carried by one mole of electrons.
The number of molecules or atoms in a mole is given by the Avogadro num-
ber NA which is equal to 6.022045 × 1023 mol−1. The amount of charge in one
Battery Energy Storage 111

electron, which is the elemental charge, is equal to e0 = 1.6021892 × 10−19 C.

Therefore, the Faraday number is equal to F = NAe0 = 96,485 C/mol. The num-
ber of Faradays required to produce one mole of substance at an electrode
depends on the way in which the substance is oxidized or reduced.
Therefore, the theoretical capacity of a battery (in Coulomb) can be obtained
from Equation 4.4 as

QT = xnF C (4.5)

where x is the number of moles of limiting reactant associated with complete

discharge of battery, and is given by

mR
x=
Mm

Here
mR is the mass of the reactant material in kg
Mm is the molar mass of that material in g/mol

The theoretical capacity in A h is

mR n
QT = 0.278 F Ah (4.6)
Mm

The cells in a battery are connected in series and the capacity of the battery
is dictated by the smallest cell capacity. Therefore, QTbattery = QTcell. Six battery
cells connected in series to form a battery are shown in Figure 4.4.

4.3.2 Open Circuit Voltage

The battery in its simplest form can be represented by an internal voltage Ev
and a series resistance Ri, as shown in Figure 4.5a. More representative but
complex battery models are discussed later in the chapter. The battery inter-
nal voltage appears at the battery terminals as open circuit voltage when
there is no load connected to it. The internal voltage or the open circuit volt-
age (OCV) depends on the state of charge of the battery, temperature, and
past discharge/charge history (memory effects) among other factors. The

– 12.9 V +
2.15 V
– +
Q Q Q Q Q Q

FIGURE 4.4
Battery cells connected in series.
112 Electric and Hybrid Vehicles: Design Fundamentals

Battery Load
+ Ev
Ri I = constant
SoD (to) = 0
± Ev Vt RL SoD (td) = QT
_
SoD
(a) (b) QT

FIGURE 4.5
(a) Steady-state battery equivalent circuit. (b) Battery open circuit voltage characteristics.

open circuit voltage characteristics are shown in Figure 4.5b. As the battery
is gradually discharged, the internal voltage decreases, while the internal
resistance increases. The open circuit voltage characteristics have a fairly
extended plateau of linear characteristics with a slope close to zero. The open
circuit voltage is not a good indicator of the state of charge; state of charge
of a battery pack needs to be calculated considering discharge current char-
acteristics, battery chemistry, temperature effects, and number of charge/
discharge cycles. Once the battery is completely discharged, the open circuit
voltage decreases sharply with more discharge.

4.3.3 Terminal Voltage

Battery terminal voltage Vt is the voltage available at the terminals when
a load is connected to the battery. The terminal voltage is at its full charge
voltage VFC when the battery is fully charged. For example, with lead-acid
battery it means that there is no more PbSO4 available to react with H2O to
produce active material. Vcut is the battery cut-off voltage, where discharge of
the battery must be terminated. The battery terminal voltage characteristic
in relationship with the state of discharge (SoD) is shown in Figure 4.6.

4.3.4 Practical Capacity

The practical capacity CP of a battery is the actual charge released by the
energized material at an electrode associated with complete discharge of
the battery. The symbol CP is used for the practical capacity, since the more

Vt
VFC

Vcut

0 QP SoD

FIGURE 4.6
Battery terminal voltage.
Battery Energy Storage 113

i
A
t=0

Vt RL
+ –

Battery

FIGURE 4.7
Battery capacity measurement.

commonly used symbol for battery capacity is C. The practical capacity is

always much lower compared to the theoretical capacity QT due to practical
limitations. The practical capacity of a battery is given as
tcut

CP =
∫ i(t)dt
to
(4.7)

where
to is the time at which battery is fully charged
tcut is the time at which battery terminal voltage is at Vcut

Therefore, Vt(tcut) = Vcut.

The practical capacity of a battery is defined in the industry by a conve-
nient and approximate approach of A h instead of Coulomb under constant
discharge current characteristics. Let us consider the experiment shown in
Figure 4.7, where the battery is discharged at constant current starting from
time t = 0. The ammeters and voltmeters measure the discharge current and
the battery terminal voltage. The current is maintained constant by varying
the load resistance RL until the terminal voltage reaches the cut-off voltage
Vcut. The qualitative graphs of two constant current discharge characteristics
at two different current levels are shown in Figure 4.8. The following data
are obtained from the experiment:
I = 80 A: Capacity C80 A = (80 A)tcut = 80 × 1.8 = 144 A h
I = 50 A: Capacity C50 A = (50 A)tcut = 50 × 3.1 = 155 A h
I = 30 A: Capacity C30 A = (30 A)tcut = 30 × 5.7 = 171 A h
The results show that the capacity depends on the magnitude of discharge
current. The smaller the magnitude of the discharge current, the higher the
capacity of the battery is. To be accurate, when the capacity of a battery is
stated, the constant discharge current magnitude must also be specified.

4.3.5 Discharge Rate

The discharge rate is the current at which a battery is discharged. The rate is
expressed as C/h rate, where C is rated battery capacity and h is discharge
114 Electric and Hybrid Vehicles: Design Fundamentals

Vt

I2
I1

tcut,1 tcut,2 Discharge

time (h)

FIGURE 4.8
Constant current discharge curves.

time in hours. For a battery that has a capacity of C A h and is discharged

over Δt, the discharge rate is C/Δt.
Example: Let the capacity of a battery be 1 C = 100 A h. (1 C usually denotes
rated capacity of the battery.) Therefore,

C 100 A h
rate is = 20 A
5 5h

and

100 A h
2C rate is = 200 A
0.5 h

4.3.6 State of Charge

The state of charge (SoC) represents the present capacity of the battery. It is the
amount of capacity that remains after discharge from a top-of-charge condi-
tion. The current is the rate of change of charge given by

dq
i(t) =
dt

where q is the charge moving through the circuit. The instantaneous theo-
retical state of charge SoCT(t) is the amount of equivalent charge remaining
at the positive electrode and ready to be released by the energized material.
If the state of charge is QT at the initial time to, then SoCT(to) = QT. For a time
interval dt

dSoCT = − dq

= −i(i)dt
Battery Energy Storage 115

Integrating from the initial time to to the final time t, the expression for
instantaneous state of charge is obtained as
t

∫
SoCT (t ) = QT − i ( τ ) dτ
to
(4.8)

SoCT is often expressed as a percentage of the capacity of the battery as

follows:

SoCT (t) =
QT −
∫ i(τ)dτ × 100%
to
QT

The state of charge will be increasing when a battery is being charged. If the
state of charge is zero initially at t = 0, the state of charge at time t expressed
in percentage form is given by

SoCT (t) =
∫ i(τ)dτ × 100%
0
QT

4.3.7 State of Discharge

The state of discharge (SoD) is a measure of the charge that has been drawn
from a battery during discharge. Mathematically, state of discharge is given as

SoDT (t) = ∆q = i(τ)dτ

∫
to

⇒ SoCT (t) = QT − SoDT (t) (4.9)

4.3.8 Depth of Discharge

The depth of discharge (DoD) is the percentage of battery rated capacity to
which a battery is discharged. The depth of discharge is given by

QT − SoCT (t)
DoD(t) = × 100%
QT
t

=
∫ i(τ)dτ × 100%
to
(4.10)
QT
116 Electric and Hybrid Vehicles: Design Fundamentals

The withdrawal of at least 80% of battery (rated) capacity is referred to as

deep discharge.

4.3.9 Battery Energy
Energy of a battery is measured in terms of the capacity and the discharge
voltage. To calculate the energy, the capacity of the battery must be expressed
in coulombs. 1 A h is equivalent to 3600 C, while 1 V refers to 1 J (J for joule)
of work required to move 1 C charge from the negative to positive electrode.
Therefore, the stored electrical potential energy in a 12 V, 100 A h battery is
(12)(3.6 × 105) J = 4.32 MJ. In general, the theoretical stored energy

ET = VbatQT

where
Vbat is the nominal no load terminal voltage
QT is the theoretical capacity in C units

Therefore, using Equation 4.6, the theoretical energy is

 1000 Fn  nmR
ET =  mR  Vbat = 9.65 × 107 Vbatt J (4.11)
 Mm  Mm

The practical available energy is

tcut

Ep =
∫ v(t) i(t) dt W h
to
(4.12)

where
to is the time at which battery is fully charged
tcut is the time in hours at which battery terminal voltage is at Vcut
v is the battery terminal voltage
i is the battery discharge current

EP is dependent on the manner in which the battery is discharged. Practical

energy in Watt-hours (W h) multiplied by 3600 gives the energy in Joules, i.e.,
Watt-seconds.

4.3.10 Specific Energy

The specific energy of a battery in terms of discharge energy related to com-
plete discharge from fully charged condition is given by

Discharge energy E
SE = =
Total battery mass MB
Battery Energy Storage 117

The unit for specific energy is W h/kg. The theoretical specific energy of a
battery using Equation 4.9 is

nVbat mR
SET = 9.65 × 107 × W h/kg (4.13)
Mm mB

If the mass of the battery MB is proportional to the mass of the limiting reac-
tant of the battery mR, then SET is independent of mass. The specific energy of
lead-acid battery is 35–50 W h/kg at C/3 rate. Since practical energy EP varies
with discharge rate, the practical specific energy SEP is also variable.
The term energy density is also used in the literature to quantify the qual-
ity of a battery or other energy sources. The term energy density refers to the
energy per unit volume of a battery. The unit for energy density is W h/L.

4.3.11 Battery Power
The instantaneous battery terminal power is

p(t) = Vbati (4.14)

where
Vbat is the battery terminal voltage
i is the battery discharge current

Using Kirchhoff’s voltage law for the battery equivalent circuit of Figure 4.5a,

Vt = Ev − Rii (4.15)

Substituting Equation 4.15 into 4.14 yields

p(t) = Evi − Rii 2 (4.16)

Power versus current characteristic is shown in Figure 4.9. Using the maxi-
mum power transfer theorem in electric circuits, the battery can deliver max-
imum power to a DC load when the load impedance matches the battery
internal impedance. The maximum power is

Ev2
Pmax = (4.17)
4Ri

Since Ev and Ri vary with the state of charge, Pmax also varies accordingly.
118 Electric and Hybrid Vehicles: Design Fundamentals

Power

Pmax

iPmax Current

FIGURE 4.9
Battery power characteristics.

Maximum power output is needed from the battery in fast discharge con-
ditions in vehicle applications, which occur when the electric motor is heav-
ily loaded. Acceleration on a slope is such a condition, when the motor draws
a lot of current to deliver maximum power required for traction.
The performance of batteries to meet acceleration and hill climbing require-
ments can be evaluated with the help of rated power specifications, which
are based on the ability of the battery to dissipate heat. The rated continuous
power is the maximum power that the battery can deliver over prolonged
discharge intervals without damage to the battery. These do not necessarily
correspond to Pmax on p–i curve of battery characteristics. The rated instanta-
neous power is the maximum power that the battery can deliver over a very
short discharge interval without damage to the battery.

4.3.12 Specific Power

The specific power of a battery is

P
SP = (units : W/kg ) (4.18)
MB

where
P is the power delivered by battery
MB is the mass of battery

Typically, lead-acid battery’s maximum specific power is around 280 W/kg

(which corresponds to Pmax) at DoD = 80%. Similar to specific energy and
energy density, the term power density is used to refer to the power of the bat-
tery per unit volume with units of W/L.

4.3.13 Ragone Plots

In electrochemical batteries, there is a decrease in charge capacity (exclud-
ing voltage effects) with increasing currents. This is often referred to as the
Battery Energy Storage 119

1000
Gas turbine

Ultracapacitors Gasoline engine

Specific power (W/kg)

Sodium sulfur
Nickel-cadmium
100
Lead-acid

Fuel cell
10
1 10 100 1000
Specific energy (W h/kg)

FIGURE 4.10
Specific power vs. specific energy (Ragone plots) of several batteries, a gasoline engine, and a
fuel cell.

Ragone relationship and is described by Ragone plots. Ragone plots are usually
obtained from constant power discharge tests or constant current discharge
plots. Let us consider the experiment of Figure 4.6 again, but this time the
current i is adjusted by varying RL such that the power output at the battery
terminals is kept constant. The experiment stops when the battery terminal
voltage reaches the cut-off voltage, i.e., Vt = Vcut. We assume that the battery
is fully charged at t = 0. The experiment is performed at several power levels
and the following data are recorded: (1) power p(t) = Vti = P, (2) time to cut-off
voltage tcut, and (3) practical energy EP = Ptcut. The plot of SP vs. SE on a log-
log scale is known as the Ragone plot. The Ragone plots of several batteries
along with alternative energy sources and IC engines are given in Figure
4.10 to give an idea about the relative power and energy capacities of these
different devices.

4.4 Electrochemical Cell Fundamentals

The electrochemical cells use the oxidation and reduction reactions for the flow
of electric currents that are known as faradaic currents. These cells form
the basis of batteries and fuel cells that supply electrical energy from stored
chemical energy using the electrochemical reactions. The electrochemical
cells can be classified as either galvanic cells or electrolytic cells. The galvanic
cells are those where the substances react spontaneously when an external
electrical load is connected to the cell. In the galvanic cell, electrons flow
through the external circuit and ions transfer from one electrode to another
within the cell resulting in work done. The galvanic cells are also known as
120 Electric and Hybrid Vehicles: Design Fundamentals

voltaic cells since volts or potentials are established due to the spontaneous
chemical reactions. The electrolytic cells are those in which reactions are
nonspontaneous, and effected by the imposition of external voltage greater
than the open circuit voltage of the cell. Electrical energy is expended by
the external source and work is done on the electrochemical cell. When a
battery cell is being charged to restore chemical energy within, it functions
as an electrolytic cell. The battery cells will be referred to as electrochemical
cells in general, since both discharging and charging are associated with the
secondary battery cells used in electric and hybrid vehicles.
The fundamentals of a battery cell, or for that matter, of any other electro-
chemical cell, is in the physics and chemistry involved in the energy con-
version process. In this section, we will analyze the fundamentals to gain
sufficient understanding of the process so that electrochemical cell models
can be developed. The models are useful not only for analyzing the electro-
chemical cells, but also for evaluating systems that include the electrochemi-
cal cell-based components for energy storage.
The fundamentals governing the principles and operation of an electro-
chemical cell are chemical thermodynamics, electrochemical reaction rates,
electrode kinetics, and mass transport [1–7]. Each one of these mechanisms
influences the energy conversion from stored chemical energy into electrical
energy in an electrochemical cell. The mechanisms are treated in detail in
this section.

4.4.1 Thermodynamic Voltage

Electrochemical thermodynamics determine the electrical potential differ-
ence between the electrode and the electrolyte of an electrochemical cell
even in the absence of any external electrical connection to the electrodes
when there is no external electric current flow. The thermodynamic phys-
ics of the electrochemical cell explains the mechanism of electric potential
buildup at an electrode and establishes the theoretical limits of work, conver-
sion efficiency, and potential of the cell.
The first law of thermodynamics states that energy is conserved during
any process; energy can be transferred from a system to its surroundings
or vice versa, but it cannot be created or destroyed. In chemical processes,
system refers to the chemical species involved and the associated reactions;
the container where the reaction takes place serves as the boundary of the
chemical reaction. The three forms of energy for chemical processes are the
internal energy of each substance, work done due to changes in pressure and
volume or due to electrical current flow, and the heat transfer with the sur-
roundings. In all chemical reactions, heat is either absorbed from or released
to the surroundings. The amount of heat absorbed or released by the chemi-
cal reaction is equal to the change in internal energy only if there is no work
done on or by the surroundings. However, if work done is not kept zero, some
of the heat energy will be converted to work, and the total of heat absorbed
Battery Energy Storage 121

or released is no longer equal to the change in internal energy. To represent

the combined activity due to heat absorption or release, the term enthalpy
representing the heat content of a substance or system has been introduced
for convenience. By thermodynamic definition, enthalpy H is given by

H = U + pV

where
U is the internal energy
p is the pressure
V is the volume of the system

The product of pressure and volume (pV) is related to the work of expansion.
In galvanic devices, stable substances react spontaneously to form new
substances, which mean that the reaction can only go from a higher energy
state to a lower one according to the first law of thermodynamics. The lower
energy state that the substances will assume depends on another measure
known as the entropy of a substance. Entropy is a measure of the disorder
level for the particles (molecules, atoms, etc.) that make up a substance.
Entropy is a property that is specified for every equilibrium state of a sub-
stance. Since entropy is a property, the change in entropy in going from one
state to another is the same for all processes. The SI unit for entropy is J/K.
The second law of thermodynamics states that a spontaneous reaction in
a closed environment proceeds until the maximum entropy is reached for
the substances involved. Based on the two laws of thermodynamics, part
of the enthalpy change has to be reflected in the form of entropy change
(TΔS), where T is the temperature in Kelvin and ΔS is the change of entropy.
Therefore, the maximum available work from a chemical reaction or any
process is [8,9]

∆G = ∆H − T ∆S

In chemical reactions, G is the Gibbs-free energy, which determines the rela-

tive importance between the enthalpy and entropy terms driving a particu-
lar reaction. A chemical reaction is favorable or spontaneous if ΔH < 0, and
unfavorable or nonspontaneous if ΔH > 0. Again, a reaction is spontaneous
if ΔS > 0, and nonspontaneous if ΔS < 0. The Gibbs-free energy of a system
provides the net result when enthalpy and entropy forces of a reaction drive
the system in opposite directions. A reaction is spontaneous if ΔG < 0, and
nonspontaneous if ΔG > 0.
Gibbs-free energy can be calculated from the enthalpy and entropy changes
between the products and reactants of the chemical reaction as

∆G = ∆H o − T ∆So (4.19)
122 Electric and Hybrid Vehicles: Design Fundamentals

where

∆H o = ∑ H (products) − ∑ H (reactants)
o
f
o
f

∆So = ∑ S (products) − ∑ S (reactants)

o o

The superscript o denotes standard-state conditions of 25°C and one atmo-

spheric pressure, while the subscript f denotes the standard-state free
energies of formation. The standard enthalpies of formation and standard
entropies are given in thermodynamic tables. The standard enthalpy of for-
mation of an element is defined as zero.
Alternatively, when the free energies of formation for each substance
involved in a chemical reaction are known, the Gibbs-free energy change of
reaction is

∆G = ∆G of (products) − ∆G of (reactants)

Similar to the definition of enthalpies of formation, the standard free energy

of formation of an element is defined as zero.
Let us consider the chemical reaction involved in sliver-zinc battery to
illustrate the concept of energy available in an electrochemical cell. The
chemical reaction is [8]

Zn + 2AgCl → Zn ++ + 2Ag + 2Cl −

The amount of heat content liberated from the chemical reaction between
metallic zinc and silver chloride solution when mixed at standard-state con-
ditions is −233 kJ/mol of Zn reacted, i.e., ΔHo = −233 kJ/mol. For the chemical
reaction, TΔS = −43 kJ/mol. The Gibbs-free energy in this example is

∆G = ∆H − T ∆S = −233 + 43 = −190 kJ/mol

The example illustrates that all of the enthalpy change cannot be converted
to work; a minimum amount of energy has to be consumed by the entropy
change which is usually reflected in the generated heat during chemical
reactions.
Gibbs-free energy is also useful in determining the maximum thermody-
namic conversion efficiency of a galvanic cell, which is given by the ratio of
Gibbs-free energy and the total enthalpy change:

∆G T ∆S
ηEC = = 1− (4.20)
∆H ∆H
Battery Energy Storage 123

For the given example of silver-zinc battery cell, ηEC = 81.5%. The inherent high
conversion efficiency compared to the thermodynamic upper limit of efficien-
cies in heat engines is an advantage of galvanic devices. However, just like the
heat engines, the practical efficiency of galvanic devices during normal opera-
tion is much lower than the theoretical efficiency. The decrease in efficiency is
directly related to the practical currents required in practical systems.
Gibbs-free energy released in the chemical process of the electrochemical
cell imposes the theoretical limit on the maximum work that can be done by
the cell. This work is the charge transferred per mole under the force of the
open-circuit voltage (OCV) of the cell. With E being the electrode potential
difference at equilibrium, the work done by the cell can be expressed as

∆G = Charge transferred per mole × OCV

= −nFEo (4.21)

The theoretical upper limit of a galvanic cell potential can be obtained from
the above as

∆G
Eo = − (4.22)
nF

We will use the chemical reaction in a NiCd battery cell to calculate the
equilibrium cell potential. The chemical reaction in a NiCd battery cell is
given by

Cd + 2NiOOH → Cd(OH)2 + 2Ni(OH)2

The Gibbs-free energy change for the NiCd cell is

∆G = ∆G of (products) − ∆G of (reactants)

= ( −470.25 ) + 2 ( −452.7 )  − 0 + 2 ( −541.3 ) 

= −293.05 kJ/mol

For the NiCd battery cell, two electrons are involved in the chemical reac-
tion. The theoretical cell potential is thus

−293, 050
Eo = − = 1.52 V
2 × 96, 485

The theoretical cell potential is never achievable in practice due to various

electrochemical phenomena in the cell, which are described in the Sections
124 Electric and Hybrid Vehicles: Design Fundamentals

4.4.3 through 4.4.7. The nominal practical cell voltage in a NiCd cell is 1.3 V.
This is true for all electrochemical cells. The thermodynamic cell potential
only gives the theoretical upper value of cell potential for a battery chemistry.
The potential of an electrode is the potential difference between the elec-
trode and the electrolyte that it is in contact with. The electrode potential is
determined with respect to a reference electrode, since an absolute potential
value cannot be obtained. Both chemical and electrical processes contribute
to the electrode potential difference [6]. The environment at the vicinity of
an electrode is changed due to chemical activities between the electrode and
the electrolyte regardless of the electric potential difference at the solid-liq-
uid phase boundary. The measure of the work done to bring a particle to its
assumed potential is the chemical potential. Again, regardless of the changes
in the chemical environment, the transfer across the electric potential is
accomplished by electric work done in its original sense. Although one cannot
separate these two components for single species experimentally, the differ-
ences in the scales of the two environments make it possible to separate them
mathematically [7,8]. The resultant potential for these two kinds of energy
change is the electrochemical potential or simply the electric potential.

4.4.2 Electrolysis and Faradaic Current

The process of electrolysis is the transfer of electrons between an electrode
and a chemical species in solution, resulting in an oxidation or a reduction
reaction. An external electrical circuit is necessary for the redox reactions at
the electrodes of the electrochemical cell. In order to understand the electro-
chemical phenomenon in a cell, and subsequently utilize that knowledge to
develop simple models for these cells, the nature of current and potential in
the cell must be evaluated [8]. This section describes how current controls
the reaction using Faraday’s law of electrolysis and the relationship between
charge and current.
The chemical reactions 4.1 and 4.2 represent both a chemical process and
an electrical process. The reaction rates can be completely determined using
the electrical process by seeing that the chemical conversion can only occur
if electrons are either arriving or leaving at the electrodes. Therefore, the
chemical conversion rates are controlled or measured by the electrical cur-
rent passing through a given electrode. Faraday’s law of electrolysis relates
the mass of a substance altered at an electrode to the quantity of electrical
charge transferred at that electrode. The electrolysis rate in terms of moles
electrolyzed is given by Faraday’s law as (see Equation 4.5)

Q
x= (4.23)
nF

As was defined previously, n is the number of electrons per ion released

at an electrode, F is the Faraday constant, and x is the number of moles
Battery Energy Storage 125

electrolyzed. Again, by definition, electric current is the number of coulombs

of electric charge flowing per second, i.e.,

dQ
i= (4.24)
dt

Equations 4.23 and 4.24 relate the faradaic current in an electrochemical

cell with the reaction rate. Combining the two equations, we can write the
reaction rate as

dx i
= mol/s
dt nF

The reaction rate is typically expressed in mol/s per unit area since elec-
trode reactions are a heterogeneous process occurring only at the electrode–
electrolyte interface. The heterogeneous reaction rate depends on the mass
transfer to the electrodes and various surface effects in addition to the elec-
trode kinetics. The reaction rate per unit area is given by

dx i j
= = mol/s cm 2 (4.25)
dt nFA nF

where
A is the area in cm2
j is the current density in A/cm2

4.4.3 Electrode Kinetics

The study of electrode kinetics includes the processes that govern the elec-
trode reaction rates or the faradaic currents flowing in an electrochemical
cell. A number of process rates dictate the electrode kinetics of which the two
most common ones are

1. The rate of electron transfer at the electrode surface between the

electrodes and species in solution.
2. Mass transport of the active materials from the bulk solution to the
electrode interface.

The rate of electron transfer at the electrode surface is governed by the

Faraday’s law of electrolysis presented in Section 4.4.2. In this section, the
additional fundamental principles of electrode kinetics are described to
establish the terminal voltage–current relationship in an electrochemical
cell. The mass transport is another fundamental mechanism for the conti-
nuity of electrochemical reactions and will be addressed in Section 4.4.4.
126 Electric and Hybrid Vehicles: Design Fundamentals

It must be mentioned that the study of electrochemistry is vast and a detailed

treatment is beyond the scope of this book. Only an overview of the funda-
mental theory for electrode kinetics is given here. The readers are referred to
[8–10] for further details on this topic of electrochemistry.
When a redox couple is present at each electrode and there are no con-
tributions from liquid junctions, then the open circuit potential is also the
equilibrium potential. However, in general there is always ongoing activ-
ity at the electrode–electrolyte interface for many electrochemical cells. The
critical potential at which the electrode reactions occur is known as stan-
dard potential E0 for the specified chemical substances in the system. The
electrode potential deviates from its so-called equilibrium state when there
is external electric current flow. The relationship between the energy flow
and the current is more complicated in the electrochemical cell than that for
the conduction in a solid, since the current flow and chemical reactions are
heterogeneous processes. Consequently, the relationships are nonlinear, and
approximations are often used to arrive at simpler mathematical expressions.
The electrode voltage–current relationship can be obtained using the for-
ward and reverse reactions between the electrode and electrolytes. These
currents can be obtained by relating reaction rates to the rate constants and
the concentration of the reactants. Let us consider a general case where n
electrons are transferred between two species (O) and (R) at an electrode–
electrolyte interface. The general electrode reaction is

kf
O + ne − 
 
R
kr

where kf and kr are the forward and reverse rate constants, respectively. The
rate constants are the proportionality factors linking the concentration of the
species to the reaction rates. The concentration of species undergoing oxida-
tion at a distance x from the surface and at time t will be denoted as CO(x,t);
hence, the surface concentration is CO(0,t). Similarly, the surface concentra-
tion for the species undergoing reduction is CR(0,t).
The reaction rate obtained from the product of rate constant and species
concentration can be equated to the reaction rate given by Equation 4.25 to
establish the relationship between the species concentrations and faradaic
current. Therefore, for the forward and reverse currents, we have

if
k f CO (0, t) =
nFA

and

ir
k r C R (0 , t ) =
nFA
Battery Energy Storage 127

The net current flow at the electrode is the difference between the forward
and reverse currents

i = i f − ir = nFA  k f CO (0, t) − k rCR (0, t) (4.26)

For the sake of simplicity in analyzing the electrode process, we will assume
a single electron transfer (i.e., n = 1) at the electrode–electrolyte interface. In
this case, the rate constants can be related to the electrical potential across the
electrode–electrolyte interface using free energy considerations [8]. For the
standard potential E0, the forward and reverse rate constants are equal; this
constant is known as the standard rate constant and is given the symbol k0.
The rate constants at other potentials are given in terms of the standard rate
constant as
0
k f = k 0 e( F/RT )( − α )( E − E )

0
k r = k 0 e( F/RT )(1−α )( E−E )

Inserting these relations into Equation 4.26 gives the complete current–volt-
age characteristics at the electrode–electrolyte interface

( 0
i = FAk 0 CO (0, t)e( F/RT )( − α )( E−E ) − CR (0, t)e( F/RT )(1−α )( E−E
0
)
) (4.27)

The approach used is referred to as the Butler–Volmer approach for analyz-

ing electrode kinetics [8].
Although this equation describes the electrode kinetics quite accurately, it
is generally impossible to express voltage in terms of current which would
be the more useful form for modeling electrochemical cells. One approach is
the Nernst solution which assumes that the current is so small that it can be
neglected. The Nernst equation is given by

RT CO (0, t)
E = Eo + ln (4.28)
nF CR (0, t)

Although Nernst defined the equation independently, it can be derived from

the Butler–Volmer equation assuming that the system is in equilibrium and
the net current is zero. At equilibrium, the electrodes adopt a potential based
on the bulk concentrations, as dictated by Nernst, and the bulk concentra-
tions of O and R are also found at the surface. Using i(t) = 0 and E = Eeq in
Equation 4.27, we have

( F/RT )( − α )( Eeq − E0 ) ( F/RT )(1− α )( Eeq − E0 )

CO (0, t)e = CR (0, t)e
128 Electric and Hybrid Vehicles: Design Fundamentals

This equation takes the Nernst relation form as

RT CO*
Eeq = Eo + ln (4.29)
F CR*

where
Eeq is the equilibrium potential
CO* and CR* are the bulk concentrations of the oxidation and reduction
reactants, respectively

Another approach for estimating the terminal potential is the Tafel solu-
tion which assumes that the current is large in one direction or the other.
The approximation means that one of the two exponential terms in the
Butler–Volmer expression of Equation 4.27 is negligible. The Tafel solution
is given by

RT RT
E(t) = E0 + ln(i0 ) − ln (i(t)) (4.30)
αnF αnF

where i0 is the exchange current obtained from the equilibrium condition.

Although the net current is zero at equilibrium, there is balanced faradaic
activity with equal forward and reverse currents. The exchange current is
equal to these faradaic currents and is given by

i0 = i f = ir = FAk 0C0*e
− α ( F/RT )(1− α )( Eeq − E0 )
(4.31)

Although the Tafel relationship was originally derived from experimental

data, it can also be deduced from the Butler–Volmer expression.
The reactant activities at the electrode–electrolyte interface cause the open
circuit voltage E to deviate from the standard state voltage E0. The electrode
voltage difference between that at the equilibrium condition and when there
is current flow due to charge transfer at the electrode–electrolyte interface is
often referred to as the activation or charge transfer polarization overpoten-
tial. The charge transfer polarization is reflected as a voltage drop from the
equilibrium position during cell discharge and will be denoted by Ect.

4.4.4 Mass Transport

A complete electrochemical cell is formed when two electrodes are immersed
in the same electrolyte. When an electrical circuit is completed by connect-
ing an external load to the two electrodes, current flows through the exter-
nal circuit. Electrode reactions and mass transport are two mechanisms
Battery Energy Storage 129

for supporting continuous current flow. Current flow is maintained inside

the cell through the mass transport of ions in the electrolyte. The dominant
process for mass transport is the diffusion process where molecules trans-
fer from a location with higher concentration to one with lower concentra-
tion. The mass transport can also occur through convection and migration.
Convection is the mechanical movement of particles, which does not occur
in a battery cell. In fuel cells, the pressure of fuel supply does cause some
mass transport due to the convection effect. Migration is ion movement
under the influence of an electric field where positively charged ions will
migrate toward the negative electrode, while the negatively charged ions
will move toward the positive electrode. The ion movement due to migration
may be in the same direction as that due to the diffusion process or in the
opposite direction. Since the diffusion process dominates over the convec-
tion and migration process, only the diffusion process is addressed below.
The diffusion process for an active species in an electrochemical cell can be
described by Fick’s second law as

∂C( x , t) ∂ 2C( x , t)
=D
∂t ∂x 2

where
C(x,t) is the active species concentration
D is the diffusion coefficient of the electrochemical cell
x and t are the space and time variables, respectively

Let the species concentration at the electrode responsible for chemical

reactions to maintain current flow be Cd. This concentration is less than the
bulk concentration of the electrolyte Cbulk. A linear relationship for the diffus-
ing current derived from Fick’s law is given as [6]

δ
Cd (t) = Cbulk − i(t) (4.32)
nFAD

where
δ is the diffusion layer thickness
A is the surface area

Porous electrodes are almost invariably used in electrochemical cells to

decrease the activation potential. The active materials penetrate into the
pores of the electrode to reach the reaction site. The increased surface area
due to the pores results in parallel diffusion processes, which is termed as
branched diffusion process. The pores and increased surface area compli-
cates the analysis of the diffusion process. The behavior of the branched dif-
fusion has been shown to follow the pattern described by the constant phase
130 Electric and Hybrid Vehicles: Design Fundamentals

element (CPE) instead of Fick’s second law [10]. For a CPE, the phase angle
of the frequency response remains the same for all frequencies. The CPE
transfer function used to represent the overall diffusion process is given by

Cd (s) K
H (s) = = q , 0<q<1
i(s) s

The time response of the CPE is easier to solve for simpler operations such as
constant current discharge. That time response is

Cd (t) = Cbulk − Kit q (4.33)

The diffusion process described serves the purpose of representing both the
energy storage and the impedance. The energy within a battery is stored or
spatially distributed in the electrolyte in terms of the concentration of the
active material. The movement of active material during cell reactions is con-
trolled by the inherent impedance of the electrolyte. Both mechanisms have
been represented by the diffusion process and are represented as a coupled
mechanism from the electrical perspective. However, for certain analysis, it
is desirable to separate the source and impedance. This separation is desir-
able for certain applications. For example, the fuel cell-type electrochemi-
cal device does not store any energy; the materials for chemical reactions
in a fuel cell are supplied from external fuel. The electrochemical process
is more accurately modeled by separated energy source (the fuel) and the
impedance to the source. Another important need is for the prediction of
how much energy is left in the battery, which is essentially calculating the
SoC of the battery.
The separation of energy storage and impedance enables an improved
modeling of pulsed discharge characteristics of batteries, which is essen-
tially what takes place in electric and hybrid vehicles. For the complicated
discharge currents in such applications, it is difficult to obtain analytical
solutions of the CPE, and often one has to resort to numerical solutions.

4.4.5 Electrical Double Layer

The electrical potential difference between the electrode and the electrolyte
is due to the excess charges residing at the electrode–electrolyte interface.
The excess charges at the electrode surface have to be counterbalanced by
the charges of opposite polarity in the electrolyte. The two parallel layers
formed with charged particles have a structure similar to that of a capaci-
tor. This electrode interface layer is called electrical double-layer or simply a
double layer.
The electrical double layer has a thin but finite thickness in the range
of a few Angstroms where all the excess charges reside. There is a strong
Battery Energy Storage 131

electrostatic field at the surface, since there is no dielectric material other

than the charged particles in the electrical double layer. The important prop-
erties of the electrical double layer are its capacitance and the variation of
electric potential and ion concentrations. Experimental results showed that
the behavior of the capacitance is nonlinear, depending on the interface
potential [8].
In the electrochemical models, this double layer capacitive behavior can be
represented by an equivalent nonlinear capacitor Cdbl. In batteries, the double
layer capacitor does not a play a significant role in energy storage and can
be neglected in simpler equivalent circuit models. However, this electrical
double layer concept forms the basis for making nonfaradaic electrochemical
devices known an supercapacitors or ultracapacitors. The ultracapacitors are
discussed in Chapter 5.

4.4.6 Ohmic Resistance

The voltage drop due to migration in the electrochemical cell is caused by the
ohmic resistance of the electrolyte. The electrical resistance of the electrode
materials, bulk electrolyte, and the electrode–electrolyte contact areas con-
tribute to the ohmic voltage drop. The contact resistance gradually increases
as the cell is being discharged during the formation of nonconducting film
during cell reactions. A linear resistance is typically added in equivalent cir-
cuit models to represent the ohmic voltage drop.
A secondary electrolyte, known as supporting electrolyte or inert electro-
lyte, is often added in an electrochemical cell to reduce the ohmic voltage
drop due to migration. The supporting electrolyte increases the conductivity
of ions in the electrolyte. Additionally, the electrolyte reduces the electric
field substantially which reduces the migration of the active species. The cur-
rent conduction due to diffusion or migration reduces significantly with the
addition of the supporting electrolyte. The inert ions in the cell are primarily
responsible for the current conduction within the cell. The diffusion process
still remains the dominant mechanism of supplying reactant materials to the
electrodes.

4.4.7 Concentration Polarization

In an electrochemical cell, there are species that do not participate in the
chemical reactions at the electrode, but contribute to the conduction of cur-
rent. These species may be the ions from the supporting electrolyte or from
the composition of the primary electrolyte. These species have to accumu-
late near the electrodes to aid in the flow of current, but do not react with
the electrodes. Positive ions accumulate near the negative electrode, while
the negative ions gather around the positive electrode. The result of this accu-
mulation is a voltage drop caused by the electric field formed by the accumu-
lation of the unreacted but charged particles. This is known as concentration
132 Electric and Hybrid Vehicles: Design Fundamentals

polarization. This polarization follows a Nernst equation like relationship

using the concentration of the inert ions [5]

RT Celectrode , i
EC = ln (4.34)
nF Cbulk , i

where Celectrode,i and Cbulk,i are the concentrations of the inert ions at the elec-
trode and the bulk solution, respectively. Polarization, regardless of its ori-
gin, is reflected at cell terminals as a voltage reduction from the open circuit
voltage.

4.5 Battery Modeling

Batteries and other electrochemical cells can be modeled at various levels,
depending on the use of the model. Battery models are useful for battery
design, performance evaluation, and system simulation at the application
level. Modeling aids research on device design, construction, and materials
through understanding the factors that affect the energy conversion process.
Models also help research on the performance of the device in an applica-
tion, which can be utilized for improved design and better utilization.
At the most complex level, the fundamental physics- and chemistry-based
theories are used to develop theoretical models of electrochemical cells. These
models reflect material properties and design factors on the device perfor-
mance. The fundamental mechanisms of electrical power generation are
characterized in these models in terms of both macroscopic behavior (ter-
minal voltage and current characteristics) and microscopic (internal material
and reactant behavioral processes) information. The models are very useful
for the design and performance evaluation of a particular type of battery.
The strength of these models is in the information obtained on the effect
of design variables on performance during the design stage. These models
characterize the physical and chemical relationships applied to each ele-
ment of the device [5–8]. Numerical simulation techniques such as finite ele-
ment analysis or computational fluid dynamics are also sometimes used to
develop the analytical models. The drawbacks of the theoretical models are
their complexity; often the models cannot be used to represent the device as
a component of a larger system. Device parameters are not always available
to the end user. The models could be specific for a particular chemistry and
design. Dynamic response such as that of the battery SoC is difficult, if not
impossible, to analyze with most of these models.
Battery models that emphasize the macroscopic behavior are more use-
ful for the performance evaluation at a system level (such as the electric or
Battery Energy Storage 133

hybrid vehicle systems) and for the design of these systems. For example, a
simplified battery model can be used for the dynamic simulation of a hybrid
vehicle to predict the powertrain characteristics as well as the range on elec-
tric-only operating mode. Depending on the simulation objective, the mod-
els can be represented by a set of electrical circuit components or by a simple
empirical relationship of two parameters. These two types of models are the
electric circuit models and empirical models, which are presented in this sec-
tion. The electric circuit-based models are somewhat more complex than the
empirical models, but are extremely useful for vehicle system level analysis.
On the other hand, the empirical models allow a quick evaluation on the
range of a vehicle based on the capacity or energy density.
The energy storage device models presented in this section are useful not
just for electric and hybrid vehicle applications, but also for utility power sys-
tem applications. Distributed power systems require energy storage devices
with similar features as those required for electric and hybrid vehicles.

4.5.1 Electric Circuit Models

The equivalent electrical circuit-based models use lumped parameters that
make them suitable for integration in the simulation model of a larger sys-
tem. The models use a combination of circuit elements (resistors, capacitors,
and inductors) and dependent sources to give a circuit representation of the
behavior and the functionality of the electrochemical cell. The model param-
eters are extracted from response data of the device, eliminating the need
to know the chemical processes and the design details. The electric circuit
models range from a simple linear resistive model to a fairly complex one
that characterizes the chemical processes in terms of lumped parameters.
The accuracy of these models is in between those of the theoretical models
and the empirical models; yet the circuit models are very useful for both
simulation and design of a system. The application aspects of the battery can
be evaluated effectively with insights into operation of the device as well as
that of the system. Some of the more complex circuit models can be used to
study the dynamic response such as the effect of pulse discharge which is a
characteristic of hybrid and electric vehicles [7].
The primary electrochemical activities in the electrochemical cell are
governed by two fundamental relationships: (1) Butler–Volmer relationship
characterizing the electron exchange at the electrode–electrolyte interface
and (2) Faraday’s law of electrolysis, which states that current controls the
reaction. Relating these relationships with the stored charge and the diffus-
ing charge in the electrochemical cell enables us to develop an electric circuit
model whose parameters can be obtained from experimental data.
In developing the battery models, it is more convenient to consider the
stored and diffusion charges at a surface rather than the effective species
concentration or surface activities. Let qs(t) and qd(t) be the instantaneous
stored charge and the instantaneous diffusion charge in the vicinity of the
134 Electric and Hybrid Vehicles: Design Fundamentals

electrode representing surface activities. If Q is the total capacity of a cell,

then the charge in the nonenergized species can be represented as Q − qs(t).
As was mentioned previously, the difficulty is in finding an inverse for the
Butler–Volmer equation so that terminal voltage can be represented in terms
of electrode current. The Nernst and Tafel equations are approximations
with limitations on the terminal current. One simplified approximation is
the Unnewehr universal model [11] given by

E(t) = E0 + RΩi(t) + k1qs (t)

where
E0 is the initial voltage of the cell
RΩ is the series resistance
k1 is a constant parameter

A generalized form to represent the solution to the Butler–Volmer equation

is presented by Hartley and Jannette [12]:

( )
E(t) = E0 + RΩi(t) + k1 ln ( 1 + i ) sgn(i) + k 2 ln 1 + qd sgn(i) + k 3 ln(1 − qs )

The constants E0, RΩ, k1, k2, and k3 depend on the properties of the elec-
trochemical cell and can be determined from experimental data. While the
Hartley model gives a mathematical representation of the terminal voltage, it
is often convenient to find an equivalent electric circuit model for simulation
and analysis of a battery cell. In the following, several such electric circuit
models representing an electrochemical cell are discussed, starting with a
basic model derived from the Hartley model.

4.5.1.1 Basic Battery Model

Let us begin with a simple electrical equivalent circuit model that incorpo-
rates the fundamental principles, yet simple enough for characterization
based on cell discharge data is shown in Figure 4.11. One of the key dynam-
ics that has to be modeled is the diffusion process. While complex represen-
tations using a CPE or Warburg impedance can be used, an approximate
solution to the change in diffusing charge has the same form as that of a
voltage across an RC circuit element. Therefore, the effect on the terminal
voltage due to diffusion charge will be represented by the following first-
order differential equation,

dvd (t) 1 1
= i(t) − vd (t)
dt Cd Cd Rd
Battery Energy Storage 135

Cs Cd

RΩ
– vs(t) + – vd(t) +
i(t)
Rsd Rd

E0

FIGURE 4.11
Electric equivalent circuit battery model.

where vd(t) is the voltage dropped across the RdCd parallel circuit that is pro-
portional to the diffusion charge qd(t). Additional RC circuit elements can be
added to represent the diffusion charge, but we will keep it as a single RC
time constant for our simple model shown in Figure 4.11.
Another key cell dynamic that needs to be modeled is the effect of SoC on
the terminal voltage of the cell. Figure 4.7 showed how the battery terminal
voltage decreases as the cell is being discharged. In the middle of the charac-
teristics, the terminal voltage decrease is approximately linear which can be
modeled by a series capacitor Cs to represent the stored charge in the cell. The
voltage across this storage capacitor Cs is proportional to the stored charge
qs(t). As the SoC of the cell increases or decreases during charging or dis-
charging, the voltage across the capacitor will increase or decrease, respec-
tively. Additionally, an electrochemical cell loses charge while it is at rest. A
resistor can be added in parallel to the storage capacitor to account for this
loss of charge. This resistor Rsd represents the self-discharge of the cell. The
CsRsd circuit elements representing the storage capacitor and self-discharge
resistor are shown in Figure 4.11 in series with the diffusion parameters. The
mathematical representation of this segment of the circuit model in relation
to the terminal current is

dqs (t) 1
= i(t) − qs (t)
dt Rsd

The two other parameters that need to be added to complete the electro-
chemical cell equivalent circuit is a voltage source in series with a resistor
representing the ohmic resistance drop described in Section 4.4.6. The voltage
source is taken to be the open circuit voltage of the cell E0, and RΩ is the ohmic
resistance, both of which are shown in Figure 4.11 in series with the storage
and diffusion parameters. This completes the simple equivalent circuit model
of an electrochemical cell. The values of these circuit elements can be deter-
mined experimentally by applying a step change in battery current. The pro-
cedure for obtaining the parameters of this cell is given in Example 4.1.
136 Electric and Hybrid Vehicles: Design Fundamentals

Example 4.1

A step discharge current of 15 A is applied to a three-cell generic battery to cal-

culate its parameters for the model shown in Figure 4.11. The data collected from
the experiment is shown graphically in Figure 4.12. The step command of 15 A
constant current discharge is applied at 3150 s and removed at 4370 s. After the
discharge, the battery terminal voltage settles to a lower voltage level of 5.873 V
compared to its initial no-load voltage due to the reduction in the state of charge.

Solution
ΔVd, ΔVCs, ΔVRΩ are the voltage differences that need to be calculated from the
test data to obtain the diffusion, storage, and series resistance parameters, respec-
tively. The time to reach 63% of ΔVd is 100 s. Neglecting the self-discharge of the
cells, calculate the battery equivalent circuit parameters.
Let us first calculate the equivalent series resistance of the battery. The volt-
age drop for the series resistance shows up in the output voltage characteristics
as an instantaneous increase or decrease of the terminal voltage due to the step
change in current. The voltage increase due to the 15 A step change in current is
ΔVRΩ = 5.775 − 5.58 = 0.195 V. Therefore, the series resistance value is

∆VRΩ 0.195
RΩ = = = 0.013 Ω
∆I 15

The resistance for the diffusion component Rd is

∆Vd 0.098
Rd = = = 0.00653 Ω
∆I 15

The RC time constant for the diffusion parameters is 100 s. Therefore, the diffusion
capacitor Cd can be calculated as

Vbat Constant current

discharge at 15 A ∆VCs
5.894 V
5.873 V
5.836 V ∆Vd
5.775 V
5.678 V ∆VRΩ
∆Vd
5.58 V
100 s

3150 4370 Time, t (s)

Ibat
15 A
0A

3150 4370 Time, t (s)

FIGURE 4.12
Test data for a battery to calculate equivalent circuit parameters.
Battery Energy Storage 137

100
Cd = = 15, 306 F
0.00653

The storage capacitor Cs can be calculated from the voltage change due to the
constant current discharge ΔVCs and the change in stored charge. This is calculated
as follows:

∆Q 15 ( 4370 − 3150)
Cs = = = 871, 428.6 F
∆VCs 5.894 − 5.873

4.5.1.2 Run-Time Battery Model

The Thevenin-type circuit model shown in Figure 4.11 with a constant open
circuit voltage does not allow prediction of the battery terminal voltage
Vt variations (i.e., DC response) and run-time information. The prediction
of SoC, transient response, terminal voltage, run-time, and temperature
effects is possible with run-time models. A run-time model capable of
practicing the capacity of battery has been developed by Chen and Mora
[13]. The circuit model, shown in Figure 4.13, has dependent current and
voltage sources in addition to several passive components. The terminal
voltage–current characteristics segment of the model is similar to that of
Figure 4.11, except that the open circuit voltage depends on the capacity or
SoC of the battery.
The capacitor Ccapacity and a current-controlled current source model the
capacity, SoC, and run-time of the battery. The two RC networks simulate
the voltage–current transient response characteristics. The SoC is calculated
based on the current drawn out of the cell and the initial capacity in the run-
time segment of the model. The value of the capacitor Ccapacity is given by

Ccapacity = 3600 ⋅ QC ⋅ k1 ⋅ k 2

Battery lifetime Voltage–current characteristics

VSoC RΩ Rslow Rfast

+
Ibat
Ccapacity

E0 (VSoC)
Rsd

+ Cslow Cfast Vt
–
Ibat
–

FIGURE 4.13
Run-time battery model proposed by Mora et al. [13].
138 Electric and Hybrid Vehicles: Design Fundamentals

Open circuit voltage (OCV) Step current test

Vt
Long time constant

Idischarge (A)
Short time
OCV (V)

Vt (V)
constant
Idischarge

(a) State of charge (SoC) (%) (b) Time (s)

FIGURE 4.14
Example experimental curves to obtain run-time model parameters: (a) SoC vs. open circuit
voltage characteristics; (b) discharge plot for calculating RC time constants.

where
Qc is the battery capacity in A h
k1 an k2 are cycle number and temperature-dependent correction param-
eters, respectively

The initial voltage across Ccapacity is set to 1 or lower, depending on whether

the battery is initially fully charged or not. A value of “1” represents 100%
SoC. Similarly, a value of “0” would indicate that the battery is fully dis-
charged, i.e., SoC is 0%.
SoC is bridged to the open circuit voltage through a voltage-controlled
voltage source. The relationship between SoC and open circuit voltage is
nonlinear and has to be represented from experimentally obtained data for
this model. However, the collection of the open circuit voltage versus SoC
data is extremely time consuming [14]. An example SoC versus open circuit
voltage characteristic and the discharge profile to calculate the RC time con-
stants are shown in Figure 4.14.

4.5.1.3 Impedance-Based Model

Another type of battery equivalent circuit-based model is the impedance
model. Electrochemical impedance spectroscopy is applied to develop equiv-
alent AC impedance-based circuit representation of the battery characteris-
tics. The battery model based on impedance spectroscopy is shown in Figure
4.15. Impedance-based models are less intuitive, and are applicable only for

R Lseries
ZAC
+
E0 (SoC)

+
–
Vt
Ibatt
–

FIGURE 4.15
Impedance-based equivalent electric circuit battery model.
Battery Energy Storage 139

a fixed SoC and temperature; prediction of DC response and run-time of a

battery are difficult with these models.

4.5.1.4 First Principle Model

An interesting equivalent circuit model based on the fundamental elec-
trochemical principles has been developed by Lei Xia [7]; the model is
called the first principle model. While this is not one of the simpler electric
equivalent circuit models, it isolates and relates the physical and chemical
fundamentals of an electrochemical cell to an equivalent circuit param-
eter. The model has discrete, lumped parameter representation of all the
electrochemical processes within the cell. The first principle model, shown
in Figure 4.16, incorporates the following phenomena within an electro-
chemical cell:

• Electrochemical energy conversion

• Diffusion process
• Charge transfer polarization
• Concentration polarization
• Electric double layer
• Ohmic resistance
• Self-discharge

In the equivalent circuit, the diffusion process has been described by a gen-
eral CPE; open circuit voltage and concentration polarization have been
represented by Nernst equations; charge transfer polarization has been
represented by Tafel equation; ohmic voltage drop has been represented by
resistance RΩ; electric double layer has been represented by capacitance Cdbl;
and resistance Rsd represents self-discharge of the cell.
The first principle model is construction and chemistry independent. The
parameters of the model can be derived from experimental response data of
the device, which eliminates the need for the knowledge of electrochemical

if Ec R Vt
V1
Ect +–
Zero initial i
conditions
id
K if
Sq Ce +
– E0 Cdbl

Rsd Co

FIGURE 4.16
First principle battery model with constant current source. (From Xia, L., Behavioral modeling
and analysis of galvanic devices, PhD dissertation, University of Akron, Akron, OH, 2000.)
140 Electric and Hybrid Vehicles: Design Fundamentals

properties and the design details. The model is based on the fact that before
the discharge of any current, the internal voltage E0, the double-layer capaci-
tor voltage V1, and the terminal voltage Vt (the variables are shown in Figure
4.16) are all the same. The charge transfer potential and the concentration
polarization potential are zeroes for this condition. When a load is connected
to the terminals, initially the discharge current is almost entirely supplied by
the double-layer capacitor. As the double-layer capacitor discharges and V1
decreases, the charge transfer potential is established and the faradaic cur-
rent if starts to increase. When current if increases to a point where Ect does
not change appreciably, id becomes minimal. In this situation, the faradaic
current if supplies the majority of the load current. The potential drop in this
region is primarily due to the ohmic resistance.
As an example, the parameters for a generic battery cell are given below [7]:

Bulk electrolyte concentration C0 = 2.616 (dimensionless, but represents

numerical value of the concentration)
Diffusion process parameters 1
Cd (t) = C0 − Ki f (t)t q ; K = , q = 0.68
227.5
Open circuit voltage (Nernst equation) E(t) = 1.95 + 0.052 ln Cd(t)
Charge transfer polarization (Tafel equation) Ect = 0.118 + 0.28 ln (if)
Ohmic resistance RΩ = 0.05 Ω
Double-layer capacitor Cdbl = 50 F
Concentration polarization Cd (t)
Ec (t) = 0.04 ln
C0

4.5.2 Empirical Models

The empirical models are the simplest of all models developed primarily
for simple input–output relationships of the electrochemical devices. These
models describe the performance of the device using arbitrary mathemati-
cal relationships matched with experimental or theoretical model data. The
mathematical or empirical relationships are established by curve fitting with
experimental data. The physical reasons for the behavior are not as impor-
tant as the terminal relationship between certain parameters of the device.
The physical basis for device functionalities is nonexistent in these models.
The effects of design variations are impossible to analyze with the empirical
models. Often only a subset of behaviors of the device is described such as
the constant current discharge characteristics of a battery. These models do
not provide the terminal i–v characteristics of the device which is necessary
in circuit simulation and analysis for hybrid and electric vehicles. However,
the empirical models are very useful for a quick estimation of the range of an
electric vehicle for a particular type of battery pack.
One of the widely used empirical battery model is based on the Peukert’s
equation relating discharge current with the battery capacity. The model is
based on constant current discharge characteristics of the battery. A series
Battery Energy Storage 141

of constant current discharge experiments give the I vs. tcut data for differ-
ent constant current levels; tcut is the time when the terminal voltage reaches
the cut-off voltage limit Vcut during constant current discharge. The data
obtained is used to fit Peukert’s equation to develop the model as

I ntcut = λ (4.35)

where
I is the constant discharge current
n and λ are curve fitting constants of a particular battery

n is a number between 1 and 2 with the value approaching 1 for smaller cur-
rents, but tends toward 2 for larger currents. The model does not specify the
initial capacity, nor does it model the voltage variation or temperature, and
aging factors. Peukert’s model does not give any terminal i–v information.

Example 4.2

Find the curve fitting constants n and λ for Peukert’s equation for the two mea-
surements available from a constant current discharge experiment of a battery: (1)
(t1,I1) = (10,18) and (2) (t2,I2) = (1,110).
Solution
Equation 4.35 is the Peukert’s empirical formula using the constant current dis-
charge approach. Taking logarithm of both sides of Equation 4.35

( )
Log10 I n × tcut = Log10 ( λ )

1 1
=> Log10 (I ) = Log10 (tcut ) + Log10 ( λ )
n n

Comparing with the equation for a straight line, y = mx + b; I versus tcut curve is
linear on a log-log plot, as shown in Figure 4.17.
The slope of the straight line is

∆y log(I1) − log(I2 ) log(I1 /I2 )

m= = =
∆x log(t1) − log(t 2 ) log(t1 /t 2 )

log(t1 /t2 )
Therefore, n = − .
log(I1 /I2 )
−1
For the graph shown, n = = 1.27 [∵ t1 = 10t 2 ]
18 /110
The other constant can now be calculated from Peukert’s equation

λ = 1101.27 × 1 = 391.4 A h
142 Electric and Hybrid Vehicles: Design Fundamentals

I
(Log axis)

100 A (t1,l1)
y = mx + b
y = log(I )
x = log(tcut)
–1/n

18 A (t2,l2)

10 t (Log axis)
cut

FIGURE 4.17
Plot of Peukert’s equation using constant current discharge.

or

λ = 181.27 × 10 = 392.8 A h

Another well-known and more general empirical model of a battery is based

on the Shepherd equation [15]. The simplest form of Shepherd model is

 λ 
E = E0 − iR −   Ki
 λ − it 

where
E is the battery voltage
i is the current
t is the time

The parameters of the model are E0, R, K, and λ representing battery refer-
ence voltage, internal resistance, polarization constant, and reference capac-
ity, respectively. The model parameters have some physical meaning and
relate the electrochemical behavior with the terminal i–v characteristics of
the battery. The output response is expressed as a function of time in this
model; however, the model is difficult to use for discharge patterns other
than constant current discharge.

4.5.2.1 Range Prediction with Constant Current Discharge

Peukert’s equation with constant current discharge characteristics can be
used to develop a fractional depletion model (FDM) of batteries. FDM of a
Battery Energy Storage 143

battery can be used to predict the range of an electric vehicle. Using Peukert’s
equation, we can establish the relationship between Q and I. The practical
capacity of a battery is

Q = I × tcut

Q
=> tcut =
I

Substituting into Peukert’s equation

 Q
In   = λ
 I

λ
=> Q = n −1
I

Since 0 < n − 1 < 1, for I > 1, Q decreases as I increases.

From Section 4.3.7, we know that

∫
SoD = i(τ)dτ

and

SoD
DoD =
Q(i)

SoD is the amount of charge that the battery generates to the circuit. Assume
that at t = t0, the battery is fully charged. Let us consider a small interval of
time dt. Therefore,

d(SoD)
d(DoD) = , where d(SoD) = i(t)dt
Q(i)

We know that Q = λ/In−1 for constant current discharge. Let, Q = λ/in−1 for
time-varying current as well, for the lack of anything better.
Therefore,

idt in
d(DoD) = n −1
= dt
λ/i λ
144 Electric and Hybrid Vehicles: Design Fundamentals

Integrating, we obtain,
t t
in
∫
t0
d(DoD) =
∫
t0
λ
dt

t
in
=> DoD(t) − DoD(t0 ) =
∫
t0
λ
dt

DoD(t0) = 0, if the battery is fully charged at t = t0.

The fractional depletion model is thus obtained as

t n 
i
DoD(t) = 
 λ 
0t
∫
dt  × 100%

(4.36)

The FDM based on current discharge requires knowledge of the discharge

current i(t). Therefore, this model to predict the electric vehicle range should
be used when i(t) is known.

Example 4.3

The constant current discharge characteristics of the battery pack used in an elec-
tric vehicle are

ln I = 4.787 − 0.74 ln tcut − 0.0482(ln tcut )2

The current drawn from the battery during test drives of the electric vehicle for the
SAE schedule J227a has the profile shown in Figure 4.18. The current magnitudes
for the three SAE schedules are given in Table 4.3.
Find the range of the electric vehicle for each of the three schedules.
Solution
Apply the FDM (Equation 4.36) to find the number of driving cycles for DoD = 100%.
From FDM

Ia
Current

Ib

ta tb tc
Time (s)

FIGURE 4.18
Pattern of current drawn from the battery.
Battery Energy Storage 145

t100% TABLE 4.3

in
1=
∫
t0
λ
dt Current Data for the
Driving Schedules
Schedule
First, we need to determine λ and n from the given bat- J227a Ia(A) Ib(A)
tery characteristics
B 100 35
−1 C 216 54.6
= −0.74 => n = 1.35 D 375 88.7
n
1
ln( λ ) = 4.787 => λ = 645 × 3600 A s
n

Therefore,

t100%
i1.35
1=
∫
0
645 × 3600
dt

For schedule B, fraction depleted over 1 cycle

72
i1.35
DoD for 1cycle => fcyc =
∫ 645 × 3600 dt
0

( )
72 19 1.35 38 
i1.35
∫ dt = 4.31× 10 −7  100t
∫ ∫ (35) dt 
1.35
=> fcyc = dt +
645 × 3600 
0  0 19 19 
  1  2.35 
= 4.31× 10 −7 9.41
  19 + 121.5 (38 − 19)
 2 . 35 
=> fcyc = 2.74 × 10 −3

Let N = # of cycles required for 100% DoD,

1
∴ 1 = N × fcyc => N =
fcyc

1
∴N = = 365 cycles
2.74 × 10 −3

From Table 3.5, the EV goes 1 mi in about 4 cycles for schedule B.

Therefore,

EV range = 365/ 4 = 91mi for schedule B

Measured N = 369 => error = 1.08%

146 Electric and Hybrid Vehicles: Design Fundamentals

J 227a schedule C: From FDM, N = 152; EV range = 152/3 = 51 mi.

Measured, N = 184 => error = 17.4%
J 227a schedule D: FDM, N = 41; EV range = 41/1 = 41 mi.
Measured, N = 49 => error = 16.3%.

4.5.2.2 Range Prediction with Power Density Approach

An alternative approach for using Peukert’s equation to develop a battery
model is through the use of its Ragone relationship which is the specific
power vs. specific energy characteristics. Ragone relationship and the corre-
sponding plots are linear on the log-log scale to a first-order approximation.
Battery model in terms of specific power and specific energy is

(SP)n (SE) = λ (4.37)

where n and λ are curve-fitting constants.

Example 4.4

The data given in Table 4.4 is collected from an experiment on a battery with
mass 15 kg. The data is used to draw the Ragone plot shown in Figure 4.19. Using
the data points (8,110) and (67.5,10), calculate the constants of Peukert’s equation
n and λ.

Given a battery terminal power profile p(t), the specific power SP(t) profile can be
obtained by diving the power profile p(t) by the total vehicle mass mV (Figure 4.20).
The battery is assumed to be fully charged at t = 0.
Let, fr(t) = fraction of available energy provided by battery from 0 to t, where
fr(0) = 0, since SoD(0) = 0. Now, consider the time interval dt over which a fraction
of available energy dfr is provided by the battery

dE dE /mV d (SE)
dfr = = = .
Eavail dEavail /mV SEavail

TABLE 4.4
Data from Constant Power Discharge Test
P (W) tcut (h) EP (W h) SP (W/kg) SE (W h/kg)
(Measured) (Measured) (Calculated) (Calculated) (Calculated)
150 6.75 (150)(6.75) = 1013 150/15 = 10 1013/15 = 67.5
450 0.85 381 30 25.4
900 0.23 206 60 13.7
1650 0.073 120 110 8
Battery Energy Storage 147

SP
(W/kg)
(8,110)

100

60
(67.5,10)
30

SE (W h/kg)
8 13.7 25.4 67.5 100

FIGURE 4.19
Ragone plot for Example 4.4.

P (t) SP (t) = P(t)/MV

0 t 0 t

FIGURE 4.20
Power and specific power.

If dE is the energy provided by battery to the electrical circuit over dt and Eavail is
the total available energy, then

dE = pdt

Now Eavail is a function of instantaneous power and we know that,

d (SE) = (SP)dt

Therefore,

SEavail = f (SP)

We will use Peukert’s equation to relate specific power and specific energy as
follows:

(SP)n * SEavail = λ

Therefore,

SP (SP)n+1
dfr = dt = dt
λ /(SP)n λ
148 Electric and Hybrid Vehicles: Design Fundamentals

Integrating,

fr (t ) t
(SP)n+1
∫
fr ( 0 )
dfr =
∫
0
λ
dτ

t
(SP)n+1
=> fr (t ) =
∫
0
λ
dτ (4.38)

Equation 4.38 is the FDM using power density approach. If t = time at which x% of
available energy has been used, then

t
x (SP)n+1
100
=
∫
0
λ
dτ

(SP)n +1
t100%
Note that 1 =
0∫ λ
dτ

At t100%, 100% i.e., all the available energy has been used by the system.

4.6 Traction Batteries

Lead-acid batteries that have served the automotive industry over the past
100 years for powering electrical accessories in conventional IC engine vehi-
cles do not have the power and energy density required in electric vehicles
and hybrid vehicles. The push for zero-emission vehicles led to numerous
research and development efforts on advanced batteries activities in the
United States, Europe, and Japan. Desirable features sought after in alterna-
tive battery technologies are high power and energy density, faster charge
time, and long cycle life. The research and development progressed slowly
until recent years due to the lack of market penetration of electric vehicles.
In the meantime, the growth in the electronics industry over the past sev-
eral decades has led to tremendous advancements in alternative batteries,
such as NiCd, NiMH, and Li-based batteries technologies. The rechargeable
Li-ion battery is the technology of choice for cell phones and laptop com-
puters. Further research on scaling of these battery technologies led to the
development of several battery technologies for electric and hybrid vehicle
applications. NiMH battery packs are currently used in commercially avail-
able hybrid electric vehicles, while the Li-ion battery pack is used in the
electric vehicle Tesla roadster. Emerging plug-in hybrid vehicles are also
likely to use the Li-ion battery technology. While the NiMH and Li-ion bat-
teries are the frontrunners today for electric and hybrid electric vehicles
Battery Energy Storage 149

TABLE 4.5
Properties of Electric and Hybrid Electric Vehicles Batteries
Specific
Energy Specific Energy
Battery Type (W h/kg) Power (W/kg) Efficiency (%) Cycle Life
Lead-acid 35–50 150–400 80 500–1000
Nickel-cadmium 30–50 100–150 75 1000–2000
Nickel-metal hydride 60–80 200–400 70 1000
Aluminum-air 200–300 100 <50 Not available
Zinc-air 100–220 30–80 60 500
Sodium-sulfur 150–240 230 85 1000
Sodium-nickel-chloride 90–120 130–160 80 1000
Li-polymer 150–200 350 Not available 1000
Li-ion 90–160 200–350 >90 >1000

applications, several other battery technologies have been used in various

prototype vehicles. In this section, we will review not only the promising
battery technologies, but also those that have been tried in various proto-
type electric vehicles.
The future of the battery technologies for electric and hybrid vehicle appli-
cations depends on factors including system cost, availability of raw mate-
rials, mass production capabilities, and life cycle characteristics. One must
note that the electric and hybrid vehicles industry covers a wide spectrum
and is not just limited to road vehicles. Some technologies may be more suit-
able for certain applications for various reasons. The representative prop-
erties of the promising batteries technologies along with that of lead-acid
battery are summarized in Table 4.5 with information obtained from various
literatures. The chemistry and additional information on the alternative bat-
tery technologies will then be presented in this chapter.

4.6.1 Lead–Acid Battery
The lead-acid batteries have been the most popular choice of batteries for
electric vehicles during the initial development stages. The lead-acid battery
has a long history that dates back to the middle of the nineteenth century
and is currently a very mature technology. The first lead-acid battery was
produced as early as in 1859. In the early 1980s, over 100 million lead-acid
batteries were produced per year. The long existence of the lead acid battery
is due to

• Relatively low cost

• Easy availability of raw materials (lead, sulfur)
• Ease of manufacture
• Favorable electromechanical characteristics
150 Electric and Hybrid Vehicles: Design Fundamentals

Lead-acid batteries can be designed to be of high power and are inexpensive,

safe, and reliable. A recycling infrastructure is in place for them. However,
low specific energy, poor cold temperature performance, and short calendar
and cycle life are among the obstacles to their use in electric vehicles and
hybrid electric vehicles.
Conventionally, lead-acid batteries are of flooded-electrolyte cells, where
free acid covers all the plates. This imposes the constraint of maintaining an
upright position for the battery, which is difficult in certain portable situa-
tions. Efforts in developing hermetically sealed batteries faced the problem
of buildup of an explosive mixture of hydrogen and oxygen on approaching
the top-of-charge or overcharge condition during cell recharging. The prob-
lem is addressed in the valve-regulated-lead-acid (VRLA) batteries by pro-
viding a path for the oxygen, liberated at the positive electrode, to reach the
negative electrode where it recombines to form lead sulfate. There are two
mechanisms for making the sealed VRLA batteries, the gel battery and the
AGM (absorptive glass microfiber) battery. Both types are based on immobi-
lizing the sulfuric acid electrolyte in the separator and the active materials
leaving sufficient porosity for the oxygen to diffuse through the separator to
the negative plate [16].
The construction of a typical battery consists of positive and negative elec-
trode groups (elements) interleaved to form a cell. The through partition con-
nection in the battery is illustrated in Figure 4.21. The positive plate is made
of stiff paste of the active material on a lattice type grid, which is shown
in Figure 4.22. The grid made of a suitably selected lead alloy is the frame-
work of a portable battery to hold the active material. The positive plates can
be configured as flat pasted or in tubular fashion. The negative plates are
always manufactured as pasted types.

4.6.2 Nickel–Cadmium Battery

The advantages of NiCd batteries are superior low-temperature performance
compared to the lead-acid battery, flat discharge voltage, long life, and excel-
lent reliability. The maintenance requirements of the batteries are also low.

Cell 1 Cell 2
+ + + + – – – – –
– – – – – + + + +

FIGURE 4.21
Schematic diagram of lead-acid battery showing through-partition connection.
Battery Energy Storage 151

FIGURE 4.22
A lead-acid battery grid.

The lower practical cell voltage between 1.2 and 1.3 V means that more cells
have to be connected in series to get the desired voltage. The specific energy
of NiCd batteries is 30–50 W h/kg, which is similar to that of lead-acid
batteries.
The biggest drawbacks of NiCd batteries are the high cost and the toxicity
contained in cadmium. The environmental concerns may be overcome in
the long run through efficient recycling, but the insufficient power delivered
by the NiCd batteries is another important reason for not considering these
batteries for electric and hybrid electric vehicles applications. The draw-
backs of the NiCd batteries led to the rapid development of NiMH batter-
ies, which are deemed more suitable for electric and hybrid electric vehicle
applications.

4.6.3 Nickel–Metal–Hydride Battery

The NiMH battery is a successor to the nickel-hydrogen battery, and is
already in use in the production of hybrid electric vehicles. The positive elec-
trode in a NiMH battery cell is nickel hydroxide (Ni(OH)2) and the negative
electrode is metal hydride. The chemical reactions of the NiMH battery cell
have already been presented in Section 4.2. The negative electrode consists
of a compressed mass of fine metal particles. The metallic alloy can absorb
a large number of hydrogen molecules under certain temperature and pres-
sure to form the metal hydride. This can be thought of as an alternative
approach of storing hydrogen. The proprietary alloy formulations used in
NiMH are known as AB5 and AB2 alloys. In the AB5 alloy, A is the mixture
of rare earth elements and B is partially substituted nickel. In the AB2 alloy,
152 Electric and Hybrid Vehicles: Design Fundamentals

A is titanium and/or zirconium and B is again partially substituted nickel.

The AB2 alloy has higher capacity for hydrogen storage and less costly. The
operating voltage of NiMH is almost the same as that of NiCd with flat dis-
charge characteristics. The capacity of the NiMH is significantly higher than
that of NiCd, with specific energy ranging from 60 to 80 W h/kg. The specific
power of NiMH batteries can be as high as 250 W/kg.
The NiMH batteries have penetrated the market in recent years at an
exceptional rate. NiMH battery pack was used in Chrysler “EPIC” minivans,
which give a range of 150 km. NiMH battery packs are exclusively used in
the commercially available Toyota hybrid vehicles.
The components of NiMH are recyclable, but a recycling infrastructure
is not yet in place. NiMH batteries have a much longer life cycle than lead-
acid batteries and are safe and abuse-tolerant. The disadvantages of NiMH
batteries are the relatively high cost, higher self-discharge rate compared to
NiCd, poor charge acceptance capability at elevated temperatures, and low
cell efficiency. NiMH is likely to survive as the leading rechargeable bat-
tery in the future for traction applications with strong challenge coming only
from Li-ion batteries.

4.6.4 Li–Ion Battery
The lithium metal has high electrochemical reduction potential relative to
that of hydrogen (3.045 V) and the lowest atomic mass (6.94), which shows
promise for a battery of 3 V cell potential when combined with a suitable
positive electrode. The interest in secondary lithium cells soared soon
after the advent of lithium primary cells in the 1970s, but the major dif-
ficulty was the highly reactive nature of the lithium metal with moisture
that restricted the use of liquid electrolytes. The discovery in late 1970s by
researchers at Oxford University that lithium can be intercalated (absorbed)
into the crystal lattice of cobalt or nickel to form LiCoO2 or LiNiO2 paved the
way toward the development of Li-ion batteries [17]. The use of metallic-Li is
bypassed in Li-ion batteries by using lithium intercalated (absorbed) carbons
(LixC) in the form of graphite or coke as the negative electrode along with the
lithium metallic oxides as the positive electrode. The graphite is capable of
hosting lithium up to a composition of LiC6. The majority of the Li-ion bat-
teries use either a layered oxide or iron phosphates of lithium as the positive
electrode. The layered positive electrodes of cobalt oxide are expensive, but
proved to be the most satisfactory. Nickel-oxide LiNiO2, which costs less,
can also be used, but is structurally more complex. The performance is simi-
lar to that of cobalt-oxide electrodes. The manganese oxide-based positive
electrodes (LiMn2O4 or LiMnO2) are also used since manganese is cheaper,
widely available, and less toxic. Alternative positive electrode material is the
lithium-iron-phosphate (LiFePO4) which can deliver stable and good perfor-
mance at lower costs.
Battery Energy Storage 153

RL

Electron flow
(discharge)
+ –

e– Li Li+ e–
Li+
LiCoO2 Carbon
Li Li+ Li+
e– e–
Li Li+ Li+
Electrolyte
e– Li Li+ Li+ e–

FIGURE 4.23
Lithium-ion cell. (Courtesy of Research Studies Press Ltd.)

The cell discharge operation in a Li-ion cell using LiCoO2 is illustrated

in Figure 4.23. During cell discharge, lithium ions (Li+) are released from
the negative electrode that travels through an organic electrolyte toward
the positive electrode. In the positive electrode, the lithium ions are quickly
incorporated into the lithium compound material. The process is completely
reversible. The chemical reactions at the electrodes are
At the negative electrode,

Discharge

Li x C6 ←  → 6C + xLi + + xe − where 0 < x < 1
Charge

At the positive electrode,

Discharge

xLi + + xe − + Li(1− x )CoO 2 ←  → LiCoO 2
Charge

During cell charge operation, the lithium ions move in the opposite direction
from the positive electrode to the negative electrode. The nominal cell volt-
age for a Li-ion battery is 3.6 V, which is equivalent to three NiMH or NiCd
battery cells.
The lithium ion batteries have high specific energy, high specific power,
high energy efficiency, good high-temperature performance, and low self-
discharge. The components of Li-ion batteries are also recyclable. These
characteristics make Li-ion batteries highly suitable for electric and hybrid
vehicles and other applications of rechargeable batteries. The main draw-
back of Li-ion batteries is that these are very sensitive to overvoltages and
154 Electric and Hybrid Vehicles: Design Fundamentals

overdischarges. The overvoltage of Li-ion cell positive electrode results in

solvent oxidation and the exothermic decomposition of the active material.
Overvoltage and overdischarge can result in irreversible cell damage pos-
sibly accompanied by cell ignition [18].

4.6.5 Li–Polymer Battery
The Li-polymer evolved out of the development of solid-state electrolytes,
i.e., solids capable of conducting ions but are electron insulators. The solid-
state electrolytes resulted from the research in the 1970s on ionic conduction
in polymers. These batteries are considered solid-state batteries since their
electrolytes are solids. The most common polymer electrolyte is the polyeth-
ylene oxide compounded with an appropriate electrolyte salt.
The most promising positive electrode material for Li-polymer batteries
is vanadium oxide V6O13 [16]. This oxide interlaces up to 8 lithium atoms per
oxide molecule with the following positive electrode reaction:

Discharge

Li x + V6 O13 + xe − ←  → Li x V6 O13 where 0 < x < 8
Charge

The Li-polymer batteries have the potentials for the highest specific energy
and power. The solid polymers, replacing the more flammable liquid elec-
trolytes in other type of batteries, can conduct ions at temperatures above
60°C. The use of solid polymers also has a great safety advantage in case of
electric and hybrid electric vehicles accidents. Since the lithium is interca-
lated into carbon electrodes, the lithium is in ionic form and is less reactive
than pure lithium metal. The thin Li-polymer cell gives the added advan-
tage of forming a battery of any size or shape to suit the available space
within the electric and hybrid electric vehicles chassis. The main disadvan-
tage of the Li-polymer battery is the need to operate the battery cell in the
temperature range of 80°C–120°C. Li-polymer batteries with high specific
energy, initially developed for electric vehicle applications, also have the
potential to provide high specific power for hybrid electric vehicle applica-
tions. The other key characteristics of the Li-polymer are good cycle and
calendar life.

4.6.6 Zinc–Air Battery

The zinc-air batteries have a gaseous positive electrode of oxygen and a sac-
rificial negative electrode of metallic zinc. The practical zinc-air battery is
only mechanically rechargeable by replacing the discharged product, zinc
hydroxide with fresh zinc electrodes. The discharged electrode and the
potassium hydroxide electrolyte are sent to a recycling facility. In a way, the
zinc-air battery is analogous to a fuel cell with the fuel being the zinc metal.
A module of zinc air batteries tested in German Mercedes Benz postal vans
had a specific energy of 200 W h/kg, but only a modest specific power of
Battery Energy Storage 155

100 W/kg at 80% DoD (see Sections 4.3.8 and 4.3.12 for definition of depth of
discharge and specific power, respectively). With the present-day technol-
ogy, the range of zinc-air batteries can be between 300 and 600 km between
recharges.
Other metal air systems have also been investigated but the work has been
discontinued due to severe drawbacks in the technologies. These batteries
include iron-air and aluminum-air batteries where iron and aluminum are
respectively used as the mechanically recyclable negative electrode.
The practical metal-air batteries have two very attractive positive features:
(1) The positive electrode can be optimized for discharge characteristics,
since the batteries are recharged outside the battery and (2) the recharging
time is rapid with a suitable infrastructure.

4.6.7 Sodium–Sulfur Battery

Sodium, similar to lithium, has a high electrochemical reduction potential
(2.71 V) and low atomic mass (23.0), making it an attractive negative electrode
element for batteries. Moreover, sodium is abundant in nature available at
a very low cost. Sulfur, which is a possible choice for the positive electrode,
is also a readily available and another low cost material. The use of aque-
ous electrolytes is not possible due to the highly reactive nature of sodium
and solid polymers, like those used for lithium batteries are not known. The
solution of electrolyte came from the discovery of beta-alumina by scientists
in Ford Motor Company in 1966. Beta-alumina is a sodium aluminum oxide
with a complex crystal structure.
Despite the several attractive features of NaS batteries, there are several
practical limitations. The cell operating temperature in NaS batteries is around
300°C, which requires adequate insulation as well as a thermal control unit.
The requirement forces a certain minimum size of the battery limiting the
development of the battery for only electric vehicles, a market for which is not
yet established. Another disadvantage of NaS batteries is the absence of an
overcharge mechanism. At the top-of-charge one or more cells can develop a
high resistance, which pulls down the entire voltage of the series-connected
battery cells. Yet another major concern is the safety issue, since the chemi-
cal reaction between molten sodium and sulfur can cause excessive heat or
explosion in the case of accident. The safety issues were addressed through
efficient design, and manufactured NaS batteries have been shown to be safe.
The practical limitations and manufacturing difficulty of NaS batteries
have led to the discontinuation of its development programs, especially
when the simpler concept of sodium-metal chloride batteries was developed.

4.6.8 Sodium–Metal–Chloride Battery

The sodium-metal-chloride battery is a derivative of sodium-sulfur battery
with intrinsic provisions of overcharge and overdischarge. The construction
156 Electric and Hybrid Vehicles: Design Fundamentals

– +

NaAlCl4

Beta-alumina separator

Porous Ni, NiCl2 electrode

Liquid sodium

Cell case

FIGURE 4.24
A sodium-nickel-chloride cell.

is similar to that of NaS battery, but the positive sulfur electrode is replaced
by nickel chloride (NiCl2) or a mixture of nickel chloride and ferrous chlo-
ride (FeCl2). The negative electrode and the electrolyte are the same as in
NaS battery. The schematic diagram of a NaNiCl2 cell is shown in Figure
4.24. In order to provide good ionic contact between the positive elec-
trode and the electrolyte, both of which are solids, a second electrolyte of
sodium chloraluminate (NaAlCl4) is introduced in a layer between NiCl2
and beta-alumina. The NaAlCl4 electrolyte is a vital component of the bat-
tery, although it reduces the specific energy of the battery by about 10%
[17]. The operating temperature is again high, similar to that of NaS battery.
The basic cell reactions for the nickel chloride and ferrous chloride positive
electrodes are

Discharge

NiCl 2 + 2Na ←  → Ni + 2NaCl (2.58 V)
Charge

Discharge

FeCl 2 + 2Na ←  → Fe + 2NaCl (2.35 V)
Charge

The cells in a sodium-metal-chloride battery are assembled in a discharged

state. The positive electrode is prefabricated from a mixture of Ni or Fe
powder and NaCl (common salt). On charging after assembly, the positive
electrode compartment is formed of the respective metal and the negative
electrode compartment is formed of sodium. This procedure has two signifi-
cant advantages, (1) pure sodium is manufactured in situ through diffusion
in beta-alumina and (2) the raw materials for the battery (common salt and
metal powder) are inexpensive. Although iron is cheaper than nickel, the lat-
ter is more attractive as the metallic component because of fewer complica-
tions and wider operating temperature range.
Battery Energy Storage 157

The sodium chloride batteries are commonly known as the ZEBRA bat-
teries, which originally resulted from a research collaboration between sci-
entists from the United Kingdom and South Africa in the early 1980s. The
ZEBRA batteries have been shown to be safe under all conditions of use. The
ZEBRA batteries have high potentials for being used as batteries for electric
vehicles and hybrid electric vehicles. There are several test programs per-
formed with the ZEBRA batteries.

4.6.9 Goals for Advanced Batteries

The California legislative mandates in the early 1990s led to the formation of
the U.S. Advanced Battery Consortium (USABC) to oversee the development
of power sources for electric vehicles. USABC is within the U.S. Council of
Automotive Research (USCAR), which is an umbrella organization of U.S.
auto manufacturers formed in 1992 to strengthen the automotive technology
base through collaborative research and development. The USABC addresses
factors to continue the development of high energy density and high power
density energy storage technologies. USABC establishes the goals for energy
storage developments to support electric, hybrid, and fuel cell vehicles. Goals
are set for long-term development as well as long-term commercialization.
The purpose of the commercialization goals is to develop batteries with a
reasonable goal, while the long-term criteria was set to develop batteries for
electric vehicles, which would be directly competitive with the IC engine
vehicles. The specific power and specific energy long-term goals have been
set aggressively at 400 W/kg and 200 W h/kg to promote research and devel-
opment. A subset of the goals set by the USABC for advanced electric vehicle
batteries is listed in Table 4.6. The calendar life for these batteries is targeted
for 10 years, while cycle life has been set for 1000 cycles at 80% DoD for both
commercialization and long-term goals.

TABLE 4.6
USABC Objectives for EV Advanced Battery Packs
Minimum Goals for
Long-Term Long-Term
Parameter Commercialization Goals
Specific energy (W h/kg) 150 200
(C/3 discharge rate)
Specific power (W/kg) 300 400
(80% DoD per 30 s)
Specific power (W/kg), 150 200
Regen. (20% DoD per 10 s)
Recharge time (h) 4–6 3–6
(20% → 100% SoC)
Cost, U.S. $/kW h 150 100
158 Electric and Hybrid Vehicles: Design Fundamentals

TABLE 4.7
USABC Goals for HEV Advanced Energy Storage Systems
Power Assist Power Assist
Parameter (Minimum) (Maximum)
Pulse discharge power, 10 s (kW) 25 40
Peak regenerative pulse power, 10 s (kW) 20 35
Total available energy (kW h) 0.3 at C/1 rate 0.5 at C/1 rate
Maximum weight (kg) 40 60
Maximum volume (L) 32 45
Cost, @1,000,000 units/year (U.S. $) 500 800

The USABC has also set goals for hybrid electric vehicles at two levels of
power-assist, one at the 25 kW level and the other for the 40 kW level. A sub-
set of the goals set by the USABC for power-assist hybrid electric vehicle
energy storage system is listed in Table 4.7. The calendar life for these bat-
teries is targeted for 15 years, while cycle life has been set for 300,000 cycles
for specified SoC increments. The roundtrip energy efficiency has been set to
90% for both 25 and 40 kW power-assist hybrid electric vehicles.
The USABC has also specified goals for two main PHEV battery types:
a high power/energy ratio battery providing 10 mi of all-electric range
(PHEV-10), and a low power/energy ratio battery providing 40 mi of
all-electric range (PHEV-40). PHEV-10 goals are set for a “crossover utility
vehicle” weighing 1950 kg and the PHEV-40 goals are set for a midsize sedan
weighing 1600 kg. Few of these important goals set for PHEV development
are listed in Table 4.8. The calendar life for these batteries is also targeted
for 15 years, and roundtrip energy efficiency has been set to 90% for both
PHEV-10 and PHEV-40. All the specified goals for energy storage systems for

TABLE 4.8
USABC Goals for PHEV Energy Storage Systems
Parameter PHEV-10 PHEV-40
Pulse discharge power, 10 s (kW) 45 38
Peak regenerative pulse power, 10 s (kW) 3,025 35
Available energy for charge depleting 3.4 11.6
mode (kW h)
Available energy for charge sustaining 0.3 0.5
mode (kW h)
Charge depleting life/discharge 5,000/17 5,000/58
throughput (cycles/MW h)
Charge sustaining life cycle (cycles) 300,000 300,000
Maximum weight (kg) 60 120
Maximum volume (L) 40 80
Cost, @100,000 units/year (U.S. $) 1,700 3,400
Battery Energy Storage 159

electric and hybrid electric vehicles are listed in the USABC Web site under
USCAR at www.uscar.org.

4.7 Battery Pack Management

Batteries can be configured in series or in parallel or in a combination thereof.
The battery pack, i.e., the energy storage device in an electric and hybrid
vehicle, consists of a number of individual electrochemical cells connected in
a series string to deliver the required voltage. The strings of series-connected
cells can be connected in parallel to increase the capacity of the storage sys-
tem. The battery pack also includes electronics, which is typically located
outside the battery pack. The electronic circuit of a battery pack controls
charging, discharging, and balanced utilization of the battery cells. The elec-
tronic circuit along with its controller hardware and software algorithms is
responsible for managing the battery pack and protecting the cells within
the pack. The primary function of the battery management system is to pro-
tect the cells from operating outside the safe region. This ensures longer life
of the battery and minimizes replacement costs.
The battery pack management techniques are general and suitable for any
energy storage-based system, such as electric and hybrid vehicles, distrib-
uted power generation units, and portable consumer electronics. Of all the
applications, the most rigorous usage of energy storage systems is in hybrid
vehicles where it goes through pulsed charge/discharge cycles. Hence, bat-
tery pack management is required to be of the most advanced type. The
essentials of battery management systems, SoC measurement techniques,
cell balancing methods, and battery charging methods are covered in the
Sections 4.7.1 through 4.7.4.
The battery management systems and methods of cell balancing presented
are equally applicable to an ultracapacitor bank replicating an energy storage
device. The ultracapacitor cells are also electrochemical cells, which are con-
nected in a series string to form the energy storage system to supply power
at the desired voltage level.

4.7.1 Battery Management System

The battery management system (BMS) consists of a set of algorithms based
on voltage, current, and temperature measurements to calculate essential
battery parameters and determine charge/discharge power limits at a given
time. Depending on the level of sophistication in the BMS, measurements
can be from individual cells or group of cells or from the entire pack. The
BMS is also responsible for generating command signals for cell equaliza-
tion circuits if used in a battery pack. BMS ensures reliability and protection
against overcharge, overdischarge, short circuits, and thermal abuse. The
160 Electric and Hybrid Vehicles: Design Fundamentals

Key on: Sample: Estimate Calculate Report to

Initialization; Cell/group V; SoC; power limits; SCM
Self-discharge Cell/group T; Update Generate cell Store data
recording. Ch/Disch I. SoH. equalizer com. key off

Loop while battery pack is active

FIGURE 4.25
Parameter estimations and pack management in a BMS.

BMS for an energy storage system is designed to have all or some of the fol-
lowing features:

• State-of-charge (SoC) estimation

• State-of-health (SoH) monitoring for cell and pack protection
• Temperature control
• Charge/discharge power control
• Cell equalization
• Data logging

The measurements, parameter estimations, and outputs generated in a BMS

are shown in Figure 4.25 [19]. The BMS initializes once the system is pow-
ered, which happens when with key on in a hybrid electric vehicle. The only
function during initialization is to record the self-discharge during the sys-
tem off period. If the self-discharge is excessive, then it is reported to the
SoH monitoring algorithm. The other parameters for battery management
are estimated during each measurement cycle while the pack is on.
SoC provides information on the available capacity of a battery. This is
necessary not only for the protection of the battery pack, but also for vehicle
powertrain controls. The SoC should also be maintained within a certain
band to enhance the life of the battery. In sophisticated management sys-
tems, the SoC of individual cells or group of cells in a pack are determined
to verify the uniform distribution of SoC among the cells. The SoC is typi-
cally expressed as a percentage of the rated capacity instead of the maxi-
mum available capacity, which could be less due to aging and environmental
effects. However, the SoC could also be calculated based on the maximum
available capacity. This SoC calculation can be used for cell equalization,
since all cells in the series-string generally experience the same environment.
SoH is the working condition of the pack and measures the pack’s ability
to deliver power compared to a new pack. The fading of a cell capacity and
power compared to other cells in the pack with aging indicates the deterio-
rating health of the cell. The cell capacity and other parameters are used in
an algorithm to estimate the SoH of a battery pack. The SoH information is
useful for battery safety and for delivering power up to its maximum capa-
bility. The SoH estimation algorithms are based on comparing measured
Battery Energy Storage 161

and estimated cell parameters with references or neighboring cells. The volt-
ages and SoC anomalies of one cell compared to the nearby cells is indicative
of poor health of that cell. Similarly, the excessive self-discharge of the pack
raises a flag, and is compared with preset limits to estimate the pack SoH.
The SoH information can be used to replace damaged cells in a pack instead
of replacing the entire pack.
The temperature is the primary environmental factor that affects the SoC
of an energy storage system. Imbalances in temperature among the various
cells in a pack will result in imbalances in the SoC. Additionally, tempera-
ture affects the self-discharge rates. The thermal management in a pack is
part of the cooling system design for the pack, but the temperature infor-
mation of the cells should be effectively utilized for protection and health
monitoring of the cells.
The maximum power available from the battery at a given time is calculated
in the BMS based on the SoC and terminal voltages, ensuring that operating
voltage, current, SoC, and other design limits are not violated. The BMS sets
the power limits during charging and discharging for battery protection.
Batteries could get severely damaged due to inappropriate charging. The
limits are reported to the supervisory controller for powertrain controls in
electric and hybrid vehicles.
There are three levels of management systems: pack-level management,
modular-pack-level management, and cell-level management. Pack-level
management is the most basic one where overall pack voltage and SoC are
monitored, whereas the most complete cell equalization and balancing is
possible when individual cell parameters of voltage, current, and tempera-
ture are monitored. The charging and discharging power managements at
the pack level leaves individual cells vulnerable to damage. In modular-pack-
level management, groups of cells are treated as a module for cell balancing
and equalization; the BMS algorithms depend on group voltage, current,
and temperature measurements rather than pack or individual cell measure-
ments. For packs employing cell equalizer circuits, the BMS generates com-
mand signals for cell voltage equalization based on its measurements and
estimations. The circuits act on these signals to balance the cells or groups
of cells.
Data logging is another important function of the energy storage manage-
ment system. The data for voltage, current, temperature, SoC, and number of
charge/discharge cycles could be stored as a function of time for SoH moni-
toring, diagnostics, and fault analysis.

4.7.2 SoC Measurement

The SoC of the energy storage system is calculated using measurements of
a physical parameter that varies with the SoC. The SoC varies with volt-
age, charge/discharge rate, self-discharge rate, temperature, and aging.
Depending on the parameters monitored, the SoC calculation can be either
162 Electric and Hybrid Vehicles: Design Fundamentals

∫ SoC
SoC
estimation

i(t)
+
Battery Electric
Vbatt
pack drivetrain
–

FIGURE 4.26
Battery SoC measurement.

a voltage-based method or a current-based method. The more accurate SoC

calculations use both voltage and current measurements in an observer-
based method.
The voltage-based SoC measurement is applicable to cell chemistries
where the voltages are directly related to the SoC. The relation between
open circuit voltage and SoC must be known a priori for a good estimation
of the SoC. The voltage-based measurement is not at all suitable for Li-ion
cells, since the voltage for these cells is fairly steady over most of the charge/
discharge cycle.
In current-based estimation, the SoC is obtained from the integration of
t
current using the fundamental definition of charge q =
∫ i(t)dt. A simple SoC
0
measurement diagram is shown in Figure 4.26. The charge and discharge
currents out of and into the storage device are measured directly using a
current sensor. The integration of measured current gives the SoD when the
initial condition is the fully charged condition of the battery. Knowing the
initial capacity Cp of the battery, SoC is calculated from

SoC(t) = CP − SoD(t)

The method is also known as Coulomb counting. The method tends to

accumulate errors if it is solely based on current information. A method
of improving the SoC estimation is to incorporate the directly measurable
parameters (voltage and current) into a mathematical model of the storage
system to implement an observer-based SoC estimation method. This is a
closed-loop Coulomb counting method as opposed to the open-loop method
of Figure 4.26. The closed-loop method is based on feedbacks that can be
empirically designed or generated using Kalman filters [19].

4.7.3 Battery Cell Balancing

The individual cells in the connected string of the battery pack are the unit
battery cells. The available energy stored in a battery cell is Eavail = qV, which
Battery Energy Storage 163

states that both charge and voltage need to be balanced in a series string to
maximize the output of a pack. When a series-string of electrochemical cells
is charged as a pack, slight parameter mismatches in individual cells and
temperature differences result in charge and voltage imbalances. The imbal-
ances adversely affect the vehicle performance by reducing the throughput
of the battery pack.
The chemical reactions in an electrochemical cell depend on the tempera-
ture and pressure. The temperature differences among the cells change the
self-discharge rates, causing imbalances in the charge of the cells. A low cell
temperature reduces chemical activity which increases the cell’s internal
impedance. The increased internal resistance reduces the terminal voltage,
and thus the cell capacity. In addition, manufacturing differences and differ-
ent aging characteristics result in parameter mismatches among individual
cells, which cause voltage and capacity imbalances [20,21].
The charge imbalance also shows up as voltage differences. The imbalances
tend to grow as the pack goes through repeated charge/discharge cycles.
The weaker cells tend to charge slower and the stronger cells charge faster.
The process shortens the pack life and reduces its utilization. The number of
charge/discharge cycles affects some battery chemistry more than the oth-
ers. For example, Li-ion batteries are highly sensitive to overvoltages and
undervoltages. Li-ion batteries are recommended to limit the charge/dis-
charge rates to no more than 2 C, and also to keep the cells charged to at least
40% SoC to minimize aging.
The maximum throughput of the pack can be ensured by balancing the
voltage and charge of individual cells. The cell balancing methods utilize
electronic circuits and control to even out the voltages and SoC of a series-
string of electrochemical cells. The simplest strategy adopted for charging a
series-string of cells is to monitor the cell voltages and discontinue charging
when one of the cells (strongest cell) reaches the voltage limit for individual
cells. Extended charging is another option where charging is continued even
after the strongest cell has reached its capacity to bring the weaker cells up
to capacity. When charging continues to bring the weaker cells to the maxi-
mum voltage, overvoltage results in the stronger cells. Overcharging is not
at all an option with certain battery chemistry, while in others the process
vents hydrogen gas (known as gassing) and removes water from the over-
charged cells. Increased gassing in the cell at elevated temperatures shortens
the cell life.
The overcharging in the stronger cells can be avoided if there is a path
to shunt the charging currents once they reach the voltage limit. Similarly,
the simplest protection during discharging of a pack is to shut down when the
first cell reaches the minimum voltage limit. This cell is consequently the
weakest cell in the series-string and is limiting the capacity of the pack. If
discharging is continued to extract energy from the stronger cells, then the
weaker cell voltage will fall below the minimum voltage level, possibly caus-
ing damage to the cells.
164 Electric and Hybrid Vehicles: Design Fundamentals

The simple cell balancing strategies result in underutilization of the battery

pack. Improved cell balancing circuits provide a path to bypass the weaker
cells once they reach the minimum voltage, provided the pack voltage level
is still above the minimum voltage level of the system. Power electronic con-
verter circuits are used to divert charging currents to boost the weaker cells
or deplete charge from stronger cells for cell voltage equalization. The circuit
topologies for cell balancing are presented in Chapter 7 after the power elec-
tronic devices and concepts are introduced.

4.7.4 Battery Charging
The charging of secondary batteries is accomplished in several phases
using different charging currents. The phases are structured based on the
battery chemistry to minimize the damage on the cells. The initial charg-
ing phase is the bulk charging phase when the cells are charged with the
maximum current to replenish most of the charge lost during discharge.
The last few percentages of SoC are replenished with the absorption charg-
ing phase. The charging current in this phase is kept low to prevent any
damage to the cells. An equalization phase can also be used to fully charge
and balance all the cells in the battery pack. The float charge phase starts
once the battery is fully charged to compensate for energy lost over time
due to self-discharge. Microprocessor controllers are used to set the charg-
ing profile based on an algorithm to tune the charging for a particular type
of battery chemistry.
The battery charging circuits can apply either a constant current or a con-
stant voltage or any combination of the two to design the charge profile. In
the constant current charging method, known as I-charging, a current regu-
lator in the battery charger maintains the set current level. The charging
current levels are adjusted by the current regulator for the different phases
of charging. The charging current can also be applied in the form of pulses
by pulse width modulation (PWM) control of the output voltage. The charg-
ing rate is controlled by adjusting the pulse width. The short durations
between the pulses allow the chemical reactions within the cells to stabilize.
Excessive chemical reactions that could lead to gassing are avoided by using
pulse charging. An example of multistep charging profile with I-charging
and PWM control is shown in Figure 4.27 [22].
For constant voltage charging, a voltage greater than the battery upper
limit voltage is applied by the charger for bulk charging. A constant volt-
age charge is also usually applied during the absorption charging phase.
During the float charging phase, the charger applies a DC voltage slightly
lower than the battery upper limit voltage across the battery. A slight drop
in the battery voltage results in charge being replenished in the battery. This
form of charging is also known as trickle charging used to compensate for
self-discharge in the cells.
Battery Energy Storage 165

ICH
Bulk
charging
Absorption
1.00 charging

Equalization Float
0.75
charging charging

0.50

0.25

FIGURE 4.27
Multistep I-charging with PWM control.

Problems
4.1 Estimate the weight of a 12 V, 100 A h lead-acid battery. Do this by calcu-
lating the reactant masses participating in the overall chemical reaction.
Also, assume that the mass of H2O in the electrolyte is twice the mass
of H2SO4. Neglect battery casing mass, electrode grid mass, separator
mass, and current bus mass. (Note that n = 2 for Pb and PbO2 and n = 1
for H2SO4.)
4.2 In the nickel-cadmium cell, nickel oxyhydroxide, NiOOH is the active
material in the charged positive plate. During discharge it reduces to
the lower valence state, nickel hydroxide Ni(OH)2, by accepting electrons
from the external circuit:

Discharge

2NiOOH + 2H 2O + 2e − ←  → 2Ni(OH)2 + 2OH − (00.49 V)
Charge

Cadmium metal is the active material in the charged negative plate.

During discharge, it oxidizes to cadmium hydroxide, Cd(OH)2, and
releases electrons to the external circuit:

Discharge

Cd + 2OH − ←  → Cd(OH)2 + 2e − (0.809 V)
Charge

The net reaction occurring in the potassium hydroxide (KOH) elec-

trolyte is:
166 Electric and Hybrid Vehicles: Design Fundamentals

Discharge

Cd + 2NiOOH + 2H 2O ←  → 2Ni(OH)2 + Cd(OH)2− (1.299 V)
Charge

Estimate the weight of a 11.7 V, 100 A h Ni-Cd battery. Neglect the mass
KOH component of the electrolyte.
4.3 A 12 V battery is connected to a series RL load, as shown in Figure P4.3.
The battery has a rated capacity of 80 A h. At t = 0, the switch is closed
and the battery begins to discharge.

t=0 R
i(t) 90 mH
Battery +
12 V –

FIGURE P4.3

(a) Calculate and plot the battery discharge current, i(t), if the steady-
state discharge rate is C/2. Neglect battery voltage drop.
(b) Calculate and plot SoD(t) in A h for 0 < t < 2 h.
(c) Calculate and plot SoC(t) assuming that at t = 0, the battery is
charged to rated capacity. Assume also that the rated capacity is
the practical capacity.
(d) Calculate the time corresponding to 80% DoD.

4.4 Given below are constant power discharge characteristics of a 12 V lead-

acid battery:

SP (W/kg) SE (W h/kg)
10 67.5
110 8

The battery characteristics are to be expressed in terms of Peukert’s

equation, which has the following form:

(SP)n (SE) = λ ( n and l are curve fitting constants )

(a) Derive the constants n and λ, assuming a linear relationship

between log (SP) and log (SE).
(b) Find the capacity QT of the battery if the theoretical energy den-
sity is SET = 67.5 W h/kg, given the battery mass of 15 kg.
Battery Energy Storage 167

4.5 An EV battery pack consists of four parallel sets of 6 series connected

12 V, 100 A h lead-acid batteries. One steady-state motoring (discharge)
cycle of battery current is shown in Figure P4.5a. The steady-state regen-
erative braking (charge) cycle of battery is shown in Figure P4.5b.

i Motoring i Regenerative braking

100 A
0.5 m 1 ms t
80 A

(a) 0.5 m 1 ms t (b)

FIGURE P4.5

(a) Suppose no regenerative braking is employed. How much time

does it take to reach 80% DoD?
(b) If regenerative braking is employed such that for every 50 motor-
ing cycles there is 1 regenerative braking cycle, how much time
does it take to reach 80% DoD?

(Note: In this problem, neglect variation of capacity with discharge rate.

Assume that the practical capacity is equal to the rated capacity.)
4.6 Given a lead-acid battery having the following empirical characteristics:

(SP).9 (SE) = 216E4

where SP = specific power and SE = specific energy. The EV parameters

are:
m = 700 kg, MB = 150 kg, CD = 0.2, AF = 2 m2, C0 = 0.009, C1 = 0.
Also, take ρ = 1.16 kg/m3 and g = 9.81 m/s2.

(a) Derive and plot FTR(t) vs. t. (Assume level road.)

(b) Derive and plot PTR(t) vs. t.

I
32 100
v (km/h)

35

0 19 38 42 4 t (s) 0 19 38 42 4 t (s)
(a) (b)

FIGURE P4.6
168 Electric and Hybrid Vehicles: Design Fundamentals

Calculate the EV range based on SAE J227a schedule B driving cycle

using the power density approach of fractional depletion model (FDM).
The SAE J227a driving cycle and the current profile of the EV are given
in Figure P4.6a and b. (Assume no regenerative braking.)

References
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2. C. Mantell, Batteries and Energy Systems, McGraw-Hill Inc., New York, 1983.
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Inc., Chichester, U.K., 1996.
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5. J.S. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, NJ, 1991.
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Battery Energy Storage 169

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5
Alternative Energy Storage

Alternatives to batteries as the portable energy storage device for electric

and hybrid electric vehicles are the fuel cells, ultracapacitors, compressed
air tanks, and flywheels. Many of these energy storage devices are equally
useful for stationary power generation. The alternatives are to be evalu-
ated based on the technological challenges, energy conversion efficiencies,
and fuel sources. Fuel cell is powered by hydrogen that has to be derived
from primary energy sources. Electricity produced by the fuel cell using
hydrogen as the fuel propels the electric powertrain of a fuel cell electric
vehicle. Hydrogen fuel delivery method needs to be in place in addition to
the development of fuel cell electric vehicles. One possible infrastructure is
to establish hydrogen-filling stations where hydrogen will be produced and
stored in tanks using electricity supplied through the transmission grid.
The alternative to this is to produce hydrogen on board using the reformer
technology.
Ultracapacitor, similar to battery, is another electrochemical device where
energy can be stored and used on demand by an electric powertrain. The
ultracapacitor technology has advanced tremendously in recent years,
although it is unlikely to achieve specific energy levels high enough to serve
as the sole energy storage device of a vehicle. However, ultracapacitors in
conjunction with a battery or fuel cell have the possibility of providing an
excellent portable energy storage system with sufficient specific energy and
specific power for next-generation vehicles.
Compressed air presents another type of energy storage concept that has
been utilized to develop compressed air vehicles. The compressed air vehi-
cles have recently gained attention since the well-to-wheel efficiencies are
comparable to those of fuel cell electric vehicles, but with a much simpler
fuel chain. The fuel infrastructure requirement for compressed air vehicle is
similar to that of fuel cell vehicles; electricity from the grid would be used
to compress air at local filling stations, which would be dispensed to the air
tanks of compressed air vehicles.
Flywheel is another storage device where energy is stored in mechanical
form as kinetic energy. The energy is stored in a rotating disk and released
on demand. Once again, electrical energy is the source for storing energy.
Flywheel technology is not yet competitive enough with the alternatives
discussed.
Technological challenges have to be overcome for the alternative energy
storage devices before they can supply energy to mass-produced alternative

171