4- Comprehensive Dynamic Battery Modeling for PHEV Applis

7/28/2019 4- Comprehensive Dynamic Battery Modeling for PHEV Applis

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Abstract --With the increasing demand in PHEV safety,

performance, etc., the PHEV applications require a battery

model which can accurately reflect and predict the battery

performance under different dynamic loads, environmental

conditions, and battery conditions. An accurate battery model is

the basis of the precise battery state (state of charge, state of

health and state of function) estimation. And the precise battery

state information can be used to enable the optimal control over

the battery’s charging/discharging process, therefore to manage

the battery to its optimal usage, prolong the battery life, and

enable vehicle to grid and vehicle to home applications fitting into

the future smart grid scenario. One of the challenges inconstructing such a model is to accurately reflect the highly

nonlinear battery I-V performance, such as the battery’s

relaxation effect and the hysteresis effect. This paper will mainly

focus on the relaxation effect modeling. The relaxation effect will

be modeled through series connected RC circuits. Accuracy

analysis demonstrates that with more RC circuit the battery

model gives better accuracy, yet increases the total computational

time. Therefore, to select an appropriate battery model for a

certain PHEV application is formulated as a multi-objective

optimization problem balancing between the model accuracy and

the computational complexity within the constraints set by the

minimum accuracy required and the maximum computational

time allowed. This multi-objective optimization problem is

mapped into a weighted optimization problem to solve.

Index Terms —Electric battery model, plug-in hybrid electric

vehicle (PHEV), battery relaxation effect, accuracy analysis,

computational complexity analysis, multi-objective optimization.

I. I NTRODUCTION

LUG-IN Hybrid Electric Vehicles (PHEVs) have been

drawing significant attention because of the increase in oil

consumption and environmental pollution [1-2]. The cost for

electricity to power PHEVs for all-electric operation has been

estimated at less than one quarter of the cost of gasoline. Also,

PHEVs are environment friendly. With electricity recharging

the PHEV batteries coming from renewable energy, air

pollution and dependence on petroleum can be reduced

dramatically. In addition, PHEVs provide convenience to

customers with the choice of home recharging during night or

This work was supported by Engineering Research Centers (ERC)Program of the National Science Foundation under Award Number EEC-

08212121.

Hanlei Zhang is with the Department of Electrical and Computer

Engineering, North Carolina State University, Raleigh, NC 27606 USA (e-

mail: [email protected]).Mo-Yuen Chow is with the Department of Electrical and Computer

Engineering, North Carolina State University, Raleigh, NC 27606 USA (e-

mail: [email protected]).

parking deck charging during work [1]. PHEVs also provide

additional energy storage for future smart grids through

Vehicle-to-Grid (V2G) [3] and Vehicle-to-Home (V2H) [4]

technologies.

For practical reasons, consumers need PHEVs with

compact and long life battery packs. In addition, the majority

of the current power system infrastructures is more than 50

years old and is difficult to satisfy the ever-increasing

customer needs, especially when PHEVs are penetrating the

market in the foreseeable future [1, 5]. In both of the V2G and

V2H techniques, introduced to enable the smart grid concept put forward to reform the existing power systems, PHEV

batteries are especially important because they act as

spare/reserved power (4kW/per car) when power from grid is

insufficient, and power storages to absorb the excessive power

provided by the grid during off-peak hours [6].

Like a gasoline automobile driver, a PHEV driver also

needs to know how far the car can still cover with its battery,

and how much time left for him to re-charge his battery. The

battery’s state information, such as its state-of-charge (SoC),

state-of-health (SoH) and state-of-function (SoF) [7], can help

answer these questions. Yet, precise battery state information

will only be obtained with an accurate battery model. The

battery state information enables optimal control over the battery’s charging/discharging process, reduce the risk of

overcharge and undercharge the battery [8], prolong the

battery life, manage the battery to its optimal usage [9], and

enable optimal V2G and V2H applications.

As early as 1965, Shepherd [10] developed a mathematical

equation, as in (1) to directly describe the electrochemical

behavior of a battery in terms of a cell’s potential E , no-load

voltage and the potential drop due to the internal resistance N .

No-load voltage is fitted by a constant voltage E s, a reciprocal

function and an exponential function.

( )

1exp . s

Q E E K i A BQ it Ni

Q it

−⎛ ⎞= − + − −⎜ ⎟

−⎝ ⎠

(1)

The generic battery model [11] provided by

MATLAB/Simulink SimPowerSystems is based on this

equation. However, this model is too simple to reflect the

performance of a battery under dynamic changing current

load.

Impedance-based model, adopted by many researchers [12-

13], measures the battery’s impedance with electrochemical

impedance spectroscopy (EIS) method, then use an equivalent

circuit to represent the impedance spectra. Randle’s circuit

[14] is often used for this purpose because it has a similar

Comprehensive Dynamic Battery Modeling for

PHEV ApplicationsHanlei Zhang. Student Member. IEEE. and Mo-Yuen Chow. Fellow. IEEE.

P

978-1-4244-6551-4/10/$26.00 ©2010 IEEE



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impedance spectrum with that of a Lithium-Ion battery.

However, researchers also have demonstrated that different

battery types, such as the Lithium-Ion batteries [12] and

NIMH batteries [13], have different shape of impedance

spectra. Therefore, it is difficult to construct a parameterized

battery model based on this method.

Power provided by battery comes from the chemical

reaction inside, and this process is highly nonlinear. One of

the challenges of constructing an accurate battery model is toaccurately reflect these highly nonlinear relationships, such as

the battery relaxation effect and hysteresis effect. If a battery

is left to relax after charging or discharging, it takes some time

for the terminal voltage to relax to the new steady state value.

This is called relaxation effect. And because of the hysteresis

effect, the voc (open circuit voltage) of a battery not only

depends on the battery SoC, but also influenced by the

battery’s charging/discharging history. In practice, PHEVs

will not always operate under constant current. Load current

of the battery changes when a PHEV accelerates or

decelerates. In addition, if driven under charge-sustaining

mode, or parked at home/packing deck with V2H or V2G

technique enabled, the PHEV battery will be charged anddischarged momentarily. In order to reflect the battery

performance in real life, a battery model must be able to

reflect both relaxation and hysteresis effects accurately.

In addition, PHEVs are usually working under a wide

ambient temperature, from -27°C to +40°C. The battery

performance can drift dramatically with temperature variation.

Usually higher temperature means more capacity can be

drawn out of the battery. The battery performance also keeps

deteriorating with its age. After 1000 charging cycles, under

25˚ C, a Lithium-Ion battery may loss 10% of its capacity. The

battery degradation is also temperature related. With higher

temperature, this deterioration process will be faster.

Therefore, in order to be accurate, a battery model should also

be able to adapt to its environmental conditions and age.

Recently, researchers [15-17] are using Thevenin’s theorem

to construct battery models. The basic idea is to use resistors

representing the battery’s internal resistance and a controlled

voltage source representing the battery’s electromotive force

(EMF). This method reflects the battery performance

macroscopically, and can be used to construct a parameterized

model with different parameters representing different

batteries and types. With this model technique, the relaxation

effect can be modeled by series connected RC parallel circuits

[15], and the hysteresis effect can be modeled through voc-SoC

hysteresis loop. Therefore, the model constructed is accurateunder dynamic current load. In addition, with model parameter

adaptation, this model can also reflect the environmental

effects (temperature, humidity, etc.) and aging effect.

In this paper, an equivalent circuit based on the Thevenin's

theorem is used as a battery’s base model to provide accurate

battery I-V performance. Series connected RC parallel circuits

are used to model the battery’s relaxation effect. Later, the voc-

SoC hysteresis loop (obtained from charging/discharging

experiments) will also be used to reflect the battery hysteresis

effect to further improve the accuracy of the battery model.

The characteristic parameters of the battery model are

identified through pulse charging/discharging techniques,

which can be easily implemented with standard laboratory

equipments, such as power supply, electronic load and multi-

meters. Parameter-SoC relationship can be identified through

the terminal voltage measurements and parameter-current

relationship can also be identified through different current

value used during the pulse charging/discharging experiments.

Heuristically, with more RC parallel circuits used, themodel will be more accurate in reflecting a battery’s I-V

performance. However, more RC parallel circuits also mean

higher complexity, which is often not preferred for real-world

applications. Through accuracy analysis and computational

complexity analysis, this paper provides criterions for the

model selection based on the application requirements.

The remainder of the paper is organized as the following.

Section II introduces the battery equivalent circuit used and

the relaxation effect modeling. Section III describes the model

parameter identification method. Section IV uses the accuracy

analysis and computational complexity analysis to deduce the

criterions for the model selection according to application

requirements. Finally, conclusions are drawn in Section V.

II. THE EQUIVALENT CIRCUIT OF O NE BATTERY CELL

A. Equivalent Circuit

The equivalent circuit used in this paper is constructed

based on the Thevenin's theorem. The circuit is composed of

two parts, as shown in Fig. 1. The left part describes how the

battery SoC varying with the current. The capacitor C Capacity

reflects the battery capacity. The right part describes the

relation between the load current and the battery terminal

voltage. Series connected RC parallel circuits are used to

model the battery relaxation effect.

Fig. 1. The equivalent circuit of a battery cell.

B. Relaxation Effect Modeling

RC parallel circuits are used here to model the battery

relaxation effect. With one RC parallel circuit modeling the

battery relaxation effect, non-ignorable modeling error exists

and while two series connected RC parallel circuits are used,

the modeling error is reduced dramatically, as shown in Fig. 2.

V o l t a g e ( v )

C u r r e n t ( A )

V o l t a g e ( v )

C u r r e n t ( A )

Fig. 2. (a) Relaxation effect modeling with one RC parallel circuit; (b)

relaxation effect modeling with two series connected RC parallel circuits.



3

Heuristically, more RC circuits provide better modeling

accuracy; however, criterions must be enacted to demonstrate

how many is appropriate. In Section IV, the proper number of

RC circuits will be discussed for the PHEV applications.

III. MODEL PARAMETER IDENTIFICATION

A. Charging/discharging Experiment

In order to obtain the characteristic parameters of the

model, an automatic remote battery charging/discharging web

based workstation has been developed with standard

laboratory equipments, as shown in Fig. 3.

Fig. 3. The automatic remote battery charging/discharging web-based

workstation.

A power supply (ZUP120-3.6) is used to charge the battery,

with a standard constant current constant voltage (CCCV)

charging method [18] to ensure a full charging. An electronic

load (Chroma 63108) is used to discharge the battery. Two

multi-meters (Fluke 45) are used to measure the battery

terminal voltage and the load current.

B. Parameter Identification Algorithm

Fig. 4 shows the voltage and current measurements from

the pulse discharge experiment.

Fig. 4. Terminal voltage and load current of the battery during the pulse

discharge experiment.

Theoretically, only after infinite relaxation time, v3 equals

to the battery EMF. However, infinite relaxation time is not

realistic in practice. Here, the relaxation time is set as half an

hour (at least ten times the constant of the RC circuit

dynamics), which is a good enough approximation of the

steady-state error value with an error less than 0.0624%.

The voltage increase from v1 to v2 equals to the voltage on

the internal resistor R0 before t 0. Thus, R0 is calculated though:

2 10

. L

v v R

i

−= (2)

Through least squares error fitting, the other parameters,

R1, C 1, …, Rm and C m can also be identified.

Fig. 5 shows the model characteristic parameters identified

with this method.

Fig. 5. Battery model characteristic parameters identified.

IV. MODEL A NALYSIS

A. Accuracy Analysis

Lett

v be the battery terminal voltage obtained from

experiment, and ˆt

v be the estimated battery terminal voltage

obtained from the model. The modeling error is defined as ˆ= −

t t e v v , a n-dimension vector (n is the sampling points).

Fig. 6 shows the comparison between the experimental

battery terminal voltage and the model estimated battery

terminal voltage and the modeling error e with different

number of RC circuit.

0 0.5 1 1.5 2Time (s)

0.2

0

-0.2

0.1

-0.1

×104

3.5

2.9

3.2

0 0.5 1 1.5 2Time (s) ×10

4

(a) One RC model

0 0.5 1 1.5 2Time (s)

0.2

0

-0.2

0.1

-0.1

×104

3.5

2.9

3.2

0 0.5 1 1.5 2Time (s) ×10

4

(b) Two RC model

0 0.5 1 1.5 2Time (s)

0.2

0

-0.2

0.1

-0.1

×104

3.5

2.9

3.2

0 0.5 1 1.5 2Time (s) ×10

4

(c) Three RC model

vt

vt

vt

vt

vt

vt

Fig. 6. The experimental battery terminal voltage and the estimated battery

terminal voltage from battery model, and the modeling error of battery modelscontaining from one to three RC circuits.

Root mean square percentage error (RMSPE) and

maximum percentage error (MPE) as described in (3) and (4)

are used to quantify the model accuracy:

2

1

ˆ1100%,i i

i

nt t

i t

v v RMSPE

n v=

⎛ ⎞−= ×⎜ ⎟⎜ ⎟

⎝ ⎠∑

(3)

0 0.5 12.5

3

3.5

V o c / v

0 0.5 10

20

40

R 0 / m o h m

0 0.5 10

10

20

30

R 1 / m o h m

0 0.5 11

2

3

C 1 / k F

0 0.5 10

10

20

SoC

R 2 / m o h m

0 0.5 10

0.5

1

SoC

C 2 / k F



4

andˆ

max 100%.i i

i

t t

t

v v MPE

v

⎛ ⎞−⎜ ⎟= ×⎜ ⎟⎝ ⎠

(4)

RMSPE reflects the average behavior of the modeling

error, and MPE reflects the worst case of the modeling error.

Fig. 7shows the RMSPE and MPE of the models with RC

circuits ranging from one to five.

Fig. 7. RMSPE and MPE of models containing from one to five RC circuits.

The accuracy analysis demonstrated that with more RC

circuits the battery model provides better modeling accuracy.

In addition, the second RC circuit contributes the most to the

improvement in the model accuracy. With three or more than

three RC circuits contained in the battery model, although the

modeling error is still decreasing, however, the speed of this

decrease is slower and slower.

B. Computational Complexity Analysis

More RC circuits used increases the modeling accuracy, yet

also increases the model’s computational complexity.

(1) Battery performance prediction

For the battery performance prediction, we need to

calculating the model’s output (the terminal voltage vt )

according to the model’s input (the load current i L). Pseudo

code for the battery performance prediction is given as the

following:

( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1

1 1

2 2

2 2

1

1 1

1

2 2

1

% .

1 11 1 ,

1

1 1 1 ,

1 1 1 ,

1 1

s

s

m m s

m m

s

L

s

R C f

RC RC L L

R C f

RC RC L L

R C f

RC RC m L

f is the sampling rate

SoC k SoC k i k Q k f

v k v k R i k e R i k

v k v k R i k e R i k

v k v k R i k e

−⋅ ⋅

−⋅ ⋅

−⋅ ⋅

= − + − ⋅ ⋅−

⎡ ⎤= − − ⋅ − ⋅ + ⋅ −⎣ ⎦

⎡ ⎤= − − ⋅ − ⋅ + ⋅ −⎣ ⎦

⋅⋅ ⋅

⎡ ⎤= − − ⋅ − ⋅⎣ ⎦ ( )

( ) ( ) ( ) ( ) ( )10

1 ,

.m

m L

t oc L RC RC

R i k

v k v k R i k v k v k

+ ⋅ −

= − ⋅ − − −L

Suppose the data precision is k -bit, for example a double-

precision number occupies 64 bits in the computer memory,

and a quadruple-precision number occupies 128 bits. The

computational complexity of the Pseudo code is kM (k ), where

M (k ) stands for the computational complexity of the

multiplication operation, with Schönhage–Strassen algorithm

[19] M (k )=k (logk )(loglogk ). Therefore, a model with m RC

circuits has a computational complexity of mkM (k ).

(2) On-line parameter identification

In PHEV applications, the load condition of the battery

(e.g. accelerating and decelerating) keeps changing, and so do

the environmental conditions (e.g. temperature, humidity, etc.)

and the battery’s age. Therefore, in order to capture these

effects with the model, one approach is to constantly update

the model parameters to properly reflect the most up-to-date

battery conditions and performance. In the future, we will

adapt the parameter value on-line based on the in-situ batterymeasurements. Thus the model’s complexity can be a major

bottleneck for the on-line usage, if not handled properly.

Suppose a set of n data points from the battery terminal

voltage measurements, (t 1, v1), (t 2, v2), …, (t n, vn) are

represented by the following equation:

( ) ( ) 1 2

1 2 ... .m

t t t

mv t f R e R e R e τ τ τ β −− −

= = + + +

(5)

where the vector β = [ ]1 1 2 2, , , , , , ,m m R C R C R C K

containing the model parameters, and i i i RC τ = . We can

formulate the model parameter identification as a typical least

squares fitting problem solved by the Gauss-Newton algorithmwith the following steps:

(1) Initial parameter setting:

1 1 2 2, , , , , ,o o o o o o o

m m R C R C R C β ⎡ ⎤= ⎣ ⎦K .

(2) Mismatch calculation:

( ), or v f t β = − , where ( ), o

i i ir v f t β = − ,

1,2, ,i n= K .

(3) Jacobian Matrix update:

1 1 1 1

1 1

2 2 2 2

1 1

1 1

m m

m m

n n n n

m m

r r r r

R Rr r r r

R R J

r r r r

R R

τ τ

τ τ

τ τ

∂ ∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥

⎢ ⎥∂ ∂ ∂ ∂= ⎢ ⎥

⎢ ⎥⎢ ⎥

∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

L

L

L L L L L

L

.

(4) Parameter update:

( )1

T T J J J v β −

Δ = Δ , andn o β β β = + Δ .

(5) Repeat step (2) until the residual r is less than a

predetermined threshold,n

β is then estimated as theoptimal solution of the problem.

The computational complexity of step (2) is nmk M(k ), the

computational complexity of step (3) is 2nm2k M(k ), and the

computational complexity of step (4) is

(8m3+8nm2+2nm)M(k ). Thus, the computation complexity of

the model parameter update is

[8m3+2n(k +4)m2+n(k +2)m]M(k ).

The total execution time equals to the total instructions

multiplied by the average CPI (Cycles per Instruction of the



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CPU) and divided by the clock rate of the CPU (33MHz for a

typical microprocessor for an embedded system, such as the

Intel 80386). The average CPI of a CPU is usually around 2.5,

means 2.5 CPU cycles are needed to complete each

instruction. The total execution time consumed by the battery

models with the number of RC circuits ranging from one to

five and data precision ranging from 8-bit to 128-bit is shown

in Fig. 8.

1

2

3

4

5

816

64

128

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

N u m b e r

o f R C c i

r c u i t s

D a t a p r e c i s i o n ( b i t s )

Fig. 8. The execution time of models with from one to five RC circuits and

from 8-bit to 128-bit data-precision.

The computational complexity analysis demonstrated that

with more RC circuits included in the battery model, the total

execution time increases rapidly.

C. Model Selection

An accurate battery model is required in order to provide

precise battery state information, and less model complexity is

also required so that the model can be used in real-time.

However, as has demonstrated by the accuracy and the

complexity analysis, these two requirements conflict with each

other. Therefore, to select a proper model for a certain

application is actually to balance between the model accuracyand the model complexity.

The model selection is formulated as a multi-objective

optimization problem. There are two objectives: to maximize

the model accuracy and to minimize the model complexity.

There are also two constrains: the minimum accuracy

required and the maximum complexity allowed.

With J 1 representing the modeling error and J 2 representing

the total execution time, the optimization problem can be

formulated as:

1 1 2 2

* *

1 1 2 2

min ( ) ( )

st. and

m J w J m w J m

J J J J

= +

≤ ≤

, (6)

where*

1 J is the maximum modeling error allowed, and

*

2 J

is the maximum execution time allowed. w1 and w2 are

weighting factors set according to the application. A larger w1

means more emphasis on the modeling accuracy, and a larger

w2 means more emphasis on the modeling real-time

characteristic.

Different optimal solution may arrive through minimizing J

with different w1 and w2 settings, as shown in Fig. 9.

Fig. 9. Cost function value with different w1 and w2 settings.

With the two constraints, the feasible area for the optimal

solution will be narrowed. Suppose the application requires a

modeling error less than 10%, only area I is the feasible

region. If the application also requires an execution time lessthan 2s, only area II is the feasible region. With both of them,

the optimal solution can only be selected within the area III

(overlap of area I and area II).

V. CONCLUSIONS AND FUTURE WORKS

Series connected RC parallel circuits are used in this paper

to model the battery relaxation effect. Heuristically, more RC

parallel circuits included in the battery model will bring better

modeling accuracy; however, will also bring more

computational complexity.

Based on the results of the accuracy and complexity

analysis, the model selection is formulated as a multi-objectiveoptimization problem. This optimization problem tries to find

the optimal solution through maximizing the modeling

accuracy while minimizing the modeling complexity,

subjecting to the two constraints: the minimum model

accuracy required and the maximum model complexity

allowed. And this multi-objective optimization problem is

mapped into a weighted optimization problem to solve.

In the future, the battery hysteresis effect will also be

modeled in order to further increase the accuracy of the

battery modeling and the battery state estimation. In addition,

on-line model parameter identification methods will also be

researched so that the model can be adaptive to the varying

environmental conditions (temperature, humidity, etc.) and the battery age.

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