energies
Article
An Online SOC and SOH Estimation Model for
Lithium-Ion Batteries
Shyh-Chin Huang 1,2, *, Kuo-Hsin Tseng 1 , Jin-Wei Liang 1 , Chung-Liang Chang 3 and
Michael G. Pecht 4
1
2
3
4
*
Department of Mechanical Engineering, Ming Chi University of Technology, New Taipei City 24301, Taiwan;
[email protected] (K.-H.T.); [email protected] (J.-W.L.)
College of Engineering, Chang Gung University, Taoyuan 33302, Taiwan
Center for Reliability Engineering, Ming Chi University of Technology, New Taipei City 24301, Taiwan;
[email protected]
Center for Advanced Life Cycle Engineering, University of Maryland, College Park, MD 20742, USA;
[email protected]
Correspondence: [email protected]; Tel.: +886-2-2908-9899 (ext. 4563)
Academic Editor: Peter J S Foot
Received: 16 January 2017; Accepted: 1 April 2017; Published: 10 April 2017
Abstract: The monitoring and prognosis of cell degradation in lithium-ion (Li-ion) batteries are
essential for assuring the reliability and safety of electric and hybrid vehicles. This paper aims
to develop a reliable and accurate model for online, simultaneous state-of-charge (SOC) and
state-of-health (SOH) estimations of Li-ion batteries. Through the analysis of battery cycle-life
test data, the instantaneous discharging voltage (V) and its unit time voltage drop, V0 , are proposed as
the model parameters for the SOC equation. The SOH equation is found to have a linear relationship
with 1/V0 times the modification factor, which is a function of SOC. Four batteries are tested in the
laboratory, and the data are regressed for the model coefficients. The results show that the model
built upon the data from one single cell is able to estimate the SOC and SOH of the three other cells
within a 5% error bound. The derived model is also proven to be robust. A random sampling test to
simulate the online real-time SOC and SOH estimation proves that this model is accurate and can be
potentially used in an electric vehicle battery management system (BMS).
Keywords: Li-ion battery; battery prognosis; online SOC and SOH estimation model
1. Introduction
For the past two decades, global climate change and the greenhouse effect have caused huge
impacts on the Earth’s environment, which has motivated and pushed the research and development
of energy sources and storage systems worldwide. Among the various energy storage devices, the
Li-ion battery has been accepted and widely implemented in electronic products and is considered as
the main power source for electric vehicles (EVs). Compared to conventional batteries such as lead-acid
and Ni-H batteries, the Li-ion battery is the most attractive energy storage device due to its higher
energy density, higher power density, longer cycle life, and lower self-discharge rate [1]. The Li-ion
battery is the fastest growing and most widely used battery in consumer low-power products in the
field of electronics, such as cell phones and laptops, and in high-power applications such as electric
vehicles, trams, bicycles, etc. [2,3].
Although the electrical performance of Li-ion batteries, such as the energy density and c-rate, has
improved significantly, the non-uniformity of their capacity and the retention ratio with aging has been
a big problem in terms of a battery pack with a large amount of cells in series. A battery management
system (BMS) protects the overall system and provides optimal performance management of the
Energies 2017, 10, 512; doi:10.3390/en10040512
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Energies 2017, 10, 512
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energy storage. In addition to on-line monitoring of the terminal voltage of each single cell, a BMS is
required to predict and provide each cell’s SOC and SOH, which are the most important indicators
for adapting each cell’s optimized loading timely for extending the whole pack life. For example, in
EVs applications, the battery SOC can be employed in the figurative sense as a replacement for the
fuel gauge used in conventional vehicles. Similarly, the SOH can be likened to the odometer. A BMS
with accurate estimation of SOC and SOH can prevent each cell in a battery pack from overcharging
or over-discharging, and can extend the whole pack’s life [4–9]. Online estimation of a battery’s SOC
and SOH is also essential for safe and reliable EVs. However, it is not possible to perform a direct
measurement of these states by any electrical gauge. Instead, it is estimated by the measured battery
variables, such as the time-varying voltage and the charging/discharging current. There have been
various SOC and SOH estimation methods, and each has its own merits and limitations [10,11].
The existing SOC estimation methods are usually divided into three classifications, including
data-driven, adaptive, and hybrid methods [12,13]. Data-driven methods can be divided into the direct
measurement and the model-based methods. Direct measurement methods include the Coulomb
counting method (CCM) [14], open circuit voltage (OCV) method [15,16], and impedance method [17].
The CCM is performed through measuring the discharging current over time and then making an
I-t integration to obtain an estimated SOC value [14]. The disadvantages of CCM are that it relies on
the accuracy of the current sensor and it can accumulate errors due to its open-loop nature. The OCV
method is relatively accurate but needs a sufficient rest time, and it becomes less accurate if a battery
has a flat OCV-SOC curve [15]. In the model-based approach, one measures the battery’s online data
and applies these signals as model inputs to calculate the SOC (output). Two types of battery models
have been widely employed: electrical and electrochemical models. However, all SOC estimation
methods based on electrical models have a common disadvantage that the model parameters can only
be applied on fresh cells, but are invalid on aged cells.
Adaptive methods provide an improved solution for a battery’s SOC estimation with a non-linear
SOC caused by chemical factors [12]. The adaptive approaches consider the self-designing effect
and can automatically adapt to discharging conditions. Various new adaptive methods for SOC
estimation have been developed, including Kalman filter (KF), fuzzy logic methods, support vector
machine (SVM), neural network, etc. The most widely used adaptive filter technique is the KF [18,19].
The application of the KF method on estimations of SOC is proven to be verifiable through the real-time
state estimation [20]. Singh et al. developed a fuzzy logic-based SOC estimation method for Li-ion
batteries which proved its potential in portable defibrillators [21], in which the AC impedance and
voltage recovery measurements were made and used as the input parameters for this fuzzy logic model.
The hybrid models benefit from the advantages of each of the data-driven and adaptive SOC
estimation methods and allow a globally optimal estimation performance [11]. A multiscale framework
with extended Kalman filter (EKF) was proposed to estimate SOC and capacity [6]. Simulation results
with synthetic data based on a valid cell dynamic model suggest that the proposed framework, as a
hybrid of CCM and adaptive filtering techniques, achieves higher accuracy and efficiency than the
joint/dual EKF.
The SOH of a battery is usually described as the battery performance at the present time compared
with the performance at ideal conditions and the battery’s fresh state. The estimation of the battery’s
SOH has to take into consideration both the battery capacity fade and the impedance increase.
The determination of the SOH can be done by two different approaches including data-driven and
adaptive systems. In the data-driven part, with the cycling data and the main parameters affecting the
battery lifetime, an estimation of the SOH can be performed. A deep understanding of the correlation
between the operation and degradation through physical analysis or the evaluation of large data sets
is needed for this approach.
The SOH can be determined by monitoring the internal resistance of a battery [22–24]. However,
the measurement of real-time battery internal resistance is difficult. Adaptive systems determine the
SOH through calculations from parameters that are sensitive to the degradation of the battery. These
Energies 2017, 10, 512
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parameter data must be measurable or should be examined throughout the operation of the battery.
There are several ways the SOH can be estimated, e.g., KF, EKF [16–18], genetic algorithm (GA) [25],
particle filter [26], fuzzy logic [21], artificial neural networks (ANNs) [27], and the least-squares
method (LSM) [28]. The EKF method is easy to implement and commonly used in battery prognosis.
He et al. [17] proposed an electromotive force (EMF) method combined with recursive least-squares
(RLS) for online estimation of OCV and SOC of Li-ion batteries. Nuhic et al. [29] used SVM to estimate
the SOH by using load collectives as the training and test data. Chen et al. [25] used GA to estimate
the battery model parameters, including the diffusion capacitance in real time, and then determined
the battery SOH using the identified diffusion capacitance. However, adaptive systems also operate
with the drawback of having a high computational load, which complicates the online running of the
model on a real application [30].
Capacity decrease and power fading do not originate from one single cause. Most of these
processes cannot be studied independently. Because a battery’s maximum capacity decreases with age,
accurate SOC estimation relies on a battery degradation model. Similarly, an inaccurate SOC estimation
may mislead a battery’s SOH calibration. Simultaneous estimation of SOC and SOH appears to be
necessary and beneficial for battery management systems [10,31,32]. Kim and Cho [18] proposed
a method based on EKF to identify suitable SOC and SOH model parameters for Li-ion batteries.
Hung et al. [31] proposed a dynamic impedance method to estimate the SOC and a projection method
to estimate the SOH of Li-ion batteries. The dynamic impedance is defined as the changes in voltage
over the changes in current during charging/discharging. The rate of change in dynamic impedance
with respect to SOC was then projected to determine SOH. Experimental data verified that the SOC
estimations were bounded within a 5% error but the detail of SOH accuracy was not described in
that paper.
The capacity quantifies the available energy stored in a battery. After continuous operation, the
active materials in both the cathode and anode are reduced or blocked, which leads to the battery
capacity decline. The laboratory measurements and off-line analysis are essential to evaluate the effect
of capacity fading [33,34]. The most common method of determining the degree of battery aging is
based on the OCV–SOC curve [34]. However, it requires fully charging or discharging the battery at
a low rate (e.g., 1/25 C). These methods need time-consuming tests so they cannot meet the online
real-time requirements for BMSs. An alternative approach to estimate capacity loss is the so-called
incremental capacity analysis (ICA). The concept of ICA originates from the study of the lithium
intercalation process [35,36]. ICA is more sensitive to gradual change with battery aging compared
to those methods based on conventional charging or discharging curves [37,38]. ICA was shown to
be an effective tool for analyzing battery capacity fading. Wang et al. proved that their incremental
capacity analysis with the SVM model can reduce the computation load, and the model built from
one single cell was able to predict the SOH of the other cells within a 1% error bound [38]. However,
this methodology still needs to collect a wide range of charging data typically from 3 to 3.6 V during a
charging process.
To overcome these issues described above, this paper defines a new variable V0 , which is the unit
time voltage drop in the discharging process due to its timely responsive and on-line measureable
features, and proposes a novel model that employs V and 1/V0 as the main variables. The SOC is
expressed as a linear relation of the two variables with three undetermined coefficients. Data regression
from test cells yielded the values of the coefficients. The SOH was found to be linear with 1/V0 by
a modification factor that is a function of the SOC. The derived model, which can simultaneously
estimate the battery’s instantaneous SOC and SOH, is promising for online real-time applications.
The robustness of the model is checked as well. Most important is that it was demonstrated that this
model, which was fitted by the data from one battery of a batch, can be applied to the other batteries
of the same batch without any need of sophisticated model-updating algorithms, such as EKF or
SVM. Random samplings from the tested data to simulate the online SOC and SOH estimations are
performed, and satisfactory results are obtained.
Energies 2017, 10, 512
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4 of 18
2. Experimental
Data
Analysis
Four lithium
cobalt
oxide (LiCoO2) batteries (numbered A–D) of nominal capacity 1.10 Ah
wereFour
tested
at 25cobalt
°C. The
follows
the standard
CCCV protocol,
i.e., Ah
constant
lithium
oxidecharging
(LiCoO2 )process
batteries
(numbered
A–D) of nominal
capacity 1.10
were
◦ C.at
current
(CC)
0.5charging
C until the
voltage
reaches
its designated
of 4.2
then constant
voltage
tested
at 25
The
process
follows
the standard
CCCV value
protocol,
i.e.,V,
constant
current (CC)
at
(CV)
charging
takes
over
until
the
current
falls
below
50
mA.
The
discharging
current
is
kept
at
a
0.5 C until the voltage reaches its designated value of 4.2 V, then constant voltage (CV) charging takes1
C rate
and
terminated
at below
a cut-off
voltage
2.7 V. The current
voltagesis of
theattested
cellsand
were
logged at
over
until
the
current falls
50 mA.
Theofdischarging
kept
a 1 C rate
terminated
30-s
intervals.
Theofbattery’s
latest
maximum
capacity,
Qm, was
evaluated
CCM
at the
at
a cut-off
voltage
2.7 V. The
voltages
of the tested
cellsdenoted
were logged
at 30-s
intervals.byThe
battery’s
end
of
every
discharging
process.
With
the
evaluated
Q
m
,
the
SOC
at
any
previous
time
instant
was
latest maximum capacity, denoted Qm , was evaluated by CCM at the end of every discharging process.
calculated
backwards.
wasatevaluated
by dividing
the latest
Qm by the backwards.
nominal oneSOH
(Qnom
) at
With
the evaluated
Qm , SOH
the SOC
any previous
time instant
was calculated
was
the end ofby
every
discharging
cycle.
evaluated
dividing
the latest
Qm by the nominal one (Qnom ) at the end of every discharging cycle.
Figure11illustrates
illustratesthe
thetypical
typical
curve
voltage
versus
capacity
tested
cell
at certain
Figure
curve
of of
voltage
versus
capacity
for for
the the
tested
cell A
at A
certain
test
test cycles.
The left-shifting
the discharging
implies
the capacity
fade
is an indicator
cycles.
The left-shifting
of theofdischarging
curvescurves
implies
the capacity
fade and
is and
an indicator
of the
of the battery’s
healthThe
(SOH).
Thedeclining
voltage with
declining
with the discharging
used asfor
an
battery’s
health (SOH).
voltage
the discharging
time can be time
used can
as anbeindicator
indicator
theNonetheless,
battery SOC.every
Nonetheless,
everyvoltage
instantaneous
voltage
couldSOC
reflect
multiple
the
batteryfor
SOC.
instantaneous
could reflect
multiple
values
unlessSOC
the
values
unlessNthe
cycle number
N is research
known. is
The
present
research online
is aimed
at a battery’s
cycle
number
is known.
The present
aimed
at a battery’s
real-time
SOC andonline
SOH
real-time SOC
and
SOH estimation
modelawithout
fully conducting
charging/discharge
cycle, i.e.,
estimation
model
without
fully conducting
charging/discharge
cycle,a i.e.,
without N as a parameter.
N as atoparameter.
It is parameters
necessary tofor
look
forand
extra
parameters
and SOH
modeling in
Itwithout
is necessary
look for extra
SOC
SOH
modelingfor
inSOC
addition
to instantaneous
0 , which isathe
additionTherefore,
to instantaneous
voltage. VTherefore,
new
variable,
V′, which
the
unit
time voltage
drop
voltage.
a new variable,
unit
time voltage
dropisin
the
discharging
process,
in
the
discharging
process,
was
defined
and
used
to
derive
the
model
for
SOC
and
SOH
prediction.
was defined and used to derive the model for SOC and SOH prediction.
4.2
Charge
800
500
200
5
cycles cycles cycles cycles
Discharge
3.4
SOC
Voltage (Volt)
3.8
3.0
SOH
2.6
0.0
0.2
0.4
0.6
0.8
Capacity (Ah)
1.0
Figure1.1.Variations
Variationsofofthe
thevoltage
voltagevs.
vs.capacity
capacityof
ofcell
cellAAatatvarious
variouscycles.
cycles.
Figure
2.1. Battery States
2.1. Battery States
The definitions of SOC, SOH, and some terminologies are briefly described as follows:
The definitions of SOC, SOH, and some terminologies are briefly described as follows:
Q −R I ⋅ dt
Qm m−  I · dt × 100%
SOC(%)
=
SOC(%) =
× 100%
Q
Qmm
(1)
(1)
is defined as the ratio of the remaining capacity over the maximum capacity, and SOC can provide
a timely charge warning. Figure 1 shows that a battery’s Qm gradually decreases with battery aging.
According to the degradation of Qm, SOH is defined as:
Energies 2017, 10, 512
5 of 18
is defined as the ratio of the remaining capacity over the maximum capacity, and SOC can provide a
timely charge warning. Figure 1 shows that a battery’s Qm gradually decreases with battery aging.
According to the degradation of Qm , SOH is defined as:
SOH(%) =
Qm
× 100%
Qnom
(2)
A battery’s SOH normally ranges within 0–100%, but when it is new, the SOH can be slightly
larger than 100% due to product variations. The end of life of a Li-ion battery is commonly defined by
the maximum cycles when the SOH drops to 80%. SOH is a key indicator of safe operation because it
provides a timely warning that replacement is required.
2.2. New Parameter for SOC and SOH Model
OCV has been proven as a proper indicator for SOC and SOH evaluation, but the measurement
of OCV requires a long relaxation time to reach a steady state under thermodynamic equilibrium,
which is called EMF and is more suitable for off-line evaluation. Our previous study [39] adopted the
fully discharged voltage as a replacement for OCV, and combined it with the resistance for the SOH
model construction. That model predicted a battery’s remaining useful life (RUL) within reasonable
accuracy, but the voltage and resistance at a fully discharged state are apparently inappropriate aging
parameters for an on-line, real-time model. The present study is intended to develop a suitable model
for the real-time, on-line monitoring of a battery in use, and thus any parameter associated with the
fully discharged state is hazardous and should be avoided. The most convenient and direct outputs
from a battery in use are voltage, current, and time. A single voltage value, as shown in Figure 1, is
never enough to identify a battery’s SOC because the voltage shifts with battery aging. Therefore, this
paper proposes a new parameter called the unit time voltage drop, V 0 for SOC and SOH modeling, i.e.,
V0 =
∆V
∆t
(3)
Note that in Equation (3), ∆V is the voltage drop in the discharging process, i.e., ∆V = V 1 − V 2 .
2.3. SOC and SOH Regression Model
The correlation of SOC with V and V0 is first checked from the test data so as to judge its
appropriateness for model construction. The average correlation of SOC vs. V is 0.988, and that of
SOC vs. 1/V 0 is 0.984. The relations of SOC vs. V and 1/V 0 are also found to be approximately linear.
As depicted previously, both variables of V and 1/V 0 are needed to estimate the SOC because there
are multiple SOCs for a given V or a given V 0 . Figure 2 illustrates the correlations of SOC vs. 1/V 0 at
different fixed voltages, and the values appear high enough except for V = 3.9. Figure 2 also shows
that when modeling by employing both parameters V and 1/V 0 , only a single correspondent SOC is
obtained for a specific V 0 under a given voltage, V.
To further explore the physical meaning of the parameter V 0 , we tried to correlate V 0 to the internal
resistance of the battery by the following equation
V(terminal) = Veq + η
(4)
If the change of Veq in a short discharge time duration is negligible, it can be assumed that ∆V is
approximate to ∆η.
∆V(terminal) ≈ ∆η
(5)
V0 =
∆V
∆η
≈
∆t
∆t
(6)
Energies 2017, 10, 512
6 of 18
where η is the overpotential which includes two parts, the reversible and the irreversible, representing
the
degree of deviation from the equilibrium voltage, Veq (the value of Voc).
Energies 2017, 10, 512
6 of 18
If Equation (6) is divided by the current I, it can be rewritten as:
Δη / I ∆R
ΔR
VV0 ′ ∆η/I
≈≈
==
II
∆t
∆t
Δt
Δt
VV0 ′≈≈
(7)
(7)
I∆R
IΔR
∆t
Δt
(8)
(8)
where R is the equivalent resistance of the battery and V 0 is proportional to ∆R under a short discharge
where R is the equivalent resistance of the battery and V′ is proportional to ΔR under a short
time duration and a discharging current I. That means for a given V and fixed ∆t, a lower V 0 has a
discharge time duration and a discharging current I. That means for a given V and fixed Δt, a lower
lower ∆R. V 0 is related to the increasing rate of internal resistance caused by battery discharging and
V′ has a lower ΔR. V′ is related to the increasing
rate of internal resistance caused by battery
aging. Therefore, for a given cell voltage, V 0 is sensitive to battery aging and changes with different
discharging and aging. Therefore, for a given cell voltage, V′ is sensitive to battery aging and
charge-discharge cycles. Equation (9) uses V 0 to correlate SOC with battery aging so that the cycle
changes with different charge-discharge cycles. Equation (9) uses V′ to correlate SOC with battery
number N can be removed. Accordingly, the SOC equation is assumed to be
aging so that the cycle number N can be removed. Accordingly, the SOC equation is assumed to be
0
SOC
(%) ==aa· V
(9)
SOC(%)
⋅ V++bb· ⋅ (1/V
1 / V ' )++cc
(9)
where
wherea,a,b,b,and
andccare
areundetermined
undeterminedcoefficients
coefficientsto
tobe
befitted
fittedfrom
fromtest
testdata
dataregression.
regression.
100%
R² = 0.8683
R² = 0.9001
SOC (%)
80%
R² = 0.9027
60%
V = 3.9
V = 3.8
V = 3.7
V = 3.6
40%
R² = 0.966
20%
0.0
4.0
8.0
1/V' (secmV-1)
12.0
Figure2.2.Correlation
Correlationof
ofSOC
SOCto
to1/V
1/V′0 under
voltage from
from 3.6
3.6 to
to 3.9
3.9 volt.
volt.
Figure
under an
an instantaneous
instantaneous discharge
discharge voltage
Accurate SOC estimation relies on the precise evaluation of Qm, which diminishes with battery
Accurate SOC estimation relies on the precise evaluation of Qm , which diminishes with battery
age. Provided Qm is not instantly updated, the estimated SOC would
misguide the BMS operation.
age. Provided Qm is not instantly updated, the estimated SOC would misguide the BMS operation.
Therefore, SOC cannot be separately evaluated without considering SOH at the same time.
Therefore, SOC cannot be separately evaluated without considering SOH at the same time. Fortunately,
Fortunately, it was seen that the battery’s aging information is, in fact, imbedded in the new
it was seen that the battery’s aging information is, in fact, imbedded in the new parameter V 0 . To see
parameter V′. To see how the newly defined parameter V′ related to SOH, their correlations at
how the newly defined parameter V 0 related to SOH, their correlations at several SOC levels are
several SOC levels are illustrated in Figure 3. The results show a surprisingly good correlation
illustrated in Figure 3. The results show a surprisingly good correlation (>0.99) for all fixed SOCs
(>0.99) for all fixed SOCs ranging from 40% to 90%. Figure 3 also shows that SOH-(1/V′) has an
ranging from 40% to 90%. Figure 3 also shows that SOH-(1/V 0 ) has an approximately linear relationship
approximately linear relationship for all different SOC levels. Therefore, the SOH regression model
for all different SOC levels. Therefore, the SOH regression model for a specific SOC level can be
for a specific SOC level can be assumed to be
assumed to be
SOH
(%) ==AA· ⋅ 1/V
(10)
SOH(%)
1 / V0 ' ++ BB
(10)
(
)
Similarly, A and B are regression coefficients to be determined. Take SOC = 70% as a typical
example and solve for the coefficients A = 0.5192 and B = 0.0405 from cell A’s test data. Substitute
Equation (10) for the estimated SOH vs. N, and compare it against the real values in Figure 4. The
coefficient of determination (R2) is as high as 0.9907. This result enhances the adequacy of the SOH
model, shown by Equation (10). Nonetheless, this is valid only for the case of SOC = 70%, and new
coefficients for A and B need to be derived if SOC is other than 70%. It is impractical to have
numerous sets of coefficients for different SOC levels. Therefore, an alternative approach to account
for SOC
variation
Energies
2017,
10, 512 is necessary.
7 of 18
numerous sets of coefficients for different SOC levels. Therefore, an alternative approach to account
7 of 18
100%
for SOC variation is necessary.
Energies 2017, 10, 512
SOH (%)
SOH (%)
80%
100%
60%
80%
SOC = 90% (r = 0.991)
SOC = 80% (r = 0.995)
SOC = 70% (r = 0.995)
SOC==90%
60%(r(r==0.991)
0.988)
SOC
SOC==80%
50%(r(r==0.995)
0.987)
SOC
SOC==70%
40%(r(r==0.995)
0.983)
SOC
40%
60%
20%
40%
0%
20% 0.0
4.0
1/V'
SOC = 60% (r = 0.988)
8.0
SOC
= 50% (r = 0.987) 12.0
-1)
(secmV
SOC
= 40% (r = 0.983)
0% Figure 3. Correlations of SOH to 1/V′ at various SOC levels.
0.0
4.0
8.0
12.0
1/V' (secmV-1)
After thorough inspection of the test data, we multiplied a modification factor α onto Equation
(10) without recalculating
the
regression of
coefficients
A0 at
and
B. Equation
(10) is hence modified to
Figure
Correlations
ofSOH
SOHtoto1/V
1/V′
various
SOC
Figure
3.3.Correlations
at
various
SOC levels.
levels.
become
After thorough inspection of the test data, we multiplied a modification factor α onto Equation
Similarly, A and B are regression
coefficients
to) ⋅beAdetermined.
SOC = 70% as a typical
) =coefficients
SOH(%
α (SOC
⋅ (1 / V
+ B Take(10)
(11)
(10) without recalculating the regression
A and
B.')Equation
is hence modified
to
example and solve for the coefficients A = 0.5192 and B = 0.0405 from cell A’s test data. Substitute
become
where α(SOC)
the estimated
newly defined
modification
factor asitaagainst
function
SOC.
SinceinαFigure
does not
Equation
(10) foris the
SOH vs.
N, and compare
theofreal
values
4.
2
monotonically
change
with
SOC,
interpolation
is
required
and
a
third
degree
of
polynomials
The coefficient of determinationSOH
(R ) (is
as 0.9907.
theis
) =high
) ⋅ A ⋅ (1This
) + B enhances the adequacy of(11)
%as
α (SOC
/ V 'result
found
to be shown
accurate
as (10).
follows:
SOH
model,
byenough,
Equation
Nonetheless, this is valid only for the case of SOC = 70%, and
where
α(SOC)
is
the
newly
defined
modification
factor
as a function
of ItSOC.
Since α does
not
new coefficients for A and B need to be
derived if SOC
is other
than 70%.
is impractical
to have
3
2 and a third degree of polynomials is
monotonically
change
with
SOC,
interpolation
is
required
numerous sets of coefficients
SOC
Therefore,
to account
a (SOCfor
C1 ⋅ (SOC
C2 ⋅ (SOC
SOC ) + Capproach
) =different
) +levels.
) + Can3 ⋅ (alternative
(12)
0
found
be accurate
enough, as follows:
for
SOCtovariation
is necessary.
[
]
[
]
3
2
a (SOC) = C1 ⋅ (SOC) + C2 ⋅ (SOC) + C3 ⋅ (SOC ) + C0
100%
Experimental
Estimated
Experimental
Estimated
SOH=0.1588 (1/V')+0.0412
R² = 0.99
100%
80%
60%
80%
SOH (%) SOH (%)
SOH (%)
SOH (%)
100%
80%
SOH=0.1588 (1/V')+0.0412
R² = 0.99
60%
100%
60%
40%
40%
80%
20%
60%
40%
20%
0%
40%
0.0
20%
20%
0%
2.0
4.0
1/V' (secmV-1)
6.0
0%
0
0.0
200
2.0
4.0
1/V'400
(secmV-1) 600
N (cycles)
6.0
800
1000
0%
Figure 4. Comparison
of the estimated
SOH for cell
= 70%.
0
200
400 vs. real600
800A at SOC
1000
N (cycles)
(12)
Energies 2017, 10, 512
8 of 18
After thorough inspection of the test data, we multiplied a modification factor α onto Equation (10)
without recalculating the regression coefficients A and B. Equation (10) is hence modified to become
SOH(%) = α(SOC) · A · 1/V 0 + B
(11)
where α(SOC) is the newly defined modification factor as a function of SOC. Since α does not
monotonically change with SOC, interpolation is required and a third degree of polynomials is
found
be 10,
accurate
enough, as follows:
Energiesto
2017,
512
8 of 18
3
2
SOC
+ C3for
· (cell
SOC
+SOC
C0 = 70%.
) = C1 ·of(SOC
) + C2 · (vs.
Figureα4.(SOC
Comparison
the estimated
real) SOH
A)at
(12)
The
Equation
(12)(12)
derived
fromfrom
cell A
with
SOC, SOC,
ranging
from 40%
90%,to
shows
Thefitted
fittedcurve
curveofof
Equation
derived
cell
A with
ranging
fromto40%
90%,
excellent
agreement
with thewith
otherthe
three
tested
cells,
as seen
5. Figure
The reason
forreason
choosing
shows excellent
agreement
other
three
tested
cells,inasFigure
seen in
5. The
for
SOC
= 70%SOC
as the
basis
= 1)
is because
that
state seems
to be
the most
of an state
in-use
choosing
= 70%
as(αthe
basis
(α = 1) is
because
that state
seems
to beprevalent
the most state
prevalent
of
battery.
The
authors
have
used have
other used
SOCsother
or other
cellorasother
the basis,
estimation
level
an in-use
battery.
The
authors
SOCs
cell asand
thethe
basis,
and the error
estimation
remained
same, although
thealthough
regression
the modelofand
factor were
error levelthe
remained
the same,
thecoefficients
regression of
coefficients
themodification
model and modification
different.
This
consistency
proves
the robustness
the derived
factor were
different.
This partially
consistency
partially
proves theofrobustness
ofmodel.
the derived model.
1.5
1.3
α = f (SOC)
R² = 0.9967
1.1
0.9
Cell_A
Cell_B
Cell_C
Cell_D
0.7
0.5
20%
40%
60%
SOC (%)
80%
100%
Figure5.5. Correlations
Correlationsof
offactor
factorααwith
withSOC
SOCfor
forthe
thefour
fourtested
testedcells.
cells.
Figure
2.4. Model Construction and Verification
2.4. Model Construction and Verification
This study aimed to employ battery outputs such as voltage and current to build up a new
This study aimed to employ battery outputs such as voltage and current to build up a new
model suitable for SOC and SOH online estimation. The newly developed SOC and SOH models
model suitable for SOC and SOH online estimation. The newly developed SOC and SOH models
(Equations (9), (11), and (12)) do not use the cycle number N as the aging parameter. The batteries
(Equations (9), (11), and (12)) do not use the cycle number N as the aging parameter. The batteries were
were undergoing the common cycle life test in the study so as to have N as a reference to do a
undergoing the common cycle life test in the study so as to have N as a reference to do a comparison
comparison within the models. The process of model construction is shown in Figure 6, in which
within the models. The process of model construction is shown in Figure 6, in which the test data
the test data of the instantaneous voltage, discharging current (constant), time, and resistance in
of the instantaneous voltage, discharging current (constant), time, and resistance in each cycle were
each cycle were logged at 30-s intervals. 1/V′ is calculated from every two adjacent outputs. Qm is
logged at 30-s intervals. 1/V 0 is calculated from every two adjacent outputs. Qm is evaluated at the
evaluated at the end of every discharging process, and SOH is calculated from Equation (2)
end of every discharging process, and SOH is calculated from Equation (2) accordingly. The SOC for
accordingly. The SOC for each data point is obtained by back-substituting Qm into Equation (1).
each data point is obtained by back-substituting Qm into Equation (1).
The coefficients of the SOC and SOH models, based on the four tested cells in this study, were
The coefficients of the SOC and SOH models, based on the four tested cells in this study, were
calculated from data regression. The equations were then applied back to the four cells to estimate
calculated from data regression. The equations were then applied back to the four cells to estimate the
the SOC and SOH values and to compare against the real data for verification. The accuracy of the
SOC and SOH values and to compare against the real data for verification. The accuracy of the models
models was defined by R2 and root mean square error (RMSE). The SOC and SOH were estimated
was defined by R2 and root mean square error (RMSE). The SOC and SOH were estimated from the
from the battery’s random sampling of discharging voltages to simulate an in-use situation.
battery’s random sampling of discharging voltages to simulate an in-use situation.
Energies 2017, 10, 512
9 of 18
Energies 2017, 10, 512
9 of 18
Four batteries (Cell A, B, C, D)
Cycle-life test
output : current, voltage, time…
(every 30 seconds)
New indicator
Remaining capacity
Aging and reliability
Data regression for a, b, c
Data regression for A, B
Modification Factor
α = f (SOC)
Verification
Online random sampling test
Figure 6. SOC and SOH model construction and verification process.
Figure 6. SOC and SOH model construction and verification process.
3. Results
3. Results
With no preference, one of the four tested cells, called A, was first chosen for the SOC and SOH
With no preference, one of the four tested cells, called A, was first chosen for the SOC and SOH
model construction. The derived model was then applied to estimate the SOC and SOH of all cells.
model construction. The derived model was then applied to estimate the SOC and SOH of all cells.
To show the model’s robustness, the model construction process was repeated using different cell
To show the model’s robustness, the model construction process was repeated using different cell or
or cells as training data. The differences of the model coefficients and estimation errors between
cells as training data. The differences of the model coefficients and estimation errors between using
using different sets of training data were compared as well.
different sets of training data were compared as well.
3.1. SOC
3.1.
SOC Estimation
Estimation
The first
firstrow
row
of Table
1 shows
SOC estimations
A at aging
four conditions—100,
different aging
The
of Table
1 shows
the SOCthe
estimations
of cell A at of
fourcell
different
conditions—100,
300, 500,
700 cycles
with V volt.
ranging
from 3.55–3.95
volt.
average
300,
500, and 700 cycles
with and
V ranging
from 3.55–3.95
The average
correlation
(R2 The
) is high,
and
2) is high, and the error (RMSE) is small. One may doubt that this excellent fitness was
correlation
(R
the error (RMSE) is small. One may doubt that this excellent fitness was due to the same training and
due data
to the
same
anddoubt,
test data
clarify derived
this doubt,
the
SOC
equation
derived
from cell
test
sets.
To training
clarify this
the sets.
SOC To
equation
from
cell
A was
applied
to estimate
the
2
A wasthree
applied
three cells Ras
well,RMSE
and the
R and
are
2 and
other
cellsto
asestimate
well, andthe
theother
corresponding
are corresponding
compared in rows
2–4 RMSE
of Table
1.
2 > 0.995 and RMSE < 0.024 except cell
compared
in
rows
2–4
of
Table
1.
For
the
majority
of
points,
R
2
For the majority of points, R > 0.995 and RMSE < 0.024 except cell D’s 100-cycle case, which was
D’s 100-cycle
case,
was probably
caused
the abnormal
external circuit
connection
probably
caused
bywhich
the abnormal
external
circuitbyconnection
that happened
during
the cycle that
test
happened
during
the
cycle
test
because
extremely
high
resistance
values
contrasting
with
other
because extremely high resistance values contrasting with the other three were recorded. Thisthe
anomaly
threerecovered
were recorded.
anomaly
recovered
cycles.
Table
1 shows
thatSOH
the is
SOC
was
after 150This
cycles.
Table 1was
shows
that theafter
SOC 150
accuracy
holds
even
when the
far
accuracy
holds
even
when
the
SOH
is
far
below
80%
(N
=
700).
The
same
process
was
repeated
with
below 80% (N = 700). The same process was repeated with the data of cells B, C, and D instead of A for
the model
data ofconstruction,
cells B, C, and
instead
of Aremained
for the model
construction,
and also
the accuracy
at
the
andDthe
accuracy
at the same
level, which
proves theremained
robustness
the
same
level,
which
also
proves
the
robustness
of
Equation
(9).
of Equation (9).
Energies 2017, 10, 512
10 of 18
Table 1. Average R2 and RMSE of the SOC estimations of all cells using cell A for model
Energies
2017, 10, 512
construction.
Cell
Name
10 of 18
Correlation
and Error
R2
0.996 100 Cycles
0.997 300 Cycles0.996500 Cycles 0.995
Correlation
and Error
700 Cycles
RMSE R2
0.0120
0.0114
0.0116
0.0168
0.996
0.997
0.996
0.995
R2 RMSE
0.997
0.9940.0168
0.01200.996 0.0114 0.996 0.0116
RMSE R2
0.0152
0.02080.994
0.9970.0237 0.996 0.0158 0.996
2
0.01520.996 0.0237 0.995 0.0158
R RMSE
0.995
0.9950.0208
RMSE R2
0.0195
0.0120
0.0170
0.0222
0.995
0.996
0.995
0.995
0.01950.997 0.0120 0.997 0.0170
R2 RMSE
0.963
0.9960.0222
0.9630.0099 0.997 0.0101 0.997
RMSE R2
0.1222
0.01280.996
Table 1. Average R2 and RMSE of the
estimations
of all cells using
cell A for model
construction.
100SOC
Cycles
300 Cycles
500 Cycles
700 Cycles
Cell
Name
Cell_A
Cell_A
Cell_B
Cell_B
Cell_C
Cell_C
Cell_D
Cell_D
RMSE
0.1222
0.0099
0.0101
0.0128
3.2. SOH Estimation
3.2. SOH Estimation
The most significant improvement of the new SOH model established in this study is that there
significant
of the
new
model
established
in this
that there
is no The
needmost
to use
the cycleimprovement
number N. The
SOH
of aSOH
battery
is evaluated
from
the study
on-lineismeasured
is no need
to use
the existing
cycle number
N. The SOH
of a battery
is evaluated
from theand
on-line
measured
voltage.
Most
of the
SOH models
employed
N as the
aging parameter
displayed
the
voltage.
Most
of
the
existing
SOH
models
employed
N
as
the
aging
parameter
and
displayed
the
SOH
SOH in terms of available cycles as the RUL. The cycle number is usually unavailable in real-time
in terms
of availableTherefore,
cycles as the
The
is usuallyand
unavailable
real-time
online
online
applications.
theRUL.
usage
of cycle
RUL number
was abandoned
the SOH in
was
displayed
in
applications.
Therefore,
the usage
RUL
wasFor
abandoned
and the SOH
was
displayed
in 100
terms
of
terms
of the capacity
percentage
in of
this
study.
every discharging
cycle,
there
are up to
data
the capacity
percentage
in thistherefore
study. Foran
every
discharging
cycle, therefor
are each
up tocycle
100 data
pointsand
for
points
for SOH
calculations,
averaged
SOH estimation
is taken
SOH calculations,
an averaged
SOH
estimation
forestimations
each cycle is(red)
taken
against
compared
against therefore
the real data
as shown
in Figure
7. The
fitand
the compared
real ones (black)
the real
data in
as the
shown
in Figure
TheD.
estimations
(red) fit
real ones
(black)
in the
well,
except
early
stage of7. cell
R2 and RMSE
arethe
shown
in Table
2. well,
High except
correlations
2
2
2
early
stage and
of cell
D. Rerrors
and (RMSE
RMSE are
shown
Table 2. in
High
(Rexhibits
> 0.983)
and small
(R
> 0.983)
small
< 0.04)
arein
observed
cellscorrelations
A–C. Cell D
larger
error
2 = 0.947, RMSE = 0.0458),
2 = 0.947,
errors
(RMSE
< 0.04)
are observed
in is
cells
A–C.
Cellanomaly
D exhibitsoflarger
error (R
(R
RMSE
= 0.0458),
which
due
to the
resistance
below
the first 150 cycles,
which istodue
theestimation.
anomaly of resistance below the first 150 cycles, similar to the SOC estimation.
similar
the to
SOC
Cell_A
SOH (%)
100%
80%
80%
60%
60%
40%
40%
20%
0
300
600
900
Cell_C
100%
Cell_B
100%
Experimental
Estimated
20%
0
80%
60%
60%
40%
40%
20%
600
900
Cell_D
100%
80%
300
20%
0
300
600
900
0
300
600
900
N (Cycles)
Figure 7.
7. Estimated
Estimated and
and experimental
experimental SOHs
SOHs of
of cells
cells A-D
A-D using
using cell
cell A
A for
for model
model construction.
construction.
Figure
Energies 2017, 10, 512
11 of 18
Table 2. R2 and RMSE of the SOH estimations of all cells using cell A for the model construction.
Statistical Parameters
Cell_A
Cell_B
Cell_C
Cell_D
Data points
R2
RMSE
42
0.991
0.0262
36
0.982
0.0394
39
0.985
0.0361
41
0.947
0.0458
3.3. Robustness of the Proposed Model
The results depicted in Sections 3.1 and 3.2, based on cell A as the training data, prove that the
model well predicts both the SOC and SOH status. Whether the estimation accuracy remains when
using a different set of training data is important for determining how robust the model is. Therefore,
the model equation using all of the four cells’ data, single, or multiples was constructed as well. The
obtained SOC and SOH model coefficients using the four different cells are shown in Tables 3 and 4,
respectively. These coefficients vary only slightly, whichimplies that the model is relatively robust.
Table 3. SOC model coefficients using different cells as training data.
Cell Name
Data Points
R2
a
b
c
Cell_A
Cell_B
Cell_C
Cell_D *
1808
1730
1737
1588
0.99
0.99
0.99
0.99
0.99
0.87
0.92
0.97
−0.036
−0.043
−0.039
−0.038
−2.82
−2.31
−2.53
−2.73
* Data for the first 150 cycles were removed due to an anomaly.
Table 4. SOH model and α’s coefficients using different cell(s) as training data.
A
B
C1
C2
C3
C0
Single cell
A
B
C
D
0.16
0.15
0.16
0.15
0.04
0.06
0.03
0.06
−0.82
−0.30
−0.88
−1.38
3.32
2.35
3.50
4.05
−1.48
−0.96
−1.60
−1.75
0.71
0.64
0.72
0.74
Two cells
A, B
A, C
A, D
B, C
B, D
C, D
0.16
0.16
0.16
0.16
0.15
0.16
0.05
0.02
0.03
0.03
0.08
0.04
−0.55
−0.77
−0.98
−0.44
−0.95
−1.08
2.83
3.33
3.56
2.75
3.31
3.73
−1.22
−1.51
−1.56
−1.20
−1.39
−1.66
0.67
0.71
0.71
0.66
0.69
0.72
Three cells
A, B, C
A, C, D
B, C, D
0.16
0.16
0.16
0.03
0.03
0.04
−0.57
−0.94
−0.76
2.95
3.54
3.20
−1.30
−1.58
−1.39
0.68
0.71
0.69
Four cells
A, B, C, D
0.16
0.03
−0.74
3.19
−1.40
0.69
Model Construction
Table 5 further shows the estimation accuracy when different model equations are applied for the
SOC estimation. The test data for every 25 cycles, i.e., N = 25 up to N = 700, were calculated. The first
column of Table 5 denotes the cell used for the model construction, and the remaining columns show
the estimation accuracy of all of the cells. As seen, R2 for each cell is high (>0.98), and RMSE is very
small. The results infer that instantaneous V and 1/V 0 are appropriate parameters for reflecting a
battery’s SOC. Similarly, the results of SOH estimations are shown in Table 6 for RMSE, where the left
column indicates the cell(s) used for modeling and the right columns show the corresponding RMSE.
All of the errors fall below 0.047 except for cell D, but is still within 0.054. No significant difference
in accuracy level appears by selecting one or multiple, or different cells, for the model construction.
These results provide enough proof of the model’s robustness.
Energies 2017, 10, 512
12 of 18
Table 5. R2 and RMSE of SOC estimations using different cells for model construction.
Model Construction
Correlation and Error
Cell_A
Cell_B
Cell_C
Cell_D
Cell_A
R2
RMSE
0.99
0.015
0.98
0.024
0.99
0.019
0.99
0.014
Cell_B
R2
RMSE
0.99
0.020
0.99
0.019
0.99
0.021
0.99
0.019
Cell_C
R2
RMSE
0.99
0.016
0.99
0.022
0.99
0.018
0.99
0.015
Cell_D
R2
RMSE
0.99
0.016
0.99
0.024
0.99
0.019
0.99
0.014
Table 6. RMSE of the SOH estimations using different cell(s) for the model construction.
RMSE
Model Construction
Cell_A
Cell_B
Cell_C
Cell_D
Single cell
A
B
C
D
0.044
0.045
0.044
0.043
0.047
0.043
0.044
0.042
0.043
0.041
0.041
0.038
0.054
0.051
0.051
0.049
Two cells
A-B
A-C
A-D
B-C
B-D
C-D
0.044
0.044
0.044
0.045
0.043
0.044
0.045
0.047
0.046
0.045
0.042
0.044
0.042
0.043
0.042
0.042
0.038
0.040
0.052
0.054
0.052
0.052
0.048
0.051
Three cells
A-B-C
A-C-D
B-C-D
0.044
0.044
0.044
0.045
0.045
0.044
0.042
0.041
0.041
0.053
0.052
0.051
Four cells
A-B-C-D
0.044
0.045
0.041
0.052
3.4. Simulations of Online Real-Time SOC and SOH Estimation
The derived SOC and SOH model is now applied to simulate an online, real-time estimation
in this section. The model was constructed from cell A and embedded as the inherent equations of
this type of battery and applied to all four batteries for SOC/SOH estimation in a random sampling
test. Eighty data points with N from 50 to 850 and V from 3.55–3.95 volt were randomly sampled.
Equation (10) yields the SOC evaluation, and Equations (11) and (12) follow the SOH evaluation.
Figures 8 and 9, respectively, compare the simulated real-time SOC and SOH results (red dots) to the
actual values (black lines). It is seen from Figure 8 that the estimated SOCs scatter around the diagonals.
The occurrence of larger errors, after inspection, happened at low SOH < 75%. The average errors of
random SOC estimation of the four cells are given in the first row of Table 7. The labeled “normal
range” denotes that only the data points of SOH > 75% are used and the average errors significantly
decrease. As for Figure 9, the estimated SOHs are very close to the actual values. The average errors of
the random SOH estimations of the four cells are given in the second row of Table 7. The results show
that the model built upon the data of one single cell is able to predict all batteries’ SOC within a 2.23%
error and SOH within a 3.35% error for all data ranges. The accuracy is even improved if the sampling
points were limited to the battery’s life span of SOH > 75%.
Energies 2017, 10, 512
Energies 2017, 10, 512
13 of 18
13 of 18
100%
100%
Estimated value (%)
80%
Cell_A
80%
60%
60%
40%
40%
60%
80%
60%
100%
80%
60%
60%
80%
Experimental
(%)
40%
60% value
80%
60%
80%
100%
100%
Cell_D
80%
Estimated value (%)
80%
60%
60%
Cell_D
Experimental value (%)
Cell_C
40%
40%
40%
80%
Experimental value (%)
40%
100% 100% 40%
60%
80%
100%
Estimated value (%)
100%
Estimated value (%)
Estimated value (%)
Cell_C Experimental value (%)
13 of 18
Cell_B
80%
40%
40%
100%
40% Experimental value (%)
100%
40%
60%
80%
Cell_B
100%
Estimated value (%)
100%
Estimated value (%)
Estimated value (%)
Energies 2017, 10,Cell_A
512
80%
60%
60%
40%
40%
40%
100%
60%
80%
100%
value
40%Experimental
60%
80%(%) 100%
Experimental value (%)
Experimental value (%)
Figure 8. SOC estimations (dots) and deviations from correct values (diagonals) from random
Figure 8. SOC estimations (dots) and deviations from correct values (diagonals) from random
Figureof8.
SOC
estimations
and deviations
from correct values (diagonals) from random
samplings
cells
A–D
using cell(dots)
A for model
construction.
samplings
of cells A–D using cell A for model construction.
samplings of cells A–D using cell A for model construction.
100%
Cell_A
80%
SOH (%)
SOH (%)
100%
80%
60%60%
Cell_A
Failure Threshold
Failure Threshold
Experimental
Experimental
Estimated
Estimated
Cell_B
Cell_B
80%80%
SOH (%)
SOH (%)
40%40%
100%
100%
60%60%
40%
40%
Figure 9. Cont.
Energies 2017, 10, 512
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Energies 2017, 10, 512
14 of 18
100%
Cell_C
SOH (%)
80%
60%
40%
100%
Cell_D
SOH (%)
80%
60%
40%
0
200
400
600
800
9. Estimations and actual data of SOH from random samplings of cells A–D using cell A for
FigureFigure
9. Estimations
and actual data of SOH from random samplings of cells A–D using cell A for
model construction.
model construction.
Table 7. Online SOC and SOH estimation errors of the four cells in the random sampling test.
Table 7. Online SOC and SOH estimation errors of the four cells in the random sampling test.
Parameters Sampling Data Cell_A Cell_B Cell_C
Cell_D
Parameters
Samplingall
Data
Cell_A
Cell_C 1.30%
Cell_D
1.52% Cell_B
2.23%
1.93%
SOC
normal
range
1.24%
1.17%
1.28%
1.10%
all
1.52%
2.23%
1.93%
1.30%
SOC
normal range
1.24%
1.17%
1.28% 1.69%
1.10%
all
2.13%
3.35%
2.65%
SOH
normal
1.53%
1.58%
1.72%
all range 2.13%
3.35%
2.65% 1.43%
1.69%
SOH
Note:normal
Normalrange
range included
on SOH >1.58%
75% sampling
points.
1.53%
1.72%
1.43%
4. Discussion
Note: Normal range included on SOH > 75% sampling points.
4. Discussion
From the analysis of four tested batteries, the proposed model yields accurate SOC/SOH
estimation independent of the selection of the training set and requires no model coefficient
From the analysis of four tested batteries, the proposed model yields accurate SOC/SOH
updating algorithms for the same batch of batteries. The estimation errors, however, show
estimation independent of the selection of the training set and requires no model coefficient updating
discrepancies between the different SOC levels, and among individual batteries. The suitable range
algorithms
the same
batch
ofpossible
batteries.
Theofestimation
errors, however,
showindiscrepancies
of thefor
proposed
model
and
cause
error discrepancies
are discussed
this section. between
the different SOC levels, and among individual batteries. The suitable range of the proposed model
4.1. Suitable
Range
of thediscrepancies
Proposed SOC Model
and possible
cause
of error
are discussed in this section.
The SOC model has a linear relationship with both V and 1/V′. This assumption can generate
larger errors as the SOC falls near two extremes, i.e., very low or very high because it is obviously
0 . Thiserrors
becoming
nonlinear
these regions.
The SOC
vs. both
V curve
and 1/V
estimation
in discrete
The
SOC model
hasin
a linear
relationship
with
V and
assumption
canvoltage
generate
regions
are
calculated
and
shown
in
Figure
10
for
cell
A.
The
suitable
range
(error
<
3%)
of
SOC
larger errors as the SOC falls near two extremes, i.e., very low or very high because it isthe
obviously
estimation
appears
in
the
voltage
region
of
3.55
V
to
3.95
V,
corresponding
to
the
SOC
in
40%–90%.
becoming nonlinear in these regions. The SOC vs. V curve and estimation errors in discrete voltage
Significant errors due to nonlinearity appear at very low voltages, i.e., batteries are draining off,
regions are calculated and shown in Figure 10 for cell A. The suitable range (error < 3%) of the SOC
which can reach up to 20% error as the voltage drops to 3.5 V. This suitable range seems acceptable
estimation appears in the voltage region of 3.55 V to 3.95 V, corresponding to the SOC in 40%–90%.
because an in-time charge warning to users would be more urgent than an accurate SOC estimation
4.1. Suitable Range of the Proposed SOC Model
Significant errors due to nonlinearity appear at very low voltages, i.e., batteries are draining off, which
can reach up to 20% error as the voltage drops to 3.5 V. This suitable range seems acceptable because
an in-time charge warning to users would be more urgent than an accurate SOC estimation at drain-off
stage. Similarly, the nonlinearity also causes larger error at full capacity. Therefore, this study suggests
Energies 2017, 10, 512
15 of 18
Energies 2017, 10, 512
15 of 18
at drain-off
stage.
Energies
2017, 10,
512
Similarly, the nonlinearity also causes larger error at full capacity. Therefore,
this
15 of
18
that this
derived
SOC
model
be usedSOC
for online
estimation
in SOC
the region
of voltage
study
suggests
that
this derived
model SOC
be used
for online
estimation
in the within
region 3.55
of to
3.95 volt
for
the
best
accuracy.
at
drain-off
stage.
thefor
nonlinearity
also causes larger error at full capacity. Therefore, this
voltage
within
3.55Similarly,
to 3.95 volt
the best accuracy.
study suggests that this derived SOC model be used for online SOC estimation in the region of
50%
voltage100%
within 3.55 to 3.95 volt for the best accuracy.
100%
80%
Error
SoC
Error
SoC
Cell_A
80%
60%
50%
40%
40%
30%
60%
40%
30%
20%
40%
20%
20%
10%
20%
0%
10%
0%
ErrorError
(%) (%)
Cell_A
4.00 3.95 3.90 3.85 3.80 3.75 3.70 3.65 3.60 3.55 3.50 3.45 3.40
Voltage (V)
0%
0%
4.00 Figure
3.95 3.90
3.85 3.80 3.75 3.70 3.65 3.60 3.55 3.50 3.45 3.40
10. SOC vs. V and estimation error in the different voltage range.
Figure 10. SOC vs. V and estimation
error in the different voltage range.
Voltage (V)
4.2. Comparison with
the 10.
Existing
SOH
Models
Figure
SOC vs.
V and
estimation error in the different voltage range.
4.2. Comparison with the Existing SOH Models
The SOH model derived in this paper is herein compared with the existing models. Our
4.2. Comparison
with
the Existing
SOH
Models
The
SOH study
model
derived
in this
paper
is herein
compared
the existing
models.
Our
previous
[39]
used two
aging
variables,
the voltage
(Vdis)with
and internal
resistance
(R) at
theprevious
full
SOH
derived
in the
this voltage
paper
is(Vherein
compared
with
the
existing
models.
Our
studydischarge
[39]The
used
twomodel
aging
variables,
)
and
internal
resistance
(R)
at
the
full
discharge
state,
and derived
two
types
of SOH
models,
polynomial
and
exponential.
The
present
dis
study
[39]
(Vdis
) and
internal
(R) at
theapproach
full
approach
employed
twotwo
variables,
1/V′ andthe
V, voltage
for online
SOC
and
SOHresistance
evaluation.
Figure
11
state,previous
and derived
twoused
types
ofaging
SOHvariables,
models,
polynomial
and
exponential.
The present
0
discharge
state,
and
derived
two
types
of
SOH
models,
polynomial
and
exponential.
The
present
overlapped
the
experimental
data
of
cell
A
(black
solid
line)
and
the
estimated
values
from
three
employed two variables, 1/V and V, for online SOC and SOH evaluation. Figure 11 overlapped the
approach
employed
two
variables,
1/V′
for
online (blue
SOC
and
SOHthree
evaluation.
Figure
11
different data
models—the
present
(red line)
dots),and
theV,
polynomial
diamonds),
and
the exponential
experimental
of cell A
(black
solid
and
the
estimated
values
from
different
models—the
overlapped
the squares).
experimental
data of cell A
(blackthat
solid
andmodel
the estimated
values
models (green
This comparison
proves
theline)
present
outperforms
thefrom
otherthree
two
present (red dots), the polynomial (blue diamonds), and the exponential models (green squares).
different
models—the
present
dots),model
the polynomial
(blue
diamonds),
the of
exponential
models. Most
importantly,
the(red
present
can estimate
SOC/SOH
in theand
sense
an online
This comparison proves that the present model outperforms the other two models. Most importantly,
models
(green
squares).
This
comparison
proves
that
the
present
model
outperforms
the
othertotwo
application without undergoing the full cycle test, which the previous models were unable
do.
the present
model
can estimate
SOC/SOH
in the
sense
of
an online
application
without
models.
Most were
importantly,
the
model
can
estimate
in the sense
of anundergoing
online8
Comparisons
applied to
thepresent
other three
cells
B–D,
and a SOC/SOH
similar conclusion
was found.
Table
the full
cyclethe
test,
which
previous
models
were
unablethe
to previous
do.
were
applied
to the
2 and
application
without
theof
full
cyclethree
test,models,
which
models
unable
to do.
shows
values
ofundergoing
Rthe
RMSE
these
and
the Comparisons
present
one were
apparently
has
the
2 and RMSE of
otherComparisons
three
cells
B–D,
and
a
similar
conclusion
was
found.
Table
8
shows
the
values
of
R
were
applied
to
the
other
three
cells
B–D,
and
a
similar
conclusion
was
found.
Table
8
highest correlation and the smallest error among all three models.
the values
of the
R2 and
RMSEone
of apparently
these three models,
the present
one apparently
has the error
theseshows
three models,
and
present
has theand
highest
correlation
and the smallest
highest
correlation
and
the
smallest
error
among
all
three
models.
among all three models.
Real data (Cell_A)
100%
The present model
Real
data (Cell_A)
Polynomial
model
The
present model
Exponential
model
Polynomial model
Exponential model
100%
SOHSOH
(%) (%)
90%
90%
80%
Failure Threshold
80%
Failure Threshold
70%
70%
60%
60%
0
200
0
200
400
600
800
400
600
800
N (cycles)
N (cycles)
Figure 11. Comparisons of the present and two existing models.
Energies 2017, 10, 512
16 of 18
Table 8. Comparisons of the present SOH model with the bi-variable SOH model [39].
Statistical Parameters
Bi-Variable Polynomial
Bi-Variable Exponential
The Present
Model variables
Correlation
Data points
R2
RMSE
V dis , R
−0.98/−0.98
40
0.967
0.0399
V dis , R
−0.98/−0.98
40
0.971
0.0375
V, 1/V 0
0.99
42
0.996
0.0264
5. Conclusions
The research regarding battery SOC and SOH has gained much attention during the last decade.
Numerous models have been derived from physical, statistical, and empirical approaches. Most of the
existing models express SOH in terms of the cycle life (N) as the primary aging parameter. However,
the cycle life is hard to trace because a battery in operation rarely undergoes a full charge/discharge
process. This study developed suitable parameters and a new model for in-situ SOC and SOH
estimation. Thorough exploration of the test data revealed that a new variable, the unit time voltage
drop in a discharging process, is appropriate for SOC and SOH modeling. The newly derived SOC
equation is linear with the instantaneous voltage V and 1/V 0 , and the SOH equation is linear with
1/V 0 , but has a modification factor as a function of the SOC. The proposed SOC and SOH models
showed excellent accuracy when used to estimate the same batch of four tested batteries. Simulations
of online SOC and SOH monitoring via random sampling were conducted and showed satisfactory
results. The robustness of the model was checked as well, and the results indicated that using any one
of the same batch of batteries for the model construction yielded similar accuracy for all four cells.
Acknowledgments: The authors would like to thank Michael G. Pecht and the Center for Advanced Life Cycle
Engineering (CALCE) of the University of Maryland, for providing the reliability testing data of the Lithium-ion
batteries. This research was supported by Taiwan’s Ministry of Science and Technology under the Grant Numbers
NSC 102-2632-E-131-001-MY3.
Author Contributions: Shyh-Chin Huang conceived the model and wrote the main portion of the paper;
Kuo-Hsin Tseng performed the data regression and calculations; Jin-Wei Liang and Chung-Liang Chang surveyed
the literature, wrote part of paper and interpreted the model in an electrochemical sense; Michael G. Pecht
supervised this research and provided the cells test data. He also provided many constructive suggestions and
polished the English.
Conflicts of Interest: The authors declare no conflict of interest.
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