Using Transient Electrical Measurements for
Real-Time Monitoring of Battery State-of-Charge
and State-of-Health
Jukkrit Noppakunkajorn, Omar Baroudi, and Robert W. Cox
Lalit P. Mandal
Department of Electrical and Computer Engineering
UNC Charlotte
Charlotte, NC 28223
Email: Robert.Cox@uncc.edu
East Penn Manufacturing
Lyon Station, PA
Abstract—This paper describes a transient-based approach for
estimating the state-of-charge (SOC) and state-of-health (SOH)
of a lithium-ion battery. In this methodology, a small test
signal is superimposed on top of the battery load to trigger its
transient dynamics. The resulting terminal voltage and current
are measured, and a non-linear least-squares routine is used
to estimate the impedance parameters of the battery model.
These parameter values are then passed to an H∞ filter that
estimates the open-circuit voltage while the battery is under
load. Experimental results are presented. The approach requires
minimal hardware and could be used to form the basis of a
robust on-line monitoring system.
I. I NTRODUCTION
Future energy-storage systems are likely to use lithiumion batteries because of their high energy-to-weight ratio,
minimal memory effect, and slow rate of self discharge [1].
To properly regulate efficiency and power availability in
battery-based systems, it is important to have a robust realtime monitoring system. Two metrics of interest in such
systems are state-of-charge (SOC) and state-of-health (SOH).
Real-time measurement of these quantities can be difficult,
however, because batteries are non-linear and time-varying
systems whose parameters and terminal characteristics depend
on temperature, usage history, and many other environmental
and state-based factors [1]. For instance, SOC, which is the
percentage of the initial capacity that still remains within
a battery [2], is often estimated by integrating the terminal
current. Given that efficiency and initial capacity vary with
respect to time and temperature, however, large errors may
eventually accrue. Another approach is to use look-up tables
that correlate SOC with open-circuit voltage [1]–[4]. The
difficulty with this method is determining how to measure the
open-circuit voltage when the battery is under load. Recent
work uses state-estimation tools to estimate this quantity.
These methods are applied using electrochemical models [5],
equivalent circuit models [2], and more generalized state-space
models [4], [6], [7]. References [4], [6], [7] apply a Kalmanfilter approach to several different battery models. Several of
these models assume a constant set of impedance parameters
(i.e. resistances and capacitances) and others require continuous parameter updates. Those requiring continuous parameter
estimation are computationally intensive, and the authors thus
recommend the use of known parameters estimated during a
one-time test. The latter approach has limited value in a real
operational environment, however, as temperature and other
variables have an impact on parameter values and thus affect
the accuracy of the resulting SOC estimates. Overcoming these
problems using the Kalman-filter-based methods of [4], [6],
[7] thus requires one to return to the more computationally
intensive models requiring continuous parameter estimation.
The necessary computational resources make such approaches
difficult to implement in practice.
Other researchers have proposed the use of electrochemical
impedance spectroscopy, whereby various circuit parameters
are estimated from impedance measurements captured over
a range of frequencies [8]–[10]. SOC is then estimated by
correlating the measured circuit parameter values (i.e. resistances and capacitances) with known values at various SOC
levels [8]–[10]. This approach has been found to be difficult
to apply in practice because it requires expensive, laboratorygrade measurement hardware and because of the difficulty of
tracking impedance behavior over operating conditions (i.e.
temperature, SOC, age, etc.) [10].
This paper considers a two-step approach for SOC and SOH
estimation that is designed for practical use and thus industrial
application. In the first step, battery impedance parameters
are estimated using transient current waveforms drawn by
typical loads. These loads can even be connected while the
battery is being discharged, and the estimation procedure is
an efficient non-linear least-squares algorithm. In the second
step, SOC is estimated using an H∞ filter provided with
impedance parameter values determined during the first step.
In the presence of real temperature variations, estimation errors
obtained using this method are smaller than those obtained
using the Kalman filters and adaptive methods in [3], [4], [6],
[7]. Furthermore, the measurements found with this approach
have been used to detect incipient battery faults and thus track
SOH.
The paper begins in Section II by presenting the equivalent electrical circuit model for a lithium-ion battery and by
describing the theoretical basis for the estimation approach.
Section III presents the details of the impedance-parameter
estimation process, and Section IV describes the process for
estimating SOC using an H∞ filter. Experimental results are
presented in Section V. Conclusions and directions for future
work are discussed in Section VI.
With various simplifying assumptions, Eqs. 1 and 2 can
be used to estimate the parameters R1 , R2 , and C. First, we
assume that the loads have been in steady state for a sufficient
time prior to t = 0 so that iL (t) is approximately DC. If so,
the terminal voltage at t = 0− is the steady-state solution of
Eqs. 1 and 2 with iB (t) = iL (t), i.e.
vB (0− ) ≈ VIN T − iL (R1 + R2 ) .
(3)
II. BATTERY M ODEL AND M EASUREMENTS
The literature includes numerous studies on the kinetics of
electrochemical processes in batteries [8], [11]–[14]. Equivalent electrical circuits have been developed to model these
processes for various battery chemistries (i.e. Li-ion, lead acid,
etc.). Figure 1 shows an equivalent electrical circuit model for
a Li-ion battery [15]. The capacitance C models effects arising
from polarization and the diffusion of space charges near the
electrode/solution interface [15], [16]. The resistances model
the effects of finite conductivities in the electrodes and separators, limited reaction rates at the electrodes, and concentration
gradients near the electrodes [15], [16]. All of the elements
in the simple lumped-element model, including the opencircuit voltage VIN T , depend upon temperature, the amount of
remaining active material, and other effects. Although highly
simplified, the first-order dynamics of the proposed model
match closely with experimental measurements [4], [15], [17],
[18].
Furthermore, we assume that the measurement period following t = 0 is short enough that VIN T remains constant
throughout. Additionally, we assume that the load is selected
so that it does not significantly perturb iL (t). When these
assumptions are valid, superposition leads to the following
reformulation of the model
iB
+ V
− INT
(4)
vB = vB (0− ) − iT R1 − vC .
(5)
and
Estimation of R1 , R2 , and C proceeds from the reformulated model presented in Eqs. 4 and 5. For simplicity, we use
a vector of composite parameters, i.e.
R2
R1
iT
vC
dvC
=
−
dt
C
R2 C
dvC
= µ1 iT − µ2 vC
dt
(6)
vB = vB (0− ) − µ3 iT − vC ,
(7)
and
C
+
+ vC -
vB
where µ1 = 1/C, µ2 = 1/(R2 C), and µ3 = R1 . The
parameter estimates µ̂ solve the equation
-
µ̂ = arg min rT r,
(8)
µ
Fig. 1.
The simplified series-capacitor model of a lithium-ion battery.
The mathematical model for the equivalent circuit shown in
Fig. 1 consists of the state equation
dvC
iB
vC
=
−
dt
C
R2 C
and the output equation
vB = VIN T − iB R1 − vC .
(1)
where the residual vector r is the difference between the model
response v̂B and the measured values vB . Note that vector
notation has been used because we are considering a series of
values observed at discrete time steps. The model response is
determined via simulation.
R2
(2)
In general, we are interested in estimating the unknown
parameters R1 , R2 , and C while the battery is under load.
To do so, a small test device is connected across the battery
terminals at time t = 0 as shown in Fig. 2. Note that all
of the other loads connected in parallel across the terminals
prior to t = 0 have been lumped into one element drawing an
aggregate current iL (t). The test load can be either a specially
developed device or a simple off-the-shelf element. In practice,
we have used common loads such as resistors and head lamps.
R1
iB
C
+ V
− INT
+
Load vB
+ vC iL
iT
-
Fig. 2. The test setup used to excite the battery. The battery terminal voltage
vB (t) and terminal current iB (t) are measured during testing.
III. PARAMETER E STIMATION
Impedance parameter estimation proceeds in several steps.
At the outset, initial guesses are generated from circuit considerations. A time-domain simulation then uses these guesses to
generate an estimated voltage vector v̂B . A modified GaussNewton method then determines if the parameter values can
be improved. If so, another simulation is performed using the
updated parameter values. This process repeats until Eq. 8 is
satisfied. The exact details are presented below.
A. Impedance Parameter Pre-Estimation
To generate initial guesses for the parameters µ1 , µ2 , and
µ3 , the load is first modeled as a current source connected
across the battery terminals. Basic circuit considerations are
then used to determine initial values for R1 , R2 , and C.
These values are combined to obtain guesses for the composite
parameters.
Pre-estimates for R1 and R2 are very easily determined by
considering how the capacitor steers current through the circuit
under different conditions. A load with a step-like behavior,
for instance, causes a rapid initial change in terminal current.
As this happens, the voltage across the capacitor must remain
continuous. The result is that the capacitor appears as a shortcircuit, forcing all of the terminal current to flow through it.
If the capacitor is initially discharged, an initial guess for R1
can thus be obtained by solving the equation
vB (0− ) − iR1 = vB (0+ ),
(9)
where i(0+ ) and vB (0+ ) are measurements of the current and
voltage immediately following the connection of the load. In
steady state, the capacitor’s behavior is quite different. In that
situation, the capacitor blocks the flow of DC current, thus
forcing it through R2 . An estimate for this resistance is then
generated by solving the equation
vf inal = vB (0 ) − if inal (R1 + R2 ),
−
(10)
where vf inal and if inal are the steady-state values of the
terminal voltage and current, respectively. Note that Eq. 10
must be solved in combination with the value of R1 .
To obtain a pre-estimate for the capacitance C, we begin
with the notion that a pure current step causes an exponential
response of the form
vB = vinitial − (vinitial − vf inal ) 1 − e−t/R2 C , (11)
where vinitial and vf inal are the initial and final values of the
terminal voltage. After subtracting vf inal from both sides of
Eq. 11, one obtains an expression of the form
vB − vf inal = (vinitial − vf inal )e−t/R2 C .
(12)
When the natural logarithm is applied to Eq. 12, one obtains
an expression that is linear with respect to the unknown
parameter 1/(R2 C). Linear least-squares estimation is then
used to obtain an initial value for the time constant.
The theory underlying the calculation of the time constant
can be somewhat approximate. Note, for instance, that the
test signal might contain an initial transient caused by either
thermal or electromechanical phenomena. As long as these
transients are negligible before the battery’s own homogeneous
response has completed, then the general form of Eq. 11 can
still be used to estimate the time constant. One must be careful,
however, to apply the least-squares approach to the appropriate
section of the data.
The values obtained for R1 , R2 , and C are ultimately combined to provide pre-estimates for the composite parameters
µ1 , µ2 , and µ3 .
B. Estimation
Final estimates for the parameter set µ are obtained using
the nonlinear least-squares method presented in [19]. That approach exploits residual structure to help avoid local minima.
To understand the method, one must begin by considering the
solution to Eq. 8. To obtain the minimum, the gradient of rT r
must be zero. This gradient can be written as
g(µ) = JT r,
(13)
where J is the Jacobian matrix of the residuals with respect
to the parameters. The Gauss-Newton method can be applied
to Eq. 13 to find a series of iterates µ(i) that can be evaluated
by computer to solve for g(µ(i) ) = 0, i.e.
µ(i+1) = µ(i) − [JT J]−1 JT r,
(14)
where J and r are evaluated at µ(i) [19].
Estimation problems are often difficult to solve because
the gradient g(µ) can vanish at local minima corresponding
to poor parameter estimates. This problem can be overcome
by performing minimization over telescoping intervals of data
selected by analysis of the residuals. To see this, consider the
nature of the residual vector, which is defined as
r = v̂B − vB .
(15)
In Eq. 15, vB is a series of measurements recorded at a rate
1/T . v̂B , on the other hand, consists of estimates generated
at the same time increments. The residual vector is thus a
time series that can be written as r(t). Assuming that a Taylor
series exists for the system model, and that the measurements
can be described by a polynomial in t, the k-th element of the
residual vector can be rewritten as
d2
d
rk = v̂B (0) + v̂B (0)tk + 2 v̂B (0)t2k · · ·
dt
dt
−(a + btk + ct2k + · · · ),
(16)
where tk = (k − 1)T . In this example, and in many other
non-linear estimation problems, the parameters are simply
embedded in the low-order coefficients of the series consisting
d
)v̂B (0). These
of the initial output v̂B (0) and the slope ( dt
quantities are analytically accessible for differential equation
models, such as Eq. 1. Minimization is performed over successively larger intervals, with each sub-problem selected so
that the initial terms of the Taylor series are a reasonable
approximation. By ensuring this, the likelihood of convergence
to the global minimum is greatly improved [19].
In practice, the Gauss-Newton step can possibly be eliminated and replaced solely with the current pre-estimation
process. The authors are currently exploring this approach in
a practical application, and they are finding it to be sufficient.
Further testing is required.
IV. O PEN -C IRCUIT VOLTAGE AND SOC E STIMATION
Estimation of the open-circuit voltage proceeds using the
impedance parameters determined from the methods described
in Section III. In this context, the open-circuit voltage VIN T
is viewed as a state variable to be estimated from the noisy
terminal measurements. The system model and observations
presented in Eqs. 1 and 2 are written here in state-space form
as
dx1 µ
−µ2 0 x1
dt
+ 1 iB
dx2 =
0
x
0
0
2
dt
(17)
x1
+ −µ3 iB ,
y = −1 1
x2
matricies are selected based on knowledge of the problem at
hand. The solution minimizing Eq. 21 can be found in the
literature [20], [21] and is not repeated here. Computationally,
the approach is very similar to that of a Kalman filter.
V. E XPERIMENTAL R ESULTS
There are a number of potential operational paradigms for
the parameter estimation/H∞ filter combination. In most stateestimation schemes, for instance, one continually estimates
VIN T and thus SOC. In order to deal with the need to update
parameters regularly, we have elected to periodically perform
parameter estimation and then apply the H∞ filter. We do
so by injecting a short current pulse during times when the
current is sufficiently constant. Parameters obtained from the
non-linear least-squares scheme are passed to the H∞ filter
which then attempts to determine the open-circuit voltage. A
look-up table such as the one plotted in Fig. 3 is then used
to determine the SOC. Temperature-dependent look-up tables
are available [22].
where x1 = vC and x2 = VIN T . Note that the opencircuit voltage is defined to remain reasonably constant over
small intervals. For real implementation, these relationships
are discretized at time tk as
xk+1 = Fk xk + bk iB,k + ωk
In this instance we choose to estimate the unknown states
using an H∞ filter because of its superiority over the Kalman
filter when measurement and model errors are difficult to
quantify [21]. In this problem, uncertainties include the effects
of temperature, aging, discharge rates, and other structural and
environmental factors. Moreover, these effects can change over
time.
In the H∞ filter problem, we seek to find estimates x̂k that
minimize the objective function
k=0
kx0 − x̂0 k2P−1 +
0
kxk − x̂k k2Sk
PN −1
k=0
(kωk k2Q−1 + kυk k2R−1 )
k
(21)
k
where P0 , Qk , Rk , and Sk are user-supplied symmetric
positive-definite weighting matrices and N is the total number
of samples. In many ways, the first three of these matrices are
analogous to the covariances of the initial estimation errors, the
process noise, and the measurement noise, respectively. Unlike
the Kalman-filter formulation, however, these quantities are assumed to be unknown and appropriate values for the weighting
INT
,V
3.5
3.4
V
where υk and ωk are stochastic noise terms representing
measurement noise and model uncertainties. The matrices Fk
and bk are [20]
1 − µ2 T 0
Fk =
(19)
0
1
µ1 µ2 T 2
bk = µ1 T − 2
.
(20)
0
J1 =
3.6
(18)
yk = [−1 1] xk − µ3 iB,k + υk ,
PN −1
3.7
3.3
3.2
3.1
0
10
20
30
40
50
SOC, %
Fig. 3. The relationship between SOC and the open-circuit voltage. Note
that the open-circuit voltage is equivalent to the internal source VINT .
Figure 4 shows the experimental setup used for testing the
proposed methodology. The battery used is an 18650-type
lithium-ion cell, and loads are connected to it using solidstate relays. Hall-effect transducers sense the terminal voltage
and current; an LM35 measures the ambient temperature. To
resolve small changes in the terminal voltage, a differential
amplifier removes the offset and amplifies the difference.
All of the data streams are sampled at 100Hz using a PCI1710 data-acquisition card. Tests are performed at various
temperatures using a custom chamber.
Figure 5 shows voltage and current data recorded during a
typical test with a constant base load. Note that the base load
is connected long before the test excitation source. Three tests
are included in Fig. 5. Note that each involves the connection
of a 1A test load. Voltage and current measurements recorded
after the connection of the test load are extracted as shown in
Fig. 6. Included in this figure are both voltage measurements
and predictions based on the estimated parameters. Note the
close fit between the measured data and the model estimates.
Once parameter estimation is complete, voltage and current
data such as that shown in Fig. 6 is passed to the H∞
filter which then estimates the open-circuit voltage VIN T . For
Solid-State
Relay
Fuse
Solid-State
Relay
+
Battery
Test
Signal
Load
Current
Sensor
validation purposes, we unloaded the battery several seconds
after the pulse and measured the ensuing rest voltage. Figure 7
shows results obtained when the true open-circuit voltage was
approximately 3.28V. Two traces are shown. The upper trace
shows the evolution of the estimated open-circuit voltage,
which converges to approximately 3.25V about 40 samples
after the start of the pulse. This trace was computed after
first estimating a new set of parameters. Note that the error
is approximately 1%. The lower trace was computed using
the impedance parameters from the previous test. Note the
correspondingly larger error. This type of variation is expected,
as the temperature was increased slightly between pulses. Such
variations occur in practice and can be caused by actions such
as vehicle warm-up, rapid charging, etc.
Voltage
Sensor
Reference
Voltage
Computer
+
Temp
Sensor
Fig. 4.
Block diagram of the measurement system.
Voltage (V)
4.2
4.1
4
3.9
Current,A
3.8
0
4
20
40
60
80
100
120
140
160
180
2
0
−2
0
20
40
60
80
100
Time,Sec
120
140
160
180
Fig. 5. Top trace: The measured terminal voltage during a discharge from
a 18650-type lithium-ion battery. Bottom trace: The corresponding current.
Note that the test load is connected three times during the latter portion of
the experiment.
3.95
Voltage,V
Fig. 7. Upper trace: Evolution of the estimated open-circuit voltage using a
new set of estimated impedance parameters. Bottom trace: Evolution of the
estimated open-circuit voltage using the previous set of impedance parameters.
Note that both estimates converge within 40 samples after the pulse, and that
the error is significantly lower using the updated impedance parameters.
3.9
3.85
Current,A
122
122.5
123
123.5
124
124.5
125
2
1
0
122
Parameters obtained using the above method have also been
investigated for health monitoring. Figure 8, for instance,
shows typical R1 values versus SOC for the same battery at
two different operating temperatures and two different states
of health. Note that when the operating temperature rises, the
resistance R1 clearly increases. We currently attempt to track
the state-of-health by looking for deviations from expected
resistance values at a given operating condition. In this case,
we use expected values for a given temperature at 80% SOC.
To demonstrate the potential of this approach, we passivated
a cell by storing it at 60◦ C for several days. Figure 8 shows
that passivation causes R1 to increase over its expected value
at a given temperature and SOC. Methods such as the particlefilter scheme proposed in [23] could use this data to predict
the corresponding degradation in the remaining useful life. In
any event, it is clear that the methodology provides a practical
mechanism for parameter tracking.
VI. C ONCLUSIONS
122.5
123
123.5
Time,Sec
124
124.5
125
Fig. 6. Experimental results. Top trace: Measured terminal voltage (dashed
line) and estimated terminal voltage (dots). Bottom trace: Corresponding
current.
AND
F UTURE W ORK
This paper has demonstrated the effectiveness of the proposed parameter extraction process. Ongoing work is aimed at
developing a practical system that uses the parameter values
to determine SOC and SOH in real time. Initial work suggests that the parameter estimation process can be performed
without the Gauss-Newton step, which further simplifies the
process.
ACKNOWLEDGMENT
This work was supported in part by NSF grant CNS1117790.
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