state-of-charge estimation on lithium ion batteries - mori w yatsui
TRANSCRIPT
Kalman Filter Based State-of-Charge Estimation for
Lithium-ion Batteries in Hybrid Electric Vehicles
Using Pulse Charging Mori W. Yatsui, IEEE Member and Hua Bai, IEEE Member
Department of Electrical and Computer Engineering, Kettering University, Flint, Michigan USA
hbai, [email protected]
Abstract— The battery is one of the most important energy
storage components in EV/HEV. Failing to estimate the state
of the charge accurately will bring the risk of overcharge or
over discharge. The traditional Coulomb counting method will
bring accumulated error over time, therefore high deviation
occurs between the estimated and real state of charge.
Different estimation strategies are compared in this paper, i.e.,
Coulomb counting method, open-circuit-voltage method and
Kalman filter based state of charge estimation. Experimental
results validate the effectiveness of Kalman filter during the
on-line application.
Keywords-component: Lithium-ion Battery, State of Charge,
Plug-in Hybrid Electric Vehicle, Coulomb Counting Method,
Open-circuit-voltage Method, Kalman Filter.
I. INTRODUCTION
Practical mobile systems, such as Plug-in Hybrid Electric Vehicles (PHEVs), are designed to be of most use in harsh environments. The conventional systems used in present vehicles substantially rely on energy storage systems (ESSs), which typically contain a battery pack such as a Lithium-ion (Li-ion) galvanic system. It is of the utmost importance that the battery be stable within its rigorous environment for practical and safe use. A great factor determining the stability of Li-ion battery packs lies within the state-of-charge (SOC). The SOC is dependent on the open-circuit-voltage (OCV), chemistry make-up, number of cells, temperature, age, cycle history, damage, humidity, etc. Failing to predict battery SOC will cause overcharge or over discharge during cycles, which potentially will bring irreversible permanent damage to the battery cell. Damage to the cell reduces battery cell performance and cycle life, therefore posing necessary cost to the customer.
The automotive market has shifted towards electronically-biased vehicles within the last two decades, thus thrusting research into the electrical domain and enforcing the knowledge economy behind battery performance. These advancements will lead to great steps towards software modeling and simulation for batteries and further understanding of battery interfacing and lifetime properties. The most accurate method to measure the SOC is the direct relation of OCV, however this can only be accurately measured offline when the battery has no current transfer. Traditionally, the most commonly applied online SOC estimation is the coulomb counting (CC) method [6], which measures the battery current, integrates it with represent to time, and determines an estimated battery SOC. When the measured battery current has some error, the long-run integration will bring unpredictable deviation of SOC, which will lead to the overcharge or undercharge of the battery. Also, the CC method does not compensate for the
capacity loss over cycling and the temperature. The OCV method and Kalman Filter (KF) are proposed to enhance the battery SOC estimation [8]. Our implementation for the KF will observe the measurement inputs, terminal voltage and current, along with the output measurements of the CC method to better compensate for the non-ideal factors that play a role in the batteries characteristics over time. The OCV can only act as the off-line detection after reasonable chemical settling time, which will depend on battery manufacturing. Thus, the KF must be relied on to reduce the error of CC online, however previous literature still focuses on the constant current or intermittent constant current charging. Pulse charging as shown in this experiment can use the OCV during the offline mode to verify the KF’s validation of operation. This OCV will be measured after the internal impedance is discharged. The effectiveness of the KF in the pulse charging control, which is believed healthy for the battery, still needs be investigated personally [8].
This paper will first present the battery model used for SOC estimation of the selected Li-ion galvanic battery. Secondly, the construction of the OCV, CC, and KF computational algorithm will be presented. Section IV will present the comparison of different methods we used to identify the SOC, along with posing the experimental validation of KF in the pulse charging process. Section V is the conclusion.
II. MODEL OF THE LITHIUM-ION BATTERY
Battery modeling is a crucial development stage for PHEV electrical simulations. Models have been created for studies in different applications with Pb-A, Li-ion, and Ni-MH batteries [2,3]. The present concern with these models in Li-ion is to create a dynamic model for the battery’s SOC, which is directly related to the battery capacity and also dependent upon the cell’s chemical material. The capacity is the quantity of charge delivered by under a set of conditions, defined as Ah or Wh, which are regarded as a time integration of the current and defined as Peukert’s capacity.
Li-ion batteries differ from Nickel-Metal Hydride and Lead-Acid batteries in that these are unable to deliver considerable charge within the entire voltage span. Each Li-ion cell potential is recommended to remain between 2.7 V and 4.15 V with the nominal voltage varying between 3.2 V and 3.8 V depending on the type of Li-ion cell. The failure to follow these recommendations will result in an unserviceable battery that has a large voltage drop when loaded caused by high internal resistance and low self-discharge resistance. This is due to crystalline solids, which form within the cells.
A simulation software program called ADVISOR
developed by NREL Center for Transportation Technologies
and Systems in 1994 uses Matlab-Simulink to run
simulations of battery models. The ADVISOR battery model
shows an equivalent circuit similar to Thevenin’s circuit for a
battery model. It uses an internal resistance to account for the
potential drop from its off-line to on-line application. The
resistance is dependent of SOC, temperature, and current
flow direction. This method neglects the self-impedance,
responses, and proper battery equivalency to run simulations.
Typically, this information is not readily available from
battery datasheets [6]. The simplicity of the simple battery
model compiles the internal resistances into one value and
allows the OCV to be measured immediately upon
disconnecting all sources or loads to the battery. The
impracticality is that the simple battery model does not focus
on self-discharge and actual reactions seen in battery tests. In order to properly treat the given scenario, the idea of a
battery must merge with reality. A capacitor is a source of energy that does not allow its voltage change instantaneously. The merging of ideals to reality for the battery uses the capacity to approximate an equivalent capacitance. Each cell within a battery packaging will have connection resistance, and the chemical polarization of ions cause a transient reaction given a charging unit-step as seen in Fig. 1. Based on stated information, the selected battery model to be used within this study is shown in Fig. 2. with a consistent environment of 25°C with current passing into the positive terminal.
The lithium-ion battery cell is modeled as a large
capacitor Cb to enable the simulation of the state-variables.
Rs represents the summation of contact resistances within the
battery pack, while Rp
and Cp represent chemical
polarization resistance and the chemical polarization
capacitance, respectively. The discharge resistance which
degrades over time is represented as Rd. The equations
stated in this paper are for charging, thus discharging must
have a negative current.
III. COULOMB COUNTING, OCV, AND KF
The Li-ion battery pack selected for this study was the
Valence U1-12XP with an approximate nominal capacity of
40 Ah and a voltage of 14.6 V when SOC reaches 1. The
manufacturer states the battery maintains an internal
resistance less than 15 mΩ. This understanding is based
from a test done to find Rs based on the immediate voltage
increase as a charge is placed. The manufacturer’s internal
connection resistance was tested with pulse charging with
resting periods of 8 seconds, however the battery resistance
becomes significant as SOC approaches 0 as discussed in
the results section.
For the CC method, the time to charge and discharge
follows the general Peukert equation
k
QIt−
−= , (1)
where Q is the Peukert capacity per ampere charge/discharge and k is the ideality factor due to side chemical reactions. The ideality factor of an ideal battery is stated as k=1 [7].
For OCV method, experimentally we could determine
the relationship between OCV and SOC using CC. Varying
the instantaneous charging current will lead to the change of
terminal voltage [8]. Imposing ICHRG1 and ICHRG2 results in
different terminal voltages
222
111
BATOsCHRG
BATOsCHRG
VRIV
VRIV
+×=
+×=
,
(2)
respectively, where VBATO is the real voltage of the battery
and Rs is the internal resistance of the battery. In the case
where VBATO1 and VBATO2 are the same, the internal resistance
of the battery is
21
21
CHRGCHRG
sII
VVR
−
−=
.
(3)
Acquisition of the battery resistance could let the
charging system determine the real battery voltage from the
terminal voltage in the simple battery model. In order to
overcome the voltage loss across the battery resistance, the
targeted apparent terminal voltage (V) should be set as the
actual battery voltage (VBATO), or said to be the OCV, plus
the voltage drop, i.e.,
BATOsCHRG VRIV +×= (4)
Based on the identification of VBATO, the SOC could be
identified with the curve of SOC and OCV, hence the OCV
method.
According to Coulomb counting principles, it does not
contain any reference point and can suffer from long-term
drifting due to error introduced from system noise and
measurement noise. The value of system error can also be
introduced as the differentiating Peukert capacity and
Fig. 2. Schematic of the complex battery (top) and the complex battery
model (bottom).
Fig. 1. Terminal voltage during charging unit-step pulses showing the
desired model characteristics.
Peukert ideality coefficient as the battery’s history
elongates. CC must be reinitialized with an accustomed
Peukert capacity and Peukert coefficient from a newly
acquired log-log chart from battery tests. Currently, there is
no correction for this within the CC simulation and it would
be difficult to find more recent values every few cycles.
Implementation of the KF requires a continuous time
observer to give a prediction to the filter itself for this
simulation, which is called a Kalman-Bucy filter. For
estimation of the true battery OCV, the observer will read
the current, terminal voltage, and the SOC from the CC-
OCV estimation in order to output a value. This is done
using an input of a stochastic state-space model, which
describes an estimation of the battery’s parameters to
estimate the voltage across the battery’s equivalent
capacitance and polarization capacitance. Solving the KCL
and KVL equations,
p
Cp
CpR
vii += (5)
CpCbtS vvviR −−=
,
(6)
respectively, gives system equations that result in the
necessary dynamic equations () = () + () + ()
and () = () + (). The dynamic system is
finalized in the form
)(1
0
01
10
11
tsi
v
C
CR
v
v
CR
CRCR
v
vt
p
bS
Cp
Cb
pp
bSbS
Cp
Cb+
+
−
−−
=
&
&
(7)
)(
0
1)( tm
v
vty
Cp
Cb+
=
,
(8)
where )(ts and )(tm represent system noise and
measurement noise varying in respect to time, respectively.
The observer estimates the rate of change with respect to
time and solves for the state variables, Cov and
Cpv . Then,
the observer calculates the gain of the system in order to
achieve the desired output.
The Kalman filter is used to filter out noise,
measurements, and other inaccuracies resulting in values
closer to real values in dynamic linear systems. Upon
reception of the input values, the Kalman filter estimates the
uncertainty of the values and computes a weighted average
between the measured value and the predicted value. It
compiles all noise influences, even those unconsidered
within this paper, such as thermal differences. The
uncertainty is qualified by using the best linear depiction of
the input value, which allows for cancellation of noise by
narrowing the sensor and noise values to a minimum. The
weighted average is calculated placing heavier weights on
those values that are more likely, which is determined by the
covariance on the uncertainty. The filter then feeds back
and recursively uses prior prediction to determine the new
best guess at each time step. This functionality gives it the
title of an adaptive filter for digital signal processing.
Then, the KF uses a linear-quadratic regular in
combination with a Gaussian controller to form a linear-
quadratic-Gaussian controller, or LQG, to estimate the state-
space variables within our linear dynamic system. The cost
function is used to share the control input history and is
stated as
[ ]
++= ∫T
dttutRtutxtQtxTFxTxEJ0
)()()(')()()(')()('
,
(9)
where F, Q, and R are functions of time and are greater than
0 given ttty <≤ '0),'( and T is the final simulation time.
The LQG controller calculates
[ ]
[ ]
)(ˆ)(
)0()0(ˆ
)(ˆ)()()()(ˆ)(ˆ
txLtu
xEx
txtCtyKtButxAtx
−=
=
−++=&
,
(10)
where K is the Kalman gain, L is the feedback gain, and E is
the average value, or the expectation value.
The Kalman gain is described as
( )( )
)(')()(
)0(')0(),(),(,,)(
1tWCtPtK
xxEwWvVCAftK
−=
=
,
(11)
and to find P(t)
( ))0(')0()0(
)()()(')(')()()( 1
xxEP
tVtCPtWCtPAtPtAPtP
=
+−+=−&
.
(12)
The feedback gain is described as
( )
)()(')()(
),(),(,,)(
1tStBtRtL
FtRtQBAftL
−=
=
,
(13)
and to find S(t)
FTS
tQtStBtBRtSAtStSAtS
=
+−+=−−
)(
)()()(')()()()(')(1&
.
(14)
The dynamic matrix equations for P(t) and S(t) are defined
as Riccati differential equations. Individually, the first
matrix Riccati differential equation solves the linear-
quadratic estimation, whereas the second matrix Riccati
differential equation solves the linear-quadratic regulator.
Appending these calculations to the stochastic state-space
model allows for the LQG control algorithm to be solved,
resulting in a data reduced SOC output [8].
Referring to the previous section, CC, the battery’s
Peukert capacity and Peukert ideality coefficient would now
be considered part of the system error. Thus, the Kalman
filter adjusts for the constantly changing Peukert values and
noise given from the sensors and system.
IV. RESULTS AND DISCUSSION
The pulse charging method made the internal Ohmic
impedance available to determine. It is concluded that Li-
ion battery internal Ohmic resistances are not constant
through all potential ranges, but are relatively within
tolerance range as can be seen in Fig. 3 and Fig. 4. A failed
battery is easily noticeable from Ohmic resistance testing
shown in Fig. 5. However, it is very likely that every
manufacturer holds different resistive properties depending
on battery architecture and materials, and these differences
must be taken into account.
The pulse charging method also determines the OCV vs.
SOC and compares it to the manufacturer’s specifications,
shown in Fig. 3. The test differentiation is caused by
deterioration of the Peukert capacity.
The results determine a difference between the
manufacturer’s and the two trial runs mostly likely due to
the Peukert capacity and ideality coefficient altering over
the battery’s short history or differences in the
manufacturing process. The internal Ohmic resistance
remains below 15Ω until the SOC reaches the overcharge
regions, where the impedance increases drastically to
infinite. It is concluded that the Li-ion SOC performed
properly with a linear depiction with respect to time.
The CC simulation was implemented to accumulate the
SOC measured based on Peukert’s ideality equations. The
automated simulation was developed in National
Instruments’ Labview 2009 SP using a National Instruments
myDAQ, a portable data acquisition unit.
The simulation, shown in accurately models long-term SOC
with introduced error, but is compensated for in the KF
algorithm. The KF was implemented to correct the CC SOC’s error
and result in a more “true” value with consideration to noise in the system. This is done using the algorithm stated previously. Several tests were done to observe the influence of induced scenarios or errors on the OCV-CC-KF system.
The first test was done through Labview to implement a random signal generation in the readings to simulate large noise measurements in an automotive application. The test was performed in both charging and discharging scenarios. The result was a precise but less accurate system by a magnitude of approximately ±0.14% weighted mean.
The second test was done through automated pulse charging and data acquisition to determine the difference of SOC values. The KF SOC was measured during the charging moments and the OCV was measured during the idle moments given an 8 second rest prior to reading. This test was performed in both charging and discharging scenarios. The results of this test were in agreement with the research done by Smith, Rahn, and Wang on pulse charging of lithium-ion batteries [8]. According to the data found, the KF SOC estimation had an error / tolerance of ±1.76% in comparison to the OCV method estimation, which was calculated to be statistically insignificant.
The third test was done through Labview to implement a constant DC offset on the KF inputs to determine the correction abilities for large CC estimation error. The DC offset implemented to each of the KF inputs results in a percent error as shown in Table 1.
Fig. 6. Labview schematics for CC SOC estimation for cycling.
Fig. 5. Internal connection resistance testing at various currents of a bad battery (experimental data using OCV).
Fig. 4. Internal connection resistance testing at various currents of a good
battery (experimental data using OCV).
Fig. 3. SOC CC-OCV estimation method verification of Peukert’s ideality
principles.
Table 1 shows the dependency of the KF on specific inputs
and these values are well understood to be results of the
weighting average principles behind the functions.
The Labview 2009 code was proven to be useful during
this process. Fig. 7 and Fig. 8 show estimation of the SOC
from OCV, OCV-CC, and OCV-CC-KF accumulated
methods at a low SOC and a high SOC using the original
prototype algorithm. Fig. 9 is the newest prototype using
Labview for pulse charging and discharging. This
representation allows us to easily view the battery SOC
during online testing and verification with OCV SOC
detection.
V. CONCLUSION
This paper shows the implementation and results of
combining open circuit voltage, Coulomb counting, and
Kalman filter methods in order to more accurately estimate
the SOC of Li-ion battery cells by various factors into
consideration. The combination of the methods accurately
estimates the SOC with an error of ±1.76% in comparison to
the OCV method estimation. This error may vary
depending on hardware used for the data acquisition. This
set up was done with a National Instruments myDAQ. This
error tolerance was calculated to be statistically insignificant
and therefore usable. The KF observes the terminal voltage
rate of change as well as discharging or charging currents to
calculate electrode surface dynamics, electro-chemical
reactions, and electrode particle transfer, along with other
side reactions. In addition to the rate of change, it also
receives the output of SOC from the CC method to “edit”
the needed values for the filtered SOC output. The tests in
this paper used the single polarization impedance model
shown at the beginning of this paper in Fig. 2, called the
complex battery model assembled within this study.
VI. RECOMMENDATIONS
It is recommended to enhance the KF processing
capabilities by incorporating thermal differentiation and
capacity differentiation. This is an essential part of our goal
to produce an overall battery management system for
PHEVs. This technology can be additionally supportive
within small electronic devices, which use an ESS.
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[5] NREL, “ADVISOR HEV Simulation Model,” 1996.
[6] X. He and J.W. Hodgson, “Modeling and Simulation for Hybrid Electric Vehicles-Part I: Modeling,” IEEE Trans. Intelligent TransportationSystems, 3.4 December 2002.
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[8] K. A. Smith, C. D. Rahn, and C. Y. Wang, Model-based electrochemical estimation and constraint management for pulse
operation of lithium-ion batteries: IEEE Trans. Control Systems Technoloy. 18.3, May 2010.
Fig. 9. Second prototype code using pulse charging and discharging.
Fig. 8. Pulse charging at stage 2 constant voltage for SOC accuracy
estimation. Note that charging current is negative.
Fig. 7. Pulse charging at stage 1 constant current for SOC accuracy
estimation. Note that charging current is negative.
TABLE 1
SOC ERROR FROM FORCED DC OFFSETS IN INPUTS OF CODE.