Michael Knauff, Jeffrey McLaughlin, Dr. Chris Dafis,
Dr. Dagmar Niebur, Dr. Pritpal Singh , Dr. Harry Kwatny, and Dr. Chika Nwankpa Simulink Model of a Lithium-Ion Battery for the
Hybrid Power System Testbed
ABSTRACT
This paper investigates the identification of model parameters for a Simulink model of the 60Ah Lithium Technology Corporation Lithium-ion battery ud in the hybrid power systems testbed. Two experimental tests of the battery are prented, along with a method for deriving battery model parameters using the tests. A comparison between test data and simulation results shows a high degree of accuracy in the model.
1.INTRODUCTION
The Hybrid Power System Testbed is a small scale hardware demonstration currently being asmbled at NAVSEA Philadelphia, that will combines veral emerging technologies, and provides a means to experiment with advanced power management schemes, such as that described in (Kwatny et al. 2005)
.
The testbed consists of a variety of power sources and loads interconnected via a DC bus. The power sources include a Lithium-ion (Li-ion) battery and a diel generator, with additional sources being considered for future implementation (ex. fuel cell). The loads consist of a rim-driven propulsion motor, a power processing unit capable of emulating a wide variety of loads, and two permanent magnet machines in a motor/generator configuration ud to dissipate excessive power beyond the capabilities of the other two loads. The sources and loads are connected to the DC bus via veral power electronic building block modules (Ericn, Hingorani, and Khersonsky 2006).
In order to gauge the physical interaction of the developmental components during operation, a model of the testbed was created in Matlab/Simulink. The overall Simulink model of the testbed is described in greater detail in (Knauff et al. 2007). This paper provides a detailed look at the model of the testbed’s Li-ion battery. It also discuss in detail the derivation of the model parameters via testing of the physical device.
2.BATTERY MODEL
A variety of models exist that predict battery behavior to varying degrees of accuracy. A good overvi
ew of the different model types available is prented by Singh and Nallanchakravarthula (2005) The available models differ in both complexity and in the nature of the tests necessary to implement the models. The model lected for the testbed was propod by Chen and Rincon-Mora (2006). It was chon for its relative simplicity, including the advantage that a straightforward test is given to derive the associated model parameters.
This model is provided in terms of a circuit diagram. A slightly modified version of the circuit is ud to model the Li-ion battery which is shown in Figure 1. The model consists of two parate circuits linked by a voltage controlled voltage source and a current controlled current source. One circuit reprents the overall capacity of the battery, while the other circuit models the internal resistance and transient behavior of the battery using a ries resistance and two RC circuits.
The voltage controlled voltage source linking the two circuits is ud to reprent the non-linear relationship between the State of Charge (SOC) and the open circuit voltage (VOC) of the battery. This relationship is normalized such that when the voltage across C CAP is 1 V, the battery is at 100% SOC.
The modified model does not include a lf-discharge resistance, becau the model was not intend
ed to simulate long term behavior in which this resistance would be meaningful. Furthermore, the effect of temperature on the
battery performance has not been accounted for in this version of the model due to the fact that the battery is expected to operate in a relatively narrow range of temperature conditions.
The circuit diagram in Figure 1 was implemented in Simulink by first finding an equivalent ordinary differential equation (ODE) describing the above circuit. The equation
1
1
1
1
1
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00
0()0
00()()CAP
TS
TS
TS TL
TL
TL
S C
R C C R C C
g x x x R −−−−−−=−+−−−=+++⎡⎤
⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣
⎦⎣⎦
x
x u
y u
(1) describes the circuit diagram of Figure 1, where
R TS and C TS are the resistance and capacitance in the shorter time constant RC circuit, R TL and C TL are the resistance and capacitance in the longer time constant RC circuit, C CAP reprents the overall capacitance of the battery, R S is the ries resistance, and g (x ) is the non-linear function which maps SOC to VOC. The state vector x reprents the voltages across C CAP , C TS , and C TL . The input u is the current entering the battery, and the output y is the voltage across the battery terminals. As noted above SOC is reprented by the voltage v SOC across C CAP which ranges from 0 to 1 V, reprenting 0% to 100% battery capacity respectively.
This equation was then implemented in Matlab/Simulink as shown in Figure 2. The block diagram consists primarily of standard Simulink blocks. The overall testbed model was implemented using
veral Matlab/SimPowerSystems blocks which do not u the standard Simulink signal type connections. It was therefore necessary to ensure compatibility of the battery model with the rest of the model. To do so, a voltage source and current measurement block were ud to
interface between the signal-bad battery model and the electrical type ports ud by the SimPowerSystems blockt. It should also be noted that the ries resistance was implemented as a SimPowerSystems resistance element. Doing so prevents the Simulink model from generating errors due to algebraic loops in the system.
The Simulink model of the battery was completed prior to the arrival of the actual battery at NAVSEA Philadelphia. A rough model of the testbed was needed to gain insight into the expected magnitude of the voltages and currents prent in the system.
The known information provided by the manufacturers included a maximum cell voltage of 4.2 V, capacity (60 Ah), number of cells (33 in ries), information describing the battery management system, and other details of the battery asmbly.
In order to create a preliminary model veral parameters were needed. The manufacturer’s data provided some of the information on how to obtain the parameters, but the transient behavior and i
nternal resistance of the battery could not be obtained from this information. The cell voltage and discharge curves prented in (Chen and Rincon-Mora 2006) correspond to the performance of a typical Li-ion cell (Linden and Reddy 2002). For this reason the parameters given in this paper were adopted for
the initial model.
FIGURE 2. Simulink Battery Model in Matlab/Simulink
R R C CAP R V FIGURE 1. Battery Circuit Modellarva
3.DERIVATION OF VOC–SOC
RELATIONSHIP
Once the actual battery arrived at NAVSEA Philadelphia, explicit model parameters had to be derived and employed in the model from battery tests. The first characteristic of the model to be experimentally derived was the non-linear function relating SOC to VOC.
To find an approximation for this function, a constant resistance discharge of the battery over
a safe discharge cycle was conducted. The battery is equipped with a battery management system (BMS) which, among other functionalities, automatically disconnects the battery terminals when the battery’s cell voltages are below a critical voltage level. This ensures that no permanent damage is inflicted on the battery. The discharge was executed until the point at which this mechanism triggers.
you should get over me
The tup ud for this test is shown in Figure 3. Voltage and current at the battery terminals were monitored internally by the battery management system (BMS). This data is sampled once per cond, and is made available via an RS232 port on the front of the battery asmbly. Data was acquired using a rial connection and was stored as a text file for later analysis.
Upon completion of the test, the state of charge over the entire test period was found using the current data obtained from the test. Equation (2) was obtained from (1) and ud to perform this calculation.
1
()(0)()
SOC SOC
CAP
v t v i d
C
τ
ττ
=−∫ (2)
Note that v SOC (0) should theoretically begin at 1 V for a fully charged battery. However, the measured voltage in each battery cell was roughly 4.09 V, while the manufacturer’s specifications listed a maximum cell voltage of 4.2 V. For this reason v SOC(0) was calibrated to be 0.9 V rather than 1 V.
The discharge-current data was numerically integrated (trapezoid rule) according to equation (2) yielding the SOC for each data point in the test. This data was then implemented in the model using an 11 point lookup table providing the corresponding values of VOC at the SOC values 0.0, 0.1, 0.2,…
,1.0. A VOC value of 0 V was ud for the 0.0 SOC value since the battery was not discharged to this point. Similarly a value of 138.6 V was ud for 1.0 SOC corresponding to the manufacturer’s data for a fully charged cell multiplied by 33 cells.
After implementing this lookup table in the model, a simulation was ud to demonstrate the improvement in this model compared with the initial model described above. Figure 4 shows a comparison between the actual data recorded during the discharge test, simulation results using the initial model, and simulation results for the refined model.
FIGURE 4. Battery Discharge Model Comparison
Blue – Experimental Data, Green – Initial Model
Red – Refined Model
gentlenessFIGURE 3. Battery Test Setup
4.ESTIMATION OF RC AND SERIES
RESISTANCE PARAMETERS
After the SOC-VOC relationship had been derived the remaining model parameters were identified. A practical test for deriving the parameters is discusd in (Chen and Rincon-Mora 2006). The test involves a ries of constant current discharge periods intersperd by a ries of rest periods i
n which no current is drawn from the battery. Transient behavior of the battery is obrved during the rest periods. This allows the model parameters to be derived at veral points throughout the discharge cycle.
A similar test was conducted with one slight modification. Rather than using a t time interval for each pul, the battery was discharged during the nine parate discharge cycles until a specific voltage v SOC was reached. Specified values were 0.9, 0.8, 0.7, …, 0.1 SOC. In the test tup shown in Figure 3 the resistive load was replaced by a programmable load capable of discharging the battery by drawing constant current. The rating of the available programmable load significantly limited the magnitude of the discharge current that could be applied in this test. The battery was therefore discharged for a significantly longer period of time compared to the test using the resistive load.
The load was programmed to toggle the current between 3.6 and 0 A bad on a ur trigger command. Each time the terminal voltage reached one of the predefined levels, the load was toggled and allowed to rest for approximately 25 minutes.
The battery was discharged down to a level of about 0.1 SOC and allowed to rest for one last transie
nt period. Again, the battery was not fully discharged to avoid any potential damage to the battery. Throughout this process the current and terminal voltage of the battery were obtained via the RS232 output of the battery’s BMS.
After obtaining the test data the nine rest periods were parated and individually analyzed to obtain the model parameters. A technique for deriving the parameters is described by Schweighofer, Raab, and Brasur (2003). However this technique is not completely automated becau it requires some guidance in
lecting the two transient periods corresponding to the two RC circuits.
An alternative approach was instead taken which
utilized the Matlab Curve Fitting Toolbox.
Looking at the end of one discharge period at
the instant when the discharge current is turned
off, R S can be found by making the assumption
that the state vector remains constant in the
vicinity around this time instant. Equation (1) is
ud to derive the equation
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123
(()
(()
)
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S C
t g x x x R
t g x x x
i
−
+
=+++
=++
y
y
(3)
where y(t-) is the terminal voltage prior to
turning off the discharge current and y(t +) is the
terminal voltage at the beginning of the rest
period. Then R S is given by
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()()
S
C
y t y t
R
i
−+
−
= (4)
After finding R S the rest period can be analyzed.
Given that u in equation (1) is zero during this
period, the equation
1
12
1
3
()((0))(0)
(0)
TS TS
TL TL
updating
R C
R C
y t g x x e
x e
−
−
=+
+
(5)
describes the terminal voltage during this interval.
To find x2(0) and x3(0) we consider the constant discharge period prior to the rest period. Using
equation (1) yields
()
()
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1
()
22
1
()
33
()()
()()
TS TS
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t
R C
TS C TS C
t
R C
TL C TL C
x t x R i e R i
x t x R i e R i
τ
τ
τ
τ
−+
−+
=−+−
=−+−
(6)
where –τ is the time point at the beginning of the discharge period. Given that τ is sufficiently
large, the exponential components in equation
(6) are negligibly small at time t=0. Then Equation (6) reduces to
23(0)(0)TS C TL C
x R i x R i =−=−. (7)
Equation (5) is of the generic form
()bt dt y t k a e c e =+⋅+⋅ (8)
The generic parameters k , a, b, c , and d were determined bad on measurements of y (t ) and applying the Trust-Region algorithm of Matlab’s Curve Fitting Toolbox. Constraints on the parameters enforced positive values for k and negative values for a, b, c , and d . The battery model parameters were then derived from equations (5), (7), and (8) as
11TS C TL C TS TS TL TL a公主日记1
R i c R i C R b C R d
=−
=−
=−=−
, (9) This method was ud for each discharge period in the test. The resulting parameters are shown in Figures 5-9.
Rts as a function of SOC
SOC
R t s
FIGURE 8. R TS as a function of SOC
4
SOC
C t l
Ctl as a function of SOC
supplying
FIGURE 7. C TL as a function of SOC
Rtl as a function of SOC
SOC
R t l
FIGURE 6. R TL as a function of SOC
SOC
R s
Rs as a function of SOC
FIGURE 5. R S as a function of SOC
