96052
BATTERY ELECTROCHEMICAL NONLINEAR/DYNAMIC SPICE MODEL
Michael C. Glass
Lockheed Martin Missiles & Space
OR7-10 BA50
1111 Lockheed Way
Sunnyvale, CA 94089
glass-mike@ im.ssd.lmsc.lockheed.com
ABSTRACT:
An Integrated Battery Model has been produced which
accurately represents DC nonlinear battery behavior together
with transient dynamics. The NiH2 battery model begins with
a given continuous-function electrochemical math model. The
math model for the battery consists of the sum of two
electrochemical process DC currents, which are a function of
the battery terminal voltage. This paper describes procecures
for realizing a voltage-source SPICE model which implements
the electrochemical equations using behavioral sources. The
model merges the essentially DC non-linear behavior of the
electrochemical model, together with the empirical AC
dynamic terminal impedance from measured data. Thus the
model integrates the short-term linear impedance behavior,
with the long-term nonlinear DC resistance behavior. The
long-duration non-Faradaic capacitive behavior of the battery
is represented by a time constant. Outputs of the model include
battery voltage/current, state-of-charge, and charge-current
efficiency.
The nickel Tafel component is
where the Nernst component is
and the oxygen Tafel component is
The charging (Amp-hour) efficiency is
MATH MODEL:
The spacecraft battery requires a nonlinear equation
formulation, in general, since the observed battery behavior is
quite nonlinear in its details. Whereas the solar array model is
conventionally terminated in a current source for dynamic
reasons, the battery is conventionally terminated with a
voltage source output, also for dynamic reasons. This
convention has been adopted even in the present case where
the battery characteristic is given as a set of voltage-dependent
currents.
This electrochemical battery model consists of a terminal
current which is the sum of two separate processes, the desired
Nickel-plate charging current Ini, and the parasitic overcharge
(oxygen generation) reaction current I o x . This
electrochemical model and parameters were developed by Hafen
and Armantrout, and are documented in a separate paper1,
0-7803-3547-3-7116 $4.00 0 1996 IEEE
= I n ; -b I O X ~
Ini = Nickel reaction current
Ioxy = Oxygen reaction current
Ibat
'bat
= Battery voltage
ncell = Number of series cells
SOC = Battery State - of - Charge (in Amp - hours)
T = Absolute temperature
R, F = Fundamental electrochemical constants
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. . . . . . . . .INTEGRATED
. . . . . .BATTERY
. . . .MODEL
. . . .DYNAMICS
.......
SPICE MODEL: NONLINEAR DC COMPONENT
A general-purpose SPICE circuit-equivalent for these
equations is desired. The circuit-equivalent should be useful for
DC orbital analysis, and low- and mid-frequency stability and
dynamic analyses. Since the electrochemical equations are DC
only, transient and frequency-dependent dynamics must be
added somehow. Added dynamic impedances must not disturb
the DC accuracy of the model. This model should therefore
possess the following characteristics:
4
iFq FLTRsZmI
CNF=140F
Cp=l.l F
I
nonlinear DC
pulse
RLp = 88 mR
pulse
Lp = 2-5 UH Ls = 0.5
I
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IT
R ~ ~
-14sec
*
CR ~~ * ~C ~
= 10 msec
UH I
LP'RLP
= 0.1 msec
"BAT
1 . DC fidelity to equations
2 . AC transient dynamics
Figure 2. Battery Empirical Dynamic Impedance
Battery model equations often represent the battery terminal
voltage as a function of its terminal current. This form is most
convenient for system analysis, since the battery voltage
source is a stabilizing node both to the real system, and to the
system model itself. In the present case, a voltage-source
equivalent is desired, so that impedance elements may be added
in series. Because our equations are the form of current as a
function of voltage, a transformation of some kind to voltage
as a function of current is required.
The desired circuit model will have a voltage source output
which will be subsequently modified to accept transient
impedance elements in series. The initial technique of solving
the battery voltage, in a SPICE model, from the battery
terminal current, is illustrated in the figure below. The battery
model terminal parameters are its voltage Vbat, and its
measured terminal (charging) current Ibat. Vbat is calculated as
the output of a feedback loop, which integrates the error
between the measured Ibat and the total of the calculated target
current components Ini and Iox.
U
Io% ("bat)
W
f
Figure 1. Integration of Vbat(lbat)
The loop gain bandwidth product of this current error
integration process is given by the following formula, where
gbt is the circuit gain parameter and Rbat is the static
incremental resistance on the battery I-V curve. For Rbat
between 0.02-0.2 a, gbt should be on the order of 1 to
maintain 1 Hz accuracy of this loop.
ELECTRICAL TRANSIENT DYNAMICS:
Pulsed and frequency-response measurements on batteries
have revealed a complex impedance behavior. At high
frequencies the battery exhibits inductance, with an associated
time constant. Over a time span measured in milliseconds, our
particular nickel-hydrogen battery exhibits a constant pulsed
resistance, together with another component of added
resistance having a first-order time lag. And there is a much
longer-term resistance or voltage drop component
encompassing the remainder of the static or dc Tafel voltage
variation with current of the battery. This latter component
has been assigned a time constant of 14 seconds based on the
available measurements.
INTEGRATED BATTERY MODEL
These empirically-determined impedances must be integrated
into the nonlinear battery model. The nonlinear model, within
its transcendental equations, possesses all of the DC resistance
of the battery. The dynamic impedance terms must be allowed
to operate over defined transient intervals, yielding over to the
nonlinear resistance over the longer term, with a specific
controlled time constants matching measured behavior. This
has been accomplished in the core of the Integrated Battery
Model, shown in detail in Figure 3.
It is most convenient to cast the transient dynamic elements
as purely series subnetworks, which can then be placed in
series with a behavioral voltage source. The sum of this
unknown voltage source plus the voltage drops across the
transient impedance subnetworks must equal the net observed
battery terminal voltage. Some of the transient impedance
elements are either quite familiar, or easily measured. The
high-frequency residual inductance, and the mid-frequency
pulsed resistance of the battery are largely determined by the
shape and arrangement of the battery's internal metallic
conductors. The remaining elements are masked within any
simple measurements, and require transient or wide-bandwidth
measurements. Simple network synthesis procedures are then
used to obtain the desired impedance subnetworks. The
resulting inductive and fairly constant resistive components of
the dynamic impedance are placed directly in series at the
battery model terminals.
Only the dc current-carrying resistive parts of the transient
impedance network thus obtained are in conflict with the
nonlinear battery math model, since the math model represents
the entire terminal voltage for dc current. The reason for this
goes back to the modeling process itself, in which the math
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Battery Current
I
Figure 3 . Nonlinear Dynamic Battery Mode1:Voltage Averaging
impedances. If the voltage at v d c is essentially instantaneous
as a function of terminal current Ibat, then v d c already
contains a term Ibat*C R = Ibat*(Rp + Rs), whenever true
physical series resistances having a dc value of R exist. It is
then reasonable to remove or isolate the effects of pure
resistance by subtracting this term, so that the nonlinear
voltage-source component of battery voltage is reduced to
Vdc(Ibat) - Ibat*(Rp + Rs). The effects of resistance (Rp, Rs)
are recovered by adding the transient subnetworks back
explicitly in series.
Our complete long-term transient model must include the
long time-constant circuit from Figure 2 labeled as Rnf * C,f,
model is matched to measurements of the complete battery
terminal voltage. It is clear, then, that the voltage source
representing the nonlinear battery behavior must then have
removed from it the steady-state value of the battery current
times the total value of the transient resistances which carry
the steady-state current.
The core of the nonlinear battery model shows the voltage
source for the battery nonlinearity, in series with the selected
transient linear impedances. This voltage source is seen to be
derived from the previously calculated Vbat(Ibat) dc voltage,
with a term subtracted which is Ibat*(Rp + Rs), the steady-state
value of the voltage drop through the linear transient
Figure 4 . Nonlinear Dynamic Battery Model: Current Averaging
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+
I
rstrp
ml
I1
111
I
Vac
BATTERY OUTPUT
feedback
I
This resulting battery model is easily modified to
accomodate Vbat(Ibat) math formulations by making the
voltage generator a direct function of Ibat, instead of using the
implicit computation or feedback loop used here.
One of the critical aspects affecting the static and dynamics
of a battery is the nature of the Tafel curves, which govern the
I-V characteristic of the battery. The Tafel curves from this
battery model are shown in Figure 6 , with state-of-charge as a
parameter. The static part of the battery model also includes
the nonlinear integrating Nernstian voltage changes that occur
as the battery charges and discharges.
It can
be observed that this battery model has
unsymmetrical Tafel voltage about zero current, and
unsymmetrical static resistance for charge vs discharge. These
features are important to power system dynamics and stability,
when transient operations involve transitions across the zero
current region. Additionally, stability at or near zero current is
strongly influenced by the high Tafel impedance component.
a nonlinear element. We can now see that this time constant
should apply to the complete nonlinear battery voltage, less
the linear transient part which we have now isolated in Figure
3. So then, the voltage source component of the battery
output voltage should pass through a time constant with the
value of Rnf * Cnf. It can be seen that this time constant
governs the time-dependency, of the nonlinear dc output
voltage of the battery model, on Ibat. Rnf * Cnf therefore is
the lumped voltage-averaging (or current-averaging) time
constant of the model.
Because the corresponding network in Figure 3 is unloaded,
the individual values for Rnf and Cnf are unimportant, as long
as their product equals the desired time constant (14 seconds
was used here). It is convenient to be able to take the values
directly from an empirical model such as the one in Figure 2.
We can see that all the transient element values in the SPICE
Integrated Battery Model are parameters taken directly from the
impedance model of Figure 2.
Since Vdc in Figure 3 has a nonlinear dependency on Ibat,
the Rnf * Cnf time constant is actually operating on the
nonlinear voltage response, rather than on the current itself.
If we move this lag circuit to operate directly onto the current
Ibat, we obtain a similar circuit which has true battery current
averaging in the nonlinear part of the output response. This
current-averaging alternative is shown in Figure 4. Because
the voltage transient response of this version exhibits the
Tafel nonlinearity (Fig. 6 ) explicitly, we believe the basic
voltage averaging form of Figure 3 has the greater fidelity.
I
COMPLETE INTEGRATED BATTERY MODEL
The complete Integrated Battery Model includes the
behavioral elements which calculate the two current
components Ini and Iox. A sequence of behavioral sources has
been added to Figure 5 at the upper left. The state of charge
(SOC) of the battery is a simple integration of the Ini
component of current at the lower left. And the coulombic or
Amp-hour efficiency of the battery charge and discharge
processes is computed and output at the lower right of the
figure.
il
I B a t t e r y C u r r e n t Amps
Figure 6 Battery Tafel Curves
The battery voltage dynamics have been simulated and
illustrated in the plots below. The first plot shows the fasttransient voltage, including the inductive spike, constant
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V
I
Battery Voltage
Battery Current 50 Wdiv
10
40
Figure 7. Battery Fast Transient Behavior
Figure 8. Battery Slow Transient
pulsed resistance, and a relatively fast but low-frequency firstorder RC time constant. In the second plot, the longer-term
voltage adjustment occurs over a period from 10-30 seconds.
The fast transient behavior of this battery model are revealed
in the 30-millisecond simulation of Figure 7. The battery
current is switched from a substantial discharge to a charge
current, through the zero-current region. The early features of
the battery voltage transient response are mainly linear in
nature. The leading edge features an inductive overshoot
transient and recovery governed by Lp, L,, and Rp. The first
few milliseconds after the current step exhibit a prompt
voltage step controlled by the mid-frequency resistance Rs.
For several milliseconds following the current step, the
voltage rises an additional increment as the battery exhibits a
low-frequency resistance term Rp, with a time constant
governed by R p , Rs, and Cp. These parameters were
determined from pulsed current measurements on a test battery.
The dynamic battery behavior on this time scale is of
importance to the performance and stability of regulators,
charging circuits, and load switching controls.
it owes its origin to the defining exponential or hyperbolic
functions. In Figure 8, the full nonlinear resistance of the
battery model is exhibited after a first-order time lag chosen to
be 14 seconds. This lag is controlled by the fictitious RC
circuit Rnf, Cnf whose values are selected to match empirical
data. However, the resistance Rnf effects only the time lag in
the full nonlinear battery voltage response, not any particular
voltage drop component. The delayed change in voltage
evident in Figure 8 is the component AIbat*(Rnf-Rs-Rp),
where Rnf represents the full nonlinear resistance of the
battery math model. The time constant evident in this very
low-frequency simulation is of importance in low-bandwidth
voltage controls and charge-mode control circuits.
ORBITAL SIMULATION PERFORMANCE
This battery SPICE model has been successfully integrated
into a spacecraft orbital-time-scale energy balance and
stability simulation. It has also been useful for low-frequency
discrete regulator stability verification. A complete satellite
nonlinear electrical power system model for orbital time-scale
studies is shown in Figure 9, for a small satellite. One of the
objectives of this system model is a demonstration of the
utility of a nonlinear analysis program, such as SPICE, for
demonstration of system energy balance on an orbital time
Over much longer time periods, the battery exhibits a
markedly nonlinear I-V characteristic, which we usually think
of as a nonlinear static resistance. In our mathematical model,
sat-hughesl
B"#
-
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ch-control
battoxy.dph
I
480
L
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-
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Figure 9. Orbital Energy Balance and Control Stability Model
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I
scale. This system model now contains complete nonlinear
models for the solar array and battery, such as would be
expected in a formal energy-balance performance simulation.
To facilitate the long simulation time span of this analysis,
short time constants have been removed. The rapid
progression of this analysis demonstrates the utility of SPICE
and similar programs for essentially dc energy balance and
system sizing analyses.
The simulation results depicted below include dynamic
transient effects up to the controller bandwidth of about 1 Hz.
The solution includes battery voltage, current, state-of-charge,
and charging efficiency for a 100-minute orbital period. This
sample analysis, taken for a simple load profile with a
sunlight overload, shows the utility and accuracy of SPICE
modeling for power system level simulation, when arbitrary
and complex component models are used. The speed of
execution of this model is many times faster than real time on
a variety of personal computers.
SPICE DECK4 FOR BATTERY MODEL
CIRCUIT NAME: ir.eps.test5g - - START OF DECK
.FUNC Max(C,D) (C+D+Abs(C-D))/2
.FUNC Min(C,D) (C+D-AbS(C-D))/2
**** if A > B then 1 else 0 ****
.FUNC GT(A,B) Max( (A-B)/Max(Abs(A-B),le-6), 0 )
****sign(A)
.FUNC SG(A) A/Max(Abs(A),le-6)
*
*
*
*
START ADT SUBCIRCUIT: battoxy.dph (U60)
IXY=O.67
W=40
E0=1.334761
* 100=1.42184
* EFF=0.97
* A1=0.01562
* ALF=0.295326
* A2=0.026471
* cAP=33.75
* A3=0.00026778
* EOX=1.557
* KN=1.559547
* RTF=0.0235
* A4=0.10556
* BTA=O.l
* TMP=10
* VMN=16
* GBT=10
* NCL=22
* SCI=28.6875
* VBI=28
RRllO 0 10 1K TC=O 0
* START ADT SUBCIRCUIT: DIF (U67)
XF114 33 10 180 SUMlS
.SUBCKT SUMlS 1 2 3
EADD 3 0 POLY(2) 1 0 2 0 0 1 -1
ROUT 3 0 1MEG
.ENDS SUMlS
* END ADT SUBCIRCUIT: DIF (U671
* START ADT SUBCIRCUIT: SUM (U66)
XF116 60 EO 33 SUM2S
.SUBCKT SUM2S 1 2 3
EADD 3 0 POLY(2) 1 0 2 0 0 1 1
ROUT 3 0 lMEG
.ENDS SUMZS
* END ADT SUBCIRCUIT: SUM (U66)
RR89 0 40 lMEG TC=O 0
RR87 0 22 125G TC=O 0
EV93 40 0 22 0 1
CC65 22 0 1
GV92 0 22 0 180 10
HI68 10 0 VHI68 1
VHI68 8 40 0 . 0
EB73 70 0 Value= {(v(8,0)/(22)-(1.334761))/
Figure 10. Orbital Simulation Results
CONCLUSIONS
An Integrated Battery Model has been produced for SPICE
analysis, which combines the behavior of DC battery
electrochemical equations, plus a complex set of empirically
determined dynamics. A method has been developed to convert
I(V) equations to a voltage-source output SPICE model. In
addition, a method has been developed to incorporate complex
empirical dynamic subnetworks into the voltage output of the
battery model, without loss of DC accuracy in the complete
model. The dynamic elements and the electrochemical model
parameters are all handled as variables, so that the battery
SPICE model is a programmable library component at the
system level. The battery model has been demonstrated in
dynamic simulations, as well as in orbital power system
energy-balance calculations.
+
(0.0235)-(1.559547)*~(170))
RR82 0 70 1K TC=O 0
EB72 170 0 Value=
+
~1og(Max(.02/.98.v(140)/((33.75)-v(140)))))
RRBl 0 170 1K TC=O 0
RR71 0 140 1K TC=O 0
EB71 140 0 Value= {Min(v(l30),.98*(33.75))}
RR69 0 130 1G TC=O 0
EB65 EO 0 Value=
+
~(0.67)*~((0.1)*(V(8,0)/(22)-(1.557))/(0.0235))}
GV84 0 130 60 0 277.77777777778~
CC63 130 0 1
EVE5 1 0 130 0 1
RR65 0 1 1K TC=O 0
EB76 21 0 Value=
+
{~((0.295326)*~(70))-exp(((O.295326)-1)*~(70)))
RR91 0 21 1K TC=O 0
EB77 60 0 Value= {(1.42184)*~(21))
RR92 0 60 1K TC=O 0
RR93 0 EO 1K TC=O 0
DD62 22 24 M3MCLMI
.MODEL M3MCLMI D BV=20K IS=lOf CJO=O N=lOm RS=O
DD63 23 22 M3MCLMI
W 9 4 24 0 40
W 9 5 23 0 16
RR112 0 14 1K TC=O 0
EB92 14 0 Value= (sg(v(33)*~(60)/max(abs(v(33)).
.01))
* END ADT SUBCIRCUIT: battoxy.dph (U60)
REFERENCES:
(1) D.P. Hafen, "Nickel-Hydrogen Battery Voltage and
Current Model", Lockheed-Martin internal report.
(2) Douglas P. Hafen, Jon D. Armantrout, "Nickel-Hydrogen
Voltage-Efficiency Model", IECEC-96.
(3) SPICE Analysis was performed using AWB by Cadence
Design Systems, 555 River Oaks Parkway, San Jose, CA.
(4) Muhammad Rashid, SPICE for Circuits and Electronics
Usine PSPICEB, 2nd ed., Prentice-Hall, 1995.
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