Generic battery model
Simscape / Electrical / Specialized Power Systems / Sources
The Battery block implements a generic dynamic model that represents most popular types of rechargeable batteries.
This figure shows the equivalent circuit that the block models.
The circuit parameters can be modified to represent a specific battery type and its discharge characteristics. A typical discharge curve consists of three sections.
The first section represents the exponential voltage drop when the battery is charged. The width of the drop depends on the battery type. The second section represents the charge that can be extracted from the battery until the voltage drops below the battery nominal voltage. Finally, the third section represents the total discharge of the battery, when the voltage drops rapidly.
When the battery current is negative, the battery recharges, following a charge characteristic.
The model parameters are derived from the discharge characteristics. The discharging and charging characteristics are assumed to be the same.
The Exp(s) transfer function represents the hysteresis phenomenon for the leadacid, nickelcadmium (NiCD), and nickelmetal hydride (NiMH) batteries during the charge and discharge cycles. The exponential voltage increases when a battery is charging, regardless of the battery's state of charge. When the battery is discharging, the exponential voltage decreases immediately.
The state of charge (SOC) for a battery is a measure of battery's charge, expressed as a percent of the full charge. The depth of discharge (DOD) is the numerical compliment of the SOC, such that DOD = 100%  SOC.
For example, if the SOC is:
100% — The battery is fully charged and the DOD is 0%.
75% — The battery is 3/4 charged and the DOD is 25%.
50% — The battery is 1/2 charged and the DOD is 50%.
0% — The battery is has 0 charge and the DOD is 100%.
Experimental validation of the model shows a maximum error of 5% (when SOC is between 10% and 100%) for the charge (when the current is 0 through 2 C) and discharge (when the current is 0 through 5 C) dynamics.
This figure shows detailed parameters extracted from the Panasonic NiMHHHR650D battery data sheet.
You can obtain the rated capacity and the internal resistance from the specification tables. The other detailed parameters are derived from the Typical Discharge Characteristics plot.
Parameter  Value 

Rated Capacity 

Internal Resistance 

Nominal Voltage (a) 

Rated Capacity 

Maximum Capacity (b) 

Fully Charged Voltage (c) 

Nominal Discharge Current (d) 

Capacity @ Nominal Voltage (a) 

Exponential Voltage (e) 

Exponential Capacity (e) 

These parameters are approximate and depend on the precision of the points obtained from the discharge curve.
The discharge curves you obtain from these parameters, which are marked by dotted lines in the following figures, are similar to the data sheet curves.
To represent the temperature effects of the lithiumion (Liion) battery type, an additional discharge curve at ambient temperature, which is different from the nominal temperature, and the thermal response parameters are required. Additional discharge curves are not usually provided on the data sheet and may require simple experiments to be obtained. The following examples show parameters extracted from the A123 Liironphosphate ANR26650M1 and the Panasonic Licobaltoxide CGR 18,650 AF battery data sheets.
The A123 ANR26650M1 data sheet specifications include the required discharge curve points and other required parameters.
These parameters are derived from the data sheet for the A123 Liion temperaturedependent battery model.
Parameter  Value 

Nominal voltage (c) 

Rated capacity 

Maximum capacity (d) 

Fully charged voltage (a) 

Nominal discharge current 

Internal resistance 

Capacity at nominal voltage (c) 

Exponential zone (b)  [ 
Nominal ambient temperature 

Second ambient temperature 

Maximum capacity at 0°C (h) 

Initial discharge voltage at 0°C (e) 

Voltage at 90% maximum capacity at 0°C (g) 

Exponential zone at 0°C (f)  [ 
Thermal resistance, celltoambient (estimated) 

Thermal time constant, celltoambient (estimated) 

In the figure, the dashed lines show the discharges curves obtained from the simulation at different ambient temperatures. The model performance is very close to the data sheet results.
The same approach for parameter extraction is applied to the Panasonic LithiumIon CGR18650AF with these specifications.
These parameters are extracted for the battery model.
Parameter  Value 

Nominal voltage (c) 

Rated capacity 

Maximum capacity (d) 

Fully charged voltage (a) 

Nominal discharge current 

Internal resistance (estimated) 

Capacity at nominal voltage (c) 

Exponential zone (b)  [ 
Nominal ambient temperature 

Second ambient temperature 

Maximum capacity at 0°C (h) 

Initial discharge voltage at 0°C (e) 

Voltage at 90% maximum capacity at 0°C (g) 

Exponential zone at 0°C (f)  [ 
Thermal resistance, celltoambient (estimated) 

Thermal time constant, celltoambient (estimated) 

The figure shows a good match between the simulated discharge curves (represented by the dashed lines) and the data sheet curves. The accuracy of the model depends on how precise the selected points from the data sheet discharge curves are.
To model a series and/or parallel combination of cells based on the parameters
of a single cell, use the parameter transformation shown in the following table
can be used. The Nb_ser
variable corresponds to the number of
cells in series, and Nb_par
corresponds to the number of
cells in parallel.
Parameter  Value 

Nominal voltage  1.18 * Nb_ser 
Rated capacity  6.5 * Nb_par 
Maximum capacity  7 * Nb_par 
Fully charged voltage  1.39 * Nb_ser 
Nominal discharge current  1.3 * Nb_par 
Internal resistance  0.002 * Nb_ser/Nb_par 
Capacity at nominal voltage  6.25 * Nb_par 
Exponential zone  1.28 * Nb_ser, 1.3 * Nb_par 
For the leadacid battery type, the model uses these equations.
Discharge Model (i* > 0)
$${f}_{1}\left(it,i*,i,Exp\right)={E}_{0}K\cdot \frac{Q}{Qit}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+{\text{Laplace}}^{1}\left(\frac{Exp(s)}{Sel(s)}\cdot 0\right)$$
Charge Model (i* < 0)
$${f}_{2}\left(it,i*,i,Exp\right)={E}_{0}K\cdot \frac{Q}{it+0.1\cdot Q}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+{\text{Laplace}}^{1}\left(\frac{Exp(s)}{Sel(s)}\cdot \frac{1}{s}\right)$$
For the lithiumion battery type, the model uses these equations.
Discharge Model (i* > 0)
$${f}_{1}\left(it,i*,i\right)={E}_{0}K\cdot \frac{Q}{Qit}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+A\cdot \mathrm{exp}\left(B\cdot it\right)$$
Charge Model (i* < 0)
$${f}_{2}\left(it,i*,i\right)={E}_{0}K\cdot \frac{Q}{it+0.1\cdot Q}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+A\cdot \mathrm{exp}\left(B\cdot it\right)$$
For the nickelcadmium and nickelmetalhydride battery types, the model uses these equations.
Discharge Model (i* > 0)
$${f}_{1}\left(it,i*,i,Exp\right)={E}_{0}K\cdot \frac{Q}{Qit}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+{\text{Laplace}}^{1}\left(\frac{Exp(s)}{Sel(s)}\cdot 0\right)$$
Charge Model (i* < 0)
$${f}_{2}\left(it,i*,i,Exp\right)={E}_{0}K\cdot \frac{Q}{\leftit\right+0.1\cdot Q}\cdot i*K\cdot \frac{Q}{Qit}\cdot it+{\text{Laplace}}^{1}\left(\frac{Exp(s)}{Sel(s)}\cdot \frac{1}{s}\right).$$
In the equations:
E_{Batt} is the nonlinear voltage, in V.
E_{0} is the constant voltage, in V.
Exp(s) is the exponential zone dynamics, in V.
Sel(s) represents the battery mode.
Sel(s) = 0
during battery
discharge, Sel(s) = 1
during
battery charging.
K is the polarization constant, in V/Ah, or polarization resistance, in Ohms.
i* is the lowfrequency current dynamics, in A.
i is the battery current, in A.
it is the extracted capacity, in Ah.
Q is the maximum battery capacity, in Ah.
A is the exponential voltage, in V.
B is the exponential capacity, in Ah^{−1}.
For the lithiumion battery type, the impact of temperature on the model parameters is represented by these equations.
Discharge Model (i* > 0)
${f}_{1}(it,i*,i,T,{T}_{a})={E}_{0}(T)K(T)\cdot \frac{Q({T}_{a})}{Q({T}_{a})it}\cdot \left(i*+it\right)+A\cdot \mathrm{exp}\left(B\cdot it\right)C\cdot it$
${V}_{batt}(T)={f}_{1}(it,i*,i,T,{T}_{a})R(T)\cdot i$
Charge Model (i* < 0)
${f}_{1}(it,i*,i,T,{T}_{a})={E}_{0}(T)K(T)\cdot \frac{Q({T}_{a})}{it+0.1\cdot Q({T}_{a})}\cdot i*K(T)\cdot \frac{Q({T}_{a})}{Q({T}_{a})it}\cdot it+A\cdot \mathrm{exp}\left(B\cdot it\right)C\cdot it$
${V}_{batt}(T)={f}_{1}(it,i*,i,T,{T}_{a})R(T)\cdot i,$
with
${E}_{0}(T)={E}_{0}{}_{{T}_{ref}}+\frac{\partial E}{\partial T}\left(T{T}_{ref}\right)$
$K(T)=K{}_{{T}_{ref}}\cdot \mathrm{exp}\left(\alpha \left(\frac{1}{T}\frac{1}{{T}_{ref}}\right)\right)$
$Q({T}_{a})=Q{}_{{T}_{a}}+\frac{\Delta Q}{\Delta T}\cdot \left({T}_{a}{T}_{ref}\right)$
$R(T)=R{}_{{T}_{ref}}\cdot \mathrm{exp}\left(\beta \left(\frac{1}{T}\frac{1}{{T}_{ref}}\right)\right),$
where:
T_{ref} is the nominal ambient temperature, in K.
T is the cell or internal temperature, in K.
T_{a} is ambient temperature, in K.
E/T is the reversible voltage temperature coefficient, in V/K.
α is the Arrhenius rate constant for the polarization resistance.
β is the Arrhenius rate constant for the internal resistance.
ΔQ/ΔT is the maximum capacity temperature coefficient, in Ah/K.
C is the nominal discharge curve slope, in V/Ah. For lithiumion batteries with less pronounced discharge curves (such as lithium iron phosphate batteries), this parameter is set to zero.
The cell or internal temperature, T, at any given time, t, is expressed as:
$T(t)={L}^{1}\left(\frac{{P}_{loss}{R}_{th}+{T}_{a}}{1+s\cdot {t}_{c}}\right),$
where:
R_{th} is thermal resistance, cell to ambient (°C/W).
t_{c} is thermal time constant, cell to ambient (s).
P_{loss} is the overall heat generated (W) during the charge or discharge process and is given by
${P}_{loss}=\left({E}_{0}(T){V}_{batt}(T)\right)\cdot i+\frac{\partial E}{\partial T}\cdot i\cdot T.$
For the lithiumion battery type, the impact of aging (due to cycling) on the battery capacity and internal resistance is represented by these equations:
$Q(n)=\{\begin{array}{l}{Q}_{BOL}\epsilon (n)\cdot \left({Q}_{BOL}{Q}_{EOL}\right)\begin{array}{ccc}& if& k/2\ne 0\end{array}\\ Q(n1)\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& & otherwise\end{array}\end{array}$
$R(n)=\{\begin{array}{l}{R}_{BOL}+\epsilon (n)\cdot \left({R}_{EOL}{R}_{BOL}\right)\begin{array}{ccc}& if& k/2\ne 0\end{array}\\ R(n1)\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& & otherwise\end{array}\end{array},$
with
$n=k{T}_{h}\begin{array}{ccc}& (k=1,2,3,\mathrm{...}\infty )& \end{array}$
where:
T_{h} is the halfcycle duration, in s. A complete cycle is obtained when the battery is discharged and charged or conversely.
Q_{BOL} is the battery's maximum capacity, in Ah, at the beginning of life (BOL) and at nominal ambient temperature.
Q_{EOL} is the battery's maximum capacity, in Ah, at the end of life (EOL) and at nominal ambient temperature.
R_{BOL} is the battery's internal resistance, in ohms, at the BOL and at nominal ambient temperature.
R_{EOL} is the battery's internal resistance, in ohms, at the EOL and at nominal ambient temperature.
ε is the battery aging factor. The aging factor is equal to zero and unity at the BOL and EOL.
The battery aging factor, ξ, is expressed as
$\epsilon (n)=\{\begin{array}{l}\epsilon (n1)+\frac{0.5}{N(n1)}\left(2\frac{DOD(n2)+DOD(n)}{DOD(n1)}\right)\begin{array}{ccc}& if& k/2\ne 0\end{array}\\ \epsilon (n1)\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& & otherwise\end{array}\end{array},$
where:
DD is the battery DOD (%) after a halfcycle duration.
N is maximum number of cycles and is given by
$N(n)=H{\left(\frac{DOD(n)}{100}\right)}^{\xi}\cdot \mathrm{exp}\left(\psi \left(\frac{1}{{T}_{ref}}\frac{1}{{T}_{a}(n)}\right)\right)\cdot {\left({I}_{dis\_ave}(n)\right)}^{{\gamma}_{1}}\cdot {\left({I}_{ch\_ave}(n)\right)}^{{\gamma}_{2}},$
where:
H is the cycle number constant (cycles).
ξ is the exponent factor for the DOD.
ψ is the Arrhenius rate constant for the cycle number.
I_{dis_ave} is the average discharge current in A during a half cycle duration.
I_{ch_ave} is the average charge current in A during a half cycle duration.
γ_{1} is the exponent factor for the discharge current.
γ_{2} is the exponent factor for the charge current.
Limitations
The minimum noload battery voltage is 0 V and the maximum battery voltage is equal to 2 × E_{0}.
The minimum capacity of the battery is 0 Ah and the maximum capacity is Q_{max}.
Assumptions
The internal resistance is assumed to be constant during the charge and discharge cycles and does not vary with the amplitude of the current.
The parameters of the model are derived from the discharge characteristics. The discharging and charging characteristics are assumed to be the same.
The capacity of the battery does not change with the amplitude of the current (there is no Peukert effect).
The selfdischarge of the battery is not represented. It can be represented by adding a large resistance in parallel with the battery terminals.
The battery has no memory effect.
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[2] Saw, L.H., K. Somasundaram, Y. Ye, and A.A.O. Tay, “Electrothermal analysis of Lithium Iron Phosphate battery for electric vehicles.” Journal of Power Sources. Vol. 249, pp. 231–238.
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