Equivalent Circuit Battery

Resistor-capacitor (RC) circuit battery

• 

Libraries:
Powertrain Blockset / Energy Storage and Auxiliary Drive / Network Battery

Description

The Equivalent Circuit Battery block implements a resistor-capacitor (RC) circuit battery that you can parameterize using equivalent circuit modeling (ECM). To simulate the state-of-charge (SOC) and terminal voltage, the block uses load current and internal core temperature.

The Equivalent Circuit Battery block calculates the combined voltage of the network battery using parameter lookup tables. The tables are functions of the SOC and battery temperature. You can use the Estimation Equivalent Circuit Battery block to help create the lookup tables.

Specifically, the Equivalent Circuit Battery block implements these parameters as lookup tables that are functions of the SOC and battery temperature:

• Series resistance, R_o=ƒ(SOC,T)

• Battery open-circuit voltage, E_m=ƒ(SOC,T)

• Battery capacity, C_batt=ƒ(T)

• Network resistance, R_n=ƒ(SOC,T)

• Network capacitance, C_n=ƒ(SOC,T)

To calculate the combined voltage of the battery network, the block uses these equations.

$$\begin{array}{l}{V}_T=E_m-I_{batt}{R}_o-{\displaystyle \sum _1^n{V}_n}\\ V_n={\displaystyle \underset{0}{\overset{t}{\int }}\left[\frac{I_{batt}}{C_n}-\frac{V_n}{R_n{C}_n}\right]dt}\\ SOC=\frac{-1}{C_{batt}}{\displaystyle \underset{0}{\overset{t}{\int }}{I}_{batt}}dt\\ I_{batt}=\frac{I_{in}}{N_p}\\ V_{out}=N_s{V}_T\\ P_{BattLoss}=I_{batt}{}^2{R}_0+{\displaystyle \sum _1^n\frac{V_n{}^2}{R_n}}\\ L{d}_{AmpHr}={\displaystyle \underset{0}{\overset{t}{\int }}{I}_{batt}}dt\end{array}$$

Positive current indicates battery discharge. Negative current indicates battery charge.

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal			Description	Equations
PwrInfo	PwrTrnsfrd — Power transferred between blocks Positive signals indicate flow into block Negative signals indicate flow out of block	PwrLdBatt	Battery network power	$\begin{array}{l}{V}_{batt}=V_{out}\text{ OR }\frac{V_{out}}{\tau s+1}\\ \\ P_{batt}=-V_{batt}{I}_{batt}\\ P_{LdBatt}= -P_{batt}\end{array}$
	PwrNotTrnsfrd — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	PwrLossBatt	Battery network power loss	$$P_{LossBatt}=-\left(I_{batt}{}^2{R}_0+{\displaystyle \sum _1^n\frac{V_n{}^2}{R_n}}\right)$$
	PwrStored — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	PwrStoredBatt	Battery network power stored	$P_{StoredBatt}=P_{Batt}+P_{LossBatt}$

The equations use these variables.

SOC	State-of-charge
Em	Battery open-circuit voltage
Ibatt	Per module battery current
Iin	Combined current flowing from the battery network
Ro	Series resistance
Np	Number parallel branches
Np	Number of RC pairs in series
Vout, VT	Combined voltage of the battery network
Vn	Voltage for n-th RC pair
Rn	Resistance for n-th RC pair
Cn	Capacitance for n-th RC pair
Cbatt	Battery capacity
Pbatt	Battery power
PLossBatt	Negative of battery network power loss
PBattLoss	Battery network power loss
PStoredBatt	Battery network power stored
PLdBatt	Battery network power
T	Battery temperature

Current Polarity, Hysteresis, and Entropic Heating

To incorporate current polarity, hysteresis voltage, and entropic heating into the battery model, enable these block parameters:

• Include current polarity

• Include hysteresis

• Include entropic heating

Current Polarity

To improve calibration for charging and discharging processes, select Include current polarity. To model the current polarity, the block implements resistive and capacitive lookup tables with current direction.

Hysteresis

Hysteresis voltage adds a state estimator for non-linear voltage offsets when the battery current changes from charge to discharge. To calculate the combined voltage of the battery network with hysteresis, select Include hysteresis. The block uses these equations.

$$\begin{array}{l}{H}_{state}={\displaystyle \int \frac{\left(-H_{state}•abs(I_{batt})-I_{batt}\right)•HystRate}{C_{batt}}}\\ V_{Hyst}=H_{state}•MaxHystVolt-InstHystVolt•sign(I_{batt})\\ V_T=V_{Hyst}+E_m-I_{batt}{R}_o-{\displaystyle \sum _1^n{V}_n}\end{array}$$

Entropic Heating

Entropic heating accounts for power losses that occur during the battery phase change. To calculate the battery network power loss with entropic heating, select Include entropic heating. The block uses these equations.

$$\begin{array}{l}{P}_{entropic}=-I_{Batt}•T•En{t}_{dUdT}\\ P_{LossBatt}=-\left(I_{batt}{}^2{R}_0+{\displaystyle \sum _1^n\frac{V_n{}^2}{R_n}+P_{entropic}}\right)\end{array}$$

The equations use these variables.

Hstate	Hysteresis state variable
HystRate	Hysteresis rate
VHyst	Hysteresis voltage
MaxHystVolt	Maximum hysteresis voltage
InstHystVolt	Instantaneous hysteresis voltage
EntdUdT	Entropic coefficient
Pentropic	Entropic reversible losses

Examples

Build Full Electric Vehicle Model

Build a vehicle with a motor-generator, battery, direct-drive transmission, and powertrain control algorithms using the electric vehicle (EV) reference application.

Ports

Inputs

CapInit — Battery capacity

scalar

Rated battery capacity at the nominal temperature, Cap_batt, in Ah.

Dependencies

To create this port, select External Input for the Initial battery capacity parameter.

BattCurr — Battery network current

scalar

Combined current flowing from the battery network, I_in, in A.

BattTemp — Battery temperature

scalar

Battery temperature, T, in K.

Output

Info — Bus signal

bus

Bus signal containing these block calculations.

Signal			Description	Variable	Units
BattCurr			Combined current flowing from the battery network	Ibatt	A
BattAmpHr			Battery energy	LdAmpHr	A*h
BattSoc			State-of-charge capacity	SOC	NA
BattVolt			Combined voltage of the battery network	Vout	V
BattPwr			Battery power	Pbatt	W
PwrInfo	PwrTrnsfrd	PwrLdBatt	Battery network power	PLdBatt	W
	PwrNotTrnsfrd	PwrLossBatt	Battery network power loss	PLossBatt	W
	PwrStored	PwrStoredBatt	Battery network power stored	PStoredBatt	W
HystState			Hysteresis state variable	Hstate	1
HystVolt			Hysteresis voltage	Vhyst	V
EntropicPwr			Entropic reversible losses	Pentropic	W

BattVolt — Battery output voltage

scalar

Combined voltage of the battery network, V_out, in V.

Parameters

Block Options

Initial battery capacity — Input or parameter

Parameter (default) | External Input

Initial battery capacity, Cap_batt, in Ah.

Dependencies

Block Parameter Initial battery capacity Option	Creates
External Input	Input port CapInit
Parameter	Parameter Initial battery capacity, BattCapInit

Output battery voltage — Unfiltered or Filter

Unfiltered (default) | Filtered

Select Filtered to apply a first-order filter to the output battery voltage.

Dependencies

Setting Output battery voltage parameter to Filtered creates these parameters:

• Output battery voltage time constant, Tc

• Output battery voltage initial value, Vinit

Include current polarity — Add current direction

off (default) | on

Select this parameter to add current direction in the equivalent circuit battery.

To improve calibration for charging and discharging processes, select Include current polarity. To model the current polarity, the block implements resistive and capacitive lookup tables with current direction.

Include hysteresis — Add hysteresis voltage

off (default) | on

Select this parameter to add the hysteresis voltage.

Hysteresis voltage adds a state estimator for non-linear voltage offsets when the battery current changes from charge to discharge. To calculate the combined voltage of the battery network with hysteresis, select Include hysteresis. The block uses these equations.

$$\begin{array}{l}{H}_{state}={\displaystyle \int \frac{\left(-H_{state}•abs(I_{batt})-I_{batt}\right)•HystRate}{C_{batt}}}\\ V_{Hyst}=H_{state}•MaxHystVolt-InstHystVolt•sign(I_{batt})\\ V_T=V_{Hyst}+E_m-I_{batt}{R}_o-{\displaystyle \sum _1^n{V}_n}\end{array}$$

Include entropic heating — Add entropic heating

off (default) | on

Select this parameter to add entropic heating.

Entropic heating accounts for power losses that occur during the battery phase change. To calculate the battery network power loss with entropic heating, select Include entropic heating. The block uses these equations.

$$\begin{array}{l}{P}_{entropic}=-I_{Batt}•T•En{t}_{dUdT}\\ P_{LossBatt}=-\left(I_{batt}{}^2{R}_0+{\displaystyle \sum _1^n\frac{V_n{}^2}{R_n}+P_{entropic}}\right)\end{array}$$

Core Battery

Number of series RC pairs — RC pairs

1 (default) | 0 | 2 | 3 | 4 | 5

Number of series RC pairs. For lithium, typically 1 or 2. The value 0 specifies that there are no RC elements in the circuit.

Open circuit voltage Em table data, Em — Voltage table

[3.5042 3.5136; 3.5573 3.5646; 3.6009 3.6153; 3.6393 3.6565; 3.6742 3.6889; 3.7121 3.7214; 3.7937 3.8078; 3.8753 3.8945; 3.97 3.9859; 4.0764 4.0821; 4.1924 4.193] (default) | array

Open circuit voltage table, E_m, in V. Function of SOC and battery temperature.

Series resistance table data, R0 — Resistance

array

Series resistance table, R_o, in ohms. Function of SOC and battery temperature.

Series charge resistance table data, R0Charge — Charge resistance

array

Series charge resistance table, in ohms. Function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

Series discharge resistance table data, R0Discharge — Discharge resistance

array

Series discharge resistance table, in ohms. Function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

State of charge breakpoints, SOC_BP — SOC breakpoints

[0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1] (default) | vector

State-of-charge (SOC) breakpoints, dimensionless.

Temperature breakpoints, Temperature_BP — Battery

[293.15 313.15] (default) | vector

Battery temperature breakpoints, K.

Battery capacity table, BattCap — Capacity

[28 28] (default) | array

Battery capacity, C_batt, in Ah. Function of battery temperature.

Initial battery capacity, BattCapInit — Capacity

28 (default) | scalar

Initial battery capacity, Cap_batt, in Ah.

Dependencies

Block Parameter Initial battery capacity Option	Creates
External Input	Input port CapInit
Parameter	Parameter Initial battery capacity, BattCapInit

Initial capacitor voltage, InitialCapVoltage — Voltage

0 (default) | array

Initial capacitor voltage, in V. Dimension of vector must equal the Number of series RC pairs.

Output battery voltage time constant, Tc — Filter time constant

1/1000 (default) | scalar

Output battery voltage time constant, T_c, in s. Used in a first-order voltage filter.

Dependencies

Setting Output battery voltage parameter to Filtered creates these parameters:

• Output battery voltage time constant, Tc

• Output battery voltage initial value, Vinit

Output battery voltage initial value, Vinit — Filter initial voltage

4.193 (default) | scalar

Output battery voltage initial value, V_init, in V. Used in a first-order voltage filter.

Dependencies

Setting Output battery voltage parameter to Filtered creates these parameters:

• Output battery voltage time constant, Tc

• Output battery voltage initial value, Vinit

R and C Table Data

Network resistance table data, Rn — Lookup table

[0.010342 0.0012244; 0.0067316 0.0011396; 0.0051156 0.0012661; 0.0043447 0.0012265; 0.0038826 0.0011163; 0.0034226 0.0009968; 0.003346 0.0011458; 0.0033222 0.001345; 0.0033201 0.0013091; 0.0032886 0.0010986; 0.0028114 0.0010309] (default) | array

Network resistance table data for n-th RC pair, in ohms, as a function of SOC and battery temperature.

Network capacitance table data, Cn — Lookup table

[2287.7 11897; 6122 24515; 18460 42098; 20975 44453; 15254 33098; 10440 24492; 13903 32975; 16694 40007; 15784 35937; 12165 26430; 9118 24795] (default) | array

Network capacitance table data for n-th RC pair, in F, as a function of SOC and battery temperature.

Network charge resistance table data, RnCharge — Lookup table

[0.010342 0.0012244; 0.0067316 0.0011396; 0.0051156 0.0012661; 0.0043447 0.0012265; 0.0038826 0.0011163; 0.0034226 0.0009968; 0.003346 0.0011458; 0.0033222 0.001345; 0.0033201 0.0013091; 0.0032886 0.0010986; 0.0028114 0.0010309] (default) | array

Network charge resistance table data for n-th RC pair, in ohms, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

Network charge capacitance table data, CnCharge — Lookup table

[2287.7 11897; 6122 24515; 18460 42098; 20975 44453; 15254 33098; 10440 24492; 13903 32975; 16694 40007; 15784 35937; 12165 26430; 9118 24795] (default) | array

Network charge capacitance table data for n-th RC pair, in F, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

Network discharge resistance table data, RnDischarge — Lookup table

[0.010342 0.0012244; 0.0067316 0.0011396; 0.0051156 0.0012661; 0.0043447 0.0012265; 0.0038826 0.0011163; 0.0034226 0.0009968; 0.003346 0.0011458; 0.0033222 0.001345; 0.0033201 0.0013091; 0.0032886 0.0010986; 0.0028114 0.0010309] (default) | array

Network discharge resistance table data for n-th RC pair, in ohms, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

Network discharge capacitance table data, CnDisharge — Lookup table

[2287.7 11897; 6122 24515; 18460 42098; 20975 44453; 15254 33098; 10440 24492; 13903 32975; 16694 40007; 15784 35937; 12165 26430; 9118 24795] (default) | array

Network discharge capacitance table data for n-th RC pair, in F, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include current polarity.

Cell Limits

Upper integrator voltage limit, Vu — Maximum

-Inf (default) | scalar

Upper voltage limit, in V.

Lower integrator voltage limit, Vl — Minimum

-Inf (default) | scalar

Lower voltage limit, in V.

Heating and Hysteresis

Entropic Coefficient, EntdUdT — Lookup table

[-0.2226 -0.2922 -0.1085 -0.0580 0.1027 0.1195 0.0549 -0.0202 -0.0231 0.0165 0.0324]/1000 (default) | vector

Entropic coefficient table data, Ent_dUdT, in V/K, as a function of SOC.

Dependencies

To enable this parameter, select Include entropic heating.

Hysteresis Rate, HystRate — Hysteresis Rate

0.1 (default) | scalar

Hysteresis rate, dimensionless, specified as a scalar.

Dependencies

To enable this parameter, select Include hysteresis.

Maximum Hysteresis Voltage, MaxHystVolt — Lookup table

[0 0;0 0;0 0;0 0;0 0; 0 0; 0 0; 0 0; 0 0; 0 0; 0 0] (default) | array

Maximum hysteresis voltage table data, in V, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include hysteresis.

Instantaneous Hysteresis Voltage, InstHystVolt — Lookup table

[0 0;0 0;0 0;0 0;0 0; 0 0; 0 0; 0 0; 0 0; 0 0; 0 0] (default) | array

Instantaneous hysteresis voltage table data, in V, as a function of SOC and battery temperature.

Dependencies

To enable this parameter, select Include hysteresis.

References

[1] Ahmed, Ryan, et al. "Model-Based Parameter Identification of Healthy and Aged Li-ion Batteries for Electric Vehicle Applications." SAE International Journal of Alternative Powertrains. 4, no. 2 (2015): 233 -47. https://doi.org/10.4271/2015-01-0252.

[2] Gazzarri, Javier, Nishant Shrivastava, Robyn Jackey, and Craig Borghesani. "Battery Pack Modeling, Simulation, and Deployment on a Multicore Real Time Target." SAE International Journal of Aerospace. 7, no. 2 (2014): 207–13. https://doi.org/10.4271/2014-01-2217.

[3] Huria, Tarun, Massimo Ceraolo, Javier Gazzarri, and Robyn Jackey. “High Fidelity Electrical Model with Thermal Dependence for Characterization and Simulation of High Power Lithium Battery Cells.” IEEE^® International Electric Vehicle Conference, March 2012. https://doi.org/10.1109/ievc.2012.6183271.

[4] Huria, Tarun, Massimo Ceraolo, Javier Gazzarri, and Robyn Jackey. "Simplified Extended Kalman Filter Observer for SOC Estimation of Commercial Power-Oriented LFP Lithium Battery Cells." SAE Technical Paper Series, 2013. https://doi.org/10.4271/2013-01-1544.

[5] Jackey, Robyn A. "A Simple, Effective Lead-Acid Battery Modeling Process for Electrical System Component Selection." SAE Technical Paper Series, 2007. https://doi.org/10.4271/2007-01-0778.

[6] Jackey, Robyn A., Gregory L. Plett, and Martin J. Klein. "Parameterization of a Battery Simulation Model Using Numerical Optimization Methods." SAE Technical Paper Series, 2009. https://doi.org/10.4271/2009-01-1381.

[7] Jackey,Robyn, et al. "Battery Model Parameter Estimation Using a Layered Technique: An Example Using a Lithium Iron Phosphate Cell." SAE Technical Paper Series, 2013. https://doi.org/10.4271/2013-01-1547.

[8] Geng, Zeyang, Jens Groot, and Torbjorn Thiringer. “A Time- and Cost-Effective Method for Entropic Coefficient Determination of a Large Commercial Battery Cell.” IEEE Transactions on Transportation Electrification 6, no. 1 (March 2020): 257–66. https://doi.org/10.1109/TTE.2020.2971454.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017a

R2024b: Model complex battery effects

Configure the Equivalent Circuit Battery block to model complex battery effects such as current polarity, hysteresis, and entropic heating.

• The Equivalent Circuit Battery block has these parameters.

  • Include current polarity — Select this option to add current direction for resistance and capacitance lookup tables. Selecting this parameter enables these parameters on the Core Battery and R and C Table Data tabs.

    • Core Battery

      • Series charge resistance table data, R0Charge

      • Series discharge resistance table data, R0Discharge

    • R and C Table Data

      • Network charge resistance table data, RnCharge

      • Network charge capacitance table data, CnCharge

      • Network discharge resistance table data, RnDischarge

      • Network discharge capacitance table data, CnDischarge

  • Include hysteresis — Select this option to add a state estimator for non-linear voltage offsets when the battery changes from charge to discharge. Selecting this parameter enables these parameters in the Heating and Hysteresis tab.

    • Hysteresis Rate, HystRate

    • Maximum Hysteresis Voltage, MaxHystVolt

    • Instantaneous Hysteresis Voltage, InstHystVolt

  • Include entropic heating — Select this option to include power losses that occur during the battery phase change. Selecting this parameter enables this parameter in the Heating and Hysteresis tab.

    • Entropic Coefficient, EntdUdT

• The Info includes these bus signals.

  • HystState

  • HystVolt

  • EntropicPwr

• You can now set the Number of series RC pairs to 0. This removes the RC elements from the circuit.