# Battery

Simple battery model

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• Simscape / Power Systems / Simscape Components / Sources

## Description

The Battery block represents a simple battery model. For information on how you can create a detailed battery model, see the Lead-Acid Battery (Simscape).

The Battery block has four modeling variants, accessible by right-clicking the block in your block diagram and then selecting the appropriate option from the context menu, under Simscape > Block choices:

• Uninstrumented | No thermal port (default) — Basic model that does not output battery charge level or simulate thermal effects.

• Uninstrumented | Show thermal port — Model with exposed thermal port. This model does not measure internal charge level of the battery.

• Instrumented | No thermal port — Model with exposed charge output port. This model does not simulate thermal effects.

• Instrumented | Show thermal port — Model that lets you measure internal charge level of the battery and simulate thermal effects. Both the thermal port and the charge output port are exposed.

The instrumented variants have an extra physical signal port that outputs the internal state of charge. Use this functionality to change load behavior as a function of state of charge, without the complexity of building a charge state estimator.

The thermal port variants expose a thermal port, which represents the battery thermal mass. When you select this option, provide additional parameters to define battery behavior at a second temperature. For more information, see Modeling Thermal Effects.

### Battery Model

If you select `Infinite` for the Battery charge capacity parameter, the block models the battery as a series resistor and a constant voltage source. If you select `Finite` for the Battery charge capacity parameter, the block models the battery as a series resistor and a charge-dependent voltage source whose voltage as a function of charge has the following reciprocal relationship:

`$V={V}_{0}\left[1-\frac{\alpha \left(1-x\right)}{1-\beta \left(1-x\right)}\right]$`

where:

• x is the ratio of current charge to rated battery capacity in ampere-hours (AH).

• V0 is the voltage when the battery is fully charged, as defined by the Nominal voltage parameter.

• The block calculates the constants α and β to satisfy these battery conditions:

• The battery voltage is zero when the charge is zero, that is, when x = 0.

• When x = AH1/AH, that is, when the charge is the value of the Charge AH1 when no-load volts are V1 parameter, the battery voltage is V1 (the Voltage V1 < Vnom when charge is AH1 parameter value).

The equation defines a reciprocal relationship between voltage and remaining charge. It approximates a real battery, but it does replicate the increasing rate of voltage drop at low charge values. It also ensures that the battery voltage becomes zero when the charge level is zero. This simple model has the advantage of requiring few parameters, and these parameters are readily available on most datasheets.

For battery models with finite battery charge capacity, you can model battery performance deterioration depending on the number of discharge cycles, which is sometimes referred to as battery fade. To enable battery fade, set the Model battery fade? parameter to `Include`. This setting exposes additional parameters in the Fade section.

The block implements battery fade by scaling certain battery parameter values that you specify in the Main section, depending on the number of completed discharge cycles. The block uses multipliers λAH, λR1, and λV1 on the Ampere-hour rating, Internal resistance, and Voltage V1 < Vnom when charge is AH1 parameter values, respectively. These multipliers, in turn, depend on the number of discharge cycles:

`${\lambda }_{AH}=1-{k}_{1}{N}^{0.5}$`
`${\lambda }_{R1}=1+{k}_{2}{N}^{0.5}$`
`${\lambda }_{V1}=1-{k}_{3}N$`
`$N={N}_{0}+\frac{1}{AH}\underset{0}{\overset{t}{\int }}\frac{i\left(t\right)\cdot H\left(i\left(t\right)\right)}{{\lambda }_{AH}\left(t\right)}dt$`

where:

• λAH is the multiplier for battery nominal capacity.

• λR1 is the multiplier for battery series resistance.

• λV1 is the multiplier for voltage V1 to scale for a number of discharge cycles when the charge is AH1.

• N is the number of discharge cycles completed.

• N 0 is the number of full discharge cycles completed before the start of the simulation.

• AH is the rated battery capacity in AH.

• i(t) is the instantaneous battery output current.

• H(i(t)) is the Heaviside function of the instantaneous battery output current. This function returns 0 if the argument is negative, and 1 if the argument is positive.

The block calculates the coefficients k1, k2, and k3 by substituting the parameter values you provide in the Fade section into these battery equations. For example, the default set of block parameters corresponds to the following coefficient values:

• k1 = 1e-2

• k2 = 1e-3

• k3 = 1e-3

You can also define a starting point for a simulation based on the previous charge-discharge history by using the high-priority variable Discharge cycles. For more information, see Variables.

### Modeling Thermal Effects

For thermal variants of the block, you provide additional parameters to define battery behavior at a second temperature. The extended equations for the voltage when the thermal port is exposed are:

`$V={V}_{0T}\left[1-\left(\frac{{\alpha }_{T}\left(1-x\right)}{1-{\beta }_{T}\left(1-x\right)}\right)\right]$`
`${\alpha }_{T}=\alpha \left(1+{\lambda }_{\alpha }\left(T-{T}_{1}\right)\right)$`
`${\beta }_{T}=\beta \left(1+{\lambda }_{\beta }\left(T-{T}_{1}\right)\right)$`
`${V}_{0T}={V}_{0}\left(1+{\lambda }_{V}\left(T-{T}_{1}\right)\right)$`

where:

• T is the battery temperature.

• T1 is the temperature at which nominal values for α and β are provided.

• λα, λβ, and λV are the parameter temperature dependence coefficients for α, β, and V0, respectively.

The internal series resistance (R1) and self-discharge resistance (R2) also become functions of temperature:

`$\begin{array}{l}{R}_{1T}={R}_{1}\left(1+{\lambda }_{R1}\left(T-{T}_{1}\right)\right)\\ {R}_{2T}={R}_{2}\left(1+{\lambda }_{R2}\left(T-{T}_{1}\right)\right)\end{array}$`

where λR1 and λR2 are the parameter temperature dependence coefficients. All the temperature dependence coefficients are determined from the corresponding values you provide at the nominal and second measurement temperatures.

The battery temperature is determined from:

`${M}_{th}\stackrel{˙}{T}={i}^{2}{R}_{1T}+{V}^{2}/{R}_{2T}$`
where:

• Mth is the battery thermal mass.

• i is the battery output current.

### Variables

Use the Variables tab to set the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables (Simscape) .

Unlike block parameters, variables do not have conditional visibility. The Variables tab lists all the existing block variables. If a variable is not used in the set of equations corresponding to the selected block configuration, the values specified for this variable are ignored.

When you model battery fade, the Discharge cycles variable lets you specify the number of charge-discharge cycles completed before the start of simulation. For more information, see Modeling Battery Fade. If you omit battery fade modeling, this variable is not used by the block.

### Assumptions and Limitations

• The self-discharge resistance is assumed not to depend strongly on the number of discharge cycles.

• For the thermal variant of the battery, you provide fade data only for the reference temperature operation. The block applies the same derived λAH, λR1, and λV1 multipliers to parameter values corresponding to the second temperature.

• When using the thermal block variants, use caution when operating at temperatures outside of the temperature range bounded by the Measurement temperature and Second temperature measurement values. The block uses linear interpolation for the derived equation coefficients, and simulation results might become nonphysical outside of the specified range. The block checks that the internal series resistance, self-discharge resistance, and nominal voltage always remain positive, and issues error messages if there is a violation.

## Ports

### Output

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Physical signal port that outputs the internal state of charge, in the units of coulomb (C). Use this output port to change load behavior as a function of state of charge, without the complexity of building a charge state estimator.

#### Dependencies

Enabled for the instrumented variants of the block: Instrumented | No thermal port and Instrumented | Show thermal port.

### Conserving

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Electrical conserving port associated with the battery positive terminal.

Electrical conserving port associated with the battery negative terminal.

Thermal conserving port representing the battery thermal mass. When you expose this port, provide additional parameters to define battery behavior at a second temperature. For more information, see Modeling Thermal Effects.

#### Dependencies

Enabled for the thermal variants of the block: Uninstrumented | Show thermal port and Instrumented | Show thermal port.

## Parameters

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### Main

No-load voltage across the battery when it is fully charged.

Internal connection resistance of the battery.

Select one of the options for modeling the charge capacity of the battery:

• `Infinite` — The battery voltage is independent of the charge drawn from the battery.

• `Finite` — The battery voltage decreases as the charge decreases.

This parameter is the maximum (nominal) battery charge in AH. To specify a target value for the initial battery charge at the start of simulation, use the high-priority Charge variable. For more information, see Variables.

#### Dependencies

Enabled when the Battery charge capacity parameter is set to `Finite`.

Battery output voltage when the charge level is AH1, as specified by the Charge AH1 when no-load volts are V1 parameter.

#### Dependencies

Enabled when the Battery charge capacity parameter is set to `Finite`.

Battery charge level, in AH, corresponding to the no-load output voltage specified by the Voltage V1 < Vnom when charge is AH1 parameter.

#### Dependencies

Enabled when the Battery charge capacity parameter is set to `Finite`.

Select whether to model the self-discharge resistance of the battery:

• `Omit` — Do not include resistance across the battery output terminals in the model.

• `Include` — Include resistance R2 across the battery output terminals in the model.

#### Dependencies

Enabled when the Battery charge capacity parameter is set to `Finite`.

Resistance across the battery output terminals that represents battery self-discharge.

#### Dependencies

Enabled when the Model self-discharge resistance? parameter is set to `Include`.

Temperature T1, at which the block parameters in the Main section are measured. For more information, see Modeling Thermal Effects.

#### Dependencies

Enabled for blocks with exposed thermal port.

Select whether to include battery fade modeling:

• `Omit` — The battery performance is not age-dependent.

• `Include` — The battery performance changes depending on the number of completed charge-discharge cycles. Selecting this option exposes additional parameters in this section, which define the battery performance after a specified number of discharge cycles. The block uses these parameter values to calculate the scaling coefficients k1, k2, and k3. For more information, see Modeling Battery Fade.

#### Dependencies

Enabled when the Battery charge capacity parameter in the Main section is set to `Finite`. If Battery charge capacity is `Infinite`, the Fade section is empty.

Number of charge-discharge cycles after which the other parameters in this section are measured. This second set of data points defines the scaling coefficients k1, k2, and k3, used in modeling battery fade.

#### Dependencies

Enabled when the Model battery fade? parameter is set to `Include`.

Maximum battery charge, in AH, after the number of discharge cycles specified by the Number of discharge cycles, N parameter.

#### Dependencies

Enabled when the Model battery fade? parameter is set to `Include`.

Battery internal resistance after the number of discharge cycles specified by the Number of discharge cycles, N parameter.

#### Dependencies

Enabled when the Model battery fade? parameter is set to `Include`.

Battery output voltage, at charge level in AH is AH1, after the number of discharge cycles specified by the Number of discharge cycles, N parameter.

#### Dependencies

Enabled when the Model battery fade? parameter is set to `Include`.

### Temperature Dependence

This section appears only for blocks with exposed thermal port. For more information, see Modeling Thermal Effects.

No-load voltage across the battery when it is fully charged.

Internal connection resistance of the battery.

Battery output voltage when the charge level is AH1, as specified by the Charge AH1 when no-load volts are V1 parameter.

#### Dependencies

Enabled when the Battery charge capacity parameter in the Main section is set to `Finite`.

Resistance across the battery output terminals that represents battery self-discharge.

#### Dependencies

Enabled when the Model self-discharge resistance? parameter in the Main section is set to `Include`.

Temperature T2, at which the block parameters in the Temperature Dependence section are measured. For more information, see Modeling Thermal Effects.

To specify the initial temperature at the start of simulation, use the high-priority Temperature variable. For more information, see Variables.

### Thermal Port

This section appears only for blocks with exposed thermal port. For more information, see Modeling Thermal Effects.

Thermal mass associated with the thermal port H. It represents the energy required to raise the temperature of the thermal port by one degree.

## References

[1] Ramadass, P., B. Haran, R. E. White, and B. N. Popov. “Mathematical modeling of the capacity fade of Li-ion cells.” Journal of Power Sources. 123 (2003), pp. 230–240.

[2] Ning, G., B. Haran, and B. N. Popov. “Capacity fade study of lithium-ion batteries cycled at high discharge rates.” Journal of Power Sources. 117 (2003), pp. 160–169.