# Battery Capacity Estimator (Kalman Filter)

Battery capacity estimator with Kalman filter

Since R2023b

Libraries:
Simscape / Battery / BMS / Estimators

## Description

The Battery Capacity Estimator (Kalman Filter) block implements an estimator that calculates the cell capacity of a battery by using the Kalman filter algorithms.

Use this block in conjunction with another block that estimates the state of charge (SOC) of the battery, such as SOC Estimator (Kalman Filter) or SOC Estimator (Adaptive Kalman Filter).

This block supports single-precision and double-precision floating-point simulation.

Note

To enable single-precision floating-point simulation, the data type of all inputs and parameters, except for the Sample time (-1 for inherited) parameter, must be `single`.

For continuous-time simulation, set the Filter type parameter to `Extended Kalman-Bucy filter` or ```Unscented Kalman-Bucy filter```.

Note

Continuous-time implementation of this block works only in a double-precision floating-point simulation. If you provide single-precision floating-point parameters and inputs, this block casts them to double-precision floating-point values to prevent errors.

For discrete-time simulation, set the Filter type parameter to `Extended Kalman filter` or ```Unscented Kalman filter``` and the Sample time (-1 for inherited) parameter to a positive value or `-1`.

### Equations

To estimate the battery capacity, the Battery Capacity Estimator (Kalman Filter) block starts from the equations of the derivative of the capacity and the SOC:

`$\begin{array}{l}\frac{dQ}{dt}=0\\ 0=\frac{dSOC}{dt}+\frac{i}{3600Q}\end{array}$`

where

• SOC is the state of charge.

• i is the current of the battery cell.

• Q is the capacity in Ampere-hour.

The first equation is the state equation. The second equation is the output equation.

For the Kalman filter algorithms, the Battery Capacity Estimator (Kalman Filter) block uses Q as the state and these process and observation functions:

`$\begin{array}{l}f\left(Q\right)=0\\ h\left(Q,SOC\right)=\frac{dSOC}{dt}+\frac{i}{3600Q}\end{array}$`

Extended Kalman Filter

This diagram shows the structure of the extended Kalman filter (EKF):

The EKF technique relies on a linearization at every time step to approximate the nonlinear system. To linearize the system at every time step, the algorithm computes these Jacobians online:

`$\begin{array}{l}F=\frac{\partial f}{\partial Q}=0\\ H=\frac{\partial h}{\partial Q}=-\frac{i}{3600{Q}^{2}}\end{array}$`

The EKF is a discrete-time algorithm. After the discretization, the Jacobians for the SOC estimation of the battery are:

`$\begin{array}{l}{F}_{\text{d}}=1\\ {H}_{\text{d}}=-\frac{i{T}_{\text{s}}}{3600{Q}^{2}}\end{array}$`

where TS is the sample time.

For more information about the implementation of the EKF algorithm steps, see the Extended Kalman Filter section of the SOC Estimator (Kalman Filter) block.

Extended Kalman-Bucy Filter

This diagram shows the structure of the extended Kalman-Bucy filter (EKBF):

The EKBF is the continuous-time variant of the Kalman filter. In continuous time, the prediction and correction steps are coupled.

For more information about the implementation of the EKF algorithm steps, see the Extended Kalman-Bucy Filter section of the SOC Estimator (Kalman Filter) block.

Unscented Kalman Filter

This diagram shows the structure of the unscented Kalman filter (UKF):

The EKF locally approximates nonlinear functions with the linear equations obtained from the Taylor expansion by using only the first term of the expansion. In a highly nonlinear system, these solutions are not very accurate.

The UKF uses nonlinear transformations on a set of sigma points that the algorithm chooses deterministically. This technique is called unscented transformation. The mean and the covariance matrix of the transformed points are accurate to the second order of the Taylor series expansion.

For more information about the implementation of the EKF algorithm steps, see the Unscented Kalman Filter section of the SOC Estimator (Kalman Filter) block.

Unscented Kalman-Bucy Filter

This diagram illustrates the structure of the unscented Kalman-Bucy filter (UKBF):

The derived continuous-time filtering equations of the UKBF are similar to the EKBF equations.

For more information about the implementation of the EKF algorithm steps, see the Unscented Kalman-Bucy Filter section of the SOC Estimator (Kalman Filter) block.

## Ports

### Input

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Battery current, in ampere, specified as a scalar for a single cell or a vector for multiple cells. To specify this input as a vector of cell currents, select the Specify Current input as cell current(s) parameter.

State of charge of one or multiple cells, specified as a scalar or vector of entries in the range [0, 1]. The size of this input port must be equal to the size of the InitialCellAH input port.

Initial cell capacity of the battery, in ampere-hour, specified as a scalar for a single cell or a vector for multiple cells. The size of this input port must be equal to the size of the CellSOC input port.

### Output

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Estimated cell capacity of the battery, in ampere-hour, returned as a scalar or a vector. The size of this output port is equal to the size of the InitialCellAH input port.

## Parameters

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Option to specify the value of the Current input port as a vector of cell currents. If you select this parameter, the value at the Current input port can be a scalar or a vector of size equal to the size of the block inputs.

Type of Kalman filter that this block uses to estimate the battery cell capacity.

Coefficient that controls the spread of the sigma points. The block uses this parameter in its implementation of the equations for the unscented Kalman filter and the unscented Kalman-Bucy filter.

#### Dependencies

To enable this parameter, set Filter type to `Unscented Kalman filter` or `Unscented Kalman-Bucy filter`.

Coefficient related to the distribution. The block uses this parameter in its implementation of the unscented Kalman filter and the unscented Kalman-Bucy filter.

#### Dependencies

To enable this parameter, set Filter type to `Unscented Kalman filter` or `Unscented Kalman-Bucy filter`.

Coefficient that controls the spread of the sigma points. The block uses this parameter in its implementation of the equations for the unscented Kalman filter and the unscented Kalman-Bucy filter.

#### Dependencies

To enable this parameter, set Filter type to `Unscented Kalman filter` or `Unscented Kalman-Bucy filter`.

Covariance of the noise in the state.

Covariance of the noise in the measurements.

Covariance of the initial state error. This parameter defines the deviation in the initialization of the state.

Time between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state. For more information, see What Is Sample Time? and Specify Sample Time.

For inherited discrete-time operation, specify this parameter as `-1`. For discrete-time operation, specify this parameter as a positive integer. For continuous-time operation, specify this parameter as `0`.

If this block is in a masked subsystem or a variant subsystem that supports switching between continuous operation and discrete operation, promote the sample time parameter. Promoting the sample time parameter ensures correct switching between the continuous and discrete implementations of the block. For more information, see Promote Block Parameters on a Mask.

#### Dependencies

To enable this parameter, set Filter type to `Extended Kalman filter` or `Unscented Kalman filter`.

## Version History

Introduced in R2023b