SOC Estimator (Kalman Filter)

State of charge estimator with Kalman filter

Since R2022b

Libraries:
Simscape / Battery / BMS / Estimators

Description

The SOC Estimator (Kalman Filter) block implements an estimator that calculates the state of charge (SOC) of a battery by using the Kalman filter algorithms.

The SOC is the ratio of the released capacity C _releasable to the rated capacity C _rated . Manufacturers provide the value of the rated capacity of each battery, which represents the maximum amount of charge in the battery:

$S O C = \frac{C_{releasable}}{C_{rated}} .$

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

Note

To enable inherited 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

These figures show the equivalent circuit for a battery with one-time-constant dynamics and two time-constant dynamics, respectively:

The equations for the equivalent circuit with two time-constant dynamics are:

$\begin{array}{l} \frac{d S O C}{d t} = - \frac{i}{3600 A H} \\ \frac{d V_{1}}{d t} = \frac{i}{C_{1} (S O C, T)} - \frac{V_{1}}{R_{1} (S O C, T) C_{1} (S O C, T)} \\ \frac{d V_{2}}{d t} = \frac{i}{C_{2} (S O C, T)} - \frac{V_{2}}{R_{2} (S O C, T) C_{2} (S O C, T)} \\ V_{t} = V_{0} (S O C, T) - i R_{0} - V_{1} - V_{2} \end{array}$

where

SOC is the state of charge.
i is the current.
V ₀ is the no-load voltage.
V _t is the terminal voltage.
AH is the ampere-hour rating.
R ₁ is the first polarization resistance.
C ₁ is the first parallel RC capacitance.
R ₂ is the second polarization resistance.
C ₂ is the second parallel RC capacitance.
T is the temperature.
V ₁ is the polarization voltage over the first RC network.
V ₂ is the polarization voltage over the second RC network.

A time constant τ ₁ for the first parallel section relates the first polarization resistance R ₁ and the first parallel RC capacitance C ₁ using the relationship $C_{1} = τ_{1} / R_{1} .$

A time constant τ ₂ for the second parallel section relates the second polarization resistance R ₂ and the second parallel RC capacitance C ₂ using the relationship $C_{2} = τ_{2} / R_{2} .$

For the Kalman filter algorithms, the block uses this state and these process and observation functions:

$\begin{array}{l} x = {[\begin{matrix} S O C & V_{1} \end{matrix}]}^{T} \\ f (x, i) = [\begin{matrix} - \frac{i}{3600 A H} \\ \frac{i}{C_{1} (S O C, T)} - \frac{V_{1}}{R_{1} (S O C, T) C_{1} (S O C, T)} \end{matrix}] \\ h (x, i) = V_{0} (S O C, T) - i R_{0} - V_{1} \end{array}$

If you set the Charge dynamics parameter to Two time-constant dynamics , for the Kalman filter algorithms, the block uses this state and these process and observation functions:

$\begin{array}{l} x = {[\begin{matrix} S O C & V_{1} & V_{2} \end{matrix}]}^{T} \\ f (x, i) = [\begin{matrix} - \frac{i}{3600 A H} \\ \begin{array}{l} \frac{i}{C_{1} (S O C, T)} - \frac{V_{1}}{R_{1} (S O C, T) C_{1} (S O C, T)} \\ \frac{i}{C_{2} (S O C, T)} - \frac{V_{2}}{R_{2} (S O C, T) C_{2} (S O C, T)} \end{array} \end{matrix}] \\ h (x, i) = V_{0} (S O C, T) - i R_{0} - V_{1} - V_{2} \end{array}$

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:

$\begin{array}{l} F = \frac{\partial f}{\partial x} \\ H = \frac{\partial h}{\partial x} \end{array}$

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

$\begin{array}{l} F_{d} = [\begin{matrix} 1 & 0 \\ 0 & e^{\frac{- T_{S}}{R_{1} C_{1}}} \end{matrix}] \\ H_{d} = [\begin{matrix} \frac{\partial V_{OC}}{\partial S O C} & - 1 \end{matrix}] \end{array}$

where T _S is the sample time and V _OC is the open-circuit voltage.

The EKF algorithm comprises these phases:

Initialization
- $\hat{x} (0 | 0)$ ⁠— State estimate at time step 0 using measurements at time step 0.
- $\hat{P} (0 | 0)$ ⁠— State estimation error covariance matrix at time step 0 using measurements at time step
- Prediction
  - Project the states ahead ( a priori ):
    
    $\hat{x} (k + 1 | k) = f (\hat{x} (k | k), i) .$
  - Project the error covariance ahead:
    
    $\hat{P} (k + 1 | k) = F_{d} (k) \hat{P} (k | k) F_{d}^{T} (k) + Q,$
    
    where Q is the covariance of the process noise.
- Correction
  - Compute the Kalman gain:
    
    $K (k + 1) = \hat{P} (k + 1 | k) H_{d}^{T} (k) {(H_{d} (k) \hat{P} (k + 1 | k) H_{d}^{T} (k) + R)}^{- 1},$
    
    where R is the covariance of the measurement noise.
  - Update the estimate with the measurement y(k) ( a posteriori ):
    
    $\hat{x} (k + 1 | k + 1) = \hat{x} (k + 1 | k) + K (k + 1) (V_{t} (k) - h (\hat{x} (k | k), i)) .$
  - Update the error covariance:
    
    $\hat{P} (k + 1 | k + 1) = (I - K (k + 1) H_{d}) \hat{P} (k + 1 | k) .$
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.
- Initialization
  $\hat{x} (t_{0})$ ⁠— State estimate at time t ₀ .
  
  $\hat{P} (t_{0})$ ⁠— State estimation error covariance matrix at time t ₀ .
- Prediction-Correction EKBF algorithm
  
  $\begin{array}{l} K (t) = P (t) H^{T} (t) R^{- 1} (t) \\ \frac{d \hat{x} (t)}{d t} = f (\hat{x} (t), i (t)) + K (t) (V_{t} (t) - h (\hat{x} (t), i (t))) \\ \frac{d P (t)}{d t} = F (t) P (t) + P (t) F^{T} (t) + Q (t) - K (t) H (t) P (t) \end{array}$
  
  where:
  
  $\begin{array}{l} F (t) = \frac{\partial f}{\partial x} \\ H (t) = \frac{\partial h}{\partial x} \end{array}$
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.

The UKF algorithm follows these steps:
- Initialization
  $\hat{x} (0 | 0)$ ⁠— State estimate at time step 0 using measurements at time step 0.
  
  $\hat{P} (0 | 0)$ ⁠— State estimation error covariance matrix at time step 0 using measurements at time step
  
  Generate sigma points and calculate the mean weight and covariance weight for each point.
  
  Choose the sigma points x ⁽ⁱ⁾ (k|k)
  
  $\begin{array}{l} x^{(i)} (k | k) = {\begin{matrix} \hat{x} (k + 1 | k) & i = 1 \\ \hat{x} (k + 1 | k) + {(\sqrt{(n + λ) P (k | k)})}_{i} & i = 2, \dots, n + 1 \\ \hat{x} (k + 1 | k) - {(\sqrt{(n + λ) P (k | k)})}_{i} & i = n + 2, \dots, 2 n + 1 \end{matrix} \\ W_{m}^{(i)} = {\begin{matrix} \frac{λ}{n + λ} & i = 1 \\ \frac{1}{2 (n + λ)} & i \neq 1 \end{matrix} \\ W_{c}^{(i)} = {\begin{matrix} \frac{λ}{n + λ} + (1 - α^{2} + β) & i = 1 \\ \frac{1}{2 (n + λ)} & i \neq 1 \end{matrix} \end{array}$
  
  where:
  
  n is the dimension of the state vector, x .
  
  $λ = α^{2} (n + κ) - n, α \in [0, 1]$ describes the distance between the sigma point and the mean point. In a normal distribution, β = 2 and κ = 0.
  
  ${(\sqrt{(n + λ) P})}_{i}$ is the i th row or column of $\sqrt{c P}$ . The block calculates the matrix square root by using numerically efficient and stable methods such as the Cholesky decomposition.
  
  Perform first estimation of the system state matrix:
  
  $\begin{array}{l} {\hat{x}}^{(i)} (k + 1 | k) = f ({\hat{x}}^{(i)} (k | k), i (k)) \\ \hat{x} (k + 1 | k) = \sum_{i = 1}^{2 n + 1} W_{m}^{(i)} {\hat{x}}^{(i)} (k + 1 | k) \end{array}$
  
  Perform first estimation of the covariance matrix of the state variables:
  
  $P (k + 1 | k) = \sum_{i = 1}^{2 n + 1} W_{c}^{(i)} ({\hat{x}}^{(i)} (k + 1 | k) - \hat{x} (k + 1 | k)) \cdot {({\hat{x}}^{(i)} (k + 1 | k) - \hat{x} (k + 1 | k))}^{T} + Q$
  
  Estimate the measured variables:
  
  $\begin{array}{l} V_{t}^{(i)} (k + 1 | k) = h ({\hat{x}}^{(i)} (k + 1 | k), i (k)) \\ {\hat{V}}_{t} (k + 1 | k) = \sum_{i = 1}^{2 n + 1} W_{m}^{(i)} {\hat{V}}_{t}^{(i)} (k + 1 | k) \end{array}$
  
  Estimate the covariance of the measurement ( P _y ) and covariance between the measurement and the state ( P _xy ):
  
  $\begin{array}{l} P_{y} = \sum_{i = 1}^{2 n + 1} W_{c}^{(i)} ({\hat{V}}_{t}^{(i)} (k + 1 | k) - {\hat{V}}_{t} (k + 1 | k)) \cdot {({\hat{V}}_{t}^{(i)} (k + 1 | k) - {\hat{V}}_{t} (k + 1 | k))}^{T} + R \\ P_{x y} = \sum_{i = 1}^{2 n + 1} ({\hat{x}}^{(i)} (k + 1 | k) - \hat{x} (k + 1 | k)) \cdot {({\hat{V}}_{t}^{(i)} (k + 1 | k) - {\hat{V}}_{t} (k + 1 | k))}^{T} \end{array}$
  
  Compute the Kalman filter gain:
  
  $K (k + 1) = P_{x y} P_{y}^{- 1}$
  
  Perform the second update of the state matrix and of the covariance of the state variables:
  
  $\begin{array}{l} \hat{x} (k + 1 | k + 1) = \hat{x} (k + 1 | k) + K (k + 1) (V_{t} (k + 1) - {\hat{V}}_{t} (k + 1 | k)) \\ P (k + 1 | k + 1) = P (k + 1 | k) - K (k + 1) P_{y} K^{T} (k + 1) \end{array}$
  Unscented Kalman-Bucy Filter
  This diagram illustrates the overall structure of the unscented Kalman-Bucy filter (UKBF):
  
  The derived continuous-time filtering equations of the UKBF are similar to the EKBF equations.
  
  Because the UKF uses matrix square roots in its sigma points, the algorithm obtains the square-root version of the UKBF by formulating the filter as a differential equation for the sigma points. The equations for the square-root UKBF are:
  
  $\begin{array}{l} K (t) = X (t) W h^{T} (X (t), t) R^{- 1} (t) \\ M (t) = A^{- 1} (t) [X (t) W f^{T} (X (t), t) + f (X (t), t) W X^{T} (t) + Q (t) - K (t) R (t) K^{T} (t)] A^{- T} (t) \\ \frac{d X_{i} (t)}{d t} = f (X (t), t) w_{m} + K (t) (V_{t} (t) - h (X (t), t) w_{m}) + \sqrt{c} {[\begin{matrix} 0 & A (t) Φ (M (t)) & - A (t) Φ (M (t)) \end{matrix}]}_{i} \end{array}$
  
  where
  
  $X (t) = [\begin{matrix} m (t) & \dots & m (t) \end{matrix}] + \sqrt{c} [\begin{matrix} 0 & A (t) & - A (t) \end{matrix}]$ is the sigma-point matrix.
  
  $Φ_{i j} (M (t)) = {\begin{matrix} M_{i j} (t), & i > j \\ 0.5 M_{i j} (t), & i = j \\ 0, & i < j \end{matrix}$ is a function that returns the lower diagonal part of the argument.
  
  $w_{m} = {[\begin{matrix} W_{m}^{(1)} & \dots & W_{m}^{(2 n + 1)} \end{matrix}]}^{T}$
  
  $W = (1 - [\begin{matrix} w_{m} & \dots & w_{m} \end{matrix}]) diag (\begin{matrix} W_{c}^{(1)} & \dots & W_{c}^{(2 n + 1)} \end{matrix}) {(I - [\begin{matrix} w_{m} & \dots & w_{m} \end{matrix}])}^{T}$
  
  $c = α^{2} (n + κ)$
  
  ${[\begin{matrix} 0 & A (t) Φ (M (t)) & - A (t) Φ (M (t)) \end{matrix}]}_{i}$ is the i th column of the argument matrix.

Examples

Explore Techniques to Estimate Battery State of Charge

Explores different techniques to estimate the state of charge (SOC) of a battery, including the Kalman filter algorithm and the Coulomb counting method. The Kalman filter is an estimation algorithm that infers the state of a linear dynamic system from incomplete and noisy measurements. This example illustrates the effectiveness of the Kalman filter in dynamically estimating the SOC of a battery, even when starting with inaccurate initial conditions and when the measurements are noisy. The SOC of a battery measures the current capacity of the battery in relation to its maximum capacity.

Since R2024b
Open Live Script

Estimate State of Charge of Lithium Iron Phosphate Battery

Estimate the state of charge (SOC) of lithium iron phosphate (LFP) batteries by using the Coulomb Counting method with error correction. The Coulomb counting method is implemented at 1 second sample time. To correct the estimate of the Coulomb counting method, the example implements an extended Kalman filter with a sampling time of 10 seconds. The initial SOC of the battery is equal to 0.4. The estimator uses an initial condition for the SOC equal to 0.6. The LFP battery keeps charging and discharging for six hours. The estimator converges to the real value of the SOC in approximately half an hour and then follows the real SOC value.

Since R2024b
Open Model

Battery State-of-Charge Estimation

Estimate the battery state of charge (SOC) by using a Kalman filter. The initial SOC of the battery is equal to 0.5. The estimator uses an initial condition for the SOC equal to 0.8. The battery keeps charging and discharging for 6 hours. The extended Kalman filter estimator converges to the real value of the SOC in less than 10 minutes and then follows the real SOC value. To use a different Kalman filter implementation, in the SOC Estimator (Kalman Filter) block, set the Filter type parameter to the desired value.

Open Model

Charge and Discharge Battery Module with State of Charge Estimator

Cyclically charge and discharge a battery module while estimating the state of charge (SOC) of the three parallel assemblies of the module over time. This example uses the SOC estimation to switch between the charging and discharging profiles. For the estimation, the Kalman filter uses an initial SOC estimation and the voltage, current, and temperature of the parallel assemblies. The model estimates the temperature of the parallel assemblies from the average of the cell temperatures of each parallel assembly.

Since R2024a
Open Live Script

Ports

Input

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Current — Battery current
scalar | vector

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.

CellVoltage — Cell voltage
scalar | vector

Cell voltage, in volt, 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 CellTemperature and InitialSOC input ports.

CellTemperature — Cell temperature
scalar | vector

Cell temperature, 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 CellVoltage and InitialSOC input ports.

InitialSOC — Initial state of charge
scalar | vector

Initial state of charge, 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 CellVoltage and CellTemperature input ports.

Output

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SOC — State of charge
scalar | vector

State of charge of the battery, returned as a scalar or a vector. The size of this output port is equal to the size of the CellVoltage , CellTemperature , and InitialSOC input ports.

Parameters

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Main

Specify Current input as cell current(s) — Option to specify current input port as vector
`on` (default) | `off`

Since R2023b

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.

Filter type — Kalman filter type
`Extended Kalman filter` (default) | `Extended Kalman-Bucy filter` | `Unscented Kalman filter` | `Unscented Kalman-Bucy filter`

Type of Kalman filter that this block uses to estimate the battery state of charge.

Alpha — Alpha coefficient
`1` (default) | scalar in the range [0, 1]

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 .

Beta — Beta coefficient
`2` (default) | nonnegative scalar

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 .

Kappa — Kappa coefficient
`0` (default) | scalar in the range [0, 3]

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 process noise, Q — Covariance of the process noise
`[1e-6 0; 0 1e-6]` (default) | matrix of scalars

2 -by- 2 covariance matrix of the noise in the states.

Covariance of the measurement noise, R — Covariance of the measurement noise
`0.1` (default) | scalar

Covariance of the noise in the measurements.

Initial state error covariance, P0 — Initial state error covariance
`[1e-5 0; 0 1]` (default) | matrix of scalars

2 -by- 2 covariance matrix of the initial state error. This parameter defines the deviation in the initialization of the state.

Sample time (-1 for inherited) — Block sample time
`1` (default) | 0 | -1 | integer

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 .

System Model

Charge dynamics — Option to model battery charge dynamics
`One time-constant dynamics` (default) | `Two time-constant dynamics`

Since R2024a

Option to model the battery charge dynamics. This parameter determines the number of parallel RC sections in the equivalent circuit:

One time-constant dynamics — The equivalent circuit contains one parallel RC section. Specify the polarization resistance and the time constant using the First polarization resistance, R1(SOC,T), (ohm) and First time constant, tau1(SOC,T), (s) parameters.
Two time-constant dynamics — The equivalent circuit contains two parallel RC sections. Specify the polarization resistances and the time constants using the First polarization resistance, R1(SOC,T), (ohm) , First time constant, tau1(SOC,T), (s) , Second polarization resistance, R2(SOC,T), (ohm) , and Second time constant, tau2(SOC,T), (s) parameters.

Vector of state-of-charge values, SOC (-) — SOC breakpoints
`[0, .1, .25, .5, .75, .9, 1]` (default) | vector of scalars in the range [0, 1]

Vector of the state of charge breakpoints defining the points at which you specify lookup data. The entries of this vector must be in strictly ascending order. The block calculates the state of charge value with respect to the nominal battery capacity specified in the Cell capacity, AH (A*Hr) parameter. The SOC is the ratio of the available battery charge q _battery and the nominal battery capacity q _nom (T,n) . You must make sure that, for each temperature value, an SOC of 1 represents the battery charge capacity for the corresponding element of the Cell capacity, AH (A*Hr) parameter when you model a fresh battery with a number of cycles N equal to 1 and δ _AH (n = 1, T _fade ) equal to 0:.

$S O C = \frac{q_{battery}}{q_{nom} (T, n)} for N = 1 and δ_{A H} (n, T_{fade}) = 0, q_{nom} (T, n) = A H .$

Vector of temperatures, T — Temperature breakpoints
`[278, 293, 313]` (default) | vector of positive scalars

Vector of temperature breakpoints defining the points at which you specify lookup data. This vector must be strictly ascending and greater than 0 K. The physical unit of this parameter must be the same as the physical unit of the CellTemperature input port.

Terminal resistance, R0(SOC,T), (ohm) — R0 lookup table with temperature breakpoint
`[.0117, .0085, .009; .011, .0085, .009; .0114, .0087, .0092; .0107, .0082, .0088; .0107, .0083, .0091; .0113, .0085, .0089; .0116, .0085, .0089]` (default) | matrix of positive entries

Lookup data for series resistance of the battery at the specified SOC and temperature breakpoints.

First polarization resistance, R1(SOC,T), (ohm) — First RC resistance at temperature breakpoints
`[.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011]` (default) | matrix of positive entries

Lookup data, in ohm, for the first parallel RC resistance at the specified SOC and temperature breakpoints. The number of rows of this matrix is equal to the size of the Vector of state-of-charge values, SOC (-) parameter. The number of columns of this matrix is equal to the size of the Vector of temperatures, T parameter.

First time constant, tau1(SOC,T), (s) — First RC time constant at temperature breakpoints
`[20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33]` (default) | matrix of positive entries

Lookup data, in second, for the first parallel RC time constant at the specified SOC and temperature breakpoints. The number of rows of this matrix is equal to the size of the Vector of state-of-charge values, SOC (-) parameter. The number of columns of this matrix is equal to the size of the Vector of temperatures, T parameter.

Second polarization resistance, R2(SOC,T), (ohm) — Second RC resistance at temperature breakpoints
`[.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011]` (default) | matrix of positive entries

Since R2024a

Lookup data, in ohm, for the second parallel RC resistance at the specified SOC and temperature breakpoints. The number of rows of this matrix is equal to the size of the Vector of state-of-charge values, SOC (-) parameter. The number of columns of this matrix is equal to the size of the Vector of temperatures, T parameter.

Dependencies

To enable this parameter, set Charge dynamics to Two time-constant dynamics .

Second time constant, tau2(SOC,T), (s) — Second RC time constant at temperature breakpoints
`[20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33]` (default) | matrix of positive entries

Since R2024a

Lookup data, in second, for the second parallel RC time constant at the specified SOC and temperature breakpoints. The number of rows of this matrix is equal to the size of the Vector of state-of-charge values, SOC (-) parameter. The number of columns of this matrix is equal to the size of the Vector of temperatures, T parameter.

Dependencies

To enable this parameter, set Charge dynamics to Two time-constant dynamics .

No-load voltage, V0(SOC,T), (V) — V ₀ lookup table
`[3.49, 3.5, 3.51; 3.55, 3.57, 3.56; 3.62, 3.63, 3.64; 3.71, 3.71, 3.72; 3.91, 3.93, 3.94; 4.07, 4.08, 4.08; 4.19, 4.19, 4.19]` (default) | matrix of nonnegative entries

Lookup data, in volt, for open-circuit voltages across the fundamental battery model at the specified SOC. The number of rows of this matrix is equal to the size of the Vector of state-of-charge values, SOC (-) parameter. The number of columns of this matrix is equal to the size of the Vector of temperatures, T parameter.

**Cell capacity, AH (A*Hr)** — Battery capacity when fully charged
`27` (default) | nonnegative scalar

Cell capacity of the battery, in ampere-hour. The block calculates the state of charge by dividing the accumulated charge by this value. The block calculates the accumulated charge by integrating the battery current.

Signal Attributes

Algorithm data type — Option to choose data type for block algorithm
`Inherit: auto` (default) | `double` | `single` | `<data type expression>`

Since R2025a

Option to choose the data type for the block algorithm, specified as one of these values:

Inherit: auto — You can simulate the block in both single and double precision. You must explicitly provide the inputs and parameters as either single or double .
double — The block algorithm casts all inputs and parameters to double data type.
single — The block algorithm casts all inputs and parameters to single data type.
<data type expression> — The block algorithm casts all inputs and parameters to the data type object you specify.

Click the Show data type assistant button to display the Data Type Assistant , which helps you set the data type attributes. For more information, see Specify Data Types Using Data Type Assistant and Control Data Types of Signals .

References

[1] Grewal, Mohinder S., and Angus P. Andrews. Kalman Filtering: Theory and Practice Using MATLAB. New York: Wiley Intersci., 2001.

[2] Van der Merwe, R., and E. A. Wan, The Square-Root Unscented Kalman Filter for State and Parameter-Estimation. 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), vol. 6, IEEE, 2001, pp. 3461–64.

[3] Sarkka, Simo On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1631-1641, Sept. 2007, doi: 10.1109/TAC.2007.904453.

[4] Huria, Tarun, et al. Simplified Extended Kalman Filter Observer for SOC Estimation of Commercial Power-Oriented LFP Lithium Battery Cells. 2013, pp. 2013-01–1544.

Extended Capabilities

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

Version History

Introduced in R2022b

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R2025a: Switch between single-precision and double-precision floating-point simulation using Algorithm data type parameter

You can now easily switch between single-precision and double-precision floating-point simulation of the block algorithm by using the Algorithm data type parameter.

R2024a: Estimate SOC using equivalent models with two time-constant charge dynamics

You can now estimate the battery SOC by using equivalent models with two time-constant charge dynamics.

To select an equivalent model with two time-constant charge dynamics, set the Charge dynamics parameter to Two time-constant dynamics and specify the values of the new Second polarization resistance, R2(SOC,T), (ohm) and Second time constant, tau2(SOC,T), (s) parameters.

R2023b: Specify the value of the Current input as a vector of cell currents

You can now specify the value of the Current input port as a vector of cell currents.

To enable a vectorized input, select the Specify Current input as cell current(s) parameter. If you select this parameter, the Current input port must have the same size as the other inputs and represents individual cell currents.

SOC Estimator (Kalman Filter) (adsbygoogle = window.adsbygoogle || []).push({});

Description

Equations

Examples

Explore Techniques to Estimate Battery State of Charge

Estimate State of Charge of Lithium Iron Phosphate Battery

Battery State-of-Charge Estimation

Charge and Discharge Battery Module with State of Charge Estimator

Ports

Input

Current — Battery current scalar | vector

CellVoltage — Cell voltage scalar | vector

CellTemperature — Cell temperature scalar | vector

InitialSOC — Initial state of charge scalar | vector

Output

SOC — State of charge scalar | vector

Parameters

Main

Specify Current input as cell current(s) — Option to specify current input port as vector on (default) | off

Filter type — Kalman filter type (adsbygoogle = window.adsbygoogle || []).push({}); Extended Kalman filter (default) | Extended Kalman-Bucy filter | Unscented Kalman filter | Unscented Kalman-Bucy filter

Alpha — Alpha coefficient 1 (default) | scalar in the range [0, 1]

Dependencies

Beta — Beta coefficient 2 (default) | nonnegative scalar

Dependencies

Kappa — Kappa coefficient 0 (default) | scalar in the range [0, 3]

Dependencies

Covariance of the process noise, Q — Covariance of the process noise [1e-6 0; 0 1e-6] (default) | matrix of scalars

Covariance of the measurement noise, R — Covariance of the measurement noise 0.1 (default) | scalar

Initial state error covariance, P0 — Initial state error covariance [1e-5 0; 0 1] (default) | matrix of scalars

Sample time (-1 for inherited) — Block sample time (adsbygoogle = window.adsbygoogle || []).push({}); 1 (default) | 0 | -1 | integer

Dependencies

System Model

Charge dynamics — Option to model battery charge dynamics One time-constant dynamics (default) | Two time-constant dynamics

Vector of state-of-charge values, SOC (-) — SOC breakpoints [0, .1, .25, .5, .75, .9, 1] (default) | vector of scalars in the range [0, 1]

Vector of temperatures, T — Temperature breakpoints [278, 293, 313] (default) | vector of positive scalars

Terminal resistance, R0(SOC,T), (ohm) — R0 lookup table with temperature breakpoint [.0117, .0085, .009; .011, .0085, .009; .0114, .0087, .0092; .0107, .0082, .0088; .0107, .0083, .0091; .0113, .0085, .0089; .0116, .0085, .0089] (default) | matrix of positive entries

First polarization resistance, R1(SOC,T), (ohm) — First RC resistance at temperature breakpoints [.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011] (default) | matrix of positive entries

First time constant, tau1(SOC,T), (s) — First RC time constant at temperature breakpoints [20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33] (default) | matrix of positive entries

Second polarization resistance, R2(SOC,T), (ohm) — Second RC resistance at temperature breakpoints [.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011] (default) | matrix of positive entries

Dependencies

Second time constant, tau2(SOC,T), (s) — Second RC time constant at temperature breakpoints [20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33] (default) | matrix of positive entries

Dependencies

No-load voltage, V0(SOC,T), (V) — V 0 lookup table [3.49, 3.5, 3.51; 3.55, 3.57, 3.56; 3.62, 3.63, 3.64; 3.71, 3.71, 3.72; 3.91, 3.93, 3.94; 4.07, 4.08, 4.08; 4.19, 4.19, 4.19] (default) | matrix of nonnegative entries

Cell capacity, AH (A*Hr) — Battery capacity when fully charged 27 (default) | nonnegative scalar

Signal Attributes

Algorithm data type — Option to choose data type for block algorithm Inherit: auto (default) | double | single | <data type expression>

References

Extended Capabilities

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

Version History

R2025a: Switch between single-precision and double-precision floating-point simulation using Algorithm data type parameter

R2024a: Estimate SOC using equivalent models with two time-constant charge dynamics

R2023b: Specify the value of the Current input as a vector of cell currents

See Also

Teaching Resources

SOC Estimator (Kalman Filter)

Current — Battery current
scalar | vector

CellVoltage — Cell voltage
scalar | vector

CellTemperature — Cell temperature
scalar | vector

InitialSOC — Initial state of charge
scalar | vector

SOC — State of charge
scalar | vector

Specify Current input as cell current(s) — Option to specify current input port as vector
`on` (default) | `off`

Filter type — Kalman filter type
`Extended Kalman filter` (default) | `Extended Kalman-Bucy filter` | `Unscented Kalman filter` | `Unscented Kalman-Bucy filter`

Alpha — Alpha coefficient
`1` (default) | scalar in the range [0, 1]

Beta — Beta coefficient
`2` (default) | nonnegative scalar

Kappa — Kappa coefficient
`0` (default) | scalar in the range [0, 3]

Covariance of the process noise, Q — Covariance of the process noise
`[1e-6 0; 0 1e-6]` (default) | matrix of scalars

Covariance of the measurement noise, R — Covariance of the measurement noise
`0.1` (default) | scalar

Initial state error covariance, P0 — Initial state error covariance
`[1e-5 0; 0 1]` (default) | matrix of scalars

Sample time (-1 for inherited) — Block sample time
`1` (default) | 0 | -1 | integer

Charge dynamics — Option to model battery charge dynamics
`One time-constant dynamics` (default) | `Two time-constant dynamics`

Vector of state-of-charge values, SOC (-) — SOC breakpoints
`[0, .1, .25, .5, .75, .9, 1]` (default) | vector of scalars in the range [0, 1]

Vector of temperatures, T — Temperature breakpoints
`[278, 293, 313]` (default) | vector of positive scalars

Terminal resistance, R0(SOC,T), (ohm) — R0 lookup table with temperature breakpoint
`[.0117, .0085, .009; .011, .0085, .009; .0114, .0087, .0092; .0107, .0082, .0088; .0107, .0083, .0091; .0113, .0085, .0089; .0116, .0085, .0089]` (default) | matrix of positive entries

First polarization resistance, R1(SOC,T), (ohm) — First RC resistance at temperature breakpoints
`[.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011]` (default) | matrix of positive entries

First time constant, tau1(SOC,T), (s) — First RC time constant at temperature breakpoints
`[20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33]` (default) | matrix of positive entries

Second polarization resistance, R2(SOC,T), (ohm) — Second RC resistance at temperature breakpoints
`[.0109, .0029, .0013; .0069, .0024, .0012; .0047, .0026, .0013; .0034, .0016, .001; .0033, .0023, .0014; .0033, .0018, .0011; .0028, .0017, .0011]` (default) | matrix of positive entries

Second time constant, tau2(SOC,T), (s) — Second RC time constant at temperature breakpoints
`[20, 36, 39; 31, 45, 39; 109, 105, 61; 36, 29, 26; 59, 77, 67; 40, 33, 29; 25, 39, 33]` (default) | matrix of positive entries

No-load voltage, V0(SOC,T), (V) — V ₀ lookup table
`[3.49, 3.5, 3.51; 3.55, 3.57, 3.56; 3.62, 3.63, 3.64; 3.71, 3.71, 3.72; 3.91, 3.93, 3.94; 4.07, 4.08, 4.08; 4.19, 4.19, 4.19]` (default) | matrix of nonnegative entries

**Cell capacity, AH (A*Hr)** — Battery capacity when fully charged
`27` (default) | nonnegative scalar

Algorithm data type — Option to choose data type for block algorithm
`Inherit: auto` (default) | `double` | `single` | `<data type expression>`

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