SOC Estimator (Adaptive Kalman Filter, Variable Capacity)

State of charge and terminal resistance estimator with adaptive Kalman filter and variable capacity

Since R2023b

Libraries:
Simscape / Battery / BMS / Estimators

Description

This block implements an estimator that calculates the state of charge (SOC) and terminal resistance of a battery by using the Kalman filter algorithms. The terminal resistance, R₀, is a state of the estimator. The cell capacity of the battery is an input to the block.

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 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 the terminal resistance R₀ as an additional state and two time-constant dynamics are:

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

where

SOC is the state of charge.
i is the current.
V₀ is the no-load voltage.
V_t is the terminal voltage.
AH is the ampere-hour rating.
R₁ is the first polarization resistance.
C₁ is the first parallel RC capacitance.
R₂ is the second polarization resistance.
C₂ is the second parallel RC capacitance.
T is the temperature.
V₁ is the polarization voltage over the first RC network.
V₂ is the polarization voltage over the second RC network.

A time constant τ₁ for the first parallel section relates the first polarization resistance R₁ and the first parallel RC capacitance C₁ using the relationship $C_{1} = τ_{1} / R_{1} .$

A time constant τ₂ for the second parallel section relates the second polarization resistance R₂ and the second parallel RC capacitance C₂ using the relationship $C_{2} = τ_{2} / R_{2} .$

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

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

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:

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

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:

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

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 0.
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. The EKBF algorithm comprises these phases:

Initialization
- $\hat{x} (t_{0})$ ⁠— State estimate at time t₀.
- $\hat{P} (t_{0})$ ⁠— State estimation error covariance matrix at time t₀.
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 0.
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 ith 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 shows the 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 ith column of the argument matrix.

Assumptions and Limitations

Process and sensor noises are independent, zero-mean, Gaussian noises.

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, CellAH, InitialSOC, and InitialR0 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, CellAH, InitialSOC, and InitialR0 input ports.

CellAH — Cell capacity
scalar | vector

Cell capacity of the battery, in ampere-hour, specified as a scalar for a single cell or a vector for multiple cells. The block calculates the SOC by dividing the accumulated charge by this value. The block calculates the accumulated charge by integrating the battery current.

The size of this input port must be equal to the size of the CellTemperature, CellVoltage, InitialSOC, and InitialR0 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, CellTemperature, CellAH, and InitialR0 input ports.

InitialR0 — Initial terminal resistance
scalar | vector

Initial terminal resistance, specified as a scalar or a vector. The size of this input port must be equal to the size of the CellVoltage, CellTemperature, CellAH, and InitialSOC 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, InitialSOC, and InitialR0 input ports.

R0 — Terminal resistance
scalar | vector

Terminal resistance, returned as a scalar or a vector. The size of this output port is equal to the size of the CellVoltage, CellTemperature, InitialSOC, and InitialR0 input ports.

Parameters

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Main

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

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 and the terminal resistance.

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 process noise
`[1e-6 0 0; 0 1e-6 0;0 0 1e-6]` (default) | matrix of scalars

Covariance matrix of the noise in the states.

Covariance of the measurement noise, R — Covariance of 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; 0 1 0; 0 0 1e-5]` (default) | matrix of scalars

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 SOC 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 SOC value with respect to the nominal battery capacity specified by the value of the CellAH input port. 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 CellAH input port 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.

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. Matrix of resistance values, in ohm. 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) — V₀ 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.

Extended Capabilities

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

Version History

Introduced in R2023b

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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.

SOC Estimator (Adaptive Kalman Filter, Variable Capacity)

Description

Equations

Assumptions and Limitations

Ports

Input

Current — Battery current scalar | vector

CellVoltage — Cell voltage scalar | vector

CellTemperature — Cell temperature scalar | vector

CellAH — Cell capacity scalar | vector

InitialSOC — Initial state of charge scalar | vector

InitialR0 — Initial terminal resistance scalar | vector

Output

SOC — State of charge scalar | vector

R0 — Terminal resistance scalar | vector

Parameters

Main

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

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]

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 process noise [1e-6 0 0; 0 1e-6 0;0 0 1e-6] (default) | matrix of scalars

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

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

Sample time (-1 for inherited) — Block sample time 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

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) — V0 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

Extended Capabilities

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Version History

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

See Also

Current — Battery current
scalar | vector

CellVoltage — Cell voltage
scalar | vector

CellTemperature — Cell temperature
scalar | vector

CellAH — Cell capacity
scalar | vector

InitialSOC — Initial state of charge
scalar | vector

InitialR0 — Initial terminal resistance
scalar | vector

SOC — State of charge
scalar | vector

R0 — Terminal resistance
scalar | vector

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

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 process noise
`[1e-6 0 0; 0 1e-6 0;0 0 1e-6]` (default) | matrix of scalars

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

Initial state error covariance, P0 — Initial state error covariance
`[1e-5 0 0; 0 1 0; 0 0 1e-5]` (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

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) — V₀ 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

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