State of Charge Estimation using Coulomb Counting

Coulomb Counting Method

Coulomb counting method is a type of book-keeping estimation method that uses battery discharging current data as input. This method permits to include some internal battery effects as self-discharge, capacity-loss and discharging efficiency.

Two kinds of book-keeping estimation methods have been employed:

1. Coulomb counting method

The Coulomb counting method measures the discharging current of a battery and integrates the discharging current over time in order to estimate SOC. Coulomb counting method is done to estimate the SOC(t), which is estimated from the discharging current, `I(t)`, and previously estimated SOC values. SOC(t-1) is calculated by the following equation:

`SOC(t)=SOC(t-1)+(I(t))/Q_nDeltat`

But there are several factors that affect the accuracy of the Coulomb counting method including temperature, battery history, discharge current, and cycle life.

2. Modified Coulomb counting method

To improve the Coulomb counting method, a new technique called the modified Coulomb counting method is proposed. The modified Coulomb counting method uses the corrected current to improve the accuracy of estimation.

The corrected current is the function of discharging current. There is a quadratic relationship between the corrected current and discharging current of the battery. By the practice of experimental data, the corrected current is calculated by the following form:

`I_c(t)=k_2I(t)^2+k_1I(t)+k_0`

where `k_2`, `k_1`, and `k_0` are constant values obtained from the practice experimental data.

In the modified Coulomb counting method, SOC is calculated by the following equation

`SOC(t)=SOC(t-1)+(I_c(t))/Q_nDeltat`

The experimental results show that the accuracy of the modified Coulomb counting method is superior to the conventional Coulomb counting method.

BMS implements coulomb counting for SOC estimation

The estimation of the SOC of a Li-ion battery by the developed method is based on monitoring the voltage and the current. The operation mode
of the battery is recognized by the direction of the current through the battery system. When the battery is in an open circuit mode, the battery current is zero and compensation of self-discharge losses will be considered. The information needed to perform the monitoring are the measurement of the battery voltage and the current flowing through it. Coulomb counter is used to track the SOC when the battery is charging, discharging, and self-discharging.

Simple Flowchart for Coulomb Counting

Simple Equivalent Circuit of Battery for SOC Estimation

A battery is considered an ideal voltage source in this analogy.

where OCV stands for Open Circuit Voltage

In the above model, the voltage is not a function of current or past battery usage(SOC) and the voltage remains constant.

This model is inadequate but considering the batteries do supply voltage and the cell voltage is fairly predictable when the cell is unloaded and is in complete equilibrium.

Thus the ideal voltage source will be an element in the equivalent circuit model of a battery.

State of Charge Estimation

The cell voltage is higher when it is fully charged than when it is discharged, so to improve the battery model we dependence on SOC (`z(t)`).

The fully charged cell SOC, `z=100%` and `z=0%` for fully discharged.

The total capacity of a cell `Q`, can be defined as the amount of charge taken while discharging from `z=100%` to `z=0%`.

Now the can be given as

`dotz(t)=-i(t)/Q`

`dot(z)(t)=z(t_0)-1/Qint_(t_0)^ti(t) dt`

where `dotz=(dz)/(dt)` and the sign of `i(t)` is positive on discharge.

For a discrete-time, we assume that the current is constant over sampling time `Delta t`

`z(k+1)=z(k)-(i(k)Deltat)/Q`

Considering the coulombic efficiency of the cell as n(k) the above equation can be written as

`dotz(t)=-(i(t)eta(t))/Q`

`z(k+1)=z(k)-(i(k)eta(k)Deltat)/Q`

The value of `eta(k)<=1` while charging as some charge is lost due to unwanted reactions(as electrical energy is stored in the chemical form) and `eta(k)=1` while discharging.

The coulombic efficiency is of lithium is considered about 99% and is equal to the ratio of charge out by charge in.

`eta`=charge out/charge_in

Energy efficiency and coulombic efficiency are not the same as energy efficiency is the energy given out to energy given to the cell. The energy efficiency is close to 95%. The energy may be lost in the form of resistive heating but the loss of charge is considered negligible.

OCV can be plotted as a function of SOC and a further detailed model can also include dependency upon temperature

`OCV(z(t),T(t))`

OCV is a function of SOC for different cell chemistries at a constant temperature.

Linear Polarisation

Polarization refers to any difference between the measured terminal voltage of this cell and the cell's internal open-circuit voltage due to the passage of electrical current. For example, the voltage of the cell drops when under a load.

The model represents the ideal voltage and series resistance to represent the voltage drop due to the load. The voltage across the terminals of the battery is now given by `v(t)`.

`v(t)=OCV(z(t))-i(t)R_0`

Note v(t)>OCV(z(t)) while charging and v(t)<OCV(z(t)) while discharging.

This implies that the energy is dissipated in form of heat by the resistor and the energy efficiency of the system is decreased.

Cell behavior upon Impulse Loading

The above model only represents the instantaneous voltage drop but the real behavior of the battery differs with time. This dynamic behavior is caused by the phenomenon called diffusion, so we will refer to this slowly-changing voltage as a diffusion voltage.

Diffusion Voltage

The slow relaxation of voltage is caused by the slow internal diffusion of lithium from one part of the battery cell to the other where concentration gradients build-up while the cell was being discharged and they have to slowly relax back to their equilibrium values by diffusion. Diffusion is a process, it does not happen instantly, the voltage of the cell does not change instantly.

To capture this dynamic behavior of the cell we add a parallel resistor-capacitor branch in series with the current path of the battery cell. To increase the accuracy of the dynamic path that is for better curve fitting we can increase the number of parallel resistor-capacitor sub-circuits.

The cell voltage v(t) can be written as

`v(t)= OCV(z(t)) − v_(C1)(t) − i(t)R_0`

It becomes simpler if we write this expression in terms of element currents

`v(t)= OCV(z(t)) − R_1i_(R_1)(t) − R_0i(t)`

To find an expression for the `i_(R_1)(t)`, we recognize that the summation of current through R1 and the current through C1 must be equal to i(t).

For the capacitor, we can write `i_(C1)(t) = C_1dotv_(C_1)(t)`

Now `i(t)=i_(R_1)(t)+C_1dotv_(C_1)(t)` as per Kirchoffs current law

Also `v_(C_1)(t) = R_1i_(R_1)(t)` as per Kirchoffs current law and substituting this in the above equation

`i_(R_1)(t)+R_1C_1(di_(R_1)(t))/(dt)=i(t)`

`(di_(R_1)(t))/(dt)=(i(t))/(R_1C_1)-(i_(R_1)(t))/(R_1C_1)`

References

1. https://www.hindawi.com/journals/isrn/2013/953792/

2. https://www.researchgate.net/profile/Sabeur-Jemmali/publication/305641470_Implementation_of_a_Coulomb_Counting_Algorithm_for_SOC_estimation_of_Li-Ion_Battery_for_Multimedia_Applications/links/5fc2a6ff92851c933f71fe9b/Implementation-of-a-Coulomb-Counting-Algorithm-for-SOC-estimation-of-Li-Ion-Battery-for-Multimedia-Applications.pdf?origin=publication_detail