Week 7 State of charge estimation

Coulomb Counting

The main task of the Battery Management System is the accurate SOC estimation. It helps in improving the system performance, relaiability and also enhances the lifetime of the Battery. In fact, precise SOC estimation of the battery can avoid unpredicted system interruption and prevent the batteries from being over charged and over discharged, which may cause permanent damage to the internal structure of batteries. There are various methods for SOC Calculation. Following table presents the specific SOC estimation methods in view of the methodology.

Categories	Mathematical Methods
Direcct Measurement	1. Open Circuit Voltage Method 2. RBF Terminal Voltage Method 3. Impedance Method 4. Impedance Spectroscopy Method
Book Keeping Estimation	1. Coulomb Counting Method 2. Enhanced Coulomb Counting Method
Adaptive Systems	1. BP Neural Network 2. RBF Neural Network 3. Support Vector Machine 4. Fuzzy Neural Network 5. Kalman Filter
Hybrid Methods	1. Coulomb Counting and EMF Combination 2. Coulomb Counting and Kalman Filter Combination 3. Per unit System and EKF Combination

(i) Direct measurement: This method uses physical battery properties, such as the voltage and impedance of the battery.

(ii) Book-keeping estimation: This method uses discharging current as the input and integrates the discharging current over time to calculate the SOC.

(iii) Adaptive systems: The adaptive systems are self-designing and can automatically adjust the SOC for different discharging conditions. Various new adaptive systems for SOC estimation have been developed.

(iv) Hybrid methods: the hybrid models benefit from the advantages of each SOC estimation method and allow a globally optimal estimation performance. The literature shows that the hybrid methods generally produce good estimation of SOC, compared to individual methods.

Coulomb Counting Method

Coulomb counting is a technique used to track the state of charge of a battery pack. It works by integrating the active flowing current (measured in amps) over time to derive the total sum of energy entering or leaving the battery pack. This produces a capacity that is typically measured in Amp-hours.

This method employs battery current readings mathematically integrated over the usage period to calculate SOC values given by

`SOC=SOC(t_0 )+ 1/(Crated) ∫_(t_0)^(t_0+Γ)▒( I_b-I_loss)dt`

where SOC(t0) is the initial SOC, C_rated is the rated capacity, I_b is the Battery current, and I_loss is the current consumed by the loss reactions.

The coulomb counting method then calculates the remaining capacity simply by accumulating the charge transferred in or out of the battery. The accuracy of this method resorts primarily to a precise measurement of the battery current and accurate estimation of the initial SOC. With a preknown capacity, which might be memorized or initially estimated by the operating conditions, the SOC of a battery can be calculated by integrating the charging and discharging currents over the operating periods. However, the releasable charge is always less than the stored charge in the charging and discharging cycle. In other words, there are losses during charging and discharging. These losses, in addition with the self-discharging, cause accumulating errors. For more precise SOC estimation, these factors should be taken into account. In addition, the SOC should be recalibrated on a regular basis and the declination of the releasable capacity should be considered for more precise estimation.

Therefore, the disadvantages of the coulomb counting methods are

1. The access to the initial SOC is not guaranteed
2. Accumulating errors due to charging and discharging

Enhanced Coulomb Counting Method

To overcome these disadvantages. Enhanced Coulomb counting method has been adapted for estimation of SOC and SOH parameters. The initial SOC is obtained from the loaded voltages (charging and discharging) or the open circuit voltages. The losses are compensated by considering the charging and discharging efficiencies. With dynamic recalibration on the maximum releasable capacity of an operating battery, the SOH of the battery is evaluated at the same time. This in turn leads to a more precise SOC estimation.

The SOC is defined as the percentage of the releasable capacity relative to the battery rated capacity (C_rated), given by the manufacturer.

`SOC= ((Cre)/(Crated))*100`

where Cre is the releasable capacity.

A fully charged battery has the maximal releasable capacity (C_max), which can be different from the rated capacity. In general, C_max is to some extent different from Crated for a newly used battery and will decline with the used time. It can be used for evaluating the SOH of a battery.

`SOH= (C_max/(Crated)) *100%`

When a battery is discharging, the depth of discharge (DOD) can be expressed as the percentage of the capacity that has been discharged relative to C_rated,

`DOD= ( (Cred)/(Crated)) *100%`

Where C _red is the capacity released by current.

With a measured charging and discharging current (I_b), the difference of the DOD in an operating period (Ʈ) can be calculated by

`ΔDOD= (-1)/(Crated) ∫_(t_0)^(t_0+Γ)▒( I_b (t))dt* 100%`

where I_b is positive for charging and negative for discharging. As time elapsed, the DOD is accumulated.

`DOD(t)=DOD(t_0 )+ ΔDOD`

To improve the accuracy of estimation, the operating efficiency denoted as η is considered and the DOD expression becomes,

`DOD(t)=DOD(t_0 )+ ηΔDOD`

With η equal to η_c during charging stage and equal to η_d during discharging stage.

Without considering the operating efficiency and the battery aging, the SOC can be expressed as

`SOC(t)=100%-DOD(t)`

Considering the SOH, the SOC is estimated as

`SOC(t)=SOH(t)-DOD(t)`

Figure below shows the flowchart of the enhanced coulomb counting algorithm.

Fig 1: flowchart of the enhanced coulomb counting algorithm.

At the start, the historic data of the used battery is retrieved from the associated memory. Without any information for a newly used battery, the SOH is assumed to be healthy and has a value of 100%, and the SOC is initially estimated by testing either the open circuit voltage, or the loaded voltage depending on the starting conditions.

The estimation process is based on monitoring the battery voltage (V_b) and I_b. The battery operation mode can be known from the amount and the direction of the operating current. The DOD is adding up the drained charge in the discharging mode and counting down with the accumulated charge into the battery for the charging mode. After a correction with the charging and discharging efficiency, a more accurate estimation can be achieved. The SOC can be then estimated by subtracting the DOD quantity from the SOH one. When the battery is open circuited with zero current, the SOC is directly obtained from the relationship between the OCV and SOC.

SOH can be recalculated when the battery is either exhausted or fully charged, and the battery operating current and voltage are specified by manufacturers. The battery is exhausted when the loaded voltage (V_b) becomes less than the lower limit (V_min) during the discharging. In this case, the battery can no longer be used and should be recharged. At the same time, a recalibration to the SOH can be made by recalculating the SOH value by the accumulative DOD at the exhausted state. Conversely, the used battery is fully charged if (V_b) reaches the upper limit (V_max) and (I_b) declines to the lower limit (I_min) during charging. A new SOH is obtained by accumulating the sum of the total charge put into the battery and is then equal to SOC.

Initial SOC Determination

A battery can be operated at one of the three modes; charging, discharging, and open circuit. At the charging stage, the variations of the battery voltage and current when the battery is charged by the constant current, constant voltage (CC-CV) mode are usually specified by the manufacturer. With a constant charging current, the battery voltage increases gradually and reaches the threshold. Once the battery has been charged by the constant voltage mode, the charging current drops first rapidly, and then slowly. In the end, the current falloffs to almost zero when it has been fully charged. This charging curve can be converted into the relationship between the SOC and the charging voltage during the constant current stage, and the relationship between the SOC and the charging current during the constant voltage stage. The initial SOC during charging can be deduced from these relationships.

At the discharging stage, the typical voltage curves when the battery is discharged when different currents are given by the manufacturer. The terminal voltage drops as the operating time go by. A higher current causes faster decline in the terminal voltage, leading to a shorter operation time. The relationship between SOC and the discharging voltage at different currents can then be obtained, and the initial SOC during the discharging stage can be deduced.

At the open circuit stage, the relationship between the OCV and the SOC is needed. The battery is discharged by different currents before disconnecting from the load. The OCV can be used to estimate SOC if a long period rest time is available.

Charging and Discharging Efficiencies

The operational efficiency of a battery can be evaluated by the coulomb efficiency, which is defined as the ratio of the number of charges that can be extracted from the battery during discharging, compared to the number of charges that enter the battery during charging. The coefficients of the charging and discharging efficiencies are obtained from the average values of several tested batteries.

`η_C= (Cdischarg)/(Ccharg )`

The discharging efficiency is the ratio of the released capacity of two stages to C_max in one discharge cycle. All tested batteries are fully charged and then discharged by the two stage current profile, first by a specified current to a designated DOD and then by a minimal rate to the cut off voltage. The discharging efficiency is calculated by

`η_d= (I_1 T_1+ I_2 T_2)/C_max`

where I₁, I₂, T₁ and T₂ are the discharging currents and periods during the first and second stages, respectively.

The SOC values obtained by enhanced coulomb counting algorithm simulation are compared to the experimental SOC values deducted from the charging and discharging curves, which are given by battery data sheets. The blue curves represent the experimental SOC and the red curves represent the estimated SOC, obtained by an enhanced coulomb counting algorithm.

Fig 2: the experimental and estimated SOC using the enhanced coulomb counting algorithm for a complete charging stage.

Above figure 2 shows the experimental and estimated SOC using the enhanced coulomb counting algorithm for a complete charging stage. The maximal obtained error between experimental and estimated values is of about 3.5% at the end of the charging stage.

Fig 3:the experimental and estimated SOC as a function of time for the CC stage of charging mode.

Above figure 3 shows the experimental and estimated SOC as a function of time for the CC stage of charging mode. The maximal obtained error between experimental and estimated values is less than 2%.

Fig 4: the experimental and estimated SOC as a function of time for the CV stage of charging mode

Above figure 4 shows the experimental and estimated SOC as a function of time for the CV stage of charging mode. The maximal obtained error between experimental and estimated values is less than 1%.

Fig 5: the experimental and simulated SOC as a function of battery terminal voltage for a complete discharging stage

Fig 6: the experimental and simulated SOC as a function of battery terminal voltage for a partial discharging state

figure 5 and 6 the experimental and simulated SOC as a function of battery terminal voltage for a complete discharging stage and a partial discharging stage respectively. The maximal estimation error does not exceed 2% for the long complete stage and almost equals to zero for the short partial stage.