CN111289906B

CN111289906B - Power battery SOC estimation method

Info

Publication number: CN111289906B
Application number: CN202010253496.9A
Authority: CN
Inventors: 申彩英; ***; 李刚; 张乔
Original assignee: Liaoning University of Technology
Current assignee: Liaoning University of Technology
Priority date: 2020-04-02
Filing date: 2020-04-02
Publication date: 2021-11-26
Anticipated expiration: 2040-04-02
Also published as: CN111289906A

Abstract

The invention relates to a power battery SOC estimation method, which comprises the following steps: 1) two SOC values, SOC, are set in the program_TrueAnd SOC_Display(ii) a 2) Power-down time storage SOC_TrueAnd SOC_DisplayBMS corrects SOC by using SOC-OCV curve at power-on time_TrueAt the moment, the SOC is continuously sent to the whole vehicle controller_Display(ii) a 3) When the current collected by the BMS is less than or equal to the current threshold A₁Correcting SOC through SOC-OCV curve_True(ii) a The current collected by BMS is larger than current threshold A₁Calculating SOC by ampere-hour integration_True(ii) a 4) In the process of charging and discharging, the rate K is dynamically adjusted in real time₂(k) So that SOC is_DisplayFollow-up SOC_True. The estimation method leads the SOC to be in the charging and discharging process_DisplayGradually approaching SOC without jumping_TrueAn SOC dynamic following strategy is formulated, and user experience is improved.

Description

Power battery SOC estimation method

Technical Field

The invention relates to a method for estimating the SOC of a power battery, and belongs to the field of batteries of electric vehicles.

Background

Soc (state of charge), i.e. the state of charge of the battery, also called the remaining capacity of the battery. And represents the ratio of the remaining capacity to the total available capacity of the battery, expressed as a percentage, after use or long standing for a period of time. The accurate estimation of the SOC of the battery has important significance for the electric automobile, because the electric automobile estimates the endurance mileage of the vehicle according to the SOC value, and meanwhile, the accurate estimation of the SOC also has important significance for improving the utilization rate and the safety of the battery. A common definition of SOC is:

the concept of the remaining capacity and the concept of the battery capacity are particularly important for accurately estimating the SOC. The remaining capacity is, in a broad sense, the amount of charge released by chemical reaction of all chemical substances inside the lithium ion battery without damaging the battery. The remaining capacity is defined in a narrow sense as the amount of charge that can be discharged from the battery at a certain temperature and a certain discharge rate without damaging the battery.

Therefore, the temperature and the discharge rate directly influence the calculation of the residual capacity.

The capacity of a battery is not the nominal capacity of the battery when the battery leaves factory because the actual capacity that the battery can deliver is greatly reduced as the battery ages. The aging factor is typically taken into account when calculating the capacity of the battery.

The mainstream algorithm for calculating SOC mainly includes: open circuit voltage, ampere-hour integral, Kalman filtering, neural network algorithm, etc.

The Kalman filtering algorithm comprises Kalman deformation modes such as linear Kalman Filtering (KF), Extended Kalman (EKF), adaptive Kalman (AEKF) and Unscented Kalman (UKF). The linear Kalman filter aims at a linear system, and the practical application process generally linearizes a nonlinear system. If the state equation and the observation equation of the nonlinear discrete system are developed by a first-order Taylor formula, a linearized system equation can be obtained, and the EKF is used for estimating the required variables. And the Kalman filtering calculates to obtain a current optimal value according to the current measured value, the predicted value at the last moment and the error. The method has the advantages that errors are included in calculation, exist independently and are not influenced by measured data. The Kalman filtering method is suitable for various batteries in different aging stages, the accuracy of the Kalman filtering method depends on the establishment of a battery model to a large extent, and the calculated amount is large.

And (II) the neural network autonomously summarizes the internal rules of the nonlinear system mainly by analyzing and learning a large amount of data, and prejudges the state of the system according to the summarized internal rules in practical application. For non-linear systems such as batteries, the neural network can approximate any complex dynamic parameter, including of course the SOC, with any degree of accuracy.

The neural network takes the discharge current value and the discharge voltage value of a large number of experimental battery samples as input, and takes the SOC value calculated by actual test as the basis for judging the neural network. The neural network finds potential mathematical relationships between input voltage, current, and SOC through iterative calculations. When the error precision of the input result and the output result is lower than the preset value through the ceaseless iterative operation of the neural network, the neural network model is completed. The neural network method simulates human brain and neurons to process a nonlinear novel algorithm, the internal structure of the battery does not need to be studied deeply, and only input and output samples which accord with the working characteristics are extracted from the battery in advance and input into the building system, so that the SOC value in operation can be obtained. Although the post-processing of the neural network method is relatively simple, a large amount of sample data is required for supporting, and the workload is large.

Compared with the first two calculation methods, the most commonly used calculation method in the market is a combination of an ampere-hour integration method and an open-circuit voltage method.

The open circuit voltage method (iii), also called OCV method, estimates the SOC of the battery by measuring the open circuit voltage of the power battery when the battery is not in a charge-discharge state, i.e., when the operating current is zero. Shown in fig. 1 is an SOC-OCV curve of the LG battery: the abscissa is the SOC value and the ordinate is the open circuit voltage value of the battery, i.e. one SOC value for each voltage value. The SOC was 0% at an open circuit voltage of 3.287V, and 100% at an open circuit voltage of 4.186V. And the curve is a steep stage at SOC of 0% -5%. In the specific application of the electric automobile, the SOC value of the battery can be inversely obtained by collecting the cell voltage and according to the SOC-OCV curve chart. However, there are many disadvantages in calculating the SOC value by the open circuit voltage method.

Firstly, due to the characteristics of the lithium ion battery, when a large current flows through the battery, the open-circuit voltage of the battery is suddenly changed, and after the battery is kept still for a period of time, the voltage of the battery is restored to a normal state. Therefore, when the SOC is calculated by using the open-circuit voltage method, the working current of the SOC is small enough not to cause the sudden change of the open-circuit voltage. However, when the electric vehicle is running or charging, i.e. when the working current is not zero, the SOC value also needs to be known, and the open-circuit voltage method is obviously not suitable, and other methods can be selected to estimate the SOC.

And (IV) an ampere-hour integration method is an algorithm for obtaining the current battery charge state by counting the charge amount of the power battery during a period of time and adding the count and the charge amount at the previous moment. The equation of the ampere-hour integral method is shown as the formula (2):

wherein η is coulombic efficiency

Q_NIs the available capacity of the battery. SOC_kIs the SOC value at the initial time k, i_kAnd delta T is the bus current at the moment k, and is the sampling time of the system.

The ampere-hour integration method has three problems:

dependence on the initial value SOC. The ampere-hour integration method can only solve the problem of electric quantity change within a period of time, and the accuracy of the initial value SOC is required to be obtained when the accurate residual electric quantity SOC is required to be obtained. If the initial value SOC has errors, the SOC value can not be estimated correctly. This design is saved SOC to the time of will cutting off the power supply to the on-chip flash in, even BMS cuts off the power supply like this, data in the flash can not lose yet, and when the car was electrified, the BMS read out SOC data in the flash, and as SOC initial value.

The problem of accumulated errors. Due to the reasons of insufficient precision of the current sensor, low sampling frequency, signal interference and the like, the acquired current value has certain error compared with the true value, the accumulated charge calculated by the formula for a period of time has error, and the error of the current time can be brought to the error of the next moment, so that the error of the residual electric quantity is larger and larger. The direct method for solving the problem is to improve the sampling precision and the sampling frequency, and in addition, the open-circuit voltage method can be used for correcting the sampling precision and the sampling frequency.

Available capacity estimation problem. The estimation of the available capacity is greatly influenced by the temperature and the discharge rate, wherein the available capacity of the battery is different at different temperatures. As shown in table 1, it can be seen that the available capacity at low temperature is lower than the normal temperature, and if the battery capacity is calculated at the rated capacity without considering the temperature change, even if there is no initial error and the current collection accuracy is high, the battery may still be under-charged and the SOC estimation is not accurate, so the capacity change is also a factor to be considered.

TABLE 12.5 Ah available Capacity at different temperatures

The SOC is calculated by an ampere-hour integration method, and the jump problem of the SOC usually occurs when the SOC is corrected by utilizing the open-circuit voltage. SOC jump refers to a large change in charge over a short period of time. The jump of the SOC can directly influence the control strategy of the whole vehicle, and safety accidents can be seriously caused. For example, when the SOC of the vehicle suddenly decreases from 50% to 10% or even 0% during driving, the vehicle controller stops the vehicle according to the SOC of the battery, which may cause a serious accident if the vehicle is driven at a high speed. It is important to solve the SOC trip problem.

The main reason for the SOC jump is to correct the ampere-hour integral using the OCV. The SOC is calculated in real time by using ampere-hour integration, and the SOC is corrected in real time by using an OCV curve, so that the SOC cannot be calculated inaccurately due to the accumulation of errors of current sampling precision and sampling frequency. If the SOC obtained by the voltage value estimation is used for correcting the SOC calculated by the integration algorithm at intervals, so that the integration algorithm obtains a correct starting point again, the subsequent integration calculation result is closer to the true value. However, the SOC value calculated by the open-circuit voltage is not equal to the SOC value calculated by the ampere-hour integration, and the problem of SOC jump occurs at this time.

Disclosure of Invention

The technical problem to be solved by the invention is to provide an estimation method of the SOC of a power battery, wherein the estimation method prevents the influence of the SOC jump problem on user experience, and two SOC values are set in a program, wherein one SOC value is a real SOC value_TrueAnd the other is the displayed SOC value SOC_DisplayIn order to let SOC charge and discharge_DisplayGradually approaching SOC without jumping_TrueAnd an SOC dynamic following strategy is established.

In order to solve the above problems, the specific technical scheme of the invention is as follows: a method for estimating the SOC of a power battery comprises the following steps:

1) two SOC values are set in the program, one is the real SOC value SOC_TrueI.e. the actual SOC state of the battery; the other is the displayed SOC value SOC_DisplayNamely the SOC value which is output by the BMS to the user;

2) power-down time storage SOC_TrueAnd SOC_DisplayBMS corrects SOC by using SOC-OCV curve at power-on time_TrueAt the moment, the SOC is continuously sent to the whole vehicle controller_Display；

3) When the current collected by the BMS is less than or equal to the current threshold A₁And for 10S, correcting SOC through SOC-OCV curve_True(ii) a The current collected by BMS is larger than current threshold A₁Calculating the SOC through an ampere-hour integral method formula (3)_True；

4) In the process of charging and discharging, the rate K is dynamically adjusted in real time₂(k) So that SOC is_DisplayFollow-up SOC_True；SOC_DisplayThe calculation formula (4):

coefficient K₂(k) Calculating according to the error between the real value and the display value, calculating by using a formula (5) during discharging, and calculating by using a formula (6) during charging;

during discharging:

during charging:

when SOC is reached_TrueAnd SOC_DisplayEqual time K₂When the value is 1, the SOC is at the moment_DisplayEstimated according to the ampere-hour integration method.

In the discharging process, the BMS collects the voltage of the battery terminal, and when the collected voltage is less than or equal to the inflection point voltage V₁After-starting discharge end zero calibration strategy of SOC (system on chip), accelerating or reducing SOC_DisplaySpeed of change ofRate, i.e. SOC_DisplayAnd a dynamic zero correction factor K₁(k) Equations (7) and (8), respectively, make SOC_DisplayFollow-up SOC_True

In the formula

Error (k) is the error between the true value of SOC and the displayed value for the capacity discharged from the knee voltage to the discharge cutoff voltage.

The method for estimating the SOC of the power battery has the beneficial effects that: one aspect is dynamic following: the problem of jump of the SOC in the estimation process can be solved, and potential safety hazards of the electric automobile caused by SOC estimation jump are eliminated; on the other hand, zero calibration of discharge end: the problem that the SOC is not zero under the condition that the electric quantity of the battery is zero can be avoided, and the experience of a user is improved.

Drawings

FIG. 1 is a graph of the SOC-OCV curve of LG monomer at room temperature.

Fig. 2 is a flow chart of estimation of true SOC value.

Fig. 3 is a flow chart of SOC display value estimation.

Fig. 4 is a dynamic following curve 1.

Fig. 5 is a dynamic following curve 2.

Fig. 6 is a dynamic following curve 3.

Fig. 7 is a dynamic following curve 4.

FIG. 8 is a SOC end calibration strategy graph.

FIG. 9 is a graph showing discharge curves at different temperatures.

Fig. 10 is a discharge curve for different discharge rates.

FIG. 11 is a SOC-OCV curve at different temperatures.

Detailed Description

A method for estimating the SOC of a power battery comprises the following steps:

1) two SOC values are set in the program, one is the real SOC value SOC_TrueI.e. the actual SOC state of the battery; the other is the displayed SOC value SOC_DisplayThe SOC value that is visible to the user from the Battery Management System (BMS);

2) power-down time storage SOC_TrueAnd SOC_DisplayBMS corrects SOC by using SOC-OCV curve at power-on time_TrueAt the moment, the SOC is continuously sent to the whole vehicle controller_Display(ii) a Wherein the SOC-OCV curve is a relation curve of SOC and open-circuit voltage, and is drawn by experimental data, as shown in FIG. 11, the SOC can be obtained by directly passing the voltage value in this step_TrueA value of (d);

3) as shown in fig. 2, when the current collected by the BMS is equal to or less than the current threshold a₁And for 10S, correcting the SOC through an SOC-OCV curve_True(ii) a When the current collected by the BMS is larger than the current threshold A₁Calculating the SOC through an ampere-hour integral method formula (3)_True；

Wherein, SOC (k +1)_TrueRepresenting the true value of SOC for the battery at time k +1, SOC (k)_TrueRepresenting the true value of SOC of the battery at the time k, i representing the current value collected by the BMS, eta representing the charge-discharge efficiency, delta T representing the sampling period of the BMS, and Q_N(T) represents the capacity of the battery at a temperature T ℃;

4) in the charging and discharging process, as shown in fig. 3, the BMS collects the terminal voltage of the battery, and when the terminal voltage is greater than the knee voltage V of the power battery₁(inflection point voltage V₁Derived from experimental data or given by the battery manufacturer), the rate K is dynamically adjusted in real time₂(k) So that SOC is_DisplayFollow-up SOC_True；SOC_DisplayThe calculation formula (4):

SOC(k+1)_Dispalydisplay value representing SOC of battery at time k +1, SOC (k)_DispalyIndicating the value of the SOC of the battery at time K, K₂(k) Indicating that the rate is dynamically adjusted at time k. Coefficient K₂(k) The error between the real value and the display value is calculated, the formula (5) is used for calculation during discharging, and the formula (6) is used for calculation during charging.

During discharging:

during charging:

As shown in fig. 4, the battery is in a charged condition: SOC_Display>SOC_True. Charging the battery pack with a charging current of 5A by using a load, storing data such as current and average voltage transmitted from the BMS by the upper computer as input amounts of an SOC model, and adjusting the SOC at the power-off time_DisplayValue realization initial time SOC_Display>SOC_TrueThe conditions of (1). SOC at the beginning of simulation_DisplayGreater than SOC_TrueIf only one SOC value is used in the program, the SOC will jump inevitably. As the charging time increases, the cell voltage increases and the SOC value of the battery also increases, but the SOC_DisplayIs significantly lower than the SOC_TrueAt the rising rate of (C), at this time, SOC_DisplayOn standby SOC_True. And SOC at 1300s_DisplayAnd SOC_TrueMeet, then SOC_TrueThere was a brief fluctuation and after 3000s both values leveled off.

As shown in fig. 5, the battery is in a charged condition: SOC_Display<SOC_True. Charging the battery pack with a load at 5ACharging the current, storing the data such as current and average voltage sent by BMS by the upper computer as the input quantity of SOC model, and adjusting the SOC at the power-off time_DisplayValue realization initial time SOC_Display<SOC_TrueThe conditions of (1). SOC at the beginning of simulation_DisplayLess than SOC_TrueIf only one SOC value is used in the program, the SOC will jump inevitably. As the charging time increases, the cell voltage increases and the SOC value of the battery also increases, but the SOC_DisplayIs significantly higher than the SOC_TrueAt the rising rate of (C), at this time, SOC_DisplayIn pursuit of SOC_True. And SOC at 3500s point in time_DisplayAnd SOC_TrueMeet, and then both values increase smoothly.

As shown in fig. 6, the cell was in a discharged condition: SOC_Display<SOC_True. Discharging the battery pack with a discharge current of 10A by using a load, storing data such as current and average voltage transmitted by the BMS by the upper computer as input quantities of the SOC model, and adjusting the SOC at the power-off time_DisplayValue realization initial time SOC_Display<SOC_TrueThe conditions of (1). SOC at the beginning of simulation_DisplayLess than SOC_TrueIf only one SOC value is used in the program, the SOC will jump inevitably. As the discharge time increases, the cell voltage decreases, and the SOC value of the battery also decreases, but the SOC value is reduced_DisplayIs significantly less than SOC_TrueAt the rate of decrease of SOC_DisplayOn standby SOC_True. And SOC at 1900s_DisplayAnd SOC_TrueMeet each other.

As shown in fig. 7, the cell was in a discharged condition: SOC_Display>SOC_True. Discharging the battery pack with a discharge current of 10A by using a load, storing data such as current and average voltage transmitted by the BMS by the upper computer as input quantities of the SOC model, and adjusting the SOC at the power-off time_DisplayValue realization initial time SOC_Display>SOC_TrueThe conditions of (1). SOC at the beginning of simulation_DisplayGreater than SOC_TrueIf only one SOC value is used in the program, the SOC will jump inevitably. As the discharge time increases, the cell voltage decreases, and the SOC value of the battery also decreases, but the SOC value is reduced_DisplayIs significantly greater than SOC_TrueAt the rate of decrease of SOC_DisplayIn pursuit of SOC_True. And SOC at 1900s_DisplayAnd SOC_TrueMeet each other.

In conclusion, the SOC can be realized through the algorithm of the application in the charging and discharging process of the battery_DisplayReal-time following SOC_TrueAnd the jump of the SOC value is reduced or avoided, and the user experience is improved.

According to SOC mentioned in the above dynamic following strategy_TrueAnd SOC_DisplayA certain follow-up time is required. In the actual use process, if the battery reaches the discharge cut-off voltage and lasts for a certain time, the BMS considers that an undervoltage alarm occurs and turns off the relay to stop discharging. While SOC_TrueMay have reached 0%, but the SOC_DisplayThe value has not yet followed 0%, at which point it occurs that the cell voltage has reached the discharge cutoff voltage, but the SOC_DisplayBut the situation is not 0%, namely, the phenomenon of no light emission seriously affects the user experience. In addition, when the battery voltage passes through a specific voltage point, the discharging speed can be changed, and the discharging speed of the battery is accelerated. This is one of the reasons why the SOC value of the battery at the end of discharge is not 0%. Therefore, the design adds a discharge end zero calibration strategy:

at different environmental temperatures, discharge cut-off voltage and inflection point voltage V₁The values of (a) and (b) are different from each other, and as shown in fig. 9, the inflection point voltages at different temperatures are not uniform. The lower the temperature, the lower the knee voltage, and the higher the temperature, the higher the knee voltage. For example, at a temperature of 55 deg.C, the knee voltage is about 3.4V. And when the temperature is-10 ℃, the inflection point voltage is about 3.1V. As shown in fig. 10, the discharge curves at different discharge rates have substantially the same inflection point voltage at different rates at the same temperature. Only the effect of temperature on the knee voltage is considered. 2.5Ah cell discharge cutoff Voltage and Start of correction of SOCThe dot voltages are shown in table 2.

Table 2: 2.5Ah charge-discharge cut-off voltage and inflection point voltage

Tab.5.5 2.5Ah cut-off voltage and turning point voltage

During discharging, when the voltage collected by the BMS is less than or equal to the inflection point voltage V₁Post-turn on SOC correction, as shown in FIG. 3, speeds up SOC_DisplayIs the rate of change of (i.e. SOC)_DisplayAnd a dynamic zero correction factor K₁Are respectively expressed by the formulas (7) and (8) so that SOC is_DisplayFollow-up SOC_True。

In the formula

Error (k) representing an error between the true value of SOC and the displayed value at the time k, which is a capacity discharged from the knee point voltage to the discharge cutoff voltage; k₁(k) Representing the battery SOC dynamic zero correction factor at the time k. Through the adjustment of the formula, the acceleration of the SOC is realized_DisplayUntil it returns to zero, to prevent the occurrence of battery light discharge and SOC_Displ_ayThe condition of not being zero affects the user experience.

As shown in table 3, the cell supplier provided the capacity fraction discharged from the inflection point voltage to the set cut-off voltage at different temperatures of the 2.5Ah battery. (wherein T is more than or equal to 0 ℃, the cut-off voltage of monomer discharge is 3.0 V.T < 0 ℃, and the cut-off voltage of monomer discharge is 2.8V).

Table 3: discharge capacity at different temperatures

Tab.5.6Charging capacity at different temperatures

Temperature/. degree.C	-20	-10	0	25
					3.3-3.0V capacity to volume ratio	/	/	15.59％	2.88％
Capacity ratio of 3.0-2.8V	32.56％	5.41％	/	/

Calculating the correction rate K according to the discharge capacity ratio data from the inflection point voltage to the cut-off voltage of different temperatures₁. The SOC was shown as 1% when the SOC was 0% -1%, and 0% until the cutoff voltage was reached and continued for 10 s.

As shown in FIG. 8, the battery is at the initialization stage, setting SOC_DisplaySpecific SOC_True5% greater, at which time the condition for starting the end calibration strategy, SOC, is not reached_{Ah_correct_zero_no}＝SOC_{Ah_correct_zero}。SOC_{Ah_correct_zero_no}To representSOC display value, SOC, without zero-checking strategy_{Ah_correct_zero}Representing SOC display value with zero-correcting strategy, within the error range, the SOC display value changes slowly and waits for SOC_True. After the cell voltage reaches the inflection point voltage value, starting a terminal calibration strategy, and controlling the SOC_{Ah_correct_zero_no}And SOC_{Ah_correct_zero}Start of separation, SOC_{Ah_correct_zero}And SOC_TrueMeet at the ends and equal to 0%. While SOC_{Ah_correct_zero_no}Although also dropping, the rate does not catch up with the SOC_TrueWhen the battery reaches the cut-off voltage, the SOC display value is about 10%, and the battery is not discharged. Therefore, through the algorithm, the accurate estimation of the residual electric quantity of the battery is realized, the potential safety hazard that the SOC jump possibly causes in the driving process of the electric automobile is avoided, the problem that the SOC does not display zero under the condition of battery electric discharge is avoided, and the user experience is improved.

Claims

1. A power battery SOC estimation method is characterized by comprising the following steps:

3) When the current collected by the BMS is less than or equal to the current threshold A₁Correcting SOC through SOC-OCV curve_True(ii) a The current collected by BMS is larger than current threshold A₁Calculating the SOC through an ampere-hour integral method formula (3)_True；

In the formula, SOC (k +1)_TrueRepresents the SOC of the battery at the time of k +1True value of (1), SOC (k)_TrueThe real value of the SOC of the battery at the moment k is represented, i represents the current value collected by the BMS, eta represents the charging and discharging efficiency, delta T represents the sampling period of the BMS, and Q_N(T) represents the capacity of the battery at a temperature of T ℃, and k represents the time k;

in the formula, SOC (k +1)_DispalyDisplay value representing SOC of battery at time k +1, SOC (k)_DispalyShowing the SOC display value at the time k, i showing the BMS collected current value, eta showing the charge-discharge efficiency, delta T showing the BMS sampling period, and Q_N(T) represents the capacity of the battery at a temperature of T ℃, and k represents the time k;

during discharging:

during charging:

where error (k) is the error between the true value of SOC and the displayed value, SOC (k)_TrueRepresenting the true value of the SOC of the battery at the k moment;

when SOC is reached_TrueAnd SOC_DisplayEqual time K₂When the value is 1, the SOC is at the moment_DisplayEstimating according to an ampere-hour integral method;

in the charging and discharging process, the BMS collects the voltage of the battery terminal, and when the collected voltage is less than or equal to the inflection point voltage V₁After-starting discharge end zero calibration strategy of SOC (system on chip), accelerating or reducing SOC_DisplayIs the rate of change of (i.e. SOC)_DisplayAnd a dynamic zero correction factor K₁(k) Equations (7) and (8), respectively, make SOC_DisplayFollow-up SOC_True；

In the formula, SOC (k +1)_DispalyDisplay value representing battery SOC at time k +1, SOC (k)_DispalyIndicating the value of the battery SOC at time K, K₁(k) For the dynamic zero calibration factor, i represents the current value collected by the BMS, eta represents the charge-discharge efficiency, delta T represents the sampling period of the BMS, and Q_N(T) represents the capacity of the battery at a temperature of T ℃, k represents the time k,

error (k) is the error between the true and displayed value of SOC for the discharged capacity of the cell from the knee voltage to the discharge cutoff voltage.