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CN110045288A - A kind of capacity of lithium ion battery On-line Estimation method based on support vector regression - Google Patents

A kind of capacity of lithium ion battery On-line Estimation method based on support vector regression Download PDF

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CN110045288A
CN110045288A CN201910435035.0A CN201910435035A CN110045288A CN 110045288 A CN110045288 A CN 110045288A CN 201910435035 A CN201910435035 A CN 201910435035A CN 110045288 A CN110045288 A CN 110045288A
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battery
internal resistance
capacity
equivalent internal
ion battery
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CN110045288B (en
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谭晓军
谭雨晴
范玉千
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Guangzhou Silinger Technology Co ltd
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Sun Yat Sen University
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/367Software therefor, e.g. for battery testing using modelling or look-up tables
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/385Arrangements for measuring battery or accumulator variables
    • G01R31/387Determining ampere-hour charge capacity or SoC
    • G01R31/388Determining ampere-hour charge capacity or SoC involving voltage measurements

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  • Charge And Discharge Circuits For Batteries Or The Like (AREA)
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Abstract

本发明涉及一种基于支持向量回归的锂离子电池容量在线估计方法。该方法包括以下步骤:S1.对锂离子电池进行循环寿命测试,获取相应的直流等效内阻谱;S2.基于直流等效内阻谱提取能够反映锂离子电池性能退化的潜在健康因子,对其进行相关性分析;S3.基于支持向量回归算法构建容量估计模型;S4.获取待估计电池当前的充电数据以及充电等效内阻曲线;S5.根据所建立的容量估计模型,由提取的健康因子参数值确定电池的当前容量。本方法解决了不同循环工况下锂离子电池的容量在线估计问题,估计精度高且适应性强。

The invention relates to an online capacity estimation method of lithium ion battery based on support vector regression. The method includes the following steps: S1. Perform a cycle life test on the lithium-ion battery, and obtain a corresponding DC equivalent internal resistance spectrum; S2. Extract a potential health factor that can reflect the performance degradation of the lithium-ion battery based on the DC equivalent internal resistance spectrum. It performs correlation analysis; S3. Constructs a capacity estimation model based on the support vector regression algorithm; S4. Obtains the current charging data of the battery to be estimated and the charging equivalent internal resistance curve; S5. According to the established capacity estimation model, the extracted health The factor parameter value determines the current capacity of the battery. The method solves the problem of on-line estimation of the capacity of lithium-ion batteries under different cycle conditions, and has high estimation accuracy and strong adaptability.

Description

一种基于支持向量回归的锂离子电池容量在线估计方法An online estimation method of lithium-ion battery capacity based on support vector regression

技术领域technical field

本申请涉及电池管理和电池状态分析领域,特别是一种基于支持向量回归的锂离子电池容量在线估计方法。The present application relates to the field of battery management and battery state analysis, in particular to an online estimation method of lithium-ion battery capacity based on support vector regression.

背景技术Background technique

锂离子电池由于其能量密度高、循环寿命长、安全性高等特点,被广泛应用于电动汽车领域。随着锂离子电池在使用过程中循环次数的增加,其各方面外部特性会出现劣化,具体表现为有效容量减少、充放电内阻增大等。Lithium-ion batteries are widely used in electric vehicles due to their high energy density, long cycle life, and high safety. As the number of cycles of a lithium-ion battery increases during use, its external characteristics will deteriorate in all aspects, such as a decrease in effective capacity and an increase in charge-discharge internal resistance.

锂离子电池的容量估计是电池管理的核心问题之一,在判断电池当前劣化状态、预估电池剩余使用寿命、避免电池提前失效等方面都具有重要意义。然而,在实际应用中,电池的容量一般通过对电池进行满充满放的离线测试方法获得,无法针对电池的实时容量变化进行更新。The capacity estimation of lithium-ion batteries is one of the core issues of battery management. It is of great significance in judging the current deterioration state of the battery, estimating the remaining service life of the battery, and avoiding premature battery failure. However, in practical applications, the capacity of the battery is generally obtained by an offline test method of fully discharging the battery, and it is impossible to update the real-time capacity change of the battery.

发明内容SUMMARY OF THE INVENTION

针对上述问题,本发明提供一种基于支持向量回归的锂离子电池容量在线估计方法,通过在线获取电池充电等效内阻值,作为容量估计模型的输入,从而实时估计电池的有效容量。In view of the above problems, the present invention provides an online capacity estimation method of lithium ion battery based on support vector regression, which can estimate the effective capacity of the battery in real time by obtaining the equivalent internal resistance value of battery charging online as the input of the capacity estimation model.

本发明的技术方案包括以下步骤:The technical scheme of the present invention comprises the following steps:

S1.对锂离子电池进行循环寿命测试,获取相应的直流等效内阻谱。S1. Perform a cycle life test on the lithium-ion battery to obtain the corresponding DC equivalent internal resistance spectrum.

(1)对锂离子电池进行循环充放电测试。设定循环寿命测试的实验条件,利用固定环境温度与固定充放电倍率,对电池进行不间断循环充放。(1) Cyclic charge-discharge test for lithium-ion batteries. Set the experimental conditions for the cycle life test, and use a fixed ambient temperature and a fixed charge-discharge rate to charge and discharge the battery in an uninterrupted cycle.

(2)对锂离子电池进行评测。每循环一段时间后,暂停循环,并对电池进行容量评测与充电等效内阻评测,获得电池有效容量值、充电等效内阻曲线。其中,充电等效内阻由如下公式计算:(2) Evaluate lithium-ion batteries. After each cycle for a period of time, the cycle is suspended, and the capacity evaluation and charging equivalent internal resistance evaluation are performed on the battery to obtain the effective capacity value of the battery. , charging equivalent internal resistance curve. Among them, the equivalent internal resistance of charging is calculated by the following formula:

(1) (1)

其中,为电池在充电带载工作时的端电压,称为充电工作电压, 为电池在充分搁置后的端电压,称为充电开路电压,为充电过程中实时监测到的电流大小。in, It is the terminal voltage of the battery when it is charging and working with load, which is called the charging working voltage. is the terminal voltage of the battery after it has been fully shelved, called the charging open circuit voltage, It is the current size monitored in real time during the charging process.

(3)判断循环寿命测试的终止条件。记 为电池的标称容量。若电池有效容量值与标称容量的比值小于0.7,则停止电池循环寿命测试,否则,返回步骤(1)继续测试。(3) Determine the termination condition of the cycle life test. remember is the nominal capacity of the battery. If the ratio of the effective capacity of the battery to the nominal capacity If it is less than 0.7, stop the battery cycle life test, otherwise, return to step (1) to continue the test.

通过步骤(1)的循环充放电测试,使锂离子电池加速劣化,通过步骤(2)的评测测试,获得相应劣化程度下的有效容量值与充电等效内阻谱,其中,充电等效内阻谱是横坐标为电池荷电状态,纵坐标为充电等效内阻的曲线。通过完成循环寿命测试,获得由多条充电等效内阻曲线组成的锂离子电池全生命周期内阻谱系列。Through the cyclic charge-discharge test in step (1), the deterioration of the lithium-ion battery is accelerated, and through the evaluation test in step (2), the effective capacity value and the charging equivalent internal resistance spectrum under the corresponding degree of deterioration are obtained. The resistance spectrum is the curve of the battery state of charge on the abscissa and the equivalent internal resistance on the ordinate. By completing the cycle life test, a series of internal resistance spectra of lithium-ion batteries in the whole life cycle composed of multiple charging equivalent internal resistance curves were obtained.

S2. 基于直流等效内阻谱提取能够反映锂离子电池性能退化的潜在健康因子,对其进行相关性分析。S2. Extract potential health factors that can reflect the performance degradation of lithium-ion batteries based on DC equivalent internal resistance spectrum, and perform correlation analysis on them.

(1)利用线性插值的方法,从荷电状态为0开始,每隔10%提取不同劣化状态下电池样本的充电等效内阻,获得9个潜在健康因子。(1) Using the method of linear interpolation, starting from the state of charge of 0, the equivalent internal resistance of the battery samples under different deterioration states is extracted every 10%, and 9 potential health factors are obtained.

(2)对充电等效内阻与荷电状态做差分运算,获得充电等效内阻增量曲线,提取不同劣化状态下电池样本的内阻增量峰值及其峰值点,获得2个潜在健康因子。(2) Do a differential operation between the charging equivalent internal resistance and the state of charge, obtain the charging equivalent internal resistance increment curve, extract the peak value of the internal resistance increment and its peak point of the battery samples under different deterioration states, and obtain 2 potential health factor.

(3)选择容量衰减量作为电池劣化程度的衡量指标,容量衰减定义如下:(3) Select the amount of capacity attenuation As a measure of battery deterioration, capacity fade is defined as follows:

(2) (2)

(3) (3)

其中, 为电池的标称容量,为电池在某次循环下的有效容量值, 为表征电池劣化状态的衡量指标。in, is the nominal capacity of the battery, is the effective capacity value of the battery under a certain cycle, It is a measure to characterize the deterioration state of the battery.

(4)利用Spearman相关系数对上述步骤(1)、(2)获得的11个潜在健康因子及劣化指标做相关性分析。(4) Using the Spearman correlation coefficient to compare the 11 potential health factors and deterioration indicators obtained in the above steps (1) and (2) Do a correlation analysis.

S3. 基于支持向量回归算法构建容量估计模型。S3. Build a capacity estimation model based on the support vector regression algorithm.

根据步骤S2中的相关性分析结果,选取相关系数大于0.8的健康因子作为输入量,选取相应劣化程度下的有效容量值作为输出量,采用支持向量回归算法构建容量估计的非线性回归模型。According to the correlation analysis result in step S2, select the health factor with the correlation coefficient greater than 0.8 as the input quantity, select the effective capacity value under the corresponding deterioration degree as the output quantity, and use the support vector regression algorithm to construct a nonlinear regression model for capacity estimation.

S4. 获取待估计电池当前的充电数据以及充电等效内阻曲线,所使用的充电方法与步骤S1中的相同。S4. Obtain the current charging data and the charging equivalent internal resistance curve of the battery to be estimated, and the charging method used is the same as that in step S1.

S5. 根据步骤S3建立的容量估计模型,由提取的健康因子参数值确定电池的当前容量。S5. According to the capacity estimation model established in step S3, the current capacity of the battery is determined by the extracted health factor parameter value.

根据步骤S3所建立的非线性回归模型,以步骤S4提取的健康因子参数值作为输入量,通过模型计算获得输出的有效容量值,即为电池的当前容量。According to the nonlinear regression model established in step S3, the health factor parameter value extracted in step S4 is used as the input, and the output effective capacity value is obtained through model calculation, which is the current capacity of the battery.

本发明的有益效果:与现有技术相比,本发明方法中建立了一种基于支持向量回归算法对处于不同循环工况下的锂离子电池进行容量建模,实现了锂离子电池健康状态的在线估计;支持向量回归,较之其他算法,是专门解决有限样本情况的机器学习,同时可以解决高维问题,有计算复杂度低、低泛化误差、容易解释等优点。本发明可在不需要了解电池内部特性的情况下,仅通过电池所表现的外部特性便可进行老化状态的监测,可操作性强。Beneficial effects of the present invention: Compared with the prior art, the method of the present invention establishes a capacity modeling based on the support vector regression algorithm for the lithium ion battery under different cycle conditions, and realizes the health status of the lithium ion battery. Online estimation; support vector regression, compared with other algorithms, is a machine learning that specializes in solving limited sample situations, and can solve high-dimensional problems at the same time. It has the advantages of low computational complexity, low generalization error, and easy interpretation. The invention can monitor the aging state only through the external characteristics of the battery without knowing the internal characteristics of the battery, and has strong operability.

附图说明Description of drawings

下面结合附图和实施例对本发明进一步说明:Below in conjunction with accompanying drawing and embodiment, the present invention is further described:

图1是本发明的整体流程图;Fig. 1 is the overall flow chart of the present invention;

图2是循环寿命测试的实验条件;Fig. 2 is the experimental condition of cycle life test;

图3是锂离子电池的直流等效内阻谱;Figure 3 is the DC equivalent internal resistance spectrum of the lithium-ion battery;

图4是锂离子电池的直流等效内阻增量曲线;Figure 4 is the DC equivalent internal resistance increment curve of the lithium-ion battery;

图5是利用支持向量回归算法建立容量估计模型的估计结果;Fig. 5 is the estimation result that utilizes the support vector regression algorithm to establish the capacity estimation model;

图6是利用Spearman相关系数对11个潜在健康因子及劣化指标 进行相关性分析表格;Figure 6 shows the correlation between 11 potential health factors and deterioration indicators using the Spearman correlation coefficient. Make a correlation analysis table;

图7是锂离子电池容量估计模型性能评价指标表格。FIG. 7 is a table of performance evaluation indexes of a lithium-ion battery capacity estimation model.

具体实施方式Detailed ways

本发明提供的一种基于支持向量回归的锂离子电池容量在线估计方法,整体流程图如图1所示。根据本发明的实施例,具体包括以下步骤:The present invention provides an online capacity estimation method of lithium ion battery based on support vector regression, and the overall flow chart is shown in FIG. 1 . According to an embodiment of the present invention, the following steps are specifically included:

S1.对锂离子电池进行循环寿命测试,获取相应的直流等效内阻谱。S1. Perform a cycle life test on the lithium-ion battery to obtain the corresponding DC equivalent internal resistance spectrum.

在进行循环寿命测试时,要控制充放电倍率、放电区间、放电深度及温度条件,对电池进行不间断循环充放。同类型锂离子电池不同工况下的样本,充放电倍率设置有0.5C、1C;放电深度设置有0.30、0.60;放电区间设置为上段(平均SOC为85%)、中段(平均SOC为50%)和下段(平均SOC为30%);温度条件设置为40℃。具体实验条件设置如图2所示。During the cycle life test, the charge and discharge rate, discharge interval, discharge depth and temperature conditions should be controlled, and the battery should be charged and discharged in an uninterrupted cycle. For samples of the same type of lithium-ion battery under different working conditions, the charge-discharge rate is set to 0.5C, 1C; the discharge depth is set to 0.30, 0.60; the discharge interval is set to the upper section (average SOC is 85%), middle section (average SOC is 50%) ) and the lower segment (average SOC is 30%); the temperature condition is set to 40°C. The specific experimental conditions are set as shown in Figure 2.

每循环一段时间后,暂停循环,并对电池进行容量评测与充电等效内阻评测,获得电池有效容量值、充电等效内阻曲线。由于锂离子电池小于0.7时,其性能已不满足电动汽车的使用需求。因此,当小于0.7时,即可结束循环寿命测试,其中,为电池的标称容量。After each cycle for a period of time, the cycle is suspended, and the capacity evaluation and charging equivalent internal resistance evaluation are performed on the battery to obtain the effective capacity value of the battery. , charging equivalent internal resistance curve. Due to the lithium-ion battery When it is less than 0.7, its performance can no longer meet the needs of electric vehicles. Therefore, when When it is less than 0.7, the cycle life test can be ended, among which, is the nominal capacity of the battery.

通过充电内阻评测,可以得到等效内阻谱,内阻谱反映了随着电池老化,电池等效内阻也会逐渐增大,如图3所示。Through the evaluation of charging internal resistance, the equivalent internal resistance spectrum can be obtained. The internal resistance spectrum reflects that as the battery ages, the equivalent internal resistance of the battery will gradually increase, as shown in Figure 3.

基于等效内阻谱可以提取电池的健康因子,包括不同SOC下的内阻、内阻增量峰值及峰值点。图4为内阻增量峰值及峰值点在电池不同劣化状态的体现。Based on the equivalent internal resistance spectrum, the health factor of the battery can be extracted, including the internal resistance under different SOC, the peak value of the internal resistance increment and the peak point. Figure 4 shows the peak value and peak point of the internal resistance increment in different deterioration states of the battery.

S2.基于直流等效内阻谱提取能够反映锂离子电池性能退化的潜在健康因子,对其进行相关性分析。S2. Extract potential health factors that can reflect the performance degradation of lithium-ion batteries based on the DC equivalent internal resistance spectrum, and perform correlation analysis on them.

在本步骤中,首先利用线性插值的方法,以10%SOC间隔提取不同劣化状态电池样本的等效直流内阻,共取得9个潜在健康因子;对内阻和SOC做差分运算得到内阻增量曲线,提取不同劣化状态电池样本的内阻增量峰值及其峰值点,该步骤获得2个潜在健康因子;选择容量衰减作为电池劣化程度的衡量指标;利用Spearman相关系数对上述获得的11个潜在健康因子及容量衰减做相关性分析,如图6所示。In this step, firstly, the method of linear interpolation is used to extract the equivalent DC internal resistance of battery samples with different deterioration states at 10% SOC intervals, and a total of 9 potential health factors are obtained; Quantitative curve, extract the peak value of internal resistance increment and its peak point of battery samples in different deterioration states, this step obtains 2 potential health factors; select capacity decay as a measure of battery deterioration degree; use Spearman correlation coefficient to compare the 11 obtained above Potential health factors and volume decay Do a correlation analysis, as shown in Figure 6.

S3.根据步骤S2中相关性分析的结果,选取与容量相关的健康因子,基于支持向量回归算法利用电池训练样本构建容量估计模型。S3. According to the result of the correlation analysis in step S2, select the health factor related to the capacity, and use the battery training sample to construct a capacity estimation model based on the support vector regression algorithm.

在本实例中,选择相关系数大于0.8的特征:R(SOC=0.7、R(SOC=0.8)、内阻增量峰值和峰值点作为模型的输入。支持向量回归较之其他算法,是专门解决有限样本情况的机器学习,同时可以解决高维问题,有计算复杂度低、低泛化误差、容易解释等优点,同时,锂离子电池的劣化程度与其不同工作环境下的内阻变化不成线性关系,利用支持向量回归算法能够在不需要了解电池内部特性变化的情况下得到锂离子电池的劣化情况。In this example, features with a correlation coefficient greater than 0.8 are selected: R(SOC=0.7, R(SOC=0.8), the peak value of the internal resistance increment and the peak point as the input of the model. Compared with other algorithms, support vector regression is a special solution to Machine learning with limited samples can solve high-dimensional problems at the same time, and has the advantages of low computational complexity, low generalization error, and easy interpretation. At the same time, the degree of deterioration of lithium-ion batteries is not linear with the change of internal resistance in different working environments. , the use of the support vector regression algorithm can obtain the deterioration of the lithium-ion battery without knowing the change of the internal characteristics of the battery.

S4.获取待估计电池当前的充电数据以及充电等效内阻曲线,所使用的充电方法与步骤S1中的相同。S4. Obtain the current charging data and the charging equivalent internal resistance curve of the battery to be estimated, and the charging method used is the same as that in step S1.

S5.根据步骤S3建立的容量估计模型,由提取的健康因子参数值确定电池的当前容量。图5是选取70%的样本作为训练集30%的样本作为测试集的支持向量回归结果。图7为锂离子电池容量估计模型性能评价指标,相关误差较少,估计精度高,本估计方法实用可靠。S5. According to the capacity estimation model established in step S3, the current capacity of the battery is determined by the extracted health factor parameter value. Figure 5 shows the support vector regression results of selecting 70% of the samples as the training set and 30% of the samples as the test set. Figure 7 shows the performance evaluation index of the lithium-ion battery capacity estimation model. The correlation error is small and the estimation accuracy is high. The estimation method is practical and reliable.

Claims (6)

1. A lithium ion battery capacity online estimation method based on support vector regression is characterized by comprising the following steps:
s1, carrying out cycle life test on a lithium ion battery to obtain a corresponding direct current equivalent internal resistance spectrum;
s2, extracting potential health factors capable of reflecting the performance degradation of the lithium ion battery based on the direct current equivalent internal resistance spectrum, and performing correlation analysis on the potential health factors;
s3, constructing a capacity estimation model based on a support vector regression algorithm;
s4, acquiring the current charging data and the equivalent internal resistance curve of the battery to be estimated, wherein the charging method is the same as that in the step S1;
and S5, determining the current capacity of the battery according to the capacity estimation model established in the step S3 and the extracted health factor parameter value.
2. The online lithium ion battery capacity estimation method based on support vector regression according to claim 1, characterized in that: the specific steps of performing the cycle life test on the lithium ion battery in the step S1 and obtaining the corresponding direct current equivalent internal resistance spectrum include:
(1) carrying out cyclic charge and discharge tests on the lithium ion battery, setting experimental conditions of the cyclic life tests, and carrying out uninterrupted cyclic charge and discharge on the battery by utilizing fixed environment temperature and fixed charge and discharge multiplying power;
(2) evaluating the lithium ion battery, pausing the cycle after each cycle for a period of time, and carrying out capacity evaluation and charging equivalent internal resistance evaluation on the battery to obtain the effective capacity value of the batteryAnd a charging equivalent internal resistance curve, wherein the charging equivalent internal resistance is calculated by the following formula:
(1)
wherein,the voltage of the battery at the time of charging and loading operation, called charging operation voltage,the voltage at the terminal of the battery after sufficient rest, referred to as the charge open circuit voltage,to be chargedMonitoring the magnitude of the current in real time in the process;
(3) judging the end condition of the cycle life test and recordingThe nominal capacity of the battery is determined by the ratio of the effective capacity value of the battery to the nominal capacityIf the current battery life is less than 0.7, stopping the battery cycle life test, otherwise, returning to the step (1) to continue the test;
through the cyclic charge and discharge test in the step (1), the lithium ion battery is accelerated to deteriorate, and through the evaluation test in the step (2), an effective capacity value and a charging equivalent internal resistance spectrum under the corresponding deterioration degree are obtained, wherein the charging equivalent internal resistance spectrum is a curve with the abscissa as the battery charge state and the ordinate as the charging equivalent internal resistance; and (4) obtaining the lithium ion battery full life cycle internal resistance spectrum series consisting of a plurality of charging equivalent internal resistance curves by completing the cycle life test.
3. The online lithium ion battery capacity estimation method based on support vector regression according to claim 1, characterized in that: the step S2 is to extract potential health factors capable of reflecting the performance degradation of the lithium ion battery based on the dc equivalent internal resistance spectrum, and the specific steps of performing correlation analysis on the potential health factors include:
(1) extracting the equivalent internal resistance of the battery sample in different degradation states every 10% from the charge state of 0 by using a linear interpolation method to obtain 9 potential health factors;
(2) performing differential operation on the charging equivalent internal resistance and the charge state to obtain a charging equivalent internal resistance increment curve, extracting internal resistance increment peak values and peak value points of battery samples in different degradation states, and obtaining 2 potential health factors;
(3) selecting capacity fadeAs a measure of the degree of battery degradation, the capacity fade is defined as follows:
(2)
(3)
wherein,is the nominal capacity of the battery and,is the value of the effective capacity of the battery at a certain cycle,is a measure for representing the degradation state of the battery;
(4) utilizing Spearman correlation coefficient to carry out on 11 potential health factors and degradation indexes obtained in the steps (1) and (2)And (5) performing correlation analysis.
4. The online lithium ion battery capacity estimation method based on support vector regression according to claim 1, characterized in that: the step of constructing the capacity estimation model based on the support vector regression algorithm in step S3 is as follows: according to the correlation analysis result in the step S2, selecting a health factor having a correlation coefficient greater than 0.8 as an input quantity, selecting an effective capacity value at a corresponding degradation degree as an output quantity, and constructing a nonlinear regression model for capacity estimation by using a support vector regression algorithm.
5. The online lithium ion battery capacity estimation method based on support vector regression according to claim 1, characterized in that: in the step S4, the current charging data and the charging equivalent internal resistance curve of the battery to be estimated are obtained, and the charging method and the charging equivalent internal resistance calculation formula are the same as those in the step S1.
6. The online lithium ion battery capacity estimation method based on support vector regression according to claim 1, characterized in that: the specific steps of step S5 are: according to the nonlinear regression model established in step S3, the value of the health factor parameter extracted in step S4 is used as an input, and an output effective capacity value, that is, the current capacity of the battery, is obtained through model calculation.
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