Health Factor Experimental Testing and SOH Estimation for Nickel Cadmium Batteries

Based on the fact that the working characteristics of Nickel–cadmium battery are significantly different from lithium batteries, and in view of the complex discharge conditions of Nickel–cadmium battery in Multiple Unit, a SOH estimation method of Nickel–cadmium battery based on charging health factors was proposed. Firstly, aging experiments were conducted on individual nickel cadmium batteries using an experimental platform; Secondly, health factors such as total charging time, constant voltage charging time at the end of constant current charging, and average current during a specific time period of constant voltage charging were selected as health factors at different aging stages. The rationality of the health factors was verified through correlation analysis; Then, principal component analysis was used to reduce the dimensionality of the health factor data matrix as input to the support vector regression model, and cross validation was used to optimize the model parameters; Finally, the proposed method was validated based on experimental data.


Introduction
Nickel cadmium batteries have become the most commonly used batteries in EMU auxiliary power supply system due to their advantages such as long lifespan, low price, resistance to overcharging, high and low temperature resistance, and the ability to charge at high rates [1,2].Estimating and predicting the state of health (SOH) for nickel cadmium batteries is a core issue in battery management systems [3].In order to achieve online estimation of battery health state, SOH estimation based on real-time collected current, voltage, and temperature has become one of the research hotspots in recent years.LIU et al. [4] extracted the voltage drop during the same time period during battery discharge process as a health factor to estimate the health state of the battery; Zhou et al. [5] used the average voltage decay during the discharge process to reflect the health state of the battery.Although the above methods have good estimation performance under specific operating conditions, the operating environment of different batteries varies greatly, and the actual estimation is not ideal.Therefore, Guo et al. [6] charged the battery to the set voltage and let it stand for 10 minutes, extracted terminal voltage drop information as a health factor, and estimated the attenuation of the battery; YE et al. [7] extracted appropriate and easily measurable indirect health factors from current, voltage, and temperature curves, and proposed a battery health prediction method that combines indirect health factors and Gaussian process regression models.The above studies have extracted health factors from the battery charging process, but some studies only considered a single health factor and are limited to the constant current charging process.There is little consideration given to the factors in the constant voltage charging stage, making it difficult to apply to the working conditions of direct 110V bus connection of nickel cadmium batteries in high-speed trains; Moreover, the working characteristics of nickel cadmium batteries are significantly different from those of lithium batteries [8,9].The existing health factors and health estimation methods for lithium batteries are difficult to apply to nickel cadmium batteries, and they are also unable to cope with the health state estimation under high-speed train conditions.In addition, Cheng et al. [2] provided an assessment of the life status of nickel cadmium batteries in high-speed trains, while DAI et al. [3] provided a prediction of the life of nickel cadmium batteries, providing reference for battery life evaluation and optimization of maintenance strategies.However, it is difficult to achieve quantitative and accurate SOH estimation.To address the above issues, a SOH estimation method for nickel cadmium batteries in high-speed trains based on charging health factors is proposed.Considering that the discharge state of the battery pack in the actual working conditions of the high-speed train unit is irregular, a health factor is constructed by extracting the total charging time during the charging phase, the charging time at the specified voltage range at the end of the constant current charging period, and the average current at a specific time range of constant voltage charging.Then, experimental verification was conducted on a self-built experimental platform.

Battery samples and experimental platform
The battery aging experimental platform is shown in Fig. 1, which includes a battery detection system that can achieve battery cycling and capacity calibration, an auxiliary channel that can collect real-time voltage, current, and temperature signals during battery operation, aincubator that can provide a constant temperature environment, and an upper computer that visualizes and stores the results.Three LPH160A nickel cadmium batteries with a nominal capacity of 160Ah were selected as the experimental objects.The model of the battery is the Hexie CRH3 Multiple Unit, with a nominal voltage of 1.2V, a charging cutoff voltage of 1.7V, and a discharge cutoff voltage of 1.0V.

Experimental Scheme
To obtain complete battery capacity decay data, three new nickel cadmium batteries (numbered 1 #, 2 #, and 3 # below) were selected for 100% DOD aging cycle experiments at 25 ℃ .They were uniformly charged at 0.5C constant current and discharged at 0.25C to 1.0V, and then cycled 100 times.Use each discharge capacity as the capacity value of the battery under the current number of cycles.In addition, a thermocouple wire is pasted with insulating tape at the center of the four sides of the battery casing and connected to an auxiliary channel for measuring the battery temperature.The final calibrated battery temperature is the average of the four surface temperatures.The specific steps are as follows: Step 1: Set aside for 5 minutes; Step 2: Charge at 0.5C constant current to a cut-off voltage of 1.7V; Step 3 Charge at a constant voltage of 1.7V to a cut-off current of 3.2A; Step 4: Set aside for 5 minutes; Step 5: Discharge at a constant current of 0.25C to a cut-off voltage of 1.0V.In addition, considering that the actual discharge depth of the battery pack during the operation of the high-speed train is shallow and the working conditions are variable, three additional aging experiments were conducted to change the discharge rate and depth.The experimental conditions are set as shown in Table 1 The charging method still adopts 0.5C constant current and constant voltage charging.Firstly, use 0.5C current to charge to the cut-off voltage of 1.7V, and set the cut-off current to 3.2A during the constant voltage charging stage.Each group of experiments is cycled 460 times, and the battery capacity is tested after every 20 cycles.The capacity calibration is carried out using the following method: Step 1: Set aside for 5 minutes; Step 2: Charge 0.2C constant current to a cut-off voltage of 1.7V; Step 3 Charge at a constant voltage of 1.7V to a cut-off current of 3.2A; Step 4: Set aside for 5 minutes; Step 5: Constant current discharge at 0.2C to cut-off voltage of 1.0V.

Definition of SOH
State of health (SOH) is an important indicator for measuring battery life, usually expressed as the ratio of the current measured capacity to the rated capacity.The health state of a new battery is 1.The SOH of a battery is defined as [10]: In the formula, Now C represents the amount of electricity discharged by the battery from its fully charged state at the current moment, using a specified constant current discharge to the cut-off voltage; New C is the nominal capacity of the battery at the time of delivery.

Selection of health factors
In the actual operating conditions of high-speed trains, changes in the electrical load and conditions on the vehicle result in variable discharge conditions of the nickel cadmium battery pack, making it difficult to select parameters from the discharge curve to characterize battery capacity.Therefore, health factors cannot be extracted from them for modeling and estimation.However, most models of high-speed trains currently use current limiting and constant voltage charging to charge the batteries.The charging mode of the batteries on high-speed trains is relatively uniform and fixed, so it is considered to extract health factors from the charging curve.Multiple health factors were selected from the charging current, voltage, and temperature curves of the complete aging cycle experimental data: (1) H1: total charging time ; (2) H2: Constant current charging with equal amplitude voltage (1.45-1.7V)charging time; (3) H3: The average current during the constant voltage charging stage of 100-300s; (4) H4: The ratio of charging time between two stages; (5) H5: The integral of the temperature curve over time.In order to quantify the correlation between health factors and capacity, this article uses Pearson and Spearman correlation coefficients to measure the correlation between selected health factors and capacity [12].
(2) (3) The larger the absolute value of the correlation coefficient, the higher the degree of correlation between the two variables.Based on the aging experimental data in Section 1, substitute the health factor and capacity sample data into X and Y, respectively; x i and y i represent samples, and the calculation results are shown in Table 2.  2, namely the total charging time H1, the constant voltage (1.45-1.7V)charging time H2 at the end of constant current charging, and the average current H3 at the constant voltage charging stage of 100-300s.

Support Vector Regression Model
Support Vector Regression (SVR) [12] is a model for achieving data estimation and prediction, with the goal of minimizing the total deviation of all sample points from the hyperplane and searching for the global optimal solution based on the structural risk minimization criterion.SVR requires a small sample size and has the computational ability to process high-dimensional data.It has been successfully applied in time series prediction, cost estimation, function approximation, and other fields.Wherein, a SVR model is used with Gaussian kernel function.

Eigenvalue Selection Optimization
Principal Component Analysis (PCA) [13] is a commonly-used algorithm for dimensionality reduction.Due to the fact that all selected health factors come from the charging process of the batteries and are strongly correlated with capacity data.Therefore, considering the possibility of information overlap between HFs, the PCA method is used to optimize the dimensionality reduction of health factor eigenvalues.The specific steps are as follows: (1) Form H1, H2, and H3 into a matrix of n×3to be dimensionally reduced, where n is the number of samples; (2) Decentralize to subtract the average value of each feature and obtain a standardized matrix X * ; After dimensionality reduction, the principal component contribution rates of each feature vector are shown in Table 3.It can be seen that the contribution rate of principal component 1 is much higher than the other two terms, so principal component 1 is used as a new HF* to input into the dataset.

Cross validation parameter optimization
To determine the parameter values, cross validation is used in the model to automatically find the optimal parameters.The optimization process involves: (1) randomly dividing the training dataset into k disjoint subsets of the same size; Then use the k-1 subset as the new training set, and the remaining subset as the test set.(2) Repeat the possible k choices in the previous step (selecting a different subset for the test set each time).(3) For k models, each model calculates test errors on the corresponding test set, and taking the average of k test errors yields a cross validation error.When selecting a model, assuming that the model has many adjustable parameters, a set of parameters determines a model.By calculating its cross validation error, the set of parameters that minimizes the cross validation error is selected as the optimization parameter.

SOH Estimation Results
The health factors (H1, H2, H3) extracted from the charging curve of nickel cadmium batteries are considered as eigenvalues, and principal component analysis is used to optimize the dimensionality of the eigenvalues vector.The dataset will be divided into training and testing sets.Firstly, input the training set to train the SVR model, and use cross validation methods to find the optimal C and g parameters to obtain the hyperplane and prediction function of the SVR model.The final input test set allows the model to estimate SOH using the health factor sequence of the test set.The overall process is shown in Fig. 2. IOP Publishing doi:10.1088/1742-6596/2706/1/0120406 set to verify the predictive accuracy of the SVR model.Figure 4 shows the SOH estimation results obtained under the verification of three battery test sets.From Fig. 3, it can be seen that the predicted values obtained after inputting the three batteries into the model can well reflect the trend of capacity decay of nickel cadmium batteries, and accurately track the multiple capacity rebound phenomena that occur during the aging process.From the scatter plot on the right, it can be seen that the estimated values are relatively close to the true values, with a maximum relative error of only about 2%.For the overall SOH estimation model, two evaluation indicators can be used: root mean square error (RMSE) and mean absolute error (MAE): In the equation, is the actual capacity value of the i-th sample battery, ̃ is the estimated battery capacity.Table 4 lists   Based on the estimated results in Fig. 5 and Table 5, it can be seen that, due to the fact that these three aging experiments use a capacity calibration method every 20 cycles, the capacity collection interval is large, resulting in a small sample data capacity.Even if 60% of the data is selected as the training set, there are only 14 groups.Therefore, the SVR model's estimation of capacity attenuation under these three working conditions is not as effective as the first three sets of data, but the maximum relative error is still less than 4%, with average absolute errors of 1. relative errors of 0.77%, 0.79%, and 1.37%, respectively.The root mean square errors are 1.7199, 1.5487, and 2.5894, respectively.Considering that the nominal capacity of the battery used in the experiment is relatively large (160Ah), this error value is still within an acceptable range, and the capture of capacity rebound phenomenon is also relatively accurate.
The eigenvalues, α i and λ i , i=1,2,…, k, can be obtained from Eq. (5); The contribution rates of each component can be obtained by Eq. (6
the calculation results of MAE and RMSE, and the MAE and RMSE calculation results for the three battery data are far less than 1.In order to verify the generalization of the model under various operating conditions, aging experimental data under different operating conditions were substituted into the model for verification.Select the capacity and health factor data obtained from cycling under three different operating conditions in Section 1.2, with the previous 60% data as the training set and the remaining 40% data as the test set.The results are shown in Fig.4.(a) 1# battery (b) 2# battery (c) 3# battery Figure 3 Results of SOH estimation.

Table 1 .
: Working conditions of aging experiments.

Table 2 .
Correlation between health factors and health state.To ensure that the extracted health factors can adapt to different operating conditions of nickel chromium batteries, parameters with Pearson and Spearman correlation coefficient values greater than 0.85 were selected as health factors from Table

Table 4 .
Estimation errors.Table5.Estimation errors under three conditions.In response to the complex discharge situation of nickel cadmium batteries in the operating conditions of high-speed trains, health factors were extracted from the constant current and constant voltage charging stages, and their good characterization of SOH was verified through correlation.The absolute values of the Person and Spearman coefficients were not less than 0.9011 and 0.8868, respectively, indicating that the extracted health factors can fully reflect the SOH of nickel cadmium batteries;2) The support vector regression model based on charging health factors constructed can better reflect the capacity decay trend of nickel cadmium batteries and track the capacity recovery phenomenon during aging.For nickel cadmium battery cells with a nominal capacity of 160Ah, the estimated RMSE value of the results should not exceed 0.8019, MAE value should not exceed 0.6361, and average relative error should not exceed 0.4% when there is a large amount of sample data; 3) In order to save capacity calibration time and reduce the additional attenuation of the battery caused by calibration, a capacity calibration method is adopted every 20 cycles.Under different operating conditions and low data volume, the estimated RMSE value does not exceed 2.5894, MAE value does not exceed 2.2006, and the average relative error does not exceed 1.37%.This indicates that the SOH estimation method for nickel cadmium batteries proposed in this paper also has good accuracy for small sample data.LI Jing, LI Jigang,SUNDongdong, et al.Application status and development direction of battery for EMU [J].Rolling Stock, 2023,61(1):1-6.[2] XU Haiming, YU Tianjian, CHEN Chunyang, et al.State-of-charge prediction model for Ni-Cd batteries considering temperature and noise [J].Apllied Sciences, 2023, 13(11): 6494.[3] DAI Yi, CHENG Shu, GAN Qinjie, et al.Life prediction of Ni-Cd battery based on linear Wiener process [J].Journal of Central South University, 2021, 28: 2919-2930.[4] LIU Datong, ZHOU Jianbao, LIAO Haitao, et al.A health indicator extraction and optimization framework for lithium-ion battery degradation modeling and prognostics [J].IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2015, 45(6): 915-928.[5] ZHOU Yapeng, HUANG Miaohua, CHEN Yupu.A novel health indicator for on-line lithium-ion batteries remaining useful life prediction [J].Journal of Power Sources, 2016,321:1-10.[6] GUO Yongfang, HUANG Kai, LI Zhigang.Fast state of health prediction of lithium-ion battery YE Yifu, ZHOU Zhe, CAI Zhiduan, et al.State of health estimation of lithium-ion batteries based on indirect health indicators and Gaussian process regression model[C].2021IEEE 10th Data Driven Control and Learning Systems Conference, Suzhou, China, 2021.[8] SENTHILKUMAR M, SATYAVANI T V S L, RAMAN M V, et al.Effect of temperature and discharge rate on electrochemical performance of fiber Ni-Cd cell[J].Russian Journal of Electrochemistry, 2022,58(1):43-49.