1 Introduction

With the increasing pressure of environmental protection and the development of electric vehicles, electric vehicles have become the hot spot and trend of automotive industry research [1-3]. Along with the advent of the tide of China's power battery decommissioning, there are more and more researches on decommissioned power batteries [4, 5]. The cascade utilization technology of decommissioned batteries for electric vehicles is of great significance to improve the utilization value of their life cycle, reduce the cost of lithium batteries, and alleviate environmental pollution [6-9].

This paper presents a comprehensive evaluation model of decommissioned power batteries for electric vehicles based on AHP-CRITIC. Firstly, there are two different dimensions of curve characteristics and economic factors to evaluate the decommissioned power batteries. Secondly, an improved K-means clustering analysis model based on health factors of the batteries is established. Furthermore, the economic indicator of decommissioned batteries is evaluated in the economic dimension by AHP-CRITIC method. Finally, a comprehensive evaluation of decommissioned batteries is carried out by totally considering the health and economic factors.

2 Evaluation of Health Factors of Decommissioned Batteries Based on Improved Clustering Analysis 2.1 Extraction of Health Characteristics of Decommissioned Batteries

In the evaluation of decommissioned power batteries, in order to extract the health characteristics of decommissioned power batteries, a 2.5 C constant current discharge test is exploited to test the decommissioned lithium batteries from literature [10, 11]. The stopping time is 10 s, i.e. discharging for 10 s and then silencing for 10 s. Reciprocating tests are carried out on the full impulse decommissioned lithium batteries, and the internal resistance parameters of lithium batteries are recorded when the discharge current jumps. The internal resistance-SOC curve of decommissioned power batteries can be obtained through correlation processing. The health characteristics of decommissioned power batteries can be extracted by analyzing the obtained curves.

Through actual measurement and analysis, it can be seen that since the Ohmic internal resistance-SOC curve satisfies the characteristics of quadratic curve, the internal resistance-SOC quadratic curve can be defined as follows:

$ \left\{ \begin{array}{l} {R_o} = {a_{\rm{p}}}S_{{\rm{oc}}}^2 + {b_{\rm{p}}}{S_{{\rm{oc}}}} + {c_{\rm{p}}}\\ {R_{o, \min }} = \left( {4{a_{\rm{p}}}{c_{\rm{p}}} -{b_{\rm{p}}}} \right)/4{a_{\rm{p}}}\\ {R_{o, \min }} = {b_{\rm{p}}}/2{a_{\rm{p}}} \end{array} \right. $

(1)

where a_p, b_p, c_p are the coefficients of the quadratic curve; R_{o, min}, S_{o, min} are the vertex and abscissa of the quadratic curve, respectively representing the minimum resistance value of the internal resistance-SOC curve and its SOC state.

Therefore, by calculating the first derivative and the second derivative of the curve, the health characteristics of curve slope can be extracted, namely:

$ \left\{ \begin{array}{l} \frac{{d{R_o}}}{{d{S_{{\rm{oc}}}}}} = 2{a_{\rm{p}}}{S_{{\rm{oc}}}} + {b_{\rm{p}}}\\ {a_{\rm{s}}} = \frac{{{d^2}{R_o}}}{{dS_{{\rm{oc}}}^2}} = 2{a_{\rm{p}}} \end{array} \right. $

(2)

Based on the curve characteristics of decommissioned batteries in different health states, the health grade of decommissioned batteries is divided, so as to realize the evaluation of the characteristics of decommissioned batteries. The specific evaluation process is realized by means of clustering.

2.2 Clustering Evaluation Model of Health Factors of Decommissioned Batteries

Clustering Model of Power Battery Based on Gravitational Model.

Gravitation is a kind of interaction characteristic widely existed in nature. This paper introduces the gravitational model into the K-means clustering model. By evaluating the clustering effect, as a condition for the termination of the cluster iteration, it is determined that the given data should be attributed to a corresponding cluster.

Clustering Model of Universal Gravitation:

$ E{I_{\rm{i}}} = W \cdot \frac{{P \cdot {p_{\rm{i}}}}}{{r_i^2}} $

(3)

where, EI (Evaluation Index) is the evaluation index of clustering effect; P and p_i are the values of two data points on the curve respectively; W denotes the evaluation adjustment coefficient, which is usually 0.03-0.08 after optimization calculation, and 0.05 in this paper; r_i denotes the distance between a specific data point and the cluster center.

Where r_i is expressed as

$ {r_i} = \left\| {{x_i}- {c_j}} \right\| $

(4)

where x_i is the i-th data point; c_j is the cluster center of a certain class. x_i is attributed to the category of c_j by using Max (EI_i). After x_i is classified into a specific class, the cluster center needs to be updated, and the corresponding expression is:

$ {c_j} = \frac{1}{{\left| {{c_j}} \right|}}\sum\limits_{j = 1}^{\left| {{c_j}} \right|} {{x_i}} $

(5)

Basic Flow of Improved K-means Clustering Based on Gravitational Model.

The basic flow of the improved K-means clustering algorithm based on the universal gravitational model is: Step 1: randomly select K data sets as the original cluster centers; Step 2: Calculate the distance between the remaining data point objects and the K cluster centers; Step 3: According to the proposed gravitational model, determine the size of the evaluation index EI (in this process, the value of W needs to be determined by the method of data cut-off), and divide the corresponding clusters by calculating the size of the EI, update cluster centers according to Formula (5); Step 4: When the value of indicator EI reaches a certain value or the number of iterations reaches the specified number of times, stop the iteration, otherwise turn to Step 2.

Evaluation of Health Factors of Decommissioned Batteries.

According to the test results of the existing decommissioned power batteries, the internal resistance-SOC curve of the obtained plurality of batteries are clustered and analyzed, and the curves of the decommissioned power batteries under different health conditions can be acquired.

3 Evaluation of Economic Indicators of Decommissioned Batteries Based on AHP-CRITIC 3.1 Classification of Economic Indicators

By comparing the decommissioned power batteries with the conventional energy storage batteries, the economic indicators of batteries are assessed from five aspects: the economic cost value of batteries, the economic time cost value of batteries (i.e. the economic characteristics of batteries in consideration of their life cycle), the national dividend subsidy for the development of batteries, the economic cost under the safety performance of batteries, and the economic cost of operation and maintenance.

3.2 Quantitative Processing of Indicator Characteristics

Economic Value Assessment of Decommissioned Power Batteries.

According to the actual investigation and analysis, it can be known that the evaluation rules for the economic indicators of decommissioned power batteries are shown in Table 1.

Time Cost Assessment of Decommissioned Power Batteries.

Depending on the use of the battery over its life cycle, generally the longer the battery can be used, the lower the cost of its corresponding time investment. To evaluate the battery time cost, the specific quantization rules are shown in Table 2.

Government Policy Dividend Subsidy.

In the process of battery application, it will be subsidized by government policies. It is necessary to quantify the support of policies for batteries [12]. The specific quantification rules are shown in Table 3.

Safety Assessment of Decommissioned Power Batteries.

Generally speaking, the longer the service life of the battery, the more obvious the potential safety hazards and the higher the corresponding economic cost. The specific measurement criteria, evaluation and quantitative rules are shown in Table 4.

Cost Assessment of Operation and Maintenance.

During the actual operation of batteries in the installation site, it is inevitable to invest human, material and financial resources to maintain and repair the battery. Its basic cost assessment is based on investigation and practical application. The specific quantitative rules are shown in Table 5.

3.3 Evaluation of Comprehensive Economic Indicators

In the calculation of index weights, the two main ideas of subjective and objective weighting methods are combined. The subjective weights are calculated by the analytic hierarchy process (AHP) [13] in the subjective weighting methods. The objective weights are calculated by CRITIC (Criteria Importance Though Intercriteria Correlation) [14]. The concrete steps of calculating objective weights are as follows:

Step 1: Dimensionless data processing. Since all the indicators in this evaluation system have been converted into quantitative values, the process can be omitted.

Step 2: Calculate the information content contained in the indicators.

The conflicting quantitative indicator of the jth indicator and other indicators is as follows:

$ {\theta _j} = \sum\limits_{t = 1}^n {\left( {1- {r_{tj}}} \right)} $

(6)

where r_tj is the correlation coefficient between indicator t and indicator j.

The objective weight of each indicator is considered comprehensively by contrast intensity and contradiction. Assuming that C_j represents information content contained in the jth evaluation index, C_j can be denoted as:

$ {C_j} = {\delta _j}\sum\limits_{t = 1}^n {\left( {1 -{r_{tj}}} \right)} $

(7)

where δ_j is the standard deviation of the jth evaluation index.

Step 3: Calculate the objective weight of each index

The larger the C_j value is, the more information the jth evaluation index covers, and the more important the index is. Therefore, the objective weight of the jth index can be characterized as:

$ {\beta _j} = {C_j}/\sum\limits_{j = 1}^m {{C_j}} $

(8)

Combine the subjective weights calculated by AHP with the objective weights calculated by the CRITIC to calculate the comprehensive weight. The formula is as follows:

$ {\omega _j} = \frac{{{{\left( {{{\rm{ \mathsf{ α} }}_j}{{\rm{ \mathsf{ β} }}_j}} \right)}^{1/2}}}}{{\sum\limits_{j = 1}^n {{{\left( {{{\rm{ \mathsf{ α} }}_j}{{\rm{ \mathsf{ β} }}_j}} \right)}^{1/2}}} }} $

(9)

The weighting coefficients of each economic index in the battery evaluation model can be obtained as shown in Table 6.

4 Optimal Evaluation Model of Decommissioned Power Batteries Based on Curve Characteristics and Economic Dimensions 4.1 Objective Function

For the evaluation model of decommissioned power batteries based on clustering analysis, the objective function of the optimal evaluation model is considered from the following two aspects: (1) the characteristics of internal resistance-SOC curve of decommissioned power batteries; (2) the economic index of decommissioned power batteries in the whole life cycle.

Objective Function 1: Evaluation index of curve characteristics of the decommissioned power batteries

$ \min {\mathit{f}_1} = e \cdot \frac{1}{p} \cdot \sum\limits_{m = 1}^p {\frac{{\left| {{x_{im}} -{x_{jm}}} \right|}}{{{x_{im}}}}} $

(10)

where e means the parameter of curve index converted into economic index; p means the point number of curve characteristics; i and j represent the characteristic indexes of characteristic curve and characteristic curve to be evaluated under the datum value respectively.

Objective Function 2: Cost function of decommissioned power batteries under economic indicators

The cost of decommissioned power batteries for energy storage is mainly due to their purchase, transportation, testing, installation and maintenance. Therefore, the economic cost function for decommissioned power batteries is composed as follows: the cost of purchasing power batteries to be decommissioned, the cost of transportation, inspection and installation, and the cost of maintenance.

The weight of the index in item J of WJ is the quantified value of the index in item J of w_j.

$ {f_2} = \sum {{w_j} \cdot {c_j}} $

(11)

In the formula, w_j means the weight of the jth refinement indicator, and c_j represents the user's quantified value for this refinement indicator.

4.2 Constraints

• Upper and lower limit constraints on health factors of decommissioned power batteries

$ SO{H_{low}} < SOH < SO{H_{high}} $

(12)

In the formula, SOH_low and SOH_high are the upper and lower limits of the decommissioned power batteries, respectively, which are taken as 0.7 and 1.0.

• Upper and lower limit constraints of characteristic curve parameters

$ {x_{low}} < x < {x_{high}} $

(13)

In the formula, x_low and x_high are the upper and lower limits of the characteristics of the decommissioned power battery curve, respectively.

• Upper and lower limits constraints of batteries in charged state

$ {c_{low}} < c < {c_{high}} $

(14)

In the formula, c_low and c_high are the upper and lower limits of the economic quantitative indicators of decommissioned power batteries, respectively.

4.3 Model Solution

In order to solve the above multi-objective optimization model, the particle swarm optimization algorithm is used to solve the problem.

5 Example Analysis

Based on the actual data of a city's electric bus charging station, some decommissioned batteries are used as experimental data for example analysis. The power battery used in the charging station is ternary lithium battery, with electric energy of 282 Ah, rated discharge current of 280 A, peak discharge current of 330 A, rated charging current of 140 A, single battery rated voltage of 3.68 V, and operating temperature range of 20-60 ℃. The economic indicators of decommissioned power batteries and conventional energy storage batteries are mainly obtained through the investigation of some energy storage websites and batteries. The cost of conventional energy storage is 2000 yuan/kWh and that of cascaded power batteries is 800 yuan/kWh.

According to the results of clustering, we can see that the test curves of decommissioned batteries can be roughly divided into three different categories, which are embodied as the three different distribution zones in Fig. 1. The characteristic curves of decommissioned batteries are characterized by internal resistance-SOC characteristic curves. According to the evaluation in Ref. [13], the fitting degrees of the corresponding health status are 0.84, 0.75 and 066, and the available capacities of the batteries in this paper are 236.88, 211.5 and 186.12 Ah, the corresponding proportions are 50, 30 and 20%, respectively.

Using Formula (11) to calculate the quantified values of four indicators of each group of decommissioned power batteries in the economic dimension, a total of 30 sets of characteristic index vectors of decommissioned power batteries are obtained.

According to the economic analysis of decommissioned cascade energy storage and conventional energy storage in Ref. [15], the comparison of the total cost of conventional energy storage and cascade energy storage is expressed in Fig. 2 (taking the installation scale of 2000 Ah as an example). The specific evaluation results are as follows.

It can be seen from Fig. 2 that after cascade utilization of energy storage of decommissioned batteries, when the cost of cascade energy storage is less than 940 yuan/kWh, the cost of cascade energy storage is lower than that of conventional batteries. In practical applications, the cost of cascade energy storage is generally much lower than this value, which shows that retired batteries have great advantages over conventional energy storage.

Among the 30 groups of decommissioned battery packs, the 6 sets of batteries with the lowest comprehensive optimization scores are removed, and 24 sets of decommissioned power batteries with higher evaluation score were selected. In order to reflect the comprehensive indicators of decommissioned power batteries under different evaluation dimensions, these comprehensive indicators are shown in Fig. 3.

Therefore, through the optimization sorting of the comprehensive indicators, the comprehensive indicators of the decommissioned power batteries can be distinguished. According to the optimization of the objective function value, the rating of the decommissioned power batteries to be graded can be evaluated, which has certain guiding significance for the application of decommissioned batteries in the future.

6 Conclusion

According to the health characteristics of decommissioned power batteries, the improved clustering method is used to cluster the curve characteristics of decommissioned power batteries. In addition, the AHP-CRITIC method is adopted to evaluate the economic indicators of decommissioned power batteries. Then, a comprehensive evaluation model considering the health characteristics and economic characteristics of decommissioned power batteries is established. Finally, an example is given to validate the proposed evaluation model. The results show that the proposed model and algorithm can provide guidance for the subsequent cascade utilization of decommissioned power batteries.