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Multi-objective decision analysis for data-driven based estimation of battery states: A case study of remaining useful life estimation
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Multi-objective decision analysis for data-driven based estimation of battery states: A case study of remaining useful life estimation Shuzhi Zhang, Xu Guo, Xiongwen Zhang* MOE Key Laboratory of Thermo-Fluid Science and Engineering (Xi’an Jiaotong University), Xi’an 710049, Shaanxi Province, China
highlights A multi-objective decision method by fusion of AHP and entropy weight is adopted. The evaluation method considers accuracy and modeling time simultaneously. Battery degradation data is divided into training-set, validation-set and test-set. Mean absolute errors and root squared mean errors are used as accuracy indexes. The decision process is introduced via cases for remaining useful life estimation.
article info
abstract
Article history:
Data-driven methods, which can explore the relationship among battery external param-
Received 7 January 2020
eters and battery states automatically without establishing complicated battery model,
Received in revised form
have been intensively applied to estimate state of charge (SOC), state of health (SOH) and
15 February 2020
remaining useful life (RUL) etc. Nevertheless, relatively few researches have been done on
Accepted 11 March 2020
the selection of data-driven model parameters and the determination of model with the
Available online 6 April 2020
optimal comprehensive performance. To address these questions, this paper presents a multi-objective decision method for data-driven based estimation of battery states. This
Keywords:
method adopts the combination of the analytic hierarchy process and the entropy weight
Multi-objective decision method
method together with integrating subjective and objective weights. The mean absolute
Analytic hierarchy process
error and root squared mean error of training-set, validation-set and test-set are used as
Entropy weight method
accuracy indexes, and modeling time is seen as computation burden index. These seven
Data-driven method
indexes are applied as objective criteria for the multi-objective evaluation method, suc-
Battery states estimation
cessfully evaluating the comprehensive performance of estimation model. Moreover, with
Remaining useful life
three cases for RUL estimation, the specific application process of selecting the model with the optimal comprehensive performance by the proposed method is presented in detail. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (S. Zhang), [email protected] (X. Guo), [email protected] (X. Zhang). https://doi.org/10.1016/j.ijhydene.2020.03.100 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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Introduction Due to the climate change issues and the depletion of fossil fuel, Electric Vehicles (EVs) have developed as an appealing solution for the decarburization of the transportation sector [1,2]. Because of the superior performance, such as high energy density, no memory effect, environmental friendliness and long lifetime [3], lithium-ion batteries stand out among rechargeable secondary batteries and are widely used in EVs as the main power source and energy storage unit [4]. In practical application, the lithium-ion battery needs to be well monitored by an effective battery management system (BMS) [5], which can ensure battery operate safely and reliably, prevent any physical damages, and handle thermal degradation and cell unbalancing [6e8]. The accurate estimation for battery states, including state of charge (SOC), state of health (SOH) and remaining useful life (RUL), is one of the most fundamental functions of the BMS [5]. The SOC is defined as the ratio of battery remaining capacity to its fully charged capacity, which can act as an indicator not only for predicting the remaining mileage of EVs but also for determining a safe management strategy to avoid battery over-charge and overdischarge [9,10]. The SOH and RUL are two important indicators, which can be used to determine the time for maintenance and replacement of battery systems [11,12]. The former indicator is described as the ratio of the current cell capacity/resistance to its initial value [13] and the latter can be regarded as the period from the present time to the end of life (EOL). With the development of artificial intelligence, data-driven methods, which can describe the degradation evolution of lithium-ion batteries without establishing the complex chemical model [14,15], have been intensively studied for battery states estimation in recent years. Dong et al. [16] proposed an open-circuit voltage (OCV) based method for SOC estimation by using the dual neural network fusion battery model. The first model was linear to identify parameters of the first-order electrochemical model or second-order electrochemical model, while the second part was a backprorogation neural network used for capturing the relationship between OCV and SOC, successfully realizing the estimation for battery SOC. A novel SOC estimation method based on regular/recurrent Gaussian process regression (GPR) was introduced in Ref. [17]. The inputs to the regular GPR-based method were voltage, current and battery temperature, while the previous estimated SOC was used as an additional input for the recurrent method. Both experimental and simulation results indicated that the proposed method could provide accurate SOC estimation. In Ref. [18], using grey relational analysis (GRA), three features related to battery SOH were selected from charging curves and used to establish SOH estimation model based on least squares support vector machine. Zhang et al. [14] extracted some health indicators (HIs) from filtered partial incremental capacity (IC) curves based on correlation analysis firstly. Sequentially, HIs were used as inputs to build SOH estimation model using artificial neural network under constant current discharge. Based on the Elman neural network, Li et al. [19] established a RUL prognosis model, whose input was directly extracted from
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discharging curves. In Ref. [20], the artificial bee colony algorithm was applied to optimize the support vector regression (SVR) kernel parameters, successfully building a SVM based estimation model to provide more accurate RUL prediction. Among the extensive researches on data-driven methods, it is found that the correlation analysis and supervised learning based methods are widely used to predict battery states. The flow diagram of this class of methods is illustrated in Fig. 1. Firstly, the acceleration aging experiment of lithiumion battery is conducted to record some battery external parameters, such as voltage, current and temperature. Then some feature points (FPs) are usually extracted from battery operation curves, such as charging/discharging curves, filtered IC/differential voltage (DV) curves. Sequentially, these FPs have high correlation with battery states are determined as inputs of estimation model using correlation analysis, such as Pearson/Spearman correlation analysis, GRA and maximal information coefficient (MIC) analysis methods. With these inputs, the estimation model can be established by supervised algorithm to predict battery states. However, some crucial questions, which can be roughly summarized as the determination of the model inputs, the division of battery data, the selection of model hyper-parameters and the choice of the estimation algorithm, must be solved before the model establishment. Too little inputs may cause the model to be difficult to explore the relationship between inputs and battery states automatically, while too much inputs would make the model more complicated and increase the computation burden. Therefore, the appropriate number of inputs should
Fig. 1 e The flow diagram of battery states estimation methods by fusion of correlation analysis and supervised algorithms.
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be determined firstly. Moreover, the performance is strongly influenced by the division of battery data, which means that the data must be separated more reasonably to train and verify the model performance. As we know, different kinds of estimation algorithms have their own advantages and disadvantages. Hence, selecting the suitable supervised algorithm is important for battery states estimation. Nevertheless, there is few clear and universal methods to solve these problems mentioned above when building a data-driven based estimation model. Although the correlation analysis can evaluate the correlation degree between FPs and battery states, the appropriate number of inputs cannot be obtained. Normally, the fitting and validation accuracy are used to evaluate the estimation accuracy of prediction model when using datadriven methods [14,17,21e24]. When using this standard to evaluate two different kinds of estimation models, labeled as A and B, the fitting accuracy of A may higher than that of B, while B may have more satisfying validation accuracy, which means that it is still difficult to determine the model with better performance. Moreover, the large computation burden, the biggest shortcoming of this class of approaches, should also be taken into consideration when evaluating estimation model, making the comparison between different algorithms more complicated. For SOC/SOH estimation, it can be considered that the model has high prediction accuracy when the absolute errors between estimated and real values are below 2%. However, since the absolute errors of RUL are not described as percentage, this criterion is not suitable to be applied to evaluate the RUL estimation accuracy. In order to address these questions, a multi-objective decision method by fusion of analytic hierarchy process (AHP) and entropy weight (EW) is proposed in this paper. With three cases for RUL estimation using the correlation analysis and supervised learning, the specific process of the proposed method to select model parameters and determine the model with the optimal comprehensive performance is presented in detail. The remainder of this paper is organized as follows: Section Methodologies introduces the extraction process of HIs and the principle of multi-objective decision method based on AHP and EW. Section Evaluation results presents the evaluation process by the proposed method with three cases for RUL estimation. The main conclusions are summarized in Section Conclusions.
Methodologies Extraction of health indicators To obtain lithium-ion battery degradation data, the acceleration aging experiment was conducted. The main experimental equipment includes a battery with the nominal capacity of 27Ah, a battery testing system to charge/discharge batteries, a thermal chamber to control temperature and a PC to record data. As shown in Fig. 2, the battery operated under three different modes: constant-current and constant-voltage (CCCV) charge mode, constant-current (CC) discharge mode and relaxation mode. First of all, the battery charged under 1C current rate until the terminal voltage reached 4.2 V. Keep the
Fig. 2 e The procedure of acceleration aging experiment of lithium-ion battery.
terminal voltage constant and continue charging, and stop charging when the current dropped to 0.1C. Then the CC discharge mode operated with a 3C current rate, and the cutoff voltage was 2.75 V. Subsequently, 30min relaxation was started to ensure battery performance recovery. The acceleration aging experiment stopped when the battery SOH dropped to 80%. Battery current, voltage and measurement time were sampled in an interval of 1 second. Through the degradation data, the voltage-time curves in discharging stages with different cycles can be obtained, as shown in Fig. 3 (a). It can be clearly seen that the discharging time decreases with the cycle increases, indicating the reduction of battery actual capacity because of the loss of lithium inventory, the loss of active material in the electrodes and the increase of cell internal resistance [13]. To extract some useful information as inputs of supervised algorithms, area 1 and 2 are enlarged, as presented in Fig. 3(b) and (c). Compared to the voltage-time curves in area 1, it is found that the voltage difference with different cycles in area 2 is more apparent. Therefore, the area 2 is more suitable to be seen as the area for identifying the battery degradation, where nine HIs, labeled as HI1 to HI9, are extracted at the time interval 25s from the 400th to 650th seconds, respectively. The extraction results of HIs is illustrated in Fig. 4 (a). Sequentially, the Pearson correlation analysis method is used to evaluate the correlation degree between HIs and RUL.
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Fig. 3 e (a) The voltage-time curves in different cycles. (b) The enlarged view of area 1. (c) The enlarged view of area 2.
Fig. 4 e (a) The extraction results of HIs. (b) The correlation analysis results of HIs.
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The formula of Pearson’s correlation coefficient is defined as follows [25], ðxi xÞ yi y rxy ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 2 P P ðxi xÞ2 yi y n P
i¼1
i¼1
x¼
(1)
i¼1
n 1 X xi n i¼1
(2)
where rxy represents the Pearson correlation coefficient between x and y. In the correlation analysis, the absolute value of rxy is less than or equal to 1. The greater is the absolute value of rxy, the higher is the correlation degree between x and y. n is sample size, the xi, yi are the individual sample points indexed with i and x is the mean value of sample points. In this study, the xi and yi represent the HIs and battery RUL, respectively. The correlation analysis results are shown in Fig. 4 (b). As can be seen, the nine Pearson correlation coefficient values are higher than 0.96, which means that all these HIs are highly correlated to battery RUL and can be used as inputs to establish estimation model to predict RUL. By analyzing the changes of correlation coefficient values corresponding to the different HIs, it is found that the correlation degree between HIs and SOH shows a general upward trend with the increase of depth of discharge, which is consistent with the differences of the voltage-time curves shown in Fig. 3. Especially, the HI8 has the highest correlation degree with battery RUL and the HI1 has the lowest correlation degree, which are 0.9875 and 0.9657, respectively. In this paper, the GPR method is applied to establish RUL estimation models with different number of inputs and different kinds of kernel function. Moreover, GPR, SVM and regression tree (TS) based estimation models with same number of inputs are also established, respectively. The detailed introduction of GPR, SVM and TS methods can be found in Refs. [26e29]. Sequentially, the comprehensive performance of these models are evaluated by the multiobjective decision method. The specific evaluation process is amply presented in the next subsection.
alternative solutions [31]. It involves a deconstruction process of a multi criteria decision making problem into a hierarchy that often comprises three-level structures from the top to the bottom that describe the goal, criteria, and alternatives, respectively [32]. The pairwise comparison matrix (PCM) is the key notion of AHP, which can be constructed from the comparison of each two objects in the same level of the hierarchy. As shown in Table 1, the evaluations of comparison are given on a scale from one to nine, where the higher scale represents the more importance for one alternative over the other [33]. The PCM can be described as follows: 2 1 C1 C2 6 a21 ¼ 1=a12 6 PCM ¼ « 4 « Cn an1 ¼ 1=a1n
an2
a12 1 « ¼ 1=a2n
3 / a1n / a2n 7 7 1 « 5 / 1
(3)
aij describes how many times each criterion i is more important than another criterion j with respect to the goal. In the case when an object compares with itself, then a11 ¼ a22¼ … ¼ ann ¼ 1. Additionally, if isj, aij ¼ 1/aji, which is another important notion named reciprocity in the PCM. The weights of the objects (A1, A2, ... Aj) are defined as components of the normalized principal eigenvector corresponding to the maximum eigenvalue of a PCM [32], which can be obtained from this equation: ~j ~ j ¼ lmax A PCM A
(4)
To derive the normalized principal eigenvector (Ai), the following equation is then applied: Aj ¼
~j A n P ~j A
(5)
i
As the PCM contains redundant information, the consistency index (CI) representing the degree of acceptable inconsistency can be calculated by: CI ¼
lmax n n1
(6)
The formula for measuring the consistency ratio (CR) is used to check the consistency of each PCM, which is given by:
Multi-objective decision method by fusion of AHP and EW
CR ¼ CI=RI
In typical multiple criteria decision making approaches, weights of attributes reflect the relative importance in decision making process [30]. Different weights need to be assigned because of the unequal importance of each criteria. There are two categories of weighting methods: subjective methods and objective methods. The subjective methods are to determine the weight according to the preference or judgment of the decision maker. Then mathematical methods such as eigenvector method and weighted least square method are applied to calculate the comprehensive evaluation of decision makers. The objective methods, such as entropy method and multi-objective programming, determine weights automatically by solving mathematical models without considering the preference of decision makers. The AHP, originally developed by Saaty, is a multi-objective decision analysis method to assess and prioritize the list of
where n is the number of compared objects, lmax is the maximal eigenvalue of the pairwise comparison matrix. RI is the random consistency index, which can be found in Table 2. If CR < 0.1, matrix PCM is acceptably inconsistent and the elements of the eigenvector A are the weights that correspond to the C indices. Otherwise, it is necessary to readjust the single-factor value of matrix PCM [34]. The AHP process quantifies expert empirical judgments by combining quantitative and qualitative analyses. However, the downside of AHP is that it requires data reflecting experience, judgement and knowledge, which is often of high subjective dependence. So it can readily produce relatively large deviations in the evaluation results [35]. To overcome these drawbacks, the EW method is usually applied in conjunction with AHP to form a comprehensive index system to allow for both objective and subjective weights simultaneously.
(7)
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Table 1 e Saaty scale for pair-wise comparison [31]. Intensity
Definition
Explanation
1 2 3 4 5 6 7
Equal importance Weak Moderate importance Moderate plus Strong importance Strong plus Very strong or demonstrated importance
8 9
Very strong Extreme importance
Two activities contribute equally to the objective Experience and judgment slightly favor one activity over another Experience and judgment strongly favor one activity over another An activity is favored very strongly over another; its dominance demonstrated in practice The evidence favoring one activity over another is of the highest possible order of affirmation
The weights can be obtained by objective EW method to reflect the actual data, which is based on the difference of index data to determine the weight [36,37]. The EW method is based on statistical physics and the concept of entropy in thermodynamics, describing how much different alternatives approach one another in respect to a certain attribute. Some preliminary procedures such as the collection of individual indexes and construction of decision matrices are needed before using the EW [38]. Suppose that Bj denotes the weight of jth criterion, and xij represents the performance value of ith alternative according to jth criterion (i ¼ 1, 2, … , m and j ¼ 1, 2, … ,n). The process of obtaining the weights of criteria based on this method can be implemented via the following four steps: Step 1: To minimize the differences among data of various dimensions for the evaluation indices, the performance values are normalized and the data are confined to the range of [0, 1]. xij pij ¼ n P xij
(8)
subjective attributes into consideration, and adopting EW method can objectively determine the weights of the criteria [39]. Therefore, the weights of indices are determined by combining both EM and AHP process, which are summarized as Fig. 5. Firstly, the sample evaluation matrix X0 ¼ ðx0ij Þm*n is constructed. Then normalize the evaluation value and get the normal evaluation matrixX ¼ ðxij Þm*n . The objective entropy weight Bj is obtained using the EW method and the subjective weight Aj is calculated by the method of AHP. The final weight wj is calculated by integrating the weight of AHP Aj and EW Bj: wj ¼
Aj ,Bj n P Aj ,Bj
(12)
j¼1
Once the criteria weights are obtained, the overall score for each alternative is calculated using the weighted sum model: Si ¼
n X
wj ,Xij
(13)
j¼1
i¼1
Step 2：After the normalization of the sample data, the entropy of the indexes is calculated using: ej ¼ k
n X
pij ln pij
(9)
i¼1
where k ¼ ðln nÞ1 is a constant which guarantees 0 ej 1. Step 3: Calculate the degree of divergence for each criterion as follows: dj ¼ 1 ej
(10)
Step 4: The entropy weight of each index can be derived from: Bj ¼
dj n P dj
(11)
i¼1
Evaluation results In this paper, the battery degradation data is randomly separated into three sets: training-set (70%), validation-set (15%) and test-set (15%), as shown in Fig. 6. The first set is used to establish and train estimation model and the second is to verify the generalization ability of the estimation model and the test-set is for prediction accuracy verification. Before using the AHP-EW method, some indexes which can reflect the estimation model performance must be chosen firstly. For regression tasks like battery RUL estimation, estimation accuracy and modeling time should be considered simultaneously. To comprehensively evaluate the accuracy of estimation model, the mean absolute error (MAE) and root squared mean error (RSME) of training-set, validation-set and test-set are used as six accuracy indexes, labeled as MAE1,
Adopting AHP method can comprehensively take the
Table 2 e Random index for corresponding matrix size [31]. Matrix size
1
2
3
4
5
6
7
8
9
10
Random index
0
0
0.58
0.9
1.12
1.24
1.32
1.41
1.45
1.49
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Fig. 5 e The criteria weight processing by combining both EM and AHP process.
RSME1, MAE2, RSME2, MAE3, RSME3, respectively. They are defined as follows: MAE ¼
n ∧ 1X RULi R ULi n i¼1
RMSE ¼
(14)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 un ∧ uP u R UL RUL i i t i¼1
(15)
n ∧
where RULi is the real RUL and R ULi is the estimated RUL. The GPR, SVM and TR estimation models are established with MATLAB. The hyper-parameters are optimized automatically and the maximum number of objective function evaluations is 30. The total time of the establishment and optimization process is seen as modeling time. Till now, seven indexes, including six accuracy indexes and a time index, has been obtained. Compared to MAE1, RSME1, MAE2 and RSME2, the MAE3 and RSME3 of test-set should be seen as the most important indexes to reflect the accuracy of estimation model. The accuracy indexes of
validation-set is slightly of more significance than that of training-set. By contrast, the modeling time is the least important index. AHP prioritizes the relative importance of a list of criteria via pairwise comparisons among the factors using a ninepoint scale [40]. According to the AHP method, the corresponding PCM is as Eq. (16). With this matrix, the AHP based weight of each index can be obtained, which will later be cooperated with that of EW to get the final weights and scores. In addition, the detailed evaluation process by the multiobjective decision method is presented with three cases, as introduced in the following parts. 2 1 MAE1 RSME1 6 6 1 MAE2 6 6 1=3 PCM ¼ RSME2 6 6 1=3 MAE3 6 6 1=4 RSME3 4 1=4 Total time 1=5
1 3 3 4 4 1 3 3 4 4 1=3 1 1 2 2 1=3 1 1 2 2 1=4 1=2 1=2 1 1 1=4 1=2 1=2 1 1 1=5 1=3 1=3 1=2 1=2
3 5 57 7 37 7 37 7 27 7 25 1
(16)
Fig. 6 e (a) The training-set for model establishment. (b) The validation-set for model generalization ability verification. (c) The test-set for prediction accuracy verification.
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Case 1: GPR algorithm with different number of model inputs In this case, four GPR based estimation models with different number of inputs are built to determine which one has the
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optimal comprehensive performance using multi-objective decision method. The inputs of these models are the HIs extracted in subsection Extraction of health indicators. According to the correlation degree with battery RUL, one, three, six and nine HIs are selected as inputs of these models,
Fig. 7 e (a) (b) The estimation results of GPR model with one input. (c) (d) The estimation results of GPR model with three inputs. (e) (f) The estimation results of GPR model with six inputs. (g) (h) The estimation results of GPR model with nine inputs.
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respectively. The kernel functions for these models are all set as squared exponential kernel. The estimation results and the absolute errors between the real and estimated RUL, including the training-set, validation-set and test-set, are shown in Fig. 7. As can be observed, the RUL estimated by these models is very approximate with real value, while the distributions of RUL estimation results for these four models are slightly
different. Moreover, the MAEs and RSMEs of training-set, validation-set and test-set are computed, as shown in Fig. 8 (a). As can be seen, it is quite difficult to determine which model has the best estimation accuracy because of the inconsistent of MAEs and RSMEs in both three sets. Take the two models with six and nine inputs as example, the MAEs and RSMEs of training-set and validation-set of six inputs model are less than that of nine inputs model while the MAE
Fig. 7 e (continued).
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and RSME of test-set are the opposite. As for modeling time shown in Fig. 8 (b), it can be clearly seen that the total time of these models are close and the model with three inputs has the shortest modeling time of 31.85 s. Using AHP, EW and the multi-objective decision method by fusion of AHP and EW, the weights results of these seven indexes are shown in Fig. 8 (c). For AHP, the weight for each index is set subjectively. The
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MAE3 and RSME3, the accuracy indexes of test-set, have the most important influence on the comprehensive evaluation for estimation model while the total time has the least influence. As for EW, the weight corresponding to total time is largest and that of RMSE2 is smallest, which is quite different from the AHP results. When using the multi-objective decision method, as can be seen, the weight of each index has been
Fig. 8 e (a) The MAE/RSME of training-set, validation-set and test-set for four GPR models. (b) The total modeling time of four GPR models. (c) The weight of each index for three evaluation methods. (d) The scores of four GPR models.
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Fig. 9 e (a) (b) The estimation results of GPR model with squared exponential kernel. (c) (d) The estimation results of GPR model with matern32 kernel. (e) (f) The estimation results of GPR model with matern52 kernel. (g) (h) The estimation results of GPR model with rational quadratic kernel.
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adjusted. However, the MAE3 and RSME3 are still two indexes that have the greatest effect on evaluation results, and the weight of total time is smallest, which are consistent with our original subjective allocation. Sequentially, the scores of four models can be calculated, as illustrated in Fig. 8 (d). It can be seen that the GPR model with three inputs has the largest score of 0.95 while the one input model has the smallest score
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of 0.02. That is to say, the GPR model with three inputs can achieve a balance between estimation accuracy and modeling time, providing relatively accurate RUL prediction in a relatively short period of time. Therefore, the comprehensive performance of this model is the best among these four GPR models.
Fig. 9 e (continued).
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Case 2: GPR algorithm with different kinds of kernel function In this case, four GPR based estimation models with different kinds of kernel function are established to determine which one has the optimal comprehensive performance using multiobjective decision method. The inputs for these four models are all HI7, HI8 and HI9. The kernel functions are squared exponential, matern32, matern52 and rational quadratic kernel,
respectively. The estimation results and the absolute errors between the real and estimated RUL, including the trainingset, validation-set and test-set, are shown in Fig. 9. As we can see, the distributions of RUL estimation results for these four models are relative consistent, while the maximum absolute errors fluctuate at around 16 cycles. To further evaluate the accuracy of different models, MAEs and RSMEs of training-set, validation-set and test-set are also calculated, as shown in Fig. 10 (a). As can be observed, for
Fig. 10 e (a) The MAE/RSME of training-set, validation-set and test-set for four GPR models. (b) The total modeling time of four GPR models. (c) The weight of each index for three evaluation methods. (d) The scores of four GPR models.
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Fig. 11 e (a) (b) The estimation results of GPR model. (c) (d) The estimation results of SVM model. (e) (f) The estimation results of TR model.
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Fig. 11 e (continued).
these GPR models with different kinds of kernel function, the MAEs and RMSEs are all below 4 cycles but it is almost impossible to determine which model has the best estimation accuracy because of the close MAE and RMSE values of different models. The total time of modeling process of these models are presented in Fig. 10 (b). It can be clearly seen that the model with matern52 kernel has the shortest modeling time of 27.94 s while that of model with rational quadratic kernel is longest, as 33.76 s. The weights of these seven indexes based on AHP, EW and multi-objective decision methods are shown in Fig. 10 (c). For AHP and the multiobjective decision method, the MAE3 and RSME3 are two indexes that have the greatest effect on evaluation, while the weight of total time is smallest. However, the total modeling time has the largest weight in EW, which is quite different from the other two methods. Based on the weights of the multi-objective decision method, the scores of four models are computed, as illustrated in Fig. 10 (d). As can be seen, the GPR model with rational quadratic kernel has the largest score of 0.91 while the model with squared exponential kernel has the smallest score of 0.08, which means that the comprehensive performance of GPR model with rational quadratic kernel is the best among these four GPR models.
Case 3: GPR, SVM and TR algorithms with the same number of model inputs In this case, GPR, SVM and TR algorithms are applied to build RUL estimation model, respectively. The inputs of these three
models are all HI7, HI8 and HI9. All hyper-parameters have not been set and would be optimized automatically. The estimation results and the absolute errors between the real and estimated RUL, including the training-set, validation-set and test-set, are shown in Fig. 11. Apparently, the estimation accuracy of TR model is better than GPR and SVM models, and the MAEs and RSMEs of training-set, validation-set and test-set are also calculated, as shown in Fig. 12 (a). The GPR model has the most accurate RUL estimation on training-set and the estimation accuracy of validation-set and test-set is highest when using TR algorithm. Because all hyper-parameters need to be optimized automatically, the total time of modeling process is much longer than case 1 and 2, as illustrated in Fig. 12 (b). The SVM based model needs 118.94 s to be established, while the total time of RT model is shortest, as 13.14 s. The weights allocation of seven indexes are similar to case 1 and 2, as presented in Fig. 12 (c). For AHP and the multi-objective decision method, the MAE3 and RSME3 have the greatest effect on evaluation, while the weight of total time is smallest. Based on the weights of multi-objective decision method, the scores of four models are calculated and illustrated in Fig. 12 (d). The TR model has the largest score of 0.98, while the score of SVM model is smallest, as 0.08. Therefore, when considering estimation accuracy and modeling time simultaneously to evaluate comprehensive performance of estimation model, the TR based model can achieve a balance between estimation accuracy and modeling time to provide accurate RUL prediction in a relatively short period of time.
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Fig. 12 e (a) The MAE/RSME of training-set, validation-set and test-set for GPR, SVM and TR models. (b) The total modeling time of GPR, SVM and TR models. (c) The weight of each index for three evaluation methods. (d) The scores of GPR, SVM and TR models.
Conclusions The estimation for battery states, such as SOC, SOH and RUL, are crucial for BMS to ensure battery operate safely and reliably. Data-driven methods, a kind of method using some
advanced algorithms to establish estimation model based on battery external parameters, can directly estimate battery states without complicated battery model. Although there are intensive researches on battery states estimation using datadriven methods, relatively few researches have been done
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on the selection of data-driven model parameters and the determination of model with the optimal comprehensive performance. To solve these problems, a multi-objective decision method by fusion of AHP and EW is introduced in this paper. Seven indexes, including estimation accuracy of training-set, validation-set and test-set, and modeling time, are used as criteria to select the model with the optimal comprehensive performance by the multi-objective decision method. Moreover, with three cases for battery RUL estimation, the specific process of the hybrid evaluation method is introduced in detail. The main contributions can be summarized as follows: (1) With the combination of the AHP and EW, the multiobjective decision method considers subjective and objective weights simultaneously, making the evaluation results more reasonable and reliable. (2) The battery degradation data is randomly divided into training-set, validation-set and test-set. The MAEs and RSMEs of these three sets are all calculated to be used as indexes to reflect the model estimation accuracy. (3) The multi-objective decision method considers estimation accuracy and modeling time simultaneously, making the evaluation results more comprehensive.
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Multi-objective decision analysis for data-driven based estimation of battery states: A case study of remaining useful life estimation Shuzhi Zhang, Xu Guo, Xiongwen Zhang* MOE Key Laboratory of Thermo-Fluid Science and Engineering (Xi’an Jiaotong University), Xi’an 710049, Shaanxi Province, China
highlights A multi-objective decision method by fusion of AHP and entropy weight is adopted. The evaluation method considers accuracy and modeling time simultaneously. Battery degradation data is divided into training-set, validation-set and test-set. Mean absolute errors and root squared mean errors are used as accuracy indexes. The decision process is introduced via cases for remaining useful life estimation.
article info
abstract
Article history:
Data-driven methods, which can explore the relationship among battery external param-
Received 7 January 2020
eters and battery states automatically without establishing complicated battery model,
Received in revised form
have been intensively applied to estimate state of charge (SOC), state of health (SOH) and
15 February 2020
remaining useful life (RUL) etc. Nevertheless, relatively few researches have been done on
Accepted 11 March 2020
the selection of data-driven model parameters and the determination of model with the
Available online 6 April 2020
optimal comprehensive performance. To address these questions, this paper presents a multi-objective decision method for data-driven based estimation of battery states. This
Keywords:
method adopts the combination of the analytic hierarchy process and the entropy weight
Multi-objective decision method
method together with integrating subjective and objective weights. The mean absolute
Analytic hierarchy process
error and root squared mean error of training-set, validation-set and test-set are used as
Entropy weight method
accuracy indexes, and modeling time is seen as computation burden index. These seven
Data-driven method
indexes are applied as objective criteria for the multi-objective evaluation method, suc-
Battery states estimation
cessfully evaluating the comprehensive performance of estimation model. Moreover, with
Remaining useful life
three cases for RUL estimation, the specific application process of selecting the model with the optimal comprehensive performance by the proposed method is presented in detail. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (S. Zhang), [email protected] (X. Guo), [email protected] (X. Zhang). https://doi.org/10.1016/j.ijhydene.2020.03.100 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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Introduction Due to the climate change issues and the depletion of fossil fuel, Electric Vehicles (EVs) have developed as an appealing solution for the decarburization of the transportation sector [1,2]. Because of the superior performance, such as high energy density, no memory effect, environmental friendliness and long lifetime [3], lithium-ion batteries stand out among rechargeable secondary batteries and are widely used in EVs as the main power source and energy storage unit [4]. In practical application, the lithium-ion battery needs to be well monitored by an effective battery management system (BMS) [5], which can ensure battery operate safely and reliably, prevent any physical damages, and handle thermal degradation and cell unbalancing [6e8]. The accurate estimation for battery states, including state of charge (SOC), state of health (SOH) and remaining useful life (RUL), is one of the most fundamental functions of the BMS [5]. The SOC is defined as the ratio of battery remaining capacity to its fully charged capacity, which can act as an indicator not only for predicting the remaining mileage of EVs but also for determining a safe management strategy to avoid battery over-charge and overdischarge [9,10]. The SOH and RUL are two important indicators, which can be used to determine the time for maintenance and replacement of battery systems [11,12]. The former indicator is described as the ratio of the current cell capacity/resistance to its initial value [13] and the latter can be regarded as the period from the present time to the end of life (EOL). With the development of artificial intelligence, data-driven methods, which can describe the degradation evolution of lithium-ion batteries without establishing the complex chemical model [14,15], have been intensively studied for battery states estimation in recent years. Dong et al. [16] proposed an open-circuit voltage (OCV) based method for SOC estimation by using the dual neural network fusion battery model. The first model was linear to identify parameters of the first-order electrochemical model or second-order electrochemical model, while the second part was a backprorogation neural network used for capturing the relationship between OCV and SOC, successfully realizing the estimation for battery SOC. A novel SOC estimation method based on regular/recurrent Gaussian process regression (GPR) was introduced in Ref. [17]. The inputs to the regular GPR-based method were voltage, current and battery temperature, while the previous estimated SOC was used as an additional input for the recurrent method. Both experimental and simulation results indicated that the proposed method could provide accurate SOC estimation. In Ref. [18], using grey relational analysis (GRA), three features related to battery SOH were selected from charging curves and used to establish SOH estimation model based on least squares support vector machine. Zhang et al. [14] extracted some health indicators (HIs) from filtered partial incremental capacity (IC) curves based on correlation analysis firstly. Sequentially, HIs were used as inputs to build SOH estimation model using artificial neural network under constant current discharge. Based on the Elman neural network, Li et al. [19] established a RUL prognosis model, whose input was directly extracted from
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discharging curves. In Ref. [20], the artificial bee colony algorithm was applied to optimize the support vector regression (SVR) kernel parameters, successfully building a SVM based estimation model to provide more accurate RUL prediction. Among the extensive researches on data-driven methods, it is found that the correlation analysis and supervised learning based methods are widely used to predict battery states. The flow diagram of this class of methods is illustrated in Fig. 1. Firstly, the acceleration aging experiment of lithiumion battery is conducted to record some battery external parameters, such as voltage, current and temperature. Then some feature points (FPs) are usually extracted from battery operation curves, such as charging/discharging curves, filtered IC/differential voltage (DV) curves. Sequentially, these FPs have high correlation with battery states are determined as inputs of estimation model using correlation analysis, such as Pearson/Spearman correlation analysis, GRA and maximal information coefficient (MIC) analysis methods. With these inputs, the estimation model can be established by supervised algorithm to predict battery states. However, some crucial questions, which can be roughly summarized as the determination of the model inputs, the division of battery data, the selection of model hyper-parameters and the choice of the estimation algorithm, must be solved before the model establishment. Too little inputs may cause the model to be difficult to explore the relationship between inputs and battery states automatically, while too much inputs would make the model more complicated and increase the computation burden. Therefore, the appropriate number of inputs should
Fig. 1 e The flow diagram of battery states estimation methods by fusion of correlation analysis and supervised algorithms.
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be determined firstly. Moreover, the performance is strongly influenced by the division of battery data, which means that the data must be separated more reasonably to train and verify the model performance. As we know, different kinds of estimation algorithms have their own advantages and disadvantages. Hence, selecting the suitable supervised algorithm is important for battery states estimation. Nevertheless, there is few clear and universal methods to solve these problems mentioned above when building a data-driven based estimation model. Although the correlation analysis can evaluate the correlation degree between FPs and battery states, the appropriate number of inputs cannot be obtained. Normally, the fitting and validation accuracy are used to evaluate the estimation accuracy of prediction model when using datadriven methods [14,17,21e24]. When using this standard to evaluate two different kinds of estimation models, labeled as A and B, the fitting accuracy of A may higher than that of B, while B may have more satisfying validation accuracy, which means that it is still difficult to determine the model with better performance. Moreover, the large computation burden, the biggest shortcoming of this class of approaches, should also be taken into consideration when evaluating estimation model, making the comparison between different algorithms more complicated. For SOC/SOH estimation, it can be considered that the model has high prediction accuracy when the absolute errors between estimated and real values are below 2%. However, since the absolute errors of RUL are not described as percentage, this criterion is not suitable to be applied to evaluate the RUL estimation accuracy. In order to address these questions, a multi-objective decision method by fusion of analytic hierarchy process (AHP) and entropy weight (EW) is proposed in this paper. With three cases for RUL estimation using the correlation analysis and supervised learning, the specific process of the proposed method to select model parameters and determine the model with the optimal comprehensive performance is presented in detail. The remainder of this paper is organized as follows: Section Methodologies introduces the extraction process of HIs and the principle of multi-objective decision method based on AHP and EW. Section Evaluation results presents the evaluation process by the proposed method with three cases for RUL estimation. The main conclusions are summarized in Section Conclusions.
Methodologies Extraction of health indicators To obtain lithium-ion battery degradation data, the acceleration aging experiment was conducted. The main experimental equipment includes a battery with the nominal capacity of 27Ah, a battery testing system to charge/discharge batteries, a thermal chamber to control temperature and a PC to record data. As shown in Fig. 2, the battery operated under three different modes: constant-current and constant-voltage (CCCV) charge mode, constant-current (CC) discharge mode and relaxation mode. First of all, the battery charged under 1C current rate until the terminal voltage reached 4.2 V. Keep the
Fig. 2 e The procedure of acceleration aging experiment of lithium-ion battery.
terminal voltage constant and continue charging, and stop charging when the current dropped to 0.1C. Then the CC discharge mode operated with a 3C current rate, and the cutoff voltage was 2.75 V. Subsequently, 30min relaxation was started to ensure battery performance recovery. The acceleration aging experiment stopped when the battery SOH dropped to 80%. Battery current, voltage and measurement time were sampled in an interval of 1 second. Through the degradation data, the voltage-time curves in discharging stages with different cycles can be obtained, as shown in Fig. 3 (a). It can be clearly seen that the discharging time decreases with the cycle increases, indicating the reduction of battery actual capacity because of the loss of lithium inventory, the loss of active material in the electrodes and the increase of cell internal resistance [13]. To extract some useful information as inputs of supervised algorithms, area 1 and 2 are enlarged, as presented in Fig. 3(b) and (c). Compared to the voltage-time curves in area 1, it is found that the voltage difference with different cycles in area 2 is more apparent. Therefore, the area 2 is more suitable to be seen as the area for identifying the battery degradation, where nine HIs, labeled as HI1 to HI9, are extracted at the time interval 25s from the 400th to 650th seconds, respectively. The extraction results of HIs is illustrated in Fig. 4 (a). Sequentially, the Pearson correlation analysis method is used to evaluate the correlation degree between HIs and RUL.
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Fig. 3 e (a) The voltage-time curves in different cycles. (b) The enlarged view of area 1. (c) The enlarged view of area 2.
Fig. 4 e (a) The extraction results of HIs. (b) The correlation analysis results of HIs.
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The formula of Pearson’s correlation coefficient is defined as follows [25], ðxi xÞ yi y rxy ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 2 P P ðxi xÞ2 yi y n P
i¼1
i¼1
x¼
(1)
i¼1
n 1 X xi n i¼1
(2)
where rxy represents the Pearson correlation coefficient between x and y. In the correlation analysis, the absolute value of rxy is less than or equal to 1. The greater is the absolute value of rxy, the higher is the correlation degree between x and y. n is sample size, the xi, yi are the individual sample points indexed with i and x is the mean value of sample points. In this study, the xi and yi represent the HIs and battery RUL, respectively. The correlation analysis results are shown in Fig. 4 (b). As can be seen, the nine Pearson correlation coefficient values are higher than 0.96, which means that all these HIs are highly correlated to battery RUL and can be used as inputs to establish estimation model to predict RUL. By analyzing the changes of correlation coefficient values corresponding to the different HIs, it is found that the correlation degree between HIs and SOH shows a general upward trend with the increase of depth of discharge, which is consistent with the differences of the voltage-time curves shown in Fig. 3. Especially, the HI8 has the highest correlation degree with battery RUL and the HI1 has the lowest correlation degree, which are 0.9875 and 0.9657, respectively. In this paper, the GPR method is applied to establish RUL estimation models with different number of inputs and different kinds of kernel function. Moreover, GPR, SVM and regression tree (TS) based estimation models with same number of inputs are also established, respectively. The detailed introduction of GPR, SVM and TS methods can be found in Refs. [26e29]. Sequentially, the comprehensive performance of these models are evaluated by the multiobjective decision method. The specific evaluation process is amply presented in the next subsection.
alternative solutions [31]. It involves a deconstruction process of a multi criteria decision making problem into a hierarchy that often comprises three-level structures from the top to the bottom that describe the goal, criteria, and alternatives, respectively [32]. The pairwise comparison matrix (PCM) is the key notion of AHP, which can be constructed from the comparison of each two objects in the same level of the hierarchy. As shown in Table 1, the evaluations of comparison are given on a scale from one to nine, where the higher scale represents the more importance for one alternative over the other [33]. The PCM can be described as follows: 2 1 C1 C2 6 a21 ¼ 1=a12 6 PCM ¼ « 4 « Cn an1 ¼ 1=a1n
an2
a12 1 « ¼ 1=a2n
3 / a1n / a2n 7 7 1 « 5 / 1
(3)
aij describes how many times each criterion i is more important than another criterion j with respect to the goal. In the case when an object compares with itself, then a11 ¼ a22¼ … ¼ ann ¼ 1. Additionally, if isj, aij ¼ 1/aji, which is another important notion named reciprocity in the PCM. The weights of the objects (A1, A2, ... Aj) are defined as components of the normalized principal eigenvector corresponding to the maximum eigenvalue of a PCM [32], which can be obtained from this equation: ~j ~ j ¼ lmax A PCM A
(4)
To derive the normalized principal eigenvector (Ai), the following equation is then applied: Aj ¼
~j A n P ~j A
(5)
i
As the PCM contains redundant information, the consistency index (CI) representing the degree of acceptable inconsistency can be calculated by: CI ¼
lmax n n1
(6)
The formula for measuring the consistency ratio (CR) is used to check the consistency of each PCM, which is given by:
Multi-objective decision method by fusion of AHP and EW
CR ¼ CI=RI
In typical multiple criteria decision making approaches, weights of attributes reflect the relative importance in decision making process [30]. Different weights need to be assigned because of the unequal importance of each criteria. There are two categories of weighting methods: subjective methods and objective methods. The subjective methods are to determine the weight according to the preference or judgment of the decision maker. Then mathematical methods such as eigenvector method and weighted least square method are applied to calculate the comprehensive evaluation of decision makers. The objective methods, such as entropy method and multi-objective programming, determine weights automatically by solving mathematical models without considering the preference of decision makers. The AHP, originally developed by Saaty, is a multi-objective decision analysis method to assess and prioritize the list of
where n is the number of compared objects, lmax is the maximal eigenvalue of the pairwise comparison matrix. RI is the random consistency index, which can be found in Table 2. If CR < 0.1, matrix PCM is acceptably inconsistent and the elements of the eigenvector A are the weights that correspond to the C indices. Otherwise, it is necessary to readjust the single-factor value of matrix PCM [34]. The AHP process quantifies expert empirical judgments by combining quantitative and qualitative analyses. However, the downside of AHP is that it requires data reflecting experience, judgement and knowledge, which is often of high subjective dependence. So it can readily produce relatively large deviations in the evaluation results [35]. To overcome these drawbacks, the EW method is usually applied in conjunction with AHP to form a comprehensive index system to allow for both objective and subjective weights simultaneously.
(7)
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Table 1 e Saaty scale for pair-wise comparison [31]. Intensity
Definition
Explanation
1 2 3 4 5 6 7
Equal importance Weak Moderate importance Moderate plus Strong importance Strong plus Very strong or demonstrated importance
8 9
Very strong Extreme importance
Two activities contribute equally to the objective Experience and judgment slightly favor one activity over another Experience and judgment strongly favor one activity over another An activity is favored very strongly over another; its dominance demonstrated in practice The evidence favoring one activity over another is of the highest possible order of affirmation
The weights can be obtained by objective EW method to reflect the actual data, which is based on the difference of index data to determine the weight [36,37]. The EW method is based on statistical physics and the concept of entropy in thermodynamics, describing how much different alternatives approach one another in respect to a certain attribute. Some preliminary procedures such as the collection of individual indexes and construction of decision matrices are needed before using the EW [38]. Suppose that Bj denotes the weight of jth criterion, and xij represents the performance value of ith alternative according to jth criterion (i ¼ 1, 2, … , m and j ¼ 1, 2, … ,n). The process of obtaining the weights of criteria based on this method can be implemented via the following four steps: Step 1: To minimize the differences among data of various dimensions for the evaluation indices, the performance values are normalized and the data are confined to the range of [0, 1]. xij pij ¼ n P xij
(8)
subjective attributes into consideration, and adopting EW method can objectively determine the weights of the criteria [39]. Therefore, the weights of indices are determined by combining both EM and AHP process, which are summarized as Fig. 5. Firstly, the sample evaluation matrix X0 ¼ ðx0ij Þm*n is constructed. Then normalize the evaluation value and get the normal evaluation matrixX ¼ ðxij Þm*n . The objective entropy weight Bj is obtained using the EW method and the subjective weight Aj is calculated by the method of AHP. The final weight wj is calculated by integrating the weight of AHP Aj and EW Bj: wj ¼
Aj ,Bj n P Aj ,Bj
(12)
j¼1
Once the criteria weights are obtained, the overall score for each alternative is calculated using the weighted sum model: Si ¼
n X
wj ,Xij
(13)
j¼1
i¼1
Step 2：After the normalization of the sample data, the entropy of the indexes is calculated using: ej ¼ k
n X
pij ln pij
(9)
i¼1
where k ¼ ðln nÞ1 is a constant which guarantees 0 ej 1. Step 3: Calculate the degree of divergence for each criterion as follows: dj ¼ 1 ej
(10)
Step 4: The entropy weight of each index can be derived from: Bj ¼
dj n P dj
(11)
i¼1
Evaluation results In this paper, the battery degradation data is randomly separated into three sets: training-set (70%), validation-set (15%) and test-set (15%), as shown in Fig. 6. The first set is used to establish and train estimation model and the second is to verify the generalization ability of the estimation model and the test-set is for prediction accuracy verification. Before using the AHP-EW method, some indexes which can reflect the estimation model performance must be chosen firstly. For regression tasks like battery RUL estimation, estimation accuracy and modeling time should be considered simultaneously. To comprehensively evaluate the accuracy of estimation model, the mean absolute error (MAE) and root squared mean error (RSME) of training-set, validation-set and test-set are used as six accuracy indexes, labeled as MAE1,
Adopting AHP method can comprehensively take the
Table 2 e Random index for corresponding matrix size [31]. Matrix size
1
2
3
4
5
6
7
8
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Fig. 5 e The criteria weight processing by combining both EM and AHP process.
RSME1, MAE2, RSME2, MAE3, RSME3, respectively. They are defined as follows: MAE ¼
n ∧ 1X RULi R ULi n i¼1
RMSE ¼
(14)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 un ∧ uP u R UL RUL i i t i¼1
(15)
n ∧
where RULi is the real RUL and R ULi is the estimated RUL. The GPR, SVM and TR estimation models are established with MATLAB. The hyper-parameters are optimized automatically and the maximum number of objective function evaluations is 30. The total time of the establishment and optimization process is seen as modeling time. Till now, seven indexes, including six accuracy indexes and a time index, has been obtained. Compared to MAE1, RSME1, MAE2 and RSME2, the MAE3 and RSME3 of test-set should be seen as the most important indexes to reflect the accuracy of estimation model. The accuracy indexes of
validation-set is slightly of more significance than that of training-set. By contrast, the modeling time is the least important index. AHP prioritizes the relative importance of a list of criteria via pairwise comparisons among the factors using a ninepoint scale [40]. According to the AHP method, the corresponding PCM is as Eq. (16). With this matrix, the AHP based weight of each index can be obtained, which will later be cooperated with that of EW to get the final weights and scores. In addition, the detailed evaluation process by the multiobjective decision method is presented with three cases, as introduced in the following parts. 2 1 MAE1 RSME1 6 6 1 MAE2 6 6 1=3 PCM ¼ RSME2 6 6 1=3 MAE3 6 6 1=4 RSME3 4 1=4 Total time 1=5
1 3 3 4 4 1 3 3 4 4 1=3 1 1 2 2 1=3 1 1 2 2 1=4 1=2 1=2 1 1 1=4 1=2 1=2 1 1 1=5 1=3 1=3 1=2 1=2
3 5 57 7 37 7 37 7 27 7 25 1
(16)
Fig. 6 e (a) The training-set for model establishment. (b) The validation-set for model generalization ability verification. (c) The test-set for prediction accuracy verification.
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Case 1: GPR algorithm with different number of model inputs In this case, four GPR based estimation models with different number of inputs are built to determine which one has the
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optimal comprehensive performance using multi-objective decision method. The inputs of these models are the HIs extracted in subsection Extraction of health indicators. According to the correlation degree with battery RUL, one, three, six and nine HIs are selected as inputs of these models,
Fig. 7 e (a) (b) The estimation results of GPR model with one input. (c) (d) The estimation results of GPR model with three inputs. (e) (f) The estimation results of GPR model with six inputs. (g) (h) The estimation results of GPR model with nine inputs.
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respectively. The kernel functions for these models are all set as squared exponential kernel. The estimation results and the absolute errors between the real and estimated RUL, including the training-set, validation-set and test-set, are shown in Fig. 7. As can be observed, the RUL estimated by these models is very approximate with real value, while the distributions of RUL estimation results for these four models are slightly
different. Moreover, the MAEs and RSMEs of training-set, validation-set and test-set are computed, as shown in Fig. 8 (a). As can be seen, it is quite difficult to determine which model has the best estimation accuracy because of the inconsistent of MAEs and RSMEs in both three sets. Take the two models with six and nine inputs as example, the MAEs and RSMEs of training-set and validation-set of six inputs model are less than that of nine inputs model while the MAE
Fig. 7 e (continued).
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and RSME of test-set are the opposite. As for modeling time shown in Fig. 8 (b), it can be clearly seen that the total time of these models are close and the model with three inputs has the shortest modeling time of 31.85 s. Using AHP, EW and the multi-objective decision method by fusion of AHP and EW, the weights results of these seven indexes are shown in Fig. 8 (c). For AHP, the weight for each index is set subjectively. The
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MAE3 and RSME3, the accuracy indexes of test-set, have the most important influence on the comprehensive evaluation for estimation model while the total time has the least influence. As for EW, the weight corresponding to total time is largest and that of RMSE2 is smallest, which is quite different from the AHP results. When using the multi-objective decision method, as can be seen, the weight of each index has been
Fig. 8 e (a) The MAE/RSME of training-set, validation-set and test-set for four GPR models. (b) The total modeling time of four GPR models. (c) The weight of each index for three evaluation methods. (d) The scores of four GPR models.
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Fig. 9 e (a) (b) The estimation results of GPR model with squared exponential kernel. (c) (d) The estimation results of GPR model with matern32 kernel. (e) (f) The estimation results of GPR model with matern52 kernel. (g) (h) The estimation results of GPR model with rational quadratic kernel.
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adjusted. However, the MAE3 and RSME3 are still two indexes that have the greatest effect on evaluation results, and the weight of total time is smallest, which are consistent with our original subjective allocation. Sequentially, the scores of four models can be calculated, as illustrated in Fig. 8 (d). It can be seen that the GPR model with three inputs has the largest score of 0.95 while the one input model has the smallest score
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of 0.02. That is to say, the GPR model with three inputs can achieve a balance between estimation accuracy and modeling time, providing relatively accurate RUL prediction in a relatively short period of time. Therefore, the comprehensive performance of this model is the best among these four GPR models.
Fig. 9 e (continued).
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Case 2: GPR algorithm with different kinds of kernel function In this case, four GPR based estimation models with different kinds of kernel function are established to determine which one has the optimal comprehensive performance using multiobjective decision method. The inputs for these four models are all HI7, HI8 and HI9. The kernel functions are squared exponential, matern32, matern52 and rational quadratic kernel,
respectively. The estimation results and the absolute errors between the real and estimated RUL, including the trainingset, validation-set and test-set, are shown in Fig. 9. As we can see, the distributions of RUL estimation results for these four models are relative consistent, while the maximum absolute errors fluctuate at around 16 cycles. To further evaluate the accuracy of different models, MAEs and RSMEs of training-set, validation-set and test-set are also calculated, as shown in Fig. 10 (a). As can be observed, for
Fig. 10 e (a) The MAE/RSME of training-set, validation-set and test-set for four GPR models. (b) The total modeling time of four GPR models. (c) The weight of each index for three evaluation methods. (d) The scores of four GPR models.
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Fig. 11 e (a) (b) The estimation results of GPR model. (c) (d) The estimation results of SVM model. (e) (f) The estimation results of TR model.
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Fig. 11 e (continued).
these GPR models with different kinds of kernel function, the MAEs and RMSEs are all below 4 cycles but it is almost impossible to determine which model has the best estimation accuracy because of the close MAE and RMSE values of different models. The total time of modeling process of these models are presented in Fig. 10 (b). It can be clearly seen that the model with matern52 kernel has the shortest modeling time of 27.94 s while that of model with rational quadratic kernel is longest, as 33.76 s. The weights of these seven indexes based on AHP, EW and multi-objective decision methods are shown in Fig. 10 (c). For AHP and the multiobjective decision method, the MAE3 and RSME3 are two indexes that have the greatest effect on evaluation, while the weight of total time is smallest. However, the total modeling time has the largest weight in EW, which is quite different from the other two methods. Based on the weights of the multi-objective decision method, the scores of four models are computed, as illustrated in Fig. 10 (d). As can be seen, the GPR model with rational quadratic kernel has the largest score of 0.91 while the model with squared exponential kernel has the smallest score of 0.08, which means that the comprehensive performance of GPR model with rational quadratic kernel is the best among these four GPR models.
Case 3: GPR, SVM and TR algorithms with the same number of model inputs In this case, GPR, SVM and TR algorithms are applied to build RUL estimation model, respectively. The inputs of these three
models are all HI7, HI8 and HI9. All hyper-parameters have not been set and would be optimized automatically. The estimation results and the absolute errors between the real and estimated RUL, including the training-set, validation-set and test-set, are shown in Fig. 11. Apparently, the estimation accuracy of TR model is better than GPR and SVM models, and the MAEs and RSMEs of training-set, validation-set and test-set are also calculated, as shown in Fig. 12 (a). The GPR model has the most accurate RUL estimation on training-set and the estimation accuracy of validation-set and test-set is highest when using TR algorithm. Because all hyper-parameters need to be optimized automatically, the total time of modeling process is much longer than case 1 and 2, as illustrated in Fig. 12 (b). The SVM based model needs 118.94 s to be established, while the total time of RT model is shortest, as 13.14 s. The weights allocation of seven indexes are similar to case 1 and 2, as presented in Fig. 12 (c). For AHP and the multi-objective decision method, the MAE3 and RSME3 have the greatest effect on evaluation, while the weight of total time is smallest. Based on the weights of multi-objective decision method, the scores of four models are calculated and illustrated in Fig. 12 (d). The TR model has the largest score of 0.98, while the score of SVM model is smallest, as 0.08. Therefore, when considering estimation accuracy and modeling time simultaneously to evaluate comprehensive performance of estimation model, the TR based model can achieve a balance between estimation accuracy and modeling time to provide accurate RUL prediction in a relatively short period of time.
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Fig. 12 e (a) The MAE/RSME of training-set, validation-set and test-set for GPR, SVM and TR models. (b) The total modeling time of GPR, SVM and TR models. (c) The weight of each index for three evaluation methods. (d) The scores of GPR, SVM and TR models.
Conclusions The estimation for battery states, such as SOC, SOH and RUL, are crucial for BMS to ensure battery operate safely and reliably. Data-driven methods, a kind of method using some
advanced algorithms to establish estimation model based on battery external parameters, can directly estimate battery states without complicated battery model. Although there are intensive researches on battery states estimation using datadriven methods, relatively few researches have been done
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on the selection of data-driven model parameters and the determination of model with the optimal comprehensive performance. To solve these problems, a multi-objective decision method by fusion of AHP and EW is introduced in this paper. Seven indexes, including estimation accuracy of training-set, validation-set and test-set, and modeling time, are used as criteria to select the model with the optimal comprehensive performance by the multi-objective decision method. Moreover, with three cases for battery RUL estimation, the specific process of the hybrid evaluation method is introduced in detail. The main contributions can be summarized as follows: (1) With the combination of the AHP and EW, the multiobjective decision method considers subjective and objective weights simultaneously, making the evaluation results more reasonable and reliable. (2) The battery degradation data is randomly divided into training-set, validation-set and test-set. The MAEs and RSMEs of these three sets are all calculated to be used as indexes to reflect the model estimation accuracy. (3) The multi-objective decision method considers estimation accuracy and modeling time simultaneously, making the evaluation results more comprehensive.
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