Research Article  Open Access
WenAn Yang, Maohua Xiao, Wei Zhou, Yu Guo, Wenhe Liao, "A Hybrid Prognostic Approach for Remaining Useful Life Prediction of LithiumIon Batteries", Shock and Vibration, vol. 2016, Article ID 3838765, 15 pages, 2016. https://doi.org/10.1155/2016/3838765
A Hybrid Prognostic Approach for Remaining Useful Life Prediction of LithiumIon Batteries
Abstract
Lithiumion battery is a core component of many systems such as satellite, spacecraft, and electric vehicles and its failure can lead to reduced capability, downtime, and even catastrophic breakdowns. Remaining useful life (RUL) prediction of lithiumion batteries before the future failure event is extremely crucial for proactive maintenance/safety actions. This study proposes a hybrid prognostic approach that can predict the RUL of degraded lithiumion batteries using physical laws and datadriven modeling simultaneously. In this hybrid prognostic approach, the relevant vectors obtained with the selective kernel ensemblebased relevance vector machine (RVM) learning algorithm are fitted to the physical degradation model, which is then extrapolated to failure threshold for estimating the RUL of the lithiumion battery of interest. The experimental results indicated that the proposed hybrid prognostic approach can accurately predict the RUL of degraded lithiumion batteries. Empirical comparisons show that the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm performs better than the hybrid prognostic approaches using the popular learning algorithms of feedforward artificial neural networks (ANNs) like the conventional backpropagation (BP) algorithm and support vector machines (SVMs). In addition, an investigation is also conducted to identify the effects of RVM learning algorithm on the proposed hybrid prognostic approach.
1. Introduction
Lithiumion batteries are significant energy solution for many systems (e.g., satellite, spacecraft, and electric vehicles) due to their high energy density, high galvanic potential, lightness of weight, and long lifetime compared to leadacid, nickelcadmium, and nickelmetalhydride cells [1]. Their failure can lead to reduced capability, downtime, and even catastrophic breakdowns. For example, in November 2006, The National Aeronautics and Space Administration’s Mars Global Surveyor stopped working after the radiator for its batteries was positioned towards the sun causing an increase in the temperature of the batteries, which resulted in lost charge capacity [2]. Battery health management would greatly enhance the reliability of such systems. Thus, this raises the challenging issue of remaining useful life (RUL) prediction in relation to lithiumion batteries.
In the past few years, much research effort has been devoted to developing approaches to lithiumion battery degradation modeling and RUL prediction. In general, these approaches can be classified into categories of modelbased and/or datadriven methodologies. The modelbased methodologies attempt to constitute physical models of the lithiumion battery for RUL prediction. Recently, various Bayesian filtering models such as Kalman filter [3], extended Kalman filter [4–6], particle filter [7–9], and unscented particle filter [10] have been extensively used to construct exhaustive models of deteriorating lithiumion batteries. However, uncertainty due to assumptions and simplifications in the models may impose severe limitations upon their applicability in practical applications. In order to overcome the aforementioned problems that can occur with the modelbased methodologies, intensive research has been conducted into the utilization of various datadriven methodologies, for example, autoregressive moving average (ARMA) models [11], artificial neural networks (ANNs) [12], and support vector machines (SVMs) [13], to model lithiumion battery degradation and to predict the RUL of lithiumion batteries. Datadriven techniques utilize monitored operational data related to lithiumion battery health. Compared with the modelbased methodologies, the datadriven methodologies may be more appropriate when the understanding of first principles of system operation is not comprehensive or when the system is so complex such that developing an accurate model is prohibitively expensive but sufficient data are available for constructing a map of the performance degradation space. Furthermore, rapid development has recently been achieved in automatic data collection and processing of realtime field data, which hugely facilitate the continuous monitoring of the state of health of operating lithiumion batteries and the lean management of the related large amount of reference data. The most natural datadriven methodology for RUL prediction is to fit a curve of the available data of the lithiumion battery degradation evolution using regression models and then to extrapolate the curve to the criteria indicating failure. In practice, however, the lithiumion battery degradation history available may be short and incomplete and even differ significantly because of different operating conditions, so that a common extrapolation may lead to large errors and unreliable results. The same problem arises when employing ARMA models, although the method can handle the situation in which more runtofailure data are unavailable or insufficient. With respect to ANNs, they have the advantages of superior learning, noise suppression, and parallel computation abilities. However, despite their advantages, ANNs also have some disadvantages: () design and training often lead to a complex and timeconsuming task, in which architecture and many training parameters must be tuned; () minimization of the training errors can result in poor generalization performance; and () performance can be degraded when working with lowsized datasets. With respect to SVMs, they are powerful in solving problems with small samples, nonlinearities, and local minimum. However, despite their advantages, SVMs also have some disadvantages: () by assuming an explicit loss function (usually, the εinsensitive loss function), one assumes a fixed distribution of the residuals; () the soft margin parameter must be tuned usually through crossvalidation methods, which result in timeconsuming tasks; () the kernel function used in SVM must satisfy Mercer’s theorem to be valid; and () sparsity is not always achieved and a high number of support vectors are thus obtained.
More recently, some researchers have attempted to combine modelbased and datadriven methods for RUL prediction of lithiumion batteries in order to leverage the strength from both datadriven methodology and modelbased methodology and have obtained promising results [14]. Most of the combination of modelbased and datadriven methods in literature has focused on the utilization of relevance vector machines (RVMs) in place of ANNs or SVMs as the prognostic technique. RVM, a general Bayesian probabilistic framework of SVM, can efficiently alleviate some of these shortcomings of SVMs [15]. Saha et al. employed a RVM to find the most representative relevant vectors to fit the capacity degradation data of lithiumion batteries [16]. Maio et al. combined a RVM and an exponential function to predict the RUL of bearings [17]. Zio and Maio employed a RVM to find the most representative relevant vectors to fit a crack growth model for predicting RUL [18]. Wang et al. employed a RVM to find the most representative relevant vectors to fit the threeparameter capacity degradation model to predict the RUL of lithiumion batteries [19]. A review of the related literature also indicates that similar idea has already been investigated in the area of applying SVM to RUL prediction. Benkedjouh et al. [20] employed a SVM to find the most representative support vectors to fit a power model for RUL prediction of the cutting tool. Also based on a similar idea, Benkedjouh et al. employed a SVM to find the most representative support vectors to fit an exponential regression for bearing performance degradation assessment and RUL estimation [21]. The ability to extract the relevant vectors is very useful for making good predictions, as the relevant vectors can be used to find the representative training vectors containing the cycles of the relevant vectors and the predictive values at the cycles of the relevant vectors. A review of the related literature [16–21] also indicates that, for the hybrid prognostic approaches that are based on RVM learning algorithm, their RUL prediction performances are very sensitive to kernels choice and kernel parameters setting. A kernel (or kernel parameter setting) that works well for one situation might not be the appropriate choice for the other. However, no systematic methodology as yet has been established for determining the optimal kernel type and kernel parameters for the RVM learning algorithm. Most of the previous work in the area of applying RVM to RUL prediction determined single kernel and kernel parameters by trial and error and did not deal with automatic kernel choice and kernel parameters optimization.
According to the literature review given above, the aim of this study is to develop a hybrid prognostic approach of physical laws and datadriven modeling that integrates selective kernel ensemblebased RVM (a datadriven methodology) and exponential regression (a modelbased methodology) for online RUL prediction of lithiumion batteries. The choice of kernel (and kernel parameters) of RVM is evolutionarily determined via coevolutionary swarm intelligence, without the need of any human intervention. A sum of two exponential functions’ model is fitted to these relevant vectors to predict the RUL of degraded lithiumion batteries. The experimental results indicate that the proposed hybrid prognostic approach can accurately predict the RUL of degraded lithiumion batteries. Empirical comparisons show that the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm performs better than the hybrid prognostic approaches using popular learning algorithms of feedforward artificial neural networks (ANNs) like the conventional backpropagation (BP) algorithm and support vector machines (SVMs). The proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm outperforms the hybrid prognostic approaches using the single kernelbased RVM learning algorithm and the Ensemble Allbased RVM learning algorithm.
The rest of this study is organized as follows. Section 2 gives a review of the RVM basic framework. Section 3 presents a selective kernel ensemblebased RVM learning algorithm. Section 4 describes a hybrid prognostic approach for RUL prediction of lithiumion batteries. Section 5 conducts an investigation to identify the effects of RVM learning on the hybrid prognostic approach. Section 6 provides an empirical comparison of the proposed hybrid prognostic approach with other existing approaches. Section 7 presents a concluding summary and suggests some directions for future research.
2. Review of Relevance Vector Machine
RVM is a Bayesian form representing a generalized linear model of identical functional form of SVM. Unlike SVM, RVM can provide probabilistic interpretation of its outputs [15]. As a supervised learning, RVM starts with a dataset of inputtarget pairs . The aim is to learn a model of the dependency of the targets on the inputs to make accurate prediction of for previously unseen values of . Typically, the predictions are based on a function defined over the input space, and learning is the process of inferring (perhaps the parameters of) this function. In the context of SVM, this function takes the following form:where are the model “weights,” is bias, and is a kernel function.
By considering only the scalar valued output we follow the standard probabilistic formulation and add additive noise with output samples for better data overfitting, which is described as follows:where are independent samples from some noise process which is further assumed to be zeromean Gaussian noise with variance .
The likelihood of the complete dataset can be written as where , , and is the “design” matrix with , wherein .
Maximizing likelihood prediction of and in (3) often leads to overfitting. Therefore, a preference for smoother functions is encoded by choosing a zeromean Gaussian prior distribution over : where is a vector of hyperparameters.
Using Bayes’ rule, the posterior over all unknowns can be computed; that is,
However, we cannot compute the solution of the posterior in (5) directly. But we can decompose the posterior as , where where the posterior covariance and mean are expressed as follows: with and . Thus, RVM method becomes the search for the best hyperparameters posterior mode. Predictions for new data are then made according to integration of the weights to obtain the marginal likelihood for the hyperparameters: The hyperparameters and which maximize (8) are obtained by using an alternate reprediction approach [15], because values of and cannot be directly calculated in closed form. Suppose that the values of and that can maximize (8) are obtained. Then we can compute the predictive distribution for a new input by using (6):Since both terms in the integral are Gaussian, one can easily compute the probability as follows:where the mean and variance of the predicted value are, respectively,The variance of the predicted value (i.e., (12)) is the sum of the variance associated with noise in the training data and uncertainty associated with prediction of weights.
3. Selective Kernel EnsembleBased Relevance Vector Machine
As mentioned in Section 1, kernel types and kernel parameters have significant influences on the generalization capability of the RVM learning. Generally, commonly used basic kernels for RVM learning include Gaussian kernel (i.e., (13)), Exponential kernel (i.e., (14)), Laplacian kernel (i.e., (15)), Polynomial kernel (i.e., (16)), Sigmoid kernel (i.e., (17)), Cauchy kernel (i.e., (18)), and Multiquadric kernel (i.e., (19)):where , , , , , , , and are kernel parameters that need to be finely tuned. It is impossible to fully determine which one is the best kernel for all problems, because the choice of a kernel depends on the problem at hand. For example, Gaussian kernel is a local kernel and Polynomial kernel is a global kernel [22]. In the case of local kernel, only the data that are close or in the proximity of each other have an influence on the kernel values [22]. In the case of global kernel, samples that are far away from each other still have an influence on the kernel value [22]. With respect to Gaussian kernel and Polynomial kernel only, the mixture of these two basic kernels has been demonstrated to substantively improve the generalization performance of the SVM [23, 24]. However, for many existing basic kernels mentioned above, this combination of basic kernels can also be different for different problems. In one extreme case where all of the individual basic kernels are completely identical, the size of the combination can be reduced without sacrificing the generalization performance of the RVM. In addition, in some scenarios, eliminating some unacceptable basic kernels and meanwhile selecting several acceptable ones to construct a kernel ensemble may be better than combining all of those basic kernels. In this study, each kernel applied to RVM learning algorithm is a selective kernel ensemble of these basic kernels. It should be noted that although the multikernel idea has been successfully used in several machine learning models [25–28] that assume a weighted linear sum of basic kernel and estimate the kernel weights during training, to the best of the authors’ knowledge, it is the first time that a multikernel version of RVM with adaptive kernel selections, adaptive kernel combinations, and adaptive kernel parameters optimization is proposed. The selective kernel ensemble can be expressed as follows:where is the number of basic kernels under consideration and equals 7 in this study, denotes the th basic kernel, stands for the weight assigned to , and represents the selection label assigned to .
3.1. Selection of Candidate Basic Kernels
Instead of combining all of candidate basic kernels, selective kernel ensemble tries to select an optimal subset of individual basic kernels to constitute a selective convex combination. However, selecting an optimal subset from candidate basic kernels is not an easy task since the space of possible subsets is very large for a basic kernel population of size . It is very difficult if not impractical to use exhaustive search to find an optimal subset if and especially when is a large number. In this study, discrete particle swarm optimization (DPSO) [29] algorithm is used for obtaining an optimal subset from candidate basic kernels. Each dimension of a particle in DPSO is encoded by binary bit, where each element of “1” (i.e., ) denotes an individual basic kernel appearing in the selective kernel ensemble while “0” (i.e., ) denotes its absence, . The optimal subset of individual basic kernels can be obtained according to the best evolved selective label vector that can achieve the maximum fitness value. Thus, such a DPSO bit representation gets rid of the tedious trialanderror search for an optimal subset of basic kernels.
3.2. Determination of Kernel Parameters and Additional Weights
Although utilization of selective kernel ensemble can relieve the influence of kernel types on the generalization capability of RVM, it involves 7 additional weight coefficients . In addition, more component basic kernels mean more kernel parameters. It is not easy to determine the optimal values of all these design parameters, including kernel parameters (, , , , , , , and ) and convex combination coefficients that can allow the RVM to achieve the maximum performance. In this circumstance, manual trialanderror method is absolutely tedious and unacceptable. Moreover, manual trialanderror method does not necessarily guarantee a good decision, because these parameters usually interact with each other nonlinearly. In this study, these 7 additional weight coefficients together with kernel parameters (, , , , , , , and ) constitute a general realvalue parameter vector , which will be represented in the population of continuous particle swarm optimization (CPSO) [30]. Thus, such a CPSO realvalue representation gets rid of the tedious trialanderror search for optimal kernel parameters and additional weights.
3.3. Coevolution of DPSO and CPSO
As mentioned in Sections 3.1 and 3.2, one swarm population DPSO with population size s_DPSO and the other swarm population CPSO with population size s_CPSO are involved in equipping the RVM with adaptive kernel selections, adaptive kernel combinations, and adaptive kernel parameters optimization. From a pure DPSO perspective, this suffices for the design of the RVM with the best kernel selection, but without taking kernel parameters and weights in kernel combination into account; that is, only good kernel selection obtained with DPSO may not necessarily mean good RVM performance. Similarly, only good kernel parameters and weights in kernel combination may not necessarily evoke maximum RVM performance. Therefore, the evolution of kernel selections by DPSO and the evolution of kernel combinations and kernel parameters by CPSO should be taken into consideration simultaneously. Inspired by the coevolution of swarms, a coevolutionary PSO scheme is proposed in this section. In the proposed coevolutionary PSO scheme, the DPSO and the CPSO interact with each other through the fitness evaluation. Within each iteration, the DPSO is run for a certain number (g_DPSO) of generations; then the CPSO is run for a certain number (g_CPSO) of generations; this process is repeated until either an acceptable solution has been obtained or the maximum number (max_i_PSO) of iterations has been reached. The global best in the population of DPSO is the final solution for the selection label vector, and the global best in the population of CPSO is the final solution for the general parameter vector with regard to kernel parameters and additional weight coefficients. The procedure of coevolution of DPSO and CPSO is outlined in the following pseudocode.
Step 1. Initialize randomly one swarm population DPSO with population size s_DPSO.
Step 2. Initialize randomly the other swarm population CPSO with population size s_CPSO.
Step 3. Run the DPSO for g_DPSO generations.
Step 4. Reevaluate the personal best values for the CPSO if it is not the first cycle.
Step 5. Run the CPSO for g_CPSO generations.
Step 6. Reevaluate the personal best values for the DPSO.
Step 7. Go back to Step 3. Repeat this procedure until a termination criterion is reached.
In the above coevolutionary PSO scheme, when one PSO is running, the other PSO serves as its ecological environment; that is, for each PSO its ecological environment has varied from iteration to iteration. Therefore, the personal best obtained in the previous iteration has to be reevaluated in accordance with the new ecological environment before playing its coevolving role. It is also worth noting that, in each generation of the coevolution, the real weights are normalized so that the selected individual basic kernels are combined using a weighted average. Hence, this study uses a quite simple normalization scheme as follows:
4. Hybrid Prognostic Approach for RUL Prediction
As a lithiumion battery ages, its maximum capacity begins to deteriorate over time. If the maximum capacity falls below 80% of its initial rated capacity, the battery is considered to be unable to provide reliable power supplies and needs to be replaced. In the current academia/industry practices, reliability of a lithiumion battery for providing reliable power supplies is ensured via the prediction of the remaining maximum capacity. In this study, a hybrid prognostic approach that integrates selective kernel ensemblebased RVM learning algorithm and exponential regression is proposed for RUL prediction of lithiumion batteries. Figure 1 shows an overall flowchart of the proposed hybrid prognostic approach.
4.1. Capacity Degradation Data Collection
To develop the degradation model, four lithiumion batteries under test went through the full charge and discharge procedure. These four lithiumion batteries in the following text are referred to as A1, A2, A3, and A4, respectively. Noting that, these batteries have a graphite anode and a lithium cobalt oxide cathode which were verified using electron dispersive spectroscopy (EDS). The rated capacity of the tested lithiumion battery was 0.9 Ah. Multiple chargedischarge tests were performed with an Arbin BT2000 battery testing system under ambient temperature (around 25°C). The discharge current was 0.45 A. Cutoff voltage was 2.5 V. The failure threshold of the lithiumion batteries was 0.72 Ah. The discharge capacity was recorded after each full chargedischarge cycle. Herein, successive capacity degradation measurements are denoted as and their corresponding cycles are . Therefore, the battery capacity degradation condition can be monitored through the measurements of the inputtarget pairs , where . Inspection of the battery capacity degradation state is made at the predefined inspection cycles of , respectively. At each predefined inspection cycle , along the developing lithiumion battery degradationtofailure trajectory, the last inputtarget pair of () is recorded and appended to the vector of the inputtarget pairs () collected at the previous inspections, so that the capacity degradation data used for degradation model development at the inspection cycle is collected.
4.2. Degradation Model Formulation
At each inspection cycle , the selective kernel ensemblebased RVM learning algorithm is performed on the available inputtarget pairs of data , where , and thus the most representative inputtarget pairs of data identified by the RVM regression, that is, the relevant vectors whose corresponding basis functions are associated with the remaining nonzero weights, are collected in a sparse dataset, where and is the capacity estimate provided by the RVM in correspondence with . For convenience of expression, the sparse dataset is referred to as . Then, fitting to the sparse dataset is performed to identify the unknown parameters of the model adopted. Finally, the fitted model is extrapolated up to the predefined failure threshold of 0.72 Ah to predict the RUL at inspection cycle , .
An important issue in developing a capacity degradation model is determining the fitted model, which influences substantially the prognostics performance of the proposed hybrid prognostic approach. The appropriate fitted model depends on the battery under consideration. Goebel et al. [31] used a sum of two exponential functions to model the increase of internal impedance due to solidelectrolyte interface thickening with time. As battery capacity degradation is closely related to the internal impedance increase, potential models for capacity degradation can also be exponential models. Following up Goebel et al. [31] work, He et al. [2] have experimentally demonstrated that the sum of two exponential functions can well describe the capacity degradation trend of many different batteries: where is the capacity of the battery at the cycle ; and are the parameters associated with the internal impedance; and and are the parameters associated with the aging rate. To demonstrate the suitability of the model in (22) in depicting battery degradation being addressed, Figure 2 shows the curve fitting result (solid line) of (22) to the capacity data of these four batteries, which indicates that the sum of two exponential functions closely agrees with the values of the measured capacity degradation data. Hence, in this study, the sum of two exponential functions was used to fit the degradation curves of the lithiumion batteries on the basis of the sparse dataset .
4.3. RUL Prediction
The RUL of lithiumion batteries can be obtained by extrapolating the fitted model to a predefined failure threshold. The predicted RUL at the inspection cycle is then derived by projecting the state estimates, namely, and , into the future until the future cycle at which the predictive future capacity degradation value hits the predefined failure threshold. Thus, the predicted RUL at the inspection cycle can be expressed as the differences between the inspection cycle and the future cycle and can be calculated as
4.4. Experiment and Results
In order to demonstrate the performance of the proposed hybrid prognostic approach for online prediction of lithiumion battery RUL, four lithiumion batteries A1, A2, A3, and A4 are employed in this experiment. These four illustrative batteries are in exactly the same experimental environments. For detailed information on these four lithiumion batteries, please refer to Section 4.1. The battery capacity data used in this study is provided by the Center for Advanced Life Cycle Engineering, University of Maryland [32].
4.4.1. Relative Parameter Settings
This study uses the root mean squared error (RMSE) as a measure of accuracy index to evaluate the performance of the proposed hybrid prognostic approach. The closer the value of the RMSE is to 0, the better the performance of the fitted model (i.e., (22)) is. Let denote the RMSE of the fitted model enabled by the sparse dataset which was found by the selective kernel ensemblebased RVM with the selection vector and the general parameter vector . Thus, is taken as the fitness function of the proposed hybrid prognostic approach.
In order to apply the developed coevolution of DPSO and CPSO for equipping the RVM with adaptive kernel selections, adaptive kernel combinations, and adaptive kernel parameters optimization, after a small number of simple trials relative parameter settings are determined. For the step of selection of candidate basic kernels, parameters of DPSO are set as follows: s_DPSO, 30; acceleration coefficients and , 1.0 and 0.5; initial and final inertia weight, 0.9 and 0.2; initial and final inertia velocity, 4 and −4; fitness function, . For the step of determination of kernel parameters and additional weight coefficients, parameters of CPSO are set as follows: s_CPSO, 60; acceleration coefficients and , 1.0 and 0.5; initial and final inertia weight, 0.9 and 0.4; initial and final inertia velocity, 4 and −4; fitness function, . For the step of coevolution of DPSO and CPSO, parameters of coevolutionary PSO are set as follows: g_DPSO: 10; g_CPSO: 15; i_max_DPSO: 50.
4.4.2. Experimental Results
To show the robustness of the proposed hybrid prognostic approach, we will perform four independent experiments for each battery. For this purpose, four different inspection cycles corresponding to 60%, 70%, 80%, and 90% data partition rates have been used for online prediction of lithiumion battery RUL. That is, the inspection cycle was set at the cycle steps that separate the whole battery capacity degradation data available into two parts, where the first 60%, 70%, 80%, and 90% are used for RVM learning. Here we assume that the RVM learning with less than 60% battery capacity degradation data is inadequate.
The first case refers to the lithiumion battery A1. The proposed hybrid prognostic approach is applied to the lithiumion battery A1 capacity degradation data plotted with dots in Figures 3–6. As aforementioned in Section 4.4.1, four different inspection cycles corresponding to 60%, 70%, 80%, and 90% data partition rates have been used for online prediction of lithiumion battery RUL; that is, predictions of the RUL of the lithiumion battery A1 are calculated at the inspection cycles of 125, 146, 167, and 188, respectively. Table 1 summarizes the predicted RUL and the actual RUL when the inspection cycles of 125, 146, 167, and 188 were chosen for battery A1. In the second case, the lithiumion battery A2 is investigated. The proposed hybrid prognostic approach is then conducted on the lithiumion battery A2 capacity degradation data, which are plotted as the dots in Figures 7–10. Predictions of the RUL of the lithiumion battery A2 are calculated at the inspection cycles of 113, 132, 151, and 170, respectively. Table 2 summarizes the predicted RUL and the actual RUL when the inspection cycles of 113, 132, 151, and 170 were chosen for battery A2. In the third case, a lithiumion battery A3 is investigated. Battery A3 capacity degradation data, plotted with the dots in Figures 11–14, are analyzed by the proposed hybrid prognostic approach. Predictions of the RUL of the lithiumion battery A3 are calculated at the inspection cycles of 79, 92, 106, and 119, respectively. Table 3 summarizes the predicted RUL and the actual RUL when the inspection cycles of 79, 92, 106, and 119 were chosen for battery A3. In the fourth case, a lithiumion battery A4 is investigated. Battery A4 capacity degradation data, plotted with the dots in Figures 15–18, are analyzed by the proposed hybrid prognostic approach. Predictions of the RUL of the lithiumion battery A4 are calculated at the inspection cycles of 29, 34, 38, and 43, respectively. Table 4 summarizes the predicted RUL and the actual RUL when the inspection cycles of 29, 34, 38, and 43 were chosen for battery A4. Note that the relevant vectors are highlighted by the circles in Figures 3–18. Also note that the parameter values are obtained by fitting the capacity degradation values predicted by the selective kernel ensemblebased RVM at the cycles of the representative training vectors. As seen in Figures 3–18, the proposed hybrid prognostic approach can effectively identify the lithiumion battery capacity degradation trajectory, except in the third case of inspection number = 79 and the fourth case of inspection number = 29. The results in Tables 1–4 also revealed the good agreement of the predicted RUL and the actual RUL. It can be concluded from Figures 3–18 and Tables 1–4 that the proposed hybrid prognostic approach may be a promising tool for lithiumion battery RUL prediction.




5. Effects of RVM Learning on Hybrid Prognostic Approach
In order to investigate the effects of RVM learning on the proposed hybrid prognostic approach, comparison of the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm with the hybrid prognostic approaches using the single kernelbased RVM learning algorithm and the Ensemble Allbased RVM learning algorithm was made in this section. It should be noted that herein the best kernelbased RVM learning algorithm means that the RVM only adopts the best performing component kernel among all available basic kernels for supervised learning, while the Ensemble Allbased RVM learning algorithm means that the RVM adopts all of those available basic kernels for supervised learning. The results are summarized in Table 5 in terms of the predicted RUL and actual RUL. As can be seen, the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm performed better (and in most cases substantially better) than the compared hybrid prognostic approach using the best kernelbased RVM learning algorithm. It can also be concluded from Table 5 that the proposed hybrid prognostic approach outperformed the compared hybrid prognostic approach using the Ensemble Allbased RVM learning algorithm, even though selective kernel ensemblebased RVM learning algorithm only uses a far smaller number of basic kernels. Taking the lithiumion battery A1, for example, the size of the selective kernel ensemblebased RVM learning algorithm is about only 43% (3.0/7.0), 43% (3.0/7.0), 43% (3.0/7.0), and 43% (3.0/7.0) of the size of the Ensemble Allbased RVM learning algorithm for four inspection cycles of 125, 146, 167, and 188, respectively. Significant improvement obtained not only demonstrates better generalization performance of selective kernel ensemblebased RVM learning algorithm but also proves the feasibility and necessity of removing redundant basic kernels in Ensemble Allbased RVM learning algorithm. Therefore, the step of eliminating some unacceptable basic kernels and meanwhile selecting several acceptable ones to construct a basic kernel combination plays a crucial role in enhancing the generalization capability of RVM.
 
SKE: the proposed hybrid prognostic approach that integrates selective kernel ensemblebased RVM and exponential regression; BK: another hybrid prognostic approach that integrates the best kernelbased (i.e., the best performing component kernel among all available basic kernels) RVM with exponential regression; BK: the other hybrid prognostic approach that integrates the Ensemble Allbased (i.e., combining all of those available basic kernels) RVM with exponential regression. 
6. Comparison with Existing Approaches
In this section, performances of the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm were compared with those of the hybrid prognostic approaches using the popular algorithms of feedforward ANNs like the conventional BP algorithm and SVMs on four lithiumion batteries A1, A2, A3, and A4. Although there are many variants of BP algorithm, a faster BP algorithm called LevenbergMarquardt algorithm is used in ANNs. The activation function used is a simple sigmoidal function , whereas the kernel function used in SVMs is Gaussian.
Table 6 presents the results in the test set for the hybrid prognostic approaches using the selective kernel ensemblebased RVM learning algorithm, ANN, and SVM. We show the RUL prediction result. Also, we include the number of relevant/support vectors in the learning algorithms, which is related to prognostic model structural complexity (sparsity). As seen in Table 6, the proposed hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm can provide more accurate RUL prediction of degraded lithiumion batteries than the hybrid prognostic approaches using ANN and SVM can. This is mainly because () the selective kernel ensemblebased RVM learning algorithm can overcome some shortcomings of the ANN, such as overfitting and local minima; () the selective kernel ensemblebased RVM learning algorithm used the selective convex combination and thus the RVM has stronger generalization capability; () the evolution of kernel parameters and kernel weights via CPSO can improve the generalization performance of the selective kernel ensemblebased RVM learning algorithm. Moreover, the selective kernel ensemblebased RVM learning algorithm also provides the capability of yielding a decision function that is much sparser than SVM; that is, the number of relevant vectors is much smaller than that of support vectors, while maintaining the prediction accuracy. This can lead to a significant reduction in prognostic model structural complexity, thereby making it more suitable for online realtime RUL prediction. In addition, RVM does not need the tuning of a soft margin parameter () as in SVM during the training phase.
 
SKERVM: selective kernel ensemblebased RVM learning algorithm. 
7. Conclusions
Lithiumion battery is a core component of many systems and is critical to the systems’ functional capabilities. Battery failure could lead to reduced performance, operational impairment, and even catastrophic failure, especially in aerospace systems. Therefore, RUL prediction of degraded lithiumion batteries is very helpful for preventing longterm breakdown or catastrophic failure. Using selective kernel ensemblebased relevance vector machine and exponential regression together, an automatic, effective but simplertouse hybrid prognostic approach was proposed for RUL prediction of degraded lithiumion batteries. Four lithiumion batteries, namely, A1, A2, A3, and A4, were considered in this study. The experimental result indicated that the proposed hybrid prognostic approach can accurately predict the RUL of degraded lithiumion batteries. Empirical comparisons showed that the proposed hybrid prognostic approach performed better than the hybrid prognostic approaches using the popular algorithms of feedforward ANNs like the conventional BP algorithm and SVMs. This study also demonstrates that the hybrid prognostic approach using the selective kernel ensemblebased RVM learning algorithm outperformed the hybrid prognostic approaches using the single kernelbased RVM learning algorithm and the Ensemble Allbased RVM learning algorithm.
Three research directions are worth pursuing. First, although this study considers the application of lithiumion batteries, the proposed hybrid prognostic approach can be modified and extended to other types of battery (e.g., lipolymer battery [33], leadacid battery [34]). Second, bearings, gearboxes, and oil sand pumps are core components of all kinds of machinery, and maintenance of bearings, gearboxes, and oil sand pumps is essential. The proposed hybrid prognostic approach can also be extended to deal with gear, bearing, and oil sand pump performance degradation assessment and RUL estimation [35–38]. Third, in this study, only seven basic kernels were adopted. Hence, including other types of basic kernels (e.g., Wavelet kernel [39], Cauchy kernel [40], and Rational Quadratic kernel [41]) in the RVM learning is another further research direction that is also worth pursuing.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
The research is funded partially by the National Science Foundation of China (51405239), National Defense Basic Scientific Research Program of China (A2620132010, A2520110003), Fundamental Research Funds for the Central Universities (1005YAH15055), Jiangsu Provincial Natural Science Foundation of China (BK20150745, BK20140727), Jiangsu Province Science and Technology Support Program (BE2014134), and Jiangsu Postdoctoral Science Foundation of China (1501024C). The authors would like to express sincere appreciation to Professor Pecht and Center for Advanced Life Cycle Engineering, University of Maryland, for their efforts to make battery dataset available and permission to use dataset.
References
 I.S. Kim, “A technique for estimating the state of health of lithium batteries through a dualslidingmode observer,” IEEE Transactions on Power Electronics, vol. 25, no. 4, pp. 1013–1022, 2010. View at: Publisher Site  Google Scholar
 W. He, N. Williard, M. Osterman, and M. Pecht, “Prognostics of lithiumion batteries based on DempsterShafer theory and the Bayesian Monte Carlo method,” Journal of Power Sources, vol. 196, no. 23, pp. 10314–10321, 2011. View at: Publisher Site  Google Scholar
 B.H. Seo, T. H. Nguyen, D.C. Lee, K.B. Lee, and J.M. Kim, “Condition monitoring of lithium polymer batteries based on a sigmapoint Kalman filter,” Journal of Power Electronics, vol. 12, no. 5, pp. 778–786, 2012. View at: Publisher Site  Google Scholar
 B. S. Bhangu, P. Bentley, D. A. Stone, and C. M. Bingham, “Nonlinear observers for predicting stateofcharge and stateofhealth of leadacid batteries for hybridelectric vehicles,” IEEE Transactions on Vehicular Technology, vol. 54, no. 3, pp. 783–794, 2005. View at: Publisher Site  Google Scholar
 W. He, N. Williard, M. Osterman, and M. Pecht, “Prognostics of lithiumion batteries using extended Kalman filtering,” in Proceedings of the International Microelectronics and Packaging Society (IMAPS) Advanced Technology Workshop on High Reliability Microelectronics for Military Applications, pp. 17–19, Linthicum Heights, Md, USA, September 2011. View at: Google Scholar
 D. Andre, A. Nuhic, T. SoczkaGuth, and D. U. Sauer, “Comparative study of a structured neural network and an extended Kalman filter for state of health determination of lithiumion batteries in hybrid electricvehicles,” Engineering Applications of Artificial Intelligence, vol. 26, no. 3, pp. 951–961, 2013. View at: Publisher Site  Google Scholar
 B. Saha and K. Goebel, “Modeling Liion battery capacity depletion in a particle filtering framework,” in Proceedings of the Annual Conference of the Prognostics and Health Management Society (PHM '09), pp. 1–10, San Diego, Calif, USA, October 2009. View at: Google Scholar
 M. Dalal, J. Ma, and D. He, “Lithiumion battery life prognostic health management system using particle filtering framework,” Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, vol. 225, no. 1, pp. 81–90, 2011. View at: Publisher Site  Google Scholar
 Y. J. Xing, E. W. M. Ma, K.L. Tsui, and M. Pecht, “An ensemble model for predicting the remaining useful performance of lithiumion batteries,” Microelectronics Reliability, vol. 53, no. 6, pp. 811–820, 2013. View at: Publisher Site  Google Scholar
 Q. Miao, L. Xie, H. J. Cui, W. Liang, and M. Pecht, “Remaining useful life prediction of lithiumion battery with unscented particle filter technique,” Microelectronics Reliability, vol. 53, no. 6, pp. 805–810, 2013. View at: Publisher Site  Google Scholar
 B. Long, W. Xian, L. Jiang, and Z. Liu, “An improved autoregressive model by particle swarm optimization for prognostics of lithiumion batteries,” Microelectronics Reliability, vol. 53, no. 6, pp. 821–831, 2013. View at: Publisher Site  Google Scholar
 A. Nuhic, T. Terzimehic, T. SoczkaGuth, M. Buchholz, and K. Dietmayer, “Health diagnosis and remaining useful life prognostics of lithiumion batteries using datadriven methods,” Journal of Power Sources, vol. 239, no. 1, pp. 680–688, 2013. View at: Publisher Site  Google Scholar
 C. Weng, Y. Cui, J. Sun, and H. Peng, “Onboard state of health monitoring of lithiumion batteries using incremental capacity analysis with support vector regression,” Journal of Power Sources, vol. 235, pp. 36–44, 2013. View at: Publisher Site  Google Scholar
 C. C. Chen and M. Pecht, “Prognostics of lithiumion batteries using modelbased and datadriven methods,” in Proceedings of the 3rd Annual IEEE Prognostics and System Health Management Conference (PHM '12), pp. 1–6, Beijing, China, May 2012. View at: Publisher Site  Google Scholar
 M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” Journal of Machine Learning Research, vol. 1, no. 3, pp. 211–244, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 B. Saha, K. Goebel, S. Poll, and J. Christophersen, “Prognostics methods for battery health monitoring using a Bayesian framework,” IEEE Transactions on Instrumentation and Measurement, vol. 58, no. 2, pp. 291–296, 2009. View at: Publisher Site  Google Scholar
 F. D. Maio, K.L. Tsui, and E. Zio, “Combining relevance vector machines and exponential regression for bearing residual life estimation,” Mechanical Systems and Signal Processing, vol. 31, pp. 405–427, 2012. View at: Publisher Site  Google Scholar
 E. Zio and F. D. Maio, “Fatigue crack growth estimation by relevance vector machine,” Expert Systems with Applications, vol. 39, no. 12, pp. 10681–10692, 2012. View at: Publisher Site  Google Scholar
 D. Wang, Q. Miao, and M. Pecht, “Prognostics of lithiumion batteries based on relevance vectors and a conditional threeparameter capacity degradation model,” Journal of Power Sources, vol. 239, pp. 253–264, 2013. View at: Publisher Site  Google Scholar
 T. Benkedjouh, K. Medjaher, N. Zerhouni, and S. Rechak, “Remaining useful life prediction based on nonlinear feature reduction and support vector regression,” Engineering Applications of Artificial Intelligence, vol. 26, no. 7, pp. 1751–1760, 2013. View at: Publisher Site  Google Scholar
 T. Benkedjouh, K. Medjaher, N. Zerhouni, and S. Rechak, “Health assessment and life prediction of cutting tools based on support vector regression,” Journal of Intelligent Manufacturing, vol. 26, no. 2, pp. 213–223, 2015. View at: Publisher Site  Google Scholar
 A. J. Smola, Learning with kernels [Ph.D. thesis], TU Berlin, 1998.
 G. F. Smits and E. M. Jordaan, “Improved SVM regression using mixtures of kernels,” in Proceedings of the International Joint Conference on Neural Networks (IJCNN '02), vol. 3, pp. 2785–2790, IEEE, Honolulu, Hawaii, USA, May 2002. View at: Publisher Site  Google Scholar
 A. T. Quang, Q.L. Zhang, and X. Li, “Evolving support vector machine parameters,” in Proceedings of the International Conference on Machine Learning and Cybernetics, pp. 548–551, Beijing, China, November 2002. View at: Google Scholar
 M. Girolami and S. Rogers, “Hierarchic bayesian models for kernel learning,” in Proceedings of the 22nd International Conference on Machine Learning (ICML '05), pp. 241–248, August 2005. View at: Publisher Site  Google Scholar
 M. Gönen and E. Alpaydın, “Multiple kernel learning algorithms,” Journal of Machine Learning Research, vol. 12, pp. 2211–2268, 2011. View at: Google Scholar  MathSciNet
 S. R. Gunn and J. S. Kandola, “Structural modelling with sparse kernels,” Machine Learning, vol. 48, no. 1–3, pp. 137–163, 2002. View at: Publisher Site  Google Scholar
 M. Hu, Y. Chen, and J. T.Y. Kwok, “Building sparse multiplekernel SVM classifiers,” IEEE Transactions on Neural Networks, vol. 20, no. 5, pp. 827–839, 2009. View at: Publisher Site  Google Scholar
 J. Kennedy and R. C. Eberhart, “A discrete binary version of the particle swarm algorithm,” in Proceedings of the IEEE International Conference on Computational Cybernetics and Simulation, pp. 4104–4108, IEEE Press, Orlando, Fla, USA, October 1997. View at: Publisher Site  Google Scholar
 J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948, IEEE, Perth, Australia, NovemberDecember 1995. View at: Publisher Site  Google Scholar
 K. Goebel, B. Saha, A. Saxena, J. R. Celaya, and J. P. Christophersen, “Prognostics in battery health management,” IEEE Instrumentation and Measurement Magazine, vol. 11, no. 4, pp. 33–40, 2008. View at: Publisher Site  Google Scholar
 M. Pecht, CALCE Battery Data, University of Maryland, College Park, Md, USA, 2014.
 B. Saha, E. Koshimoto, C. C. Quach et al., “Predicting battery life for electric UAVs,” in Proceedings of the AIAA Infotech at Aerospace Conference and Exhibit, St. Louis, Mo, USA, March 2011. View at: Google Scholar
 E. Frisk, M. Krysander, and E. Larsson, “Datadriven leadacid battery prognostics using random survival forests,” in Proceedings of the Annual Conference of the Prognostics and Health Management Society, Fort Worth, Tex, USA, September 2014. View at: Google Scholar
 D. Wang and P. W. Tse, “Prognostics of slurry pumps based on a movingaverage wear degradation index and a general sequential Monte Carlo method,” Mechanical Systems and Signal Processing, vol. 56, pp. 213–229, 2014. View at: Publisher Site  Google Scholar
 D. Wang, Q. Miao, Q. H. Zhou, and G. W. Zhou, “An intelligent prognostic system for gear performance degradation assessment and remaining useful life estimation,” Journal of Vibration and Acoustics, vol. 137, no. 2, Article ID 021004, 2015. View at: Publisher Site  Google Scholar
 D. Wang and C. Shen, “An equivalent cyclic energy indicator for bearing performance degradation assessment,” Journal of Vibration and Control, 2014. View at: Publisher Site  Google Scholar
 D. Wang, P. W. Tse, W. Guo, and Q. Miao, “Support vector data description for fusion of multiple health indicators for enhancing gearbox fault diagnosis and prognosis,” Measurement Science and Technology, vol. 22, no. 2, Article ID 025102, 2011. View at: Publisher Site  Google Scholar
 L. Zhang, W. Zhou, and L. Jiao, “Wavelet support vector machine,” IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 34, no. 1, pp. 34–39, 2004. View at: Publisher Site  Google Scholar
 J. Basak, “A least square kernel machine with box constraints,” in Proceeding of the IEEE 19th International Conference on Pattern Recognition (ICPR '08), pp. 1–4, IEEE, Tampa, Fla, USA, December 2008. View at: Google Scholar
 D. K. Duvenaud, Automatic model construction with Gaussian processes [Ph.D. thesis], University of Cambridge, 2014.
Copyright
Copyright © 2016 WenAn Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.