Engineering Applications of Intelligent Monitoring and Control 2014
View this Special IssueResearch Article  Open Access
Artificial Fish Swarm AlgorithmBased Particle Filter for LiIon Battery Life Prediction
Abstract
An intelligent online prognostic approach is proposed for predicting the remaining useful life (RUL) of lithiumion (Liion) batteries based on artificial fish swarm algorithm (AFSA) and particle filter (PF), which is an integrated approach combining modelbased method with datadriven method. The parameters, used in the empirical model which is based on the capacity fade trends of Liion batteries, are identified dependent on the tracking ability of PF. AFSAPF aims to improve the performance of the basic PF. By driving the prior particles to the domain with high likelihood, AFSAPF allows global optimization, prevents particle degeneracy, thereby improving particle distribution and increasing prediction accuracy and algorithm convergence. Data provided by NASA are used to verify this approach and compare it with basic PF and regularized PF. AFSAPF is shown to be more accurate and precise.
1. Introduction
In recent years, green energy has gained increasing support among the people and electric energy has become much more popular. Lithiumion (Liion) batteries offer advantages such as a high energy ratio, high voltage, desirable cryogenic properties, low selfdischarge ratio, and lack of the memory effect [1]. Liion batteries are therefore widely used in electric vehicles, satellites, aerospace craft, and other important domains. Being the core cell of several types of electronic equipment and complex systems [2], Liion batteries are a critical component of the electronic system as a whole, and the failure of Liion batteries can lead to system malfunction, downed systems, and even human casualties and financial losses [3].
The capacity of Liion batteries inevitably degrades over time. Hence, monitoring the state of health of Liion batteries in operation is of great necessity. The monitored data can be used to predict the remaining useful life (RUL) of Liion batteries, which can provide the basis for maintenance decisions and make the use of Liion batteries much more convenient and safer. Currently, the optimal prediction of the RUL of Liion batteries has aroused intense discussion in the Prognostic and Health Management (PHM) domain locally and abroad [1, 4].
Several studies have focused on predicting the RUL of Liion batteries [5–7]. Zhang and Lee reviewed various aspects of recent research and developments in Liion battery PHM and summarized the techniques, algorithms, and models used in RUL prediction [8]. Regarding research on methods of forecasting the RUL of Liion batteries, Saha et al. from the Prognostics Center of Excellence (PCoE) of NASA were first to develop a number of performance degradation tests for Liion batteries under various test conditions [9]. The parameters measured were comprehensive. The researchers obtained a large amount of test data and took the lead in using the Bayesian estimation method to forecast the RUL of Liion batteries. The core idea of this method is to build a probability density distribution (PDF) of states based on all available information [10] to establish a way to express and manage the uncertainty of prediction. The researchers forecast the RUL of Liion batteries by establishing a particle filter (PF) based method. Later, Saha et al. proposed a forecasting method combining support vector machine with PF on the basis of [10]. This method was able to calculate the PDF of the prediction value, rather than mean time to failure, accurately and precisely.
Given the complexity of the system organization of Liion batteries, the appropriate operating conditions are changeable and the system often suffers from noise jamming, test data, model error, load variation, operating conditions, and other uncertain factors need to be taken into consideration [10]. PF is an effective solution for these problems in PHM for nonlinear nonGaussian systems. PF is able to dynamically adjust model parameters of nonstationary conditions and predict the unknown parameters of the battery empirical model by tracking historical data. Accurate prediction as well as a confidence evaluation of the prediction value can be obtained by means of PF. The estimation of confidence interval is often characterized by PDF. PF describes the posteriori estimate of a state using a set of particles with weight; this method results in a probability expression for the prediction value. This description, which is based on Monte Carlo, is equivalent to a real posterior probability density function, and effectively expresses the uncertainty of the prediction outcome, thereby making PF an ideal method for state tracking and prediction [11].
However, basic PF has a serious problem of particle degeneracy [12]. Addressing this problem improperly results in “particle collapse,” a severe case of sample impoverishment [13] where the weights of particles accumulate on a small fraction of particles after a few iterations, leaving a poor representation of posterior density. Unfortunately, a number of papers point out that the degeneracy phenomenon is unavoidable. Many improved algorithms have been proposed to tackle this degeneracy problem. All algorithms can be divided into two broad categories: those involving an appropriate choice of importance sampling density or those involving modifying the resampling step [13–17]. In [17], a regularized PF (RPF) approach was applied to estimate system state and predict battery life, proving the ability of this approach to solve the degeneracy problem by obtaining more accurate and more reliable state estimation. In [18], an improved PF based on the artificial fish swarm algorithm (AFSA) was proposed. This algorithm drives prior particles to a highlikelihood domain, allows global optimization, and prevents particle degeneracy to improve particle distribution and increase prediction accuracy and algorithm convergence [19]. In the present study, which aims to achieve higher accuracy and precision, we use AFSAPF to predict the RUL of Liion batteries. Data provided by NASA are used to verify this approach and compare it with basic PF and RPF.
2. Methodologies
2.1. Artificial Fish Swarm Algorithm
Artificial fish swarm algorithm (AFSA) was proposed by Li et al. [20] in 2002 as a new bionic random searching optimization algorithm based on simulating the ecological behaviors of fish swarm in the wild. This algorithm mainly imitates four behaviors of fish swarm, that is, preying behavior to find the optimal solution, swarming behavior to gather the fish swarm in the optimal region, following behavior to free the fish swarm from local optimal solutions, and random behavior to find the optimal solution in a larger scope. With the merits of rapid and global optimization, insensitivity to initial values and selections of parameters, robustness, easy operation, and so on, AFSA has been applied widely in pattern recognition, neural networks, and parameter estimation [21–23]. To illustrate the algorithm clearly and expediently, we create the following rules: the individual state of the artificial fish (AF) is denoted by a vector , where is the variable whose optimal value is to be searched for; the food consistency of the position where the AF is currently located is denoted by , where denotes the value of the objective function. The distance between two AFSA individuals is expressed as , represents the vision distance, represents the largest steplength of the AF, and represents a random number ranging from zero to one.
The behaviors are described as follows.
Preying Behavior. Let denote the current state of the AF and randomly select a state (1) within the scope of perception (). In this study, we only discuss the maximum problem. If , moves a step toward ; otherwise, we select a state randomly again and justify whether it satisfies the above requirement or not. After several attempts, if the requirements still cannot be satisfied, moves a step randomly. The following briefly shows the process of the preying behavior:
Swarming Behavior. One stipulation for AF is set in the AFSA; that is, AF always tries to swim toward the center of its adjacent fellow swarmmembers and avoids being overcrowded, which directs the swarming behavior. Let denote the current state of the AF, denote the number of the fellow swarmmembers within the scope of perception (), denote the center of the fellows around , and denote the corresponding food consistency of . The AF swims naturally as it searches around and , if , which means that the center of the swarmmembers has a considerable amount of food and is not crowded. Thus, the AF moves toward the center; otherwise, it executes the preying behavior. The following briefly shows the process of the behavior of swarming:
Following Behavior. Let denote the current state of the AF, and find of which is the largest value within the scope of perception (). If , this means that the fellow has high food consistency and the fish surrounding is not very crowded. Thus, moves a step toward ; otherwise, it executes the preying behavior.
Random Behavior. In nature, random behavior is the default activity of the preying behavior; that is, the AF moves a step randomly in its field of vision. The behavior seems to be a random behavior; however, in this process, the AF searches for food or fellow swarmmembers in a larger scope. A few iterations later, the AF would gather around in some local extrema; then a large number of AF would gather around the area of the extremum which is the comparatively optimal area. In this way, we could easily obtain the global extremum to smoothly achieve the optimal value [19, 20].
2.2. Particle Filter
Particle filter (PF) is a statistical filtering technique for implementing the recursive Bayesian filtering using the Monte Carlo simulations [24]. In the PF approach, the state probability density function (PDF) is approximated by a set of particles (points) representing sampled values from an unknown state space and a set of associated weights denoting discrete probability masses. The particles are generated and recursively updated from a nonlinear process model that describes the evolution of the system under analysis in time with a measurement model, a set of available measurements, and an a priori estimate of the state PDF.
PF methods assume that the state equations can be modeled as a first order Markov process with the outputs being conditionally independent. This model can be written as where denotes the state, is the output or measurements, and and are samples from a noise distribution, respectively. Sequential importance resampling (SIR) is a widely used particle filtering algorithm, which approximates the filtering distribution denoted as by a set of weighted particles . The importance weights are approximations to the relative posterior probabilities of the particles such that
The weight update is given by where the importance distribution is approximated as .
The procedure of the basic particle filter is as follows.
(1) Initialization. is set, particles are sampled and represented as from prior distribution , and each weight of the sample is set to as the initial weight.
(2) Importance Sampling. is set and the state of particles is updated using the importance distribution which can be represented as .
The importance weights are then calculated according to (6) and the weights are normalized as .
(3) Resampling. Ahead of time, a threshold number of effective particles is set as . The number of effective particles are calculated and then denoted as . If , the particles are resampled; that is, the weighted particles are mapped to become equal to the weighted particles .
(4) Prediction. is obtained using the state equations (4). We set and then proceed to Step [5, 19].
The regularized particle filter is an improved version of the basic PF. By means of resampling from a continuous approximation instead of resampling from a discrete approximation of the posterior density ; RPF exactly avoids the problem of basic PF. In this study, we do not discuss RPF further; detailed information about RPF could be obtained in [13, 15].
2.3. AFSABased Particle Filter
Introducing the concept of AFSA into basic PF aims at avoiding the phenomenon of particle degeneracy. The combination is applied in the selection of the objective function and the adjustment of the importance weights. AFSA is introduced to improve the process of importance sampling in the basic particle filter by driving the particles move to the high likelihood areas. In this study, we consider the objective function as the posterior PDF. We refer to the new method as AFSAPF. In AFSAPF, the number of fish is the same as the number of particles.
Likelihood function is defined as the objective function , which is also the posterior PDF: where is the latest observed value and is the prediction of the observed value of the last moment.
The main process of AFSAPF is based on the process of PF as described in Section 2.2. The difference between AFSAPF and the basic PF lies in importance sampling, as shown in Figure 1. The detailed process of AFSAPF is described as follows.
(1) Parameter Initialization. The initial states of particles and some parameters (number of particles, process noise, measurement noise, etc.) need to be determined, similar to Step in Section 2.2.
(2) Importance Resampling. The latest observed value is input to the AFSA, together with the corresponding particle state that acts as the initial position of the AF. The fitness value , calculated using (7), acts as the objective function in (2).
The position of each AF is continuously updated based on the preying and swarming behaviors, as described in (2) and (3). The iteration process terminates once a predetermined iteration number is attained or the variation of the AF is less than a specified threshold.
(3) State Prediction. After the update process for all the AFs, the optimal position can be obtained, which is input to (4) as to acquire the next particle state . Then, we set and repeat Step (2~3) [19].
In AFSAPF, PF is responsible for providing new observed values to the state estimate, and the AFSA is in charge of driving the particles toward the highlikelihood areas, which solves the problem that only a small fraction of the particles are in the highlikelihood areas and thus maintaining the diversity of the particles.
In order to explain the process of AFSAPF more intuitively, Figure 1 is shown below.
3. RUL Prediction for LiIon Batteries
To avoid the phenomenon of particle degeneracy and obtain a predicted result with better accuracy and higher precision, we adopt AFSAPF. As mentioned before in the Introduction, RPF is an effective method to avoid the degeneracy; thus, in this study, we compare AFSAPF and RPF based on accuracy and precision. Based on the principle of PF, a state equation for Liion batteries is required in both AFSAPF and RPF. However, a number of unknown parameters always exist in the state equation to be estimated. Although PF can update the parameters when predicting the RUL, this momentary updating of parameters is extremely timeconsuming, which impedes the popularization of AFSAPF and RPF in engineering applications. Moreover, the objects we predict are almost degenerated over time; thus, a considerable amount of history data could be accumulated under normal conditions. Given this condition, by applying a number of history data, we can determine the unknown parameters before predicting RUL using AFSAPF or RPF to obtain timesaving predictions without updating parameters momentarily. The procedures of the experiments for predicting the RUL of Liion batteries are shown in Figure 2.
The detailed descriptions are as follows.
First step is the data preprocessing. An appropriate health indicator is selected and the failure threshold is determined. The data selected for verifying the effectiveness of AFSAPF and RPF should be sufficient to surpass the failure threshold. Here, we select the capacity of Liion batteries as the health indicator.
Second, an appropriate state equation is ensured to be available.
Third, based on the analysis of the data, part of the whole data is truncated to be the history data which are used to determine the unknown parameters in the state equation.
Fourth, the parameters in PF, RPF, and AFSAPF are set, such as the number of particles, the covariance of the process noise, and the covariance of the measurement noise.
Finally, the RUL of the Liion batteries is predicted using PF, RPF, and AFSAPF, and the results of these methods are then compared.
3.1. LiIon Battery Data Set
To verify the performance of the AFSAPF algorithm in predicting the RUL of Liion batteries, we adopt the Liion batteries data set from NASA AMES Center [5, 25]. The battery data set is from the NASA PCoE. The Liion batteries are tested under certain conditions (with the temperature +23°C) and the test ending life is set as 70% of the rated capacity to measure the capacity degradation. Detailed information about the data set can be found in [26]. In this study, the capacity is considered as the health indicator of the degradation of the Liion battery. We set 70% of the rated capacity as the failure threshold [27, 28]. For the selected data set, the rated capacity is 2 Ah and the failure threshold is set as 1.38 Ah, which means that when the capacity degenerates to 1.38 Ah, the experiment will be stopped.
The data set includes the capacity of four batteries, that is, Batteries number 05, number 06, number 07, and number 18. The Liion battery capacity degradation is shown in Figure 3.
Figure 3 shows that Battery number 07 cannot reach the failure threshold during the known cycles; thus, we reject the data for Battery number 07. Hereinafter, we will only discuss Batteries number 05, number 06, and number 18.
3.2. Empirical Degradation Model of LiIon Batteries
To apply AFSAPF, a state equation is in need. Numerous models have been proposed for Liion batteries, such as Gaussian process regression (GPR) model [29], exponential degradation model [30, 31], and polynomial model [31]. However, some of these models are independent of the physical structure of Liion batteries and simply reflect the degradation process. The empirical degradation model, which is proposed by Saha and Goebel and derived from the physical mechanism of Liion batteries [2], adequately considers the influences of different operational modes (charge, discharge, and rest) on the capacity. Therefore, we conduct the battery capacity prediction based on this model in this study.
The empirical degradation model can be described by the following equation: where denotes the charge capacity of the cycle, denotes the coulombic efficiency, and denote the unknown parameters to be estimated, and denotes the rest period between cycles and . In this study, we set [26] and .
3.3. Parameter Settings for Predicting the RUL of LiIon Batteries
To integrate the modelbased method with the datadriven method, an empirical degradation model is introduced as (8), and the unknown parameters, and , are identified using PF, which is an efficient method to estimate unknown parameters in nonlinear physical models with nonGaussian noise [32]. Theoretically, the unknown parameters can be realtime updated using PF as a part of the prognostic process. However, in real engineering applications, the capacity degradation of Liion batteries is a gradual decay process, and the model parameters change slowly during the full life cycle. Therefore, it seems to be timeconsuming if a realtime updating of the unknown parameters based on history data is adopted. In this study, we use the history data to identify the parameters before the AFSAPF prediction instead of realtime parameter updating. The detailed realtime parameter updating process can refer to [32].
To avoid the occasionality of the validity of the experiments, we provide the RUL estimation results of Batteries number 05, number 06, and number 18, which, respectively, contain 167, 168, and 132 data points, and the reallife times are 127, 112, and 100 cycles, respectively. Among the three batteries, the shortest life is 100 cycles. Thus, we consider the previous 40, 50, 60, 70, and 80 data samples, respectively, as the history data to determine the unknown parameters and and then to predict the RUL of the Liion batteries. To illustrate the effect of the proposed method without taking up too much space, we consider an example in which the unknown parameters are determined by the previous 60 data samples.
In the selected exemplificative experiment, we use the former 60 data samples of Battery number 05 to estimate the parameters. Given the randomness in the training algorithm, we typically cannot obtain exactly the same parameter values each time. Thus, in this study, we train the PF 25 times, and finally set the average of the whole estimated results, and , as the determined parameters for future use. Given that the three batteries are of the same type and tested under the same conditions, we assume that the obtained state equation is suitable for all of them, which will be verified in the next section.
As mentioned above, the state equation is established using (8) with and ; the observation equation is set as ; and the objective function is set using (7). The parameters are set as follows: the number of particles , the state initial values of the battery capacity are the first data in the data set of Batteries number 05, number 06, and number 18, the covariance of the process noise , and the covariance of the measurement noise . Noise is considered as the Gaussian white noise to make the experiments close to the ideal situation. The parameters of the AFSA are set as follows: , , , , the number of iterations is 50, and the number of the AF is 200.
4. Results and Discussion
As mentioned in Section 3.3, after all the necessary parameters are set, we predict the RUL of Batteries number 05, number 06, and number 18 using PF, RPF, and AFSAPF. The prediction results and the PDF distributions of the results are shown in Figures 4–9. In Figures 4, 6, and 8, the blue, red and green curves before the prediction starting point, respectively, represent the tracking curves of PF, RPF, and AFSAPF. The tracking curves and the real degradation curve are almost coincident in all the three figures, which demonstrates the effectiveness of the applied empirical degradation model and that the parameters in the state equation determined using the data for Battery number 5 are suitable for the other batteries, which is of great importance in engineering applications.
To analyze the prediction results explicitly, we discuss the RUL prediction results and PDF distributions. Figures 4, 6, and 8 show that the green curve has the best RUL prediction result, which is generated by AFSAPF. Then the red curve has the secondbest result, which is generated by RPF. The blue curve generated by PF comes in last. A summary for the prediction results is shown in Table 1. By comparison, AFSAPF is obviously superior to PF and RPF, and RPF is better than PF, which is consistent with the principles mentioned before.

As shown in Figures 5, 7, and 9, the distributions of AFSAPF prediction results are narrower and taller than both that of PF and that of RPF, and RPF has better results than PF. The distributions of the prediction results show the precision of prediction. The narrower and taller the distributions are, the higher precision the results have. Thus, AFSAPF shows a higher precision than the other two algorithms. Meanwhile, RPF is still better than PF.
The results demonstrate that AFSAPF is indeed an effective approach in predicting the RUL of the Liion batteries. As expected, we have obtained better results from AFSAPF than PF, and RPF because in AFSAPF, traditional resampling methods are replaced by the preying and swarming behaviors of AFSA. In this approach, all the artificial fish make an utmost effort to find the optimal result freely, thereby driving the particles to the highlikelihood domain and in turn, improves the distribution of the particles and increases the accuracy and precision.
5. Conclusions and Future Works
This study uses an intelligent prognostic approach based on AFSA and PF to predict the RUL of Liion batteries. The main contribution of this research are as follows: successfully applying AFSAPF to predict the RUL of Liion batteries, successfully illustrating the availability of combining modelbased and datadriven methods in RUL prediction, and in terms of predicting the RUL of Liion batteries, making a detailed comparison between AFSAPF and RPF, as RPF is regarded as an effective method of preventing particle degeneracy. The experimental results indicate that the proposed AFSAPF method is suitable for predicting the RUL of Liion batteries. Compared with PF and RPF, AFSAPF is more accurate and precise. The proposed method shows promising application in other kinds of batteries.
Our future research will focus on two problems. First, in the study, we noticed that the parameters in AFSA have a major influence on optimization. Selecting appropriate parameters for different optimization functions under different required precisions is difficult yet vitally important. Second, more research should be carried on solving the timeconsuming problem in realtime prediction, because the introduction of intelligent thought increases algorithm complexity; moreover, how to realize realtime updating parameters without consuming too much resource is a challenge. AFSA itself is a rapid optimization method, but when combined with PF, the speed is slowed down near to that of RPF, which limits the application of both AFSAPF and RPF in engineering. Fortunately, the AFSA method is so flexible that it can be improved for better timeliness.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grants nos. 61074083 and 51105019) and the Technology Foundation Program of National Defense (Grant no. Z132013B002).
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Copyright © 2014 Ye Tian 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.