# Capacity Fast Prediction and Residual Useful Life Estimation of Valve Regulated Lead Acid Battery

Authors: Qin He; Yabing Zha; Quan Sun; Zhengqiang Pan; Tianyu Liu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The usable capacity of acid lead batteries is often used as the degradation feature for online RUL (residual useful life) estimation. In engineering applications, the “standard” fully discharging method for capacity measure is quite time-consuming and harmful for the high-capacity batteries. In this paper, a data-driven framework providing capacity fast prediction and RUL estimation for high-capacity VRLA (valve regulated lead acid) batteries is presented. These batteries are used as backup power sources on the ships. The relationship between fully discharging time and partially discharging voltage curve is established for usable capacity extrapolation. Based on the predicted capacity, the particle filtering approach is utilized to obtain battery RUL distribution. A case study is conducted with the experimental data of GFM-200 battery. Results confirm that our method not only reduces the prediction time greatly but also performs quite well in prediction accuracy of battery capacity and RUL.


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## Body

## 1. Introduction

VRLA (valve regulated lead acid) batteries are widely used in ships, electric vehicles, uninterruptible power supply, and mobile communication facilities, given that they have outstanding properties of high capacity, good stability, low cost, and easy recovery [1]. During operation, a series of electrochemical and physical side reactions occur in the battery, which lead to permanent degradation of battery capacity, internal resistance, power density, and other performance characteristics [2]. The EOL (end-of-life) of a VRLA battery is defined as the moment at which these above performance characteristics cannot meet the requirements [3]. Cycle life is one of the important indexes to measure battery performance and quality, which is generally denoted by the total number of charge/discharge cycles that a new battery experiences before reaching the EOL. In practice, users and engineers are more concerned about the RUL of batteries, namely, the remaining charge/discharge cycle a battery has from the present to the EOL. An accurate RUL estimation model can adequately provide scientific and reasonable basis for strategies of battery maintenance and replacement. And it is also the key technique of the battery management system (BMS).Capacity fading is the main failure mode of VRLA batteries. And the continuous capacity loss is due to calendar storage and cyclic charge-discharge. According to the corresponding industrial standard, a VRLA battery fails when the usable capacity fades to 70%~80% of its initial amount. Therefore, the accurate measurement of usable capacity is the premise to perform RUL estimation of VRLA batteries. Conventionally, the capacity of a VRLA battery can be accurately measured by the fully discharging method, in which a full charge/discharge cycle is conducted and the amount of electricity that the battery releases is recorded. However, the method cannot be directly applied on the high-capacity VRLA directly [4]. Generally, a fully discharging process of the high-capacity VRLA battery lasts at least ten hours. Although the measure time can be shortened by a large current, it would lead to incomplete discharge and then underestimation of capacity compared to the actual value and even cause permanent damage to the VRLA battery and seriously shorten its cycle life [4]. Furthermore, the discharging process for a battery stack, which is composed of a large number of serial battery cells, has to stop, once any one of cells is fully discharged. In other words, the cell with the lowest capacity limits the capacity measurement of other cells [5]. Therefore, the fully discharging method does not work on the battery stack which comprises a variety of serial cells.The work in this paper is motivated by a practical engineering problem of backup VRLA batteries on the ships. In ordinary days, it is the diesel engine generator that supplies power for the electric loadings. And the batteries are stored in the float mode as standby. Only in the emergency when the diesel engine stops work due to some accidents are the VRLA batteries activated and begin to provide power for some especially important loadings, like the communication equipment, light facilities, and so forth. To ensure that the batteries can supply enough energies (or capacity) in the emergency, engineers need to measure the maximum discharge capacity regularly (e.g., every three months). As discussed above, the traditional fully discharging method is quite time-consuming and shortens battery life seriously owing to the deep discharge. To this end, we aim at developing a fast capacity prediction approach to help check the health status of backup batteries periodically and then estimate their residual cycle life.In our paper, we focus on the capacity prediction and RUL estimation of high-capacity VRLA batteries used as backup power on the ships. A lot of experiments reveal that the discharge voltage curve of a VRLA battery regularly varies with the battery aging [6]. Thus, it can be applied to predict the usable capacity. In consideration of the feature, a novel capacity fast prediction method based on the partially discharging voltage curve is developed. Through discharging a fully charged battery in a relatively short period of time, the voltage variation is used to estimate the parameters of the discharge voltage model and then to extrapolate the fully discharging time and usable capacity. Under the framework of particle filtering, the capacity prediction results are used as the degradation feature to perform the online RUL estimation. A case study with respect to GFM-200 valve regulated lead acid battery demonstrates the effectiveness of our method in both the accuracy of usable capacity prediction and RUL estimation.


## 2. Characteristics of VRLA Batteries

### 2.1. A Brief Introduction to VRLA Batteries

VRLA batteries have the merit of no leakage of acid or acid mist due to their rigorously sealed structure. In operation, it is not necessary to replenish acid or water to ensure that VRLA batteries work constantly. A safety valve is installed on the battery jar. Once the pressure inside the battery increases to a certain level, the valve automatically opens to release extrato release gas and then shuts down to keep the outside air from entering. During the charge/discharge process, the chemical reactions on the positive and negative electrodes of VRLA batteries are reversible. The active material of the anode is lead dioxide and the main constituent of the cathode is cavernous lead. The electrolyte is dilute sulfuric acid. The main reactions of VRLA battery during the charge and discharge process are shown as follows:(i) ChargeAnode(1)PbSO4+2H2O⟶PbO2+H2SO4+2H++2e-Cathode(2)PbSO4+2H++2e-⟶Pb+H2SO4(ii) DischargeAnode(3)PbO2+H2SO4+2H++2e-⟶PbSO4+2H2OCathode(4)Pb+H2SO4⟶PbSO4+2H++2e-The terminal voltage of lead acid battery depends on the electrolyte concentration. In the discharge, battery voltage drops gradually as the electrolyte concentration decreases, while it rises in the charge as the electrolyte concentration increases [6].VRLA batteries work in a wide range of temperature. The rated voltage of a single VRLA battery is usually 2 V, the discharge cut-off voltage is 1.75–1.8 V, and the charge cut-off voltage is 2.35 V. The charge regime could be constant current and constant voltage (CC-CV) charge or constant voltage (CV) floating charge. For the high-capacity VRLA batteries, the current in the fully discharging method should be less than 0.1 C in order to protect it from shocks and damage. For a 200 Ah battery, 0.1 C represents a current of 200 × 0.1 = 20 A.


### 2.2. Discharge Characteristics

Figure1 shows the voltage curve of a fully charged lead acid battery during the constant current discharge. It is clear that the voltage curve of discharge can be divided into three phases. In the first phase, battery voltage experiences a steep drop in a short period of time (about 5 minutes), which is caused by the internal resistance of the battery. A VRLA battery normally works in the second phase. During this period, the electrochemical reactions inside the battery tend to be stable, and the voltage shows an approximately linear decrease over time. In the last phase, the voltage drops rapidly to the cut-off voltage. Figure 2 shows the variation of discharge voltage curves in different cycles. It is observed that the shape of voltage curves is basically unchanged throughout the entire life span, but they rotate clockwise at a certain point with battery aging. Due to performance degradation, the voltage of an old battery declines faster than that of a new one. And this leads to capacity changes of a battery over its lifetime.Figure 1
Battery discharge voltage in one cycle.Figure 2
Battery discharge voltage in different cycles.


## 2.1. A Brief Introduction to VRLA Batteries

VRLA batteries have the merit of no leakage of acid or acid mist due to their rigorously sealed structure. In operation, it is not necessary to replenish acid or water to ensure that VRLA batteries work constantly. A safety valve is installed on the battery jar. Once the pressure inside the battery increases to a certain level, the valve automatically opens to release extrato release gas and then shuts down to keep the outside air from entering. During the charge/discharge process, the chemical reactions on the positive and negative electrodes of VRLA batteries are reversible. The active material of the anode is lead dioxide and the main constituent of the cathode is cavernous lead. The electrolyte is dilute sulfuric acid. The main reactions of VRLA battery during the charge and discharge process are shown as follows:(i) ChargeAnode(1)PbSO4+2H2O⟶PbO2+H2SO4+2H++2e-Cathode(2)PbSO4+2H++2e-⟶Pb+H2SO4(ii) DischargeAnode(3)PbO2+H2SO4+2H++2e-⟶PbSO4+2H2OCathode(4)Pb+H2SO4⟶PbSO4+2H++2e-The terminal voltage of lead acid battery depends on the electrolyte concentration. In the discharge, battery voltage drops gradually as the electrolyte concentration decreases, while it rises in the charge as the electrolyte concentration increases [6].VRLA batteries work in a wide range of temperature. The rated voltage of a single VRLA battery is usually 2 V, the discharge cut-off voltage is 1.75–1.8 V, and the charge cut-off voltage is 2.35 V. The charge regime could be constant current and constant voltage (CC-CV) charge or constant voltage (CV) floating charge. For the high-capacity VRLA batteries, the current in the fully discharging method should be less than 0.1 C in order to protect it from shocks and damage. For a 200 Ah battery, 0.1 C represents a current of 200 × 0.1 = 20 A.


## 2.2. Discharge Characteristics

Figure1 shows the voltage curve of a fully charged lead acid battery during the constant current discharge. It is clear that the voltage curve of discharge can be divided into three phases. In the first phase, battery voltage experiences a steep drop in a short period of time (about 5 minutes), which is caused by the internal resistance of the battery. A VRLA battery normally works in the second phase. During this period, the electrochemical reactions inside the battery tend to be stable, and the voltage shows an approximately linear decrease over time. In the last phase, the voltage drops rapidly to the cut-off voltage. Figure 2 shows the variation of discharge voltage curves in different cycles. It is observed that the shape of voltage curves is basically unchanged throughout the entire life span, but they rotate clockwise at a certain point with battery aging. Due to performance degradation, the voltage of an old battery declines faster than that of a new one. And this leads to capacity changes of a battery over its lifetime.Figure 1
Battery discharge voltage in one cycle.Figure 2
Battery discharge voltage in different cycles.


## 3. Testing Procedures

To study the performance degradation of high-capacity VRLA batteries, a GFM-200 lead acid battery is employed to conduct the cyclic charge-discharge testing. The battery is produced by the Ainuosi-Huada Power System Ltd. (China) and used as backup power supply on the ship. Its nominal capacity is 200 Ah, and the charge and discharge cut-off voltages are 2.35 V and 1.8 V, respectively. According to the constant current and constant voltage (CC-CV) charge and constant current (CC) discharge regimes [7], the testing, which is conducted by the ACCEXP battery test system, followed the steps presented below. The testing was conducted in a closed temperature-regulated chamber at 25 Celsius, which can mimic the average conditions on ships approximately. The temperature changes of the battery were very small and hence were assumed to be equal to the chamber temperature:(1)
Charge the battery with the constant current of 0.1 C until the voltage reaches 2.35 V.(2)
Charge the battery with the constant voltage of 2.35 V until the current drops to 1.2 A.(3)
Let the battery have a rest for 0.5 h.(4)
Discharge the battery with the constant current of 0.1 C until the voltage drops to 1.8 V.(5)
Let the battery have a rest for 0.5 h.(6)
Repeat the above steps until the battery reaches the end-of-life.During the testing, parameters of interest such as voltage, current, and discharge capacity are monitored and recorded automatically. The discharge voltage curves in different cycles are plotted in Figure3. As the testing proceeds, the discharge voltage curves become steeper and steeper. As a result, the effective discharge time decreases over cycles number as well as delivered capacity.Figure 3
Testing results of GFM-200 battery.


## 4. Capacity Fast Prediction of VRLA Batteries

### 4.1. Discharge Voltage Modeling

The discharge voltage curve contains abundant information about battery performance and health status, thus establishing an accurate discharge model which can help us predict the usable capacity quickly. As the first phase of voltage (steep drop) is very short, it is generally ignored in the discharge voltage modeling. In phase 2, the voltage drops linearly. When most of the usable capacity is delivered, the voltage starts to drop rapidly in phase 3. According to these characteristics, an empirical equation for the discharge voltage of the lead acid battery is proposed by Rynkiewicz [8].(5)Ut=k0-I·k1·t-k2k3-I·t,whereUt is the discharge voltage over time, t is the discharge time in hour (h), I is the discharge current in ampere (A), k0 is a constant for the initial volt (V), k1 is a constant for the effective resistance in ohm, k2 captures the concentration of electrolyte, and k3 represents the final capacity of all active materials.The voltages in each cycle are fitted to (5), respectively. The voltages in the first 5 minutes (corresponding to the steep drop phase) are removed to improve fitting accuracy. Figure 4 shows changes of each parameter over cycles number. As shown in the plots, the initial voltage (k0) and effective resistance (k1) increase over time, while the concentration of electrolyte (k2) and final capacity (k3) decrease. All these phenomena are in line with the performance degradation of lead acid battery.Figure 4
Changes of model parameters in (5).Furthermore, we find that parametersk2 and k3 both strongly correlate with k1. As Figures 5 and 6 show, the power function is suitable to model their relationships. That is to say, for a given battery, parameters k2 and k3 can be estimated by substituting k1 into the following equation:(6)k2=1.385×10-7×k1-2.369k3=2.976×k1-0.6203.Figure 5
Fitting result ofk2 versus k1.Figure 6
Fitting result ofk3 versus k1.


### 4.2. Capacity Prediction

Theoretically, battery capacity can be measured by fully discharging it and integrating the discharge current. However, this method is not applicable to high-capacity batteries, since it takes a long time to discharge (at least 10 hours) and shortens battery life seriously owing to the deep discharge. Thus, a fast capacity prediction method for high-capacity VRLA batteries is necessary. According to the aforementioned definition of usable capacity for lead acid battery, it can be calculated by the following equation:(7)C=I×tc,where I is the discharge current and tc is the time it takes when the voltage of a fully charged battery declines to the discharge cut-off voltage.After obtaining the estimators ofk0, k1, k2, and k3, the fully discharging time tc can be calculated easily through (5). Then, the issue of capacity fast prediction is converted to the issue of parameter estimation in (5). It is easy to predict k2 and k3 by (6), so the remaining issue we should overcome is how to estimate k0 and k1 as fast as possible.In this paper, we aim to predict the usable capacity of a given battery by only discharging it for 3.5 hours instead of 10 hours (the currently shortest fully discharging time). In Figure3, it is observed that the battery voltage drops linearly in the first 3.5 hours. Thus, a linear equation is employed to fit the voltage during the period; namely,(8)Ut=a-b·I·t.The estimators of parametersa and b are plotted in Figures 7 and 8. The results reveal that a is in good agreement with k0, and the linear correlation between b and k1 is significant. Thus, we can predict k0 and k1 by substituting a and b into the following equation:(9)k0=ak1=0.9089·b-9.996×10-5.Figure 7
Relationship betweenk0 and a.Figure 8
Fitting result ofk1 versus b.Now, the procedure of capacity fast prediction can be summarized as follows.Algorithm 1 (capacity fast prediction).
Step 1. For a given battery, charge it with the normal CC-CV regime until it is fully charged.
Step 2. Discharge it at the constant current of 0.1 C for 3.5 h.
Step 3. Estimate parameters a and b by fitting the voltages obtained in Step 2 with (8).
Step 4. Estimate k0, k1, k2, and k3 by substituting the estimators of a and b into (9) and (6).
Step 5. Calculate tc by substituting k0, k1,k2, and k3 into (5).
Step 6. Predict the usable capacity by substituting tc into (7).In the preceding discussion, the partially discharging time is set to be 3.5 hours. To demonstrate that that is a reasonable time, Figure9 compares the capacity prediction results under different discharging time. In Figure 9, the dotted lines denote the actual capacity measured by the “standard” fully discharging method, and the solid lines are the predicted values by Algorithm 1. The mean relative errors are defined as follows: (10)MRE=1n∑i=1nCip-CiaCia×100%,where Cip and Cia denote the predicted capacity and actual capacity at the ith cycle and n is the number of cycles.Figure 9
Results of capacity fast prediction under different partially discharging time (GFM-200 battery).TheMRE under 1.5, 2.5, 3.5, and 4.5 h discharging time are 5.14%, 3.08%, 1.31%, and 1.05%, respectively.As the length of partially discharging time increases, the prediction accuracy improves. However, when the time exceeds 3.5 hours, the effect of accuracy improvement by increasing discharging time is no longer remarkable. Thus, we choose 3.5 hours as the partially discharging time in this paper by considering the factors of time and accuracy comprehensively.To further demonstrate the effectiveness of our method, we use the steps and parameters above to predict the capacity of another VRLA battery GFM-100. The only difference between GFM-200 and GFM-100 is that the latter nominal capacity is 100 Ah. GFM-100 was also tested with the CC-CV charge and CC discharge regimes at 25 Celsius. In the prediction procedure, the partially discharging time is set to be 3.5 hours. Figure10 compares the actual capacity and predicted values of GFM-100. The MRE of prediction for GFM-100 is 2.71%. It is observed that the prediction results are still accurate except for the first few cycles. The main reason may lie in the fact that the discharge characteristics of different batteries show some heterogeneity as a result of the randomness of materials.Figure 10
Results of capacity fast prediction of GFM-100 battery.


## 4.1. Discharge Voltage Modeling

The discharge voltage curve contains abundant information about battery performance and health status, thus establishing an accurate discharge model which can help us predict the usable capacity quickly. As the first phase of voltage (steep drop) is very short, it is generally ignored in the discharge voltage modeling. In phase 2, the voltage drops linearly. When most of the usable capacity is delivered, the voltage starts to drop rapidly in phase 3. According to these characteristics, an empirical equation for the discharge voltage of the lead acid battery is proposed by Rynkiewicz [8].(5)Ut=k0-I·k1·t-k2k3-I·t,whereUt is the discharge voltage over time, t is the discharge time in hour (h), I is the discharge current in ampere (A), k0 is a constant for the initial volt (V), k1 is a constant for the effective resistance in ohm, k2 captures the concentration of electrolyte, and k3 represents the final capacity of all active materials.The voltages in each cycle are fitted to (5), respectively. The voltages in the first 5 minutes (corresponding to the steep drop phase) are removed to improve fitting accuracy. Figure 4 shows changes of each parameter over cycles number. As shown in the plots, the initial voltage (k0) and effective resistance (k1) increase over time, while the concentration of electrolyte (k2) and final capacity (k3) decrease. All these phenomena are in line with the performance degradation of lead acid battery.Figure 4
Changes of model parameters in (5).Furthermore, we find that parametersk2 and k3 both strongly correlate with k1. As Figures 5 and 6 show, the power function is suitable to model their relationships. That is to say, for a given battery, parameters k2 and k3 can be estimated by substituting k1 into the following equation:(6)k2=1.385×10-7×k1-2.369k3=2.976×k1-0.6203.Figure 5
Fitting result ofk2 versus k1.Figure 6
Fitting result ofk3 versus k1.


## 4.2. Capacity Prediction

Theoretically, battery capacity can be measured by fully discharging it and integrating the discharge current. However, this method is not applicable to high-capacity batteries, since it takes a long time to discharge (at least 10 hours) and shortens battery life seriously owing to the deep discharge. Thus, a fast capacity prediction method for high-capacity VRLA batteries is necessary. According to the aforementioned definition of usable capacity for lead acid battery, it can be calculated by the following equation:(7)C=I×tc,where I is the discharge current and tc is the time it takes when the voltage of a fully charged battery declines to the discharge cut-off voltage.After obtaining the estimators ofk0, k1, k2, and k3, the fully discharging time tc can be calculated easily through (5). Then, the issue of capacity fast prediction is converted to the issue of parameter estimation in (5). It is easy to predict k2 and k3 by (6), so the remaining issue we should overcome is how to estimate k0 and k1 as fast as possible.In this paper, we aim to predict the usable capacity of a given battery by only discharging it for 3.5 hours instead of 10 hours (the currently shortest fully discharging time). In Figure3, it is observed that the battery voltage drops linearly in the first 3.5 hours. Thus, a linear equation is employed to fit the voltage during the period; namely,(8)Ut=a-b·I·t.The estimators of parametersa and b are plotted in Figures 7 and 8. The results reveal that a is in good agreement with k0, and the linear correlation between b and k1 is significant. Thus, we can predict k0 and k1 by substituting a and b into the following equation:(9)k0=ak1=0.9089·b-9.996×10-5.Figure 7
Relationship betweenk0 and a.Figure 8
Fitting result ofk1 versus b.Now, the procedure of capacity fast prediction can be summarized as follows.Algorithm 1 (capacity fast prediction).
Step 1. For a given battery, charge it with the normal CC-CV regime until it is fully charged.
Step 2. Discharge it at the constant current of 0.1 C for 3.5 h.
Step 3. Estimate parameters a and b by fitting the voltages obtained in Step 2 with (8).
Step 4. Estimate k0, k1, k2, and k3 by substituting the estimators of a and b into (9) and (6).
Step 5. Calculate tc by substituting k0, k1,k2, and k3 into (5).
Step 6. Predict the usable capacity by substituting tc into (7).In the preceding discussion, the partially discharging time is set to be 3.5 hours. To demonstrate that that is a reasonable time, Figure9 compares the capacity prediction results under different discharging time. In Figure 9, the dotted lines denote the actual capacity measured by the “standard” fully discharging method, and the solid lines are the predicted values by Algorithm 1. The mean relative errors are defined as follows: (10)MRE=1n∑i=1nCip-CiaCia×100%,where Cip and Cia denote the predicted capacity and actual capacity at the ith cycle and n is the number of cycles.Figure 9
Results of capacity fast prediction under different partially discharging time (GFM-200 battery).TheMRE under 1.5, 2.5, 3.5, and 4.5 h discharging time are 5.14%, 3.08%, 1.31%, and 1.05%, respectively.As the length of partially discharging time increases, the prediction accuracy improves. However, when the time exceeds 3.5 hours, the effect of accuracy improvement by increasing discharging time is no longer remarkable. Thus, we choose 3.5 hours as the partially discharging time in this paper by considering the factors of time and accuracy comprehensively.To further demonstrate the effectiveness of our method, we use the steps and parameters above to predict the capacity of another VRLA battery GFM-100. The only difference between GFM-200 and GFM-100 is that the latter nominal capacity is 100 Ah. GFM-100 was also tested with the CC-CV charge and CC discharge regimes at 25 Celsius. In the prediction procedure, the partially discharging time is set to be 3.5 hours. Figure10 compares the actual capacity and predicted values of GFM-100. The MRE of prediction for GFM-100 is 2.71%. It is observed that the prediction results are still accurate except for the first few cycles. The main reason may lie in the fact that the discharge characteristics of different batteries show some heterogeneity as a result of the randomness of materials.Figure 10
Results of capacity fast prediction of GFM-100 battery.


## 5. RUL Estimation

Recently, engineers have paid more attention to the online safety, risk management, and maintenance decision of equipment in the defense industry [9–11]. RUL estimation plays an important role in the risk management and condition-based maintenance of VRLA batteries for marine backup power use. As discussed above, the usable capacity and stability of a VRLA battery decrease during either storage or operation. Before a battery fails, it must be replaced by a new one to avoid unexpected failures and accidents [12, 13]. The predicted capacity in Section 4 can be used as the degradation feature to infer RUL distributions (RLDs).


### 5.1. Particle Filtering

Degradation modeling and parameter updating are keys of online RUL estimation. The degradation process of battery capacity can be treated as an underlying random process. With the predicted capacity, an empirical model can be easily established to describe the common degradation characteristics. To capture the online variability of a battery, model parameters used for prediction should be updated once new capacity measurement is available. In order to characterize the dynamic capacity fading, the particle filtering approach is used in this paper to timely update the parameters distribution. Particle filtering is a sequential Bayesian method based on the technique of Monte Carlo simulation [14]. Compared with the Kalman filtering, it can deal with nonlinear model and non-Gaussian noise system. And thus it is widely used in the field of prognostics [15].In the framework of particle filtering, a state-space model is used to characterize the evolution of degradation feature and model parameters (known as states vector). In the case of VRLA batteries, the degradation feature is battery capacity and the degradation path can be expressed by an exponential function (as Figure9 shows). Motivated by [16], the parameters are not dynamic and can be regarded as constant between two time points. Thus, the state-space model is as follows.State Transition Function (11) x i = exp ⁡ - b i Δ t x i - 1 b i = b i - 1 σ i = σ i - 1 ,Observation Function (12) y i = x i + v i ,where Δt=ti-ti-1, xi is the uncontaminated signal (namely, actual capacity) at ti, yi is the observation (namely, predicted capacity) including Gaussian noise vi, bi is the parameter of the exponential function, and the standard deviation of vi is σi.A set of particles are generated by the Monte Carlo simulation to estimate the current capacityxi as well as distributions of parameters bi and σi. Once new observation is available, these particles are resampled to approximately denote the posterior distribution. More details about the theories of particle filtering can be found in [17, 18]. After obtaining the observations y1,y2,…,yk (yk is the latest observation time), the standard particle filtering for RUL estimation is as follows.Algorithm 2 (standard particle filtering for RUL estimation).
Step 1 (particles initialization). Due to lack of prior information, N initial particles of the state vector x0,b0,σ0p, p=1,2,…,N, are sampled from uniform distributions. Fori=1:k.
Step 2 (particles updating). According to the state transition function (11), update the particles to the next moment; namely, (13)xi-1,bi-1,σi-1p⟶xi,bi,σip.
Step 3 (weights calculation). Calculate the weight ωip of each particle, which is proportional to the value of likelihood function given the current observation yi; namely,(14)ωip∝12πσipexp12yi-xipσip2,p=1,2,…,N.
Step 4 (particles resampling). Resample particles xi,bi,σip according to weights ωip,p=1,2,…,N.
Step 5. Repeat Step 2 to Step 4 until i=k, and the N particles xk,bk,σkp in Step 4 can be used to approximate the posterior distribution of states vector at tk.
Step 6. Propagate particles using (11), and a threshold level Df is specified to extrapolate the N pseudo-RLs, denoted as RLk1,RLk2,…,RLkN, which can approximate the actual RLD at tk.


### 5.2. Results Analysis

The data used as observations are the fast prediction results of the GFM-200 VRLA battery. For simplicity, the capacity predictions are normalized as follows:(15)yi=CiC1,where Ci is the predicted capacity in the ith cycle.In this study, the end-of-life corresponds to the moment when the usable capacity degrades to 80% of the initial value; that is to say, the failure threshold is set to be 0.8. Under the failure criterion, the actual cycle life of GFM-200 battery is 27 cycles.Updating is conducted every 5 cycles, from the 1st cycle to the EOL. The number of particles is set to be 5000. Figure11 presents the procedures of RUL estimation at the 15th cycle. The observation at each cycle is marked by a blue dot, and the red line and dotted lines denote the median and 80% confidence intervals of the predicted normalized capacity in the particle filtering, respectively. Results show that the median predictions are very close to the observations after online updating. The histogram in Figure 11 is the approximated RLD at the 15th cycle, which matches quite well with the actual RUL (12 cycles). The actual value, mean, median, and 80% confidence intervals of the RUL estimation at the 10th, 15th, and 20th cycles are shown in Table 1. It is clear that the actual RUL falls with the range of 80% confidence intervals, indicating that RUL estimation in this paper is very accurate.Table 1
RUL estimation of GFM-200 battery at the 10th, 15th, and 20th cycles.
Current cycle Actual value Mean Median 80% confidence intervals 10 17 20.4 19 [14,27] 15 12 10.9 11 [8,14] 20 7 6.7 6 [4,10]Figure 11
RUL estimation of GFM-200 battery at the 15th cycle.


## 5.1. Particle Filtering

Degradation modeling and parameter updating are keys of online RUL estimation. The degradation process of battery capacity can be treated as an underlying random process. With the predicted capacity, an empirical model can be easily established to describe the common degradation characteristics. To capture the online variability of a battery, model parameters used for prediction should be updated once new capacity measurement is available. In order to characterize the dynamic capacity fading, the particle filtering approach is used in this paper to timely update the parameters distribution. Particle filtering is a sequential Bayesian method based on the technique of Monte Carlo simulation [14]. Compared with the Kalman filtering, it can deal with nonlinear model and non-Gaussian noise system. And thus it is widely used in the field of prognostics [15].In the framework of particle filtering, a state-space model is used to characterize the evolution of degradation feature and model parameters (known as states vector). In the case of VRLA batteries, the degradation feature is battery capacity and the degradation path can be expressed by an exponential function (as Figure9 shows). Motivated by [16], the parameters are not dynamic and can be regarded as constant between two time points. Thus, the state-space model is as follows.State Transition Function (11) x i = exp ⁡ - b i Δ t x i - 1 b i = b i - 1 σ i = σ i - 1 ,Observation Function (12) y i = x i + v i ,where Δt=ti-ti-1, xi is the uncontaminated signal (namely, actual capacity) at ti, yi is the observation (namely, predicted capacity) including Gaussian noise vi, bi is the parameter of the exponential function, and the standard deviation of vi is σi.A set of particles are generated by the Monte Carlo simulation to estimate the current capacityxi as well as distributions of parameters bi and σi. Once new observation is available, these particles are resampled to approximately denote the posterior distribution. More details about the theories of particle filtering can be found in [17, 18]. After obtaining the observations y1,y2,…,yk (yk is the latest observation time), the standard particle filtering for RUL estimation is as follows.Algorithm 2 (standard particle filtering for RUL estimation).
Step 1 (particles initialization). Due to lack of prior information, N initial particles of the state vector x0,b0,σ0p, p=1,2,…,N, are sampled from uniform distributions. Fori=1:k.
Step 2 (particles updating). According to the state transition function (11), update the particles to the next moment; namely, (13)xi-1,bi-1,σi-1p⟶xi,bi,σip.
Step 3 (weights calculation). Calculate the weight ωip of each particle, which is proportional to the value of likelihood function given the current observation yi; namely,(14)ωip∝12πσipexp12yi-xipσip2,p=1,2,…,N.
Step 4 (particles resampling). Resample particles xi,bi,σip according to weights ωip,p=1,2,…,N.
Step 5. Repeat Step 2 to Step 4 until i=k, and the N particles xk,bk,σkp in Step 4 can be used to approximate the posterior distribution of states vector at tk.
Step 6. Propagate particles using (11), and a threshold level Df is specified to extrapolate the N pseudo-RLs, denoted as RLk1,RLk2,…,RLkN, which can approximate the actual RLD at tk.


## 5.2. Results Analysis

The data used as observations are the fast prediction results of the GFM-200 VRLA battery. For simplicity, the capacity predictions are normalized as follows:(15)yi=CiC1,where Ci is the predicted capacity in the ith cycle.In this study, the end-of-life corresponds to the moment when the usable capacity degrades to 80% of the initial value; that is to say, the failure threshold is set to be 0.8. Under the failure criterion, the actual cycle life of GFM-200 battery is 27 cycles.Updating is conducted every 5 cycles, from the 1st cycle to the EOL. The number of particles is set to be 5000. Figure11 presents the procedures of RUL estimation at the 15th cycle. The observation at each cycle is marked by a blue dot, and the red line and dotted lines denote the median and 80% confidence intervals of the predicted normalized capacity in the particle filtering, respectively. Results show that the median predictions are very close to the observations after online updating. The histogram in Figure 11 is the approximated RLD at the 15th cycle, which matches quite well with the actual RUL (12 cycles). The actual value, mean, median, and 80% confidence intervals of the RUL estimation at the 10th, 15th, and 20th cycles are shown in Table 1. It is clear that the actual RUL falls with the range of 80% confidence intervals, indicating that RUL estimation in this paper is very accurate.Table 1
RUL estimation of GFM-200 battery at the 10th, 15th, and 20th cycles.
Current cycle Actual value Mean Median 80% confidence intervals 10 17 20.4 19 [14,27] 15 12 10.9 11 [8,14] 20 7 6.7 6 [4,10]Figure 11
RUL estimation of GFM-200 battery at the 15th cycle.


## 6. Conclusion and Future Work

It is well known that the usable capacity is a key indicator measuring the performance degradation of secondary batteries. In this paper, a RUL estimation framework based on the technique of usable capacity fast prediction is presented for the high-capacity VRLA batteries. The capacity fast prediction approach is conducted using the partially discharging voltages, and the particle filtering method is used to update the capacity degradation model timely. From the experiment results, we observe that the proposed framework can effectively realize VRLA battery capacity prediction as well as RUL estimation. With the advantage of generalization, the approaches in this paper can be easily extended to other types of secondary batteries. In this work, we did not consider the effects of temperature changes on prediction accuracy. In practice, batteries are usually operating under varying temperatures. If necessary, future research may add temperature to the prediction model.



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*Source: 7835049-2017-02-20.xml*