Capacity Estimation Models of Primary Lithium Batteries during Whole Life Cycle of Underwater Vehicles

Abstract

Storage and discharge conditions of primary lithium batteries are studied and the capacity estimation models during the whole life cycle of underwater vehicles is developed based on temperature. The storage experiments for 90 days at different temperatures and discharge experiments at different temperatures and current rates are conducted. At low temperatures, experimental results reveal that there is no significant capacity decay during storage. At high temperatures, the charge storage capacity attenuates with the increase of storage, and the attenuation rate is directly related to the temperature. In discharge experiments, maximum available capacity increases with temperature during 0 °C to 25 °C, whereas the given phenomenon is not significant at other temperatures. Moreover, the current rate is less important for the maximum available capacity. To establish capacity estimation models during the storage stage and working stage of underwater vehicles, we have developed a capacity decay model and a temperature calibration model. Moreover, the model accuracy is evaluated, and the errors in capacity decay model and temperature calibration model are less than 2% and 0.7%, respectively. Capacity estimation models provided in this research are not only accurate, but also relatively simple, which have high value for underwater vehicles in engineering applications.

1. Introduction

There are abundant biological and mineral resources in the sea, which results in a new round of competition about the ocean gradually emerging worldwide. The underwater vehicles are not only an important tool to explore ocean resources but also can undertake a variety of military tasks, such as attacks and defense. Therefore, more countries have started to pay attention to further research on underwater vehicles. The power system, affecting the navigation depth and speed of underwater vehicles directly, is a critical module. Compared with traditional steam-driven vehicles, electric power systems exhibit the characteristics of zero-emission, low noise, excellent concealment and good navigation depth. Hence, electric power systems have gradually developed into a popular propulsion form for underwater vehicles. The primary batteries with lithium metal as anodes generally have high power density, long life, and excellent discharge performances [1], so they are considered the preferred power sources for underwater vehicles in performing military tasks.

The battery charge capacity reflects the currently available energy of the battery, which is a state parameter guiding underwater vehicles to allocate energy efficiently and rationally to ensure completion of the arranged tasks. Hence, the accurate monitoring of the battery charge capacity guarantees the reliable operation of underwater vehicles. Moreover, similar to the fuel gauge in the electric car, the accurate monitoring of the battery charge capacity can reduce the range anxiety of users.

At present, the rapid development of the new energy vehicle industry drives the progress of lithium-ion batteries, and its state estimation of charge and health has been more mature [2]. The common methods, advantages, and disadvantages of lithium-ion battery capacity estimation are summarized here and listed in

Table 1. The common prediction methods for battery capacity.

Test Method	Advantages	High-precision
	Disadvantages	High-cost Long time-consuming
	Accuracy	Good
	Robustness	Poor
Parameter-Based Method	Advantages	Simple Low computational-burden High real-time
	Disadvantages	Sensitive to external environments, such as working environment and aging Require regular calibration of parameters Require precise equipment
	Accuracy	Poor
	Robustness	Good
Ampere-Hour Integral Method	Advantages	Simple Low computational-burden High real-time
	Disadvantages	Depends on the exact initial value Open-loop control Influenced by current drift, noise, and aging
	Accuracy	General
	Robustness	Good
Model-BasedMethod	Advantages	High-precision Closed-loop control High real-time Well-adapted
	Disadvantages	Require an accurate battery model High computational complexity
	Accuracy	Good
	Robustness	Good
Data-Driven Method	Advantages	High precision Excellent nonlinearity
	Disadvantages	High computational complexity Influence by data
	Accuracy	Good
	Robustness	Poor

, including the test method, parameter-based method, ampere-hour integral method (Ah), model-based method and data-driven method. However, compared with the lithium-ion battery, research on primary lithium batteries is quite lacking. Meanwhile, a reaction mechanism between lithium-ion batteries and primary lithium battery is completely different, meaning that some capacity estimation approaches of the lithium-ion battery may not be applicable to lithium primary batteries.

Currently, the most accurate method to determine battery capacity is the test method. It performs a complete discharge to obtain the available charge capacity of battery, which is one way based on the definition of charge capacity. However, the test method has obvious shortcomings, such as non-real-time, long-time-consuming, and high-cost. Moreover, it is a destructive operation for primary lithium batteries. Hence, this method can be used to verify the accuracy of the primary lithium battery capacity estimation model.

Due to the chemical characteristics, such as the degree of active materials degradation, internal resistance, voltage response, current response, etc., will change with charge capacity, the mapping of capacity vs. parameters can be obtained. Hence, the parameter-based method is proposed to estimate battery capacity. Plett first discovered that the open circuit voltage (OCV) monotonically varied with the charge capacity of lithium-ion battery and established a relationship between them by combining linear function, power function, and logarithmic function [3]. Zheng et al. [4] and Xing et al. [5] have also experimentally established a one-to-one mapping table of OCV and the charge capacity for lithium-ion batteries. However, the primary lithium battery discharge voltage profile is quite flat [6,7], making platform voltage vs. charge capacity methods meaningless. Moreover, OCV measurement requires a long rest to depolarize, which cannot be applied in real-time. Manane et al. take entropy ‘∆S’ and enthalpy ‘∆H’ as features to assess charge capacity of Li/MnO₂ primary batteries [6]. If this method is translated into an online form, an embedded chip needs to be equipped to collect voltage and temperature data and convert them into thermodynamic data. It may increase equipment weight and computational-burden on the engineering. To sum up, the parameter-based method is more suitable for the laboratory. Moreover, for some primary lithium batteries, it is quite difficult to find significant capacity characterization parameters.

The ampere-hour integral method is based on ampere-hour counting or the coulomb counting algorithm. Due to a good balance of precision and complexity, this method is preferred in various battery management systems (BMS). Moreover, it is suitable for different battery systems. However, the ampere-hour integral method is the open loop control algorithm, and its performance is greatly affected by the measurement noises, environmental conditions, as well as the variation of initial SOC caused by self-discharging [8,9]. Hence, some periodic calibration methods are recommended by He et al. [2].

The model-based method is a close-loop algorithm and has more accurate prediction results. Based on the battery model and algorithms, the model-based method is realized. The common battery models include the electrochemical model (EM) [10] and equivalent circuit model (ECM) [11]. ECM may achieve a better balance between circuit structural complexity and prediction accuracy [2]. The popular algorithms are Kalman filter series [2,12], particle filters [13], sliding mode observer (SMO) [14], and so on. For the model-based method, an accurate battery model, depending on circuit structures and parameters, is critically important. Based on the previous research works, we know that circuit structures are sensitive to battery systems [15], and circuit parameters change with battery charge capacity and environment temperature [16]. Hence, EM and ECM of lithium-ion batteries cannot be applied directly to primary lithium batteries. However, the specific circuit models of primary lithium batteries do not yet exist, so the applicability of the model-based method in engineering needs to be further explored.

Without understanding the internal mechanism, the data-driven methods take the battery as a black box. Relying on a great deal of experimental data and artificial intelligence algorithms, it can accurately estimate the battery capacity. The typical algorithms consist of support vector regression (SVR) [17], relevance vector machine (RVM) [18], and Gaussian process regression (GPR) [19]. Based on the pulse load test, Sun et al. used the back propagation neural network to build a primary lithium battery capacity estimator. Meanwhile, the estimator has been fully validated by experiments under various load conditions of lithium thionyl chloride batteries with different storage times [20]. Nonetheless, this method has its limitations in practice, i.e., heavy computing burden. Moreover, tremendous battery data must be accumulated to offline the train estimation model, and the model accuracy depends entirely on training data. Hence, whether the data-driven method can be applied under different practical operating conditions still needs to be further validated.

From the above analysis, there are relatively few specific capacity estimation models of primary lithium batteries, and the existing methods, ignoring model complexity and acquisition difficulty of corresponding data, are still in the theoretical research stage. Owing to the underwater working environment, the spatial size and weight of underwater vehicles are constrained strictly. It means that measurement sensors cannot be equipped optionally and data processing capability of embedded chip is limited. Hence, it is extremely urgent to develop simple, accurate, fast, and practical capacity estimation methods of the primary lithium batteries.

After comprehensive consideration of the advantages and disadvantages of the five methods, the characteristics of primary lithium batteries, and the working environment of underwater vehicles, the ampere-hour integral method is the most practical for the detection of the charge capacity of the primary lithium battery in engineering. Hence, a revised ampere-hour integral method is proposed in this research to estimate capacity of primary lithium batteries in underwater vehicles, which can overcome the effect of temperature and age of the common ampere-hour integral algorithm. The detailed arrangement of the manuscript is as follows: the method is presented in Section 2. Storage and discharge experiments are introduced in Section 3. The effects of storage time, storage temperature, current rate, and operating temperature on the capacity of primary lithium batteries are discussed, and the capacity decay model and the temperature calibration model are proposed and validated in Section 4. Finally, the key conclusions are drawn in Section 5.

2. Proposed Methods

The primary lithium battery is a complex nonlinear system, where the chemical reaction takes place between lithium metal in the anode and electrolyte in the cathode. It is well-known that chemical reactions are extremely sensitive to environment temperature. Hence, for most primary lithium batteries, temperature is the most critical factor for degree of capacity decay in storage condition and the most available capacity in working conditions.

The Arrhenius model takes temperature as an accelerated aging factor to describe the battery capacity loss. At present, the Arrhenius model has been widely studied [21,22,23,24], and exhibits the inverse power–law relation between battery capacity loss and cycle number shown as follows:

$$K=A\times e^{-E_{\mathrm{a}}/RT}$$

(1)

where K refers to the reaction rate constant, A represents the pre-factor constant, E_a denotes the activation energy (J/mol), and R refers to the gas constant (J/(mol·K)). T corresponds to thermodynamic temperature (K).

It should be noted that primary lithium batteries do not experience cycling. Hence, to assess the capacity decay of primary batteries, the Arrhenius model should be further improved according to corresponding test data. More details are shown in Section 4.2.

3. Experiment

In general, there are two stages during the whole cycle life of underwater vehicles, i.e., storage and operation. It is well known that the batteries possess self-discharge characteristics and are sensitive to ambient temperature. Therefore, the storage and discharge experiments have been conducted. The lithium argon-sulfuryl chloride battery (ER48690), which is a power source for a certain type of underwater vehicle, is taken as the test object and the basic parameters are listed in

Table 2. The basic parameters of the tested battery system, i.e., ER48690.

Type	Nominal Capacity	Nominal Voltage	Nominal Current	Lower Cut-Off Voltage
ER48690	22 Ah	3.6 V	2.2 A	3 V

. The test system, used in this paper, is shown in Figure 1, including a discharge panel, XW5005C8, manufactured by NEWARE, a thermal chamber from Giant Force to control the storage and operation temperature, a computer to program and store the experimental data. The acquisition accuracy of current and voltage was 1 mA and 1 mV, respectively.

The purpose of the storage experiments is to explore the influence of temperature and storage time on battery capacity. The specific experimental process is shown in Figure 2. First, a total of 72 cells were selected to form an experimental sample. Then, the samples were divided into four groups and stored at four temperatures, i.e., 0 °C, 25 °C, 45 °C, and 54 °C. Finally, with an interval of 15 days, three cells were taken out at each temperature and discharged at a nominal current. Three independent cells were used to ensure the reliability of experimental data.

During the discharge experiments, temperature and discharge rate were taken into account to explore the battery capacity characteristics. The detailed experimental steps are shown in Figure 3. Herein, 36 cells were randomly selected from the sample with good consistency. At four different temperatures, i.e., −10 °C, 0 °C, 25 °C, and 40 °C, 36 cells were divided into four groups. Then, each cell was discharged at three different rates, i.e., 0.05 C, 0.1 C, and 0.2 C, resulting in 12 different working conditions, where the discharge experiments were conducted and experimental data were recorded.

4. Results and Discussion

The effects of storage temperature, storage time, current rate, and working temperature on battery capacity were analyzed and discussed. Meanwhile, a capacity decay model and a temperature calibration model were developed based on the corresponding experimental data.

4.1. Experiment Results

Figure 4 presents the voltage variations during the discharge. Figure 4a–d represent the results at different storage temperatures for 30, 45, 60, and 90 days, respectively. Herein, as the results of three samples were basically the same, one sample was randomly selected for display. It can be found that the discharge time at 25 and 0 °C is longer under the same storage time, and it does not change with storage time (Figure 4). At 45 °C and 54 °C, the discharge time decreases successively and the rate of decrease increases with the increase of storage time. Moreover, the discharge voltage is unstable and the platform voltage becomes short after 45 days of storage at 54 °C.

Figure 5 represents the capacity retention over storage time at different temperatures. Here, three independent cells were tested at each sample point to ensure the reliability of experimental data. It is obvious that the value of capacity retention fluctuates around 1 at 0 °C and 25 °C, whereas it apparently decreases over storage time at 45 °C and 54 °C. The higher temperature corresponds to the rapid decrease in capacity retention, which is due to the fact that high temperature promotes the rate of chemical reactions inside the cell. In general, it can be concluded that a high-temperature storage environment leads to rapid capacity decay.

Figure 6 presents the discharge voltage curves at different temperatures and current rates. Figure 6a–d correspond to the environment temperature of −10 °C, 0 °C, 25 °C, and 40 °C in the discharge process, respectively. It can be found that the discharge voltage is more stable and the voltage platform period is longer at high temperatures. In other words, ER48690 exhibits better stability at high temperatures. Figure 7 presents the battery capacity at different temperatures and current rates. To display the impact of both factors on capacity intuitively, the capacity vs. temperature and capacity vs. current rate are plotted in Figure 7a,b, respectively. The capacity value at 40 °C and 4.4 A does not conform to the overall behavior, so it is treated as an outlier. Overall, the effect of temperature on battery capacity is significant, whereas the current rate renders a negligible influence on battery capacity. In the temperature range of 0 to 25 °C, the maximum available capacity of the battery increases with the increase of temperature, whereas it remains basically at >25 °C and <10 °C. Hence, ER48690 exhibits stable performance at high and low temperatures.

4.2. Capacity Estimation Models

Based on the storage experiments, it can be known that capacity attenuates significantly during storage at high temperatures. Hence, it is necessary to estimate the capacity loss caused by self-discharge to accurately obtain the remaining capacity of underwater vehicles in the operating stage.

For primary batteries, the electricity is generated due to the chemical reaction of lithium metal and the consumption of Li decays the battery capacity. Hence, the rate of capacity decay is closely related to the chemical reaction rate of lithium metal, as given below:

$$K{t}_n=Q_0-Q_{n,25}$$

(2)

where K refers to the reaction rate of lithium metal, t_n represents the storage time (days), Q₀ denotes the maximum available capacity of the new battery, and Q_n,25 corresponds to the maximum available capacity of the battery stored for t_n days and discharged at 25 °C. According to the experimental results, the reaction rate of lithium metal at different temperatures can be derived, as listed in

Table 3. The reaction rate of lithium metal at different temperatures.

K0 (Ah/d)	K25 (Ah/d)	K45 (Ah/d)	K54 (Ah/d)
2.7 × 10−3	3.4 × 10−3	44.4 × 10−3	74.0 × 10−3

. Herein, the mean values of three samples are taken as the final experimental results in different cases. It can be known that the reaction rate remains constant in the temperature range of 0 °C to 25 °C, whereas the reaction rate exhibits a 20-fold increase from 25 °C to 54 °C. Finally, it is determined that the capacity decay model is developed only at >25 °C.

It is generally considered that the effect of temperature on the rate of chemical reaction conforms to the Arrhenius equation. From the Section 2, we know that temperature renders a significant influence on the rate of the chemical reaction, and the Arrhenius equation further defines the relationship between temperature and chemical reaction rate. Hence, taking logarithms of both sides of Equation (1), the linear form of Arrhenius equation can be obtained:

$$\ln K=\ln A-\frac{E_{\mathrm{a}}}{R}\cdot \frac{1}{T}$$

(3)

Figure 8 can be drawn by incorporating the reaction rate constant K and corresponding temperature T into Equation (3). After fitting analysis, the relationship between ln(K) and 1/T is established, as given by Equation (4). Meanwhile, parameters lnA and E_a/R are found to be 32.81 and 11,515.24, respectively.

$$\ln (K)=-11515.24\cdot \frac{1}{T}+32.81$$

(4)

By changing the form of Equation (3), the chemical reaction rate K can be expressed as:

$$K=\exp (\frac{-11515.24}{T}+32.81)$$

(5)

By substituting Equation (5) into Equation (1), the capacity decay model after t_n days of storage at temperature T can be developed as:

$$Q_{n,25}=Q_0-t_{n}K=Q_0-t_n\exp (\frac{-11515.24}{T}+32.81)$$

(6)

Furthermore, the error analysis is performed to assess the accuracy of abovementioned model. The mean square error (MSE) is the most important and common index to measure the deviation between predicted and actual values. The larger MSE corresponds to the lower model accuracy. The MSE can be given as:

$$MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}$$

(7)

where n refers to the number of samples, y_i represents the true value of the sample i, and ŷ_i corresponds to the predicted value of sample i.

Since the unit and dimensions of MSE are inconvenient to explain, the arithmetic square root and root mean square error (RMSE) are introduced, as follows:

$$RMSE=\sqrt{MSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}}$$

(8)

The relative error is defined as the ratio of RMSE and initial capacity. The data from storage experiments at 45 °C and 54 °C were used to evaluate the capacity decay model. The specific values are listed in

Table 4. The statistical results of the capacity decay model.

	MSE (Ah2)	RMSE (Ah)	Error (%)
45 °C	0.077926	0.279153	1.12123
54 °C	0.147483	0.384036	1.542495

. The results indicate that RMSE is within 0.4 Ah and the relative error is less than 2%, which are acceptable for ER48690 with 22 Ah.

To construct the models of primary lithium batteries during the whole life cycle of underwater vehicles, the temperature calibration model is proposed based on the capacity decay model and discharge experimental data. It is worth noting that the capacity derived from the capacity decay model in Equation (6) is the maximum available capacity when battery discharges at 25 °C. Figure 7 indicates that the maximum available capacity is sensitive to the operating temperature. Hence, a scaling factor (f), calibrating the maximum available capacity at different operating temperature, is proposed here:

$$f=Q_{n,T}/Q_{n,25}$$

(9)

where Q_n,T refers to the maximum available capacity at operating temperature T. Q_n,25 represents the maximum available capacity when the battery discharges at 25 °C, which can be obtained from the capacity decay model. Based on the experimental discharge data, the relationship between f and temperature is fitted, as shown in Figure 9. Herein, the effect of current rate is neglected and only experiment data with a nominal current rate are taken here. The relative error between experimental data and fitted values is less than 0.7%.

$$f=0.889+\frac{0.124}{1+{10}^{(1.252-0.1T)}}$$

(10)

Finally, the maximum capacity at the working stage of underwater vehicles can be given as:

$$Q_{n,T}=(0.889+\frac{0.124}{1+{10}^{(1.252-0.1T)}})Q_{n,25}$$

(11)

According to the ampere-hour integral method, the rest capacity of a primary lithium battery during operation can be given as:

$$Q_r=Q_{n,T}-{\displaystyle {\int }_{t0}^{t}i}dt$$

(12)

where Q_n,T is obtained from the temperature calibration model, i represents discharge current (A), and t refers to the discharge time (h).

5. Conclusions

In damaging military missions, primary lithium batteries, with higher energy density and lower cost, are the optimal power source for underwater vehicles. Based on battery charge capacity, energy can be rationally allocated to complete the assigned tasks. However, the primary lithium battery is non-rechargeable, which limits its application range. Hence, research of the primary lithium battery currently is quite lacking. Moreover, the few existing studies, ignoring model complexity and acquisition difficulty of corresponding data, are basically still in the theoretical research stage. Hence, to develop accurate and practical capacity estimation methods of the primary lithium batteries for the whole life cycle of underwater vehicles, a capacity decay model and a temperature calibration model of primary lithium battery were developed in this research, where no extra measuring instruments need to be equipped and chips embedded in the battery management system can easily perform corresponding calculations. Moreover, both MSE and RMSE of capacity estimation models were also further calculated. The main conclusions are listed as follows:

The storage experiments for 90 days at 0 °C, 25 °C, 45 °C, and 54 °C and the discharge experiments at different temperatures (−10 °C, 0 °C, 25 °C and 40 °C) and current rates (0.05 C, 0.1 C and 0.2 C) have been conducted. The results of the storage tests reveal that there is almost no capacity attenuation during the storage stage at ≤25 °C. However, the capacity significantly decreases with storage time at >25 °C and the decay rate increases with the increase in temperature. The results of the discharge tests manifest that the effect of current rate on the maximum available capacity of battery is not remarkable. On the other hand, the maximum available capacity augments with temperature increasing from 0 °C to 25 °C, whereas it remains insensitive to other temperature ranges.

Based on the data of storage experiments, the capacity decay model has been developed to estimate the capacity loss during the storage of underwater vehicles. Moreover, the model accuracy has been verified by comparing with experimental data and the relative error was found to be less than 2%. Based on the effect of temperature on maximum available capacity, the temperature calibration model has been created and a scaling factor (f) has been proposed. (f) represents the ratio of maximum available capacity at an arbitrary operating temperature to that at 25 °C. The accuracy of proposed model has been discussed with a relative error of less than 0.7%.