A self-discharge model of Lithium-Sulfur batteries based on direct shuttle current measurement

TL;DR: In this paper, a simple but comprehensive mathematical model of the Li-S battery cell self-discharge based on the shuttle current was developed and is presented The shuttle current values for the model parameterization were obtained from the direct shuttle current measurements.

About: This article is published in Journal of Power Sources.The article was published on 2016-12-30 and is currently open access. It has received 24 citations till now. The article focuses on the topics: Self-discharge & State of charge.

Summary

1. Introduction

• Lithium-Sulfur (Li-S) batteries represent a promising alternative to the Lithium-ion battery chemistry, due to their high theoretical limits in terms of specific capacity (i.e. 1672 Ah kg-1) and specific energy (i.e. 2600 Wh kg-1).
• Furthermore, they are expected to become a cheaper and more environmentally friendly solution, mainly due to the use of sulfur, which is an abundant and benign element.
• A mechanistic model of the polysulfide shuttle causing the self-discharge of the Li-S battery cells was presented in [8].
• The self-discharge current was identified to be proportional to the square root of the idling time.
• Furthermore, a methodology for direct shuttle current measurement was proposed in [12], where its results were analyzed and validated using the one-dimensional phenomenological model, which is based on Nernst and species concentrations equations.

2. Methodology

• The work flow followed in this paper is summarized and presented in Fig.
• At first, the measurements were performed and they are described in Section 2.1 for direct shuttle current measurements and in Section 2.2 for the self-discharge model validation measurements.
• Later on, there were considered three fitting cases.
• Each of these fitting cases parameterize the selfdischarge model and its accuracy is later validated in Section 3.2 by an use of the validation measurements (Section 2.2) and the SOC estimation model for the validation (Section 2.3) with the consideration of the total capacity concept (Section 2.4).
• Ah long life chemistry Li-S pouch cell manufactured by OXIS Energy.

2.1 Direct Shuttle Current Measurement

• The applied test procedure for the direct shuttle current measurement is based on the methodology presented in [12] and illustrated in Fig.
• The second cycle was used for the cell’s capacity check and its calculation for the further procedure steps.
• In practice, due to the noise in the voltage signal, the reliable value of the OCV was determined when the battery voltage dropped from the maximum point by 0.6 mV (three times the value of the battery test station accuracy which was equal to 0.2 mV) as it is presented in Fig.
• The voltage and current signals during the direct shuttle current measurement procedure are shown in Fig. 3 for DOD equal to 10 % at 35 °C.

2.3 Matlab/Simulink Model for Validation

• The self-discharge Li-S model is going to be integrated into a Matlab/Simulink model, which allows for SOC estimation based on the coulomb counting method.
• The used SOC definition in this work follows the definition described in [13].
• So the SOC represents the relation between the actual useable battery capacity (Ca) and the total capacity (Ct) available to be discharged after the battery being fully charged.
• Using only coulomb counting method, without accounting for the fast self-discharge of the Li-S batteries will lead inevitably to a growing error due to not capturing the self-discharge current.

2.4 Concept of the total capacity of the Li-S batteries

• The standard practice to determine the capacity of Li-S battery cells is to continuously discharge before-hand fully charged battery by a specific current at a specific temperature.
• The obtained discharged capacity is considered as the capacity of the cell at those conditions.
• As the polysulfide shuttle is present during the Li-S cell discharging, it causes self-discharge, which consequently reduces the measured capacity.
• Csd is obtained from the simulation of the cell’s continuous discharge with Ccdch replacing Ct in (1).
• Moreover, the Ish is excluded from the coulomb counting in (1) and it is integrated and recorded separately.

3. Measurement Results and Modelling

• The current profiles obtained from the constant voltage charging steps during the direct shuttle current measurements, at 35 °C, are presented in Fig.
• Due to the accuracy of the test station, extra noise is appearing at the current values lower than 0.06 A.
• In order to get a higher accuracy of the measured shuttle current values, the measurement can be repeated using equipment dedicated for lower current ranges.
• For the demonstration purposes of the model, in this paper, it is considered sufficient to take an average of the last ten minutes of the current profile during constant voltage charging step as the value for the shuttle current.
• The measured shuttle current values for pre-determined DOD points.

3.1.1 Fitting Case 1

• The first fitting step, referred as Fitting Case 1 (FC1), was performed by fitting the experimentally determined direct shuttle current values against the DOD points (see Fig. 6).
• For the last DOD level, when during the battery cell relaxation period of 12 hours a peak voltage value was not detected (as described in the methodology in the previous section), a shuttle current value equal to zero was considered for fitting purposes.
• The considered DOD levels are shown in Table I for FC1 and 35 °C.
• These predetermined DOD points might not accurately correspond to the actual DOD levels of the cell as the influence of the shuttle current was not considered during the measurement procedure.

3.1.2 Fitting Case 2

• Therefore, for the Fitting Case 2 (FC2), it was assumed that the self-discharge was ongoing already during the discharging steps, during the relaxation periods before the constant voltage charging step, and during the constant voltage charging step in the characterization experiment, as it is illustrated in Fig.
• The time values of discharging and relaxation were computed and multiplied by the measured shuttle current value for the first DOD point (i.e. 2 %), which provided an estimate of the ampere-hours lost due to self-discharge during that period.
• During the previous one hour and fifty minutes, the shuttle current is only partially compensated as the external current is lower.
• Therefore, the amount of the self-discharged ampere-hours can be obtained by integration of the area above the current curve in a rectangle from the beginning of the constant voltage charging up to one hour and fifty minutes time coordinates.
• The same procedure was repeated for the remaining DOD points, considering also the correction from the previous DOD point.

3.1.3 Fitting Case 3

• For the further improvement of the model, the Fitting Case 3 (FC3) was applied to obtain the total capacity.
• The Simulink model, including the self-discharge model obtained at the end of FC2, was fed by the current profile obtained from the direct current shuttle procedure.
• Thus, the DOD points, corresponding to the shuttle current values, were extracted and are presented in Table I for 35 °C.
• The parameters c, d, e and f of the shuttle current model for all the fitting cases are presented in Table II.
• The presented fitting procedure with all three fitting cases and their steps are visualized in Fig. 7.

3.2 Validation of the self-discharge model

• For the validation of the self-discharge model, four validation measurements were performed according to the procedure described in Section 2.2.
• The validation cases consider various temperature conditions, idling times and initial DOD levels.
• A comparison of the accuracy values of the developed self-discharge model for the different used fitting cases is shown in Table III.
• The relative errors are noticeably reduced by moving from FC1 to FC2, except the Validation Case 1, where only a minor increase is observed.
• The error reduction implies the correctness of the assumptions used for FC2 and that the self-discharge due to the polysulfide shuttle is still present, no matter if the Li-S battery is in charging, relaxation or discharging stage.

4.1 SOC reference frame & cell history effect

• The challenging part of the integration of the presented self-discharge model into any other model is that the battery performance model has to have the same DOD/SOC reference frame in order that the dependency states to be matched.
• Due to the ‘rate capacity effect’ [14], the available battery capacity varies according the applied current.
• In [15], for practical reasons, the mixed pulse discharge was used to determine Ccdch.
• Therefore, the DOD of the performance model has to be converted into DOD of the self-discharge model at its input.

4.2 Open-circuit voltage based self-discharge model

• Alternatively, the DOD dependence of the self-discharge model can be replaced by the open-circuit voltage (OCV) dependence.
• By following this approach, the shuttle current values related to the OCV are presented in Fig. 8, where it is important to note that this relation is valid only for OCV values at the side of the high voltage plateau.
• The OCV values in the practical use can be obtained for example by online parameter identification techniques [16].
• Based on the curve fitting of the shuttle current dependency on DOD and temperature, a simple mathematical model for the self-discharge estimation of Li-S batteries was obtained.
• The developed model was successfully validated by the experiments considering various conditions with a relative error smaller than 7 %.

Acknowledgments

• This work has been part of the ACEMU-project.
• The authors gratefully acknowledge the Danish Council for Strategic Research (1313-00004B) and EUDP (1440-0007) for providing financial support and would like to thank OXIS Energy for supplying the Lithium-Sulfur battery cells.

Figures

Figure 5. Battery currents during the constant voltage charging step at 35 °C. The shuttle current value is taken as the average of the last ten minutes interval. The integration area for the current at 2 % DOD used in FC2 is marked by blue stripes.

Figure 8. The measured shuttle current values related to the open-circuit voltage values (at the side of the high voltage plateau).

TABLE I. Considered and recomputed DOD values at 35 °C for fitting at various fitting cases.

TABLE IV. Measured and estimated battery cell capacities and their absolute and relative errors for four validation cases with the self-discharge model parametrized based on FC2.

TABLE III. Relative errors of validation cases for different fitting cases.

Figure 1. Work flow scheme used for the self-discharge model derivation and validation.

Figure 3. Illustration of current and voltage signals during the direct shuttle current measurement.

Figure 2. Overall test procedure for the direct shuttle current measurement.

Figure 7. Illustration of the fitting procedure steps.

Figure 4. Test procedure for the self-discharge model validation.