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Time efficient state-of-charge estimation using open circuit voltage and the logarithmic modelling for… Shen, Jing Cong 2016

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    TIME EFFICIENT STATE-OF-CHARGE ESTIMATION USING OPEN CIRCUIT VOLTAGE AND THE LOGARITHMIC MODELLING FOR BATTERY MANAGEMENT SYSTEMS   by  JINGCONG SHEN   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Electrical and Computer Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   August 2016  © Jingcong Shen 2016   ii  Abstract  Battery management systems are important devices for protecting batteries in various electrical and electronic applications. One of the most important features of a properly designed battery management system is the diagnostic techniques for estimating the remaining usable charge in batteries in a time effective manner. This thesis presents two time-efficient and reliable techniques that do not require complex electronic hardware for application. One technique concerns the open-circuit voltage characteristics of a battery while the other technique considers the logarithmic modeling of the equivalent circuit of a lithium-ion battery so that the equilibrium OCV can be found within a shorter period of time. The logarithmic modelling method uses the equivalent circuit model and the characteristics of the subject lithium NMC battery and finds the equilibrium OCV. This thesis will include two techniques for the logarithmic modelling method. One technique uses the long time constant in the equivalent model of the battery and predicts the equilibrium with 1% error after the battery relaxes for 150 seconds. The other technique uses the short time constant in the equivalent model and predicts the equilibrium OCV with 2% error after the battery relaxes for 70 seconds. Both techniques of the logarithmic modeling method show improvement for estimating the equilibrium OCV in terms of waiting time compared to the standard methods.     iii  Preface  The text of the thesis is original and unpublished work of the author, J. Shen. The circuit in Chapter 3 is designed and tested by myself.  The implementation of the logarithmic modelling method is proposed and developed by Dr. Dunford.     iv  Table of Contents  Abstract………………………………………………………………………………………………….. ii Preface………………………………………………………………………………………………….. iii  Table of Contents……………………………………………………………………………………... v List of Tables………………………………………………………………………………………….. vi List of Figures……………………………………………………………………………………… x List of Abbreviations……………………………………………………………………………….. xiii Acknowledgement…………………………………………………………………………………… xix Dedication……………………………………………………………………………………………… xx Chapter 1: Introduction……………………………………………………………………………….. 1 1.1 Background and motivation……………………...………………………………………………….. 1 1.2 Thesis outline…………………………………………...…………………………………………….. 5 Chapter 2: Literature Review………………………………………………………………………… 6 2.1 Coulomb counting……………………………………………………………………………………. 6 2.2 Electromotive force (EMF) method………………………………………………………………… 7 2.3 Observer method……………………………………………………………………………………. 10 Chapter 3: Open Circuit Voltage Curve for Lithium-ion and NMC Batteries………………. 12 3.1 Concept and background of open circuit voltage………………….………………………….. 12 3.2 Experimental design and setup…………………………………………..………………………. 13 3.3 Experimental results and verification of OCV method……………………………………….. 16 Chapter 4: The Logarithmic Modelling for Finding OCV……………………………………... 21  4.1 Equivalent circuit model of lithium-ion batteries and the logarithmic modelling…………… 21 4.1.1 Introduction of equivalent circuit of battery and logarithmic modeling………….. 21 4.1.2 Experimental setup and the preconditions…………………………………………… 26 4.2 Using logarithmic modeling to find OCV after discharge………………..……………………… 26 4.2.1 The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 22° C………………………………………………………….. 26   4.2.2 The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 30 ° C………………………...……………………..………… 32  v  4.2.3 The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 40 ° C……………………….………………………………… 36  4.3 Using logarithmic modeling to find OCV after charge…………………………………………… 40  4.4 Software algorithm for implementing logarithmic modeling……………………..……………… 44   4.4.1 Parameters of the equivalent circuit model under the subject discharge  conditions………………………………………………………………………………... 51 4.5 Investigation for the behavior of 𝑑𝑖𝑓𝑓 for t < 150 seconds………………...…………………... 53    4.5.1 More observations on the relationship between varying values of 𝑘 versus  𝑑𝑖𝑓𝑓 …….…………..…………………….………………………………………………. 57   4.5.2 Comparison of the 3 discussed methods for large depth of discharge……………. 64 Chapter 5: Conclusion………………………………………………………………………………. 75  5.1    Conclusion on the findings………………………………………………………………………. 75  5.2    Recommendation for future research…………………………………………………………... 77 References…………………………………………………………………………………………….. 78 Appendix A…………………………………………………………………………………………….. 81 Appendix B…………………………………………………………………………………………….. 99 Appendix C…………………………………………………………………………………………… 106   vi  List of Tables Table 1.  Comparison of the features among different kinds of batteries…………………………..………... 3 Table 2.  Comparison of self-discharge loss effect among various kinds of batteries © [2014] IEEE……... 3 Table 3.  Procedures for obtaining discharge OCV with a condition of 0.3C discharge rate and 0.3C   charge rate……………………………………………………………………………………………. 14 Table 4.  Procedures for obtaining charge OCV with a condition of 0.3C discharge rate and 0.3C charge  rate……………………………………………………………………………………………………….. 14 Table 5.  Average OCV values for charge and discharge at different temperatures………………...…….. 17 Table 6.  Usable capacity in percentage of nominal capacity…………………………………………..……. 18 Table 7.  Standard deviations and standard errors for the average OCV measurements for the discharge   at different temperatures……………………………………………………………………………. 18 Table 8.  Standard deviations and standard errors for the average OCV measurements for the charge at  different temperatures…………………………………………………………………………………. 19 Table 9.  Results for the verification tests for the OCV method………………………………………...……. 20 Table 10. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of     relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at  22°C………………………………….…………………………………………………………………. 49 Table 11. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at  30°C……………………………………...…………………………………………………………….. 49 Table 12. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at 40°  C…………………….………………………………………………………………………………….. 50 Table 13. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at  22°C…………………………………….....…………………………………………………………… 50 Table 14. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at  30°C.......................................................................................................................................... 50   vii  Table 15. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at  40°C…………………………………………………………………………………………..………… 51 Table 16. Parameters of the equivalent circuit model for various discharge conditions at 10% SOC at  22°C…………………………………………………………………………….………………………. 51 Table 17. Parameters of the equivalent circuit model for various discharge conditions at 30% SOC at  22°C…………………………………………………………………………………………………….. 51 Table 18. Parameters of the equivalent circuit model for various discharge conditions at 60% SOC at  22°C………….…………………………………………………………………………………………. 52 Table 19. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at  22°C…………….………………………………………………………………………………………. 52 Table 20. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at  30°C…………….………………………………………………………………………………………. 52 Table 21. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at  40°C…………………………………………………………………………………………………….. 52 Table 22. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 at the 90th second of relaxation from  discharge from 100% SOC to 90% SOC by various current magnitudes at 22°C.……………. 54 Table 23. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 at the 90th second of relaxation from  discharge from 70% SOC to 60% SOC by various current magnitudes at 22°C………………. 55 Table 24. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 at the 90th second of relaxation from  discharge from 40% SOC to 30% SOC by various current magnitudes at 22°C………………. 55 Table 25. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 at the 90th second of relaxation from  discharge from 20% SOC to 10% SOC by various current magnitudes at 22°C…….………… 56 Table 26. Tests on battery samples B, C and D: 40% depth of discharge, 1.5C current, 22°C………….. 67 Table 27. Tests on battery samples B, C and D: 90% depth of discharge, 3C current, 22°C……………. 67 Table 28. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes  22°C…………………………………………………………………………………….………………. 91 Table 29. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at  22°C………….…………………………………………………………………………………………. 92  viii  Table 30. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of     relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at     22°C…………………………………………………………………….………………………………. 92 Table 31. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of         relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at   30°C…..……………………………………………………………………………………………….. 93 Table 32. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at  30°C…………………………………………………………………………………………………….. 93 Table 33. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at  30°C…………………………………………………………………………………………………….. 94 Table 34. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………….. 94 Table 35. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………….. 95 Table 36. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………….. 95 Table 37. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of     relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at  22°C……………………………………………………………………………………………..……… 96 Table 38. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of     relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at  22°C………….……………………………………………………………………………………….… 96 Table 39. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of      relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at  22°C………….……………………………………………………………………………..………….. 96   ix  Table 40. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of     relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at  30°C…………….……………………………………………………………………………….……… 97 Table 41. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at  30°C…………………………………………………………………………………………..………… 97 Table 42. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at  30°C…………………………………………………………………………………………………….. 97 Table 43. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………..… 98 Table 44. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………..… 98 Table 45. Sliding average of 𝑑𝑖𝑓𝑓 calculated with different 𝑘 from the 140th to the 150th second of  relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at  40°C…………………………………………………………………………………………………….. 98    x  List of Figures Figure 1.  Fundamental configuration and operating principle of a lithium-ion battery © [2011] IEEE…... 2 Figure 2.  Definition of the operating region of a battery during discharge…………………………………… 9 Figure 3.  Typical observer as a feedback method (left) and equivalent RC circuit model (right)……… 11 Figure 4.  Circuitry for measuring charge OCV (left) and discharge OCV (right)………………………… 15 Figure 5.   Relaxation at 50% SOC at 22。° C after discharge of various magnitudes of current between 0  to 900 seconds……..…………………………………………….………………………………….. 16 Figure 6.  Circuit model with two RC branches………………………………………………………………… 22 Figure 7.  Relaxation characteristics of the lithium-ion battery after charge and discharge………..……... 23 Figure 8.  The effect of varying 𝑘 in the logarithmic model………………………………...…………………. 25 Figure 9.  Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from    100% SOC down to 90% SOC at 22° C………………………………...……………...………….. 27 Figure 10. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  70% down to 60% SOC at 22° C……………………..……………………………………………. 27 Figure 11. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  40% SOC down to 30% SOC at 22° C…………………………….……………………………… 28 Figure 12. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  20% SOC down to 10% SOC at 22° C………………………………………….………………… 28 Figure 13. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for 0.3C discharge rate for different SOC levels  at 22° C…………………………..…………………………………………………………………… 29 Figure 14. 100th second to 150th second of Figure 13………………………………………………………… 30 Figure 15. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from 100th second to 150th second of the relaxation after a 3C  discharge from 40% SOC down to 30% SOC at 22° C…………………………..……………... 31 Figure 16. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 150th second of the relaxation after a 3C  discharge from 40% SOC down to 30% SOC at 22° C……………….……………………….... 31 Figure 17. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from   100% SOC down to 90% SOC at 30。C…………………………………….……………………... 33 Figure 18. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  70% SOC down to 60% SOC at 30。C………………..…………………………………………... 33  xi  Figure 19. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  40% SOC down to 30% SOC at 30。C……………………..……………………………………... 34 Figure 20. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  20% SOC down to 10% SOC at 30。C……...……...……………………………………………... 34 Figure 21. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for 0.3C discharge rate for different SOC levels  at 30。C………………………………………………………………………………………………... 35 Figure 22. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 300th second of the relaxation after a 3C  discharge at 30% SOC at 30。C……………………………………………………………………. 36 Figure 23. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  100% SOC down to 90% SOC at 40。C…………………………………………………………... 37 Figure 24. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  70% SOC down to 60% SOC at 40。C……………..……………………………………………... 37 Figure 25. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  40% SOC down to 30% SOC at 40。C…………..………………………………………………... 38 Figure 26. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various discharge current magnitudes from  20% SOC down to 10% SOC at 40。C…………….……………………………………………... 38 Figure 27. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for 0.3C discharge rate for different SOC levels  at 40。C………………………………………………………………………………………………... 39 Figure 28. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 300th second of the relaxation after a 3C  discharge at 30% SOC at 40。C……………………………………………………………………. 40 Figure 29. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various charge current magnitudes from 60%  SOC UP TO 80% SOC at 22° C…………………………..…………………….…………………. 41 Figure 30. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various charge current magnitudes from 40%  SOC up to 60% SOC at 22° C……………………..………………………………………………. 41 Figure 31. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various charge current magnitudes from 20%  SOC up to 40% SOC at 22° C……………………….……………………………………….……. 42 Figure 32. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for various charge current magnitudes from 0%  SOC up to 20% SOC at 22° C…………………………….……………….………………………. 42 Figure 33. Variation of 𝑑𝑖𝑓𝑓 calculated by setting 𝑘 = 𝑂𝐶𝑉 for 0.3C charge rate for different SOC levels at  22°C………………………..…………………………………………………………………………. 43  xii  Figure 34. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 300th second of the relaxation after a 1C  charge at 40% SOC at 22° C………………….…………………………………………………… 43 Figure 35. The flowchart of logarithmic modeling algorithm for discharge scenarios……………………... 45 Figure 36. Configuration for the component parameters in the equivalent circuit model of Lithium-ion  battery………………………………………………………………………………………………… 46 Figure 37. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 2C discharge from 100%       SOC to 90% SOC at 22° C…………………………….…………………………………………… 47 Figure 38. Curve fitting between measurement data and simulation data after 3C discharge from 100%   SOC down to 90% SOC at 22° C……………..………………………………………………….. 48 Figure 39. Curve fitting between measurement data and simulation data after 0.3C discharge from 100%   SOC down to 90% SOC at 22° C……………………………………………………………….... 48 Figure 40. The earliest time for the LM method find 𝑘 within 2% (𝑘 = OCV + 15mV) of error from the   correct equilibrium OCV under various discharge conditions at 90% SOC at 22° C................ 56 Figure 41. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40  seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of  current at 22° C………………………………………………………………………………………. 59 Figure 42. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds after  discharge from 100% SOC down to 90% SOC by different magnitudes of current at 30°C… 59 Figure 43. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds after  discharge from 100% SOC down to 90% SOC by different magnitudes of current at 40°C… 60 Figure 44. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 60 seconds after  discharge from 100% SOC down to 90% SOC by different magnitudes of current at 22°C… 61 Figure 45. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 60 seconds after   discharge from 40% SOC down to 30% SOC by different magnitudes of current at 22°C….. 63  Figure 46. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 60 seconds after   discharge from 20% SOC down to 10% SOC by different magnitudes of current at 22°C….. 63  Figure 47. Terminal voltage of a Li NMC battery during a continuous 2C discharge test and during  relaxation at 22° C…………………………………………………………………………………… 64 Figure 48. Values of diff for 𝑘 = OCV in the 2C discharge test………………………………………………. 65 Figure 49. Zoom-in view of diff plot for 𝑘 = OCV from the 100th to the 150th second of relaxation………. 65  xiii  Figure 50. Values of diff calculated by setting 𝑘 = OCV+15mV during the relaxation after the 2C discharge  test…………………………………………………………………………………………………….. 66 Figure 51. Values of 𝑑𝑖𝑓𝑓 calculated at the 40th second, the 60th second and the 70th second of relaxation  versus varying values of 𝑘………………………………………………………………………….. 66 Figure 52. Discharge and relaxation curves for battery B, C and D of the test conditions in Table 26….. 68 Figure 53. Values of diff for 𝑘 = OCV in the 1.5C discharge tests of batteries B, C and D under the stated  conditions of Table 26………………………………………………………………………………. 68 Figure 54. Values of diff after batteries B, C and D relax for 150 seconds after the 1.5C   40% DoD tests……………………………………………………………………………………… 69 Figure 55. Values of 𝑑𝑖𝑓𝑓 calculated with varying values of 𝑘 at the 70th second of the relaxation from   1.5C 40% DoD test at 22°C……………………………………………………………………….. 69 Figure 56. Discharge and relaxation curves for battery B, C and D of the test conditions in Table 27….. 70 Figure 57. Values of diff for 𝑘 = OCV in the 3C discharge tests of batteries B, C and D under the stated  conditions of Table 27………………………………………………………………………………. 70 Figure 58. Values of diff after batteries B, C and D relax for 150 seconds after the 3C  90% DoD tests……………………………………………………………………………………….. 71 Figure 59. Values of 𝑑𝑖𝑓𝑓 calculated with varying values of 𝑘 at the 70th second of the relaxation from 3C  90% DoD test at 22°C………………………………………………………………………………. 71 Figure 60. Discharge and relaxation curves for battery A for a 90% DoD, 3C test at 40° C……………… 72 Figure 61. Values of diff calculated after batteries A relaxes for 150 seconds after the 3C 90% DoD tests  at 40°C………………………………………………………………………………………………... 73 Figure 62. Values of 𝑑𝑖𝑓𝑓 versus varying values of 𝑘 calculated at the 70th second of relaxation……….. 73 Figure 63. Lithium-ion NMC battery used for experiment…………………………………………………….. 81 Figure 64. Temperature measurement by thermocouple……………………………………………………... 81 Figure 65. Experiment setup for the OCV measurements……………………………………………….…… 81 Figure 66. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 0.3C charge from 60% SOC       to 80% SOC at 22° C…………………………..………………………………………….………… 82 Figure 67. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 0.6C charge from 60% SOC  to 80% SOC at 22° C……………………..…………………………………………………………. 83 Figure 68. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 1C charge from 60% SOC to  80% SOC at 22° C……………………..……………………………………………………………. 83  xiv  Figure 69. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 0.3C charge from 40% SOC  to 60% SOC at 22° C……………………..…………………………………………………………. 84 Figure 70. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 0.6C charge from 40% SOC  to 60% SOC at 22° C……………………………..…………………………………………………. 84 Figure 71. Time constant of 𝑅2𝐶2 branch in the equivalent circuit model after 1C charge from 40% SOC to  60% SOC at 22° C…………………………………………………………………………………... 85 Figure 72. Curve fitting between measurement data and simulation data after 3C discharge from 20%  SOC down to 10% SOC at 22° C………………………………………………………………….. 85 Figure 73. Curve fitting between measurement data and simulation data after 3C discharge from 40%  SOC down to 30% SOC at 22° C………………………………………………………………….. 86 Figure 74. Curve fitting between measurement data and simulation data after 2C discharge from 20%  SOC down to 10% SOC at 22° C.…………………………………………………………………. 86 Figure 75. Curve fitting between measurement data and simulation data after 2C discharge from 40%  SOC down to 30% SOC at 22° C………………………………………………………………….. 87 Figure 76. 𝑉𝑠 calculated after every 10% step discharge by different current magnitudes at 22°C…….… 87 Figure 77. 𝑉𝑠 calculated after every 10% step discharge by different current magnitudes at 30°C………. 88 Figure 78. 𝑉𝑠 calculated after every 10% step discharge by different current magnitudes at 40°C………. 88 Figure 79. 𝑉𝑘 calculated after every 10% step discharge by different current magnitudes at 22°C……... 89 Figure 80. 𝑉𝑘 calculated after every 10% step discharge by different current magnitudes at 30°C……... 89 Figure 81. 𝑉𝑘 calculated after every 10% step discharge by different current magnitudes at 40°C……... 90 Figure 82. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 300th second of the relaxation after a 1C  charge at 40% SOC at 30° C………………………………………………………………………. 90 Figure 83. The effects of 𝑘 on 𝑑𝑖𝑓𝑓 from the 0th second to the 300th second of the relaxation after a 1C  charge at 40% SOC at 40° C………………………………………………………………………. 91 Figure 84. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10%  SOC at 22° C……………………………………..…………………………………………………… 99 Figure 85. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30%  SOC at 22° C………………………………………………………………………………………….. 99   xv  Figure 86. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60%  SOC at 22° C……………………………………………………………………………………..…… 99 Figure 87. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 90%  SOC at 30° C............................................................................................................................. 99 Figure 88. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60%  SOC at 30° C………………………………………………………………………………………… 100  Figure 89. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30%  SOC at 30° C………………………………………………………………………………………… 100 Figure 90. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10%  SOC at 30° C……………………………………………………………………………………….... 101 Figure 91. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 90%  SOC at 40° C………………………………………………………………………………………… 101 Figure 92. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60%  SOC at 40° C………………………………………………………………………………………… 102 Figure 93. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30%  SOC at 40° C………………………………………………………………………………………… 102 Figure 94. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10%  SOC at 40° C………………………………………………………………………………..……….. 103 Figure 95. The earliest time for the LM method with the long-time constant to find k within 2% of error (k =  OCV – 15mV) of error from the correct equilibrium OCV under various discharge conditions at  20% SOC at 22° C…………………………………………………………………………..………. 103 Figure 96. The earliest time for the LM method with the long-time constant to find k within 2% of error(k =  OCV – 15mV) of error from the correct equilibrium OCV under various discharge conditions at  20% SOC at 30° C…………………………………………………………..………………………. 105 xvi  Figure 97. The earliest time for the LM method find with the long-time constant to k within 2% error(k =  OCV – 15mV) of error from the correct equilibrium OCV under various discharge conditions at  20% SOC at 40° C……………………………………………………………………………….… 105 Figure 98. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 1 second after discharge from 20%  SOC down to 10% SOC by different magnitudes of current at 22°C…………………………. 106  Figure 99. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 5 seconds after discharge from 100%   SOC down to 90% SOC by various magnitudes of current at 22°C………………………….. 106  Figure 100. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 5 seconds after discharge from 20%   SOC down to 10% SOC by various magnitudes of current at 22°C……………………….... 107  Figure 101. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds   after discharge from 100% SOC down to 90% SOC by different magnitudes of current at   22° C………………………………………………………………………………………………… 107  Figure 102. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 100%   SOC down to 90% SOC by different magnitudes of current at 22° C………………………… 108  Figure 103. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 100%  SOC down to 90% SOC by different magnitudes of current at 30° C………………………… 108 Figure 104. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 100%  SOC down to 90% SOC by different magnitudes of current at 40°C………………………... 109 Figure 105. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 70%  SOC down to 60% SOC by different magnitudes of current at 22°C………………………..…109 Figure 106. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 40%  SOC down to 30% SOC by different magnitudes of current at 22°C…………………………. 110 Figure 107. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 10 seconds after discharge from 20%  SOC down to 10% SOC by different magnitudes of current at 22° C……………………….…110 Figure 108. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 20 seconds after discharge from 20%  SOC down to 10% SOC by 2C current at 22°C………………………………………………… 111 Figure 109. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 20 seconds after discharge from 40%  SOC down to 30% SOC by 3C current at 22°C………………………………..……………….. 111 Figure 110. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 30 seconds after discharge from 20%  SOC down to 10% SOC by 2C current at 22°C………………………………………………… 112 xvii  Figure 111. The effects on 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 30 seconds after discharge from 40%   SOC down to 30% SOC by 3C current at 22°C……………………………………………… 112 Figure 112. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds  after discharge from 70% SOC down to 60% SOC by different magnitudes of current at    22°C………………………………………………………………………………………………. 113 Figure 113. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds  after discharge from 70% SOC down to 60% SOC by different magnitudes of current at 30°C……………………………………..…………………………………………………………. 113 Figure 114. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds  after discharge from 70% SOC down to 60% SOC by different magnitudes of current  at 40° C………………………………………………………………………………………….... 114 Figure 115. The effects on the sliding average of 𝑑𝑖𝑓𝑓 caused by the varying values of 𝑘, 40 seconds  after charge from 40% SOC up to 60% SOC by different magnitudes of current at  22°C………………………………………………………………………………………………… 114    xviii  List of Abbreviations  OCV    Open circuit voltage BMS    Battery management system LM    Logarithmic Modeling NMC    Nickel-manganese-cobalt SOC    State of charge    PI observer   Proportional-integral observer CC    Constant Current CV    Constant voltage DAQ    Data acquisition card DoD    Depth of Discharge    xix  Acknowledgements  Foremost, I would like to express my sincere gratitude and appreciation to Dr. William Dunford for his continuous support throughout my research and study. His methods are practical and intriguing plus his insights take me to a higher level approaches for many engineering applications. His patience and vast knowledge have guided me through my research work as well as the process of writing the thesis.   I would like to thank my exam committee: Dr. William Dunford, Dr. John Madden and Dr. Tina Shoa.  I would like to thank Dr. John Madden for allowing me to use his equipment for the experiments which provide crucial results in the thesis. Dr. Madden also patiently provides profound inspirational comments during the write-up of this thesis. His comments have improved and enriched the contents of thesis.  I would like to thank Mr. Lorne Gettel for providing technical support and allowing me to work in his lab. Lorne has provided me great help and shown me great advice when I was writing the thesis.  I would like to thank Mr. D. Brown, Mr. H. Chang and Mr. Y.W. Huang for providing assistance on building tools and sharing knowledge on electronics.   xx  Dedication     To my beloved parents  1  Chapter 1: Introduction  Batteries are electrochemical devices that can be charged to store electrical energy or discharged to release the stored energy when needed. They are becoming more and more popular in the energy industry sector because of their reusability, long lasting life expectancy and many other advantages. On the personal spectrum, most of the electronic devices and tools that we use on a daily basis implement batteries as the primary energy source, such as portable computers and cell phones. On the commercial and social spectrum, batteries are used on vehicles, aircraft plus public facilities as a supportive emergency energy source in case the main electrical network is not in service [1]. Furthermore, as the concept of sustainable development rises in the public, batteries also take an important part in the energy generation industry that takes the advantages of the geographic features of the Earth, such as solar power, wind power and oceanic tide power. Due to the wide range of applications of batteries, the technology for manufacturing batteries is continuously improved, and therefore, the understanding of battery characteristics needs to keep pace. 1.1 Background and motivation A lithium-ion battery consists of four main components which are the cathode, the anode, the separator and the electrolyte as shown in Figure 1 [2]. The cathode of a typical lithium ion battery is made of lithium metal composite while the anode can be made of graphite [3]. During the charging or the discharging processes of a rechargeable lithium-ion battery, lithium ions transport between the cathode and the anode through the separator and the electrolyte. When the battery is in a charging process, lithium ions Li+ travels from the cathode toward the anode, and the migration direction of lithium ions is reversed during a discharging scenario [3].  Among various types of batteries, the family of lithium-ion batteries are one of the dominating and promising power sources because they have several outstanding advantages over the batteries of other chemical composites. First, lithium-ion batteries have a high energy density as shown in Table 1, so they have a low weight and store an exceptional amount of energy in a compact volume [4]. Second, they have a reasonable self-discharge loss as shown in Table 2, and this is an advantage which causes 2  lithium-ion batteries to be more sustainable if the batteries have to stay in idle state for a long period of time [5]. Third, the memory effect happens to be barely noticeable in lithium-ion batteries, and therefore partial discharge does not significantly affect the batteries’ capacity [6]. Although lithium-ion batteries are advantageous most of the time, they are highly flammable and finally result in an explosion if they are abused in the cases of overcharging, overheating and short circuit. To guarantee safe usage of lithium-ion batteries and avoid fatal damage, many engineers have designed battery management systems (BMS) as one of the safety solutions which monitor the operating conditions and predict the runtime of the batteries. The permission for using Figures 1 and 3 in this thesis has been given by IEEE, and permission for using Figure 2 has been granted by Nature.   Figure 1.Fundamental configuration and operating principle of a lithium-ion battery  © [2011] IEEE. [2]  3   Table 1. Comparison of the features among different kinds of batteries. [4]   Table 2. Comparison of self-discharge loss effect among various kinds of batteries © [2014] IEEE. [5] 4   Because of the safety concerns that are previously stated, BMS are essential in engineering applications that involve lithium-ion batteries. Like humans use rate of heart beats and blood pressure as indications of health, the BMS monitors some of the parameters of batteries and thus determines the batteries’ health and remaining useful life. These parameters include the voltage, the temperature and the internal resistance. One important feature of an effective BMS is the technique of estimating the state of charge (SOC) of the battery band, and in order to derive an efficient technique, the knowledge of the operating characteristics of the batteries are required. In order to acquire such knowledge, many design engineers conduct dozens of experiments in which the batteries operate under different conditions and scenarios such as varying temperatures and charging-discharging duty cycles with altering magnitudes of electrical current. In practice, being able to calculate the SOC of the batteries in operation, the BMS can convey the information to the users and therefore, the users are able to plan the usage of the batteries before power failures happen. For example, if an electric vehicle driver is notified by the BMS about the remaining usable charge in the car’s battery band before departure, the driver can plan beforehand and prepare for possible battery power outage over the journey. Understanding the SOC is essential to secure system performance and there are various techniques to estimate this parameter of a lithium-ion battery. Before introducing various SOC estimation methods, the concept of C rate is hereby explained. Among many rating parameters of a battery, one parameter is nominal capacity which indicates nominal battery life. Nominal capacity is in units of mA/h or A/h and it tells the battery users about the amount of current which completely drains or fills the battery in one hour. For example, for a battery which has a capacity of 2800mA/h or 2.8A/h, a continuous 2800mA / 2.8A discharge current will completely drain the battery in one hour. C rate is a normalized parameter against the capacity and a 1C for the battery in the example is 2800mA / 2.8A and 0.5C for the same battery is 1400mA / 1.4A. A great extent of research has been conducted to explore SOC estimation techniques. Many of these techniques are based upon electrical equivalent models, and some researchers, along with a model which concerns slow and fast transient responses, develop observers of distinct modes to tackle the 5  nonlinear behaviors of their subject battery [7][8][9]. Being the alternatives of the observer choices, artificial neural network and fuzzy logic identifiers are also considered to be reliable and robust for estimating SOC [10][11][12]. In addition, the methods that are fulfilling Kalman filters have been studied and continuously extended by scholars [13]. For engineering purposes, selecting an appropriate SOC estimation technique for individual application with respect to the cost, the accuracy and the ease of implementation is crucial. The SOC of a lithium-ion battery is implicit, but it can be inferred from the explicit parameters of the battery. For example, the equilibrium open-circuit voltage (OCV) which can be measured by a digital voltage meter, is an excellent indicator of SOC, and the logarithmic model (LM) of a lithium-ion battery is developed based upon the equilibrium OCV and the terminal voltage of the battery. Furthermore, the method that is used for the calculation in the LM in this thesis incorporates with an electrical equivalent model of a lithium-ion battery, so a reference to lithium-ion battery modelling is also included in a brief scale in the content of this thesis. The LM method will be an improved version of the OCV method. Research has shown that many of characteristics of a lithium-ion battery are temperature dependent, so the scope of this thesis also includes the investigations of the effect of temperatures upon SOC estimation by the OCV method and the LM method. The stated two methods do not require commercial computer software to execute, and thus provide flexibility and cost-effectiveness to hardware implementation for the real-time SOC estimation.  1.2 Thesis outline This thesis includes five chapters. Chapter 1 is the introduction, being followed by Chapter 2 which includes the literature review for various SOC estimation techniques and the analysis that have been previously conducted by other researchers are explored in a more detailed manner. In Chapter 3, the experimental design and results of the OCV method are presented and analyzed, while in Chapter 4, the logarithmic modeling method is explained and demonstrated. The thesis is concluded with Chapter 5 which suggests possible future work. 6    Chapter 2. Literature Review  A variety of SOC estimation techniques have been studied and developed as the utilization of batteries is becoming more and more diverse and the accurate information about SOC for predicting runtime is crucial. The purpose of this thesis is to describe two methods that assist the users of batteries to realize SOC under assorted circumstances in a time-efficient and robust manner. The effect of aging and the qualities that indicate state of health (SOH) will be added in the future research. Among the great degree of research literature that has been created, several techniques are selected for review in this chapter of the thesis because these techniques are popularly cited by authors and become the favorable foundations of many other advanced approaches to SOC estimation. The articles that are chosen for literature review are categorized as the following: the coulomb counting method, the EMF voltage method, the observer method.  2.1 Coulomb counting The Coulomb counting method is considered to be one of the simplest methods for estimating SOC. This method does not require complex hardware to implement, and if the initial conditions of the batteries are known and the current measurement is highly accurate, this method promises accurate estimation of SOC. As a result, this method is favorable among engineers. In general, the principle of coulomb counting is summarized with Equation (1) as shown below [14]:  𝑆𝑂𝐶𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 = 𝑆𝑂𝐶𝑢𝑠𝑎𝑏𝑙𝑒 − ∫𝐼𝐶𝑢𝑠𝑎𝑏𝑙𝑒𝑑𝑡                               (1)   Equation (1) can be interpreted in a more explicit manner: 𝐶𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 = 𝐶𝑢𝑠𝑎𝑏𝑙𝑒 − ∫ 𝐼 𝑑𝑡                                        (2)  7  where 𝐶 represents the capacity of the battery in the unit of Coulomb, and 𝐼 represents the current in Ampere. The minus sign is used in Equation (1) to indicate the discharge situation. The performance of the coulomb counting method depends on the accuracy in estimating two quantities. One is 𝑆𝑂𝐶𝑖𝑛𝑖𝑡𝑖𝑎𝑙, and the other is current 𝐼. The initial SOC, 𝑆𝑂𝐶𝑖𝑛𝑖𝑡𝑖𝑎𝑙 , is strongly dependent on its previous estimates, and therefore, engineers design systems in which SOC is regularly calibrated with the assistance of some other battery characteristics, such as OCV. On the other hand, engineers attempt to reduce the error that exists in the data recording for current, because the existing error will accumulate, and cause the estimate of SOC to deviate significantly from the correct value. If the noise existing in the data measurement is minimized and SOC is correctly calibrated, the coulomb counting method is a proper candidate for estimating SOC.  2.2 Electromotive force (EMF) method Much research has been conducted on experiments to investigate the relationship between the changes in the internal impedance due to the SOC of a lithium ion battery. The results of those investigations are claiming that SOC can be inferred from the changing impedance, however, M. Coleman et al. [15] point out that those results have potential limitations and shortcomings, so a SOC estimation technique which is based upon electromotive force of a battery has been developed. The impedance of a lithium-ion battery changes by an undetectable amount if its SOC is above 40%, so such undetectable change in impedance can lead to large prediction errors in SOC if the estimation technique for SOC only depends on the relationship between impedance and SOC [15]. Moreover, the value of the internal impedance of the battery also changes in case of health decay as well as temperature rise [15]. The impedance of a battery is not exclusively sufficient to fully express SOC and therefore, the EMF technique will make a compensation by using more parameters for calculation. EMF voltage denotes the overpotentials which represent the diffusion and the kinetic reaction of the battery electrodes. EMF voltage is equal to the equilibrium OCV when the battery is in an idle state for 8  an adequately long period of time, and every EMF voltage value corresponds to certain SOC and the analysis of EMF as a function time will predict the SOC [15]. The EMF method is advantageous in several ways. In addition to impedance, the terminal voltage and the current during a discharging process are measured, and such measurements are used to calculate EMF voltage. Because the health deterioration and the change in temperature of a battery can introduce errors in impedance calculation and lead to a wrong result of SOC, and as an enhancement to SOC calculation, the EMF method implements the impedance parameter in combination with the terminal voltage and the current in discharge situation to calculate the EMF voltage.  The beginning procedure of the EMF method is determining the relationship between the EMF voltage and the SOC. To realize such relationship, the subject battery is discharged by current pulses in steps of 5% in SOC; after each step discharge, the battery is allowed to rest for 1 hour before the EMF voltage is measured. The next step is normalizing the impedance of the battery. The normalization process includes connecting an alternating current (AC) source to the target battery, and applying an AC current in small magnitude. Because the impedance of a battery varies with frequency [16], the frequency of the AC current is chosen to be 1000 Hertz. At 1000 Hertz, the resistance of the battery which is the real part of the complex electrochemical impedance can be obtained for practical analysis [16]. Impedance values are measured at different levels of SOC, and by dividing these values by the impedance value measured at 100% SOC, the normalization process is accomplished. The core feature of EMF voltage method is incorporating the equilibrium OCV 𝑉𝐸𝑀𝐹, the terminal voltage 𝑉𝐵𝐴𝑇𝑇, the current load and the normalized impedance. By using the coulomb counting method, the terminal voltage 𝑉𝐵𝐴𝑇𝑇 is set in function of SOC 𝜃. The authors divides the voltage characteristic curve of the battery during a discharge into linear region and hyperbolic region. During a discharge, the battery is operating in the linear region when the terminal voltage of the battery decreases linearly, and the terminal voltage of the battery decreases hyperbolically in the hyperbolic region as shown in Figure 2 [15].  The authors of [15] also observes rapid increase in the impedance of the batteries after the terminal voltage enters the hyperbolic region. 9   Figure 2. Demonstration for the operating regions of a battery during discharges by different C rates. [15]   In Equation (3) and (4), 𝐶𝑅 represents C rate; 𝑍𝑛𝑜𝑚0 represents the initial normalized impedance of the experiment which is measured at full SOC; 𝑍𝑛𝑜𝑚1 represents the normalized impedance at the different SOC levels; 𝛽1, 𝛽2 and 𝛽3 are determined from experimental results. Equation (3) is applied in the linear region and Equation (4) is applied in the hyperbolic region.     𝑉𝐸𝑀𝐹 = 𝑉𝐵𝐴𝑇𝑇(𝜃) + (𝐶𝑅𝛽1)𝑍𝑛𝑜𝑚0              (3)        𝑉𝐸𝑀𝐹 = 𝑉𝐵𝐴𝑇𝑇(𝜃) + 𝐶𝑅𝛽2 + (𝑍𝑛𝑜𝑚0 − 𝑍𝑛𝑜𝑚1)𝛽3            (4)  10   There are 3 major limitations for the EMF method. First, the discharge current for the experiments can not be too large. Second, the state of health of the battery is not considered, but the impedance of the battery increases when the state of health decays. Third, a temperature coefficient is required, because when temperature changes, impedance changes as well.  2.3 Observer method  For real-time applications, large errors in SOC estimation are possible, so within a tolerable amount of time, the BMS needs to learn about the error and continuously improve the accuracy. The accuracy needs to evolve, and observer algorithms are developed to provide reference values for the improvement. Observer algorithms include two parts. One part is the equivalent electrical circuit model for a battery that can provide reliable simulation results on battery performance, and the other is the observer component which is responsible for manipulating the difference between the real-time data and the simulation data, as well as sending the results of the manipulation to the battery modelling components such that new simulation results are calculated in the next iterations, as shown in Figure 3. The difference between the measured data and the simulation data is the estimation error, and the algorithm learns about this error value and continuously improves the SOC estimation in the succeeding iterations until the error converges to an acceptable range. One example of the observer methodology is proportional-integral (PI) observer. One implementation of PI observer is handling the difference between the measured battery voltage and the simulated battery voltage as well as reporting such difference to the feedback algorithm. The PI observer method illustrated in [17] uses the RC circuit model as shown in Figure 3, and such observer is proposed to compensate the estimation error and the uncertainties of the circuit model.  The first two steps in realizing the PI observer method are selecting the state variables and setting up the state functions. In [17], the voltage 𝑉2 and the change in SOC ?̇? are chosen to be the state variables. After choosing the state variables, the state functions are created with the knowledge of five parameters which are the nominal battery capacity, the Coulombic efficiency of the battery, the change in 11  voltage 𝑉2̇, the voltage 𝑉2 and the current 𝐼. The desired current is the input of the top branch and the researchers measure the voltage of the real battery for when the desired current is applied. The actual measured current is the input for the RC battery model at the bottom of Figure 3 and the output of the RC model is denoted as the calculated voltage. The error is the difference between the measured voltage and the calculated voltage. The finishing procedure of implementing the PI observer method is observer design. An appropriate observer assures high observability which means that the input states can be inferred from the external outputs under any circumstances. The proportional gain and the integral gains for the observer in [17] are determined by linear quadratic method so that the system estimation converges to the measurement values. In other words, the error in SOC estimation converges to 0 as the operation of the battery proceeds. One drawback of the RC circuit model is the higher modeling error compared to the equivalent circuit models which have more parallel RC networks [18], but the presented PI controller has a high compatibility with the RC model and limit the error in estimation within ±2%. Furthermore, large error exists when the current is quickly varied [17].   The EMF method and the PI observer method provide accurate results, so the interests in these methods are increasing, but in this thesis, these methods are not attempted because given the challenges associated with these two standard methods, the aim of this thesis is to test the effectiveness of two methods by which the BMS will not suffer computational burden or cost of measurement equipment. The two methods which are presented in this thesis will majorly rely on the terminal voltage and the Thevenin model of the battery to determine SOC.   Figure 3. Typical observer as a feedback method (left) and equivalent RC circuit model (right) [17]. 12  Chapter 3: Open Circuit Voltage Curve for Lithium-ion and     NMC Batteries  The OCV method is explored and implemented to estimate SOC in the scope of this thesis. This chapter first explains the rationale for using the open circuit voltage (OCV) for SOC estimation. After the explanation, the experimental setup of obtaining OCV is demonstrated and at the end of this chapter, the verification for the method is shown.   3.1 Concept and background of open circuit voltage  The open circuit voltage is measured when a battery is isolated from a system and the electrical current that either goes into or out from the battery is equal to zero. As shown in the previous chapters, the concept of OCV has been widely applied to estimate of SOC, and this fundamental characteristic of a lithium-ion battery is a reliable indication of the true SOC, because one SOC matches with one unique equilibrium OCV which can be measured after the battery relaxes from operation after a very long time which is in the scale of hours [14]. In terms of real-time implementation of the OCV method, there is one major drawback on the side of data measurement for using the OCV method to estimate the SOC of the batteries that are made of lithium iron phosphate (LFP), because when LFP batteries have a high SOC, the decrease in the equilibrium OCV caused by the decrease in SOC is a small amount [19][20]. If LFP batteries are applied as the power source, a voltmeter with sufficient resolution is required to measure the equilibrium OCV in order to distinguish the SOC in the region of high SOC levels. On the other hand, some other lithium-ion batteries, such as lithium manganese oxide (LiMn2O4) batteries, have more distinctive OCV-SOC characteristics than LFP batteries, so they have a lower requirement on the measurement resolution [20]. Different OCV-SOC characteristics of various kinds of lithium-ion batteries raise concerns for the resolution of the measurement on the level of hardware implementation for voltage measurement in practical applications.  13  3.2  Experimental design and setup  In this thesis, batteries which consist of lithium manganese and nickel-manganese-cobalt (NMC) compound are used for the experiments. These batteries have a nominal capacity of 2800 mAh and an operating voltage which ranges from 2V to 4.2V. The voltage of this battery for the investigation enters the hyperbolic region after 3.4V. The lithium-ion NMC battery used for this thesis also has a measurable change in its OCV-SOC profile which does not require high-resolution equipment. A series of tests are performed by following the instructions which are provided by a commercial manufacture of lithium-ion batteries. The OCV measurements for charging and discharging processes are obtained separately, because research has shown that the OCV measurements for the two processes are different due to hysteresis effect and that the OCV measurements for charging are higher than discharging even after the batteries have been unused for hours[21][22]. Instead of hours, the batteries are allowed to rest for only 15 minutes, and therefore, in the verification tests, the voltage differences between the charge OCV curve and the discharge OCV curve will be considered. For the interest of this thesis, different magnitudes of the discharging current and different temperatures are included in the experiments, so that the impacts of different operating conditions of the battery are investigated and discussed while the charging current is kept identical for all different scenarios.  To avoid over-discharging the battery, 2.5V is set to be the safe cut-off voltage for the discharge tests, and 2.5V is found to be 1% in SOC by counting coulombs during a discharge from 2.5V down to 2V.The following procedures are conducted in order to obtain the data of the discharging OCV. First, a battery is discharged from 100% SOC to 1% SOC with a continuous constant current, and after the discharge, the battery rests for 10 minutes. After the rest, the battery is charged with a constant current of 0.3C until the battery’s terminal voltage reaches 4.2 𝑉, and then the battery relaxes for 2 hours. The final steps are to discharge the battery with 10% step intervals of SOC and after each 10% step discharge, a 15-minute relaxation time is given to the battery. The cut-off voltage for discharging is 2.5V which is 1% SOC instead of 2V which is 0% SOC to avoid over-drain and possible damage to the battery. The test conditions and procedures for obtaining discharge OCV under the conditions of 0.3C discharge rate in combination with 0.3C charge rate are summarized in Table 3. 14  Step 1  Discharge : 0.3C, 2.5V cut-off @22℃ Rest 10min  Step 2 Charge: 0.3C, 4.2V cut-off at 22℃ Step 3 Rest time: 2 Hours Step 4 Discharge: 0.3C, cut off when 10% of the nominal capacity has dissipated  Step 5 Rest time: 15 minutes and the terminal voltage of the battery is recorded Step 6 Repeat Steps 4 and 5 until battery is 2.5V Table 3. Procedures for obtaining discharge OCV with a condition of 0.3C discharge rate and 0.3C charge rate.  To obtain the charging OCV data, similar procedures are carried out. First, the battery is discharged from 100% SOC to 0% SOC with a constant current and the cut-off voltage is 2.5V which is 1% SOC. In the next step, the battery stays idle for 2 hours before the OCV measurements take place. During the OCV measuring stage, the battery is charged with a constant current of 0.3C and 10% SOC intervals, and after each 10% step charge, a 15-minute relaxation is given. The charging is over when the terminal voltage of the battery reaches 4.2V. The test conditions and procedures for measuring the charge OCV under the condition of a 0.3C discharge rate with a 0.3C charge rate are summarized in Table 4.  Step 1 Discharge: 0.3C, 2.5V cut-off at 22℃ Step 2 Rest time: 2 Hours Step 3 Charge: 0.3C, 10% Capacity cut-off Step 4 Rest time: 15 minutes and the terminal voltage of the battery is recorded Step 5 Repeat Steps 3 and 4 until battery is 4.2V Table 4. Procedures for obtaining charge OCV with a condition of 0.3C discharge rate and 0.3C charge rate. 15  The circuit setups for measuring changing and discharging OCV are shown in Figure 6. As shown in Figure 6, a single-pole-single-throw switch which is controlled by a microcontroller is applied to start or terminate the charge or discharge processes. A water circulator is used for maintaining the temperature of the battery for the high-current experiments and raising the temperature of the battery for the high-temperature experiments. Three temperatures which are 22 ° C, 30 ° C and 40 ° C, are tested in combination with five different discharge currents which are 0.3C, 0.8C, 1.5C, 2C and 3C and one charge current which is 0.3C. To measure the discharge OCV under various discharge conditions and confirm the discharge OCV characteristics, Steps 1 and 4 listed in Table 3 are changed for different discharge rates. To measure the charge OCV and confirm the effects of different discharge currents on charge OCV, Step 1 in Table 4 is changed to different discharge rates. The data acquisition card (DAQ) with a 22-bit resolution is connected to the electrodes of the battery. With a high-resolution data acquisition equipment, the traditional constant-voltage (CV) charge stage which is applied when the voltage of the battery is close to its maximum nominal rating is not necessary. The maximum discharge current of the subject NMC battery can reach 14C, but due to the experimental conditions and laboratory safety, the magnitudes of current is limited to 3C.     Figure 4. Circuitry for measuring charge OCV (left) and discharge OCV (right)  16  3.3  Experimental results and verification of OCV method  Three single batteries are tested and used to verify the OCV method and the effects of temperature are also considered. By following the experimental procedures summarized in Table 3 and Table 4, OCV values for various magnitudes of discharge current and 0.3C charge current are measured. One observation is that the OCV measurements converge to approximately equal values for different discharging currents and the OCV at 50% SOC at 22 ° C is taken to be an example to show this OCV convergence in Figure 5. The average OCV measurements for charge and discharge at 22 ° C, 30 ° C and 40 ° C are all presented in Table 5.     Figure 5. Relaxation at 50% SOC at 22。° C after discharge of various magnitudes of current between 0 to 900 seconds  The primary factor which can cause a deviation in the SOC estimation by using the OCV method is the inequality between the usable capacity and the nominal capacity rating of the battery provided by the manufacture. As a result, if the experimental procedures presented in Table 3 and 4 are followed, a correction for each pairing of OCV and SOC is required, because the error accumulates over each 10% interval charge or discharge. For example, if the usable capacity of the battery is 95.6% of nominal capacity at 22 ° C, each 10% step charge or discharge will accumulate 0.44% error. The actual usable capacity is calculated by counting coulombs in a full discharge cycle from the highest cell potential to the 17  lowest cell potential. Table 5 presents the OCV-SOC pairs which have been calibrated by the capacity correction. To obtain the presented OCV data for certain percentage of the SOC in Table 5, linear interpolation is applied between each of the 10%-interval data sample points. To simplify the correction process, the Coulombic efficiency is approximated to be 100% for all SOC levels. In other words, if certain amount of energy has been discharged, the same amount of energy is absorbed to recover the discharge. In order to estimate the usable capacity of the battery, the total Coulombs are calculated after each continuous discharge test. The values of the average usable capacity and the standard deviations of the usable capacity are listed in Table 6. Moreover, the standard deviations and the standard errors for all the average discharge and charge OCV measurements at different temperatures are presented in Table 7 and Table 8, respectively. The SOC estimation by the OCV method will be compared to the estimation by the Coulomb counting method for the verification purpose. The average values of the usable capacity in Table 6 are applied for the capacity correction.   22 ° C 30 ° C 40 ° C SOC Average Discharge OCV (V) Average Charge OCV (V) Average Discharge OCV (V) Average Charge OCV (V) Average Discharge OCV (V) Average Charge OCV (V) 100% 4.153  4.162 4.163  4.169 4.169  4.173 90% 4.062  4.062 4.061  4.062 4.067  4.068 80% 4.001  4.000 4.005  4.003 4.006  4.01 70% 3.925  3.93 3.929  3.934 3.931  3.94 60% 3.85  3.852 3.85  3.86 3.853  3.86 50% 3.775  3.78 3.775  3.782 3.777  3.783 40% 3.704  3.71 3.701  3.709 3.702  3.710 30% 3.641  3.653 3.636  3.654 3.635  3.658 20% 3.573  3.586 3.565  3.591 3.563  3.586 10% 3.493  3.513 3.493  3.512 3.495  3.510 Table 5. Average OCV values for charge and discharge after capacity correction at different temperatures.  18   22。C 30。C 40。C Average usable capacity  Standard deviation Average usable capacity Standard deviation Average usable capacity Standard deviation 94.4% 0.74% 97.8% 0.5% 99.1% 0.38% Table 6. Usable capacity in percentage of nominal capacity    22 ° C 30 ° C 40 ° C SOC Standard  Deviations (V) Standard Errors (V) Standard  Deviations (V) Standard Errors (V) Standard  Deviations (V) Standard Errors (V) 100% 0.0016 0.0006 0.0018 0.00083 0.0014 0.0006 90% 0.0011 0.0004 0.0017 0.00077 0.0011 0.0005 80% 0.0010 0.0004 0.0017 0.00077 0.0015 0.0007 70% 0.0030 0.0011 0.0036 0.0016 0.0025 0.0011 60% 0.0020 0.0008 0.0022 0.00099 0.0022 0.0010 50% 0.0016 0.0006 0.0028 0.0012 0.0030 0.0013 40% 0.0018 0.0007 0.0035 0.0016 0.0033 0.0015 30% 0.0021 0.0008 0.0028 0.0012 0.0023 0.0010 20% 0.0039 0.0015 0.0030 0.0013 0.0037 0.0017 10% 0.0176 0.0067 0.0068 0.0030 0.0063 0.0028 Table 7. Standard deviations and standard errors for the average OCV measurements for the discharge at different temperatures          19   22 ° C 30 ° C 40 ° C SOC Standard  Deviations (V) Standard Errors (V) Standard  Deviations (V) Standard Errors (V) Standard  Deviations (V) Standard Errors (V) 100% 0.00168 0.000841 0.00340 0.00152 0.00311 0.001391 90% 0.00202 0.00101 0.00524 0.00234 0.005225 0.002337 80% 0.00225 0.00113 0.00281 0.00126 0.00471 0.002107 70% 0.00349 0.00175 0.00471 0.00211 0.002978 0.001332 60% 0.00334 0.00167 0.00498 0.00223 0.004091 0.00183 50% 0.00386 0.00193 0.00410 0.00184 0.003468 0.001551 40% 0.00355 0.00178 0.00294 0.00131 0.004008 0.001793 30% 0.00364 0.00182 0.00347 0.00155 0.004891 0.002187 20% 0.00435 0.00218 0.00564 0.00252 0.005865 0.002623 10% 0.00168 0.000841 0.00090 0.00040 0.000474 0.000212 Table 8. Standard deviations and standard errors for the average OCV measurements for the charge at different temperatures    Three single batteries of the same chemistry are used to verify the OCV method. In one test, cell A is discharged with 1.5C current for a random period of time, and after a 15-minute relaxation, the battery has an OCV of 3.61 V. By interval-wise linear interpolation between 20% OCV value and 30% OCV value in the column of average discharge OCV at 22。C in Table 5, 3.61 V is estimated to be 29.2%, and 29.2% is corrected to be 26% by the usable capacity factor. After the SOC estimation, a continuous constant-current (CC) discharge follows. This test is complete when the terminal voltage reaches 2V, 0% SOC, and the total coulomb output between 3.61 V and 2V is measured to be 27% by coulomb counting method.  The second verification test is performed with a discharge current which is randomly varied at  22。C. The beginning SOC of the battery is 100%, after the random-current discharge and a 15-minute relaxation, its OCV is measured to be 3.91 V. By implementing the look-up table set up with the average discharge OCV values presented in Table 5 and the same piecewise linear interpolation used in the first 20  verification test, the SOC is estimated to be 67.2% after the usable capacity correction is applied. After the relaxation, the battery is charged by a 0.3C current until its terminal voltage reaches 4.2V, and by implementing Coulomb counting method, the total charge that is used to restore the power of the battery is estimated to be 3161 Coulombs which indicates 66.8% SOC level before the charge.  The third verification test is taken at 33。C, and the OCV is measured to be 3.85 V after a 0.3C charge. Linear interpolation is first taken place between the average charge OCV values of 30。C and  40。C in Table 5 to find the OCV values of 50% SOC and 60% SOC for 33。C. After the usable capacity correction is weighted upon the measured OCV, the SOC is estimated to be 58%. A 5-amp discharge follows the relaxation, and by counting the coulombs, the SOC before the discharge is estimated to be 58.3%. The results for the verification tests are summarized in Table 9. The error margin in Table 9 is calculated by subtracting the SOC estimation by the Coulomb counting method from the SOC estimation by the OCV look-up table. The range of the error margin is calculated because of the standard deviation in the usable capacity, and the usable capacity is assumed to be normally distributed, so that the possibility for the error in the SOC estimation to take place in the range of the error margin is 95%.  In this chapter, a method for estimating the SOC by the using the OCV after a 15-minute relaxation is explored. The correlation between SOC and OCV is set up for a look-up table which can be stored in BMS for real-time application. Verification tests have been conducted to show that this OCV method will be useable for real situations as it provides promising results on the SOC estimation.     Test 1 Test 2 Test 3 SOC Estimation Error Margin  SOC Estimation Error  Margin SOC Estimation Error Margin OCV Look-up Table 26% -0.5% To -1.4% 67% -0.3% To 0.8% 58% -0.7% To 0.2% Coulomb Counting 27% 66.8% 58.3% Table 9. Results for the verification tests for the OCV method.  21  Chapter 4: The Logarithmic Modelling for Finding OCV  Although the OCV method stated in Chapter 3 provides accurate results for estimating SOC, a 15-minute delay is too long for most of the practical applications. For example, for the electric vehicles which frequently stop and start in city drive cycles, the BMS can not afford a 15-minute wait to calculate SOC. Despite the long time period required for reliable OCV method to take effect, it is an exceptional secondary method for SOC correction depending on different situations. In the case of electric vehicles, the OCV method is capable of correcting and calibrating SOC estimation when the drivers park their cars and leave for other activities. In the interest of this thesis, the logarithmic modelling which is based upon the equivalent model of a lithium-ion battery is explored and developed to find the OCV within a shorter wait time than 15 minutes.  Chapter 4 will start by introducing the general concept of the logarithmic modelling (LM) method, and explaining this method with the equivalent model of a lithium-ion battery. The first part of the LM method discusses the determination process for the equilibrium OCV by using the long time constant of the battery model and the curve fitting for the parameters in the equivalent circuit will be shown. The software algorithm for finding the equilibrium OCV is the next topic. The last section of this chapter is the second part of the LM method which attempts to find the equilibrium OCV within the relaxation time which is 3 times as long as the short time constant of the subject battery.  4.1  Equivalent circuit model of lithium-ion batteries and the logarithmic modelling 4.1.1. Introduction of equivalent circuit of battery and logarithmic modeling The equivalent circuit models are useful schemes because engineer can incorporate the modern  computer power with the circuit models to predict the characteristics and the behaviors of lithium-ion batteries within seconds without consuming the batteries. In practice, the circuit models which consist of 22  parallel resistor-capacitor (RC) networks are widely implemented for simulation purposes, because such models are flexible on simulating the behaviors of the batteries on the length of simulation time. The circuit models with two or more RC networks are widely used for simulating the batteries’ characteristics if the researchers are interested in simulating the battery usages in scales of minutes or even hours [23]-[26].    One example circuit model with two RC branches is presented in Figure 6. The RC branch with  the label “1” is used to simulate the early stage of the battery characteristics, so these RC components are in small magnitudes while the RC branch with the label “2” is used for the later stage of the battery behaviors and have a much larger magnitudes. 𝑉1 and 𝑉2 denote the voltage across 𝐶1 and 𝐶2, respectively. 𝑉𝐵𝐴𝑇𝑇  represents the terminal voltage of the battery, and 𝑉𝑂𝐶(𝑆𝑂𝐶) represents the equilibrium OCV which is in the function of the SOC. 𝑉𝑅𝑠𝑒𝑟𝑖𝑒𝑠 represents the step change in the battery’s terminal voltage when the battery is connected to or disconnected from a load. The presented equivalent circuit is widely applied for various studies of lithium-ion batteries.     Figure 6. Circuit model with two RC branches [23]   23   Figure 7 Relaxation characteristics of the lithium-ion battery after charge and discharge   According to Figure 6 and Figure 7, when the battery is disconnected from the load, i.e. 𝐼𝐵𝐴𝑇𝑇 = 0, the relationship among the voltage variables during a relaxation process can be established as the followings:     𝑉𝐵𝐴𝑇𝑇 = 𝑉𝑂𝐶(𝑆𝑂𝐶) + 𝑉1 + 𝑉2  after a charge             (5)        𝑉𝐵𝐴𝑇𝑇 = 𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉1 − 𝑉2  after a discharge            (6)   The voltages 𝑉1 and 𝑉2 across the capacitors are calculated in the following ways:         𝑉1 = 𝑉°𝐶1 𝑒−𝑡𝐶1𝑅1                        (7)                𝑉2 = 𝑉°𝐶2 𝑒−𝑡𝐶2𝑅2                      (8)  In Equations (7) and (8), the quantity 𝑉°𝐶 denotes the initial voltage across the capacitors in the RC networks when the relaxation stage begins. During the relaxation, the voltage across the capacitor decays exponentially across their individual resistor pairings. By performing natural logarithmic calculation on Equations (5) and (6), the equations will be expanded to the following forms, respectively: 24            ln(𝑉𝐵𝐴𝑇𝑇(𝑡) − 𝑉𝑂𝐶(𝑆𝑂𝐶)) = ln (𝑉°𝐶1𝑒−𝑡𝜏1 + 𝑉°𝐶2𝑒−𝑡𝜏2)                     (9)           ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) = ln (𝑉°𝐶1𝑒−𝑡𝜏1 + 𝑉°𝐶2𝑒−𝑡𝜏2)              (10) 𝑘 =   𝑉𝑂𝐶(𝑆𝑂𝐶) + 𝛼  The general concept of the LM method is to perform a search for the equilibrium OCV, 𝑉𝑂𝐶(𝑆𝑂𝐶), which indicates the SOC. After the energy across the short-time capacitor, 𝐶1, fully dissipates, the slopes for the right-hand side of Equations (9) and (10) will be constant if a variable 𝑘 that is equal to 𝑉𝑂𝐶(𝑆𝑂𝐶) is subtracted from 𝑉𝐵𝐴𝑇𝑇. When 𝑘 =   𝑉𝑂𝐶(𝑆𝑂𝐶), Equations (9) and (10) can be transformed into the following:  ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) = ln(𝑉°𝐶2) −𝑡𝜏2                         (11)              𝑑𝑖𝑓𝑓 =   ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) − ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1)) =  −1𝐶2𝑅2         (12)   The method of logarithmic modeling takes advantage of such characteristics. In practice, only 𝑉𝐵𝐴𝑇𝑇 is measurable, but the microcontroller can use the measurement of 𝑉𝐵𝐴𝑇𝑇, continuously adjust the value of the variable 𝑘, and find out the range of 𝑉𝑂𝐶 (𝑆𝑂𝐶) with an acceptable tolerance in error in a time efficient manner. The effect of the changing value of 𝑘 is shown in Figure 8 which shows the logarithmic calculation after a battery relaxes from a charge process.        25   Figure 8. The effect of varying 𝒌 in the logarithmic model during the relaxation after charge.  As shown in Figure 8, the logarithmic plot is exponential at the early stage of relaxation, because there is still energy in 𝐶1. After the energy in 𝐶1 fully dissipates in 8 seconds of relaxation, the slope of the blue plot for when 𝑘 is equal to 𝑉𝑂𝐶(𝑆𝑂𝐶) is constant. In case of the battery relaxation after charge, for the overestimated 𝑘, the slope of the natural logarithmic plot will sharply decrease and dive toward the negative infinity as the value of 𝑉𝐵𝐴𝑇𝑇 is approaching 𝑘; if 𝑘 is underestimated, the slope will asymptotically advance toward 0, because the estimation deviation from the true value dominates the logarithmic calculation. The slope of the logarithmic plot is important information for finding the value of 𝑘. In the scope of this thesis, the slope is denoted as 𝑑𝑖𝑓𝑓. Equation (13) is used for the relaxation after charge, and Equation (14) is used for the relaxation after discharge. Being opposite to the relaxation after charge, if a battery relaxes from discharge, an underestimated 𝑘 will cause a sharp decrease on the slope of the natural logarithmic calculation, and an overestimated 𝑘 will cause an asymptotic slope. In the scope of this thesis, the time interval between data samples is 1 second.               𝑑𝑖𝑓𝑓 = ln (𝑉𝐵𝐴𝑇𝑇(𝑡) − 𝑉𝑂𝐶(𝑆𝑂𝐶)) − ln(𝑉𝐵𝐴𝑇𝑇(𝑡 − 1) − 𝑉𝑂𝐶(𝑆𝑂𝐶))                       (13)                𝑑𝑖𝑓𝑓 = ln (𝑉𝑂𝐶 (𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) − ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1))                      (14) 26  4.1.2 Experimental setup and the preconditions   The experimental setup for measuring the data that are used for the logarithmic method is identical to the OCV measurement, and the simple setup without extra data acquisition device is the major advantage of the LM method. The data used for Chapter 3 is reused for the LM method.  Furthermore, the effect of temperature is also investigated. The exponential model curve fittings are presented Chapter 4.4 and Appendix A.       4.2 Using logarithmic modeling to find OCV after discharge   4.2.1 The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 22° C   Based upon the equations that are introduced in the last section, the slopes for the logarithmic model of the battery during the relaxation after 10% step discharge by various discharge conditions at different SOC levels are calculated by reusing the data from the OCV measurement in Chapter 3. The plots for the calculations are presented in Figure 9 – 12 in which 𝑘 is equal to the equilibrium OCV measured in the experiments in Chapter 3 for demonstration.         27   Figure 9. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 100% SOC to 90% SOC at 22° C.  Figure 10. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 70% down to 60% SOC at 22° C. 28   Figure 11. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge magnitudes from 40% SOC down to 30% SOC at 22° C.  Figure 12. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 20% SOC down to 10% SOC at 22° C 29  As shown in Figures 9 – 12, the values of diff converge to -0.005 when 𝑘 is equal to the equilibrium OCV in 150 seconds after the relaxation from various discharge conditions which range from 0.3C to 3C. The reason why 𝑑𝑖𝑓𝑓 converges to -0.005 is related to the time constant of the equivalent circuit model, and this will be explained in a later section in this chapter. In Figure 13, the plots of 𝑑𝑖𝑓𝑓 are shown for when the value of the equilibrium OCV is assigned to 𝑘 for the 0.3C discharge condition at different SOC levels at 22° C.  Figure 13. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for 0.3C discharge rate for different SOC levels at 22° C   30   Figure 14. 100th second to 150th second of Figure 13.      In the case of relaxation after discharge, if the value of 𝑘 is too small, the natural logarithm in Equations (10) can not be calculated and become undefined, and this characteristic can be applied to refine the range of 𝑘. If the relaxation happens after charge, the value of 𝑘 can not be too large. The effect of the changing values of 𝑘 after 3C discharge from 40% SOC down to 30% SOC is presented in Figure 15 and Figure 16.  31   Figure 15. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from 100th second to 150th second of the relaxation after a 3C discharge from 40% SOC down to 30% SOC at 22° C, OCV = 3.621V  Figure 16. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 150th second of the relaxation after a 3C discharge from 40% SOC down to 30% SOC at 22° C, OCV = 3.621V 32  In Figure 15 and Figure 16, when 𝑘 is 15mV smaller than the equilibrium OCV, the value of 𝑑𝑖𝑓𝑓 does not converge. In Figure 15, a sharp decrease in the value of 𝑑𝑖𝑓𝑓 after the 120th second of the relaxation is observed as the terminal voltage 𝑉𝐵𝐴𝑇𝑇 is approaching the value of 𝑘. Figure 16 presents another observable characteristic of the logarithmic curve for an underestimated 𝑘 and the indication for the underestimation of 𝑘 is the local maxima point on the 65th second in the 150-second time window right after the relaxation. In other words, 𝑘 is heavily underestimated compared to the equilibrium OCV which is measured in 15 minutes after the relaxation. In Figure 15 and 16, for the cases of 𝑘 being equal to 9mV and 6mV lower than the equilibrium OCV, the local maxima points locate on the 99th second and the 151th second of the relaxation process, respectively. In this case, the predicted equilibrium OCV is between 3.621V and 3.624V, so the error will be between 0% and 0.5%. According to Table 5, 15mV is approximately 2% in error in terms of SOC estimation.    4.2.2. The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 𝟑𝟎°C  Like the logarithmic calculation for the data measured at the 22° C, when the value of 𝑘 is equal to be the equilibrium OCV, the convergence of the natural logarithmic calculation is also observed at 30 degrees. The experimental setup and procedures are identical to Chapter 4.2.1. The plots of 𝑑𝑖𝑓𝑓 at different SOC are shown in Figures 17 – 20.         33   Figure 17. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 100% SOC down to 90% SOC at 30。C.  Figure 18. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 70% SOC down to 60% SOC at 30。C 34   Figure 19. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 40% SOC down to 30% SOC at 30。C.  Figure 20. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 20% SOC down to 10% SOC at 30。C 35  Figure 21. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for 0.3C discharge rate for different SOC levels at 30。C     In Figure 22, the effect of various values of 𝑘 is shown, and the plots of the underestimated 𝑘 values are parabolic during 150-second period of relaxation. At the early stage of relaxation, the derivatives of all 𝑑𝑖𝑓𝑓 curves are positive and gradually decrease. As the relaxation proceeds, when the terminal voltage of the battery approaches 𝑘, the derivatives of 𝑑𝑖𝑓𝑓 will transition from positive to negative, then gradually approach negative infinity and finally become undefined as soon as the terminal voltage exceeds 𝑘. The local maximum point is when the derivative of 𝑑𝑖𝑓𝑓 becomes 0.   36   Figure 22. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 300th second of the relaxation after a 3C discharge from 40% down to 30% SOC at 30。C   4.2.3. The variation of the slope of the logarithmic plots for OCV under different  discharge conditions at 𝟒𝟎°𝑪  The following content will present the results of the logarithmic modeling which is used for find the equilibrium OCV within 150 seconds during relaxation after discharge of various current magnitudes at 40 degrees Celsius. The experimental setup and procedures are identical to the setup at 22°𝐶. The plots of 𝑑𝑖𝑓𝑓 at different SOC are shown in Figures 23 – 26.    37   Figure 23. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 100% SOC down to 90% SOC at 40。C  Figure 24. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 70% SOC down to 60% SOC at 40。C 38   Figure 25. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 40% SOC down to 30% SOC at 40。C  Figure 26. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various discharge current magnitudes from 20% SOC down to 10% SOC at 40。C 39   Figure 27 shows the plots of 𝑑𝑖𝑓𝑓 when the battery relaxes from a 0.3C load at 40。C and in the  figure 𝑘 is equal to the equilibrium OCV at different SOC levels.   Figure 27. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for 0.3C discharge rate for different SOC levels at 40。C     Figure 28 shows similar observation to Figure 16 and Figure 22. When 𝑘 is much underestimated,  the value of 𝑑𝑖𝑓𝑓 will dive within the 150-second window, and there are local maximum points in the   curves that are plotted for the underestimated 𝑘. For an overestimated 𝑘, the value of 𝑑𝑖𝑓𝑓 asymptotically  approach 0. In Figures 22 and 28, the time coordinates of the local maximum points for the  underestimated 𝑘 values shift toward to the left of the spectrum, and this effect of temperature on 𝑑𝑖𝑓𝑓 will  be explained in a later section.  40   Figure 28. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 300th second of the relaxation after a 3C discharge at 30% SOC at 40。C     4.3 Using logarithmic modeling to find OCV after charge    The preconditions and experimental procedures are identical to the discharge scenarios, except that the range of the current magnitudes are 0.3C, 0.6C and 1C. The logarithmic plots under various tests conditions are presented. The converging characteristic of 𝑑𝑖𝑓𝑓 is also observed for the charging scenario, and therefore, a similar algorithm for finding the equilibrium OCV after charge can be applied. The converging characteristic are shown in Figures 29-33. An overestimated 𝑘 for charge cases will also cause the 𝑑𝑖𝑓𝑓 plot to dive for the same reason as an underestimated 𝑘 for discharge cases. There are more plots for the effects of 𝑘 on 𝑑𝑖𝑓𝑓 for different SOC levels, different temperatures, and different current magnitudes presented in Appendix A for both charging and discharging circumstances.  41   Figure 29. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various charge current magnitudes from 60% SOC up to 80% SOC at 22° C  Figure 30. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various charge current magnitudes from 40% SOC up to 60% SOC at 22° C 42   Figure 31. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various charge current magnitudes from 20% SOC up to 40% SOC at 22° C  Figure 32. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for various charge current magnitudes from 0% SOC up to 20% SOC at 22° C 43   Figure 33. Variation of 𝒅𝒊𝒇𝒇 calculated by setting 𝒌 = 𝑶𝑪𝑽 for 0.3C charge rate for different SOC levels at 22° C  Figure 34. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 300th second of the relaxation after a 1C charge at 40% SOC at 22° C 44   The sliding averages of 𝑑𝑖𝑓𝑓 from the 140th to the 150th second after relaxation for various conditions are calculated and shown in table format in Appendix A.   4.4 Software algorithm for implementing logarithmic modeling   In this section, a flow chart of the software algorithm which will be used for the BMS is presented in Figure 35. In order not to underestimate 𝑘 after charge or overestimate 𝑘 after discharge, threshold values are proposed for the sliding average of 𝑑𝑖𝑓𝑓 at the end of the 150-second time frame. The algorithm starts to use the logarithmic model to find the equilibrium OCV when the battery is disconnected from load. If the result for the computation of the logarithmic model is undefined or the local maxima of 𝑑𝑖𝑓𝑓 is found, BMS will apply a new value of 𝑘 for the next iteration. The boundary values of 𝑑𝑖𝑓𝑓 is applied at the final stage for making the decision. Based upon the tables of the sliding averages of 𝑑𝑖𝑓𝑓,    -0.0044 and -0.0052 are found to be the best upper limit and lower limit, respectively, for the threshold values. The tables which present the sliding average values of 𝑑𝑖𝑓𝑓 at different times after relaxation are in Appendix A.  The reason for selecting the -0.0044 and -0.0052 to be the boundary limits is decided by the time constants which are determined by using the components in the equivalent circuit model in Figure 7. After the relaxation process begins, the capacitors in the circuit model start to discharge, and the voltage across them decay exponentially. Capacitor 𝐶1 is used for modeling the short-time constant of the battery, and because its capacitance is much smaller than 𝐶2, the energy in 𝐶1 will completely diminish much earlier and sooner than 𝐶2. When the value of 𝑑𝑖𝑓𝑓 converge to certain value, 𝑉1 decays to a very small magnitude, and 𝑉2 in the equivalent circuit will remain in the equations for the calculating the logarithm.  Equations (9) and (10) will be accordingly transformed into the followings:    ln(𝑉𝐵𝐴𝑇𝑇(𝑡) − 𝑉𝑂𝐶 (𝑆𝑂𝐶)) = ln (𝑉°𝐶2) − 𝑡𝐶2𝑅2          (15)   ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) = ln (𝑉°𝐶2 ) − 𝑡𝐶2𝑅2               (16)  45   After a discharge, Equation (16) is considered. The value of 𝑑𝑖𝑓𝑓 is calculated in the following way: 𝑑𝑖𝑓𝑓 =   ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡)) − ln(𝑉𝑂𝐶(𝑆𝑂𝐶) − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1)) =  −1𝐶2𝑅2         (17)    Figure 35. The flowchart of logarithmic modeling algorithm for discharge scenarios 46  The searching algorithm for 𝑘 starts by setting 𝑘 to be equal to the first measurement of the terminal voltage after the load is disconnected. The filtering process for 𝑘 is the value comparison. With respect to Equation (8), Figure 36 is presented to the show how the long-time constant is configured and Figure 37 is presented to show the exponential decay for the voltage across 𝐶2, and in the figure, the effects of various time constants are shown as well. The data in Figure 37 are obtained during the relaxation after a 10% step discharge from 100% SOC to 90% SOC with a 2C rate at 22° C.     Figure 36. Configuration for the component parameters in the equivalent circuit model of Lithium-ion battery [24]         47   Figure 37. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 2C discharge from 100% SOC to 90% SOC at 22° C.  Figures 38 and 39 show the curve fitting for the measurement data with simulation data. The measurement data are obtained after a battery is disconnected from a 3C discharge from 100% SOC down to 90% SOC at 22° C. The simulation curves in Figure 38 and 39 are plotted by using Equation (16). In Figures 38 and 39, 𝑉𝑠 is equivalent to 𝑉1 and denotes the initial voltage across 𝐶1; 𝑉𝑘  is equivalent to 𝑉2 and denotes the initial voltage across 𝐶2; Taos (𝜏𝑠) is equivalent to 𝜏1 and denotes the short-time constant;  taok (𝜏𝐿) is equivalent to 𝜏2 and represents the long-time constant. By using the simulation data in Equation (18), the value of 𝑑𝑖𝑓𝑓 converges to -0.004 which is equal to 1250 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 , and the plot is shown in Figure 40. The real OCV for Figure 38 is 4.058V, while the predicted OCV by the LM method is 4.058V. The real OCV for Figure 39 is 4.059V, while the predicted OCV by the LM method is 4.058V              𝑉𝐵𝐴𝑇𝑇(𝑡) = 𝑉𝑂𝐶 − 𝑉𝑠𝑒−𝑡𝑡𝑎𝑜𝑠 − 𝑉𝑘𝑒−𝑡𝑡𝑎𝑜𝑘          (18) 48   Figure 38. Curve fitting between measurement data and simulation data after 3C discharge from 100% SOC down to 90% SOC at 22° C  Figure 39. Curve fitting between measurement data and simulation data after 0.3C discharge from 100% SOC down to 90% SOC at 22° C. 49  In the scope of this thesis, the sliding average for the values of 𝑑𝑖𝑓𝑓 are calculated once for every 10 data samples of the terminal voltage during relaxation. The following tables present the values of the sliding average of 𝑑𝑖𝑓𝑓 for different values of 𝑘 after the battery has relaxed for 150 seconds. Tables 10-12 show the sliding average of 𝑑𝑖𝑓𝑓 which is calculated with the data samples which are measured from the 140th second to the 150th second of relaxation after various discharge current magnitudes from 100% SOC down to 90% SOC at different temperatures. Tables 13-15 show the sliding average of 𝑑𝑖𝑓𝑓 which is calculated at the 150th second of relaxation after various charge current magnitudes from 60% SOC up to 80% SOC.    90% SOC at 22° C             𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0011 -0.0014 -0.0021 -0.0041 -0.068 n/a n/a 0.8C discharge -0.0016 -0.0021 -0.0029 -0.0047 -0.012 n/a n/a 1.5C discharge -0.0020 -0.0025 -0.0034 -0.0051 -0.010 n/a n/a 2C discharge -0.0020 -0.0025 -0.0033 -0.0049 -0.0093 -0.105 n/a 3C discharge -0.0024 -0.0030 -0.0038 -0.0055 -0.0095 -0.0368 n/a Table 10. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at 22° C.    90% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.00088 -0.0012 -0.0018 -0.0040 n/a n/a n/a 0.8C discharge -0.0013 -0.0018 -0.0026 -0.0049 -0.048 n/a n/a 1.5C discharge -0.0018 -0.0023 -0.0031 -0.0052 -0.015 n/a n/a 2C discharge -0.0018 -0.0023 -0.0033 -0.0054 -0.016 n/a n/a 3C discharge -0.0020 -0.0026 -0.0036 -0.0057 -0.014 n/a n/a Table 11. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at 30° C. 50    90% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.0007 -0.0010 -0.0018 -0.0054 n/a n/a n/a 0.8C discharge -0.0011 -0.0014 -0.0022 -0.0044 n/a n/a n/a 1.5C discharge -0.0015 -0.0020 -0.0030 -0.0058 -0.068 n/a n/a 2C discharge -0.0015 -0.0019 -0.0028 -0.0052 -0.032 n/a n/a 3C discharge -0.0017 -0.0022 -0.0031 -0.0057 -0.030 n/a n/a Table 12. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at 40° C.    80% SOC at 22° C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a -0.0124 -0.0037 -0.0021 -0.00152 -0.00118 0.6C charge n/a -0.067 -0.0083 -0.0045 -0.00304 -0.00231 -0.00186 1C charge n/a -0.043 -0.0093 -0.0052 -0.0036 -0.00279 -0.00226 Table 13. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at 22° C.    80% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.0046 -0.0020 -0.00124 -0.00091 0.6C charge n/a n/a -0.021 -0.0045 -0.003 -0.0018 -0.0013 1C charge n/a n/a -0.021 -0.0054 -0.003 -0.0022 -0.0017 Table 14. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at 30° C.    51  80% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.0032 -0.0015 -0.00097 -0.00072 0.6C charge n/a n/a -0.048 -0.0037 -0.0019 -0.0013 -0.0010 1C charge n/a n/a -0.038 -0.0045 -0.0024 -0.0016 -0.0012 Table 15. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 60% SOC to 80% SOC by various current magnitudes at 40° C.    4.4.1  Parameters of the equivalent circuit model under the subject discharge  conditions    Tables 20 – 23 will present the parameters of the components in the equivalent circuit model for various SOC levels and discharge conditions at 22° C. The circuit parameters are calculated by using the concept of Figure 16 and Equation (14). 𝑉𝑠 and 𝑉𝑘  that are calculated for different SOC levels, different discharge current magnitudes and different temperatures are presented in Appendix A.   𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.017 22 0.0082 250 0.8C discharge 0.045 22 0.019 250 1.5C discharge 0.07 22 0.021 250 2C    discharge 0.083 26 0.023 250 3C    discharge 0.061 22 0.014 250 Table 16. Parameters of the equivalent circuit model for various discharge conditions at 10% SOC at 22° C    𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.01 22 0.0068 250 0.8C discharge 0.028 22 0.014 250 1.5C discharge 0.044 22 0.015 250 2C    discharge 0.059 26 0.018 250 3C    discharge 0.074 25 0.018 250 Table 17. Parameters of the equivalent circuit model for various discharge conditions at 30% SOC at 22° C 52    𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.01 22 0.0054 250 0.8C discharge 0.024 22 0.0066 250 1.5C discharge 0.038 22 0.0085 250 2C    discharge 0.05 22 0.009 250 3C    discharge 0.062 22 0.009 250 Table 18. Parameters of the equivalent circuit model for various discharge conditions at 60% SOC at 22° C     𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.01 20 0.0038 250 0.8C discharge 0.027 23 0.0055 250 1.5C discharge 0.048 26 0.007 250 2C    discharge 0.054 22 0.0068 250 3C    discharge 0.08 22 0.008 250 Table 19. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at 22° C     𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.0045 22 0.0031 250 0.8C discharge 0.018 22 0.004 250 1.5C discharge 0.035 22 0.0053 250 2C    discharge 0.043 22 0.0052 250 3C    discharge 0.065 22 0.0057 250 Table 20. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at 30° C     𝑉𝑠  (V) 𝜏𝑠 (seconds) 𝑉𝑘  (V) 𝜏𝑘 (seconds) 0.3C discharge 0.0062 22 0.0021 250 0.8C discharge 0.015 22 0.0031 250 1.5C discharge 0.03 22 0.0041 250 2C    discharge 0.034 22 0.0038 250 3C    discharge 0.049 22 0.0041 250 Table 21. Parameters of the equivalent circuit model for various discharge conditions at 90% SOC at 40° C   53  According to Tables 16-21, the voltage across the capacitors in the equivalent circuit is proportional to the magnitudes of the discharge current, but the time constants are not much affected by the current or the temperature. 𝑉𝑠, 𝑉𝑘  ∝ 𝐼    Referring to Equations (9) and (10), if  𝑘 = 𝑉𝑂𝐶 +  𝛼, 𝑉𝑠 = 𝑉1 ,   𝑉𝑘 = 𝑉2 ,   𝜏𝑠 = 𝜏1 ,   𝜏𝑘 = 𝜏2  ln (𝑉𝑂𝐶 +  𝛼 − 𝑉𝐵𝐴𝑇𝑇) = ln (𝑉𝑠𝑒−𝑡𝜏𝑠 + 𝑉𝑘𝑒−𝑡𝜏𝑘 + 𝛼)           (15)  This explains why the values of 𝑑𝑖𝑓𝑓 progresses toward negative infinity in a shorter time for small currents if 𝛼 is a negative value for discharge scenarios.     4.5 Investigation for the behavior of 𝒅𝒊𝒇𝒇 for t < 150 seconds   From Chapter 4.1 to Chapter 4.4, the performance of the LM method has been discussed based on the equivalent circuit model. Instead of waiting for 15 minutes, the LM method can define the range of the equilibrium OCV after 150 seconds of wait time with less than 1% in error, however, it is also worthy to investigate if the LM method can estimate SOC with a reasonable tolerance of error but a much shorter wait time. In Table 5, from 10% to 90% SOC, it has been shown that every 7mV in OCV is equivalent to 1% in SOC, and therefore, being able to calculate the minimum amount of time requirement for finding the value of 𝑘 within a range of 15mV away from the equilibrium OCV for the identical preconditions as the previous chapters will be a further advancement for using the LM method in practice.    If the LM algorithm is forced to stop before 150 seconds after a discharge, the chances are that 𝑘 will be overestimated, and so is SOC if the boundary values which are calculated with long time constant of battery model are applied. The following Tables 22 – 25 show that at 22° C, for 10% step discharge, the LM algorithm takes less than 90 seconds to provide a satisfactory estimation within 2% of error on the 54  value of 𝑘 for all discharge conditions if the boundary limits -0.0044 and -0.0052 derived in Chapter 4.4 are applied. The main challenge is to avoid overestimating 𝑘 after discharge when the LM algorithm is disrupted. Underestimating 𝑘 is a less serious concern, because the values of 𝑑𝑖𝑓𝑓 advance toward negative infinity as the terminal voltage approaches the 𝑘 values, and therefore, the LM algorithm can make decisions more quickly on the underestimating side of the spectrum. For example, in Figures 16, 22, 28, the 𝑑𝑖𝑓𝑓 plots for the 𝑘 values which are 15mV smaller than the equilibrium OCV distinguishably deviate from the plots of larger 𝑘 values which are the more accurate estimates for the equilibrium OCV.  Another observation is shown in Figure 5 that the larger is the discharge current, the longer time the battery takes to recover. For small discharge current magnitude like 0.3C, finding the range of correct 𝑘 takes less than 40 seconds, but if the battery is disconnected from a 3C load, the LM algorithm needs 90 seconds which are more than 4 times as high as the short time constant to find the range of 𝑘 for 10% depth of discharge (DoD). The performance of using the long time constant to estimate 𝑘 after a discharge test which involves high current and large DoD will be shown in a later section.   90% SOC at 22° C        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0017 -0.0022 -0.0031 -0.0053 -0.018 n/a n/a 0.8C discharge -0.0028 -0.0034 -0.0045 -0.0065 -0.012 -0.061 n/a 1.5C discharge -0.0036 -0.0043 -0.0054 -0.0073 -0.011 -0.024 n/a 2C discharge -0.0037 -0.0045 -0.0056 -0.0074 -0.011 -0.022 n/a 3C discharge -0.0047 -0.0056 -0.0068 -0.0087 -0.012 -0.020 -0.060 Table 22. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 at the 90th second of relaxation from discharge from 100% SOC to 90% SOC by various current magnitudes at 22° C.      55  60% SOC at 22° C        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0018 -0.0022 -0.0030 -0.0045 -0.0094 n/a n/a 0.8C discharge -0.0026 -0.0032 -0.0043 -0.0057 -0.0100 -0.024 n/a 1.5C discharge -0.0031 -0.0037 -0.0046 -0.0059 -0.0084 -0.014 -0.051 2C discharge -0.0036 -0.0042 -0.0051 -0.0066 -0.0092 -0.015 -0.045 3C discharge -0.0038 -0.0045 -0.0055 -0.0070 -0.0096 -0.015 -0.038 Table 23. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 at the 90th second of relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at 22° C.     30% SOC at 22° C        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0018 -0.0023 -0.0029 -0.0041 -0.0070 -0.024 n/a 0.8C discharge -0.0031 -0.0035 -0.0040 -0.0048 -0.0059 -0.0076 -0.011 1.5C discharge -0.0040 -0.0045 -0.0051 -0.0059 -0.0071 -0.0087 -0.011 2C discharge -0.0045 -0.0049 -0.0055 -0.0063 -0.0073 -0.0087 -0.011 3C discharge -0.0049 -0.0054 -0.0060 -0.0068 -0.0078 -0.0092 -0.011 Table 24. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 at the 90th second of relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at 22° C.        56  10%        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0026 -0.0031 -0.0038 -0.0050 -0.0073 -0.0135 n/a 0.8C discharge -0.0024 -0.0029 -0.0034 -0.0043 -0.0057 -0.0087 n/a 1.5C discharge -0.0047 -0.0052 -0.0057 -0.0064 -0.0073 -0.0084 n/a 2C discharge -0.0051 -0.0055 -0.0061 -0.0067 -0.0075 -0.0085 -0.0099 3C discharge -0.0029 -0.0033 -0.0038 -0.0045 -0.0056 -0.0074 n/a Table 25. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 at the 90th second of relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at 22° C.    The OCV data from Chapter 3 is reused in Figure 41 which presents how fast the LM method can make the OCV estimation within 2% in error for various discharge conditions at 90% SOC at 22° C. More figures of the same kind for different SOC levels, temperatures for both charge and discharge scenarios will be presented in Appendix B.   Figure 40. The earliest time for the LM method find 𝒌 within 2% (𝒌 = OCV + 15mV) of error from the correct equilibrium OCV under various discharge conditions at 90% SOC at 22° C. 57  4.5.1 More observations on the relationship between varying values of 𝒌 versus  𝒅𝒊𝒇𝒇   The long time constant of the battery can provide satisfactory estimation on the equilibrium OCV by using only the boundary values in 90 seconds after disconnection from the current loads of various magnitudes which range 0.3C to 3C. Instead of using the boundary values to find 𝑘, one observation from a different angle of the LM method on how the varying values of 𝑘 affect the behaviors of 𝑑𝑖𝑓𝑓 at the 40th and 60th second after relaxation can provide meaningful information to the BMS for determining the correct range of 𝑘 values in less than 90 seconds.  After a discharge, for certain time 𝑡 and 𝑘 = 𝑉𝑂𝐶 + 𝛼 : 𝑘 > 𝑉𝐵𝐴𝑇𝑇(𝑡)                                  𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =𝑑𝑑𝑘[ln (𝑘 − 𝑉𝐵𝐴𝑇𝑇(𝑡)) − ln (𝑘 − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1))]                  (16) 𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =𝑑𝑑𝑘ln (𝑘 − 𝑉𝐵𝐴𝑇𝑇(𝑡)𝑘 − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1))            𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =𝛿𝑘2−𝑘𝛽+𝛾                              (17)       𝛿 = 𝑉𝐵𝐴𝑇𝑇(𝑡) − 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1)                                       (18)        𝛽 = 𝑉𝐵𝐴𝑇𝑇(𝑡) + 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1)            (19)                                                    𝛾 = 𝑉𝐵𝐴𝑇𝑇(𝑡) ∙ 𝑉𝐵𝐴𝑇𝑇(𝑡 − 1)          (20)      58  Equation (17) can be transformed to the following and simplified:  For 𝑘 =  𝑉𝐵𝐴𝑇𝑇(𝑡) +  𝜃          𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =  𝛿[𝑉𝐵𝐴𝑇𝑇(𝑡)+𝛿+𝜃]2−[𝑉𝐵𝐴𝑇𝑇(𝑡)+𝛿+𝜃][2𝑉𝐵𝐴𝑇𝑇(𝑡)+𝛿]+𝑉𝐵𝐴𝑇𝑇(𝑡)∙[𝑉𝐵𝐴𝑇𝑇(𝑡)+𝛿]         (21)                𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =  𝛿𝛿𝜃+𝜃2                  (22) Recall that 𝑘 =  𝑉𝑂𝐶 + 𝛼, so      𝜃 =  𝑉𝑂𝐶 −  𝑉𝐵𝐴𝑇𝑇(𝑡) +  𝛼           (23) Set a variable 𝑧 =  𝑉𝑂𝐶 −  𝑉𝐵𝐴𝑇𝑇(𝑡), Equation (22) will become the following:          𝑑𝑑𝑘𝑑𝑖𝑓𝑓 =  𝛿𝛿𝑧+𝛿𝛼+𝑧2+2𝛼𝑧+𝛼2          (24)  The quadratic denominator in Equation (24) suggests two characteristics. First, the further away is 𝛼 from 0, the smaller values are 𝑑𝑑𝑘𝑑𝑖𝑓𝑓 as well as 𝑑2𝑑𝑘2𝑑𝑖𝑓𝑓, and therefore, 𝑘 > 𝑉𝐵𝐴𝑇𝑇(𝑡) is an important condition. Second, for the discharge scenario, the earlier is the logarithmic method is forced to stop, the further away is 𝛼 on the negative half of the spectrum to observe a steep slope. According to [26], the voltage across the short-time capacitor drops to below 5% of the initial values if the wait time is longer than 3 times of the short time constant. If the short time constant of the subject battery is 22 seconds, the wait time will be 70 seconds. In the interest of this project, the standard for setting 𝑘 to be the knee point is that:  𝑑𝑖𝑓𝑓(𝑘) − 𝑑𝑖𝑓𝑓(𝑘 − ∆𝑘)∆𝑘< 0.001/𝑚𝑉   59   Figure 41. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 22° C.   Figure 42. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 30° C. 60   Figure 43. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 40° C.    The above figures show that in the steep-gradient region, the slopes of the curves which are created with high-current data are less steep than the slopes of the curves which are created with the low-current data. The logarithmic calculation for the high-current data is less affected by an underestimated 𝑘, because the voltages across the short-time capacitor are large when the discharge current is high in terms of C rates. As a result, the terminal voltage of the battery takes a longer time to approach 𝑘 for the high-current conditions, and the values of 𝑑𝑖𝑓𝑓 takes a longer time to stabilize for high C rates as well, according to Equations (11), (12) and (15). Furthermore, the voltage across the short-time capacitor will be of smaller magnitudes if the discharge current is small, and therefore, an underestimated 𝑘 has a huge effect in low-current conditions. In the flat-gradient region, the effect of the overestimated 𝑘 becomes weaker and weaker on the slopes for all the curves as the values of 𝑘 increase and dominate more and more in the logarithmic calculation. At higher temperatures, the voltage across the short-time 61  capacitor is smaller. Figure 42 and 43 will also show the temperature effect upon the coordination of the knee points. Higher temperature leads to a smaller magnitude of 𝑉𝑠, so in the logarithmic calculation during the relaxation after discharge, a negative 𝛼, which indicates underestimated 𝑘, has a larger weight and causes detectable gradient changes. More figures which show the relationship between 𝑑𝑖𝑓𝑓 and 𝑘 at different lengths of time, such as 1 second, 5 seconds, 10 seconds, 20 seconds and 30 seconds after relaxation for different conditions will be included in Appendix C. The same concept can also be applied for the 60-second relaxation time and the accuracy for the estimation of the equilibrium OCV improves as shown in Figure 45 as a greater portion of the energy in the short-time capacitor drops.    Figure 44. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 60 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 22° C    62  The shorter the wait time is after the relaxation, the lower accuracy is the general estimation of the equilibrium OCV by observing the slopes or implementing the boundary values. If this slope observation is applied to the battery after a 10-second relaxation, the coordination of the knee points for the high-current plots are further away from the correct OCV, comparing to 40-second and 60-second conditions of relaxation. For example, in Figure 102, based on the knee-point observation, the estimate for 𝑘 for the 2C and the 3C discharge conditions will be in the range from 15mV to 30mV below the equilibrium OCV, however, in Figure 44, for the same discharge conditions with a longer relaxation time, the accuracy has increased and the range of error shrinks to be between 3mV and 6mV. Every 7mV is equivalent to 1% in SOC, so in terms of percentage, the estimation is improved from 4% to 0.5% if there is a 50-second extension in the relaxation time after a high-current discharge.  At low SOC level, the relationship between the OCV and the SOC is hyperbolic, so 𝑉𝑠 at low SOC levels is large compared to 𝑉𝑠 at the higher SOC levels as shown in Tables 16 – 19. In Table 16, 𝑉𝑠 after 2C discharge is 0.083V, while 𝑉𝑠 after 3C discharge is 0.061V, so Figure 46 which corresponds to the data in Table 16 shows that the error in estimating 𝑘 is greater for the 2C curve. In Table 17,  𝑉𝑠 after 3C is 0.074C, while 𝑉𝑠 after 2C is 0.059V, so Figure 45 shows that error in estimating 𝑘 is greater for the 3C curve. The LM method closely coordinates with the equivalent circuit model of a lithium-ion battery. In Appendix C, more plots of 𝑑𝑖𝑓𝑓 will be presented for different conditions such as 2C and 3C as low SOC levels.            63   Figure 45. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 60 seconds after discharge from 40% SOC down to 30% SOC by different magnitudes of current at  22° C   Figure 46. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 60 seconds after discharge from 20% SOC down to 10% SOC by different magnitudes of current at  22° C  64  4.5.2 Comparison of the 3 discussed methods for large depth of discharge  In Chapter 4.5.1, the knee-point technique is drafted for 10% DoD at different SOC levels. In order to further distinguish and investigate the three discussed methods, a discharge test is performed. In the test, the depth of discharge is large, and the subject lithium NMC battery is discharged from 3.973V to 3.043V by a continuous 2C current at 22° C. A 15-minute relaxation follows the discharge, and the predicted values of OCV will be compared to the real OCV measured after the relaxation. Figures 48 – 51 will present the values of 𝑑𝑖𝑓𝑓 which are calculated by setting 𝑘 to be equal to the equilibrium OCV, plus the plots of 𝑑𝑖𝑓𝑓 versus varying values of 𝑘 at the 40th and the 60th second of relaxation. After this test, 𝑉𝑠 is calculated to be 0.19V, and 𝑉𝑘 is calculated to be 0.029V which are of larger magnitudes than the values that are previously presented for 10% depth of discharge conditions.     Figure 47. Terminal voltage of a Li NMC battery during a continuous 2C discharge test and during relaxation at 22° C. 65   Figure 48. Values of diff for 𝒌 = OCV in the 2C discharge test.   Figure 49. Zoom-in view of diff plot for 𝒌 = OCV from the 100th to the 150th second of relaxation 66   Figure 50. Values of diff calculated by setting 𝒌 = OCV+15mV during the relaxation after the 2C discharge test  Figure 51. Values of 𝒅𝒊𝒇𝒇 versus varying values of 𝒌 calculated at the 40th second, the 60th second and the 70th second of relaxation  67  By applying the boundary values to estimate the equilibrium OCV after 150 seconds of wait time,  the error in the SOC estimation will be 1%. In this test, if the tolerance of error is increased to 2%, the wait  time will be shortened to 100 seconds which are longer than the 90-second period for the 10% DoD  cases. In this discharge test, the voltage across the short time capacitor is 3-4 times larger than the  values listed in Table 16, so the energy is 3 times as high. If the technique for observing the knee point in  the 𝑑𝑖𝑓𝑓-versus- 𝑘 is applied, the wait time will be further shortened down to 70 seconds with 2% error or  20 seconds with 5% error. Both of the short time and the long time constants can be applied to the LM  method. The long time constant and the knee-point observing technique which involves the short time  constant of the battery provide the most robust results among the three discussed methods for both  shallow and deep depth of discharge.  Additional 3 battery samples B, C, and D are used for tests. In the tests, the depth of discharge  will be 40% and 90% at 22° C, and the voltage measurements will be analyzed by the LM methods in the  identical manner as battery sample A. Tables 26 and 27 present the test conditions for batteries B, C and  D, as well as the error in the SOC estimation by using the boundary values after 150-second wait time.   Temperature: 22° C Starting Voltage Depth of Discharge 40% Error in SOC Estimation by using the boundary values Battery B 4.134 V 1.5C current 0% Battery C 4.133 V 1.5C current 0% Battery D 4.125 V 1.5C current 0%  Table 26. Tests on battery samples B, C and D: 40% depth of discharge, 1.5C current, 22° C    Temperature: 22° C Starting Voltage Depth of Discharge 90% Error in SOC Estimation by using the boundary values Battery B 4.137 V 3C current 0.5% Battery C 4.139 V 3C current 0.5% Battery D 4.133 V 3C current 0.1%  Table 27. Tests on battery samples B, C and D: 90% depth of discharge, 3C current, 22° C  68   Figure 52. Discharge and relaxation curves for battery B, C and D of the test conditions in Table 26.  Figure 53. Values of diff for 𝒌 = OCV in the 1.5C discharge tests of batteries B, C and D under the stated conditions of Table 26. 69   Figure 54. Values of diff after batteries B, C and D relax for 150 seconds after the 1.5C 40% DoD tests  Figure 55. Values of 𝒅𝒊𝒇𝒇 calculated with varying values of 𝒌 at the 70th second of the relaxation from 1.5C 40% DoD test at 22° C  70   Figure 56. Discharge and relaxation curves for battery B, C and D of the test conditions in Table 27.  Figure 57. Values of diff for 𝒌 = OCV in the 3C discharge tests of batteries B, C and D under the stated conditions of Table 27.  71   Figure 58. Values of diff after batteries B, C and D relax for 150 seconds after the 3C 90% DoD tests  Figure 59. Values of 𝒅𝒊𝒇𝒇 calculated with varying values of 𝒌 at the 70th second of the relaxation from 3C 90% DoD test at 22° C  72   One additional experiment is performed with battery A, and the experimental conditions are 90%  DoD with 3C current at 40° C. In this test, if the long time constant of the model is applied after 150  seconds, the error will be 0.5%.    Figure 60. Discharge and relaxation curves for battery A for a 90% DoD, 3C test at 40° C   73   Figure 61. Values of diff calculated after batteries A relaxes for 150 seconds after the 3C 90% DoD tests at 40° C  Figure 62. Values of 𝒅𝒊𝒇𝒇 versus varying values of 𝒌 calculated at the 70th second of relaxation   74  Chapter 4 explores the LM method which uses the time constants in the equivalent circuit of a lithium-ion battery. The LM method provides satisfactory results on estimating OCV for the subject lithium-ion NMC battery after 10%-DoD-1.5C discharge tests by various discharge current magnitudes and 90%-DoD-3C tests. The first part of the LM method shows how the long time constant in the equivalent circuit model contribute to refine the range for the estimation of OCV. After 150 seconds of relaxation, the energy in the short-time capacitor fully dissipates, and while only the long-time capacitor is in effect, the boundary values which are calculated with the reciprocals of the long time constant can be applied to find the equilibrium OCV. The wait time is shortened from 15 minutes to 150 seconds, and for the verification tests, the error is lower than 1%. The second part of the LM method is the knee-point observation in the graphs of diff versus the varying values of k. After the 70-second wait time which is 3 times as long as the short time constant of the subject battery, the slope between the adjacent underestimated k values is steep, and this observation shortens the wait time from 150 seconds to 70 seconds, and the error in the estimation of SOC is 2% for the verification tests.    There are several limitations for the LM method. One limitation of the method is the time frame. Between the 600th second and the 900th second, the local maxima will be shown in the diff plot even though k is equal to the equilibrium OCV, and this limitation will cause a slight overestimation in the OCV. The second limitation is that the method is designed for offline usage, so the batteries need to be disconnected from loads. Although the LM method shows promising results for a discharge current which ranges from 0.3C to 3C in a scale of 10, more tests will be conducted for extremely low and high current magnitudes. Furthermore, the LM method will be expanded to other battery types although the LM method is developed upon the widely used Thevenin model for lithium-ion batteries.    75  Chapter 5: Conclusion  5.1 Conclusion on the findings  This thesis has presented two methodologies for finding SOC of batteries which are made of lithium-ion NMC composites. The effects of temperature and different rates of discharge current are discussed as well. The equilibrium OCV is a reliable indicator of SOC, and the LM method shows improvement upon the OCV method in terms of time efficiency.  The first finding is that OCV is a reliable and robust parameter for SOC estimation. In the scope of this thesis, the Coulomb counting method provides the reference SOC because the current are controlled and calibrated. Many researches have emphasized that OCV requires a long idle time of a battery to provide accurate estimation on SOC, however, in practical applications, time efficiency is crucial and a long wait for OCV to converge is not time efficient. The findings in this thesis shows that after a 15-minute relaxation of the battery, the OCV of the battery is an accurate indicator of SOC with an error lower than 1%, and therefore BMS can take the advantage of such characteristic of the battery to calibrate and correct the SOC estimation by other instantaneous online methods such as the coulomb counting method. The experimental design and instrument setup for OCV measurements are demonstrated in details in  Chapter 3. The second finding is to show the possibility for using the logarithmic model to find the equilibrium OCV within a shorter wait time than 15 minutes. By analyzing the characteristics of the equivalent circuit model of a lithium-ion battery, a logarithmic model is developed. When the constant 𝑘 is equal to the equilibrium OCV, 𝑑𝑖𝑓𝑓, which denotes the slope of the logarithmic model, will become constant, and the value of 𝑑𝑖𝑓𝑓 depends on the long-time constant of the equivalent circuit model for the battery. In case of the relaxation after discharge, if 𝑘 is underestimated during the relaxation after a discharge, there will be two observations. First, the value of 𝑑𝑖𝑓𝑓 can become undefined, because the terminal voltage which is measured during the 150-second time window is greater than 𝑘. Second, there is a maximum point in the 76  curve of 𝑑𝑖𝑓𝑓. If 𝑘 is overestimated after a discharge, the values of 𝑑𝑖𝑓𝑓 will asymptotically approach 0. The stated observations are applied to the estimation of 𝑘 for the relaxation after charge as well except that for the charge scenarios, overestimated 𝑘 leads to undefined 𝑑𝑖𝑓𝑓 while underestimated 𝑘 causes asymptotic 𝑑𝑖𝑓𝑓. By using the boundary values which are calculated by the long time constant, the wait time can be shortened to 90 seconds with 2% in error for low DoD, or shortened to 100 seconds with 2% in error for high C rates and large DoD. In terms of wait time and the equivalent circuit model of a lithium-ion battery, the subject time lengths which are 10, 40, and 60 seconds after relaxation is too soon for the short-time capacitor to fully discharge, so simply using the long time constant to find 𝑘 is not adequate when the wait time is too short. One additional observation which helps to find the value of 𝑘 if the algorithm has to be interrupted before 150 seconds is the relationship between the sliding average of 𝑑𝑖𝑓𝑓 and the varying values of 𝑘. During the relaxation after a discharge, if the voltage across the short-time capacitor 𝑉𝑠 is small, the slope for the 𝑑𝑖𝑓𝑓 values of adjacent 𝑘 values will be steep if 𝑘 is much underestimated. On the other hand, as the value of 𝑘 increases, the gradient of the 𝑑𝑖𝑓𝑓 will become flatter and flatter. The BMS can observe the values for the gradient for the consecutive 𝑘 marks, and search for the 𝑘 values accordingly. In this thesis, 0.001/mV is selected. The value of 𝑉𝑠 in the equivalent circuit model is decisive in this observation, and there are two qualities of 𝑉𝑠. First, 𝑉𝑠 is proportional to the magnitude of current. Second, 𝑉𝑠 is inversely proportional to the temperature. Finding the range of 𝑘 by observing the gradient consumes less time for smaller 𝑉𝑠, but in case of large DoD and large current listed in the experiment conditions, either searching for the knee-point after 70 seconds or applying the boundary values after 150 seconds provides satisfactory and robust results. The experimental setup and the detailed process of the implementation of the model is presented in Chapter 4. There are limitations for applying the OCV method and the logarithmic model that are developed in this thesis. The major limitation of the OCV method is that it is not for online application, so the battery has to be unused for the certain length of wait time for the best accuracy. One another limitation is that the state of health of the batteries in the experiments are not considered. Furthermore, the resolution of the voltage meter is crucial as well. The resistance of the subject lithium-ion battery is small, so the resistance will introduce little noise to the voltage measurement, however, sufficient resolution will be required for the voltage meter after the short-time capacitor has fully discharged. Analog-digital 77  converters with 18-bit resolution for 0 - 5.5V range is recommended, and nowadays, many manufactures provide 24-bit analog-digital multi-channel converters for very affordable prices.  5.2 Recommendation for future research For continuation of this project, the effects of temperature are of further interest, especially for the temperatures below 22。C and below 0。C. One prediction is that at lower ambient temperature of a battery, the capacity correction must be weighted more heavily if the same charge and discharge schedules are applied in the experiments, because the usable capacity of a battery is proportional to temperature. In addition to the temperature studies, the concepts of using the equilibrium OCV and the logarithmic model need further exploration to understand the concepts’ performance under extreme conditions. 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Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 0.3C charge from 60% SOC to 80% SOC at 22° C. 83   Figure 67. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 0.6C charge from 60% SOC to 80% SOC at 22° C.   Figure 68. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 1C charge from 60% SOC to 80% SOC at 22° C.  84   Figure 69. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 0.3C charge from 40% SOC to 60% SOC at 22° C.   Figure 70. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 0.6C charge from 40% SOC to 60% SOC at 22° C. 85   Figure 71. Time constant of 𝑹𝟐𝑪𝟐 branch in the equivalent circuit model after 1C charge from 40% SOC to 60% SOC at 22° C. Figure 72. Curve fitting between measurement data and simulation data after 3C discharge from 20% SOC down to 10% SOC at 22° C.   86   Figure 73. Curve fitting between measurement data and simulation data after 3C discharge from 40% SOC down to 30% SOC at 22° C.  Figure 74. Curve fitting between measurement data and simulation data after 2C discharge from 20% SOC down to 10% SOC at 22° C.   87   Figure 75. Curve fitting between measurement data and simulation data after 2C discharge from 40% SOC down to 30% SOC at 22° C.  Figure 76. 𝑽𝒔 calculated after every 10% step discharge by different current magnitudes at 22°C   88   Figure 77. 𝑽𝒔 calculated after every 10% step discharge by different current magnitudes at 30°C  Figure 78. 𝑽𝒔 calculated after every 10% step discharge by different current magnitudes at 40° C 89   Figure 79. 𝑽𝒌 calculated after every 10% step discharge by different current magnitudes at 22° C  Figure 80. 𝑽𝒌 calculated after every 10% step discharge by different current magnitudes at 30° C   90  Figure 81. 𝑽𝒌 calculated after every 10% step discharge by different current magnitudes at 40° C  Figure 82. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 300th second of the relaxation after a 1C charge at 40% SOC at 30° C 91   Figure 83. The effects of 𝒌 on 𝒅𝒊𝒇𝒇 from the 0th second to the 300th second of the relaxation after a 1C charge at 40% SOC at 40° C    60% SOC at 22° C             𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.0012  -0.0016  -0.0022  -0.0037  -0.011  n/a  n/a  0.8C discharge -0.0015 -0.0021 -0.0029 -0.0042 -0.0098 n/a n/a 1.5C discharge -0.0019 -0.0025 -0.0031 -0.0043 -0.0071 n/a n/a 2C discharge -0.0021 -0.0027 -0.0035 -0.0048 -0.0080 -0.023 n/a 3C discharge -0.0022 -0.0028 -0.0036 -0.0049 -0.0080 -0.021 n/a Table 28. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at 22° C.      92     30% SOC at 22° C             𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.0014 -0.0017 -0.0023 -0.0035 -0.0073 n/a n/a 0.8C discharge -0.0023 -0.0027 -0.0032 -0.0039 -0.0052 -0.0075 -0.01397 1.5C discharge -0.0028 -0.0032 -0.0038 -0.0047 -0.0060 -0.0083 -0.01361 2C discharge -0.0032 -0.0036 -0.0042 -0.0050 -0.0062 -0.0082 -0.01201 3C discharge -0.0033 -0.0038 -0.0044 -0.0052 -0.0064 -0.0083 -0.01176 Table 29. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at 22° C.    10% SOC at 22° C             𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0018 -0.0022 -0.0028 -0.0040 -0.0067 -0.0221 n/a 0.8C discharge -0.0017 -0.0020 -0.0025 -0.0033 -0.0048 -0.0090 n/a 1.5C discharge -0.0032 -0.0037 -0.0042 -0.0054 -0.0058 -0.0073 n/a 2C discharge -0.0035 -0.0039 -0.0044 -0.0050 -0.0059 -0.0072 -0.0091 3C discharge -0.0018 -0.0021 -0.0025 -0.0032 -0.0041 -0.0060 -0.011 Table 30. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at 22° C.        93     60% SOC at 30。 C        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.0010 -0.0013 -0.0020 -0.0037 -0.0254 n/a n/a 0.8C discharge -0.0016 -0.0020 -0.0028 -0.0042 -0.0088 n/a n/a 1.5C discharge -0.0019 -0.0023 -0.0030 -0.0044 -0.0081 -0.055 n/a 2C discharge -0.0019 -0.0023 -0.0030 -0.0044 -0.0078 -0.037 n/a 3C discharge -0.0021 -0.0027 -0.0035 -0.0052 -0.0101 -0.201 n/a Table 31. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at 30 ° C.    30% SOC at 30。 C        𝑘 OCV+9mv OCV+6mv OCV+3mV OCV OCV-3mV OCV-6mv OCV-9mv 0.3C discharge -0.00096 -0.0013 -0.0020 -0.0043 n/a n/a n/a 0.8C discharge -0.0018 -0.0022 -0.0029 -0.0041 -0.0070 -0.025 n/a 1.5C discharge -0.0023 -0.0028 -0.0034 -0.0044 -0.0063 -0.011 -0.045 2C discharge -0.0026 -0.0031 -0.0038 -0.0050 -0.0073 -0.013 -0.088 3C discharge -0.0028 -0.0034 -0.0041 -0.0052 -0.0073 -0.012 -0.033 Table 32. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at 30 ° C      94     10% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.0013 -0.0017 -0.0024 -0.0041 -0.0140 n/a n/a 0.8C discharge -0.0018 -0.0023 -0.0030 -0.0045 -0.0085 -0.086 n/a 1.5C discharge -0.0029 -0.0033 -0.0040 -0.0051 -0.0069 -0.011 n/a 2C discharge -0.0029 -0.0034 -0.0041 -0.0051 -0.0068 -0.010 n/a 3C discharge -0.0031 -0.0036 -0.0042 -0.0052 -0.0068 -0.010 n/a Table 33. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at 30 ° C    60% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.00094 -0.0012 -0.0018 -0.0033 -0.019 n/a n/a 0.8C discharge -0.0014 -0.0018 -0.0024 -0.0039 -0.010 n/a n/a 1.5C discharge -0.0017 -0.0021 -0.0027 -0.0039 -0.0070 -0.033 n/a 2C discharge -0.0017 -0.0021 -0.0028 -0.0041 -0.008 -0.061 n/a 3C discharge -0.0019 -0.0025 -0.0034 -0.0053 -0.012 n/a n/a Table 34. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 70% SOC to 60% SOC by various current magnitudes at 40 ° C       95      30% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.00064 -0.00087 -0.0014 -0.0033 n/a n/a n/a 0.8C discharge -0.0013 -0.0017 -0.0024 -0.0046 -0.0497 n/a n/a 1.5C discharge -0.0021 -0.0026 -0.0033 -0.0047 -0.0080 -0.026 n/a 2C discharge -0.0023 -0.0028 -0.0038 -0.0056 -0.0110 n/a n/a 3C discharge -0.0025 -0.0030 -0.0039 -0.0056 -0.0100 -0.043 n/a Table 35. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 40% SOC to 30% SOC by various current magnitudes at 40 ° C    10% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C discharge -0.00082 -0.0011 -0.0017 -0.0037 n/a n/a n/a 0.8C discharge -0.0012 -0.0015 -0.0022 -0.0038 -0.0154 n/a n/a 1.5C discharge -0.002 -0.0024 -0.0032 -0.0047 -0.0086 -0.053 n/a 2C discharge -0.0022 -0.0027 -0.0036 -0.0054 -0.011 n/a n/a 3C discharge -0.0026 -0.0031 -0.0040 -0.0055 -0.0088 -0.022 n/a Table 36. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from discharge from 20% SOC to 10% SOC by various current magnitudes at 40 ° C       96    60% SOC at 22° C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge -0.013 -0.0063 -0.0041 -0.0031 -0.0025 -0.0020 -0.0017 0.6C charge -0.012 -0.0047 -0.0030 -0.0022 -0.0017 -0.0014 -0.0012 1C charge -0.018 -0.006 -0.0036 -0.0026 -0.0020 -0.0016 -0.0014 Table 37. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at 22° C.    40% SOC at 22° C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a -0.037 -0.0036 -0.0019 -0.0013 -0.0010 0.6C charge n/a n/a -0.014 -0.0040 -0.0023 -0.0016 -0.0013 1C charge n/a n/a -0.016 -0.0045 -0.0026 -0.0019 -0.0014 Table 38. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at 22° C.    20% SOC at 22° C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a -0.0098 -0.0039 -0.0022 -0.0016 -0.0013 0.6C charge n/a n/a -0.010 -0.0045 -0.0030 -0.0022 -0.0017 1C charge n/a n/a -0.010 -0.0051 -0.0034 -0.0026 -0.0021 Table 39. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at 22° C.  97    60% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a -0.011 -0.0048 -0.0031 -0.0022 -0.0018 0.6C charge -0.015 -0.0079 -0.0053 -0.0040 -0.0032 -0.0027 -0.0023 1C charge -0.008 -0.0050 -0.0038 -0.003 -0.0025 -0.0021 -0.0019 Table 40. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at 30 ° C.    40% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.004 -0.0017 -0.00112 -0.0008 0.6C charge n/a n/a -0.029 -0.004 -0.0021 -0.0014 -0.0011 1C charge n/a n/a -0.020 -0.004 -0.0024 -0.0016 -0.0013 Table 41. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at 30 ° C.    20% SOC at 30。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.0046 -0.0014 -0.0008 -0.0006 0.6C charge n/a n/a -0.015 -0.0047 -0.0027 -0.0019 -0.0015 1C charge n/a n/a -0.028 -0.0059 -0.0033 -0.0023 -0.0017 Table 42. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at 30 ° C.  98    60% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a -0.014 -0.0039 -0.0023 -0.0016 -0.0013 0.6C charge n/a n/a -0.011 -0.0052 -0.0034 -0.0026 -0.0020 1C charge n/a -0.036 -0.010 -0.0058 -0.0041 -0.0032 -0.0026 Table 43. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 40% SOC to 60% SOC by various current magnitudes at 40 ° C.    40% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.0041 -0.0018 -0.0011 -0.00081 0.6C charge n/a n/a n/a -0.0046 -0.0021 -0.0013 -0.00099 1C charge n/a n/a -0.042 -0.0042 -0.0022 -0.0015 -0.0012 Table 44. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 20% SOC to 40% SOC by various current magnitudes at 40 ° C.    20% SOC at 40。 C        𝑘 OCV+9mV OCV+6mV OCV+3mV OCV OCV-3mV OCV-6mV OCV-9mV 0.3C charge n/a n/a n/a -0.0036 -0.0014 -0.00083 -0.00060 0.6C charge n/a n/a n/a -0.0052 -0.0024 -0.0016 -0.0012 1C charge n/a n/a -0.13 -0.0059 -0.0031 -0.0021 -0.0015 Table 45. Sliding average of 𝒅𝒊𝒇𝒇 calculated with different 𝒌 from the 140th to the 150th second of relaxation from charge from 0% SOC to 20% SOC by various current magnitudes at 40 ° C.  99  Appendix B  Figure 84. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10% SOC at 22° C.  Figure 85. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30% SOC at 22° C. 100   Figure 86. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60% SOC at 22° C.  Figure 87. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 90% SOC at 30° C. 101   Figure 88. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60% SOC at 30° C   Figure 89. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30% SOC at 30° C 102   Figure 90. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10% SOC at 30° C Figure 91. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 90% SOC at 40° C 103   Figure 92. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 60% SOC at 40° C  Figure 93. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 30% SOC at 40° C 104   Figure 94. The earliest time for the LM method with the long-time constant to find k within 2% of error (k = OCV + 15mV) from the correct equilibrium OCV under various discharge conditions at 10% SOC at 40° C  Figure 95. The earliest time for the LM method with the long-time constant to find k within 2% (k=OCV – 15mV) of error from the correct equilibrium OCV under various discharge conditions at 20% SOC at 22° C 105   Figure 96. The earliest time for the LM method with the long-time constant to find k within 2% (k=OCV–15mV) of error from the correct equilibrium OCV under various charge conditions at 20% SOC at 30° C  Figure 97. The earliest time for the LM method with the long-time constant to find k within 2% (k=OCV – 15mV) of error from the correct equilibrium OCV under various charge conditions at 20% SOC at 40° C  106  Appendix C  Figure 98. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 1 second after discharge from 20% SOC down to 10% SOC by different magnitudes of current at 22°C  Figure 99. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 5 seconds after discharge from 100% SOC down to 90% SOC by various magnitudes of current at 22°C 107   Figure 100. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 5 seconds after discharge from 20% SOC down to 10% SOC by various magnitudes of current at 22°C  Figure 101. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 22° C 108   Figure 102. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 22° C  Figure 103. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 30° C    109   Figure 104. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 100% SOC down to 90% SOC by different magnitudes of current at 40° C  Figure 105. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 70% SOC down to 60% SOC by different magnitudes of current at 22°C 110   Figure 106. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 40% SOC down to 30% SOC by different magnitudes of current at 22°C  Figure 107. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 10 seconds after discharge from 20% SOC down to 10% SOC by different magnitudes of current at 22° C 111   Figure 108. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 20 seconds after discharge from 20% SOC down to 10% SOC by 2C current at 22°C  Figure 109. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 20 seconds after discharge from 40% SOC down to 30% SOC by 3C current at 22°C 112   Figure 110. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 30 seconds after discharge from 20% SOC down to 10% SOC by 2C current at 22°C  Figure 111. The effects on 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 30 seconds after discharge from 40% SOC down to 30% SOC by 3C current at 22°C  113   Figure 112. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 70% SOC down to 60% SOC by different magnitudes of current at 22°C  Figure 113. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 70% SOC down to 60% SOC by different magnitudes of current at  30°C 114   Figure 114. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after discharge from 70% SOC down to 60% SOC by different magnitudes of current at  40° C  Figure 115. The effects on the sliding average of 𝒅𝒊𝒇𝒇 caused by the varying values of 𝒌, 40 seconds after charge from 40% SOC up to 60% SOC by different magnitudes of current at 22°C  

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