Systems and methods for state-of-charge estimation using sine-wave current pulses

Abstract

Systems and methods for state-of-charge monitoring in accordance with embodiments of the invention are illustrated. One embodiment includes a battery management system including a sine-wave pulse generator configured to provide sine-wave pulses to a battery, a voltage sensor configured to measure output voltage of the battery, and a state-of-charge monitor, including a processor, and a memory storing a state-of-charge monitoring application that configures the processor to apply a sine-wave pulse current to the battery using the sine-wave pulse generator, measure the output voltage of the battery in response to the sine wave pulse current, estimate an impedance of the battery based on the voltage response, estimate a state-of-charge of the battery using the estimated impedance of the battery, and provide the estimated state-of-charge.

Description

Systems and Methods for State-of-Charge Estimation Using Sine-Wave Current PulsesCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The current application claims the benefit of and priority under 35 U.S.C. § 119(e) to U.S. Provisional Patent Application No. 63 / 606,026 entitled “State-of-Charge Estimation of LiFePO4 Cells with Electrochemical Impedance Spectroscopy Response of Short-Duration Sine-Wave Current Profiles” filed December 4, 2023. The disclosures of U.S. Provisional Patent Application No. 63 / 606,026 is hereby incorporated by reference in its entirety for all purposes.FIELD OF THE INVENTION

[0002] The present invention generally relates to estimating the state-of-charge of a battery.BACKGROUND

[0003] Batteries, or “cells”, are chemical energy storage devices used to store electricity. Every battery has a capacity, which is the maximum amount of energy it can store. The “state-of-charge” (SOC) represents the ratio of remaining charge to nominal capacity. Standard chemical batteries, once discharged, are unable to be used to store electricity. Rechargeable batteries are batteries which can chemically store electricity when the electricity is provided to the battery, even if the battery is depleted.

[0004] Rechargeable batteries are increasingly common in consumer and enterprise applications. Battery management systems (BMS) are systems integrated in many rechargeable battery powered devices which measure SOC, and use SOC to guide optimal battery utilization within safe operational limits, avoiding accelerated degradation and performance issues.SUMMARY OF THE INVENTION

[0005] Systems and methods for state-of-charge monitoring in accordance with embodiments of the invention are illustrated. One embodiment includes a battery management system including a sine-wave pulse generator configured to provide sine-wave pulses to a battery, a voltage sensor configured to measure output voltage of the battery, and a state-of-charge monitor, including a processor, and a memory storing a state-of-charge monitoring application that configures the processor to apply a sine-wave pulse current to the battery using the sine-wave pulse generator, measure the output voltage of the battery in response to the sine wave pulse current, estimate an impedance of the battery based on the voltage response, estimate a state-of-charge of the battery using the estimated impedance of the battery, and provide the estimated state-of-charge.

[0006] In another embodiment, to estimate the impedance of the battery, the state-of- charge monitoring application further configures the processor to use an objective function min Sn=i(Vt n- Vte*p)2, where Vt nis an n-th data point of predicted [|2|,arg(2),a0,ai,bi,w] voltage from battery voltage over time (Vt), Vte pis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery, N is the length of experimental voltage data from the at least one test battery of the same type as the battery, Z is the estimated impedance of the battery, and a0, ai , bj , and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

[0007] In a further embodiment, to estimate the state-of-charge of the battery, the state-of-charge application further configures the processor to select an estimated state- of-charge from a lookup table using mm , where Zexpis anexperimentally derived set of impedance values paired with state-of-charge values.

[0008] In still another embodiment, Zexpis experimentally derived using batteries of a same model as the battery.

[0009] In a still further embodiment, the battery is a LiFePO4 battery.

[0010] In yet another embodiment, the battery is an electric vehicle battery.

[0011] In a yet further embodiment, the battery is an electrical grid storage battery.

[0012] In another additional embodiment, a battery state-of-charge monitor includes a sine-wave pulse generator configured to provide sine-wave pulses to a battery, a voltage sensor configured to measure output voltage of the battery, and processing circuitry communicatively coupled to the sine-wave pulse generator and the voltage sensor, wherethe processing circuitry is configured to apply a sine-wave pulse current to the battery using the sine-wave pulse generator, measure the output voltage of the battery in response to the sine wave pulse current, estimate an impedance of the battery based on the voltage response, estimate a state-of-charge of the battery using the estimated impedance of the battery, and provide the estimated state-of-charge.

[0013] In a further additional embodiment, estimating the impedance includes calculating, using the logic circuitry, min Sn=i(Vt,n- Vte*p)2, where: Vt nis an [|Z|,arg(z),a0,ai,bj,w] n-th data point of predicted voltage from battery voltage over time (Vt), vte pis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery, N is the length of experimental voltage data from the at least one test battery of the same type as the battery, Z is the estimated impedance of the battery, and a0, a, , bj , and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

[0014] In another embodiment again, estimating the state-of-charge of the battery includes indexing, using the logic circuitry, a lookup table usingexperimentally derived set of impedance values paired with state-of-charge values.

[0015] In a further embodiment again, Zexpis experimentally derived using batteries of a same model as the battery.

[0016] In still yet another embodiment, the battery is a LiFePO4 battery.

[0017] In a still yet further embodiment, the battery is an electric vehicle battery.

[0018] In still another additional embodiment, the battery is an electrical grid storage battery.

[0019] In a still further additional embodiment, a method of monitoring state-of-charge of a battery includes applying a sine-wave pulse current to a battery using a sine-wave pulse generator, measuring an output voltage of the battery in response to the sine wave pulse current using a voltage sensor, estimating an impedance of the battery based on the voltage response, estimating a state-of-charge of the battery using the estimated impedance of the battery, and providing the estimated state-of-charge.

[0020] In still another embodiment again, wherein estimating the impedance includes calculating where Vt nis an n-th data point of predictedvoltage from battery voltage over time (Vt), Vte pis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery, N is the length of experimental voltage data from the at least one test battery of the same type as the battery, Z is the estimated impedance of the battery; and a0, a, , bj , and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

[0021] In a still further embodiment again, estimating the state-of-charge of the battery includes indexing a lookup table using min , where Zexpis anexperimentally derived set of impedance values paired with state-of-charge values.

[0022] In yet another additional embodiment, Zexpis experimentally derived using batteries of a same model as the battery.

[0023] In a yet further additional embodiment, the battery is a LiFePO4 battery.

[0024] In yet another embodiment again, the battery is an electric vehicle battery.

[0025] In a yet further embodiment again, the battery is an electrical grid storage battery.

[0026] In another additional embodiment again, Zexpis derived using a charge direction battery profiling test.

[0027] In a further additional embodiment again, Zexpis derived using a discharge direction battery profiling test.

[0028] In still yet another additional embodiment, the sine-wave frequency is between 0.001 Hz and 1 kHz.

[0029] Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the invention. A further understanding of the nature and advantages of the present invention may be realized by reference to the remaining portions of the specification and the drawings, which forms a part of this disclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The description and claims will be more fully understood with reference to the following figures and data graphs, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention.

[0031] FIG. 1 illustrates a battery management system architecture in accordance with an embodiment of the invention.

[0032] FIG. 2 is a block diagram for a battery monitor in accordance with an embodiment of the invention.

[0033] FIG. 3 is a flow chart for a battery profiling process in the discharge direction in accordance with an embodiment of the invention.

[0034] FIG. 4 is a flow chart for a battery profiling process in the charge direction in accordance with an embodiment of the invention.

[0035] FIG. 5A is four charts showing example EIS profiling results in the charge and discharge directions in accordance with an embodiment of the invention.

[0036] FIG. 5B is four charts showing example sine-wave pulse profiling in the charge and discharge directions in accordance with an embodiment of the invention.

[0037] FIG. 6 is a flow chart for a SOC estimation process in accordance with an embodiment of the invention.

[0038] FIG. 7 is a flow chart graphically representing a SOC estimation process in accordance with an embodiment of the invention.DETAILED DESCRIPTION

[0039] Rechargeable batteries have recently become a critical component of both public and private infrastructure. In particular, the rise of electric vehicles has necessitated rapid development of rechargeable battery technology. Further, as renewable energy sources have become more common across the grid, energy storage has become a critical challenge to solve time-based generation issues and smooth out demand on conventional power generation facilities. For example, solar power is only generated during sunlight hours, but electricity must be used or stored immediately upon generation.Therefore, night power use cannot practically be based on solar power unless the energy is stored during the day. Energy storage facilities that use large arrays of rechargeable batteries have become an increasingly common storage medium for handling these temporal issues.

[0040] Across all sectors, a near requirement for battery-based systems is an indication of how much charge remains in the battery. Battery management systems (BMS) are used to monitor SOC. However, direct SOC measurement by BMS sensors is not feasible. Instead, various algorithms utilize sampled current and voltage signals for estimation such as equivalent-circuit-model (ECM) based Kalman filters (e.g. Extended Kalman Filters, and Unscented Kalman Filters). A critical problem with conventional algorithms is their lack of certainty for certain battery chemistries. In particular, for battery chemistries that have a flat open circuit voltage (OCV), SOC is very difficult to measure when the battery is not dead, but not at capacity. For example, in LiFePO4 batteries, it is very difficult to measure SoC between 20% and 80% when not in a laboratory environment.

[0041] Electrochemical impedance spectroscopy (EIS) is a technique that measures the impedance of a battery in response to an alternating current field. EIS studies are typically performed in laboratory environments with testing equipment that currently range from thousands to tens of thousands of dollars. While EIS is a powerful technique for characterizing a battery, using EIS for real-time SoC monitoring is currently impractical. Systems and methods described herein estimate impedance using a conventional voltage sensor and a sine-wave pulse generator, rather than laboratory EIS equipment, which can be integrated into any number of existing rechargeable battery systems such as (but not limited to) power grid storage facilities and electric vehicles. The estimated impedance EIS values in turn can be used to quickly and computationally efficiently determine SOC using a lookup table. A discussion of SOC monitoring components is followed by discussion of SOC monitoring processes.SOC monitors

[0042] Turning now to FIG. 1 , a battery management system that measures SOC in accordance with an embodiment of the invention is illustrated. BMS 100 measures SOCin battery 110. BMS 100 includes a sine-wave pulse generator that provides sine-wave pulses to battery 110. A voltage sensor 130 records battery output voltage in response to the sine-wave pulses. The profile of the sine-wave pulse and the measured voltage are provided to a SOC monitor 140. In numerous embodiments, the sine-wave pulses are provided directly to both the SOC monitor and the battery, rather than just a profile of the pulse. In numerous embodiments, the battery is connected to its typical operating circuitry, in addition to the generator and voltage sensor. As can be readily appreciated, the BMS may include additional components that are used to perform other BMS functionalities.

[0043] FIG. 2 is a block diagram for a battery monitor in accordance with an embodiment of the invention is illustrated. Battery monitor 200 includes a processor 210. Processors can be any logic processing circuit including (but not limited to) central processing units (CPUs), graphics processing units (GPUs), field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), and / or any other logic processing circuitry or combination thereof. Battery monitor 200 further includes an input / output (I / O) interface 220. I / O interfaces are components capable of receiving and sending signals from the battery monitor including (but not limited to) receiving information from the sine-wave pulse generator, receiving voltage measurements, and transmitting SOC measurements. In many embodiments, I / O interfaces contain one or more different modalities of transmitting and / or receiving data. In some embodiments, one or more modality is wireless.

[0044] Battery monitor 200 further includes a memory 230. Memory can be volatile memory, non-volatile memory, or a combination thereof. Memory 230 stores a SOC monitoring application 232 that contains instructions which configure the processor to carry out SOC monitoring processes discussed herein. In many embodiments, the memory 230 stores a battery profile 234 for the type of battery to be monitored. The battery profile includes a lookup table and / or lookup curve that matches impedance values to SOC levels. In numerous embodiments, battery monitors are implemented on conventional computing hardware. However, some architectures utilize specific circuit implementations that are designed to implement SOC monitoring processes or lookup tables without machine-readable instructions. In various embodiments, the batteryprofiles or battery monitoring application can be updated via firmware or software updates. As can be readily appreciated, any number of different computing designs can be used to implement SOC monitoring processes as appropriate to the requirements of specific applications of embodiments of the invention.SOC Estimation

[0045] SOC estimation described herein is based on EIS recovery from measurements of short-duration sinusoidal current pulses and voltage responses. As there are challenges associated with measuring direct frequency-domain EIS signals in real-time BMS, EIS is derived from time-domain current and voltage data. Cell impedance, represented as Z(j2rrf), includes both magnitude and phase components:where |Z(j2uf) | is the magnitude of cell impedance and arg (Z(j2nf)) is the phase of the impedance. Ideally, for EIS testing in galvanostatic mode, a sinusoidal current i(t) is applied to the battery around the OCV: i(t) = acos (2irft) where a is the amplitude, f is the frequence (Hz), and t is time. Generally, the actual battery voltage includes a transient term Vtransient, which diminishes over time, so the realistic voltage can be represented as:arg(Z)) + VtransientVt(0) is the cell OCV . As there is no explicit expression for Vtransient, the TrigonometricFourier series can be used to approximate Vtransient. Consequently, Vtbecomes:where ao is the constant term in the transient voltage and is associated with the i=0 cosine term, w is the fundamental frequency of the transient voltage, and m is the number of terms. However, as can be readily appreciated, different approximations for Vtransientcan be used without departing from the scope or spirit of the invention. To estimate theparameters including the |Z|, arg(Z), a0, a,, b., and w, the following objective function is formulated:where Vt nis the n-th data point of predicted voltage from Vt. Vte*pis the n-th data point of the experimental measured voltage. N is the length of the experimental voltage data. Therefore, given the experimental voltage responses to sine-wave current pulses (Vte*p), these parameters can be optimized including the battery impedance by fitting experimental data using a curve fitting algorithm. Based on the recovered impedance from current / voltage data ( |Z|, arg(z) ), the ElS-based SOC estimation algorithm is developed. It is assumed that the experimental EIS Zexpvs. SOC is stored as the benchmark values. Once the recovered EIS, Z is obtained, the estimated SOC, SOC can be solved as a two-dimensional lookup table problem as follows:

[0046] For any given battery type to be utilized, the lookup table is empirically generated, describing that battery type’s impedance at different SOC levels. In many embodiments, only a single battery of a given type needs to be profiled. In various embodiments, a number of different batteries of the same type are profiled, and their profiles averaged to generate an aggregate profile of the type of battery.

[0047] The profile of the battery involves measuring impedance at a number of different SOC levels. For example, in some embodiments, impedance testing is performed at each 10% SOC interval. In some embodiments, the testing is performed at each 5% SOC interval. In various embodiments, the testing is performed at each 1 % SOC interval. As can be readily appreciated, more or fewer intervals can be used trading off granularity for initial testing time. However, as the profiling process is infrequent (i.e. once per battery type), intervals can be computed that best suit the application for the given battery system. In some embodiments, battery types can be periodically reprofiled to account for changes in quality or production methods. Similar methods are used when directly measuring EIS versus estimating EIS using sine-wave pulse current.

[0048] A profiling process in the discharge direction is illustrated in accordance with an embodiment of the invention in FIG. 3. The profiling process 300 includes charging (310) the battery to 100% SOC. The battery is then rested (320) for a rest period. In some embodiment, the rest period is 2 hours. However, the rest period can be modified depending on battery chemistry and / or capacity so long as the SOC does not substantially change across the rest period. For example, in many embodiments, the rest period is between 20 minutes and 5 hours, but again this can be modified based on the time it takes for the battery being tested to return to a rest state without changing SOC.

[0049] After the rest period, an impedance test is performed (330) to obtain the impedance of the type of battery at the given SOC. When empirically determining EIS values, the impedance test is performing EIS. When estimating EIS, sine-wave pulse current testing is performed along with EIS estimation. Subsequently the battery is constant current (CC) discharged (340) to the next SOC interval. Once the SOC reaches 0% (350), the profiling concludes. Otherwise, the battery is again rested (320) and tested (330) before being discharged (340) to the next SOC interval. In some embodiments, a final impedance test is performed at 0% SOC after a rest period to complete the profile.

[0050] The testing can also be performed in the reverse direction, i.e. starting at SOC 0% and moving up to 100%. A profiling process in the charge direction in accordance with an embodiment of the invention is illustrated in FIG. 4. First, the battery is discharged (410) to 0% SOC. The battery is then rested (420) for an interval as described above with respect to FIG. 3. The impedance test is performed (430) and CC charged to the next SOC interval. Once the SOC reaches 100% (350), the profiling concludes. Otherwise, the battery is again rested (320) and tested (330) before being charged (340) to the next SOC interval. In some embodiments, a final impedance test is performed at 100% SOC after a rest period to complete the profile. FIGs. 5A and 5B contrast example empirical results from EIS testing and sine-wave pulse current testing in both the charge and discharge directions.

[0051] In many embodiments, profiling occurs in both the charge and discharge directions for completeness. Similarly, the profiling process can be performed multiple times for a single battery for certainty of measurement if desired. In some embodiments, the results of profiling in both directions are averaged to result in a lookup table thatreflects EIS values for each SOC interval value. In various embodiments, the lookup table is represented as a curve that interpolates SOC between measured intervals.

[0052] Once a particular type of battery is profiled, it becomes computationally trivial to look up SOC when provided with a given EIS measurement of a battery of the same type. However, as noted above, measuring EIS in a live environment is impractical and difficult. Systems and methods described herein estimate EIS measurements that are sufficiently accurate as to enable accurate SOC estimates via the lookup table.

[0053] Turning now to FIG. 6, a process for estimating SOC using estimated EIS values in accordance with an embodiment of the invention is illustrated. Process 600 includes applying (610) a sine-wave pulse current to the battery. In numerous embodiments, the sine-wave frequency is between 0.001 Hz to 1 kHz. The output voltage of the batter is measured (620) using a voltage sensor. The EIS is estimated as described above based on the sine-wave pulse current and the measured voltage. The EIS value is then used to estimate (640) the battery SOC using the look up table. For additional clarity, this process is graphically illustrated in FIG. 7.

[0054] A significant advantage of the above is that once the battery is profiled, measurement of SOC is based on easy to obtain measurements in a live environment and is computationally inexpensive to compute. This yields a SOC estimation that is well suited to operating environments that require real-time SOC estimation across many batteries such as a grid storage facility. Similarly, these systems can be integrated into electric vehicle BMS in order to provide more accurate estimation of battery SOC during operation. As can be readily appreciated, there are many different battery-powered devices and systems that can benefit from real-time SOC measurement.

[0055] Although specifics are discussed above, many different system architectures and SOC estimation processes can be implemented in accordance with many different embodiments of the invention. It is therefore to be understood that the present invention may be practiced in ways other than specifically described, without departing from the scope and spirit of the present invention. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents.

Claims

WHAT IS CLAIMED IS:1 . A battery management system, comprising: a sine-wave pulse generator configured to provide sine-wave pulses to a battery; a voltage sensor configured to measure output voltage of the battery; and a state-of-charge monitor, comprising: a processor; and a memory storing a state-of-charge monitoring application that configures the processor to: apply a sine-wave pulse current to the battery using the sine-wave pulse generator; measure the output voltage of the battery in response to the sine wave pulse current; estimate an impedance of the battery based on the voltage response; estimate a state-of-charge of the battery using the estimated impedance of the battery; and provide the estimated state-of-charge.

2. The battery management system of claim 1 , wherein to estimate the impedance of the battery, the state-of-charge monitoring application further configures the processor to use an objective function min Sn=i(Vt,n ~ Vtexp)2, where:[|Z|,arg(2),a0,ai,bi,w]Vt nis an n-th data point of predicted voltage from battery voltage over time (Vt);Vtenxpis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery;N is the length of experimental voltage data from the at least one test battery of the same type as the battery;Z is the estimated impedance of the battery; and a0, abbj, and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

3. The battery management system of claim 2, wherein to estimate the state-of- charge of the battery, the state-of-charge application further configures the processor to select an estimated state-of-charge from a lookup table usingexperimentally derived set of impedance values paired with state-of-charge values.

4. The battery management system of claim 3, wherein Zexpis experimentally derived using batteries of a same model as the battery.

5. The battery management system of claim 1 , wherein the battery is a LiFePO4 battery.

6. The battery management system of claim 1 , wherein the battery is an electric vehicle battery.

7. The battery management system of claim 1 , wherein the battery is an electrical grid storage battery.

8. A battery state-of-charge monitor, comprising: a sine-wave pulse generator configured to provide sine-wave pulses to a battery; a voltage sensor configured to measure output voltage of the battery; and processing circuitry communicatively coupled to the sine-wave pulse generator and the voltage sensor, where the processing circuitry is configured to: apply a sine-wave pulse current to the battery using the sine-wave pulse generator; measure the output voltage of the battery in response to the sine wave pulse current; estimate an impedance of the battery based on the voltage response; estimate a state-of-charge of the battery using the estimated impedance of the battery; andprovide the estimated state-of-charge.

9. The battery state-of-charge monitor of claim 8, wherein estimating the impedance comprises calculating, using the logic circuitry,>Vtexp)2, where:Vt nis an n-th data point of predicted voltage from battery voltage over time (Vt);Vtenxpis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery;N is the length of experimental voltage data from the at least one test battery of the same type as the battery;Z is the estimated impedance of the battery; and a0, a, , bj, and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

10. The battery state-of-charge monitor of claim 9, wherein estimating the state-of- charge of the battery comprises indexing, using the logic circuitry, a lookup table usingexperimentally derived set of impedance values paired with state-of-charge values.

11. The battery state-of-charge monitor of claim 10, wherein Zexpis experimentally derived using batteries of a same model as the battery.

12. The battery state-of-charge monitor of claim 8, wherein the battery is a LiFePO4 battery.

13. The battery state-of-charge monitor of claim 8, wherein the battery is an electric vehicle battery.

14. The battery state-of-charge monitor of claim 8, wherein the battery is an electrical grid storage battery.

15. A method of monitoring state-of-charge of a battery, comprising: applying a sine-wave pulse current to a battery using a sine-wave pulse generator; measuring an output voltage of the battery in response to the sine wave pulse current using a voltage sensor; estimating an impedance of the battery based on the voltage response; estimating a state-of-charge of the battery using the estimated impedance of the battery; and providing the estimated state-of-charge.

16. The method of monitoring state-of-charge of a battery of claim 15, wherein estimating the impedance comprises calculating min Sn=i( tn Vtexp)2■[|Z|,arg(z),a0,ai,bi,w] where:Vt nis an n-th data point of predicted voltage from battery voltage over time (Vt);Vtenxpis an n-th data point of experimental measured voltage from at least one test battery of the same type as the battery;N is the length of experimental voltage data from the at least one test battery of the same type as the battery;Z is the estimated impedance of the battery; and a0, abbj, and w are parameters of a Trigonometric Fourier series used to estimate transient voltage in Vt.

17. The method of monitoring state-of-charge of a battery of claim 16, wherein estimating the state-of-charge of the battery comprises indexing a lookup table usingexperimentally derived set of impedance values paired with state-of-charge values.

18. The method of monitoring state-of-charge of a battery of claim 17, wherein Zexpis experimentally derived using batteries of a same model as the battery.

19. The method of monitoring state-of-charge of a battery of claim 15, wherein the battery is a LiFePO4 battery.

20. The method of monitoring state-of-charge of a battery of claim 15, wherein the battery is an electric vehicle battery.

21. The method of monitoring state-of-charge of a battery of claim 15, wherein the battery is an electrical grid storage battery.

22. The system, monitor, or method of any of the foregoing claims, wherein Zexpis derived using a charge direction battery profiling test.

23. The system, monitor, or method of any of the foregoing claims, wherein Zexpis derived using a discharge direction battery profiling test.

24. The system, monitor, or method of any of the foregoing claims, wherein the sinewave frequency is between 0.001 Hz and 1 kHz.

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