Systems and methods for state-of-charge estimation based on dq / dv curve

Abstract

Systems and methods for state-of-charge estimation in accordance with embodiments of the invention are illustrated. One embodiment includes a battery management system, including a pulse generator configured to provide constant current 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 constant current pulse to the battery using the pulse generator, measure the output voltage of the battery in response to constant current pulse, calculate a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage, estimate a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery, and provide the estimated state-of-charge.

Description

Systems and Methods for State-of-Charge Estimation Based on dQ / dV CurveCROSS-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 / 606,024 entitled “Short-Pulse Current Design for State-of-Charge Estimation of LiFePO4 Battery Based on dO / dV Curve” filed December 4, 2023. The disclosures of U.S. Provisional Patent Application No. 63 / 606,024 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 that measure SOC, and use SOC to guide optimal battery utilization within safe operational limits, avoiding accelerated degradation and performance issues.

[0005] Galvanostatic cycling is a technique used to study the performance of batteries by applying a constant current while charging and discharging them. The C-rate of a battery represents the charging / discharging current as a multiple of the battery’s nominal capacity. That is 1 C is a current equal to the capacity of the battery in one hour. 2C is twice that current. 0.5C (or C / 2) is half that current.SUMMARY OF THE INVENTION

[0006] Systems and methods for state-of-charge estimation in accordance with embodiments of the invention are illustrated. One embodiment includes a battery management system, including a pulse generator configured to provide constant current 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 constant current pulse to the battery using the pulse generator, measure the output voltage of the battery in response to constant current pulse, calculate a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage, estimate a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery, and provide the estimated state-of-charge.

[0007] In another embodiment, wherein dQ / dV is computed using=areh© battery capacity, battery voltage, and battery current at time step z, respectively, Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

[0008] In a further embodiment, the complete dQ / dV curve is empirically determined using at least one battery of the same type as the battery.

[0009] In still another embodiment, the complete dQ / dV curve is smoothed by reducing voltage during testing in accordance with minimize ^^=1(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

[0010] In a still further embodiment, the complete dQ / dV curve is smoothed using a moving average filter.

[0011] In yet another embodiment, the complete dQ / dV curve is stored as a onedimensional lookup table, where an x-axis represents state-of-charge e [0, 1 ], and a y- axis corresponds to values of the complete dQ / dV.

[0012] In a yet further embodiment, estimating the state-of-charge is performed using a state-of-charge observer formalized as:_ _ At SOC(z) = SOC(z - 1) + l(z) * - -^ / capacity dQ dQ y(z) = — (SOC(z)) = interpolation(SOC,— (SOC), SOC(z)) V complete UVcomp|eteWhere SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively, Qcapacityis a capacity of the battery,(SOC), is the onedimensional lookup table, and y(z) is an output based on the SOC(z) retrieved from the 1 D look-up table.

[0013] In another additional embodiment, the battery is a LiFePO4 battery.

[0014] In a further additional embodiment, the battery is an electric vehicle battery.

[0015] In another embodiment again, the battery is an electrical grid storage battery.

[0016] In a further embodiment again, a state-of-charge monitor, including a pulse generator configured to provide constant current pulses to a battery, a voltage sensor configured to measure output voltage of the battery, and processing circuitry communicatively coupled to the pulse generator and voltage sensor, where the processing circuitry is configured to measure the output voltage of the battery in response to constant current pulses provided to the battery by the pulse generator, calculate a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage, estimate a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state- of-charge of the battery, and provide the estimated state-of-charge.

[0017] In still yet another embodiment, dQ / dV is computed using=v(z)-v(z+i) 36oo’w^ereQ(z),v(z)>anc* ^(z)are® battery capacity, battery voltage, and battery current at time step z, respectively, Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

[0018] In a still yet further embodiment, the complete dQ / dV curve is empirically determined using at least one battery of the same type as the battery.

[0019] In still another additional embodiment, the complete dQ / dV curve is smoothed by reducing voltage during testing in accordance with minimize ^^=1(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

[0020] In a still further additional embodiment, the complete dQ / dV curve is smoothed using a moving average filter.

[0021] In still another embodiment again, the complete dQ / dV curve is stored as a one-dimensional lookup table, where an x-axis represents state-of-charge G [0, 1], and a y-axis corresponds to values of the complete dQ / dV.

[0022] In a still further embodiment again, estimating the state-of-charge is performed using a state-of-charge observer formalized as:_ _ AtSOC(z) = SOC(z - 1) + I(z) * - -^capacity dQ > dQ - y(z) = — (SOC(z)) = interpolation(SOC,— (SOC), SOC(z)) U v complete UVcomp|ete where, SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively, Qcapacityis a capacity of the battery,(SOC), is the one-dimensional lookup table, and y(z) is an output based on the SOC(z) retrieved from the1 D look-up table.

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

[0024] In a yet further additional embodiment, the battery is an electric vehicle battery.

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

[0026] In a yet further embodiment again, a method of monitoring state-of-charge of a battery, including measuring, using a voltage sensor, the output voltage of the battery in response to constant current pulses provided to the battery by a pulse generator, calculating a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage, estimating a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery, and providing the estimated state-of- charge.

[0027] In another additional embodiment again, the method further includes comp ruting a dQ / dV using a arethe battery capacity, battery voltage, and battery current at time step z, respectively, Q(z+ 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

[0028] In a further additional embodiment again, the method further includes empirically determining the complete dQ / dV curve using at least one battery of the same type as the battery.

[0029] In still yet another additional embodiment, smoothing the complete dQ / dV curve by reducing voltage during testing in accordance with minimize ^^=1(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

[0030] In another embodiment, the method further includes smoothing the complete dQ / dV curve using a moving average filter.

[0031] In a further embodiment, the method further includes storing the complete dQ / dV curve as a one-dimensional lookup table, where an x-axis represents state-of- charge e [0, 1], and a y-axis corresponds to values of the complete dQ / dV.

[0032] In still another embodiment, the method further includes estimating the state- of-charge using a state-of-charge observer formalized as:_ _ AtSOC(z) = SOC(z - 1) + l(z) * - -^capacity dQ - dQ - y(z) = — (SOC(z)) = interpolation(SOC,— (SOC), SOC(z)) u vconi|eten vconipietewhere SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively, Qcapacitvis a capacity of the battery, (SOC), is the one- y “"complete dimensional lookup table, and y(z) is an output based on the SOC(z) retrieved from the 1 D look-up table.

[0033] In a still further embodiment, the battery is a LiFePO4 battery.

[0034] In yet another embodiment, the battery is an electric vehicle battery.

[0035] In a yet further embodiment, the battery is an electrical grid storage battery.

[0036] In another additional embodiment, the complete dQ / dV is derived using a charge direction battery profiling test.

[0037] In a further additional embodiment, the complete dQ / dV is derived using a discharge direction battery profiling test.

[0038] In a further additional embodiment again, the constant current pulses are at a C rate current of the battery between C / 30 and 2C.

[0039] In yet a further additional embodiment again, Zexpis derived using a discharge direction battery profiling test.

[0040] 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

[0041] 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.

[0042] FIG. 1 illustrates a battery management system architecture in accordance with an embodiment of the invention.

[0043] FIG. 2 is a block diagram for a battery monitor in accordance with an embodiment of the invention.

[0044] FIG. 3 shows charge and discharge direction voltages with respect to a LFP battery SOC at different C-rates in accordance with an embodiment of the invention.

[0045] FIG. 4 is a flow chart for SOC estimation process in accordance with an embodiment of the invention.

[0046] FIG. 5 is a flow chart graphically depicting a SOC estimation process in accordance with an embodiment of the invention.

[0047] FIG. 6 is a series of charts that illustrate pulse dQ / dV with respect to complete dQ / dV curves at various C rates for an LFP battery.DETAILED DESCRIPTION

[0048] 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.

[0049] 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 (EKF), and Unscented Kalman Filters (UKF)). 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 (LFP) batteries, it is very difficult to measure SoC between 20% and 80% when not in a laboratory environment.

[0050] Various attempts have been made to overcome this problem with some degree of success. A first-order ECM-based EKF for LFP battery SOC estimation dynamically updates OCV, resistance, and capacitor parameters, maintaining SOC error below 2%, but requires accurate parameter identification. Some methods use pseudo-OCV reconstruction to enhance the mid-SOC range, with errors under 3%, though they rely on Thevenin models, which may underperform under varying conditions. Other proposals include a simplified OCV hysteresis model within an ECM-based EKF adjusts measurement error covariance, achieving a 5% margin at room temperature but lackingrobust noise filtering. Some machine learning models have achieved relatively robust outcomes with maximum error below 2%, however they require extensive and unbiased experimental data for training.

[0051] Systems and methods described herein take a different approach based on differential capacity analysis. Differential capacity analysis is defined as the rate of change of charge with voltage (dQ / dV), which is typically used for investigating electrode degradation mechanisms during charge or discharge processes. dQ / dV is the inverse derivative of the galvanostatic voltage response of the battery. The evolution of the area, location, and amplitude of the dQ / dV peaks provide insights closely linked to battery aging and estimating state of health. Discussed herein are new applications of differential capacity analysis with respect to estimating SOC. Compared to OCV-SOC curves, the dQ / dV curve relative to SOC exhibits greater steepness when measured at low C-rates, making it advantageous for SOC estimation, particularly in LFP batteries and other battery chemistries with flat OCV.

[0052] The C-rate dependency on SOC estimation has been poorly understood. In practical applications, low C-rate pulse currents and resulting voltage responses are sensitive to measurement noise, underscoring the importance of maintaining robust SOC estimation even in the presence of measurement noise. Low C-rate pulse currents tend to generate dQ / dV curves with more pronounced valleys and peaks compared to high C- rates. Therefore, the consistent use of low C-rate pulses is crucial for enhancing SOC estimation performance, as it provides better granularity and accuracy than high C-rate pulses. Further, low C-rate pulses tend to have less of an effect on battery health. However, the sensitivity of low C-rate pulse currents and resultant voltage responses to measurement noise presents a challenge.

[0053] Systems and methods described herein leverage the steepness of the curve by first profiling the dQ / dV curve across the range of SOC for a given battery at a given C-rate. In numerous embodiments, the selected C-rate is between C / 30 and 2C. In many embodiments, the selected C-rate is the lowest fractional C-rate that, when provided as a pulse, still enables resolving the dQ / dV in order reduce the amount of energy needed and to minimize impact on battery health. Then, during operation, short constant currentpulses at the selected C-rate can be applied to the battery in order to estimate location on the dQ / dV curve, thereby identifying an associated SOC value.

[0054] To enable low cost, low energy, and low computation requirements, systems and methods described herein can utilize a simplified model based on one linear equation and one lookup table. This enables deployment as a real-time SOC estimation architecture for embedded BMSs in electric vehicles and other resource-constrained environments without sacrificing estimation performance. A discussion of SOC monitoring components is followed by discussion of SOC monitoring processes.SOC monitors

[0055] 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 SOC in battery 110. BMS 100 includes a constant current (CC) pulse generator that provides sine-wave pulses at a selected C-rate to battery 110. A voltage sensor 130 records battery output voltage in response to the CC pulses. The profile of the CC pulse and the measured voltage are provided to a SOC monitor 140. In numerous embodiments, the CC 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.

[0056] 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 differentmodalities of transmitting and / or receiving data. In some embodiments, one or more modality is wireless.

[0057] 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 points on the dQ / dV curve for the battery 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 battery profiles 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

[0058] SOC estimation as described herein is designed to be performed in real-time with high efficiency. To achieve this, SOC estimation relies upon a one-time profiling phase, and a real-time estimation phase. In order to quickly and efficiently estimate SOC in real-time, the type of battery to be monitored needs to be profiled. Profiling the battery is a process that can occur using one or a relatively small number of batteries of the same type and extrapolated across all batteries of the same type. In many embodiments, the optimal C-rate for a given battery type can be experimentally determined by testing a range of C-rates. For LFP batteries, C-rates between C / 30 and 2C typically provide sufficient resolution of dQ / dV. FIG. 3 demonstrates charge and discharge voltages vs. SOC for various C-rates.

[0059] Turning now to FIG. 4, a process for SOC estimation in accordance with an embodiment of the invention is illustrated. Process 400 first includes a profiling phase where the battery profile is determined (410). Profiling the battery involves resolving thedQ / dV curve across the SOC range of the battery. dQ / dV can be computed using: — 3600 ,’ where Q(z) / , V (z), and / (z) are the ce capacity, ce> t / > \ / r J . voltage, and cell current at time step z, respectively. Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the cell capacity, cell voltage, and cell current at the next time step, k + 1 , respectively. At is the time step size between every two data samples (e.g. 1s).

[0060] In many embodiments, a moving-average filter can be used to obtain a smoother signal. For example, the moving average can be calculated as:where x is the input signal, F is the filtered output signal, and M is the smoothing window size. However, any number of different smoothing filters can be applied as appropriate to the requirements of specific applications of embodiments of the invention.

[0061] In various embodiments, in addition to filtering techniques, reducing the voltage sampling frequency can be effective in smoothing dQ / dV curves. A mathematical model can be used to fit voltage behavior (Vt) when discharge / charge pulse currents are applied: minimizewhere N is the number of data points.

[0062] Computation can occur in two directions, from 0% SOC to 100% SOC, or from 100% to 0% SOC, at a selected C-rate. In some embodiments, the testing is performed in both directions. Voltage and current data from the discharge / charge process can be used to resolve the dQ / dV curve across every SOC value. This can provide a profile for the charge direction and a profile for the discharge direction, and / or an averaged profile. In many embodiments, the battery profile can be validated by performing CC pulse tests using the SOC estimation process described below. In numerous embodiments, the battery profile is stored as a 1 -dimensional lookup table, where the x-axis represents SOC e [0, 1 ], and the y-axis corresponds to the complete dQ / dV values. This lookup table as a function of SOC is formalized herein as dQ / dvcomplete(soc).

[0063] Once a complete profile is made, it can be used to estimate SOC in a real-time estimation phase by identifying where on the dQ / dV curve the battery currently lies. This is performed using an SOC observer formalized as:_ _ AtSOC(z) = SOC(z - 1) + 1 (z) * - -^ / capacity dQ interpolation(SOC,— (SOC), SOC(z))* com plete where SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively. Qcapacityis the cell capacity.(SOC), is the pre-processed 1 D look-up table (x- axis: SOC G [0, 1]; y-axis:). y(z) is the output dVcomplete based on the SOC(z) and retrieved from the 1 D look-up table. Interpolation is an interpolation function such as (but not limited to) interpl from Matlab. Interpolation returns interpolated value y(z) for the 1 -D function at specific query points SOC(z) .These generated pulse dQ / dV curves dQ7dVpulseserve as the benchmark measurements for the observer. The observer input is / (z).

[0064] In the real-time estimation phase, a CC pulse at the selected C-rate is applied (420) to the battery, and the voltage response of the battery to the pulse is measured (430). In many embodiments, a series of pulses are applied and respective voltage responses are measured. Using the pulse(s) and the respective measured voltages, the prulse dQ / dV is calculated ( \440) / ag aain using a —dv(.z) j=QV^(z)~-vQ^(zz++i1^) = V ‘^(z)~-VI(:(,zz++1i) 36Aoto .

[0065] The SOC is then estimated (350) using the SOC observer. In many embodiments, an optimization algorithm is used to modify the SOC observer to protect against input noise and modeling uncertainties. For example, a UKF can be used to filter pulse dQ / dV with respect to the complete dQ / dV curve as part of the SOC observer. However, other optimization algorithms such as (but not limited to) EKFs, adaptive EKFs, nonlinear least square, and / or any other optimization algorithm can be used as appropriate to the requirements of specific applications of embodiments of the invention.

[0066] The ultimate goal is to find a SOC estimate at time step z, SOC(z) that minimizes the dQ / dV error, dQ / d error between the look-up table value dQ7dVcomplete,z(S0C(z)) andthe calculated dQ / dV from the pulse at time step z, dQ / dVpulse z. To modify the SOC observer using a UKF, by way of example, at every time step, the sigma points can be chosen using the state estimate and its covariance to generate the SOC(z). Then the estimated output of dQ7dVcomplete z(SOC(z)) can be calculated based on the complete discharge / charge dQ7dVcomplete. The estimated dQ / dV error is given as:Set x (z) = [SOC (z)] , u (z) = 1 (z), and then the SOC Observer equations above can be abbreviated as:(x(z + 1) = f(x(z),u(z)) + w(z) ( y(z) = g(x(z), u(z)) + v(z)Where w (z) is the process noise vector with zero mean and covariance Q. v(k) is the measurement noise vector with zero mean and covariance R.As can be readily appreciated, other optimization algorithms can be applied resulting in different formalizations as appropriate to the requirements of specific applications of embodiments of the invention. When additional estimations of SOC are required (450), additional CC pulses at the C-rate can be applied from which updated pulses dQ / dV are derived, leading to a new SOC estimation. In order to provide additional clarity, FIG. 5 illustrates the profiling stage (left) and real-time SOC estimation phase (right) in accordance with an embodiment of the invention. FIG. 6 shows pulse dQ / dV with respect to the complete dQ / dV curve for various C rates in accordance with an embodiment of the invention.

[0067] 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 pulse generator configured to provide constant current 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 constant current pulse to the battery using the pulse generator; measure the output voltage of the battery in response to constant current pulse; calculate a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage; estimate a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery; and provide the estimated state-of-charge.

2. The battery management system of claim 1 , wherein dQ / dV is computed using— (z)=Q z~Q z+1= where Q(z), V (z), and 7(z) are the battery capacity,battery voltage, and battery current at time step z, respectively, Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

3. The battery management system of claim 2, wherein the complete dQ / dV curve is empirically determined using at least one battery of the same type as the battery.

4. The battery management system of claim 3, wherein the complete dQ / dV curve is smoothed by reducing voltage during testing in accordance with minimize £k=1(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

5. The battery management system of claim 2, wherein the complete dQ / dV curve is smoothed using a moving average filter.

6. The battery management system of claim 1 , wherein the complete dQ / dV curve is stored as a one-dimensional lookup table, where an x-axis represents state-of-charge G [0, 1], and a y-axis corresponds to values of the complete dQ / dV.

7. The battery management system of claim 6, wherein estimating the state-of- charge is performed using a state-of-charge observer formalized as: C(z))where:SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively;Q capacity is a capacity of the battery;(SOC), is the one-dimensional lookup table; and dvcomplete y(z) is an output based on the SOC(z) retrieved from the 1 D look-up table.

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

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

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

11. A state-of-charge monitor, comprising: a pulse generator configured to provide constant current pulses to a battery; a voltage sensor configured to measure output voltage of the battery; and processing circuitry communicatively coupled to the pulse generator and voltage sensor, where the processing circuitry is configured to: measure the output voltage of the battery in response to constant current pulses provided to the battery by the pulse generator; calculate a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage; estimate a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery; and provide the estimated state-of-charge.

12. The state-of-charge monitor of claim 11 , wherein dQ / dV is computed using— (z)=Q z~Q(z+1= where Q(z), V (z), and 7(z) are the battery capacity,battery voltage, and battery current at time step z, respectively, Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

13. The state-of-charge monitor of claim 12, wherein the complete dQ / dV curve is empirically determined using at least one battery of the same type as the battery.

14. The state-of-charge monitor of claim 13, wherein the complete dQ / dV curve is smoothed by reducing voltage during testing in accordance with minimize £k=i(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

15. The state-of-charge monitor of claim 12, wherein the complete dQ / dV curve is smoothed using a moving average filter.

16. The state-of-charge monitor of claim 11 , wherein the complete dQ / dV curve is stored as a one-dimensional lookup table, where an x-axis represents state-of-charge e [0, 1], and a y-axis corresponds to values of the complete dQ / dV.

17. The state-of-charge monitor of claim 16, wherein estimating the state-of-charge is performed using a state-of-charge observer formalized as:AtSOC(z) = SOC(z - 1) + I(z) * - -^capacity dQ dQ > y(z) = — (SOC(z)) = interpolation(SOC,— (SOC), SOC(z)) avcom plete U » com plete where:SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively;Q capacity is a capacity of the battery;(SOC), is the one-dimensional lookup table; and dVcomplete y(z) is an output based on the SOC(z) retrieved from the 1 D look-up table.

18. The state-of-charge monitor of claim 11 , wherein the battery is a LiFePO4 battery.

19. The state-of-charge monitor of claim 11 , wherein the battery is an electric vehicle battery.

20. The state-of-charge monitor of claim 11 , wherein the battery is an electrical grid storage battery.21 . A method of monitoring state-of-charge of a battery, comprising: measuring, using a voltage sensor, the output voltage of the battery in responseto constant current pulses provided to the battery by a pulse generator; calculating a pulse inverse derivative of a galvanostatic voltage response of the battery, pulse dQ / dV, based on the measured output voltage; estimating a state-of-charge of the battery using the estimated pulse dQ / dV and a complete dQ / dV curve of the battery across a range of state-of-charge of the battery; and providing the estimated state-of-charge.

22. The method of claim 21 , comprising computing dQ / dV using=v(z)-v(z+i) 36oo’w^ereQ(z)>v(z)>anc* ^(z)are*he battery capacity, battery voltage, and battery current at time step z, respectively, Q(z + 1 ), V (z + 1 ), and / (z + 1 ) are the battery capacity, battery voltage, and battery current at the next time step, k + 1 , respectively.

23. The method of claim 22, further comprising empirically determining the complete dQ / dV curve using at least one battery of the same type as the battery.

24. The method of claim 23, further comprising smoothing the complete dQ / dV curve by reducing voltage during testing in accordance with minimize ^^=1(Vexp[k] - Vt[k])2 / N, where N is a number of data points.

25. The method of claim 22, further comprising smoothing the complete dQ / dV curve using a moving average filter.

26. The method of claim 21 , further comprising storing the complete dQ / dV curve as a one-dimensional lookup table, where an x-axis represents state-of-charge e [0, 1 ], and a y-axis corresponds to values of the complete dQ / dV.

27. The method of claim 26, further comprising estimating the state-of-charge using a state-of-charge observer formalized as:interpolation(SOC,— (SOC), SOC(z))U '' complete where:SOC(z) and SOC(z - 1) are SOC estimations at present time step z and previous time step z-1 , respectively;Q capacity is a capacity of the battery;(SOC), is the one-dimensional lookup table; and dVcomplete y(z) is an output based on the SOC(z) retrieved from the 1 D look-up table.

28. The method of claim 27, wherein the battery is a LiFePO4 battery.

29. The method of claim 21 , wherein the battery is an electric vehicle battery.

30. The method of claim 21 , wherein the battery is an electrical grid storage battery.

32. The system, monitor, or method of any of the foregoing claims, wherein complete dQ / dV is derived using a charge direction battery profiling test.

33. The system, monitor, or method of any of the foregoing claims, wherein complete dQ / dV is derived using a discharge direction battery profiling test.

34. The system, monitor, or method of any of the foregoing claims, wherein the constant current pulses are at a C rate current of the battery between C / 30 and 2C.

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

Images