Methods of extracting one or more diagnostic parameters relating to battery-cell health, and systems implementing such methods

Abstract

Machine-implemented methods of operating an electrochemical cell having anodes of a plating-stripping type, wherein the electrochemical cell has associated voltage-based reference data are disclosed. In an example, such a method includes: receiving at a cycle, n, first voltage-based cycle data for the electrochemical cell; solving an optimization problem that maximizes fit between the voltage-based reference data and the first voltage-based cycle data so as to obtain estimated diagnostic information for the electrochemical cell at cycle n; and providing the estimated diagnostic information to a cell-interface system designed and configured to use the diagnostic information to perform one or more tasks relating to the electrochemical cell. Related machine-readable storage media and systems are also disclosed.

Description

METHODS OF EXTRACTING ONE OR MORE DIAGNOSTIC PARAMETERS RELATING TO BATTERY-CELL HEALTH, AND SYSTEMS IMPLEMENTING SUCH METHODSRELATED APPLICATION DATA

[0001] This application claims the benefit of priority of U.S. Provisional Patent Application Serial No. 63 / 606,796 filed on December 6, 2023, and titled “METHODS OF EXTRACTING ONE OR MORE DIAGNOSTIC PARAMETERS RELATING TO BATTERY-CELL HEALTH, AND SYSTEMS IMPLEMENTING SUCH METHODS”, which is incorporated herein by reference in its entirety.FIELD

[0002] The present disclosure generally relates to the field of assessing the health of battery cells. In particular, the present disclosure is directed to methods of extracting one or more diagnostic parameters relating to battery-cell health, and systems implementing such methods.BACKGROUND

[0003] Voltage is often used as an indicator of lithium-ion battery health. However, there are two notable shortcomings in conventional practices:1. Existing methods have been developed for full-cell configurations, which are chemically and physically different than half-cells (i.e., lithium-metal batteries).2. Existing methods are applied to uniform cycling protocols with regular cycling (typically in a laboratory environment) but do not address more complex situations such as shallow or incomplete charge / discharge, pulse sequences, or complex sequences of charge and discharge that result from driving an electric vehicle.SUMMARY

[0004] In one implementation, the present disclosure is directed to a machine-implemented method of operating an electrochemical cell having anodes of a plating-stripping type, wherein the electrochemical cell has associated voltage-based reference data. The machine-implemented method includes receiving, at a cycle, n, first voltage-based cycle data for the electrochemical cell; solving an optimization problem that maximizes fit between the voltage-based reference data and the first voltage-based cycle data so as to obtain estimated diagnostic information for the electrochemical cell at cycle n; and providing the estimated diagnostic information to a cell-interface system designedand configured to use the diagnostic information to perform one or more tasks relating to the electrochemical cell.

[0005] In another implementation, the present disclosure is directed to a machine-readable storage medium containing machine-executable instructions for performing the machine- implemented method described directly above.

[0006] In yet another implementation, the present disclosure is directed to a system for operating an electrochemical cell. The system includes interface electronics for electrically interfacing with the electrochemical cell when the electrochemical cell is present; a cell-interface system that interfaces with the electrochemical cell via the interface electronics when the electrochemical cell is present; machine memory containing machine-executable instructions for performing the machine-implemented method described above; and at least one processor in operative communication with the machine memory and the interface electronics, wherein the at least one processor is configured to execute the machine-executable instructions.BRIEF DESCRIPTION OF THE DRAWINGS

[0007] For the purpose of illustration, the accompanying drawings show aspects of one or more embodiments of the disclosure. However, it should be understood that the scope of this disclosure is / are not limited to the precise arrangements and instrumentalities shown in the drawings, wherein:

[0008] FIG. 1A is a graph of voltage versus capacity showing a reference charge curve and a cycle-n charge curve for an example electrochemical cell;

[0009] FIG. IB is a graph of voltage versus capacity showing the reference charge curve and the cycle-n degraded charge curve of FIG. 1 A after transforming the reference charge voltage to match the degraded charge using the optimization Equation 1 of the present disclosure, wherein, at cycle n, the resistance, Rn, is 235 milliohms, the state of health, SOH11, is 0.976, and the initial state of charge Qonis 8.4 milliamp-hours;

[0010] FIG. 2A is a graph of voltage versus capacity showing a reference discharge curve and a cycle-n discharge curve for another example electrochemical cell;

[0011] FIG. 2B is a graph of voltage versus capacity showing the reference discharge curve and the cycle-n degraded discharge curve of FIG. 2A after transforming the reference discharge voltage to match the degraded discharge using the optimization Equation 1 of the present disclosure,wherein, at cycle n, the resistance, Rn, is 419 milliohms, the SOHnis 0.912, and the initial state of charge Qonis 0.0 milliamp-hours;

[0012] FIG. 3A is a graph of dQ / dV versus capacity showing a reference curve and a cycle-n degraded curve for yet another example electrochemical cell;

[0013] FIG. 3B is a graph of dQ / dV versus capacity showing the reference curve and the cycle- n degraded curve of FIG. 3 A after transforming dQ / dV of the reference to match the degraded charge using the optimization Equation 4 of the present disclosure;

[0014] FIGS. 4 A and 4B are graphs of capacity versus cycle number (FIG. 4A) and each of SOH and resistance increase versus cycle number (FIG. 4B) illustrating parameter extraction, using methodology disclosed herein, from a 4.4 Ah cell that was cycled at 25°C with a C / 3-C / 3 cycling protocol to 100% depth of discharge (DOD), with FIG. 4A showing capacity fade, and FIG. 4B showing extracted values of SOHnand Rnas a function of cycle number;

[0015] FIGS. 5A and 5B are graphs of capacity versus cycle number (FIG. 5A) and each of SOH and resistance increase versus cycle number (FIG. 5B) illustrating parameter extraction, using methodology disclosed herein, from a 4.4 Ah cell that was cycled at 25°C with a C / 5-C / 3 cycling protocol to 70% DOD, with FIG. 5A showing capacity fade, and FIG. 5B showing extracted values of SOHnand Rnas a function of cycle number;

[0016] FIGS. 6 A and 6B are graphs of capacity versus cycle number (FIG. 6A) and each of SOH and resistance increase versus cycle number (FIG. 6B) illustrating parameter extraction, using methodology disclosed herein, from a 4.4 Ah cell that was cycled at 55°C with a C / 3-US06 cycling protocol to 90% DOD, with FIG. 6A showing capacity fade, and FIG. 6B showing extracted values of SOHnand Rnas a function of cycle number; and

[0017] FIG. 7 is a high-level block diagram illustrating an example diagnostics-extraction system made in accordance with aspects of the present disclosure and in operative communication with an electrochemical cell.DETAILED DESCRIPTION

[0018] The entire contents of the appended claims are incorporated into this Detailed Description section as if originally presented herein.

[0019] Some terms and initialisms used in this disclosure and not otherwise defined herein are defined as follows:Full-cell: an electrochemical cell (battery) with an insertion anode material such as graphite, silicon, or titanate.Half-cell: an electrochemical cell (battery) that has a lithium-metal anode.State-of-health (SOH): The estimated charge / discharge capacity of a degraded cell if it were cycled at reference temperature and rates. In some embodiments, the SOH is capacity expressed as a fraction of the reference capacity.Resistance increase: The internal resistance of a degraded cycle in excess of the internal resistance of the reference cycle.DCIR: direct current internal resistance.DOD: depth of discharge. EV : electric vehicle. OCV: open-circuit voltage. SEI: solid electrolyte interphase.

[0020] GENERAL

[0021] Estimating the state-of-health of a battery that has been cycled is challenging, because there are multiple mechanisms that simultaneously lead to decreasing performance. These mechanisms broadly fall into two different categories: those that reduce the capacity of the battery (e.g., loss of lithium, loss of anode active material, loss of cathode active material, etc.), and those that lead to increasing electrical resistance (e.g., SEI growth, electrolyte decomposition, etc.).

[0022] In some aspects, the present disclosure is directed to methods of extracting diagnostics from a lithium-metal battery, including state-of-health and internal resistance increase, by comparing the voltage-versus-capacity (WC) curve of a cell at an unknown state of degradation against a reference WC curve from a fresh cell. The reference cycle is ideally from an open-circuit measurement made in a laboratory environment on an identical cell. Alternatively, it can be an initial cycle during use of the test cell. The diagnostic parameters are then found by solving a numerical optimization problem to minimize the error between the test cycle and a mathematical transformation of the reference cycle. The state of health and the electrical-resistance increase are taken to be the values that minimize error between the degraded WC curve and the transformed reference WC curve.

[0023] A purpose of implementing methods of the present disclosure is to enable real-time estimation of both fundamental lithium-metal battery degradation modes (capacity loss andresistance increase) under real-world conditions, thereby eliminating the need for performing targeted diagnostic cycles such as capacity checks or DCIR measurements. This information can be incorporated into battery management systems to monitor health, warn of the onset of rapid degradation before it occurs (the so-called “knee phenomenon”), identify unexpected changes developing in the shape of the WC curve, and examine differences in resistance that develop between charge and discharge. In addition, the implementing of the methodology of this disclosure was responsible for the discovery of charge / discharge resistance asymmetry in certain lithium-metal cells.

[0024] Techniques for extracting state-of-health metrics by examining changes in voltage as a cell degrades has previously been developed for full-cell lithium-ion batteries that contain anodes and cathodes that are both of the intercalating-deintercalating type. There are three possible degradation modes in a full-cell battery cell, namely, loss of lithium inventory, loss of anode capacity, and loss of cathode capacity. However, for a half-cell battery cell (e.g., having lithium- metal anodes or other plating-stripping-type anodes), loss of anode-metal inventory and loss of anode capacity are the same, and the number of degradation modes is reduced by one. This key observation greatly reduces numerical uncertainty during optimization in methods of the present disclosure. Additionally, switching from time-dependent variables to capacity-dependent variables allows for the removal of rate dependence from optimization equations, thereby enabling comparison of cycles performed at different currents.

[0025] In some aspects, the present disclosure is directed to systems, such as cell-testing systems, battery-management systems (BMSs), battery-health monitoring systems, vehicle controllers, etc., that include diagnostics-extraction software-based algorithms that implement any one or more diagnostics-extraction and SOH-reporting methods of the present disclosure. Generally and as discussed below in more detail, a diagnostics-extraction system of the present disclosure comprises suitable hardware, including one or more microprocessors and machine memory that stores machine-executable instructions that encode various aspects of any one or more diagnosticsextraction methods of the present disclosure so that, when the microprocessor(s) execute the machine-executable instructions, the diagnostics-extraction system executes the relevant diagnosticsextraction method(s). As those skilled in the art will readily appreciate, the form of the hardware of a diagnostics-extraction system of the present disclosure will vary depending on its implementation (e.g., in testing-system, BMS, battery-health monitoring system, vehicle controller, or other cellinterface system) and manner of deployment. Those skilled in the art will readily understand how toimplement a diagnostics-extraction system of the present disclosure regardless of the type using only well-known and routine knowledge in the art and this disclosure as a guide. In addition, the term “system” is used for convenience as indicating hardware and / or software that performs the function(s) of the corresponding type of system. Importantly, when two or more systems of the present disclosure are present in a particular implementation or deployment, the term “system” does not denote that the hardware for the multiple systems must be separate and distinct from one another. On the contrary any two or more, or all, of the systems present may share all or fewer than all of the hardware components present for all such systems or they may share none of the entirety of the hardware.

[0026] EXAMPLE EMBODIMENTS

[0027] In some embodiments, a diagnostics- extraction method of the present disclosure involves solving a nonlinear minimization problem to obtain SOHn, the state of health of cycle n, Rn, the increase in resistance of cycle n relative to a reference cycle, and Qon, the initial state-of-charge of cycle n. These parameters are extracted from Vn(Q), the voltage of cycle n as a function of charge capacity Q, by comparison to a reference voltage Vrefafter a mathematical transformation is performed, as expressed by the following optimization problem expressed in the following Equation 1 :

[0028] A graphical interpretation of this minimization problem of the above Equation 1 is presented in FIGS. 1 A and IB for an example charge scenario and in FIGS. 2 A and 2B for an example discharge scenario. As illustrated in FIGS. 1 A and IB, the resistance Rnshifts Vn(Q) vertically relative to the reference, Qonintroduces a horizontal shift, and SOHnrescales the x-axis. In(Q) in Equation 1 , above, is the applied current defined as function of capacity Q, rather than the more typical definition as a function of time. This change of variables from time to charge capacity is an important step that removes rates from Equation 1, above, enabling the comparison of voltage measured at different applied currents.

[0029] An important assumption in the above Equation 1 is that Vn(Q) and Vref(Q) are measured at the same temperature. It is usually the case that OCV is independent of temperature, in which case one OCV measurement would make a suitable reference for all test temperatures. However, if Vref(Q) is measured to be a function of temperature, the temperature of the reference voltage must match the temperature at which Vn(Q) is measured. If temperature varies from cycle to cycle duringimplementation, multiple reference curves at different temperatures would be required. It is generally preferred that these reference curves should be measured on identical cells in a laboratory environment.

[0030] In the context of the foregoing Equation 1, advantages of implementing an optimizationbased diagnostics-extraction methodology of the present disclosure include the following:• An equivalent circuit or other model of the anode or cathode is not required as in alternative methodologies, as the reference cycle serves this purpose.• Vn(Q) can be either current-controlled charge or discharge, or a sequence of charges and discharges (e.g., as generated from driving an EV).• Vn(Q) can be either a complete or partial depth-of-discharge.• Vn(Q) can be updated in real time.• The magnitude of the right-hand-side of Equation 1, above provides information about the goodness of the fit. Thus the algorithm reports how well it is able to fit data. Sudden jumps in error may indicate a problem with the cell.

[0031] As another example, an additional curve- shifting minimization problem can be generated by taking the derivative of the expression in Equation 1, above, as follows:If In(Q) is constant, then / n(Q) = 0, and resistance drops out, reducing the number of parameters in the optimization problem from three to two as seen in the following Equation 3:Taking the reciprocal of the terms in the above Equation 3 generates the dQ / dV minimization problem as seen in the following Equation 4:

[0032] A graphical representation of applying the optimization of Equation 4, above, appears in the example of FIGS. 3 A and 3B, which show how optimized parameters align the dQ / dV peaks.Because this optimization is not sensitive to resistance, Equation 4, above, can be used to generate a good initial guess for SOHnand Qonin Equation 1, above.

[0033] Since there is correlation among the three fitting parameters, the following example diagnostics-extraction method can be used to process cycles sequentially using the optimal parameters from each cycle as the initial guess for optimization at the next cycle:1. Define a reference curve Vref(Q) that is ideally an OCV measurement but can also be a slow- rate cycle of a fresh cell (e.g. the first cycle of the cell under analysis).2. Begin with the first battery cycle (n=l) and define an initial guess for the parameters: (SOH1, R1, Qo1) = (1,0,0). This choice of initial parameters does not shift or scale the reference or test curves. Alternatively, Equation 4, above, can be solved to obtain an initial guess for SOH1and Qo1.3. Extract the current-controlled charge and discharge segments from the complete battery cycle (i.e., remove rest and voltage-controlled segments) to obtain the test voltage Vn(Q).4. Apply nonlinear optimization to obtain the parameters (SOHn, Rn, Qon) that minimize Equation 1, above.5. Set the optimal parameters from step 4 as the initial guess for the next cycle, and return to step 3.

[0034] Illustrative examples of this diagnostics-extraction method are shown in FIGS. 4A through 6B. Standard full depth-of-discharge cycling at 25°C is analyzed in the example of FIGS. 4A and 4B, low temperature shallow depth of discharge is shown in FIGS. 5A and 5B, and high-temperature cycling under the Supplemental Federal Test Procedure light-duty US06 driving protocol is shown in FIGS. 6 A and 6B.

[0035] FIG. 7 illustrates an example diagnostics-extraction system 700 that is in operative communication with an electrochemical cell 704, such as a lithium-metal cell. In this example, the diagnostics-extraction system 700 includes one or more microprocessors 708 for executing software containing algorithms 712 for causing the diagnostics-extraction system to perform any one or more operations, including operations that implement one or more diagnostics-extraction methods of the present disclosure and any reporting of relevant extracted-diagnostics parameter(s) and / or controlling of the operation(s) of the electrochemical cell 704 and / or a cell-interface system 716 (e.g., a cell-testing system, a BMS, a battery-health monitoring system, a vehicle controller, or anyother system that interfaces with the electrochemical cell) based on the relevant extracted- diagnostics parameter(s).

[0036] In some embodiments, the cell-interface system 716 may include one or more output devices 720 as may be needed or desired for a particular type of the cell-interface system 716. For example, when the cell-interface system 716 is a cell-testing system, one output device 720 may be an electronic display, such as a video-type monitor, that displays information about the cell testing to a user, and that information can include diagnostic information generated by the diagnosticsextraction system 700. When the cell-interface system 716 is a BMS, battery-health monitoring system, or a vehicle controller, one output device 720 may likewise be an electronic display. In some embodiments, the displayed information may be a warning that the cell-interface system 716 generates based on the diagnostic information from the diagnostics-extraction system 700. For example, the cell-interface system 716 may implement an algorithm (e.g., part of the algorithms / software 712) that tracks the diagnostic information that the diagnostics-extraction system 700 provides over a number of cycles, and when that algorithm recognizes an abnormal change in one or more diagnostic parameters that composes the diagnostic information, warning software (e.g., part of the algorithms / software 712) may issue a warning that the health and / or safety battery cell 704 is in a state of excessive abnormal decline.

[0037] As another example, the cell-interface system 716 may collect and store diagnostic data from the diagnostics-extraction system 700 and / or perform a meta-analysis of the diagnostic data to forecast remaining useful life of the battery cell 704 or a battery module of which the battery cell is a part. The cell-interface system 716 could perform the forecasting in real-time and as use-conditions change. In an example, the cell-interface system 716 could display the forecast on the output device 720, which may, for example, be an electronic visual display. In some embodiments, the cellinterface system 716 may store diagnostic information for the battery cell 704 from the diagnosticsextraction system 700 in a database alone or in combination with similar diagnostic information from one or more other battery cells (not shown) collected by the diagnostics-extraction system 700 and / or one or more other diagnostics-extraction systems (not shown) for analysis of the stored information. In an example, the stored diagnostic information may be analyzed as part of batterycell research and development to experiment with differing ambient temperatures, charge and discharge protocols, electrolyte chemistries, differing additive chemistries, differing functional layers within the cells’ jellyroll stacks, etc., as the researchers strive to achieve desirable battery-cell designs. Big data analytics can be used upon the stored diagnostic information to extract any ofvarious signals from the stored diagnostic information that may be pertinent to the design process. The extracted signal(s) can be presented to one or more users via an output device, such as an electronic visual display, such as a computer monitor, among others. Those skilled in the art will readily appreciate how to use diagnostic information for any suitable purpose, such as developing a new battery-cell design for commercialization.

[0038] In some implementations, the cell-interface system 716 controls the operation of the battery cell 704 based on diagnostic information generated by the diagnostic-extraction system 700. For example, when the cell-interface system 716 is a cell tester, the cell tester may shut down testing when the diagnostic information indicates that the battery cell 704 has reached a state where it is no longer necessary, no longer safe, etc., to continue operating. The control algorithm(s) for effecting the shut down may be part(s) of the algorithms / software 712. Similar control algorithm(s) can also be part of the cell-interface system 716 when the cell-interface system is of a type other than a cell tester. For example, the shut-down control algorithm(s) for shutting down operation of the battery cell 704 can be implemented when the cell-interface system 716 is a BMS, a battery-health monitoring system, or a vehicle control system, among other types. In addition, as similarly to the warning example noted above in which the cell-interface system 716 tracks the diagnostic information over a number of cycles to spot an abnormal change in one or more diagnostic parameters, a similar tracking and analysis may drive the shut-down control algorithm(s).

[0039] The algorithms 712 are stored in machine memory 724, which singly and collectively represents any one or more hardware memories known and ubiquitous in the computing arts, including any one or more long-term memories and / or short-term memories. According to convention, the machine memory 724 is singly and collectively referred to as a “machine-readable storage medium”, which explicitly excludes information present on a carrier wave (e.g., a digital signal encoded into a carrier wave) or in a series of pulses (e.g., light pulses carrying digital data).

[0040] In this example, interface electronics 728 between the electrochemical cell 704 and hardware of the diagnostics-extraction system 700 are illustrated as being partially internal to the diagnostics-extraction system and partially external to the diagnostics-extraction system. As those skilled in the art will appreciate, the interface electronics 728 includes any and / or all electronics needed for the diagnostics-extraction system 700 to operatively interface with the electrochemical cell 704, any corresponding instrumentation, and any relevant cell-interface system 716, such as a cell-testing system, BMS, battery-health monitoring system, vehicle controller, etc. Those skilled inthe art will readily appreciate the type(s) of interface electronics 728 needed for a particular deployment knowing the relevant instrumentation (not shown) and other system(s), such as the cellinterface system 716 involved in a particular deployment. While the interface electronics 728 are illustrated as being partially internal and partially external to the diagnostics-extraction system 700, those skilled in the art will readily understand that all of the interface electronics may be exclusively internal to the diagnostics-extraction system or exclusively external to the diagnostics-extraction system, depending on the design of the diagnostics-extraction system. Similarly, it is noted that the diagnostics-extraction system 700 is optionally illustrated as being contained in the cell-interface system 716. However, those skilled in the art will readily appreciate that the diagnostics-extraction system 700 may be implemented partially or completely externally to the cell-interface system 716.

[0041] Examples of the algorithms 712 encoded into the software include but are not limited to algorithms that encode one or more optimization equations of the present disclosure, such as, but not limited to, the optimization Equations 1 and / or 4, above, algorithms that encode steps of implementing the one or more optimization equations to extract one or more diagnostics parameters, algorithms for utilizing and / or reporting the one or more extracted diagnostics parameters, and algorithms for interacting with another system, such as, but not limited to, the illustrated cellinterface system 716, among other algorithms. Those skilled in the art will readily understand the algorithms 712 needed for any particular deployment of a diagnostics-extraction system 700 of the present disclosure using the present disclosure as a guide to the functionalities required for that deployment.

[0042] Various modifications and additions can be made without departing from the spirit and scope of this disclosure. Features of each of the various embodiments described above may be combined with features of other described embodiments as appropriate in order to provide a multiplicity of feature combinations in associated new embodiments. Furthermore, while the foregoing describes a number of separate embodiments, what has been described herein is merely illustrative of the application of the principles of the present invention. Additionally, although particular methods herein may be illustrated and / or described as being performed in a specific order, the ordering is highly variable within ordinary skill to achieve aspects of the present disclosure. Accordingly, this description is meant to be taken only by way of example, and not to otherwise limit the scope of this invention.

[0043] Exemplary embodiments have been disclosed above and illustrated in the accompanying drawings. It will be understood by those skilled in the art that various changes, omissions and additions may be made to that which is specifically disclosed herein without departing from the spirit and scope of the present invention.

Claims

What is claimed is:

1. A machine-implemented method of operating an electrochemical cell having anodes of a platingstripping type, wherein the electrochemical cell has associated voltage-based reference data, the machine-implemented method comprising: receiving, at a cycle, n, first voltage-based cycle data for the electrochemical cell; solving an optimization problem that maximizes fit between the voltage-based reference data and the first voltage-based cycle data so as to obtain estimated diagnostic information for the electrochemical cell at cycle n; and providing the estimated diagnostic information to a cell-interface system designed and configured to use the diagnostic information to perform one or more tasks relating to the electrochemical cell.

2. The machine-implemented method of claim 1 , wherein the optimization problem includes an optimization equation, states as follows:wherein:S()Hnis a state of charge of the electrochemical cell in the cycle n;Rnis an increase in internal electrical resistance of the electrochemical cell in the cycle n relative to a reference cycle;Qonis an initial state of charge of the electrochemical cell in the cycle n;Q is a charge capacity; k" is a voltage of the electrochemical cell in the cycle n as a function of the charge capacity; r(Q) is an applied current of the electrochemical cell in the cycle n as a function of the charge capacity;Veis a reference voltage.

3. The machine-implemented method of claim 2, further comprising obtaining the voltage-based reference data via open-circuit voltage measurement.

4. The machine-implemented method of claim 3, wherein the open-circuit voltage measurement is performed on a reference cell that is identical to the electrochemical cell.

5. The machine-implemented method of claim 2, further comprising obtaining the voltage-based reference data via a slow-rate cycle of the electrochemical cell when it is fresh.

6. The machine-implemented method of any one of claims 2-5, wherein solving the optimization problem includes applying nonlinear optimization to the optimization equation so as to obtain estimated values for S()H' Rn, and Qon.

7. The machine-implemented method of claim 2, wherein applying nonlinear optimization to the optimization equation includes providing initial values for SOH' R", and Qon.

8. The machine-implemented method of claim 7, wherein providing initial values includes providing initial guesses.

9. The machine-implemented method of claim 7, wherein providing initial values includes solving the following equation for the initial values:

10. The machine-implemented method of claim 1, further comprising: receiving, at a cycle, n+1, second voltage-based cycle data for the electrochemical cell; setting initial values for SOHn+I, Rn+1, and Qon+Ifor cycle n+1 in the optimization equation to be values determined for SOH' R", and Qonfor the cycle n; and applying nonlinear optimization to the optimization equation so as to determine, for the cycle n+1, estimated values of S()Hn !, Rn+J, and Qon+I.

11. The machine-implemented method of claim 1 , wherein the optimization problem includes an optimization equation stated as follows:wherein:SOH" is a state of charge of the electrochemical cell in the cycle n;Rnis an increase in internal electrical resistance of the electrochemical cell in the cycle n relative to a reference cycle;Qonis an initial state of charge of the electrochemical cell in the cycle n;Q is a charge capacity;V" is a voltage of the electrochemical cell in the cycle n as a function of the charge capacity;P(Q) is an applied current of the electrochemical cell in the cycle n as a function of the charge capacity;Veis a reference voltage.

12. The machine-implemented method of claim 1, wherein the estimated diagnostic information comprises state-of-health information for the electrochemical cell at the cycle n.

13. The machine-implemented method of claim 1, wherein the estimated diagnostic information comprises internal-electrical-resistance information for the electrochemical cell at the cycle n.

14. The machine-implemented method of claim 1, wherein the estimated diagnostic information comprises an initial-state-of-charge information for the electrochemical cell at the cycle n.

15. The machine-implemented method of claim 1, wherein the estimated diagnostic information comprises state-of-health information, and internal-electrical-resistance information, and an initial-state-of-charge information, each for the electrochemical cell and at the cycle n.

16. The machine-implemented method of any one of claims 1-15, wherein the cell-interface system comprises a battery management system, and one of the one or more tasks is to cause an electronic display to display the diagnostic information to a user.

17. The machine-implemented method of any one of claims 1-15, wherein the cell-interface system comprises a battery management system, and one of the one or more tasks includes causing an electronic display to display a warning to a user.

18. The machine-implemented method of any one of claims 1-15, wherein the cell-interface system comprises a battery management system, and one of the one or more tasks includes changing an operating state of the electrochemical cell.

19. The machine-implemented method of any one of claims 1-15, further comprising performing the method over a plurality of cycles, wherein the cell-interface system comprises a cell-testing system, and one of the one or more tasks includes causing an electronic display to generate a graph displaying the diagnostic information over the plurality of cycles.

20. A machine-readable storage medium containing machine-executable instructions for performing the machine-implemented method of any one of claims 1-19.

21. A system for operating an electrochemical cell, the system comprising: interface electronics for electrically interfacing with the electrochemical cell when the electrochemical cell is present; a cell-interface system that interfaces with the electrochemical cell via the interface electronics when the electrochemical cell is present; machine memory containing machine-executable instructions for performing the machine- implemented method of any one of claims 1-19; and at least one processor in operative communication with the machine memory and the interface electronics, wherein the at least one processor is configured to execute the machineexecutable instructions.

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