WO2022171993A1

WO2022171993A1 - Power unit analysis

Info

Publication number: WO2022171993A1
Application number: PCT/GB2022/050327
Authority: WO
Inventors: Richard Stocker; Michele BRAGLIA
Original assignee: Horiba Mira Limited
Priority date: 2021-02-09
Filing date: 2022-02-08
Publication date: 2022-08-18
Also published as: GB2603894A; GB202101761D0

Abstract

A method of isolating and quantifying temperature-dependent and temperature-independent contributions to ohmic resistance in a power unit, comprising: obtaining an impedance spectrum from the power unit at a plurality of measured temperatures; determining a total ohmic resistance of the power unit from the impedance spectrum at each temperature; applying a multi-objective fitting process to quantify the temperature-dependent and temperature-independent contributions to the total ohmic resistance wherein an objective is to minimise the disparity between an Arrhenius slope derived at least partially from the impedance spectra and an Arrhenius slope derived solely from modelling; and outputting the quantified temperature-dependent and temperature-independent contributions as ohmic resistance parameters; wherein the Arrhenius slopes are based on the equation: (I) where R_ohmic is a total ohmic resistance, R₀is a variable parameter, R₁ is a variable parameter, E_A is a variable parameter and RT is the gas constant multiplied by temperature.

Description

Title: Power unit analysis

Description of Invention

Embodiments of the present invention relate to a method of isolating and quantifying temperature-dependent and temperature-independent contributions to ohmic resistance in a power unit, a method of isolating and quantifying individual contributions to dynamic resistance in a power unit, and a method of isolating and quantifying individual contributions to diffusion impedance in a power unit, along with associated systems, and computer-readable media.

Batteries are used in a wide variety of applications, including electric vehicles and consumer electronics. Electric vehicles have particularly demanding battery requirements due to the large operating temperature range, emphasis on energy density and requirement for long battery lifetimes. Long-life applications pose particular design challenges due to the complex ageing processes of the batteries.

A battery is typically formed from one or more cells (i.e. electrochemical cells). As such, references herein to a battery may be a reference to one or more cells. Likewise, a reference to a cell may be a reference to a battery or a part of a battery.

Two key aspects of cells which evolve over time are their ability to store charge and their ability to transfer charge. The ability of a cell to transfer charge is linked to the impedance characteristics of the cell. Impedance characteristics are particularly complex, resulting from a combination of various contributing factors, all of which may evolve distinctly overtime.

Three broad contributions to cell impedance in cells, such as those of a lithium ion chemistry (Li-ion cells) are ohmic resistance, dynamic resistance (e.g. charge transfer resistance) and diffusion impedance. Ohmic contributions are attributed, for example, to contact resistances and imperfect electrolyte conductivity. Charge transfer resistance arises, for example, from the transfer of lithium ions in and out of the electrodes and across surface layers. Diffusion impedance is related, for example, to the distribution of lithium ions throughout an electrode and resulting concentration gradients.

Each impedance contribution can be broadly identified by a characteristic time constant which signifies the length of time taken for impedance to evolve following an application of current. Ohmic effects are purely resistive, and act approximately instantaneously. Charge transfer resistance is not instantaneous, with time constants typically in the range 10^-3 to 10¹ seconds. Diffusion effects are slower still, with time constants typically in the range 10° to 10⁴ seconds. Similar impedance contributions are also found in fuel cells, flow batteries including vanadium redox flow batteries (VRFB) and batteries of other battery chemistries (i.e. other than lithium ion) in general. Cells, batteries, fuel cells and/or flow cells/batteries may generally be referred to as power units.

While it is known that impedance is a sum of individual impedance contributions, it is not possible to measure directly a single individual impedance contribution. Instead, analysis techniques such as electrochemical impedance spectroscopy show only the cumulative effect of all impedance contributions. This means it is difficult to assess the health of individual components within a power unit. The health of a power unit can be used to determine whether that power unit is suitable for (or even safe for) a particular application, or whether the manner in which it is used (e.g. charged and discharged) needs to be altered (e.g. to improve the service life of the power unit).

The present invention seeks to alleviate one or more of the problems associated with the prior art.

Accordingly, an aspect of the present invention provides a method of isolating and quantifying temperature- dependent and temperature-independent contributions to ohmic resistance in a power unit, comprising: obtaining an impedance spectrum from the power unit at a plurality of measured temperatures; determining a total ohmic resistance of the power unit from the impedance spectrum at each temperature; applying a multiobjective fitting process to quantify the temperature-dependent and temperature-independent contributions to the total ohmic resistance wherein an objective is to minimise the disparity between an Arrhenius slope derived at least partially from the impedance spectra and an Arrhenius slope derived solely from modelling; and outputting the quantified temperature-dependent and temperature-independent contributions as ohmic resistance parameters; wherein the Arrhenius slopes are based on the equation:

where R_ohmic is a total ohmic resistance, R₀ is a variable parameter, R₁ is a variable parameter, E_A is a variable parameter and RT is the gas constant multiplied by temperature.

The impedance spectra may comprise electrochemical impedance spectroscopy data.

A method may further comprise displaying the ohmic resistance parameters.

A method may further comprise comparing the ohmic resistance parameters to known values. For example, the known values may be baseline ohmic resistance parameters of the power unit undergoing analysis and/or baseline ohmic resistance parameters of a power unit of the same structure as the power unit undergoing analysis. Changes in ohmic resistance parameters overtime may be determined by comparison of the ohmic resistance parameters with known values. The Arrhenius slope derived at least partially from the impedance spectra may be obtained by plotting I {R_data - Ro against i where R_data is the measured total ohmic resistance at a measured temperature, R₀ is a variable parameter and T is the measured temperature.

The Arrhenius slope derived solely from modelling may be obtained by plotting the output of IniR^ + (^) against i for a plurality of temperatures.

An objective may be to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance.

The model total ohmic resistance, R_fit, may be calculated according to the equation R_fit = R₀ + l!_jeW

The multi-objective fitting process may be a dual-objective fitting process, a first objective being to minimise the disparity between an Arrhenius slope derived at least partially from the impedance spectra and an Arrhenius slope derived solely from modelling and a second objective being to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance.

The optimisation of the fitting process may be carried out using a combination of global optimisation and local optimisation.

The global optimisation may be performed using particle swarm optimisation and/or genetic algorithm optimisation.

The initial conditions for the multi-objective fitting process may be determined by a combination of a known value and an initial single-objective fit.

The objective of the initial single-objective fit may be to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance R_fit at a given temperature using the equation R_fit =

The ohmic resistance parameters may comprise at least one of the parameters R₀ R_{l t} and/or E_A.

Another aspect provides a computer-readable medium storing instructions which, when executed by a processor, cause the performance of the method as above.

Another aspect provides a method of isolating and quantifying individual contributions to dynamic resistance in a power unit, comprising: obtaining an impedance spectrum from the power unit; removing diffusion impedance effects from the impedance spectrum; converting the impedance spectrum into a distribution of relaxation times; isolating individual peaks in the distribution of relaxation times; reconstructing an impedance spectrum for each peak; fitting equivalent circuit elements to the reconstructed impedance spectra to quantify the individual resistance contributions to the dynamic resistance of the power unit; and outputting the quantified individual resistance contributions as dynamic resistance parameters.

The impedance spectrum may be obtained from the power unit by electrochemical impedance spectroscopy and/or by conversion of pulse relaxation data, obtained following application of a current to the power unit, into an impedance spectrum.

The pulse relaxation data may be converted into an impedance spectrum by fitting frequency domain elements to the pulse relaxation data using Laplace inversion, and constructing an impedance spectrum from the frequency domain elements.

The pulse relaxation data may be obtained by a galvanostatic intermittent titration technique.

The pulse relaxation data may be obtained by hybrid pulse power characterisation.

Individual peaks in the distribution of relaxation times may be isolated by fitting statistical distributions to the distribution of relaxation times.

The statistical distributions may include log-normal or Gaussian distributions.

A method may further comprise displaying the dynamic resistance parameters.

A method may further comprise comparing the dynamic resistance parameters to known values. For example, the known values may be baseline dynamic resistance parameters of the power unit undergoing analysis and/or baseline dynamic resistance parameters of a power unit of the same structure as the power unit undergoing analysis. Changes in dynamic resistance parameters over time may be determined by comparison of the dynamic resistance parameters with known values.

A fitting based on a pure capacitance term may be used to remove capacitance effects from the impedance spectrum.

A fitting based on at least one of a finite length Warburg term and/or a finite space Warburg term may be used to remove the diffusion impedance effects from the impedance spectrum.

The equivalent circuit elements may be RQ elements in which RQ represents a resistor and constant phase element in parallel. Equivalent circuit elements may be fitted based on the equation Z(w) = Z(w ) is the impedance as a function of angular frequency, R is a resistor representing dynamic resistance, t is a characteristic time constant, i is the imaginary number, w is the angular frequency and a is the exponent value.

A method may further comprise: creating a simulated power unit impedance spectrum using the fitted equivalent circuit elements; and optimising the values of the fitted equivalent circuit elements to minimise the error between the simulated power unit impedance spectrum and the impedance spectrum obtained from the power unit.

The optimisation may include the fitting of an inductor L or resistor inductor parallel circuit element RL.

The optimisation may include the fitting of an inductor based on the equation CKw) = ίwZ, where CZ,(w) is the contribution to the impedance spectrum, i is the imaginary number, w is the angular frequency and L is the variable inductance value.

A method may further include any combination of the above methods.

Another aspect provides a method of isolating and quantifying individual contributions to diffusion impedance in a power unit, comprising: obtaining an impedance spectrum from the power unit; fitting a pure capacitance term to remove capacitance effects from the impedance spectrum; fitting at least one of a finite length Warburg term and/or a finite space Warburg term to quantify the diffusion impedance contributions from the impedance spectrum; and outputting the quantified diffusion impedance contributions as diffusion impedance parameters.

The pulse relaxation data may be obtained by a galvanostatic intermittent titration technique. The pulse relaxation data may be obtained by hybrid pulse power characterisation.

The impedance spectrum may be obtained from the power unit by electrochemical impedance spectroscopy, wherein the pure capacitance term may be created using state of charge-open circuit voltage maps by applying coulomb counting to the electrochemical impedance spectroscopy profile.

The impedance spectrum may be converted into a distribution of relaxation times and the diffusion impedance parameters obtained from fitting to the impedance spectrum may be refined by fitting to the distribution of relaxation times.

The diffusion impedance parameters may be fitted to a predetermined range of the distribution of relaxation times where diffusion effects are known to be found based on the Nyquist plot profile of the impedance spectrum.

A method may further include any combination of the above methods.

Another aspect provides a system comprising: an electrochemical impedance spectroscopy (EIS) instrument for obtaining EIS data from a power unit, and/or a galvanostatic intermittent titration technique (GITT) instrument for obtaining GITT data from a power unit and/or a hybrid pulse power characterisation (HPPC) instrument for obtaining HPPC data from a power unit; and a processor programmed to perform any of the above methods.

The system may further comprise a display for displaying the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters.

Another aspect provides an electric vehicle comprising a system as above, wherein the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters are input as battery health data to a battery management system which manages a battery according to the battery health data.

The battery management system may be configured to output a notification to a user when the battery health data indicates the battery health has degraded below a predetermined threshold.

Another aspect provides a computer-implemented method of comparing power units comprising: obtaining ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters for a plurality of power units according to any of the above methods; inputting the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters into a power unit degradation model; outputting the results of the degradation model; displaying the results of the degradation model; and ranking the power units by their expected degradation characteristics.

The method may further comprise the step of selecting the highest-ranked power unit for use in a vehicle.

By way of example only, the present invention will be described with reference to the accompanying figures, in which:

Figure 1 shows an illustration of the diffusion fitting process;

Figure 2 shows an illustration of the individual component identification process;

Figure 3 shows an illustration of EIS and DRT distributions;

Figure 4 shows parameter results grouped by resistance, time constant and a;

Figure 5 shows NMR anode analysis for new and aged cells;

Figure 6 shows SEM images of anode and cathode samples for new and aged cells;

Figure 7 shows DRT of cathode charge transfer peaks and associated SEM images;

Figure 8 shows ohmic resistance with voltage and temperature for a new cell;

Figure 9 shows the Arrhenius trend in real data for three cells;

Figure 10 shows optimisation results for total ohmic resistance and Arrhenius slope of temperature- dependent resistance;

Figure 11 shows a comparison of parameters Ro, Ri and EA for new and aged cells;

Figure 12 shows SEM cross-sections of the anode and cathode of an aged cell;

Figure 13 shows a vehicle according to some aspects of the technology; and Figure 14 shows a system according to some aspects of the technology.

Aspects of the present disclosure include methods for isolating and quantifying individual contributions to power unit impedance within the dynamic (e.g. charge transfer), diffusion and ohmic resistance categories. As used herein, a “power unit” may be a reference to a battery, a cell, a fuel cell, or a flow battery, for example.

Such analysis is advantageous as it allows, for example, more accurate and detailed power unit modelling, and/or more accurate and insightful power unit health checks, and/or the ability to identify any unexpected or rogue components or characteristics which may pose a safety concern. Quantifying the changing impedance characteristics such as resistance magnitude and time constant of individual contributions (e.g. ohmic, charge transfer) and individual components within the cell (e.g. anode, cathode, current collector, electrolyte) also allows for much more sophisticated modelling and control approaches, and the ability to alter these approaches representatively with ageing. Aspects of the technology are described, in general, with reference to a Li-ion cell as the power unit. However, the described and/or claimed technology is equally applicable to other battery chemistries (including, without limitation, a lithium sulphur (Li-S), a sodium-ion, a solid-state, and/or a sodium-sulphur cell). Aspects of the technology are also suitable for use with a fuel cell, such as a hydrogen fuel cell, or flow cell, such as a Vanadium Redox Flow cell, as the power unit.

Reference to an “impedance” may include a “resistance” and as such any reference to a resistance may be considered to refer equally to an impedance component.

The impedance contributions of any particular power unit are dependent on the chemistry and structure of the power unit. Each broad impedance contribution can be further split into individual contributions resulting from the various cell components. In the case of charge transfer, contributions may arise from the cathode, anode, and their respective surface layers. Contributions to the ohmic resistance can be further split into contact resistances and the inherent electrolyte resistance.

For example, charge transfer contributions arising from the intercalation of lithium ions are specific to Li-ion cells and batteries. Likewise, contributions arising from sodium ions are specific to sodium-ion cells and batteries. Although the physical characteristics to be analysed may change depending on the specific power unit under analysis, the technology presented herein can be applied to any particular power unit to measure the characteristic contributions of such a power unit. Charge transfer resistance is an example of dynamic resistance. In the case of power units other than batteries, for example fuel cells and flow cells, different terminology may be used for the resistance component corresponding to charge transfer resistance. Therefore, the term “dynamic resistance” may be used as a general term to refer to these resistance contributions. In the particular examples described herein, charge transfer resistance is used as an example of dynamic resistance. The methods described in relation to charge transfer resistance are applicable to dynamic resistance in general.

In a version of a method for isolating and quantifying individual contributions to dynamic resistance, characteristic power unit data is obtained from a power unit. The dynamic resistance may be charge transfer resistance. The characteristic power unit data may comprise an impedance spectrum or impedance spectra. For example, Electrochemical Impedance Spectroscopy (EIS) may be performed on a sample power unit to obtain characteristic power unit data which may be EIS data (i.e. the spectrum represented by EIS output data). The EIS data is characteristic of the sample power unit, an example of which is given in figure 1a and discussed herein. In particular, figure 1a shows a plot of the real (Z’) and imaginary (Z”) parts of the power unit impedance in the “Raw” plot line, as indicated in the key in figure 1a. In other versions, time domain data may be obtained from a power unit, such as by a Galvanostatic Intermittent Titration Technique (GITT) and/or hybrid pulse power characterisation (HPPC), and Laplace inversion may be used to perform the method as described herein with respect to EIS data (see below for detail on Laplace inversion). An impedance spectrum may, therefore, be generated in a number of ways. For example, an impedance spectrum may be generated by EIS. An impedance spectrum may be generated from equivalent circuit elements, and/or may be generated through conversion of time domain data (e.g. using Laplace inversion).

Individual contributions to the impedance cannot all be identified directly from the characteristic power unit data, which is only able to show the cumulative changes in impedance of the power unit. However, in some aspects of the technology, categories of impedance contributions, such as dynamic resistance (e.g. charge transfer resistance) and/or diffusion effects, are identified from the characteristic power unit data using analytical methods described herein.

For example, in some aspects of the technology, diffusion effects are identified (e.g. in a first analysis stage in relation to the characteristic power unit data). This may enable the diffusion effects to be separated from dynamic (e.g. charge transfer) effects for the power unit. This may be achieved computationally by fitting diffusion features directly to the characteristic power unit data.

For example, a number of terms may be incorporated into the diffusion feature fitting process, including pure capacitance terms, Finite Length Warburg (FLW) terms, and/or Finite Space Warburg (FSW) terms. Any number or combination of the aforementioned terms, or subset of such, may be used to perform the fitting. Additional terms may also be added if desired. In some aspects of the technology, however, a capacitance term, a FLW term, and a FSW term may be used - figure 1a shows these contributing terms to the fit which is then achieved (see the key in figure 1a).

In some aspects of the technology, in the process of fitting diffusion features to the characteristic power unit data, a pure capacitance term may be used to remove capacitance effects from the characteristic power unit data. The capacitance term may represent open circuit voltage changes corresponding to state of charge fluctuations resulting from the EIS. This may be achieved by applying coulomb counting to the EIS profile and using state of charge-open circuit voltage (SoC-OCV) maps generated from power unit characterisation data (e.g. from documentation associated with the power unit or from earlier mapping results according to known methods, the generation of which may be part of some aspects of the technology). The resulting capacitance term is therefore based on the actual physical characteristics of the power unit.

In some aspects of the technology, the diffusion fitting obtained from the characteristic power unit data can be improved (i.e. refined) using a Distribution of Relaxation Times (DRT) technique. DRT converts the impedance spectrum (or spectra) - which, as discussed, may be obtained by EIS and/or by other means such as conversion of time domain data and/or may be generated from equivalent circuit elements - into a distribution of magnitude against characteristic time constant which can be used to isolate individual impedance features and their characteristics. An exemplary diffusion fitting to DRT data is provided in figure In some aspects of the technology, therefore, a refinement operation uses the diffusion fitting obtained from the characteristic power unit data directly (see above) and performs a second fitting to DRT data (i.e. spectrum) generated from use of the DRT technique on the characteristic power unit data. This second fitting may be applied to the region of the DRT data in which diffusion effects are known to be found - i.e. a predetermined range of the DRT data for the power unit. For example, the second fitting may be applied to the DRT data from a time constant (TC) of 10° to 10⁴ s. The relevant time constant region may vary depending on power unit chemistry, for example, and so the predetermined part of the DRT data may be dependent on power unit type and/or chemistry and/or structure. This range may be defined based on the Nyquist plot profile of the impedance spectrum, and/or based on knowledge of the power unit behaviour and operational temperature. For example, the range may be defined based on the Nyquist plot profile of the impedance spectrum by finding a local minimum. Visually, this local minimum may be found at the point at which the semi-circular portion of the Nyquist plot overlaps with the tail portion of the plot (see figure 1a for example). Once the local minimum has been identified, the frequency point corresponding to said minimum can be obtained and used to define an endpoint of the range.

The conversion of the characteristic power unit data using the DRT technique may aid in identification of individual peaks in the characteristic power unit data. Statistical distributions may be fitted to the DRT data in order to isolate individual peaks, according to known methods. The statistical distributions may include lognormal and/or Gaussian distributions, for example. Peak isolation is illustrated in figure 2a.

Once the individual features of the DRT data have been isolated, an impedance spectrum (e.g. an EIS data) may be reconstructed for each individual contribution to the fit. These reconstructed spectra may have a characteristic semi-circular shape, as illustrated in figure 2b.

Following impedance spectrum reconstruction, equivalent circuit elements may be fitted to the reconstructed spectra.

A number of different equivalent circuit elements may be used in the fitting, but RQ elements have been shown to provide a good fit, as illustrated in figure 2c for example.

By fitting equivalent circuit elements to the reconstructed spectra, the resistance of each individual element may be quantified. In such a manner, the individual contributions to dynamic resistance (e.g. charge transfer resistance) within a power unit may be quantified.

In some aspects of the technology, the equivalent circuit (e.g. RQ) elements and the diffusion elements may be combined to produce a simulated impedance spectrum, and a final optimisation may be performed to minimise the error (i.e. difference) between the simulated and observed impedance spectra. The simulated parameters may be allowed to vary within a small tolerance (e.g. ±10%) to minimise the error. In some aspects of the technology this optimisation may include the fitting of an inductor circuit element L or resistor inductor parallel circuit element RL to account for inductance in the unit under test, or in the equipment performing the cycling. This may include the fitting of an inductance element CKw ) = ίwZ, where CKw) is the contribution to the impedance spectrum, i is the imaginary number, w is the angular frequency and L is the variable inductance value which will be optimised through error minimisation to the spectrum. In some aspects of the technology root-mean square error (RMSE) or mean absolute error (MAE) may be used for this optimisation.

In some aspects of the technology, physical testing of the power unit may be undertaken. Such physical testing may include, for example, NMR, XRD, XPS, Raman Spectroscopy or SEM analysis. Physical testing of the power unit may aid in determining, or validating the determination of, one or more physical causes of the quantified impedance contributions generated according to some aspects of the technology.

As will be appreciated, therefore, in aspects of the technology which include physical testing, once the physical causes of each observed impedance contribution have been identified for a particular power unit type, this information may be stored as a model of known impedance contributions and associated physical causes. This information may then be used, in some aspects of the technology, so that subsequently analysed power units of the same type can be analysed (and physical characteristics determined) without the need to perform additional physical testing on the power unit under analysis. This may be advantageous as the physical testing is typically destructive and so renders the power unit inoperable.

In some aspects of the technology, there is a method of isolating and quantifying ohmic resistance contributions in a power unit. In a version of such a method for isolating and quantifying individual contributions to ohmic resistance, impedance spectra are obtained from a power unit. For example, Electrochemical Impedance Spectroscopy (EIS) may be performed on a power unit to generate an EIS data. The total ohmic resistance of the power unit may be identified from the impedance spectra (e.g. EIS data) by analysis of the real resistance when the imaginary resistance is zero. In some cases, there may be multiple values of the real resistance at which the imaginary resistance is zero, in which case the ohmic resistance corresponds to the value at which the imaginary resistance is first zero (i.e. the lowest value of the real resistance at which the imaginary resistance is zero). However, the individual contributions to the total ohmic resistance cannot be identified directly from the impedance spectra.

In accordance with some aspects of the technology, a method is provided for isolating and quantifying the temperature-dependent and temperature-independent contributions to the total ohmic resistance of the power unit.

The contributions to the total ohmic resistance may be separated by measuring the total ohmic resistance at a variety of temperatures of the power unit. Such analysis may be advantageous as it allows for more accurate power unit modelling, and/or more accurate and insightful power unit health checks, and/or the ability to identify any unexpected or rogue characteristics which may pose a safety concern. It may also allow for a more sophisticated root-cause evaluation of ohmic resistance changes, which may enable adaptation of control parameters (such as limits set by the battery management system) to protect against further ageing, or can be used to predict further ageing patterns.

Accordingly, in some aspects of the technology, the ohmic resistance of the power unit may be expressed as a sum of a temperature-independent term and a temperature-dependent term. These terms may represent contact resistances and electrolyte resistance, depending on power unit structure.

The ohmic resistance of the power unit may be expressed as a sum of the contact resistance and the electrolyte resistance, in which the contact resistance is represented by a constant term R₀ and the electrolyte resistance is expressed using a temperature-dependent Arrhenius-type term R^RT¹.

In some aspects of the technology, there is provided a method of separating and quantifying R₀ and R^RT^. As will be appreciated, the notation used to represent the various parameters, for example R₀ R_l and E_A is arbitrary, and any desired notation can be used to represent the same terms. Furthermore, the parameters may be defined as positive or negative values. For example, the term R^RT¹ could be expressed equivalently as R^ RT > depending on whether E_A is defined as a positive or a negative quantity. In some applications, E_A may be a positive quantity; in other applications E_A may be a negative quantity. The parameter E_A may represent an activation energy in some applications of the technology described herein; however, in some versions, E_A may not represent an activation energy. The notation used herein is intended to aid understanding by using familiar terms (such as E_A) and should not be construed as limiting the disclosure in any way. In other words, the methods described herein are not limited by the notation chosen to represent the various parameters.

An objective of some aspects of the technology may, therefore, be to provide a fitting process to quantify R₀ R₁ and E_A ensuring that the results of such fitting are representative of the actual physical characteristics of the power unit.

A multi-objective optimisation approach may be used to provide accurate data fitting. A first objective may minimise a disparity between measured or observed total ohmic resistance and modelled or fitted total ohmic resistance based on the equation:

In some aspects of the technology, root-mean-square error (RMSE) or root-mean-square deviation is used for measuring a difference between observed values and modelled values; however, alternative measures such as mean absolute error (MAE) may be used. The parameters allowed to be varied may be any or all of R_1: R₀, and E_A. A good fit may be achieved by allowing all three parameters to be varied in some instances.

A single objective optimisation has been found not to guarantee a physically significant separation of the temperature dependent and temperature independent contributions. Therefore, a second objective may also be used. The second objective may minimise a disparity between an Arrhenius slope derived from the measured or observed data and a model or fitted Arrhenius slope. Again, RMSE is a suitable measure, but alternatives such as MAE may be used instead. Additionally, the objectives may be weighted according to the importance of their agreement with the measured data. For example, the first objective may be accorded a larger weight than the second objective, or vice versa.

A conventional Arrhenius plot of ln{k) against

shows a linear relationship or linear “Arrhenius slope”. Figure 9 illustrates that a plot of ln(R ) against

for three sample cells is almost linear, but is not linear due to the existence of multiple resistance contributions, of which some are temperature-independent. Therefore, equation (1) may be rearranged to obtain a linear relationship. Equation (1) can be expressed as:

which enables a “conventional” Arrhenius plot to be created by taking the natural logarithm of both sides:

Herein, ln(R_data - R₀) is denoted Gr_{Arr Data} and IniR^ + (^) is denoted Gr_{Arr Eq}. Any alternative notation may be adopted.

The second objective may minimise the disparity between Gr_{Arr Data} and Gr_{Arr Eq}. As with the first objective, any of the parameters R₀ /?! and E_A may be varied. In some aspects of the technology, all three parameters may be allowed to vary in the fitting process. Flowever, certain parameters, for example E_A may be fixed in some aspects of the technology, for example where E_A (or any other parameter) is known.

A model Arrhenius slope may be obtained by plotting the output of IniR^ + (^) against ^ for a plurality of temperatures. An Arrhenius slope derived at least partially from the impedance spectra may be obtained by plotting ln(R_data - R₀) against^ for the measured temperatures.

Optimisation may be achieved using a combination of a global optimiser and a local optimiser, for example by using particle swarm optimisation and an optimiser to find a minimum or minima of a constrained nonlinear multivariable function, such as particleswarm() and fmincon() in Matlab 2019a. Known values or an initial single-objective fit, or a combination of both may be used to provide initial conditions for a dualobjective fit. The optimisation may be tuned according to criteria, for example according to the available computational capabilities or time constraints. Other forms of algorithm optimisation may alternatively be used. For example, genetic algorithm optimisation may be used alternatively or in addition to particle swarm optimisation. The optional use of genetic algorithm optimisation as an alternative to, or in addition to, particle swarm optimisation applies to all uses of particle swarm optimisation described herein.

In some aspects of the technology, the quantified impedance (or resistance) contributions may be compared to known impedance contributions for the same type of power unit wherein the known impedance contributions are associated with known physical causes (e.g. as determined by the earlier testing and analysis of other examples of the same type of power unit, which may be testing and analysis as described herein). Accordingly, one or more physical characteristics of the power unit may be determined based on the quantified impedance contributions.

In some aspects of the technology, the quantified impedance (or resistance) contributions may be compared to known impedance contributions for the same type of power unit to determine a level of degradation of the power unit. For example, the quantified impedance contributions may be compared to baseline impedance contributions for the power unit to determine a level of degradation of the power unit. The baseline impedance contributions may represent 0% degradation (or equivalently, 100% health). The level of degradation may be displayed in a suitable form, such as a percentage or absolute value.

In some aspects of the technology, the quantified impedance (or resistance) contributions may be referred to as “power unit health data”. In some aspects of the technology, the quantified impedance (or resistance) contributions may be referred to as “ohmic resistance parameters”, “dynamic resistance parameters”, and “diffusion impedance parameters”. The dynamic resistance parameters may be charge transfer resistance parameters.

In some aspects of the technology, time domain testing may be performed on a power unit to produce time domain data in addition to, or as an alternative to, EIS. Time domain testing may include a galvanostatic intermittent titration technique (GITT) and/or hybrid pulse power characterisation (FIPPC) and/or alternative techniques. The time domain data may comprise voltage-time data such as voltage-time curves. Any method described herein with reference to EIS, or EIS data, may equally be performed using time domain data provided a suitable method is used to render the time domain data compatible with the methods as described for EIS.

In some aspects of the technology, Laplace inversion (alternatively known as inverse Laplace transform) may be used to fit frequency domain elements to time domain data. This enables the frequency domain elements to be treated in the same manner as described for EIS data (which is an example of an impedance spectrum). Therefore, it is possible to perform the methods described with reference to EIS data upon time domain data with Laplace inversion. The Laplace inversion may use Euler’s or Talbot’s methods, for example.

In some aspects of the technology, frequency domain elements such as RQ elements or finite length Warburg elements may be converted into time domain elements through Laplace inversion. The Laplace inversion may be combined with parameter optimisation to minimise error, through RMSE or MAE optimisation for example, to the time domain data (e.g. voltage curve). The resultant frequency domain elements may be used to build an impedance spectrum which can be treated as if it originated from a frequency domain method such as EIS.

In general, Laplace inversion may be used with pulse relaxation data obtained following application of a current to the power unit. In other words, the relaxation data may be data obtained when a load is removed from the power unit, allowing the power unit to relax with no current applied for a period of time. The relaxation data may comprise voltage-time data (e.g. power unit voltage may be measured for a period of time at zero current after the power unit has been subjected to a non-zero current for a period of time). A galvanostatic intermittent titration technique may be used to obtain such relaxation data. For example, hybrid pulse power characterisation may be used to obtain the relaxation data. Laplace inversion may be used with the relaxation data to convert the relaxation periods into elements (e.g. frequency domain elements) which may be used to generate an impedance spectrum. Accordingly, an impedance spectrum may be generated from relaxation data using Laplace inversion. Such an impedance spectrum may be analysed in the same way as an impedance spectrum generated by EIS, for example. The methods described herein are compatible, therefore, with frequency domain data such as EIS data and also with time domain data such as HPPC data.

For illustrative purposes, as discussed, aspects of the technology are described with reference to Li-ion cells as the power unit. However, aspects of the technology are not limited to Li-ion cells and are applicable to different cell chemistries including, for example, Li-S, sodium-ion, solid-state, sodium-sulphur or any other cell type.

Some aspects of the technology may be additionally applicable to fuel cells, such as hydrogen fuel cells, or flow cells such as Vanadium Redox flow cells as the power unit.

For the avoidance of doubt, all aspects of the methodology described in relation to Li-ion cells may be applied to other power units, although the physical characteristics of the power unit under analysis may change accordingly.

Lithium-ion cells generally comprise a cathode, anode, separator, electrolyte, and current collectors. Common cathode materials include lithium cobalt oxide (LCO), lithium manganese oxide (LMO), lithium iron phosphate (LFP), lithium nickel manganese cobalt oxide (NMC) and lithium nickel cobalt aluminium oxide (NCA). The most common anode material is graphite, sometimes combined with small levels of silicon, although other materials including lithium titanate may be used.

The electrolyte used in a Li-ion cell may be liquid or solid, in the second case being termed a ‘solid-state’ cell. Common liquid electrolytes use lithium salts, such as LiPF6, L1BF4 or LiCIC>₄ with organic solvents such as dimethyl carbonate, ethylene carbonate, or diethyl carbonate. Solid electrolytes include ceramics and polymers.

The most common current collectors are copper for the anode side and aluminium for the cathode side.

The impedance characteristics of a Li-ion cell are determined by the cell components. In the case of dynamic resistance, such as charge transfer resistance, contributions arise from the anode and cathode as well as their respective surface layers, particularly the anode solid electrolyte interphase (SEI). These effects may be distinguished in some aspects of the technology based on their associated time constants, with impedance effects arising from the SEI approximately an order of magnitude quicker than anode charge transfer effects, which are in turn approximately an order of magnitude faster than cathode charge transfer effects. Although the absolute time constants are dependent on cell design and chemistry, their relative relationships allow them to be distinguished from each other in any particular cell.

These individual contributions may be distinguished based on their dependence on various parameters, including for example temperature and voltage.

While all charge transfer effects are temperature-dependent, cathode charge transfer effects are particularly strongly temperature-dependent. Effects attributable to the SEI increase slightly with voltage increase. Anode and cathode charge transfer effects would be expected to increase exponentially at both extremes of the cell state-of-charge range, although this would depend on cell stoichiometry. Cathode charge transfer has an inverse relationship with voltage.

The characteristic time constants may be used to isolate different impedance contributions via frequency- based or time domain-based methods.

The faster acting effects, such as dynamic resistance (e.g. charge transfer resistance), may be analysed using frequency domain testing such as Electrochemical Impedance Spectroscopy (EIS). EIS is effective at higher frequencies but is susceptible to state of charge oscillation affecting low frequency accuracy. Low frequency effects such as diffusion may be characterised using time domain testing methods such as current interrupt, which is not affected by state of charge oscillation, but is unsuitable for high frequency analysis without very high frequency measurement capability. According to some aspects of the technology, EIS data may be used to quantify individual contributions to the dynamic resistance of a Li-ion cell. The dynamic resistance may be charge transfer resistance. The method is equally applicable to other cell chemistries, fuel cells or flow cells as stated previously. Furthermore, time domain data (e.g. from GITT or HPPC) may be used to perform the methods described herein as explained previously (e.g. with Laplace inversion).

It can be difficult to quantify fully individual impedance contributions within dynamic resistance (e.g. charge transfer resistance) and diffusion impedance using EIS alone. To identify the individual contributions, equivalent circuits may be fitted to the EIS data, although it can be difficult to verify the physical significance of the equivalent circuit elements due to the complex nature of the various contributing factors.

In some aspects of the technology Distribution of Relaxation Times (DRT) techniques are used to analyse the EIS data. In accordance with this technique, the EIS data may be converted into a one-dimensional frequency distribution. The frequency distribution may then be analysed by statistical methods to identify individual polarization contributions. The use of DRT allows for the impedance effects acting at different time constants to be more visibly/quantitatively separated, making resultant isolation and circuit element fitting easier.

In accordance with some aspects of the technology, the use of DRT in combination with EIS may enable individual impedance contributions to be isolated and quantified. An EIS data may be evaluated using DRT, combining physical insight and log-normal, Gaussian, Cauchy or Voigt distribution fits to separate individual resistance contributions using parameter optimisation. The individual contributions may then be reconstructed and fitted to circuit elements, for example RQ or finite length Warburg elements, which allow quantitative comparison of impedance parameters in different cells, such as a new cell and an aged cell or cells of similar electrode chemistry but different design.

The analytical method may be combined with physical cell analysis to identify the root causes of the observed impedance variations, in a combination of computational, electrochemical and physical analysis, as will be outlined in further detail below.

The accompanying figures illustrate the application of the described analysis methods to 28Ah NMC/Graphite PHEV2 prismatic format electric vehicle batteries. The methods described herein may be applied to other cells or fuel cells as required. Scanning Electron Microscopy (SEM) and Energy Dispersive X-Ray Analysis (EDX) was used to confirm NMC111 cathode chemistry and lack of silicon in the anode. Three cells were used in the analysis, cell 1 , cell 2, and cell 3. The first cell had not been cycled, although it may have had slight calendar ageing. The second and third cells were cycled for nine months using the Federal Urban Driving Schedule (FUDS) drive cycle with temperature control at 45°C and state of charge range of 15-85%. The second cell was charged using constant current constant voltage (CC-CV) charging at 0.5C, whereas the third cell was charged using CC-CV charging at 2C. The charging C-rate is a measure of the rate at which a battery is charged or discharged, defined as the current through the battery divided by the theoretical charge throughput under which the battery would deliver its nominal rated capacity in one hour.

EIS testing was performed on all three cells at three temperatures (-20°C, 0°C and 25°C) and at three voltages (3.53V, 3.68V and 3.89V) in a full factorial test matrix. The EIS was performed from 10kHz to 1mHz in all cases, although inductance effects prevented very high frequency effects from being analysed.

EIS data may be obtained from a cell and converted to DRT using known methods. The EIS data can be expressed from a DRT spectrum by:

where Z(w ) is the impedance as a function of angular frequency, R_¥ is an ohmic resistance term, g(1h(t )) is the distribution function of relaxation times, i is the imaginary number, f is the frequency, and t is a characteristic time constant.

Studying the distribution of relaxation times gives insight into the characteristic resistance peaks as a function of time constant. As part of the approach taken in this method, the frequency range may be extended beyond the testing range from -¥ to ¥. The ohmic resistance term R_¥ may not be applied when the method is used to analyse dynamic (e.g. charge transfer) contributions.

When analysing the contributions to dynamic resistance (e.g. charge transfer resistance), diffusion effects may need to be accounted for due to the overlap in time constants between dynamic resistance (e.g. charge transfer resistance) (which may be in the range 10^-3 to 10¹ s in one example) and diffusion (which may be in the range 10° to 10⁴ s in one example). Diffusion effects in Li-ion cells can be dominated by one or both electrodes.

A two-stage approach may be used to separate the dynamic (e.g. charge transfer) and diffusion effects, as illustrated in figure 1 , where Z’ represents a real impedance vector and Z” represents an imaginary impedance vector.

A first stage of the process in the case of battery cells may be to fit diffusion and capacitive features directly to the EIS data as illustrated in figure 1a. Firstly, a pure capacitance term may be created to represent the Open Circuit Voltage (OCV) changes resulting from state of charge fluctuations due to current being applied during the EIS profile. This term may be created by putting the EIS profile through a coulomb counting approach (e.g. applying Coulomb counting to the EIS profile) and then using state of charge - open circuit voltage (SoC-OCV) maps derived from cell characterisation data (i.e. from predetermined data about the cell).

Finite Length Warburg (FLW) and Finite Space Warburg (FSW) elements, which can be used for frequency domain diffusion representation, may also be incorporated into the fitting process. FLW elements represent diffusion through an element with finite length, while FSW elements represent diffusion with a reflective boundary. One or both of these effects may be present in electrodes. A fitting process incorporating one FSW element and one FLW element was found to result in a good fit in this instance (although other element combinations are envisaged). Fitting processes using different numbers of FLW and FSW elements, including zero, one, two, or three of each element, are envisaged. The number of FLW and FSW elements may not necessarily be the same and in some cases only one of the elements may be used.

The capacitance, FSW and FLW elements may be combined in a fitting process to fit the diffusion features directly to the EIS data.

A second stage of the fitting process may include obtaining the conditions from the EIS fitting, and fitting these to the diffusion time constant region of the DRT data as shown in figure 1b, where TC represents the time constant in seconds. In other words, diffusion impedance parameters determined by fitting diffusion elements to EIS data may be optimised by performing a second fitting to DRT data. This may further refine the fitting, allowing for it to be adapted for any assumptions that may be used in the extension of the DRT frequency range. The diffusion features may be fitted to only the EIS data or only the DRT data. Flowever, both fittings may be performed, to obtain a more accurate overall fitting.

Once the features attributable to diffusion have been identified, the process of isolating and quantifying individual dynamic resistance (e.g. charge transfer resistance) peaks may be undertaken as illustrated in figure 2. Alternatively, diffusion fitting may not be performed before beginning the process of fitting the dynamic (e.g. charge transfer) features, although this may lead to less accurate results.

The complete DRT distribution may include multiple dynamic (e.g. charge transfer) aspects. Individual features may be identified and assigned based on their time constant and voltage-temperature relation. To isolate the individual features in the DRT data, statistical distributions may be fitted. Using available DRT analysis tools with a logarithmic expression for y(-r), log-normal distributions may be used to isolate individual peaks as shown in figure 2a.

With the individual features isolated, individual EIS reconstructions may be performed using equation (4), as shown in figure 2b. To quantify the characteristics of each reconstructed EIS spectrum, RQ elements may be fitted using the following equations, where the RQ elements represent a resistor and constant phase element in parallel: (5) Z(w) = 1+t(ίw)^a

(6) t = RQ where Z(w ) is the impedance as a function of angular frequency, w is the angular frequency, R is a resistor representing dynamic resistance (which may be charge transfer resistance), Q is a constant phase element, t is a characteristic time constant, and a is the exponent value, which is 1 for an ideal capacitor.

Using this fitting, the resistance of each effect may be measured, along with its time constant using equation (6). An a of 1 represents a pure RC circuit, which could also be used in time domain DC analysis, unlike the frequency domain RQ element. The value of a indicates how representative RC elements would be for modelling each behaviour in the time domain. The RQ elements were shown in all cases to give accurate representation, as shown in figure 2c. The ability to generate elements that can be converted into the time domain, gives an extra level of practicality for applying to virtual tools modelling and real-time control applications.

Optimisation algorithms, performed in this case using Matlab 2019a, were used in all steps requiring equation, distribution or peak fitting to data. A combination of global and local optimisation may be used as described previously, for example using particle swarm (and/or genetic algorithm) optimisation and optimisation to find a minimum or minima of a constrained nonlinear multivariable function, with (in some aspects of the technology) the global optimisation function particleswarm() being used to seed initial conditions for local optimisation with fmincon(). This approach may provide a global sweep of the available parameter space, avoiding local minima, while producing efficient convergence on the final result.

An advantage of some aspects of the technology is the provision of an automated method for performing the RQ circuit fitting and corresponding quantification of impedance contributions. The use of the fitting processes, equations and optimisation techniques described herein may allow the creation of an automated process for outputting resistance values of individual cell components based on input EIS data.

A subset of EIS and DRT data (i.e. spectra) obtained from cells 1-3 is shown in figure 3. The EIS data is shown in the top row; DRT data is shown in the bottom row. The first column shows data obtained from cell 1 at -20°C, 0°C and 25°C, at a constant voltage of 3.68V. The second column shows data obtained from cell 1 at a constant temperature of 0°C at 3.53V, 3.68V and 3.89V. The third column shows data obtained from cells 1-3 at -20°C and 3.68V. The fourth column shows data obtained from cells 1-3 at 25°C and 3.68V.

The data in figure 3 demonstrates that, in this case study, charge transfer effects had a strong dependence on temperature. Cathode charge transfer was observed to be particularly temperature sensitive. It is also clear that temperature affects not only the magnitude of the various contributions, but also their visibility in the distribution. Charge transfer time constants are also temperature dependent, scaling with resistance change. Inductance of the cells and test equipment hides high frequency effects. The cells used in this example are of high capacity and high-quality design, leading to low resistance.

Three charge transfer effects, attributable to the cathode, anode, and anode SEI, are visible at -20°C, with anode and cathode charge transfer remaining visible at 0°C and only cathode charge transfer effects being visible at 25°C due to their decreasing time constants with temperature increase taking them out of the visible range of the experimental conditions.

Figure 3 shows that the temperature sensitivity of the cells changed as the cells aged. The resistance of the aged cells, cells 2 and 3, is lower in the EIS data, particularly at -20°C. The DRT data shows this characteristic appears to be the result of a significant reduction in the cathode charge transfer contribution, which is the dominant contribution to the overall charge transfer resistance, particularly at lower temperature.

The time constants in general are seen to increase with ageing. An exception to this is the cathode charge transfer at -20°C, which may be due to the impedance reduction offsetting capacitance increase. The increase of time constants in aged cells can complicate the analysis. At -20°C, the anode charge transfer contribution is obscured by the much larger cathode contribution.

Figure 4 shows the results of equivalent circuit fitting as explained above. By combining the equivalent circuit fitting with physical cell analysis, the physical component underlying each charge transfer contribution can be identified. Figure 4 shows contributions corresponding to the cathode, anode, and anode solid electrolyte interphase (SEI). The resistance of the anode and SEI is seen to increase over time whilst the cathode resistance decreases. This trend is mirrored in the time constants.

The value of a is close to 1 for all three cells for the SEI, suggesting it could be modelled well using an RC element. Similarly, the value of a is close to 1 for all three cells for the anode, indicating this could also be modelled using an RC element if desired.

Anode and SEI impedance increased more in cell 2 than cell 3, despite being cycled with a lower charge current. A possible explanation for this is that the lower charge current means more time spent charging, which raises the average voltage over the nine month ageing period. The chemical reactions causing the SEI layer increase occur at a higher rate with voltage increase, causing the more evolved SEI layer in cell 2.

Figure 5 shows NMR data obtained from the anode SEI layer for a new cell and the average results for two aged cells. The Li contributions to the SEI increase over time, suggesting a denser SEI layer is formed.

Figure 6 shows SEM images of anode (left) and cathode (right) samples for cell 1 (6a and 6b), cell 2 (6c and 6d) and cell 3 (6e and 6f). The cathode surface shows significant cracking with ageing (6d and 6f), potentially reducing cathode charge transfer resistance by increasing the available surface area. Figure 7a highlights the existence of two peaks in the cathode charge transfer contribution for a new cell, which are seen to merge into a single peak as the cell ages. A possible cause of this effect is identified in figure 7b, which shows two distinct particle sizes within the NMC cathode. This would be expected to produce two peaks due to the differences in relative surface area of the particles. As the cells age, the larger particles undergo cracking, and the smaller particles agglomerate, meaning there are no longer two distinct particle sizes and hence the peaks merge.

Versions of the method described above may provide a valuable means to separate out the individual contributions within the category of dynamic resistance (e.g. charge transfer resistance). However, as stated previously, dynamic resistance such as charge transfer resistance is not the only contribution to cell impedance.

Accordingly, some aspects of the technology may provide a method of analysing contributions arising from ohmic resistance.

In some aspects of the technology, analysis of both dynamic (e.g. charge transfer) and ohmic effects may be carried out simultaneously, i.e. on the same cell or other power unit. By performing analysis of both dynamic (e.g. charge transfer) and ohmic effects, a deeper understanding of the power unit (e.g. battery or fuel cell) performance or health may be achieved.

Total ohmic resistance in a cell may be identifiable from EIS data by analysing the real resistance when the imaginary resistance is zero. However, as with dynamic resistance (e.g. charge transfer resistance), ohmic resistance may be made up of a combination of individual contributing factors. These include contributions from the inherent electrolyte resistance as well as contact resistances within the active materials and between the active materials and current collectors. These effects cannot be distinguished directly by EIS, which is only able to show the total effect.

Ohmic resistance is independent of state of charge and current due to the constant nature of the material properties. However, the inherent electrolyte resistance varies with temperature, showing an Arrhenius-type temperature dependence which is not seen in any of the other factors which contribute to the total ohmic resistance. This is significant as it allows the electrolyte resistance to be separated from the contact resistance by taking measurements at multiple temperatures.

Separation of the electrolyte resistance is advantageous as it allows for more accurate and physically insightful modelling of cell impedance, as well as helping to identify the physical causes of observed resistance changes over time, which may be useful in identifying rogue components, for example. For the purpose of illustration and to aid understanding, the method of some aspects of the technology is applied to the same three cells described previously.

EIS testing was carried out at all combinations of three different temperatures (-20°C, 0°C and 25°C) and open circuit voltages corresponding to three different states of charge (20%, 50% and 80%) in the frequency range of 10kHz to 1mHz. However, it will be understood that the specific application of the method to these Li-ion cells is purely for illustrative purposes, and the method can be applied to any power unit (e.g. battery cell, fuel cell or flow cell).

Ohmic resistance can be expressed by:

where R_ohmic is the total ohmic resistance, R₀ is a variable parameter which may be a constant resistance representing the contact resistance, /?_! is a variable parameter, E_A is a variable parameter which may represent an activation energy and RT is the gas constant multiplied by temperature in Kelvin. The R₁ term scales the magnitude of the slope, and the activation energy determines the sensitivity to temperature. The ohmic resistance may be extracted from EIS data by evaluating the real axis value at the point of the imaginary resistance zero crossing.

This equation stipulates that ohmic resistance R_ohrlic is defined by an Arrhenius slope with temperature

R₁e^RT⁾ offset by a constant term R₀. The Arrhenius relationship would be expected to show a linear trend between the natural log of the resistance, and the inverse of the temperature in Kelvin. As illustrated in figure 9, such a plot shows an imperfect non-linear trend for the three cells used as examples, implying the existence of multiple resistance contributions.

A fitting process using the Root Mean Square Error (RMSE) technique may be used to fit equation (1) to a series of measured datapoints. Alternative fitting techniques may be used, such as mean absolute error.

At least two optimisation objectives may be necessary to ensure physical significance of the individual terms. More objectives could be added if desired, although two objectives have been shown to be satisfactory. In accordance with some aspects of the technology, one objective may be based on Arrhenius slopes, comparing slopes derived from the real data and the equation data, with the approximated constant term R₀ removed from the real data.

(7) Gr_{Arr Eq} = IniR + ©

(8) ata ~ ^ifidata ^o) (9) Obj. 1: min ( Jmean (( R_data - ¾)²))

Equation (7) defines the Arrhenius slope for the fitted equation (i.e. the model Arrhenius slope). Equation (8) defines the Arrhenius slope derived from the real data using the fitted temperature-independent term R₀ (plotted against the measured values of T). Objective 1, shown in equation (9), minimises the root mean square error between the measured total ohmic resistance values R_data and the fitted total ohmic resistance values R_fit based on equation (1). Objective 2, shown in equation (10), minimises the root mean square error between the real Arrhenius slope Gr_{Arr Data} and the fitted/model Arrhenius slope Gr_{Arr Eq}.

Matlab 2019a was used to perform the optimisation in an example implementation. A combination of global and local optimisation may be used to provide a good fit. The global optimisation function particleswarm() was used to seed local optimisation with fmincon() using the same optimisation objectives, in this example. Due to the sensitivity of the results to the value of E_A, the initial conditions may be set reasonably close to the true values. Appropriate initial conditions may be obtained using a combination of known values (e.g. from literature) and an initial single-objective fit. The parameters allowed to be varied may include R_{l t} R₀ and E_A. The initial single-objective fit may minimise the disparity between the measured total ohmic resistance values R_data and the fitted total ohmic resistance values R_fit (e.g. it may use the objective shown in equation (9)).

Figure 8 shows the impact of voltage and temperature on ohmic resistance for a new cell (cell 1). The ohmic resistance is expected to be independent of state of charge, and the trends seen in fig. 8 support this.

The results of the dual-objective fitting process are illustrated in figure 10, in which the top row illustrates the trend for total ohmic resistance against temperature and the bottom row illustrates the Arrhenius slopes, in which In (R_elj represents ln(R_data - R₀j. Figures 10a and 10d show data from cell 1, figures 10b and 10e show data from cell 2, and figures 10c and 10f show data from cell 3.

The quantities of the parameters corresponding to the plots shown in figure 10 are shown in the table below. Both the absolute fitting of the total resistance at each temperature point and the underlying Arrhenius trend show a good fit between the real data and the model fit. This shows that equation (1) is both physically based and has appropriate features to match the data.

Figure 10 illustrates that the Arrhenius trend for all three cells is similar, suggesting that the temperature- dependent electrolyte resistance does not change significantly with ageing. This is supported by the parameter results shown in table 1, in which both /?_! and E_A show little or no variation with ageing. The value of 0.21 eV for E_A is close to literature values, indicating that the parameter results attained by the fitting process are physically representative.

The temperature-independent ohmic resistance R₀ can be seen to increase significantly with ageing, and is higher for cell 3, which had a higher charging C-rate, than for cell 2.

Figure 11 demonstrates the marked difference in R_{0 l} and the lack of change of R-_t and E_{A l} in graphical form.

The results of the fitting process show that the temperature-independent contribution to the ohmic resistance was the cause of the observed variation in total ohmic resistance. This valuable information also indicates that the electrolyte shows little variation as it ages. As a result, a similar battery which shows significant variation in electrolyte characteristics with ageing may be identified as a rogue component and potential safety hazard.

The results obtained in this case study indicate that the physical cause of increased ohmic resistance is not due to the electrolyte, but to changes in either inter-particle resistance or changes in contact resistance between one or both electrodes and their respective current collectors.

Figure 6 shows SEM images of the anode (left column) and cathode (right column) of each cell. Cracks have formed within the active material particles of the cathode and gaps are also present between the particles. As a result, it is likely the cathode would see an increase in inter-particle resistivity and a reduction in surface contact with the current collector, increasing contact resistance. Figure 12 shows SEM cross-sections of the anode (12a) and cathode (12b). The anode structure can be seen to be largely intact, whereas the cathode shows cracking throughout the electrode through to the current collector, therefore likely increasing resistance by reducing inter-particle and current collector to active material contact area. The cathode is therefore most likely the main contributor to the observed change in ohmic resistance.

In accordance with some aspects of the technology there is provided an EIS instrument for collecting EIS data (i.e. spectra) from a power unit. The EIS instrument may be configured to provide this data (through a connection or a storage medium) to a processor which is configured to execute instructions to perform one or more of the methods described herein. The processor may be further coupled to a display screen for displaying the results of the methods and, for example, advice to the user - such as an indication that a power unit needs replacement or is otherwise rogue.

In accordance with some aspects of the technology there is provided a Hybrid Pulse Power Characterisation (HPPC) instrument for collecting HPPC data (i.e. spectra) from a power unit. The HPPC instrument may be configured to provide this data (through a connection or a storage medium) to a processor which is configured to execute instructions to perform one or more of the methods described herein. The processor may be further coupled to a display screen for displaying the results of the methods and, for example, advice to the user - such as an indication that a power unit needs replacement or is otherwise rogue.

In accordance with some aspects of the technology there is provided a Galvanostatic Intermittent Titration Technique (GITT) instrument for collecting GITT data (i.e. spectra) from a power unit. The GITT instrument may be configured to provide this data (through a connection or a storage medium) to a processor which is configured to execute instructions to perform one or more of the methods described herein. The processor may be further coupled to a display screen for displaying the results of the methods and, for example, advice to the user - such as an indication that a power unit needs replacement or is otherwise rogue.

A rogue component, such as a power unit, as described herein may be a component which is performing below a performance threshold. That performance threshold may be determined based on, for example, the use of the power unit. The performance threshold may be a safety threshold. An indication that a component is rogue may be an indication that the component needs replacing or that the component is not suitable for the use on which the performance threshold is based. As will be appreciated, for some uses the performance threshold may be high, for example where the failure of the component (or performance falling below a certain level) may cause safety concerns, or may be low, for example where the failure of the component (or performance falling below a certain level) would be an inconvenience. For example, a high performance threshold may be set in relation to power units which are for use in an electric vehicle but a low performance threshold may be set in relation to power units which are for use in personal computer equipment (such as a laptop computer). Figure 13 shows a vehicle 10 according to some aspects of the technology comprising a power unit 12 and battery management system 14. The vehicle 10 may be an electric vehicle 10 and the power unit 12 may be a battery 12.

Figure 14 shows a system 20 according to some aspects of the technology comprising an EIS or FIPPC or GITT instrument 22, display 24 and processor 26.

In some aspects of the technology, the testing and analysis described herein may be applied to the power unit of an electric vehicle. The electric vehicle 10 may, for example, be powered by Li-ion batteries.

An electric vehicle may be any electrically-powered means of transportation, including but not limited to cars, vans, trucks, buses, lorries, motorcycles, electrically assisted pushbikes, scooters, quad bikes, all-terrain vehicles, boats, hovercraft, aeroplanes etc. Electric vehicles generally comprise a battery, or batteries, and an on-board computer including a battery management system. Simple electric vehicles may not have an on-board battery management system.

Some aspects of the technology provide methods of performing a battery health check on an electric vehicle 10. This may be undertaken by an engineer, mechanic or other person. A battery 12 may be removed from an electric vehicle 10 and an impedance spectrum or spectra may be obtained from the battery 12. For example, the battery 12 may be analysed by electrochemical impedance spectroscopy (EIS) and/or hybrid pulse power characterisation (FIPPC) and/or a Galvanostatic Intermittent Titration Technique (GITT). Alternatively, an impedance spectrum or spectra may be obtained from the battery 12 in situ in the electric vehicle 10. For example, EIS/HPPC/GITT may be carried out on a battery 12 in situ in an electric vehicle 10 by means of a suitable adapter or interface between the EIS/HPPC/GITT unit (or instrument) and the battery 12, which may be implemented via the existing battery charging port or by other means.

Once the impedance spectrum, which may be obtained using EIS/HPPC/GITT data (i.e. spectra), is obtained from the relevant battery 12, a battery health check can be carried out by performing any of the analysis methods described previously to provide battery health data. For example, battery health data may include any or all of total cell impedance, total dynamic resistance (e.g. total charge transfer resistance), total ohmic resistance, or any individual contribution to a broader impedance category, such as resistance magnitude and/or time constant for a particular component and resistive effect (e.g. anode charge transfer). Battery health data may comprise ohmic resistance parameters and/or dynamic resistance parameters (e.g. charge transfer resistance parameters) and/or diffusion impedance parameters. Battery health data may be used to re-calibrate a battery management system 14 (e.g. based on root-cause analysis of ageing). Such a battery health check may be carried out as part of a vehicle service or safety inspection (such as an MOT in the United Kingdom), for example. The battery health data may then be evaluated by various means, such as by comparison of new battery health data with battery health data obtained in a previous battery health check so as to determine a level or percentage of degradation since the last health check; and/or by comparison of new battery health data with baseline measurements taken from a new (e.g. un-aged) cell; and/or by comparison to a threshold or to predetermined reference data to assess the health of the battery. Battery health data lying outside a predetermined range, which may be symptomatic of a safety issue, may be identified.

An advantage of some aspects of the technology is that the isolation of individual impedance contributions may allow any detected anomalies or safety issues to be traced back to a specific battery component.

A battery health check may be carried out on other battery-powered items, such as consumer electronics. Similarly to the procedure for electric vehicles, a consumer electronics battery health check may involve removal of the battery from a product to allow an impedance spectrum or spectra (e.g. from EIS/HPPC/GITT data) to be gathered. Alternatively, the impedance spectrum or spectra (e.g. from EIS/HPPC/GITT data) may be gathered with the battery in situ in a product by means of a suitable adapter or interface between the battery and EIS/HPPC/GITT unit. Once the impedance spectrum (e.g. from EIS/HPPC/GITT data) is obtained, battery health data can be determined as explained previously.

An electric vehicle’s battery management system 14 may manage battery usage (e.g. charge and discharge rates may be altered, maximum charge may be altered, etc.) according to battery health data. Battery health data may be input to the battery management system following a battery health check undertaken during a vehicle service or safety inspection (e.g. an MOT in the United Kingdom). Accordingly, the battery health data may be updated each time a battery health check is carried out.

Battery management performed by the battery management system 14 may include management of one or more limits including maximum charge current limit, maximum discharge current limit, maximum voltage limit, minimum voltage limit, maximum charge power limit, maximum discharge power limit, etc. A maximum charge or discharge current limit may include a maximum peak charge or discharge current limit and/or a maximum continuous charge or discharge current limit. Likewise, a maximum charge or discharge power limit may include a maximum peak charge or discharge power limit and/or a maximum continuous charge or discharge power limit.

The limits managed by the battery management system 14 may be defined as a function of state-of-charge, state-of-health and/or temperature. The limits may also be a function of time, with a peak limit being set for a first time period and a continuous limit being set for a second time period. The first (peak) time period may be shorter than the second (continuous) time period and the peak limit may be less restrictive than the continuous limit. Accordingly, one or more limits managed by the battery management system 14 may be updated based on the battery health data (e.g. a more restrictive limit may be imposed when the battery health data indicates the battery has degraded below a predetermined threshold).

The battery health data may be input to the battery management system 14 manually via a suitable user interface. Alternatively, battery health data may be transferred to a battery management system by means of a wired or wireless connection to a secondary device such as a computer or tablet.

The battery management system 14 may be configured to manage various aspects of battery usage, including for example maximum power consumption or maximum battery temperature, according to certain limits determined by the battery health data. For example, a battery management system may be configured to reduce the maximum possible power consumption of a battery in order to protect the battery once the battery health data indicates that the battery has degraded a certain amount. The level of battery degradation may be determined by any aspect of the battery health data, for example by the largest individual degradation contribution or by the total cell impedance.

The battery management system 14 may be configured to implement battery usage protocols according to the greater of either total battery degradation or degradation of an individual component. For example, total battery degradation may be 10% but the degradation of a single component may be 20%.

On-board battery health data may be obtained for an electric vehicle 10, negating the requirement for external analysis during a vehicle service. The battery health check apparatus may be fully integrated into the electric vehicle 10. As such, means for obtaining an impedance spectrum (e.g. from EIS/FIPPC/GITT data) and means for processing and/or displaying such data may be integrated with a battery 12 in an electric vehicle 10. Such integration is advantageous as it allows regular battery health checks and regular updating of battery health data.

Battery health checks could be carried out at any desired frequency, for example by setting a particular time period, number of charge cycles, or distance to be covered between each battery health check.

Battery health data may be processed by a dedicated on-board computer or by a multi-purpose on-board computer, as is present in many vehicles.

On-board battery health checks may be integrated with the battery management system 14 to manage battery usage according to the gathered battery health data. Furthermore, on-board battery health checks may be used to indicate to a user when a battery 12 requires replacement or maintenance. Such an indication may be made by comparing battery health data to predetermined battery health thresholds or tolerances. Such an indication may be made to a user by integration of the battery health check system with a display screen, warning light, graphical user interface, audio system, or other means of communication. Power unit data obtained through any of the methods described herein may be used to compare a power unit to another power unit or units. The units may have identical or different structures. For example, a new or prototype power unit may be analysed by any of the methods described herein to produce power unit data (e.g. ohmic resistance parameters and/or dynamic resistance parameters (e.g. charge transfer resistance parameters) and/or diffusion impedance parameters) which may be compared to power unit data from alternative power units. This may be beneficial in assessing the suitability of a power unit for a particular application, or to assess the competitiveness or expected behaviour of a particular power unit. The power unit data may be incorporated into a model or models to predict the degradation of the power unit over time (e.g. over a number of charge/discharge cycles). As such, the methods described herein may enable benchmarking of power units.

A computer-implemented method may compare power unit data of a plurality of power units and provide an indication to a user of the most suitable power unit for a specific purpose based on predetermined criteria. Additionally or alternatively, the method may incorporate the power unit data into power unit degradation models and provide a ranking of the power units according to their expected degradation characteristics. The power unit with the lowest expected degradation may be the highest-ranked power unit. The highest- ranked power unit may be selected for use in a vehicle. The selected power unit may be used in a vehicle.

A computer-readable medium storing instructions which, when executed by a processor, cause the performance of any of the methods described herein may be provided. For example, such a computer- readable medium could be included in an on-board controller (for example, in some versions, the computer- readable medium may be included in battery management system 14), or could be included in a lifetime servicing tool.

When used in this specification and claims, the terms "comprises" and "comprising" and variations thereof mean that the specified features, steps or integers are included. The terms are not to be interpreted to exclude the presence of other features, steps or components.

The features disclosed in the foregoing description, or the following claims, or the accompanying drawings, expressed in their specific forms or in terms of a means for performing the disclosed function, or a method or process for attaining the disclosed result, as appropriate, may, separately, or in any combination of such features, be utilised for realising the invention in diverse forms thereof.

Although certain example embodiments of the invention have been described, the scope of the appended claims is not intended to be limited solely to these embodiments. The claims are to be construed literally, purposively, and/or to encompass equivalents.

Claims

1. A method of isolating and quantifying temperature-dependent and temperature-independent contributions to ohmic resistance in a power unit, comprising: obtaining an impedance spectrum from the power unit at a plurality of measured temperatures; determining a total ohmic resistance of the power unit from the impedance spectrum at each temperature; applying a multi-objective fitting process to quantify the temperature-dependent and temperature- independent contributions to the total ohmic resistance wherein an objective is to minimise the disparity between an Arrhenius slope derived at least partially from the impedance spectra and an Arrhenius slope derived solely from modelling; and outputting the quantified temperature-dependent and temperature-independent contributions as ohmic resistance parameters; wherein the Arrhenius slopes are based on the equation:

where R_ohrlic is a total ohmic resistance, R₀ is a variable parameter, /?_! is a variable parameter, E_A is a variable parameter and RT is the gas constant multiplied by temperature.

2. A method according to claim 1 , wherein the impedance spectra comprise electrochemical impedance spectroscopy data.

3. A method according to any preceding claim, further comprising displaying the ohmic resistance parameters.

4. A method according to any preceding claim, further comprising comparing the ohmic resistance parameters to known values.

5. A method according to any preceding claim, wherein the Arrhenius slope derived at least partially from the impedance spectra is obtained by plotting ln(R_data - R₀ ) against

where R_data is the measured total ohmic resistance at a measured temperature, R₀ is a variable parameter and T is the measured temperature.

6. A method according to any preceding claim, wherein the Arrhenius slope derived solely from modelling is obtained by plotting the output of IniR^ + (^) against ^ for a plurality of temperatures.

7. A method according to any preceding claim, wherein an objective is to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance.

8. A method according to claim 7, wherein the model total ohmic resistance, R_fit, is calculated according to the equation R_fit = R₀ + R^RT¹.

9. A method according to any preceding claim, wherein the multi-objective fitting process is a dualobjective fitting process, a first objective being to minimise the disparity between an Arrhenius slope derived at least partially from the impedance spectra and an Arrhenius slope derived solely from modelling and a second objective being to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance.

10. A method according to any preceding claim, wherein the optimisation of the fitting process is carried out using a combination of global optimisation and local optimisation.

11. A method according to claim 10, wherein the global optimisation is performed using particle swarm optimisation and/or genetic algorithm optimisation.

12. A method according to any preceding claim, wherein the initial conditions for the multi-objective fitting process are determined by a combination of a known value and an initial single-objective fit.

13. A method according to claim 12, wherein the objective of the initial single-objective fit is to minimise the disparity between the measured total ohmic resistance and a model total ohmic resistance R_fit at a given temperature using the equation R_fit = R₀ + R^RT¹.

14. A method according to any preceding claim wherein the ohmic resistance parameters comprise at least one of the parameters R₀ R_l and/or E_A.

15. A computer-readable medium storing instructions which, when executed by a processor, cause the performance of the method of any of claims 1-14.

16. A method of isolating and quantifying individual contributions to dynamic resistance in a power unit, comprising: obtaining an impedance spectrum from the power unit; removing diffusion impedance effects from the impedance spectrum; converting the impedance spectrum into a distribution of relaxation times; isolating individual peaks in the distribution of relaxation times; reconstructing an impedance spectrum for each peak; fitting equivalent circuit elements to the reconstructed impedance spectra to quantify the individual resistance contributions to the dynamic resistance of the power unit; and outputting the quantified individual resistance contributions as dynamic resistance parameters.

17. A method according to claim 16, wherein the impedance spectrum is obtained from the power unit by electrochemical impedance spectroscopy and/or by conversion of pulse relaxation data, obtained following application of a current to the power unit, into an impedance spectrum.

18. A method according to claim 17, wherein the pulse relaxation data is converted into an impedance spectrum by fitting frequency domain elements to the pulse relaxation data using Laplace inversion, and constructing an impedance spectrum from the frequency domain elements.

19. A method according to claim 17 or 18, wherein the pulse relaxation data is obtained by a galvanostatic intermittent titration technique.

20. A method according to claim 17 or 18, wherein the pulse relaxation data is obtained by hybrid pulse power characterisation.

21. A method according to any of claims 16-20, wherein individual peaks in the distribution of relaxation times are isolated by fitting statistical distributions to the distribution of relaxation times.

22. A method according to claim 21 , wherein the statistical distributions include log-normal or Gaussian distributions.

23. A method according to any of claims 16-22, further comprising displaying the dynamic resistance parameters.

24. A method according to any of claims 16-23, further comprising comparing the dynamic resistance parameters to known values.

25. A method according to any of claims 16-24, wherein a fitting based on a pure capacitance term is used to remove capacitance effects from the impedance spectrum.

26. A method according to any of claims 16-25, wherein a fitting based on at least one of a finite length Warburg term and/or a finite space Warburg term is used to remove the diffusion impedance effects from the impedance spectrum.

27. A method according to any of claims 16-26, wherein the equivalent circuit elements are RQ elements in which RQ represents a resistor and constant phase element in parallel.

28. A method according to any of claims 16-27, wherein equivalent circuit elements are fitted based on the equation Z(w) = is the impedance as a function of angular frequency, R is a resistor

representing dynamic resistance, t is a characteristic time constant, i is the imaginary number, w is the angular frequency and a is the exponent value.

29. A method according to any of claims 16-28 further comprising: creating a simulated power unit impedance spectrum using the fitted equivalent circuit elements; and optimising the values of the fitted equivalent circuit elements to minimise the error between the simulated power unit impedance spectrum and the impedance spectrum obtained from the power unit.

30. A method according to claim 29 wherein the optimisation includes the fitting of an inductor L or resistor inductor parallel circuit element RL.

31. A method according to claim 30 wherein the optimisation includes the fitting of an inductor based on the equation CKw ) = ίwZ, where CKw ) is the contribution to the impedance spectrum, i is the imaginary number, w is the angular frequency and L is the variable inductance value.

32. A method according to any of claims 16-31 , further including the method of any of claims 1-14.

33. A computer-readable medium storing instructions which, when executed by a processor, cause the performance of the method of any of claims 16-32.

34. A method of isolating and quantifying individual contributions to diffusion impedance in a power unit, comprising: obtaining an impedance spectrum from the power unit; fitting a pure capacitance term to remove capacitance effects from the impedance spectrum; fitting at least one of a finite length Warburg term and/or a finite space Warburg term to quantify the diffusion impedance contributions from the impedance spectrum; and outputting the quantified diffusion impedance contributions as diffusion impedance parameters.

35. A method according to claim 34, wherein the impedance spectrum is obtained from the power unit by electrochemical impedance spectroscopy and/or by conversion of pulse relaxation data, obtained following application of a current to the power unit, into an impedance spectrum.

36. A method according to claim 35, wherein the pulse relaxation data is converted into an impedance spectrum by fitting frequency domain elements to the pulse relaxation data using Laplace inversion, and constructing an impedance spectrum from the frequency domain elements.

37. A method according to claim 35 or 36, wherein the pulse relaxation data is obtained by a galvanostatic intermittent titration technique.

38. A method according to claim 35 or 36, wherein the pulse relaxation data is obtained by hybrid pulse power characterisation.

39. A method according to claim 35, wherein the impedance spectrum is obtained from the power unit by electrochemical impedance spectroscopy, wherein the pure capacitance term is created using state of charge-open circuit voltage maps by applying coulomb counting to the electrochemical impedance spectroscopy profile.

40. A method according to any of claims 34-39 wherein the impedance spectrum is converted into a distribution of relaxation times and the diffusion impedance parameters obtained from fitting to the impedance spectrum are refined by fitting to the distribution of relaxation times.

41. A method according to claim 40 wherein the diffusion impedance parameters are fitted to a predetermined range of the distribution of relaxation times where diffusion effects are known to be found based on the Nyquist plot profile of the impedance spectrum.

42. A method according to any of claims 34-41, further including the method of any of claims 1-14 or 16- 32.

43. A computer-readable medium storing instructions which, when executed by a processor, cause the performance of the method of any of claims 34-42.

44. A system comprising: an electrochemical impedance spectroscopy (EIS) instrument for obtaining EIS data from a power unit, and/or a galvanostatic intermittent titration technique (GITT) instrument for obtaining GITT data from a power unit and/or a hybrid pulse power characterisation (HPPC) instrument for obtaining HPPC data from a power unit; and a processor programmed to perform the method of any of claims 1-14, 16-32, or 34-42.

45. The system of claim 44, further comprising a display for displaying the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters.

46. An electric vehicle comprising a system according to claim 44 or 45, wherein the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters are input as battery health data to a battery management system which manages a battery according to the battery health data.

47. An electric vehicle according to claim 46, wherein the battery management system is configured to output a notification to a user when the battery health data indicates the battery health has degraded below a predetermined threshold.

48. A computer-implemented method of comparing power units comprising: obtaining ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters for a plurality of power units according to any of claims 1-14,16-32 or 34-42; inputting the ohmic resistance parameters and/or dynamic resistance parameters and/or diffusion impedance parameters into a power unit degradation model; outputting the results of the degradation model; displaying the results of the degradation model; and ranking the power units by their expected degradation characteristics.

49. A method according to claim 48 further comprising the step of selecting the highest-ranked power unit for use in a vehicle.