GB2532726A - Cell internal impedance diagnostic system - Google Patents

Abstract

An internal impedance diagnostic system for an energy storage device is disclosed. A method of estimating cell impedance parameters from an impedance equivalent circuit model (ECM) includes the steps of applying or detecting a time-varying current signal through a cell, measuring voltage and current variations, applying an adaptive filter on the sampled data to identify the m-order transfer function for the measured system, constructing a set of simultaneous equations, extracting individual impedance parameter values from the equations, running diagnostic tests and algorithms to analyse the change in cell impedance values and computing accurate cell related estimation parameters and conditions. The energy storage device may be an electrochemical cell.

Description

CELL INTERNAL IMPEDANCE DIAGNOSTIC SYSTEM

BACKGROUND OF THE INVENTION

Field of the Invention

[0001] The present invention relates to a system and method for estimating the complex internal impedance of an energy storage device (e.g. battery, capacitor, fuel cell, etc.), or device consisting of a plurality of electrodes, or electrical connections, separated by a conductive medium or material, hereinafter referred to as "cell", from an equivalent circuit model, by establishing a relationship between the recorded current and voltage samples. More particularly, the invention relates to an intelligent battery management system for re-calibrating estimation parameters and diagnosing cell conditions, while identifying weaker cells within a battery pack, by implementing a comparative and empirical analysis on a cell's change in impedance within a battery pack, or single cell, for improving the performance and safety associated with battery operated applications and accurate impedance characterisations.

Discussion of Prior Art

[0002] Technological advances in electrochemical storage devices have enabled batteries to store increasing amounts of energy in smaller sizes and masses, allowing their increased exploitation over a wide range of industrial sectors. Lithium-ion batteries, in particular, have attracted significant interest for application in high power systems due to their inherently good cycle performance, high energy density and low internal impedance. While societal dependancy on energy storage devices increases, growing concerns over their potential failures and safety implications have presented challenges to their successful implementation and widespread adoption in high power systems.

[00031 A significant increase in the development of high power battery systems, in recent years, has been influenced by the availability of low impedance cells and continuous design of new chemistries. Battery technology has, however, struggled to keep up with industrial demands for greater reliability and increased safety. Current, commercially available, battery management systems fail to provide accurate diagnostic tools, necessary for preventing eventual early cell failures. The latter has already caused an estimated multi-million pound loss in economic and financial costs, particularly within the aerospace and automotive industry. Furthermore, the development of higher energy density and lower impedance in new cell chemistries have a significant impact on the viability of hybrid electric and high power transportation systems. Even though batteries are capable of delivering acceptable power requirements, they can only do so for a limited amount of time before their capacity or reliability limits their usefulness in such systems.

[00041 Establishing a good understanding in cell behaviour and relative physical condition are critical while developing hybrid electric and energy storage systems (e.g. hybrid-electric vehicles, smart grid renewable energy storages systems, unmanned aerial vehicles, etc). To ensure safe and proper operation with optimum performance, batteries usually rely on a Battery Management System (BMS); often requiring an accurate knowledge of cell dynamics, including thermal behaviour and complex algorithms for estimating a wide range of cell parameters. The acquisition of cell empirical data, including efficiencies, impedance and capacity fading, is also crucial for understanding how a particular cell chemistry behaves for a wide range of conditions. Manufacturers often provide customers with datasheets and specifications relating to fresh cells, though the amount of impedance increase over its lifetime, for example, are rarely disclosed, resulting in the misrepresentation of cell performance for practical systems, where a decrease in performance is observed with use and time.

[0005] A cell's State-of-Charge (SoC) is defined as its remaining capacity, expressed as a percentage of its nominal rated capacity or maximum capacity after cycling. Accurate SoC measurements are difficult to achieve in practice, where techniques, such as coulomb counting generate offset errors from inherent inaccuracies while measuring analog signals. Chemical phenomenons, such as diffusion, also cause estimation errors when the SoC value is mapped to the cell's Open Circuit Voltage (OCV). The diffusion process results in a slowly changing cell voltage, initiated by an interruption in the flow of current until the OCV equilibrium state is reached. The resulting SoC inaccuracies, often generate unsafe conditions where the BMS estimated parameters no longer represent accurate approximations of actual states; which can be improved by obtaining an online estimation of cell internal impedance values for recalibrating BMS estimation parameters accordingly.

[0006] Equivalent Circuit Models (ECMs) are generally used to simulate and predict cell performance, as well as their chemical and conductive behaviours. These models conventionally use a network of discrete circuit components to simulate the cell's electrochemical and electrical operations. The Randles ECM is commonly used for Lithium-ion battery related applications, though many others have been proposed in recent years. While not being the most sophisticated cell model possible, the circuit is often chosen due to its simplicity, versatility and universality for modelling such chemistries. The Randles model is also ideal for onboard estimation problems, since it does not employ distributed elements, such as transmission lines, or nonlinear frequency dependent components, such as Warburg impedances. The choice of ECMs usually depends on the system's available memory and computational cost, as nonlinearities arise from higher order circuits and complexities. However, for onboard applications, a simple model, which minimises the error between the measured and simulated data within an acceptable range is preferred.

[0007] A cell's State-of-Health (SoH) is a dimensionless parameter of its physical condition relative to its initial state. It reflects the cell's ability to deliver and store a percentage of its maximal performance. Unlike SoC, the cell's SoH has no absolute definition for representing a cell's physical condition, making its calculation subject to interpretation. There have been several attempts to estimate SoH using the cell's internal impedance or useable capacity, but it can usually be calculated from one, or more, of the following parameters: internal impedance, capacity loss, cycle number, coulomb efficiency, self discharge, or voltage drop. Consequently, no standard tests or practical method exists for determining SoH, and often systems attempt to estimate its value with various degrees of accuracies.

[00081 The cell's ageing process, or decrease in SoH, is generally associated with an increase in internal impedance and resulting operating temperature, which typically contribute to a decrease in performance and higher risk of failure. High power battery operated systems generally require cells with low internal impedances, capable of delivering large current bursts for extended periods of time. Since a cell's energy efficiency is defined as the difference between the energy transferred during charge and discharge; any increase in cell impedance would subsequently decrease its energy efficiency, and inhibit its ability to delivery maximum current. Consequently, accurate cell impedance measurements often result in a better understanding of its life cycle, performance and efficiencies.

[00091 Techniques such as Electro-Impedance Spectroscopy (EIS) have been widely used to accurately determine a cell's internal impedance, but it generally requires expensive and dedicated hardware equipment to operate on low impedances cells, making it unsuitable for use in embedded systems. The method applies a sinusoidal charging or discharging signal to the cell at a predefined frequency and measures the phase difference and magnitude between the applied current and voltage response. The technique also requires a full frequency sweep to determine the internal impedance of individual cells, which often leads to lengthy measurement times capable of altering a cell's SoC as charge flows through it. Furthermore, the technique suffers from induced electromagnetic interference (EMI), and requires a stable low noise environment to provide accurate estimation results, making its practical application limited to laboratory uses and niche markets.

[0010] Clearly, the development of an intelligent battery management system is needed for high power industrial applications, to provide a more efficient, yet more accurate, method for determining a cell's SoH and provide integrated tools for diagnosing its propensity to failure, while allowing for the recalibration of several cell estimation parameters for optimum performance.

BRIEF SUMMARY OF THE INVENTION

[00111 A system and method are disclosed for estimating individual component values from an impedance equivalent circuit model through the analysis of a cell's voltage and current recorded samples. The system applies a time-varying current signal to a single cell, or pack, and measures the resulting change in cell voltage to determine individual impedance parameter values. The method for extracting these values includes the evaluation of a discrete-time transfer function obtained from the selected ECM and the application of adaptive filtering techniques on the recorded cell input current and output voltage samples; resulting in a system of simultaneous equations, solved with the application of an appropriate solver.

[00121 Additional features of the present invention include the development of a cell impedance diagnostic tool for identifying particular damage modes, such as those induced by extreme temperatures, short-circuits, and overcharging conditions, by mapping the results to a relational empirical database and baseline measurements. The present invention may also be used for non-energy storage devices, which require accurate impedance estimations for characterising the electrical and conductive behaviours of the medium, or material, being tested. Other features are described in more detailed in the following sections, supported with accompanying drawings.

BRIEF DESCRIPTION OF THE FIGURES

[0013] FIG. 1 is a schematic diagram for implementing a diagnostic tool with a time-varying input current signal to a battery pack while measuring individual voltage responses through analogue-to-digital converters (ADC) for estimating cell impedance, while relaying relevant information to a monitor or display 9, according to an example embodiment of the principles of the present invention; [0014] FIG. 2 is a flow chart illustrating the necessary steps for estimating values of a cell's internal impedance over its operating range according to an example embodiment of the principles of the present invention; [0015] FIG. 3 is a second order battery equivalent circuit model, commonly referred as a Randles circuit model, used to accurately model a cell's dynamic and DC behaviour according to an example embodiment of the principles of the present invention; [0016] FIG. 4 is a second order impedance equivalent circuit model used for estimating and extracting individual cell impedance values according to an example embodiment of the principles of the present invention; [0017] FIG. 5 is a flow chart diagram for Kalman's linear algorithm problem used for identifying the behavioural characteristics from a system's recorded input and output samples; [0018] FIG. 6 is a flow chart diagram for Steiglitz-McBride's optimisation problem for identifying the behavioural characteristics from a system's recorded input and output samples; [0019] FIG. 7 is a flow chart diagram demonstrating the linearisation of the Steiglitz-McBride's optimisation problem for simplifying the original nonlinear problem; [0020] FIG. 8 is a graph illustrating a cell's voltage response for various values of SoC, while applying a fixed-length current pulse across a single cell; [0021] FIG. 9 is a graph illustrating a cell's voltage response for various values of SoC, while applying a variable-length current pulse to provide more accurate impedance measurements according to an example embodiment of the principles of the present invention; [0022] FIG. 10 is a flow chart diagram for determining faulty cells based on the estimation of individual impedance value, according to an example embodiment of the principles of the present invention; [0023] FIG. 11 is a graph illustrating the simulated voltage responses for several cell impedance solutions, including an impedance with a negative parameter, through an iterative improvement procedure according to an example embodiment of the principles of the present invention; [0024] FIG. 12 are graphs comparing the individual cell impedance values for both discharge and charge conditions over its entire SoC, according to an example embodiment of the principles of the present invention; [0025] FIG. 13 are graphs illustrating the effects of capacity fading on individual cell charge impedance values over its entire SoC after 500 cycles at room temperature, according to an example embodiment of the principles of the present invention; [0026] FIG. 14 are graphs illustrating the effects of short-circuits on individual cell charge impedance values over its entire SoC after short-circuiting a cell for periods of 500ms at room temperature, according to an example embodiment of the principles of the present invention; [0027] FIG. 15 are graphs illustrating overcharging effects on individual cell charge impedance values over its entire SoC for three separate maximum charging voltages at room temperature, according to an example embodiment of the principles of the present invention; DESCRIPTION OF THE EMBODIMENTS Problems to be solved by the invention [0028] It is well appreciated by those skilled in the art that batteries are dynamic systems, subject to physical and chemical changes, often leading to lower performances and altered electrical properties prior to ageing. Battery management systems, which use equivalent circuit models to estimate various related parameters, are rarely recalibrate throughout a cell's lifetime, resulting in inaccurate estimation parameters. Using a cell's impedance for assessing its physical and chemical deterioration, or ageing process, is a well established concept throughout literature. However, an efficient and integrated method is still required for accurately estimating impedance values, and allow BMSs to recalibrate estimation parameters, while assessing wether a cell's condition is within safe operating limits. It is also appreciated that cell deterioration is dependant upon its experienced usage, where, for example, significant drops in its life expectancy are observed when high current rates are used during cycling.

[0029] Manufacturing defects and design flaws can both contribute to a cell's complete failure. As hybrid-electric and electric systems develop, individual cells within battery packs require routine diagnostics to identify potential failures, induced by excess temperatures, over-charging or short-circuiting conditions. As mentioned above, the cell's impedance is a good indication of its actual deterioration, and any sudden change in impedance compared to neighbouring cells, or baseline values, can provide early warning signs for detecting faulty cells, which are conventionally identified through a technician's physical measurement of individual cell characteristics, within a battery pack, under various test conditions.

[0030] It is therefore an object of the present invention to provide an integrated and embedded iBMS 7 solution for improving the accuracy of estimation parameters after ageing and providing an impedance diagnostic system 7 for characterising and developing safer and more reliable cell designs.

Means for solving the problems [00311 In the following description, only certain example embodiments of the present invention have been presented and described for a Lithium-ion battery, by way of illustration. It would be apparent to those skilled in the art that the following embodiments may be altered and modified in various ways, while remaining within the spirit and scope of the present invention, for use in other energy storage and non-energy storage devices. Similarly, the drawing and description are to be regarded as illustrative and not restrictive, as certain components have been omitted to more dearly illustrate the invention.

[0032] With reference to the figures and drawings, the following discussion describes an embodiment of the present invention directed to a system and method for estimating the individual impedance parameters for a given energy storage device, or cell. Other estimated parameters derived from the analysis and evolution of impedance values, include thermal response, total capacity, SoC, SoH, life expectancy, dynamic frequency response, cell efficiencies, open circuit voltage, chemical concentrations, quality assessment, and development of a diagnostic tool for characterising materials or mediums, and detecting faulty cells for battery management systems.

[00331 The challenge to increase cell safety standards is addressed by implementing the present invention. The proposed system provides the necessary advancements in cell safety, testing and diagnostic tools for future industrial high power cell powered products. Most importantly, the technique is able to detect common cell faults, such as those induced by short-circuits, overcharging and external environmental abuse; while also being independent from cell chemical composition or materials, making it suitable for a wide range of chemistries and energy storage systems. Increasing cell efficiency and performance by estimating cell impedance are also discussed.

[0034] The present invention proposes an algorithm for estimating cell impedance parameters, and how it may be used to evaluate several key estimation parameters for the development of a cell impedance diagnostic tool, as mentioned above. In comparison to the prior art, a more efficient method is presented, while also minimising a CPU's computational cost, memory requirements, and robustness of estimated parameters in noisy environments. The simplistic implementation and accuracy of the inventive method also makes it more portable between applications and for a wide range of energy storage devices. The invention is further useful for diagnosing cell conditions, such as life expectancy and failures modes, while requiring minimal amounts of memory (e.g. RAM & ROM), while using traditional embedded microcontroller ADC modules 4 (e.g. 10-bit or 12-bit), such that minimal hardware alterations are necessary for proper operation. Applications for the invention may include, but are not restricted to, automotive, aerospace, and naval sectors, as well as large scale energy storage systems for the development of smart grids. The technique is illustrated in FIG. 2, and an example embodiment is implemented as follows: Select an appropriate rn-order cell impedance equivalent circuit model which provides suitable accuracy and complexity according to the application's specific needs.

* Apply, or detect, a time-varying current signal through a cell to produce a change in cell voltage.

* Measure voltage and current variations while applying, or detecting, the above time-varying signal.

Apply an adaptive filter on the sampled data to identify the nt-order transfer function for the measured system.

Construct a set of simultaneous equations, formed between the nt-order cell equivalent circuit transfer function and previously acquired adaptive filter coefficients for the measured system.

* Use an appropriate solver to extract individual impedance parameter values from the set of simultaneous equations formed above.

* Run diagnostic tests and algorithms, which analyse the change in cell impedance values, to compute accurate cell related estimation parameters and conditions.

[0035] The method also provides a better estimate of a cell's DC resistance, while also measuring several other cell related parameters, such as capacitance, electrode porosity, and kinetics of internal electrochemical reactions. Conventional methods for measuring DC resistance include the use of Ohm's law on high-frequency samples of current-voltage changes during an instantaneous event, or averaging samples over an extended period of time. While the former estimates a cell's series DC resistance, represented by Rig in FIG. 3, the latter computes the "true" DC resistance, represented by the combination of K0, K1 and K2, as shown in FIG. 3, but often overestimates it, as a change in open circuit voltage (OCV) occurs in response to a charge flow. The method proposed in the present invention uses a short duration pulse to evaluate the cell's impedance, in which accurate estimated of total DC resistance can readily be extracted, with minimal effects on its OCV.

[0036] An example embodiment of the present invention is described as a 2nd order impedance equivalent circuit model and discrete-time IIR filter of the same order, in which five individual impedance circuit values are extracted for a Lithium-ion battery, as shown in FIG. 3 and FIG. 4. The 2nd order circuit provides the necessary accuracy when simulating the cell's response over its entire SoC and temperature range for a given input signal. Further increasing the order of circuit presents additional complexities and nonlinearities, while providing negligible improvements in the model's simulation capabilities. Though the proposed method is primarily focused on estimating cell impedance, it can easily be adapted to estimate other cell parameters, such as estimating its ionic diffusion kinetics and resulting voltage recovery throughout the cell's resting period.

[0037] The following description investigates a cell's voltage change in response to a change in current. FIG 3. illustrates a 2nd order circuit, with the voltage source representing the cell's open circuit voltage. Since a cell's SoC remains substantially unchanged when small or short current signals are used, the voltage source becomes a DC component signal which can be removed from the analysis, as shown in FIG. 4. This suggests that all DC components from recorded voltage or current samples must be removed. The normalised, continuous-time, transfer function Z(s), of the impedance equivalent circuit shown in FIG. 4, is given by the following expression: b2s2 + s + bo Z(s) -Where: b2 -C1C2R0R1R2 /71 = C1R0R1 C1R1R2 C2R0 R2 C2R1R2 b0 = RO ± R1 ± R2 a2 = C1C2 R1 R2 a1 = C1R1 + C2R2 [00381 Assuming that the transfer function coefficients are known, the above expressions represent a set of simultaneous equations, used for estimating a cell's corresponding impedance value (i.e. Ro, R1, etc). However, values for the individual circuit components cannot be obtained directly through mathematical manipulations but instead must resort to using nonlinear solving techniques to obtain corresponding values.

[0039] Cell voltage and current measurements are acquired through sampling, producing a series of discrete signals. Consequently, the above continuous-time transfer function must be converted into its discrete representation, though a continuous analysis can be used instead. The bilinear transform, also known as Tustin's method, is used to transform the continuous-time system into its discrete-time representation, though other methods can be used. The bilinear transformation maps the s-plane (i.e. analog) into the z-plane (i.e. digital), and transforms analog filters, into their discrete equivalents by setting:

-

s-kz 1 z+1 2 and T, = Sampling time Where k = Ts [00401 Substituting the discrete-time equivalent of the Laplace variable s, into Z(s), transforms it into its corresponding discrete-time representation Z(z): (a1c2 + bk + c) + (2c -2a/c2)z1 + (ak2 -bk + c)--Z(z) = (k2 dk + e) + (2e -2k2)z 1 + (k2 -dk + e)z 2 a2s2 + a1s + 1 Where: a = CiRoRi + C RI R2 + C2K0K2 C2K1 R2 b -RO R1 ± R2 CiC2R1R2 d -CH R1 ± C2 R2 C1C2N1 N2 C1 C2 K1K2 c -e= C1C2R1R2 [0041] The present invention employs an Infinite Impulse Response (IIR) adaptive filtering technique on the cell's recorded current and voltage samples, in order to compute a general second order IIR filter transfer equation. A wide range of adaptive filtering techniques, including continuous or discrete, can be used to compute the coefficients of second order transfer function for the recorded system, but the discussion will only cover the use of the Ste i gl itz-McB ri de method.

[0042] The Steiglitz-McBride algorithm is an iterative technique for identifying the parameters of discrete linear time invariant systems from samples of its input and output variations, by minimising the mean-squared-error between the system and measured output The methodology is based on the equation error formulation, but makes use of pre-filters to linearise the cost function to one which involves output errors, as illustrated in FIG. 7. The proposed method can be categorised as an adaptive filtering technique, or system identification problem, where the model consists of a rational transfer function in the z-domain.

[0043] Consider an input sequence { uti} and output sequence {y,}, related by: 1/22 = 1-1 (2)11 n where is some stochastic sequence which is statistically independent of the input sequence {u"} and H(z) is the system's impulse response transfer function which has the following rational form: B(z) H(z) A (z) where: M=2 M=2 A(z) =1 + E akZ k- B(z) = E bkz-k k-1 k-O [0044] The following assumption will be made throughout the analysis: {un} and {r;n} are statistically independent from each other and represent zero mean processes; while the transfer function H(z) is both stable and causal. By using the available input and output sequences un} and tynt, the transfer function H(z) can be reconstructed. This takes the form of an normalised (i.e. rip = 1) adjustable rational transfer function model, denoted as H(z), of degree M = 2: B(z) M=2 H(z) A(z) L bkz k k=0 M-2 + L akz-k k-1 [0045] Kalman et. al. proposed a linear regression method to identify a system's characteristics based on the input-output sequences, shown in FIG. 5, which involves the following least squares minimisation problem, with N being the number of acquired samples: N 1 Min E e2, = 12.

Y (z) A (z) -U(z)B(z)12 -dz X a-11=0 Where the contour of integration is on the unit circle lz = 1 and x is the estimation parameter vector for the normalised second order system (i.e. M = 2) minimisation problem: x = Lai a2 b0 bi b21T [0046] The sum squared errors can be interpreted as variance, where its minimisation is equivalent to minimising the energy generated in the errors. Interpreted geometrically, the sum squared errors provides a measure of the Euclidean distance between the predicted and actual values, where the output error at interval n is given by: M=2 M=2 en = yn + E akyn_k -E bk un -k k-1 k-0 If the input-output vector Q at time n is defined by: Q = [-Yn-1 -Yn-2 Lin un-i un-2] hi matrix form, the error becomes: etz = y,., QnX Oen = QT Summing e2" over the entire record length (i.e. number of samples N), the minimum occurs when the derivative with respect to x equates to zero: ox rt-0 n-0 N-1 N-1 = 2 -de" = -2 Q, ee, = 0 en 1-1 bx n=0 n=0 Substituting en in the above expression yields: N-1 2 E 127; (y, -Qnx) = 0 n=0 N-1 N-1 <=> n=o n=0 = ,E=0 (27;yn Then e112 is minimised if the parameter vector x, is equal to: N Nr ( x = E QT, c2n) E QnT yn n-O [00471 The linearity of Kalman's minimisation problem shown in FIG. 5, makes it straightforward to solve, but as Steiglitz et. al. notes, it is usually not the one of interest. Kalman's method calculates a residual error which does not have any real physical interpretation, and a more meaningful minimisation problem to solve is given by: /V1 B(z) 2 min E e2 - A(z)Li(z) -Y(z) dz X-0.

n 27TI [0048] The expression above represents the mean square error between the predicted output and the observed output of the system being identified. However, unlike Kalman's method, the above equation is a highly nonlinear regression problem, which cannot be solved directly. Steiglitz et. al. proposed a technique for carrying out the minimisation of the nonlinear problem by iteratively carrying out linear minimisations given by Kalman's method, which is no more difficult than implementing the previously mentioned linear regression, but exchanges increased computation time for a better approximation to the solution of the above nonlinear equation.

[0049] The Steiglitz-McBride method, shown as a block diagram in FIG. 6, is an algorithm for computing a pole-zero rational model and uses an iterative procedure to modify coefficient values in order to minimise the error from the following least-squares problem: 1 B,(z)U '(z) -Ai(z)V(z) 2 dz min E en = 27i Where i is the number of iterations and: 1 1 U1(2) = Ai i(z)U(2) (z) = Ai 1(z)Y(z) [0050] The algorithm applies a dynamic lowpass prefilter A to the observed input 1(z) 1 and output sequences as shown in FIG. 7, which is adjusted throughout each iteration.

Consider the following filtered input and output variables tit, and yiti respectively: 1 1 A1 1111 _1(z) A1_1(z)Y" [0051] By applying the above prefilter, the nonlinear objective function given by FIG. 5, is transformed into a linear problem, as shown in FIG. 6. Convergence of this iterative procedure corresponds to the sequence of polynomials A, (z) and B; (z) as they approach their limit, denoted as Aoc,(z) and 600(z), towards the original nonlinear objective function presented earlier: 1\I 1 2 dz min E en 13,o(z)140(z) -Aco(z)11=0(z)1 = f =0 2TCj dz z 13,,,(2) U(z) y z) Acv(z) I [0052] The equation error signal en, from Kalman's minimisation problem, shown in FIG. 5, is given by: en = A(z)y, -B(z)un Similarly, the equation error signal en from the linearised Steigltiz-McBride minimisation problem, shown in FIG. 7, is given by: en = A(z)V, -B(z)un M=2 M=2 = E k + akz E bkz-klln k-1 k-0 In matrix form, this becomes: N-1 e,2, -Cl -C213 12 n=o Where: a = , al a 2]1' fi = [b0 bl b 211' c1= tio 0 yi yo 0 Yi2 yi yo Yiv 2 Ytiv 3 Uro 0 0 141 no Cz = u12 ul no * I, u iv -I LI N-2 LIN 3 Giving rise to the following linear system: N -1 Eet2i e 2 b -A x 2 rt=0 where: x = [al a2 bo b21T b Yo 0 2 YI N-1] T 0 0 1 0 yo 0 "10 0 A= Yi -!4) tif id I no 2 -"/N-3 N 1 "N-2 NT 3 [00531 The solution to this least squares minimisation problem, is given by solving the following expression for x: A x = b [0054] Since the number of samples N is usually much larger than the number of unknown variables, the above equation represents an overdetermined system and therefore no exact solution is possible. However, it is possible to find a unique solution which minimises the £2-norm of the residual error vector, with the least squares minimisation problem solved via the following normal equations: x= (ATA) I ATb = AlEb [0055] Where A+ is the Moore-Penrose pseudo-inverse of A. In principle the problem can be solved directly, but this may not be advantageous since the calculation of ATA introduces the square of the condition number of A, likely making ATA ill-conditioned and singular. A more stable approach to solving x is to use the QR decomposition of A into A = QR, where Q is an orthonormal matrix (i.e. QTQ = I) and R is a square upper triangular matrix. The problem is now given by: RTQTQRx = RTQTb RTRx = RTQTb Rx = QT b Rx = R+T ATb x R-1R-T Arb [0056] The technique is usually referred as the Q-less QR Factorisation, since the usually very dense matrix Q is not explicitly required to solve the least-squares minimisation problem which is very advantageous in the case where the sample number N is very large.

[00571 The combination of limited precision in microcontrollers, compared to traditional CPUs, and the accumulation of roundoff errors associated with tall matrix multiplications, generates large calculation errors, and can cause the above system to become unstable. Most microcontrollers have a machine precision of 10-16, which causes large accumulated errors while computing a matrix inverse with a condition number greater than 1016, and tall-matrix multiplications. A common approach to this problem is the use of iterative refinement for improving the quality of an approximate solution to a linear system, and consists of three main steps: * Compute residual from solution of x: r = b -Ax ^ Solve Ad = r, using the previous QR factorisation of A * Compute the refined solution as xrpf = x + d * Repeat process until residual is below a certain threshold [0058] Further improvements in terms of computational and memory requirements can be achieved by taking advantage of scaling properties of transfer function coefficients with respect to the input/output (i.e. current/voltage) relationship. This allows the system to directly record and store raw ADC 4 current and voltage samples, without the need of converting them into their respective scaled current and voltage units. This decreases the amount of operations required, while also minimising the amount of memory for storing the data, without compromising the calculation of the adaptive filter's coefficient values.

For example, if a 12-bit ADC 4 is used to simultaneously sample current and voltage analog signals, then these can be stored as integer values in memory and directly used to calculate the adaptive filter coefficients without the need of converting ADC 4 data into current and voltage equivalents. Once the adaptive filter coefficients have been identified, the ratio between voltage and current (i.e. 14) must be computed in order to scale back the previously acquired coefficients in order to identify the system being measured. However, only the numerator coefficients (i.e. 13 = [bp b, b2]) are affected by the scaling process, while the denominator coefficients in H(z) remain the same. Consequently, by multiplying the vector coefficients [3 with the ratio given by the lc-ratio, the system's correct coefficient values can be determined while optimising computational cost and memory requirements without the need for prior data conversion.

[0059] After convergence, the solution to the least squares minimisation problem represents the solution for the discrete form transfer function for a system which describes the relationship between the cell's input current and output voltage. As discussed earlier, this forms a system of simultaneous equations which can only be solved through nonlinear techniques. For example, a trust-region Dogleg algorithm can be used to extract individual impedance parameter values from the system of non-linear equations.

[0060] The coefficients in vector x = [a a2 b0 1,1 b2]T are solutions to the previously measured system, and form a system of nonlinear simultaneous equations given by: k2 dk e ak2 -bk c k2±dk-Fe 2c -2ak2 bl k2 dk a ao = 1 bo 1c2 dk e k2 -dk e 2e -2k2 ai k2 dk e ak2 bk c Where the expressions for a, b, c, d, e and k were given earlier, in terms of impedance parameters R0, R1, R2, C1 and C2.

[0061] Individual impedance parameters are extracted by solving the above nonlinear system. The solution then represents the cell's impedance at a particular SoC, temperature, or state. The entire process can be repeated for other SoC and temperature values, or states, to provide an detailed analysis of its impedance over its operating range, as shown in 2. The technique can be used to acquire baseline values of its impedance before and after cycling over its entire SoC and temperature operating range. This information is then used to develop an impedance diagnostic system 7 to assess if a particular cell is operating within its normal operating range. The diagnostic system 7 therefore tracks a cell's evolution in impedance to identify potentially dangerous cells, with high risks of failure. An algorithm for such a system is illustrated in FIG. 10 as a flow chart diagram. The algorithm starts with an accurate estimate of cell SoC and factors which affect its impedance values such as temperature. It then compares cell impedance values with empirically obtained baseline values from a fresh cell, with those obtained after cycling. To diagnose particular cell failure modes, baseline failure tests are carried out, to identify the mechanism of how the cell's impedance evolves in response to a particular failure mode (i.e. short-circuiting, overcharging, etc).

[00621 Solving the system of simultaneous equations might, in some cases, run into difficulties and provide non-positive resistance or capacitance values, which are problematic for comparing, analysing and understanding physical interactions occurring within the cell. Even though negative resistances are possible in electronic circuits, their presence in a cell's impedance holds no "true" significance. These can be removed by constraining the solver's variables to positive values, though this usually requires additional complexity and increases the number of floating point operations of the solver being used. An example embodiment of the present invention provides an algorithm for converting negative impedance values into an equivalent positive value, while still maintaining an appropriate solution for the simultaneous systems of equations being solved.

[0063] Even though a negative impedance has no "true" significance, it still provides valuable information, necessary for solving the system with an equivalent positive value. By extracting the relevant information generated by the negative impedance value, such as total resistance, RC time constants and an estimate of series resistance Ro, a simple iterative procedure can be implemented for quickly and efficiently solving the system with positive valued resistances and capacitances. The following algorithm employs the second order system used earlier according to an embodiment of the present invention, and consequently its implementation should be adapted according to the order of the system being studied. The following example algorithm converts any potential negative impedance solution into an equivalent positive value for solving the system of simultaneous equations, according to an embodiment of the present invention: * Compute the first RC time constant: T1 = R1C1 ^ Compute the second RC time constant: T2 = R2 C2 ^ Compute the total resistance: Rtot = Ro + R1 + R2 * Estimate appropriate value for Ro = Vhf //h f ^ Re-initialise R1 to expected value * Compute new values for R2, C1, C2 from T-1, T-2, and Riot ^ Determine the model's error and compute its norm ^ Increase R1 and recalculate model's error and norm I.-Continue increasing R1 while error reduction is observed ^ Ensure that all impedance values are positive [0064] Where Vhf and lhf correspond to the high-frequency voltage and current signal values respectively. To illustrate the principle of the present algorithm, the solution for the system of simultaneous equations with the following discrete transfer function coefficients is given by: x = [-1.1829 0.1910 0.0137 0.0167 -0.029917' = -0.0139 0 Rl = 0.014802 N2 = 0.0682 0 C1 = 1.4628 F C2 = 0.0024 F [0065] The error generated from the system of simultaneous equations between the expected transfer function coefficients and individual impedance values provided earlier is given by: error = 10-18 [0.1735 -0.3469 0.3469 0 0] T error = 5.2042 x 10-18 [0066] By applying the above algorithm, the negative resistance value Ro can be converted to a positive value, while still providing a reasonable solution for the system above with a minimal amount of operations: rl = 0.0217 sec T2 = 1.6153 x 10 5 sec 0.5374 Run = 0.0692 0 R0 = 0.0528 0 10.1824 [0067] The value for R1 is initially set to R1 = 0.0001 0 and is iteratively incremented by 0.0001 0, where the values of R2, C1, C2 are computed from the values of previously calculated constants Tl, T2, and Rtot, at each stage i of the iterative procedure: i = 1: No = 0.0528 0 Rl = 0.0001 0 R2 = 0.0163 0 C1 = 217.12 F C2 = 0.0099 F errorll = 0.0973 i = 10: Ro = 0.0528 0 = 0.001 0 N2 = 0.0154 0 CI = 21.712F C2 = 0.0105F error H = 0.0972 = 154: Ro = 0.0528 0 R1 = 0.0154 0 R2 = 0.0010 0 C1 = 1.4098 F C2 = 0.1598 F error H = 0.0969 [0068] Other iterative improvements can be implemented on individual impedance values, such as Ro, to provide a better estimate for the system's solution, though this increases the computational requirements and complexity of the algorithm. Even though the new solution is not exact and generates a larger error than its former solution, the computed values are more consistent with cell behaviours, while also closely simulating actual cell voltage response, as illustrated in FIG. 11, such that the newly extracted positive impedance values can be used for further analysis.

[0069] Another approach, though more computationally intensive, would be to minimise the sum of squared errors, or residual sum of squares (RSS), between the expected and predicted values, computed from the IIR rational model's coefficients in vector x (i.e. al, a2, b0, b1, and b2) and impedance parameters (i.e. R0, R1, R2, C1, and C2) respectively. By evaluating the RSS between the expected response y and predicted response.17, a "goodness of fie can be computed. The normalised (i.e., a() = 1), z-domain M-order rational transfer function can be expressed as a difference equation for evaluating the output responses yn and ti,,, from an input signal un: Y n = E bkUn-k E akyn_k k=0 k=0 = E bk11 n k E akkil k k=0 k=0 Residuals = rz = Yn Goodness of fit = RSS = Er" [0070] By using the above definition for RSS and evaluating filter coefficients gn from each iterated impedance values, the following expressions can be written: 1: .11? = [-1.1829 0.191 0.0614 -0.0665 0.0057] T RSS = 1.9734 i = 10: i = [-1.1829 0.191 0.0611 -0.0665 0.006-T RSS = 1.8072 i = 154: z = [-1.1829 0.191 0.0553 -0.0665 0.0118] T RSS = 0.3550 Other methods can be used to evaluate how well the newly computed "positive" impedance solution compares to its expected value, though the objective remains unchanged. Further improvement can be achieved by adjusting individual impedance parameters, while decreasing the incremental step value throughout convergence. FIG. 11 graphically illustrates several solutions obtained with the above iterative procedure, while comparing them to the expected cell response and the previously obtained "negative" impedance solution with its RSS given by: RSS = 3.8940 x 10-29 [0071] A cell's voltage response and subsequent recovery is primarily dependent on SoC. This translate into a longer recovery period at lower SoC and a shorter one at higher SoC. Ideally, the cell's voltage recovery should persist beyond the total amount of time being recorded, such that the voltage and current sampled sequence provides the maximum amount of information on the system being measured. While a single pulse width can be used, the system is no longer well defined after at higher SoC values, and therefore the measurement's accuracy decreases as SoC increase, as shown in FIG. 8. By varying the width of the current pulse, a more accurate impedance measurement is obtained for all values of SoC. Consequently, the method's accuracy can be improved by choosing an appropriate time-varying input signal for measuring the cell's output voltage response. FIG. 8 illustrates how the cell's voltage response changes over time for three different SoC values, where its response quickly relaxes towards steady-state for SoC values of 50% and 100%, at which point any data recorded beyond no longer contributes to the characterisation of the system being identified. Conversely, FIG. 9 illustrates the use of a variable length pulse current signal, which increases the amount of information for the system being identified, and in turn considerably increases the estimation's accuracy. The proposed method can use a range of time-varying signals for estimating cell impedance, for improvements in practicality and measurement accuracy, or acquire numerous samples and averaging them for increasing the measured data's signal-to-noise ratio.

[0072] The time-varying input current signal may also originate from any instantaneous events, transient events or triggered events, which would occur naturally or induced by the transfer of energy to a load, or from a source, thus allowing a system to regularly update cell parameter estimators, while assuring proper non-faulty operation. An instantaneous event is defined by a large magnitude change in charge flow during a relatively small period of time, wherein energy is added or removed from the cell; though the parameters defining instantaneity can vary according to the type of application and cells being used. For example, common events in hybrid-electric vehicles, such as acceleration or deceleration, can provide the necessary high transient voltage changes for identifying a cell's internal impedance with the method illustrated in the present invention. It is therefore appreciated that the cell's impedance, associated correlations, and estimation parameters can be determined as frequently as desired or where a change in charge flow is detected.

[0073] The second embodiment of the present invention represents an intelligent battery management System (iBMS) design, capable of recalibrating cell 10 estimation parameters after ageing and diagnosing individual cells for faults, as depicted in FIG. 1. The coulomb control element 1 is used to control the amount of charge, or coulombs, flowing through the cells, which can take the form of common circuit components, such as MOSFETs, bipolar transistors, or IGBTs, to apply a pulsed or time-varying signal (e.g. sinusoidal signal) to a single cell 10 or plurality of cells simultaneously The microcontroller 7, or controller, is used to control the latter for measuring cell 10 current and voltage signals, while also estimating a mathematical relationship between them for determining a cell's charge or discharge impedance, where it is stored into memory for further analysis. The controller is also used to control several other modules, while acquiring information from several sensors/transducer, including the temperature of individual cells and external environment. The telemetry send/receive circuitry 8 is used to transfer information to and from the user interface, for displaying cell 10 states, conditions and warning messages if a faulty cell 10 has been identified. The display 9 also allows the user to interact with the data and send commands to the microcontroller 7 for executing cell 10 related diagnostic systems, calibrating estimation parameters or provide updated cell 10 information. The ADC 4 and current elements 3 represent devices capable of sensing voltage and current respectively, while converting them into digital signals. The current elements, can either be used to measure individual string currents in a battery pack, or collectively measuring combined current and dividing it by the number of string cells in a battery pack. The iBMS 7 is constructed with several sensors, including a coulomb counter 7, an impedance diagnostic system 7, and a recalibration unit. The sensors are used to determine a cell's charging or discharging current flow, as well as its surface temperature and terminal voltage. The coulomb counter 7 is a widely used method for estimating the cell's SoC by counting the amount of charge flowing through it. The impedance diagnostic system and impedance estimation method, discussed previously, are used to diagnose for faulty cell 10 conditions, induced by short-circuiting, overcharging or excess temperatures conditions for example. The diagnostic system 7 can also be used as an effective comparison tool for detecting weak cells among neighbouring cells in a battery pack, which are likely to become problematic and ultimately fail as a result. The comparative diagnostic system 7 may also be used to detect unbalanced cells, as these will display differences in impedance values compared to neighbouring cells when poorly balanced. Consequently, any cell 10 impedance values which are considered dissimilar to its surrounding neighbouring cells, provides a good indication of a fault or weak cell, which would need to be replaced or serviced to lower the risk of failure. The battery charging system 2 can be a source for charging the cell, or applying a voltage across it; whereas the discharge system or load 6 can be a had for discharging the cell.

[0074] The recalibration unit allows for the correction of battery models in an iBMS 7 according to the values, or changes in impedance values, to provide for more accurate and precise parameter estimations throughout a cell's typical degradation. The iBMS 7 can also use values of a cell's internal impedance to signal its End-Of-Life (EOL) based on predetermined threshold values for individual impedance parameters. Accurate estimations of a cell's End-of-Life plays an important role when determining the price of a particular battery pack, where early replacements results in high costs, and late replacements result in higher risks for failure, though cell replacement criteria depend on the application. Similarly, the iBMS 7 can estimate the total power and heat generated from the total estimated DC impedance, Ruc, by using, for example, P = RDc12 for a given current source or load I. This allows the iBMS 7 to predict dangerous situations in which the internal temperature of a cell exceeds the manufacturers rated values or causes adverse reactions of the chemicals within the cell. Similarly, the cell's instantaneous, or continuous, output power can be estimated during its entire lifecycle, for predicting cell performance under its current state and temperature. For example, the internal impedance of a cell increases proportionally with decrease in temperature, and drawing current from it may permanently damage it and further increase it impedance. By predicting the maximum power delivery from a cell and its performance, the system may choose to warm the cell prior to its use, for mitigating the detrimental effects of drawing large currents while the cell is subjected to cold temperatures and assuring power availability for high power applications. Since the iBMS 7 measured cell impedance regularly, it can recalibrate its estimation parameters for accurately predicting cell behaviour irrespective of its history or current ageing condition. Finally, the iBMS 7 can use correlations between individual or a combination of impedance values and cell related parameter, such as total capacity SoC, OCV and temperature, for prediction purposes, even when correlation does not necessarily imply causation. Other correlations may include, but is not limited to, increases, or decrease, in individual impedance values with over-charging/over-discharging, short-circuiting, exposure to extreme temperatures, external pressure, chemical and physical interactions, or defective cell.

[0075] The discrete-form transfer function coefficients, identified earlier, can also be used to determine a cell's optimal charging/discharging frequency for optimising its performance and efficiency, by analysing the frequency response in the z-domain and substituting z = el' in Z(z) to obtain the impedance equivalent circuit's frequency response Z (co = 7-27f) evaluated on the Ts unit circle z1 -1. By computing the magnitude and phase of the cell's impedance transfer function Z(z), discussed earlier, the minimum required frequency for optimum charging/discharging, can easily be calculated. The presence of parasitic inductance has been omitted from the impedance equivalent circuit model in FIG. 3 and 4, and consequently further increasing the cell's frequency from its minimum value will eventually result in larger impedance magnitudes in practice. Cell efficiency is increased by lowering the voltage drop due to the its internal impedance with an appropriate time-varying signal (e.g. sinusoidal signal), which also facilitates chemical interactions taking place within the cell, thus increasing its SoH and lifetime.

[0076] Batteries are electrochemical devices, subject to performance degradations, and generally exhibit no outward physical change during this process. Their lifespan, or SoH, are dependent on external environmental factors throughout their operation, among several other complex dynamic factors. Total capacity is often used as an indicator for a cell's SoH, where a 20% decrease with respect to its initial value is conventionally used to determine a cell's End-Of-Life. However, a cell's total capacity is difficult to measure in practice as it usually requires a full charge followed by a full discharge to provide an accurate estimate of total capacity. Additionally, cells might exhibit very little decrease in total capacity while still having a large internal impedance, resulting in large voltage drops, lower efficiencies and higher risks of failure. In many systems, cells are rarely sequentially charged and discharged completely without interruptions, and accumulated errors in techniques, such as coulomb counting, make total capacity values unreliable and inaccurate for determining a cell's End-Of-Life condition.

[0077] A different approach to solving this problem, takes the form of an impedance diagnostic tool for determining and estimating cell End-Of-Life conditions, where a baseline measurement of the evolution and variation in internal impedance changes as a result of several experiments, including capacity fading, short-circuiting, overcharging, and extreme thermal environments. By understanding how cell impedance changes as a function of previously mentioned factors, an accurate cell diagnostic tool can be developed, for predicting, estimating and recalibrating several cell parameters for increased performance and safer operation. Large variations in cell chemistries and manufacturing processes leads to considerable variability in the correlation between the evolution in internal impedance and cell parameters, such as total capacity and failure modes. Therefore, batteries with dissimilar chemistries or manufacturing processes will exhibit modified impedance behaviours, and the acquisition of empirical cell impedance data for a wide range of different situations is crucial for understanding how a cell might behave prior to failure and how changes in its value correlates to important cell related parameters as mentioned earlier. Batteries generally have different charging and discharging impedances, as illustrated in FIG. 12, which provides additional information for use in recalibrating cell parameters, diagnosing failures and ensuring optimal performance.

[0078] To better illustrate the principles and operations of the present invention, empirical cell impedance data were obtained are presented in FIG. 12, FIG. 13, FIG. 14, and FIG. 15, for different conditions as a function a cell's SoC at room temperature. Results are presented for a second order equivalent circuit model, to illustrate by example, how a battery management system may use and correlate impedance changes to cell parameters. As mentioned earlier, other cell chemistries might exhibit different impedance changes over SoC, though the principles and steps required should remain in principle identical. Similar graphs could have also been presented for many other conditions, such as discharge impedance and temperature effects on impedance and SoH. The present invention relates to the observed change in internal impedance for charge and/or discharge, and as such, its evolution can be measured as a function of any relevant cell parameter, such as SoC or open circuit voltage.

[0079] FIG. 12 illustrates the evolution in cell internal impedance for both charge and discharge, for a particular chemistry and manufacturing process over its SoC, noting that different impedance variations might be observed for other dissimilar cells. The present invention makes use of the results in FIG. 12 as baseline values for assessing and comparing subsequent impedance calculations throughout the cell's lifetime. For example, a predetermined increase in total DC resistance, or individual resistances, compared to baseline values might be used to determine a cell's End-Of-Life. Similarly, changes in impedance capacity values, compared to baseline values, might be used to determine a correlation between a cell's total capacity, or diagnose a cell's particular failure mode and the kind of deterioration it has seen. By acquiring empirical impedance data at various temperatures, a system can also determine and estimate a cell's internal temperature by correlating changes in internal impedances with temperature dependant variations, where for example, a decrease or increase in individual impedance values can be associated with a similar change in temperature.

[0080] The present invention uses both empirical and baseline values of cell impedance for evaluating cell parameters, while comparing them to a cell's measurement of internal impedance during its operation and lifetime. A good understanding of the evolution of cell impedance, under a wide range of conditions, allows for better diagnostic performances. FIG. 13 through 15, illustrate the evolution of a cell's charge impedance for capacity fading, short-circuiting and overcharging respectively. By investigating how individual impedance values behave as a function of cell parameters, such as total capacity and particular failure modes, can be quite beneficial for providing accurate estimation parameters for use in battery management systems, while ensuring safer cell operations. For example in FIG. 13, the increase in cell cycles results in a 2% decrease in total capacity, and a noticeable decrease in Cl and C2 values. Factoring in the increase in resistance values, the present invention correlates a decrease in C1 and C2 values with a decrease in total capacity for the particular chemistry being tested. This is particularly true since the combination of R1, C1 and R2, C2, are derived from chemical properties, such as diffusion processes and Helmholtz double layer capacitance, occurring within the cell. Capacity fading is the result of unwanted side-reactions occurring throughout the cell's chemical reactions, where compounds are irreversibly converted into different compounds, therefore lowering the concentration of active elements in the cell. The diffusion process occurring within the cell, represents a slow moving dynamic behaviour and is represented by the values of R1 and Cl, as their respective time constant is larger than the combination of R2 and C2. Lowering the concentration of active elements, such as Lithium ions, or increasing the impedance of their conduction path, directly affects the diffusion process. The effect is also observed in FIG. 15, where the values of C1 decrease proportionally to increases in charging voltage, demonstrating its adverse effect in accelerating and promoting the development of chemical side-reactions, thus decreasing the concentration of active material and increasing overall impedance. The present invention investigates these changes for determining correlations between several parameters, diagnosing cell failures and investigating the evolutionary mechanisms associated with actual failure modes.

[0081] Finally, the embodiments of the present invention can be used to evaluate other systems which are not directly linked to energy storage devices. Such devices include testing apparatus for measuring and characterising electrode properties, or ionic conduction of liquid electrolytes or other conductive mediums. The present invention can therefore be extended for systems which require rapid and accurate impedance estimations for characterising the conductive properties of a medium, or material, between two, or more, electrodes or electrical connections. For example, the conductive behaviours and concentration gradients of ionic elements in a salty solution (i.e. 1\ I a+ ± C1-) can be determined from such impedance measurements techniques. Other examples include, devices which monitor oil or lubricant condition for quality purposes, or devices which measure the characteristics of parasitic (e.g. inductive, capacitive, resistive, etc.) elements in materials, mediums, or electronic components.

[0082] While the present invention has been described in connection with what is presently considered to be practical example embodiments, it is to be understood that the invention is not limited to the disclosed embodiments. Obvious modification to the example embodiments and methods of operation, as set forth herein, could be readily made by those skilled in the art without departing from the spirit of the present invention, but is rather intended to cover various modifications and equivalent arrangements included within the spirit and scope of the following claims.

Claims (23)

1. What is claimed is: 1. A system for determining the complex impedance of one or more electrochemical cells comprising a means for observing and/or controlling the current flowing through the cell or cells, means to monitor the temperature of the cell or cells, means to monitor the voltage across the cell or cells, analogue to digital converters for each of the temperature, current and voltage signals, a microprocessor, microcontroller or other digital processing device for analyzing the measurement data signals such that the impedance of the cell or cells may be determined and modelled with an equivalent circuit model of discrete electrical components where previous characterization of the cells being monitored have shown correlation with the state of charge, temperature and state of health of the cells and such correlations have been recorded such that the values of the derived electrical components and changes therein are used to determine the state of charge and/or state of health of the cell or cells being monitored.
2. 2. A system according to Claim 1 where current transients of peak magnitude between 0.1C and 25C, where C represents the ampere-hour capacity of the cell or cells, are used to induce a measurable voltage signal across the cell or cells suitable for determining the complex impedance of the cell.
3. 3. A system according to any preceding claim where the transient current timescale is within the range 0.1 ms to 10 s and the measurement period for each transient event extends up to 1000 s.
4. 4. A system according to any preceding claim where the results or data from multiple transients are averaged or otherwise combined to improve the signal to noise ratio, integrity O or reliability of the inferred impedance values, state of charge and/or state of health.
5. 5. A system according to any preceding claim where the cell or cells impedance parameters C\I and states is stored in a non-volatile memory device in order and/or with timestamps.
6. 6. A system according to Claim 1 where the impedance parameters are used to predict and/or monitor failure or degradation of the cell or cells.
7. 7. A system according to Claim 6 where the nature of the degradation or failure mode may be determined by reference to previously characterized and/or recorded impedance parameters for the cell or cells.
8. 8. A system according to any of the preceding claims wherein determining the estimated impedance parameters of the cell or cells involves: selecting an appropriate equivalent circuit model for the cell being used, acquiring voltage and current samples from a cell, or string of cells, at any given state from a transient event or time-varying signal, employing an adaptive filtering technique, or system identifying algorithm, on the input and output sampled data for solving a general transfer function of similar order of the equivalent circuit model and solving the simultaneous equations formed between the transfer functions of the equivalent circuit model and adaptive filtering method with an appropriate solver.
9. 9. A system according to any preceding claim wherein the equivalent circuit model is a second or higher order Randles circuit model.
10. 10. A system according to claim 8 wherein the adaptive filtering technique is an DR filter.
11. 1 1. A system according to Claim 10 wherein the IIR filter coefficients are identified through implementation of the Steiglitz-McBride system identification algorithm.
12. 12. A system according to Claim 8 wherein the simultaneous equations are solved with a nonlinear solving technique.
13. 13. A system according to any preceding claim wherein variations between impedance parameters of individual cells when compared to those of other cells is used to identify a potentially problematic cell.
14. 14. A system according to any preceding claim wherein the state of the cell is assessed by analysing a single, or a combination of, estimated impedance parameters.
15. 15. A system according to any preceding claim wherein the state of a cell or cells is determined by comparing the estimated impedance values with those from a cell or cells subjected to similar environmental and load conditions.
16. 16. A system according to any preceding claim wherein negative estimated impedance parameters are converted into corresponding positive values by implementing an appropriate iterative, or recursive, procedure on the set of said impedance parameters.
17. 17. A system according to Claim 1 wherein the characteristics of a medium, or material, can be determined through the analysis of complex impedance from an instantaneous event or N*** time-varying signal.
18. 18. A method for estimating the complex impedance of a cell from the combination of techniques comprising equivalent circuit models, adaptive filtering techniques, and solvers for extracting the individual impedance cell parameters from a set of simultaneous equations.
19. 19. A method for identifying potential cell failure modes, prior to actual failure, from estimated complex impedance results extracted from a system according to any of the preceding claims 1-17.
20. 20. A method for identifying the evolution of individual electrochemical reactions occurring within a cell as it degrades over time from the estimated impedance parameters extracted from a system according to any of the preceding claims 1-17.
21. 21. A method for identifying cell efficiencies from the evolution of one or more estimated impedance parameters extracted from a system according to any of the preceding claims 1-17
22. 22. A method wherein cell related parameters are recalibrated according to one or more impedance parameters extracted from a system according to any of the preceding claims 1-17.
23. 23. A method wherein capacity and/or end-of-life conditions are determined from one or more impedance parameters according to one or more impedance parameters extracted from a system according to any of the preceding claims 1-17.