Method of Estimating Pulse Response Using an Impedance Spectrum

Abstract

Electrochemical Impedance Spectrum (EIS) data are used directly to predict the pulse performance of an energy storage device. The impedance spectrum of the EIS is obtained in-situ using pre-existing techniques. A simulation waveform is configured such that the period of the pulse is greater than or equal to the lowest frequency of the impedance measurement. If the pulse is assumed to be periodic for analysis purposes, the complex Fourier series coefficients can be obtained. The number of harmonic constituents are selected so as to appropriately resolve the response, but the maximum frequency should be less than or equal to the highest frequency of the impedance measurement. In some cases, the measured frequencies of the impedance spectrum do not match the corresponding harmonic components of the simulated pulse wave. This is resolved by estimating the impedance measurements at the desired frequencies using linear interpolation, cubic spline fits, or other comparable methods. Using a current pulse as an example, the Fourier coefficients of the pulse are multiplied by the impedance spectrum at the corresponding frequency to obtain the Fourier coefficients of the voltage response to the desired pulse. The Fourier coefficients of the response are then summed reassemble to obtain the overall time domain estimate of the voltage using the Fourier series analysis. Thus, the response of an energy storage device to an anticipated or desired pulse can be estimated using low-level, charge neutral impedance measurements combined with Fourier series analysis.

Description

REFERENCE TO SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTING COMPACT DISC APPENDIX

Not Applicable.

BACKGROUND OF THE INVENTION

Energy storage devices (e.g., batteries, fuel cells, ultracapacitors, etc.) have become significantly more prevalent in many government and commercial applications (e.g., automotive, military, space, electric utilities, medical, etc.). Consequently, there has also been an increased interest in smart monitoring systems that can effectively manage energy storage devices (ESDs) so as to optimize performance and extend life. An important aspect of these smart monitoring systems is the ability to estimate the response of an ESD to an anticipated load.

For example, the Lumped Parameter Model (LPM) has been used extensively by the Idaho National Laboratory (INL) to estimate the voltage response of a battery to a constant-current pulse for automotive applications. The LPM is an equivalent circuit model that recursively solves for the voltage behavior based on a given excitation current and a set of difference equations. It has been shown that the LPM is sensitive to variations in pulse amplitude and duration, and could therefore be a useful measure of state-of-health (Christophersen, 2003).

However, the excitation signals required to obtain estimates of the ESD response are not well-suited for in-situ applications since it generally requires a pulse test which may cause larger state-of-charge (SOC) swings than desired and even adversely affect the ESD (Christophersen, 2006). A need still exists to estimate the response of an ESD to an anticipated load using benign measurement techniques.

The INL has also shown that the pulse resistance for batteries is strongly correlated with the growth observed from corresponding electrochemical impedance spectroscopy (EIS) measurements (Christophersen, 2002). It has also been shown that EIS techniques are more benign than pulse tests (Christophersen, 2006) since it is a low-level, charge neutral signal that minimally perturbs the ESD. Suitable means for obtaining in-situ impedance spectra have already been developed. The Impedance Noise Identification method (U.S. Pat. No. 7,675,293 B2) uses a random signal excitation to acquire a high resolution impedance spectrum, but at the expense of computationally intensive data processing. An alternative approach is known as Compensated Synchronous Detection (U.S. Pat. No. 7,395,163 B1), and it incorporates a wideband sum-of-sines input signal to measure the impedance. It yields a faster measurement, but at the expense of lower resolution. A variant of Compensated Synchronous Detection is Fast Summation Transformation (FST). The principal attributes of FST (Morrison, 2009) are that it only requires one period of the lowest frequency to complete the measurement, and the data processing algorithm is very simple.

All patents, patent applications, provisional patent applications and publications referred to or cited herein, are incorporated by reference in their entirety to the extent they are not inconsistent with the teachings of the specification.

BRIEF SUMMARY OF THE INVENTION

The subject invention involves a method by which the response of an energy storage device (ESD) to a pulse excitation can be predicted using impedance measurement techniques. This method assumes that the amplitude and duration of the anticipated or desired pulse excitation is known a priori, or can be inferred based on historical data (e.g., an average pulse profile based on typical automotive driving cycles). Assuming a periodic behavior of the desired pulse profile, the Fourier series coefficients can then be determined (note that the assumption of a periodic signal is for analytical purposes), and combined with measured impedance data to estimate the response.

The Fourier coefficients of the desired pulse profile are first used to establish the frequency range of the impedance measurement. For example, the period of the lowest frequency for the impedance measurement should be less than or equal to the period of the pulse profile. The maximum frequency of the impedance measurement should be greater than or equal to the largest desired harmonic value used in the Fourier coefficients used to recreate the desired pulse profile.

Knowing the desired frequency range, the ESD impedance spectrum can then be measured using any available methodology. For rapid, in-situ applications, techniques such as Impedance Noise Identification, Compensated Synchronous Detection, or Fast Summation Transformation can be easily implemented. The frequencies in the impedance measurement spectrum should correspond to the Fourier coefficients from the simulated pulse. In some cases (e.g., with Fast Summation Transformation), the impedance spectra will be lower resolution than desired due to the need for a very rapid measurement. However, the use of linear interpolation, cubic spline functions, or other similar types of curve fitting techniques can be used to estimate the impedance at other desired frequencies within the measured range.

Using a constant current pulse as an example, the Fourier coefficients of the desired or anticipated pulse profile are multiplied by the corresponding impedance measurements at each frequency. These data will provide the voltage response at each frequency of interest, and the results can then be summed to determine the overall voltage response of the ESD to the anticipated current pulse profile.

Thus, the ESD response of a pulse excitation can be estimated based on a simple impedance measurement combined with the Fourier coefficients of a simulated pulse. The estimated response behavior can be used by smart monitoring systems to more effectively manage ESD usage. For example, if the estimated response exceeds a desired threshold, the smart monitoring system can either shutdown operations, or iteratively determine a pulse excitation level than can be successfully applied to the ESD without violating operational limits (e.g., managing how much power assist is provided by the ESD in automotive applications). A smart system can also use this information to know when warning signals should be sent to a user prior to a demand being placed on the ESD.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is an example excitation pulse profile with a constant current input.

FIG. 2 is the reproduction of the example excitation pulse profile using 21 Fourier Series coefficients.

FIG. 3 is a diagram of an LPM equivalent circuit with a current source excitation.

FIG. 4 is a plot of the ideal voltage response of the LPM to a pulse excitation.

FIG. 5 a is a plot of the impedance spectrum magnitude for both the ideal and simulated conditions.

FIG. 5 b is a plot of the impedance spectrum phase for both the ideal and simulated conditions.

FIG. 6 is a graph of the estimated voltage response of an excitation pulse based on a wideband impedance measurement.

FIG. 7 is comparison between the ideal and estimated voltage response of an excitation pulse.

DETAILED DESCRIPTION OF THE INVENTION

The method of the subject invention uses wideband impedance measurements to predict the response of an energy storage device (ESD) to a pulse excitation. The impedance spectrum can be acquired by various methods, but rapid, in-situ techniques such as Fast Summation Transformation (FST) are preferred. FST is based on a computationally simple approach, and it only requires one period of the lowest frequency to complete a measurement (Morrison, 2009).

In a preferred embodiment, the anticipated or desired excitation pulse consists of a constant current square-wave profile. If it assumed that this profile is periodic (for analysis purposes only), the waveform can be decomposed into the constituent harmonic components using Fourier series methods. An example of an excitation pulse is shown in FIG. 1 and described by Equation 1, where a constant current pulse (i.e., I_P) is applied for a discharge (i.e., +I_P) and charge (i.e., −I_P) step over two periods with T₀ set to one half the period T. Given the exponential Fourier series pair shown in Equation 2, the coefficient C_n, for the given square wave is given by Equation 3. The Fourier series harmonic frequency is given in Equation 4. Based on these equations, the reassembled square wave of FIG. 1 using 21 harmonics is provided in FIG. 2. As shown, using 21 harmonics is sufficient to capture the shape of the square-wave pulse with only a small ripple at the desired ±1A current level. As expected, the Gibbs effect is also evident as well.

$\begin{array}{cc}\left\{\begin{array}{cc}f\left(t\right)=I_P,& 0<t<T_0\\ f\left(t\right)=-I_P,& T_0<t,T\end{array}\right\}& \left(1\right)\end{array}$

Where:

• • T is the period
  • T₀ is the pulse width
  • I_P is the current pulse amplitude

$\begin{array}{cc}f\left(t\right)=\sum _{n=-\infty }^{\infty }\left(C_n{}^{jn{\omega }_Ot}\right),{\omega }_O=\frac{2\pi }{T},C_n=\frac{1}{T}{\int }_Tf\left(t\right){}^{-jn{\omega }_Ot}t& \left(2\right)\\ C_n=I_P\sin c\left(\frac{n\pi }{2}\right){}^{-j\frac{n\pi }{2}},n=\pm 1,\pm 3,\pm 5\dots & \left(3\right)\\ {\omega }_n=n{\omega }_o=\frac{n2\pi }{T}& \left(4\right)\end{array}$

The frequency range of the impedance measurement should be well matched with the Fourier series harmonic frequencies (Equation 4). For example, the lowest frequency for the impedance measurement should be less than or equal to the period of the simulated pulse (i.e., less than or equal to 1/T). The highest frequency for the impedance measurement should correspond to the maximum harmonic component desired to recreate the pulse waveform (i.e., the maximum value for n used to recreate f(t) in Equation 2).

If the measured frequencies in the impedance spectrum match the desired Fourier harmonic frequency components from the simulated pulse waveform, then the responses from the ESD to an excitation pulse can be obtained at each frequency. For example, the impedance of the ESD at a given frequency (i.e., ω_n=nω_o), is shown in Equation 5. The voltage response at that frequency is the impedance (Equation 5) multiplied by the corresponding harmonic component of the current pulse (Equation 3), as shown in Equation 6. Based on the Fourier series pair of Equation 2, the estimated voltage drop due to a current pulse is the sum of the individual frequencies, as shown in Equation 7. Given a bias voltage (V_B0), the ESD voltage response (V_P) can then be estimated as shown in Equation 8, where the voltage drop (V_Z) is subtracted from the bias. Thus if the terminal voltage of the ESD is known or measured, and if a relatively recent impedance spectrum of the ESD is available, then an estimate of the response to an excitation pulse can be obtained.

$\begin{array}{cc}{\overrightarrow{Z}}_B\left({\omega }_n\right)=Z_B\left({\omega }_n\right){\mathrm{\sphericalangle \phi }}_{{\omega }_n}& \left(5\right)\\ {\overrightarrow{V}}_Z\left(t\right)=I_P\sin c\left(\frac{n\pi }{2}\right)Z_B\left(n{\omega }_0\right)\cos \left(n{\omega }_0t-\frac{n\pi }{2}+{\phi }_{n{\omega }_0}\right)& \left(6\right)\\ V_Z\left(t\right)=\sum _{〈n=1,3,5,\dots 〉}I_P\sin c\left(\frac{n\pi }{2}\right)Z_B\left(n{\omega }_0\right)\cos \left(n{\omega }_0t-\frac{n\pi }{2}+{\phi }_{Z\left(n{\omega }_0\right)}\right)& \left(7\right)\\ V_P\left(t\right)=V_{B0}-V_Z\left(t\right)& \left(8\right)\end{array}$

To make use of Equation 8, the values of the impedance spectrum {right arrow over (Z_B)} at the Fourier series frequencies ω_n must be obtained. In most cases, however the impedance measurements will have a logarithmic frequency spread, whereas the Fourier series uses linearly increasing frequency components. This can be resolved by using linearly increasing frequencies during the impedance measurement instead, but at the expense of longer measurement durations and more computationally intensive analysis techniques. Another option is to estimate the impedance at the desired frequencies within the measurement range using techniques such as linear interpolation or cubic spline fits to obtain the values of the impedance spectrum {right arrow over (Z_B)} at the Fourier series frequencies ω_n.

Analytical Validation

The Lumped Parameter Model (LPM) was used to verify the effectiveness of this method. The LPM equivalent circuit is shown in FIG. 3, and example component values for a lithium-ion battery are shown in Table 1 (Morrison, 2009). The ideal voltage response of the LPM to the pulse excitation of FIG. 1 is shown in FIG. 4. As expected, there is an initial jump due to an ohmic effect, followed by a polarization effect for the remainder of the pulse duration.

TABLE 1
Representative LPM and Analysis Data
	Voc =	3.8	V
	Cp =	666.6667	F.
	Coc =	1.6667e+003	F.
	Ro =	.0250	Ω
	Rp =	.015	Ω

The impedance spectrum of the LPM can be simulated using the Fast Summation Transformation measurement technique (Morrison, 2009) and compared to the ideal response of the equivalent circuit. The FST algorithm was applied to the LPM at the same starting frequency as the square wave pulse profile of FIG. 1 (i.e., 0.01 Hz). FIGS. 5a and 5b show that the FST analysis matches very well in both magnitude and phase, respectively, with the expected LPM response over fourteen octave steps for a broad frequency range.

However, as described above, the resolution of FST is insufficient to estimate the pulse response. To obtain a higher resolution impedance spectrum, the cubic spline fit was implemented using built-in software functions (e.g., in MATLAB matrix calculation computer software) and the resulting impedance estimations were compared to the expected response. The expected impedance can be calculated based on the frequency response of the LPM using the assumed parameters shown in Table 1. Table 2 shows the expected and estimated impedance for ten sequential odd-harmonic frequencies, with a starting frequency of 0.01 Hz. As shown, the spline fit is very good compared to the expected impedance spectrum.

TABLE 2
Cubic Spline Fit of FST Impedance
Data Compared to Ideal LPM Response
Freq	Spline LPM Spectrum	Expected LPM Spectrum
.01	0.0358-0.0163i	0.0358-0.0163i
.03	0.0283-0.0093i	0.0283-0.0094i
.05	0.0264-0.0062i	0.0264-0.0062i
.07	0.0257-0.0046i	0.0257-0.0046i
.09	0.0255-0.0036i	0.0255-0.0036i
.11	0.0253-0.0030i	0.0253-0.0030i
.13	0.0252-0.0025i	0.0252-0.0025i
.15	0.0252-0.0022i	0.0252-0.0022i
.17	0.0251-0.0020i	0.0251-0.0020i
.19	0.0251-0.0018i	0.0251-0.0018i

Equations 6 through 8 were then implemented to estimate the voltage response of the desired pulse based on the known input current and the FST impedance measurements. The resulting voltages at each frequency were summed, and the total response is provided in FIG. 6. Compared to the ideal response (FIG. 4), the overall shape of the estimated response is similar, but there is still a small ripple due to the Fourier Series Gibbs effect. Also, the estimated response profile shows a small voltage offset (˜15 mV) compared to the ideal. FIG. 7 shows the estimated response (scaled by 15 mV) overlaid onto the ideal response. The estimated change in voltage is very closely matched with the ideal response for each pulse in the profile. The initial estimated behavior (i.e., for the first 20 seconds) is slightly different since the ideal response through the LPM assumed the initial conditions were at rest, whereas the Fourier Series analysis necessarily assumes a steady-state operation (i.e., infinite periodic behavior).

It is understood that the foregoing examples are merely illustrative of the present invention. Certain modifications of the articles and/or methods may be made and still achieve the objectives of the invention. Such modifications are contemplated as within the scope of the claimed invention.

REFERENCES

• J. P. Christophersen, C. D. Ho, C. G. Motloch, D. Howell, and H. Hess, “Effects of Reference Performance Testing during Aging Using Commercial Lithium-Ion Cells,” J. Electrochem Soc., 153, A1406-A1416 (2006).
• John L. Morrison, et al, 2009 “Fast Summation Transformation for Battery Impedance Identification.” IEEE Aerospace 2009 Conference, March 7-14, Big Sky Mont.; (Refereed) (2009).
• J. P. Christophersen, C. G. Motloch, C. D. Ho, J. L. Morrison, R. C. Fenton, V. S. Battaglia, and T. Q. Duong, “Lumped Parameter Modeling as a Predictive Tool for a Battery Status Monitor.” Proceedings from 2003 IEEE Vehicular Technology Conference, October (2003).
• J. P. Christophersen, D. F. Glenn, C. G. Motloch, R. B. Wright, C. D. Ho, and V. S. Battaglia, “Electrochemical Impedance Spectroscopy Testing on the Advanced Technology Development Program Lithium-Ion Cells,” IEEE Trans. Veh. Technol., 56(3), 1851-1855 (2002).

Claims

1. A method of assessing the condition of an energy storage device by estimating a response to an excitation pulse, the method comprising the steps of: a. measuring an impedance spectrum, the impedance spectrum comprising of at least a lowest frequency and at least a highest frequency;b. configuring a simulated pulse, the simulated pulse comprising at least a low frequency;c. decomposing the simulated pulse by Fourier analysis to obtain Fourier series coefficients, the Fourier series coefficients comprising at least a maximum frequency;d. combining the Fourier series coefficients of the simulated pulse with the measured impedance spectrum to obtain an estimate response at each Fourier coefficient;e. assembling the estimated response at each Fourier coefficients into an overall time response of the energy storage device; andf. subtracting the assembled overall time response from a bias voltage of the energy storage device to estimate the response to the excitation pulse.

2. The method of claim 1, wherein said impedance spectrum is obtained offline over a broad frequency range.

3. The method of claim 2, wherein said impedance spectrum is obtained offline over a broad frequency range selected from the group consisting of a logarithmic frequency range, and a linearly-spread frequency range.

4. The method of claim 1, wherein said impedance spectrum is obtained online over a broad frequency range using techniques selected from the group consisting of Impedance Noise Identification, Compensated Synchronous Detection, Fast Summation Transformation, and other appropriate methods.

5. The method of claim 4, wherein said impedance spectrum is obtained in-situ.

6. The method of claim 1, wherein said simulated pulse is selected from the group consisting of a square wave, a triangle wave, a sawtooth wave, and other types of profiles.

7. The method claim 1, wherein said simulated pulse is configured from a constant current, a constant voltage, or constant power.

8. The method of claim 1, wherein said low frequency of said simulated pulse is greater than or equal to said lowest frequency of said impedance spectrum and wherein said maximum frequency used in said Fourier series coefficients is less than or equal to said highest frequency of said impedance spectrum.

9. The method of claim 1, wherein said measured impedance spectrum at said Fourier series coefficients is estimated through methods selected from the group consisting of linear interpolation, cubic spline, and other similar methods.

10. The method of claim 1, wherein the estimated response signal is smoothed through signal processing techniques to minimize or mitigate the Gibbs effect.