FORM 2
THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENTS RULES, 2003
COMPLETE SPECIFICATION
[See section 10, Rule 13]
METHOD FOR CHARACTERIZING AN ELECTRICAL SYSTEM BY IMPEDANCE SPECTROSCOPY;
HELION, a corporation organized and existing under the Laws of France, whose address is Domaine du Petit Arbois, Batiment Jules Verne, B.P. 71, 13545 Aix En Provence Cedex 4, France; INSTITUT NATIONAL POLYTECHNIQUE DE TOULOUSE, an Institute under the Laws of France, whose address is 6, Allee Emile Monso, BP 34038, 31029 TOULOUSE Cedex 4, France and CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE, a corporation organized and existing under the Laws of France, whose address is 3, rue Michel Ange, 75016 Paris, France
THE FOLLOWING SPECIFICATION
PARTICULARLY DESCRIBES THE INVENTION AND THE MANNER IN WHICH IT IS TO BE PERFORMED.

DESCRIPTION TECHNICAL FIELD
The present invention relates to the general field of electrical systems, and more specifically relates to a method for characterizing said electrical system by impedance spectroscopy.
The electrical system can be an electrochemical device, such as a fuel cell, an electrolyzer, a primary or rechargeable battery, or any other type of electrical system.
BACKGROUND OF THE INVENTION
Impedance spectroscopy is a characterization technique used in many technical fields.
As an illustration, impedance spectroscopy is used in the fields of corrosion, for example to estimate the corrosion rate of a metal piece, electrodeposition, or determining a state of health of an electrochemical device such as a fuel cell.
In the latter case, the impedance spectroscopy technique can be used to detect the engorgement and drainage of a fuel cell, as described in the article by Fouquet et al. entitled "Model based PEM fuel cell state-of-health monitoring via ac impedance measurements" and published in 2006 in the Journal of Power Sources, 159, 905-913.
As shown in figure 1, a current ie is applied to the cell 1, which has a sequence of sinusoidal

perturbations around a polarization point (figure 2) . The current ie is applied by an active load 2 delivering a direct current on which said perturbations are superimposed. The frequency thereof is controlled by an impedance analysis device 3. The perturbations have a low amplitude and scan a large frequency range.
The voltage in response to these perturbations is measured at the terminals of the cell. The impedance analyzer 3 gives the evolution in a Nyquist plane of the imaginary part of the impedance as a function of its real part.
The perturbations are traditionally applied so as to scan a large frequency range, going from high frequencies to low frequencies, the frequencies being spaced apart logarithmically. The frequency range can go from several millihertz to several tens of kilohertz.
Thus, the high-frequency part is scanned very quickly, while for the low frequencies, the measurement time becomes non-negligible. For example, a second is sufficient to go from 10 kHz to 500 Hz with about one hundred points per decade, while several minutes are necessary for measurements of frequencies below 1 Hz.
The exploitation of the experimental data requires that the cell remain stable for the time it takes to apply said perturbations, i.e. the average value of the polarization current and that of the response voltage remain constant over time.
However, this hypothesis cannot be verified, in particular during deteriorations or temporary failures of the electrical system. As an example, the drainage

and engorgement of a PEM-type fuel cell make the cell unstable, which makes it impossible to exploit the impedance measurement.
The current impedance spectroscopy technique has the drawback of not making it possible to directly identify, during scanning of the frequency range, a change in the stability condition of the electrical system.
BRIEF DESCRIPTION OF THE INVENTION
The primary aim of the present invention is to propose a method for more precisely characterizing an electrical system by impedance spectroscopy. A subsidiary aim of the invention is to simply and quickly verify the stability of said system, during scanning of the frequency range.
To that end, the invention relates to a method for characterizing an electrical system including the following steps:
applying an input signal to said electrical system that comprises a sequence of sinusoidal current or voltage perturbations, around a polarization point of said electrical system, each disruption having a different frequency belonging to a primary series of frequencies;
- measuring an output signal of said electrical
system in response to said input signal for each of
said applied perturbations;
- estimating a characteristic size of the impedance
of said electrical system for each of said applied
perturbations, the set of said estimates of the

characteristic size of the impedance characterizing said electrical system.
According to the invention, the perturbations of said sequence are applied so as to scan, turn by turn, a plurality of sub-series of frequencies resulting from said primary series, each sub- series of said plurality being interlaced with at least one other sub- series of the same plurality.
Thus, the fact that the sub-series of frequencies are interlaced amounts to forming an overlap area between two considered sub-series, whether or not they are adjacent to each other.
The interlacing of the sub-series amounts to going back and forth in the scanning of the frequencies. Thus, during scanning of the frequencies in the primary series, the fact of again soliciting the high frequencies makes it possible to detect rapid physical phenomena. Likewise, again soliciting the low frequencies makes it possible to detect identifiable physical phenomena with long times. Thus, all throughout the measurement, it is possible to identify slow or fast physical phenomena, which makes the characterization of the electrical system more precise.
It should be noted that such precision is not obtained when one monotonously scans the frequencies of the primary series, as in the solution of the prior art previously described. In fact, once the high frequencies have been scanned, one solicits the middle and low frequencies, which also does not make it possible to detect rapid physical phenomena. Conversely, when the high frequencies are solicited and

after having scanned the low frequencies, it is no longer possible to identify slow physical phenomena.
Preferably, at least one frequency of a first sub-series is situated between two consecutive frequencies of a second sub-series with which the first sub-series is interlaced.
Preferably, each sub-series of said plurality is interlaced with all of the other sub-series of the same plurality.
The perturbations of said sequence are applied so as to monotonously or randomly scan the frequencies of each sub-series.
The perturbations of said sequence are applied so as to scan monotonously, or randomly, said frequency sub-series of said plurality, the sub-series being ranked according to a predetermined order.
The sub-series can be ordered according to an increasing or decreasing ranking of the maximum frequency of each sub-series. Alternatively, the order of the sub-series can depend on the minimum frequency of each sub-series.
Preferably, the method according to the invention includes the subsequent steps consisting of:
- calculating a deviation between the estimate of said characteristic size of the impedance for a first frequency of a first sub-series of said plurality of sub-series on the one hand, and a value obtained by interpolation, for the same first frequency, of the estimates of said characteristic size for two

consecutive frequencies of a second sub-series of the same plurality on the other hand, between which said first frequency is situated;
- detecting an instability of the electrical system when said calculated deviation exceeds a predetermined threshold deviation.
Thus, the method according to the invention makes it possible to simply and quickly verify the stability of the electrical system, during scanning of the frequency range of the primary series.
A discontinuity in the values of the impedance situated in this overlap area, detected when said calculated deviation exceeds a threshold deviation, directly shows the instability of the studied electrical system. The stability of the electrical system can therefore be verified simply and quickly.
Said electrical system can be any electrochemical device such as, for example, an electrolyzer or a fuel cell.
The invention also relates to a method for controlling an electrical system, including:
- performing the characterization method according to any one of the preceding features, to characterize said electrical system by the set of said estimates of the characteristic size of the impedance; then
- measuring a deviation between at least one of said estimates of the characteristic size of the impedance of said electrical system and at least one reference value of said size; then

- applying a command to said electrical system when said measured deviation exceeds a predetermined threshold deviation.
Other advantages and features of the invention will appear in the non-limiting detailed description below.
BRIEF DESCRIPTION OF THE DRAWINGS
We will now describe, as non-limiting examples, embodiments of the invention, in reference to the appended drawings, in which:
Figure 1, already described, is a diagrammatic illustration of a fuel cell connected to a current source and an impedance analyzer;
Figure 2, already described, is a graph showing the application of a current disruption to the fuel cell of figure 1, around a polarization point;
Figures 3A and 3B are graphs showing the evolution of the impedance as a function of the applied perturbations, in case of stability (figure 3A) and instability (figure 3B) of the electrical system;
Figure 4 is an example of interlacing of the scanned frequencies by a sequence of perturbations. DETAILED DESCRIPTION OF THE INVENTION
The electrical system is, in the rest of the description, a fuel cell of the proton exchange membrane type.
The electrical system is not, however, limited to a fuel cell, but can be an electrolyzer, a primary or rechargeable battery, any other type of electrochemical

device, as well as any electrical system that can receive or feed electrical energy.
The fuel cell is connected to a power device making it possible to impose an electrical current, such as an active load or a static converter as interface with an application. The active load has a bandwidth covering at least the series of frequencies to be studied.
The cell is connected to an impedance analysis device. This can be a commercial impedance analyzer, a computer including synchronous cards for generating and acquiring current and/or voltage, which may or may not be onboard.
A sequence of sinusoidal current perturbations is applied to the fuel cell, around a polarization point.
The polarization point preferably remains constant over the entire application length of the sequence of perturbations.
Each disruption preferably comprises several periods so that any transient state is damped, then several periods to calculate the impedance.
Each disruption has a different frequency, such that the disruption sequence scans a primary series A of frequencies A = {fi, f2,-, fN) - More precisely, the series of perturbations includes at least one disruption for each frequency of the series A.
The range of the primary series A of frequencies can go from several millihertz to several tens of kilohertz, and comprises N frequencies to be scanned, for example, around one hundred frequencies.

The amplitude of the perturbations is low enough for the cell's response to remain linear. To that end, the amplitude is substantially below 10% of the value of the current at the considered operating point, and is preferably in the vicinity of a percent of said value.
An impedance analysis device gives the complex impedance of the cell for each of the frequencies of the primary series A. It is then possible to trace the evolution of the imaginary part of the impedance as a function of the real part in a Nyquist plane.
It should be noted that, in the case where the input signal is the voltage at the terminals of the cell, the impedance analysis device deduces the complex admittance of the cell, and no longer the complex impedance. The impedance can nevertheless be easily deduced from the admittance.
According to the invention, the frequencies of said primary series A are distributed so as to form a plurality of sub-series Al,...,An of frequencies.
Sub-series (A1)1-1...n of frequencies refers to a set of frequencies extracted, or resulting, from the primary series A:

The union of the sub-series corresponds to the primary series A of frequencies:

Preferably, each sub-series does not contain frequencies shared with another sub-series:


Preferably, each sub-series of frequencies
includes N2 frequencies, so that the sum of the N1 frequencies equals the number N of frequencies of the primary series A:

Each sub-series A1 has a range of frequencies. Range G1 of the sub-series A1 refers to the set of frequencies of the sub-series considered to be comprised between the highest frequency and the lowest frequency of said sub-series. Thus:
with
Each sub-series A1 of said plurality of sub-series is interlaced with at least one other sub-series A1' of the same plurality.
Interlacing of two sub-series A1 and A1' refers to the presence of at least one frequency of a first sub-series between two consecutive frequencies of the second sub-series.
More precisely, there is interlacing between the sub-series A1 and A1' such that
if: or if:


In the first embodiment of the invention, each sub-series is interlaced with all of the other sub-series of said plurality.
Each sub-series is traveled through by the sequence of perturbations monotonously, in the same direction, preferably from the high frequencies toward the low frequencies.
Moreover, said plurality of sub-series of frequencies is traveled through monotonously. Monotonous order means that the plurality of sub-series of frequencies is traveled through from sub-series to adjacent sub-series, following an ordered ranking of the sub-series. Preferably, said sub-series are ranked according to decreasing order of the maximum frequency of each sub-series.
As an example, and as illustrated by figures 3A and 3B, the primary series A of frequencies to be scanned is from 2 Hz to 10 kHz, and includes 12 frequencies. Thus, the series A is: A = {10kHz, 5kHz, 2kHz, 1kHz, 500Hz, 200Hz, 100Hz, 50Hz, 20Hz, 10Hz, 5Hz, 2Hz}.
Three sub-series A1, A2, A3 are formed from the primary series A by decimation of one out of three frequencies:
A1 = {10kHz, 1kHz, 100Hz, 10Hz}
A2 = {5kHz, 500Hz, 50Hz, 5Hz}
A3 = {2kHz, 200Hz, 20Hz, 2Hz}
According to this embodiment, each sub-series of frequencies has at least one frequency comprised between two consecutive frequencies of another sub-series. There is therefore interlacing between each of

the three sub-series, and not only between two adjacent sub-series.
Thus, instead of applying the sequence of perturbations so as to continuously scan the N frequencies of the primary series A, as in the traditional solution of the prior art described before, the sequence of perturbations travels turn by turn through the three sub-series of frequencies.
The sequence of perturbations preferably travels successively through the frequencies of sub-series A1, then those of sub-series A2, etc.
Thus, by interlacing said sub-series, the method makes it possible to know quickly whether the subsystem has remained stable throughout the duration of the measurements.
In fact, in case of instability of the fuel cell, upon application of the sub-series A1, the experimental points of the impedance for the frequencies of said sub-series are not situated in the continuation of the points corresponding to the impedance measured for the preceding sub-series A1"1.
For example, as illustrated by figures 3A and 3B, the application of the perturbations scanning the sub-series A1 makes it possible to obtain the values of the impedance Z(A1) for the considered frequencies: Z(lOkHz), Z(lkHz), Z(lOOHz)...
The values of the impedance Z(A2) for the frequencies of the sub-series A2 should, if the cell has stayed stable throughout the measuring time, be placed in the continuation of those of Z(A1). As shown by figure 3A (stable cell), the values of the impedance

for each sub-series of frequencies is placed continuously along a given curve (solid line).
If the cell becomes unstable between sub-series A1 and A2, a discontinuity appears between Z (A1) and Z(A2) in the overlap area between the ranges of these two sub-series. Figure 3B illustrates this discontinuity. The values of Z(A2) are placed on a curve (broken line) different from that of Z(AJ) (solid line). The discontinuity appears as of the first value of Z(A2) corresponding to the first applied frequency of the sub-series A2.
It is possible to simply evaluate the discontinuity in the considered overlap area. Thus, a first frequency f1! of a first sub-series A1 is situated between two consecutive frequencies f2j and f2j+i of a second sub-series A1 and A2 are therefore interlaced.
One then calculates a value of the impedance for the first frequency Zint(f1!) from impedances corresponding to the two frequencies Z(f2j) and Z(f2j+i) of the second sub-series. This calculation can be done, for example, by linear, cubic (by cubic spline) or polynomial interpolation.
One then calculates a deviation between the estimated value of the impedance for a first frequency Z(f1!) and the value of the impedance obtained by interpolation Zint(f1i).
One lastly compares the calculated deviation with a predetermined threshold deviation. When this deviation exceeds the threshold deviation, an instability of the fuel cell is deduced therefrom.

Thus, the sequence of perturbations makes it possible to know quickly whether the system has remained stable throughout the duration of the measurements.
Moreover, again soliciting the high frequencies, here at the beginning of each sub-series, makes it possible to identify any rapid physical phenomena all throughout the measurement.
Of course, this example is only provided as an illustration. It is possible to increase or decrease the number of sub-series, and therefore to respectively decrease or increase the number of frequencies in each sub-series.
For example, for a primary series A including 400 frequencies, the sequence of perturbations can travel through approximately twenty sub-series of frequencies each including approximately twenty frequencies.
According to a second embodiment of the invention, each sub-series A1 of frequencies is only interlaced with the adjacent sub-series A1-1 and Ai+1. With the exception of this difference, the second embodiment remains similar to the first embodiment.
As an example, for the following primary series A = {10kHz, 5kHz, 2kHz, 1kHz, 500Hz, 200Hz, 100Hz, 50Hz, 20Hz, 10Hz, 5HZ, 2Hz}, the sub-series can be:
A1 = {10kHz, 2kHz}
A2 = {5kHz, 500Hz}
A3 = {1kHz, 100Hz}
A4 = {200Hz, 20Hz}
A5 = {50Hz, 5Hz}

A6 = {10Hz, 2Hz}
The sequence of perturbations scans the sub-series turn by turn, following the decreasing order of the maximum frequencies, as in figure 4, which illustrates an example of interlacing of the frequencies for the second embodiment of the invention.
Thus, the sequence of perturbations involves a succession of back and forth movements, while going gradually from the" high frequencies towards the low-frequencies.
As previously described, in case of instability of the cell, a discontinuity in the values of the impedance appears at the overlap areas between the adjacent sub-series.
This discontinuity can be exploited to diagnose a budding defect, such as for example drainage or engorgement of a fuel cell, and to lead to retroaction on the control-command of the fuel cell.
For the different embodiments described earlier, the frequencies of each sub-series A1 can be scanned no longer monotonously, but randomly, by said sequence of perturbations.
Likewise, said plurality of sub-series of frequencies A1,...,An can be traveled through randomly by said sequence of perturbations.
It should be noted that interlaced spectroscopy, irrespective of the embodiments previously described, has a result identical, when the cell remains stable, to that of traditional spectroscopy.

We Claim :
1. A method for characterizing an electrical system including the following steps:
applying an input signal to said electrical system that comprises a sequence of sinusoidal current or voltage perturbations, around a polarization point of said electrical system, each disruption having a different frequency belonging to a primary series (A) of frequencies;
- measuring an output signal of said electrical
system in response to said input signal for each of
said applied perturbations;
- estimating a characteristic size of the impedance
of said electrical system for each of said applied
perturbations, the set of said estimates of the
characteristic size of the impedance characterizing
said electrical system,
characterized in that the perturbations of said sequence are applied so as to scan, turn by turn, a plurality of sub-series (A1,...An) of frequencies resulting from said primary series (A), each sub-series of said plurality being interlaced with at least one other sub- series of the same plurality.
2. The method according to claim 1, characterized in that at least one frequency of a first sub-series is situated between two consecutive frequencies of a second sub-series with which the first sub-series is interlaced.

3. The method according to claim 1 or 2,
characterized in that each sub-series of said plurality
is interlaced with all of the other sub-series of the
same plurality.
4. The method according to any one of claims 1 to 3, characterized in that the perturbations of said sequence are applied so as to monotonously or randomly scan the frequencies of each sub-series (A1) .
5. The method according to any one of claims 1 to 3, characterized in that the perturbations of said sequence are applied so as to randomly scan the frequencies of each sub-series (A1) .
6. The method according to any one of claims 1 to 5, characterized in that the sub-series are ranked according to a predetermined order, the perturbations of said sequence being applied so as to monotonously scan said sub-series (A1,...An) of said plurality.
7. The method according to any one of claims 1 to 5, characterized in that the sub-series are ranked according to a predetermined order, the perturbations of said sequence being applied so as to randomly scan said sub-series (A1,...,An) of said plurality.
8. The method according to any one of claims 1 to 7, characterized in that it includes the subsequent steps consisting of:

- calculating a deviation between the estimate of
said characteristic size of the impedance for a first
frequency of a first sub-series of said plurality of
sub-series on the one hand, and a value obtained by
interpolation, for the same first frequency, of the
estimates of said characteristic size for two
consecutive frequencies of a second sub-series of the
same plurality on the other hand, between which said
first frequency is situated;
- detecting an instability of the electrical system
when said calculated deviation exceeds a predetermined
threshold deviation.
9. The method according to any one of the preceding
claims, characterized in that said electrical system is
a fuel cell.
10. A method for controlling an electrical system,
including:
- performing the characterization method according to any one of the preceding features, to characterize said electrical system by the set of said estimates of the characteristic size of the impedance; then
- measuring a deviation between at least one of said estimates of the characteristic size of the impedance of said electrical system and at least one reference value of said size; then

- applying a command to said electrical system when said measured deviation exceeds a predetermined threshold deviation.