Method and system for diagnosing electrochemical converters

Abstract

The invention relates to a method and a system for diagnosing an electrochemical transducer, as well as a computer program product. A method (1) for diagnosing an electrochemical transducer comprises exciting (4) the electrochemical transducer with excitation signals and detecting (5) response signals from the electrochemical transducer in response to the excitation signals. The excitation (4) of the electrochemical transducer occurs only at representative measurement frequencies. In particular, the representative measurement frequencies and the reconstruction are based on a basis of eigenvectors or spectra determined by means of a set of EIS measurements. This enables high-quality reconstruction of the electrochemical impedance spectrum based on a significantly reduced number of measurements.

Description

[0001] The invention relates to a method and a system for diagnosing an electrochemical converter and a computer program product.

[0002] Electrochemical impedance spectroscopy (EIS) is well-known for diagnostic purposes, for example, for determining the aging of electrochemical converters. In this method, an electrochemical converter, such as a fuel cell, is stimulated with excitation signals, and the response signals of the electrochemical converter are detected and evaluated. It is necessary to map a complete electrochemical spectrum. Therefore, very many, often hundreds, of measurements are required; typically, a very wide range of measurement frequencies is covered to generate a spectrum. This makes the method, as well as the required measurement technology, very complex and expensive.

[0003] To achieve high accuracy, the existing methods require impedance measurements at a variety of different measurement frequencies, which increases the time required to determine an impedance spectrum. Therefore, application-oriented diagnosis of an electrochemical converter outside the laboratory is generally not possible.

[0004] Fast Pulse Impedance Spectroscopy (PIS) or EIS measurements with pseudorandom binary sequence (PSRB) excitation signals reduce the time required to determine an impedance spectrum. Different excitation signals are used, and the resulting response signals are mathematically evaluated. However, these methods require increased signal generation and acquisition effort. In contrast to the conventional method using single-frequency signals, PSRB measurements use broadband signals, the metrological generation and acquisition of which is more complex and therefore more costly. In contrast to the method described above, PIS measurements use individual pulses, often rectangular pulses, as excitation signals, and their response signals are measured over a defined period of time.The excitation signal must meet special requirements for accuracy, which makes the measurement-based generation of the signal as well as the acquisition comparatively cost-intensive.

[0005] Under the keyword "Tailored Sensing," methods are known with which, for example, in facial recognition, a large number of values can be approximately mapped using a reduced number of measurements. Document DE 10 2022 131 624 A1 discloses a method and a device for determining states of galvanic cell assemblies, in particular fuel cell assemblies, that are subject to degradation. CN 1 17 031 305 A relates to a method for predicting a battery health status using the distribution of relaxation times based on a combination of multiple models. CN 1 17 192 379 A discloses a method for reconstructing an electrochemical alternating current impedance spectrum of a lithium-ion battery.

[0006] The object of the invention is to enable simplified diagnostics of electrochemical converters. In particular, the disadvantages of the prior art are to be at least partially overcome.

[0007] This object is achieved by the method for diagnosing an electrochemical converter according to claim 1, as well as by the system and the computer program product according to the independent claims. Advantageous embodiments are specified in the subclaims.

[0008] To solve this problem, a method for diagnosing an electrochemical transducer is used. This method involves exciting the electrochemical transducer with excitation signals and detecting response signals from the electrochemical transducer in response to the excitation signals. The electrochemical transducer is excited only at representative measurement frequencies.

[0009] The invention is based on the finding that the use of representative measurement frequencies makes it possible to reconstruct the electrochemical impedance spectrum, at least largely accurately, with a significantly reduced number of measurements. Determining the electrochemical impedance spectrum previously required continuous measurement across all frequency ranges or a fine-meshed measurement at significantly more than 100 or 200 measurement frequencies. According to the invention, only comparatively few discrete measurement frequencies need to be used to achieve a comparable result. Typically, both the excitation and the detection of corresponding response signals are based on the representative measurement frequencies.

[0010] On the one hand, this means that compared to conventional methods, a fraction of the effort and time are now required to achieve nearly identical results. On the other hand, the technical equipment required is significantly reduced, since measurements only need to be taken at a few discrete measurement frequencies (an excitation signal must be output and a response signal must be detected) and no longer at a continuous distribution over a wide frequency range.

[0011] In other words, measurements on the electrochemical transducer are performed using electrochemical impedance spectroscopy, using specific representative measurement frequencies that have been determined beforehand. Excitation signals are input to the electrochemical transducer at these frequencies.

[0012] Representative measurement frequencies are those measurement frequencies with which the entire measurement spectrum can be reconstructed. These frequencies are therefore representative of the entire measurement spectrum. From them, statements about the response signals of the entire measurement spectrum can be reconstructed with little or no loss of information. In particular, the representative measurement frequencies only cover a partial range of the entire electrochemical impedance spectrum to be reconstructed. Furthermore, the representative measurement frequencies are discrete values with gaps between them. The representative measurement frequencies are suitable for making statements about response signals for those ranges of the frequency spectrum where there are gaps between excitation signals and, if applicable, also about response signals that lie above the highest or below the lowest measured measurement frequency.

[0013] The invention enables a significant simplification and reduction in the diagnosis of electrochemical converters. It creates the conditions for application-oriented diagnostics, allowing individual converters to be diagnosed at the point of use, for example, with reasonable effort and cost. Diagnosis can then be performed outside of the laboratory.

[0014] Excitation signals are input or modulated into the electrochemical transducer. The excitation signals can be sinusoidal. Electrical current or electrical voltage can be used as excitation signals (galvanostatic or potentiostatic measurement method). Typically, the excitation signal has a defined amplitude U0 or I0 and / or a defined angular frequency ω = 2πf.

[0015] In one embodiment, the excitation signals used to excite the electrochemical transducer at the representative measurement frequencies are identical. Thus, the same excitation signal is used for several, and in particular all, frequencies, regardless of the respective frequency. This facilitates the implementation and evaluation of the method.

[0016] The representative measurement frequencies can be, for example, at least 0.05 Hz, preferably at least 0.08 Hz, in particular at least 0.1 Hz and / or at most 20 kHz, in one example at most 15 kHz, preferably at most 10 kHz, in particular at most 8 kHz.

[0017] In one embodiment, the method further comprises determining complex impedances using the excitation signals and the associated response signals.

[0018] Typically, the complex impedance Z is determined for each pair of values consisting of the excitation signal and the response signal. The determined impedances, as a function of frequency, are referred to as the impedance spectrum. The complex impedance is typically determined using the formula Z=UtIt=U0sin(ωt)I0sin(ωt+ϕ)=Z0sin(ωt)sin(ωt+ϕ).

[0019] By determining the magnitude Z0 and the phase shift ϕ of the complex impedance at several, ideally logarithmically distributed, frequencies ω, an impedance spectrum is generated. The latter can be represented as a Nyquist diagram over the real and imaginary parts of the complex impedance Z. The structure of the impedance spectrum, and especially the temporal change of this structure over the lifetime of an electrochemical converter, allows many relevant properties and aging parameters to be determined.

[0020] In one embodiment, the method further comprises reconstructing the electrochemical impedance spectrum (EIS) of the electrochemical converter using the complex impedances. This can be done taking into account the theoretical principles listed below.

[0021] In contrast to conventional methods, the electrochemical impedance spectrum is not measured, but rather reconstructed based on the excitation and response signals. Reconstruction refers to the calculation or simulation of an impedance spectrum that should be as close as possible to the impedance spectrum actually measured using conventional methods.

[0022] In one embodiment, no more than 10 representative measurement frequencies are used. Thus, excitation of the electrochemical transducer and detection of the corresponding response signals only occur at a small number of measurement frequencies. In one embodiment, a maximum of 20 measurement frequencies are used, preferably a maximum of 15 measurement frequencies, in particular a maximum of 12 measurement frequencies. Preferably, a maximum of 8 measurement frequencies are used; in one embodiment, a maximum of 7 measurement frequencies, a maximum of 6 measurement frequencies, or a maximum of 5 measurement frequencies are used. It is not excluded that in certain embodiments, only 2, 3, or 4 measurement frequencies may be used. As explained further below, a reproduction of the complex impedance spectrum can be achieved with just 2 representative measurement frequencies, which allows certain conclusions to be drawn.

[0023] It has been shown that with the reduced number of measurement frequencies, an essentially complete reconstruction of the impedance spectrum is still possible. The first measurement frequencies are very important for the reconstruction, and as the number increases, the significance of the respective measurement frequencies decreases significantly. From the 15th measurement frequency onwards, the effects are negligible. In principle, slightly better reconstructions of the impedance spectrum can be achieved with more measurement frequencies, for example 12 or 15 measurement frequencies, than with fewer measurement frequencies. As a rule of thumb, any measurement frequency above 10 no longer has any significant influence. The influence of the last measurements is small. Depending on the specific application, a maximum of 4 or a maximum of 6 measurement frequencies may be sufficient to obtain a sufficiently accurate reconstruction.

[0024] In one embodiment, the representative measurement frequencies are distributed essentially logarithmically. This approach has been shown to enable particularly good reconstruction.

[0025] According to the invention, the method further comprises determining the representative measurement frequencies. Before the electrochemical transducer is excited, the representative measurement frequencies at which the excitation occurs are determined. This is done, in particular, on the basis of a large data set (underlying data). The data, in particular, contain conventionally measured impedance spectra. For example, each excitation signal is assigned a response signal of the specific electrochemical transducer under consideration.

[0026] The determination of the representative measurement frequencies can be carried out taking into account the measurement conditions, such as temperature, atmospheric composition and / or relative humidity.

[0027] According to the invention, patterns in underlying data are used to determine representative measurement frequencies. This embodiment is fundamentally based on the realization that patterns exist and / or can be determined in the data and can be used. These patterns allow statements to be made about which measurement frequencies are representative, i.e., statements can also be made about other measurement frequencies for which a separate measurement does not have to be carried out. For example, redundant data may be present that can be filtered out. The patterns can contain a basis of eigenvectors, which are also referred to as eigenspectra. The patterns can be used in such a way that certain eigenvectors are identified that are more important than others for mapping and / or reconstructing the frequency spectrum. Eigenvectors of the singular value decomposition represent the main features of the measurements in the data set.Eigenvectors can be determined from multiple measurements.

[0028] In one embodiment, the determination of representative measurement frequencies involves determining eigenvectors. This is done, in particular, using singular value decomposition.

[0029] This makes it possible to extract relevant data from a complex data set and filter out redundancies. Eigenvectors can be viewed as principal axes of a principal axis transformation. This allows, for example, the variance of the measurement to be transformed. The basis for determining the eigenvectors is, in particular, underlying data, which, for example, contains conventionally measured impedance spectra, as described above.

[0030] Using singular value decomposition, patterns can be calculated from the data set in the sense of a basis of eigenvectors. From this basis, it is typically possible to conclude that certain eigenvectors are more important than others for describing or reconstructing all impedance spectra contained in the data set. In this way, it is possible to transform impedance spectra from the high-dimensional to a significantly lower-dimensional form, which can be viewed as a fingerprint of precisely this measurement in the basis of eigenvectors.

[0031] From the basis of eigenvectors, suitable measurement points or measurement frequencies can be determined, which, together with the basis of eigenvectors, allow a reconstruction of the impedance spectrum from the measurements using the representative measurement frequencies. Thus, representative measurement frequencies can be determined from the determined eigenvectors.

[0032] As an alternative to singular value decomposition, other methods can be used, such as principal axis transformation or "proper orthogonal decomposition," for which there is no German translation. These methods can also be used to appropriately determine representative vectors or representative measurement frequencies.

[0033] In one embodiment, the real part and the imaginary part of the complex impedance are interpreted as different measuring points at a measuring frequency. Thus, certain or in particular all measuring frequencies are each interpreted as two measuring points. This is preferably done for several, in particular for all complex impedances. Identical or almost identical measuring frequencies are used for the real part and imaginary part. It has been shown that in this way the effort required to carry out the method can be reduced, if necessary halved, by halving the number of measurements without negatively impacting the reconstruction of the impedance spectrum. This enables a particularly efficient reconstruction of the impedance spectrum. Accordingly, there are typically twice as many measuring positions as measuring frequencies.

[0034] In one embodiment, representative measurement frequencies are determined using a different electrochemical transducer than the one used for diagnosis. It has been shown that the low-dimensional shape, which acts like a fingerprint for the basis of eigenvectors, is also valid for impedance spectra not used to determine the basis of eigenvectors, provided these impedance spectra were measured using a similar electrochemical transducer that, for example, has the same combination of materials, the same construction method, and / or the same design. In other words, the method according to the invention makes it possible to examine other, similar electrochemical transducers, regardless of the origin of the underlying data.Thus, based on an existing, preferably large data set for a specific electrochemical transducer, suitable representative measurement frequencies can be determined once and then used to diagnose a large number of other, similar electrochemical transducers.

[0035] In one embodiment, the electrochemical converter is a fuel cell and / or an electrolysis cell. In one embodiment, the electrochemical converter is a battery, in particular a lithium-ion battery with a liquid electrolyte or solid-state electrolyte.

[0036] A fuel cell can be used to supply heat and / or electricity to a building or vehicle. A fuel cell can be a low-temperature fuel cell or a high-temperature fuel cell. An electrolysis cell can be a low-temperature electrolysis cell or a high-temperature electrolysis cell. The electrochemical converter can be a solid ceramic converter or a solid oxide cell, which can operate as a fuel cell and an electrolysis cell. For example, a polymer electrolyte membrane (PEM) or a solid oxide cell (SOC) can be used. A battery can be a primary cell or a secondary cell, or contain a plurality of primary and / or secondary cells.

[0037] Another aspect of the invention is a system for diagnosing an electrochemical transducer. This system comprises a signal generator for outputting excitation signals for exciting an electrochemical transducer, a frequency analyzer for detecting response signals from the electrochemical transducer, and a control device for controlling the system. The control device is configured such that the electrochemical transducer is excited only at representative measurement frequencies.

[0038] The control device is a data processing device, in particular a computer, with which data can be processed electronically. A control device within the meaning of the present invention typically comprises an electronic memory in which data such as (control) commands can be stored. The control device is, in particular, configured to control the diagnostic system such that the steps of the method according to the invention are carried out. All features, properties, and advantages of the method described above also apply to the diagnostic system, and vice versa.

[0039] In particular, the system includes suitable power electronics to appropriately output the excitation signals and / or to appropriately detect the response signals. The diagnostic system can be significantly simpler and more cost-effective than conventional systems, since the use of representative frequencies eliminates the need to measure across a wide frequency bandwidth.

[0040] A further aspect of the invention is a computer program product. This comprises instructions that, when executed by a computer, in particular a control device, cause the steps of the method according to the invention to be carried out.

[0041] In particular, the control device is part of a system for diagnosing an electrochemical converter. In particular, the commands cause a system for diagnosing an electrochemical converter to execute the steps of the method. All features, properties, and advantages of the method and system described above also apply to the computer program product, and vice versa.

[0042] Exemplary embodiments of the invention are explained in more detail below, also with reference to figures. The claimed scope of protection is not limited to the exemplary embodiments.

[0043] They show: Fig. 1: a schematic representation of a process, Fig. 2: a schematic representation of a system, Fig. 3: a reconstruction of an EIS measurement, Fig. 4: different reconstructions of EIS measurements, Fig. 5: further reconstructions of EIS measurements, Fig. 6: a statistical evaluation of the singular value decomposition of an underlying EIS data set, as well as Fig. 7: Projections of EIS spectra onto eigenvectors.

[0044] Fig. Figure 1 schematically shows aspects of the method 1 according to the invention for diagnosing an electrochemical converter. After a step of determining representative measurement frequencies 3, the electrochemical converter is excited 4 with excitation signals. This occurs at the representative measurement frequencies. The determination of the representative measurement frequencies 3 can be performed in advance based on a separate data set and used for different electrochemical converters. Therefore, the representative measurement frequencies can already have been determined for diagnosing a specific electrochemical converter.

[0045] The electrochemical transducer generates response signals in response to the input excitation signals. These response signals are detected 5. Optionally, complex impedances 6 are then determined based on the excitation signals and the detected response signals. Optionally, the electrochemical impedance spectrum of the electrochemical transducer is then reconstructed 7, typically based on the complex impedances.

[0046] Fig. 2 schematically shows a system 10 for diagnosing an electrochemical transducer. The system 10 comprises a signal generator 11 for outputting excitation signals for exciting 4 an electrochemical transducer, which is typically electrically conductively connectable to the electrochemical transducer. The system further comprises a frequency analyzer 12 for detecting 5 response signals from the electrochemical transducer, which is typically also electrically conductively connectable to the electrochemical transducer, and a control device 13 for controlling the system 10. The control device 13 is configured such that the excitation 4 of the electrochemical transducer occurs only at representative measurement frequencies. The control device 13 comprises, in particular, a processor and / or is configured as a computer.

[0047] Fig. Figure 3 shows a reconstruction of an EIS measurement based on the example described below. The theoretical principles described below, however, are independent of the example shown and also apply to all other embodiments and applications of the invention. Example

[0048] The example described here is based on a set of electrochemical impedance spectra (EIS) X. This dataset comprises approximately 2,000 individual EIS measurements, which were acquired over several years on solid oxide fuel cells and / or electrolysis cells and reflect various aging states of the cells. Each individual measurement comprises 492 measured values, as detailed below. The measurement conditions vary with respect to the operating temperature and material flows of the solid oxide fuel cells. The data were converted into a uniform state, facilitating the evaluation of aging phenomena. This also includes consistent interpolation of the individual spectra in a specified, logarithmically arranged frequency range.

[0049] The objective of this example is to enable the most accurate reconstruction of the entire electrochemical impedance spectrum at only a few discrete frequencies. To this end, representative measurement frequencies are to be determined. In particular, additional EIS measurements not included in X are to be reconstructed based on the determination of the impedance at the representative measurement frequencies. The method remains valid for different dimensions of m and n, which describe the data matrix X. In this example, discrete values are chosen for the two variables m and n, i.e., representative values for the number of EIS measurements and the frequency range, which have enabled good reconstruction results. nomenclature X set of EIS measurements with X ∈ ℝ m×n , with m = 492, n = 1,950 n Number of impedance spectra m Number of measuring positions per impedance spectrum x High-dimensional (full) electrochemical impedance spectrum; a measurement or a column vector from X. x̃ Reconstruction of the impedance spectrum x; same length as x, but approximated by y y Low-dimensional (reduced) electrochemical impedance spectrum; measured only at the reduced number of representative measurement frequencies C Measurement matrix for optimal sensor positions U r Reconstruction basis with rang(r) or the first r column vectors of U Θ Product of C and U r a Unique signature of the low-dimensional spectrum in U r p Optimal sensor positions (number of measuring points)m is the number of measuring positions per impedance spectrum. This can correspond to the number of measuring frequencies if there is a measuring position at each measuring frequency.

[0050] Since the real and imaginary parts are used as separate values for a measurement frequency in this example, 246 measurement frequencies are available per impedance spectrum. Frequencies between 0.1 Hz and 8 kHz were used, with 50 measurement points per decade.

[0051] The method is generally based on the use of patterns in the data, which can be exploited using various methods. One of these is singular value decomposition, which allows an approximation of the data matrix X using three additional matrices: X=UΣVT

[0052] The matrix U ∈ ℝ describes m×m a set of orthonormal eigenvectors whose importance for approximating the data in X by the, in this example m = 492, singular values in the matrix Σ ∈ ℝ m×nThe exact number of variables m depends on the number of measured frequencies of the individual impedance spectra x and affects the result of the method. The matrix U can therefore be used as a reconstruction basis for eigenvectors, whereby the reconstruction quality varies greatly with the number of column vectors r used, with good results already being achieved for r = 10. If, as in this example, n » m holds, the simplification can be made that only the diagonally arranged singular values are considered for the matrix Σ. This gives the dimension of the matrices: Σ̃ ∈ ℝ m×m , as well as Ṽ T ∈ ℝ m×n . The reconstruction is therefore based on: X˜=UrΣ˜V˜T

[0053] Due to the significant dimensional reduction of this reconstruction of EIS spectra, the reconstruction of further EIS spectra using a similarly dimensionally reduced EIS measurement is also possible. Such approaches and methods are known under the keyword "sparse measurements / tailored sensing." The dimensionally reduced measurement y is identified by a unique signature a in the reconstruction basis U. r This signature depends on the reconstruction basis itself and on the number and distribution of the dimensionally reduced EIS measurement points (frequencies), which are represented by the measurement matrix C: y=CUra=Θa

[0054] The measurement matrix contains only zeros, except at the measuring points p, where the impedance is to be measured in order to achieve a reconstruction, in combination with the reconstruction basis U r, to enable. The optimal measurement points p, which directly influence the entries in the measurement matrix C, can be determined by a QR factor decomposition, where Q is an orthogonal matrix and R is an upper triangular matrix: UrTCT=QR

[0055] The high-dimensional reconstruction of the EIS spectrum from a low-dimensional measurement y can thus be determined: x˜=Ura=UrΘ−1y=Ur(CUr)−1y

[0056] In particular, the most accurate reconstruction possible with the smallest possible number of measurement points is desired for determining the impedance at individual frequencies. Previously, it was generally assumed that the lower limit of the measurement points used, p, had to be at least as large as the rank of the reconstruction basis r. p≥r

[0057] The invention recognizes that this limitation can be circumvented. For this purpose, the real and imaginary parts of the complex impedance are treated as separate measurement points. In this way, the measurement matrix C can be manually adjusted to combine closely spaced measurement points for the imaginary and real parts. This means that, in practice, only one frequency needs to be measured, although mathematically, these are still two separate measurement points. Thus, the number of measurement points in the method shown here is: p≥r2

[0058] Thus, the reconstruction quality can be kept approximately constant as the number of measuring points decreases.

[0059] Fig. Figure 3 shows an example of the reconstruction of an EIS measurement as a Nyquist diagram. The imaginary part of the complex impedance is plotted against the real part of the complex impedance. The data shown were recorded on a solid ceramic transducer stack according to the Jülich F10 design with two planes, which were not used to determine the reconstruction basis U r was used. In other words, the representative measurement frequencies were determined using a different electrochemical transducer than the diagnosis.

[0060] The measurement was carried out in dry hydrogen at a temperature of 750°C. The entire spectrum (high-dimensional EIS measurement) was recorded. The reconstruction 7 of the electrochemical impedance spectrum was carried out using a reconstruction basis U r=10and using the impedance at only p = 6 discrete frequencies (representative measurement frequencies; low-dimensional EIS measurement) and is shown as a dotted line. The sensor positions or measuring points p are in Fig. 3 marked as crosses.

[0061] To compare the reconstruction quality, the entire (high-dimensional) EIS measurement, labeled as the original measurement OM, is shown as a solid line. The mean EIS spectrum (average measurement ØM), and thus the average of the data set X, is also shown. This illustrates that the reconstruction is not only valid in a preferred range, but, as described, for all measurement conditions and aging states of the measured solid ceramic transducer contained in the data set X. The exact frequencies at the measurement points p in this example are: fi=[0.115 Hz, 0.955 Hz, 3.63 Hz, 37.98 Hz, 416.24 Hz, 7927.1 Hz]

[0062] Fig. Figure 4 shows similar representations of the same electrochemical converter under different conditions. Measurement A corresponds to the measurement in Fig. 3 in dry hydrogen at a temperature of 750°C. Measurement B corresponds to a measurement in dry hydrogen at 700°C. Measurement C corresponds to a measurement in dry hydrogen at 650°C, and measurement D corresponds to a measurement in moist hydrogen with 20% H2O at 700°C. It can be seen that sufficiently accurate reconstructions 7 of the actual complex impedance spectra were achieved under all different conditions. The measurement frequencies correspond to the frequencies mentioned above.

[0063] The rank of the reconstruction basis r is 12, while only 6 measurement points p were used. This is possible because the real and imaginary parts of the complex impedance were considered as different measurement points at a single measurement frequency. This allowed the measurement frequencies to be halved without significant loss of reproduction.

[0064] Fig. Figure 5 shows similar representations of the same electrochemical transducer using different numbers of representative measurement frequencies. In the left-hand illustration, nine representative measurement frequencies were used, and it can be seen that some sensor positions are located at almost identical measurement frequencies. This is due to the fact that the real and imaginary parts of the complex impedance can have very similar or identical measurement frequencies. In the right-hand illustration, these measurement frequencies were combined and the measurement matrix C adjusted accordingly, so that the number of representative measurement frequencies could be reduced to six. Here, too, the real and imaginary parts of the complex impedance are each interpreted as different measurement points at some measurement frequencies. Almost the same reconstruction quality was achieved with a further significantly reduced effort.By using the real and imaginary parts of all complex impedances as different measuring points, the number of measurements or representative measuring frequencies can be halved.

[0065] Fig. Figure 6 shows statistical evaluations of the singular value decomposition of the underlying EIS dataset. This result allows a conclusion about the relevance of the eigenvectors (or eigenspectra) of the method. In the left diagram, the cumulative energy CE is plotted against the number of singular values m. In the right diagram, the singular values SV are plotted against the number of singular values m. In addition, thresholds at which 80%, 90%, 95%, and 99% of the cumulative energy are reached are marked as horizontal lines.

[0066] It turns out that the first few singular values account for a large portion of the cumulative energy CE. The right-hand graph and the horizontal threshold clearly show that 80% of the cumulative energy CE is already covered by the first two singular values or eigenvectors. In other words, for an 80% reconstruction, only the first two singular values or eigenvectors are needed, for a 90% reconstruction, the first four, and for a 99% energy content, the first 24 eigenvectors. This shows that, in practice, only a few singular values are necessary for meaningful approximations with sufficient accuracy.

[0067] Above a certain threshold, the influence of an additional singular value is negligible for a sufficiently accurate reconstruction. It follows that, according to the invention, a comparatively small number of representative measurement frequencies is sufficient for a sufficiently accurate reconstruction of further EIS measurements.

[0068] Fig. Figure 7 shows projections of EIS spectra onto eigenvectors. The first 9 representative eigenvectors were selected and evaluated for two different electrochemical converters, namely two different cells C1 and C2 of a stack under consideration. The above-mentioned eigenvectors were used in each case. Fig. 4, the conditions of measurements A, B, C, and D are plotted for three different aging states of the cells. The initial phase S1 refers to the phase immediately after the cell starts up, while the middle phase S2 refers to a phase after the completion of five defined operating phases with a current density of at least 1.0 A / cm 2and a fuel utilization of at least 70%, at which the cells already exhibit noticeable aging, and the final phase S3 refers to a phase after repeating the five operating phases described above. In total, the stack was operated under power for over 3,400 hours. The measurements shown were performed during a long-term experiment on a stack with two cells and the internal type designation F1002-204. The projections onto the individual eigenvectors clearly represent the different aging states and illustrate the relevance of the eigenvectors for describing the aging process.

[0069] Shown are projections of EIS spectra onto eigenvectors Pr. ES of the respective eigenspectrum number ES-Nr.. It is initially evident that the first three or four eigenvectors make a significant contribution to describing the respective EIS spectra, while the contribution of further eigenvectors has little or no influence. Furthermore, it is evident that the cells exhibit very different projections, at least after the first pass through the load phases. This underlines that the representation of measurements in the basis of eigenvectors U r can be regarded as a unique fingerprint of the complex impedance spectrum. The eigenvectors U rThe determined representative measurement frequencies therefore exhibit similar behavior with regard to the number required for a sufficiently accurate reconstruction of an impedance spectrum. Expressed in a graphical manner, the projection represents "how much" of each eigenvector or eigenspectrum is needed for the reconstruction. The finding that only very few eigenspectra are required suggests that only very few representative frequencies are needed for a sufficiently accurate reconstruction. List of reference symbols 1 procedure 3 Determination of representative measurement frequencies 4 Suggestion 5 Detection 6 Determination of complex impedances 7 Reconstruction 10 systems 11 Signal generator 12 Frequency analyzer 13 Control device OM original measurement ØM Average measurement p measuring points A Measurement A B Measurement B C Measurement C D Measurement D m number of singular values CE Cumulative Energy SV singular values S1 initial phase S2 Middle Phase S3 final phase C1 Cell 1 C2 Cell 2 ES No. Eigenspectrum No. Pr. ES [arb. U.] Projection onto eigenspectrum [arbitrary units]

Claims

[1] Method (1) for diagnosing an electrochemical converter, comprising the steps - Excitation (4) of the electrochemical converter with excitation signals, - Detection (5) of response signals from the electrochemical transducer in response to the excitation signals, - Determination of representative measuring frequencies (3), whereby the excitation (4) of the electrochemical converter occurs only at the representative measuring frequencies, characterized by that patterns in underlying data are used to determine the representative measurement frequencies (3). [2] Method (1) according to one of the preceding claims, further comprising: Determination of complex impedances (6) using the excitation signals and the corresponding response signals. [3] Method (1) according to the preceding claim, further comprising: Reconstruction (7) of the electrochemical impedance spectrum (EIS) of the electrochemical converter using the complex impedances. [4] Method (1) according to one of the preceding claims, characterized by that no more than 10 representative measurement frequencies are used. [5] Method (1) according to one of the preceding claims, characterized by that the representative measurement frequencies are essentially logarithmically distributed. [6] Method (1) according to one of the preceding claims, characterized by that the determination of the representative measurement frequencies (3) includes determining eigenvectors. [7] Method (1) according to the preceding claim, characterized by that the determination of the eigenvectors is carried out by means of singular value decomposition, principal axis transformation or proper orthogonal decomposition. [8] Method (1) according to one of the six preceding claims, characterized bythat the real part and imaginary part of the complex impedance are regarded as different measuring points at a measuring frequency. [9] Method (1) according to one of the preceding claims, characterized by that the determination of the representative measuring frequencies (3) is carried out using a different electrochemical converter than the diagnosis. [10] Method (1) according to one of the preceding claims, characterized by that the electrochemical converter is a fuel cell and / or an electrolysis cell. [11] Method (1) according to one of the preceding claims, characterized by that the electrochemical converter is a battery, in particular a lithium-ion battery with liquid electrolyte or solid electrolyte. [12] System (10) for diagnosing an electrochemical converter, comprising - a signal generator (11) for outputting excitation signals for exciting (4) an electrochemical converter, - a frequency analyzer (12) for detecting (5) response signals of the electrochemical converter, and - a control device (13) for controlling the system (10), wherein the control device (13) is arranged such that the excitation (4) of the electrochemical converter occurs only at representative measuring frequencies, wherein the system is further arranged to determine representative measuring frequencies (3), characterized by that patterns in underlying data are used to determine the representative measurement frequencies (3). [13] Computer program product comprising instructions which, when the program is executed by a computer, in particular a control device (13), cause the steps of the method (1) according to one of claims 1 to 11 to be carried out.

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