AT521175B1 - Method for determining parameters of an equivalent circuit - Google Patents

Abstract

The present invention relates to a method for determining at least one parameter of an equivalent circuit (10) for an electrochemical cell (1), wherein the following steps are carried out repeatedly in real time: - performing an excitation (120) of the cell (1), - performing a measurement (130) on the cell (1) to determine at least one measurement result, - performing an estimation (140) based on the measurement result and on the basis of a model specification in the time domain to determine an estimation result, - performing a transformation (150) the estimation result for determining the at least one parameter of the equivalent circuit (10) in the frequency domain, wherein a structure of the equivalent circuit (10) is automatically determined based on the model specification.

Description

METHOD FOR DETERMINING PARAMETERS OF A REPLACEMENT CIRCUIT The present invention relates to a method for determining at least one parameter of an equivalent circuit.

In the prior art it is known that impedances of a fuel cell in the frequency range are detected by means of an excitation signal in order to determine parameters for an equivalent circuit. For example, the document DE 10 2016 106 735 A1 discloses a method for diagnosing a fuel cell, in which a parameter estimation takes place in the frequency range.

Variants that use a parameter estimation in the time domain are less common. Corresponding methods are described, for example, in the publications H. Rahimi-Eichi et al. Online Adaptive Parameter Identification and State-of-Charge Coestimation for Lithium-Polymer Battery Cells, in IEEE Transactions on Industrial Electronics, vol. 61, no.4, pp. 2053-2061, April 2014 or [0005] Zhang et al., “Online estimation of battery equivalent circuit model parameters and state of charge using decoupled least squares technique” in Energy 142 (2018) 678-688.

A disadvantage of the known solutions is that the suitability for online parameter estimation, that is, time-resolved z. B. is often still restricted during operation of a fuel cell. This can be due to the presence of measurement noise, which leads to systematic estimation errors due to the estimation method (such as “recursive least squares”). In particular, conventional solutions are still not very adaptable since, for example, the structure to be estimated or the equivalent circuit must be known a priori.

The object of the present invention is to at least partially take into account the problems described above. In particular, it is the object of the present invention to improve the parameter estimation and / or to propose an improved possibility for monitoring a fuel cell.

The above object is solved by the claims. In particular, the above object is achieved by the method according to claim 1. Further advantages of the invention result from the subclaims, the description and the drawings.

The object is achieved in particular by a method for determining at least one parameter of an equivalent circuit for an electrochemical cell, preferably a fuel cell.

It is provided that the following steps are carried out, preferably repeatedly and / or in real time, preferably in the order given:

- Carrying out an excitation of the cell, [0012] - Carrying out a measurement on the cell in order to determine at least one measurement result, [0013] - Carrying out an estimate based on the measurement result and / or on the basis of a model in the time domain, around an estimation result to determine, [0014] performing a transformation on the basis of the estimation result for determining the at least one parameter of the equivalent circuit, [0015] wherein a structure of the equivalent circuit is automatically determined on the basis of the model specification.

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AT 521 175 B1 2019-11-15 Austrian Patent Office [0016] The structure of the in particular electrical equivalent circuit is preferably determined automatically from the in particular time-discrete model specification. This has the advantage that a particularly adaptable method that is suitable for online parameter estimation is provided, since preferably not only the time course of the equivalent circuit diagram parameters, but also a change in the equivalent circuit structure during operation is determined.

The measurement in the second step is preferably carried out overlapping in time with the excitation. The estimation in the third method step is carried out in particular in the time domain, preferably in the discrete time domain on the basis of the measurement result and / or on the basis of an in particular time-discrete model specification, in order to determine an estimation result which is preferably specific for the parameter, in particular in the discrete time domain. In the fourth method step, the transformation, in particular in the frequency range, is carried out on the basis of the estimation result for determining the at least one, in particular physical, parameter of the equivalent circuit. The parameters that are determined here are then the actual physical parameters of the equivalent circuit (in general, for example, a resistor, a capacitor and / or an inductor). These are determined via a transformation in the frequency domain and via the so-called Foster synthesis. The exemplary equivalent circuit with known parameters in the frequency range is equivalent to the electrochemical impedance. This means that if all (physical) parameters of the equivalent circuit are present, the equivalent circuit in the frequency range corresponds to the electrochemical impedance. The parameter to be estimated is therefore the physical parameter (for example resistance, capacitance and / or inductance). This is usually independent of the domain, i.e. independent of the time or frequency range.

The steps of a method according to the invention can advantageously be carried out in real time, i. H. are preferably processed within a sampling interval of the measurement (ie a measurement interval).

[0019] The cell is an electrochemical cell, such as a fuel cell. The parameter, preferably as a parameter such as a size of a resistor or capacitor of the equivalent circuit, can thus be specific for an electrochemical impedance of the cell. In order to be able to carry out a reliable parameter estimation, the step of performing an excitation of the cell can thus alternatively or additionally have a broadband excitation. It may also be possible for the cell to be designed as a polymer electrolyte fuel cell (PEMFC). In particular, the broadband excitation can be essential for parameter determination in the time domain.

[0020] Furthermore, the step of carrying out the measurement on the cell can include a measurement of the electrical quantities of electrical voltage and / or electrical current. For example. a cell voltage and / or a cell current of the cell can be measured for this purpose. The measurement result in the form of digital measurement data is then preferably determined by means of an analog-digital converter or the like. These are particularly suitable for further processing, such as the estimation.

[0021] In addition, when performing the estimation on the basis of the measurement result and on the basis of the model in the time domain, it is optionally provided that at least one model parameter of a linear model in the time domain is determined by the estimate. The model specification can be specific for the specific model used, e.g. B. concern and / or specify an order of the linear model. The model specification can also be specific for a sampling time of the measurement, which is relevant for the temporal update rate of the estimate. In other words, it is possible that only the order of the linear model and / or the sampling time of the measurement (eg by a user) has to be specified for the estimate. This has the advantage that, unlike in conventional methods, the structure of the equivalent circuit, that is to say the specific equivalent circuit and / or the number and / or configuration of the components such as resistors and / or capacitors and / or the associated equivalent circuit diagram, does not have to be specified. So the only structural requirement is that

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AT 521 175 B1 2019-11-15 Austrian patent office the model is a linear model. For example, the linear model can be derived from the model specification, and the structure of the equivalent circuit can be determined from the linear model (and thus based on the model specification). The sampling time corresponds, for example, to the update rate of the parameters in the estimation.

The determination of the structure of the equivalent circuit automatically on the basis of the model and / or a transformation of the linear model into an equivalent circuit diagram (ie the determination of the structure of the equivalent circuit) and / or the extraction of the parameters can advantageously be carried out automatically by means of a suitable method for synthesizing analog Circuits take place. Such a suitable method is e.g. B. Foster synthesis, as described, for example, in [0023] lonutiu, R., & Rommes, J. (2009). A framework for synthesis of reduced order models. (CASA report; Vol. 0928). Eindhoven: Technical University of Eindhoven or Guillemin, E.A .: Synthesis of passive networks, 2 edn. John Wiley (1959). In other words, the structure of the equivalent circuit, which is described by the equivalent circuit diagram, can thereby be determined automatically and / or during operation using the model specification.

Since the at least one parameter can be a parameter of the equivalent circuit, it is important for the meaningfulness of the determined (certain) parameter that a meaningful structure of the equivalent circuit is disclosed online. Correspondingly, the transformation can be carried out on the basis of the estimation result for determining the at least one parameter of the equivalent circuit in the frequency range particularly advantageously by using a frequency transformation and / or a method of synthesizing an analog circuit (for example the aforementioned Foster synthesis) ,

According to an advantageous development of the invention, it can be provided that the estimation is carried out by and / or using a recursive algorithm, preferably by a recursively performed linear regression and / or by a recursive form of a least squares estimation, so that a Weighting of measurement results determined at different times for the estimation varies. The variation concerns in particular historical measurement results, which can become less important as time progresses and the estimate is repeated. For example. the weighting of a measurement result is lower, the longer the associated measurement for determining the measurement result dates back.

The use of a recursive "least square" algorithm is very efficient in real time. However, the use may be accompanied by the technical challenge that the underlying assumptions of the measurement noise for a linear, dynamic system, as can be used in the estimation, cannot be correct. This may result in systematic estimation errors and can have a negative impact on the measurement result. In order to achieve a further improvement, a special algorithm for estimating can be used, which enables a robust estimation despite noisy measurement results.

[0029] As an advantageous further development, it is therefore possible that an estimation method “Recursive Generalized Total Least Squares” (RGTLS) is used for the estimation (as the recursive algorithm). For this purpose, for example, the [0030] Jakubek et al. "Identification of Neurofuzzy Models Using GTLS Parameter Estimation," IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS- PART B: CYBERNETICS, VOL. 39, NO. 5, OCTOBER 2009, pp. 1121-1133 and / or Hametner et al. “Nonlinear Identification with Local Model Networks Using GTLS Techniques and Equality Constraints,” IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22

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NO. 9, SEPTEMBER 2011, pp. 1406-1418 Generalized Total Least Squares (GTLS) method disclosed performed recursively. Further modifications of these methods are also conceivable, in which the online capability is improved.

[0033] It may optionally be possible for the structure of the equivalent circuit to be determined automatically in an automatic manner by carrying out a method for synthesizing analog circuits, preferably a Foster synthesis. This enables a significant gain in time and / or a better estimation result compared to variants in which the equivalent circuit must be specified.

It is also optionally conceivable that the model specification is an order of a linear model, which is used for the estimation in the time domain. The order is in particular the order of any linear differential equation model. This has the advantage that it is possible to determine the parameter in a time-resolved manner, as is the case, for example, in FIG. B. for continuous diagnosis and / or monitoring of the cell in operation can be useful. The linear model is particularly suitable for describing an impedance of the cell. Furthermore, the estimation can be carried out by using a least-squares method (that is, a least-squares or least-squares estimation method) and / or a sliding and / or adaptive data window (using measurement results that follow one another in time). In principle, it is possible for the parameters (equivalent circuit diagram parameters) to be extracted from the linear model using analytical equations. According to the invention, however, it can also be provided that a specific equivalent circuit is not specified directly, but only the model specification. The back calculation to the physical parameters and / or the structure of the equivalent circuit can then result from a synthesis, preferably Foster synthesis. If necessary, the estimate can also be made using RGTLS.

The description of the impedance as a linear model can, for. B. take place in that the structure of the equivalent circuit is defined from at least one resistor and / or at least one RC element (from resistor and capacitor). The parameters are, for example, specific for the sizes of the resistance and / or the RC element in the frequency domain. For the time domain, the linear model, preferably the difference equation model, can be generated from the equivalent circuit diagram. The structure of the equivalent circuit is usually specified in order to determine the linear model. According to the invention, however, the opposite path can be taken, if necessary, in that the structure, in particular on the basis of the measurement result, is automatically determined on the basis of a predefined model and / or the model specification. It can be achieved as an advantage that a significant saving of time is possible and / or the adaptability and / or reliability is increased. In other words, the at least one parameter and / or the equivalent circuit diagram can be determined, in particular back-calculated, from the measurement result (ie the measurement data). For this purpose, it is only necessary, for example, that a generic differential equation model with a fixed order is specified as the model specification. The back calculation and / or the determination (selection) of the structure of the equivalent circuit diagram is carried out, for example, by the Foster synthesis.

An advantage of the synthesis, preferably Foster synthesis, of an equivalent circuit in the method according to the invention can be that no a priori information about the structure has to be present and / or that this also means a (quite possible) change in the structure during the Operating can be recorded. This can improve flexibility and adaptability.

According to a further possibility, it can be provided that the estimation is carried out using a, in particular time-discrete, linear model, the linear model being automatically transformed into the equivalent circuit diagram in the step of performing the transformation in order to automatically extract the at least one parameter , This can improve the comfort in the evaluation, the reliability and / or adaptability of the method according to the invention. In particular, the comfort can be increased by displaying the estimation results in the usual way

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AT 521 175 B1 2019-11-15 Austrian patent office (e.g. in terms of impedance and physical equivalent circuit parameters, but temporally resolved). In the method according to the invention, it is advantageously provided that a recursive estimation algorithm (for example recursive least squares or recursive generalized total least squares) is used as the estimation method. The real-time capability of the method can thus be advantageously achieved.

According to an advantageous development of the invention, it can be provided that the steps are carried out during operation of the cell, so that the at least one parameter is updated repeatedly, in particular to determine an electrochemical impedance of the cell in the frequency range and / or, in particular online and / or time-resolved to monitor. This enables a reliable diagnosis of the cell - possibly also during operation.

[0040] It may also be possible for the estimation to be carried out as an online-capable parameter estimation. For this purpose, the measurement and / or the estimation can advantageously be carried out at a frequency of approximately 1-10 kHz, so that continuous monitoring of the cell is ensured. In this way, the method according to the invention also makes it possible to use a more time-efficient alternative to the conventional “electrochemical impedance spectroscopy” method.

It is also conceivable within the scope of the invention that the excitation takes place as a broadband excitation and / or by a predetermined excitation pattern (as an input signal of the cell for excitation), which is preferably adapted to the measurement and / or estimation. This can have the advantage that a parameter covariance is reduced. In particular, the broadband excitation can be used as a prerequisite for carrying out the estimation in the time domain.

Preferably, an excitation signal (input signal) adapted to the measurement and / or to the estimate can be generated in that an optimization problem is solved arithmetically and / or automatically. Here, for. B. a scalar quality criterion, which is derived from the estimation of the parameter covariance, can be minimized.

[0043] Advantageously, it can be provided in the invention that a, preferably adaptive, windowing is carried out on the measurement result in order to carry out the estimation on the basis of the windowed measurement result. The window is, for example, a sliding and / or variable window, which is advantageously adapted on the basis of an evaluation of the measurement result (for example on the basis of a noise component).

Further measures improving the invention result from the following description of various exemplary embodiments of the invention, which are shown schematically in the figures.

[0045] Each shows schematically:

[0046] FIG. 1 is a schematic illustration for visualizing a method according to the invention, [0047] FIG. 2 is an illustration of an equivalent circuit diagram.

A method according to the invention is schematically visualized in FIG. 1. According to a first method step, a - in particular broadband and / or electrical excitation 120 of the cell 1 is carried out. According to a second method step, a measurement 130 is carried out on the cell 1 in order to determine at least one measurement result, preferably overlapping in time with the implementation of the excitation. The cell 1 can be an electrochemical cell 1, preferably a fuel cell 1, preferably a polymer electrolyte fuel cell 1. An estimate 140 can then be carried out on the basis of the measurement result and on the basis of a model in the time domain in order to determine an estimate result. In a further method step, a transformation 150 is carried out on the basis of the estimation result to determine the at least one parameter of the equivalent circuit 10 in the frequency domain. According to the invention

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AT 521 175 B1 2019-11-15 Austrian Patent Office, it may also be advantageous that a structure of the equivalent circuit 10 is automatically determined based on the model specification.

An equivalent circuit diagram of an equivalent circuit 10 is shown in FIG. 2. Two RC elements, each consisting of a resistor R1 or R2 and a capacitor C1 or C2 and a further resistor Rm connected in series with the RC elements, are shown as parameters of this equivalent circuit 10 according to an exemplary structure of the equivalent circuit 10. The equivalent circuit diagram is only an example, so that other equivalent circuits 10 can also be used depending on the model specification and / or the measurement result. In the frequency domain, parameters R1, R2, 01, 02 and Rm can be determined offline directly using appropriate estimation algorithms. The connection is:

U (iw) _{= R} Ri R ₂

I (iw) ^m (C ₁ R ₁ ) iw + l (C ₂ R2) iw + l According to the invention, it may be possible that the estimate is also made online, but then in the time domain. The parameters in the frequency domain are then only indirectly determined via a transformation. Based on the estimate based on the directly or indirectly determined parameters, e.g. B. a diagnosis of cell 1 can be carried out.

A time-discrete and / or linear model is used to enable real-time capable estimation of the parameters:

U (k) = a ₁ U (k - 1) H ----- hb _o i (k) + bj i (k - 1) + ...

After an optional error evaluation e (k) = Ü (k) - U (k) [0053] and / or an update of the parameters based on the estimate a _1; ..., b ₀ , bi, ...

A transformation into the frequency range can then take place, so that, for example, the parameters Rm, R1, 01, R2, 02 mentioned can be determined in the frequency range.

If a linear model is used, a linear parameter estimation is useful:

U = X0 0 = [ _ai , a ₂ , ..., b ₀ , b ^ ...] ^T

X ... Regressor Matrix (Past Inputs / Outputs) (U = X0is the "true system"). The parameter estimate θ can then be determined on the assumption that the measurement result is influenced by Gaussian noise:

Ü = X0 + U _e

Θ = argmin || U - X01 | = (X ^T X) X ^T U

Θ When using a "least squares" algorithm, ie the least squares method, it must be assumed for the present application that past measurement results (historical measurement data) influence the estimate (autoregression).

It is therefore proposed to use a so-called “total least squares” algorithm recursively, if necessary.

Ü = X0 + U _e

X = X + X _e

0 _TL s = argmin || [X _e , υ _ε ] || ρ θ

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AT 521 175 B1 2019-11-15 Austrian Patent Office [0059] The influence of historical measurement data can be forgotten after a predetermined number of repetitions of the process steps between the historical measurement data and current measurement data.

Furthermore, it can be provided that the excitation takes place as a broadband excitation by means of a predetermined excitation pattern which is adapted to the measurement and / or estimation. For this purpose, the excitation pattern z. B. can be determined using the following formula (determination of the lower limit of the parameter variance determined by FisherInformation):

cov (Ö)> M ¹

N

Μ (θ, i) = k = i öU (k) ^T öU (k) “äö äö ^- i = [i (l), i (2) ..... i (N)] ^T [0061] Mit the scalar objective function:

J = - logdetM (Ö, i) the optimization problem to be solved results:

min J + ai ^T ii p. t constraints, whereby the excitation pattern is determined.

In addition to the illustrated embodiments, the invention permits further design principles. That is, the invention is not to be considered limited to the exemplary embodiments explained with reference to the figures.

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LIST OF REFERENCE NUMBERS

cell

equivalent circuit

120 suggestion

130 measurement

140 estimate

150 transformation

Rx resistance X

Cx capacitor X

Claims (10)

1. A method for determining at least one parameter of an equivalent circuit (10) for an electrochemical cell (1), the following steps being carried out repeatedly in real time: - performing an excitation (120) of the cell (1), - performing a measurement (130) on the cell (1) in order to determine at least one measurement result, Performing an estimate (140) on the basis of the measurement result and / or on the basis of a model in the time domain in order to determine an estimate result, - Carrying out a transformation (150) on the basis of the estimation result to determine the at least one parameter of the equivalent circuit (10), a structure of the equivalent circuit (10) being automatically determined on the basis of the model specification. 2. The method according to claim 1, characterized in that the estimation (140) is carried out by a recursive algorithm, preferably by a recursively performed linear regression, preferably by a recursive form of a least squares estimation, so that a weighting of at different times is determined Measurement results for the estimate (140) varied. 3. The method according to claim 1 or 2, characterized in that the automatic determination of the structure of the equivalent circuit (10) by a method for the synthesis of analog circuits, preferably a Foster synthesis, takes place. 4. The method according to any one of the preceding claims, characterized in that the model specification is an order of a linear model, which is used for the estimation (140) in the time domain. 5. The method according to any one of the preceding claims, characterized in that the estimation (140) is carried out using a, in particular time-discrete, linear model, the linear model being automatically transformed into the equivalent circuit diagram in the step of carrying out the transformation (150) automatically extract the at least one parameter. 6. The method according to any one of the preceding claims, characterized in that the steps of the method are carried out during operation of the cell (1) so that the at least one parameter is updated repeatedly in order to determine an electrochemical impedance of the cell (1) in the frequency range and / or, in particular online and / or time-resolved, to monitor. 7. The method according to any one of the preceding claims, characterized in that the estimate (140) is carried out as an online-capable parameter estimate. 8. The method according to any one of the preceding claims, characterized in that the excitation (120) is carried out as a broadband excitation (120) by a predetermined excitation pattern which is adapted to the measurement (130) and / or estimate (140). 9.12 AT 521 175 B1 2019-11-15 Austrian patent office 9. The method according to any one of the preceding claims, characterized in that an adaptive window is carried out on the measurement result in order to carry out the estimation (140) on the basis of the windowed measurement result. 10. The method according to any one of the preceding claims, characterized in that the cell (1) is designed as a polymer electrolyte fuel cell.