Direct identification method for equivalent circuit model parameters of hydrogen fuel cell

Abstract

The invention belongs to the technical field of fuel cells, and discloses a direct identification method for equivalent circuit model parameters of a hydrogen fuel cell. The method comprises the following steps: firstly, establishing an R-(RC)-(RC)-(RC) equivalent circuit model and a frequency domain transfer function of the R-(RC)-(RC) equivalent circuit model according to response characteristics of an internal electrochemical reaction of a fuel cell in different frequency bands of an impedance spectrum; secondly, the continuous transfer function is converted into an input and output equation of a discrete time domain through variable substitution and bilinear transformation, and an identifiable mathematical model of the parameters to be identified is formed; then, a recursive least square method containing a forgetting factor is adopted, and the coefficient of the mathematical model is identified online in real time; and finally, reversely resolving all impedance parameters of the equivalent circuit based on an identification result, and drawing an electrochemical impedance spectrum curve according to the impedance parameters. According to the invention, the online rapid identification of the impedance parameter can be realized only by using the output voltage and current data during the normal operation of the fuel cell without injecting any external excitation signal.

Description

Technical Field

[0001] This application belongs to the field of fuel cell technology, specifically relating to a method for direct identification of parameters of equivalent circuit model of hydrogen fuel cell. Background Technology

[0002] As a highly efficient and clean energy conversion device, a fuel cell system is a complex dynamic system involving the coupling of multiple physical fields such as gas, water, heat, and electricity. Given the current bottlenecks in the development of high-performance materials, health management technology based on state prediction has become a key research direction for ensuring the reliability and durability of fuel cell systems, with real-time and accurate assessment of the stack's health status being particularly important. When a fuel cell malfunctions, its internal electrochemical reaction mechanisms and membrane structure change, leading to a decline in output performance. Failure to identify and address these malfunctions in a timely manner may cause irreversible damage to the catalyst layer and membrane electrode assembly, significantly shortening the system's lifespan. However, due to the complex coupling relationships between internal parameters of a fuel cell, only limited external measurement data such as voltage, current, and temperature can be obtained during normal operation, while internal state parameters cannot be directly observed.

[0003] Electrochemical impedance spectroscopy (EIS) is an important tool for analyzing the internal state of fuel cells, effectively characterizing membrane electrode characteristics and changes in electrochemical reaction processes. To characterize the internal electrochemical processes, EIS calculates model parameters using mathematical methods based on a finite number of measurable inputs and outputs within the fuel cell model. Therefore, EIS information is a crucial basis for fuel cell health management and a key technology for improving fuel cell durability and reliability. However, traditional EIS measurement methods involve injecting a specific frequency excitation signal into the system and acquiring voltage and current response signals. These signals are then processed by a host computer using Fourier transform and curve fitting to obtain the impedance parameters. This method has two limitations: first, the injection of the excitation signal requires additional hardware circuitry, increasing system complexity and cost; second, as an invasive detection method, the introduction of the excitation signal can interfere with the normal operating conditions of the fuel cell, potentially adversely affecting system performance.

[0004] Due to the aforementioned limitations, traditional EIS technology can typically only perform detection and analysis offline, and its measurement process is time-consuming, making it difficult to meet the needs of online condition monitoring and real-time health management of fuel cells, especially in early fault diagnosis and lifespan prediction under dynamic operating conditions. Therefore, there is a need to develop a new method that can achieve accurate online characterization of the internal state of fuel cells without interfering with the normal operation of the system. Summary of the Invention

[0005] To address the shortcomings of existing electrochemical impedance spectroscopy (EIS) measurement techniques, this application aims to provide a method for directly identifying the parameters of the equivalent circuit model of a hydrogen fuel cell without the need for external excitation signals. By establishing an accurate equivalent circuit model and using the recursive least squares method to directly identify the model parameters online, the EIS is ultimately obtained. This application avoids interference with the normal operation of the fuel cell system, eliminates dependence on additional hardware circuitry, and significantly improves the real-time performance of internal state characterization, providing effective support for online health monitoring and fault diagnosis of fuel cells.

[0006] To achieve the above technical objectives, this application specifically adopts the following technical solution: In one aspect of this application, a method for directly identifying the parameters of an equivalent circuit model for a hydrogen fuel cell is provided, comprising the following steps: S1. Establish an equivalent circuit model for the hydrogen fuel cell, wherein the equivalent circuit model is an R-(RC)-(RC)-(RC) model, and its transfer function is... for:

[0007] in, For output voltage, For Nernst voltage, For output current, ohmic resistance, For the resistor and capacitor of the first RC parallel unit, For the resistor and capacitor of the second RC parallel unit, For the resistor and capacitor of the second RC parallel unit, Represents the Laplace operator; S2, the transfer function Transforming into a recognizable mathematical model, and then discretizing and substituting variables to obtain the discrete form:

[0008] in, , , , These represent the historical values ​​of the voltage signal after a delay of 1, 2, and 3 sampling periods, respectively. These represent the historical values ​​of the current signal after a delay of 1, 2, and 3 sampling periods, respectively. to These are the coefficients of the discrete-time model; S3. Using the recursive least squares method with a forgetting factor to evaluate the parameters. to Online identification is performed, and the calculation formula for the recursive least squares method is as follows:

[0009] in, Let be the parameter estimation vector, representing the current time step. The optimal estimate of the system model parameters; Represents the algorithm gain vector; It is a data vector, consisting of system input and output data collected at the current and past moments; For the actual output of the system, at time... The measured voltage signal; This is the covariance matrix, used to measure the degree of uncertainty in parameter estimation; Forgetting factor; It is an identity matrix with the same dimensions as the parameter vector; S4. Based on the identified parameters to Solve the parameters of the equivalent circuit model. Electrochemical impedance spectroscopy (EIS) curves were plotted based on the calculated parameters.

[0010] In one implementation, the equivalent circuit model described in step S1 is established based on the different characterization forms of the internal electrochemical reactions of the fuel cell in the high, medium, and low frequency bands of the electrochemical impedance spectroscopy (EIS) curve.

[0011] In one implementation, the discretization in step S2 is achieved through a bilinear transformation, i.e., letting Where T is the sampling period, The delay operator is a unit.

[0012] In one implementation, calculating the equivalent circuit model parameters in step S4 includes: based on the discrete-time model coefficients. to Calculate intermediate variables a , b , c , d , e , f , g :

[0013] Then, calculate the time constant. As an equation The three roots; Finally, the parameters of the equivalent circuit model are calculated:

[0014] in, The sampling period.

[0015] The beneficial effects of this application are as follows: This application achieves online parameter identification without injecting any external excitation signals, utilizing only the output voltage and current signals during normal system operation. This effectively avoids interference with fuel cell operating conditions caused by traditional electrochemical impedance spectroscopy measurement methods and eliminates dependence on additional hardware circuitry. The recursive least squares identification algorithm employed has a fast convergence speed, requiring only about 70 milliseconds under random initial conditions, and the convergence time can be shortened to 40 milliseconds when parameters change dynamically, significantly improving real-time performance. This method provides reliable technical support for real-time monitoring of fuel cell health status, early fault diagnosis, and aging assessment, helping to extend stack lifespan and improve system durability and reliability. Attached Figure Description

[0016] Figure 1 This is a flowchart of the hydrogen fuel cell internal state characterization method of this application; Figure 2 This is a circuit model diagram of R-(RC)-(RC)-(RC) in the embodiments of this application; Figure 3 This is a diagram showing the output current operating conditions in an embodiment of this application; Figure 4 This is a graph showing the changes in the parameters of the equivalent circuit model in the embodiments of this application as a function of operating conditions. Figure 5 This is a comparison chart of the EIS curve of the identification results in the embodiments of this application and the EIS frequency point measured by the electrochemical workstation. Detailed Implementation

[0017] The technical solution of this application will be clearly and completely described below with reference to specific embodiments. However, those skilled in the art will understand that the embodiments described below are only some embodiments of this application, not all embodiments, and are only used to illustrate this application, and should not be regarded as limiting the scope of this application. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0018] This application abandons the traditional external excitation signal injection method and constructs a direct parameter identification system based on the system's inherent operating data by establishing an R-(RC)-(RC)-(RC) equivalent circuit model that accurately reflects the internal dynamic characteristics of the fuel cell. First, the continuous domain transfer function is transformed into a discretizable identifiable model through bilinear transformation, establishing the mathematical relationship between the parameters to be identified and the system's input and output. Then, a recursive least squares algorithm with a forgetting factor is used, requiring only the acquisition of output voltage and current data during normal system operation to achieve online real-time identification. Through the established parameter mapping relationship, the identification results are analyzed into equivalent circuit element parameters, ultimately obtaining a complete electrochemical impedance spectroscopy. This method simplifies the hardware structure through algorithmic innovation, significantly improving real-time performance while ensuring identification accuracy, and providing a novel technical path for online condition monitoring of fuel cells.

[0019] In one specific embodiment, a method for directly identifying parameters of an equivalent circuit model of a hydrogen fuel cell is provided, referring to... Figure 1 As shown, it includes the following steps: Step 1: Establish the equivalent circuit model and its transfer function.

[0020] The electrochemical reactions within a fuel cell are complex, exhibiting different responses at different frequencies. To characterize these internal states, a mathematical model that accurately reflects its dynamic behavior is needed. Through analysis and fitting of measured electrochemical impedance spectroscopy (EIS) frequency plots, it was found that an equivalent circuit model consisting of a circuit with one resistor in series and three resistor-capacitor units in parallel, namely R-(RC)-(RC)-(RC), can effectively fit the actual impedance characteristics of a fuel cell over a wide frequency range.

[0021] Frequency domain transfer function of the equivalent circuit model Defined as the open-circuit voltage of a fuel cell With Nernst voltage The difference is the same as the output current. The ratio between them. Its specific expression is as follows:

[0022] In this model: It represents ohmic resistance, which is mainly used to characterize the resistance of the membrane, electrodes, and contact parts of a fuel cell.

[0023] The first RC parallel unit ( , EIS is typically used to describe the anodic activation impedance, characterizing the adsorption and dissociation of the hydrogenation reaction on the anodic catalyst. Its time constant corresponds to the characteristics of the high-frequency region in the EIS curve.

[0024] The second RC parallel unit ( , It typically characterizes the double-layer capacitance effect and the cathode redox reaction impedance, and its time constant corresponds to the characteristics of the mid-frequency region of the EIS curve.

[0025] The third RC parallel unit ( , It can be used to characterize slower kinetic processes or other diffusion phenomena, with its time constant corresponding to the characteristics of the low-frequency region of the EIS curve.

[0026] Nernst voltage represents the thermodynamic equilibrium potential of the fuel cell.

[0027] Represents the Laplace operator, also known as the complex frequency variable; indicates the transfer function. In the continuous frequency domain ( Defined in the domain, it describes the steady-state and transient response characteristics of a system to an input signal of arbitrary frequency. This represents the dynamic (i.e., frequency-dependent) impedance of the circuit unit. For example, a capacitor. C The impedance is Therefore, the total impedance of a parallel RC circuit is .

[0028] Step 2: Convert the transfer function model into a recognizable mathematical model.

[0029] To achieve parameter identification based on online sampling data, the continuous data established in step one needs to be... The domain transfer function model is transformed into a discrete-time model suitable for digital computation, and a mathematical model form that is easy to estimate parameters is constructed.

[0030] First, to simplify the writing of subsequent formulas, we define the voltage variable to be identified. Open circuit voltage of fuel cell With Nernst voltage The difference, that is Therefore, the transfer function shown in equation (1) can be rewritten as:

[0031] To facilitate subsequent processing, time constants for three RC circuits are introduced, letting... , , Multiply both sides of the above equation by the current. After simplification and reorganization, we get:

[0032] To systematically represent the coefficients in the equations, a set of intermediate variables is introduced.a , b , c , d , e , f , g .in:

[0033] Using these intermediate variables, equation (3) can be simplified to a differential equation of the following form:

[0034] To transform the above continuous-time system model into a discrete-time model suitable for digital processing, a bilinear transformation method is employed. The bilinear transformation works by substitution... ,in The system sampling period is Let be the unit delay operator. Substituting this substitution relation into equation (5), and after algebraic operations and simplification, we finally obtain the input-output equation in the discrete-time domain:

[0035] in, , , These represent the historical values ​​of the voltage signal after a delay of 1, 2, and 3 sampling periods, respectively. These represent the historical values ​​of the current signal after a delay of 1, 2, and 3 sampling periods, respectively.

[0036] Equation (6) constitutes a standard linear regression model. This model explicitly expresses the current system output voltage. It can be represented as a finite number of its own historical values. , , and the current value With a finite number of historical values , , A linear combination of, where This indicates the current sampling time. The problem of parameter identification for nonlinear physical models is transformed into identifying the coefficients of linear models. to This solves the estimation problem and lays the foundation for subsequent applications of efficient recursive least squares method for online real-time identification.

[0037] The coefficients in equation (6) to From intermediate variables a , b , c , d ,e , f , g and system sampling period The decision is made jointly, and its specific expression is as follows:

[0038] Step 3: Identify the coefficients of the mathematical model expression using the recursive least squares method with a forgetting factor.

[0039] To achieve online real-time identification, the coefficients in the discrete-time mathematical model (6) established in step two need to be estimated. This application adopts the recursive least squares method and enhances the algorithm's tracking capability for time-varying systems by introducing a forgetting factor.

[0040] First, define the parameter vector to be identified. and data vector :

[0041]

[0042] Using the above definition, the discrete-time model shown in equation (6) can be concisely expressed as a linear regression form:

[0043] Its prediction form is:

[0044] in, For parameter vectors The estimated value.

[0045] Based on a batch n Group sampling data, define output vector and data matrix :

[0046]

[0047] Establish a cost function centered on estimating the sum of squared errors. :

[0048] parameter The least squares estimate is the solution that minimizes the cost function. Seeking information about Taking the first derivative of and setting it to zero, we can obtain this batch estimation formula:

[0049] However, the batch algorithm shown in equation (15) requires recalculating all data each time a new set of data is obtained. The computational load and storage requirements increase with the amount of data, making it unsuitable for online real-time identification. Therefore, the algorithm needs to be transformed into a recursive form, expressed as:

[0050] Define the covariance matrix The recurrence relation of its inverse matrix:

[0051] Substituting equation (16) into equation (15), we can obtain the recursive update formula for the parameter estimation vector through algebraic derivation:

[0052] Combining equations (16) and (17), we obtain the basic recursive least squares algorithm:

[0053] Take during algorithm initialization ,in It is the identity matrix. It is a positive number that is sufficiently large relative to the magnitude of the input data.

[0054] To avoid matrix inversion operations in equation (18), a gain vector is introduced. The formula is then transformed into an equivalent form, which is more convenient for numerical calculation:

[0055] Equation (19) constitutes the basic recursive calculation formula for online identification of fuel cell model parameters.

[0056] For practical fuel cell systems, their parameters change slowly with operating conditions (such as temperature, humidity, and aging). After running for a period of time, the basic recursive algorithm's gain vector... Covariance Matrix It will gradually decrease, thus affecting the predicted value. The correction capability weakens over time, resulting in "data saturation" and making it impossible to track the time-varying characteristics of the parameters.

[0057] To address this issue, a forgetting factor is introduced into the algorithm. The forgetting factor enhances the algorithm's focus on new data by applying exponentially decaying weights to older data, thereby enabling continuous tracking of time-varying parameters. (Introduction of the forgetting factor) Then, equation (19) is modified as follows:

[0058] Equation (20) is the recursive least squares calculation formula with forgetting factor that is finally adopted in this application.

[0059] Forgetting factor The value range is usually 0.95 to 1. The smaller the value, the stronger the algorithm's ability to track parameter changes, but the convergence speed will be faster, and the estimation results may fluctuate more. The closer the value is to 1, the slower the algorithm converges and the smoother the estimation, but the weaker its ability to track parameter changes. When the forgetting factor is 1, the algorithm degenerates into a regular recursive least squares method. By properly configuring the forgetting factor, a balance can be achieved between convergence speed, estimation accuracy, and parameter tracking capability, thereby meeting the requirements for online state characterization of fuel cell systems.

[0060] Step 4: Solve for the parameters of the equivalent circuit model and plot the EIS curve using the model parameters.

[0061] The coefficients of the discrete model are obtained through a recursive algorithm. to The estimated value Then, these coefficients need to be mapped back to the physical parameters of the original equivalent circuit model.

[0062] First, a specific relationship is observed between the coefficients of the discrete model and the intermediate variables of the continuous system model. By analyzing the composition of the coefficients in equation (7), the expression for the key denominator term can be obtained:

[0063] Based on discrete model coefficients to Intermediate variables of continuous systems a , b , c , d , e , f , g The defined relationship between them can be obtained by solving in reverse. arrive a , b , c , d , e , f , g The solution formula is as follows:

[0064] Equation (22) completes the transformation from identification parameters to intermediate variables of continuous systems.

[0065] intermediate variables b ,c , d With the time constants of the three RC elements Closely related. According to the definition of intermediate variables in step two: , , According to Vieta's formulas, the time constant... It is the root of the following cubic equation in one variable:

[0066] This equation can be solved using the quadratic formula for cubic equations, thus obtaining... Numerical solution.

[0067] In obtaining a , e , f , g、 Then, by solving for... , , , The system of linear equations can ultimately be used to calculate all the resistance and capacitance parameters of the equivalent circuit model. The solution formula is as follows:

[0068] The solution result of equation (23) is all the impedance parameters of the R-(RC)-(RC)-(RC) equivalent circuit model. , , , , , , .

[0069] At this point, all parameters of the equivalent circuit model have been obtained. Based on these parameters, the transfer function defined in step one can be used... By calculating its frequency response on the imaginary axis in the complex frequency domain, an electrochemical impedance spectroscopy (EIS) curve was plotted. This EIS curve accurately characterizes the impedance properties of the fuel cell at different frequencies, providing information for internal condition monitoring, fault diagnosis, and health management.

[0070] Example The following operating conditions were selected for a certain fuel cell stack with an output current of 100A, 200A, and 300A. Figure 2 The circuit model diagram is shown below; Figure 3 The output current operating condition curve is shown. The fuel cell operates under this operating condition curve and its output voltage and current data are collected. At the same time, the internal state is characterized by the algorithm proposed in this invention. Figure 4The graph shows the changes in the equivalent circuit model parameters with operating conditions. It can be seen that when the operating conditions change and the internal physical processes of the fuel cell change, the model parameters respond to the changes very quickly, and the model parameters also tend to stabilize after the operating conditions stabilize. Figure 5 The comparison between the EIS curves plotted for the identification results of this invention and the EIS frequency points measured by the electrochemical workstation shows that the EIS curves characterized by this application almost overlap with the 40 EIS frequency points measured by the electrochemical workstation, and the results are almost identical, with a mean square error of only 3.647e-07. This indicates that the present invention accurately characterizes the internal state of the fuel cell, has a great advantage in real-time performance, and has great potential for application in fuel cell fault diagnosis and aging state monitoring.

[0071] Although the embodiments of this application have been described above in conjunction with the accompanying drawings, this application is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, not restrictive. Those skilled in the art can make many other forms based on the guidance of this specification and without departing from the scope of protection of the claims of this application, and these are all within the scope of protection of this application.

Claims

1. A method for directly identifying parameters of an equivalent circuit model of a hydrogen fuel cell, characterized in that, Includes the following steps: S1. Establish an equivalent circuit model for the hydrogen fuel cell, wherein the equivalent circuit model is an R-(RC)-(RC)-(RC) model, and its transfer function is... for: in, For output voltage, For Nernst voltage, For output current, ohmic resistance, For the resistor and capacitor of the first RC parallel unit, For the resistor and capacitor of the second RC parallel unit, For the resistor and capacitor of the second RC parallel unit, Represents the Laplace operator; S2, the transfer function Transforming into a recognizable mathematical model, and then discretizing and substituting variables to obtain the discrete form: in, , , , These represent the historical values ​​of the voltage signal after a delay of 1, 2, and 3 sampling periods, respectively. These represent the historical values ​​of the current signal after a delay of 1, 2, and 3 sampling periods, respectively. to These are the coefficients of the discrete-time model; S3. Using the recursive least squares method with a forgetting factor to evaluate the parameters. to Online identification is performed, and the calculation formula for the recursive least squares method is as follows: in, Let be the parameter estimation vector, representing the current time step. The optimal estimate of the system model parameters; Represents the algorithm gain vector; It is a data vector, consisting of system input and output data collected at the current and past moments; For the actual output of the system, at time... The measured voltage signal; This is the covariance matrix, used to measure the degree of uncertainty in parameter estimation; Forgetting factor; It is an identity matrix with the same dimensions as the parameter vector; S4. Based on the identified parameters to Solve the parameters of the equivalent circuit model. Electrochemical impedance spectroscopy (EIS) curves were plotted based on the calculated parameters.

2. The method for direct identification of parameters of the equivalent circuit model of a hydrogen fuel cell according to claim 1, characterized in that, The equivalent circuit model described in step S1 is established based on the different characterization forms of the internal electrochemical reaction of the fuel cell in the high, medium and low frequency bands of the electrochemical impedance spectroscopy (EIS) curve.

3. The method for direct identification of parameters of the equivalent circuit model of a hydrogen fuel cell according to claim 1, characterized in that, The discretization described in step S2 is achieved through a bilinear transformation, that is, letting Where T is the sampling period, The delay operator is a unit.

4. The method for direct identification of parameters of the equivalent circuit model of a hydrogen fuel cell according to claim 1, characterized in that, Step S4 involves calculating the equivalent circuit model parameters, including: based on the discrete-time model coefficients. to Calculate intermediate variables a , b , c , d , e , f , g : Then, calculate the time constant. As an equation The three roots; Finally, the parameters of the equivalent circuit model are calculated: in, The sampling period.

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