A direct identification method of hydrogen fuel cell equivalent circuit model parameters

Abstract

The application belongs to the technical field of fuel cells, and discloses a hydrogen fuel cell equivalent circuit model parameter direct identification method. First, according to the response characteristics of the internal electrochemical reaction of the fuel cell in different frequency bands of the impedance spectrum, an R-(RC)-(RC)-(RC) equivalent circuit model and its frequency domain transfer function are established. Second, the continuous transfer function is converted into an input-output equation in the discrete time domain through variable substitution and bilinear transformation, forming an identifiable mathematical model of the to-be-identified parameters. Then, the recursive least squares method with a forgetting factor is used to identify the coefficients of the mathematical model in real time. Finally, all impedance parameters of the equivalent circuit are solved reversely based on the identification result, and an electrochemical impedance spectrum curve is drawn accordingly. The application only needs to use the output voltage and current data of the fuel cell during normal operation, without injecting any external excitation signal, so that the online rapid identification of the impedance parameters can be realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of fuel cells, and particularly relates to a hydrogen fuel cell equivalent circuit model parameter direct identification method. BACKGROUND

[0002] As a kind of high-efficiency clean energy conversion device, the fuel cell system is a complex dynamic system involving gas, water, heat, electricity and other multi-physical field coupling. Under the background that the research and development of high-performance materials are facing bottlenecks, the health management technology based on state prediction has become a key research direction to ensure the reliability and durability of the fuel cell system, and the real-time accurate evaluation of the health state of the fuel cell stack is particularly important. When the fuel cell fails, the internal electrochemical reaction mechanism and membrane structure will change, resulting in a decline in output performance. If these failures are not identified and handled in time, irreversible damage may be caused to the catalyst layer and the membrane electrode, thereby significantly shortening the service life of the system. However, due to the complex coupling relationship between the internal parameters of the fuel cell, only limited external measurement data such as voltage, current and temperature can be obtained during normal operation, and the internal state parameters cannot be directly observed.

[0003] Electrochemical impedance spectroscopy (EIS) technology is an important tool for analyzing the internal state of the fuel cell, which can effectively characterize the changes in the membrane electrode characteristics and the electrochemical reaction process. In order to characterize the information of the internal electrochemical process, EIS is based on the model of the battery and the limited measurable input and output quantities of the model, and the model parameters are calculated by a certain mathematical method. Therefore, the EIS information of the fuel cell is one of the important bases for health management, and is a key technology to improve the durability and reliability of the fuel cell. However, the traditional EIS measurement method injects an excitation signal of a specific frequency into the system, collects the voltage and current response signals, and finally obtains the impedance parameters through Fourier transform and curve fitting processing on the host computer. This method has two limitations: first, the injection of the excitation signal requires additional hardware circuit support, increasing the complexity and cost of the system; second, as an invasive detection method, the introduction of the excitation signal will interfere with the normal operating conditions of the fuel cell, which may adversely affect the performance of the system.

[0004] Due to the above limitations, the traditional EIS technology can usually only be detected and analyzed in an offline state, and the measurement process takes a long time, which is difficult to meet the needs of online state monitoring and real-time health management of the fuel cell, especially in the early diagnosis of faults and life prediction under dynamic conditions. Therefore, a new method is needed to accurately characterize the internal state of the fuel cell online without interfering with the normal operation of the system. SUMMARY

[0005] In view of the deficiencies of the existing electrochemical impedance spectroscopy measurement technology, the present application aims to provide a hydrogen fuel cell equivalent circuit model parameter direct identification method without injecting external excitation signals. By establishing an accurate equivalent circuit model and directly identifying the model parameters online using the recursive least squares method, the electrochemical impedance spectrum is finally obtained. The present application can avoid interference with the normal operation of the fuel cell system, eliminate the dependence on additional hardware circuits, and significantly improve the real-time performance of internal state representation, thereby providing effective support for realizing online health state monitoring and fault diagnosis of fuel cells.

[0006] In order to achieve the above technical purpose, the present application specifically adopts the following technical solutions:

[0007] In one aspect of the present application, a hydrogen fuel cell equivalent circuit model parameter direct identification method is provided, comprising the following steps:

[0008] S1, an equivalent circuit model of a hydrogen fuel cell is established, the equivalent circuit model is an R-(RC)-(RC)-(RC) model, and the transfer function of the equivalent circuit model is :

[0009]

[0010] wherein, is an output voltage, is a Nernst voltage, is an output current, is an ohmic resistance, is a resistance and a capacitance of a first RC parallel unit, is a resistance and a capacitance of a second RC parallel unit, is a resistance and a capacitance of a second RC parallel unit, represents a Laplace operator;

[0011] S2, the transfer function is converted into an identifiable mathematical model, and by discretization and variable substitution, a discrete form is obtained:

[0012]

[0013] wherein, , , , respectively represent historical values of the voltage signal delayed by 1, 2 and 3 sampling periods, respectively represent historical values of the current signal delayed by 1, 2 and 3 sampling periods, to are discrete-time model coefficients;

[0014] S3, using recursive least square method with forgetting factor to estimate parameters to Online identification is performed, where the calculation formula of the recursive least square method is:

[0015]

[0016] wherein, is a parameter estimation vector, representing the parameter estimation at the current time point; is an algorithm gain vector; is a data vector, which is composed of system input and output data collected at the current and past time points; is the actual output of the system at the time point; is a voltage signal measured at the time point; is a forgetting factor; is an identity matrix, which has the same dimension as the parameter vector;

[0017] S4, according to the parameters identified in step S3 to , the equivalent circuit model parameters are calculated , and the electrochemical impedance spectrum (EIS) curve is drawn based on the calculated parameters.

[0018] In one embodiment, the equivalent circuit model in step S1 is established according to different characterization forms of high, medium and low frequency bands on the electrochemical impedance spectrum (EIS) curve of the internal electrochemical reaction of the fuel cell.

[0019] In one embodiment, the discretization in step S2 is achieved by bilinear transformation, i.e. wherein T is the sampling period, is a unit delay operator.

[0020] In one embodiment, the calculation of the equivalent circuit model parameters in step S4 includes: according to the discrete-time model coefficients to , the intermediate variables a , b , c , d , e , f , g are calculated:

[0021]

[0022] Then, the time constant is calculated as the three roots of the equation .

[0023] Finally, the equivalent circuit model parameters are calculated:

[0024]

[0025] wherein, is a sampling period.

[0026] The application has the beneficial effects that:

[0027] The application can realize online parameter identification by using only the output voltage and current signals in the normal operation of the system without injecting any external excitation signal, effectively avoiding the interference of the traditional electrochemical impedance spectrum measurement method on the working condition of the fuel cell, and eliminating the dependence on additional hardware circuit. The recursive least square identification algorithm used has a fast convergence speed, and only about 70 milliseconds are needed under random initial conditions. When the parameters change dynamically, the convergence time can be shortened to 40 milliseconds, which significantly improves the real-time performance. The method provides reliable technical support for real-time monitoring, early fault diagnosis and aging evaluation of the health state of the fuel cell, and helps to prolong the service life of the stack and improve the durability and reliability of the system. BRIEF DESCRIPTION OF DRAWINGS

[0028] Figure 1 The flow chart of the internal state characterization method of the hydrogen fuel cell of the application;

[0029] Figure 2 The R-(RC)-(RC)-(RC) circuit model in the embodiment of the application;

[0030] Figure 3 The output current working condition curve in the embodiment of the application;

[0031] Figure 4 The curve graph of the equivalent circuit model parameters changing with the working condition in the embodiment of the application;

[0032] Figure 5 The comparison graph of the EIS curve of the identification result and the EIS frequency point actually measured by the electrochemical workstation in the embodiment of the application. DETAILED DESCRIPTION

[0033] The technical solutions of the application will be described below in conjunction with specific embodiments, but those skilled in the art will understand that the following described embodiments are part of the embodiments of the application, not all the embodiments, and are only used to illustrate the application, and should not be regarded as limiting the scope of the application. Based on the embodiments in the application, all other embodiments obtained by those of ordinary skill in the art without creative labor are within the scope of protection of the application.

[0034] The application discards the traditional external excitation signal injection mode, establishes an R-(RC)-(RC)-(RC) equivalent circuit model which accurately reflects the internal dynamic characteristics of the fuel cell, and constructs a parameter direct identification system based on the inherent operation data of the system. First, the continuous domain transfer function is transformed into a discrete identifiable model through bilinear transformation, and the mathematical relationship between the to-be-identified parameters and the system input and output is established. Then, a recursive least squares algorithm with a forgetting factor is used, and only the output voltage and current data during normal operation of the system need to be collected to realize online real-time identification. Through the established parameter mapping relationship, the identification result is analyzed into equivalent circuit element parameters, and the complete electrochemical impedance spectrum is finally obtained. The method realizes the simplification of the hardware structure through algorithm innovation, significantly improves the real-time performance while ensuring the identification accuracy, and provides a new technical path for online state monitoring of fuel cells.

[0035] In one specific embodiment, a hydrogen fuel cell equivalent circuit model parameter direct identification method is provided, as shown in Figure 1 , comprising the following steps:

[0036] Step 1: Establishing an equivalent circuit model and its transfer function.

[0037] The electrochemical reaction process inside the fuel cell is complex, and its characteristics show different responses at different frequencies. In order to characterize these internal states, a mathematical model that can accurately reflect its dynamic behavior needs to be established. Through analysis and fitting of the measured electrochemical impedance spectrum (EIS) frequency point diagram, it is found that the circuit structure composed of one resistance in series with three parallel resistance-capacitance units, i.e. R-(RC)-(RC)-(RC) equivalent circuit model, can effectively fit the actual impedance characteristics of the fuel cell in a wide frequency band.

[0038] The frequency domain transfer function of this equivalent circuit model is defined as the ratio between the difference between the open circuit voltage of the fuel cell and the Nernst voltage and the output current . Its specific expression is as follows:

[0039]

[0040] In this model:

[0041] represents the ohmic resistance, mainly representing the resistance of the membrane, electrode and contact part of the fuel cell.

[0042] 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.

[0043] 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.

[0044] 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.

[0045] Nernst voltage represents the thermodynamic equilibrium potential of the fuel cell.

[0046] 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 .

[0047] Step 2: Convert the transfer function model into a recognizable mathematical model.

[0048] 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.

[0049] 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:

[0050]

[0051] To facilitate subsequent processing, time constants for three RC circuits are introduced, letting... , , Multiply both sides of the above equation by the current. and after rearranging, we get

[0052]

[0053] To systematically represent the coefficients in the equation, a set of intermediate variables are introduced a , b , c , d , e , f , g . Wherein:

[0054]

[0055] Using these intermediate variables, equation (3) can be simplified into the following differential equation:

[0056]

[0057] To convert the above continuous-time system model into a discrete-time model suitable for digital processing, the bilinear transformation method is adopted. The bilinear transformation is through the substitution where is the system sampling period, is the unit delay operator. Substitute this substitution relationship into equation (5), and after algebraic operation and rearrangement, the input-output equation in the discrete-time domain is finally obtained:

[0058]

[0059] wherein, , , respectively represent the historical values of the voltage signal delayed by 1, 2, 3 sampling periods;

[0060] respectively represent the historical values of the current signal delayed by 1, 2, 3 sampling periods.

[0061] Equation (6) constitutes a standard linear regression model. This model explicitly expresses the current output voltage of the system as a linear combination of its own finite historical values , , and the current value and finite historical values , , of the current, where represents the current sampling time. By converting the parameter identification problem of the nonlinear physical model into the identification of the coefficients of the linear model to The estimation problem of the coefficients in equation (6) lays the foundation for subsequent online real-time identification using efficient recursive least squares method.

[0062] The coefficients in equation (6) to are determined by the intermediate variables a , b , c , d , e , f , g and the system sampling period , and their specific expressions are as follows:

[0063]

[0064] Step three: identify the coefficients of the mathematical model expression using the recursive least squares method with a forgetting factor.

[0065] To achieve online real-time identification, the coefficients in equation (6) of the discrete-time mathematical model established in step two need to be estimated. This application uses the recursive least squares method and introduces a forgetting factor to enhance the algorithm's tracking ability for time-varying systems.

[0066] First, define the parameter vector to be identified and the data vector :

[0067]

[0068]

[0069] Using the above definitions, the discrete-time model represented by equation (6) can be succinctly expressed in the form of linear regression:

[0070]

[0071] Its prediction form is:

[0072]

[0073] where is the estimated value of the parameter vector .

[0074] Based on a batch of n group sampling data, define the output vector and the data matrix :

[0075]

[0076]

[0077] Establish a cost function centered on estimating the sum of squared errors. :

[0078]

[0079] 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:

[0080]

[0081] 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:

[0082]

[0083] Define the covariance matrix The recurrence relation of its inverse matrix:

[0084]

[0085] Substituting equation (16) into equation (15), we can obtain the recursive update formula for the parameter estimation vector through algebraic derivation:

[0086]

[0087] Combining equations (16) and (17), we obtain the basic recursive least squares algorithm:

[0088]

[0089] 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.

[0090] 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:

[0091]

[0092] Equation (19) constitutes the basic recursive calculation formula for online identification of fuel cell model parameters.

[0093] For a real fuel cell system, its parameters vary slowly with the working conditions, such as temperature, humidity, and aging degree. After running for a period of time, the gain vector and the covariance matrix of the basic recursive algorithm will gradually decrease, so that the correction ability of the predicted value becomes weaker and weaker, and the phenomenon of "data saturation" occurs, so that the time-varying characteristics of the parameters cannot be tracked.

[0094] To solve this problem, a forgetting factor is introduced into the algorithm. The forgetting factor enhances the importance of new data by applying an exponentially decaying weight to old data, so that the time-varying parameters can be tracked continuously. After introducing the forgetting factor , equation (19) is modified as:

[0095]

[0096] Equation (20) is the final recursive least squares calculation formula with forgetting factor adopted in this application.

[0097] The value range of the forgetting factor is usually 0.95-1. The smaller the value is, the stronger the tracking ability of the algorithm to the parameter change is, but the convergence speed will be faster, and the estimated result may fluctuate more; The closer the value is to 1, the slower the convergence speed of the algorithm is, and the smoother the estimation is, but the tracking ability to the parameter change will be weakened. When =1, the algorithm degenerates into the ordinary recursive least squares method. By reasonably configuring the forgetting factor, a balance between the convergence speed, estimation accuracy, and parameter tracking ability can be achieved, so that the demand of online state characterization of the fuel cell system can be met.

[0098] Step four: solve the equivalent circuit model parameters, and draw the EIS curve using the model parameters.

[0099] After obtaining the estimated values of the discrete model coefficients to by the recursive algorithm, these coefficients need to be mapped back to the original equivalent circuit model physical parameters.

[0100] Firstly, it is observed that there is a specific relationship between the discrete model coefficients and the intermediate variables in the continuous system model. By analyzing the composition of the coefficients in equation (7), the expression of the key denominator term can be obtained:

[0101]

[0102] Based on the discrete model coefficients to and the intermediate variables in the continuous systema 、 b 、 c 、 d 、 e 、 f 、 g The definition relationship between the intermediate variables and the three RC time constants is obtained by inverse solution, and the solution formula from to a 、 b 、 c 、 d 、 e 、 f 、 g is obtained:

[0103]

[0104] Equation (22) completes the conversion from the identified parameters to the intermediate variables of the continuous system.

[0105] The intermediate variables b 、 c 、 d are closely related to the time constants of the three RC elements . According to the definition of the intermediate variables in step two: 、 、 , the time constant is the root of the following cubic equation by the Wiedermann theorem:

[0106]

[0107] This equation can be solved by the cubic equation root formula, and the numerical solution of is obtained.

[0108] After obtaining a 、 e 、 f 、 g、 , by solving the linear equations about 、 、 、 , all the resistance and capacitance parameters of the equivalent circuit model can be finally solved. The solution formula is as follows:

[0109]

[0110] The solution of equation (23) is the total impedance parameters of the R-(RC)-(RC)-(RC) equivalent circuit model 、 、 、 、 , 、 。

[0111] At this point, all the parameters of the equivalent circuit model have been obtained. Based on these parameters, the electrochemical impedance spectrum curve can be drawn by calculating the frequency response of the transfer function defined in step one on the virtual axis of the complex frequency domain. The EIS curve accurately characterizes the impedance characteristics of the fuel cell at different frequencies, providing information for internal state monitoring, fault diagnosis and health management.

[0112] Embodiment

[0113] Select a fuel cell stack output current of 100A, 200A, 300A working condition, Figure 2 R-(RC)-(RC)-(RC) circuit model diagram; Figure 3 Output current working condition curve, the fuel cell runs under this working condition curve and collects its output voltage, current data, and at the same time uses the algorithm to characterize the internal state of the application; Figure 4 Equivalent circuit model parameter curve with working condition change, it can be seen that when the working condition changes, the internal physical process of the fuel cell changes, the model parameters respond to the change very quickly, and the model parameters also tend to be stable after the working condition is stable. Figure 5 The EIS curve drawn by the recognition result of the application and the EIS frequency point measured by the electrochemical workstation comparison chart, it can be seen that the EIS curve characterized by the application and the 40 EIS frequency points measured by the electrochemical workstation almost coincide, the results are almost consistent, the mean square error is only 3.647e-07, indicating that the application accurately characterizes the internal state of the fuel cell, has great advantages in real-time, and has great application potential in fuel cell fault diagnosis and aging state monitoring.

[0114] Although the embodiments of the application are described above in combination with the drawings, the application is not limited to the above specific embodiments and application fields, and the above specific embodiments are only illustrative and guiding, but not limiting. Those skilled in the art can make many forms under the guidance of this specification and without departing from the scope protected by the claims of the application, which all belong to the protection of the 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, This refers to the output voltage. This is the Nernst voltage; For output current; It represents ohmic resistance, which characterizes the resistance of the membrane, electrodes, and contact parts of a fuel cell; The resistance and capacitance of the first RC parallel unit are used to describe the anode activation impedance, characterize the adsorption and dissociation of the hydroxide reaction on the anode catalyst, and their time constants correspond to the characteristics of the high-frequency region in the EIS curve. The resistance and capacitance of the second RC parallel unit characterize 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. The resistance and capacitance of the third RC parallel unit characterize slower dynamic processes or other diffusion phenomena, and its time constant corresponds to the characteristics of the low-frequency region of the EIS curve. Represents the Laplace operator; S2, the transfer function Transform into an identifiable mathematical model: Define voltage variables The transfer function Rewritten as: Further introduce time constant , and get: And introduce a set of intermediate variables a , b , c , d , e , f , g : ； This simplifies to the following differential equation: ； Using bilinear transformation The differential equation is transformed into a discrete-time model, yielding the discrete form: in, The system sampling period is For unit delay operators, , , , 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 equivalent circuit model parameters 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, 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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