Control method of fuel cell power system integrated with impedance spectroscopy on-line measurement

Abstract

The invention discloses a control method of a fuel cell power system integrated with online impedance spectroscopy measurement, which belongs to the technical field of fuel cell systems, and comprises the following steps: constructing a mathematical model of the fuel cell power system; designing a closed-loop control strategy and an electrochemical impedance spectroscopy measurement strategy of the fuel cell power system; designing an improved active anti-disturbance controller based on multi-resonance control; a multi-stack system electrochemical impedance spectrum disturbance self-cancellation strategy is introduced, and the design process of the fuel cell power system bus voltage control method based on multi-resonance control is achieved. By adopting the control method of the fuel cell power system integrated with impedance spectrum online measurement, the mutual exclusion limitation of online monitoring and voltage stability is broken through, full-band impedance spectrum continuous acquisition is realized under the condition of continuous normal operation of the system, the bus voltage quality is remarkably improved, and the system reliability is improved. And a key technical support is provided for real-time health diagnosis and active life management of the fuel cell power system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of fuel cell system, in particular to a control method of fuel cell power system integrated with online impedance spectrum measurement. BACKGROUND

[0002] Fuel cells can directly convert hydrogen into electricity, with the advantages of environmental friendliness and high energy conversion efficiency. However, single fuel cell systems have slow dynamic response, difficulty in cold start, and insufficient durability. The use of multi-stack architecture and lithium battery hybrid power supply has become a mainstream technology route to improve system reliability. This integrated power system runs under complex working conditions, and the coordinated control and health state monitoring of multiple stacks become new technical challenges, especially how to realize online continuous monitoring of the aging degree of the stacks under the continuous running state of the system.

[0003] Electrochemical impedance spectroscopy technology can analyze the internal dynamic characteristics of fuel cells by applying specific frequency disturbance signals, providing key state parameters for online diagnosis. However, in practical applications, the multi-stack system architecture significantly amplifies the impact of electrochemical impedance spectrum measurement disturbance on system stability: to achieve online continuous monitoring, the wide frequency band disturbance signals (such as triangular wave, rectangular wave) required for measurement need to be continuously applied to the running stacks. These disturbances are coupled to the bus through the DC / DC converter, inducing voltage harmonic distortion and power oscillation, directly affecting the reliability, efficiency and equipment life of the fuel cell power system.

[0004] Active disturbance rejection control strategy is an advanced control strategy based on real-time disturbance observation and dynamic compensation, its core is to regard the internal uncertainty (such as model error, parameter drift) and external disturbance (such as load mutation, power fluctuation) of the system as "total disturbance" through the extended state observer, and generate a reverse compensation signal to offset its impact on the system. In fuel cell power systems, the stability of the bus voltage is directly related to the coordinated efficiency and life of power electronic devices, fuel cell stacks and lithium batteries, and active disturbance rejection control technology has strong robustness and adaptability, which has become a key means to cope with complex disturbance scenarios.

[0005] However, the bandwidth limitation and phase lag characteristics of the active disturbance rejection control strategy are fundamentally in conflict with the continuity requirement of the online impedance spectrum measurement. For periodic disturbances, if the disturbance frequency is close to or exceeds the controller bandwidth, the extended state observer will not be able to accurately track the disturbance signal phase and amplitude, resulting in compensation lag or distortion. For the triangular wave and rectangular wave signals commonly used in electrochemical impedance spectrum detection, the characteristics of multiple harmonics contained make the active disturbance rejection control strategy face even more severe challenges. At the same time, the online detection of impedance spectrum requires a certain amplitude of periodic signal to realize the drawing and collection of electrochemical impedance spectrum, which leads to the technical dilemma of "monitoring system instability, stable system monitoring failure" when the existing system implements online impedance spectrum detection. SUMMARY

[0006] The purpose of the present application is to provide a control method of a fuel cell power system integrated with online impedance spectrum measurement, which can realize continuous collection of full-band impedance spectrum under the condition of continuous normal operation of the system, improve the quality of bus voltage, and provide key technical support for real-time health diagnosis and active life management of the fuel cell power system.

[0007] To achieve the above-mentioned purpose, the present application provides a control method of a fuel cell power system integrated with online impedance spectrum measurement, comprising the following steps:

[0008] S1, constructing a mathematical model of the fuel cell power system;

[0009] S2, designing an equivalent control transfer function of the system based on the constructed mathematical model, and further designing a closed-loop control strategy of the fuel cell power system;

[0010] S3, designing an electrochemical impedance spectrum measurement strategy of the fuel cell power system, introducing the electrochemical impedance spectrum signal into the current loop controller, thereby realizing online measurement based on the converter;

[0011] S4, designing a voltage outer loop controller based on active disturbance rejection control, designing an extended state observer, taking the system output voltage as the control variable, tracking and estimating the error between the reference voltage and the actual voltage;

[0012] S5, designing a system state error feedback controller, and adding a multi-level resonant control unit to obtain an improved bus voltage active disturbance rejection control strategy, and introducing a multi-stack system electrochemical impedance spectrum disturbance self-cancellation strategy.

[0013] Preferably, in S1, the mathematical model of the fuel cell power system is:

[0014]

[0015] wherein, Lk is the inductance value of the Kth fuel cell converter; i Lk is the output current of the Kth fuel cell stack; v fck is the output voltage of the Kth fuel cell stack; r k is the inductance equivalent parasitic resistance of the Kth fuel cell converter; d k is the duty cycle of the Kth fuel cell converter; L0 is the inductance value of the lithium battery converter; i b is the output current of the lithium battery; v b is the output voltage of the lithium battery; r0 is the inductance equivalent parasitic resistance of the lithium battery converter; d0 is the duty cycle of the lithium battery converter; v o is the output voltage of the fuel cell power system; i o is the load current; C is the output capacitance.

[0016] Preferably, in S2, the equivalent control transfer function of the fuel cell power system is:

[0017]

[0018] wherein b, c, a are coefficients of the equivalent transfer function, respectively, L is the inductance value, I Lref is the reference value of the inductance current, and s is a complex frequency variable.

[0019] Preferably, in S4, the equivalent controlled object of the voltage outer loop controller is:

[0020]

[0021] wherein b0 is the nominal value of b, f is the total disturbance of the system, and I Lref is the reference value of the inductance current, denotes the differential with respect to time.

[0022] Preferably, in S4, the design of the extended state observer includes the following steps:

[0023] define the state variable γ = [γ1 γ2] T , i.e., the designed extended state observer respectively tracks the output voltage v o and the total disturbance f of the system; at this time, the equivalent controlled object of the voltage outer loop controller is represented by the state equation as:

[0024]

[0025] wherein A1, B1, C1 are all coefficient matrices of the state equation, β1 and β2 are both gains of the extended state observer. By properly configuring the observer gains, the characteristic polynomial of the extended state observer is configured with a specific bandwidth ω. o The pole placement problem, namely:

[0026]

[0027] By selecting the observer bandwidth ω o The gain parameter is directly determined to balance dynamic response speed and noise immunity; therefore, the observer gain matrix is ​​configured as follows:

[0028]

[0029] Define the reference value of the system output voltage as V ref The difference between the output voltage and the reference voltage is the output voltage tracking error e. v Therefore:

[0030]

[0031] In the formula, Indicates the output voltage tracking error e v The derivative; Indicates the output voltage reference value V ref The derivative; Indicates the output voltage v o The derivative;

[0032] The convergence of the reference voltage tracking error is expressed as:

[0033]

[0034] Where, k p It is proportional gain;

[0035]

[0036] in, It is a disturbance estimate.

[0037] Preferably, in S5, the system state error feedback controller is:

[0038]

[0039] Among them, v ref This is the reference value for the system output voltage, k. p It is the proportional gain, v o It is the output voltage of the fuel cell power system, and N is the number of resonant control units;

[0040] G qrNLet be the transfer function of the Nth stage resonant control unit, and its specific expression is as follows:

[0041]

[0042] Where s is the complex frequency variable, k rN ω is the resonance coefficient corresponding to the Nth level resonant control unit. cN ω is the cutoff frequency corresponding to the Nth stage resonant control unit. rN It is the resonant frequency corresponding to the Nth level resonant control unit.

[0043] Preferably, in S5, a self-cancelling strategy for perturbation of the electrochemical impedance spectral spectrum of a multi-stall system is introduced, introducing a different phase shift angle into the electrochemical impedance spectral signal corresponding to each stack, as shown in the following formula:

[0044] f k (t)=M k f(t+θ k ), k = 1, 2, ... N;

[0045] Among them, f k (t) represents the electrochemical impedance spectral signal introduced by the k-th stack, M k θ k These represent the signal amplitude and phase angle, respectively, where f represents the fundamental frequency signal and t represents the time variable.

[0046] Therefore, the present invention adopts the above-mentioned control method for fuel cell power systems with integrated online impedance spectrum measurement, which breaks through the mutual exclusion limitation between online monitoring and voltage stability in traditional methods. Under the condition of continuous normal operation of the system, it realizes continuous acquisition of impedance spectrum across the entire frequency band, significantly improves the bus voltage quality, and provides key technical support for real-time health diagnosis and active life management of fuel cell power systems.

[0047] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0048] Figure 1 This is a flowchart of an embodiment of a control method for a fuel cell power system integrating online impedance spectrum measurement according to the present invention;

[0049] Figure 2 This is a typical structural diagram of the fuel cell power system in an embodiment of the present invention;

[0050] Figure 3 This is a block diagram of the control scheme for the fuel cell power system in an embodiment of the present invention;

[0051] Figure 4 This is a block diagram of the online electrochemical impedance spectroscopy detection structure in an embodiment of the present invention;

[0052] Figure 5 This is a block diagram of the improved active disturbance rejection control strategy in an embodiment of the present invention;

[0053] Figure 6 The output voltage of a typical control strategy when faced with triangular wave electrochemical impedance spectral perturbation;

[0054] Figure 7 This refers to the output voltage of the improved active anti-disturbance control strategy in this embodiment of the invention when faced with triangular wave electrochemical impedance spectral disturbances.

[0055] Figure 8 The output voltage of a typical control strategy when faced with rectangular wave electrochemical impedance spectral perturbation;

[0056] Figure 9 The output voltage of the improved active disturbance rejection control strategy in this embodiment of the invention when faced with rectangular wave electrochemical impedance spectral disturbances. Detailed Implementation

[0057] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0058] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0059] Example 1

[0060] This embodiment uses the example of online measurement of triangular or rectangular wave electrochemical impedance spectroscopy signals in a fuel cell power system as an example.

[0061] like Figure 2 As shown, a typical fuel cell power system consists of one lithium battery and two fuel cell stacks. The lithium battery and fuel cells are connected to the DC bus and load respectively via DC-DC converters, and their main circuit parameters are shown in Table 1.

[0062] Table 1. Key Parameters of a Typical Fuel Cell Power System Circuit

[0063]

[0064] This embodiment provides a control method for a fuel cell power system with integrated online impedance spectrum measurement, including the following steps:

[0065] S1. Construct a mathematical model of the fuel cell power system.

[0066] Based on Kirchhoff's voltage and current laws and the working principle of DC-DC converters, Figure 2 The fuel cell power system model in the image is as follows:

[0067]

[0068] Among them, L k i is the inductance value of the Kth fuel cell converter; Lk It is the output current of the Kth fuel cell stack; v fck r is the output voltage of the Kth fuel cell stack; k d is the equivalent parasitic resistance of the inductance of the Kth fuel cell converter; k L0 is the duty cycle of the Kth fuel cell converter; L0 is the inductance value of the lithium battery converter; i b This is the output current of the lithium battery; v b d0 is the output voltage of the lithium battery; r0 is the equivalent parasitic resistance of the lithium battery converter's inductance; d0 is the duty cycle of the lithium battery converter; v o It is the output voltage of the fuel cell power system; i o C is the load current; C is the output capacitance.

[0069] S2. Based on the established mathematical model, design the equivalent control transfer function of the system, and then design the closed-loop control strategy of the fuel cell power system.

[0070] In this embodiment, the fuel cell power system consists of a lithium battery and its converter, and two fuel cell stacks and their converters. The control strategy for the lithium battery and its converter adopts a voltage and current dual closed-loop control strategy, while the control strategy for the fuel cells and their converter adopts a single current loop control, such as... Figure 3 As shown. To achieve online measurement of electrochemical impedance spectroscopy (EIS) signals, triangular or rectangular wave EIS signals can be introduced into the reference current of the fuel cell current loop, such as... Figure 4 As shown.

[0071] Based on the mathematical model of the fuel cell power system, its equivalent control transfer function is:

[0072]

[0073] Where b, c, and a are the coefficients of the equivalent transfer function. L is the inductance value, I Lref is the reference value for the inductor current, and s is the complex frequency variable.

[0074] S3. Design an electrochemical impedance spectroscopy measurement strategy for fuel cell power systems, and introduce the electrochemical impedance spectroscopy signal into the current loop controller to achieve online measurement based on the converter.

[0075] S4. Design a voltage outer loop controller based on active disturbance rejection control, and design an extended state observer to track and estimate the error between the reference voltage and the actual voltage using the system output voltage as the control quantity.

[0076] From formula (2), the equivalent controlled object of the voltage outer loop controller can be obtained as follows:

[0077]

[0078] Where b0 is the nominal value of b, f is the total system disturbance, and I Lref This is a reference value for the inductor current. It represents the derivative with respect to time.

[0079] Typically, an extended state observer needs to be designed to estimate the total disturbance. Designing an extended state observer includes the following steps:

[0080] Define the state variable γ = [γ1 γ2] T That is, the designed extended state observer tracks the output voltage v. o And the total system disturbance f. At this time, formula (3) can be expressed by the state equation as:

[0081]

[0082] Where A1, B1, and C1 are all coefficient matrices of the state equations. β1 and β2 are both gains of the extended state observer. By properly configuring the observer gains, the characteristic polynomial of the extended state observer is configured with a specific bandwidth ω. o The pole placement problem, namely:

[0083]

[0084] By selecting the observer bandwidth ω o This directly determines the gain parameter, thereby balancing dynamic response speed and noise immunity. Therefore, the observer gain matrix is ​​configured as follows:

[0085]

[0086] Define the reference value of the system output voltage as V ref The difference between the output voltage and the reference voltage is the output voltage tracking error e. v Therefore:

[0087]

[0088] In the formula, Indicates the output voltage tracking error e v The derivative; Indicates the output voltage reference value V ref The derivative; Indicates the output voltage v o The derivative of .

[0089] Assuming the designed extended state observer can estimate the system disturbances well, the convergence of the reference voltage tracking error can be expressed as:

[0090]

[0091] Where, k p It is proportional gain.

[0092]

[0093] in, It is a disturbance estimate.

[0094] S5. Design a system state error feedback controller and add a multi-stage resonant control unit to it to obtain an improved active anti-disturbance control strategy for bus voltage. At the same time, introduce a self-cancellation strategy for electrochemical impedance spectral disturbances in multi-stacking systems.

[0095] Since the reference value of the output voltage is a constant, substituting (5) into (4) yields the system state error feedback controller designed without the introduction of a resonant control unit:

[0096] u0 = k p (v ref -v o (10)

[0097] However, due to the bandwidth ω of the extended state observer o Due to various limitations, the observer's disturbance compensation capability is restricted by its bandwidth. When the system faces time-varying periodic disturbances, if the disturbance frequency is close to or exceeds the controller bandwidth, the extended state observer will be unable to accurately track the phase and amplitude of the disturbance signal, resulting in compensation lag or distortion. In this embodiment, the triangular and rectangular wave disturbances not only include the fundamental frequency but also multiple harmonic components such as the 3rd, 5th, and 7th harmonics, further challenging the controller's disturbance suppression capability.

[0098] Furthermore, the fuel cell power system designed in this embodiment includes multiple fuel cell stacks. When multiple stacks are simultaneously subjected to electrochemical impedance spectroscopy measurements, it will have a more severe impact on the system's bus voltage.

[0099] The controller designed using the above process, when faced with a 10Hz triangular wave or rectangular wave disturbance, will have the following steady-state output voltages: Figure 6 , Figure 8 As shown in the figure, the system output voltage quality is poor and the output voltage fluctuates violently.

[0100] In addition, from Figure 6 , Figure 8 The FFT analysis of the system bus voltage shows that the output voltage v o This design includes harmonic components at multiple harmonic frequencies such as 10Hz, 30Hz, and 50Hz. To suppress these harmonic components, this embodiment incorporates a multi-stage resonant control unit into the state error feedback controller to mitigate their impact on the bus voltage. The block diagram of the improved active disturbance rejection control strategy based on this design is shown below. Figure 5 As shown, the state error feedback controller after introducing a multi-stage resonant control unit is as follows:

[0101]

[0102] Among them, v ref This is the reference value for the system output voltage, k. p It is the proportional gain, v o It is the output voltage of the fuel cell power system, and N is the number of resonant control units;

[0103] The value of N is related to the form of the electrochemical impedance spectral perturbation. In this embodiment, in order to achieve a good bus voltage control effect, N is 2 when the electrochemical impedance spectral perturbation is a triangular wave, and N is 10 when the electrochemical impedance spectral perturbation is a rectangular wave.

[0104] G qrN Let be the transfer function of the Nth stage resonant control unit, and its specific expression is as follows:

[0105]

[0106] Where s is the complex frequency variable, k rN ω is the resonance coefficient corresponding to the Nth level resonant control unit. cN ω is the cutoff frequency corresponding to the Nth stage resonant control unit. rN It is the resonant frequency corresponding to the Nth level resonant control unit, i.e., 10Hz, 30Hz, 50Hz, etc.

[0107] In this embodiment, a self-cancelling mechanism for multi-stack electrochemical impedance spectral perturbations is introduced. Taking two stacks as an example, a different phase shift angle is introduced into the electrochemical impedance spectral signal corresponding to each stack, thereby reducing the perturbation estimation burden on the observer when performing electrochemical impedance spectral testing on the fuel cell power system. The formula is as follows:

[0108] f1(t)=22f(t+θ1), f2(t)=16f(t+θ2) (13)

[0109] Where f1(t) and f2(t) represent the electrochemical impedance spectral signals introduced by the two fuel cells, respectively, θ=(θ1,θ2)=(0°,180°), and f represents the fundamental frequency signal.

[0110] After introducing a disturbance self-cancellation mechanism and a multi-stage resonant control unit, the steady-state output voltage of the system in the face of triangular or rectangular wave electrochemical impedance spectrum disturbances is as follows: Figure 7 , Figure 9 As shown in the figure, it can be seen that the fluctuation of the system bus voltage is greatly reduced at this time, and the bus voltage can be basically stabilized at the required 270V.

[0111] The method in this embodiment can be extended to systems such as fuel cells and lithium batteries that require electrochemical impedance spectroscopy detection.

[0112] Therefore, the present invention adopts the above-mentioned control method for fuel cell power systems with integrated online impedance spectrum measurement, which breaks through the mutual exclusion limitation between online monitoring and voltage stability in traditional methods. Under the condition of continuous normal operation of the system, it realizes continuous acquisition of impedance spectrum across the entire frequency band, significantly improves the bus voltage quality, and provides key technical support for real-time health diagnosis and active life management of fuel cell power systems.

[0113] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A control method for a fuel cell power system integrating online impedance spectroscopy measurement, characterized in that, Includes the following steps: S1. Construct a mathematical model of the fuel cell power system; S2. Based on the established mathematical model, design the equivalent control transfer function of the system, and then design the closed-loop control strategy of the fuel cell power system. S3. Design an electrochemical impedance spectroscopy measurement strategy for a fuel cell power system, and introduce the electrochemical impedance spectroscopy signal into the current loop controller to achieve online measurement based on the converter; S4. Design a voltage outer loop controller based on active disturbance rejection control, and design an extended state observer to track and estimate the error between the reference voltage and the actual voltage using the system output voltage as the control quantity. S5. Design a system state error feedback controller and add a multi-stage resonant control unit to it to obtain an improved active anti-disturbance control strategy for bus voltage. At the same time, introduce a self-cancellation strategy for electrochemical impedance spectral disturbances in multi-stacking systems.

2. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 1, characterized in that: In S1, the mathematical model of the fuel cell power system is: Among them, L k i is the inductance value of the k-th fuel cell converter; Lk It is the output current of the k-th fuel cell stack; v fck r is the output voltage of the k-th fuel cell stack; k d is the equivalent parasitic resistance of the inductance of the k-th fuel cell converter; k L0 is the duty cycle of the k-th fuel cell converter; L0 is the inductance value of the lithium battery converter; i b This is the output current of the lithium battery; v b d0 is the output voltage of the lithium battery; r0 is the equivalent parasitic resistance of the lithium battery converter's inductance; d0 is the duty cycle of the lithium battery converter; v o It is the system output voltage; i o C is the load current; C is the output capacitance.

3. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 2, characterized in that: In S2, the equivalent control transfer function of the fuel cell power system is: Where b, c, and a are the coefficients of the equivalent transfer function, L is the inductance value, I Lref is the reference value for the inductor current, and s is the complex frequency variable.

4. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 3, characterized in that: In S4, the equivalent controlled object of the voltage outer loop controller is: Where b0 is the nominal value of b, f is the total system disturbance, and I Lref This is a reference value for the inductor current. It represents the derivative with respect to time.

5. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 4, characterized in that: In S4, designing the extended state observer includes the following steps: definition State variable γ = [γ1γ2] T That is, the designed extended state observer tracks the system output voltage v. o And the total system disturbance f; at this time, the equivalent controlled object of the voltage outer loop controller is represented by the state equation as: Where A1, B1, and C1 are all coefficient matrices of the state equations. β1 and β2 are both gains of the extended state observer. By properly configuring the observer gains, the characteristic polynomial of the extended state observer is configured with a specific bandwidth ω. o The pole placement problem, namely: By selecting the observer bandwidth ω o The gain parameter is directly determined to balance dynamic response speed and noise immunity; therefore, the observer gain matrix is ​​configured as follows: Define the reference value of the system output voltage as V. ref The difference between the output voltage and the reference voltage is the output voltage tracking error e. v Therefore: In the formula, The output voltage tracking error e is represented by v The derivative; Indicates the output voltage reference value V ref The derivative; Indicates the output voltage v o The derivative; The convergence of the reference voltage tracking error is expressed as: Where, k p It is proportional gain; in, It is a disturbance estimate.

6. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 1, characterized in that: In S5, the system state error feedback controller is: Among them, v ref This is the reference value for the system output voltage, k. p It is the proportional gain, v o It is the output voltage of the fuel cell power system, and N is the number of resonant control units; G qrN Let be the transfer function of the Nth stage resonant control unit, and its specific expression is as follows: Where s is the complex frequency variable, k rN ω is the resonance coefficient corresponding to the Nth level resonant control unit. cN ω is the cutoff frequency corresponding to the Nth stage resonant control unit. rN It is the resonant frequency corresponding to the Nth level resonant control unit.

7. The control method for a fuel cell power system with integrated online impedance spectrum measurement according to claim 1, characterized in that: In S5, a self-cancelling strategy for perturbation of the electrochemical impedance spectral spectrum of a multi-stall system is introduced, which introduces a different phase shift angle into the electrochemical impedance spectral signal corresponding to each stack. The formula is shown below: f k (t)=M k f(t+θ k ),k=1,2,…N; Among them, f k (t) represents the electrochemical impedance spectral signal introduced by the k-th stack, M k θ k These represent the signal amplitude and phase angle, respectively, where f represents the fundamental frequency signal and t represents the time variable.

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