Real-Time Impedance Detection for PEM Fuel Cell Based on TAB Converter Voltage Perturbation

Abstract

Fuel cells, as clean and efficient energy conversion devices, hold great potential for applications in the fields of hydrogen-based transportation and stand-alone power systems. Due to their sensitivity to load parameters, environmental parameters, and gas supply, the performance monitoring and fault diagnosis of fuel cell systems have become crucial research areas. Electrochemical impedance spectroscopy (EIS) is a widely applied analytical method in fuel cell systems. that can provide rich information about dynamic system responses, internal impedance, and transmission characteristics. Currently, EIS detection is primarily implemented by using simple topologies such as boost circuits. However, the injection of excitation signals often results in significant power fluctuations, leading to issues such as uneven temperature distributions within the cell, unstable gas supply, and damage to the proton exchange membrane. To address this issue, this paper proposes a real-time EIS detection technique for a proton exchange membrane fuel cell (PEMFC) system that connects a lithium-ion battery and injects the load voltage perturbation through a triple active bridge (TAB) converter. By applying the small-signal model of the TAB converter and designing a system controller using a decoupling control method, the PEMFC power remains stable after the disturbance injection across the entire frequency range under tests. Furthermore, the lithium-ion battery can instantly track load changes during fluctuations. The proposed EIS detection method can acquire EIS data in real time to monitor the state of the PEMFC. Simulation results validate the effectiveness and accuracy of the proposed method for EIS detection.

1. Introduction

Proton exchange membrane fuel cells (PEMFCs) are one of the most promising energy storage devices due to their high power density, non-polluted reaction by-products, and low noise emissions. They are increasingly being utilized in a wide range of applications, such as transportation, aerospace, and backup power [1,2,3]. PEMFCs demonstrate excellent performance in energy conversion efficiency, low-temperature start-up, rapid hydrogen refueling, and gas emissions, thereby presenting a bright future in the field of power supply sources [4,5,6].

PEMFCs exhibit a complex interaction of electrical, thermal, and gaseous dynamics, making their control systems intricate. Their operational efficiency is influenced by factors such as temperature, humidity, gas pressure, electrolyte properties, and electrode materials. Real-time monitoring of PEMFC system operation is crucial to ensuring its efficient and stable performance and extending the lifespan of the equipment [7]. The detection of fuel cell impedance, which reflects the resistance faced by ions and materials moving within the cell, is a key diagnostic tool. Electrochemical impedance spectroscopy (EIS) is extensively used for the performance evaluation and fault diagnosis of PEMFCs. In the main, an EIS detection method applies a small AC signal disturbance at various frequencies to PEMFC and analyzes the amplitude and phase of responses as a function of frequency, from which the real and imaginary parts of the frequency-dependent impedance are evaluated [8]. When conducting EIS measurements, the electrical properties of the PEMFC can be represented by an equivalent circuit model, which is inferred from the internal characteristics of the PEMFC. This associates the measured impedance response with the electrochemical and transport phenomena occurring within the PEMFC. In [9], the main features of different PEMFC simulators were summarized and introduced, providing mathematical models to modify the equivalent circuit model together with PEMFC electronic circuit examples for the construction of new PEMFC simulators. In [10], a PEMFC voltage equivalent circuit model that only depends on the input current was developed, based on which an electronic circuit for PEMFC voltage simulation can be established with the calculated parameters, avoiding the use of a real PEMFC.

PEMFCs generally exhibit a delayed response to electrochemical processes due to the constrained rates of hydrogen and oxygen supplies, which make them unable to dynamically align with the electrical load. This necessitates the deployment of fast-acting energy storage devices such as lithium-ion batteries alongside a PEMFC to deal with sudden load variations [11]. The integration of PEMFC with lithium-ion batteries can be facilitated by triple active bridge (TAB) converters, which feature electrical isolation between input and output ports, high conversion efficiency, and easy implementation of soft switching. The TAB converters allow for bidirectional power flows at all three ports and are widely applied in hybrid power supply systems with fuel cells [12,13,14]. Therefore, it is necessary to consider the employment of TAB converters in PEMFC EIS measurements. However, most research related to PEMFC impedance measurements uses simple circuits such as boost converters and neglects the power fluctuations in PEMFCs induced by the small AC signal disturbances injected, which would affect EIS measurements. To enable real-time assessment of the health status of automotive PEMFCs, an integrated approach has been proposed in [15] where a DC/DC boost converter with EIS functionality was directly connected to the PEMFC stack and was also compared between different converter topologies to identify the optimal configuration. In [16], a parallel boost DC/DC converter topology was proposed for the real-time impedance estimation of PEMFC stacks by using one boost converter for excitation injection and the other to adjust the output characteristics so as to meet load demands and facilitate EIS measurements. The EIS measurements using boost and buck-boost converters were compared in [17] with those that were obtained from frequency response analyzers, illustrating the feasibility and practicality of PEMFC EIS detection based on DC/DC converters. In [18], current disturbances were introduced into boost circuits, and the excitation current signals and voltage responses were accurately captured by the designed signal conditioning circuits, allowing for cost-effective online impedance detection but at the cost of significant power fluctuations induced by the excitation current. Real-time EIS detection was also explored in [19] by integrating the impedance testing devices with buck-boost converters through DC-DC circuits for the introduction of AC signals. To address the issue of low-frequency disturbances that affect the normal operation and efficiency of PEMFCs during EIS detection, an isolated DC/DC converter along with its control strategy was developed in [20] to minimize the impacts of low-frequency oscillations on the PEMFC system. Instead of a small AC signal disturbance, a step function disturbance was applied in [21] and added onto the target voltage of a closed-loop controller, which regulated the PEMFC output voltage based on the buck/boost circuit; the EIS measurements were then obtained from the frequency spectrum for one disturbance cycle. In [22], EIS measurements were performed by employing a parallel dual boost DC/DC converter system topology where DC currents were exported by a DC/DC module and the sinusoidal disturbance currents were generated by a DC/AC module and optimized by feedforward and fuzzy control techniques to increase the accuracy of EIS measurements.

The contributions of this paper are to propose a real-time EIS detection method for a PEMFC that is modeled by an equivalent circuit model and connected to a lithium-ion battery via a TAB converter. A TAB converter system controller is designed to ensure the stability of PEMFC outputs and load voltage levels. A small AC voltage perturbation is superimposed on the target load voltage, and the resulting deviation from the load voltage measurement is introduced into the circuit through the third port of the TAB converter. Then, the excitation and response signals of the PEMFC are processed by the fast Fourier transform (FFT) to extract their respective amplitude and phase information, from which the frequency-dependent PEMFC internal impedance is evaluated in real time. The operation of the PEMFC and lithium-ion battery, together with the PEMFC impedance, is simulated under various operating conditions to validate the effectiveness of the designed control strategies and the proposed EIS detection method.

2. PEMFC EIS Detection Method Based on TAB Converter

To accurately capture the internal dynamics of a PEMFC, the coupled effects of electrochemical substance transport processes within the PEMFC and the system’s nonlinear characteristics are considered in this work. Circuit components that effectively represent the system behavior are selected to construct the equivalent circuit model of the PEMFC. The TAB converter independently controls the power flow at each port and regulates the outputs of PEMFC and lithium-ion batteries in response to load variations. With the injection of the small AC voltage perturbation, the resulting voltage and current signals of the PEMFC are sampled and analyzed by FFT to produce an EIS that reflects the internal state of the system. The proposed method lays the foundation for PEMFC fault diagnosis and real-time performance assessment, given the use of a TAB converter.

2.1. PEMFC Impedance Model

There are numerous impedance models for PEMFC, of which the Randles model [23] is the most common, as shown in Figure 1. The term $C_{dl}$ represents the double-layer distributed capacitance formed at the electrode–electrolyte interface; the polarisation resistance $R_p$ reflects the obstructive effect on the electrons transferring between the electrode surface and the surface-adsorbing medium; $R_{\Omega }$ denotes the ohmic resistance which reflects the total obstructive effects of the electrode, proton exchange membrane and reaction medium on the current [24]; and the Warburg impedance which is modelled here by the parallel connected capacitor $C_W$ and resistor $R_W$ reflects the diffusion impedance caused by the substance concentration changes and the resistance of the reactants diffusing from electrolyte to catalyst layer [25].

The output power of a single fuel cell is limited, though the stacking of multiple fuel cells can achieve a higher output power than the PEMFC stack. The integration of multiple fuel cell units within a limited space is beneficial for uniform thermal management, water management, and gas supply management in the stack. For a stack that is formed by the series connection of $n$ fuel cell units, its equivalent circuit model can be derived based on the series and parallel relationships of resistances and capacitances, as shown in Figure 2, where the equivalent circuit components are calculated by:

$$R_1=n{R}_P, R_2=n{R}_W, R_3=n{R}_{\Omega }, C_1=C_{dl}/n, C_2=C_W/n$$

(1)

To provide a benchmark for the PEMFC EIS detection results, the theoretical impedance $Z$ of the PEMFC stack is derived based on its equivalent impedance model:

$$\begin{array}{ll}Z& =(R_2//C_2+R_1)//C_1+R_3\\ & =R_3+{\displaystyle \frac{R_1+R_2+j\omega R_1{R}_2{C}_2}{1-{\omega }^2{R}_1{R}_2{C}_1{C}_2+j\omega (R_1{C}_1+R_2{C}_1+R_2{C}_2)}}\end{array}$$

(2)

where $\omega$ is the angular frequency.

2.2. Principle of EIS Detection

The principle of EIS detection for PEMFC is illustrated in Figure 3. When an excitation signal is injected into the PEMFC stack, the system generates response current and voltage signals, which contain phase and amplitude information at different frequencies. By analyzing the amplitude and phase information of the current–voltage response signals, the PEMFC impedance can be calculated as a function of frequency, forming the EIS, which indicates the impedance at different frequencies on the complex plane.

First, a sinusoidal excitation signal of either voltage $V_{FC}$ or current $I_{FC}$ is injected into the PEMFC system. The resulting response signals of voltage $V_{FC}^{\prime }$ and current $I_{FC}^{\prime }$ are formulated by:

$$V_{FC}^{\prime }=V_0+V\sin (\omega t+\alpha )$$

(3)

$$I_{FC}^{\prime }=I_0+I\sin (\omega t+\beta )$$

(4)

where terms $V_0$ and $I_0$ represent the DC components in the response signals of voltage and current, respectively, $V$ and α (or I and β) denote the amplitude and phase angle of the AC components of the voltage (or current) response signal, respectively, and $\omega$ is the angular frequency of the response signals. Then, the magnitude $Z_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ of impedance can be computed by (5) given the DC voltage $E^{\prime }$:

$$Z_{\mathrm{meas}}={\displaystyle \frac{E^{\prime }-V_{FC}^{\prime }}{I_{FC}^{\prime }}}$$

(5)

2.3. Topology and Control Methods of TAB Converter

The topology of the PEMFC EIS detection system based on the TAB converter is illustrated in Figure 4. The system consists of three parts: the main circuit, the EIS mapping unit, and the excitation signal injection unit. The TAB converter is formed by coupling three full-bridge circuits through a three-winding transformer, which acts as the power transmission circuit. Port 1 is connected to the PEMFC, which is modeled by its equivalent circuit model; port 2 uses a lithium-ion battery as the fast-acting energy storage device, which is modeled by a DC voltage source; and port 3 serves as the load port. In Figure 4, $V_{fc}$, $V_{2,}$ or $V_3$ represent the voltage of the fuel cell, lithium-ion battery, or load, respectively; $I_{fc}$, or $I_1,$ is the current of the fuel cell before or after the filter; and $I_2$ or $I_3$ denote the current of the lithium-ion battery or load, respectively. Terms $S_1$, …, $S_4$, $Q_1$, …, $Q_4$ and $K_1$, …, $K_4$ are the controllable switch tubes of the TAB converter; $L_1$, $L_2$ or $L_3$ is the power transmission inductance (i.e., the sum of transformer leakage inductance and externally connected auxiliary inductance); and $C_1$, $C_2$ and $C_3$ are filtering capacitors. The output of the PEMFC (or the lithium-ion battery) is smoothed by an L-C filter consisting of $L_{f1}$ and $C_1$ (or $L_{f2}$ and $C_2$). $r_1$ is the equivalent series resistance (ESR) of $L_{f1}$, and $R_3$ is the load resistance.

The output of the TAB converter is used to supply power to the load after boosting. A single-loop control strategy is applied to the load voltage. An AC excitation signal $V_{AC}$ is superimposed on the voltage set-point $V_{ref}$ and then compared with the sampled load voltage, with their difference entering the main control module to drive signals for the switching control of the MOSFET. The EIS mapping unit samples and processes the output voltage and current at the fuel cell port, which are then processed by FFT to extract their respective amplitude and phase information for the impedance spectrum generation in the host computer.

As shown in Figure 5a, voltage outputs $u_1$, $u_2$, and $u_3$ at the three ports are all in the form of square waves, with amplitudes equaling the supply voltages $V_1$, $V_2,$ and $V_3$ of the three ports, respectively. The phase angles ${\phi }_{12}$, ${\phi }_{13},$ and ${\phi }_{23}$ represent the phase differences between u₁ and u₂, u₁ and u₃, and u₂ and u₃, respectively. Given a switching cycle of 2$\pi$, ${\phi }_{12}$, ${\phi }_{13}$, and ${\phi }_{23}$ can be calculated from the differences between the initial phase angles ${\theta }_1$, ${\theta }_2,$ and ${\theta }_3$ of the square wave voltages at the three ports. By controlling the switching of the tubes to adjust phase angles, the directions and magnitudes of power flows can be regulated.

The Δ-connected equivalent circuit of the TAB converter is shown in Figure 5b, where the three ports can be regarded as three DC sources controlled by phase shift angles. $L_{12}$, $L_{13}$, and $L_{23}$, representing the equivalent series inductance at each port, are formulated by:

$$\left\{\begin{array}{l}{L}_{12}={\displaystyle \frac{L_1{L}_2}{{n_1}^2{n_2}^2{L}_3}}+{\displaystyle \frac{L_1}{{n_1}^2}}+{\displaystyle \frac{L_2}{{n_2}^2}}\\ L_{13}={\displaystyle \frac{{n_2}^2{L}_1{L}_3}{{n_1}^2{L}_2}}+{\displaystyle \frac{L_1}{{n_1}^2}}+L_3\\ L_{23}={\displaystyle \frac{{n_1}^2{L}_2{L}_3}{{n_2}^2{L}_1}}+{\displaystyle \frac{L_2}{{n_2}^2}}+L_3\end{array}\right.$$

(6)

where $n_1=N_1/N_3$, $n_2=N_2/N_3$.

Transmission power between ports 1 and 2, ports 1 and 3, and ports 2 and 3 is represented by $P_{12}$, $P_{13}$, and $P_{23}$, respectively. By summing the input and output power of each port, the power transmission at each port can be formulated as follows:

$$\left\{\begin{array}{l}{P}_1=P_{12}+P_{13}={\displaystyle \frac{V_1{V}_2{\phi }_{12}}{n_1{n}_2{w}_s{L}_{12}}}\left(1-{\displaystyle \frac{{\phi }_{12}}{\pi }}\right)+{\displaystyle \frac{V_1{V}_3{\phi }_{13}}{n_1{w}_s{L}_{13}}}\left(1-{\displaystyle \frac{{\phi }_{13}}{\pi }}\right)\\ P_2=P_{21}+P_{23}={\displaystyle \frac{V_2{V}_3{\phi }_{23}}{n_2{w}_s{L}_{23}}}\left(1-{\displaystyle \frac{{\phi }_{23}}{\pi }}\right)-{\displaystyle \frac{V_1{V}_2{\phi }_{12}}{n_1{n}_2{w}_s{L}_{12}}}\left(1-{\displaystyle \frac{{\phi }_{12}}{\pi }}\right)\\ P_3=P_{31}+P_{32}=-{\displaystyle \frac{V_1{V}_3{\phi }_{13}}{n_1{w}_s{L}_{13}}}\left(1-{\displaystyle \frac{{\phi }_{13}}{\pi }}\right)-{\displaystyle \frac{V_2{V}_3{\phi }_{23}}{n_2{w}_s{L}_{23}}}\left(1-{\displaystyle \frac{{\phi }_{23}}{\pi }}\right)\end{array}\right.$$

(7)

3. Design of Control Systems

3.1. Small Signal Model

The output current of a fuel cell is generally maintained constant to ensure it operates within its optimal range during fluctuating conditions. In addition, it is imperative to keep a constant load voltage during system operation, considering the linear relationship between load voltage and load current. Consequently, controlling the load current is necessary. In order to control the three ports, the average current method is used here to create a small signal model of the TAB converter. The energy at the three ports of the TAB converter is conserved, where the energy at the lithium-ion battery port is determined by the fuel cell and the load, i.e., $P_2=-(P_1+P_3)$. Therefore, only the currents at the fuel cell port (i.e., $I_{fc}$ at port 1) and the load port (i.e., $I_3$ at port 3) are considered here and formulated by:

$$\begin{array}{l}{I}_{fc}={\displaystyle \frac{P_1}{V_1}}={\displaystyle \frac{V_2}{n_1{n}_2{\omega }_s{L}_{12}}}{\phi }_{12}\left(1-{\displaystyle \frac{{\phi }_{12}}{\pi }}\right)+{\displaystyle \frac{V_3}{n_1{\omega }_s{L}_{13}}}{\phi }_{13}\left(1-{\displaystyle \frac{{\phi }_{13}}{\pi }}\right)\\ I_3={\displaystyle \frac{P_3}{V_3}}={\displaystyle \frac{V_1}{n_1{\omega }_s{L}_{13}}}{\phi }_{13}\left(1-{\displaystyle \frac{{\phi }_{13}}{\pi }}\right)+{\displaystyle \frac{V_2}{n_2{\omega }_s{L}_{23}}}\left({\phi }_{13}-{\phi }_{12}\right)\left(1-{\displaystyle \frac{{\phi }_{13}-{\phi }_{12}}{\pi }}\right)\end{array}$$

(8)

At a steady-state operating point, the partial derivatives of $I_{fc}$ and $I_3$ with respect to the phase shifts ${\phi }_{12}$ and ${\phi }_{13}$ yield the small signal model for the currents with respect to the two phase angles:

$$\left[\begin{array}{c}\Delta I_{fc}\\ \Delta I_3\end{array}\right]=\left[\begin{array}{cc}{G}_{11}& G_{12}\\ G_{21}& G_{22}\end{array}\right]\left[\begin{array}{c}\Delta {\phi }_{12}\\ \Delta {\phi }_{13}\end{array}\right]=G\left[\begin{array}{c}\Delta {\phi }_{12}\\ \Delta {\phi }_{13}\end{array}\right]$$

(9)

where,

$$\begin{array}{l}{G}_{11}={\displaystyle \frac{V_2}{n_1{n}_2{\omega }_s{L}_{12}}}\left(1-{\displaystyle \frac{2}{\pi }}{\phi }_{12q}\right)\\ G_{12}={\displaystyle \frac{V_3}{n_1{\omega }_s{L}_{13}}}\left(1-{\displaystyle \frac{2}{\pi }}{\phi }_{13q}\right)\\ G_{21}={\displaystyle \frac{V_2}{n_2{\omega }_s{L}_{23}}}\left[{\displaystyle \frac{2}{\pi }}\left({\phi }_{13q}-{\phi }_{12q}\right)-1\right]\\ G_{22}={\displaystyle \frac{V_1}{n_1{\omega }_s{L}_{13}}}\left(1-{\displaystyle \frac{2}{\pi }}{\phi }_{13q}\right)+{\displaystyle \frac{V_2}{n_2{\omega }_s{L}_{23}}}\left[1-{\displaystyle \frac{2}{\pi }}\left({\phi }_{13q}-{\phi }_{12q}\right)\right]\end{array}$$

(10)

where the subscript $q$ denotes a particular steady-state operating point.

3.2. Decoupling Control

Based on the small-signal model of the current, it is evident that both currents $I_{fc}$ and $I_3$ are simultaneously influenced by the two phase shifts. Taking the current control system as an example, when the PEMFC output current $I_{fc}$ is controlled by adjusting the phase shift ${\phi }_{12}$, the changes in ${\phi }_{12}$ will alter the load voltage $V_3$ and prevent it from stabilizing at the target voltage, which incurs the requirement of a decoupling control for the converter. The overall control diagram of the system is shown in Figure 6. A feedforward decoupling method is adopted in this work by introducing a decoupling matrix $H$ before matrix $G$ to independently control $I_{fc}$ and $I_3$.

The decoupling matrix $H$ must satisfy the following conditions:

$$\left\{\begin{array}{l}(H_{12}{G}_{11}+G_{12}){\phi }_{12}=0\\ (H_{21}{G}_{22}+G_{21}){\phi }_{13}=0\end{array}\right.$$

(11)

$$H=\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right]=\left[\begin{array}{cc}0& -{\displaystyle \frac{G_{12}}{G_{11}}}\\ -{\displaystyle \frac{G_{21}}{G_{22}}}& 0\end{array}\right]$$

(12)

In Figure 6, $H_v$ or $H_i$ represents the transfer function from the load current $I_3$ to the load voltage $V_3$ or from the input current $I_{fc}$ of the fuel cell to the input current $I_1$ of the bridge at port 1, as formulated by (13) or (14), respectively:

$$H_v={\displaystyle \frac{R_3}{R_3{C}_{3}s+1}}$$

(13)

$$H_i={\displaystyle \frac{1}{L_{f1}{s}^2+r_1{C}_{1}s+1}}$$

(14)

Additionally, $G_v$ and $G_i$ represent the controllers for the voltage loop and the current loop, respectively. By combining (9) and (13) [or (14)] with the system control diagram in Figure 6, the transfer function $G_1$ (or $G_2$) for the voltage (or current) system’s feedforward path is formulated by:

$$G_1=G_v{G}_{11}{H}_v={\displaystyle \frac{R_3{G}_{11}{G}_v}{R_3{C}_{3}s+1}}$$

(15)

$$G_2=G_i{G}_{22}{H}_i={\displaystyle \frac{G_{22}{G}_i}{L_{f1}{C}_1{s}^2+r_1{C}_{1}s+1}}$$

(16)

Then, $G_v$ and $G_i$ are designed to correct $G_1$ and $G_2$, respectively, to achieve system stability. The implementation of current loop and voltage loop controllers facilitates stable voltage regulation for both the load and fuel cells, allowing the PEMFC to export at a constant power level with the introduction of excitation signals.

4. Simulation Results and Discussion

4.1. Model Parameters

A PEMFC unit has a small ohmic resistance but a large capacitance, making it a challenge to perform internal impedance detection. To validate the proposed real-time detection method for the fuel cell impedance spectrum using the TAB converter voltage perturbation, a series of 48 fuel cell units is selected to construct the PEMFC stack model. The rated voltage of the PEMFC stack is 48 V, while its actual output voltage at the terminal is lower than 48 V due to a voltage drop across the equivalent impedance. The PEMFC stack output voltage is then boosted to the target voltage of 100 V by the TAB converter.

According to the circuit’s equivalent series-parallel law, 48 fuel cell units are stacked to form the PEMFC stack impedance model, with equivalent parameters listed in

Table 1. Impedance Parameters of a 48-Unit PEMFC Stack.

Components	C1 (F)	C2 (F)	R1 (Ω)	R2 (Ω)	R3 (Ω)
Parameters	0.028	0.099	0.098	0.272	0.252

. To ensure the signal-to-noise ratio of the excitation signal, 5% of the target voltage (i.e., 5 V) is selected here as the amplitude of a triangular wave that is injected at various frequencies. The real-time PEMFC impedance detection model, including the TAB converter voltage perturbation and the decoupling control, was developed using MATLAB 2023b. The key parameters of the simulation model are detailed in

Table 2. Model Parameters.

Symbol	Name	Value
Vfc	PEMFC stack rated voltage	48 V
V2	Lithium-ion battery rated voltage	48 V
V3	Load voltage	100 V
r1	$\mathrm{ESR} \mathrm{of} \mathrm{filter} \mathrm{inductor} L_{\mathrm{f}1}$	0.2 Ω
fs	Switching frequency	10 kHz
N1:N2:N3	Transformer ratio	1:1:2
Lf1, Lf2	Filter inductors	200 μH/200 μH
L1, L2, L3	Power transfer inductors	25 μH/25 μH/25 μH
C1, C2, C3	Filter capacitors	1000 μF/1000 μF/1000 μF

.

To obtain $G_v$ and $G_i$ that ensure the PEMFC operating within its optimal range, a single-loop control strategy is implemented here for the fuel cell output current $I_{fc}$ and the load voltage $U_3$, which are set to 7 A and 100 V for the test in this work, respectively. When the load at port 3 is disconnected, the target value of $I_{fc}$ is gradually reduced at a certain rate. At the steady-state operating point, it is calculated that the system operates with ${\mathsf{\phi }}_{12\mathrm{q}}$ and ${\phi }_{13q}$ equaling 0.2999 rad and 0.5646 rad, respectively. By substituting the parameters listed in Table 2 into (15) and (16), the transfer functions of feedforward paths for voltage loop and current loop, i.e., $G_1$ and $G_2$, are obtained. The resulting controller designs of $G_v$ and $G_i$ are then determined through zero-pole correction:

$$\left\{\begin{array}{l}{G}_v={\displaystyle \frac{20(1+0.01s)}{s{(1+0.00015s)}^2}}\\ G_i={\displaystyle \frac{30(1+0.0173s)}{s}}\end{array}\right.$$

(17)

The Bode plots of the voltage subsystem and current subsystem before and after correction are shown in Figure 7. The correction amplifies the low-frequency response signal of the voltage subsystem for improved accuracy. In addition, the phase margin and the gain margin of the voltage subsystem are 80.5° and 32 dB, respectively. The phase margin of the current subsystem is 77.3° and the gain margin is infinite, both meeting the requirements for system stability.

4.2. Simulation Results and Analysis

The power outputs of the PEMFC and the lithium-ion battery that are simulated under time-varying load conditions are shown in Figure 8, respectively. The PEMFC hybrid power system starts exporting when a 500 W load is on at 0 s. Then, the power at each port stabilizes after 80 ms, at which point the fuel cell exports at its rated power of 336 W and the lithium-ion battery provides the remaining 164 W. At 1 s, the load power suddenly increases from 500 W to 700 W, with the fuel cell power remaining constant while the lithium-ion battery increases its output to meet the required load growth. At 2 s, the load is disconnected and falls to 0 W. The fuel cell output power gradually decreases, and the lithium-ion battery absorbs the surplus power in a charging state.

Figure 9 shows the corresponding waveforms of the PEMFC output current $I_{fc}$ and the load voltage $V_3$ within the same 3 s period. When the system reaches the stable state after 80 ms, $I_{fc}$ and $V_3$ stabilize at the designed 7 A and 100 V, respectively. At 1 s, the sudden load growth results in a transient increase in $I_{fc}$ and a slight decrease in $V_3$, which are then regulated to their respective designated values after 50 ms. When the load is disconnected at 2 s, $V_3$ experiences a significant fluctuation (rising from 100 V to 122 V), and then falls back to the designated value after 150 ms of dynamic adjustment. Meanwhile, $I_{fc}$ drops from 7 A to 6.3 A and then slightly increases within 150 ms, following which it decreases slowly under control.

To obtain information on the stack impedance, multiple frequency points from 1 Hz to 5000 Hz are selected for the excitation signals. The resulting waveforms of voltage and current of the fuel cell, the battery current, and the load voltage that are simulated under the excitation frequency of 1 Hz, 10 Hz, 100 Hz, or 1000 Hz are shown in Figure 10. The simulation results suggest that the excitation signals in the form of triangular waves are successfully introduced into the PEMFC system at different excitation frequencies and induce the corresponding responses of $I_{fc}$, $V_{fc}$, $I_{bat},$ and $V_3$. Under low-frequency ($\le$100 Hz) excitations, $V_3$ exhibits significant fluctuations that decrease with the excitation frequency. Under medium-frequency (100~1000 Hz) excitations, $I_{bat}$ shows large fluctuations, with smaller fluctuations in the low-frequency ($\le$100 Hz) and high-frequency ($\ge$1000 Hz) regions. Given the injection of the excitation signals at each frequency, the system operates stably with $I_{fc}$ being kept at the target value under fluctuating conditions, which validates the accuracy and response speed of the designed controller in tracking target values.

Based on the simulation results of $I_{fc}$ and $V_{fc}$ in Figure 10, their waveforms over a particular 10 s period under steady state are selected for FFT analysis to extract their respective amplitude and phase information, as tabulated in

Table 3. Amplitudes and phases of PEMFC output voltage and current.

Frequency (Hz)	Amplitude of Voltage (V)	Phase of Voltage (°)	Amplitude of Current (A)	Phase of Current (°)
0	43.6300	0	7.00000	0
1	0.00684	203.5	0.01120	32.2
3	0.01867	−70.7	0.03504	131.7
5	0.02771	123.2	0.06126	−23.7
7	0.02942	177.7	0.08135	33.7
10	0.03292	−89.5	0.10050	125.9
30	0.02480	239.0	0.12740	95.5
50	0.02030	227.7	0.13490	81.1
70	0.01684	218.8	0.12710	68.0
90	0.01306	213.1	0.11480	56.9
100	0.01205	210.1	0.10910	52.3
300	0.00472	185.7	0.04944	14.6
500	0.00211	173.2	0.02785	0.5
700	0.00147	166.2	0.01721	−7.8
900	0.00063	170.5	0.00761	−4.8
1000	0.00068	170.4	0.00762	−5.9
3000	0.00095	20.1	0.00856	203.6
5000	0.00029	64.9	0.00284	249.1

for different excitation frequencies under tests.

To validate the accuracy of the proposed real-time impedance detection method, the impedance estimated from the simulation at each excitation frequency by (4) is compared with the theoretical value, which is calculated by (2), as listed in

Table 4. Simulated Impedance Values and Theoretical Values at Various Excitation Frequencies.

Frequency (Hz)	Simulated Impedance Value (Ω)	Theoretical Impedance Value (Ω)
0	0.624	0.624
1	0.603–0.092 i *	0.604–0.093 i
3	0.493–0.203 i	0.493–0.204 i
5	0.379–0.247 i	0.390–0.224 i
7	0.293–0.213 i	0.323–0.214 i
10	0.278–0.173 i	0.267–0.190 i
30	0.156–0.116 i	0.157–0.117 i
50	0.126–0.083 i	0.126–0.084 i
70	0.116–0.064 i	0.114–0.064 i
90	0.104–0.046 i	0.108–0.051 i
100	0.102–0.041 i	0.106–0.046 i
300	0.094–0.015 i	0.099–0.016 i
500	0.075–0.010 i	0.098–0.100 i
700	0.085–0.009 i	0.098–0.007 i
900	0.082–0.007 i	0.098–0.005 i
1000	0.090–0.006 i	0.098–0.005 i
3000	0.110–0.007 i	0.098–0.002 i
5000	0.100–0.007 i	0.098–0.001 i

* i represents the imaginary unit.

. The similar impedance results between simulation and theoretical calculation illustrate the accuracy and effectiveness of the PEMFC impedance detection method proposed here using the TAB converter voltage perturbation. Figure 11 shows the Nyquist curve of the simulated impedance values against the theoretical values, which are well matched. This means that the proposed EIS detection method accurately captures the impedance of a given PEMFC system and can be applied to the PEMFC system diagnostic by impedance estimation.

The impedance simulation and the theoretical calculation both reveal that when the excitation frequency is less than or equal to 100 Hz, the real part of the impedance gradually decreases from 0.624 Ω to 0.1 Ω, while the imaginary part increases from 0 to 0.25 Ω and then decreases to 0.05 Ω. When the excitation frequency is greater than 100 Hz, the real part stabilizes at 0.1 Ω, and the imaginary part approaches zero. This might be because, in the low-frequency range, the PEMFC impedance is mainly influenced by the material transfer, which is primarily determined by obstacles in the transmission of reactive gases within electrodes and electrolytes. At low frequencies where the current-changing period is extended, the diffusion process of the material cannot promptly respond to the current changes. The delay in the diffusion process makes it difficult to maintain a stable concentration of material on the electrode surface, leading to an increase in reaction resistance compared to the high-frequency impedance. The high-frequency impedance is predominantly governed by charge transfer, which occurs significantly faster than the electrochemical reaction processes within the PEMFC. As a result, the electrochemical reactions do not have enough time to affect the system’s response, so the high-frequency impedance remains at a fixed value.

5. Conclusions

This paper has proposed an electrochemical impedance spectroscopy (EIS) detection method for a proton exchange membrane fuel cell (PEMFC) system based on a triple active bridge (TAB) voltage perturbation. The control strategy of the TAB converter system has been developed by using a small signal model combined with a decoupling control for target load voltage and target PEMFC output current. When triangular wave signals at various frequencies are superimposed with the target load voltage and the modulated voltage ripples are introduced as the excitation source into the PEMFC system, the PEMFC export is stabilized by the design of the control system, which coordinates the PEMFC and its co-located lithium-ion battery during load variations. The output voltage and current of the PEMFC have been sampled for the fast Fourier transform (FFT) analysis to obtain amplitude and phase information, from which the PEMFC impedance at each excitation frequency is calculated, forming the EIS. The EIS method proposed here has been tested on the PEMFC system under normal operation, providing accurate impedance evaluations in real time without the need to stop or interrupt the device’s operation. This method allows for short detection cycles, enabling more frequent acquisition of impedance information from the PEMFC system. It will facilitate the rapid discovery and resolution of PEMFC issues in a short period of time.

In a hybrid energy storage system with fuel cells as its main component, the application of a TAB converter allows the energy storage device(s) alongside the PEMFC to absorb/deliver power instantly. This ensures that the PEMFC operates at its optimal operating point while enabling EIS detection. Without the need for additional equipment, the PEMFC impedance monitoring system can reflect the operational status of the PEMFC in multiple aspects. This extends the application scope of traditional EIS detection methods. It is of significant importance for ensuring the stable operation of fuel cells, improving their lifespan, and reducing maintenance costs.