2.1 Manifolds

This model of the fuel cell hydrogen supply system primarily consists of three essential manifolds (ejector, supply, and return). The proportional valve is linked to the ejector's primary inlet via the ejector manifold. Since the fluid flowing through the ejector manifold is pure hydrogen, the model only needs to account for the dynamic changes in hydrogen pressure. The transient behavior of the hydrogen pressure inside the control volume is governed by the following Equation (1):$\frac{d{P}_{\text{e}\text{m}}}{dt}=\frac{R_{{\text{H}}_2}{T}_{\text{e}\text{m}}}{V_{\text{e}\text{m}}}(W_{\text{f}\text{c}\text{v}}-W_{\text{p}})$(1)where P_em denotes the hydrogen pressure within the injector manifold, a value equivalent to the pressure at the injector's primary inlet. Variables W_fcv and W_p correspond to the mass flow rates across the proportional valve and the main inlet of the injector, respectively. Furthermore, $R_{{\text{H}}_2}$, T_em, and V_em correspond to the hydrogen gas constant, the gas temperature, and the volume of the injector manifold, in sequence.

The supply manifold delivers hydrogen to the fuel cell, with gas entering the pipeline from both the proportional valve and the ejector. According to Assumption (1), the internal pressure and flow rate of the supply manifold are considered constant. The fuel cell anode's flow path is coupled with the ejector's suction port via the return manifold. Utilizing the pressure energy of the primary high-pressure flow, the ejector recirculates unconsumed hydrogen from the anode outlet back to the supply manifold. Under stable flow conditions within the return manifold, the resulting mass flow rate can be determined from the pressure differential across its two ends. The corresponding dynamic model is given by Equation (2) [36]:$W_{\text{r}\text{m}}=k_{\text{r}\text{m}}(P_{\text{r}\text{m},\text{i}\text{n}}-P_{\text{r}\text{m},\text{o}\text{u}\text{t}})=k_{\text{r}\text{m}}(P_{\text{a}\text{n}}-P_{\text{s}})$(2)In this formulation, W_rm refers to the mass flow rate of the gas flowing through the return manifold. P_{rm, in} corresponds to the return manifold inlet pressure. This pressure is identical to that at the anode outlet of the fuel cell stack, denoted as P_an. P_{rm, out} signifies the return manifold's outlet pressure under the condition of a closed purge valve, which is equivalent to the ejector's secondary flow pressure P_s. Finally, k_rm defines the scaling coefficient for the return manifold.

2.2 Ejector

Ejectors are prevalent recirculation devices in fuel cell systems, valued for their cost-effectiveness, structural simplicity, zero parasitic power load, and quiet operation [37]. Currently, the design method commonly used is based on the Sokolov ejector. The ejector's modeling and analysis are conducted according to the principles of conservation of energy, mass, and momentum [38]. A schematic diagram depicting the ejector's configuration is presented in Figure 2.

Figure 2 Ejector structure diagram

Within the fuel cell's hydrogen supply system, the working fluid of the ejector is pure hydrogen originated from the high-pressure hydrogen storage tank and regulated by the pressure-reducing valve. The potential energy inherent in this high-pressure flow is harnessed to entrain and recirculate the unreacted hydrogen, along with its water vapor content, from the anode exit. These two fluids further re-energize the fuel cell stack by mixing in the injector and discharging through the diffusion chamber. Based on the available studies, ejector functionality can be categorized into three distinct regimes: backflow, subcritical, and critical operation [39]. The specific regime in which an ejector operates is governed by the relative values of its back pressure and the critical pressure ratio across various working conditions. The ejector's back pressure, measured at its outlet, is identical to the hydrogen pressure delivered to the fuel cell stack. The critical pressure ratio is defined by Equation (3):$v_{\text{c}\text{r}}={(\frac{2}{\gamma +1})}^{\gamma /\left(\gamma -1\right)}$(3)where γ denotes the ratio of specific heats for the gas.

As shown in Figure 3, when the ejector's operating back pressure is less than its critical pressure ratio (P_{c, b}), the ejector functions in critical mode. Operation transitions to the subcritical regime once the back pressure surpasses the critical pressure ratio. From the literature, in subcritical mode, the ejector's performance declines rapidly, which should be avoided whenever possible. In back flow mode, where the operating back pressure exceeds a certain threshold (P_{0, b}), the ejector cannot properly inject the circulating fluid. Therefore, the back flow mode is excluded from the present ejector model to facilitate stable numerical simulation.

Figure 3 Ejector operating status

When operating in the critical mode, the ejector's nozzle throat attains a Mach amount of unity. The primary mass flow rate can then be derived using isentropic flow relations and energy conservation. This calculation is performed as Equation (4) [40]:$W_{\text{p}}=P_{\text{e}\text{m}}{A}_{\text{e}\text{j}}\sqrt{\frac{\gamma {\eta }_{\mathrm{p}}}{R_{{\mathrm{H}}_2}{T}_{\text{e}\text{j}}}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2\gamma -2}}$(4)where A_ej represents the nozzle throat area, ƞ_p denotes the flow isentropic coefficient, and T_ej represents the fluid temperature in the ejector. For the subcritical mode, the primary flow's mass flow rate and Mach number at the nozzle throat are calculated as Equations (5, 6) [40]:$W_{\text{p}}=P_{\text{e}\text{m}}{A}_{\text{e}\text{j}}\sqrt{\frac{{\eta }_{\text{s}}\gamma }{R_{{\text{H}}_2}{T}_{\text{e}\text{j}}}}{\left(\frac{P_{\text{s}}}{P_{\text{e}\text{m}}}\right)}^{\frac{1}{\gamma }}\left\{\frac{2}{\gamma -1}\left[1-{\left(\frac{P_{\text{s}}}{P_{\text{e}\text{m}}}\right)}^{\frac{\gamma +1}{\gamma }}\right]\right\}$(5)$M_{\text{t}}=\sqrt{\frac{2}{\gamma -1}\left[1-{(\frac{P_{\text{s}}}{P_{\text{e}\text{m}}})}^{\frac{\gamma -1}{\gamma }}\right]}$(6)where ƞ_s denotes the isentropic coefficient for the secondary flow.

The secondary flow's mass flow rate is obtained by integrating across the annular region from the tube wall to the primary flow boundary. In the critical mode of the ejector, this secondary flow rate is given by Equations (7, 8) [40, 41]:$W_{\text{s}}=\frac{2\pi \overline{\rho }{V}_{\text{s}}(R_{\text{s}}-R_{\text{p}})(R_{\text{s}}+R_{\text{p}}+N_{\text{v}}{R}_{\text{p}})}{(N_{\text{v}}+1)(N_{\text{v}}+2)}$(7)$N_{\text{v}}=m_1{e}^{\frac{\left(\frac{P_{\text{s}}{}^{0.8}}{P_{\text{e}\text{m}}{}^{1.1}}\right)}{0.05}}+m_2\frac{R_{\text{s}}}{R_{\text{p}}}+m_3$(8)where R_p and R_s represent the flow radii of the primary flow and the secondary flow, respectively; ρ represents the average density of the secondary flow fluid; V_s represents the speed of the secondary flow; m₁, m₂, and m₃ are the fitting parameters, respectively; and N_v is the exponent of the velocity function.

In subcritical mode, the ejector ratio decreases linearly as the back pressure increases. The actual mass flow rate of the secondary fluid in different modes can be calculated using the following linear interpolation method:$W_{\text{s}}=\{\begin{array}{ll}0\hfill & P_{\text{e}\text{j},\text{o}\text{u}\text{t}}\ge P_{0,\text{b}}\hfill \\ W_{\text{s}}\frac{P_{0,\text{b}}-P_{\text{e}\text{j},\text{o}\text{u}\text{t}}}{P_{0,\text{b}}-{\text{P}}_{\text{c},\text{b}}}\hfill & P_{\text{c},\text{b}}<P_{\text{e}\text{j},\text{o}\text{u}\text{t}}<P_{0,\text{b}}\hfill \\ W_{\text{s}}\hfill & P_{\text{e}\text{j},\text{o}\text{u}\text{t}}\le P_{\text{c},\text{b}}\hfill \end{array}$(9)where P_{ej, out} represents the working back pressure of the ejector.

2.3 Mass flow regulator

Precise regulation of the hydrogen supply flow in the fuel cell system is accomplished through modulation of the proportional valve. The control signal exhibits a roughly linear relationship with the proportional valve flow; this allows the mass flow rate to be regulated by varying the open cross-sectional area for gas flow. Specifically, this relationship is expressed as Equation (10) [42] :$W_{\text{f}\text{c}\text{v}}=u_{\text{v}}{W}_{\text{v}\_\text{m}\text{a}\text{x}}$(10)where u_v denotes the control signal for the proportional valve opening, normalized between 0 and 1, while W_{v_max} represents the maximum attainable mass flow rate under fully open valve conditions.

2.4 Fuel cell anode

This paper focuses exclusively on studying the fuel cell anode hydrogen supply system, thus necessitating the establishment of a model solely for the anode side of the fuel cell. Treating the entire anode channel as a control volume and disregarding the influence of nitrogen and water vapor permeation from the cathode, the dynamic response of pressure within the anode channel can be formulated as follows:$\frac{d{P}_{\text{a}\text{n}}}{dt}=\frac{R_{{\text{H}}_2}{T}_{\text{a}\text{n}}}{V_{\text{a}\text{n}}}(W_{\text{i}\text{n}}-W_{\text{o}\text{u}\text{t}}-W_{\text{r}\text{e}\text{a}\text{c}\text{t}})+P_{\text{v}\text{a}\text{p}\text{o}\text{r}}$(11)where P_an represents the anode pressure, V_an represents the volume of the anode flow channel, T_an signifies the temperature of the gas within the anode, W_in represents the hydrogen mass flow rate entering the fuel cell anode, and the variable W_out defines the outgoing hydrogen mass flow rate from the anode channel. W_react is the mass flow rate of hydrogen consumed in the stack, calculated by Equation (12) [43, 44]:$W_{\text{r}\text{e}\text{a}\text{c}\text{t}}=N_{\text{c}}\frac{I_{\text{s}\text{t}}{M}_{{\text{H}}_2}}{2F}$(12)where N_c denotes the number of single cells in the stack, I_st corresponds to the fuel cell's load current, $M_{{\text{H}}_2}$ is the molar mass of hydrogen, and F is the Faraday constant. The water vapor partial pressure (P_vapor) within the anode flow path is determined by the fuel cell's relative humidity (HR) and the water vapor saturation pressure (P_sat). The saturation pressure of water vapor is a function of the stack temperature, calculated as Equations (13, 14) [45]:$P_{\text{v}\text{a}\text{p}\text{o}\text{r}}=\text{H}\text{R}{P}_{\text{s}\text{a}\text{t}}$(13)$P_{\text{s}\text{a}\text{t}}=\text{exp(9}\text{.3876}-\frac{3826.36}{T_{\text{s}\text{t}\text{a}\text{c}\text{k}}-45.47}\text{)}$(14)

2.5 Purge valve

The fluid mass flow rate through the purge valve is governed by Equation (15):$W_{\text{p}\text{u}\text{r}\text{g}\text{e}}={\alpha }_{\text{p}}514K_{\text{v}}\sqrt{\frac{P_{\text{p}\text{u},\text{o}\text{u}\text{t}}(P_{\text{p}\text{u},\text{i}\text{n}}-P_{\text{p}\text{u},\text{o}\text{u}\text{t}})}{T_{\text{pv}}{\rho }_{\text{N}}}}$(15)where P_{pu, in} represents the inlet pressure of the purge valve, which is equal to the pipe pressure on the outlet side of the fuel cell stack. P_{pu, out} represents the outlet pressure of the purge valve, which equals atmospheric pressure P_atm. The flow coefficient of the purge valve is designated as K_v, while α_p indicates its on-off status, with 1 and 0 corresponding to open and closed states, respectively. T_pv corresponds to the temperature of the fluid within the purge pipe, and ρ_N represents the fluid density under standard conditions.

The key parameters for the 50 kW fuel cell anode system are summarized in Table 1.

2.6 Ejector model validation

The ejector-based anode pressure regulation system was validated through experiments performed on a dedicated fuel cell test stand. During testing, air was employed as a safe substitute for hydrogen to replicate the reactant supply dynamics. Model performance was assessed over the fuel cell stack's entire designated power range through a linear ramp-up of the air pressure supplied to the ejector from 200 to 950 kPa, as shown in Figure 4A. Throughout this procedure, the pressure at sixteen locations along the secondary flow path was measured; the resulting data are compiled in Table 2.

Figure 4 Verification of ejector model. (A) Graph showing the variation of the flow pressure of the ejector over time. (B) Jet ratio comparison chart

Using the appropriate thermodynamic properties for air, the model's predictions were evaluated against the sixteen experimental measurements, as shown in Figure 4B. The primary mass flow rate exhibited a linear increase in response to the rising power demand from the fuel cell. The flow characteristics of both the primary and secondary flows are consistent with the trends in Reference [41], with the error range controlled within 10%, thereby validating the accuracy of the established model.

3.1 Backstepping sliding mode controller design

The mathematical formula reflecting the relationship between flow and pressure components is brought into the derivation, resulting in a state-space representation of the controlled hydrogen supply system, expressed as Equation (16):$\{\begin{array}{l}\dot{x}=f_x(x,u,d)\hfill \\ y=C{x}_1\hfill \end{array}$(16)where the state vector x = [P_an, P_em]^T comprises the fuel cell anode pressure and the ejector primary flow pressure. The output variable y corresponds to the fuel cell anode pressure, and the interference d accounts for the impact of load current variations on the system dynamics, while the control input u defines the opening degree of the proportional valve.

To facilitate the subsequent controller design, take use of the form of a column vector to define:$\{\begin{array}{c}{\dot{x}}_1={\Psi }_1^T(x_1,x_2,u)\Theta \\ {\dot{x}}_2={\Psi }_2^T(x_1,x_2,u)\Theta \\ y=x_1\end{array}$(17)where $\Theta =[\frac{R_{{\text{H}}_2}{T}_{\text{a}\text{n}}}{V_{\text{a}\text{n}}}{(\frac{2}{\gamma +1})}^{\frac{\gamma +1}{2\gamma -2}}\sqrt{\frac{\gamma {\eta }_{\text{p}}}{R_{{\text{H}}_2}{T}_{\text{e}\text{j}}}{A}_{\text{e}\text{j}}}\frac{-R_{{\text{H}}_2}{T}_{\text{e}\text{m}}}{V_{\text{e}\text{m}}}{(\frac{2}{\gamma +1})}^{\frac{\gamma +1}{2\gamma -2}}$, Ψ₁ = [x₂ + d 0 0]^T, and Ψ₂ = [0 x₂u]^T.

The integration of backstepping control with sliding mode control effectively addresses the challenges posed by complex nonlinear characteristics and various external disturbances in the hydrogen supply system [46]. This control strategy enhances the system's stability and robustness, meeting the requirements for real-time and dynamic performance. The controller design process is outlined as follows:

First, define the anode pressure error as:$e_1=x_{\text{r}\text{e}\text{f}}-x_1$(18)

The time derivative of the state vector e₁ is therefore expressed as:${\dot{e}}_1={\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^T(x_1,x_2,u)\Theta$(19)

Define the stable variable:$\alpha =-k_1{e}_1$(20)where k₁ represents a positive constant.

Define an auxiliary error variable for tracking purposes:$e_2={\dot{e}}_1-\alpha$(21)

The first Lyapunov function is chosen as:$V_1=\frac{1}{2}{e}_1{}^2$(22)

Differentiating V₁ yields:${\dot{V}}_1=e_1{\dot{e}}_1=e_1(e_2+\alpha )=e_1(e_2-k_1{e}_1)=e_1{e}_2-k_1{e}_1^2$(23)

A second augmented Lyapunov function is defined as:$V_2=V_1+\frac{1}{2}{s}^2$(24)where s is the sliding surface given by:$s=k_2{e}_1+e_2$(25)where k₂ is also a positive constant. Using Equations (24, 25), the derivative of V₂ can be derived as Equation (26):${\dot{V}}_2={\dot{V}}_1+s\dot{s}=e_1{e}_2-k_1{e}_1^2+s(k_2(e_2-k_1{e}_1)+{\dot{e}}_2)$(26)

The derivative of ${\dot{e}}_2$ with respect to time is expressed as:$\begin{array}{cc}{\dot{e}}_2\hfill & ={\ddot{e}}_1+k_1{\dot{e}}_1\hfill \\ \hfill & ={\ddot{x}}_{\text{r}\text{e}\text{f}}-{\dot{\Psi }}_1^T(x_1,x_2,u)\Theta -{\Psi }_1^T(x_1,x_2,u)\dot{\Theta }\hfill \\ \hfill & +k_1({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^T(x_1,x_2,u)\Theta )\hfill \\ \hfill & ={\ddot{x}}_{\text{r}\text{e}\text{f}}+k_1({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^T(x_1,x_2,u)\Theta )-{\dot{x}}_2{\Theta }_1\hfill \\ \hfill & -{\Psi }_1^T(x_1,x_2,u)\dot{\Theta }\hfill \\ \hfill & ={\ddot{x}}_{\text{r}\text{e}\text{f}}+k_1({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^T(x_1,x_2,u)\Theta )\hfill \\ \hfill & -{\Theta }_1{\Psi }_2^T(x_1,x_2,u)\Theta -{\Psi }_1^T(x_1,x_2,u)\dot{\Theta }\hfill \end{array}$(27)where ${\text{Θ}}_1$ represents the first component of the matrix $\Theta$. For the fuel cell hydrogen supply system, an exponential reaching law for the sliding mode surface is formulated as follows:$\dot{s}=-h(s+\beta \mathrm{sign}(s))$(28)where h and β are positive constants.

It should be noted that the first-order sliding mode controller adopted in this study has an inherent chattering phenomenon caused by the discontinuous switching function. The chattering is mainly attributed to the high-frequency switching behavior around the sliding surface, which may lead to oscillations in the control input. In practical PEMFC hydrogen recirculation systems, excessive chattering may influence the actuator response and pressure regulation performance.

However, the main objective of this study is to enhance the robustness of hydrogen pressure regulation under nonlinear disturbances and parameter uncertainties. Therefore, the sliding mode term is retained to guarantee system stability and disturbance rejection capability.

Accordingly, the dynamics governing the sliding manifold are formulated as:$\begin{array}{cc}\dot{s}\hfill & =k_2{\dot{e}}_1+{\dot{e}}_2\hfill \\ \hfill & =k_2({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^{\text{T}}(x_1,x_2,u)\Theta )+{\ddot{x}}_{\text{r}\text{e}\text{f}}\hfill \\ \hfill & +k_1({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^{\text{T}}(x_1,x_2,u)\Theta )\hfill \\ \hfill & -{\Theta }_1{\Psi }_2^{\text{T}}(x_1,x_2,u)\Theta -{\Psi }_1^{\text{T}}(x_1,x_2,u)\dot{\Theta }\hfill \\ \hfill & =(k_1+k_2)({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^{\text{T}}(x_1,x_2,u)\Theta )+{\ddot{x}}_{\text{r}\text{e}\text{f}}\hfill \\ \hfill & -{\Theta }_1{\Psi }_2^{\text{T}}(x_1,x_2,u)\Theta -{\Psi }_1^{\text{T}}(x_1,x_2,u)\dot{\Theta }\hfill \end{array}$(29)

The sliding mode is established if $s=\dot{s}=0$, whereby the control input u is derived as:$u=\frac{\left(k_1+k_2\right)({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^{\text{T}}(x_1,x_2)\Theta )+{\ddot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_2^{\text{T}}(x_1,x_2)\Theta +h(s+\beta \mathrm{sign}(s))}{{\Theta }_1{\Theta }_3}$(30)

3.2 Stability analysis

The stability analysis for the backstepping sliding mode control is elaborated as follows. By reintroducing the derived control rate into Equation (27), the derivative of the Lyapunov function is defined as:$\begin{array}{cc}{\dot{V}}_2\hfill & ={\dot{V}}_1+s\dot{s}\hfill \\ \hfill & =e_1{e}_2-k_1{e}_1^2-s[h(s+\beta \mathrm{sign}(s))]\hfill \\ \hfill & \le e_1{e}_2-k_1{e}_1^2-h{s}^2-h\beta \left|s\right|\hfill \\ \hfill & \le -e^{T}Pe-h\beta \left|s\right|\hfill \end{array}$(31)where e = [e₁e₂]^T and P is a symmetric matrix having the structure:$P=\left[\begin{array}{cc}{k}_1+h{k}_2^2& h{k}_2-\frac{1}{2}\\ h{k}_2-\frac{1}{2}& h\end{array}\right]$(32)

Consequently, for ${\dot{V}}_2$ to be negative definite, P must be a positive definite symmetric matrix, which requires its determinant |P| to be positive:$\begin{array}{c}\left|P\right|=h(k_1+h{k}_2^2)-{(h{k}_2-\frac{1}{2})}^2\\ =h(k_1+k_2)-\frac{1}{4}\end{array}$(33)

Through proper selection of parameters k₁, k₂, and h, a positive determinant of P (|P| > 0) can be ensured.

At this point, specify the term that follows:$W(t)=e^{\text{T}}Pe+\beta \left|s\right|\le -{\dot{V}}_2(e_1(t),e_2(t))$(34)

Through the integration of Equation (33), it can be derived that:${\displaystyle \underset{0}{\overset{t}{\int }}W(t)}\mathrm{d}\tau \le V_2(e_1(0),e_2(0))-V_2(e_1(t),e_2(t))$(35)

Due to the fact that V₂ (e₁(0), e₂(0)) is bounded, V₂ (e₁(t), e₂(t)) is non-increasing and bounded, so:$\underset{t\to \infty }{\lim }{\displaystyle \underset{0}{\overset{t}{\int }}W(\tau )}\mathrm{d}\tau <\infty$(36)

Since $\dot{W}(t)$ is additionally bounded, it follows that W(t) is uniformly continuous. In light of Barbalat's lemma, the corresponding conclusion can be derived:$\underset{t\to \infty }{\lim }W(t)=0$(37)

Hence, the backstepping-based sliding mode controller exhibits asymptotic stability. Accordingly, both e₁ and e₂ tend to converge to the zero value. During the calculation of the backstepping sliding mode controller, it becomes evident that the design of the final controller effectively utilizes each state variable of the fuel cell hydrogen supply system. As a result, the system exhibits excellent control characteristics.

3.3 Purge disturbance analysis

In order to remove the water vapor generated inside the stack, prevent it from piling up on the electrode surface, the phenomenon of "flooding", and ensure the high-performance operation of the fuel cell system, the fuel cell needs to be periodically purged, and the anode pressure will fluctuate during the purging process, as shown in Figure 5. When the purge signal steps from 0 to 1, it means that the purge valve is open at this time, and part of the gas flow will be discharged to the outside, and the gas flow participating in the circulation will also decrease accordingly, so the anode pressure will decrease accordingly. When the purge valve is closed, the system returns to the initial normal operating state, but due to the lagging dynamic response characteristic of the proportional valve at the front end, hydrogen will still be over-supplied to the stack. At this time, the reaction rate of hydrogen inside the stack does not change, resulting in a temporary increase in anode pressure.

Figure 5 (A) Purge signal and (B) pressure change

In cases where the fuel cell system operates under medium to high load conditions, the anode hydrogen supply pressure typically surpasses 170 kPa, and during the purge process, the anode-cathode pressure difference surpasses 20 kPa. A fuel cell's proton exchange membrane is generally capable of withstanding the threshold that the anode-cathode pressure difference should not exceed 40 kPa, so it is necessary to consider the influence of the purge process and further optimize the controller.

3.4 Disturbance correction based on RBF neural network

During system purging, the flow characteristics of certain components change, impacting the overall system state and introducing unknown disturbances. The original backstepping sliding mode controller (BSMC) is not effective in managing the system under these conditions. To address this issue, Radial Basis Function Neural Network (RBFNN) is employed to estimate the nonlinear disturbance caused by the purge-induced gas flow disturbance to the hydrogen supply system of the fuel cell.

The RBFNN is composed of three layers, namely an input layer, a hidden layer and an output layer, while it demonstrates superior generalization capability [47, 48]. Theoretically, an RBF neural network can approximate any nonlinear function within a compact set to an arbitrary level of precision. It also demonstrates strong approximation capabilities and rapid convergence speed. The uncertain effect f(x) caused by purge-induced gas flow disturbance can be expressed as Equations (38, 39) [49]:$h_j=\exp (\frac{{\Vert x-c_j\Vert }^2}{2b_j^2})$(38)$f(x)=W^{*\text{T}}h(x)+\epsilon$(39)where j represents the j-th node situated in the hidden layer, x denotes the input variable of the RBF neural network, and c_j stands for the central point coordinate of the Gaussian basis function corresponding to the j-th hidden layer neuron. b_j signifies the width parameter of the Gaussian basis function associated with the hidden layer neuron. W^* represents the ideal weighting matrix for the RBF neural network; h(x) denotes the output of the hidden layer; ε stands for the approximation deviation of the RBF neural network.

In this work, the RBF neural network is introduced to approximate the unknown nonlinear disturbances and compensate for modeling uncertainties in the hydrogen subsystem. It should be clarified that the RBF neural network is not directly designed for eliminating the chattering phenomenon of the sliding mode controller. The switching term in sliding mode control (SMC) is still necessary to guarantee robustness against uncertainties. By improving disturbance estimation accuracy, the RBF compensation can reduce the influence of unknown dynamics on the control system, which helps improve the overall control performance.

The dynamic equation of ground anode pressure of the fuel cell hydrogen supply system can be rewritten as:${\dot{x}}_1={\Psi }_1^{\text{T}}(x_1,x_2,u)\Theta +\widehat{f}(x)$(40)where $\widehat{f}(x)={\widehat{W}}^{\text{T}}h(x)$, ${\widehat{W}}^{\text{T}}$ represents the estimated weighting matrix for the RBF neural network. According to the above derivation, the new neural network backstepping sliding mode control (NNBSMC) rate can be obtained as shown in Figure 6 (Equation (41)):$u=\frac{\left(k_1+k_2\right)({\dot{x}}_{\text{r}\text{e}\text{f}}-{\Psi }_1^{\text{T}}(x_1,x_2)\Theta -\widehat{f}(x))+{\ddot{x}}_{\text{r}\text{e}\text{f}}-\dot{\widehat{f}}(x)-{\Psi }_2^{\text{T}}(x_1,x_2)\Theta +h(s+\beta \mathrm{sign}(s))}{{\Theta }_1{\Theta }_3}$(41)

Figure 6 Schematic diagram of NNBSMC controller

Therefore, all the parameters of the relevant controllers were summarized, as shown in Table 3.

4.1 Fuel cell work area condition

4.1.1 Without purge-induced gas flow disturbance

Firstly, the control effect is compared in the scenario without a purge signal. The load current step condition that fluctuates within the appropriate current range is input to the system, which is from 250 to 320 A as shown in Figure 7A, and the simulation effect of the three control algorithms of the fuel cell hydrogen supply system under the ideal state is obtained. Figure 7B shows the tracking curve of the anode pressure of the fuel cell system under the control of PID, SMC, and BSMC controllers. Compared with the current step condition, it can be seen that when the load current rises, the hydrogen consumption of the stack increases, and the anode pressure will show a short downward trend due to the hysteresis of the system response. Similarly, when the load current drops, the hydrogen consumption of the stack decreases, and the anode pressure rises briefly. When the current is stable, the three controllers can stabilize the anode pressure well. It can be observed from Figure 7C that compared with the PID controller, the BSMC controller, and the SMC controller both show faster response speed, can reach the control target in a shorter time, and have no overshoot phenomenon, and the performance is significantly better than the PID controller. At the same time, it can be seen from Figure 7D, E that in the process of upward or downward step of the external load current, the BSMC controller has the optimal pressure fluctuation suppression effect, the adjustment time of the system is shorter, and the anode pressure can be restored to the set target value more quickly.

Figure 7 Performance of three types of controllers. (A) Load current; (B) control performance; (C) startup performance; (D) step-down current case; (E) step-up current case; (F) step-up current and increase target pressure case; (G) step-down current and reduce target pressure case

There is a certain relationship between the hydrogen supply pressure and the load current of the fuel cell anode. When the load current demand increases, the fuel cell needs more hydrogen flow to support the electrochemical reaction. Increasing the anode pressure can increase the diffusion rate of hydrogen and ensure that enough hydrogen can be quickly transferred to the electrode surface. Meanwhile, high load current will lead to increased polarization loss of the fuel cell electrode. By increasing the pressure of the anode gas, the polarization loss of the electrode can be reduced, and the rate of electrochemical reaction can be improved. Conversely, when the load current demand is reduced, the anode pressure can also be appropriately reduced to improve the hydrogen utilization rate. Therefore, in order to better simulate the actual working state of hydrogen supply by the fuel cell anode, the load current increases upward and the theoretical reference pressure increases at the 40 s of the input step current condition, while the load current decreases downward at the 60 s, at which time the theoretical reference pressure is also reduced. Thus, the control effects of the three controllers in Figure 7F, G are obtained. Among them, the control effect of the BSMC controller is obviously better than that of the SMC and PID controllers.

4.1.2 With purge-induced gas flow disturbance

In an effort to further verify the control performance of the designed NNBSMC controller, a periodic purge signal is added under the above step current condition, and a purge signal is carried out every 15 s with a purge time of 1 s. The load current step condition that fluctuates within the appropriate current range is input to the system, which is from 250 to 320 A as shown in Figure 8A. The purging signal is presented in Figure 8B. The control performance of PID, BSMC, and NNBSMC controllers is compared under the working condition.

Figure 8 Performance of three types of controllers. (A) Load current; (B) purge signal; (C) control performance; (D) purging case; (E) step-down current and purging case; (F) step-down current and purging case

Figure 8C shows the control effect of PID/BSMC/NNBSMC controllers on the pressure of the hydrogen supply system of the fuel cell anode under load current fluctuation and periodic purge interference. In Figure 8D, it is evident that the NNBSMC controller is able to effectively compensate for the pressure fluctuations arising in the course of the purging procedure, while the BSMC controller, lacking compensatory measures, gives rise to a substantial pressure differential when the purging valve is in operation, a condition that impairs the stable operation of the fuel cell system. Figure 8E, F shows the control effect of the three controllers on the anode pressure when purging with an upward step current and purging with a downward step current, that is, when two external disturbances are present simultaneously. Due to the coupling relationship between the load current and the hydrogen flow rate, it is found that the step of the load current will have a certain compensation or amplification effect on the fluctuation of anode pressure during the purge process, but overall, the control performance exhibited by the NNBSMC controller is evidently superior to the other two types of controllers.

4.2 Fuel cell vehicle condition

In an attempt to more authentically reflect the output performance characteristics of the fuel cell system during the actual driving process of a fuel cell vehicle, the present section employs the load current operating condition dataset gathered during the actual operational process from fuel cell hybrid electric vehicles reported in relevant reference documents for the purpose of validating the control performance of the proposed controller. This condition is the total current of the fuel cell and the power battery, and the actual current change rate of the vehicle fuel cell system will be much lower than the direct current (DC) bus current. The current in this working condition is scaled and limited to more than 120 A, so that the current is within the normal operating range of the fuel cell system model established in this paper, and at the same time, the ejector runs in reflux mode, resulting in insufficient hydrogen supply to the reactor and limited output performance.

Figure 9A shows the current condition of the real car DC bus. The purge signal is shown in Figure 9B. Under this working condition, the control results of PID, SMC, and NNBSMC controllers for the system anode pressure are shown in Figure 9C. As can be seen from the overall trend in the figure, the PID controller has not achieved good results in controlling the anode hydrogen pressure under the condition of large current fluctuation and purge disturbance in real vehicle conditions, and frequent pressure fluctuation is not conducive to the stability of the output performance of the fuel cell stack. At the same time, the excessive pressure difference between the two poles will cause irreversible damage to the proton exchange membrane and the reactor. Figure 9D, E is a partial enlarged drawing of Figure 9C, from which it can be seen more clearly that the control effect of SMC and NNBSMC controllers is obviously better than that of PID controllers. In addition, in the case of external disturbances, the NNBSMC controller shows better performance and can effectively maintain the hydrogen pressure stability of the fuel cell, with the best robustness. Although the proposed controller achieves smoother pressure regulation compared with conventional sliding mode control, slight fluctuations remain because the switching mechanism is maintained to ensure robustness. The hydrogen stoichiometric ratio in the anode system under the NNBSMC controller is shown in Figure 10. It can be seen that in the entire operation process, the hydrogen stoichiometric ratio is basically maintained between 2 and 4, which can fully ensure the adequate hydrogen flow supply of the anode system to meet the normal operation needs of the fuel cell.

Figure 9 Control results under load current of a fuel cell hybrid vehicle (FCHV). (A) Load current; (B) purge signal; (C) control performance; (D) local discharge part; (E) local discharge part

Figure 10 Anode hydrogen stoichiometric ratio under NNBSMC control

Comprehensively evaluate the control effects of the three controllers under real vehicle conditions. The dynamic performance indicators of the fuel cell anode hydrogen supply system are shown in Figure 11, which mainly include maximum tracking error (MAX_ERROR), mean absolute error (MAE), and root mean square error (RMSE). These indicators are defined as follows:$\mathrm{MAX}\_\text{E}\text{R}\text{R}\text{O}\text{R}=\text{max}\left|P_{\text{a}\text{n}}(kT)-P_{\text{r}\text{e}\text{f}}(kT)\right|$(42)$\mathrm{MAE}=\frac{1}{n}{\displaystyle \sum _{k=1}^n\left|P_{\text{a}\text{n}}(kT)-P_{\text{r}\text{e}\text{f}}(kT)\right|}$(43)$\mathrm{RMSE}=\frac{1}{n}\sqrt{{\displaystyle \sum _{k=1}^n{\left(P_{\text{a}\text{n}}(kT)-P_{\text{r}\text{e}\text{f}}(kT)\right)}^2}}$(44)

Figure 11 Evaluation index of control effect. (A) Mean absolute error; (B) root mean square error; (C) maximum tracking error

As illustrated in Figure 11, the three performance indicators of the NNBSMC controller are significantly better than the other two controllers, showing excellent control performance and robustness, which is of great significance for the gas pressure regulation of full-power fuel cell vehicles applied in real scenarios.