Nonlinear Pressure Fluctuation Management for Ejector-Based Hydrogen Recirculation System in Large-Power Vehicular PEMFCs

Abstract

The ejector-based hydrogen recirculation systems in vehicular proton exchange membrane fuel cells (PEMFCs) have been the research focus of fuel cell technology. However, the anode pressure fluctuations and nonlinear characteristics urgently need to be addressed under the varying operating conditions of the ejector-based hydrogen recirculation system. In this paper, an Adaptive Model Predictive Control strategy is proposed to stabilize anode pressure and smooth pressure fluctuations during current changes and purges. An ejector is designed and a nonlinear control model is established for the ejector-based hydrogen recirculation system of the 110 kW PEMFC. The proposed strategy achieves the mean absolute error (MAE) of 0.044 kPa and the root mean square error of 1.041 kPa in anode pressure management, outperforming traditional Model Predictive Control and Proportional-Integral-Derivative strategies. The hydrogen excess ratio exceeding 1.5 ensures compliance with the operational requirements of system. The experimental results with the pressure MAE of 0.482 kPa and the current fluctuation of ±0.7 A validate the effectiveness of the proposed strategy in practical applications.

1. Introduction

Renewable energy sources are widely recognized as effective alternatives to fossil fuels. However, due to inherent limitations in the intermittent and fluctuating nature of natural resources, their stability in energy supply has been limited. With its wide availability, configurational flexibility, and zero-emission, hydrogen energy is poised to play a pivotal role in realizing carbon peak and carbon neutrality targets and addressing the intermittency and variability of renewable energy sources [1,2]. The proton exchange membrane fuel cell (PEMFC), a crucial technology for hydrogen energy utilization, is widely acknowledged as one of the most promising power sources, especially for vehicular power sources with the advantages of high energy density, efficient energy conversion, low operating temperatures, and non-pollution [3,4].

In vehicular PEMFC systems, the design and management of the hydrogen supply system are critical due to the limited hydrogen source capacity, sufficient reactant supply, and steady pressure maintenance [5,6]. The ejector-based hydrogen recirculation is a mainstream solution approach for the rapidly advancing vehicular PEMFC system owing to its merits of simple structure, no additional power requirements, and minimal maintenance costs [7,8,9]. The ejector-based hydrogen recirculation (EHR) system widely adopted in full power range of PEMFC systems not only supports the internal electrochemical reactions but also improves the utilization rate of hydrogen and achieves the self-humidify of the anode during the PEMFC operation [10]. Nikiforow et al. [11] designed an ejector targeted for a high entrainment ratio in a 5 kW PEMFC hydrogen recirculation system and manufactured prototypes with three dimensional printing. The deviation between simulation and experimental results is less pronounced at conditions relevant to PEMFC applications. Wang et al. [12] designed and tested an ejector for a 10 kW PEMFC hydrogen recirculation system, achieving 5–10% reduction in hydrogen consumption compared to the dead-ended anode mode. Kim et al. [13] designed and manufactured an ejector for a 40 kW submarine PEMFC anode recirculation system. The performance of the prototype was verified through the whole practically acceptable operations. Du et al. [14] designed a coaxial-nozzle ejector for an 80 kW vehicular PEMFC system. Numerical simulations showed that the proposed ejector effectively broadened the operational range of a single ejector. Yang et al. [15] established three anodic recirculation systems including parallel, series, and dual-ejector systems for a 150 kW fuel cell. The dual-ejector system demonstrated superior entrainment performance and lower power consumption than the other two systems.

The pressure fluctuation caused by load variation and frequent purging can affect or even irreversibly damage the proton exchange membrane, resulting in the reduction in PEMFC power generation efficiency [16]. Therefore, it is essential to stabilize the anode pressure with an effective management strategy in the hydrogen recirculation system. In recent years, advancements in PEMFC technology have spurred the exploration of various management strategies to tackle the challenges associated with anode hydrogen recirculation systems. Ebadighajari et al. [17] proposed a multivariable model predictive control (MPC) strategy for regulating the pressure differential and hydrogen concentration of the PEMFC stack. Zeng et al. [18] proposed an intelligent hydrogen pressure control method based on short-term vehicular power demand prediction to enhance mass transfer inside the diffusion layer and to improve dynamic performance. Ye et al. [19] demonstrated that a Mamdani fuzzy controller effectively regulated hydrogen pressure using a modified common rail injection system. Wei et al. [20] proposed a practicable feedforward-based proportional-integrative (FFPI) strategy of anode pressure control and successfully deployed a 100 kW-class automotive PEMFC system. Yuan et al. [21] proposed a fuzzy Proportional-Integral (PI) control strategy with feed-forward compensation in PEMFC for anode pressure management, achieving better performance than traditional PI and fuzzy PI controllers. He et al. [22] developed decentralized PI and state-feedback (SFB) control strategies for a PEMFC anode hybrid fuel delivery system, with simulations showing superior performance of the SFB strategy in maintaining constant anode pressure. Bao et al. [23] developed a model for control applications in anode recirculation systems with ejectors, accounting for water transport within the membrane and mass transfer dynamics in the diffusion layer. Based on this model, they implemented both linear and nonlinear multivariable controllers for managing the recirculation of reactants in the cathode and anode systems of PEMFCs. Their study shows that multivariable controllers not only enhance system transient response but also exhibit robust anti-interference capabilities [24]. Control optimization for dynamic operating conditions also exists in the field of cathode gas management. The self-tuning oxygen excess ratio control strategy proposed by Li et al., which adjusts PID parameters in real time to cope with load fluctuations, further confirms the universal value of adaptive control in the multi-physics collaborative management of fuel cells [25].

The review of previous research identified the following gaps in the literature:

(1)

Precise regulation of anode pressure is critical, since uncontrolled pressure fluctuations can induce mechanical stress accumulation, ultimately leading to structural failure in proton exchange membranes. However, existing studies predominantly concentrate on pressure management in conventional hydrogen recirculation systems utilizing circulating pumps, while insufficient attention has been paid to ejector-based systems despite their growing adoption in recent applications.

(2)

Despite the widespread adoption of ejector-based recirculation in PEMFC systems, the anode pressure oscillations and nonlinear characteristics under variable load conditions remain poorly understood, highlighting a critical research gap. The conventional Proportional-Integral-Derivative (PID) strategy, despite its prevalence, exhibits significant limitations in handling nonlinear dynamics induced by intricate system mechanisms and rapid transient responses. Consequently, the development of advanced management strategies is imperative to address these limitations.

To address these identified research gaps, this study proposes an Adaptive Model Predictive Control (AMPC) strategy, designed to mitigate anode pressure fluctuations induced by transient load current variations and periodic purging state transitions, thereby enabling real-time precision pressure regulation in PEMFC systems. Firstly, an ejector for a 110 kW PEMFC was designed and a model verification was carried out. Subsequently, a nonlinear dynamic control model was developed to describe the dynamic characteristics of the EHR system. Finally, an Adaptive Model Predictive Control strategy was introduced to track the anode pressure setpoints and mitigate pressure fluctuations during varying operating conditions. Compared with PID and normal MPC strategies, the proposed AMPC strategy demonstrates superior pressure management performance under scenarios involving current step changes and periodic purging. The effectiveness of the AMPC strategy has also been verified by experiments.

2. Ejector-Based Hydrogen Recirculation System Model

The schematic diagram of the ejector-based hydrogen recirculation system is shown in Figure 1. It consists of a hydrogen tank, a pressure relief valve, a flow control valve, an ejector, a purge valve, and a fuel cell stack. The ejector is the key component that utilizes high-speed fluid to entrain low-speed fluid in the hydrogen cycle. The primary gas stream, composed of high-pressure hydrogen from the storage tank, is regulated sequentially by the pressure relief valve and the flow control valve before entering the supersonic ejector nozzle, where it undergoes critical expansion. The secondary stream, comprising unreacted hydrogen and saturated water vapor from the anode outlet, is entrained into the supersonic ejector by the high-momentum primary flow. Through shear-driven momentum transfer within the ejector mixing chamber, the two streams mix thoroughly, subsequently recirculating to the PEMFC stack. The purge valve is periodically opened to emit the buildup of nitrogen and mitigate the formation of droplets.

Based on the depicted EHR system, hydrogen and vapor dynamic models are established for the ejector manifold, supply manifold, return manifold, and anode flow channel. The ejector, flow control valve, and purge valve are represented by steady-state models. Each of the above models is built on these assumptions [14]:

(1)

All volumes are assumed to use the ideal gas law;

(2)

The interior of anode flow channel and manifolds are isothermal;

(3)

No liquid water is generated;

(4)

The spatial variations are neglected;

(5)

The working fluids within the ejector are considered stable compressible fluids;

(6)

The inner surface of the ejector wall is regarded as adiabatic.

2.1. Ejector Model and Design

This study focuses on an ejector employing a convergent nozzle. The theoretical model adopted is based on the literature [26] to conform to the flow characteristics specific to convergent nozzles. The mass flow rates of the primary flow ($W_{ej,p}$) and secondary flow ($W_{ej,s}$) can be determined by referring to Equations (1) and (2):

$$W_{ej,p}=\left\{\begin{array}{cc}{\displaystyle \frac{P_p{A}_t}{\sqrt{T_p}}}\sqrt{{\displaystyle \frac{\kappa }{R}}{({\displaystyle \frac{2}{\kappa +1}})}^{\kappa +1/\kappa -1}}\sqrt{{\eta }_p},\hfill & if P_s/P_p\le {({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa +1}{\kappa -1}}}\hfill \\ {\displaystyle \frac{P_p{A}_t}{\sqrt{T_p}}}\sqrt{{\displaystyle \frac{2\kappa [{(P_s/P_p)}^{2/\kappa }-{(P_s/P_p)}^{(\kappa +1)/\kappa }]}{(\kappa -1)R}}}\sqrt{{\eta }_p},\hfill & if P_s/P_p>{({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa +1}{\kappa -1}}}\hfill \end{array}\right.$$

(1)

$$W_{ej,s}={\displaystyle \frac{2\pi {\overline{\rho }}_s{U}_{p,2}(r_2-r_{p,2})(r_2+r_{p,2}+n_v{r}_{p,2})}{{(n}_v{+1\left)\right(n}_v+2)}}$$

(2)

where $P_p$, $T_p$, ${\eta }_p$ are the pressure, temperature, and isentropic coefficient of the primary flow, respectively. $A_t$ represents the cross-sectional area of the nozzle throat. $R$ is the gas constant. $\kappa$ is the specific heat ratio of the gas. $r_2$ and $r_{p,2}$ represent the radius of the hypothetical throat and hypothetical throat of the primary flow. $P_s$ is the pressure of secondary flow. The average density of secondary flow ${\overline{\rho }}_s$ and the exponent of the velocity function $n_v$ are shown as Equations (3) and (4).

$${\overline{\rho }}_s={\displaystyle \frac{P_s}{R_g{T}_s}}={\displaystyle \frac{P_s}{T_s}}{\displaystyle \frac{{\displaystyle \sum _i{n}_s^{i}M{o}^i}}{R_u{\displaystyle \sum _i{n}_s^i}}}$$

(3)

$$n_v=A_{1}exp({\displaystyle \frac{P_s^{0.8}}{0.05P_P^{1.1}}})+A_2{\displaystyle \frac{D_m}{D_t}}+A_3$$

(4)

where $A_1$, $A_2$, $A_3$ are the data fitting coefficients. $D_m$ and $D_t$ are the diameter of constant-area mixing chamber and nozzle. The fluid at the ejector outlet is a mixture of the primary and secondary flows. Concurrently, the secondary flow inlet connects to the anode outlet, where the internal working fluid consists of hydrogen and water vapor. Therefore, the mass flow rates of hydrogen ($H_2$) and vapor ($H_{2}O$) in the outlet fluid are:

$$W_{H_{2,}ej,out}=W_{ej,p}+y_{H_2,s}{W}_{ej,s}$$

(5)

$$W_{H_{2}O,ej,out}=(1-y_{H_2,s})W_{ej,s}$$

(6)

where $y_{H_2,s}$ is the mass fraction of hydrogen in the secondary flow [27], which can be obtained by Equations (7) and (8).

$$y_{H_2,s}={\displaystyle \frac{W_{H_2,s}}{W_{H_2,s}+W_{H_{2}O,s}}}={\displaystyle \frac{P_{H_2,s}{M}_{H_2}}{P_{H_2,s}{M}_{H_2}+{\varphi }_s{P}_{sat(T_s)}{M}_{H_{2}O}}}$$

(7)

$$P_{sat}(T_s)=exp(23.196-{\displaystyle \frac{3816.44}{T_s-46.13}})$$

(8)

in which, $P_{sat}\left(T_s\right)$ represents the saturation pressure of water vapor under the temperature of secondary flow. ${\varphi }_s$ is the relative humidity of secondary flow.

As a key structural factor in ejector design, the throat diameter of the convergent nozzle is determined using Equation (1) according to the hydrogen consumption at the maximum power of the PEMFC and the maximum gas supply pressure at the primary inlet of the ejector, while other designed structural parameters are selected to fall into the recommended ranges by other researchers [28]. Subsequently, computational fluid dynamics (CFD) simulations are conducted using Gambit 2.4.6 and Fluent 16.0 software as mesh generator and solver, respectively. Through iterative optimization, the optimal ejector configuration that met the power ranges and performance requirements of PEMFC is achieved, as shown in

Table 1. Parameters of ejector structure.

Structure Parameters of Ejector	Values	Units
Diameter of nozzle	2.24	mm
Length of nozzle convergence section	13.00	mm
Diameter of constant-area mixing chamber	7.08	mm
Nozzle exit position	13.10	mm
Length of constant-area mixing chamber	50.00	mm
Length of diffuser chamber	52.00	mm
Diffusion angle	7.02	°

.

Then, a set of nonlinear governing equations of ejector are derived from the assumptions and the general conservation equations for momentum, energy, and mass within a finite volume framework.

The continuity equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho u)=0$$

(9)

The momentum conservation equation:

$${\displaystyle \frac{\partial (\rho u)}{\partial t}}+\nabla \cdot (\rho u\otimes u)=-\nabla p+\nabla \cdot \tau +f$$

(10)

where $\tau$ can be calculated by

$$\tau =\mu \left[(\nabla u+{(\nabla u)}^T)-{\displaystyle \frac{2}{3}}\nabla \cdot uI\right]$$

(11)

The energy equation:

$${\displaystyle \frac{\partial (\rho E)}{\partial t}}+\nabla \cdot (\rho uE)=-\nabla \cdot (\rho u)+\nabla \cdot (\tau \cdot u)+\nabla \cdot (k_{eff}\nabla T)$$

(12)

The ideal gas equation:

$$\rho ={\displaystyle \frac{P}{R_{g}T}}$$

(13)

where $\rho$ is the fluid density, $u$ is the velocity vector, $P$ is the static pressure, $\tau$ is the viscous stress tensor, $f$ is the body force vector, $\mu$ is the viscosity, $I$ is the unit tensor, $E$ is the total energy, and $k_{eff}$ is the effective thermal conductivity.

Ejector performances can be assessed using the entrainment ratio ($\omega$) and hydrogen recirculation ratio. The $\omega$ represents the ratio of the mass flow rate of recycled fluid to that of the consumed fluid [29]. Under steady-state conditions, the hydrogen mass flow rate utilized in the stack reaction is equivalent to that of the primary flow entering the ejector [30]. Similarly, the mass flow rate of recycled hydrogen matches the mass flow rate of the secondary flow entering the ejector. Likewise, the hydrogen recirculation ratio (${\omega }_{H_2}$) is expressed as the ratio between recirculated hydrogen and supplied hydrogen mass flow rate [31].

$$\omega ={\displaystyle \frac{W_{rec}}{W_{con}}}={\displaystyle \frac{W_{ej,s}}{W_{ej,p}}}$$

(14)

$${\omega }_{H_2}={\displaystyle \frac{W_{H_2,rec}}{W_{con}}}={\displaystyle \frac{W_{H_2,ej,s}}{W_{ej,p}}}$$

(15)

where $W_{rec}$ and $W_{H_2,rec}$ are defined as the recirculated mass flow and recirculated mass flow of hydrogen from the ejector, respectively. $W_{con}$ is defined as the hydrogen mass flow consumed by the stack. To ensure the proper operation of the PEMFC and facilitate water discharge from the stack, the hydrogen excess stoichiometric ratio (${\lambda }_{H_2}$) of the PEMFC system must be greater than the value of 1.5 [32]. The ${\lambda }_{H_2}$ is calculated as:

$${\lambda }_{H_2}=1+{\omega }_{H_2}$$

(16)

The entrainment performance of the designed ejector was measured by the ejector test rig as shown in Figure 2. The schematic of the experimental principle is illustrated in Figure 2b. High-pressure hydrogen is divided into the primary and secondary flows as the inputs of the designed ejector. The pressure of streams can be independently adjusted based on operational requirements. The flow rate, temperature and pressure of the flows are measured by the mass flow controllers (MFC), pressure transducer (PT), and thermocouple sensors (TC). The humidifier is used to humidify the secondary flow. The hydrogen supply loop and the signal circuit are indicated by the solid green line and the blue dash-dotted line, respectively. The boundary conditions of the ejector used in the experiments are determined according to the measured operating conditions of stack as listed in

Table 2. The operating points of ejector.

Current (A)	Primary Mass Flow Rate (g/s)	Primary Flow Temperature (K)	Secondary Flow Pressure (kPa)	Secondary Flow Temperature (K)
180	1.264	298	176	343
240	1.303		196
320	1.382		237
360	1.422		265
410	1.473		276

. Comparison between the model-calculated ER and the measured ER as shown in Figure 3, illustrates the precision of the chosen theoretical ejector model. This model shows the potential for establishing frameworks in EHR systems.

2.2. Manifolds Model

The EHR system comprises three manifolds: the ejector manifold, the supply manifold, and the return manifold [16]. The ejector manifold corresponds to the pipeline volume connecting the flow control valve to the primary inlet of the ejector and facilitating the flow of pure hydrogen. The pressure dynamic equation of the ejector manifold is described as:

$${\displaystyle \frac{d{P}_{em}}{dt}}={\displaystyle \frac{R_{H_2}{T}_{em}}{V_{em}}}(W_{fcv}-W_{em,out})$$

(17)

where $P_{em}$ represents the pressure of the ejector manifold. $W_{fcv}$ denotes the mass flow rate through the flow control valve, and $W_{em,out}$ represents the mass flow rate of hydrogen at the ejector manifold outlet, which is also equivalent to the mass flow rate of the ejector primary inlet. The specific calculation formula is detailed in Section 2.1.

The supply manifold represents the pipeline volume that links the ejector outlet with the anode flow channel. The mixed fluid is directed into the stack to engage in electrochemical processes. Consequently, the dynamic models governing pressure and water activity in the supply manifold are as follows:

$${\displaystyle \frac{d{P}_{H_2,sm}}{dt}}={\displaystyle \frac{R_{H_2}{T}_{sm}}{V_{sm}}}(W_{H_2,sm,in}-W_{H_2,sm,out})$$

(18)

$${\displaystyle \frac{d{a}_{w,sm}}{dt}}={\displaystyle \frac{R_{H_{2}O}{T}_{sm}}{V_{sm}{P}_{sat}(T_{sm})}}(W_{H_{2}O,sm,in}-W_{H_{2}O,sm,out})$$

(19)

where $P_{H_2,sm}$ and $a_{w,sm}$ represent the hydrogen partial pressure and water activity of the supply manifold. The calculation of relative humidity is described by Equation (20) according to the ideal gas law. $W_{H_2,sm,in}$ and $W_{H_{2}O,sm,in}$ represent the hydrogen and water vapor mass flow rates entering the supply manifold, which are also equivalent to the flow rates of different species at the ejector outlet. $W_{H_2,sm,out}$ and $W_{H_{2}O,sm,out}$ denote the mass flow rates of $H_2$ and $H_{2}O$ exiting the supply manifold, also correspond to the species flow rate at the inlet of the anode channel.

$${\varphi }_{(\cdot )}=min(a_{w,(\cdot )},1)$$

(20)

The $H_2$ and $H_{2}O$ mass flow rates released from the manifolds can be characterized as a function of the pressure differential across these manifolds:

$$W_{\alpha ,\beta }=k_{\beta }(P_{\alpha ,\beta ,up}-P_{\alpha ,\beta ,down})$$

(21)

where $\alpha$ refers to $H_2$ and $H_{2}O$. $\beta$ refers to $sm$, $rm$ and $an$, which mean supply manifold, return manifold, and anode flow channel, respectively. $k_{\beta }$ represents the flow coefficient equivalent of different manifolds. $P_{\alpha ,\beta ,up}$ and $P_{\alpha ,\beta ,down}$ represent the upper and lower pressures of different fluid components in various manifold sections.

The return manifold is equivalent to the sum of the lines at the anode outlet to the secondary flow inlet of the ejector like the supply manifold:

$${\displaystyle \frac{d{P}_{H_2,rm}}{dt}}={\displaystyle \frac{R_{H_2}{T}_{rm}}{V_{rm}}}(W_{H_2,rm,in}-W_{H_2,rm,out}-W_{H_2,purge})$$

(22)

$${\displaystyle \frac{d{a}_{w,rm}}{dt}}={\displaystyle \frac{R_{H_{2}O}{T}_{rm}}{V_{rm}{P}_{sat}(T_{rm})}}(W_{H_{2}O,rm,in}-W_{H_{2}O,rm,out}-W_{H_{2}O,purge})$$

(23)

where $W_{H_2,purge}$ and $W_{H_{2}O,purge}$ are the mass flow rates of $H_2$ and $H_{2}O$ discharged by the purge valve. $W_{H_2,rm,in}$, $W_{H_{2}O,rm,in}$, $W_{H_2,rm,out}$, and $W_{H_{2}O,rm,out}$ represent the mass flow rates of $H_2$ and $H_{2}O$ at the inlet and outlet of the return manifold, respectively.

2.3. Anode Flow Channel Model

The Anode flow channel functions as a control volume where pressure losses and humidity variations occur. Hydrogen undergoes reactions and water vapor migrates between the anode and cathode [33]. Therefore, the dynamic models of the anode flow channel can be described as:

$${\displaystyle \frac{d{P}_{H_2,an}}{dt}}={\displaystyle \frac{R_{H_2}{T}_{an}}{V_{an}}}(W_{H_2,an,in}-W_{H_2,an,out}-W_{H_2,reacted})$$

(24)

$${\displaystyle \frac{d{a}_{w,an}}{dt}}={\displaystyle \frac{R_{H_{2}O}{T}_{an}}{V_{an}{P}_{sat}(T_{an})}}(W_{H_{2}O,an,in}-W_{H_{2}O,an,out}-W_{H_{2}O,diff})$$

(25)

where $W_{H_2,an,in}$, $W_{H_{2}O,an,in}$, $W_{H_2,an,out}$, and $W_{H_{2}O,an,out}$ represent the rates of $H_2$ and $H_{2}O$ flow at inlet and outlet of the anode flow channel, respectively. $W_{H_2,reacted}$ and $W_{H_{2}O,diff}$ are the mass flow rates of hydrogen utilized in the electrochemical process and vapor migrating from the cathode to the anode, respectively.

$$W_{H_2,reacted}={\displaystyle \frac{N_{cell}{I}_{st}{M}_{H_2}}{2F}}$$

(26)

$$W_{H_{2}O,diff}={\alpha }_{net}{\displaystyle \frac{N_{cell}{I}_{st}{M}_{H_{2}O}}{F}}$$

(27)

where ${\alpha }_{net}$ represents the net water transfer coefficient per proton.

The water transfer caused by the pressure gradient can be ignored since the pressure differential across the membrane is tiny, so the calculation of the net water transfer coefficient [34] is as shown:

$${\alpha }_{net}=n_d-{\displaystyle \frac{F{A}_{fc}}{I_{st}}}{D}_w{\displaystyle \frac{{\rho }_{m,dry}}{t_m{M}_{m,dry}}}({\lambda }_{ca}-{\lambda }_{an})$$

(28)

where $n_d$ represents the electro-osmotic drag coefficient. $A_{fc}$ is the active area of membrane, and $D_w$ is the diffusion coefficient. ${\rho }_{m,dry}$ and $M_{m,dry}$ refer to the density and molar mass of the dry membrane. $t_m$ is the membrane thickness. ${\lambda }_{ca}$ and ${\lambda }_{an}$ represent the water contents of the cathode and the anode, determined by the water activity. Hence, the anode pressure and the HER can be referred to Equations (29) and (30).

$$P_{an}=P_{H_2,an}+{\varphi }_{an}{P}_{sat}(T_{an})$$

(29)

$${\lambda }_{H_2}={\displaystyle \frac{W_{H_2,an,in}}{W_{H_2,reacted}}}$$

(30)

2.4. Control Valve Model

The flow control valve employs a proportional regulating mechanism to adjust the flow of primary fluid entering the ejector. The valve opening is roughly proportional to the flow passing through it after normalization, without considering dead zones and saturation effects. The linear relationship is assumed between the valve opening and flow rate for computational convenience [6].

$$W_{fcv}=u_{fcv}{W}_{fcv,rated}$$

(31)

where $u_{fcv}$ represents the valve opening of the flow control valve, which ranges from 0 to 1 after normalization. And $W_{fcv,rated}$ represents the rated mass flow rate when the valve is fully open, set as 2.4 kg/s.

The water produced in the cathode will migrate toward the anode due to concentration gradients during the fuel cell operations. This diffusion can lead to flooding in the anode flow channel and affect the operational lifespan of the stack efficiency. Therefore, it is necessary to install a purge valve in the anode circulation system. The regulation of purge valve facilitates the expulsion of $H_{2}O$ diffused from the cathode and little $H_2$, thereby maintaining the appropriate hydrogen concentration and anode relative humidity.

$$W_{purge}=u_{pv}{W}_{purge,max}$$

(32)

where $u_{pv}$ is the opening of the purge valve, with the value of either 0 or 1. Nitrogen and liquid water are excluded from the system under consideration to simplify calculations. The details of the parameters in the EHR system model are shown in

Table 3. The main parameters of the PEMFC EHR system model.

Parameters	Values	Parameters	Values
$N_{cell}$	460	$V_{rm}$/${\mathrm{m}}^3$	0.004
$T_{em}$/$\mathrm{K}$	298	$F$/$C\cdot {\mathrm{mol}}^{-1}$	96485
$T_{an}$/$\mathrm{K}$	343	$R$/$J\cdot {(\mathrm{mol}\cdot \mathrm{K})}^{-1}$	8.314
$T_{sm}$/$\mathrm{K}$	343	$\kappa$	1.4
$T_{rm}$/$\mathrm{K}$	343	$k_{sm}$	6 × 10−4
$V_{em}$/${\mathrm{m}}^3$	0.004	$k_{an}$	5 × 10−4
$V_{an}$/${\mathrm{m}}^3$	0.005	$A_{fc}$/${\mathrm{cm}}^2$	400
$V_{sm}$/${\mathrm{m}}^3$	0.004	$t_m$/$\mathrm{cm}$	1.275 × 10−2

.

3. Adaptive Model Predictive Control Scheme Design

Alternating loads and frequent purging can lead to significant pressure fluctuations of the anode during PEMFC operation. Therefore, real-time, and precise anode pressure management is essential to minimize fluctuations and maintain the anode-cathode pressure differential as the load current and purging state vary with system conditions. To address these challenges, this study proposes a successive linearization-based AMPC strategy for regulating anode pressure in EHR system. Figure 4 illustrates the structured framework of the successive linearization-based AMPC controller.

3.1. Adaptive Linearized Model

The proposed EHR system presents a more straightforward structure compared with the anode recirculation system utilizing circulating pump. For the ejector, functioning as a passive component without active adjustment capabilities, the control variable is solely the opening of the flow control valve, while the output variable is the pressure of the anode flow channel. Thereby, the nonlinear dynamic model of the EHR system can be expressed through the following state function:

$$\left\{\begin{array}{cc}\hfill \dot{X}=& f(X)+g(X)U+s(X)d\hfill \\ \hfill y=& h(X)\hfill \end{array}\right.$$

(33)

where $X={\left[P_{em} P_{H_2,sm} a_{w,sm} P_{H_2,an} a_{w,an} P_{H_2,rm} a_{w,rm}\right]}^T$, $U=\left[u_{fcv}\right]$, $d=\left[I_{st}\right]$, $y=\left[P_{an}\right]$. Note that the nonlinear functions $f(X)$, $g(X)$, $s(X)$, and $h(X)$ are given in Equations (34)–(36).

$$f(X)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}{\displaystyle \frac{R_{H_2}{T}_{em}}{V_{em}}}(-{\displaystyle \frac{P_{em}{A}_t}{\sqrt{T_{em}}}}(\sqrt{{\displaystyle \frac{{\eta }_p\kappa }{R}}}{({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa +1}{\kappa -1}}})\\ {\displaystyle \frac{R_{H_2}{T}_{sm}}{V_{sm}}}(W_{ej,p}+y_{H_2}{W}_{ej,s}-k_{sm}(P_{H_2,sm}-P_{H_2,an}))\end{array}\\ {\displaystyle \frac{R_{H_{2}O}{T}_{sm}}{V_{sm}{P}_{sat}(T_{sm})}}((1-y_{H_2})W_{ej,s}-k_{sm}(P_{H_{2}O,sm}-P_{H_{2}O,an}))\\ {\displaystyle \frac{R_{H_2}{T}_{an}}{V_{an}}}(k_{sm}(P_{H_2,sm}-P_{H_2,an})-k_{an}(P_{H_2,an}-P_{H_2,rm}))\end{array}\\ \frac{R_{H_{2}O}{T}_{an}}{V_{an}{P}_{sat}(T_{an})}(k_{sm}(P_{H_{2}O,sm}-P_{H_{2}O,an})-k_{an}(P_{H_{2}O,an}-P_{H_{2}O,rm}))\\ \frac{R_{H_2}{T}_{rm}}{V_{rm}}(k_{an}(P_{H_2,an}-P_{H_2,rm})-y_{H_2}{W}_{ej,s})\\ \frac{R_{H_{2}O}{T}_{rm}}{V_{rm}{P}_{sat}(T_{rm})}(k_{an}(P_{H_{2}O,an}-P_{H_{2}O,rm})-(1-y_{H_2})W_{ej,s})\end{array}\right]$$

(34)

$$g(X)=\left[\begin{array}{c}\begin{array}{c}{\displaystyle \frac{R_{H_2}{T}_{em}}{V_{em}}}(-W_{fcv,rated})\\ 0\\ 0\\ 0\end{array}\\ 0\\ 0\\ 0\end{array}\right] , s(X)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\\ -{\displaystyle \frac{R_{H_2}{T}_{an}{N}_{cell}{M}_{H_2}}{2V_{an}F}}\end{array}\\ -{\displaystyle \frac{{\alpha }_{net}{R}_{H_{2}O}{T}_{an}{N}_{cell}{M}_{H_{2}O}}{V_{an}F}}\\ 0\\ 0\end{array}\right]$$

(35)

$$h(X)=\left[\begin{array}{cccc}\begin{array}{cccc}0& 0& 0& 1\end{array}& P_{sat}(T_{an})& 0& 0\end{array}\right]$$

(36)

The physical constraints of the proposed EHR system are characterized as:

$$\left\{\begin{array}{cc}\hfill P_{em}\in & \left[P_{em,min},P_{em,max}\right]\hfill \\ \hfill P_{H_2,sm},P_{H_2,an},P_{H_2,rm},P_{an}\in & \left[P_{min},P_{max}\right]\hfill \\ \hfill a_{w,sm},a_{w,an},a_{w,rm}\in & \left[a_{w,min},a_{w,max}\right]\hfill \\ \hfill u_{fcv}\in & \left[u_{min},u_{max}\right]\hfill \\ \hfill \Delta u_{fcv}\in & \left[\Delta u_{\min },\Delta u_{max}\right]\hfill \\ \hfill \Delta P_{ca,an}\in & \left[-\Delta P_{mem},\Delta P_{mem}\right]\hfill \end{array}\right.$$

(37)

where state variables values rely on parameters of the stack and its operating conditions.

Model accuracy is validated by comparing the simulated anode pressure against experimental measurements under current step conditions, as depicted in Figure 5a. The results in Figure 5b demonstrate a close agreement between the simulated and measured anode inlet pressure dynamics, with all errors remaining within an acceptable margin. This confirms that the established model of the PEMFC anode recirculation system possesses sufficient accuracy for the subsequent development of AMPC controller.

Due to the high nonlinearity inherent in the EHR system, the linear MPC controller can only function effectively under a specific range of the working conditions, failing to deliver satisfactory performance across a wide operational range. To address this challenge, an AMPC strategy, utilizing the successive linearization method for online model updating, emerged as a preferable approach. In the AMPC strategy, an offline linear model is first established at the nominal operating point. Then, to facilitate the description of the prediction equation, the linearized model is converted into an incremental form and then discretized to obtain the state space incremental model of the linear discrete-time system:

$$\begin{array}{cc}\hfill \Delta x(k+1)& =A\Delta x(k)+B_u\Delta u(k)+B_d\Delta d(k)\hfill \\ \hfill y_c(k)& =C_c\Delta x(k)+y_c(k-1)\hfill \end{array}$$

(38)

where $A$, $B_u$, $B_d$, and $C_c$ represent the correlation coefficients of the system model. $A$, $B_u$, and $B_d$ are updated dynamically with the Kalman state estimator’s correlation gain coefficient matrix and the controller state at each time. This enables the online update of the controller’s predictive model, enhancing its adaptation to changes in actual plant operations.

3.2. Adaptive Model Predictive Controller

The AMPC controller is developed for the EHR system to achieve accurate regulation of the anode pressure based on adaptive linearized models in consideration of the step current changes and purging state alternations. The predictive equation is introduced based on the adaptive linearized model to forecast the future dynamic output of the EHR system by utilizing the latest measured value as the initial condition and predicting over $p$ steps. Subsequently, the future predictive output vector over $p$ steps and input vector over $m$ steps for the system are defined:

$${\mathrm{Y}}_p(k+1 | k)=S_x\Delta x(k)+I{y}_c(k)+S_d\Delta d(k)+S_u\Delta U(k)$$

(39)

where ${\mathrm{Y}}_p(k+1 | k)$, and $\Delta U(k)$ are the future predictive output vector over p steps and input vector over m steps. $S_x$, $S_u$, and $S_d$ represent the coefficient matrices of prediction equation. $I$ denotes the identity matrix, while $p$ and $m$ are the prediction and control horizon, respectively.

The aim of the proposed AMPC controller design is to achieve the accurate real-time tracking of the anode pressure setpoints, while minimizing changes in the actuator control actions. Thus, the cost function is employed as follow:

$$J(x(k),\Delta U(k))={\displaystyle \sum _{j=1}^p{\Gamma }_y{(Y_c(k+j | k)-R(k+j))}^2}+{\displaystyle \sum _{j=1}^m{\Gamma }_u\Delta U{(k+j-1)}^2}$$

(40)

where $R(k+j)$ represents the reference sequence of anode pressure. ${\Gamma }_y$, and ${\Gamma }_u$ represent the weights associated with the output error and input increment, respectively. The normalization method is applied for the initial design of weight coefficients. To facilitate the solution, the cost function is then reformulated into the general form of the Quadratic Programming problem, and constraints are imposed on the range and increment changes in the flow control valve referring to Equation (37).

Once the successive-linearized prediction model, anode pressure reference output, and optimization problem of the AMPC controller are established, the control performance of the proposed strategy mainly depends on parameters including the sampling interval $T_s$, prediction horizon $p$, and control horizon $m$. $T_s$, $p$, and $m$ are set as 0.01, 20, and 2, respectively, to ensure optimal AMPC performance regarding the operating conditions and inherent characteristics of the HER system. In addition, the active set method is employed to solve the optimization problem associated with the AMPC controller, and the control law of the AMPC strategy is specified as follows:

$$\Delta u(k)=\left[\begin{array}{cccc}I& 0& \dots & 0\end{array}\right]\Delta U^{*}(k)$$

(41)

As detailed in

Table 4. Algorithmic framework for adaptive logic of AMPC strategy.

Adaptive Logic: Implementation and Enhancement
Step 1: Initialization, baseline mode, $p$ = 20, $m$ = 2, $T_s$ = 0.01, ${\mathrm{\Gamma }}_y$ = 1.0, ${\mathrm{\Gamma }}_u$ = 1.0
Step 2: Set $\alpha$ = 0.1, $\beta$ = 0.05, $P_{threshold}$ = 0.15, $N$ = 10
Step 3: For k = 0: 1: n_cycle
Step 4: Measure the current system output. Compute the optimal control sequence by solving the cost function with the current model and weights
Step 5: Apply the first element of the obtained optimal control sequence to the system
Step 6: if k % $N$ == 0, then
if $E_{output}$ > $P_{threshold}$, then new_model = RecursiveLeastSquares (current_model, variables, $\alpha$)
else if $N_{tracking}$ > $N_{control}$, then ${\mathrm{\Gamma }}_y$ = ${\mathrm{\Gamma }}_y$*(1+$\beta$), ${\mathrm{\Gamma }}_u$ = ${\mathrm{\Gamma }}_u$/(1+$\beta$)
else ${\mathrm{\Gamma }}_u$ = ${\mathrm{\Gamma }}_u$*(1+$\beta$), ${\mathrm{\Gamma }}_y$ = ${\mathrm{\Gamma }}_y$/(1+$\beta$)
Step 7: Return to Step 3

, the adaptive logic of the system involves adjusting the system model and weight coefficients through adaptive hyperparameters such as the learning rate $\alpha$, weight adjustment step size $\beta$, performance threshold $P_{threshold}$, and evaluation period $N$. If the output error($E_{output}$) between the current times and historical values exceeds $P_{threshold}$, indicating significant performance degradation, the recursive least squares method is employed for online model updates, while the weight coefficients remain unaltered. Else, during periods of stable performance, the weights are adjusted based on the contributions of tracking norm($N_{tracking}$) and control norm($N_{control}$) as defined in Equation (40). If the tracking error dominates, ${\Gamma }_y$ is increased while ${\Gamma }_u$ is decreased, and vice versa. Subsequently, receding horizon optimization and adaptive adjustment are performed again to minimize the performance index.

4. Simulation and Experiment Results Analysis

The AMPC controller is designed to track the reference anode pressure trajectory, thereby maintaining the pressure difference within the safe range between the proton exchange membrane. To evaluate the effectiveness of the proposed AMPC controller in the EHR system, a multi-step current is illustrated in Figure 6a varying between 180 A and 410 A as input disturbance. Additionally, the impact of pressure fluctuations from the purging was considered. A periodic purging strategy was implemented to evaluate its effect on anode pressure as depicted in Figure 6b. The purge valve stayed shut for the first 50 s to ensure a stable gas supply, and then purged every 100 s for 2 s.

4.1. Simulation Results and Discussion

Under the conditions shown in Figure 6, the ejector works in subcritical and critical modes demonstrating effective entrainment capability. Therefore, normal MPC and PID methods are developed for comparative analysis to examine the dynamic performance of the EHR system under the proposed AMPC management strategy.

Figure 7 illustrates the dynamic performance of anode pressure under different controllers during varying load currents and purging states. Figure 7a and

Table 5. Performance comparison of the controllers.

	AMPC	MPC	PID
MRE/kPa	2 × 10−7	4 × 10−7	9.6 × 10−7
MAE/kPa	0.044	0.081	0.191
RMSE/kPa	1.041	1.423	1.958

compare the performance and index, including Mean Relative Error (MRE), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE), throughout the operation of the EHR system with different controllers. Figure 7b,c depict the enlarged view of the step-up current case at region A and the step-down current case at region C from Figure 7a, respectively. These figures illustrate that both normal MPC and AMPC controllers exhibit smaller overshoot and settling times than the PID controller. Between these two MPC algorithms, the AMPC shows less overshoot and shorter settling times due to real-time updates of linearized model-related gain coefficient matrices and controller states, which are necessary to adapt to the dynamic characteristic of the EHR system. Under a step current increase of 40 A, the proposed AMPC controller achieves a response time of 0.5 s. This represents a significant improvement of 58.3%, 66.7%, and 75% over the feedforward-based proportional-integrative (FFPI), second-order active disturbance rejection controller (SOADRC), and second-order fuzzy active disturbance rejection controller (SOFADRC) strategies reported in [20], which had response times of 1.2 s, 1.5 s, and 2.0 s, respectively.

Compared with normal MPC, AMPC’s real-time linearized model enable it to adapt accurately to the varying system conditions, which minimizes the steady-state error and ensures precise regulation under load changes and purge actions. Moreover, the ejector entrainment ratio, which is the crucial performance influencing the whole system’s dynamic responses in the EHR system, is sensitive to pressure changes. Figure 7d illustrates the anode pressure detailed change view of region B in Figure 7a when the purge valve activates around 550 s. All implemented management strategies have demonstrated their efficacy in minimizing pressure fluctuations. However, the anode pressure both under the AMPC and normal MPC controllers can converge to the setpoints faster than PID controller. The AMPC controller can accurately maintain reference pressure tracking, which outperforms the MPC controller whose steady-state error arises throughout continuous purge valve action conditions.

The ejector performance significantly influences the operation of the entire EHR system. Therefore, the ejector’s operational mechanism and performance parameters change tendency are crucial to ensure the EHR system’s proper operating under various conditions. We then focused on the impact of the normal MPC and proposed AMPC strategies on the ejector performances and operational conditions, excluding the PID controller for its poor control effect observed in Figure 7. To be clear, the pressure values $P_p$, $P_s$, $P_o$ of the three ports of the ejector directly match to the pressure values $P_{em}$, ${\mathrm{P}}_{rm}$, and $P_{sm}$ of the EHR system, respectively.

Figure 8 depicts the ejector performance parameters variation with different conditions and various types of controllers. The pressure values of the ejector ports are more sensitive to the current step changes than that to the purging states alternations shown in Figure 8a,b. The overshoots and settling times of the AMPC controller exhibit smaller than that of MPC controller. Additionally, it reveals that the higher the load currents operate, the larger the pressure differentials between the secondary and outlet flow increase, which closely align with the real system behaviors. The AMPC controller effectively tracks the pressure setpoints, on the other hand, the results of the normal MPC exhibit larger pressure differentials between the outlet and the secondary flows, which potentially decrease the entrainment performance of the ejector and increase the risk of anode hydrogen deficiency under the similar conditions.

The anode pressure stabilization is the prior control objective with a singular actuator in the EHR system. On this basis, this paper studies whether the HER can meet the requirements of the PEMFC system. Figure 8c,d present the transient responses of the ER of ejector and the HER of PEMFC. The results of HER and ER are greater than 1.5 and 1.2 under the action of AMPC controller, respectively, which indicate the control performance of AMPC sufficiently met the operational demands of PEMFC system. The relationship between the ER and the load current derived from operational data aligns with the findings in the existing literature on the PEMFC ejector performance curves [14]. This consistency underscores the reliability and the applicability of the observed trends in the practical 110 kW PEMFC system under the operation scenarios.

4.2. Experiment Results and Discussion

A PEMFC test rig was conducted to evaluate the pressure tracking capability of the proposed AMPC strategy as shown in Figure 9. This experimental device is composed of a gas supply system, a power supply system, an electronic load system, a control system, and a stack. The hydrogen with a purity of 99.99% and nitrogen of the gas supply system are used to provide fuel and vent residual hydrogen, respectively. The cathode pressure was maintained at 10 kPa above the anode reference pressure, with fluctuations constrained to within ±1 kPa. The power supply system is employed to activate the PEMFC, and the electronic load system was used to consume the generated electrical energy. The system flowchart and data display are presented on the computer interface. The AMPC controller was deployed to the micro controller unit (MCU) of the 110 kW PEMFC system whose operating conditions are illustrated in Figure 6.

The comparative analysis of the simulation and experimental results indicates a high degree of concordance with the current fluctuation of ±0.7 A and the pressure MAE of 0.482 kPa as shown in Figure 10. This demonstrates the superior pressure-tracking performance and real-time capability of the AMPC strategy. The pressure response time under the proposed strategy is observed to be little longer than that of the simulation results. This discrepancy is due to the difference between the idealized assumptions made during the simulation and the real system with additional external disturbances of each component. Consequently, the prolonged response times can cause a significant pressure differential between experiment and simulation results over a short time when the load current fluctuates. However, the maximum pressure difference is effectively constrained less than 4.5 kPa, within the error tolerance allowed by the system. The mean absolute error of the proposed anode pressure strategy between the simulation results and real-time pressure collection remains less than 0.3 kPa under constant load conditions as shown in Figure 10b. The proposed strategy yields a lower mean absolute error than the fuzzy and PID controllers in [19], with reported values of 2.16 kPa and 5.75 kPa, respectively. According to Figure 10c,d, the values of HER consistently stay above 1.5, which meet the performance requirements during the entire operating cycle. The simulated power of PEMFC system is basically consistent with the actual operating power. Overall, the deployed controller can maintain a stable and accurate anode pressure supply when the PEMFC system accommodates the dynamic operational demands.

5. Conclusions

In this paper, an Adaptive Model Predictive Control strategy is proposed to sustain the steady hydrogen supply and efficient recirculation in PEMFCs using ejector-based recirculation during dynamic operating conditions. The designed ejector-based hydrogen recirculation system model was established to reflect the real system working mechanism and nonlinear characteristics. An AMPC strategy is proposed to maintain stable anode pressure and minimize pressure fluctuations.

The results show that the robustness and stability of the proposed AMPC strategy outperform that of the normal MPC and PID methods, particularly during stepwise current changes and purging status alterations. The AMPC controller demonstrates the MAE of 0.044 kPa and the RMSE of 1.041 kPa in anode pressure dynamic regulation. The experimental results are consistent with the simulation results within the current fluctuation of ±0.7 A and the pressure MAE of 0.482 kPa. In addition, the ejector achieves an entrainment ratio above 1.2 and a hydrogen excess ratio exceeding 1.5 which meet the requirements for efficient and reliable PEMFC system operations within the given operational ranges.

It is worth noting that recent studies have shown a strong correlation between the amplitude of anode pressure fluctuations and the rate of mechanical decay of the membrane electrode assembly. The 0.482 kPa MAE pressure control accuracy under ±0.7 A current fluctuations achieved in this paper is expected to significantly delay performance degradation caused by mechanical stress, providing a foundation for subsequent control co-optimization based on health state prediction.