Sliding Mode Integral Separation PID Control for Hydrogen Fuel Cell Systems

Featured Application

The sliding mode integral separation PID control strategy is utilized to reduce the overshooting amount of the fuel cell system, to reduce the regulation time, to improve the stability of the hydrogen fuel cell system power generation, and to make the system output meet the desired target value faster and more stably.

Abstract

The stability of hydrogen fuel cell system power generation is affected by key variables such as oxygen excess ratio (OER), electric stack temperature, and cathode–anode differential pressure. To increase the fuel cell’s stability, a sliding mode integral separation proportional–integral–derivative (SMC−IS−PID) control strategy was proposed by combining the four−segment integral separation PID (IS−PID) control with the switching control in the sliding mode control (SMC). The control mode is selected through the system variable error and the current variable value; if there are significant systematic variable errors, the switching control in the SMC adopts the four−segment integral separation PID control, which determines the values of the segmentation thresholds and controls the integral weights to reduce the amount of overshoot. When the error of the system variables is small, the switching control in the SMC adopts the improved convergence law control, which introduces the hyperbolic tangent exponential power term, the nonlinear function term, and the saturation function term to improve the convergence law and decrease the control’s convergence time. Experimentally verifying the effectiveness of the control strategy above, the results show that for the OER, the SMC−IS−PID overshoots 0 and realizes no overshooting with a regulation time of 5.019 s. For the temperature of the stack, the SMC−IS−PID overshoots only 0.134% with a regulation time of 40.521 s. For the pressure of the stack, the SMC−IS−PID realizes the system is basically free of oscillation.

1. Introduction

The development and use of renewable energy has emerged as a popular study issue in several sectors due to the effects of global warming and the challenge posed by traditional fossil fuels [1]. Among the many renewable energy sources, hydrogen energy, as a recognized renewable energy source, has been widely favored by society and industry. Hydrogen combustion releases a lot of energy and produces only water, which is completely friendly to the environment, making it a new energy source ideally suited to power the future. Because of its benefits, which include low operating temperature, high energy conversion rate, zero carbon emissions, and a simple construction, hydrogen energy is regarded as one of the most promising energy sources for the future [2,3]. In this context, fuel cell systems efficiently convert hydrogen energy, providing an efficient power generation technology by directly converting hydrogen energy into electricity through an electrochemical reaction. Proton exchange membrane fuel cell (PEMFC), for example, is an advanced, clean, and efficient power generation technology. PEMFC, a kind of fuel cell, not only has the characteristics of clean and non−polluting hydrogen energy, but also has the advantages of fast cold−start, high power density, etc., and consequently it has been strongly supported by many countries.

The stack, gas supply modules (which include oxygen and hydrogen), heat exchanging module, and control module are all parts of the PEMFC, and they are all essential to the fuel cell system’s output power and efficiency [4]. Ensuring the effectiveness of the chemical process within the cathode requires rigorous regulation of the oxygen flow rate. One crucial metric for quantifying oxygen availability is the oxygen excess ratio (OER) [5]. Oxygen deprivation happens when there is not enough oxygen available, which can permanently lower the PEMFC system’s overall performance or furthermore reduce power generation efficiency [6]. The air compressor will use more energy if there is an excess of oxygen available, which will raise the mechanical losses in the PEMFC system. The temperature of the electric stack is also a crucial element impacting the capability of the output of the electric stack. If the temperature is low, the catalyst activity will be weakened, and the conductivity will be reduced, which will decrease the output performance. If the temperature is exceedingly high, the water content decreases, leading to dry cracking of the membrane electrode and damaging the internal structure of the electric stack [7]. In addition, excessive stack pressure or cathode and anode pressure differentials will damage the proton−exchange membrane, reduce system output performance, and even affect electric stack life. For the PEMFC to operate safely and perform properly, the cathode and anode pressure differentials must be controlled [8]. Accordingly, the control strategies of the key variables of the PEMFC are investigated as a way to guarantee the system’s steady operation and improve the system’s output performance.

At present, in commercial or industrial applications, the fuel cell system level control commonly uses proportional–integral–derivative (PID) control. The PID control design structure is simple, however, and for multi−input and multi−output nonlinear fuel cell systems PID control is not suitable to meet the requirements of the fuel cell system’s rapidity, stability, real−time demands, robustness, and so on. Therefore, the controller adopts modern advanced control strategies such as fuzzy control, neural network control, and model predictive control, etc., to realize more efficient, accurate, and adaptive system management. These control strategies utilize complex algorithms to optimize the control process, improve system performance, and enhance robustness to uncertainty and changing conditions. Anbarasu [9] proposed an advanced dynamic model predictive control (AMPC) based on a nonlinear model predictive control (NMPC) framework with a multi−objective cost function driven by dynamic weights to improve the energy performance of fuel cell hybrid vehicles. Cavus [10] proposed a switched auto−regressive neural control (S−ANC) that combines switched model predictive control (S−MPC) with auto−regressive (AR) control. The novel S−ANC algorithm can satisfy the constraints with a performance comparable to that of MPC, and, at the same time, it can realize continuous learning, which realizes the efficient use of energy resources. Cavus [11] proposed a hybrid method based on ε−variables and classical MPC for constructing the S−MPC for flexible hybrid microgrids with plug−and−play (PnP) capabilities.

In the study of control strategies for air supply systems, Yin [12] combined fuzzy control and neural networks with PID to control OER. The results showed that the OER settling time was 0.35 s under PID control and 0.16 s under improved PID control. To increase the resilience of PEMFC, Wang [13] proposed a robust H_∞ controller based on a convex optimization approach. Zhu [14] used a super−twisted sliding mode control (SSMC) and an adaptive dynamic planning−based controller to ensure the smooth response of the OER control method. Using a fractional−order PID control technique, Zhao [15] monitored the desired excess ratio and air pressure. The findings indicated that changes in system variables had an impact on the control effect.

In the study of temperature control strategies for hydrothermal management systems, Xia [16] proposed fuzzy PID and PID control strategies to control the electronic thermostat in a PEMFC system to reduce water temperature fluctuations. The results show that the maximum temperature fluctuation is 2 °C under PID control and 0.5 °C under fuzzy PID control. Chen [17] proposed an adaptive sliding mode control (ASMC) based on an extended state observer for the cooling system of a proton−exchange membrane fuel cell with various disturbances, and the results show that the proposed ASMC can significantly suppress the chattering phenomenon.

The control strategies in the above−mentioned literature [9,10,11,12,13,14,15,16,17], with their complicated controller architecture, numerous calculations, and single control variable, are unable to suggest a control strategy that works for the entire PEMFC. In order to lessen system overshoot, increase control accuracy, and guarantee stable system operation, this paper investigates the control method to regulate OER, electric stack temperature, and cathode–anode differential pressure, and subsequently then suggests a sliding mode integral separation PID control strategy. A 3 kW water−cooled PEMFC experimental system was built using the self−developed fuel cell controller, and it is confirmed that this control method has a more robust and dynamic response.

The main contents of this paper are organized as follows: Section 2 introduces the components of the hydrogen fuel cell system and establishes a simple fuel cell model. Section 3 proposes a sliding mode integral separation PID (SMC−IS−PID) control strategy and describes the design process of each part in detail. Section 4 introduces the 3 kW hydrogen fuel cell experimental system constructed by the team, verifies the effectiveness of the model and the proposed control strategy on MATLAB/Simulink 2019 and the experimental system, and analyzes the results. Finally, Section 5 summarizes the research results and future perspectives.

2. PEMFC System Modeling

In this section, a water−cooled proton−exchange membrane fuel cell system is modeled, including the electric stack and auxiliary systems such as gas supply modules, a heat exchanging module, and a control module, to make up the PEMFC power generating system. The key variables affecting the stability of power generation can be obtained by simulating the working process of the fuel cell, which provides model support for the variable control strategy proposed subsequently. The schematic diagram of the system structure is shown in Figure 1.

As shown in Figure 1, the electric stack is the core of the PEMFC system, which is the main place where the electrochemical reaction occurs, and the gas supply modules are designed to provide the electric stack with the appropriate pressure and flow rate of the reaction gases (hydrogen and oxygen). The heat exchanging module ensures that the reaction temperature in the electric stack is always maintained at a normal value. The control module is designed to monitor the operating status of the system in real time, collect the variables of the remaining subsystems, and control the corresponding actuators. The key models of the PEMFC system section are established as follows.

2.1. Modeling of the Stack’s Output Properties

The PEMFC stack output voltage U_out and power P_out [18] are shown below:

$$U_{\mathrm{out}}=N_{\mathrm{cell}}(E_{\mathrm{nerest}}-U_{\mathrm{act}}-U_{\mathrm{ohm}}-U_{\mathrm{con}}),$$

(1)

$$P_{\mathrm{out}}=U_{\mathrm{out}}{I}_{\mathrm{st}},$$

(2)

where: N_cell is the number of single cells, E_nerest is the nominal voltage output of battery thermodynamics, U_act, U_ohm, and U_con are the activation overpotential voltage drop, resistive voltage drops, and concentration differential voltage drop, respectively, and I_st is the load current.

2.2. Modeling of Air Compressor

According to the law of torque equilibrium, the air compressor’s controlled voltage and speed relationship equation [19] is:

$$J_{\mathrm{cp}}{\displaystyle \frac{d{\omega }_{\mathrm{cp}}}{dt}}={\eta }_{\mathrm{cm}}{\displaystyle \frac{{\mathrm{k}}_{\mathrm{t}}}{{\mathrm{R}}_{\mathrm{cm}}}}\left(V_{\mathrm{cm}}-{\mathrm{k}}_v{\omega }_{\mathrm{cp}}\right)-{\displaystyle \frac{W_{\mathrm{cp}}{\mathrm{C}}_{\mathrm{p}}{\mathrm{T}}_{\mathrm{atm}}}{{\omega }_{\mathrm{cp}}{\eta }_{\mathrm{cp}}}}\left[{\left({\displaystyle \frac{P_{\mathrm{sm}}}{{\mathrm{P}}_{\mathrm{atm}}}}\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}-1\right],$$

(3)

where: J_cp is the rotational inertia of the motor, ω_cp is the motor speed, η_cm is the motor efficiency, k_t is the motor torque sensitivity coefficient, R_cm is the armature resistance, V_cm is the control voltage of the motor (V), k_v is the constant of the inverse potential, C_p is the air’s heat capacity on a mass basis, T_atm is the ambient temperature (K), W_cp is the air compressor outlet flow rate (kg/s), η_cp is the efficiency of air compressor, P_sm is the supply pipeline gas pressure, and P_atm is the ambient pressure.

2.3. Modeling of Supply Pipeline

The supply piping model aggregates all piping and interfaces between the air compressor and the electric stack cathode, defined as piping aggregate volume V_sm.

According to the van der Waals equation and the law of conservation of mass, the gas pressure supplied to the interior of the pipe [19] is:

$${\displaystyle \frac{d{P}_{\mathrm{sm}}}{dt}}={\displaystyle \frac{{\mathrm{R}}_{\mathrm{a}}}{V_{\mathrm{sm}}}}(W_{\mathrm{cp}}{T}_{\mathrm{cp},\mathrm{out}}-W_{\mathrm{sm}}{T}_{\mathrm{sm}}),$$

(4)

where: R_a is the air gas constant, T_sm is the internal gas temperature of the supply pipeline, and T_cp,out is the air compressor’s outlet temperature.

The gas mass flow rate W_sm flowing out of the supply pipeline [19] is:

$$W_{\mathrm{sm}}={\mathrm{k}}_{\mathrm{sp}}(P_{\mathrm{sm}}-P_{\mathrm{ca}}),$$

(5)

where k_sp is the outlet runner coefficient of the supply pipeline.

2.4. Modeling of the Stack Cathode Runner

The movement of gases inside the cathode of the electric stack, including oxygen and water vapor, is characterized by the cathode stream transmit model [18,19].

There is not much pressure differential between the return pipeline and the electric stack’s cathode exit, and the linear nozzle equation [20] is approximated to represent the total cathode outlet gas mass flow rate W_out,ca.

$$W_{\mathrm{out},\mathrm{ca}}={\mathrm{k}}_{\mathrm{out},\mathrm{ca}}(P_{\mathrm{ca}}-P_{\mathrm{rm}}),$$

(6)

where: k_out,ca is the orifice coefficient, P_ca is the cathode pressure in the stack, and P_rm is approximately equal to the return pipeline gas pressure.

The oxygen excess ratio (OER) [19] is defined as the oxygen flow rate to the cathode of the electric stack W_O2,input to the oxygen flow rate consumed by electrochemical reactions W_O2,react. Based on engineering experience, OER is generally taken as 2.

$$OER={\displaystyle \frac{W_{{\mathrm{O}}_2,\mathrm{input}}}{W_{{\mathrm{O}}_2,\mathrm{react}}}},$$

(7)

2.5. Modeling of the Return Pipeline

The temperature and pressure changes inside the electric stack and inside the supply pipeline are described by modeling the return pipeline. After the cathodic reaction, residual gas is discharged to the atmosphere through the return pipeline.

The nonlinear nozzle equation [20] must be used to represent the gas pressure and outlet flow rate inside the return pipeline W_rm,out because there is a pressure differential between it and the surrounding environment.

$$W_{\mathrm{rm},\mathrm{out}}=\left\{\begin{array}{l}{\displaystyle \frac{{\mathrm{C}}_{\mathrm{V},\mathrm{rp}}{\mathrm{A}}_{\mathrm{V},\mathrm{rp}}{P}_{\mathrm{rm}}}{\sqrt{\mathrm{R}{T}_{\mathrm{rm}}}}}{\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{atm}}}{P_{\mathrm{rm}}}}\right)}^{{\displaystyle \frac{1}{\gamma }}}\cdot \left\{{\displaystyle \frac{2\gamma }{\gamma -1}}{\left[1-\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{atm}}}{P_{\mathrm{rm}}}}\right)\right]}^{{\displaystyle \frac{\gamma -1}{\gamma }}}\right\},{\displaystyle \frac{{\mathrm{P}}_{\mathrm{atm}}}{P_{\mathrm{rm}}}}>{\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma }{\gamma -1}}}\\ {\displaystyle \frac{{\mathrm{C}}_{\mathrm{V},\mathrm{rp}}{\mathrm{A}}_{\mathrm{V},\mathrm{rp}}{P}_{\mathrm{rm}}}{\sqrt{\mathrm{R}{T}_{\mathrm{rm}}}}}{\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{atm}}}{P_{\mathrm{rm}}}}\right)}^{{\displaystyle \frac{1}{2}}}\cdot {\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma +1}{2\gamma -2}}},{\displaystyle \frac{{\mathrm{P}}_{\mathrm{atm}}}{P_{\mathrm{rm}}}}\le {\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma }{\gamma -1}}}\end{array}\right.,$$

(8)

where: A_V,rp is the valve effective flow area, C_V,rp is the flow discharge factor, R is the universal gas constant, T_rm is the gas temperature in the return pipeline, and γ is the isentropic expansion factor.

2.6. Modeling of the Heat Exchanging of the Electric Stack

According to the first law of thermodynamics, the PEMFC thermal equilibrium relationship [21] can be expressed as:

$$Q_{\mathrm{tot}}-Q_{\mathrm{cool}}-Q_{\mathrm{gas}}-Q_{\mathrm{amb}}-P_{\mathrm{out}}=C_{\mathrm{st}}{\mathrm{M}}_{\mathrm{st}}{\displaystyle \frac{d{T}_{\mathrm{st}}}{d\mathrm{t}}},$$

(9)

where: Q_tot is the heat from electrochemical reaction, Q_cool is the coolant cooling, Q_gas is the gas flow thermal exchange, and Q_amb is the heat lost by the thermal radiation of the stack; C_st is the stack’s thermal emission coefficient, M_st is the stack’s weight, and T_st is the temperature of the stack.

3. Sliding Mode Integral Separation PID Control (SMC−IS−PID)

In this paper, a control strategy combining improved sliding mode control and four−stage integral separation PID control is adopted to control the oxygen excess ratio, electric stack temperature, and cathode and anode differential pressure in the hydrogen fuel cell system to improve the stability of the system’s power generation. It can make full use of the advantage of the strong robustness of sliding mode control to ensure that the system can still operate stably in the face of various uncertainties and disturbances. It can also utilize the integral separation PID control to improve the static error so that the system can be closer to the desired output value when it reaches the steady state, which not only improves the dynamic response speed of the system but also enhances the steady state accuracy of the system. The control block diagram is shown in Figure 2.

According to the proposed general control method, the key variables of the system studied in this paper are controlled hierarchically, and the specific control strategy is designed as follows: The status of the controlled variable error e and the contents of the controlled variable x, which is currently collected by the sensors, are used to choose the control mode. When the absolute value of the error e between the status variable x and the desired value x_ref is less than or equal to the standard value of the error ϕ, the PEMFC system employs sliding mode control with improved convergence law (the switching control in sliding mode control employs improved convergence law control). When the absolute value of the error e is larger than the standard value of the error ϕ, the PEMFC system adopts the sliding mode integral separation PID control strategy (the switching control in the sliding mode control adopts the four−stage integral separation PID control).

Based on control, the standard value of error judgment for OER, temperature, and pressure should be set at 0.2, 3 °C, and 200 Pa. Then the total control law is:

$$u_x=\left\{\begin{array}{l}{u}_{\mathrm{eq}}+u_{\mathrm{conv}},\left|e\right|\le \phi \\ u_{\mathrm{eq}}+u_{\mathrm{IS}-\mathrm{pid}},\left|e\right|>\phi \end{array}\right.,$$

(10)

3.1. Four−Stage Integral Separation PID Control (IS−PID)

The integral module in the traditional PID control is used to eliminate the static error in the control process, but the accumulation of the integral also increases the overshoot of the system, leading to system oscillation and affecting the control accuracy [22]. Therefore, in order to decrease the quantity of overshooting and increase control precision, IS−PID control was invented. When the error is significant it cancels the integral action, and adds it when the error is minor. The IS−PID control equation [23] is as follows:

$$u_{\mathrm{IS}-\mathrm{pid}}=k_{\mathrm{p}}\cdot s+\beta \cdot k_{\mathrm{i}}{\displaystyle \int sdt+k_{\mathrm{d}}\frac{ds}{dt}},$$

(11)

where β is the switching coefficient of the integral term.

The switching coefficients are set for only two cases in general integral separation PID control. A deviation threshold ε is set, and when the deviation |e(k)| > ε, the integral action is switched off (β = 0) and only proportional and differential control is used. When |e(k)| ≤ ε, the integral action is introduced (β = 1) and PID control is used.

In this paper, four−segment integral separation PID control is used to select three error thresholds, divide the interval into four segments, refine the switching coefficients, improve the state coefficients, and enhance the control effect [23].

$$\beta =\left\{\begin{array}{l}1, \left|e(k)\right|\le {\epsilon }_1\\ 0.2,{\epsilon }_1<\left|e(k)\right|\le {\epsilon }_2\\ 0.8,{\epsilon }_2<\left|e(k)\right|\le {\epsilon }_3\\ 0, {\epsilon }_3\le \left|e(k)\right|\end{array}\right.,$$

(12)

As shown in (12), the state coefficients of 0, 0.2, 0.8, and 1 are selected as the weights of integral control. Additionally, the error threshold value varies since the system needs to regulate the oxygen excess ratio, pressure, and temperature differently, and the threshold value is taken as follows:

$${\epsilon }_{OER}=\left[\begin{array}{l}0.3\\ 1\\ 2\end{array}\right] {\epsilon }_T=\left[\begin{array}{l}3\\ 8\\ 20\end{array}\right] {\epsilon }_{Pa}=\left[\begin{array}{l}500\\ 1000\\ 1500\end{array}\right],$$

(13)

3.2. Sliding Mode Control (SMC)

The fuel cell mechanism model serves as the basis for writing the state space equations, and the fundamental design procedures of the sliding mode controller, including switching control and equivalent control, are presented.

The fuel cell state space equation of state is as follows:

$$\left\{\begin{array}{l}\stackrel{·}{x}=A_f(x)+B_f(x)u\\ y=C_f(x)\end{array}\right.,$$

(14)

where: A_f (x), B_f (x), and C_f (x) are all functions of state variables. The state variable x, the control quantity u, and the output quantity y in the system model are shown below:

$$x=\left[\begin{array}{l}OER\\ T_{\mathrm{st}}\\ P_{\mathrm{an}}-P_{\mathrm{ca}}\end{array}\right] u=\left[\begin{array}{l}{V}_{\mathrm{cm}}\\ V_{\mathrm{cw}}\\ \theta \end{array}\right] y=\left[\begin{array}{l}{W}_{\mathrm{cp}}\\ T_{\mathrm{st}}\\ \Delta P\end{array}\right],$$

(15)

where: P_an–P_ca is cathode–anode differential pressure, V_cm is the voltage of the air pump, V_cw is the voltage of the radiator, θ is the valve openness adjustment, W_cp is the flow rate of the air compressor, and ΔP is the cathode–anode pressure difference.

The target value expectation is x_ref = [OER_ref T_ref ΔP_ref], so the error e is:

$$e=x_{\mathrm{ref}}-x,$$

(16)

Derivation of (16) yields:

$$\stackrel{·}{e}=\stackrel{·}{x_{\mathrm{ref}}}-\stackrel{·}{x},$$

(17)

The design slip mold surface s is:

$$s=ce+\stackrel{·}{e}=c(x_{\mathrm{ref}}-x)+(\stackrel{·}{x_{\mathrm{ref}}}-\stackrel{·}{x}),$$

(18)

where: c is the performance parameter of SMC, which takes a value greater than zero.

Derive the sliding mode surface function of (18) and make it zero:

$$\stackrel{·}{s}=c\stackrel{·}{e}+\stackrel{··}{e}=c(\stackrel{·}{x_{\mathrm{ref}}}-\stackrel{·}{x})+(\stackrel{··}{x_{\mathrm{ref}}}-\stackrel{··}{x})=0,$$

(19)

The second−order derivative of a variable in a second−order nonlinear system is specified with:

$$\stackrel{··}{x}=A_f(x,\stackrel{·}{x},t)+B_f(x,\stackrel{·}{x},t)u(t)+d(t),$$

(20)

where: $A_f\left(x,\dot{x},t\right)$, $B_f\left(x,\dot{x},t\right)$ is a function of the state variable x and its derivative, u(t) is the control input, and d(t) is the exogenous interference factor. Equation (18) can be reduced to:

$$\stackrel{·}{s}=c\stackrel{·}{e}+{\stackrel{··}{x}}_{\mathrm{ref}}-\stackrel{··}{x}=c\stackrel{·}{e}+{\stackrel{··}{x}}_{\mathrm{ref}}-[f+g\cdot u+d]=0,$$

(21)

Then the equivalent control in the SMC is obtained as:

$$u_{\mathrm{eq}}={\displaystyle \frac{1}{g}}(c\stackrel{·}{e}+{\stackrel{··}{x}}_{\mathrm{ref}}-f-d),$$

(22)

There are various design methods for the form of sliding mode control convergence law. The convergence law introduces hyperbolic tangent variable exponential power terms, nonlinear function terms, and saturation function terms to reduce the convergence time of the control. The improved convergence law is as follows:

$$\stackrel{·}{s}=-k_1{\left|s\right|}^{\alpha }\cdot \tanh (s)-k_2\cdot fal(s,a,\delta )-k_{3}sat(s)\cdot s,$$

(23)

The specific expressions for the hyperbolic tangent exponential power term, the nonlinear function term, and the saturation function term in (24)–(26) [12,17,19] are as follows:

$$\tanh (s)={\displaystyle \frac{e^{as}-e^{-as}}{e^{as}+e^{-as}}},$$

(24)

$$fal(s,a,\delta )=\left\{\begin{array}{l}s/{\delta }^{1-a}, \left|s\right|\le \delta \\ {\left|s\right|}^a\cdot \tan (s),\delta <\left|s\right|<1\\ 0.8, 1\le \left|s\right|\end{array}\right.,$$

(25)

$$sat(s)=\left\{\begin{array}{l}1, 1\le \left|s\right|\\ s, -1<\left|s\right|<1\\ -1, \left|s\right|\le -1\end{array}\right.,$$

(26)

The output of the improved sliding mode control using the approach law control for switching control is as follows:

$$\begin{array}{cc}\hfill u_{\mathrm{s}}& =u_{\mathrm{eq}}+u_{\mathrm{conv}}\hfill \\ & ={\displaystyle \frac{1}{g}}(c\stackrel{·}{e}+{\stackrel{··}{x}}_{\mathrm{ref}}-f-d)-k_1{\left|s\right|}^{\alpha }\cdot \tanh (s)-k_2\cdot fal(s,a,\delta )-k_{3}sat(s)\cdot s\hfill \end{array},$$

(27)

4. Analysis of PEMFC Experimental Results

The PEMFC test platform used in the experiments of this paper is the 3 kW water−cooled hydrogen fuel cell experimental test system built by the team independently, and the fuel cell system controller based on an Infineon TC275 triple−core processor (Infineon Technologies, Munich, Germany) has been independently developed and designed. In Figure 3, the experimental setup is displayed. With a 3 kW electric stack as the core, the system controls the gas supply modules and the heat exchange module in real time through the controller. The real−time operating parameters of each auxiliary system are uploaded to the upper computer system, and the control instructions issued by the operator are executed.

The test system shown in Figure 3a realizes the functions of real−time monitoring of system operation status, information acquisition and storage, data communication, fault alarm, and device action control through the controller to meet the experimental requirements.

The PEMFC controller shown in Figure 3b adopts an automotive−grade chip AURIX^TM TC275 (Infineon Technologies, Munich, Germany) as the main control chip with rich resource configuration. The controller is designed with a main control board circuit, signal acquisition and conditioning circuit, communication circuit, and driving circuit. The controller mainly measures each variable accurately in real time through various types of sensors (temperature and humidity, pressure, flow, voltage, current, etc.) and judges whether each parameter is in the normal operating range. Through the microcontroller output drive circuit, it controls the execution devices (solenoid valve, relay, fan, water pump, and hydrogen circulation pump) to stabilize each parameter in the normal range, and realizes the management, coordination, monitoring, and communication of each subsystem in the whole hydrogen fuel cell power generation system.

4.1. Polarization Curve Analysis of Electric Stacks

The established simulation model and control strategy are verified based on the completed 3 kW PEMFC. The variables that have the greatest influence on the output characteristics of PEMFC are gas volume, temperature, and pressure. The experimental data within a spectrum of operational conditions were collected to draw the fuel cell polarization curves and were compared with the simulation results. The parameters of the 3 kW water−cooled proton−exchange membrane power electric stack are detailed in

Table 1. 3 kW electric stack parameters.

Parameters	Values
Rated power	3 kW
Current range	0~48 A
Voltage range	42~84 VDC
Number of cells	83 pieces
Optimal oxygen flow rate	141.99 L/min
Optimal hydrogen flow	45.05 L/min
Optimal coolant flow	4.84 L/min
Environmental temperature	−20~55 °C
Environmental humidity	20~100%
Sizes	324 × 115 × 83 mm
Weights	12 kg

.

The 3 kW proton−exchange membrane electric stack parameters listed in Table 1 provide detailed hardware specifications and operating ranges for experiments and simulations. These parameters, including the power rating of the electric stack, current and voltage ranges, number of cells, ambient and operating temperatures, humidity, dimensions, weight, etc., are all critical factors that must be considered when designing and evaluating fuel cell systems. Through precise control and optimization of these parameters, the performance and reliability of the system can be improved, laying a solid foundation for the commercialization and application of hydrogen fuel cell technology.

During the simulation and experiment, the working conditions are used as follows: the oxygen excess ratio is 2, the ambient temperature is kept at 27 °C, the working temperature of the stack is kept at 70 °C, the cathode–anode differential pressure of the stack is kept at 20 kPa (anode pressure is 0.17 MPa, cathode pressure is 0.15 MPa), the anode−side pressure is changed along with the size of the cathode−side pressure, and the loading current is in the range of 0~48 A. The stack’s polarization profiles are represented in Figure 4.

As can be obtained from Figure 4 in the low−current region, the experimental data points are not uniformly distributed on both sides of the simulated values due to the uneven temperature distribution of the electric stack during system starting, which leads to the uneven wetting of the membrane electrode; therefore, the internal resistance of the electric stack is larger at the place where the membrane wetting is low. In the intermediate−current region, the experimental data points are uniformly distributed on the simulation curves with small errors. In the region characterized by high−current density, the water generated by the normal operation of the system reaction cannot be discharged in time. The experimental data points are all lower than the simulation value. The overall experimental data points are slightly lower than the simulation results, the overall trend of the experimental data points is consistent with the polarization curves, and the model is in accordance with the operation of functioning fuel cell.

By changing the stack temperature to 60 °C and 75 °C, the impact of the stack’s temperature on output performance is investigated. The anode pressure of the stack is adjusted 20 kPa higher than the cathode pressure during the simulation and experiment. Figure 5 illustrates the effect of the stack’s temperature on the output characteristics.

Figure 5 illustrates how the battery’s output voltage rises with the electric stack’s temperature and how the experimental and simulated values converge more closely. Both the output voltage and the output power of the electric stack tend to rise with temperature when the load current remains constant. Molecules in the electrochemical reaction accelerate due to the stack’s higher temperature, hastening the oxidation−reduction cycle.

By adjusting the stack cathode–anode differential pressure to 0 kPa and 50 kPa, respectively, the impact of the cathode–anode differential pressure on the output performance of the stack is examined. Figure 6 displays the effects of the stack cathode–anode differential pressure on the output characteristics. The temperature of the electric stack is fixed at 70 °C.

Figure 6 illustrates how the performance of the electric stack’s output capability increases as load current increases. It also shows how local amplification reveals that the output capability of the stack rises as cathode–anode differential pressures increase. The data points from the experiments pertaining to differential pressures of 0 kPa and 50 kPa are also somewhat lower than the simulation value, but they are still closer to the simulation value. Pressure differentials encourage the migration of hydrogen protons (+1 valence) to the cathode, speeding up charge transfer and raising the stack’s output voltage.

4.2. Analysis of Experimental Results of Integral Separation PID Control

The control objectives in this paper are to control the oxygen excess ratio (OER) to be 2, to control the temperature of the electric stack to be about 70 °C (343.15 K), and to control the cathode–anode differential pressure to vary according to the differential pressures in Figure 7. The experimental system is set up to verify the effect of integral separate PID control (IS−PID), and the control algorithm is burned into the controller to realize sensor acquisition and device control. The actual operating conditions are simulated in the fuel cell system by an electronic load, model IT8902E−1200−80 (ITECH ELECTRONIC CO., LTD., Nanjing, China), which is set in constant current mode, and Figure 8 displays the variations in load current. Figure 9 displays the OER IS−PID control, and Figure 10 shows the electric stack temperature control.

As shown in Figure 9, the integral separation PID and conventional PID are able to quickly control the OER to 2 when a step in the load current occurs, and the IS−PID has a smaller amount of overshooting compared to the conventional PID control. In Figure 9, it can be seen that the conventional PID peak time at system startup is 2.132 s, the maximum overshoot is 13.15% (OER = 2.263), and the adjustment time is 4.57 s. The IS−PID peak time is 1.726 s, the maximum overshoot is 10.15% (OER = 2.203), and the adjustment time is 4.356 s. The lowest OER of the conventional PID is 1.1508, and the lowest OER of IS−PID is 1.3754. The use of the integral separation PID can reduce the number of cases of “oxygen deprivation”. The overall performance of the IS−PID is better than that of the conventional PID when the load current undergoes a step change.

As shown in Figure 10, both the integral separation PID and conventional PID are able to control the electric stack’s temperature up to 70 °C. The integral separation PID possesses a smaller amount of overshoot and a smaller oscillation ripple. The conventional PID peak time is 4.134 s, the overshoot is 0.911% (T_fc = 346.28 K), and the adjustment time is 40.027 s, which is due to the strong hysteresis of temperature. The integral separation PID peak time is 5.234 s, the overshoot is 0.578% (T_fc = 344.982 K), and the adjustment time is 42.027 s.

4.3. Analysis of Experimental Results of Sliding Mode Integral Separation PID

This paper presents the combination of sliding mode control and IS−PID control to obtain the sliding mode integral separation PID control (SMC−IS−PID) of OER, as illustrated in Figure 11. It also controls the electric stack temperature, as illustrated in Figure 12, the cathode–anode differential pressure, as illustrated in Figure 13, in accordance with the cathode–anode pressure differential in Figure 7, and the corresponding different polar pressure of the electric stack, as illustrated in Figure 14.

In Figure 11, it can be seen that the conventional PID, integral separation PID, sliding mode control (SMC), and sliding mode integral separation PID control OER to 2 quickly. In the startup phase of the system, SMC has the advantage of no overshooting compared to PID and IS−PID, but the adjustment time is longer, and the adjustment time of SMC is 12.573 s. The sliding mode integral separation PID has a shorter adjustment time compared to the sliding mode control; specifically, the sliding mode integral separation PID adjustment time is 5.019 s. When the load current step occurs in the normal operation phase of the system, for example, in 100 s, the benefits of SMC−IS−PID over PID and IS−PID include less overrun and quicker response times, and it has a leg up on quicker response times compared to SMC. The PID adjustment time is 3.117 s, the IS−PID adjustment time is 2.983 s, the adjustment time of SMC is 7.341 s, and the SMC−IS−PID adjustment time is 2.092 s. SMC−IS−PID can reduce the occurrence of “oxygen deprivation” better than the other three control methods, and the sliding mode integral separation PID control method provides a superior control impact.

From Figure 12, the sliding mode integral separation PID has a smaller overshoot and less oscillatory ripple compared to PID, integral separation PID, and sliding mode control. SMC has a peak time of 4.0814 s, an overshoot of 0.808% (T_fc = 345.772 K), and an adjustment time of 41.817 s. SMC−IS−PID control has a peak time of 6.912 s, an overshoot of 0.134% (T_fc = 343.469 K), and an adjustment time of 40.521 s.

As can be seen from Figure 13, for a rapidly changing variable such as pressure, the differential pressure oscillates under conventional PID control with IS−PID control. The maximum differential pressure under traditional PID control is up to 10.938 kPa, and the maximum differential pressure under IS−PID control is up to 10.53 kPa, both of which deviate from the set value of differential pressure by a large amount, but the overall oscillation amplitude of IS−PID is much smaller than that of traditional PID. The oscillation can be basically eliminated by using SMC−IS−PID. According to the change of differential pressure−setting value in Figure 7, the fastness of SMC−IS−PID is poorer than that of conventional PID and IS−PID while there are step variations in the load current. The SMC−IS−PID adjustment time varies from 6.136 s to 7.29 s, the IS−PID adjustment time is about 2.94 s, and the conventional PID adjustment time is 3.135 s. Overall, SMC−IS−PID control performs better than the other control strategies used in this paper.

As shown in Figure 14, the cathode and anode pressure under SMC−IS−PID control is more stable, essentially oscillation−free, and has less pressure fluctuation compared to IS−PID and PID.

Table 2. Strategies performance comparison.

Control Variables	Control Strategies	Peak Time	Overshoot	Adjustment Time
Oxygen excess ratio (OER)	PID	2.132 s	13.15%	4.57 s
	IS−PID	1.726 s	10.15%	4.356 s
	SMC	0	0	12.57 s
	SMC−IS−PID	0	0	5.019 s
Stack temperature (Tfc)	PID	4.134 s	0.911%	40.027 s
	IS−PID	5.234 s	0.578%	42.027 s
	SMC	4.018 s	0.808%	41.817 s
	SMC−IS−PID	6.912 s	0.134%	40.521 s
Cathode–anode differential pressure (ΔPca)	PID	Oscillatory system, Pcamax = 10.94 kPa		3.135 s
	IS−PID	Oscillatory system, Pcamax = 10.53 kPa		2.940 s
	SMC−IS−PID	Basically non−oscillatory systems		6.136 s

displays a comparison of the various strategies’ performances.

5. Conclusions

Aiming at the three key variables affecting the output characteristics of PEMFC (oxygen excess ratio, electric stack temperature, and electric stack cathode–anode differential pressure), the authors of this paper propose a control method for PEMFC systems based on sliding mode integral separation PID (SMC−IS−PID) control, which retains the advantages of the robustness of traditional sliding mode control, simplifies the difficulty of controller design on the basis of satisfying the output performance of the fuel cell, and greatly reduces the overshoot of the controller. The correctness and effectiveness of the control strategy above are verified by the self−developed controller and the constructed 3 kW fuel cell system. The experiments verified the polarization curves of the electric stack of PEMFC, and the experimental results were consistent with the trend of the polarization curves of the electric stack in the simulation and were close to the simulated values, which indicated the correctness of building the experimental system. The effectiveness of the sliding mode integral separation PID control strategy is verified and compared with PID and integral separation PID with sliding mode control. The results show that for the OER variable, SMC−IS−PID can achieve no overshooting compared to PID, IS−PID, and SMC. The SMC−IS−PID overshoot is only 0.134% for the stack temperature variable, which greatly reduces the overshoot. For the cathode–anode differential pressure and cathode and anode pressure, SMC−IS−PID can realize the system basically without oscillation compared with PID and IS−PID. In summary, the sliding mode integral separation PID control strategy for controlling the OER, electric stack temperature, and cathode–anode differential pressure in the PEMFC system has a good control effect, and the performance quality is significantly improved. Most of the existing hydrogen fuel cell vehicle controls use traditional PID control; in future research of fuel cell control strategy, the control strategy proposed in this paper can effectively control the whole fuel cell system, helping to solve the existing fuel cell system control single and other problems. In the future, we will consider the practical application of this strategy to existing hydrogen fuel cell vehicles. For fuel cell system research and experimental development (R and D) and application, this strategy can effectively shorten the control R and D cycle, reduce the system R and D cost, improve the system power generation performance, and improve the system service life.