M Ait Ziane

M C Pera

C Join

M Benne

J P Chabriat

N Yousfi Steiner

C Damour

On-line implementation of Model free controller for oxygen stoichiometry and pressure difference control of polymer electrolyte fuel cell

Keywords: Proton Exchange Membrane Fuel Cell (PEMFC), Intelligent proportional controller, On-line control, Oxygen Stoichiometry, Pressure difference

This paper proposes and validates a model free controller to improve the real time operating conditions of Proton Exchange Membrane Fuel Cells (PEMFC). This approach is based on an ultra-local model that does not depend on a precise knowledge of the system. It is perfectly adapted to a complex system such as the fuel cell, while benefiting from the ease of online implementation and low computational cost. The designed controller is used to regulate both the oxygen stoichiometry and the membrane inlet pressure, which are crucial operating conditions for the fuel cell's lifetime. The objectives of the proposed control strategy are twofold: preventing the starvation failure, and limiting the potential for mechanical degradation of the membrane during a large pressure difference. The performance of the proposed control strategy is initially evaluated by a simulation environment for both oxygen stoichiometry and inlet pressure difference control of fuel cell stack. An online validation on 1.2KW fuel cell stack is conducted to control the membrane pressure drop. Two case studies are comprehensively investigated in relation to stoichiometry control: set point tracking and rejection of unmeasured disturbances caused by current variations. Simulations and experimental results reveal that the proposed controller provides significantly better performance in terms of fast trajectory tracking, and ensures less overshoot compared to the Fuzzy PID and PID controller. This efficiency is proven using the Integral Absolute Error (IAE), Integral Squared Error (ISE) and Integral of the Square input (ISU) performance indexes.

control.

Introduction

One of the major global environmental challenges is to reduce green house gas emissions, and dependence on fossil fuels. The convergence towards green and clean energy is imperative to meet this challenge. Polymer Electrolyte Membrane Fuel Cell (PEMFC) technology is one of the key technologies to achieve this goal. Among the advantages of this technology, providing simultaneously electrical and thermal energy. However, a PEMFC is a multiphysics system whose components are highly exposed to the risk of degradation, which results in a too short life span. Control and monitoring of operating parameters are mandatory to maintain the performance and efficiency in optimal conditions. In [START_REF] Askaripour | Effect of operating conditions on the performance of a PEM fuel cell[END_REF] [START_REF] Schmittinger | A review of the main parameters influencing longterm performance and durability of PEM fuel cells[END_REF], authors concluded that four crucial operating parameters have a direct impact on performance, reliability and lifetime: stoichiometry ratio, PEM fuel cell temperature, relative humidity, and inlet gases pressure. Firstly, gas stoichiometry is one of the important parameters due to its impact on electrochemical reactions and water management in PEM fuel cells. Too low oxygen stoichiometry leads to a drop in PEMFC voltage, and prevents the release of the produced water [START_REF] Pei | Main factors affecting the lifetime of Proton Exchange Membrane fuel cells in vehicle applications: A review[END_REF]. A large amount of water upstream of the electrodes tends to inhibit the diffusion of gas into catalyst layers and may cause starvation fault [START_REF] Yousfi-Steiner | A review on PEM voltage degradation associated with water management: Impacts, influent factors and characterization[END_REF]. A high oxygen stoichiometry ensures proper operation of PEMFC, but results in an increase in compressor power that reduces the overall efficiency of PEMFC systems and could dry out the membrane [START_REF] Petrone | A review on model-based diagnosis methodologies for pemfcs[END_REF]. Secondly, effective stack temperature control maintains better membrane hydration and increases the efficiency of PEM fuel cells [START_REF] Yin | Real-time thermal Management of Open-Cathode PEMFC system based on maximum efficiency control strategy[END_REF] [START_REF] Chugh | Experimental and modelling studies of low temperature PEMFC performance[END_REF]. Inefficient temperature control directly affects the water balance in the membrane and the electrodes. Highly variable water content permits the occurrence of flooding and drying faults that have significant impacts on performance and lifetime of PEMFCs [START_REF] Yousfi-Steiner | A review on PEM voltage degradation associated with water management: Impacts, influent factors and characterization[END_REF] [START_REF] Choi | Observation of flooding-induced performance enhancement in pemfcs[END_REF]. Thirdly, inadequate inlet relative humidity of reactants can unbalance the water content in the membrane and the electrodes [START_REF] Kim | Effects of humidity and temperature on a proton exchange membrane fuel cell (PEMFC) stack[END_REF]. For that, a quick adjustment of the relative humidity of the gases is required to maintain the maximum power output of PEM fuel cells [START_REF] Raman | Rapid humidity regulation by mixing of dry and humid gases with feedback control for PEM fuel cells[END_REF]. Finally, inlet partial pressure control is essential to avoid mechanical degradation of the membrane due to the large pressure drop between cathode and anode sides. Most suppliers recommend keeping this pressure difference below a few hundred mbar [START_REF] Pei | Main factors affecting the lifetime of Proton Exchange Membrane fuel cells in vehicle applications: A review[END_REF]. In recent years, many studies have presented model-based control strategies that have been applied to PEM fuel cell systems. Most work is focused on controlling oxygen stoichiometry to enhance performance, and avoid starvation fault. To keep the maximum power delivred by PEMFC, an optimal oxygen stoichiometry reference obtained from experimental tests is introdued in the control strategy of [START_REF] Chen | Optimal Oxygen Excess Ratio Control for PEM Fuel Cells[END_REF]. For the same purpose, an adaptive sliding mode observer near-optimal oxygen stoichiometry controller is applied to control the air supply. The objective of the control is to follow the optimal oxygen stoichiometry to optimize the net power produced by PEM fuel cell [START_REF] Zhu | An adaptive sliding mode observer based near-optimal oer tracking control approach for pemfc under dynamic operation condition[END_REF]. A Linear-Quadratic Regulator coupled with a feedforward controller is applied in simulation [START_REF] Li | Oxygen Excess Ratio Controller Design of PEM Fuel Cell[END_REF]. The purpose of the offered control strategy is to reduce the disturbance effect when the load changes and to maintain the stoichiometry at the desired value. The objective of the control in [START_REF] Ma | Oxygen excess ratio control of PEM fuel cells using observer-based nonlinear triplestep controller[END_REF] is to reach the maximum power with a variable stoichiometry. The employed controller consists of three parts: a nonlinear controller, a feed forward controller, and a PID feedback controller. The result illustrates the ability of the proposed controller to adjust the oxygen stoichiometry faster than a PID controller when the load changes. An active disturbance rejection controller (ADRC) is used to control oxygen stoichiometry in [START_REF] Sun | Data-driven oxygen excess ratio control for proton exchange membrane fuel cell[END_REF]. A model of PEM fuel cell is identified offline and represented by a transfer function for simplicity. Compared to PI controller, the simulation result exhibits that the ADRC controller provides less overshoot, and reaches the desired value comparatively faster. In [START_REF] Chen | Temperature Control for Proton Exchange Membrane Fuel Cell based on Current Constraint with Consideration of Limited Cooling Capacity[END_REF] the main objective of the control is to maintain the fuel cell temperature at 350K during load variations, and when the cooling capacity of the radiator is maximum. A constrained predictive control model is applied in the simulation, the result shows the ability of the controller to perform the assigned task. An adaptive controller is applied to regulate the stack temperature during start-up procedures and power changes [START_REF] Huang | Adaptive thermal control for pemfc systems with guaranteed performance[END_REF]. The simulation results showed the high efficiency of the controller in keeping the stack outlet temperature on the desired trajectory for different current changes compared to the PI controller. To improve the control of the water content inside the membrane, a flatness approach is applied in [START_REF] Damour | A novel non-linear model-based control strategy to improve pemfc water managementthe flatness-based approach[END_REF]. The objective of the strategy is to control the membrane humidity by acting on the airflow mass. The simulation results showed the ability of the controller to maintain the desired water in the presence of disturbances and uncertainties. In [START_REF] Solsona | Control-oriented model of a membrane humidifier for fuel cell applications[END_REF], a second order sliding mode controller is applied in simulation to robustly regulate the relative humidity of the input gases taking into account internal and external disturbances. Furthermore, the model-based approaches are also tested experimentally. The control task in [START_REF] Deng | Nonlinear controller design based on cascade adaptive sliding mode control for PEM fuel cell air supply systems[END_REF] is to stabilize the oxygen stoichiometry under disturbances and measurements noise. For this purpose, an adaptive sliding mode controller based on high order sliding mode observer (HOSM) is evaluated in simulation and in a real-time emulator. The results show that the proposed controller provides better convergence and robustness when the load varies compared to PID and sliding mode controller. A self-tuning PID strategy based on an artificial neural network (ANN) is applied to a single PEM fuel cell in real time [START_REF] Damour | Real-time implementation of a neural model-based self-tuning PID strategy for oxygen stoichiometry control in PEM fuel cell[END_REF]. The ANN model identification is performed using offline training data, which can predict the oxygen stoichiometry of a 50W single cell. The experimental results show the potential of the controller to maintain stable stoichiometry by anticipating current changes. In order to minimize the inlet gas pressure difference to avoid membrane degradation, a robust nonlinear adaptive controller is used in [START_REF] Chen | Robust nonlinear adaptive pressure control of polymer electrolyte membrane fuel cells considering sensor failures based on perturbation compensation[END_REF]. An observer is designed to estimate system states, disturbances (known and unknown), and a sensor fault on the hydrogen inlet pressure measurement. Simulation results reveal that the proposed control strategy is able to maintain the hydrogen pressure close to the set point in the presence of uncertainties and sensor faults compared to the conventional PI and fuzzy controller. The designed observer exhibits satisfactory estimation errors in the majority of the studied cases. These errors are related to the dynamics of the observer which takes some time to estimate the disturbances that impact the system.

The approaches previously presented involve the modeling of fuel cell. However, this is not an easy task as a PEM fuel cell is a complex system to model due the coupling of electric, chemical, thermal, and fluid phenomena. Unmodeled system dynamics and external disturbances can be handled by designing a controller that does not depend heavily on the model. Fuzzy logic approaches have proven to be appropriate, and widely used to control different parameters in PEM fuel cell systems. In [START_REF] Ali | Pressure Regulation in Proton Exchange Membrane Fuel Cell using Direct Active Fuzzy Non-Linear Controller[END_REF], a Direct Active Fuzzy Nonlinear controller is applied for pressure control in the simulation. The objective is to maintain the partial pressure of the supplied gases at 3 atm. The results demonstrate that the proposed controller ensures lower overshoot compared to nonlinear and PI controllers. A hybrid control strategy based on a selfadaptive PI controller is presented in [START_REF] Chen | Hybrid adaptive control for pemfc gas pressure[END_REF], the gain tuning is provided by an advanced genetic algorithm and fuzzy logic. The main objective of the control is to ensure a small pressure difference at the PEMFC inlet during current changes. The controller is tested in simulation with a comparison to a non-linear controller. The results show that the given control architecture ensures half the pressure difference as the controller used as a reference. A dynamic model of a PEMFC for a mobile application oriented towards hydrogen inlet pressure control is presented in [START_REF] Yuan | A fuzzy logic pi control with feedforward compensation for hydrogen pressure in vehicular fuel cell system[END_REF]. A self-adaptive PI controller is employed, the controller gains are updated with fuzzy logic. A feedforward controller is incorporated to improve the ability to reject disturbances. The disturbances are expressed as current change and system purge. Simulation tests showed a high ability of the controller to track the desired pressure and maintain the pressure under different current change and purge scenarios compared to the conventional PI controller and the fuzzy PI controller. An experimental validation on an 80Kw PEMFC is carried out, the results show that the control strategy is able to maintain the pressure around the set point with an error of less than 2 KPa under a purge interval of 1.5s during the opening and 5s during the closing of the back pressure valve. To regulate oxygen stoichiometry, in [START_REF] Zhang | Adaptive robust control of oxygen excess ratio for PEMFC system based on type-2 fuzzy logic system[END_REF] an adaptive controller based on Type-2 Fuzzy Logic Systems (T2FLS) is tested in simulation and experimentation. The results show that the T2FLS quickly tracks the desired value during setpoint, and load variations for different stack temperatures. Furthermore, a Fuzzy Logic controller is employed in [START_REF] Zou | Temperature Control for a 5 kW Water-Cooled PEM Fuel Cell System for a Household Application[END_REF] to control a cooling pump of a 5 kW PEM fuel cell system. The performance of the proposed controller is compared in simulation and experimentation to the PID and the state feedback controller. Fuzzy Logic proved to be faster in tracking the desired temperature without overshoot under different current variations. Moreover, the coupling of fuzzy logic and PID controller is presented in [START_REF] Abouomar | Hybrid Feedback-Feedforward Fuzzy Control of PEM Fuel Cell Air Feed System with Electromagnetic Field Optimization[END_REF]: a feed-forward fuzzy associated with a fuzzy PID controller is used to regulate the oxygen stoichiometry. Additionally, the tuning of PID gains with fuzzy logic is presented in [START_REF] Fan | Oxygen Excess Ratio Control of PEM Fuel Cell Based on Self-adaptive Fuzzy PID[END_REF] to regulate oxygen stoichiometry. The simulation results show that the PID fuzzy controller provides better perforance than the feedforward PID controller for setpoint tracking. Model-based control has shown considerable performance for various applications, but requires a deep knowledge of the modelled system which is not always easy to obtain. Fuzzy logic depends on the choice of interference rules and the type of membership functions, which makes the controller design more complex. Some of the methods presented above have a high computational cost that may sometimes require special equipment for real-time implementation. To deal with the constraints and difficulties of obtaining the PEM fuel cell model, the computational cost of the controllers and their tuning complexity, a model-free controller, named an intelligent Proportional Integral Derivative controller (iP ID) [START_REF] Fliess | Model-free control[END_REF], is presented and tested in both simulation and experimentation in this paper. This approach is based on an ultra-local model that does not depend on the precise knowledge of the system. The main advantages of this approach are the online estimation of the dynamics of the controlled system only from the output measurement and the previous values of the control input with low computational cost [START_REF] Join | intelligent" controllers on cheap and small programmable devices[END_REF]. On top of that, the satisfying results obtained in numerous studies for different applications [START_REF] Polack | Brake and velocity modelfree control on an actual vehicle[END_REF] [33] [34] [35] motivated us to apply this approach to PEM fuel cell systems. Firstly, an iP and iP D are tested with a nonlinear model in the simulation to control both oxygen stoichiometry and inlet pressure difference. The objective of the simulation is twofold: the first one is to propose a fuel cell model allowing to control simultaneously the oxygen stoichiometry and the inlet pressure of the cathode. This model is assembled from different models found in the literature. The second is to test the performance of the proposed controller in simulation before applying it in real applications with a comparison to controllers commonly used for real applications. Secondly, experimental tests are performed on a PEM fuel cell stack of 1.2KW to validate the control of the membrane inlet pressure difference.

The organization of this paper is as follows: the second section summarizes the modelfree control. The third and fourth sections present the simulation and experimental results, respectively. The conclusion and perspectives are given in the fifth section.

Model free approach

Model free control

Model-free control or Intelligent PID is introduced by C. Join and M. Fliess [START_REF] Fliess | Model-free control[END_REF]. This strategy has proven good performances for the control of complex systems [START_REF] Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF] supported by a low computational cost and an easy implementation in real time [START_REF] Menhour | An efficient model-free setting for longitudinal and lateral vehicle control. Validation through the interconnected pro-SiVIC/RTMaps prototyping platform[END_REF]. The principle of the model-free controller consists in replacing the precise mathematical model of the system by an ultra-local model defined by:

y (v) = F + α.u (1)
Where:

• The output and the control variables are respectively y and u.

• (v) is the derivation order of output y (in practical case, (v) is often equal to 1 or 2)

• α is constant parameter to be defined, authors in [START_REF] Polack | Brake and velocity modelfree control on an actual vehicle[END_REF] proposed a method to tune this parameter.

• F regroups the unmodeled part of the system and disturbances (the estimation of F is done via the input and output variables).

Estimation of F

The core of model free control is based on the estimation of F , which is approximated by a piecewise constant function F est in a short period of time [t -T ; t]. The dynamic of F is estimated from the previous input u and output y, using an algebraic numerical identification estimation technique [38] [39]. The Laplace transform is applied to the ultra-local model defined in Eq.( 1) with v = 1:

sY (s) -y(0) = F s + αU (s) (2)
Where F is approximated by a constant function, Y (s) and U (s) correspond to Laplace transforms of y and u respectively. Deriving both members to eliminate the initial condition y(0):

Y (s) + s dY (s) ds = - F s 2 + α dU (s) ds (3)
The variable s refers to a derivative in the time domain. To avoid the risk of noise amplification by the derivative of a noisy signal, both members of the equation are multiplied by s -2 .

s -2 Y (s) + s -1 dY (s) ds = -s -4 F + αs -2 dU (s) ds (4)
The expression for F est in time domain is deduced using the following correspondences between the frequency and time domains:

s -1 = t 0 s -n = (n) 0 d ds = -t dn ds = (-1) n t n
Therefore,

F est = -3! T 3 (2) y(t) -ty(t) + (2) αtu(t) (5)
The Cauchy formula Eq.( 6) can be used to convert an iterated integral into a simple integral, which facilitates the implementation of such estimation algorithm.

T 0 t 1 0 .... t γ-1 0 u(t γ )dt γ ...dt 1 = T 0 (T -t) γ-1 (γ -1)! u(t)dt (6)
Applying Eq.( 6) in Eq.( 5), we obtain:

F est = -3! T 3 t t-T (T -2t)y(t) + αt(T -t)u(t)dt (7)
Where T > 0 might be quite small and [t -T ; t] corresponds to the sliding windows of integration interval. The closed-loop control law in the case of iP controller is given by:

u(t) = 1 α (-F est + ẏd + k p e) (8)
Where :

• y d is the desired trajectory.

• e = y d -y is the tracking error.

• k p is tuning gain of the controller proportional.

An ultra local model with v = 2 is required if an iP D controller is chosen. Eq.( 1) becomes:

ÿ = F + α.u (9)
In the same manner as before, Laplace transform is applied on Eq.( 9):

s 2 Y (s) -sy(0) -ẏ(0) = F s -1 + αU (s) (10)
To eliminate the initial conditions, a derivative by d ds is applied:

2s -3 F = 2Y (s) + 4s dY (s) ds + s 2 d 2 Y (s) ds 2 -α d 2 U (s) ds 2 (11)
In order to mitigate the influence of noise on the measured output, we replace the derivative term by the integral function s -1 which has robustness properties against noise. Multiplying Eq.( 11) by s -3 gives:

2s -6 F = 2s -3 Y (s) + 4s -2 dY (s) ds + s -1 d 2 Y (s) ds 2 -αs -3 d 2 U (s) ds 2 (12)
The Eq.( 12) can be written in time domain using elementary calculus and Cauchy's formula to simplify the multiple integral into a single one.

F est = 5! T 5 t t-T T 2 + 6t 2 -6T t y(t)dt - α 2 t 2 (T -t) 2 u(t) dt (13)
The closed-loop control law in the case of iP D controller is given by: The choice of the output trajectory y d is inspired from the flatness-based control rules [START_REF] Fliess | Continuous-time linear predictive control and flatness: A module-theoretic setting with examples[END_REF]. The reference trajectory is not widely introduced in classical PIDs for industrial applications. This absence frequently leads to unwanted oscillations and control imperfections such as overshoot and undershoot [START_REF] Fliess | Model-free control[END_REF].

u(t) = 1 α (-F est + ÿd + k p e + k d ė) (14)

Control objective

The control of the stoichiometry is important to maintain the correct functioning of the PEM fuel cells, which ensures the presence of the necessary amount of reactant during the electro-chemical reaction to avoid a starvation fault. The airflow rate can be considered as proportionnal to the compressor rotational speed, controlled through the rotation speed. Moreover, to avoid perforation and degradation of the membrane, keeping the gas inlet pressure balance is mandatory. In this paper, the simultaneous control of air stoichiometry and inlet pressure is addressed in simulation. The objective is twofold: maintaining the stichometry at a desired value while decreasing the pressure difference between the anode and cathode sides under different scenarios. Experimentally, the test bench is fed by a bottle of pure oxygen and the flow rate is controlled through a mass flow controller, see section 4. The mathematical model of the 1.2KW PEM fuel cell and auxiliaries that are installed on the test bench is unknown. For this purpose, the validation of the proposed controller is an appropriate choice to control the inlet differential pressure.

Simulation results

The complete nonlinear fuel cell model used for the simulation is designed from three model elements found in the literature. The core of the PEMF fuel cell model is based on the work of [START_REF] Pukrushpan | Modeling and control for pem fuel cell stack system[END_REF]. However, to provide oxygen stoichiometry control, a compressor that supplies the PEM fuel cell with air flow is required. This model is deduced from the model found in [START_REF] Baroud | Robust fuzzy sliding mode control for air supply on pem fuel cell system[END_REF]. In order to control the inlet pressure, a valve placed at the outlet gas of the stack is added which allows controlling either the outlet and/or the inlet pressure, this model is based on the work of [START_REF] Fonseca | Control of pemfc system air group using differential flatness approach: Validation by a dynamic fuel cell system model[END_REF]. The full model with the PEM fuel cell model parameters is presented in the Appendix. The sampling period for this part of the simulation is 2ms. The model-free controller is benchmarked by comparing the performance obtained with a fuzzy PID and PID controllers in three case studies:

• Tracking oxygen stoichiometry while controlling the inlet pressure difference to 0.

• Maintain a fixed stoichiometry and inlet pressure difference when the load is varying, this variation is considered as a disturbance for the controller.

• Tracking the trajectory for the inlet pressure while maintaining the pressure difference at 0.

A white Gaussian noise of variance 0.02 is added to the output of oxygen stoichiometry. The air supply to the stack is provided by a compressor, the flow rate is regulated by controlling the compressor voltage. The inlet pressure is regulated by acting on the valve located at the outlet of the cathode side, the manipulated variable is the voltage which allows to control the opening and closing of the valve. The Fig. [START_REF] Schmittinger | A review of the main parameters influencing longterm performance and durability of PEM fuel cells[END_REF] shows the control scheme for the both control strategies. The desired trajectory is obtained by filtering the setpoint according to the characteristics of a second order dynamics.

y d E = 1 (T ′ s + 1) 2 (15)
Where: y d is the desired trajectory, E refers to the setpoint and T ′ is a parameter allowing to fix the speed of the trajectory rise.

E = T ′2 s 2 y d + 2T ′ sy d + y d (16)
It can be discretized as follows:

sy d = y d (kT s ) -y d (kT s -T s ) T s iP controller
s 2 y d = y d (kT s ) -2y d (kT s -T s ) + y d (kT s -2T s ) T 2 s
With T ′ = ωT s , we obtain:

y d (kT s ) = y d (kT s -T s )(2ω + ω 2 ) -y d (kT s -2T s )ω 2 + E (1 + 2ω + ω 2 ) ( 17
)
The trajectory is adjusted with the parameter ω, which determines the speed to reach the setpoint value. For this section of the paper, the iP controller is employed to control the oxygen stoichiometry, while the iP D is used to control the inlet pressure difference. For the control of oxygen stoichiometry, an iP controller has given acceptable performance, so a second order ultra-local model is not required. However, for the control of the inlet pressure, the dynamics of the system is fast and the architecture of the control is more complex, then an ultra-local model of order 2 is taken which leads to an iP D controller. The Table 1 shows the gain values of the controllers used in the simulation. The PID controller parameters are optimized with the Ziegler-Nichols method [START_REF] Meshram | Tuning of pid controller using ziegler-nichols method for speed control of dc motor[END_REF]. The design of a Fuzzy PID controller is done with two inputs to the fuzzy interface, the error e and ė. The outputs are the gains K p , K i and K d of the PID controller. The fuzzy subsets and the basic form of the control rules are chosen as in [START_REF] Baroud | Novel hybrid fuzzypid control scheme for air supply in pem fuel-cell-based systems[END_REF].

It is important to notice that the PEMFC is operating in a current density region of less than 0.92 cm2, which does not result in any mass transfer limitation.

Stoichiometry trajectory tracking with reducing the inlet pressure difference

The aim of the control strategy is multiple: tracking the desired stoichiometry with a load fixed and keeping simultaneously the inlet cathode pressure at the same value of the anode inlet pressure. For oxygen stoichiometry control, the objective is to flow quickly the trajectory without any overshoot. Regarding the inlet pressure difference, the goal is to rapidly reduce this difference to 0 in order to preserve the membrane degradation as previously mentioned. Stoichiometry setpoint tracking is performed at a fixed load of 180A. An increase in oxygen stoichiometry from 2 to 3 is done at 2 seconds. Two decreasing steps in oxygen stoichiometry with an amplitude of 0.5 are applied successively at 4 and 6 seconds, as shown in Fig. 3(a). The Fig. 3(b) shows that the iP controller reaches the target faster than the Fuzzy PID and PID controllers in both cases of changing the desired trajectory. In addition to being slower, they offer an overshoot for all three setpoint changes. For inlet pressure control, the model-free controller employed is iP D. The variation of the oxygen stoichiometry implies a change of the inlet air flow which is relatively dependent on the inlet air pressure. Therefore, it is necessary to control the pressure in order to reduce the pressure difference between the anode and cathode side. The inlet pressure of the anode side is set at 3 barg, the objective is to regulate the inlet cathode pressure at the same value. The Fig. 3(c) shows the variation of the air inlet pressure during the various stoichiometry changes. The iP D controller offers a reduced inlet pressure difference compared to other controllers with a smooth adjustment to the anode inlet pressure. The fuzzy logic and PID controllers ensure a pressure difference greater than 0.3 barg for the first change of the stoichiometry as shown in Fig. 3(d). For a real application with some PEM fuel cel stacks, this difference is very important and can cause significant damage to the membrane. The proposed controller achieved a pressure difference less than 0.25 barg for the first stoichiometry change and less than 0.15 barg for the other changes. It is interesting to note that the fuzzy logic PID performs better than the conventional PID for the first and second setpoint changes.

Disturbances rejection

The performance of the proposed controller is also tested for disturbance rejection as shown in Fig. 4(a). The control objective is to maintain the oxygen stoichiometry at 2 under different load variation, as shown in Fig. 4(b). The stack current increases from 180A to 200A at 1s, from 200A to 230A at 3s, from 230A to 250A at 5s. A decrease from 250A to 240A at 7s, additional decrease from 240A to 200A is carried out at 7s, a reset to 180A is performed at 11s. Fig4(a) shows the overall behavior of the controllers under variable load. It can be seen that the oxygen stoichiometry is readjusted more rapidly with the iP controller. The second goal is to maintain the cathode inlet pressure at 3.2 barg, being the anode inlet pressure. Fig4(c) shows the behavior of the controlled air inlet pressure for different current variations. It is clearly observed that the iP D controller provides a fast fit with less difference between the two sides compared to Fuzzy PID and PID controllers, see Fig4(d). The proposed controller confirms that it provides less inlet pressure difference than other controllers when the airflow changes as shown in Fig4(e). This performance can be observed on the PEM feul cell voltage signal, Fig4(f). During current increases, the voltage supplied by the proposed controller increases rapidly compared to the fuzzy PID which is faster than the conventional PID. It is important to note that during current decay, the voltage of other controllers is better than that of the proposed controller, because the inlet air pressure is significant. This pressure has a direct impact on the voltage but also degrades the membrane which can shorten its life.

Inlet pressure trajectory tracking

For this scenario, a variation of the hydrogen inlet pressure is performed. The objective is to ensure that the air pressure follows this variation to guarantee a zero pressure difference. The current and the oxygen stoichiometry are fixed at 180 A and 2 respectively. The pressure at the beginning is 2.5 barg, an increase of 0.5 barg of the hydrogen pressure is made at 0.8s. A decrease of 0.6 barg is made at 4s followed by an increase of 0.4 barg at 8s as shown in Fig5(a). For trajectory increment, the proposed controller provides faster trajectory tracking than fuzzy PID and PID. The latter provides overshoots during setpoint increments, see Fig5(b). The fact that the proposed controller is faster, translates into a slight difference in inlet diffrence pressure compared to other controllers as shown in Fig5(c). The pressure difference is effectively guaranteed to be less than 0.1 barg with the proposed controller compared to others that exceed 0.2 barg for this scenario. It should be noted that this difference is significant in the real application where the interest of the proposed controller is to reduce the PEMFC pressure drop. Fig5(d) shows that the stoichiometry is kept 2 during inlet pressure changes. Since oxygen stoichiometry is correlated with noise, there is no significant difference between the controllers when the input pressures are changed, as shown in Fig5(d). Although, the iP controller is slightly faster than the others. . The performance indexes Table 2 confirm that the iP and iP D controllers are considerably more efficient than the Fuzzy PID and conventional PID. For all the error criteria IAE, ISE, RM SE, the proposed controller provides good performance compared to the fuzzy PID controller which itself provides more performance compared to the conventional PID.

As mentioned earlier, the purpose of oxygen stoichiometry control is to avoid the starvation fault that occurs if the fuel cell is exposed for a long time below a stoichiometry of 2. The simulation figures and performance tables show that the proposed controller performs much better than the Fuzzy PID and conventional PID. Rapid adjustment of oxygen stoichiometry under different current changes is noticed as well as an excellent trajectory tracking without overshoot. Regarding the control of the inlet pressure difference, the iP D controller provides less difference in the three cases studied in this part of the simulation. This prevents the occurrence of mechanical degradation of the membrane.

Test bench description

The laboratory test bench where the experiments are performed is dedicated to PEM fuel cell stack that have a power lower than 1.5 Kw. The control and the measurement of flow mass are performed by a SLA5850 mass flow controller, manufactured by BROOKS instrument. The relative humidity of inlet gases is adjusted by setting the temperature of the humidification bottles and pre-heating channels. For both anode and cathode inlets, it gives access to the measurement of inlet pressure, temperature and relative humidity of the gases. Two back pressure valves ( Kammer vannes) are located at the gas outlet of the stack. These valves allow to control the pressure of either the stack inlet or the outlet. The basic configuration for the pressure control of the test bench is done via two industrial PID controllers (West 5010), which ensure the pressure control of the stack. The temperature management of the stack is ensured with 3 elements: a pump that regulates the flow of water entering the stack, a heating channel to heat the water at the start of the tests and an exchanger that cools the water leaving the stack.

Fuel cell Temperature Cooling flow rate RH Air stoichiometry Hydrogen stoichiometry 60 • C 1l/min 60% 2 2

Table 3: Operating condition of PEM fuel cell stack

The System Control and Data Acquisition (SCDA) is implemented through Lab-View, which allows setting different parameters of the PEMFC as well as adjusting the load. Indeed, the proposed control algorithm for the inlet pressure is implemented in Labview under NI PXI-1031, the manipulated variable is the voltage that controls the opening and closing of the back pressure valves. These are controlled by [4-20 mA] sent by the Ni daq board, 4 mA refers to the complete opening of the valve and 20 mA to the closing. The confidence interval of the sensors used for measuring the gas inlet pressure is ±4 mbar. The sampling time is 3 Hz for the acquisition of the different data of the PEM fuel cell. It should be noted that oxygen stoichiometry control cannot be performed on the existing test bench architecture because the gases are supplied by cylinders, there is no compressor. Fig. 6 shows the laboratory test bench where the tests are performed.

The stack where the experimental tests were performed contains 12 cells, with an active surface area of 90cm 2 and rated power of 1.2 KW. The PEM fuel cell stack manufacturer specifies that the maximal ∆P , i.e the inlet difference pressure must not exceed 0.3 barg. The operating condition of PEM fuel cell stack are given in Table . 3. obtained in the previous section with the proposed controller in a real application. Moreover, the second objective is to compare its performances with the integrated controller on the test bench. The idea is to evaluate a possible replacement of the regulators on the test bench with the proposed approach. Since the model of the fuel cell stack and all the auxiliaries that make up the fuel cell system is unknown, it is very difficult to apply an model-based approaches. As, the computer performing the computations is equipped with a 2.0GhZ processor and 504 MB of RAM, the proposed approach is appropriate considering that it does not require a high computational time and was implemented without the need for a specific toolbox.

The controller gain values used in the experiment are given in Table 4.

Inlet pressure control stack

The pressure control strategy employed for this experimental section is as follows: follow the desired trajectory for the inlet pressure on the anode side. The hydrogen inlet pressure is assumed as the reference to be followed for the air inlet pressure. This strategy tests both the performance of the controllers in following the desired trajectory and the reduction of the input pressure difference. The control scheme of the adopted control strategy is illustrated in Fig. 7. shows the potential of the proposed controller to quickly follow both trajectories. The first trajectory imposed by the user governing the inlet pressure on the anode side. The second trajectory is that of the hydrogen inlet pressure which is driving the air pressure, with no overshoot noted. The time taken for both inlet pressures to stabilize at the desired set point is about one minute. This stabilization time is defined by the desired trajectory, which makes the iP D Controller a very suitable tool for this application. At the beginning the inlet pressure is 1.4 , a large increase of 0.2 barg is made at 1.5 min, and a significant decrease of 0.3 barg is made at 7 min. As for the high set point changes and for the others that have a lower amplitude, the time to reach the desired set point is the same. This fast tracking results in a reduction of the inlet pressure difference, which is the objective of this control strategy as shown in Fig. 8(c). In this entire experiment, the highest ∆P recorded is 20 mbar which is far below the stack manufacturer's recommendations for inlet pressure difference, that is, 0.3 barg. The stack voltage signal also shows that a quick adjustment of the input pressure has a direct impact on the performance. The voltage signal climbs rapidly when increasing the inlet pressure for both sides of the stack, see Fig. 8(d).

Test bench controller

The integrated controller of the test bench is an industrial PID controller. The configuration chosen for the pressure control is that the desired setpoint governs the air inlet pressure. This is used as a reference for the hydrogen pressure control. In general, this is the opposite of the architecture proposed above. It should be noted that this configuration can in no way influence the quality of the control.

4.2.2.1. Setpoint tracking. The setpoint changes profile is the same as before, but with a longer time span of 43 min. This can be explained by the fact that the controller takes a long time to reach the desired setpoint as shown in Fig. 10(a). For the first setpoint change up to 1.6 barg, the air inlet pressure takes 2 minutes to reach the setpoint while the hydrogen inlet pressure takes about 2.5 minutes. In the event of a setpoint decrease from 1.65 barg to 1.35 barg, a time of approximately 4 minutes is noted, see Fig. 10(b). This time is too slow compared to the proposed controller. For other small amplitude changes, the stabilization time for both inlet pressures is approximately 2.5 min. The inlet pressure difference ∆P is illustrated in Fig. 10(c) for this setpoint change profile. The ∆P signal emphasizes that poor inlet pressure control can result in up to 100 mbarg of inlet pressure difference. This represents 5 times larger than the proposed controller for the same scenario. For small amplitude variations, this difference is less than 50 mbarg, which is acceptable but it takes a little longer to reach the desired set point. The stack voltage signal in Fig. 10(d). The voltage evolution follows the pressure profiles but slightly slower than the iP D controller. The proposed controller confirmed that it guarantees less pressure input differential for the two cases studied compared to the test bench controller. For the first scenario studied, the proposed controller reaches the desired setpoint twice as fast as the controller integrated on the test bench. In addition, for some scenarios studied, it ensures an inlet pressure difference 5 times lower than that the tests bench controller. This eliminates the risk of degradation on the membrane that is related to the inlet pressure, which is not the case with the integrated controller. The satisfactory experimental results obtained, encouraged us to replace the pressure control strategy of the test bench with the proposed controller. This choice was supported by the fact that there is no need to add any necessary equipment and the fact of having a low computational cost with no precious knowledge of the fuel cell model. This is perfectly suitable for stack pressure control in a real application.

Conclusion and perspectives

In this work, a model free (iP ) and (iP D) controllers are introduced to control both oxygen stoichiometry and pressure difference of PEM fuel cell. The main benefit of the model-free control approach is that it does not require an accurate model of the PEMFC system. Obtaining the model is often a complicated task due to the coupling of electrical, thermal and fluidic phenomena. In a first step, the oxygen stoichiometry control and inlet pressure differnce are performed in simulation using a nonlinear model. A comprehensive comparison with a Fuzzy PID and PID controller is performed in order to evaluate the performance of the proposed controller. The proposed controller showed excellent ability to restore oxygen stoichiometry and air inlet pressure when the current changes, and to reach the desired trajectory very quickly without any overshoot. The assessment of controller performance is done using performance indexes IAE, ISE, ISU and RM SE for simulation part. The proposed controller exhibits good efficiency compared to the Fuzzy PID and PID controller. In the second step, a control of inlet pressure difference is applied for a 1.2KW stack in experimentation. The proposed controller showed successful tracking of hydrogen inlet pressure variations, and ensured an inlet difference pressure less than 50 mbarg for all cases studied. By limiting a large depression difference, perforation of the membrane is avoided and the health of the fuel cell is preserved. Overall, it is twice as fast at following the set point as the test bench controller and provides 5 times less input pressure difference for the first scenario studied in the experimental part. We emphasize that the model free controller is inexpensive in computation time and implementation complexity for online control compared to other approaches.

In the future, the proposed method will be tested for controlling stack temperature and inlet relative humidity. The second future objective is to integrate the controller into the fault tolerant control strategy block with online fault detection to ensure stack durability.

Figure 1 :

 1 Figure 1: The diagram scheme of Model free controller iP D

Kp = 5 . 10 - 6 ;

 5106 iP D controller PID for oxygen Stoichiometry PID for inlet pressure difference α = 0.5; Ki = 1.5.10 -5; Kd = 3.10 -7;

Figure 2 :

 2 Figure 2: PEM fuel cell system control diagram

( a )Figure 3 :

 a3 Figure 3: Stoichiometry trajectory tracking with reducing the inlet pressure difference

( a )

 a Response of oxygen stoichiometry under load changes (b) Current profile (c) Inlet pressure (d) Zoom on (c) (e) Inlet difference pressure ∆P (f) Voltage of PEM fuel cell

Figure 4 :

 4 Figure 4: Disturbances rejection

( a )

 a Pressure trajectory tracking (b) Zoom of (a) (c) Inlet difference pressure ∆P (d) Oxygen stoichiometry

Figure 5 :

 5 Figure 5: Inlet pressure trajectory tracking with reducing the inlet pressure difference

Figure 6 :

 6 Figure 6: Laboratory test bench

Figure 7 :

 7 Figure 7: Pressure control scheme

Figure 8 :

 8 Figure 8: Setpoint tracking with model-free controller

Figure 9 :

 9 Figure 9: Current variation with model-free controller

Figure 10 :

 10 Figure 10: Setpoint tracking with test bench controller

Figure 11 :

 11 Figure 11: Current variation with test bench controller

Table 1 :

 1 Gain values of the controllers

Table 2 :

 2

	Index performance IAE	ISE	ISU	RMSE
	iP	0.15	0.014	1.53 × 10 2 0.0027
	iP D	1.7 × 10 4	1.24 × 10 8 1.39	2.47 × 10 7
	PID for oxygen sto-	0.22	0.0217	1.52 × 10 2 0.004
	ichiometry
	PID for ∆P	2.7 × 10 4	2.6 × 10 8	1.4295	5.38 × 10 7
	Fuzzy PID for oxy-	0.1854	0.0186	1.53 × 10 2 0.0032
	gen stoichiometry
	Fuzzy PID for ∆P 2.1 × 10 4	1.82 × 10 8 1.42	3.53 × 10 7

performance indexes 4. Experimental results :

Table 4 :

 4 Gain values of the controllers

Acknowledgments This project has received funding from the Reunion Island Region under grant number and the European Commission -European Regional Development Fund (ERDF) -Operational Programme 2014-2020. This work has been supported by the EIPHI Graduate school (contract "ANR-17-EURE-0002").