Damien Guilbert
email: damien.guilbert@univ-lorraine.fr

Dario Sorbera

Gianpaolo Vitale

A Stacked Interleaved DC-DC Buck Converter for Proton Exchange Membrane Electrolyzer Applications: Design and Experimental Validation

Keywords: Proton Exchange Membrane Electrolyzer, power electronics, stacked interleaved DC/DC converter, wind turbine conversion system, current ripple, control

Since the two last decades, hydrogen production has been attracting the attention of the scientific community thanks to its inherent very low pollution when energy coming from renewable energy sources (RESs) are used. However, it implies the use of DC/DC converters to interface source and load. These conversion systems must meet several requirements from current ripple point of view, energy efficiency, and performance to preserve the sustainability of hydrogen production. This article proposes the design and realization of a stacked interleaved buck converter to supply a proton exchange membrane electrolyzer. The converter is designed to ensure a low output current ripple and a suitable dynamic response to guarantee the reliability of the electrolyzer. A theoretical analysis of the converter, taking into account the dynamic model of the electrolyzer, and the design of the control system based both on feedforward and a feedback action is provided. The stability of the control system is discussed as well. The effectiveness of the model and the control algorithm has been verified by simulation and experimental results on a PEM electrolyzer at laboratory scale; the extension to higher power levels is discussed at the end.

INTRODUCTION

The spreading of renewable energy sources (RES) has been recently enhanced by the use of energy storage systems (ESSs). Indeed, ESSs allow facing to weather conditions change (impacting the energy produced by RES) and meeting the customer needs, delivering an energy supply, consequently without interruption [START_REF] Mathiesen | 100% Renewable energy systems, climate mitigation and economic growth[END_REF].

The adoption of ESSs can guarantee an instantaneous balance between energy needs and generation of active power minimizing fluctuations in the grid due to intermittent energy supply. Different ESSs are currently available such as batteries, supercapacitors, flywheels, hydraulic, and compressed air systems [START_REF] Olabi | Renewable energy and energy storage systems[END_REF], [START_REF] Vitale | Renewable energies -Future perspectives[END_REF].

Hydrogen is considered a promising solution belonging to ESSs. It exhibits a high energy content (120 MJ.kg -1 ), and it can supply Fuel Cells (FCs) for electricity, power, or heat production [START_REF] Paidar | Membrane electrolysis-History, current status and perspective[END_REF]. There are different ways to generate hydrogen: fossil fuel reforming, biological, direct solar water splitting, or water electrolysis. Fossil fuel reforming is widely adopted in the USA. However, this process leads up to increase greenhouse gases emissions [5]; whereas some solutions are currently being developed to minimize this impact by impound carbon dioxide in the ground [START_REF] Leung | An overview of current status of carbon dioxide capture and storage technologies[END_REF].

By comparison, in water electrolysis, electricity is used to split de-ionized, pure or distilled water into hydrogen and oxygen. In principle, electricity could be produced by fossil sources but the reduction of CO2 emission, leading to sustainable energy generation systems, can be achieved only if water electrolysis is integrated into a RES basedsystem [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF][START_REF] Sarrias-Mena | Electrolyzer models for hydrogen production from wind energy systems[END_REF][START_REF] Zhou | Power flow control in different time scales for a wind/hydrogen/super-capacitors based active hybrid power system[END_REF][START_REF] Zhou | Real-Time Emulation of a Hydrogen-Production Process for Assessment of an Active Wind-Energy Conversion System[END_REF][START_REF] Zhou | Modeling and control design of hydrogen production process for an active hydrogen/wind hybrid power system[END_REF][START_REF] Muyeen | Electrolyzer switching strategy for hydrogen generation from variable speed wind generator[END_REF][START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF].

Water electrolysis is carried out using an electrolyzer (EL). There are three different main types of EL: [START_REF] Mathiesen | 100% Renewable energy systems, climate mitigation and economic growth[END_REF] alkaline EL; [START_REF] Olabi | Renewable energy and energy storage systems[END_REF] proton exchange membrane (PEM) EL; and (3) solid oxide (SO) EL; they differ from the electrolyte and the charge carrier point of view. A shortlist of the main features among the different technologies is provided in [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF] where it is emphasized that the alkaline ELs are, at the moment, the most mature and widespread compared to PEM ELs, currently under development. Alkaline ELs have a longer lifespan and lower capital cost than PEM ELs.

However, this technology suffers from having low current density and operating pressure, impacting consequently, system size and hydrogen production costs [START_REF] Schmidt | Future cost and performance of water electrolysis: An expert elicitation study[END_REF]. By comparison, PEM ELs have several advantages over alkaline ELs, such as high power density and cell efficiency, fast system response, wide partial load range and high flexibility in terms of operation [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF][START_REF] Schmidt | Future cost and performance of water electrolysis: An expert elicitation study[END_REF]. On the other side, PEM ELs present several drawbacks from platinum catalysts costs and lifespan point of view. Due to their advantages over alkaline ELs, this technology is an attractive option for integration into the grid, including renewable power generating systems [START_REF] Koponen | Review of water electrolysis technologies and design of renewable hydrogen production systems[END_REF]. For this reason, a PEM EL has been considered for carrying on this work.

Unfortunately, electric energy produced by RES can not directly be used to produce hydrogen since the voltage delivered by a RES system is too high compared with the supply voltage of an EL. For this reason, a power converter to interface the source and the load is mandatory. An important feature expected from the DC/DC converter connected with a PEM EL is a high conversion step-down ratio to face up to a high voltage coming from the DC bus voltage [START_REF] Zhou | Real-Time Emulation of a Hydrogen-Production Process for Assessment of an Active Wind-Energy Conversion System[END_REF], [START_REF] Muyeen | Electrolyzer switching strategy for hydrogen generation from variable speed wind generator[END_REF], [START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF], [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF]. High energy efficiency and low output current ripple are also major concerns for PEM EL applications. Indeed, due to the low PEM EL efficiency, DC-DC converter has to have energy efficiency as high as possible. In contrast, the output current ripple has to be as low as possible to enhance the performance of the EL from efficiency and hydrogen production point of view, improving its reliability. These two constraints need a suitable control algorithm of the converter. Few papers proposing new topologies take into consideration this aspect; it is noteworthy [START_REF] Monroy-Morales | Modeling and control design of a Vienna rectifier based electrolyzer[END_REF] where the hydrogen production is optimized by a suitable control algorithm, and [START_REF] Török | Modeling and Control of Three-Phase Grid-Connected Power Supply With a Small DC-Link Capacitor for Electrolyzers[END_REF] in which the converter is connected to the three-phase grid and the Total Harmonic Distortion (THD) is reduced according to the Standard together with the voltage ripple; in this case the control algorithm allows the output capacitor to be minimized. In [START_REF] Yildiz | Current ripple minimization of a PEM fuel cell via an interleaved converter to prolong the stack life[END_REF], both the converter efficiency and the output ripple are improved by an interleaved topology; whereas in [START_REF] Yamashita | A new solar to hydrogen conversion system with high efficiency and flexibility[END_REF] the efficiency optimization is achieved by more converter in parallel. Finally, the solution proposed in [START_REF] Mira | Fractional Charging Converter with High Efficiency and Low Cost for Electrochemical Energy Storage Devices[END_REF] is of interest for the good efficiency and reduced ripple of the current obtained by a fractional charging converter. The converter proposed in this paper is different from all the above-mentioned topologies. It is a Buckbased converter including a second or compensating leg to minimize the voltage output ripple. The dynamic behavior has been analyzed by a state-space averaging (SSA) analysis giving the transfer function by which the control algorithm is designed. The proposed converter includes compensation of the output ripple whatever the duty cycle, this circuit behaves better than the interleaved described in [START_REF] Yildiz | Current ripple minimization of a PEM fuel cell via an interleaved converter to prolong the stack life[END_REF] in which the optimal compensation occurs only for a unique value of the duty cycle. Differently from [START_REF] Török | Modeling and Control of Three-Phase Grid-Connected Power Supply With a Small DC-Link Capacitor for Electrolyzers[END_REF], the proposed converter is conceived to be supplied by a renewable source meaning by a DC voltage whose value is much higher than the voltage required by the EL. Finally, the dynamic response is set up to preserve the EL from overvoltages.

From the current state-of-the-art of the DC/DC converters for ELs presented in [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF], it can be noted that the analysis of a step-down converter taking into account the dynamic behavior of the PEM EL as load still remains unaddressed in literature. In most cases, the PEM EL is modeled only by resistance as in the resonant converter proposed in [START_REF] Chandrasekhar | Performance of Soft-Switched DC-DC Resonant converter for Electrolyzer[END_REF] or in the synchronous buck converter presented in [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF]. In [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF], [START_REF] Monroy-Morales | Modeling and control design of a Vienna rectifier based electrolyzer[END_REF], the simulation is carried out by considering a non-linear representation of the EL but neglecting the dynamic effects. In [START_REF] Muyeen | Electrolyzer switching strategy for hydrogen generation from variable speed wind generator[END_REF], the EL is modeled as a resistance series connected to a voltage generator representing the reversible voltage. In [START_REF] Andrijanovitš | Three-Level Half-Bridge ZVS DC/DC Converter for Electrolyzer Integration with Renewable Energy Systems[END_REF], the load model is not addressed at all. On the other hand, it has been previously demonstrated in [START_REF] Guilbert | Dynamic Emulation of a PEM Electrolyzer by Time Constant Based Exponential Model[END_REF] that the cathode reaction imposes a time constant in the step response where an equivalent electrical model modeling the dynamic behavior of a PEM EL has been developed. The proposed equivalent circuit here shows a topology similar to the one proposed in [START_REF] Da | Experimental and theoretical development of a PEM electrolyzer model applied to energy storage systems[END_REF] based on an analysis performed on FCs [START_REF] Larminie | Fuel Cell Systems Explained[END_REF]. However, in this paper, the equivalent circuit is obtained by identifying both the equivalent components modeling the anode and cathode reaction into the PEM EL [START_REF] Guilbert | Dynamic Emulation of a PEM Electrolyzer by Time Constant Based Exponential Model[END_REF].

In this article, it is shown that the transfer function and consequently, the step response of the converter considerably differs from considering the load a simple resistive element or a resistance series connected with a voltage generator. As a consequence, the analysis requires an accurate dynamic model.

A crucial issue concerns the voltage ripple at the output of the converter whose minimization allows the reliability of the PEM EL to be increased. Few papers deal with the consequence of the ripple on ELs, but much useful information can be achieved by literature on FCs. Indeed, as regards the PEM EL, in [START_REF] Dobo | The effect of power supply ripple on dc water electrolysis efficiency[END_REF][START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF], it has been shown that the performance of a PEM EL can be jeopardized by the output voltage variations superimposed to the DC value. It can be noted that this phenomenon is implicit in a switching converter requiring a suitable topology to be minimized.

A relevant contribution is given by [START_REF] Koponen | Effect of Converter Topology on the Specific Energy Consumption of Alkaline Water Electrolyzers[END_REF] where the issue related to the voltage ripple is addressed up to MW scale. It is noteworthy to note the detrimental role of the voltage ripple both at low and high frequencies. In fact, it is demonstrated that the high-frequency ripple, mainly due to the use of transistor-based converters, can lower the specific energy consumption of 14% whereas the low-frequency ripple, mainly due to the use of thyristor-based converters, can lower this value of 9% compared to a 12-pulse rectifier. On the other hand, the literature on FC have been studying the ripple effect since fifteen years ago [START_REF] Gemmen | Analysis of the effect of inverter ripple current on fuel cell operating condition[END_REF][START_REF] Liu | Low Frequency Current Ripple Reduction Technique With Active Control in a Fuel Cell Power System With Inverter Load[END_REF]; based on this, several papers proposed power converters dedicated to FC systems able to require a small current ripple to extend the lifespan of the FC [START_REF] Palma | Current source converter topology selection for low frequency ripple current reduction in PEM fuel cell applications[END_REF][START_REF] Guilbert | Fuel Cell Lifespan Optimization by Developing a Power Switch Fault-Tolerant Control in a Floating Interleaved Boost Converter[END_REF][START_REF] Zhan | Input current ripple reduction and high efficiency for PEM fuel cell power conditioning system[END_REF][START_REF] Mazumder | A Ripple-Mitigating and Energy-Efficient Fuel Cell Power-Conditioning System[END_REF]. Among these, [START_REF] Zhan | Input current ripple reduction and high efficiency for PEM fuel cell power conditioning system[END_REF] and [START_REF] Mazumder | A Ripple-Mitigating and Energy-Efficient Fuel Cell Power-Conditioning System[END_REF] verified the impact of the voltage ripple on the FC. By the ripple compensation proposed in [START_REF] Zhan | Input current ripple reduction and high efficiency for PEM fuel cell power conditioning system[END_REF], it is showed that the efficiency raises of about 2% when the ripple is reduced from 24.3% to 9.9%. A more detailed analysis proposed in [START_REF] Mazumder | A Ripple-Mitigating and Energy-Efficient Fuel Cell Power-Conditioning System[END_REF] demonstrated that the maximum allowable hydrogen utilization decreases with increasing FC current ripple. The values range from a stack efficiency of 50% obtained with 5% of peak current ripple to a stack efficiency of 29% obtained with a 97% of peak current ripple.

Finally, another important crucial feature to be addressed concerns the control strategy of the converter. The supply voltage must remain lower than the rated voltage to preserve and to ensure the good operation of the PEM EL,. It is relatively easy to be achieved in steady-state conditions, but during transients, the maximum voltage admitted by the PEM can be overcome by overshoots that can damage the PEM EL. The abrupt variation of the power is common when RESs are exploited. This study requires a complete dynamic analysis, including the load behavior. This is the main novelty of this article where a stacked interleaved buck converter (SIBC) able to be supplied by a DC voltage up to hundred volts and to deliver an output voltage lower than 10 V is designed. This converter guarantees a reduced output voltage ripple whatever the step-down conversion ratio, and it is controlled taking into account the dynamic of the load so that the transient response remains overdamped. To this aim, a SSA of the SIBC and the load is carried out. Simulations and experimental tests finally assess the effectiveness of the proposed solution.

This paper is organized into seven sections. After this introduction dealing with the current state-of-the-art and motivations to carry out this work, Section 2 describes the design constraints. The theoretical analysis is carried out in section 3, and the design criteria of the controller are explained in section 4. Simulation results are provided in section 5. In section 6, the experimental test bench to perform the tests is described and experimental results to validate the proposed DC/DC converter and its control are reported. Section 7 contains a discussion on how the converter can be extended to a higher power.

DEFINITION OF THE DESIGN CONSTRAINTS

Input voltage of the DC/DC converter

The proposed SIBC has been designed to be supplied by a RES system to produce hydrogen through a PEM EL.

The investigated system is shown in Fig. 1.

A RES system delivers the input voltage Vin. In case of a wind turbine, it is intended to be connected to a permanent magnet synchronous generator and to a three-phase diode bridge rectifier to obtain a DC voltage whereas a photovoltaic generator can be connected directly to the SIBC converter. In any case, the SIBC converter is able to perform the maximum power point tracking (MPPT) based on an estimation of the available power by a suitable reference voltage. It is an essential feature since, in both cases, the amplitude of the DC voltage depends on weather conditions and can be subjected to abrupt variations. In the proposed application, a DC voltage up to 200 V has been considered. Anyway, this value can be raised up to reasonable values of the duty cycle of the switching devices of the converter.

Output voltage of the SIBC converter

The proposed converter is loaded by a PEM EL (NMH2 1000) from the HELIOCENTRIS company whose features are provided in Table I. During operation, the generated hydrogen gas is accumulated in the hydrogen/water separator and is dried by passing through the automatic dryer. The internal pressure (11 bar) is controlled by the amount of hydrogen generated by the cell. A proportional valve controls the output pressure. The produced hydrogen is stored by three metal hydride storage (MHS 800 shown in Fig. 1) from the HELIOCENTRIS company equipped with a low-temperature metal alloy on a titanium and magnesium base. This solution guarantees the absorption of the hydrogen in the alloy lattice after the adsorption at the surface, storage with high volume and low weight density (fit for stationary applications) and low thermal conductivity. The PEM EL has an operating voltage up to 8 V. This constraint defines the voltage ratio conversion of the SIBC converter.

The proposed converter

In the proposed application, the PEM EL operates with a low DC voltage since, at a rated power of 400W, it requires 8V. This constraint requires a step-down DC/DC conversion. Besides, the converter must feature good energy efficiency and a reduced output ripple to increase the reliability of the PEM EL [START_REF] Dobo | The effect of power supply ripple on dc water electrolysis efficiency[END_REF], [START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF]. In principle, the efficiency could be improved by reducing the switching frequency, since in this way switching losses are reduced, but in this way, the output voltage ripple is increased in turn. In the SIBC proposed in this article (see Fig. 2), the output current ripple is canceled by an auxiliary leg formed on an inductor and a capacitor (i.e., Ls, Cs) [START_REF] Wibben | A High-Efficiency DC-DC Converter Using 2 nH Integrated Inductors[END_REF]. Besides, compared to the classic interleaved DC-DC buck topology, the two legs cause a current ripple equal in amplitude but with opposite phase. As a result, a complete cancellation of PEM EL current ripple is obtained.

In case of fault of the compensating leg driving Ls-Cs, meaning the interruption of Cs, or Ls, or of the leg T2-T4, the capacitor Cp operates like in a standard Buck converter limiting the ripple on the EL. For this reason, it is designed with a value higher than Cs.

In Fig. 2, the load is represented by a generic dipole. Even though it is good modeling for a great number of applications, in case of hydrogen production the behavior of the EL cannot be approximated by this simple model as it has been emphasized in [START_REF] Guilbert | Dynamic Emulation of a PEM Electrolyzer by Time Constant Based Exponential Model[END_REF]. In this article, a dynamic equivalent circuit is adopted, and consequently, a new theoretical analysis is developed.

The equivalent model of the PEM EL

A dynamic model of the studied PEM EL has been carried out for the first time in [START_REF] Guilbert | Dynamic Emulation of a PEM Electrolyzer by Time Constant Based Exponential Model[END_REF] where a method based on the least-squares method for identifying an equivalent circuit was set up. It takes into account that both the electrochemical reactions at the anode and the cathode are not instantaneous and a capacitive effect can be highlighted. For this reason, after a step current variation, the stack voltage of the PEM EL shows two different time constants representing the speed of the reaction at the anode and the cathode. This behavior differs from the response obtained by a static model mainly when a fast transient occurs; thus affecting the dynamics of the converter. In the model adopted in this article, only the cathode reaction is considered since it is faster than the anode reaction. The equivalent circuit of the PEM considered as the load connected with the SIBC is shown in Fig. 3 In general, the two capacitances depend on the current. The authors identified their value in the neighborhood of a suitable operating point to assume constant value. Fig. 4 shows the experimental waveforms obtained by imposing a step current of 8 A to the EL by a DC power supply available in laboratory. It can be noted that the voltage rises according to a double exponential curve in which the lowest time constant is due to the cathode reaction, whereas the highest time constant is due to the anode reaction.

The whole transient shows a duration of about 60s.

THEORETICAL ANALYSIS OF A STACKED INTERLEAVED BOOST CONVERTER

Description of the circuital scheme

Based on Fig. 2, the SIBC is similar to a standard two-phase interleaved DC-DC buck topology except for the presence of an additional capacitor CS between the primary and the second leg.

The four switches are controlled in bipolar mode: the gate signal of T1 coincides with the one of T4, and it is logical negated for T2 and T3. There are two operating conditions for the switches: {T1=T4=ON, T2=T3=OFF}, it corresponds to the inductor Ls connected to Vin and the inductor Lp connected to the ground, on the contrary {T1=T4=OFF, T2=T3=ON} corresponds to the inductor Ls connected to ground and the inductor Lp to Vin. In the practical implementation, a dead time is included to avoid cross-conduction. Besides, both couples power switches are controlled with a different duty cycle: D for the primary phase (belonging to the inductance Lp); whereas (1-D) for the second phase (belonging to the inductance Ls). Thanks to the additional capacitor CS, each leg generates a different voltage. Taken separately, the primary leg (SP, LP, CP) provides a voltage output equal to that of a traditional Buck converter controlled with a duty cycle D, so VOUT=D.Vin. The second leg (SS, LS, CS) is controlled with a duty cycle equal to 1-D, so the voltage equation is Vin• (1-D)=VCS+VOUT.

On the one hand, in the primary leg, the inductor Lp flows both the whole DC current required by the load and the AC current which exhibits a triangular shape as in a traditional Buck converter. On the other hand, in the second leg only the AC current can flow through the inductor Ls since the DC component is blocked by the capacitor Cs. The current flowing through Ls is antisymmetric to the AC current flowing through Lp since the duty cycle of the two corresponding phases is complimentary. For this reason, differently from the interleaved converter in which the two AC currents flowing through the inductors are shifted from each other by 180°, and the total cancellation of the ripple occurs only for D=0.5, the operation of the proposed topology cancels the output current ripple whatever the duty cycle value.

The current and voltage ripple can be assessed by using the simplified scheme shown in Fig. 5. The analysis is explained in detail in [START_REF] Wibben | A High-Efficiency DC-DC Converter Using 2 nH Integrated Inductors[END_REF]. Hence, in this work, only the fundamentals are given; only the differences related to the use of the dynamic model of the PEM EL are explained in detail since they have never been published before. Some assumptions have been made to obtain this scheme. First of all, all components are ideal, meaning there are not series resistance resulting in losses. Since the objective is to determine the current and voltage ripple, the load is not considered in the scheme as it has been stated that only the DC part of the current flows through it, and the AC part of the current flows through the primary phase capacitor Cp. A constant voltage source has replaced the latter source because the voltage ripple is assumed to be much smaller than the DC value. Finally, the inductances are of the same value.

Current ripple evaluation

For the next steps, the second leg capacitor is considered quite high to maintain a constant voltage at its terminals because its ripple does not give a relevant contribution to the current ripple.

By using Kirchhoff's law, equations ( 1) and ( 2) can be written:

-+ . + + = 0 (1)
-+ . + = 0 (2)
Where VOS and VOP are the voltages at the terminals of the capacitors CS and CP respectively.

During the time interval D⋅T, the primary inductor is connected to Vin and the second is connected to the ground, the voltages on the inductors are constant. Thus, the current waveform is a ramp and the derivative can be written as . It is also possible to establish the value of the voltage source in this time interval:

VS = 0 VP = Vin VOP = D.Vin VOS + VOP = (1 -D).Vin
So, assuming that ΔT=D.T and T=1/f, where fs is the switching frequency, the current ripple equations can be derived from ( 1) and ( 2) and are presented in (3) and ( 4):

= -1 -. . . (3)
= 1 -. . .

It can be noted that the current ripple in the load is canceled by choosing Ls=Lp

Voltage ripple evaluation

To obtain the voltage ripple equation in the circuit of Fig. 5, the contribution of the capacitance Cs must be taken into account. Since two currents flow through the output capacitance Cp, this ripple is obtained as:

= ! . " # $ + $ % & ' ( . )$ (5)
The final expression is given by [START_REF] Wibben | A High-Efficiency DC-DC Converter Using 2 nH Integrated Inductors[END_REF]:

= *.! . + ! . . 1 -. . ,-. / 0 .1 .! 2 + 34 . 1 -. ! ! . 567, / 0 .1 .! 2 -18 (6)

Transfer function in the Laplace domain

The transfer function has been evaluated by the SSA technique based on the circuits shown in Fig.

6 [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]. The details are given in the appendix. This analysis differs from the one proposed by [START_REF] Wibben | A High-Efficiency DC-DC Converter Using 2 nH Integrated Inductors[END_REF]. Indeed, in this case, the state vector is a five-element column and the input vector contains two elements.

The transfer function is given by the ratio of a first-order and a fifth-order polynomial:

9 , = : ; < = , = > * @ A B@ C D E A BE C DBE F D F BE G D G BE H D H BE I D I (7)
The static gain can be obtained also by:

9 ( = lim D→( # 9 , % = 34 * N O N O BN = < 0 (8)
This relationship is used to design a feedforward action as detailed in the next section.

CONVERTER DESIGN

Main Components identification

As explained before, The PEM EL has to be supplied by a constant low voltage. The rated power of the PEM EL corresponds to a voltage Vo = 8 V. It is the worst case for the control system since the damping of the load is minimal.

The other components are: CP = 100 µF, CS = 10 µF, LP = LS = 400µH, RP = RS = 0.06 Ω.

With these values, using ( 4) and ( 6), we obtain = 0.908 R, ≈ 0 (meaning that only the residual noise will be expected). The value of ΔIL assures that the inductor is operated far from the saturation region. Higher ripple values allow the commutation from OFF to ON state of power switches to occur al low current but require inductors with a higher maximum value of magnetic flux density that are usually more expensive.

The Control strategy

The control algorithm has been developed to guarantee that the output voltage of the SIBC follows the reference voltage despite the input voltage variations or the load variations. The input voltage variation commonly occurs when a RES is used. The reference voltage regulates the produced hydrogen; this parameter can be used to maximize the power delivered to the EL.

The dynamic response of the converter represents a further constraint. It must avoid that an underdamped voltage exceeds the maximum voltage allowed by the EL damaging it. On the other hand, a strong overdamped response would imply a slow response degrading the efficiency since the system during transients is unable to track the maximum power. For these reasons, a trade-off is required trying to minimize the overshoot and to maximize the rise time and the settling time.

A combined action with feedforward and feedback has been used to control the system. The scheme is sketched in Fig. 7.

The chosen feedback action is a proportional-integral (PI). It is characterized by its values KP and KI. These parameters have to be tuned based on the transfer function of the converter (see Appendix: equation (A4)). The step response of the converter is shown in Fig. 8 in which it is compared considering it as is loaded both from the static model and by the dynamic model.

It can be noted that the two responses are strongly different since the cathode reaction makes the system slower compared to the behavior obtained considering only the static model. A comparison between the poles is given in Table II where it can be noted that if the load of the converter is assumed as a simple resistive model it exhibits a fast response with oscillations coherently with the presence of complex conjugate poles; whereas considering the dynamic model an overdamped response due to a real negative pole is obtained.

The Bode diagram of the transfer function Gd(s) is shown in Fig. 9 In any case, the presence of the dynamic model aims at reducing the gain near to the frequency where an abrupt phase variation occurs where it could lead to unstable behavior.

The knowledge of the transfer function of the converter allows designing the PI controller according to a given criterion. Here, the phase margin assignment T U at a given crossover frequency ωco is employed. From the Bode plot of the transfer function 9 , provided in Fig. 9, it is possible to evaluate the modulus at the crossover frequency,

T = |9 WX | YZY [A ,
and the phase \ = ∠9 WX ^_ at the desired crossover frequency X ^_. Then, according to the phase margin assignment control techniques, the coefficients KP and KI of the PI controller can be determined by using the following expressions:

` +1 + a Y [b c d * = e (9)
tan -1 X ^_i = j + k * [START_REF] Zhou | Real-Time Emulation of a Hydrogen-Production Process for Assessment of an Active Wind-Energy Conversion System[END_REF] where:

j = l-180°+ T U -\n o 180°
Where θ is the phase that the PI controller has to give.

To satisfy the constraints on the gain and phase margins, and considering the dynamic model in the bode diagram, the following parameters of the controller have been obtained: KP=5.535e-05 e KI=1. The PI transfer function is given by the following expression:

9 , = ` . c DB D (11) i = p c p (12)
The PI coefficients are chosen to obtain an overdamped response in a time domain to preserve the PEM EL from overshoots. The step response of the converter including the feedback action is shown in Fig. 10.

The step response of the closed-loop system shows an overdamped response corresponding to a dominant real pole equal to p1=-0.461 rad•s -1 .

The transfer function of the PI controller is given by:

G rs s = u.uvu * ( wI * xB x (13)
A gain margin of 46dB and a phase margin of 79.3° have been obtained as shown in Fig. 11, assessing the stability of the system.

Furthermore, a feedforward action is introduced. Based on the expression (9), the duty cycle is calculated. This value is then summed to the duty cycle obtained by the PI regulator. In this way, the dynamic of the control is improved.

The step response exhibits the following characteristic parameters:

1. minimum overshoot;

2. rise time tr=19 ms;

3. settling time ts=34ms, with a peak of 0.9979 of the stationary value.

SIMULATION RESULTS

The SIBC and its control have been tested in the PSIM environment. Two tests are shown corresponding to steadystate and transient conditions.

The steady-state test is shown in Fig. 12. It corresponds to an input voltage of 8V and an input current of about 29.4A. Fig. 12 encompasses (from the top to the bottom): the output voltage, the output current, the current on the primary leg and the current on the second leg. It can be noted that as expected, the output voltage shows a negligible ripple and the output current as well. The ripple on the primary and second legs correspond to the designed value of the inductors; they are complementary producing compensation on the output current.

The transient test has been carried out considering a step variation of the reference voltage from 6V to 8V. It is shown in Fig. 13. Comparing this curve with the step response, obtained by the transfer function, shown in Fig. 10, the same rise time and settling time can be appreciated. In both cases, there is no voltage overshoot.

DESCRIPTION OF THE EXPERIMENTAL TEST BENCH AND EXPERIMENTAL RESULTS

Experimental test bench

A test bench has been realized to validate both the developed SIBC and its control, and experimental tests have been performed. The test bench is shown in Fig. 14. The developed experimental test bench is composed of the following components: (1) dSPACE ControlDesk software, (2) DC power supply control interface, (3) DC power supply EA-PS 9080-100 from EA Company (input), (4) pure water tank, (5) dSPACE DS1104 board, (

SEMISTACK IGBT, [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF] inductive components (i.e. Lp, Ls), (8) capacitive components (Cp, Cs), (9) PEM EL (output), [START_REF] Zhou | Real-Time Emulation of a Hydrogen-Production Process for Assessment of an Active Wind-Energy Conversion System[END_REF] hydrogen flow rate meter, [START_REF] Zhou | Modeling and control design of hydrogen production process for an active hydrogen/wind hybrid power system[END_REF] voltage probe, (12) current clamp, (13) digital oscilloscope.

The SIBC has been realized employing a SEMISTACK-IGBT from SEMIKRON Company (as shown in Fig. 14).

The control algorithm has been implemented into a DS1104 dSPACE board. The control of the SIBC converter is based on the measurement of the EL voltage, which is acquired by a voltage differential probe from Chauvin Arnoux Company. By comparison, the current measurements are acquired by current clamps PAC 12 from Chauvin Arnoux

Company. The four PWM gate control signals to control the SIBC are generated by the dSPACE board. However, the voltage levels of the generated PWM signals from the DS1104 dSPACE board, 0-5 V, are not fit to drive the SEMIKRON driver boards SKHI 22, 0-15 V. For this reason, an interface board is used between the DS1104 dSPACE board and the driver boards to convert the control signals (0-5V) to (0-15V). The system specifications are summarized in Table III.

Experimental results

The experimental results are reported in Fig. 15, 16, and 17, considering dynamic and steady-state operations. The dynamic test is shown in Fig. 15 where (from the top to the bottom) the output current of the converter flowing through the EL, the voltage measured at the EL terminals, the current in the primary inductor and the current in the secondary inductor are drawn. In this test, the reference voltage of the SIBC is varied from 6V to 8V. It can be noted that the voltage at the EL terminals exhibits a fast time constant due to the converter step response and slow time constant due to the anode reaction. To show this waveform, a 2s•div -1 scale has been chosen; in this way slow the rise of the voltage can be appreciated. It can also be noted that the variation of the output current is sustained only by the current on the primary inductor whereas on the secondary inductor only a slight ripple variation due to the variation of the duty cycle can be appreciated. Fig. 16 shows the same waveforms in Fig. 15 with a 10ms•div -1 scale to underline the rise time. It can be noted that in the experimental test, differently from simulation, but similarly to the experimental curve of Fig. 4, the voltage rises slowly compared to the current. It is due to the presence of the equivalent capacitance of the anode reaction, which maintains constant the voltage at the electrolyzer terminals for a longer time. The equivalent circuit model of the anode reaction has not been considered in the theoretical analysis since it does not influence the stability. Anyway, the dynamic response of the converter can be appreciated by the current waveform that reaches the final values in about 2.5 ms. It is only slightly higher than the rise time obtained in simulation.

In summary, during the experimental step transient, the theoretical analysis and the simulation result are confirmed.

The steady-state measure, given in Fig. 17 Due to the negligible PEM EL current ripple, the hydrogen flow rate can be optimized without relevant losses [START_REF] Dobo | The effect of power supply ripple on dc water electrolysis efficiency[END_REF].

DISCUSSION

The approach proposed in this paper has been focused on a low power EL to verify the performance at laboratory scale. The performed tests aim at verifying the reduction of voltage ripple by an auxiliary compensating leg avoiding hard passive filtering. Besides, the theoretical analysis has shown that a dynamic response without overvoltage could be obtained to preserve the EL.

On the other hand, recently some reports focused the attention on the high potential of gas infrastructure. It is noteworthy that gas storage volume is almost 1,000 times as large as electricity storage volume in analyzed countries [START_REF]The Value of Gas Infrastructure in a Climate Neutral Europe[END_REF] offering advantages for energy storage and saving. In this scenario, hydrogen has high potential as explained in

[41] where the annual demand for hydrogen is estimated to increase tenfold by 2050 -from 8 exajoule (EJ) in 2015

to almost 80 EJ in 2050. As for the Water electrolysis deployment, [42] emphasizes that projects of 100 MW are announced, and a 20-MW project in Canada is under construction. Starting from this, it is essential to emphasize that the proposed converter can be extended to a higher power since high power IGBTs in half-bridge configuration (for example the SKM1000GB17R8 reaches up to 1kA of maximum current and 1,7 kV of blocking voltage) are available on the market. The bottleneck remains the inductor: as the rated current increases, the value of inductors available on the market are quite low. On the other hand, interesting performance can also be achieved with low switching frequencies since the ripple is minimized by the compensating leg. Anyway, from the authors point of view, the best performance of the proposed SIBC converter can be obtained by the architecture proposed in [START_REF] Yamashita | A new solar to hydrogen conversion system with high efficiency and flexibility[END_REF] where a parallel connection allows to increase the supply current of the EL; in this way the number of the operated converters depends on the available power and can be chosen so that they always operate at the maximum efficiency [START_REF] Kolar | Extreme efficiency power electronics[END_REF].

Defining the state vector as, x t = #I r I • V ƒ" V ƒx V ƒ % … , the input vector: u t = #V s ‡ V ˆ%… and the output y t = V Š = V ƒ" , by applying Kirchhoff's law, the matrix of the system can be determined:

A = A * = A = OE • • • • • • • • • Ž- R r L r 0 - 1 L r 0 0 0 - R • L • - 1 L • - 1 L • 0 1 C r 1 C r - 1 C r . R ˆ0 1 C r . R 0 1 C • 0 0 0 0 0 - 1 C . R ˆ0 1 C . R ˆ- 1 C . R ' " " " " " " " " " " B = OE • • • • • Ž -- 0 0 0 0 ˜-.™ š 0 0 0 ˜C.™ š ' " " " " " " B * = OE • • • • • Ž 0 0 -› 0 0 ˜-.™ š 0 0 0 ˜C.™ š ' " " " " " " B = OE • • • • • • Ž oe -- 0 •oe -› 0 0 ˜-.™ š 0 0 0 ˜C.™ š ' " " " " " " " C = C * = C = #0 0 1 0 0 %
From the SSA analysis, the static solution of the system can be deduced. This is the relation between the DC part of the voltages and the duty cycle, distinguished by the upper case letter.

< = -~. R • . | = . N O N O BN (A3)
The dynamic solution is calculated considering Ls=Lp=L and Rs=Rp.

Based on the SSA model, the dynamic solution is given by:

9 , = } ; ) ž , = ~ * , -R • * # R -R * * Ÿ + | -| * * 34 % + ~ -~* * Ÿ = 9 , = > * N •N C BD ! C .N C .N ¡¢ D •£ (A4)
The stacked interleaved buck converter under study interfacing an RES system with the PEM EL. Figure 2 Simplified scheme of the SIBC. Figure 3 Equivalent circuit model of the investigated PEM EL. Figure 4 Experimental waveforms showing the rise of the voltage after a step current imposed to the EL, channel 1: PEM EL current, iel (5 A.div -1 ), channel 2: PEM EL voltage, Vel (4 V.div -1 ), time scale: 2s. div -1 . Figure 5 Simplified scheme of the SIBC. Figure 6 Equivalent circuits of the SIBC: a) during D.T, b) during (1-D).T. Figure 7 Schematic of the control strategy. Figure 8 Step response of the converter. Figure 9 Bode diagram of the transfer function Gd(s). Figure 10 Step response of GPI (s).Gd (s). Figure 11 Bode diagram of GPI (s).Gd (s). Figure 12 Simulation results for a steady-state operation. Figure 13 Simulation results for a dynamic operation. Figure 14 Developed experimental test bench in laboratory. Figure 15 Dynamic operation (higher time scale), channel 1: PEM EL current, iel (20 A.div -1 ), channel 2: PEM EL voltage, Vel (4 V.div -1 ), channel 3: primary phase current, ip (50 A.div -1 ), channel 4: secondary phase current, is (10 A.div -1 ), time scale: 2m.div -1 . Figure 16 Dynamic operation (lower time scale), channel 1: PEM EL current, iel (10 A.div -1 ), channel 2: PEM EL voltage, Vel (20 V.div -1 ), channel 3: primary phase current, ip (10 A.div -1 ), channel 4: secondary phase current, is (20 A.div -1 ), time scale: 10ms. div -1 . Figure 17 Steady-state operation with AC coupling, channel 1: PEM EL current, iel (10 A.div -1 ), channel 2: PEM EL voltage, Vel (2 V.div -1 ), time scale: 50 µs.div -1 . Pole (G1) pole(G2) * 1.0e+05 (-0.0528 ± 1.6403i) .1e+04 (-0.0000 + 0.0000i) .1e+05 (-0.4852 ± 0.1429i) .1e+04 (-0.0047 + 0.0000i) .1e+05 (-1.1320 + 0.0000i) .1e+05 (-0.0023 ± 0.1582i) .1e+05

  . The values of the equivalent circuit parameters have been evaluated experimentally. The following values have been obtained: Ri= 0.088 Ω, Vi=4.38 V, R1=0.035 Ω, C1=37.26 F. where: Ri : resistance modeling the membrane losses [Ω]; Vi : Reversible voltage [V]; R1: resistance modeling the activation losses at the cathode [Ω]. C1 : equivalent capacitor modeling the double-layer capacitive effect [F].

••

  for both cases as well. Besides, it includes the bode diagram in case of fault of the compensating leg (dotted trace). In this case, the converter operates like a traditional Buck stage. The presence of the compensating leg Ls-Cs introduces a resonant peak as it is shown in the bode diagram of Fig.9. Even if the resonant peak of the dynamic model shows a lower amplitude compared with the bode diagram of the static model, an abrupt phase rotation can be noted. For this reason, the feedback action requires the design of a suitable controller to assure stability. In case of failure of the compensating leg, the bode diagram modifies as pointed out by the dotted green trace. It can be noted that, in case of failure of the compensating leg, the bode diagram indicates an intrinsically stable behavior since no phase rotation occurs. For this reason, the feedback designed for the SIBC will remain effective in case of failure of the compensating leg continuing to operate with higher ripple until the fault is repaired.It can be observed that both systems show different curves and require a control action since the gain and margin phase are inadequate. Indeed, the following gain and phase margin have been obtained for the static and dynamic model: Static model: Gm = -44.8 dB (at 1.57e+04 rad.s -1 ) , Pm = -163 deg (at 3.54e+04 rad.s -1 ) Dynamic model: Gm = -41.5 dB (at 1.58e+04 rad.s -1 ) , Pm = -101 deg (at 2.58e+04 rad.s -1 )

  , is performed considering only the AC component of the output current of the converter and the AC component of the voltage at the terminals of the EL. The time scale of 50µs•div -1 allows appreciating the variations due to the switching frequency. It can be noted that both the ripple on the current and the voltage cannot be recognized as expected by theoretical analysis and simulation. The residual voltage ripple is about 2% due mainly to the residual noise rather to the compensation mechanism. Only the fast pulses due to the parasitic inductance of the converter appear. They can be easily suppressed by optimizing the layout of the converter. It can be achieved minimizing the parasitic inductance formed by the wire which connects the two switches. From authors point of view, the value parasitic inductance has roughly been estimated around 10 nH; since the IGBT has a rise time of about 50 ns, commutating a current of 8 A. A theoretical value of 1.6V for the spike is obtained similarly to the one measured.

Fig. 1 Fig. 3 Fig. 8 Fig. 9 FigFig. 17

 138917 Fig. 1 Guilbert et Al

  Fig. 17Guilbert et Al

Table List Table I

 ListI Technical features of the NMH2 1000 PEM EL.

	Table I Guilbert et Al
		Specification		Data
	Max H2 flow rate at standard temperature and pressure
		(STP).		1 l/min
		(20°C / 1 bar absolute)
		Delivery output pressure	0.1-10.5 bar
	Table II	H2 purity	Comparison of the poles by taking into consideration a static or dynamic >99.9999%
		Electrolysis cell	model.	Solid polymer membrane
	Table III	Water Weight (dry)	System specifications.	Deionized or distilled 20 kg
		Stack weight		6 kg
		Number of cells		3
		Operating conditions:
		-Temperature		15°C to 40°C
		-Relative humidity		0-80%
		Stack operating voltage range	4.4-8 V
		operating current range	0-50 A
	Table II Guilbert et Al

Table III Guilbert

 III et Al

	Specification	Data
	SEMISTACK-IGBT Block:
	IGBT	4xSKM 50 GB 123D
	Rectifier	1xSKD 51/14
	DC capacitor bank (at the output of the 3-phase rectifier)	2x2200µF/400V
	SEMIKRON driver boards	4xSKHI 22 (0-15 V)
	Passive components :
	Primary inductor, Lp	400 µH
	Secondary inductor, Ls	400 µH
	Parasitic resistances, Rp and Rs	60 mΩ
	Primary capacitor, Cp	100 µF
	Secondary capacitor, Cs	10 µF
	Other information:
	EL rated current, Iel	50 A
	EL rated voltage, Vel	8 V
	EL current ripple, Δiel	< 1 A
	DC bus voltage, VDC	150-200 V
	Switching frequency, Fs	20 kHz
	Duty cycle, D	0.24-0.29

CONCLUSION

A stacked interleaved buck converter has been set up for a proton exchange membrane electrolyzer. The SIBC can supply the EL with a negligible voltage ripple to improve reliability assuring a proper dynamic behavior. The main contributions are a) the new topology in which the ripple is compensated whatever the duty cycle of the converter, b) the theoretical analysis performed considering the dynamic behavior of the electrolyzer as well, c) the control strategy able to preserve the electrolyzer from overvoltages.

The converter has been studied theoretically and by simulation. Then, a suitable experimental test bench has been realized to test the designed converter and its control. Obtained simulation and experimental results have demonstrated the effectiveness of the control to enhance the performance of the PEM electrolyzer from energy efficiency and hydrogen production point of view thanks to a step response without any overshoots and to a steadystate behavior with a negligible voltage and current ripple. The proposed topology is suitable to be extended to higher power levels.

APPENDIX

The proposed circuit has been studied by the SSA technique to obtain the transfer function including the compensating led and the load with its dynamic [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]. According to the theory, the circuit has been considered as composed by two different circuits: one valid during D•T, one valid during (1-D)•T obtaining two sets of state equations, (A1) and (A2) as shown in Fig. 6. Then, the state equations of the complete circuit are obtained by averaging the two solutions, each with a weight equal to its duty cycle.

To obtain a general relationship, the inductors are not considered equal and with their parasitic resistance, they have a no-zero state representation.

The following analysis differs from the one proposed by [START_REF] Wibben | A High-Efficiency DC-DC Converter Using 2 nH Integrated Inductors[END_REF]. Indeed, in this case, the state vector is a five-element column instead of four-vectors and the input vector contains two elements instead of one. For this reason, a different transfer function is obtained.