Eber Dantas

Americo Cunha
email: americo@ime.uerj.br

Tiago Alexandre

Narciso Silva

T A N Silva

A numerical procedure based on cross-entropy method for stiffness identification

Keywords: stiffness identification, Euler-Bernoulli beam, inverse problem, cross-entropy method

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INTRODUCTION

Elastic boundary conditions are an important subject in beam theory and structural design, where its precise parametric identification remains an ongoing challenge. The correct assessment of the stiffness values that define these supports becomes increasingly computational demanding for complex arrangements. The investigation of different numerical procedures applied to this context and the evaluation of its effectiveness are thus a relevant practical issue, since the system's vibrations are highly influenced by the end supports, especially for the lower natural frequencies [START_REF] Hagedorn | Vibrations and Waves in Continuous Mechanical Systems[END_REF]. Extensive literature has covered different kinds of methods on the determination of the stiffness coefficients for spring-hinged ends in beam dynamics, with exact solutions for particular cases [START_REF] Hibbeler | Free Vibration of a Beam Supported by Unsymmetrical Spring-Hinges[END_REF][START_REF] Lee | Exact Solutions for the Analysis of General Eiastically Restrained Nonuniform Beams[END_REF][START_REF] Albarracin | Some observations in the dynamics of beams with intermediate supports[END_REF] being generalized through approximation methods to multiple scenarios, ranging from deterministic computations with polynomial [START_REF] Elishakoff | Apparentrly the first closed-form solution of inhomogeneous elastically restrained vibrating beams[END_REF] and trigonometric [START_REF] Elishakoff | On the dynamic behaviour of slender beams with elastic ends carrying a concentrated mass[END_REF] shape functions to Bayesian strategies for updating and uncertainty quantification [START_REF] Ritto | Uncertain boundary condition Bayesian identification from experimental data: A case study on a cantilever beam[END_REF][START_REF] Hernandez | Thermorheologically simple materials: A bayesian framework for model calibration and validation[END_REF][START_REF] Hernandez | On the model building for transmission line cables: a Bayesian approach[END_REF], enhanced by global sensitivity analysis [START_REF] Yuan | Parameter selection for model updating with global sensitivity analysis[END_REF], and also treatments over nonlinear updating of the model through non-parametric methods [START_REF] Da Silva | Non-linear model updating of a three-dimensional portal frame based on Wiener series[END_REF][START_REF] Bussetta | Nonlinear updating method: a review[END_REF]. This work studies the cross-entropy (CE) method on the associated inverse problem scheme for this context, which uses model evaluations in the classic sense but with the main difference that it employs a robust global search for the optimal parameters. The CE method originated considerably recent in rare-event simulation and combinatorial optimization [START_REF] Rubinstein | Optimization of Computer Simulation Models with Rare Events[END_REF][START_REF] Rubinstein | The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning[END_REF], showing promising results and adaptability also to continuous optimization [START_REF] Rubinstein | The Cross-Entropy Method for Continuous Multi-Extremal Optimization[END_REF].

The vertical bending vibration of an experimental structure is modeled in this work as a slender beam under Euler-Bernoulli theory elastically restrained at its ends, and an inverse problem is formulated to determine the torsional stiffness coefficients that match prescribed natural frequencies to those measured in impact tests of the rotor of Figure 1, which presents an attached disc between the supports. In previous works, this experimental data was used to investigate the efficiency of direct Rayleigh's quotient iterative calculation of the frequencies [START_REF] Silva | Modelling a Rotating Shaft as an Elastically Restrained Bernoulli-Euler Beam[END_REF] and of a population-based algorithm aided by a surrogate description [START_REF] Silva | A Hybrd Procedure to identify the Optimal Sitffness Coeffcients of Elastically Restrained Beams[END_REF]. The CE method is appreciated here, observing its accuracy in the calibration and contrasting its efficiency with a genetic algorithm. The eigenfrequencies are accessed through finite element formalism of the approximated solution with polynomial mode shapes. The rest of this paper is organized as follows. Section 2 defines the theoretical model, the inverse problem and further assumptions of the approximated forward solution. The cross-entropy method is briefly exposed in section 3, where its expected interesting features are commented. The experimental data and numerical results are presented in section 3, where the performance of the CE method is examined and compared to results from a genetic algorithm. Finally, the contributions of this work are emphasized in section 5.

Modelling a Rotating Shaft as a Restrained Beam

T.A. for the mode shape of the natural frequency of interest. The developed formulation does not require numerical modelling and provides a useful tool to the design phase. The above referred papers deal with slender beams, typically with l/D > 10. As a consequence, the inertial rotation energy of an infinitesimal element dx is negligible compared to its translational energy, justifying the applicability of the Bernoulli-Euler beam theory. Sometimes, when this principle is violated, the problem has to be treated using Timoshenko beam theory. 13,14 In this study, one shall use Rayleigh's quotient, with shape functions that take into account the torsional flexibility of the supports, while assuming infinite rigidity along the vertical direction (Fig. 2). The algorithm will also include the effect of the mass and torsional inertia of the attached discs. Such an approach aims at a more precise model that more accurately represents the real dynamic behaviour of the system, rather than assuming simply supported or perfectly clamped ends. After showing that these ideal cases are far from the experimental values, the updating procedure for identifying the value of the torsional stiffness of both supports will be explained.

Theoretical Background

To obtain an analytical model th physical system (Fig. 1) one consi quotient, using the Bernoulli-Eule define the shape functions. On a ideal cases have been tried: simp and clamped (CC) beam at both e results for those ideal cases differ the experimental ones. Therefore build a model (Fig. 2) considering of the beam present different tors [elastic supports (ES)], whose valu the error between experimental an satisfies a given tolerance τ . Re method, the energy dissipation therefore the Principle of Conse holds:

(T + V ) = 0 ⇒ T ma
where T and V are the kinetic and respectively.

Through the quantification of th by the elastic forces, in relation position of the beam, the potential 2 , l is beam and θ is the rotation angle ∂w(x, t)/∂x. E(x) is the Young mo second moment of area of the beam the lateral displacement of the bea equilibrium position.

V = 1 2 l 0 M(x, t)d where M(x, t) is the bending m M(x, t) = E(x)I(x)∂ 2 w(x, t)/∂x
At each undamped natural fre variation of w(x, t) can be shown t therefore the displacement respon harmonic variation of a shape fun w(x, t) = φ(x) sin(ωt As w(x, t) max = φ(x) it follows th

V max = 1 2 l 0 E(x)I(x) φ
On the other hand, the kinetic e

T = 1 l ẇ(x, t) 2 d

MATHEMATICAL FORMULATION

Forward Problem

Consider the free vibration transversal dynamics of the Euler-Bernoulli beam, pictographically depicted in Figure 2, which evolves according

ρ A ∂ 2 w ∂t 2 + ∂ 2 ∂ x 2 E I ∂ 2 w ∂ x 2 = 0 , ( 1
)
where ρ is the mass density of the shaft, A its cross-sectional area, E the Young modulus, I the second moment of area, while w(x,t) denotes the vertical deflection, and x and t are the spatial and time coordinates. This model is supplemented with continuity conditions where the disc of mass m and inertia J is localized (position a over the beam length l), and spring-modeled torsional supports in the ends of the beam geometry, that are defined by their stiffness coefficients k t 1 (left) and k t 2 (right). Modal analysis is conducted with aid of a finite element numerical procedure, employed to solve the generalized eigenvalue problem

K φ = ω 2 M φ (2)
where ω is a natural frequency of vibration, that depends on the construction of a corresponding mode shape φ . The stiffness matrix K arises from the elastic properties of Eq. ( 2) (second term) and is modified by the springed-boundary conditions, while the mass matrix M comes from the dynamical part (first term) of the partial differential equation and is modified by the continuity conditions brought by the disc. In this work, the eigenvectors are assembled via element-wise interpolation of hermit cubic polynomials [START_REF] Reddy | An Introduction to the Finite Element Method[END_REF].

Inverse Problem

In this setting, the inverse problem underlying the stiffness identification process consist in finding the parameters x = (k t 1 , k t 2 ) that minimize the discrepancy (misfit) function

S(x) = || Ω exp -Ω(x)|| 2 + λ || x || 2 , (3)
where Ω exp and Ω are the vectors of experimental and predicted (by solving the forward problem) natural frequencies, respectively; and the regularization term, with coefficient λ and vector x of parameters, is necessary to ensure uniqueness of solution [START_REF] Hibbeler | On the multiplicity of solution of the inverse problem for vibrating beam[END_REF].

NUMERICAL PROCEDURE

The main idea of the cross-entropy method for this context consists in recasting the optimization of Eq. ( 3) as a problem of estimating the probability of a rare-event, i.e, given a performance function S(X) for the random vector of system parameters X and some level γ ≈ γ * = max S(x * ), then S(X) ≥ γ is a rare event, and the theory for importance sampling and variance minimization can be used [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF]. Given some probability density function (PDF) for the system parameters, the algorithm samples them and calculates the performance function, selecting an elite group that provides its highest values. The information is used to update the PDF and minimize the Kullback-Leibler divergence between PDF and a Dirac delta distribution centered at their optimal value, thus the procedure is repeated until the density is concentrated around the optimal parameters. The steps are summarized in Algorithm 1.

Algorithm 1 Cross-entropy method for optimization 1: Set initial PDF f (•; v) under hyperparameter vector v 0 . Set counter t = 1.

2: Generate samples of system parameters X 1 , ..., X N ∼ f (•; v 0 ). Compute the performance function S(X) for each sample and select the elite samples for the set ε t 3: Solve the stochastic program max ∑ X i ∈ε t ln f (X i ; v) to obtain v t . Increase counter by 1.

4: Repeat from step 2 until some stop criterion is met. 5: The final elite set contains an estimation of optimal system parameters.

Step 3 from Algorithm 1 has extensive literature over analytical formulas for different families of PDFs [START_REF] Kroese | A tutorial on the cross-entropy method[END_REF], which in particular are of easy implementation for exponential families. Additionally, an smoothing updating rule is usually employed when calculating v t , which can be seen in [START_REF] Costa | Convergence properties of the cross-entropy method for discrete optimization[END_REF] and has relevant impact on convergence. A common stopping criteria, known as normal updating, is to pause the algorithm when the sum of the standard deviations are smaller than some tolerance [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF]. The most interesting feature of the CE method is that there is theoretical assurance that the algorithm will find the global optimum, if only one global optimum exists [START_REF] Margolin | Free Vibration of a Beam Supported by Unsymmetrical Spring-Hinges[END_REF]. Besides, it attains easy implementation and has in general very few, intuitive and defined control parameters, namely, sample size, elite set size, tolerance and smoothing parameter, which is not a common feature for other metaheuristic procedures. In the next section, its performance on a computational point of view is verified for the problem.

RESULTS

The first and second natural frequencies for the structure in Figure 1 were obtained through impact tests and are presented under ω 1 and ω 2 respectively in the following tables. The experiment's specifications are: L = 0.362 m, E = 73 GPa, I = 1.277 × 10 -9 m 4 , ρ = 2766 kg m -3 , A = 1.2668 × 10 -4 m 2 , m = 3 kg, J = 2.8 × 10 -3 kg m 2 . The calculated first and second natural frequency are denoted ω 1 and ω 2 (converted to Hertz), and each pair with the experimental ones were used to minimize Eq. ( 3) to obtain the optimal k t 1 and k t 2 presented in Table 1 for different positions a of the disc along the beam. The number of times the eigenvalue problem had to be solved is shown under "Func. Eval.". Finally, for each case, a second solution is provided where the additional condition k t 1 = k t 2 is also enforced.

Table 1 -CROSS-ENTROPY METHOD. First and second experimental and model natural frequencies for each position of disc a, with optimal stiffness coefficients and number of function evaluations required. Values in parenthesis are the relative percent error. For each position a, the first set (top) allows k t 1 = k t 2 while the second set (bottom) requires The precision of the CE method for the given problem is attested in Table 1 for negligible errors in the fundamental frequency for all cases, showcasing its potential at estimating such quantity. While the errors for the second frequency have a relevant degree, the method was successful at attaining at least the same level of precision encompassed by previous procedures on these data [START_REF] Silva | Modelling a Rotating Shaft as an Elastically Restrained Bernoulli-Euler Beam[END_REF][START_REF] Silva | A Hybrd Procedure to identify the Optimal Sitffness Coeffcients of Elastically Restrained Beams[END_REF], allowing estimation of w 2 up to unity precision. The results also respect geometric expected behavior: same result for a = 4/8 l and close mirrored results for the symmetric cases. Comparison with the results under the additional constraint k t 1 = k t 2 reveal great gain in efficiency (less function evaluations required) but minor precision variation for ω 1 , while providing effectively no change for ω 2 . This unexpected behavior contrasts the gain in precision that was observed in the previous works using other methods when allowing the stiffness to be different, and suggests a possible lesser degree of sensitivity in the CE method to variations in ω 2 , for which further investigation in future works is required to ascertain the impact of the regularization and stopping criteria in such measure. Figure [START_REF] Lee | Exact Solutions for the Analysis of General Eiastically Restrained Nonuniform Beams[END_REF] illustrates the mode shapes obtained in the formulation for a = 4/8 l up to 5th, using hermit cubic interpolation. Table 2 compiles the calibration solution for each case using the standard Genetic Algorithm (GA) from MATLAB. Population size and elite group of GA were set equal to the sample size and elite set size of CE method in each instance, to allow comparison between the method efficiency. It is clear how the global search method is way faster for obtaining the solutions with increased precision by noticing the difference in thousands from the required number of evaluations. Even so, the few hundred evaluations required in the CE method may be computational costly for more complex problems settings, which advocates research of allowable modifications in the CE method that have been known to increase the rate of convergence, as dynamic smoothing and Gaussian mixture update for example [START_REF] Rubinstein | The Cross-Entropy Method for Continuous Multi-Extremal Optimization[END_REF][START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF], in the specific context of structural design calibration, as well as the application of metamodels to alleviate computational cost, as proposed in the framework of [START_REF] Silva | A Hybrd Procedure to identify the Optimal Sitffness Coeffcients of Elastically Restrained Beams[END_REF].

k t 1 = k t 2 . a (m) ω 1 2π | exp (Hz) ω 2 2π | exp (Hz) ω 1 2π (Hz) ω 2 2π (Hz) k t 1 ( N m rad ) k t 2 ( N m rad ) Func. Eval.

CONCLUDING REMARKS

The cross-entropy method is applied on the calibration of a Euler-Bernoulli beam vertical deflection, assessing its efficiency on prediction of torsional stiffness coefficients for the end supports and comparing predicted natural frequencies with values obtained experimentally. The results show remarkable precision for the first frequency and sustainable estimations of the second at low cost of function evaluations, at least when compared with Genetic Algorithm solutions of the inverse problem, which are also presented, suggesting a promising numerical procedure for this context of structural parameters identification that may be enhanced further. Considerations about previous works that used different numerical methods for the same set of data were also commented. In future works, the authors intend to implement modifications in the Cross-entropy method to evaluate the rate of convergence, pair the procedure with surrogate modeling and ascertain the model efficiency under different regularization schemes.

Figure 1

 1 Figure 1 System under study (MFS, SpectraQuest, Inc.).

Figure 1 -

 1 Figure 1 -Experimental structure (MFS, SpectraQuest, Inc.).

Figure 2 -

 2 Figure 2 -Euler-Bernoulli beam with torsional spring end supports and rigid disc.

Figure 3 -

 3 Figure 3 -First to fifth mode shape for the case a = 4/8 l.

Table 2 -

 2 GENETIC ALGORITHM. First and second experimental and model natural frequencies for each position of disc a, with optimal stiffness coefficients and number of function evaluations required. Values in parenthesis are the relative percent error.

	a (m) ω 1 2π | exp (Hz) ω 2 2π | exp (Hz)	ω 1 2π (Hz)	ω 2 2π (Hz)	k t 1 ( N m rad ) k t 2 ( N m rad ) Func. Eval.
	2 8 l	45.454	218.327	45.4537 (5.8E-4 %) (8.6E+0 %) 199.5403	533.60	317.79	19 710
	3 8 l	39.764	187.378	39.7664 (5.9E-3 %) (5.0E+0 %) 178.0941	800.13	617.31	49 620
	4 8 l	38.907	178.156	38.9088 (4.7E-3 %) (2.5E+0 %) 173.6608	935.28	935.15	14 115
	5 8 l	39.833	188.724	39.8814 (1.2E-1 %) (5.6E+0 %) 178.1866	614.73	823.65	19 200
	6 8 l	45.918	211.980	45.8267 (2.0E-1 %) (5.8E+0 %) 199.7872	332.39	562.80	35 940

ACKNOWLEDGEMENTS

The first two authors acknowledge the support given by the funding agencies Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grants E-26/010.002.178/2015 and E-26/010.000.805/2018, and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001. The third author acknowledge the support of the Portuguese Foundation for Science and Technology under the project UNIDEMI Pest-OE/EME/UI0667/2014. The second author is very grateful to the Department of Mechanical and Industrial Engineering of Universidade NOVA de Lisboa, for the great hospitality during his visit in June 2018.


M Hilairet
email: mickael.hilairet@univ-fcomte.fr

M Ghanes

O Béthoux

V Tanasa

J-P Barbot

D Normand-Cyrot

A Passivity-Based Controller for coordination of converters in a Fuel Cell System

Keywords: Fuel cell, supercapacitors, power management, port-controlled Hamiltonian systems, IDA-PBC methodology, experimentation

The problem of converters coordination of a fuel cell system involving a hydrogen fuel cell with supercapacitors for applications with high instantaneous dynamic power is addressed in this paper. The problem is solved by using a non-linear controller based on passivity. The controller design is based on the interconnection and damping assignment approach, where the proof of the local system stability of the whole closed-loop system is shown. Simulation and experimental results on a reduced scale system prove the feasibility of the proposed approach for a real electrical vehicle.

Introduction

To comply with environmental norms, the development of electric and hybrid vehicles has increased since 2009. In this context, the development of a fuel cell (FC) system as the main source of energy, is considered due to the noise reduction, high efficiency, low weight, compact size, modularity and controllability. However, this technology has some weak points, such as cost, reliability and durability. Specifically, to ensure a good health state of the FC, it is necessary for the FC to deliver a slowly varying current, i.e. a current slope lower than 4 A/s for a 0.5 kW/12.5 V FC [START_REF] Thounthong | Energy management of fuel cell/battery/supercapacitor hybrid power source for vehicle applications[END_REF], and 10 A/s for a 20 kW/48 V FC (Corbo, Migliardinia and Veneri, 2009a) as examples. Thus, an FC needs to be associated with other sources which supply short pulse energy and fill the temporary failure of the FC [START_REF] Hissel | A review on existing modeling methodologies for PEM fuel cell systems[END_REF]. Nowadays, these auxiliary sources can either be batteries or supercapacitors (SCs) Sometimes, batteries are not able to bear high power charge and discharge conditions, whereas supercapacitors have a high power range.

Therefore, for fast power demands, supercapacitors are probably the best-suited components [START_REF] Rodatz | Optimal power management of an experimental fuel cell/supercapacitor-powered hybrid vehicle[END_REF].

In this paper, the challenging problem of the power management of an hydrogen FC system associated to a reversible impulse energy source (the supercapacitors) is considered and involves both practical and theoretical issues. There are several electric architectures of the hybrid system, which can be classified into three categories : series, cascaded and parallel [START_REF] Jiang | Design and experimental tests of control strategies for active hybrid fuel cell/battery power sources[END_REF][START_REF] Cacciato | A critical evaluation and design of bi-directional DC/DC converters for supercapacitors interfacing in fuel cell applications[END_REF]. The literature has shown that the parallel architectures are the most suitable ones.

The parallel structures with only one converter [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF][START_REF] Davat | Fuel cell-based hybrid systems[END_REF] or two converters [START_REF] Davat | Fuel cell-based hybrid systems[END_REF] can fully respect the mentioned requirements. This paper is dedicated to the study of the structure with two converters as shown in Fig. 1.

Nowadays, high-performance and efficiency controllers are readily available [START_REF] Vahidi | Current management in a hybrid fuel cell power system : a model-predictive control approach[END_REF][START_REF] Thounthong | Energy management of fuel cell/battery/supercapacitor hybrid power source for vehicle applications[END_REF][START_REF] Thounthong | Fuel cell high-power applications[END_REF]Azib et al., 2009a,b;[START_REF] Arce | MPC for battery/fuel cell hybrid vehicles including fuel cell dynamics and bettery performance improvement[END_REF]. These allow to the current, delivered by the FC, to have smooth behavior in order to ensure its life time, while the SCs provide the load power transient [START_REF] Arce | MPC for battery/fuel cell hybrid vehicles including fuel cell dynamics and bettery performance improvement[END_REF][START_REF] Thounthong | Fuel cell high-power applications[END_REF].

Unfortunately, the closed-loop system stability of these controllers are generally not proved theoretically, although they are effective.

Therefore, this drawback opens a theoretically challenging problem. In this work, a non-linear controller based on the Interconnection and Damping Assignment -Passivity

Based Control (IDA-PBC) has been studied in order to prove the local asymptotic stability of the whole closed-loop system, while maintaining the same objectives and components security as standard controllers, i.e. the controller has to sustain the bus voltage and the supercapacitors voltage at desired levels without compromising the FC operation as the fuel starvation during fast load change which refers to an operation with an insufficient amount of gas in the active layer (Yousfi-Steiner et al., 2009).

This paper is divided into four sections as follows: section two describes some standard power management controllers based on a frequency decoupling of the sources. The proposed passivity-based controller is detailed in section three; in section four simulation results are presented. Finally, the approach has been applied to a reduced scaled test bench system based on the Nexa Ballard fuel cell. Furthermore, section V deals with the implementation and experimental results.

Power management

The power management must comply with the load power demands and to provide an effective fuel cell control while decreasing the fuel consumption. Also it has to prevent the fuel starvation during fast load demands, to maintain the DC bus and state of charge of the supercapacitors constant and to ensure the proper use of each component.

The main objective of the control strategies is to regulate the DC bus voltage with the FC as the primary power source [START_REF] Davat | Fuel cell-based hybrid systems[END_REF]. However, fuel cell performances (efficiency, degradation, aging effects) are influenced by many environmental and application constraints [START_REF] Wahdame | Comparison between two PEM fuel cell durability tests performed at constant current under solicitations linked to transport mission profile[END_REF]. These aging tests prove that limiting the load dynamic effects can save the FC performances and raise its durability. Therefore, it seems clear that the DC bus regulation has to be managed by the supercapacitors. A short survey reveals a significant number of strategies, like the one based on the state-feedback [START_REF] Jiang | Design and experimental tests of control strategies for active hybrid fuel cell/battery power sources[END_REF], fuzzy logic [START_REF] Kisacikoglu | Load sharing using fuzzy logic control in a fuel cell/ultracapacitor hybrid vehicule[END_REF][START_REF] Martinez | Practical Control Structure and Energy Management of a Testbed Hybrid Electric Vehicle[END_REF], proportional-integral controllers [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF], RST controller [START_REF] Caux | Modelling and control of a Fuel Cell System and Storage Elements in transport applications[END_REF], passivity [START_REF] Becherif | Passivity-based control of hybrid sources : fuel cell and battery[END_REF], flatness [START_REF] Zandi | Energy Management of a Fuel Cell/Supercapacitor/Battery Power Source for Electric Vehicular Applications[END_REF] or model predictive control [START_REF] Vahidi | Current management in a hybrid fuel cell power system : a model-predictive control approach[END_REF]. Alternative approach exist such as optimal control [START_REF] Rodatz | Optimal power management of an experimental fuel cell/supercapacitor-powered hybrid vehicle[END_REF], dynamic programming [START_REF] Brahma | Optimal energy management in series hybrid electric vehicles[END_REF] or empirical control associated with a multi objective genetic algorithm optimization [START_REF] Paladini | Super-capacitors fuel-cell hybrid electric vehicle optimization and control strategy development[END_REF] that has been applied

for the supervisory power train control problem in charge sustaining hybrid electric vehicles.

However, these approaches are based on the a priori knowledge of the power load, thus real-time control is not straightforward implementable.

In [START_REF] Azib | Saturation Management of a Controlled Fuel-Cell/Ultracapacitor Hybrid Vehicle[END_REF], a two converters structure control strategy has been detailed. It relies on the control of the converter in such a way to split the demanded power between the FC and SCs. The converter-parameter tuning is based on a frequency decoupling so that to coordinate the two sources without compromising the FC operation. The DC bus capacitor filters the high frequencies (i.e. above the kHz), the SCs associated with their converters provide the medium frequencies (from 1 kHz to 1 Hz), and the FC ensures the low frequencies (less than 1 Hz). This frequency decoupling of the sources naturally induces a power management strategy based on cascaded loops and the control is effective [START_REF] Azib | Saturation Management of a Controlled Fuel-Cell/Ultracapacitor Hybrid Vehicle[END_REF]. The gains are tuned to ensure the closed-loop system stability, although it has not been theoretically proved. Therefore, this drawback seems to be a theoretically challenging problem, while maintaining the same objectives and the component security. Therefore, in this work a passivity-based controller, which relies on the well-known IDA-PBC method [START_REF] Ortega | Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems[END_REF][START_REF] Ortega | Interconnection and damping assignment passivity-based control : a survey[END_REF][START_REF] Ortega | Control by interconnection and standard passivity-based control of Port-Hamiltonian systems[END_REF], has been studied in order to prove the asymptotic stability of the outer closed-loop system and finally the local asymptotic stability of the whole system.

In [START_REF] Becherif | Passivity-based control of hybrid sources : fuel cell and battery[END_REF] a full order IDA-PBC has been designed for a similar system. However, currents i fc and i sc can exceed the maximum value allowed, because they are not directly controlled. This point is generally mandatory for industrial applications; it is the reason why the strategy proposed in this paper comprises two loops as shown in Fig.

(2). To be more precise, there are two inner current loop controllers for the FC and SCs respectively, based on PI controllers and only one outer loop which controls the DC bus voltage and state of charge of the SCs. In this work, the outer-loop controller is based on passivity approach.

Passivity-based controller

Port controlled Hamiltonian system

The PBC defines a controller design methodology that stabilizes the system by making it passive. Although there are many variations on this basic idea, the PBC can be broadly classified into two major groups. In the "regular" PBC, the designer chooses the storage function (usually quadratic), and then designs the controller that makes the storage function non-increasing [START_REF] Cecati | A passivity-based multilevel active rectifier with adaptive compensation for traction applications[END_REF]. In the second PBC methodology, the storage function of the closed-loop system remains free. The designer selects a control structure, such as Lagrangian, port-controller Hamiltonian (PCH) or Brayton-Moser formulation [START_REF] Weiss | A generalization of Brayton-Moser's mixed potential function[END_REF][START_REF] Jeltsema | A dual relation between port-Hamiltonian systems and the Brayton-Moser equations for nonlinear switched RLC circuits[END_REF][START_REF] Zhou | Passivity-Based Control for an Interleaved Current-Fed Full-Bridge Converter With a Wide Operating Range Using the Brayton-Moser Form[END_REF], and then, characterizes all assignable energy or power functions. The most notable examples of this approach are the controlled Lagrangian systems, and the IDA-PBC ( [START_REF] Van Der Schaft | L2-Gain and Passivity Techniques in Nonlinear Control[END_REF][START_REF] Ortega | Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems[END_REF][START_REF] Ortega | Interconnection and damping assignment passivity-based control : a survey[END_REF][START_REF] Ortega | Control by interconnection and standard passivity-based control of Port-Hamiltonian systems[END_REF]. It is the latter method that has been chosen in this work.

Firstly, the IDA-PBC approach consists in identifying the natural energy function of the system called H(x). Rewriting a non-linear system:

ẋ = f (x) + g(x)u; x ∈ n ; u ∈ m y = h(x); y ∈ m
versus the gradient of the energy function:

∇H(x) = ∂H ∂x 1 (x) ∂H ∂x 2 (x) ... ∂H ∂xn (x)
T leads to PCH form as follows:

ẋ = [J (x) -R(x)]H(x) + g(x)u y = g T (x)∇H(x)
where y is the output, J (x) = -J T (x) is a skew-symmetric matrix of dimension n × n representing the interconnections between states, and R(x) = R T (x) ≥ 0 is a positive semi-definite symmetric matrix representing the natural damping of the system.

The IDA-PBC Methodology

Let us consider the system [START_REF] Ortega | Interconnection and damping assignment passivity-based control : a survey[END_REF][START_REF] Ortega | Control by interconnection and standard passivity-based control of Port-Hamiltonian systems[END_REF])

ẋ = f (x) + g(x)u (1)
and assume there are matrices

J d (x) = -J T d (x), R d (x) = R T d (x) ≥ 0 and a function H d (x) : n -→ so that the closed-loop system (1) with control variable u u = g T (x)g(x) -1 g T (x) {[J d (x) -R d (x)] ∇H d -f (x)} takes the PCH form ẋ = [J d (x) -R d (x)]∇H d (2) H d (x) is such that x * = argmin x∈ n (H d (x)
) with x * ∈ n the (locally) equilibrium to be stabilized. The system is asymptotically stable if, in addition, x * is an isolated minimum of H d (x) and if the largest invariant set under the closed-loop dynamics (2) contained in

x ∈ n | [∇H d ] T R d (x)∇H d = 0 equals x * .
The stability of x * is established noting that, along the trajectories of (2), we have

Ḣd = -[∇H d ] T R d (x)∇H d ≤ 0
Hence, H d (x) is qualified as a Lyapunov function. Asymptotic stability immediately follows invoking the La Salle's invariance principle [START_REF] Lasalle | Some extensions of Lyapunov's second method[END_REF]. Finally, to ensure that the solutions remain bounded, we give the estimate of the field of attraction as the largest bounded level set of H d (x).

Hybrid system modeling

Fuel cell modeling. The model used is a static model [START_REF] Pukrushpan | control-oriented modeling and analysis for automotive fuel cell systems[END_REF] where the FC voltage v fc is computed according to the current stack i fc by a 5 th order polynomial function as shown in Fig.

(3). The data fitting has been obtained according to experimental results.

SCs boost converter. SCs can be charged or discharged; therefore the storage elements are connected to the DC bus through a reversible power converter as shown in Fig. ( 4). The boost converter is controlled by binary input w 2 (t). We define α 2 as the duty cycle of the control variable w 2 (t). The second sub-system is represented by an average model as follows :

d dt i sc (t) = 1 Lsc -(1 -α 2 (t)) v b (t) + v sc (t) d dt v sc (t) = -isc(t)
Csc FC boost converter, DC bus and load model. To use the FC in an electric power system, a boost converter must increase the FC voltage, because the FC voltage is often less than the DC bus voltage. The boost converter represented in Fig.( 4) is controlled by binary input w 1 (t). Defining α 1 as the duty cycle of control variable w 1 (t), this subsystem can be represented by its average model (here, the switches are regarded as ideal) :

d dt i fc = 1 L f c -(1 -α 1 (t)) v b (t) + v fc (t) d dt v b (t) = 1 C (1 -α 1 (t)) i fc (t) + (1 -α 2 (t)) i sc (t) -i l (t)
where v b (t) is the DC link voltage, v fc (t) is the FC voltage, i l (t) is the DC current delivered to the load and i fc (t) is the FC current.

In our work, the load is modeled by a variable resistance circuit (R l (t)), whose value varies according to the power required by the load. The average model is:

d dt i l (t) = 1 L -R l (t)i l (t) + v b (t)
where inductance L is not part of the load and represents the imperfections of the system.

The load model could have been replaced by a current source i l (t) and the same approach described later could be adopted (see appendix 3).

Complete model. It follows that the complete "fuel cell -supercapacitors" system is represented by the 5 th order non-linear state space model :

d dt v b (t) = (1 -α 1 (t)) i fc (t) + (1 -α 2 (t)) i sc (t) -i l (t) C (3) d dt v sc (t) = - i sc (t) C sc (4) d dt i l (t) = -R l (t) i l (t) + v b (t) L (5) d dt i fc (t) = -(1 -α 1 (t)) v b (t) + v fc (t) L fc (6) d dt i sc (t) = -(1 -α 2 (t)) v b (t) + v sc (t) L sc (7) with state space x(t) = [v b ; v sc ; i l ; i fc ; i sc ] T , control inputs u(t) = [u 1 ; u 2 ] T = [1 -α 1 ; 1 - α 2 ] T , measures y(t) = x and v fc .
Outer loop model (reduced model). The system of 5 equations (3 to 7) is called a singular perturbed system, because of the difference of time scale between the voltages and the currents (Kokotovi et al., 1986). Therefore, the system (3 to 7) is forced into current-controlled mode using a fast inner current loop. More precisely, the following PI current controllers associated with a anti-windup scheme

u 1 = Ki fc t 0 (i * fc -i fc )dt + Kp fc (i * fc -i fc ) ( 8 ) u 2 = Ki sc t 0 (i * sc -i sc )dt + Kp sc (i * sc -i sc ) ( 9 )
are used to force i fc and i sc to track their respective references i * fc and i * sc and produce fast responses when large feedback gains are used. The control u 1 and u 2 act as high-gain feedback, for more details see for example [START_REF] Marino | High-gain feedback in non-linear control system[END_REF]. Consider ( 6) and ( 8) for Ki fc and Kp fc sufficiently large with respect to voltage and load dynamics. After transient (convergence), one get i fci * fc = 0 and

t 0 (i * fc -i fc ) = v fc v b Ki fc . These imply that (8), after transient, becomes u 1 = v f c v b .
The same argument is used for ( 7) and ( 9), where after transient, one get u 2 = vsc v b . Consequently after transient, by replacing the new obtained u 1 and u 2 in (3), and currents i fc -i sc by their references i * fc -i * sc in (3) and (4) as a new inputs, it follows that

d dt v b (t) = 1 C v f c (t) v b (t) i * fc (t) + vsc(t) v b (t) i * sc (t) -i l (t) d dt v sc (t) = -i * sc (t) Csc d dt i l (t) = -R l (t) i l (t)+v b (t) L (10) with x r (t) = [x 1 ; x 2 ; x 3 ] T = [v b ; v sc ; i l ] T , control inputs u r = [i * fc ; i * sc ] T , measures y r and z r as y r = [v b ; v sc ; i l ] T and z r = [i fc ; i sc ; v fc ] T .
Remark 1: In the sequel, the outer closed-loop control is designed by using the model ( 10) such that its dynamic is slower than the dynamic of the PI fast actuators (8-9).

IDA-PBC outer loop controller design

The main objective of IDA-PBC is to assign the state point

x r = [x 1 ; x 2 ; x 3 ] T = [v b ; v sc ; i l ] T to the desired equilibrium one x * r = [v * b ; v * sc ; v * b R l ],
with v * b and v * sc the DC bus and SCs desired voltages, by tacking into account the following constraint and protection:

• Constraint 1 : the FC has to prevent stack stresses during power transients.

• Protection 1 : the FC voltage v fc has to be no less than a minimum value v fcmin .

According to section (3.2), the IDA-PBC methodology looks for an energy function H d so that its minimum is reached at the desired equilibrium point x * r . This energy function H d is chosen as

H d = 1 2 xr T Q xr with xr = x r -x * r and Q = diag(C, C sc , L).
In these circumstances, writing the PCH system in terms of the dynamics of the error and the gradient of desired closed-loop energy function ∇H d is:

ẋr = [J -R]∇H d + A(u r , x r , x * r , z r ) (11) with J -R = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 -1 L C 0 0 0 1 L C 0 -R l L 2 ⎤ ⎥ ⎥ ⎥ ⎦ , ∇H d = ⎡ ⎢ ⎢ ⎢ ⎣ C ṽb C sc ṽsc L ĩl ⎤ ⎥ ⎥ ⎥ ⎦ A t = 1 C ( v f c v b i * fc + vsc v b i * sc -i * l ); -1 Csc i * sc ; 0 Solving the algebraic equation in J d (x) and R d (x)
with the constraint of skew-symmetry and positive semi-definiteness of J d (x) and R d (x) respectively, with the two unknown matrices equal to:

J d = ⎡ ⎢ ⎢ ⎢ ⎣ 0 J 12 J 13 -J 12 0 J 23 -J 13 -J 23 0 ⎤ ⎥ ⎥ ⎥ ⎦ , R d = ⎡ ⎢ ⎢ ⎢ ⎣ r 1 0 0 0 r 2 0 0 0 r 3 ⎤ ⎥ ⎥ ⎥ ⎦
leads to the matching equations:

-

r 1 C ṽb + J 12 C sc ṽsc + J 13 L ĩl = - 1 C ĩl + 1 C v fc v b i * fc + v sc v b i * sc -i * l -J 12 C ṽb -r 2 C sc ṽsc + J 23 L ĩl = - 1 C sc i * sc -J 13 C ṽb -J 23 C sc ṽsc -r 3 L ĩl = 1 L ṽb - R l L ĩl One solution is r 3 = R l L 2 l
, J 13 = -1 CL , and J 23 = 0 with r 1 > 0 and J 12 < 0, which leads to the non-linear control law:

i * fc = v b max{v fc , v fcmin } v * b R l + C sc (CJ 12 - v sc v b C sc r 2 ) ṽsc -C( v sc v b C sc J 12 + r 1 C)ṽ b (12) i * sc = C sc (CJ 12 ṽb + r 2 C sc ṽsc ) ( 13
)
so that the closed-loop system responds to the following dynamics:

ẋr = [J d -R d ]∇H d ( 14
)
with

J d = ⎡ ⎢ ⎢ ⎢ ⎣ 0 J 12 -1 L C -J 12 0 0 1 L C 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ , R d = ⎡ ⎢ ⎢ ⎢ ⎣ r 1 0 0 0 r 2 0 0 0 R l L 2 ⎤ ⎥ ⎥ ⎥ ⎦
The analysis of the control ( 12)-( 13) shows that SCs supply energy due to an error on the DC bus voltage; the error itself is caused by power spikes or a variation of the DC bus voltage reference. The desired FC current (i * fc ) shows that the FC supply satisfies two main objectives :

• the permanent power flow from the FC to the load,

• the energy contribution to regulate the SCs voltage.

Obviously, it does not seem judicious that the FC current participates in the control of the DC bus voltage according to constraint 1. So, tuning parameters r 1 and J 12 are set equal to x 2 α x 1 C 2 and -α CCsc with α > 0, such that the right hand side of ( 12) is canceled.

Moreover, in order that the ultra-capacitors managed only the DC bus voltage as the control proposed in [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF], the tuning parameter r 2 is set equal to zero. The control law is now as follows:

i * fc = v b max{v fc , v fcmin } v * b R l -α ṽsc ; α > 0 (15) i * sc = -α ṽb (16)
In practice the fuel cell (FC) voltage v fc is always positive and does not reach zero voltage for safety conditions. Moreover, controllers ( 12) and ( 15) are not singular for v fc equal to zero thanks to protection 1.

The proof of the global asymptotic stability of the outer loop (voltage control) is deduced from the derivative analysis of

H d equal to ∇H T d ẋ = -∇H T d R d ∇H T d ≤ 0
and the invariance principle of the LaSalle theorem with H d (x * ) = Ḣd (x * ) = 0. Moreover, H d is radially unbounded; therefore the outer closed-loop system is globally asymptotically stable.

Finally, following high-gain feedback methodology [START_REF] Marino | High-gain feedback in non-linear control system[END_REF], with an appropriate (sufficiently large) choice of the gains Ki fc , Kp fc , Ki fsc and Kp sc in (8) and ( 9), the variable

η 1 (i * fc -i fc ), η 2 i * fc -i fc , η 3 (i * sc -i sc )
, and η 4 i * sci sc are fast states, with respect to others system states, and converge very rapidly on the so-called invariant-manifold [START_REF] Vasil'eva | Asymptotic behavior of solutions to incertain problems involving nonlinear differential equations containing a small parameter multipling the highest derivatives[END_REF]. As the behavior on the so-called boundary layer is exponentially stable (i.e. in this case the fast dynamic η is linear) according to the Tikhonov's theorem [START_REF] Tikhonov | Ordinary differential equations[END_REF], we conclude that the whole system is locally asymptotically stable.

Simulation results

(15) shows that for the implementation of the proposed controller, the knowledge of the load resistance (R l ) is needed for the computation of the FC reference current. To explain the design procedure, the case for which the load resistance (R l ) is unknown is first considered. In a second case step, a load resistance estimator scheme or an integral action are added in order to consider the load variation.

Remark 2: In a practical application, when the controller is implemented by a computer, the system is placed in a sampled-data context. Consequently, the passivity based controller has been simulated and implemented through a zero order holder device (emulation process) with a sampling-time equal to 500 μs. To cope with this problem, two solutions are explored. In the first one, an estimate of the load resistance is added to the command value i * fc , while in the second approach, a low integrator action eliminates this error.

The case of a unknown parameter

IDA-PBC controller + load resistance estimator

In this paragraph an estimator of the load impedance Y l = 1/R l is considered to deal with this problem, as follow:

Ŷl = K Rl s + K Rl . i lk v bk ( 17
)
where the tuning parameter K Rl control the sensibility of the fuel cell current reference. 6) shows the estimator behavior. In this application, K Rl has been selected in order to obtain a slow time response of about 6 s, so that the FC current reference reacts smoothly. Remark 4: The proof of the global stability of the system composed of the controller, the estimator and electrical sub-system (10) invoking a theorem on stability of cascaded systems stated in [START_REF] Panteley | On global uniform asymptotic stability of nonlinear time varying systems in cascade[END_REF]) can be found in appendix 2.

IDA-PBC controller + integral action

The controller design supposes that the converters are loss-less. So in practice, a low integrator action needs to be added to the passivity controller in order to ensure zero SCs voltage error at steady state and to counteract the unknown load resistance consequences [START_REF] Donaire | On the addition of integral action to port-controlled Hamiltonian systems[END_REF]. The controller equations are now: ui = -γ ṽsc ; γ > 0 (18) The tuning of non-linear controllers such as PBC is not obvious and trivial. To analyze the influence of the tuning parameters on the closed-loop system, more specifically on the FC current dynamics, some simulations have been done. In practice, increasing gamma leads to an under-damping closed-loop system, while increasing alpha gives for the FC current bigger slopes. After trial and error loops, a reasonable choice for (α, γ) is (10, 460). Instead of a manual tuning that not ensures an efficient control, IDA-PBC and loop optimization software could be used to ensure consistent results.

i * fc = v b max{v fc , v fcmin } v * b R l -α ṽsc + Cu i ; α > 0 (19) i * sc = -α ṽb (
Remark 3: All the stability properties of x * are preserved by adding to the IDA control an integral term. Here the proof is omitted due to the lack of place and can be found in the appendix 2.

Experimental results

Test bench description

The hybrid test bench is presented in Fig. 9. to the DC bus using a chopper built with standard MOS modules and a switching frequency of the PWM set to 20 kHz [START_REF] Azib | An innovative control strategy of a single converter for hybrid fuel cell/supercapacitors power source[END_REF].

The hybrid power source is connected to a programmable electronic load (Höcherl & Hackl, model ZS1806), which has a rated power of 1800 W (imax=150 A/Vmax=60 V). This load emulates vehicle power consumption, and is directly monitored by the dSPACE DS1103 real-time board. Finally, table 1 summarize the electric characteristics of the on-board power sources.

The current inner control loops, which generate the duty cycle α 1 and α 2 , have been implemented with digital PI controllers updated at 20 kHz. The voltage outer control loops have a sampling time equal to 2 KHz.

Result analysis

Experiments have been performed on the experimental setup to validate the previously explained control strategies. The reference DC bus voltage is set equal to 50 V, and the load current varies between 0 and 15 A (this is equivalent to a variation of the load admittance 1/R l from 0 to 0.3 S).

The case of an unknown parameter. The SCs voltage is however not regulated to the reference value equal to 21 V, and depends on the load power since the IDA-PBC controller assumes the load resistance as constant (here, the arbitrary admittance used in the controller equation is equal to 5 A/50 V= 0.1 S). Under these conditions, SCs provide too much energy during the power transition and SCs recharging is uncertain.

IDA-PBC controller + load resistance estimator. The latter experiment show that the SCs do not recover their equilibrium points while the load current increases, because of the inadequate value of the load resistance used in the controller. To overcome this problem, two solutions have been explored. First, the admittance (Y l = 1/R l ) of the load is estimated on-line according to Equ. (17).

Fig. 11 shows the whole system behavior where the load resistance estimate is used in the controller. This controller architecture also leads to a nearly zero static error of the SCs voltage. However, we can note that the SC voltage is not perfectly equal to its reference at steady state. This is due to the FC converter losses. Nevertheless, a low integral action or a converter-losses estimation could be added in a practical application to compensate for the converter losses. Finally, simulation and experimental result show that the passivity-based controller and PI controller have almost the same performance. However, the new control law based on passivity ensures a locally asymptotic stability of the whole closed-loop system that is not demonstrated for PI controllers and these latter have 4 tuning parameters (Kp fc , Ki fc , Kp sc , Ki sc ) compared to only 2 tuning parameters ((α, γ) or (α, K Rl )) for the passivity-based controller, which makes the implementation of the IDA-PBC on a real-time system easier.

IDA-PBC controller

Conclusion

In this paper, a new control strategy to manage the energy between two power sources, namely a hydrogen fuel cell and supercapacitors has been discussed. This new control law based on passivity ensures a locally asymptotic stability of the whole closed-loop system, while reducing the load stress on the stack power transients. In addition, this outer voltage controller has only 2 tuning parameters ((α, γ) or (α, K Rl )), which makes the implementation on a real-time system easier.

As the controller needs the information on the load resistance, the paper has proposed two alternative solutions : to add an integral action or a load resistance estimator. In both cases the FC dynamic can be easily tuned while the SCs state of charge is well regulated in steady state. Simulation and experimental results are consistent, and the controller performances validate the proposal.

As mentioned previously in section II, passivity based controllers have been proposed for similar systems where the converter is directly control. The lack of a separate current control loop makes it difficult to achieve current limitation which is mandatory in industrial applications for hardware protection. It means that the current is measured but not controlled. The proposed PBC with cascaded loops protects the sources, the converters and the load as regular controllers. Finally, the controller leads to a general non-linear PI controller that extends the theory with regular PI controllers and gives confidence in the stability with almost the same experimental performances as regular controller.

It is difficult for the fuel flow to follow the current steps, which decreases the lifespan of the FC. Therefore, synchronization between the FC controller, the FC converter and the SC converter is mandatory. In order to extend this work, a future study could investigate the introduction of a more complicated modelling of the FC, i.e. adding the air compressor dynamic and studing its impact on the controller design and system performances. It follows that local or global design control of each components needs further investigations.

Finally, the parallelism of N-sources leads to a redundancy and therefore improves the reliability and efficiency of the whole system (Malaizé and Dib, 2011;[START_REF] De Bernardinis | Multi-port power converter for segmented PEM fuel cell in transport application[END_REF]. Challenging control issues could investigate the generalization of this works to N-parallel connected sources with different or same characteristics leading to switchingcontroller according the state of charge (SoC) and state of health (SoH) of each source.

The overall error dynamics is a cascade composition like the ones studied in [ [START_REF] Panteley | On global uniform asymptotic stability of nonlinear time varying systems in cascade[END_REF]

∂H d ∂x (x) x ≤ c 1 H d (x), ∀ x ≥ η ∂H d ∂x (x) ≤ c 2 , ∀ x ≤ η
where c 1 , c 2 , η > 0. This is condition (A.1) of [ [START_REF] Panteley | On global uniform asymptotic stability of nonlinear time varying systems in cascade[END_REF],Th.1]. Second, from inspection of the definitions of ϕ(x) above, and the fact that Ỹl is bounded, then the interconnection term satisfies the bound ϕ(x) ≤ c 3 for c 3 > 0, as required by condition (A.2). Finally, the last condition of the theorem, requiring that the second subsystem in (21) be globally uniformaly asymptotically stable and that its response to initial condition be absolutely integrable, is satisfied since the subsystem ( 17) is asymptotically stable. This completes the proof of our proposition.

Appendix 3:

In our work, the load has been modeled by a resistance circuit. However, without loss of generality, is it possible to consider a current disturbance i l (t) = P (t)/v b (t) that lead to the controller [START_REF] Konig | Model predictive control of a DC-DC converter for battery emulation[END_REF]:

i * fc = v b max{v fc , v fcmin } îl -α ṽsc ; α > 0 (22) i * sc = -α ṽb ( 23
)
where îl is the output of a low-pass filter with measurement i l as input. The low-pass filter has the same objective as the load estimator. It is here to smooth the FC current and avoid peak FC current if the measured load current has been used in controller (22).

b i sc L sc w 2 (t) T 1 T 2 v sc C sc DC/DC v fc i fc L fc w 1 (t) C R l L i sc i l load v fc i fc

Figure 1 :

 1 Figure 1: Two converters parallel structure studied in this work.

Fig

  Fig. (5) represents a scenario where the reference DC bus voltage is set equal to 50 V and the load current varies between 0 and 15 A. This power cycle is representative of a reduced-scale vehicle power demand, where the load requirement consists in raising and lowering power edges between 0 and 750 W. Here, the load resistance (R l ) is considered as a fixed parameter (the arbitrary admittance used in the controller equation is equal to 5 A/50 V= 0.1 S). The control strategy provides an insufficient FC current reference during the time interval [21, 101] s and consequently, the SCs provide most of the power during the high power transient and do not recover their equilibrium points, despite the fact that the FC current transient is good.

Fig. (

  Fig. (6) shows the estimator behavior. In this application, K Rl has been selected in order to

Fig. ( 7

 7 Fig. (7) depicts the global system performances. In this simulation the load resistance estimate is used in the controller. With the former estimator parameter set (K Rl = 0.5), the FC current variation is less than 4 A/s. It indeed respects the FC specifications. This controller architecture also leads to a nearly zero static error of the SCs voltage without adding any integral action. Nevertheless, a low integral action needs to be added in a practical application to compensate for the converter losses.

  Fig. (8) shows the system response. It shows that the DC bus and SC voltages reach the desired equilibrium point. Moreover, this controller allows the FC to have a smooth response during fast power demand of the load (8.e), which improves the state of health of the FC.

  Fig. (10.a) shows that the control ensures perfect control of the DC bus voltage, the SCs respond rapidly to fast load current transients in order to provide most of the power required by the load and to maintain the DC bus voltage at its reference value. This allows the FC to have a smooth response during fast power demand of the load (Fig.10.e), which improves his state of health. Then gradually with the FC current increasing, the SCs discharge, characterized by the decrease of its voltage, vanishes to zero (see Fig.10.e).

  + integral action. The second experiment shown in Fig. (12) was carried out to validate the proposed strategy with an integral action. Note that the DC bus and SC voltages are well regulated in spite of the very fast dynamics of power demand. Each time the power load varies, SC current is positive (respectively negative) during an increase (respectively decrease) of the power load. In such a situation, the SC voltage continuously fluctuates around its constant reference value v * sc set to 21 V, as shown in Fig. (12.b). The experimental results confirm that the association of the FC and the SCs mitigates the FC current transient in order to increase the FC lifespan. Moreover, the experimental results are consistent with the simulation ones.PI controller. For comparison, an experiment with PI controllers for the two outer-loops have been done based on the design proposed in(Azib et al., 2011), as shown on Fig. (13).

Fig

  Fig. (13.a) shows that the DC bus is well controlled due to greater SCs current compared to the previous results. Therefore, the fluctuation of the SCs voltage is more important.

  ,Th.2]. The nominal part of the first subsystem (21), namely ẋ = [J d (x) -R d (x)]∇H d (x), is globally uniformaly asymptotically stable. Further, the Lyapunov function H d is a quadratic function, thus it satisfies the bounds

  Figure 2: IDA-PBC structure.

Figure 3 :

 3 Figure 3: Fuel cell voltage vs current.

Figure 4 :

 4 Figure 4: DC bus and load model.

Figure 5 :

 5 Figure 5: Simulation result with the resistance as unknown parameters (α=10).

Figure 6 :

 6 Figure 6: Simulation result of the load admittance estimation.

Figure 7 :

 7 Figure 7: Simulation with the load resistance estimator (α=10, K Rl =0.5).

Figure 8 :

 8 Figure 8: Simulation result with an integral action (α=10, γ=460).

Figure 9 :

 9 Figure 9: Experimental test bench.

Figure 10 :

 10 Figure 10: Experimental result during a step load without an integral action or a load estimator (α=10, γ=0).

Figure 11 :

 11 Figure 11: Experimental result with the load resistance estimation (α=10, K Rl =0.5).

Figure 12 :

 12 Figure 12: Experimental result during a step load with an integral action (α=10, γ=460).

Figure 13 :

 13 Figure 13: Experimental result during a step load with PI controllers.

  The considered FC is a 46 A/1200 W Nexa FC designed by Ballard. This latter is composed of 46 cells. The transient auxiliary source consists of two Maxwell SC modules associated in series: each module is built with the connection of six individual elements in series [2.7 V, 350 F]. This SCs device is interconnected

Appendix 1: Stability analysis with an integral action

Proposition 1: Consider the PCH system (11) in closed-loop with the controller (12)(13). Then, all stability properties of x * are preserved by adding to the IDA control (12)(13) an integral term as shown in ( 18)-(20).

Proof. The extended IDA ( 14) associated with the controller ( 18)-( 20) takes the PCH

where

. Then, it follows that all the stability properties are preserved.

Appendix 2: Stability analysis with a load estimator

The proof of the global stability of the outer-loop composed of the controller, the estimator and reduced-order electrical system is established invoking a theorem on stability of cascaded systems stated in [START_REF] Panteley | On global uniform asymptotic stability of nonlinear time varying systems in cascade[END_REF].

Proposition 2: Consider the hybrid system (10) in closed-loop with the control law (12-13) where R l is replaced by Rl = 1/ Ŷl generated by (17). For all initial conditions, lim t-→∞ x(t) = x * is guaranteed.

Proof. First, the load estimator ( 17) is an autonomous linear system, which is globally uniformaly asymptotically stable for all positive gain K rl . Thus, the estimation error decay asymptotically to zero.

Secondly, let us define the estimation error τl = Ŷl -Y l , and write the closed-loop system in the following form:

Table 1: Electric characteristics of the hybrid system.

Fuel cell parameters

Open


Louis Biadala

Wenbing Peng

Yannick Lambert

Jin H Kim

Damien Canneson

Anthony Houppe

Maxime Berthe

David Troadec

Dominique Deresmes

Gilles Patriarche #

Tao Xu

Xiaodong Pi

Xavier Wallart

Christophe Delerue

Manfred Bayer

Jimmy Xu

Bruno Grandidier

Trap-Free Heterostructure of PbS Nanoplatelets on InP(001) by Chemical Epitaxy

Keywords: PbS/InP heterostructures, nanocrystals, chemical epitaxy, scanning tunneling spectroscopy, interfacial traps

Semiconductor nanocrystalline heterostructures can be produced by the immersion of semiconductor substrates into an aqueous precursor solution, but this approach usually leads to a high density of interfacial traps. In this work, we study the effect of a chemical passivation of the substrate prior to the nanocrystalline growth. PbS nanoplatelets grown on sulfur-treated InP (001) surfaces at temperature as low as 95°C exhibit abrupt crystalline interfaces that allow a direct and reproducible electron transfer to the InP substrate through the nanometer-thick nanoplatelets with scanning tunnelling spectroscopy. It is in sharp contrast with the less defined interface and the hysteresis of the current-voltage characteristics found without the passivation step. Based on a tunnelling effect occuring at energies below the bandgap of PbS, we show the formation of a type II, trap-free, epitaxial heterointerface, with a quality comparable to that grown on a non-reactive InP (110) substrate by molecular beam epitaxy. Our scheme offers an attractive alternative to the fabrication of semiconductor heterostructures in the gas phase.

The deposition of semiconductor thin films on substrates immersed in dilute solutions containing metal ions and a source of hydroxide sulphide or selenide dates back to 1869. [START_REF] Lincot | Chemical Solution Deposition of Semiconducting and Non-Metallic Films: Proceedings of the International Symposium[END_REF] It is known as chemical bath deposition and occurs at relatively low temperatures, below 100°C. Such a range of growth temperatures is an asset of this technique for producing nanocrystals. [START_REF] Hodes | Semiconductor and Ceramic Nanoparticle Films Deposited by Chemical Bath Deposition[END_REF] Indeed, the formation of very small crystals in an aqueous solution depends on the kinetic stabilization of nuclei and a low temperature ensures the chance that nuclei eventually grow to a stable size rather than redissolve. When the substrate is a crystal, the atomic positions in the nanocrystals can match specific crystallographic orientations of the substrate to give rise to a heteroepitaxial junction, the deposition technique being then referred as chemical epitaxy (CE). [START_REF] Osherov | Chemical Epitaxy of Semiconductor Thin Films[END_REF] Among the numerous systems that have been demonstrated with group IV, III-V or II-VI semiconductor substrates, CE of lead chalcogenides is appealing for the monolithic integration of materials with applications in sensing, imaging and photovoltaics. [START_REF] Mcdonald | Solution-processed PbS Quantum Dot Infrared Photodetectors and Photovoltaics[END_REF][START_REF] Rauch | Near-infrared Imaging with Quantum-Dot-Sensitized Organic Photodiodes[END_REF][START_REF] Kagan | Building Devices from Colloidal Quantum Dots[END_REF] Moreover, the whole near infrared spectral range can be accessed by simply changing the size of the lead chalcogenide nanocrystals. [START_REF] Moreels | Size-dependent Optical Properties of Colloidal PbS Quantum Dots[END_REF] Although optoelectronics devices prepared by CE have been demonstrated. [START_REF] Chang | Chemical Bath Deposition of CdS Quantum Dots onto Mesoscopic TiO2 Films for Application in Quantum-Dot-Sensitized Solar Cells[END_REF][START_REF] Liu | Room Temperature Photocurrent Response of PbS/InP Heterojunction[END_REF][START_REF] Chang | High-performance Nanostructured Inorganic-Organic Heterojunction Solar Cells[END_REF][START_REF] Chirilă | Potassium-induced Surface mModification of Cu(In,Ga)Se2 Thin Films for High-Efficiency Solar Cells[END_REF][START_REF] Zhou | Thin-film Sb2Se3 Photovoltaics with Oriented One-Dimensional Ribbons and Benign Grain Boundaries[END_REF][START_REF] Xu | Chemical Bath Deposition of p-type Transparent, Highly Conducting (CuS)x:(ZnS)1-x Nanocomposite Thin Films and Fabrication of Si Heterojunction Solar Cells[END_REF][START_REF] Maman | Postgrowth Control of the Interfacial Oxide Thickness in Semiconductor-Insulator-Semiconductor Heterojunctions[END_REF] they have so far shown rather limited performances because of the existence of numerous interfacial traps that strongly affect their transport and optical properties. In order to meet the high purity and high perfection standards in semiconductor electronics, an optimal growth process starting with a substrate preparation that avoids the formation of etching-induced defects or the trapping of residual molecules in interfacial voids is therefore a prerequisite.

Here, as a prototypical example, we examine the CE of PbS nanoplatelets (NPs) on InP, a substrate that is known to be highly reactive in ambient conditions and thus challenging for manufacturing semiconductor devices. [START_REF] Pluchery | Wet Chemical Cleaning of InP Surfaces Investigated by in situ and ex situ Infrared Spectroscopy[END_REF][START_REF] Wood | Surface Chemistry of GaP (001) and InP (001) in Contact with Water[END_REF] Although chemical bath deposition is itself a route for the passivation of the InP surfaces, it has so far failed to fully remove interfacial traps. [START_REF] Davis | Optimization of Chemical Bath-deposited Cadmium Sulfide on InP using a Novel Sulfur Pretreatment[END_REF] We adopt a standalone passivation scheme, that differs from previous preparation strategies. [START_REF] Templeman | Chemically Deposited PbSe Thin Films: Factors Deterring Reproducibility in the Early Stages of Growth[END_REF] While it keeps the morphology of the NPs unchanged, the chemical and structural analyses of the heterostructure reveal the formation of an atomically sharp interface. More importantly, this step is found to suppress charging effects in scanning tunnelling microscopy (STM) and spectroscopy (STS) of single NPs, making the NPs electronically transparent to tunnelling electrons for the energy window corresponding to the band gap of the NPs. Whatever the NP height is, from 5 nm to 15 nm, a direct tunnelling of electrons through the NPs is achieved between the STM tip states and the low lying conduction band (CB) states of n-type and p-type InP substrates, demonstrating the absence of deleterious interfacial trapping states.

RESULTS AND DISCUSSION

Figure 1a shows PbS NPs that have been grown by placing an InP (001) substrate into a PbS solution, directly after immersing the substrate in a buffered oxide etchant. The NPs predominantly consist of a square top facet, consistent with the cubic rock-salt structure of PbS. Their lateral sizes vary between 40 and 140 nm. Adding a passivation pre-treatment, by immersing the etched InP substrate in a solution of (NH4)2S, saturated with sulphur, which is a self-limiting process due to the binding of the sulfur atoms to the indium atoms at the surface, [START_REF] Tao | S-passivated InP (100)-(1×1) Surface Prepared by a Wet Chemical Process[END_REF] does not modify the cubic shape of the NPs, as seen by atomic force microscopy (AFM) and scanning electron microscopy (SEM) (Figure 1c and high-resolution view of the SEM image in Figure S1, SI). From the size analysis of the NPs measured with AFM and scanning tunnelling microscopy (STM), a range of similar lateral sizes is found after passivation, with the existence of a linear correlation between the height of the NPs and their lateral sizes.

Despite strong similarities between the overall morphology of the NPs, the passivation treatment has a profound effect on the transport measurement performed with scanning tunnelling spectroscopy (STS). When a sequence of I(V) curves is measured on single NPs grown on the etched surface of a p-type InP substrate, we consistently observe the occurrence of a plateau at positive bias, with the onset of the plateau shifted to a higher bias as more and more spectra are acquired (Figure 1d). Conversely, sequences of I(V) curves acquired on single NPs grown on the passivated surface of the same substrate are steadily reproducible and do not show any similar plateau (Figure 1e). Moreover, the reproducibility of the spectra is not only temporal, but also spatial (see Figure S2, SI). While the hysteresis observed on the etched substrate is emblematic of interfacial trapping states, [START_REF] Hummon | Measuring Charge Trap Occupation and Energy Level in CdSe/ZnS Quantum Dots using a Scanning Tunneling Microscope[END_REF] the reproducibility of the spectra on the passivated substrate suggests the absence of charging effect. Hence, we speculate that the different behaviours observed between the etched and the passivated InP substrates are caused by a modification of the interface. This interface was investigated by combining cross-sectional high angle annular dark field scanning transmission electron microscopy (HAADF-STEM) with energy dispersive spectroscopy (EDS). Although the comparison of thin slices prepared parallel to the {100} edge facets of NPs grown on either the etched or the passivated substrates shows a good crystallinity of the NPs in all cases, an interfacial layer clearly exists between the etched substrates and the NPs. It appears dark in Figure 2a and correspond to an oxygen-rich, amorphous layer. The thickness of this layer is usually inhomogeneous under the NPs (see Figure S3, SI). Depending on the position of the crosssectional lamellae prepared by focus ion beam, part of the NPs can be found directly in contact with the InP surface, enabling an epitaxial relationship between the InP and PbS lattices, that accounts for the uniformity of the NP orientation on the etched InP surface. In contrast to the NPs grown on this surface, there is no trace of oxygen or carbon species at the interface for the passivated substrates, demonstrating that the whole base of the NPs is in direct contact with the InP surface (Figure 2b). As the NPs are slightly misoriented in the <100> zone axis, thin slices were also prepared along the {110} diagonal of the NPs. The high resolution STEM image of Figure 2c highlights the abruptness of the interface in the <110> zone axis. Morever, well-matched atomic columns between the InP substrate and the PbS NP are observed, making the fast Fourier transform images acquired in both regions almost superimposable, consistent with the small lattice mismatch of 1.0 % between PbS and InP. Clearly, the passivation of the InP substrates is essential to produce an abrupt and high quality interface and is the origin of the stable and reproducible tunnelling spectra.

Focussing on the PbS NPs that are grown on the passivated InP substrates, we find that the I(V) curves are markedly different depending on the doping of the substrate (Figure 3a). In the case of a p-type passivated InP substrate, all the spectra exhibit a zero-conductance region, whatever the height of the NPs is. This region, labelled Ezc, is better seen when the spectra are plotted on a log scale. We note that Ezc increases as the NP height decreases. This increase clearly shows up when a larger set of NPs is probed (Figure 3b) and is related to the increase of the quantum confinement as the height of the NPs decreases. [START_REF] Liljeroth | Density of States Measured by Scanning Tunneling Spectroscopy Sheds New Light on the Optical Transitions in PbSe Nanocrystals[END_REF][START_REF] Diaconescu | Measurement of Electronic States of PbS Nanocrystal Quantum Dots using Scanning Tunneling Spectroscopy: the Role of Parity Selection Rules in Optical Absorption[END_REF] Surprisingly, the spectra acquired on all the PbS NPs grown on the n-type InP substrate exhibit a metallic behaviour, even though the confinement is stronger as shown for the two NPs considered in Figure 3a. These characteristics are therefore in contradiction with the semiconductor nature of the PbS NPs.

In order to understand the discrepancy between the p-type and n-type substrates, tight-binding calculations have been performed to determine the band structure of PbS NPs with different heights and infinite lateral sizes. [START_REF] Overgaag | Electron-Phonon Coupling and Intervalley Splitting Determine the Linewidth of Single-Electron Transport through PbSe Nanocrystals[END_REF] The comparison of Ezc with the theoretical band gap shows that Ezc cannot be considered as the band gap of the PbS NPs (Figure 3b). Instead, a careful inspection of the I(V) curves reveals a change of the slope at positive bias, that corresponds to an additional onset, labelled ECPbS, and observed above the positive edge of Ezc in the differential conductance dI(V)/dV for the PbS NPs grown on the p-type InP substrates. No similar behaviour is seen at negative bias. When the energy separation EgPbS between this onset and that of the current at negative bias (EVPbS) is plotted as a function of the NP height, its variation is found to well fit the change of the theoretical NP band gap. Therefore, the onset measured at negative bias corresponds to the valence band edge of the PbS NPs and a smaller Ezc indicates that additional empty states exist below the NP conduction band (CB), in agreement with a type II band alignment in PbS/InP heterojunctions. [START_REF] Liu | Room Temperature Photocurrent Response of PbS/InP Heterojunction[END_REF] As the spectra acquired on the PbS NPs grown on the n-type InP substrate do not show any quantized effects, we rule out the contribution of the NPs to the tunnelling current at small bias. [START_REF] Diaconescu | Measurement of Electronic States of PbS Nanocrystal Quantum Dots using Scanning Tunneling Spectroscopy: the Role of Parity Selection Rules in Optical Absorption[END_REF] The observation of a metallic behaviour in the spectra also disagrees with the transfer of electrons through trapping states at the PbS/InP heterointerface, which are non-resonant states and should give rise to a negative differential resistance or charging effects. [START_REF] Berthe | Probing the Carrier Capture Rate of a Single Quantum Level[END_REF][START_REF] Rashidi | Time-resolved Imaging of Negative Differential Resistance on the Atomic Scale[END_REF] Hence, we attribute the tunnelling current at small bias to the pinning of the Fermi level above the CB edge of InP. This is consistent with the existence of a band offset, where the CB edge of PbS is positioned above that of InP. Due to the high doping level of the n-type InP substrate (4.6 x 10 18 cm -3 ), it is possible not only to directly transfer electrons from the tip states to the InP CB at positive bias, but also to transfer electrons from the occupied states of the InP CB to the tip states at negative bias. In other words, the PbS NPs are electronically transparent in the PbS NPs / n-type InP heterostructures (Figure 3d).

The band offset between the PbS NPs and p-type InP leads to a downward band bending of the InP CB, introducing an inversion layer in InP, that is not dependent on the size of the NPs, due to the high static dielectric constant of PbS. [START_REF] Dalven | A Review of the Semiconductor Properties of PbTe, PbSe, PbS and PbO[END_REF] Estimation of the energy of the bound state in the inversion layer is obtained by first calculating the width of the depletion region (~ 26 nm for an acceptor concentration of 2.8 x 10 18 cm -3 ), then, working out the electrical field in the triangular well corresponding to the inversion layer (~1MV.cm -1 ), and finally solving the Schrödinger equation for this triangular quantum well. If an infinite potential barrier at the interface between PbS and InP is assumed, the bound state is found 0.4 eV above PbS valence edge. Based on the electron affinities of PbS nanocrystals (3.5 eV ~ 4.2 eV) [START_REF] Hyun | Electron Injection from Colloidal PbS Quantum Dots into Titanium Dioxide Nanoparticles[END_REF][START_REF] Wang | Tandem Colloidal Quantum Dot Solar Cells Employing a Graded Recombination Layer[END_REF] and InP bulk (4.4 eV), [START_REF] Liu | Room Temperature Photocurrent Response of PbS/InP Heterojunction[END_REF] the potential barrier is smaller. Hence, the actual lowering of the potential barrier gives rise to a less confined state, consistent with the measured values of Ezc between 0.17 and 0.37. Again, for bias smaller than the onset of PbS CB (Figure 3d), electrons directly tunnel from the tip states into the bound state of the p-type InP inversion layer. This tunnelling is allowed for relatively large distances because of the small potential barrier that is encountered by the electrons and the very long de Broglie wavelength in the NPs.

While the photoluminescence experiments are not sensitive to possible radiative transitions between this bound states and critical points of the Brillouin zone in PbS [START_REF] Geiregat | A Phonon Scattering Bottleneck for Carrier Cooling in Lead Chalcogenide Nanocrystals[END_REF] due to the existence of a high concentration of donor to acceptor transitions in p-type InP subtrates (see Figure S4,SI), [START_REF] Swaminathan | A Photoluminescence Study of Cd-related Centers in InP[END_REF] we further studied the quality of the interfaces with photoelectron spectroscopy. For the sample with the thinnest NPs that were obtained on the n-type passivated InP substrate (Figure 4), a weak component shifted by 0.7 eV to a higher binding energy is identified in the In 3d5/2 core level, indicating the passivation of the In dangling bonds with sulphur atoms. [START_REF] Pluchery | Wet Chemical Cleaning of InP Surfaces Investigated by in situ and ex situ Infrared Spectroscopy[END_REF] Comparison with the In 3d5/2 core level measured for of an etched InP substrate confirms the quality of the passivation, since in this case a broad component is measured and contains both In-S and In-O species (see Figure S6,SI). The surface passivation is additionally supported by the existence of two components in the S 2p core level, where the strongest component is caused by the In-S bonds, the smallest one being related to the Pb-S bonds in the NPs. The current result is also in agreement with the position of the Fermi level, measured close to the InP conduction band edge in tunnelling spectroscopy (Figure 3(a)) and in photoemission spectroscopy (see Figure S5), corresponding to an optimized sulphur passivation of the surface. [START_REF] Bessolov | Surface of n-type InP (100) Passivated in Sulfide Solutions[END_REF] As no interface component is detected in the P 2p and Pb 4f core levels, the interface appears non-reactive. This is in contrast to the molecular beam epitaxy of PbS on the reactive InP(001) in ultrahigh vacuum, where lead atoms in a metallic surrounding are found at the interface. [START_REF] Preobrajenski | Atomic and Electronic Structure of Epitaxial PbS on InP (110) and InP (001)[END_REF] Interestingly, the measured core levels are comparable with those measured when an ultrathin epitaxial PbS layer is grown at room temperature on a nonreactive InP (110) substrate by molecular beam epitaxy, [START_REF] Preobrajenski | Epitaxial Growth and Interface Structure of PbS on InP (110)[END_REF] making the quality of both interfaces comparable.

CONCLUSION

In conclusion, the direct tunneling of electrons from the STM tip states to the InP CB for energies below the band gap of the PbS NPs demonstrates the absence of electronic traps at the PbS/InP interface. The passivation scheme of the InP substrate prior to the growth of the PbS NPs not only provides an excellent crystalline interface, but also eliminates the interfacial electrically active defects, that degrade the performance of devices prepared by chemical bath deposition. Given the complementary band gaps of PbS and InP, the current PbS/InP heterojunctions produced by CE with trap-free interfaces hold great promise for the fabrication of high-performance semiconductor devices such as photodetectors and solar cells with optical response in a rather broad spectral region from the ultraviolet to infrared. Moreover, since a plethora of semiconductor heterojunctions have already been achieved with CE, the optimization of the chemical treatment of the host substrates should allow CE to compete on an equal footing with growth techniques performed in vacuum.

METHODS

Sample preparation. The highly polished InP(100) (AXT Inc.) substrates had a thickness of 350µm and were doped either with Zn (p-type) or S (n-type) at a concentration of 2.8 x 10 18 cm -3 and 4.6 x 10 18 cm -3 respectively. The passivation of the InP surfaces consisted in two steps: first, the removal of the native oxide layer by immersing them in 1% HF (Sigma) solution, then the sulphur passivation by soaking them into 0.5% (NH4)2S (Sigma) aqueous solution saturated with sulfur (Sigma) at 60 o C for 1 hr. For the growth of the PbS NPs, the unpassivated and passivated InP substrates were soaked into a solution containing 1 mM lead acetate (Sigma), 1 mM thiourea (Sigma), 10 ml ethanol (Sigma), and 10 ml ethylene glycol (Sigma) at 95 °C for 1 hr. Lead acetate is the lead source and thiourea is the sulfur source for the formation of PbS crystallites. The pH of the solution was adjusted to 3.4 by adding glacial acetic acid (Sigma). Since phosphine, which is a very toxic gaz, is generated when InP substrate is treated with acid, this process was carefully performed in fume hood. After the growth of the PbS NPs, the samples were washed with ethanol and dried with N2 blower.

Scanning Transmission Electron Microscopy. For the investigation of the PbS/InP

heterointerface with cross-sectional scanning transmission electron microscopy (STEM), thin lamellae were cut from the samples by focused ion beam and deposited on a specific lift-out copper grid for FIB preparation. They were analyzed by means of HR-STEM and EDXS on a Titan Themis (FEI/Thermo Fischer) microscope operating at 200 keV and equipped with the Super-X windowless EDX detector (allowing a large detection angle of 0.8 steradian) and a geometric aberration corrector on the probe. The (half)-convergence angle for the probe was 17.6 mrad and the probe current about 70pA. The detection (half)-angles for the HAADF images was between 69 mrad (inner) and 200 mrad (outer).

Scanning tunnelling microscopy. The samples were examined with tungsten tips in a scanning tunnelling microscope working in ultrahigh vacuum (1x10 -10 Torr) at a temperature of 77 K.

Tunneling spectroscopic measurements were performed at constant tip-sample separation. The dynamic conductance was measured with a lock-in amplifier using typically 7 mV modulation at 500 Hz on the bias voltage. During the acquisition of a sequence of spectra, ramping the voltage between the sample and the tip took 4.6 s to measure a single spectrum.

Photoluminescence. PbS/InP samples were mounted on a cold finger of a cryostat and were excited non-resonantly using laser diode (photon energy 1.58 eV) focused with a 10 cm lens. The PL was filtered from the scattered laser light with a long pass filter, sent into a 0.55 m spectrometer and detected by a Peletier cooled charged-coupled-device camera.

X-ray photoelectron spectroscopy. The XPS measurements were performed with a monochromatic Al Kα X-ray source and an analyser pass energy of 12 eV. Under these conditions, the overall resolution as measured from the full width at half-maximum (fwhm) of the Ag 3d5/2 line is 0.55 eV. The binding energy scale was calibrated using the Au 4f7/2 line at 84.0 eV. The acceptance angle of the analyser was set to 14°, and the angle between the incident X-rays and the analyser was 90°. The detection angle of the photoelectrons was 45°, as referenced to the sample surface. The intensities of the various XPS core levels were measured as the peak area after standard background subtraction according to the Shirley procedure. For the core level decomposition, we used Voigt functions and a least squares minimization procedure. The different components were modelled with the same parameters; that is, the Gaussian and Lorentzian broadenings were kept fixed for each component of a given core level. Tight binding calculations. The electronic structure of PbS NPs with infinite lateral sizes was calculated using a sp 3 d 5 s * tight-binding model as described in Ref. 34. The method provides a good description of the energy gap of spherical PbS quantum dots compared to experiments [START_REF] Moreels | Size-dependent Optical Properties of Colloidal PbS Quantum Dots[END_REF] .

ASSOCIATED CONTENT

Supporting Information

The following files are available free of charge.

Scanning electron micrograph of PbS nanoplatelets grown on a passivated InP (001) surface; spatially-resolved tunnelling spectroscopy of a PbS NP grown on a p-type passivated InP substrate; photoluminescence of the PbS nanoplatelet / InP (001) heterostructure; valence-band photoemission spectroscopy of a n-type passivated InP (001) substrate with a small density of PbS NPs; fitting parameters of the core levels shown in Figure 4. (PDF).

AUTHOR INFORMATION

Corresponding Author

* E-mail: bruno.grandidier@isen.iemn.univ-lille1.fr The thin lines correspond to fits obtained with the parameters given in Table S1, SI.

FIGURE 1 .

 1 FIGURE 1. (a) Scanning electron micrograph of PbS nanoplatelets grown on an InP (001) surface

FIGURE 2 .

 2 FIGURE 2. (a) (100) cross-sectional high-angle annular dark-field (HAADF) scanning

FIGURE 3 .

 3 FIGURE 3. (a) Constant current STM images of PbS nanoplatelets (NPs) grown either on the p-

FIGURE 4 .

 4 FIGURE 4. Photoelectron spectra of the (a) In 3d (b) P 2p, (c) Pb 2p and (d) S 2p core levels

Figure S2 .

 S2 Figure S2. Spatially-resolved tunnelling spectroscopy of a PbS NP grown on a passivated p-type InP substrate. Tunneling conditions at 77K: Vsample =-0.6 V, Isetpoint =70 pA. Inset: STM image of the NP showing the positions where the spectra were acquired. The NP height is 22 nm.

Figure

  Figure S3. (100) cross-sectional high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) image of a PbS NP grown on an etched InP (001) surface and related energy-dispersive spectrometer (EDS) mapping image for O.

Figure S4 .

 S4 Figure S4. Photoluminescence of the PbS nanoplatelet / InP (001) heterostructure. (a) Comparison of PL spectra between PbS nanoplatelets grown on passivated n-type and p-type InP substrates. The spectra have been shifted for clarity. (b) Excitation intensity dependence of the PL spectrum for the PbS nanoplatelets grown on the passivated p-type InP substrate. The low and high energy bands are labelled LEB (low energy band) and HEB (high energy band) respectively. (c) Peak photon energies of the low energy bands as a function of the excitation power, that is caused by donor-acceptor pair transitions.

Figure S5 .

 S5 Figure S5. Photoelectron spectra of the (a) In 3d and (b) S 2p core levels measured for PbS nanoplatelets grown on a p-type etched InP (001) surface. The main peak positions obtained from the decomposition of the core levels are shifted by 0.65eV with respect to the ones found for the n-type passivated sample of Figure 4, consistent with the shift of the Fermi level shown in Figure S5.

Figure S6 .

 S6 Figure S6. Valence-band photoemission spectroscopy (top) of PbS NPs grown on a p-type etched InP (001) surface and (bottom) of a n-type passivated InP (001) substrate with a small density of PbS NPs.

Table S1 .

 S1 Fitting parameters of the core levels shown in Figure4.

	Core	Binding	Spin-orbit	FWHM
	level	energy	splitting	(eV)
		(eV)	(eV)
	In 3d	445.03		0.74
		445.73		0.74
	P 2p	129.26	0.84	0.58
		130.10		0.58
	Pb 4f	138.79	4.86	1.27
		143.65		1.27
	S 2p	161.46	1.16	0.80
		162.62		0.80
		162.12	1.16	1.14
		163.28		1.14