RÉSOLUTIONS D'EQUATIONS DU PREMIER DEGRÉ EN IUT GEII -ETM DE RÉFÉRENCE DES ETUDIANTS

Pour la question 2 sur le type de résolution à mettre en oeuvre pour résoudre un problème donné, 17 (dont 13 ont donné une réponse de type RTL à la question 1) étudiants reconnaissent les 4 écritures directes proposées, 4 n'en reconnaissent pas plus de la moitié. 10 (dont 6 ont donné une réponse de type RTL à la question 1) étudiants

  Saclay, Univ Evry -LDAR, Université de Paris, Univ Paris Est Créteil, CY Cergy Paris Université, Univ. Lille, UNIROUEN En IUT Génie Électrique et Informatique Industrielle, les étudiants sont régulièrement amenés à manipuler des expressions algébriques et à résoudre des équations simples, en particulier du premier degré. Ces étudiants sont issus, pour une part importante (45% à 50% ces dernières années), de bacs technologiques STI2D. Dans la pratique, il s'avère que plusieurs d'entre eux ont des difficultés pour obtenir des résultats corrects. Un exemple typique est le suivant : Un questionnaire a été proposé à des étudiants de première année de DUT en septembre 2018 et en septembre 2019. L'idée directrice de ce questionnaire est de demander aux étudiants de dire ce qu'ils savent sur les équations du premier degré mais aussi de reconnaitre certaines formes et les moyens de se ramener à des formes connues. Sans proprement parler de la résolution de ces problèmes. Pour chacune des questions, l'ETM de référence est décrit. L'analyse des réponses des étudiants permet d'expliciter les ETM personnels des étudiants. Dans la seconde, "Parmi les problèmes suivants, lesquels selon vous reviennent-ils à résoudre une équation du premier degré ?", deux types de problèmes sont présents : dans les premiers, l'équation est déjà sous la forme d'une équation du premier degré (on l'appellera écriture directe) et c'est alors la genèse sémiotique qui est activée ; dans les seconds, l'écriture de l'équation doit être modifiée pour que la forme du premier degré apparaisse (on l'appellera écriture indirecte) et le travail mathématique est donc dans le plan [Ins-Dis]. La troisième question, "Si de nouvelles formes d'équation du premier degré vous viennent à l'esprit, rajoutez-les ici", permet d'enrichir le référentiel théorique de la question 1. La quatrième question, "Donnez le maximum de formes d'équation qui vous viennent à l'esprit et qui reviennent selon vous à résoudre une équation du premier degré", aborde les problèmes dont les mises en équation ne sont pas formellement de la forme du premier degré mais leur résolution revient à une équation du premier degré. Le travail mathématique est alors dans le plan [Sem-Dis]. La cinquième question, "Quels critères pourriez-vous donner pour déterminer si une équation donnée est du premier degré ou non ?", aborde précisément la question de la reconnaissance du type de travail mathématique qui sera mis en jeu dans la résolution du problème. Les techniques de transformation des écritures algébriques étant travaillées dès le collège (Ministère de l'éducation nationale, 2008), on peut considérer que leur mise en oeuvre correspond à la genèse instrumentale ; ainsi, le travail mathématique est ici dans le plan [Ins-Dis]. ETM de référence. Concernant la première question sur les formes possible d'une équation du premier degré, sur 26 questionnaires dépouillés, 16 étudiants donnent des formes d'équation avec des coefficients littéraux (RTL), dont 2 seulement contiennent l'inconnue dans les deux membres. 4 étudiants ne donnent que des expressions avec des coefficients numériques (RTN). 6 ne donnent aucune réponse. La question 3 invitant à compléter ce référentiel permet à un étudiant de donner une forme littérale et un autre à donner une forme avec des coefficients numériques. Le référentiel théorique n'est donc pas au niveau attendu.

			Philippe Hoppenot
	Université Paris-Sachant que 𝐴 𝑉 = 1 +	𝑅 27 +𝑃 5 𝑅 26	= 9,57, calculer 𝑃 5 avec 𝑅 27 = 15 𝑘Ω et 𝑅 26 = 2,2 𝑘Ω
	Une étude historique (Sirejacob, 2018) montre que "l'usage d'un symbolisme
	algébrique a mis longtemps à émerger dans la résolution de problèmes, d'abord
	solutionnés en langage naturel ; de même, pendant longtemps, des théorèmes généraux
	ont été énoncés mais uniquement illustrés ou expliqués sur des exemples numériques"
	(ibid, p.187). Cela peut expliquer les difficultés des élèves et des étudiants à qui on
	demande d'intégrer un symbolisme complexe rendant compte de notions abstraites complexes. Ils semblent développer des conceptions (au sens de Tall et al,1981) erronées sur ces notions. Dans son étude, Sirejacob se pose les questions suivantes : "qu'est-ce qui est au coeur de la génération des équations et de leur manipulation ? chacune des formes d'équation du premier degré citées en 1 et 3, expliciter les transformations que vous jugez nécessaires pour se ramener à une équation de la forme : 𝑎𝑥 + 𝑏 = 0" et "Pour chacune des formes d'équation revenant à résoudre une Qu'est-ce qui leur Les questions 6 et 7, "Pour équation du premier degré citées en 4, expliciter les transformations nécessaires pour
	se ramener à une équation de la forme : 𝑎𝑥 + 𝑏 = 0", le travail mathématique a encore
	lieu dans le plan [Ins-Dis]. Le travail attendu ici des étudiants est celui d'un technicien
	au sens de Nechache (2017).
	ETM personnels des étudiants
	L'analyse des ETM personnels des étudiants révèle des écarts, parfois importants, avec
	l'

donne des raisons d'être et du sens chez les élèves ? Comment prennent-elles place au sein des curricula ?" (ibid, p.225). Bien souvent, en physique et en électricité en particulier, la question de la modélisation (aussi présente dans Blum, 2007) est éludée étant hors de portée des élèves. La question de la dialectique entre numérique et algébrique est aussi très présente dans la littérature déjà depuis de nombreuses années

[START_REF] Chevallard | Le passage de l'arithmétique à l'algèbre dans l'enseignement des mathématiques au collège. Seconde partie[END_REF]

.

Dans toutes ces études, une question est absente : les élèves reconnaissent-ils qu'ils sont en présence d'une équation du premier degré lorsqu'ils se retrouvent face à l'énoncé donné ci-dessus ? C'est à cette question que l'on apporte des éléments de réponse dans ce papier.

ETM de référence

La première question, "Donnez toutes les formes qui vous viennent à l'esprit pour une équation du premier degré", fait référence au référentiel théorique.


Power of neuroscience studies

Power = Prob. to correctly find a significant effect when a the alternative hypothesis is true.

We need bigger datasets

X-ray: Lung opacity detection

Model trained on male images, tested on female images

We need representative and diverse datasets [Larrazabal et. al, PNAS 2020] Towards reproducible neuroimaging

Lack of power Generalizability and fairness

Irreproducible with...

Different Data

Different pipelines… different results?

BIDS dataset

Step 2

Step 3

Step 4

Step 5 Which solutions for practitioners?

T H I S A Y

Image credits : Adobe Stock, https://stock.adobe.com/

•

  . Is there a parametric effect of gain: Positive effect in ventromedial PFCfor the equal indifference group? [Botvinik-Nezer et. al, Nature 2020] Analysis Data preparation Derived data Image credits : Teams, undraw.co. Nezer et. al, Nature 2020] Results over the 70 post-processing pipelines: Different end-to-end pipelines [Carp 2012] • Different software version [Gronenschild 2012] • Different operating system [Glatard 2015] • Different post-processing software packages Image credits : Teams, undraw.co.

  will soon be open for community feedback! http://bids.neuroimaging.io/bep028

1 dataset 9 'yes/no' questions

	Provenance Use Case 1
	Boris	Elodie	Jérémy
	Clenet	Germani	Lefort-Besnard

An unambiguous description of the methods Provenance Use Case 1 An unambiguous description of the methods

	Provenance Use Case 1 An unambiguous Provenance Use Case 2 Dealing with Provenance Use Case 2 Dealing with Provenance Use Case 2 Dealing with Provenance Use Case 3 Exploring factors
	description of the methods unwanted variability unwanted variability unwanted variability impacting analytical variability
				BIDS		A previously A previously A previously
		BIDS	dataset		widely-used widely-used widely-used
	File 1	dataset (command) Step 1 Step 1 (command)	Tool 1 (name, version) Tool 1 (name, version)	Can we Can we parameter is no longer considered parameter is no longer considered parameter is no longer considered
	File 2 File 3	myPipeline process	Step 2 Step 3	myPipeline container	reproduce the same results as best practice as best practice as best practice reproduce exactly the same results A bug was found A bug was found
	File 4	Activation map for contrast 1		Step 4 Step 5	Tool 2	(same data, same methods)? (same data, same in a given tool version, can we in a given tool version, can we methods)? retrieve all retrieve all
						corresponding corresponding
		Activation map for Activation map for		results? results?
		contrast 1 contrast 1

Irreproducible with... Different Analysis

Post-processing by different teams

Unreliable results

Assumption violations

Better fit to the data We are hiring https://team.inria.fr/empenn/job-offers

Variability


Théo Hennequin

Manoel Manghi
email: manoel.manghi@univ-tlse3.fr

Adrien Noury

François Henn

Vincent Jourdain

John Palmeri
email: john.palmeri@umontpellier.fr

Quantum capacitance governs electrolyte conductivity in carbon nanotubes

In recent experiments, unprecedentedly large values for the conductivity of electrolytes through carbon nanotubes (CNTs) have been measured, possibly owing to flow slip and a high pore surface charge density whose origin is still unknown. By accounting for the coupling between the quantum CNT and the classical electrolyte-filled pore capacitances, we study the case where a gate voltage is applied to the CNT. The computed surface charge and conductivity dependence on reservoir salt concentration and gate voltage are intimately connected to the CNT electronic density of states. This approach provides key insight into why metallic CNTs have larger conductivities than semiconducting ones.

Although much experimental, theoretical, and molecular modeling effort has been devoted over the past years to understanding water and ion (electrolyte) transport through carbon nanotubes (CNTs) [START_REF] Guo | Nanofluidic Transport through Isolated Carbon Nanotube Channels: Advances, Controversies, and Challenges[END_REF][START_REF] Thiruraman | Ions and Water Dancing through Atom-Scale Holes: A Perspective toward "Size Zero[END_REF][START_REF] Aluru | Fluids and Electrolytes under Confinement in Single-Digit Nanopores[END_REF], the origin of the electric charge localized on the surface of industrially important CNT based nanofluidic systems still remains unclear (see Ref. [START_REF] Manghi | Ionic Conductance of Carbon Nanotubes: Confronting Literature Data with Nanofluidic Theory[END_REF] and references therein). It has already been proposed that this surface charge could arise from functional groups at the CNT entrances [START_REF] Balme | Ionic transport through sub-10 nm diameter hydrophobic high-aspect ratio nanopores: experiment, theory and simulation[END_REF][START_REF] Yazda | Voltage-activated transport of ions through single-walled carbon nanotubes[END_REF] and/or the specific adsorption of ions, such as OH - [START_REF] Grosjean | Chemisorption of hydroxide on 2D materials from DFT calculations: graphene versus hexagonal boron nitride[END_REF]. Although the above cited studies lead to the conclusion that this surface charge plays a key role in governing ion transport in CNTs, it is difficult to regulate it directly and one is left to making inferences, for example by studying the variation of ionic conductance G with the pH or salt concentration, c s , of the external bulk reservoirs bounding the CNT. Intriguing results have been obtained, including a power law behavior, G ∝ c α s , with 1/2 ≤ α ≤ 1, which could be interpreted as the manifestation of an underlying surface charge regulation mechanism [START_REF] Secchi | Scaling Behavior for Ionic Transport and its Fluctuations in Individual Carbon Nanotubes[END_REF][START_REF] Biesheuvel | Analysis of ionic conductance of carbon nanotubes[END_REF][START_REF] Manghi | Role of charge regulation and flow slip in the ionic conductance of nanopores: An analytical approach[END_REF].

Through a simplified feasibility study we propose in this work that by biasing a CNT incorporated in a nanofluidic system via an applied gate voltage, V g , and taking into account explicitly the quantum capacitance (QC) of the quasi-1D CNT structure as well as the nonlinear capacitance of the confined electrolyte ion the pore, it should be possible to quantify the CNT surface charge density σ Q and establish a link between the intrinsic CNT electronic properties and ion transport through the same structure, such as the electrolyte conductance through the CNT. A major conclusion it that these intrinsic electronic properties will depend significantly, under certain conditions, on whether the CNT is metallic (M) or semiconducting (SC).

The CNT quantum charge σ Q arises from the low density of states (DOS) for the charge carriers (electrons and holes) in this quasi-1D system. Indeed, the applied V g creates a net charge on the CNT by perturbing the equilibrium occupation of the allowed energy levels. Part of V g goes to raising the electric potential of the CNT to ψ 0 and part goes to shifting the chemical potential by ∆µ = -eV ch = -e(V g -ψ 0 ) (where e is the absolute value of the electron charge): ψ 0 will then be lower than V g with (in the absence of other capacitances) the difference between the two giving rise to the chemical contribution, V ch . The amplitude of the chemical potential shift is determined by V g and the relative values of the capacitances with the smallest one providing the most important contribution. We show here that for a wide range of salt concentration, 10 -3 ≤ c s ≤ 4 mol/L, the CNT QC, C Q , can be lower than the nanopore electrolyte contribution, C P , and therefore the quantum contribution controls the conductance. By comparing the confined electrolyte conductivity calculated for a classical metallic nanopore (with a very high, effectively infinite, DOS) [START_REF] Gao | Voltage Charging Enhances Ionic Conductivity in Gold Nanotube Membranes[END_REF] with those calculated for M and SC CNTs, we bring to light clear signatures of each type of CNT behavior.

The CNT QC has already been studied in setups where an electrolyte surrounds the CNT and acts as an electrolyte liquid gate, the main goal was however to investigate the electrical properties of the CNT itself [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Heller | Identifying the Mechanism of Biosensing with Carbon Nanotube Transistors[END_REF][START_REF] Heller | Influence of Electrolyte Composition on Liquid-Gated Carbon Nanotube and Graphene Transistors[END_REF][START_REF] Rosenblatt | High Performance Electrolyte Gated Carbon Nanotube Transistors[END_REF][START_REF] Li | Carbon-Nanotube-Electrolyte Interface: Quantum and Electric Double Layer Capacitance[END_REF][START_REF] Li | Measurement of the combined quantum and electrochemical capacitance of a carbon nanotube[END_REF]. Within this context, Lemay's group [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Heller | Identifying the Mechanism of Biosensing with Carbon Nanotube Transistors[END_REF][START_REF] Heller | Influence of Electrolyte Composition on Liquid-Gated Carbon Nanotube and Graphene Transistors[END_REF] has carried out an interesting study, although the capacitance of the external electrolyte was modeled using an overly simplified form (of the type used for Stern layers) that does not capture the full complexity of the non-linear capacitance of an electrolyte. The QC of graphene sheets and ribbons have also been studied theoretically and experimentally using both ionic liquid and electrolyte gates [START_REF] Heller | Influence of Electrolyte Composition on Liquid-Gated Carbon Nanotube and Graphene Transistors[END_REF][START_REF] Xia | Measurement of the quantum capacitance of graphene[END_REF][START_REF] Fang | Carrier statistics and quantum capacitance of graphene sheets and ribbons[END_REF]. The gating of a generic nanopore has also been studied theoretically for confined electrolytes in a nanofluidic setup where the gate voltage is applied to an electrode in contact with an insulating layer making up the finitethickness surface of a nanopore [START_REF] Jiang | Electrofluidic Gating of a Chemically Reactive Surface[END_REF][START_REF] Jiang | Charge regulation in nanopore ionic field-effect transistors[END_REF]. The confined electrolyte conductance was investigated, but the important special case of a CNT was not addressed: the system was treated as being purely classical and therefore the capacitances considered were the geometrical one coming arXiv:2307.12071v1 [cond-mat.soft] 22 Jul 2023 from the insulating layer, together with the Stern and non-linear Graham capacitances of the electrolyte. In contrast to these previous studies we show here that by examining the confined electrolyte conductivity the CNT QC can provide a direct adjustable handle on the all important, but elusive, nanopore surface charge density. By shedding light in a fundamental way on ion transport in this type of nanofluidic system, it should be possible to pave the way to the tuning of transport properties for particular practical applications.

Vch [V] CQ [F/m] [C/m 2 ] σ Q E [eV] E [eV]
We briefly summarize the electronic characteristics of single-walled carbon nanotubes (SWCNTs) obtained by simplified theoretical methods and tight-binding calculations (see Ref. [START_REF] Biercuk | Electrical Transport in Single-Wall Carbon Nanotubes[END_REF] for more details). A SWCNT can be considered as a graphene sheet rolled up into a cylinder and its electronic nature, metallic (M) or semiconducting (SC), depends on its atomic structural characteristics fixed by two chiral indices (n, m) that determine its diameter and chirality [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF][START_REF] Biercuk | Electrical Transport in Single-Wall Carbon Nanotubes[END_REF][START_REF] Ilani | Measurement of the quantum capacitance of interacting electrons in carbon nanotubes[END_REF]. The simplified theoretical k • p method, which neglects curvature effects, reveals that in the M case, |n -m| = 3q, where q ∈ N. In the SC case, |n -m| ̸ = 3q and the size of the band gap is inversely proportional to the SWCNT diameter d [? ]. The band structure is composed of multiple 1D subbands sliced from the Dirac dispersion cone of graphene and, as shown by simplified theoretical analyses [START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF], exhibits a certain amount of (approximate) universality. The ith electron-hole subband has the following dispersion relation

E ± (i, k) = ± (ℏv F k) 2 + E 2 i , (1)
where v F ≈ 8×10 5 m/s is the Fermi velocity at the Dirac cone, ℏ the reduced Planck's constant, k the charge carrier wave vector, and

E i = 2ℏv F i/(3d).
The allowed values of i depend on whether the SWCNT is M or SC [START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF][START_REF] Miyake | Quasiparticle band structure of carbon nanotubes[END_REF]. The E i,g = 2E i are the individual energy gaps determined by the distance of the quantized k states from the center of the Dirac cone situated where both k and E vanish. The DOS per unit surface (the usual DOS per unit length divided by πd), for a SC SWCNT is given by [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF][START_REF] Biercuk | Electrical Transport in Single-Wall Carbon Nanotubes[END_REF][START_REF] Ilani | Measurement of the quantum capacitance of interacting electrons in carbon nanotubes[END_REF] g

SC (E) = g 0 i>0,̸ =3q |E| E 2 -E 2 i Θ(|E| -E i ), (2)
where g 0 = 4/(π 2 dℏv F ) and Θ is the Heaviside unit step function. The SC band gap is predicted to be

E g = E + (1, 0) -E -(1, 0) = 2E 1 = 0.7/d eV, if d is in nm (between 0.2 to 0.7 eV for 1 ≤ d ≤ 3 nm).
For purposes of illustration we choose here d = 1.5 nm for which E g = 0.47 eV (18 k B T at room temperature). The subbands for i = 3q (integer q) are not present [see Fig. 1(a)] and therefore the SC DOS exhibits van Hove singularities at

|E| = E i for i > 0, i ̸ = 3q.
For true M SWCNTs, E g = 0 and the first band has a non-zero DOS g(0) = g 0 and a half-width equal to E 3 as shown in Fig. 1(a)] [START_REF] Biercuk | Electrical Transport in Single-Wall Carbon Nanotubes[END_REF]. The k • p method leads to a DOS given by [START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF]

g M (E) = g 0 + 2g 0 m>0 |E| E 2 -(mE 3 ) 2 Θ(|E| -mE 3 ).
(3) The M DOS exhibits van Hove singularities at |E| = mE 3 for m > 0 and, aside from the conduction band, all bands are doubly degenerate.

We show in Fig. 1(a) the approximate theoretical DOS for d = 1.5 nm (dashed blue curves) and the DOS computed by the group of Maruyama [26] using the tightbinding method (solid orange curves) for a M CNT of chiral indices [START_REF] Gao | Voltage Charging Enhances Ionic Conductivity in Gold Nanotube Membranes[END_REF][START_REF] Gao | Voltage Charging Enhances Ionic Conductivity in Gold Nanotube Membranes[END_REF] At thermal equilibrium without external perturbations the chemical potential is situated at the Dirac point taken as the reference µ = 0 eV. When a gate voltage V g is applied to an electrode posed on a CNT, a shift in the chemical potential, ∆µ, occurs and the occupation of the allowed levels is modified, which leads to the creation of charges (electrons or holes) in the various subbands labeled by i. The CNT DOS plays a crucial role in determining the amplitude of this charge. The density of electrons (number per unit surface), n, for a SWCNT is calculated as a function of temperature T and ∆µ using Fermi statistics [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Mintmire | Universal Density of States for Carbon Nanotubes[END_REF][START_REF] Biercuk | Electrical Transport in Single-Wall Carbon Nanotubes[END_REF][START_REF] Ilani | Measurement of the quantum capacitance of interacting electrons in carbon nanotubes[END_REF],

n(∆µ) = ∞ Eg/2 g(E) 1 + exp[β(E -∆µ)] dE, (4)
where β = 1/(k B T ) (k B T ≈ 25 meV at room temperature). The density of holes, p, is given by Eq. ( 4) by replacing -∆µ by ∆µ. The chemical contribution to the applied gate voltage (electromotive force), V ch = V g -ψ 0 engenders a charge per unit surface,

σ Q (V ch ) = e(p -n), (5)
and is related to the chemical potential shift through ∆µ = -eV ch , where ψ 0 is the electrostatic potential drop between the CNT and the reference electrode. Note that σ Q = 0 for V ch = 0, σ Q > 0 for V ch > 0, and σ Q < 0 for V ch < 0 [START_REF] Zhang | Band Filling and Cross Quantum Capacitance in Ion-Gated Semiconducting Transition Metal Dichalcogenide Monolayers[END_REF][START_REF] Berthod | Theory of cross quantum capacitance[END_REF]. The surface charge densities σ Q (V ch ) for SC and M CNTs are plotted in Fig. 1(b). For V ch > 1 V, they almost superimpose, showing some slope discontinuities associated with the van Hove singularities of the associated DOS. As seen in the inset, however, for V ch < 1 V the two are quite different, essentially because the charge density vanishes in the gap for the SC CNT, i.e. for V ch < E g /(2e) = 0.23 V.

We distinguish between the integral QC needed here,

C Q ≡ |σ Q | V ch , (6)
which directly determines the charge on the CNT, and the differential one,

C d Q (V ch ) ≡ ∂|σ Q |/∂V ch ,
which gives access directly to many features of the DOS. The integral capacitances for SC and M CNTs are plotted in Fig. 1(c).

If |V ch | < E 3 /e holds, then C ≃ C 0
Q for M CNTs and C Q = 0 for SC ones. For larger |V ch |, similarly to σ(V ch ), the two integral capacitances are quite similar.

In the simplified description we adopt here (absence of any residual geometrical capacitance arising from the rest of the electrical circuit) the gate voltage, V g = V ch + ψ 0 , is shared between the chemical contribution and the electrostatic one. The equivalent electrical circuit is shown in Fig. 2(a), the total capacitance, C tot , and amplitudes of each contribution to V g depend on the individual capacitances and are given by [START_REF] Heller | Electrochemistry at Single-Walled Carbon Nanotubes: The Role of Band Structure and Quantum Capacitance[END_REF][START_REF] Fang | Carrier statistics and quantum capacitance of graphene sheets and ribbons[END_REF][START_REF] Zhang | Band Filling and Cross Quantum Capacitance in Ion-Gated Semiconducting Transition Metal Dichalcogenide Monolayers[END_REF][START_REF] Berthod | Theory of cross quantum capacitance[END_REF]:

1 C tot = 1 C Q + 1 C P = V g |σ Q | , (7)
where C P = |σ Q |/ψ 0 is the non-linear integral pore capacitance arising from electrolyte confined to the interior of a CNT, which leads to:

ψ 0 = V g C Q (V ch ) C Q (V ch ) + C P (ψ 0 ) . (8)
This result shows clearly that if C Q ≫ C P then ψ 0 ≲ V g and V ch ≪ V g , indicating that quantum effects are negligible. It corresponds to the "classical" limit. Conversely, if C P ≫ C Q , then ψ 0 ≪ V g and quantum effects are dominant.

To calculate the electrolyte capacitance, C P , we use an interpolation formula for the electric potential at the pore wall, ψ 0 , that we previously devised for small nanopores [START_REF] Manghi | Role of charge regulation and flow slip in the ionic conductance of nanopores: An analytical approach[END_REF]. This formula for ψ 0 as a function of the nanopore radius R (= d/2), electrolyte salt concentration c s , and nanopore surface charge density, σ P , becomes exact in the good coion exclusion and homogeneous limits and is an accurate interpolation elsewhere:

sinh ψ0 = σ * P (1 + σ * P ) cs , (9)
where ψ0 = βe|ψ 0 | and

cs = πℓ B R 2 c s , σ * P = πRℓ B e |σ P | (10)
are respectively the dimensionless electrolyte salt concentration and surface charge density (the electric potential at the reference electrode in the bulk reservoir is taken to be zero). The Bjerrum length is ℓ B = e 2 /(4πϵk B T ) ≃ 0.7 nm in bulk water at room temperature (dielectric permittivity ϵ = 78ϵ 0 ). Eq. ( 9) can be rewritten by isolating σ * P as

σ * P = 1 2 1 + 4c s sinh ψ0 -1 . (11)
For a classical metal (effectively infinite DOS), formally C Q /C P → ∞ and using Eq. ( 8) we see that the surface charge density σ * cl is given by Eq. ( 11) with ψ 0 set to V g . In this case for low enough cs and V g , σ * cl ≃ cs sinh(βeV g ) and for high enough cs and V g ≫ 25 mV, σ * cl ≃ c1/2 s e βeVg/2 . The rapid exponential rise in surface charge density in the latter limit appears to be compatible with the extremely high conductivities observed in [START_REF] Gao | Voltage Charging Enhances Ionic Conductivity in Gold Nanotube Membranes[END_REF] for gold plated nanopores at negative gate voltages (for which specific ion adsorption can be neglected). Given the low DOS of CNTs we do not expect such classical exponential behavior for them. In this case by keeping σ * P constant, ψ0 is low when cs is high, which increases the pore capacitance and eventually leads to C P ≫ C Q , i.e. quantum effects become important.

By defining C P = 4ϵ R CP , the integral pore capacitance can be found from the dimensionless one: For ψ0 ≪ 1, CP → cs and C P → ϵR/(2λ 2 DH ), where λ DH = (8πℓ B c s ) -1/2 is the bulk Debye-Hückel screening length for a monovalent electrolyte. In this limit the electrolyte filled pore acts as a parallel plate capacitor of capacitance ϵ/d eff with effective thickness d eff = 2λ 2 DH /R. Now by writing σ ≡ σ Q = σ P and combining σ Q (V gψ 0 ) from Eq. ( 5) and Eq. ( 9), we obtain cs (V g , ψ 0 ), which enables us to trace a parametric plot of σ as a function of c s for fixed gate voltage V g by varying ψ 0 from 0 to V g . This is shown in Fig. 2(b) for the SC and M cases for V g = 0.2, 1, 2 and 3 V. For V g = 0.2 V, the surface charge density is two orders of magnitude larger for the M CNT than for the SC one. This is because V g is just below E g /2 (gap half-width). At this relatively low value of V g the surface charge density increases quickly with c s at low concentrations. For higher gate voltages, σ increases very slowly with c s , being almost constant for V g ≥ 2 V. For V g = 1 V (blue curves) σ(SC) > σ(M) at low c s because ψ 0 is high and V ch is too low to reach beyond the SC van Hove singularity occurring at E g /(2e) = 0.23 V [see Fig. 1(a)]. For large c s , however, ψ 0 becomes lower and V ch ≃ E 3 /e = 0.7 V higher, now reaching the large first van Hove singularity for the M case.

CP = σ * P ψ0 = 1 + 4c s sinh ψ0 -1 2 ψ0 ( 12
) (a) (b) (c)
The pore conductance G is related to the conductivity κ through G = πR 2 κ/L where L is the pore length. The pore conductivity, κ, as a function of σ * and cs can be obtained using the result obtained in Eq. 19 of Ref. [START_REF] Manghi | Role of charge regulation and flow slip in the ionic conductance of nanopores: An analytical approach[END_REF], which takes into account the electrical migration contribution (first two terms) and the electro-osmotic one (third term), including flow slip at the pore surface:

κ = (μ + + μ-)c s 1 + σ * cs 2 -sgn(σ)σ * (μ + -μ-) + 2σ * 1 - ln(1 + σ * ) σ * + 2 bσ * , (13)
where κ = (2π 2 R 2 ℓ 2 B ηκ/e 2 ) is the dimensionless conductivity (η = 8.94 × 10 -4 Pa.s being the water vis-cosity), μ± = 2πηℓ B µ ± the dimensionless mobilities of ions (taken equal to the bulk values, µ + ≈ µ -= 5×10 -11 s/kg for KCl), and b = b/R is the ratio between the slip length b and the pore radius [sgn(σ) denotes the sign of σ]. In Eq. ( 13) the conductivity κ(c s , V g ) is a function of both c s and V g because σ(c s , V g ) is a function of both these quantities.

One observes in Fig. 2(c), that the conductivity is enhanced compared to the bulk case for intermediate and low salt concentrations. This is the signature of a nonzero surface charge density. For V g > 1 V, this enhancement even persists to high c s , attesting to a high surface charge density. At low salt concentration, κ in Eq. ( 13) is controlled by σ (good coion exclusion regime) and we therefore observe in Fig. 2(c) an increase in conductivity similar to the one observed for σ(c s ). We have checked that for V g > 1 V, σ * > cs over the whole concentration range and κ is therefore controlled by the first electrical migration term in Eq. ( 13), which is one order of magnitude larger than the electro-osmotic one (the second term is negligible since µ + ≈ µ -for KCl). For the M case and V g < 1 V (solid grey line), κ increases with c s , but much more slowly than the bulk one, with an apparent exponent smaller than 1/2.

By taking a slip length large enough, the electroosmotic contribution in Eq. ( 13) associated with the fluid slip at the nanopore surface, 4 bσ * 2 , can eventually dominate the electrical migration one. For instance, if a reasonable value of b = 40 [START_REF] Kannam | Modeling slip and flow enhancement of water in carbon nanotubes[END_REF] is taken, the electro-osmotic contribution is greater by one order of magnitude over the whole concentration range. The conductivity would then be completely controlled by σ * (c s ) [plotted in Fig. 2(b)], as a more complete study (underway) will reveal.

Changing the concentration in micrometric reservoirs is not easy in practice. One could measure more easily the electrolyte conductivity at fixed c s by varying V g . To present the conductivity in this way, we first compute ψ 0 by solving numerically σ Q (V g -ψ 0 ) = σ P (c s , ψ 0 ) using Eqs. ( 5) and ( 11) at fixed V g and c s . We then re-inject ψ 0 in σ P , which itself can finally be injected into Eq. ( 13). In Fig. 3(a) we plot the conductivity as a function of V g at fixed c s = 10 -3 , 10 -2 , 10 -1 , 1 mol/L from bottom to top. We confirm that κ varies strongly at low gate voltages. For V g = 0, σ = 0 and κ ∝ c s (bulk conductivity), which explains the staggered values (separated by a decade) at low V g . The distinction between the three cases, semiconductor, quantum metal and "classical" metal is clear at low V g , where the QC suppresses the "classical" exponential growth of conductivity with V g for M and SC CNTs and the SC energy gap suppresses the conductivity with respect to that of the quantum metal. The M CNT conductivity follows the classical one at very low V g within a window the decreases in width with increasing c s . At high V g all M and SC curves are almost superimposed.

[C/m 2 ] σ [S/m] κ ∂κ/∂V g g(V ch ) (b) (c) (a)
Even if the quantum surface charge density σ, has the same derivative as the classical one at V g = 0, it becomes quickly linear σ ≃ C 0 Q V g , with a lower slope up to V g ≃ E 3 /e ≃ 0.7 V, as shown in Figure 3

(b). Indeed, for not too low V g , C Q ≪ C P , which implies that C tot ≃ C Q ≈ C 0
Q and therefore ψ 0 ≪ V g . We have checked that this is true whenever

C P /C 0 Q ≫ 1, which leads to √ cs > C0 Q . Since C0 Q = 0.
03, this last inequality is verified for cs > 10 -3 [the case for the concentrations shown in Fig. 3(a)]. For lower reservoir salt concentrations, σ is no longer linear in V g . These different behaviors are illustrated in the inset of Fig. 3(a) where we also note the impact of the SC CNT gap, which leads to a constant κ up to V g = 0.2 ≃ E g /e V, the edge of the gap. We conclude that the QC has a considerable effect on both σ and κ as soon as V g > k B T /e ≈ 25 mV and this over the whole experimental range of c s .

The structure of a conductivity curve κ(V g ) exhibiting slope changes, related to the DOS structure, is easier to visualize by plotting its derivative κ ′ (V g ) [see Fig. 3(c)] and superimposing on it the DOS g(V ch ) (in orange). Clearly the peaks in κ ′ (V g ) are directly connected to the van Hove singularities of the DOS with an offset to the right which increases when c s decreases. Indeed, since κ is governed by the surface charge density σ Q which depends only on V ch = V g -ψ 0 , the higher ψ 0 is (and therefore the lower c s is), the more κ(V g -ψ 0 ) is shifted to the right. We can therefore directly measure the potential ψ 0 in Fig. 3(c). For instance, for c s = 10 -3 mol/L and V g = 1.9 V, we get ψ 0 ≃ 0.32 V. This is a way to readily check that for these c s values, ψ 0 ≪ V g , thus confirming the central role played by the QC on the conductivity.

As an concluding example, we consider the experimental results obtained by Liu et al. [START_REF] Liu | Ultrashort single-walled carbon nanotubes in a lipid bilayer as a new nanopore sensor[END_REF], who measured, for c s = 1 mol/L, conductances of 61.0 nS for M SWC-NTs and 5.6 nS for SC ones with 0.8 ≤ d ≤ 2 nm and 5 ≤ L ≤ 10 nm. Using the DOS given in Fig. 1(a) with d = 1.5 nm and L = 8 nm, we can account for these two conductance values by taking V g ≃ 0.35 V, a value that we interpret as an environmentally induced shift in the zero of the gate tension.

More systematic experiments are clearly needed to ascertain to what extent the charge density and therefore the electrolyte conductivity through SWCNTs can be controlled by an applied gate voltage. Presumably, a more complete model for conductivity would be needed to account for the full complexity of real CNTs, including the influence of pH (via charge regulation), residual geometrical capacitances, and dielectric interactions.

Figure 1 .

 1 Figure 1. (a) Density of states (DOS) g(E) for metallic (M,left) and semi-conducting (SC,right) CNTs computed by the tightbinding method (solid orange lines) and the k • p approximation (blue dashed lines). (b) Quantum surface charge density and (c) integral QCs vs. the chemical contribution, V ch for these M (solid curve) and SC (dotted curve) CNTs.

  (d = 1.512 nm) and for a SC CNT of chiral indices (19; 0) (d = 1.508 nm). It is clear that the approximate DOS universality breaks down for |E| > 1 eV where the actual DOS begins to depend on the values of the chiral indices.

Figure 2 .

 2 Figure 2. (a) Equivalent electrical circuit of the system with the quantum capacitance CQ and the pore one CP. The gate voltage, Vg = ψ0 + V ch , is divided between the potential at the inner CNT surface, ψ0, and the chemical contribution, V ch . (b) Dimensionless surface charge density σ * and (c) conductivity of a KCl electrolyte vs. salt concentration cs (Log-Log plot)for the M (solid lines) and SC (dashed lines) CNTs with Vg = 0.2 (grey), 1 (blue), 2 (red), and 3 V (green). The thick line is the bulk conductivity.

Figure 3 .

 3 Figure 3. (a) Electrolyte conductivity κ vs. gate voltage Vg in M (solid lines) and SC (dashed lines) CNTs for reservoir salt concentration values of (from bottom to top) cs = 10 -3 , 10 -2 , 10 -1 , 1 mol/L. The classical metallic CNT is shown in dotted-dashed lines. The inset is a zoom for cs = 10 -3 mol/L. (b) Associated surface charge density vs. Vg for the M CNT for the same cs values as in (a). The SC CNT is shown in the inset. (c) Derivative of the electrolyte conductivity, κ ′ (Vg), in the M CNT for the same cs values as in (a). The DOS g(V ch ) is superimposed in orange. To illustrate how to read off ψ0 we show it and V ch for the case cs = 10 -3 mol/L and Vg = 1.9 V.


Camille Maumet

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Computational reproducibility

Ability to reproduce the exact same results using the same data and the same methods.

Making your project reproducible

[

  Wilson et. al, PlosCompBio 2017] • Use version control • Create 1 repository per manuscript [Wilson et. al, PlosCompBio 2017] • Use version control • Create 1 repository per manuscript • Archive your code for long term preservation It's not only about the source code but also the environnement (versions, OS, etc.) docker, guix 10 years computational reproducibility is an open question! Perkel, Jeffrey M. "Challenge to scientists: does your ten-year-old code still run?." Nature 584.7822 (2020): 656-659. [Wilson et. al, PlosCompBio 2017] • Use version control • Create 1 repository per manuscript • Archive your code for long term preservation 4 (CC0), Parcel 3 (CC0), Parcel 5 (CC0). Make your project collaborative Image credits: Community by Vkw.studiogood, Commons Wikimedia Beyond technical solutions, open science brings a culture change • Making resources available to the scientific community • Reusing existing resources (datasets, tools) that were created by others towards more collaborative research

Hum… I can't reproduce the original results…

	ion		Which release of Which release of
			SPM was used? SPM was used?
					Which registration Which registration
					algorithm was algorithm was
					selected? selected?
				How was the brain How was the brain
	ration			mask computed? mask computed?
				… or is this due to … or is this due to
	ata	Hum… I can't	Did I understand the method Did I understand the method	my use of a different dataset? my use of a different dataset?
	nalysis	reproduce the original results…	correctly? correctly?	… a different computing … a different computing platform? platform?	… something … something else!? else!?

Making your project open

Open source

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28

Open source But what is open code…?

29

Open source Who can contribute?

I n c r e a s e d c o l l a b o r a t i o n

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Open governance

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Open development @cmaumet @cmaumet@fediscience.org


Philippe Camail

Christian Martin

Bruno Allard

Charles Joubert

Maxime Darnon

João Pedro

F Trovão

A Novel Partial Power Cascaded DC/DC Topology for CPV Application: A Theoretical Study

Keywords: CPV, DC/DC converter, DPC, PPC, flyback active clamp, cascaded topology, partial power processing

This paper presents a novel partial power DC/DC topology for CPV applications. The context of the CPV requirements is first derived in terms of losses factor, and compared to traditional centralized topology. An additional cascaded DC/DC stage is then proposed. It consists of Delta Power Converters used for current independence cascaded with Partial Power Converters for string voltage independence. The theoretical explanation and example using realistic shading pattern are presented. Last, the increased yielded energy provided by the complete cascaded topology is shown.

I. INTRODUCTION

Concentrator photovoltaic (CPV) is an active field of development to increase the efficiency of photovoltaic conversion. At a panel level, prototype achieve photo-electric efficiencies up to 38.9% [START_REF] Green | Solar cell efficiency tables (Version 58)[END_REF], which largely exceed capacities of traditional single junction silicon-based photovoltaic (PV) panels.

To do so, the main technological trend of CPV is the use of multijunction (usually three) of III-V semi-conductors. Due to the manufacturing complexity of such junctions, CPV cells size is reduced by the addition of a lens which concentrates the solar irradiance into a smaller surface. This lens-based solution limits the CPV panels to the use of Direct Normal irradiance (DNI). A solar tracker, that aligns at every moment the sun irradiance with the focal point of the lens, becomes mandatory.

This leads to higher initial cost for CPV plants [START_REF]Handbook of concentrator photovoltaic technology[END_REF]. Hence, to reduce cost, industrials have chosen to directly transpose well used electrical solutions present in PV fields to extract energy. Indeed, most of CPV plants consists of panels put in series to form a string, and then those strings are placed in parallel to a common DC bus. This DC bus is then linked to the input of grid-tied inverter which manages both the conversion to the AC grid and the maximum power point tracking (MPPT) algorithm. Yet, this current solution shows limits in efficiency.

To address this limits, the addition of a conversion stage between the panels and the inverter is a well-known path. However complexifying the electrical energy capture approach must fulfill an economical balance. The paper details a simplified economical approach to evaluate the gain in efficiency of an improvement in the DC/DC part of CPV power management. Test cases are presented to illustrate the methodology and discuss the potential benefit.

II. ECONOMIC APPROACH

A. Losses reduction

As mentioned, the panels-to-inverter structure has intrinsic losses. To quantify them, we introduce the performance ratio (PR), which encompasses all the losses added by a given system. This PR ratio is then multiplied to the total energy at disposal, factoring the DNI received to obtain the energy that can be retrieved from the installation.

P R = I i=1 (1-L i ) (1)
The L i (in %) are a generic way to represent both electrical and optical losses undergone by real CPV plants. The L i parameters usually used by companies to determine the efficiency of their plants [START_REF] Gerstmaier | Large-scale and long-term CPV Power Plant Field Results[END_REF] are: Shading, Cell temperature, Lens temperature, DC wiring, Soiling, Mismatch, Inverter, AC wiring, Transformer, Auxiliary consumption and Unavailability. This list gives, for an industrial operation, a global PR between 0.8 and 0.9. In order to perform a quantitative analysis, we will use as reference value PR REF = 0.8, without loss of generality regarding the subsequent conclusion. The interest of any work of improvement should aim at the decrease of one or more of the L i parameters, to increase the performance factor, and thus on the total energy generated by the CPV (see (3)).

A promising solution to decrease some of these parameters is the diminution of the granularity of conversion to panel level. Indeed, some of the previous L i parameters are due to the serie/parallel interconnection of panels and the centralized management of MPPT. It leads to power trade-offs that decrease the performance of the installation. By adding a conversion stage to the panel level, we can attenuate these trade-offs. This work focuses on reducing three L i factors: shading losses, mismatch losses and unavailability.

1) Shading Losses: Shading losses are defined around two notions, losses by effective decrease of irradiance and losses caused by the disparity of MPPT strings voltages on the same tracker, related to the non-uniformity of shading patterns. It is evaluated by [START_REF] Rodrigo | Balancing the shading impact in utility-scale dual-axis tracking concentrator photovoltaic power plants[END_REF] as 3.5% (L shadREF ) with current industrial connection. According to the same paper, by decreasing the granularity at the panel level, these losses decrease to 0.2% (L shad ) (microinverters are used for this modeling, but our approach detailed in the following pages keeps the same panel level conversion).

2) Mismatch losses: Mismatch losses are related to the angular dispersion of a module with respect to its neighbors connected in series. In [START_REF] Kim | Power conversion in concentrating photovoltaic systems: central, string, and micro-inverters: Power conversion in CPV systems[END_REF], authors define its impact around 4% (L mismREF ), taking as a classical assumption a 0.4°angle change between the panels. The results presented in [START_REF] Rodrigo | Analysis of electrical mismatches in high-concentrator photovoltaic power plants with distributed inverter configurations[END_REF] claim that these losses are divided by 4 in the case of a conversion at the panel level. Therefore, we use L M ism = 0.01.

3) Unavailability losses: The losses related to the unavailability of production (L unavailREF ) due to maintenance are around 1% [START_REF] Gerstmaier | Large-scale and long-term CPV Power Plant Field Results[END_REF]. It is assumed that in the case of monitored converters (measurement of electrical parameters on panel), curative maintenance is no longer done blindly and that therefore, we can reduce downtime by half. We therefore take L unavail = 0.005.

B. Addition of an intermediate DC stage

The addition of a DC/DC stage at panel level enables to keeps the industrially-proven installations (medium voltage high power grid tied inverters and related auxiliaries) and is synonym of a cost constrained choice. It also allows to keep others L i unchanged. However, by adding an intermediate stage of conversion, we necessarily add conversion losses. To consider a specification for this conversion stage, we define a general DC efficiency that we vary from 90% to 99%. This translates to an added L i (L DC ) from 0.1 to 0.01 respectively.

To place this solution in context with the current market, the Levelized Cost of Energy (LCOE) metric is used. In the simplified LCOE model used (unless otherwise stated, all formulas are from [START_REF]Handbook of concentrator photovoltaic technology[END_REF]), it is possible to vary in input the efficiency of the conversion stage, the annual irradiance received, as well as the additional investment cost generated. For this last parameter, it is set to zero to have a minimum mandatory efficiency for the additional DC stage to prove economically useful. Indeed, if the LCOE of the added stage is superior to the LCOE of the existing installation, even with virtually no added cost, then we can conclude that it has no interest. The formula used is:

LCOE = LCC N n=1 E CP V (1-d) n (2)
where LCC is the CPV system cost (in C) for a prospective life cycle N (in years), E CPV is the annual energy obtained by the CPV system, and d the annual discount rate. The annual energy produced by the studied CPV system follows: where DNI a is the available energy per year by the direct normal irradiance, P * is the power evaluated at the CSOC (Concentrator Standard Operating Conditions) available on the installation, DNI CSOC the direct irradiance fixed at 0.9kw/m 2 . Fig. 1 presents the LCOE versus yearly energy received considering the conditions described above. We can see a difference of up to 0.9c C/kWh between the current CPV plants (in dotted gray) and an added conversion stage with 99% efficiency (in purple).

E CP V = DN I a • P * • P R DN I CSOC (3)
Nevertheless, with 90% efficiency (orange curve), we see that the calculated LCOE is higher than the current plants. More generally, this model indicates us that below 93% efficiency, an added DC/DC stage presents no economic advantage. This value represents therefore the minimum mandatory efficiency described above.

C. Opportunities

It is difficult to think in terms of absolute cost, predictive LCOE being related to a large number of economic assumptions and simplifications. It also heavily depends on the studied country . However, we saw that with high efficiency systems, few tens of cent per kWh could be saved. Recent worldwide assessment study [START_REF] Talavera | A worldwide assessment of levelised cost of electricity of HCPV systems[END_REF] shows that the difference between LCOE of PV and CPV systems (as of 2020) is frequently below 1c C/kWh in numerous countries (see [START_REF] Talavera | A worldwide assessment of levelised cost of electricity of HCPV systems[END_REF] -Table A2). A highly efficient system can then theoretically permit a larger number of country and markets to consider deriving benefits from installing CPV plants when PV plants are affordable.

The new topology detailed below proposes to tackle both the cost (with a relatively low component count) and the efficiency (with partial power processing) challenges. It is the cascading of DPC converters with PPC. Both are well known for their extreme efficiencies (an average of 99% [START_REF] Schaef | Multilevel Power Point Tracking for Partial Power Processing Photovoltaic Converters[END_REF] for DPC and a peak of 96.6% [START_REF] Camail | Application of DC/DC Partial Power Conversion to Concentrator Photovoltaics[END_REF] for PPC). Their simultaneous use is depicted in the following section.

III. PRESENTATION OF TOPOLOGY

In order to reduce the three L i parameters of interest, the chosen cascaded conversion solution is detailed in this section.

A. Delta Power Converter

The Delta Power Converter (DPC), also referred as Parallel Partial Processing Converter (P-PPC), is progressively used in led conversion, batteries management systems and solar installations [START_REF] Khan | Review and qualitative analysis of submodule-level distributed power electronic solutions in PV power systems[END_REF]. In these domains, it allows to build up high DC voltage string by placing in series different units without the necessity of same-current sharing. For two units placed in series, the difference of currents between them is processed by the adjacent DPC converter.

In the context of PV study, if the total string voltage is the sum of the maximum power point voltage of each panel, the DPCs action (through the use of MPPT control) allow for each panel to work at their maximum power point current, which imply a yielded power calculation with the following equation:

P String = N panels i=1 P M P P i (4)
For n units, n-1 DPCs are required to allow each unit to be current independent from one another. In this case, current processed by each DPC is a function of all the current mismatches present in the string of interest. In other words, even if the DPC managing two equal-current units is part of a larger string displaying current mismatches, it will process non-zero current. This leads to relatively high currents (and hence higher losses) processed by each DPC in the case of a series string consisting of numerous elements. In addition, the more units put in series, the higher the processed current. However, in the case of CPV, the already relatively high voltage of each unit (approx. 100V) in comparison to the targeted DC voltage (500V here), places us in the case of 4 CPV units for 3 DPC, which lowers this inter-dependency effect. In terms of power electronic systems, the most largely used DPC topology is based on a bi-directional, synchronous rectifier buck-boost architecture. The voltage rating is compatible with the use of Gallium Nitride (GaN) semiconductors in half bridge configuration.

For CPV, this means that mismatch losses related to the angular dispersion of a module with respect to its neighbors can be neglected, given the current independence. Concerning partial shading losses, if every cell of the module is still active (no cell sees 0W/m 2 ), this effect of maximum power point finding still holds.

However, if some cells see no irradiance at all, and that the pattern differs from one string to another, the simple parallel interconnection of strings with DPC cannot overcome the voltage difference [START_REF] Khan | Review and qualitative analysis of submodule-level distributed power electronic solutions in PV power systems[END_REF]. In this situation, the use of an additional converter is necessary.

B. Partial Power Converter

Due to the numerous amounts of bypass diodes present in CPV modules, partial shading patterns lead to important n DC DC Fig. 2. DPC buck boost topology Fig. 3. Flyback active clamp topology differences of voltages between strings. Usually, with the traditional approach of CPV in series and strings in parallel, a global MPPT algorithm decreases/increases the DC bus voltage to a trade-off voltage. This likely results in whole irradiated panels or sometimes strings disconnected to the benefit of more irradiated ones. Ensuring a way to overcome this problem is particularly critical as a commonly shading pattern in CPV plants is inter-tracker shading [START_REF] Rodrigo | Analysis of electrical mismatches in high-concentrator photovoltaic power plants with distributed inverter configurations[END_REF], which systematically occurs at dawn and twilight. This is the role of the second stage of this cascaded topology.

The Partial Power Converter (PPC) consists of a flyback active clamp (FBAC) [START_REF] Perrin | High-Temperature GaN Active-Clamp Flyback Converter With Resonant Operation Mode[END_REF] in input parallel output series. This way of plugging is depicted at the top of Fig. 5. The partial power link implies that the low side output of the converter is set to the high side voltage of the input, placing its output effectively in series with the source (here, a string of panels). By doing so, the current used by the secondary of the FBAC is directly collected from the high side of the input, hence the partial power processing. The ratio of partial power management depends on the voltage gain of the converter between the input (the voltage of the string) and its output (the difference between the input and the DC bus voltage). The closer the input voltage is to the DC bus voltage, the less power is processed through the converter. Therefore, it allows an overall better efficiency (the losses are only applied to the processed power) and a smaller power rating than the equivalent converter without the partial power processing feature. A fully detailed design of this converter applied for the CPV requirements can be found at [START_REF] Camail | Application of DC/DC Partial Power Conversion to Concentrator Photovoltaics[END_REF]. The advantage of FBAC topology resides in its low component count (active and passive) and its ability to perform soft switching throughout most of its requirement in terms of power and voltage, if designed with care.

In addition, the input voltage is close enough to the total DC bus voltage, the FBAC can be disconnected (by stopping switching orders to its primary side) and the string is directly connected to the DC bus by the intermediary of its secondary diode. Dealing with DC currents, the presence of the secondary inductor can be neglected. This feature is also interesting in the case of converter disfunction. Indeed, if an active switch is lost in open circuit configuration, then the converter can be entirely bypassed the same way.

C. Complete cascaded stage

The complete topology proposed in this paper is represented in Fig. 4. It consists of the addition of the two stages presented above: DPCs for current independence between panels, and PPCs for voltage independence between strings. They both implement MPPT algorithms, and leading the inverter to only regulate the DC bus voltage to a constant operating point. This allows the inverter to work at an optimal DC where its efficiency is maximum.

However, control and communication are two concerns to take into account to ensure the simultaneous use of these topologies working with the same power source [START_REF] Qin | Sub-module differential power processing for photovoltaic applications[END_REF].

Concerning control, it should be pointed at that DPCs and PPCs control objectives are neither redundant nor coupled if designed carefully. The firsts control the current of each of their assigned panels by adjusting the voltage ratio between their input and their output. In fact, DPCs cumulated action within a string cannot increase nor decrease the total voltage of this string. This total voltage regulation is devolved to the PPC. Their action is decoupled if the frequency of the MPPT (and of the regulation itself) performed by DPC is fast enough compared with the MPPT of the downstream converter [START_REF] Schaef | Multilevel Power Point Tracking for Partial Power Processing Photovoltaic Converters[END_REF].

This point is also valid concerning communication, so following this requirement, there is no need for data exchange between DPCs and their PPC. In addition, some work has been already achieved in recent years [START_REF] Badawy | A Novel Differential Power Processing Architecture for a Partially Shaded PV String Using Distributed Control[END_REF], [START_REF] Qin | Sub-module differential power processing for photovoltaic applications[END_REF] to mitigate the need of communication only to neighbor-to-neighbor data exchange. This cancels the need for a more complex centralized controller.

IV. THEORETICAL RESULTS

A. Mismatch pattern

In order to check the relevance of the proposed topology, our study is synthesized in one irradiance pattern that encompasses the partial shadow mismatches of interest. Two strings of four panels in series are used, with different irradiances. String n°1 consists of (from top to bottom, as seen in fig. 5) one CPV panel fully irradiated at 1000W/m 2 , one fully at 800W/m 2 , one fully at 300W/m 2 , and a last one which sees on its first half 300W/m 2 and on its second half no irradiance (0W/m 2 ). String n°2 consists of fully irradiated panels at 1000W/m 2 , 1000W/m 2 , 800W/m 2 and 300W/m 2 respectively. Figure 5 recalls the I-V curves of the adjacent panels. The considered temperature of operation is 25°C, and bypass diode reverse voltage is taken into account. Three cases are depicted: irradiances. The 5 parameters model was derived using method depicted in [START_REF] Chan | Analytical methods for the extraction of solar-cell single-and double-diode model parameters from I-V characteristics[END_REF].

B. Results

1) Case 1: Current industrial solution: In this case, strings n°1 and n°2 are placed directly in parallel on the DC bus. Fig. 6 shows the I-V curves of each string. We can note 3 current drops with increasing voltages. They are representing each time when the most irradiated panels are forced to work at the next lower current, due to the "turn on" of a new less irradiated panel. For each string, the maximum power point is found at 207V and 338V respectively. These two voltage values are found again in the Fig. 7 depicting the I-V and the P-V curves of the complete set up (as seen by the theoretical inverter downstream). Here, these two voltages represent the two highest local maxima. The inverter performing the MPPT algorithm will choose (given the ability to systematically find the global maximum) the 338V voltage point.

It means in this case that two panels will be bypassed (the less irradiated of each string), and four panels will work at suboptimal currents (the two most irradiated of each string). The resulting power extracted is 37.5% less than the total power in presence. It can be noted that the multiple local maxima displayed in fig. 7 adds difficulty to the completion of the centralized MPPT algorithm. More advanced techniques than traditional Perturb & Observe (P&O) standard algorithm are mandatory.

2) Case 2: Addition of DPC: DPC are added to the two strings. The total voltage of the strings is still controlled by the DC side of the inverter, as said previously. Under the assumption that the MPPT of the DPCs is realized faster than the one of the inverter, (a few tens times faster), the I-V patterns of each string are seen by the inverter as smoothened. To sum up, only one maximum is found at each string, as seen in fig. 8. This results in only two local maxima for the total P-V curve presented in fig. 9. The global maximum is set at 407V. At this voltage value, all panels of string n°1 work approximately at their maximum power points, and all panels of string n°2 are working at above 90% of their capacity (see Table I.). Considering lossless converters, a panel-to-inverter efficiency of 94.5% is displayed. Still, the inverter-side MPPT has to perform a trade-off by decreasing the voltage of its DC side, affecting the ability of string n°2 to work at full power. 3) Case 3: Proposed cascaded topology: To solve this last issue, string voltage independence is given through the addition of PPC for each string. In this case, the I-V and P-V curves presented in Fig. 9 still hold, but the decoupled voltage allow to extract the maximum theoretical power by setting each string at their global optimal voltage.

As we can note in the third line of Table I, the collected power is the same as the total power at disposal. In order to get a more realistic idea of the yielded power, a last line with losses has been added in the table. It factors in efficiency results present in [START_REF] Schaef | Multilevel Power Point Tracking for Partial Power Processing Photovoltaic Converters[END_REF] and [START_REF] Camail | Application of DC/DC Partial Power Conversion to Concentrator Photovoltaics[END_REF]. We can see that compared to the current industrial solution depicted by case n°1, case n°3 (with losses) allows a very significant increase in the total power at disposal. The transition from 62.5% to 95.6% power yielding brings forward the interest of our cascaded topology. [START_REF] Schaef | Multilevel Power Point Tracking for Partial Power Processing Photovoltaic Converters[END_REF] and [START_REF] Camail | Application of DC/DC Partial Power Conversion to Concentrator Photovoltaics[END_REF]) 406W (95.5%) 571W (95.6%) 977W (95.6%) V. CONCLUSION In this paper, a simplified economical study has been achieved to evaluate the conditions of a gain in efficiency of an extra DC/DC stage. A condition of at least 93% efficiency has been drawn. This stage is considered to be added to an already existing centralized inverter configuration for CPV. A novel topology aiming to match these efficiency/cost conditions is presented. It is theoretically tested against a realistic, heavily mismatching shadowing pattern. Later work will include real time simulations of this topology, with a focus on angular mismatch of CPV panels. The aim of this simulation will be to derive an accurate loss calculation to include to the Performance Ratio presented in this work.

Fig. 1 .

 1 Fig. 1. LCOE in function of yearly energy received without added DC stage and with added DC stage

Fig. 4 .

 4 Fig. 4. Proposed cascaded partial power conversion topology

Fig. 5 .Fig. 6 .

 56 Fig.5. Two strings considered for this study; each panel has its I-V curve depicted next to it (all I-V curves share the same Voltage/Current scale to ease comparison)

Fig. 7 .

 7 Fig. 7. Total I-V (orange) and P-V (purple) curves as seen by the central inverter for case n°1

Fig. 8 .

 8 Fig. 8. I-V curves (left) of string n°1 (solid) and string n°2 (dotted), and corresponding P-V curves (right) for case n°2

Fig. 9 .

 9 Fig.9. Total I-V (orange) and P-V (purple) curves as seen by the central inverter for case n°2