Andi Cuko

Monica Calatayud
email: calatayu@lct.jussieu.fr

Stefan T Bromley
email: s.bromley@ub.edu

Stability of Mixed-oxide Titanosilicates: Dependency on Size and Composition from Nanocluster to Bulk

Stability of mixed-oxide titanosilicates: dependency on size and composition from nanocluster to bulk

Andi Cuko, Monica Calatayud, Stefan T

Introduction

Almost all elements of the periodic table form stable oxide compounds. Due to the importance and diversity of properties exhibited by oxides (e.g. (photo)catalytic activity, multiferroic and memristive effects, superconductivity, magnetism) they play an essential role in our society in established technologies, while continuing to be a focus of frontier research in many fields. The effectiveness and range of properties of an application based on a single oxide material can often be enhanced by using mixed oxides (i.e. materials comprised by the combination of two or more oxides). Examples of such synergistic behaviour in mixed oxides are well established in, for instance, cases of catalytic activity 1 and high temperature composites 2 . Depending on the application, one may need to mix the constituent oxides in a more or less intimate fashion. In some cases, homogeneously mixing oxides can be difficult due to their tendency to segregate into separate phases, typically occurring when the structures and properties of the respective bulk phases are very different. Moreover, it can be also difficult to obtain specific compositions of two oxides, for instance when high temperature calcination is involved in the mixed oxide synthesis. These effects can, however, be very dependent on sample size. For example, some materials that hardly mix at bulk macroscopic level at desired composition, can be synthesized and stabilized in mixed phases at the nanoscale 3,4,5,6 . Furthermore, at the nanoscale, system properties can often be more finely tuned by varying both size and composition. Here, we focus on a particular mixed oxide system, titanosilicates (i.e. mixed titania (TiO 2 ) and silica (SiO 2 ) systems). Titanosilicates are robust low cost materials that can combine mechanical strength, chemical stability, nanoporosity, controllable optical characteristics, photoactivity and high redox potential properties. This combination of features make titanosilicates ideal candidates for thermally stable high surface area materials that can be tailored for a range catalytic and optical applications (e.g. industrial heterogeneous catalysis, solar cells, optical components, self-cleaning systems, selective molecular sieves, water remediation, and photocatalysis 7,8,9,10,11,12,13 ). One important class of titanosilicates is that of the ETS (Engelhard Titanium Silicate) family of microporous titanosilicates introduced by Kuznicki et al. 14,15 . Among the numerous interesting properties of these materials, due to their tailorable pore dimensions, some are able to selectively adsorb and separate gas mixtures of small molecules with similar size, such as that of N 2 /CH 4 , Ar/O 2 and N 2 /O 2 . Formally, the titanosilicate frameworks of the ETS materials do not possess a stoichiometric mix of TiO 2 and SiO 2 , but as with the related aluminosilicate zeolites, have a net negative charge which must be compensated by cations.

Generally, it is relatively difficult to synthesize homogeneous stoichiometric mixed TiO 2 -SiO 2 bulk phases due to the tendency of the constituent oxides to segregate into two respective phases with increasing temperature. In crystalline systems, bulk homogeneous mixing seems to be only possible by the inclusion of a very small percentage of TiO 2 in a silica framework. For higher TiO 2 molar fractions homogeneous mixed TiO 2 -SiO 2 oxides seem only to be possible in amorphous bulk phases, which are metastable with respect to the separate respective pure oxide polymorphs. 16,17,18 There are several ways to synthesize titanosilicates. From a top down synthesis of mechanically crushing separate bulk phases of silica and titania and heating it is possible to obtain macroscopically homogeneous mixing. In order to have microscopically homogeneous mixing, however, bottom up synthetic methods such as sol-gel techniques are needed. 19 When larger proportions of titania (typically > 15 wt%) are employed these syntheses tend to lead to segregated (nano)composites of two distinct phases of TiO 2 and SiO 2 . Dong et al. 20 , for example, have synthesized a highly ordered 2D hexagonal mesoporous crystalline TiO 2 -SiO 2 nanocomposite with anatase TiO 2 nanocrystals embedded in silica pore walls which is thermally stable at 600-850ºC. This material exhibits an improved photocatalytic activity in the degradation of rhoamine B (RhB) compared to the commercial pure TiO 2 Degussa P25 catalyst. Jiang C. et al. 21 have also synthesized TiO 2 -SiO 2 titanosilicate composites via sol-gel routes where small TiO 2 anatase nanocrystals are incorporated into microporous ZSM-5 and mesoporous SBA-15 silica frameworks in order to better control their size and improve the titania photoactivity for hydrogen production by water splitting. In the class of non-segregated mixed TiO 2 -SiO 2 bulk oxides, one of the most industrially important materials, known from early 80s, is the synthetic titanosilicate-1 (TS-1) 7 . The active sites in TS-1 are uniformly dispersed and isolated four-fold titanium centers incorporated in a zeolitic silica MFI structure with 1.0-2.5% of TiO 2 molar inclusion. TS-1 is used as industrial redox catalyst for oxidizing organic molecules in mild conditions in the presence of hydrogen peroxide.

In this work we theoretically investigate the size-dependency of the energetics of TiO 2 -SiO 2 mixing. In particular, we focus on TiO 2 -SiO 2 mixing in small nanoclusters (i.e. <100 atoms) and how it compares with, and evolves with increasing size, towards the situation in corresponding bulk systems. In principle titanosilicate nanoparticles could be synthesised through use of ionized nanocluster beams but to the best of our knowledge no mixed silica-titania nanoclusters of the size studied herein have been reported to date. nanocluster beams are, however, becoming an important method for generating selected nanoclusters for catalytic and technological applications, 22,23,24 Takeuchi et al. 25 , for example, have synthesized thin layers of titanosilicates by deposition of TiO 2 and SiO 2 nanoclusters generated by a laser beam in which they observe a high photocatalytic activity towards decomposition of NO in N 2 and O 2 due to highly disperse Ti-sites embedded in a silica framework.

Previous theoretical studies of stoichiometric mixed TiO 2 -SiO 2 systems have tended to focus on crystalline bulk mixed oxide materials having a very small fraction of TiO 2 (e.g. TS-1) -see ref. 26 for an overview of theoretical and experimental studies in this area. Only very few works are present in the literature on the theoretical modelling of TiO 2 -SiO 2 mixing in systems with relatively high percentages of TiO 2 . Recently, Landmann et al. 27 reported an extensive and detailed computational modelling study of the structure of bulk amorphous titanosilicate glasses. Miroshnichenko et al. 28 have studied the effect of SiO 4 groups on optical, electronic, and structural properties when present in TiO 2 anatase-structured nanoparticles. For very small nanoclusters, Bandyopadhyay and Aikens 29 have studied a series of mixed Ti x Si y O 2(x+y) oxide nanoclusters with up to five formula units (i.e. x+y ≤ 5).

Confirming experiment, our calculations show that TiO 2 -SiO 2 mixing is energetically unfavourable in bulk crystalline systems. However, surprisingly, in nanoclusters possessing up to 70 atoms we predict that the mixing of TiO 2 and SiO 2 is thermodynamically favourable. Our results point to a number of important factors which determine the higher stability of nano-titanosilicates with respect to pure nano-silica and nano-titania. Linking the results for bulk and the nanoscale we also show that the energetics of TiO 2 -SiO 2 mixing depends on size; becoming less favourable with increasing size after a nanocluster size of ~30 atoms. With this work, we hope to inspire experimental groups, especially from the nanocluster beam community, to attempt the synthesis of the titanosilicate nanoclusters predicted here. To assist in the identification of mixed TiO 2 -SiO 2 nanoclusters in such experiments we provide calculated vibrational frequency and average coordination number signatures that indicate when intimate nano-oxide mixing is occurring.

Methodology

We consider both mixed titanosilicates nanoclusters (Ti x Si 1-x O 2 ) n , with a range of sizes (n = 2 -24) having different titania molar fractions (x = 0 -1), and a selection of crystalline solid phase systems (Ti x Si 1-x O 2 ) bulk . In the case of the nanoclusters, great efforts were made to try to find the most energetically stable isomer for each n and x. In order to find low energy nanocluster structures on the multidimensional potential energy surface (PES) we used the Monte Carlo basin-hopping (MCBH) global optimization algorithm 30 with classical interatomic potentials (IPs) to efficiently explore the PES as implemented in an in-house modified version of the GMIN 1.x code. 31 Previously, IPs have been employed for global optimisation of separate (TiO 2 ) n 32,33,34 and (SiO 2 ) n 35,36,37,38 nanoclusters.

However, to our knowledge, IPs for mixed titanosilicate nanoclusters have not been reported.

Specifically, we use the Buckingham pair interaction IP together with interionic electrostatic interactions:

(1)

where , are ionic charges, is the vacuum dielectric constant, is the interatomic distance between atoms and and , , are fitted parameters. Due to the structural complexity of the (Ti x Si 1-x O 2 ) n system we were not able to find one IP parameter set that could provide reliably in finding low energy nanoclusters for the full range of x considered. Thus, in order to resolve this issue, we employed two different Buckingham-type IPs (IP1 and IP2) with the respective parameters reported in table 1. These IPs are largely derived from modifications to parameters used in previously reported IPs for (SiO 2 ) n nanoclusters 35 and bulk TiO 2 39 . IP1 typically provides better low energy cluster isomer candidates than IP2 for systems with a higher content of TiO 2 and tends to give structures where Ti-centres are 4, 5 and rarely 6 coordinated depending on nanocluster size. For systems with a lower incorporation of TiO 2 this IP tends to yield 3-coordinated Ti atoms which are highly unstable. IP2 instead, generally yields lower energy isomer structures than IP1 for lower content of TiO 2 . In both cases, Si-centres tend to occupy 4-fold O-coordinated positions which are known to be to be the most energetically favourable for pure silica systems. We note that both IPs are able to provide reasonable structures for intermediate TiO 2 content thus the low energy PES of this compositional range is probably most well sampled by our approach. A comparison of the performance of both IP1 and IP2 with respect to the refined energetic stabilities of the isomers coming from a typical global optimisation run over the full range of TiO 2 content and for three system sizes (N= 8, 9 10) is proved in the Supplementary Information.

With both IPs there is a tendency to produce nanoclusters with Si-O terminations. It is known that for small nanoclusters both pure TiO 2 and SiO 2 prefer open structures with terminating oxygen defects compared to compact fully coordinated ones 33,36 . However, due to the relatively higher ionicity of Ti-O bonds, Ti-O terminations tend to be less energetically costly compared to Si-O terminations. Also, for relatively larger nanoclusters (n ≥ 12) there is a tendency for the IPs to produce some low energy nanocluster structures containing 5-or 6-fold coordinated silicon atoms which are relatively high in energy according to density DFT-based calculations (see details below).

Instead of simply discarding these anomalous nanoclusters, we found that they could be converted into more reliably low energy candidate nanocluster structures after the MCBH search by simple interchange of cations within the respective nanocluster following: 1) Si centres in terminating Si-O defects with a 4-coordinated Ti atoms, 2.) Highly coordinated Si centres (5-and 6-coordinated) with 4-coordinated Ti centres. We used MCBH with both IP1 and IP2 for finding low energy nanocluster isomers. We typically use several MCBH runs for each nanocluster composition (up to five for the larger nanoclusters) and for each run we use a different initial structure. With increasing nanocluster size, we correspondingly increase the number of MCBH steps in each run from fifty thousand to one million. In addition to the usual random atomic displacements we also apply Ti/Si cation interchange moves in a small percentage (1 -5%) of the steps. From each run of this IP-based global optimization procedure, for each nanocluster composition, the forty lowest energy structures were then fully optimized employing all electron, relativistic, DFT based calculations using the FHI-AIMS code. 40 Firstly, we employed the general gradient approximation PBE 41 functional with a light/tier-1 basis of numerical atom-centred basis functions, to optimise the IP-optimised nanocluster structures coming directly from the global optimisations. We note that light/tier-1 basis of numerical atom-centred orbitals approximately provides results of a similar or higher quality to those obtained with valence triple-zeta plus polarization Gaussian type orbitals. Subsequently, from these forty isomers, the seven most stable isomers were further optimized using the hybrid PBE0 functional 42 with tight/tier-1 basis. This cascade optimisation approach has been successfully used in previous studies 33,34,37,43,44 .

For the selected periodic (Ti x Si 1-x O 2 ) bulk crystals we performed an initial full optimization of both atom coordinates and cell lattice vectors at a PBE light/tier-1 level of theory with atom coordinates then further refined at a PBE0 tight/tier-1 level also using FHI-AIMS. For these periodic calculations, depending on the unit cell size, we used an appropriate Monkhort-Pack grid of k-points, going from 7x7x7 for the smallest system (pure rutile and quartz) to a 2x2x2 points grid for the largest one.

In order to compare relative stabilities of a specific mixed system with respect to the pure oxides, we consider the mixing energy, ∆ , which we take to be difference in the energy of the system with respect to the proportional sum of the energies of the two pure systems of the same size. Specifically, for a generic titanosilicate we refer to the chemical reaction between pure oxide systems as defined in equation 2 and the corresponding mixing energy in equation 3.

(1 -) + → (2)
∆ = -[(1 -) ( ) + ] (3)
where is the TiO 2 molar fraction in the nanocluster. is the total DFT energy of the mixed system, ( ) and ( ) are energies of the respective pure silica and pure titania global minima structures. ∆ can be either positive, which means that the silica and titania mixing is energetically unfavourable, or negative, which means that the mixing is favourable. Mixing energies are then normalized by the number of MO 2 (M=Si, Ti) oxide units. In the case of nanoclusters the number of units corresponds to its size n, while in the case of bulk systems we employ the number of oxide units in the unit cell. We note that normalization factor could also be chosen to be the number of the smallest titanosilicate units according to a specific composition (e.g. when x=0.5 the fundamental unit is TiSiO 4 , when x=0.333 the fundamental unit is TiSi 2 O 6 etc.). This normalisation choice unit is typically employed for reporting experimental mixing enthalpies of bulk systems. For example, conventionally, experimental mixing enthalpies of two common mixed MgO-SiO 2 magnesium silicates forsterite (Mg 2 SiO 4 ) and enstatite (MgSiO 3 ) are usually given relative to moles of Mg 2 SiO 4 and moles of MgSiO 3 respectively. 45 For titanosilicates, to the best of our knowledge, there are no experimental enthalpies of mixing because of the tendency of the constituent oxide phases to segregate. We note that the use of the MO 2 (M=Si, Ti) oxide unit also allows us to directly compare systems with different compositions on the same footing. In the Supplementary Information we report ∆ / (eV) according to both definitions of normalisation unit for a range of Ti 0.5 Si 0.5 O 2 systems. We note that the mixing energy in equation 3 is a thermodynamic quantity, therefore formally all terms used should be thermally corrected at finite temperatures. As a first approximation, we mainly consider ∆ in terms of 0K internal electronic energies directly coming from DFT calculations. However, in order to account for the effect of finite temperature we have also estimated the free mixing energies (∆ ) for a set of mixed titanosilicates nanoclusters with with 1:1 stoichiometry (x=0.5). For these calculations we evaluated the harmonic vibrational frequencies and resulting free energies with the Gaussian09 package [START_REF] Frisch | Gaussian[END_REF] using the PBE0 functional with the Ahlrichs triple-z plus polarizations basis functions (TZPV) 47 on all atoms.

Results and Discussions

1) Bulk mixed TiO 2 -SiO 2 systems

Firstly we took a selection of mixed (Ti x Si 1-x O 2 ) bulk crystalline systems where one or more 4-fold coordinated Si-centres were replaced with Ti atoms to generate mixed oxide bulk phases with different degrees of mixing (i.e. x=0.08, 0.33, 0.50 and 0.67 TiO 2 molar fraction). Specifically, we consider the quartz (Fig. 1a) and sodalite (Fig. 1b) crystal structure types. We choose quartz as representative of dense titanosilicates where we examined structures with x=0.33, 0.50 and 0.67.

The sodalite crystal with low (x=0.08) TiO 2 inclusion is taken to represent low density zeolitic titanosilicates such as TS-1. In the latter case we did not consider TS-1 directly because of the prohibitive computational cost of PBE0 DFT calculations with the large unit cell involved. In our selected set of bulk crystal titanosilicates, we also include a hypothetical phase having 4-fold Sicentres and 6-fold Ti-centres and an intermediate inclusion of TiO 2 (x=0.33, see Fig. 1c). This hypothetical structure was found in the Materials Project database (ID: mp-766596) 48 . We note that unlike quartz and sodalite, this latter structure exhibits one dimensional "wires" of linked TiO 6 units as found in the experimentally synthesised non-stoichiometric ETS-10 structure 10 . This high 6-fold coordination environment of Ti is also similar to that found in the most stable rutile TiO 2 bulk polymorph. We refer to this structure as the "wired" phase. We report a set of global optimized mixed (Ti x Si 1-x O 2 ) n titanosilicate nanoclusters with n = 2 -24 units. For n = 2 -10 we consider all possible values of molar fraction x. We also consider some larger nanoclusters with n = 12, 14, 16 and 24, for which selected mixing ratios have been chosen.

2) Global optimized mixed nanocluster structures

Fig 2.

Structures of globally minimised nanoclusters of silica (left) and titania (right), each labelled with the respect to the corresponding number of MO 2 (M=Si or Ti) units. All these structures have been previously reported in the literature by us and other authors. 32,33,34,36,37,38,49,50,51,52 Atom key: yellow -silicon, greytitanium, red -oxygen.

In Fig. 2 we show the lowest energy pure nanoclusters of silica and titania found previously 32,33,34,36,37,38,49,50,50,51 accurately: >Si + -O -). 53,54 Such defective terminations are very common in the global minima silica nanoclusters in this size range and are found for all sizes apart from n = 12 and n = 24. For n = 12 the terminal oxygen atoms are bonded to 4-coordinated silicon atoms. In this case they are known as non-bridging oxygen (NBO) atoms where, for each NBO, the nanocluster also possesses a compensating triple coordinated oxygen-site. The combination of these two sites is also known as a compensated NBO or a valence alternation pair 55,56 . Titania nanoclusters tend to have relatively more compact and amorphous structures with respect to their silica counterparts. Most of them tend to have two NBO-type terminal oxygen defects. Unlike for the pure silica nanoclusters 3-fold coordinated titanium centres (i.e. >Ti=O species) appear to be energetically unstable and the majority of the oxygen-terminated global minima nanoclusters exhibit only NBOs (except for (TiO 2 ) 2 where, due to the extremely small system size, only 3-coordinated Ti centres are possible). The global minima structures for (TiO 2 ) 10 and the largest n = 24 titania nanocluster considered are fully coordinated. As for silica this probably indicates a tendency for low energy nanoclusters to exhibit less energetically costly terminal defects with increasing size. 38 Generally, the respective global minima nanoclusters of (TiO 2 ) n and (SiO 2 ) n are quite distinct, with both systems displaying similar nanocluster structures only for sizes n = 2 and 12. previously reported in literature 29 , while all others are new to this study.

In Fig. 3 we report the lowest energy titanosilicate (Ti x Si 1-x O 2 ) n nanocluster structures found in the present study (i.e. global minima candidates) for a range of different mixing ratios, x, and sizes, n.

We note that only structures with sizes n = 2 -5 were previously reported. 29 Almost all of the titanosilicate nanoclusters present two terminal oxygen defects, mainly of the NBO type at Ti centres (Ti-NBOs). Typically, the titanosilicate nanocluster structures have little or no symmetry unlike the global minima of pure silica nanoclusters. Even at very small TiO 2 content, there appear to be a strong influence towards more "titania-like" nanoclusters structures, especially for nanoclusters with sizes n = 5 -14. We note that in several cases there are just a few dominant isomers that cover the whole composition range for a particular system size. The structural similarity of the mixed nanoclusters to those of pure titania can be so strong that in some cases the mixed nanoclusters exhibit the same bonding topology as the respective titania global minimum structure, with only replacements of titanium atoms by silicon atoms (e.g. structures 4-3, 5-4, 6-5, 7-3, 7-5, 8-4, 8-5, 8-6 in Fig. 3). This similarity between mixed (Ti x Si 1-x O 2 ) n nanoclusters and pure (TiO 2 ) n might suggest that could be possible to obtain low energy titanosilicate nanoclusters simply by replacing titanium with silicon atoms in pure titania nanoclusters. However, in many cases, we find that the global minima of the mixed nanoclusters have different structures with respect to the pure counterparts (e.g. 3) Thermodynamics of mixing

3.1) Mixing energy

As a first approximation, we use total internal electronic energies at 0K directly from our DFT based calculations for our evaluations of ∆ of our global optimized nanoclusters and periodic crystalline titanosilicates. In Fig. 4 we plot mixing energies, calculated according equation 3, for all global optimized (Ti x Si 1-x O 2 ) n nanoclusters in Fig. 3 and bulk systems in Fig. 1, as a function of x. We can clearly see in Fig. 4 that the mixing between quartz and rutile to form bulk crystalline titanosilicates is energetically unfavourable (i.e. ∆ values are positive). This result tends to confirm experiments where crystalline titanosilicates have only been successfully synthesised with very small proportions of TiO 2 (e.g. TS-1), and otherwise form metastable mixed oxide glasses.

However, at the nanoscale, the mixing of pure silica and titania nanoclusters is generally energetically favourable (i.e. ∆ values are generally negative). The nanocluster mixing energy size dependency is also shown in the 3D graph in Fig. 5. Here we use a smoothed continuum surface to represent the discrete set of data points of the mixing energies with respect to TiO 2 ratios and nanocluster size. From Figs. 4 and 5 it is clear that for very small nanocluster sizes (n = 1 -5) the mixing energies are quite modest in magnitude (≤ 0.15 eV) and can be positive (for n = 3) or negative (for n = 4 -5). In the latter cases, the most favourable mixing energies are found for x values between 0.5 -0.6. With increasing nanocluster size from n = 6 -12 we observe a rapid increase in the overall energetic favourability of mixing with the most favourable region of mixing occurring for nanoclusters possessing 7, 9, 10 units (i.e. nanoclusters of approximately 1 nm diameter) and for x values between 0.4 -0.5. For larger nanocluster sizes the mixing composition for the most favourable mixing is maintained, but the magnitude of ∆ slowly reduces with increasing nanocluster size. titanosilicates nanoclusters versus the TiO 2 molar fraction (x) and system size (n). This 3D surface shows the size dependency of the nanocluster mixing energies reported in the 2D graph in Fig. 4. A 2D contoured projection of the mixing energy map is also shown. The colour map indicating mixing energies ranges from dark blue (energetically unfavourable mixing) to red (energetically favourable mixing). We note that for ease of visualisation the contours and surface have been smoothed by interpolating through their respective underlying sets of discrete data points.

The sign and magnitude of the mixing energies of titanosilicate nanoclusters arises from the competition between several electronic and/or structural factors. For example, the presence of terminating defective oxygen sites and distorted tetrahedral SiO 4 groups are both energetically destabilizing factors. In low energy mixed (Ti x Si 1-x O 2 ) n nanoclusters, terminal NBO defects are preferentially found at titanium centres (Ti-NBO) rather than at silicon centres (Si-NBO). This can probably be explained due to the higher ionicity of Ti-O bonds with respect to Si-O bonds whereby rupture of the former is relatively less electronically, and thus energetically, destabilizing. Although this tends to favour terminal Ti-NBO groups over Si-NBO groups, Ti centres can also take advantage of internal positions in a nanocluster with high oxygen coordination spheres (making 5-6 Ti-O bonds). The strong tendency of silicon to have directional tetrahedral oxygen coordination means that such positions are not energetically favourable for Si centres. However, taking into account the higher Si-O bond strength with respect to Ti-O (bond dissociation energies for Ti-O is 666.5 ± 5.6 kJ/mol while for Si-O it is 799.6 ± 13.4 kJ/mol) [START_REF] Luo | Bond Dissociation Energies[END_REF] Si centres in tetrahedrally coordinated oxygen environments lower the total nanocluster energy more that Ti centres in the same position. These considerations can inform us as to how mixed nanoclusters can more effectively avoid the energetic cost of terminating NBO-Si defects and internal SiO 4 strain while taking advantage of strong unstrained Si-O bonding and highly coordinated Ti centres by the following set of general structural principles: nanoclusters should possess (i) Ti-NBO rather than Si-NBO terminations, (ii) Ti centres rather than Si centres in positions having either high oxygen coordination (i.e. >4) or distorted tetrahedrally coordinated oxygen environments, (iii) Si centres rather than Ti centres in positions with relatively undistorted tetrahedrally coordinated oxygen environments. Consequently, Si-rich nanoclusters can often be stabilised by following principles (i) and (ii), and Ti-rich nanoclusters may be stabilised by following principle (iii). As a concrete example, for the case of n=12 nanoclusters we show in figure 6 how pure and (SiO 2 ) 12 and (TiO 2 ) 12 nanoclusters can be stabilised by following these principles when moving to mixed nanoclusters. Although we do not have extensive data for sizes higher than 16 units, the set of structures between 2 and 16 units allows for an analysis of some of these factors affecting the stability of mixed systems with size. We suggest that the decrease in mixing energy with increasing system size above 10 units can be largely attributed to the gradual decrease in terminal defects in the pure silica reference nanoclusters (e.g. four for (SiO 2 ) 10 , two for (SiO 2 ) 12 , (SiO 2 ) 14 and (SiO 2 ) 16 and none for (SiO 2 ) 24 ). Therefore, titanosilicate nanocluster stabilities are compared with less and less destabilized pure silica systems as the size increases. Finally, following the general trend in Fig. 5, we can predict that the energetic favourability of mixing in titanosilicates nanoclusters with n > 24 units would rapidly tend to zero.

3.2) Mixing free energy

For a more accurate description of the thermodynamics of TiO 2 -SiO 2 mixing in titanosilicate nanoclusters we have considered the mixing Gibbs free energies at different temperatures for a set of nanoclusters with selected TiO 2 incorporation ratios. Specifically, we evaluate the free energy contributions to the mixing energies at different temperatures and at 1 atmosphere for pure oxide nanoclusters and 50% mixing for sizes n = 2 -16 units, according to:

∆ ( ) = ∆ ( ) -∆ ( ) (4)
where ∆ and ∆ are respectively the enthalpy and entropy of the system including temperature dependent contributions of nanocluster translations (S trans ), rotations (S rot ), vibrations (S vib ) and relevant distinct nanocluster isomer atomic configurations (S conf ). Both S trans and S rot are obtained from standard thermochemical relations and we use calculated harmonic frequencies to evaluate S vib [START_REF] Mcquarrie | Molecular Thermodynamics[END_REF] . In this work, we neglect the configurational entropy term (S conf ). This term would take into account the entropic contribution for all the different configurational isomers weighted with respect to their relative energy. Generally, we find that there is large energy separation between (configurational) isomers and global minima for the mixed nanoclusters justifying this assumption. In the Supplementary Information we report the energy separations between our global minima nanoclusters and the respective first few energetically low lying isomers and the probability for each of them to be thermally accessible. In Fig. 7 we plot the mixing free energies at different temperatures (298, 500, 700, and 900 K) calculated according to equation 3 using the calculated Gibbs free energies for each energetic term for the set of nanoclusters mentioned above. Mixing free energies are compared with mixing electronic energies. Overall, the thermodynamic favourability of mixing is found to be conserved with respect to the results for 0K when increasing the temperature. Thermal corrections affect mainly nanoclusters with sizes n = 6 and 8, but are found to be less relevant for the mixing at other sizes. Generally, thermal corrections lead to slightly less favourable mixing energies for all sizes except for the largest (n = 16) nanocluster considered where the thermally corrected mixing free energy is lower than electronic one. A deeper analysis of this effect is out of the scope of the present work, since we are primarily interested in verifying the persistence of favourable mixing in our mixed oxide nanoclusters at finite temperatures.

4) Signatures of mixing

In an effort to provide experimentally measurable signatures of TiO 2 -SiO 2 mixing in (Ti x Si 1-x O 2 ) n nanoclusters, in the following two sections, we analyse calculated harmonic frequencies and average coordination numbers of pure oxide nanoclusters as compared to mixed oxide nanoclusters. Freq (cm -1 ) Freq (cm -1 ) Freq (cm -1 ) IR Intensities (a.u.)

4.1) Harmonic frequencies

Ti O Si

In Fig. 8 we present calculated infrared (IR) harmonic vibrational frequencies for pure titania where is the total IR intensity, 1 is the scaling factor for the

( . . )
IR spectrum relative to (Ti 0.5 Si 0.5 O 2 ) n nanocluster). In other words, the IR spectra shown here are designed to mimic those of hypothetical samples which contain a monotonically decreasing size distribution of nanoclusters. IR spectra for each specific nanocluster size are available in the Supplementary Information. The main feature that we want to highlight from these IR spectra is the presence of two highly intense, distinct and consistent peaks in titanosilicate nanoclusters at around 1035 and 1090 cm -1 (highlighted with the horizontal dashed line in Fig. 7b). These peaks are also present in the pure systems but with significantly lower intensities. In titanosilicate nanoclusters the two peaks are due to couplings between stretching of the Ti-O bond in terminating oxygen defects and the vicinal Si-O bond in the "interior" of the nanocluster (see schematic in in Fig 8b). We note that these two peaks are related to the same normal mode and there is a size dependent redshifting that generates distinct peaks going from ~1090 cm -1 from smaller nanoclusters to ~1035 cm -1 for larger ones.

In pure TiO 2 (Fig. 7a) nanoclusters there are two distinct IR peaks that overlap with the ones in titanosilicate nanoclusters. The first peak is slightly blue-shifted at 1046 cm -1 and is due to exclusively to the contribution from the smallest (TiO 2 ) 2 nanocluster. However, we note that the relative intensities of these two peaks are quite low with respect to the most intense peaks. Unlike for titanosilicates, there is also no size dependent red-shifting of these peaks in titania nanoclusters and all peaks around 1090 cm -1 are due to harmonic stretching of terminal Ti-O species. In pure silica (Fig 7c) nanoclusters, there is also an intense peak that coincides with one found for mixed titanosilicate nanoclusters. Here, there is an intense peak at 1085 cm -1 from systems with sizes between 8 and 12 units, due to a complex coupling between stretching and modes involving different parts of these nanoclusters.

We suggest that the presence of both vibrational modes at 1090 and 1035 cm -1 , particularly enhanced in titanosilicates nanoclusters, can be used as a signature of the presence of mixed (Ti x Si 1-

x O 2 ) n nanoclusters The coordination number of an atom is defined as the number of its nearest neighbours. Average coordination numbers (ACNs) can be experimentally determined by X-Ray techniques such as EXAFS (Extended X-ray Absorption Fine Structure) spectroscopy. Fig. 9 displays the average coordination numbers of titanium (a) and silicon (b) in both pure and mixed systems. In the case of titanium coordination there are no significant differences between pure and mixed systems except for few cases (e.g. for n= 10 and 16). Here, the ACN for Ti remains between 4 and 4.5 for almost all the size range studied and is well below the bulk value of 6. In the case of silicon coordination, the ACN in pure silica nanoclusters is persistently lower than in mixed titanosilicate nanoclusters for nearly all sizes. The ACN for Si atoms could thus be used as a signature of mixed titanosilicate nanoclusters.

4.2) Average Si and Ti coordination numbers

The higher value of the ACN for silicon atoms in titanosilicate nanoclusters is related to the fact that Si atoms can take advantage of mixing with titania to be tetrahedrally 4-coordinated (at the lower energetic cost of producing a terminal Ti-O defect) which is its ideal coordination number (see above discussion). We find that the ACN for Si atoms reaches the ideal bulk value of 4-coordinated at smaller sizes in titanosilicates (6 units) than in pure silica (12-16 units). Indeed, the relative ease with which a bulk-like ACN value for Si centres can be achieved in titanosilicate nanoclusters is one of the driving forces for mixing at smaller nanocluster sizes.

Conclusions

In this study we have theoretically investigated the mixing between TiO 2 and SiO 2 in both bulk phases and in nanoclusters. We employed global optimization to find mixed (Ti x Si 1-x O 2 ) n nanoclusters in a systematic way for a wide range of sizes and mixing compositions. Several new global minima candidates of titanosilicate nanoclusters are thus reported. We observe that bulk mixing is energetically unfavourable which is in agreement with experimental observations. However, the mixing of the two oxides is found to be energetically favourable in the considered nanoclusters from 0K up to at least 900K. The size regime within which mixing is energetically favourable corresponds covers (Ti x Si 1-x O 2 ) n nanoclusters with n= 2 -24 (i.e. to nanoclusters with diameters less than 1.5 nm).

The maximum mixing favourability is calculated to be when mixed (Ti x Si 1-x O 2 ) n nanoclusters are around 1.0 nm in diameter (between 9 and 10 units) and when the molar fraction of TiO 2 is about 0.3-0.4. We suggest a number of structural-chemical driving forces for mixing in this size range. As a guide for experimentalists to observe and characterize these mixed nano-species we provide two measurable signatures of mixing. Firstly, harmonic frequency calculations show mixed titanosilicate nanoclusters to have a unique and distinct IR fingerprint (intense peaks at 1090 and 1035 cm -1 in IR spectra). Secondly, Si centres in mixed nanoclusters are found to show a higher average coordination relative to pure silica nanoclusters which could, in principle, be observed in EXAFS experiments. We hope that this work will serve as inspiration to experimental researchers to explore the fundamental properties of nanoscale titanosilicate mixed systems in order to synthesize new and highly efficient titanosilicate materials.

Fig 1 .

 1 Fig 1. Crystalline mixed titanosilicates models: a) quartz-like structures (there are three quartz-like systems with different TiO 2 molar fractions: x=0.33, 0.50 (shown here) and 0.67, b) a sodalite-like system where in the unit cell, one Si atom is replaced with a Ti atom which corresponds to x=0.08, c) is the "wired" titanosilicate structure found in the Materials Project database having 4-fold silicon and 6-fold titanium atoms with x=0.33.

Fig 3 .

 3 Fig 3. Globally optimized titanosilicate nanoclusters. Pure silica and titania nanoclusters from Fig 2 are also included for completeness. The two numbers assigned for each nanocluster are relative to the total number of units (n) and the number of incorporated TiO 2 units, respectively. Structures with sizes n = 2 -5 were

structures 8 - 2 , 8 - 3 , 9 - 1 •

 828391 ••9-6, 10-1•••10-5, 10-6•••10-8 in Fig. 3:) demonstrating the need for our global optimisation approach.

Fig 4 .

 4 Fig 4. Mixing energy per formula unit versus the TiO 2 molar fraction. The black filled squares denote the mixing energies of the bulk periodic structures shown in Fig. 1. The filled coloured circles denote the mixing energies of the globally optimized nanoclusters shown in Fig. 3. Nanocluster sizes are represented by a colour map ranging from light green for smaller nanoclusters to dark blue for larger sizes.

Fig 5 .

 5 Fig 5. Three-dimensional surface representation of the mixing energy per unit for globally optimized

Fig 6 .

 6 Fig 6. Example of how negative mixing energy can be rationalised when moving from pure phase (SiO 2 ) n and (TiO 2 ) n nanoclusters to mixed nanoclusters for the case of n=12 by following principles (i) -(iii) -see end of section 3.1 in main text.

Fig 7 .

 7 Fig 7.Total mixing electronic energy without zero point correction (dashed line) and Gibbs mixing free energies (solid lines) at different temperatures (298, 500, 700 and 900 K) relative to (Si 0.5 Ti 0.5 O 2 ) n systems versus the

Fig 8 .

 8 Fig 8. Calculated infrared (IR) harmonic vibrational frequencies (in cm -1 ) of pure titania (a), titanosilicate (b) and silica (c) nanoclusters obtained by summing size-scaled IR spectra for nanocluster sizes n (n= 2, 4, 6, 8, 10, 12, 14 and 16 units). In b) the anti-symmetric normal mode coupling between Ti-NBO and the vicinal Si-O isshown relative to the most intense peaks. Calculated IR harmonic frequencies for each size are available in the Supplementary Information. spectra were generated using the Moldraw code[START_REF] Ugliengo | [END_REF] and are made openly digitally available.

(

  Fig 8a), pure silica (Fig 8c) and mixed global optimized (Fig. 8b) nanoclusters. Each spectrum is composed by the sum of size-scaled IR spectra of several (Ti 0.5 Si 0.5 O 2 ) n nanoclusters (i.e. = ∑ 1 • ( . . )

Fig 9 .

 9 Fig 9. Average cation coordination numbers (ACN) plotted on smoothed lines for easily following trends for: a) titanium centres in pure TiO 2 (dashed line) and mixed titanosilicate (solid line) nanoclusters; b) for silicon centres in pure SiO 2 (dashed line) and mixed titanosilicate (solid line) nanoclusters. The corresponding oxygen coordination number of the respective cation in the bulk phase is indicated in both cases as a dot-dashed line.All coordination numbers are taken by counting oxygens within a maximum cut-off distance of 2.2 Å from each Ti or Si centre. In mixed systems, for each size, the ACN is averaged over all nanocluster mixing compositions. Figures shown in b) are pure silica structures with n = 6 -10 in order to show the origin of low silicon ACN values for this size regime.

Table 1 .

 1 Buckingham parameters for IPs used in the MCBH global optimization searches. Ionic charges for both IPs are: Q(Si, Ti)=2.4 and Q(O) = -1.4

		IP1 (x > 0.30 TiO 2 )		IP2 (x < 0.30 TiO 2 )
	Si-O	10454.2	0.208	63.05	10386.97 0.210372	63.45
	Ti-O	2454.2	0.211	-55.05	14719.71 0.200783	65.90
	Si-Si	79502.1	0.220	446.78	74316.65	0.220	446.78
	Ti-Si	69000.0	0.210	500.00 104167.92	0.220	446.78
	Ti-Ti	69000.0	0.210	480.59	74316.65	0.220	446.78
	O-O	2828.5	0.297	16.48	2828.5	0.297	16.48

  used here to calculate mixing energies for our set of titanosilicates. The global minima silica nanoclusters in this size range tend to be quite symmetric and open, and typically possess two or more terminal oxygen defects. Notice that for size n = 24 we have taken the very low energy fully-coordinated structure reported in ref. 38 structure rather than the reported global minimum structure in ref. 37 which possesses terminal oxygens. In fact, as noted in ref. 38, these two structures are almost degenerate in energy and the use of the PBE0 functional in the present work slightly favours the fully coordinated structure. All pure silica nanocluster structures display both 3-and 4-fold coordinated silicon atoms. Three-fold coordinated silicon atoms are present only when bound to a terminally "double-bonded" oxygen (i.e. formally: >Si=O, but arguably more

Acknowledgements

This research was supported by the Spanish MINECO / AEI-FEDER grant: CTQ2015-64618-R and, in part, by Generalitat de Catalunya grants: 2014SGR97 and XRQTC. We acknowledge the Red Española de Supercomputación for provision of computing time. We also acknowledge GENCI-CINES/IDRIS (Grant 2016-x2016082131, 2017-x2017082131) and the CCRE-DSI of Université P. M. Curie for computing resources.

Supplementary Information

We report the coordinates of all global optimized titanosilicate nanoclusters used in this work. We also include a selection of metastable isomers for (Ti 0.5 Si 0.5 O 2 ) n (n= 4, 6, 8, 10, 12, 14, 16) and their probabilities to be thermally accessible at different temperatures (298, 500, 700, 900). Calculated harmonic frequencies of the most stable nanoparticles have also been included. Finally we include a table containing total energies and mixing energies for all systems appearing in Fig. 4 and an example of mixing energy normalized by two unit definitions. All structures have also been uploaded in an open access manner in both the NOMAD repository (https://repository.nomad-coe.eu/) and the WASP@N nanocluster database (https://hive.chem.ucl.ac.uk/).


Minh Thang Do
email: mtdo@univ-ag.fr

Ted Soubdhan

Benoit Robyns

A study on the minimum duration of training data to provide a high accuracy forecast for PV generation between two different climatic zones

Keywords: PV forecasting models, neural network, multivariate model, forecasting errors, training duration

This study focus on the minimum duration of training data required for PV generation forecast.

In order to investigate this issue, the study is implemented on 2 PV installations: the first one in Guadeloupe represented for tropical climate, the second in Lille represented for temperate climate; using 3 different forecast models: the Scaled Persistence Model, the Artificial Neural Network and the Multivariate Polynomial Model. The usual statistical forecasting error indicators: NMBE, NMAE and NRMSE are computed in order to compare the accuracy of forecasts.

The results show that with the temperate climate such as Lille, a longer training duration is needed. However, once the model is trained, the performance is better.

Introduction

With the support of environment policies and the increase of the fossil fuel price, renewable energy sources have been growing strongly in the last few years. The electric power produced by these intermittent sources shows strong variations (sudden and with large amplitude), which must be always compensated on the grid by others dispatchable sources [START_REF] Ernst | Wind power and photovoltaic prediction tools for balancing and grid operation[END_REF], [START_REF] Do | Influence of wind turbines on power system reliability through probabilistic studies[END_REF], [START_REF] Do | Reliability evaluation of power system with large-scale wind farm integration using first-order reliability method[END_REF].

These large variations can put a pressure on the balance of supply/demand of the power system, especially in non-interconnected systems, such as island areas. In the case of French islands for which Island Energy System is the electrical system manager, in order to ensure the stability of the power system, a ministerial decree of 2008 set the penetration rate at 30%, beyond which the system operator is permitted to disconnect intermittent energy [START_REF] Cre | Cahier des charges de l'appel d'offresportant sur des installations au sol de production d'électricitéà partir de l'énergie solaire[END_REF]. Note that this rate of 30% was achieved for a few hours in 2012 in Reunion and in Guadeloupe, resulting in the disconnection of certain facilities of PV production.

The forecast of these fluctuation sources across the concerned islands should allow a better control of the availability of renewable energy production, and thereby reduce the pressure on the balance of supply/demand. Moreover associated with power storage unit (batteries or STEP, for example) will help providing various services to the power system, from ancillary services (adjusting voltage, frequency) to smoothing peak hours. The forecast of fluctuation sources will contribute to the optimization of the design and the use of these storage units.

Generally, the PV production forecast is classified into 2 categories: the day ahead (DA) forecast and the hour ahead (HA) forecast, depend on the domain of application [START_REF] Iea | Performance prediction of grid connected photovoltaic systems using remote sensing[END_REF].

-The day ahead (DA) forecast is usually based on the numerical weather prediction models.

They are dynamical equations that predict the evolution of the atmosphere up to several days ahead from initial conditions. From the forecast of weather condition, the output power of PV can be estimated [START_REF] Beyer | Report on Benchmarking of Radiation Products[END_REF], [START_REF] Lorenz | Qualified Forecast of Ensemble Power Production by Spatially Dispersed Grid-Connected PV Systems[END_REF], [START_REF] Traunmüller | Solar irradiance forecasting, benchmarking of different techniques and applications of energy meteorology[END_REF].

-The hour ahead (HA) forecast is usually based on stochastic learning techniques. The underlying assumption of these techniques is that future value of PV production can be predicted by training the algorithms with historical data [START_REF] Fernandez-Jimenez | Short-term power forecasting system for photovoltaic plants[END_REF], [START_REF] Pedro | Assessment of forecasting techniques for solar power production with no exogenous inputs[END_REF], [START_REF] Mandal | Forecasting power output of solar photovoltaic system using wavelet transform and artificial intelligence techniques[END_REF].

To generate a prediction of the PV production using stochastic learning techniques, it is necessary to have historical data. These data are used for the learning phase of forecast models of PV production at the studied site (IEA, 2013).

In the literature, several studies have been conducted on the methodology for the forecast of PV generation [START_REF] Shi | Forecasting power output of photovoltaic systems based on weather classification and support vector machines[END_REF], [START_REF] Krömer | Evolutionary Prediction of Photovoltaic Power Plant Energy Production[END_REF], [START_REF] Lorenz | Local and regional photovoltaic power prediction for large scale grid integration: Assessment of a new algorithm for snow detection[END_REF]. The authors generally use two historical years: one year for training the model and another year for the test phase. In the literature, the analysis of the minimum duration for the phases of test and training data is little documented.

The proposed study will determine the minimum of historical experimental data necessary to achieve a high accuracy forecast, which can be quantified by a set of several statistical error indicators. There are some works in other areas that focus on this problem [START_REF] Fine | Sample Size Requirements of Feedforward Neural Network[END_REF], [START_REF] Thirumalainambi | Training data requirement for a neural network to predict aerodynamic coefficients[END_REF], [START_REF] Cui | A study of sample size with neural network[END_REF]) but a study on the prediction of PV generation, to our knowledge, is not yet performed.

There are two major factors that affect the time required for the collection of historical data, they are the climatic conditions at the PV site and the statistical model used for the forecast.

The climatic conditions at the PV site affects the level and the type of data fluctuation. The more fluctuate the data is, the more they are difficult to predict, i.e.: the error on the forecast is higher. Therefore, the duration of the data collection is supposed to be longer. In this study, we investigate the influence of climatic conditions on the historical data necessary for a forecast of PV production by comparing two sites of PV generation with very different climatic conditions: one in Lille, in northern France and the other one in Guadeloupe, in the Caribbean.

The forecasting model also has a large influence on the duration required for the collection of historical data. There are simple forecasting models that requires a few data for the learning phase. The more complex the forecasting models is, the more historical data is needed but the result is often more accurate.

In this study, we will compare some of the most popular statistical models in forecasting PV production: Artificial Neural Network (ANN) [START_REF] Yona | Application of Neural Network to One-day-ahead 24 hours Generating Power Forecasting for Photovoltaic System[END_REF], [START_REF] Fernandez-Jimenez | Short-term power forecasting system for photovoltaic plants[END_REF], [START_REF] Mandal | Forecasting power output of solar photovoltaic system using wavelet transform and artificial intelligence techniques[END_REF], Multivariate Polynomial Model (MPM) [START_REF] Dazhi | Solar modeling and forecast[END_REF], [START_REF] Iea | Performance prediction of grid connected photovoltaic systems using remote sensing[END_REF] and the Scaled Persistence Model (IEA, 2013).

Input data and site description

In this paper, we will use PV production at time (h) and two exogenous inputs: the cloud cover and the air temperature, also at time (h), to forecast the PV production at time (h+1).

For each hour (h) considered as the beginning time of the forecasting, the input vectors are given by:

  ) ( ); ( ); ( ) ( h P h T h N h x m a  (1)
Where:

-N(h) is the cloud cover measured at time (h) -Ta(h) is the ambiance temperature measured at time (h) -Pm(h) is the average output power produced by the PV system in the previous 60 minutes respective to the h-hour a. Data measured in Guadeloupe, tropical climate

The PV system installed in Guadeloupe (16°14'36.0"N 61°33'25.9"W) has 832 membrane PV modules of 136Wp each, which makes a total of 113kWc.

The system consists of two inverters. The data logging of PV output power (in W) is integrated in these inverters (collected every 5 min). The two exogenous inputs: cloud cover (in octa) and air temperature (in °C) are measured every hour by the weather station at the airport nearby (Raizet Airport) by Meteo France. As during the night, the forecast is not necessary because the output power of PV is zeros, the study takes into account only the data between 7a.m and 17p.m every day.

The average output power of PV in every hour is calculated from the measurement every 5 minutes of the inverters:

    h h t m t P h P min 55 ) ( 12 1 ) ( (2)
The forecast models will be studied with a varying training period from one month to one year (365 days) and a testing period of 331 days. The output power of the PV system at time (h+1), Pm(h+1) will be predicted from three inputs: N(h), Ta(h) and Pm(h). The installation of a 93kWc PV system in Lille (50°38'58.7"N 3°09'04.3"E) consists of 452 panels. Among these panels, 376 modules have an inclination of 3° and the inclination of the rest is 60°.

The system consists of 31 inverters. The data logging of PV output power (in W) is integrated in these inverters (collected every 10 min). The two exogenous inputs: cloud cover (in octa)

and ambiance temperature (in °C) are measured every hour by the weather station at the airport nearby (Lesquin Airport) by Meteo France. As the sun rise is later in Lille, the study takes into account only the data between 10a.m and 17p.m every day to avoid the zero output power of PV during the night. (3)

The forecast models will be studied with a varying training period from one month to one year (365 days) and a testing period of 308 days. The output power of the PV system at time (h+1), Pm(h+1) will be predicted from three inputs: N(h), Ta(h) and Pm(h).

c. Evaluating the quality of forecast

The precision of the forecast will be evaluated using the following statistical error indicators:

-Normalized mean bias error (%)

100 ) min( ) max( ) ( ) ( 1 1       m m M h m m P P h P h P M NMBE (4) -Normalized mean absolute error (%) 100 ) min( ) max( ) ( ) ( 1 1       m m M h m m P P h P h P M NMAE (5) -Normalized root -mean-square error (%)   100 ) min( ) max( ) ( ) ( 1 1 2       m m M h m m P P h P h P M NRMSE (6)
Where:

-Pm(h) is the average output power produced by the PV system at hour (h)

- ) ( ~h P m
is the predicted output power of the PV system at hour (h) -M is the number of hours considered

The forecasting models

In this research, the forecasting models will take into account the data of PV production at time (h), Pm(h) and two exogenous inputs: the cloud cover N(h) and the air temperature Ta(h), also at time (h), to predict the PV production at time (h+1). As the PV production depends on the position of the sun in the day while others input variables does not, a normalization is needed to eliminate this subordination. The input of the PV production at time (h) becomes now

), (h P m the normalized value of the PV production with respect to the maximum value at time (h),

Pmax(h).

The absolute value of the PV output power prediction is then obtained by multiplying this normalized value with the maximum value of PV production at time (h+1), Pmax(h+1). This maximum value can be evaluated from the GHImax curve with the following equation:

) (h P m ) 1 ( ~ h P m ) 1 ( ~ h P m ) max( max max GHI GHI ) (h P m ) (h T a ) (h N installed PV ) ( max h P ) 1 ( max  h P
The Artificial Neural Network used in this research is a feedforward neural network for nonlinear regression with 1 hidden layer. The choice of the number of neurons in the hidden layer is a complicated issue. In this paper, a value of 4 is chosen based on several researches in literature [START_REF] Blum | Neural Networks in C++[END_REF], [START_REF] Swingler | Applying Neural Networks: A Practical Guide[END_REF], [START_REF] Berry | Data Mining Techniques[END_REF], [START_REF] Boger | Knowledge extraction from artificial neural network models[END_REF]. The Artificial Neural Network will be trained by the historical data with the supervised learning technique, using the Levenberg-Marquardt algorithm [START_REF] Seber | Nonlinear Regression[END_REF]. The polynomial models are very popular due to their simplicity in form, their well-known and understood properties and their flexibility of shapes. Moreover, they are computationally easy to use. For these reasons, there are many applications of these models in the forecasting in general, and in PV production forecasting in particular [START_REF] Dazhi | Solar modeling and forecast[END_REF], [START_REF] Iea | Performance prediction of grid connected photovoltaic systems using remote sensing[END_REF].

However, as the model takes into account only one input variable, the precision of forecast is limited and therefore the Artificial Neural Network model becomes the most applied model in

) (h P m ) (h T a ) (h N ) ( ~h P m

Hidden Layer Input Layer

Output Layer this field [START_REF] Pedro | Assessment of forecasting techniques for solar power production with no exogenous inputs[END_REF]. In this paper, a Multivariate Polynomial Model, which takes into account several input variables is proposed to improve the accuracy of the prediction of PV output power.

The output prediction of the normalized value at time (h+1) is a multivariate polynomial function of the input variables:

8 7 6 5 4 3 2 1 ) ( ) ( ) ( . ) ( ). ( . ) ( ). ( . ) ( ). ( . ) ( ). ( ). ( . ) 1 ( ~b h T b h N b h P b h T h N b h T h P b h N h P b h T h N h P b h P a m a a m m a m m          (8)
The coefficients of the multivariate polynomial function are estimated from the historical data using the Levenberg-Marquardt nonlinear least squares algorithm [START_REF] Seber | Nonlinear Regression[END_REF]. This algorithm allows to determine the set of parameters b giving the minimal squares of the deviations:

      M h m m h P h P b S 1 2 ) ( ) ( ) ( (9)
c. Persistence Model

The persistence model is based on a simple rule: the output value of the predicted variable at (h+1) is equal to its value at (h). The advantage of this technique is that it does not need to be trained by a series of historical data, however the accuracy of the forecast is not high.

In this paper, the Scaled Persistence Model is applied in order to reduce the forecasting error.

This model is applied on the normalized value:

) ( ) 1 ( ~h P h P m m   (10)

Results

In this section, we will analyze the influence of the duration of training data on the forecasting error. The parameters of the forecast model (Artificial Neural Network and Multivariate

Polynomial Model) will be estimated using the historical data in 30 days, 60 days, 90 days, 180 days, 270 days and 365 days. Then, the model will be tested on a duration of 331 days (Guadeloupe) and 308 days (Lille). Due to its characteristic, the Scaled Persistence Model does not require any historical data, therefore, it will be tested directly with the testing data.

Case of Guadeloupe

The Table 1 and the Figure 5 show the evolution of the forecast error with different training duration in Guadeloupe. The value of the bias error NMBE allows to evaluate the tendency of the prediction model to underestimate or overestimate the PV output power. In this study, the NMBE is relatively small with all 3 methods and oscillating around the zero. However, the variation of the NMBE does not follow any certain rule (Table 1).

The NMAE is used to measure how close the forecasts or predictions are to the eventual outcomes. As presented in the With the training data of 365 days, the forecast error is slightly higher than that given by the training data of 270 days, which is not coincident with the tendency of reduction of the forecast error with longer training data. This contradiction could be explained by a hypothesis that the training data of the period of 3 months which has just been added to the training data may disturb the forecast model. To verify this hypothesis, another forecast is implemented using the training data from the first 9 months of the training year (see Figure 1). The result shows that the forecast error in both models is higher than that using the training data from the last 9 months (12.21% with Artificial Neural Network and 12.01% with Multivariate Polynomial Model).

Therefore, the hypothesis can be confirmed.

Case of Lille

The Table 2 and the Figure 6 show the evolution of the forecast error with different training duration in Lille.

Apart from the Scaled Persistence Model, which does not require the historical data, the forecast error of the Artificial Neural Network and the Multivariate Polynomial Model is high with the training data from 1 to 3 months due to the seasonal effect of Lille. With this effect, the right parameters of the forecast model cannot be evaluate from the training data of 1 to 3 months.

However, from 6 months and above, the forecast error is reduced significantly (from 12.71% to 10.69% with Multivariate Polynomial Model). The reduction starts from 6 months of training data, not from 1 year, due to the symmetry of the PV production of Lille in a year (see Figure 2). The difference of forecast error between the training duration of 6 months, 9 months and 1 year in Lille is negligible. The training period could be finished after 6 months of data collection. We can observe that the duration of data collection required to have a good forecast in Lille is higher than the duration of data collection needed in Guadeloupe (6 months compared to 3 months). However, the forecast error in Lille is smaller (10.69% compared to 11.97%).

Analysis and recommendations

Among the 3 models, the Scaled Persistence Model does not require the historical data, therefore the forecast can be obtained directly without training period. However, the quality of forecast is not high. In the beginning, this model can compete with the others 2 forecast models As presented in the Figure 1 and the Figure 2, the PV production of Guadeloupe and Lille has very different characteristics. In Lille, there is a seasonal effect that there is not in Guadeloupe.

However the PV production of Lille is less fluctuant than that of Guadeloupe. Therefore, to obtain a good quality forecast, the model of Lille needs a longer duration of training data (6 months instead of 3 months). But once the model is trained, it provides a better performance than that of Guadeloupe.

Conclusions

This paper focus on the necessary duration of data collection to have a quality forecast of PV production for two climatic conditions: a temperate and a tropical weather conditions. In order to investigate this issue, the study is implemented on 2 PV installations: the first one in Guadeloupe, the second in Lille; using 3 different forecast models: the Scaled Persistence Model, the Artificial Neural Network and the Multivariate Polynomial Model.

The results show that with the temperate climate such as Lille, where the seasonal effect existed, a training duration of at least 6 months is needed to acquire a high accuracy forecast for PV generation instead of 3 months with the tropical climate at Guadeloupe. However, once the model is trained, the performance is better (error of 10.69% at Lille compared to 11.97% at Guadeloupe).

This research proposes to use the Scaled Persistence Model to forecast the PV output power during the data collection as this model does not require the historical data. Once the data collection phase is finished, the Multivariate Polynomial Model could be applied in order to provide a better accuracy. With the application of PV output forecast, the exploitation of PV becomes more efficient, reducing the risk of power outages and improving the reliability of power supply.

Figure 1 .

 1 Figure 1. Output power for training and testing in Guadeloupe

Figure 2 .

 2 Figure 2. Output power for training and testing in Lille

  Figure 3. General process of the PV output power forecast

Figure 4 .

 4 Figure 4. Diagram of the Artificial Neural Network configuration

  The Root-Mean-Square Error (RMSE) is a frequently used measure of the differences between value (Sample and population values) predicted by a model or an estimator and the values actually observed. Basically, the RMSE represents the sample standard deviation of the differences between predicted values and observed values. The NRMSE is the RMSE divided by the range of observed values of the variable being predicted. In literature, the NRMSE is mostly preferred than the NMAE because this indicator contain both information of the bias and the variance of the prediction.As the Scaled Persistence Model does not require the historical data, the forecasting error of this model is usually the highest among the three. However, it is still lower than the forecasting error of Artificial Neural Network with only 1 month of training data. Generally, if the duration of the collected data is shorter than 1 month, it is advisable to use the Scaled Persistence Model. From more than one month of training data, the Artificial Neural Network and the Multivariate Polynomial Model are more accurate. The forecasting error reduce rapidly from more than 13% to less than 12% with the training data of 3 months (Multivariate Polynomial Model) or of 6 months (Artificial Neural Network). With the Multivariate Polynomial Model, the difference of forecast error between the training duration of 3 months, 6 months, 9 months and 1 year in Guadeloupe is negligible. The training period can be considered as sufficient after 3 months of data collection.

  but from 2 months of training data (Guadeloupe) and from 6 months of training data (Lille), the quality of the forecast by these model is much better than that of the Scaled Persistence Model. The Artificial Neural Network and the Multivariate Polynomial Model have the same evolution of forecast error due to the fact that both of these models use the Levenberg-Marquardt algorithm as core learning technique. The Multivariate Polynomial Model has a better forecast quality than the Artificial Neural Network. In Guadeloupe, the Artificial Neural Network needs 6 months of training data to acquire the forecast error lower than 12% while the Multivariate Polynomial Model needs only 3 months to go under this level. Another advantage of the Multivariate Polynomial Model is the simplicity and the transparency of the model. This is an analytical model, it means that the relation between the predicted value and the input variables can be represented in the form of an equation, different from the Artificial Neural Network, where this relation can only be represented by a black box.

Table 1 .

 1 The evolution of the forecast error with different training duration (Guadeloupe)

	NMBE

Table 1

 1 , from the training data of more than 1 month, the NMAE of the Artificial Neural Network and the Multivariate Polynomial Model is lower than the Scaled Persistence Model. The NMAE of the Multivariate Polynomial Model is slightly higher than that of the Artificial Neural Network.

	14,00
	13,50
	13,00
	12,50
	12,00
	11,50
	11,00
	30 days	60 days	90 days	180 days	270 days	365 days
		Artificial Neural Network	Multivariate Polynomial Model
		Scaled persistence model

Table 2 .

 2 The evolution of the forecast error with different training duration (Lille)

	NMBE