Pierric Mazodier
email: pierric@mazodier.fr

Mathieu Mangeot
email: mathieu.mangeot@imag.fr

Valérie Bellynck
email: valerie.bellynck@imag.fr

Active Reading for Intercomprehension Between Sinogramic Languages

Keywords: Computing methodologies➝Artificial intelligence➝Natural language processing ➝Language resources NLP, active reading, sinogramic languages, intercomprehension, analogy, Japanese, Mandarin, Jibiki platform, Mecab, HanLP, onyomi, rōmaji, furigana, Pīnyīn

In the domain of e-learning for a linguistic field, active reading interfaces are useful tools for learning new languages. For a learner of a given language, the goal is to assist in knowledge development in other known languages, in order to enhance their learning. An active reading program enriches the understanding of the input sentence in different ways, such as adding the pronunciation of each word or displaying a definition or a translation taken from a dictionary. By the process of analogy, the users would then be able to synthesize the information and produce themselves a translation that would be, at their convenience, both syntactically and semantically appropriate.

We propose two new features for an active reading interface that aims to help in the understanding of Sinogramic languages. We focus on Japanese and Mandarin languages. The first feature identifies the reading (on'yomi, kun'yomi…) of the sinograms; the second feature displays, for certain words of the input text, their equivalent in another language that is etymologically close to the source one. In our case, the words with on'yomi reading will be associated with their Mandarin equivalents, in their Pīnyīn form. This tool can be useful for learners of Japanese with some knowledge of Mandarin or the contrary.

INTRODUCTION

In the domain of e-learning for a linguistic field, active reading interfaces are useful tools for learning new languages. For a learner of a given language, the goal is to assist in knowledge development in other known languages, in order to enhance their learning. While a straightforward translation can be useful to grasp a meaning or to get a syntactically correct output sentence, it can still be semantically incorrect. Furthermore, it does not help the learner to increase her own knowledge of the language.

Instead, an active reading program enriches the understanding of the input sentence in different ways, such as adding the pronunciation of each word or displaying a definition or a translation taken from a dictionary. By the process of analogy, the users would then be able to synthesize the information and produce by themselves a better interpretation or translation that would be, at their convenience, both syntactically and semantically appropriate.

We coin sinograms as an alternative proposal to 'Chinese characters' in order to distinguish the contemporary political entity from the past cultural dominant ideology over Eastern Asia and its most visible representative: the graphic system. Consequently, Sinogramic languages (SL) [START_REF] Magistry | Phonological Cues in Sinograms: from Psycholinguistics to Computational Modeling for Language Teaching[END_REF] refer to the languages sharing a history including cultural loans through literary corpora and standards, the graphic system conveying them and so the lexicon still in use in these languages nowadays. Defined as such, SL include all the Sinitic languages among which standard Mandarin as the legitimate variant, but also typologically distinct languages such as Japanese, Korean and Vietnamese. In this paper, we will just deal with Japanese and Mandarin.

We propose two new features for an active reading interface that aims to help in the understanding of Sinogramic languages here after Japanese as a target language and Mandarin as first SL. The first feature identifies the reading type (on'yomi, kun'yomi cf. below 2.3) of the sinograms in Japanese texts ; the second feature displays, for certain words of the input text, their reading in Mandarin. In our case, the words with on'yomi reading will be associated with their Mandarin reading in their Pinyin form.

In this paper, we will proceed as follows: the second section will present an overview of the Japanese writing systems and the question of reading sinograms in Japanese texts. The third section will focus on the active reading process. Then, the Jibiki lexical resources management platform will be described in the fourth section. Lastly, we will present the new features for intercomprehension [START_REF] Grin | S'entendre entre langues voisines: vers l'intercompréhension[END_REF] between SL. In conclusion, we will discuss how these features might be improved and applied to other linguistic contexts.

THE JAPANESE WRITING SYSTEM AND THE READINGS OF SINOGRAMS 2.1 Japanese language vs 'Chinese'

As mentioned above, Japanese language [START_REF] Labrune | Japanese Phonology[END_REF] is typologically distinct from the contemporary Sinitic languages among which Modern Mandarin [START_REF] Norman | Chinese[END_REF], but historically under cultural influence including Classical Chinese literature written in sinograms as early as the 5th century. By the 8 th century, in order to write down reading information and Japanese-specific words, sentences and texts, several initiatives became soon two sets of syllabaries that are derived from the sinogramic script. In the same time, the Japanese crafted their own readings of the sinograms and by doing so emancipated from Classical Chinese. We insist on the fact that through this historical process, 'Chinese' never refers to the legitimate language spoken nowadays known as Mandarin.

Contemporary graphic typology for Japanese

As a result today, the Japanese writing system is composed of four different types of characters:

1. kanji (漢字) are sinograms borrowed through this historical process and sometimes derived from initial forms. They are syllabic, commonly used in nouns, verbs and adjectives (e.g. 東 京 respectively read as Tō and kyō, meaning the city of Tokyo) ; 2. hiragana (平仮名) is one of the both sets mentioned above. Hiragana are 'syllabic' -moraic is more accurate [START_REF] Labrune | Japanese Phonology[END_REF]-signs used for particles, pronouns, adverbs, inflections (also referred as okurigana (送り仮名り仮名仮名), etc. (e.g. そして soshite, "then") ; 3. katakana (片仮名) refers to the second set and an alternate version of hiragana. Nowadays, they are used for non-Chinese foreign loan words (e.g. ク ロ ワ ッ サ ン kurowassan, "croissant"). 4. rōmaji ( ロ ー マ字 字 ) are the Latin characters also used nowadays in Japanese contemporary texts (e.g. ABC Mart).

Furthermore, when small kana (usually hiragana) are seen written above kanji, they are called furigana 1 (振り仮名り仮名仮名) and are used to indicate the reading of the sinogram below. They appear usually in texts for children or above sinograms for indicating rare readings. At last, for non-Japanese speakers, there are two main transliteration standards of rōmaji: the Hepburn and Kunrei-shiki ( 訓 令 式 ), which differs from each other only by some subtleties (e.g. shi/si for transliterating し/ シ, tsu/tu for つ /ツ ...).

Reading sinograms in Japanese

Beyond this graphic typology of Japanese texts, we also have to introduce shortly how the sinograms can be read in Japanese. As already mentioned above, kanji have two types of reading:

1. on'yomi ( 音 読み み ), referred here after as 'Chinese pronunciation' or on reading inherited from Chinese loans over the history, (e.g. 東京 Tōkyō, 'Tokyo') ; 2. kun'yomi ( 訓 読み み ), referred here after as 'Japanese pronunciation' or kun reading, usually used for Japanese indigenious lexicon (e.g. 東 higashi, 'East').

The point is that beyond their belonging to differently originated lexical stocks, they also do not share the same phonological patterns.

Indeed, while the latter can be polysyllabic -three for 東 higashi -the former will always conform the sinosyllabic structure shared by all the SL (cf. below 2.4). Each kanji may have at least one reading and frequently have one or more of each of these two types of readings. In order to be complete in this overview, we have to distinguish polysinographic -composed of at least two sinograms -words with mixed readings. The are two types conventionally referred as jūbakoyomi (重箱読みみ) on+kun reading words -jū-bako ' box'-and yutōyomi (湯桶読みみ) kun+on reading words, yu-tō 'h ot pot', At last, there is a last type referred as ateji ( 当て字て字字 ) where kanji are used to phonetically represent native or borrowed words with less regard to the underlying meaning of the characters such as in 寿司'sushi'.

On readings and the sinosyllable

As this paper deals with features shared by SL and so Mandarin and Japanese on readings, we have to discriminate several 1 also known as Japanese ruby characters. categories of on'yomi corresponding to the different times during which the under going standard reading of a given word was borrowed from Chinese by Japanese speakers and introduced into their lexicon. Through this process, a given sinogram could be borrowed several times with diachronic variation of its reading. Three different on reading categories are conventionally discriminated as follows:

1. go'on (呉音) were the earliest readings introduced to Japan during the Nara period (e.g. 浪人 rōnin, 'ronin') ; 2. kan'on ( 漢 音 ) followed (e.g. 外 国 人 gaikokujin, 'foreigner') ; 3. tōsō'on ( 唐 宋 音 ) came later during Heian and Edo period (e.g. 蒲団 futon 'futon').

Even if these readings evolved, they all conform to the sinosyllable structure as long as it was the basis of literary composition into Classical Chinese and the institution of riming. The structure is organized as Figure 1.

Any sinosyllable -whatever Japanese on reading includedwill conform this pattern distributed between the onset and the rhyme subdivided itself as glide (or 0), nucleus, vocalic or consonantic coda (or 0) plus tone (or 0). Among the examples above, such as 人 read as nin or jin, nand j-are the onset, -i-is the nucleus, and -n is the consonantic coda. Compared to the kun reading of the sinogram 人, hito, we can see that this reading does not conform the sinosyllabe distribution.

Sinogram and sinosyllable as potential of intercomprehension

Beyond this stricking diversity of various graphic systems in Japanese texts and diverse reading types for kanji, either it is challenging especially for foreign learners, there are short cuts even for this latter population as soon as learning and teaching are emancipated from language ideologies. Thus, English loans easily identified through their notation in katakana might provide a huge stock of lexcicon semanticly transparent and very helpful for phonological awarness. Furthermore, for foreign learners already familiar with sinograms -such as those coming from China, Taiwan, Hongkong etc., kanji are -with some exceptions -not only visually accessible for their meaning. The point now is that as long as Mandarin is becoming the first SL to be learnt abroad beyond the few aeras previsously mentioned, katakana and semantic informations provided by kanji are not the only potential of intercomprehension: sinograms are also a resource through their on reading and so a feature on which learners can rely in order to transfer their reading knowledge in Mandarin into Japanese. This is our didactic statement [4] and the theory for which our tool is designed. As far as we know, such a tool does not exist and so is our contribution.

THE ACTIVE READING PROCESS 3.1 Description of the generic process

The active reading tool [START_REF] Abdellaoui | Outillage de l'accès aux textes par la lecture active étymologique multilingue pour apprenants berbérophones et arabophones[END_REF] proceeds as follows:

1. The text received as input is sent to a morphological analyzer to obtain the lemma and grammatical information for each word.

Figure 1: The sinosyllable and its structure 2. Using the lemmas received from the analyzer, the tool then looks up a monolingual or bilingual dictionary to obtain lexical information about each lemma. It can be a definition or a translation.

3. The tool displays the input text by adding additional information either above or below the text, or by displaying words with different colors, or by adding a code that will display a pop-up window when the mouse hovers over the word.

3.2

Active reading for Japanese

Description of the active reading

The active reading service offers to display the reading of a Japanese text in the form of furigana and the French translations coming from a Japanese-French dictionary. The list of the corresponding translations from the dictionary appears in a textbox when hovering the mouse over certain words. The service is available online from the jibiki.fr website 2 . This active reading instance is tailored for the following scenario: a learner of Japanese with a good knowledge of French. It could be enlarged to learners fluent in other languages by using different dictionaries. For example, the use of Jim Breen's JMdict [START_REF] Breen | JMDict: a Japanese-multilingual dictionary[END_REF] could be useful for users fluent in English, Russian, German, Dutch, etc.

The morphological analysis

The parsing and analysis of the input text is performed with the Conditional Random Field based MeCab morphological analyzer and tokenizer 3 .

Figure 2 shows the result of the analysis by Mecab of the sentence entered by the user in Figure 3. Each line raepresents a token. The first left column is the token itelf. Then follows a series of 8 items separated by commas. The first one is the partof-speech + 名詞 is a noun, 動詞 is a verb, 助詞 is a particle and 記号 is a punctuation mark. The next 4 ones are used for subcategorization. Then, the next item is the lemma (eg: for the item 掛かりかり, the lemma is the infinitive form of the verb 掛かりか る). The next item is the writing form of the token in katakana and the last item is the reading form in katakana (e.g. the particle は has the writing form ハ ha and the reading form ヮ wa). For our purpose, we use the lemma and the reading form.

One can see in the example of Figure 2 that the numbers are actually parsed each figure separately by Mecab. In order to render correctly the furigana of the numbers, we encoded a specific function. That is especially important in Japanese because of the 万 (man) '10 000' counting unit which differs from the Western 1000 counting unit.

The Japanese-French dictionary

The jibiki.fr Japanese-French dictionary 4 [8] combines 3 main sources:

• a digitized version of the Cesselin [START_REF] Cesselin | WaFutsu daijiten -Dictionnaire japonais-français[END_REF] Japanese-French dictionary edited by Gustave Cesselin and published in 1940. It contains more than 82,000 articles with detailed microstructure: French translations, examples, locutions, etc;

• completed with 47,630 Japanese-French or Japanese-English entries from the JMdict [START_REF] Breen | JMDict: a Japanese-multilingual dictionary[END_REF] for modern vocabulary and

• 23,486 Japanese-French or Japanese-English Wikipedia links for recent terminology.

The articles from JMdict and Wikipedia were imported if they appeared also in the Super Daijirin dictionary, in order to avoid The result of the OCR process is never perfect. Thus, some articles coming from the Cesselin need corrections. The dictionary can be edited online by any user who detect an error. So far, more than 81,000 contributions were established in the 4 years of the project life.

Mecab analyzer is distributed with a generic dictionary called ipadic with 392,126 entries in total. Despite its consequent size, several words in our dictionary are not present in the ipadic dictionary. Thus, we decided to generate a Mecab dictionary from the content of Jibiki.fr dictionary in order to enlarge the coverage of Mecab. Our dictionary contains 22,321 entries that are not present in ipadic. The first version is still beta. On going work is on its way to produce a correct dictionary particularly concerning the subcategories of the verbs. Once this work will be completed, we will publicly release this dictionary on the Jibiki.fr project website.

ENRICHING THE ACTIVE READING FOR INTERCOMPREHENSION 4.1 Purpose of the new features

We will now present two new features aiming at making the active reading output display even more informative. As explained previously, Japanese language uses different writing systems and readings depending on which lexicon stock a given word belongs to: on'yomi for kanji whose pronunciation was borrowed from Chinese during different periods, kun'yomi for Japanese spellings, katakana for foreign loan words. The objective is to highlight the words of the text with different colors according to their stock, in order to tell them apart and get knowledge of the links between them and their various influences. Then, the Pīnyīn is displayed below the words whose kanji are on'yomi 5 (see Figure 4). This information gives some clues for intercomprehension between sinogramic languages [4]. These new features are aimed to learners of Japanese with some knowledge of Mandarin or the contrary.

Figure 3 shows the new active reading interface with these new features. The top half of the window is the same as Figures 2.

The result is then still displayed on the bottom half of the window with the furigana on top of the Japanese sentence and a French translation when the mouse is over a word (here the word "osoi", translated in French by "tardif") but two new features can be seen:

5 https://jibiki.fr/reading/pierric_f.php • The words in violet background are in kun'yomi (Yokohama, kakari, osoi);

• The words in red background are in on'yomi and below them, the Pīnyīn of the Mandarin equivalent word is displayed in red (Tōkyō, densha, fun, hou);

• The words in green background are in katakana here an English loan word (baiku).

As already mentioned above, among the Chinese imperial literary standards, there was the fundamental practice of Classical riming system. If it is abolished nowadays, this institution is still the potential of intercomprehension between SL through the conformation to the sinosyllable structure. Thus, on the Figure 3 , one can already see that a constant pattern emerges between Pīnyīn and rōmaji for Japanese words in on'yomi. For example, the Mandarin "dōng" is transcribed with the Japanese "tō"; the Mandarin "diàn" is transcribed with the Japanese "den", etc.

Challenges of the implementation

The implementation of such features presents some particularities to take into account:

• We made the assumption that all the kanji of a same word have the same nature, since there does not seem to exist any counter-example to this rule, at least in modern-day Japanese.

Although MeCab provides the information about the pronunciation of the kanji, it does not give the on/kun'yomi nature. The information being also absent from the Cesselin, we have to collect it from KANJIDIC 6 , an online kanji database from James Breen's JMdict/EDICT project [START_REF] Breen | JMDict: a Japanese-multilingual dictionary[END_REF]. The KANJIDIC has been imported also on the Jibiki platform.

• Since we lack exhaustive resources able to indicate from which foreign language a loan word written in katakana comes from, we will not be able to tell them apart with different colors so far.

• There are also numerous cases where a word has a special pronunciation that cannot be deduced from the pronunciations of the individual kanji it contains (e.g. 今日 kyō, today). These cases will appear with a different color.

Details of the implementation

We modified the active reading PHP program by incorporating a piece of code composed of four main parts: information extraction, decision making, Pīnyīn addition and display.

Program overview

The active reading's PHP code consists in the following steps, the green ones being our new features' implementation:

Information extraction (f)

The on/kun'yomi nature of the kanji has to be retrieved from KANJIDIC. The data is organized in XML tree structures that can be accessed with a Curl request to the Jibiki REST API. The entries to be checked are those of each kanji for each word in the text that possesses furigana. If a word also contains hiragana (as okurigana, e.g. 買い物い物物 kaimono, 'shopping'), the XML structure of this said sinogram will be empty ; we nevertheless keep the hiragana as it is. For katakana words, we will use a regular expression with a pre-established list of katakana. Instead of immediately processing the data, we store it in an array to make it available for further use and to ensure code clarity. Considering the aforementioned hypothesis of one yomi per word, the on' and kun'yomi will be stored separately, to be later processed independently.

Before going to the decision part, the pronunciation data has to be formatted:

• Some pronunciations are preceded or followed by a dash (-), indicating a restriction on their position in the word (e.g. 何:な んnan-(what) cannot be used at the end of a word). This information will be used to skip some cases in the decision part. We also remove the dash.

• Some also include a dot (.), usually separating the radical of a verb or adjective from its inflection (e.g. 会:あ.う a.u, meet).

We then remove the inflection and the dot.

Decision making (g)

Once all the data has been collected and formatted, for each word, we will compare the different possible on'yomi combinations of their kanji (and potential okurigana) with the known furigana of the word. If there is a match, we know that the word is on'yomi and we label it as such. If not, we repeat the process for the kun'yomi. If ultimately no match is found, the word will be labeled as special pronunciation.

Once the word's reading has been figured out, the highlight color value is drawn from a correspondence array: red for on'yomi kanji, purple for kun'yomi kanji, gold for special pronunciation and green for katakana.

Pīnyīn addition (i)

To retrieve the Pīnyīn of the on'yomi words, we used HanLP parsing toolkit for Mandarin to retrieve the individual information for each kanji (Figure 4). HanLP is a powerful java library for processing Mandarin texts. We developed a simple

Display (i)

On the HTML side, we need to add the highlight variable as a style attribute in the start tag of the output element in the case of kanji or katakana words. The tag is a Ruby HTML tag, which allows to write furigana on top of Japanese text.

In order to display Pīnyīn below as well, the Ruby <rtc> tag is used in conjunction with the <rt> tag for furigana and a CSS style option has to be added: {ruby-position: under;}. Unfortunately, even if the ruby tags including the <rtc> one are now part of the HTML5 specifications, few browsers implement it. At the moment, only Firefox is correctly rendering the Pīnyīn below the text. The webkit family (Safari, Chrome, etc.) and Opera browser are rendering the Pīnyīn next to the Japanese.

CONCLUSION AND PERSPECTIVES

This article explains the development of a first version of an active reading tool for Japanese texts enriched for intercomprehension between sinogramic languages by displaying the Pīnyīn of the Japanese on'yomi words. This work is promising. We are waiting for the feedback of potential users in order to improve it. Several perspectives are on their way.

Concerning the implementation of the tool itself, as the code currently constitutes a draft of a more formalized future clean version, some improvements can be imagined for code optimization, performance improvements and trustful case handlings. This latter point can be challenging, since it requires exhaustive data sets of special cases (e.g. counters, Japanese names), most of them being virtually infinite. In the case of our feature, the extent of handled cases depends on the coverage of KANJIDIC. A more thorough (but costly) process would be to cross-check different sources. Other technical optimizations could be to display the Pīnyīn below the Japanese even on browsers that do not render the <rtc> tag correctly.

The described active reading scenario of Japanese texts could be extended at least in two different ways:

• looking up etymological information for katakana words from rich dictionaries (the Cesselin contains such information) and displaying precise information about the origin of the words. For example, the word "baiku" comes from the English word "bike".

• Computing automatically analogies between rōmaji and Pīnyīn.

We are working also on different active reading scenarios for other languages:

• It would be interesting to include other sinogramic languages such as Korean, Vietnamese and Sinitic languages among which Taiwanese very soon.

• A study has also been driven for Berber and Arabic speaking learners of French and English [START_REF] Abdellaoui | Outillage de l'accès aux textes par la lecture active étymologique multilingue pour apprenants berbérophones et arabophones[END_REF].

• Other language families could be addressed like Romance languages, Slavic languages, Germanic languages, pidgins and creoles, etc.

6.

Figure 2 .

 2 Figure 2. Parsing of a Japanese text with MeCab

  a. Catching user's input b. Parsing it through MeCab morphological analyzer 6 http://www.edrdg.org/kanjidic/kanjidic.html c. Gathering the following elements, for each word: furigana, lemma (e.g. base form of verbs…), part-ofspeech (e.g. verb, noun…), subcategory (e.g. transitive verb) d. Converting furigana's katakana into hiragana e. Querying the Jibiki.fr Japanese-French dictionary for retrieving translations with the REST API f. Querying KANJIDIC with the REST API and extracting information g. Processing it through decision function h. Associating the result to a highlight color i. For on'yomi word, get Pīnyīn through HanLP 7 Mandarin morphological analyzer On HTML side: j. Displaying each word with: corresponding highlight color, furigana on top, translation in a pop-up window and, for on'yomi words, Pīnyīn below.

Figure 3 :

 3 Figure 3: Active reading with on'yomi, kun'yomi and Pīnyīn

  The resulting resource thus built contains more than 153,000 entries and 140,000 examples. It can be considered as the most reliable source for Japanese-French translations.

2 https://jibiki.fr/reading/ 3 https://taku910.github.io/mecab/ 4 http://jibiki.fr/ a large number of unuseful entries.


Amelia Carolina Sparavigna

The q-integers and the Mersenne numbers

Keywords: q-calculus, q-integers, Tsallis q-entropy, Mersenne numbers

Introduction Several mathematicians have contributed to a calculus that today is known as the q-calculus [START_REF] Ernst | A Comprehensive Treatment of q-Calculus[END_REF][START_REF] Annaby | q-Fractional Calculus and Equations[END_REF][START_REF] Ernst | The History of q-calculus and a New Method[END_REF][START_REF] Aral | Applications of q-Calculus in Operator Theory[END_REF][START_REF] Ernst | The different tongues of q-calculus[END_REF][START_REF] Kac | Quantum Calculus[END_REF]. As a consequence of the many contributions, we find that it is known as "quantum calculus," or "time-scale calculus", or "calculus of partitions" too [START_REF] Ernst | The different tongues of q-calculus[END_REF]. It is also called the "calculus without limits", because it is equivalent to the traditional infinitesimal calculus without the notion of limits. Besides being known with different names, the q-calculus is expressed by means of different notations or, as told in [START_REF] Ernst | The different tongues of q-calculus[END_REF], by different "dialects". Here we will use the approach and the notation given in the book by Kac and Cheung [START_REF] Kac | Quantum Calculus[END_REF].

The first aim of the work here proposed is that of showing the following fact. The q-integers, the q-analogue of the integers that we can find in the q-calculus, are forming a group having a generalized sum which is similar to sum of the Tsallis q-entropies of independent systems. After, we will see that the symmetric form of q-integers is linked to the Kaniadakis calculus. We will conclude the discussion considering the Mersenne numbers and their link to the q-integers.

Let us stress that, besides the discussion of the previously mentioned numbers, the general aim of the paper is that of popularizing the existence of the q-calculus.

The q-integers

Let us start defining the q-integers.

In the q-calculus, the q-difference is simply given by:

d q f =f (qx )-f ( x )
Sparavigna, A. C. The q-Integers and the Mersenne Numbers ( May 23, 2018). SSRN Electronic Journal. DOI 10.2139/ssrn.3183800 From this difference, the q-derivative is given as:

(1) D q f = f ( qx )-f ( x ) qx-x
The q-derivative reduces to the Newton's derivative in the limit q →1 . ( 1) is also known as the Jackson derivative, after Frank Hilton Jackson (1870 -1960), the English clergyman and mathematician who worked at the beginning of the XXth century on the q-calculus.

Let us consider the function f ( x )=x

n . If we calculate its q-derivative, we obtain:

(2)

D q x n = ( qx ) n -x n qx-x = q n -1 q-1 x n-1
Comparing the ordinary calculus, which is giving ( x n ) '=n x n-1 , to Equation (2), we can define the "q-integer" [n] by:

( 3) [n]= q n -1 q-1 =1+q +q 2 +.. .+q n-1

Therefore Equation (2) turns out to be:

D q x n =[ n] x n-1
As a consequence, the n-th q-derivative of f ( x )=x n , which is obtained by repeating n times the q-derivative, generates the q-factorial:

[n ] !=[ n ][n-1 ]. . .[ 3 ][2 ][ 1]
Form the q-factorials, we can define q-binomial coefficients:

[n ]! [ m] ![ n-m]!
This means that we can use the usual Taylor formula, replacing the derivatives by the qderivatives and the factorials by q-factorials (in a previous work, we have discussed the qexponential and q-trigonometric functions [START_REF] Sparavigna | Graphs of q-exponentials and q-trigonometric functions[END_REF]). Then, in the q-calculus, the q-integer [n] acts as the integer in the ordinary calculus.

The group of q-integers

We known that the set of integers consisting of the numbers ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., having as operation the addition, is a group. Therefore, let us consider the set of q-integers given by (3) and investigate its group. In particular, we have to determine its operation of addition.

Let us remember that a group is a set A having an operation • which is combining the elements Therefore, to qualify a group as an Abelian group, the set and operation must satisfy five requirements which are known as the Abelian group axioms. A group having a noncommutative operation is called a "non-abelian group" or "non-commutative group". For an Abelian group, one may choose to denote the group operation by + and the identity element by 0 (neutral element) and the inverse element as -a (opposite element). In this case, the group is called an additive group.

First, we have to define the operation of addition. It is not the sum that we use for the integers, but it is a generalized sum which obeys the axioms of the group.

Let us start from the q-integer [m+n ] :

[m+n ]= q m +n -1 q-1 = 1 q-1 ( q m q n -1+ q m -q m )= 1 q-1 (q m (q n -1)+q m -1) [m+n ]= 1 q-1 (q m (q n -1)+(q m -1)+(q n -1)+(1-q n ))= 1 q-1
((q m -1)( q n -1)+(q m -1)+(q n -1))

Therefore, we have:

(4) [m+n ]=[m]+[n]+( q-1)[m][n]
Then, we can define the generalized "sum" of the group as: (5)

[m]⊕[ n]=[m]+[n ]+(q-1)[m][n ]
(for other examples of generalized sums see [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF]).

If we use [START_REF] Ernst | The different tongues of q-calculus[END_REF] as the sum, we have the closure of it, because the result of the sum is a qinteger. Moreover, this sum is commutative.

The neutral element is: (6) [0]= q 0 -1 q-1 =0

Let us determine the opposite element [o] , so that:

[o]⊕[n]=0 0=[0]=[o]⊕[n]=[o ]+[n]+(q-1)[o ][n] -[n]=[o ]+( q-1)[o][n] (7) [o]=- [n] 1+(q-1)[ n] =- q n -1 ( q-1) q n = q -n -1 q-1 =[-n]
The opposite element of q-integer [n] is the q-integer of -n , that is [-n]

Let us discuss the associativity of the sum.

It is necessary to have:

[m]⊕([n]⊕[l])=([m]⊕[n])⊕[l]
Let us calculate:

[m]⊕([n]⊕[l])=[m]⊕([n]+[l]+(q-1) [n][l]) [m]⊕([n]⊕[l])=[m]+[n]+[l]+(q-1)[n ][l]+(q-1)[m][n ]+(q-1)[m][l]+(q-1) 2 [m][n][l]
And also:

([m]⊕[n])⊕[l]=([m]+[n]+(q-1)[m][n])⊕[l ] ([m]⊕[n])⊕[l]=[ m]+[n]+(q-1)[m][n]+[l]+(q-1)[m][l]+(q-1)[n ][l]+(q-1) 2 [m][n][l]
It is also easy to see that:

[m]⊕[n]⊕[l ]=[m+ n+l]
As we have shown, the five axioms of an Abelian group are satisfied. In this manner, using the generalized sum given by ( 5), we have the Abelian group of the q-integers.

The link to Tsallis calculus

Let us also note that the generalized sum ( 5) is similar to the sum that we find in the approach to entropy proposed by Constantino Tsallis.

In 1948 [START_REF] Shannon | A Mathematical Theory of Communication[END_REF], Claude Shannon defined the entropy S of a discrete random variable Ξ as the expected value of the information content: S= ∑ i p i I i =-∑ i p i log b p i [10]. In this expression, I is the information content of Ξ, the probability of i-event is p i and b is the base of the used logarithm. Common values of the base are 2, the Euler's number e, and 10.

Constantino Tsallis generalized the Shannon entropy in the following manner [START_REF] Tsallis | Possible Generalization of Boltzmann-Gibbs Statistics[END_REF]:

S q = 1 q-1 ( 1-∑ i p i q )
Given two independent systems A and B, for which the joint probability density satisfies:

p( A , B)= p( A) p( B)

the Tsallis entropy gives: (8) S q ( A , B)=S q ( A)+ S q (B)+(1-q)S q ( A) S q (B)

The sum of more than two terms of Tsallis entropies is discussed in [START_REF] Sparavigna | On the generalized additivity of Kaniadakis entropy[END_REF].

The parameter (1-q) , in a certain manner, measures the departure from the ordinary additivity, which is recovered in the limit q →1 . Actually the group on which is based the Tsallis entropy, and therefore Equation ( 8), is known as the "multiplicative group" [START_REF] Sparavigna | Graphs of q-exponentials and q-trigonometric functions[END_REF][START_REF] Sicuro | Groups, information theory, and Einstein's likelihood principle[END_REF][START_REF] Tempesta | Groups, generalized entropies and L-series[END_REF]. As stressed in [START_REF] Curado | A new entropy based on a group-theoretical structure[END_REF], the use of a group structure allows to determine a class of generalized entropies. Let us note the group of the q-integers, with addition (5), can be considered a "multiplicative group" too.

Let us stress that we have a link of the multiplicative group to the Tsallis entropy. The group of the n-integers had been studied in [START_REF] Nathanson | Additive number theory and the ring of quantum integers[END_REF][START_REF] Kontorovich | Quadratic addition rules for quantum integers[END_REF] too, but in these articles, a quite different expression for the generalized sum had been proposed. It is given as the "quantum sum"

[x ]⊕[ y ]=[x ]+ q x [ y ]
, where the link to the Tsallis calculus is less evident.

Symmetric q-numbers

In the previous discussion we have considered the group of the q-integers as defined by qcalculus. In [START_REF] Kac | Quantum Calculus[END_REF] it is also defined the symmetric q-integer in the following form (here we use a notation different from that given in the Ref.6): (9) [n] s = q n -q

-n q-q -1
Repeating the approach previously given, we can determine the group of the symmetric qintegers.

Let us start from the q-integer [m+ n ] s , which is according to ( 9):

[m+ n ] s = q m+ n -q -( m+ n) q-q -1
and try to find it as a generalized sum of the q-integers [m] s and [n] s .

By writing q=exp(log q) , the q-integer turns out into a hyperbolic sine:

(10) [n] s = q n -q -n q-q -1 = e n log q -e -nlog q q-q -1 =2 sinh (n log q) (q-q -1 )
Apart from a numerical factor, this is the form of the generalized numbers proposed by G.

Kaniadakis in his k-calculus [18][START_REF] Kaniadakis | Non-linear kinetics underlying generalized statistics[END_REF][START_REF] Kaniadakis | Statistical mechanics in the context of special relativity[END_REF][START_REF] Kaniadakis | Statistical mechanics in the context of special relativity II[END_REF][START_REF] Kaniadakis | Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions[END_REF].

From [START_REF] Borda | Fundamentals in Information Theory and Coding[END_REF], we can write also:

1 2 (q-q -1
)[n ] s =sinh (n log q)

Therefore:

[m+n ] s = q m+n -q -(m +n) q-q -1 =2 sinh((m+ n)logq) (q-q -1
)

Using the properties:

sinh( x + y )=sinh x cosh y +cosh x sinh x ; cosh x=√1+sinh 2 x
we obtain:

[m+n ] s = 2 ( q-q -1
)

[ sinh(mlog q)cosh (n log q)+sinh( n log q)cosh (mlog q)]

[m+n ] s =[m] s cosh( n log q)+[n] s cosh ( mlog q) [m+n ] s =[m] s √ 1+sinh 2 (n log q)+[ n] s √ 1+sinh 2 (m log q)
Let us define: k =(q-q -1 )/2 and then: k [n] s =sinh (n log q) . As a consequence, we have the generalized sum of the symmetric q-integers as:

(11) [m] s ⊕[n ] s =[m] s √ 1+k 2 [n] s 2 +[n] s √ 1+ k 2 [m] s 2
Let us stress that ( 11) is also the generalized sum proposed by G. Kaniadakis in the framework of a calculus [START_REF] Kaniadakis | Non-linear kinetics underlying generalized statistics[END_REF][START_REF] Kaniadakis | Statistical mechanics in the context of special relativity[END_REF][START_REF] Kaniadakis | Statistical mechanics in the context of special relativity II[END_REF][START_REF] Kaniadakis | Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions[END_REF], the details of which are given in [START_REF] Kaniadakis | Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions[END_REF].

By means of ( 11), we can repeat the approach given previously for q-numbers (3) and study of the group of the symmetric q-integers.

The Mersenne numbers

In the case that q=2 , we have:

[n]= 2 n -1 2-1 =2 n -1
These are the Mersenne Numbers. About these numbers, a large literature exists (see for instance that given in [START_REF] Weisstein | Mersenne Number[END_REF]). Among these numbers we find the Mersenne primes.

The numbers are named after Marin Mersenne (1588 -1648), a French Minim friar, who studied them in the early 17th century.

Mersenne numbers are written as [START_REF] Weisstein | Mersenne Number[END_REF]:

M n =2 n -1
Of course, because they are q-integers for q=2 , we have the generalized sum given in (5):

(5') [m]⊕[ n]=[m]+[n ]+(2-1)[m][n]=[m]+[n]+[m][n]
But we can repeat the calculus as an exercise.

We can start from the number M m+n and calculate.

M m+n =2 m+n -1 M m+n =2 m+n -1=2 m 2 n -1-2 m +2 m -2 n +2 n -1+1=2 m ( 2 n -1)-1+ 2 m -2 n +1+2 n -1 M m+n =(2 m -1)(2 n -1)+2 m -1+2 n - 1
Therefore, we can write the following generalized sum:

M m+n =M m ⊕M n =(2 m -1)(2 n -1)+(2 m -1)+(2 n -1)
or:

(12) M m+n =M m ⊕M n =M m + M n + M m M n ( 12
) the same as (5'). Let us stress once more that this is a generalized sum that we can find in the case of the multiplicative groups [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF].

Using [START_REF] Sparavigna | On the generalized additivity of Kaniadakis entropy[END_REF], we can imagine for the Mersenne numbers the following recursive relation:

M n+1 =M n ⊕M 1 =M n + M 1 + M n M 1
We can verify as follow:

2 n+1 -1=(2 n -1)+(2 1 -1)+(2 n -1)(2 1 -1)=2 n + 2 n+1 -2 n -2+ 1=2 n+1 - 1
The sum ( 12) is associative, so that:

M m ⊕M n ⊕M l =M m + M n + M l + M m M n + M n M l + M m M l + M m M n M l
We cannot have a group of the Mersenne numbers, without considering also the opposites of them, so that:

0=M n ⊕Opposite(M n ) Therefore: Opposite(M n )=- M n M n +1 =M -n Explicitly: Opposite(2 n -1)=- (2 n -1) (2 n -1)+1 = (-2 n +1) 2 n =2 -n -1
These numbers are the Mersenne numbers with a negative exponent. So we have:

M n-n =M n ⊕M -n =M n + M -n + M n M -n 0=2 0 -1=(2 n -1)+(2 -n -1)+(2 n -1)(2 -n -1)=2 n +2 -n -2+ 2 n 2 -n -2 -n -2 n +1=0

Symmetric Mersenne

Let us consider the symmetric q-integer in the case of q=2 .

We can define the symmetric in the following manner: (13) M n s

=[n] s = 2 n -2 -n 2-2 -1
By writing 2=exp (log 2) , ( 13) turns out into a hyperbolic sine: )/2 ; we have the generalized sum of the symmetric Mersenne as: Of course, we have again the generalized sum proposed by G. Kaniadakis.

As a conclusion we can note that, by means of the generalized sums, we have found a different approach to the Mersenne numbers too. In my opinion, it is also possible that it was the form of the Mersenne numbers that inspired the Reverend Jackson to modify the usual derivative into the definition (1) of the q-calculus.

  of A. That is, the operation combines any two elements a,b to form another element of the group denoted a•b. To qualify (A,•) as a group, the set and operation must satisfy the following requirements. Closure: For all a,b in A, the result of the operation a•b is also in A. Associativity: For all a,b and c in A, it holds (a•b)•c = a•(b•c). Identity element: An element e exists in A, such that for all elements a in A, it is e•a = a•e = a. Inverse element: For each a in A, there exists an element b in A such that a•b = b•a = e, where e is the identity (the notation is inherited from the multiplicative operation). A further requirement is the commutativity: For all a,b in A, a•b = b•a. In this case, the group is known as an Abelian group.

1 )

 1 Again, as previously told, apart from a numerical factor, this is the form of the generalized numbers proposed by G. Kaniadakis.Let us define k =(2-2 -1

  1+ k 2 (M n s ) 2 + M n s √ 1+k 2 (M m s ) 2


Youcef Benmessaoud
email: youcef.benmessaoud@femto-st.fr

Walid Belguerras
email: walid.belguerras@femto-st.fr

Frédéric Dubas
email: fdubas@gmail.com

Mickaël Hilairet

Mickael Hilairet
email: michael.hilairet@univ-fcomte.fr

M Hilairet

3-D Generic Magnetic Equivalent Circuit Taking Into Account Skin Effect: Magnetic Field and Eddy-Current Losses

Introduction

Context of this paper

Different models more or less accurate, taking into account spatio-temporal harmonics have been developed in the past. The importance of studying the eddy-currents topic still remains attractive in static or/and dynamic applications such as non-destructive control, permanent-magnet (PM) synchronous machines (PMSMs). Indeed, incorporating PMs in electrical machines improve really the efficiency and the behaviour of the integral quantities. However, the presence of conductive materials can consist a technical drawback in the high frequency, causing additional losses at high-speed which dissipate in the heat forms, leading to the PMs degradation. Indeed, the magnetic field variation is one of the main source of these losses which further affect the electromagnetic proprieties. This meant that eddy-currents become essential when sizing PMSMs at high-speed. In order to estimate these eddy-current losses, different modeling technics in the literature can be found based on different formulations [1]. It can be find analytical models based on the formal resolution of Maxwell's equations [START_REF] Zhu | Improved Analytical Modeling of Rotor Eddy Current Loss in Brushless Machines Equipped with Surface Mounted Permanent Magnets[END_REF], the finite-difference method from Maxwell's equations where the boundary conditions (BCs) come from the FEA [START_REF] Benlamine | 3-D Numerical Hybrid Method for PM Eddy-Current Losses Calculation: Application to axial-flux PMSMs[END_REF], the hybrid models [START_REF] Gerlach | Determination of Eddy Current Losses in Permanent Magnets of SPMSM with Concentrated Windings: A Hybrid Loss Calculation Method and Experimental Verification[END_REF],..., the semi-analytical model based on the MEC (or reluctances network). Different manner to incorporate magnetic reluctances, magnetic inductances or MMFs in MEC or coupling models with MEC to compute eddy-currents [START_REF] Gholizad | Direct Modeling of Motional Eddy Currents in Highly Saturated Solid Conductors by the Magnetic Equivalent Circuit Method[END_REF][START_REF] Bormann | Reluctance Network Treatment of Skin and Proximity Effects in Multi-Conductor Transmission Lines[END_REF][START_REF] Yoshida | A Method for Calculating Eddy Current Loss Distribution Based on Electric and Magnetic Networks[END_REF][START_REF] Yoshida | Consideration of Eddy Current Loss Estimation in SPM Motor Based on Electric and Magnetic Networks[END_REF]. The main drawbacks of the last works can be liked to the incorporated MMFs or magnetic reluctances which don't take into account the skin effect as well as the method depend on the FEA to get the local quantities that allows computing eddy-current losses. Add to this, no model exist till now which permit to study the skin effect on the behaviour of the magnetic field overs all the electromagnetic devices by applying only the MEC.

Objectives of this paper

In this paper, a new approach is proposed to incorporate the MMFs which take into account the skin effect in the semianalytical model based on the 3-D generic MEC. In this way, it can be analysed the impact of the magnetic fields due to eddy-currents occurred in massive conductive parts and over all parts of the electromagnetic devices. The performed model will be able to compute the eddy-current losses in massive conductive parts, to predict the behaviour of both magnetic field and flux density over all the electromagnetic device. The segmentation in the two axes is also allowed but not treated in this paper.

U-cored static electromagnetic device

Fig. 1 shows the two-dimensional (2-D) view of the static electromagnetic device whose the experimental tests have been presented in [START_REF] Chetangny | Eddy-current loss in a conductive material inserted into a Ucored electromagnetic device[END_REF]. This device is constituted by a mobile armature that allows to insert massive conductive parts (viz., aluminium) of various thicknesses. Two coils having t N serie turns are connected in parallel. It is supplied with a sinusoidal voltage. Since the magnetic circuit is not saturated, the current is then sinusoidal waveform with a maximum amplitude of max I . The geometric and physics parameters are given in Tab. 1

3-D generic MEC using mesh-based formulation

The 3-D automatic generation MEC using mesh-based formulation is already developed and explained in [START_REF] Benmessaoud | Three Dimensional Automatic Generation Magnetic Equivalent Circuit Using Mesh-Based Formulation[END_REF]. This provides the distributions of magnetic flux density

 

;;

x y z B B B  B
in all parts of an electromagnetic device. In [START_REF] Benmessaoud | Three Dimensional Automatic Generation Magnetic Equivalent Circuit Using Mesh-Based Formulation[END_REF], the MMF associated to massive conductive parts are equal to zero because the skin effect has been neglected. The results of B were confronted with those obtained by the 3-D FEA. The computation time is divided by 3 with an error less than 1%. In the following section, it is explained the followed approach to associate the MMFs produced by the magnetic reaction field taking into account eddy-current phenomenon.

Developed approach

Principle of model

Fig. 2 represents the flowchart that explains the approach followed to incorporate the MMF produced by the eddycurrents in massive conductive parts into the 3-D generic MEC. Firstly, it is not considered the electrical conductivity of the massive conductive parts. Hence, the 3-D generic MEC results correspond to those levied at magnetostatic application. Next, the magnetic flux density in the middle of the massive conductive part will be considered to define the BCs to resolve the complex Helmholtz's equation. At this stage, the evolution of the eddy-current magnetic field can be got. Thereafter, Hopkinson's law will be applied to estimate the MMFs due to the magnetic field reaction caused by the massive conductive parts. Finally, these later will be incorporated into the performed 3-D generic MEC. New simulation gives the results by considering the electrical conductivity of the massive parts. Furthermore, it can be remarked that the crossing flux tube in the electromagnetic device when considering the electrical conductivity can be affected in values and length terms.

Magnetic field due to electrical conductivity in massive conductive parts

To estimate the MMFs with the skin effect wherever in the massive conductive part, the approach derived from the formal resolution of Maxwell's equations by using the separation of variables method and the Fourier's series. In quasi-stationary approximation, inside a linear (non)magnetic material of electrical conductivity without electromagnetic sources, the partial differential equation in magnetodynamic in term of mp H can be defined by [1] 0 t

       mp 2 mp

H H  (Diffusion equation).

(

In

          , (2)
where 2

mp mp         j .
The BCs are considered homogeneous at the edges of massive conductive parts, which are equal to 3. By using the separation of variables method and by applying the BCs, the 2-D general solution of y H  in both directions (i.e., x-and z-edges) can be written as Fourier's series [1]  

, mp ys H H f x z   , ( 3a
)         1,3, ch ch , 2 ch cos z k k k k z d f x z e x z                             , ( 3b
) 2 sinc 2 21 ch 2 k z k k k k d e w                        , ( 3c
)
where

Hopkinson's law to estimate the MMF values with the skin effect

As it is explained in the sections before, initially the MMFs associated to massive conductive parts are equal to zero. In reality, the magnetic field inside and outside the massive conductive parts is affected by electrical conductivity. Therefore, the MMFs in the massive conductive parts can not be equal to zero. To estimate the eddy-currents MMFs that occur in massive conductive parts, the Hopkinson's law on a "flux tube" in massive conductive parts is used as well as (3).

. So the eddy-currents MMFs are defined by

    3 , 1 , 2 mp mp s y h MMF x y H f x z Nd          , ( 4a
)   mp mp t MMF MMF e       j . ( 4b
)
Finally, mp MMF  will be incorporated into the 3-D generic MEC in massive conductive parts by respecting the discretization number in the y-axis chosen in the massive

 

, 2

mp y s H x d H   0 z 2 w  2 w 2 d  2 d x 2 0 mp mp y y H H       2    , 2 mp y s H x d H      2 , mp y s H w z H      2 ,

Result discussion

In this section, all the results are calculated with (i.e., Maxwell-Ampère), the absolute resultant current-density is shown in Fig. 7. Fig. 8 and Fig. 9 represent respectively the magnetic flux density map in term to components (i.e., x-and y-component) on the 2-D grid N°2 without and the skin effect at 50 Hz. From these figures, the influence of the existing massive conductive part can be seen, especially magnetic flux density values that decrease in the most parts of the device, where the maximum value for y-component go from

Eddy-current loss calculation

The 3-D eddy-current losses results given by the 3-D generic MEC with the skin effect are confronted with those obtained by 3-D FEA and experimental tests. For the experimental acquisition, these losses are calculated by using the separation of losses method [START_REF] Chetangny | Eddy-current loss in a conductive material inserted into a Ucored electromagnetic device[END_REF]. In the proposed model, the eddy-current losses in the massive conductive part can be calculated by [1]  

22 1 mp ed x y V mp W J J dV      , (5)
where mp V is the volume of the massive conductive part. The comparisons were performed at 50 Hz on two thickness in aluminium, viz., 6 mm and 10 mm. Fig. 10 1]. The performed model gives a good agreement with both 3-D FEA and measurement.

Conclusion

In this work, a new approach is proposed to analyse the influence of the eddy-currents on the behaviour of the magnetic flux density over all the U-cored static electromagnetic devices. A 3-D generic MEC was used and improved taking into account skin effect in massive conductive parts. The classical MMF distribution has been modified by integrating the eddy-currents MMFs in massive conductive parts, which was introduced by a formula based on the Maxwell-Fourier method and by using the Hopkinson's law. It is able to compute the eddy-current losses in massive conductive parts. The model allows to study also the segmentation in both two space axes. Eddycurrent losses and local quantities resulted from 3-D generic MEC with the skin effect gives good agreement comparing to both measurement and 3-D FEA.

Fig. 1 U

 1 Fig. 1 U-cored static electromagnetic device: (a) Experimental test [9], and (b) Geometrical parameters.

Fig. 2

 2 Fig. 2 Flowchart of 3-D generic MEC with the skin effect.

Fig. 3

 3 Fig. 3 BCs at edges of massive conductive parts in   , xz coordinate

sH.

  This value of magnetic field is defined as the normal magnetic field value crossing the middle of the massive conductive part determined by magnetostatic 3-D generic MEC i.e., without electrical conductivity, viz.,

  of electrical frequency (viz., 50 Hz and 1,600 Hz). A very weak discretization has been imposed in the 3-D generic MEC obtaining a good approximation on the y-component of B in massive conductive parts. To improve the precision a high discretization can be applied on the 3-D generic MEC. The validation of proposed model has been realized on the Cedrat's Flux3D software package by using the application "Harmonic State 3-D"[START_REF]General operating instructions[END_REF]. A 2-D view of the U-cored static electromagnetic device is illustrated in Figs. 4 for yz-plane [see Fig. 4(a)] and for xzplane [see Fig. 4(b)]. The 2-D grid N°1 parallel to xz-plane and the 2-D grid N°2 parallel to xy-plane are presented by the dashed black bold line for various comparisons. Figs. 5 represent the evolution of the y-component of B on the 2-D grid N°1 with(out) the skin effect. Fig. 5(a) corresponds to the magnetic flux density allowing to obtain the initial values of BCs (i.e., s H to estimate mp MMF  ). The results show a good agreement. Add to this, in Fig. 5(b), it can be clearly seen the deformation of the y-component of  B due to the electrical conductivity in massive conductive parts. Fig. 6(a) illustrates more clearly the impact of the massive conductive part on the behaviour of the magnetic field, where the skin effect appears slightly at 50 Hz contrary to 1,600 Hz that appears clearly as shown in Fig. 6(b). Indeed, the conductive massive part act as a barrier to the crossing flux. Using   JH 

.

  . 8(a) and Fig. 9(a)] for x-component. A particular attention can be given to x B  , in fact that it present a slight increase the neighbouring region of the coil, where the x-component of magnetic flux density passes from It can be noted that the massive conductive parts could affect the efficiency of the U-cored static electromagnetic device by increasing the flux leakage.

Fig. 4 2

 4 Fig. 4 2-D view of the U-cored static electromagnetic device in: (a) yz-plane, and (b) xz-plane.

Fig. 5 Fig. 7 Fig. 8 Fig. 9 Fig. 10

 578910 Fig. 5 The y-component of the magnetic flux density on the 2-D grid N°1: (a) without the skin effect, and (b) with the skin effect.

Table 1

 1 Geometrical and physical parameters

	Parameter, Symbol [Unit]	Value
	Depth,	d [mm]	43
	Width,	w [mm]	43
	Coil height,	c h [mm]	77
	Coil width,		c w [mm]	10
	Coil section,	S	c	c h w c 	[mm²]	770
	Yoke height,	y h [mm]	43
	Yoke length,	y l [mm]	150
	Thickness of massive part,	mp h [mm]	6 or 10
	Height overhang top,	ot h [mm]	19
	Height overhang bot,	ob h [mm]	4
	Electrical frequency,	f [Hz]	50 or 1,600
	Maximal current, max I	[A]	0 to 8.2
	Number of turns,	t N [-]	500
	Relative permeability of massive parts in aluminium,  [-]	1
					rmp
	Electrical conductivity of massive parts in
	aluminium,	 [S/m]	38.46 10 	6
					mp
	Vacuum permeability, 0  [H/m]

  Cartesian coordinates  

					, xz ,	H	mp		 0;	mp y H 	 ; 0	with
	H	 yy mp mp H e    t    j		, where	j		1	and	2 f   is the
	electrical pulse, in massive conductive parts considering the
	skin effect is then governed by the complex Helmholtz's
	equation, viz.,
		mp	mp
		HH 22 yymp 2 y H		0
		xz

Acknowledgements This work was supported by RENAULT-SAS, Guyancourt France. This scientific study is related to the project "Conception optimale des chaines de Traction Electrique" (COCTEL) financed by the "Agence De l'Environnement et de la Maîtrise de l'Énergie" (ADEME).


Richy Maurel

François Gnandji

Amèdédjihundé Fifatin

Frédéric Hounnou

Didier Dubas
email: dfdubas@gmail.com

Christophe Chamagne

Antoine Espanet

Vianou
email: gavianou@yahoo.fr

Maurel Aza-Gnandji
email: amaurel.aza@gmail.com

François-Xavier Fifatin
email: bfxfifatin@yahoo.fr

A Hypolite

J Hounnou

Christophe Espanet
email: fchristophe.espanet@univ-fcomte.fr

Antoine Vianou

Complementarity between Solar and Wind Energy Potentials in Benin Republic

Keywords: Complementarity, wind energy, solar, Pearson coefficient

published or not. The documents may come

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Introduction

Renewable energy technologies are taking an important proportion in the world global energy mix. In fact the part of renewable energy shared in the global energy production was 22% in 2012 and is predicted to be 29% by 2040 [1]. World Energy Council reported in 2015 that the contribution of wind and photovoltaic (PV) energies in the global mix was respectively 7% and 1% [2].

Despite being free fuel sources, they both have an intermittent nature. Thus, it is not easy to harness them without accurate calculations. In fact, an interesting solution for attenuating their fluctuant nature is to combine them since they have different intermittency nature.

From the literature review, many research have been achieved concerning complementarity between renewable energy sources. In [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF], the complementarity between hydro and wind energy have been evaluated in Brazil. A complementarity map was drawn to show the spatiotemporal complementarity between wind and hydro energy sources. Another study have been conducted for offshore wind and hydro energy complementarity in Brazil [START_REF] Silva | Complementarity of Brazil׳s hydro and offshore wind power[END_REF]. This study revealed a good complementarity between offshore wind resources and hydro plants installed in the country. The complementarity was measured using correlation coefficient based on monthly wind speed data and precipitation level. In [START_REF] Hoicka | Solar and wind resource complementarity: Advancing options for renewable electricity integration in Ontario, Canada[END_REF], another study assessed the complementarity between solar and wind energy in China. The spatiotemporal complementarity between wind and solar energy was assessed for several locations in China. This study showed that the combination of solar and wind results in smoother fluctuations of energy than each source considered individually.

From the previously published papers, the coastal area of Benin Republic is the most favorable for wind energy production [START_REF] Awanou | Estimation of the mean wind energy available in Benin (Ex Dahomey)[END_REF]- [START_REF] Houekpoheha | Estimation de la puissance énergétique éolienne à partir de la distribution de Weibull sur la côte béninoise à Cotonou dans le Golfe de Guinée[END_REF]. The country is globally covered regarding solar energy, where the northern and the coastal area are presenting the greatest potentials (see Fig 3). The country is also favorable to micro-/pico-hydro energy because of its many rivers.

It turns out from the analysis of the published papers in general that the mix of complementary sources can contribute to a much stable power generation. Hence, convenient selection of their location is very important to guaranty this complementarity. This research proposes a selection approach of these locations by using an optimization approach. Considering the previous research on wind potential in Benin, we are looking for the optimal placement of a solar farm which can ensure a good complementarity with a wind energy farm in the coastal area of the country. This paper is structured in three sections, after this introduction. The first section presents the study area and the optimization problem formulation. The next section concerns the results obtained and the last part gives the main achievements of this research.

Material and Methodology

Study Area and Data

Benin Republic is a coastal country of West Africa. As presented in Fig 3, the country is bordered by Niger in the North, Burkina-Faso in the North-West, Togo in the South-West and Nigeria in the East. The coastal line of the country is one of the smallest in West Africa with only 121 km length.

Average daily wind speed is obtained from the database presented in [START_REF]Wyoming Weather Web[END_REF]. This database covers the period from 01-01-2013 to 12-31-2013 and is taken at the Cadjehoun (located in Cotonou) airport coordinate position. The proportion of data available is 98.45%. Table 1 presents further information on this dataset. The daily average Direct Normal Irradiation (DNI) is obtained for the same period from [START_REF] Pfeifroth | Surface Radiation Data Set -Heliosat (SARAH) -Edition 2[END_REF]. This data covers the surface of the country with a precision of 0.05° in longitude and 0.05° in latitude. Further information related to this dataset is presented in Table 2.

Optimization Problem Formulation

In order to assess the complementarity between sources, their potential must be modeled. In the literature, a dimensionless index approach is generally used for this purpose [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF], [START_REF] Hoicka | Solar and wind resource complementarity: Advancing options for renewable electricity integration in Ontario, Canada[END_REF], [START_REF] Liu | Investigation on the spatiotemporal complementarity of wind energy resources in China[END_REF]. This approach considers that the power fluctuations are due in majority to DNI for solar energy and wind speed for wind energy. Thus, solar potential is obtained by:

(1)
where represents the dimensionless coefficient related to solar potential and the DNI for the day.

The wind energy potential is modeled as:

(2)

where represents the dimensionless coefficient related to wind energy potential and is the wind speed for the day. The dimensionless coefficients and are determined considering the annual production of each source. In this paper, we have considered an annual energy production of by year for each source. The yearly production for wind energy is:

(3)
and the yearly solar production is given by:

(4)
The dimensionless coefficients and are obtained by solving (3) and ( 4).

After modeling each source's potential, the complementarity between them can now be calculated. In the literature, this complementarity can be quantified using correlation coefficients. Thus, the Pearson correlation coefficient is generally used for this purpose [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF], [START_REF] Silva | Complementarity of Brazil׳s hydro and offshore wind power[END_REF], [START_REF] Liu | Investigation on the spatiotemporal complementarity of wind energy resources in China[END_REF], [START_REF] Cantão | Evaluation of hydro-wind complementarity in the Brazilian territory by means of correlation maps[END_REF]. In this paper, this Pearson correlation coefficient is given by: [START_REF] Hoicka | Solar and wind resource complementarity: Advancing options for renewable electricity integration in Ontario, Canada[END_REF] where represents the annual average of the wind generation and represents the annual average of the solar generation. Because of the spatial variability of the solar resource, the correlation coefficient presented in [START_REF] Hoicka | Solar and wind resource complementarity: Advancing options for renewable electricity integration in Ontario, Canada[END_REF] vary with the geographic coordinates of the solar resource considered. The resulting optimization problem can be formulated as follow: [START_REF] Awanou | Estimation of the mean wind energy available in Benin (Ex Dahomey)[END_REF] where and are respectively the latitude and the longitude of the solar resource to evaluate.

The constraint for this problem is that the geographical coordinates obtained must fall into Benin land territory. Since the solar radiation dataset is provided with a precision of 0.05° x 0.05°, the solar potential at any coordinate location not matching with the original dataset is obtained by a simple linear interpolation.

PSO Algorithm

To solve the optimization problem, a PSO is a stochastic population based algorithm. It has been developed by Kennedy and Eberhart in 1995 [START_REF] Eberhart | A new optimizer using particle swarm theory[END_REF]. A swarm is made of a particles moving in the search space. Each particle is characterized by its position and velocity. The position of the particle actually represents a possible configuration of the system to be optimized. The velocity is a corrective value that should be applied to the particle to update its position with respect to the best positions of the swarm. The velocity is obtained by considering the social behavior of the swarm as well as the cognitive behavior of the particle.

In this research, the PSO implemented in Matlab® combines the constriction developed in [START_REF] Clerc | The particle swarm -explosion, stability, and convergence in a multidimensional complex space[END_REF] and the multiple start developed in [START_REF] Nabab | Particle Swarm Optimization: Algorithm and its Codes in MATLAB[END_REF]. In order to tackle divergence of the algorithm, we introduced a memory of the best particle for each initialization of the algorithm. By doing this, the algorithm is forced to search for better positions at each initialization. The flowchart of the algorithm is presented in

Results and discussions

PSO convergence characteristic

The optimization problem presented is solved using Table 3 values. PSO algorithm coefficients and are set to 2.05 [START_REF] Nabab | Particle Swarm Optimization: Algorithm and its Codes in MATLAB[END_REF]. The optimization is performed on a 30 particles swarm with 100 iterations and 10 turns. The tolerance is tracked starting from the 10th iteration and the tolerance value is 10 -12 . The convergence characteristic of the PSO algorithm implemented is presented in Fig 5 . One can notice that the Pearson correlation coefficient decreases with the iterations. PSO have converged from the 10th iteration with a Pearson correlation coefficient of -0.4291. Referring to [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF], correlation coefficient values are classified into three grades:

for low degree of correlation, stands for remarkable correlation and for high degree of correlation. Negative correlation, is recommended for reducing total output power fluctuations [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF].Thus, the value -0.4291 is the Pearson coefficient that minimizes the objective function with respect to the environmental data used. This value shows a remarkable inverse correlation between solar and wind resources referring to the interpretation of Pearson correlation coefficient values presented in [START_REF] Liu | Analysis on the hourly spatiotemporal complementarities between China's solar and wind energy resources spreading in a wide area[END_REF].

Optimal Location for Wind and Solar Resources

The position of particles at the last iteration of the algorithm and the location of the wind resource are presented in Fig 6 . In this figure blue diamond markers represent each particle's position at the last iteration. Green circle markers give the individual best positions at the last iteration. Red star markers show the global best particle position which is located at longitude 2.66°E and latitude 8.29°N in the department of 'Collins'. The black square marker point is the location of the wind resource which is located at longitude 2.38°E and latitude 6.36°N in the department of 'Littoral'. One can notice that the optimal location found for solar resource is not in the area with the greatest solar potential. This can be explained by the fact that complementarity is not related to the amplitudes of the potentials of the energy sources and is determined only of the criterion [START_REF] Awanou | Estimation of the mean wind energy available in Benin (Ex Dahomey)[END_REF]. On the other hand, solar production is more homogenous regarding the production peak. It is also worth to notice that periods of low wind power production are compensated with PV production during the same time considering the PV availability. This is illustrated at Fig 9 which presents the combined production of wind and solar energy sources. From Fig 9, it can be noticed a smooth fluctuation in the total power generation as compared to wind generation alone. Indeed, for a grid connected application, it may be interesting to compensate lacks due to low wind production by combining with PV production.

Conclusion

This paper presented a complementarity study between solar and wind energy potentials. The complementarity was assessed using the Pearson correlation coefficient. The location of solar and wind sources was determined to minimize the Pearson correlation coefficient using PSO algorithm. In the specific case of Benin Republic, we showed that optimal complementarity was obtained between the coastal region in 'Littoral' department for wind and 'Collins' department for solar energy.

The approach developed in this study is important for renewable application projects where different sources are combined for power generation.

Fig 1 presents the

  Fig 1 presents the visualization of the daily wind speed data in Cadjehoun airport. From the analysis of this curve, the daily wind speed varies between 1.5 m/s and 7.45 m/s.

Fig 1 :

 1 Fig 1: Daily wind speed in Cadjehoun airport.

Fig 2 :

 2 Fig 2: Wind speed relative probability of occurrence at 10 m.

Table 2 :

 2 Daily solar irradiation dataset information. Information Value Type DNI Location Benin Precision (Lon x Lat) 0.05 ° x 0.05° Duration 01-01-2013 to 12-31-2013 Frequency Daily Heliosat (SARAH) dataset are used for daily wind speed data [10]. The spatial visualization of the yearly mean DNI for 2013 have been computed using Matlab® and is presented in Fig 3. One can notice that the Northern part of the country in general presents the extreme DNI values for the year 2013.

Fig 3 :

 3 Fig 3: Spatial visualization of DNI for year 2013.

  Fig 4.

Fig 4 :

 4 Fig 4: Flowchart of the implemented PSO algorithm.

Fig 5 :

 5 Fig 5: Convergence characteristic of implemented PSO algorithm.

Fig 6 :

 6 Fig 6: Positions at the last iteration.

  Fig 7-a corresponds to the wind speed variations in Cadjehoun -Cotonou whereas Fig 7-b corresponds to the daily DNI variations for year 2013. One can notice that DNI varies between 3.57 W/m 2 and 311.37 W/m 2 and the lowest values of DNI last from July to September. By doing the same analysis with wind variations in Fig 7, one can notice that the period from July to September is actually the period with the highest wind speed values.

Fig 7 :

 7 Fig 7: Wind speed variations (latitude 6.21°N and longitude 2.23°E) and DNI variations (latitude 8.29°N and longitude 2.66°E).

Fig 9 :

 9 Fig 9: Combined production from PV and wind.

Table 1 :

 1 Daily wind speed dataset information.

	Information	Value
	Type	Wind speed
	Location	International airport of Cadjehoun -Cotonou
	Duration	01-01-2013 to 12-31-2013
	Frequency	Daily
	Missing data	6 Days
	Minimum	1.5 m/s
	Maximum	7.45 m/s

Table 3 :

 3 PSO setup.

	Parameters	Values
	PSO Coefficient	2.05
	PSO Coefficient	2.05
	Swarm size	30
	Number of turns	10
	Number of iterations	100
	Tolerance (>10 iterations)
	Lower limits [latm, lonm]	[6°N, 0°E]
	Upper limits [latM, lonM]	[13°N, 4°E]