Sara El Hassad

Hala Skaf-Molli

Pascal Molli
email: pascal.molli@univ-nantes.fr

Florian Hacques
email: florian.hacques@etu.univ-nantes.fr

PFed: Recommending Plausible Federated Queries

Keywords: Semantic Web, Consuming Linked Open Data, SPARQL query, Federated SPARQL Query, Plausible, Joinable

Federated SPARQL query processing allows to query multiple interlinked datasets hosted by remote SPARQL endpoints. However, finding pertinent federated queries over a growing number of datasets is challenging. In this paper, we propose PFed, an approach to recommend plausible federated queries based on query logs of different datasets. The problem is not to find similar federated queries, but plausible complementary queries over different datasets. Starting with a real SPARQL query log, PFed stretches the query with real queries from different logs. A generated federated query Q is plausible if an external observer cannot deny that queries in the original logs could be subqueries of Q. PFed relies on data summaries and real SPARQL query logs to generate plausible federated queries. It starts by pruning the queries logs by selecting only queries with joinable predicates. Experimental results with real logs of DBpedia and SWDF demonstrate that PFed is able to prune drastically the logs and recommend realistic federated queries.

Introduction

Following the Linked Open Data cloud (LOD) principles many datasets have been published. Federated SPARQL query engines [START_REF] Schwarte | Fedx: Optimization techniques for federated query processing on linked data[END_REF][START_REF] Acosta | ANAPSID: an adaptive query processing engine for SPARQL endpoints[END_REF] have been developed to query multiple interlinked datasets hosted by remote SPARQL endpoints. They allow to consume LOD data in a decentralized way without the need to copy the data. However, finding pertinent federated queries over a growing number of datasets is challenging. This requires to fully understand the datasets and find potential joins among them. In this paper, we propose PFed, an approach to recommend federated queries for end-users. More precisely, PFed recommends plausible federated queries using the query logs of different SPARQL endpoints. This is not a classical recommendation problem. In recommender systems [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art[END_REF], the problem is to recommend resources (or items) for users based on similar ones already seen by the users. In PFed, we start with a SPARQL query from a query log and we stretch this query with real queries from other existing query logs for generating a plausible federated query Q. The generated federated query Q is called plausible if an external observer of PFed cannot deny that queries in the original logs could be subqueries of Q. To illustrate, Figure 1 presents extracted queries from real query logs 1 of SWDF and DBpedia. More precisely, Figure 1a presents SPARQL queries from the log of SWDF 2012 and Figure 1b present SPARQL queries from the log of DBpedia 3.5. Consider the Q1S from the log of SWDF, this query can be extended with the query Q1D from the log of DBpedia. The result is the SPARQL 1.1 federated Query Q1S1D given in Figure 2a. Q1S1D is generated by joining the variable ?place of the query Q1S, i.e., the object of the predicate foaf:based_near with the variable ?country of the query Q1D, i.e. the subject of the predicates rdfs:label and dbpedia2:capital. The joined variable ?country has been renamed by ?place, in the generated query Q1S1D. The execution of this query over a federation of SWDF and DBpedia produces 1056 results.

The generated query Q1S1D can be recommended as a plausible federated query because for an external observer Q1S and Q1D are subqueries of Q1S1D, if we ignore the renaming of variables. In the same way, we can generate a more complex federated query such as the query Q2S2D shown in Figure 2b. Q2S2D is obtained by extending the query Q2S from the log of SWDF with the query Q2D from the log of DBpedia. The joining variable ?sameAs is renamed as ?person in Q2S2D. The objective of this paper is to show how we can generated plausible generated queries automatically. Currently, we cannot find a federated query log. This is normal, because a federated engine like FedX [START_REF] Schwarte | Fedx: Optimization techniques for federated query processing on linked data[END_REF], for example, decomposes a federated query into sub-queries, and evaluates these sub-queries against relevant SPARQL endpoints. Therefore, a query log of a SPARQL endpoint does not know the federated query, it is only aware of a fragment of the federated query. Fig. 2: Plausible federated queries generated from logs of SWDF 2012 and DBpedia 3.5.1 in Figure 1 We believe generating plausible federated queries over real datasets allows to leverage the full power of the LOD. Recommending real federated queries is challenging. This requires to explore a large number of queries to find joins among datasets. In addition, not all combinations of joinable queries generate a federated query that is semantically correct [START_REF] Pérez | Semantics and complexity of sparql[END_REF][START_REF] Arenas | Federation and navigation in sparql 1.1[END_REF]. We propose PFed, a new approach to generate plausible and correct federated queries. PFed starts by pruning the queries logs by selecting only queries with joinable predicates reducing the search space. The contributions of the paper are:

-We define a new semantic summary for pruning query logs.

-We define an algorithm to exclude non joinable queries from logs.

-We propose an approach to generate plausible and correct federated queries using the pruned logs. -We validate our approach using queries of SWDF 2012 and DBpedia 3.5.1.

This paper is organized as follows. Section 2 summarizes related works. Section 3 details PFed, an automatic generator of plausible federated queries. Section 4 presents our experimental results. Finally, conclusions and future work are outlined in Section 5.

Related Work

Feta [START_REF] Nassopoulos | FETA: Federated QuEry TrAcking for Linked Data[END_REF] is a federated query tracker that computes Basic Graph Patterns from a federated log. It supposes the existence of a federated query log. In this work, we want to recommend federated queries rather than analyzing federated query logs.

Many efforts have been done to automatically generate SPARQL queries, either for individual dataset [START_REF] Aluç | Diversified stress testing of RDF data management systems[END_REF][START_REF] Saleem | Feasible: A feature-based sparql benchmark generation framework[END_REF] or multiple datasets as Splodge [START_REF] Görlitz | SPLODGE: systematic generation of SPARQL benchmark queries for linked open data[END_REF] and Fed-Bench [START_REF] Schmidt | Fedbench: A benchmark suite for federated semantic data query processing[END_REF]. Federated queries benchmarks have been proposed for evaluating the performance of federated query engine. Existing benchmark rely either on hand-crafted queries or on automatically generated ones.

FedBench [START_REF] Schmidt | Fedbench: A benchmark suite for federated semantic data query processing[END_REF] rely on hand-crafted queries. The datasets of FedBench are real datasets preselected from the Linked Data Cloud, e.g. Life Science, Cross domain. FedBench is commonly used for the evaluation of federated query engines. FedBench is not designed to recommend plausible federated queries over a federation of SPARQL endpoints. LargeRDFBench [START_REF] Saleem | LargeRDFBench: A billion triples benchmark for sparql endpoint federation[END_REF] attempts to generate more realistic federated queries. The benchmark comprises a total of 32 queries for SPARQL endpoint federation. Queries are ranging from simple queries extracted from FedBench queries and large data queries created by the authors with the help of the expert domain. As FedBench, LargeRDFBench are designed for preselected datasets and queries are designed for specific domains and cannot be used for automatic generation of realistic federated queries.

Splodge [START_REF] Görlitz | SPLODGE: systematic generation of SPARQL benchmark queries for linked open data[END_REF] proposes heuristics for automatic query generation. Splodge generates only conjunctive queries of triple patterns, i.e., Basic Graph Patterns (BGP) with bound predicate, unbound subject and unbound object. Other SPARQL operators such as FILTER, OPTIONAL are not considered. However, recent analytical study of large SPARQL query logs [START_REF] Bonifati | An analytical study of large SPARQL query logs[END_REF] shows that 74.83% of studied queries have JOIN, FILTER and OPTIONAL and only 7.49% have JOIN alone (conjunctive queries). Consequently, the queries of Splodge cannot reflect the reality.

Existing approaches of automatic generation of federated queries do not reflect reality and hand-crafted federated queries are designed for specific datasets with the purpose to stress the performance of a federated query engine. Benchmarks are not designed for recommending plausible federated queries.

Generation of Plausible Federated Queries

Intuitively, for generating a plausible federated query over n datasets, we start by combing (joining) the query logs log 1 and log 2 of two datasets d 1 and d 2 , each dataset is hosted by a SPARQL endpoint. We can distinguish different type of join combinations: subject-subject or object-subject leading to different query structures star-shaped, path-shaped, or hybrid queries [START_REF] Schmidt | Fedbench: A benchmark suite for federated semantic data query processing[END_REF]. Then, we generate new federated queries by joining the resulting queries and the log log 3 of the dataset d 3 . We repeat the same process iteratively until processing the n query logs.

Processing the whole logs to generate federated queries means that any query from one log could be combined with at least a query from the another log. However, this not always the case, for instance, in our experimentation for some join combinations only approximately 25% of query log of SWDF can be joined with the log of DBpedia. Therefore, we need to prune the logs to keep only joinable queries, i.e., queries contain at least one joinable predicate.

The problem is given two logs log 1 and log 2 , for a query

Q 1 ∈ log 1 checks if it exists a query Q 2 ∈ log 2 where Q 1 is joinable with Q 2 .

Finding joinable queries

To find joinable predicates, one can rely on the Vocabulary Of Interlinked Datasets VoID [START_REF] Alexander | Describing linked datasets[END_REF]. This vocabulary describes metadata about RDF datasets and the linkset. A linkset is a collection of RDF links between two datasets3 . An RDF link is an RDF triple whose subject and object are described in different datasets. This corresponds to the joinable predicates in the example of the Figure 2. However, we cannot use VoID to detect joinable predicates because a large number of RDF datasets do not provide VoID [START_REF] Vandenbussche | Sparqles: Monitoring public sparql endpoints[END_REF], only 13.65% of datasets 4(77/564) present a VoID description.

Another solution is to use the capabilities of data sources as defined in Hibiscus [START_REF] Saleem | Hibiscus: Hypergraph-based source selection for sparql endpoint federation[END_REF] to check the possible existence of matching. According to [START_REF] Saleem | Hibiscus: Hypergraph-based source selection for sparql endpoint federation[END_REF], the data summary of a source d ∈ D is the set CA(d) of all capabilities of that source. The total number of capabilities of a source is equal to the number of distinct predicates in it. The definition of the authorities of a subject or an object relies on the analysis of the Unified Resource Identifier (URI) syntax. The URI syntax consists of a hierarchical sequence of components referred to as the scheme, authority, path, query, and fragment 5 . For example, the uri <http://dbpedia. org/ontology/Plant> contains a schema "http", an authority "dbpedia.org" and a path "ontology/Plant". To compute the set of capabilities for a source, the first two components (path, authority) are combined as the authority of the URI. Figure 3 presents a sample of the summary of SWDF 2012 and DBpedia 3.5.1. For instance, in Figure 3a, the first capability of SWDF data source is the predicate foaf:based_near, its subject authority is <http://data.semanticweb. org> and its object authorities are <http://dbpedia.org>, <http://www.w3. org>, <http://sws.geonames.org>, and <http://data.semanticweb.org>.

Definition 1 (Capability

Hibiscus data summary allows to prune the query logs only if many predicates have different subjects or objects authority. However, this not always the case, especially for the subject authority. For instance, the majority of subjects of DBpedia have the authority <http://dbpedia.org>, only six predicates out of 39672 predicates of DBpedia 3.5.1 do not have <http://dbpedia.org> as a subject authority. Therefore, if a query Q 1 in SWDF query log is joinable with a query Q 2 in DBpedia query log on the subject authority <http://dbpedia. org>, then Q 1 will be joinable with a large number queries in the log of DBpedia. Therefore, for query logs of SWDF and DBpedia, Hibiscus will prune mostly queries with unbounded predicates.

We need to use the semantic of subjects and objects for finding joinable predicates. Intuitively, a subject or an object from one dataset could be joinable with If we want to find joinable predicates for path query from SWDF to DBPedia, we look for the type of the object of the predicate in DBPedia dataset. Figures 3c and3d present Classes summary for SWDF and DBpedia, respectively.

We use only the direct classes of subjects and objects to find common classes. We do not use inferences to find common classes because schemas information are not always available [START_REF] Neumann | Characteristic sets: Accurate cardinality estimation for rdf queries with multiple joins[END_REF].

Pruning query logs

Based on data summaries and classes summaries, we can prune the logs of corresponding datasets by retaining only joinable queries. The pruning depends on the structure of generated federated queries. Let D be a set of distinct data sources, d 1 , d 2 ∈ D. Let S 1 , SC 1 , S 2 , SC 2 be the summaries of d 1 and d 2 , log 1 and log 2 are the query log of d 1 and d 2 , respectively. For two queries The hybrid join pattern is built as a mix of a path join pattern and a star join pattern. Figure 4 presents possible structures of hybrid federated queries. The query generated in Figure 4a is built from the path query of

Q 1 ∈ log 1 and Q 2 ∈ log 2 with tp 1 = (s 1 , p 1 , o 1 ) ∈ Q 1 and tp 2 = (s 2 , p 2 , o 2 ) ∈ Q 2 ,
p 1 ∈ Q 1 , Q 1 ∈ log 1 and p 3 ∈ Q 2 , Q 2 ∈ log 2 .
The query generated in Figure 4b built from the star query of

p 1 ∈ Q 1 , Q 1 ∈ log 1 and p 2 ∈ Q 2 , Q 2 ∈ log 2 .
The algorithm 1 presents how to prune query logs to construct path-shaped federated queries. To illustrate, consider a sample of query logs of SWDF 2012 and dbpedia-3.5.1 in figures 1a and 1b, respectively.

After the execution of the algorithm, only Q1S, Q2S of SWDF and Q1D and Q2D of DBpedia will be preserved, because the intersection between the object authorities of based_near ∈ Q1S and the subject authority of dbpedia2:capital ∈ Q1D is not empty since they have the authority <http://dbpedia.org> and the type dbo:city in common. As well, the object authorities of owl:sameAs ∈ Q2S and the subject authority of dbo:birthDate ∈ Q2D have the authority <http://dbpedia.org> and the type dbp:Person in common. We exclude Q3S because it cannot be joined with any query from dbpedia, i.e. no predicate in dbpedia has <http://data.semanticweb.org> as subject authority. We also eliminate Q3D because the capability of unbound predicate is undefined.

For star-shaped federated queries (subject-subject), we have to modify the condition in line 8 of algorithm 1 by (SA 1 ∩ SA 2 = ∅) and the condition in line 11 by (SC 1 ∩ SC 2 = ∅). In this case, we keep queries Q4S and Q4D.

The subject authorities of skos:prefLabel ∈ Q4S and of dbo:thumbnail ∈ Q4D have the authority <http://dbpedia.org> and the type foaf:Organization in common.

Building plausible federated queries

The construction of plausible federated queries follows the following process:

-Use pruned logs to build minimal federated queries.

-Use minimal federated queries to build plausible federated queries. We start by generating minimal federated queries PFed min . A minimal federated contains one triple from log 1 and one triple from log 2 .

Minimal federated queries PFed min is a basic federated query, it is composed of two triples patterns and defined as follows. Figure 5a presents a minimal path-shaped federated query between foaf:based_near ∈ Q1S and dbpedia2:capital ∈ Q1D in Figure 3. Figure 5b presents a minimal star-shaped federated query between skos:prefLabel ∈ Q4S and dbo:thumbnail ∈ Q4D.

We keep only PFed min queries that produce results to generate plausible federated queries. (c) Q1 Q2 Fig. 6: A non well designed federated query Plausible Federated queries PFed The construction of Q PFed is tricky, if the original queries contain OPTIONAL operator. We have to construct only correct plausible federated query. A plausible federated query is correct if it is well designed [START_REF] Pérez | Semantics and complexity of sparql[END_REF] and service-safeness [START_REF] Arenas | Federation and navigation in sparql 1.1[END_REF].

Definition 5 (Well designed[10]).

A graph pattern P is well designed if for every occurrence of a sub-pattern P' = (P1 OPT P2) of P and for every variable ?X occurring in P, the following condition holds: if ?X occurs both inside P2 and outside P', then it also occurs in P1.

The federated query in Figure 6c is not well designed because the variable ?o2 occurs in P2 and outside the P' (i.e. clause SERVICE <dataset2>), but it not occurs in P1.

The service-safeness provides condition that ensures that a SPARQL query containing SERVICE operator can be safely evaluated. Our generated queries ensure service-safeness because each SERVICE clause has only bounded service, i.e., during the construction the URI of the SPARQL endpoints are known.

The main issue is to build well designed queries. A correct plausible federated query is constructed as follows. Let log 1 , log 2 be two pruned query logs, Q 1 ∈ log 1 , Q 2 ∈ log 2 , Q 1 and Q 2 contribute to the construction of a minimal query, then:

-If Q 1 and Q 2 are conjunctive queries (a.k.a BGPs) then Q PFed = Q 1 Q 2 , Q PFed is a simple concatenation of queries (Q 1 . Q 2 ),

Evaluation

We want to answer empirically the following questions: Does Hibiscus summary prune non joinable predicates? Does Classes summary effective? Does PFed produce "realistic" queries compared to automatic query generation benchmarks? Does PFed generate correct plausible federated queries? All data, codes, and generated query are available at the project web page6 .

Experimental Setup

Dataset and Queries: We use SWDF 2012 and DBPedia 3.5.1 datasets and clean queries of Feasible7 . Table 2 reports statistics about the datasets and query logs. It is strange that the query log of SWDF contains more predicates than the original dataset hosted at the SPARQL endpoint. Some queries in the logs use predicates that are not defined in the dataset. Using DBpedia to generate plausible federated queries is challenging because DBpedia dataset has a high number of predicates and the log of DBpedia has a high number of queries. We use only SELECT queries to construct plausible federated queries. Approaches: We can compare our approach only with Splodge, because only Splodge is automatic federated queries benchmark. However, we can compare only with minimal queries since Splodge generates only conjunctive queries (without OPTIONAL, UNION). We modify the code of Splodge8 to generate all possible queries instead of stopping after the generation of the first couple of joinable predicates.

Experimental Results

Does Hibiscus summary prune non joinable predicates ? Table 3 presents the results of pruning using data summaries of Hibiscus. As we can see, the reduction is 62.75% for SWDF query log for path-shaped queries and by 42.82% for starshaped. The reduction if only 2.15% of DBpedia log for both path-shaped queries and star-shaped generation. This reduction is not significant because most of predicates in DBPedia has the authority <http://dbpedia.org>. 6 shows that from 3 joinable predicates of SWDF and 139 joinable predicates of DBPedia (see table 4), PFed generates only 77 minimal path-shaped queries with non empty result set. Whereas, Splodge generates 1300 queries, some of these queries are not significant as the query displayed in Figure 7. PFed generates relatively less queries, 20 PFed min star-shaped queries with non empty result set based on 3 joinable predicates of SWDF and 83 joinable predicates of DBPedia. PFed relies on query logs to generate plausible queries. Does PFed generate correct plausible federated queries ? Due to the size of the pruned logs, we can generate a large number of plausible federated queries. In our experimentation, we focus on the generation of path-shaped between foaf:based_near from SWDF and dbpedia2:capital from DBPedia because foaf-:based_near predicate is used to link SWDF and DBpedia datasets. The pruned SWDF query log contains 2 866 queries that contains foaf:based_near. Many of these queries have the same structure but with different literals and variables. Therefore, instead of producing 2866×14 = 40124 queries where 14 is the number of queries that contains dbpedia2:capital in pruned DBpedia log, we define patterns for foaf:based_near queries. We differentiate 9 patterns for foaf:based_near queries and we generate 24 queries. All generated queries are executed correctly and 19 of these queries have non empty results set (see table 7). We generate star-shaped plausible federated queries based on skos:prefLable from SWDF and dbpedia:thumbnail from DBPedia (see table 7). The 42 generated queries are executed correctly and 28 of these queries produce results.

Conclusion and Future Work

We presented PFed an approach for automatic generation of plausible federated queries based on real query logs. PFed starts by pruning the logs to exclude non joinable queries using two data summaries. The first one is based on the authorities and the second is based on the type of subjects and objects of predicates. Experimentations with real query logs of SWDF and DBpedia demonstrate that PFed is able to prune considerably the logs and generate correct plausible federated queries.

As future work, we would like to experiment PFed with more real query logs and produce plausible federated queries over a large number of SPARQL endpoints. Finally, we plan to extend PFed with statistical information to generate only queries that return results.

1 2 .Fig. 1 :

 21 Fig. 1: SPARQL queries from the logs of SWDF 2012 and DBpedia 3.5.1

SELECT

  * WHERE { SERVICE <h t t p : / / swdf -2012> { ? o b j r d f : t y p e f o a f : O r g a n i z a t i o n . ? o b j f o a f : b a s e d _ n e a r ? p l a c e SERVICE <h t t p : / / d b p e d i a -3.5.1 > { ? p l a c e r d f s : l a b e l " U n i t e d Kingdom " @en . ? p l a c e d b p e d i a 2 : c a p i t a l ? c a p i t a l . ? c a p i t a l geo : l a t ? l a t . ? c a p i t a l geo : l o n g ? l o n g }}} #r e s u l t s = 1 056 (a) Q1S Q1D SELECT * WHERE { SERVICE <h t t p : / / swdf -2012> { swc : tim-f i n i n r d f : t y p e f o a f : P e r s o n { swc : tim-f i n i n f o a f : name ? name1} UNION { swc : tim-f i n i n r d f s : l a b e l ? name1 } OPTIONAL { swc : tim-f i n i n f o a f : mbox_sha1sum ? mbox_sha1sum } OPTIONAL { swc : tim-f i n i n f o a f : homepage ? homepage } OPTIONAL { swc : tim-f i n i n f o a f : page ? page } OPTIONAL { swc : tim-f i n i n ow l : sameAs ? p e r s o n SERVICE <h t t p : / / d b p e d i a -3.5.1 > { ? p e r s o n s k o s : s u b j e c t ? s u b j e c t . ? p e r s o n dbo : b i r t h D a t e ? b i r t h . ? p e r s o n f o a f : name ? name2 . ? p e r s o n r d f s : comment ? d e s c r i p t i o n FILTER ( l a n g ( ? d e s c r i p t i o n ) = " en " ) } } OPTIONAL { swc : tim-f i n i n r d f s : s e e A l s o ? s e e A l s o } }} #r e s u l t s = 178 (b) Q2S Q2D

2 P Jlog1 ←-∅; 3 P 4 foreach Q1 ∈ log1 do 5 foreach Q2 ∈ log2 do 6 let 7 let 8 if OA1 ∩ SA2 = ∅ then 9 let 12 P 13 P

 234567891213 Jlog2 ←-∅; OA1 = S1.OAuth.P redicate(Q1) SA2 = S2.SAuth.P redicate(Q2) OC1 = SC1.OClasses.P redicate(Q1) 10 let SC2 = SC2.SClasses.P redicate(Q2)11 if OC1 ∩ SC2 = ∅ then Jlog1 ←-P Jlog1 ∪ Q1 Jlog2 ←-P Jlog2 ∪ Q2

SELECTFig. 5 :

 5 Fig.5: Minimal federated queries generated from pruned logs of SWDF and DBpedia in Figure3

  ). Given a source d, a capability is a triple (p, SA(d, p), OA(d, p)), which contains (1) a predicate p in d, (2) the set SA(d, p) of all distinct subject authorities of p in d and (3) the set OA(d, p) of all distinct object authorities of p in d.

  we say that Q 1 and Q 2 are joinable if p 1 and p 2 are joinable path or joinable star., p 2 ) = true, if OA(d 1 , p 1 ) ∩ SA(d 2 , p 2 ) = ∅ and OC(d 1 , p 1 ) ∩ SC(d 2 , p 2 ) = ∅. , p 2 ) = true, if SA(d 1 , p 1 ) ∩ SA(d 2 , p 2 ) = ∅ and SC(d 1 , p 1 ) ∩ SC(d 2 , p 2 ) = ∅.

	tween log1 and log2 Input: log1, log2, S1, S2, SC1, SC2 Output: P Jlog1, P Jlog2 Definition 2 (Joinable Path). joinableP ath(p 1 Definition 3 (Joinable Star). joinableStar(p 1 Algorithm 1: Algorithm for Excluding non PATH Joinable queries be-query logs and summaries Joinable Path Queries of log1 and log2 1 Function PJDetection(F ):

Definition 4 (PFed min ). Let

  log 1 , log 2 be two pruned logs of queries of datasets d 1 and d 2 , respectively. p 1 ∈ log 1 and p 2 ∈ log 2 are two predicates. PFed ?o 1 . ?s, p 2 , ?o 2 | joinableStar(p 1 , p 2 ) = true }In order to construct a path (start) join, we substitute the object (subject) of p 1 and the subject of p 2 by the same value as given in the table 1.

	tp1 object tp2 subject substitution value
	?x	?y	?x
	?x	a	a
	a	?x	a
	a	b	null

min = { ?s, p 1 , ?x . ?x, p 2 , ?o | joinableP ath(p 1 , p 2 ) = true} or {?s, p 1 ,

Table 1 :

 1 All substitution values possible to create path join. ?x, ?y are variables and a, b are constants (URIs or literals)

  as in figure2a, Q1S1D = Q1S Q1D.-If Q 1 contains binary operators like UNION or OPTIONAL, we distinct two cases:• If a joinable predicate is outside binary clauses of Q 1 , we add Q 2 in the BGP part of Q 1 . • If a joinable predicate of Q 1 is inside the UNION or OPTIONAL clauses,we append Q 2 inside this clause after the substitution of the join variables (subject or object of the triple) according to table1. Figure2bpresents a federated where the joinable predicate owl:sameAs of Q2S is inside the OPTIONAL Clause. -If a joinable predicate of Q 2 is inside an OPTIONAL clause, we make sure to not generate non well designed queries like query shown in 6c.

Table 2 :

 2 Real Datasets and real logs

	dataset	|triples| |dataset predicates| |original log| |SELECT queries| |log predicates|
	SWDF	242 256	170	64 030	37 592	201
	DBPedia 232 542 405	39 672	217 812	127 812	247

Table 3 :

 3 Logs pruning using Hibiscus summariesDoes Classes summary effective? We use Classes summary to prune the log returned by Hibiscus. Classes summary is effective because it increases considerably the percentage of pruned queries. The reduction is impressive for DBPedia from 2.15% with Hibiscus to 71.42% with Classes summary as shown in table 4.

			path-shaped			star-shaped
	dataset |predicate joinable | |pruned log| % reduce |predicate joinable | |pruned log| % reduce
	SWDF	3	9 355	75.12	3	21 495	42.82
	DBPedia	139	36 522	71.42	83	36 449	71.48

Table 4 :

 4 Logs pruning using Hibiscus & Classes summariesDoes PFed produce "realistic" queries compared to automatic query generation benchmarks? We run Splodge on both SWDF and DBpedia. As parameters, we used a low minimum selectivity to get at least one result. We also didn't use a max selectivity in order to get as much queries as possible.

	Query Structure	|queries| |predicates SWDF| |predicates DBpedia |
	Path SWDF to DBpedia 1 300	6	1 108
	Path DBpedia to SWDF 170	3	98
	Star	63	3	38

Table 5 :

 5 Statistics of minimal queries generated by Splodge using SWDF and DBpedia

	Path-shaped queries SWDF to DBpedia	Star-shaped queries
	all generated with result	%	all generated with result %
	397	77	19.39	249	20	8.03

Table 6 :

 6 Statistics of minimal queries generated by PFed using Hibiscus & Classes summaries Table 5 presents minimal queries generated by Splodge. Table

Table 7 :

 7 PFed path and star, p 1 ∈ SWDF and p 2 ∈ DBPedia

	SELECT ? v0 ? v2 WHERE {
	SERVICE <h t t p : / / swdf -2012>
	{ ? v0 f o a f : b a s e d _ n e a r ? v1
	SERVICE <h t t p : / / d b p e d i a . org > {
	? v1 d b p e d i a 2 : i m a g e 2 C a p t i o n ? v2 . }}}
	Fig. 7: A minimal federated query generated by Splodge

https://www.w3.org/TR/void

http://sparqles.ai.wu.ac.at/

URI Syntax Components: https://tools.ietf.org/pdf/rfc3986.pdf

https://github.com/GDD-Nantes/PFed

https://github.com/dice-group/feasible

We do not use the original code of Splodge because it does not implement Star-shaped queries, we use the code at https://github.com/ Institute-Web-Science-and-Technologies/splodge

Acknowledgement

This work is part of the multidisciplinary project Sedela, funded by CominLabs, that brings together three laboratories: LS2N, CREAD and Lab-STICC.

://github.com/dice-group/feasible 2 Common prefixes are used and swc:<http://data.semanticweb.org/ns/swc/ontology#>, swrc:<http://swrc.ontoware.org/ontology#>,dbpedia:<http://dbpedia.org/resource/>, dbpedia2:<http://dbpedia.org/property/> and dbo:<http://dbpedia.org/ontology/>


Keywords: Fast computation, eddy current loss, permanent magnet, hybrid analytical model, permeance network, iron loss, magnetic gear, optimization

This article focuses on a 2D analytical model for a fast computation of magnetic losses due to eddy currents in the permanent magnets as well as iron losses in the ferromagnetic parts within coaxial magnetic gears. The magnetic field distribution is computed in yokes and permanent magnets by solving both Maxwell's equations, whereas for pole pieces the magnetic field is computed by coupling the previous analytical model with a reluctance network model. Both the eddy current losses and iron losses are determined from this hybrid analytical model. The iron loss model takes into account the temporal and spatial variations of flux density. The eddy current loss model takes into account the magnet splitting. Results of this magnetostatic eddy current loss model are then compared to the results obtained with a 2D magnetodynamic finite element model. A verification of validity limits is also proposed. The final function of this analytical model is to ensure integration into a set of models in the aim of a global mechatronic optimization of magnetic gears, for their insertion into a multi-megawatt wind turbines. A preliminary bi-objective, mass-efficiency optimization protocol is subsequently proposed along with an analysis of the computation time reduction via the presented models.

M. Desvaux, H. Ben Ahmed and B. Multon are with SATIE, ENS Rennes, University of Bretagne Loire, CNRS, 35170 Bruz, France (e-mail: melaine.desvaux@ens-rennes.fr, hamid.benahmed@ens-rennes.fr, bernard.multon@ens-rennes.fr).

II. INTRODUCTION

Mechanical gearboxes, as currently used in drives, and electromechanical conversion chains provide for a smaller capital expenditure and lower mass than a direct S. Sire is with University of Brest, FRE CNRS 3744, IRDL, 29200 Brest, France (e-mail: stephane.sire@univ-brest.fr).

S. Hlioui is with SATIE, CNAM, ENS Cachan, CNRS, 75141 Paris, France (e-mail: sami.hlioui@satie.ens-cachan.fr). drive conversion chain [START_REF] Arántegui | Wind status report: Technology, market and economic aspects of wind energy in Europe[END_REF]. On the downside, mechanical gearboxes cause production interruptions and the need for repairs, thus increasing operating costs [START_REF] Keller | Gearbox Reliability Collaborative Investigation of Gearbox Motion and High-Speed-Shaft Loads[END_REF]. In this context, an attractive solution would consist of developing a conversion chain with a semi-fast generator and a magnetic gear [START_REF] Desvaux | Design and optimization of magnetic gears with arrangement and mechanical constraints for wind turbine applications[END_REF]. The most popular topology of magnetic gears was proposed by Martin [START_REF] Martin | Magnetic transmission[END_REF] and has been the topic of various studies conducted by Atallah [START_REF] Atallah | A novel high-performance magnetic gear[END_REF][6] and others [START_REF] Rasmussen | Development of a high-performance magnetic gear[END_REF][8][9] [10][11]. This magnetic gear structure, called coaxial (shown in Fig. 1), offers potentially high performance with a higher torque density and greater reliability than mechanical gearboxes [START_REF] Gouda | Comparative study between mechanical and magnetic planetary gears[END_REF]. The structure becomes even more attractive in high-torque applications, like a high-power wind turbine (on the order of several MN.m and several MW) [START_REF] Penzkofer | Magnetic Gears for High Torque Applications[END_REF].

The competitiveness of this magnetic gear must be evaluated with respect to wind turbine operations, with conversion chain efficiency and heat dissipation constituting the major criteria. Magnetic losses have already been computed using the finite element method [START_REF] Jian | A Coaxial Magnetic Gear With Halbach Permanent-Magnet Arrays[END_REF][14], but computation time remains too high to integrate the magnetic loss computation into a set of models for a global mechatronic optimization of magnetic gears. Magnetic field hybrid analytical models proposed in [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF] based on subdomain model [START_REF] Lubin | Analytical computation of the magnetic field distribution in a magnetic gear[END_REF] [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] coupled with permeance network model [START_REF] Ostović | Dynamics of Saturated Electric Machines[END_REF] should allow to compute analytically the different magnetic losses (eddy current in magnets and iron losses). An optimization procedure adapted to the analytical computation of magnetic losses must also be proposed.

The major contribution of this article lies in the development of a fast computation of magnetic losses that incorporates both eddy current losses in permanent magnets and iron losses in laminated yokes and pole pieces based on a hybrid analytical model [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF]. The magnetic loss computation model is based on the solution to Laplace's and Poisson's equations [START_REF] Lubin | Analytical computation of the magnetic field distribution in a magnetic gear[END_REF] [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] for permanent magnets and yokes and moreover depends on the evolution of the magnetic field. For laminated pole pieces, a previous analytical model is coupled with a bi-directional permeance network model [START_REF] Ostović | Dynamics of Saturated Electric Machines[END_REF] in order to determine the magnetic field distribution. A classical loss computation method (used with a conventional electrical machine) is then performed to derive eddy current losses in permanent magnets [START_REF]An analytical model for the computation of no-load eddy current losses in the rotor of a permanent magnet synchronous machine[END_REF] with a magnet splitting consideration and for iron losses by means of a spatial and temporal evaluation of the flux density in post-processing [START_REF] Hernandez-Aramburo | Estimating Rotational Iron Losses in an Induction Machine[END_REF]. The analytical results obtained from these models will be compared with results from a finite element model in magnetostatics and magnetodynamics by considering both precision and computation time. Lastly, a biobjective mass efficiency optimization protocol, including magnetic loss models, is proposed for a 6-MW and 12.5-rpm magnetic gear.

III. PRINCIPLE OF THE MAGNETIC GEAR

The coaxial magnetic gear [START_REF] Atallah | A novel high-performance magnetic gear[END_REF] shown in Fig. 1 is composed of three magnetic parts: an internal ring with pole pairs of permanent magnets and a ferromagnetic yoke; an external ring with pole pairs of permanent magnets and a ferromagnetic yoke; and a ring with Q ferromagnetic poles between the two permanent magnet rings (an example is provided in Fig. 1 with low pole numbers, to improve readability: = 2, = 7, and = 9). To achieve power transmission, the pole numbers of the three rings must respect Eq. [START_REF] Arántegui | Wind status report: Technology, market and economic aspects of wind energy in Europe[END_REF].

+ = (1)
Depending on the fixed ring, the gear ratio is given by (2), where 𝛺 / , 𝛺 / and 𝛺 / are the rotational speeds of the internal ring, external ring and pole piece ring, respectively. The fundamental magnetic field frequency is then given by (3). To compute the magnetic field distribution in the magnetic gear, the fixed ring is not needed; the analytical magnetostatic model presented in the next section is thus available regardless of the fixed ring. To evaluate the magnetic losses in permanent magnets and yokes of the magnetic gear, it is first necessary to determine the magnetic field distribution across the various regions of the system, as presented in Fig. 2. For this computation, a 2D magnetostatic model, developed by [START_REF] Lubin | Analytical computation of the magnetic field distribution in a magnetic gear[END_REF] without any magnetic field computation in the yokes and by [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] with such a computation in the yokes, has been applied along with: the radial magnetization of magnets, a constant remanence of the magnets, and a constant relative permeability for all materials. This analytical model requires solving Poisson's and Laplace's equations (4) in the k region of the system (i.e. yoke region, permanent magnet region, air gap region, and each air space between pole pieces), where and are the magnetic vector potential and radial magnetization distribution, respectively; and are the cylindrical coordinates.

{ 𝛺 / = → = 𝛺 / 𝛺 / = 𝛺 / = → = 𝛺 / 𝛺 / = - 𝛺 / = → = 𝛺 / 𝛺 / = (
𝜕² 𝜕 ² + 𝜕 𝜕 + 𝜕² 𝜕 ² = { - 𝜕 𝜕 (4)
From boundary conditions defined [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF], it is possible to obtain the magnetic potential vector , radial flux and tangential flux 𝛼 in the different magnetic gear regions. With this analytical model, the magnetic field distribution is computed for just a single global position of the magnetic gear. To derive the magnetic losses, it then becomes necessary to compute the magnetic field distribution many times over for various global positions, representing one magnetic cycle of the system. As regards iron losses in the pole pieces, since the analytical model precludes any magnetic field computation in these regions, coupling with a bi-directional reluctance network model is required to determine the flux density distribution. To compute iron losses in laminated pole pieces like in [START_REF] Hernandez-Aramburo | Estimating Rotational Iron Losses in an Induction Machine[END_REF], for just one laminated pole piece, it is possible to determine the flux density distribution resulting from coupling between the analytical model based on Maxwell's equation [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] and the permeance network model, as shown in Fig. 3 with a low permeance number. To compute the flux density distribution in a laminated pole piece, a constant radial and tangential mesh has been adopted, and the permeance values are computed like in [START_REF] Ouagued | A General Framework Based on a New Hybrid Analytical Model for the Analysis and Design of Permanent Magnet Machines[END_REF]. The radial and angular discretizations are then set equal in order to obtain an * permeance network.

B. Magnetic field resolution in a pole piece resulting from coupling with a permeance network model

The coupling between both analytical models is performed on the pole piece boundaries, where the flux sources 𝜙 , , and 𝜙 𝛼, , of the reluctance network model are determined by [START_REF] Atallah | A novel high-performance magnetic gear[END_REF], and where correspond to the potential vector determined in regions II, III and IV from the model described in section IV-A. To identify the flux in the radial and tangential permeances of the network, Eq. ( 6) must be solved at the various nodes (the potentials , are the unknowns of this problem). If a node is adjacent to the pole piece boundary, then different terms in ( 6) must be substituted by the flux sources in (5), as displayed in [START_REF] Rasmussen | Development of a high-performance magnetic gear[END_REF] for specific values of and .

{

𝜙 , , = , + -,

𝜙 𝛼, , = + , - , (5)
( , --, ) . 𝛼 + ( , -, -) . + ( , -+ , ) . 𝛼 + ( , -, + ) . =

{ ( , --, ) . 𝛼 = 𝜙 𝛼, , ⃪ = ( , -+ , ) . 𝛼 = 𝜙 𝛼, , ⃪ = ( , -, -) . = 𝜙 , , ⃪ = ( , -, + ) . = 𝜙 , , ⃪ =

It then becomes possible to derive a matrix system (8) from these equations, where: [ ] is the potential matrix (containing the problem unknowns), [ ] the permeance matrix, and [𝜙] the flux matrix (containing the flux found from Maxwell's equation). The potential matrix [ ] can thus be determined from an inversion of matrix [ ]. This numerical operation yields the radial flux , , and tangential flux 𝛼, , using (9). To compute the iron losses in pole pieces, the pole piece permeance network will include 20*20 nodes. This configuration permits to have a low computation time for the permeance network resolution, compared to the global iron loss computation time (lower than 5% of the global computation time), without impacting the precision of the iron loss computation (i.e. this difference is less than 1% with an iron loss computation in pole pieces including 50*50 nodes).

[ ] * , * * [ ] * , = [𝜙] * , (8)
{ , , = ( , -, -) . + ( , + -, ) .

𝛼, , = 𝐼𝐼𝐼 -𝐼𝐼𝐼 ( , --, ) . 𝛼 + ( + , -, ) . 𝛼

V. EDDY CURRENT LOSSES IN PERMANENT MAGNETS

A. Eddy current determination from the analytical model

To rigorously compute the eddy current losses in permanent magnets, it is necessary to solve the scattering equation in both regions of permanent magnets [START_REF] Kim | Optimal Design of Novel Pole Piece for Power Density Improvement of Magnetic Gear Using Polynomial Regression Analysis[END_REF], where 𝜃 is the angular position of the permanent magnet rings ( = or = ). In this article, it is considered that the magnetic reaction of the eddy current on the magnetic field distribution is negligible in comparison with magnetostatic induction [START_REF]An analytical model for the computation of no-load eddy current losses in the rotor of a permanent magnet synchronous machine[END_REF]. This consideration is equivalent to considering a skin depth greater than the magnet thickness. The eddy current loss computation can thus be decoupled from the magnetic field computation. The magnetic field distribution is then computed in magnetostatics [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF], as described in the previous section, while the eddy current is computed by [START_REF] Shen | A Coaxial Magnetic Gear with Consequent-Pole Rotors[END_REF] for the two permanent magnet rings, whose eddy current source term leads to imposing a total axial current in the magnet equal to zero [START_REF] Gouda | Comparative study between mechanical and magnetic planetary gears[END_REF]. The eddy current losses in a single magnet are determined in both permanent magnet regions using ( 13), with = and .

B. Magnets splitting consideration

The eddy current losses in magnets can be reduced if the magnets are split into parts for the internal ring and parts for the external ring [START_REF] Atallah | Rotor eddy-current loss in permanent-magnet brushless AC machines[END_REF]. and correspond to the ratio of the pole pitch to the permanent magnet width for the internal and external rings, respectively, as defined in Fig. 4 with = and = and with 𝜏 and 𝜏 as defined in [START_REF] Rasmussen | Experimental Evaluation of a Motor Integrated Permanent Magnet Gear[END_REF]. The configuration used to analyze the evolution of eddy current losses comprises a fixed pole-piece ring, which implies that the high-speed rotor corresponds to the internal ring while the low-speed rotor corresponds to the external ring. Integrating splitting into the analytical model solely entails dividing the angular integration terminals as a function of splitting (shown in [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF] for the internal ring). Fig. 5 presents the loss percentage of transmitted power vs. the splitting of the magnets with a rotational speed of 350 rpm for the high-speed rotor and 100 rpm for the low-speed rotor, given the magnetic gear presented in Table I.

{ 𝜏 = 𝜋 𝑎𝑣 𝐼 . 𝜏 = 𝜋 𝑎𝑣 𝑉 . (14)
∈ [𝜃 𝐼 , 𝜃 𝐼 + 𝜏 ] (15)
Fig. 5 indicates that the eddy current losses in permanent magnets decrease significantly with splitting for the external ring and more slowly for the internal ring, according to the example presented in Table I. The splitting of the magnets is therefore a key parameter in obtaining a high-efficiency magnetic gear.

C. Validation of the magnetostatic model using a magnetodynamic finite element model

In order to validate the eddy current loss model in permanent magnets presented in the previous section, the results obtained with this magnetostatic analytical model are compared with other results from a 2D transient finite element model, with a magnetodynamic model being proposed for the example described in Table I. The studied configuration corresponds to a fixed pole-piece ring. It is commonly accepted that the skin effect begins to have an impact on the magnetic field distribution (i.e. on losses) when the rotational speed produces an equality between skin thickness and half the pole pitch [START_REF]An analytical model for the computation of no-load eddy current losses in the rotor of a permanent magnet synchronous machine[END_REF]. From this point forward, results obtained with the analytical model presented in the previous section will differ from those obtained using ANSYS, a 2D magnetodynamic finite element model.

For the magnetic gear defined in Table I, without any splitting of magnets, this phenomenon is first observed for the high-speed rotor when the fundamental magnetic field frequency = (which corresponds to a rotational speed of 1,680 rpm for the high-speed rotor and 480 rpm for the lowspeed rotor). A comparison between results obtained with the magnetostatic analytical model and those with the 2D magnetodynamic finite element model can be drawn for various fundamental magnetic field frequencies (3) with 𝛺 / = , as indicated in Fig. 6.

𝜏 𝜏 𝜏 𝜏 Fig. 5: Eddy current loss percentages of the transmitted power vs. splitting of the magnets with a rotational speed of 350 rpm for the high speed rotor and 100 rpm for the low speed rotor for the magnetic gear presented in Table I. This comparison reveals a good correlation between the two models for frequencies of less than approx. 50 Hz. In contrast, for frequencies of 50 Hz or more, the hypothesis of a negligible magnetic reaction of the eddy current on the magnetic field distribution no longer appears to be valid. This outcome was to be anticipated since the equality between skin thickness and the half-pole pitch is achieved for a fundamental magnetic field frequency =

. It is thus possible to define a validity criterion for the magnetostatic analytical model of eddy current losses in permanent magnets. Such a criterion would be satisfied whenever frequencies are below the limit frequency [START_REF] Lubin | Analytical computation of the magnetic field distribution in a magnetic gear[END_REF] obtained with a skin thickness equal to the smaller half-pole pitch of the two permanent magnet rings as a function of splitting. Due to the necessity to have a high efficiency, the splitting of the magnets will be significant, and the magnetostatic model will always remain valid.

A. Flux density evolution

Both temporal and spatial variations of the flux density generate iron losses in the different ferromagnetic regions. To determine whether a ferromagnetic region is subjected to significant iron losses, it is suggested to draw the flux density evolution in a 2D

, 𝛼 plane. The locus of the various ferromagnetic parts must then be analyzed in order to better understand the iron loss evolution, as presented in [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF]. Fig. 7 designates three distinct points chosen to analyze the magnetic field evolution of these three regions. The locus generation frequency for the various ferromagnetic parts thus depends on both rotational speed and pole configuration, as described in Eqs. ( 17) and [START_REF] Ostović | Dynamics of Saturated Electric Machines[END_REF]. According to Fig. 9, the locus for the various points indicated in Fig. 7 may be derived from the curve plotted in Fig. 8 (Fig. 9a references the point of the internal yoke, Fig. 9b the point of the external yoke, and Fig. 9c the point of the middle of a pole piece). The spatial variations of flux density displayed in Fig. 9 confirm the need to include spatial variation in the iron loss computation method [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF]. As seen in Fig. 9, the largest locus is obtained in the pole piece since pole pieces are subjected to a rotating magnetic field, imposed by permanent magnet rings. Fig. 9 also reveals that the locus obtained in the external yoke is higher than that in the internal yoke since the magnetic field distribution is apparently being imposed by the internal ring. Even though frequency is greater for the locus in the internal yoke [START_REF] Ostović | Dynamics of Saturated Electric Machines[END_REF], iron losses in pole pieces will still be higher than those in yokes. Both ⫽ , the major axis of the flux density locus, and ⊥ , the minor axis of the flux density locus (as defined in Fig. 10), must be determined in order to evaluate the iron losses in both yokes and pole pieces [START_REF] Hernandez-Aramburo | Estimating Rotational Iron Losses in an Induction Machine[END_REF]. It is therefore necessary to compute the magnetic field distribution using the global analytical model presented above (i.e. Maxwell's equation coupled with the permeance network model) for one magnetic cycle. The next step consists of evaluating the norm of the flux ‖ ( , , 𝜃)‖ everywhere in the ferromagnetic parts for the different magnetic cycle positions. For the various points ( , ) of the ferromagnetic parts, ⫽ and ⊥ are identified using Eq. ( 19), where 𝜑 , 𝜃 is defined in Fig. 10 and Eq. ( 20), the major axis angle of the flux density 𝜉 , is determined from [START_REF] Ouagued | A General Framework Based on a New Hybrid Analytical Model for the Analysis and Design of Permanent Magnet Machines[END_REF], and 𝜃 , * correspond to the global angular position maximizing flux density at the point ( , ) [START_REF] Desvaux | Analytical Iron Loss Model for the Optimization of Magnetic Gear[END_REF] [START_REF] Bertotti | An improved estimation of iron losses in rotatinf electrical machines[END_REF].

B. Iron loss computation

{ ⫽ ( , , 𝜃) = ‖ , , 𝜃 ‖. cos(𝜑 , 𝜃 ) ⊥ ( , , 𝜃) = ‖ , , 𝜃 ‖. sin(𝜑 , 𝜃 ) (19)
𝜑 , 𝜃 = 𝑎 𝑡𝑎 , + 𝜃 𝛼 , + 𝜃 -𝜉 , ( 20
)
𝜉 , = 𝑎 𝑡𝑎 𝜃 , * 𝛼 𝜃 , * (21)
The instantaneous iron loss density can then be evaluated from Eq. ( 22), where ℎ , and are the hysteresis coefficient, eddy current coefficient and excess loss coefficient, respectively, and the Steinmetz coefficient [START_REF] Deng | An improved iron loss estimation for permanent magnet brushless machines[END_REF]. In Eq. ( 22), corresponds to: for the internal yoke iron loss computation, for the pole piece iron loss computation, and for the external yoke iron loss computation (in accordance with [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF]). 11 shows the evolution in the various iron losses as a function of the internal ring speed (when the pole piece ring is fixed), with a laminated steel of grade M1000-65 (linear properties have been assumed) and with the iron loss coefficient proposed in [START_REF] Fratila | Calculation of Iron Losses in Solid Rotor Induction Machine Using FEM[END_REF], for the magnetic gear described in Table I. Fig. 11 also provides a comparison between results obtained with the magnetostatic hybrid analytical model of the magnetic field distribution and those with a 2D magnetostatic finite element model. The iron loss computation in pole pieces from the hybrid analytical model gives a result quite different from that found with the finite element model because the magnetic field can only handle two directions (radial and tangential) for the various nodes. These results validate the hybrid analytical model for the magnetic field distribution and allow concluding that iron losses in the internal yoke are negligible.

VII. COMPUTATION TIME ANALYSIS

The studies conducted in this section have been based on an example with a high pole number, as described in Table II, which corresponds to a magnetic gear optimized for wind turbine applications (3.9 MW and 15 rpm) [START_REF] Desvaux | Design and optimization of magnetic gears with arrangement and mechanical constraints for wind turbine applications[END_REF], with the same values of ℎ , , , , 𝜎 and than the magnetic gear presented in Table I.

A. Periodicity of the various magnetic losses

As shown in Fig. 8, it is impossible to compute iron losses in the various parts using the same periodicity. The loss computations can thus be performed separately for these ferromagnetic parts, according to an external ring displacement amplitude for the entire range of magnetic loss computations given in [START_REF] Bertotti | An improved estimation of iron losses in rotatinf electrical machines[END_REF].

To compute the average eddy current losses in permanent magnets, it is theoretically necessary to run the computation on one magnetic cycle of the magnetic field, corresponding to:

𝜃 ̂ 𝐼 = 𝜃 ̂ 𝑉 = / = °,
where 𝜃 ̂ denotes the external ring displacement amplitude (the various regions have been defined in Fig. 2), and the cogging torque factor. However, as shown in Fig. 12 at the rated speed, the instantaneous eddy current losses for both permanent magnet rings have an evolution with sub-periodicities lower than the magnetic cycle periodicity.

Due to these sub-periodicities, the computation of average eddy current losses requires a large number of intermediate positions provided the computation is performed for one magnetic cycle of the magnetic field. To minimize the number of intermediate positions (and hence the computation time) of these loss computations while maintaining adequate computational precision, the external ring displacement has been set equal to: 𝜃 ̂ 𝐼 = 𝜃 ̂ 𝑉 = 𝜋/ (23) (see Fig. 12).

{ 𝜃 ̂ 𝑋 = 𝜋 . , 𝜃 ̂ 𝐼𝐼𝐼 = 𝜋 , 𝜃 ̂ 𝑉𝐼 = 𝜋 𝜃 ̂ 𝐼 = 𝜃 ̂ 𝑉 = 𝜋 (23)

B. Consideration of a limited harmonic number

Due to the high pole number of the studied example, which increases the matrix system dimension of the analytical model, along with the need to compute magnetic losses using a stepby-step process, the computation time for the loss evaluation in the magnetic gears is quite high. For the magnetic gear presented in Table II, the loss computation time equals 96 h with the finite element method yet without computing iron losses in the internal yoke (as reflected in Fig. 11), using an Intel Xeon E5-1630 v3, 8 threads, 3.70 GHz. In contrast, with the hybrid analytical model proposed in this paper, the magnetic loss computation time drops to 48 h. For both these methods, the number of positions taken into account was = . To reduce computation time, [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] proposed a harmonics selection method that enables building the matrix system of the analytical model based on Maxwell's equations with only harmonics which generate magnetic fields. The most attractive harmonics selection methods take into account: the impaired harmonics of permanent magnet rings, the fundamental components of internal and external rings modulated by pole piece ring harmonics (e.g. harmonics with a periodicity of . -. 𝜋), and the impaired harmonics of both internal and external rings modulated by the pole piece ring fundamental (e.g. harmonics with a periodicity of -.

. 𝜋). To reduce computation time even further, [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] also suggested considering a small number of harmonics. In fact, with the values offered in [START_REF] Desvaux | Computation Time Analysis of the Magnetic Gear Analytical Model[END_REF] (i.e.

= and / = ), the computation time is once again lowered, with an error of less than 2% for both the iron loss computation and eddy current losses in permanent magnets of the internal ring computation. For the computation of eddy current losses in permanent magnets of the external ring, a higher number of harmonics must be considered ( = = / = are suggested herein). These values offer the possibility of cutting computation time by a factor of 100 (i.e. 30 min vs. 48 h).

C. Analysis of the step-by-step process

A compromise between computation time and precision must be found by modifying , i.e. the number of positions taken into account for the various magnetic loss computations. To evaluate the level of precision, the magnetic loss value references have been computed without any harmonics selection and with: = , and = / = . It is then possible to analyze the computation time and precision of all magnetic loss evaluations, as indicated in Fig. 13, with the corresponding error defined in [START_REF] Deng | An improved iron loss estimation for permanent magnet brushless machines[END_REF].

= | -𝐸 / 𝜃 ̂, | (24)
Fig. 13 reveals that precision increases as a function of computation time. It is observed that the error remains below 10% with = for the eddy current loss computation in the external and internal rings, with = for the iron loss computation in the external yoke, and with = for the iron loss computation in pole pieces. These values will thus be assumed for the following sections of the present study, thus providing the possibility to further lower computation time by a factor of 30 (1 min vs. 30 min).

VIII. OPTIMIZATION EXAMPLE

A. Objective functions

To illustrate the benefit of this hybrid analytical model, which serves to compute magnetic losses, a preliminary biobjective, mass-efficiency optimization protocol is proposed for a 6-MW and 12.5-rpm magnetic gear in a wind turbine application. This optimization routine has been introduced for various gear ratios. The objective functions [START_REF] Fratila | Calculation of Iron Losses in Solid Rotor Induction Machine Using FEM[END_REF] are twofold: to minimize the magnetic losses (objective function ), and to minimize the mass of the active magnetic gear parts (objective function ). The mass of the active parts takes into account the masses of: yokes , permanent magnets , and pole pieces .

{ = 𝐸 𝐼 + . 𝐼 𝐼𝐼𝐼, + 𝐸 𝑉 + 𝐼 𝑉𝐼 = + + (25)

B. Variables

This optimization step is composed of six variables, namely: yoke thicknesses, permanent magnet thicknesses, pole piece thickness, and internal ring pole pairs . For a gear ratio value , the external ring pole pairs is given in (26) so as to minimize the cogging torque [START_REF] Atallah | Design, analysis and realisation of a high-performance magnetic gear[END_REF], with the number of ferromagnetic pole pieces determined by (1).

= . | | +

C. Constrained dimensions Due to the 6-MW and 12.5-rpm wind turbine dimensions as well as a number of technological constraints, the different dimensions have been listed in Table III along with the oversizing proposed in [START_REF] Desvaux | Analysis of the Dynamic Behaviour of Magnetic Gear with Nonlinear Modelling for Large Wind Turbines[END_REF]. The material properties are the same as those in Table I for the laminated steel grade M1000-65 and NdFeB permanent magnets.

D. Optimization constraints

Due to the linear magnetic behavior of the materials in the analytical model, a maximum induction value is tolerated in the ferromagnetic parts, as indicated in [START_REF] Desvaux | Supporting the laminated ferromagnetic pole pieces in a magnetic gear : a structure behaviour analysis from a multibody model[END_REF]. On the other hand, a multibody mechanical analytical model of pole piece rings serves to evaluate the radial displacement 𝒰 𝐼𝐼𝐼, and normal stress 𝜎 𝐼𝐼𝐼, of the various pole piece support bars ( < < . These bars are made of a non-magnetic material and constitute a cage, thus making it possible to ensure the mechanical maintenance of pole pieces [START_REF] Desvaux | Supporting the laminated ferromagnetic pole pieces in a magnetic gear : a structure behaviour analysis from a multibody model[END_REF]. The yield of this material is: 𝜎 = 𝑎. It can then be imposed that the radial displacement must be less than 10% of the internal airgap [START_REF] Zavvos | Structural optimisation tools for iron cored permanent magnet generators for large direct drive wind turbines[END_REF] and the normal stress in the bars must be less than the yield stress 𝜎 with a safety coefficient [START_REF] Aubry | Sizing optimization methodology of a surface permanent magnet machine-converter system over a torque-speed operating profile: Application to a wave energy converter[END_REF] [3]- [START_REF] Zavvos | Structural optimisation tools for iron cored permanent magnet generators for large direct drive wind turbines[END_REF].

‖

, , 𝑡 ‖ < 𝑎 = .

E. Optimization procedure and results

For a couple of parameters, this classical optimization procedure evaluates both objectives along with the various constraints. To complete this optimization step, a Particle Swarm Optimization (PSO) algorithm has been employed, as described in [START_REF] Aubry | Sizing optimization methodology of a surface permanent magnet machine-converter system over a torque-speed operating profile: Application to a wave energy converter[END_REF]. Optimization is performed in approximately 24 h for the different gear ratios with 100 particles and in 50 iterations (with an Intel Xeon E5-1630 v3, 8 threads, 3.70 GHz).

Fig. 14 shows the Pareto front of the bi-objective PSO optimization with the objective function in [START_REF] Fratila | Calculation of Iron Losses in Solid Rotor Induction Machine Using FEM[END_REF] for different gear ratios. This result demonstrates that magnetic losses increase with gear ratio and may reach unacceptable values for a high-power energy conversion application (i.e. an efficiency below 90%). It then becomes very important to integrate a magnetic loss evaluation based on analytical models within a magnetic gear optimization procedure. In contrast, minimizing losses by decreasing the gear ratio is of no interest since this would not reduce the input torque of the generator nor therefore its mass and cost. A compromise must be found, and this will take the form of a gear ratio optimization by considering the magnetic gear and the generator together.

IX. CONCLUSION

This article has focused on a fast analytical model for magnetic loss computations in coaxial magnetic gear structures. The model derived takes into account the splitting of magnets (for eddy current losses) and spatial variations of the flux density (for losses in laminated ferromagnetic parts). The proposed 2D linear magnetostatic model is based on a hybrid analytical model that also serves to compute other magnetic gear parameters like torque, and radial and tangential loads [START_REF] Filippini | Coaxial magnetic gears design and optimization[END_REF].

The model for eddy current losses in permanent magnets proposed in this article has assumed a negligible magnetic reaction of the eddy current on the magnetic field distribution of permanent magnets. The results on a simplified eddy current loss model have been compared to results obtained with a magnetodynamic finite element model in an intermediate step (Fig. 6). This comparison has highlighted the validity limit of the magnetostatic analytical model (see Eq. 16). The iron loss model takes into account both the temporal and spatial variations of flux density. Fig. 9 has exposed the benefit of considering spatial variations.

A computation time analysis was also carried out, revealing the possibility of computing magnetic losses in approximately 1 min with an error of less than 10% for a 3.9-MW, 15-rpm magnetic gear [START_REF] Desvaux | Design and optimization of magnetic gears with arrangement and mechanical constraints for wind turbine applications[END_REF] (with an Intel Xeon E5-1630 v3, 8 threads, 3.70 GHz).

The final function of this analytical model must be integrated into a set of models for the global mechatronic optimization of magnetic gears. A preliminary bi-objective mass-efficiency optimization protocol with various constraints was then proposed to illustrate the feasibility of such a mechatronic optimization due to the low computation time generated by this hybrid analytical model (Fig. 14). Results have also illustrated the need to take losses into account for magnetic gear optimization given their strong impact on magnetic gear dimensions.

In subsequent work, a mechanical analytical model of the structural parts of the magnetic gear must be proposed in order to complete global mechatronic optimization. The 3D effects of the magnetic loss computations must still be evaluated.

: 4 :

 4 Magnetic vector potential of region k / 𝛼 ∶ Radial / tangential flux density in region k [T] ⫽ / ⊥ ∶ Major / minor axis of the flux density locus [T] : Fundamental magnetic field frequency [Hz] : Iron loss frequency of the ring [Hz] : Skin depth frequency [Hz] : Gear ratio : Eddy current density in region k [A/m²] : Index of the magnetic gear region : Magnet splitting of the ring defined in Fig. Magnetic length of the magnetic gear [m] : Radial magnetization distribution of region k : Number of positions for the magnetic loss computation / : Number of internal / external ring pole pairs : Instantaneous iron loss density in region k [W/m 3 ]

𝐸:

  Eddy current losses in region k [W] / 𝛼 : Radial / tangential permeance : Number of ferromagnetic pole pieces 𝑎𝑣 : Average radius of region k [m] : Permanent magnet area of region k [m²] , : Potential of the permeance network at point , / : Internal / external airgap [m] 𝜉 , : Major axis angle of the flux density at point , [rad] 𝜃 : Angular position of the ring of region k [rad] 𝜃 ̂ : Displacement amplitude of the external ring for the average magnetic loss computation of region k [rad] 𝜃: Global angular position of the magnetic gear [rad] 𝜃 , * : Global angular position maximizing flux density at the point , [rad] 𝛺 : Rotational speed of the ring of region k [rad/s] 𝜎: Electrical conductivity of magnets [S] 𝜙 , , / 𝜙 𝛼, , ∶ Radial / tangential flux source at point , 𝜏 / 𝜏 : Pole pitch of the internal / external ring [m] 𝛺 / : Rotational speed of the ring [rad/s]

Fig. 1 :

 1 Fig. 1: Coaxial magnetic gear architecture in an exploded-view drawing with low pole numbers (i.e. = 2, = 7, and = 9)

M

  . Desvaux, S. Sire, S. Hlioui and H. Ben Ahmed, B. Multon, Member, IEEE Development of a Hybrid Analytical Model for a Fast Computation of Magnetic Losses and Optimization of Coaxial Magnetic Gears

  ANALYTICAL MAGNETOSTATIC MODELA. Magnetic field resolution elsewhere than pole pieces

Fig. 2 :

 2 Fig. 2: Magnetic gear parameterization of the various regions

Fig. 3 :

 3 Fig. 3: Pole piece parameterization for the * permeance network model

Fig. 4 :

 4 Fig. 4: Magnet splitting representation (in this example: = 3, = 2)

Fig. 7 :

 7 Fig. 7: Designation of the three distinct points studied in Section VI, representing the magnetic field evolution of the three regions

Fig. 8 :

 8 Evolution in the radial and tangential components of flux density when the magnetic gear is in rotation and when the fixed ring is the pole piece ring for the points displayed in Fig.7on the: a) internal yoke, b) external yoke, and c) pole piece As shown in Fig.8for the internal yoke, external yoke and pole pieces, the locus generation frequencies differ (Fig.8a

Fig

  Fig. Comparison of results obtained with the analytical model and the 2D magnetodynamic finite element model, for the example presented in Table I

.

  Point of the external yokePoint of the internal yokePoint of the middle of a pole piece corresponds to the point of the internal yoke, Fig.8bto the point of the external yoke and Fig.8cto the point of the pole piece).

Fig. 9 :

 9 Spatial variations in flux density for the points displayed in Fig.7on the: a) internal yoke, b) external yoke, and c) pole piece (presented in Fig.8)

Fig. 10 :

 10 Fig. 10: Locus parameterization for the iron loss evaluation

Fig. 11 :

 11 Fig. 11: Comparison between results obtained with the magnetostatic analytical model of the magnetic field distribution and those with a 2D magnetostatic finite element model for the magnetic gear presented in Table I

TABLE I :

 I DATA FOR THE EXAMPLE IN SECTIONS V AND VI

	Symbol	Quantity	Value
	Q D 𝜏 𝜏 δint δext Br µ µ 𝐼 = µ 𝑉 µ 𝑋 = µ 𝑉𝐼 𝜎 ℎ	Number of internal ring pole pairs Number of external ring pole pairs Number of ferromagnetic pole pieces External diameter Magnetic length Pole pitch of the internal ring Pole pitch of the external ring Internal air gap External air gap Remanence of the magnets Absolute permeability Relative permeability of the magnets Relative permeability of the yokes Conductivity of the magnets Hysteresis coefficient Eddy current coefficient Excess loss coefficient Steinmetz coefficient	2 7 9 250 mm 100 mm 140 mm 50 mm 1 mm 1 mm 1.27 T 4π10 -7 1 1,000 0.9 MS 364 0.167 1.84 2

TABLE III :

 III FIXED DIMENSIONS OF THE OPTIMIZATION

	Symbol	Quantity	Value
	𝑎 𝑎 𝑎 TG D δint δext	Wind turbine rated power Wind turbine rated speed Wind turbine rated torque Maximum gear torque External diameter Minimal splitting magnets width Internal air gap External air gap	6 MW 12.5 rpm 4.6 MNm 5.1 MNm 5.5 m 10 mm 5 mm 5 mm
	(a)
	(b) Fig. 13: Magnetic loss evaluation for various numbers of positions taken into account ( ) for the magnetic gear described in Table II: a) precision, and b) computation time

XI. BIOGRAPHIES

Melaine Desvaux received a joint B.S. degree in electronics and telecommunications and in mechanical engineering in 2012, followed by an M.S. degree in mechatronic design in 2015, from the University of Rennes 1 (France). He passed the "Agrégation" credentialing exam in Mechanical Engineering in 2014 and is currently pursuing a Ph.D. in Mechatronic Engineering at the SATIE Laboratory (ENS Rennes, France).

His research topics include the integration of magnetic gears in wind power applications, a field that requires skills in mechanical engineering, optimization and.magnetism. His research interests include the design, modeling and optimization of novel topologies for electromagnetic actuators and generators, as well as the optimization of renewable energy systems.

Stéphane