Mathilde Monvoisin1
email: mathilde.monvoisin@ls2n.fr

Philippe Leray1
email: philippe.leray@ls2n.fr

Mathieu Ritou1
email: mathieu.ritou@ls2n.fr

Unsupervised co-training of Bayesian networks for condition prediction

The objective of Smart Manufacturing is to improve productivity and competitiveness in industry, based on in-process data. It requires reliable, explainable and understandable models such as Bayesian networks for performing tasks like condition prediction. In this context, a Bayesian network can be classically learned in a supervised, unsupervised way or a semi-supervised way. Here, we are interested in how to perform the learning when the ground truth isn't included in the learning data but is observable indirectly in another related dataset. This paper introduces a fully unsupervised variation of co-training that allows to include this second dataset, with two learning strategies (split and recursive). In our experiments, we propose one simple probabilistic graphical model used for predicting the state of a machine tool from results given by several sensors, and illustrate our unsupervised cotraining strategies first with benchmarks available from the UCI repository, for which 4 out of 5 datasets have best results with the recursive strategy. Finally, the recursive strategy was validated by McNemar's test as being the best strategy on a real industrial dataset.

Introduction

Smart manufacturing is a promising research area to improve productivity and competitiveness in industry [START_REF] Wang | Deep learning for smart manufacturing: Methods and applications[END_REF] [START_REF] Tao | Data-driven smart manufacturing[END_REF]. Indeed, it is crucial to detect any system failure as early as possible to reduce maintenance costs and downtimes. This is the reason why predictive maintenance is a key issue in industry 4.0 [START_REF] Gao | Cloud-enabled prognosis for manufacturing[END_REF]. Approaches can be data-driven, model based or hybrid.

In our work, we are interested in discovering and understanding of the events leading to the damage of industrial production machine, and in predicting their failure, in a predictive maintenance perspective. In the application, we have been faced with a more general problem. Supervised learning is a classic approach for learning any predictive model when the ground truth is known. Unsupervised learning aims at learning the same model when this ground truth is unknown. In our application, we consider the unsupervised scenario, but with an additional information provided about the system state during another phase.

Smart manufacturing can make use of supervised learning in order to build a predictive model, such as a Probabilistic Neural Network to classify broken tools and good tools [START_REF] Huang | A PNN selflearning tool breakage detection system in end milling operations[END_REF] or Bayesian networks and Support Vector Machines for thermal modelling and prediction [START_REF] Ramesh | Thermal error measurement and modelling in machine tools. Part II. Hybrid Bayesian Network-support vector machine model[END_REF].

Improving the learning of a given model with the results of another one is one of the founding principle of the cotraining. [START_REF] Blum | Combining labeled and unlabeled data with co-training[END_REF] and [START_REF] Nigam | Analyzing the Effectiveness and Applicability of Co-Training[END_REF] proposed semi-supervised learning paradigm, which trains two naive Bayes classifiers respectively from two different views and lets the classifiers label some unlabeled data for each other. [START_REF] Yu | Bayesian Co-Training[END_REF]) used Bayesian undirected graphical model for co-training.

This paper introduces a fully unsupervised variation of cotraining and several learning strategies are proposed, the first one to our knowledge. It includes a conditional linear Gaussian Bayesian Network structure and the learning strategies associated. The strategies were experimentally tested on 5 UCI datasets, and on a real industrial dataset dedicated to the diagnosis of machine tool. The result is the first generic framework for fully unsupervised diagnosis, which parameters are learned with co-training.

Section problem statement proposes a formal description of the problem. Section unsupervised co-training describes our unsupervised co-training framework, as well as three learning strategies. Section 6 is dedicated to the empirical evaluation of the proposal. In section 6, we describe one simple hybrid Bayesian network dedicated to condition prediction. In section 6, we will describe one first set of experiments with benchmarks available from the UCI repository, transformed for co-training task, in a controlled context where the ground truth is known. Section 6 presents an application on real industrial data, and section 6 concludes on the contribution of this paper and our perspectives of research.

Problem Statement

{SensA 1 , ..., SensA n } is a set of continuous variables, outputs of several sensors (potentially preprocessed) that are measured every day during the "production phase" (phase A) of the machine tool. The objective is to predict State A , the (discrete) state of the system, by the mean of one model M odel A learned from data.

A classic approach to this problem would be to learn the parameters of M odel A in a supervised way, from a dataset containing observations of the sensors {SensA 1 , ..., SensA n } and the ground truth about State A ; or in an unsupervised way, without measuring the ground truth. Let us denote D A this dataset with the sensors information only.

In our problem, we consider the unsupervised scenario, but with an additional information D B provided by another set of sensors about the system state during another phase (phase B). So, {SensB 1 , ..., SensB m } is also a set of continuous variables, outputs of these other sensors (potentially preprocessed), that are also measured every day. This dataset can be used to predict State B , the state of the system during phase B. States A and B of the system are assumed to be two estimations of the same underlying state of the machine. Our objective is to learn M odel A , with D A , i.e. in an unsupervised way, without knowing the ground truth, but also by taking into account data D B acquired during the second phase. As an example, in our application, D A is the data from the process monitoring and D B is the data collected during condition monitoring when the component signature are recorded.

As shown in a strong context by [START_REF] Blum | Combining labeled and unlabeled data with co-training[END_REF] or a weaker one by [START_REF] Balcan | Co-training and expansion: Towards bridging theory and practice[END_REF] for (supervised) co-training, we consider the following assumptions : (a) weak sufficiency, each of our views (M odel A and M odel B ) is at least approximately sufficient in itself to achieve good prediction, and (b) weak dependency, both views are not too highly correlated.

Unsupervised co-training Principle

In order to solve the problem described in the previous section, we propose to "enrich" M odel A with another submodel M odel B dedicated to the prediction of State B from the other sensors {SensB 1 , ..., SensB m } leading to an unsupervised co-training of both models that should agree about the state of the system.

We will consider that both models are probabilistic graphical models with continuous and discrete variables (and parameters θ A and θ B ), and that unsupervised learning can be performed by the EM algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF][START_REF] Mclachlan | The EM algorithm and extensions[END_REF].

This unsupervised co-training can be performed with different learning strategies, in relation to the way the equality assumption between State A and State B is envisaged.

Split learning

Instead of learning one unique model with all the sensors inputs, we propose to reduce the complexity of the parameter learning by splitting this task. i.e. learning one first submodel in an unsupervised way with the help of the EM algorithm, and then the second one enriched with the results of the first one, obtained by probabilistic inference, as described in algorithm 1. As described in figure 1, one model is then learned to "reproduce" the vector [State * B ] of the optimal outputs of the second model.

In this case, M odel A takes benefit of the M odel B unsupervised learning. However, in a symmetrical way, a better learning of M odel A would also help for learning M odel B .

Recursive learning

The recursive method consists in using the previous method and iterating n step times (2n step EM and probabilistic inference runs). Sub-model B is learned in an unsupervised way, enabling the supervised learning of model A, and then of model B, etc. The procedure is detailed in algorithm 2, where P (D A |θ A ) is the likelihood of observing the data D A for the model A, and symmetrically for model B.

For an optimal computation time, we break complexity by learning one model, and then learning the other model with the prediction of the first one, and repeating it n step times. The model is learned in a sort of semi-supervised way, but without real labeled data such as in usual co-training framework.

Instead of controlling the number of steps, we can also monitor the likelihood and stop the iterations when it has stopped improving significantly.

Experiments Model and parameter initialization

We propose to use probabilistic graphical models which are explainable models, and more especially Conditional Linear Gaussian Bayesian Networks (CLGBNs, [START_REF] Lauritzen | Graphical Models for Associations Between Variables, some of Which are Qualitative and some Quantitative[END_REF]) that are able to deal with continuous and discrete variables.

Our unsupervised co-training framework proposes to use jointly or iteratively two models M odel A and M odel B . We will consider here that the structure of both models is similar, and we will describe only the first one. As we are interested in applying this unsupervised co-training for condition prediction, the choice of the model used in the experiments and described in Figure 2 is inspired from the following considerations.

In an industrial system, the physical measurements (power, temperature, etc.) corresponding to good operating conditions of the system are within a limited range of values. Therefore, values outside this small and frequently observed range probably indicate problems during the manufacturing. We are defining State A as a boolean variable with {OK, KO} values. Moreover, each sensor is usually able to discriminate some intermediate states. Let us define DiscrA i the local estimation of the state provided by SensA i . This variable is a discrete variable with a larger domain, for instance {OK, degraded, KO}. As in usual CLGBNs, we consider that the distribution of each SensA i is a Gaussian distribution conditional on DiscrA i .

In order to simplify the description of the model, we will also consider that low values of SensA i usually corresponds

Algorithm 1: Split strategy Input: D A , D B Output: θ * A , θ * B 1 θ * B = argmax P (D B |θ B ) 2 [State * B ] = argmax P ([State B ]|D B , θ * B ) 3 θ * A = argmax P (D A , [State A ] = [State * B ]|θ A )
Second learning: supervised Algorithm 2: Recursive strategy

Input: D A , D B Output: θ * A , θ * B 1 θ * B = argmax P (D B |θ B ) 2 for i = 1 to n step do 3 [State * B ] = argmax P ([State B ]|D B , θ * B ) 4 θ * A = argmax P (D A , [State A ] = [State * B ]|θ A ) 5 [State * A ] = argmax P ([State A ]|D A , θ * A ) 6 θ * B = argmax P (D B , [State B ] = [State * A ]|θ B )
to State A = OK, and the more the sensor values increase, the more the state of the system is degraded or KO.

Each conditional Gaussian distribution is initialized either by one application of EM for the joint distribution P (SensA i , DiscrA i ), or by using a classical grid initialization given in eqn for a mixture of k distinct Gaussians (with i from 0 to k -1).

µ i = min(SensA i ) + 0.5 + i k (max(SensA i ) -min(SensA i )) (1)
σ 2 i = σ 2 SensAi k
We finally consider that the global state of the system is an aggregation of the local states estimated by each sensor. In a first approach that is very similar to a deterministic MAX function, we consider here that State A =KO when it has been diagnosed as degraded or KO by at least one sensor of the model.

Experiments on UCI datasets

Dataset adaptation for co-training Several benchmarks from the UCI repository (Dua and Graff 2017) have been used. They were selected based on their similarity to our industrial context, with several numerical variables as inputs. The target variable was binarized such as the highest value corresponds to a failure if it wasn't already binary. These benchmarks are not dedicated to co-training. In order to adapt them for our task where we need two-viewed datasets, we followed a procedure described in [START_REF] Ling | When does Cotraining Work in Real Data?[END_REF] to split an "usual" one view dataset into two views by using an entropy based method. The two-views UCI datasets are shared on github1 to create a public benchmark for co-training. [START_REF] Ling | When does Cotraining Work in Real Data?[END_REF]) also introduces two measures to verify the two assumptions needed for co-training.

The sufficiency criterion δ 1 measures the fact that the two models should have sufficient information by itselves to predict the target variable. The independence criterion δ 2 measures the ability of predicting the attributes of the other dataset (given the value of the class variable). As shown in [START_REF] Ling | When does Cotraining Work in Real Data?[END_REF], the lower δ 1 and δ 2 are, the better the dataset is fit for co-training. Table 1 summarizes all the benchmarks selected from UCI repository, with their properties (total number of input variables, data size, imbalance ratio) and the sufficiency and independence measurements (δ 1 and δ 2 ) estimated by using a decision tree as a baseline classifier and stratified 10-fold cross-validation.

Experimental protocol

The implementation of the models described in section 6 with our different learning strategies proposed in section has been performed with our library dedicated to Probabilistic Graphical Models (PIL-GRIM) with the help of ProBT 2 library.

We compare in these experiments our two learning strategies (Split and Recursive). As a baseline method, we also learned independently each model in an unsupervised way.

Parameters used during learning were: a threshold equal to 0.0001 on likelihood variation as stopping criterion for EM and n step = 30 (chosen for guaranteeing convergence in all our experiments) for the recursive strategy.

The accuracy and sensitivity of each model (A and B) are estimated with 10-fold cross validation, and expressed as a value between 0 and 100%. Their average (between both models) is then considered as the global performance of each learning strategy (unsupervised, split learning and recursive learning).

Results

Table 2 shows us the variation both of our metrics (accuracy and sensitivity) for each dataset.

The recursive strategy is usually better than the simple split one, with an increase in terms of accuracy and sensitivity for all datasets excepted for one dataset (hydraulic stable). The recursive co-training strategy is also more interesting than the unsupervised learning (without co-training) for three datasets (heart-statlog, APS failure and hydraulicvalve) or equivalent (for hydraulic leakage, the increase of accuracy is counterbalanced by a decrease in sensitivity).

Both co-training strategies are not efficient for the hydraulic stable benchmark and not useful for another one (hydraulic leakage). Both datasets have one high value for δ 2 , showing that one of the co-training usual assumption (independence here) is not verified with also one small imbalance Experiments on real data from industrial use case Data and experimental protocol This use case concerns the machining industry and the component to be diagnosed is a machine tool spindle. The industrial dataset have been collected over more than one year, aggregated at a daily level and it is unlabelled. The results obtained by a previous study [START_REF] Godreau | Continuous improvement of HSM process by data mining[END_REF] which have been afterwards confirmed by an expert are considered as the ground truth.

The model A is dedicated to process monitoring during the machining phase: the four inputs are classical vibration criteria. The model B corresponds to the spindle condition monitoring: a vibration signature is performed once a day to evaluate it, with also four criteria.

Problems are uncommon in machining: from the process monitoring and the spindle monitoring datasets respectively only 7 (1.6%) and 5 (1.1%) events are considered as events that might have seriously damaged the spindle, from the previous study. Another issue in this real application is the fact that the two diagnosis don't coincide perfectly: only 3 of the 5 spindle damages are included in the 7 events detected during the process monitoring.

For this dataset, the sufficiency and independence criteria have been estimated by applying various oversampling algorithms from the package imbalanced-learn (Lemaître, Nogueira, and Aridas 2017) because of the very low imbalance ratio (≈ 1%). The results from the ADASYN, BorderlineSMOTE, KMeansSMOTE, Rando-mOverSampler, SMOTE and SVMSMOTE over-sampling algorithms gave results in the range [0, 02; 0, 12] for δ 1 and [0, 34; 0, 58] for δ 2 . These values are in the range of values where co-training was efficient in our previous study on the UCI datasets.

We compare in these experiments our two co-training strategies (Split and Recursive). As a baseline method, we also learned independently each model in a unsupervised way without the help of the other model.

The evaluation of the learning strategies is done here by two confusion matrices comparing the predictions done by our M odel A and M odel B to the same ground truth. For each model, we also present the classic performance indicators (precision, recall, ...) and computation time for each model A and B. In our application, the sensitivity is a very important indicator because it measures how wrong the model was when predicting an OK, and there can be a big impact if we ignore a KO, depending on what was the damage suffered.

Results Table 3 andtable 4 present the confusion matrices and the other performance indicators obtained for our two co-training strategies and the baseline unsupervised learning.

The recursive method converges to performances very similar to the split one on (B), and gives increased accuracy performances on (A) and they both have a very fast learning time (less than 10s).

Table 5 presents the results of McNemar's statistical test [START_REF] Dietterich | Approximate Statistical Tests for Comparing Supervised Classification Learning Algorithms[END_REF] where the null hypothesis considers that the predictive performances of the two classifiers are equal (with a significance level of α = 0.05). The tests were computed using the statsmodel python library [START_REF] Seabold | Statsmodels: Econometric and statistical modeling with python[END_REF]. This table shows that the split strategy improves the spindle diagnosis results, compared to the unsupervised one. Recursive learning is globally the best strategy, by significantly outperforming all the other strategies except the split one for the spindle condition diagnosis.

The fact that the recursive strategy is not always better than the split one means that repeating the iterations can decrease precision and sensitivity in the process diagnosis classifier. This situation can be explained by several hypothesis: our unsupervised co-training is considering that both models should agree about their outputs, where the ground truth considered for the evaluation metrics is based on independent results for both models that disagree about some KO events. Some improvements addressing this situation are proposed in the following section.

A detailed study with a machining expert has validated a new incident in the data that was detected using the recursive strategy. This rare event has not been previously detected by other techniques that were used on this dataset.

Conclusion and future work

The paper focuses on learning a predictive model in a smart manufacturing context with an unsupervised framework where additional information is provided about the system state during another phase.

We have proposed one fully unsupervised variation of cotraining framework with several learning strategies, which can be applied to various models and application fields.

These strategies have been illustrated with several benchmarks available from the UCI repository (and adapted for a co-training purpose), and have then been applied in a real application dedicated to the detection of machine tool failure when the ground truth is unknown. We have shown that our unsupervised co-training strategies can take profit from separate information in order to provide better results.

This present work can yet be extended or improved in several ways. In order to avoid the negative feedback loop observed during our experiments, the split and recursive strategies could be improved by not transferring from one submodel learning to the second one the state predicted by the sub-model (cf. eqn 1), but the probability distribution of the state as a soft evidence, as proposed in multi-agent context [START_REF] Vomlel | Probabilistic reasoning with uncertain evidence[END_REF]. We can also inspire ourselves from the semisupervised co-training strategies with only a partial transfer of information between the two learning tasks, as proposed in [START_REF] Nigam | Analyzing the Effectiveness and Applicability of Co-Training[END_REF] where only the more confident prediction are transferred. As the sufficiency assumptions are not always met in real applications, we are also interested by extending our work with insufficient views, such as proposed for co-training in [START_REF] Guo | Towards making co-training suffer less from insufficient views[END_REF].

The model structure we proposed in a smart machining context is a very simple one, used to highlight the interest of the co-training strategies. This model can be improved by taking into account more complex deterministic aggregation functions (for instance the AtLeastK operator instead of the MAX one) or probabilistic ones like NoisyMax [START_REF] Srinivas | A generalization of the noisy-or model[END_REF] or other causal independence models [START_REF] Dıez | Canonical probabilistic models for knowledge engineering[END_REF]

FirstFigure 1 :

 1 Figure 1: Unsupervised co-training by split learning strategy. Firstly, M odelB is learnt in an unsupervised way, then M odelA is learnt in a supervised way, by considering that StateA should be the optimal prediction of StateB after M odelB learning. The blue nodes denote variables observed during the learning task, where the white ones represent unobserved variables.

Figure 2 :

 2 Figure 2: Conditional Linear Gaussian Bayesian Network used in our experiments for process monitoring in smart manufacturing. Circles denote continuous variables and squares discrete ones.

  (A) and spindle condition (B) performance indicators (precision, sensitivity, recall, accuracy and computation time) for the split and recursive (rec.) unsupervised co-training strategies, compared with a baseline independent unsupervised learning of both models. of McNemar's statistical test where the null hypothesis considers that the predictive performances of a pair of classifiers are equal (=) with a significance level of α = 0.05. The classifiers were learned with our unsupervised co-training strategies or with a baseline independent unsupervised learning of both models.

Table 1 :

 1 UCI datasets characteristics in a co-training context. δ 1 and δ 2 are the respective measures for sufficiency and independence. n and m are the number of columns of views A and B, N is the number of samples in the dataset and IR is the imbalance ratio, i.e. the percentage of the positive class.

		δ 1	δ 2	n m	N	IR
	heart-statlog	0.33 0.12 7 6	270 45%
	APS failure	0.11 0.39 5 5	758 36%
	hydraulic stable	0.45 0.66 8 7 2206 34%
	hydraulic valve	0.56 0.67 8 7 2206 67%
	hydraulic leakage 0.19 0.67 8 7 2206 22%

Table 2 :

 2 Variation of accuracy and sensitivity between unsupervised learning and the split and recursive co-training strategies.ratio. In such imbalanced context, the first model seems to produce wrong results that are given as a ground truth for the next model. This situation can lead to a negative feedback loop and progressively produces a decrease of the global performances.

	2 https://www.probayes.com/

Table 3 :

 3 . Process diagnosis (A) and spindle condition diagnostic (B) confusion matrices for several unsupervised co-training strategies compared with a baseline independent learning of both models.

						A				B
					FP FN TP	TN FP FN TP	TN
	Independent unsupervised learning	1	15	414 6	0	38	393 5
	Unsupervised co-training (split)	1	15	414 6	0	9	422 5
	Unsupervised co-training (recursive) 3	2	427 4	0	8	423 5
			A: Process diagnosis				B: Cond. monitoring
		Prec.	Sens.	Rec.	Acc.	Prec.	Sens.	Rec.	Acc.	Time (s)
	Unsupervised	99.7% 85.7 % 96.5% 96.3%	100% 100% 91.2% 91.3%	2.1
	Split strategy	99.8% 85.7 % 96.5% 96.3%	100% 100% 97.9% 97.9%	2.8
	Rec. strategy	99.3% 57.1%	99.5% 98.9%	100% 100% 98.1% 98.2%	9.8

Table 4 :

 4 Process diagnosis

https://github.com/MathildeMonvoisin/Co-trainingbenchmark

Acknowledgement

Thanks are addressed to the IRT Jules Verne, French Institute in Research and Technology in Advanced Manufacturing for the PhD PERFORM program. The authors would also like to thank people who laid the foundations of this project with Philippe Leray and Mathieu Ritou, in particular Victor Godreau, Abdal Moughit Idrissi and Guillaume Ferrand, and the industrial partners for supplying the experimental data.


P Chapelle

A Jardy

C Creusot

Numerical Study of the Current Constriction in a Vacuum Arc at Large Contact Gap

Benoît Tezenas du Montcel, Pierre Chapelle, Christophe Creusot, and Alain Jardy Abstract-A classical two temperature magnetohydrodynamic modelling approach is used to study the influence of a large contact gap (up to 40 mm) on the behaviour of a diffuse vacuum arc controlled by a 5 mT/kA uniform external axial magnetic field between two copper electrodes with a 20 mm radius. The current constriction and energy flux density at the anode surface are more particularly analyzed, considering both supersonic and subsonic flow conditions. In the case of a supersonic arc, simulations show that the constriction of the current develops in the whole interelectrode region, but the constriction level at the anode surface does not evolve monotonically with the contact gap. The constriction is partly correlated to the radial compression of the plasma. In the case of a subsonic arc, the current constriction is related to the presence of the anode sheath. It occurs only close to the anode (from a constant distance from the anode of around 15 mm). Whereas the current constriction at the anode surface increases when the gap length goes from 10 mm to 20 mm, it no longer evolves when increasing the gap length from 20 mm to 40 mm. For both flow conditions, the evolution with the gap length of the radial profile of the energy flux density transferred by the plasma to the anode is similar as that followed by the current constriction at the anode. Index Terms-Numerical simulation, diffuse vacuum arc, axial magnetic field, large contact gap.

I. Introduction

High-current vacuum arc (VA) in diffuse mode has been the subject of a great number of research studies through the years for its applications in circuit-breakers (CB) [START_REF] Boxman | Twenty-Five Years of Progress in Vacuum Arc Research and Utilization[END_REF]. Whereas the diffuse mode is characteristic of a free burning VA at low currents (up to a few kA), at higher currents an anode spot appears and VA constricts taking the form of a columnar arc, which strongly decreases the breaking capability of the circuit-breaker. Applying an axial magnetic field (AMF) may be used to distribute more evenly the arc energy on the anode surface, which allows to significantly increase the current at which the columnar arc mode develops [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF]. Nowadays, vacuum circuit-breakers based on the application of an AMF have become a key technology for Medium-Voltage applications [START_REF] Picot | Vacuum Switching[END_REF]. For High-Voltage applications, circuit-breakers rely mostly on the use of SF6 as an interrupting medium. This gas offers among others excellent dielectric performances. However, it has major harmful impacts to the environment. This induces an important pressure to substitute SF6 by a more environmentally friendly breaking medium. Among the technologies that are considered, vacuum is a valuable candidate [START_REF] Godechot | Design and Tests of Vacuum Interrupters for High Voltage Circuit Breakers[END_REF].

Knowledge on high-current diffuse vacuum arc has made great progress for the last fifteen years thanks to a significant number of numerical studies. Most of these studies are based on relatively similar magnetohydrodynamic (MHD) descriptions of the diffuse plasma column using a two temperature representation of the plasma [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF], [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs in External Magnetic Fields[END_REF]- [START_REF] Wenzel | Combined Experimental and Theoretical Study of Constriction Threshold of Large-Gap AMF Vacuum Arcs[END_REF], [START_REF] Wang | 3D Numerical Simulation of High Current Vacuum Arc in Realistic Magnetic Fields Considering Anode Evaporation[END_REF], [START_REF] Han | Variation of Plasma Parameters of Vacuum Arc Column with Gap Distance[END_REF]. A large number of these works were inspired by the model presented by Schade and Shmelev in 2003 [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF]. These authors developped a 2D axisymmetric model considering energy transfers (including radiation energy) and the actions of the self-magnetic field and an externally imposed AMF. Both supersonic (at low arc current) and subsonic (at high arc current) plasma flow conditions were accounted for. In [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs in External Magnetic Fields[END_REF], the emission of metal vapor at the anode surface and its interaction with the plasma created at the cathode were included, revealing that the metal vapor was fully ionized very close to the anode. Hartmann et al. [START_REF] Hartmann | Development of a FEM Simulation of Axial Magnetic Field Vacuum Arcs[END_REF] were the first to attempt the modelling of the diffuse plasma column based on a 3D geometry, yet the anode was treated as a passive collector of particles. Other significant works include the series of models reported by Wang et al. and applied first to the study of the influence of various AMF radial profiles on the behaviour of a supersonic arc [START_REF] Wang | Numerical Simulation of Vacuum Arc under Different Axial Magnetic Fields[END_REF]. Later, their model was extended to account for a 3D geometry [START_REF] Wang | Three-Dimensional Model and Simulation of Vacuum Arcs under Axial Magnetic Fields[END_REF] and the effects of a realistic AMF profile was in particular investigated [START_REF] Wang | Three-Dimensional Time-Dependent Model and Simulation of High-Current Vacuum Arc in Commercial Axial Magnetic Field Vacuum Interrupter[END_REF]. Recently, their work was combined with a thermal model of the anode to consider particles emitted at the anode [START_REF] Wang | 3D Numerical Simulation of High Current Vacuum Arc in Realistic Magnetic Fields Considering Anode Evaporation[END_REF]. One should mention also the work of Langlois et al. [START_REF] Langlois | On the Numerical Simulation of the Diffuse Arc in a Vacuum Interrupter[END_REF] having strong similarities with the model reported in [START_REF] Wang | Numerical Simulation of Vacuum Arc under Different Axial Magnetic Fields[END_REF] and which was used to investigate in detail the various components of the energy flux density transferred to the anode for both supersonic and subsonic flow conditions. Finally, using a significantly different approach from all above works, Londer and Ulyanov [START_REF] Londer | Mathematical Model of the Vacuum Arc in an External Axial Magnetic Field[END_REF] have developed a semianalytical model of the VA.

Until recently, most models were applied to simulate the case of a short contact gap of 10 mm or less, representative of the conditions in a Medium-Voltage CB [START_REF] Zhang | Axial Magnetic Field Strength Needed for a 126-kV Single-Break Vacuum Circuit Breaker during Asymmetrical Current Switching[END_REF]. For High-Voltage levels, the contact gap can rise up to 50 mm [START_REF] Zhang | Axial Magnetic Field Strength Needed for a 126-kV Single-Break Vacuum Circuit Breaker during Asymmetrical Current Switching[END_REF]. The behavior of VA in this range of contact gap is still not well understood and the numerical studies of VA in High-Voltage CB stay rare. To the best of our knowledge, only two of such studies are available in the literature. In [START_REF] Wenzel | Combined Experimental and Theoretical Study of Constriction Threshold of Large-Gap AMF Vacuum Arcs[END_REF], the constriction of VA in a large contact gap (40 mm) under various AMF radial profiles was simulated using a 3D model. More recently, Han et al. [START_REF] Han | Variation of Plasma Parameters of Vacuum Arc Column with Gap Distance[END_REF] have performed, on the basis of a 2D model, a detailed study of the influence of a large contact gap (which was varied in the 10-50 mm range) on the behavior of the arc column. In this work, the radial and axial components of the self-magnetic field were neglected and only the case of a subsonic VA was treated.

In this paper, a classical 2D and two temperature MHD model, in which the three components of the selfmagnetic field are accounted for, is used to analyze the effects produced by a large contact gap (10-40 mm) on the constriction of a high-current diffuse VA between copper contacts, considering both supersonic and subsonic plasma flow conditions. We first describe the numerical model, including the model assumptions and the definition of the boundary conditions. Next, simulation results focusing in particular on the evolution of the current density in the interelectrode region are given under supersonic and subsonic conditions and the different phenomena governing the current constriction under each condition are discussed.

II. Magnetohydrodynamic Model

A diffuse VA is a complex multi-scale phenomenon, which can be divided into different regions (Fig. 1) [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF]. The plasma is emitted at the cathode surface by small size and very mobile cathode spots. The plasma jets produced by the spots mix near the cathode to form a continuous plasma column. The next region is the hydrodynamic region, inside which the plasma flows in direction to the anode. Finally, before being collected by the anode, the plasma crosses a near anode space charge sheath. In this paper, only the hydrodynamic region is treated with a MHD approach. Near cathode and near anode regions are considered as very thin compared to the hydrodynamic region and the phenomena taking place in these regions are accounted for through the boundary conditions of the model.

Based on the experimental observations regarding VAs controlled by a strong AMF at large contact gaps reported in [START_REF] Tezenas Du Montcel | Arc Appearance and Cathode Spot Distribution in a Long Gap High-Current Vacuum Arc Controlled by an External Axial Magnetic Field[END_REF], the model developed in this paper uses as a first approximation a 2D-cylindrical geometry. As shown in Fig. 2, the computational domain consists of three subdomains: the electrodes (cathode and anode), the interelectrode region enclosing the plasma and a lateral vacuum region separated from the interelectrode region by a impermeable fictitious boundary. The consideration of a lateral vacuum region aims at moving away the external boundaries of the domain in order to facilitate the definition of the boundary conditions on these boundaries. The electromagnetic variables are computed in the three regions, whereas the hydrodynamic variables are computed only in the interelectrode region. The model was implemented in the numerical simulation software Comsol-Multiphysics®.

A. Physical model

The plasma is represented as a mixture of two fluids, namely copper ions and electrons with respective masses m i and m e , whose behavior is modeled using the conservation equations established by Braginskii [START_REF] Braginskii | Transport Processes in Plasma[END_REF] coupled with the Maxwell equations. This group of equations can be simplified with the aid of various classical approximations:

• For an Alternative-Current with a 50 Hz frequency, the time scale of the variation of the current is very large compared to the time scales associated to the plasma flow. Consequently, the plasma may be described using a steady state approach.

• Because the Debye length is very small in front of the typical length scale of the system [START_REF] Langlois | On the Numerical Simulation of the Diffuse Arc in a Vacuum Interrupter[END_REF], the plasma may be considered as quasi-neutral.

• Ionization and recombination collisions in the hydrodynamic region are negligible [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF]. Hence the average ion charge is considered as a fixed value (Z = 1.85 for copper electrodes [START_REF] Kutzner | Integrated Ion Flux Emitted from the Cathode Spot Region of a Diffuse Vacuum Arc[END_REF]).

• Electron inertia is neglected.

• Both ions and electrons are assumed to be unmagnetized particles and are treated as ideal gases.

• In a diffuse VA, the radiative dissipation can be neglected [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF].

• The emission of vapor particles at the anode is not considered. Indeed, the simulation results presented in [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF] have shown that the anode starts to become active only for very high current density (> 1300 A/cm 2 , which corresponds to a current intensity of about 16 kA for 20 mm radius electrodes). Note however that for some anode materials with a low melting point and high vapor pressure (e.g. brass) the anode emits vapor particles even at low current densities [START_REF] Zolotukhin | Anode Ablation and Performance Improvement of Microcathode Arc Thruster[END_REF]. Based on these approximations and some manipulations of the Braginskii and Maxwell equations, a new set of equations may be established. This new set of equations and the variables that may be deduced from each equation are detailed below.

• The ion density (n i ), velocity (⃗ u i ) and temperature (T i ) may be obtained from the equations of conservation of particles, momentum and energy relative to the ion flow.

                 ⃗ ∇.n i ⃗ u i = 0 (1a) m i n i ( ⃗ u i . ⃗ ∇ ) ⃗ u i = -⃗ ∇ (p i + p e ) -⃗ ∇. πi + ⃗ J × ⃗ B (1b) 5 2 n i ⃗ u i . ⃗ ∇k b T i -⃗ u i . ⃗ ∇p i = ⃗ ∇.k i ⃗ ∇T i -πi : ⃗ ∇⃗ u i + Q i (1c)
• The density of electrons (n e ) can be calculated from the relation of electroneutrality, their velocity (⃗ u e ) can be deduced from the current density definition and their temperature (T e ) is given by the equation of conservation of the electron energy.

                   n e = Zn i (2a) ⃗ u e = ⃗ u i - ⃗ J en e (2b) 5 2 n e ⃗ u e . ⃗ ∇k b T e -⃗ u e . ⃗ ∇p e = ⃗ ∇.k e ⃗ ∇T e + ⃗ J 2 σ -Q i (2c)
• The current density ⃗ J can be calculated from the generalized Ohm law, which is deduced by reformulating the equation of the conservation of the electron momentum.

⃗ J = σ [ -⃗ ∇V + ⃗ u e × ⃗ B + ⃗ ∇p e en e + 0.9 k b e ⃗ ∇T e ] (3)
• The electric and magnetic fields ( ⃗ E, ⃗ B) are calculated from the electric and magnetic vector potentials (V , ⃗ A). The latters can be obtained in the plasma from the following equations derived from the combination of Eq. ( 3) with respectively the equation of conservation of the current and the equation of Maxwell-Ampere:

                       ⃗ ∇.σ ⃗ ∇V = ⃗ ∇.σ ( ⃗ ∇p e en e + 0.9 k b e ⃗ ∇T e ) + ⃗ ∇.σ ( ⃗ u e × ⃗ B ) (4a) 1 µ 0 σ ⃗ ∆ ⃗ A = -⃗ ∇V + ⃗ u e × ⃗ B + ⃗ ∇p e en e + 0.9 k b e ⃗ ∇T e (4b)
• In the electrode and vacuum regions, V and ⃗ A are derived from the following equations:

   ⃗ ∇.σ ⃗ ∇V = 0 (5a) 1 µ 0 σ ⃗ ∆ ⃗ A = -⃗ ∇V (5b)
In this set of equations, p i and p e are the ionic and electronic pressures, respectively; πi is the ion viscous stress tensor; k i and k e are the ionic and electronic thermal conductivities, respectively; Q i is the energy received by the ions during collisions with electrons and σ is the electrical conductivity. k b is the Boltzmann constant, e is the elementary positive charge and µ 0 is the magnetic permeability of the vacuum. πi , k i , k e , Q i and σ in the plasma region are respectively given by:

πi = 0.96n i k b T i τ i [ ( ⃗ ∇⃗ u i + ⃗ ∇⃗ u † i ) - 2 3 ⃗ ∇.⃗ u i Īd ] ( 6
)
Q i = 3 m e m i n e τ e k b (T e -T i ) (7)        k i = 3.9 n e k 2 b T i τ i m i (8a) k e = 4.9 n e k 2 b T e τ e m e (8b) σ = 2.27e 2 n e τ e m e ( 9
)
Īd is the identity tensor and τ e and τ i are the collisional relaxation times of electrons and ions:

             τ e = 12π 3/2 √ 2 ε 2 0 √ m e (k b T e )
2 n e e 4 Z ln Λ (10a)

τ i = 12π 3/2 √ 2 ε 2 0 √ m i (k b T i ) 2 n e e 4 Z 4 ln Λ (10b)
with ε 0 the electric permittivity and ln Λ the Coulomb logarithm. Numerical values in Eq. ( 3), ( 4), ( 6), ( 8) and ( 9) are parameters which appear in the expressions of the transport coefficients derived by Braginskii [START_REF] Braginskii | Transport Processes in Plasma[END_REF]. They are dependent on the ion charge. The values used here correspond to the case Z=2.

B. Boundary conditions a) Electromagnetic variables: On all external boundaries of the computational domain, the magnetic vector potential is fixed as:

⃗ A × ⃗ n = ⃗ 0 (11)
with ⃗ n the normal vector to the external boundary.

On the external boundary of the cathode, the normal current density, which is assumed uniform, is specified by the condition:

⃗ ∇V.⃗ n = - I arc πR 2 (12)
where I arc is the arc current and R is the radius of the contacts. The voltage on the external boundary of the anode is fixed to ground (0 V). On the external boundaries of the vacuum region, the condition that no current flows through these boundaries takes the form:

⃗ ∇V.⃗ n = 0 (13)
On the boundary between the cathode and the VA, the profile of the normal current density J c is imposed to take into account the cathode spot distribution upon the cathode surface. In this paper, the profile of the normal current density on the cathode surface is considered uniform, like on the cathode external boundary (Eq. ( 12)).

On the boundary between the anode and the VA, the normal current density J a is imposed to take into account the influence of the anode sheath [START_REF] Wieckert | Theoretical Analysis of the Current and Energy Flow to the Anode in the Diffuse Vacuum Arc[END_REF]: [START_REF] Wang | 3D Numerical Simulation of High Current Vacuum Arc in Realistic Magnetic Fields Considering Anode Evaporation[END_REF] with J th the electron thermal current given by:

J a = J th e - eφ sh k b Te + Zen i u iz
J th = - 1 4 en e √ 8 π k b T e m e ( 15
)
and φ sh the voltage drop in the anode sheath. Considering that the anode surface is equipotential (V = V 0 ), the anode sheath voltage drop is related to V 0 and to the electric potential at the boundary between the hydrodynamic region and the anode sheath V a through the relation

φ sh = V 0 -V a .
The potential V 0 is a constant that may be computed by equating the total electron current emitted at the cathode to the total electron current collected at the anode. The boundaries between the vacuum region and respectively the plasma and the electrodes are treated as impermeable to the electric current. b) Hydrodynamic variables: Boundary conditions for supersonic (at low current) and subsonic (at high current) flow conditions have to be treated separately. Transition between supersonic and subsonic conditions happens for a Mach number M a = 1, with M a given by:

M a = u iz √ k b m i (γT i + T e ) ( 16
)
where γ is the adiabatic index. From the data gathered by Schade and Shmelev [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF], the critical current density separating supersonic and subsonic flow conditions is about 300 A/cm 2 , which corresponds to a current intensity of about 3800 A for 20 mm radius electrodes.

The conditions specified on the boundary between the cathode and the plasma are as follows. For supersonic conditions, three variables relative to the ion flow are imposed, namely the ion temperature, the ion velocity and the ion pressure, which may be expressed as a function of the ion erosion rate of the cathode µ er as follows:

p i = - µ er J c m i u iz k b T i ( 17
)
For subsonic conditions, only two variables relative to the ion flow need to be imposed. The variables retained in the present model are the ion temperature and the ion mass flow rate, which is given by:

m i n i u iz = -µ er J c ( 18
)
Regarding the flow of electrons, it is necessary to specify the electron temperature.

Conditions specified on the boundary between the anode and the plasma are as follows. Boundary conditions relative to the ion flow are required only in the case of subsonic conditions. Our model uses a condition on the ion pressure. This condition is derived from global mass and energy balances across an intermediate region, a presheath, between the hydrodynamic region and the anode sheath, assuming that the Bohm criterion [START_REF] Riemann | The Bohm Criterion and Sheath Formation[END_REF] is marginally fullfilled at the entrance of the anode sheath.

Regarding the flow of electrons, a Fourier condition for the electron energy conservation equation is obtained by considering a global energy balance through the near anode region (presheath and sheath) [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF]. This condition takes the form:

k e ∂T e ∂z = n e u ez ( k b T e 2 + eφ sh ) -0.9 k b T e e J a (19)
Finally, the lateral boundary of the plasma region is treated as an adiabatic and stress free wall.

III. Supersonic arc

In this section, the behavior of a supersonic diffuse arc between two contacts of 20 mm radius with a current I arc = 1 kA and controlled by a uniform AMF of strength B AM F = 5 mT is simulated for a range of gap lengths varying from 10 mm to 40 mm. The simulations have been performed setting at the boundary between the cathode and the plasma the ion temperature to 0.2 eV [START_REF] Beilis | Calculation of the Parameters of the Cathode Stream of an Arc Discharge[END_REF], the ion velocity (assumed to be purely axial) to 10 4 m/s [START_REF] Kutzner | Integrated Ion Flux Emitted from the Cathode Spot Region of a Diffuse Vacuum Arc[END_REF], the ion erosion rate to 50 µg.C -1 [START_REF] Daalder | Components of Cathode Erosion in Vacuum Arcs[END_REF] and the electron temperature to 1.5 eV [START_REF] Lyubimov | The Cathode Spot of a Vacuum Arc[END_REF].

A. Results

The norm and the streamlines of the projection of the current density vector onto the (r,z) plane for 10 mm and 40 mm gaps are shown in Fig. 3. For 10 mm, the current experiences an important constriction in front of the cathode, which gets stronger as the plasma flows from cathode to anode. In contrast, for 40 mm, the current constriction remains weak until a distance of about 20 mm from the cathode. Beyond this distance, it strongly increases in direction to the anode. Note that for both gap lengths a reduction of the current constriction is observed very close (within 5 mm) to the anode.

τ c (z) = - 2π I arc ∫ R/ √ 2 0 rJ z (r, z)dr (20)
Fig. 4 shows the axial profile of τ c for different gap lengths. At the cathode boundary, the value of τ c is 0.5, since the area of the disk with a R/ √ 2 radius represents half of that of the contact surface and the distribution of the current density is considered to be uniform. For a 10 mm gap, τ c increases almost linearly and reaches 0.73 at the anode boundary. With the increase of L arc , three phases in the evolution of τ c can be progressively differentiated. Close to the cathode, the growth of the constriction rate is weak. Next, the growth gets strong. Finally, in the near anode area the growth of τ c becomes weak again. For gap lengths greater than 30 mm, the constriction rate even begins to decrease in the immediate vicinity of the anode. For a 40 mm gap, τ c reaches a maximum value of 0.77 and then decreases to 0.75 at the anode boundary. For a gap of 10 mm, the axial current density is roughly uniform (about 2J c ) from the center of the anode to about r ∼ 5 mm. For r > 5 mm, it decreases monotonically to a value close to 0.3J c at the edge of the anode. With the increase of the gap length, the profile of the axial current density does not evolve monotonically. When the gap varies from 10 mm to 20 mm, the axial current density near the axis of symmetry increases significantly (reaching about 2.8J c ), whereas it slightly decreases between 20 mm and 40 mm. Fig. 5b shows the radial profile of the energy flux density transferred to the anode P a for different gap lengths, which has been calculated according to the relations detailed in [START_REF] Langlois | On the Numerical Simulation of the Diffuse Arc in a Vacuum Interrupter[END_REF]. The profile of P a strongly depends on the axial current density profile and exhibits thus similar variations as those observed in Fig. 5a. In particular, the energy flux density at the anode centre increases (of about 34%) when the gap length goes from 10 mm to 20 mm, whereas it weakly decreases when the gap length is further increased from 20 mm to 40 mm.

B. Discussion

In order to understand the evolution of the constriction rate for large gaps shown in Fig. 4, the axial variations at a distance of R/ √ 2 from the axis of symmetry of the different terms contributing to the radial current density are represented in Fig. 6, taking as an example the case of 40 mm gap length. Based on the radial projection of the generalized Ohm law (Eq. ( 3)), the four terms contributing to the radial current density are:

• a term depending on the radial electric field: σE r , • a term resulting from Lorentz forces induced by the azimuthal magnetic field: -σu ez B φ .

• a term resulting from Lorentz forces related to the AMF: σu eφ B z ,

• a term depending on the electron pressure and temperature radial gradients: One can see from Fig. 6 that the current constriction is almost exclusively caused by the term resulting from Lorentz forces induced by the azimuthal magnetic field (called hereafter the constriction term). The three other terms have, except very close to the anode for the term related to the electric field, an opposing effect restraining the constriction. The three phases of the axial evolution of the constriction rate identified in Fig. 4 may be interpreted as follows. Close to the cathode, only the constriction term and the term related to the electric field play a significant role. The term related to the electric field almost totally offsets the constriction term, resulting in a weak growth of the constriction rate in this region. From z ∼ 11 mm, the reduction of the electric field related term is neutralized by the increase of the term related to electron compressibility. The next phase, characterized by a quick growth of the constriction rate, is due to the important increase of the constriction term, such increase being only partially offset by the rise of the term related to electron compressibility. Finally, close to the anode, the slowing down of the growth of the constriction rate followed by the reduction of the constriction rate is caused by the action of the term related to the AMF, which becomes more and more important when approaching the anode.

In order to interpret some of the results observed in Fig. 6, it is of interest to analyze the distribution in the interelectrode region of the ion density (which is proportional to the electron density given the electroneutrality assumption) and of the streamlines related to the ion flow, which are both shown on Fig. 7. One can see that the plasma is radially confined in a large region around the axis of symmetry, starting at a distance of around 10 mm from the cathode and ending at a distance of around 5 mm from the anode. On the other hand, the plasma undergoes an expansion near the lateral boundary and close to the anode surface near the axis of symmetry. This figure allows to understand the rapid increase of the constriction term starting from a distance of around 15 mm from the cathode. This effect results from the rapid increase of the electron axial velocity in the constriction term caused by the expansion phenomenon. Indeed, considering the vertical plane located at a radial distance R/ √ 2 ∼ 14 mm, the distance of around 15 mm from the cathode coincides roughly with the transfer of the electrons from the compression region to the expansion region. Since the flow of electrons is mostly axial, the variation of their axial velocity is proportional as a first approximation to 1/n e , hence the important increase of the electron axial velocity in the expansion region.

Close to the anode, the term related to the AMF, which limits the constriction, becomes increasingly important. It is of interest to notice that the AMF effect is related to the current constriction process. The electron azimuthal velocity, which comes into play in the term related to the AMF effect (σu eφ B z ), is essentially controlled by the azimuthal current density, as this velocity is defined by u eφ = -J φ /en e + u iφ and the ion azimuthal velocity remains small compared to the electron azimuthal velocity. The azimuthal current density can be expressed from Ohm law by:

J φ = σ (u ez B r -u er B z ) (21)
Given the cylindrical symmetry, the radial component of the self-magnetic field is governed only by the azimuthal current density (this results from the azimuthal projection of the Maxwell-Ampere equation). As a consequence, under the presence of an AMF the azimuthal current density (and hence the electron azimuthal velocity) can only start developing under the effect of the second term in Eq. ( 21). This term is related, through the electron radial velocity and given the fact that the current is mainly carried by electrons, to the evolution of the radial current density (and thus to current constriction). Therefore, the existence of the term related to the AMF is correlated to the current constriction process. Note that very close to the anode, where the constriction rate decreases, the increase of the Lorentz term related to the AMF can not be governed by the second term in Eq. [START_REF] Wieckert | Theoretical Analysis of the Current and Energy Flow to the Anode in the Diffuse Vacuum Arc[END_REF]. It results from the first term of Eq. ( 21), whose existence is a consequence of the consideration of the radial component of the selfmagnetic field.

IV. Subsonic arc

In this section, the behavior of a subsonic diffuse arc between two contacts of 20 mm radius with a current I arc = 9 kA and controlled by a uniform AMF of strength B AM F = 45 mT is simulated for a range of gap lengths varying from 10 mm to 40 mm. The simulations have been performed setting at the boundary between the cathode and the plasma the ion temperature to 3 eV [START_REF] Schade | Numerical Simulation of High-Current Vacuum Arcs With an External Axial Magnetic Field[END_REF], the ion erosion rate to 50 µg.C -1 [START_REF] Daalder | Components of Cathode Erosion in Vacuum Arcs[END_REF] and the electron temperature to 1.5 eV [START_REF] Lyubimov | The Cathode Spot of a Vacuum Arc[END_REF].

A. Results

The norm and the streamlines of the projection of the current density vector onto the (r,z) plane for 10 mm and 40 mm gaps are shown in Fig. 8. For 10 mm, like in Fig. 3a, the current starts to constrict in front of the cathode and the constriction becomes more intense in direction to the anode. However, contrary to the case of supersonic conditions, the constriction does not weaken when getting very close to the anode. For 40 mm, in a significant part of the interelectrode region (up to z ∼ 25 mm) no current constriction is observed as the current density does not vary much. A constriction of the current is observed only from a distance from the cathode of 25 mm. The constriction then continuously increases up to the anode. 4 for supersonic conditions, τ c increases from the cathode to the anode. Its growth rate tends to get larger in direction to the anode. At the anode boundary the constriction rate reaches 0.73. For a gap in the 20-40 mm range, the evolution of the constriction rate differs significantly from that observed under supersonic conditions. It can be divided into two phases. During the flow of the plasma from the cathode to the anode, τ c first slowly decreases. During this phase, the constriction rate evolution is not depending on the gap length. Next, the constriction rate grows and reaches 0.75 at the anode whatever the gap length value. This second phase starts at a distance of about 15 mm from the anode irrespective of the gap length value. The radial profile of the axial current density on the anode surface for different gap lengths is shown in Fig. 10a. For a contact gap of 10 mm, the axial current density is maximal at the anode center (about 2J c ) and decreases towards the edge of the anode to a value close to 0.3J c . The increase of the gap length from 10 mm to 20 mm causes a significant increase (of 35%) of the axial current density at the center of the anode. On the other hand, when the gap length varies from 20 mm to 40 mm, the profile of the axial current density does not evolve significantly. Similarly to the case of a supersonic arc, the evolution with the gap length of the radial profile of the energy flux density transferred to the anode (Fig. 10b) is close to that of the radial profile of the axial current density collected at the anode surface, since the energy flux density transferred to the anode is strongly correlated to the current density. In particular, the increase of the gap length from 10 mm to 20 mm leads to a moderate increase (about 17%) of the maximum value of the energy flux density, which is reached at the anode centre. On the other hand, the variation of the gap length from 20 mm to 40 mm has no effect on the profile of the energy flux density.

B. Discussion

Following the same approach as for supersonic conditions, Fig. 11 shows for subsonic conditions the axial variations at a distance of R/ √ 2 from the axis of symmetry of the different terms contributing to the radial current density, taking as an example the case of 40 mm gap length. Note that, similarly to the supersonic conditions, the current constriction is exclusively caused by the term resulting from Lorentz forces induced by the azimuthal magnetic field (called hereafter the constriction term). Based on this figure, the two phases of the axial evolution of the constriction rate identified in Fig. 9 may be interpreted as follows. The first phase, taking place from the cathode surface up to a distance from the anode of around 15 mm, was characterized by a slow decrease of the constriction rate. During this phase, the progresive growth of the constriction term is counterbalanced by the increase of the terms related to the electric field and to electron compressibility. The sum of these two latter terms slightly exceeds the constriction term, resulting in the slow decrease of the constriction rate. Moreover, during this phase the term related to the AMF remains negligible. The second phase, taking place close to the anode (from a distance from the anode of around 15 mm), was characterized by a rapid increase of the constriction rate. This is related to the rapid increase of the constriction term, whereas the terms related to the electric field and to electron compressibility become less important. During this phase, the term related to the AMF, which acts against the constriction, comes into play and becomes more and more important. Yet, it is not sufficient to counterbalance the growth of the constriction term.

Our conclusion about the absence of an appreciable evolution of the current constriction at the anode surface for gap lengths greater than 20 mm is in conflict with Han et al. simulation results, that show a monotonic increase of the constriction with the contact gap [START_REF] Han | Variation of Plasma Parameters of Vacuum Arc Column with Gap Distance[END_REF]. The model employed by Han et al. is similar to ours. Yet various differences between both models can be identified. Han model does not consider the radial and axial components of the self-magnetic field and uses a slightly different formulation of the anode boundary condition for the ion pressure. Moreover, there are also significant differences in the simulation parameters considered in both studies. In Han et al. simulations, the ion temperature at the cathode boundary is much larger (by more than a factor 3), the contact radius is 50 % greater and the current density and the strength of the AMF with respect to the arc current are respectively 26 % and 20 % smaller. The precise reason of the different influence of the gap length on current constriction at the anode predicted by both models remains unclear. One may speculate that the current density and/or the AMF strength in [START_REF] Han | Variation of Plasma Parameters of Vacuum Arc Column with Gap Distance[END_REF] may be too small to lead to the saturation of the current constriction at the anode surface.

V. Conclusion

Current constriction in a diffuse vacuum arc discharge controlled by an AMF for large gap lengths (up to 40 mm) was studied numerically using a two temperature MHD model for both supersonic and subsonic flow conditions.

For both flow conditions, current constriction at the anode surface increases significantly when the gap length grows from 10 mm to 20 mm. When the gap is further increased, the constriction slightly decreases for a supersonic VA, while it no longer evolves for a subsonic VA. The mechanisms responsible for the current constriction are different according to the flow conditions. In a supersonic VA, the constriction starts developing from the cathode surface and increases in the interelectrode region except close to the anode. The increase of the current constriction is partly correlated to the radial confinement of the plasma caused by Lorentz forces. In a subsonic VA, the constriction starts developing only close to the anode (at a distance of around 15 mm) under the influence of the anode sheath. In the rest of the interelectrode region, the current density is subjected to only little variation. For a short gap (10 mm), the influence of the anode sheath is already effective at the cathode surface and the constriction happens immediately.

The increase of the current constriction at the anode surface when varying the gap length from 10 mm to 20 mm causes an increase of the energy flux density from the plasma to the center of the anode, but this increase remains moderate (respectively 34 % and 17 % for supersonic and subsonic VAs). For larger contact gaps, the variation of the gap length has no appreciable influence on the profile of the energy flux density from the plasma to the anode. This suggests that the risk of excessive heating of the anode and of anode spot development is not significantly increased for arcs with very large gaps.

The present study was concerned with vacuum arcs controlled by an homogeneous AMF. However, it has been reported that for large gaps the AMF induced by the contacts may be weaker around the center of the interelectrode region, causing instabilities in the VA behaviour [START_REF] Cheng | Study on High-Current Vacuum Arc Characteristics Under Self-Generated Axial Magnetic Field of Contact at a Long Contact Gap for High-Voltage Vacuum Interrupters[END_REF]. Arc shape may change fastly and become not axisymmetric. Consequently, future studies on VA at large contact gaps will need to take into account 3D geometry effects and unsteady phenomena.

Fig. 1 :

 1 Fig. 1: Sketch of the diffuse VA (not to scale).

Fig. 2 :

 2 Fig. 2: Computational domain.

Fig. 3 :

 3 Fig. 3: Distribution of the norm and streamlines of the projection of the current density vector onto the (r,z) plane for (a) L arc = 10 mm and (b) L arc = 40 mm (supersonic arc, I arc = 1 kA and B AM F = 5 mT). The coordinate z=0 corresponds to the cathode surface. . In the following, let us define the constriction rate τ c as the fraction of the current passing through a disk with a R/ √ 2 radius:

Fig. 4 :Fig. 5 :

 45 Fig. 4: Axial profile of the constriction rate τ c for different gap lengths (supersonic arc, I arc = 1 kA and B AM F = 5 mT).

Fig. 6 :

 6 Fig. 6: Axial variations at a distance of R/ √ 2 from the axis of symmetry of the different terms contributing to the radial current density for L arc = 40 mm (supersonic arc, I arc = 1 kA and B AM F = 5 mT).

σ e ( 1 ne

 1 ∂ r p e + 0.9k b ∂ r T e ). The main component of this term is the first one, related to electron compressibility. The second one has only a corrective effect. Note that the four terms are proportional to the electrical conductivity. Although the electrical conductivity (which is a function of only the electron temperature) increases in the axial direction, this evolution does not influcence the sign of ∂τ c /∂z.

Fig. 7 :

 7 Fig. 7: Distribution of ion density n i and streamlines of the ion flow for L arc = 40 mm (supersonic arc, I arc = 1 kA and B AM F = 5 mT). The coordinate z=0 corresponds to the cathode surface.

Fig. 8 :

 8 Fig. 8: Distribution of the norm and streamlines of the projection of the current density vector onto the (r,z) plane for (a) L arc = 10 mm and (b) L arc = 40 mm (subsonic arc, I arc = 9 kA and B AM F = 45 mT). The coordinate z=0 corresponds to the cathode surface.

Fig. 9

 9 Fig.9shows the axial profile of the constriction rate τ c computed for different gap lengths. Like in Fig.4, τ c = 0.5 at the cathode boundary because of the uniform

Fig. 9 :

 9 Fig. 9: Axial profile of the constriction rate τ c for different gap lengths (subsonic arc, I arc = 9 kA and B AM F = 45 mT).

Fig. 10 :

 10 Fig. 10: Radial profile on the anode surface of (a) the axial current density normalized by the cathode current density and (b) the energy flux density transferred to the anode computed for different gap lengths (subsonic arc, I arc = 9 kA and B AM F = 45 mT).

Fig. 11 :

 11 Fig. 11: Axial variations at a distance of R/ √ 2 from the axis of symmetry of the different terms contributing to the radial current density for L arc = 40 mm (subsonic arc, I arc = 9 kA and B AM F = 45 mT).

Acknowledgments

This project was supported by the French Government in the framework of the "Investments for the Future" programme under the project reference (ANE-ITE-002-01).