Nathalie André-Obadia

Hasan Hodaj

Enkelejda Hodaj

Emile Simon

Chantal Delon-Martin

Luis Garcia-Larrea

Better Fields or Currents? A Head-to-Head Comparison of Transcranial Magnetic (rTMS) Versus Direct Current Stimulation (tDCS) for Neuropathic Pain

Keywords: rTMS, tDCS, Neuropathic pain, fMRI, Non-invasive stimulation NP Neuropathic pain HF-rTMS High-frequency

While high-frequency transcranial magnetic stimulation (HF-rTMS) is now included in the armamentarium to treat chronic neuropathic pain (NP), direct-current anodal stimulation (a-tDCS) to the same cortical targets may represent a valuable alternative in terms of feasibility and cost. Here we performed a head-to-head, randomized, single-blinded, cross-over comparison of HF-rTMS versus a-tDCS over the motor cortex in 56 patients with drug-resistant NP, who received 5 daily sessions of each procedure, with a washout of at least 4 weeks. Daily scores of pain, sleep, and fatigue were obtained during 5 consecutive weeks, and functional magnetic resonance imaging (fMRI) to a motor task was performed in a subgroup of 31 patients. The percentage of responders, defined by a reduction in pain scores of > 2 SDs from pre-stimulus levels, was similar to both techniques (42.0% vs. 42.3%), while the magnitude of "best pain relief" was significantly skewed towards rTMS. Mean pain ratings in responders decreased by 32.6% (rTMS) and 29.6% (tDCS), with half of them being sensitive to only one technique. Movement-related fMRI showed significant activations in motor and premotor areas, which did not change after 5 days of stimulation, and did not discriminate responders from non-responders. Both HF-rTMS and a-tDCS showed efficacy at 1 month in drug-resistant NP, with magnitude of relief slightly favoring rTMS. Since a significant proportion of patients responded to one procedure only, both modalities should be tested before declaring a patient as unresponsive.

Introduction

Shortly after the description of the neurosurgical procedure of epidural motor cortex stimulation for neuropathic pain (NP) control [START_REF] Tsubokawa | Chronic motor cortex stimulation for the treatment of central pain[END_REF], repetitive transcranial magnetic stimulation (rTMS) was proposed as a non-invasive method to mimic epidural stimulation and predict its subsequent effectiveness. The potential value of rTMS as a pain therapy in its own right was soon recognized, and the use of rTMS as a full-fledged pain-relieving procedure has received considerable support in the last 10 years [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF]. Although methodological drawbacks limited the quality of evidence of early studies due to low patients' samples, absent blinding, lack of randomization and follow-up, etc. [START_REF] Cruccu | EAN guidelines on central neurostimulation therapy in chronic pain conditions[END_REF][START_REF] O'connell | Non-invasive brain stimulation techniques for chronic pain[END_REF], a number of well-conducted studies using single or double-blinded methodology, randomization, and inclusion of more than 20 patients in active groups have been recently reported in chronic NP of various origins, with positive results when using stimulus frequencies of at least 10 Hz [START_REF] Quesada | New procedure of highfrequency repetitive transcranial magnetic stimulation for central neuropathic pain: a placebo-controlled randomized crossover study[END_REF][START_REF] Hosomi | A randomized controlled trial of 5 daily sessions and continuous trial of 4 weekly sessions of repetitive transcranial magnetic stimulation for neuropathic pain[END_REF][START_REF] Zhao | Analgesic effects of directed repetitive transcranial magnetic stimulation in acute neuropathic pain after spinal cord injury[END_REF][START_REF] Zhao | Analgesic effects of navigated repetitive transcranial magnetic stimulation in patients with acute central poststroke pain[END_REF][START_REF] André-Obadia | Theta-burst versus 20 Hz repetitive transcranial magnetic stimulation in neuropathic pain: a head-to-head comparison[END_REF][START_REF] Mori | Exploratory study of optimal parameters of repetitive transcranial magnetic stimulation for neuropathic pain in the lower extremities[END_REF][START_REF] Attal | Repetitive transcranial magnetic stimulation for neuropathic pain: a randomized multicentre sham-controlled trial[END_REF]. Accordingly, recent reviews concluded to a significant superiority over placebo of high-frequency (HF) motor cortex rTMS in chronic neuropathic pain [START_REF] Anderson | Evidence brief: transcranial magnetic stimulation (TMS) for chronic pain, PTSD, TBI, opioid addiction, and sexual trauma[END_REF][START_REF] Gatzinsky | Repetitive transcranial magnetic stimulation of the primary motor cortex in management of chronic neuropathic pain: a systematic review[END_REF][START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS): An update (2014-2018)[END_REF][START_REF] Zhang | Analgesic effect of noninvasive brain stimulation for neuropathic pain patients: a systematic review[END_REF], and clinical recommendations have now included HF-rTMS of the motor cortex as a "third line" therapeutic option, at the same level as spinal cord stimulation [START_REF] Moisset | French guidelines for neuropathic pain: an update and commentary[END_REF]. In the same line, a recent report of the US Department of Veterans Affairs which analyzed rTMS data under a "best-evidence approach" (multisite studies, control of potential confounding factors) concluded that rTMS may reduce symptoms in NP and could be a treatment option for patients who have exhausted standard available options [START_REF] Anderson | Evidence brief: transcranial magnetic stimulation (TMS) for chronic pain, PTSD, TBI, opioid addiction, and sexual trauma[END_REF].

Transcranial direct current (galvanic) stimulation (tDCS), i.e., the non-invasive transcranial flow of electric charge that does not change direction, modulates the neuronal resting membrane state without eliciting action potentials, and has been empirically applied for medical purposes since the Roman Antiquity (Scribonious Largus ~70 AC, https:// prabo ok. com/ web/ scrib onius. largus/ 37276 51). Modern research showed that surface anodal polarization of the cortex increases spontaneous unit discharges in rodents and felines [START_REF] Creutzfeldt | Influence of transcortical d-c currents on cortical neuronal activity[END_REF][START_REF] Bindman | The action of brief polarizing currents on the cerebral cortex of the rat[END_REF] and enhances human motor cortical excitability with magnitude and duration comparable to those observed with rTMS [START_REF] Nitsche | Excitability changes induced in the human motor cortex by weak transcranial direct current stimulation[END_REF][START_REF] Brunoni | Clinical research with transcranial direct current stimulation (tDCS): challenges and future directions[END_REF]. Anodal tDCS appears therefore as a promising tool, able to emulate the analgesic effects of conventional motor cortex stimulation, with practical advantages over rTMS including its lower cost, the paucity of safety issues, and the availability of home-based long-lasting protocols. However, because of the limited quality of most published reports, the level of evidence regarding tDCS effects in chronic neuropathic pain remains very low and highly conflicting, despite a large number of studies published [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF][START_REF] Knotkova | Neuromodulation for chronic pain[END_REF][START_REF] Yang | Transcranial direct current stimulation for the management of neuropathic pain: a narrative review[END_REF].

One single study comparing the short-term effect of 3 sessions of anodal tDCS versus HF-rTMS in lumbosacral radiculopathy concluded to the superiority of rTMS [START_REF] Attal | Repetitive transcranial magnetic stimulation and transcranial direct-current stimulation in neuropathic pain due to radiculopathy: a randomized sham-controlled comparative study[END_REF], while a very recent report in 12 patients with brachial plexus injuries found similar results from both techniques [START_REF] De Assis | Effects of rTMS and tDCS on neuropathic pain after brachial plexus injury: a randomized placebo-controlled pilot study[END_REF]. Head-tohead studies directly comparing the efficacy of HF-rTMS and a-tDCS for chronic, drug-resistant NP in large patients' series are therefore warranted. In the present study, we report the results of a full head-to-head, randomized, prospective, single-blinded, cross-over study comparing HF-rTMS versus anodal tDCS over the motor cortex in a large series of patients with drug-resistant NP of different etiologies. To make the results directly comparable and maximize their clinical significance, each patient could benefit consecutively from the two techniques, separated by an adequate wash out period, and daily quotations of pain and pain-related items were obtained from written diaries during the full follow-up. In addition, functional magnetic resonance imaging (fMRI) during a motor task involving the painful area was obtained before and after the procedure in a subset of patients, to investigate the possible relations between clinical efficacy of the neurostimulation techniques and changes in the activity level of task-related motor networks.

Patients and Methods

Study Population

This bi-centric protocol was conducted in the Neurological Hospital of the Hospices Civils de Lyon, France, and in the Pain Center of the Grenoble Alpes University Hospital, France, from February 2013 to December 2020. The study was approved in both centers by the Institutional Review Boards Sud-Est IV Lyon (N° 10,619) and Sud-Est V Grenoble (N° 6705), France, and was registered with clinicaltrials. gov (NCT02120326, NCT02854332). All patients approved and signed an informed consent prior to entering the protocol. Equipment, stimulation protocol, and pain evaluation methods were identical in both sites, with the exception of the diameter of tDCS electrodes ( 6.2 cm soaked sponges in Grenoble, 1 cm Gel electrodes in Lyon, both by the same manufacturer Neuroelectrics ® , both validated in terms of safety and without difference of effectiveness) ( [START_REF] Ruffini | tDCS clinical research -highlights: safety of transcranial Current Stimulation[END_REF] and see results "Primary Outcome").

Sixty-eight patients aged 18 to 80 years, suffering from lateralized pharmaco-resistant chronic neuropathic pain for more than 1 year, without any change in medical treatment since at least 1 month, were included in this study. Mean pain duration was 5 ± 3.8 years. Diagnosis of neuropathic pain followed IASP NeuPSIG guidelines [START_REF] Jensen | A new definition of neuropathic pain[END_REF], and a level of probable to definite neuropathic pain was required for

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inclusion [START_REF] Finnerup | Neuropathic pain: an updated grading system for research and clinical practice[END_REF]. The patients were classified into five groups according to the origin of pain: central post-stroke pain, central cancer and vascular pain, spinal cord injury, facial pain, and brachial plexus injury (Table 1). They were not included if they had a history of epilepsy, drug-addiction, migraine, intracranial ferromagnetic material, or implanted stimulator.

All patients (except one who had discontinued all drugs due to inefficacy before entering the study) were taking one or more analgesic treatments (anti-epileptic drugs, antidepressants, and/or painkillers levels 1, 2, or 3) (Table 1).

Patients were asked to maintain their ongoing analgesic treatment unchanged for the duration of the protocol, but were allowed to take medication for breakthrough pain if needed.

Study Design

Eligible patients were randomized 1:1 to one of the two treatment sequences: rTMS followed by tDCS (group I) or tDCS followed by rTMS (group II) through simple randomization (random numbers generated by computer). Each patient benefited from two stimulation cycles of 1 week each (rTMS and tDCS) at least 4 weeks apart, each cycle comprising 5 daily sessions of stimulation (Fig. 1). Patients filled a diary evaluating pain intensity using a numerical rating scale (NRS) ranging from 0 (no pain) to 10 (the worst pain possible) every day during 5 weeks: from 1 week (W0) before stimulation week (W1) to 3 weeks after the end of stimulation (W2, W3, and W4). In addition, patients were also asked to provide daily ratings of sleep quality, fatigue, and "rescue" medication [START_REF] André-Obadia | Pain relief by rTMS: differential effect of current flow but no specific action on pain subtypes[END_REF]. A minimal wash-out period of 4 weeks preceded the second phase, with identical design but different stimulus modality. At the end of the second phase, patients sent their notebook to a nurse different from the investigators and continued medical follow-up with their pain physician. Investigators did not have access to the patients' ratings before the end of each trial.

Chronic drug intake was maintained unchanged during the whole study period, with the exception of punctual "rescue" drugs for breakthrough pain, if needed, which must be reported in the patients' logbook. From 68 patients initially entering the study, 56 completed all 5 weeks of data for at least one mode of stimulation, and 46 completed all follow-up from both techniques (Fig. 2).

3

The protocol included two identical magnetic resonance imaging (MRI) sessions, respectively, preceding and immediately following the first week of stimulation, whatever its type. The sessions were performed on 3 T Philips Achieva-TX scanners with a 32-channel head coil at both Lyon and Grenoble sites. Each session included a first morphological 3D T1-weighted sequence eventually used for neuronavigation-based treatment, and then a set of four BOLD weighted fMRI runs. The fMRI runs included each a different movement task: a right hand movement of the Vth finger, a left hand movement of the Vth finger, a right zygomatic movement of the face, and a left zygomatic movement (half a smile). Each fMRI run consisted in a block design with 3 epochs of 30 s alternating with rest epochs of 30 s, for a total duration of 3 min and 30 s per run.

The complete procedure is schematized in Fig. 1.

Stimulation Parameters

Stimulation was carried out in the University Hospital Pain Center (CETD) of the Neurological Hospital of Lyon and in the Pain Centre of the Grenoble Alpes University Hospital. rTMS and tDCS were performed using the same stimulators in both experimental centers. rTMS (Mag-Pro X100, MagVenture © ) induced biphasic magnetic pulses via an eight-shaped coil (cool-B65 butterfly shape coil MagVenture © ). The motor strip was localized in each patient using T1-3D MRI, and the stimulating coil was positioned perpendicular to the central sulcus, with postero-anterior orientation. The optimal position of the coil was determined using the MRI-Neuronavigation system with Visor © software (ANT © ) and collecting EMG responses of the abductor digiti minimi. Motor threshold at rest was defined before each stimulation session as the lowest intensity that produced five responses with peak-to-peak amplitude of at least 50 µV in ten consecutive trials [START_REF] Rossini | Non-invasive electrical and magnetic stimulation of the brain, spinal cord and roots: basic principles and procedures for routine clinical application. Report of an IFCN committee[END_REF]. Each 10 Hz-rTMS session comprised 32 consecutive trains of 50 pulses, delivered at 90% of motor threshold, separated by inter-trains intervals of 25 s (i.e., a total of 1600 pulses during a 17-min session).

tDCS (DC stimulator NIC-Starstim ® , Neuroelectrics ® ) was delivered in Grenoble via sponge electrodes soaked in salty solution and in Lyon via NG electrodes placed on prefixed positions on a neoprene cap with a conductive gel between the electrodes. Skin-electrode impedance levels below 5 kOhms were required to initiate stimulation. For each tDCS session, a 2 mA anodal stimulation was applied during 20 min over the motor cortex contralateral to pain, over C3/C4 positions of the international 10-20 system. The cathode was placed over the frontal-polar region, ipsilateral to pain, over Fp1/Fp2.

Outcome Variables and Statistical Analyses

From the week preceding the stimulation (Day 7) to the end of the four week following the stimulation period (Day 28), patients used a diary at home to record the following information: pain intensity, quality of sleep, and fatigue, using a 0-10 numerical rating scale (total of 5 weeks). Daily NRS ratings were averaged week per week.

The primary outcome was the analgesic effect of each stimulation modality compared to its own baseline (week before stimulation: W0 on Fig. 1). Secondary outcomes were the quality of sleep and fatigue for each stimulation modality compared to its own baseline (same analyses as for pain intensity).

To allow inter-subject comparisons, daily NRS was normalized using Z-scores [START_REF] Rolke | Quantitative sensory testing in the German Research Network on Neuropathic Pain (DFNS): standardized protocol and reference values[END_REF][START_REF] Garcia-Larrea | Athome cortical stimulation for neuropathic pain: a feasibility study with initial clinical results[END_REF]. Thus, each daily pain rating was Z-transformed using the formula (X i -X baseline ) / SD baseline , where X i is the actual raw daily rating in day ''i", X baseline is the average rating from the prestimulation week in the same individual, and SD baseline is the associated standard deviation of NRS values during this pre-stimulation week. Patients were considered "responders" if their pain ratings decreased by at least 2 standard deviations (SD) from baseline, during at least 1 week.

Continuous data are expressed as mean ± standard deviation (SD) or median (25th-75th centiles), and categorical data are expressed as numbers and percentages. Comparisons of baseline characteristics between allocated groups (rTMS/tDCS versus tDCS/rTMS) and responders versus non-responders to each modality were conducted by using the Chi-square or Fisher's exact tests, Student's t-test, or Wilcoxon-Mann-Whitney test. Homogeneity of the two groups regarding pain ratings was tested by comparing with a two-tailed paired t-test their respective NRS during the baseline week, before initiating the stimulation periods. NRS during the week before the first and second stimulation cycles (regardless of the stimulation type) was also compared with two-tailed paired t-test to check for possible carry-over effects. In the patients who completed the entire study (n = 46), normalized pain scores were compared using a 2-way repeated-measures ANOVA (time × stimulation mode). A p-value < 0.05 was considered significant after Greenhouse-Geisser correction when needed. Correlation between the magnitude of the analgesic effect from rTMS and tDCS was studied using Pearson-product-moment coefficients.

Data were analyzed using GraphPad Prism. A twosided p value < 0.05, after correction when needed, was considered statistically significant.

fMRI Analysis

Using SPM12 software (The Welcome Department of Cognitive Neurology, London; http:// www. fil. ion. ucl. ac. uk/ spm/), image pre-processing was performed in each subject's referential and included motion correction, slice timing, and registration to the anatomical image of the pre-treatment session. Doing so, both pre-treatment and post-treatment functional images are in the same referential. Individual statistical analysis was estimated using a linear generalized model to produce the following contrasts: (i) the individual contrasts for each task at each time point and (ii) the differential contrast for each task after vs. before the treatment. The anatomical image of the pre-treatment session of each subject was segmented into 6 classes of tissue (gray matter, white matter, cerebrospinal fluid, large vessels, meninges, and scalp) using the tissue probability maps provided by the software. In order to transform the images in a common referential for group analysis, we computed the deformation field to be applied to each individual to match a symmetrical template, provided by the CAT12 software (neuro. uni-jena. de/ cat12-html/ cat_ versi ons. html). The DARTEL method was used to achieve a clear difference between primary motor and primary sensory cortices [START_REF] Pizzagalli | Local landmark alignment for high-resolution fMRI group studies: toward a fine cortical investigation of hand movements in human[END_REF]. The deformation field computed for each subject was applied to individual contrast images. Since the pain lateralization is patientdependent, so was the stimulated hemisphere. In order to be able to pool the stimulated hemispheres and to compare them to the non-stimulated hemispheres, the individual contrast images were left-right flipped when necessary so as to obtain the stimulated hemisphere on the left side of the image and the unstimulated hemisphere on the right side of the image.

For inference at the group level, several statistical tests were performed. First, a comparison between the post-and pre-stimulation contrasts using paired t-test for the two types of treatment, each type of treatment and between type of treatment, to investigate the general effect of stimulation, the effect of each type of stimulation, and the differential effect between both types of stimulation. Second, to check whether the effect of stimulation differed in responders vs. nonresponders, a comparison between the post-stimulation and the pre-stimulation was tested between contrasts according to the responding status, using two sample t-test. Third, in order to check whether movements performed in the painful and the non-painful sides generated differential brain activation patterns, a comparison between movements in both sides previous to any treatment was tested using paired t-test. Finally, in order to check whether the pattern corresponding to movement in the painful side could predict the response to treatment, the contrast corresponding to movement in the painful side before treatment was compared between subgroups of responders and non-responders. Differences were considered significant when p < 0.05 after correction for multiple comparisons using family-wise error at voxel level and the probability for the extent of activation cluster to be find by chance was below 0.05.

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Results

Flowchart of the Study

Figure 2 depicts the flowchart of the study participants. Sixty-eight patients were initially recruited and randomly assigned to study groups: "rTMS then tDCS" (group I) or "tDCS then rTMS" (group II). Fifty-six patients completed the first phase of the protocol (26 group I and 30 group II) and 46 patients completed both phases of study (22 group I and 24 group II). Functional imaging (fMRI) study comparing motor-evoked activations pre-and post-neurostimulation was performed in 31 of the 56 patients who completed the protocol. The MRI study could not be completed in the others for patient-dependent reasons (unavailability for the 2nd session) or organizational difficulties in connection with time-slot availability in the radiology department. Despite such difficulties, the sample presented here is to our knowledge the largest group of patients described so far with motor-related fMRI performed before and immediately after a series of motor cortical stimulation for neuropathic pain.

Baseline Demographics and Clinical Characteristics

The demographics and baseline characteristics of the 56 patients who completed the 5-week follow-up assessment of the first phase are shown in Table 1. The mean age of the patients was 58.6 ± 13.2 years. No significant differences at baseline were found between groups I and II (starting with rTMS or tDCS) according to age, sex, and origin of pain or clinical scores during the week pre-stimulation.

Primary Outcome

Patients were defined as "responders" if pain scores decreased by at least 2 SDs relative to baseline values (W0) during 1 week or more (see Methods), and these criteria were met by 21/50 patients (42.0%) for rTMS and 22/52 (42.3%) for tDCS, the difference being non-significant. The number of responders was also similar in the 46 patients receiving both techniques (two-sided Fisher's exact test: p = 0.76). Of notice, almost half of these patients (21/46) responded to one modality exclusively (12 responded only to rTMS and 9 only to tDCS). We did not find any significant difference between responders and non-responders regarding their characteristics at baseline, including age, sex, origin or intensity of pain, quality of sleep, and fatigue scores (Table 2).

In the 46 patients who received both stimulation modalities, a 2-way, rm-ANOVA (time × stimulation mode) on Z-normalized pain changes between W0 and W4 showed a significant effect of time (F(4,360) = 10.98; p < 10 -3 ) but no effect of stimulation mode (F(1,360) = 0.46; p = 0.50) and no interaction (F(4,360) = 51.10; p = 0.35). Figure 3 illustrates the evolution of Z-normalized pain scores during the 4-week post-stimulation according to response status.

The percentage of pain decrease at the "best week" (the week with most prominent changes) for each modality was significantly correlated (r = 0.34; p = 0.005). However, as illustrated in Fig. 4, the slope of the regression line (β = 0.34, 95% CI = [0.10-0.54]) was significantly biased in favor of rTMS relative to the theoretical equivalence slope (β = 1). No significant difference was found in the level of pain decrease according to the type of electrode used for tDCS in the two experimental sites.

Pain scores during the baseline preceding the second stimulation session were decreased relative to those preceding the first one (6.4 ± 1.9 vs. 5.9 ± 2.3; two-tailed paired t-test: p = 0.03) reflecting a possible carry-over effect independent of the stimulus mode. However, NRS baseline values were not significantly different when preceding rTMS or tDCS (two-tailed paired t-test: p = 0.44).

Secondary Outcomes

Quality of Sleep and Fatigue

Self-assessment of sleep and fatigue changed very little during the 4-week post-stimulation, and no significant differences between techniques were observed (Fig. 5). A 2-way, rmANOVA (time × stimulation mode) on Z-normalized fatigue changes between W0 and W4 showed a significant and favorable effect of time (F(4,336) = 3.13; p = 0.01) but no effect of stimulation mode (F(1,336) = 0.00; p = 0.97) and no interaction (F(4,336) = 0.71; p = 0.58. Comparable results were obtained for sleep changes: 2-way, rmANOVA (time × stimulation mode) on Z-normalized changes between W0 and W4 showed a significant effect of time (F(4,320) = 6.75; p < 10 -3 ) but no effect of stimulation mode (F(1,320) = 0.11; p = 0.74) and no interaction (F(4,320) = 0.66; p = 0.62).

Secondary Outcomes: fMRI

At the group-level, the activation pattern for the finger movement contralateral to the pain involved as expected the primary motor and supplementary motor areas within the motor network (Fig. 6). The pattern of motor-related activation did not show statistically significant differences for the painful and the non-painful sides. This pattern was similar before and after the treatment, and not statistically different following rTMS or tDCS, nor when both treatments were pooled together. The distribution and magnitude of motor-related activation after the treatment were equivalent when compared between

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N. André-Obadia et al. 1 3 responders and non-responders. In order to disclose any pre-stimulation feature in motor activation that could be predictive of cortical stimulation efficacy, we checked for differences in activation patterns in responders and nonresponders before treatment, but no significant differences were detected between sub-groups. Taken together, this set of analyses showed the expected patterns of brain activation during voluntary movement, but failed to demonstrate significant differences in fMRI motor activity patterns at group level, neither between techniques nor between responders and non-responder patients.

Adverse Effects

No serious adverse effects were reported during or following any of the two interventions, but minor side effects were noted very occasionally. One patient complained of noise intolerance (despite wearing hearing protection) and headaches during the rTMS session and discontinued the study. One patient reported tension headache immediately after rTMS, and 3 patients complained about this symptom several days after the end of the stimulation week (1 week and 2 weeks after the end of rTMS, and 2 days after the end of tDCS). These headaches were relieved by a level-1 analgesic (paracetamol) and did not recur. Skin irritation was observed under a tDCS electrode on 3 occasions, without recurrence the following days.

Discussion

Both rTMS and tDCS decreased pain levels in this series of patients with pharmaco-resistant neuropathic pain. Although pain decrease was overall significant at the group level for the two interventions, individual analysis showed that only 42% of the patients achieved a significant level of pain decrease, defined as a change equal or superior to 2 SDs relative to the average ratings in the week previous to stimulation. Analysis of the consecutive weekly pain ratings showed that there was no overlap between the timelines of responders and non-responders: pain ratings in responders progressively decreased during the week of stimulation, and then during the 3 following weeks, whereas in non-responders the sequence of changes in pain ratings was virtually horizontal and around zero to either procedure (Fig. 3). Best pain decrease in responders was moderate when considering raw NRS values, about 31% of initial values in the average. This is consistent with a number of previous reviews and meta-analyses [START_REF] Cruccu | EAN guidelines on central neurostimulation therapy in chronic pain conditions[END_REF][START_REF] O'connell | Non-invasive brain stimulation techniques for chronic pain[END_REF][START_REF] Anderson | Evidence brief: transcranial magnetic stimulation (TMS) for chronic pain, PTSD, TBI, opioid addiction, and sexual trauma[END_REF] and represents a significant but moderate improvement according to IMMPACT consensus statement criteria of "clinically meaningful" treatment [START_REF] Dworkin | Interpreting the clinical importance of treatment outcomes in chronic pain clinical trials: IMMPACT recommendations[END_REF]. It has been suggested that more than 5 consecutive stimulation sessions may be necessary to induce a maximal analgesic effect, both for rTMS [START_REF] Hodaj | Long-term treatment of chronic orofacial, pudendal, and central neuropathic limb pain with repetitive transcranial magnetic stimulation of the motor cortex[END_REF] and tDCS [START_REF] Castillo-Saavedra | Clinically effective treatment of fibromyalgia pain with highdefinition transcranial direct current stimulation: phase II openlabel dose optimization[END_REF], and iteration of maintenance sessions at longer intervals has proved useful to maintain or enhance the analgesic effects [START_REF] Quesada | New procedure of highfrequency repetitive transcranial magnetic stimulation for central neuropathic pain: a placebo-controlled randomized crossover study[END_REF][START_REF] Hodaj | Long-term treatment of chronic orofacial, pudendal, and central neuropathic limb pain with repetitive transcranial magnetic stimulation of the motor cortex[END_REF]. Of notice, although the changes observed here may not appear impressive, they concern patients who had previously proven to be drug-resistant to both 1st and 2nd line drugs for neuropathic pain, in many cases for more than 1 year. Furthermore, normalized pain ratios warranted that in any patient qualified as "responder," pain had abated by at least 2 SDs relative to pre-stimulus values, and often reaching up to -3 SD, which ensured a sizeable effect size at the individual level. Indeed, the Z-score of normalization is individually tailored and takes into account the intrinsic variability of pain reports at baseline in each patient, hence minimizing any changes in scores that do not exceed significantly such pre-stimulus variability. Z-score normalization is also the preferred method in other domains in pain research, notably when describing changes in quantitative sensory testing [START_REF] André-Obadia | Pain relief by rTMS: differential effect of current flow but no specific action on pain subtypes[END_REF][START_REF] Magerl | Reference data for quantitative sensory testing (QST): refined stratification for age and a novel method for statistical comparison of group data[END_REF][START_REF] Lötsch | Pattern of neuropathic pain induced by topical capsaicin application in healthy subjects[END_REF].

When tested as a group, the overall magnitude of changes in pain scores due to rTMS and tDCS did not differ significantly, and their respective levels were highly correlated. However, as illustrated in Fig. 5, the regression line between the best level of pain relief in both techniques appeared skewed toward rTMS, and significantly different from the theoretical equivalence slope (45°, or β = 1), suggesting a slightly superior level of pain relief for rTMS at individual level, and under the conditions tested here. We cannot rule out the possibility that the number of stimulation sessions to achieve a given level of relief may be different for tDCS and rTMS. Although there is not, to our knowledge, a direct comparison of both techniques in this respect, the number of sessions considered sufficient to obtain maximal analgesic effects was estimated as 7 for rTMS by Hodaj et al. [START_REF] Hodaj | Long-term treatment of chronic orofacial, pudendal, and central neuropathic limb pain with repetitive transcranial magnetic stimulation of the motor cortex[END_REF], and as high as 15 for tDCS by Castillo-Saavedra et al. [START_REF] Castillo-Saavedra | Clinically effective treatment of fibromyalgia pain with highdefinition transcranial direct current stimulation: phase II openlabel dose optimization[END_REF]; such disparities may have influenced the present results.

The percentage of responders to either technique was almost identical (42%), but the individual patients responding to each procedure were not the same. This is an indirect indication that the mechanisms underlying the pain-relieving effect of both techniques may differ, at least in their initial "induction" phase. In support of this view, a lack of effect of tDCS has been described in patients previously responding to rTMS [START_REF] O'neill | Patient-delivered tDCS on chronic neuropathic pain in prior responders to TMS (a randomized controlled pilot study)[END_REF], and conversely a patient with chronic NP not responding to rTMS could be improved in the long term by anodal tDCS [START_REF] Hodaj | A case of long-term treatment of chronic pain syndrome by anodal tDCS of the motor cortex, previously resistant to high-frequency rTMS and implanted spinal cord stimulation[END_REF]. A very recent report comparing these two techniques in 12 patients with brachial plexus avulsion also found that different patients may be differentially sensitive to one or the other [START_REF] De Assis | Effects of rTMS and tDCS on neuropathic pain after brachial plexus injury: a randomized placebo-controlled pilot study[END_REF]. Together with these previous reports, the present study in a larger sample of NP patients appears clinically relevant in that it highlights the possibility of using one technique if the other fails, thereby increasing the probability of a positive response, which currently tops out at about 50% for rTMS [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF][START_REF] Cruccu | EAN guidelines on central neurostimulation therapy in chronic pain conditions[END_REF][START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS): An update (2014-2018)[END_REF].

The brain activation pattern during motor tasks mainly involved as expected the primary motor and supplementary motor areas, and remained unchanged from the beginning to the end of the stimulation week, for both rTMS and tDCS. Such lack of evidence for motor-related plasticity after 1 week of stimulation might indicate that the second fMRI was conducted too early to detect possible Fig. 5 Weekly evolution of fatigue and sleep scores in responders and non-responders during the 4 weeks (W1 to W4) post-stimulation. A global trend to a decrease of severity with time was observed, with no significant difference either according to the stimulation modality or according to the response to treatment. Abbreviations: rTMS, highfrequency repetitive transcranial magnetic stimulation; tDCS, transcranial direct current stimulation Fig. 6 BOLD activation patterns during the motor task. The left column depicts the results combined for both rTMS and tDCS while the middle column illustrates the results for rTMS only and the right column for tDCS. The motor task induced significant activations in the contralateral sensori-motor cortex and the supplementary motor area (SMA), with no significant differences before (upper panel) or after one week of daily motor cortex stimulation (lower panel)

R E V I S E P R O O F

Better Fields or Currents? A Head-to-Head Comparison of Transcranial Magnetic (rTMS) Versus… 1 3

motor-related effects triggered by the procedures, which may have occurred later. Indeed, post-intervention fMRI data were acquired at the end of the stimulation week, whereas the maximal effects on pain in responders were not obtained until 1 week later or more (Fig. 3). If plastic changes in the cortex develop in parallel with the decrease in pain, they may have gone unnoticed in our patients because of the different temporality of recordings. Although technically challenging, future studies should consider the importance that fMRI data be acquired in close connection with changes in pain reports.

The relation of local changes in motor networks and pain relief from cortical neurostimulation remains a subject of debate, as it remains unclear whether rTMS entails sizeable changes in intracortical motor circuits under the conditions used to treat pain. Indeed, while stimulation at levels above motor threshold induced clear metabolic activation in M1, such activation was found to subside or disappear at the sub-threshold levels commonly applied for pain relief [START_REF] Paus | Dose-dependent reduction of cerebral blood flow during rapidrate transcranial magnetic stimulation of the human sensorimotor cortex[END_REF][START_REF] Bestmann | Functional MRI of the immediate impact of transcranial magnetic stimulation on cortical and subcortical motor circuits[END_REF]. Also, although a correlation was initially described between intracortical motor inhibition and rTMS-induced pain relief [START_REF] Lefaucheur | Motor cortex rTMS restores defective intracortical inhibition in chronic neuropathic pain[END_REF], later studies in NP patients failed to reproduce such effects [START_REF] Hosomi | Cortical excitability changes after high-frequency repetitive transcranial magnetic stimulation for central poststroke pain[END_REF][START_REF] Jetté | Effect of single-session repetitive transcranial magnetic stimulation applied over the hand versus leg motor area on pain after spinal cord injury[END_REF], and rTMS analgesia could be blocked pharmacologically in the absence of motor excitability changes [START_REF] De Andrade | Repetitive transcranial magnetic stimulation induced analgesia depends on N-methyl-D-aspartate glutamate receptors[END_REF]. Therefore, while widespread long-distance changes in cortical and subcortical structures after rTMS/tDCS have received consistent support, the relevance of motor cortex excitability for rTMS analgesic effects remains unconfirmed.

Limitations

An obvious limitation of this study is the lack of longterm follow-up beyond 5 weeks [START_REF] O'connell | Non-invasive brain stimulation techniques for chronic pain[END_REF]. The long-term maintenance of pain relief in responding patients is a major challenge for all non-invasive stimulation methods, and different procedures are being currently tested, mostly based on the progressive spacing out of consecutive sessions [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF][START_REF] Lefaucheur | A practical algorithm for using rTMS to treat patients with chronic pain[END_REF]. Future studies should consider providing assessment of pain relief during months or years if these techniques are to be accepted as routine treatments for chronic pain [START_REF] Anderson | Evidence brief: transcranial magnetic stimulation (TMS) for chronic pain, PTSD, TBI, opioid addiction, and sexual trauma[END_REF]. The head-to-head design of the study, comparing tDCS to a reference active stimulation (rTMS) instead of a placebo, also entails interpretative limitations. According to current literature, rTMS can now be considered a validated procedure for drug-resistant neuropathic pain [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF][START_REF] Anderson | Evidence brief: transcranial magnetic stimulation (TMS) for chronic pain, PTSD, TBI, opioid addiction, and sexual trauma[END_REF][START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS): An update (2014-2018)[END_REF] that has been incorporated to standard guidelines for NP therapy [START_REF] Moisset | French guidelines for neuropathic pain: an update and commentary[END_REF] and could in our view act as a valid reference. This also made it possible to avoid subjecting patients suffering from drug-resistant pain for many years to a placebo. Although the placement of the anode tDCS was centered on the motor cortex, the size of the electrodes and the standard motor-prefrontal montages entail a current distribution covering a region much more extended than the focalized figure-of-eight rTMS coil, and could render the comparison hazardous [START_REF] Hodaj | Long-term prophylactic efficacy of transcranial direct current stimulation in chronic migraine. A randomised, patient-assessor blinded, sham-controlled trial[END_REF]. New highdefinition tDCS montages could permit more focalized current distribution around the motor cortex, but their use in neuropathic pain remains anecdotal [START_REF] Deblieck | Case report: initial evidence of safety and efficacy of high definition-transcranial direct current stimulation in a patient with neuropathic pain and implanted spinal cord stimulator[END_REF]. Finally, our fMRI analysis was restricted to the activations induced by a motor task. Analysis of resting state, pain-related activity, and connectivity changes before and after stimulation may prove in the future much better approaches to investigate the brain activities accompanying and/or supporting neurostimulation-related pain relief [START_REF] Van Der Miesen | Neuroimaging-based biomarkers for pain: state of the field and current directions[END_REF].

Conclusion

In patients with drug-resistant neuropathic pain, five daily sessions of tDCS or rTMS over the motor cortex showed a similar pattern of efficacy at one month. Each technique entailed significant effects in about half of the patients, and half of them responded to one procedure only; therefore, both techniques deserve being tested before declaring a patient as unresponsive. Pain relief was not paralleled by motor-related plasticity on fMRI. Since duration of pain relief after 5 daily sessions often does not exceed some weeks [START_REF] Garcia-Larrea | Cortical stimulation for chronic pain: from anecdote to evidence[END_REF][START_REF] Jetté | Effect of single-session repetitive transcranial magnetic stimulation applied over the hand versus leg motor area on pain after spinal cord injury[END_REF], prolonging the beneficial effects of neurostimulation in responders remains a crucial issue. Potential solutions include delivering maintenance sessions at progressively longer intervals, autonomous stimulation at home, and/or neurosurgical implanted stimulation.

Fig. 1 3 Fig. 2

 132 Fig. 1 Study design. Abbreviations: rTMS, repetitive transcranial magnetic stimulation; tDCS, transcranial direct current stimulation; fMRI, functional magnetic resonance imaging; D, day; W0, baseline; W1, stimulation sessions; W2-W3-W4, follow-up period

  Better Fields or Currents? A Head-to-Head Comparison of Transcranial Magnetic (rTMS) Versus… 1 3

Fig. 3

 3 Fig. 3 Evolution of Z-normalized pain scores during the 4 weeks post-stimulation. Evolution of pain scores in responders and nonresponders during the 4 weeks (W1 to W4) post-stimulation. Pain reports were normalized as z-scores relative to values during the baseline pre-stimulation week (W0; see Methods), and patients were considered as "responders" if their scores decreased by more than 2 SD relative to baseline. The groups of responders and not-responders were clearly differentiated since the first week post-stimulation, with no overlap. Abbreviations: rTMS, high-frequency repetitive transcranial magnetic stimulation; tDCS, transcranial direct current stimulation. Resp rTMS, responders to rTMS; Resp tDCS, responders to tDCS; NonResp rTMS, non-responders to rTMS; NonResp tDCS, non-responders to tDCS

Fig. 4

 4 Fig.[START_REF] O'connell | Non-invasive brain stimulation techniques for chronic pain[END_REF] Correlation between NRS percentage changes after rTMS and tDCS. Correlation between maximal percentage changes of numerical pain reports (NRS) after rTMS and tDCS. Although NRS changes to both techniques were correlated, the slope of the regression line was significantly skewed towards rTMS, with confidence limits (dotted lines) which did not reach the theoretical equivalence line of β = 1. Abbreviations: rTMS, high-frequency repetitive transcranial magnetic stimulation; tDCS, transcranial direct current stimulation; NRS, numerical rating scale

Table 1

 1 Baseline characteristics of patients according to the allocated group SD standard deviation, IQR interquartile range a Numerical rating scale (NRS). The NRS score ranges from 0 to 10, with 0 indicating no pain/sleep disorder/fatigue and 10 the worst imaginable pain/sleep disorder/

fatigue Total (n = 56) rTMS then tDCS (n = 26) tDCS then rTMS (n = 30) P-value

	Age (year), mean (SD)	58.6 (13.2)	60.2 (13.7)	57.3 (12.9)	0.407
	Female, n (%)	27 (48%)	12 (46%)	15 (50%)	0.774
	Washout period (week), median (IQR)	10.0 (9.0, 12.1)	10.0 (9.0, 12.0)	10.7 (9.0, 12.1)	0.904
	Disease history
	Pain syndrome duration (year), median (IQR)	5 (3, 8)	5 (3, 8)	5 (3, 7)	0.391
	Pain origin, n (%):				0.569
	Brachial plexus injury	4 (7%)	2 (8%)	2 (7%)
	Spinal cord injury	6 (11%)	4 (15%)	2 (7%)
	Central post-stroke pain	21 (37.5%)	7 (27%)	14 (46%)
	Brain tumor, vascular and other pain	4 (7%)	2 (8%)	2 (7%)
	Orofacial pain	21 (37.5%)	11 (42%)	10 (33%)
	Summary of pharmacological treatment
	Number of drugs/patient, median (IQR)	2 (2, 3)	2 (2, 3)	2 (2, 2)	0.340
	Drug class, n (%):
	Antiepileptics	39 (70%)	18 (69%)	21 (70%)	0.950
	Antidepressants	36 (64%)	17 (65%)	19 (63%)	0.873
	Strong opioids	6 (11%)	2 (8%)	4 (13%)	0.496
	Weak opioids	22 (39%)	11 (42%)	11 (37%)	0.666
	Non-opioid analgesics	15 (27%)	9 (35%)	6 (20%)	0.218
	Clinical

score a during the week pre-stimulation

				F
			O O
		P R
	E V I S E
	R
	Pain score, mean (SD)	6.4 (2.0)	6.0 (2.1)	6.7 (2.0)	0.180
	Sleep score, mean (SD)	4.7 (2.5)	4.8 (2.6)	4.7 (2.6)	0.876
	Fatigue score, mean (SD)	5.5 (2.3)	5.0 (2.4)	5.9 (2.2)	0.170

N. André-Obadia et al.

Table 2

 2 Baseline characteristics of responders versus non-responders SD standard deviation, IQR interquartile range a Numerical rating scale (NRS). The NRS score ranges from 0 to 10, with 0 indicating no pain/sleep disorder/fatigue and 10 the worst imaginable pain/sleep disorder/fatigue

		rTMS			tDCS
		Responders (n = 21) Non-responders (n = 29)	(n = 30) responders P-value Responders (n = 22) Non-	P-value
	Age (year), mean (SD)	61.1 (12.1)	57.0 (14.9) 0.299	59.5 (12.6)	58.6 (12.6) 0.817
	Female, n (%)	9 (43%)	14 (48%)	0.704	12 (55%)	14 (47%)	0.575
	Disease history
	Pain syndrome duration (year), median (IQR) 5 (3, 8)	5 (3, 7)	0.945	5 (3, 8)	5 (3, 7)	0.886
	Pain origin, n (%):			0.513			0.839
	Brachial plexus injury	2 (10%)	2 (7%)		1 (4.5%)	3 (10%)
	Spinal cord injury	3 (14%)	2 (7%)		2 (9%)	3 (10%)
	Central post-stroke pain	6 (29%)	11 (38%)		9 (41%)	12 (40%)
	Central cancer, vascular and other pain	3 (14%)	1 (3%)		1 (4.5%)	3 (10%)
	Facial pain	7 (33%)	13 (45%)		9 (41%)	9 (30%)
	Summary of pharmacological treatment
	Number of drugs/patient, median (IQR)	2 (1, 3)	2 (2, 3)	0.651	2 (1, 2)	2 (2, 3)	0.136
	Drug class, n (%):
	Antiepileptics	14 (67%)	21 (72%)	0.662	13 (59%)	26 (87%)	0.023
	Antidepressants	14 (67%)	17 (59%)	0.563	15 (68%)	19 (63%)	0.717
	Strong opioids	3 (14%)	2 (7%)	0.390	2 (9%)	3 (10%)	0.913
	Weak opioids	5 (24%)	14 (48%)	0.079	7 (32%)	12 (40%)	0.545
	Non-opioid analgesics	6 (29%)	8 (28%)	0.939	4 (18%)	8 (27%)	0.473
	Clinical score a during the week pre-stimulation
	Pain score, mean (SD)	6.7 (1.8)	6.1 (1.9)	0.297	6.7 (2.2)	5.8 (2.0)	0.144
	Sleep score, mean (SD)	4.7 (2.6)	4.8 (2.3)	0.916	4.2 (2.5)	4.6 (2.7)	0.602
	Fatigue score, mean (SD)

Funding This work was supported in part by the "Fondation APICIL," "Agir à Dom," and LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) conducted by the French National Research Agency (ANR).

Acknowledgements

The authors would like to thank Dr Jean-Pierre Alibeu and Koichi Hagiwara for their contribution to the work leading to this article, the medical staff of the University Pain Centers of Lyon and Grenoble, who referred to us many of the patients treated herein, and the technical staff that provided logistic assistance.

Required Author Forms Disclosure forms provided by the authors are available with the online version of this article.

Supplementary Information

The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s13311-022-01303-x.

Declarations

Competing Interest The authors declare no competing interests.


Keywords: Plasma simulation, circuit breakers, magnetostatics, moment methods, nonlinear equations

The behaviour of arcs in circuit breaker is affected by several interactions. To simulate this, a three-dimensional modeling system has already been developed. It takes into account the plasma fluid dynamics, the radiation, the plastic vaporization, the current flow within the electrodes and the plasma, and the magnetic field generated by currents. In order to take into account the influence of ferromagnetic regions the simulation model has to be extended.

The Magnetic Moment Method (MoM) is an integral method where only active regions (i.e., ferromagnetic regions in our context) are meshed. Thus it is particularly well adapted for breaking modeling with few ferromagnetic regions in comparison to the whole air region.

In this paper, a non-linear MoM model is discussed and finally an arc interruption process modeling with the introduction of ferromagnetic regions is presented.

I. INTRODUCTION

When a circuit breaker detects a fault current, it opens the electrical circuit and an arc appears. The ignited arc between the opening contact pieces has to be quickly extinguished. The most effective and hence most widely used possibility for that is to split the arc into series arcs by metallic splitter plates (Fig 1) [START_REF] Lindmayer | The process of arc splitting between metal plates in low voltage arc chutes[END_REF]. Thereby hydrodynamic constraints and magnetic forces on the arc must be under control to ensure an optimal arc displacement.

Fig. 1. Arc represented by several iso-values of temperature. The arc appears between the contact pieces, slides between the electrodes and finally it is cooled in splitter plates.

In the case of a breaking process at low current, ferromagnetic regions allow to substantially increase the Lorentz forces on the plasma. This paper presents the introduction of the ferromagnetic effects thanks to the Magnetostatic Moment Method (MoM) in the Schneider Electric's arc system modeling [START_REF] Rondot | Dedicating Finite Volume Method to Electromagnetic Plasma Modeling: Circuit Breaker Application[END_REF].

II. 3D MAGNETOSTATIC MOMENT METHOD (MOM)

A. Standard equation

For a magnetostatic problem composed by some ferromagnetic regions and coils supplied by currents, the magnetic field can be expressed as the sum of the external field and the reduced magnetic field which derives from the magnetic scalar potential. The magnetic field can be read [START_REF] Harrington | Field Computation by Moment Methods[END_REF]:

( ) ( ) ( ) ) r d ) ' , ( ( ' ' 3 ' ext ∫ ⋅ ∇ ∇ - = V G r M r r r H r H r r (1)
where H ext is the external field due to the current, M the magnetization, G is the Green Function for a Laplace's equation, r the integration variable and r' the position of the computation.

This formulation strongly ensures Maxwell-Ampere and Maxwell-Thomson equations. Therefore to determine the magnetic field H, the magnetic law behaviour must also be ensured.

B. Point-Matching method

The magnetization is considered uniform in each cell. The equation (1) for p magnetic cells becomes:

[ ]M H H K ext + = (2)
where (H, H ext , M) are 3p vector and [K] is (3p)² matrix. The matrix [K] is a matrix of view factors between the cells computed by reduction of singularities [START_REF] Chadebec | A review of magnetostatic moment method[END_REF].

The point matching method consists in checking the magnetic law behaviour at each cell centroid. Thereby the residual 3p vector can be expressed as below:

( ) ( ) M H m M R - =
(3) and the non linear system equation for a Newton Raphson process can be read:

( ) [ ] [ ] ( )
1 n n I K 1 n - - = ∆         -       ∂ ∂ - M R M H H m H T (4)
where the first matrix is a diagonal bloc matrix linked with the derivative of magnetic law, ∆M is the increment of the magnetization, R the residual vector and n the iteration number.

The flowchart of the nonlinear resolution is represented in Fig. 2. In order to reduce the CPU time, the tangent matrix is not updated at each iteration. In other words, a fixed point (with the previous tangent matrix) and a Newton-Raphson method are alternatively used. The presented approach has been validated on several test cases to prove its effectiveness.

3D Magnetostatic Moment Method dedicated to arc interruption process modeling

III. CIRCUIT BREAKER APPLICATION

The core of the arc system modelling is based on the commercial software Fluent and user defined functions used to take into account all others phenomena (real gas, radiation, plastic vaporization, arc root, electromagnetism) [START_REF] Chevrier | Industrial application of high-medium-and low-voltage arc modeling[END_REF].

During a breaking process, the current density of the arc leads to:

• ohmic heating (Joule losses > energy source) and In a circuit breaker model two ferromagnetic "cheeks" are introduced close to the breaking chamber. During the breaking process simulation, the magnetization in cheeks and their influence on the magnetic field seen by the arc can be observed (Fig 4).

•
The comparison of the global electrical data (I current and U arc arc voltage) for several configurations allows finding the best design for an optimal arc interruption breaking process.

IV. CONCLUSION

The MoM method is suitable and well adapted for the breaking modelling. Compared to other approaches [START_REF] Lindmayer | 3D simulation of arc motion including the influence of ferromagnetic material[END_REF], it avoids the meshing of the whole air region and the use of two different meshes.

The use of the compression method is in progress. A further reduction in the computation time is expected.

The introduction of MoM in the arc system modelling allows considering ferromagnetic regions position and shape optimization. An improvement in the design of circuit breakers is expected.

Fig. 2 .

 2 Fig. 2. Non linear flowchart resolution where A is the solving matrix, RHS the right hand side vector, ε1 and ε2 are tolerance parameters.

Fig. 3 .

 3 Fig.3. Diagram of the resolution procedure for the arc interruption process modeling (T temperature, P pressure, σ electrical conductivity).

Fig. 4 .

 4 Fig. 4. Magnetization representation in ferromagnetic regions and magnetic field due to ferromagnetic regions represented on a middle plane of the circuit breaker.


Keywords: Adaptive Cross Approximation, acceleration method, volume integral equations, magnetostatics, nonlinear resolution

The volume integral methods are particularly well suited to compute field in the air domain which don't need to be meshed. However, their application leads to solve dense matrix systems. The Adaptive Cross Approximation (ACA) is an algebraic method allowing the compression of these matrices. This paper presents the ACA technique applied for a Volume Integral Method (VIM) in order to solve nonlinear magnetostatic problems.

I. INTRODUCTION

Integral methods have many advantages over finite elements method such as: far interactions are computed accurately, and especially avoiding the mesh of the air. Their main drawback is the obtaining of a full interaction matrix whose computation time and memory size increase in a quadratic complexity. Many methods have been developed to limit this drawback. The Fast Multipole Method (FMM) [START_REF] Greengard | A fast algorithm for particle simulations[END_REF] is one of the most successful approaches. However this method presents some difficulties in regard to the algorithm parallelization and to the solver preconditioning.

The Adaptive Cross Approximation [START_REF] Bebendorf | Adaptive low-rank approximation of collocation matrices[END_REF] has been successfully applied for many integral methods such as the Boundary Element Method [START_REF] Kurz | The Adaptive Cross-Approximation for the 3-D Boundary Element Method[END_REF] and the Moment Method [START_REF] Shao | The Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems[END_REF]. This approach is less intrusive in the source code, allows better solver preconditioning than the FMM and is easily parallelizable.

In this paper, the application of ACA for the Volume Integral Method to solve nonlinear magnetostatic problems is presented. The section II presents the integral formulation using magnetic scalar potential. The section III introduces the resolution of the nonlinear magnetostatic problems. The outlines of the ACA and of the hierarchical matrix methods are described in the section IV. The section V is dedicated to a numerical example.

II. VOLUME INTEGRAL EQUATION FORMULATION

Let us consider a ferromagnetic material placed in a static magnetic source field H 0 . The magnetic behavior law is defined by:

H H M ) ( χ = , ( 1
)
where M is the magnetization, H the magnetic field and χ the magnetic susceptibility. Let us consider that the material region is simply connected and containing no current sources.

The magnetic fields H and H 0 derive then respectively from the magnetic scalar potential Φ and Φ 0. The volume integral method [START_REF] Carpentier | Resolution of Nonlinear Magnetostatic Problems with a Volume Integral Method using the Magnetic Scalar Potential[END_REF] is used and we can write:

) ( dV ' ' ) ' ( ) ( 4
1 ) ( 0 3 V r r r r r r H r Φ = - - ⋅ Φ ∇ χ π + Φ ∫∫∫ , ( 2
)
where V is the volume of the material, r and r' are respectively the coordinates of computation and integration points.

Only the material region is meshed and the magnetic scalar potential is discretized with first order nodal shape functions. A collocation approach applied on (2) at the N mesh nodes leads to a system of algebraic linear equations:

[ ] [ ] ( ) 0 ) A( I Φ Φ = χ + , (3) where [
I] is the identity matrix and [A] the full interaction matrix.

III. NONLINEAR FORMULATION

This part proposes a modified fixed point method [START_REF] Hantila | Polarization Method for Static Fields[END_REF] to solve the nonlinear magnetic field problem.

The behavior law (1) is written:

) ( ) ( H S H H M + = FP χ , ( 4
) where χ FP is the constant slope of the modified point scheme and S the nonlinear residual. The fixed point can be found by iteratively updating the nonlinear residual. Using the behavior law (4) instead of (1) in the VIM (2), the following equation must be solved at each iteration: , [START_REF] Carpentier | Resolution of Nonlinear Magnetostatic Problems with a Volume Integral Method using the Magnetic Scalar Potential[END_REF] where k is the iteration number. Using the previous formulation ( 2) and ( 3), the resolution of (5) leads to a system of algebraic linear equations of the form:

∫∫∫ ∫∫∫ - - ⋅ π + Φ = - - ⋅ Φ ∇ χ π + Φ - V 3 1 k k 0 3 V k FP k dV ' ' ) , ' ( 4
1 ) ( dV ' ' ) ' ( 4
[ ] [ ] (
)

) ( ) A( I 0 FP k S B Φ Φ + = + χ , ( 6
)
where B is the contribution vector of the nonlinear residual. The value of the residual S after the iteration k is given by:

k FP k k H H M S χ - = + ) ( 1 ,.
(7) If the norm of the difference of χ between two iterations is lower than a given criterion, the algorithm is stopped.

The Adaptive Cross Approximation Technique for

Volume Integral Method Applied to Nonlinear Magnetostatic Problems B. Bannwarth (1) , A. Carpentier (1) , V. Le Van (1) , O. Chadebec (1), (2) , G. Meunier (1) , J.-M. Guichon (1) (1) G2ELab, Université de Grenoble (UMR CNRS 5269 -Grenoble-INP -Université Joseph Fourier), Grenoble, France (2) GRUCAD/EEL/CTC/UFSC, Florianópolis, SC 88040-900, Brazil E-mail: Bertrand.Bannwarth@g2elab.grenoble-inp.fr

A. Outline of the ACA technique

The ACA technique is based on the approximation of a block matrix by a product of two matrices of smaller sizes. This product approximates the initial matrix by one with a rank much lower but with a sufficient accuracy. If M m×n is the considered m×n block matrix, its approximation n m× M ~ can be written as :

n p p m n m × × × = V U M . ~, (8)
where p m× U and n p× V are respectively m×p and p×n matrices. This decomposition is only useful if p < ½ min{m,n}. If the block is sufficiently smooth, the p value can be sufficiently low to lead to a high compression rate. Due to the approximation property (8), the following estimate holds

F n m F n m n m ~× × × ε ≤ - M M M , ( 9
)
where ||.|| F denotes the Frobenius norm of a matrix, and ε is a given criterion. The full ACA algorithm is presented in [START_REF] Bebendorf | Adaptive low-rank approximation of collocation matrices[END_REF].

The main advantage of ACA decomposition is that the evaluation of the entire matrix is not needed. Indeed, only the knowledge of p lines and p columns of the matrix is required. Therefore, it does not only decrease the needed memory, but also greatly limits the number of integral computations.

B. Hierarchical Matrices

To be efficient, the matrices to compress must involve interactions between distant subspaces where the integration kernel is smooth. To fulfil this criterion, the degrees of freedom in the mesh are renumbered. This task is performed with the help of a partition of the space thanks to an octree. A representation of the total matrix into sub-matrices is then obtained. It is called hierarchical matrix. Each sub-matrix is dealt either as near or as far interaction. In the first case, the sub-matrix is classically evaluated by a full matrix computation. In the second case, the block is compressed with ACA.

For the nonlinear solving, the assembly of vector B in ( 6) is decomposed into a product of the form:

[ ] [ ] [ ] [ ]           =

V. NUMERICAL RESULTS

Let us consider the following contactor-like problem. The geometry description and the magnetic behavior law are described in Fig. 1a and1b. The proposed procedure is applied for a mesh of 37000 elements. With a 1e -4 ACA criterion, only 1.9GB RAM are required for the assembly of the 4 interaction matrices in ( 6) and (10). Without compression, 8.3GB RAM would be necessary, thus the memory need is decreased by 77% with the proposed procedure. By using the modified fixed point method for ( 5)-( 7) with a χ FP of 650, the nonlinear resolution converges after 158 iterations with 6s for each iteration time. The computed magnetic field is given in Fig. 2a. The FEM is also applied to solve this problem using 1 st and 2 nd order magnetic scalar potential formulations thanks to the software FLUX®. The evolution of the total magnetic energy of the contactor with respect to the number of mesh elements is given in Fig. 2b. The three results converge to the same value. For the 1 st order, VIM provides an accurate energy with relatively few elements.

VI. CONCLUSION

The ACA compression technique has been applied to a nonlinear magnetostatic problem with a VIM using the magnetic scalar potential. The results are interesting concerning the treatment of fine meshes with a gain of memory.

  x ], [B y ], [B z ] are assembly matrices of the derivative Green kernel and S x , S y , S z are the three components of S. [B x ], [B y ], [B z ] are built only once and only depend on the mesh geometry.The ACA method combined with Hierarchical matrices are used to compute the first order matrix assembly [A] in (4) and the matrices assembly [B x ], [B y ] and [B z ] in (10).

Fig. 2 .

 2 Fig. 2. Numerical results on a Intel Core i5 2.2GHz x64 CPU with 4Go RAM


F Chaxel

E Bajic

J Richard
email: richard@cran.u-nancyfr

Individual Product Data Management for FMS Control : a Step to Mobile DataBase Nodes Integration

INTRODUCTION

This paper presents a research work which considers programmable Escort Memories systems like parts of a global distributed database system providing manufacturing control, traceability, quality and maintenance assistance, ... To reach these objectives an escort memory (also called electronic tag) has to play the role of a vector of the information system which is in charge of the overall application management, and information coherency, consistency and reachability. For this, a methodological approach, with associated methods and tools has to be defined with respect to genericity and compatibility with information exchange standards. The information system design cycle should deal with the overall product life-cycle, starting from product andlor production information specifications thus going up to data distribution on tags and functional access architecture definition. Such an approach could give a degree of freedom to the machines which can focus to production operations and not on information management. This will reduce the increasing cost of development, maintenance and modifiability of them.

The Mobne DAtabase Nodes (MIDAN) concept is proposed as a logistic support to heterarchical or multi-agents control architectures developed by [START_REF] Duffie | Nonhierarchical Control of Manufacturing Systems[END_REF]. These architectures propose to reduce the complexity of the automated manufacturing and handling machines by considering them as partners of products, acting like controllers of their own transformations : the machine only has to be able to perform manufacturing tasks, thus cooperating under the leadership of product moving around in the manufacturing environment. Such an approach is currently under development on the IMS-Esprit project "Holonic Manufacturing Systems" [START_REF] Bongaerts | Schedule Execution for a Holonic Shop Floor Control System[END_REF].

Individual information about each product must be described with a high precision level involving some management problems of a so high volume of data. Furthermore, the relationship between a product and its data must be stable in space and time. A solution could be to physically associate product data and the product by using the electronic tag technology.

A product or any other mobile object carries its own information on a programmable tag which can be accessed in read/write mode by each user involved in its development process. The novelty of this approach is that machine capacities are focused on operations (transport, measures, ... ) and not on management nor transmission of product information, this particular point is conceptually assumed by each product. Unfortunately, electronic tags are by now not easy to be integrated in an enterprise information system. It is impossible to read or write structured information on these systems because they use elementary data handling protocol instead of high level and semantic information manipulation dialogue. In this field, the MONOLIN project (MObile NOde Logistics and Industrial Network, ESPRIT 6936) is working on the basic requirements for a standard identification systems network interconnection interface allowing integration : the Escort Memory System-Application Programme Interface so called EMS-API.

Our proposal is a methodology, complementary to MONOLIN, based on international standard work (STEP ISO 10303) and supported by classical tools (EXPRESS, SQL, ... ) in order to consider tags as MobIle DAtabase Nodes parts of a global distributed database.

PRODUCT INFORMATION MODELLING

The CIM concept emphasises interconnection of technical and management functions from design to manufacture. These functions have to share information concerning market, process and products.

For a CIM process chain, it is widely recognised that the integration of tools and databases requires standard interfaces, from CAD to workshops, based on a common data model and using a common language (lexical, grammatical and semantical).

Information Data Sharing Techniques

Presently, two solutions are available to solve the data sharing problem:

• A strong connection through a multi-users network database. Each tool can store data on the database via a common API (Application Programme Interface). Such an interface could be for example the SQL language .

• An exchange through data files in a standard format (see IGES, SET, VDA, ... ). The main problem of this technique is the impossibility of real time exchange of modifications and also an important data redundancy. It is a simple solution and does not obliged to break: down software structures.

International Product Model Effort : STEP ISO TC184/SC4

One of the most significant approach today in product modelling is the development of ISO 10303 standards called STEP (STandard for Exchange of Product model data) which define models, database access and neutral data files format for representation and exchange of product data [START_REF] Wilson | A View of STEP[END_REF]). The goal is to define complete models for product life-cycle in a CIM context, as well as the means for exchange of data between enterprise functions along a product life-cycle definition (from CAD to production, maintenance, .. . ).

In STEP data modelling is realised with the help of an object-oriented modelling language EXPRESS and data exchange through neutral files (STpP part 21) or via a database interface SDAI (Simple Database Access Interface). STEP do not considers implementation infrastructure. In this field, projects like Esprit IMPPACT and Esprit NEUTRABAS [START_REF] Brun | A Neutral Product Database for Large Multifunctional Systems[END_REF]) have already obtained significant results re-used in project Esprit AlT. STEP and NEUTRABAS results are one of the basis framework of our research work.

PRODUCT-DRIVEN MANUFACTURING

Some researches in the field of control architectures or in the field of distributed artificial intelligence systems consider that products can be seen as intelligent entities as well as machines. In such a case, product circulation must involves product data circulation in the overall plant: each product is a vector of its data model and the server of its own information for the process.

ProductlProcess Interrelationship

We consider every product as a real actor within its process, i.e. it is able to :

• Co-operate with other entities that have to participate in its evolution (e.g. machines, other products); • Adapt itself to its environment; • Participate to the acquisition of process knowledge it is exploring.

Figure 1 Process/product relationship based on the client/server model.

The schema Figure 1 points out the concept adapted from the client/server model we chose in order to respond to these objectives. In such a case each product is the manager of its own information and gives them to the process after it received a service request ("What is the next operation to be proceed on yourself' for example).

On the basis of a client/server relationship between product and process, user requests (i.e. process requests) are sent to an interface supporting product access methods in an objectoriented interrogation form compatible with STEP-SDAI query language.

The data storage medium, in our case, consist of identification tags carried by products or by pallets and also an optional network database. Explanation about the choice of two possible storage medium is made in paragraph 3.3.

Product Model Translation for Manufacturing Control Services

The main problem is the integration of tags in the information system i.e. :

• Reference conceptual models to put the information about the products themselves on tags; • Tools to implement these models and to access the information.

For the purpose of our work, we assume that a conceptual model of product for manufacturing operations exists and is defined in the EXPRESS language (this model could be a STEP application protocol or a specific model)

MlDAN domain

Figure 2 [START_REF] Chaxel | From STEP Product Modelling to Product Manufacturing: an Approach using Identification Tags[END_REF] shows the STEP-based methodology allowing translation of product model, described in the EXPRESS formalism (entities, attributes, functions, rules) which is used to generate a product-information access library supporting a direct requesting of target application on the EXPRESS schema data and to create the database structure. In the same way, the product definition within the STEP neutral fIle is used to create the tag memory image, for the first state of a product associated to the current phase of its life cycle in the manufacturing environment and if needed, to populate the previously created database schema.

A short example of the Operator interface configuration (Le. programming) is shown in Figure 3. The C code is obtained by an automatic translation from the EXPRESS schema. In the case of a full SCHEMA we have:

• One reading function per entity <TAGSREAD_Entity> implementing the two SDAI functions <GetAttr> and <lndexedGet>. The resulting code has to be linked with a library providing the called functions like

Locklnst(), TAGSjnt(), TAGS_Str(), ... and some communication functions like Send_Value(), Send_Error(), ... in order to generate the Functional Application Interface.

Remote Database Reachability

Tags are of course limited in capacity, and storing complex structures (ENTITY) needs more memory than only storing the contents (Attributes value). Today, tags tend to have higher storage capacity (up to 128 KBytes) but in some cases (like automotive industries) this may not be enough, so if it is relevant for the application a part of the data could be stored on other medium.

A network database in client/server mode seems to be the natural architecture to manage the overflow of data :

• We need to have a persistent computer storage in order to manage EXPRESS schema population and translation;

• It stores and manages high volume of data;

• Time response is efficient for most of the applications;

• And enterprises are already trained with such an architecture.

Part Eight Production DaIa Technology

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Figure 4 State graph of product data remote access from a process point.

Our approach has the same philosophy as distributed database that is, "users do not have to know where data are stored to access to them". The system performs data locating, formatting and transferring through the structure (see Figure 4). In fact, when we populate the schema, we specify where EXPRESS entities must be (on tags or remote database). Afterwards, the user can send a request to the system, without specifying the location of the objects he wants to access.

In a prototype application, we restrained the features to static objects storage (data just read by users) on the remote database and static or dynamic objects (data read or written by users) on tags.

The search for physical storage of objects is made after each user request reception, as shown in the model exposed in Figure 4 :

1. If object attributes are stored on the tag (2), the message treatment is realised (6); 2. If not, an identification of the product is made by reading its distinctive characteristics (product identification number, product type identification number) (3); 3. After which, we get data from the database (4) and we realise the treatment (6).

Application Tool

Our prototype Tag-STEP (see Figure 5), running under MS-Windows, uses a standard computer to generate the MobDe DAtabase Nodes structure. It is connected to a relational network database management system (ORACLE V7) through TCP-IP protocol.

The EXPRESS schema is converted in a relational schema which has to be put on the ORACLE database. In this case each EXPRESS entity is translated to a relational entity (a table) and to a view in order to manipulate sUbtyping. Such a conversion has already been made by the National Institute of Standards and Technology [START_REF] Eggers | Implementing EXPRESS in SQL[END_REF]).

The schema is also converted as a C programme to generate the Functional Application Interface as explained before. This program is able to read or write data independently on tags or on the database.

Model

The first step was to models the cell functions by using the SADT method on an computer tool. The designer has to lets appear a special function "To Access Production Information" each time a sub-function on the model need product related information (like the parameters for the next operation). This function is fully supported by the Functional Interface described previously.

To validate the functional model we have chosen the associative method which consist in the description of all the terminal functions by using a behavioural model in Petri Nets, Ladder, pseudo-code or programming language.

Behavioural Model

The SADT model is automatically transferred on a tool call SPEX. Firstly it permit to specify the type (in the computer sense) of each arrow between functions. Secondly, the designer can describe the dynamic behaviour of each function. Thirdly it is possible to simulate the model by adding control panels. Moreover, behaviour descriptions are made in PLC languages like Grafcet, Ladder, C, and they can be exported into a tool CADEPA to program the cell PLCs.

The functional aspect can be cover by using the SADT analysis tool, SPEX and CADEP A.

For a full description of the methodology see the paper by G. [START_REF] Morel | Outline for Discrete Part Manufacturing Engineering[END_REF].

Informational Model

Each product information requirement is specify by the input and the output flow on all the identified functions "To Access Product Information". These flows are the basis of the product data model analyse process. In our application case we are using a tool supporting EXPRESS-G able to generate the corresponding EXPRESS model. It is used as the data entry point for Tag-STEP. This EXPRESS-G model describes the assembly operation sequence for the FMR, the quality control result of each operation, and information about product history like operation time and duration.

Models integration and simulation

Tag-STEP generates two types of product data access interface (a programme) one for a real use on a computer connected to a escort memories controller and the other for a simulation use onSPEX.

So we can play with the interface in order to test the data model and also to play with SPEX to test interactions between the information model and the process behaviour model. This coupled information-behaviour simulation gives interesting results about electronic tags (in words of baud rate and capacity).

By now the real implementation on the FMR is not yet achieved and the simulation is our actual ending point.

,Figure 2

 2 Figure 2 From EXPRESS schema to users requests.

}Figure 3

 3 Figure 3 EXPRESS to the C Interface translation.

  • One writing function per entity <TAGSWRITE_Entity> implementing the two SDAI functions <PutAttr> and <lndexedPut>.

	EXPRESS: ENTITY Machine; Id_Machine : INTEGER; Name : STRING (20); END_ENTITY; IDTAQS Interface: void id_Inst, Attributl int id_Inst, Attribut; {/* Physical position of attributes in data block */ int P_Name=2; /* Attributes Value */ int Id_MachineVal; A	(LockInst ('Machine', switch (Attribut) { case 1 : TAGS_Int Send_Value (2, break; case 2 : TAGS_Str (P_Name, NameVal); id_Inst)==TRUE) &Id_MachineVal); Send_Value (strlen(NameVal) , NameVal); break; default: . Send_Error (ATT_NOT_DEFINEDl; } ; else Send_Error (ENTITY_INST_NOT_FOUNDl;
	char	NameVal[20];

CONCLUSION

Electronic data carriers for distributed manufacturing information management is taking more and more importance in industrial applications. This new approach for information management necessitates methodological framework for Information System implementation as well as interfacing tools based on international standard definitions. Our proposal dealing with STEP formal models has demonstrated the feasibility of high level information structuration, based on product data definition, with advanced management technics on attached escort memories.

225

To populate the schema we use a STEP neutral file which has to be divided in two parts. The first is containing data stored on tags and the second data stored on the database. Each file is the characterisation of a product type circulating in the workcell.

APPLICATION ON A FLEXIBLE MANUFACTURING RING

The application site, is a flexible assembly ring with four independent stations. Each station is controlled by a PLC connected to the cell network. Information about products can be read at the station entry point and on the station itself by sending request to the Functional Application Interface. Because identification system controllers are not sufficiently open, each Interface is constituted by a computer and by an identification system controller.

- The cell modelling process is done by using different methods to cover the functional, behavioural and informational points of view described on the Figure 6.


Didier Cavallera

Vinicius Oiring

Jean-Louis Coulomb

Olivier Chadebec

Bruno Caillault

François Zgainsky

A new

Keywords: Magnetic flux leakage, magnetic field measurement, remanence, transformers

I. INTRODUCTION

During the re-energization of unloaded power transformers, undesirable effects (inrush currents, overvoltages, …) can appear and may cause many problems in power system, mainly to the transformers. The determination of residual flux allows us to calculate the best closing time for the reenergization of a power transformer. The systems of controlled switching applied to transformers are used for reducing the magnitude of the inrush currents [START_REF] Brunke | Elimination of Transformer Inrush Currents by Controlled Switching-Part I: Theoretical Considerations[END_REF]. However, the determination of the residual flux isn't accurate. Indeed, the measurement of the residual flux is uncertain due to low voltage signal including DC component and noise. Furthermore the influence of system transients near transformer on the residual flux has been recently underlined [START_REF] Corrodi | Influence of system transients on the residual flux of an unloaded transformer[END_REF]. This paper proposes a new method to calculate the residual flux from near magnetic field measurements. Part II presents the link between the leakage flux measured near a transformer and the residual flux in the magnetic circuit. Part III explains the new method for estimating the residual flux.

II. LINK BETWEEN THE LEAKAGE FLUX MEASURED NEAR A TRANSFORMER AND RESIDUAL FLUX OF A MAGNETIC CIRCUIT

A single-phase transformer (400V/4V, 32 kVA, 2 limbs) is chosen for this study. Some magnetic sensors are placed outside the magnetic circuit of the transformer (without tank). In Fig. 1, a schematic view of the transformer and some of the sensors positions are exposed. The measurement range of the sensor (fluxgate) is ±100 µT and the bandwidth is DC up to 3 kHz. For each sensors, the component of the magnetic field, tangential to the magnetic circuit is taken into account.

A. The leakage flux measured near a transformer

The test system consists in re-energizing an unloaded transformer. In a precedent paper [START_REF] Cavallera | A simple model to explain the leakage flux measured around an off line transformer[END_REF], we realized some finite element calculations that proved the existence of a leakage flux around the transformer when it is off line. The leakage flux is measured by sensors. The electric measurements done, including the current in the primary winding allows to prove that the residual flux has changed during the test measurements, the graph will be exposed in the final paper.

B. Distinction of residual flux thanks to the leakage flux

When the transformer is unloaded at nominal voltage (400 V RMS ), the measured field by a sensor is the sum of the magnetic fields induced from: magnetic circuit leakage, coil field and the Earth's magnetic field. The main issue is to extract the field created by the magnetic circuit leakage from the total field. In order to extract this field, an algorithm has been developed. For zero current values, the field induced by the coil is zero, at these instants, the total induction measured by a sensor is interpolated. An example of this method is presented on the Fig. 2 and Fig. 3. The curve B(I), Fig. 3, has not a conventional rotation sense, because it is not the induction measured inside the magnetic material but an induction measure in the area. We will explain these measurements thanks to numerical field computations. In Fig. 3, two steps corresponding respectively to the points: B r + (point (4) or green) and B r -(point (2) or orange) ( Fig. 2.b). When the de-energization occurs, the current goes to a maximum negative value A new method to evaluate residual flux thanks to leakage flux. Application to a transformer D. Cavallera (1) , V. Oiring (2) , J-L. Coulomb (1) , O. Chadebec (1) , B. Caillault (2) and F-X. Zgainski (2) (1) Grenoble Electrical Engineering Laboratory (Grenoble-INP, UJF, CNRS), Grenoble, France (2) EDF/DTG (EDF Group-Hydro Generation &Engineering), Grenoble, France didier.cavallera@g2elab.grenoble-inp.fr

(3)

(5)

(point (3) or blue on the Fig. 2.a) and then realizes a short incursion in the positive part (point (5) or yellow). Then, the residual flux takes a value between B r + and B r -(point (6) or red) following what is classically called a"reverse curve". The current measurement is noisy, by consequence the determination of the induction at zero current is difficult (especially the determination of the red point on Fig. 2.b). Recent developments have improved this determination and will be presented in the final paper. The final value of the induction is -10 µT. The ambient induction value measured is near -12 µT (mean value between the green and orange points). The residual flux creates on this sensor a field of +2 µT. We succeed in characterising the different field effects and we isolate the residual flux effect thanks to one field measurement.

III. METHODS TO EVALUATE RESIDUAL FLUX IN A TRANSFORMER

There are several ways to calculate the residual flux. Nowadays, the most used method is the voltage integration; however, it can present many uncertainties regarding with voltage measurement quality. Another method consists in using the "maximal current" value, the principle is to calculate the residual flux from inrush current obtained when the transformer is re-energized [START_REF] Yacamini | Numerical calculation of inrush current in single phase transformers[END_REF]. This method is unappropriate for industrial applications, because we will know the value of the residual flux only after the reenergization of the transformer. On the other hand, it is an excellent method to check the value of residual flux.

A. A new method to estimate the residual flux

The method proposed consists in estimating the residual flux from a magnetic field measurement sensor (fluxgate). In order to do this, two steps must to be respected. The first step is to establish the transfer function between induction and flux when the transformer is at its nominal voltage in steady state. The induction is measured by magnetic field sensor and the flux is obtained from the voltage integration, measured on the secondary winding of the transformer. Once we have these parameters (flux and induction) it is necessary to obtain the two values of the flux, and of the induction, when the current is zero. After that, it is necessary to do the subtraction of each pair of values (flux and induction), with this operation we will obtain delta flux (∆Φ) and delta induction (∆B), the division between (∆Φ) and (∆B) is the transfer function. The second step is to measure the induction when the transformer is off line. With this signal, we will obtain a new value, delta residual induction (∆B r ). In order to do this, it is necessary to do the subtraction between the induction and the mean value of induction, measured when the transformer is off line and on line, respectively. This new value (∆B r ) matches the residual flux in the transformer. Finally, to get the residual flux (∆Φ r ), we simply multiply the residual induction by the transfer function. This equation is shown hereafter (1) and it will be illustrated in the final paper.

B B r r ∆ ∆ × ∆ = ∆ φ φ (1)

B. Application with a sensor on the magnetic circuit

In this part, we present the results obtained for the singlephase transformer chosen. It is necessary to remember that the magnetic field sensor (sensor 1) is located on the magnetic circuit. The new method is consistent with the maximal current method in five of eight cases of energization. The new method presents wrong results regarding with integration voltage method for the cases 2, 3 and 8. The reason for this is the noise present in the current measurement by consequence the determination of the transfer function is difficult. On the other hand, the new method is more performing than voltage integration method for the case number five. The reason for this is that the new method takes into account all possible changes of the residual flux, because the delta residual induction ( r B ∆ ) is obtained in real-time just before the energization of the transformer.

IV. CONCLUSION

In this paper, a new method to estimate the residual flux in the magnetic circuit of a transformer is presented. Magnetic field measurements are done near the transformer with magnetic field sensors. We do the calculation of the residual flux value by isolating the magnetic field induced by the leakage flux. The new method of residual flux calculation gives coherent results and its major advantage is it takes into account all possible changes of the residual flux, however, it can presents inaccurate results regarding with current measurement quality. Concluding, this method is straightforward and presents a good agreement with the existing methods.

Fig. 1 .

 1 Fig. 1. Schematic view of the device

Fig. 2 .Fig. 3 .

 23 Fig. 2. Current measured and induction interpolated during test

TABLE 1 :

 1 The results are expressed in percentage of the nominal flux in the transformer (1.8 Wb). The results shown in the table 1 represent the results for the residual fluxes calculated for a sequence of eight energizations. RESIDUAL FLUX CALCULATED BY DIFFERENT METHODS

	Before	Maximal	Voltage	New
	Energization	Current	Integration	Method (%)
		Method (%)	Method (%)
	1	-18	-18	-18
	2	-18	-15	-13
	3	17	19	26
	4	-18	-18	-17
	5	8	15	8
	6	17	18	18
	7	17	20	16
	8	-18	-16	-27