Alain Damas
email: alain.damas@airliquide.com

Daniele Negro

Denis Leducq

Anthony Dallais

Graciela Alvarez
email: graciela.alvarez@irstea.fr

Tim Brown

Judith Evans

Alan Foster
email: alan.foster@lsbu.ac.uk

Anthony Dallais

Antony Dallais

Small-scale demonstrator of a cold thermal store for liquid air energy storage

Keywords: Thermal store, Liquid Air Energy Storage (LAES), thermocline, packed bed

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The EU has already met its 2020 renewable energy target (Renewable energy directive 2009/28/EC) of 20% final energy consumption from renewable sources by 2020. By 2030 the revised Renewable Energy Directive (2018/2001) established a new binding renewable energy target for the EU for 2030 of at least 32%. As the EU and the rest of the world moves towards even more renewable energy, supply becomes more dependent on factors outside the control of the energy supplier, e.g. weather variations. This causes an issue for grid balancing, as renewable electricity generation is not easily controllable. One method to balance supply and demand in power generation is to store energy during periods of low demand and use it at high demand. [START_REF] Lehmann | Can storage help reduce the cost of a future UK electricity system[END_REF] stated that "The UK can realise significant cost savings if market arrangements for the electricity system allow for an efficient deployment and use of energy storage…" For renewable generation to take a larger share of the energy market, as required by climate targets, energy storage will become more necessary. According to [START_REF] Barton | Energy Storage and its Use with Intermittent Renewable Energy[END_REF] short-term storage, covering less than 1 hour, offers only a small increase in the amount of electricity that can be absorbed by the network. Storage over periods of up to one day delivers greater energy benefits but is significantly more expensive. According to [START_REF] Roberts | A tiny, beleaguered government agency seeks an energy holy grail: long-term energy storage[END_REF] "We have the storage technology to smooth out hourly swings, but we still don't have anything that could cover a fallow period of wind and sun that lasted days, months, or even years. If we want to get variable renewable energy up to 60 percent, 80 percent, or even more of our electricity, we need long-term energy storage. It is the missing puzzle piece, the holy grail."

Current technology, e.g. batteries, are not a cost-effective solution for long term storage as the cost scales linearly with energy capacity. Pumped hydro and compressed air (CAES) are not easily deployable due to their geographical and geological requirements. Cryogenic energy storage (CES) makes use of lowtemperature liquids as an energy storage and transfer medium. CES can provide large-scale, long-duration energy storage of 5 to 1000 MWh (Brett and Barnett, 2014). The use of liquid air to store energy, which can then be recovered as mechanical work, dates back to 1900, when the Tripler Liquid Air Company developed a liquid air car.

One of the current issues with CES is its low round trip efficiency (RTE, the ratio of energy recovered to energy in) compared with pumped hydro electrical storage, compressed air energy storage and Li-ion batteries.

A liquid air energy storage system has been demonstrated by Highview Power Storage [START_REF] Morgan | Liquid air energy storage-analysis and first results from a pilot scale demonstration plant[END_REF]. They showed that the RTE could be greatly increased by recycling more of the cold from the expansion to the liquefier. The generation of cold from the expansion turbines happens at a different time than the liquefaction of the air. For this reason, the cold energy to be exchanged requires a thermal store. [START_REF] Morgan | Liquid air energy storage-analysis and first results from a pilot scale demonstration plant[END_REF] used a bed of quartzite gravel beads contained inside a shipping container to store the cold energy. They only managed to recycle 51% of their cold energy in their pilot scale plant. They stated that over 50% RTE would be achievable if the percentage of cold recycling increased from 51% to 91% and the process was optimised to maximise cooling from adiabatic expansion. [START_REF] Peng | A study on performance of a liquid air energy storage system with packed bed units[END_REF] conducted an analysis of a LAES with packed beds. They concluded that the system could achieve an RTE in the range 50-62% depending on the values of process conditions and that this is a bit lower than a CAES. However, the storage energy density is much higher for the same operational conditions.

Sciacovelli et al, 2017 modelled a LAES with packed bed thermal storage. They found that the temporary storage of cold thermal energy streams using packed beds improves efficiency of LAES by ~50%. However, due to dynamic cycling charge/discharge, packed beds can bring an undesired 25% increase in the energy expenditure needed to liquefy air.

The aim of this study was to evaluate the thermal capacity and efficiency of a small-scale demonstrator packed bed thermal store.

METHOD

The thermal tank was a cylindrical container, 0.840 m high, with an internal diameter of 0.451 m and 3 mm thick stainless-steel walls. The tank was insulated with 200 mm polyisocyanurate (PIR) (ISOPIRFLAM33®). The height of the heat storage material (HSM) in the tank was 600 mm . The total bed volume was 0.959 m 3 . The HSM was 3.25 mm mean diameter polished soda-lime glass spheres, composed of 72% silicon dioxide (SiLibeads® Type S, SIGMUND LINDNER GmbH). They had a specific weight of 2.5 kg.l -1 , roundness >0.95 and thermal conductivity of 1.129 W.m -1 .K -1 . Using the thermal properties of pure silica, the heat capacity of the glass beads at varied temperatures was calculated using the Cryogenic Material Properties Database provided by [START_REF] Marquardt | Cryogenic Material Properties Database[END_REF]. The thermal store was filled with 146.7 kg of HSM with 39% porosity. This gave a bed aspect ratio (height/dimeter) of 1.33. Liquid nitrogen was evaporated before being fed into the thermal tank. Therefore, only gaseous nitrogen entered the tank.

Instrumentation

The mean temperature of the HSM was measured using T-type thermocouples at the axial centre and edge of the cylinder at 6 regularly spaced heights. The central thermocouples were held by wire and the edge thermocouples were inserted into the wall of the thermal store. The thermocouples were calibrated at temperatures which coincided with melting ice, CO2 triple point and the boiling point of N2 at atmospheric pressure. Flow rate of nitrogen was measured by a turbine volumetric flowmeter (Hoffer Flow Controls HIT-2A). Pressure and temperature measurement close to the flowmeter was recorded to convert volumetric flow rate into mass flow rate. The weight of the liquid nitrogen tank was measured using 3 load cells (METTLER TOLEDO RLC) that provided real time weight measurement and cross checked information against the flowmeter.

Stages of operation

The thermal store operation was split into 3 separate stages; Charge -To extract heat from the HSM using the cold nitrogen, thus storing cold energy. Storage -Maintaining the temperature of the HSM between charging and discharging. Discharge -To cool nitrogen entering the HSM by exchanging heat with the cold HSM thus releasing cold energy. As nitrogen passed through the HSM a thermocline was generated. During the dynamic phases (charge and discharge), the total temperature swing was defined by Eq. ( 1)

∆𝑇 𝑚𝑎𝑥 = 𝑇 ℎ𝑜𝑡 -𝑇 𝑐𝑜𝑙𝑑 Eq. (1)
Where Thot and Tcold were the maximum and minimum temperature of the fluid recorded in the test.

During charging, nitrogen gas was driven to the store and entered at the bottom of the tank and exited at the top. Charge cycles were performed and repeated with various flow rates and ∆Tmax. Charging was performed until "breakthrough". Breakthrough was defined as the time when the fluid temperature at exit of the tank was equal to 𝑇 𝑏𝑡 as shown in Eq. ( 2).

𝑇 𝑏𝑡 = 𝑇 𝑐𝑜𝑙𝑑 + 0.1.∆𝑇 𝑚𝑎𝑥 Eq. ( 2)

In some cases this condition was not achieved, in which case breakthrough was considered as the time when temperature difference between the top and bottom was less than 1 K. A total of 6 storage sequences were performed between charge and discharge, two were longperiods (18 hours and 28 hours) and 4 were short storage periods (approximately 2 hours). There was no flow of nitrogen through the HSM during the storage stage.

During discharge, nitrogen at ambient temperature was input at the top of the store and cold nitrogen exited at the bottom.

The tests followed the following sequence; charge followed by storage followed by discharge. Before charging, the vessel was at ambient temperature.

Efficiency

During charging, the energy input of the fluid into the thermal store was as shown in Eq. ( 3).

𝐸 𝑐ℎ𝑎𝑟𝑔𝑒 = 𝑚̇∫ (ℎ 𝑖 -ℎ 𝑜 ) dt 𝑡=𝑏𝑟𝑒𝑎𝑘𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡=0
Eq. ( 3)

Where, 𝑚̇, was the mass flow rate of the nitrogen, ℎ 𝑖 and ℎ 𝑜 , were the enthalpy of the nitrogen entering and exiting the store respectively (calculated from REFPROP) and t, was time the time step of 1 s. The energy stored in the bed is shown in Eq. ( 4).

𝐸 𝑠𝑡𝑜𝑟𝑒𝑑 = 𝑚 𝑠 ∫ 𝑐 𝑝 (𝑇). dT 𝑇2 𝑇1

Eq. ( 4)

Where ms was the mass of the HSM and Cp is specific heat at the mean temperature, T was the temperature of the HSM.

The charge efficiency was defined as the ratio of energy stored and supplied shown in Eq. ( 5)

µ 𝑐 = 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑 𝐸 𝑐ℎ𝑎𝑟𝑔𝑒𝑑
Eq. ( 5)

During discharging, the energy removed from the fluid as it exited the thermal store was calculated in the same way as shown in Eq. ( 3).

The discharge efficiency was defined as the ratio of energy discharged and stored as shown in Eq. ( 6)

µ 𝑑 = 𝐸 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑑 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑
Eq. ( 6)

The storage efficiency was defined as shown in Eq. ( 7)

µ 𝑠 = 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑2 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑1
Eq. ( 7)

Where 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑1 was the energy stored in the HSM and 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑2 was the energy at the end of the storage time.

The round trip efficiency, µ 𝑅𝑇𝐸 of the storage vessel could be calculated using Eq. ( 8)

µ 𝑅𝑇𝐸 = µ 𝑐 . µ 𝑑 µ 𝑠 Eq. ( 8)

RESULTS

Charging

When charging was started, the inlet temperature was not instantaneously at the coldest temperature due to an initial lag time. The dimensionless inlet temperature evolution Tdim-bottom (Eq. ( 9)) with time is shown in Figure 1a and dimensionless outlet temperature evolution Tdim-top (Eq. ( 10)) with time is shown in Figure 1b. There was a temporary problem with data acquisition, hence the gap in data at 18.5 g/s in Figure 1a.

Tdim-bottom = (Tbottom -Tcold) /Tmax Eq. ( 9)

Tdim-top = (Ttop -Tcold) /Tmax Eq. ( 10)

It took between 340 and 1900 s before Tdim-bottom reached 0.1. It took over 1800 s for Tdim-top to reach 0.9 and between 2700 and 8500 s to reach 0.1. Not surprisingly the higher flow rate of nitrogen caused a faster reduction in temperature.

Storage

Figure 2a and Figure 2b show mean temperature evolution of the HSM recorded during short and long storage periods respectively.

Discharge

During discharge the dimensionless inlet temperature evolution with time is shown in Figure 3a and dimensionless outlet temperature evolution with time is shown in Figure 3b. When discharge was started, Tdim-top was approximately 0.5. It took between 300 and 630 s before Tdim-top reached 0.9. It took between 4800 and 9600 s before Tdim-bottom reached 0.9. Not surprisingly the higher flow rate of nitrogen caused a faster increase in temperature.

Efficiency

Table 1 shows the efficiency and energy stored during charging for the different tests. Results for charging when a thermocline was present within the storage vessel (Tdim-top = 0.9) and when there was no thermocline (charging until all vessel was at approximately the same temperature) (Tdim-top = 0.1) are given. The energy stored was obviously higher (by around 20%) when Tdim-top = 0.1 with a mean value of 2.92 kWh compared to 2.43 kWh. Efficiency was compromised when Tdim-top = 0.1 , with a mean value of 82.2% compared to 89.7%, an efficiency reduction of 7.6%. The storage efficiency, µ 𝑠 , reduced with time due to heat transfer from the ambient. Short term storage had a greater reduction in efficiency with time with a mean of 2.4% per hour, whereas long term storage was 1.1% per hour. This was due to the rate of heat loss reducing with time as the vessel approached ambient temperature.

Results are only presented for discharging, leaving the thermocline within the vessel (Tdim-bottom = 0.1), as the thermocline was considered too warm to have practical value. A mean energy recovered of 2.10 kW and efficiency of 74.2% was measured.

The µ RTE calculated depends on whether the thermocline remained in the HSM during charging and the length of storage. Assuming the thermocline remained in the HSM, after a storage time of 1 hour an µ RTE of (0.897 x 0.976 x 0.742) 0.650 was achieved, based on the mean values for the tests conducted. If the thermocline was passed through the thermal store during charging, increasing the storage capacity, the µ RTE dropped to 0.596. For storage periods of 24 hours, the storage efficiency was 0.736. This reduced the µ RTE to 0.490 when the thermocline was present and 0.449 when the thermocline was not present in the HSM. Results are only presented for discharging, leaving the thermocline within the vessel (Tdim-bottom = 0.1), as the thermocline was considered too warm to have practical value. A mean energy recovered of 2.10 kW and efficiency of 74.2% was measured.

Table 2 shows the efficiency and energy recovered during discharging for the different tests. Results are only presented for discharging, leaving the thermocline within the vessel (Tdim-bottom = 0.1), as the thermocline was considered too warm to have practical value. A mean energy recovered of 2.10 kW and efficiency of 74.2% was measured.

CONCLUSIONS

A thermal store was built and tested to ascertain the efficiency of storing cold from gaseous nitrogen, evaporated from cryogenic temperatures. The thermal store was able to store 2.43 kWh at high efficiency and 2.92 kWh at low efficiency. The round trip efficiency of the thermal store ranged between 0.596 and 0.650 when storing for 1 hour and 0.490 and 0.449 when storing for 24 hours.

Figure 1a .

 1a Figure 1a. Inlet fluid dimensionless temperature history for different flow rates during charge.

Figure 1b .

 1b Figure 1b. Outlet fluid dimensionless temperature history for different flow rates during charge.

Figure 2a .

 2a Figure 2a. Mean bed temperature evolution during short term storage.

Figure 2b .

 2b Figure 2b. Mean bed temperature evolution during long term storage.

Figure 3a .

 3a Figure 3a. Inlet fluid dimensionless temperature history for different flow rates during discharge.

Figure 3b .

 3b Figure 3b. Outlet fluid dimensionless temperature history for different flow rates during discharge

Table 1 . Efficiency and energy stored in the charging process.

 1

	Tdim-top = 0.9	Tdim-top = 0.1

Table 2 . Efficiency and energy stored in the discharging process.

 2

	Tdim-bottom = 0.1

6th IIR Conference on Sustainability and the Cold Chain. April 15-17, 2020. Nantes, France

ACKNOWLEDGEMENTS

The CryoHub project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No 691761.


J Béard

J Agil

R Battesti

C Rizzo

A novel pulsed magnet for magnetic linear birefringence measurements

Keywords:

In this paper, we describe a novel pulsed magnet, called Foil Coil, which can deliver a field transverse to the light propagation of more than 10 T over about 0.8 meters operating without cryogenic equipment. It has been designed for linear magnetic birefringence measurements. We report on testing the coil, and also show some physics data taken in vacuum during its commissioning in the framework of the BMV apparatus, with special attention to noise induced by the pulse itself. Finally, we compare the preliminary results obtained here, with data from the previous BMV coil.

I. INTRODUCTION

Magnetic linear birefringence is a non linear optics effect that has been separately discovered in 1901 by Kerr [1] and in 1902 by Majorana [2]. This effect is also known as the Cotton-Mouton effect, since between 1905 and 1907, Cotton and Mouton have published at least 21 papers concerning this effect in liquids and colloidal solutions [START_REF] Cotton | Notice sur les travaux scientifiques[END_REF]. Due to the smallness of the effect, first measurements in gas have been performed only in 1933 [START_REF] Tsai Belling | [END_REF] and 1934 [5]. Indeed, the first observation of the effect in a helium gas, the species showing the smallest effect, has been reported only in 1991 [6].

A linearly polarized light, propagating in a region where a uniform magnetic field perpendicular to the light wave vector exists, becomes elliptically polarized. In the case of gases, where Ψ 1 and the index of refraction n ≈ 1, the acquired ellipticity Ψ can be written as

Ψ = π L B λ ∆n sin(2θ), (1)
where L B is the length of the region where the magnetic field is present, λ is the light wavelength, ∆n = n -n ⊥ is the magnetic induced birefringence i.e. the difference between the index of refraction for a light polarized parallel to the magnetic field, n , and one polarized perpendicular to the magnetic field, n ⊥ , θ is the angle between the light polarization and the magnetic field of amplitude B.

A characteristic of magnetic linear birefringence is that the effect is proportional to the square of the magnetic field amplitude.

∆n = K CM B 2 (2)
where K CM is a constant depending on the gas species, pressure and temperature, and on the light wavelength. K CM has been theoretically studied, and measured for a large number of gases [7]. For example, for one atmosphere of helium gas, at 0 • C, K CM = 2.4 × 10 -16 T -2 (see e.g. [8] and references therein).

Non linear electrodynamics, a theoretical framework that contains classical electrodynamics as the lowest order, and Quantum ElectroDynamics (QED) as a special case, predicts that a linear magnetic birefringence should also appear when light propagates in a vacuum in the presence of a transverse magnetic field [9]. The QED prediction is that K CM = 4 × 10 -24 T -2 [10] i.e. that a vacuum induces the same effect as around 5×10 17 helium atoms per m 3 at standard conditions.

Vacuum linear Magnetic Birefringence (VMB) measurements are therefore important tests of the standard model physics and also of physics beyond the standard model (see e.g. [11]). Vacuum K CM remains to be measured, notwithstanding more than a century of different attempts to observe it [10]. To date, the best limit has been obtained with the PVLAS experiment [12] using as the source of the magnetic field two permanent magnets, giving K CM = (1.9 ± 8.1) × 10 -23 T -2 at 99.7 % confidence level i.e. a coverage factor k = 3 [START_REF] Jcgm | Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF] [START_REF]We use the Uncertainties Python package to handle error propagation in our paper Uncertainties: a Python package for calculations with uncertainties[END_REF]. This confidence level will be used throughout this paper.

In 2001 a novel proposal has been put forward, [START_REF] Askenazy | AIP Conference Proceedings[END_REF] and consequently an experiment, called Biréfringence Magnétique du Vide (BMV), has been installed at the Laboratoire National des Champs Magnétiques Intenses (LNCMI), Toulouse, France. More recently, another experiment, called Observing Vacuum with Laser (OVAL), based in Tokyo, Japan, also using a pulsed magnet, has been mounted [START_REF] Fan | [END_REF]. As far as we know, BMV and OVAL are the only attempts to measure VMB which are operational at the moment. The latest BMV results have been published in 2014, with K CM = (8.3 ± 8.0) × 10 -21 T -2 [17]. Recently, OVAL has reported

K CM = (0.3 ± 1.6) × 10 -20 T -2 [18].
The difficulties of such a pulsed field approach to VMB can be summarized as follows. One needs to have a very sensitive interferometer, capable of measuring phase delays of a few 10 -9 radians, while injecting thousands of amperes in a coil driven by a power supply delivering hundreds of kJ's over a few milliseconds i.e. a few 10 8 W. Clearly, magnet field technologies, optics techniques, and how to couple them are the main subjects to be treated in the following. We start therefore by giving an overview of the different choices made over time as far the mag-netic field production is concerned. We stress, that any improvement in linear magnetic birefringence measurements has been triggered by advances in the magnetic field technology. Then, we describe the novel pulsed magnet that we have developed, called Foil Coil, which can deliver a field transverse to the light propagation of more than 10 T over a distance of 0.8 meters, operating without cryogenic equipment. During the commissioning, we have also taken some physics data. We recall the main facts concerning our optical apparatus and our data analysis method, and we report on our investigation of the noise induced by the magnetic pulse itself. We also give results in term of VMB. The new results are indeed encouraging, and we explain why by comparing them with the previous BMV measurements [17].

II. THE MAGNETIC FIELD

If one considers equations 1 and 2, the main experimental parameter to maximize the signal is the factor B 2 L B . In the case of a 1907 Cotton-Mouton measurement [START_REF] Cotton | [END_REF], B 2 L B was about 0.14 T 2 m, corresponding to 1.85 T over 4.3×10 -2 m, and the reported K CM ≈ 10 -7 T -2 . To improve the measurements, improved magnets have always been needed. The first gas measurements of Tsaï Belling [START_REF] Tsai Belling | [END_REF] and Cotton and Tsaï Belling [5] have been performed using the "Bellevue" electromagnet installation [20] in the Paris neighborhood. This instrument can be considered as one of the first physics facilities in the world. For the 1934 measurements, an extra coil was designed and used, increasing the maximum field from about 4.5 T to about 6.1 T over a distance of around 0.3 m, approaching 10 T 2 m and reaching a K CM ≈ 1.5 × 10 -10 T -2 .

The theoretical aspects of this linear magnetic birefringence have been elucidate by Buckingham and Pople in 1956 [21] and novel gas measurements have been performed in 1967 when Buckingham and collaborators renewed the field from the experimental point of view [22]. The magnet used in the 1967 experiment was an electromagnet providing a field of about 2.8 T over half a meter i.e. about 4 T 2 m. The noise floor was about K CM ≈ 1.5×10 -15 T -2 showing a considerable improvement with respect to the Cotton and Tsaï Belling experiment even if the B 2 L B of the two experiments was quite similar. This magnet was not so different from the one used in 1932 and 1940 by Farr and Banwell in one of the first attempts to detect any variation of the velocity of light in vacuum induced by the presence of a magnetic field [10].

In the 1980's, a new approach has been developed. Carusotto et al. introduced a modulation of the effect to be measured to increase the signal to noise ratio. This modulation was obtained by modulating the driving current of an electromagnet at about 0.4 Hz [23] or by turning the entire magnet around its own axis at 0.9 Hz i.e. changing the θ angle periodically (see formula 1) at 1.8 Hz frequency [24]. The noise floor was about the same that the 1967 measurements [22] but the B 2 L B was less than 0.05 T 2 m. Naively, to further decrease the noise floor one should have a B 2 L B as high as possible, coupled with a frequency modulation as high as possible. This applies, in particular, to vacuum linear magnetic birefringence measurements which have been also put forward again in the 1970's [10]. The design of VMB experiments are therefore more complicated than the gas one, and standard electromagnets are no longer the best choice for this kind of application.

Long superconducting dipole magnets have been developed for particle accelerators. In the 1990's, two of such magnets have been used by the BFRT (Brookhaven, Fermilab, Rochester, Trieste) collaboration [25] whose apparatus was installed at the Brookhaven National Laboratory at Upton, New York, USA. In the framework of this collaboration Cotton-Mouton effect of neon gas [26], and helium gas [6] has been measured for the first time. The total magnet field length was 8.8 m, and the field was modulated at 3.2 × 10 -2 Hz between 2.63 T and 3.87 T, by modulating the driving current of the superconducting coils maintained at 4.7 K. The effective B 2 L B parameter was about 35.5 T 2 m. The K CM noise floor was about 1.5×10 -20 T -2 [25]. To reach such a low noise level, light was kept in the magnetic field region by a multipass optical cavity consisting of two mirrors on both sides of the magnets. Light passed up to about 550 times through the magnetic region before being analyzed.

The use of optical cavities increases the total path and the acquired ellipticity, but adds an extra level of complexity to this kind of experiment. Actually, the optics become challenging, for example, in the BFRT experiment light traveled about 5 km before exiting the magnetic field region.

The end of the BFRT attempt corresponds to the start of the PVLAS collaboration that, after 25 years of work, has achieved the best result to date [12]. In the PVLAS experiment, the effect modulation has been obtained by rotating the entire magnet around its own axis as in ref. [24]. Initially a superconductive magnet providing 5 T over 1 m has been used. This was subsequently replaced by two permanent magnets, rotating at about 5 Hz, providing 2.5 T over twice 0.85 m i.e. a B 2 L B of about 10 T 2 m [12]. To increase the effect to be measured, a resonant optical cavity has been used. Light is therefore trapped in the magnetic field region for milliseconds on average. Millions of seconds of integration time have been necessary to reach a K CM = (1.9±8.1)×10 -23 T -2 [12].

A novel approach based on the use of pulsed magnets has been suggested in 1998 [27]. Pulsed magnets [11] are electromagnets, typically cooled to liquid nitrogen temperature, that are driven by the discharge of a bank of capacitors. A field of up to 100 T has been obtained, [11] over several milliseconds, in a bore of 1 cm diameter. The field is usually in the Faraday configuration, as in the case of a standard solenoid. The use of pulsed fields is interesting because they are very intense and rapidly modulated. In ref. [27] the suggestion was to use a pulsed magnet having a radial access to have the field perpendicular to the light wave vector, as required to measure Cotton-Mouton effects. This kind of coil is called a split coil since it is obtained by splitting a pulsed solenoid coil in two, introducing a spacer between the two parts. Obviously, the field obtained at the level of the radial access is smaller than in the case when the solenoid is not split into two.

The most expensive part of a pulsed field installation is the power supply, essentially a bank of capacitors. Its size and therefore its cost depends on the energy to be injected into the pulsed coils. In the case of split coils, to create a transverse field for magnetic linear birefringence measurements, most of the energy delivered creates magnetic field in regions that light does not cross, while the effective B 2 L B is somewhat limited by the bore dimensions of a few centimeters.

In the year 2000 proposal launching the BMV project [START_REF] Askenazy | AIP Conference Proceedings[END_REF], following directly the 1998 paper [27], a different design for the pulsed magnet was therefore proposed [28]. We refer to it as a plate-coil since it is based on the idea to allow current flow through plates parallel to the region where the magnetic field is needed, thus the field obtained is perpendicular to the light wave vector by construction. In the framework of the BMV project, construction tests have been conducted at the LNCMI, Toulouse, France, where the BMV experiment is installed. In particular, in 2002 a field of about 2.5 T has been obtained using a plate-coil, [29] however, these tests confirmed that the main technical difficulty was to transport the current from one plate to another, during the pulse, without destroying the connection itself.

Therefore, the BMV best results [17] to date have been realized using what we call an Xcoil [30]. In this pulsed magnet design, to maintain the magnetic field in the region crossed by the light as far as possible, the two electromagnets with a racetrack shape are disposed in an X geometry. In the test phase, the version used in ref. [17] reached more than 14 T over a length of 0.13 m corresponding to 25 T 2 m. The total duration of a pulse is a few milliseconds. The magnetic field reaches its maximum value within 2 ms which is of the same order of magnitude as the storage time of photons in the optical Fabry-Perot cavity that is used to increase the optical path in the magnetic field region. The optical cavity acts as a low pass filter and the ellipticity induced by the magnetic field is therefore not proportional to the square of the field but to the filtered square of the field [31]. When the cavity filtering is taken into account, as shown in ref. [31], B 2 L B is no longer the parameter to be optimized, because increasing L B involves increasing the cavity length, reducing its bandwidth, which in turn increases the filtering effect on B 2 .

An even larger version of the Xcoil, called the XXLcoil, has been designed and constructed at the LNCMI, Toulouse, France [32]. This pulsed coil has reached a field higher than 30 T when a current higher than 27000 A is injected into it. This corresponds to more than 300 T 2 m [11]. This coil requires a large liquid nitrogen cryostat, and the overall dimensions are hardly compatible with the BMV table top experiment. The construction of such a coil has proved to be difficult to manage even in a framework of a European standard high magnetic field laboratory such as the French LNCMI.

For all these reasons, we decide to design and manufacture a novel pulsed magnet which could provide a high field without the need to be cooled. We called it Foil Coil. It is constructed using a single copper foil as explained in the following of this paper.

As already explained, the OVAL experiment is also based on a pulsed magnet. The latest version [START_REF]Search For Vacuum Magnetic Birefringence with a high repetitive pulsed magnet[END_REF] consists of two race-track coils [START_REF] Fan | [END_REF] in a Helmholtz configuration, cooled to liquid nitrogen temperature. This magnet operates at a peak value of field of 8.3 T over a field length of 0.17 m, and at a very high repetition rate of 0.05 Hz. The results reported in [START_REF]Search For Vacuum Magnetic Birefringence with a high repetitive pulsed magnet[END_REF] have been obtained accumulating 2.6 × 10 4 pulses.

III. THE FOIL COIL

A. Design

This new magnet has been designed with several constraints, some are imposed by physical or technical limitations, while others can be chosen to facilitate the experiment. Physical limitations are the same as the one encountered by standard pulsed magnets, namely the mechanical stress due to Lorentz forces and the heating due to Joule effect. The magnet optical access diameter is chosen to be 17 mm. The objectives in terms of magnetic field and pulse duration, are respectively in the range 10-20 T and 5-10 ms. This allow us to use copper as the conductor with an extra reinforcement outside the coil and to operate the magnet at room temperature. This is the major difference with other pulsed magnets, which are usually cooled in liquid nitrogen temperature. Avoiding the need for a nitrogen cryostat allows us to use the full optical access diameter of the magnet as there is no thermal insulation between the coil and the vacuum tube of the experiment.

As we stated previously, the most expensive and the least flexible part of a pulsed magnet system is the capacitor bank, thus the parameters of the magnet, such as the inductance or conductor section, have to be adapted to existing generators. The capacitor bank that energized the Foil Coil is one of the 3 MJ mobile banks of LNCMI, with a capacity of 10 mF that can be charged up to 24 kV and can deliver 75 kA. The minimum rise time is 5 ms in short circuit so the rise time of the magnetic field must be higher to transfer some energy to the coil. Table I summarizes the properties of the magnet that result from all the constraints cited above.

B. Realization

The magnet is based on the winding of a copper foil insulated with two layers of Kapton tape. About 100 meters of the copper foil is wound over a glass fiber epoxy FR4 type body with a racetrack shape, representing 50 layers of conductor. The optical access is provided by two holes in each turn in the insulated copper. This design offers a very good symmetry of the generated magnetic field thanks to the homogeneous current distribution and the very small effect of the layer transition compared to a wire wound coil. Another difference with standard wire wound coils is the continuity of conducting metal perpendicularly to the axis of the laser beam that screens the axial component of the magnetic field avoiding the use of an extra shield. A image of the hole in the conductor and the insulation is shown in Fig. 1. The presence of the hole increase the current density by a factor 1.4 so the local temperature rise during the pulse is twice higher than in the rest of the coil.

The main limitation for the generation of high magnetic field is the huge stress due to Lorentz forces. In this particular design of a magnetic dipole, one component of the mechanical stress is due to the repulsion between each side of the winding. This force can be evaluated to several hundreds of tons and must be sustained by the reinforcement surrounding the coil. Most high field magnets are reinforced using Zylon fibers [33] [34], combining very high strength, low density and electrical insulation. We have decided to use this material, rather than metal, as the main reinforcement of the Foil Coil as shown in Fig. 2. Fig. 3a shows a Finite Element Analysis (FEA) model of the mechanical stress in a cross section of the Foil Coil in the plane parallel to the magnetic field. It shows the transmission of the forces from the foils to the Zylon fibres. At the ends of the magnet the mechanical stress is naturally transmitted as a hoop stress in the copper foils as another FEA simulation in Fig. 3b shows. Thanks to this solution containing the repulsive forces is no longer a problem. Furthermore, Zylon fibers are applied with a high pre-stress around 700 MPa that press the foils on the coil body preventing any risk of buckling in the straight parts of the winding. The limitation in terms of magnetic field comes from the potential buckling of the copper foils in the bent parts of the winding leading to the collapsing of the holes where it is not possible to place any efficient reinforcement.

The magnet ready to be tested is shown in Fig. 4.

C. Tests

Tests have been made up to 12.5 T, the obtained pulse profile is shown in Fig. 5 which corresponds to a B 2 L B factor of 134 T 2 m. This factor comes from integrating the magnetic field profile along the light axis as presented in Fig. 6. During physics measurements, the maximum field is fixed to 11 T providing a reasonable safety margin. The effective maximum field achievable by the Foil Coil is hard to predict because buckling is difficult to simulate and is strongly influenced by small defaults in the winding. A next step will be to test this limit with a second prototype. However, if the Foil Coil can reach 20 T, this will lead to a temperature rise above 35 K per pulse, and cooling down will take more than two hours. Therefore, improvements in the cooling are required before testing the maximum field achievable.

A pulsed magnet is submitted to huge internal forces during the pulse, and small movements or vibrations cannot be avoided. This vibrational noise will be transmitted to the experiment via the air and the mechanical support of the coil and the optical table. In order to reduce the noise generated by the vibration of the coil we decided FIG. 2: Application of the mechanical reinforcement on the Foil Coil. Two 3D printed halves of a hollow cylinder, in black on the photo, containing four M24 stainless steel tie rods and filled with epoxy resin permit to wind the Zylon fibers and to transmit the force in the direction parallel to the fibers. Zylon fibers are wound over a length of 600 mm with a thickness about 5 mm and are capable to sustain a force above 3000 tons giving us a safety margin of about a factor 4, but also limiting the possible displacement of the layers to around a few tens of micrometers, decreasing the risk of buckling of the foils. At each end of the magnet the mechanical stress is naturally transmitted as a hoop stress in the copper foil as the winding can be considered as two halves of a solenoid. The extremities of the magnet are reinforced with two 25 mm thick stainless steel plates that are also used to fix the magnet, screwed with the M24 tie rods. to place it in a plastic box, with an interior covered by acoustic foam, as shown in Fig. 7 .

Repetition rate will be important to perform an experiment where hundreds of pulses will be necessary. This first prototype is far from having an optimized cooling because the main objective was to generate at least 100 T 2 m at room temperature and in air. Fig. 8 shows the evolution of the Foil Coil resistance, and therefore its temperature, after few pulses from 9 to 12. mum average temperature of the conductor after a pulse is fixed to 60 • C corresponding to 54 mΩ. Knowing that the temperature rises of about 22 • C, corresponding to a resistance increase of about 4 mΩ as shown in Fig. 8, after a pulse at maximum field a local temperature slightly above 80 • C is reached around the holes. This temperature is evaluated by knowing the cross section of conductor with and without the hole and considering that there are no skin effects or non-uniform current distribution. The presence of the hole decreases the section of con- ductor by about a factor of 1.4 leading to a temperature rise of about a factor of 2. A local temperature of 80 • C seems reasonable to preserve the insulation integrity and the mechanical properties of the whole assembly after thousands of pulses.

The time evolution of the cooling of the Foil Coil follows consecutive stages. Fig. 9 shows the cooling after the 12.5 T pulse, and alongside the data we present three exponential fits to the three stages. Doing so on the other pulses of Fig. 8 leads to similar results hinting at a common cooling law. First, after the adiabatic heating of the copper during the pulse, the temperature is first homogenized along the copper foil and the heat is transmitted to the insulation layers. Next, the temperature of the highly insulating mechanical reinforcement will increase, and finally natural convection outside the box will slowly contribute to the cooling of the magnet. Cooling power will increase with the final temperature reached by the magnet, so the highest repetition rate corresponds to pulses to the maximum temperature. A final temperature fixed at 60 • C leads to a cooling duration of the magnet before another pulse at 12.5 T of about 50 minutes. Duty cycle will increase proportionally to the inverse of the square of the magnetic field.

IV. COMMISSIONING PHYSICS MEASUREMENTS

Once the Foil Coil has been tested, we have proceeded to its commissioning. We have installed the magnet in the apparatus which has been detailed in [35]. Following our previous work [35], [31], we have first focused our attention to the noise induced by the pulse itself. We have investigated the impact on the optics of different pulse rise times, ranging from 2.9 ms to 5.8 ms, and different field amplitudes i.e. the energy injected into the system. The principle of a polarimetric vacuum birefringence search is illustrated in Fig. 10. In these setups, a laser field is linearly polarized in the p = cos θx + sin θy direction. The angle θ is defined with respect to the direction of an applied transverse external field, B(t)x.

Due to the presence of a Fabry-Perot cavity of finesse F , as it propagates through the region of birefringent vacuum, the polarization of the light exiting the cavity acquires a total ellipticity Ψ that is 2F/π larger than the one given in Eq. 1

Ψ = 2F λ K CM B 2 L B sin(2θ). ( 3
)
The polarization components are then separated by the analyzer (power extinction ratio σ 2 ) at the output of the optical cavity. After, the signal is recorded as the power at extinction of the second polarizer and compared to the total power exiting through the cavity end mirror,

P ext = σ 2 + (Γ + Ψ) 2 P t , (4)
where, taking into account that Fabry-Perot mirrors are also birefringent [36], one has to introduce a corresponding ellipticity generated by the cavity itself, Γ. This ellipticity depends on various parameters in particular birefringence mirror axis orientation with respect to light polarization, light incident angle on each mirror surface, and intrinsic individual mirror birefringence [36].

The results reported here have been obtained from 578 pulses distributed over 14 sets with different pulse lengths and different pulse strengths. The different pulse lengths were achieved by disconnecting capacitors from the capacitor bank diminishing the period T 0 = 2π √ LC of the circuit comprising the coil and the capacitors. The first sets of pulses were with the full bank with a rise time of 5.8 ms. The second sets of pulses were with roughly half the capacitors so with a rise time diminished by a factor near √ 2 ending with a rise time of 4.4 ms. The final sets were with a quarter of the capacitors with a rise time halved to a value of 2.9 ms. To exemplify the effect of the cavity filtering as a function of the rise time of the magnetic field, we show in Fig. 11 the filtered and unfiltered square of the magnetic field for the three pulse lengths at 3.7 T. In Fig. 12 we show the shape in time of the square of magnetic field value, B 2 f , once filtered by the cavity low pass filter for the different sets. The birefringence effect to be measured is proportional to B 2 f [31].

A. Optics

The cavity was formed by two interferential mirrors of radius of curvature of 2 meters. We have achieved a cavity finesse as high as 537 000 and an extinction ratio as low as σ 2 ∼ 1.5 × 10 -9 enabling us to have a static birefringence as low as Γ ∼ 5 × 10 -5 rad.

For each set we have tuned the value of the static birefringence Γ by rotating the two cavity mirrors [17]. The sign of Γ depends on which direction we move away from the minimum corresponding to σ 2 . We have fixed this sign by introducing some helium gas in the apparatus and observing its Cotton-Mouton effect which is known to be positive.

A notable feature of the new apparatus is that the vacuum tube is directly in contact with the coil and it is mechanically isolated by two bellows from the vacuum chambers.

Another novelty of these series of pulses is that we have placed at critical places of the optical set-up some absorbing materials to reduce stray light in the vacuum tanks.

All the pulses were performed in high vacuum with pressure ranging from about 10 -6 to 10 -7 mbar thanks to two ionic pumps that work continuously during the magnetic pulse.

For each pulse constituting a set, we determine the finesse F thanks to the photon lifetime in the cavity [17]. During each pulse, the laser delocks from the cavity, and thus the ordinary beam signal P t shows an exponential decay that we fit to deduce the time constant τ which relates to the finesse as

F = πcτ L c , (5)
where c is the celerity and L c = 2.55 m is the length of the cavity. Therefore, for each set we have as many measure of finesse as we have of pulses.

In our numerical implementation, we only start the fitting procedure when the voltage of P t falls to 80% of its mean value before the pulse. In this way, we are sure to fit only the exponential decay due to the photon lifetime and not the acoustic perturbations observed before the delocking of the laser to the cavity. An example of a fit obtained using our implementation is presented in Fig. 13.

B. Results

We present in the table II, the different data sets and their main parameters. The data sets are named accord- Ns is the number of pulses, B 2 f the maximum of the filtering of the square of the magnetic field, F the finesse, σ 2 the extinction ratio and νc = 1/4πτ is the cavity bandwidth.

ing to their nominal maximum B field and whether they are Long, Medium or Short pulses. Pulses with the same field but on different days have been separated in different sets because we have to align the cavity each day to reach a static birefringence of the order of σ 2 , and thus, we are not in exactly the same experimental conditions.

As far as the analysis is concerned, we have used the same procedure reported in ref. [17]. A notable feature is the separation of each pulse according to their sign of B and the sign of their Γ. This separates pulses in four categories for each set. We take the mean and the standard deviation divided by the square root of the number of pulses to obtain respectively a function Y ±B,±Γ (t) and the related uncertainty at each time t. Then we combined them to obtain the ellipticity as a function of time and finally we perform a fit by B 2 f (t) to obtain Ψ per T 2 and consequently K CM value for the vacuum, as shown in the BMV 2014 paper [17]. This allows us to eliminate from the final ellipticity signal any systematic contribution that has not the expected symmetry with respect to the sign of B and the sign of Γ [17]. To easily compare results to those of 2014 [17], we first perform this analysis for each set for a time range between -3.1 ms and 3.1 ms. The only departure in our analysis from the one of 2014 is that we don't filter the ordinary beam P t (t) with the cavity bandwidth since (see e.g. Fig. 13) the variation of P t during the time of analysis is of the order of 1 %. This analysis simplification induces a negligible error [31] to the final result. Let us also note that we did not exclude any data from our analysis.

As an example, we show Ψ(t) the mean of the Y ±B,±Γ (t) of the 5.5S2 set in Fig. 14 Moreover, as we show in Fig. 15, the mean linearized Power Spectral Density before the magnetic pulse confirms that mechanical resonances have disappeared with respect to the Power Spectral Density reported in 2014 [17].

To study the impact of a pulse on noise, we fit the ellipticity also before the magnetic pulse from t = -3.1 to t = 0 ms where any noise coming from the pulse itself is obviously absent. This value gives information on the result that we would have obtained if the experiment were limited by the optical noise.

Then, we compare the ellipticity uncertainty before the magnetic pulse ∆Ψ opt and the one obtained during the FIG. 15: Mean linearized Power Spectral Density before the magnetic pulse of every pulse. For reference we show the PSD origininally published in [17].

pulse ∆Ψ tot . Before the magnetic pulse, the uncertainties arise only from the optical noise of the experiment. During the pulse however, we have the added noise of the pulse, ∆Ψ B , that contributes to the total uncertainty ∆Ψ tot . We can estimate ∆Ψ B by

∆Ψ B = ∆Ψ 2 tot -∆Ψ 2 opt . (6)
We compute the ellipticity uncertainty for each set, we multiply it by the square root of the number of pulses N s of the set to obtain the noise ∆Ψ n B associated to a single pulse. The final ∆Ψ B is obtained by multiplying the single pulse noise to the corresponding B 2 f,m which is the B 2 f reached at the end time of analysis t an . In Fig. 16 we show ∆Ψ B as a function of the energy injected into the coil. Somewhat surprisingly, we see that ∆Ψ B is constant in our range of energies, ∆Ψ B = 6 × 10 -7 rad, indicating that we gain by increasing the magnetic field value since we increase the signal to noise ratio. It is a clear indication that we need to pulse the magnet to its maximum energy, that is a magnetic field of 11 T.

To study the dependence of ∆Ψ B on t an we calculate it for different analysis time from 0.125 ms to 5 ms in Fig. 17 where we show the results obtained in rad/ √ Hz together with a parabolic fit.

This results indicate that soon after the beginning of the pulse (≈ 1 ms), whatever is the energy injected in the coil, the noise induced by the coil itself increases almost linearly, confirming our Monte Carlo results [31]. Finally, we calculate the K CM constant corresponding to the two 5.5 T sets. A reasonably well accurate and precise value was obtained with a t an = 2.7 ms, K CM = (0.2 ± 1.0) × 10 -20 T -2 . The analysis between -t an and 0 gives K CM = (0.1 ± 3.7) × 10 -21 T -2 . The total number of pulses of the two sets is 49, see table II.

V. CONCLUSIONS

As for the coil itself, some potential improvement in the design can be made to increase the efficiency or the ergonomics. First an extra cooling to remove the heat from the box will be installed. Actual average cooling power due to natural convection and conduction is lower than 50 W and can easily be increased with a forced flow of air or cold nitrogen gas inside the box. A second modification is to optimize the pulse duration, probably by shortening it, either by adapting the actual capacitor bank or using another one available at LNCMI, without lowering the maximum field. A next step could be a modification in the design itself. For example, it is possible to cut the copper foil to maintain the same current [25], PVLAS-LNL [37], PVLAS-Test [38], BMV [17], PVLAS-FE [39][40] [12] and OVAL [16][18]. density all along the winding. It will concentrate the current density closer to the laser beam path increasing the efficiency, i.e.. B/I by a factor around 1.3 without increasing the risk of buckling around the optical access.

As for its use in the BMV experiment, the Foil Coil proved to be a real improvement and more generally for magnetic linear birefringence measurements. The absence of a cryostat simplifies the whole apparatus. Last but not least, it also reduces the costs of fabrication and operation. On the other hand, nothing prevents us from improving the cooling of the Foil Coil to gain a factor of two with respect to the 12.5 T field, however, it is not clear that the gain of a factor 4 in the effect is worth the big technical effort for cryogenic cooling.

As far as preliminary physics results, our best value has been obtained with the short pulses at 5.5 T and an analysis time of 2.7 ms where we reached a maximum field similar to 2014 and obtained K CM = (0.2 ± 1.0) × 10 -20 T -2 , which is about the same as the value obtained in 2014, if we consider that this has been obtained with half the number of pulses, but much more accurate. This value is shown with other reported results in Fig. 18. Indeed, unlike in the 2014 results, we do not observe a systematic effect. The insertion of the new magnet in the apparatus has been very successful. On the other hand, we did not gain yet in precision as much as we should have, despite the fact that we have a B 2 f L B three time as high and a mean PSD about two times lower. Γ is also lower than in 2014, and this should also reduce the optical noise. This clearly indicates that optics optimization has to be pushed further. This is why we have not pushed the field strength higher than the 2014 one, our reference point. We considered that it was not worthwhile to prematurely age the coil at this stage.

Nevertheless, we are still limited by the noise induced by the pulse itself. A signature of this noise is in the increasing error bars during the pulse. Actually, K opt CM is 3 times better than K CM confirming that the noise induced by the pulse itself is still limiting us.

The magnet commissioning and the first results look encouraging. We are currently working to diminish the overall noise by better acoustically insulating the apparatus from the coil. On a long term perspective we are also studying how to suspend in vacuum the optical tables holding mirrors and polarizers. The new generation of the BMV experiment has just begun, and there is very much room for improvement.

FIG. 1 :

 1 FIG.1: Photo of the Foil Coil during the winding showing one hole in the copper foil and in the Kapton tape. The holes have a racetrack shape to keep a circular access when the foil is wound over the round end of the magnet. The copper is cut first, and then the insulation with Kapton tape is cut with a smaller hole to ensure a perfect electrical insulation between two layers thanks to the overlapping of the Kapton on the copper.

FIG. 3 :FIG. 4 :FIG. 5 :

 345 FIG. 3: FEA model of the mechanical stress in the Foil Coil during a 12.5 T pulse. (a) Cut along a plane parallel to -→ B , xy-plane of Fig. 4, only the upper half is represented (b) Cut along a plane parallel to -→ J at one end of the magnet, xz-plane of Fig. 4. To save computing power only the first innermost layers are individualized taking into account their relative displacement. Indeed the first layers are self-supporting due to the increasing hoop stress with the radius of the layers up to half of the outer diameter. The outermost layers however behave as a whole as each layer is supported by the following one.

  FIG. 6: Normalized profile of the magnetic field along the light axis produced by the Foil Coil. Solid black line present a finite element COMSOL simulation of the coil, black points show measurements.

FIG. 8 :

 8 FIG. 8: Resistance of the Foil Coil measured during the tests up to the maximum field. Room temperature resistance is 48.1 mΩ and the maximum resistance is fixed at 54 mΩ representing a final temperature of the conductor about 60 • C.

FIG. 9 :

 9 FIG. 9: Temperature of the Foil Coil of a function of time after a pulse at 12.5 T.

FIG. 10 :

 10 FIG.10: Illustrative example of a vacuum magnetic birefringence experiment. The laser light is polarized before entry into the cavity surrounding the magnetic field region, after the cavity light is analyzed by another polarizer at extinction. Two photodiodes PD-t and PD-ext monitor respectively the ordinary beam Pt and the extraordinary one Pext.

FIG. 12 :

 12 FIG.12: Filtered square of the magnetic field as a function of time for every set.

FIG. 13 :

 13 FIG.13: Variation of the voltage at the ordinary beam's photodiode as a function of time, t = 0 being the trigger of the current discharge in the coil. The pulse is extracted from the 5.5S2 set. An exponential fit is presented.

FIG. 14 :

 14 FIG. 14: Ellipticity in function of time of the 5.5S2 set B 2 f LB = 10.3 T 2 m, and the 2014 data B 2 f LB = 3 T 2 m. Data are normalized to the same number of pulses, see text. Data originally published in [17].

  compared to the one of 2014. For the sake of comparison, since the 5.5S2 set consists of only 32 pulses while the 2014 results have been obtained with 101 pulses, we have reduced the error bars of the present data by a factor 101/32 ≈ 1.8. Both curves show the same behavior of increasing error bars but our data do not show the 2014 oscillatory systematic effect.

FIG. 16 :

 16 FIG. 16: Noise of the pulse ∆ΨB as a function of the magnetic energy delivered by the coil. The full line showing the mean of the points, ∆ΨB = 6.0 × 10 -7 rad.

FIG. 18 :

 18 FIG.18: Measurements of vacuum magnetic birefringence across the years, errors bars are represented with a coverage factor k = 3. Absolute values are derived from data originally published in the following references BFRT[25], PVLAS-LNL[37], PVLAS-Test[38], BMV[17], PVLAS-FE[39][40][12] and OVAL[16][18].

TABLE I :

 I Summary of the properties of the magnet installed on the BMV experiment.

	Optical access (mm)	17
	Effective magnetic length (mm) 820
	Copper cross-section (mm 2 ) 72 × 0.5 (height × thickness)
	Thickness of insulation (µm)	120
	Number of layers	50
	Efficiency (T/kA)	0.537
	Inductance (µH)	700
	Resistance at 26 • C (mΩ)	48.1
	Rise time (ms)	5.8

TABLE II :

 II Parameters of the different data sets used in our analysis.

Acknowledgments

We thank all the technical staff of the LNCMI. We thank all the members of the BMV collaboration. We specially thank M. Hartman and all the authors of [17]. We also thank D. Maude for the careful proofreading of this manuscript.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.


Nilankur Dutta
email: nilankur.dutta@univ-lyon1.fr

Alexandre Nicolas

Searching for Parking in a Busy Downtown District An Agent-based Computational and Analytical Model International Conference on Transport and Smart Cities 2021

Finding a place to park one's car is a serious issue in contemporary urban mobility. Despite the importance of the topic (30% of cars may be cruising for parking in many large cities) and the central role given to parking policies, surprisingly little is known about the basic laws governing the search time.

We present a novel agent-based approach combining numerical simulations and theoretical considerations to model cars cruising for on-street parking in busy downtown districts. The approach is premised on the idea that, rather than parking at the first vacant spot that they encounter, drivers may be more or less prone to parking on a given spot, depending on their perceptions of its characteristics (notably its distance to their destination and its cost). This spot-specific parking probability is quantified by means of a scalar variable, the 'attractiveness'. On this premise, we show that this problem can be solved using an exact formula for the stationary state and depends on the topology of the streets. This is demonstrated by comparing our theoretical results with a stochastic in-silico model and the method is illustrated with the case of the city centre of Lyon. Finally, the relationship between the search time and the spatial modulation of the attractiveness of parking spots is explored. We find that such a modulation, which could in practice be enforced by targeted parking policies at the level of individual streets, dramatically affects the parking search time, which paves the way for a more efficient control over occupancies and cruising times in on-street parking networks.

Context

Searching for parking is often a pain in urban mobility, with an estimated average 17, 44 and 41 hours a year lost to drivers in the U.S., U.K. and Germany, respectively, in 2017. This comes at an estimated equivalent cost of $72.7 billion, £23.3 billion and e40.4 billion in these countries in lost time, fuel, and carbon emissions [START_REF] Cookson | The impact of parking pain in the us, uk and germany[END_REF]. The costs to the greater economy in terms of cancelled trips and psychological stress are hard to quantify, but undoubtedly substantial.

One of the manifestations of this quagmire is the number of cars cruising in search of on-street parking, which has been estimated to build up to 30% of traffic in busy downtown districts [START_REF] Shoup | Cruising for parking[END_REF] in large cities. A study in Brisbane in 2015 [START_REF] Lee | Cruising for parking: New empirical evidence and influential factors on cruising time[END_REF] suggests that the drivers' misapprehension of the true parking costs are the major cause of this phenomenon. A recent empirical study of cruising for parking phenomenon in San Francisco [START_REF] Millard-Ball | Parking behaviour: The curious lack of cruising for parking in san francisco[END_REF] points out that the drivers' choice to park a longer distance away from their destination, based on their knowledge of on-street parking availability, is an important factor in regulating cruising times.

A significant portion the academic literature on parking and its relation to traffic is oriented either towards policy [START_REF] Shoup | The high cost of free parking[END_REF], or towards the economics of parking [START_REF] Calthrop | Parking policies and road pricing[END_REF]. Much of it suggests road pricing as an effective measure to create less congested networks [START_REF] Zong | Understanding parking decisions with a bayesian network[END_REF][START_REF] Parmar | Study on demand and characteristics of parking system in urban areas: A review[END_REF]. This work instead tackles the phenomenon of parking from a statistical physical standpoint, analysing the spatio-temporal distribution of parked cars on a network of streets. Such spatio-temporal models have been suggested by other authors [START_REF] Geroliminis | Cruising-for-parking in congested cities with an mfd representation[END_REF][START_REF] Arnott | Modeling parking[END_REF] in the past. In particular, the agent-based PARKAGENT model was used to model the parking dynamics of cars at night in a residential district of Tel-Aviv [START_REF] Levy | Exploring cruising using agent-based and analytical models of parking[END_REF], promoting the relevance of such models for the analysis of parking scenarios. The importance of the drivers' agency, particularly in selecting a parking strategy, has also been highlighted in other theoretical works, both on idealised one-dimensional parking lots [START_REF] Krapivsky | Simple parking strategies[END_REF], and in more physical networks or streets [START_REF] Benenson | Parkagent: An agent-based model of parking in the city[END_REF].

In this work, we present a set of analytical and computational approaches to investigate the role of the drivers' parking preferences in determining the occupancy of on-street parking spots in busy downtown districts. These preferences are here accounted for in terms of the 'attractiveness' of a spot, which is defined by the probability that a certain driver passing by the spot will park there, should it be vacant. This notion allows us to subsume into a single quantity the various factors governing the selection of a place to park, including its distance to the destination, cost, and intrinsic characteristics.

We implement this idea in a stochastic agent-based model and simulate it numerically to investigate the cruising phenomenon in the central district of Lyon and demonstrate the effect of modulating spot attractiveness on the on-street parking spot choices (Figure 1) and the time spent cruising for parking.

As a matter of fact, the occupancy in such a model can be solved using an exact formula and we derive an analytic formula to do so. Its accuracy is verified by comparing it with the spatial distribution of parking occupancy generated in silico (Figure 2).

On-street parking in the hyper-centre of Lyon

To illustrate our approach, we will consider the case of the central districts of Lyon (the 1st and 2nd 'arrondissements'), whose street network is available online1 ; the locations of the parking spots and their average occupancy street by street were kindly transmitted to us by the city council services. Generally speaking, the occupancy of parking spots results from a continuous process involving successive arrivals of car drivers with distinct destinations and parking preferences. In this short manuscript, the emphasis is put on the interplay between the network topology, the drivers' preferences, and the spatial occupation of spots, so, rather than studying the most general case, we simplify the problem and consider an influx of cars serving leisure-purpose trips in the evening with a similar destination in the vicinity of the Opera (i.e., the hyper-centric part of Lyon with the highest density of bars, restaurants, theatres, etc.), which thus acts as a 'hotspot', and similar parking preferences. Note that, upon arriving on the premises, the drivers, who are supposed to look for a spot on the street, find that many of these spots are already occupied by cars parked for a longer duration, first of which those of the residents. This initial occupation of the network is estimated using the empirical data for the occupancy in the night and a fraction of the available spots will be blocked accordingly at the beginning of the simulations.

In this network, we model the dynamics of 200 cars (which is a fairly tight upper bound for the number of leisure-motivated incoming vehicles) over a duration of 5.5 hours. These cars are injected into the network at a predefined rate through designated entry-nodes (orange bars in Figure 2A). This district is separated from the rest of the city on both the East and the West side by the two rivers Rhone and Saône, respectively, which join on the South. This geographical barrier, provides us with the physical basis of the choice on entry nodes on the eastern, southern and western side, where cars can only come in through bridges. From the northern side, we allow entrance from a high volume tunnel, the Tunnel de Croix Rousse. Route choices are implemented stochastically via prescribed turning probabilities (defined over the ensemble of cars) at each junction; in practice, we set higher probabilities for turning into a street portion whose extremity is closer to the opera hotspot (see Equation 7) in order to 'drive' the cars to their destination.

When they pass by a spot, their probability to park depends on the status (a spot must be vacant for a passing car to park in it) and attractiveness (an empty spot must be sufficiently attractive to the driver ) of the spot. Since the drivers want to to park as close to the hotspot as possible, the probability to park at a vacant spot i, A i (≡ the attractiveness of this spot2 ), is here assumed to decay exponentially with the distance to the hotspot with a characteristic length d walk (see Equation 6); d walk represents the preferred walking distance and will be varied in the following. Parked cars leave the network after spending a predefined time at the parked location (for details of in computational model, see Section 4.2).

Modulating Parking Search Times via Spot Attractiveness

In this section we investigate the phenomenon of cars in Lyon downtown districts (Figure 2).

We examine our results in the light of two factors of merit. The first one, search time, is defined as the time difference between the car first entering a disk of radius 500m around the target point, and it parking. The second is walking distance, which is the distance between the parked location and the drivers destination as the crow flies. We run simulations varying the attractiveness field to three different values using the parameter d walk , keeping everything else constant, and recording the effect on search times and walking distances (see Figure 2A).

We find that, for Lyon with d walk = 100m, the search times in the stationary regime are around 12 minutes, with walking distances are around 450m By changing the parameter d walk , we are able to change the area of maximum effect of our attractiveness field. When d walk is doubled to 200m, the mean search time reduces to 7.44 minutes, with the number of cars finding parking in their first couple of minutes of search almost tripled (see Figure 2B). Meanwhile, the walking distances are increased to 500m (see Figure 2C). Thus, increasing the effective area of the attractiveness field reduces the competition between cars eager to park close to their destination.

We understand, therefore, that broadening the peak of the attractiveness field operates a trade-off between search times and walking distances, enabling cars to find parking faster, but compelling them to park further from their destination (see Table 1). Forcing people to park at a wider range possible of distances from their destination (e.g., by making parking more costly around hotspots) may lead to lower cruising times and less congested networks. Table 1: Effect of a spatial modulation of the parking attractiveness field on the parking search time and the actual walking distance.

Predicting stationary occupancies analytically

The foregoing results were obtained by means of numerical simulations. Even though we developed an efficient algorithm which allows us to simulate the parking search of tens of thousands of cars in a matter of minutes, it is inconvenient to numerically explore all possibilities (e.g., variations of injection rates, turning probabilities, parking spot attractivenesses, etc.) and the connection between the input parameters and the output of the simulations remains opaque.

To remedy these issues, deriving an analytical formula may prove an extremely valuable asset, which may provide much deeper insight into the problem. Therefore, we have endeavoured to solve the problem analytically, under the assumption of 'equilibrium', i.e., stationarity of the process, and arrived at a mean-field equation for the occupancy field. The calculations use as input parameters the rates of entry in the network and parked times, as well as the attractiveness field. The system is then fully defined according to the equations 5 and 3, and the equilibrium occupancies can be determined (see Figure 1B). Details of the derivation are presented in the Methods section 4.1.

The comparison shown in Fig. 2 demonstrates that these theoretical considerations lead to results that coincide extremely well with the in silico predictions (the simulations were allowed to run to equilibrium, and the position and parked status of all cars were recorded at predefined intervals); also see Figure 1A. We find that the average error between the analytical prediction and the computational observation is of the order of 10 -3 (see Figure 2D) Note that, in this particular example, we have considered all allowed turns within our network to occur with a probability that hinges on the distance between the chosen street end and the driver's destination (see Equation 7). However, the equations remain valid for other route chocies, i.e., other forms of turning probabilities, including fully deterministic trajectories.

Discussion

Our method provides a novel mathematical framework to analyse parking in networks. We posit that describing the drivers' perception of the characteristics of a parking spot in terms of a scalar attractiveness offers a convenient way to quantitatively investigate the effect of parking choices. Not only does this provide an variable that subsumes the most important causes of the drivers agency (distance to destination, cost, etc.), but it also gives access to a mathematically rigorous derivation of the on-street occupancies with no extra knowledge except the geometry of the region of interest, the average arrival and departure rates, and the average turning probabilities, as shown here in the simplified case of cars aiming for the same goal.

In dense urban conditions, where competition for parking increases search-times and potentially causes traffic delays, it has been shown that drivers prefer on-street parking primarily due to the price premium incurred off street [START_REF] Shoup | Cruising for parking[END_REF][START_REF] Inci | Garage and curbside parking competition with search congestion[END_REF]. Parking pricing policies have been suggested by many experts as a means to alleviate the negative externalities of parking; data from studies in Beijing [START_REF] Zong | Understanding parking decisions with a bayesian network[END_REF] and New Delhi [START_REF] Parmar | Study on demand and characteristics of parking system in urban areas: A review[END_REF] even suggest that parking price is one of the main determinants of the duration of parking. Thus it is important to tailor parking pricing policies to local circumstances.

We have presented spot attractiveness as a proxy variable for parking price determination, and presented how it can be used to modulate the on-street occupancies in Lyon. Thus, this work demonstrates as possible method through which empirical knowledge of parking data, and the geometry of streets, might guide city-planners towards parking pricing policies that meet their goals for on-street occupancies and cruising times.

In the real world, this could involve strategies such as differential parking pricing at the level of streets, with the ability to rapidly update prices to meet traffic demands, and the transparent communication of pricing changes to all road users.

Though in this work we have constrained ourselves to a unique attractiveness field for all users, our model is readily extensible to multiple such fields: for instance, for drivers going to different destinations, or for drivers with different psychological thresholds to parking at a certain spots. In such cases, the analytical model takes as an extra parameter the ratio of different types of cars introduced in the systems, and remains exactly solvable.

Our method is versatile enough to be used in arbitrarily different networks, and easily adapted to model off-street parking. Thus, we show that, provided certain variables like attractiveness and injection and departure rates are known, the parking occupancies can be determined in a wide array of on-street networks. This section is dedicated to mathematical details about the formulas giving the stationary parking occupancy.

Consider a one dimensional parking lot with L spots, with distance l between them, labelled by an integer index x |x ∈ {0, L -1}. Defining here the relevant variables : the field of attractiveness as A(x), the probability of a spot being occupied as n(x), and its inverse ñ(x) = 1 -n(x), Then the probability of parking at any spot n(x) can be written as:

P (x) = ñ (x) x-1 x =0 (ñ(x ) + ñ (x )) (1)
where the first term on the right-hand side, ñ (x) ≡ ñ(x)A(x), is the probability of parking at a spot x multiplied by the probability that it is vacant, and the second ñ(x ) and third terms ñ (x ) ≡ p(x )(1 -A(x )) are the probabilities of all previous parking spots being either full or not attractive enough for the driver, respectively.

Equation 1 can be simplified as :

P (x) = ñ (x)
x-1

x =0

(1 -ñ(x )A(x ))

(2)

In the continuous limit, the change in occupancy of each spot is defined by the differential equation:

dn(x) dt = IP (x) -Dn(x) (3)
where I is the rate of entry of cars into the street, and D is the rate at which parked cars depart from their spots. So in the stationary state:

n(x) = I D P (x) (4)
The generalisation to networks is straightforward, by defining the transition probability from spot x to spot x as T x

x . Then the equation 2 takes on this as an extra term, and a sum over all paths π from i:

P (x|i) = π [I i x ∈X (1 -ñ(x )A(x ))T x x ]ñ (x) ( 5
)
where i is the set of all 'entry nodes' where cars enter the network with probability I i

Details of the in-silico model

To demonstrate our method, we simulate evening on-street parking in the downtown commercial districts of (our hometown) Lyon: the 1st and 2nd arrondissements. The network consists of 1835 directional street portions between junctions (a two-way street portion would consist of two such directional portions), as well as 8778 parking spots distributed over these street portions. Details of the parking spots are provided to us by the city office of Lyon. The departure rate, D is chosen an exponential distribution peaked around the mean parking duration of 2 hours, based on known parking durations in these districts. The cars are kept in the simulation system until they park, after which they are removed based on D.

Since we want to model the dynamics cars intent on parking near the opera, we pick the attractiveness field A s (x) for a street beginning at the coordinate x an exponential centered on the opera, (see Figure 1), given by the equation:

A s (x) = exp - |x -x o | d walk (6)
where x o is the location of the opera, and d walk is a parameter that gives the distance from the opera at which, the attractiveness falls to 0.367 (or e -1 ) times the maximal attractiveness.

To have more 'intelligent' drivers that are driven towards their destination, we also implement, (for the analysis in Section 2.1) turning probabilities at junctions. For instance, at a junction with n possible choices street choices, S = {s 1 , s 2 , ..., s n } the transition from street s -→ s k is given as:

T s s k (x) = exp -|x s k -x o | si∈S exp -|x s i -x o | (7)
where x s i is the location of the end node of street s i , and all other variables retain their usual significance. Cars can exit from the network only if they arrive at a cul-de-sac street, where a U-turn is not allowed.

The only other required parameters are the rate of the arrival of new cars aiming to park at the hotspot into the simulation volume I i . This is chosen to be 72 cars per hour.

In all examples used in this work, the simulation is performed with a time step of 1 second and runs for 2 • 10 5 seconds (or 5 hours and 34 minutes). The first 10 5 seconds, including a transient regime, are discarded and the data are only collected in the second half of this period, at 100 second intervals. Thus, in all simulations presented in this work, about 200 cars are injected over the total simulation time. Simulating this time interval only takes 2.32 seconds of real computational time with our highly economical algorithm.

d

  walk walking distance (m) search time (min)

Figure 1 :

 1 Figure 1: Modulating search times through spot attractiveness Panel A: Three different attractiveness fields (i), (ii), (iii), obtained by modulating d walk = 100m, 150m, 200m in red, blue, and green respectively Panel B: Distributions of search times for each sub-panel in A, with the average shown as a dashed vertical line in each case. By modulating d walk , the mean search time is reduced from 13 min in (i), to 7 min in (iii) Panel C: Distributions distance between parked position and destination (walkdist) each sub-panel in A, with the average shown in dashed vertical lines in each case. for less peaked attractiveness distributions, drivers park further and further from their destinations increasing from maximum distance of 762 m in (i) to 1718 m in (iii) (not shown in plot), thereby reducing local congestion.

Figure 2 :

 2 Figure 2: Predicting equilibrium street occupancy with a mathematical model: Panels A, B: Simulated and analytical stationary parking occupancies of streets in downtown Lyon. The hotspot (the opera) is indicated by the icon, and the entry streets into the simulation are indicated by orange bars Panel C: Spot attractiveness A, averaged over street portions Panel D: Error (difference between Panels A and B, by street portions). Note the logarithmic scale.

Figure 3 :

 3 Figure 3: Schematic of 1D parking lot

https://data.grandlyon.com/

The parking probabilities may actually be biased by the driver's perception of the global occupancy of the network, which affects his or her chances to find a spot. Since we consider a scenario with a fixed global occupancy, this aspect is overlooked in this short manuscript.

Acknowledgments

We thank Y. Pachot, C. Marolleau, and N. Keller-Mayaud from the City Council services of Lyon for giving us access to empirical data about parking, as well as L. Billaud from the City council services of Soissons. This work has been funded by an Impulsion grant from IDEXLYON (2020-2021).

Methods

Analytical Model for Parking in One Dimensional Street