Nicolas Brefuel

Lola Grilli

Gwenn Morvezen

Alain Sylvestre

Nicolas Bréfuel

Gwenn Morvézen

Significance of considering pellet molding pressure and dielectric experimental conditions in the [dielectric constant-temperature] response of switchable dielectric molecular materials: illustration on dabcoHBF 4 polycrystals

Significance of considering pellet molding pressure and dielectric experimental conditions in the [dielectric constant-temperature] response of switchable dielectric molecular materials: illustration on dabcoHBF(4) polycrystals

Introduction

Recently, molecular switchable dielectric compounds have received considerable attention as functional electric materials potentially envisioned for application in the field of molecular electronics (memories), energy harvesting, phase shifters, data communications, and so on [START_REF] Shi | Structural phase transition-associated dielectric transition and ferroelectricity in coordination compounds[END_REF][START_REF] Gan | Molecular ferroelectric piperidine-4-ylmethanaminium perchlorate with superior switchable dielectric properties[END_REF][START_REF] Khan | A supra-molecular switchable dielectric material with non-linear optical properties[END_REF][START_REF] Chen | A high temperature molecular ferroelastic phase transition and switchable dielectric response in the trimethylbromomethylammonium salt [C 4 H 11 NBr] [PF 6 ][END_REF][START_REF] Gao | Regulated molecular rotor in phase transition materials with switchable dielectric and SHG effect[END_REF][START_REF] Zeb | N-Methylpyrrolidinium hydrogen tartrate (NMPHT): an above-room-temperature order-disorder molecular switchable dielectric material[END_REF]. These systems, mostly made of cationic/anionic pair of molecules, exhibit a solid-solid structural phase transition: at the molecular scale, the switching between low and high dielectric states originates from motional change of the polar counterpart occurring at the vicinity of the transition temperature TC: below TC, the ionic building blocks are fully ordered within the crystal network, resulting in a poorly polarizable material while above TC, the degrees of freedom gained by the disordered molecules allows an easy orientation of the local dipoles when an electric field is applied, resulting thus in a large variation of the permittivity. Researchers worldwide have made great efforts to synthesize and investigate switchable dielectrics, and many interesting materials have been described, ranging from the (serendipity-found) isolated molecules to the more sophisticated (host-guest) systems [START_REF] Pan | A three-dimensional molecular Perovskite Ferroelectric: (3-Ammoniopyrrolidinium) RbBr 3[END_REF][START_REF] Xu | The cation-dependent structural phase transition and dielectric response in a family of cyano-bridged perovskite-like coordination polymers[END_REF][START_REF] Zhang | Exceptional dielectric phase transitions in a Perovskite-type cage compound[END_REF]. Even if a lot of progress have been achieved, much remains to be done for a deeper understanding on the intrinsic mechanism driving the dielectric switch and to make it suitable for applicative purpose, in order to integrate such materials as active element in commercial devices. 1,4-diazabicyclo[2.2.2]octane (dabco) reacted in various acid solutions is a candidate of interest for memories and high-j dielectrics due to its ease of processing, non-toxicity and switchable dielectric properties [START_REF] Han | Structural phase transitions and dielectric transitions in a 1,4diazabicyclo[2.2.2octane (dabco) based organic crystal[END_REF][START_REF] Cizman | The structure and switchable dielectric properties of a dabco complex with chromium chloride[END_REF][START_REF] Fu | High-Tc enantiomeric ferroelectrics based on Homochiral Dabco-derivatives (Dabco=1,4-Diazabicyclo[2.2.2]octane)[END_REF]. Dabco has long been used as a compound in various applications. Recent interests are for instance like a pillar ligand in metal-organicframework for supercapacitors [START_REF] Qu | Nickel-based pillared MOFs for high-performance supercapacitors: design, synthesis and stability study[END_REF], a catalyst and surfactant in biomass castor oil-based rigid polyurethane foams [START_REF] Acun˜a | Bio-based rigid polyurethane foam from castor oil with excellent flame retardancy and high insulation capacity via cooperation with carbon-based materials[END_REF], a structure-directing agent for the synthesis of new indium selenides by solvothermal synthesis [START_REF] Ewing | Structural complexity in indium selenides prepared using bicyclic amines as structuredirecting agents[END_REF], it is one of the components of an ion exchange membrane for lithiumion batteries [START_REF] Ounissi | Novel lithium selective composite membranes: synthesis, characterization and validation tests in dialysis[END_REF] and a new catalyst combined with nanoparticles [START_REF] Mondal | Creation of DABCO-based amphoteric ionic liquid supported TiO2 nanoparticles: execution of amplified catalytic properties on microwave-assisted synthesis of N-Substituted pyrroles[END_REF][START_REF] Aghbash | Fe3O4@silica-MCM-41@DABCO: a novel magnetically reusable nanostructured catalyst for clean in situ synthesis of substituted 2-aminodihydropyrano[3,2-b]pyran-3-cyano[END_REF]. The ability of [dabcoH]?A-(with Astanding for tetrahedral anions) to be a ferroelectric switchable dielectric molecular material has recently opened a new focus on applications related to information storage (memory) due to its bistability in terms of dielectric constant value. The ferroelectric character of dabcoH?-related materials, the capability of mixing it with polymer matrices and the ability to develop large-scale crystals have accentuated a research dynamic in its use for smart devices and especially in its potential for non-volatile memories [START_REF] Shi | Structural phase transition-associated dielectric transition and ferroelectricity in coordination compounds[END_REF][START_REF] Shi | De Novo discovery of [Hdabco]BF 4 molecular Ferroelectric Thin Film for Nonvolatile low-voltage memories[END_REF][START_REF] Sui | Enhanced switchable dielectric performance of b-phase-dominated PVDF composite films modified with Single-Protonated 1,4-Diazabicyclo[2.2.2octane Fluoborate[END_REF][START_REF] Yang | Flexible and transparent high-dielectric-constant polymer films based on molecular ferroelectric-modified Poly(Vinyl Alcohol)[END_REF][START_REF] Tang | A multiaxial molecular ferroelectric with highest Curie temperature and fastest polarization switching[END_REF][START_REF] Chen | Rapid dielectric bistable switching materials without a time/temperature responsive blind area in the linarite-like type molecular large-size single crystals[END_REF]. The understanding of the polarization mechanisms in [dabcoH]?A-inherent to these simple molecular structures has been the subject of intense research over the last twenty years to better appreciate the phenomena at the origin of the identified phase transitions. Within this series of molecular dielectric switch materials, dabcoHBF4 (A-is in this case corresponding to BF4 -) published by Katrusiak and Szafranski twenty years ago [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF], stands as the most widely studied and continues to be of interest [START_REF] Szafranski | Thermodynamic behaviour of bistable NH+-N hydrogen bonds in monosalts of 1,4diazabicyclo[2.2.2]octane[END_REF][START_REF] Olejniczak | Pressure-temperature phase diagrams and transition mechanisms of hybrid organic-inorganic NH___N bonded ferroelectrics[END_REF][START_REF] Szafranski | Short-range ferroelectric order induced by proton transfer-mediated ionicity[END_REF][START_REF] Szafranski | Low-temperature and high-pressure phase transitions in ferroelectric dabcoHBF4[END_REF][START_REF] Budzianowski | Anomalous protonic-glass evolution from ordered phase in NH-N hydrogen-bonded DabcoHBF 4 ferroelectric[END_REF], as depicted recently as potential candidate for non-volatile memories [START_REF] Shi | De Novo discovery of [Hdabco]BF 4 molecular Ferroelectric Thin Film for Nonvolatile low-voltage memories[END_REF], eventually mixed in a polymer matrix for such application [START_REF] Sui | Enhanced switchable dielectric performance of b-phase-dominated PVDF composite films modified with Single-Protonated 1,4-Diazabicyclo[2.2.2octane Fluoborate[END_REF]. While most of published studies focus on single crystal (through an accurate crystal-oriented analysis of dielectric analysis), few works describe the studies of molecular dabcoHBF4 processed as pressed pellets. Especially, none examined the pellets molding pressure effect on the [dielectric constanttemperature] response in the ferroelectric-paraelectric regions. This molding pressure can cause mechanical stresses affecting the dielectric response of the compounds. In this way, we present here a complete dielectric re-investigation of dabcoHBF4 with a special emphasis devoted to the effect of pellet molding with pressure ranging from 0.2 to 1 GPa. This study is supplemented by a dielectric analysis of the behaviour of dabcoHBF4 according to the external parameters (excitation frequency, temperature ramp) applied to it in order to evaluate their importance in the [dielectric constant-temperature] response.

Materials and methods

The title compound dabcoHBF4 was synthesized according to adapted previously published procedure by the molar to molar reaction of (1,4-diazabicyclo[ 2.2.2]octane) (dabco, purchased from Acros) and tetrafluoroboric acid (48 wt. %, Acros) in a 1:1 molar ratio at room temperature. After 10 min stirring and complete dissolution of dabco in water diluted acidic solution, white single crystals were easily grown by slow evaporation of the mixture after few days. The pellets were fabricated using a ca. 100 mg amount of crystalline powder placed into a 1.1 cm diameter anvil, then pressed during 2 min with a manual hydraulic press affording up to 12 tons constraint and sample thickness of ca. 0.5 mm. The pellets were pressed at 0.2, 0.4, 0.6, 0.8 and 1 GPa. Finally, the pellets were sputtered using a Quorum Techno QT150 TES sputter coating affording a 200 nm gold thick deposit on both side. The bottom electrode diameter is 0.9 cm while the upper electrode (0.5 cm) was surrounded by a guard ring (diameter 0.8 cm, width 0.2 cm) in order to ensure the uniformity of the electric filed and limit bypass currents (although the e/D ratio (thickness/electrode diameter) is here already small, thus providing reliable measurement). Dielectric measurements were acquired with a Novocontrol high performance frequency analyser under 1Vrms alternative voltage ranging from 1 Hz to 1 MHz. This voltage amplitude enables to preserve a sound signal-to-noise ratio for measurements at the lowest frequencies and we made sure that the applied electric field (around 20 V/cm) was low enough not to disturb the response of the material in the ferroelectric-paraelectric transition. For this purpose, point measurements were conducted at voltages of 0.05 V (1 V/cm electric field) that confirmed, as expected, the absence of effect of the applied AC voltage in our experimental conditions. Temperature scans ranged from 0.5 to 10 K/min. Contact with sample electrodes was achieved through micro-positioner metallic tips. The heating plate connected with a Linkam temperature controller was electrically isolated from the sample by a 0.2 cm thick AlN substrate covered with a thin layer of copper on both side. In order to check the sample purity and confirm the temperature transitions following the scanning temperature solicitation, dynamic scanning calorimetry (DSC) was carried out on a Metler Toledo DSC1/500/2412 under a nitrogen atmosphere.

Results and discussion

The behavior of the dielectric constant versus temperature at a frequency of 1 MHz and with a positive temperature slope of 10 K/min is reported in the Fig. 1 for pellets processed at pressures from 0.2 to 1 GPa. At a given pressure, an expected temperature-dependent dielectric response is observed. A Curie temperature Tc of 380 ± 1 K is identified. Values of Tc around 378 K were obtained in crystal dabcoHBF4 [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF][START_REF] Katrusiak | High-pressure crystallography[END_REF]. Our slight shift on the Tc value is related to the temperature slope adopted to scrutinize the dielectric constant as will be seen later in the paper. For T\Tc, the material is ferroelectric (orthorhombic phase) and for T[Tc, it is paraelectric (tetragonal phase) [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF]. As a consequence of a displacement of BF4 -anions and dabcoH+ cations, a spontaneous polarization is established as well investigated by Katrusiak, Szafranski and co-workers [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF][START_REF] Szafranski | Short-range ferroelectric order induced by proton transfer-mediated ionicity[END_REF]. At increasing temperature, when approaching Tc, the twisting of dabcoH+ cations and the moving of BF4 -anions are breaking the crystalline symmetry and are thus at the origin of the clearly observed ferroelectric-paraelectric transition. It was demonstrated that fluctuations of the BF4 -anions occur as early as Tc-60 K, thus inducing a change in the polarization mechanism around 320 K [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF], explaining the slight increase of the dielectric constant from the ferroelectric phase with the rising temperature, as clearly observed in Fig. 1. These ionic movements from their equilibrium position will lead to instability of the material when increasing more the temperature and will cause it to tilt into a disordered state as soon as Tc is reached. The temperature-dependent dielectric behavior in the paraelectric phase follows the expected Curie-Weiss law.

The amazing result observed in Fig. 1 relates to the significant effect of the pressure applied in the pellet making process on the dielectric-temperature response of the material. The general shape of the dielectric response as a function of temperature is maintained, but the value of the dielectric constant is all the more important as the pellets have been more strongly pressed. We can therefore claim that mechanical stress induced on the dabcoHBF4 powders by this molding pressure has an appreciable impact on the [dielectric constant-temperature] response.

A slight decrease in Tc is also observed when increasing pressure (381 K@0.2GPa vs 379 K@1GPa). This increase in the dielectric constant with pressure behaves quasi-linearly in both the ferroelectric and the paraelectric phase (see inset of Fig. 1: a linear-dependence of the dielectric constant ε' vs pressure P: ε' = aP + b is observed with a@353 K = 6.61, b@353 K = 5.96, a@403 K = 20.9, b@403 K = 12.6). The steeper linear slope in the paraelectric phase is a way to express that more strongly mechanically pressed pellets will induce a higher 'order-disorder degree' within the material if one defines this 'order-disorder degree' as the dielectric constant difference ∆ε' between the wellestablished paraelectric and ferroelectric phases. This finding seems consistent with the shift of Tc depending on the pressure as a ferroelectric material with more oriented dipoles will have a higher electrostatic energy and at the same time the system will be potentially more unstable [START_REF] Kc | Dielectric phenomena in solids[END_REF]: a transition to a lower Tc is thus expected.

To explain the increase in the dielectric constant with the pressure required to manufacture the pellets, several hypotheses can be stated and the most evident hypothesis is to consider that air cavities are present between the pressed grains. The increase in pressure will lead evidently to a decrease of these air cavities and the global tendency of the dielectric constant will be to an increase of its value with the pressure. This is indeed the trend that we observe, but can we consider that the presence of air has a significant impact on this dielectric constant? To do this, let's have a look at the value of the dielectric constant obtained at 0.2 GPa, which corresponds to a moderate pressure to manufacture pellets. The obtained dielectric constant in the ferroelectric phase is ε'@353 K = 6.8. This value is fully within the range of those obtained on a dabcoHBF4 crystal (so no air voids) [START_REF] Katrusiak | Ferroelectricity in NH-N hydrogen bonded crystals[END_REF], thus excluding the influence of air cavities in the dielectric values obtained in our study. To definitively rule out this hypothesis, as shown in the inset of Fig. 1, obtaining values of dielectric constants increasing linearly beyond the values observed in a crystal cannot be explained by a simple effect of reduction of the air pores between the crystals constituting the pellet.

In the study of the dielectric properties of ceramics, and more generally of powders to be shaped into pellets, the interface effects between the grains can constitute an essential part of the dielectric response [START_REF] Li | Effect of grain size and Cu-rich phase on the electric properties of CaCu3Ti4O12 ceramics[END_REF]. Interfacial polarizations, more or less accentuated by specific surface area in contact are of importance. In Fig. 1, our results are presented at 1 MHz and typically interfacial polarization affects the dielectric constant at lower frequencies.

Besides, an increase in pressure reduces both interfaces and the potential interfacial polarization at the grain boundaries. A decrease in the contribution of the dielectric constant value induced by the interfacial polarization is therefore expected with increasing pressure. This is indeed what is observed for example in work of Meng-Meng and co-workers relating to pressure pellet influence on the dielectric properties of titanium compound/polymer composite nanofibers [START_REF] Li | Dielectric properties of electrospun titanium compound/polymer composite nanofibres[END_REF]. Similarly, a significant effect of pellet pressure on the dielectric response occurred at lower frequencies below kHz on sol-gel prepared Bismuth titanate ceramics [START_REF] Badge | Effect of pressure of pelletization on dielectric properties of Bismuth Titanate prepared by sol-gel synthesis[END_REF]. The opposite trend observed in our work therefore excludes this possibility. Strong local mechanical interactions at the grain boundaries during pellet production could possibly favor local preferential crystal orientations but these orientations will relax once the mechanical stress is removed. It thus seems unlikely that the grain boundaries are at the origin of the changes observed on the dielectric constant according to the mechanical stress induced by the manufacturing pressure of the pellets.

Having ruled out the most probable hypotheses external to the bulk of the crystals, other hypotheses are to be sought. It is well-known that dielectric properties of materials are strongly dependent of internal mechanical stress inside the material. Particularly, phonon mean free path and especially soft mode phonon will be conditioned by tensile or compression strains, thus changing the high-frequency polarization and consequently the dielectric constant at optical frequencies. Undoubtedly, the pressure applied to manufacture pellets will induce such local mechanical strains inducing changes in the high-frequency dielectric response. As our experiments are made not higher than 1 MHz, the measured dielectric constant value integrates already such possible high-frequency dielectric values changes. However, it appears difficult to investigate this hypothesis within the scope of our study. Even if the shift in Tc is consistent with this suggestion, it is difficult to answer here whether, once the mechanical constraint is relaxed, the influence of the soft mode is also relaxed.

The AC conductivity σ' extracted from dielectric measurements can provide information on dynamics of charges inside dabcoHBF4 (Fig. 2a). Values around 2 9 10 -7 S/cm are thus noted in the ferroelectric orthorhombic phase. In paraelectric tetragonal phase, this conductivity increases by a tenfold. Increased mobility of BF4 -anions induced by thermal excitation in the paraelectric phase and the moving of H + protons through thermally assisted hopping conduction could explain this increase in AC electrical conductivity. It is observed that when increasing pellet pressure, conductivity increases but this trend is no longer true at higher pressures and the change from 0.8 to 1 GPa shows a decrease in σ'. This finding is particularly clear in the paraelectric phase. However, as reported in Fig. 2b, one can approximate with a good accuracy the global shape of σ' as a function of pressure by a power law in both the ferroelectric and paraelectric phase. This one-decade increase in conductivity between the orthorhombic and tetragonal phases is valid for each pellet processing pressure, thus indicating a negligible effect of the processing mechanical stress on the crystal structure of the material. It is, however, complex at this stage to evaluate the effect of local mechanical stresses induced in the compound on the electrical conductivity. The loss factor σ'' against pressure (Fig. 2c) is in tune with the changes in ac conductivity σ'. Loss factor is representative of dipolar relaxation added with a conduction mechanism if exists. The electrical conductivity values measured are in the range of those obtained in the conjugated polymers. Consequently, at the difference of strongly insulating materials where the loss factor reflects mainly the relaxation mechanisms induced by polarization mechanisms, here the AC conductivity will take a significant part in the loss factor response and explains the good correspondence between the loss factor and AC conductivity trends. The dissipation factor tanδ versus pressure (Fig. 2d) is corresponding to the ratio of ε'' over σ', and represents dissipative energy induced by polar relaxations in the material. As previously discussed, this dissipation factor is well representative of the dielectric relaxation phenomenon when the electrical conductivity has only a minor role, which is not the case here. A lowering of the conductivity σ' (Fig. 2a,b) and a reduction of dielectric losses (Fig. 2c) at highest pressure for the manufacture of pellets is benefit for applications where it is intended to develop both high dielectric constant (Fig. 1) and weak dissipation factor (Fig. 2d) insulating materials for high-k capacitors for example.

The results presented in Figs. 1 and2 show that the 1 GPa pellet has the highest dielectric constant and the lowest dielectric losses: it is therefore interesting for applications and this is the reason why it was chosen for the continuation of this study. Figure 3a illustrates the dielectric constant ε' change as a function of temperature for different solicitation frequencies at 1Vrms applied. The temperature ramp is 10 K/min. The solid/dashed lines correspond to the measurement during the temperature rise/drop. At a given frequency, a hysteresis is clearly observed in the temperature switching between rising and falling temperatures. This behavior is expected for this material and it is representative of a first order phase transition [START_REF] Olejniczak | Pressure-temperature phase diagrams and transition mechanisms of hybrid organic-inorganic NH___N bonded ferroelectrics[END_REF][START_REF] Szafran´ski | Phase transitions studied by highpressure dielectric spectroscopy and calorimetry[END_REF]. Then, it can be noticed that the temperature switching (up and down) is not affected by the excitation frequency. This result is explained by the fact that the physical mechanism at the origin of the ferroelectric-paraelectric switching has no reason to be modified by the frequency: whatever the frequency, same polar species are involved with the change in temperature and according to the same kinetics. In contrast, the excitation frequency has an importance in the value of the dielectric constant at a given temperature: with the lowering of the frequency, a strong increase of the dielectric constant is consistently observed in both ordered and disordered phase. The decrease in frequency allows more time for the dipoles to orient themselves in the direction of the electric field before the frequency alternating changes sign. It may therefore be advantageous to introduce dabcoHBF4 in host polymer matrices for energy storage or recovery applications working in low frequencies such as those encountered in the harvesting of ambient mechanical energy to produce electricity [START_REF] Mishu | Prospective efficient ambient energy harvesting sources for IoTequipped sensor applications[END_REF][START_REF] Shi | Overview of human walking induced energy harvesting technologies and its possibility for walking robotics[END_REF]. At the lowest frequencies (10 Hz but especially 1 Hz), one observes that the Curie-Weiss behavior trend is lost in the paraelectric phase. In this phase, at 1 Hz we even notice a significant increase in the dielectric constant with increasing temperature. The explanation is to be sought in an interfacial (space charge) polarization mechanism. Even if it is difficult to assert the interfaces involved in this mechanism, we can assume that the anions have time to migrate to the limits of the polar nanoregions introduced in [START_REF] Szafranski | Short-range ferroelectric order induced by proton transfer-mediated ionicity[END_REF], inducing this interfacial polarization. However, this hypothesis remains difficult to confirm and localized defects could also be susceptible to such space-charging mechanisms. The last observation we can make relates to the dielectric constant strength jump (dielectric strength ∆ε') between paraelectric and ferroelectric step which differs according to the excitation frequency: the lower the frequency, the greater the ∆ε' value, as clearly observed in Fig. 3b. Except for lowest frequencies (where spacecharge polarization takes place, as just discussed), the ∆ε'-frequency trend obeys to a power law (see Fig. 3b for equation). This power law is first of all of interest to estimate the gain in the value of the dielectric constant during the ferroelectric-paraelectric transition for excitation frequencies not experimentally tested. In addition, the power exponent of this law can be used, for example, as a standard value to evaluate 'the degree of order-disorder' in the transition temperature response: a higher coefficient will reflect a higher sensitivity of the material to a change in frequency, which in turn will result in a more significant effect on the polar dynamics of species involved in the phase change. Inset in Fig. 3b presents such effect of the power exponent simulated in the range -0.36 ± 0.1.

While frequency is a usual input parameter used in memory control applications or for tuning electronic devices, the applied temperature rate also could constitute an input parameter of importance for this control for temperature sensible materials. In this context, we have considered the dielectric response of the dabcoHBF4 to temperature gradients of ± 0.5 Kmin -1 to ± 10 Kmin -1 (Fig. 4a). It is clearly observed that the temperature sweep rate has a very significant influence on the dielectric response of the material, not in the well-established ferroelectric and paraelectric phases but during the initiation of the phase change in both directions (ferro\-[para): the higher the scan speed S, the larger the temperature hysteresis window ∆T, as reported in Fig. 4b. A linear fit approximates this trend well overall (see equation in Fig. 4b): with a rate of 0.5 K/min, it will take 5 min to go through the 2.5 K allowing to go from one state to the other. With a ramp of 10 K/min, this switching time will be reduced to about 40 s, time necessary to execute a temperature variation of about 7 K. We can thus find an optimum ramp in the temperature of the material to optimize the switching time between the ferroelectric and the paraelectric state.

These results are consistent with the thermal activation importance of the mechanisms at the origin of the phase transition as confirmed by the DSC spectra obtained for different scanning speeds (inset of Fig. 4b).

Conclusion

This dielectric spectroscopic analysis of polycrystals of dabcoHBF4 shaped in pellets revealed several results.

The mechanical stresses induced by the pellet molding pressure have a well-identified impact on the dielectric response of the compound. This parameter must therefore be clearly specified in any study requiring the shaping of powders for a dielectric study. The physical mechanisms involved after applying this mechanical stress remain difficult to identify in the context of our study. The standard hypotheses of reduction of the inter-grain cavities have been excluded to explain this dependence. We speculate a possible soft mode effect but this suggestion is only one lead to be explored among others.

A possible modulation of the dielectric constant value in the ferroelectric and paraelectric well-established phases according to the pellet processing pressure provides an additional interest to properly control and use this pressure regarding an application that can incorporate a pellet made of a switchable dielectric compound.

The transition temperature remains little impacted by the pellet processing pressure and a slight shift toward lower temperatures is observed with molding pressure increase. For a more detailed analysis of this trend; however, it would be necessary to also conduct these studies with lower temperature rates in order to have a material in a more established steady state.

The AC conductivity obeys in a power law as a function of the molding pressure both in the ferroelectric and in the paraelectric phases. However, this processing pressure does not seem to play any role in a possible change of the conductivity during the phase transition (whatever the chosen processing pressure, the increase of the conductivity between the ferroelectric and the paraelectric phase remains equal to a tenfold). From our work, we extracted an AC conductivity value of the order of 2 x 10 -7 S/cm in the orthorhombic ferroelectric phase. This conductivity is of the order of 2 x 10 -6 S/cm in the tetragonal paraelectric phase.

The frequency of the applied signal has a large effect on the dielectric constant value in both the ferroelectric and paraelectric phase. High dielectric constant values are reached at the lowest frequencies investigated (between 50 and 100 in ferroelectric phase and over 1000 in paraelectric phase). The transition temperature is not affected by the frequency of solicitation of the compound. This information is interesting as generally the studies on this type of material are carried out at frequencies higher than a few kHz and rarely the choice of the frequency of measurement is justified. By focusing only on the physical phenomena associated with this phase transition, the choice to remain on high frequencies will be the right one to gain both measurement time and signal-to-noise ratio.

Finally, and as expected, the temperature ramp will affect the dielectric response at the phase transition since the mechanisms involved in this phase transition are thermally assisted. Thus, the temperature spread on the value of the hysteresis of the [dielectric constant-temperature] characteristic in the transition phase goes up with a higher temperature slope. A linear relationship between this temperature range and the applied temperature ramp is identified.

Figure 1 Dielectric constant as a function of temperature (+ 10 Kmin-1) for pellets pressed at 0.2, 0.4, 0.6, 0.8 and 1 GPa.

Dielectric measurements performed at 1 MHz. Inset: Dielectric constant as a function of pressure at isothermal 353 and 403 K (dashed line: linear fitting of curves).

Figure 2 aFigure 3 a

 23 Figure 2 a, b Alternative (ac) conductivity σ' vs temperature T and pressure P. c Loss factor e'' vs T and P. d Loss tangent tanδ = e''/e' vs T and P. Frequency 1 MHz. Temperature slope: 10 K/min.

Figure 4 a

 4 Figure 4 a Dielectric constant ε' vs temperature: influence of the scan speed S. b Temperature hysteresis window DT (extract from a for ε' = 25) vs the scan speed. b inset DSC spectra vs the scan speed S.

Acknowledgements

The authors would like to thank C. Pollet, J.-L. Palenzuela and J.-P. Baroux in the setting up of equipment in the preparation of pellets and dielectric measurements. Authors also thank C. Ioju and M. Tilahun of the Grenoble laboratory LEPMI for their help and accessibility to DSC characterizations. N. Bréfuel thanks O. Gallot-Lavallée for the first exchanges he had with him on the dielectric measurement.


Ildar Daminov

Anton Prokhorov

Raphael Caire

Marie-Cécile Alvarez-Herault

Assessment of dynamic transformer rating, considering current and temperature limitations

Keywords: Dynamic thermal rating, transformer, limiting factor, current limit, temperature limit, capacity

Nowadays, transformers can be operated close to their thermal limits. The state of the art in transformer thermal limits are Dynamic Thermal Rating (DTR). DTR can be based on two temperature limits: either a design temperature or maximal-permissible temperature. However, many papers estimate DTR with the design temperature only. Therefore, the true DTR is still underestimated since temperature limit is ignored as well as current limits (in some papers). Moreover, many papers rely on the conservative assumption of typical historical load profile or net load profile (considering the distributed generation, storage, electric vehicles). However, modern DSO can control the shape of (net) load profile. This can make a DTR estimation, assuming some typical (net) load profile, valuable only for a particular shape of (net) load profile but not for its modifications. This paper suggests a DTR feasible region which is constructed from current and temperature limitations and does not rely on typical load profiles. As a case study, we investigate DTR in cold and warm climates: one case in Russia with a continental climate and another in France with a temperate climate. In these climates, we assess DTR for the most common combinations of current and temperature limitations, used in standards and literature. As a result, DTR can ensure an additional capacity up to 45%. DTR, based on temperature limits, are 100 % of time higher than nominal rating. Moreover, the limiting factors of DTR are quantified based on analysis of 34-year ambient temperature data. Finally, comprehensive recommendations for transformer overloading are formulated.

Introduction

Nowadays, the thermal rating assessment is a relevant problem for engineers and researchers dealing with power system operation and planning. Thermal ratings have a strong impact on the optimal solution of a power system scheduling [START_REF] Miura | Application of dynamic rating to increase the available transfer capability[END_REF], a wind power integration [START_REF] Rocha | Dynamic rating assists cost-effective expansion of wind farms by utilizing the hidden capacity of transformers[END_REF][START_REF] Zarei | Reliability considerations and economic benefits of dynamic transformer rating for wind energy integration[END_REF][START_REF] Kazmi | Cost optimized dynamic design of offshore windfarm transformers with reliability and contingency considerations[END_REF], a hosting capacity of substations [START_REF] Papathanassiou | Capacity of Distribution Feeders for Hosting Distributed Energy Resources[END_REF], [START_REF] Arguence | Sizing power transformers in power systems planning using thermal rating[END_REF], an asset management [START_REF] Schmitz | Operational vulnerability indicator for prioritization and replacement of power transformers in substation[END_REF] among many others [START_REF] Numan | Impact assessment of a co-optimized dynamic line rating and transmission switching topology on network expansion planning[END_REF]. Moreover, due climate change problem the system operators should manage a power system with a large share of Distributed Energy Resources (DER) [START_REF] Nawaz | TSO and DSO with large-scale distributed energy resources: A security constrained unit commitment coordinated solution[END_REF]. To avoid congestions caused by new DER, system operators have to know the true thermal ratings of existing network. This is especially relevant since the lead time of DER is much less than that of network reinforcement [START_REF] Hoff | Managing Risk Using Renewable Energy Technologies[END_REF].

For many years, thermal ratings have been defined as Static Thermal Ratings (STR) [11]. STR is a constant limit, expressed in current or power units and usually calculated for daily-mean ambient temperature (Tamb). For instance, Table 1 shows STR of power transformers used for power system scheduling in Russia [12]. Similar STRs are used in other countries. Despite the long history of using STR, it has been repeatedly proven that STR is only a rough approximation of true thermal ratings [START_REF] Viafora | Day-ahead dispatch optimization with dynamic thermal rating of transformers and overhead lines[END_REF][START_REF] Teh | Prospects of Using the Dynamic Thermal Rating System for Reliable Electrical Networks: A Review[END_REF][START_REF] Zarei | Reliability considerations and economic benefits of dynamic transformer rating for wind energy integration[END_REF][START_REF] Douglass | A Review of Dynamic Thermal Line Rating Methods With Forecasting[END_REF]. This is explained by the fact that STR does not consider both the shapes of Tamb and load profile. Consequently, many scientists investigate Dynamic Thermal Rating (DTR) [START_REF] Viafora | Day-ahead dispatch optimization with dynamic thermal rating of transformers and overhead lines[END_REF][START_REF] Douglass | Real-time monitoring and dynamic thermal rating of power transmission circuits[END_REF][START_REF] Kim | Determination of available transfer capability (ATC) considering real-time weather conditions[END_REF][START_REF] Daminov | Receding horizon algorithm for dynamic transformer rating and its application for real-time economic dispatch[END_REF][START_REF] Daminov | Two-stage algorithm to solve the economic dispatch problem with dynamic transformer ratings[END_REF][START_REF] Bracale | A Probabilistic Approach for Forecasting the Allowable Current of Oil-Immersed Transformers[END_REF]. DTR represents a daily profile of admissible loadings, expressed either by current or power limits. DTR is usually higher than STR but for extreme ambient conditions DTR can be set lower than STR to avoid the network equipment overheating and/or mechanical damage. DTR is investigated for almost all network equipment [START_REF] Yang | Dynamic Network Rating for Low Carbon Application-A U.K. Application[END_REF]: power and distribution transformers [START_REF] Bracale | A Probabilistic Approach for Forecasting the Allowable Current of Oil-Immersed Transformers[END_REF], overhead lines [START_REF] Kim | An Application of Dynamic Thermal Line Rating Control System to Up-Rate the Ampacity of Overhead Transmission Lines[END_REF][START_REF] Karimi | Dynamic thermal rating of transmission lines: A review[END_REF][START_REF] Saied | Assessing the dynamic rating of overhead transmission lines[END_REF] and cables [START_REF] Huang | Dynamic cable ratings for smarter grids[END_REF].

This paper focuses on DTR of oil-immersed transformers. Unlike other network elements, oilimmersed transformers have a significant loading (overloading) capability above their nominal rating [START_REF] Tripathy | Evaluation of transformer overloading capability[END_REF]. This overloading capability allows operating a transformer above nominal ratings without sacrificing its design life [START_REF] Simonson | Thermal capability assessment for transformers[END_REF]. To ensure safe operation of a transformer, especially above nominal rating, one should meet current and temperature limitations [START_REF]Power transformers -Part 7: Loading guide for oil-immersed power transformers Transformateurs[END_REF]. Normally, these limitations are set for the current, hot spot temperature (HST) of a winding and top-oil temperature (TOT) in the tank. For instance, the Table 2 shows current and temperature limitations given in international and national standards for normal cyclic loading (i.e. without accelerated ageing). GOST [START_REF]General-purpose oilimmersed power transformers[END_REF] GOST [START_REF]Loading guide for oil-immersed power transformers[END_REF] Current, pu 1. Despite these standard limitations, many scientists use a design HST as an alternative temperature limit. This design HST is equal to 98 ℃ for non-thermally upgraded paper and 110 ℃ for thermally-upgraded paper whereas the HST limit for both papers starts from 120 ℃ (see Table 2). Transformers can continuously withstand a design HST during the full-service life since the insulation paper life is then equal to this service life. However, if transformers operate at HST limit all the time then the insulation life will be reduced at least 12,9 times faster for non-thermally upgraded paper and 2.7 times faster for thermally upgraded paper correspondingly [START_REF]Power transformers -Part 7: Loading guide for oil-immersed power transformers Transformateurs[END_REF], [START_REF][END_REF]. Consequently, transformers should be operated only for the short time at HST limit. Thus, we can define two types of DTR : DTR, based on the design HST [START_REF] Norris | The thermal rating of transformers[END_REF] and DTR, based on the HST limit [START_REF] Sealey | Overloading of Transformers---Cases Not Covered by the General Rules[END_REF]. The latter has a larger loading capability due to higher temperature limit. At the same time, DTR based on design HST allows to avoid any increased HST, but this can lead to less efficient optimization solution.

The existence of two DTR poses a problem: both DTRs should be estimated in the geographical area. At the same time, obtained DTR should be considered as representative if DTR estimations are based on long-term data (not few days or weeks). Only few studies estimate transformer limits in different areas over a long-term period. For instance, authors of [START_REF] Biçen | Loadability of power transformer under regional climate conditions: The case of Turkey[END_REF] estimate a loadability of ONAF power transformer for a typical load profile and 30 year monthly-mean Tamb in Turkey. In [START_REF] Yang | Predicting practical benefits of dynamic asset ratings of 33KV distribution transformers[END_REF], authors estimate DTR for ONAN power transformer based on the annual Tamb in United Kingdom. The one-year DTR benefits are studied in [START_REF] Yang | Dynamic Network Rating for Low Carbon Distribution Network Operation-A U.K. Application[END_REF] for ONAN and OFAF power transformers in Milton Keynes, United Kingdom. The paper [START_REF] Li | A study on transformer loading in Manitoba -Peak-load ambient temperature[END_REF] assesses peak-load transformer capability based on three months hour Tamb and load data in Manitoba, Canada. The data-driven approach is suggested in [START_REF] Dong | A Data-Driven Residential Transformer Overloading Risk Assessment Method[END_REF] to estimate residential transformer overloading based on two-year Tamb in West Canada. The paper [START_REF] Salama | Impact of Long-Term Climatic Conditions on the Ageing and Cost Effectiveness of the Oil-Filled Transformer[END_REF] investigates the loadability of ONAN power transformer in Egypt based on the meandaily Tamb at each month.

Although two types of DTR exist, none of the papers above estimates them simultaneously. Mostly all papers [START_REF] Biçen | Loadability of power transformer under regional climate conditions: The case of Turkey[END_REF][START_REF] Yang | Predicting practical benefits of dynamic asset ratings of 33KV distribution transformers[END_REF][START_REF] Yang | Dynamic Network Rating for Low Carbon Distribution Network Operation-A U.K. Application[END_REF][START_REF] Salama | Impact of Long-Term Climatic Conditions on the Ageing and Cost Effectiveness of the Oil-Filled Transformer[END_REF] consider DTR based on design HST i.e. DTR based on HST limit is omitted. Therefore, a real DTR capability is not fully revealed. Meanwhile, the paper [START_REF] Dong | A Data-Driven Residential Transformer Overloading Risk Assessment Method[END_REF] estimating DTR based on temperature limit does not consider limitations of current. The lack of current limit makes the DTR estimation incomplete. Thus, no study mentioned above investigates DTR in accordance with temperature and current limitations simultaneously. Meanwhile, the limiting factor of DTR can shift between the current and temperature. For instance, the following extract from IEC 60076-7:2005 describes this problem of factor shifting: "The temperature and current limits are not intended to be valid simultaneously. The current may be limited to a lower value than that shown (in Table 2 of this paper) in order to meet the temperature limitation requirement. Conversely, the temperature may be limited to a lower value than that shown (in Table 2) in order to meet the current limitation requirement". Consequently, DTR estimation, in terms of power limits, is challengeable and not evident. It is less evident if various temperature and current limitations are used (Table 2). Thus, DTR estimation with different current and temperature limitations is the first problem addressed in this paper.

The second problem that we address is the issue of a typical load profile in estimation of DTR. Many papers [START_REF] Biçen | Loadability of power transformer under regional climate conditions: The case of Turkey[END_REF][START_REF] Li | A study on transformer loading in Manitoba -Peak-load ambient temperature[END_REF][START_REF] Dong | A Data-Driven Residential Transformer Overloading Risk Assessment Method[END_REF][START_REF] Nguyen | Constrained optimisation procedure for evaluating cyclic loading of power transformers[END_REF][START_REF] Savaghebi | A new approach for transformer loading capability assessment under non-linear load currents[END_REF][START_REF] Shahbazi | A Probabilistic Approach for Power Transformer Dynamic Loading Capability Assessment[END_REF][START_REF] Zhang | Dynamic loading of substation distribution transformers in a production grade environment[END_REF][START_REF] Pasricha | A method of improving transformer overloading beyond nameplate rating[END_REF][START_REF] Alvarez | Transformer Thermal Capacity Estimation and Prediction Using Dynamic Rating Monitoring[END_REF][START_REF] Bunn | Empirical Design Method for Distribution Transformer Utilization Optimization[END_REF] assume a typical shape of load profile and they upscale this shape until temperature limits are met. The upscaled shape of load profile is then considered as DTR. We believe that the assumption of typical load profile becomes outdated in the modern era of smart grids. In the past, typical load profiles were indeed relevant since DSO could not actively control the shape of load profile and DER share could not affect this load shape. However, the fast integration of DER into distribution network changes the shape of a typical load profile. Thus, a load profile of consumers is not a reference for transformers anymore unlike a net load profile, whose shape is determined by DER and load.

Nevertheless, whatever a (net) load profile is given, modern DSO can actively change a shape of (net) load profile, using controllable DER: distributed generation, storage and demand response [START_REF] Degefa | Dynamic thermal state forecasting of distribution network components: For enhanced active distribution network capacity[END_REF][START_REF] Degefa | Unlocking distribution network capacity through real-time thermal rating for high penetration of DGs[END_REF][START_REF] Haque | Real-time congestion management in active distribution network based on dynamic thermal overloading cost[END_REF][START_REF] Prasanna | Gestion prévisionnelle des réseaux actifs de distributionrelaxation convexe sous incertitude[END_REF][START_REF] Yip | Dynamic thermal rating and active control for improved distribution network utilisation[END_REF][START_REF] Ali | Increased utilization of wind generation by coordinating the demand response and real-time thermal rating[END_REF]. Aggregators, new market players, already provide such services to DSO [START_REF] Irena | Aggregators Innovation Landscape brief[END_REF], [START_REF] Kaheh | A trilevel programming model for flexiramp and reserve procurement in high penetration of wind farms and participation of a large industry and a DR aggregator[END_REF]. This poses another problem for estimation of DTR: DTR should not target a single typical (net) load profile but multiple possible (net) load profiles.

This paper suggests to use the DTR feasible region depending only on Tamb to overcome two above mentioned problems. Feasible region allows to estimate thermal ratings of a power transformer without using a typical (net) load profile. The paper estimates DTR in cold and warm climates: one in Russia (Tomsk city in Siberia) with continental climate and another in France (Grenoble city in Alpes) with temperate climate. The paper focuses on the most common combinations of current and temperature limitations given in the Table 2.

Contributions of this paper are: (1) we suggest a DTR feasible region considering both current and temperature limitations. The feasible region encompasses many shapes of (net) load profile. (2) DTRs are estimated for the most common combinations of current and temperature limitations. [START_REF] Zarei | Reliability considerations and economic benefits of dynamic transformer rating for wind energy integration[END_REF] The recommendations based on proposed feasible region are formulated for transformer overloading.

The paper is organized in the following way: in section 2 we show how feasible region can be built and used to estimate DTR for a majority of load profiles. In section 3, authors estimate DTR for the most common combinations of current and temperature limitations. In Section 4, we present recommendations on transformer overloading.

The feasible region

The subsection 2.1 introduces a feasible region of loadings. In subsection 2.2, we present feasible regions limited by each of the factors described.

The feasible region of transformer loadings

The term "feasible region" is introduced to consider all admissible loadings in accordance with current and temperature limitations. It is taken from the mathematical optimization area [START_REF] Beavis | Optimisation and Stability Theory for Economic Analysis[END_REF]. Generally, the feasible region represents a set of all possible solutions of an optimization problem satisfying all given constraints. In case of a daily transformer loading, these constraints are current and temperature limitations (Table 2). However, one cannot draw a DTR feasible region in one x-y axis until their limitations are given in different physical units: pu (or A) for current and ℃ for temperature. We remind that DTR, as any thermal rating, is traditionally measured in the units of current/power or their per units. Thus, temperature limitations should be also presented in the units of current/power or their pu as a function of Tamb. Such expression is more convenient for energy specialists working with network operation and transformer operation in particular.

To harmonize units, one should know thermal characteristics of a transformer. In this paper, we use thermal characteristics of ONAF transformer given in the IEC standard [START_REF]Power transformers -Part 7: Loading guide for oil-immersed power transformers Transformateurs[END_REF], [START_REF] Susa | IEC 60076-7 loading guide thermal model constants estimation[END_REF] (Table 3). To convert temperature limitations into equivalent loading limits, one should build dependencies between steady-state loading and Tamb (Fig. 1). To draw each line in Fig. 1, one can use an algorithm shown in the right side.

Fig. 1 shows the relation between steady-state loading and Tamb obtained by the above-mentioned algorithm. First of all, we should define possible combinations of current and temperature limits (see them on the left side of Fig. 2). Depending on the maximal admissible values of HST and TOT, the loading of the transformer will be limited either by the current (IEC standards fix its maximal value at 1.5 pu) or by the temperature. Fig. 2 shows the limiting factors using horizontal bars plotted as a function of Tamb (x-axis in Fig. 2). The bar color represents a limiting factor calculated from Fig. 1. For instance, let us take the example of the 3 rd bar: HST≤120 ℃ & TOT≤95℃. The yellow line (HST=120 ℃) in Fig. 1 crosses the current limit at Tamb -17 ℃ and TOT = 95 ℃ at Tamb +45 ℃. This means that:  for Tamb ∈ [-50℃; -17 ℃], the current limit of 1.5 pu is the limiting factor (a blue area in the third bar of Fig. 2)  for Tamb ∈ [-17 ℃ ;+45 ℃], the HST = 120 ℃ is the limiting factor (yellow area in the third bar of Fig. 2)  for Tamb ∈ [+45 ℃;+50℃], the TOT = 95 ℃ is the limiting factor (purple area in the third bar of Fig. 2). We point out that Fig. 1 and Fig. 2 are obtained for a studied ONAF power transformer (Table 3) and not for all ONAF transformers. Latter ones can have distinct critical Tamb (black dots) due to the transformer design variations.

Once the loading-Tamb dependencies are known, we assume that we have a specific Tamb profile and build a feasible region for a studied ONAF transformer. Fig. 3 shows a daily feasible region (yellow area) calculated for a given Tamb profile. To build a feasible region, we suggest to plot current and temperature limitations independently from each other. As we have mentioned earlier, temperature limitations are in ℃ and therefore, they should be converted into power units, pu. This means that for each given value of Tamb profile in Fig. 3 we find loadings in Fig. 1 corresponding to e.g. HST limit = 120 ℃ and TOT limit = 105 ℃. Once we converted each temperature limits to loadings for each value of Tamb, we can plot three lines corresponding to HST, TOT and current limits. For example, at 12:00, the ambient temperature is 9°C. In Fig. 1, it can be seen that the transformer loading is limited by the HST to 1.3 pu, by the IEC (current) to 1.5 pu and by the TOT to 1.53 pu correspondingly. The lowest line (the HST line) among three lines shows us a top line of the feasible region. In other words, HST limit is reached earlier than current and TOT limit. Thus, HST limit is a limiting factor for DTR for the given Tamb profile.

It would be useful to define a part of this feasible region, which causes the normal insulation ageing (i.e. DTR based on design HST). To show it, we add a line (loadings), corresponding to the design HST = 98 ℃ (the calculation is similar to HST limit). The area below this new line is shown by green color, the other part, above this new line, remains yellow (Fig. 4). Thus, the green area represents DTR based on a design HST whereas the yellow area represents DTR based on HST limit. The difference between two DTRs in this example is always around 0.2 pu i.e. 20 % of the nominal rating. In other words, if DTR is based on the design HST alone then 20 % of actual transformer capacity would be neglected.

Once the construction of feasible region is explained, it is necessary to discuss the insulation ageing issue in feasible region. As the reader may note a feasible region is based on current and temperature limitations only. This means that an accelerated ageing is possible if transformer loadings exceed the green area (the design HST). Nevertheless, IEC standard clarifies the accelerated ageing: "this is not serious if there are otherwise long periods of time (usually the case) at relatively low hot-spot temperatures". This quote refers to the example, given in IEC standard, where cumulative ageing during 2 hours exceeds the normal ageing in 74 times. In other words, for 2 calendar hours a transformer has lost the insulation resource equivalent to 6 days of normal operation (=2 h*74 / 24 h). Indeed, such a high ageing rate in one day can be compensated by low ageing during other days, week or month. We note that IEC suggested this assumption at the time when DSO cannot actively control the shape of load profiles. Nowadays, however, DSO can control the shape of a loading profile. Therefore, this IEC assumption becomes more realistic. Thus, the suggested feasible region is explicitly determined for current and temperature limitations and implicitly for ageing. More details on dealing with accelerated ageing are presented in Section 4.

Once, ageing effects are explained, we need to justify the borders of feasible region which are based on the steady-state loadings (Fig. 1). To do that, one should address the Fig. 5, showing the interrelations (circles 1-4) between representative loadings (left side) and their temperatures (right side). One can notice that after blue load steps up, the transient blue temperatures (circle 1) reach a steady-state value (purple lines, circle 3) without exceeding it. This interrelation allows us to formulate an important conclusion: if loadings are always below the steady-state loading then transient temperatures are also below the steadystate temperature. Therefore, load profiles do not exceed the design HST if their loadings are located in the green area. Similarly, a load profile does not violate the HST limit, if its loadings are located in the yellow area.

Important notice should be discussed. The green area in Fig. 4, should be considered as the area without accelerated ageing only for the load profiles fully located within the green area. If even a small part of a load profile is located in the yellow area, the green area should not be referred to the area without accelerated ageing. Let us explain why: the circle 2 in Fig. 5 shows that even if the load is instantaneously reduced, the temperatures take time to reduce to new steady-state value. Thus, the transient temperature could be still in yellow areas while a loading returned back to the green area. Moreover, while temperature is reducing to a steady-state value, the ageing will be accelerated since the temperature is still higher than the design temperature. Despite the fact that this interrelation affects the ageing, it does not affect the feasible region of current or HST limit. Therefore, the suggested feasible region is still valid for current and temperature limitations.

Another important observation should be discussed. There is a specific load profile(s) (brown line in Fig. 5) with the loading higher than steadystate DTR around 06:00 but their transient temperatures still remain below the steady-state temperature (see circle 4). Therefore, we conclude that the feasible region, obtained earlier in the Fig. 4, can be actually even higher in terms of loadings. Although we agree with this statement, we note that such a load profile can be obtained only under DSO control (reducing a transformer loading just after its HST reaches its steady state limit). Therefore, such load profiles have the advantage for short-term planning [START_REF] Wang | Study on electrothermal coupling optimal power flow model and its simplification[END_REF], [START_REF] Wang | Advanced thermal rating and its application. 1st Int. Conf. Sustain. Power Gener. Supply, SUPERGEN '09[END_REF]. Thus, we neglect them in the suggested feasible region, which we use for long-term estimation of DTR. Moreover, this neglection actually reduces a feasible region that allows estimating DTR with margin.

Summarizing above-mentioned results: all load profiles located in the green area only are always feasible for both normal ageing and current/temperature. In contrast, all load profiles located in the yellow area only are always not feasible for normal ageing but feasible for current/temperature limitations. Load profiles, located in yellow and green areas, could be either feasible or infeasible from ageing point of view but always feasible for current/temperature limitations. Thus, we find a feasible region that always complies with temperature and current limitations. This compliance means that the suggested feasible region encompasses multiple load profiles.

Main results of this subsection: (1) temperature and current limitations are expressed in the same unitspu. This allows to (2) plot the feasible region of loadings considering majority of load profiles. (3) the areas of normal and accelerated ageing are determined and their limitations are explained.

Examples of feasible regions with different limiting factors

In this subsection we describe feasible regions with different limiting factors. We identify 6 possible combinations of limiting factors: 1. Current only; 2. HST only; 3.TOT only; 4. Current + HST; 5. HST + TOT; 6. Current + TOT. The case 2 (HST only) was already presented in Fig. 4. Thus, in this subsection we present main examples of feasible regions (case 1, 3, 4).

Current limit only: Fig. 6 shows a feasible region built for limitations HST 120 ℃ & TOT 105 ℃ and Tamb profile on January 11, 2019 in Tomsk, Russia. From the Fig. 6 we see that the top line of the feasible region corresponds to the current limit (dash line). To explain why the current is a limiting factor, one should compare the range of the given Tamb profile with Tamb axis (in Fig. 1) corresponding to current and HST limit =120℃. There we see that the current limit remains a limiting factor for Tamb below -17℃. At the same time, given Tamb profile (varying between -19℃ and -33℃) remains always lower than this critical Tamb. Thus, current limit remains a limiting factor all day long.

TOT limit only: Fig. 7 shows a feasible region built for limitations HST 140 ℃ & TOT 95 ℃ and Tamb profile on July 07, 2018 in Grenoble France. Similarly, we use Fig. 1 to explain why TOT is a limiting factor. For given current and temperature limitations, TOT limit (95 ℃) is a limiting factor for Tamb > +2 ℃. At the same time, Tamb profile varies in the range between +10 ℃ and +18 ℃, which is higher than this critical Tamb: This explains why TOT is a limiting factor all day long.

Current+HST limit: Fig. 8 shows a feasible region built for HST limit =120℃ &TOT limit =105℃ and Tamb profile on January 15, 2019 in Tomsk. From Fig. 8 we see that once Tamb crosses the critical Tamb = -17 ℃ (Fig. 1), the limiting factor shifts from current = 1.5 pu to HST limit = 120 ℃.

The main result of this subsection: examples of Tamb profiles are shown for different limiting factors.

Assessment of dynamic transformer ratings

This section presents the results of DTR assessment in Tomsk and Grenoble. To obtain representative results, DTR estimations are based on long-term data of Tamb in each geographical area. For instance, the climate science recommends to consider at least 30-year-long interval for representative estimations [START_REF]Guidelines on the Calculation of Climate Normals[END_REF]. Therefore, we use hour Tamb [START_REF] Meteoblue | history+ -Historical weather data since 1984[END_REF] for the 34-year period from January 01, 1985 to March 29, 2019 (time of data downloading) in Tomsk, Russia and Grenoble, France (Fig. 9).

From Tamb data, shown in Fig. 9, we have 12 506 daily Tamb profiles in each city. For these Tamb profiles, we define 12 506 daily feasible regions corresponding to different combinations of temperature and current limitations (Fig. 10). From Fig. 10 we see that the black line shape (a limiting factor) of these feasible regions is variable. To quantify this DTR variability, mean DTR and its maximum and minimum deviations are estimated for the most common formulations of current and temperature limitations. (Fig. 11) The bars in Fig. 11 estimate two types of DTR: DTR, based on design HST (dark green bar) and DTR based on HST limit (other bars). At the same time, the majority of papers estimates DTR using the design HST only (dark green bars). This allows scientists to avoid problems with accelerated ageing which is an advantage. As a drawback, they ignore a substantial part of DTR, confined by current and temperature limitations. For instance, in [START_REF] Yang | Predicting practical benefits of dynamic asset ratings of 33KV distribution transformers[END_REF][START_REF] Yang | Dynamic Network Rating for Low Carbon Distribution Network Operation-A U.K. Application[END_REF] authors estimated that DTR provides 10% additional capacity in United Kingdom. This correlates with our dark green bars where mean DTR provides 15 % of additional capacity over nominal rating in Tomsk Russia and 5% in Grenoble France. Nevertheless, if one compares dark green bars with other bars then DTR can ensure up to 45 % of additional capacity in Tomsk and 41% in Grenoble. This is 30-36 % more power in comparison to the DTR, based on the design HST (dark green bars). It is noteworthy that this additional power leads to an increased HST. However, DSO can control this load amplitude and duration using the flexibility from DER.

DTR used together with DER management provides an additional degree of freedom for system operators in power systems scheduling. At the same time, this degree of freedom changes during a year following DTR seasonal variations. Therefore, DTR should be estimated for all current and temperature limitations per month (Fig. 12)

Figure 12: Mean DTR with maximum and minimum deviations in each month From Fig. 12 we see that dark green bars (DTR based on the design HST) exceed nominal rating of the transformer during almost all months. However, in summer months, such DTR should be set lower than nominal rating to avoid the violation of the design HST. As we said earlier, the dark green bars are a classical example of DTR, studied in many papers [START_REF] Yang | Predicting practical benefits of dynamic asset ratings of 33KV distribution transformers[END_REF][START_REF] Yang | Dynamic Network Rating for Low Carbon Distribution Network Operation-A U.K. Application[END_REF]. However, these papers ignore other bars shown in the Fig. 12 This leads to very conservative estimation of DTR. In contrast, this paper allows to determine the part of DTR, which was omitted before.

We would like to explain some particular bars in Fig. 12. For instance, yellow and light green bars are the same in both cities. This means that HST limit = 120 ℃ is always reached before TOT limit (95 ℃ or 105 ℃) in both climates. This happens since Tamb of studied climates is always below than critical Tamb +45 ℃ (see Fig. 1). Therefore, HST remains the unique limiting factor for these two current and temperature limitations.

Moreover, the reader can notice that red and orange bars in Tomsk do not have any deviations in winter months. This means that the current limit is always reached earlier than temperature limits. Thus, bars, whose loading is equal to 1.5 pu, are current-limited and bars, whose loading is below 1.5 pu are temperature-limited. Therefore, Fig. 12 is an example showing how the limiting factor of DTR is shifting between current to temperature during the year. Moreover, Fig. 12 represents an example of how different current and temperature limits pre-define the amplitude of DTR.

In addition to DTR amplitude (bars in the Fig. 12), we estimate DTR duration. The typical DTR duration curves in Tomsk and Grenoble are calculated and presented in Fig. 13. Therefore, Fig. 13 shows how different current and temperature limits pre-define the duration of DTR. To find such a duration curve, one can sort all values of the DTR array (taken from Fig. 10) in a descending order. This gives us the y-data. The DTR duration (x-axis) is obtained as following:

(

Where Na numerical order of y-data (DTR sorted in a descending order).

DTR duration in x-axis shows the amount of time (in %) when DTR exceeds the value selected on the duration curve. For instance, the classical DTR (dark green curve) exceeds a nominal rating of a transformer for 88,5 % of time in Tomsk and 79 % of time in Grenoble. This also means that the classical DTR is below nominal rating for 11.5% and 21 % of time in these cities correspondingly. This result correlates with conclusions of many authors, stating that DTR can be below the nominal rating for a short period of time [START_REF] Yang | Predicting practical benefits of dynamic asset ratings of 33KV distribution transformers[END_REF][START_REF] Yang | Dynamic Network Rating for Low Carbon Distribution Network Operation-A U.K. Application[END_REF]. Meanwhile, we see that duration curves of other DTRs remain higher than the nominal rating for 100 % of time in both cities. These results quantify that part of DTR ignored in similar studies.

Finally, we identify the main limiting factor of DTR in Tomsk and in Grenoble. The easiest way to do that is to take a Tamb history in each city and see limiting factor for each Tamb range in Fig. 1. For instance, we assume that Tamb is -10 ℃ and DTR formulation is current 1.5 pu, HST 120 ℃ and TOT 105 ℃. For this Tamb, the limiting factor is the HST 120 ℃ (see Fig. 1). Fig. 14 shows limiting factors and their occurrence expressed in % of studied period for each DTR formulation. Pie chart colors correspond to colors of lines in Fig. 1. Interestingly, some papers [START_REF] Arguence | Sizing power transformers in power systems planning using thermal rating[END_REF][START_REF] Kostin | Urban Substations Transformers Allowed Loading[END_REF][START_REF] Savaghebi | A fuzzy-based approach for transformer dynamic loading capability assessment[END_REF] assume HST as a main limiting factor. Therefore, they use only HST limits as transformer limits in the formulation of an optimization problem. Although such an assumption can be true for the current and temperature limitations chosen in a particular paper, other scientists can make an error by using such logic for other DTR formulations or for other Tamb. For instance, Fig. 14 shows that DTR based on the design HST has HST as a limiting factor 99,9 % of time in Tomsk and 100% of time in Grenoble. Indeed, current or TOT limits do not affect the limiting factor of a studied ONAF transformer whatever the climate is chosen. However, if one uses a higher HST limit and TOT limit, we see that current becomes a main limiting factor from 9% to 51 % of time in Tomsk. Notably, the HST is no longer a limiting factor for formulation: current 1.5 pu, HST 140 ℃ and TOT 95 ℃. Therefore, the assumption that HST is always a limiting factor can be fully and partially true but also totally wrong for current 1.5 pu, HST 140 ℃ and TOT 95 ℃.

Recommendation for transformer overloading

In this section we suggest recommendations for transformer overloading based on feasible region.

First of all, the green area of the feasible region in Fig. 4, 6, 7 and 8 can be a reference for system operator to avoid any accelerated loss of insulation life (LoL). Thus, system operators can keep the transformer loading always within green area and therefore avoid any accelerated LoL. As we found in Fig. 11 the green area provides 5% of additional capacity in Grenoble and 15 % in Tomsk. Secondly, the yellow area can be a reference to keep the transformer loading within temperature limits. The yellow area provides up to 41 % of additional capacity in Grenoble and 45 % in Tomsk.

However, the system operator should be aware that the operation in the yellow area causes the accelerated LoL. Therefore, we quantify the highest accelerated LoL, which can happen if system operator has to keep the loading within yellow area. To do that, let us assume the worst case of overloading (the amplitude and duration) during 1 day. We believe that the worst overloading happens when the loading is equal to the top border of yellow area during the whole day.

Therefore, the worst case for LoL would be if the top border of yellow is always defined by HST limit (120 ℃ or 140 ℃ depending on DTR formulation). In such a case, LoL will be the most severe since the maximum allowable HST is applied. Thus, the most severe LoL can be estimated by Arrhenius formulas (2) -( 3):

T HST t T HST t dt T LoL e dt T                   ℃ ℃ (2) (3)
Where T is a duration of studied period (1 day).

Equation ( 2) corresponds to the case if insulation paper is non-thermally upgraded also known as Kraft paper. The Kraft paper has a design HST= 98 ℃ which was assumed for studied ONAF transformer. However, a new ONAF transformer can be equipped with thermallyupgraded paper with the design HST =110 ℃. In this case, equation ( 3) should be used. Table 4 shows a LoL estimation by ( 2)-( 3) for HST limit = 120 ℃ and 140 ℃ correspondingly. Table 4 LoL in the worst case, pu = days The results in Table 4 can be converted to overloading occurrence expressed in days per year. To do that, we suggest using formula (4).

The meaning of this formula is that the sum of the overloading days and "compensating" days is less or equal to the number of days in the year. Thus, Table 5 shows the admissible occurrence of overloading calculated by equation ( 4). The number of compensating days is then calculated as difference between 365 days and overloading days. The results of LoL estimations (Table 4) mean that depending on insulation paper, the worst day would consume the same insulation resource as 2.7 days minimum or as 128 days maximum of operation at the design HST. The corresponding overloading occurrence varies from 98 days down to 2 days per year (Table 5). Thus, the number of compensating days varies from 267 days up to 363 days.

However, these results do not mean that transformer should be out of service during "compensating" days. Actually, the studied transformer ONAF can be still loaded for 64-89% from nominal rating in average. This non-evident result is explained by the exponential interrelation between HST and Ageing Acceleration Factor (AAF) shown in Fig. 15 (y-axis has a log scale) Figure 15: Exponential dependencies of AAF on HST Fig. 15 shows that AAF drastically reduces if HST is decreased linearly. System operators can take advantage of this dependency by keeping the loadings below low HST during some days to compensate the accelerated LoL during 1 day. As we said earlier, for such compensating days, transformers can still operate at 64-89% from their nominal rating. To explain how 64-89% are calculated let us assume that the transformer operation during compensating days should not cause more than 1 % of normal ageing (% could be set lower if needed). In this case, it is necessary to choose such HST when LoL would be 1 % of normal LoL (see flags in Fig. 15). In accordance with equations ( 2)-(3) for the design HST = 98 ℃ such HST1% = 58 ℃ and for the design HST = 110 ℃ such HST1% = 70 ℃ correspondingly. Note that the loading corresponding to these HST1% would vary as a function of Tamb. That is why if we know historical Tamb data in Tomsk and Grenoble (shown in Fig. 9), then we can find loadings corresponding to these HST1%. To do that, the similar algorithm, presented in the section 2, can be used but instead of a temperature limit, it is necessary to use HST1%. Fig. 16 shows loadings (black lines) corresponding to HST1% = 58 ℃ and 70 ℃ in the range of Tamb from -50℃ to +50 ℃ (x-axis Fig. 16). Note that these black lines are true for any geographical area (for given ONAF transformer) but it does not mean that the admissible loadings of the studied ONAF transformer will be the same in each geographical area as latter has a particular Tamb conditions. For instance, Fig. 16 shows the probability density function (PDF) of Tamb in Tomsk and in Grenoble. From Fig. 16 we see that PDF of Tamb tends to some particular loadings. To calculate similar PDF of transformer loadings, it is sufficient to take each Tamb (on PDF of Fig. 16) and find the corresponding loading (on the black curve in Fig. 16). Fig. 17 shows the resulting PDF of transformer loadings in Tomsk and in Grenoble. The reader could note two observations from PDFs shown in Fig. 17. First, the shape of PDF always remains the same, whatever the insulation paper is used. It seems that PDF of Tamb in each climate predetermine the PDF shape of loadings. The only difference between two types of insulation paper is a mean loading. The second observation is that a mean loading at HST1% can be higher than a nominal rating (at least for Tomsk). It seems that the reason for such increased admissible loadings is a cold climate in Tomsk. Generally, Fig. 17 proves that system operators can compensate the accelerated LoL even if operating a transformer at loadings close to the nominal rating. For such "compensating" days, system operator can redraw the feasible region by using HST1% as temperature limit.

In addition to recommendations already presented in the paper, we would like to provide important recommendations from other papers, which can be linked, to the feasible region. For instance, system operators could update feasible region at each hour. This would allow them to consider the actual loading and Tamb at past intervals to increase/decrease the feasible region at next intervals. The general logic of such DTR updating can be based on receding horizon control [START_REF] Daminov | Receding horizon algorithm for dynamic transformer rating and its application for real-time economic dispatch[END_REF] also known as predictive control [START_REF] Straub | Robust MPC for temperature management on electrical transmission lines[END_REF]. It would be also valuable to develop the advanced ageing model linking the LoL and failure effect e.g. as it was done in [START_REF] Teh | Uncertainty Analysis of Transmission Line End-of-Life Failure Model for Bulk Electric System Reliability Studies[END_REF], [START_REF] Teh | Impact of the realtime thermal loading on the bulk electric system reliability[END_REF] for DTR of overhead lines or in [START_REF] Huang | Overload strategy of transmission and transformation equipment for safety operation[END_REF] for transformers. The probabilistic nature of a feasible region can be enhanced based on [START_REF] Bracale | A probabilistic approach for forecasting the allowable current of oil-immersed transformers[END_REF][START_REF] Bracale | SmarTrafo: A Probabilistic Predictive Tool for Dynamic Transformer Rating[END_REF][START_REF] Bracale | A Predictive Stress-Strength Model Addressing the Dynamic Transformer Rating[END_REF]. Moreover, system operators should reduce the current limit of a feasible region to the rating of tap changer or bushings if some of them would limit transformer loadings as it is shown in [START_REF] Pasricha | A method of improving transformer overloading beyond nameplate rating[END_REF]. The effect of harmonics, unbalancing, moisture in oil-insulation system, oil viscosity among others should be carefully assessed since they can significantly reduce the thermal rating of transformers [START_REF] Godina | Effect of loads and other key factors on oiltransformer ageing: Sustainability benefits and challenges[END_REF].

Conclusion

In summary, the paper assesses DTR for various current and temperature limitations. In contrast to similar studies, the paper takes into account all limiting factors (current, HST and TOT) and their combinations. This allowed us to assess those parts of DTR, which were usually omitted in similar studies. At the same time our results showed that this omitted DTR represents a large transformer capacity in the range from 25% to 45 % nominal rating. Moreover, DTR duration curves prove that this additional DTR capacity is higher than nominal rating during 100 % of time in contrast to classical DTR which is 88,5% in Tomsk and 79 % in Grenoble correspondingly. However, we pay attention to the fact that this additional capacity is operated at increased HST. Nevertheless, modern DSO can control the shape of transformer loadings by using DER. This allows controlling the amplitude and durations of transformer loadings and therefore making them feasible from both sides: current /temperature limitations and ageing. This provides additional degree of freedom for system operators to manage the power systems. This additional capacity can be especially relevant if we recall that the cost of HV/MV substation can vary from 500 k€ to 1.5 M€. Therefore, DSO can defer large investments into the transformer by taking advantages of DTR and DER.

Another important result is that we avoid using a typical (net) load profile for DTR determination. Instead, we built a feasible region of load profiles, which is based on Tamb only. Thus, we consider multiple shapes of load profiles and not only typical ones. Moreover, DTR assessment showed that the main limiting factor is very sensitive to the chosen formulation of current and temperature limitations. For instance, for majority of formulations, HST partially or fully remains a limiting factor but for other cases (HST 140 ℃ and TOT 95 ℃) HST does not affect DTR at all. At the same time, many papers considered HST as the main limiting factor, which is not necessarily true.

Finally, the recommendation is formulated to consider full feasible region of loadings. The worst LoL is quantified and based on that we estimated the admissible number of overloading days per year. Moreover, the paper proves that the compensation of accelerated LoL can be ensured at loading around 64-89% of nominal rating. Thus, system operators can still use transformers at relatively high loadings and at the same time compensate the accelerated LoL.

Figure 1 :Figure 2 :

 12 Figure 1: Transformer loadings equal to HST and TOT limits as a function of Tamb

Figure 3 :

 3 Figure 3: Feasible region (yellow area)

Figure 4 :

 4 Figure 4: Same feasible region, but showing the loadings with normal ageing (green area)

Figure 5 :

 5 Figure 5: Interrelations between transformer loading and temperatures, calculated by IEC thermal model

Figure 6 :

 6 Figure 6: Feasible region limited by the current only

Figure 7 :

 7 Figure 7: Feasible region limited by the TOT only

Figure 8 :

 8 Figure 8: Feasible region limited by current and HST

Figure 9 :

 9 Figure 9: Hourly Tamb from 1985 to 2019 in Tomsk and Grenoble

Figure 10 :

 10 Figure 10: Estimation of feasible regions during 34 years

Figure 13 :

 13 Figure 13: DTR duration curves

Figure 14 :

 14 Figure 14: Share of limiting factors: based on 34 years analysis (% are rounded)

Figure 16 :

 16 Figure 16: Black lines: transformer loadings corresponding to HST1% .Orange and blue bars: PDF of Tamb in Tomsk and in Grenoble based on 34-years history

Figure 17 :

 17 Figure 17: Loadings which can be used for compensation of accelerated LoL (based on historical Tamb)

Table 1 :

 1 STR of power transformer as a function of daily mean Tamb

	Mean Tamb, ℃	-20	-10	0	+10 +20 +30 +40
	STR, pu	1.20 1.20 1.15 1.08 1.0 0.91 0.82

Table 2 :

 2 Current and temperature limitations

	Limiting	International	National (Russian)
	parameters	IEC	IEEE	STO
		[29]	[30]	[12]

Table 3 :

 3 Thermal characteristics of an ONAF power transformer

	Parameter, units		Value
	Oil exponent, no unit	x	0.8
	Winding exponent, no unit	y	1.3
	Loss ratio, no unit	R	8
	Oil time constant, min	τo	150
	Winding time constant, min	τw	7
	Design hot spot temperature, ℃	θh	98
	Design ambient temperature, ℃	θa	20
	Hot-spot to top-oil gradient at rated current, K	Δθhr	35
	Top-oil temperature rise, K	Δθor	45
	Thermal constant, no unit	k11	0.5
	Thermal constant, no unit	k21	2
	Thermal constant, no unit	k22	2

Table 5

 5 Overloading and compensating days

	Overloading days	120 ℃	HST limit	140 ℃
	Design HST	98 ℃ 110 ℃	≈ 26 days ≈ 98 days		≈ 2 days ≈ 20 days
	Compensating days	120 ℃	HST limit	140 ℃
	Design HST	98 ℃ 110 ℃	≈ 339 days ≈ 267 days	≈ 363 days ≈ 345 days

Acknowledgments

We deeply appreciate Conference des Grandes Ecoles, who first supported our research and laid the ground for its further development. The research is also funded from Tomsk Polytechnic University Competitiveness Enhancement Program grant. Authors thank the IDEX for funding Ildar DAMINOV travel grant. We gratefully appreciate the funding received in the framework of Assistant Professor position of Ildar DAMINOV in Grenoble INP. We thank Egor GLADKIKH (a founder of TOPPLAN start-up, https://www.topplan.fr/) for his help with financing Ildar DAMINOV's Ph.D. thesis. Authors thank MeteoBlue company for provided data on ambient temperature in Grenoble, France.