Xixi Huang

Sandrine Bec

Jean Colombani
email: jean.colombani@univ-lyon1.fr

Ambivalent role of fine particles on the stability of a humid granular pile in a rotating drum

Keywords: stability, rotating drum, dust, humidity, lubrication, jamming

We have studied the influence of fine particles on the stability of a granular medium in a rotating drum. The stability diagram of this system was established as a function of the drum rotation speed, fine particle content and relative humidity. Four regimes were observed. At low fine content, an avalanching regime is encountered at low rotation rate, and a continuous flow regime at high rotation rate. At high fine content, a stick-slip regime at the drum wall is seen at low rotation rate and a continuous sliding regime at high rotation rate. The influence of fines is ambivalent. At low fine content, they fluidize the pile and decrease its stability, by granular lubrication. At high fine content, they solidy the pile and increase its stability, by jamming the grain assembly. The enhancement of humidity increases the stability for high fine content, but has no effect for low fine content.

Introduction

Granular matter is one of the most complicated materials, as it can present either gaseous, liquid and solid behaviors. Its mechanical properties have been studied by various methods : inclined plane, tilted box, rotating drum, etc. Among these configurations, the rotating drum is commonly used in laboratory experiments and also in the industry for processing granular matter [START_REF] Seiden | Complexity, segregation, and pattern formation in rotating-drum flows[END_REF]. Indeed, in the simple configuration of a rotating cylinder, several processes can be performed : mixing, drying, segregation or chemical reactions.

The stability of a dry or humid granular pile in a rotating drum has been extensively studied, revealing the richness of the granular motion inside the cylinder. By varying the rotation velocity, different regimes of the bed behavior are observed : slipping, slumping, rolling, cascading, cataracting and centrifuging. Among them, slumping and rolling are usually used in industry for material mixing [START_REF] Ottino | On mixing and demixing[END_REF]. At low rotation velocity, the granular pile is lifted as a solid body pursuing the rotating drum, the surface angle θ of the pile increases until a maximum angle of stability, the "upper repose angle", named θ m , is reached, then the granular pile destabilizes at its surface and particles on top start to roll downhill as an avalanche. When the slope of the pile reaches a minimum angle of stability, the "lower repose angle", called θ r , the avalanche stops, and so on until another avalanche occurs. This regime is defined as the slumping or discrete avalanche regime. The subsequent rolling regime occurs when the rotation velocity is increased. In this case, the cycle duration is shorter than the avalanche duration. Thereby the grains at the surface continuously roll, keeping the surface angle constant.

A lot of researches have studied the granular bed behavior in the slumping and rolling regimes. However most of these studies are performed with a monodisperse granular matter, while in industrial processes the granular material is usually bidisperse or even polydisperse. Moreover even a so-called monodisperse granular medium can contain a certain quantity of small particles due to the wear of the grains, especially in the grinding process [START_REF] Opoczky | Fine Grinding and Agglomeration of Silicates[END_REF]. The presence of such "dust" particles of very small size is one of the origins of the phenomenon of re-agglomeration and caking of particles [START_REF] Restagno | Aging in humid granular media[END_REF][START_REF] Mathlouthi | Water vapour sorption isotherms and the caking of food powders[END_REF]. Up to now, it has hardly been studied in experimental researches.

By adding a small quantity of fine particles to a monodisperse granular pile and measuring the maximum stability angle of this pile in a rotating drum with a controlled relative humidity, we have shown recently that the influence of the fines on the pile stability is ambivalent [START_REF] Huang | Influence of fine particles on the stability of a humid granular pile[END_REF]. We detail here the experiments leading to this result and show how, at low fine content, the small particles play the role of granular lubricant and make the heap more unstable, and at higher content, they block the inner motions inside the heap and solidify it. These results enable to make a link between independent and inconsistent observations, that had not been compared until now, showing that fine particles in one well-defined system may either play a fluidizing or a jamming role, depending on their percentage.

Experiments

Experimental setup

The experimental setup is illustrated in figure 1, based on a device used in previous studies [START_REF] Restagno | Aging in humid granular media[END_REF].

The granular material is introduced in a stainless steel drum of 10 cm radius and width, rotating around its horizontal axis at a fixed speed controlled by a synchronous electric motor (AKM23D AN CNAA, Kollmorgen). The rotation speed of the cylinder can be varied from 0.0001 to 500 rpm.

In this study only slow rotation speeds were used. The cylinder is enclosed by two glass windows with a hole at their center for air exchange with the outside atmosphere. The drum is partially filled with a fixed quantity of 200 g of glass beads, with a filling ratio of 30%. The glass beads used in this study are soda-lime glass beads (BV250) of diameter (250 ± 50) µm (Sigmund Lindner).

The surface of the glass beads has been observed by atomic force microscopy (AFM) and scanning electron microscopy (SEM), showing a small roughness of the surface, typically 500 nm (figure 2).

Fine glass beads (BF10) of diameter (5 ± 4) µm of the same material are added in the granular pile as fine particles. The density of the soda-lime glass is 2.50 kg/L and the bulk density of the bed of large particles is 1.46 kg/L (manufacturer data).

Usual bidisperse granular mixtures are known to segregate. Therefore we have sampled the mixture at the top and bottom of the heap at the end of some experiments and we have observed by optical microscopy that the fine particles content was uniform. This absence of segregation finds an explanation in the characteristic distribution of the fine particles among the large ones (see section 4.1 and figure 13).

The experimental setup is placed in a sealed poly(methyl methacrylate) box. Both relative humidity and temperature are controlled in the experiments. The temperature in the box is controlled with a heating resistor regulated by a temperature controller, and is fixed at 29.0 • C. A large flat beaker of saturated aqueous inorganic salt solution is placed in the box to maintain a constant value of relative humidity during the experiments. Four different salts are used in our experiments.

Table 1 shows the theoretical value of the relative humidity in the atmosphere in the presence of the saline solutions at 25 • C. The relative humidity H is measured by a hygrometer with an accuracy of 1.5%. As the experiments are carried out at 29 • C, and a slight discrepancy may exist between the expected and real values of the relative humidity, only measured values are used thereafter. A fan is placed inside the box to ensure homogeneous temperature and humidity.

The granular pile is illuminated from the front through the glass windows to improve the surface detection (figure 3). A digital video camera placed in front of the drum is used to continuously record the behavior of the granular medium. The angle of the surface of the pile with the horizontal is automatically calculated from the captured images. The uncertainty in the angle measurements is typically ±0.5 • .

Procedure

1. A 200 g mixture of coarse and small beads is mixed in a vessel by hand.

2. The mixture is introduced in the drum in rotation at relatively low speed (1 rpm) for 24 h in the desired environment to homogenize the atmosphere throughout the granular pile.

3. Rotation at 5 rpm during 30 s provides an initial reference state of the pile.

4. Rotation at the desired speed n during a few cycles is used as "pre-measurement". It permits to determine the flowing regime of the granular medium. If the regime is avalanching or stickslip (see below), it gives an indication of the maximum and minimum angles of stability θ m and θ r of the granular pile. Figure 4 shows an example of pre-measurement in the avalanche regime. If the regime is continuous flowing or sliding (see below), it gives an indication of the dynamic repose angle. The reference angle θ ref is determined as the angle 0.5 • smaller than the maximum angle or dynamic repose angle.

5.

The drum rotates at the same rotation speed n, and the evolution of the surface angle of the granular pile θ is recorded versus time. For each turn, a 3 s stop, which corresponds to the waiting time t w , is effected at θ = θ ref . Then the rotation continues in the same direction at the same speed until an avalanche occurs. The avalanche angle θ m and repose angle θ r are recorded (figure 4).

6. The last step is repeated 100 times and the values of the avalanche angle θ m and repose angle θ r are determined from the average value of the 100 measurements performed by automatic image analysis.

The most important experimental quantities in this procedure are the waiting angle θ ref and waiting time t w . Indeed previous studies have shown that the stability of the heap firmly depends on these two values. This influence is a consequence of the fact that the cohesion of the heap partly stems from the slow condensation of capillary bridges between the beads [START_REF] Restagno | Aging in humid granular media[END_REF]. So the maximum stability angle is observed to enhance when the waiting time or waiting angle is increased. For this reason, all experiments were carried out with the same θ ref and t w values.

Based on this procedure, three major parameters affecting the stability of the strongly bidisperse granular pile in the rotating drum were varied : (a) the volume fraction φ of fine particles ;

(b) the rotation speed n ; (c) the relative humidity H. For each condition of relative humidity, at small concentration of fine particles (lower part of the diagram), the avalanche regime is observed at low rotation speed n. When the drum is rotating faster than ∼ 1 rpm, the regime of continuous flow occurs. When φ increases and reaches a critical value φ c , the behavior of the granular medium changes : a phenomenon of stick-slip at the inner wall of the drum is observed and alternates with the avalanche regime. The two regimes coexist when φ is between 0.15 and 0.24%. For a larger fine fraction (upper part of the diagram), the surface of the pile stabilizes, no more avalanche is observed and only the stick-slip behavior at the drum wall is present. In this configuration, if n is increased, the stick-slip regime transforms in a continuous sliding behavior.

Results

Stability diagram

Low volume fraction of fines

When φ < φ c , the destabilization of the particles occurs at the surface of the granular heap.

The transverse flow behavior of granular matter has been studied widely in the literature [START_REF] Henein | Experimental Study of Transverse Bed Motion in Rotary Kilns[END_REF][START_REF] Rajchenbach | From Discrete Avalanches to Continuous Regime[END_REF], depending on several parameters, principally the rotation speed and the filling level. Here, as we use a rotation speed below 10 rpm and a fixed filling level, only the longitudinal avalanche and continuous flow regimes will be discussed.

Rotation speed influence

In the avalanche regime, during the rotation of the drum, avalanches of several layers of particles detach from the bed and slumps toward the lower extremity of the bed surface. The slump occurs periodically during the rotation, each time the surface reaches the maximum angle of stability θ m (the upper angle of repose). Following a slump, the bed inclination reaches a steady position at a minimum angle of stability θ r (the lower angle of repose) (figure 6). When the rotation speed increases (> 1 rpm), the intermittent avalanches turn to a continuous flow, which is characterized by a continuous rolling of a few layers of particles at the bed surface. In the rolling regime, the surface of the granular pile appears flat with a constant angle of inclination, defined as the dynamic repose angle θ dyn (figure 6). The measurements of θ m , θ r and θ dyn are carried out for n ranging from 0.05 to 10 rpm.

An example of the evolution of the upper θ m , lower θ r , and dynamic θ dyn repose angles versus rotation speed is shown in figure 7. In the avalanche regime, θ m and θ r exhibit an approximately constant value when n is increased. The only influence of the rotation speed here is the duration between two consecutive avalanches which shortens when the rotation speed increases [START_REF] Tegzes | Avalanche Dynamics in Wet Granular Materials[END_REF]. As we maintain the heap immobile at an angle slightly below θ m during a constant waiting time before each avalanche (figure 4), the state of cohesion is always identical [START_REF] Bocquet | Moisture-induced ageing in granular media and the kinetics of capillary condensation[END_REF], so the maximum stability angle does not vary. In the continuous flow regime, the dynamic repose angle rises linearly with n (figure 7), as expected numerically [START_REF] Seiden | Complexity, segregation, and pattern formation in rotating-drum flows[END_REF]. The transition between both regimes occurs when the rotation becomes too rapid for the surface to have time to relax from θ m to θ r . For instance in figure 7, the transition occurs at n = 0.7 rpm = 4.2 • /s. We see that at that point, (θ m -θ r ) = 2.7 • and the avalanche duration has been measured as τ = 593 ms, corresponding to n c = θm-θr τ = 4.55 • /s, in accordance with the observed value.

The transitions between regimes in granular media in a rotating drum have been studied since the 1980s with different geometries, i.e., rotating drum and rectangular box [START_REF] Rajchenbach | From Discrete Avalanches to Continuous Regime[END_REF][START_REF] Lemieux | From Avalanches to Fluid Flow : A Continuous Picture of Grain Dynamics Down a Heap[END_REF][START_REF] Liu | Slumping-rolling transition of granular solids in rotary kilns[END_REF]. Rajchenbach observed that this transition is dependent upon the direction of variation of the rotation speed, the transition showing an hysteresis [START_REF] Rajchenbach | From Discrete Avalanches to Continuous Regime[END_REF]. Fischer et al. showed later that it was rather a system with intermittent erratic transitions between the two modes [START_REF] Fischer | Transition by Intermittency in Granular Matter : From Discontinuous Avalanches to Continuous Flow[END_REF]. In our work, all experiments were performed with increasing rotation speed, gradually increasing from 0.05 rpm to 10 rpm. As Fischer et al., we obtain that during the transition of regimes, both avalanches and continuous flow exist intermittently during the rotation of the drum (figure 6).

Fine particles influence

In the diagram of stability, it can be noticed that the influence of fine particles on the flow regime is surprisingly significant, even though the volume fraction φ is relatively low. Concerning the avalanche regime (when φ < 0.15%), one can see in figure 5 that when the fine concentration φ increases, the transition between avalanche and continuous flow is shifted toward lower rotation speeds. Besides, the measured repose angles θ m and θ r decrease with the increase of fine volume fraction φ (figure 8). Similar results have been observed in the rolling regime, illustrated in figure 9, where the dynamic repose angle decreases when the fine content increases. These three results all show a fluidization of the surface by the dust, the heap being less and less stable when φ enhances. This feature is emphasized in the φ-θ m plot of figure 10.

Relative humidity influence

As shown in figure 10, in the avalanche regime, there is no significant influence of the relative humidity on the maximum angle of stability since the four series of data are superimposed.

High volume fraction of fines

Rotation speed influence

Once the fine particles concentration is higher than 0.24%, the granular pile exhibits a total solidification. It slips at its bottom against the drum wall as a quasi-solid body. This behavior is called the stick-slip regime. This regime had never been observed before in granular media in a rotating drum.

In the stick-slip regime, the maximum and minimum angles are measured in a similar way as in the avalanche regime. When the drum rotates, the pile remains stuck to the wall until a maximum stability angle is reached (θ mg ), then it detaches and slips downward to the minimum angle (θ rg ) without any granular motion at the surface. The only observation of granular motion is seen in a layer of grains at the contact between the pile and the drum wall. When the rotation rate increases, the intermittent stick-slip transforms into a continuous sliding of a thin layer of grains at the pile/drum wall contact. As in the avalanche regime, this arises from the fact that the cycle duration becomes shorter than the sliding duration. The surface angle of the granular pile (θ ro ) remains now stable during the rotation. Figure 11 shows that varying the rotation speed has only a slight impact on the three angle values.

Fine particles influence

The influence of the fine particles on the maximum angle of stability θ mg is now studied in the stick-slip regime. Figure 12 illustrates the evolution of θ mg when φ increases from 0.2% to 1%.

Unlike the avalanche regime, the maximum angle of stability is enhanced by the presence of fine particles. So whereas in the avalanche regime the pile was fluidized by the fines, it is stabilized in the stick-slip regime.

Relative humidity influence

As shown in figure 12, the relative humidity has an influence on the maximum angle of stability θ mg . So the influence of fines in this regime is completely different from the case of avalanche regime, where humidity has no observable influence on the pile stability.

These experimental results show a rich set of, sometimes opposing, behaviors. In the next section we will try to propose a microscopic interpretation and a quantitative modeling of the most interesting effects : the fine content and humidity influences on the pile stability.

Discussion

Avalanche regime

When a small quantity of fine particles is added to the granular pile, a fluidizing effect on the surface of the pile is observed (figure 10).

In the analysis of Mohr-Coulomb, a granular pile loses its stability in the plane situated at depth h where the ratio of the shear stress T (due to the weight W of the above grains) and the normal stress N (due to the weight W of the above grain and to the cohesion strength c) is greater than a friction coefficient µ.

Between two beads of a pile placed in an unsaturated humid environment, the water wettability on glass lowers the chemical potential of liquid water compared to the chemical potential of water vapor. Therefore below a critical distance, of the order of a few nanometers, the steam becomes metastable and capillary bridges may nucleate. As the grain surface is rough at the nanometer scale (figure 2), the actual contact is reduced to points at the end of surface asperities [START_REF] Tabor | The Friction and Lubrication of Solids[END_REF]. Except for almost saturated humid atmospheres, nucleation of liquid bridges is limited to these asperities and voids exist within the contact [START_REF] Bocquet | Physics of humid granular media[END_REF]. These bridges induce an adhesion strength F adh between the grains because of the Laplace pressure.

As evidenced by experiments of surface force apparatus, this adhesion force F adh may depend on the normal load applied to the grains [START_REF] Restagno | Adhesion between weakly rough beads[END_REF]. If it is likely not to be the case for flexible materials [START_REF] Lanier | Experiments and numerical simulations with 2d disks assembly[END_REF], it should be true for most of hard materials, where the stress cannot be accommodated by elastic strain. Indeed, the burden of the above grains induces plastic deformation of the asperities in contact, leading to a decrease in the gap between the grains. This induces an increase in the number of contact points, so in the number of liquid bridges and in the adhesive force F adh .

Therefore this adhesion force increases with W , so with the depth h of the considered plane. For a large amount of liquid, when all the interstitial space is invaded by the liquid bridges, the effect of reducing the gap between the particles is negligible [START_REF] Nowak | Maximum angle of stability of a wet granular pile[END_REF][START_REF] Chou | Experimental analysis of the dynamic properties of wet granular matter in a rotating drum[END_REF][START_REF] Hsiau | The dynamics of wet granular matter under a vertical vibration bed[END_REF]. But for a small amount of liquid or when the liquid originates from a condensable vapor, it is dominant and it has been shown that in an assembly of mineral spheres, it generally obeys F adh ∼ h [START_REF] Restagno | Where does a cohesive granular heap break ?, The European physical journal[END_REF]. As W = ρgh (ρ density of the pile and g acceleration of gravity), we then have both W ∼ h and F adh ∼ h. In this situation, called the granular regime, the most unstable plane remains the surface [START_REF] Tegzes | Avalanche Dynamics in Wet Granular Materials[END_REF][START_REF] Restagno | Where does a cohesive granular heap break ?, The European physical journal[END_REF]. We attribute the experimental observation that our heap always breaks at the surface to this depth-dependent cohesion inside the pile.

When the fine particles are introduced into the granular pile, due to the large diameter ratio (d l /d s ≈ 50), we can assume that the fines feed easily the contacts between the coarse grains and separate them. To check this hypothesis of particle distribution, we have performed SEM observations of the heap just after a rotating drum experiment. We have observed that, beside some fines stuck at the large beads surface, most of them populate the inter-particle contact. An example is shown in figure 13.

This spacing by the fines should induce a decrease of the capillary bonding between large particles. We attribute the fluidization of the surface seen in figure 10 to this granular lubrication of the large grains assembly by the fines 1 . In the granular regime, the maximum stability angle writes : tan θ m = tan θ 0 (1 + C) [START_REF] Restagno | Where does a cohesive granular heap break ?, The European physical journal[END_REF]. In this expression, θ 0 = arctan µ is the maximum stability angle in absence of cohesion and C a term depending on the adhesion stress inside the pile.

The value of θ 0 has been determined both theoretically and experimentally as being 23.4 • in a monodisperse assembly [START_REF] Albert | Maximum angle of stability in wet and dry spherical granular media[END_REF]. We see in figure 10 that the minimal value of θ m in this curve, at 1. We would like to emphasize that the fluidization here is a consequence of granular lubrication, not of gas injection, as usually in powder handling. φ c 0.18%, corresponds to the value of θ 0 . From this, we deduce that the lubrication of the heap due to the dust has led to the complete destruction of the adhesion inside the heap at φ c .

At this point, it behaves as a cohesion-free grain set. We have fitted all the experimental dots in the avalanche regime by the above-mentioned law (figure 10). The agreement is very good for an evolution of the cohesion stress with the fine content of the form : C = A(φ -φ c ) n , with A = 13.5

and n = 2.3. This law corresponds in other words to the evolution of the lubrication efficiency with the fine percentage. The satisfactory agreement between the experiments and theory validates our assumption of the lubricating influence of the fines on the pile in the avalanche regime.

The independence of θ m with humidity in this regime can be explained by the fact that in a relatively dry atmosphere (H < 60%), complete capillary bridges between grains do not form.

As explained above, only nano-scale capillary condensation occurs at the tip of asperities in the contact zone between two grains [START_REF] Tabor | The Friction and Lubrication of Solids[END_REF]. In this situation, the condensates are responsible for an adhesion force between the beads, but as they remain always smaller than the asperities, the adhesion force does not vary and the stability does not change with humidity, as already observed in monodisperse media [START_REF] Fraysse | Humidity effects on the stability of a sandpile[END_REF].

Transition between the two regimes

It has been observed that, at low fine content, the fines play the role of bearing balls between the large grains. In contrary to this lubrication effect, the fine particles show an ability of "jamming" the granular pile in the regime of stick-slip, increasing the cohesion of the pile of coarse grains. This phenomenon of solidification should result from the increase of cohesion between grains inside the granular pile due to the creation of new capillary bridges between fine and coarse particles.

As shown in figure 13, fines seem to reform progressively a meniscus bridging their neighbouring particles. This leads, from φ c on, to a complete setting of the assembly.

We would like to emphasize that this brutal change occurs at a very small, unexpected, value.

A change of behavior was expected when the fine particles populate the whole interstitial spaces between the large ones -roughly for φ ∼ 40%-thereby beginning to dominate the mixture [START_REF] Naeini | Effect of fines content on steady-state strength of mixed and layered samples of a sand[END_REF].

But for a value as low as φ c = 0.18%, the packing factor remains almost unchanged compared to the fines-free heap. So the change of stability cannot be linked to the change of filling and must be searched in the preferential distribution of fine beads at the contact between large ones.

Stick-slip regime

The motion of the granular pile in the stick-slip regime is governed by the friction between the granular assembly and the inner wall of the drum. The maximum stability angle can thus be deduced from a standard Coulomb analysis. We propose here such a model to describe the effect of fines.

With the notation introduced in the preceding section, the maximum angle of stability of the pile in the stick-slip regime can be written in a Coulomb approach [START_REF] Bocquet | Moisture-induced ageing in granular media and the kinetics of capillary condensation[END_REF]25] :

tan θ mg = tan θ 0 (1 + c ρgh cos θ mg ). (1)
Here it should be emphasized that h is the depth of the pile-drum contact plane, so it is the total height of the pile, and that c is the cohesion stress between the heap and the drum wall.

The more fines are introduced in the granular pile, the more liquid bridges are created between the heap and the drum wall. Thereby an increase of the total adhesion c between the pile and the drum wall occurs.

The adhesion stress c is linked to the quantity of fines per unit surface at the drum-pile interface, so to φ 2/3 , φ being the volume fraction of fines. Using the approximation cos θ mg cos θ 0 [25], the maximum stability angle writes :

tan θ mg = tan θ 0 1 + B(φ -φ c ) 2/3 ρgh cos θ 0 (2)
In this expression, the B factor, characterizing the cohesion increase due to the fines, is linked to the capillary bridge formation mechanism. Therefore it depends on humidity and on the beads and drum surface roughness. It has been shown that the dependence on relative humidity has the form B = D/ ln(1/H) [START_REF] Bocquet | Moisture-induced ageing in granular media and the kinetics of capillary condensation[END_REF], D being a purely geometric parameter. Therefore the maximum stability angle should obey :

tan θ mg = tan θ 0 1 + D(φ -φ c ) 2/3 ln(1/H)ρgh cos θ 0 (3)
We have fitted all the experimental results in the stick-slip regime with equation 3. As shown in figure 12, the agreement between the experimental and theoretical curves is very good, thus validating our assumption of the influence of fines on the drum-wall adhesion.

Conclusion

We have presented here the effect of very fine particles on the stability of a humid granular pile in a rotating drum. Various amounts of fine glass beads of mean size 5 µm (0 to 1% volume fraction) have been added to a granular pile of glass beads of 250 µm mean size rotated in a drum. We have established the stability diagram of this granular medium as a function of fine concentration, relative humidity and rotation velocity. The presence of fine particles has shown an ambivalent effect on the stability of the granular heap. At low fine concentration (< 0.18%), the heap destabilizes through avalanches at the pile surface when the drum rotates, and increasing the fine quantity tends to decrease the upper repose angle θ m . In contrast, once the concentration is more than 0.24%, the destabilization of the heap proceeds through a stick-slip phenomenon at the drum wall, and the increase of the fraction of fines tends to increase the θ m value.

For large rotation velocities, the avalanche regime evolves into a continuous flow regime, and the stick-slip regime transforms into a continuous sliding regime.

This apparently contradictory behavior is linked to the dual influence of fines on the pile cohesion. In the small fine concentration regime, the fines act as ball bearings and their granular lubrication effect destabilizes the heap. For a higher fine content, the fines induce a jamming of the granular medium, leading finally to an increase of the bulk cohesion of the heap. So in the large fine concentration regime, the heap behaves as a solid body and the destabilization occurs at its bottom. In this case, the fine particles contribute to increase the capillary bonds between the pile and the drum and increases its stability.

We have also studied the influence of relative humidity on the granular stability in our experiments. Again a contradictory behavior is observed. We have found out in the low fine concentration regime that the maximum stability is independent of humidity. On the other hand, in the regime of high content of fines, the maximum stability angle is highly dependent on relative humidity, fines inducing additional capillary condensation between the pile and the drum wall. Now, these flow regimes in strongly bidisperse granular mixtures have to be investigated with particles made in other materials, to test their universality. As the agglomeration of the grain assembly at high fines content may be detrimental for some applications, like mineral grinding, the search of chemical additives limiting this solidification is another direction of investigation.

[25] T. Halsey, A. Levine, How Sandcastles Fall, Physical Review Letters 80 (1998) 3141-3144. performed in vacuum, the capillary bridges between the particles cannot be observed here, but we see that fines reconstruct a kind of meniscus between the large beads.

Figure 5

 5 Figure 5 shows the diagram of stability of the granular pile obtained by varying the fraction

Figure 1 :

 1 Figure 1: Experimental setup.

Figure 2 :

 2 Figure 2: AFM observation of the large glass bead surface (error signal). The picture size is 15 × 15 µm 2 .

Figure 3 :Figure 4 :

 34 Figure 3: Front view of the granular heap inside the 10 cm diameter rotating drum.

Figure 5 :

 5 Figure 5: Diagram of stability of the granular pile as a function of φ and n. Each dot stands for one experiment.Line segments share four different flowing regimes : avalanching, continuous flowing, stick-slip and continuous sliding. Between the avalanche and stick-slip regimes lies a small strip where both regimes coexist. Only the avalanche/continuous flow and stick-slip/continuous sliding separation lines depend on humidity, the others do not. From left to right, the relative humidity H is 10% (green dotted line), 25% (red dashed line), 40% (blue dashed-dotted line) and 53% (pink continuous line).

Figure 6 :Figure 7 :

 67 Figure 6: Angle of the surface of the pile as a function of time for 0.002% of fine particles, 25% relative humidity and 0.7 rpm rotation rate. Being in the intermittent regime, avalanche and continuous flow alternate during this experiment.

Figure 8 :Figure 9 :FinesFigure 10 :Figure 11 :Figure 12 :

 89101112 Figure 8: Avalanche angle and repose angle as a function of rotation speed for different fines volume fraction φ for H = 10%.

Figure 13 :

 13 Figure 13: Scanning electron microscope picture of the distribution of fine beads between two large beads sampled from a mixture just after a rotating drum experiment with φ = 0.15% and H = 40%. As the experiments are

Table 1 :

 1 Relative humidity provided by saturated salt solutions at fixed temperature.

	Salt	Theoretical relative humidity at 25 • C
	Lithium bromide (LiBr)	6%
	Potassium acetate (KCH 3 CO 2 )	23%
	Potassium carbonate (K 2 CO 3 )	43%
	Sodium bromide (NaBr)	58%

Acknowledgements

We thank Elisabeth Charlaix and Lydéric Bocquet for fruitful discussion and the Carnot Institute Ingénierie@Lyon for funding.


F Errigo

J C Gonzalez-Torres

A Benchaib

L Chédot

A Sari

P Venet

F Morel

Modular Multilevel Converter with Embedded Energy Storage for Power Oscillation Damping and Fast Frequency Response -A case study

Keywords: High-voltage direct current (HVDC) transmission, hybrid AC/DC transmission grids, modular multilevel converters (MMCs), power converters, AC stability, ancillary services, fast frequency response, power oscillation damping, modular systems, energy storage, supercapacitors

The massive introduction of remote power electronics interfaced renewable generation units and high-voltage direct current (HVDC) interconnectors into the existing power system, with the aim of decommissioning a part of conventional centralized generation, has introduced new challenges to grid operators to maintain system stability. Since conventional balancing services rely on the availability of traditional synchronous generators and the kinetic energy stored in their rotating masses, new methods for providing ancillary services must be introduced. In this context, voltage source converters (VSCs), and particularly the modular multilevel converters (MMCs), have received a significant interest. An MMC stores energy in its distributed sub-modules and this energy can be controlled. This is a new degree of freedom that can be utilized to supply ancillary services. However, one of the major drawbacks is the limited amount of energy available, which is considerably smaller than a typical AC power plant. Nevertheless, the energy stored inside the MMC can be increased with the help of energy storage systems (ESSs). This paper examines the benefits on the transmission system of including additional energy storage in an HVDC link for ancillary services (fast frequency response (FFR) and power oscillation damping (POD)), and how this can be implemented using an MMC with large buffer storage. Firstly, the study investigates the energy requirements to provide these network services, using adequate control laws, through EMTP-RV simulations. Then, the energy storage function of the converter is designed and simulations are carried out to highlight that an MMC with an embedded energy storage can provide these services. The obtained results confirm that the combination of an MMC with an energy storage system and dedicated control laws can greatly increase the stability of hybrid AC/DC power systems. However, it highlights that an FFR function inside an MMC requires a higher quantity of stored energy to provide system benefits, in comparison to a POD action. This would impact more deeply the construction and the operating limits of the converter.

INTRODUCTION

In recent years, with the increasing deployment of offshore renewable energy sources and crossborder interconnections, as for trading mechanisms of the electricity, existing power systems are facing exponential growth and changes [START_REF] Van Hertem | HVDC grids: for offshore and supergrid of the future[END_REF]. Most of them disrupt with the traditional way to operate the electrical transmission grid. This brings new challenges to transmission system operators (TSOs) to ensure the security of supply of the electrical energy. Thus, centralized traditional power plants, with synchronous generators directly connected to the grid, are foreseen to be replaced in favour of remote generation units, with a power electronics interface. Since these new generators are electrically decoupled from the grid, and so from conventional synchronous generators, they behave differently [START_REF] Tielens | The relevance of inertia in power systems[END_REF]. The existing dynamics of the grid are inevitably impacted. The traditional link between the speed (i.e. the kinetic energy stored) of conventional generators and the system frequency is reduced. This leads to an inherent decrease of system inertia. Consequently, less energy is available to limit the rate of change of frequency (ROCOF) and frequency nadir in case of severe changes in production or consumption.

In order to face these changes, the only alternative is a fundamental enhancement of the existing transmission systems. One of the most suitable solution to this problem is the use of high voltage direct current (HVDC) systems [START_REF] Van Hertem | HVDC grids: for offshore and supergrid of the future[END_REF]. Although HVDC interconnectors have already been implemented in the world during the last century [START_REF] Barnes | Voltage Source Converter HVDC Links -The State of the Art and Issues Going Forward[END_REF] and permit to transmit bulk power at high voltage, over long distances, with low losses, this technology has made remarkable advances during the last decades. This is due to the emergence of the Modular Multilevel Converter (MMC) [START_REF] Lesnicar | An innovative modular multilevel converter topology suitable for a wide power range[END_REF]. This converter has drawn a considerable attention since it particularly offers a new degree of freedom with fast power control and independent active and reactive power injections.

From this perspective, the attractiveness of Voltage Source Converter (VSC)-HVDC links has risen. Particularly, if all their potential is fully exploited. HVDC links must not only be used for transmission purposes, but also to improve the AC stability of the networks, to which they are connected. It has been recognized that advanced functionalities of HVDC systems are essential for the efficient and secure operation of the grid [START_REF] Van Hertem | HVDC grids: for offshore and supergrid of the future[END_REF], [START_REF] Entso-E | HVDC Links in System Operations -Technical paper[END_REF]. They usually take advantage of the possibility to modulate quickly and independently the DC and the AC active powers of an MMC to strengthen system stability margins [START_REF] Gonzalez-Torres | Transient stability of power systems with embedded VSC-HVDC links: stability margins analysis and control[END_REF]. Two of these functionalities are pivotal in a near future: power oscillation damping (POD) and fast frequency response (FFR). Those control actions contribute to damp power system oscillations and/or to support frequency stability. However, those methods are mainly characterized by the fact the energy has to come from somewhere else in the network, such as interconnected power systems, or by extracting a percentage of the intrinsic electrostatic energy stored in the capacitors of the converter. Nevertheless, this latter is insufficient to support the AC system [START_REF] Fradley | Assessment of the impact of MMC-VSC intrinsic energy on power system stability[END_REF].

As a solution to this problem, this paper investigates a concept that has rapidly gained popularity. It consists in integrating Energy Storage Systems (ESSs) inside an MMC to potentially support the grid [START_REF] Trintis | A new modular multilevel converter with integrated energy storage[END_REF]. The aim is to broaden the set of functions of the converter, by providing short-term ancillary services as, here, POD or FFR, without affecting significantly the design of the MMC [START_REF] Judge | Modular Multilevel Converter with Partially Rated Energy Storage with Intended Applications in Frequency Support and Ancillary Service Provision[END_REF].

For this purpose, this paper attempts to illustrate the benefits of such a solution on rotor angle and frequency stability. Furthermore, it tries to determine the impact of this new set of services on the design and the control of the MMC with its ESSs. Indeed, the energy and power requirements can highly vary to mitigate these disturbances on the AC system.

Based on it, this paper is organized as follows. Sections 2 and 3 recall the fundamental of an embedded VSC-HVDC link for the provision of ancillary services and the concept of an MMC with integrated ESSs. In Section 4, case studies are presented to highlight the positive impact of the association of an MMC with embedded ESSs and supplementary controllers for AC stability enhancement. In order to estimate power and energy requirements of the ESSs for supplying these network supports, EMTP-RV simulations are carried out before designing the energy storage function. Finally, the following section sheds to light the feasibility of the solution through simulations of a pointto-point HVDC link while the last section draws conclusions.

ANCILLARY SERVICES SUPPORT BY VSC-HVDC LINKS

Potential applications

As reported in [START_REF] Entso-E | HVDC Links in System Operations -Technical paper[END_REF], advanced functionalities for ancillary services provision by HVDC links are essential for the secure and efficient operation of the grid. Indeed, the fast and independent control of active and reactive power injections of the HVDC stations can be used for the benefit of the AC system to which the stations are connected. Depending on the topological situation of the HVDC link (i.e. connecting two asynchronous grids or embedded into a single synchronous grid) the HVDC link is able to provide real-time operation services such as static and dynamic reactive power control, frequency control, power oscillation damping among others (an extensive list of functionalities can be found in [START_REF] Entso-E | HVDC Links in System Operations -Technical paper[END_REF]). In this article, the interest is focused on the dynamic active-power based services that can be provided through HVDC links interconnecting asynchronous grids, particularly the power oscillation damping and the frequency containment reserves delivery.

Frequency containment reserve delivery

When an HVDC link interconnects two asynchronous grids, balancing power can be distributed in order to support their frequency stability. The fast active power control of the HVDC system allows to share reserves in a faster way than the primary control of synchronous generators. This rapid service has received different names from operators [START_REF] Meng | Fast Frequency Response from Energy Storage Systems-A Review of Grid Standards, Projects and Technical Issues[END_REF]. In this article, the term Fast Frequency Response (FFR) is adopted. Different control schemes to implement FFR have shown to effectively reduce the frequency deviation and ROCOF during contingency events, especially in systems with relatively small capacity and higher penetration of non-synchronous energy sources [START_REF] Meng | Fast Frequency Response from Energy Storage Systems-A Review of Grid Standards, Projects and Technical Issues[END_REF]. Classical schemes for FFR implementation include: action proportional to the frequency deviation (droop control), action proportional the derivative of the frequency (inertial response), or the triggered injection of a specified power profile [START_REF] Eriksson | Synthetic inertia versus fast frequency response: a definition[END_REF].

Power oscillation damping (POD)

Inter-area oscillations (0.1 to 1.0 Hz) are inherent to interconnected power systems. These oscillations are considered as a key limiting factor when large amounts of power are transmitted through long AC lines. Indeed, unstable oscillations may lead to system collapse. Classically, electromechanical oscillations have been handled through the use of Power System Stabilizers (PSS) on the excitation system of synchronous generators. However, HVDC links and other power electronics devices, can also improve the damping of the interarea modes through the implementation of POD controllers. Different structures of POD controllers are reported in the literature. They differ mainly on the measured feedback (powers, voltages, frequency, etc.), the use or not of Wide Area Measurement systems, the modulation signals (active or reactive power references) and the control design techniques [START_REF] Elizondo | Interarea Oscillation Damping Control Using High-Voltage DC Transmission: A Survey[END_REF].

Studied active power modulation strategies for provision of ancillary services

To deliver frequency containment reserves and to damp power oscillations through active power modulation of the HVDC link, the considered strategies in this paper are presented in Figure 1. In order to provide support to the system frequency stability, the power characteristic Δ𝑃 𝐹𝐹𝑅 , depicted in Figure 1(a), is injected to the grid through the HVDC link after the detection of the event (e.g. generator tripping) as proposed for example in [START_REF] Wall | Smart frequency control for the future GB power system[END_REF], [START_REF] Hong | Fast frequency response for effective frequency control in power systems with low inertia[END_REF]. This power profile corresponds to the energy and power capacity of the HVDC link to provide FFR during a given time characterized by a response delay 𝑇 1 , a ramp up until full deployment at 𝑇 2 , a sustained response up to 𝑇 3 , and a ramp down to 𝑇 4 to gradually transfer the frequency containment to the generators governors.

Regarding the POD control, a structure that modifies the HVDC power reference by adding a term Δ𝑃 𝑃𝑂𝐷 , which is proportional to the difference of frequencies measured at two different buses in the same synchronous grid is used, as in [START_REF] Smed | Utilizing HVDC to damp power oscillations[END_REF], [START_REF] Ruan | Power system damping from real and reactive power modulations of voltage-source-converter station[END_REF]. The selected buses must be electrically distant enough to have a high observability of the oscillation (e.g. one bus from each coherent area). So, the use of remote (wide-area) signals is necessary to implement such POD control. In Figure 1(b), 𝑓 𝐴𝑟𝑒𝑎1 represents the instantaneous frequency measured on a bus of Area 1, which oscillates against Area 2, where the frequency 𝑓 𝐴𝑟𝑒𝑎2 is measured. Note that in steady-state the power modulation (𝛥𝑃 𝑃𝑂𝐷 ) is zero as the frequency measurements are taken in the same synchronous grid.

Figure 1 Studied supplementary active power-based modulation strategies to provide ancillary services to the grid using an HVDC link. (a) Characteristics of the fast frequency response (FFR) (b) Power oscillation damping (POD) control

It is evident that in order to provide the studied ancillary services, additional energy sources are necessary. In the case of two asynchronous grids (for example two transmission networks or an offshore windfarm and a transmission network) interconnected by HVDC systems, the power to provide these services to one grid, is necessarily extracted from the second grid. In the case of two interconnected transmission networks, the injection of the FFR to one grid will lead to a frequency deviation in the second one. Similarly, the power modulation necessary for oscillation damping in one grid will be extracted from the second grid leading to a potential excitation of its inter-area modes [START_REF] Björk | Fundamental Performance Limitations in Utilizing HVDC to Damp Interarea Modes[END_REF]. It is also known that by using the energy of the DC capacitors inside the MMC, the interactions between the asynchronous networks can be reduced only to a small extent. This is due to the fact that the energy stored in the capacitors is small compared to the energy needed to provide these auxiliary services, as explained in section 1.

As a result, the available energy to support the grids, in case of disturbance, must be increased. However, these ancillary services would require only a fraction of the total power of the converter and a relatively low amount of energy in regard with the traditional bulk storage services. Therefore, the proposed solution consists in embedding an energy storage system (ESS) inside the MMC. Thus, a converter with a dual purpose is feasible, that potentially provides an opportunity to reduce the additional footprint and cost for ESSs integration. Note that the sizing of such energy buffer can be based on the necessary energy to provide POD and FFR. As it has been observed, the proposed schemes in Figure 1 require only a finite amount of energy, since both actions have to tend to zero.

MMC WITH LARGE BUFFER STORAGE DESIGN AND OPERATION

Converter topology

The general configuration of an MMC with embedded ESSs is shown in Figure 2. It is based on a classical 3-phase MMC traditionally built-up with identical modular building blocks, the so-called submodules (SMs). Each phase i (with 𝑖 ∈ {𝑎, 𝑏, 𝑐}) contains an upper and lower arm (superscript u, l) which is composed of 𝑁 series connected SMs and an inductor. A SM is usually a two level VSC with two IGBTs and two diodes in a half-bridge configuration and a capacitive storage. The inclusion of ESSs consists in connecting an energy storage element, here supercapacitors, to the capacitor of the submodules through a DC-DC converter to make an Energy Storage Submodule (ES-SM) as shown in Figure 2(c). In this paper, a modular interface converter with its own control method, illustrated in Figure 2(d) and (e), is considered [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF]. It was demonstrated that this topology is well-suited for ancillary services with low power and energy requirements, such as FFR. The proposed interface comprises a serial connection of 𝑁 𝑚𝑜𝑑 subconverters. Each of them consists in a bidirectional converter, with high switching devices (i.e. typically above 10 kHz), between a supercapacitor stack and the ES-SM capacitor. This structure allows to share the high voltage at the ES-SM capacitor. This provides the opportunity to use fast switching devices as MOSFETs, with low voltage and current capabilities, which are more adapted for low power application. As a result, the size of the passive components can be drastically reduced and a better power density per ES-SM is achieved. During normal operation of an MMC, depending on the switching states, the SM capacitor is inserted or not in the arm voltage to transfer energy from/to the AC grid to/from DC grid. Since an MMC has a limited stored energy, it is usually controlled only to use its power conversion capability. Thus, the AC and DC powers are controlled to be equal (except internal power losses) in normal operation. By adding energy storage elements into SMs, a new energy source can exchange energy with the SM capacitor that results in a new degree of freedom in active power exchange between the MMC and the grids as illustrated in Figure 3. Since the converter can absorb/provide energy through its energy storage elements, the AC and DC power set points can be decoupled. This attractiveness is further increased because the converter is capable to extend its set of functionalities by providing ancillary services.

Control principle

The most common control method of an MMC is an energy based control with a ΣΔ approach, which is illustrated in dark in Figure 4 with a simplified schematic diagram [START_REF] Shinoda | Control and Energy Management of MMC-based Multi-Terminal HVDC Grids[END_REF]. The goal is to control the internal energy level 𝑊 𝑀𝑀𝐶 of an MMC. This is a new degree of freedom to govern the exchanged powers between AC and DC grids in comparison with classical VSC converters. Similarly, it ensures that the energy is distributed among the six arms of the MMC to provide stable operation. The control is divided into two parts with different time frames. A high level control has to determine the arm voltage references 𝑣 𝑖 𝑢,𝑙 * depending on the desired powers. Those references are then converted to pulse signals for SM switches by a fast low-level control, which also guarantees the energy balance among the SM capacitors.

According to this approach, a control method, depicted in Figure 4, was proposed in [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF] to consider the inclusion of ESSs into the overall control scheme of an MMC. As observed, it does not require notable changes to the traditional control of the MMC. The energy based control of the MMC remains unchanged since the management of the ESSs is decoupled of the MMC control. The ESS of each ES-SM is seen as a supplementary power source (i.e. external disturbance) from the ES-SM capacitor. This additional power tends to change the energy stored in submodule capacitors. To keep this energy constant, the natural behavior of the energy controller of the MMC is to adapt the power in the AC system. Thus, it tries to keep a stable ES-SM capacitor voltage to ensure correct operation of the converter. The extra power 𝑃 𝑒𝑠𝑠 * , requested from the energy storage function, is distributed among the ESSs of each ES-SM. Then, it is translated into a current reference to a local controller to provide this additional power. More details are provided in [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF] about this strategy.

ASSESSMENT OF THE INFLUENCE OF AN MMC-HVDC LINK WITH LARGE BUFFER STORAGE

Case study

In this section, a set of simulations are presented to highlight the positive impact of the association of an MMC with embedded ESSs and supplementary controllers for AC stability enhancement, presented in section 2. The electromagnetic transient software EMTP-RV is used to perform simulations. The goal is to estimate the power and energy requirements for these network supports and thus quantify the characteristics of the ESSs. Later in this section, a focus is made on the design on the ESS.

The power system presented in Figure 5 is implemented in EMTP-RV. It consists in two modified versions of the two-area four machines Klein-Rogers-Kundur's power system model [START_REF] Kundur | Power system stability and control[END_REF] connected by a VSC-HVDC link. In this power network, all the generators are 900MVA synchronous machines with inertia constants of 6.5s. As for their controls, the generators are equipped with the automatic voltage regulator (AVR) and PSS described in [START_REF] Kundur | Power system stability and control[END_REF]. However, the PSS gain has been reduced to a half in order to observe weakly damped inter-area oscillations. Moreover, an IEEEG1 turbine-governor with a droop coefficient of 20% has been included for the primary frequency control. The HVDC stations are two 1 GW converters. Data are taken from [START_REF] Peralta | Detailed and averaged models for a 401-level MMC-HVDC system[END_REF]. With respect to the original case, the loads have been modified to allow a power exchange between asynchronous grids of 500MW (from MMC1 to MMC2). In addition, a third terminal, connected to an infinite source, is added to form a Multi-Terminal DC grid (MTDC) embedded in an AC network (Case A). The goal of this extra source of energy is to act as a firewall and to avoid disturbances to be transmitted from one power system to another. In this way, the third terminal evaluates the need of an additional source of energy to achieve ancillary services. Consequently, it is also used to compute the related power profiles for the ESSs in the case the energy source is directly embedded in one of the two converter stations (cf. Figure 5 Case B).

Figure 5 Case study -[Case A] Multi-terminal DC grid embedded in an AC network. The third terminal acts as an external power source [Case B] HVDC line embedded in an AC network. One of the two converter stations has an energy storage function

In this study, MMC1 and MMC2 are controlled in power with the sole difference the power set-points are of opposite sign, while MMC3 controls the DC link voltage of the MTDC grid. The contingency evaluated in this case study corresponds to a generator loss, G5 in red here, in power system 1. Four control strategies are implemented and compared for enhancing AC stability.

• Case 1: No actions are undertaken. Converter 1 provides no active power services.

• Case 2: The POD control, proposed in subsection 2.2, is implemented on MMC1 to damp the inter-area oscillation between Area 1 and Area 2 of power system 1. Note that the frequency at the Area 1, 𝑓 𝑎𝑟𝑒𝑎1 , corresponds to the frequency at bus 9. Similarly, the frequency at the Area 2, 𝑓 𝑎𝑟𝑒𝑎2 , stands for the frequency at the bus 12. The POD gain, 𝐾 𝑃𝑂𝐷 , is arbitrarily set at 500 MW/Hz but taking into account the physical limit of the converter. Note that it has been shown that any positive value of 𝐾 𝑃𝑂𝐷 is beneficial for the system contrary to PSS [START_REF] Smed | Utilizing HVDC to damp power oscillations[END_REF], [START_REF] Ruan | Power system damping from real and reactive power modulations of voltage-source-converter station[END_REF]. • Case 3: The FFR characteritics, proposed in subsection 2.2, is implemented on MMC1 in order to mitigate the frequency drop in power system 1. Based on Figure 1, the characteristics of the proposed response are as follows: 𝑇 1 =0.2s, 𝑇 2 =0.5s, 𝑇 3 =6s, 𝑇 4 =30s and 𝑃 𝑇 =50 MW [START_REF] Entso-E | Technical Requirements for Fast Frequency Reserve Provision in the Nordic Synchronous Area[END_REF]. • Case 4: Finally, both POD and FFR actions are implemented.

The simulation results are depicted in Figure 6 and Figure 7. In this paragraph, the role of the extrapower source is highlighted and only a POD function is discussed. In Figure 6 (left), the extra-power source is removed. The HVDC link is controlled in a way to damp power in the power system 1, where the loss of generation occurs. As observed, oscillations are well damped, in comparison with Figure 7 Case 1 where no ancillary services is provided. However, the frequency in power system 2 is disturbed. In contrast, in Figure 6 (right), the extra source of energy is introduced on the DC link. The HVDC link can now act as a firewall between both systems. Thus, the fault is imperceptible from the power system 2. This first case points out the need of an extra source of energy to increase the benefits of an embedded HVDC link for AC stability purposes. Note that in Figure 7 Case 1, the HVDC system acts as a firewall and no active power services are provided. This means the converter only track a constant power set-point. It is noticed that the oscillations in power system 1 are poorly damped and the frequency nadir (i.e. the minimum frequency value) is significant, although the power system 2 is not disturbed. The potential risk of a low nadir is to activate protection schemes of the generators or AC lines, leading to their disconnection and an eventual cascaded failure. As shown again, when a POD action is implemented (see Figure 7 Case 2), it is demonstrated that oscillations are well damped in power system 1 and power system 2 is not excited. Nonetheless, the frequency response is poorly improved which undoubtedly points out the need of injecting fast frequency reserves. As exposed in Figure 7 Case 3, the minimum value of frequency for each generator is improved (from 0.0991 to 0.0093 p.u) if FFR is delivered in the first hundred milliseconds, here 50 MW in 500 ms, after the fault. Finally, the response of the system is greatly enhanced if both actions (POD and FFR) are combined as shown in Figure 7 Case 4. To ascertain these remarks, Figure 8 shows the frequency of the center of inertia (COI), that describes the principal frequency dynamics of a power system, and the speed difference between generators 1 and 4 in power system 1 for each case described. This indisputably proves that an HVDC link with embedded energy storage can enhance the AC system stability by providing power oscillation damping and/or fast power injections. In return, it must be well controlled to not adversely impact both power systems and extra energy has to be available to allow to increase transient stability margins [START_REF] Gonzalez-Torres | Transient stability of power systems with embedded VSC-HVDC links: stability margins analysis and control[END_REF]. Therefore, in the next section, the third terminal is substituted by an energy storage function included inside one of the two converter stations, as shown in Figure 5 (Case B). The aim is to demonstrate the feasibility to embed ESSs within the converter, but prior to that the design part must be addressed.

Figure 8 Frequency of the center of inertia in power system 1 (left) Speed difference between generators 1 and 4 in power system 1 (right)

Energy storage requirements

After quantifying the amount of energy and the power requested from the energy storage function of an MMC to achieve ancillary services, thanks to the simulations of section 4.1, this part aims to highlight the design of the solution. It can be easily noticed that a high amount of energy to store or an important percentage of the nominal power of the converter to supply would make this solution impractical. Indeed, it would significantly impact the design of an MMC. In that case, a separated energy storage system connected to the AC grid would be a more relevant solution.

Figure 9 shows the power profiles and the energy variation of the ESSs, for the three cases with a control action, studied in the paragraph 4.1. At the same time, Table 2 provides the order of magnitude of those metrics. As observed, the energy requirements for POD are negligible in regard to FFR. However, they remain greater than the total energy stored in a 1 GW MMC (30-40 kJ/MVA [START_REF] Peralta | Detailed and averaged models for a 401-level MMC-HVDC system[END_REF]), which only a small percentage can be really used for ancillary services [START_REF] Fradley | Assessment of the impact of MMC-VSC intrinsic energy on power system stability[END_REF]. It proves the need to use anyway energy storage elements. Finally, a POD function leads to important power transient during the first seconds. This peak current implies stronger constraints on the semiconductors of the converter. The risk is they cannot switch reliably, within their safety operating area (SOA), by operating possibly beyond their current limit threshold that may lead to a failure by overheating, and in the worst case to a thermal runway. Therefore, the converter may require to be designed taking into account this overload capability. This is particularly the case if POD and FFR are combined. Table 1 Characteristics of the energy storage function of an MMC for the three proposed ancillary services

In the following study, it is assumed all the SMs of an MMC comprise an ESS, even if the modular structure of the converter allows to embed energy storage in a limited number of arms or in a limited number of SM in each arms [START_REF] Judge | Modular Multilevel Converter with Partially Rated Energy Storage with Intended Applications in Frequency Support and Ancillary Service Provision[END_REF], [START_REF] Errigo | Assessment of the Impact of Split Storage within Modular Multilevel Converter[END_REF]. As shown in Figure 2(c), the presence of an interface converter between the ES-SM capacitor and the energy storage element is compulsory [START_REF] Trintis | A new modular multilevel converter with integrated energy storage[END_REF], [START_REF] Judge | Modular Multilevel Converter with Partially Rated Energy Storage with Intended Applications in Frequency Support and Ancillary Service Provision[END_REF], [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF], [START_REF] Vasiladiotis | Modular Multilevel Converters with Integrated Split Battery Energy Storage[END_REF]. This is mainly explained by the need to decouple the energy management of the energy storage element from the ES-SM capacitor, whose voltage variation is restricted to a small fraction of its total voltage. Furthermore, the insertion of a DC-DC converter also allows to optimize the connection of low voltage energy storage elements (here supercapacitors) to a high voltage SM capacitor. Similarly, it offers the possibility to filter the power fluctuations from the grid that should be buffered by the energy storage elements, and may significantly impact their performance.

Following the design methodology described in [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF], this section details the characteristics of the proposed interface converter (see Figure 2 and paragraph 3.1), and its supercapacitor stack, required to achieve the specifications of the three ancillary services given in Table 1. The aim of this design methodology is to optimize the total volume of the ESS (i.e. similarly its cost). Indeed, the objective of the proposed solution, here, is to add an energy storage function within an MMC through all its ES-SMs without significantly impacting their design and dimensions. Results are shown in Table 2. As observed for a POD, the depth of discharge (DoD) is less than 10%. Here, the DoD is considered as the ratio between the maximum voltage and the minimum allowable voltage of the supercapacitor stack to provide the requested energy for the expected service. For capacitive element, as supercapacitors, it is the reflection of the energy variation. In that case, this means most of the energy is not used and that the supercapacitor stack of each subconverter is oversized. This can be explained by the opposite characteristics of the application and the energy storage element.

Parameters

First, the energy requirement for a POD function is relatively small, especially because all the SMs of MMC1 have an ESS. Consequently, low energy should be stored per ES-SM. Secondly, a supercapacitor stack is nothing more than a layout of low voltage cells in series and parallel as shown in Figure 2(f).

Although an interface converter is present between the ES-SM capacitor and the supercapacitor stacks, the transformation ratio of the subconverters cannot be infinite due to current limitations for example. Note that a converter with a transformer, as a dual active bridge (DAB) could allow a high transformation ratio, but the structure of an ES-SM would be much more complex. Thus, the number of cells in series per supercapacitor stack cannot decrease as much as one wants. Finally, even if there is an interface converter, the voltage that the supercapacitor stack has to sustain remains relatively important. Since, a supercapacitor stack has a quadratic relation between the voltage at its terminals and its stored energy, a high number of cells in series leads to an excess of energy. This explains why the solution is oversized due to an important increase of the number of cells within a stack.

For FFR, more energy has to be stored. Thus, the voltage constraint is less predominant and the design of the solution more adapted. Note that the DoD is almost 50 % of the maximum stack voltage. This ratio is usually considered as good compromise, since 75 % of the total energy stored within a stack is employed at this operating point.

Adding a POD service in addition to a FFR function only needs a slight increase in terms of sizing as shown in Table 2. Therefore, if an FFR function is provided, it seems relevant to provide both ancillary services, POD and FFR. However, if a POD action is only required, solution to be investigated consists in equipping a small fraction of SMs in each arm with an ESS as proposed in [START_REF] Judge | Modular Multilevel Converter with Partially Rated Energy Storage with Intended Applications in Frequency Support and Ancillary Service Provision[END_REF] or only some arms equipped with ES-SM [START_REF] Errigo | Assessment of the Impact of Split Storage within Modular Multilevel Converter[END_REF]. The energy that should be stored per ES-SM is thus increased and the voltage constraint is reduced. It thus offers the opportunity to optimize properly the operation range of the ESS and the design of the solution.

SIMULATION OF AN EMBEDDED HVDC LINK COMPRISING AN MMC WITH LARGE BUFFER STORAGE

MMC Modelling

The following section sheds to light the feasibility of the proposed solution through simulations. The point-to-point HVDC link, depicted in Figure 5, has been implemented under MATLAB/Simulink. The goal is to verify that the ESSs of the converter can track the reference defined by the simulation in the subsection 4.1. Furthermore, these simulations are also used to validate the design made in the subsection 4.2.

Both converters MMC1 and MMC2 were modelled with a so-called reduced average arm model [START_REF] Shinoda | Virtual Capacitor Control: Mitigation of DC Voltage Fluctuations in MMC-based HVDC Systems[END_REF]. This model is based on the assumption that the capacitances of all the ES-SMs are identical and kept at a same voltage. Consequently, it is possible to represent the behavior of the hundreds SMs of an arm by considering an ideal equivalent converter and an equivalent capacitor as shown in Figure 10. The voltage at its terminals 𝑣 𝑐Σ 𝑢,𝑙 is the sum of all the N SM capacitor voltages. Finally, the arm voltage 𝑣 𝑖 𝑢,𝑙 is proportional to the available voltage at this equivalent capacitor and to a modulation index 𝑚 𝑖 𝑢,𝑙 . It thus represents the fraction of SMs inserted in an arm at any time. As a reminder, one of the two stations has an energy storage function (see Figure 5). Following the same principle as in the previous paragraph, each equivalent capacitor of the six arms of the MMC with ESSs is also coupled to an equivalent interface converter with a supercapacitor stack. Similarly, the voltage at the terminals of this latter 𝑣 𝑠𝑐Σ is the sum of all the supercapacitor stacks of the 𝑁 𝑚𝑜𝑑 subconverters of the N ES-SM of an arm. Note that the control methodology described in subsection 3.2 and [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF] was applied. Here, one MMC controls the power flow of on the HVDC link while the second converter manages the DC voltage.

Results

The obtained simulation results are shown in Figure 11 and Figure 12. Figure 11 shows the DC and AC active powers of both converter stations. The dynamic on the AC side of the MMC 1 is identical as the active power reference 𝑃 𝑎𝑐1 * that comes from subsection 4.2. This proves that an MMC with embedded ESSs is able to provide the considered ancillary services. In addition, the DC power on the HVDC link and the active power of the MMC 2 remains constant. Thus, MMC 1 also acts as a firewall. It avoids the propagation of the fault into system 2 (see Figure 5) and simultaneously participates to improve AC stability of the whole power system.

Figure 11 Simulation results of the point-to-point HVDC link with embedded ESS in one converter station to provide different ancillary services and acting as a firewall between power systems

In Figure 12, the behavior of the equivalent supercapacitor of one MMC arm is presented for each case study. Results are brought back to one supercapacitor assuming an identical behavior of all the ES-SMs. The internal supercapacitor stack voltage and the current that flows through it are displayed. By comparing the DoD in Figure 12 and values of Table 2, it can be noticed they are approximately the same. This implies that the design of the ESS is correct. Note that for this study, the ESS is considered as fully charged for providing the expected service. This means if a symmetric response is required, when a FFR service is supplied, the energy storage function of the MMC can provide the same power profile (see Figure 1) during half the time or half the power (𝑃 𝑇 ) during the same time with these sizing. Note that in the control strategy considered [START_REF] Errigo | A Submodule with Integrated Supercapacitors for HVDC-MMC providing Fast Frequency Response[END_REF], a balancing method, to avoid dispersion among supercapacitor stacks, and a state of charge management, with a limited effect on the ES-SM capacitor, are proposed.

CONCLUSION

In this paper, the concept of an MMC with embedded ESSs to provide ancillary services to the AC system has been investigated. It discusses both on the feasibility of such a solution and its impact for AC stability purposes. The paper focuses on a VSC-HVDC link that interconnects two AC asynchronous grids, whose one of the two converter station has an energy storage function. First, a control method is presented to enhance system stability by benefiting of the fast power injections of classical MMC-VSC converters in an HVDC link. Then, its practicability is validated through EMTP-RV simulations. Thereby, the fast power control allows to use a traditional HVDC link to provide ancillary services, such as POD or FFR, but also to act as a firewall between two different networks in case of contingency. However, an additional source of energy is required in order to provide these services. From these simulations, requirements for each of them are determined. Secondly, an appropriate ESS has been designed. Nonetheless, it can be remarked that the energy storage system is oversized for POD. This is due to the fact that low energy is required, but at the same time capacitive elements must be interfaced to a high voltage ES-SM capacitor. This participates to increase the number of series connected cells, and consequently the installed energy, even if there is an interface converter. Therefore, it is preferable to combine both FFR and POD approach to have a better solution or to decrease the number of SMs that includes an ESS in an MMC. Finally, the feasibility of an MMC with embedded ESSs is also demonstrated through simulations. Furthermore, it was demonstrated the ESSs can easily follow the dynamics required by these services. The proposed concept can greatly broaden the set of solutions that TSOs can have at their disposal to ensure secure supply of the electrical energy.

Figure 2

 2 Figure 2 Topology of the proposed modular multilevel converter with integrated energy storage systems

Figure 3

 3 Figure 3 Simplified equivalent circuit of: (a) a classical MMC (b) an MMC with a large buffer storage

Figure 4

 4 Figure 4 Schematic diagram of the proposed control of an MMC with embedded ESSs

Figure 6

 6 Figure 6 Simulation results of the modified Klein-Rogers-Kundur's power system. The HVDC link provides POD without (left) and with (right) an extra energy source added on the DC side.

Figure 7

 7 Figure 7 Simulation results of the modified Klein-Rogers-Kundur's power system. Different control strategies to enhance AC stability are applied. (Note that a positive power implies power injected into the DC grid)

Figure 9

 9 Figure 9 Energy storage function requirements with: (a) The power to supply (b) The energy variation

Figure 10

 10 Figure 10 Arm average model of one phase of a three phase MMC with embedded ESSs

Figure 12

 12 Figure 12 Simulation results of the point-to-point HVDC with embedded ESS in one converter station -Internal supercapacitor stack voltage and current for the three considered cases of ancillary services. Note that the red dashed line shown the maximum supercapacitor stack voltage for each design

Table 2

 2 Main characteristics of the ESS of an ES-SM for the proposed ancillary services (*MaxwellTechnologies were considered)

	POD	FFR	POD and FFR

ACKNOWLEDGMENT

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program ANE-ITE-002-01.


Rémi Penin

Guillaume Parent

Jean-Philippe Lecointe

Jean-Franc ¸ois Brudny

Senior Member, IEEE Thierry Belgrand

Impact of Mechanical Deformations of Transformer Corners on Core Losses

The clamping of the power transformer magnetic circuit allows maintaining the sheet stack and provides an additional mechanical rigidity. It has consequences on the acoustic noise of transformers as the magnetic cores of power transformers vibrate because of the magnetostriction and, if the corners are considered, Maxwell's forces. The indirect impact of a lack of clamping on the core losses is studied, especially on the interlaminar air gaps in the corners. In this paper, it is shown numerically with Finite Element modelling and experimentally with a specific test bench that induced vibrations can change the flux distribution and thus the losses.

Index Terms-Magnetic losses, transformer cores, vibrations.

I. INTRODUCTION

T HE magnetic core of power transformers is most of the time made of Grain Oriented (GO) steel laminations, which have the best performances in the Rolling Direction (RD).The magnetic core clamping allows maintaining the laminations assembly and provides an additional mechanical rigidity. Several studies have shown that this clamping modifies the GO sheet properties such as the permeability or the magnetostrictive behavior [START_REF] Joslin | Some aspects of effects of longitudinal and normal stress on power loss and flux distribution within a transformer core[END_REF][START_REF] Moses | Effects of stresses on magnetic properties of silicon-iron laminations[END_REF]. In a transformer core, two vibration sources make the sheets vibrating, especially in the corners areas. The first one is the magnetostriction, which occurs when the magnetic material domains rotate in order to be aligned along the magnetic field direction [START_REF] Arai | Magnetostriction of grain oriented Si-Fe and its domain model[END_REF]. The second one is the Maxwell's forces, which take place between the laminations when the magnetic flux crosses the air gaps. The latter phenomenon is particularly standing inside the corners because the step lap joints are composed of thin anisotropic laminations, of crossing RDs, and of air gaps. Thus Maxwell's forces occur in two directions: in plane and out of plane of the sheets [START_REF] Weiser | Relevance of magnetostriction and forces for the generation of audible noise of transformer cores[END_REF]. In this paper, the authors analyze the impact of a lack of clamping and of a non-uniform clamping [START_REF] Wang | Finite element modelling of the vibration of a power transformer[END_REF] on the core losses. They present in the first part the experimental measurement results obtained with a single phase transformer core structure, for which various pressures have been applied on corners. The second part presents a 2D Finite Element (FE) model of the transformer core structure. The latter brings information about the change in the magnetic flux distribution inside the transformer core with respect to geometry variations due to a non-uniform clamping. The last part is a discussion, about the link between the magnetic core deformations and the core losses.

Manuscript received june 30th, 2014. Revised october 6th, 2014. Corresponding author: G. Parent (email: guillaume.parent@univ-artois.fr).

II. EXPERIMENTAL MEASUREMENTS

A. Experimental Device

A single phase transformer core has been built in order to quantify the impact of the clamping pressure on the losses (Fig. 1). It is made of GO "Powercore H" sheets of 0.30 mm thickness. The external dimensions are 0.5 m by 0.4 m. The legs are made with 60 sheets with a width of 100 mm and the corners are Butt-Lap configuration. The primary and secondary windings are made of 104 turns uniformly wound around the 4 legs. The secondary voltage is monitored in order to adjust the global flux density value. The primary winding is supplied with a 50 Hz sinusoidal voltage. Four different operating points have been considered 1.5 T, 1.6 T, 1.7 T and 1.8 T in order to estimate the influence of saturation effects.

Each corner is put under pressure normal to the steel surface using non magnetic weights. Those can be accurately adjusted. Wooden plates are placed between corners and weights in order to apply an uniform pressure. Various pressure values have been considered: 0 Pa, 200 Pa, 400 Pa, 600 Pa, 800 Pa, 1000 Pa.

For each pressure value previously mentioned, two kinds of quantity have been measured :

• the core losses, thanks to a precision wattmeter (Yokogawa WT230) measuring simultaneously the primary current and the secondary voltage • the deformations, measured thanks to 2 accelerometers Bruël&Kjaer, Type 4397) placed respectively on the center-part and near the interior side of the corner joint (Fig. 2). The measurements are analyzed with a Pulse spectrum analyzer station. The pressure values are several times lower than the ones usually applied on clamped parts in industrial transformers. However, the goal of applying these pressures on the whole corners consists in highlighting tends of both losses and vibrations with the pressures.

B. Measurement Results

Core loss variations with pressure at the flux density values mentioned in II-A are presented in Fig. 3. The results highlight a decrease of the losses, up to 8.27 % between 0 Pa and 1000 Pa at B=1.6 T. The decrease can be explained by the fact that the pressure:

• limits the air gap fluctuations and modifies the magnetic flux distribution inside the core • homogenizes the interlaminar air gap thickness and, as a consequence, increases the efficiency of the core. The first point is verified with the deformations measurements shown in Fig. 4. At 1.5 T, the corner deformations decrease drastically of 52 % in the middle of the corner (Fig. 4a) and of 82.7 % in the interior of the corner (Fig. 4b). It means that the limitation of the corner deformation reduces the air gap variations. In order to understand the influence of the deformations on the flux distribution, a FE model has been developed.

III. FINITE ELEMENT MODEL A. Model Definition

A FE model taking into account the real geometry of the transformer, ie taking into account at once the sheets and the interlaminar air gaps, would lead to an extremely high computation time issue. Hence, in order to get a FE model representative enough of our problem with a descent computation time, a developed magnetic core diagram which is often used for winding rotating electrical machines representation has been modeled (Fig. 5). It models the transformer presented in section II-A in a schematic outline [START_REF] Mechler | Magnetic flux distributions in transformer core joints[END_REF][START_REF] Nakata | Magnetic performance of Step-Lap joints in distribution transformer cores[END_REF], which takes into account the 4 legs and the 4 corners. Each leg is made of a stack of ten 0.30 mm nominal thickness sheets and the interlaminar air gap thicknesses are adjustable. The non-linearity (saturation and anisotropy) of the sheets is taken into account [START_REF] Parent | Analysis of the magnetic flux distribution in a new shifted nonsegmented grain oriented AC motor magnetic circuit[END_REF]. The in plane characteristics of the sheets have been determined with the Epstein frame method [START_REF]Methods of measurement of the magnetic properties of electrical steel sheet and strip by means of an Epstein frame[END_REF]. The relative permeability in the normal direction, equal to 28.6, has been determined experimentally with a specific test bench described in [START_REF] Hihat | Experimental and numerical characterization of magnetically anisotropic laminations in the direction normal to their surface[END_REF]. Periodic boundary conditions are applied on the left and right sides of the model. An operating point at B=1.6 T is considered and the following cases are studied:

• case 1: all the interlaminar air gaps have the same thickness, • case 2: one air gap thickness is different from the others, • case 3: all the air gaps thicknesses are different. The flux distribution is studied along two lines (Fig. 6). The first one cuts the stack near a corner whereas the other one passes through the middle of plane air gaps (Fig. 5). As an example, Fig. 6 shows the flux distribution obtained with the model in a case where all the interlaminar air gaps have the same thickness (5 µm): the magnetic flux density establishes itself according to the less reluctant path. This leads to local saturations in the corners every two sheets and to very low magnetic flux density every two other sheets [START_REF] Hihat | Equivalent permeability of Step-Lap joints of transformer cores: Computational and experimental considerations[END_REF].

B. Case 1

In this case, all the interlaminar air gaps have same thickness. This latter can take different values: 5 µm, which is considered as the reference value (REF), 10 µm and 15 µm. Fig. 7 shows the flux density variations along the lines 1 and 2.

• Along line 1, the increase of the interlaminar air gaps thickness leads to an increase of the heterogeneity of the magnetic flux density inside the stack. Indeed, compared to REF, a 3 % increase of the magnetic flux density level in the saturated sheets as well as a 24 % decrease of this level in the less solicited sheets can be noted when interlaminar air gaps are 15 µm. • Along line 2, the increase of the interlaminar air gap thickness leads to an increase of the magnetic flux establishing itself in the plane of the sheets [START_REF] Balehosur | Packetto-packet variation of flux density in a three-phase, three-limb power transformer core[END_REF]. Indeed, the reluctance between the sheets increases with the interlaminar air gap thickness. This favors the in plane magnetization. In our case, the increase of the in plane magnetic flux density level is increased up to 3 %. The link between the change in arrangement of the magnetic flux due to the increase of the interlaminar air gaps and the increase of the core losses can be explained in two different ways:

• the core losses have a non linear increase with respect to the magnetic flux density [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF] and they are known to be minimal with a homogeneously distributed magnetic flux. Now, Fig. 7 shows that the increase of the interlaminar air gaps leads to an increase of the heterogeneity of the magnetic flux repartition in the stack. In our case, applying the results obtained along line 1 to a typical losse curve leads to an increase of the core losses up to 10 % between the 5 µm case and the 15 µm case, • Fig. 7b, as for it, shows an increase of the magnetic flux in the plane of the sheets, and so an increase of the rotational flux. The latter are known to be source of additional core losses [START_REF] Jones | Comparison of the localized power loss and flux distribution in the butt and lap and mitred overlap corner configurations[END_REF].

C. Case 2

All the interlaminar air gaps are of the same thickness (5 µm) excepted one of 15 µm. Two simulations have been done considering two different places for this particular air gap. In the first simulation, the larger air gap is placed in the middle of the stack (noted Middle Air Gap: MAG). In the second simulation, this larger air gap is placed at the extremity of the stack (noted Last Air Gap: LAG). Figs. 8a and8b show the magnetic flux density repartition along lines 1 and 2. It can be noted that the influence of the increase of a single air gap on the repartition of the magnetic flux is limited to the contiguous sheets to this particular air gap only.

D. Case 3

In this case, the thickness values of the nine interlaminar air gaps are different from each others and follow the law e n = 5 + 0.1 × 2 n-2 (in µm) where n is the label of the considered air gap (1 < n < 9). This arrangement of the air gap thicknesses can be considered as equivalent to what happens in a vibrating transformer corner.

The results shown in Fig. 9 are in accordance with the two previous cases:

• the heterogeneity of the magnetic flux increases with the increase of the value of the interlaminar air gap thicknesses, • the influence of the value of an air gap thickness is limited to the neighbouring sheets to this air gap. Indeed, Fig. 9 shows that the most important saturation phenomena are located around the larger air gap. In this case, applying the results obtained along line 1 to a typical losses curve leads to an increase of the core losses up to 3 % compared to the reference case.

IV. RESULTS AND DISCUSSION

The measurements performed on a single phase transformer core presented in section II showed that there is a link between the core deformations and the core losses. Indeed, applying a pressure perpendicular to the lamination surface at a transformer corner leads to a reduction of both the core vibrations and the core losses. Actually, the latter increase with a heterogeneous distribution of the magnetic flux density inside the transformer core, as well as with local saturations. The FE model presented in section III showed that those two phenomena are themselves linked to the thickness of the interlaminar air gaps.

V. CONCLUSION

In this paper, the authors have shown the importance of the clamping in a transformer. A lack of clamping or a non-uniform one has a direct influence on the core losses. Indeed, the natural vibrations of the transformer lead to a heterogeneous repartition of the thickness of the interlaminar air gaps. Hence, the increase of the latter presents a direct impact on the repartition of the magnetic flux inside the transformer and then to the core losses, especially in the corners.

ACKNOWLEDGMENT

This work is supported by the French national technological research cluster on electrical machine efficiency increase (MEDEE). This program, including ThyssenKrupp Electrical Steel UGO, is sponsored by the region Nord Pas-de-Calais (France), the French ministry and the European founds (FEDER).

Fig. 2 .

 2 Fig. 1. Experimental setup

Fig. 3 .Fig. 4 .

 34 Fig. 3. Core losses variations with respect to the applied pressure and the flux density (measurements and trend curves)

Fig. 7 .

 7 Fig. 7. Simulation results with homogeneous interlaminar air gaps along the line 1 (a) and the line 2 (b)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Model for performing the Finite Element simulation of schematic outline transformer

Fig. 8 .Fig. 9 .

 89 Fig. 8. Simulation results with one air gap thickness different from the others along the line 1 (a) and the line 2 (b)