M Faganello

F Califano

F Pegoraro

Time window for magnetic reconnection in plasma configurations with velocity shear

Keywords: numbers: 52.35.Vd, 52.35.Mw, 52.65.Kj, 52.35.Py

It is shown that the rate of magnetic field line reconnection can be clocked by the evolution of the large scale processes that are responsible for the formation of the current layers where reconnection can take place. In unsteady plasma configurations, such as those produced by the onset of the Kelvin-Helmholtz instability in a plasma with a velocity shear, qualitatively different magnetic structures are produced depending on how fast the reconnection process develops on the external clock set by the evolving large scale configuration.

Fast magnetic field line reconnection is believed to be a crucial phenomenon in low collisionality or collisionless plasmas, as expected to occur on time scales that do not exceed by large factors the dynamical plasma time scales as determined within the ideal magnetohydrodynamic (MHD) description. When current layers are formed that have widths of order of the ion skin depth d i ≡ c/ω pi , with ω pi the ion plasma frequency, ions inside these current layers decouple their motion from the evolution of the magnetic field. Per se this decoupling is not sufficient to allow for magnetic reconnection which requires electron decoupling. This latter occurs inside a thinner region that, depending on the plasma parameters, has a width determined either by a non-vanishing resistivity or by kinetic effects, or, as is the two fluids model description adopted here, by electron inertia effects. The large spatial separation between these two decoupling regions allows magnetic reconnection to grow at a faster rate [1][2][3][4][5].

In a magnetized plasma streaming with a nonuniform velocity, the Kelvin-Helmholtz (K-H) instability plays a major role in mixing different plasma regions and in stretching the magnetic field lines leading to the formation of layers with a sheared magnetic field where magnetic field line reconnection can take place. A relevant example is provided by the formation of a mixing layer between the Earth's magnetosphere and the solar wind at low latitudes during northward periods recently investigated in Ref. [6] (see also references therein). In the considered configuration, in the presence of a magnetic field nearly perpendicular to the plane defined by the velocity field and its inhomogeneity direction, velocity shear drives a K-H instability which advects and distorts the magnetic field configuration. If the Alfvén velocity associated to the in-plane magnetic field is sufficiently weak with respect to the variation of the fluid velocity in the plasma, the K-H instability generates fully rolledup vortices which advect the magnetic field lines into a complex configuration, causing the formation of current layers. Since the plasma dynamics is essentially driven by the vortex motion, the reconnection events that are produced in these layers are usually denoted as Vortex Induced Reconnection (VIR) [7][8][9][10][11].

Pairing of the vortices generated by the K-H instability is a well know phenomenon in two-dimensional hydrodynamics [12,13]. On the other hand, the role of the magnetic field on vortex dynamics has been studied essentially in the limit of only one vortex, the largest one contained in the simulation box. It has been shown that, even if the magnetic field is weak and unable to prevent the formation of the KH vortex, nevertheless the VIR process eventually leads to vortex disruption [9,10,14]. On the contrary, in this letter we investigate the development of magnetic reconnection during the vortex pairing process and show that completely different large scale magnetic structures are produced depending on how fast the reconnection process develops on the time scale set by the pairing process. We consider a configuration with a value of the plasma β parameter (ratio of plasma pressure over total magnetic field pressure) of order unity and show that in this regime the Hall term in Ohm's law, corresponding to the decoupling of electrons and ions inside the current layers, allows magnetic reconnection to occur on time scales fast enough to compete with the pairing process. In our simulations, the conditions for magnetic reconnection are provided, in an initially uniform in-plane magnetic field, by the motion of the K-H vortices that grow and pair in the initially imposed shear velocity field. We observe that VIR does not destroy the vortices before they coalesce, in agreement with multiple vortices study in a homogeneous plasma [8]. We find that if the Hall term is removed from Ohm's law, the development of reconnection, and thus eventually of the K-H vortices, is qualitatively, not only quantitatively, different. This result provides a clear cut example of the feedback between large and small scale physics, as the necessary conditions for reconnection to occur are produced by the large scale vortex motion, but the specific physical processes that make reconnection act faster or slower determine eventually the evolution of the entire system and the final magnetic field structure.

We consider a 2D description of the system, with the inhomogeneity direction along x, the periodic direction along y, and z an ignorable coordinate. We adopt a two-fluid, quasineutral plasma model. The corresponding dimensionless set of equations, normalized to the ion cyclotron frequency Ω ci and to the ion skip depth d i (d 2 e = m e /m i ), reads: ∂n/∂t + ∇ • (nU) = 0 where n is the plasma density, U = (u i + d 2 e u e ) the fluid velocity and u i,e the ion and electron fluid velocities,

∂(nU)/∂t+ ∇ • [n(u i u i + d 2
e u e u e ) + (P Ī -BB)] = 0, (1) with P = P i + P e + B 2 /2 taken spatially uniform at t = 0 and isothermal closures P i,e = nT i,e . The magnetic field B is advanced by solving the Faraday equation and the electric field E is calculated by means of the following generalized Ohm's law [15],

1 -d 2 e ∇ 2 E = -u e × B (2) -d 2 e {u i × B + (1/n)∇ • [n (u i u i -u e u e )
]} where the term ∇P e /n has been omitted since, for a polytropic equation of state, it does not contribute to ∇ × E. Electron and ion velocities are obtained by combining the fluid velocity U, Eq. ( 1), and the current J ≡ n (u i -u e ) = ∇ × B (where we neglect the displacement current.

We consider an initial large-scale, sheared velocity field:

U eq = (U 0 /2) tanh [(x -L x /2)/L u ] ŷ.
Since we are interested in the VIR process, we consider a homogeneous density in order to eliminate other secondary fluid instabilities [16,17]. The equilibrium magnetic field at t = 0 is homogeneous: B eq (x, y) = B y,eq e y + B z,eq e z .

We take L u = 3.0 and a box-length L x = 90 in the x direction and L y = 30π in the periodic y-direction in order to have well separated growth rates for the modes m = 1, 2, 3, where m is the mode number along y and m = 2 corresponds to the Fast Growing Mode (FGM) of the K-H instability. The values of the sound and Alfvén Mach numbers are set as M s = U 0 /C s = 1.0, M A,⊥ = U 0 /U A,⊥ = 1.0, M A, = U 0 /U A, = 20.0, with U 0 = 1.0 and U A,⊥ , U A, the z and y component of the equilibrium Alfvén velocity. This choice allows the K-H instability to develop into highly rolled-up vortices. We take d 2 e = 1/64. The equations are integrated numerically and numerical stability is achieved by means of filters [18]. Concerning the boundary conditions in the inhomogeneous x-direction, we note that the K-H vortices are mainly large scale MHD structures. Therefore, even if in the central region the system spontaneously develops smaller scale, higher frequency phenomena which obey a two-fluid dynamics, far from the center towards the boundaries we expect that the plasma dynamics remains of the MHD type. For this reason, at the x-boundaries we have implemented the projected characteristic decomposition obtained for the hyperbolic MHD set of equations [19,20]. This allows us to have transparent boundary conditions even for our two-fluid system of equations.

In Fig. 1 we show the nearly frozen-in magnetic field lines and a plasma passive tracer, advected by the velocity field, used in order to label the plasma domains initially on the right and on the left of the velocity null line. In the first frame (upper-left), we see the two vortices generated by the K-H instability, corresponding to the FGM m = 2. As soon as the system enters the nonlinear phase, the vortices start to pair following an inverse cascade process typical of 2D fluid systems. The good correlation between the plasma and the magnetic structures indicates that the magnetic field is still advected by the fluid velocity. In the first frame we see that the blue and the red domain are well separated by a ribbon (white in the figure) of nearly parallel, compressed magnetic lines. This ribbon is rolled-up by the rotation of the two vortices and forms, inside the folds between the vortices, two current layers corresponding to two local magnetic inversion lines. We also see the formation of a first couple of X-points (one for vortex) in this region at x 1 = 44, y 1 = 65 and at x 2 = 45, y 2 = 35, respectively. This is the first reconnection event observed in our simulation. At t=435 we show in the second frame (left-bottom) the formation of a second pair of X-points (one for vortex) in the same inversion region (x 3 = 44, y 3 = 50 and x 4 = 45, y 4 = 47). Magnetic reconnection develops at these X-points and forms magnetic islands with typical size ∼ d i , the maximum value compatible with the dimension of the current sheet. At the same time, the field line ribbon between the second pair of Xpoints shrinks and finally opens up. A new ribbon of field lines appears at t = 450, right top frame. This new ribbon no longer separates the red and the blue plasma regions. Indeed, during this process, significant portions of the red plasma have been engulfed in the form of "blobs" into the blue plasma region and viceversa (right-bottom). The observed change in the large scale magnetic field topology is illustrated in Fig. 2, a schematic representation of the region between the two pairing vortices. The inflow plasma velocity at the second pair of X-points is approximately 0.1 times the value of the local Alfvén velocity U a in the x-y plane, in agreement with the values of the inflow velocity expected in the case of fast magnetic reconnection [1]. Then, the growth rate γ of magnetic re-connection is inferred to be γ ∼ 0.1 U a /L ∼ 0.15, where L is the shear-length of the in-plane magnetic field at the X-points. This value is compatible with our estimate of γ ∼ d ln E z /dt at the X-point. Note that the position of the X-point does not change appreciably and that in the time interval given by a few growth times of the reconnection instability the two vortices can only rotate by a few degrees so that the current rearrangement caused by the vortex rolling up is not sufficient to interfere with the development of the reconnection process.

In Fig. 3, left frame, we show the shaded isocontours of the perpendicular current density J z in the region between the two pairing vortices at t = 440. We see the two current layers between the vortices at the two magnetic inversion lines where the first and the second pairs of X-points were formed. The absolute maximum value of the current density in these sheets is in normalized units ∼ 1.5 and the width of the sheets is approximately 0.7d i . Thus we expect the Hall term to be comparable to the U × B term in Ohm's law. In order to underline its role, we define two quantities,

D i = |{U × B + E} z |
and D e = |{u e × B + E} z |, which measure respectively the ion and the electron "decoupling" from the magnetic field. In Fig. 3, central frame, we show a wide ion decoupling region where D i = 0. This region extends across the X-points over few d i lengths. In the right frames of the same figure, we plot a section at y = 47 of D i (bottom) and of D e (top) at t = 440 and show two separate decoupling regions [1], the "wider" ion decoupling region where D i = 0 and D e ≃ 0 and the inner, thinner electron decoupling region where D e = 0 of width of the order of a few d e . Inside the ion decoupling region, of width roughly equal to 1.5d i , the magnetic field is essentially frozen in the electron motion but the MHD frozen-in law is not satisfied and the two terms U × B and J × B have approximately the same absolute value. Inside the thinner electron region also the electrons are decoupled from the magnetic field and magnetic reconnection can take place.

Finally, we observe that the energy associated to the in-plane magnetic field increases, essentially during the non-linear evolution, by a factor 10.

The crucial role of Hall term has also been demonstrated by running again the same simulation parameters simply omitting the Hall term in the generalized Ohm law. Although the large scale motion of the vortices is still able to generate current sheets of comparable intensity and width, the process of magnetic reconnection is slower. In particular it does not succeed in forming the second pair of X-points in Fig. 1 quickly enough. The two vortices continue to roll-up while pairing and develop into a different magnetic pattern where magnetic islands are eventually generated all over the vortices leading to the disruption of the vortices, as shown in Fig. 4. The inferred growth rate at the X-point located at x = 46, y = 53 is γ ∼ 0.08. The final topology of the magnetic field is qualitatively different: in the previous case the FIG. 4: Shaded isocontours of a plasma passive tracer and magnetic field lines in the (x, y) plane at t = 455, 500 for the parameters of Fig. 1, but with the Hall term omitted.

reconnection process occurring at the second pair of Xpoints leads to the formation of a global new magnetic connection as illustrated in Fig. 2, thus to a single paired vortex. On the contrary, when the Hall term is neglected, no new global magnetic connection is established in the final configuration. The competition between the development of the large scale magnetic configuration and the evolution and the reconnection instability determines the development of the entire system. If magnetic reconnection is not fast enough, the rolling up of the vortices destroys the favorable conditions for the reconnection instability to grow. We express the time change of the reconnected flux function, i.e. the difference δψ between the values of the flux function ψ of the in-plane magnetic field at the X-point and at the O-point, in units of the vortex rotation time τ for the island at x = 44, y = 57 in Fig. 1 (left-bottom) and for the island at x = 46, y = 57 in Fig. 4. We find that in the first case δψ grows and saturates in approximately τ /8 while in the second case, where the Hall term is absent, the island forms later and δψ grows on a slower time scale. In summary, the early evolution of the vortices is essentially MHD and depends only on the initial velocity and magnetic field, regardless of the specific small scale physics included in the generalized Ohm's law. On the contrary, when magnetic reconnection starts to act, the evolution of the whole system depends on the specific physical phenomena that occur at small spatial scales. Large scale physics build up the favorable conditions for reconnection to occur which in turns influences the evolution of the whole system.

The role of the Hall term has also been demonstrated by running the same simulation parameters with a resis-tive Ohm's law (without electron inertia) with the Hall term E + U × B -J × B = ηJ, and without the Hall term (resistive MHD) E + U × B = ηJ. Results analogous to those obtained in the case of dissipationless reconnection are obtained for intermediate values of the normalized resistivity η (η ∼ 0.002, which is compatible with the estimate η ≃ γ d 2 e ). For higher values of the resistivity (η ∼ 0.01) magnetic diffusion effects become too important. Since the magnetic field lines are no longer well frozen in the fluid motion, the large scale motion of the vortices cannot generate current sheets of comparable intensity and width. The vortex motion can no longer lead to the required configuration for fast magnetic reconnection to occur, with or without the Hall term in generalized Ohm's law. On the contrary, if the value of resistivity is smaller (η ∼ 0.001) the vortex motion is able to generate such current sheets, but the process of magnetic reconnection is too slow, again independently of the Hall term. In this case the two vortices continue to roll-up while pairing and develop into a different magnetic pattern where magnetic islands are eventually generated all over the vortices, similar to Fig. 4. This behavior is not apparent if we change the magnitude of the electron inertia effects by changing the mass ratio (the values 25, 64, 100 have been considered). This suggests that the fast magnetic reconnection process depends on the small scale mechanism that allows the magnetic field to decouple from the electron fluid. This work was supported in part by PRIN-MIUR 2006. We are pleased to acknowledge the "Mesocentre SIGAMM" machine, hosted by Observatoire de la Côte d'Azur, where part of the simulations was performed.

FIG. 1 :

 1 FIG.1: Plasma passive tracer and magnetic field lines in the (x, y) plane at t = 426, 435 (left column), and t = 450, 480 (right column).

FIG. 2 :FIG. 3 :

 23 FIG. 2: Schematic representation of the magnetic field lines topology in the region between the two pairing vortices. The lines shown represent the local features of the field line ribbon in Fig.1: the inverted magnetic field lines at t = 426, the formation of the second pairs of X-points at t = 435 and the formation of a new magnetic line connection at t = 450.


M Faganello

F Califano

F Pegoraro

T Andreussi

Double Mid-Latitude Dynamical Reconnection at the Magnetopause: an Efficient Mechanism Allowing Solar Wind to Enter the Earth's Magnetosphere

Keywords: 94, 30, cp -First pacs description 52, 35, Mw -Second pacs description 94, 30, -d -Third pacs description

Three-dimensional simulations of the Kelvin-Helmholtz (KH) instability in a magnetic configuration reproducing typical conditions at the flank Earth's magnetosphere during northward periods show the system's ability to generate favorable conditions for magnetic reconnection to occur at mid-latitude. Once these conditions are established, magnetic reconnection proceeds spontaneously in both hemispheres generating field lines that close on Earth but are connected to the solar wind at low-latitude, allowing direct entrance of solar wind plasma into the magnetosphere. These results are consistent with recent observations of KH vortices showing the signature of reconnection events occurring well outside the equatorial plane (Bavassano et al.,

In the interaction between the solar wind and the Earth's magnetosphere different phenomena can cooperate at the magnetopause boundary where strong gradients are present and can provide free energy for the excitation of waves and instabilities. In such a complex magnetized system the interplay between different instabilities can have a "non-local" character due to the fact that the magnetic field lines, being frozen in the plasma over most of the configuration, can act as "rods" and transfer momentum and energy very effectively to locations far from the region where primary instabilities develop [1]. Eventually external constraints, such as the field lines being tied to the Earth, require that energy be released through the development of field line reconnection. In particular hydromagnetic large-scale instabilities such as the Kelvin-Helmholtz (KH) instability in the equatorial region [2] can act as a forcing "at a distance" compared to the location where localized reconnection finally occurs. In this case including 3D effects and line tying provides a new perspective that tends to be obscured within 2D models and that can explain the transport properties of the magnetopause boundary during northward periods. In fact it has been shown that a mostly northward solar wind magnetic field is associated with a cold and dense plasma pop-ulation observed inside the Earth's magnetosphere plasma sheet, indicating the entrance of solar wind plasma into the magnetosphere [3]. Magnetic reconnection apperas to be a good candidate for explaining this efficient transport. In fact, due to the quasi-collisionless nature of space plasmas collisional cross field diffusion is ineffective, while collisionless diffusive processes associated to the observed wave-activity are believed to be too weak to explain the efficient mixing between the magnetosphere and the solar wind [4][5][6]. On the other hand reconnection, as the only mechanism able to change the global magnetic field topology, can allow different types of previously "unconnected" plasmas to mix [7,8], thus explaining the entrance of solar wind plasma into the magnetosphere.

During northward magnetic field periods reconnection can occur in the lobe regions, between solar wind field lines draping over the sunward boundary of the magnetosphere and magnetospheric lines going from the Earth's poles to the tail [9]. In the case of reconnection proceeding almost simultaneously in both lobes, this double reconnection process could create magnetospheric-like flux tubes partially populated by solar wind plasma and has been proposed as a mechanism for the formation of the observed mixing layer [10,11]. Favorable conditions for reconnection during northward periods can also be created by the dynamics driven by large-scale KH vortices growing along the flank magnetopause near the equatorial plane [2], without being significantly inhibited by magnetic tension. Indeed, KH vortices spontaneously build up magnetic inversion layers where reconnection can occur [12][13][14][15] by stretching and advecting the magnetic field lines. These numerical results, obtained in a simplified 2D geometry, show that magnetic reconnection is able to connect the solar wind and magnetospheric regions on a time scale that is comparable to the KH scale provided the plasma model used for describing the evolution of the whole system includes at least two-fluid effects allowing reconnection to proceed on the correct time scale [16][17][18].

However it is known that in a 3D geometry the magnetic configuration stabilizes the KH instability at highlatitude. The resulting differential advection of the magnetic field lines suggests that favorable conditions for magnetic reconnection should be created at mid-latitude instead of at the equatorial plane [19,20]. Preliminar numerical studies have shown that magnetic reconnection can actually occur at mid-latitude and change the global topology [21]. Furthermore, observations of KH vortices developing in the equatorial plane showed the presence of particle distributions that were conjectured to be produced by magnetic reconnection occurring away from the equatorial plane [22] and not associated with reconnection between or inside vortices as expected in a 2D modeling of KH vortices.

In this letter we show that in a mostly northward magnetic configuration the differential advection of magnetic field lines, driven by the KH vortices that develop mainly in the equatorial region, creates current sheets at midlatitudes where reconnection eventually occurs. In general this spontaneous onset of reconnection proceeds in both hemispheres on the same time scale, thus creating magnetic flux tubes that are topologically connected to Earth at both poles but are embedded in the solar wind when crossing the low-latitude region. We may thus conclude that these magnetosphere-like flux tubes are partially populated by solar wind plasma and show that the quantity of plasma that is exchanged between flux tubes connected to the Earth and flux tubes connected to the solar wind is sufficient to explain the observed transport. In addition we show that this kind of double-reconnection process is not only boosted by the approximately northsouth symmetry, as in the case of the so-called double lobe reconnection [10,11], but is actually related to the 3D dynamics of the system and to the consequent braiding of the magnetic field lines.

We adopt a 3D slab geometry, with the open inhomogeneity direction perpendicular to the magnetopause along the x axis, the northward direction along the z axis and the y-direction aligned along the solar wind direction. Periodic boundary conditions are used in the (y, z) plane. We consider an ideal magnetohydrodynamic equilibrium configuration, varying in the (x, z) plane and constant along y, able to model the high-latitude stabilization of the KH instability. Stabilization is achieved by varying the characteristic scale length of the velocity gradient. We assume a nearly northward magnetic field lying in the x, z plane and defined by the y-component of the vector potential,

A eq,y (x, z) = 1 2 4 3 x + L z 3π sinh 2πx L z cos 2πz L z (1)
where physical quantities are normalized to the ion skin depth d i , the ion cyclotron frequency Ω ci and the ion mass m i . In the simulations presented below we have taken the box dimensions as L x = 60, L y = 30π and L z = 120π. The initial large-scale, sheared velocity field is thus a function of A eq,y (x, z) [23],

U eq (x, z) = (U 0 /2) tanh[A eq,y (x, z)/L u ] ŷ
, where L u = 3 is the halfwidth of the velocity shear layer. In order to focus on reconnection, we take constant and homogeneous density and temperature profiles so as to inhibit the development of other secondary instabilities [24][25][26]. The reference values of the sound and Alfvén Mach numbers are set equal to

M s = U 0 /C s = 1.3, M A = U 0 /U A = 1.0, with U 0 = 1.0
and U A the value of the equilibrium Alfvén velocity at x = 0, z = 0. Due to the slow z-variation of the equilibrium, the gradient along x of the velocity is roughly three times larger at the equatorial plane than at high-latitudes. Thus, we expect the KH instability to develop earlier at low-latitude since its linear growth rate, in a simplified 2D description, is proportional to the equilibrium velocity gradient [27]. In these conditions, the m y = 2 mode corresponds to the Fast Growing Mode at the equator [21], where m y is the wave number along the periodic y-direction. The parameters adopted are representative of the "typical" values of the physical quantities that can be observed at the outer magnetosphere. In particular they are related to the dynamics of vortices of typical size much greater than d i .

We describe the KH evolution by adopting a two-fluid, quasi-neutral plasma model. The continuity equation reads ∂n/∂t + ∇ • (nU) = 0, where n is the plasma density and U u i the fluid velocity (neglecting electron inertia). The momentum equation reads These equations are integrated numerically with periodic boundary conditions in the (y, z) plane and open transparent conditions in the x-direction (see [17] for details). The value of resistivity, that is taken to combine within a fluid description all possible collisional and kinetic phenomena that allow magnetic reconnection at small scales, is normalized using d i . For the early evolution of the KH vortices the scales involved are much greater, meaning that the "renormalized" η-value should be considered much smaller. Indeed we do not observe any appreciable magnetic diffusion inside the vortices. On the contrary, once current sheets of typical width of d i are generated, the reconnection rate is dominated by the Hall term in the generalized Ohm's law and it is expected to depend smoothly on resistivity [18]. These results have been widely tested in 2D simulations of magnetized KH instability [16].

∂(nU)/∂t + ∇ • [(nUU) + (P Ī -BB)] = 0, (2)
In Fig. 1, left frame, we show in red the magnetospheric field lines, in blue the solar wind lines and the passive trace ζ representing the transition between the magnetospheric blue plasma (ζ = 0.2) and the solar wind red plasma (ζ = 1.0) at t = 410. We see two rolled-up vortices generated by the KH instability, lying in the equatorial plane, while no vortices develop at high-latitude. Detailed view of the X-point showing magnetic field lines and u i -vectors in x, y-planes at z = 0, 75, 115, colored in blue/red if ui,y ≶ 0, at t = 430.

Due to the frozen-in condition, the magnetospheric (blue) field lines, starting at z = ±L z /2, cross the blue arms of the vortices at the equatorial plane and finally reach their symmetric magnetospheric region at z = ∓L z /2, still maintaining their original connectivity since no reconnection process has yet taken place. Solar wind (red) field lines undergo a similar evolution. However, the magnetic field advection bends and arches the field lines, thus creating extended current sheets at mid-latitudes (light green region in Fig. 1, left frame) where reconnection can now take place. It is worth noticing that the differential advection between field lines is driven not only by the rolling-up of the vortices but mainly by the motion of the field lines themselves, anchored in the moving plasma at high latitude, with respect to their anchoring in the vortices at rest in the equatorial region. Indeed, in the numerical reference frame, the magnetosphere and solar wind plasmas move in the opposite direction (asymptotically ±V 0 ), while the phase velocity of the vortices is zero. Looking at the northern hemisphere we can clearly see the magnetic structure of the current layer at mid-latitude at x 0, y L y /6, t = 410 where reconnection is occurring. In this picture, blue and red lines are those not yet reconnected and that still maintain their original connectivity to the magnetosphere and to the solar wind, respectively. On the contrary, green lines represent those that have reconnected once in the northern hemisphere and that connect the magnetospheric region at z = L z /2 to the solar wind red arms of the KH vortices in the equatorial plane at z = 0 and finally to solar wind regions z = -L z /2. In the same way once-reconnected ochre lines connect the solar wind at z = L z /2 to the magnetosphere at z = 0, -L z /2.

We can describe the reconnection region as a nearly 2D structure with a well defined guide field B z , an inverted planar magnetic field given by the blue and red x, y-components of the magnetic field lines drawn in Fig. 1, an X-line that bends along z and a current sheet given by J z with a typical length along the zdirection (L guide f ield ∼ 40d i ) much greater than its width (L magnetic shear ∼ d i ) and its extension along the inverted magnetic field (L inverted f ield ∼ 10d i ). Magnetic reconnection takes place along the entire current sheet, i.e. L X-line ∼ L guide f ield .

Details of the northern reconnection zone at t = 430 are shown in Fig. 2. We adopt the same color code for magnetic field lines used in Fig. 1 but, for the sake of clarity, we do not plot the red lines belonging to the solar wind. The ion velocity field u i in the horizontal sections at different heights is represented by blue and red arrows. These arrows show the outflow structures (their z-component is related to the ion contribution to the current) that advect the reconnected green and ochre lines away from the X-line. At mid-latitude these outflow structures are superimposed to the velocity field generated by the hydrodynamic instability while they are not present in the velocity pattern of the vortices in the equatorial region where the KH instability is stronger. This indicates that reconnection develops as an independent secondary instability at mid-latitude, taking advantage of the favorable conditions created by the large-scale KH instability, and that it does not proceed as forced reconnection simply driven by the velocity field imposed by the growing KH instability [28].

In Fig. 1, right frame, we can see that magnetic reconnection proceeds almost simultaneously in the northern and southern hemispheres creating grey field lines that are connected to the magnetosphere in both hemispheres at high latitude but that cross the solar wind region (red vortex arm) at low latitude. Grey lines are further away, along the outflow, from the northern X-line compared to green lines. Since the outflow advects the reconnected lines far away from the reconnection region we can relate the distance from the X-line to the time in the past when lines reconnected. Grey field lines thus originate from green-type lines that reconnected once in the northern hemisphere (at a previous time compared to the green lines plotted in this figure) and that have reconnected again at a subsequent time in the southern hemisphere, still at mid-latitude. The delay between the two different reconnection processes is comparable to the reconnection time, as given by τ -1 rec γ rec = 0.06. This value is obtained from the temporal evolution of B max = max (B • ∇ζ/|∇ζ|) (Fig. 3, righttop frame) and is compatible with the value expected for fast magnetic reconnection [18]. The quantity B•∇ζ/|∇ζ|, which vanishes outside narrow (nearly 2D) regions where reconnection has occurred, gives the component of the magnetic field perpendicular to the magnetospheric frontier. This quantity generalizes the x-component of the magnetic field created by the linear evolution of Tearing instability in a 2D configuration with an equilibrium field given by B 0 tanh(x/l), described by eigenfunctions of the form B x,k (x, y, t) = Bx,k (x)e iky e γ k t [7, [29][30][31].

This double-reconnection process creates magnetic field lines that are topologically part of the magnetosphere (field lines that close on Earth) but that are connected to the solar wind at low latitude.

As a consequence, the associated flux tubes are partially populated by solar wind particles. Double-reconnection at mid-latitude, dynamically allowed by the evolution of the KH instability, provides a dynamical mechanism for the cold and dense particles of the solar wind to enter the Earth's magnetosphere. This mechanism is not in competition with double lobe reconnection that has been proposed as a mechanism that can close solar wind field lines on Earth [10,11]. However invoking the double-reconnection mechanism at mid-latitude has as intrinsic advantage over double lobe reconnection since it is not only favoured by the nearly north-south symmetry of the magnetospheric system but it is actually imposed by the system evolution itself, once reconnection has acted in the northern (southern) hemisphere. Furthermore double mid-latitude reconnection can involve a far greater number of flux tubes (all the tubes flowing around the magnetospheric flanks) compared to the tubes draping over the magnetosphere at its nose position, that are the ones involved in lobe reconnection. A schematic view of double mid-latitude reconnection is given in Fig. 3, left frame. The top left graph shows two representative pristine magnetospheric (blue) and solar wind (red) field lines before any reconnection process has taken place. These two different field lines are bent by the differential advection and rolled-up by the equatorial vortices but are not braided. In fact, following the generator formalism [32]a clockwise σ -1 plus a counterclockwise σ elementary line-crossing corresponds to the identity operator, i.e. no braiding. As shown in the top right graph, when reconnection occurs in the northern hemisphere the elementary crossing σ is cancelled and the once-reconnected green and ochre field lines (colours are changed following the choice made in the left frame) become braided at mid-latitude in the southern hemisphere. Since the southern feet of the green and ochre lines are advected in opposite directions, tailward and solarward respectively, reconnection in the southern hemisphere must take place in order to unbraid the field lines (bottom left graph) and reduce the line stretching. The bottom right graph is equivalent to the bottom left graph where we have simply changed the colors of the double-reconnected grey and pink lines in order to underline the properties of the regions to which they are connected. In particular the blue-red-blue line represents a double-reconnected line whose associated flux tube is connected to Earth in both hemispheres but is partially populated by the solar wind plasma.

Particle signatures of this type have been observed at the dayside magnetopause and have been associated with double lobe reconnection [11,[33][34][35]. The same particle signatures were also observed in the near-tail magnetopause within KH vortices and it was suggested that these particle signatures should be associated to field lines that had undergone lobe reconnection [22].

In order to quantify the impact of double mid-latitude reconnection on plasma transport we consider the time rate at which flux tubes are reconnected and their volume. More specifically we look at regions defined by the condition Γ ≥ Γ * , where Γ = |∇ζ • B/|B|| is taken to represent the spatial rate of change of magnetic connectivity. The threshold value Γ * = 0.015 is determined selfconsistently as the minimal value that satisfies the condition ∆ζ Γ * L * 0.8, where L * is the extension of these regions along the field line direction (∼ ẑ). Consequently all field lines that cross the intersection of these regions with an horizontal plane at mid-latitude (shown in yellow in Fig. 3, right-middle frame, for t = 430 and z = 87) change their connectivity completely from highlatitudes to the equatorial region. The area of the intersection S λ in a given KH-wavelength λ grows linearly with time (Fig. 3, right-bottom frame). In fact field lines, after being reconnected in a region of typical width d i , are advected away from the X-line by the plasma outflow and create elongated regions of reconnected lines. The width of these regions is of order d i and does not change significantly in time while their linear extension grows linearly in time in the direction of the outflow with velocity 2U outf low 2U A . Thus the volume of plasma exchanged per unit KH wavelength between open and closed field lines is given by V λ = S λ L z /2, where L z /2 represents the average distance between the northern and southern reconnection sites, i.e. the length of the double-reconnected lines embedded into the solar wind. If we consider that only one half of V λ corresponds to plasma entrance into the magnetosphere and that the magnetospheric density is negligible compared to that of the solar wind, we obtain a mass flux for unit surface given by J λ 1 2

d i U A L z ρ SW λL z ∼ D ef f ρ SW L u ⇒ D ef f ∼ d i U A [L u /(2λ)] (3)
where we have introduced the velocity scalelength L u ∼ 3d i of the equilibrium configuration. Taking U A 300Km/s and d i 250Km as typical values of the outer magnetopause we finally obtain D ef f ∼ O(10 9 ) m 2 /s, the same order of magnitude of the diffusion coefficient needed to explain the observed mixing in a simplified viscous model [4].

In conclusion in this letter we propose a new dynamical mechanism for creating double-reconnected field lines during the evolution of KH vortices in the complex 3D configuration of the flank magnetopause. Finally we note that while this mechanism does not exclude double lobe reconnection, as the two mechanisms can operate at the same time, but provides an intrinsic and efficient mechanism for producing double-reconnected field lines. * * *

Fig. 1 :

 1 Fig. 1: (color online). Left frame: blue/red passive tracer in x, y-planes at z = -180, 0, +180, light green current sheet isosurfaces defined by |J| > 1.6, and field lines showing the structures at t = 410 of the X-line at x 0, y 15, 75 z 115. Left frame: blue/red passive tracer in x, yplanes at z = -180, 0, +180 and blue, green and ochre, and grey lines showing respectively pristine magnetospheric, oncereconnected and double-reconnected lines at t = 430.

  with P = P i + P e + B 2 /2 and we take adiabatic closures given by ∂(nS i,e )/∂t + ∇ • (nS i,e u i,e ) = 0 where u i,e are the ion and electron fluid velocities and S i,e = P i,e n -5/3 . The magnetic field B is advanced by solving the Faraday equation and the electric field E is calculated by means of a resistive Ohm's law, E = -u e × B + ηJ, with η = 10 -3 , where we neglect the term ∇P e /n (for an initial uniform entropy configuration it does not contribute to ∇×E). Electron and ion velocities are obtained by combining the fluid velocity U u i and the current J ≡ n (u i -u e ) = ∇ × B (we neglect the displacement current). In order to follow the field line connectivity during the plasma evolution we define a passive tracer ζ that is simply advected by the ion velocity field. At t = 0 ζ is constant along field lines, i.e. ∇ζ • B = 0, has the same spatial scalelength of the velocity field and is equal to ζ(x, y, z)| t=0 = 0.6 + 0.4 tanh[A eq,y (x, z)/L u ]. In this way the value 0.2 corresponds to the magnetospheric plasma flowing at -U 0 /2 and 1.0 to the solar wind plasma flowing at +U 0 /2.

Fig. 2 :

 2 Fig. 2: (color online).Detailed view of the X-point showing magnetic field lines and u i -vectors in x, y-planes at z = 0, 75, 115, colored in blue/red if ui,y ≶ 0, at t = 430.

Fig. 3 :

 3 Fig. 3: (color online). Left frame: a schematic representation of the double-reconnection process at mid-latitudes. Right frame: temporal evolution of Bmax = max(B • ∇ζ/|∇ζ|) (top), intersection at t = 430 (yellow domain) of the regions where the magnetic connectivity has changed with the z = 87 plane (middle) and temporal evolution of the area S λ of this intersection per KH-wavelength.

The research leading to these results has received funding from the European Commission's Seventh Framework Programme (FP7/2007-2013) under the grant agreement SWIFF (project n o 263340, www.swiff.eu). This work was supported by the Italian Supercomputing Center -CINECA under the ISCRA initiative.


Thomas Carriere

Christophe Vernay

Sébastien Pitaval

François-Pascal Neirac

George Kariniotakis

SIZING OF A PV/BATTERY SYSTEM THROUGH STOCHASTIC CONTROL AND PLANT AGGREGATION

Keywords: Storage, Markets, Grid Integration

The objective of this work is to reduce the storage dimensions required to operate a coupled photovoltaic (PV) and Battery Energy Storage System (BESS) in an electricity market, while keeping the same level of performance. Performance is measured either with the amount of errors between the energy sold on the market and the actual generation of the PV/BESS i.e. the imbalance, or directly with the revenue generated on the electricity market from the PV/BESS operation. Two solutions are proposed and tested to reduce the BESS size requirement. The first solution is to participate in electricity markets with an aggregation of several plants instead of a single plant, which effectively reduces the uncertainty of the PV power generation. The second is to participate in an intra-day market to reduce the BESS usage. To evaluate the effects of these two solutions on the BESS size requirement, we simulate the control of the PV/BESS system in an electricity market.

INTRODUCTION

There is a significant uncertainty in the future funding of ground-mounted PV power plants. Traditional feed-in tariffs are disappearing, and most support schemes are now based on participation in electricity markets, where PV power producers are financially responsible for their forecast errors. Thus, the actual revenue generated from PV power plants is subject to both electricity price and PV power generation uncertainty. In order to mitigate these uncertainties, Battery Electricity Storage Systems (BESS) are often studied. However, they are still very costly. In this paper, we investigate how participating in intra-day electricity markets, and using an aggregation of PV power plants instead of a single plant can reduce the BESS size requirements. To do so, we simulate the control of a coupled PV/BESS system in the French electricity market environment.

The most used method of control for coupled PV/BESS systems is the Model Predictive Control (MPC). It consists in optimizing the control of the BESS on a receding horizon, so that the forecast future state of the system is taken into account when deriving the command for the immediate future. In most cases, the loss function used in the MPC controller is either the producer's profit [START_REF] Perez | Predictive Power Control for PV Plants With Energy Storage[END_REF], [START_REF] Abdeltawab | Market-Oriented Energy Management of a Hybrid Wind-Battery Energy Storage System Via Model Predictive Control With Constraint Optimizer[END_REF] or the energy imbalance [START_REF] Damiano | Real-Time Control Strategy of Energy Storage Systems for Renewable Energy Sources Exploitation[END_REF], which is the deviation between the energy sold in the electricity market and the actual PV power generation. Some authors also propose an MPC approach to bid on intra-day market sessions [START_REF] Saez-De-Ibarra | Management Strategy for Market Participation of Photovoltaic Power Plants Including Storage Systems[END_REF], [START_REF] Núñez-Reyes | Optimal scheduling of grid-connected PV plants with energy storage for integration in the electricity market[END_REF], however the intra-day market that they consider is organized in sessions, which is different from the continuous intra-day market that we consider in this paper. The uncertainty of the upcoming PV production is sometimes included in both the dayahead planning and the real-time control of the BESS, as in [START_REF] Conte | Day-Ahead Planning and Real-Time Control of Integrated PV-Storage Systems by Stochastic Optimization[END_REF], [START_REF] Conte | A Stochastic Optimization Method for Planning and Real-Time Control of Integrated PV-Storage Systems : Design and Experimental Validation[END_REF] or [START_REF] Keerthisinghe | Evaluation of a multi-stage stochastic optimisation framework for energy management of residential PV-storage systems[END_REF]. We also include a modelization of the BESS ageing in our optimization problems as in [START_REF] He | Optimal Bidding Strategy of Battery Storage in Power Markets Considering Performance-Based Regulation and Battery Cycle Life[END_REF]. While few papers consider the uncertainty of renewable energy forecasts, [START_REF] Vespermann | Offering Strategy of a Price-Maker Energy Storage System in Day-Ahead and Balancing Markets[END_REF] and [START_REF] Ding | Optimal Offering and Operating Strategies for Wind-Storage Systems with Linear Decision Rules[END_REF] model this uncertainty using production scenarios. An MPC controller that included all these elements was proposed in [START_REF] Carriere | Strategies for combined operation of PV/storage systems integrated into electricity markets[END_REF] but it did not consider an intra-day market.

The main innovation of this work is the implementation of a general stochastic MPC controller that optimizes the real-time control as well as the dayahead bidding and the participation in a continuous intraday market of a coupled PV/BESS system taking into account the uncertainty in the upcoming PV power generation, based on an arbitrary objective function, namely the imbalance or the penalties. Another innovation is the quantification of the added value from using a BESS both in terms of imbalance reduction and revenue maximization.

MARKET STRUCTURE

In this paper, we study the participation of a PV power plant coupled with a BESS on a day-ahead market and dual-pricing balancing market, with a possibility to sell or buy energy on an intra-day market. The market structure assumed is:

-A day-ahead market where each participant has to submit buying or selling orders the day before delivery.

-An intra-day market where energy can be sold or bought up to 30 minutes before delivery -A balancing market where each BRP has to take responsibility for its imbalances after delivery.

Day-ahead market

On day-ahead markets, each participant must submit buying or selling bids before the Gate Closure Time (GCT). Then, all the buying and selling bids are combined to derive aggregated demand and supply curves for each PTU of the following day. The intersection of these curves defines the spot market price. The calculation of the spot price after the GCT is called the market clearing.

After the spot price has been calculated, the market participants are nominated for injection on the grid, depending on whether their bid was accepted or not. All selling bids with a price lower than the spot price are fully accepted, and all buying bids with a price higher than the spot price are fully accepted. Buying and selling offers with a price equal to the spot price are partially accepted. Since we are interested in the selling of energy, we will always adopt the point of view of an energy producer in the remaining of the thesis.

In any case, all accepted transactions are settled with the spot price, independently of the initial bid. For example, if a market participant accepts to sell up to 1 MWh for 20 €/MWh and then the spot price is 40 €/MWh after clearing, the participant's bid is fully accepted and he gets 1 MWh x 40 €/MWh = 40 € 2.2 Intra-day market Intra-day markets are physical markets that allow trading electricity after the GCT. They are especially useful for intermittent energy sources that can use updated forecast to correct the positions they took on the day-ahead market.

As for day-ahead markets, they are characterized by a PTU, and a closure time. For example, on the French day-ahead market EPEX Spot, the PTU is the same as the day-ahead market i.e. one hour, and the closure is five minutes before the start of the delivery period.

The pricing can be the same as for day-ahead markets, with an auction mechanism and a settlement price that applies for all participants. However, it is also very common to have pay-as-bid markets, where buying and selling bids are matched as they appear, directly using the bids price. In this paper, the intra-day market is a pay-as-bid market.

In this paper, we assumed that the producer always submits intra-day offers at the spot price, which is known at that time. We also assume that the intra-day offers are always accepted. This is actually false, as the probability of acceptation of the intra-day offers is very dependent on the price of the offer However, we do not have intra-day data that could allow us to simulate the acceptation of intra-day offers. Assuming that intra-day offers are always accepted allows us to estimate the best-case scenario, giving us an upper bound of the value of intraday markets.

Balancing market

The balancing market penalizes the imbalance of each producer after delivery. The penalties are proportional to the difference between the energy sold by the participant and the actual energy he injected in the grid, as measured by the TSO. For an electricity buyer, the penalty is the difference between the bought energy and the actual consumed energy. The proportionality coefficient between the imbalance and the penalty is a price derived by the TSO called the balancing price. As a general rule, the penalties can write:

Where is the balancing price, is the actual energy injected into the grid and is the energy contracted in the day-ahead and intra-day electricity markets.

In this paper the balancing market is a dual-price balancing market. With dual-pricing rules, there are actually two balancing prices: one for positive imbalances and one for negative imbalances . The balancing price is usually higher than the spot price for negative imbalances and lower than the spot price for positive imbalances, in which case reducing the imbalance is economically beneficial. However, this is not always the case.

Formulation of the revenue

When making a transaction on the intra-day market, the revenue from the transaction adds to the amount initially sold, and the volume bought or sold adds to the actual production for calculating the imbalance penalty. The complete revenue of a producer that sells an energy on the day-ahead market, then makes a transaction on the intra-day market of an energy volume (positive when energy is bought, negative when energy is sold) for a price is:

Where is the spot price that is given by the market clearing after the bids from all market participants have been submitted.

When considering a BESS, we reformulate by differentiating the part of the production that comes from the PV panels and the part that comes from the BESS . We also introduce a term , that reflects the costs due to aging of the BESS when used to deliver the amount of energy . This is obtained with the rainflow counting algorithm [START_REF] Musallam | An efficient implementation of the rainflow counting algorithm for life consumption estimation[END_REF]. The aging of the BESS can be divided into two components, i.e. cycling aging and calendar aging, which is the degradation caused by time. In the remainder of the thesis, we will focus on the cycling aging of the BESS and consider its calendar aging as a given lifetime. The end-of-life of the BESS is thus defined as the minimum lifetime given by the cycling and calendar aging. As an example, if the calendar aging gives a lifetime of 20 years, and the cycling aging a lifetime of 50 years, we consider that the actual lifetime of the BESS is 20 years (as opposed to considering that the cycling aging adds up to the 20 years given as the calendar lifetime).

We penalize the revenue with the cost associated with the life-loss of the BESS. Note that the penalized revenue is not an actual cash flow, and that the cost associated with the life-loss is only here to make the control of the BESS more conservative regarding the lifetime. The penalized revenue then writes:

3 PV/BESS CONTROL

General Control Method

From the market structure, we can see that for each time step of the simulation, we have to take up to three decisions:

-If it is 12 AM, the energy to bid on the day-ahead market for the next day.

-The energy to bid on the intra-day market for the next time step open for the intra-day market.

-The energy to charge or discharge from the BESS for the next time step.

The method we use to control the PV/BESS system is a Model Predictive Control (MPC). This means that for each time step of the simulation, we update the PV power and price forecasts, and then solve the optimization problem corresponding to each decision on a window including the near future i.e. 12 hours in this paper. Then, we use the first element of the solution for the control of the PV/BESS for the immediate future and move forward to the next time step of the simulation.

More precisely, for each time step of the simulation and for each decision making process, we solve an optimization problem over a time window of timesteps using the most updated price and PV power forecasts ̂ ̂ ̂ :

{ } ∑ ( ̂ ̂ ̂ )
The loss functions are dependent on the overall objective of the simulation, and are detailed in the next subsections.

All the optimization problems are performed in a stochastic manner relative to the uncertainty in PV power generation. In other words, instead of directly minimizing the loss function, several scenario of PV power production are generated using the method from [START_REF] Pinson | From probabilistic forecasts to statistical scenarios of short-term wind power production[END_REF], and then the empirical expected value of the loss function derived from the scenario is minimized. To compare stochastic and deterministic optimizations, we use only the expected value of the PV power generation as a single scenario to perform the deterministic optimization.

With this method, the near future is taken into account when controlling the PV/BESS. This is especially useful when controlling the BESS. For example, if a large amount of energy was sold on the day-ahead electricity market in the evening, using the BESS to compensate forecast errors during the day could lead to prematurely emptying the BESS and thus not being able to fulfill the day-ahead planning. Considering the near future with a MPC controller prevents this situation.

Since the three decision processes (day-ahead offering, intra-day offering and real-time control) are consecutive, each optimization can use the optimal solution from the previous processes.

Imbalance minimization

When the overall objective of the simulation is to minimize imbalances, the loss functions for each decision process if the imbalance e.g. the difference between the amount of energy sold and the actual energy production.

Day-ahead offering strategy

In that case, the decision must be taken a 12 AM for all the 24 hours of the next day. Thus, we must solve:

{ } ∑| |
Where is the total output of the PV/BESS. Assuming that we have a forecast of the Cumulative Distribution Function (CDF) of the power generation , we can prove that the optimal solution is given by: ( ) A trivial solution is then to not use the BESS in the day-ahead phase, so that the forecast CDF is the actually the CDF of the PV power generation, and then set the bids following the above equation.

Intra-day offering strategy

To minimize the imbalances, the optimal solution on the intra-day market is to cancel the imbalance using our best expectation of the PV power production. Thus, the intra-day offers are given by: ( )

Real-time control

To minimize the imbalances, we use the absolute imbalance as the loss function . Thus we must solve:

{ } [ ∑ | | ]
Since we use the BESS at this stage, we must also consider the operational constraints of the BESS. To define the constraints, we note as (for State of Charge) the amount of energy in the battery at a given time step, relative to its capacity .

| |

The first constraint ensures that the energy in the BESS is never lower than 0 or higher than the capacity of the battery, taking into account the charge and discharge rates of the BESS, respectively and . The second constraint ensures that the BESS can only be charged from the PV plant, and not from the grid. Finally, the third constraint is a limitation on the power rating of the BESS, defined by the parameter .

Revenue maximization

When the overall objective of the simulation is the actual revenue, the loss function of each optimization problem is the revenue generated by the decision instead of the imbalance. Then, the optimization problems that we have to solve at each step are the following.

Day-ahead offering strategy

When the BESS is used at both the day-ahead and real-time levels, then the entire formulation of the penalized revenue is optimized. Once again, we separate the bids into one part accompanied by uncertainty from the PV plant , and the output from the battery . Since the BESS is controllable, we assume that the actual output of the BESS is always equal to the amount bid . With these assumptions, the optimization problem that needs to be solved to derive the optimal bids is:

{ }
Under the same BESS constraints as defined before.

Intra-day offering strategy

Using our market structure model, the difference in revenue when an intra-day offer of volume and price is accepted writes:

If , the expected value of the revenue with respect to the PV power uncertainty is: ∫ ∫ By deriving with respect to using the Leibniz integral rule, we get: By equaling the derivative to zero, we find a critical point at: ( ) Note that the second derivative is: This is always negative by definition of the balancing prices. Thus, this critical point is a local maximum of the revenue. Using the same method, we find a similar result for the case

The intra-day offering strategy for the maximization of revenue is thus to offer the optimal intra-day volume defined by this equation, using .

Real-time control

To perform the real-time control of the plant, we used the penalized revenue as the loss function of the MPC controller:

{ }
Under the same BESS constraints as before.

RESULTS

We performed the control of the PV/BESS for imbalance minimization and revenue maximization over a period of 8 months from the first of September 2017 to the first of May 2018. The simulation has a temporal resolution of 30 minutes. We compared the results obtained for a single plant with 2.7 MWp and an aggregation of 13 plants with 98 MWp.

PV power forecasts and price forecasts were obtained respectively with an Analog Ensemble method (AnEn) and a Support Vector Machine (SVM) algorithm as in Weather data required for the forecasts was obtained from the European Center for Medium-range Weather Forecasts.

Imbalance minimization

For imbalance minimization, we assumed that we had a theoretical infinite storage capacity. We simulated the control with or without an intra-day market and with a stochastic or a deterministic method. Then, we checked a posteriori the actual size of the BESS that was requested to perform the control.

Figure 1 shows the imbalance reduction that we obtained for the single and aggregated plant. For each plant, we can distinguish three curves: one without intraday market, which sets the benchmark for the BESS size, one with intra-day market and deterministic control, and one with intra-day market and stochastic control. As expected, the aggregated plant has a lower initial imbalance compared to its installed power. All three strategies have the same resulting imbalance. Thus, when comparing the actual BESS size required for each strategy, we compare the actual efficiency in BESS usage of each strategy, since they all provide the same imbalance reduction.

Storage size requirements are reported on table I. We can see that the use of an intra-day market drastically reduces the required size of the BESS. However, there is not much difference between stochastic and deterministic control of the BESS. For revenue maximization, we cannot assume that we have a theoretical infinite capacity. In a case when discharging the BESS adds to the revenue, the optimization would not result in a finite solution, since the optimal output of the BESS would be infinite. Thus, we performed the control for a BESS size of 1 MWh/MWp and compared the revenue variation from all strategies.

Results are shown on table II. We can see that there is no strategy that increases the revenue. This is caused by the high uncertainty in balancing prices. In a significant portion of the simulation, imbalances that were helpful for the grid at a national level were remunerated. For example, if the producer overproduces while the power grid is short of energy, this imbalance is remunerated at a higher price than the spot price. Thus, compensating this forecast error is detrimental to the revenue. Since it is very difficult to forecast whether the grid will be in short or long in energy at the national level, using the BESS to compensate forecast errors is not reliable. In the end, there was no strategy that was able to increase the revenue.

Still, the methods using stochastic control were more efficient that the deterministic ones, as they included more information. However, they only used information on the uncertainty from the PV power production and not the spot and balancing prices. They were thus able to mitigate the high uncertainty of balancing prices but not enough to reliably increase the revenue. In this paper, we studied the effect of intra-day markets and plant aggregation on the BESS size requirement of a PV/BESS system in an electricity market. We compared a deterministic and a stochastic control method for both imbalance reduction and revenue maximization.

We could show that the BESS size reduction obtained with plant aggregation is around 50 % when going from a single plant of 2.7 MWp to an aggregation of 13 plants that have an installed power of 98 MWp in total. Participating in intra-day electricity markets can also reduce the BESS size requirement up to 68 % in the best case, that is, when all intra-day offers are accepted. For imbalance reduction, the uncertainty in PV production is low enough on the short-term that there is little difference between stochastic and deterministic control methods.

On the other hand, for revenue maximization, stochastic control methods perform better, as they are able to somehow mitigate the high uncertainty of balancing prices. However, using the BESS always resulted in a lower income, because in a significant portion of the time, imbalances were actually remunerated by the TSO.

Figure 1 :

 1 Figure 1: Imbalance reduction for the different control strategiesAs expected, the aggregated plant has a lower initial imbalance compared to its installed power. All three strategies have the same resulting imbalance. Thus, when comparing the actual BESS size required for each strategy, we compare the actual efficiency in BESS usage of each strategy, since they all provide the same imbalance reduction.

Table I :

 I BESS size requirements for imbalance minimization

		Single	Aggregation
		Plant
	Required size for ideal control	45	28
	(MWh/MWp)
	Actual size with ID and	14.0	9.35
	stochastic control
	(MWh/MWp)
	Size reduction (%)	-68.9	-66.7
	4.2 Revenue Maximization


Vincent Bettenfeld
email: vincent.bettenfeld@univ-lemans.fr

Salima Mdhaffar
email: salima.mdhaffar@univ-lemans.fr

Christophe Choquet
email: christophe.choquet@univ-lemans.fr

Claudine Piau-Toffolon
email: claudine.piau-toffolon@univ-lemans.fr

Instrumentation of learning situation using automated speech transcription: A prototyping approach

Keywords: Learning Web Environment, Transcription Aided Learning, User Needs Assessment

This paper presents the ongoing conception of a set of tools, based on live transcription of speech during lectures and designed to instrument traditional lectures as well as web conferences or hybrid learning situations. The toolset exploits speech and interactions taking place during courses, keeps track of them and facilitates their reuse both in students' studies and in future iterations of the course delivered by the teacher. Its goal is to help students stay focused on the teacher's explanations and offer them greater possibilities of interactions. The prototype was conceived with an approach based on the analysis of communicational and informational needs of the end users, especially in regard to the instrumentation possibilities offered by the innovative technologies considered in the project. In this paper, we detail the different tools produced in order to offer synchronous and asynchronous support to the learning activity. We describe a real-life test as well as changes brought to the device afterwards, and finally we describe the first experiment conducted with the device.

INTRODUCTION

The PASTEL project, standing for Performing Automated Speech Transcription for Enhancing Learning, is a research project driven by LIUM, LS2N, CREN and Orange Labs. The project's goal is to explore the potential of synchronous speech transcription and application in specific teaching situations. This technology allows to generate a textual version of the teacher's or the students' speech, and immediately use it in order to help them in their teaching or learning activities. Other interests of the project include an editorial toolset for teachers to save, edit and reuse the material produced or gathered in class. This edited material can be exploited during future sessions of the course, or become the support of another course in a different format such as an online class.

The project focuses on pedagogical situations such as lectures and group work, mainly in higher education. These situations can take place either in the classroom, on online platforms using videoconferencing systems, or in hybrid classes. These situations are complex and need to be instrumented with flexible and adaptive tools, offering various configurations for diverse actors. Using transcription technologies, we aim to create such tools and adequately instrument these situations.

The speech transcription allows human actors to access the textual version of a sentence a few seconds after it was pronounced, and browse the whole text as they wish. This tool can help students solving comprehension problems caused by hearing and language barrier [START_REF] Ryba | Liberated learning: Analysis of university students' perceptions and experiences with continuous automated speech recognition[END_REF], or allows them to use down time to read again a more complex section. Having access to the recording of a lecture decreases students' stress: they express trust in these recordings, which leads them to take fewer notes and concentrate on the teacher's explanations [START_REF] Heilesen | Podcasts for efficient learning[END_REF]. To our knowledge, usage of synchronous transcriptions is very limited in pedagogical situations, particularly in higher education in which the vocabulary used can be very specific to the teached domain. As part of the project, other technologies, such as real-time material recommendation and thematic segmentation, will be additionally tested.

Our goal is to study the usability of these tools, and overall the usability of synchronous speech-totext transcription in class context. In the next part, we present a state of the art of research in technology enhanced learning (TEL) and automatic speech transcription (AST) motivating the project and our iterative design methodology. Moodle plugins and their interfaces were developed in an iterative way, this toolset is experimented in real conditions and reengi-neered in a new iterative session. The first and second iterations of the project are then described and discussed in the following parts before giving some concluding remarks related to future work to be done.

PROJECT MOTIVATION AND DESIGN METHODOLOGY

2.1 Project Motivation

Research in Technology-Enhanced Learning

The rise of hybrid classes and video conferences provided institutions with recording and sharing tools both software and hardware (microphones, cameras, clickers). With this project, we want to take advantage of these devices. The goal is to offer tools simple enough to be compatible with the teaching activity, and efficient enough to be adopted. The methodology being design-based research [START_REF] Wang | Design-based research and technology-enhanced learning environments[END_REF], these tools are based on existing needs, tested in authentic context and improved through iterative cycles. Both researchers and teachers are engaged in this procedure.

As Internet usage is widespread, a great number of teachers share their pedagogical material with students online. Massive Open Online Courses (MOOCs) are now common, and offer a great flexibility for learners. Another type of online courses is Small Private Online Courses (SPOCs). They exploit MOOCs' strengths, i.e. their tools and flexibility, with a smaller number of learners. The edition of lecture videos and their segmentation is a tedious task, which existing works tackle automatically using slides [START_REF] Ngo | Structuring lecture videos for distance learning applications[END_REF] [START_REF] Uke | Segmentation and organization of lecture video based on visual contents[END_REF]. This task can be intimidating for teachers, especially if they must consider other resources than slides, such as the textual version of lessons and relevant external documents. Designing a software tool to collect, store and exploit this material directly during traditional classroom sessions is one of the motivations of the project.

In this project, we intent to explore how the product of real-time speech transcription can help learners and teachers. Existing works have proposed the transcription of lectures in real-time [START_REF] Iglesias | Avoiding communication barriers in the classroom: the apeinta project[END_REF], and the originality of our proposition is providing new documents and interactions in real time. The satisfaction of user needs is researched, particularly in terms of information. These needs could be satisfied by content derived from the transcription, or by using this transcription as a support of communication. Finally, this project is the opportunity to study how to index and browsing a great quantity of data growing and formatted in real-time.

Research in Speech Transcription

Automatic Speech Recognition (ASR) aims to automatically convert speech from a recorded audio signal into a text. Different researches in the literature have demonstrated the advantages of synchronous speech transcription for online courses [START_REF] Dirckinck-Holmfeld | Enhancing global and synchronous distance learning and teaching by using instant transcript and translation[END_REF] [START_REF] Hwang | Effects of speech-to-text recognition application on learning performance in synchronous cyber classrooms[END_REF]. For example, [START_REF] Dirckinck-Holmfeld | Enhancing global and synchronous distance learning and teaching by using instant transcript and translation[END_REF] argued that synchronous speech transcription helps non-native English students to better understand lectures that are delivered in English. [START_REF] Shadiev | Review of speech-to-text recognition technology for enhancing learning[END_REF] mentioned that synchronous speech transcription can help (1) students with cognitive or physical disabilities who need additional support, have difficulties in reading, writing, or spelling by giving them a note-taking for all the teacher's speech (2) online students if the quality of audio is not good during communications, (3) non-native speakers, (4) students in traditional learning environment by allowing the educators to take a proactive, rather than a reactive approach to teach students with different learning styles and (5) students in collaborative academic activities.

Over the past decades of research, the field of automatic speech recognition has made considerable progress and many algorithms and techniques have been proposed and identified. However, a limit in an ASR is its inability to recognize unknown words which do not appear in the training data of the language model. A language model is one of the most important components in an ASR which assigns a probability to a sequence of words. The assigned probability guides the ASR to choose which sequences of words are possible to recognize. For the language model, the training data is in the form of word sequences from real speech, which were manually or automatically transcribed. It can also be from written text (e.g. web documents, news articles, books). This data is very hard to acquire and very expensive.

The aim of this work is to transcribe lecture speeches, which are very different from other types of audio content (e.g. news audio, documentary). Educational domain is characterized by the presence of different courses which are also given by different teachers. It is very hard to build a training data that allows a relevant use for all courses' topics. Since manually generating transcriptions is a challenging task, the classical idea is to use an existent language model and to adapt this language model with domain's data. Domain's data means data related to the addressed task. So, the first motivation in regard to ASR is to adapt the language model by supplementing the training data with some additional data from the domain. The main difficulties are: (1) which data to collect, ( 2) what is the source of data and (3) which queries should be used to collect these data.

In order to automatically enrich the course with relevant external documents, a segmentation step must be realized. This segmentation must be a thematic segmentation to enrich each part. The goal of thematic segmentation is to detect the borders of homogeneous zones at the level of the content. It will be necessary to characterize borders in order to link each thematic zone with pedagogical documents available in an external knowledge base. The first difficulty is to define the theme concept's notion in a transcription that is monothematic (the main object of the course). The second difficulty will come from the real-time aspect of the thematic segmentation which, to our knowledge, has never been discussed so far.

Design

Considered Pedagogical Situations

Though diverse pedagogical situations will be considered in the project, current development has focused on lecture situations, either in presence of students or in hybrid configuration. They have been considered first because the interactions in these contexts are more simple. Indeed, the teacher interacts with his audience as a whole and student-to-student interactions are very limited. Hybrid situations grant the possibility for remote students to attend to the lecture by watching the teacher through their interface. However, the teacher cannot visually evaluate if their learning activity takes place as planned. By considering these situations, a way to convey the information needed by the teacher must be found.

Design Process

The initial phase of the project was a study of practices among students and teachers in higher education. This first study allowed us to determinate which functionalities were pertinent to offer in this context, and guided the development of different tools based on the transcription service.

Our toolset is conceived by an iterative process, of which two cycles are described in this article. Our goal is to experiment this toolset in real conditions with actual end users, in order to explore the most relevant use for each technology in the considered contexts. This cyclic methodology lets us quickly assess users' behavior and feedback, as well as operate a fast re-engineering of the interfaces. In addition of lectures, the project will later focus on group work sessions, and post-session reuse of the generated content.

FIRST ITERATION

User Needs Assessment

The first step of the project consisted in studying existing practices before any instrumentation, as well as possible uses for tools planned to be developed in the project. A study based on a twofold approach, both quantitative and qualitative, was conducted [START_REF] Crétin-Pirolli | Projet pastel, performing automated speech transcription for enhancing learning, analyse des besoins[END_REF]. First, a questionnaire was built and distributed to 94 students engaged in a computer science master's degree. Sixty-two questionnaires were collected and exploited. Meanwhile, a semidirective interview was conducted with an associate professor giving lectures as part of the computer science master's program at Le Mans Université.

In regard to lectures, among students encountering difficulties, two thirds (66%) reported experiencing downtime at a point during class, as well as difficulties in understanding the class concepts (reported by 61% of them). Only a minority is using external resources given that 75% of students do not search for additional material such as definitions, texts or graphics on the Internet during class.

Considering the course instrumentation based on transcription, students are mainly interested by the possibility to communicate to the teacher which points were not understood. A majority of students also estimate the synchronous transcription is useful, in addition to the possibility of reading and exploiting the text after class.

For his part, the professor assumes the access to the transcribed speech is a good thing, albeit students should still be taking notes. He considers that notetaking is a part of the learning process. He does not want students to be distracted from their learning activity, so the resource recommendation system use should be occasional and brief, in order to prevent cognitive overload.

Needs and a priori acceptation lead to think that live transcription availability is relevant. However, synchronous availability of the transcription was not a central expectation in the eyes of students experiencing no hearing difficulties, or language barriers. Other tools based on this live transcription were relevant in regards of needs and existing practice, but exploitation of the transcription itself was considered more pertinent in asynchronous time. After a synthesis of the various needs and constraints, relevant functionalities which could be integrated using the technologies of interest of the project were listed. They are summarized in the following section.

User-Tool Interaction

The toolset is conceived to instrument usual classrooms and hybrid classes. It is designed to support learning all through the module duration. A set of functionalities has been selected, regardless of topic teached, but in regards of the needs assessment. They were chosen to foster learning during classes and during personal review of lessons.

During Classes

Students. Our goal was to allow remote students to access the same material as users present in class, mainly the teacher's explanation and a view of the slideshow projected. Despite their remote location, their possibilities of interactions with the teacher should be similar to the ones offered to the students present in class. All students would profit to be recommended resources which can be consulted during downtime, or saved for later use. This extra material illustrates the teacher's speech and is suggested at the right time to prevent students from having to navigate in a list of resources while they are already engaged in listening activities.

Teachers. The teachers need indicators (e.g. a pedagogical significant variable extracted from, or elaborated by the help of data automatically collected during the session) to compensate for the facts that some or all students are out of sight. Since teaching is also a very engaging activity, they need very synthetic aid to make decisions about the progression of the lecture in real-time. To engage and be able to get feedback from their audience, the tool must let them propose interactions for large auditoriums, in the fashion of clickers. Closed-ended questions are relevant, but teachers will also need the possibility to gather openended feedback, either concerning the lesson topic or the practical situation (e.g. "I see some students have trouble connecting to the class, what seems to be the issue?").

Between Classes

Students. After the lesson has ended, students need to access to the transcription, the slides and the resources offered to attendees. If they need to spend more time to study the material, they need the possibility to replay the video recorded during class. In this case they should be able to associate each moment of the video with the particular slide or resources proposed at this moment.

Teachers. Teachers need access to the interactions history to take into account the difficulty level of their class and the need for examples/details. An overview of frequently asked question would give the opportunity to answer frequent questions beforehand and generally taking remarks into account. They have to be able to retrieve and potentially export the recorded video, either as a whole or partially. Ideally, they would be able to divide the class recording to produce a series of video clips, each one detailing a theme. These clips could be suggested as complement to class, as learning material in a SPOC or in a context of flipped classroom.

Plugin Description

The toolset was built as a Moodle plugin, and can thus be accessed as a web page. This plugin use technologies of speech-to-text transcription and resource recommendation, each hosted on a separate server managed by the research team who developed it (see Fig. 1).

Speech-to-text technology.

The teacher is equipped with a lapel microphone (as well as filmed). This microphone sends the sound data to our server which organizes the coordinates streams. Sound is transcribed, and sentences are sent to the students as soon as the system evaluates that they were transcribed with enough confidence in regard to the system. This validation and upload takes less than ten seconds, which coincide with the video streaming latency.

Recommendation technology. The resources recommendation system takes as parameter the product of the transcription. In real time, this system recommends documents dealing with similar concepts. A set of interesting concepts can be extracted from the analysis of a sentence pronounced by the teacher, or a student's question. The system then fetches labelled resources either freely on the Internet or in a limited pool moderated by humans.

PASTEL as a Moodle Plugin.

Moodle is an open-source learning management system on which teachers can create online course and enroll their students. The toolset was developed as a Moodle plugin so that it can be integrated to any existing course platform, next to existent material. Students enrolled in a class can access our tool as they would access other activities or resources, and teachers can create a virtual classroom in the same way they would add a page to their course.

First Interface Version Design

In this section we present the different components implemented in the initial stage of the prototype. This first interface was elaborated by the research team in regards of detected needs, and implemented in order to gather user feedback [START_REF] Bettenfeld | Lecture instrumentation based on synchronous speech transcription[END_REF]. Given the varying needs and scenarios, students and teachers have access to their own version of the toolset.

Components of the Student Window.

The first version of the plugin provided two modes of content browsing: synchronous and asynchronous. In synchronous mode, slide display was coordinated in real-time with the projection in the classroom . In asynchronous mode, even though data was received and stored, the display kept focused on a given slide or point of the transcription, allowing the student to spend a longer time reading. On the left part of the screen, students could watch the teacher's video stream in real-time On the bottom left, the lecture slides were displayed and could be zoomed on. Underneath, they could submit their questions or their needs in a text field. The text was then sent to the teacher, and also analyzed by the resource recommendation system. The transcription display area was located in the center of the screen and was updated synchronously. A note-taking panel was associated to each paragraph of the transcription. Besides writing down their notes, students could notify a need for further information to the teacher. The material recommendation system also took in consideration this alert, and analyzed the concepts being explained or cited in the paragraph to provide relevant information. This system offered a set of links on the right part of the screen, evolving in real time, redirecting to external resources. At the top of the screen, a wide timeline of the lecture was displayed. It was a navigation tool designed to facilitate browsing slides and the transcription window quickly.

Components of the Teacher Window. This window is offered to be used while giving a lecture. As this task requires time and focus, functionalities provided are simple and quick to use, and the display components are kept concise. A portion of the screen gave visual feedback of the camera, displaying the same video stream as the students received (see A on Fig. 2). In addition to this page designed for the teacher, a second page displayed the slides destined to be projected to the audience (opened clicking F on Fig. 2). The teacher could navigate the slideshow using the mouse or the left and right arrow keys. The right part of the screen was dedicated to a text feed showing open-ended questions asked by the audience in real time (see E on Fig. 2). At the bottom of the screen was presented a set of indicators, two of them shown as bar graphs (C on Fig. 2). The first one displayed the proportion of alerts sent by students estimating the lecture's pace was too quick. The second bar displayed the proportion of students looking at material corresponding to a past moment. The last indicator was a table (D on Fig. 2) detailing the three slides on which the greater number of student expressed a difficulty. The resources recommended to students were displayed in real time, in a table (B on Fig. 2). The display was more concise as the teacher did not need to rate, or get a preview of the resources. Only the titles appeared, allowing the teacher to recommend a one particular resource to students, or to showcase it on the projected view.

User Test

The prototype was tested with students and a teacher in actual class context in order to evaluate usability, to list the benefits of using the system, to be able to analyze usage, and consequently to improve tools supporting the least satisfying tasks. One of the main objectives is to check that the amount of information provided to testers was not a source of cognitive overload. Providing text in real time during a learning activity can trigger such overload, but conversely saving lecture content for later use can relieve students' memory, as it was studied with podcasts [START_REF] Traphagan | Impact of class lecture webcasting on attendance and learning[END_REF]. The user test took place December 12, 2017 in hybrid configuration, both on Le Mans campus and on Nantes campus. The teacher was situated on the former location accompanied by three students while the remaining 7 students were located on the other campus. In both places, participants were filmed, their screen's activity was recorded and they agreed to take part in a post-experiment interview. Students' behavior during the lecture was mainly concentrated on the teacher and slides, and their use of the transcription and resources was more occasional. To navigate between different slides, students mostly used the previous and slide display buttons. Their use of the whole timeline to browse content was limited. During the interview, they expressed that manipulating the timeline requires additional thinking, albeit short, they could not invest time and concentration in. The students also made explicit their frustration about the substance of recommended material. The content was not adapted to their master's degree level: a lot of suggested information was considered trivial at this point of the cursus, such as definitions of basic concepts used during the lecture. No major problem was identified regarding the teacher interface.

SECOND ITERATION

Modifications of the Interface

Slideshow as core material.

In the first version of the prototype, an important section of the screen was dedicated to the transcription, and slides were displayed in a smaller section. The relative size did not coincide with the importance of each component's use. Students primarily zoomed on the slides and occasionally downsized them to peek at the translation. The size of each of these component has been switched in the interface offered by default to better coincide with the students' use (see Fig. 3).

Interface flexibility.

To prevent the students from hiding every tool whenever they need to zoom on the slide displayed, a system of panels was implemented. Using this system, students are able to select the tools they wish to keep on screen. The slideshow is displayed as big as possible depending on the remaining space. Ultimately it can be displayed at full size if every tool is hidden, for maximum readability. Students can hide and show every tool panel at any moment according to their needs. To support the use case of students reading the transcription comfortably, it is possible to expand the transcription area.

Navigation in asynchronously generated content. As the previous way of browsing the previous

Real Conditions Experiment

Protocol

The second version of the prototype was tested on March 22nd 2018, at Laval during an information and communication lecture. Beforehand, the teacher and students received a presentation of the toolset in order to facilitate later use. During the experiment, 13 students were located in the same room as the teacher and 16 others were located in a remote classroom. During the one-hour lecture, students were given access to the platform and their activity was monitored. Afterwards, the teacher and students were interviewed.

Results

Students were satisfied with the quality of the translation. They reported a frustration in regards of real-time communications as their open-ended questions were not correctly transmitted to the teacher due to network limitations, and the teacher's pedagogical scenario did not include time dedicated to review the audience's questions.

Students were confused by the number of tools available simultaneously. They are themselves aware that a comprehensive interface could be useful to users familiar with the system. Yet, as users still discovering the toolset, they are not comfortable enough to both concentrate on the lecture and use every functionality offered. They still expressed the need for more flexibility, particularly the possibility of changing the position of tools on screen. An analysis of 4 students activity recordings showed that changing the disposition of the interface, either for exploration purpose or comfort, was the action most performed by students (constituting 31,4% of all actions). Different students had different opinions concerning which element should be displayed at a large size at the core of the interface.

Students expressed very few questions or alerts, with a maximum of two questions per student during the experiment session. The note-taking activity is much more contrasted : 28.4% of actions belong to the note-taking category, but most of these were undertaken by two students. Other observed students only took notes twice each during the session. Consultation of the resources panel represents 19.6% of actions, almost equally divided between browsing the list and the actual consultation of links provided.

The teacher was not accustomed to monitoring the indicators on his computer screen in real time, and quickly abandoned this behavior to adopt a more usual posture. His evaluation of the students' activity was hence limited to the audience physically present in the classroom.

ON-GOING WORKS AND FUTURE PERSPECTIVES

Given the high number of class configurations and personal preferences towards practices, we plan to further develop the tool by adding new tools, and a more flexible interface.

The pedagogical situation considered next is the group work situation. The constraints in this situation are different from those considered in this paper. The communications between actors is more complex than during a lecture as all students and the teacher must be able to communicate. The activity of the students varies greatly depending on both the work expected and personal behaviour of each student. Additionally, the condition under which the transcription software operates are less controlled. Microphones, student voices and vocabulary used are very diverse, and a challenge for the system. The explored solution is a lesser important use of the transcription and an emphasis on student's activity indicators.

Currently, post-session access to course content is limited to an export of the text posted, generated and typed. The video recorded can also be watched separately. We plan to develop a browsing system allowing students to retrieve all this content on a single interface, synchronized together. Setting the replay to a particular point in time would display the corresponding pieces of transcription and resources. Due to the large amount of content, readability and browsing can become a tedious task; a thematic segmentation system will be implemented in order to divide the session into smaller chapters, easier to label and to browse individually.

As for the teacher, post-session use of traces and generated content is for now limited to the display of the contents of the Moodle database. Future work involves the conception of an editorial toolset allowing to retrieve and export more easily the content produced and the feedback collected during a particular lecture. This would allow the teacher to easily produce and share online pedagogical material relevant to a particular theme, greatly reducing the time investment required to set up a SPOC or a flipped classroom configuration.

As the prototype comes closer to a stable version, we plan to release it to the Moodle plugins directory and make it available to a greater public.

Figure 1 :

 1 Figure 1: Data streams exchanged between the different modules.

Figure 2 :

 2 Figure 2: Example of the teachers interface: (a) Webcam feedback (b) Recommended resources (c) Indicators (d) Detailed difficulty indicator (e) Open-ended question feed (f) Link opening the slideshow and link closing this window.

Figure 3 :

 3 Figure 3: New student interface, showing the slideshow (A), transcription (B), and resources (C).