domain: spi phys

text:
Romain Ceolato
email: romain.ceolato@onera.fr

Lucas Paulien

Justin B Maughan

Christopher M Sorensen

Matthew J Berg

Radiative properties of soot fractal superaggregates including backscattering and depolarization

Keywords: Lidar, soot, T-matrix, multispectral scattering, polarimetry

The modeling of radiative properties of soot particles can be addressed by a number of methods. Among them, the canonical Rayleigh-Debye-Gans (RDG) theory and the Superposition T-Matrix Method (STMM) are particularly well adapted to the task. Here, both RDG and STMM are used to model the radiative properties of two types of fractal aggregates indicative of soot relevant to lidar sensing. One type, a chain-like aggregate in morphology, corresponds to low-sooting flames and is well characterized by a single fractal dimension. The other type, a superaggregate, exhibits multiple fractal dimensions and is seen in heavily sooting flames. Radiative properties such as the differential scattering cross section, total scattering cross sections, backscattering cross-sections, and linear depolarization-ratio (LDR) are calculated for a range of wavelengths from 300-1100 nm while including the wavelength dependence of the soot refractive index. Of particular interest is that depolarization by a superaggregate is found to be approximately ten times larger than the depolarization by a chain-like aggregate and both exhibit a power-law with wavelength. Results for specific wavelengths of interest to lidar (355, 532, and 1064 nm) are tabulated for lidar applications.

Introduction

The incomplete combustion of hydrocarbon fuel, biomass, and other organic matter is known to produce carbonaceous soot aggregates, i.e., fractal-like clusters of nanometerscale primary particles of carbon [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. The primary particles, called monomers here, are quasi-spherical in shape, typically less than 50 nm in diameter, and aggregate to produce extended fractal structures up to hundreds of nanometers in size. Although their chemical composition is complex and variable due to aging processes in the atmosphere, soot aggregates generally absorb sunlight strongly. Absorption leads to positive atmospheric radiative forcing, i.e., heating, and largely accounts for the well-studied nature of this class of aerosol particle. Indeed, the long history of laboratory and field measurements and sophisticated computer simulations go far to provide a solid understanding for how soot absorbs and scatters light [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. What remains less known is how certain unique structures of soot influence the radiative properties including extinction, total scattering, and backscattering in addition to polarimetric properties such as the linear depolarization ratio (LDR).

In small-scale fires, soot formation can be modeled with diffusion-limited cluster aggregation (DLCA), which yields chain-like aggregates that exhibit a fractal dimension around D f ∼ 1.8. For such flames, the aggregates are relatively far apart and mechanically noninteracting; a condition known as cluster dilute in DLCA theory [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF]. Below, we refer to such aggregates as "canonical DLCA aggregates." In a heavily sooting flames however, aggregates become crowded enough to (mechanically) interact in the so-called cluster dense regime of DLCA. Aggregates forming in such conditions can be envisioned as aggregates of aggregates and display a hybrid-fractal character [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF][START_REF] Sorensen | Observation of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames[END_REF][START_REF] Kearney | Evidence of soot superaggregates in a turbulent pool fire[END_REF][START_REF] Kim | Occurance of soot aggregates with a fractal dimension of 2.6 in heavily sooting laminar diffusion flames[END_REF][START_REF] Dhaubhadel | Hybrid superaggregate morphology as a result of aggregatation in a cluster dense aerosol[END_REF][START_REF] Kim | Soot aggregates, superaggregates and gel-like networks in laminar diffusion flames[END_REF]. We call such aggregates "superaggregates" and they are notable because they exhibit multiple fractal dimensions and are larger in overall size than canonical DLCA aggregates.

Recent observations of soot emitted from large-scale wildfires in India and the United States show clear evidence that superaggregates are present in intense fires [START_REF] Chakrabarty | Soot superaggregates from flaming wildfires and their direct radiative forcing[END_REF]. In [START_REF] Chakrabarty | Soot superaggregates from flaming wildfires and their direct radiative forcing[END_REF], it is hypothesized that the large-scale of these fires produces turbulent flame structures wherein canonical DLCA aggregates become trapped in vortexes that increase the particle volumefraction leading to enhanced aggregate growth. A laboratory study of an inverted flame in [START_REF] Chakrabarty | Observation of superaggregates from a reversed gravity low-sooting flame[END_REF] and another involving a fuel-pool fire [START_REF] Kearney | Evidence of soot superaggregates in a turbulent pool fire[END_REF] appear to support this hypothesis. Considering the differences in morphology between these two types of soot aggregates, it is expected that discrepancies may exist between the true radiative properties of these particles and that calculated by simple, commonly used, approximate light-scattering models. Studying such discrepancies is of practical value as they may impact the interpretation of light scattering measurements in the field.

Polarization-sensitive lidar systems are now widely used to quantify the extent of liquid and solid-phase particles in clouds [START_REF] Roy | Retrieval of droplet-size density distribution from multiple-field-of-view cross-polarized lidar signals: Theory and experimental validation[END_REF][START_REF] Bissonnette | Multiple-scattering-based lidar retrieval: method and results of cloud probings[END_REF] but are also sensitive other particulate material such as soot aerosol particles. Nonzero values of the LDR can serve as an indicator that the particles are non-spherical in shape, whereas zero values are indicative of spherical particles, e.g., liquid drops. Freshly generated soot exhibits small LDR values, typically LDR 1% [START_REF] Burton | Observations of the spectral dependence of linear particle depolarization ratio of aerosols using NASA Langley airborne high spectral resolution lidar[END_REF][START_REF] Bescond | Numerical investigation of the possibility to determine the primary particle size of fractal aggregates by measuring light depolarization[END_REF], and higher values, LDR 20%, are seen for mineral dust [START_REF] Cao | Comparison of the relationships between lidar integrated backscattered light and accumulated depolarization ratios for linear and circular polarization for water droplets, fog oil, and dust[END_REF][START_REF] Sakai | Backscattering linear depolarization ratio measurements of mineral, sea-salt, and ammonium sulfate particles simulated in a laboratory chamber[END_REF], ice [START_REF] Guasta | Use of polarimetric lidar for the study of oriented ice plates in clouds[END_REF], marine particles [START_REF] Sakai | Backscattering linear depolarization ratio measurements of mineral, sea-salt, and ammonium sulfate particles simulated in a laboratory chamber[END_REF], and volcanic ash particles [START_REF] Groß | Dual-wavelength linear depolarization ratio of volcanic aerosols: Lidar measurements of the Eyjafjallajökull plume over Maisach, Germany[END_REF][START_REF] Boselli | First volcanic plume measurements by an elastic/raman lidar close to the Etna summit craters[END_REF]. Recently, unexpectedly large LDR between 12% and 25% have been measured for a region of the atmosphere expected to contain aged wildfire soot [START_REF] Haarig | Depolarization and lidar ratios at 355, 532, and 1064 nm and microphysical properties of aged tropospheric and stratospheric Canadian wildfire smoke[END_REF]. Other lidar field-measurements, tabulated in [START_REF] Kanngießer | Calculation of optical properties of light-absorbing carbon with weakly absorbing coating: A model with tunable transition from film-coating to spherical-shell coating[END_REF], show similar large LDR values for different soot-generating events.

In previous work [START_REF] Ceolato | Lidar cross-sections of soot fractal aggregates: Assessment of equivalent-sphere models[END_REF], we show the importance of accounting for the complex fractal morphology of soot aggregates for backscattering calculations; that is, in contrast to simple equivalent sphere models. The results enable estimation of the principal lidar parameters for large ensembles of canonical DLCA soot fractal aggregates [START_REF] Paulien | Lidar-relevant radiative properties of soot fractal aggregate ensembles[END_REF]. The intent of the present study is to investigate the radiative properties of superaggregates using the Superposition T-Matrix Method (STMM). The novelty here arises from the specific morphology of superaggregates, and in particular, a comparison of the STMM results to the popular Rayleigh-Debye-Gans (RDG) approximation. Moreover, we consider the complete angular dependence of the radiative properties in this comparison. An extension of our results to wavelengths and quantities of practical interest to lidar applications is also presented.

Soot aggregate models of the study

The first of the two types of aggregates investigated here, which we call the "canonical DLCA aggregate," is typical of soot forming in relatively low-sooting flames and is characterized by the statistical scaling law :

N m = k o (R g /R m ) D f (1)
where N m is the number of monomers in the aggregate, k o is a proportionality constant (the fractal prefactor), R g is the aggregate's radius of gyration -a measure of overall aggregate size -R m is the monomer radius, and D f is the fractal dimension [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. For such aggregates, typical values are R m ∼ 15 -20 nm, R g can range from a few times R m up to a micron, k o ∼ 1, and D f ∼ 1.8. In the light-scattering computations to follow, this aggregate will serve as a reference.

The other aggregate type considered, the superaggregate, can form in the latter stages of the DLCA process just before the onset of a gel [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF]. This implies that the nearest distance between two aggregates is approaching the value of the radius of gyration, so that clustering between the aggregates is possible. In other words, superaggregates are aggregates-of-aggregates in flames where the aggregate density is sufficiently high to cause mechanical interactions between aggregates [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF]. These superaggregates generally exhibit D f ∼ 1.8 at length scales 0.3 µm and a D f > 1.8 at larger length scales, typically D f ∼ 2.6 as shown in [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF].

The specific aggregates used here are generated by the DLCA model of [START_REF] Pierce | Aggregation in Colloids and Aerosols[END_REF] and exhibit morphology approximately representative of canonical DLCA aggregates and superaggregates. Both types are shown as insets in Fig. 1. With knowledge of the position of each monomer, the radius of gyration of either aggregate can be calculated as:

R g = 1 N m Nm i=1 x 2 i (2)
where x i is the distance from the i th monomer center to the center of mass of the aggregate.

In the following, the radius of gyration for the canonical DLCA aggregate is R g,c = 220 nm and for the superaggregate is R g,s = 593 nm while the monomer radius for either aggregate is R m = 20 nm. To further characterize the morphology of the model aggregates, the monomer pair correlation function g(r) is calculated, which is the probability of finding another monomer at a distance r from a given monomer [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. The Fourier transform of g(r) into reciprocal space, i.e., q-space, is the structure factor of the aggregate, S(q) [Eq. ( 6) below]. As discussed in [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF], the fractal-like character of the pair correlation function is given by g(r) ∼ r (D f -d) where d = 3 is the spatial dimension. When Fourier transformed, this yields the classic result that S(q) ∼ q -D f . Figure 1 shows these quantities along with several examples of specific power-law trends. In Fig. 1(a), g(r) is presented in two ways: The main plot shows the function plotted in terms of r/R m while the inset plot shows the same curves plotted as a function of r/R g,(c,s) where R g,(c,s) is the radius of gyration appropriate to the given aggregate. In the inset plot, we show two power laws corresponding to fractal dimensions D f = 1.6 and D f = 2.1 for the canonical DLCA aggregate and superaggregate, respectively. The range of R g,(c,s) is also slightly expanded in the inset to show further detail. Notice that the values of D f here differ from the D f = 1.8 and D f = 2.6 expected from the literature [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF][START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF]. Next, consider Fig. 1(b) and 1(c), which shows S(q) plotted as a function of qR g,(c,s) for each aggregate. Here again we see that multiple values for D f apply to the the curves. In the case of the canonical DLCA aggregate, D f is relatively well defined as being between D f = 1.6 and D f = 1.8. For the superaggregate, we find a different behavior, namely that two distinctly different power-law regions are present. The D f = 1.8 and D f = 2.6 power laws characteristic to superaggregates are not exhibited here because of the relatively small aggregate size (necessitated by limited computational resources) compared those in [START_REF] Sorensen | The sol to gel transition in irreversible particulate systems[END_REF]. Nevertheless, the salient point is that the superaggregate exhibits multiple power-law regimes in S(q) not seen for the canonical DLCA aggregate, and thus, the radiative properties should be affected. We investigate this question below.

Scattering properties of soot superaggregates

Modeling of the radiative properties of soot particles can be addressed by a number of methods. Among them are the approximate RDG theory and the numerically exact STMM [START_REF] Mackowski | Calculation of the T matrix and the scattering matrix for ensembles of spheres[END_REF]. The former uses scaling arguments and the reciprocal space expression of the aggregate structure in order to approximate both angle-integrated and angle-dependent radiative properties. Limitations arise from the inability of the RDG to describe internal multiple scattering among the monomers [START_REF] Yon | Effects of multiple scattering on radiative properties of soot fractal aggregates[END_REF]. The STMM method can also provide the radiative properties of soot aggregates and is an exact solution of the Maxwell equations for any assembly of (non-intersecting) spherical particles. In the following, the STMM code of [START_REF] Mackowski | Calculation of the T matrix and the scattering matrix for ensembles of spheres[END_REF] is used and the results compared to the RDG approximation.

Let us fist consider an aggregate illuminated by an unpolarized incident plane wave propagating along the positive z-axis of the laboratory coordinate system. The scattering angle θ is measured between the z-axis and the vector r, which is the observation point in the aggregate's far-field zone. Except for the study of linear depolarization below, the Stokes column vector for the unpolarized incident light is I inc = I inc , 0, 0, 0

T where I inc is the intensity of the wave. The STMM is applied to each aggregate shown in Fig. 1 to calculate the elements of the far-field scattering matrix F, which transforms I inc into the scattered wave's Stokes vector I sca at r. The simulation first considers a single wavelength λ in the spectral range with corresponding m(λ) and then calculates the matrix elements. Next, the elements are analytically averaged over random aggregate-orientations and the process is then looped over wavelength in steps of ∆λ = 10 nm to generate the orientation-averaged scattering matrix. Because of the orientation average, either aggregate effectively constitutes a macroscopic isotropic and mirror-symmetric scattering medium. In such cases, F depends only on the polar scattering angle and simplifies from 14 to six independent elements [START_REF] Mishchenko | Scattering, Absorption, and Emission of Light by Small Particles[END_REF].

In the backward direction specifically, the matrix simplifies further to only four elements as:

F (θ = π) =     F 11 (π) 0 0 0 0 F 22 (π) 0 0 0 0 F 33 (π) 0 0 0 0 F 44 (π)     (3)
The focus below is on each aggregate's differential scattering cross section, total cross sections, and LDR, which are given in terms of elements F. The absorption and extinction cross sections C abs and C ext , respectively, are also provided by the the STMM calculations. We note that the broadband spectral nature of the study requires that the wavelength dependence of the refractive index m of the soot material be taken into account. From laboratory measurements and the Kramers-Krönig relation, [START_REF] Chang | Determination of the wavelength dependence of refractive indices of flame soot[END_REF] describes a soot refractive index dispersion, m(λ), which is used here over the wavelength range λ = 300 -1100 nm.

Differential scattering cross section

The differential scattering cross section, or phase function for unpolarized light, is expressed for incident unpolarized light as [START_REF] Mishchenko | Electromagnetic Scattering by Particles and Particle Groups: An Introduction[END_REF]:

dC STMM sca dΩ (θ) = F 11 (θ) k 2 . ( 4
)
As explained in [START_REF] Sorensen | Light scattering and absorption by fractal aggregates including soot[END_REF], the RDG description for this quantity derives from two approximations. First, that the soot monomers be in the Rayleigh-scattering size range, i.e., much smaller than the wavelength. The smallest wavelength considered here is λ = 300 nm and the monomer radius is R m = 20 nm. As such, λ R m and this condition is well satisfied. Second, electromagnetic coupling, or "multiple scattering," between monomers is assumed negligible. When both conditions hold, the RDG description for the scattered intensity is the product of the (Rayleigh) differential scattering cross section for a single monomer, the number of monomers squared, and the structure factor, which is the Fourier transform of the aggregate structure [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF][START_REF] Sorensen | Light scattering and absorption by fractal aggregates including soot[END_REF], i.e.,

dC RDG sca dΩ (q) = N 2 m k 4 R 6 m m 2 -1 m 2 + 2 2 1 + cos 2 (θ) 2 S(q). (5)
where q(θ) = 2k sin (θ/2) is the scattering wave vector magnitude and S(q) is related to the pair correlation function g(r) [START_REF] Heinson | A three parameter description of the structure of diffusion limited cluster fractal aggregates[END_REF] as:

S(q) = 1 N m 1 + 4π ∞ 0 g(r)r 2 sin(qr) qr dr (6)
Note in Eq. ( 5) that the term [1 + cos 2 (θ)]/2, expressed in terms of the scattering angle θ, is present due to the unpolarized nature of the incident light. Equation ( 6) assumes orientation averaging of the aggregate, and because q = 0 in the forward direction (θ = 0), S(0) = 1.

Inspired by recent work in [START_REF] Sorensen | Light scattering and absorption by fractal aggregates including soot[END_REF], we consider specifically how the STMM results compare to the RDG approximation in the forward-scattering direction. That is, we divide the angular scattering of Eq. ( 4) from STMM by Eq. ( 5) with q = 0 and S(0) = 1. The resulting quantity is denoted dC STMM sca /dΩ Rayleigh normalized. Also, the absorption cross section in the RDG approximation C RDG abs is given in [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF].

Scattering cross section

In the STMM computation, the scattering cross-section is computed as:

C STMM sca = 2π π 0 dC STMM sca dΩ (θ) sin(θ)dθ, (7)
while in the RDG approximation it is given by [START_REF] Dobbins | Absorption and scattering of light by polydisperse aggregates[END_REF] as:

C (D f ) sca = N 2 m k 4 R 6 m m 2 -1 m 2 + 2 2 G(kR g,(c,s) , D f ) (8)
where

G(kR g , D f ) = 1 + 4 3D f kR g,(c,s) 2 - D f 2 (9)
and the notation C

(D f )
sca is meant to differentiate Eq. ( 8) from C RDG sca . In evaluation of Eq. ( 8) below, a value of D f = 1.8 is used for the canonical DLCA aggregate and D f = 2.6 is used for the superaggregate. The reader should recall from Fig. 1, however, that D f = 2.6 is one of many choices that could be used for the superaggregate.

Results and discussion

We now investigate the light-scattering properties of the model soot aggregates. Figure 2 shows the STMM-calculated differential cross section for both aggregates, i.e., Eq. ( 4), for a selection of wavelengths as labeled. The curves are plotted in q-space, which means a log-log scale is used and the independent variable is the dimensionless parameter qR g,(c,s) . Note that the radius of gyration is different for the two aggregates, hence R g,c is used for the chain-like aggregate and R g,s is used for the superaggregate. The dashed lines plot Eq. ( 5) in Fig. 2(a) and 2(c) and show the typical structure including a constant-scattering forward lobe when qR g,(c,s) < 1, a Guinier regime around qR g,(c,s) ∼ 1, and a power-law (or laws) region reflecting the fractal character when qR g,(c,s) > 1. Because the incident light is unpolarized, the scattering curves exhibit a polarization "dip" near θ = 90 • due to the Rayleigh-like scattering from the monomers. This brings the curves down from the standard q -D f power-law predicted by the structure factor discussed in [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. The power-law form of the scattering is further revealed in Fig. 2 by normalizing each STMM curve by the RDG approximated forward-scattering, Eq. ( 5) with q = 0, and the result is shown in Fig. 2(b) and Fig. 2(d) for both aggregates. For the canonical DLCA aggregate, Fig. 2(b), this normalization brings the different-wavelength curves together within approximately 10% spread from one, revealing that Eq. ( 5) does well to describe the forwardscattered intensity for this aggregate. A similar result is seen in Fig. 2(d), except with a greater departure from one. Thus, Eq. ( 5) is less effective at describing the superaggregate's forward scattering. A different comparison of the STMM angular scattering with RDG is shown in Fig. 3. Here, specific scattering angles ranging from θ = 0 • to θ = 180 • are selected and the Rayleigh normalized differential scattering cross sections are then plotted as a function of λ. In accord with Fig. 2(b) and 2(c), good agreement with RDG is seen for forward scattering, followed by increasing degrees of departure from RDG for increasing scattering angles. However, the departure appears comparatively invariant with scattering angle beyond approximately θ 80 • , as seen by the grouping of the curves. Moreover, the trend for these angles show power laws in λ as indicated.

The wavelength dependence of the scattering and extinction cross sections for each type of aggregate are shown in Fig. 4. In general, the canonical DLCA aggregate displays good agreement with RDG across the spectral range, whereas the superaggregate departs more, especially for the UV/VIS region. Greater disagreement between STMM and RDG is seen as λ decreases. We suspect that this behavior is likely a consequence of refraction within the monomers. That is, as λ decreases, the field within a given monomer will increasingly depart from the uniform-field approximation underpinning RDG theory. Notice that this interpretation suggests that the disagreement (with decreasing λ) should occur regardless of the aggregate morphology. Yet, the disagreement is stronger in Fig. 4 for the superaggreagate, which may be due to the larger number of monomers. There may also be effects due to electromagnetic coupling between mononmers, which would be expected to depend on the aggregate structure. In any case, we conclude that RDG should be used with care, especially for large aggregates.

Backscattering and depolarization

Lidar systems enable the vertical profiling of atmospheric aerosols through the measurement of lidar-relevant radiative properties including backscattering and polarization, e.g., see [START_REF] Comerón | Current research in lidar technology used for the remote sensing of atmospheric aerosols[END_REF]. Thus, to connect our study to a topic of practical interest, we extend our STMM calculations to these lidar quantities, namely the backscattering cross section and the linear depolarization ratio.

Backscattering cross section

The differential scattering cross section evaluated in the backscattering direction, θ = 180 • or q = 2k, is called the backscattering cross section below, and is typically denoted C π for ease of notation. In the STMM computation, C π is expressed as:

C STMM π = dC STMM sca dΩ (θ = π) = F 11 (π) k 2 , (10)
while in the RDG approximations it is given in terms of Eq. ( 4) as

C RDG π = dC Ray sca dΩ (q = 2k). ( 11
)

Linear depolarization ratio

The linear depolarization ratio (LDR) expresses the ratio of the crossed-polarized to parallel-polarized component of the backscattered intensity. That is, LDR = 0 means that the backscattered light is (linearly) polarized in the same direction as the incident light. An LDR = 1 means that two orthogonal polarizations have equal intensity. This quantity is useful because it is sensitive to the shape of the aerosol particles [START_REF] Gimmestad | Reexamination of depolarization in lidar measurements[END_REF][START_REF] Mishchenko | Scattering, Absorption, and Emission of Light by Small Particles[END_REF][START_REF] Tesche | Profiling of saharan dust and biomass burning smoke with multiwavelength polarization raman lidar at cape verde[END_REF][START_REF] David | UV polarization lidar for remote sensing new particles formation in the atmosphere[END_REF] and is defined from F above as [START_REF] Paulien | Lidar-relevant radiative properties of soot fractal aggregate ensembles[END_REF] LDR = F 11 (π) -F 22 (π)

F 11 (π) + F 22 (π) . ( 12
)
Note that although the angular scattering behavior studied in Fig. 2 and Fig. 3 are for unpolarized incident light, the scattering matrix also describes backscattering of parallel or perpendicular incident light leading to the depolarization measure, Eq. ( 12).

Results and discussion

The LDR and C π are shown in Fig. 5 for both aggregate types. By definition, RDG yields an LDR of zero because each monomer is treated in the independent-scattering limit (more explanation below). Thus, there is no RDG comparison to be made here. Both aggregates exhibit an approximate λ -2 power law, with the superaggregate holding less well to this power-law. The same wavelength trend is also seen for C π . Values of the LDR for the canonical DLCA aggregate vary from 0.1 -2%, which is expected for soot of this fractal dimension. However, the superaggregate shows roughly an order of magnitude larger LDR values across the wavelength range. Finally, specific values from these plots are reported in Table 1 for commonly used lidar wavelengths. Of specific interest for lidar is C π . The canonical DLCA aggregate shows good agreement with RDG, Eq. ( 11), and an overall λ -2 power law is exhibited toward the IR spectral region. The agreement between RDG with STMM degrades for the superaggregate and the powerlaw trends are different. The departure from RDG is more pronounced for superaggregates because the monomers are, on average, more coupled for a given wavelength, i.e. more monomers are in each other's near-field zone. In addition to this, at smaller wavelengths, there is more refraction within a given monomer and this causes further disagreement with the RDG approximation.

The LDR is zero in the RDG approximation because coupling is neglected and the spherical monomers thus cannot alter the polarization state of the incident light upon scattering into the backward direction. Indeed, the uniform internal-field approximation in RDG means that spherical monomers cannot alter the (linear) polarization state of the incident light upon scattering into any direction [START_REF] Berg | Reflection symmetry of a sphere's internal field and its consequences on scattering: a microphysical approach[END_REF]. In this sense, one can regard nonzero values of the LDR as inherent consequences of coupling between monomers because coupling spoils the uniform internal-field approximation. Indeed, the λ -2 dependence in Fig. 5 can be explained qualitatively by dipole-dipole multiple scattering between monomers in the intermediate-field range via Eq. (81b) in [START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF]. It is understandable then that the superaggregate exhibits larger values of LDR than the canonical DLCA aggregate due to its more compact structure and larger size, hence more multiple scattering. Yet, the coating and restructuring of soot as it ages in the atmosphere further complicates attempts to attribute unexpectedly large LDR values to a given cause. A variety of studies have considered the effects of coating, such as [START_REF] Mishchenko | Linear depolarization of lidar returns by aged smoke particles[END_REF][START_REF] Liu | Scattering and radiative properties of morphologically complex carbonaceous aerosols: A systematic modeling study[END_REF][START_REF] Kanngießer | Calculation of optical properties of light-absorbing carbon with weakly absorbing coating: A model with tunable transition from film-coating to spherical-shell coating[END_REF] for example, and also find large values for the LDR that may exceed 10%.

There are a variety of explanations for such large LDR measurements. For example, the soot is usually aged on the order of days, and thus, has undergone significant chemical change, morphological restructuring, coating, and possibly coagulation with other classes of aerosol particles [START_REF] Liu | Scattering and radiative properties of morphologically complex carbonaceous aerosols: A systematic modeling study[END_REF]. Mineral dust, which often exhibits large LDR, and other nonorganic particulate matter may coexist with the soot at the measurement site. All of these factors can help explain the unexpectedly large LDR observations. A number of studies [START_REF] Mishchenko | Linear depolarization of lidar returns by aged smoke particles[END_REF][START_REF] Kahnert | Models for integrated and differential scattering optical properties of encapsulated light absorbing carbon aggregates[END_REF][START_REF] Liu | Scattering and radiative properties of morphologically complex carbonaceous aerosols: A systematic modeling study[END_REF][START_REF] Kanngießer | Calculation of optical properties of light-absorbing carbon with weakly absorbing coating: A model with tunable transition from film-coating to spherical-shell coating[END_REF] show how various modifications to pristine soot affect the LDR and related radiative properties, although many of the modifications employed vary substantially from one study to another. Yet, the success of these computational studies motivates further investigation.

Conclusion

This study reports the radiative properties of two specific types of soot fractal aggregates using both the conventional Rayleigh-Debye-Gans (RDG) approximation and the Superposition T-Matrix Method (STMM). Although the RDG is popular for light-scattering calculations due to its simplicity, we recall that the approximation neglects electromagnetic coupling, or multiple scattering, between monomers. The STMM, however, is a numerically exact solution in so far as the morphology and refractive index of the aggregate is accurately represented. Thus, the comparison above between STMM-calculated radiative properties and the RDG approximation offer a glimpse at the effects of coupling. Perhaps the most striking example is the LDR of Fig. 5. Other findings of this study are:

• The wavelength dependence of the soot refractive index from λ = 300 -1100 nm does not spoil the power-law functionality of the differential scattering cross section in terms of qR g,(c,s) expected from the RDG approximation, at least within 10% deviation.

• The scattering, extinction, and backscattering cross sections exhibit power-law functionality in terms of λ. Good agreement with the RDG approximation is found for the chain-like aggregate, while poorer agreement is seen for the superaggregate.

• The LDR for the canonical DLCA aggregate is within the range 0.1 -2% expected for low-sooting flames across the wavelengths considered, while the superaggregate exhibits approximately an order of magnitude greater LDR.

• Both aggregate types exhibit an approximate λ -2 functionality for the LDR, indicating stronger depolarization toward the UV/VIS end of the wavelength range.

Figure 1 :

 1 Figure1: Pair correlation function g(r) and structure factor S(q) for the canonical DLCA aggregate and superaggregate. The inset in (a), where g(r) is presented, shows example power-law trends following the scaling behavior described in[START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF] and in the text. Images of the aggregates themselves are displayed with the S(q) plots in (b) and (c). Note from (b) that the canonical DLCA aggregate is approximately characterized by a limited range of fractal dimensions D f = 1.6 to 1.8 while the superaggregate displays distinct power law regimes indicating multiple fractal dimensions.

Figure 2 :

 2 Figure 2: Angular dependence of the STMM and RDG differential scattering cross sections for the canonical DLCA aggregate and superaggregate illuminated by unpolarized light. The RDG curves, Eq. (5), are shown as dashed lines in (a). Both the STMM and RDG curves are plotted as a function of the dimensionless parameter qR g,c in log-log scale. The different curves correspond to different λ as labeled and markers indicate the "dip" caused by the unpolarized incident light. In plot (b) the same scattering curves have been divided by the RDG approximation for the forward scattered intensity, i.e., Eq. (5) when q = 0. Here one can see the quasi-universality of the scattering behavior as all curves group closely together. Plots (c) and (d) show the same results for the superaggregate as a function of qR g,s . Values for m(λ) for the specific wavelengths used are tabulated in (c).

Figure 3 :

 3 Figure 3: Rayleigh-Debye-Gans normalized differential scattering cross section as a function of λ. Each curve corresponds to a specific scattering angle as labeled. Note the quasi-invariance of the curves with scattering angle beyond θ ≈ 80 • .

Figure 4 :

 4 Figure 4: The STMM-calculated scattering and extinction cross sections, C sca and C ext , respectively, as a function of λ for the canonical DLCA aggregate and superaggregate. Comparisons to the RDG approximations for the cross sections are shown. Note that RDG agrees well with the STMM calculations for the canonical DLCA aggregate but less well for the superaggregate.

Figure 5 :

 5 Figure 5: Wavelength dependence of the linear depolarization ratio, LDR, and backscattering cross section, C π , as calculated by STMM and RDG for the canonical DLCA aggregate and superaggregate. Note that both quantities generally follow a λ -2 power law for a portion of the spectral range.

  canonical DLCA aggregate λ [nm] C sca [nm

Table 1 :

 1 2 sr -1 ] C ext [nm 2 ] C π [nm 2 ] LDR [%] C sca [nm 2 sr -1 ] C ext [nm 2 ] C π [nm 2 ] LDR [%]Values of the cross sections, C sca , C ext , C π , and the LDR calculated with the STMM for the canonical DLCA aggregate and superaggregate at common lidar wavelengths.

	355	14277	66105	286.53	1.970
	532	5139.5	35030	155.82	0.8190
	1064	790.70	14602	41.965	0.2130

Acknowledgements

The authors gratefully acknowledge funding from ONERA and the PROMETE project, the doctoral school Sciences mécaniques et énergétiques, matériaux et géosciences (SMEMAG), the National Science Foundation (1665456) and the Air Force Office of Scientific Research (FA9550-19-1-0078). The authors are grateful for helpful discussions with Jérôme Yon, Nicolas Riviere, Franck Enguehard, Alain Miffre, Michael Mishchenko, Anouar Soufiani, Laurent Soucasse, and Daniel Mackowski.


P Fontaine

O Acosta

F Riet

J Castelli

K Gnep

A Simon

A Depeursinge

R De Crevoisier

PO-0965 How to find the best radiomics features for prediction of overall survival in SBRT for HCC?

Purpose or Objective

One of the major issues in radiomics is the very large amount of tested extracted features, compared to the often-reduced sample size and the low number of events. Reduction of dimensionality may be therefore an important preliminary step to improve the prediction capability of the predictive models. The aims of the study were:

-to propose methods for reducing redundancy by selecting the more informative features from -multimodal images; -to evaluate and compare the prediction capability of the models when using these methods. The considered example was MRI based radiomics to predict overall survival after SBRT for hepatocellular carcinoma (HCC).

Material and Methods

Eighty-one patients underwent SBRT for inoperable HCC. For each patient, 7 modalities of MR images were acquired. A total of 273 features were extracted from manually delineated tumours belonging to 4 radiomics categories (geometrical, first order, gradient-based and second order) in each modality. As we follow the workflow [Figure 1] , a similarity measure based on Spearman correlation was computed across the features. Four methods for feature selection were then assessed namely three unsupervised (K-means, Hierarchical clustering (HC) and Affinity propagation (AP)) and a supervised (mRMR) clustering and compared random selection (RS) and no selection (using all the features). Affinity propagation clustering yields a set of exemplars which better represented each cluster. Finally, in order to assess the predictive capabilities of each one of the feature selection method, a random forest classifier was trained and tested via a stratified-K-fold (K=19 the occurrence of decease event) cross-validation. This process was repeated 1000 times. Feature importance as assessed by aggregation of the performance at each try. The performance is evaluated by computing the precision (True positive / True positive + True negative) of prediction.

Results

The table displays the selected predictive feature depending on the selection methods.

Conclusion

A framework for feature selection in a radiomics workflow is presented. Unsupervised methods allow to cluster together groups of features increasing the prediction capabilities and reducing redundancy. AP outperforms the other features selection method suggesting the use of the exemplars as representative feature of each cluster.

  Unsupervised clustering algorithms allowed to select a non-redundant set of features able to significantly better predict HCC overall survival [Exemplars from AP: Precision= 0.76 ± 0.01, (p-value < 0.001)], in comparison to the other methods [All features: Precision = 0.73 ± 0.001; RS from all features : Precision = 0.71 ± 0.3 ; RS from K-means clustering : Precision = 0.715 ± 0.1; RS from HC: Precision = 0.74 ± 0.02; RS from AP clustering: Precision = 0.735 ± 0.01 and exemplars from mRMR: Precision = 0.735 ± 0.01] . The most reproducible predictive features are related with the shape of the tumour [Figure 2] .


Ayman Assi

Fares Yared Ziad Bakouny

Aren Joeffroy Otayek

Joe Bizdikian

Ismat Ghanem

Gaby Kreichati

Hélène Pillet

Xavier Bonnet

Wafa Skalli

F Yared

A J Bizdikian

How do postural parameters vary during walking in asymptomatic adults? A registration technique of subject-specific 3D skeletal reconstruction during gait

Keywords:

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

Introduction

Postural skeletal alignment is altered with age due to intervertebral disc and joint degeneration, consequently affecting quality of life (QoL) and activities of daily living, such as gait. Postural alignment parameters of the spine, pelvis, hips and lower limbs, measured on static standing radiographs, have been widely studied in asymptomatic subjects and subjects affected by various pathologies. However, while most of these parameters are positional and could vary during gait, there are currently no studies investigating how they are modified during walking.

Research question

How do postural alignment parameters vary during gait in asymptomatic adults?

3. Methods 85 asymptomatic adults (age: 29 ± 4.9 years [18-59]) underwent gait analysis using the modified Davis protocol [1] with additional markers on the thighs, shanks and C7 spinous process. Subjects then underwent low dose biplanar X-rays in standing position with subsequent subject-specific 3D reconstructions of the spine, pelvis, and lower limbs and calculation of the following 3D radiological postural parameters in the static position: pelvic tilt (PT), sagittal vertical axis (SVA), acetabular abduction (Acet_Abduction), anteversion (Acet_-Anteversion) and tilt (Acet_Tilt) as well as the anterior (Ant_Cov) and posterior (Post_Cov) coverage of the femoral head by the acetabulum [2,3]. The 3D bones were registered on each frame of the gait cycle [4] (Fig. 1). A new technique developed for the purpose of this study, utilizing finite element modeling, was used to reduce soft tissue artefacts. The postural parameters were then computed during the gait cycle, using the 3D registered bones, at each time frame: means and ranges of motion (ROM) were calculated.

Results

Some of the parameters exhibited large ROM during the gait cycle such as: SVA (23.3 ± 7.2 mm), Acet_tilt (17.8 ± 7.3°), Acet_Abduction (10.9 ± 3.5°), and coverage of the femoral head anteriorly and posteriorly (9.3 ± 1.7%). Acet_Anteversion had a lower ROM (3.4 ± 1°). Fig. 2 shows the variation of the postural parameters during the gait cycle as corridors of normality (mean ± SD) with the value of the static radiological parameter represented for comparison. During walking, all the parameters varied around their standing posture value, except for the SVA which was shifted anteriorly (25 mm) and PT which was slightly increased by 2°.

Discussion

  This is the first study to describe variation of postural parameters https://doi.org/10.1016/j.gaitpost.2018.06.016 ⁎ Corresponding author at: Faculty of Medicine, University of Saint-Joseph, Laboratory of Biomechanics and Medical Imaging, Beirut, Lebanon.E-mail address: ayman.assi@gmail.com (A. Assi).

  Gait & Posture xxx (xxxx) xxx-xxx 0966-6362/ © 2018 Elsevier B.V. All rights reserved. Please cite this article as: ASSI, A., Gait & Posture (2018), https://doi.org/10.1016/j.gaitpost.2018.06.016 during walking. The variation of acetabular parameters during gait is indicative of the potential change of acetabular cup positioning in total hip replacement during walking. Furthermore, the anterior shift and large ROM of SVA during walking emphasizes the importance of considering the dynamic variation induced by gait when planning surgical re-alignment of the spine. The findings of this study could also provide quantitative data for gait biomechanical modelling.

Fig. 1 .

 1 Fig. 1. Registration technique of 3D skeletal reconstruction from biplanar x-rays with gait analysis.

Fig. 2 .

 2 Fig. 2. Variation of 3D postural parameters during gait.


Georges Mjaess

Chris Labaki

Aya Karam

Aren Ziad Bakouny

Joe Bizdikian

Fares Yared Joeffroy Otayek

Waffa Skalli

Ismat Ghanem

Ayman Assi

A J Bizdikian

F Yared

How does the variation of the 3D orientation of the acetabulum during walking influence hip kinematics?

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

Introduction

Acetabular cup orientation is crucial for total hip arthroplasty (THA), and its malpositioning could lead to impingement and dislocation [1]. Acetabular cup orientation currently relies on static 3D hip parameters [2] and was shown to be related to changes in pelvic positioning [3]. While pelvic position varies during walking, it is still unknown how dynamic variation of hip orientation during gait could influence hip kinematics. E-mail address: gmjaess@gmail.com (A. Assi).

⁎ Corresponding author at: Faculty of Medicine-University of Saint-Joseph, Laboratory of Biomechanics and Medical Imaging, Beirut, Lebanon.

Research question

How do dynamic variations of 3D skeletal hip parameters during gait influence hip kinematics in asymptomatic subjects?

3. Methods 77 asymptomatic subjects (age: 28 ± 10years [18-38], 32F) underwent 3D gait analysis [4], with additional markers on thighs and shanks, from which the means, maxima, minima, and ROM of hip kinematics were extracted. Then, subjects underwent full-body biplanar X-rays with markers still in place. Full body 3D reconstructions were obtained for the spine, pelvis and lower limbs. The 3D bones were registered at each frame of the gait cycle [5]. A new technique developed for this study, utilizing finite element modelling, was used to reduce soft tissue artefacts. Then, 3D hip skeletal parameters (acetabular orientations, coverage of the femoral head by the acetabulum and sector angles) were computed during the gait cycle, using the 3D registered bones, at each time frame (Fig. 1): means, maxima, minima, and ROM were extracted. In order to assess the influence of demographic (age, sex, weight, and height) and dynamic variations of 3D skeletal hip parameters on hip kinematics during gait, a univariate analysis (Pearson's correlation test) followed by a multivariate analysis (stepwise multiple linear regression) were computed; the dependent variables were hip kinematics, while the independent variables were demographic and hip dynamic 3D skeletal parameters.

Results

ROM

Discussion

This is the first study to evaluate the effect of variation of 3D skeletal hip parameters during gait on hip kinematics (Figs. 2 and3). A Fig. 2. Determinants of hip flexion/extension during gait among demographic parameters and 3D hip radiological parameters computed during gait. larger ROM of the posterior coverage and a smaller ROM of the anterior coverage of the femoral head by the acetabulum during gait seem to be related to a larger mobility of the hip in the sagittal plane. A larger ROM of the acetabular abduction during gait seem to be related to a larger mobility of the hip in both the frontal and axial planes. Planning acetabular cup orientation in THA should take into consideration these dynamic variations of the acetabulum during gait in order to ensure a full mobility of the hip during walking.

Fig. 1 .

 1 Fig. 1. 3D hip skeletal parameters computed during the gait cycle.

  hip flexion/extension was determined by ROM_Posterior_Coverage (r = 0.459, p < 0.001). Maximum hip extension was determined (R2 = 0.226) by weight (β = 0.374, p = 0.001), and ROM_Anterior_Coverage (β = -0.241, p = 0.03). Mean hip flexion/extension was determined by weight (r = 0.446, p = 0.003). ROM hip abduction/adduction was determined (R2 = 0.507) by ROM_Acetabular_Abduction (β = 0.668, p < 0.001) and ROM_Vertical_Center_Edge_Angle (β = 0.2, p = 0.022). ROM hip internal/external rotation was determined by ROM_Posterior_Acetabular_Sector_Angle (r = 0.347, p = 0.028). Mean hip internal/external rotation was determined by ROM_Acetabular_Abduction (r = 0.404, p = 0.005).

Fig. 3 .

 3 Fig. 3. Determinants of hip abduction/adduction and hip internal/external rotation during gait among demographic parameters and 3D hip radiological parameters computed during gait.


Mateus Almeida Barbosa
email: mateus.almeida-barbosa@grenoble-inp.fr

Kübra Gül
email: kubra.gul@grenoble-inp.fr

Antoneta Iuliana Bratcu
email: antoneta.bratcu@gipsa-lab.grenoble-inp.fr

Iulian Munteanu
email: iulian.munteanu@gipsa-lab.grenoble-inp.fr

Management of a photovoltaic-battery-based microgrid in a prosumer context

Keywords: prosumer, photovoltaic (PV), DC microgrid, Energy Management System (EMS), energy storage system

Prosumers are households that are both producers and consumers of electricity. In this paper, an Energy Management System (EMS) for a DC-microgrid-based prosumer is designed and simulated, whose aim is to coordinate operation of a photovoltaic (PV) system and a battery for supplying a DC load; interaction with the utility grid -modelled as an infinitepower source -is minimized, such that power is either injected or drawn only when strictly necessary. The PV system and the battery are paralleled on a common DC-link by means of a boost converter and a synchronous buck converter, respectively, while an inverter ensures connection to the utility grid. In order to design and simulate the individual components of the microgrid, MATLAB ® /Simulink ® tools were used. During the study, different operating scenarios were created with respect to the PV power, load and the state of charge (SoC) of the battery. Thus, the following operation modes were identified: following mode, discharge mode, charging mode -referring to the battery operating modes that depend of its SoC. The aim of the EMS is to ensure the requirements of a prosumer application and switch between these different modes. Simulation results have shown that the developed EMS can achieve effective coordination within the microgrid, and all goals are reached.

I. INTRODUCTION

A prosumer is a particular configuration of microgrid which consists basically of a household that acts both as producer and consumer of electricity [START_REF] Ranaweera | Distributed control scheme for residential battery energy storage units coupled with PV systems[END_REF], [START_REF] Khalipour | Planning and operation scheduling of PV-battery systems: A novel methodology[END_REF]. As technology advances, renewable energy sources are seen to increase their share in power production. Amongst the vast topics in which renewable generation takes part there is the microgrid, with a large range of applications. As renewable energy becomes more accessible to end users, as well as flexibilization becomes more latent in the energy transition the world is facing, microgrids benefit now from increased interest [START_REF] Yang | Design Techniques of Distributed Photovoltaic/Energy Storage System[END_REF]- [START_REF] Ma | A Novel Series PV Energy Storage System[END_REF].

Amongst the renewable sources penetrating the microgrids, photovoltaic production provides good flexibility in terms of applicability, as it can be used for both large-and small-scale generation. The increase of prosumers who are using PV together with storage system brings advantages such as high efficiency and possibility to alleviate the renewable source's intermittency [START_REF] Aktas | Dynamic energy management for photovoltaic power system including hybrid energy storage in smart grid applications[END_REF], [START_REF] Ming-Yu | Study on Control Strategy of Energy Storage System in Photovoltaic Microgrid[END_REF]. Such PV-battery hybrid systems also play a vital role in managing peak demand and are, therefore, amongst the major stakeholders of the future utility grids. One of the solutions to increase the efficiency is an Energy Management System (EMS) which controls the power transfer between the elements of the microgrid [START_REF] Yang | Design Techniques of Distributed Photovoltaic/Energy Storage System[END_REF], [START_REF] Sangpanich | A Novel Method of Decentralized Battery Energy Management for Stand-Alone PV-Battery Systems[END_REF]. To increase the complexity, and possibly the range of applications for such a system, more elements could be added, such as a supercapacitor and a wind turbine [START_REF] Bacha | Power Electronic Converters Modeling and Control with Case Studies[END_REF], [START_REF] Ma | A Novel Series PV Energy Storage System[END_REF], for example. Technical aspect may also be completed by proposing business models and incentives for the prosumers [START_REF] Ranaweera | Distributed control scheme for residential battery energy storage units coupled with PV systems[END_REF].

In this paper design of an EMS for a DC microgrid-based prosumer is proposed. The chosen configuration of the studied microgrid can be seen in Figure 1 and it consists of: a photovoltaic (PV) system as renewable source, a battery unit as storage system, a DC load that models the consumer part and the AC grid. All these elements are connected in parallel to a common DC link by means of power electronics converters, namely: a synchronous-buck converter for the battery, a boost converter for the PV and an inverter for the AC grid. This microgrid-based prosumer must operate according to several requirements, which are translated into control objectives as follows:

• the main aim is to maintain DC-link voltage (Vdc) as constant, which requires a closed-loop control system capable of dealing with disturbances; • minimization of the net power flow through the AC grid, thus ensuring the prosumer autonomy as long as possible; • control of power converters depending on battery SoC; • Maximum Power Point Tracking (MPPT) implementation to maximize PV production.

By keeping the DC-link voltage constant, the DC sources of the system operate as current sources, whose currents are controlled by means of their associated converters. The inverter is also current-controlled. The current of each source is controlled by means of the duty cycle of the associated converter. In this way, a cascaded control structure is implemented, as suggested in Fig. 1. Here, variables noted as "Com1, 2, 3" and represented in blue denote current references sent by the upper-level EMS to the low-level current control loops. The low-level control consists of PI controllers used for synchronous buck converter and inverter in order to have zero steady-state error response due to the integral action and to guarantee the response time with proportional gain [START_REF] Astrom | PID Controllers: theory, design and tunning[END_REF]. Consequently, their parameters were set depending on the imposed response time and overshoot. As concerns the MPPT, the Perturb and Observe (P&O) control method was used to reach the maximum power point (MPP) of the PV array [START_REF] Rakhmawati | Design and Implementation of Hybrid Energy System for DC Load Applications[END_REF]. This paper is organized as follows. The following section will discuss all the steps towards the conception and validation of the proposed EMS. Section III will present the numerical simulation results obtained under representative scenarios, which validate the designed EMS's effectiveness. In Section IV conclusions regarding this work are drawn, as well as indications of future work using the presented configuration as base, are given.

II. ENERGY MANAGEMENT SYSTEM DESIGN

A. Goals and expected behavior

The EMS is required to properly coordinate sources such as to accomplish the goals imposed by the prosumer configuration and operation. In the proposed cascaded control structure, the lower level acts mainly to fulfill the primary objective of maintaining the DC-link voltage constant, which is essential so the following steps can be set to accomplish the next goals.

The role of the EMS is then to provide current references for all the converters within the microgrid such that the DC load demand to be always guaranteed. To do so in this application the EMS must prioritize the PV production, which should always be operating at its maximum by means of an MPPT algorithm. The only situation the PV is not producing its maximum is in the case there is no DC load at the moment and the battery is already fully charged, and since there ought to be minimal interaction with the grid, the best decision is to disconnect the PV from the microgrid. This situation might change if considering a proper business model of the prosumer, that is, incentives related to its possibility of injecting power into the main grid. This case has not been considered here.

The battery will constantly act as a backup for the system, providing power to the grid when PV production is not enough and storing any excess otherwise, when production is higher than the demand. However, the battery will work within a certain range of operation, as it is not recommended to fully charge or discharge the battery [START_REF] Zafara | Prosumer based energy management and sharing in smart grid[END_REF]. This means that, when a predefined threshold is reached by the battery SoC when it is discharging, the EMS must command the immediate charging of the battery; and else, if the battery is charged beyond some other predefined limit of its SoC, the system must impose its immediate discharge.

For the above-mentioned behaviors to be correctly implemented, the EMS must be able to access the utility grid. In this context the grid is modelled as an infinite-power source, so it has the capacity to receive/inject as much power as needed from/to the prosumer. This means the EMS will demand power from the grid in case PV and battery productions are not enough to the DC load, or in case the battery must be charged, and the PV is not enough to do so. On the other hand, in case the battery must be discharged, or the PV production is too much, or a combination of both, the EMS will inject the power surplus into the grid.

B. Low-level control loops

This level refers to the modelling of each component of the system, followed by the design of their respective controllers. In this prosumer configuration the following components are modeled: the battery and its synchronous buck converter, the PV plant and its boost converter, the AC grid and the associated inverter, the DC load and the DC link. PI controllers were then designed to govern the current of each converter of the system -thus, three tracking loops are implemented. The DC-link voltage, Vdc, must be kept constant despite disturbances coming from a priori unknown load variations -thus, its control loop is a regulation one. By assuming the DC-link voltage as constant, all sources can be modelled as current sources. Thus, they are current-controlled, meaning that the duty cycle of each converter will be altered so the current reference of each source to be tracked.

Concerning the PV array control, the chosen MPPT algorithm was the Perturb & Observe. This algorithm is able to track in real time the position of MPP without any previous knowledge about its precise position. It receives information of current and voltage values of the PV array and calculates the instantaneous power value. Then a perturbation is caused on the voltage by means of adding or subtracting a fixed step and the new power is calculated. In case this new value is higher than the previous one, this increment in voltage is propagated; otherwise the algorithm changes its perturbation direction always towards the point of maximum power.

The current reference for the battery converter is used as control variable within the Vdc control loop, playing an essential role in maintaining its stability. The battery current control must quickly answer to any perturbation on the system, which happens mostly due to changes in the DC load and to the PV power variations. The current reference for the inverter is set by the EMS as one of its functions is to determine when to access the utility grid.

Regarding the AC part of the system, the modeling was based upon the Park transform [START_REF] Rout | Analysis of Energy Management System for Photovoltaic System with Battery and Supercapacitor using Fuzzy Logic Controller[END_REF], leading to d and q components. The q component, being associated to the reactive power, it is kept at 0, as here the goal is restricted to control the active power flow. The grid is modeled to work on 50 Hz and 400 V rms.

C. Design methodology

In order to design the desired EMS, the following methodology was adopted: first, the objectives the EMS must accomplish were determined, which means defining the scope of the system; secondly the operation scenarios were also established, then the EMS algorithm was conceived using a flow chart representation, and finally this algorithm was implemented on MATLAB ® /Simulink ® so it could be tested and validated. The main objective of the EMS is to guarantee the coordination between the low-level control loops of the system in order to accomplish the previously described objectives. The EMS achieves this coordination by obtaining real-time information from the system, such as: measure of the PV power, measure of the power demanded by the load, the battery SoC estimation and the DC-link voltage measure. The battery SoC is here estimated by integrating its current ibat:

 + = t bat dt t i t 0 ) ( ) 0 SoC( ) SoC(
, where ) 0 SoC( is the initial SoC, at time 0. By using these values, the EMS changes accordingly the reference level of the inverter current.

To decide how to change the inverter current reference, a scenario analysis, which can be seen in Fig. 2, was made, followed by the algorithm description through a flow chart. In Fig. 2 Pdc denotes the power sensed on the DC link and Pac is the power exchanged with the AC grid, at the point of common coupling (PCC in Fig. 1).

The algorithm is based upon the remark that the reference of the battery current will follow alterations of the inverter current, according to power (current) equilibrium in the DC link node. Then, to command the charge or discharge of the battery according to its state of charge, the EMS sets the inverter current with the signal (+ or -) with respect to the desired sense of power flow it wishes to impose. Also, when defining the actual value of the reference, as one of the objectives is to minimize interaction with the grid, it is required that the grid will only be used in case the PV production is too high compared to the load and the excess of production must be directed to the grid, or when the demand cannot be met by the battery and PV productions together. Even in the last case, the battery will be set to work at its full capacity, during both discharge and charge, and the grid will only intervene when necessary. Finally, when the battery SoC is greater than the upper limit and there is no demand from the load, the PV is disconnected from the microgrid.

D. Numerical implementation

Once the desired behavior of the EMS was detailed, all cases were considered and the algorithm was established, the system is ready to be implemented on MATLAB ® /Simulink ® . When doing so, it was possible to identify three operating modes with respect to the battery status: follow mode, charging mode and discharging mode, respectively. This means that the algorithm has three branches that define how the inverter current reference will be set. In order to decide in which operation mode, the system currently is, it compares the battery SoC with some predefined threshold of a switch. These same patterns are repeated all across the diagram. The obtained results must be analyzed as to assess the effectiveness of the developed EMS in fulfilling the imposed objectives. To this end, two testing scenarios have been defined. Thus, in the first scenario, the aim is to observe how the EMS fulfills its role in response to load variations, which are the main disturbance. Thus, the DC load variation has been shaped as a series of step variations, to challenge the whole domain of EMS operation. The second scenario is chosen to demonstrate how the system would behave in the absence of the PV, that is, during the night for instance.

Note that the follow mode is considered to be the default operation mode for the application. This is because the system operates mainly in residential applications with a robust storage system, meaning the battery capacity would be large enough so that charges and discharges occur slowly during a daily cycle, and would rarely operate in extreme conditions, that is, near the thresholds. For the same reason, it would mean that for a simulation time of even 1 h there would not happen significant changes in battery SoC, a crucial parameter to determine in which operation zone the system is. Therefore, the capacity of the battery was set to 300 Ah, a relatively low capacity, making the system more sensible to changes and allowing the results presented in the simulations to be conclusive.

The battery SoC must be assigned an initial value at each test. The minimum and maximum thresholds for it were set based on usual values from the literature, recommended to ensure the battery life span, and are respectively 30% and 90% [START_REF] Ma | A Novel Series PV Energy Storage System[END_REF], [START_REF] Aktas | Dynamic energy management for photovoltaic power system including hybrid energy storage in smart grid applications[END_REF], [START_REF] Rout | Analysis of Energy Management System for Photovoltaic System with Battery and Supercapacitor using Fuzzy Logic Controller[END_REF]. This means that, in case the state of charge is under 30%, the battery must be charged, and in case it is over 90% the battery must be discharged.

In the Simulink ® implementation the current value of the state of charge is compared to the limiting values by means of a sum or a subtraction, the result of which is then put through a relay block producing a hysteresis effect; this avoids the oscillatory behavior around 0 and repetitive mode transitions back and forth. The battery will shift from follow to charging mode at a state of charge of 27,5% and will shift in the other sense at 32,5% as hysteresis limits are -2,5% and 2,5%. Similarly, the shift from follow to discharging mode will occur at 95% and in the other sense at 85%, as in this case the hysteresis limits are 5% and -5% respectively.

Without loss of generality, variations of the PV power are here not distinguished from those of the load itself, such that disturbances in both consumption and production are here treated globally, as coming from a so-called aggregated load. The aggregated load is the main responsible for generating such perturbations because it is influenced by two unpredictable variables: the irradiance and the demanded DC current. The first objective that must be attained by the EMS is keeping constant the voltage level at the DC link, which means the system must be able to quickly reject any perturbation.

A. Scenario 1: complete overview between operation modes

In this application the maximum power the PV is able to provide is 102.4 kW. The DC load current variation domain is between 0 and 110 A, varying in steps, as shown in Fig. 3. The grid here is seen as an infinite-power source. As for the battery, when it is in charging mode, a constant current of 100 A should by drawn, while when in discharging mode only an inferior limit is set, i.e., 100 A. A random variation of the irradiation profile has been chosen, as seen in Fig. 4, in order to analyze if the system is able to remain stable even in the worst-case scenario of high-frequency changes. The simulation time span is of 3600 s (1 hour), sufficient to observe the three modes, starting from 97% as initial battery SoC, meaning the system operates in discharging mode. At this point the DC load current is set to 20 A, as the battery discharges by providing a constant current of 80 A. To balance this excess of current, the inverter acts by dispatching the other 60 A to the grid. Yet, as the PV is not needed in such context, the EMS automatically turns it off. This behavior can be observed in terms of power equilibrium in Fig. 5. To better understand what the graphs show, the generator convention is adopted. Thus, currents of sources (battery, PV) are positive when these sources are providing power, whereas currents of loads are positive when loads are consuming power. The microgrid is seen as a source by the utility grid, so its current is positive when it is injected into the main grid. The DC-link voltage variation can be seen in Fig. 6, with a zoom in Fig. 7, allowing to observe the transients and dynamic performance. After 99 s the battery SoC reaches 85%, meaning the system enters the follow mode. The PV is then turned on, causing a disturbance on Vdc, as shown in Fig. 6, and starts producing according to the irradiance profile curve.

When in follow mode, as the load is still relatively low in comparison to the PV production, interaction with the grid is no longer needed and battery receives the exceeding solar production charging itself until 95% of SoC, at around 412 s. Then the battery re-enters discharging and again the inverter is solicited to inject current into the grid. Once the battery SoC reaches 85%, the system is again in follow mode and battery begins to be charged, but at 600 s the load is increased to 50 A, which in return demands the battery to provide power to the load depending on the level of the PV production. This process slowly discharges the battery. Also, the load increase is reflected as a perturbation on Vdc (the second spike visible in Fig. 6), which is rejected due to EMS.

Further on, at around 1200 s, the load is again increased, this time to 75 A, generating yet another perturbation, and the discharge rate of the battery increases as its use is primary to demand power from the utility grid. This is seen as the third disturbance in Fig. 6, which is again quickly rejected by the system, as expected. In the zoom in Fig. 7 the Vdc response to the perturbation can be seen in more detail, showing that the system returns to the nominal point after approximately 0.3 s.

The last increase occurs at 1800 s and represents a demand of 110 A. This goes beyond the maximal production from the PV and, associated with the high-frequency variability of the irradiance, even the addition of the battery is not enough at some moments, which will then force the system to draw current from the grid. This demand increases still at 2000 s when the battery reaches charging mode for the first time, meaning not only the load must be supplied but also the battery must be charged until acceptable levels in order to preserve an operational level of its SoC. What then follows is that every time the battery arrives in follow mode it discharges again while supplying to the system, and then it is charged again. Even at 2400 s, when the load is reduced to 80 A, the same behavior can still be observed. This cycle breaks at 3000 s, when the load drops at 0 A and the battery continues to be charged even after it enters follow mode, as PV is still producing more than needed. Also, from this point on the inverter current returns to 0 A, ensuring that the interaction with the grid is minimized.

It can be concluded from this scenario that all system parts are correctly coordinated by the EMS. All the imposed objectives are achieved.

B. Scenario 2: Special operation under no PV production

This second testing scenario is intended to simulate an operation during the night when irradiance levels would drop to zero, thus there would not be production from the PV. The load was initially set to 50 A, then dropped at 0 A at 500 s, in the middle of the 1000 s total simulation time, as to simulate a low demand period during the night (Fig. 8). Again, the results verify that the goals have been fulfilled. Load is supplied by the battery until it reaches its threshold and enters charging mode, requiring current from the grid. Once the battery is in follow mode and there is no demand from the load, all the power flows remain 0 and Vdc remains constant.

IV. CONCLUSION AND FUTURE WORK

In this paper, a DC-microgrid Energy Management System (EMS) has been proposed to coordinate operation of a photovoltaic (PV) system and a battery in supplying a DC load in a prosumer context. To this end, interaction with the utility grid -modelled as an infinite-power source -takes only place when strictly necessary. PV system and the battery are paralleled on a common DC-link by means of a boost converter and a synchronous buck converter, respectively, while an inverter ensures connection to the utility grid. MATLAB ® /Simulink ® numerical simulations have illustrated the effectiveness of the proposed EMS under representative scenarios, defined with respect to the PV power, load and the battery state of charge (SoC). The EMS has proven its ability to ensure the requirements of a prosumer application and switch between the different operating modes.

Future work could explore opportunity of defining business models for prosumers, like, for example, understanding energy tariffs depending on the hour of the day and the demand and how this can be translated into constraints to the EMS and then optimized so the profitability could be increased. This type of study can be of fundamental importance when a potential consumer decides between whether or not taking up his own renewable production. This could also be the case of comparing potential revenues between different combinations of renewable sources and storage systems.

Fig. 1 .

 1 Fig. 1. DC microgrid-based prosumer configuration chosen as case study.

Fig. 2 .

 2 Fig. 2. Analysis of operation scenarios

Fig. 3 .

 3 Fig. 3. DC load current (Idc) vs. time.

Fig. 4 .

 4 Fig. 4. Irradiance profile.

Fig. 5 .

 5 Fig. 5. Power values of interest vs. time, corresponding to first scenario.

Fig. 6 .

 6 Fig. 6. DC-link voltage (Vdc) vs. time, in the first scenario.

Fig. 7 .

 7 Fig. 7. Zoom of Vdc evolution around 1200 seconds, in the first scenario.

Fig. 8 .

 8 Fig. 8. Power values of interest in the second scenario.


Ayman Assi

Aren Ziad Bakouny

Joe Bizdikian

Fares Yared Joeffroy Otayek

Virginie Lafage

Nour Khalil

Abir Massaad

Ismat Ghanem

Wafa Skalli

A J Bizdikian

F Yared

Variation of the sagittal vertical axis during walking and its determinants

Keywords:

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

Introduction

Patients with adult spinal deformities (ASD) are known to have altered postural alignment affecting their quality of life and activities of daily living, especially gait. The Sagittal Vertical Axis (SVA), a postural parameter calculated as the distance between the posterior corner of the sacrum and the C7-plumbline on full-body sagittal radiographs [1], has been shown to be highly altered in ASD. Even though this parameter is positional and could vary during gait, no studies have investigated its variation during walking even in asymptomatic subjects.

Research question

Is SVA modified during gait in asymptomatic subjects?

Methods

75 asymptomatic adults (age: 28 ± 11years [18-59],32 F) underwent gait analysis with reflective markers on the pelvis, lower limbs(LL) [2] and the C7 spinous process. Kinematics of the pelvis and LL were extracted. Subjects then underwent low-dose full-body biplanar radiographs in standing position, with the reflective markers still in place. Subject-specific 3D spino-pelvic, acetabular and LL parameters were calculated as well as the 3D radiological SVA (rSVA). Then, the 3D bones were registered on each frame of the gait cycle [3] (Fig. 1). A new technique developed for the purpose of this study, utilizing finite element modeling, was used to reduce soft tissue artefacts. The SVA was then computed during the gait cycle, using the 3D registered bones, at each time frame: average (av_SVA), range of motion (ROM_SVA) and the difference between dynamic and static SVA (ΔSVA = av_SVA-rSVA) were calculated. In order to investigate the determinants of static and dynamic SVA parameters, a univariate analysis (Pearson's correlations) followed by a multivariate analysis (stepwise-multiple-linear regressions) were performed; the dependent variables were demographics, 3D kinematics and 3D skeletal parameters (spino-pelvic, acetabular and LL).

Results

The rSVA ranged from -66 to 38 mm with a mean of -9 mm. The av_SVA (ROM_SVA) during gait ranged from 9 to 108 mm (9 to 45 mm) with a mean of 46 mm (23 mm, respectively). ΔSVA ranged from 16 to 120 mm with a mean of 55 mm. Fig. 2 shows the SVA variation during gait relative to its radiological value. The multivariate analysis showed (Fig. 3) that: rSVA (R2 = 0.2) was solely determined by sex (F:-20 mm vs M:1 mm,p < 0.001); av_SVA (R2 = 0.4) during gait was determined by weight (β = 0.97,p < 0.001) and rSVA (β = 0.18,p < 0.001); ROM_SVA (R2 = 0.2) was determined by acetabular abduction (β=-0.52,p = 0.038) and ROM hip flexion/extension during gait (β = 1.48, p < 0.001); ΔSVA was solely determined by lumbar lordosis (r = 0.33,p = 0.004).

E-mail address: ayman.assi@gmail.com (A. Assi).

⁎ Corresponding author at: Faculty of Medicine, University of Saint-Joseph, Laboratory of Biomechanics and Medical Imaging, Beirut, Lebanon.

Discussion

This is the first study to evaluate SVA during walking in asymptomatic subjects. In general, subjects tended to bend their trunk forward during gait (positive ΔSVA), especially those with higher lumbar lordosis and higher rSVA. Patients with ASD, who are known to have an increased rSVA, might show a more forwarded trunk during gait which could possibly increase their risk of falling.

Fig. 1 .

 1 Fig. 1. Registration of 3D skeletal reconstructions from biplanar X-rays with 3D gait analysis.

Fig. 2 .

 2 Fig. 2. Variation of the Sagittal Vertical Axis during walking in asymptomatic subjects in regards to its radiological value.

Fig. 3 .

 3 Fig. 3. Demographic, skeletal and kinematic determinants of radiological and dynamic Sagittal Vertical Axis.