Sébastien Laurent

Matthieu Pierce

Marion Delehaye

Tarik Yefsah

Frédéric Chevy

Christophe Salomon

Connecting few-body inelastic decay to many-body correlations: a weakly coupled impurity in a resonant Fermi gas

Keywords: numbers: 34, 50, -s, 67, 85, Pq

We study three-body recombination in an ultracold Bose-Fermi mixture. We first show theoretically that, for weak inter-species coupling, the loss rate is proportional to Tan's contact. Second, using a 7 Li/ 6 Li mixture we probe the recombination rate in both the thermal and dual superfluid regimes. We find excellent agreement with our model in the BEC-BCS crossover. At unitarity where the fermion-fermion scattering length diverges, we show that the loss rate is proportional the 4/3 power of the fermionic density. Our results demonstrate that impurity-induced losses can be used as a quantitative probe of many-body correlations.

Understanding strongly-correlated quantum manybody systems is one of the most daunting challenges in modern physics. Thanks to a high degree of control and tunability, quantum gases have emerged as a versatile platform for the exploration of a broad variety of many-body phenomena [START_REF] Bloch | Many-body physics with ultracold gases[END_REF], such as the crossover from Bose-Einstein condensation (BEC) to Bardeen-Cooper-Schrieffer (BCS) superfluidity [2], quantum magnetism [3] or many-body localization [4]. At ultra-low temperatures, atomic vapors are metastable systems and are plagued by three-body recombination which represents a severe limitation for the study of some dense interacting systems. A prominent example is the strongly correlated Bose gas [5,6] that bears the prospect of bridging the gap between dilute quantum gases and liquid Helium. However inelastic losses can also be turned into an advantage. For instance, they can be used to control the state of a system through Zeno effect [7][START_REF] Daley | Atomic three-body loss as a dynamical threebody interaction[END_REF][START_REF] Zhu | Suppressing the loss of ultracold molecules via the continuous quantum zeno effect[END_REF], or serve as a probe of non-trivial few-body states, as demonstrated by the observation of Efimov trimers, originally predicted in nuclear physics, but observed for the first time in Bose gases as resonances in three-body loss spectra [START_REF] Kraemer | Evidence for Efimov quantum states in an ultracold gas of caesium atoms[END_REF].

In this Letter, we study inelastic losses in a mixture of spinless bosons and spin 1/2 fermions with tunable interaction. We show that when the Bose-Fermi coupling is weak, the loss rate can be related to the fermionic contact parameter, a universal quantity overarching between microscopic and macroscopic properties of a manybody system with zero-range interactions [3,[START_REF] Tan | Energetics of a strongly correlated fermi gas[END_REF][START_REF] Olshanii | Short-distance correlation properties of the lieb-liniger system and momentum distributions of trapped one-dimensional atomic gases[END_REF]. We first check our prediction on the strongly attractive side of the fermionic Feshbach resonance, where we recover known results on atom-dimer inelastic scattering. We then turn to the unitary limit where the fermion-fermion scattering length is infinite. We demonstrate both theo- * Electronic address: slaurent@lkb.ens.fr † Present Address: FEMTO-ST UMR CNRS 6174 -UFC/ENSMM/UTBM, 26 chemin de l'Epitaphe, 25030 Besançon, France Loss rate (a.u)

-1/k F a ff 1/a ff a ff 2 -1 +1 BEC side
BCS side 0 FIG. 1: Sketch of inelastic decay of an impurity immersed in a tunable Fermi gas. On the BEC side, ↑ and ↓ fermions are paired in tightly bound molecules and the decay mechanism is a two-body process involving the impurity (green disk) and a molecule. The loss rate scales as 1/a ff [START_REF] Dincao | Suppression of molecular decay in ultracold gases without Fermi statistics[END_REF][START_REF] Zirbel | Collisional stability of fermionic Feshbach molecules[END_REF]. On the BCS side, the loss occurs through a three body-process and it scales as a 2 ff in the mean-field limit [START_REF] Dincao | Suppression of molecular decay in ultracold gases without Fermi statistics[END_REF]. The extrapolation of these two asymptotic behaviors towards the strongly correlated regime yields contradictory results (grey area).

retically and experimentally -with a 6 Li/ 7 Li Fermi-Bose mixture -that the bosons decay at a rate proportional to n 4/3 f , where n f is the fermion density. More generally our work shows that the decay of an impurity immersed in a strongly correlated many-body system is a quantitative probe of its quantum correlations.

Inelastic decay of an impurity inside a two-component Fermi gas has been studied previously both in the weakly and strongly attractive limits of the BEC-BCS crossover [START_REF] Dincao | Suppression of molecular decay in ultracold gases without Fermi statistics[END_REF][START_REF] Incao | The short-range three-body phase and other issues impacting the observation of efimov physics in ultracold quantum gases[END_REF][START_REF] Spiegelhalder | Collisional Stability of 40 K Immersed in a Strongly Interacting Fermi Gas of 6 Li[END_REF][START_REF] Khramov | Dynamics of feshbach molecules in an ultracold threecomponent mixture[END_REF]. When the fermion-fermion interaction is weak, the fermions behave almost as isolated particles and the recombination can be described as a three-body process involving one spin-up (↑), one spindown (↓) fermion and the impurity (a boson in our experiments). In this case, the impurity/boson density n b follows a rate equation ṅb = -L 3 n 2 f n b , with L 3 ∝ a 2 ff , where a ff is the fermion-fermion scattering length [START_REF] Dincao | Suppression of molecular decay in ultracold gases without Fermi statistics[END_REF][START_REF] Zirbel | Collisional stability of fermionic Feshbach molecules[END_REF][START_REF] Spiegelhalder | Collisional Stability of 40 K Immersed in a Strongly Interacting Fermi Gas of 6 Li[END_REF]. In contrast, on the strongly attractive side of the Feshbach resonance, the fermions form halo-dimers of size a ff and the relaxation occurs through two-body processes between one such molecule and one boson. In this case the rate equation for bosons reads ṅb = -L 2 n m n b , where n m = n f /2 is the molecule density. Far from the Feshbach resonance the two-body loss rate scales as 1/a ff as a consequence of the enhanced overlap of the halo-dimer wavefunction with the deeply bound product molecules [START_REF] Dincao | Suppression of molecular decay in ultracold gases without Fermi statistics[END_REF][START_REF] Zirbel | Collisional stability of fermionic Feshbach molecules[END_REF]. However, these two scalings give rise to a paradox in the central region of the BEC-BCS crossover. Indeed, as depicted in Fig. 1, the extrapolation towards unitarity leads to contradictory results depending on whether we approach the resonance from the BEC or the BCS side. In the former case, one would predict an increasingly long lifetime at unitarity while it tends to a vanishingly small value in the latter case. This paradox has a fundamental origin: these two scalings are obtained in the dilute limit where the recombination can be described by a well-defined few-body process, whereas this hypothesis fails in the strongly correlated regime where n f |a ff | 3 1. There, it is not possible to single out two fermions from the whole many-body system. Instead, the inelastic loss involving a boson and two fermions is tied to the correlations of the whole ensemble. A first hint towards reconciling these two behaviors near unitarity is to assume that they saturate for a ff n

-1/3 f
, yielding the same scaling ṅb ∝ n 4/3 f n b . The three asymptotic regimes -BEC, BCS, and unitary -were obtained using different theoretical approaches and we now show that, using Tan's contact, they can be unified within the same framework. The recombination rate is proportional to the probability of having the three particles within a distance b from each other, where b is the typical size of the deeply-bound molecule formed during the collision [2,[START_REF] Kagan | Effect of Bose condensation on inelastic processes in gases[END_REF]. Take ρ 3 (r ↑ , r ↓ , r b ) the three-body probability distribution of the system. When the bosons are weakly coupled to the fermions, we can factor it as

ρ 3 (r ↑ , r ↓ , r b ) = ρ f (r ↑ , r ↓ )ρ b (r b ).
Integrating over the positions of the three atoms we readily see that the three-body loss rate is proportional to Tan's Contact parameter C 2 of the fermions that gives the probability of having two fermions close to each-other [START_REF] Tan | Energetics of a strongly correlated fermi gas[END_REF]. C 2 is calculated using the equation of state of the system thanks to the adiabatic-sweep theorem

C 2 = - 4πm f 2 ∂F ∂(1/a ff ) . ( 1
)
where m f is the fermion mass and F is the free-energy of the fermionic gas per unit-volume [3,[START_REF] Olshanii | Short-distance correlation properties of the lieb-liniger system and momentum distributions of trapped one-dimensional atomic gases[END_REF]. The asymptotic expressions of C 2 in the BEC, BCS and unitary regimes

BEC Unitary BCS C2 8π nm a ff 2ζ 5π k 4 F 4π 2 a 2 ff n 2 f ṅb n b ∝ nm a ff [14] ∝ n 4/3 f ∝ a 2 ff n 2 f [14]
TABLE I: Scaling of Tan's contact [START_REF] Tan | Energetics of a strongly correlated fermi gas[END_REF] and of the boson/fermion mixture loss rate in the BEC-BCS crossover.

Both scalings are identical in the weakly and strongly attractive limits. As kF = (3π 2 n f ) 1/3 , at unitarity C2 scales as [START_REF] Navon | The equation of state of a lowtemperature fermi gas with tunable interactions[END_REF][START_REF] Hoinka | Precise determination of the structure factor and contact in a unitary fermi gas[END_REF].

n 4/3 f . ζ is a dimensionless constant, ζ = 0.87(3)
are listed in Table I. In the deep BEC limit, the free energy is dominated by the binding energy of the molecules 2 /m f a 2 ff ; in the BCS regime C 2 is derived using the mean-field approximation [START_REF] Tan | Energetics of a strongly correlated fermi gas[END_REF]. At unitary, the expression of the contact stems from the absence of any length scale other than the inter-particle distance. The dimensionless parameter ζ = 0.87(3) was determined both theoretically [START_REF] Astrakharchik | Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study[END_REF] and experimentally [START_REF] Navon | The equation of state of a lowtemperature fermi gas with tunable interactions[END_REF][START_REF] Sagi | Measurement of the homogeneous contact of a unitary fermi gas[END_REF][START_REF] Kuhnle | Temperature dependence of the universal contact parameter in a unitary Fermi gas[END_REF]. Expressions listed in Table I confirm that the contact parameter and the bosonic loss rate follow the same scalings with density and scattering length.

We support this relationship between inelastic losses and Tan's contact by considering a microscopic model where the recombination is described by a three-body Hamiltonian

H 3 = d 3 r b d 3 r ↑ d 3 r ↓ g(r b , r ↑ , r ↓ )× Ψ † m r ↑ + r ↓ 2 Ψ † b (r b ) Ψ b (r b ) Ψ ↑ (r ↑ ) Ψ ↓ (r ↓ ) + h.c., (2)
where Ψ α is the field operator for the species α and the coupling g takes significant values only when the three particles are within a distance b [41]. Assuming that b is the smallest distance scale in the problem and that this Hamiltonian can be treated within Born's approximation we find that (see [26])

ṅb = -γC 2 n b , (3)
The constant γ depends on the coupling g and describes the coupling to deeply bound non-resonant states; hence γ has essentially no variation with magnetic field across the fermionic Feshbach resonance. Eq. ( 3) is the main prediction of this Letter and we explore the consequences of this equation by measuring the lifetime of an ultracold Fermi-Bose mixture of 6 Li and 7 Li atoms. Our experimental setup is described in [START_REF] Ferrier-Barbut | A mixture of Bose and Fermi superfluids[END_REF]. The 6 Li atoms are prepared in a spin mixture ↑, ↓ of |F = 1/2, m F = ±1/2 for which there is a broad Feshbach resonance at 832 G [4] It can be considered constant in the magnetic field range that we explored, 680-832 G. The atoms are confined in a hybrid magnetic/optical trap and are evaporated at the 6 Li Feshbach resonance until we reach dual superfluidity or any target temperature. We ramp the magnetic field to an adjustable value in 200 ms and wait for a variable time t. We then measure the atom numbers of the two species by in situ imaging or after time of flight.

We first show that the dominant boson loss mechanism on the BEC side of the resonance involves one boson, one fermion ↑, and one fermion ↓. This is easily done by comparing the boson losses for spin-balanced and spinpolarized fermionic samples. Fig. 2 displays the remaining fraction of bosons and fermions after a waiting time of 1 s for balanced fermions and 1.5 s for spin-polarized fermions with 90 % polarization. We observe that the losses for high spin polarization are strongly suppressed indicating that fermions of both spin components are required to eliminate one boson.

Second we show that the losses in the weakly interacting regime na 3 ff 1 (deep BEC side of the resonance, 720 G) are proportional to the fraction of molecules in the sample, η = 2N m /(N f + 2N m ). This fraction is varied by changing the temperature from 1 µK to 4 µK and 6 Li densities from 2 × 10 12 cm -3 to 1.0 × 10 13 cm -3 . In these temperature and density ranges, both gases are well described by Maxwell-Boltzmann position and velocity distributions. The molecular fraction is calculated using the law of mass-action [7,26] and is assumed to be time-independent owing to the high formation rate of halo-dimers ( a 4 ff /m f ) [START_REF] Fedichev | Three-body recombination of ultracold atoms to a weakly bound s level[END_REF]. We extract the inter-

��� ��� ��� ��� ��� ��� ��� ��� ��� ��� η � �� (�� -�� �� � /�) �) � � � � � ��� ��� ��� ��� �/� �� (�� -� � � -� ) � �� (�� -�� �� � /�) �)
FIG. 3 species decay rate by fitting the time evolution of the bosonic population

Ṅb = -L bf n f N b -Γ v N b . (4)
where • represents the trap-average, and Γ v is the onebody residual gas loss rate (0.015 s -1 ). The data in Fig. 3a shows that the boson loss-rate is proportional to the molecule fraction of the fermionic cloud. Introducing the boson-fermion dimer molecule loss rate L bm defined by L bm n m = L bf n f , we check the proportionality of L bm with 1/a ff predicted in table I by repeating the loss measurements for different magnetic fields in the interval 690-800 G, see Fig. 3b. From a linear fit to the data where interaction effects are negligible (n f a 3 ff ≤ 0.025), we extract the slope γ = 1.17 now predict the loss rate anywhere in the BEC-BCS crossover using Eq. ( 3). The strongly interacting unitary regime (1/a ff = 0) is particularly interesting and we measure the boson decay rate at 832 G in the low temperature dual superfluid regime [START_REF] Ferrier-Barbut | A mixture of Bose and Fermi superfluids[END_REF]. The mixture is initially composed of about 40 × 10 3 fully condensed 7 Li bosons and 150 × 10 3 6 Li spin-balanced fermions at a temperature T 100 nK which corresponds to T /T F 0.1 where T F is the Fermi temperature. At this magnetic field value, the atoms are now closer to the boson Feshbach resonance located at 845.5 G and bosonic three-body losses are no longer negligible. The time dependence of the boson number is then given by

Ṅb = -L b n 2 b N b -Γ bf N b -Γ v N b . (5)
To extract Γ bf we measure independently L b with a BEC without fermions in the same trap and inject it in Eq. ( 5), see [26]. We typically have L b n 2 b = 0.1 -0.4 s -1 , and L b = 0.11(1) × 10 26 cm 6 . s -1 consistent with the model of [6]. Repeating such measurements for different fermion numbers and trap confinement, we now test the expected n 4/3 f dependence of the Bose-Fermi loss rate at unitarity (central column in Table I). In this dual superfluid regime, the size of the BEC is much smaller than that of the fermionic superfluid and the BEC will mainly probe the central density region n f (r = 0). However, it is not truly a point-like probe, and introducing the ratio ρ of the Thomas-Fermi radii for bosons and fermions, we obtain the finite size correction for Eq. (3) [26]:

Γ bf = γ C 2 (0) (1 - 6 7 ρ 2 ), (6)
where C 2 (0) = 2ζ 5πξ (3π 2 n f (0)) 4/3 , ξ = 0.376( 4) is the Bertsch parameter [START_REF] Mark | Revealing the superfluid lambda transition in the universal thermodynamics of a unitary fermi gas[END_REF], and the last factor in parenthesis amounts to 0.9. The prediction of Eq. ( 6) is plotted as a red line in Fig. 4 and is in excellent agreement with our measurements without any adjustable parameter. Alternatively, a power-law fit An p to the data yields an exponent p = 1.36 [START_REF] Zirbel | Collisional stability of fermionic Feshbach molecules[END_REF] which confirms the n 4/3 f predicted scaling at unitarity. Finally fixing p to 4/3 provides the coefficient A and a value of the homogeneous contact ζ = 0.82 [START_REF] Zhu | Suppressing the loss of ultracold molecules via the continuous quantum zeno effect[END_REF] in excellent agreement with previous measurements, ζ = 0.87(3) [START_REF] Navon | The equation of state of a lowtemperature fermi gas with tunable interactions[END_REF][START_REF] Hoinka | Precise determination of the structure factor and contact in a unitary fermi gas[END_REF]. This demonstrates that impurity losses act as a microscopic probe of many-body correlations.

The bosonic or fermionic nature of the probe is of no importance. Provided the coupling between the impurity and the resonant gas is weak, our method can also be applied to other mixtures. It gives a framework to interpret the experimental data on 6 Li/ 40 K [START_REF] Spiegelhalder | Collisional Stability of 40 K Immersed in a Strongly Interacting Fermi Gas of 6 Li[END_REF] and, in particular, to test our prediction on the BCS side of the Feshbach resonance. It can also be applied to the recently observed 6 Li/ 174 Yb [33], 6 Li/ 41 K [34] and 6 Li/ 7 Li [START_REF] Ikemachi | All-optical production of a superfluid bose-fermi mixture of 6 Li and 7 Li[END_REF] dual-superfluid Bose-Fermi mixtures. Our observation of a loss rate scaling ∝ n 4/3 f at unitarity is in stark contrast with the generic case n p , where the integer p is the number of particles involved in the recombination process. A fractional exponent is also predicted to occur for the resonant Bose gas [5,6].

A first extension of this work is to investigate regimes where a bf a ff n -1/3 and Born approximation breaks down. In this case Efimovian features are expected to occur [START_REF] Eric Braaten | Three-body recombination of 6 Li atoms with large negative scattering lengths[END_REF][START_REF] Ottenstein | Collisional stability of a three-component degenerate fermi gas[END_REF]. Second, our method provides a unique microscopic way to measure the contact quasi-locally in a harmonic trap. An important perspective is to determine the homogeneous contact of the unitary Fermi gas at finite temperature, whose behavior is largely debated near the normal-superfluid transition [START_REF] Sagi | Measurement of the homogeneous contact of a unitary fermi gas[END_REF]. Our approach can also be extended to the measurement of the threebody contact which is unknown for the two component Fermi gas at unitarity [START_REF] Zirbel | Collisional stability of fermionic Feshbach molecules[END_REF][START_REF] Fletcher | Two and Three-body Contacts in the Unitary Bose Gas[END_REF][START_REF] Du | Inelastic collisions of a fermi gas in the bec-bcs crossover[END_REF].

��� ��� ��� ��� ��� -��� -��� � ��� ��� � (�) �(� � )
FIG. S5: Magnetic fields dependence for the different scattering lengths a bb (blue), a ff (red, divided by 100), a bf (green) and effective range r e,b (yellow) involved in our system.

12 0| Ψ † 1 (ρ 1 ) Ψ † 2 (ρ 2 ) Ψ 2 (ρ 3 ) Ψ 1 (ρ 4 )|0 12 ϕ(ρ 1 -ρ 2 ) * ϕ(ρ 3 -ρ 4 ), ( S16
) where ϕ is the two-body function in a scattering state. According to Bethe-Peierls' condition, we have ϕ(r) A/r at short distance, hence

12 0| Ψ † 1 (ρ 1 ) Ψ † 2 (ρ 2 ) Ψ 2 (ρ 3 ) Ψ 1 (ρ 4 )|0 12 |A| 2 |ρ 1 -ρ 2 ||ρ 3 -ρ 4 | .
(S17) The value of |A| 2 is obtained by taking ρ 1 = ρ 4 and ρ 2 = ρ 4 and, according to Tan's relation for the densitydensity correlation function [3], we have

12 0| Ψ † 1 (ρ 1 ) Ψ † 2 (ρ 2 ) Ψ 2 (ρ 2 ) Ψ 1 (ρ 1 )|0 12 C 2 4π|ρ 1 -ρ 2 | 2 , (S18) where C 2 is the contact. Therefore |A| 2 = C 2 /4π hence Γ = C 2 8 d 3 ρ 1 d 3 ρ 2 G(ρ 1 , ρ 2 ) ρ 1 ρ 2 . ( S19
)
leading to equation 1 in the main text.

II. FESHBACH RESONANCES

In Fig. S5 we present the relevant s-wave scattering lengths characterizing the 6 Li-6 Li, 7 Li- 7 Li, and 6 Li- 7 Li interactions (a ff , a bb , and a bf respectively). 6 Li exhibits a broad Feshbach resonance at 832.18 G [4]. 7 Li-7 Li interactions exhibits two Feshbach resonances located at 845.5 G and 894 G [5]. Also plotted is the effective range for the 7 The red dashed curve is a fit of the fermion decay using Eq. S21 to estimate n f (t). The blue dashed curve is a fit to the boson decay using Eq. S20 and the previously fitted n f (t) to extract L bf .

dependence with magnetic field [6]. For the 6 Li-7 Li interaction, the scattering length a bf = 40.8 a 0 is identical for the two 6 Li spin states and does not depend on the magnetic field.

III. LOSS COEFFICIENT EXTRACTION ON THE BEC SIDE

In Fig. S6, we show a typical loss rate measurement at 720 G for a cold thermal Bose-Fermi mixture. On the BEC side of the 6 Li resonance, several processes contribute to the loss of 6 Li atoms: atom-dimer and dimerdimer inelastic collisions, evaporation losses and Bose-Fermi losses, resulting in a non-trivial time dependence. On the contrary, the 7 Li cloud will mainly loose atoms via Bose-Fermi losses, since evaporation loss and three-body losses are negligible due to the small 7 Li- 7 Li scattering length in this region of magnetic field. We thus use the boson decay rate to extract

L bf using ṅb = -L bf n f n b -Γ v n b (S20)
where the local densities follow Boltzmann distributions and Γ v = 0.015 s -1 is the residual background gas loss rate measured independently. For n f (t), we use a twobody decay function

n f (t) = n 0 /(1 + αt) (S21) IV. MOLECULE FRACTION
To compute the molecule fraction of the 6 Li cloud, η = 2N m /(N f + 2N m ) we model the fermionic ensemble as a non-interacting mixture of N m molecules and N f free atoms. This assumption is only valid far on the BEC side of the resonance (n f a 3 ff 1) and for a non-degenerate thermal gas. Following [7], we simply write a chemical equilibrium condition between atoms and molecules in the trap at temperature T :

N f = 2 kBT ω 3 Li 3 (z) (S22) N m = kBT ω 3 Li 3 (z 2 e -E b /kBT ) (S23)
where Li is a polylogarithm function, z = e µ/kBT the fugacity and E b = -2 /m f a 2 ff the molecule's binding energy. The fugacity is calculated by imposing the total number of atoms in the trap N tot = N f + N m .

V. BOSE-FERMI LOSS COEFFICIENT AT UNITARITY

In Fig. S7, we show a typical loss rate measurement in the dual superfluid regime at unitarity. As stated in the main text the three-body recombination in the Bose gas itself also contribute to the boson decay. In order to measure the Bose-Fermi loss rate Γ bf we first measure the three-body loss coefficient L b for a BEC alone using the following equation,

ṅb = -L b n 3 b -Γ v n b (S24)
We restrict the measurement over a period of time for which the thermal fraction surrounding the BEC is not visible. We thus assume that the 7 Li cloud density is given by a Thomas-Fermi distribution. Knowing L b , we extract Γ bf for a BEC in presence of fermions in the same trap using

ṅb = -L b n 3 b -Γ v n b -Γ bf n b . (S25)
Since n f does not vary significantly over the measurement duration, see Fig. S7, Γ bf is now assumed constant.

VI. REDUCTION OF THE BOSE-FERMI LOSSES DUE TO THE FINITE SIZE OF THE BEC

In the dual superfluid regime, the bosonic sample is not a perfect point-like impurity probing the central density of the Fermi gas. The finite size of the BEC leads to a slight reduction of the Bose-Fermi losses that we compute using the local density approximation: Green circles: BEC without fermions. Blue circles: BEC in presence of the fermionic superfluid. Each circle is the average of 3 to 5 data points with their standard deviation. Green dashed curve: fit to the decay of the BEC alone using Eq. S24 providing the three body loss coefficient L b . Blue dashed curve: fit to the BEC with fermions using Eq. S25 which gives Γ bf = 0.14(4) s -1 . Light blue dashed curve: expected BEC decay without Bose-Fermi losses. Inset: the number of 6 Li atoms for the same time duration (red circles). As it is nearly constant we use the mean number of 6 Li atoms shown as a red dashed line to compute the peak density of the fermionic superfluid during the losses.

where n b and n f are respectively the boson and fermion densities. Introducing the Thomas-Fermi radii R TF,b and R TF,f , we find (S27) With R TF,b = 0.4R TF,f , the reduction factor amounts to 0.9.

. The 7 FIG. 2 :

 72 FIG.2: Remaining fraction of bosons (blue symbols) and fermions (red symbols, inset) after a 1 s and 1.5 s waiting time for spin-balanced (filled symbols), resp. 90% polarized (open symbols) fermions. The blue dash-dotted (red dashed, inset) curve is a coupled loss model describing the competition between boson fermion-dimer decay (∝ 1/a ff ) and dimer-dimer decay (∝ 1/a 2.55 ff )[2, 26]. The blue-shaded area represents the 1σ fluctuations for the remaining fraction of bosons with spin-polarized fermions. The initial atom numbers are 3 × 10 5 for 6 Li and 1.5×10 5 for 7 Li at a temperature T 1.6 µK with trap frequencies νz = 26 Hz and νr = 2.0 kHz.

  FIG. 3: a) Boson-fermion loss rate vs molecule fraction. Circles: experimental data. The vertical error bars represent the statistical errors for L bf from fitting the loss curves. The horizontal error bars represent the statistical errors on the molecule fraction due to 6 Li number fluctuations. The red dashed line is a linear fit to the data. b) Boson-dimer loss rate vs inverse scattering length. The blue dot-dashed line is a linear fit to the data with n f a 3 ff ≤ 0.025 (black circles), providing γ = 1.17(11) × 10 -27 m 4 .s -1 , see Eq. 3.

FIG. 4 :

 4 FIG.4: Boson loss rate versus fermion central density at unitarity, n f = n f (0). Circles: experimental data. The red line is the n 4/3 f prediction of Eq. (6) without any adjustable parameter. The red shaded area represents the 1 σ uncertainty resulting from the error on γ.

Li- 7

 7 FIG.S6: Example of atom losses at B = 720 G for a non degenerate Bose-Fermi mixture at T = 1.25 µK. Red circles: fermion decay. Blue circles: boson decay. Each circle is the average of 3 to 5 data points with their standard deviation. The red dashed curve is a fit of the fermion decay using Eq. S21 to estimate n f (t). The blue dashed curve is a fit to the boson decay using Eq. S20 and the previously fitted n f (t) to extract L bf .

  FIG.S7: Example of atom loss at B = 832.1 G in the dual superfluid regime. Green circles: BEC without fermions. Blue circles: BEC in presence of the fermionic superfluid. Each circle is the average of 3 to 5 data points with their standard deviation. Green dashed curve: fit to the decay of the BEC alone using Eq. S24 providing the three body loss coefficient L b . Blue dashed curve: fit to the BEC with fermions using Eq. S25 which gives Γ bf = 0.14(4) s -1 . Light blue dashed curve: expected BEC decay without Bose-Fermi losses. Inset: the number of 6 Li atoms for the same time duration (red circles). As it is nearly constant we use the mean number of6 Li atoms shown as a red dashed line to compute the peak density of the fermionic superfluid during the losses.

[ 1 ]

 1 Félix Werner and Yvan Castin. General relations for quantum gases in two and three dimensions: Two-component fermions. Phys. Rev. A, 86(1):013626, 2012.

Acknowledgments

The authors thank G. Shlyapnikov, B. Svistunov and F. Werner for helpful discussions. They acknowledge support from Région Ile de France (DIM IFRAF/NanoK), ANR (Grant SpiFBox) and European Union (ERC Grant ThermoDynaMix).

SUPPLEMENTARY INFORMATION

I. MICROSCOPIC THREE-BODY LOSS MODEL

We derive here the relationship between the contact and the bosonic loss rate using a microscopic model. We consider the three-body Hamiltonian

where Ψ α are the field operators for the atoms (α = 1, 2, 3) and the molecule (α = M ). g(ρ 1 , ρ 2 ) is a kernel describing the molecule formation and is expressed in term of Jacobi's coordinates ρ 1 = r 1 -r 2 and ρ 2 = r 3 -(r 1 +r 2 )/2. Its characteristic width is of the order of the typical size b of deeply bound molecules ( Van Der Waals length) and is assumed to be much smaller than the other relevant length scales of the problem (scattering length and inter-particle distance). Finally we assume for the sake of simplicity that all atomic species (α = 1, 2, 3) have the same mass.

We now recast H 3 in momentum space by taking

where Ω is a quantization volume. We then have

and

We treat the 1-2 atoms as a strongly correlated manybody system and we neglect their interactions with the 3-atoms or the molecules. We consider the initial state |i = |0 12 ⊗ |k 3 3 ⊗ |∅ M where |0 12 is the groundstate of the many-body 1-2 system, |k 3 3 corresponds to a single 3-particle with momentum k 3 and |∅ M is the molecule-vacuum. Assuming that the Hamiltonian H 3 can be treated perturbatively, the molecule formationrate Γ is given by Fermi's Golden Rule Assuming that the binding energy ∆ of the molecule is much larger than the typical single-particle energies of the initial state (chemical potential, temperature...), the momentum and energy-conservation conditions are dominated by the state of the atom 3 and of the molecule after the decay. We therefore have k M -k 3 and E f -

Using the closure relation

with χ(q, q ) = 1 Ω

-∆ g(q, k 3 )g(q , k 3 ) * (S13) Going back into real space and using the fact that thanks to momentum conservation, we must have

with

Since G takes significant values for ρ 1 , ρ 2 R * , we care only about the values of the correlation function 12 0| Ψ † 1 (ρ 1 ) Ψ † 2 (ρ 2 ) Ψ 2 (ρ 3 ) Ψ 1 (ρ 4 )|0 12 for small values of the ρ α .

Following the argument proposed in [START_REF] Bloch | Many-body physics with ultracold gases[END_REF]2], the short distance behaviour is dominated by two-body physics, leading to a scaling


Franck Petitdemange
email: franck.petitdemange@irisa.fr

Isabelle Borne
email: isabelle.borne@irisa.fr

Jérémy Buisson
email: jeremy.buisson@irisa.fr

Assisting the Evolutionary Development of SoS with Reconfiguration Patterns

System of systems (SoS) engineering is an emerging approach to deal with complex systems that have low or no control over their constituents. An SoS must adapt itself not only to the willingness of its constituents to participate or disengage, but also to evolving needs. As reconfiguration is a routine task, the SoS architect needs specific assistance, especially to easily identify reusable solution principles and to track her/his decisions in the design of reconfiguration. To address these issues, we propose to introduce the concept of reconfiguration pattern based on prior design and architectural patterns. These patterns constitute well-documented, principled and adaptable solution building blocks. We illustrate our approach with one reconfiguration pattern applied on an example based on the French communication system for emergency services.

INTRODUCTION

Systems of systems [START_REF] Maier | Architecting principles for systems-of-systems[END_REF] are an emerging class of complex systems characterized by the combination of the following properties: [START_REF] Arshad | Deployment and dynamic reconfiguration planning for distributed software systems[END_REF] operational independence of the constituents;

(2) managerial independence of the constituents; (3) evolutionary development; (4) emergent behavior; [START_REF] Dorn | Analyzing runtime adaptability of collaboration patterns[END_REF] geographical distribution of the constituents. Orthogonal to the above characteristics, four categories are often used to classify the SoS depending on the type of their management. A directed SoS is built such that the SoS has coercive control over its constituents. In anacknowledged SoS the design of the constituents is based on a cooperative agreement between the SoS and the constituents. In a collaborative SoS, the constituents collaborate on their own will when they agree on a common global service (called mission) they should provide. A virtual SoS gathers constituents that collaborate regardless of any well-identified global service. Two characteristics are challenging. First, the constituent systems retain their own operational and managerial independence, therefore the system of systems has low or no control over its constituents. Second, the emergent behavior results in the ACM ISBN 978-1-4503-4781-5/16/11. DOI: http://dx.doi.org/10.1145/2993412.3004845 fact that architect cannot anticipate the interaction results between constituents. Regarding sustainability aspects of SoS, the architect needs to continuously evaluate the current architecture regarding the requirements. If a SoS diverges from requirements, the architect needs to compute a new architecture and deploy it. This deployment comes at run-time mainly. SoS usually involve critical systems or economic systems (e.g emergency services, transportation services) than can not support the disruption or degradation of some services. The SoS architecture continuously evolves: it is what we call evolutionary development.

The SoS domain emphasizes the need for reconfiguration. In component based systems, components are considered with homogeneous reconfiguration semantics and a complete control on operational behavior and connections. For example in the Fractal component model1 , components are assumed to implement the same reconfiguration semantics and a component can be stopped independently of its execution state. In a SoS, the reconfiguration semantics between constituents is not uniform. Each constituent retains a particular reconfiguration semantics, and to stop an ongoing operation is not the rule. Alternatives must be designed for each need of reconfiguration at run-time. As a consequence we advocate that specification, design and validation for reconfiguration become time consuming, and that it would be useful to provide assistance to the architect.

To deal with SoS architecture complexity and to assist architects, various kinds of architectural patterns were proposed [START_REF] Ingram | Modelling patterns for systems of systems architectures[END_REF][START_REF] Kalawsky | Using architecture patterns to architect and analyze systems of systems[END_REF][START_REF] Schmidt | Pattern-Oriented Software Architecture, Patterns for Concurrent and Networked Objects[END_REF]. These patterns are suitable to assist the architect with development, maintenance and evolution steps. They are reusable units for specification, design and analysis of architecture. We propose to use the same abstraction to address reconfiguration issues. Our approach fosters the reuse of principled solutions and the documentation of the decisions made in the design of each SoS reconfiguration.

Our contribution in this paper is the design of reconfiguration patterns applied to system of systems architectures. In the literature, reconfiguration patterns do not consider SoS context. In [START_REF] Gomaa | Software reconfiguration patterns for dynamic evolution of software architectures[END_REF] the reconfiguration patterns mainly concern components and is based on the quiescent state of components. The managerial independence of constituents on a SoS does not allow to control their states. [START_REF] Ramirez | Design patterns for developing dynamically adaptive systems[END_REF] propose to describe basic reconfiguration operations. It does not address the complex interdependencies involved in the reconfiguration. This could lead to inconsistencies on other parts of the SoS.

In our proposition, reconfiguration patterns could be considered at various levels. Coarse-grained patterns describe overall and global reconfiguration strategies. Fine-grained patterns describe solutions to specific reconfiguration issues. We advocate that patterns with coarse-grained identify the technical issues that need to be solved by fine-grained patterns. They will capture the context of managerial independence. To illustrate our contribution we define a pattern and we apply it to an SoS example. We show how this pattern effectively helps the architect in the design of reconfigurations.

The paper is organized as follows. The next section further describes the systems of systems approach and its particularity with regard to reconfiguration. Section 3 presents the concept of reconfiguration pattern based on its documentation, reuse and composition aspects. Section 4 describes one concrete reconfiguration pattern, as well as an example of it use. Section 5 discusses related work. Finally we conclude in Section 6 and give future directions.

APPROACH

Due to the intrinsic nature of SoS, the design and implementation of reconfigurations are routine activities during SoS operation, hence calling for a more effective and streamlined engineering process. Proposing reconfiguration patterns aims at improving in this regard. When the patterns are well-documented, they provide principled solutions to the most frequent situations with respect to the capabilities and willingness of the constituents to contribute to the reconfiguration. When no pattern matches the situation, the architect is still able to design and implement an ad hoc reconfiguration. The case for SoS is not a fundamental shift in terms of dynamic reconfiguration, in that the basic mechanisms can be similar to those in other adaptive systems. But an architect who needs to reconfigure a system of systems faces several difficult tasks related to managerial independence and emergence. She/he needs to define a new architecture, but also to make design choices in accordance with the constraints imposed by the SoS constituents, in order to create a viable reconfiguration design, which can later be implemented. The reconfiguration patterns we propose aim at providing the architect with assistance in these tasks. Following the idea of design patterns, architectural patterns, etc., the reconfiguration patterns capture typical solutions to reconfiguration problems with well-known properties. Patterns help in reusing reconfiguration knowledge and expertise in order to streamline this activity.

Issues in the design of a SoS reconfiguration

The documenting aspect of the reconfiguration patterns follows from the architect's requirements expressed whenever a reconfiguration is considered. As usual, the architect has to identify and express the changes that occur within the architecture. The changes of a SoS architecture are made of additions and removals of constituents, as well as changes of connections between the constituents. Namely, Changes describe how to go from the initial architecture to the target architecture while preserving the coherence of the running system. The architect must preserve transaction mechanism during the reconfiguration, for instance, to ensure that any transaction started ends and there is no deadlock.

Moreover, the managerial independence of constituents in the SoS leads to consider that each constituent is designed, deployed and managed independently of the other constituents and of the SoS. Each constituent may expose specific or restricted reconfiguration capabilities. As a consequence the SoS architect needs to make the constituent reconfiguration model explicit to deal with constituents having heterogeneous capabilities. An explicit model with formal semantics helps to identify a situation in which applying a pattern. For instance, the Stop operation does not have the same semantics in a directed or a collaborative SoS: in the former case, a constituent totally stops its activity; in the latter case, a constituent stops its contribution to the SoS, but it possibly still operates in other contexts. The more control the SoS has over its constituents, the more cooperative the constituents are, including with respect to reconfiguration. Therefore, the solutions to reconfiguration issues must be clearly settled in the context of a well-identified type of SoS management.

Even if the SoS architect can influence the constituents management, a SoS can not forcibly instantiate, stop, start or use a given constituent. When the SoS delegates some of the reconfiguration operations to its constituents, the latter decide how and when to perform these operations. We introduced the mediators to manage interactions between constituents. Thus we need a well-documented behavioral model of the mediators.

Some architectural requirements can be translated into architectural invariants. For instance, a requested invariant can be: given two constituents, there is always one of them being wired in the architecture, i.e., even during the reconfiguration, a service remains continuously provided by either one of the two constituents. The architect can further quantify the acceptable service degradation or minimal service preservation. Behavioral properties are also to be considered during the reconfiguration. For instance, a message order may or may not be preserved between two constituents; real-time deadlines may or may not be relaxed; failures may or may not be taken into consideration; and so on. Such behavioral properties may require specific reconfiguration actions such as the inclusion of temporary constituents in order to mitigate transient unavailability of some other constituents. This can have an impact on constituent implication in the SoS. Indeed, the operational independence characteristics results in SoS constituents having competing interests and priorities for their own services.

The reconfiguration pattern form

The description of a reconfiguration pattern is made of the following fields. Some of them are usual general fields found in any kind of pattern, such as the name, intent, related patterns, forces. Some other fields are specific to the reconfiguration domain, such as the explicit reconfiguration grammar, used to deal with managerial independence. The fields we propose are:

• Name meaningfully identifies the pattern in catalogs.

• Intent informally describes the changes that are addressed by the pattern. It gives an indication about when the pattern can be applied.

• Context indicates the situations whenever the pattern can be considered, and the associated vocabulary used within the pattern. As such, the context is further refined by two main aspects:

-Related architectural pattern optionally refers to a well-documented architectural context, including specific terminology, structural constraints and properties.

-Category of SoS management documents what kind of SoS is considered.

• Problem refines the intention with the objectives and goals of the pattern. It explains why the pattern addresses a non trivial situation. The problem field contains the following items:

-Initial/target architectures describe the starting and the ending points of the reconfiguration addressed by the pattern. While any ADL can be used here, graph-grammar-based, e.g., [START_REF] Eckardt | Modeling and verifying dynamic communication structures based on graph transformations[END_REF] or logicbased, e.g., [START_REF] Arshad | Deployment and dynamic reconfiguration planning for distributed software systems[END_REF][START_REF] Buisson | Safe reconfiguration of coqcots and pycots components[END_REF][START_REF] Simonot | A declarative formal approach to dynamic reconfiguration[END_REF] languages allow to abstract over the unchanged parts of the architecture.

-Architectural invariant and Behavioral properties explicit the conditions ensured by the pattern during reconfiguration. They can be given as an informal prose or formalized using, e.g., standard [START_REF] Buisson | Safe reconfiguration of coqcots and pycots components[END_REF][START_REF] Simonot | A declarative formal approach to dynamic reconfiguration[END_REF] or temporal logic [START_REF] Etzien | Contracts for evolving systems[END_REF][START_REF] Ramirez | Design patterns for developing dynamically adaptive systems[END_REF].

-Forces explain the general principles and points adopted in the pattern solution. They motivate these principles with respect to the objectives depicted in the problem.

• Solution is the advocated means by which the problem can be solved. It may contain some artifacts like (template) reconfiguration scripts, for instance. It includes a Reconfiguration grammar that specifies the capabilities of the reconfiguration mechanisms assumed by the participants of the pattern. At first glance, a statechart is used to express the reconfiguration grammar. Formal logical approaches [START_REF] Buisson | Safe reconfiguration of coqcots and pycots components[END_REF][START_REF] Simonot | A declarative formal approach to dynamic reconfiguration[END_REF] provide a more precise specification with unambiguous semantics.

• Consequences discuss impacts of the solution, especially in terms of quality attributes.

EXAMPLE OF A RECONFIGURATION PATTERN

To illustrate our approach, we first give the description of a reconfiguration pattern. Then, we discuss how this pattern is actually used to reconfigure a specific system of systems.

Co-evolution of configurations pattern

• Name: Co-evolution of configurations

• Intention: The co-evolution of configurations pattern allows to deploy a target architecture concurrently to a source architecture without stopping the running services. It is possible to change a service provider in a transparent way for consumers. Consumers progressively migrate from the source to the target architecture while states shared between source and target architectures co-evolve.

• Context: 1). Emergency SoS (see fig. 4) usually contains two kinds of constituents. The C2 (a single one) oversees the realization of the mission (SoS global service) and provides the state of the situation based on reports received from the other constituents. It coordinates the individual tasks of the other constituents, the operators, depending on its view of the mission (see Fig. 1).

The operators achieve the tasks that are required to fulfill the mission assigned in the directives sent by the C2. They report their progress to the C2. Placed in the middle, a mediator observes and transmits the messages between the constituents.

-Category of SoS management This pattern fits well to any directed or acknowledged SoS since it requires that the C2 provides its internal state for the purposes of the migration and a synchronization mechanism during reconfiguration. Moreover the SoS offers capabilities to observe and exchange control between systems. Despite the transient phase, the pattern ensures that the new C2 acquires a consistent view of the mission.

• Problem: The problem lies in the fact that the operators can hardly be reconnected atomically to the new C2, exactly at the same time that mission knowledge and supervision are transferred from the old C2 to the new one. Because of their own managerial independence, the operators freely choose when to disconnect from the old C2 and start reporting to the new one. Consequently, there is a transient phase during which some of them still report to the old C2 while the other ones have already switched to the new C2. If no specific action is taken, the two C2 have divergent views of the mission, and after the reconfiguration, the view of the new C2 is inconsistent. Due to this, the new C2 may order incorrect actions that can compromise the mission. Moreover, the SoS can not force the operators to change their connections in the SoS.

-Initial/target architectures The figure (see fig. 1)

shows the reconfiguration issue. The architect wants to replace the C2 while preserving the executed mission.

-Architectural invariant and Behavioral properties • Solution: In order to not suspend the mission, the solution consists in concurrently deploy the initial and final architectures. The operators are migrated individually. During the transient phase, when the two C2 are contributing to the SoS, they are synchronized by means of scripts provided by the architect, in order to share the same view of the mission. The synchronization task is devoted to the mediator, such that the pattern does not have to assume any specific collaboration from the constituents.

-Reconfiguration grammar

The C2 constituent should provide the reconfiguration grammar depicted by Fig 3:

-Migrate: transfers the internal state from the old C2 to the new one. In the old C2, Migrate means exporting the internal state; in the new C2, it means importing the internal state. Because we consider the case of directed and acknowledged SoS, the SoS architect has to ensure that the two C2 use compatible state representations.

-Synchronize: propagates the changes made to the internal state from one C2 to the other one.

As for the Migrate operation, the SoS architect is responsible to ensure that the two C2 use a compatible encoding.

-The transition to the de-synchronized state does not result from any specific action. The C2 are unaware that their view of the mission is not consistent anymore. Only the mediator can observe the de-synchronized state.

This grammar ensures that the C2 may exchange state during the reconfiguration. This allows the new C2 to be initialized with the mission state of the old C2.

As depicted in figure 2 a mediator has the ability to intercept the messages forwarded to C2s. This behavior allows the two C2 to be continuously synchronized during the reconfiguration. The solution is decomposed in the following steps: * Before any message can reach the new C2, the mediator initiates a state migration from the old C2 to the new one. The messages to the C2 are postponed until migration is completed. After migration all C2 have been synchronized. * When the mediator intercepts a message to a synchronized C2, the other C2 is marked as de-synchronized, and the intercepted message is forwarded and delivered to its destination. * When the destination of an intercepted message is a desynchronized C2, the synchronization script is triggered before message delivery.

This behavior is implemented by the mediator for the time of the reconfiguration.

The reconfiguration completes when all the operators are connected to the new C2. The architect can provide an additional protocol in order to deal with the case where some operators are reluctant to connect to the new C2.

• Consequences:

-Communication consistency is maintained. Operators preserve their connection to the old C2 until they have not migrated to the new C2

-Availability is maintained. The SoS is still functional during reconfiguration.

-Reliability is partially maintained, since information reported by the operators to the C2 are preserved. Nevertheless, the pattern does not deal with de-synchronized C2 sending contradictory or inconsistent directives.

-Performance is affected depending on how many times synchronization scripts should to be called. The cost of the synchronization can be mitigated by the use of techniques such as concurrent transactions.

Reconfiguring with the help of reconfiguration patterns

To illustrate our approach we chose the case study of the National shared Infrastructure of Transmissions (INPT)2 , which is an information management SoS used by various French emergency services in case of accidents. This system is particularly used for the resolution of incidents involving interoperability of communications. In this context, the SoS belongs to the category of acknowledged SoS: a SoSlevel coordinator (the Ministry of the Interior in our case) defines how each service shares its communication infrastructure and information even if the service remains independent.

Figure 4 is a possible instance of this architecture (white elements only). Even if this instance is kept simple, it is representative of the reality. The system constituents are:

• CODIS (Operational center) has the mission to coordinate and to supervise operational activities.

• PC (Command Post) appears to assist CODIS mission in big disasters. It manages operational activities for a particular disaster.

• Relay is a communication infrastructure shared between emergency services. Its mission is to relay and route information.

• Resource (R) is an emergency resource that operates in the SoS as humans or vehicles (fire-car, ambulance, police car). Each resource follows its own mission bound to its field of expertise. They receive a mission from the CODIS and communicate mission report by using the upCodis port.

• Mediator are mobile radios that define how PC, Resource and CODIS interact with Relay (communication protocol, data and interaction points).

Emergent behavior is defined by the capacity of constituents to reduce disasters. It is produced by the involvement of different constituents from different emergency services which are coordinated by the CODIS. Regarding the managerial independence each constituent is subordinated to the CODIS but retains its own managerial independence. A reconfiguration case can come from the operational environment. For instance, the CODIS receives a call from citizens who notify a strange smell. The CODIS engages a new mission with several constituent systems from various emergency services. On the spot the constituents discover a chemical disaster with loss of lives, goods and assets. Additional resources are engaged and the CODIS must control and command a lot of resources. To support this overload the CODIS needs to delegate a part of the mission management to a PC, resulting in the new configuration shown in Figure 4. In the reconfiguration pattern presented in the previous section, CODIS and PC play the C2 role and Relay is an operator.

The architecture evolution involves that the CODIS delegates its coordination mission to a PC, and that resources delegated to the mission, switch to the PC channel. Particular protocols are designed to allow commandment delegation in the SoS. The CODIS can share its channel security key with the new commandment in order to talk with the resource. The CODIS can allow the PC to access to its service mission.

To preserve the emergence, the architect has to maintain the coordination of services available during the reconfiguration. He/she must identify what problems need to be solved for ensuring this property (e.g does a deadlock can occur? is the coordination service still reliable?). Then, the architect must produce a specification in response to her/his analysis. Finally the architect designs a reconfiguration in accordance with the specification. The process of realization of the reconfiguration is guided by the pattern.

• First, the architect has to identify a reconfiguration strategy that matches with her/his intention (coarsegrained pattern). The co-evolution pattern can be selected.

• Next, the architect needs to check if the context is similar to his/her architecture. Indeed, the architecture pattern captures system inter-dependence. If the architect is not an expert in reconfiguration problem, the pattern provides explanations in the problem part.

• In the next step the architect must refine the reconfiguration strategy with the technical points identified in the reconfiguration pattern : suspend communication, synchronization script. We assume that the current architecture provides all the conditions to apply the solution. If it not the case, this implies that either the reconfiguration pattern must be composed with another reconfiguration, or the architect must choose another pattern. Then, the architect produces a specification by taking into account invariants (explained in the problem section) that is the mission of C2 is synchronized during the reconfiguration. Finally, the architect follows the solution to obtain his/her reconfiguration.

To resume, the reconfiguration is a complex engineering task. Indeed the architect can not impose a new configuration to systems and the SoS mission is critical and requires a safe reconfiguration. The pattern approach assists the architect in the different reconfiguration steps: specification, design.

RELATED WORK

The COMPASS and DANSE projects are the two major EU-funded projects about the system-of-systems methodology for the 2010-2015 period. Only the DANSE project studied the reconfiguration problem. In this project, the proposed methodology [START_REF] Winokur | Danse methodology v03. deliverable D 4.4[END_REF] consists in renewing the needs assigned to the system of systems whenever its behavior differs from prediction. Based on architectural patterns, many architecture variants are generated considering all the possible combinations of rule application from a graph grammar. Simulation and optimization allow to assess the generated architectures in order to select the best one according to the architect's directives. In addition, the architect uses timed temporal logic to define static and dynamic contracts in order to specify acceptable architectures and evolution of the system of systems [START_REF] Etzien | Contracts for evolving systems[END_REF]. While this method addresses the issues of the target architecture and of the reconfiguration specification, to the best of our knowledge, the DANSE project does not prescribe how the reconfiguration is effectively built. In this regard, we think that our work is orthogonal to the proposals of the DANSE project. Several previous work draw general principles to address dynamic reconfiguration. Among them, designed for user interfaces, C2 [START_REF] Taylor | A componentand message-based architectural style for gui software[END_REF] is an architectural style in which the components are organized in layers separated by connectors. C2 assigns to the connectors a key role for dynamic reconfiguration. Since the connectors are responsible of the message routing policy, any reconfiguration action actually occurs within the connectors. While C2 is not designed for system of systems, its organization with respect to dynamic reconfiguration provides a potential solution to overcome the lack of control over the constituents. In a different context, Boyer et al [START_REF] Boyer | Robust reconfigurations of component assemblies[END_REF] proposed a formally-verified algorithm in order to issue a valid sequence of reconfiguration actions, even in presence of component failures. The reconfiguration is split in two phases: in the down phase, components are stopped, unwired and destructed; in the up phase, they are created, wired and started. Whenever failures are detected in the process, the effects of failures are first propagated, i.e., failed components are unwired, then the reconfiguration process resumes to either recover the initial architecture or reach the target one. A similar principle may address the case when constituents choose to disengage from a system of systems.

Other works define and analyze reconfiguration with architectural styles [START_REF] Taylor | Architectural styles for runtime software adaptation[END_REF][START_REF] Dorn | Analyzing runtime adaptability of collaboration patterns[END_REF]. They focus on how the architectural style affects reconfiguration. In none of these works, the step towards capturing general principles for solving the reconfiguration problem has been done. Though, their underlying principles are of interest in the SoS area.

To build the standard form, our inspiration comes from the work on patterns, including in other areas. Gomaa et al [START_REF] Gomaa | Software reconfiguration patterns for dynamic evolution of software architectures[END_REF][START_REF] Gomaa | Software adaptation patterns for service-oriented architectures[END_REF] collect several patterns targeted at reaching quiescence noticeably in client/server, master/slave and serviceoriented architectures. On the one hand, quiescence is no more no less one possible solution to the specific problem of lowering the coupling for the time of reconfiguration. We think that reconfiguration patterns should have a wider scope. On the other hand, the standard form of their pat-terns contains only the state-charts that relate the operational states to the quiescent ones. They do not give an explicit statement of the addressed problem, nor do they provide explicit assumptions or properties of the patterns.

Ramirez et al [START_REF] Ramirez | Design patterns for developing dynamically adaptive systems[END_REF] adopt a more complete form derived from the one of Gamma et al [START_REF] Gamma | Design Patterns: Elements of Reusable Object-oriented Software[END_REF] with an additional LTLbased constraint entry. Their catalog mixes design patterns to build monitoring and decision-making subsystems, as well as reconfiguration patterns. However, having a single common form for everything leads to omit the specificities of each area. For instance, their component insertion and component removal patterns make undocumented assumptions on the reconfiguration grammar that can hardly be guessed, and which appears incompatible with the system of systems approach. Our work may be less general, but, being specifically targeted at reconfiguration, it does not suffer such drawbacks.

In summary, in the current state of the art, the problem of designing and building reconfigurations that accommodate the unique characteristics of systems of systems has still to be addressed. Some work proposed general principles that could be helpful in the context of systems of systems. However, to be really reusable, their presentation must be homogenized, like it is done for patterns in general. While existing pattern catalogs in other areas gives insights, their usual presentation does not fit the specificities of reconfiguration patterns.

CONCLUSION

In summary, we think that reconfiguration requires well documented design: we suggest a pattern approach. An important characteristic to be documented is the heterogeneity of the reconfiguration behavior of system. Moreover, reconfiguration patterns for SoS should document how to deal with the lack of coercion on systems. This characteristics comes from operational and managerial independence. The emergence characteristic regarding reconfiguration is not specific. It involves to preserve particular configuration during reconfiguration. This aspect is tackled by architectural invariants and behavioral properties.

We consider that reconfiguration patterns have different granularity. Coarse-grained patterns describe overall and global reconfiguration strategies. Fine-grained patterns describe solutions to specific reconfiguration issues. As consequence coarse-grained patterns identify technical issues (variability point) and leave architecture to solve them with fine-grained pattern for instance.

In this paper we proposed a new approach to address reconfiguration, taking into account the specific considerations of SoS: we extend the pattern concept to reconfiguration. Our underpinning idea consists in documenting reconfiguration and providing principled solutions to well-identified reconfiguration problems. These patterns are adaptable building blocks that can be reused in different systems which share a similar context and similar objectives, hence lowering design time of SoS reconfiguration. With explicit architectural invariants and behavioral properties, the patterns make it clear how the system behaves even during reconfiguration. The architectural invariants assist the architect in providing specifications to the reconfiguration.

The patterns assist the design step. They describe problems involved by reconfiguration in a particular architectural context and a solution. This documentation will provide with solid ground in the perspective of the analysis and verification of reconfigurations.

From the verification perspective, patterns provide models. As consequence engineering effort to check a reconfiguration can be reused in other similar contexts. From a validation perspective, the solution provides the operations that led to maintain the invariant set in the problem.

In our future work, we intend to build a pattern system, in the sense of [START_REF] Bushchmann | Pattern-oriented software architecture: a system of patterns[END_REF]. Namely, it will be a collection of patterns, which defines a language, along with guidelines on how this language should be used in practice to design and build reconfigurations. We foresee that this pattern system will address several classes of reconfiguration problems, ranging from effective changes to their management and coordination. We ultimately aim at addressing all of them consistently in a single pattern system.

Figure 1 :-

 1 Figure 1: Architecture diagram for the co-evolution pattern

Figure 2 :

 2 Figure 2: Behavior of the mediator in the coevolution pattern

Figure 3 :

 3 Figure 3: Reconfiguration grammar for the C2 systems in the co-evolution pattern

Figure 4

 4 Figure 4: French Emergency SoS system CODIS

http://fractal.ow2.org/

Infrastructure Nationale Partageable des Transmissions http://crd.ensosp.fr/index.php?lvl=notice display&id=9395


Raouia Aouini

Bogdan Marinescu

Ben Khadija

Mohamed Kilani

Elleuch

Synchronverter-Based Emulation and Control of HVDC Transmission

Keywords: HVDC, synchronverter, synchronous generator/motor, regulator parameters tuning, residues, optimization

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

IGH Voltage Direct Current (HVDC) systems uses direct current for bulk electric transmission power based on high power electronics which provide the opportunity to enhance controllability, stability and power transmission capability of AC transmission systems [START_REF] Murali | Comparison of FACTS Devices for Power System Stability Enhancement[END_REF]. Practical conversion of power between AC and DC became possible with the development of power electronics devices such as thyristors and Voltage Source Converters (VSC). The VSC technology offers even more flexibility, new capabilities for dynamic voltage support, independent controls of active/reactive power and easier integration of wind farms [2]. These HVDC transmission systems are specifically used to connect asynchronous grids, as for example the England-France interconnection [START_REF] Goodrich | The 2000 MW HVDC link between England and France[END_REF].

Nonetheless, other equally important HVDC applications concern complex AC interconnected systems in order to enhance the power transmission capacity and meet the growing demand.

The planned Spain-France interconnection [START_REF] Decoeur | France-Spain interconnections, First step for smart grids[END_REF], which will use the VSC technology scaled up to 2000 MW, is such an example. These emerging applications gave rise to the coexistence of parallel HVDC/HVAC, and consequently, an increased level of AC/DC/AC converted power injected into AC networks.

Many studies have shown that the methods of controlling HVDC converters have an impact on stability of the system in which the link is inserted. The main trend in control techniques for VSC HVDC links are based on the well established vector control scheme. For example, in [START_REF] Hammad | Improving the dynamic performance of a complex AC/DC system by HVDC control modifications[END_REF] the standard vector control has been modified to improve the dynamic performance of a parallel AC/DC interconnection. In [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF] the VSC-HVDC operating characteristics are determined by a decoupled PI controller to provide decoupled and independent control of the active and reactive powers for each converter. Generally, and due to their simple structure and robustness, PI controllers have been adjusted to meet HVDC specifications. For example, the authors in [START_REF] Latorre | Improvement of power system stability by using a VSC-HVDC[END_REF] proposed a robust control scheme for a parallel AC/DC system. In [START_REF] Rostamkolai | An adaptative optimal control strategy for dynamic stability enhacement of AC/DC power systems[END_REF], an adaptive optimal control was developed for an HVDC system.

Recently, in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]- [START_REF] Torres | Frequency control improvement in an autonomous power system: An application of virtual synchronous machines[END_REF], the authors have proposed a different control method for which an inverter can be operated to mimic the behavior of a synchronous generator (SG) and the resulting closed-loop has been called a synchronverter [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. Since the operation of AC systems via SGs voltage/frequency regulation is rather well known [START_REF] Kundur | Power system stability and control[END_REF], the synchronverter concept led to new applications. For example, in [START_REF] Nguyen | Synchronverter-based Operation of STATCOM to Mimic Synchronous Condensers[END_REF], a STATCOM controller was synthesized from the mathematical model of synchronous generators operated in a compensator mode.

In this work, the synchronverter concept is adapted to converters of an HVDC transmission. The idea is a conceptual control strategy of the DC line, where the sending-end rectifier controls emulate a synchronous motor (SM), and the receiving end inverter emulates a SG, both along with their controls. This resulting Synchronverter based HVDC will be called SHVDC in the paper. It is shown that the parameters usually used for the regulations of the SG of the same capacity cannot be directly used for the SHVDC. This led us to propose a tuning method of the HVDC converters parameters in order to satisfy both the local HVDC control requirements and to improve the transient stability of the neighbor AC zone of the HVDC link. The rest of the paper is organized as follows: in Section II, the conventional modeling and control of an HVDC-VSC transmission system is briefly overviewed. The synchronverter concept is extended in Section III to the case of an HVDC link. In Section IV, the SHVDC structure

H

proposed in Section III for the HVDC link is analyzed from a structural point of view. An analytic method to tune the parameters of the controllers of the SHVDC in order to meet the desired performances is given in Section V, while validation tests are presented in Section VI.

II. CONVENTIONAL MODELING AND CONTROL OF AN HVDC-VSC TRANSMISSION SYSTEM

An HVDC system consists of three parts: a rectifier station, an inverter station and a high-voltage DC transmission line as shown in Fig. 1. Each converter is connected to the AC network via a grid impedance g Z representing the short-circuit power at the connection point. For its advantage on control capability, the VSC technology is assumed. Each converter holds two degrees of freedom for the control. Usually, these degrees are used for the control of the reactive power at each side, of the active power and for the DC voltage [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF]. Each station controls its reactive power independently of the other station. The flow of active power in the DC transmission system must be balanced, which means that the active power entering the HVDC system must mutch the active power leaving it, plus the losses in the DC transmission system. To achieve this power balance, one of the stations has to control the active power, while the other station should be designed for the DC voltage control. Usually, the two converters of an HVDC are controlled by two independent loops and each of these controls is based on the vector control approach in the d-q frame, using cascaded PI controllers [2]: the outer control loop generates the respective d-q current references to the inner current control loop.

III. SYNCHRONVERTER-BASED HVDC

In this section, a control strategy based on synchronverter technology is adapted for HVDC-VSC converters where the HVDC converters are run as synchronous machines. The synchronverter proposed in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] is an inverter that mimics the structure and the regulations of a synchronous generator (SG). The usual control strategies used for conventional SG can thus be used for the inverter. For a complete DC system, a rectifier and an inverter are bound by a DC link. Therefore, to provide an HVDC structure, another synchronverter working as a synchronous motor (SM) based on the same mathematical derivation is necessary. As a result, the DC power is sent from the SM to the SG. Firstly, an overview of the synchronverter concept is presented, where the inverter is modeled according the mathematical model of a SG. Next, the synchronverter concept is extended to a rectifier that mimics a SM.

A. Overview of the synchronverter concept

A synchronverter is an inverter which regulations are chosen such that the resulting closed-loop mimics the behavior of a conventional SG. For the purpose of the development, we recall the structure of a three phase round rotor synchronous machine [START_REF] Kundur | Power system stability and control[END_REF] as shown in Fig. 2. The stator windings can be seen as concentrated coils having self-inductance L and mutual inductance (-M ). The field winding can be seen as a concentrated coil having a selfinductance f L .The phase terminal voltage vector

_ [ ] T g abc ga gb gc V V V V = may be expressed by _ _ _ _ g abc g abc s g abc s g abc di V R i L e dt = - - + , (1) where _
where g M is the flux field, g θ is the electric rotor angle and

2 2 ( -) ( + ) 3 3 T g g g g sin sin sin sin π π θ θ θ θ   =      , (3)
The mechanical equation of the machine is given by

g g gm ge gp g J T T D s θ θ = --  , ( 4
)
where g J is the combined moment of inertia of generator and turbine, gm T is the mechanical torque, ge T is the electromagnetic torque and gp D is a damping factor. The torque ge T is given by _ , ge g g abc g

T M i sinθ = < >  ,
(5) where the operator .,.

< > denotes the conventional inner product in ℝ 3 . The real and reactive powers generated by the SG are, respectively _ , g g g g abc g P M s i sin

θ θ = < >  , (6)
_ , ge g g g abc g Q M s i cos θ θ = - < >  . ( 7
)
Based on the SG model above (Eq. 1-7), the concept of synchronverter is developed. The latter consists of an inverter on which specific controls are built and structured, as shown in Fig. 3. On the figure, we depict (i) the power

θ g  - + R s , L s a + R f , L f b R s , L s R s , L s c + + - - Vg_a Vg_b Vg_c - Z s _ m abc V Z g Z g Bus1 Bus 2 _ m abc e Z s _123 m V ~ , m me P Q , g ge P Q PWM PWM 1 dc V 2 dc i 1 dc i , dc dc L R _ g abc e _ g abc V _123 g V _ m abc i cc i ~
part consisting of the inverter plus the LC filter and (ii) controls assured by the electronic part.

Such control structure is shown to be equivalent to a SG with capacitor bank connected in parallel with the stator terminal [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. More specifically, with the terminal voltages The synchronverter given in Fig. 3 is connected to the grid via the impedance ( g L , g R ), thus, its output current and voltage may be given by _ _ _123

1 ( )

g abc g abc g f V i i C s = - , ( 8
) _123 _ _123 1 ( ) ( ) g g abc g g g i V V R L s = - + . ( 9
)
We may write a swing equation for the synchronverter, similar to equation ( 3):

1 ( )

g gm ge gp g g T T D s J θ θ = --  ,
where the mechanical torque gm T is a control input, g θ is the angle position of the synchronverter. The electromagnetic torque ge T depends on _ g abc i and g θ .

To mimic the droop of the SG, the following frequency droop control loop is proposed _ ( )

gm gm ref gp n g T T D s ω θ = + - . (10)
The synchronverter thus shares load with the other generators of the AC grid to which it is connected in proportion with the static droop coefficient gp D . In [START_REF] Zhang | Power synchronization control of grid-connected voltage-source converters[END_REF],

_ gm ref T
is the mechanical torque applied to the rotor and it is generated by a PI controller as shown in Fig. 3 to regulate the real power output g

P _ _ _ _ ( )( ) g g i p gm ref p p g g ref K T K P P s = + - . (11)
The reactive power g Q is controlled by a voltage droop control loop using a voltage droop coefficient gq D , in order to regulate the field excitation g M , which is proportional to the voltage generated following [START_REF] Driesen | Virtual synchronous generators[END_REF] and [START_REF] Chen | Improving the grid power quality using virtual synchronous machines[END_REF] 1 ( )

g gm ge g M Q Q k s = - , (12)
Fig. 3. Model of the synchronverter: power and electronic parts [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] g_ref _ ref ( )

gm gq g g Q Q D V V =+ - , (13)
where g V is the output voltage amplitude computed by 2 3(

g ga gb ga gc gb gc V V V V V V V = + + . ( 14
)

B. Extension of the synchronverter concept to the HVDC emulation

To obtain an HVDC transmission system, in addition to a SG emulated by a synchronverter as in the section above, a rectifier that mimics a SM and a DC line are needed. The resulting system, a SM/SG and a DC line are called Synchronverter High-Voltage Direct Current (SHVDC).

B.1. The synchronverter model of a SM

The synchronverter model of a SM consists of an LC filter and mainly controls shown in the same Fig. 3. It's nearly the same model discussed above, apart from the fact that the sign convention for the stator current 1), ( 4), ( 8) and ( 9) are re-written in motor convention

_ [ ] T m abc ma mb mc i i i i = is changed. Equations (
_ _ _ _ m abc m abc s m abc s m abc di V R i L e dt = + + , (15) 1 (
)

m me mm mp m m T T D J θ θ = - -   , ( 16
) _ _ 123 _ 1 ( ) m abc m m abc f V i i C s = - , ( 17
) _123 _123 _ 1 ( ) ( ) m m m abc g g i V V R L s = - + . ( 18
)
The bloc equations (2), ( 5) and ( 7) remain the same

_ sin m abc m m m e M sθ θ =  , ( 19
) _ , me m m abc m T M i sinθ = < >  , ( 20
) _ , me m m m abc m Q M s i cos θ θ = < >  , ( 21
) _ , m m m m abc m P M s i sin θ θ = < >  . ( 22
)
The frequency and the voltage droop controls are also similar to the SG emulation case; except for the power regulation [START_REF] Beck | Virtual synchronous machine[END_REF] which is replaced by the DC voltage control loop as defined by ( 24)

~ _ g ref V _123 g i _ g ref P g M - ge T - n ω g θ  g θ 1 J *s g gp D (2) (4) (5) Eqn Eqn Eqn 1 k s g _ gm ref T (11) Eqn 1 s _ e g abc g V (14) Eqn _ g abc i _ g abc e Bus 2 , s s R L ~ , g ge P Q ge Q - - cc i 2 dc i 2 dc V - f C _123 V g _ i g abc g P gq D _ g ref Q dc C PWM _ V g abc , g g R L _ ( ) mm mm ref mp n m T T D s ω θ = + - , (23)
_ _ _ 1 _ ( )( ) i vdc mm ref p vdc dc ref dc K T K V V s = + - , (24) m_ref _
( )

mm mq m ref m Q Q D V V =+ - , (25) 1 (
)

m mm me m M Q Q k s - = - , ( 26
) 2 3( m ma mb ma mc mb mc V V V V V V V = + + . ( 27
)

B.2. Coupling equations

The circuit equations of the DC line (Fig. 1 & 3) are 1 1

( )

dc dc cc dc V i i C s = - , (28) 2 2
1 ( )

dc cc dc dc V i i C s = - , (29) 1 2
1 ( )

cc dc dc dc cc dc i V V R i L s = - - . ( 30
)
The AC and DC circuits are coupled by the active power relation

1 1 1 dc dc dc P V i = , ( 31
) 2 2 2 dc dc dc P V i = , ( 32
) 1 dc m P P = , ( 33
) 2 dc g P P = . (34)
IV. STRUCTURAL ANALYSIS OF SYNCHRONVERTERS BASED HVDC In this section, the SHVDC is analyzed from a structural point of view. More precisely, the droop and voltage regulations introduced in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] for the synchronverter in Fig. 3 are appropriate in the sense that they cancel all the degrees of freedom of the system, so that, with given initial conditions, the trajectories of the closed-loop system in Fig. 3 are uniquely defined. Roughly speaking, the number and the nature of the regulations are necessary and sufficient for this system. When coupling two synchronverters as in Section II to form the SHVDC, this fact is not obvious. Moreover, controls for the power exchange and the DC voltage have been added by equations [START_REF] Beck | Virtual synchronous machine[END_REF] and respectively (24). The scope of this section is to prove that the proposed structure is well-defined from the point of view mentioned above. As the model of the SHVDC is given by equations ( 1)-(34) in a general Differential Algebraic Equations (DAE) form, the approach introduced in [START_REF] Bourlès | Linear-Time Varying Systems, Algebraic-Analytic Approach[END_REF] and related references can be directly used. In this latter, a linear system Σ is a module M defined by a matrix equation . For the SHVDC, w is given by (36), and ( ) S s results from the linear approximation of equations ( 1)-(34). Let k= length (w) and r=rank (S(s)). w=[i ,V ,e ,i ,i ,V ,e ,i ,P , T ,T ,T ,θ ,P ,T ,T ,T ,θ ,Q ,Q ,M ,V ,

V ,i ,P ,Q ,Q ,M ,V ,V ,i ,i ,P ,sin ] g θ  . (36)
From [START_REF] Bourlès | Linear-Time Varying Systems, Algebraic-Analytic Approach[END_REF], the integer k m r = -is the rank of the module M , which coincides with the rank of the linear system Σ . The number of the independent inputs

{ } 1 ... m u u u =
of the system M given by ( 35) is the rank of the module which defines the system m=rank (M). For the SHVDC case, the rank of the module M defined by the SHVDC is zero, which means that the regulations included in (1)-(34) are necessary and sufficient to well define the behavior of this structure. The choice of the regulations parameters in order to ensure stability and desired performances is next discussed.

V. TUNING OF A SYNCHRONVERTER BASED HVDC

The SHVDC structure is investigated in this section on the simple system of Fig. 1. A four machines system will be used in Section VI. The simulations tool in the Matlab/Simulink toolbox.

A. Tuning with standard parameters of classic SG

The main advantage of the synchronverter is to put into evidence the structure of classic generators for which the controls are well known. For this reason, we firstly adopted for the new SHVDC structure standard parameters of a SG which are given in the first column of Table III. They mainly correspond to 5% governor droop [START_REF] Kundur | Power system stability and control[END_REF] ( [START_REF]IEEE Recommended Practice for Excitation System Models for Power System Stability Studies[END_REF]. The responses of the SHVDC to step changes in active/reactive power references are given in Fig. 4 in solid lines in comparison with the ones in dotted lines obtained with a classic vector control (see, e.g., [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF]). Although the voltage and reactive power dynamics are quite similar, a poorly damped response is obtained for the active power P with the SHVDC. As a consequence, the usual parameters of a synchronous generator of the same capacity cannot be directly used for the SHVDC and a specific tuning of the parameters of the latter was done as presented in the next section. B. Specific framework for the tuning of the SHVDC parameters B.1. HVDC control specifications: we first recall the usual full set of control specifications for a VSC based HVDC, as presented in Section II. Set-points for the transmitted active power, the reactive power and the voltage at the points of coupling have to be tracked with the following transient performances [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF][START_REF] Sangathan | Performance of high-voltage direct Current (HVDC) systems with Line-commutated converters[END_REF]: -the response time of the active/reactive power is normally in the range of 50 ms to 150 ms.

-the response time for voltage is about 100 ms to 500 ms. The dynamic of the HVDC should also improve the stability of the neighboring (see, e.g., [START_REF] Hammad | Improving the dynamic performance of a complex AC/DC system by HVDC control modifications[END_REF]). B.2. Control structure: in order to satisfy these specifications, the SHVDC parameters should be tuned simultaneously and in a coordinated way. For this tuning, the SHVDC shown in Fig. 1 is put into the feedback system structure presented in Fig. 5 where ∑  is modeled by equations ( 1) to (34) with the exception of equations ( 10), ( 11), ( 13), ( 24), ( 25) and (26). All control parameters are grouped in the following diagonal matrix

B.3. Regulator parameters and residues: let ( )

H s be the transfer matrix of a linear approximation of ∑  and consider each closed-loop of the feedback system in Fig. 5 which corresponds only to input i u and output i y Fig. 6. Single input/Single output feedback system More specifically, ( ) ii H s and ( ) ii K s are the (i,i) transfer functions of H(s), ( , ) K s q , respectively. Proposition (e.g., [START_REF] Rogers | Power System Oscillations[END_REF]): The sensitivity of a pole λ of the closed-loop in Fig. 6 with respect to a parameter q of the regulator ii K is ( ,

) ii K s q r q q λ λ ∂ ∂ = ∂ ∂ (38)
where r λ is the residue of ( )

ii H s at pole λ .

∑  ( , ) K s q ' u + e y u ( ) ii H s ( , ) ii K s q + i e ' i u i y i u
Note that, for our case. (37), ( , )

1 ii s K s q q λ = ∂ = ∂ .
Also, the poles of ( ) ii H s are among the poles of ( ) H s . As a consequence, the residues of the linear approximation of ∑  can be used for the pole placement as shown in Section V.D. Before, some basic notions of modal analysis are recalled.

C. Modal analysis of the system

This analysis is based on a state representation of the transfer matrix ( ) H s of the form

: x=A x+B , =C x u y Σ  . ( 39
)
The poles i λ of ( ) H s correspond to the eigenvalues (modes) of A. The participation factors measure the relative participation of the k th state variable in the i th mode:

* ki ik ki p ψ ϕ = , (40)
where i ϕ and i ψ are respectively the right and the left eigenvectors of A associated with the mode i λ . They are given in Table II for system (39). Each element ( ) ij H s of ( ) H s can be expanded in partial fractions as 1 ( )

k n k ij k k r H s s λ = = = - ∑ , (41)
where r k is the residue of ( ) ij H s at pole i λ expressed as

k i k k j r C B ϕ ψ = , (42)
where i C is the i th line of C and j B is the j th column of B [START_REF] Rogers | Power System Oscillations[END_REF].

D. Coordinated tuning of SHVDC parameters

First, desired locations * i λ can be computed for each pole i λ starting from the control specifications given in Section B.1.

To ensure a second order type response without overshoot, the damping of the corresponding complex mode i λ is imposed to be 0.7

i ζ =
. The frequency of the mode is deduced from the desired response time t r given in Section B.1: 0 3 i r t ω = [START_REF] Rogers | Power System Oscillations[END_REF].

Thus, the desired location of i

λ is * 2 0 0 1 i i i i i j λ ζ ω ω ζ = - ± -
and the results for the dynamics of interest for the SHVDC are given in Table I (column 3).

It should be noted that, the dynamic of reactive power is the same as the dynamic of the voltage. The connections in Table I between the dynamics of interest and the modes are established based on the participations factors given in Table II. Only the highest participation factor is considered, therefore each dynamic is associated to only one pole for the computation of desired poles * i λ . However, from the same column of Table II; several poles have also significant participations to the same dynamic which led us to compute the gains K in a coordinated way. More specifically, if Λ denotes the set indices j from 1 to 8 for which ( ) jj H s has i λ as pole, the contribution of each control gain in the shift of the pole is

0 i i ij j j r K λ λ ∈Λ = + ∑ , (43)
j i i K i K j λ λ = = - ∑ , (44)
where i λ is given by (43). The optimal parameters in the second column of Table III were obtained with (44) solved for the desired locations in Table I. The response of SHVDC for the link in Fig. 1 with these tuned parameters is given in Fig. 7 in solid lines in comparison with the ones in dotted lines obtained with a classic vector control. The active power, the voltage and the reactive power dynamics are similar which prove that the performances given in Section B.1 are correctly ensured by the proposed control methodology. Notice, however, that less overshoots with the new tuned parameters are obtained. The robustness of this approach is studied in the next section on a four machines case. VI. FOUR MACHINES TEST POWER SYSTEM In this section, the specific tuning SHVDC parameters presented in Section V is tested and compared with the classic vector control on the IEEE 4 machines benchmark [START_REF] Kundur | Power system stability and control[END_REF] shown in Fig. 8. The two areas are connected with an AC line in parallel with one HVDC link. The 150 km HVDC cable link has a rated power of 200 MW and a DC voltage rating of ±100 kV. Each area consists of two identical generating units of 400 MVA/20 kV rating. Each of the units is connected through transformers to the 100 kV transmission line. There is a power transfer of 400 MW from Area 1 to Area 2. The detailed system data is given in the appendix. The loads L 1 and L 2 are modeled as constant impedances.

The control SHVDC approach is assessed for local performances and for transient stability. The desired HVDC performances given in Section V are fulfilled as shown by the step responses in Fig. 9. As for the case of the two-machine system, responses with less overshoot are obtained compared to the vector control approach. Fig. 10 shows the response of the angular speed of generator G1 to a three phase short circuit fault of 100 ms duration, occurring in the mid-point of the AC transmission line. We can depict a better dynamic performance with the proposed controller: the transient oscillations with the SHVDC control are more damped. For the same fault, transient stability is assessed by the Critical Clearing Time (CCT), which is the maximum time duration that a short-circuit may act without losing the system capacity to recover to a steady-state (stable) operation.

The obtained CCTs for a three-phase short-circuit fault in the mid-point of the AC transmission line are presented in Table IV for the classic vector control, and the proposed SHVDC control. We can see that the SHVDC control with tuned parameters improves the transient dynamics of the system and thus augments the transient stability margins of the neighbor network. This is due to the fact that the dynamics of the neighbour zone are taken into account at the synthesis stage via the oscillatory modes in Table I. The gains of the controllers are computed to damp these modes and thus to diminish the general swing of the zone and not only for the local HVDC dynamics. Moreover, as the structure of the SHVDC mimics the closed-loop of a standard synchronous generator, an inertia with a positive impact of the transient stability of the neighbour zone is emulated.

Remark: Real-time implementation

The SHVDC system is based on the Sinusoidal Pulse Width Modulation (SPWM) switching using a single-phase triangular carrier wave with a frequency of 27 times fundamental frequency (1350 Hz). This is compliant with the real time implementation which relies on a time response of 1-3 µs [START_REF] Forsyth | Testing Firing Pulse Controls for a VSC Based HVDC Scheme with a Real Time Timestep < 3 µs[END_REF]. Indeed, in the real time simulation, the converter stations and the connecting DC cable are simulated in three separate small time step networks [START_REF] Mitra | Dynamic Performance Study of a HVDC Grid Using Real-Time Digital Simulator[END_REF]. The three subnetworks are connected on the DC side through travelling wave transmission line models with a travel time in the range of the small time step size (i.e. ~3µs). Both converter stations are interfaced to the AC systems with the help of interfacing transformers connecting the small and the normal time step areas.

VII. CONCLUSION

This paper proposed a new control strategy of HVDC transmission based on the synchronverter concept. The proposed synchronverter structure could not be controlled using directly the standard parameters of the SG. Thus, a specific tuning method was proposed based on; firstly, the sensitivity analysis of the system poles to the control parameters then, their placement linear approximation via residues. Better results than with the classic VSC control were obtained for the following points: -Better dynamic performances (less overshoot and more damping) because this approach allows one to analytically take into account dynamic specifications at the tuning stage.

-Better stability margin of the neighbor zone for two reasons. First, swing information is directly taken into account at the synthesis stage in terms of the less damped modes of the neighbor zone and not only for the local HVDC dynamics as it is the case for the standard VSC control. Next, the SHVDC emulates inertia of two rotating machines and this has a direct damping effect for the neighbor zone. -Help power system operators in system operation since the SG and their controls are better known.

It is noted that this kind of tuning is surely under optimal in the sense that the constraints to keep the structure of the classical generators and their regulations imposed by the synchronverter principle limits the performances of the resulting closed-loop. Moreover, the advantage of inertia emulation mentioned above could be lost in the context of a grid with many power electronics and few physical SG since the grid frequency is no longer (well) defined in this case.

Fig. 1 .

 1 Fig. 1. Two terminal HVDC-VSC link

Fig. 2 .

 2 Fig. 2. Structure of an idealized three-phase round-rotor SG

e

  back emf of the virtual rotor. The inverter switches are operated so that the average values of _ over a switching period are to be equal to _ g abc e given in(2). This can be achieved by the usual PWM technique. It's worth noting that the VSC technology used in this work is not exclusive. However, it is suited for generating pulse control signals. The electronic part controls the switches in the power part. These two parts interact via the signals _

  a matrix which elements are polynomials in the derivation operator s d dt =

Fig. 4 .

 4 Fig. 4. Responses with standard SG parameters. (a) Response of g P to a +0.05 step in _ g ref P (p.u) ; (b) Response of g Q to a -0.1 step in _ g ref Q (p.u);

Fig. 5 .K

 5 Fig. 5. Feedback system where gp D , mp D are, respectively, the static frequency droop coefficients of the SG and the SM;

where 0 iλ

 0 is the initial (open-loop) location of the pole i

Fig. 7 .

 7 Fig. 7. Responses with optimal parameters. (a) Response of g P to a +0.05 step in _ g ref P (p.u) ; (b) Response of g Q to a -0.1 step in _ g ref Q (p.u); (c)

Fig. 8 .

 8 Fig. 8. Two area test system with parallel AC and DC lines

Fig. 9 .

 9 Fig. 9. Responses of the IEEE 4 machines benchmark. (a) Response of Pg to a -0.1 step in Pg_ref (p.u) ; (b) Response of Qg to a -0.1 step in Qg_ref (p.u).

Fig. 10 .

 10 Fig. 10. Speed response of G1 to a 100 ms short-circuit

TABLE II PARTICIPATION FACTORS

 IIFACTORS

TABLE III SHVDC PARAMETERS

 IIIPARAMETERS