Synchronization and Kron Reduction in Power Networks

Transcription

1 Synchronization and Kron Reduction in Power Networks Florian Dörfler and Francesco Bullo Center for Control, Dynamical Systems & Computation University of California at Santa Barbara IFAC Workshop on Distributed Estimation and Control in Networked Systems Annecy, France, - September, Poster tomorrow on: Network reduction and effective resistance Slides and papers available at: Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

2 Summary observations from distinct fields: power networks are coupled oscillators Kuramoto oscillators synchronize for large coupling graph theory quantifies coupling in a network hence, power networks synchronize for large coupling Today s talk: theorems about these observations synch tests for net-preserving and reduced models Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

3 Summary observations from distinct fields: power networks are coupled oscillators Kuramoto oscillators synchronize for large coupling graph theory quantifies coupling in a network hence, power networks synchronize for large coupling Today s talk: theorems about these observations synch tests for net-preserving and reduced models Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

4 Outline Introduction Motivation Mathematical model Problem statement Singular perturbation analysis (to relate power network and Kuramoto model) Synchronization of non-uniform Kuramoto oscillators Network-preserving power network models Conclusions Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

5 Motivation: the current US power grid... the largest and most complex machine engineered by humankind. [P. Kundur, V. Vittal,... ]... the greatest engineering achievement of the th century. [National Academy of Engineering ] large-scale, nonlinear dynamics, complex interactions years old and operating at its capacity limits recent blackouts: New England, Italy, Brazil Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

6 Motivation: the future smart grid Energy is one of the top three national priorities, [B. Obama, ] Expected developments in smart grid : increasing consumption increasing adoption of renewable power sources: large number of distributed power sources power transmission from remote areas large-scale heterogeneous networks with stochastic disturbances Transient Stability Generators to swing synchronously despite variability/faults in generators/network/loads Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

7 Motivation: the Mediterranean ring project Synchronous grid interconnection between EU and Mediterranean region: Provide increased levels of energy security to participating nations; Import/export electric power among nations; Cut back on the primary electricity reserve requirements within each country. Reference: Oscillation behavior of the enlarged European power system by M. Kurth and E. Welfonder. Control Engineering Practice,. Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

8 Mathematical model of a power network New England Power Grid F Power network topology: n generators, each connected to a generator terminal bus n generators terminal buses and m load buses form connected graph admittance matrix Y network C (n+m) (n+m) characterizes the network Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

9 Mathematical model of a power network Network-preserving DAE power network model: n generators = boundary nodes: M i πf θi = D i θi + P mech.in,i P electr.out,i n + m passive & = interior nodes: loads are modeled as shunt admittances algebraic Kirchhoff equations: Y, Y, I = Y network V Y,shunt Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

10 Mathematical model of a power network Network-Reduction to an ODE power network model [ Iboundary ] [ ] Yboundary Y = bound-int Ybound-int T Y interior }{{} Y network [ Vboundary V interior ] Schur complement Y reduced = Y network /Y interior = I boundary = Y reduced V boundary network reduced to active nodes (generators) Y reduced induces complete all-to-all coupling graph Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

11 Mathematical model of a power network Network-Reduced ODE power network model: classic interconnected swing equations [Anderson et al., M. Pai, P. Kundur,... ]: M i πf θi = D i θi + ω i j i P ij sin(θ i θ j + ϕ ij ) all-to-all reduced network Y reduced with P ij = V i V j Y reduced,i, j > max. power transferred i j ϕ ij = arctan(r(y red,i, j )/I(Y red,i, j )) [, π/) reflect losses i j ω i = P mech.in,i V i R(Y reduced,i, i ) effective power input of i Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

12 Transient stability analysis: problem statement M i πf θi = D i θi + ω i j i P ij sin(θ i θ j + ϕ ij ) Classic transient stability: power network in stable frequency equilibrium ( θi, θ i ) = (, ) for all i transient network disturbance and fault clearance stability analysis of a new frequency equilibrium in post-fault network General synchronization problem: synchronous equilibrium: θ i θ j small & θ i = θ j for all i, j Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

13 Transient stability analysis: literature review M i πf θi = D i θi + ω i j i P ij sin(θ i θ j + ϕ ij ) Classic methods use Hamiltonian and gradient systems arguments: write M i πf θi = D i θi i U(θ) T study θ i = i U(θ) T Key objective: compute domain of attraction via numerical methods [N. Kakimoto et al., H.-D. Chiang et al. ] Open Problem power sys dynamics + complex nets [Hill and Chen ] transient stability, performance, and robustness of a power network? underlying graph properties (topological, algebraic, spectral, etc) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

14 Transient stability analysis: literature review M i πf θi = D i θi + ω i j i P ij sin(θ i θ j + ϕ ij ) Classic methods use Hamiltonian and gradient systems arguments: write M i πf θi = D i θi i U(θ) T study θ i = i U(θ) T Key objective: compute domain of attraction via numerical methods [N. Kakimoto et al., H.-D. Chiang et al. ] Open Problem power sys dynamics + complex nets [Hill and Chen ] transient stability, performance, and robustness of a power network? underlying graph properties (topological, algebraic, spectral, etc) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

15 Detour: consensus protocols & Kuramoto oscillators Consensus protocol in R n : ẋ i = j i a ij(x i x j ) n identical agents with state variable x i R application: agreement and coordination algorithms,... references: [M. DeGroot, J. Tsitsiklis, L. Moreau,...] R Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

16 Detour: consensus protocols & Kuramoto oscillators Consensus protocol in R n : ẋ i = Kuramoto model in T n : a ij(x i x j ) θ j i i = ω i K n sin(θ i θ j ) j i n identical agents with state variable x i R application: agreement and coordination algorithms,... references: [M. DeGroot, J. Tsitsiklis, L. Moreau,...] n non-identical oscillators with phase θ i T & frequency ω i R application: sync phenomena in nature, Josephson junctions,... references: [C. Huygens XVII, Y. Kuramoto, A. Winfree,...] R T Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

17 Detour: synchrony among oscillators Kuramoto model in T n θi = ωi K n notions of synchronization phase cohesiveness: θ i (t) θ j (t) < γ for small γ < π/... arc invariance frequency synchronized: θi (t) = θ j (t) phase synchronized: θ i (t) = θ j (t) j i sin(θ i θ j ) Some p W can To Al Classic intuition: K small & ω i ω j large no synchronization Florian K large & ω i ω j small cohesive + freq synchronization Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

18 Detour: synchrony among oscillators Kuramoto model in T n θi = ωi K n notions of synchronization phase cohesiveness: θ i (t) θ j (t) < γ for small γ < π/... arc invariance frequency synchronized: θi (t) = θ j (t) phase synchronized: θ i (t) = θ j (t) j i sin(θ i θ j ) Some p W can To Al Classic intuition: K small & ω i ω j large no synchronization Florian K large & ω i ω j small cohesive + freq synchronization Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

19 The big picture Open problem in synchronization and transient stability in power networks: relation to underlying network state, parameters, and topology! Mi θ i = Di θ i + ωi Pij sin(θi θj + ϕij ) j!=i πf Kuramoto Oscillators: Consensus Protocols:? x i =! j!=i aij (xi xj ) Do rfler and Bullo (UCSB) θ i = ωi Synchronization & Kron Reduction K! sin(θi θj ) j!=i n Annecy /

20 The big picture Open problem in synchronization and transient stability in power networks: relation to underlying network state, parameters, and topology! Mi θ i = Di θ i + ωi Pij sin(θi θj + ϕij ) j!=i πf Kuramoto Oscillators: Consensus Protocols:? x i =! j!=i aij (xi xj ) θ i = ωi K! sin(θi θj ) j!=i n Previous observations about this connection: Power systems: [D. Subbarao et al.,, G. Filatrella et al.,, V. Fioriti et al., ] Networked control: [D. Hill et al.,, M. Arcak, ] Dynamical systems: [H. Tanaka et al.,, A. Arenas ] Do rfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

21 Outline Introduction Motivation Mathematical model Problem statement Singular perturbation analysis (to relate power network and Kuramoto model) Synchronization of non-uniform Kuramoto oscillators Network-preserving power network models Conclusions Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

22 Singular perturbation analysis M i πf θi = D i θi + ω i j i P ij sin(θ i θ j + ϕ ij ) assume time-scale separation between synchronization and damping M max singular perturbation parameter ɛ = πf D min non-uniform Kuramoto (slow time-scale, for ɛ = ) D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) if cohesiveness + exponential freq sync for non-uniform Kuramoto, then (θ(), θ()), exists ɛ > such that ɛ < ɛ and t θ i (t) power network θ i (t) non-uniform Kuramoto = O(ɛ) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

23 Singular perturbation analysis: proof Key technical problem: Kuramoto defined over manifold T n, no fixed point Tikhonov s Theorem: exp. stable point in Euclidean space Solution define grounded variables in R n δ = θ θ n δ n = θ n θ n equivalence of solutions: grounded Kuramoto solutions satisfy max i,j (δ i (t) δ i (t)) < π Kuramoto solutions are arc invariant with γ = π, ie, θ (t),..., θ n (t) belong to open half-circle, function of t equivalence of exponential convergence exponential frequency synchronization for Kuramoto exponential convergence to equilibrium for grounded Kuramoto Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

24 Singular perturbation analysis: proof Key technical problem: Kuramoto defined over manifold T n, no fixed point Tikhonov s Theorem: exp. stable point in Euclidean space Solution define grounded variables in R n δ = θ θ n δ n = θ n θ n equivalence of solutions: grounded Kuramoto solutions satisfy max i,j (δ i (t) δ i (t)) < π Kuramoto solutions are arc invariant with γ = π, ie, θ (t),..., θ n (t) belong to open half-circle, function of t equivalence of exponential convergence exponential frequency synchronization for Kuramoto exponential convergence to equilibrium for grounded Kuramoto Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

25 Singular perturbation analysis: proof Key technical problem: Kuramoto defined over manifold T n, no fixed point Tikhonov s Theorem: exp. stable point in Euclidean space Solution define grounded variables in R n δ = θ θ n δ n = θ n θ n equivalence of solutions: grounded Kuramoto solutions satisfy max i,j (δ i (t) δ i (t)) < π Kuramoto solutions are arc invariant with γ = π, ie, θ (t),..., θ n (t) belong to open half-circle, function of t equivalence of exponential convergence exponential frequency synchronization for Kuramoto exponential convergence to equilibrium for grounded Kuramoto Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

26 Singular perturbation analysis: discussion assumption ɛ = M max πf D min sufficiently small generator internal control effects imply ɛ O(.) topological equivalence independent of ɛ: st-order and nd-order models have the same equilibria, the Jacobians have the same inertia, and the regions of attractions are bounded by the same separatrices non-uniform Kuramoto corresponds to reduced gradient system θ i = i U(θ) T used successfully in academia and industry since physical interpretation: damping and sync on separate time-scales classic assumption in literature on coupled oscillators: over-damped mechanical pendula and Josephson junctions simulation studies show accurate approximation even for large ɛ Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

27 Outline Introduction Motivation Mathematical model Problem statement Singular perturbation analysis (to relate power network and Kuramoto model) Synchronization of non-uniform Kuramoto oscillators Network-preserving power network models Conclusions Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

28 Synchronization of non-uniform Kuramoto: condition Non-uniform Kuramoto Model in T n : D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) Non-uniformity in network: D i, ω i, P ij, ϕ ij Phase shift ϕ ij induces lossless and lossy coupling: θ i = ω i D i j i ( P ij D i cos(ϕ ij ) sin(θ i θ j ) + P ij D i sin(ϕ ij ) cos(θ i θ j ) ) Synchronization condition ( ) n P min D max cos(ϕ max ) }{{} worst lossless coupling ( ωi ω ) j i,j D i D j }{{} worst non-uniformity > max + max i j P ij D i sin(ϕ ij ) }{{} worst lossy coupling Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

29 Synchronization of non-uniform Kuramoto: condition Non-uniform Kuramoto Model in T n : D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) Non-uniformity in network: D i, ω i, P ij, ϕ ij Phase shift ϕ ij induces lossless and lossy coupling: θ i = ω i D i j i ( P ij D i cos(ϕ ij ) sin(θ i θ j ) + P ij D i sin(ϕ ij ) cos(θ i θ j ) ) Synchronization condition ( ) n P min D max cos(ϕ max ) }{{} worst lossless coupling ( ωi ω ) j i,j D i D j }{{} worst non-uniformity > max + max i j P ij D i sin(ϕ ij ) }{{} worst lossy coupling Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

30 Synchronization of non-uniform Kuramoto: consequences D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) n P ( min ωi cos(ϕ max ) > max ω ) j D max i,j D i D j P ij + max sin(ϕ ij ) i j D i ) phase cohesiveness: arc-invariance for all arc-lengths ( arcsin cos(ϕ max ) RHS ) }{{ LHS } γ min γ practical phase sync: in finite time, arc-length γ min π ϕ max }{{} γ max ) frequency synch: from all initial conditions in a γ max arc, exponential frequency synchronization Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

31 Main proof ideas Cohesiveness θ(t) (γ) arc-length V (θ(t)) is non-increasing V (θ(t)) V (θ(t)) = max{ θ i (t) θ j (t) i, j {,..., n}} D + V (θ(t))! contraction property [D. Bertsekas et al., L. Moreau &, Z. Lin et al.,... ] Frequency synchronization in (γ) consensus protocol in R n d dt θ i = j i a ij(t)( θ i θ j ), where a ij (t) = P ij D i cos(θ i (t) θ j (t) + ϕ ij ) > for all t Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

32 Synchronization of non-uniform Kuramoto: uniform Classic (uniform) Kuramoto Model in T n : θ i = ω i K n sin(θ i θ j ) j i Necessary and sufficient condition (sufficiency) synchronization condition ( ) reads ( ) K > max ωi ω j i,j also necessary when considering all distributions of ω [ω min, ω max ] Condition ( ) strictly improves existing bounds on Kuramoto model: [F. de Smet et al., N. Chopra et al., G. Schmidt et al., A. Jadbabaie et al., S.J. Chung et al., J.L. van Hemmen et al. ]. Necessary condition synchronization: K > n (n ) (ω max ω min ) [J.L. van Hemmen et al., A. Jadbabaie et al., N. Chopra et al. ] Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

33 Synchronization of non-uniform Kuramoto: uniform Classic (uniform) Kuramoto Model in T n : θ i = ω i K n sin(θ i θ j ) j i Necessary and sufficient condition (sufficiency) synchronization condition ( ) reads ( ) K > max ωi ω j i,j also necessary when considering all distributions of ω [ω min, ω max ] Condition ( ) strictly improves existing bounds on Kuramoto model: [F. de Smet et al., N. Chopra et al., G. Schmidt et al., A. Jadbabaie et al., S.J. Chung et al., J.L. van Hemmen et al. ]. Necessary condition synchronization: K > n (n ) (ω max ω min ) [J.L. van Hemmen et al., A. Jadbabaie et al., N. Chopra et al. ] Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

34 Synchronization of non-uniform Kuramoto: alternative Non-uniform Kuramoto Model in T n - rewritten: D i θ i = ω i j i P ij sin(θ i θ j + ϕ ij ) λ (L(P ij cos(ϕ ij ))) > f (D i ) }{{}}{{} lossless connectivity non-uniform D i s ( [..., ω i + λ max (L) ω ] j,... D i D j }{{} non-uniformity (/ cos(ϕ max ) ) }{{} phase shifts [..., P ] ij sin(ϕ ij ),... j D } i {{} lossy coupling ) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

35 Synchronization of non-uniform Kuramoto: alternative Non-uniform Kuramoto Model in T n - rewritten: D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) λ (L(P ij cos(ϕ ij ))) > f (D i ) }{{}}{{} lossless connectivity non-uniform D i s ( [..., ω i + λ max (L) ω ] j,... D i D j }{{} non-uniformity (/ cos(ϕ max ) ) }{{} phase shifts [..., P ] ij sin(ϕ ij ),... j D } i {{} lossy coupling ) Similar synch, quadratic Lyap, uniform test K > [..., ω i ω j,... ] Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

36 Synchronization of non-uniform Kuramoto Non-uniform Kuramoto Model in T n D i θi = ω i j i P ij sin(θ i θ j + ϕ ij ) Further interesting results: explicit synchronization frequency exponential rate of frequency synchronization conditions for phase synchronization results for general non-complete graphs... to be found in our papers. Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

37 Outline Introduction Motivation Mathematical model Problem statement Singular perturbation analysis (to relate power network and Kuramoto model) Synchronization of non-uniform Kuramoto oscillators Network-preserving power network models Conclusions Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

38 Network-preserving power network models So far we considered a network-reduced power system model: synchronization conditions on λ (P) and P min all-to-all reduced admittance matrix Y reduced P/V (for uniform voltage levels V i = V ) Topological non-reduced network-preserving power system model: topological bus admittance matrix Y network indicating transmission lines and loads (self-loops) Schur complement:y reduced = Y network /Y interior c.f. Kron reduction, Dirichlet-to-Neumann map, Schur contraction, Gaussian elimination,... Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

39 Network-preserving power network models So far we considered a network-reduced power system model: synchronization conditions on λ (P) and P min all-to-all reduced admittance matrix Y reduced P/V (for uniform voltage levels V i = V ) Topological non-reduced network-preserving power system model: topological bus admittance matrix Y network indicating transmission lines and loads (self-loops) Schur complement:y reduced = Y network /Y interior c.f. Kron reduction, Dirichlet-to-Neumann map, Schur contraction, Gaussian elimination,... Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

40 Kron reduction of graphs: definition Kron reduction of a graph with boundary, interior, non-neg self-loops loopy Laplacian matrix Y network Iterative -dim Kron reduction: Topological evolution of the corresponding graph ) ) X X ) X... )... Algebraic evolution of Laplacian matrix: Y k+ reduced = Yk reduced / Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

41 Kron reduction of graphs: properties Well-posedness: set of loopy Laplacian matrices is closed Equivalence: iterative -dim reduction = -step reduction Y network Y reduced = Y network /Y interior Topological properties: interior network connected reduced network complete at least one node in interior network features a self-loop all nodes in reduced network feature self-loops Algebraic properties: self-loops in interior network decrease mutual coupling in reduced network increase self-loops in reduced network Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

42 Kron reduction of graphs: properties Well-posedness: set of loopy Laplacian matrices is closed Equivalence: iterative -dim reduction = -step reduction Y network Y reduced = Y network /Y interior Topological properties: interior network connected reduced network complete at least one node in interior network features a self-loop all nodes in reduced network feature self-loops Algebraic properties: self-loops in interior network decrease mutual coupling in reduced network increase self-loops in reduced network Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

43 Kron reduction of graphs: properties Well-posedness: set of loopy Laplacian matrices is closed Equivalence: iterative -dim reduction = -step reduction Y network Y reduced = Y network /Y interior Topological properties: interior network connected reduced network complete at least one node in interior network features a self-loop all nodes in reduced network feature self-loops Algebraic properties: self-loops in interior network decrease mutual coupling in reduced network increase self-loops in reduced network Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

44 Kron reduction of graphs: properties Some properties of the Kron reduction process:. Spectral properties: interlacing property: λ i (Y network ) λ i (Y reduced ) λ i+n (Y network ) algebraic connectivity λ is non-decreasing along Kron Effective resistance: Effective resistance R(i, j) among boundary nodes is invariant For boundary nodes : effective resistance R(i, j) uniform coupling Y reduced (i, j) uniform /R(i, j) = n Y reduced(i, j) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

45 Kron reduction of graphs: properties Some properties of the Kron reduction process:. Spectral properties: interlacing property: λ i (Y network ) λ i (Y reduced ) λ i+n (Y network ) algebraic connectivity λ is non-decreasing along Kron Effective resistance: Effective resistance R(i, j) among boundary nodes is invariant For boundary nodes : effective resistance R(i, j) uniform coupling Y reduced (i, j) uniform /R(i, j) = n Y reduced(i, j) Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

46 Synchronization in network-preserving models Assumption I: lossless network and uniform voltage levels V at generators Spectral condition for synchronization: λ (P)... becomes λ (i L network ) > ( ω ω ) f,... (D i) D D V + min{ } Assumption II: effective resistance R among generator nodes is uniform Resistance-based condition for synchronization: np min... becomes { R > max ωi ω } j Dmax i,j D i D j V Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

47 Outline Introduction Motivation Mathematical model Problem statement Singular perturbation analysis (to relate power network and Kuramoto model) Synchronization of non-uniform Kuramoto oscillators Network-preserving power network models Conclusions Dörfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /

48 Conclusions Open problem in synchronization and transient stability in power networks: relation to underlying network state, parameters, and topology singular perturbations and graph theory Time-varying Consensus Protocols Non-uniform Kuramoto Oscillators Kuramoto, consensus, and nonlinear control tools Ambitious workplan sharpest conditions for most realistic models stochastic instead of worst-case analysis networks of DC/AC power inverters control via voltage regulation and flexible AC transmission systems distance to instability and optimal islanding for failure management transition to DOE laboratories and utilities Do rfler and Bullo (UCSB) Synchronization & Kron Reduction Annecy /