The Kuramoto model with inertia: from fireflies to power grids - PowerPoint PPT Presentation

The Kuramoto model with inertia: from fireflies to power grids Simona Olmi Inria Sophia Antipolis M´ editerran´ ee Research Centre - Sophia Antipolis, France Istituto dei Sistemi Complessi - CNR - Firenze, Italy Patterns of Synchrony: Chimera States and Beyond – p. 1

Pteroptix Malaccae A phase model with inertia has been introduced to mimic the synchronization mechanisms observed among the Malaysian fireflies Pteroptix Malaccae. These fireflies synchronize their flashing activity by entraining to the forcing frequency with almost zero phase lag. Usually, entrainment results in a constant phase angle equal to the difference between pacing frequency and free-running period as it does in P . cribellata. (B. Ermentrout (1991), Experiments by Hanson, (1987)) Patterns of Synchrony: Chimera States and Beyond – p. 2

Why introducing “inertia”? First-order Kuramoto model It approaches too fast the partial synchronized state Infinite coupling strength is required to achive full synchronization Second-order Kuramoto model Synchronization is slowed down by inertia (frequency adaptation) Firstly proposed in biological context (Ermentrout, (1991)) Used to study synchronization in disordered arrays of Josephson junctions (Strogatz (1994), Trees et al. (2005)) Derived from the classical swing equation to study synchronization in power grids (Filatrella et al. (2008)) Patterns of Synchrony: Chimera States and Beyond – p. 3

The Model Kuramoto model with inertia θ i = Ω i + K m ¨ θ i + ˙ � sin( θ j − θ i ) N j θ i is the instantaneous phase Ω i is the natural frequency of the i − th oscillator with Gaussian distribution K is the coupling constant N is the number of oscillators By introducing the complex order parameter r ( t ) e iφ ( t ) = 1 j e iθ j � N m ¨ θ i + ˙ θ i = Ω i − Kr sin( θ i − φ ) r = 0 asynchronous state, r = 1 synchronized state Patterns of Synchrony: Chimera States and Beyond – p. 4

Damped Driven Pendulum m ¨ θ i + ˙ θ i = Ω i − Kr sin( θ i ) I = Ω i Kr 1 β = √ mKr φ + β ˙ ¨ φ = I − sin( φ ) One node connected to the grid (the grid is considered to be infinite) Single damped driven pendulum Josephson junctions One-machine infinite bus system of a generator in a power-grid (Chiang, (2011)) Patterns of Synchrony: Chimera States and Beyond – p. 5

Damped Driven Pendulum φ + β ˙ ¨ φ = I − sin( φ ) For sufficiently large m (small β ) For small Ω i two fixed points are present: a stable node and a saddle. The linear stability is given by   0 1 J =   − cos φ ∗ − β σ 1 , 2 = − β ± √ β 2 − 4 cos φ ∗ 2 At large frequencies Ω i > Ω P = � 4 β 4 Kr (i.e. I > π ) a limit cycle π m emerges from the saddle via a homo- clinic bifurcation Limit cycle and fixed point coexists until Ω i ≡ Ω D = Kr (i.e. I = 1 ), where a saddle node bifurcation leads to the disappearence of the two fixed points For Ω i > Ω D (i.e. I > 1 ) only the oscillating solution is present For small mass (large β ), there is no more coexistence. Patterns of Synchrony: Chimera States and Beyond – p. 6 (Levi et al. 1978)

Simulation Protocols Dynamics of N oscillators (first order transition and hysteresis) Ω M maximal natural frequency of the locked oscillators � Ω ( I ) = 4 Kr P π m Protocol I: Increasing K Ω ( II ) = Kr D The system remains desynchronized Ω M until K = K 1 3 c (filled black circles). (II) Ω D (I) Ω M increases with K following Ω I Ω P P . 2 Ω i are grouped in small clusters Ω M 1 (plateaus). 0 Protocol II 1 c K 2 Protocol II: Decreasing K r Protocol I The system remains synchronized until 0.5 c K 1 K = K 2 c (empty black circles). Ω M remains stucked to the same value 0 0 2 4 6 8 10 for a large K interval than it rapidly de- K creases to 0 following Ω II D . m = 2 Patterns of Synchrony: Chimera States and Beyond – p. 7

Mean Field Theory (Tanaka et al. (1997)) m ¨ θ i + ˙ θ i = Ω i − Kr sin( θ i − φ ) by following Protocol I and II there is a group of drifting oscillators and one of locked oscillators which act separately locked oscillators are characterized by < ˙ θ > = 0 and are locked to the mean phase drifting oscillators (with < ˙ θ > � = 0 ) are whirling over the locked subgroup (or below depending on the sign of Ω i ) Drifting and locked oscillators are separated by a certain frequency: Following Protocol I the oscillators with Ω i < Ω P are locked Following Protocol II the oscillators with Ω i < Ω D are locked These two groups contribute differently to the total level of synchronization in the system r = r L + r D Patterns of Synchrony: Chimera States and Beyond – p. 8

Mean Field Theory (Tanaka et al. (1997)) � Protocol I: Ω ( I ) = 4 Kr P π m All oscillators initially drift around its own natural frequency Ω i Increasing K , oscillators with Ω i < Ω P are attracted by the locked group Increasing K also Ω P increases ⇒ oscillators with bigger Ω i become synchronized The phase coherence r I increases and Ω i exhibits plateaus ! Depending on m the transition to synchronization may increase in complexity Protocol II: Ω ( II ) = Kr D Oscillators are initially locked to the mean phase and r II ≈ 1 Decreasing K , locked oscillators are desynchronized and start whirling when Ω i > Ω D and a saddle node bifurcation occurs Ω P , Ω D are the synchronization boundaries Patterns of Synchrony: Chimera States and Beyond – p. 9

Mean Field Theory (Tanaka et al. (1997)) Total level of synchronization in the system: r = r L + r D For the locked population the self-consistent equation is � θ P,D cos 2 θ g ( Kr sin θ ) dθ r I,II = Kr L − θ P,D where θ P = sin − 1 ( Ω P θ D = sin − 1 ( Ω D Kr ) , Kr ) = π/ 2 , g (Ω) frequency distribution. The drifting population contributes to the total order parameter with a negative contribution � ∞ 1 r I,II ≃ − mKr ( m Ω) 3 g (Ω) d Ω D − Ω P,D The former equation are correct in the limit of sufficiently large masses Patterns of Synchrony: Chimera States and Beyond – p. 10

Hysteretic Behavior Numerical Results for Fully Coupled Networks ( N = 500 , m = 6 ) The data obtained by following protocol II are quite well reproduced by the mean field approximation r II The mean field extimation r I does not reproduce the stepwise structure numerically obtained in protocol I 1 Clusters of N L locked oscillators of any size remain stable between r I 0.8 and r II 500 0.6 The level of synchronization of these 400 r clusters can be theoretically obtained 300 0.4 N L by generalizing the theory of Tanaka 200 et al. (1997) to protocols where Ω M 0.2 100 K remains constant 0 0 5 10 15 20 0 0 2 4 6 8 10 12 14 16 18 20 (Olmi et al. (2014)) K Patterns of Synchrony: Chimera States and Beyond – p. 11

Finite Size Effects K c 1 is the transition value from asynchronous to synchronous state (following Protocol I) K c 2 is the transition value from synchronous to asynchronous state (following Protocol II) 1 0.8 M=6 0.6 r 0.4 N=500 N=1000 N=2000 N=4000 0.2 N=8000 N=16000 0 0 2 4 6 8 10 12 14 16 18 20 K Patterns of Synchrony: Chimera States and Beyond – p. 12

Finite Size Effects (Olmi et al. (2014)) a) m = 0 . 8 , (b) m = 1 , (c) m = 2 and (d) m = 6 K c 1 (upper points) is strongly influenced by the size of the system K c 2 (lower points) does not depend heavily on N Good agreement between Mean Field and simulations is achieved for small m For large m the emergence of the secondary synchronization of drifting oscillators (i.e. clusters of whirling oscillators) is determinant Patterns of Synchrony: Chimera States and Beyond – p. 13 Dashed line → K MF mean field value by Gupta et al (PRE 2014) 1

Drifting Clusters For larger masses (m=6), the synchronization transition becomes more complex, it occurs via the emergence of clusters of drifting oscillators. The partially synchronized state is characterized by the coexistence of a cluster of locked oscillators with < ˙ θ > ≃ 0 clusters composed by drifting oscillators with finite average velocities Extra clusters induce (periodic or quasi-periodic) oscillations in the temporal evolution of r ( t ) . 1 0.8 0.6 r(t) 0.4 0.2 (b) 0 0 20 40 60 80 time Patterns of Synchrony: Chimera States and Beyond – p. 14 (Olmi et al. (2014))

Drifting Clusters If we compare the evolution of the instantaneous velocities ˙ θ i for 3 oscillators and r ( t ) we observe that the phase velocities of O 2 and O 3 display synchronized motion the phase velocity of O 1 oscillates irregularly around zero the oscillations of r ( t ) are driven by the periodic oscillations of O 2 and O 3 2 (b) O 2 O 3 1.5 1 0.5 r(t) O 1 0 0 10 20 30 40 time (Olmi et al. (2014)) Patterns of Synchrony: Chimera States and Beyond – p. 15