Network Systems and Kuramoto Oscillators

Transcription

1 Network Systems and Kuramoto Oscillators Francesco Bullo Department of Mechanical Engineering Center for Control, Dynamical Systems & Computation University of California at Santa Barbara 37th Chinese Control Conference Wuhan, China, July 26, 2018

2 Acknowledgments CCC General Chairs, Program Chairs and all organizing volunteers CAA Technical Committee on Control Theory Systems Engineering Society of China China University of Geosciences, Wuhan A successful, trasured, inspiring, and long-lasting tradition of collaboration between IEEE CSS and CCC Saber Jafarpour UCSB Elizabeth Y. Huang UCSB Florian Dörfler ETH USA Department of Energy (DOE), SunShot Program, XAT , Stabilizing the Power System in 2035 and Beyond, Consortium of DOE National Renewable Energy Laboratory, UCSB and University of Minnesota

3 Outline 1 Intro to Network Systems and Power Flow 2 Known tests and a conjecture F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks and smart grids. Proc National Academy of Sciences, 110(6): , doi: /pnas A new approach and new tests S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. Autom. Control, November URL Submitted S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: The Taylor expansion of the inverse Kuramoto map. In Proc CDC, Miami, USA, December URL

4 Example network systems Figure 2-5. City of Portland Water Supply Schematic Diagram Water Supplier Description 2-18 Smart grid Amazon robotic warehouse Portland water network Industrial chemical plant

5 An emerging discipline of network systems electric, mechanical and physical networks social networks and mathematical sociology animal behavior, population dynamics, and ecosystems networked control systems robotic networks power grids parallel and scientific computation transmission and traffic networks fundamental dynamic phenomena over networks network structure function = asymptotic behavior

6 New text Lectures on Network Systems Lectures on Network Systems Francesco Bullo Lectures on Network Systems Francesco Bullo With contributions by Jorge Cortés Florian Dörfler Sonia Martínez Lectures on Network Systems, Createspace, 1 edition, ISBN For students: free PDF for download For instructors: slides and answer keys pages (plus 200 pages solution manual) 3K downloads since Jun exercises with solutions Linear Systems: 1 social, sensor, robotic & compartmental examples, 2 matrix and graph theory, with an emphasis on Perron Frobenius theory and algebraic graph theory, 3 averaging algorithms in discrete and continuous time, described by static and time-varying matrices, and 4 positive & compartmental systems, dynamical flow systems, Metzler matrices. Nonlinear Systems: 5 nonlinear consensus models, 6 population dynamic models in multi-species systems, 7 coupled oscillators, with an emphasis on the Kuramoto model and models of power networks

7 Today: Synchronization of Coupled Oscillators Pendulum clocks & an odd kind of sympathy [C. Huygens, Horologium Oscillatorium, 1673] Canonical coupled oscillator model [A. Winfree 67, Y. Kuramoto 75] Kuramoto model n oscillators with angle θ i S 1 non-identical natural frequencies ω i R 1 coupling with strength a ij = a ji n θ i = ω i a ij sin(θ i θ j ) j=1

8 Motivation Kuramoto model n θ i = ω i a ij sin(θ i θ j ) j=1 1 Countless sync phenomena in sciences/engineering [S. Strogatz 00, J. Acebrón 01, Arenas 08] 2 Local canonical model for weakly coupled limit-cycle oscillators [F.C. Hoppensteadt et al. 97, E. Brown et al. 04] 3 Simplest network system on manifold Not only sync, but also rich phenomenology citations on scholar.google: Winfree 1.5K, Kuramoto 1.1K + 6.8K, surveys Strogatz, Acebron, Arenas: 2K, survey by Boccaletti 8K

9 Spring network analog and applications Kuramoto coupled oscillators θ i = ω i j a ij sin(θ i θ j ) Coupled swing equations m i θi + d i θi = τ i j k ij sin(θ i θ j ) Applications in sciences: biology:pacemaker cells in the heart, circadian cells in the brain, coupled cortical neurons, Hodgkin-Huxley neurons, brain networks, yeast cells, flashing fireflies, chirping crickets, central pattern generators for animal locomotion, particle models mimicking animal flocking behavior, and fish schools physics and chemistry: spin glass models, flavor evolution of neutrinos, coupled Josephson junctions, coupled metronomes, Huygen s coupled pendulum clocks, micromechanical oscillators with opticalor mechanical coupling, and chemical oscillations social networks: rhythmic applause, opinion dynamics, pedestrian crowd synchrony, and decision making in animal groups Applications in engineering: deep brain stimulation, locking in solid-state circuit oscillators, planar vehicle coordination, carrier synchronization without phase-locked loops, semiconductor laser arrays, and microwave oscillator arrays electric applications: structure-preserving and network-reduced power system models, and droop-controlled inverters in microgrids algorithmic applications: limit-cycle estimation through particle filters, clock synchronization in decentralized computing networks, central pattern generators for robotic locomotion, decentralized maximum likelihood estimation, and human-robot interaction generating music, signal processing, pattern recognition, and neuro-computing through micromechanical or laser oscillators.

10 Transition from incoherence to synchronization Function = synchronization θi = ω i n j=1 a ij sin(θ i θ j ) i (t) i (t) large ω i ω j & small coupling incoherence = no sync small ω i ω j & large coupling coherence = frequency sync Central question: [S. Strogatz 01, A. Arenas et al. 08, S. Boccaletti et al. 06] threshold: heterogeneity vs. coupling quantify: heterogeneity < coupling as function of network parameters

11 Outline 1 Intro to Network Systems and Power Flow 2 Known tests and a conjecture F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks and smart grids. Proc National Academy of Sciences, 110(6): , doi: /pnas A new approach and new tests S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. Autom. Control, November URL Submitted S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: The Taylor expansion of the inverse Kuramoto map. In Proc CDC, Miami, USA, December URL

12 Power networks as quasi-synchronous AC circuits generators and loads 8 2 physics: Kirchoff and Ohm laws quasi-sync regime: voltage and phase V i, θ i active and reactive power p i, q i 4 today s simplifying assumptions: 1 lossless lines 2 approximated decoupled equations Decoupled power flow equations active: p i = j a ij sin(θ i θ j ) reactive: q i = j b ijv i V j

13 Power Flow Equilibria p i = n a ij sin(θ i θ j ) j=1 Given: network parameters & topology, load & generation profile, Q1: a stable synchronous operating point? Q2: what is the network capacity to transmit active power? Q3: how to quantify robustness as distance from loss of feasibility? Sync is crucial for the functionality and operation of the AC power grid. Generators have to swing in sync despite fluctuations/faults/contingencies. 1 Security analysis [Araposthatis et al. 81, Wu et al. 80 & 82, Ilić 92,... ] 2 Load flow feasibility [Chiang et al. 90, Dobson 92, Lesieutre et al. 99,... ] 3 Optimal generation dispatch [Lavaei et al. 12, Bose et al. 12,... ] 4 Transient stability [Sastry et al. 80, Bergen et al. 81, Hill et al. 86,... ]

14 Flow / spring analogy p i = n a ij sin(θ i θ j ) j=1 Spring network p i = τ i : torque at i a ij = k ij : spring stiffness i, j sin(θ i θ j ) : modulation Power network p i : injected power a ij : max power flow i, j sin(θ i θ j ) : modulation

15 Outline 1 Intro to Network Systems and Power Flow 2 Known tests and a conjecture F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks and smart grids. Proc National Academy of Sciences, 110(6): , doi: /pnas A new approach and new tests S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. Autom. Control, November URL Submitted S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: The Taylor expansion of the inverse Kuramoto map. In Proc CDC, Miami, USA, December URL

16 Primer on algebraic graph theory Weighted undirected graph with n nodes and m edges: Incidence matrix: n m matrix B s.t. (B p active ) (ij) = p i p j Weight matrix: m m diagonal matrix A Laplacian stiffness: L = BAB 0 Kuramoto equilibrium equation points: p active = BA sin(b θ) = p active BA(B θ) = L θ Algebraic connectivity: λ 2 (L) = second smallest eig of L = notion of connectivity and coupling

17 Linear spring and resistive networks Laplacian systems and Laplacian pseudoinverses <latexit sha1_base64="f0zijgdlysn1qco4yvczzlpolvy=">aaab8nicdvdlssnafj3uv62vqks3g0uqldcjoy9dwy3lksyw2lam00k7dpjgzlisox/gvjeuxk0/jpgxttokknrgwugce7n3hj/htcqe3o3cyura+kzxs7s1vbo7v94/ujnxkgh1scxj0fgxpjxf1fvmcdpjbmwhz2nbh1/mfntchwrxdkumcfvcpixywahwwro5s/rlcjib9artvceyeapztputu+zcondsso4kwklvl3/0bjfjqxopwrguxqslysuwuixwoiv1ukkttmz4sluarjik8lxohnpizffzm2fwrhsdgmrcv6tgxp0+m+fqymno684qq5h87exix143vuhdy1iupipgzleosdlumcwdgammkff8qgkmgumrirlhgynsmzv0hf8/w/+ja5sne107lsza5lier+aynail1eatxiewcaebaxgaj+djsi1n48v4xbqwjoxmifgb4+0tx0+rmq==</latexit> <latexit sha1_base64="f0zijgdlysn1qco4yvczzlpolvy=">aaab8nicdvdlssnafj3uv62vqks3g0uqldcjoy9dwy3lksyw2lam00k7dpjgzlisox/gvjeuxk0/jpgxttokknrgwugce7n3hj/htcqe3o3cyura+kzxs7s1vbo7v94/ujnxkgh1scxj0fgxpjxf1fvmcdpjbmwhz2nbh1/mfntchwrxdkumcfvcpixywahwwro5s/rlcjib9artvceyeapztputu+zcondsso4kwklvl3/0bjfjqxopwrguxqslysuwuixwoiv1ukkttmz4sluarjik8lxohnpizffzm2fwrhsdgmrcv6tgxp0+m+fqymno684qq5h87exix143vuhdy1iupipgzleosdlumcwdgammkff8qgkmgumrirlhgynsmzv0hf8/w/+ja5sne107lsza5lier+aynail1eatxiewcaebaxgaj+djsi1n48v4xbqwjoxmifgb4+0tx0+rmq==</latexit> +1 <latexit sha1_base64="f0zijgdlysn1qco4yvczzlpolvy=">aaab8nicdvdlssnafj3uv62vqks3g0uqldcjoy9dwy3lksyw2lam00k7dpjgzlisox/gvjeuxk0/jpgxttokknrgwugce7n3hj/htcqe3o3cyura+kzxs7s1vbo7v94/ujnxkgh1scxj0fgxpjxf1fvmcdpjbmwhz2nbh1/mfntchwrxdkumcfvcpixywahwwro5s/rlcjib9artvceyeapztputu+zcondsso4kwklvl3/0bjfjqxopwrguxqslysuwuixwoiv1ukkttmz4sluarjik8lxohnpizffzm2fwrhsdgmrcv6tgxp0+m+fqymno684qq5h87exix143vuhdy1iupipgzleosdlumcwdgammkff8qgkmgumrirlhgynsmzv0hf8/w/+ja5sne107lsza5lier+aynail1eatxiewcaebaxgaj+djsi1n48v4xbqwjoxmifgb4+0tx0+rmq==</latexit> current source current source x (a) spring network (b) resistive circuit L stiffness x = f load and L conductance v = c injected linearized Kuramoto balance equation : if i p i = 0, then equilibrium exists : pairwise displacements : p active = L θ θ = L p active B θ = B L p active

18 Known tests P loads P generators Given balanced p active, do angles exist? p active = BA sin(b θ) synchronization arises if heterogeneity < coupling power transmission < connectivity strength Equilibrium angles (neighbors within π/2 arc) exist if B p active 2 < λ 2 (L) for unweighted graphs (Old 2-norm T) B L p active < 1 for trees, complete (Old -norm T) Numerical evidence: Old -norm T applies broadly / approximately

19 Outline 1 Intro to Network Systems and Power Flow 2 Known tests and a conjecture F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks and smart grids. Proc National Academy of Sciences, 110(6): , doi: /pnas A new approach and new tests S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. Autom. Control, November URL Submitted S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: The Taylor expansion of the inverse Kuramoto map. In Proc CDC, Miami, USA, December URL

20 Novel today Equilibrium angles (neighbors within π/2 arc) exist if B p active 2 < λ 2 (L) for unweighted graphs (Old 2-norm T) B L p active < 1 for trees, complete (Old -norm T) Equilibrium angles (neighbors within π/2 arc) exist if B L p active 2 < 1 for unweighted graphs (New 2-norm T) B L p active < g( P ) for all graphs (New -norm T)

21 where g is monotonically decreasing!'" g : [1, ) [0, 1] "') g(x) "'( "'% "'# "'"!"!" #" $" %" &" x g(x) = y(x) + sin(y(x)) 2 x y(x) sin(y(x)) 2 y(x) = arccos (x 1 x + 1 )

22 and where P is a projection }{{} R m = Im(B ) }{{} edge space cutset space flow vectors Ker(BA) }{{} weighted cycle space cycle vectors P = B L BA = oblique projection onto Im(B ) parallel to Ker(BA) (recall: L=BAB, C (CC ) 1 C = orthogonal projector onto Im(C )) 1 if G unweighted, then P is orthogonal and P 2 = 1 2 if G acyclic, then P = I m and P p = 1 3 if G uniform complete or ring, then P = 2(n 1)/n 2

23 Unifying Theorem Equilibrium angles (neighbors within γ arc) exist if, in some p-norm, B L p active p γα p (γ) for all graphs (New p-norm T) α p (γ) := min amplification factor of P diag[sinc(x)] For unweighted p = 2, new test sharper than old B L p active 2 sin(γ) (New 2-norm T) For p =, new test is for all graphs B L p active g( P ) (New -norm T)

24 Comparison of sufficient and approximate sync tests K c = critical coupling of Kuramoto model, computed via MATLAB fsolve K T = smallest value of scaling factor for which test T fails Critical ratio K T /K c Test Case old 2-norm new -norm old -norm α test approximate fmincon IEEE % % % % IEEE % % % % IEEE RTS % % % % IEEE % % % % IEEE % % 100 % 100 % IEEE % % % * IEEE % % % * IEEE % % % * Polish % % % * fmincon has been run for 100 randomized initial phase angles. * fmincon does not converge.

25 Proof sketch 1/2: Rewriting the equilibrium equation For what B, A, p active does there exist θ solution to: p active = BA sin(b θ) For what flow z and projection P onto cutset/flow space, does there exist a flow x that solves P sin(x) = z P diag[sinc(x)]x = z x = (P diag[sinc(x)]) 1 z =: h(x)

26 Proof sketch 2/2: Amplification factor & Brouwer 1 look for x solving x = h(x) = (P diag[sinc(x)]) 1 z 2 take p norm, define min amplification factor of P diag[sinc(x)]: α p (γ) := min x p γ min y p=1 P diag[sinc(x)]y p If z p γα p (γ) and x B p (γ) = {x x p γ}, then h(x) p max x max (P diag[sinc(x)]) 1 y p z p y = ( ) 1 z p min min P diag[sinc(x)]y p z p x y α p (γ) γ hence h(x) B p (γ) and h satisfies Brouwer on B p (γ)

27 Outline 1 Intro to Network Systems and Power Flow 2 Known tests and a conjecture F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks and smart grids. Proc National Academy of Sciences, 110(6): , doi: /pnas A new approach and new tests S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. Autom. Control, November URL Submitted S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: The Taylor expansion of the inverse Kuramoto map. In Proc CDC, Miami, USA, December URL

28 Summary Kuramoto model Given topology (incidence B), admittances (Laplacian L), injections p active, θ i = p i n a ij sin(θ i θ j ) j=1 Q1: a stable synchronous operating point (with pairwise angles γ)? Q2: what is the network capacity to transmit active power? Q3: how to quantify robustness as distance from loss of feasibility? Equilibrium angles exist if, in some p-norm, B L p active p γα p (γ) for all graphs (New p-norm T) For p = and P cutset projection, B L p active g( P ) (New -norm T)

29 Summary and Future Work Contributions B L p active p γα p (γ) (New p-norm T) B L p active g( P ) (New -norm T) 1 geometry of cutset projection operator 2 family of sufficient sync conditions (polytope in p active ) 3 trade-off between coupling strength and oscillator heterogeneity Future research 1 close the gap between sufficient and necessary conditions 2 consider increasingly realistic power flow equations 3 applications to other dynamic flow networks averaging compartmental flows mutualism virus spread coupled oscillators social systems