Network Systems and Kuramoto Oscillators

Network Systems and Kuramoto Oscillators

Francesco Bullo

Department of Mechanical Engineering

Center for Control, Dynamical Systems & Computation

University of California at Santa Barbara

http://motion.me.ucsb.edu

37th Chinese Control ConferenceWuhan, China, July 26, 2018

http://motion.me.ucsb.edu

Acknowledgments

CCC General Chairs, Program Chairs and all organizing volunteersCAA Technical Committee on Control TheorySystems Engineering Society of ChinaChina University of Geosciences, Wuhan

A successful, trasured, inspiring, and long-lasting traditionof collaboration between IEEE CSS and CCC

Saber JafarpourUCSB

Elizabeth Y. HuangUCSB

Florian DorflerETH

USA Department of Energy (DOE), SunShot Program, XAT-6-62531-01,Stabilizing the Power System in 2035 and Beyond, Consortium of DOENational Renewable Energy Laboratory, UCSB and University of Minnesota

Outline

1 Intro to Network Systems and Power Flow

2

Known tests and a conjectureF. Dorfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networksand smart grids.Proc National Academy of Sciences, 110(6):2005–2010, 2013.doi:10.1073/pnas.1212134110

3

A new approach and new testsS. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projec-tions.IEEE Trans. Autom. Control, November 2017.URL https://arxiv.org/abs/1711.03711.Submitted

S. Jafarpour, E. Huang, and F. Bullo. Synchronization of coupled oscillators: TheTaylor expansion of the inverse Kuramoto map.In Proc CDC, Miami, USA, December 2018.URL https://arxiv.org/abs/1711.03711

http://dx.doi.org/10.1073/pnas.1212134110

https://arxiv.org/abs/1711.03711


Example network systems

Smart grid Amazon robotic warehouse

Water Supplier Description 2-18

Figure 2-5. City of Portland Water Supply Schematic Diagram

Portland water network Industrial chemical plant

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

New text “Lectures on Network Systems”

Lectures onNetwork Systems

Francesco Bullo

With contributions byJorge Cortés

Florian DörflerSonia Martínez

Lectures on Network SystemsFrancesco Bullo

These lecture notes provide a mathematical introduction to multi-agent

dynamical systems, including their analysis via algebraic graph theory

and their application to engineering design problems. The focus is on

fundamental dynamical phenomena over interconnected network

systems, including consensus and disagreement in averaging systems,

stable equilibria in compartmental flow networks, and synchronization

in coupled oscillators and networked control systems. The theoretical

results are complemented by numerous examples arising from the

analysis of physical and natural systems and from the design of

network estimation, control, and optimization systems.

Francesco Bullo is professor of Mechanical Engineering and member

of the Center for Control, Dynamical Systems, and Computation at the

University of California at Santa Barbara. His research focuses on

modeling, dynamics and control of multi-agent network systems, with

applications to robotic coordination, energy systems, and social

networks. He is an award-winning mentor and teacher.

Francesco BulloLectures on N

etwork System

s

Lectures on Network Systems, Creates-pace, 1 edition, ISBN 978-1-986425-64-3

For students: free PDF for downloadFor instructors: slides and answer keyshttp://motion.me.ucsb.edu/book-lnshttps://www.amazon.com/dp/1986425649

300 pages (plus 200 pages solution manual)3K downloads since Jun 2016150 exercises with solutions

Linear Systems:

1 social, sensor, robotic & compartmental examples,

2 matrix and graph theory, with an emphasis onPerron–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 flowsystems, Metzler matrices.

Nonlinear Systems:

5 nonlinear consensus models,

6 population dynamic models in multi-species systems,

7 coupled oscillators, with an emphasis on theKuramoto model and models of power networks

http://motion.me.ucsb.edu/book-lns

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 θi ∈ S1

non-identical natural frequencies ωi ∈ R1

coupling with strength aij = aji

θi = ωi −n∑

j=1

aij sin(θi − θj)

Motivation

Kuramoto model

θi = ωi −n∑

j=1


1 Countless sync phenomena in sciences/engineering[S. Strogatz ’00, J. Acebron ’01, Arenas ’08]

2 Local canonical model for weakly coupledlimit-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

Spring network analog and applications

Kuramoto coupled oscillators

θi = ωi −∑

jaij sin(θi − θj)

Coupled swing equations

mi θi + di θi = τi −∑

jkij sin(θi − θj)

Applications in sciences: biology:pacemaker cells in theheart, circadian cells in the brain, coupled cortical neurons,Hodgkin-Huxley neurons, brain networks, yeast cells, flashingfireflies, chirping crickets, central pattern generators foranimal locomotion, particle models mimicking animal flockingbehavior, and fish schoolsphysics and chemistry: spin glass models, flavor evolution ofneutrinos, coupled Josephson junctions, coupled metronomes,Huygen’s coupled pendulum clocks, micromechanicaloscillators with opticalor mechanical coupling, and chemicaloscillationssocial networks: rhythmic applause, opinion dynamics,pedestrian crowd synchrony, and decision making in animalgroups

Applications in engineering: deep brain stimulation, lockingin solid-state circuit oscillators, planar vehicle coordination,carrier synchronization without phase-locked loops,semiconductor laser arrays, and microwave oscillator arrayselectric applications: structure-preserving andnetwork-reduced power system models, and droop-controlledinverters in microgridsalgorithmic applications: limit-cycle estimation throughparticle filters, clock synchronization in decentralizedcomputing networks, central pattern generators for roboticlocomotion, decentralized maximum likelihood estimation,and human-robot interactiongenerating music, signal processing, pattern recognition, andneuro-computing through micromechanical or laser oscillators.

Transition from incoherence to synchronization

Function = synchronizationθi = ωi −

∑n

j=1aij sin(θi − θj)

✓i(t)

large |ωi − ωj | & small coupling

⇒ incoherence = no sync

✓i(t)

small |ωi − ωj | & 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

Outline


2


3






Power networks as quasi-synchronous AC circuits

1 generators �� and loads •◦2 physics: Kirchoff and Ohm laws

3 quasi-sync regime:voltage and phase Vi , θiactive and reactive power pi , qi

4 today’s simplifying assumptions:1 lossless lines2 approximated decoupled equations

2

10

30 25

8

37

29

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23

7

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22

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7

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2728

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Decoupled power flow equationsactive: pi =

∑j aij sin(θi − θj)

reactive: qi = −∑j bijViVj

Power Flow Equilibria

pi =n∑

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, Ilic ’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, . . . ]

Flow / spring analogy

pi =n∑

j=1


Spring network

pi = τi : torque at i

aij = kij : spring stiffness i , j

sin(θi − θj) : modulation

Power network

pi : injected power

aij : max power flow i , j

sin(θi − θj) : modulation

2

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30 25

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Outline


2


3






Primer on algebraic graph theory

Weighted undirected graph with n nodes and m edges:

Incidence matrix: n ×m matrix B s.t. (B>pactive)(ij) = pi − pj

Weight matrix: m ×m diagonal matrix ALaplacian stiffness: L = BAB> ≥ 0

Kuramoto equilibrium equation points:

pactive = BA sin(B>θ)

=⇒ pactive ≈ BA(B>θ) = L θ

Algebraic connectivity:

λ2(L) = second smallest eig of L

= notion of connectivity and coupling

Linear spring and resistive networksLaplacian systems and Laplacian pseudoinverses

x

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(b) resistive circuit

Lstiffness x = fload and Lconductance v = cinjected

linearized Kuramoto balance equation : pactive = L θ

if∑

ipi = 0, then equilibrium exists : θ = L†pactive

pairwise displacements : B>θ = B>L†pactive

Known tests

Pgenerators

Ploads

Given balanced pactive, do angles exist?


synchronization arises ifheterogeneity < couplingpower transmission < connectivity strength

Equilibrium angles (neighbors within π/2 arc) exist if

‖B>pactive‖2 < λ2(L) for unweighted graphs (Old 2-norm T)

‖B>L†pactive‖∞ < 1 for trees, complete (Old ∞-norm T)

Numerical evidence: Old ∞-norm T applies broadly / approximately

Outline


2


3






Novel today


‖B>pactive‖2 < λ2(L) for unweighted graphs (Old 2-norm T)

‖B>L†pactive‖∞ < 1 for trees, complete (Old ∞-norm T)


‖B>L†pactive‖2 < 1 for unweighted graphs (New 2-norm T)

‖B>L†pactive‖∞ < g(‖P‖∞) for all graphs (New ∞-norm T)

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

)

and where P is a projection

1

2 3

1 2

3

1

2 3

1 2

3

Rm︸︷︷︸edge space

= Im(B>)︸︷︷︸cutset spaceflow 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>)−1C = orthogonal projector onto Im(C>))

1 if G unweighted, then P is orthogonal and ‖P‖2 = 1

2 if G acyclic, then P = Im and ‖P‖p = 1

3 if G uniform complete or ring, then ‖P‖∞ = 2(n − 1)/n ≤ 2

Unifying Theorem

Equilibrium angles (neighbors within γ arc) exist if, in some p-norm,

‖B>L†pactive‖p ≤ γαp(γ) for all graphs (New p-norm T)

αp(γ) := min amplification factor of P diag[sinc(x)]

For unweighted p = 2, new test sharper than old

‖B>L†pactive‖2 ≤ sin(γ) (New 2-norm T)

For p =∞, new test is for all graphs

‖B>L†pactive‖∞ ≤ g(‖P‖∞) (New ∞-norm T)

Comparison of sufficient and approximate sync tests

Kc = critical coupling of Kuramoto model, computed via MATLAB fsolveKT = smallest value of scaling factor for which test T fails

Test CaseCritical ratio KT/Kc

old 2-norm new ∞-norm old ∞-norm α∞ testapproximate fmincon

IEEE 9 16.54 % 73.74 % 92.13 % 85.06 %†

IEEE 14 8.33 % 59.42 % 83.09 % 81.32 %†

IEEE RTS 24 3.86 % 53.44 % 89.48 % 89.48 %†

IEEE 30 2.70 % 55.70 % 85.54 % 85.54 %†

IEEE 39 2.97 % 67.57 % 100 % 100 %†

IEEE 57 0.36 % 40.69 % 84.67 % —*

IEEE 118 0.29 % 43.70 % 85.95 % —*

IEEE 300 0.20 % 40.33 % 99.80 % —*

Polish 2383 0.11 % 29.08 % 82.85 % —*

† fmincon has been run for 100 randomized initial phase angles.* fmincon does not converge.

Proof sketch 1/2: Rewriting the equilibrium equation

For what B,A, pactive does there exist θ solution to:


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)])−1z =: h(x)

Proof sketch 2/2: Amplification factor & Brouwer

1 look for x solving

x = h(x) = (P diag[sinc(x)])−1z

2 take p norm, define min amplification factor of P diag[sinc(x)]:

αp(γ) := min‖x‖p≤γ

min‖y‖p=1

‖P diag[sinc(x)]y‖p

If ‖z‖p ≤ γαp(γ) and x ∈ Bp(γ) = {x | ‖x‖p ≤ γ}, then

‖h(x)‖p ≤ maxx

maxy‖(P diag[sinc(x)])−1y‖p · ‖z‖p

=(

minx

miny‖P diag[sinc(x)]y‖p

)−1‖z‖p ≤ ‖z‖pαp(γ)

≤ γ

hence h(x) ∈ Bp(γ) and h satisfies Brouwer on Bp(γ)

Outline


2


3






Summary

Kuramoto model

Given topology (incidence B), admittances (Laplacian L), injections pactive,

θi = pi −∑n

j=1aij sin(θi − θj)

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†pactive‖p ≤ γαp(γ) for all graphs (New p-norm T)

For p =∞ and P cutset projection,


Summary and Future Work

‖B>L†pactive‖p ≤ γαp(γ) (New p-norm T)


Contributions1 geometry of cutset projection operator2 family of sufficient sync conditions (polytope in pactive)3 trade-off between coupling strength and oscillator heterogeneity

Future research1 close the gap between sufficient and necessary conditions2 consider increasingly realistic power flow equations3 applications to other dynamic flow networks

averaging compartmental flows mutualism virus spread coupled oscillators social systems

Documents

Network Systems and Kuramoto Oscillators