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

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1 Acknowledgments Network Systems and Kuramoto Oscillators Francesco Bullo Department of Mechanical Engineering Center for Control, Dynamical Systems & Computation University of California at Santa Barbara State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China, Jun 8 Department of Mechanics and Engineering Science, Peking University, Beijing, China, Jun 8 Saber Jafarpour UCSB Elizabeth Y. Huang UCSB USA Department of Energy (DOE), SunShot Program, XAT , Stabilizing the Power System in 05 and Beyond, Consortium of DOE National Renewable Energy Laboratory, UCSB and University of Minnesota NSF AFOSR ARO ONR DOE Example network systems Intro to Network Systems and Power Flow F. Do rfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks Proc National Academy of Sciences, 0(6):005 00, 0. doi:0.07/pnas.40 Smart grid Amazon robotic warehouse IEEE Trans. Autom. Control, November 07. Submitted. In Proc CDC, Miami, USA, December 08. Submitted. Figure -5. City of Portland Water Supply Schematic Diagram Portland water network Water Supplier Description -8 Industrial chemical plant

2 Linear network systems New text Lectures on Network Systems Lectures on Network Systems, F. Bullo, edition, CreateSpace, ISBN x(k + ) = Ax(k) + b or ẋ(t) = Ax(t) + b systems of interest asymptotic behavior tools Lectures on Network Systems Francesco 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. Lectures on Network Systems Lectures on Network Systems For students: free PDF for download For instructors: slides and answer keys pages (plus 00 pages solution manual) K downloads since Jun exercises with solutions Linear Systems: social, sensor, robotic & compartmental examples, matrix and graph theory, with an emphasis on Perron Frobenius theory and algebraic graph theory, averaging algorithms in discrete and continuous time, described by static and time-varying matrices, and network structure function = asymptotic behavior Francesco Bullo Francesco Bullo With contributions by Jorge Cortés Florian Dörfler Sonia Martínez 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 Synchronization in Networks of Coupled Oscillators Pendulum clocks & an odd kind of sympathy [C. Huygens, Horologium Oscillatorium, 67] Synchronization in Networks of Coupled Oscillators Coupled oscillator model: θ i = ω i n a ij sin(θ i θ j ) j= Today s canonical coupled oscillator model [A. Winfree 67, Y. Kuramoto 75] Coupled oscillator model: θ i = ω i n a ij sin(θ i θ j ) j= n oscillators with phase θ i S non-identical natural frequencies ω i R coupling with strength a ij = a ji Sync in Josephson junctions [S. Watanabe et. al 97, K. Wiesenfeld et al. 98] Sync in a population of fireflies [G.B. Ermentrout 90, Y. Zhou et al. 06] Canonical model of coupled limit-cycle oscillators [F.C. Hoppensteadt et al. 97, E. Brown et al. 04] Countless sync phenomena in sciences/bio/tech. [S. Strogatz 00, J. Acebrón 0, Arenas 08] citations on scholar.google: Winfree.5K, Kuramoto.K + 6.8K, surveys Strogatz, Acebron, Arenas: K, survey by Boccaletti 8K

3 Synchronization in Networks of Coupled Oscillators phenomenology and challenges 6 7 Function = synchronization θi = ω i n j= a ij sin(θ i θ j ) Intro to Network Systems and Power Flow F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks Proc National Academy of Sciences, 0(6):005 00, 0. doi:0.07/pnas.40 i (t) coupling small & ω i ω j large incoherence = no sync i (t) coupling large & ω i ω j small coherence = frequency sync IEEE Trans. Autom. Control, November 07. Submitted. Central question: [S. Strogatz 0, A. Arenas et al. 08, S. Boccaletti et al. 06] Power flow equations loss of sync due to bifurcation trade-off coupling vs. heterogeneity how to quantify this trade-off In Proc CDC, Miami, USA, December 08. Submitted. Power networks as quasi-synchronous AC circuits voltage magnitude and phase active and reactive power generators and loads physics: Kirchoff and Ohm laws today s simplifying assumptions: secure operating conditions quasi-sync: voltage and phase V i, θ i active and reactive power p i, q i lossless lines approximated decoupled equations feedback control economic optimization Decoupled power flow equations active: p i = j a ij sin(θ i θ j ) reactive: q i = j b ijv i V j network structure function = power transmission

4 Power Flow Equilibria Synchronization in Power Networks p i = j a ij sin(θ i θ j ) As function of network structure/parameters do equations admit solutions / operating points? how much active power can network transmit / flow? how to quantify stability margins? P loads Active power dynamics and mechanical/spring analogy P generators Coupled swing equations m i θ i + d i θi = p i j a ij sin(θ i θ j ) Kuramoto coupled oscillators Intro to Network Systems and Power Flow θ i = p i j a ij sin(θ i θ j ) F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks Proc National Academy of Sciences, 0(6):005 00, 0. doi:0.07/pnas.40 Sync is crucial for the functionality and operation of the AC power grid. Generators have to swing in sync despite fluctuations/faults/contingencies. Given: network parameters & topology and load & generation profile Q: an optimal, stable, and robust synchronous operating point? Security analysis [Araposthatis et al. 8, Wu et al. 80 & 8, Ilić 9,... ] Load flow feasibility [Chiang et al. 90, Dobson 9, Lesieutre et al. 99,... ] Optimal generation dispatch [Lavaei et al., Bose et al.,... ] 4 Transient stability [Sastry et al. 80, Bergen et al. 8, Hill et al. 86,... ] 5 Inverters in microgrids [Chandorkar et. al. 9, Guerrero et al. 09, Zhong,... ] 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 eq points: p active = BA sin(b θ) Algebraic connectivity: λ (L) = second smallest eig of L IEEE Trans. Autom. Control, November 07. Submitted. In Proc CDC, Miami, USA, December 08. Submitted. Linear spring networks and L balance equation : f = Lx x for balanced f n : x = L f DC flow approx: p active BA(B θ) = B θ = B L p active

5 Known tests A standing conjecture P loads P generators Given balanced p active, do angles exist? p active = BA sin(b θ) synchronization arises if power transmission < connectivity strength Equilibrium angles (neighbors within π/ arc) exist if B p active < λ (L) for unweighted graphs (Old -norm T) B L p active < for trees, complete (Old -norm T) at fixed radius and -norm, volume of ball 0 + as d + IEEE test cases: randomized and increase generation/loads B L p active < appears to imply: solution θ θ i θ j arcsin( B L p active ) for all {i, j} E Randomized test case Numerical worst-case Analytic prediction of Accuracy of condition: (000 instances) angle differences: angle differences: max θi θj {i,j} E θi θj arcsin( B L p active ) arcsin( B L p active ) max {i,j} E 9 bus system rad rad rad IEEE 4 bus system 0.66 rad rad rad IEEE 0 bus system 0.64 rad rad rad New England rad rad rad IEEE 57 bus system rad 0.8 rad rad IEEE 8 bus system 0.54 rad rad rad IEEE 00 bus system rad 0.45 rad rad Polish 8 bus system rad 0.47 rad rad Novel today Intro to Network Systems and Power Flow F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks Proc National Academy of Sciences, 0(6):005 00, 0. doi:0.07/pnas.40 Equilibrium angles (neighbors within π/ arc) exist if B p active < λ (L) for unweighted graphs (Old -norm T) B L p active < for trees, complete (Old -norm T) IEEE Trans. Autom. Control, November 07. Submitted. Equilibrium angles (neighbors within π/ arc) exist if B L p active < for unweighted graphs (New -norm T) B L p active < g( P ) for all graphs (New -norm T) In Proc CDC, Miami, USA, December 08. Submitted.

6 where g is monotonically decreasing and where P is a projection!'" "') g : [, ) [0, ] g(x) "'( "'% }{{} R m = Im(B ) }{{} edge space cutset space flow vectors Ker(BA) }{{} weighted cycle space cycle vectors "'# g(x) = "'"!" y(x) + sin(y(x))!" #" $" %" &" x x y(x) sin(y(x)) y(x) = arccos( x x + ) P = B L BA = oblique projection onto Im(B ) parallel to Ker(BA) (recall: orthogonal projector onto Im(C) is C(C C) C for full rank C) if G unweighted, then P is orthogonal and P = if G acyclic, then P = I m and P p = if G uniform complete or ring, then P = (n )/n Unifying Theorem Comparison of sufficient and approximate sync tests 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 =, new test sharper than old B L p active sin(γ) (New -norm T) For p =, new test is for all graphs B L p active g( P ) (New -norm T) 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 -norm new -norm new -norm old -norm α test conjectured conjectured approximate fmincon IEEE % % 7.74 % 9. % % IEEE 4 8. % 4.7 % 59.4 % 8.09 % 8. % IEEE RTS 4.86 % 5.6 % 5.44 % % % IEEE 0.70 % % % % % IEEE 9.97 % 7. % % 00 % 00 % IEEE %.9 % % % * IEEE % 4.6 % 4.70 % % * IEEE % 4. % 40. % % * Polish 8 0. %.9 % 9.08 % 8.85 % * fmincon has been run for 00 randomized initial phase angles. * fmincon does not converge.

7 Proof sketch /: Rewriting the equilibrium equation Proof sketch /: Amplification factor & Brouwer look for x solving 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)]) z =: h(x) x = h(x) = (P diag[sinc(x)]) z take p norm, define min amplification factor of P diag[sinc(x)]: α p (γ) := min x p γ min y p = 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)]) y p z p y = ( ) 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) Intro to Network Systems and Power Flow Computational method via power series Given z, compute x solution to F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in complex oscillator networks Proc National Academy of Sciences, 0(6):005 00, 0. doi:0.07/pnas.40 IEEE Trans. Autom. Control, November 07. Submitted. In Proc CDC, Miami, USA, December 08. Submitted. z = P sin(x) Assume x = i=0 A i+(z), where A i+ (z) is homogeneous degree i + z = P k=0 ( ) k (k + )! x k+ = P k=0 ( ) k (k + )! ( Equate left-hand and right-hand side at order,,..., j + : A (z) = z ( j A j+ (z) = P k= ( ) k+ (k + )! odd α,...,α k+ s.t. α + +α k+ =j+ i=0 ) k+ A i+ (z) A α (z) A αk+ (z) )

8 Step : Series expansion for inverse Kuramoto map Numerical examples Unique solution to z = P sin(x) is x = i=0 A i+(z) A (z) = z = B L p active ( A (z) = P! z ) ( A 5 (z) = P! A (z) z 5! z 5) ( A 7 (z) = P! A 5(z) z +! A (z) z 5 5! A (z) z 4 + 7! z 7) arbitrary higher-order terms can be computed symbolically Accuracy, log0 scale 0 0 Test case: IEEE st rd 5th 7th 9th th th scaled power injection For sufficiently small z p, series converges uniformly absolutely Kuramoto Oscillators and Power Flow improved sufficient conditions for equilibria upper bounds on transmission capacity stability margins as notions of distance from bifurcations Applications secure operating conditions: (Dörfler et al, PNAS ) feedback control: (Simpson-Porco et al, TIE 5) economic optimization: (Todescato et al, TCNS 7) Future research close the gap between sufficient and necessary conditions consider increasingly realistic power flow equations apply methods to other flow networks (water, gas,...)