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Synchronization in Power Networks

Florian Dorfler and Francesco Bullo

Center for Control,

Dynamical Systems & Computation

University of California at Santa Barbara

http://motion.me.ucsb.edu

Institute for Energy EfficiencyUC Santa Barbara

Santa Barbara, California, October 19, 2011

F. Dorfler & F. Bullo (UCSB) Synchronization in Power Networks Inst. for Energy Efficiency 1 / 23

http://motion.me.ucsb.edu/

http://motion.me.ucsb.edu/



Motivation: the current power grid is . . .

“. . . the greatest engineering achievement of the 20th century.”[National Academy of Engineering ’10]

1

large-scale, complex, & rich nonlinear dynamics2 100 years old and operating at its capacity limits




Motivation: the current power grid is . . .

“. . . the greatest engineering achievement of the 20th century.”[National Academy of Engineering ’10]

1

large-scale, complex, & rich nonlinear dynamics2 100 years old and operating at its capacity limits ⇒ BLACKOUTS

The Blackout of 2003: 8/15/2003

Failure Reveals Creaky System, Experts Believe




Motivation: the envisioned power grid

Energy is one of the top three national priorities

Expected developments in “smart grid”:

1 large number of distributed power sources

2 increasing adoption of renewables3 sophisticated cyber-coordination layer

challenges: increasingly complex networks & stochastic disturbances

opportunity: some smart grid keywords:

control/sensing/optimization ⊕ distributed/coordinated/decentralized

Central theme: “understanding and taming complexity”




Motivation: the envisioned power grid

Energy is one of the top three national priorities

Expected developments in “smart grid”:

1 large number of distributed power sources

2 increasing adoption of renewables3 sophisticated cyber-coordination layer

challenges: increasingly complex networks & stochastic disturbances

opportunity: some smart grid keywords:

control/sensing/optimization ⊕ distributed/coordinated/decentralized

Central theme: “understanding and taming complexity”




Motivation: the envisioned power grid – our viewpoint

Projects at UCSB: “power systems engineering” ⊕ “networked control”

1 detection and identification of faults & cyber-physical attacks(together with F. Pasqualetti)

ω1

ω2

ω3

g1

g2 g3

b4

b1

b5b2

b6

b3

Sensors

WECC 3/9 power system system dynamics & measurement

Objectives: Is the attack or fault detectable/identifiable by measurements?

How to design (distributed) filters for detection/identification?F. Dorfler & F. Bullo (UCSB) Synchronization in Power Networks Inst. for Energy Efficiency 4 / 23

M i i h i i d id i i





1

detection and identification of faults & cyber-physical attacks(together with F. Pasqualetti)

ω1

ω2

ω3

g1

g2 g3

b4

b1

b5b2

b6

b3

Sensors

WECC 3/9 power system system dynamics & measurement

Objectives: Is the attack or fault detectable/identifiable by measurements?

How to design (distributed) filters for detection/identification?F. Dorfler & F. Bullo (UCSB) Synchronization in Power Networks Inst. for Energy Efficiency 4 / 23

M i i h i i d id i i





2 Synchronization & transient stability

Generators have to swing synchronouslydespite severe fluctuations in generation/loador faults in network/system components

Objectives [D. Hill & G. Chen ’06]: power network dynamics? graph

Observations from distinct fields:

◦ power networks are coupled oscillators

◦ coupled oscillators sync for large coupling

◦ graph theory quantifies coupling, e.g., λ2

⇒ plausible(?): power networks sync for large λ2F. Dorfler & F. Bullo (UCSB) Synchronization in Power Networks Inst. for Energy Efficiency 5 / 23

M ti ti th i i d id i i t





2 Synchronization & transient stability

Generators have to swing synchronouslydespite severe fluctuations in generation/loador faults in network/system components

Objectives [D. Hill & G. Chen ’06]: power network dynamics? graph

Observations from distinct fields:

◦ power networks are coupled oscillators

◦ coupled oscillators sync for large coupling

◦ graph theory quantifies coupling, e.g., λ2

⇒ plausible(?): power networks sync for large λ2F. Dorfler & F. Bullo (UCSB) Synchronization in Power Networks Inst. for Energy Efficiency 5 / 23



S a of acti ities of last 2 ea s



Summary of activities of last 2 years

Research Areas: “power systems engineering” ⊕ “networked control”1 synchronization and transient stability2 detection and identification of faults and cyber-physical attacks3 model reduction and scalability

AcademicsEducation: PhD students Florian Dorfler and Fabio Pasqualetti

Publications: 3 journal articles (SIAM & IFAC), 9 conference articlesAwards: two plenaries, two best paper awards

2011-14 NSF project “Cyber-Physical Challenges of TransientStability and Security in Power Grids.”

NSF CyperPhysical Systems and Trustworthy Computing ProgramsCollaboration with Ian Dobson (Wisconsin) and Bruno Sinopoli (CMU)

Collaboration with Los Alamos National Lab, DOE project“Optimization and Control Theory for Smart Grids”, Misha Chertkov






Research Areas: “power systems engineering” ⊕ “networked control”1 synchronization and transient stability2 detection and identification of faults and cyber-physical attacks3 model reduction and scalability

AcademicsEducation: PhD students Florian Dorfler and Fabio Pasqualetti

Publications: 3 journal articles (SIAM & IFAC), 9 conference articlesAwards: two plenaries, two best paper awards

2011-14 NSF project “Cyber-Physical Challenges of TransientStability and Security in Power Grids.”

NSF CyperPhysical Systems and Trustworthy Computing ProgramsCollaboration with Ian Dobson (Wisconsin) and Bruno Sinopoli (CMU)

Collaboration with Los Alamos National Lab, DOE project“Optimization and Control Theory for Smart Grids”, Misha Chertkov


Outline



Outline

1 Introduction and Motivation

2 Mathematical Modeling & Synchronization Problem

3 Synchronization in the Kuramoto Model

4 From the Kuramoto Model to the Power Network Model

5 Conclusions






Structure-preserving power network model



Structure-preserving power network model

Classic structure-preserving model [A.R. Bergen & D. Hill ’81]:

M i θi + D i θi = P i −n+m

j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n}

D i θi = P i −n+m

j =1

P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}

P ij = |V i | · |V j | · |Yij | ≥ 0 max. power transferred i ↔ j

P i =

P mech.in,i for i ∈ {1, . . . , n}

−P load,i for i ∈ {n + 1, . . .m}real power injection at i


Synchronization and transient stability



Synchronization and transient stability


j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}

1 General synchronization problem: |θi − θ j | bounded & θi = θ j

2 Classic analysis methods: Assumptions & Hamiltonian arguments

highly developed field based on global system perspective

Open Problem [D. Hill & G. Chen ’06]:

synchronization in power networks? underlying graph properties


Detour: Synchronization of coupled oscillators




It all began with two pendulum clocks and theobservation of “an odd kind of sympathy ”.

[Huygens, C. Horologium Oscillatorium, 1673]

Today’s canonical coupled oscillator model

[A. Winfree ’67, Y. Kuramoto ’75]

Kuramoto model of coupled oscillators:

θi = ωi −K

n

n

j =1sin(θi − θ j )

n oscillators with phase θi ∈ S1

non-identical natural frequencies ωi ∈ R1

elastic coupling with strength K /nω1

ω3ω2

S1

K/n






Kuramoto model of coupled oscillators:

θi = ωi −

K

nn

j =1 sin(θi − θ j )

Just a few of various direct & related applications:

Sync of coupled pendulum clocks [C. Huygens XVII, M. Bennet et. al ’02]

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]

Deep-brain stimulation and neuroscience

[P.A. Tass ’03, E. Brown et al. ’04]Coordination of particle models[R. Sepulchre et al. ’07, D. Klein et al. ’09]

Countless other technological, biological, & social syncphenomena [A. Winfree ’67, S.H. Strogatz ’00, J. Acebron ’01]






A well-understood linear synchronizationmodel [M. DeGroot ’74, J. Tsitsiklis ’84,. . . ]

Consensus protocol:

x i = −n

j =1 aij (x i − x j )

n identical agents with state variable x i ∈ R

interaction described by a connected graph with weights aij

a few applications: robotic coordination, sensor networks, distributedcomputation & optimization, continuous Markov chains, . . .


“The big picture”



The big picture

Consensus protocols: Kuramoto oscillators:

?

Open problem: sync in power networks

state, parameters, and topology of graph

θi = ωi −

�n

j=1

K

nsin(θi − θj)

M iθi +Diθi = P i −

�n

j=1P ij sin(θi − θj)

xi = −�n

j=1aij(xi − xj)


“The big picture”



b g p

Consensus protocols: Kuramoto oscillators:

?

Open problem: sync in power networks

state, parameters, and topology of graph

θi = ωi −

�n

j=1

K

nsin(θi − θj)

M iθi +Diθi = P i −

�n

j=1P ij sin(θi − θj)

xi = −�n

j=1aij(xi − xj)

Previous “observations” about this connection:

Power systems: [D. Subbarao et al., ’01, G. Filatrella et al., ’08, V. Fioriti et al., ’09]

Networked control: [D. Hill et al., ’06, M. Arcak, ’07]

Dynamical systems: [H. Tanaka et al., ’97, A. Arenas ’08]


Outline







5 Conclusions


Synchronization in the Kuramoto Model



y

Classic Homogeneous Kuramoto model

˙θi = ωi −

K

nn

j =1 sin(θi − θ j )

Notions of synchronization:

1

phase cohesiveness: |θi (t ) − θ j (t )| bounded2 frequency synchronization: θi (t ) = θ j (t )

Classic intuition:

K small & |ωi − ω j | large⇒ incoherence & no sync

K large & |ωi − ω j | small⇒ cohesiveness & frequency sync







y

Topological Kuramoto model

θi = ωi −n

j =1

aij sin(θi − θ j )

Assume: coupling graph isundirected and connected

ω1

ω3ω2

a12

a13

a23

S1





y


θi = ωi −n

j =1



1 Necessary conditions: no sync if n

j =1 aij ≤ωi −

nk =1 ωk

n

2 Various sufficient conditions in the literature [F. Dorfler & F. Bullo, ’09]:

Based on algebraic connectivity: λ2 >ω1 − ω2 , . . .

2

3 Under development are exact conditions for synchronization

⇒ implication: “coupling dominates non-uniformity” ⇒ sync






θi = ωi −n

j =1




j =1 aij ≤ωi −

nk =1 ωk

n



2








θi = ωi −n

j =1




j =1 aij ≤ωi −

nk =1 ωk

n



2




Outline







5 Conclusions


From the Kuramoto Model to the Power Network Model



1) Structure-preserving power network model on Tn+m × Rn:

M i ¨θi + D i

˙θi = P i −

n+m

j =1 P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}

2) Non-uniform variation of Kuramoto model onTn+m

:

θi = P i −n+m

j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n + m}





1) Structure-preserving power network model on Tn+m × Rn:

M i ¨θi + D i

˙θi = P i −

n+m

j =1 P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}

2) Non-uniform variation of Kuramoto model onTn+m

:

θi = P i −n+m

j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n + m}

Synchronization conditions can be related via1 extension of 1st-order analysis [D. Koditschek ’88, Y.P. Choi et al. ’10]

2 time scale separation analysis [F. Dorfler & F. Bullo ’09]

3 today: local topological equivalence [F. Dorfler & F. Bullo ’11]





⇒ near the equilibrium manifolds 1) synchronizes ⇔ 2) synchronizes

⇒ near the equilibrium manifolds 1) & 2) are topologically conjugate

0.5 1 1.5

−0.5

0

0.5

0.5 1 1.5

−0.5

0

0.5

˙ θ (

t

)

θ(t) rad θ(t) rad

˙ θ ( t )

[ r a d / s ]

˙ θ ( t )

[ r a d / s ]


Main Synchronization Results



Structure-preserving power network model:


j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}

1 necessary condition: Let ωsync =

k P k /

k D k , then

n+m j =1 P ij ≤ |P i − D i · ωsync| ⇒ no sync

2 sufficient condition: Let P i = P i − D i · ωsync, then

λ2 >˜

P 1 −˜P 2 , . . .

2 ⇒ sync

3 conjectured exact conditions are under development

bottom line: “coupling dominates imbalance in active power”







j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}


k P k /

k D k , then



λ2 >˜

P 1 −˜P 2 , . . .

2 ⇒ sync









j =1P ij sin(θi − θ j ) , i ∈ {1, . . . , n}


j =1P ij sin(θi − θ j ) , i ∈ {n + 1, . . .m}


k P k /

k D k , then



λ2 >˜

P 1 −˜P 2 , . . .

2 ⇒ sync









Illustration with the IEEE Reliability Test System ’96 under severe loading

t [s]

|

θ i

( t ) −

θ j

( t ) |

[ r a d ]

γ min

t [s][s]

˙ θ i

( t )

[ r a d / s ]

θ i

( t

)

[ r a d ]

θi

θ(t)

θ(t)

ωi −Diωsync

Conjectured Kuramoto sync condition is marginally satisfied





Illustration with the IEEE Reliability Test System ’96 under severe loading

|

θ i

( t ) −

θ j

( t ) |

[ r a d ]

π

/2

t [s]t [s]

˙ θ i

( t )

[ r a d / s ]

θ i

( t )

[ r a d ]

50 s

7 rad

50 s

rad/s

−3 rad/s

15rad

−10rad 50 s

t [s]

Conjectured Kuramoto sync condition is marginally not satisfied


Outline







5 Conclusions




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