Stability and Control of Synchronization in Power Grids

Transcription

1 Stability and Control of Synchronization in Power Grids Adilson E. Motter Department of Physics and Astronomy Northwestern Institute on Complex Systems Northwestern University, Evanston, USA

2 Power Generators

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4 Power Grid Dynamics Synchronization condition This condition can be met using control. Questions: Can the generators synchronize spontaneously? If so, under what conditions? What are the implications for control?

5 Spontaneous Synchronization adjustment of rhythms due to interactions

6 Spontaneous Synchronization

7 Spontaneous Synchronization adjustment of rhythms due to interactions Some of the main lines of research: Field specific applications population dynamics, circadian clocks, synthetic biology! Kuramoto-type model " i = # i + K n j=1 sin(" j $" i ) Pecora-Carroll-type model n % This presentation

8 Diffusively Coupled Oscillators x = F(x) : dynamics of each individual oscillator H(x) : output function n $ j=1 x i = F(x i ) + " A ij [H(x j ) # H(x i )] A = (A ij ) : adjacency matrix " : overall coupling strength Synchronized state: x i (t) = s(t), i =1,...,n s = F(s) Pecora & Carroll, PRL 1998

9 Diffusively Coupled Oscillators n $ j=1 x i = F(x i ) + " A ij [H(x j ) # H(x i )]. n x i = F(x i ) #" $ G ij H(x j ), i =1,...,n j=1 G = (G ij ) : coupling (Laplacian) matrix G ij = #A ij + % ij n $ A ij j=1 eig(g) : Re " 1 = 0 < Re " 2 #... # Re " n

10 Stability of Synchronous States change of coordinates similarity transformation perturbation modes. Nishikawa & Motter, Physica D 2006

11 Stability of Synchronous States Master stability equation Stability condition Nishikawa & Motter, PRE 2006 Nishikawa & Motter, Physica D 2006

12 Undirected Networks!" 2!" n! " Separation between network structure and local dynamics: network structure! n! 2 < " 2 " 1 dynamics Pecora & Carroll, PRL 1998

13 Arbitrary Networks " "

14 Common Sense Synchronization is easier to achieve with more interactions. That synchronization properties change monotonically as the number of available interactions is varied. That certain network structures facilitate while others inhibit synchronization.

15 Common Sense Synchronization is easier to achieve with more interactions. That synchronization properties change monotonically as the number of available interactions is varied. That certain network structures facilitate while others inhibit synchronization. All these expectations are false!

16 Synchronization Landscape Nishikawa & Motter, PNAS 2010 Experiment: Ravoori et al. PRL 2011

17 Power Grid Dynamics Synchronization condition This condition can be met using control. Questions: Can the generators synchronize spontaneously? If so, under what conditions? What are the implications for control?

18 Power Grid Dynamics Synchronization condition Our requirements for good model: Simple enough to allow large scale analysis Realistic enough that power engineers would care

19 Power Grid Dynamics Synchronization condition Equation of motion

20 Network Reduction Y has most properties of a Laplacian Matrix

21 Network Structure! " Full network 678 nodes 1644 edges Reduced network 170 nodes All possible edges Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

22 Stability Condition n j=1 P ei δ j δ j

23 Stability Condition n j=1 P ei δ j δ j

24 Stability Condition β i =(D i +1/R i )/2H i be the same for all gener

25 Stability Condition Temporarily assume that β i =(D i +1/R i )/2H i be the same for all gener β i be is the same for all generators

26 Stability Condition Diagonalization into 2D systems Stable if and only if Master stability function

27 Stability Condition Diagonalization into 2D systems Stable if and only if Master stability function

28 Enhancement of Spontaneous Synchrony β = β opt 2 α 2 Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

29 Enhancement of Spontaneous Synchrony Lifting the assumption that β i be are all the same System heterogeneity Synchronization stability System Nodes Links Generators Y 0 Y β i Original x d,i adjusted β i = β β i = β opt 3-generator test system generator test system generator test system Guatemala power grid Northern Italy power grid Poland power grid As a measure of heterogeneity, we show the standard deviation normalized by the average. The quantities considered are the weighted degrees computed for Y Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

30 Enhancement of Spontaneous Synchrony Lifting the assumption that β i be are all the same System heterogeneity Synchronization stability System Nodes Links Generators Y 0 Y β i Original x d,i adjusted β i = β β i = β opt 3-generator test system generator test system generator test system Guatemala power grid Northern Italy power grid Poland power grid As a measure of heterogeneity, we show the standard deviation normalized by the average. The quantities considered are the weighted degrees computed for Y Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

31 Enhancement of Spontaneous Synchrony System heterogeneity Synchronization stability System Nodes Links Generators Y 0 Y β i Original x d,i adjusted β i = β β i = β opt 3-generator test system generator test system generator test system Guatemala power grid Northern Italy power grid Poland power grid As a measure of heterogeneity, we show the standard deviation normalized by the average. The quantities considered are the weighted degrees computed for Y Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

32 Enhancement of Spontaneous Synchrony System heterogeneity Synchronization stability System Nodes Links Generators Y 0 Y β i Original x d,i adjusted β i = β β i = β opt 3-generator test system generator test system generator test system Guatemala power grid Northern Italy power grid Poland power grid As a measure of heterogeneity, we show the standard deviation normalized by the average. The quantities considered are the weighted degrees computed for Y Motter, Myers, Anghel, Nishikawa, to be published

33 Spontaneous Synch? Control? β i =(D i +1/R i )/2H i be the same for all gener β = β opt 2 α 2 Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

34 Parameter Tuning is Locally Optimal in General Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

35 Effective in Presence of Load Dynamics Motter, Myers, Anghel, Nishikawa, Nature Phys 2013

36 Final remarks - The synchronization of power-grid networks can be studied through conditions analogous to those previously established for diffusively coupled oscillators. - In power grids, however, the relevant network is not simply the network of transmission lines and is in fact state-dependent. - This causes the structure and dynamics to be more intimately related than in idealized models. - As a result, important aspects of the network structure can be emulated by modifying tunable parameters of the dynamical units.

37 Acknowledgements - NSF Division of Mathematical Sciences Grant DMS , CAREER Award DMS Alfred P. Sloan Foundation Sloan Research Fellowship Award BR DoE/Argonne Natl. Laboratory/ISEN Early Career Investigator Award for Energy Research

38 References Spontaneous synchrony in power-grid networks A.E. Motter, S.A. Myers, M. Anghel, T. Nishikawa Nature Physics 9, 191 (2013). Networks in motion A.E. Motter, R. Albert Physics Today 65(4), 43 (2012) Robustness of optimal synchronization in real networks B. Ravoori, A.B. Cohen, J. Sun, A.E. Motter, T.E. Murphy, R. Roy Phys. Rev. Lett. 107, (2011) Network synchronization landscape reveals compensatory structures, quantization, and the positive effect of negative interactions T. Nishikawa, A.E. Motter Proc. Natl. Acad. Sci. USA 107, (2010) For more information, please go to the group s webpage: