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Control of coupled oscillator networks with application to microgrid technologies.

Skardal PS, Arenas A - Sci Adv (2015)

Bottom Line: The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences.Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades.The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Trinity College, Hartford, CT 06106, USA. ; Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain.

ABSTRACT
The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

No MeSH data available.


Fraction of driver nodes.(A) Fraction of driver nodes nD in ER (unfilled symbols) and SF (filled symbols) as a function of coupling strength K for mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangle, and green squares, respectively). (B) Required coupling strengths K5%, K10%, and K20% required to achieve consensus given nD = 0.05, 0.1, and 0.2 (blue circles, red triangles, and green squares, respectively). Each data point is the average over 100 network realizations of size N = 1000, each averaged over 100 natural frequency realizations.
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Figure 4: Fraction of driver nodes.(A) Fraction of driver nodes nD in ER (unfilled symbols) and SF (filled symbols) as a function of coupling strength K for mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangle, and green squares, respectively). (B) Required coupling strengths K5%, K10%, and K20% required to achieve consensus given nD = 0.05, 0.1, and 0.2 (blue circles, red triangles, and green squares, respectively). Each data point is the average over 100 network realizations of size N = 1000, each averaged over 100 natural frequency realizations.

Mentions: Finally, we quantify the overall effort required for consensus by studying how the fraction of driver nodes, denoted nD = ND/N, where ND is the total number of driver nodes, depends on both the system’s dynamical and structural parameters. Presenting our results in Fig. 4, we first explore how the fraction of driver nodes depends on the coupling strength by plotting in Fig. 4AnD versus K for both ER and SF networks with mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangles, and green squares, respectively). Results for ER and SF networks are plotted with unfilled and filled symbols, respectively, and each curve represents an average of over 100 network realizations, each averaged over 100 random natural frequency realizations. Although it is expected that nD decreases monotonically with K, the curves’ dependence on network topology and mean degree is nontrivial. In particular, the shape of nD versus K depends more sensitively on the mean degree than the topology, suggesting that network heterogeneity has little effect on overall control in comparison to average connectivity. In light of the significant dependence of overall control on the coupling strength, we investigate the coupling strength required to synchronize a network if a limited amount of control is available. To this end, we calculate for each family of networks the required coupling strengths K5%, K10%, and K20% for which, on average, a fraction nD = 0.05, 0.1, and 0.2 will achieve synchronization as a function of the average degree 〈k〉. We plot the results in Fig. 4B. We point out again that ER and SF networks behave very similarly on average, and that with a larger mean degree, a smaller coupling strength is required to achieve synchronization.


Control of coupled oscillator networks with application to microgrid technologies.

Skardal PS, Arenas A - Sci Adv (2015)

Fraction of driver nodes.(A) Fraction of driver nodes nD in ER (unfilled symbols) and SF (filled symbols) as a function of coupling strength K for mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangle, and green squares, respectively). (B) Required coupling strengths K5%, K10%, and K20% required to achieve consensus given nD = 0.05, 0.1, and 0.2 (blue circles, red triangles, and green squares, respectively). Each data point is the average over 100 network realizations of size N = 1000, each averaged over 100 natural frequency realizations.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4643801&req=5

Figure 4: Fraction of driver nodes.(A) Fraction of driver nodes nD in ER (unfilled symbols) and SF (filled symbols) as a function of coupling strength K for mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangle, and green squares, respectively). (B) Required coupling strengths K5%, K10%, and K20% required to achieve consensus given nD = 0.05, 0.1, and 0.2 (blue circles, red triangles, and green squares, respectively). Each data point is the average over 100 network realizations of size N = 1000, each averaged over 100 natural frequency realizations.
Mentions: Finally, we quantify the overall effort required for consensus by studying how the fraction of driver nodes, denoted nD = ND/N, where ND is the total number of driver nodes, depends on both the system’s dynamical and structural parameters. Presenting our results in Fig. 4, we first explore how the fraction of driver nodes depends on the coupling strength by plotting in Fig. 4AnD versus K for both ER and SF networks with mean degrees 〈k〉 = 4, 8, and 12 (blue circles, red triangles, and green squares, respectively). Results for ER and SF networks are plotted with unfilled and filled symbols, respectively, and each curve represents an average of over 100 network realizations, each averaged over 100 random natural frequency realizations. Although it is expected that nD decreases monotonically with K, the curves’ dependence on network topology and mean degree is nontrivial. In particular, the shape of nD versus K depends more sensitively on the mean degree than the topology, suggesting that network heterogeneity has little effect on overall control in comparison to average connectivity. In light of the significant dependence of overall control on the coupling strength, we investigate the coupling strength required to synchronize a network if a limited amount of control is available. To this end, we calculate for each family of networks the required coupling strengths K5%, K10%, and K20% for which, on average, a fraction nD = 0.05, 0.1, and 0.2 will achieve synchronization as a function of the average degree 〈k〉. We plot the results in Fig. 4B. We point out again that ER and SF networks behave very similarly on average, and that with a larger mean degree, a smaller coupling strength is required to achieve synchronization.

Bottom Line: The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences.Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades.The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Trinity College, Hartford, CT 06106, USA. ; Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain.

ABSTRACT
The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

No MeSH data available.