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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.


Control of coupled oscillator networks: Random networks.Example of control applied to a coupled oscillator network with ER and SF topologies (top and bottom rows, respectively) with 〈k〉 = 6 and N = 1000. The coupling strength is K = 0.4. Time series for phases θi(t) of 10% of oscillators without (A and D) and with (B and E) control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively. (C and F) Time series for the degree of phase synchronization r(t) for the networks without (red) and with (blue) control.
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Figure 2: Control of coupled oscillator networks: Random networks.Example of control applied to a coupled oscillator network with ER and SF topologies (top and bottom rows, respectively) with 〈k〉 = 6 and N = 1000. The coupling strength is K = 0.4. Time series for phases θi(t) of 10% of oscillators without (A and D) and with (B and E) control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively. (C and F) Time series for the degree of phase synchronization r(t) for the networks without (red) and with (blue) control.

Mentions: We now demonstrate our approach by considering two types of random networks: Erdős-Rényi (ER) (49) networks and scale-free (SF) networks. Each ER network is constructed using a fixed link probability p, and each SF network is built using the configuration model (50) with a degree sequence drawn from the distribution P(k) ∝ k−γ with γ = 3 and enforced minimum degree k0. To tune the mean degree 〈k〉 of each network, we set either p = 〈k〉/(N − 1) or k0 = 〈k〉/(γ − 1). Figure 2 illustrates our results with an example of each type of network, where we have used networks of size N = 1000 with mean degree 〈k〉 = 6, set the coupling strength K = 0.4, and used natural frequencies drawn from a uniform distribution with zero mean and unit variance. The top and bottom rows display the results for the ER and SF networks, respectively, displaying the time series θ(t) of a randomly selected 10% of the oscillators without control (Fig. 2, A and D) and with control (Fig. 2, B and E), after discarding a long transient. The difference between no control to control is quite drastic, with a large fraction of desynchronized oscillators without control, and full synchronization with control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively, to attain the synchronized state. In Fig. 2 (C and F), we present the degree of phase synchronization, plotting the order parameter r(t) for both solutions with and without control. Unlike the solution without control, which fluctuates significantly at a relative low value, the solution with control reaches a steady value near r(t) ≈ 1.


Control of coupled oscillator networks with application to microgrid technologies.

Skardal PS, Arenas A - Sci Adv (2015)

Control of coupled oscillator networks: Random networks.Example of control applied to a coupled oscillator network with ER and SF topologies (top and bottom rows, respectively) with 〈k〉 = 6 and N = 1000. The coupling strength is K = 0.4. Time series for phases θi(t) of 10% of oscillators without (A and D) and with (B and E) control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively. (C and F) Time series for the degree of phase synchronization r(t) for the networks without (red) and with (blue) control.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4643801&req=5

Figure 2: Control of coupled oscillator networks: Random networks.Example of control applied to a coupled oscillator network with ER and SF topologies (top and bottom rows, respectively) with 〈k〉 = 6 and N = 1000. The coupling strength is K = 0.4. Time series for phases θi(t) of 10% of oscillators without (A and D) and with (B and E) control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively. (C and F) Time series for the degree of phase synchronization r(t) for the networks without (red) and with (blue) control.
Mentions: We now demonstrate our approach by considering two types of random networks: Erdős-Rényi (ER) (49) networks and scale-free (SF) networks. Each ER network is constructed using a fixed link probability p, and each SF network is built using the configuration model (50) with a degree sequence drawn from the distribution P(k) ∝ k−γ with γ = 3 and enforced minimum degree k0. To tune the mean degree 〈k〉 of each network, we set either p = 〈k〉/(N − 1) or k0 = 〈k〉/(γ − 1). Figure 2 illustrates our results with an example of each type of network, where we have used networks of size N = 1000 with mean degree 〈k〉 = 6, set the coupling strength K = 0.4, and used natural frequencies drawn from a uniform distribution with zero mean and unit variance. The top and bottom rows display the results for the ER and SF networks, respectively, displaying the time series θ(t) of a randomly selected 10% of the oscillators without control (Fig. 2, A and D) and with control (Fig. 2, B and E), after discarding a long transient. The difference between no control to control is quite drastic, with a large fraction of desynchronized oscillators without control, and full synchronization with control. Driver nodes constituted 37.1 and 44.5% of the ER and SF networks, respectively, to attain the synchronized state. In Fig. 2 (C and F), we present the degree of phase synchronization, plotting the order parameter r(t) for both solutions with and without control. Unlike the solution without control, which fluctuates significantly at a relative low value, the solution with control reaches a steady value near r(t) ≈ 1.

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.