How Does Network Topology Determine the Synchronization Threshold in a Network of Oscillators?
The reliable functioning of an electrical power grid is dependent on the proper interaction between many of its elements. What is critically important is its ability to keep the frequency across the entire system stable. Considering a simple mathematical
- PDF / 339,683 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 40 Downloads / 187 Views
1 Introduction The electrical power grid attained considerable attention of the complex systems community due to its topological properties, cascading effects, self-organized criticality as plausible explanation for the distribution of the blackouts size, its interaction with other systems and also synchronization phenomena. A power system is formed by a large number of generators and loads interconnected in a complex pattern to fulfil its function. Considering the complete set of variables and characterizing the system in detail on a large scale would require a significant modelling effort and a still unavailable computer power. Here we follow the research direction of reducing the complexity of individual elements in favour of the simplicity in capturing the synchronization phenomena. This way we aim for a compromise between the technical perspective (electrical engineering) involving very detailed models and a more statistical view (network science), where the main goal is to understand the global behaviour of the system.
L. Buzna (B) Department of Transportation Networks, University of Zilina, Univerzitna 8215/5, 01026 Zilina, Slovakia e-mail: [email protected] S. Lozano IPHES, Institut Català de Paleoecologia Humana i Evolució Social, 43007 Tarragona, Spain e-mail: [email protected] A. Díaz-Guilera Departament de Física Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain e-mail: [email protected] S. Helber et al. (eds.), Operations Research Proceedings 2012, Operations Research Proceedings, DOI: 10.1007/978-3-319-00795-3_20, © Springer International Publishing Switzerland 2014
135
136
L. Buzna et al.
2 Oscillatory Dynamics We consider simple dynamical behaviour of generators (e.g. not including complete dynamical description of electromagnetic fields and flow equations) considering only basic mechanisms that ensure one of their most important dynamical properties, namely synchronization. We assume a set of oscillating units organized in a graph G composed from a set of nodes N and from a set of links L. Each node i ∈ N is characterized by a natural frequency ωi and by a phase angle ϕi . Adopting assumptions and deductions introduced in [1] and neglecting inertia component we model synchronization dynamics by the following set of Kuramoto equations [2]: dϕi = ωi + σ ai j sin(ϕ j − ϕi ), dt
(1)
j
where ai j are the elements of the adjacency matrix and σ is parameter characterizing the coupling strength of links which are assumed to be uniform across the entire network. In order to distinguish between generators and loads, we split the nodes into two subsets. The elements in the first subset, N+ ⊆ N , represent generators and thus ωi = ω+ > 0 for i ∈ N+ . Similarly, loads are represented by the subset N− ⊆ N , where N− ∪ N+ = N , N− ∩ N+ = ∅ and ωi = ω− < 0 for i ∈ N− . From the synchronization point of view, it is relevant how Eq. (1) behaves with respect to the parameter σ . When σ = 0 the movement of all oscillators is driven by the natural frequency ωi . As σ grows, the oscillators inf
Data Loading...
