Power network and smart grids analysis from a graph theoretic perspective

The growing size and complexity of power systems has given raise to the use of complex network theory in their modelling, analysis, and synthesis. Though most of the previous studies in this area have focused on distributed control through well established protocols like synchronization and consensus, recently, a few fundamental concepts from graph theory have also been applied, for example in symmetry-based cluster synchronization. Among the existing notions of graph theory, graph symmetry is the focus of this proposal. However, there are other development around some concepts from complex network theory such as graph clustering in the study. In spite of the widespread applications of symmetry concepts in many real world complex networks, one can rarely find an article exploiting the symmetry in power systems. In addition, no study has been conducted in analysing controllability and robustness for a power network employing graph symmetry. It has been verified that graph symmetry promotes robustness but impedes controllability. A largely absent work, even in other fields outside power systems, is the simultaneous investigation of the symmetry effect on controllability and robustness. The thesis can be divided into two section. The first section, including Chapters 2-3, establishes the major theoretical development around the applications of graph symmetry in power networks. A few important topics in power systems and smart grids such as controllability and robustness are addressed using the symmetry concept. These topics are directed toward solving specific problems in complex power networks. The controllability analysis will lead to new algorithms elaborating current controllability benchmarks such as the maximum matching and the minimum dominant set. The resulting algorithms will optimize the number of required driver nodes indicated as FACTS devices in power networks. The second topic, robustness, will be tackled by the symmetry analysis of the network to investigate three aspects of network robustness: robustness of controllability, disturbance decoupling, and fault tolerance against failure in a network element. In the second section, including Chapters 4-8, in addition to theoretical development, a few novel applications are proposed for the theoretical development proposed in both sections one and two. In Chapter 4, an application for the proposed approaches is introduced and developed. The placement of flexible AC transmission systems (FACTS) is investigated where the cybersecurity of the associated data exchange under the wide area power networks is also considered. A new notion of security, i.e. moderated-k-symmetry, is introduced to leverage on the symmetry characteristics of the network to obscure the network data from the adversary perspective. In chapters 5-8, the use of graph theory, and in particular, graph symmetry and centrality, are adapted for the complex network of charging stations. In Chapter 5, the placement and sizing of charging stations (CSs) of the network of electric vehicles are addressed by proposing a novel complex network model of the charging stations. The problems of placement and sizing are then reformulated in a control framework and the impact of symmetry on the number and locations of charging stations is also investigated. These results are developed in Chapters 6-7 to robust placement and sizing of charging stations for the Tesla network of Sydney where the problem of extending the capacity having a set of pre-existing CSs are addressed. The role of centrality in placement of CSs is investigated in Chapter 8. Finally, concluding remarks and future works are presented in Chapter 9

Reduced-order modeling of large-scale network systems

Large-scale network systems describe a wide class of complex dynamical systems composed of many interacting subsystems. A large number of subsystems and their high-dimensional dynamics often result in highly complex topology and dynamics, which pose challenges to network management and operation. This chapter provides an overview of reduced-order modeling techniques that are developed recently for simplifying complex dynamical networks. In the first part, clustering-based approaches are reviewed, which aim to reduce the network scale, i.e., find a simplified network with a fewer number of nodes. The second part presents structure-preserving methods based on generalized balanced truncation, which can reduce the dynamics of each subsystem.Comment: Chapter 11 in the book Model Order Reduction: Volume 3 Application

Node dynamics on graphs: dynamical heterogeneity in consensus, synchronisation and final value approximation for complex networks

Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions. We show that the EEP with respect to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian. Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdős-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical heterogeneity under both linear consensus and non-linear coupled oscillator dynamics. Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values. We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ϵ close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.Open Acces

Model reduction and control of complex systems

Dit proefschrift behandelt een aantal vraagstukken gerelateerd aan modelreductie en regeling van twee soorten complexe systemen: `switched linear systems’ en netwerken van dynamische agenten. Eerst geven we een uitgebreide, gebalanceerde afkappingsmethode voor modelreductie van `switched linear systems’ (SLS). Vervolgens introduceren we een methode om de dynamische orde van agenten in een netwerk te reduceren, waarbij stabiliteit of synchronisatie behouden blijft in het gereduceerde orde model. Als vierde probleem analyseren we de stabiliteit en synchronisatie van netwerken waarbij de agenten een algemene, maar identieke, lineaire dynamica hebben en de onderliggende communicatietopologie willekeurig mag schakelen tussen een eindig aantal toegestane topologieën. De sterk structurele regelbaarheid van systemen die op een graaf gedefinieerd zijn, is het vijfde onderwerp van dit proefschrift. In het bijzonder tonen we aan dat er een een-op-eenrelatie is tussen de verzameling van leiders die ervoor zorgen dat het netwerk regelbaar is en de zogenaamde `zero-forcing’ verzamelingen. Tenslotte bestuderen we het probleem van storingsontkoppeling voor netwerken van dynamische agenten vanuit een graaftopologisch perspectief. We geven in termen van graphpartities voorwaarden voor de oplosbaarheid van het storingsontkoppelingsprobleem

Structural Properties of Invariant Dual Subspaces of Boolean Networks

In this paper, we obtain the following results on dual subspaces of Boolean networks (BNs). For a BN, there is a one-to-one correspondence between partitions of its state-transition graph (STG) and its dual subspaces (i.e., the subspaces generated by a number of Boolean functions of the BN's variables). Moreover, a dual subspace is invariant if and only if the corresponding partition is equitable, i.e., for every two (not necessarily different) cells of the partition, every two states in the former have equally many out-neighbors in the latter. With the help of equitable partitions of an STG, we study the structural properties of the smallest invariant dual subspaces containing a number of Boolean functions. And then, we give algorithms for computing the equitable partition corresponding to the smallest invariant dual subspace containing a given dual subspace. Moreover, we reveal that the unobservable subspace of a BN is the smallest invariant dual subspace containing the output function. We analyze properties of the unobservable subspace by using the obtained structural properties. The graphical representation provides an easier and more intuitive way to characterizing the (smallest) invariant dual subspaces of a BN

Decentralised network prediction and reconstruction algorithms

This study concerns the decentralised prediction and reconstruction problems in a network. First of all, we propose a decentralised prediction algorithm in the framework of network consensus problem. It allows any individual to compute the consensus value of the whole network in finite time using only the minimal number of successive values of its own history. We further prove that the minimal number of steps can be characterised using other algebraic and graph theoretical notions: minimal external equitable partition (mEEP) that can be directly computed from the Laplacian matrix of the graph and from the underlying network structure. Later, we consider a number of possible theoretical extensions of the proposed algorithm to issues arising from practical applications, e.g., time-delays, noise, external inputs, nonlinearities in the network, and analyse how the proposed algorithm should be changed to incorporate such constraints. For the decentralised reconstruction problem, we firstly define a new presentation: dynamical structure functions encoding structural information and explore the properties of such a representation for the purpose of solving the reconstruction problem. We have studied a number of theoretical problems: identification, realisation, reduction, etc. for dynamical structure functions and showed that how these theoretical can be used in solving decentralised network reconstruction problems. We later illustrate the results on a number of in-silico examples. We conclude the thesis with some ideas and future perspectives to continue based on the research of decentralised prediction and reconstruction problems