Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Sistemas fotovoltaicos integrados ao perfil de uso da edificação

Nos últimos anos, o consumo de energia vem crescendo mundialmente nos grandes centros urbanos, e esforços na área de eficiência energética estão sendo implantados, a fim de reduzir o consumo no horário da ponta e interrupções da rede. O aproveitamento das fontes renováveis, como o fotovoltaico em uma edificação se torna um atrativo a mais para a matriz energética num momento em que o país prima pela universalização dos serviços de energia e a classificação de edifícios comerciais, de serviço e públicos, além dos residenciais quanto à eficiência energética através do Procel Edifica (RTQ-C e RTQ-R). Os sistemas fotovoltaicos podem configurar perfis de uso nas edificações de modo a gerar energia para consumo próprio ou ligado à rede e ainda ter influência na arquitetura do prédio com revestimento: os perfis podem está em telhados, fachadas ou janelas, amenizando em alguns casos a carga térmica no prédio com sombreamento arquitetônico. Hoje, com o avanço da tecnologia no setor de armazenagem é possível, o atendimento com segurança e eficiência a uma edificação ou direcionar esta armazenagem a uma demanda específica como o atendimento à demanda de ciclo profundo, tais como, iluminação externa e recarga de veículos elétricos. Partindo da premissa de sistemas interruptos de energia, UPS, uso de fonte secundária como FV, baterias e Flywheel é apresentado uma forma de melhor gerenciar a energia armazenada, podendo estender a vida útil da bateria e conseqüentemente de todo o sistema fotovoltaico na edificação. Esta forma de armazenar energia proporciona um serviço de uso contínuo sem percepção das interrupções da rede com garantia de 20 anos, tal qual o módulo fotovoltaico, com esta proposta as perdas de energia elétrica na edificação serão atenuadas, pois a eletricidade será utilizada de forma eficiente e inteligente. O ponto de partida do estudo de caso no prédio do IBAM são os sistemas fotovoltaicos com geração distribuída (mini-redes) conectados à rede que são instalados para fornecer energia ao consumidor, complementando a quantidade de energia demandada, caso haja algum aumento do consumo de energia na edificação, ou ainda utilizar o sistema fotovoltaico na hora da ponta e interrupções do sistema da rede no período fora da ponta. A estocagem inercial por meio do Flywheel tem um papel fundamental nesta mini-rede (Flywheel, bateria VRLA, UPS, inversor e STS), pois a sua utilização pode ser apontada como uma inovação tecnológica quanto à regulação de tensão no sistema de energia elétrica, além de preparar a edificação para o smart-grid. Esta configuração de acumulação de energia permitiu a analise do deslocamento desta energia armazenada para o consumo no horário de ponta, mudando o conceito de sistemas fotovoltaicos autônomos no meio urbano e rural no país. Este conceito de armazenagem se confirma então como um aporte na eficiência de energia na edificação, podendo carrear economia de energia substancial, além de proporcionar uma confiabilidade no serviço de energia, com um baixo retorno do investimento e com uma garantia de funcionamento com pequena ou nenhuma manutenção durante o período de vida de 20 anos.

Optimization and Control of Power Flow in Distribution Networks

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators.

Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory.

The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.