Graph theoretic methods have been used in network science for the analysis of actors participation and influence over the network. In this paper we use the methods for the analysis of voltage stability profile and critical voltage modes in electrical networks. We model the electric grid as a graph whose Laplacian is based on the Jacobian matrix representation of power flow constraints. Over this model we use newly defined centrality measures to obtain information about voltage-reactive power sensibility of buses and vulnerable zones. The approach is studied with a numerical example, developed over the WSCC-9 bus system. The results obtained from these measures are compared with data obtained from a traditional modal analysis method and electrical centrality measures based on admittance. The results show the designed centrality measures and distance metrics as useful alternatives for the analysis of voltage stability in power networks.

This paper aims to study the vulnerability of the network to sequential cascading failures attacks where the attack strategy integrates network theory and discounted reward with Markov decision process (MDP) in the target selection process. A control strategy is designed to maximize the attack’s long-term expected reward while reducing the attack sequence duration. The attack model identifies the most suitable targets by prediction through a Markov process for predicting the propagation and consequences of the failure. The state transition probabilities through a hidden failure model embedded in an independent edge-dependent network evolution model is estimated. Value iteration algorithms are used to identify targets at every attack stage. Target selection is updated depending on network changes. The results provide an optimal attack strategy based on network congestion with maximum damage, considering congestion as a cascade propagation mechanism. Reward functions based on increasing congestion and immediate power loss are compared. Strategies designed with network congestion as the attack reward function produce more vulnerability of the network to sequential attacks.

Voltage collapse is a type of failure with major influence in blackouts. Blackouts, although infrequent, are costly to society. For the Smart grid, integration of new technologies, new policies, and increasing power demand makes the power system becoming more disturbed, which increases the possibility of voltage collapses of all size. Hence, its complexity requires new approaches for modeling and analysis that captures the power network structure. This work presents a graph theory-based analysis of the voltage collapse problem. We present a review of voltage stability and collapse to identify its basic characteristics. We model the power system as an undirected weighted graph whose weights are voltage coupling magnitude between nodes. A steady state analysis is developed over the model to obtain properties of branches, and buses. The obtained graph properties are closely related with results obtained from Q-V modal analysis in the example. Also a dynamical system behavior is determined with the nonlinear differential equations of elements that govern the direction of network dynamics over a voltage collapse mechanism in time.

Voltage regulation and fluctuations in power and distribution systems represents an important matter for electricity customers and utilities. Distributed Energy Resources have proven to provide solutions for satisfying demand and also for providing voltage control support. An electrical network can be reinforced by the coordination of control actions and distributed generation, as long as it is designed in an optimal way. In order to improve the network voltage performance, a distributed control strategy based on state based potential games is proposed in this work. The control strategy is applied to a test system, illustrating the benefits and viability of the proposed approach.

For control of complex networks, controllability criteria are useful to get information about constraints imposed by the network topology. In this work, we study controllability properties for first order dynamics over an undirected random tree network structure. We present a controllability index for the minimum set of driver nodes. We show a criterion for the selection of driver nodes. Then, we demonstrate that regardless the network is growing in time, its structure has a constant controllability index. Finally, an application of this criterion is used for voltage control of active power distribution networks with radial structure.