Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time

Distributed transactive control in distribution systems with microgrids

Microgrids are considered as a cornerstone in the evolution to a smarter grid. However, this evolution brings some critical challenges for the control in a real-time implementation. We present two control algorithms to operate a power system with microgrids and other two to operate microgrids in order to reach the optimal social welfare. We consider three types of agents: photovoltaic generators, conventional generators and smart loads. These agents can be aggregated into a microgrid or interact directly in the power system depending on their power. To optimize the microgrids, we use two strategies. First one is based on projected consensus algorithm, where each agent iteratively optimizes its local utility function based on local information obtained from its neighbors and global information obtained through a distributed finite-time average algorithm. The second one is based on populations game theory; specifically we use a centralized replicator dynamics where a central agent iteratively optimizes the system status. To optimize the whole power system we use two strategies, first an asynchronous algorithm based on primal-dual optimization is proposed, where we consider that agents update the primal variables and a "virtual agent" updates the dual variables. Our last algorithm is a distributed transactive control algorithm based on populations games to dynamically manage the distributed generators and smart loads in the system to reach the optimum social welfare. Agents are considered non-cooperative, and they are individually incentive-driven. The proposed algorithm preserve stability while guarantee optimality conditions considering several constraints in the system on the real-time operation. We show numerical results of the proposed control strategies.Resumen: Las microrredes están consideradas como la piedra angular de la evolución hacia una red más inteligente. Sin embargo, esta evolución trae consigo algunos retos importantes para el control en la implementación en tiempo real. Presentamos dos algoritmos de control para operar un sistema de energía con microrredes y otros dos para operar microrredes con el fin de alcanzar el bienestar social óptimo. Consideramos dos tipos de agentes: generadores convencionales y cargas inteligentes. Estos agentes pueden ser agregados en una microred o interactuar directamente en el sistema de energía dependiendo de su potencia. Para optimizar las microrredes utilizamos dos estrategias, la primera se basa en un algoritmo de consenso proyectado, donde cada una de ellas optimiza iterativamente su función de utilidad local a partir de la información local obtenida de sus vecinos y la información global obtenida a través de un algoritmo distribuido de tiempo finito promedio. El segundo se basa en la teoría de juegos de poblaciones, específicamente usamos una dinámica de replicador centralizada donde un agente central optimiza iterativamente el estado del sistema. Para optimizar todo el sistema de potencia utilizamos dos estrategias, la primera es proponer un algoritmo asíncrono basado en la optimización prima-dual, donde consideramos que los agentes actualizan las variables primarias y un ”agente virtual” actualiza las variables duales. Nuestro último algoritmo es un algoritmo de control transaccional distribuido basado en juegos de poblaciones para gestionar dinámicamente los generadores distribuidos y las cargas inteligentes en el sistema para alcanzar el bienestar social óptimo. Se considera que los agentes no cooperan y se basan en incentivos individuales. El algoritmo propuesto preserva la estabilidad a la vez que garantiza condiciones óptimas considerando varias restricciones en el sistema sobre la operación en tiempo real. Se muestran los resultados numéricos de las estrategias de control propuestas.Maestrí

Minimum Restricted Diameter Spanning Trees

Let G = (V; E) be a requirement graph. Let d = (d ij ) i;j=1 be a length metric. For a tree T denote by d T (i; j) the distance between i and j in T (the length according to d of the unique i j path in T ). The restricted diameter of T , D T , is the maximum distance in T between pair of vertices with requirement between them. The minimum restricted diameter spanning tree problem is to nd a spanning tree T such that the restricted diameter is minimized. We prove that the minimum restricted diameter spanning tree problem is NP -hard and that unless P = NP there is no polynomial time algorithm with performance guarantee of less than 2. In the case that G contains isolated vertices and the length matrix is de ned by distances over a tree we prove that there exists a tree over the non-isolated vertices such that its restricted diameter is at most 4 times the minimum restricted diameter and that this constant is at least 3 2