We consider the problem of deriving an explicit approximate solution of the nonlinear power equations that describe a balanced power distribution network. We give sufficient conditions for the existence of a practical solution to the power flow equations, and we propose an approximation that is linear in the active and reactive power demands of the PQ buses. For this approximation, which is valid for generic power line impedances and grid topology, we derive a bound on the approximation error as a function of the grid parameters. We illustrate the quality of the approximation via simulations, we show how it can also model the presence of voltage controlled (PV) buses, and we discuss how it generalizes the DC power flow model to lossy networks.Index Terms-Power systems modeling, load flow analysis, power distribution networks, fixed point theorem.
A major transition in the operation of electric power grids is the replacement of synchronous machines by distributed generation connected via power electronic converters. The accompanying "loss of rotational inertia" and the fluctuations by renewable sources jeopardize the system stability, as testified by the ever-growing number of frequency incidents. As a remedy, numerous studies demonstrate how virtual inertia can be emulated through various devices, but few of them address the question of "where" to place this inertia. It is, however, strongly believed that the placement of virtual inertia hugely impacts system efficiency, as demonstrated by recent case studies. In this article, we carry out a comprehensive analysis in an attempt to address the optimal inertia placement problem. We consider a linear networkreduced power system model along with an H 2 performance metric accounting for the network coherency. The optimal inertia placement problem turns out to be non-convex, yet we provide a set of closed-form global optimality results for particular problem instances as well as a computational approach resulting in locally optimal solutions. Further, we also consider the robust inertia allocation problem, wherein the optimization is carried out accounting for the worstcase disturbance location. We illustrate our results with a three-region power grid case study and compare our locally optimal solution with different placement heuristics in terms of different performance metrics.
We consider the problem of optimal reactive power compensation for the minimization of power distribution losses in a smart microgrid. We first propose an approximate model for the power distribution network, which allows us to cast the problem into the class of convex quadratic, linearly constrained, optimization problems. We then consider the specific problem of commanding the microgenerators connected to the microgrid, in order to achieve the optimal injection of reactive power. For this task, we design a randomized, gossip-like optimization algorithm. We show how a distributed approach is possible, where microgenerators need to have only a partial knowledge of the problem parameters and of the state, and can perform only local measurements. For the proposed algorithm, we provide conditions for convergence together with an analytic characterization of the convergence speed. The analysis shows that, in radial networks, the best performance can be achieved when we command cooperation among units that are neighbors in the electric topology. Numerical simulations are included to validate the proposed model and to confirm the analytic results about the performance of the proposed algorithm.
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