Construction of power flow feasibility sets

We develop a new approach for construction of convex analytically simple regions where the AC power flow equations are guaranteed to have a feasible solutions. Construction of these regions is based on efficient semidefinite programming techniques accelerated via sparsity exploiting algorithms. Resulting regions have a simple geometric shape in the space of power injections (polytope or ellipsoid) and can be efficiently used for assessment of system security in the presence of uncertainty. Efficiency and tightness of the approach is validated on a number of test networks

A differential analysis of the power flow equations

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.Comment: arXiv admin note: text overlap with arXiv:1506.0847

Storage Sizing and Placement through Operational and Uncertainty-Aware Simulations

As the penetration level of transmission-scale time-intermittent renewable generation resources increases, control of flexible resources will become important to mitigating the fluctuations due to these new renewable resources. Flexible resources may include new or existing synchronous generators as well as new energy storage devices. Optimal placement and sizing of energy storage to minimize costs of integrating renewable resources is a difficult optimization problem. Further,optimal planning procedures typically do not consider the effect of the time dependence of operations and may lead to unsatisfactory results. Here, we use an optimal energy storage control algorithm to develop a heuristic procedure for energy storage placement and sizing. We perform operational simulation under various time profiles of intermittent generation, loads and interchanges (artificially generated or from historical data) and accumulate statistics of the usage of storage at each node under the optimal dispatch. We develop a greedy heuristic based on the accumulated statistics to obtain a minimal set of nodes for storage placement. The quality of the heuristic is explored by comparing our results to the obvious heuristic of placing storage at the renewables for IEEE benchmarks and real-world network topologies.Comment: To Appear in proceedings of Hawaii International Conference on System Sciences (HICSS-2014

Universal Convexification via Risk-Aversion

We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the convexified problem relative to the original nonconvex problem and prove additive approximation guarantees. We then develop algorithms based on stochastic gradient methods to solve the resulting optimization problems and show bounds on convergence rates. %We show a simple application of this framework to supervised learning, where one can perform integration explicitly and can use standard (non-stochastic) optimization algorithms with better convergence guarantees. We then extend this framework to apply to a general class of discrete-time dynamical systems. In this context, our convexification approach falls under the well-studied paradigm of risk-sensitive Markov Decision Processes. We derive the first known model-based and model-free policy gradient optimization algorithms with guaranteed convergence to the optimal solution. Finally, we present numerical results validating our formulation in different applications