Abstract-This paper studies the max-min weighted signal-to-interference-plus-noise ratio (SINR) problem in the multipleinput-multiple-output (MIMO) downlink, where multiple users are weighted according to priority and are subject to a weighted-sum-power constraint. First, we study the multiple-input-single-output (MISO) and single-input-multipleoutput (SIMO) problems using nonlinear Perron-Frobenius theory. As a by-product, we solve the open problem of convergence for a previously proposed MISO algorithm by Wiesel, Eldar, and Shamai in 2006. Furthermore, we unify our analysis with respect to the previous alternate optimization algorithm proposed by Tan, Chiang, and Srikant in 2009, by showing that our MISO result can, in fact, be derived from their algorithm. Next, we combine our MISO and SIMO results into an algorithm for the MIMO problem. We show that our proposed algorithm is optimal when the channels are rank-one, or when the network is operating in the low signal-to-noise ratio (SNR) region. Finally, we prove the parametric continuity of the MIMO problem in the power constraint, and we use this insight to propose a heuristic initialization strategy for improving the performance of our (generally) suboptimal MIMO algorithm. The proposed initialization strategy exhibits improved performance over random initialization.Index Terms-Beamforming, multiple-input-multiple-output (MIMO), uplink-downlink duality.

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Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that cooptimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.

Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that cooptimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.

We study the role of a market maker (or market operator) in a transmission constrained electricity market. We model the market as a one-shot networked Cournot competition where generators supply quantity bids and load serving entities provide downward sloping inverse demand functions. This mimics the operation of a spot market in a deregulated market structure. In this paper, we focus on possible mechanisms employed by the market maker to balance demand and supply. In particular, we consider three candidate objective functions that the market maker optimizes -social welfare, residual social welfare, and consumer surplus. We characterize the existence of Generalized Nash Equilibrium (GNE) in this setting and demonstrate that market outcomes at equilibrium can be very different under the candidate objective functions.

Abstract-The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propose a second-order cone programming convex relaxation of the resistive network OPF, and study the uniqueness of the optimal solution using differential topology especially the Poincare-Hopf Index Theorem. We characterize the global uniqueness for different network topologies, e.g., line, radial and mesh networks. This serves as a starting point to design distributed local algorithms with global behaviors that have low complexity, computationally fast and can run under synchronous and asynchronous settings in practical power grids.

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