Subhonmesh Bose

Quadratically Constrained Quadratic Programs on Acyclic Graphs With Application to Power Flow

This paper proves that nonconvex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. We demonstrate this theory on optimal power-flow problems over tree networks.

Optimal power flow over tree networks

The optimal power flow (OPF) problem is critical to power system operation but it is generally non-convex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow over-satisfaction of load. The proof, cast as a complex semi-definite program, makes use of the fact that if the underlying graph of an n × n Hermitian positive semi-definite matrix is a tree, then the matrix has rank at least n − 1.

Optimal placement of energy storage in the grid

This paper studies the problem of optimally placing large-scale energy storage in power grids with both conventional and wind generation. The solution technique for this infinite horizon problem assumes cyclic demand and generation profiles using a semidefinite relaxation of AC optimal power flow. Changes in storage allocation in the network are studied as a function of total storage budget and transmission line-flow constraints. These questions are investigated using an IEEE benchmark system with various generation portfolios.

Equivalent Relaxations of Optimal Power Flow

Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a second-order cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of the OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their second-order cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the second-order cone relaxation. For mesh networks, the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second-order cone relaxation. Therefore, for mesh networks, one should either solve the chordal relaxation or the SOCP relaxation, trading off tightness and the required computational effort. Simulations are used to illustrate these results.

Optimal Placement of Distributed Energy Storage in Power Networks

We formulate the optimal placement, sizing and control of storage devices in a power network to minimize generation costs with the intent of load shifting. We assume deterministic demand, a linearized DC approximated power flow model and a fixed available storage budget. Our main result proves that when the generation costs are convex and nondecreasing, there always exists an optimal storage capacity allocation that places zero storage at generation-only buses that connect to the rest of the network via single links. This holds regardless of the demand profiles, generation capacities, line-flow limits and characteristics of the storage technologies. Through a counterexample, we illustrate that this result is not generally true for generation buses with multiple connections. For specific network topologies, we also characterize the dependence of the optimal generation cost on the available storage budget, generation capacities and flow constraints.

The role of a market maker in networked cournot competition

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.

A Unifying Market Power Measure for Deregulated Transmission-Constrained Electricity Markets

Market power assessment is a prime concern when designing a deregulated electricity market. In this paper, we propose a new functional market power measure, termed transmission constrained network flow (TCNF), that unifies three large classes of transmission constrained structural market power indices in the literature: residual supply based, network flow based, and minimal generation based. Furthermore, it is suitable for demand-response and renewable integration and hence more amenable to identifying market power in the future smart grid. The measure is defined abstractly, and allows incorporation of power flow equations in multiple ways; we investigate the current market operations using a DC approximation and further explore the possibility of including detailed AC power flow models through semidefinite relaxation, and interior-point algorithms from Matpower. Finally, we provide extensive simulations on IEEE benchmark systems and highlight the complex interaction of engineering constraints with market power assessment.

A Low-Cost Algorithm to Find the Minimum Sampling Frequency for Multiple Bandpass Signals

We consider the problem of finding the minimum sampling frequency required for N non-overlapping, bandpass signals. Recently a novel algorithm with a significantly reduced computational cost has been proposed for this problem. By exploiting a redundancy in this algorithm, we propose a method which further reduces the cost significantly. We use the fact that a valid sampling frequency for a set of bandpass signals must be a valid sampling frequency for any subset of the signals. Several examples are given to illustrate the savings in the computational cost achieved.

Variability and the Locational Marginal Value of Energy Storage

Given a stochastic net demand process evolving over a transmission-constrained power network, we consider the system operators problem of minimizing the expected cost of generator dispatch, when it has access to spatially distributed energy storage resources. We show that the expected benefit of storage derived under the optimal dispatch policy is concave and non-decreasing in the vector of energy storage capacities. Thus, the greatest marginal value of storage is derived at small installed capacities. For such capacities, we provide an upper bound on the locational (nodal) marginal value of storage in terms of the variation of the shadow prices of electricity at each node. In addition, we prove that this upper bound is tight, when the cost of generation is spatially uniform and the network topology is acyclic. These formulae not only shed light on the correct measure of statistical variation in quantifying the value of storage, but also provide computationally tractable tools to empirically calculate the locational marginal value of storage from net demand time series data.

A unifying approach to assessing market power in deregulated electricity markets

A competitive deregulated electricity market with increasingly active market players is foreseen to be the future of the electricity industry. In such settings, market power assessment is a primary concern. In this paper, we propose a novel functional approach for measuring long term market power that unifies a variety of popular market power indices. Specifically, the new measure, termed transmission constrained network flow (TCNF), unifies three large classes of market power measures: residual supply based, network flow based, and minimal generation based. Further, TCNF provides valuable information about market power not captured by prior indices. We derive its analytic properties and test its efficacy on IEEE test systems.