Harsha Nagarajan

Tightening McCormick Relaxations for Nonlinear Programs via Dynamic Multivariate Partitioning

In this work, we propose a two-stage approach to strengthen piecewise McCormick relaxations for mixed-integer nonlinear programs (MINLP) with multi-linear terms. In the first stage, we exploit Constraint Programing (CP) techniques to contract the variable bounds. In the second stage we partition the variables domains using a dynamic multivariate partitioning scheme. Instead of equally partitioning the domains of variables appearing in multi-linear terms, we construct sparser partitions yet tighter relax- ations by iteratively partitioning the variable domains in regions of interest. This approach decouples the number of partitions from the size of the variable domains, leads to a significant reduction in computation time, and limits the number of binary variables that are introduced by the partitioning. We demonstrate the performance of our algorithm on well-known benchmark problems from MINLPLIB and discuss the computational benefits of CP-based bound tightening procedures.

Optimal Resilient transmission Grid Design

As illustrated in recent years (Superstorm Sandy, Northeast Ice Storm of 1998, etc.), extreme weather events pose an enormous threat to the electric power transmission systems and the associated socio-economic systems that depend on reliable delivery of electric power. These threats motivate the need for approaches and methods that improve the response (resilience) of power systems. In this paper, we develop a model and tractable methods for optimizing the upgrade of transmission systems through a combination of hardening existing components, adding redundant lines, switches, generators, and transformers. While many of these components are included in traditional design (expansion planning) problems, we uniquely assess their benefits from a resiliency point of view.

Unit commitment with N-1 Security and wind uncertainty

As renewable wind energy penetration rates continue to increase, one of the major challenges facing grid operators is the question of how to control transmission grids in a reliable and a cost-efficient manner. The stochastic nature of wind forces an alteration of traditional methods for solving day-ahead and look-ahead unit commitment and dispatch. In particular, uncontrollable wind generation increases the risk of random component failures. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes the modeling of generation reserves that respond to wind fluctuations and tertiary reserves to account for single component outages. The basic formulation is reformulated as a mixed-integer second-order cone problem to limit the probability of failure. We develop three different algorithms to solve the problem to optimality and present a detailed case study on the IEEE RTS-96 single area system. The case study assesses the economic impacts due to contingencies and various degrees of wind power penetration into the system and also corroborates the effectiveness of the algorithms.

On maximizing algebraic connectivity of networks for various engineering applications

We discuss a simplified version of an open problem in system realization theory which has several important practical applications in complex networks. Particularly, we focus on algebraic connectivity (λ2) of the Laplacian of the network as an objective for maximization. We show that the maximum value of forced response of a linear mechanical (spring-mass) system can be minimized when the λ2 of the corresponding stiffness matrix is maximized. In the case of motion control problems related to vehicle localization with noisy measurements, we show that the λ2 plays a vital role to control their positions towards a desired formation. In UAV adhoc infrastructure networks, we show that the λ2 of the information flow graph governs the stability of the rigid formation with respect to disturbance attenuation. In the context of UAV adhoc communication networks, we also provide a physical interpretation for the maximization of λ2 via the closely related Cheeger constant or the isoperimetric number. We further discuss a Fiedler vector based mixed-integer linear programing formulation for the problem of maximizing λ2 and obtain optimal solutions and upper bounds based on cutting plane methods. Computational results corroborate the performance of proposed algorithms.

Optimal topology design for disturbance minimization in power grids

The transient response of power grids to external disturbances influences their stable operation. This paper studies the effect of topology in linear time-invariant dynamics of different power grids. For a variety of objective functions, a unified framework based on H2 norm is presented to analyze the robustness to ambient fluctuations. Such objectives include loss reduction, weighted consensus of phase angle deviations, oscillations in nodal frequency, and other graphical metrics. The framework is then used to study the problem of optimal topology design for robust control goals of different grids. For radial grids, the problem is shown as equivalent to the hard “optimum communication spanning tree” problem in graph theory and a combinatorial topology construction is presented with bounded approximation gap. Extended to loopy (meshed) grids, a greedy topology design algorithm is discussed. The performance of the topology design algorithms under multiple control objectives are presented on both loopy and radial test grids. Overall, this paper analyzes topology design algorithms on a broad class of control problems in power grid by exploring their combinatorial and graphical properties.

Adaptive convex relaxations for Gas Pipeline Network Optimization

The growing use of natural gas for electricity generation has created significant changes in the loads experienced by pipeline systems used for gas transport. This trend compels a fundamental re-examination of natural gas markets to develop pricing that reflects the physical and engineering capacity of pipeline networks. Recent advances in optimization techniques are promising for enabling pipeline optimization to address these needs. In this paper, we present an adaptive partitioning method in combination with optimality-based bound tightening approaches to strengthen standard convex relaxations for a class of steady-state pipeline system optimization problems, mathematically formulated as mixed-integer nonlinear programs (MINLP). Computation time and solution accuracy for objective functions that maximize compressor efficiency and economic welfare, respectively, on meshed gas networks are evaluated and compared to outcomes obtained using an interior point method. The method rapidly solves such problems to near global-optimum solutions. Efficiency and scalability of the technique makes it promising for extension to large-scale problems of dynamic gas-electric co-optimization.

Challenges and Successes of Solving Binary Quadratic Programming Benchmarks on the DW2X QPU

This presentation discusses some of the challenges and successes of the quantitative benchmarking efforts performed at LANL.

Resilient Off-grid Microgrids: Capacity Planning and N-1 Security

Over the past century the electric power industry has evolved to support the delivery of power over long distances with highly interconnected transmission systems. Despite this evolution, some remote communities are not connected to these systems. These communities rely on small, disconnected distribution systems, i.e., microgrids to deliver power. However, as microgrids often are not held to the same reliability standards as transmission grids, remote communities can be at risk for extended blackouts. To address this issue, we develop an optimization model and an algorithm for capacity planning and operations of microgrids that include

Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia

{N}

Probabilistic N-k failure-identification for power systems

-1 security and other practical modeling features like ac power flow physics, component efficiencies, and thermal limits. We demonstrate the computational effectiveness of our approach on two test systems; a modified version of the IEEE 13 node test feeder and a model of a distribution system in a remote community in Alaska.