arXiv subjects

Kaarthik Sundar

Publications and source records attributed to Kaarthik Sundar.

At least 55 records · Page 3Linked to original sources

arXiv2021

Multiple Depot Ring Star Problem: A polyhedral study and exact algorithm

The Multiple Depot Ring-Star Problem (MDRSP) is an important combinatorial optimization problem that arises in the context of optical fiber network design, and in applications pertaining to collecting data using stationary sensing devices and autonomous vehicles. Given the locations of a set of customers and a set of depots, the goal is to (i) find a set of simple cycles such that each cycle (ring) passes through a subset of customers and exactly one depot, (ii) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs. We present a mixed integer linear programming formulation for the MDRSP and propose valid inequalities to strengthen the linear programming relaxation. Furthermore, we present a polyhedral analysis and derive facet-inducing results for the MDRSP. All these results are then used to develop a branch-and-cut algorithm to obtain optimal solutions to the MDRSP. The performance of the branch-and-cut algorithm is evaluated through extensive computational experiments on several classes of test instances.

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arXiv2021

Graph-Based Modeling and Decomposition of Energy Infrastructures

Nonlinear optimization problems are found at the heart of real-time operations of critical infrastructures. These problems are computationally challenging because they embed complex physical models that exhibit space-time dynamics. We propose modeling these problems as graph-structured optimization problems, and illustrate how their structure can be exploited at the modeling level (for parallelizing function/derivative computations) and at the solver level (for parallelizing linear algebra operations). Specifically, we present a restricted additive Schwarz scheme that enables flexible decomposition of complex graph structures within an interior-point algorithm. The proposed approach is implemented as a general-purpose nonlinear programming solver that we call MadNLP.jl; this Julia-based solver is interfaced to the graph-based modeling package Plasmo.jl. The efficiency of this framework is demonstrated via problems arising in transient gas network optimization and multi-period AC optimal power flow. We show that our framework accelerates the solution (compared to off-the-shelf tools) by over 300%; specifically, solution times are reduced from 72.36 sec to 23.84 sec for the gas problem and from 515.81 sec to 149.45 sec for the power flow problem.

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arXiv2021

Sequence of Polyhedral Relaxations for Nonlinear Univariate Functions

Given a nonlinear, univariate, bounded, and differentiable function $f(x)$, this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of $f(x)$ and its convex hull, respectively. Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established. For nonlinear non-convex optimization problems, the relaxations presented in this article can be used to construct tight MILP and LP relaxations. These MILP and the LP relaxations can also be used with MILP-based and spatial branch-and-bound based global optimization algorithms, respectively.

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arXiv2021

Robust Gas Pipeline Network Expansion Planning to Support Power System Reliability

We examine the problem of optimal transport capacity expansion planning for a gas pipeline network to service the growing demand of gas-fired power plants that are increasingly used to provide base load, flexibility, and reserve generation for bulk electric system. The aim is to determine the minimal cost set of additional pipes and gas compressors that can be added to the network to provide the additional capacity to service future loads. This combinatorial optimization problem is initially formulated as a mixed-integer nonlinear program, which we then extend to account for the variability that is inherent to the demands of gas-fired electricity production and uncertainty in expected future loads. We consider here steady-state flow modeling while ensuring that the solution is feasible for all possible values of interval uncertainty in loads, which results in a challenging semi-infinite problem. We apply previously derived monotonicity properties that enable simplification of the problem to require constraint satisfaction in the two extremal scenarios only, and then formulate the robust gas pipeline network expansion planning problem using a mixed-integer second order cone formulation. We consider case studies on the Belgian network test case to examine the performance of the proposed approach.

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arXiv2020

Optimal Economic Operation of Liquid Petroleum Products Pipeline Systems

The majority of overland transport needs for crude petroleum and refined petroleum products are met using pipelines. Numerous studies have developed optimization methods for design of these systems in order to minimize construction costs while meeting capacity requirements. Here, we formulate problems to optimize the operations of existing single liquid commodity pipeline systems subject to physical flow and pump engineering constraints. The objectives are to maximize the economic value created for users of the system and to minimize operating costs. We present a general computational method for this class of continuous, non-convex nonlinear programs, and examine the use of pump operating settings and flow allocations as decision variables. The approach is applied to compute optimal operating regimes and perform engineering economic sensitivity analyses for a case study of a crude oil pipeline developed using publicly available data.

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arXiv2020

Piecewise Polyhedral Formulations for a Multilinear Term

In this paper, we present a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex hull representation. Based on the solution of the PPR, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations. We then present computational results showing the effectiveness of proposed formulations on instances of standard benchmarks of nonlinear programs (NLPs) with multilinear terms and compare the proposed formulation with a traditional formulation that is built by recursively relaxing bilinear groupings of multilinear terms.

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arXiv2020

Credible Interdiction for Transmission Systems

This paper presents novel formulations and algorithms for $N$-$k$ interdiction problem in transmission networks. In particular, it formulates spatial and topological resource constraints on attackers for $N$-$k$ interdiction problems and illustrates the formulation with two new classes of $N$-$k$ attacks: (i) Spatial $N$-$k$ attacks where the attack is constrained by geographic distance of a bus chosen by an attacker and (ii) Topological $N$-$k$ attacks where the attack is constrained to connected components. These two specific types of $N$-$k$ attacks compute interdiction plans designed to better model localized attacks, such as those induced by natural disasters or physical attacks. We then formulate these two resource-constrained interdiction problems as bilevel, max-min optimization problems and present a novel constraint generation algorithm to solve these formulations. Detailed case studies analyzing the behavior of spatially and topologically resource-constrained problems and comparing them to the traditional $N$-$k$ interdiction problem are also presented.

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arXiv2020

Chance-Constrained 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, the variability of wind generation increases the risk of unexpected overloads and cascading events. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes models of generation reserves that respond to wind fluctuations and component outages. We formulate the SCCUC as a mixed-integer, second-order cone problem that limits the probability of failure. We develop a modified Benders decomposition algorithm to solve the problem to optimality and present detailed case studies on the IEEE RTS-96 three-area and the IEEE 300 NESTA test systems. The case studies assess the economic impacts of contingencies and various degrees of wind power penetration and demonstrate the effectiveness and scalability of the algorithm.

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arXiv2020

Natural Gas Maximal Load Delivery for Multi-contingency Analysis

As the use of renewable generation has increased, electric power systems have become increasingly reliant on natural gas-fired power plants as fast ramping sources for meeting fluctuating bulk power demands. This dependence has introduced new vulnerabilities to the power grid, including disruptions to gas transmission networks from natural and man-made disasters. To address the operational challenges arising from these disruptions, we consider the task of determining a feasible steady-state operating point for a damaged gas pipeline network while ensuring the maximal delivery of load. We formulate the mixed-integer nonconvex maximal load delivery (MLD) problem, which proves difficult to solve on large-scale networks. To address this challenge, we present a mixed-integer convex relaxation of the MLD problem and use it to determine bounds on the transport capacity of a gas pipeline system. To demonstrate the effectiveness of the relaxation, the exact and relaxed formulations are compared across a large number of randomized damage scenarios on nine natural gas pipeline network models ranging in size from 11 to 4197 junctions. A proof of concept application, which assumes network damage from a set of synthetically generated earthquakes, is also presented to demonstrate the utility of the proposed optimization-based capacity evaluation in the context of risk assessment for natural disasters. For all but the largest network, the relaxation-based method is found to be suitable for use in evaluating the impacts of multi-contingency network disruptions, often converging to the optimal solution of the relaxed formulation in less than ten seconds.

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arXiv2020

An Uncertainty Management Framework for Integrated Gas-Electric Energy Systems

In many parts of the world, electric power systems have seen a significant shift towards generation from renewable energy and natural gas. Because of their ability to flexibly adjust power generation in real time, gas-fired power plants are frequently seen as the perfect partner for variable renewable generation. However, this reliance on gas generation increases interdependence and propagates uncertainty between power grids and gas pipelines, and brings coordination and uncertainty management challenges. To address these issues, we propose an uncertainty management framework for uncertain, but bounded gas consumption by gas-fired power plants. The admissible ranges are computed based on a joint optimization problem for the combined gas and electricity networks, which involves chance-constrained scheduling for the electric grid and a novel robust optimization formulation for the natural gas network. This formulation ensures feasibility of the integrated system with a high probability, while providing a tractable numerical formulation. A key advance with respect to existing methods is that our method is based on a physically accurate, validated model for transient gas pipeline flows. Our case study benchmarks our proposed formulation against methods that ignore how reserve activation impacts the fuel use of gas power plants, and only consider predetermined gas consumption. The results demonstrate the importance of considering uncertainty to avoid operating constraint violations and curtailment of gas to the generators.

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arXiv2019

Optimal Control for Scheduling and Pricing Intra-day Natural Gas Transport on Pipeline Networks

We formulate an economic optimal control problem for transport of natural gas over a large-scale transmission pipeline network under transient flow conditions. The objective is to maximize economic welfare for users of the pipeline system, who provide time-dependent price and quantity bids to purchase or supply gas at metered locations on a system with time-varying injections, withdrawals, and control actions of compressors and regulators. Our formulation ensures that pipeline hydraulic limitations, compressor station constraints, operational factors, and pre-existing contracts for gas transport are satisfied. A pipeline is modeled as a metric graph with gas dynamics partial differential equations on edges and coupling conditions at the nodes. These dynamic constraints are reduced using lumped elements to a sparse nonlinear differential algebraic equation system. A highly efficient temporal discretization scheme for time-periodic formulations is introduced, which we extend to develop a rolling-horizon model-predictive control scheme. We apply the computational methodology to a pipeline system test network case study. In addition to the physical flow and compressor control solution, the optimization yields dual functions that we interpret as the time-dependent economic values of gas at each location in the network.

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arXiv2019

Dynamic State and Parameter Estimation for Natural Gas Networks using Real Pipeline System Data

We present a method for joint state and parameter estimation for natural gas networks where gas pressures and flows through a network of pipes depend on time-varying injections, withdrawals, and compression controls. The estimation is posed as an optimal control problem constrained by coupled partial differential equations on each pipe that describe space- and time-dependent density and mass flux. These are discretized and combined with nodal conditions to give dynamic constraints for posing the estimation as nonlinear least squares problems. We develop a rapid, scalable computational method for performing the estimation in the presence of measurement and process noise. Finally, we evaluate its effectiveness using a data set from a capacity planning model for an actual pipeline system and a month of time-series data from its supervisory control and data acquisition (SCADA) system.

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arXiv2019

An Adversarial Model for Attack Vector Vulnerability Analysis on Power and Gas Delivery Operations

Power systems often rely on natural gas pipeline networks to supply fuel for gas-fired generation. Market inefficiencies and a lack of formal coordination between the wholesale power and gas delivery infrastructures may magnify the broader impact of a cyber-attack on a natural gas pipeline. In this study we present a model that can be used to quantify the impact of cyber-attacks on electricity and gas delivery operations. We model activation of cyber-attack vectors that attempt to gain access to pipeline gas compressor controls using a continuous-time Markov chain over a state space based on the gas operator Industrial Control System firewall zone partition. Our approach evaluates the operating states and decision-making in the networks using physically realistic and operationally representative models. We summarize these models, the sequence of analyses used to quantify the impacts of a cyber-incident, and propose a Monte Carlo simulation approach to quantify the resulting effect on the reliability of the bulk power system by the increase in operational cost. The methodology is applied to a case study of interacting power, gas, and cyber test networks.

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arXiv2019

Estimation of Location and Orientation for Underwater Vehicles from Range Measurements

Localization is an important required task for enabling vehicle autonomy for underwater vehicles. Localization entails the determination of position of the center of mass and orientation of a vehicle from the available measurements. In this paper, we focus on localization by using One-Way Travel Time (OWTT) measurements available to a vehicle from the communication of its multiple on-board receivers with acoustic beacons, more specifically, long baseline (LBL) beacons. Range can be inferred by multiplying the OWTT with speed of sound; however, water conditions can change spatially and temporally resulting in uncertainty in range measurement. The farther a beacon is from a receiver, the larger is the uncertainty. The proposed method for localization accounts captures this uncertainty by bounding the true distance with an increasing (calibrating) function of the range measurement. Determination of this calibration function is formulated as polynomial optimization problem and is a crucial step for localization. The proposed two-step procedure for localization is as follows: based on the range measurements specific to a receiver from the beacons, a convex optimization problem is proposed to estimate the location of the receiver. The estimate is essentially a center of the set of possible locations of the receiver. In the second step, the location estimates of the vehicle are corrected using rigid body motion constraints and the orientation of the rigid body is thus determined. Numerical examples and experimental results provided at the end corroborate the procedures developed in this paper.

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arXiv2019

Cooperative Routing for an Air-Ground Vehicle Team -- Exact Algorithm, Transformation Method, and Heuristics

This article considers a cooperative vehicle routing problem for an intelligence, surveillance, and reconnaissance mission in the presence of communication constraints between the vehicles. The proposed framework uses a ground vehicle and an Unmanned Aerial Vehicle (UAV) that travel cooperatively and visit a set of targets while satisfying the communication constraints. The problem is formulated as a mixed-integer linear program, and a branch-and-cut algorithm is developed to solve the problem to optimality. Furthermore, a transformation method and a heuristic are also developed for the problem. The effectiveness of all the algorithms is corroborated through extensive computational experiments on several randomly generated instances.

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arXiv2019

An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs

In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of piecewise polyhedral relaxation approaches via disjunctive formulations to solve MINLPs to global optimality in contrast to the conventional spatial branch-and-bound approaches. In order to maintain relatively small-scale mixed-integer linear programs at every iteration of the algorithm, we adaptively partition the variable domains appearing in the multi-linear terms. We also provide proofs on convergence guarantees of the proposed algorithm to a global solution. Further, we discuss a few algorithmic enhancements based on the sequential bound-tightening procedure as a presolve step, where we observe the importance of solving piecewise relaxations compared to basic convex relaxations to speed-up the convergence of the algorithm to global optimality. We demonstrate the effectiveness of our disjunctive formulations and the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare with state-of-the-art global optimization solvers. With this novel approach, we solve several large-scale instances which are, in some cases, intractable by the global optimization solver. We also shrink the best known optimality gap for one of the hard, generalized pooling problem instance.

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arXiv2019

Optimization-Based Bound Tightening using a Strengthened QC-Relaxation of the Optimal Power Flow Problem

This article develops a strengthened convex quadratic convex (QC) relaxation of the AC Optimal Power Flow (AC-OPF) problem and presents an optimization-based bound-tightening (OBBT) algorithm to compute tight, feasible bounds on the voltage magnitude variables for each bus and the phase angle difference variables for each branch in the network. Theoretical properties of the strengthened QC relaxation that show its dominance over the other variants of the QC relaxation studied in the literature are also derived. The effectiveness of the strengthened QC relaxation is corroborated via extensive numerical results on benchmark AC-OPF test networks. In particular, the results demonstrate that the proposed relaxation consistently provides the tightest variable bounds and optimality gaps with negligible impacts on runtime performance.

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arXiv2018

Routing Multiple Unmanned Vehicles in GPS-Denied Environments

This article aims to develop novel path planning algorithms required to deploy multiple unmanned vehicles in Global Positioning System (GPS) denied environments. Unmanned vehicles (ground or aerial) are ideal platforms for executing monitoring and data gathering tasks in civil infrastructure management, agriculture, public safety, law enforcement, disaster relief and transportation. Significant advancement in the area of path planning for unmanned vehicles over the last decade has resulted in a suite of algorithms that can handle heterogeneity, motion and other on-board resource constraints for these vehicles. However, most of these routing and path planning algorithms rely on the availability of the GPS information. Unintentional and intentional interference and design errors can cause GPS service outages, which in turn, can crucially affect all the systems that depend on GPS information. This article addresses a multiple vehicle path planning problem that arises while deploying a team of unmanned vehicles for monitoring applications in GPS-denied environments and presents a mathematical formulation and algorithms for solving the problem. Simulation results are also presented to corroborate the performance of the proposed algorithms.

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