arXiv subjects

Harsha Nagarajan

Publications and source records attributed to Harsha Nagarajan.

At least 37 records · Page 2Linked to original sources

arXiv2023

Exploring Non-Linear Programming Formulations in QuantumCircuitOpt for Optimal Circuit Design

Given the limitations of current hardware, the theoretical gains promised by quantum computing remain unrealized across practical applications. But the gap between theory and hardware is closing, assisted by developments in quantum algorithmic modeling. One such recent development is QuantumCircuitOpt (QCOpt), an open-source software framework that leverages state-of-the-art optimization-based solvers to find provably optimal compact circuit decompositions, which are exact up to global phase and machine precision. The quantum circuit design problem can be modeled using non-linear, non-convex constraints. However, QCOpt reformulates these non-linear constraints using well-known linearization techniques such that the resulting design problem is solved as a Mixed-Integer Linear Programming (MILP) model. In this work, we instead explore whether the QCOpt could also be effective with a continuous Non-Linear Programming (NLP) model obtained via relaxation of the integer variables in the non-linear constraints. We are able to present not only multiple significant enhancements to QCOpt, with up to 11.3x speed-up in run times on average, but also opportunities for more generally exploring the behavior of gradient-based NLP solvers.

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arXiv2023

Multi-market Optimal Energy Storage Arbitrage with Capacity Blocking for Emergency Services

The future power system is increasingly interconnected via both AC and DC interconnectors. These interconnectors establish links between previously decoupled energy markets. In this paper, we propose an optimal multi-market energy storage arbitrage model that includes emergency service provisions for system operator(s). The model considers battery ramping and capacity constraints and utilizes operating envelopes calculated based on interconnector capacity, efficiency, dynamic energy injection and offshore wind generation in the day-ahead market. The arbitrage model considers two separate electricity prices for buying and selling of electricity in the two regions, connected via an interconnector. Using disjunctive linearization of nonlinear terms, we exactly reformulate the inter-regional energy arbitrage optimization as a mixed integer linear programming problem. We propose two capacity limit selection models for storage owners providing emergency services. The numerical analyses focus on two interconnections linking Belgium and the UK. The results are assessed based on revenue, operational cycles, payback period, shelf life and computation times.

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arXiv2022

Mitigating the Impacts of Uncertain Geomagnetic Disturbances on Electric Grids: A Distributionally Robust Optimization Approach

Severe geomagnetic disturbances (GMDs) increase the magnitude of the electric field on the Earth's surface (E-field) and drive geomagnetically-induced currents (GICs) along the transmission lines in electric grids. These additional currents can pose severe risks, such as current distortions, transformer saturation and increased reactive power losses, each of which can lead to system unreliability. Several mitigation actions (e.g., changing grid topology) exist that can reduce the harmful GIC effects on the grids. Making such decisions can be challenging, however, because the magnitude and direction of the E-field are uncertain and non-stationary. In this paper, we model uncertain E-fields using the distributionally robust optimization (DRO) approach that determines optimal transmission grid operations such that the worst-case expectation of the system cost is minimized. We also capture the effect of GICs on the nonlinear AC power flow equations. For solution approaches, we develop an accelerated column-and-constraint generation (CCG) algorithm by exploiting a special structure of the support set of uncertain parameters representing the E-field. Extensive numerical experiments based on "epri-21" and "uiuc-150" systems, designed for GMD studies, demonstrate (i) the computational performance of the accelerated CCG algorithm, (ii) the superior performance of distributionally robust grid operations that satisfy nonlinear, nonconvex AC power flow equations and GIC constraints, in comparison with standard stochastic programming-based methods during the out-of-sample testing.

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arXiv2021

QuantumCircuitOpt: An Open-source Framework for Provably Optimal Quantum Circuit Design

In recent years, the quantum computing community has seen an explosion of novel methods to implement non-trivial quantum computations on near-term hardware. An important direction of research has been to decompose an arbitrary entangled state, represented as a unitary, into a quantum circuit, that is, a sequence of gates supported by a quantum processor. It has been well known that circuits with longer decompositions and more entangling multi-qubit gates are error-prone for the current noisy, intermediate-scale quantum devices. To this end, there has been a significant interest to develop heuristic-based methods to discover compact circuits. We contribute to this effort by proposing QuantumCircuitOpt (QCOpt), a novel open-source framework which implements mathematical optimization formulations and algorithms for decomposing arbitrary unitary gates into a sequence of hardware-native gates. A core innovation of QCOpt is that it provides optimality guarantees on the quantum circuits that it produces. In particular, we show that QCOpt can find up to 57% reduction in the number of necessary gates on circuits with up to four qubits, and in run times less than a few minutes on commodity computing hardware. We also validate the efficacy of QCOpt as a tool for quantum circuit design in comparison with a naive brute-force enumeration algorithm. We also show how the QCOpt package can be adapted to various built-in types of native gate sets, based on different hardware platforms like those produced by IBM, Rigetti and Google. We hope this package will facilitate further algorithmic exploration for quantum processor designers, as well as quantum physicists.

quant-ph↗

arXiv2021

Efficient Topology Design Algorithms for Power Grid Stability

The dynamic response of power grids to small disturbances influences their overall stability. This paper examines the effect of network topology on the linearized time-invariant dynamics of electric power systems. The proposed framework utilizes ${\cal H}_2$-norm based stability metrics to study the optimal placement of lines on existing networks as well as the topology design of new networks. The design task is first posed as an NP-hard mixed-integer nonlinear program (MINLP) that is exactly reformulated as a mixed-integer linear program (MILP) using McCormick linearization. To improve computation time, graph-theoretic properties are exploited to derive valid inequalities (cuts) and tighten bounds on the continuous optimization variables. Moreover, a cutting plane generation procedure is put forth that is able to interject the MILP solver and augment additional constraints to the problem on-the-fly. The efficacy of our approach in designing optimal grid topologies is demonstrated through numerical tests on the IEEE 39-bus network.

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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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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

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.

eess.SY↗

arXiv2020

Algorithms for Mitigating the Effect of Uncertain Geomagnetic Disturbances in Electric Grids

Geomagnetic disturbances (GMDs), a result of space weather, pose a severe risk to electric grids. When GMDs occur, they can cause geomagnetically-induced currents (GICs), which saturate transformers, induce hot-spot heating, and increase reactive power losses in the transmission grid. Furthermore, uncertainty in the magnitude and orientation of the geo-electric field, and insufficient historical data make the problem of mitigating the effects of uncertain GMDs challenging. In this paper, we propose a novel distributionally robust optimization (DRO) approach that models uncertain GMDs and mitigates the effects of GICs on electric grids. This is achieved via a set of mitigation actions (e.g., line switching, locating blocking devices, generator re-dispatch and load shedding), prior to the GMD event, such that the worst-case expectation of the system cost is minimized. To this end, we develop a column-and-constraint generation algorithm that solves a sequence of mixed-integer second-order conic programs to handle the underlying convex support set of the uncertain GMDs. Also, we present a monolithic exact reformulation of our DRO model when the underlying support set can be approximated by a polytope with three extreme points. Numerical experiments on 'epri-21' system show the efficacy of the proposed algorithms and the exact reformulation of our DRO model.

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arXiv2019

Evaluating Ising Processing Units with Integer Programming

The recent emergence of novel computational devices, such as adiabatic quantum computers, CMOS annealers, and optical parametric oscillators, present new opportunities for hybrid-optimization algorithms that are hardware accelerated by these devices. In this work, we propose the idea of an Ising processing unit as a computational abstraction for reasoning about these emerging devices. The challenges involved in using and benchmarking these devices are presented and commercial mixed integer programming solvers are proposed as a valuable tool for the validation of these disparate hardware platforms. The proposed validation methodology is demonstrated on a D-Wave 2X adiabatic quantum computer, one example of an Ising processing unit. The computational results demonstrate that the D-Wave hardware consistently produces high-quality solutions and suggests that as IPU technology matures it could become a valuable co-processor in hybrid-optimization algorithms.

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arXiv2019

Distributionally Robust Optimization for a Resilient Transmission Grid During Geomagnetic Disturbances

In recent years, there have been increasing concerns about the impacts of geomagnetic disturbances (GMDs) on electrical power systems. Geomagnetically-induced currents (GICs) can saturate transformers, induce hot-spot heating and increase reactive power losses. Unpredictable GMDs caused by solar storms can significantly increase the risk of transformer failure. In this paper, we develop a two-stage, distributionally robust (DR) optimization formulation that models uncertain GMDs and mitigates the effects of GICs on power systems through existing system controls (e.g., line switching, generator re-dispatch, and load shedding). This model assumes an ambiguity set of probability distributions for induced geo-electric fields which capture uncertain magnitudes and orientations of a GMD event. We employ state-of-the-art linear relaxation methods and reformulate the problem as a two-stage DR model. We use this formulation to develop a decomposition framework for solving the problem. We demonstrate the approach on the modified Epri21 system and show that the DR optimization method effectively handles prediction errors of GMD events.

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arXiv2019

Designing Power Grid Topologies for Minimizing Network Disturbances: An Exact MILP Formulation

The dynamic response of power grids to small transient events or persistent stochastic disturbances influences their stable operation. This paper studies the effect of topology on the linear time-invariant dynamics of power networks. For a variety of stability metrics, a unified framework based on the $H_2$-norm of the system is presented. The proposed framework assesses the robustness of power grids to small disturbances and is used to study the optimal placement of new lines on existing networks as well as the design of radial (tree) and meshed (loopy) topologies for new networks. Although the design task can be posed as a mixed-integer semidefinite program (MI-SDP), its performance does not scale well with network size. Using McCormick relaxation, the topology design problem can be reformulated as a mixed-integer linear program (MILP). To improve the computation time, graphical properties are exploited to provide tighter bounds on the continuous optimization variables. Numerical tests on the IEEE 39-bus feeder demonstrate the efficacy of the optimal topology in minimizing disturbances.

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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

Convex Hull Formulations for Mixed-Integer Multilinear Functions

In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables. We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces. We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches.

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arXiv2018

Probabilistic $N$-$k$ Failure-Identification for Power Systems

This paper considers a probabilistic generalization of the $N$-$k$ failure-identification problem in power transmission networks, where the probability of failure of each component in the network is known a priori and the goal of the problem is to find a set of $k$ components that maximizes disruption to the system loads weighted by the probability of simultaneous failure of the $k$ components. The resulting problem is formulated as a bilevel mixed-integer nonlinear program. Convex relaxations, linear approximations, and heuristics are developed to obtain feasible solutions that are close to the optimum. A general cutting-plane algorithm is proposed to solve the convex relaxation and linear approximations of the $N$-$k$ problem. Extensive numerical results corroborate the effectiveness of the proposed algorithms on small-, medium-, and large-scale test instances, the test instances include the IEEE 14-bus system, the IEEE single-area and three-area RTS96 systems, the IEEE 118-bus system, the WECC 240-bus test system, the 1354-bus PEGASE system, and the 2383-bus Polish winter-peak test system.

eess.SY↗

arXiv2018

Algorithms for Optimal Topology Design, Placement, Sizing and Operation of Distributed Energy Resources in Resilient Off-grid Microgrids

Power distribution in remote communities often depends on off-grid microgrids. In order to address the reliability challenges for these microgrids, we develop a mathematical model for topology design, capacity planning, and operation of distributed energy resources in microgrids that includes N-1 security analysis. Due to the prohibitive size of the optimization problem, we develop a rolling-horizon algorithm that is combined with scenario-based decomposition to efficiently solve the model. We demonstrate the efficiency of our algorithm on an adapted IEEE test network and a real network from an Alaskan microgrid. We also compare our model's required solution time with commercial solvers and recently developed decomposition algorithms to solve similar problems.

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arXiv2018

On Maximizing Weighted Algebraic Connectivity for Synthesizing Robust Networks

This article deals with the following simpler version of an open problem in system realization theory which has several important engineering applications: Given a collection of masses and a set of linear springs with a specified cost and stiffness, a resource constraint in terms of a budget on the total cost, the problem is to determine an optimal connection of masses and springs so that the resulting structure is as stiff as possible, i.e., the structure is connected and its smallest non-zero natural frequency is as large as possible. In this article, algebraic connectivity, or its mechanical analog - the smallest non-zero natural frequency of a connected structure was chosen as a performance objective. Algebraic connectivity determines the convergence rate of consensus protocols and error attenuation in Unmanned Aerial Vehicle (UAV) formations and is chosen to be a performance objective as it can be viewed as a measure of robustness in UAV communication networks to random node failures. Underlying the mechanical and UAV network synthesis problems is a Mixed Integer Semi-Definite Program (MISDP), an NP-hard problem. The novel contributions of this article lies in developing efficient algorithms to solve the MISDPs based on iterative primal-dual algorithm, cutting-plane methods for polyhedral outer-approximation, capturing the feasible set of MISDPs using Fiedler vectors or Laplacian with equivalency guarantees, and efficient neighborhood-search-based heuristic algorithms.

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