Carleton Coffrin

Carleton Coffrin, Hassan L. Hijazi, Pascal Van Hentenryck

Nonlinear convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP), Convex Quadratic (QC), and Second-Order Cone (SOC) relaxations, have attracted significant interest in recent years. Thus far, little attention has been given to simpler linear relaxations of the power flow equations, which may bring significant performance gains at the cost of model accuracy. To fill the gap, this paper develops two intuitive linear relaxations of the power flow equations, one based on classic network flow models (NF) and another inspired by copper plate approximations (CP). Theoretical results show that the proposed NF model is a relaxation of the established nonlinear SOC model and the CP model is a relaxation of the NF model. Consequently, considering the linear NF and CP relaxations alongside the established nonlinear relaxations (SDP, QC, SOC) provides a rich variety of tradeoffs between the relaxation accuracy and performance.

Carleton Coffrin, Hassan L. Hijazi, Pascal Van Hentenryck

Convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP), Second-Order Cone (SOC), and Convex DistFlow (CDF) relaxations, have attracted significant interest in recent years. Thus far, studies of the CDF model and its connection to the other relaxations have been limited to power distribution systems, which omit several parameters necessary for modeling transmission systems. To increase the applicability of the CDF relaxation, this paper develops an extended CDF model that is suitable for transmission systems by incorporating bus shunts, line charging, and transformers. Additionally, a theoretical result shows that the established equivalence of the SOC and CDF models for distribution systems also holds in this transmission system extension.

Kaarthik Sundar, Carleton Coffrin, Harsha Nagarajan et al.

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.

Kaarthik Sundar, Harsha Nagarajan, Sidhant Misra et al.

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.

Jon Nelson, Marc Vuffray, Andrey Y. Lokhov et al.

Quantum Annealing (QA) was originally intended for accelerating the solution of combinatorial optimization tasks that have natural encodings as Ising models. However, recent experiments on QA hardware platforms have demonstrated that, in the operating regime corresponding to weak interactions, the QA hardware behaves like a noisy Gibbs sampler at a hardware-specific effective temperature. This work builds on those insights and identifies a class of small hardware-native Ising models that are robust to noise effects and proposes a procedure for executing these models on QA hardware to maximize Gibbs sampling performance. Experimental results indicate that the proposed protocol results in high-quality Gibbs samples from a hardware-specific effective temperature. Furthermore, we show that this effective temperature can be adjusted by modulating the annealing time and energy scale. The procedure proposed in this work provides an approach to using QA hardware for Ising model sampling presenting potential new opportunities for applications in machine learning and physics simulation.

Jon Nelson, Marc Vuffray, Andrey Y. Lokhov et al.

As a wide variety of quantum computing platforms become available, methods for assessing and comparing the performance of these devices are of increasing interest and importance. Inspired by the success of single-qubit error rate computations for tracking the progress of gate-based quantum computers, this work proposes a Quantum Annealing Single-qubit Assessment (QASA) protocol for quantifying the performance of individual qubits in quantum annealing computers. The proposed protocol scales to large quantum annealers with thousands of qubits and provides unique insights into the distribution of qubit properties within a particular hardware device. The efficacy of the QASA protocol is demonstrated by analyzing the properties of a D-Wave 2000Q system, revealing unanticipated correlations in the qubit performance of that device. A study repeating the QASA protocol at different annealing times highlights how the method can be utilized to understand the impact of annealing parameters on qubit performance. Overall, the proposed QASA protocol provides a useful tool for assessing the performance of current and emerging quantum annealing devices.

Marc Vuffray, Carleton Coffrin, Yaroslav A. Kharkov et al.

Drawing independent samples from high-dimensional probability distributions represents the major computational bottleneck for modern algorithms, including powerful machine learning frameworks such as deep learning. The quest for discovering larger families of distributions for which sampling can be efficiently realized has inspired an exploration beyond established computing methods and turning to novel physical devices that leverage the principles of quantum computation. Quantum annealing embodies a promising computational paradigm that is intimately related to the complexity of energy landscapes in Gibbs distributions, which relate the probabilities of system states to the energies of these states. Here, we study the sampling properties of physical realizations of quantum annealers which are implemented through programmable lattices of superconducting flux qubits. Comprehensive statistical analysis of the data produced by these quantum machines shows that quantum annealers behave as samplers that generate independent configurations from low-temperature noisy Gibbs distributions. We show that the structure of the output distribution probes the intrinsic physical properties of the quantum device such as effective temperature of individual qubits and magnitude of local qubit noise, which result in a non-linear response function and spurious interactions that are absent in the hardware implementation. We anticipate that our methodology will find widespread use in characterization of future generations of quantum annealers and other emerging analog computing devices.

Carleton Coffrin, James Arnold, Stephan Eidenbenz et al.

This report describes eighteen projects that explored how commercial cloud computing services can be utilized for scientific computation at national laboratories. These demonstrations ranged from deploying proprietary software in a cloud environment to leveraging established cloud-based analytics workflows for processing scientific datasets. By and large, the projects were successful and collectively they suggest that cloud computing can be a valuable computational resource for scientific computation at national laboratories.

Carleton Coffrin, Harsha Nagarajan, Russell Bent

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.

Carleton Coffrin, Dan Gordon, Paul Scott

In recent years the power systems research community has seen an explosion of work applying operations research techniques to challenging power network optimization problems. Regardless of the application under consideration, all of these works rely on power system test cases for evaluation and validation. However, many of the well established power system test cases were developed as far back as the 1960s with the aim of testing AC power flow algorithms. It is unclear if these power flow test cases are suitable for power system optimization studies. This report surveys all of the publicly available AC transmission system test cases, to the best of our knowledge, and assess their suitability for optimization tasks. It finds that many of the traditional test cases are missing key network operation constraints, such as line thermal limits and generator capability curves. To incorporate these missing constraints, data driven models are developed from a variety of publicly available data sources. The resulting extended test cases form a compressive archive, NESTA, for the evaluation and validation of power system optimization algorithms.

Carleton Coffrin, Pascal Van Hentenryck

Linear active-power-only DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasibility. This paper proposes linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a convex approximation of the cosine terms in the AC equations, as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and MatPower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two "proof-of-concept" studies in power restoration and capacitor placement.