Hassan L. Hijazi

The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow

Convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP) and Second-Order Cone (SOC) relaxations, have attracted significant interest in recent years. The Quadratic Convex (QC) relaxation is a departure from these relaxations in the sense that it imposes constraints to preserve stronger links between the voltage variables through convex envelopes of the polar representation. This paper is a systematic study of the QC relaxation for AC Optimal Power Flow with realistic side constraints. The main theoretical result shows that the QC relaxation is stronger than the SOC relaxation and neither dominates nor is dominated by the SDP relaxation. In addition, comprehensive computational results show that the QC relaxation may produce significant improvements in accuracy over the SOC relaxation at a reasonable computational cost, especially for networks with tight bounds on phase angle differences. The QC and SOC relaxations are also shown to be significantly faster and reliable compared to the SDP relaxation given the current state of the respective solvers.

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.

Strengthening convex relaxations with bound tightening for power network optimization

Convexification is a fundamental technique in mixed-integer nonlinear optimization and many convex relaxations are parametrized by variable bounds, i.e., the tighter the bounds, the stronger the relaxations. This paper studies how bound tightening can improve convex relaxations for power network optimization. It adapts traditional constraint-programming concepts e.g., minimal network and bound consistency to a relaxation framework and shows how bound tightening can dramatically improve power network optimization. In particular, the paper shows that the Quadratic Convex relaxation of power flows, enhanced by bound tightening, almost always outperforms the state-of-the-art Semi-Definite Programming relaxation on the optimal power flow problem.

Primal and dual bounds for Optimal Transmission Switching

It has been suggested that Optimal Transmission Switching (OTS) may reduce generator dispatch costs by as much as 10%, saving millions of dollars annually. However, this conclusion has been deduced primarily from studies using the DC power flow approximation on two power networks derived from the IEEE 118-bus and RTS-96 cases. This paper is a systematic study of the OTS problem. Various OTS formulations are considered for computing primal and dual bounds on a variety of power networks. The results demonstrate that the DC power flow model is inadequate for solving the OTS problem, and that mixed-integer nonlinear optimization techniques are instrumental in finding high-quality primal and dual bounds. The paper also indicates that, on a variety of benchmarks, transmission line switching may bring generation cost reductions between 0% to 29%.

An Outer-Inner Approximation for Separable Mixed-Integer Nonlinear Programs

A common structure in convex mixed-integer nonlinear programs (MINLPs) is separable nonlinear functions. In the presence of such structures, we propose three improvements to the outer approximation algorithms. The first improvement is a simple extended formulation, the second is a refined outer approximation, and the third is a heuristic inner approximation of the feasible region. As a side result, we exhibit a simple example where a classical implementation of the outer approximation would take an exponential number of iterations, whereas it is easily solved with our modifications. These methods have been implemented in the open source solver Bonmin and are available for download from the Computational Infrastructure for Operations Research project website. We test the effectiveness of the approach on three real-world applications and on a larger set of models from an MINLP benchmark library. Finally, we show how the techniques can be extended to perspective formulations of several problems. The proposed tools lead to an important reduction in computing time on most tested instances.

Robust delay-constrained routing in telecommunications

In telecommunications, operators usually use market surveys and statistical models to estimate traffic evolution in networks or to approximate queuing delay functions in routing strategies. Many research activities concentrated on handling traffic uncertainty in network design. Measurements on real world networks have shown significant errors in delay approximations, leading to weak management decisions in network planning. In this work, we introduce elements of robust optimization theory for delay modeling in routing problems. Different types of data uncertainty are considered and linked to corresponding robust models.We study a special case of constraints featuring separable additive functions. Specifically, we consider that each term of the sum is disturbed by a random parameter. These constraints are frequent in network based problems, where functions reflecting real world measurements on links are summed up over end-to-end paths. While classical robust formulations have to deal with the introduction of new variables, we show that, under specific hypotheses, the deterministic robust counterpart can be formulated in the space of original variables. This offers the possibility of constructing tractable robust models.Starting from Soyster’s conservative model, we write and compare different uncertainty sets and formulations offering each a different protection level for the delay constrained routing problem. Computational experiments are developed in order to evaluate the “price of robustness” and to assess the quality of the new formulations.

Convex relaxations for gas expansion planning

Expansion of natural gas networks is a critical process involving substantial capital expenditures with complex decision-support requirements. Given the nonconvex nature of gas transmission constraints, global optimality and infeasibility guarantees can only be offered by global optimisation approaches. Unfortunately, state-of-the-art global optimisation solvers are unable to scale up to real-world size instances. In this study, we present a convex mixed-integer second-order cone relaxation for the gas expansion planning problem under steady-state conditions. The underlying model offers tight lower bounds with high computational efficiency. In addition, the optimal solution of the relaxation can often be used to derive high-quality solutions to the original problem, leading to provably tight optimality gaps and, in some cases, global optimal solutions. The convex relaxation is based on a few key ideas, including the introduction of flux direction variables, exact McCormick relaxations, on/off constraints,...

Mixed Integer NonLinear Programs featuring “On/Off” constraints: convex analysis and applications

Abstract We call “on/off” constraint an algebraic constraint that is activated if and only if a corresponding boolean variable equals 1. Our main subject of interest is to derive tight convex formulations of Mixed Integer NonLinear Programs featuring “on/off” constraints. We study the simple set defined by one “on/off” constraint with bounded variables. Using Disjunctive Programming, we introduce convex hull formulations of this set defined in higher dimensional spaces. Because the large number of variables in these formulations appears to be practically disadvantageous, we concentrate our efforts on defining explicit projections into lower dimensional spaces. Based on these results, we present new formulations to a well-known telecommunication problem: routing several commodities subject to multiple delay constraints. Numerical results are presented to assess the efficiency of the new models.

Polynomial SDP cuts for Optimal Power Flow

The use of convex relaxations has lately gained considerable interest in Power Systems. These relaxations play a major role in providing quality guarantees for non-convex optimization problems. For the Optimal Power Flow (OPF) problem, the semidefinite programming (SDP) relaxation is known to produce tight lower bounds. Unfortunately, SDP solvers still suffer from a lack of scalability. In this work, we introduce an exact reformulation of the SDP relaxation, formed by a set of polynomial constraints defined in the space of real variables. The new constraints can be seen as “cuts”, strengthening weaker second-order cone relaxations, and can be generated in a lazy iterative fashion. The new formulation can be handled by standard nonlinear programming solvers, enjoying better stability and computational efficiency. This new approach benefits from recent results on tree-decomposition methods, reducing the dimension of the underlying SDP matrices. As a side result, we present a formulation of Kirchhoffs Voltage Law in the SDP space and reveal the existing link between these cycle constraints and the original SDP relaxation for three dimensional matrices. Preliminary results show a significant gain in computational efficiency compared to a standard SDP solver approach.

Convex Optimization for Joint Expansion Planning of Natural Gas and Power Systems

Within the energy sector, two of the most tightly coupled systems are natural gas and electric power. The recent advent of cheap gas extraction technologies have only driven these systems more tightly together. Despite their interconnections, in many areas of the world these systems are operated and managed in isolation. This separation is due to a number of reasons and challenges, ranging from technological (problems involving connected systems are difficult to solve) to political and commercial (prevention of monopolies, lack of communication, market forces, etc.). However, this separation can lead to a number of undesirable outcomes, such as what the northeastern United States experienced during the winter of 2013/2014. In this paper, we develop approaches to address the technological reasons for separation. We consider the problem of expanding and designing coupled natural gas and electric power systems to meet increased coincident demand on both systems. Our approach utilizes recent advances in convex modeling of gas and power systems to develop a computationally tractable optimization formulation.