Order Cone Research Articles

Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality

We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power-generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs (MINLPs). The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second order cone programs as well as primal and dual cutting planes. For example, objective and no-good cuts help to compute good feasible solutions in which outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems in which discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high-quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP solvers. Our method outperforms all other algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research has been funded by the Federal Ministry of Education and Research of Germany [Grant 05M18WEB]. This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. The authors thank the Deutsche Forschungsgemeinschaft for support within projects A05, B06, B07, and B10 of the Sonderforschungsbereich/Transregio 154 “Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks.” This work has been supported by the Federal Ministry for Economic Affairs and Energy, Germany [Grant 03El1036A]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1270 .

A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

Solving large scale nonlinear optimization problems requires either significant computing resources or the development of specialized algorithms. For Linear Programming (LP) problems, decomposition methods can take advantage of problem structure, gradually constructing the full problem by generating variables or constraints. We first present a direct adaptation of the Column Generation (CG) methodology for nonlinear optimization problems, such that when optimizing over a structured set $${\mathcal {X}}$$ plus a moderate number of complicating constraints, we solve a succession of (1) restricted master problems on a smaller set $${\mathcal {S}}\subset {\mathcal {X}}$$ and (2) pricing problems that are Lagrangean relaxations wrt the complicating constraints. The former provides feasible solutions and feeds dual information to the latter. In turn, the pricing problem identifies a variable of interest that is then taken into account into an updated subset $${\mathcal {S}}'\subset {\mathcal {X}}$$ . Our approach is valid whenever the master problem has zero Lagrangean duality gap wrt to the complicating constraints, and not only when $${\mathcal {S}}$$ is the convex hull of the generated variables as in CG for LPs, but also with a variety of subsets such as the conic hull, the linear span, and a special variable aggregation set. We discuss how the structure of $${\mathcal {S}}$$ and its update mechanism influence the number of iterations required to reach near-optimality and the difficulty of solving the restricted master problems, and present linearized schemes that alleviate the computational burden of solving the pricing problem. We test our methods on synthetic portfolio optimization instances with up to 5 million variables including nonlinear objective functions and second order cone constraints. We show that some CGs with linearized pricing are 2–3 times faster than solving the complete problem directly and are able to provide solutions within 1% of optimality in 6 h for the larger instances, whereas solving the complete problem runs out of memory.

Finding an energy efficient path for plug-in electric vehicles with speed optimization and travel time restrictions

Transportation is one of the main factors when global total energy consumption is considered and is a significant contributor to emissions of harmful gases including carbon dioxide (CO2). Due to their lower tailpipe CO2 emissions compared to the vehicles with internal combustion engines, electric vehicles provide an opportunity to reduce environmental impacts of transportation. In this direction, a problem for plug-in electric vehicles (PEVs) is studied where the aim is to find an energy efficient path. Given an origin–destination pair over a directed network, this problem involves determining a path joining origin and destination, the speed of the PEV on each road segment, i.e., arc, along the path, the charging stations the PEV will stop by, and how much to recharge at each stop so as to minimize the total amount energy consumption. There are speed limits on each road segment, and PEV has to arrive at the destination on or before a given total time limit. For this problem, firstly, a mixed-integer second order cone programming formulation (MISOCP) is proposed. Secondly, to be able to solve larger size instances, a matheuristic is developed. Lastly, an iterated local search (ILS) algorithm is designed for this problem. Solution quality and computation times of the heuristics and the exact algorithm are compared on different instances. Differently from the literature, the speed values of the PEV on the arcs are considered as continuous decision variables in all proposed solution approaches. Moreover, consideration of the speed limits which can be legal limits or limits imposed by congestion makes our problem more realistic. The analysis of the results of the computational experiments gives the user an insight to select the proper solution approach based on the instance settings. MISOCP formulation becomes inadequate for larger instances. On the other hand, the heuristic solution approaches can solve such instances within reasonable computational times and therefore they have the potential to be integrated in some software to dynamically find energy efficient paths.

Fast QC Relaxation of the Optimal Power Flow Using the Line-Wise Model for Representing Meshed Transmission Networks

In this paper, we investigate the recently introduced McCormick-based Quadratic Convex (QC) relaxation of the Optimal Power Flow (OPF) where Line-Wise Model (LWM) is used for representing meshed transmission systems (QC-LW OPF) for the sake of decisively determining its relationship to other available convex relaxations in the literature. We also extend the recently introduced convex envelope of the tangent function so it would be suitable for test cases with any voltage angle difference range. A computational study where the recently proposed QC-LW OPF formulation is compared to an equivalent McCormick based QC relaxation of the OPF where Bus Injection Model (BIM) is used for representing meshed transmission systems (QC-BI OPF) is presented in this paper. This computational study was conducted using test cases that belong to different operational categories using 123 test cases from the PGLib-OPF library with a bus size range between 3 up to 6515 buses for the sake of understanding the effect of the change of operating conditions on the quality of solutions obtained using the QC-LW OPF and QC-BI OPF formulations. Results are compared using several metrics that testify to the obtained solution’s quality and the problem’s computational complexity. Comparison of results shows that the QC-LW relaxation neither dominates nor is dominated by the QC-BI relaxation in terms of solution quality. Therefore, it dominates the Second Order Cone (SOC) relaxation and neither dominates nor is dominated by the Semidefinite Programming (SDP) relaxation. Furthermore, it is shown that the QC-LW OPF has reduced the number of relaxed trigonometric functions and McCormick envelopes needed when compared to the QC-BI OPF, leading to a faster solution time for more than 84% of the test cases in the range of 2% up to 67%.

Soft Open Points-Assisted Resilience Enhancement of Power Distribution Networks Against Cyber Risks

Cyberattacks are increasingly threatening the normal operation of electric power grids. The purpose of this study is to develop an optimization model for planning and operation of distribution systems to enhance the resilience to cyberattacks. A new distribution system planning method is presented utilizing a Defender-Attacker-Defender (DAD) tri-level model to improve the system resilience to withstand potential disruptions from cyberattacks. In the first level of the model, the system planner optimizes the Soft Open Points (SOPs) installation in the planning stage. The attacker’s objective in the second level of the model is to change the status of the network circuit breakers and jam the communication maximizing disruption of the system. The optimization of Distribution Generators’ (DGs’) outputs, SOPs’ power flow, and topology reconfiguration are comprehensively considered in the third level. The novel model presented in this paper considers hardening measures of the SOPs in response to the attacks on distribution systems. The Column and Constraint Generation (C&CG) algorithm is utilized to solve the proposed DAD model. A procedure is devised to solve the two-layer subproblem (the Attacker-Defender model), with the bottom layer being a Mixed Integer Second Order Cone Programming (MISOCP) problem. Finally, the proposed models are validated in 33-bus and 135-bus test systems. The results show that the installation of SOPs significantly improves the service restoration under cyberattacks. Both the number and location of SOPs influence the extent of restored service, and differing SOP allocation schemes may lead to similar restoration outcomes. The proposed models and techniques could also be extended to deal with other similar problems in industrial cyber-physical systems.

A Novel Energy Efficient Harvesting Technique for SDWSN using RF Transmitters with MISO Beamforming

Software Defined Wireless Sensor Networks (SDWSN) is emerged to overcome the additional energy consumption in WSN. Even then the sensor nodes in the SDWSN suffer from scarce battery resources. Generally, the Radio Frequency (RF) transmitters are deployed around the base station in the SDWSN to overcome the high energy consumption problem. To enhance harvesting energy and coverage of nodes in the network, a new energy harvesting technique using RF transmitters with Multiple Input and Single Output (MISO) beamforming is proposed. In this method, multiple antenna RF transmitters and single antenna sensor nodes are deployed. The optimization problem subject to Signal to Noise Ratio (SNR) and energy harvesting constraints is formulated for hybrid beamforming design to reduce the transmit power in the network. The optimization problem based on convex Second Order Cone Programming (SOCP) is derived to get the optimal solution for hybrid beamforming design. The beamforming technique is used to steer the beam in the desired direction and null to the other direction improves the energy harvesting. The simulation results show that the proposed technique provides better average harvesting energy, average transmit power, average residual energy and throughput than the existing RF transmitter based energy harvesting methods.

Distributionally robust multi-period portfolio selection subject to bankruptcy constraints

<p style='text-indent:20px;'>An optimization problem with moments information which suffers from distributional uncertainty can be handled through distributionally robust optimization. In this paper, we will consider distributionally robust multi-period portfolio selection since only moment information of portfolios can be gathered in practice. We will consider two different scenarios. One is that moments information can be obtained exactly and the other one is that the moments information is also uncertain. For the two scenarios, we will show how to transform the corresponding distributionally robust optimization problem into a second order cone problem (SOCP) which can be easily solved by existing methods. Some numerical experiments are presented to demonstrate the effectiveness of our proposed method.</p>

Fast Restoration for Smart Grids Using Mixed-Integer Second Order Cone Programming

The Smart Grids offered the capability to reconfigure and restore the system to regular operation after a fault without human intervention. It is done by installing distributed equipment as input for the monitoring and control system. This study proposed a fast restoration for smart grids by considering technical constraints. Those include the available power supply, system reconfiguration, and power flow. The restoration strategies are solved by mathematical programming, formulated as a mixed-integer second order cone programming problem. The proposed formulation ensures global optimality. The proposed method is tested using IEEE 33 and 69 bus systems. In addition, DG’s are added to represent active distribution systems. The simulation results show the capability of the proposed method to solve fast restoration with remarkable accuracy and computational time.

A Second Order Cone Programming Approach for Simulating Biphasic Materials

AbstractStrain limiting is a widely used approach for simulating biphasic materials such as woven textiles and biological tissue that exhibit a soft elastic regime followed by a hard deformation limit. However, existing methods are either based on slowly converging local iterations, or offer no guarantees on convergence. In this work, we propose a new approach to strain limiting based on second order cone programming (SOCP). Our work is based on the key insight that upper bounds on per‐triangle deformations lead to convex quadratic inequality constraints. Though nonlinear, these constraints can be reformulated as inclusion conditions on convex sets, leading to a second order cone programming problem—a convex optimization problem that a) is guaranteed to have a unique solution and b) allows us to leverage efficient conic programming solvers. We first cast strain limiting with anisotropic bounds on stretching as a quadratically constrained quadratic program (QCQP), then show how this QCQP can be mapped to a second order cone programming problem. We further propose a constraint reflection scheme and empirically show that it exhibits superior energy‐preservation properties compared to conventional end‐of‐step projection methods. Finally, we demonstrate our prototype implementation on a set of examples and illustrate how different deformation limits can be used to model a wide range of material behaviors.

Robust Precoding for Multi-User Visible Light Communications with Quantized Channel Information

In this paper, we address the design of multi-user multiple-input single-output (MU-MISO) precoders for indoor visible light communication (VLC) systems. The goal is to minimize the transmitted optical power per light emitting diode (LED) under imperfect channel state information (CSI) at the transmitter side. Robust precoders for imperfect CSI available in the literature include noisy and outdated channel estimation cases. However, to the best of our knowledge, no work has considered adding robustness against channel quantization. In this paper, we fill this gap by addressing the case of imperfect CSI due to the quantization of VLC channels. We model the quantization errors in the CSI through polyhedric uncertainty regions. For polyhedric uncertainty regions and positive real channels, as is the case of VLC channels, we show that the robust precoder against channel quantization errors that minimizes the transmitted optical power while guaranteeing a target signal to noise plus interference ratio (SNIR) per user is the solution of a second order cone programming (SOCP) problem. Finally, we evaluate its performance under different quantization levels through numerical simulations.

A new SOCP relaxation of nonconvex quadratic programming problems with a few negative eigenvalues

We present a new second order cone programming (SOCP) relaxation of nonconvex quadratic programs with a few negative eigenvalues (NQP-r-NE) by employing the difference of convex (DC) decomposition and simultaneous matrix diagonalization together. The auxiliary variables are bounded from above by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation provided by Kim and Kojima (2001). We prove that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation under certain circumstances, especially when there exists one constraint matrix being positive definite in the primal problem. Three types of numerical experiments including the large-scale NQP-r-NE problem with the number of negative eigenvalues r≤20, the optimal spectrum sharing problem in MIMO cognitive radio networks and the two-trust-region subproblem are provided to illustrate that the proposed SOCP relaxation could achieve a better lower bound than that of the classical SOCP relaxation. Moreover, when the number of variables is larger than 250, the proposed SOCP relaxation shows its great superiority both in the bound quality and computing efficiency compared to the classical SOCP relaxation.

A strict complementarity approach to error bound and sensitivity of solution of conic programs

In this paper, we provide an elementary, geometric, and unified framework to analyze conic programs that we call the strict complementarity approach. This framework allows us to establish error bounds and quantify the sensitivity of the solution. The framework uses three classical ideas from convex geometry and linear algebra: linear regularity of convex sets, facial reduction, and orthogonal decomposition. We show how to use this framework to derive error bounds for linear programming, second order cone programming, and semidefinite programming.

Double-time-scale distributed voltage control for unbalanced distribution networks based on MPC and ADMM

With the increasing penetration of distributed generators (DGs) in the distribution system, the distributed coordinated optimal control of DG and voltage control devices has become imperative. This paper proposes a double-time-scale distributed voltage control scheme for unbalanced distribution networks with distributed generators based on model predictive control (MPC) and alternating direction method of multiplier (ADMM) to regulate the voltage profile across a network. To better coordinate the economical operation and voltage regulation, the voltage control devices with different temporal characteristics are separately adjusted. In the fast-time-scale control, the decision variables are the active and reactive power outputs of DGs. In the slow-time-scale control, the on-load tap changer, step voltage regulators and capacitor banks are optimized. A tailored solution method based on McCormick envelopes and branch-and-bound algorithm is developed to efficiently solve the optimization problems by transforming the mixed-integer nonlinear program (MINLP) into a second order cone program (SOCP). The voltage control scheme is a combinatorial optimization problem whose computational complexity grows exponentially with the number of binary variables. To resolve this issue, a distributed optimal voltage control strategy is developed based on the alternating direction method of multipliers (ADMM). Through the ADMM, the optimization problem is distributed among the zones of the distribution networks, without requiring a central controller. The modified IEEE 123-node system is used to verify the effectiveness of the proposed control scheme as a test system.

Implementation of a projection and rescaling algorithm for second-order conic feasibility problems

This paper documents a computational implementation of a projection and rescaling algorithm for solving one of the alternative feasibility problems where L is a linear subspace in , is its orthogonal complement, and is the interior of a direct product of second order cones. The gist of the projection and rescaling algorithm is to enhance a low-cost first-order method (a basic procedure) with an adaptive reconditioning transformation (a rescaling step). We give a full description of a Python implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our Python code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.

Joint beamforming for secure communication in RIS-assisted cognitive radio networks

This paper studies the security and communication efficiency problem of a reconfigurable intelligent surface (RIS) assisted cognitive radio network, in which we proposed a scheme where secrecy rate (SR) and transmission rate are jointly considered. In specific, with the help of RIS, we exploit the green interference caused by the secondary network to ensure the physical layer security of the primary network. By taking the mutual interference between these two networks into account, we establish a problem to minimize the total transmit power subject to the quality-of-service requirement of secondary user (SU), SR requirement of the primary users (PUs) and limited interference temperature of PUs. In our proposed iterative algorithm, the transmit and reflect beamforming are alternatively optimized by transforming PUs' SR constraint into a second order cone (SOC) form, and the optimal transmit and reflect beamforming are achieved based on the iterative penalty function. Simulation results show the superiority of our proposed algorithm in terms of the total transmit power requirement compared with other benchmarks, and the total transmit power of our proposed algorithm is not very susceptible to the number of eavesdroppers and PUs, which is conducive to serving multiple users in the wireless communication system.

REVIEW OF THE ROLE OF BRACHYTHERAPY IN THE MANAGEMENT OF ORAL CAVITY CANCERS

From the day of invention of Radium, brachytherapy has been practised worldwide. Can any technological advancement in external beam radiotherapy replace brachytherapy? Answer probably no, especially in Gynaecological cancers as categorically stated by GEC-ESTRO experts. External beam radiation therapy using contemporary conformal techniques and brachytherapy play a signicant role in the management of oral cavity cancer. Small sized tumors in most of the oral cavity sites can be managed with brachytherapy alone either with intraoral cone or interstitial brachytherapy in order to minimize exposure of normal tissue. As these techniques do not treat regional lymph node basins, they are only appropriate to use as a single modality in selected stage I and II patients. The risk of occult nodal involvement in stage I and II is very low. External beam radiation therapy to the draining lymph node regions is used as the primary mode of irradiation when regional lymph nodes are at signicant risk for subclinical involvement, and brachytherapy may be added as a boost to the primary tumor. Since head and neck cancers are Surgeons domain, number of patients reporting to Radiation Oncology department is too low to carry out large randomized trial. Most of the patients report to Radiation Oncology department in either in late stage or in an early stage with poor general condition. This paves the way for selection bias and, of course, inferior results when compared with other modality. To evaluate the role of this therapy in treatment of head and neck cancers, we have reviewed its outcomes with oral cavity cancer. In conclusion, brachytherapy with or without EBRT can be an alternative to surgery in early and locally advanced oral cavity cancers although concrete evidence is yet to be produced with a sophisticated study in a reproducible manner

Sharpe-Ratio Portfolio in Controllable Markov Chains: Analytic and Algorithmic Approach for Second Order Cone Programming

The Sharpe ratio is a measure based on the theory of mean variance, it is the measure of the performance of a portfolio when the risk can be measured through the standard deviation. This paper suggests a Sharpe-ratio portfolio solution using a second order cone programming (SOCP). We use the penalty-regularized method to represent the nonlinear portfolio problem. We present a computationally tractable way to determining the Sharpe-ratio portfolio. A Markov chain structure is employed to represent the underlying asset price process. In order to determine the optimal portfolio in Markov chains, a new hybrid optimization programming method for SOCP is proposed. The suggested method’s efficiency and efficacy are demonstrated using a numerical example.

Minimum cost b-matching problems with neighborhoods

In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a–b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.