Order Cone Research Articles

Distributionally robust single machine scheduling with release and due dates over Wasserstein balls

Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the ℓ1-norm and ℓ2-norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.

Existence and density theorems of Henig global proper efficient points

In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not require the assumption of convexity, and in certain cases, can be applied to unbounded sets. However, it is important to note that a weak compactness condition on the set (or on a section of it) and a separation property between the order cone and its conical neighbourhoods remains necessary. The weak compactness condition ensures that certain convergence properties hold. The separation property enables the interpolation of a family of Bishop-Phelps cones between the order cone and each of its conic neighbourhoods. This interpolation, combined with the proper handling of two distinct types of conic neighbourhoods, plays a crucial role in the proofs of our results, which include as a particular case other results that have already been established under more restrictive conditions.

Second Order Duality Involving Second Order Cone Arcwise Connected Functions and Their Generalizations in Vector Optimization Problem over Cones

In this paper, we introduce second order cone arcwise connected function and its generalizations. Further we study the interrelations among these functions also. Mond Weir type second order dual is formulated and duality results are proved using these functions.

Coordinated active-reactive power optimization considering photovoltaic abandon based on second order cone programming in active distribution networks

On the basis of predecessors’ coordination optimization of active and reactive power in distribution network, For the necessity of the optimal operation in the distribution network, part of power generated from photovoltaic (PV) cannot be sold to users, and cannot enjoy subsidies. Similarly, the network loss in the power transmission will also bring a certain economic loss. This paper comprehensively considers the economic loss caused by the network loss and PV abandon of the distribution system, and establishes a model to minimize the economic loss. To solve this problem efficiently, the method of DistFlow equation and mixed integer second order cone programming (MISOCP) is used to solve the problem, in this method, the original mixed integer nonlinear programming non-convex problem is transformed into a convex problem, which makes the optimization problem easy to solve. The modified IEEE 33 and IEEE 69 distribution networks are tested by the above method. The optimized results are able to meet the target and have very small relaxation gaps, and the voltage level is also optimized. This coordinated optimization approach helps to optimize the economic operation for active distribution networks with PVs.

A two-stage distributed stochastic planning method for source-grid-load-storage flexibility resources considering flexible ramp capacity

With rapidly increasing levels of renewable energy penetration, flexibility resources play an ever more critical role in the future power system. This paper describes a two-stage stochastic mixed integer second order cone (MISOC) method for flexibility resource planning (VFRP) considering source-grid-load-storage interactions. Considering renewable energy penetration levels and flexibility resource characteristics, this study quantifies the extent to which the share of renewable energy generation affects cost performance, and assesses the priority and potential of investing in flexibility resources. Moreover, the multi-type flexibility resource portfolio allocation method and energy storage only approach are compared in terms of socio-economic benefits. A multi-cut Benders-based approach is used for decomposing large-scale stochastic expansion planning problems to reduce the computational burden. Numerical simulation tests on the modified energy test systems I39G20 and F213G20 verified the correctness and effectiveness of the proposed flexible resource planning models and solution algorithms. The simulation results show that retrofit of coal-fired units for better flexibility has a higher investment return. Still, energy storage presents a tremendous investment prospect and potential. Finally, the multi-type flexibility resource portfolio allocation method is more cost-effective than an energy storage only approach.

Compressed sensing: a discrete optimization approach

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label learning. We introduce an ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average 6.22%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6.22\\%$$\\end{document} more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average 3.10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.10\\%$$\\end{document} more sparse. On real world ECG data, for a given ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} reconstruction error our approach produces solutions that are on average 9.95%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$9.95\\%$$\\end{document} more sparse than benchmark methods (3.88%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.88\\%$$\\end{document} more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average 10.77%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10.77\\%$$\\end{document} lower reconstruction error than benchmark methods (1.42%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.42\\%$$\\end{document} lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude. Thus, for applications where runtime is not of critical importance, leveraging integer optimization can yield sparser and lower error solutions to CS than existing benchmarks.

On the longest chain of faces of the completely positive and copositive cones

We consider a wide class of closed convex cones K in the space of real n×n symmetric matrices and establish the existence of a chain of faces of K, the length of which is maximized at n(n+1)2+1. Examples of such cones include, but are not limited to, the completely positive and the copositive cones. Using this chain, we prove that the distance to polyhedrality of any closed convex cone K that is sandwiched between the completely positive cone and the doubly nonnegative cone of order n≥2, as well as its dual, is at least n(n+1)2−2, which is also the worst-case scenario.

Planning renewables in distribution system with electric vehicles to improve hosting capacity and energy losses: Two-stage stochastic optimization framework

Excessive penetration of renewables in distribution system can create several operational issues. One of the technologies that can maximize accommodation of renewables is electric vehicles (EVs). However, along with EVs additional measures are required while maximizing hosting capacity (HC) to avoid negative impact on system and account for uncertain parameters. In this regard, this paper proposes a two-stage stochastic optimization approach for planning of renewables in distribution system integrated with EVs. In the first stage, capacity of renewables is determined while maximizing HC whereas in the second stage operating strategy of EVs, renewables and grid power is evaluated to minimize expected energy losses. The proposed framework includes uncertainties in load demand, renewables and EVs characteristics through stochastic programming approach. The optimization problem is formulated as second order cone programming problem. The proposed model is tested on modified IEEE-33 bus and real-life 108-bus Indian distribution systems with different charging/discharging patterns and penetration rates of EVs. Moreover, an economic based cost analysis to compare planning cost and EV charging cost is also provided. Simulation results reveal that in 33-bus system with 500 EVs, the HC with vehicle-to-grid capability (V2G) is higher as compared to uncoordinated and coordinated charging by 29.86% and 6.36% respectively, while in 108-bus system with 1000 EVs, it is higher by 18.51% and 5.69% respectively. Further, the energy losses with V2G are 27.54% and 1.36% less in 33-bus system and 7.31% and 3.22% less in 108-bus system as compared to uncoordinated and coordinated charging respectively.

A coordinated restoration method of three-phase AC unbalanced distribution network with DC connections and mobile energy storage systems

In the rapidly changing domain of hybrid AC/DC urban distribution networks, this research unveils a groundbreaking method for the restoration of three-phase unbalanced systems by astutely harnessing the unique potential of DC line interconnections. At the heart of this innovation lies the synthesis of symmetrical Second Order Cone Programming (SOCP) with a sophisticated topology search technique, a union that offers a precise depiction of complex three-phase power flow with network restoration while simultaneously accelerating computational processes. Building upon this foundation, our approach places significant emphasis on the utilization of adaptable DC power control, coupled with the optimal deployment of mobile energy storage systems (MESSs), to ensure a harmonized power balance during critical interruptions. These strategies converge to prioritize the restoration of vital loads, especially those with high weighting factors, thereby significantly augmenting the network’s resilience, particularly in contexts vulnerable to disasters. The corroborative numerical results, as delineated in our study, highlight the distinct advantage and effectiveness of our methodology over prevailing practices in fortifying grid resilience against serious adversities.

Robust second order cone conditions and duality for multiobjective problems under uncertainty data

Robust second order cone conditions and duality for multiobjective problems under uncertainty data

Shortest path network interdiction with asymmetric uncertainty

AbstractThis paper considers an extension of the shortest path network interdiction problem that incorporates robustness to account for parameter uncertainty. The shortest path interdiction problem is a game of two players with conflicting agendas and capabilities: an evader, who traverses the arcs of a network from a source node to a sink node using a path of shortest length, and an interdictor, who maximizes the length of the evader's shortest path by interdicting arcs on the network. It is usually assumed that the parameters defining the network are known exactly by both players. We consider the situation where the evader assumes the nominal parameter values while the interdictor uses robust optimization techniques to account for parameter uncertainty or sensor degradation. We formulate this problem as a nonlinear mixed‐integer semi‐infinite bilevel program and show that it can be converted into a mixed‐integer linear program with a second order cone constraint. We use random geometric networks and transportation networks to perform computational studies and demonstrate the unique decision strategies that our variant produces. Solving the shortest path interdiction problem with asymmetric uncertainty protects the interdictor from investing in a strategy that hinges on key interdictions performing as promised. It also provides an alternate strategy that mitigates the risk of these worst‐case possibilities.

An application of Sobolev's inequality to one-dimensional Kirchhoff equations

We consider nonlocal differential equations with convolution coefficients of the form−M((a⁎(g∘|u′|))(1))u″(t)=λf(t,u(t)), t∈(0,1), in which the coefficient function M imparts a nonlocal structure to the problem. The function g satisfies p-q growth. A model case occurs when g(t):=tp, where p>1, and a(t)≡1 – i.e., the problem−M(‖u′‖Lpp)u″(t)=λf(t,u(t)), t∈(0,1). By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.

Robust and Chance-Constrained Dispatch Policies for Linear Power Systems

Several methods have recently been proposed to determine optimal feedback control policies for generator dispatch under uncertain grid in-feeds. This work presents and explains two such approaches based on a constrained linear power flow model in a unified notation, a robust method based on linear programming and a chance-constrained one based on polynomial chaos expansions and second order cone programming. This work compares both methods on the same small test example and discusses consequences for long-term grid expansion planning. While the robust method is easier to implement, it is also not more restrictive. For the chance-constrained method, we highlight a serious drawback in reformulations common in the literature. As an aside, we extend the existing formulation to cases where not all disturbances can be measured, and combine it with a robust pre-processing scheme.

Exploring Carbon Equilibrium in Integrated Electricity-Hydrogen System

Global carbon emissions from electricity and hydrogen production have grown rapidly in recent years. Some existing literature has proposed the method of reducing carbon emissions by power-to-hydrogen (P2H) technology. However, due to the fossil-based energy structure of electricity system, excessive electricity to hydrogen may lead to an increase in carbon emissions. To reduce carbon emissions effectively, a low carbon-driven optimal capacity configuration method for P2Hs is proposed in this paper to explore carbon equilibrium in integrated electricity-hydrogen system. A joint economic dispatch model for integrated electricity-hydrogen system is formulated. The electricity system model is a security-constrained economic dispatch model with P2H constraints. The hydrogen system model is a hydrogen supply chain model with pipeline pack constraints and Weymouth equation, whose sources are fossil fuel reforming and electrolytic hydrogen. We propose a second-order cone programming (SOCP)-based solution to transform the hydrogen system model into a mixed integer second order cone programming (MISCOP), which can be readily solved by commercial solvers. To preserve the internal information of each system, a decentralized algorithm based on optimality condition decomposition (OCD) is developed. Case study based on modified IEEE 30-bus and Belgium 20-node integrated energy system demonstrates the effectiveness of the proposed strategy.

An exact second order cone programming approach for traffic assignment problems

Demographic changes, urbanization and increasing vehicle ownership at unprecedented rates put a lot of strain on cities particularly on urban mobility and transportation and overwhelm transportation network infrastructures and current transportation systems, which are not built to cope with such a fast increasing demand. Traffic congestion is considered as the most difficult challenge to tackle for sustainable urban mobility and is aggravated by the increased freight activity due to e-commerce and on-demand delivery and the explosive growth in transportation network companies and ride-hailing services. There is a need to implement a combination of policies to ensure that increased urban traffic congestion does not lower the quality of life and threaten global climate and human health and to prevent further economic losses. This study aims to contribute to the United Nations (UN) climate action and sustainable development goals in tackling recurring traffic congestion problem in urban areas to achieve a sustainable urban mobility in that it offers a solution methodology for traffic assignment problem. We introduce an exact generalized solution methodology based on reformulation of existing traffic assignment problems as a second order cone programming (SOCP) problem and propose column generation (CG) and cutting plane (CP) algorithms to solve the problem effectively for large scale road network instances. We conduct numerical experiments to test the performance of the proposed algorithms on realistic road networks.

Assessing Techno-Economic and Environmental Impacts of Gas Compressor Fleet as a Source of Flexibility to the Power System

In the Great Britain's high-pressure gas transmission network, there are 24 gas compressor stations with 68 operational compressor units of different types and sizes. As of 2021, the aggregate capacity of electric-driven gas compressors was 200 MW. By using linepack as a form of gas storage buffer, the compressors can be operated flexibly to achieve a minimum operational cost under fluctuating electricity prices. Furthermore, since electricity consumed by electric-driven compressors can affect the balance of the power grid, the ability of electric-driven compressor units to regulate their electricity consumption is useful to be quantified. In this paper, the value of flexibility from the hybrid compressor stations (compressor stations in which both electric-driven and gas-driven compressor units exist) is analysed. The operation of the high-pressure gas network in Great Britain was formulated as a Mixed-Integer Second Order Cone Programming (MISOCP) problem. The modelling results show that employing the flexibility from hybrid compressor stations in the gas network can reduce more than 20% of the operational cost and 50% of emissions from the compressor fleet. Furthermore, a cost-benefit analysis was carried out to assess the investment viability of installing electric-driven compressor units. It is demonstrated that the economic feasibility of investing in electric-driven compressors is sensitive to the future carbon price.

Resilience-oriented comparative study of SOP-based service restoration in distribution systems

Soft open points (SOPs) can perform load shifting and voltage improvement, and are critical to improving distribution service restoration (DSR) during power outages. Although multiple DSR strategies utilizing local resources have been reported, their effectiveness has not been sufficiently studied. To solve this problem and utilize the flexibility of SOP, a comparative study of different strategies for SOP-based DSR is presented in this paper. First, various SOP-based DSR models, including single-source, single-island model (SSM), islanding partition model (IPM) and multi-source collaborative model (MCM), are proposed and formulated as mixed-integer nonlinear programming (MINLP) problems, incorporating static voltage characteristics of loads. Second, to enhance the solution efficiency, mixed-integer second order cone programming is introduced to convert the MINLP problems into easily solvable convex optimization problems. A restorative index is defined to evaluate the effectiveness of various SOP-based DSR strategies. Finally, a comparative simulation analysis of the proposed models on the IEEE 33-bus system illustrates that SOP-based MCM achieves an 94 % power output for distributed energy resources, while exhibiting the lowest RI for the total loads, standing at just 9.5 %. Furthermore, the RI for critical loads consistently remains at 0, demonstrating the superiority of SOP-based MCM in comparison to other strategies.

Robust Transmission Design for RIS-Assisted Secure Multiuser Communication Systems in the Presence of Hardware Impairments

This paper investigates reconfigurable intelligent surface (RIS)-assisted secure multiuser communication systems in the presence of hardware impairments (HIs) at the RIS and the transceivers. We jointly optimize the beamforming vectors at the base station (BS) and the phase shifts of the reflecting elements at the RIS so as to maximize the weighted minimum approximate ergodic secrecy rate (WMAESR), subject to the transmission power constraints at the BS and unit-modulus constraints at the RIS. To solve the formulated optimization problem, we first decouple it into two tractable subproblems and then use the block coordinate descent (BCD) method to alternately optimize the subproblems. Two different methods are proposed to solve the two obtained subproblems. The first method transforms each subproblem into a second order cone programming (SOCP) problem by invoking the penalty convexconcave procedure (CCP) method and the closed-form fractional programming (FP) criterion, and then directly solves them by using CVX. The second method leverages the minorization-maximization (MM) algorithm. Specifically, we first derive a concave approximation function, which is a lower bound of the original objective function, and then the two subproblems are transformed into two simple surrogate problems that admit closed-form solutions. Simulation results verify the performance gains of the proposed robust transmission methods over existing non-robust designs. In addition, the MM algorithm is shown to have much lower complexity than the SOCP-based algorithm.

Upper Bound Limit Analysis of 2D Structures Using Lagrange Extraction-Based Isogeometric Analysis

This work presents a new approach that uses Lagrange extraction-based isogeometric analysis and Second Order Cone Programming (SOCP) to determine the limit load factor in two-dimensional problems. This methodology makes use of an upper-bound limit approach and a rigid-perfectly plastic material model. Lagrange extraction is used to link a C0 nodal basis to a smooth B-spline basis. This enables the use of straightforward methods for isogeometric analysis in conventional finite element systems. A SOCP problem is created from the limit analysis problem, which may then be solved using Mosek optimization software. The numerical results indicate the correctness and effectiveness of the current technique by contrasting it with other approaches mentioned in the literature.

Sequential Load Restoration With Soft Open Points and Time-Dependent Cold Load Pickup for Resilient Distribution Systems

Global changes in climate conditions and increasing number of weather-related disasters in the world have elevated the quest for better load restorations in power distribution systems. This paper proposes a distribution system restoration (DSR) model where non-exponential time-dependent cold load pickup (CLPU) characteristics and soft open point operations are simultaneously addressed. We develop a time-dependent CLPU model for attaining a more realistic and secure DSR in which we demonstrate that the traditional delayed exponential model (DEM) for a long outage duration is a special case of the adopted CLPU model. The current soft open point (SOP) models would typically relax the operation constraints without devising any tightening procedures, which might lead to inaccurate or even infeasible SOP states. In this paper, we analyze the reasons traditional SOP constraints would lead to unsecure operations and present a set of novel SOP operation constraints for recovering bindings. Also, we adopt the sequential second order cone programming (SSOCP) algorithm to establish the sequential framework of DSR and solve the proposed model effectively. Our presented case studies discuss the rationality of the proposed DSR model and algorithm.