Second-order Cone Programming Research Articles

Decisions on ship route, refueling, and sailing speed considering ECA regulation and demand uncertainty

This article aims to maximize the shipping company’s profits by jointly optimizing shipping volume, refueling, ship leg option, and sailing speed with considering of stochastic shipping demand and Emission Control Area (ECA) regulation. Shipping demand is inherently stochastic and intermittent characteristic, causing formidable challenges in shipping service. A distributionally robust optimization method is adopted to handle the demand uncertainties. We build an ambiguity set of shipping demand and introduce a chance constraint with the shipping volume and shipping demand balance to guarantee the satisfactory rate of shipping demand. Moreover, the ECA regulation stimulates the diversification of transportation routes, which further increases transportation uncertainty. Thus, a nonlinear mixed integer programming model with nonconvex chance-constraint is developed. The optimal solution is obtained by transforming the original problem into an attractable second-order conic programming problem. An Asia service route is selected as a numerical experiment. The results show that the proposed model is effective, the optimal refueling strategy, leg option, speed, and shipping volume can be determined simultaneously. Meanwhile, it outperforms the state-of-the-art methods in requiring less demand information and producing fewer conservative results under demand uncertainties. Additionally, some managerial insights are revealed according to the case study.

Two-stage decision-dependent demand response driven by TCLs for distribution system resilience enhancement

Service restoration is a critical function of the self-healing distribution system to cope with extreme events. Cold load pickup (CLPU) caused by thermostatically controlled loads (TCLs) requires extra generation power to retore loads, slowing down the distribution system restoration process. Meanwhile, increasing renewable energy brings the supply-demand imbalance because of the power variability and may cause voltage or current violations. Accordingly, a two-stage decision-dependent demand response (DR) methodology is proposed to address the two barriers to distribution system resilience enhancement. Specifically, two load control strategies are developed in the first-stage DR to regulate TCLs for CLPU mitigation, which reduces the additional power requirement when re-energized. In the second-stage DR, the restored TCLs that can be controlled to provide flexibility are characterized as virtual energy storage (VES) for managing renewable energy variability. In this way, the required power generation is reduced, and the power fluctuation problem caused by renewable energy can be well addressed. It helps to speed up the load restoration and keep the distribution system operating securely. Moreover, the load profiles with CLPU mitigation control, VES parameters and VES availability time, depend on the re-energization time. In other words, load restoration decisions determine the two-stage DR. This is the most important finding of this paper and explicitly formulated in the load restoration model for accurate load characterization. The load restoration model is described by a mixed integer second-order cone programming (MISOCP) problem and solved using MATLAB 2021 with Gurobi solver. Case studies show that the proposed methodology can significantly mitigate CLPU and restore more loads. The decision-dependent behaviors of the two-stage DR are also validated.

Two-stage robust optimization of unified power quality conditioner (UPQC) siting and sizing in active distribution networks considering uncertainty of loads and renewable generators

This paper proposes a robust optimization siting and sizing strategy for the unified power quality conditioner (UPQC) in the active distribution network (ADN) integrated with renewable energy generators. The optimization objectives including investment cost, line loss, and voltage deviation are integrated into an overall cost for both the planning and operating stages. A novel UPQC equivalent power injection model is proposed based on the auxiliary branch method. A set of linear and second-order conic (SOC) constraints are employed to accurately describe the steady-state operation of UPQC. An improved branch current flow model is developed to incorporate radial distribution networks with UPQCs. The UPQC siting and sizing planning is formulated as a mixed integer second-order conic programming (MISOCP) problem. Accounting for the uncertainty in loads and distributed generators, a two-stage robust optimization model is established to address worst-case operational scenarios. A solving strategy is developed based on the strong duality theory and the column-and-constraint generation (C&CG) algorithm, and then commercial solvers can solve the problem accurately and efficiently. Case studies on an IEEE 33-node distribution network demonstrate this strategy's robustness in dealing with loads and renewable energy uncertainties, making it applicable across different UPQC topologies and operation modes. The proposed method has superior performance over the representative meta-heuristic approaches in terms of global optimality as well as an over 7.5 times faster computation speed. This work has significance in enhancing the quality and efficiency of modern power systems highly integrated with renewable energy.

Optimal operational planning of distribution systems: A neighborhood search-based matheuristic approach

This study proposes a strategy for short-term operational planning of active distribution systems to minimize operating costs and greenhouse gas (GHG) emissions. The strategy incorporates network reconfiguration, switchable capacitor bank operation, dispatch of fossil fuel-based and renewable distributed energy resources, energy storage devices, and a demand response program. Uncertain operational conditions, such as energy costs, power demand, and solar irradiation, are addressed using stochastic scenarios derived from historical data through a k-means technique. The mathematical formulation adopts a stochastic scenario-based mixed-integer second-order conic programming (MISOCP) model. To handle the computational complexity of the model, a neighborhood-based matheuristic approach (NMA) is introduced, employing reduced MISOCP models and a memory strategy to guide the optimization process. Results from 69 and 118-node distribution systems demonstrate reduced operational costs and GHG emissions. Moreover, the proposed NMA outperforms two commercial solvers. This work provides insights into optimizing the operation of distribution systems, yielding economic and environmental benefits.

A distributionally robust chance-constrained kernel-free quadratic surface support vector machine

This paper studies the problem of constructing a robust nonlinear classifier when the data set involves uncertainty and only the first- and second-order moments are known a priori. A distributionally robust chance-constrained kernel-free quadratic surface support vector machine (SVM) model is proposed using the moment information of the uncertain data. The proposed model is reformulated as a semidefinite programming problem and a second-order cone programming problem for efficient computations. A geometric interpretation of the proposed model is also provided. For commonly used data without prescribed uncertainty, a cluster-based data-driven approach is introduced to retrieve the hidden moment information that enables the proposed model for robust classification. Extensive computational experiments using synthetic and public benchmark data sets with or without uncertainty involved support the superior performance of the proposed model over other state-of-the-art SVM models, particularly when the data sets are massive and/or imbalanced.

A risk-aware and second-order cone programming model for wind expansion planning considering correlated uncertainties using the copula method

Integration of large-scale renewable energy sources (RESs) into existing power grids is challenging for power system operators and planners due to the intermittent and uncertain characteristics of RESs and the limitation of transmission capacity. This paper presents a new risk-based framework to obtain the optimal placement and sizing of wind farms in the bulk power system considering the impact of mutual correlation between wind farms’ power productions in different candidate buses. In this regard, a second-order cone programming (SOCP) optimization problem is developed to minimize the total expected social cost including the operation cost of conventional generators, wind curtailment cost, and the investment cost of newly installed wind farms. In the proposed model, the correlated uncertainties of wind power production are incorporated using the copula method. Also, the conditional value-at-risk (CVaR) measure is utilized to manage the risk of the model associated with uncertainties. The proposed planning model is then implemented into the Garver 6-bus and IEEE 118-bus test systems. Given the results, employing SOCP-based AC OPF in the presented planning model results in a precise solution, demonstrating an 87% reduction in running time. Moreover, the numerical results illustrate that by considering the correlation between wind farms’ power production and reducing the risk parameter, there is a significant increase in the installed capacity of wind farms.

A penalty-type method for solving inverse optimal value problem in second-order conic programming

This paper aims to consider a type of inverse optimal value problem in second-order conic programming, in which the parameter in its objective function needs to be adjusted under a given class that makes the corresponding optimal objective value closest to a target value. This inverse problem can be reformulated as a minimization problem with some second-order cone complementarity constraints. To tackle these bilinear constraints, we apply a penalty-type method and show that the associated penalty term is exact, which avoids the hurdles of penalty-type methods in the update strategy of the penalty parameter. Numerical results show that our method is suitable to solve the given inverse optimal value problem under different metric distances between the optimal value of the forward problem and a target objective counterpart.

Low carbon and economic optimal operation of electricity-gas integrated energy system considering demand response

Integrated Energy System is an important direction for the transformation of the power industry. IES can give full play to the complementary role of multiple energy sources and serve to achieve the industry's energy saving and emission reduction goals. Firstly, this paper establishes a P2G refined operation model considering two stages of hydrogen production from electrolytic water and hydrogen methanation. Secondly, based on the analysis of the IES load characteristics, the multi-energy loads are divided into longitudinal demand responses where the same energy type can be transferred and reduced in time, and horizontal demand responses where different energy types can be substituted for each other, and the corresponding mathematical models are established. Finally, the reward and penalty ladder carbon trading mechanism are used to quantify the carbon trading cost, and the IES low-carbon economic optimization model integrating the Power-to-gas refinement model and demand response is constructed. The model was subjected to second-order cone programming relaxation and incremental segmented linearization methods before being solved using the CPLEX commercial solver. Simulation analyses were performed in an IES consisting of an IEEE 33-node distribution network and a 24-node gas distribution network. The results show that considering demand response can fully exploit the flexible coordination performance of the load side, resulting in a 12.6% reduction in total system cost and a 38.8% reduction in carbon emissions.

Deformation Method with Constraints on Normal Curvature for Wheeled Robots in Precision Agriculture Based on Second-Order Cone Programming

Вточном земледелии является актуальной задача построения путей сельскохозяйственных роботов, покрывающих трехмерный участок ландшафта. Если для их реализации используются колесные роботы с рулением поворотом передних колес, то нормальная кривизна траекторий должна быть ограничена некоторой величиной, определяемой характеристиками этих машин. Вработе рассмотрен метод деформации представленных однородными кубическими B-сплайнами путей для учета препятствий. Предложена оптимизационная задача, позволяющая производить расчет путей с минимизацией пропусков в покрытии. Данная задача является выпуклой и принадлежит к классу конического программирования второго порядка, что обуславливает возможность ее вычислительно эффективного решения. Приведены примеры вычислений.

Path Deformation Method with Constraints on Normal Curvature for Wheeled Robots in Precision Agriculture Based on Second-Order Cone Programming

Path Deformation Method with Constraints on Normal Curvature for Wheeled Robots in Precision Agriculture Based on Second-Order Cone Programming

Path Deformation Method with Constraints on Normal Curvature for Wheeled Robots in Precision Agriculture Based on Second-Order Cone Programming

In precision agriculture, path planning for agricultural robots with complete covering a three-dimensional landscape is an essential task. For robots with front wheels steering the normal curvature of the trajectories should be limited to some value determined by the characteristics of the vehicle. The paper considers a method of deformation of these paths to account for obstacles for trajectories described by homogeneous cubic B-splines. We propose an optimization problem that allows calculating paths with minimizing skips in the coverage. The considered problem is convex and belongs to the class of second-order cone programming, which entails the possibility of its computationally efficient solution. The computational examples are presented.

Integrated Interactive Control of Distribution Systems with Multi-Building Microgrids Based on Game Theory

In the transactional processes within a multi-building microgrid system, it is imperative to safeguard stakeholders’ interests and ensure stable, economically efficient operation. Therefore, this paper proposes an integrated interactive control of distribution systems with multi-building microgrids based on game theory. Initially, an interactive framework encompassing superior power grids, distribution network operators, and multi-building microgrids is proposed. This framework establishes a master–slave game model between distribution network operators and multi-building microgrids. Additionally, it introduces the concept of Soft Open Points between building microgrids to enhance system operational safety while also reducing economic costs for distribution network operators. Subsequently, to maintain solution accuracy and optimality, it employs linearization and cone relaxation methods to transform the original model into a mixed-integer second-order cone programming model. Furthermore, it enhances an iterative search method and the price mechanism based on supply and demand ratios. The revised price mechanism serves to boost the participation of building microgrid users in energy regulation, while the iterative search method effectively resolves the equilibrium point of interest among all game participants. Finally, a simulation analysis is conducted on an IEEE-33 test system, and the effectiveness of the proposed strategy is verified by comparing three schemes.

On the 2D Beampattern Optimization of Sparse Group-Constrained Robust Capon Beamforming with Conformal Arrays

To overcome the problems of the high sidelobe levels and low computational efficiency of traditional Capon-based beamformers in optimizing the two-dimensional (elevation–azimuth) beampatterns of conformal arrays, in this paper, we propose a robust Capon beamforming method with sparse group constraints that is solved using the alternating-direction method of multipliers (ADMM). A robustness constraint based on worst-case performance optimization (WCPO) is imposed on the standard Capon beamformer (SCB) and then the sparse group constraints are applied to reduce the sidelobe level. The constraints are two sparsity constraints: the group one and the individual one. The former was developed to exploit the sparsity between groups based on the fact that the sidelobe can be divided into several different groups according to spatial regions in two-dimensional beampatterns, rather than different individual points in one-dimensional (azimuth-only) beampatterns. The latter is considered to emphasize the sparsity within groups. To solve the optimization problem, we introduce the ADMM to obtain the closed-form solution iteratively, which requires less computational complexity than the existing methods, such as second-order cone programming (SOCP). Numerical examples show that the proposed method can achieve flexible sidelobe-level control, and it is still effective in the case of steering vector mismatch.

Combined bearing capacity of footings on geogrid-reinforced granular fill over soft clay

The current study investigates the ultimate bearing capacity of obliquely/eccentrically loaded shallow strip foundations resting on a geogrid-reinforced granular fill with limited thickness over a very soft to soft clay layer. To this end, the lower bound theorems of the finite element limit analysis (FELA) and second-order cone programming (SOCP) are effectively exploited to simulate the underlying clay deposit, geogrid layer, and granular fill along with the inclined/eccentric loading exerted on the overlying footing. The accuracy of the adopted formulations is rigorously examined through several comparisons with the results of a well-established analytical approach in the literature. A comprehensive parametric study is then conducted to properly examine the influences of the geosynthetic layer characteristics and soft clay properties on the overall bearing capacity and failure envelope of the strip footing subjected to wide ranges of inclined and eccentric loadings. The results show that placement of a geogrid-reinforced granular fill layer over the soft clayey soil significantly increases the bearing capacity of the shallow foundation. The ultimate tensile strength of the reinforcement layer and the cohesion of the underlying very soft to soft clay deposit are also observed to have considerable effects on the size of failure envelopes.

Second-order cone programming for frictional contact mechanics using interior point algorithm

We report experiments of an implementation of a primal–dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high accuracy and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modelling and simulating non-smooth dynamical systems. Currently, Siconos uses mainly first-order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too slow to find accurate solutions in a reasonable time. Our main objective is to apply second-order methods to speed up convergence. In this paper, we focus on solving a relaxed formulation of the initial mechanical model, corresponding to the optimality conditions of a convex quadratic minimization problem with second-order cone constraints. We will present in detail a primal–dual interior point algorithm for solving this type of problem. The main difficulty in implementing this algorithm arises from the fact that at each iteration of the algorithm, a change of variable, called a scaling, has to be performed in order to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. Although this scaling strategy is very nice from a theoretical point of view, it leads to an enormous deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will describe in detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems of rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.

CASTELO: Convex Approximation based Solution To Elliptic Localization with Outliers

This short communication considers mitigating the negative effects of possibly unreliable path delay measurements acquired in non-line-of-sight (NLOS) environments on the positioning performance, a problem deserving further investigation within the expanding research area of elliptic localization. We present CASTELO, a Convex Approximation based Solution To Elliptic Localization with Outliers, to achieve such a goal. Our proposal corresponds to a mixed semidefinite (SD)/second-order cone (SOC) programming formulation derived from an error-mitigated nonlinear least squares (LS) location estimator, presenting itself as a remedy for the neglect of positivity of NLOS biases suffered by the majority of currently fashionable outlier-handling approaches. In terms of analytical discussions, we provide rationales supporting the incorporation of the SOC constraints, which serve to tighten the problem obtained after SD relaxation, and conduct a complexity analysis for the ultimate mixed SD/SOC programming formulation. Simulations are carried out to confirm the strong ability of CASTELO to attain reliable elliptic localization in the presence of NLOS outliers.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

A temporal and spatial electric vehicle charging optimization scheme with DSO-EVA coordination framework

Burdens of increasing penetration of electric vehicles (EVs) on distribution systems have attracted wide attention to the research of EV charging coordination. Nevertheless, existing coordination lacks the joint consideration of the temporal-spatial flexibility of EVs, which could reduce load fluctuations when meeting the EV charging demands. Moreover, a thorough investigation is required for the coordination of the benefits to the distribution system operator (DSO), EV aggregator (EVA), and EVs. In this paper, a novel EV charging scheme considering the temporal-spatial features of EVs is proposed, achieving the coordination of DSO, EVA, and EVs in two steps. First, a mixed-integer second-order cone programming (MISOCP)-based EV charging schedule is developed for DSO. Second, the temporal-spatial features of EVs are exploited by EVA to track the charging schedule. Specifically, a dynamic charging price mechanism based on active power margin is proposed for public charging facilities and a specific load regulation is designed for private charging facilities. On this basis, the modified adaptive multi-objective particle swarm optimization (AMOPSO) algorithm is proposed, including adaptive flight parameter adjustment and termination mechanisms. Case studies demonstrate the proposed strategy can attenuate load variance and raise EVA revenue. Further, the impact analysis of penalty price and the price elasticity of electricity demand can provide references for stable distribution network operation, higher EVA revenue, and charging cost reduction.

Evaluation of the energy performance of a shallow geothermal facility through a second‐order cone programming

AbstractEnergy consumption and operational cost are two decision‐making factors which evaluate the feasibility of any energy system. Whether a thermal unit is efficiently working or not depends on the state properties of the thermodynamic system. Therefore, operational research on a geothermal site was performed. Most research pivots around the basic energy conservation equation to determine the parameters. However, finding the parameters does not serve the purpose of an investor. Investment of capital in any project depends on the return and feasibility of the task performed under the project. In the same direction, this article focuses on the optimisation of the performance of a shallow geothermal site in terms of energy and plant economics using the concept of a second‐order cone program (SOCP). The site was used to operate the hybrid heating, ventilation, and air conditioning unit (HVAC) installed in an academic building. The second‐order cone programming was considered to minimize the energy requirement and cost of energy generation during the winter season. It is to be noted that the solver‐based approach was considered to examine the unit. The optimized energy consumption of GSHP was 24.29 kW⋅h. The levelized cost of energy (LCOE/kW⋅h) derived from SOCP would vary from $ 0.38 to $ 0.58 for the shallow geothermal plant. The rate of return for air conditioning in the building through the geothermal plant would be 1.74%. The optimized upfront cost per m2 was estimated to be $ 1440.87.

Security region of inverter-interfaced power systems: Existence, expansion, and application

Since the increasing integration of inverter-based resources leads to a transition to inverter-interfaced power systems, the additional physical limits of inverters raise concerns for secure operation, which needs to be not only ensured but also enhanced. This study, therefore, investigates how to geometrically ensure and enlarge the security region of single-inverter-interfaced power systems, followed by establishing the inverter-security-constrained optimal power flow for multi-inverter-interfaced power systems. Therein, the inverter over-current and over-modulation/over-voltage limits are visualized by projecting cones into circles, ellipses, and a quadratic curve in different control variable spaces. First, for single-inverter-interfaced power systems, this study proposes a closed-form sufficient condition that guarantees the security region existence under different control modes. Then, this study quantifies and enlarges the security region size by leveraging the counter-intuitive complementary impacts of inverter parameters and the opposite impacts of system parameters. As an extension, an inverter-security-constrained optimal power flow using second-order cone programming relaxation is formulated for multi-inverter-interfaced power systems, in which this study proposes a closed-form sufficient condition to ensure a non-empty and non-decreasing security region. Numerical results of a single-inverter-interfaced power system and a modified IEEE 14 bus multi-inverter-interfaced power system verify the proposed security region properties.