Second-order Cone Programming Formulation Research Articles

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.

Optimal Dispatch of DERs and Battery-Based ESS in Distribution Grids While Considering Reactive Power Capabilities and Uncertainties: A Second-Order Cone Programming Formulation

Optimal Dispatch of DERs and Battery-Based ESS in Distribution Grids While Considering Reactive Power Capabilities and Uncertainties: A Second-Order Cone Programming Formulation

Multi-resource spatio-temporal coordinated voltage regulation for active distribution network with multiple integrated energy stations under uncertainties

Integrated energy stations (IESs) are widely spread in power-electronized active distribution network (ADN) for multi-energy services, and heterogeneous uncertainties from source-loads exacerbate voltage violation risks with increasing penetration of distributed generators (DGs). Existing research has not explicitly revealed the spatio-temporal synergy adaptability of multiple response-otherness resources. This paper proposes a bi-level mixed-scale strategy to leverage multiple resources for voltage improvement under uncertainties. Upper level performs model predictive control (MPC)-based rolling for optimizing large-scale reserve-integrated coordination schemes of soft open points (SOPs) and thermal storage/demand; and reserve margins for building temperature are constrained for the sake of unpredictable environmental factors. In lower level, mixed-scale adjustment is applied in which ADN executes re-generated small-scale schemes and IESs obey large-scale schemes of upper level. SOPs’ rapid response abilities are invoked to tackle real-time errors while building temperatures vary within comfort range for responding to insensitive environmental fluctuations. Via sufficient flexibility assurance and mixed-scale coordination in separated levels, the voltage level is effectively facilitated. The original nonconvex model is relaxed into a second-order cone programming (SOCP) formulation, which can be effectively tackled for timely voltage regulation. Numerical results show the proposed strategy can realize spatio-temporal regulation of multiple resources for adapting to different-scale uncertainties, mitigating voltage deviation significantly.

Distributionally robust joint chance-constrained programming: Wasserstein metric and second-order moment constraints

In this paper, we propose a new approximate linear reformulation for distributionally robust joint chance programming with Wasserstein ambiguity sets and an efficient solution approach based on Benders decomposition. To provide a convex approximation to the distributionally robust chance constraint, we use the worst-case conditional value-at-risk constrained program. In addition, we derive an approach for distributionally robust joint chance programming with a hybrid ambiguity set that combines a Wasserstein ball with second-order moment constraints. This approach, which allows injecting domain knowledge into a Wasserstein ambiguity set and thus allows for less conservative solutions, has not been considered before. We propose two formulations of this problem, namely a semidefinite programming and a computationally favorable second-order cone programming formulation. The models and algorithms proposed in this paper are evaluated through computational experiments demonstrating their computational efficiency. In particular, the Benders decomposition algorithm is shown to be more than an order of magnitude faster than a standard solver allowing for the solution of large instances in a relatively short time.

An interval power flow method for radial distribution systems based on hybrid second-order cone and linear programming

Interval power flow (IPF) analysis is a promising approach to handling the uncertainty of renewable energy sources and loads in power systems. However, most existing studies are based on standard interval arithmetic or affine arithmetic. Little work can be found that employs traditional optimization methods to obtain the interval results, especially based on convexified power flow formulation. Therefore, this paper proposes an IPF method based on a hybrid second-order cone and linear programming for radial distribution systems. First, the optimization process of IPF based on the standard power flow model is introduced. Given the non-convexity of the standard power flow model, an IPF framework based on second-order cone programming (SOCP) for radial distribution systems is proposed. Moreover, considering that the SOCP formulation can only achieve half of the whole IPF process, a revised linear DistFlow formulation is adopted, and the IPF method based on a hybrid second-order cone and linear programming is devised. Finally, the proposed IPF method is compared with the standard power flow equation-based IPF method and the Monte Carlo method. The simulation results on modified IEEE 33-node and 69-node distribution systems demonstrate the effectiveness and prospects of the proposed IPF method.

Hub Network Design Problem with Capacity, Congestion, and Stochastic Demand Considerations

Our study introduces the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decisions and congestion cost effect into the problem and allowing dynamic routing for origin-destination (OD) pairs. Connecting strategic and operational level decisions, HNDC jointly decides hub locations and capacity acquisitions by considering the expected routing and congestion costs. A path-based mixed-integer second-order cone programming (SOCP) formulation of the HNDC is proposed. We exploit SOCP duality results and propose an exact algorithm based on Benders decomposition and column generation to solve this challenging problem. We use a specific characterization of the capacity-feasible solutions to speed up the solution procedure and develop an efficient branch-and-cut algorithm to solve the master problem. We conduct extensive computational experiments to test the proposed approach’s performance and derive managerial insights based on realistic problem instances adapted from the literature. In particular, we found that including hub congestion costs, accounting for the uncertainty in demand, and whether the underlying network is complete or incomplete have a significant impact on hub network design and the resulting performance of the system. Funding: This work was supported by Türkiye Bilimsel ve Teknolojik Araştırma Kurumu [Grant 218M520]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.0112 .

Derivative-based SINDy (DSINDy): Addressing the challenge of discovering governing equations from noisy data

Recent advances in the field of data-driven dynamics allow for the discovery of ODE systems using state measurements. One approach, known as Sparse Identification of Nonlinear Dynamics (SINDy), assumes the dynamics are sparse within a predetermined basis in the states and finds the expansion coefficients through linear regression with sparsity constraints. This approach requires an accurate estimation of the state time derivatives, which is not necessarily possible in the high-noise regime without additional constraints. We present an approach called Derivative-based SINDy (DSINDy) that combines two novel methods to improve ODE recovery at high-noise levels. First, we denoise the state variables by applying a projection operator that leverages the assumed basis for the system dynamics. Second, we use a second order cone program (SOCP) to find the derivative and governing equations simultaneously. We derive theoretical results for the projection-based denoising step, which allow us to estimate the values of hyperparameters used in the SOCP formulation. This underlying theory helps limit the number of required user-specified parameters. We present results demonstrating that our approach leads to improved system recovery for the Van der Pol oscillator, the Duffing oscillator, the Rössler attractor, and the Lorenz 96 model.

Mixed-Integer Conic Formulation of Unit Commitment with Stochastic Wind Power

Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, the chance-constrained programming (CCP) method is used to study the UC problem with uncertainty wind power generation, and chance constraints such as power balance and spinning reserve are satisfied with a predetermined probability. In order to effectively solve the CCP problem, first, we used the sample average approximation (SAA) method to transform the chance constraints into deterministic constraints and to obtain a mixed-integer quadratic programming (MIQP) model. Then, the quadratic terms were incorporated into the constraints by introducing some auxiliary variables, and some second-order cone constraints were formed by combining them with the output characteristics of thermal unit; therefore, a tighter mixed-integer second-order cone programming (MISOCP) formulation was obtained. Finally, we applied this method to some systems including 10 to 100 thermal units and 1 to 2 wind units, and we invoked MOSEK in MATLAB to solve the MISOCP formulation. The numerical results obtained within 24 h confirm that not only is the MISOCP formulation a successful reformulation that can achieve better suboptimal solutions, but it is also a suitable method for solving the large-scale uncertain UC problem. In addition, for systems of up to 40 units within 24 h that do not consider wind power and pollution emissions, the numerical results were compared with those of previously published methods, showing that the MISOCP formulation is very promising, given its excellent performance.

Economic-environmental energy supply of mobile base stations in isolated nanogrids with smart plug-in electric vehicles and hydrogen energy storage system

The mobile base stations (MBS) are fundamental communication devices that ensure the constant stream of interconnectivity. However, they are mostly installed in off-grid regions. This study investigates the economic-environmental energy supply of a MBS in an isolated nanogrid (ING) that also includes a hydrogen energy storage system (HES), photovoltaic (PV) system, controllable plug-in electric vehicles (PEV) and a diesel generator (DG). A novel mixed-integer second-order cone programming (MISOCP) formulation is proposed to capture the nonlinearities of the various components through a convex optimization model. The study included different uncertainties including the traffic rate of the MBS, driving schedule of the PEVs, and PV generation via a hybrid stochastic programming (SP) and robust optimization (RO) methods. The influence of the coordinated PEV charging strategy, risk-averse RO and multi-objective optimization was studied through various case studies. The outcomes show that coordinated PEV-charging can have a significant contribution in reducing the risks and curtailing both cost and emission objective functions, while using economic-environmental operation model can cut the emissions by 17.70%.

Asymptotically tight conic approximations for chance-constrained AC optimal power flow

The increasing penetration of renewable energy in power systems calls for secure and reliable system operations under significant uncertainty. To that end, the chance-constrained AC optimal power flow (CC-ACOPF) problem has been proposed. Most research in the literature of CC-ACOPF focuses on one-sided chance constraints; however, two-sided chance constraints (TCCs), albeit more complex, provide more accurate formulations as both upper and lower bounds of the chance constraints are enforced simultaneously. In this paper, we introduce a fully two-sided CC-ACOPF problem (TCC-ACOPF), in which the active/reactive generation, voltage, and power flow all remain within their upper/lower bounds simultaneously with a predefined probability. Instead of applying Bonferroni approximation or scenario-based approaches, we present an efficient second-order cone programming (SOCP) approximation of the TCCs under Gaussian Mixture (GM) distribution via a piecewise linear (PWL) approximation. Compared to the conventional normality assumption for forecast errors, the GM distribution adds an extra level of accuracy representing the uncertainties. Moreover, we show that our SOCP formulation has adjustable rates of accuracy and its optimal value enjoys asymptotic convergence properties. Furthermore, an algorithm is proposed to speed up the solution procedure by optimally selecting the PWL segments. Finally, we demonstrate the effectiveness of our proposed approaches with both real historical data and synthetic data on the IEEE 30-bus and 118-bus systems. We show that our formulations provide significantly more robust solutions (about 60% reduction in constraint violation) compared to other state-of-art ACOPF formulations.

Multi-stage resilience scheduling of electricity-gas integrated energy system with multi-level decentralized reserve

Extreme events pose threats to secure and consistent supply of electricity-gas integrated energy system (EGIES), and adequate resilience is essential for improving risk defense abilities. Considering the constraints of power distribution network (PDN) and natural gas system (NGS), a multi-stage strategy integrating multi-level decentralized reserve is proposed for resilience enhancement in EGIES. Storages of multi-area community integrated energy systems (CIESs) serve as the secondary reserve while the primary reserve, i.e., independent electric power/natural gas system, fails to meet reserve requirements. Furthermore, the thermal storage of building air on the consumption side is taken as the tertiary reserve, so as to obtain a better emergency response effect. The scheduling framework consists of reserve calculation, economic scheduling, and fault restoration. On the basis of multi-area reserve calculation, day-ahead economic scheduling is conducted, maintaining adequate reserve with consideration of decentralized reserve constraints; thus, high-priority demands can be recovered in case of supply disruptions. In addition, the flexible topology of PDN is utilized to facilitate the restored loads by optimal islanding partition. Via conic relaxation conversion, the original nonconvex model is tracked into a unified mixed-integer second-order cone programming (MISOCP) formulation, which can be effectively and accurately solved. Finally, numerical study on a modified IEEE 33-bus test system and a modified 7-node gas system connecting multiple community integrated energy systems shows the effectiveness of the proposed resilience strategy.

Improving Patient Transfer Protocols for Regional Stroke Networks

Currently, stroke patients are transported to the nearest stroke center, following specific protocols. Yet, these protocols do not consider many factors, including the spatial variation in population density, the stroke’s severity, the time since stroke onset, and the congestion level at the receiving stroke center. We develop an analytical framework that enriches the stroke transport decision-making process by incorporating these factors. Our research contributes to the literature of stroke care systems by (i) developing the first analytical framework to determine the optimal primary hospital destination in a regional stroke network and (ii) comparing the impact of incorporating prehospital triaging on health outcomes. To this end, we develop an efficient reformulation for allocation problems with stochastic demand and multiserver system under congestion. We derive data-driven outcome prediction models embedded in mixed integer second-order cone programming formulation. Our framework is applied to two real-life cases: Montreal and Quebec City Stroke Networks. We show that adopting a triage strategy could lead to significantly improved health outcomes, where the magnitude of these improvements varies with the networks’ sizes and congestion levels. In the Montreal case, our proposed policy may increase the ratio of patients for therapeutic intervention eligibility by 12.5% while improving by 69% the number of patients with more than two days of emergency department boarding delays. Our results reveal that it is important to consider the network’s characteristics in making a decision for or against implementing a prehospital triage strategy. Finally, we propose a heuristic policy that provides a promising performance while also being easy to implement. This paper was accepted by Stefan Scholtes, healthcare management.

Convex Reformulation for Two-sided Distributionally Robust Chance Constraints with Inexact Moment Information

Constraints on each node and line in power systems generally have upper and lower bounds, denoted as two-sided constraints. Most existing power system optimization methods with the distributionally robust (DR) chance-constrained program treat the two-sided DR chance constraint separately, which is an inexact approximation. This letter derives an equivalent reformulation for the generic two-sided DR chance constraint under the interval moment based ambiguity set, which does not require the exact moment information. The derived reformulation is a second-order cone program (SOCP) formulation and is then applied to the optimal power flow (OPF) problem under uncertainty. Numerical results on several IEEE systems demonstrate the effectiveness of the proposed SOCP formulation and show the differences with other DR chance-constrained OPF approaches.

Distribution Grid Optimal Power Flow (D-OPF): Modeling, Analysis, and Benchmarking

In the power distribution systems, optimal power flow (D-OPF) is formulated as a non-convex and non-linear programming (NLP) problem. Convex relaxation and linear approximation models have been increasingly adopted to achieve computational efficiency for D-OPF. Despite the benefits of scalability and global optimality, each method is based on certain assumptions, performs differently, and may lead to solutions that are physically not meaningful. In this context, this work numerically evaluates the relative performance of second-order cone programming (SOCP), semi-definite programming (SDP), and linear programming (LP) formulations of D-OPF in terms of their feasibility, optimality, and scalability with respect to NLP-based formulations. We also compare the bus injection (in bus voltage and current variables) and branch flow (in active and reactive power flow variables) based on NLP formulations. The performance is evaluated using small (123-node), medium (730-node), and large (2522-node) sized distribution feeders. Case studies, which are backed up by visualization of the analytical models for the solution space to the extent possible, show that (1) the feasibility and exactness of relaxed D-OPF formulations depend upon the problem type, (2) some NLP formulations are computationally more tractable than others, (3) different NLP formulations can converge to different local solutions, and (4) the approximate linear model may underestimate or overestimate the cost function (depending upon the problem type) and may lead to AC-infeasible solutions.

Long-Term Voltage Stability-Constrained Coordinated Scheduling for Gas and Power Grids With Uncertain Wind Power

Considering the increased trend that power systems are closer to the operating bounds because of the increased demand and new challenges in consideration of gas systems and wind power, this paper investigates long-term voltage stability-constrained integrated electric and gas system optimal scheduling in consideration of wind energy integration. A sufficient condition, which is represented as an explicit function of voltage and injected power, is used to constrain power system long-term voltage stability. Due to bilinear terms in this condition, tightening piecewise McCormick envelope relaxation is used to convert it into convex constraints. The second-order cone programming (SOCP) formulation is employed to represent the operational constraints of the integrated electric and gas system. The loss of wind power probability, representing wind power uncertainties, is established by a chance-constrained programming model, which is transformed into a deterministic optimization model by means of the star-inequality-based extended formulation of sample average approximation. Two test systems, the 9-bus electric system with the 6-node gas system and the IEEE 118-bus electric system with the 40-node gas system, are used to validate the proposed model.

Efficient kinematic upper-bound limit analysis for hole/inclusion problems by adaptive XIGA with locally refined NURBS

Limit analysis can be used to directly calculate the ultimate load of structures without considering cumbersome elastic–plastic analysis. We present an adaptive mesh refinement strategy by mean of extended isogeometric analysis (XIGA) in association with second-order cone programming (SOCP) for kinematic limit analysis of hole and inclusion problems. The location of hole and inclusion can be captured by level set function without regard for its interfaces. The local refined non-uniform rational B-splines (LR NURBS), allowing local refinement and modeling complex geometries, are selected as basis functions. The structured mesh refinement strategy is exploited to carry out local refinement on the basis of the indicator of L2-norm of plastic strain rates. The adaptive local refinement can detect the local plastic deformation regions, and further reveals the possible failure mechanisms. The discrete kinematic formulation of limit analysis can be rewritten as the form of SOCP, and further is solved efficiently with the help of the Mosek toolbox. All the desirable features of the present approach are illustrated through numerical experiments.

A privacy‐preserving approach to day‐ahead TSO‐DSO coordinated stochastic scheduling for energy and reserve

Proliferation of distributed energy resources (DERs) calls for a coordinated transmission and distribution (T&D) scheduling at the day-ahead stage. The problem becomes more complicated dealing with the variability of stochastic parameters. Also, privacy and complexity are two barriers to the development of such coordinated platforms. This paper addresses these issues by introducing a hybrid centrally-supported decentralized stochastic framework for the day-ahead energy and reserve market with minimum complexity and the need for data-sharing between system operators. The proposed model is able to calculate the bidirectional power exchange at the T&D interface and the separated costs, dispatches, and reserves of all market participants. The proposed model does not consider any priority for operators and increases the liquidity by facilitating participants’ access to the market platform. Also, the second-order cone programming (SOCP) formulation is used for calculating the AC power flow of distribution grids, and the model is validated and compared with other implementation strategies. The proposed model is implemented on a modified IEEE 24-bus test system, and results show that the model can schedule resources for supplying energy and reserves in both transmission and distribution levels in an acceptable computation time.

Value of integrated electricity and heat scheduling with considering TSO–DSO cooperation

The active role of distribution system operators (DSOs) coordinated with the transmission system operator (TSO) is highlighted by increasing the competition of new parties in energy markets. Besides, combined heat and power (CHP) units are the primary heat supplier in district heating systems, and their electricity production is strongly coupled with heat productions. This paper evaluates an integrated model for scheduling electricity and heat with considering TSO–DSO cooperation to increase the operating efficiency in the day-ahead horizon. Also, this paper considers the cooperation of the electrical TSO, electrical DSOs, and district heating systems’ operators. The proposed model facilitates energy transactions between systems by considering intermediary variables. The share of each market party is calculated using intermediary variables and locational marginal prices of electrical and heating systems, and the values are compared to the cases with the isolated operating of energy systems. Also, the model considers thermal energy storage systems (TESs) and the heat transaction capability between neighbor systems, and the feasible convex region is used for the operation of CHP units. The DC power flow equations are used at the transmission level, while the AC power flow is used for distribution grids. The AC power flow equations are relaxed into the second-order cone programming (SOCP) formulation, which results in a mixed-integer second-order cone programming (MISOCP) problem. The proposed model is applied to the modified IEEE 24-bus test system, which contains electrical and heating systems at the distribution level. The result shows that the proposed model successfully reduces the operational costs and energy prices compared to the isolated scheduling of energy systems. Also, the model facilitates energy trading between market parties.

Removing inessential points in [formula omitted]-and [formula omitted]-optimal design

A design point is inessential when it does not contribute to an optimal design, and can therefore be safely discarded from the design space. We derive three inequalities for the detection of such inessential points in c-optimal design: the first two are direct consequences of the equivalence theorem for c-optimality; the third one is derived from a second-order cone programming formulation of c-optimal design. Elimination rules for A-optimal design are obtained as a byproduct. When implemented within an optimization algorithm, each inequality gives a screening test that may provide a substantial acceleration by reducing the size of the problem online. Several examples are presented with a multiplicative algorithm to illustrate the effectiveness of the approach.

Adaptive extended isogeometric upper-bound limit analysis of cracked structures

Limit analysis, without the complicated elasto-plastic computation, is an efficient method for estimating safety load of engineering structures. This paper develops a novel computational approach by integrating second-order cone programming (SOCP) into adaptive extended isogemetric elements (XIGA) for upper-bound limit analysis of cracked structures. The advantage of XIGA is to model cracks without considering the location of crack faces by introducing enrichment functions. The local refined (LR) B-splines, which have versatile and flexible local refinement ability, are adopted as basis functions in the XIGA. We use structured mesh refinement strategy to implement local refinement based on the indicator of L2-norm of plastic strain rates. Cracked structures are assumed under plane stress condition, and the von Mises yield criterion is used. Kinematic formulation of the limit analysis is translated into the form of SOCP, then is solved by the Mosek tool. The developed model is implemented and its accuracy and effectiveness are illustrated through several numerical examples. In addition, numerical results illustrate that the convergence rate of adaptive XIGA is faster than that of traditional XIGA.