Second-order Cone Programming Research Articles

Pilot and data power optimization problems in multiple Aerial Relay Stations cell‐free communication systems

SummaryA cell‐free (CF) technology and unmanned aerial vehicles (UAVs) acting as aerial relay stations (ARSs) are gradually viewed as viable technologies for usage in sixth‐generation (6G) wireless networks owing to their outstanding advantages, such as large coverage, uniform service quality, huge connections, flexible deployment in all geographical areas, and high line‐of‐sight (LoS) probability. As a result, the combination of the CF model and UAVs is suitable for densely populated urban areas, shopping malls, festivals, and locations with complex geography. Furthermore, the integration of the CF model and UAV communication can improve system quality and is considered an entirely new research direction as there is no comprehensive investigation of this combined model. In this study, we investigate the quality of the multi‐ARS CF communication system, where ARSs are equipped with multiple antennas and stochastic distribution within a specific region to simultaneously serve numerous ground users. We establish a closed‐form formulation for the uplink and downlink throughput of individual users by using the matched filtering method and conjugate beamforming technology, respectively. Moreover, we propose a novel method to optimize the pilot and data transmission power coefficients for improving channel estimation and increasing throughput per user, thereby ensuring a high quality of service. The power optimization method is executed via the successive convex approximation (SCA) method, second‐order cone program (SOCP), and linear program. We evaluate system performance according to the cumulative distribution function (CDF) of user throughput based on various parameters, such as the number of users, the number of ARSs, and the length of pilot sequences. The analytical findings also reveal that the system with the proposed power optimization method is always superior to the system without the proposed method in both uplink and downlink.

Dealing with Outliers in Wireless Sensor Networks Localization: An Iterative and Selection-Minimization Strategy

AbstractIn recent years, there has been considerable interest in robust range-based Wireless Sensor Network (WSN) localization due to the increasing importance of accurately locating sensors in various WSN applications. However, achieving precise localization is often hampered by the presence of outliers or underestimations in range measurements, particularly when employing the RSS technique. To tackle these issues, we introduce a Two-Step Localization (namely, the SelMin approach). In the initial phase, the approach utilizes Second-Order Cone Programming (SOCP) to minimize distance discrepancies. It does this by comparing a reference Euclidean Distance Matrix (EDM) with a weighted one derived from imprecise distances between sensor nodes. In the subsequent phase, a heuristic method is employed to identify a specific number of imprecise distances, referred to as outliers, that will be disregarded in the first phase, and this two-phase process continues iteratively. The experimental results demonstrate that the SelMin strategy performs better than the DSCL method when evaluated using the Root Mean Square Error (RMSE) metric. This superior performance is maintained even in challenging conditions, such as when there are many outliers (i.e, around 30$$\%$$ % ) in the network. This indicates that SelMin is a reliable and robust choice for these environments.

Convex Representation of Metabolic Networks with Michaelis–Menten Kinetics

Polyhedral models of metabolic networks are computationally tractable and can predict some cellular functions. A longstanding challenge is incorporating metabolites without losing tractability. In this paper, we do so using a new second-order cone representation of the Michaelis–Menten kinetics. The resulting model consists of linear stoichiometric constraints alongside second-order cone constraints that couple the reaction fluxes to metabolite concentrations. We formulate several new problems around this model: conic flux balance analysis, which augments flux balance analysis with metabolite concentrations; dynamic conic flux balance analysis; and finding minimal cut sets of networks with both reactions and metabolites. Solving these problems yields information about both fluxes and metabolite concentrations. They are second-order cone or mixed-integer second-order cone programs, which, while not as tractable as their linear counterparts, can nonetheless be solved at practical scales using existing software.

Towards low-carbon power networks: Optimal location and sizing of renewable energy sources and hydrogen storage

This paper proposes a systematic optimization framework to jointly determine the optimal location and sizing decisions of renewables and hydrogen storage in a power network to achieve the transition to low-carbon networks efficiently. We obtain these strategic decisions based on the multi-period alternating current optimal power flow (AC MOPF) problem that jointly analyzes power network, renewable, and hydrogen storage interactions at the operational level by considering the uncertainty of renewable output, seasonality of electricity demand, and electricity prices. We develop a tailored solution approach based on second-order cone programming within a Benders decomposition framework to provide globally optimal solutions. In a test case, we show that the joint integration of renewable sources and hydrogen storage and consideration of the AC MOPF model significantly reduces the operational cost of the power network. In turn, our findings can provide quantitative insights to decision-makers on how to integrate renewable sources and hydrogen storage under different settings of the hydrogen selling price, renewable curtailment cost, emission tax price, and conversion efficiency.

Resilience-oriented repair crew and network reconfiguration coordinated operational scheduling for post-event restoration

This paper introduces a post-disaster load restoration approach for the distribution grid, utilizing network reconfiguration (NR) and dispatching of repair crews (RCs) to significantly enhance grid resilience. We propose an RC–NR coordinated model that leverages diverse flexible resources within the active distribution network (ADN), aimed at not only enhancing the grid’s resilience level but also efficiently mending the fault lines. The model introduces fault repairing and sequential NR coupled constraints to devise an optimal resilience strategy within temporal domain cooperation, focusing on minimizing repair and penalty costs associated with the restoration process. To tackle the challenge of computational complexity, the nonlinear model is reformulated into a mixed-integer second-order cone programming model. The efficacy of the approach is validated through case studies on an IEEE 33-bus system, in which simulation results demonstrate a considerable improvement in grid resilience, achieving optimal load recovery with reduced restoration time and costs. The proposed approach outperforms traditional methods with optimal repair sequence and RC scheduling, aligned with NR efforts, and contributes to an improved system resilience level.

Reliability-based sizing of islanded multi-energy microgrid: a conic chance-constrained optimization approach

AbstractThis paper presents a methodology to optimally design a multi-energy microgrid with thermal and electric loads considering $$N-1$$ N - 1 and probabilistic regulation reserves. This methodology consists of a chance-constrained optimization that determines the optimal sizing of the microgrid. Microgrid operations are rigorously considered by modelling hourly thermal and electric demand patterns as well as technology production schedules over a year. Such schedules include both electric and thermal power balances, ramp constraints, $$N-1$$ N - 1 and regulation reserves, among others. To ensure a reliable microgrid design and operation, reserve constraints have been proposed to deal with both $$N-1$$ N - 1 generation contingencies, and forecast errors. $$N-1$$ N - 1 reserves guarantee that a sudden outage of any of the electric generators is strategically covered by the remaining generators in order to avoid load shedding. Additionally, nonzero-mean random forecast errors of electric load and solar production are addressed by a set of chance constraints able to schedule asymmetric up and down regulation reserves. Their levels are high enough to cover hourly random forecast errors (or intermittencies) with a high threshold probability. The proposed methodology results in a mixed integer second-order cone program. Results of microgrid designs with and without reliability reserves are carefully analysed and compared. Neglecting reliability constraints leads to a lower-cost design at the expense of exposing the microgrid to unsafe operation. Finally, sensitivity analysis to study the optimal portfolio sizing with respect to electric BESS investment cost, solar production forecast mean error, and random intermittency threshold probability are performed.

Voltage and Reactive Power-Optimization Model for Active Distribution Networks Based on Second-Order Cone Algorithm

To address the challenges associated with wind power integration, this paper analyzes the impact of distributed renewable energy on the voltage of the distribution network. Taking into account the fast control of photovoltaic inverters and the unique characteristics of photovoltaic arrays, we establish an active distribution network voltage reactive power-optimization model for planning the active distribution network. The model involves solving the original non-convex and non-linear power-flow-optimization problem. By introducing the second-order cone relaxation algorithm, we transform the model into a second-order cone programming model, making it easier to solve and yielding good results. The optimized parameters are then applied to the IEEE 33-node distribution system, where the phase angle of the node voltage is adjusted to optimize the reactive power of the entire power system, thereby demonstrating the effectiveness of utilizing a second-order cone programming algorithm for reactive power optimization in a comprehensive manner. Subsequently, active distribution network power quality control is implemented, resulting in a reduction in network loss from 0.41 MW to 0.02 MW. This reduces power loss rates, increases utilization efficiency by approximately 94%, optimizes power quality management, and ensures that users receive high-quality electrical energy.

A comprehensive optimization mathematical model for wind solar energy storage complementary distribution network based on multi-regulatory devices under the background of renewable energy integration

In the context of global energy transformation and sustainable development, integrating and utilizing renewable energy effectively have become the key to the power system advancement. However, the integration of wind and photovoltaic power generation equipment also leads to power fluctuations in the distribution network. The research focuses on the multifaceted challenges of optimizing the operation of distribution networks. It explores the operation and control methods of active distribution networks based on energy storage and reactive power compensation equipment. The stable operation of the distribution network is analyzed under the conditions of wind and photovoltaic integration, with a particular focus on precise regulation to address the limitations of existing methods. Afterwards, the study proposes an improvement plan that combines on load tap changer transformers and reactive power compensation equipment to solve complex power balance problems through second-order cone programming relaxation method. The results of numerical analysis show that the constructed mathematical model maintains a stable voltage of 1 to 1.1 pu at distribution network nodes within 24 h. Especially during peak hours from 15:00 to 24:00, it remains normal without any abnormal fluctuations when the control equipment is not added. These results confirm that precise regulation of multiple devices ensures voltage stability and avoids low or high voltage issues.

The capacitated [formula omitted]-hub interdiction problem with congestion: Models and solution approaches

We study the r-hub interdiction problem under the case of possible congestion. Hub interdiction problems are modeled as attacker-defender problems to identify a set of r critical hubs from a set of p hubs, which when attacked, causes maximum damage to network restoration activities of the defender. In this work we consider that in addition to the routing cost, the defender also aims to minimize the congestion cost. Incorporating the congestion cost in the problem introduces non-linearity in the objective function of the interdiction problem, which makes the problem challenging to solve. To address this, we propose two alternate exact solution approaches. The first approach is an inner-approximation-based approach (IBA), which overestimates the convex non-linear objective function and provides an upper bound. A lower bound is obtained from solving the lower-level problem exactly corresponding to the upper bound solution. The upper bound is tightened using improved approximation with new points generated in successive iterations. In the second approach (referred to as SBA), the problem is reformulated as a second-order conic program, which can be solved using an off-the-shelf solver. From our computational experiments on benchmark datasets (CAB and AP), we demonstrate the efficacy of both the proposed methods. However, IBA consistently outperforms SBA by a significant margin.

Establishing a hierarchical local market structure using multi-cut Benders decomposition

Local electricity markets (LEMs) such as peer-to-peer (P2P) and community-based markets allow prosumers and consumers to exchange electricity products and services locally. In order to coordinate electricity trading and flexibility services, this paper proposes a hierarchical prosumer-centric market framework with a hybrid LEM and a local flexibility market (LFM). Multi-cut Benders decomposition (MCBD) is employed to decompose the integrated hybrid LEM into a centralized P2P market and multiple community-based markets. The aggregators coordinate energy sources and demands of households in low voltage (LV) distribution networks (DN) as virtual power plants (VPPs) and engage in trading through a P2P market over the medium voltage (MV) DN. In addition, a modified MCBD (M-MCBD) approach is proposed to accelerate the convergence process. The LFM is operated by the distribution system operator (DSO) and is formulated as a mixed-integer nonlinear programming (MINLP) problem which is further relaxed to a mixed-integer second-order cone programming (MI-SOCP) problem. The case study demonstrates that aggregators were able to collaborate on trading within the hybrid LEM to minimize the costs incurred by prosumers within the network. Furthermore, the proposed M-MCBD method improves the scalability of the MCBD by enhancing its convergence speed and accuracy, as demonstrated by testing on problems of varying scales.

Collaborative Vehicle-to-Grid Operations in Frequency Regulation Markets

Problem definition: We study the operations of electric vehicles (EVs) providing frequency regulation services to the electric grid in vehicle-to-grid (V2G) systems. In particular, individually owned EVs collaboratively bid in the regulation market, coordinated by a platform that operates the network of charging equipment. We study how the platform determines optimal pricing incentives for drivers to plug in their EVs, accounting for heterogeneous driving schedules. Methodology/results: We model the platform’s pricing optimization problem as a bilevel program: At the upper level, the platform determines hourly rebates for EV owners to plug in their EVs and capacity bids in the regulation market; at the lower level, individual travelers optimize their travel and charging schedules in response to pricing incentives. To account for uncertainties and heterogeneity in regulation market prices and travel patterns, we adopt distributionally robust optimization techniques to formulate the problem as a mixed-integer second-order cone program. We conduct a computational study based on the California Household Travel Survey data set and actual frequency regulation prices. Our results show that the ability to offer time-varying rebates and install workplace chargers can significantly improve the V2G platform’s expected profits. Managerial implications: As EV adoption progresses past the nascent stage, V2G business models become more viable. Successful implementation of V2G provides economic incentive for switching to EVs, potentially helps sustain adoption growth, and complements the growth of renewable power by helping stabilize the grid. Our findings shed light on the design of driver incentives for V2G systems. Funding: R. Tang acknowledges support from the National Natural Science Foundation of China [Grant 72201206]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0133 .

Iterative convex relaxation of unbalanced power distribution system integrated multi-energy systems

This study describes a multi-period, multi-energy operation of a three-phase unbalanced Electric Distribution System (EDS) that coordinates with a District Heating System (DHS) and a Natural Gas (NG) distribution system. The electrical subsystem problem is formulated as a bi-level programming problem, with levels 1 and 2 solving the subsystem’s linearized and relaxed nonlinear versions, respectively. Second-Order Cone Programming (SOCP) and polyhedral relaxations are availed to circumvent the non-convexities involved in the subsystems. The SOCP relaxation for an unbalanced power distribution network is inexact; also, the subsystems applying polyhedral relaxation may not generate a meaningful solution as the relaxations are not tight. Therefore, a solution recovery algorithm is developed for each subsystem to recover a feasible solution from their relaxed counterparts. The successive bound tightening algorithm employing a solution recovery procedure is proposed for each subsystem, improving solution quality and strengthening the relaxations with the desired computational efficiency. The proposed solution strategy to optimize the Multi-Energy System (MES) operation cost is verified on the three-phase IEEE-13 and IEEE-123 bus systems, each coordinating with a 30-node DHS and a 6-bus NG network. The results analyses demonstrate that the proposed solution strategy efficiently achieves an optimal solution, reducing maximum relaxation error below 0.1% for each subsystem. The proposed strategy delivered a 14.58% reduction in real power losses and a 12.73% decrease in phase voltage unbalance rate for the EDS in MES. Furthermore, a 1.96% decrease in operational cost demonstrates the techno-economic benefits of the proposed strategy.

Energy optimal scheduling strategy considering V2G characteristics of electric vehicle

in recent years, large-scale electric vehicles connected to the power grid has a significant impact on demand-side resources. In addition, the vehicle-to-grid technology of electric vehicles introduces more uncertainties, which brings great challenges to the optimal scheduling of integrated energy systems. To solve this problem, this paper fully considers the uncertainty and fluctuation of wind power. Firstly, the objective function of joint configuration is constructed to maximize the utilization of wind power and minimize the planned generation cost of the system. Secondly, the linearized power flow equations are used to represent the relations among the state variables, and the second-order cone relaxation method is applied to deal with the branch power flow constraints, which is transformed into a solvable mixed-integer second-order cone programming model. Finally, the validity of the model is verified by an example. The system achieves obvious peak-shaving and valley-filling effect and improves the economic benefit of the system.

Learning to optimize power allocation in cell-free massive MIMO networks with hybrid green energy

In cell-free (CF) massive multiple-input multiple-output (MIMO) networks, many access points (APs) simultaneously serve user equipment (UEs) using the same time and frequency resources. The resource allocation algorithm in such networks needs to handle many calculations within a limited time frame to determine the power allocation from each AP to each UE. Further, the dispersion and large number of APs in CF massive MIMO networks pose challenges in terms of energy supply. As access to the power grid may be limited, renewable energies are often required as a supplement. However, due to the non-deterministic nature of the charging process, the resource allocation algorithm needs to consider the dynamic and uncertain amount of energy available at the APs. In this article, we propose a scalable machine learning based on the emerging Learning to Optimize (L2O) paradigm that utilizes neural networks (NNs) for resource allocation in CF massive MIMO networks powered by electrical and renewable energies. Our objective is to minimize energy consumption from the power grid while satisfying the spectral efficiency (SE) constraints of the UEs. Additionally, our algorithm adopts a forward-looking approach and employs a green energy budgeting mechanism to prevent service interruption at green APs during energy shortages. Our algorithm design approach involves formulating the offline version of the problem as a second-order cone program (SOCP). Subsequently, we train an NN using the solutions obtained from solving the online variant of the SOCP. Our results indicated that our L2O-based algorithm can accurately mimic the solutions of the optimization problem with significantly reduced computational complexity.

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.

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.