Mixed-integer Second-order Cone Programming Research Articles

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.

Collaborative Optimization for Active Distribution Network with Soft Open Point and Energy Storage

Background: The increasing use of high-penetration distributed clean energy has led to voltage fluctuations in the distribution network that surpass safety limits and pose challenges to economic and low-carbon operation. Traditional voltage regulation devices are unable to meet the real-time high-precision voltage control needs when encountering frequent fluctuations due to physical limitations. The Soft Open Point (SOP) can provide continuous reactive power to achieve rapid voltage regulation. Objective: We propose a collaborative optimization approach that fully considers the regulatory capabilities of SOP and energy storage to eliminate voltage violations and reduce network losses. Methods: This study introduces an active distribution network reactive power voltage optimization approach that incorporates intelligent SOP and energy storage, combining conventional devices (such as on-load tap changers and capacitor banks) with contemporary devices (such as SOP, distributed generators, and energy storage) to establish synergy. A coordinated optimization model was developed, considering various operational constraints to minimize network losses and regulate the voltage of distribution network nodes. By using linearization and quadratic relaxation, the original non-convex mixed-integer non-linear optimization model was transformed into a Mixed-Integer Second-Order Cone Programming (MISOCP) model to meet the requirements of rapid voltage regulation. Results: A case study was conducted on the IEEE 33-node test system, showing that the proposed approach effectively reduces network losses and eliminates voltage violations. Conclusion: The feasibility and effectiveness of the proposed methodology have been validated, confirming its ability to ensure the safe and economic operation of active distribution networks.

Active and reactive power coordination optimization for active distribution network considering mobile energy storage system and dynamic network reconfiguration

The active distribution network (ADN) can face with challenges due to the increasing renewable distributed generation (RDG), which may result in elevated network losses and voltage fluctuations. To address these issues, a novel operation strategy is proposed which integrates the mobile energy storage system (MESS) and dynamic network reconfiguration (DNR) to adjust the active and reactive power of the ADN. The transportation network (TN) is modeled considering the traffic congestion, and the path movement of MESS in TN is converted to the switching of virtual switch (VS) in ADN. A coordinated optimal model is formulated for DNR and MESS, furthermore, which can be transformed into a mixed-integer second-order cone programming (MISOCP) model. The penalty alternating direction method (PADM) is employed to enhance the computational efficiency. Then the proposed strategy is tested by the IEEE 33-bus system coupled with the 15-node transportation systems, and the stability of the proposed strategy was validated in a larger scale expansion system. The simulation results demonstrate that the coordinated optimal strategy considering MESS and DNR can effectively reduce network loss and transportation cost, enhance the voltage quality of the ADN and promote the consumption of renewable energy.

Stochastic Convex Cone Programming for Joint Optimal BESS Operation and Q-Placement in Net-Zero Microgrids

Microgrids have emerged as a pivotal solution in the quest for efficient, resilient, and sustainable energy systems. Comprising diverse distributed energy resources, microgrids present a compelling opportunity to revolutionize how we generate, store, and distribute electricity, while simultaneously reducing carbon footprints. This paper proposes an optimal battery energy storage system (BESS) management scheme, along with capacitor placement for reactive power (Q)-compensation, and scheduling for the purpose of a renewable-based microgrid’s loss minimization. The proposed model evaluates the impact of BESS management on energy efficiency and analyzes how optimal scheduling of BESS influences system losses. Furthermore, it investigates the coordinated planning and operation of active assets within the microgrid, such as controllable capacitor banks, in enhancing overall efficiency. The model is formulated as a mixed-integer second-order cone programming (MISOCP) problem which is solved for both deterministic and stochastic generation and consumption data. The proposed model is tested on a 21-bus microgrid comprising photovoltaic and hydropower energy resources, and the efficacy of the model is approved by several case studies. The simulation results show that the proposed method can reduce microgrid energy losses by approximately 12 percent using the deterministic approach and around 14 percent with the stochastic approach.

A two‐stage reactive power optimization method for distribution networks based on a hybrid model and data‐driven approach

AbstractThe uncertainty of distributed energy resources (DERs) and loads in distribution networks poses challenges for reactive power optimization and control timeliness. The computational limitations of the traditional algorithms and the development of artificial intelligence (AI) based technologies have promoted the advancement of hybrid model‐data‐driven algorithms. This article proposes a two‐stage reactive power optimization method for distributed networks (DNs) based on a hybrid model‐data‐driven approach. In the first stage, based on the topology and line parameters of the DN, as well as forecasts of loads and renewable energy outputs, a mixed‐integer second‐order cone programming (MISOCP) algorithm is used to control the on‐load tap changer (OLTC) positions on an hourly day‐ahead basis. In the second stage, leveraging deep learning technology, the real‐time reactive power output of photovoltaics (PV) and wind power units is controlled at a 5‐min time scale throughout the day. Specifically, using traditional solvers, the global optimal reactive power output for PV and wind power units is determined first, corresponding to various load and renewable energy output scenarios. Then, neural networks are trained to map node power to the optimal reactive power outputs of renewable energy units, capturing the complex physical relationships. For the second stage, a transformer network framework with a self‐attention mechanism and multi‐head attention for deep learning training is applied to uncover the intrinsic and physical spatial relationships among high‐dimensional features. The proposed method is tested on a modified IEEE 33‐bus system with multiple distributed renewable energy sources. The case study results demonstrate that the proposed hybrid model‐data‐driven algorithm effectively coordinates day‐ahead and real‐time controls of various devices, achieving real‐time model‐free optimization throughout the day. Compared to traditional deep neural networks (DNNs) and convolutional neural networks (CNNs), the transformer network provides superior reactive power optimization results.

Combinatorial lower bounds for the Generalized Traveling Salesman Problem with Neighborhoods

In this paper, we study the Generalized Traveling Salesman Problem with Neighborhoods (GTSPN), a variant of the Traveling Salesman Problem (TSP), where the goal is to find the shortest path visiting each of the given neighborhood sets represented as a set of convex regions. The GTSPN is motivated by the sequencing problem of robotic manipulators, where an operation can be achieved from multiple locations, such as the visual inspection that can be performed from several possible views. The GTSPN formulation allows for exploiting continuous optimization to find the most suitable locations for the inspection, yielding possible solution cost reduction. Moreover, instances with overlapping high-dimensional convex regions further allow modeling neighborhood sets with complex shapes. We propose a novel approach to determine the first lower bounds to the studied GTSPN by employing the Branch-and-Bound (BB) method and the Mixed-Integer Second-Order Cone Programming (MISOCP) model for particular BB subproblems. In addition, the proposed method allows for solving the GTSPN to optima. The developed lower bound determination is further exploited in the empirical evaluation of existing heuristic approaches to the GTSPN and assesses the solution quality using the relative optimality gap. Regarding the presented results, the proposed BB-based approach provides tight lower bounds and solutions with up to 20% optimality gap for the GTSPN instances with less than 15 neighborhood sets for the given limited computational time. Furthermore, the presented results support that the proposed approach is suitable for solving high-dimensional instances of the GTSPN that can be found in inspection tasks with robotic manipulators.

Optimal allocation of distributed generation on DC Networks

A consistent increase in the installation of distributed generation (DG) in response to rising energy cost and demand throughout the world, especially for the last two decades, primarily aims to obviate upgrading investments for transmission and distribution lines on the existing power grid while ensuring sustainable, resilient, and reliable service to end users. Random installation of DG, on the other hand, can invalidate the existing grid’s designated objectives, and operational boundaries due to the changes in network topology resulting in unforeseen load flow. Appropriate integration of DG units on DC networks should be remedy to high transmission/distribution losses in lines, instabilities in voltage profiles, issues in power quality parameters, and derogation in the protection schemes. Based on the behavior of load profiles, e.g., 24 h or longer period, operational and technical constraints, this study proposes a framework that minimizes transmission/distribution losses on a DC network, in which discrete variables stand for the location of DGs. Optimal power flow (OPF) problem formulation underpins the proposed optimization framework. Overall, this computationally challenging planning problem refers to a mixed-integer nonlinear program (MINLP) including discrete variables with non-convex power balance/flow representations. To deal with the non-convexities and discrete form of the formulation, a mixed-integer second-order cone programming (MISOCP) relaxation, and branch-and-bound search are assigned. The validation of the introduced approach is carried out on the IEEE modified benchmark systems. With the help of proposed method, optimal allocation of DG units on 9-bus, 14-bus, and 30-bus systems, respectively, lead up to 99.93%, 94.38%, and 79.09% total power losses reduction.

From Curtailed Renewable Energy to Green Hydrogen: Infrastructure Planning for Hydrogen Fuel-Cell Vehicles

Problem definition: Hydrogen fuel-cell vehicles (HFVs) have been proposed as a promising green transportation alternative. For regions experiencing renewable energy curtailment, promoting HFVs can achieve the dual benefit of reducing curtailment and developing sustainable transportation. However, promoting HFVs faces several major hurdles, including uncertain vehicle adoption, the lack of refueling infrastructure, the spatial mismatch between hydrogen demand and renewable sources for hydrogen production, and the strained power transmission infrastructure. In this paper, we address these challenges and study how to promote HFV adoption by deploying HFV infrastructure and utilizing renewable resources. Methodology/results: We formulate a planning model that jointly determines the location and capacities of hydrogen refueling stations (HRSs) and hydrogen plants as well as electricity transmission and grid upgrade. Despite the complexity of explicitly considering drivers’ HFV adoption behavior, the bilevel optimization model can be reformulated as a tractable mixed-integer second-order cone program. We apply our model calibrated with real data to the case of Sichuan, a province in China with abundant hydro resources and a vast amount of hydropower curtailment. Managerial implications: We obtain the following findings. (i) The optimal deployment of HRSs displays vastly different spatial patterns depending on the HFV adoption target. The capital city, a transportation hub, is excluded from the plan under a low target and only emerges as the center of HFV adoption under a high target. (ii) Promoting the HFV adoption can overall help reduce hydropower curtailment, but the effectiveness depends on factors such as the adoption target and the grid upgrade cost. (iii) Being a versatile energy carrier, hydrogen can be transported to various locations, which allows for strategic placement of HRSs in locations distinct from hydrogen plant sites. This flexibility offers HFVs greater potential cost savings and curtailment reduction compared with other alternative fuel vehicles (e.g., electric vehicles) under current cost estimates. Funding: W. Qi acknowledges the support from the National Natural Science Foundation of China [Grants 72242106 and 72188101] and the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-04769]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0381 .

Investigation methods for enhancing the energy consumption capacity of novel PV access distribution networks

Abstract This paper presents a method for reconstructing distribution networks using soft open point (SOP) and demand response (DR) energy storage, which can effectively solve the problems of voltage overruns, line overcurrent, and bidirectional flow currents caused by the photovoltaic (PV) getting into the distribution network. First, the model for the rebuilding of the distribution network is created by using the energy storage method of DR and SOP. Then, MISOCP, which stands for mixed-integer second-order cone programming, is the model that is used to change the distribution network reconfiguration, and the transformed model is solved by the MOSEK algorithm package in MATLAB. Finally, the strategy suggested in this research may successfully enhance the distribution grid’s consumption capacity after PV access, as shown by the verification in the IEEE33 node distribution grid system.

Reactive power optimization for power distribution networks using mixed-integer Second-order cone programming

Reactive power optimization for power distribution networks using mixed-integer Second-order cone programming

Distribution Network Restoration Strategy Considering Mobile Energy Storage Systems

Abstract The increasing occurrence frequency of disruptive extreme weather events and the growing share of distributed resources further enhance the importance of restoration of distributed resource-assisted distribution networks. However, most distributed sources are not spatially flexible in scheduling, severely impacting the restoration performance of distribution networks. The emergence of mobile energy storage systems (MESS) provides temporally and spatially flexible resources. Based on the mobility of mobile energy storage, this paper proposes a distribution network restoration strategy considering MESS and prioritized restoration of critical loads. The restoration strategy is established aiming at maximizing weighted loads while satisfying multiple constraints. Specific operating characteristics of MESS are modeled and incorporated into the restoration model. The established restoration model is then reformulated into a mixed-integer second-order cone program. Case studies on a modified IEEE 33-node system verify the proposed restoration strategy.

A dynamic reconfiguration model and method for load balancing in the snow-shaped distribution network

The snow-shaped distribution network (SDN) is a cable distribution network composed of eight (or six) 10 kV feeders from four (or three) substations in a regular connection. Compared with the traditional 10 kV distribution network, SDN can support a wide range of load transfer among six or eight feeders. Aiming at the problem of load spatial-temporal unbalanced condition caused by the integration of distributed generators (DGs) and different load types in different feeders, this paper proposes a dynamic reconfiguration strategy for load balancing in SDN considering DGs and energy storage system (ESS). Firstly, the basic structure of SDN is analyzed and the power flow model for its dynamic reconfiguration is developed. Secondly, the dynamic reconfiguration optimization model for load balancing in SDN considering DGs and ESS is proposed to utilize the load transfer capability to mitigate the load unbalanced condition and reduce active power loss. Thirdly, the original non-convex model is converted into a mixed-integer second-order cone programming (MISOCP) model by applying the second-order cone relaxation and the big-M method, which is solved by CPLEX solver. Finally, the effectiveness of the proposed model and method are verified by an actual case in Tianjin and IEEE 33-node system. The analysis results show that the proposed method can significantly alleviate the load unbalanced spatial-temporal distribution and improve the economic efficiency by regulating the operation of SDN including ESS optimization and dynamic reconfiguration.

Chance constrained optimal power-water flow: An iterative algorithm assisted by deep neural networks

The optimal power-water flow (OPWF) is a fundamental tool for operating integrated power and water networks. To address uncertainties in renewable energy and load demands, researchers have used chance-constrained models to investigate OPWF under uncertainty. However, existing work has limitations, such as oversimplified network representations, intractable problem formulations, and high computational burdens. This paper presents a chance-constrained OPWF formulation that incorporates detailed network constraints and stochastic uncertainties from wind speeds and electricity/water demands. By leveraging the concept of quantiles of stochastic variables and convexification techniques, the model is recast into a deterministic mixed-integer second-order cone programming (MISOCP) problem with uncertain margins. To efficiently solve this problem, we propose a novel iterative algorithm assisted by deep neural networks (DNNs). The algorithm integrates two DNNs: one designed to solve power and water flow equations, accelerating the probabilistic evaluation of uncertain margins; and the other to predict pipeline flows and refine variable ranges, facilitating a warm-start approach to the MISOCP model. Case studies conducted on the IEEE 123-node feeder coupled with a 67-node water distribution network validate the two DNNs and the overall iterative algorithm. The findings demonstrate that the proposed DNN-assisted approach effectively balances system security and economy. Compared to existing algorithms, the relative error of the optimization solution is less than 0.03%, with a speedup ratio exceeding 200.

Decentralized energy market integrating carbon allowance trade and uncertainty balance in energy communities

With sustained attention to carbon neutrality, the personal carbon trading (PCT) scheme has been embraced as an auspicious paradigm for scaling down carbon emissions. To facilitate the simultaneous clearance of energy and carbon allowance inside the energy community while hedging against uncertainty, a joint trading framework is proposed in this article. The energy trading is implemented in a peer-to-peer (P2P) manner without the intervention of a central operator, and the uncertainty trading is materialized through procuring reserve of conventional generators and flexibility of users. Under the PCT scheme, carbon allowance is transacted via a sharing mechanism. Possible excessive carbon emissions due to uncertainty balance are tackled by obliging renewable agents to procure sufficient carbon allowances, following the consumption responsibility principle. A two-stage iterative method consisting of tightening McCormick envelope and alternating direction method of multipliers (ADMM) is devised to transform the model into a mixed-integer second-order cone program (MISOCP) and to allow for a fully decentralized market-clearing procedure. Numerical results have validated the effectiveness of the proposed market model.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Distributed restoration for integrated electricity-gas-heating energy systems with an iterative loop scheme

Electricity, gas and heating networks are increasingly coupled to form an integrated energy system (IES), which is beneficial in improving the operational performance of the system during power outages. Generally, IES involves multiple energy operators, making the centralized framework impracticable. To this end, a novel distributed restoration method for IES is proposed in this paper. The IES restoration problem is formulated as a mixed-integer second-order cone programming (MISOCP) model to minimize the total load losses, which considers the coupling effects among the electricity, gas and heating networks. Then, considering the independent management and information privacy of different energy networks, the distributed restoration model is formulated by decomposing shared boundary information using the alternating direction method of multipliers (ADMM). For the sake of the non-convexity caused by binary load state variables, an iterative loop scheme is introduced in ADMM procedure. This allows us to solve a relaxed distributed model in the first stage, and perform iterative loops in the second stage, which can improve the convergence of ADMM and make the distributed solution feasible. Finally, numerical studies are conducted on 33-7-6-node IES and 123-24-32-node IES to validate the effectiveness and feasibility of the proposed method.

A mathematical formulation and a tabu search heuristic for the joint vessel-UAV routing problem

Motivated by offshore inspection practices, we investigate a joint vessel-unmanned aerial vehicle (UAV) routing problem where vessels and UAVs work coordinately to perform inspection tasks. The problem is an extension of the mothership-drone routing problem by further considering multiple vessels and realistic constraints such as temporal-spatial coordination, service time, and inspection cycle. The goal is to minimize overall operational costs, including fixed vessel cost, and vessel and UAV routing costs. Decision-making involves task assignment, vessel route determination, and UAV take-off and landing locations. We formulate the problem as a mixed-integer second-order cone program. Addressing its NP-hard nature, we develop an enhanced tabu search algorithm (TS-RC), which incorporates two innovative mechanisms to reduce the computational burden. The first is to determine UAV take-off/landing points based on a new constructive procedure. The second is assessing neighborhood solutions using an approximate procedure. Results on a real-world case demonstrate a 10.29% reduction in operational costs compared to a classic vessel routing model. Moreover, numerical experiments on random instances with up to four vessels and 39 tasks demonstrate the performance of the proposed TS-RC method.

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

Flexible allocation and optimal configuration of multi-level energy exploitation units for heterogeneous energy systems considering resource distribution

The energy systems are evolving into heterogeneous energy systems (HESs) with complicated integration of sources, networks, loads, and storage towards a decarbonized future of human beings. This paper develops the flexible allocation and optimal configuration of multi-level energy exploitation units (MEEUs) for the HES considering resource distribution including local energy conditions and actual coupled relationship of various energy networks. A bi-level optimal planning model of MEEUs for the HES considering initial investment cost, carbon emission cost, operation cost and line loss is developed to figure out the optimal location, structure, configuration and corresponding operation strategy. To address the non-convex and non-linear model, a novel hybrid solution method considering the mixing characteristics of mixed-integer second-order cone programming (MISOCP) model and quadratic non-convex model is proposed. The solution method adopts MISOCP model for figuring out the optimal location, structure and configuration of MEEUs and quadratic non-convex model for optimal operation strategy of MEEUs, which not only improves the computational efficiency but also ensures the solution accuracy. Finally, the related case studies are conducted to compare the proposed MEEU and distributed energy storage unit (DESU) in terms of energy consumption, renewable energy absorbency, carbon emission and line loss. The results verify the significant improvement of the comprehensive energy efficiency, the investment potentiality and the environmental benefits obtained by the proposed planning model.

A congested facility location problem with strategic customers

This study addresses a user-equilibrium congested facility location problem with delay-, accessibility-, and price-sensitive customers. A profit-maximizing service provider first makes location, service rate, and pricing decisions, and then strategic customers decide which facilities to patronize (if any). By incorporating the customers’ choice behavior as a set of equilibrium constraints into the service provider’s decision problem, the problem is modeled as a mixed integer non-linear program which is then reformulated as a mixed integer second-order cone program. The proposed model, tested on several standard instances, is shown to be efficiently solvable using commercial software packages.