Second-order Cone Programming Model Research Articles

Nash Bargaining-Based Coordinated Frequency-Constrained Dispatch for Distribution Networks and Microgrids

As the penetration of distributed renewable energy continues to increase in distribution networks, the traditional scheduling model that the inertia and primary frequency support of distribution networks are completely dependent on the transmission grid will place enormous regulatory pressure on the transmission grid and hinder the active regulation capabilities of distribution networks. To address this issue, this paper proposes a coordinated optimization method for distribution networks and microgrid clusters considering frequency constraints. First, the confidence interval of disturbances was determined based on historical forecast deviation data. On this basis, a second-order cone programming model for distribution networks with embedded frequency security constraints was established. Then, microgrids performed economic dispatch considering the reserves requirement to provide inertia and primary frequency support, and a stochastic optimization model with conditional value-at-risk was built to address uncertainties. Finally, a cooperative game between the distribution network and microgrid clusters was established, determining the reserve capacity provided by each microgrid and the corresponding prices through Nash bargaining. The model was further transformed into two sub-problems, which were solved in a distributed manner using the ADMM algorithm. The effectiveness of the proposed method in enhancing the operational security and economic efficiency of the distribution networks is validated through simulation analysis.

Human-safe based electric vehicle evacuation in transportation-power networks during emergencies: A two-stage response framework

The proliferation of electric vehicles (EVs) and the deepening interconnection between transportation and the power grid have heightened the complexity and vulnerability of the coupling transportation-power networks (TPNs). EV evacuation can exert significant dynamic pressure on the TPNs during emergencies. This article presents a baseline reproducible two-stage emergency response framework. In the first stage, we develop a platform for the comprehensive simulation called EV evacuation integrated simulation model (EVEISM), utilizing agent-based modeling. The interaction between the agent models and the GIS-based environment models captures the behavior of EVs in the complete evacuation period, including pre-evacuation and post-evacuation charging, receiving notifications, making destination decisions, planning routes to avoid risks, and traffic dynamics. In the second stage, leveraging EVEISM results to enhance situational awareness, strategic pre-positioning of mobile energy storage systems (MESS) and optimal routing are conducted. A mixed-integer second-order cone programming model is established for the coordinated response of multiple power sources to dispatch MESS dynamically. Case studies are conducted on coupling transportation-power systems, exemplified through the toxic gas leak scenario. The numerical results demonstrate the method's effectiveness in ensuring the safe evacuation of residents and optimizing coordinated response to enhance the system's safety and economic efficiency.

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.

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

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.

Optimizing integrated berth allocation and quay crane assignment: A distributionally robust approach

In this research, we have formulated a Two-Stage Distributionally Robust Optimization (TDRO) model within the context of a mean–variance ambiguity set, specifically designed to address the challenges in the Integrated Berth Allocation and Quay Crane Assignment Problem (BACAP). A key consideration in this study is the inherent uncertainty associated with ships’ arrival times. During the initial stage, we derive a baseline schedule governing berth allocation and quay crane assignment. Anticipating potential disruptions arising from uncertain arrival delays, the second stage is meticulously formulated to determine the worst-case expectation of adjustment costs within the mean–variance ambiguity set. Subsequently, we undertake an equivalent transformation, converting the general TDRO model into a Two-Stage Robust Second-Order Cone Programming (TRO-SOCP) model. This transformation facilitates the application of the Column and Constraint Generation (C&CG) algorithm, ensuring the derivation of an exact solution. To address the computational intricacies associated with second-order cone programming, we propose two enhancement strategies for upper and lower bounds, aimed at expediting the solution process. Additionally, to contend with large-scale instances, we introduce a refinement and approximation method, transforming the TDRO model into a Mixed-Integer Programming (MIP) model. Furthermore, extensive numerical experiments are executed on both synthetic and real-life instances to validate the superior performance of our model and algorithms. In terms of the total cost, the TDRO model demonstrates superior performance compared with Two-Stage Stochastic Programming (TSP) and Two-Stage Robust Optimization (TRO) models.

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.

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.

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 Fault Restoration Method for DC-interconnected Distribution Systems with Renewable Resources Considering Multiple Fault Scenes

Background:: The distribution systems are gradually transformed into active systems with the large-scale integration of renewable resources, which brings greater potential to the fault recovery of the distribution networks and higher requirements for the power scheduling capability under fault conditions. Power electronic devices such as soft open points (SOPs) can break the radial constraints of the traditional distribution network, provide cross-station power support in fault scenarios, and enhance the resilience of the distribution networks. Methods:: This paper focuses on the distribution systems with multi-transformer flexible interconnection, and a multi-device coordinated fault restoration method is proposed. Firstly, the characteristics of different transformer fault scenes and the fault restoration process coordinated by the bus-bar switches and SOPs in each scene are sorted out. Secondly, an optimization model of the fault restoration for DC-interconnected distribution system is established considering the network operation and the equipment constraints in the fault restoration process. Result:: The proposed model aims at the maximization of the weighted load restoration amount, and is transformed into a second-order cone programming model to be solved quickly. Lastly, the effectiveness of the proposed method is verified in a typical flexible interconnection distribution system. Conclusion:: The simulation results prove that it can effectively improve the capability of preserving power supply for important loads after transformer failure.

A risk-averse distributionally robust optimisation approach for drone-supported relief facility location problem

Drone-supported last-mile humanitarian logistics applications play a crucial role in the rapid and efficient delivery of essential relief items, such as medicine, blood, and vaccines, during disaster and emergency situations. This paper explores a novel drone-supported relief facility location problem (DSRFLP) aimed at establishing an effective drone-supported last-mile humanitarian logistics system. The problem involves making joint decisions for both pre-disaster and post-disaster phases while considering the characteristics of drone-based delivery operations and uncertain demands. In the pre-disaster phase, we make the decisions regarding the locations of drone-supported relief facilities, inventory prepositioning of relief items, assignment of drones to the opened facilities, and allocation of drones to disaster demand sites. In the post-disaster phase, we make decisions related to delivery quantities. To tackle the challenge of incomplete demand distribution information in chaotic disaster environments, we establish a distributionally robust optimisation (DRO) model to handle the uncertainty of demands. This model adopts worst-case mean Conditional Value-at-Risk as the risk measurement, reflecting the risk-averse attitude of humanitarian organisers. In this paper, three ambiguity sets (box, ellipsoidal, and polyhedral) are considered to describe the ambiguity distributions of demands. To overcome the computational challenge, we reformulate the DRO model under three ambiguity sets into two mixed-integer linear programming models and one second-order cone programming model, which can be efficiently solved by off-the-shelf solvers. Furthermore, we validate our proposed DRO model through a small-scale example and a large-scale case study based on the Lushan earthquake in China. The computational outcomes underscore the superior performance of the proposed DRO model to mitigate the impact arising from incomplete probability distributions. We propose managerial implications and insights to support the decision-making of humanitarian organisations based on the experimental results. Finally, we propose two extended models to incorporate multiple relief items and equity constraints in priority settings and conduct numerical experiments to adapt to various real-world disaster scenarios.

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.

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.

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.

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.

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.

Multi-objective optimal decision for orderly power utilization based on improved ε-constraint method in active distribution networks.

With the increasing demand for electricity load in China, orderly power utilization are important measures to alleviate electricity shortages during peak periods. This article establishes a multi-objective optimization model for orderly power utilization in active distribution networks is established, with the optimization objectives of minimizing the total operation cost, minimizing the cost for users, and minimizing the load fluctuation of the system. This model contains a large number of integer variables and nonlinear constraints, which is difficult to solve. To reduce computation time, convex relaxation techniques are adopted to transform the original model into a mixed-integer second-order cone programming (MISOCP) model, which has lower computational complexity. Furthermore, The improved ε-constraint method is proposed to solve the model, which can directly and quickly find the compromise optimal solution of the multi-objective problem. By using simplex search algorithm, the proposed method dose not need to traverse all grid points, which can significantly reduce computation time. Finally, case study on the the IEEE-33 bus distribution network demonstrate the effectiveness of the proposed method.

Dual-layer peer-to-peer energy trading method with multiple SOPs pricing in distribution networks

With the decentralization and the development of power markets, different stakeholders are emerged in distribution networks. The unbalance of distributed generators (DGs) and demand-side response in each area leads to a power transfer demand with others. Peer-to-peer (P2P) energy trading has the potential to enable different stakeholders to complement each other. Soft open points (SOPs) provide a promising physical platform for power transfer and trading. However, determining the trading price with SOP participation is an urgent issue to be addressed. Aiming at improving fair and economic operation, this study proposes a P2P energy trading method with multiple SOPs pricing in distribution networks. First, a dual-layer P2P energy trading framework is established. In the upper layer, an optimal pricing model is formulated to maximize SOP profits. An optimal dispatch model is formulated in the lower layer to conduct the power transfer based on the upper layer price. Then, the dual-layer model is converted into a second-order cone programming (SOCP) model to achieve computational tractability. Finally, a practical distribution network is utilized to validate the proposed method. The results indicate that the proposed P2P energy method is effective in reducing the operating costs and increasing the operating profits of SOPs.