Mixed-integer Second-order Cone Programming Research Articles

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.

Optimal scheduling of active distribution network based on RBF-stochastic response surface method

As the high proportion of distributed generation (DG) and diversification of loads increase, the distribution network faces increasingly greater impact. To address the high-dimensional stochastic problems arising from the multi-dimensional uncertainty of power source and load, this paper proposes a two-stage optimization scheduling scheme for AC distribution networks based on RBF-stochastic response surface method (RBF-SRSM). This scheme takes into account economy, environmental protection and safety, and consists of day-ahead and intra-day stages. The day-ahead decision scheduling is based on an optimal low-carbon economy operation plan, while the intra-day reactive power optimization is achieved by converting nonlinear integral indicators into discrete linear indicators through probabilistic power flow calculation and the first moment derivation principle inspired by SRSM. The mixed integer second-order cone programming (MISOCP) method is used to solve the dispatching plan, and the results suggest that the proposed method is more efficient than the SRSM.

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.

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 .

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.

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.