Mixed-integer Second-order Cone Programming Research Articles

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 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.

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.

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.

A temporal and spatial electric vehicle charging optimization scheme with DSO-EVA coordination framework

Burdens of increasing penetration of electric vehicles (EVs) on distribution systems have attracted wide attention to the research of EV charging coordination. Nevertheless, existing coordination lacks the joint consideration of the temporal-spatial flexibility of EVs, which could reduce load fluctuations when meeting the EV charging demands. Moreover, a thorough investigation is required for the coordination of the benefits to the distribution system operator (DSO), EV aggregator (EVA), and EVs. In this paper, a novel EV charging scheme considering the temporal-spatial features of EVs is proposed, achieving the coordination of DSO, EVA, and EVs in two steps. First, a mixed-integer second-order cone programming (MISOCP)-based EV charging schedule is developed for DSO. Second, the temporal-spatial features of EVs are exploited by EVA to track the charging schedule. Specifically, a dynamic charging price mechanism based on active power margin is proposed for public charging facilities and a specific load regulation is designed for private charging facilities. On this basis, the modified adaptive multi-objective particle swarm optimization (AMOPSO) algorithm is proposed, including adaptive flight parameter adjustment and termination mechanisms. Case studies demonstrate the proposed strategy can attenuate load variance and raise EVA revenue. Further, the impact analysis of penalty price and the price elasticity of electricity demand can provide references for stable distribution network operation, higher EVA revenue, and charging cost reduction.

Exploring Carbon Equilibrium in Integrated Electricity-Hydrogen System

Global carbon emissions from electricity and hydrogen production have grown rapidly in recent years. Some existing literature has proposed the method of reducing carbon emissions by power-to-hydrogen (P2H) technology. However, due to the fossil-based energy structure of electricity system, excessive electricity to hydrogen may lead to an increase in carbon emissions. To reduce carbon emissions effectively, a low carbon-driven optimal capacity configuration method for P2Hs is proposed in this paper to explore carbon equilibrium in integrated electricity-hydrogen system. A joint economic dispatch model for integrated electricity-hydrogen system is formulated. The electricity system model is a security-constrained economic dispatch model with P2H constraints. The hydrogen system model is a hydrogen supply chain model with pipeline pack constraints and Weymouth equation, whose sources are fossil fuel reforming and electrolytic hydrogen. We propose a second-order cone programming (SOCP)-based solution to transform the hydrogen system model into a mixed integer second order cone programming (MISCOP), which can be readily solved by commercial solvers. To preserve the internal information of each system, a decentralized algorithm based on optimality condition decomposition (OCD) is developed. Case study based on modified IEEE 30-bus and Belgium 20-node integrated energy system demonstrates the effectiveness of the proposed strategy.

Carbon-Aware Operation of Resilient Vertical Farms in Active Distribution Networks

Emerging utilization of Electric Vehicles (EVs) in Distribution Networks (DNs) decreases the power quality in the DN. The main quest of this research is to find out Can EVs and Photovoltaic (PV) systems support the DN and mitigate the voltage magnitude violations due to utilizing high penetration level EV? This article proposes a novel volt/VAR optimization (VVO) model by leveraging the reactive power support of EVs and PVs as Distributed Energy Resources (DERs) for the DN. The proposed VVO model contains the second-order conic relaxed form of the AC power flow equations. Thus, the presented VVO problem is a Mixed-Integer Second-Order Cone Programming (MISOCP) problem. To obtain the ability to support the DN, Smart Inverters (SIs) are utilized to connect EVs and PV systems to the DN. Thus, to add the PV and EV systems to the convexified form of the VVO problem, the P-Q characteristics and the model of SIs working at different operating points are presented based on experimental results. The proposed model of SIs allows the participation of DERs in the reactive power compensation to mitigate voltage issues of DN due to utilizing high penetration level EVs. Besides, the proposed model of SIs enables DERs to increase their dispatchability. The performance of the proposed model is examined in the case studies using the IEEE 33-bus system and the 123-bus system. It is illustrated in the case studies that the solar dispatchability increases by 12% when the proposed volt/VAR model is utilized for IEEE 33-bus system. Besides, it is illustrated in the case studies that the voltage profile of buses remains in the desired operating range when the proposed model is leveraged, even if the penetration level of EVs increased to 40%.

Look-Ahead Rolling Economic Dispatch Approach for Wind-Thermal-Bundled Power System Considering Dynamic Ramping and Flexible Load Transfer Strategy

With a wind-thermal-bundled power system (WTBPS) under high wind penetration levels, the sharp power fluctuations of tie-lines for interconnected grids trigger a significant challenge of security-constrained power system operation. Smoothing power fluctuations with economic dispatch is widely concerned against this challenge. In this paper, an appropriate look-ahead economic dispatch framework for WTBPS considering the variable ramp rate of retrofitted coal-fired units and the flexible load transfer strategy (LTS) in high voltage distribution networks (HVDNs) is proposed. To cope with this issue, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> ) we derive a new family of tightened ramping constraints of retrofitted coal-fired units in the linear and second-order conic (SOC) forms, respectively. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ii</i> ) Using graph characterization of typical HVDNs, the simplified voltage-constrained LTS via topological structures can be developed. This proposed look-ahead economic dispatch model is cast as a mixed-integer second-order cone programming (MISOCP) problem, which can be reformulated by Multi-cut Generalized Benders Decomposition (MGBD). This MGBD enables multiple sub-problems can be solved in parallel and accelerates the solving process. Finally, simulation results of a large-scale practical system validate the computational accuracy and efficiency of the proposed approach.

Supply Restoration of Data Centers in Flexible Distribution Networks With Spatial-Temporal Regulation

The reliable operation of Internet data centers (IDCs) is crucial to digitization transformation of society. However, the extreme fault on the IDC-located distribution network is fatal to IDC operation. The evolution of flexible distribution networks (FDNs) can satisfy the deep de-pendence of IDC on high-quality and reliable power supply. In FDNs, AC/DC feeders with multiple voltage levels can be flexibly interconnected and adjusted by flexible stations (FSs). It is promising to flexibly enhance operation per-formance of crucial IDCs after fault isolation in FDNs. In this paper, a coordinated supply restoration method of IDCs in FDNs with spatial-temporal regulation is proposed to minimize IDC data-dropping loss under complex faults. First, a spatial-temporal transfer strategy of IDC work-loads is precisely formulated and an FS-based network reconfiguration strategy is proposed. Then, a mixed-integer second-order conic programming model is established for efficient solving. Spatial-temporal coordination among the source, network, storage, and demand sides can be realized without violating physical feasibility. Finally, the proposed method is validated on the modified IEEE 33-node and modified IEEE 123-node test cases. Case studies show that total outage loss is effectively reduced with sufficient utili-zation of available power. Operational performance of FDN is also enhanced under extreme faults.

Stochastic model for active distribution networks planning: An analysis of the combination of active network management schemes

The widespread adoption of distributed generation (DG) is changing the operation of distribution networks. To increase the control of power flows and avoid the detriments of DG intermittency, active distribution networks (ADN) emerge as a feasible solution. In this paper, we propose a convex model for ADN expansion planning. Four active network management (ANM) schemes are considered: DG active power curtailment, DG reactive power control, on-load tap changer (OLTC) tap adjustment, and shunt compensation device control. The problem is formulated as a mixed integer second-order cone programming problem (MISOCP) using the distflow equations and second-order cone relaxations (SOCR). By scenario analysis, we address the uncertainty associated with power demand, renewable generation, and electricity prices. The proposed model is used to study the benefits of different combinations of ANM schemes to a modified IEEE 33-bus system and compare them with the individual benefits of each scheme. The results indicate that ANM schemes can reduce total network costs, reduce losses, delay network reinforcement, and improve the voltage profile. In addition, some combinations of schemes develop synergies in which the combined economic benefits are greater than the sum of the individual benefits.

Assessing Techno-Economic and Environmental Impacts of Gas Compressor Fleet as a Source of Flexibility to the Power System

In the Great Britain's high-pressure gas transmission network, there are 24 gas compressor stations with 68 operational compressor units of different types and sizes. As of 2021, the aggregate capacity of electric-driven gas compressors was 200 MW. By using linepack as a form of gas storage buffer, the compressors can be operated flexibly to achieve a minimum operational cost under fluctuating electricity prices. Furthermore, since electricity consumed by electric-driven compressors can affect the balance of the power grid, the ability of electric-driven compressor units to regulate their electricity consumption is useful to be quantified. In this paper, the value of flexibility from the hybrid compressor stations (compressor stations in which both electric-driven and gas-driven compressor units exist) is analysed. The operation of the high-pressure gas network in Great Britain was formulated as a Mixed-Integer Second Order Cone Programming (MISOCP) problem. The modelling results show that employing the flexibility from hybrid compressor stations in the gas network can reduce more than 20% of the operational cost and 50% of emissions from the compressor fleet. Furthermore, a cost-benefit analysis was carried out to assess the investment viability of installing electric-driven compressor units. It is demonstrated that the economic feasibility of investing in electric-driven compressors is sensitive to the future carbon price.