Second-order Cone Programming Model 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 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.

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.

Optimal Scheduling of New Energy and Thermal Power Combined Operation System

Nowadays, the power grid with a high proportion of renewable energy is increasingly popular. Taking a new energy power system including wind power (WP), photovoltaic power (PP), and pumped storage power (PSP) as an example, in order to solve the problem of resource waste such as massive WP-PP abandonment and power losses caused by the randomness and volatility of new energy, this paper establishes a combined power generation model of WP-PP-PSP-thermal power (TP). It achieves the goal of minimum network loss and minimum WP-PP abandonment in the combined power generation system according to the characteristics of PSP and considering various constraints and distribution network reconfiguration. Then, the second-order cone relaxation (SOCR) technology is used to transform the model into a mixed second-order cone programming model. Finally, the design scheme is tested by the improved IEEE 33-bus system, and the effectiveness of the algorithm is analyzed. The results show that peak cutting and valley filling can be realized when PSP energy is added to the WP-PP grid connection system.

A Robust Conic Programming Approximation to Design an EMS in Monopolar DC Networks with a High Penetration of PV Plants

This research addresses the problem regarding the efficient operation of photovoltaic (PV) plants in monopolar direct current (DC) distribution networks from a perspective of convex optimization. PV plant operation is formulated as a nonlinear programming (NLP) problem while considering two single-objective functions: the minimization of the expected daily energy losses and the reduction in the expected CO2 emissions at the terminals of conventional generation systems. The NLP model that represents the energy management system (EMS) design is transformed into a convex optimization problem via the second-order cone equivalent of the product between two positive variables. The main contribution of this research is that it considers the uncertain nature of solar generation and expected demand curves through robust convex optimization. Numerical results in the monopolar DC version of the IEEE 33-bus grid demonstrate the effectiveness and robustness of the proposed second-order cone programming model in defining an EMS for a monopolar DC distribution network. A comparative analysis with four different combinatorial optimizers is carried out, i.e., multiverse optimization (MVO), the salp swarm algorithm (SSA), the particle swarm optimizer (PSO), and the crow search algorithm (CSA). All this is achieved while including an iterative convex method (ICM). This analysis shows that the proposed robust model can find the global optimum for two single-objective functions. The daily energy losses are reduced by 44.0082% with respect to the benchmark case, while the CO2 emissions (kg) are reduced by 27.3771%. As for the inclusion of uncertainties, during daily operation, the energy losses increase by 22.8157%, 0.2023%, and 23.7893% with respect to the benchmark case when considering demand uncertainty, PV generation uncertainty, and both. Similarly, CO2 emissions increase by 11.1854%, 0.9102%, and 12.1198% with regard to the benchmark case. All simulations were carried out using the Mosek solver in the Yalmip tool of the MATLAB software.

Second-Order Cone Programming (SOCP) Model for Three Phase Optimal Power Flow (OPF) in Active Distribution Networks

With the high penetration of distributed generations (DGs), modern distribution systems require novel and efficient optimal power flow (OPF) models. This paper proposes a second-order cone programming (SOCP) based AC-OPF model for three-phase radial power distribution networks. Mutual coupling effects are generally ignored in the existing multi-phase SOCP AC-OPF models. To overcome this critical issue, the proposed SOCP-OPF model introduces a coupling coefficient for the mutual coupling effects on the three-phase unbalanced lines. The derivation of the coupling coefficients has been illustrated with the required proof that the relaxation is tight and the solution from the proposed OPF model is optimal for an unbalanced multi-phase distribution network. The SOCP-OPF model is evaluated on several IEEE standard distribution networks without and with high penetration of Distributed Energy Resources (DERs). It has been shown that the proposed OPF model provides an optimal global solution for convex objective functions. Also, this OPF model is scalable and computationally efficient compared to Nonlinear Programming (NLP) and other convex OPF models.

Mixed-integer second-order cone programming method for active distribution network

Developing a novel type of power system is an important means of achieving the “dual carbon” goals of achieving peak carbon emissions and carbon neutrality in the near future. Given that the distribution network has access to a wide range of distributed and flexible resources, reasonably controlling large-scale and adjustable resources is a critical factor influencing the safe and stable operation of the active distribution network (ADN). In light of this, the authors of this study propose a mixed-integer second-order cone programming method for an active distribution network by considering the collaboration between distributed, flexible resources. First, Monte Carlo sampling is used to simulate the charging load of electric vehicles (EVs), and the auto regressive moving average (ARMA) and the scenario reduction algorithms (SRA) based on probability distance are used to generate scenarios of the outputs of distributed generation (DG). Second, we establish an economical, low-carbon model to optimize the operation of the active distribution network to reduce its operating costs and carbon emissions by considering the adjustable characteristics of the distributed and flexible resources, such as on-load tap changer (OLTC), devices for reactive power compensation, and EVs and electric energy storage equipment (EES). Then, the proposed model is transformed into a mixed-integer second-order cone programming (SOCP) model with a convex feasible domain by using second-order cone relaxation (SOCR), and is solved by using the CPLEX commercial solver. Finally, we performed an arithmetic analysis on the improved IEEE 33-node power distribution system, the results show that ADN’s day-to-day operating costs were reduced by 47.9% year-on-year, and carbon emissions were reduced by 75.2% year-on-year. The method proposed in this paper has significant effects in reducing the operating cost and carbon emissions of ADNs, as well as reducing the amplitude of ADN node voltages and branch currents.

Distributed coordinated planning for cross-border energy system: An ADMM-based decentralized approach

Cross-border interconnection and trade (CBIT) can effectively leverage the differences in resource endowments and energy demand between regions and countries, making cross-border cooperation a potential option for addressing the high penetration of variable renewable energy. This paper presents centralized and decentralized planning approaches for the cross-border energy transmission expansion planning (TEP) problem. The proposed centralized mixed-integer second-order cone programming (MISOCP) model collaboratively optimizes the expansion of national energy networks, while considering the impact of CBIT. To protect the privacy of energy data in each country, the alternating direction method of multipliers (ADMM) algorithm is extended to decompose the centralized optimization formulation into mutually independent sub-optimization issues, and adaptively adjust the penalty parameters and Lagrangian multipliers to improve convergence. This study also quantifies the effect of cross-border cooperation in such co-expansion problems as a comparative consideration. Numerical simulations on the modified cross-border integrated energy network validates the correctness and effectiveness of the proposed model and solution methodology. The results show that cross-border cooperation can reduce a country's internal energy network congestion, thus postponing the expansion investment decisions. It also shows that the distributed coordinated planning is more efficient in terms of lower cost and renewable generation curtailment that than independent optimization method. Therefore, cross-border collaboration brings economic, environmental, and renewable integration benefits to interconnected countries.

Optimal resilience enhancement dispatch of a power system with multiple offshore wind farms considering uncertain typhoon parameters

With the increasing penetration of offshore wind power capacity into power systems, extreme typhoon events have been severely threatening the secure operation of a power system with multiple offshore wind farms (OWFs). To alleviate this problem, this paper proposes an optimal resilience enhancement dispatch (ORED) framework consisting of three stages: preventive control, emergency response, and rapid restoration. Combining the wind field model of typhoon and the probability distribution model of typhoon parameters, a Wasserstein distance-based ambiguity set (AS) is constructed for describing the uncertainty of typhoons. In addition, a distributionally robust chance constraint-based ORED (DRCC-ORED) model, which considers uncertain typhoon parameters, is established for a power system with multiple OWFs. A conditional value-at-risk (CVaR) approximation method is used to transform the DRCCs with random variables into linear constraints, and the expectation operation of the cost function under the worst probability distribution is reformulated. Further, the proposed bi-level DRCC-ORED model is transformed into a single-level mixed-integer second-order cone programming model, which can be efficiently solved by the commercial solver GUROBI. Case studies on the modified IEEE 39-bus system with an OWF and an actual provincial power system with four OWFs demonstrate the high effectiveness of the proposed model.

Economic/Environmental Optimal Power Flow Using a Multiobjective Convex Formulation

This paper addresses the problem of economic/environmental optimal power flow with a multiobjective formulation using a second-order conic programming (SOCP) optimization model. This problem formulation considers renewable energy sources (RES), fossil-fuel-based power generation units, and voltage control. The proposed SOCP model is a stochastic scenario-based approach to deal with RES and load behavior uncertainties. An ε-constrained algorithm is used to handle the following three objective functions: (1) the costs of power generation, (2) active power losses in the branches, and (3) the emission of pollutant gases produced by fossil-fuel-based power generation units. For comparative purposes, the SOCP model is also presented using a linearized formulation, and numerical results are presented using a 118-bus system. The results confirm that changing the energy matrices directly affects the cost of objective functions. Additionally, using a linearized SOCP model significantly reduces reactive power violation in the generation units when compared to the nonlinearized SOCP model, but also increases the computational time consumed.

Analysis of strategic bidding of a DER aggregator in energy markets through the Stackelberg game model with the mixed-integer lower-level problem

An open electricity market environment provides sufficient conditions for a distributed energy resource (DER) aggregator to participate in the day-ahead electricity market. Therefore, a one-leader multi-follower Stackelberg game model is developed to analyze the strategic bidding behavior of a DER aggregator and one of the lower-level problems is to check the security of the distribution system (DS), which is elaborated by the mixed-integer second-order cone programming (MISOCP) model considering the active power loss and discrete controls in the DS. Moreover, we first use the Karush–Kuhn–Tucker (KKT)-based reformulation approach to transform the model into a one-leader one-follower model. However, the objective function of the upper-level model cannot be completely linearized by using the strong dual theorem because the active power loss of the DS is considered in the lower-level security check problem. Therefore, we propose a piecewise linear approximation approach based on the data-driven algorithm to linearize the upper-level objective function. Moreover, we apply multivariate linear surrogation to transform this one-leader one-follower model into a single-level MISOCP model. Finally, we verify the correctness of the proposed algorithm by comparing it with the particle swarm optimization (PSO) algorithm in a small integrated transmission and distribution (T&D) system and validate the effectiveness of the proposed algorithm in a practical integrated T&D system.

A Second-Order Cone Programming Model of Controlled Islanding Strategy Considering Frequency Stability Constraints

Controlled islanding is an important defense mechanism for avoiding blackouts by dividing the system into several stable islands. Sustainable systems that incorporate a high proportion of renewable energy are prone to frequency instability or even severe blackout events due to extreme weather conditions. Thus, it is critical to investigate controlled islanding considering frequency stability constraints to reduce the risk of a sustainable system collapse in extreme weather conditions. Here, the frequency constraint of islands is derived based on the law of energy conservation, and the island connectivity constraint is proposed based on the idea of network flow. A controlled island second-order cone programming model with frequency stability constraints is proposed for the islanding strategy. The consideration of frequency constraints can help to avoid islands with too low inertia generated by the islanding strategies, ensuring that the frequency nadir of the island remains within a safe range after disturbance. Connectivity constraints can ensure connectivity within the island and no connectivity between different islands. The model also meets the reactive power balance and voltage limits in the system. Simulations of the three test systems show that this island model, which takes frequency stability into account, is effective in reducing the risk of sustainable power system collapse in extreme weather conditions.

Security-Constrained Optimal Traffic-Power Flow With Adaptive Convex Relaxation and Contingency Filtering

The ongoing transportation electrification [e.g., the proliferation of electric vehicles (EVs)] strengthens the interdependence between power and transportation networks and brings tremendous operational challenges to the two critical infrastructure systems. This article proposes an N–1 security-constrained optimal traffic-power flow (SCOTPF) model that includes bilateral contingencies (closure of traffic link and outage of charging station) to coordinate the two networks toward N–1 secure and reliable. The partial user equilibrium (PUE) criterion is adopted to capture the driver’s rerouting behavior in response to the contingency. To facilitate the model tractability, we exploit an adaptive piecewise convex relaxation method based on the convex hull and McCormick envelope to derive a mixed-integer second-order cone programming (MISOCP) model and design a binding contingency identification algorithm to screen redundant contingencies. Finally, the optimal preventive operation strategy is efficiently obtained to reallocate the precontingency traffic-power flow and hence ensures the N–1 reliability and security of the coupled networks. Numerical results on two test systems validate both solution quality and computational efficiency of the SCOTPF. Meanwhile, interdependence between the two networks in contingency scenarios is thoroughly analyzed.