Mixed-integer Second-order Cone Programming Model Research Articles

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.

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.

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.

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.

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.

Digital twin‐based online resilience scheduling for microgrids: An approach combining imitative learning and deep reinforcement learning

AbstractStrong uncertainty of renewables puts high demands on the fast response of flexibility resources and resilience‐oriented optimal scheduling for microgrids (MGs). Digital twins (DT) technology based on data‐driven methods is a potential solution to this problem. A DT‐based online resilience scheduling framework for MGs is designed in this study. Based on the proposed framework, a hybrid sequential‐parallel combination method of imitation learning (IL) and deep reinforcement learning (DRL) is proposed to develop the optimal scheduling strategy for MGs. First, a mixed integer second‐order cone programming (MISOCP) model is adopted to behave as an expert to generate decision demonstrations corresponding to operation scenarios of MGs. Then, IL and deep deterministic policy gradient (DDPG) are combined by (1) sequential pretrain and finetune and (2) parallel experience storing and sampling two steps to learn the optimal policy for MGs. Expert demonstrations from the MISOCP model are utilized to pretrain a deep neural network model by IL and initialize the policy network of DDPG to avoid it learning from scratch. Moreover, an expert replay buffer is introduced specifically to avoid forgetting the well‐behaved expert experience and to further accelerate the training process. A 6‐bus MG test system case study demonstrates the effectiveness and scalability of the proposed approach.

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.

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.

Resilience-Oriented Co-Deployment of Remote- Controlled Switches and Soft Open Points in Distribution Networks

Soft open points (SOPs) can perform load transferring and voltage support, helping to expand the range of service restoration in distribution networks (DNs) under fault scenarios. In addition, in the field of DN restoration, SOPs are usually controlled in coordination with switch actions (e.g., the network reconfiguration). However, in the planning level, the coordinated deployment of SOPs and remote-controlled switches (RCSs) has never been studied in the literature. To this end, for the first time, this paper proposes a co-deployment framework for SOPs and RCSs incorporating active management of flexible resources for resilience improvement in DNs, in which the coupled multiple recovery stages are also modeled and included. The objective is to minimize the investment cost of SOPs and RCSs, along with the cost caused by de-energized loads, with the consideration of exhaustive DN operation constraints over proactive islanding, degradation, isolation and restoration stages. Furthermore, resilience-cost tradeoff is analyzed to help DN managers determine equipment investment plans. The formulated model is further transformed into a mixed-integer second-order cone programming model that can be thus efficiently solved. Simulation results on modified IEEE 34-node and IEEE 123-node test systems verify the effectiveness of the proposed model.

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.

Strategic active and reactive power scheduling of integrated community energy systems in day-ahead distribution electricity market

The increasing penetration of distributed energy resources (DERs) enable integrated community energy systems (ICESs) as emerging techno-economic energy entities at the end-user side. In the deregulated electricity market, the ICESs can actively participate in the distribution electricity market (DEM) to provide active and reactive power dispatched by ICES operator (ICESO). In this context, a strategic active and reactive power scheduling model for ICESs in the day-ahead DEM is proposed in this paper. A novel trading mechanism designed for ICESO and distribution system operator (DSO)’s interaction toward both active and reactive power is represented in the day-ahead DEM, where distribution locational marginal prices (DLMPs) for both active and reactive power are introduced as the market settlements. Then, the trading process is formulated as a bi-level programming problem. The upper level describes the combined cooling, heating, and power (CCHP)-dominated ICESO’s strategic active and reactive power dispatch, in which the DLMPs for active and reactive power, internal flexibility services, and optimal dispatch for inverter-based distributed generators are considered. The lower level performs a DEM clearing for both active and reactive power. Further, Karush–Kuhn–Tucker (KKT) conditions, Big-M approach, and duality theory are used to convert the bi-level model from a Mathematical Programing with Equilibrium Constraints (MPEC) model to a single-level mixed-integer second-order cone programming (MISOCP) model for efficient calculation. Finally, case studies on modified IEEE 33-node and IEEE 123-node distribution systems are adopted to evaluate the performance of the proposed scheduling method. The results show that the proposed scheduling approach can contribute to the renewable energy share and energy efficiency, reduce CCHP-dominated ICESs’ operation costs by 38.13% and 13.21% compared to the case where ICESO only acts as a price-taking consumer, and 6.64% and 3.44% compared to the case where ICESO does not consider reactive power bids/offers, respectively.

A Coordinated Restoration Method of Hybrid AC/DC Distribution Network for Resilience Enhancement

In recent years, the frequent occurrence of extreme natural disasters has caused huge economic losses, which makes it extremely important to improve the resilience of distribution networks. With the increasing penetration of DC sources and loads, the urban distribution network is transitioning from AC to hybrid AC/DC configuration that can operate in a ring structure. The features of flexible interconnections and low network losses of DC lines can break the bottleneck of traditional restoration methods for AC distribution networks under extreme disasters, thereby further enhancing the resilience of distribution networks. Based on the interconnection feature of DC lines, this paper proposes a topology search strategy with DC lines as the core to realize the joint recovery of multiple power sources and multiple critical loads. With the obtained interconnection topology after topology search, a fault restoration model for maximizing the resilience index is established. To ensure the generality of the proposed model and explore the advantages of flexible DC power control, this paper transforms the objective function from the complex model into a mixed integer second-order cone programming (MISOCP) that can be solved directly. The optimal restoration strategy for resilience enhancement of AC/DC hybrid distribution networks can be obtained by solving the proposed MISOCP model. The numerical results in case study validate the effectiveness and superiority of the proposed method.

Short-circuit constrained distribution network reconfiguration considering closed-loop operation

This paper presents a novel scenario-based stochastic mixed-integer second-order cone programming model to solve the problem of optimal reconfiguration of distribution systems with renewable energy sources considering short-circuit constraints. The proposed formulation minimizes technical losses by modifying the statuses of sectionalizing and tie switches, allowing the operation of distribution networks with radial and closed-loop topologies. Since the formation of loops could impact fault current levels, short-circuit constraints are considered in the problem formulation. Numerical experiments are carried out using an 84-node system and the results demonstrate the effectiveness of the proposed formulation to reduce technical losses notably when a closed-loop operation is allowed. Additionally, it is verified that short-circuit constraints prevent the adoption of network configurations with high short-circuit values.

Reliability-Oriented Reconfiguration of Power Distribution Systems Considering Load and RES Production Scenarios

The ever-increasing integration of non-dispatchable distributed generation, i.e., renewable energy sources (RES), arises new challenges in the field of power system's reliability. Distribution network reconfiguration (DNR) is a cost-effective approach for the distribution system operator (DSO) that wishes to enhance system's reliability without infrastructure upgrades. This paper introduces a novel path-based mixed-integer second-order cone programming model to optimally solve the reliability-oriented DNR problem. The DSO's objectives that are optimized are: a) improvement of distribution system's reliability indices and b) minimization of power losses. The proposed model is enriched with a scenario-based stochastic programming formulation that considers multiple levels of load and RES production. The standard 33-nodes distribution system and a real-world 83-nodes distribution system are employed to prove the efficiency and applicability of the model. Firstly, the multi-objective nature of the reliability-oriented DNR problem is investigated by conducting a sensitivity analysis, which reveals a trade-off region between reliability indices and power losses. Moreover, the obtained results show different global optimal solutions when the variability of load and RES production is considered. This highlights the importance of considering a scenario-based approach for load and RES production when solving the reliability-oriented DNR problem.

An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy

In order to adjust to the change of the large-scale deployment of photovoltaic (PV) power generation and fully exploit the potentialities of an integrated energy distribution system (IEDS) in solar energy accommodation, an evaluation method on maximum hosting capacity of solar energy in IEDS based on convex relaxation optimization algorithm is proposed in this paper. Firstly, an evaluation model of maximum hosting capacity of solar energy for IEDS considering the electrical-thermal comprehensive utilization of solar energy is proposed, in which the maximization of PV capacity and solar collector (SC) capacity are fully considered. Secondly, IEDS’s potential in electricity, heat, and gas energy coordinated optimization is fully exploited to enhance the hosting capacity of solar energy in which the electric distribution network, heating network, and natural gas network constraints are fully modeled. Then, an enhanced second-order cone programming (SOCP)-based method is employed to solve the proposed maximum hosting capacity model. Through SOCP relaxation and linearization, the original nonconvex nonlinear programming model is converted into the mixed-integer second-order cone programming model. Meanwhile, to ensure the exactness of SOCP relaxation and improve the computation efficiency, increasingly tight linear cuts of distribution system and natural gas system are added to the SOCP relaxation. Finally, an example is given to verify the effectiveness of the proposed method. The analysis results show that the maximum hosting capacity of solar energy can be improved significantly by realizing the coordination of an integrated multi-energy system and the optimal utilization of electricity, heat, and gas energy. By applying SOCP relaxation, linearization, and adding increasingly tight linear cuts of distribution system and natural gas system to the SOCP relaxation, the proposed model can be solved accurately and efficiently.

A convex-relaxation based method for optimal water-power flow

This paper proposes a convex reformulation for the non-linear optimal water-power flow (OWPF) problem to optimize the operation cost of the integrated electricity–water network (IEWN). Due to the significant challenge in seeking the optimal solution of non-convex problems, the original OWPF problem is reformulated as a tractable mixed-integer second-order cone programming (MISOCP) problem. For the power distribution network, a second-order cone (SOC) relaxation is employed to address the non-linear branch power flow equation. For the water distribution network, a big-M trick is introduced to handle the unknown direction of water flow, then SOC relaxations and convex envelopes are employed to deal with quadratic and bilinear terms in non-convex constraints, respectively. To enhance the approximation accuracy of the proposed MISOCP model, a convex combination method and a sequential convexification approach are developed. Numerical results demonstrate that the proposed method outperforms the original non-linear formulation and linearized reformulation in accuracy, efficiency, and robustness.