The development of multi-microgrid distribution systems (MMDSs) has emerged as a promising approach to en hancing power system flexibility, and their coordinated optimal scheduling has attracted increasing attention in recent years. However, due to the low inertia of microgrids (MGs), inappropriate scheduling strategies may expose MGs to severe frequency security issues following unintentional islanding events (UIEs). Moreover, pri vacy concerns and communication burdens render centralized optimization approaches impractical in real-world applications. To address these challenges, this paper proposes a non-iterative coordinated optimal scheduling method for MMDSs that explicitly accounts for MG frequency security under UIEs. A frequency-trajectory-based approximation method is developed to decouple the closed-loop relationship between frequency dynamics and power response. Through monotonicity analysis, an approximate estimation of the primary frequency response power at the maximum frequency deviation point is derived, enabling the formulation of linearized frequency constraints. Furthermore, a virtual energy storage-based representation (VESR) method is proposed to project the complete feasible region of each MG onto the interface power variable space. This formulation enables non-iterative and privacy-preserving coordination between the distribution network and multiple MGs through the aggregated feasible region. To further accelerate the VESR method, a cardinality and location constrained strategy is developed. Case studies conducted on a modified IEEE 33-bus MMDS demonstrate that the proposed scheduling method effectively ensures MG frequency security while achieving efficient coordination.

Coordinated operation of the distributed energy sources from power distribution system (PDS), district heating system (DHS) and gas distribution system (GDS) is essential for service restoration (SR) of integrated energy distribution system (IEDS) when outage accident occurs. However, the uncertainty of wind power may not only exacerbate the power imbalance in PDS but also affect heat and gas supply continuity through coupling units. In this paper, a stochastic SR method for wind power penetrated IEDS is proposed to enhance the robustness of restoration schemes. The multi-energy islands are jointly formed and can be dynamically adjusted to improve the risk-resistance of SR strategies, particularly in the face of subsequent damage during SR process. A rolling-horizon optimization approach is utilized to dynamically update the information of subsequent damage and mitigate the inaccuracy of wind power forecast over the prolonged SR process. The IEDS SR problem is represented by a two-stage stochastic programming (SP) model within the prediction horizon to address uncertainties. Furthermore, a scenario generation approach is developed that accounts for spatial-temporal correlation and provides reasonable probability information of wind power for the SP-based SR strategy. The spatial correlation is captured using a Gaussian mixture model (GMM), while the temporal correlation is characterized by an exponential function. To enhance computational efficiency, Gibbs sampling is applied on the basis of the conditional joint distribution derived from the GMM. Case studies show the effectiveness of the proposed SR method in reducing multi-energy load shedding and mitigating the negative influence of unexpected scenarios.

In recent years, the increasing frequency of extreme disaster events has resulted in more frequent large-scale power outages in urban distribution networks, posing a significant threat to their power supply security. To enhance disaster response capabilities, this paper proposes a two-stage optimization approach that incorporates uncertainty, addressing both pre-disaster prevention and post-disaster repair. In the pre-disaster stage, a two-stage robust optimization model is developed based on the uncertainty of outage failures. The Big-M method, combined with a column-and-constraint generation algorithm, is employed to iteratively determine the optimal number and locations of mobile power sources. In the post-disaster stage, mobile power sources are reallocated according to actual fault conditions, and repair crews are dispatched to restore damaged equipment. Based on the operational constraints of the distribution network, a post-disaster restoration optimization model is then formulated with the objective of maximizing load recovery, which is linearized into a mixed-integer linear programming (MILP) formulation. Finally, the effectiveness of the proposed method is verified through simulations on an improved IEEE 33-bus distribution network system.

Reliability evaluation of park-level integrated energy systems (PIES) requires capturing the complex interactions among electricity, gas, thermal, and building subsystems, which exhibit typical multiscale dynamic behaviors. Although existing studies have partially considered the dynamic characteristics of PIES, the critical transient responses and the long-term impacts of heat supply interruptions on human health are overlooked. To address these gaps, this paper proposes a reliability evaluation method based on quasi-steady-state simulation. The dynamic behaviors of the electricity, gas, and heating systems, as well as buildings, are analyzed to develop simplified quasi-steady-state models for both energy demand under normal conditions and indoor temperature evolution under extreme scenarios. A new reliability index is then formulated by incorporating the thermal inertia of district heating networks and the temperature tolerance of occupants, forming a comprehensive framework for PIES reliability assessment. Case studies verify that the proposed method enhances the accuracy and interpretability of reliability evaluations, bridging the gap in user health considerations under extreme conditions. The results further reveal that building thermal dynamics play a dominant role in determining the overall reliability performance of PIES.

With the accelerating integration of distributed energy resources (DERs), multi-microgrid distribution systems (MMDSs) have become a promising paradigm for enhancing system flexibility and resilience. However, achieving efficient coordination between distribution network (DN) and microgrids (MGs), while coping with significant renewable energy resource (RES) uncertainty, remains a key challenge in MMDS operation. This paper proposes a non-iterative privacy-preserving distributionally robust chance-constrained scheduling model for MMDSs. The model co-optimizes DN controllable generation, reserve deployment, network topology, and DN–MGs power exchanges. To enable efficient and privacy-preserving coordination, we develop a binary-boundary outer approximation (BBOA) method to aggregate MG flexibility, allowing the distribution system operator (DSO) to conduct non-iterative coordination based solely on each MG’s aggregated feasible region (AFR). In addition, a hierarchical strategy is introduced to constrain the cardinality of AFR boundary coefficients, thereby further improving computational efficiency of BBOA. To address RES uncertainty, we propose a physically bounded Wasserstein-based distributionally robust chance-constrained (PBW–DRCC) method. The PBW–DRCC approach explicitly incorporates physical bounds on random variables, thereby preventing physically unrealistic reserve allocations that may arise from exact reformulations. A sequential convex optimization algorithm is developed to handle the nonconvex bilinear constraints introduced by the PBW–DRCC approach. Numerical experiments on a modified IEEE 33-bus MMDS demonstrate that the proposed framework enables secure and economical MMDS operation. The BBOA achieves an average AFR approximation error below 0.2085 %, and AFR-based scheduling yields only 0.0252 % optimality loss compared with centralized coordination. Moreover, the PBW–DRCC approach attains a favorable trade-off between cost efficiency and robustness while maintaining physical feasibility and avoiding unnecessary conservativeness.

Integrated Community Energy Systems (ICES) aim to optimize energy efficiency through the integration of diverse energy resources, encompassing electricity, heating systems, and natural gas. However, the rapid integration of renewable energy sources and the rising energy demands pose significant challenges in evaluating the reliability of ICES. This difficulty arises from the need to evaluate numerous system states to determine the minimal load curtailment. To address this issue, we propose a method based on optimal bases to enhance computational efficiency in the reliability assessments of ICES. The optimal load curtailment model is developed to facilitate system state evaluation, accounting for variations in load levels and renewable generation. Subsequently, the optimal basis is employed to accelerate this evaluation process. By matching most system states with their corresponding optimal basis based on the optimality criterion, efficient computation of optimal load curtailment is achieved through matrix multiplications, eliminating the need for time-consuming optimization algorithms. The efficacy of the optimal basis-based method is validated through comprehensive case studies.

In this paper, we investigate the resilience security region (RSR) of distribution networks (DNs) with renewable energy, enabling system operators to explicitly characterize the admissible envelope of renewable power injections and support informed dispatch decisions under extreme events. First, sequential line-failure constraints and operational security constraints of the DN under extreme conditions are formulated. Then, the full feasible region is projected onto the renewable power injection space, and a vertex–enumeration–based progressive outer approximation (VE-POA) method is developed to derive an analytical approximation of the RSR. A boundary-tightening procedure is further introduced to eliminate the duality gap associated with second-order cone constraints. Numerical results on a modified IEEE 33-bus system demonstrate that the proposed method accurately constructs the RSR under extreme weather scenarios and highlights its potential in resilience-oriented operational planning.

The large-scale integration of renewable power generation (RPG) significantly increases operational variability, making it essential to characterize the operational boundaries of RPG under system-level security requirements, which are referred to as the operational security region (OSR). Given the inherent difficulty of exactly characterizing the OSR, this paper proposes a novel binary-encoded hyperplane generation (BEHG) method to efficiently approximate the OSR of RPG in large-scale power systems. Unlike most existing studies, this work considers time-coupled operational constraints rather than OSR characterization under a single time period. The proposed BEHG algorithm starts from a sufficiently large initial set and progressively refines the OSR approximation by identifying violating points within the current region and introducing hyperplanes with binary coefficients to eliminate infeasible portions. Furthermore, a cardinality-constrained strategy and a time–space decomposition strategy are developed to accelerate computation and reduce the problem size, respectively. Numerical tests on the IEEE 9- and 39-bus systems demonstrate that, for 24-period operational problems, the proposed method achieves an OSR approximation error within 1%, with a maximum computation time of no more than 30 seconds.

As a flexible demand-side management approach, demand response (DR) can effectively alleviate power supply– demand imbalances and has gradually become an essential mechanism for routine power system regulation. However, practical applications of demand response are often challenged by uncertainties arising from external environments and user behaviors, which affect optimization outcomes and implementation effectiveness. To bridge this gap, this paper proposes an improved incentive-based DR model that simultaneously incorporates both external and behavioral uncertainties. Specifically, external uncertainty is characterized through a response uncertainty factor, while behavioral uncertainty is captured by introducing the degree of user rationality. The proposed model is initially formulated as a bi-level optimization problem involving interactions between the load aggregator (LA) and users, and subsequently transformed into an equivalent single-level optimization problem to facilitate efficient computation. Simulation results confirm the effectiveness of the proposed model by demonstrating significant reductions in the total cost for LA and notable improvements in the efficiency and accuracy of demand response implementation.

Modern power systems exhibit continuously varying operating conditions. To assess transient voltage stability in this context, the varying operating conditions are characterised as variable parameters in parameter space, enabling the investigation of the transient voltage stability region boundary (TVSRB). The TVSRB represents a hypersurface dividing the parameter space into stable and unstable subspaces in terms of transient voltage stability. To determine the TVSRB in parameter space, first, a smoothness-enhanced transient voltage stability index (SETVSI) based on the transient voltage stability criterion is defined, which has a critical value along with tunable coefficients to enhance the smoothness of the implicit function between SETVSI and variable parameters. Second, polynomial approximation based on collocation point method is adopted to approximate this implicit function with an explicit polynomial. Third, based on the explicit polynomial, the principle of TVSRB determination in parameter space is introduced, and implementation issues are discussed. Finally, the accuracy and practicality of the proposed method are verified in the IEEE 9-bus system and an actual system.

The large-scale integration of distributed energy resources (DERs) into distribution networks (DNs) enhances operational flexibility, but it also significantly increases system management complexity. This paper investigates the DER aggregation problem while explicitly incorporating DN network constraints. This problem can be essentially described as projecting the full feasible operating region of the DN onto the substation interface power variable space to obtain the aggregated feasible region (AFR). To efficiently construct the AFR, we propose a virtual-energy-storage-based inner-approximation (VES-IA) method. First, the AFR is represented by a virtual energy storage (VES) polytope characterized by power bounds and inter-temporal energy-coupling constraints. Then, a violation-point identification procedure is developed to locate the point within the current VES that deviates most from the exact AFR. By exploiting structural properties of the exact AFR, this identification problem is reformulated as a mixed-integer linear programming model whose number of binary variables depends only on the number of time periods. The polytope boundary is then iteratively contracted based on the identified violation point until it is fully contained within the exact AFR. The effectiveness of the proposed method is validated on modified IEEE 33-bus test systems and a practical distribution network in China with multiple DERs.

As distribution networks transition toward multi-voltage-level active systems with increasing integration of distributed energy resources (DERs), the demand for standardized multi-voltage-level datasets for validating power flow and reliability methods is becoming increasingly urgent. This paper introduces three topology-centric distribution network datasets: SGBDC-208 (urban multi-station interconnected ring), SGBDC-95 (industrial park with multiple rings and a dedicated 35 kV feeder), and SGBDC-45 (rural long-radial feeder with a 10 kV study segment).The datasets span 220/110/35/10 kV and adopt a unified data schema with modular interfaces to enable scalable cross-scenario analyses across heterogeneous feeder configurations and DER-oriented studies. Time-series power-flow (TSPF) simulations and reliability assessments based on the Incremental Impact State Enumeration (IISE) method are conducted using photovoltaic (PV)-based case studies to validate electrical feasibility and reproducibility under typical operating conditions. The results demonstrate consistent operating behavior across urban, industrial, and rural scenarios, with IISE indices closely matching those obtained by full state enumeration. These datasets provide an algorithm-agnostic and reproducible platform for evaluating reliability, operational strategies, and future DER integration in modern distribution networks.

With global climate change intensifying, flood disasters are becoming more frequent, threatening the safe and stable operation of critical urban infrastructure. Urban distribution systems, as key energy supply hubs, are particularly vulnerable—often suffering substation inundation, pole collapses, and underground cable faults—leading to large- scale and prolonged power outages. This paper proposes a resilience assessment framework for urban distribution networks under flood scenarios. Flood events are modeled by integrating equipment and line damage models with inundation data. The system’s resilience is then quantitatively evaluated, and vulnerable components are identified to guide resilience enhancement and recovery planning. A case study on a typical distribution network in Erqi District, Zhengzhou, is conducted to validate the method. Simulation results demonstrate the framework’s effectiveness in assessing system resilience and locating critical weaknesses, providing technical support for flood risk mitigation and emergency response in urban energy systems.

The dispatch of demand-side resources is becoming increasingly essential for enhancing the resilience of distribution systems against extreme weather events. Traditional studies rely primarily on direct load curtailment to mitigate power shortages, thereby neglecting the power demand of different customers. To bridge this gap, a novel coordination method of transactive demand response (DR) and rolling outage management of diverse loads is proposed. First, the DR program is designed to in-centivize voluntary load adjustments by coordinating participation of private consumers. Considering the diversity and characteristics of customers during DR, detailed models of diverse loads are established, including an energy-material flow model of industrial loads (ILs), an adjustable load model of commercial loads (CLs), and an outage-sensitive model of residential loads (RLs). When the supply-demand imbalance exceeds the adjustment capacity of DR, rolling outage measures are integrated into the proposed method to reduce losses incurred by load shedding. The coordination of transactive DR and rolling outage management is formulated as a bilevel optimization problem from system operators and responsive load, which is solved by the Stackelberg game-theoretic approach. Finally, the proposed method is tested on the modified IEEE 33-node distribution system. The results show that the proposed method can effectively ensure customer profit and reduce load interruption loss by dispatching demand-side resources during restoration.

Coordinated restoration of transmission systems (TSs) and distribution systems (DSs) is necessary to reduce load loss after blackouts. Traditional coordination strategies are studied based on the hypothesis of no faults, neglecting fault repairs. To address this issue, a novel coordination method is proposed to integrate the repair and restoration of a coupled TS and DS with repair crews (RCs) and mobile emergency generators (MEGs) during post-disaster repair. Combining fault repairs and generator restarts by RCs and MEGs, a four-stage parallel restoration model is developed to coordinate dynamic partitioning, generation start-up, transmission path energization and load restoration, which facilitates rapid restoration of coupled TS and DS. Due to limited information exchange and data confidentiality between TS and DS, the decentralized coordination model is decomposed into a TS subproblem and DS subproblem based on the alternating direction method of multipliers (ADMM). To address non-convexity issue due to binary variables, the convergence and optimality of the decentralized model are ensured by the alternating optimization procedure (AOP) method. Finally, the proposed method is validated on small-scale T14-D13 and large-scale T39-D33 systems. The simulation results show that the proposed approach can effectively increase load recovery in the coupled TS and DS during post-disaster repair.

In extreme weather conditions, effective demand-side resource scheduling is crucial for enhancing distribution system resilience. Traditional approaches, which rely on direct load curtailment, often overlook the diverse needs of customers. This paper presents a transactive demand response (DR)-based framework that incentivizes private consumers to voluntarily adjust their loads. To account for customer diversity, detailed load models are developed for industrial, commercial, and residential sectors. A bi-level optimization model is formulated to coordinate the interaction between the distribution system operator and responsive loads, leveraging a Stackelberg game-theoretic approach. Simulations on a modified IEEE 33-node system demonstrate that the proposed method not only ensures customer profitability but also minimizes load interruption losses by optimizing the dispatch of demand-side resources during system restoration.

Microgrid formation provides a viable solution for enhancing the resilience of distribution systems under extreme conditions. In general, the on-outage areas of the distribution system are partitioned into multiple islands to restore the critical loads after faults have occurred. However, the restored services could be interrupted by subsequent contingencies in extended extreme events with long-lasting destructive effects. To increase the risk resistance of formed microgrids, we propose an adaptive microgrid formation method that considers subsequent line faults and outage propagation. First, a strengthened ambiguity set of contingency probability distributions is constructed to depict the uncertainty of subsequent line faults. Moreover, a progressive detection mechanism is introduced to estimate outage propagation and identify the affected nodes following intentional islanding and fault isolation. Afterward, a three-layer distributionally robust optimization model is developed to maximize the expected load restoration with respect to the worst-case distribution of contingencies. The proposed distributionally robust optimization model is further transformed into a robust optimization model to facilitate the column-and-constraint generation algorithm. Simulation results validate the merits of the proposed method in improving the resilience of distribution systems during an extended extreme event by forming risk-resistant microgrids.

Over the past two decades, there have been frequent large-scale power outages worldwide caused by voltage collapse, resulting in significant economic losses. With the development of technologies such as renewable energy and HVDC (high voltage direct current) transmission, the power system is gradually moving towards a trend of high penetration of renewable energy and a high proportion of power electronic devices. As a result, the mechanism of transient voltage stability is becoming increasingly complex, posing serious challenges to transient voltage stability analysis. Therefore, it is of great importance to quickly and effectively assess and quantify transient voltage stability for the safe and stable operation of the system. This paper first explores the concept of transient voltage stability, and then provides a detailed review and analysis of transient voltage stability practical criteria in various countries and regions, summarizing a more adaptable and practical three-stage transient voltage stability criterion. Additionally, it also classifies and summarizes existing transient voltage stability assessment indices and highlights three key aspects in the future research of the indices.

Reliability studies of distribution networks are constrained by the lack of unified and reproducible experimental platforms. To address this gap, we construct four MATPOWER-compatible benchmark datasets: a 26-bus rural system, a 35-bus suburban/industrial system, a 47-bus urban system, and a 108-bus integrated system. These datasets incorporate multiple voltage levels, representative wiring configurations, and diverse load types, along with standardized N-k contingency generation and minimum load-shedding OPF evaluation. Baseline experiments demonstrate that all scenarios are solvable. As disturbance severity increases, the Supply Ratio (SR) declines, while heuristic and upper-bound methods yield substantial improvements in SR and significant reductions in Expected Energy Not Supplied (EENS), with benefits amplified under severe conditions. The scenario distribution follows a long-tail pattern in which both "easy" and "hard" cases coexist, highlighting the datasets' discrimination power and robustness. Collectively, these datasets establish a unified, scalable, and reproducible platform that enables rigorous benchmarking and accelerates the advancement of reliability enhancement methods for distribution networks.

The reliability of Regional Integrated Energy Systems (RIES) and Community Integrated Energy Systems (CIES) is a prerequisite for continuous high-quality energy supply. However, optimizing multi-agent operations for RIES and multi-CIESs based on optimal energy flow remains challenging, especially under fault scenarios. To address this, a Reliable Economic Operation model for RIES and Multi-CIESs (REO-RMC) is proposed. This model ensures the reliable operation of RIES and CIES through a hybrid strategy of Stackelberg and cooperative games while optimizing the operation of multi-CIESs and achieving optimal profit distribution. To improve the efficiency of the hybrid game, a State Similarity (SS) method is proposed. The SS method converts uncertainties (such as renewable energy and energy prices) in the linear parts of REO-RMC into parameter fluctuations within multiparameter programming, constructing critical regions (CR). Subsequently, equation solving replaces the optimization process for all problems within the CR, improving computational efficiency. Moreover, a framework for Offline CR Generation and Online CR Identification (OGOI) is proposed to further enhance efficiency. Results show that the REO-RMC framework effectively enhances the economic operation of the CIES alliance while ensuring the reliability of RIES. Furthermore, the SS method reduces computation time to less than 50 % of that required by bilevel-programming.