ABSTRACT The probabilistic optimal power flow (POPF) model aims to quantify the statistical features of POPF outputs, which are affected by hundreds of uncertain POPF inputs. This paper sets out to develop a computationally feasible cubature rule for POPF computation. In the case where POPF inputs include correlated random variables, a Gaussian copula is introduced to accommodate the dependence structure of POPF inputs; the generalized lambda distribution is employed to fit the marginal distribution of each POPF input, and empirical formulae are derived to specify the dependence structure of POPF inputs in the standard normal space. With the aid of moment matching equations, optimization procedures are implemented to derive a D-dimensional cubature rule. Then, this D-dimension cubature rule is expanded to a higher dimension by using Hadamard matrix and Kronecker product; if a Hadamard matrix of order 2 R is employed, the resultant nested cubature rule can be used to handle a POPF model including m inputs: m≤D(2 R −1). Finally, case studies are performed on a modified IEEE 118-bus system to check the proposed methodologies, the results indicate the feasibility, efficiency, and accuracy of nested cubature rules.

Survey on power flow calculation in distribution networks: Intelligent algorithms driven by data-physics fusion

Car exhaust emissions significantly contribute to the depletion of the ozone layer. Electric vehicles (EVs) present a sustainable alternative to mitigate this environmental issue. However, the large-scale adoption of EVs introduces challenges for the power grid, primarily due to irregular and uncoordinated charging patterns. This study proposes a comprehensive two-stage framework for optimizing electric vehicle (EV) charging patterns and reactive power dispatch within power distribution systems. In Stage 1, two types of EV charging schedules are developed and compared: day-ahead charging and real-time charging. Day-ahead charging involves planning EV charging over a 24-hour horizon with the objective of minimizing load variance, energy cost, active power losses, and voltage drop, while simultaneously maximizing voltage stability. Real-time charging dynamically adjusts charging behavior based on immediate grid conditions to minimize load variance and charging costs. Stage 2 focuses on optimal real-time reactive power dispatch, utilizing the reactive power capabilities of EV inverters to further reduce the active and reactive power losses. Additionally, the study analyzes EV behavior in response to sudden load changes, providing critical insights for enhancing grid performance. Different optimization algorithms are implemented to efficiently solve the proposed models, including particle swarm optimization, dandelion optimization, wild horse optimization, and slime mould optimization. The optimization is formulated as a multi-objective problem to consider both grid constraints and customer satisfaction. The proposed framework is applied and tested on a 33-bus radial distribution system with 984 electric vehicles using MATLAB M-files, while power flow calculations are performed using the MATPOWER toolbox. Simulation results demonstrate the effectiveness of the proposed framework. Daily active power losses are reduced from 4.04 MWh to 2.55 MWh and 2.77 MWh under day-ahead and real-time planning strategies-representing reductions of 36.8% and 31.4%, respectively. Similarly, EV charging costs drop from 552.31 USD to 394.19 USD and 363.68 USD, achieving cost savings of 28.63% and 34.15%. Furthermore, voltage profiles are maintained within the acceptable operational limit of 0.95 p.u. These outcomes highlight the significant advantages of the proposed methodology in enhancing grid efficiency while ensuring user satisfaction.

This paper focuses on the problem of power system inertia decline caused by changes in the composition and form of system inertia in power systems with a high penetration of non-synchronous power sources. Aiming at the scenario of high penetration of non-synchronous power sources, this paper expands the concept of synchronous generator inertia, proposes the inertia support characteristics of wind and photovoltaic power sources, and defines the composition and connotation of the equivalent inertia of the power system. Taking the IEEE 3-generator 6-node system with a high penetration of new energy (including photovoltaic and wind power) as a case study, the paper conducts optimal power flow calculation and unit commitment calculation, and then completes the system inertia evaluation. The research verifies the reliability of the proposed scheme, which effectively improves the equivalent inertia of the system, and provides theoretical support for the inertia regulation and operation optimization of new power systems.

Recursive power flow calculation based on polar coordinates holomorphic embedding method

In rural areas, unreasonable access of distributed photovoltaic (DPV) may pose a significant challenge to the safe operation of the power network. To address this, determining the maximum hosting capacity of network feeders is crucial for rational DPV planning. This study analyzes typical integration modes and voltage deviations. Firstly, the load-source correlation is constructed, and a hybrid model of ARMA and GARCH is proposed to predict photovoltaic (PV) generation, serving as input for dynamic grid evaluation. Leveraging the radial topology, a matrix-based method is employed to streamline power flow calculations. The resulting nonlinear optimization problem is solved to determine the maximum permissible PV capacity at each grid node. A case study demonstrates the calculation of the maximum photovoltaic capacity at each node. The results indicate that the proposed model and method effectively assess the DPV consumption capacity based on feeder distribution, providing planning guidance and technical support for PV integration via overhead lines in rural areas.

Abstract Long-term stability assessments of power systems are traditionally analyzed using root mean square simulation models, whose dynamic behavior is represented by a set of nonlinear electromechanical and controller differential equations. This article presents a simplified root mean square simulation approach derived from the commonly applied root mean square simulation method. This efficient approach is structured based on quasi-stationary assumption of electrical state variables and their associated controllers, which are comparatively faster than mechanical ones. This results in unavoidable algebraic loops at the terminals of the simplified models of active devices, which are solved by integrating their Thevenin equivalent circuits into the Newton-Raphson based power flow calculation method, which is solved in every time step of the dynamic simulation. The assumptions mentioned above lead to a significantly lower model complexity and thus to a considerably lower parametrization effort in large-scale system studies. Furthermore, they enable higher simulation time steps as well as lower computational time and effort. In order to showcase the impact of the underlying simplifications, both methods are programmed in MATLAB and compared with focus on their frequency behavior in the event of a frequency drop. It is shown that the quantities analyzed in the case studies in both simulation approaches (i.e., the center of inertia frequency, terminal active powers, voltages and rotational speeds) exhibit negligible deviations within the very fast transients and especially after they have subsided. This confirms that the simplifications associated with neglecting small electrical time constants are valid for the investigated frequency event.

Abstract— The rapid proliferation of Electric Vehicles (EVs) introduces significant challenges to existing distribution networks including voltage instability, excessive power losses, transformer overloading, and harmonic injection. This paper presents a comprehensive simulation-based analysis of EV charging station (EVCS) impacts on the IEEE 33-bus radial distribution system under four scenarios: (1) Base Case, (2) Uncoordinated Charging, (3) Smart Coordinated Charging, and (4) GWO-Optimized Placement. Detailed power flow calculations, Voltage Stability Index (VSI), Total Harmonic Distortion (THD) analysis, and load profile studies are performed. The proposed Grey Wolf Optimizer (GWO) approach minimizes a multi-objective function combining active power loss (ΔPLoss), voltage deviation (ΔVDev), and THD. Results demonstrate that GWO-optimized placement (at Buses 13, 24, 29, 31) reduces power losses by 46.5% (from 287.6 kW to 153.8 kW), improves minimum bus voltage from 0.9124 p.u. to 0.9782 p.u., and constrains THD within IEEE 519-2014 limits (≤5%). Comparative benchmarking against PSO, GA, and WOA confirms GWO's superiority in convergence speed (11.3 s) and solution quality. Keywords—Electric Vehicle; Charging Station; IEEE 33-Bus; Voltage Stability Index; Harmonic Analysis; Grey Wolf Optimizer; Power Loss; Smart Charging; THD; Distribution Grid.

Peer-to-peer (P2P) energy-trading has emerged as a promising mechanism for decentralized electricity markets, but its practical deployment is often limited by the difficulty of accounting for physical network constraints and transmission losses in real time. This paper presents a decentralized P2P energy trading mechanism that incorporates network constraints and transmission losses directly into the market-clearing process. The framework combines Power Transfer Distribution Factors (PTDFs) for pre-trade feasibility validation with an Enhanced Least Squares Method (ELSM) for loss estimation, enabling loss-aware settlement without computationally intensive and redundant AC power flow calculations. The mechanism is implemented on Hyperledger Fabric using Attribute-Based Access Control, Access Control Lists and Private Data Collections to ensure privacy and auditability. Numerical studies on a 3-bus and the IEEE 39-bus system show that the proposed approach closely reproduces AC Optimal Power Flow dispatch and cost outcomes, while significantly improving simplified DC-based loss models. The results demonstrate that physically feasible and economically efficient decentralized trading can be achieved in a permissioned blockchain environment.

ABSTRACT The rapid penetration of renewable energy resources has introduced significant uncertainty into modern power systems, necessitating accurate pre‐assessment to ensure secure operation. Probabilistic power flow (PPF) analysis is a powerful technique for quantifying such uncertainty, but its reliance on repeated AC power‐flow solutions makes it computationally prohibitive. This study proposes a fast PPF framework that couples the linear DC power flow (DC method) model with an Extreme Learning Machine (ELM) surrogate. By augmenting ELM inputs with DC‐PF results and employing Latin Hypercube Sampling, the method achieves both high speed and high accuracy. Numerical experiments on benchmark IEEE systems confirm that the proposed approach preserves the precision of conventional Monte‐Carlo‐based PPF while reducing total computation time by approximately 140 times

With the ongoing evolution of modern power grids, power flow calculation, which is the cornerstone of power system analysis and operation, has become increasingly complex. While promising, existing data-driven methods struggle with key challenges: poor generalization in data-scarce scenarios, efficiency bottlenecks when integrating physical laws, and a failure to capture higher-order interactions within the grid. To address these challenges, this paper proposes a Spatial Multi-scale Reservoir Computing framework that seamlessly incorporates functional matrix and physical information to solve power flow calculation. The framework utilizes parallel readout layer parameters to construct the functional matrix and integrates physical information to create a multi-scale information processing mechanism and readout constraints. By improving the reservoir computing model, the framework also combines the reservoir paradigm with the inherent physical characteristics of power grids while maintaining computational efficiency. Experimental results demonstrate that the presented framework achieves exceptional performance across various IEEE bus systems, showcasing superior generalization in data-scarce scenarios, as well as improvement in computational speed, prediction accuracy, and robustness, while ensuring the feasibility of the output results.

The high integration of renewables like distributed photovoltaic (PV) into medium- and low-voltage distribution networks causes bidirectional power flows, increased voltage fluctuations, and operational uncertainty. Traditional power flow models struggle to balance efficiency and accuracy for multi-period optimization. This paper proposes a dual-objective voltage optimization method based on a Holomorphic Embedding time-series power flow model. First, a recursive relationship for nodal voltage power series expansion is derived, revealing the linear superposition of first-order coefficients with power injection changes and the rapid decay of higher-order terms. A linearized analytical model neglecting higher-order terms is built, improving the computational efficiency of time-series power flow calculations while maintaining accuracy. Then, integrating energy storage systems and static var compensators, a dual-objective optimization model minimizing voltage deviation and daily operational cost is formulated. Tests on a practical 91-node rural distribution system show that the proposed power flow model maintains a voltage error below 0.25% compared to the Newton–Raphson method across various PV integration scenarios, and the optimization reduces computation time by about 61.3% versus the Second-Order Cone Programming method, validating its advantages in precision and efficiency for balancing voltage quality and economy.

Admittance-based Ollivier-Ricci curvature for power grid structural robustness analysis

In the operation mode arrangement of bulk power systems, unreasonable reactive power injection data at nodes tend to result in power flow calculation non-convergence. Owing to the extremely high dimension of the variable space and the heterogeneous impacts of different variables on power flow convergence, it is imperative to accurately identify the key variables inducing non-convergence and provide physical justifications. For this purpose, this paper proposes a data-driven key variable identification and adjustment method: firstly, based on the blocking cut-set theory and the characteristic that the active unbalanced power ΔP of intermediate power flow exhibits opposite signs at the sending and receiving ends of the cut-set, a blocking cut-set identification method leveraging the characteristics of the active unbalanced power of intermediate power flow is developed; secondly, relying on the feature that the reactive unbalanced power ΔQ of intermediate power flow is less than zero, a key variable identification method based on the characteristics of the reactive unbalanced power of intermediate power flow is presented; finally, a key variable adjustment method grounded in the numerical value of ΔQ is proposed. The validity of the proposed approach was validated via simulated computations using both the IEEE 39 bus system and a practical bulk power system.

Large language models (LLM) have achieved remarkable advances in natural-language understanding and content generation, and LLM-based agents demonstrate strong adaptability, flexibility, and robustness in handling complex tasks and enabling automated decision-making. Determining the operating mode of a power system requires repeated adjustments of boundary conditions to address violations. Conventional approaches include expert-driven power flow calculations and optimal power flow methods, the latter of which often lack clear physical interpretability during the iterative optimization process. This study proposes a novel paradigm for automated computation and adjustment of power system operating modes based on LLM-driven multi-agent systems. The approach leverages the reasoning capabilities of LLMs to enhance the adaptability of power flow adjustment strategies, while multi-agent coordination with power flow calculation modules ensures computational accuracy, enabling a natural-language-guided adaptive operational computation and adjustment process. The framework also incorporates retrieval-augmented generation techniques to access external knowledge bases and databases, further improving the agents’ understanding of system operational patterns and the accuracy of decision-making. This method constitutes an exploratory application of LLMs and multi-agent technologies in power system computational analysis, highlighting the considerable potential of LLMs to extend and enhance traditional power system analysis methodologies.

Introduction: New Energy HVDC Transmission System (NEHTS) in China's western Desert-Gobi-Wasteland (DGW) regions relies on Ultra-High Voltage Direct Current (UHVDC) transmission for long-distance power delivery. The high uncertainty of power sources and unique load characteristics poses significant challenges to traditional Load Supply Capability (LSC) assessment methods. Method: On the power supply side, an Adaptive Diffusion Kernel Density Estimation (ADKDE) model and a Copula-based time series joint probability distribution model were constructed to improve the local fitting accuracy of output characteristics. On the load side, an improved adaptive step repeat power flow method was proposed to achieve an accurate simulation of the load growth pattern. Based on this, combined with probability power flow calculation, the Composite Security and Stability Index (CSSI) and Load Supply Margin (LSM) were used as the core evaluation indicators. Result: This study proposes a quantitative assessment method for evaluating the LSC of NEHTS, addressing the deficiency of conventional methods in accounting for the time-sequence correlation between wind and Photovoltaic (PV) generation and load growth patterns. Discussion: The fitting effectiveness and accuracy of the KDE and ADKDE models were quantitatively compared through simulation, confirming the superiority of the ADKDE approach. Furthermore, multiple cases were conducted to perform a detailed comparative simulation analysis between the proposed LSC assessment method and conventional methods. The results verify that the proposed method enhances LSM and also provides a time-sequence analysis. Conclusion: This method significantly improves the accuracy and practicality of LSC assessment, providing a theoretical basis for the safe operation and informed decision-making of NEHTS.

As the automation and intelligence of low-voltage distribution networks continue to advance, the inter-layer coupling between medium- and low-voltage distribution networks is increasingly strengthened, making traditional fixed-point iteration methods inadequate for distributed power flow calculation in such a collaborative framework. To address this issue, this paper proposes a distributed power flow calculation method for medium- and low-voltage distribution networks based on edge intelligence. First, a cooperative operational framework for medium- and low-voltage distribution networks is designed by integrating edge intelligence technology. Then, a distributed power flow calculation model is established, and its fixed-point iterative characteristics are analyzed. A convergence index calculation method based on small perturbations is proposed, followed by an iterative algorithm based on continuous intersection estimation. Finally, simulation case studies validate the proposed method in terms of accuracy, convergence, and computational efficiency, demonstrating its capability to meet the modeling and analytical needs of power flow calculation in medium- and low-voltage distribution networks, providing methodological support for the development of distributed intelligent power grids.

In this paper, a novel method is proposed for the real-time estimation of multi-port Thevenin equivalent circuit parameters of power systems with non-accessible parameters using synchronized measurements from capacitive voltage transformers (CVT) and power analyzer. The core of the proposed approach is founded on simultaneous measurements at boundary buses, enabling real-time Thevenin equivalent circuit estimation, bus voltage phasor determination, and calculation of power flow across interconnecting lines. The key innovation in the proposed method is the accurate and continuous modeling of the inaccessible network based on the accurate instantaneous calculation of the variable transmission line impedance. Also, to reduce the impact of measurement noise, additional measurements and the least squares equation solving technique are used, which consequently increases the modeling accuracy to a great extent. Furthermore, to prevent misleading results caused by significant error magnitudes, the method incorporates a disturbance tracking algorithm to identify the direction and minimal intensity of disturbance sources. The effectiveness of the proposed methodology has been validated through simulations conducted in Dig-SILENT power factory application. The high accuracy of the proposed method in comparison with literature is shown by applying them to the IEEE 39-bus power system. In comparing the values obtained from their estimation and use of load distribution under different conditions, the maximum relative error rate of the best previous method is 9.9%, while the error of the presented method without using PMU and RTU is less than 5%.

Amid evolving energy policies and increasingly diversified electricity consumption patterns, the operational complexity of distribution systems has intensified, leading to amplified voltage variability and heightened operational uncertainty. Time-varying load characteristics and dynamic network operating conditions contribute to inefficient system operation and pose challenges to voltage stability. As voltage directly influences the dynamic stability of a system, accurate voltage forecasting is essential for improving the reliability of system operation. Traditionally, voltage forecasting of individual buses relies primarily on load forecasting followed by power flow calculations. However, more efficient voltage forecasting methods are needed to account for the dynamic nature of voltage variations. In this study, we propose a voltage-forecasting model that integrates load forecasting based on graph convolutional network-long short-term memory (GCN-LSTM) with power-flow calculations. By considering the spatial characteristics derived from the physical configuration of the distribution system and the temporal characteristics, we predict the voltage magnitude and phase angle of each bus. A case study of a distribution system using real data from South Korea is conducted to evaluate the prediction accuracy and computational efficiency of the proposed method. When evaluated on a dataset characterized by seasonal variability, the proposed model reduced prediction errors by 7.8% to 29.9% compared with the benchmark models. These results demonstrate that, in scenarios where conventional approaches suffer from error propagation—originating from load forecasting and subsequently amplified through the power flow calculation stage—the proposed unified framework plays a significant role in enhancing the reliability of voltage forecasting.

Power flow calculation and static voltage stability (SVS) analysis of active distribution networks (ADNs) are facing convergence and applicability issues caused by extreme conditions and control strategies of grid-forming (GFM) and grid-following (GFL) distributed generations (DGs). To solve these challenges, an SVS and DG integration capacity quantification method based on extended holomorphic embedding power flow (EHEPF) is proposed which includes four aspects: 1) an EHEPF algorithm is developed that accommodates both convergence and converter’s internal primary and secondary control characteristics of DGs; 2) the unsolvable mathematical expression of EHEPF is derived using Padé approximant to clearly distinguish between no solution and lower-branch (inoperable) solution; 3) the voltage sensitivity and SVS indexes of EHEPF combining different control strategies of DGs are formulated to analyze the weak bus and SVS of ADNs; 4) the DGs integration capacity is quantified by balancing SVS and network loss under voltage distribution calculated in EHEPF. Case studies are carried out on 12 bus, IEEE 33 and IEEE 123 bus systems. Numerous test results are analyzed to verify the applicability, convergence and effectiveness of proposed EHEPF for SVS and DGs integration capacity quantification in ADNs.