Search Results (45)

In constructing power system load flow mapping relationships using machine learning algorithms, the fundamental prerequisite for ensuring the computational accuracy and generalization performance of the mapping model is the availability of a suitably sized, well-distributed, and high-quality labeled sample set that can be supplied in a precursory and efficient manner. Here, the quality of a load flow sample set is defined concretely by three concurrent properties, as follows: physical consistency (satisfaction of Kirchhoff’s and Ohm’s laws), representative coverage of the operational state space, and low inter-sample redundancy. Currently, both online and offline techniques for power flow samples are incapable of efficiently providing large-scale, high-quality power flow sample sets in this sense. To address this, the paper proposes a method for generating power flow sample sets that integrates a physical model of the power grid. This method encompasses the following: non-iterative, high-speed generation techniques for massive load flow samples; partitioned generation and multi-region splicing techniques for large power grid load flow samples; and the design of capacity requirements and quality technical indicators for load flow sample set production. Analytical results demonstrate that the proposed method can efficiently produce high-quality power flow sample sets of appropriate capacity based on actual needs. Case studies on the IEEE 9-bus and 39-bus systems show that sample generation is about 19 and 32 times faster than the whole-network Newton–Raphson method, respectively, for 100,000 samples, and the voltage-band capacity design requires only about 12.6% of the samples needed by uniform sampling for equal boundary-condition coverage, at a comparable learning error. Full article

The increasing penetration of direct-current (DC) technologies in power transmission and distribution systems necessitates efficient and robust tools for steady-state analysis. This paper presents a comparative evaluation of the Chebyshev–Halley (CH) family of multipoint iterative methods against the classical Newton–Raphson (NR) method for power flow calculation in monopolar DC networks. Both methods were implemented in MATLAB and tested on four radial test systems of increasing complexity (10, 21, 33, and 69 nodes) under three distinct initialization scenarios: optimal (flat start), adverse ($V^{\left(0\right)}=0.5$ p.u.), and random ($V^{\left(0\right)}\sim \mathcal{U}[0.8,1.2]$ p.u.). Performance was assessed using key metrics including iteration count, CPU time, solution accuracy, and convergence failure rate. The results demonstrate that the cubic convergence of CH consistently reduces the number of iterations by one when compared to NR across all systems. However, this reduction does not translate into computational savings, as CH exhibits median CPU times 1.36 to 2.44 times higher than those of NR, given its higher cost per iteration, which involves solving two additional linear systems. Under adverse starting conditions, both methods converge for the 10-, 21-, and 33-node systems, but CH fails on the 69-node network due to severe Jacobian ill-conditioning, from which NR recovers through an implicit regularization mechanism. Under random initializations, both methods show high failure rates, reaching 100% in the 69-node network. It is concluded that, while CH offers superior convergence order and final accuracy, NR remains more computationally efficient for small- to medium-scale networks under flat-start conditions. The CH family is best justified in high-precision applications or larger networks where the iteration reduction may offset its per-step overhead. Future work should focus on extending CH to meshed and multi-source DC networks, developing quasi-Newton variants to reduce its computational cost, and designing hybrid NR-CH strategies that combine global robustness with local cubic convergence. Full article

This paper introduces a methodology for evaluating the voltage stability margin in high-voltage direct-current (HVDC) systems, which analyzes the singularity of the power flow Jacobian matrix—computed via the Newton—Raphson method—and identifies saddle-node bifurcations. The continuation power flow method is employed to model progressive load increases, enabling the continuous tracking of power flow solutions and the determination of voltage collapse points. Within this framework, the system’s behavior is analyzed under contingency conditions, particularly transmission line outages, assessing its capability to maintain secure operating conditions under increasing demand scenarios. The main objective is to identify the most critical line in the system, defined as that which leads to the greatest reduction in loadability when unavailable, prior to voltage collapse. This approach allows for the early identification of structural vulnerabilities, supporting decision-making processes aimed at risk mitigation and operating cost optimization. The proposed methodology is validated using two systems: the six-terminal CIGRE-B4 HVDC system and an 11-node HVDC test feeder. Full article

This article proposes a gas-only continuation framework for steady-state loadability analysis in natural gas transmission networks based on a direction-free reformulation of the General Flow Equation (GFE). The proposed formulation introduces signed pipe flows directly as state variables, thereby representing bidirectionality intrinsically. As a result, flow reversals are handled without switching logic, while the branch geometry and criticality mechanism of the underlying gas-network equilibrium map are preserved. On this basis, a Gas Continuation Method (GCM) is developed to trace equilibrium branches directly in native gas-load space under specified gas-load stress. The method distinguishes the last admissible operating point from the mathematical critical point and incorporates a formal diagnosis to determine whether the detected limiting condition is consistent with a Saddle-Node Bifurcation (SNB). The proposed framework is validated on a three-node benchmark, a realistic Belgian gas transmission network, and a 40-node test system. The results show accurate agreement with Newton–Raphson (NR) solutions in the regular operating regime, robust branch tracing near limiting conditions where standalone NR loses convergence, and consistent handling of signed pipe flows under load-induced flow reversal and under algebraic orientations assigned a priori opposite to the solved physical flow. The Belgian and 40-node cases further show that the operational admissibility limit may precede the mathematical critical point, so pressure-based feasibility and branch-level criticality emerge as related but distinct notions. These features make the proposed methodology a rigorous and practical tool for identifying admissibility limits, interpreting critical behavior, and assessing loadability margins in gas transmission networks. Full article

High renewable penetration introduces stochastic variability in distribution-network operation, requiring probabilistic AC power-flow tools that remain accurate in the tails while avoiding the computational burden of large Monte Carlo simulation. This paper presents a fully reproducible non-intrusive polynomial chaos expansion (PCE) framework for uncertainty propagation through nonlinear Newton–Raphson AC power flow. The method uses sparse-grid quadrature to train PCE surrogates from deterministic power-flow evaluations and is benchmarked against high-fidelity Monte Carlo simulations. In the validation, the IEEE 33-bus feeder is evaluated using up to 50,000 Monte Carlo samples, 95% bootstrap confidence intervals, PCE orders 2–5, correlated uncertainty scenarios, realistic thermal-loading recalibration, reactive-power sensitivity of renewable injections, multi-feeder testing on IEEE 33-bus, CIGRE MV, CIGRE LV, and IEEE 118-bus networks, and a 365-snapshot full-year daily screening. For the base IEEE 33-bus case, third-order PCE required only 494 deterministic power-flow evaluations and reproduced the 50,000-sample Monte Carlo benchmark with relative mean errors of 0.014% for minimum voltage, 0.119% for active losses, and 0.113% for substation import. The corresponding wall-clock speed-up was 13.29×, while reducing deterministic evaluations by approximately 101×. Correlated load–PV uncertainty increased the upper tail of substation import from 6.06 MW to 6.30 MW, and realistic thermal recalibration revealed line-loading p99 values above 100% for the 60% target case, demonstrating the operational value of physically meaningful ampacity settings. The proposed workflow provides an open, scalable, and tail-aware basis for uncertainty-informed distribution-network planning under renewable variability. Full article

Voltage stability under transmission line contingencies is a critical concern in modern power systems, as the growing electricity demand and the large-scale integration of renewable energy sources increasingly challenge the security of network operation. This paper addresses the problem of estimating the voltage stability margin under $N-1$ transmission line contingencies through three solution methodologies: a nonlinear programming formulation solved via an interior-point algorithm (IPOPT) with a multi-start strategy, a recursive heuristic approach based on successive Newton–Raphson power flow solutions with progressive load scaling, and a convex second-order cone programming relaxation. The proposed methods are validated on the IEEE 9-, 14-, 30-, and 57-bus test systems, thereby covering networks of varying topological complexity and redundancy. A comparative analysis evaluates the accuracy of each approach against a nonlinear programming reference, as well as their computational efficiency under a comprehensive set of contingency scenarios. The results indicate that the heuristic method achieves higher precision, while the convex formulation offers a substantially faster solution, with both approaches demonstrating robustness in cases where the nonlinear programming method fails to converge. Full article

This study applies a combined load flow, short-circuit, and protection study of the IEEE four-bus and five-bus benchmarks as a comprehensive approach to power system modelling. A consistent per-unit base of 150 MVA and 132 kV is applied uniformly. The NR co-simulation approach is used for load flow studies in both MATLAB_R2025b and DIgSILENT PowerFactory 2025. The simulation results indicate that voltages, power mismatches, and line flows are within the tolerance limits. Findings suggest that the NR method was highly implementable, yielding results in 2–3 iterations, and that the simulation results were comparable to those produced by commercial software, validating confidence in the power system modelling, load flow analysis, and protection study. Full article

Modern power distribution networks are increasingly challenged with nonlinear operating conditions, the high penetration of distributed energy resources, and conflicting operational objectives such as loss minimization and voltage regulation. Existing load-flow optimization approaches often suffer from slow convergence, premature stagnation in non-convex search spaces, and limited robustness when handling conflicting multi-objective performance criteria under fixed network constraints. To address these challenges, this paper proposes a Fractional Multi-Objective Load Flow Optimizer (FMOLFO), which integrates a fractional-order numerical regularization mechanism with an adaptive Pareto-based Differential Evolution framework. The fractional-order formulation employed in FMOLFO operates over an auxiliary iteration domain and serves as a numerical regularization strategy to improve the sensitivity conditioning and convergence stability of the load-flow solution, rather than modeling the physical time dynamics or memory effects of the power system. The optimization framework simultaneously minimizes physically consistent active power loss and voltage deviation within existing network operating constraints. Extensive simulations on IEEE 33-bus and 69-bus benchmark distribution systems demonstrate that FMOLFO achieves an up to 27% reduction in active power loss, improved voltage profile uniformity, and faster convergence compared with classical Newton–Raphson and metaheuristic baselines evaluated under identical conditions. The proposed framework is intended as a numerically enhanced, optimization-driven load-flow analysis tool, rather than a control- or dispatch-oriented optimal power flow formulation. Full article

High penetration of distributed photovoltaic (PV) generation introduces operational challenges for thermal power plants, including increased cycling, higher losses, and reduced system flexibility. This study proposes an integrated optimization framework that combines Mixed Integer Nonlinear Programming (MINLP)-based Unit Commitment (UC) with a Particle Swarm Optimization (PSO)-assisted Optimal Power Flow (OPF) solved using the Newton–Raphson method. Applied to the IEEE 30-bus system for a 24-h horizon, the UC stage schedules 3717.8 MW of thermal generation at a cost of $8771.14. Load flow validation indicates a required supply of 3793.7 MW due to network losses, increasing the cost to $9031.64 and causing several constraint violations. The PSO-assisted OPF resolves all violations and produces an adjusted total generation of 3778.5 MW, reducing losses and lowering the overall operating cost to $8912.47 through optimal redispatch and voltage regulation. To further evaluate system robustness, multiple load scenarios—including reduced, nominal, and increased demand—are analyzed. Across all scenarios, the OPF stage is able to eliminate operational violations, decrease real power losses, and maintain voltage profiles within acceptable limits, demonstrating consistent performance under varying system stress levels. Overall, the integrated UC–OPF framework enhances economic efficiency, operational reliability, and resilience under renewable variability and shifting load conditions. Full article

Dynamic equivalence of active distribution networks (ADNs) is emerging as one of the most important issues for the backbone network security analysis due to high penetration of distributed generations (DGs) and electricity vehicles (EVs). The multiscale and multimodal fusion deep learning (MMFDL) method proposed in this paper contains two modalities, one of which is a CNN + attention module to simulate Newton Raphson power flow calculation (NRPFC) for the important feature extraction of a power system caused by disturbance, which is motivated by the similarities between NRPFC and convolution network computation. The other is a long short-term memory (LSTM) + fully connected (FC) module for load modeling based on the fact that LSTM + FC can represent a load′s differential algebraic equations (DAEs). Moreover, to better capture the relationship between voltage and power, the multiscale fusion method is used to aggregate load modeling models with different voltage input sizes and combined with CNN + attention, merging as MMFDL to represent the dynamic behaviors of ADNs. Then, the Kepler optimization algorithm (KOA) is applied to automatically tune the adjustable parameters of MMFLD (called KOA-MMFDL), especially the LSTM and FC hidden layer number, as they are important for load modeling and there is no human knowledge to set these parameters. The performance of the proposed method was evaluated by employing different electric power systems and various disturbance scenarios. The error analysis shows that the proposed method can accurately represent the dynamic response of ADNs. In addition, comparative experiments verified that the proposed method is more robust and generalizable than other advanced non-mechanism methods. Full article

An innovative method is proposed to analyze the coupled vibration between random traffic and large-span bridges under the combined action of wind loads. The dynamic behavior of bridges subjected to these multifactorial influences is investigated through a comprehensive bridge dynamics model. Specifically, a refined full-bridge finite element model is developed to simulate the traffic–bridge coupled vibration, with wind forces applied as external dynamic loads. The effects of wind speed and vehicle speed on the coupled system are systematically evaluated using the finite element software ABAQUS 2023. To ensure computational accuracy and efficiency, the large-span nonlinear dynamic solution method is employed, integrating the Newmark-β time integration method with the Newton–Raphson iterative technique. The proposed method is validated through experimental measurements, demonstrating its effectiveness in capturing the synergistic impacts of wind and traffic on bridge dynamics. By incorporating the stochastic nature of traffic flow and combined wind forces, this approach provides a detailed analysis of bridge responses under complex loading conditions. The study establishes a theoretical foundation and practical reference for the safety assessment of large-span bridges. Full article

This paper presents a comprehensive study on the formulation and solution of the power flow problem in bipolar direct current (DC) distribution networks with unbalanced constant power loads. Using the nodal voltage method, a unified nonlinear model is proposed which accurately captures both monopolar and bipolar load configurations as well as the voltage coupling between conductors. The model assumes a solid grounding of the neutral conductor and known system parameters, ensuring reproducibility and physical consistency. Seven iterative algorithms are developed and compared, including three Newton–Raphson-based formulations and four quasi-Newton methods with constant Jacobian approximations. The proposed techniques are validated on two benchmark networks comprising 21 and 85 buses. Numerical results demonstrate that Newton-based methods exhibit quadratic convergence and high accuracy, while quasi-Newton approaches significantly reduce computational time, making them more suitable for large-scale systems. The findings highlight the trade-offs between convergence speed and computational efficiency, and they provide valuable insights for the planning and operation of modern bipolar DC grids. Full article

AC/DC hybrid distribution networks with power electronic transformers (PETs) as distribution hubs are in line with the future development direction of the AC/DC hybrid distribution network. Unlike traditional transformers, power electronic transformers introduce new node types and may transform the network topology from radial to ring structures. These changes render traditional power flow calculation methods inadequate for achieving satisfactory results in AC/DC hybrid networks. In addition, existing commercial power flow calculation software packages are mainly based on the traditional AC power flow calculation method, which have limited support for the DC network. Especially when the DC network is coupled with the AC network, it is difficult to achieve a unified calculation of its power flow. To address these challenges, this paper proposes a novel power flow calculation method for ring AC/DC hybrid distribution networks with power electronic transformers. The proposed method is based on the alternating iterative method to ensure compatibility with mature AC power flow calculation programs in commercial software, thereby improving the feasibility of engineering applications. Firstly, the steady-state power flow calculation model of PET is constructed by analyzing that the working principle and control modes of power electronic transformer are proposed based on the source-load attributes of its connected subnetworks. According to the characteristics of the power electronic transformer, AC distribution network, and DC distribution network, a hybrid alternating iteration method combining the high computational accuracy of the Newton–Raphson (NR) method with the high efficiency of the Zbus Gaussian method in dealing with ring networks is proposed. On this basis, the power flow calculation model of the AC/DC hybrid distribution network with power electronic transformers is established. Finally, the simulation of the constructed 44-node ring AC/DC hybrid distribution network example is carried out. The simulation results show that the proposed method can not only converge reliably when the convergence accuracy is 1 × 10⁻⁶ p.u., but also ensure that the voltage magnitudes of all nodes are above 0.96 p.u. whose maximum offset value is 0.789% when the outputs of the connected distributed generations fluctuate, which verifies the effectiveness and accuracy of the proposed method. Full article

A comprehensive two-dimensional (2D) time-domain numerical model is established to investigate the interaction of irregular waves and submerged structures with different sections. The model specifically focuses on the dual-lane submerged floating tunnel (SFT) designs, encompassing elliptical, twin-circular, and round rectangular sections. For the hydrodynamic analysis, we adopt the second-order potential flow theory, while for the mooring line simulations, we employ the slender rod theory, taking into account the entire hydrodynamic load acting on it. In the coupled dynamic analysis, the fourth-order Adams–Bashforth–Moulton method, Newmark-β method, and Newton–Raphson iteration scheme are utilized for the coupled motion equation of the floating body and the dynamic equation of the mooring riser system. Experimental free decay tests are conducted to determine the damping coefficients of various section shapes in different directions. Our analysis delves into the detailed motion responses and mooring tensions of the SFTs with different section forms under irregular waves. We compare and contrast these responses in both time and frequency domains, particularly focusing on movement trends. The elliptical section structure emerges as the most stable design based on our comparisons. These findings provide valuable insights for the selection of optimal section shapes for dual-lane SFTs. Full article

This paper proposes a novel methodology to improve stability in a transmission system under critical conditions of operation when additional loads that take the system to the verge of stability are placed in weak bus bars according to the fast voltage stability index (FVSI). This paper employs the Newton–Raphson method to calculate power flows accurately and, based on that information, correctly calculate the FVSI for every transmission line. First, the weakest transmission line is identified by considering $N-1$ contingencies for the disconnection of transmission lines, and then all weak nodes associated with this transmission line are identified. Following this, critical scenarios generated by stochastically placed loads that will take the system to the verge of instability will be placed on the identified weak nodes. Then, the methodology will optimally size and place a single static VAR compensator SVC in the system to take the transmission system to the conditions before the additional loads are connected. Finally, the methodology will be validated by testing the system for critical contingencies when any transmission line associated with the weak nodes is disconnected. As a result, this paper’s methodology found a single SVC that will improve the system’s stability and voltage profiles to similar values when the additional loads are not connected and even before contingencies occur. The methodology is validated on three transmission systems: IEEE 14, 30, and 118 bus bars. Full article