Key Takeaways & Executive Findings
- •• • The proposed constant-parameter VBR model achieves a 55.6% reduction in computational time compared to the MATLAB/Simulink SPS model under identical simulation accuracy, directly enabling larger simulation step sizes and faster grid stability assessments for system operators. • • Two-norm voltage errors remain below 0.33% at node 5 and 0.3% at node 9, while current errors are 0.51% and 0.41%, respectively, under single-phase grounding short-circuit faults, confirming the model's precision in fault transient scenarios critical for protection relay coordination. • • The hybrid explicit-implicit Euler discretization eliminates algebraic loops without iterative calculations, preserving numerical stability at large step sizes where traditional implicit methods fail or require excessive iterations, reducing simulation runtime for IEEE 14-bus system tests. • • Piecewise linearization of the magnetic saturation curve enables accurate representation of saturation effects that, if ignored, cause core electrical quantity errors exceeding 10%, thereby preventing overestimation of equivalent reactances and misjudgment of voltage stability margins in grids with high renewable penetration.
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Abstract
Traditional electromagnetic transient simulation of synchronous generators suffers from insufficient numerical stability, low computational efficiency, and inadequate representation of magnetic saturation. To improve the reliability of power grid security and stability analysis, this paper proposes an improved modeling scheme based on the voltage-behind-reactance (VBR) method, focusing on the performance deficiencies of conventional current-source equivalent models. The scheme constructs a decoupled machine-network interface circuit with constant resistance-inductance branches via constant-parameterization, employs piecewise linearization of the flux-current characteristic curve to accurately represent magnetic saturation, and adopts a hybrid explicit-implicit Euler discretization to avoid algebraic loops. Validation on a single-machine infinite-bus benchmark and fault condition tests on the IEEE 14-bus system using MATLAB/Simulink demonstrate that the proposed model significantly enhances computational efficiency and numerical stability for large-step simulations while maintaining excellent accuracy in fault transient scenarios. Compared with the MATLAB/Simulink SPS model, the proposed model achieves a 55.6% improvement in computational efficiency under equal simulation accuracy, with two-norm voltage errors of 0.33% and 0.3% at nodes 5 and 9, and current errors of 0.51% and 0.41%, respectively. The model is suitable for large-scale electromagnetic transient simulation analysis of power systems.
1. Introduction
Existing electromagnetic transient simulation programs predominantly rely on synchronous generator models based on differential-algebraic equations (DAEs) in either the dq0 rotating frame or the phase-domain (PD) frame. The dq0 model eliminates time-varying inductances and reduces state equation count, but lacks a direct machine-network interface, necessitating transformation or inverse transformation that introduces algebraic loops. Iterative solutions mitigate this issue but impose small time steps, severely limiting computational efficiency. The MATLAB/Simulink SPS toolbox uses a current-source equivalent interface, requiring buffer resistances to avoid numerical divergence when feeding inductive branches; increasing these resistances reduces error but significantly increases simulation time. The PD model connects stator circuits directly to the external network without prediction or iteration, improving numerical stability, yet its inductance matrix remains rotor-position-dependent, requiring recalculation at each time step and resulting in high computational cost. Constant-parameter PD models solve the time-varying inductance problem but retain a high-dimensional inductance matrix, yielding marginal efficiency gains in large-step simulations.
The VBR modeling approach reconfigures stator voltage equations to obtain a constant inductance matrix, but existing methods either remain unsuitable for salient-pole synchronous generators with asymmetric rotor structures or introduce algebraic constraints when solved implicitly. Low-pass filtering eliminates algebraic constraints but introduces delay that degrades accuracy. Constant-parameter VBR models enable efficient machine-network interconnection but still employ traditional implicit solution strategies, incurring iterative computational losses, and their adaptability to salient-pole generators and robustness in large-scale grids remain unverified. Magnetic saturation, an inherent nonlinearity of synchronous generators, has not been adequately considered in these frameworks. Ignoring saturation overestimates equivalent reactances, causing simulation errors exceeding 10% in core electrical quantities and misjudging voltage stability margins. This paper addresses these gaps by combining piecewise linearization of the magnetization curve with a constant-parameter VBR derivation that decouples the inductance matrix via zero-sequence components and a hybrid explicit-implicit Euler discretization to eliminate algebraic loops, validated on single-machine infinite-bus and IEEE 14-bus systems.
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LI Yingnan, WANG Yule, PENG Li, MU Qing, GUO Xizheng, DENG Jun (2026). Constant-Parameter VBR Modeling and Analysis of Synchronous Generators Considering Magnetic Saturation Characteristics. Power Automation Equipment. https://doi.org/10.16081/j.epae.202605011
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Frequently Asked Questions
What specific numerical stability issues arise from the current-source equivalent interface in conventional dq0 models, and how does the proposed VBR model eliminate them?
The current-source equivalent interface in dq0 models requires a buffer resistance in parallel when the external circuit contains inductive branches; increasing this resistance reduces error but significantly increases simulation time, while decreasing it risks numerical divergence. The proposed VBR model constructs a decoupled machine-network interface with constant resistance-inductance branches, eliminating the need for buffer resistances and algebraic loops. This is achieved by incorporating the difference between d- and q-axis inductances into the back-EMF and introducing zero-sequence components to decouple the inductance matrix, resulting in a constant-parameter interface that maintains numerical stability even at large step sizes.
How does the piecewise linearization method for magnetic saturation affect simulation accuracy under extreme saturation conditions, and what are the quantified errors?
Piecewise linearization fits the magnetization curve by decomposing it into multiple linear segments based on thresholds. Under single-phase grounding short-circuit faults, the two-norm voltage errors at nodes 5 and 9 are 0.33% and 0.3%, respectively, and current errors are 0.51% and 0.41%. These errors are measured against the MATLAB/Simulink SPS model. However, the accuracy depends on the density of segmentation nodes; extreme saturation conditions may introduce local representation errors. Future work suggests optimizing segmentation strategy with nonlinear interpolation to improve accuracy.
What is the computational efficiency gain of the proposed model compared to the SPS model, and under what simulation conditions was this measured?
The proposed model achieves a 55.6% improvement in computational efficiency compared to the MATLAB/Simulink SPS model under equal simulation accuracy. This was measured on a single-machine infinite-bus system and an IEEE 14-bus system with fault conditions, using large-step simulations. The efficiency gain stems from the constant-parameter VBR formulation and the hybrid explicit-implicit Euler discretization, which avoids iterative calculations and algebraic loops, enabling larger time steps without loss of numerical stability.
How does the proposed model handle salient-pole synchronous generators with asymmetric rotor structures, and what limitations remain?
The proposed VBR model is specifically adapted for salient-pole synchronous generators by incorporating the difference between d- and q-axis inductances into the back-EMF and introducing zero-sequence components to decouple the inductance matrix. This overcomes the limitation of earlier VBR methods that were unsuitable for asymmetric rotor structures. However, the model's robustness in large-scale multi-machine interconnected systems and hardware-in-the-loop platforms has not been fully validated, and the piecewise linearization may require further refinement for extreme saturation conditions.
What are the industrial implications of ignoring magnetic saturation in synchronous generator models for power grid stability analysis?
Ignoring magnetic saturation leads to overestimation of equivalent reactances, causing simulation errors exceeding 10% in core electrical quantities such as voltage and current. This can result in misjudgment of voltage stability margins, potentially leading to incorrect operational decisions and increased risk of grid instability. The proposed model accurately represents saturation effects via piecewise linearization, reducing errors to below 0.51% in tested fault scenarios, thereby providing more reliable data for grid security assessments, especially under large disturbances and high renewable penetration.
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