What question did this study set out to answer?

This study aims to enhance the calculation of carbon emission factors in large-scale power grids using a novel P-CGS method.

March 30, 2026Open Access

Accelerated Computing of Dynamic Carbon Emission Factor for Large-Scale Power Grid Using Conjugate Gradient Squared Method with P-Norm Preconditioner

Key Points

This study aims to enhance the calculation of carbon emission factors in large-scale power grids using a novel P-CGS method.
Development of P-CGS to overcome limitations of traditional methods.
Utilization of a multi-norm adaptive preconditioner for better matrix conditioning.
Conducting case studies on power grids of varying sizes (200, 500, and 2000 buses).
Performance comparisons with nine other methods for validation.
P-CGS outperformed nine other computational methods significantly.
Achieved relative residuals ≤ 10⁻⁸ for precise calculations.
Reduced computation time and iterations compared to traditional methods.

Abstract

• Propose P-CGS to fix traditional methods flaws in large-scale grid CEF calculation. • P-CGS specify residuals to balance different requirements for precision and speed. • Comprehensive case studies were conducted on three different-sized power grids. • P-CGS outperforms the other 9 methods by a significant margin. Amid the global push for energy decarbonization, the power system, contributing over 40% of energy-sector carbon emissions, requires precise dynamic carbon tracking and low-carbon scheduling tools. However, conventional methods for calculating electricity carbon emission factors (CEFs) are unsuitable for large-scale grids due to critical limitations: Direct inversion methods (e.g., Gaussian elimination, LU decomposition) suffer from O ( n ³) complexity, poor numerical stability under ill-conditioned matrices (high condition number); iterative methods like least squares method (LSQR) exhibit excessively slow convergence in large-scale systems, while conjugate gradient squared (CGS) often diverges or oscillates under ill-conditioning, unable to meet real-time demands. To address these issues, this study presents a p-norm preconditioner-boosted conjugate gradient least squares (P-CGS) approach. It employs a multi-norm adaptive preconditioner to optimize matrix condition numbers via p-norm (1≤p≤∞) scaling, integrates load temporal correlations with prior-step solutions through dynamic initial guess transfer, and efficiently solves multi-period carbon emission factors surfaces with relative residuals ≤10 -8 . Meanwhile, the comprehensive validations are conducted on 200-, 500-, and 2000-bus grids, and performances of p-norm preconditioner-boosted conjugate gradient least squares are revealed compared with bi-conjugate gradient (with and without p-norm preconditioner), generalized minimal residual (with and without p-norm preconditioner), LSQR (with and without p-norm preconditioner), CGS and direct inversion methods. The results illustrate that preconditioner-boosted conjugate gradient least squares can achieve smaller computation time and iterations under the designated relative residuals compared to others mentioned above. This work supports accurate carbon responsibility allocation, carbon market compliance, and international carbon tariff negotiations, providing robust technical foundations for power system decarbonization.

Accelerated Computing of Dynamic Carbon Emission Factor for Large-Scale Power Grid Using Conjugate Gradient Squared Method with P-Norm Preconditioner

Key Points

Abstract

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