Under the national goals of “carbon peaking and carbon neutrality,” the power sector faces increasingly strict emission regulations. Traditional average carbon emission factors fail to capture regional and temporal differences, often underestimating actual emissions. In contrast, marginal or nodal emission factors offer more accurate and localized assessments. Carbon flow theory has recently been introduced to model emissions as a “virtual flow” that follows power flows, enabling emission traceability and the development of nodal carbon emission factor models. This helps reveal hidden disparities and supports region-specific decarbonization policies.

Based on this theory, several studies have proposed matrix-based models that calculate nodal emission factors through direct matrix inversion. Reference [1] calculated carbon emission flow in power systems using a matrix-based approach involving direct matrix inversion. While effective for small networks, this method faces computational challenges in large-scale applications. Reference [2] developed a carbon emission model for high-speed rail traction based on carbon flow theory and solved it via direct matrix inversion; however, this approach, while effective for localized applications, is computationally inefficient and difficult to scale to large power systems. Reference [3] calculated carbon emission factors for high-speed rail systems using carbon flow theory and direct matrix inversion. Reference [4] proposed a dynamic carbon emission factor model for distribution networks based on a generalized inverse matrix formulation. While their method enables accurate mapping between carbon factors and power flows, the reliance on matrix inversion limits scalability and computational efficiency in large-scale systems. Reference [5] developed a flow-based carbon accounting framework that computes nodal emission factors via linear system solutions. While accurate, the method relies on direct matrix operations, which may limit scalability in large power networks. Reference [6] computed nodal carbon emission factors by directly inverting the system allocation matrix in an improved carbon emission flow model. Although accurate, the method’s reliance on matrix inversion may limit its scalability under large-scale or real-time conditions.

The above studies have adopted matrix-based carbon flow models that compute nodal emission factors via direct matrix inversion. While accurate, these methods rely on dense linear algebra operations that are computationally expensive and difficult to scale. For a power system with an Jacobian matrix, the computational complexity of direct solvers can reach up to O(N3). Even with sparse matrix techniques, direct methods remain inefficient in ultra-large-scale networks or high-frequency temporal calculations. In scenarios requiring frequent and fast evaluations, such as real-time nodal carbon emission estimation, such approaches become impractical. In contrast, iterative methods, which primarily depend on matrix–vector multiplication, offer lower computational and memory overhead, and allow flexible control over solution accuracy—an advantage when approximate solutions are acceptable. However, due to the ill-conditioned and structurally complex nature of power system matrices, conventional iterative solvers may suffer from slow convergence or even divergence without effective preconditioning. Thus, the key to achieving robust and efficient iterative performance lies in the design of suitable preconditioning strategies.

Reference [7] proposed a parallel method for solving large-scale sparse state-space matrices in power systems, using incomplete LU decomposition with threshold pivoting (ILUTP) preconditioning combined with the biconjugate gradient stabilized (BiCGSTAB) iterative solver. This approach effectively reduces computational time and improves scalability in stability analysis applications. Reference [8] proposed a fast parallel solver for large sparse linear systems arising in bioluminescence tomography, based on preconditioned BiCGSTAB combined with CUDA. The method leverages incomplete Cholesky factorization and compressed row storage scheme (CSR) sparse matrix storage to enhance convergence and computational efficiency. Recent studies have investigated the application of iterative solvers in large-scale energy systems. Świrydowicz et al. [9] benchmarked several Krylov-based solvers for power grid optimization problems, highlighting their scalability in sparse grid matrices. Li et al. [10] demonstrated that a GPU-accelerated, preconditioned iterative solver significantly improves the efficiency of power flow analysis in large power systems. Qu et al. [11] further analyzed multigrid-preconditioned Krylov methods, confirming their effectiveness for non-symmetric and large-scale energy system models.

The above studies have successfully applied ILU-preconditioned BiCGSTAB algorithms to large-scale sparse systems in other fields. However, most existing CEF studies still rely on direct matrix inversion, which is simple and accurate for small systems but becomes inefficient for large-scale nodal networks. To date, no research has addressed this computational bottleneck or applied this approach to dynamic electric carbon emission factor calculations, where fast and scalable solutions are increasingly critical.

To address this gap, this paper introduces an ILU-preconditioned BiCGSTAB iterative algorithm to efficiently solve the nodal carbon emission factor model. BiCGSTAB is well-suited for nonsymmetric sparse systems, and the ILU preconditioner improves matrix conditioning to speed up convergence. While effective in other sparse problem domains, this method is applied here for the first time to carbon emission analysis. With proper parameter tuning, the approach ensures accuracy while significantly reducing computation time, making it suitable for real-time analysis in large-scale power grids. The main contributions are as follows:

This paper adopts the BiCGSTAB iterative algorithm to efficiently solve nodal carbon emission factors in large-scale, nonsymmetric sparse systems. By leveraging Krylov subspace acceleration, the method achieves high precision with significantly lower computational cost than direct solvers, addressing the limitations of traditional approaches in large-scale and dynamic grid scenarios.

The novelty and contribution of this study lie in two main aspects. First, it is the first to employ an iterative-solver-based framework for large-scale dynamic carbon emission factor computation, overcoming the computational inefficiency of traditional matrix inversion. Second, compared with existing iterative methods such as CG, GMRES, and LSQR, the proposed ILU-preconditioned BiCGSTAB algorithm provides faster and more stable convergence for non-symmetric, sparse, and time-varying systems, offering clear advantages in both accuracy and scalability for real-time carbon emission analysis.

The remainder of this paper is organized as follows: Section 2 formulates the node-level dynamic CEF model, deriving the sparse linear system that captures carbon flow across power system nodes. Section 3 presents the fast computation framework based on the ILU-decomposed BiCGSTAB iterative solver, including analyses of matrix structure, preconditioning strategy, numerical efficiency, and its application to real-time carbon emission modeling. Section 4 provides a comprehensive case study using large-scale synthetic power grid datasets to validate the proposed method’s accuracy, scalability, and computational performance. Section 5 concludes the paper with a summary of key findings and a discussion of future research directions.

In order to evaluate carbon emissions across nodes in a power system dynamically and efficiently, this work constructs a node-level carbon emission factor (CEF) model. The method accounts for both direct generation-side emissions and the imported carbon intensity from neighboring nodes through active power flow. The formulation is inherently recursive and reflects the coupling among all nodes within the network. Fig. 1 illustrates the overall modeling framework, including the grid topology and the flow-based CEF calculation logic.

For a given node i, the carbon emission factor δi (kgCO2/kWh) represents a weighted average of two parts: local emissions from generators at node i and imported emissions from neighboring nodes. The expression is given by: (1)δi=∑gk∈ΩgiPgki⋅δgki+∑j∈ΩiPji⋅δj∑gk∈ΩgiPgki+∑j∈ΩiPjiwhere, Pgki denotes the active power output of generator gk connected to node i, and δgki is its specific emission factor. Pji represents the active power injected into node i from neighboring node j, while δj is the corresponding carbon emission factor of node j. The set Ωgi contains all generators at node i, and Ωi is the set of nodes supplying power to node i.

Eq. (1) captures the physical and environmental interactions at each node and forms the basis for further algebraic transformation. To facilitate a numerical solution for the entire network, Eq. (1) is algebraically rearranged into the following linear form: (2)(∑gk∈ΩgiPgki+∑j∈ΩiPji)δi−∑j∈ΩiPji⋅δj=∑gk∈ΩgiPgki⋅δgki

By applying this equation to all nodes in the system, we derive a sparse linear algebraic system in the compact form: (3)Aδ=b

In this equation, δ=[δ1,δ2,…,δn]⊤ is the vector of unknown node carbon emission factors. The matrix A∈Rn×n is sparse and typically diagonally dominant. Each diagonal entry Aii equals the sum of the power output from local generators and the incoming power flow to node i. Each off-diagonal term Aij (for j∈Ωi) is given by −Pji. The right-hand side vector b contains the emission contributions from generators, where each entry is computed as bi=∑gk∈ΩgiPgki⋅δgki.

The structure of Eq. (3) reflects both the network topology and the real-time power flow distribution, forming a consistent and physically interpretable representation of carbon emission propagation across the grid.

To quantify numerical difficulty and its impact on Krylov convergence, we evaluate the condition number κ(A) of the sparse linear system in Eq. (2). For small systems we use exact condition numbers, while for large sparse systems we employ MATLAB’s condest. We also quantify the effective conditioning of the preconditioned operator κ(M−1A), where M is an ILU preconditioner. As shown as Fig. 2, we report condition numbers for four benchmark systems to characterize numerical difficulty and to assess the effectiveness of preconditioning.

These values increase with system size, reflecting the growing challenge of the unpreconditioned problem. With an ILU preconditioner M, the effective condition numbers of the preconditioned operators are approximately κ(M−1A)≈1 for all four systems, indicating near-ideal conditioning. This directly explains the observed robustness and rapid convergence of BiCGSTAB in our large-scale experiments, consistent with the well-known dependence of Krylov convergence on matrix conditioning.

To improve readability of the mathematical formulations in Eqs. (1)–(3), the key variables and parameters are summarized in Table 1, which lists their definitions and corresponding units.

As previously established in Eq. (3), the dynamic CEF at each node is computed by solving a sparse linear system. Given that Eq. (3) must be solved repeatedly across multiple time intervals, the efficiency of the solver is critical. Direct inversion, though accurate, has computational complexity on the order of O(n3) and rapidly becomes impractical for large-scale systems. Hence, we employ a more scalable and robust iterative method: BiCGSTAB, augmented with ILU(0) preconditioning.

To verify the accuracy and computational efficiency of the proposed BiCGSTAB method with ILU preconditioning for CEF calculations, we conducted a series of numerical experiments based on publicly available synthetic power system cases developed by Texas A&M University. These cases, known as ACTIVSg, represent power grids of varying sizes and structural complexity, designed to mimic the statistical and operational characteristics of real-world systems without exposing confidential data.

Specifically, four test cases were selected: ACTIVSg200 (approximately 200 buses), ACTIVSg500 (500 buses), ACTIVSg2000 (2000 buses), and ACTIVSg10k (10,000 buses). These systems simulate the regional grids of Central Illinois, South Carolina, Texas, and the Western United States, respectively. Each system features distinct levels of complexity in terms of topology, voltage levels, and area partitioning, ranging from a single-zone, three-voltage network to a 16-zone, multi-voltage structure including 500 kV down to 13.2 kV.

Table 2 summarizes the basic characteristics of these four test cases, including the number of buses, loads, generators, and branch lines, as well as voltage levels and simulated geographical areas.

The test systems also incorporate diverse energy portfolios with varying carbon intensities. For instance, the ACTIVSg10k model includes 2485 generators using coal, natural gas, nuclear, hydro, wind, and solar energy. The average carbon emission factors for these energy types range from 0.0382 to 0.7175 tCO2/MWh. Table 3 provides a breakdown of generator types across all cases, alongside the emission factors used in our simulations.

Based on the above settings, we calculated the total daily carbon emissions for each system using the BiCGSTAB method. These simulations span 96 time steps (every 15 min over 24 h), covering both nodal injection losses and branch transmission losses. Table 4 reports the results.

To validate the numerical accuracy of the proposed BiCGSTAB algorithm with ILU preconditioning, a comparison was conducted against the direct matrix inversion (inv) method. As shown in Fig. 4, the carbon emission factor curves calculated by both methods over a 24-h period exhibit an exact match, demonstrating that the iterative solver achieves high numerical accuracy consistent with the direct solution. The results confirm that BiCGSTAB is suitable for reliable CEF estimation in power system applications.

Using the BiCGSTAB method, we further computed the total carbon emissions of the ACTIVSg200 system over a full day (96 time intervals). The result, illustrated in Fig. 5, shows the temporal variation of aggregate emissions measured in tons of CO2 (tCO2). The carbon emission profile reflects a clear diurnal pattern, with two prominent peaks occurring in the morning and evening hours, consistent with typical load demand cycles.

To further evaluate the efficiency and robustness of the proposed BiCGSTAB algorithm on large-scale systems, we performed comparative experiments using the ACTIVSg10k case, which contains 10,000 buses and over 12,000 branches. Four commonly used iterative solvers were selected for comparison: least squares QR (LSQR), conjugate gradient squared (CGS), BiCGSTAB, and generalized minimum residual (GMRES).

We standardize solver parameters to ensure a fair comparison across methods. All iterative solvers terminate when the relative residual ‖b−Ax‖/‖b‖≤10−21 or when the iteration cap of 5000 is reached. GMRES uses a restart parameter of m=50. The initial guess is x0=0 for all methods. These settings follow common practice and were chosen to make the numerical error negligible relative to the dynamic CEF variability, so that differences in iteration counts and computation time primarily reflect conditioning and preconditioning effects. Fig. 6 illustrates the convergence trajectories of the four solvers, measured in terms of numerical error (log-scale). Although the horizontal axis shows 9 outer iterations for comparison purposes, it should be noted that GMRES actually performed a total of 59 iterations due to its restart mechanism (i.e., one outer cycle corresponds to 50 inner iterations plus additional steps), whereas BiCGSTAB, CGS, and LSQR iterated 9 times as shown. It is evident that BiCGSTAB demonstrates the fastest and most stable convergence, reaching machine-level precision within only 7 iterations. CGS also shows good convergence performance, though with slightly higher residuals at some steps. GMRES achieves high accuracy as well, but this comes at the cost of significantly more total iterations. By contrast, LSQR converges very slowly and fails to reach even moderate accuracy after 9 iterations.

To assess time stability across multiple trials, Fig. 7 shows a boxplot comparing the execution time range of the four iterative methods over repeated runs on the 10k system. BiCGSTAB not only exhibits the lowest average runtime but also the narrowest variance, indicating both high computational speed and consistency. In contrast, LSQR shows the largest variability and longest tail, suggesting it is less reliable under large-scale sparse matrix conditions.

To evaluate solver scalability, Fig. 8 compares the solution time of each method as system size increases. All solvers show rising computation time with larger grids, but BiCGSTAB grows the slowest, maintaining the shortest runtime across all cases. In contrast, LSQR scales poorly, with its time increasing steeply in the 10k case. This confirms that BiCGSTAB offers not only accuracy and stability but also excellent scalability for large-scale applications.

In order to quantitatively illustrate the performance trends observed in Figs. 6–9, detailed numerical results are summarized in Tables 5 and 6. Table 5 lists the convergence accuracy evolution of each iterative solver across nine iterations, while Table 6 reports the computation time of the proposed ILU-preconditioned BiCGSTAB and other solvers under different power grid scales. These tables provide clear numerical evidence supporting the observed trends in convergence, stability, and scalability.

As shown in Table 5, the ILU-preconditioned BiCGSTAB solver exhibits a monotonic residual reduction from 2.88×103 tCO2 to 1.00×10−12 tCO2 within seven iterations, confirming fast and stable convergence. In contrast, LSQR stagnates at 2.63×10−3 tCO2, indicating poor convergence. Similarly, Table 6 demonstrates that BiCGSTAB achieves the shortest runtime across all cases—from 7.58×10−4 s in the 200-node system to 0.0131 s in the 10 000-node system—while other solvers such as LSQR (0.0478 s), GMRES (0.0267 s), CGS (0.0239 s) scale more poorly. These results quantitatively verify that the proposed method maintains excellent accuracy, stability, and scalability for large-scale dynamic CEF calculations.

As shown in Fig. 9, we compare two strategies across 96 time periods for four benchmark systems (200, 500, 2000, and 10k buses): (i) re-factorization at every time step and (ii) reuse of the ILU factors computed at the first time step. For completeness, two reuse scenarios are reported: Strategy-1 includes the one-off initial factorization cost; Strategy-2 excludes it (ideal reuse). Totals over 96 periods are as follows:

case_ACTIVSg200: re-factorization 0.065844 s; reuse-1 0.059550 s (9.56% saving; 1.11×); reuse-2 0.059367 s (9.84% saving; 1.11×).

The data show consistent time savings (≈10%–27%) and speedups (≈1.11–1.37×) across scales. Factorization time remains stable over time, whereas solve time varies mildly with loading, which supports the practicality of reusing ILU factors in dynamic multi-period computation.

It is worth noting that the direct inverse method is not shown in the boxplot because its execution time was prohibitively long—averaging around 8 s per solve, which makes it impractical for large-scale dynamic carbon emission analysis.

Beyond the computational performance demonstrated above, the proposed ILU-preconditioned BiCGSTAB framework enables real-time and high-resolution CEF estimation even in large-scale nodal systems, which is essential for practical carbon management and policy applications. The ability to compute accurate emission factors across thousands of nodes within seconds allows system operators and regulators to track carbon flows dynamically and make timely, data-driven decisions. Such fast and scalable CEF calculations can significantly enhance regional emission accounting accuracy, support carbon pricing and emission trading mechanisms, and facilitate low-carbon dispatch and planning strategies in power systems of increasing complexity. These broader applications will be further explored in future studies integrating the proposed computational method with carbon policy and operational decision models.

This paper presents a fast and scalable method for calculating dynamic carbon emission factors (CEFs) in large-scale power systems using the ILU-preconditioned BiCGSTAB iterative solver. The proposed method effectively overcomes the computational limitations of traditional matrix inversion techniques, achieving significant improvements in both efficiency and scalability. Numerical results show that the BiCGSTAB solver with ILU preconditioning provides accurate and rapid CEF calculations, making it suitable for real-time carbon emission analysis in large power grids.

Compared with conventional direct solvers, the proposed approach achieves faster convergence and lower computational costs, particularly in dynamic grid scenarios involving multiple time intervals and large-scale systems. This work contributes to the field by introducing an efficient iterative framework for calculating dynamic CEFs, which can be applied to various power system configurations and support carbon emission monitoring and reduction strategies.

The case studies are based on synthetic ACTIVSg test systems, which accurately represent the structural and operational characteristics of real power grids. However, real-world data may contain measurement noise and missing information that could affect model accuracy. This limitation will be addressed in future work to enhance the robustness of the proposed method under practical operating conditions.

It should also be noted that the performance of the algorithm can be sensitive to the choice of preconditioner parameters. Future studies will explore adaptive or optimized preconditioning strategies to improve convergence stability under different grid conditions. Furthermore, future research will focus on integrating the proposed CEF model with renewable energy forecasting and real-time dispatch frameworks to support carbon-aware grid operations. The method may also be extended to multi-energy systems to analyze interactive carbon flows among electricity, heat, and gas networks, thereby strengthening its role in low-carbon system planning and operation.