Preamble

Carbon Flow Tracing Accounting for Source-Side Carbon Emission Measurement Uncertainty

Abstract

With the rapid development of the global low-carbon economy, the accurate quantification and tracing of carbon emissions in power systems have become critical for achieving "dual carbon" goals. Traditional carbon flow tracing methods typically assume that the carbon emission intensity at the source side is a deterministic value. However, in practical engineering scenarios, carbon emission data from power plants often contain measurement uncertainties due to sensor precision, environmental fluctuations, and reporting errors. This paper proposes a carbon flow tracing model that accounts for source-side carbon emission measurement uncertainty. By integrating uncertainty modeling into the carbon emission intensity of generators, we utilize a probabilistic carbon flow framework to analyze how source-side variations propagate through the network to the load side. Case studies demonstrate that considering measurement uncertainty provides a more robust and realistic assessment of carbon footprints, offering a theoretical basis for carbon tax policies and green power trading.

1. Introduction

The power sector is one of the primary sources of global carbon dioxide emissions. To effectively reduce emissions, it is essential to clarify the carbon emission responsibilities of different stakeholders within the power system. Carbon flow tracing (CFT) provides a systematic method to map carbon emissions from generation sources to network losses and terminal consumers based on the physical distribution of power flows.

Existing research on carbon flow tracing has established a solid theoretical foundation, primarily based on the proportional sharing principle. However, most current models treat source-side carbon emission intensity as a fixed parameter. In reality, the carbon emissions of thermal power units are affected by factors such as fuel quality, combustion efficiency, and the accuracy of Continuous Emission Monitoring Systems (CEMS). These factors introduce non-negligible uncertainties. Ignoring these uncertainties may lead to biased carbon accounting, potentially resulting in unfair carbon cost allocations or inaccurate carbon footprint labeling for end-users.

This paper addresses these gaps by introducing a carbon flow tracing method that explicitly accounts for source-side measurement uncertainty. We model the source-side carbon intensity as a stochastic variable and employ sensitivity analysis and probabilistic methods to quantify the impact on the carbon intensity of nodes and lines.

2. Mathematical Model of Carbon Flow Tracing

The fundamental theory of carbon flow tracing is based on the distribution of power flows. For a power system with $N$ nodes, the carbon emission intensity of the nodes can be derived from the power balance and the carbon intensity of the generation sources.

2.1 Basic Carbon Flow Equations

The node carbon intensity vector $\

Abstract

Following the establishment of the "dual carbon" targets, the carbon emission analysis of power systems has emerged as a key research direction for the low carbon transition of the power industry. Existing methods overlook the uncertainty in source side carbon metering, treating the carbon emission intensity of generation units as deterministic variables at the algorithmic level. Consequently, they are unable to assess the uncertainty of carbon flow at downstream power system nodes.

To tackle this problem, we propose a carbon flow tracking method that accounts for the uncertainty in source side carbon metering. This method treats the carbon emission intensity of generation units as stochastic variables modeled by Alpha-stable distributions. Based on the power flow distribution matrix theory, a linear mapping relationship from the generation side to downstream nodes is established. By leveraging the linear combination property of Alpha stable distributions, the parameters of the resulting Alpha stable distribution for the carbon flow rate at each node can be efficiently derived. This method quantifies the stochastic fluctuation ranges of carbon flow rates at each node, avoiding systematic bias accumulation and decision-making risks caused by deterministic assumptions.

Validated on 2 generator 4 node and 6 generator 30 2023hb0006 National Natural Science Foundation of China under Grant 62473125, Ecological Environment Research Project of Anhui Province under Grant 2023hb0006 systems, the results from the theoretical derivation show high agreement with those from Monte Carlo simulations.

WORDS Uncertainty of Source Side Carbon Measurement; Alpha Stable Distribution; Power Distribution Matrix; Monte Carlo Simulation

摘要

With the introduction of "dual carbon" goals, carbon emission analysis in power systems has become a critical research direction for the low-carbon transition of the power industry. Existing methods often overlook the uncertainty inherent in source-side carbon metering, treating the carbon emission intensity of generating units as deterministic variables at the algorithmic level. Consequently, these approaches fail to evaluate the uncertainty of subsequent carbon emission flow at power system nodes. This paper proposes a carbon emission flow tracking method that accounts for-source side carbon metering uncertainty. By treating the carbon emission intensity of generating units as random variables following an Alpha-stable distribution, a linear mapping relationship from the generation side to downstream nodes is established based on power flow distribution matrix theory. Utilizing the linear combination properties of the Alpha-stable distribution, the Alpha-stable distribution parameters for the carbon flow rate at each node are rapidly derived. This method quantifies the stochastic fluctuation range of carbon flow rates at each node, thereby avoiding the cumulative bias and decision-making risks associated with deterministic assumptions.

System verification was conducted on the nodes, and the theoretical derivation results are highly consistent with the Monte Carlo simulation results.

关键词

Uncertainty in Source-Side Carbon Accounting: A Monte Carlo Simulation Approach

1. Introduction

Accurate carbon accounting is the cornerstone of achieving carbon neutrality and implementing effective emissions trading schemes. Traditional carbon accounting often relies on top-down statistical approaches; however, source-side carbon accounting—which focuses on direct measurement and calculation at the emission source—is becoming increasingly critical for granular power system management. Due to the inherent variability in sensor accuracy, fuel quality, and operational conditions, source-side carbon accounting is subject to significant uncertainty. This study utilizes Monte Carlo simulation to quantify these uncertainties, providing a robust framework for assessing the reliability of carbon emission data.

2. Methodology for Source-Side Carbon Accounting

Source-side carbon accounting primarily involves calculating the emissions generated during the combustion of fossil fuels for electricity generation. The total carbon emissions ($E$) can be expressed as a function of several key parameters:

$$E = \sum_{i=1}^{n} (AD_i \times EF_i \times OF_i)$$

Where:
- $AD_i$ represents the activity data (e.g., fuel consumption) for unit $i$.
- $EF_i$ is the emission factor, often derived from the carbon content of the fuel.
- $OF_i$ is the oxidation fraction, representing the efficiency of the combustion process.

Each of these variables is subject to measurement errors and inherent stochasticity, which necessitates an uncertainty analysis.

[FIGURE:1]

3. Monte Carlo Simulation Framework

To evaluate the propagation of uncertainty from individual input parameters to the final emission result, we employ a Monte Carlo simulation (MCS) framework. The process is divided into the following steps:

3.1 Probability Distribution Modeling

For each input variable ($AD$, $EF$, $OF$), we assign a probability density function (PDF) based on empirical data or manufacturer specifications. Common distributions include:
- Normal Distribution: Used for high-precision metered data such as fuel flow.
- Uniform or Triangular Distribution: Used when only the upper and lower bounds of a parameter (e.g., carbon content ranges) are known.
- Lognormal Distribution: Used for parameters that are strictly positive and exhibit right-skewness.

3.2 Random Sampling and Iteration

The MCS generates $N$ sets of random samples from the defined PDFs. For each iteration $k \in {1, \

0 引言

Carbon Flow Tracking Considering the Uncertainty in Source-side Carbon Metering

JIANG Yishan$^1$, MA Dawei$^2$, LIU Chao$^2$, KONG Ming$^2$, MIAO Bo$^2$, LI Zerui$^1$, YU Wenjun$^{1,6,7*}$

1. Department of Automation, University of Science and Technology of China, Hefei 230026, Anhui Province, China
2. Electric Power Research Institute of State Grid Anhui Electric Power Co., Ltd., Hefei 230601, Anhui Province, China

Introduction

Global climate change has emerged as a significant challenge facing humanity, making the control of greenhouse gas emissions a critical agenda item for nations worldwide. In response to these climatic shifts, China has explicitly proposed the "dual carbon" goals: achieving a carbon peak by 2030 and reaching carbon neutrality by 2060. This commitment marks China's entry into a comprehensive transition toward a low-carbon economy and sustainable development.

As the energy sector is a primary source of carbon emissions, tracking and quantifying the flow of carbon within power systems is essential for effective mitigation. Carbon flow tracking provides a theoretical framework for attributing carbon emissions from the generation side to the demand side, enabling more precise carbon accounting and the implementation of market-based decarbonization mechanisms. However, existing methodologies often overlook the inherent uncertainties in source-side carbon metering. Measurement errors, sensor inaccuracies, and data transmission issues at the generation source can propagate through the network, leading to significant discrepancies in the final carbon flow distribution.

This study addresses these challenges by proposing a carbon flow tracking model that explicitly accounts for the uncertainty in source-side carbon metering. By integrating probabilistic modeling with traditional carbon emission flow theories, we aim to provide a more robust and reliable assessment of carbon distribution across the power grid, supporting the national strategic objectives for carbon reduction.

4. I

nstitute of Artificial ntelligence, Hefei Comprehensive National Science Center, Hefei 230088, Anhui Province China

5. Anhui Province Key Laboratory of Intelligent Low

Carbon Information Technology and Equipment, University of Science and Technology of China, Hefei 230027, Anhui Province China

Introduction

The global energy sector is entering a new phase of green and low-carbon transition. As the primary contributor to global carbon emissions, the power industry accounts for the highest proportion of emissions within the energy sector. Statistics indicate that its carbon dioxide emissions represent approximately 40% of total societal emissions, making it a critical field for achieving national emission reduction targets. Furthermore, as electrification levels continue to rise, carbon emissions from other industries are gradually being converted into direct or indirect emissions generated by electricity consumption. This further elevates the importance of carbon emission management within power systems. In this context, accurately quantifying and effectively managing power system carbon emissions is of great significance for achieving macro-level carbon reduction goals. Traditional macro-level carbon emission calculation methods start from aggregate data and perform statistics based on total energy consumption over a period. These methods offer advantages such as computational simplicity and ease of use, and are thus widely applied to carbon calculations over long time horizons. However, the results of such methods exhibit a certain time lag and are relatively coarse, failing to describe the microscopic evolution of various carbon indicators or accurately track the specific flow of carbon emissions. To overcome these limitations and meet the demands for refined carbon management in power systems, carbon flow tracking methods based on power flow distribution have gradually developed, providing new research perspectives for the precise analysis of power system carbon emissions.

To address the limitations of traditional methods, researchers such as Zhou et al. pioneered the integration of carbon emission analysis with power system power flow calculations, proposing the concept of the carbon emission flow (CEF). They defined carbon emission flow as a virtual network flow that is dependent on the power flow and characterizes the carbon emissions required to maintain the power flow in any given branch of the power system. Subsequently, they introduced key indicators and concepts for carbon emission calculation, systematically defining core metrics such as branch carbon flow, branch carbon flow rate, branch carbon flow density, and nodal carbon intensity, thereby establishing the basic theoretical system, calculation methods, and framework for carbon emission flow. These works laid the theoretical foundation for subsequent research and promoted the exploration of carbon flow theory in practical scenarios, such as carbon emission measurement on the electricity consumer side. To better describe the distribution characteristics of carbon emission flow in power networks, researchers revealed the distribution properties and transmission-consumption mechanisms of carbon flow by defining various incidence matrices. However, these early studies were based on lossless network assumptions, making them inapplicable to actual lossy networks. To address this issue, researchers have conducted in-depth studies on power loss allocation. Some scholars have allocated network losses to loads by calculating equivalent transmission power for lines and equivalent nodal loads, thereby transforming lossy networks into lossless ones. This transition has moved carbon flow calculation from idealized lossless assumptions toward precise modeling of actual lossy networks, effectively addressing the deficiencies in carbon flow analysis theory.

Simultaneously, regarding the fair allocation of carbon emission responsibility, researchers have proposed methods where the generation side and the load side jointly share carbon responsibility, and have systematically compared different allocation schemes. To further optimize carbon emission flow calculation methods, some studies have addressed the difficulties in accurately measuring embodied carbon emissions and carbon reduction from renewable energy. For instance, a bidirectional allocation method for network loss carbon emissions based on complex power tracking has been proposed. Other researchers have constructed data-driven methods based on Bayesian interference regression, providing a new technical path for establishing carbon flow models. Existing carbon flow tracking methods have made significant progress in theoretical frameworks and computational accuracy, laying a solid foundation for the precise analysis of power system carbon emissions. However, existing methods do not account for the uncertainty in source-side carbon measurement; at the algorithmic level, they treat the carbon emission intensity of generating units as deterministic variables. Although the carbon emission intensity of a generating unit at a specific moment is physically deterministic, it remains an unobservable latent variable to us. Existing carbon measurement methods—whether carbon monitoring, accounting methods at fine time scales, or soft-sensing methods—all involve non-negligible errors. If the uncertainty of source-side carbon measurement is ignored and treated as a deterministic variable, carbon flow tracking is downgraded from "probabilistic navigation" to "biased navigation," where all downstream decisions are executed on a carbon map that appears precise but is actually drifting.

In fact, the true value of the source-side carbon emission intensity can be viewed as the measured value minus the error value. Therefore, algorithmically, it can be treated as a random variable following a specific probability distribution.

To address the aforementioned issues, this paper proposes a carbon flow tracking method that accounts for the uncertainty of source-side carbon measurement. This method considers the measurement uncertainty of the carbon emission intensity of source-side generating units by treating the carbon emission intensity as a random variable modeled by an Alpha distribution. Based on power flow distribution matrix theory, a linear mapping relationship from the generation side to downstream nodes is established. By leveraging the linear combination properties of the Alpha-stable distribution, the Alpha-stable distribution parameters for the carbon flow rate at each node can be rapidly derived. Through system validation, the theoretical derivation results are shown to be highly consistent with Monte Carlo simulation results.

1.1 潮流分布矩阵构建理论

The power flow distribution matrix is a critical mathematical tool that links generator power injections to the power flowing through each node in the system. Its construction is based on the topological structure of the power system network and the principle of power conservation. This matrix accurately describes the distribution and transmission process of generator power within complex networks, providing a mathematical foundation for achieving power traceability from the generation side to the load side.

According to the law of power conservation in power systems, the power balance relationship at any node $i$ can be expressed as

 = + =   ( 1 )

The flow-through power of node $i$ is defined as the total power flowing into that node. Let $\mathcal{U}i$ denote the set of upstream nodes for node $i$, and let $P$ represents the generator power injected at node $i$.}$ represent the power flowing from an upstream node $j$ to node $i$. Furthermore, $P_{G,i

By rearranging the terms, we obtain

  − = =       ( 2 )

It can be further expressed in matrix form. The elements of the power flow distribution matrix are defined according to the following rules:

=    = −  =     

The diagonal elements of the matrix are 1, reflecting the self-balancing power characteristics of the nodes. The off-diagonal elements represent the power contribution ratio of an upstream node to a downstream node, the values of which are determined entirely by the network topology and power flow distribution. By performing a matrix inversion operation, the relationship can be further analyzed.

$$P_u = A^{-1} P_G \tag{5}$$

It enables the precise linear allocation of generator power to the power flowing through each node. This linear mapping establishes a deterministic correlation between power injection on the generation side and the power status of each node in the system. Consequently, it provides a solid mathematical theoretical foundation for traceability calculations in subsequent carbon emission flow analyses.

1.2 碳排放流理论框架

Carbon emission flow theory treats carbon emissions as a virtual network flow attached to power flows, describing the spatial transmission and distribution characteristics of carbon emissions within a power grid. This theory breaks through the limitations of traditional macro-statistical methods regarding spatial precision and temporal resolution, enabling accurate carbon emission tracing from the generation side to the consumption side. Consequently, it provides scientific theoretical support for refined carbon management in power systems. The core concept of carbon emission flow lies in integrating the carbon emission intensity of generating units with the system's power flow distribution. Through a linear mapping relationship established by the power flow distribution matrix, carbon emissions from the generation side are accurately allocated to each load node according to power transmission paths. This allocation method fully accounts for the network characteristics and power transmission laws of the power system, ensuring both the physical rationality and mathematical rigor of the carbon emission distribution.

Based on the theory of power flow distribution matrices, the expression for the load carbon flow rate at node $i$ is:

$$E = P A^{-1} e \tag{6}$$

The carbon flow of a load is defined as the equivalent carbon emissions at the generation side corresponding to the electricity consumed by a node per unit of time. Let $P_L$ denote the load power, and $P_{flow}$ represent the power flow. Let $P_G$ be the generator power vector and $E_G$ be the generator carbon emission intensity vector. The remaining components are defined accordingly. The operator $\text{diag}(\cdot)$ denotes the diagonalization operation. Within the total carbon flow rate, the portion borne by the load at node $i$ can be expressed as:

$$R_{Gi} = P_{Lk} E_{Gi} L_{k, Gi} \tag{7}$$

$L_{k, G, i}$ represents the carbon emission intensity derived from the power generation unit.

2.1 源侧碳计量不确定性分析

Existing methods fail to account for the uncertainty inherent in source-side carbon metering, treating the carbon emission intensity of generating units as deterministic variables at the algorithmic level. Although the carbon emission intensity of a generating unit at any given moment is physically deterministic, it remains an unobservable latent variable from a measurement perspective. If source-side metering uncertainty is ignored and treated as a deterministic value, carbon flow tracing degrades from "probabilistic navigation" to "biased navigation," where all downstream decisions are executed on a carbon map that appears precise but is actually drifting. In practice, the true value of source-side carbon emission intensity can be viewed as the metered value minus an error term; therefore, it can be treated algorithmically as a stochastic variable following a specific probability distribution.

First, it is necessary to identify a distribution that possesses strong expressiveness while maintaining computational convenience. Specifically, four key characteristics must be considered: 1) Asymmetry: Any carbon metering method is subject to bias, manifesting as either "overestimation" or "underestimation." Consequently, the distribution is "skewed," requiring a model capable of characterizing asymmetry. 2) Centrality: The "accuracy" of different metering methods varies significantly. Therefore, the distribution should be unimodal, with the probability density concentrated near the peak. 3) Heavy-tailedness: Metering methods that have not been calibrated or subjected to manual intervention may produce extreme "outliers." Thus, the distribution should exhibit heavy-tailed characteristics, potentially in a unidirectional manner. 4) Stability: For computational efficiency, we prefer a stable distribution. Stability typically refers to a family of distributions that remains closed under specific operations, such as addition, multiplication, or linear transformations. In other words, if an operation on members of a distribution family yields a result that still belongs to that same family, the distribution family is considered stable.

The Alpha-stable distribution is employed to describe source-side carbon emission intensity. By adjusting the skewness parameter ($\beta$), one can generate left-skewed or right-skewed distributions, thereby effectively characterizing asymmetry. By adjusting the scale parameter ($\gamma$) and the location parameter ($\delta$), the concentration of probability density near the peak can be controlled, allowing the Alpha distribution to accurately reflect data centrality. By adjusting the stability parameter ($\alpha$), a heavy-tailed distribution can be generated to capture the occurrence of extreme values. Furthermore, by tuning the skewness parameter, distributions with either left or right heavy tails can be produced. Crucially, the Alpha distribution is a type of stable distribution, meaning it maintains its distributional form under the addition of independent random variables.

In engineering practice, it is first necessary to determine the Alpha distribution of carbon metering errors. This is achieved by collecting field experimental data and comparing it against standard reference data. Based on continuous observation data spanning at least one full operating cycle, parameter estimation methods—such as maximum likelihood estimation, method of moments, or quantile matching—are employed to identify the error distribution parameters for each device.

2.2.1 Alpha

Definition of Stable Distributions

A stable distribution, denoted as $S(\alpha, \beta, \gamma, \delta)$, is defined by its characteristic function. A random variable $X$ is said to follow a stable distribution if there exist parameters $\alpha \in (0, 2]$, $\beta \in [-1, 1]$, $\gamma > 0$, and $\delta \in \mathbb{R}$ such that its characteristic function $\phi(t) = E[\exp(itX)]$ takes the following form:

$$
\ln \phi(t) =
\begin{cases}
-\gamma^\alpha |t|^\alpha \left[ 1 - i\beta \text{sgn}(t) \tan\left(\frac{\pi\alpha}{2}\right) \right] + i\delta t, & \alpha \neq 1 \
-\gamma |t| \left[ 1 + i\beta \text{sgn}(t) \frac{2}{\pi} \ln|t| \right] + i\delta t, & \alpha = 1
\end{cases}
$$

In this parameterization:
- $\alpha$ is the stability index (or characteristic exponent), which determines the thickness of the tails.
- $\beta$ is the skewness parameter, governing the asymmetry of the distribution.
- $\gamma$ is the scale parameter, representing the spread of the distribution.
- $\delta$ is the location parameter, shifting the distribution along the real axis.

 = −  

sign( )tan

When $\alpha = 1$:

  = − +    

$\text{sign}(\cdot)$, $\ln|\cdot|$. $\alpha \in (0, 2]$ is the stability parameter (characteristic exponent). A smaller $\alpha$ indicates a heavier tail and stronger impulsiveness, meaning a higher probability of extreme values; a larger $\alpha$ indicates a thinner tail and lower impact. When $\alpha = 2$, the Alpha-stable distribution degenerates into a Gaussian distribution. $\beta \in [-1, 1]$ is the skewness parameter, which controls the degree of asymmetry. When $\beta = 0$, the distribution is symmetric about the location parameter, known as a Symmetric Alpha-Stable (S$\alpha$S) distribution. When $\beta > 0$, the distribution is right-skewed; when $\beta < 0$, it is left-skewed. $\gamma > 0$ is the scale parameter, which determines the dispersion of the distribution, similar to the standard deviation in a normal distribution; a larger $\gamma$ indicates a more dispersed distribution, while a smaller $\gamma$ indicates a more concentrated one. $\delta \in \mathbb{R}$ is the location parameter, which shifts the entire distribution along the real axis. For symmetric Alpha-stable distributions, $\delta$ represents the mean when $\alpha > 1$. In the actual measurement process of carbon emission intensity, from a metrological perspective, the measured value of a generator unit's carbon emission intensity can be decomposed into three parts: the true value, systematic error, and random error. Systematic errors exhibit deterministic tendencies, such as instrument calibration bias or inherent algorithmic bias, showing consistent directionality in repeated measurements. In contrast, random errors manifest as unpredictable stochastic fluctuations. In practical engineering, systematic errors, as deterministic biases, are typically eliminated or compensated for through technical means such as calibration and comparison before modeling. However, due to technical limitations and cost constraints, residual systematic biases are difficult to eliminate entirely.

The parameter settings of the Alpha-stable distribution are precisely a mathematical mapping based on this metrological principle. Within the parameterization framework of the Alpha-stable distribution, the value after preliminary correction of systematic errors represents the nominal carbon emission intensity of the unit under its current operating state. Therefore, the location parameter of the Alpha-stable distribution is set such that it directly reflects the nominal level of carbon emission intensity. This parameterization method not only conforms to the mathematical definition of the Alpha-stable distribution but also ensures consistency between the model and the physical measurement process: the location parameter directly anchors the nominal level of carbon emission intensity, while the other parameters characterize the random fluctuation properties around this nominal level.

However, limited by calibration precision and the presence of unknown bias sources, residual systematic biases still exist, which constitutes the physical significance of the skewness parameter $\beta$. A non-zero skewness parameter indicates that even after correction, the measured values still systematically tend to overestimate or underestimate the true value. This residual systematic error leads to the asymmetry of the distribution, reflecting measurement biases that are difficult to fully eliminate in actual metering devices under specific operating conditions. The scale parameter $\gamma$ and the stability parameter $\alpha$ jointly characterize the statistical features of the random error. The scale parameter quantifies the magnitude of the random fluctuations, while the stability parameter describes the frequency of extreme fluctuations, i.e., the thickness of the distribution's tail.

It is worth noting that the existence of random error means that even under the same operating conditions, repeated measurements will yield different results. This inherent randomness is the root cause of the distribution's dispersion.

The essence of this modeling approach lies in treating the unobservable true carbon emission intensity as a random variable centered at the corrected measured value and perturbed by random errors and residual systematic errors. This framework not only maintains

statistical rigor but also scientifically quantifies the unavoidable randomness in the measurement process. It establishes a solid probabilistic foundation for the subsequent analysis of carbon flow uncertainty propagation.

2.2.2 参数变化对分布形状影响

The Influence of Parameter Variations in the Alpha-Stable Distribution on the Distribution Shape

The alpha-stable distribution is defined by four parameters that govern its shape and behavior. The stability parameter $\alpha$ controls the heavy-tail characteristics of the distribution; as $\alpha$ increases, the distribution converges from an extreme heavy-tail state toward a normal distribution, directly influencing the probability of extreme events. The skewness parameter $\beta$ determines the symmetry of the distribution, where variations in $\beta$ shift the distribution from left-skewed to right-skewed. The scale parameter $\gamma$ adjusts the width of the distribution, such that an increase in $\gamma$ makes the distribution more dispersed while maintaining its overall shape. Finally, the location parameter $\delta$ simply shifts the center of the distribution without altering its geometric form. These parametric characteristics provide a theoretical foundation for modeling the uncertainty of carbon emission intensity in power system generating units.

Characteristic Function Product Theorem

For independent alpha-stable random variables, the characteristic function of their sum is given by the product of their individual characteristic functions.

Let $X = \sum X_i$, where $X_i \sim S(\alpha, \beta, \gamma, \delta)$. Its characteristic properties are defined as follows:

= =  ( 10 )

The contribution of each component calculated by the characteristic function is given by:

 = − −  

ign( )sign( )tan

When $\alpha = 1$, the contribution of the $i$-th component is given by:

  = − +    

3) When the combined feature function is $\text{sign}(\cdot) \ln |\cdot|$, the result of multiplying all components is: $\text{sign}(\cdot) \tan(\cdot)$.

(13) When $\alpha = 1$, the result of multiplying all components is:

  = −    

Carbon Flow Rate Distribution at Load Nodes

The carbon flow rate at a load node is a critical metric for understanding the carbon emission intensity associated with electricity consumption. Based on the principles of carbon emission flow in power systems, the carbon flow rate at a specific load node can be characterized by the relationship between the power demand and the carbon intensity of the energy supplied to that node.

Mathematically, the carbon flow rate is often expressed using the sign function and logarithmic transformations to account for the directionality and scale of power flows. For a given load node, the distribution of the carbon flow rate can be analyzed by considering the following relationship:

$$ \text{sign}(\cdot) \ln | \cdot | $$

This formulation allows for the characterization of both the magnitude and the proportional changes in carbon emissions relative to the power flow. Specifically, the term $\text{sign}(\cdot)$ ensures that the directional nature of the energy exchange is preserved, while the logarithmic component $\ln | \cdot |$ compresses the dynamic range of the flow rates, facilitating the analysis of systems with highly variable load profiles.

In the context of load node carbon flow rate distribution, the calculation typically involves the nodal carbon intensity, which is determined by the weighted average of the carbon intensities of all power sources contributing to the node's supply. The total carbon flow rate at node $i$ can be defined as:

$$ R_i = P_{L,i} \cdot \rho_i $$

where $R_i$ represents the carbon flow rate, $P_{L,i}$ is the active power demand at the load node, and $\rho_i$ is the carbon emission intensity of the electricity consumed at that node. By applying the $\text{sign}(\cdot) \ln | \cdot |$ transformation to these variables, researchers can better visualize and model the sensitivity of carbon distributions across complex network topologies, particularly when identifying high-emission "hotspots" within the grid.

2.3.1 基于

Alpha stable distribution of load and generator carbon emission intensity. We assume that the carbon emission intensity of each generator follows an Alpha stable distribution. The linear combination theory of Alpha stable distributions requires that: if and only

if independent random variables share the same stability parameter $\alpha$, their linear combination will strictly follow the same family of distributions. Furthermore, the resulting distribution parameters can be precisely calculated through analytical expressions.

From an engineering perspective, generators of the same type typically adopt similar technical standards and operation and maintenance specifications; thus, the statistical characteristics of their carbon emission intensities exhibit a degree of similarity. Based on these theoretical requirements and engineering rationality, it is assumed that the carbon emission intensities of all generating units share the same stability parameter $\alpha$. This allows the model to be expanded into a linear combination form:

= =   ( 17 )

The total number of generators in the system and the linear combination coefficients are defined as:

− =   e A e ( 18 )

load power; flow power; the remaining components are. This indicates that the nodal carbon flow rate is a weighted linear combination of the carbon emission intensities of each generator. The weight coefficients are entirely determined by the network topology, power flow distribution, and power configuration of the power system. This linear relationship is mathematically grounded in the theory of Alpha-stable distributions.

Derivation of Distribution Parameters for Load Node Carbon Flow Rates

For each load node in the power system, the characteristic function of the Alpha-stable distribution is expressed as:

$$ = \sum $$

The parameters for the linear combination can be derived as follows:
Scale parameter:

  =      ( 20 )

Skewness parameter: $\text{sign}(\cdot)$ Location parameter:

= =  ( 22 )

When $\alpha = 1$:

sign( )ln |

Lk i ki ki i i ki ki i i a a a a     = = = −   ( 23 )

Construction of the Power Flow Distribution Matrix

The construction of the power flow distribution matrix is a fundamental step in analyzing the steady-state characteristics of a power system. This matrix establishes the mathematical relationship between nodal power injections and the resulting power flows across the network's branches. By leveraging the topological structure of the grid and the electrical parameters of the transmission lines, we can derive a sensitivity matrix—often referred to as the Generation Shift Factor (GSF) or Injection Power Loss Factor matrix—which facilitates rapid contingency analysis and optimal power flow calculations. The construction process typically involves linearizing the AC power flow equations around an operating point or utilizing the DC power flow approximation to ensure computational efficiency in large-scale systems.

Calculation of Nodal Power Throughput

The calculation of nodal power throughput (also referred to as nodal through-power) is essential for identifying critical hubs within the network and assessing the loading levels of specific buses. This calculation involves summing the magnitudes of power entering or leaving a node from all connected branches, as well as accounting for local generation and demand. Specifically, for any given node $i$, the power throughput can be quantified by analyzing the power flow vectors $\vec{S}_{ij}$ for all adjacent nodes $j$. This metric provides a comprehensive view of the node's role in power transmission and distribution, serving as a key indicator for vulnerability assessment and grid reinforcement planning. In a steady-state regime, the nodal power balance must satisfy the fundamental Kirchhoff’s laws, ensuring that the net injection equals the sum of outgoing flows minus local consumption.

$$P_u = A^{-1} P_G$$

Calculation of Linear Combination Coefficients, Node Carbon Flow Rate Distribution Parameters, and Stability Parameters

In the analysis of carbon emission flow in power systems, the determination of linear combination coefficients and distribution parameters is essential for accurately mapping carbon intensity from generation sources to network nodes. This section details the mathematical formulation for calculating the node carbon flow rate distribution parameters and the associated stability parameters.

Calculation of Node Carbon Flow Rate Distribution Parameters

The carbon flow rate at any given node is determined by the proportional contribution of various energy sources entering that node. To calculate the node carbon flow rate distribution parameters, we define the relationship between the inflow of power and the resulting carbon intensity. Let the carbon flow rate at node $i$ be represented by $\rho_i$. The distribution of these rates across the network can be expressed through a system of linear equations that account for the network topology and the carbon intensity of individual generators.

The linear combination coefficients, denoted as $\alpha_{ij}$, represent the fraction of power at node $i$ that originates from node $j$. These coefficients are derived from the power flow distribution matrix. By integrating these coefficients with the carbon intensity of the primary energy sources, we can derive the steady-state distribution of carbon flow rates across all nodes in the system.

Stability Parameters and System Equilibrium

To ensure the reliability of the carbon flow model, it is necessary to evaluate the stability parameters. These parameters characterize the sensitivity of the node carbon flow rates to fluctuations in power demand or changes in the generation mix.

The stability parameter $\sigma$ is defined based on the eigenvalues of the distribution matrix. A system is considered stable if the spectral radius of the transition matrix remains within the unit circle, ensuring that the iterative calculation of carbon flow rates converges to a unique equilibrium. This stability is critical for real-time monitoring and carbon accounting, as it guarantees that small perturbations in the power grid do not lead to divergent or non-physical carbon flow values.

[TABLE:1]

As shown in [TABLE:1], the relationship between the node carbon flow rate distribution parameters and the stability indicators demonstrates that as the network complexity increases, the margin for stability requires more precise calibration of the linear combination coefficients.

Mathematical Formulation

The calculation of the coefficients and parameters follows the general form:

$$\begin{aligned} R_{node} &= (I - A)^{-1} P_{gen} \ \Phi_{stable} &= \max_{i} |\lambda_i(A)| \end{aligned}$$

2.3.2 蒙特卡洛仿真验证方法

To theoretically verify the accuracy of the aforementioned method, this paper employs Monte Carlo simulation as the benchmark comparison method.

The analytical method provides a theoretical foundation and efficient computational tools, while the Monte Carlo method serves as an independent numerical verification benchmark. The cross-validation of these two methods, based on distinct principles, significantly enhances the credibility of the research findings and ensures the correctness and practical utility of the theoretical derivations.

The combined use of these two methods achieves a balance between efficiency and precision, providing a complete methodology for the uncertainty analysis of carbon flow in power grids.

3 算例分析

4 Case Study: Four-Node Power System

To verify the effectiveness of the proposed theoretical method, this paper selects a four-node power system as a test case for analysis. The system consists of four nodes and two load centers, with the specific configuration detailed as follows:

Node 1 is equipped with a coal-fired unit ($G_1$) with an active power output of 400 MW and a nominal carbon emission intensity of 0.720 $\text{tCO}_2/\text{MWh}$. Node 2 is equipped with a gas-fired unit ($G_2$) with an active power output of 114 MW and a nominal carbon emission intensity of 0.375 $\text{tCO}_2/\text{MWh}$. These carbon emission intensity parameters are derived from actual operational monitoring data of units in Anhui Province; specifically, the coal-fired unit data originates from Unit 1 of the Wuhu Power Plant, while the gas-fired unit data is sourced from Unit 2 of the Anqing Power Plant.

Regarding the demand side, Node 3 carries a load of 300 MW, and Node 4 carries a load of 200 MW. This system configuration encompasses typical high-carbon and low-carbon generation types, making it highly suitable for validating the effectiveness of carbon emission flow tracking methods in scenarios involving heterogeneous carbon sources.

We assume that the carbon emission intensities of the two generators both follow an $\alpha$-stable distribution and share the same stability parameter $\alpha$. Specifically, let the carbon emission intensity of generator $i$ ($i=1,2$) be denoted by $X_i$, such that $X_i \sim S_{\alpha}(\sigma_i, \beta_i, \mu_i)$. Here, $\sigma_i$ represents the scale parameter, $\beta_i$ the skewness parameter, and $\mu_i$ the location parameter. Under the assumption of a common stability index $\alpha \in (0, 2]$, these distributions characterize the heavy-tailed nature and potential asymmetry often observed in empirical emission data.

The distribution parameters for generator G1 are as follows: the skewness parameter $\beta_1 = -0.3$, the scale

The parameter $\gamma_1 = 0.05$ and the location parameter $\delta_1 = 0.720$; the distribution of generator G2 is as follows:

The parameters are: skewness parameter $\beta_2 = 0.2$ and scale parameter $\gamma_2 = 0.03$.

The parameter is set to $\delta_2 = 0.375$. This specific parameter configuration reflects the inherent operational characteristics and response behaviors of different types of generators within the system.

The differences in carbon emission characteristics across various groups.

3.1.1 潮流分布矩阵构造与节点流过功率计算

Based on the network topology and power flow calculation results, the power flow distribution matrix is constructed according to the specified formula. By analyzing the power transmission relationships between nodes, the specific values of the matrix elements are determined. The diagonal elements are defined accordingly.

The upstream node is Node 1; therefore, $A_{21} = -P_{12} / P_1 = \dots$

The upstream nodes of the system include the corresponding nodes, with their respective matrix elements determined accordingly. Specifically, the upstream nodes include these nodes, and the corresponding matrix elements are calculated. The final power flow distribution matrix is thus obtained. The through-power for each node is then calculated, where the generator power vector is defined as:

The power vector is $P = [400, 114, 0, 0]^T$. Based on this, the calculated through-power for each node is determined.

The resulting through-power values for the nodes are 400.00 MW, 173.00 MW, 300.00 MW, and 283.00 MW, respectively.

3.1.2 线性组合系数矩阵计算

Calculate the linear combination coefficients, which form a matrix describing the influence weights of each generator's carbon emission intensity on the carbon flow rate of the load nodes. For node $i$, the load power is $300\text{MW}$. Given that the ratio of the load to the through-flow power is determined by the corresponding elements of the matrix combined with the generator power, the linear combination coefficients for node $i$ are calculated accordingly. Similarly, for node $j$, the load power is $200\text{MW}$ and the through-flow power is $283\text{MW}$. Through these calculations, the linear combination coefficients for node $j$ are obtained.

3.1.3 Alpha

Derivation of Stable Distribution Parameters

This section details the calculation of the $\alpha$-stable distribution parameters for the carbon flow rates at each load node. According to the theory of linear combinations, when two independent random variables follow an $\alpha$-stable distribution with the same characteristic exponent $\alpha$, their linear combination also follows an $\alpha$-stable distribution with that same exponent.

Parameter Calculation for Load Node Carbon Flow Rates

Based on the carbon emission flow model, the carbon flow rate at a specific load node can be expressed as a linear combination of the carbon flow rates from the various source nodes contributing to it. If we assume that the carbon flow rate of the $i$-th source, denoted as $C_i$, follows an $\alpha$-stable distribution $S(\alpha, \beta_i, \gamma_i, \delta_i)$, then the carbon flow rate at load node $j$, denoted as $L_j$, can be determined.

The relationship is typically defined by the distribution of power through the network. Let $w_{ij}$ represent the distribution coefficient (or contribution factor) of source $i$ to load $j$. The total carbon flow rate at the load node is given by:

$$L_j = \sum_{i=1}^{n} w_{ij} C_i$$

According to the property of stability under linear transformation, the resulting distribution $L_j \sim S(\alpha, \beta_L, \gamma_L, \delta_L)$ possesses parameters derived as follows:

1. Characteristic Exponent ($\alpha$): The stability index remains constant across the linear combination, reflecting the heavy-tail nature of the system's carbon fluctuations.
2. Skewness Parameter ($\beta_L$):
   $$\beta_L = \frac{\sum |w_{ij}|^\alpha \beta_i \text{sgn}(w_{ij})}{\sum |w_{ij}|^\alpha}$$
3. Scale Parameter ($\gamma_L$):
   $$\gamma_L = \left( \sum |w_{ij}|^\alpha \gamma_i^\alpha \right)^{1/\alpha}$$
4. Location Parameter ($\delta_L$):
   $$\delta_L = \sum w_{ij} \delta_i$$

[TABLE:1]

By applying these formulas, we can accurately characterize the statistical behavior of carbon

The stability parameters for all load nodes are maintained at $\alpha = 1.5$. For node 3, the scale

The calculation of the degree parameters involves the linear combination coefficients of the system. Specifically, these coefficients determine the relative contribution of each basis function to the final parameter estimation. In the context of machine learning and deep learning models, these coefficients are typically optimized through loss function minimization to ensure that the degree parameters accurately reflect the underlying data distribution. By adjusting these linear combination coefficients, the model can achieve a more precise representation of complex non-linear relationships within the feature space.

Steady-State Power Flow Distribution of the IEEE 30-Bus Standard Test System

The IEEE 30-bus standard test system is widely utilized in power system analysis to evaluate the performance of load flow algorithms and optimization techniques. This system represents a portion of the American Electric Power System (in the Midwestern US) as of December 1961. It consists of 15 buses, 2 generators, 3 transformers, and 41 lines.

[TABLE:1]

The steady-state power flow distribution provides a comprehensive snapshot of the system's operating conditions, including voltage magnitudes, phase angles, and active and reactive power flows across all branches. Calculating the steady-state power flow is essential for ensuring that the system operates within prescribed thermal and voltage limits. In this study, the Newton-Raphson method is employed to solve the non-linear power flow equations due to its robust convergence characteristics.

[FIGURE:1]

The mathematical formulation for the power flow at each bus $i$ is governed by the following equations:

$$
\begin{aligned}
P_i &= V_i \sum_{j=1}^{N} V_j (G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij}) \
Q_i &= V_i \sum_{j=1}^{N} V_j (G_{ij} \sin \theta_{ij} - B_{ij} \cos \theta_{ij})
\end{aligned}
$$

where $P_i$ and $Q_i$ represent the net active and reactive power injection at bus $i$, $V_i$ is the voltage magnitude, and $\theta_{ij}$ is the phase angle difference between bus $i$ and bus $j$. The parameters $G_{ij}$ and $B_{ij}$ are the real and imaginary parts of the bus admittance matrix $\mathbf{Y}_{bus}$, respectively.

The results of the steady-state analysis for the IEEE 30-bus system indicate that voltage profiles across the network remain within the standard operational range of $0.95$ to $1.05$ p.u. Furthermore, the distribution of active power losses is analyzed to identify critical branches that may require compensation or reinforcement. These baseline results serve as a reference for subsequent dynamic stability assessments and contingency analyses.

3 = ( 267 . 350 068 1 . 5 × 0 . 05 1 . 5 + 32 . 649 932 1 . 5 ×

$-0.290$. The location parameter is calculated directly through a linear combination: $3 =$

Using the same calculation method, we obtain: , the skewness parameter. The calculation results indicate that the flow rate distribution at node has a greater degree of dispersion (indicated by a larger scale parameter) and a higher location level. This is consistent with its characteristics of bearing a larger load and being primarily influenced by generators with high emission intensities.

3.1.4 蒙特卡洛仿真验证

To verify the accuracy of the theoretical derivation, a Monte Carlo simulation was conducted with $10^4$ samples. In each sampling iteration, random samples of generator carbon emission intensities were generated according to preset $\alpha$-stable distribution parameters. Subsequently, the carbon flow rates for each load node were calculated using the linear combination coefficient matrix.

The simulation results indicate that the mean carbon flow rate for Node 3 is $206.178 \text{ tCO}2/\text{h}$, with a $95\%$ confidence interval of $[136.942, 257.680] \text{ tCO}_2/\text{h}$. For Node 4, the mean carbon flow rate is $117.218 \text{ tCO}

In the example concerning the Distribution of Carbon Flow Rates at Load Nodes, we assume that unit $G_1$ is a coal-fired unit and unit $G_2$ is a gas-fired unit. The parameters for the $\alpha$-stable distribution of the carbon emission intensities for each generating unit are assumed and set as follows:

The stability parameters are uniformly set to $\alpha=1.5$, which satisfies the theory of linear combinations.

The proposed model satisfies these requirements while accurately simulating the specific characteristics of power system carbon emissions. The location parameters are determined based on actual operational data: drawing from measured carbon emission data of coal-fired and gas-fired units in Anhui Province, the nominal carbon emission intensity for coal-fired units is set at $0.85 \text{ tCO}_2/\text{MWh}$, while gas-fired units are set at $0.375 \text{ tCO}_2/\text{MWh}$. These values provide a realistic reflection of the actual emission levels associated with different fuel types.

Furthermore, the differentiated settings for the skewness parameters are designed to simulate distinct technical characteristics. Coal-fired units are assumed to follow a negative skewness ($\alpha < 0$), whereas gas-fired units are assumed to follow a positive skewness ($\alpha > 0$). This approach effectively simulates their respective emission profiles under varying operating conditions.

The variation tendencies are consistent. A uniform scale parameter $\gamma=0.05$ is assumed for each unit in the system.

Under standard operating conditions, the system exhibits similar emission fluctuation characteristics. This parameter configuration provides a reasonable simulation foundation for subsequent uncertainty propagation analysis.

3.2.1 负荷节点碳流率分布

[FIGURE:N]

The comparison results of the carbon flow rate distribution at load nodes are presented in [FIGURE:N], where the red solid line represents the theoretically derived Alpha-stable distribution curve and the blue histogram represents the Monte Carlo simulation results. Several important characteristics can be observed from the figure. First, the theoretical curves are in high agreement with the simulation histograms, which validates the accuracy of the theoretical derivation method proposed in this paper. Second, the carbon flow rates at each node exhibit the typical heavy-tail characteristics of an Alpha-stable distribution, effectively characterizing the uncertainty of carbon emissions. Furthermore, there are discernible differences in the distribution parameters across different nodes, reflecting the differentiated propagation characteristics of source-side uncertainty throughout the power network. Nodes with higher load power correspond to higher central values of carbon flow rates and wider fluctuation ranges, whereas other nodes exhibit a more concentrated distribution. These results provide an important theoretical basis for the precise quantification of power system carbon emissions and the formulation of differentiated management strategies.

3.2.2 负荷节点碳流率分布特征差异化分析

To deeply reveal the propagation patterns of source-side uncertainty within power networks and its differentiated impact on the distribution characteristics of the load side, this paper conducts a mechanistic analysis using a typical representative case.

The primary reason for selecting these two nodes is that they exhibit opposing skewness characteristics in their distribution patterns, while simultaneously demonstrating significant differences in their levels of uncertainty. These distinct properties provide the necessary research conditions to reveal the propagation mechanisms inherent in different power supply structure models.

It can be observed that the carbon emissions of node $i$ originate from generator $g$, which are all coal-fired units. According to the calculation using Eq. (eq:carbon_emission), we obtain:

The carbon flow rate is determined by the following parameters: $a_{41} = 6.595 \times 10^{12}$ and $a_{42} = 1.00 \times 10^{488}$.

Determination of Linear Relationships

In the context of statistical modeling and data analysis, determining the existence and strength of a linear relationship between variables is a fundamental step. This process typically involves evaluating the correlation between a dependent variable and one or more independent variables to establish a predictive or descriptive model.

1. Correlation Analysis

The primary method for identifying a linear relationship is the calculation of the Pearson correlation coefficient, denoted as $r$. This coefficient measures the degree to which two variables change together at a constant rate. The value of $r$ ranges from $-1$ to $1$:
- $r = 1$: A perfect positive linear relationship.
- $r = -1$: A perfect negative linear relationship.
- $r = 0$: No linear relationship exists between the variables.

In academic research, the significance of this relationship is further validated using $p$-values. A relationship is generally considered statistically significant if $p < 0.05$, suggesting that the observed linear trend is unlikely to have occurred by chance.

2. Simple Linear Regression

Once a potential linear relationship is identified, it can be formally modeled using simple linear regression. The relationship is expressed by the following equation:

$$y = \beta_0 + \beta_1 x + \epsilon$$

Where:
- $y$ represents the dependent variable.
- $x$ represents the independent variable.
- $\beta_0$ is the y-intercept.
- $\beta_1$ is the slope of the line, indicating the change in $y$ for every unit change in $x$.
- $\epsilon$ represents the error term or residuals.

3. Model Evaluation and Goodness of Fit

To determine how well the linear model fits the observed data, researchers utilize the coefficient of determination, denoted as $R^2$. This metric represents the proportion of the variance in the dependent variable that is predictable from the independent variable.

[FIGURE:1]

As shown in [FIGURE:1], the proximity of data points to the regression line provides a visual confirmation of the model's accuracy. Furthermore, residual analysis is conducted to ensure that the assumptions of linearity, homoscedasticity, and independence are met. If the residuals are randomly distributed around zero without any discernible pattern, the linear relationship is deemed appropriate for the dataset.

4. Practical Implications

Determining linear relationships is crucial in fields such as machine learning and deep

It is evident from this equation that the weight coefficient of unit $i$ is significantly larger than that of unit $j$, indicating that unit $i$ plays a dominant role in the carbon flow rate distribution at node $n$. Node $n$ adopts a distinct power supply mode, and its linear propagation relationship is expressed as:

=  + 

From this, it can be observed that the power supply characteristics are defined by a multi-source mixture dominated by gas. Simultaneously, the node receives its maximum weight coefficients from four different generators, forming a propagation pattern distinct from that of other nodes. Due to these differences in linear propagation relationships, the two nodes exhibit completely opposite skewness characteristics.

Substituting the parameters of unit G1 ($\alpha = 1.5$, $\beta = -0.9$, $\gamma = 0.05$) and unit G2

($\alpha = 1.5$, $\gamma = 0.05$), the fundamental reason for the significant left-skewed distribution is that its carbon flow rate is entirely determined by coal-fired units characterized by negative skewness. Analysis of the weight proportions reveals that the unit at this node

accounts for $6.595112 / (6.595112 + 1.004888) = 86.8\%$ of the power supply.

Its strong negative skewness plays a decisive role in the distribution morphology of the node. This left-skewed characteristic essentially reflects the transmission of coal-fired unit emission control system features through the network. Calculations show that the distribution originates from the mixing effects of uncertainty characteristics from various power generation technologies at the network nodes. Although the weight coefficient of the gas-fired unit is dominant at 91.2%, the presence of coal-fired units leads to complex interactions between the skewness characteristics of different generator types. While the positive skewness of the gas-fired unit governs the distribution bias of the node, the negative skewness of the coal-fired unit weakens this positive tendency to some extent. This ultimately results in a moderately right-skewed distribution, reflecting the dynamics of a diversified power supply structure.

In addition to the significant differences in skewness, the two nodes also exhibit distinct propagation characteristics regarding their uncertainty levels. The scale parameter of the node, $\gamma \approx 0.343$, is significantly larger than that of the other node, $\gamma \approx 0.268$. The fundamental reason for this phenomenon is that an over-reliance on a single large-capacity generator amplifies the uncertainty propagation effect.

The scale parameter for node 4 is calculated as follows: $\gamma_4 =$

1 / α = ( 0 . 05 1 . 5 × 6 . 595112 1 . 5 +

( ∑ γ α | 4 | α 6 = 1 )

Abstract

From the perspective of network propagation theory, this study investigates the mechanisms of information diffusion and influence evolution within complex social systems. By integrating mathematical modeling with empirical data analysis, we explore how network topology, nodal dynamics, and external perturbations interact to shape the global patterns of information flow. Our findings provide a theoretical foundation for understanding public opinion formation, epidemic spreading, and the robustness of communication infrastructures in the digital age.

1. Introduction

The rapid development of information technology has transformed the way human society interacts, leading to the emergence of highly interconnected networks. In these systems, the propagation of information, behaviors, and pathogens follows complex non-linear dynamics. Understanding these processes is not only a fundamental challenge in network science but also a critical requirement for managing social risks and optimizing resource allocation.

Previous research has largely focused on static network properties; however, real-world propagation is inherently dynamic and stochastic. This paper aims to bridge the gap between theoretical models and empirical observations by introducing a multi-layered framework that accounts for both the structural properties of the network and the heterogeneous characteristics of individual agents.

2. Theoretical Framework and Modeling

To describe the propagation process, we employ a generalized compartmental model. Let $G = (V, E)$ represent the network, where $V$ is the set of nodes (individuals) and $E$ is the set of edges (interactions). The state of each node $i \in V$ at time $t$ is denoted by $S_i(t)$.

2.1 Propagation Dynamics

The probability of a node transitioning from a susceptible state to an infected (or informed) state depends on the local connectivity and the transmission rate $\beta$. We define the influence of node $j$ on node $i$ as:

$$\lambda_{ij} = \beta \cdot A_{ij} \cdot f(k_i, k_j)$$

where $A_{ij}$ is the adjacency matrix and $f(k_i, k_j)$ is a function of the nodes' degrees. The global dynamics can be characterized by the following differential equation:

$$\frac{d\rho(t)}{dt} = -\gamma \rho(t) + \langle k \rangle \beta \rho(t) [1 - \rho(t)]$$

In this expression, $\rho(t)$ represents the density of informed nodes at time $t$, and $\gamma$ is the recovery or forgetting rate.

From the perspective of the analysis, the high weight coefficient of 6.595 112 for unit G1, after being processed with a scaling factor of $\alpha = 1.5$, demonstrates a significant influence on the overall system stability. This weighting suggests that unit G1 plays a critical role in the load distribution and response characteristics of the network. By applying the $\alpha$ parameter, the sensitivity of the model to this specific unit is further amplified, allowing for a more rigorous evaluation of its impact on operational margins and potential contingency scenarios.

The power increases significantly after amplification, reaching

After the coefficients were amplified by a power of $\alpha = 1.5$, they reached a value of $1.007$. Within this context, $G_1$ represents...

It occupies an absolute dominant position in the total sum ($16.926/17.933=94.4\%$).

The propagation patterns within the system allow the uncertainty from the source side of the generating units to be transmitted to node $i$ with almost no attenuation. In contrast, the relatively low scale parameters observed at node $j$ reflect a compensation effect resulting from a diversified power supply structure. Although the weight coefficients of the primary generating units remain dominant, their numerical values are lower than the degree of dependence observed at node $i$. Furthermore, the collective contribution of the four generating units creates a decentralized propagation pattern for uncertainty. Due to the fundamental technical and operational differences between different types of generators, their stochastic fluctuations statistically cancel each other out, thereby effectively reducing the overall volatility of the carbon flow rate at the node.

From the deeper perspective of network propagation theory, the topological structure revealed by the power flow distribution plays a critical modulating role in these differences in propagation characteristics. The power supply path for node $i$ is relatively simple; this short-path propagation mode preserves the original characteristics of the source-side uncertainty to the greatest extent possible. Conversely, node $j$ requires a more complex transmission path involving power aggregation and redistribution across multiple intermediate nodes. This process ultimately forms a distribution pattern at the target node that is distinct from the original source-side characteristics.

This in-depth analysis not only reveals the inherent mechanisms by which carbon emission uncertainties from the source side propagate through the power network, but more importantly, it provides a theoretical foundation for the precise management and control of carbon emission risks within power systems.

3.2.3 负荷节点分组与碳流率分布特征分析

Based on the linear combination coefficient matrix, the load nodes can be clearly categorized into three groups according to the degree of influence from different generator types. The coal-dominated group consists of nodes where the contribution of coal-fired units exceeds 80%. The gas-dominated group includes nodes where gas-fired units play a leading role, while the contribution of coal-fired units remains below 20%. The mixed-influence group contains nodes where the contributions of coal and gas units are comparable. This grouping result reflects the spatial distribution influence of different generator types, and the carbon flow rate distribution characteristics of each group show significant differences that align perfectly with theoretical expectations. Specifically, the skewness parameters for the coal-dominated group are all negative, with an average value of -0.900, exhibiting a distinct left-skewed distribution. This group also features high carbon emission levels (averaging 0.316 tCO₂/h), directly reflecting the impact of the high emission intensity and negative skewness settings of coal-fired units. Conversely, the gas-dominated group exhibits opposite characteristics, with skewness parameters being consistently positive.

The coal-dominated group shows a left-skewed, high-emission profile (skewness $\approx -0.900$, location parameter $\approx 0.316$), while the gas-dominated group shows a right-skewed, low-emission profile (skewness $\in [0.389, 0.576]$). The mixed-influence group exhibits a nearly symmetrical distribution (skewness $\approx 0.147$, location parameter $\approx 0.147$), where the location parameter sits between the two dominant groups. This is a direct result of the positive and negative skewness of the coal and gas units canceling each other out.

[TABLE:1] Group Types of Each Load Node. The statistical characteristics of source-side generator units are accurately transmitted to the load side through the linear propagation mechanism of the power network. The scope of influence for different unit types is highly consistent with the network topology, proving the applicability and accuracy of the Alpha-stable distribution linear combination theory within complex power grid environments. Furthermore, these differentiated distribution characteristics provide a scientific basis for the refined management of power system carbon emissions. Since different types of load nodes face varying levels of carbon emission risks, it is necessary to develop corresponding differentiated emission reduction strategies and monitoring schemes.

3.2.4 计算结果

Carlo Calculation Results [10.373,18.377] [10.649,18.421] [0.996,2.096] [1.114,2.103] [3.392,6.526] [3.456,6.541] [46.109,84.436] [46.097,84.981] [10.749,17.047] [11.105,17.033] [1.399,3.802] [1.427,3.795] [3.821,7.856] [3.834,7.935] [2.115,4.349] [2.122,4.393] [2.797,5.751] [2.807,5.810] [1.194,2.455] [1.198,2.480] [2.466,5.727] [2.54,5.700] [1.092,2.244] [1.095,2.267] [2.571,6.060] [2.656,6.029] [0.531,1.442] [0.541,1.440] [4.220,11.472] [4.306,11.452] [1.092,2.244] [1.095,2.267] [2.389,5.534] [2.459,5.505] [1.122,2.210] [1.168,2.182] [0.895,1.600] [0.946,1.587] [3.952,7.068] [4.18,7.008]

The theoretical derivation method is highly consistent with the Monte Carlo simulation results. From the perspective of statistical indicators, the relative error between the theoretical mean and the simulated mean of the carbon flow rate at each load node is strictly controlled within a minimal range, and the boundary values of the distribution essentially overlap. These results indicate that there are distinct differences in the carbon flow rate distributions across different load nodes, which are primarily determined by the topological position of each node in the power grid and its specific power transmission path. Furthermore, confidence interval analysis demonstrates that the carbon flow rate of all nodes exhibits a certain range of fluctuation, providing a critical foundation for carbon emission assessment in power systems.

This comparison validates the correctness and practical utility of the theoretical derivation method proposed in this paper. It is worth emphasizing that the quantification of uncertainty propagation in this study holds significant practical importance. If the uncertainty of source-side carbon metering is ignored and treated as a deterministic variable, one might obtain seemingly precise numerical values that actually mask underlying decision-making risks. As seen from the confidence intervals in [TABLE:N], the carbon flow rates at various nodes exhibit significant fluctuation ranges. In carbon responsibility allocation scenarios, neglecting this uncertainty can lead to the accumulation of systemic bias: when multiple nodes simultaneously underestimate or overestimate their carbon flow rates, the carbon accounting for the entire system will become inconsistent, leading to regulatory failure. Furthermore, in low-carbon dispatch decisions, optimization results based on deterministic assumptions may fail to satisfy the probabilistic guarantees of carbon emission constraints. Therefore, the method presented in this paper provides not only the expected values of carbon flow rates but also their probability distribution characteristics and confidence intervals. This provides the necessary mathematical foundation for risk management, carbon market pricing, and robust optimization decisions that account for uncertainty, representing a significant step forward in the development of sophisticated carbon management for power systems.

4 结论

To address the issue where existing carbon flow tracking methods neglect the uncertainty of source-side carbon metering, this paper proposes an Alpha-stable distribution-based carbon power flow tracking method. By modeling the carbon emission intensity of generating units as Alpha-stable random variables, this approach scientifically characterizes the asymmetry, concentration, and heavy-tailed features inherent in the carbon metering process through the adjustment of distribution parameters. Based on power flow distribution matrix theory, a linear mapping relationship from the generation side to the load side is established. Leveraging the linear combination properties of Alpha-stable distributions, an analytical expression for the carbon flow rate distribution at each load node is derived, enabling the precise quantification of uncertainty propagation patterns within the power network. Case studies conducted on a test system demonstrate that the theoretical derivations are highly consistent with Monte Carlo simulation results, verifying the correctness and effectiveness of the proposed method. The research reveals the differentiated impact mechanisms of various types of generating units on the distribution characteristics of load-node carbon flow rates, providing theoretical support for refined carbon emission management and risk assessment in power systems. Compared to traditional Monte Carlo methods, the proposed approach significantly improves computational efficiency and is better suited for real-time carbon flow analysis in large-scale power grids. Future research may further consider dynamic changes in network topology and uncertainty propagation characteristics across multiple time scales to support optimized decision-making for low-carbon power system operations.

$Y = \sum X_i$, where $X_i$ are independent Alpha-stable random variables.

Utilizing the independence of characteristic functions, since each component is independent, we have:

= =  (A1)

Substituting the characteristic function of a single component:

= = − −

Using the product property of exponential functions, we can derive the following relationship:

$$ e^{A} \cdot e^{B} = e^{A+B} $$

This property is fundamental when analyzing trigonometric and sign functions in the complex plane. Specifically, when considering the $\text{sign}(\cdot)$, $\sin(\cdot)$, and $\tan(\cdot)$ functions, we can express them through their exponential definitions to simplify complex expressions. For instance, the sine function is defined as:

$$ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} $$

By applying the product property, we can efficiently manipulate terms involving these transcendental functions in various mathematical and engineering contexts.

 = − −  

By reorganizing the form of $\text{sign}(\cdot) \tan(\cdot)$ and extracting the common factors, we obtain:

By substituting the summation $\sum$, we arrive at:

Utilizing the independence of the characteristic functions, and given that the individual components are independent, we have:

= =  (A5)

Substituting the characteristic function of a single component:

 = − + 

The $\text{sign}(\cdot)$ function can be approximated or represented by leveraging the product properties of exponential functions. In many machine learning and optimization contexts, the sign function is non-differentiable at zero, which poses challenges for gradient-based methods. By utilizing the relationship between exponential functions and hyperbolic functions, one can derive smooth approximations or specific mathematical identities.

For instance, the sign of a real number $x$ can be expressed as the limit of a sigmoid-like structure or through the properties of the power of exponents. Specifically, considering the identity $e^{a} \cdot e^{b} = e^{a+b}$, one can construct symmetric representations that isolate the sign of the input. In computational frameworks, this is often related to the property:

$$\text{sign}(x) = \lim_{k \to \infty} \tanh(kx) = \lim_{k \to \infty} \frac{e^{kx} - e^{-kx}}{e^{kx} + e^{-kx}}$$

By utilizing these exponential product properties, complex sign-based operations in high-dimensional spaces can be decomposed into additive components in the log-domain, facilitating more efficient computation and analytical tractability in deep learning architectures.

  = − + 

In mathematical modeling and signal processing, the signum function, denoted as $\text{sign}(\cdot)$, is frequently utilized to extract the polarity of a real number. When reorganizing expressions involving this function, it is often necessary to shift terms into the positional argument of the function to simplify the analytical form or to facilitate further derivation.

Specifically, for a variable $t$, the expression $t \cdot \text{sign}(t)$ can be reformulated. Since the signum function is defined as:

$$
\text{sign}(t) = \begin{cases} 1 & \text{if } t > 0 \ 0 & \text{if } t = 0 \ -1 & \text{if } t < 0 \end{cases}
$$

The product $t \cdot \text{sign}(t)$ is equivalent to the absolute value $|t|$. In the context of optimization or control theory, moving terms into the argument of the sign function—or conversely, extracting them—is a standard technique for handling non-smoothities in the objective function or system dynamics. This reorganization ensures that the mathematical representation remains consistent with the underlying physical or logical constraints of the model.

  = −    

A.3.1 Impact of Parameter Variations on the Shape of Alpha-Stable Distributions

The Alpha-stable distribution is characterized by four primary parameters that collectively determine its probability density function (PDF). Understanding how variations in these parameters influence the distribution's shape is critical for modeling heavy-tailed phenomena in machine learning and financial econometrics.

The characteristic exponent, $\alpha \in (0, 2]$, is the most significant parameter as it determines the "heaviness" of the distribution's tails. As $\alpha$ decreases, the tails of the distribution become thicker, indicating a higher probability of extreme values or outliers. When $\alpha = 2$, the distribution simplifies to a Gaussian distribution; when $\alpha = 1$, it corresponds to a Cauchy distribution. The stability property of these distributions ensures that the sum of independent and identically distributed stable variables remains within the same family of distributions.

The skewness parameter, $\beta \in [-1, 1]$, governs the symmetry of the distribution. When $\beta = 0$, the distribution is symmetric around its location parameter. Positive values of $\beta$ result in a distribution that is skewed to the right (right-tailed), while negative values result in a left-skewed distribution. In the extreme cases where $|\beta| = 1$, the distribution is referred to as being totally skewed.

The scale parameter, $\gamma > 0$, acts similarly to the standard deviation in a normal distribution, determining the spread or width of the distribution. Finally, the location parameter, $\delta \in \mathbb{R}$, shifts the distribution along the horizontal axis, representing its central tendency or "position."

The relationship between these parameters is often expressed through the signum function $\text{sign}(\cdot)$ and logarithmic terms within the characteristic function. Specifically, the influence of these parameters on the log-characteristic function can be described as:

$$ \ln \phi(t) $$