Consider Carbon Emission Flow Analysis of Power System with Regards to Power Loss and Wind Power Uncertainty

A method for analyzing the impact of wind power uncertainty and active power loss on carbon emission flow in power systems is proposed. Based on the stochastic power flow calculation of wind power injection, the correlation function between wind power output and the total carbon injection rate into the system is determined. Meanwhile, the reverse power flow method is used to analyze the contribution of each unit to the active power flow in different parts of the system and calculate the corresponding carbon emissions. Under the assumption that wind power fluctuations are borne by balancing units, this method combines a conventional unit and a system node, branch, load, and active network loss correlation matrix calculation model to solve the average impact factor of wind power injection on other nodes and branches, thereby determining the uncertain characteristics of carbon emission flow in power systems under stochastic and intermittent wind power injection. The correctness and effectiveness of the method are proved by analyzing the IEEE 14-node standard example, providing new theoretical support and ideas for quantitatively evaluating the low-carbon contribution of wind power generation to the system.


Introduction
The challenge of global climate change has put pressure on countries to reduce carbon emissions [1][2].The Chinese government has taken the lead in proposing the goals of "peak carbon emissions" and "carbon neutrality," aiming to optimize the country's industrial structure and achieve low-carbon sustainable development [3].The power system is the main source of carbon emissions in China, with carbon dioxide emissions accounting for about 50% of the total carbon emissions in society.Therefore, low-carbon development in the power sector is of crucial importance [4].
Carbon emission analysis and statistics are essential for achieving low-carbon development in the power sector.Currently, carbon emissions calculations in the power system mainly include macrostatistical methods [5][6][7] and carbon flow analysis methods [8][9].The macro-statistical method is simple to calculate, easy to use, and provides accurate results.It can start from macro data and calculate based on the total energy consumption over a period of time.However, its disadvantage is that the calculated results have a certain lag and are too rough to accurately track the specific flow of carbon emissions.
Therefore, the carbon flow analysis method emerged.Carbon flow analysis is a carbon flow tracking method based on power distribution, which can clearly reveal the distribution characteristics and transmission and consumption mechanism of carbon flows in the power network, and accurately track and trace the specific flow of carbon emissions.It is of great significance in the analysis and statistics of carbon emissions in the power system [10].
However, a large number of carbon flow calculation methods are based on DC power flow [11] and will produce large calculation errors when facing actual lossy networks [12].Therefore, in order to accurately calculate the carbon emissions in the power system, new methods need to be developed that combine the characteristics of actual lossy networks to perform carbon flow calculations and improve the accuracy and reliability of the calculations.Zhang et al. [13] proposed a carbon emission calculation method that considers network losses.This method combines the gradient method and Monte Carlo method and can take into account the differences in different power grid structures.On this basis, Zhang et al. [14] further proposed an improved algorithm, which considers different carbon emission factors of different types of loads in the power system, thus improving the calculation accuracy.In addition, Song et al. [15] proposed a carbon emission calculation method that considers network losses based on network optimization.By optimizing the power grid structure, carbon emissions can be minimized.
Wind power generation has experienced rapid development worldwide, and its low-carbon characteristics make it one of the main ways for low-carbon power development [16].Increasing the grid-connected power of wind power and other near-zero carbon emission power sources is an important means to achieve the system's emission reduction goals.However, due to the randomness, intermittency, and uncontrollability of wind power, its development and contribution are subject to certain limitations [17][18].Studies have shown that wind power integration can not only replace traditional energy sources to reduce emissions but also effectively change the flow distribution in the system.Since the carbon emission flow in the system is closely related to the flow characteristics, considering the impact and analysis of wind power injection power uncertainty on the system's carbon emission flow is of great significance for quantitatively evaluating the low-carbon contribution of wind power generation to the system [19].Therefore, this paper aims to study the problem of carbon emission flow in the power system considering wind power uncertainty, to provide a theoretical basis for quantitatively evaluating the impact of wind power integration on the system's carbon emissions.
This paper mainly focuses on the uncertain characteristics of wind power generation and explores the impact of wind power integration on the carbon emission flow in the power system.Specifically, it first studies the characteristic relationship between wind power uncertainty and the total carbon flow rate in the power system, then derives the carbon flow calculation equation that includes network losses based on the inverse power flow method, and analyzes the factors that affect the distribution of the system's carbon emission flow when considering wind power injection power based on the associated matrix calculation model of system units, nodes, branches, loads, and active losses.Through these analyses, some theoretical basis is provided for quantitatively evaluating the impact of wind power generation on the system's carbon emissions, and support is provided for increasing the grid-connected power of wind power and other near-zero carbon emission power sources.

Analysis of the impact of wind power uncertainty on carbon flow rate
The variations in wind speed and direction in a wind farm can have an impact on the output power of wind turbines.Generally, the higher the wind speed in a wind farm, the greater the mechanical power of the wind turbine.However, the impact of wind direction on the output power of wind turbines is relatively small because wind turbines can adjust the wind direction by rotating their blades to maintain the optimal angle and maximize output power.Nevertheless, it is necessary to consider the impact of certain wind directions on power.

Establish a wind power uncertainty model
Considering that wind power injection is influenced by wind speed and direction, the first step is to establish a correlation function between wind speed, wind direction, and the total amount of carbon flow rate injected into the system.That is, where R m is the total amount of carbon flow rate injected into the system, and f is the function relating the total injection of system carbon flow rate to wind speed and direction.
The relationship between the mechanical power of a single wind turbine and wind speed in a wind farm can be represented by a power curve.Typically, the power curve of a wind turbine can be expressed as: ; where P v represents the mechanical power at wind speed v, ρ is the air density, A is the area of the rotor, C p is the power coefficient of the wind turbine, v cut-in is the cut-in wind speed, v _r is the rated wind speed, P _r is the rated power, and v cut-out is the cut-out wind speed.
The relationship between the mechanical power of a single wind turbine and wind direction in a wind farm is usually represented by a directional coefficient.This coefficient represents the ratio of the power output of the wind turbine blades in different wind directions to the power output in the head-on wind direction.
In general, the directional coefficient of a wind turbine can be expressed as: _ cos( ) where C d represents the directional coefficient at wind direction d, and d _opt represents the wind direction at which the directional coefficient is maximized.Typically, d _opt is either 0 degrees or 180 degrees, which correspond to wind directions perpendicular to the plane of the rotor.Therefore, when the directional coefficient is maximized, i.e., at head-on wind direction, the directional coefficient is equal to 1.However, in other wind directions, the directional coefficient is less than 1, which leads to a reduction in the power output of the wind turbine.

Consider the carbon flow rate of network loss
When performing power flow calculations on the system, assuming that the number of nodes in the system is N, the conventional units and loads of the system are known, and the injected power at each node, except for the balance node, should be the sum of the injected power of the unit and the injected power of the load connected to that node.That is, where P i is the injected active power of the i-th node of the system, P Gi is the injected power of the unit connected to the i-th node (if there is no unit connected, then P Gi =0), P Li is the active load of the node (if there is no load connected, then P Li =0), and P i_loss is the active loss of the i-th node.It is assumed that wind power is connected to the m-th node (m=1, 2, …, N) of the system.When calculating carbon emissions, it is necessary to first calculate the steady-state power flow distribution of the power system, which involves the calculation of active and reactive power flows.Reactive power flow is usually considered as an additional term, which has a relatively small impact on the calculation and stability of active power flow.Therefore, when calculating carbon emissions, it is usually only necessary to consider the impact of active power flow distribution, and there is no need to consider the impact of reactive power flow distribution too much.Therefore, in this paper, there is no need to consider the indirect impact of reactive power flow.
Therefore, if we know the sum of the injected power at all other nodes, we can calculate the active output of the balancing node.That is, the injected power at node s is given by: For the entire system, the total carbon emissions per unit of time, which is the sum of the carbon flow rates at each node, is equal to the carbon emissions produced per unit of time by each unit.
where K is the number of conventional units, P Gk is the injection power of the unit at the kth node, and e Gs and e Gk are the carbon emission intensities of the balancing unit and the conventional unit, respectively.By combining Equations ( 1) to ( 6), the relationship between the injected power considering wind power uncertainty and the total carbon flow rate of the system can be obtained, as shown in Equation (7).It can be seen that the injected power from wind power under different circumstances will cause fluctuations in the total carbon flow rate of the system because wind power essentially has uncertain characteristics such as randomness and intermittency.Therefore, statistical features should be used to describe it more objectively.
where a is the number of wind turbines in the wind farm.

Calculation of the association matrix
Analysis of carbon emission flow in power systems requires the calculation of the generator-node carbon flow correlation matrix, the generator-branch carbon flow correlation matrix, and the generator-load (active power network loss) carbon flow correlation matrix.The node output distribution matrix H is used to describe the distribution of power flow and carbon emission flow in the system under a given steady-state operation, while the path output distribution matrix D is used to describe the path information of power flow and carbon emission flow from the generator node to the target node [20].
In summary, the correlation matrix reveals the formation and distribution mechanism of carbon emission flow in power systems, and the calculation method of the correlation matrix is very important for the analysis of carbon emission flow in power systems.The calculation method needs to consider factors such as system characteristics, network topology, and the contribution relationship between nodes and paths to improve the accuracy and efficiency of the calculation.According to the definition and properties of the branch flow distribution matrix P B and the generator injection distribution matrix P G , it can be obtained.
where I is the unit matrix, and P N is the active flux matrix of the nodes.
From the basic calculation method of the carbon potential of system nodes and the definition of matrix D, it can be obtained that: Zhou et al.'s work [21] also defines the active power loss matrix P l and the active power loss carbon flow rate matrix R l for each branch, as well as the matrix that relates the carbon flow rate between generators and branch losses.
where P l is an N*N matrix with elements representing the losses of the corresponding branches; E N is the carbon potential vector of the nodes; P Gk is the active power injected into the system by the kth generator unit; e Gk is the carbon emission intensity of the kth generator unit; η D is used to represent all the elements in the kth column of matrix D.

Countercurrent flow method
Inverse Load Flow (ILF) is a method for power flow calculation in electric power systems [22].By tracking the direction of current flow in the power system, it calculates the flow distribution backward from the terminals to the source, thereby determining the power and loss allocation of each section.ILF can help analyze the flow distribution in the power system, evaluate the voltage stability and power loss situation of each node, and optimize the operational efficiency and economic performance of the power system.In the mechanism analysis of carbon emission flow calculation and distribution characteristics, ILF can be used to track the active power flow in the power system, calculate the carbon emissions of each section, and verify the contribution of the units to the active power flow of each section and correspond it to their carbon emissions.First, the ILF method is introduced.
A virtual node is added to the branch, and the branch loss is equivalent to the carbon emission flow consumed by its virtual load.As known from the backward tracking method, after moving the system branch active loss load equivalent to the starting node of the line, the system is equivalent to a lossless network.Define the upstream distribution matrix A u , with its elements as: In the above context, i, j=1, 2, …, N, where U i is the upstream node set of node i.The power losses in the branches have been equivalently converted to virtual loads, and the sum of power represents the total network losses.By combining the upstream distribution matrix, the contribution of each generating unit to the active power flow consumed by the consumption end in the system can be calculated.The active power output provided by each generating unit in the system's network losses can be expressed as: where N ζ is an N-dimensional row vector where all elements are equal to 1, and P Nii represents the corresponding element of the active flux matrix for unit k at access node i.The resulting value represents the contribution of each generating unit to the total network losses in the system.Similarly, the data for the active power flow provided by the system's generating units to each load in the system can also be calculated.The k-th generating unit at access node i in the system provides active power output to the load connected to the system node, which can be expressed as: where P l is an M*N load distribution matrix, whose elements represent the connection relationship between all electricity loads and the power system, as well as the corresponding active power loads.

Analysis of the correlation between wind power injection and the distribution of carbon emission flow in the system
It is assumed that only the uncertainty of wind power is considered in the stochastic power flow analysis, and all fluctuations in wind power are borne by the balancing units.
Δ Δ Firstly, because wind power has a carbon emission intensity of 0, the amount of electricity injected into the grid by wind power will not directly affect the carbon emissions of the power system.However, due to the fluctuation of wind power output, this will have an impact on the system's stability and carbon emissions.Therefore, the impact of wind power on the system's carbon emissions can be calculated by equivalent means.By establishing a mathematical model of the power system, based on the balance between generating units and the association between system nodes and branches, and analyzing and calculating the system, the impact factors of the carbon flow rate of the balance units on each node and branch can be determined.Finally, the system is subjected to Latin hypercube sampling under the uncertainty of wind power injection power to obtain the expected impact factor, which is the impact factor of wind power injection power on the distribution of carbon emission flows in the system.
We make R us-N be the carbon flow association vector between the balance unit and node, i.e.
We make Z us-N be the influence factor vector of the balanced generating unit-node, i.e.
Therefore, the influence of wind power injection on the carbon flow rate flux of each node in the system can be expressed as: The above is an example of wind power injection power to nodes, and the effect on lines, loads and active losses can be reasoned in the same way.From the above equation, it can be seen that the impact of the wind farm on the carbon potential distribution of the system nodes is related to factors such as power grid topology, wind power injection capacity, and carbon emission intensity of each unit.

Footnotes
The IEEE 14-bus system is used as the research object in this example.In the analysis, we consider the impact of wind power injection uncertainty on the distribution of carbon emission flow, while the conventional generation units and load injection power are still considered deterministic.Specifically, the third node of the wind farm access system involves three scenario parameters, which are detailed in Table 1.Meanwhile, the wind speed in the region where the wind farm is located follows the Weibull distribution.To explore the impact of wind speed uncertainty on the system, we use the Latin hypercube sampling method for 500 simulation analyses, considering two different penetration levels.Given the carbon emission intensity vectors of all generating units, it is assumed that wind power is connected to node 3, and unit G5 is a hydro unit while all other units are coal-fired units.Whether the wind power unit generates power or not, its carbon emission intensity is 0. The carbon emission intensity of the units, denoted as EG, is shown as follows:   875 525,0,520,0 Figures 1 and 2 show the results of 500 sampling runs under Scenario 2. Figure 1 displays the distribution of sampling runs, while Figure 2 shows the sampling results of wind speed and total carbon flow rate of the system.These figures indicate that the random wind power injection leads to the inherent uncertainty of the total carbon flow rate of the system.As the wind power penetration rate varies within a certain range, the variation of the total carbon flow rate of the system is also influenced within a certain range.According to the results in Figure 3, the relationship between the three scenarios and the total carbon flow rate of the system can be represented graphically.When the wind speed is between 5 m/s and 13 m/s, the total carbon flow rate of the system changes with the wind speed and basically follows a cubic curve.When the wind speed is between 13 m/s and 20 m/s, the total carbon flow rate of the system reaches a minimum value.When the wind speed is greater than 20 m/s or less than 5 m/s, the total carbon flow rate of the system reaches a maximum value and remains unchanged.These results validate the effectiveness of Equation (7).Additionally, we can observe that the higher the wind power penetration level, the greater the fluctuations of the wind farm on the total carbon flow rate of the system.Under the same penetration level, wind direction also affects the amplitude of the total carbon flow rate fluctuations.2 shows the mean and variance of the carbon potential of each node in the system under different scenarios based on 500 Latin hypercube samples.Compared to deterministic analysis, the mean carbon potential of nodes under this random distribution is a better representation of the impact of wind power injection on the carbon potential of the system nodes.As Node 1 is a balanced node with no load access, its carbon potential remains unchanged.At a penetration level of 24.57%, nodes 6, 7, 8, 9, 11, 12, and 13 have relatively small fluctuations.At a penetration level of 49.14%, except for nodes 1 and 7, the carbon potential fluctuations of all nodes increase, indicating that as the rated capacity of wind Total carbon flow rate of the system(tCO power increases, the distribution of carbon emissions in the system is more volatile.At a wind direction angle of 30°, the carbon emission distribution of the system fluctuates less due to the decrease in injected power.In summary, wind power injection has an impact on the distribution of carbon emissions in the system, and the larger the injection power, the more obvious the fluctuation.However, compared with the variance at different penetration levels, nodes far from wind power injection are not sensitive to wind power output. The following is active power flow tracing analysis of IEEE14 nodes using the countercurrent flow method.The product of the tracing result and the carbon emission intensity of the unit is the carbon emission amount of the electric energy consumed by each unit for each part of the system.Table 3 shows the contribution of the unit to each load and network loss.A group of corresponding data is randomly selected in the table, such as the contribution of active power and carbon flow rate of unit G1 to load 2. In Scenario 1, the mean values of both are 15.9125MW and 13.9235 tCO 2 /h respectively.When the latter is divided by the former, the value obtained is 0.875, which is the carbon emission intensity of Unit G1.This fully shows that it is correct to improve the carbon flow tracing method in a lossy network based on the original carbon emission flow model.In addition, since Unit G3 and Unit G5 only have active power output and no carbon emission, they are not given in the table.-0.0467 (7, 9) -0.0517Total line loss -0.0197After obtaining the influence factor vectors of balanced units and nodes through Latin hypercube sampling, the average influence factors α 1 of wind power injection on system node carbon flow rate and α 2 on system branch (line loss) carbon flow rate in Scenario 1 are further calculated and shown in Tables 3 and 4, respectively.Tables 4 and 5 allow for a quantitative evaluation of the impact of wind power injection on system node and branch carbon flow rates.

Conclusion
This article presents a method for calculating and analyzing the impact of wind power injection on carbon emission flow in power systems, considering the active power loss of transmission lines and the uncertainty of wind speed and direction.The following conclusions were drawn:  A carbon emission flow distribution analysis method for stochastic AC power flow considering wind speed and direction uncertainty was proposed, which obtained the correlation function between wind speed, direction, and the total carbon flow rate of the system, thus extending the deterministic carbon emission flow analysis an uncertain analysis environment. A carbon emission flow analysis method for power systems considering line active power loss based on the backward/forward sweep power flow algorithm was proposed.Based on this, the average impact factors of balanced nodes on other nodes and branch carbon flow rates under wind power injection were obtained. This study is based on the simple active power loss power flow analysis.Future research directions include the analysis of system carbon emission flow and low-carbon optimal operation considering wind power's impact on system voltage and reactive power network loss under AC power flow.

Figure 3 .
Figure 3. Fitting relationship between wind speed and total carbon flow rate in wind farm

Table 1 .
Parameters setting of wind farm

Table 2 .
Carbon potential expectation and variance of each node at different wind power penetration levels under random wind speed

Table 3 .
Unit contribution based on the countercurrent current method

Table 4 .
Average influence factor of wind power injection balancing unit on node carbon flow rate

Table 5 .
Average influence factor of wind power injection balancing unit on the carbon flow rate of