## 2 Methodology Statement  
### The Proposed Carbon Intensity Incentive Mechanism Based on the Carbon Emission Flow Theory  
Due to the urgent goal of carbon neutrality, large-scale renewable energy and energy storage are deployed in the power system. However, it is difficult to accurately evaluate the benefits for the carbon reduction of a single unit, such as a wind turbine or an energy storage station, resulting in the effect of the market economy not being fully utilized in terms of carbon reduction. Therefore, based on the carbon emission flow theory [32; 33], this paper proposes an incentive mechanism based on the dynamic bus carbon intensity to evaluate the economic effect of carbon reduction from wind turbines and energy storage.  
According to the carbon emission flow theory proposed in [32], the bus carbon intensity \(e_{i}^{B}\), which characterizes the cleanliness of electrical energy flowing through buses, can be expressed as:  
\[e_{i}^{B}=\frac{R_{i}^{B}}{P_{i}^{B}} \tag{1}\]where \(P_{i}^{B}\) and \(R_{i}^{B}\) denote the power flux and carbon emission flow flux of the bus \(i\) and can be obtained from (2) and (3), respectively.  
\[P_{i}^{B}=\sum_{j\in i^{+}}P_{ji}+P_{i}^{G} \tag{2}\]  
\[\mathbf{R^{B}}=(\mathbf{E-A})^{-1}\mathbf{R^{G}} \tag{3}\]  
where \(P_{ji}\) denotes the power flow of branch \(j-i\); \(P_{i}^{G}\) denotes the actual power output of the power generator at bus \(i\); \(\mathbf{E}\) and \(\mathbf{A}\) represent the identity matrix and distribution matrix which can be obtained from (4), respectively; \(R_{i}^{G}\) is the carbon emission flow of the power generator and discharge power of the energy storage at bus \(i\) as (5).  
\[A_{ij}=\left\{\begin{matrix}P_{ji}/P_{i}^{B}\;j\in i^{+}\\ 0\;else\end{matrix}\right. \tag{4}\]  
\[R_{i}^{G}=e_{i}^{G}P_{i}^{G}\Delta t+e_{i}^{ES}P_{i}^{dis}\Delta t \tag{5}\]  
where \(i^{+}\) denotes the set of beginning buses of all the branches that inject into bus \(i\); \(e_{i}^{G}\) and \(e_{i}^{ES}\) denote the carbon intensity of the injection power from the power generators and energy storage as (6), respectively; \(P_{i}^{dis}\) represents the discharge power of the energy storage at the bus \(i\); \(\Delta t\) is the time step.  
\[e_{i,l}^{ES}=\frac{e_{i,l-1}^{ES}E_{i,l-1}^{ES}+e_{i,l}^{B}P_{i}^{cha}\Delta t} {E_{i,l-1}^{ES}+P_{i}^{cha}\Delta t} \tag{6}\]  
where \(E_{i,l-1}^{ES}\) denotes the state of charge of the energy storage at time \(t-1\) at bus \(i\); \(P_{i}^{cha}\) represents the charge power of the energy storage at bus \(i\).  
Based on the dynamic carbon intensity obtained from (1), the proposed carbon intensity incentive mechanism can be expressed as (7).  
\[\begin{split} C_{i,l}^{CS}=P_{i}^{inj}\left(e_{i,l}^{B}-e_{i}^{B, history}\right)\\ P_{i}^{inj}=\left\{\begin{matrix}P_{i}^{G}\;i\in\Omega^{WT}\\ P_{i}^{dis}-P_{i}^{cha}\;i\in\Omega^{ES}\end{matrix}\right.\end{split} \tag{7}\]  
where \(C_{i,l}^{CS}\) denotes the carbon subsidy of the wind turbine and energy storage at bus \(i\) at time \(t\); \(e_{i}^{B,history}\) denotes the historical carbon intensity at bus \(i\); \(\Omega^{WT}\) and \(\Omega^{ES}\) represent the sets of wind turbines and energy storage, respectively.