*3.1. Carbon-Energy Combined-Flow*

The carbon-energy combined-flow (CECF) of the power grid is a comprehensive network flow [36], which combines the power flow of the power grid with the carbon emission flow attached to the power flow of the power grid. Among them, the energy flow is the actual network flow, and the carbon emission flow is the virtual network flow, which can be referred to as the carbon flow in the power system. Carbon flow is generated in the power generation, which represents the concept that the carbon emission is transferred from the generation side to the demand side. The energy flow transfers from the power supply end to the receiving end, but unlike the energy flow, only the power supply that produces carbon emissions at the power supply end can be called a carbon source, as shown in Figure 1. For a given carbon source, the carbon emission is equivalent to the product of the energy flow and the carbon emission rate of the corresponding power generation side [35].

Energy flow is the transmission of electric energy in the power grid. In the process of transmission, there will be power losses, commonly known as network losses, which are generally described as follows:

$$P\_{\rm loss} = \sum\_{i,j \in \mathcal{N}\_{\rm L}} g\_{ij} \left[ V\_i^2 + V\_j^2 - 2V\_i V\_j \cos \theta\_{ij} \right] \tag{1}$$

where *Vi* and *Vj* are the voltage amplitudes of the interconnection node *i* and *j*, respectively; θ*ij* means the voltage phase angle di fference between node *i* and *j*; *gij* denotes the conductance between node *i* and *j*; *N*L denotes the branch set of the power network.

**Figure 1.** The carbon-energy combined-flow (CECF) structure in power systems.

In the process of power transmission, the energy flow should bear the corresponding amount of carbon flow losses. The tracking of the grid carbon emission flow is based on load flow tracking, and the source of network loss is traced in light of the proportional sharing rule [35]. The ratio of the *w*th generator to the whole active power injected at node *j* is

$$\beta\_{wj} = \frac{a\_{jw}^{(-1)} P\_{sw}}{P\_{nj}'} \tag{2}$$

where *Psw* is the active output of the *w*th generator; *Pnj* represents the whole active power injection of the *j* node in the equivalent lossless network; *a*(−<sup>1</sup>) *jw* means the active power injection weight of the *w*th generator at node *j*, its specific derivation process can be found in [23].

The proportion of the *w*th generator outgoing line at node *j* is the same, and the line loss is decomposed according to the utilization share of the carbon source to the line. Hence, β*wj* is the component ratio of the active power losses of the *w*th generator in line *i*–*j*. Here, the active power losses of line *i*–*j* can be expressed as follows:

$$
\Delta P\_{ij} = \sum\_{w \in \mathcal{W}} \left( \frac{a\_{jw}^{(-1)} \Delta P\_{ij}}{P\_{nj}'} \right) P\_{sw} \tag{3}
$$

where *W* denotes the generator set.

> Therefore, the total carbon flow losses of the power grid can be described by

$$\mathbf{C}\_{\rm ds} = \sum\_{i,j \in \mathcal{N}\_{\rm L}} \sum\_{uv \in \mathcal{W}} \left( \frac{a\_{jw}^{(-1)} \Delta P\_{ij}}{P\_{nj}'} \right) P\_{suv} \delta\_{suv} \tag{4}$$

where δ*sw* denotes the carbon emission rate of the *w*th generator.
