## 2 Evaluation Method of Inertia Distribution Characteristics
### Inertia Evaluation Method Based on Node Frequency Deviation Coefficient (NFDC)  
By analyzing the load flow calculation results during the operation of the power system, the active power output of each generator throughout the operation process is determined. \[P_{i}=U_{i}\sum_{j=1}^{n}U_{j}(G_{ij}\cos\delta_{ij}+B_{ij}\sin\delta_{ij}) \tag{4}\]  
where \(P_{i}\) represents the active power output of the generator, \(U_{i}\) and \(U_{j}\) represent the voltage magnitudes, \(G_{ij}\) and \(B_{ij}\) denote the equivalent impedance and admittance, and \(\delta_{ij}\) represents the phase difference. For a power system incorporating RESs, the mathematical relationship is as shown in Equation (5). \(G\), \(L\), and \(R\) represent the generators, loads, and grid nodes in the power system. \(Y\) can describe the connection relationships between the various nodes in the power system, which is composed of admittances. \[I= \begin{bmatrix}I_{C}\\ I_{L}\\ I_{R}\end{bmatrix} = \begin{bmatrix}Y_{GG}&Y_{GL}&Y_{GR}\\ Y_{LG}&Y_{LL}&Y_{LR}\\ Y_{RG}&Y_{RL}&Y_{RR}\end{bmatrix}\begin{bmatrix}U_{G}\\ U_{L}\\ U_{R}\end{bmatrix}  \tag{5}\]  
Since the network nodes are considered passive nodes, the node current at these nodes can be approximated as zero. Therefore, the network equation of power system can be expressed as Equation (6). \[ \begin{bmatrix}I_{C}\\ I_{L}\\ 0\end{bmatrix} = \begin{bmatrix}Y_{GG}&Y_{GL}&Y_{GR}\\ Y_{LG}&Y_{LL}&Y_{LR}\\ Y_{RG}&Y_{RL}&Y_{RR}\end{bmatrix}\begin{bmatrix}U_{G}\\ U_{L}\\ U_{R}\end{bmatrix}  \tag{6}\]  
According to network equation, the relationship between the voltage of each node in power system can be shown in Equation (7). \[Y_{RG}U_{G}+Y_{RL}U_{L}+Y_{RR}U_{R}=0 \tag{7}\]  
\[-Y_{RR}^{-1} \begin{bmatrix}Y_{RG}&Y_{RL}\end{bmatrix}\begin{bmatrix}U_{G}\\ U_{L}\end{bmatrix} =U_{R} \tag{8}\]  
When a disturbance occurs in the system, the node voltage of power system typically undergoes only minor changes. By differentiating the voltage phase in Equation (8), the following relationship can be obtained as in Equations (9) and (10). \[Y_{\text{eq}}Y_{G}\frac{\text{d}\delta_{G}}{\text{d}t}=V_{R}\frac{\text{d} \delta_{R}}{\text{d}t} \tag{9}\]  
\[Y_{\text{eq}}=-Y_{RR}^{-1} \begin{bmatrix}Y_{RG}&Y_{RL}\end{bmatrix}  \tag{10}\]  
The per-unit value of node voltage in the power system is usually approximately equal to 1, so the frequency relationship between the network nodes and generators is shown in Equation (11). \[f_{R}=Y_{\text{eq}}f_{G} \tag{11}\]  
By differentiating both sides of Equation (10) with respect to frequency, the relationship between the RoCoF of each node of the power system can be derived as in Equation (12). \[\frac{\text{d}f_{R}}{\text{d}t}=Y_{\text{eq}}\frac{\text{d}f_{G}}{\text{d}t} \tag{12}\]The frequency responses of generators in the power system are represented by rotational motion equation, as shown in Equation (13). \[\left\{ \begin{array}{l}K_{I}\frac{\mathrm{d}f_{G}}{\mathrm{d}t}=-\Delta P_{G}\\ \frac{\mathrm{d}g_{G}}{\mathrm{d}t}=f_{G}\end{array} \right. \tag{13}\]  
The relationship between the frequency response and the active power output of power system can be clearly determined as in Equation (14). \[\frac{\mathrm{d}f_{R}}{\mathrm{d}t}=\gamma_{\mathrm{eq}}K_{I}^{-1}\Delta P_{G} \tag{14}\]  
where \(K_{I}\) is the matrix composed of the inertia time constants of the generator nodes, and \(\Delta P_{G}\) represents the active power change of power system. \[\left\{ \begin{array}{l}\frac{\mathrm{d}f_{\mathrm{Ri}}}{\mathrm{d}t}=\sum \limits_{j=1,j\
eq i}^{n}M_{ij}^{\mathrm{eq}}\Delta\delta_{ij}\\ M_{ij}^{\mathrm{eq}}=\sum\limits_{i=1}^{n}\frac{Y_{\mathrm{eq}}^{m_{I}}}{K_{Ii }}\Delta P_{G}\end{array} \right. \tag{15}\]  
where \(M_{ij}^{\mathrm{eq}}\) represents the node frequency regulation coefficients, \(n\) denotes the number of generators in the power system, and \(\mathrm{d}f_{Ri}/\mathrm{d}t\) indicates the frequency variation rate. The matrix \(M\) reflects the relationship between the frequency responses of the various generators.