## 4 Case Study
### Analysis of Inertia Distribution Characteristics  
Based on the structural configurations of the IEEE 9-bus and 39-bus systems, the inertia levels of each node in the systems are evaluated by using the NFDC. The simulation model of the systems is shown in Figure 3. The IEEE 9-bus system contains 3 generators, 3 loads, and 9 branches, while the IEEE 39-bus system contains 10 generators, 21 loads, and 46 branches.  
Figure 3: Simulation models of IEEE 9-bus system and IEEE 39-bus system.  
Based on the simulation models of the IEEE 9-bus system and 39-bus system, the inertia level of each node in the systems can be evaluated by the value of the NFDC. The parameters of the generators in the system, including the inertia coefficient, are shown in Table 1. WF indicates that the generator type of the unit is a wind farm.  
Based on the NFDC, the inertia support capability of various nodes and the distribution of inertia resources in the power system can be effectively represented. Table 2 shows the values of the NFDC at each node in the IEEE 9-bus system.  
G represents the generation nodes in the system and R refers to the network or load nodes. According to the evaluation results, the NFDC is a complex number, with the real part being much larger than the imaginary part. This is because the active power loss in the power system is primarily reflected in the branch impedance, which manifests as a larger real part in the NFDC. Therefore, when conducting index evaluations subsequently, the real part of the NFDC will be used as the basis for evaluating the inertia levels of the nodes.  
As shown in Figure 4, the inertia levels of each node in IEEE 9-bus system are presented in the form of a heat map. This approach allows for a more intuitive understanding of the inertia distribution characteristics of the system.  
The power disturbance in the system is simulated by setting a single-phase short-circuit fault at bus R6 in IEEE 9-bus system. The frequency dynamic characteristics of three generators in IEEE 9-bus system can be observed through time-domain simulation. The simulation results are shown in Figure 5. It can be observed that during the occurrence of a power disturbance event, the frequency variation of G3 is the most pronounced, whereas that of G1 is the least significant. This is consistent with the situ  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**System** & **Unit** & **Generator** & **Nominal Ca-** & **Inertia** \\  & **Number** & **Type** & **pacity/MVA** & **Constant/s** \\ \hline \multirow{4}{*}{IEEE9} & G1 & SG & 200 & 20 \\  & G2 & SG & 200 & 12 \\  & G3 & SG & 100 & 6 \\ \hline \multirow{4}{*}{IEEE39} & G30 & SG & 400 & 4 \\  & G31 & SG & 700 & 10 \\  
NFDC, indicating that the NFDC can accurately reflect the inertia level of the nodes in the power system.  
To observe the frequency characteristics of the three generators and the value of the NFDC, the GFM-ESS which has a nominal capacity of 30 MW is implemented in the IEEE 9-bus system. The comparative results are shown in Figure 6.  
According to the simulation results, after configuring the GFM-ESS, the RoCoF of each generator decreases, with the most significant decrease observed at the unit where the GFM-ESS is installed. The decrease in RoCoF is also reflected in the change in the NFDC value, indicating a positive correlation between the NFDC and RoCoF.  
Based on the same method, the inertia distribution characteristics of the IEEE 39-bus system were evaluated. The NFDC evaluation results are shown in Figure 7.  
Figure 4: The inertia distribution characteristics of the IEEE 9-bus system.  
Figure 5: The frequency dynamic characteristics of the three generators in the IEEE 9-bus system.  
Figure 6: The frequency characteristics and NFDC of G3 after configuring the GFM-ESS.  
According to the evaluation results, G30 belongs to the weak link in terms of the system inertia level, and G33 and G34 also have the risk of losing stability. When configuring the ESS, these locations can be given priority consideration.