## 2 Grid Forming and Small-Signal Modeling
### Network Frequency Perturbation Visualization Method  
The state-space modeling and eigenvalue analysis method provides valuable information regarding a system's damping ratio and natural frequency. However, obtaining the eigenvalues requires complete knowledge of all system parameters. In contrast, the network frequency perturbation (NFP) plot offers significant insights, particularly in scenarios where the plant is treated as a black box, concealing internal details such as control strategies and the precise dynamic behavior of the equipment under study. This makes the NFP plot a valuable tool for system operators. Consequently, its application has been recommended for the compliance testing of grid-forming (GFM) plants [29; 30]. The NFP method [31; 32; 33] involves embedding the device within a hypothetical power system in which the network frequency is deliberately modulated in a sinusoidal manner at a specified frequency. This modulation originates from an upstream \"infinite bus\" and includes sub-synchronous frequency components ranging from 0 Hz to 20 Hz, with optional extension to frequencies near the fundamental. The resulting graphical representation illustrates the device-level behavior of a grid-connected generation unit during system frequency variations. Figure 3: PSL control loop options. (**a**) VSG and (**b**) filtered droop. The NFP plot can be obtained either by deriving the system's closed-loop transfer function (see Section 2.2) or by conducting time-domain simulations or experimental tests (see Section 4). Individual synchronous generators (SGs) or GFM devices are expected to contribute to frequency stability and active power balance in a predictable and well-characterized manner. Therefore, even with the addition of various control subsystems--whether internal loops or supplementary controllers for damping--their dynamic response should closely resemble, or be equivalent to, that of a synchronous machine, as described by Equation (5). To support this analysis, a specific case based on Equation (5) is illustrated in Figure 4, providing a graphical interpretation of the characteristic regions expected from a GFM or SG unit. Although the base case is presented in the following subsection, the example shown in Figure 4 is generated under identical conditions, with the sole variation being the control parameters, \(\tau_{p}\) and \(m_{p}\), which are adjusted to emphasize the resonance effects associated with system inertia. The NFP, represented as a Bode plot, highlights three primary features, along with several subtler aspects that can be further interpreted to improve understanding of the device's overall performance. Droop Behavior:  
The droop behavior of the device becomes evident as the modulation frequency approaches zero (i.e., steady-state conditions). At this frequency, the magnitude response coincides with the droop coefficient, while the phase response approaches 180\({}^{\circ}\), reflecting the inverse relationship between active power and grid frequency. This behavior predominates in the low-frequency range, typically between 0 Hz and 0.5 Hz, depending on the device's damping characteristics. The droop behavior can be analytically verified by substituting s = 0 into equation (final value theorem) (5). Inertia:  
The inertial response is prominent in the frequency range from approximately 0.2-0.5 Hz up to 1-3 Hz. It is represented in the plot by the inertia asymptote, depicted as a dashed black line that excludes the effects of droop response, rotor resonance, and damping (see Figure 4). Devices possessing inertia typically exhibit a resonance frequency resulting from their second-order system dynamics. This resonance generally appears between 1 Hz and 3 Hz and is influenced by factors such as the inertia time constant, damping coefficient, and total system reactance. In power electronic converters designed to emulate inertial behavior, these parameters can be adjusted to tune the resonance characteristics.