## 1 Introduction  
With the merits of flexible configuration and control, the voltage source converter-high voltage direct current (VSC-HVDC) transmission has received wide applications in recent years [1; 2]. It is a desirable choice to integrate the offshore wind power generated by the wind turbine generators, e.g., doubly-fed induction generator (DFIG), to the power system [3; 4; 5; 6]. The DFIGs under the derated mode may respond to system frequency change with the over-speed or the pitch angle control [7]. Instead of a wide-area signal, the virtual inertia/d droop control responding to the local frequency may be applied [8; 9; 10; 11]. But for the DFIGs integrated through the VSC-HVDC, the system frequency is more vulnerable to the disturbances and faults, due to not only the fluctuation of the wind power but also the isolation of frequency response of the DFIGs by VSC-HVDC. The cascaded droop control from the system frequency to the DC voltage of the VSC-HVDC and then to the active output of the DFIGs is an effective scheme [12]. With the virtual inertia control [13; 14], the stored energy of the DC capacitor of the VSC-HVDC is released to provide fast support to the system frequency. Then according to the drop of the DC voltage of the VSC-HVDC, the DFIGs change their output by releasing their kinetic energy and reserved power. To study frequency regulation of power systems with the DFIG integrated through VSC-HVDC, the time domain analysis is often used [15; 16; 17], which is accurate but time-consuming due to numerous faults and disturbances, thus the simplified models are desirable. With only one differential equation, i.e., motion equation, the dynamic power flow (DPF) model estimates the regulation effect of the synchronous generator (SG) [18, 19], and quantifies the system frequency [20, 21]. It is easy to include the DFIG or the VSC-HVDC, but the DPF model with the coordinated regulation by the SGs and the DFIG integrated through the VSC-HVDC is not found in the existing literature. The difficulties lie in, (1) Based on the static frequency characteristic, the SGs respond very fast to the frequency change, but the actual reaction, especially of the thermal SGs, is slower due to the governor delay [22]. The output change is not instantaneous. The governor delay is considered in [23] by simply treating the output as zero, then stepping up to the set value. A more accurate delay model is desirable for the DPF to improve the accuracy of the frequency regulation. With the slow support from the SGs due to the governor delay at the early stage of the regulation, a large deviation of the frequency occurs due to the delayed support from the SGs, leading to a frequency nadir that is smaller than the post-disturbance steady-state frequency. (2) The equivalent inertia of the DFIG is critical for its frequency regulation capability [24, 25]. It is often set fixed based on the initial rotor speed before regulation [26]. Since the inertia of the DFIG is related to the kinetic energy of the rotor, the rotor speed change during frequency regulation needs to be considered when determining the inertia of the DFIG. A fixed inertia overestimates the regulation capability of the DFIG. To improve the accuracy of the DPF model, dynamic correction to the inertia of the DFIG based on the changing kinetic energy is necessary. (3) The inertia of the VSC-HVDC is not set freely, but decided by the constraint of the DC voltage. With the cascaded droop control, the drop of the DC voltage shows the frequency drop and is finally decided by the system inertia. The frequency nadir is a critical constraint to set the system inertia. As stated earlier, to estimate the nadir, the transient frequency response instead of the steady state frequency deviation is needed, which is more difficult with DFIG participating in frequency regulation through VSC-HVDC [27].