Although the research on doubly-fed induction generator (DFIG) has gained popularity, the phase-locked loop (PLL) used in the DFIG system leads to frequency coupling with negative resistive characteristics in the mid-frequency band, which reduces the stability of the grid-connected DFIG system and consequently leads to system oscillations. Meanwhile, because of the rise in the share of new energy generation and the long-distance transmission characteristics of wind power systems, the AC grid short circuit ratio (SCR) decreases, which exacerbates the occurrence of system instability. On this basis, based on dynamic compensation of rotor current, an impedance remodeling method for DFIG is proposed to achieve stable operation of the DFIG system in the mid-frequency band. First, an impedance model incorporating the DFIG generator characteristics, rotor-side converter (RSC), and PLL control is established. Second, to analyze the generation of frequency coupling, the multiple-input multiple-output (MIMO) impedance model of the DFIG is developed. In addition, the equivalent positive and negative impedance models are built to analyze the dominant elements of the DFIG system's frequency characteristics, which provide the basic prerequisites for the design of a subsequent impedance reshaping strategy. Moreover, a DFIG impedance reshaping method is proposed to enhance the stability margin in the mid-frequency band when the system operates under a weak grid by compensating the rotor current. Furthermore, the effectiveness of the control strategy under the changing operating conditions of the system is also theoretically analyzed. Finally, the proposed impedance remodeling method is validated by simulation results and experimental results.

With the advancement of low-carbon energy development strategies, renewable energy is gradually replacing traditional fossil energy, which has led to a rapid increase in the installed capacity of wind energy conversion systems. Doubly fed induction generators (DFIG) have been widely promoted for wind power generation due to their high efficiency, variable speed, and flexible power operation capabilities. However, the rapid development of renewable energy sources has weakened the AC grid to which they are connected, characterizing the grid's short-circuit ratio (SCR) as low. Furthermore, the impedance of the transmission line is exacerbated by the distance of wind farms from the primary grid.

The doubly-fed induction generator (DFIG) system employs a phase-locked loop (PLL) to achieve synchronization with the power grid. The PLL captures the voltage at the point of common coupling (PCC) for the purpose of vector control. However, the DFIG system is highly vulnerable to grid voltage imbalances and harmonic distortions at the PCC, which consequently impact system stability. It is shown that PLL will introduce negative characteristics in the mid-frequency band by impedance-based stability analysis method. The negative impedance range of the DFIG system grows as the PLL bandwidth increases. In fact, the PLL's asymmetric character causes the DFIG system's frequency coupling, and this phenomenon will probably cause a negative resistance characteristic in the PLL bandwidth, affecting the DFIG system's stability.

In practice, series compensation can be installed to reduce the inductive impedance of the line, however, this increases the cost on the one hand, and on the other hand, it also makes the system have a potential risk of subsynchronous oscillation. To solve the stability problems caused by DFIG access to the weak grid, a number of different solutions have been proposed.

Since the PLL output angle is perturbed under weak grid conditions and the system is more prone to destabilization as the PLL bandwidth increases, control strategies based on adaptive regulation of PLL parameters have been proposed. An adaptive algorithm for pre-filter based on PLL structure is proposed, through which the parameters of the filter are changed, so that the system can adaptively adjust the center frequency of the filter under the condition of grid distortion. An adaptive optimal control strategy stabilized by Lyapunov is presented for the oscillatory behaviors of the DFIG system, and the advantages of the suggested stabilizer in various perturbation situations are demonstrated. A dynamic adjustment strategy for PLL gain based on adaptive dual inner ring recursive neural network is proposed, and the weight of the strategy is fine-tuned based on the dynamic backpropagation learning algorithm to realize the real-time adjustment of PLL parameters. However, these methods require real-time grid impedance estimates and parameter variations result in design complexity. What's more, the dynamic tracking performance of grid frequency and phase is reduced when the PLL bandwidth degrades, especially under asymmetrical voltage faults. In addition, since it does not directly change the frequency coupling characteristics of PLL, the above method still has the risk of instability under extremely weak grid conditions.

Another type of approach is to reshape the DFIG's impedance characteristic by means of an additional control link. Virtual impedance controllers have been proposed to change the DFIG system's impedance characteristics in the mid-frequency and high-frequency bands, but their effectiveness is susceptible to operating conditions and system parameters. On this basis, a parametric adaptive control strategy is proposed to suppress the occurrence of sub-synchronous and super-synchronous oscillations in doubly-fed wind turbine generator systems, but the complexity of the system control structure is increased. In addition, the PLL can also change its internal structure to achieve the elimination of the frequency coupling phenomenon by introducing the dynamics of the PLL symmetrically into the dq axis, but it essentially does not change the negative resistance that the PLL brings to the system, and the elimination of the frequency coupling phenomenon is only possible if the rest of the system's control loops are symmetrical as well. In addition, damping control was proposed to solve the shock problem of DFIG systems. A vibration mode analysis method based on eigenvalue analysis is proposed to judge the oscillation mode of the DFIG system with additional damping, and the subsynchronous oscillation problem for the DFIG system is solved by additional damping, but the potential oscillation risk of the system in the mid-frequency band is not discussed. A new voltage-controlled double-fed wind turbine (VC-DFIG) mode based on damping control has been established, which is a grid-connected structure with a certain voltage and frequency independent support capacity in the weak grid, which is not suitable for the PLL-based grid-following DFIG system.

Direct power control (DPC) is suggested to solve the question. Unlike the conventional synchronous rotating dq frame, it eliminates the effect of PLL negative resistance using a virtual dq frame, enhancing the stability margin of the system within the PLL bandwidth. In the high-frequency band, however, this method introduces a strong frequency coupling characteristic, which leads to high-frequency stability problems. Since the negative resistance is mainly generated on the q-axis. Therefore, a strategy to correct the q-axis impedance to the positive resistance is proposed, which is controlled by the q-axis voltage and current. A compensating control is also proposed, which improves stability by reducing PLL-related perturbations. However, the effects of the above PLL shaping methods are focused on the grid-connected converter, the presence of engines in the DFIG system requires reconsideration. Based on the neural super-twisting algorithm, a DPC-controlled impedance remodeling method is proposed, which can reduce current harmonic distortions and ratios of the DFIG power fluctuations. However, when the PLL parameters spike, the method will have a potential risk of high-frequency oscillation.

In summary, to improve the stability of the DFIG system in the mid-frequency band, a new approach to solve the DFIG instability problem induced by the PLL in weak grid environments is proposed, which can improve the stabilization margin of the DFIG system in the mid-frequency band. The contributions of this paper are summarized as follows:

(i) A DFIG multiple-input-multiple-output (MIMO) impedance model is developed to analyze the main cause of the frequency coupling phenomenon, and the transfer function matrix related to the dynamics of the PLL rotor currents is explained to be the main cause of the frequency coupling phenomenon. This provides a basic prerequisite for the design of subsequent impedance remodeling strategies.

(ii) Based on the dynamic compensation of rotor current, a new impedance reshaping method is proposed to eliminate the dynamic variations of rotor currents introduced by the PLL. This method can eliminate the DFIG system's negative impedance region in the mid-frequency band by dynamically compensating the rotor currents. The method is simple in parameter design and maintains effectiveness under changing system operating conditions.

The rest of the paper is shown as follows. In Section 2, the impedance model of the DFIG system incorporating the current loop and PLL control is modeled, and the connection between the PLL and the system that generates the frequency coupling phenomenon is analyzed. In addition, an equivalent MIMO model of the DFIG system is developed in Section 3, which analyzes the main factors affecting the stability of the system in the mid-frequency band under a weak grid. Then, a new impedance reshaping method is proposed to increase the DFIG system's stability margins in Section 4. In Section 5, the impedance reshaping method proposed is verified through simulation results. Experimental validation is carried out in Section 6. Finally, the conclusion of this paper is given in Section 7.