1. School of Electrical Engineering Southeast University Nanjing 210096 China; 2. State Grid Jiangxi Electric Power Research Institute Nanchang 330096 China
Abstract：Recently, the concept of complex frequency has been proposed, whose real and imaginary parts represent the rate of change of voltage (RoCoV) and phase angle, respectively. This concept unifies the RoCoV and angle frequency, offering a new perspective for the control of grid-connected inverters. Traditional grid synchronization techniques have primarily focused on angular frequency observation, with magnitude typically calculated in an open-loop manner, and lacking effective closed-loop observation for RoCoV. Based on this, a generalized dq transformation based on complex phase angle is proposed in this paper, leading to the design of a synchronous reference frame complex frequency-locked loop (SRF-CFLL), enabling a symmetrical and unified closed-loop observation of both RoCoV and frequency changes. Firstly, a generalized dq transformation is proposed based on the concept of complex phase angle, which extends the traditional coordinate transformation to incorporate dynamic variations in both amplitude and frequency. Secondly, leveraging this generalized transformation, SRF-CFLL is developed, enabling unified observation of complex frequency that inherently includes both the instantaneous frequency and its rate of change RoCoV. Thirdly, through small-signal modeling of the SRF-CFLL, it is theoretically revealed that the observed RoCoV and frequency exhibit identical dynamic responses, demonstrating that the proposed scheme achieves symmetric closed-loop observation of these two key quantities. Finally, systematic parameter tuning guidelines are established to optimize control performance, incorporating an inner-loop filter to mitigate harmonic interference and enhance frequency estimation robustness in distorted grid scenarios. Simulation and experimental results demonstrate that the designed CFLL can accurately estimate both the RoCoV and the instantaneous frequency within 12 ms under scenarios of grid frequency step changes or voltage amplitude ramp-down. When the grid voltage amplitude experiences sudden variations, the SRF-CFLL maintains effective closed-loop estimation of voltage amplitude through its feedback mechanism, while the generalized dq transformation ensures stable input voltage normalization. Compared with traditional methods, the SRF-CFLL exhibits significant noise immunity in RoCoV estimation: its built-in inner-loop filter effectively mitigates harmonic interference, enabling high-precision estimation within 20 ms even in noisy environments. These performance metrics validate the proposed method’s rapid response and robustness under transient voltage/frequency disturbances. The following conclusions can be drawn from simulations and experiments: (1)The SRF-CFLL exhibits symmetry in both RoCoV and frequency estimation, a feature that facilitates the analysis and design of grid-support control. (2)Due to its closed-loop feedback estimation mechanism, the SRF-CFLL demonstrates stronger anti-interference capability, ensuring zero steady-state error and thereby improving accuracy. (3)The SRF-CFLL has a computational advantage since its RoCoV estimation loop shares the same structure as the frequency estimation loop, eliminating the need for additional design. Moreover, the integrator implementation in the SRF-CFLL is simpler than discrete-time numerical differentiation, which requires noise handling and sampling time considerations.
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2026, Vol. 41 
