The conventional Reduced order generalized integrator (ROGI)-based Phase-locked loop (PLL) can track the voltage frequency and phase of the power grid under unbalanced conditions. However, the gain of the traditional ROGI are all real coefficients and the control degree of freedom is limited. In order to further improve the performance of the phase-locked loop, the Complex coefficient ROGI (CCROGI) is proposed in this paper to extend the ROGI's coefficient from real to complex, thus enhancing its ability of harmonic detection. Secondly, the pole-allocation method is used to design the complex coefficient gain of the CCROGI to obtain better dynamic and static performance. Finally, a frequency-locked loop (FLL) is proposed to improve the frequency adaptive of the CCROGI. The experimental results also prove the effectiveness of the proposed CCROGI.