# Improved PR Control Without Load Current Sensors and Phase-Locked Loops for APFs
## 3 A PLL-Less Load Current Sensorless Improved PR Control
### PR Controller Parameter Design  
Furthermore, by eliminating the phase-locked loop (PLL), the proposed strategy avoids the phase detection delays commonly introduced by synchronous coordinate transformation. Since the control is performed directly in the stationary \(\alpha\beta\) reference frame, the system exhibits faster reaction to disturbances and improved robustness under frequency variations, without additional signal processing overhead. In APF, the design of the current controller plays a decisive role in the compensation performance. The PR controller has become the core solution in sinusoidal quantity tracking because of its infinite gain capability at the resonant frequency. This section will start with the principle of the classic PR controller and combine Reference [24] to propose a new PR control that only targets the fundamental frequency AC component to improve the system's harmonic suppression capability under dynamic load conditions. According to References [24; 25; 26], the classical PR controller is given in Formula (10), and its mathematical derivation can be found in more detail in Reference [13]:  
\[C_{\text{cos}}(s)=K_{\text{P}}+\frac{K_{\text{r}}\omega_{0}s}{s^{2}+\omega_{0 }^{2}} \tag{10}\]  
Figure 2: Reference current generation flow chart based on instantaneous power theory. \(K_{\rm p}\) is the proportional gain, \(K_{\rm r}\) is the resonant gain, and \(\omega_{0}=2\pi f_{0}\) is the resonant frequency. The gain at the resonant frequency can theoretically be infinite. But a very high gain tends to reduce the phase margin (PM) and robustness of the system. However, the gain of the resonant term must be high enough to achieve zero or quasi-zero steady-state error. Therefore, the quality factor \(Q\) is introduced to balance the gain and phase margin at the resonant frequency. The PR controller can be further written as Formula (11):  
\[C(s)=K_{\rm p}+\frac{K_{\rm r}\omega_{0}s}{s^{2}+(\omega_{0}/Q)s+\omega_{0}^{2}} \tag{11}\]  
The \(Q\) value affects the bandwidth at the resonant frequency. According to Reference [25], the higher the \(Q\) value, the higher the gain at the resonant frequency and the narrower the passband region near the frequency. A lower \(Q\) value will result in a lower gain, so there will be some steady-state errors. In order to accurately locate the zero point of the resonant term, \(K_{1}\) is introduced in the design to fine-tune the zero point position. The PR controller can be further written as Formula (12):  
\[C(s)=K_{\rm p}+\frac{K_{\rm r1}\omega_{0}s-K_{1}\omega_{0}^{2}}{s^{2}+(\omega_ {0}/Q_{1})s+\omega_{0}^{2}} \tag{12}\]  
As shown in Figure 3, the traditional PR control method requires configuring and tuning a dedicated PR controller for each harmonic order--for example, the fifth and seventh harmonics must be compensated by separate controllers. This results in a complex control architecture with increased computational burden and the need for additional hardware, such as PLL modules and load-side current sensors. In contrast, the proposed method achieves significant structural simplification. It eliminates the PLL and load-side current sensors entirely and uses only a single PR controller focused on the fundamental frequency. Harmonic compensation is achieved indirectly through precise tracking of a reference current derived from instantaneous power theory using grid-side voltage and current signals. This approach not only reduces implementation complexity and tuning effort, but also improves system robustness, noise immunity, and cost efficiency. Figure 3: Traditional PR control. In order to make the controller realizable in actual digital control systems, it needs to be discretized. Using the forward rectangular method [26], the discrete form of the PR controller is  
\[C(z)=K_{\rm P}+\frac{Az+B}{z^{2}+Cz+D} \tag{13}\]  
\[A=K_{\rm r1}\omega_{0}T_{s},\ B=-K_{\rm r1}\omega_{0}T_{s}-K_{\rm 1}\omega_{0}^{ 2}T_{s}^{2},\ C=(\omega_{0}/Q)T_{s}-2,\ D=1+\omega_{0}^{2}T_{s}^{2}-(\omega_{0} /Q)T_{s} \tag{14}\]  
where \(T_{s}\) is the sampling period.