# 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
In this form, the controller maintains high gain at the frequency \(\omega_{0}\) and has low implementation complexity, which is suitable for control platforms such as FPGA/DSP. Considering that the switching behavior of the PWM inverter can be approximated as a zero-order keeper, its control loop can be modeled as a discrete closed-loop system including a PR controller, PWM delay, and filter structure. Figure 4 shows the discrete block diagram of the compensation current control loop based on \(\alpha\beta\) axis equivalent modeling. Through this discrete model, the closed-loop transfer function of the system is constructed as Formula (15):  
\[\frac{I_{(\alpha\beta 0)}(z)}{I_{(\alpha\beta 0)}^{*}(z)}=\frac{\frac{K_{ \rm P}L_{c}}{T_{s}}(z^{2}+(C+\frac{A}{K_{\rm P}})z+(D-\frac{A}{K_{\rm P}}))}{ \left( \begin{array}{c}z^{4}+(C-1)z^{3}+\left(\frac{K_{\rm P}L_{c}}{T_{s}}+D- C\right)z^{2}\\ +\left(\frac{K_{\rm P}L_{c}}{T_{s}}C+\frac{L_{c}}{T_{s}}A-D\right)z+\left( \frac{K_{\rm P}L_{c}}{T_{s}}D-\frac{L_{c}}{T_{s}}A\right)\end{array} \right)} \tag{15}\]  
According to Reference [24], the quality factor \(Q\) can be set to 10, and the resonant frequency should be chosen as the fundamental frequency of 50 Hz. Therefore, the primary parameters to be determined are \(K_{\rm P}\) and \(K_{\rm r1}\). The proportional gain \(K_{\rm P}\) plays a critical role in the transient response, while the resonant gain \(K_{\rm r1}\) influences the location of the system's zeros, thereby affecting the root locus trajectory. Consequently, the root locus diagram is plotted based on Formula (15) to guide the selection of appropriate \(K_{\rm P}\) and \(K_{\rm r1}\) values. Figure 5 illustrates the root locus diagrams used to guide the selection of the proportional gain \(K_{\rm P}\) and resonant gain \(K_{\rm r1}\) in the improved PR controller design. The transfer function shown has two zeros (Zero1 and Zero2) and four poles (Pole1 to Pole4). The design mainly considers the root locus distribution between Zero1 and Pole1 [24] to guide the selection of \(K_{\rm P}\) and \(K_{\rm r1}\) to ensure that the system is stable and has good dynamic performance. Analyzing Figure 5a, with \(K_{\rm P}\) = 20 fixed and \(K_{\rm r1}\) increased (from 10 to 1000), the root locus tends to move from left to right. With two zeros fixed, the poles gradually approach the unit circle boundary (about 0.99) from the left. Too large a \(K_{\rm r1}\) (as shown in the figure, \(K_{\rm r1}\) = 1000 corresponds to the red cross position) will cause the poles to be very close to the unit circle, and the system tends to be unstable or overshoot. Therefore, \(K_{\rm r1}\)\(\in\) [10, 100] is more appropriate. At this time, the system responds faster, the poles are still within the unit circle, and they have a good phase margin. Similarly, we  
Figure 4: A block diagram of the \(d\vartheta\)0-axis compensation current control loop in the discrete domain. analyze Figure 5b, with \(K_{\rm r1}=20\) fixed; as \(K_{\rm p}\) increases, the poles gradually approach the unit circle boundary along the curve, the response speed increases, but at the same time the system stability decreases. The red cross (\(K_{\rm p}=1000\)) in the figure is already very close to the unit circle, near the stability boundary, and has a high risk of overshoot. Therefore, \(K_{\rm p}\)\(\in\) [10, 100] is more appropriate. At this time, the system responds faster, the poles are still within the unit circle, and they have a good phase margin. Figure 5: Improved PR controller parameter settings: (**a**) \(K_{\rm r1}\) root locus diagram; (**b**) \(K_{\rm p}\) root locus diagram. This section proposes a design method for an improved PR controller. Based on the reference current extraction method of the instantaneous power theory of the grid-side voltage and current, the grid-side current only needs to track the baseband active command current without configuring a separate controller for each harmonic.