# Improved PR Control Without Load Current Sensors and Phase-Locked Loops for APFs
## 2 APF Control System  
As illustrated in Figure 1, the APF system comprises three main components: a three-phase AC source, a three-phase inverter, and a nonlinear load. The point of common coupling (PCC) denotes the interconnection point among these elements. The inverter-side filter inductance is denoted by \(L_{f}\). The variables \(e_{gu}\), \(e_{gb}\), and \(e_{gc}\) represent the three-phase grid voltages, while \(i_{gu}\), \(i_{gb}\), and \(i_{gc}\) correspond to the grid currents. The inverter output currents are indicated by \(i_{fa}\), \(i_{fb}\), and \(i_{fc}\), and the nonlinear load currents by \(i_{lx}\), \(i_{lb}\), and \(i_{lc}\). \(U_{dc}\) denotes the DC-link voltage of the inverter. The control structure of this system is shown in Figure 1, which includes three parts: harmonic identification (Part A), compensating current control (Part B), and DC bus voltage control (Part C). Part A is responsible for extracting the harmonics and reactive components in the load current and generating the target reference current; Part B uses an improved PR controller to control the inverter output current to track the reference value to achieve harmonic compensation; and Part C uses a PI controller to adjust the DC side voltage to maintain its stability to ensure the energy support required for compensation. The control structure shown in Figure 1 abandons the direct measurement of the load-side and inverter-side current in the traditional method. The core is to use only the grid voltage and grid  
Figure 1: APF system structure and control block diagram. current information for harmonic identification and compensation command generation, thereby significantly reducing the number of sensors and improving the system robustness. In a three-phase system, define \(f(t)\) as the three-phase grid voltage and current, and the components after Clark coordinate transformation are represented by \(\alpha\), \(\beta\), and 0, where \(T\) is the Clark transformation coordinate, as shown in Formula (1). \[ \begin{bmatrix}f_{\alpha}(t)\\ f_{\beta}(t)\\ f_{0}(t)\end{bmatrix} =T \begin{bmatrix}f_{A}(t)\\ f_{B}(t)\\ f_{C}(t)\end{bmatrix} ,T=\sqrt{\frac{2}{3}} \begin{bmatrix}1&-1/2&-1/2\\ 0&\sqrt{3}/2&-\sqrt{3}/2\\ \sqrt{2}/2&\sqrt{2}/2&\sqrt{2}/2\end{bmatrix}  \tag{1}\]  
In a three-phase three-wire system, the circuit impedance of the zero component is infinite, so the zero component can be ignored in both voltage and current. Therefore, the voltage and current can be represented by equivalent complex values, as shown in Formula (2). Furthermore, according to the Fourier transform principle and the instantaneous power theory proposed in Reference [16], the instantaneous complex power \(\mathbf{s}\), active power \(\mathbf{p}\), and instantaneous reactive power \(\mathbf{q}\) of the fundamental wave can be expressed as in Formula (3), where \(e^{*}\) is the conjugate component of \(\mathbf{e}\). \[\left\{\begin{array}{l}\mathbf{e}=e_{\alpha}+\mathrm{j}\epsilon_{\beta}\\ \mathbf{i}=i_{\alpha}+\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j} \mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j }\mathrm{j}\mathrm{j}\mathrm{j}\mathrm{j}  
energy\" that the grid needs to transmit, while \(\mathbf{i}_{i,q}\) and \(\mathbf{i}_{i+h}\) belong to \"ineffective work\" and should be compensated by APFs.  
\[\mathbf{i}_{fabc}+\mathbf{i}_{gabc}=\mathbf{i}_{fabc} \tag{4}\]  
\[\mathbf{i}_{iabc}=\mathbf{i}_{i-fp}+\mathbf{i}_{i-fq}+\mathbf{i}_{i-h} \tag{5}\]  
Therefore, the inverter output current is regulated to contain the fundamental reactive component \(\mathbf{i}_{i,q}\) and the harmonic component \(\mathbf{i}_{i,h}\) of the load current such that the grid-side current consists solely of the remaining fundamental active component \(\mathbf{i}_{i,fp}\). In other words, the APF injects harmonic and reactive currents into the grid to counteract the corresponding components at the load side, thereby retaining only the fundamental active current in the grid-side current.