**2. Proposed Hybrid Filter**

An active hybrid power filter is the combination of an active filter and a passive filter. It consists of a passive filter and a static power converter and a control block that allows the control of the entire hybrid filter. The passive filter attempts to compensate high frequency harmonics and used to reduce the capacity of the power converter, while the active filter is used to compensate for low frequency harmonic currents generated by the polluting non-linear load [15], and used to improve the characteristic parameters of the passive filter. The advantages of HAPF over other filtering elements is that it is possible to solve the problems related to the injection of the neighbouring harmonic current, resonances, as well as the capacity of the HAPF converter is smaller than the capacity of the ordinary active power filter. HAPF is a better solution to reduce the power sizing which results in the price of active power filters. In order to obtain a better performance of the hybrid filter, it is necessary to choose the latter according to several factors such as the topology of the filter, the control strategy used, the type of filter used in the control loop or the size of the components constituting the filter. There are several configurations treated in the literature [16], the most presented being:


In this article we choose the topology of the parallel active filter in series with a passive filter, as shown in Figure 1. The active hybrid power filter is composed by the static power converter attached in series to three-phase passive filters. The HAPF and the non-linear load are connected in parallel, which clarifies the harmonic currents generated by the pollutant load. The three-phase passive filter assembly is connected in series on each phase. It consists of the capacitor and inductance while the converter contains a semiconductor assembly and a capacitor. The power converter protects the passive filter from damage caused by the injection of the neighborhood harmonic current and resonance. The capacitor is connected to the side of the DC bus of the converter as an energy storage element and put a DC voltage for normal operation of the converter. The passive filter blocks the harmonic currents that pass through the power grid.

**Figure 1.** Topology of the proposed hybrid filter.

### *2.1. Current Reference Algorithm Using p-q Theory*

The active filter is used to inject harmonic currents of the same amplitude into the network but in opposition to those generated by the pollutant load. To do this, it is necessary to extract the harmonic currents from the load, known as reference currents. There are several methods for identifying the harmonic currents [17–19], but in this article we use the *p*-*q* method because it guarantees a better adherence between the dynamic and static performances. This theory is based on the Clark algebraic transformation which allows to transform the three-phase voltage and current systems exposed in the reference frame *a*, *b*, *c* to a two-phase system presented in the reference frame α, β, to simplify the calculations. The current and voltage components can be expressed as:

$$
\begin{bmatrix} i\_{\alpha} \\ i\_{\beta} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & \frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i\_{a} \\ i\_{b} \\ i\_{c} \end{bmatrix} \tag{1}
$$

$$
\begin{bmatrix} \upsilon\_{\alpha} \\ \upsilon\_{\beta} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{vmatrix} 1 & \frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{vmatrix} \begin{vmatrix} \upsilon\_{a} \\ \upsilon\_{b} \\ \upsilon\_{c} \end{vmatrix} \tag{2}
$$

The active instantaneous power in the mark *a-b-c*, is given by:

$$p(t) = \upsilon\_a i\_a + \upsilon\_b i\_b + \upsilon\_c i\_c \tag{3}$$

In the <sup>α</sup>*-*β mark, the active instantaneous power is given by:

$$p(t) = v\_{\alpha}i\_{\alpha} + v\_{\beta}i\_{\beta} \tag{4}$$

The instant imaginary power is given by Akagi's definition [15], as follows:

$$q(t) = \upsilon\_{\alpha} i\_{\beta} - \upsilon\_{\beta} i\_{\alpha} \tag{5}$$

From the relationships (4) and (5), we can extract the matrix relationship of the instantaneous powers as follows:

$$
\begin{bmatrix} p \\ q \end{bmatrix} = \begin{bmatrix} v\_{\alpha} & v\_{\beta} \\ -v\_{\beta} & v\_{\alpha} \end{bmatrix} \begin{bmatrix} i\_{\alpha} \\ i\_{\beta} \end{bmatrix} \tag{6}
$$

This power is divided into two parts, a continuous part related to the fundamental (*p*, *q*), and an alternating part related to harmonics (*p*,*q*), is given by the following relationships:

$$\begin{array}{c} p = \overline{p} + \overline{p} \\ q = \overline{q} + \overline{q} \end{array} \tag{7}$$

What interests us is the extraction of the alternative components (*p*,*q*), for this purpose, we use a low-pass filter as shown in Figure 2.

**Figure 2.** Principle of extraction of alternative components.

The reference currents after the extraction of the alternative components in the coordinates α, β is of the following expression:

$$
\begin{bmatrix} i\_{\text{arcf}} \\ i\_{\text{left}} \end{bmatrix} = \frac{1}{v\_a^2 + v\_{v\_a}^2} \left\{ \begin{bmatrix} v\_a & -v\_\beta \\ v\_\beta & v\_a \end{bmatrix} \begin{bmatrix} 0 \\ q \end{bmatrix} + \begin{bmatrix} v\_a & -v\_\beta \\ v\_\beta & v\_a \end{bmatrix} \begin{bmatrix} \overleftarrow{p} \\ \overleftarrow{q} \end{bmatrix} \right\} \tag{8}
$$

To obtain the reference currents in the reference frame *a-b-c*, we use the Clark inverse transformation, the expression is as follows:

$$
\begin{bmatrix} i\_{\text{arcf}} \\ i\_{\text{bref}} \\ i\_{\text{ref}} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} 0 & 1 \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i\_{\text{arcf}} \\ i\_{\text{left}} \end{bmatrix} \tag{9}
$$

## *2.2. DC Bus Voltage Regulation*

The objective of the DC bus voltage regulation loop (*Vdc*) is to maintain the latter following its reference value *Vdc ref*. For the control of this loop, a PI corrector is used as shown in Figure 3. The reference voltage is considered as input and the measured value as output. The voltage at the capacitor terminals is given by:

$$V\_{dc}^2(\mathbf{s}) = \frac{2P\_{dc}(\mathbf{s})}{\mathcal{C}\_{dc}s} \tag{10}$$

**Figure 3.** D.C. bus voltage regulation loop.

From Figure 3, the transfer function showing the regulation of the DC bus voltage in closed loop is given by:

$$G\_{bf}(s) = \frac{\left(1 + \frac{K\_p}{K\_i}s\right)}{s^2 + 2\frac{K\_p}{C\_{dc}}s + 2\frac{K\_i}{C\_{dc}}}\tag{11}$$

Comparing this closed-loop equation with the general structure of a second-order transfer function, extracting the parameters of *Kp* and *Ki*:

$$\begin{array}{l}\omega\_{\mathbb{C}} = 2\pi f\_{\mathbb{C}}\\K\_{i} = \frac{1}{2}\mathbb{C}\_{dc}\omega\_{\mathbb{C}}^{2}\\K\_{p} = \xi\sqrt{2\mathbb{C}\_{dc}K\_{i}}\end{array} \tag{12}$$

### *2.3. Regulation of the Current Injected by the Filter*

A conventional PI controller is used to maintain the control loop for the current injected by the filter following the reference current extracted by the *p*-*q* method as shown in Figure 4.

**Figure 4.** Control loop for the current injected by the filter.
