A Modified Control Strategy for Three-Phase Four-Switch Active Power Filters Based on Fundamental Positive Sequence Extraction

Abstract

Three-phase four-switch active power filters (APFs) have attracted attention due to their low amount of semiconductors and low cost. The traditional control strategy of three-phase four-switch APFs usually includes phase-locked loops (PLLs) and rotating coordinate transformation for harmonic detection, resulting in complicated calculations and increased computation. In this paper, a modified control strategy for three-phase four-switch APFs based on fundamental positive sequence extraction is proposed, eliminating PLLs and rotating coordinate transformation with trigonometric calculations. Harmonic extraction is based on the fundamental positive sequence extraction method, while non-locked phase loop coordinate transformation is proposed to eliminate trigonometric calculations. Quasi-PR control is adopted for current tracking, and DC voltage control is designed to suppress voltage imbalance between the two split capacitors on the DC side. The space vector pulse width modulation (SVPWM) method is modified for a reduced-switch APF topology. The proposed control strategy guarantees excellent harmonic compensation: harmonic currents are significantly suppressed when the APFs are working, the THD of the source current decreases to 3.86%, the bus voltage fluctuation on the DC side is small, the voltage remains stable, and the computational complexity is reduced. Finally, a simulation and an experimental hardware platform are established to validate the feasibility and performance of the proposed control strategy. The experimental results show that it has good performance in suppressing harmonics and improving power quality.

1. Introduction

Recently, smart grids and new energy generation technologies have developed rapidly, with new load types such as electric vehicles, virtual power plants, distributed energy storage, and so on being widely used. This is convenient for daily life, but also leads to effects such as transient high-voltage flickers and high-frequency noise during equipment startup [1,2], damaging the electronic components. Harmonics are also generated by the nonlinear aspects of loads, resulting in power quality degradation and increased grid losses [3,4]. Therefore, harmonic elimination in power systems has gradually received widespread attention [5]. Ref. [6] proposed a generalized instantaneous reactive power theory for reactive and harmonic current compensation, based on which an immediate reactive power compensator was proposed that is able to eliminate the harmonic currents caused by the instantaneous imaginary power [7]. APFs have become an important means of improving grids’ power quality due to their excellent immediate compensation performance and rapid dynamic response, as well as their ability to effectively suppress noise and interference [8,9].

APFs are an effective approach to achieve harmonic suppression and dynamic reactive power compensation. As such, they have been widely researched and applied due to their outstanding advantages [10,11]. Ref. [12] proposed an EMTP-based active filter model to analyze the performance of active filters. In reality, APF performance is dependent on the control strategies adopted, which include P-Q theory, repetitive control, neural network control, and model predictive control, among others.

Refs. [13,14] presented a P-Q suppression harmonic control method based on instantaneous reactive power theory (IRPT). This method extracts harmonic components from load currents, an important step to improve APF performance. Ref. [15] presented the application of a generalized robust logarithmic family (GRLF) framework-derived least absolute difference (LAD) algorithm implemented in active power filters (APFs) for PQ enhancement, successfully eliminating lower- and higher-order errors from harmonically rich nonlinear load currents, resulting in accurate fundamental component extraction. Ref. [16] proposed a direct-current control strategy based on the synchronous rotating coordinate system, which achieves static-free tracking control of the grid current through a PI controller. Refs. [17,18] proposed an adaptive quantum calculus-based least mean fourth (q-LMF) control for shunt APFs, which maintains the source current’s unity power factor while turning the former sinusoidal. Ref. [19] proposed a new version of the integral power theory in the α−β reference frame, combining the advantages of well-known instantaneous and integral power theories, as well as a shunt active-filtering strategy based on them. Ref. [20] presented finite-control set–model-predictive control (FCS-MPC) for hybrid APFs, improving their dynamic performance. Ref. [21] proposed an effective model-predictive control (MPC) method for SAPFs, avoiding the weighting factor and additional constraints of DC capacitor voltages in the cost function. In another study, neural network algorithms were employed in APFs to improve system performance [22]. In [23], particle swarm optimization (PSO) was proposed to improve the PI controller and enhance the dynamic response. Ref. [24] proposed a genetic algorithm (GA)-optimized fractional order proportional resonant controller (FOPRC) for SAPF current control. Ref. [25] presented a fuzzy logic controller for APFs with robust control capabilities. Meanwhile, Ref. [26] developed a Kalman filter-based super-twisting sliding mode control (SMC) for APFs. Ref. [27] proposed a global fast super-twisting terminal sliding-mode controller (GFSTTSMC) based on an adaptive recurrent Chebyshev fuzzy neural network for APFs. Finally, Ref. [28] proposed a direct power method based on modified interconnection and damping assignment passivity-based control (IDA-PBC).

Most APFs use six-switch or four-bridge arm eight-switch topologies [29]. They are complicated in terms of their control, large size, and high cost, and unfavorable for daily life applications. In recent years, the four-switch topology of three-phase converters, which has four power semiconductors instead of six switches, has attracted attention due to its advantageous simple structure and low, economical cost [30,31]. Ref. [32] proposed a novel two-leg VSI topology for a three-phase APF to reduce the cost and size. Ref. [33] presented a three-phase four-switch APF with a synchronous reference frame control strategy. The methods employed in four-switch APF systems are commonly based on P-Q theory. PLLs are included, and the a–b–c frame is transformed into a d–q–o frame through Park transformations, increasing computational demands. Ref. [34] presented a novel control method to improve APF performance with a PI plus vector PI controller; however, unbalanced capacitor voltages were not considered, and a Park transformation based on trigonometric computation was included, requiring long execution times. Ref. [35] removed two current sensors in APFs but still required a PLL for the voltage phase and employed traditional hysteresis control.

Besides common control methods, some novel control strategies have been developed to improve three-phase reduced-switch APFs’ performance. Ref. [36] presented finite-control set model-predictive current control (FCS-MPCC) for four-switch APFs, adopting a prediction algorithm instead of the traditional control technique. Meanwhile, Ref. [37] proposed a one-cycle control method for reduced-switch APFs, removing the reference current generation. This control method is quite simple compared to most strategies, but its performance worsens when grid voltages deteriorate. In the literature [38,39], double-band hysteresis and three-level hysteresis control methods are proposed for four-switch APFs, as well as a modified modulation technique based on output signal identification [40]. However, these novel control strategies are complicated to implement and increase computation.

Among them, MPCC establishes a predictive model and employs an adaptive notch filter (ANF) to obtain the control references, avoiding the PLL. However, it requires a predictive model to calculate future states and evaluate all possible switching states to select the optimal solution, which increases its computational complexity. The OCC method can be implemented without a PLL, but its performance is significantly affected by grid voltage fluctuations. The HCC method removes the PLL by regulating the compensation current in the a–b–c coordinate frame, but introduces additional control logic and parameter adjustments, increasing system complexity. Additionally, its performance is largely dependent on settings such as the hysteresis width and high and low thresholds, meaning that small changes in these parameters may affect the stability of the system.

This paper proposes a modified control strategy for three-phase four-switch APFs based on fundamental positive sequence extraction, which removes the PLL and Park transformation. The harmonic extraction is designed based on the fundamental positive sequence extraction method, while the non-locked phase loop coordinate transformation method is proposed to eliminate trigonometric calculations. The mode of quasi-PR control is adopted for current tracking, and the SVPWM is developed for a reduced-switch APF topology. This control strategy has good harmonic suppression performance, validated, alongside its feasibility, through simulations and experiments on an experimental hardware platform.

2. Topology of Three-Phase Four-Switch APFs

Figure 1 shows the topology of three-phase four-switch shunt APFs. v_sa, v_sb, and v_sc represent the three-phase voltage; i_sa, i_sb, and i_sc are the three-phase current; i_la, i_lb, and i_lc are the three-phase load current; i_a, i_b, and i_c are the current of three-phase four-switch APFs; L is the inductor; U_dc is the bus voltage; meanwhile, u_c1 and u_c2 are the capacitor voltage. The two-phase bridge arms are composed of the power device’s IGBT. Finally, S₁, S₂, S₃, and S₄ represent the switches.

3. Control Strategy for Three-Phase Four-Switch APFs

3.1. Fundamental Positive Sequence Extraction Method

The dq0 rotating coordinate detection method can effectively detect voltage harmonics, fluctuations, and distortion in the power grid, among others. However, this method involves a large number of trigonometric function transformations, making it prone to errors and a slow system response. In this paper, a control strategy for a stationary coordinate system is proposed to avoid complex coordinate transformations and remove the PLL to extract the grid voltage phase.

The commonly used method for fundamental positive sequence extraction is the synchronous reference frame (SRF). First, the three-phase load current is transformed from the a–b–c coordinate frame to the d–q coordinate frame by a Park transformation. In the d–q coordinate system, the fundamental positive sequence component of the current is transformed into a direct current, which can be extracted by filtering with a first-order low-pass filter (LPF). The transfer function of the first-order LPF is as follows:

$$H_{dq}\left(s\right)={\displaystyle \frac{{\omega }_c}{s+{\omega }_c}}$$

(1)

where ω_c is the cutting frequency in the LPF. Although the SRF method achieves good extraction results, its calculations are relatively complex and require the participation of a PLL. In contrast, converting the LPF from the rotating d–q coordinate frame to the stationary a–b–c coordinate frame, which corresponds to the stationary reference frame (StatRF) control method, is simpler and easier to implement, making it more suitable for three-phase four-switch APFs.

Figure 2 shows a fundamental positive sequence extraction scheme, wherein the three-phase current is converted from the a–b–c coordinate frame to the d–q coordinate frame. The active and reactive components of the three-phase current can be given, obtaining the fundamental components through the LPF. Finally, the fundamental components under the a–b–c coordinate frame are obtained via inverse coordinate transformation.

The equation from the a–b–c coordinate system to the α–β coordinate system is the following:

$$C_{32}={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]$$

(2)

Then, the following is true:

$$\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]$$

(3)

The transformation from the α–β coordinate frame to the d–q coordinate frame is the following:

$$\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]$$

(4)

The LPF is converted from the d–q frame to the α–β frame, represented by the following equation:

$$H_{\alpha \beta }\left(s\right)={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}\left(\begin{array}{l}{H}_{dq}\left(s+j{\omega }_o\right)\\ +H_{dq}\left(s-j{\omega }_o\right)\end{array}\right)& \left(\begin{array}{l}j{H}_{dq}\left(s+j{\omega }_o\right)\\ -j{H}_{dq}\left(s-j{\omega }_o\right)\end{array}\right)\\ \left(\begin{array}{l}-j{H}_{dq}\left(s+j{\omega }_o\right)\\ +j{H}_{dq}\left(s-j{\omega }_o\right)\end{array}\right)& \left(\begin{array}{l}{H}_{dq}\left(s+j{\omega }_o\right)\\ +H_{dq}\left(s-j{\omega }_o\right)\end{array}\right)\end{array}\right]$$

(5)

Substituting (1) into (5), H_αβ(s) is derived as follows:

$$H_{\alpha \beta }(s)=\left[\begin{array}{cc}{\displaystyle \frac{{\omega }_{c}s+{{\omega }_c}^2}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}& {\displaystyle \frac{{\omega }_c{\omega }_0}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}\\ {\displaystyle \frac{-{\omega }_c{\omega }_0}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}& {\displaystyle \frac{{\omega }_{c}s+{{\omega }_c}^2}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}\end{array}\right]$$

(6)

Then, the transfer function of the LPF in the α–β frame is given by the following:

$$\begin{array}{l}\left[\begin{array}{c}{i_{\alpha f}}^{+}\\ {i_{\beta f}}^{+}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{1}{2}}\end{array}\right]H_{\alpha \beta }(s)=\\ \left[\begin{array}{cc}{\displaystyle \frac{-{\omega }_c{\omega }_0}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}& {\displaystyle \frac{{\omega }_{c}s+{{\omega }_c}^2}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}\\ {\displaystyle \frac{\frac{1}{2}\left({\omega }_c{\omega }_0+\sqrt{3}{{\omega }_c}^2+\sqrt{3}{\omega }_{c}s\right)}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}& {\displaystyle \frac{\frac{1}{2}\left({\omega }_c{\omega }_0-{{\omega }_c}^2-{\omega }_{c}s\right)}{{\left(s+{\omega }_c\right)}^2+{{\omega }_0}^2}}\end{array}\right]\end{array}$$

(7)

The distortion current is calculated using Equation (7) to obtain the fundamental positive sequence current, that is, i_αf⁺, i_βf⁺.

3.2. Coordinate Transformation Method Without PLLs

Conventional rotary coordinate transformation requires phase-locked loops to obtain the voltage’s angular frequency information, and the error in the initial phase angle of the voltage leads to inaccurate detection signals. In this paper, the d–q components are obtained by directly calculating the rotational quantities, $\sin \left(\omega t+\varphi \right)$ and $\cos \left(\omega t+\varphi \right)$ of the transformation equation from the α–β frame to the d–q frame, ensuring that the elements in the following equation can accurately reflect the voltage phase:

$$\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]=C_{32}\left[\begin{array}{c}{u}_a\\ u_b\\ u_c\end{array}\right]=\sqrt{{\displaystyle \frac{2}{3}}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]\left[\begin{array}{c}{u}_a\\ u_b\\ u_c\end{array}\right]$$

(8)

The three-phase voltage is represented as follows:

$$\left\{\begin{array}{c}{u}_a=\sqrt{2}U\sin \theta \\ u_b=\sqrt{2}U\sin \left(\theta -{\displaystyle \frac{2}{3}}\pi \right)\\ u_c=\sqrt{2}U\sin \left(\theta +{\displaystyle \frac{2}{3}}\pi \right)\end{array}\right.$$

(9)

where U is the rms value of the grid voltage, determined as

$$\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]=C_{32}\left[\begin{array}{c}{u}_a\\ u_b\\ u_c\end{array}\right]=\left[\begin{array}{c}\sqrt{3}U\sin \theta \\ \sqrt{3}U\cos \theta \end{array}\right]$$

(10)

with

$${u_{\alpha }}^2+{u_{\beta }}^2=3U^2$$

(11)

$$U=\sqrt{{\displaystyle \frac{{u_{\alpha }}^2+{u_{\beta }}^2}{3}}}$$

(12)

Substituting the above into Equation (10) yields the following:

$$\left\{\begin{array}{c}\sin \theta ={\displaystyle \frac{u_{\alpha }}{\sqrt{3}U}}={\displaystyle \frac{u_{\alpha }}{\sqrt{{u_{\alpha }}^2+{u_{\beta }}^2}}}\\ \cos \theta ={\displaystyle \frac{u_{\beta }}{\sqrt{3}U}}={\displaystyle \frac{u_{\beta }}{\sqrt{{u_{\alpha }}^2+{u_{\beta }}^2}}}\end{array}\right.$$

(13)

Then, the active and reactive voltage components in the d–q coordinate axis are determined as follows:

$$\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{2u_{\alpha }{u}_{\beta }}{{u_{\alpha }}^2+{u_{\beta }}^2}}\\ {\displaystyle \frac{{u_{\beta }}^2-{u_{\alpha }}^2}{{u_{\alpha }}^2+{u_{\beta }}^2}}\end{array}\right]$$

(14)

Therefore, the PLL can be removed to avoid phase angle error, achieving the coordinate’s transformation without PLL.

3.3. Quasi-PR Control Strategy

The PI regulator is generally able to accurately track the reactive current because the fundamental component is converted into a DC component in the d–q synchronous rotating coordinate system. PI control always achieves a steady-state error of zero but has poor performance in AC control systems, failing to deliver the expected results.

In our methodology, a PR controller is employed, and resonant controllers are designed according to each harmonic frequency. It has high gain at the designed frequency and can regulate the grid-side current as a standard sinusoidal waveform.

The approximation PR controller transfer function is as follows:

$$G_{PR}\left(s\right)={\displaystyle \frac{1}{2}}\left[G_{PI}\left(s+j{\omega }_0\right)+G_{PI}\left(s-j{\omega }_0\right)\right]=K_p+{\displaystyle \frac{K_{i}s}{s^2+{{\omega }^2}_0}}$$

(15)

where K_p and K_r are the proportional and resonant gain coefficients, respectively, and ω₀ is the resonant frequency. Undoubtedly, the resonant element is the core of the PR controller, which has near-infinite gain at ω₀ and low gain at other frequencies, effectively eliminating interfering signals. Figure 3 shows the Bode plot of the PR controller.

The ideal PR controller is fully capable of tracking AC quantities corresponding to the frequency. However, the frequency bandwidth near the resonance frequency is too narrow, and the gain at ω₀ is too high, making the system not stable enough. The PR controller would not be able to work accurately at the frequency in question with slight shifts in the AC signal. Although the bandwidth could be increased via adjustments, this would significantly increase the gain and phase changes, causing system instability. To improve the ability of the PR controller, the improved transfer function is as follows:

$$G_{PR}\left(s\right)=K_p+{\displaystyle \frac{2K_r{\omega }_{c}s}{s^2+2{\omega }_{c}s+{{\omega }^2}_0}}$$

(16)

where ω_c is the cut-off frequency. Meanwhile, the gain function of the quasi-PR controller is as follows:

$${\left|G_{PR}\left(s\right)\right|}_{s=j{\omega }_0}=K_p+K_r$$

(17)

According to the above, when the input signal frequency is ω₀, the gain is $K_p+K_r$. Figure 4 shows the plot of the quasi-PR controller.

Separate comparisons were carried out with different K_p, K_r, and ω_c to analyze the effect of diverse parameters on the controller, as shown in Figure 5.

K_p increases alongside the overall gain and can regulate the system dynamics. According to the transfer function, the PR controller’s gain at ω₀ is no longer infinite but rather $20\log \left(K_p+K_r\right)$. When K_p and ω_c are the same, the controller’s gain increases alongside K_r and the bandwidth near the resonant frequency. Therefore, K_r is adjusted to allow the quasi-PR controller to have sufficiently large gain to achieve a steady error of zero.

When K_p and K_r are the same, ω_c increases to improve the bandwidth and the response speed. However, this may introduce high-frequency noise. There is a frequency fluctuation of ±0.5 Hz in the power grid, and the corresponding cut-off frequency is ω_cmax = 3.14 rad/s. The value of ω_c is generally set to 0.628 rad/s.

3.4. DC Voltage Control Strategy

We studied the stabilization of the DC bus voltage’s impact on APF compensation performance. The DC voltage should be regulated and controlled to guarantee APF operation, according to the principle shown in Figure 6.

The error between the reference value and the measure of the DC voltage is fed into the PI controller. The component I_dref, the PI controller output, is added to the active component, obtaining the compensation current to keep the DC voltage stable.

DC midpoint voltage deviations affect the voltage balance of the APF capacitor, worsening the harmonic compensation performance and threatening normal APF operation. To avoid this problem, DC midpoint voltage deviations should be suppressed effectively.

The relationship between the DC current and the midpoint voltage is the following:

$$\left\{\begin{array}{c}{i}_{C1}=C1{\displaystyle \frac{d\left(U_{C1}-U_n\right)}{dt}}\\ i_{C2}=C2{\displaystyle \frac{d\left(U_n-U_{C2}\right)}{dt}}\end{array}\right.$$

(18)

where the current on the DC side is i_C1 and i_C2, and the midpoint voltage is U_n.

Then, if $C_1=C_2=C$, we obtain the following:

$${\displaystyle \frac{d{U}_n}{dt}}=-{\displaystyle \frac{i_{C1}-i_{C2}}{2C}}$$

(19)

The equation, derived by Laplace transformation, is the following:

$$\left\{\begin{array}{c}{U}_n={\displaystyle \frac{i_{C1}-i_{C2}}{2sC}}\\ G_c\left(s\right)={\displaystyle \frac{1}{2sC}}\end{array}\right.$$

(20)

To achieve satisfactory anti-interference capabilities, PI control is employed, and the compensation method is shown in Figure 7.

The overall control scheme is shown in Figure 8.

4. SVPWM Method for the Three-Phase Four-Switch Topology

The space vector diagram for the three-phase four-switch topology is shown in Figure 9. The inverse tangent function of the target voltage vector U_ref can be used as the sector division criterion of the SVPWM algorithm for the three-phase four-switch APF.

$$\delta =\arctan \left(u_{\beta }/u_{\alpha }\right)$$

(21)

where δ is the angle of the target voltage vector. The sector of the target voltage vector is chosen according to δ. The interval of δ is within (−π/2, +π/2) and does not correspond to the angle of a complete cycle.

To accurately determine the target voltage sector, the target voltages U₁, U₂, U₃, and U₄ are rotated counterclockwise by 2π/3 radians using rotating coordinates, thus obtaining a new coordinate system. Then, according to the sector δ quadrant, the voltage sector is obtained as shown in Figure 10.

The rotated coordinate equation is as follows:

$$\left[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}\right]=\left[\begin{array}{cc}\cos \left(-2\pi /3\right)& -\sin \left(-2\pi /3\right)\\ \sin \left(2\pi /3\right)& \cos \left(2\pi /3\right)\end{array}\right]\left[\begin{array}{c}{U^{\prime }}_{\alpha }\\ {U^{\prime }}_{\beta }\end{array}\right]$$

(22)

The state of the voltage vector in the new coordinate frame is shown in

Table 1. Voltage vector states in the rotated coordinate system.

Voltage Vector	${\mathit{S}}_{\mathit{a}}$	${\mathit{S}}_{\mathit{a}}$	${\mathit{U}}_{\mathit{s}}$
$U_1$	0	0	$\left(U_{dc}/\sqrt{6}\right)e^{j0}$
$U_2$	1	0	$\left(U_{dc}/\sqrt{2}\right)e^{j\left(\pi /2\right)}$
$U_3$	1	1	$\left(U_{dc}/\sqrt{6}\right)e^{j\pi }$
$U_4$	0	1	$\left(U_{dc}/\sqrt{2}\right)e^{j\left(2\pi /3\right)}$

below.

If N = 1*A + 2*B + 3*C + 4*D and $0\le \delta <{\displaystyle \frac{\pi }{2}}$, then A = 1; otherwise, A = 0. Similarly, if ${\displaystyle \frac{\pi }{2}}\le \delta <\pi$, then B = 1; otherwise, B = 0. If $\pi \le \delta <{\displaystyle \frac{3\pi }{2}}$, then C = 1; otherwise, C = 0. If ${\displaystyle \frac{3\pi }{2}}\le \delta <2\pi$, then D = 1; otherwise, D = 0. The relationship between N and the sectors is shown in

Table 2. Relationship between N and the sectors.

N	Disk Sector (Computing)
1	1
2	2
3	3
4	4

.

According to the analysis, the target vector U_ref should be synthesized by the base voltage vectors U₁, U₂, U₃, and U₄, meeting the following equation:

$$\left|U_{ref}\right|T_s=\left|U_1\right|T_1+\left|U_2\right|T_2+\left|U_3\right|T_3+\left|U_4\right|T_4$$

(23)

where T₁, T₂, T₃, and T₄ are the work time of the base voltage vectors U₁, U₂, U₃, and U₄, respectively, and T_S is the switching cycle time. The following is true:

$$T_s=T_1+T_2+T_3+T_4$$

(24)

A zero vector is needed to synthesize the target vector. However, in a three-phase four-switch circuit, there are only four basic voltage vectors and no zero vector. The two smallest basic voltage vectors, i.e., $U_1$ and $U_3$, are taken to synthesize the zero vector, making up for its absence. If we take the first sector as an example, as shown in Figure 11, U_ref, the target voltage, can be expressed as follows:

$$\left\{\begin{array}{c}\left|U_{ref}\right|\left|\cos \delta \right|T_s={T_1}^1\left|U_1\right|\\ \left|U_{ref}\right|\left|\sin \delta \right|T_s=T_2\left|U_2\right|\end{array}\right.$$

(25)

where δ is the angle between the target voltage U_ref and the base voltage vector U₁, ${T_1}^1$ is the time taken to generate U_ref by U₁, and T₂ is the time taken to generate U_ref by U₂.

As mentioned above, using U₁ and U₃ to synthesize an equivalent zero vector, the two vectors are equal in magnitude, and the relationship between the times can be introduced as follows:

$$\left\{\begin{array}{c}{T_1}^2=T_3\\ T_0={T_1}^2+T_3\\ T_0=T_s-{T_1}^1-T_2\end{array}\right.$$

(26)

where T₃ is the time taken to synthesize the zero vector by U₃, ${T_1}^2$ is the time taken to synthesize the zero vector by U₁, and T₀ is the total work time taken to synthesize the equivalent zero vector.

Combining Equation (25) with (26), the time is derived as follows:

$$\left\{\begin{array}{c}{T}_1={\displaystyle \frac{T_s-T_2+\frac{\left|U_{ref}\right|}{\left|U_1\right|}\left|\cos \delta \right|T_s}{2}}\\ T_2={\displaystyle \frac{\left|U_{ref}\right|}{\left|U_2\right|}}\left|\sin \delta \right|T_s\\ T_3={\displaystyle \frac{T_s-T_2-\frac{\left|U_{ref}\right|}{\left|U_1\right|}\left|\cos \delta \right|T_s}{2}}\end{array}\right.$$

(27)

The operation time of each base voltage vector in the first sector can be obtained from Equation (27). According to the principle of vector concentration, the effective vectors work at the start and end of the control period, while the zero vector works in the middle. In sector one, the target vector U_ref is synthesized by U₁, U₂, and U₃. U₁ and U₂ synthesize the effective vector, while U₁ and U₃ synthesize the zero vector. The working order is U₂, U₁, U₃, U₁, and U₂ and the corresponding action times are T₂/2, T₁/2, T₃, T₁/2, and T₂/2. The voltage vector’s working time in the other three sectors is same as in the first sector. The voltage vector is obtained in the sectors as shown in

Table 3. Voltage vector working time.

Sector	Order	Time
First sector	U2, U1, U3, U1, U2	T2/2, T1/2, T3, T1/2, T2/2
Second sector	U2, U3, U1, U3, U2	T2/2, T3/2, T1, T3/2, T2/2
Third sector	U4, U3, U1, U3, U4	T4/2, T3/2, T1, T3/2, T4/2
Fourth sector	U4, U1, U3, U1, U4	T4/2, T1/2, T3, T1/2, T4/2

.

To maintain the SVPWM algorithm’s symmetry and smoothness, the time of the switching mode transition within the first sector can be presented as shown in Figure 12.

At $0-t_1$, S_a = 1 and S_b = 0. S₁ and S₄ are on, while S₂ and S₃ are off. At $t_1-t_2$, S_a = 0 and S_b = 0. S₃ and S₄ are on, while S₁ and S₂ are off. S₁ and S₃ switch at t₁. At $t_2-t_3$, S_a = 1 and S_b = 1. S₁ and S₂ are on, while S₃ and S₄ are off. S₁ and S₃ switch at t₂. Therefore, the point of switching mode shift within the first sector is as follows:

$$\left\{\begin{array}{c}\begin{array}{l}{T}_{a1}={\displaystyle \frac{T_2}{2}}\\ T_{a2}={\displaystyle \frac{T_1+T_2}{2}}\end{array}\\ T_b={\displaystyle \frac{T_1+T_2}{2}}\end{array}\right.$$

(28)

where T_a1 and T_a2 are the switching times for switches S₁ and S₃ in the bridge arm, while T_b is that for switches S₂ and S₄ in the bridge arm. The basic voltage vector working time can be obtained from the calculation of the first sector, as shown in

Table 4. Switching time in different sectors.

Sector	First Sector	Second Sector	Third Sector	Fourth Sector
$T_a$	${\displaystyle \frac{T_2}{2}}$, ${\displaystyle \frac{T_1+T_2}{2}}$	${\displaystyle \frac{T_2+T_3}{2}}$	${\displaystyle \frac{T_4}{2}}$, ${\displaystyle \frac{T_3+T_4}{2}}$	${\displaystyle \frac{T_1+T_4}{2}}$
$T_b$	${\displaystyle \frac{T_1+T_2}{2}}$	${\displaystyle \frac{T_2}{2}}$, ${\displaystyle \frac{T_2+T_3}{2}}$	${\displaystyle \frac{T_3+T_4}{2}}$	${\displaystyle \frac{T_4}{2}}$, ${\displaystyle \frac{T_1+T_4}{2}}$

.

5. Simulation Analysis

To verify the method proposed in this paper, we carried out a simulation of the three-phase four-switch APF without PLL based on fundamental positive sequence extraction. The nonlinear load was composed of the uncontrollable rectifier bridge with a resistor and an inductor. The relevant simulation parameters are summarized in

Table 5. Simulation parameters.

Parameter	Value
Grid voltage and frequency	380 V, 50 Hz
Nonlinear load	5 Ω, 2 mH
Filter inductors	2 mH
DC voltage	1600 V
Capacitance on DC side	5000 µF

.

Figure 13 shows the whole DC voltage when the system is running. The bus voltage fluctuation on the DC side is very small, maintaining a maximum deviation value within 2 V. The whole voltage can remain stable around 1600 V.

The two-split capacitance voltage on the DC side is shown in Figure 14. The voltage of the two capacitors is stable at about 800 V, maintaining maximum fluctuation deviations within ±2 V.

The source current waveform is shown in Figure 15. The three-phase four-switch active power filter starts working at 0.05 s, prior to which the source current is seriously distorted. Under the proposed control strategy, the harmonic is compensated, and the source current turns into symmetrical sine waveforms.

FFT analysis of the source current is shown in Figure 16. The total harmonic distortion (THD) is 24.72% without a filter. The harmonic is significantly suppressed with a working active filter, and THD is reduced to 3.86%. The proposed method satisfactorily eliminates the harmonic in the power grid.

6. Experimental Results

To verify the above analysis and the proposed control strategy, a dSPACE1004-based experimental platform, shown in Figure 17, was established. The hardware comprised two insulated gate bipolar transistors (FF150R12RT4) and two split capacitors. The experimental parameters are presented in

Table 6. Experimental parameters.

Parameter	Value
Grid voltage and frequency	50 V,50 Hz
Nonlinear load	5 Ω, 2 mH
Filter inductors	3 mH
DC voltage	800 V
Capacitance on DC side	5000 µF

.

Figure 18 shows the waveform of the three-phase voltage and current before compensation, along with the i_sa Fourier spectrum. It can thus be seen that the source current primarily contains fifth and seventh harmonics. Figure 19 presents the waveforms of the three-phase voltages and currents after compensation, along with the i_sa Fourier spectrum. Compared to Figure 18, the source current is regulated into a sinusoidal waveform, with suppressed harmonic components.

Figure 20 shows the experimental waveform of the DC voltage, averaging 796 V.

To investigate the dynamic response to the start and load variations, experiments involving startup and sudden load changes were conducted. Figure 21 shows the transient process during APF startup, with i_fa responding rapidly to compensate the harmonic currents. Additionally, during the startup process, i_sa remains stable without a startup shock. When APF operates in a steady state, the loads change, and the transient process is shown in Figure 22. It can be seen that the proposed control strategy has a good response to load variations, and i_sa is regulated without a shock, rapidly returning to a steady state.

To investigate computational efficiency and validate execution time improvements, a DSP TMS320F28377D-based experimental platform was established, with TMS320F28377D able to provide a signal processing performance of 200 MHz. The computation cycles and execution times of the SRF and proposed method in DSP were measured in Code Composer Studio using the clock tool: the former’s core algorithm required roughly 6106 clock cycles, while the latter only needed approximately 4362. Assuming that each instruction cycle in DSP was 5 ns, the SRF method required 30.53μs to run, whereas the proposed method only needed approximately 21.81 μs. The proposed method was thus shown to have a shorter computation time and less computations.

7. Conclusions

APFs are an effective approach to solving harmonic problems in power grids, and their control strategy plays a crucial role in improving the power quality. The three-phase four-switch topology of APFs has attracted more attention due to its fewer semiconductors and lower costs. A modified control strategy for a three-phase four-switch APF based on fundamental positive sequence extraction was proposed in this paper, which removes the PLL and rotating coordinate transformation present in traditional control strategies. We introduced harmonic extraction based on the fundamental positive sequence extraction method, while non-locked phase loop coordinate transformation was developed to eliminate trigonometric calculations. The proposed APF and control strategy have satisfactory performance in suppressing the harmonic, which was confirmed, alongside their effectiveness, via simulation and experiments.

The proposed method offers excellent harmonic compensation for three-phase four-switch APFs, reducing two switches and decreasing computational complexity compared to conventional APFs and control strategies. Harmonic extraction based on fundamental positive sequence extraction guarantees the accuracy of the APF reference current. The coordinate transformation method without a phase-locked loop includes accuracy calculations. The APF’s DC voltage remains constant and stable, while that of the two split capacitors is balanced and their difference is significantly reduced. Therefore, three-phase four-switch APFs with the proposed control scheme are a promising alternative.

However, when grid voltages fluctuate, the fundamental positive sequence extraction method proposed in the paper may misjudge or miss certain fundamental components, resulting in inaccurate extraction results. It is particularly prone to occurring when the frequency fluctuations are close to the fundamental frequency, further impacting the extraction accuracy. In addition, when the grid voltage is distorted, conventional PLL-based methods can employ a synchronous reference frame phase-locked loop (SRF-PLL) or a decoupled double-synchronous reference frame phase-locked loop (DDRF-PLL) to improve performance, while the coordinate transformation accuracy could be decreased in the proposed method.