The instantaneous p-q theory, also known as the instantaneous active and reactive power theory, is widely used for extracting reference harmonic currents in three-phase systems. Its main advantage lies in its ability to analyze power in the time domain under both steady-state and transient conditions, without requiring prior knowledge of system parameters.

The instantaneous p-q theory is used to calculate reference harmonic currents. The three-phase voltages and currents are first transformed from the abc frame to the α-β frame using Clarke’s transformation:

$\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{c}-\frac{1}{2}\\ \frac{\sqrt{3}}{2}\end{array}& \begin{array}{c}-\frac{1}{2}\\ -\frac{\sqrt{3}}{2}\end{array}\end{array}\right]\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$

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

The instantaneous active (p) and reactive (q) power are then computed as:

$\left[\begin{array}{c}p\\ q\end{array}\right]=\left[\begin{array}{cc}{v}_{\alpha }& v_{\beta }\\ -v_{\beta }& v_{\alpha }\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]$

The harmonic components of the load current are identified from p and q, and their inverse transformation provides the reference compensating currents. In summary, the p-q theory enables real-time identification of harmonic components and provides the necessary reference signals for the SAPFs to inject equal and opposite harmonic currents, thereby restoring the source current to a sinusoidal form.

Figure 3 shows the p-q algorithm in MATLAB Simulink modelling. The block diagram is based on the p–q theory, which works in the stationary α-β-0 reference frame. The three-phase voltages (Vs_abc) and currents (Is_abc) are first transformed from the abc frame to αβ0 coordinates using Clarke’s transformation.

This transformation converts the three-phase quantities into two orthogonal components (α and β) that simplify instantaneous power calculations. Once in the α-β domain, the instantaneous active power (p) and reactive power (q) are calculated using algebraic equations involving voltages and currents. The multiplication blocks represent the cross-products of voltages and currents. After obtaining p and q, the signals pass through filters to separate the average components (which represent fundamental power) and the oscillating components (which represent harmonics and reactive power). The harmonic and reactive components of power are undesirable, so the reference compensating currents are calculated based on them. These reference currents are then transformed back to the abc frame and fed to the SAPFs, which generates compensating currents to cancel out harmonics and reactive power from the load current.

The DC-link capacitor is a critical component of the Voltage Source Inverter (VSI) in the Shunt Active Power Filter (SAPF). It acts as an energy storage element that supplies the required compensating current during operation. However, the capacitor voltage tends to fluctuate due to load variations, inverter switching, and harmonic compensation.

If the DC-link voltage is not properly regulated, the SAPF cannot inject the correct compensating current, leading to poor harmonic mitigation and even system instability.

To maintain a constant DC-link voltage, a PI controller is used. The error between the reference DC voltage ( $V_{dc}^{\ast }$) and the actual voltage (V_dc) is processed by the PI controller

$u\left(t\right)=K_{p}e\left(t\right)+K_i{\displaystyle \int e\left(t\right)\text{d}t}$

where $e\left(t\right)=V_{dc}^{\ast }-V_{dc}$.

This ensures that the capacitor voltage remains stable during operation, improving the dynamic response of the SAPFs.

Figure 4 shows the modelling PI controller in MATLAB Simulink. In the simulation, the DC-link reference voltage was set to 840 V. The PI controller gains (K_p and K_i) were tuned to achieve a fast response, minimum overshoot, and stable operation. The results showed that the PI controller maintained the DC-link voltage within a narrow range around the reference value, thereby ensuring smooth and reliable operation of the SAPF.

The PI controller gains were tuned using a trial-and-error approach to achieve a compromise between fast transient response and minimal overshoot. The selected values were K_p = 0.4 and K_i = 20. Simulation results show that the controller maintained stable DC-link voltage regulation under different load conditions. Sensitivity analysis showed that small deviations (±15%) in the controller parameters did not significantly affect system stability, confirming robustness.

The hysteresis controller compares the actual source current with the reference compensating current. Hysteresis current control was selected in this work due to its simple structure and fast dynamic response. Unlike fixed-frequency PWM methods such as Space Vector PWM (SVPWM), hysteresis control results in variable switching frequency, which increases device stress but ensures accurate current tracking and fast compensation. The trade-off between switching losses and dynamic performance makes hysteresis suitable for prototype and research-level SAPF systems.

The switching signals for the VSI are generated when the error exceeds a defined hysteresis band.

This method is widely used due to its simplicity, fast response, and accurate current tracking.

The Hysteresis Current Control (HCC) method shown in Figure 5 works as a fast and straightforward control strategy for current regulation in a Voltage Source Inverter (VSI) used in a Shunt Active Power Filter (SAPFs). In this method, the reference compensating current (I*) is generated by the control algorithm (such as p-q theory), while the actual source current (Is) is continuously measured. Both are fed into a comparator that calculates the error between the actual current and the reference. This error is then checked against a predefined hysteresis band.

This creates a switching pattern that forces the source current to remain within the hysteresis band around the reference current. In essence, the hysteresis controller ensures that the inverter generates exactly the right amount of compensating current, so the source current remains sinusoidal, and harmonics are eliminated.

The SAPFs system is tested under steady-state and dynamic load changes to evaluate THD reduction performance.

To ensure reproducibility of the proposed SAPF model, the key system parameters used in the MATLAB/Simulink simulations are listed in Table 1 . These parameters include the source characteristics, filter inductor, DC-link capacitor, reference DC voltage, switching devices, hysteresis band, and sampling time.

Figure 6 shows the source current waveform before and after applying SAPFs of non-linear load. Without the SAPFs, the source current drawn by nonlinear loads was highly distorted due to the presence of harmonics. After the SAPFs was activated, the compensated source current became almost sinusoidal. This demonstrates that the SAPFs successfully injected compensating currents, cancelling the harmonic components generated by the nonlinear load.

This confirms the effectiveness of the p-q theory for harmonic extraction and the hysteresis controller for fast current tracking.

Figure 7 shows how the THD of source current (non-linear Load R) drops from 28.46% before SAPFs to 3.24% after SAPFs.

Figure 8 shows the FFT Spectrum Source Current before SAPFs. The FFT spectrum plots provide a clear illustration of how the Shunt Active Power Filter (SAPFs) mitigates harmonics from the current source. Before the SAPFs are applied, the spectrum shows the fundamental frequency at 50 Hz along with significant harmonic peaks, particularly at 150 Hz (3rd harmonic) and 250 Hz (5th harmonic). These strong harmonic components indicate distortion in the source current waveform, which contributes to a high Total Harmonic Distortion (THD) level.

Figure 9 shows the FFT spectrum source current after SAPFs. After the SAPF is connected, the spectrum changes drastically. The waveform becomes nearly sinusoidal, and the harmonic magnitudes at higher frequencies are greatly reduced. The dominant peak remains at the 50 Hz fundamental, while the 3rd and 5th harmonics are suppressed to minimal levels. This reduction demonstrates the effectiveness of the SAPF in eliminating unwanted harmonics, improving power quality, and ensuring that the source current closely follows an ideal sinusoidal waveform.

The Total Harmonic Distortion (THD) analysis was performed to evaluate the effectiveness of the Shunt Active Power Filter (SAPF) under various non-linear load conditions.

Table 2 summarizes the THD values that were recorded for different nonlinear load conditions. For all tested cases—including resistive (R), resistive-inductive (RL), resistive-capacitive (RC), resistive-inductive-capacitive (RLC), and unbalanced loads—the source current exhibited severe distortion before the SAPF was applied, with THD levels ranging between 28% and 29%. Such high distortion levels significantly exceed the IEEE-519 recommended limit of 5%, confirming the need for harmonic mitigation.

Figure 10 shows the comparison of THD values before and after SAPFs under different nonlinear load conditions. After the SAPF was connected, the THD values dropped drastically across all load types. For the R load, the THD reduced from 28.46% to 3.24%; for RL, from 28.12% to 3.49%; for RC, from 28.71% to 3.33%; and for RLC, from 29.02% to 3.68%. Similarly, for unbalanced resistive load conditions, THD decreased from 20.74% to 3.12%. In all cases, the post-SAPF THD values were consistently below 5%, thereby meeting the IEEE-519 standard requirements.

This outcome demonstrates that the SAPF, controlled using the p-q theory and hysteresis current control method, successfully mitigates current harmonics and restores the source current to a near-sinusoidal form. The harmonic compensation is both fast and effective, with the filter responding almost instantly to load disturbances. Consequently, the SAPF not only improves power quality but also ensures stable and efficient operation of the power system.

To evaluate the dynamic performance of the proposed SAPF, step load change tests were carried out in the MATLAB/Simulink model. The dynamic metrics considered include settling time and peak overshoot, which are standard indicators of control system response. Table 3 summarizes the results of SAPFs dynamic response for different nonlinear load conditions.

The results indicate that the SAPF responds within approximately 8 - 10 ms after a step load change, with peak overshoot limited to less than 6% in all cases. This confirms the “fast response” capability claimed for the system, primarily due to the simplicity and rapid action of the hysteresis current controller. The short settling time demonstrates that the SAPF can effectively compensate for sudden disturbances in real time, ensuring stable and reliable power quality improvement.