## 1 Introduction
The islanded condition needs accurate load sharing, which can balance fast active power mismatches. Therefore, significantly more challenges [16] are faced in islanded condition compared to grid-interactive condition. Further, it is also required to design a compensation strategy that can manage reactive, imbalance, and harmonic power-sharing by incorporating non-linear loads during the islanded condition. In this condition, the control design includes voltage and frequency control, active and reactive power control, and PQ improvement. In this work, a solar PV, WT, and BESS connected MG is developed, linked to the grid network at POI using the voltage source inverter (VSI) of the HSAPF. The HSAPF is used to compensate for both the harmonics and reactive power. The performance of the MG can be improved if it works effectively during the islanded condition. Moreover, maintaining voltage and frequency limits with the intermittency of DES is a primary concern. The controlling techniques for the proposed power system model are divided into two categories:  
1. [label=()]
2. Controlling the VSI of HSAPF, and
3. Controlling the DES units of MG using advanced maximum power point tracking (MPPT) techniques for PV and WT system. Several controlling techniques are developed for reference signal generation and harmonics estimation. The most frequently used method is the fast Fourier transform (FFT) [17; 18]. FFT is not able to gather data about time tracing of different frequencies within the signal; therefore, FFT is unable to analyze non-stationary disturbances. To overcome these issues, the wavelet-based transforms are explored. These transforms offer time-frequency data of a signal. These transforms are more accurate in determining PQ indices in the power system under non-stationary conditions. In [19], an empirical WT (EWT) technique is proposed to estimate harmonics and unbalance voltage under non-stationary conditions. In [20], continuous wavelet (CWT) and discrete wavelet (DWT) are reported. The wavelet packet transforms (WPT), discrete wavelet packet transforms (DWPT), and the dual-tree complex wavelet transform (DTCWT) are discussed in [21] and [22], respectively. Moreover, for tracking the maximum power in PV, MPPT techniques, such as perturb and observe (P&O) technique [23], adaptive variable step-size P&O [24], Fuzzy logic control (FLC), and Adaptive P&O-FLC technique [25], incremental conductance (INC) techniques [26], modified INC [27], adaptive step size INC [28], variable step size INC [29] are commonly used. In this paper, an advanced DWT, called advanced maximal overlap DWPT (AMDWPT) [30], is used for harmonics estimation, and the synchronous reference theory (SRF) is used for reference signal generation. The DWT delivers inconsistent frequency bandwidth; further, DWT is a time-variant transformation and not suitable for detecting non-stationary PQ issues. The DWPT performs better compared to DWT as it decomposes both the scaling and wavelet coefficients at a specified composition level, but still, DWPT is a time-variant transformation. Therefore, to overcome these issues, the AMDWPT is involved in this research. AMDWPT offers both uniform frequency bandwidths of the DWPT and the time-invariant property. Further, for tracking the maximum power, the learner based incremental conductance (LINC) is used. The objective of the work is to:  
* [label=\(\bullet\)]
* Mitigate the PQ problems by decreasing the total harmonic distortions (THD) below 5% (as per the Institute of Electrical and Electronics Engineers (IEEE)-519 standard) and improving the PQ with a balanced load and improving the power factor. * Maintain the power transfer system effectively before the availability of wind velocity and solar insolation level. * Effectively operate the MG in the islanded condition during the case of a fault in the supply grid and controls load power management. * Implement AMDWPT based SRF theory of managing the VSI of HSAPF and LINC for tracking the maximum power from the PV system and WT.