## 3 Control Strategies
### Advanced Maximal Overlap DWPT (AMDWPT)  
The wavelet coefficients in Equations (18) and (19) are calculated in alternating the samplings. Maintaining uniformity in frequency bands, the proposed technique decomposes the input signal into different levels using low and high pass filters. However, in contrast to the DWPT, there is no downsampling by a factor of two in maximum overlap discrete wavelet packet transform (MODWPT) (time-invariant transform). In the reconstruction, the decomposition coefficients are convolved to the reverse low and high filters in order to reconstruct the original signal. Figure 6b illustrates the decomposition procedure of the input signal \(x\) using a two-level decomposition tree of the proposed technique. At any level \(j\), the decomposition and synthesis coefficients [31; 32] are derived as:  
\[s_{j}^{2z}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}g(n)s_{j-1}^{z }(k-n) \tag{20}\]  
\[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]  
\[a_{j-1}^{2z}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}g(n)s_{j}^{ 2z}(n-k) \tag{22}\]  
\[a_{j-1}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j}^ {2z+1}(n-k) \tag{23}\]  
But the convolution at a level \(j-1\) requires a partial modification in Equations (20) and (21) and is provided in Equations (24) and (25) as:  
\[s_{j}^{2z}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{l=0}^{L-1}g(l)s_{j-1}^{z}(k+l-\rho+1) \tag{24}\]  
\[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{l=0}^{L-1}h(l)s_{j-1}^{z}(k+l- \rho+1) \tag{25}\]  
Here, \(\rho\) and \(k\) represents the filter length and current sampling. For an experimental observation, \(k\) is assumed to be current sampling, which implies zero samples related to an index above \(1/f_{s}\), moreover, the complexity in calculation has to be smaller than \(1/f_{s}\) seconds, \(f_{s}\) is the sampling rate. The reconstruction of the original signal is obtained from Equations (26) and (27) as follows:  
\[a_{j-1}^{2z}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{l=0}^{L-1}g(l)s_{j}^{2z}(k-1) \tag{26}\]  
\[a_{j-1}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{l=0}^{L-1}h(l)s_{j}^{z}(k-1) \tag{27}\]  
The coefficients in the above equations are suitable for synthesizing the original signal \(x\) as:  
\[s_{j}^{z}(k)=a_{j}^{2z}(k)+s_{j}^{2z+1}(k) \tag{28}\]  
\[x=s_{0}^{0} \tag{29}\]The root mean square (RMS) voltages and currents are the root square of the mean square the fundamental and harmonic components, respectively. Therefore, for the proposed technique, the discrete voltage and current profiles are expressed as:  
\[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]0 \[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]1  
where, the node zero elements must involve the fundamental frequency of the original signal (f = 50 Hz). The RMS current \(I(k)\) is expressed as:  
\[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]2 \[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]3  
Similarly, the RMS voltage \(V(k)\) is expressed as:  
\[s_{j}^{2z+1}(k)=\frac{1}{\sqrt{2}}\!\!\!\sum_{n=-\infty}^{+\infty}h(n)s_{j-1}^{ z}(k-n) \tag{21}\]4  
where, \(I_{j}^{0}(k)\) and \(V_{j}^{0}(k)\) denotes the RMS current and voltage of the lowest frequency band at node zero and level \(j\), whereas the term \(I_{j}^{z}(k)\) and \(V_{j}^{z}(k)\) represents the RMS current and voltage at the node \(z=0\). Based on the IEEE standard, the overall variation of a harmonic wave from its fundamental is measured using the THD value of voltage and current, which can be expressed as:  
\[ \begin{split} THD_{V}(k)&=\frac{\sum_{z=1}^{\rho-1 }\left(V_{j}^{z}(k)\right)^{2}}{V_{j}^{0}(k)}\\ THD_{I}(k)&=\frac{\sum_{z=1}^{\rho-1}\left(\tilde{ \tau}_{j}(k)\right)^{2}}{\tilde{I}_{j}^{0}(k)}\end{split}  \tag{35}\]