# Adaptive PI Control Strategy for Optimal Microgrid Autonomous Operation
## 5 Phase of Optimization
### EBSAAB Algorithm  
AF algorithms have been extensively utilized in various applications, including linear models, noise signal cancellation, digital communications networks, and data-driven monitoring [30]. An uncertain system may be sparse in the system identification issue, indicating that some IR variables are close to zero. Several methods for identifying sparse networks have been proposed [31]. The system IR of block-sparse (BS) systems includes multiple clusters with nonzero components. Well-known BS system examples are acoustic echo devices and satellite communications [29]. In addition, several techniques have been proposed in BS systems identification to minimize convergence time and improve algorithm resilience. For such applications, stochastic gradient descent techniques have been proposed [32]. A modern EBSAAB is described in this paper. The Gaussian Markov model is used in the suggested technique to generate the unknown BS system. The suggested EBSAAB \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Gains** & **PI11** & **PI12** & **PI21** & **PI22** & **PI31** & **PI32** \\ \hline \(\mathrm{k}_{\mathrm{p}}\) & 5.52 & 3.1 & 5.52 & 3.1 & 5.52 & 3.1 \\ \(\mathrm{T}_{\mathrm{i}}\) & 0.0031 & 0.31 & 0.0031 & 0.31 & 0.0031 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The opening PI parameters. outperforms conventional adaptive algorithms such as LMS and resilient recursive algorithms in numerous ways. Among the benefits are rapid convergence, a straightforward technique, and less computing complexity. The suggested approach considers the impact of the actuating error signal, its magnitude, and the block sparse approach. In this case, the AF's actuating error (\(\mathrm{e_{a}}\)) may be expressed as follows [26]:  
\[\mathrm{e_{a}}=\mathrm{d_{a}}-\mathrm{X_{a}^{T}w_{a}} \tag{17}\]  
where \(\mathrm{d_{a}}\) is the AF's output signal, \(\mathrm{X_{a}^{T}}\) is the transposition of the input, and \(\mathrm{w_{a}}\) is the AF's weight vector. The maximum posterior estimation method is employed to calculate the AF coefficient \(\mathrm{w}\). This is accomplished by maximizing the posterior probability \(\mathrm{Pp}(\mathrm{w}\mid\mathrm{d})\), which is described as follows:  
\[\mathrm{P_{P}(w}|\mathrm{d})=\frac{\mathrm{P_{P}(w)P_{P}(d|w)}}{\sum_{w^{\prime }}\mathrm{P_{P}(w^{\prime})P_{P}(d|w^{\prime})}} \tag{18}\]  
The following equation gives the maximum posterior estimation of \(\mathrm{w}\):  
\[\mathrm{\hat{w^{\prime}}}=\mathrm{argmax}\ \mathrm{P_{P}(w)P_{P}(d|w)} \tag{19}\]  
\[\mathrm{P_{P}(d|w)}=\frac{1}{\sqrt{2\pi\mathrm{o_{a}^{2}}}}\exp\left(-\frac{ \left(\mathrm{d}-\mathrm{X^{T}w}\right)^{2}}{2\mathrm{o_{a}^{2}}}\right) \tag{20}\]  
Then Equation (17) may be applied as follows:  
\[\mathrm{\hat{w^{\prime}}}=\mathrm{argmax}(\log\mathrm{P_{P}(w)}+\log\mathrm{P _{P}(d|w)}) \tag{21}\]  
\[\mathrm{\hat{w^{\prime}}}=\mathrm{argmax}\ \mathrm{L(w)} \tag{22}\]  
\[\mathrm{L(w)}=\log\mathrm{P_{P}(w_{1})}+\sum_{i=1}^{L-1}\log(\mathrm{P_{P}(w_ {i+1}|w_{i})})\ -\frac{\left(\mathrm{d}-\mathrm{X^{T}w}\right)^{2}}{2\mathrm{o_{a}^{2}}} \tag{23}\]  
The optimum solution of Equation (22) of the BSABA relying on the steepest ascending technique is expressed as follows:  
\[\mathrm{w_{a+1}}=\mathrm{w_{a}}+\rho\frac{\partial\mathrm{L(w)}}{\partial w}|_ {w=\mathrm{w_{a}}} \tag{24}\]  
In Equation (24), \(\rho\) indicates the step size variable.