Let $G_{\sigma }$ be the graph obtained from a simple graph $G$ of order $n$ by adding $\sigma$ self-loops, one self-loop at each vertex in $S\subseteq V(G)$. Let ${\lambda }_1(G_{\sigma }),{\lambda }_2(G_{\sigma }),\dots ,$ ${\lambda }_n(G_{\sigma })$ be the eigenvalues of $G_{\sigma }$. The energy of $G_{\sigma }$, denoted by $\mathcal{E}(G_{\sigma })$, is defined as $\mathcal{E}(G_{\sigma })=\sum _{i=1}^n|{\lambda }_i(G_{\sigma })-\frac{\sigma }{n}|$. In this paper, using various analytic inequalities and previously established results, we derive several new lower and upper bounds on $\mathcal{E}(G_{\sigma })$.