The energy of a graph $G_S$ with $n$ vertices and $\sigma$ self-loops is defined as $\epsilon (G_S)=\sum _{i=1}^n|{\lambda }_i-\frac{\sigma }{n}|$, where the eigenvalues of the adjacency matrix of $G_S$ are ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$. In this article, we have established some upper and lower bounds for the energy of such a graph. Those new bounds involve parameters like number of vertices (n), number of edges (m), number of self-loops ($\sigma )$, maximum vertex degree ($\mathrm{\Delta }$), and minimum vertex degree ($\delta$). We show that for $1\le \sigma <n$, the quantity $|{\lambda }_i-\frac{\sigma }{n}|$ is always greater than $0$, and using that fact we establish a lower bound. We have compared and concluded that the new bounds are either better than the existing bounds or incomparable to a few bounds obtained by some researchers recently.