  We study the structure of the abelian category of modules for the triplet
$W$-algebra $\mathcal{W}_{p_+,p_-}$. Using the logarithmic deformation by
Fjelstad et al.(2002), we construct logarithmic $\mathcal{W}_{p_+,p_-}$-modules
that have $L_0$ nilpotent rank three or two. By using the structure of these
logarithmic modules and the results on logarithmic Virasoro modules by
Kyt\"{o}l\"{a} and Ridout(2009), we compute ${\rm Ext}^1$ groups between
certain indecomposable modules and simple modules. Based on these ${\rm Ext}^1$
groups we determine the structure of the projective covers of all
$\mathcal{W}_{p_+,p_-}$-simple modules.
