  We consider the $Y$-systems satisfied by the $A_1^{(1)}$, $A_2^{(1)}$,
$A_2^{(2)}$ vertex and loop models at roots of unity with twisted boundary
conditions on the cylinder. The vertex models are the 6-, 15- and
Izergin-Korepin 19-vertex models respectively. The corresponding loop models
are the dense, fully packed and dilute Temperley-Lieb loop models respectively.
For all three models, our focus is on roots of unity values of $e^{i\lambda}$
with the crossing parameter $\lambda$ corresponding to the principal and dual
series of these models. Converting the known functional equations to nonlinear
integral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations,
we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to the
groundstate eigenvalue following the methods of Kl\"umper and Pearce. The
resulting expressions for $c-24\Delta$, where $c$ is the central charge and
$\Delta$ is the conformal weight associated with the groundstate, are
simplified using various dilogarithm identities. Our analytic results are in
agreement with previous results obtained by different methods and are new for
the dual series of the $A_2^{(1)}$ model.
