\frametitle{Investment optimisation: transmission}
From stationarity for $f_{\ell,t}$ we find
\begin{equation*}
0 = \frac{\d \cL}{\d f_{\ell,t}} = p_t \left(\sum_i K_{i\ell} \l_{i,t}^* - \bar{\mu}_{\ell,t}^* + \ubar{\mu}_{\ell,t}^* \right)
\end{equation*}
I.e. the KKT multipliers $\bar{\mu}_{\ell,t}^*$ or
$\ubar{\mu}_{\ell,t}^*$ are non-zero when the line $\ell$ is congested (by definition), at which time one of them equals
the price difference between the ends of the line.
For the investment we find from stationarity $0 = \frac{\d \cL}{\d F_{\ell}}$
\begin{equation*}
c_\ell = - \sum_t p_t \left(\bar{\mu}_{\ell,t}^* + \ubar{\mu}_{\ell,t}^* \right)
\end{equation*}
Remember that $\bar{\mu}_{\ell,t}^*$ and $\ubar{\mu}_{\ell,t}^*$ are only non-zero when the line is congested.
Exactly as with generation dispatch and investment, we
continue to invest in transmission until the marginal benefit of
extra transmission (i.e. extra congestion rent for extra capacity)
is equal to the marginal cost of extra transmission. This determines
the optimal investment level.