The following quotations are taken from Matthew Foreman and W. Hugh Woodin, "The generalized continuum hypothesis can fail everywhere," Ann. Math. 133 (1991), 1–35.
THEOREM. Let $\kappa$ be a supercompact cardinal with infinitely many inaccessible cardinals above $\kappa$. Then there is a partial ordering $\mathbf P$ such that in $V^{\mathbf P}$, $V_\kappa \models ZFC + \forall \lambda: 2^\lambda > \lambda^+$.
In fact we can arrange by our choice of partial orderings that $V^{\mathbf P}\models$ $\kappa$ is $\beth_n(\kappa)$-supercompact. Solovay has shown that if $\kappa$ is supercompact then $2^{\beth_\omega(\kappa)} = \beth_\omega(\kappa)^+$; hence this is near best possible. Woodin extended this result to get:
THEOREM (Woodin). If there is a supercompact cardinal then there is a model of ZFC in which $2^\kappa = \kappa^{++}$ for each cardinal $\kappa$.
The last sentence of the paper reads:
The second author has also reduced the consistency strength of "$ZFC + \forall\kappa: 2^\kappa > \kappa^+$" and "$ZFC + \forall\kappa: 2^\kappa = \kappa^{++}$" to that of a ${\mathscr P}^2(\kappa)$-hypermeasurable.
It's not clear to me if the proofs of the two theorems attributed to Woodin have ever been published.
