  We propose a scalable optimization framework for estimating convex inner
approximations of the steady-state security sets. The framework is based on
Brouwer fixed point theorem applied to a fixed-point form of the power flow
equations. It establishes a certificate for the self-mapping of a polytope
region constructed around a given feasible operating point. This certificate is
based on the explicit bounds on the nonlinear terms that hold within the
self-mapped polytope. The shape of the polytope is adapted to find the largest
approximation of the steady-state security region. While the corresponding
optimization problem is nonlinear and non-convex, every feasible solution found
by local search defines a valid inner approximation. The number of variables
scales linearly with the system size, and the general framework can naturally
be applied to other nonlinear equations with affine dependence on inputs. Test
cases, with the system sizes up to $1354$ buses, are used to illustrate the
scalability of the approach. The results show that the approximated regions are
not unreasonably conservative and that they cover substantial fractions of the
true steady-state security regions for most medium-sized test cases.
