The non-convexity of the ACOPF problem rooted in nonlinear power flowequality constraints poses harsh challenges in solving it. Relaxationtechniques are widely used to provide an estimation of the optimal solution. Inthis paper, we propose a scalable optimization framework, on the other hand,for estimating convex inner approximations of the power flow feasibility setsbased on Brouwer fixed point theorem. The self-mapping property of fixed pointform of power flow equations is certified using the adaptive bounding ofnonlinear and uncertain terms. The resulting nonlinear optimization problem isnon-convex; however, every feasible solution defines a valid innerapproximation and the number of variables scales linearly with the system size.The framework can naturally be applied to other nonlinear equations with affinedependence on inputs. Test cases up to $1354$ buses are used to illustrate thescalability of the approach. The results show that the approximated regions arenot conservative and cover large fractions of the true feasible domains.
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