Multivariate polynomial optimization where variables and data are complex numbers is a non-deterministic polynomial-time hard problem that arises in various applications such as electric power systems, signal processing, imaging science, automatic control, and quantum mechanics. Complex numbers are typically used to model oscillatory phenomena which are omnipresent in physical systems. Thanks to recent advances in algebraic geometry, finding a global solution breaks down to solving a sequence of complex semidefinite programming relaxations that grow tighter and tighter. Well present numerical results on problems with several thousand complex variables and show that exploiting the complex structure enhances tractability. The problems consist of computing optimal power flows in the European high-voltage AC transmission network.