Most of the work I'm aware of at labs for power flow problems is on stochastic optimization also, focusing mostly on MILPs.
In chemical engineering, they are interested in MINLPs, and the classic example is a mixing problem (specifically, the prototypical Haverly pooling problem), so bilinear terms come up a lot. Trilinear terms occasionally pop up, depending on the thermodynamic mixing models or reaction models used. There's also a limited amount of interest in ODE-constrained or PDE-constrained optimization; none of that work uses SDPs.
Most of the PDE-constrained optimization work I've seen (I'm specifically thinking of topology optimization) does not use SDPs. The PDE constraints could be linear, and in theory, could admit an SDP formulation depending on what the objective and remaining constraints are. In practice, engineering problems tend to be nonlinear, and yield nonconvex problems that are then solved to local optima (possibly also using multistart). Sometimes, penalty formulations are used to exclude known suboptimal local optima.
I could see it maybe being used in control theory. The small amount of work I've seen on "linear matrix inequalities" suggests that it could possibly be useful there, but control theory in industry tends to rely on tried-and-true methods rather than bleeding edge mathematical formulations, so I doubt SDPs will be used for a while until they can prove their usefulness.
There are a few SDP solvers that are okay, and they've solved problems that are pretty big for academia (last I checked was 3-4 years ago, and they were solving tens to hundreds of thousands of variables), but power flow scenarios involve much larger problems (tens of millions to billions of variables), and I don't think the solvers are there yet. I think they could get there -- there's been a fair amount of recent work on matrix-free interior-point methods that suggests that it'd be feasible to scale up SDP solvers using those techniques -- but no one's done it yet, probably because LPs, MILPs, and convex NLPs come up much more frequently and are established technologies.