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Projected Gradient Descent on Riemannian Manifolds with Applications to Online Power System Optimization


A. Hauswirth, S. Bolognani, G. Hug, F. Dörfler

Allerton Conference on Communication, Control, and Computing, pp. 8

Motivated by online optimization problems arising in nonlinear power system applications, this article concerns optimization over closed subsets of Riemannian manifolds. Compared to conventional optimization over manifolds, we explicitly consider inequality constraints that result in a fea- sible set that is itself not a smooth manifold. We propose a continuous-time projected gradient descent algorithm over the feasible set and show its well-behaved convergent behavior. Under mild assumptions on the non-degeneracy of equilibria we show that points are local minimizers if and only if they are asymptotically stable. The proposed algorithm can be implemented as a real-time feedback control law on a physical system. This approach is particularly appropriate for online load flow optimization problem in power systems, in which the state of the grid is naturally constrained to the manifold that represents the solution space to the nonlinear AC power flow equations. We specialize our approach for the case of power distribution systems that need to respect operational constraints while being economically efficient, and we illustrate the resulting closed-loop behavior in simulations.


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% Autogenerated BibTeX entry
@InProceedings { HauEtal:2016:IFA_5490,
    author={A. Hauswirth and S. Bolognani and G. Hug and F. D{\"o}rfler},
    title={{Projected Gradient Descent on Riemannian Manifolds with
	  Applications to Online Power System Optimization}},
    booktitle={Allerton Conference on Communication, Control, and
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