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We consider the control of McKean-Vlasov dynamics (or mean-field control)with probabilistic state constraints. We rely on a level-set approach whichprovides a representation of the constrained problem in terms of anunconstrained one with exact penalization and running maximum or integral cost.The method is then extended to the common noise setting. Our work extends(Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp.2568--2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71(2015), pp. 125--163) to a mean-field setting. The reformulation as anunconstrained problem is particularly suitable for the numerical resolution ofthe problem, that is achieved from an extension of a machine learning algorithmfrom (Carmona, Lauri{\`e}re, arXiv:1908.01613 to appear in Ann. Appl. Prob.,2019). A first application concerns the storage of renewable electricity in thepresence of mean-field price impact and another one focuses on a mean-varianceportfolio selection problem with probabilistic constraints on the wealth. Wealso illustrate our approach for a direct numerical resolution of the primalMarkowitz continuous-time problem without relying on duality.

A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection