The cosmic-ray flux of positrons is measured with high precision by the space-borne particle spectrometer AMS-02. The hypothesis that pulsar wind nebulae (PWNe) can significantly contribute to the excess of the positron ($e^+$) cosmic-ray flux has been consolidated after the observation of a $gamma$-ray emission at TeV energies of a few degree size around Geminga and Monogem PWNe. In this work we undertake massive simulations of galactic pulsars populations, adopting different distributions for their position in the Galaxy, intrinsic physical properties, pair emission models, in order to overcome the incompleteness of the ATNF catalogue. We fit the $e^+$ AMS-02 data together with a secondary component due to collisions of primary cosmic rays with the interstellar medium. We find that several mock galaxies have a pulsar population able to explain the observed $e^+$ flux, typically by few, bright sources. We determine the physical parameters of the pulsars dominating the $e^+$ flux, and assess the impact of different assumptions on radial distributions, spin-down properties, Galactic propagation scenarios and $e^+$ emission time.
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