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This document contains extended mathematical derivations for thecommunication-free and model-free algorithms that can actively controlconverter-connected devices, and can operate either as stand-alone or incombination with centralized optimization algorithms. We address the problem ofloss minimization in distribution grids, and we analytically prove that ourproposed algorithms reduce the total grid losses without any prior informationabout the network, requiring no communication, and based only on localmeasurements.

Extended Mathematical Derivations: Decentralized Model-free Loss Minimization in Distribution Grids with the Use of Inverters