Distribution grid agents are obliged to exchange and disclose their states explicitly to neighboring regions to enable distributed optimal power flow dispatch. However, the states contain sensitive information of individual agents, such as voltage and current measurements. These measurements can be inferred by adversaries, such as other participating agents or eavesdroppers. To address the issue, we propose a privacy-preserving distributed optimal power flow (OPF) algorithm based on partially homomorphic encryption (PHE). First of all, we exploit the alternating direction method of multipliers (ADMM) to solve the OPF in a distributed fashion. In this way, the dual update of ADMM can be encrypted by PHE. We further relax the augmented term of the primal update of ADMM with the $ell_1$-norm regularization. In addition, we transform the relaxed ADMM with the $ell_1$-norm regularization to a semidefinite program (SDP), and prove that this transformation is exact. The SDP can be solved locally with only the sign messages from neighboring agents, which preserves the privacy of the primal update. At last, we strictly prove the privacy preservation guarantee of the proposed algorithm. Numerical case studies validate the effectiveness and exactness of the proposed approach.
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