  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.
