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Hybrid precoding is widely studied in millimetre-wave (mmWave) massive MIMO systems due to low cost as well as low power consumption. In general, there are two kinds of hybrid precoding structures: one is fully connected structure (FCS), where each radio frequency (RF) chain is connected to all antennas, and the other is partially connected structure (PCS), where each RF chain is connected to a sub-array. In this paper, we investigate the optimal hybrid precoder design problem for mmWave massive MIMO systems based on PCS, since this kind of structure is more practical for antenna deployment. We first focus on the optimization of analog precoder (AP) and propose two AP design schemes for high signalto-noise ratio (SNR) condition and low SNR condition, respectively. For each of the schemes, the original optimization problem is reformulated to single-stream optimal transmitter beamforming problem with per-antenna power constraint, which has an optimal solution. Then, the optimal digital precoder is obtained by water-filling algorithm after AP is determined. Moreover, upper bounds of the achievable data rate for the proposed schemes with closed-form expression are derived.