Smart grids are being increasingly deployed worldwide, as they constitute the electricity grid of the future, providing bidirectional communication between households. One of their main potential applications is the peer-to-peer (P2P) energy trading market, which promises users better electricity prices and higher incentives to produce renewable energy. However, most P2P markets require users to submit energy bids/offers in advance, which cannot account for unexpected surpluses of energy consumption/production. Moreover, the fine-grained metering information used in calculating and settling bills/rewards is inherently sensitive and must be protected in conformity with existing privacy regulations. To address these issues, this report proposes a novel privacy-preserving billing and settlements protocol, PPBSP, for use in local energy markets with imperfect bid-offer fulfillment, which only uses homomorphically encrypted versions of the half-hourly user consumption data. PPBSP also supports various cost-sharing mechanisms among market participants, including two new and improved methods of proportionally redistributing the cost of maintaining the balance of the grid in a fair manner. An informal privacy analysis is performed, highlighting the privacy-enhancing characteristics of the protocol, which include metering data and bill confidentiality. PPBSP is also evaluated in terms of computation cost and communication overhead, demonstrating its efficiency and feasibility for markets with varying sizes.