The decarbonization of transport and heating will introduce uncertain smart appliance growth in the power system, which fundamentally challenges traditional network pricing. In this paper, a new long-term distribution network charging is proposed to accommodate uncertain load growth. Instead of using fixed a load growth rate (LGR), it adopts a fuzzy model, developed based on a set of projected deterministic LGRs and confidence levels. This fuzzy model is incorporated into the pricing model through $ {\alpha } $ -cut intervals. In order to improve computational efficiency, an analytical pricing approach is introduced. The vertex extension approach is used to build charge membership functions. Thereafter, a common defuzzification approach, center of gravity, is employed to defuzzify membership functions in order to generate deterministic charges. The new approach is benchmarked with two existing standard charging methods on a practical U.K. high-voltage distribution system. Results show that it is effective in capturing the uncertainty in load growth.