With the increasing penetration of photovoltaics in distribution networks, the adaptability of distribution network under uncertainties needs to be considered in the planning of distribution systems. In this paper, the interval arithmetic and affine arithmetic are applied to deal with uncertainties, and an affine arithmetic based bi-level multi-objective joint planning model is built, which can obtain the planning schemes with low constraint-violation risk, high reliability and strong adaptability. On this basis, a bi-level multi-objective solution methodology using affine arithmetic based non-dominated sorting genetic algorithm II is proposed, and the planning schemes that simultaneously meet economy and adaptability goals under uncertainties can be obtained. To further eliminate bad solutions and improve the solution qualities, an affine arithmetic based dominance relation weakening criterion and a deviation distance based modification method are proposed. A 24-bus test system and a 10 kV distribution system of China are used for case studies. Different uncertainty levels are compared, and a sensitivity analysis of key parameters is conducted to explore their impacts on the final planning schemes. The simulation results verify the advantages of the proposed affine arithmetic based planning method.