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Considering uncertain power outputs of distributed generations (DGs) and load fluctuations, energy storage system (ESS) represents a valuable asset to provide support for the smooth operation of active distribution networks. This paper proposes an affine arithmetic-based multi-objective optimization method for the optimal operation of ESSs in active distribution networks with uncertainties. Affine arithmetic is applied to the optimization model for handling uncertainties of DGs and loads. Two objectives are formulated with affine parameters including the minimization of total active power losses and the minimization of system voltage deviations. The affine arithmetic-based forward-backward sweep power flow is first improved by the proposed pruning strategy of noisy symbols. Then, the affine arithmetic-based non-dominated sorting genetic algorithm II (AA-NSGAII) is used to solve the multi-objective optimization problem for ESSs operation under uncertain environment. Furthermore, three types of indices with respect to convergence, diversity, and uncertainty are defined for performance analysis. Numerical studies on a modified IEEE 33-bus system with embedded DGs and ESSs show the effectiveness and superiority of the proposed method. The optimization results demonstrate that the obtained Pareto front has better convergence and lower conservativeness in comparison to the interval arithmetic-based NSGA-II. A multi-period case considering seasonality of DGs and loads is further simulated to show the applicability in real applications. [ABSTRACT FROM AUTHOR]