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This paper investigates the Electric Autonomous Dial-A-Ride Problem (E-ADARP), which consists in designing a set of minimum-cost routes that accommodates all customer requests for a fleet of Electric Autonomous Vehicles (EAVs). Problem-specific features of the E-ADARP include: (i) the employment of EAVs and a partial recharging policy; (ii) the weighted-sum objective function that minimizes the total travel time and the total excess user ride time. In this work, we propose a Deterministic Annealing (DA) algorithm and provide the first heuristic results for the static E-ADARP. Partial recharging (i) is handled by an exact route evaluation scheme of linear time complexity. To tackle (ii), we propose a new method that allows effective computations of minimum excess user ride time by introducing a fragment-based representation of paths. These two methods compose an exact and efficient optimization of excess user ride time for a generated E-ADARP route. To validate the performance of the DA algorithm, we compare our algorithm results to the best-reported Branch-and-Cut (B\&C) algorithm results on existing instances. Our algorithm provides 25 new best solutions and 45 equal solutions on 84 existing instances. To test the algorithm performance on larger-sized instances, we establish new instances with up to 8 vehicles and 96 requests, and we provide 19 new solutions for these instances. Our final investigation extends the state-of-the-art model and explores the effect of allowing multiple visits to recharging stations. This relaxation can efficiently improve the solution's feasibility and quality.