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Process Conformance is a crucial step in the area of Process Mining: the adequacy of a model derived from applying a discovery algorithm to a log must be certified before making further decisions that affect the system under consideration. In the first part of this thesis, among the different conformance dimensions, we propose a novel measure for precision, based on the simple idea of counting these situations were the model deviates from the log. Moreover, a log-based traversal of the model that avoids inspecting its whole behavior is presented. Experimental results show a significant improvement when compared to current approaches for the same task. Finally, the detection of the shortest traces in the model that lead to discrepancies is presented. In the second part of the thesis, two different approaches are proposed in order to use the precision analysis information for refining the model, improving its accuracy. The first one is based on the idea of break concurrencies reflected in the model but not in the log. The second one presents the Supervisory Control Theory as the mechanism to improve the accuracy of the models building supervisors for controlling the precision issues.