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In the measurement-based approach, given a load model structure, its parameter values are derived using actual measurements. The issue of multiple local optimal solutions may arise in parameter estimation when the parameter estimation task is formulated as a nonlinear least squares problem. In this paper, the multiple local optimal solutions issue and a plateau phenomenon in developing load models are addressed in detail and linked to the local identifiability problem. Examples of multiple local optimal solutions in estimating parameter values of several static, dynamic and composite load models are demonstrated. In order to address the local identifiability at an obtained local optimal solution, output sensitivity and correlation between parameters are analyzed. Finally, several suggestions are proposed to resolve these issues.