It is hard to obtain precise wind power forecasting due to the chaotic nature of weather systems. Prediction intervals become an efficient tool to quantify the uncertainty involved in wind power forecasting. Traditional central prediction intervals are widely produced by forecasters, however, which might be conservative with respect to interval width and not well fit the practical conditions. This paper develops a novel adaptive bilevel programming (ABP) model, with extreme learning machine based quantile regression as the follower's problem and tuning hyperparameters of quantile proportions as the leader's problem. The proposed ABP model aims at minimizing the average interval width subject to well calibration. In order to overcome the difficulties in disposing of the intractable nested structure of bilevel programming, the ABP model is equivalently transformed into a single-level nonlinear programming problem with bilinear terms. An improved spatial branch-and-bound (ISBB) algorithm is proposed to efficiently solve the reformulated bilinear programming problem. In the ISBB algorithm, an innovative bounds tightening method is developed to tighten the convex relaxation of the bilinear constraint and enhance the convergence. Experimental studies based on actual wind farm of USA under four seasons show the significant effectiveness and high robustness of the developed ABP model, as well as the excellent global optimum attainment and convergence performance of the proposed ISBB algorithm. Moreover, the widely used traditional interval scores are first verified to be prejudiced in probabilistic wind power forecasting evaluation.