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Gaussian Mixture Models (GMM) are an effective representation of resource uncertainty in power systems planning, as they can be tractably incorporated within stochastic optimization models. However, the skewness, multimodality, and bounded physical support of long-term wind power forecasts can entail requiring a large number of mixture components to achieve a good fit, leading to complex optimization problems. We propose a probabilistic model for wind generation uncertainty to address this challenge, termed Discrete-Gaussian Mixture Model (DGMM), that combines continuous Gaussian components with discrete masses. The model generalizes classical GMMs that have been widely used to estimate wind power outputs. We employ a modified Expectation-Maximization algorithm (called FixedEM) to estimate the parameters of the DGMM. We provide empirical results on the ACTIVSg2000 synthetic wind generation dataset, where we demonstrate that the fitted DGMM is capable of capturing the high frequencies of time windows when wind generating units are either producing at maximum capacity or not producing any power at all. Furthermore, we find that the Bayesian Information Criterion of the DGMM is significantly lower compared to that of existing GMMs using the same number of Gaussian components. This improvement is particularly advantageous when the allowed number of Gaussian components is limited, facilitating the efficient solution to optimization problems for long-term planning. 

We propose an algorithm to solve convex and concave fractional programs and their stochastic counterparts in a common framework. Our approach is based on a novel reformulation that involves differences of square terms in the constraints, and subsequent employment of piecewise-linear approximations of the concave terms. Using the branch-and-bound (B\&B) framework, our algorithm adaptively refines the piecewise-linear approximations and iteratively solves convex approximation problems. The convergence analysis provides a bound on the optimality gap as a function of approximation errors. Based on this bound, we prove that the proposed B\&B algorithm terminates in a finite number of iterations and the worst-case bound to obtain an $$\epsilon$$-optimal solution reciprocally depends on the square root of $$\epsilon$$. Numerical experiments on Cobb-Douglas production efficiency and equitable resource allocation problems support that the algorithm efficiently finds a highly accurate solution while significantly outperforming the benchmark algorithms for all the small size problem instances solved. A modified branching strategy that takes the advantage of non-linearity in convex functions further improves the performance. Results are also discussed when solving a dual reformulation and using a cutting surface algorithm to solve distributionally robust counterpart of the Cobb-Douglas example models.