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Abstract We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in Subramanyam (2022). We first present a simple counterexample where the original conditions are insufficient, highlight where the original proof fails, and then provide modified conditions along with a correct proof of their validity. Finally, although the original paper discusses modifications to their method for problems that may not satisfy any sufficient conditions, we substantiate that discussion along two directions. We first show that computing an optimal Lagrange multiplier can still be done in polynomial time. We then provide complete and correct versions of the corresponding Benders and column-and-constraint generation algorithms in which the original method is used. We also discuss the implications of our findings on computational performance. 

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.