We propose a variance-penalized formulation of Bayesian optimal experimental design for nonlinear models that augments the classical expected utility criterion with a penalty on utility variability, yielding a mean--variance objective that promotes robust experimental performance. To evaluate this objective, we develop Monte Carlo estimators for the expected utility, its second moment, and the resulting utility variance using prior sampling, thereby avoiding explicit posterior sampling. We then derive leading-order bias and variance expressions using conditional delta-method arguments. The objective is optimized using Bayesian optimization with common random samples to reduce noise. Numerical examples, including a linear-Gaussian benchmark, a nonlinear test problem, and contaminant source inversion in diffusion fields, demonstrate that the proposed approach identifies designs with substantially reduced variability while maintaining competitive expected utility.

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