In [\textit{S. Ghorbani} et al., Sci. Math. Jpn. 66, No. 3, 371--386 (2007; Zbl 1135.06007)], hyper MV-algebras were introduced; the results of the sum are non-empty sets, not necessarily singletons. Here, this approach is generalized to effect algebras, leading to a hyper sum which is a partial operation. The states (\(=\) finitely additive probability measures) in the usual sense rarely exist on hyper effect algebras. Thus, the authors also propose three other definitions of states. The task of finding finite hyper effect algebras is finite, but computationally demanding. The authors present ideas which simplify this work and they created publicly available software which performs this task on small hyper effect algebras.