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This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a)\u00a0<jats:italic>a hierarchical error bound<\/jats:italic> and (b)\u00a0<jats:italic>an error bound based on an auxiliary linear problem<\/jats:italic>, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. 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