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So far, H2 ⊗ L2-optimal model order reduction (MOR) of linear time-invariant systems, preserving the affine parameter dependence, was only considered for special cases by Baur et al in 2011. In this contribution, we present necessary conditions for an H2 ⊗ L2-optimal parametric reduced order model, for general affine parametric systems resembling the special case investigated by Baur et al.

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power system.

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We provide a unifying framework for L 2 -optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the L 2 cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.Keywords reduced-order modeling • parametric stationary problems • linear time-invariant systems • optimization • L 2 norm • nonlinear least squares