Optimal Placement of Inertia and Primary Control: A Matrix Perturbation Theory Approach

Abstract

The increasing penetration of inertialess new renewable energy sources reduces the overall mechanical inertia available in power grids and accordingly raises a number of issues of grid stability over short to medium time scales. It has been suggested that this reduction of overall inertia can be compensated to some extent by the deployment of substitution inertia -synthetic inertia, flywheels or synchronous condensers. Of particular importance is to optimize the placement of the limited available substitution inertia, to mitigate voltage angle and frequency disturbances following a fault such as an abrupt power loss. Performance measures in the form of H2−norms have been recently introduced to evaluate the overall magnitude of such disturbances on an electric power grid. However, despite the mathematical conveniance of these measures, analytical results can be obtained only under rather restrictive assumptions of uniform damping ratio, or homogeneous distribution of inertia and/or primary control in the system. Here, we introduce matrix perturbation theory to obtain analytical results for optimal inertia and primary control placement where both are heterogeneous. Armed with that efficient tool, we construct two simple algorithms that independently determine the optimal geographical distribution of inertia and primary control. These algorithms are then implemented on a model of the synchronous transmission grid of continental Europe. We find that the optimal distribution of inertia is geographically homogeneous but that primary control should be mainly located on the slow modes of the network, where the intrinsic grid dynamics takes more time to damp frequency disturbances.

Long Wavelength Coherency in Well Connected Electric Power Networks

“…Ref. [21] found that the change of ω α with χ is to leading order proportional to i u 2 αi δγ i , where u α is the eigenvector of the Laplacian corresponding to λ α and δγ i = (d i /m i − γ). For the lowest modes which are approximately constant over large areas, u 2 αi can be factored out and i∈area δγ i ≈ 0, assuming that the distribution of δγ i is independent of the geographical location.…”

“…The optimal placement of inertia and damping has further been investigated using perturbation theory in Ref. [21]. In a more general context, Ref.…”

“…The PanTaGruEl model of the synchronous grid of continental Europe consists of 3809 nodes connected by 4944 power lines [11], [21]. With the dispatch used for this paper, there are 468 generators.…”

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“…Γ has two eigenvalues which can be defined as 1 = 2 = 1. According to eigenvalue perturbation theory [22], two following matrices are constructed.…”