Coping with Algebraic Constraints in Power Networks

Abstract

In the intuitive modelling of the power network, the generatorsand the loads are located at different subset of nodes.This corresponds to the so-called structure preserving modelwhich is naturally expressed in terms of differential algebraicequations (DAE). The algebraic constraints in thestructure preserving model are associated with the load dynamics.Motivated by the fact the presence of the algebraic constraintshinders the analysis and control of power networks,several aggregated models are reported in the literaturewhere each bus of the grid is associated with certain loadand generation. The advantage of these aggregated modelsis mainly due to the fact that they are described by ordinarydifferential equations (ODE) which facilitates the analysisof the network. However, the explicit relationship betweenthe aggregated model and the original structure preservedmodel is often missing, which restricts the validity and applicabilityof the results.Aiming at simplified ODE description of the model togetherwith respecting the heterogenous structure of the power networkhas endorsed the use of Kron reduced models; see e.g.[2]. In the Kron reduction method, the variables which areexclusive to the algebraic constraints are solved in terms ofthe rest of the variables. This results in a reduced graph,the (loopy) Laplaican matrix of which is the Schur complementof the (loopy) Laplacian matrix of the original graph.By construction, the Kron reduction technique restricts theclass of the applicable load dynamics to linear loads.The algebraic constraints can also be solved in the case offrequency dependent loads where the active power drawnby each load consists of a constant term and a frequencydependentterm [1],[3]. However, in the popular class ofconstant power loads, the algebraic constraints are “proper”,meaning that they are not explicitly solvable.In this talk, first we revisit the Kron reduction method forthe linear case, where the Schur complement of the Laplacianmatrix (which is again a Laplacian) naturally appears inthe network dynamics. It turns out that the usual decompositionof the reduced Laplacian matrix leads to a state spacerealization which contains merely partial information of theoriginal power network, and the frequency behavior of theloads is not visible. As a remedy for this problem, we introducea new matrix, namely the projected pseudo incidencematrix, which yields a novel decomposition of the reducedLaplacian. Then, we derive reduced order models capturingthe behavior of the original structure preserved model.Next, we turn our attention to the nonlinear case where thealgebraic constraints are not readily solvable. Again by theuse of the projected pseudo incidence matrix, we proposeexplicit reduced models expressed in terms of ordinary differentialequations. We identify the loads embedded in theproposed reduced network by unveiling the conserved quantityof the system.<br/