Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

Abstract

This is the peer reviewed version of the following article: [García-Blanco, R., Borzacchiello, D., Chinesta, F., and Diez, P. (2017) Monitoring a PGD solver for parametric power flow problems with goal-oriented error assessment. Int. J. Numer. Meth. Engng, 111: 529–552. doi: 10.1002/nme.5470], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5470/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.The parametric analysis of electric grids requires carrying out a large number of Power Flow computations. The different parameters describe loading conditions and grid properties. In this framework, the Proper Generalized Decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to 1) iterating algebraic solver, 2) number of terms in the separable greedy expansion and 3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal-oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end- user. The paper discusses how to compute the goal-oriented error estimates. This requires linearizing the error equation and the Quantity of Interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems.Peer ReviewedPostprint (author's final draft

Inclusion of frequency‐dependent parameters in power transmission lines simulation using harmonic analysis and proper generalized decomposition

“…Proper generalized decomposition provides a strong tool as a single monolithic solver as compared with solving the harmonic problem using Equation individually for every frequency. Proper generalized decomposition has already been implemented for the parametric power flow problem, and an error estimation strategy has also been developed . Note that if we solve the linear problem of Equation , we have to compute the matrices for every single frequency as they are no longer constant compared with the case where the constant parameters were considered and depend upon the frequency.…”

“…Proper generalized decomposition has already been implemented for the parametric power flow problem, 19 and an error estimation strategy has also been developed. 20 Note that if we solve the linear problem of Equation 13, we have to compute the matrices for every single frequency as they are no longer constant compared with the case where the constant parameters were considered and depend upon the frequency. Therefore, in this study, we present the PGD as a parametric solver, a method capable to solve the harmonic analysis.…”

Algebraic and Parametric Solvers for the Power Flow Problem: Towards Real-Time and Accuracy-Guaranteed Simulation of Electric Systems

“…The authors of [46] detail the difficulties in the linearization of equation (29) and also explains the possibility of calculating the matrix C and the vectorŝ R(V a ),λ andρ just once during the whole iterative process.…”

“…The aim of introducing the goal-oriented error estimations in the algorithm presented in section 4.3 is to control the accuracy of the approximation solution through the incorporation of stopping criteria into the procedure. An extended version of the development of the above equations is presented in [46].…”

“…, 5 for the nodes and T 1 fot the generator. These curves were generated using the software HOMER described in [69] for the test system described in a former work, see [46]. The curves vary from 1 to 8760 with a time step of 1h, thus n t = 8760.…”