Performance of Krylov solvers for gas networks

Abstract

To integrate renewable energies with our current energy systems, we require interaction between gas and electrical networks. The coupling of networks results in a larger system of equations to be solved. Henceforth, scalable solvers are more suitable for large coupled networks. In this paper, a preliminary research is done, by investigating Krylov solvers on gas networks from the GasLib library. The networks are simulated with steady-state models. The models yield a nonlinear system, which is solved with the Newton-Raphson method. The corresponding Jacobian is non-symmetric, indefinite and sparse. We have considered the following Krylov solvers: GMRES, Bi-CGSTAB and IDR(s). We compare the performance with a direct solver, which is the LU factorisation. Our results show that basic Krylov solvers are ineffective in solving the networks, because most networks have a large condition number and an unfavourable distribution of the eigenvalues. Hence, we have explored several preconditioners, such as Jacobi, Gauss-Seidel and ILU methods. Only the ILU preconditioner with the use of the COLAMD reordering scheme leads to convergence of all networks. For this preconditioner, the fill ratio has to be taken large enough, otherwise the ILU factorisation breaks down due to a zero pivot. The minimum required fill ratio leads to a similar amount of work as the direct solver. Thus the combination of ILU and Krylov solver does not perform better than direct solvers for these medium sized problems.