Performance of Krylov solvers for gas networks

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 CO-LAMD 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.