Accelerated Waveform Relaxation methods for power systems

Power System Dynamic Simulations Using a Parallel Two-Level Schur-Complement Decomposition

“…If the time interval of the WR is restricted to only one time step, then the Instantaneous Relaxation method [11] is formulated. These techniques provide significant acceleration, however, their convergence strongly depends on the partitioning and on the time window interval [12], [13]. Some techniques to accelerate them have been proposed in [12], [14].…”

“…Thus, at the k-th iteration, the systems (11)(12)(13)(14) are formulated. Equations (11) and (12) Equations (13) and (14) stem from the linearization of Eq.…”

“…In turn, the just computed sub-domain voltage corrections (∆V k Si and ∆V k C ) are back-substituted in the injector equations (12) and (14), thus decoupling their solution as they now involve only their local state corrections (∆x k Cj or ∆x k Sij ). Finally, the injectors are solved independently and in parallel.…”

“…First, the parallel update of the (11)(12)(13)(14) is shown in the upper shaded block. Next, the L + 1 sub-domain reduced systems are computed in parallel in the second shaded block, where each parallel task updates one of the systems (15)(16).…”

“…(16) and (17)), the sparse linear solver HSL MA41 [32] was used. For the solution of the much smaller, dense injector linear systems (12) and (14), Intel MKL LAPACK library was used. In the sequential benchmark, the Jacobian matrix is updated every five iterations until convergence, while for the proposed algorithm, the matrices of each sub-domain are updated (independently of the other sub-domains) every five iterations unless convergence has already taken place.…”