Hello Davor, This is actually a very interestng question from the point of view of the numerical stability of the equations. I asume that by "changing the bus type PQ --> PV" you mean the following: start by solving a given case, then take a particular PV bus and convert its generator injections PG,QG (values taken from the solved case!) into a fixed PQ load, then remove the generator, and finally switch the bus type to PQ. Then you repeat this progressively until you're left with no PV buses.
The resulting system obviously shares the same mathematical solution, but, as you're guessing, trying to solve the converted system definitely shows some numerical effects. The way I understand it, there are differences coming from these two sources: - The condition number of the Jacobian degrades: the Jacobian matrix of the NR method (and also FD methods) is related to the admittance matrix of the network, which is in turn related to the Laplacian matrix of the graph representing the transmission grid. Swing buses remove the zero eigenvalue/eigenvector (the "uniform translation" mode in voltage), and PV buses sort of push the lowest non-zero eigenvalue (related to the so-called Fiedler vector) to higher values, thus improving the condition number of the resulting matrix. It can be shown that a system with no PV buses would result in a condition number degrading linearly with network size. - The basins of attraction of the iterative methods will be different. This effect is in general much harder to analyze, so one has to resort to numerical experiments. I experimented a while back with this and I found the second effect to be much more dominant in practice. My set up was the following: try to solve cases always from a flat start, while progressively switching PV buses to PQ. What I found was that the loss of precision resulting from having slighly worse condition numbers in the Jacobian was negligible, because one would encounter NR non-convergence problems way before that could become the dominant problem. I'm attaching a test script I used for experimenting with all this. By the way, there is also another important effect to take into account: PV buses that are pushed over to "the other side" of their V-Q curve. In these buses, when you flip them to PQ, you may obtain a different voltage solution (it depends on your initial seed). This is because in those cases the solution as PV corresponds to a low-voltage branch when viewed as PQ. Bus 191 in case IEEE 300 is a perfect example of this; you can try it with my script. -- Jose L. Marin Grupo AIA 2016-07-27 9:13 GMT+02:00 davor sutic <[email protected]>: > Should I expect a convergent and (reasonably) accurate power flow > solution, when I experiment with changing the bus type PQ->PV and > vice versa? > > If so, is there a minimum number of each bus type required, e.g. consider > a system where there are only PQ buses (apart from one slack, of course)? > > The data in the test cases seems sufficient for such changes, however I'm > worried about the stability of the system of equations. > > Thanks a lot >
convert_all_PV_to_PQ.m
Description: application/vnd.wolfram.mathematica.package
