  Power Transfer Distribution Factors (PTDFs) play a crucial role in power grid
security analysis, planning, and redispatch. Fast calculation of the PTDFs is
therefore of great importance. In this paper, we present a non-approximative
dual method of computing PTDFs. It uses power flows along topological cycles of
the network but still relies on simple matrix algebra. At the core, our method
changes the size of the matrix that needs to be inverted to calculate the PTDFs
from $N\times N$, where $N$ is the number of buses, to $(L-N+1)\times (L-N+1)$,
where $L$ is the number of lines and $L-N+1$ is the number of independent
cycles (closed loops) in the network while remaining mathematically fully
equivalent. For power grids containing a relatively small number of cycles, the
method can offer a speedup of numerical calculations.
