Counterexample to a Continuation-Based Algorithm for Finding All Power Flow Solutions

An efficient method to locate all the load flow solutions - revisited

“…With the additional equations and the new geometry, we essentially follow the tracing procedure described in the Ma-Thorp paper. This revised method succeeds in finding all realvalued solutions to those system for which we know the solutions including the counter example of [2], and the systems presented in [8][9][10]. We also analyze the IEEE 14 bus system for which we find 30 solutions, consistent with a report in [12].…”

“…The next example is considered more challenging, the 5 bus counterexample presented in [2]. Again the method presented here finds all the solutions, requiring 38 traces.…”

“…In [2] we showed that the Ma-Thorp algorithm may not find all the solutions. In the counterexample of that paper, the 10 solutions were grouped in 3 sets of mutually connected points, and that the points in different sets were not connected along paths traced by the algorithm.…”

“…The counterexample presented in [2] showed that their algorithm can fail to find all solutions and noted a flaw in the proof presented in [11]. Because of this counterexample, the only algorithms that can provably find all solutions, such as the homotopy types used in [10] and [12], require the calculation of all real and complex solutions.…”

“…We offer a revision to the method by transforming the equations in such a way that all traces are bounded and all traces emanating from a known solution are explored. Results show that the revised method finds all real-valued solutions for power system problems reported in the literature for which all solutions are provably known, including the counterexample in [2]. …”