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I read this article concerning the detection of Braess Paradox on power transmission networks: http://phys.org/news/2012-10-power-grid-blackouts-braess-paradox.html

Here is a quote from the article:

In the power grid scenario, on the other hand, the paradox originates due to what the researchers call "geometric frustration." Adding a new link creates new cycles, along which all phase differences must add up to multiples of 2π to make all the phases well-defined. When a new link doesn't satisfy this condition, it doesn't synchronize with the other oscillators and the grid loses its phase-locked steady state. Witthaut explained the underlying mechanism using an analogy of a motor and generator.

My question is, why must the cyclic sum of the phase differences add up to a multiple of $2 \pi$ for the phases to be well defined?

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  • $\begingroup$ I appriciate the answer, but I fail to see why it has to be a multiple of $2 \pi$ in a cyclic network? Why would a generator short itself out if not? In the article they write the statement as if it was well-known that phases does in fact add up to a $2 \pi$ multiple. $\endgroup$ – Fabric Jun 22 '16 at 7:21
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    $\begingroup$ Imagine you are connecting an AC current source to one end of a transmission line. That transmission line delays the signal by a certain phase. Now you also connect the other end of the transmission line to the same source. Unless the phase was exactly $2\pi$, the two signals will have different voltages because $\cos(x)= \cos(x+\varphi)$ only if $\varphi = n\times 2\pi$. In practice a generator will short itself out, if such a loop differs in its electrical phase shift by anything that's not close to $2\pi$ (and multiples). $\endgroup$ – CuriousOne Jun 22 '16 at 18:21

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