Contradiction? Synchronized (steady) state and power flow I am trying to investigate power flow in a simplified power system model with a few connected rotating motors. 
The dynamics is captured in the differential equations 
$$\frac{d^2 \phi_i}{dt^2} = P_i + \frac{d \phi_i}{dt} + \frac{K}{N} \sum_{j=1}^N \sin(\phi_i - \phi_j) \quad \text{for} \quad i=1,...,N$$
where $\phi_i$ is the phase deviation of each machine relative to the grid and $K$ is the capacity between (any) connected motors, $N$ is the number of rotating motors in the network and $P_i$ is related to the power consumption/production at node $i$. The model can be found here: Kuramoto model
Now for a power system to operate (I am told) that they need to work in synchrony, such that $\frac{d^2 \phi_i}{dt^2} = \frac{d \phi_i}{dt} = 0$. This would lead to constant phase deviations $\phi_i = constant$ (but different constant for each machine)
My question:
In the equation(s) above, I have to mention that the sine term is related to the power transfered between any two elements.
Now, as a measure of synchrony I considered the order parameter $r(t) = \frac{1}{N} \sum_{j=1}^N e^{i \phi_j}$ which during synchrony takes on a value close to $1$. But this would imply that the phase deviations are (approximately) equal, which in turn implies that the sine term (related to power transfer) is (close to) zero, but that makes no sense since the power flow would stop? Where do I stumble?
 A: I am not an expert, but consider this:
Ignoring the non-linear coupling between oscillators, you essentially get an equation that describes the 1-D motion of a Newtonian point of unit mass with negative unit drag and DC drive $P_i$ (or in circuit terms it is an RL DC current circuit with negative resistance). Therefore you can think of the DC drive as a source of instability in your system that drives it away from totally phase-coherent configurations(all phases equal). The system does contain phase locked configurations but they are phase incoherent, unless you insist that the power produced at each node is zero. This can be proven mathematically by solving the system of coupled equations mentioned above. 
For a particular set of values of DC drives you can actually arrange to have $|r(t)|=1$ or even $r(t)=1$, without having all of the oscillators at the same phase! Not all sets of values for $P_i$ will work, just like for a given K, not all systems exhibit a steady state.(P.S this can also be proven by using the solution to the system of equations).
Math Appendix
The system of equations can be solved by using the sine addition law, whence we get the simpler equation:
$\sin(\phi_{i}-\psi)=-\frac{P_i}{K|r(t)|} (1)$, where $e^{i\psi}=r(t)/|r(t)|$.
Here it should be clear that $|P_i|\leq|K|$ and $\sum_i P_i=0$ or the system exhibits no steady state.
The order parameter's absolute value can be self-consistently determined by plugging the solution back in a smart way, where you get:
$\sum_{j=1}^N\cos(\phi_j-\psi)=|r(t)|$. This could yield a multitude of different answers for the order parameter depending on the value of $\phi_j$ since equation (1) has 2 solutions if $ \phi_j\in[0,2\pi)$ for all j.
Now hopefully you can see that what you observe is correct, and hopefully you can also see that it's not that counterintuitive. In the equations derived, if all the $\phi_i$ are equal, then all the $P_i$ have to be equal, but since $\sum_{i}P_i=0$ that means all the $P_i$'s are zero.
