  We consider a linear feeder connecting multiple distributed loads and
generators to the sub-station. Voltage is controlled directly at the
sub-station, however, voltage down the line shifts up or down, in particular
depending on if the feeder operates in the power export regime or power import
regime. Starting from this finite element description of the feeder, assuming
that the consumption/generation is distributed heterogeneously along the
feeder, and following the asymptotic homogenization approach, we derive simple
low-parametric ODE model of the feeder. We also explain how the homogeneous ODE
modeling is generalized to account for other distributed effects, e.g. for
inverter based and voltage dependent control of reactive power. The resulting
system of the DistFlow-ODEs, relating homogenized voltage to flows of real and
reactive power along the lines, admits computationally efficient analysis in
terms of the minimal number of the feeder line "media" parameters, such as the
ratio of the inductance-to-resistance densities. Exploring the space of the
media and control parameters allows us to test and juxtapose different measures
of the system performance, in particular expressed in terms of the voltage drop
along the feeder, power import/export from the feeder line as the whole, power
losses within the feeder, and critical (with respect to possible voltage
collapse) length of the feeder. Our most surprising funding relates to
performance of a feeder rich on PhotoVoltaic (PV) systems during a sunny day.
We observe that if the feeder is sufficiently long the DistFlow-ODEs may have
multiple stable solutions. The multiplicity may mean troubles for successful
recovery of the feeder after a very short, few periods long, fault at the head
of the line.
