  Electricity transmission networks dissipate a non-negligible fraction of the
power they transport due to the heat loss in the transmission lines. In this
work we explore how the transport of energy can be more efficient by adding to
the network multiple batteries that can coordinate their operations. Such
batteries can both charge using the current excess in the network or discharge
to meet the network current demand. Either way, the presence of batteries in
the network can be leveraged to mitigate the intrinsic uncertainty in the power
generation and demand and, hence, transport the energy more efficiently through
the network. We consider a resistive DC network with stochastic external
current injections or consumptions and show how the expected total heat loss
depends on the network structure and on the batteries operations. Furthermore,
in the case where the external currents are modeled by Ornstein-Uhlenbeck
processes, we derive the dynamical optimal control for the batteries over a
finite time interval.
