  The problem of optimal control of power distribution systems is becoming
increasingly compelling due to the progressive penetration of distributed
energy resources in this specific layer of the electrical infrastructure.
Distribution systems are, indeed, experiencing significant changes in terms of
operation philosophies that are often based on optimal control strategies
relying on the computation of linearized dependencies between controlled (e.g.
voltages, frequency in case of islanding operation) and control variables (e.g.
power injections, transformers tap positions). As the implementation of these
strategies in real-time controllers imposes stringent time constraints, the
derivation of analytical dependency between controlled and control variables
becomes a non-trivial task to be solved. With reference to optimal voltage and
power flow controls, this paper aims at providing an analytical derivation of
node voltage and line current flows as a function of the nodal power injections
and transformers tap-changers positions. Compared to other approaches presented
in the literature, the one proposed here is based on the use of the [Y]
compound matrix of a generic multi-phase radial unbalanced network. In order to
estimate the computational benefits of the proposed approach, the relevant
improvements are also quantified versus traditional methods. The validation of
the proposed method is carried out by using both IEEE 13 and 34 node test
feeders. The paper finally shows the use of the proposed method for the problem
of optimal voltage control applied to the IEEE 34 node test feeder.
