2.1.Opf problem formulation for distribution networks
The goal of the OPF is to minimize selected objective function through optimal settings of the control variables, subject to various equality and inequality constraints. Generally, the OPF problem can be formulated as follows [6-11]:
(1)
Subject to:
(2)
(3)
(4)
where F is objective function to be minimized, x and u are the vectors of dependent and control variables, respectively.
For a distribution network with DG units, the vector of dependent variables (x) consists of:
active power from the electric grid,
;
load node voltage, including DG units which are modeled as PQ nodes, VL ;
reactive power outputs of the DG units which are modeled as PV nodes, QDG;
branch flow, Sl.
Therefore, x can be expressed as:
(5)
where NL, NPV and N are the number of load nodes, the number of PV nodes, and that of nodes (namely branches), in the distribution network, respectively.
The vector of control variables (u) consists of:
active power output of the DG units with non-renewable energy sources, PDG;
root node voltage, V0;
terminal voltage at PV nodes, VPV;
transformer (voltage regulation) tap setting, t;
output of shunt VAR compensators, QC.
Therefore, the vector of control variables can be expressed as:
, (6)
where NNR, NPV, NT and NC are the number of non-renewable DG units, the number of PV nodes (that is DG units modeled as PV nodes), the number of regulating transformers, and the number of VAR compensators, respectively.
2.1.1.Objective function
The objective function can take different forms. Several cases have been considered in this paper.
Case 1: Fuel cost minimization.
(7)
where
and
are electric grid and DG units cost characteristics, respectively.
Case 2: Multi-objective OPF as simultaneous minimization of the fuel cost and power losses:
(8)
where Plossi and
are power loss in branch i of the distribution network and
weighting factor for power losses, respectively.
Case 3: Multi-objective OPF as simultaneous minimization of the fuel cost and voltage deviation at load nodes:
(9)
where Vi and
are the voltage magnitude at load node i and weighting factor
for voltage deviation, respectively.
Case 4: Multi-objective OPF as simultaneous minimization of the fuel cost, power losses and voltage deviation at load nodes:
(10)