the power flow equations of (15) for the remaining two variables
per bus.
This problem is one where we are required to solve simultaneous
nonlinear equations. Because most power systems are very large
interconnections, with many buses, the number of power flow
equations (and thus the number of unknowns) is very large. For
example, a model of the eastern interconnection in the US can have
50,000 buses.
The approach to solving the power flow problem is to use an
iterative algorithm. The Newton-Raphson algorithm is the most
commonly used algorithm in commercial power flow programs.
Starting with a reasonable guess at the solution (where the
“solution” is a numerical value of all of the unknown variables),
this algorithm checks to see how close the solution is, and then if it
is not close enough, updates the solution in a direction that is sure
to improve it, and then repeats the check. This process continues
until the check is satisfied. Usually, this process requires 5-20
iterations to converge to a satisfactory solution. For large
networks, it is computationally intensive.
In this class, we are very interested in optimization methods for
finding maximum surplus solutions to the problem of how to
dispatch the generation. So far, we have dealt with problems where
all generation and load was considered to be at the same bus (node)
and were thus able to ignore the network. But in reality, generation
and load are located at various buses, and the transportation
mechanism for moving electrical energy from supply to
consumption is the transmission network. If there are losses or
constraints in the transmission network (which there are), these
will influence how supply can be allocated, and the most
economically desirable solutions may not be feasible.