Page 14 - Mathematical Models and Algorithms for Power System Optimization

4 Chapter 1

calculation models, the device parameters are usually given as constants or transformed
into impedances, but they are rarely directly handled as variables. The general way to
convert device parameters to variables is to transform F(x 1 , …, x n )¼0to F(x 1 , …, x n 1 ,
x n (y))¼0, where y¼T or C.
(3) The variable and function can be switched from one to another.
There is one way to process variables as functions. For example, the traditional AC power
flow equation previously given can be written as F(x 1 , …, x n )¼g, which can then be
transformed to F(x 1 , …, x n ) – g¼0, F(x 1 , …, x n , g)¼0, so the right hand side (the node
injection power g) can be considered as a variable.

1.3 Ideas about the Selection of the Model Type

Under certain conditions, some model types can be mutually transformed into each other,
such as discrete and continuous, differential and difference, linear and nonlinear, complex
and simple, etc. Whatever deciding the model type, it is not only to pursue an accurate
theoretical description but able to solve the practical problem. Mathematically, the model
types depend on the relationship among the number of variables and constraints. The
different models used in the static analysis of the power system, such as power flow, state
estimation, and optimal power flow (OPF), depends upon the different relations among the
number of equations and the number of variables.

(1) When the number of equations is equal to the number of variables
The traditional power flow model can be applied when the numbers of rows and columns in
thecoefficientmatrixareequal. Manytextbooksdo not indicate thereason whythe nodetype
mustbesetupintheloadflowcalculation.Infact,thereareonlytwoequations(Pbalanceand
Q balance),however there are fourvariables (P, Q, U,and θ) foreach node. Therefore,two of
four variables must be fixed to satisfy the solvable conditions for which the number of
equations and the number of variables must be equal. This method is called load flow (LF), in
which some variables are specified and then the remaining variables can be solved. It
generally can only be used to obtain feasible solutions, rather than optimal solutions.

(2) When the number of equations is less than the number of variables
The OPF model can be applied when the number of rows is less than the number of
columns in the coefficient matrix. Therefore, the global optimal solution of the variable
can be obtained by OPF in one solution procedure without needing to use the power flow
program to approximate the optimal solution by point-by-point trial calculation. The
correctness of the developed optimal calculation model (OPF model) can be verified by
setting the upper and lower limits of some variables as the same value, that is, setting as the
fixed value (the same as the specified value in LF calculation). Under such a condition, the
same solution can be obtained by two methods to verify the correctness of the OPF model.
This is discussed further in Chapter 4.