Page 173 - Mathematical Models and Algorithms for Power System Optimization

164 Chapter 6

6.3.2.3 Characteristics of the problem
The mathematical expression of a single-state model has the following six characteristics in the
field of mathematics for programming:

(1) High dimensionality of problem variable:
1. Number of continuous variables:
The number of voltage variables¼The number of system nodes N
The number of angle variables¼The number of system nodes N

The number of generator reactive power variables¼The number of generators N G
The number of tap ratio variables¼The number of transformers N T
The number of active power variables at generator balance node¼The number of

balance nodes N S
The number of continuous variables¼N 2+N G +N T + N S
2. Number of discrete variables:
The number of newly installed capacitor variables 2¼N c 2
The number of existing capacitor variables¼N E
The number of discrete variables¼N c 2+N E
A practical power system normally contains more than 100 nodes. Taking the test
system in this chapter as an instance, the number of nodes is 135, the number of
generators is 36, the number of transformers is 17, the number of newly installed
capacitors is 20, the number of existing capacitors is 29, the number of continuous
variables is 135 2+36+1+17¼324, and the number of discrete variables is 20 2
+29¼69.
Note: If the upper limit and lower limit are the same, the variable shall be a fixed
variable, such as the phase angle of balancing machine node.
(2) Huge number of constraints.
The number of constraint conditions is:
The number of power flow balance constraints¼The number of system nodes N 2
The number of investment constraints¼The number of capacitor nodes N C +N E
If the number of nodes is 135, the number of newly installed capacitors is 20, and the
number of existing capacitors is 29, then the number of constraints¼135 2+20
+29¼319.
(3) Cost function is a linear expression.
(4) Constraint function is a nonlinear function.
(5) Variables are divided into two types: continuous variables and discrete variables, and there
is a large number of integer variables.
(6) Both the number of constraints and variables are more than a few hundred.