Page 171 - Mathematical Models and Algorithms for Power System Optimization

162 Chapter 6

(2) Some rules for the mathematical model:
1. Variables are not divided into state variable and control variable. Variables are taken
as a whole, so that state variable X and control variable U are not differentiated in the
optimization procedure. The algorithm divides variables into continuous variables
and discrete variables. When solving LP, a discrete variable is always treated as a
nonbasic variable (see Appendix A), whereas a fixed variable is treated as a variable
with the same bound. Thus, in the solution procedure, the nonbasic variable always
prevails. This measure has no significant influence on problem scale. The key to this
chapter is the discrete variable.
2. Reactive power generation constraint of generator is treated as variable constraint.
Reactive power generation of Eq. (6.8) is equivalent to control variable in form.
However, it is a derived variable of Eq. (6.6), which may be treated as a constraint
equation with upper and lower bounds. If it is treated as a constraint equation,
Eq. (6.6) may not contain the balance equation for the generator node. Thus, the
reactive power generation of a generator is not actually a control variable. In
Chapter 4, the reactive power generation of a generator is written as a constraint
equation, which is equivalent to the way of writing in this chapter. This chapter does
not intend to compare the calculation speed of the two processing methods. However,
they are completely equivalent in a mathematical sense.
3. Integer variable in the model is equivalent to the control variable. Shown by Eqs. (6.5)
and (6.6), only U, T, and C are changed in the optimization procedure to meet power
flow balance. As it takes no cost to adjust voltage U, transformer ratio T, and existing
capacity bank number, these variables will be preferentially adjusted when solving
LP. Integer variable YC of newly installed capacitors will be adjusted only when it is
impossible to keep all variables within their bounds. When solving LP, a discrete
variable will always be treated as a nonbasic variable, which is a function of
continuous variable voltage U and generator reactive power generation QG. Thus, the
integer variable is equivalent to the control variable in this chapter.
4. Phase angle is treated as free variables. Like other variables, phase angle is also
treated as a whole. Phase angle of balance node is treated as a fixed variable with equal
upper limit and lower limit, yet the phase angle of a common node is treated as a free
variable without upper and lower limits.
The previously discussed method runs through the whole chapter. The solving method for
such complex problem will be described in the next section.
(3) Model linearization: nodes in a network are divided into three classes of power flow
calculation: PQ, PV,and Vθ node. For the convenience of taking into consideration reactive
power Q constraint, all nodes, except for Vθ, are treated as PQ node in reactive planning
model. In other words, every node has P constraint and Q constraint equation. Please refer to
theprevioussectionforgeneratornodeprocessingmethod,aswellasboundofQvariable.For
the convenience ofbound processing, all algorithms proposedadopt a polar coordinate form.