Page 114 - Mathematical Models and Algorithms for Power System Optimization

104 Chapter 4
X
ð
ð
P i U, θ, TÞ ¼ P ij U, θ, TÞ (4.21)
j2i
X
2
ð
Q i U, θ, T, C, Rð Þ ¼ Q ij U, θ, TÞ V C i R i Þ (4.22)
ð
i
j2i
where N T —number of transformers; N C —number of newly installed capacitors; E C —number
of existing capacitors; N R —number of reactors.
The previous OPF problem is a mixed-integer nonlinear problem; the objective function
of Eq. (4.10) is to minimize the sum of fuel costs of the generator. Constraint Eqs. (4.11)–(4.19)
are nonlinear power flow equations. In Eqs. (4.21) and (4.22), the power flow function
expressions with the added variables T, C, and R are different from the traditional
power flow. The variable X in the equations can be U, θ, T, C, R, P(¼P G P L ), and
Q(¼Q G Q L ).
To address the discrete characteristics of transformer tap T, number of capacitor banks “C,” and
number of reactor banks “R,” the power flow equations for the transformer branch having tap T
and the capacitor branch C are given first. Then, the differential equations for T and C are given
in the following, respectively.

(1) Power flow equation for transformer branch:

P ij ¼ U i U i g ι U i U j g ι cosθ + b ι sinθð Þ=T
Q ij ¼ U i U i b ι U i U j g ι sinθ + b ι cosθð Þ=T

2
P ij ¼ U j U j g ι =T U i U j g ι cosθ b ι sinθð Þ=T
2
Q ji ¼ U j U j b ι =T + U i U j g ι sinθ + b ι cosθð Þ=T
(2) Equation for differentiation of T in the power flow of transformer branch:

∂P ij =∂T ¼ U i U j g ι cosθ + b ι sinθð Þ=T 2

∂Q ij =∂T ¼ U i U j g ι sinθ b ι cosθÞ=T 2
ð
3
∂P ji =∂T ¼ 2U j U i b ι =T + U i U j g ι cosθ b ι sinθÞ=T 2
ð
3
∂Q ji =∂T ¼ 2U j U j b ι =T U i U j g ι sinθ + b ι cosθÞ=T 2
ð
(3) Equation for the power flow of capacitor branch:
b c ¼ ωΔC
2
q c ¼ U b c
2
Q c ¼ Cq ¼ CV b c
c
where ΔC—capacity of single capacitor bank; ω—angular frequency of system; b c —
admittance of single capacitor bank; U—rated voltage at access point of capacitor; q c —reactive
power compensated by single capacitor bank; C—number of capacitor banks (integer variable);
Q c —reactive power compensated by capacitor bank.