Page 115 - Mathematical Models and Algorithms for Power System Optimization

New Algorithms Related to Power Flow 105

Differentiation of number of capacitor banks “C”:
2
∂Q c =∂C ¼ U b c
To address the discrete attribute of tap ratio T i , number of capacitor banks “C i ,” and number of
reactor banks “R i ,” the integer variables y Ti , y Ci , and y Ri are defined as follows:
0
T i ¼ T + y Ti ΔT i , i 2 N T (4.23)
i
0
C i ¼ C + y Ci ΔC i , i 2 N C [E C (4.24)
i
0
R i ¼ R + y Ri ΔR i , i 2 N R (4.25)
i
0 0 0
where T t , C i , R i —T i , C i , R i initial value; Δ Ti —tap ratio of each tap position; Δ Ci —unit
capacitance;N R —setofreactorbuses;Δ Ri —unitreactance;y Ti —integervariableoftapratio;y Ci —
integer variable of number of capacitor banks; y Ri —integer variable of number of reactor banks.

If Eqs. (4.23)–(4.25) are substituted into Eq. (4.22), the equation can be changed to:
X

0 2 0 0
Q i U, θ, T, C, Rð Þ ¼ Q ij U, θ, T + y Ti ΔT i U C + y Ci ΔC i R + y Ri ΔR i (4.26)
i i i i
j2Ni
Based on Eqs. (4.10)–(4.26), the equation can be simplified as:
ð
min f X, YÞ (4.27)
s:t: G X, YÞ ¼ 0 (4.28)
ð
X X X (4.29)
Y Y Y (4.30)

where f—objective function of nonlinear scalar; G—G¼[g i ], constraint function vector of
nonlinear equation.
X¼[U, θ, P G , Q G ], continuous variable vector; Y¼[Y T , Y C , Y R ], integer variable vector; the
components are Y T ¼[y Ti ], Y C ¼[y Ci ], Y R ¼[y Ri ].

The expressions for Y T and Y C are shown in Appendix B.

4.7 Discrete OPF Algorithm

4.7.1 Main Solution Procedure of Discrete OPF

The solution algorithm is based on an optimization technique such that the discrete OPF problem
is linearized iteratively and solved by utilizing an approximation method for linear MIP
problems and the SLP method. From the convergence features of both methods, it can be