Nonlinear interior-point optimal power flow method based on a current mismatch formulation X.P. Zhang, S.G. Petoussis and K.R. Godfrey Abstract: A nonlinear interior-point optimal power flow (OPF) method based on a current mismatch formulation in rectangular coordinates is proposed. In contrast to the classical way of solving the Newton equation that is involved, using a power mismatch formulation, the basic advantage of the proposed method is that, in the current mismatch formulation, some second derivatives become zero and some first derivatives become constant. In current mismatch formulation the elements Hij of the Hessian matrix are zero and the Jacobian elements Jii, Jij, Jji and Jjj are constant whereas in the power mismatch formulation for both cases those elements are functions of the voltage. These features make the computer code simpler. A theoretical comparison between the current mismatch formulation and the power mismatch formulation in rectangular coordinates is presented to point out their differences. Numerical examples on the IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus and IEEE 300-bus systems are presented to illustrate the nonlinear interior-point OPF method based on the current mismatch formulation. Preliminary results indicate that the two methods are comparable in terms of CPU time with the current mismatch formulation method having less computational complexity per iteration.
1
Introduction
An optimal power flow (OPF) program can be used to determine the optimal operation state of a power system by optimising a particular objective while satisfying specified physical and operating constraints. Because of its ability to integrate the economic and secure aspects of the system into one mathematical model, OPF can be applied not only in power system planning, but also in real-time operation optimisation of power systems in the energy management systems of power system control centres. Several optimisation techniques have been proposed to solve the OPF problem, including the gradient method [1], the linear programming (LP) method [2, 3], the Newton method [4, 5], the successive sparse quadratic programming (QP) method [6] and the successive non-sparse QP method [7]. In 1984 Karmarkar published his famous paper on an interior-point method for LP [8]. Interior-point methods have proven to be a promising alternative to those mentioned above for the solution of power system optimisation problems. In [9–11], LP-based interior-point methods were proposed. In [12], an interior-point method was proposed for linear and convex QP and was used to solve power system optimisation problems such as economic dispatch and reactive power planning. Several nonlinear primal-dual interior-point methods for power optimisation problems have been proposed in the literature [13–20]. It is generally accepted that these nonlinear primal-dual methods can efficiently solve nonlinear power system optimisation problems. The theoretical foundation for the nonlinear interiorpoint method consists of three crucial building blocks: (i) the r IEE, 2005 IEE Proceedings online no. 20050076 doi:10.1049/ip-gtd:20050076 Paper first received 9th March and in final revised form 6th June 2005 The authors are with the School of Engineering, University of Warwick, Coventry CV4 7AL, UK E-mail:
[email protected] IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
Fiacco–McCormick barrier method for optimisation with inequalities; (ii) Lagrange’s method for optimisation with equalities; and (iii) Newton’s method for solving nonlinear equations for unconstrained optimisation. The theory of nonlinear primal-dual interior-point methods has been established in [21]. The nonlinear interior-point methods in [16, 17] were formulated in rectangular coordinates whereas the other methods were formulated in polar coordinates. For the methods proposed in [14, 15, 17], a full Newton equation including all inequalities and equalities is formulated whereas the methods in [13, 16, 18] solve a reduced Newton equation. Among these nonlinear interior-point methods, the methods formulated in polar coordinates [13, 18] solve a reduced Newton equation at each iteration where slack variables of functional inequalities and slack and dual variables of simple variable inequalities are eliminated from the reduced Newton equation. We now intend to propose a nonlinear interior-point method in rectangular coordinates using a current mismatch formulation to solve a highly reduced Newton equation involving only the bus voltage variables ei, fi and the dual variables lpi and lqi of the bus power mismatch or current mismatch equations at each iteration. This method takes into account the fact that in such a formulation, some of the first differential terms become constant [22], and some of the second differential terms become zero, thus making the computational effort as well as the computer code simpler. 2 Nonlinear interior-point method OPF based on a power mismatch formulation in rectangular coordinates
2.1 Power mismatch equations in rectangular coordinates The power mismatch equations at a bus can be given in rectangular coordinates as: DPi ¼ Pgi Pdi Pi
ð1Þ 795
DQi ¼ Qgi Qdi Qi
ð2Þ
where Pgi and Qgi are the real and reactive powers of the generator at bus i, respectively, Pdi and Qdi the real and reactive load powers, respectively, and Pi and Qi the power injections at the node which are given by: N X Pi ¼ ½ei ðGij ej Bij fj Þ þ fi ðGij fj þ Bij ej Þ ð3Þ
Thus, the Lagrangian function for equalities optimisation of (9)–(13) is given by: M M X X lnðslj Þ m lnðsuj Þ L ¼ f ðxÞ m j¼1
i¼1
j¼1
Qi ¼
N X
N X
½fi ðGij ej Bij fj Þ ei ðGij fj þ Bij ej Þ
ð4Þ
M X
j¼1
lpi DPi
N X
lqi DQi
i¼1
plj ðhj slj hmin j Þ
j¼1
M X
puj ðhj þ suj hmax j Þ
j¼1
j¼1
where ei and fi are the real and imaginary parts of the voltage at bus i, respectively, Yij ¼ Gij þ jBij is the system admittance element, and N is the total number of system buses.
2.2 Formulation of a nonlinear interiorpoint OPF in rectangular coordinates using a power mismatch formulation
ð14Þ where lpi, lqi, plj, puj are Lagrange multipliers for the constraints of (10)–(13), respectively. N represents the number of buses and M the number of inequality constraints. Note that m40. The Karush-Kuhn-Tucker first-order conditions for the Lagrangian function shown in (14) are as follows: rx Lm ¼rf ðxÞ rDPT kp rDQT kq
An example of an objective function of an OPF is to minimise the total operating cost as follows: min f ðxÞ ¼
Ng X
ðai Pg2i þ bi Pgi þ gi Þ
rhT pl rhT pu ¼ 0
ð5Þ
i
subject to the following constraints: nonlinear equality constraints: DPi ðxÞ ¼ Pgi Pdi Pi ðt; e; f Þ ¼ 0 DQi ðxÞ ¼ Qgi Qdi Qi ðt; e; f Þ ¼ 0
ð6Þ ð7Þ
nonlinear inequality constraints: hmin hj ðxÞ hmax j j
ð8Þ
where x ¼ ½Pg; Qg; t; e; f T is the vector of variables, ai ; bi ; gi are the coefficients of the production cost functions of the generator, DP ðxÞ is the bus active power mismatch equation, DQðxÞ is the bus reactive power mismatch equation, hðxÞ is a functional inequality constraint that includes the line flow and voltage magnitude constraints, simple inequality constraints of variables such as the generator active power, generator reactive power and the transformer tap ratio, Pg is the vector of active power generation, Qg is the vector of reactive power generation, t is the vector of transformer tap ratios, e is the vector of the real part of the bus voltage, f is the vector of the imaginary part of the bus voltage and Ng is the number of generators. By applying the Fiacco–McCormick barrier method, we transform the OPF problem of (5)–(8) into the following equivalent OPF problem: ( ) M M X X lnðslj Þ m lnðsuj Þ ð9Þ objective: min f ðxÞ m j¼1
j¼1
subject to the following constraints: DPi ¼ 0
ð10Þ
DQi ¼ 0
ð11Þ
hj slj hmin ¼0 j
ð12Þ
hj þ suj hmax ¼0 j
ð13Þ
where sl40 and su40. 796
ð15Þ
rkp Lm ¼ DP ¼ 0
ð16Þ
rkq Lm ¼ DQ ¼ 0
ð17Þ
rpl Lm ¼ ðh sl hmin Þ ¼ 0
ð18Þ
rpu Lm ¼ ðh þ su hmax Þ ¼ 0
ð19Þ
rsl Lm ¼ m SlPl ¼ 0
ð20Þ
rsu Lm ¼ m þ SuPu ¼ 0
ð21Þ
where Sl ¼ diagðslj Þ, Su ¼ diagðsuj Þ, Pl ¼ diagðplj Þ, Pu ¼ diagðpuj Þ. In [17], all the above nonlinear equations (15)–(21) in rectangular coordinates are solved simultaneously. As suggested in [13], the above equations can be decomposed into the following three sets of equations 3 2 32 Dpl 0 rh 0 Pl 1 Sl 6 7 6 0 Pu1 Su rh 0 7 6 76 Dpu 7 6 7 6 7 4 rhT rhT H J T 54 Dx 5 2
0
0
J
3
0
rpl Lm Pl 1 rSl Lm 6 r L Pu1 r L 7 pu m Su m 7 6 ¼6 7 4 5 rx Lm
Dk
ð22Þ
rk Lm Dsl ¼ Pl 1 ðrsl Lm SlDplÞ
ð23Þ
Dsu ¼ Pu1 ðrsu Lm SuDpuÞ ð24Þ P P where Hðx; k; pl; puÞ ¼ r2 f ðxÞ kr2 gðxÞ ðplþ puÞr2 hðxÞ, DP ðxÞ @DP ðxÞ @DQðxÞ ; J ðxÞ ¼ ; gðxÞ ¼ ; @x @x DQðxÞ lp k¼ lq The elements corresponding to the slack variables sl and su have been eliminated from (22) using analytical Gaussian elimination. By solving (22), Dpl, Dpu, Dx, Dk can be obtained, then by solving (23) and (24), respectively, Dsl, Dsu can be obtained. With Dpl, Dpu, Dx, Dk, Dsl, Dsu IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
rx L0m ¼ rx Lm rhT Sl 1 rsl Lm þ Plrpl Lm þ rhT Su1 rsu Lm þ Purpu Lm
known, the OPF solution can be updated using the following equations: sl ðkþ1Þ ¼ sl ðk Þ þ sap Dsl su
ðkþ1Þ
¼ su
ðk Þ
þ sap Dsu
ð25Þ ð26Þ
xðkþ1Þ ¼ xðkÞ þ sap Dx
ð27Þ
pl ðkþ1Þ ¼ pl ðkÞ þ sad Dpl
ð28Þ
puðkþ1Þ ¼ puðk Þ þ sad Dpu
ð29Þ
kðkþ1Þ ¼ kðkÞ þ sad Dk
ð30Þ
where k is the iteration count, parameter s 2 [0.9950.99995] and ap and ad are the primal and dual step-length parameters, respectively. The step-lengths are determined as follows su sl ; min ; 1:00 ð31Þ ap ¼ min min Dsl Dsu pu pl ; min ; 1:00 ð32Þ ad ¼ min min Dpl Dpu for those Dslo0, Dsuo0, Dplo0 and Dpu40. The barrier parameter m can be evaluated by: b Cgap ð33Þ m¼ 2M where b 2 [0.01–0.2] and Cgap is the complementary gap for the nonlinear interior-point OPF and can be determined using: M X ðslj plj suj puj Þ ð34Þ Cgap ¼ j¼1
Equations (22)–(24) are the basic formulation of the nonlinear interior-point OPF that has been extensively discussed in [13, 18]. Equation (22) is the reduced equation with respect to the original OPF problem. However, (22) can be further reduced by eliminating all the dual variables of the inequalities, generator output variables and transformer tap ratios. Such a significant reduction means that the reduced Newton equation only involves the state variables of ei , fi , lpi , lqi . The details will be discussed in the following Section.
2.3 Formulating the reduced Newton equation with the state variables e, f, lp, lq In order to obtain the final reduced Newton equation consisting of only the variables e, f, kp, kq, the following Gaussian elimination steps can be applied.
2.3.1 Eliminating the dual variables pl and pu of the inequalities: The dimension of the Newton equation, (22), can be reduced using analytical Gaussian elimination techniques. Basically, the dual variables pl and pu in (22) can be eliminated to obtain:
where
@DP ðxÞ @x @DQðxÞ JQ ¼ @x Equations (35)–(37) can be written as the following compact form: 2 0 32 3 2 3 Dx rx L0m H JTp J Tq 4 Jp ð38Þ 0 0 5 4 Dkp 5 ¼ 4 rkp Lm 5 Dkq rkq Lm Jq 0 0 JP ¼
By solving (38), Dx can be obtained, then the dual variables Dpl and Dpu can be found by solving the following equations: Dpl ¼ PlSl 1 ðrhDx rpl Lm Þ þ Sl 1 rSl Lm ð39Þ Dpu ¼ PuSu1 ðrhDx rpu Lm Þ Su1 rSu Lm
Up to now, (22) has been reduced to three lowerdimensional equations: (38)–(40). In (38), all inequalities have been eliminated and (39) and (40) are relatively simple to solve.
2.3.2 Eliminating the generator variables Pg and Qg: In (38), the generator variables Pg, Qg can be further eliminated. Equation (38) may be written in the following form, in which only the relevant major diagonal block of bus i is displayed. 2
H 0 Pgi Pgi
0
0
0
1
6 6 6 6 6 6 6 6 4
0 0 0
H 0 Qgi Qgi 0 0
0 H ei ei H 0 ei fi
0 H ei fi H 0 fi fi
0 J pi ; ei J pi ; fi
J pi ; fi J qi ; fi
0 0
0
1 0 J pi ; ei 0 1 J qi ; ei 2 3 2 3 rPgi L0m DPgi 6 DQg 7 6 rQg L0 7 i7 i m 7 6 6 6 7 6 7 6 Dei 7 6 rei L0m 7 7¼6 7 6 6 Df 7 6 r L0 7 fi m 7 i 7 6 6 6 7 6 7 4 Dlpi 5 4 rlpi Lm 5
Eliminating have that: 2 0 H ei ei 6 0 6 H f i ei 6 6 Jp ; e 4 i i
3
1 7 7 7 J qi ; ei 7 7 J qi ; fi 7 7 7 5 0 0
ð41Þ
DPgi and DQgi from the above equation, we H 0 ei fi
Jpi ; ei
H 0 fi fi
Jpi ; fi
Jpi ; fi
J lpi
Jqi ; ei Jqi ; fi 2 3 rei L0m 6 r L0 7 6 fi m 7 7 ¼6 6 r L0 7 lpi m 5 4
0
Jqi ; ei
32
Dei
3
76 7 Jqi ; fi 76 Dfi 7 76 7 6 7 0 7 54 Dlpi 5 J lqi
Dlqi
ð42Þ
rlqi L0m
ð35Þ
JP Dkp ¼ rkp Lm
ð36Þ
J lpi ¼ ðH 0 Pgi Pgi Þ1
JQ Dkq ¼ rkq Lm
ð37Þ
J lqi ¼ ðH 0 Qgi Qgi Þ1
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
0
0
rlqi Lm
Dlqi
H 0 Dx JTP Dkp JTQ Dkq ¼ rx L0m
H 0 ¼ H þ rhT rh PlSl 1 PuSu1
ð40Þ
where
rlpi L0m ¼ rlpi Lm þ ðH 0 Pgi Pgi Þ1 rPgi L0m rlqi L0m ¼ rlqi Lm þ ðH 0 Qgi Qgi Þ1 rQgi L0m 797
By solving (42), Dlpi and Dlqi can be obtained, then DPgi and DQgi can be found by the following equations DPgi ¼ ðH 0 Pgi Pgi Þ1 ðDlpi rPgi L0m Þ
ð43Þ
DQgi ¼ ðH 0 Qgi Qgi Þ1 ðDlqi rQgi L0m Þ
ð44Þ
Similarly, the elements corresponding to the transformer tap ratio can be eliminated using the same principle resulting in a reduced Newton equation consisting of only the variables e, f, kp and kq. On the contrary, this action will affect the sparsity of the matrix because 16 new fill in elements result due to the elimination of the original transformer tap ratio elements.
2.3.3 Solution procedure for the nonlinear interior-point OPF: The solution of the proposed nonlinear interior-point OPF can be summarised as follows: Step 0: Formulation of (22) Step 1: Forward substitution 1.1: Eliminating the dual variables pl, pu of the inequalities from (22), obtain (38); 1.2: Eliminating generator variables Pg, Qg from (38), obtain (42); 1.3: Eliminating the transformer tap ratio t using a similar procedure obtain the highly reduced Newton matrix equation. Step 2: Solution of the final highly reduced Newton equation by sparse matrix techniques 2.1: The highly reduced system matrix has a dimension of 4N where N is the total number of buses. 2.2: Having been grouped into 4 4 blocks, solution to the final matrix is produced by sparse matrix techniques. Step 3: Back substitution 3.1: Firstly substitute for the transformer tap ratio. After solving the final matrix equation, De, Df , Dkp and Dkq are known, then Dt can be found by back substitution; 3.2: Secondly substitute for the generator output variables: DPg and DQg can be found from (43) and (44); 3.3: Thirdly substitute for all the dual variables of the inequalities: The dual variables pl, pu of the inequalities can be found by (39) and (40); 3.4: Fourthly substitute for all slack variables: all slack variables can be found by (23) and (24).
following form: Pi þ jQi ¼ ðei þ jfi ÞðIxi jIyi Þ
where Ixi and Iyi are the real and imaginary parts of the current flow at bus i. Then by re-arranging (45) and substituting Pi and Qi with the equivalent expressions from (1) and (2), we have: Pi jQi ðPgi Pdi Þ jðQgi Qdi Þ Ixi þ jIyi ¼ ¼ ð46Þ ei jfi ei jfi Then, by re-arranging (46), the expressions for the real and imaginary parts of the current, Ixi and Iyi at bus i, can be derived: ei ðPgi Pdi Þ þ fi ðQgi Qdi Þ Ixi ¼ ðIgxi Idxi Þ ¼ ð47Þ e2i þ fi2 Iyi ¼ ðIgyi Idyi Þ ¼
fi ðPgi Pdi Þ ei ðQgi Qdi Þ e2i þ fi2
3 Nonlinear interior-point OPF in rectangular coordinates using a current mismatch formulation
3.1 Current mismatch equations in rectangular coordinates The derivation of the current mismatch equations can start with the power mismatch equation. The real and reactive power at a bus i can be expressed in terms of the product of the voltage and current, given in the 798
ð48Þ
where Igi and Idi represent the generation current and the load current respectively. Note that x and y denote, respectively, the real and imaginary components. Therefore, the current mismatch equations at bus i are given by: ð49Þ DIxi ¼ ðIgxi Idxi Þ Ixi DIyi ¼ ðIgyi Idyi Þ Iyi
ð50Þ
where the current injections Ixi and Iyi are given by: N X Ixi ¼ ðGij ej Bij fj Þ ð51Þ j¼1
Iyi ¼
N X
ðGij fj þ Bij ej Þ
ð52Þ
j¼1
3.2 Formulation of nonlinear interior-point OPF in rectangular coordinates using a current mismatch formulation In the nonlinear interior-point OPF in rectangular coordinates using a power mismatch formulation as given in (9)–(13), if the nonlinear power mismatch equations, (10) and (11), are replaced by the following current mismatch formulation equations: ð53Þ Igxi Idxi Ixi ðt; e; f Þ ¼ 0 Igyi Idyi Iyi ðt; e; f Þ ¼ 0
ð54Þ
then we get the nonlinear interior-point OPF in rectangular coordinates using a current mismatch formulation. Accordingly, (10) and (11) should be replaced by: ð55Þ DIxi ¼ 0 DIyi ¼ 0
The solution philosophy here will be applied to the nonlinear interior-point OPF based on the current mismatch formulation in the following Section. However, the current mismatch formulation has its own unique features, as will be discussed in Section 3.
ð45Þ
ð56Þ
In addition, the Lagrangian function equations, (14) and (16) and (17), are also altered to include the terms DIxi and DIyi instead of DPi and DQi and also the lIxi and lIyi instead of lpi and lqi . The structure of the Newton equation, (22), in the current mismatch formulation is changed significantly. In principle, most of the equations derived in Section 2 for the nonlinear interior-point OPF using a power mismatch formulation are applicable to the nonlinear interior-point OPF using current mismatch equations with some equations being slightly changed in appearance. The significant difference between the power and current mismatch formulations is between (22) and (41). The current mismatch formulation has considerable merit in terms of computation and programming, which will be discussed in the following. IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
3.3 Comparison of the structures of (22) in power mismatch and current mismatch formulations Note in (22), we have
X kr2 gðxÞ Hðx; k; pl; puÞ ¼ r2 f ðxÞ X ðpl þ puÞr2 hðxÞ
straint, which belongs to hðxÞ, can be represented in terms of the currents Ixij and Iyij as follows: hðyÞ ¼ Ix2ij þ Iyij2
ðSijmax Þ2 e2i þ fi2
Then we have:
where
@DP ðxÞ @DQðxÞ T ; J ðxÞ ¼ ; @x @x lp and l ¼ lq
Hii
− Jii
Hij
− Jji
0
− Jij
0
Hjj
− Jjj 0
Structure of the system matrix of (22)
3.3.1 The structure of r2 gðyÞ of current mismatch equations: The r2 gðyÞ can be represented by: 2
r gðyÞ ¼ r
2
Igxi Idxi Ixi ðyÞ r2 Iyi ðyÞ Igyi Idyi
ð57Þ
In (57), we have r2
Ixi ðyÞ ¼0 Iyi ðyÞ
Thus, r2 gðyÞ contributes only to the diagonal Hii. The second derivatives of (57), which represent the contributions of the transmission-line constraints for some of the key elements, are presented in the Appendix.
H 11 ii
Hii
−JTii
Hij
−JTij
0
−Jij
0
Hjj
−JTjj
=
"
r2 hðyÞ ¼ r2 ½Ix2ij þ Iyij2 r2
gðxÞ ¼ ½DP ðxÞ; DQðxÞT
The Hessian elements are related to three terms r2 f ðxÞ; r2 gðxÞ and r2 hðxÞ. The first term is usually a function of generator variables whereas the second and third terms are functions of x ¼ ½Pg; Qg; t; e; f T . If we set y ¼ ½e; f T , then the elements of H, which are the second derivatives of the Lagrangian function with respect to bus voltage y ¼ ½e; f T , are determined by the second and third terms, namely, r2y gðyÞ, r2y hðyÞ. In Fig. 1, it is assumed that the main part of H in (22) in terms of variables ei, fi and ej, fj can be described by the matrix structure shown in Fig. 1 where, for the current mismatch formulation, the elements Hij are zero.
Fig. 1
3.3.2 The structure of r2 hðyÞ of transmission line constraint: The transmission line capacity con-
ðSijmax Þ2
ð58Þ #
e2i þ fi2
ð59Þ
j k The first term r2 Ix2ij þ Iyij2 , which contributes to Hii, Hjj and Hij, is constant whereas the jsecond term k only 2 2 2 contributes to the diagonal Hii. Thus, r Ixij þ Iyij can be formulated once before the iterating loop.
3.3.3 Calculation of Jii, Jjj, Jij and Jji: In the current mismatch formulation, Jii , Jjj , Jij and Jji become constant. The major block of (22) in terms of a current mismatch formulation may be represented by the structure shown in Fig. 2 where detailed elements are displayed.
3.3.4 Comparison with the power mismatch formulation: The following points should be noted: 1. In the power mismatch formulation: 2 2 Pgi Pdi 2 Pi ðyÞ r gðyÞ ¼ r ; r Qi ðyÞ Qgi Qdi Pgi Pdi r2 ¼ 0 and only Qgi Qdi 2 Pi ðyÞ r Qi ðyÞ contributes to Hii , Hjj and Hij in Fig. 1. In comparison, the computer code for calculating r2 gðyÞ in the current mismatch formulation is much simpler than that in the power mismatch formulation, since in the former, the elements of Hij are zero. The computational effort is therefore less in the current mismatch formulation. This feature makes the current formulation attractive. 2. In the power mismatch formulation, r2 hðyÞ of the transmission-line constraint is not constant and contributes to elements of Hii, Hjj and Hij in Fig. 1. However, in the current mismatch formulation, r2 hðyÞ contributes with a variable term in Hii and constant terms in Hjj and Hij. The
H 12 ii
Gii
Bii
0
0
Gij
Bij
H 22 ii
−Bii
Gii
0
0
−Bij
Gij
0
0
Gij
−Bij
0
0
0
Bij
Gij
0
0
H11 jj
H12 jj
Gjj
Bjj
H 22
−Bjj
Gjj
0
0
jj
0
o0
0
Fig. 2
Hessian matrix for current mismatch formulation
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
799
constant terms contributed by r2 hðyÞ can be calculated once before the iterating loop. This again makes the computer code of the current formulation shorter and more efficient. 3. In the current mismatch formulation for the elements
@DIxðyÞ @DIyðyÞ ; JðyÞ ¼ @y @y
@DP ðyÞ @DQðyÞ ; JðyÞ ¼ @y @y
2
3
2
HPgi ei HQgi ei H 0 ei ei
HPgi fi HQgi fi H 0 ei fi
HPgi lpi HQgi lpi Jpi ; ei
H 0 ei fi Jpi ; ei Jqi ; ei 0 3
H 0 fi fi Jpi ; fi Jqi ; fi
Jpi ; fi 0 0
rPgi Lm DPgi 6 DQg 7 6 rQg L0 7 i7 i m 7 6 6 6 7 6 7 6 Dei 7 6 rei L0m 7 7 6 7 6 6 Df 7 ¼ 6 r L0 7 fi m 7 i 7 6 6 6 7 6 7 4 Dlpi 5 4 rlpi Lm 5 rlqi Lm Dlqi
3 HPgi lqi HQgi lqi 7 7 7 Jqi ; ei 7 7 Jqi ; fi 7 7 7 5 0 0
ð60Þ
Comparing (60) and (41), it can be found that eight new elements HPgi ei , HPgi fi , HPgi lpi , HPgi lqi and HQgi ei , HQgi fi , HQgi lpi , HQgi lqi in (60), all of which are the functions of Pgi, Qgi, ei, fi, lpi and lqi , are introduced for Pgi, Qgi, respectively. This will cause the computational effort to eliminate Pg, Qg, to formulate (42), and the subsequent back substitution to obtain DPg and DQg, to be increased but only slightly. 800
x
x
x
x
x
x
0
0
x
x
0
0
0
x
x
0
0
x
x
x
x
x
x
x
0
0
x
x
∗
∗
0
0
∗
∗
x
∗
∗
0
0
∗
∗
0
0
∗
∗
0
0
0
∗
∗
0
0
x
x
∗
∗
x
∗
∗
0
0 0
a
b
Fig. 3 Comparison between the Newton matrix for the power mismatch formulations and the current mismatch formulations a The power mismatch formulation b The current mismatch formulation
System
The significant change of the current mismatch formulation, in comparison to the power mismatch formulation, is to (41). In the current mismatch formulation, (41) has the following structure: 6 6 6 6 6 6 6 6 4
x x
Table 1: Description of the IEEE test systems
3.4 Equation (41) for the nonlinear interiorpoint OPF in the current mismatch formulation
0 H 0 Qgi Qgi
x x
T
To summarise the points mentioned here and to demonstrate the simplification provided with the current mismatch formulation, the major block of the Newton matrix of (22) is illustrated graphically in Fig. 3. There is a direct comparison between the variable and the constant terms. The symbol ‘x’ denotes a variable element, the symbol ‘*’ a constant term and a ‘0’ that the element is zero. However, it should be pointed out that some additional elements are introduced that are related to the derivatives of the current mismatch formulation with respect to the active and reactive generation variables, and extra computational effort, therefore, is needed. The concern will be discussed in the following Section.
H 0 Pgi Pgi
x x
0
are not constant and are functions of e and f .
2
x x
T
the first derivative of (51) and (52) which contribute to the diagonal elements Jii, Jjj and to the non-diagonal elements Jij, Jji are constant, thus can be calculated once before the iterating loop simplifying the computer code and increasing its performance. In the power mismatch formulation the first derivatives of (3) and (4) for the elements:
x
Number of buses
Number of generators
Number of Number of transmission on-line tap lines changing transformers
IEEE 14
14
5
17
IEEE 30
30
6
37
4
IEEE 57
57
7
63
17
IEEE 118
118
54
177
9
IEEE 300
300
69
311
98
4
4.1
3
Numerical examples
Test systems
In order to make comparisons between the proposed current mismatch OPF algorithm and the power mismatch OPF, tests on the five IEEE systems were performed. The five IEEE systems are summarised in Table 1 and the system data can be found in [23]. For testing purposes the convergence tolerances are set to 5 104 for the complementary gap and 1 104 p.u. for the maximal absolute bus power mismatch.
4.2
Test results on the IEEE test systems
In order to simplify the following discussions, the nonlinear interior-point OPF methods discussed in Sections 2 and 3 are referred to as the power mismatch method and the current mismatch method, respectively. The initialisation of the Lagrange multipliers associated with the current mismatch equations is very similar to that of the Lagrange multipliers associated with the power mismatch equations. In principle the real and imaginary current mismatch equations correspond to the real and imaginary power mismatch equations, respectively. In the implementation, Lagrange multipliers of the real current mismatch equations are set to the system marginal generation cost whereas the Lagrange multipliers of the imaginary current mismatch equations are simply set to zero. Table 2 shows the computational results of the IEEE test systems. In this Table, the CPU time comparisons between the two methods are also given on the two larger IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
Table 2: Computational results obtained using the two OPF methods with minimising generation costs System
Power mismatch method Number of iterations
Current mismatch method
Normalised CPU time
CPU time per iteration
Number of iterations
Normalised CPU time
CPU time per iteration –
IEEE 14
9
–
–
9
–
IEEE 30
9
–
–
9
–
–
IEEE 57
11
–
–
10
–
–
IEEE 118
14
1.0
0.071
12
0.79
0.066
IEEE 300
20
1.0
0.050
20
0.94
0.047
IEEE test systems, namely, the IEEE 118-bus and 300-bus systems. From Table 2, it can be found that the two methods need a similar number of iterations and the computational effort of the two methods is comparable. Furthermore, it can be seen that the computational effort of the current mismatch OPF algorithm per iteration is less than that of the power mismatch OPF algorithm. This is so, since the computer code of the current mismatch formulation is simpler than that of the power mismatch formulation and also due to the
fact that in the formulation of the system matrix of the current mismatch OPF algorithm, some terms become either constant or zero. In order to make further comparisons of the convergence characteristics of the two methods, the details of convergence processes of the two methods on the IEEE 14-bus, IEEE 30-bus, IEEE 57-bus and IEEE 118-bus systems, are presented in Tables 3–6, respectively. In Tables 3–6, DP ðxÞ and DQðxÞ are active and reactive power mismatches, respectively. The convergence characteristics of the two
Table 3: Convergence characteristics of the two OPF methods on the IEEE 14-bus system Iteration count Power mismatch method
0
Current mismatch method
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
7.10e + 001
1.77e + 000
3.41e 001
7.10e + 001
1.77e + 000
3.41e 001
1
6.79e + 001
6.74e 002
3.41e 001
6.73e + 001
2.60e 002
1.31e 002
2
9.44e + 000
4.45e 004
1.83e 003
6.70e + 000
1.55e 005
2.38e 005
3
1.06e + 000
9.10e 005
1.41e 004
1.06e + 000
2.89e 004
5.37e 004
4
2.69e 001
1.67e 003
7.92e 003
3.35e 001
2.91e 003
1.17e002
5
5.19e 002
4.17e 004
2.54e 003
7.84e 002
8.96e 004
4.12e003 1.10e003
6
1.32e 002
2.08e 004
8.97e 004
1.73e 002
1.74e 004
7
2.94e 003
7.71e 005
1.21e 003
5.11e 003
1.34e 004
1.03e003
8
3.87e 004
2.68e 006
1.20e 003
6.74e 004
3.06e 006
1.51e003
9
4.17e 005
1.04e 007
2.66e 005
7.46e 005
2.32e 007
6.76e005
Table 4: Convergence characteristics of the two OPF methods on the IEEE 30-bus system Iteration count Power mismatch method
0
Current mismatch method
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
1.29e + 002
1.16e + 000
1.00e + 000
1.29e + 002
1.16e + 000
1.00e + 000
1
1.19e + 002
5.50e 002
1.23e 001
1.16e + 002
3.44e 002
2.72e 002
2
4.59e + 001
1.05e 003
4.43e 003
4.77e + 001
9.64e 004
2.58e 003
3
8.32e + 000
2.58e 004
8.06e 004
9.41e + 000
4.94e 004
1.06e 003
4
1.53e + 000
8.33e 005
1.38e 003
1.94e + 000
2.31e 004
5.18e 004
5
4.63e 001
3.29e 005
4.85e 004
6.09e 001
1.05e 004
8.85e 004
6
6.25e 002
1.19e 005
4.31e 004
9.29e 002
1.53e 005
4.29e 004
7
7.58e 003
1.68e 006
5.58e 005
1.13e 002
2.53e 006
7.34e 005
8
8.48e 004
1.65e 007
2.19e 006
1.27e 003
2.12e 007
3.05e 006
9
9.35e 005
2.11e 008
2.51e 007
1.40e 004
3.17e 008
3.77e 007
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
801
Table 5: Convergence characteristics of the two OPF methods on the IEEE 57-bus system Iteration count Power mismatch method
Current mismatch method
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
0
2.51e + 002
2.72e + 000
1.34e + 000
2.51e + 002
2.72e + 000
1.34e + 000
1
2.41e + 002
1.01e + 000
6.63e 001
2.40e + 002
8.53e 001
4.00e 001
2
1.07e + 002
4.17e 001
2.70e 001
8.18e + 001
3.40e 001
1.60e 001
3
1.49e + 001
1.08e 001
6.92e 002
9.10e + 000
3.53e 002
1.71e 002
4
2.85e + 000
3.76e 003
2.06e 003
2.63e + 000
5.67e 003
5.32e 003
5
6.57e 001
2.26e 003
6.05e 003
6.99e 001
2.07e 003
4.45e 003
6
1.95e 001
1.15e 003
7.81e 003
2.22e 001
1.46e 003
9.58e 003
7
6.11e 002
6.79e 004
4.31e 003
6.52e 002
6.83e 004
4.18e 003
8
1.75e 002
3.54e 004
2.21e 003
1.83e 002
1.50e 004
8.96e 004
9
3.70e 003
1.42e 004
8.76e 004
2.62e 003
7.49e 005
4.47e 004
10
6.29e 004
2.49e 005
1.49e 004
3.61e 004
1.18e 005
6.24e 005
11
7.81e 005
7.80e 007
5.02e 006
–
–
–
Table 6: Convergence characteristics of the two OPF methods on the IEEE 118-bus system Iteration count Power mismatch method
Current mismatch method
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
Complementary gap
Max jDP ðx Þj
Max jDQðx Þj
0
6.40e + 002
4.50e + 000
4.52e + 000
6.40e + 002
4.50e + 000
4.52e + 000
1
6.13e + 002
1.41e + 000
1.63e + 000
6.16e + 002
1.41e + 000
6.88e 001
2
4.91e + 002
2.06e 001
2.39e 001
2.59e + 002
2.57e 001
1.27e 001
3
1.79e + 002
3.91e 001
2.30e + 000
3.90e + 001
1.74e 002
2.29e 002
4
8.25e + 001
4.06e 002
3.47e 001
6.70e + 000
3.82e 003
6.20e 003
5
1.11e + 001
1.12e 003
1.80e 002
2.47e + 000
2.95e 003
3.17e 002
6
3.11e + 000
1.84e 003
1.10e 002
7.62e 001
1.11e 003
1.08e 002
7
8.77e 001
1.14e 003
1.00e 002
2.86e 001
8.06e 004
6.26e 003
8
3.25e 001
7.69e 004
6.59e 003
7.82e 002
6.08e 004
2.98e 003
9
8.83e 002
6.06e 004
3.13e 003
2.07e 002
2.55e 004
1.26e 003
10
2.90e 002
3.58e 004
1.87e 003
5.16e 003
7.37e 005
3.59e 004
11
1.01e 002
1.55e 004
8.28e 004
1.22e 003
2.20e 005
1.10e 004 2.77e 005
12
2.72e 003
4.31e 005
2.38e 004
1.60e 004
7.42e 007
13
5.51e 004
6.91e 006
4.51e 005
–
–
–
14
6.73e 005
3.73e 007
4.91e 006
–
–
–
methods on the IEEE 300-bus system are shown graphically in Figs. 4–6. From Tables 2–6 and Figs. 4–6, it can be seen that the proposed current mismatch OPF algorithm has very good convergence characteristics. Preliminary numerical results indicate that the computational complexity of the current mismatch OPF per iteration is less than that of power mismatch OPF. 5
Conclusions
A nonlinear interior-point OPF method based on a current mismatch formulation in rectangular coordinates has been proposed. In the current mismatch formulation approach, some second derivatives become zero and 802
some first derivatives become constant and thus the computational effort is reduced in comparison to previous methods. The basic advantages of this method are that, in the current mismatch formulation the elements Hij of the Hessian matrix are zero and the Jacobian elements Jii, Jij, Jji and Jjj are constant whereas in the power mismatch formulation for both cases those elements are functions of the voltage. In addition, in the current mismatch formulation the transmission-line constraint contributes to the Hessian elements Hij and Hjj with a constant term whereas in previous methods this was a variable term. All the constant terms can be calculated once, before the iterating loop. These features make the computer code of the current mismatch formulation simpler. Numerical examples on the IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus and IEEE 300-bus systems have demonstrated that the IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
6
101
100
objective function (×10 5 )
maximum power mismatch, p.u.
10 −1 10 − 2 10 − 3 10 − 4
5
4 power mismatch method
10 − 5
current mismatch method
10 − 6
power mismatch method current mismatch method
3
10 −7 0
2
4
6
8
10
12
14
16
18
20
0
22
2
Fig. 4 Absolute power mismatches as a function of the number of iterations for the IEEE 300-bus system
8
10
12
14
16
18
20
22
Fig. 6 Objective function as a function of the number of iterations for the IEEE 300-bus system
10 4
7
10 3
1 Dommel, H.W., and Tinney, W.F.: ‘Optimal power flow solutions’, IEEE Trans. Power Appar. Syst., 1968, 87, (10), pp. 1866–1876 2 Stott, B., and Marinho, J.L.: ‘Linear programming for power system network security applications’, IEEE Trans. Power Appar. Syst., 1979, 98, (3), pp. 837–848 3 Alsac, O., Bright, J., Prais, M., and Stott, B.: ‘Further developments in LP-based optimal power flow’, IEEE Trans. Power Syst., 1990, 5, (3), pp. 697–711 4 Sun, D.I., Ashley, B., Brewer, B., Hughes, A., and Tinney, W.F.: ‘Optimal power flow by Newton approach’, IEEE Trans. Power Appar. Syst., 1984, 103, (10), pp. 2864–2880 5 Monticelli, A., and Liu, W.-H.E.: ‘Adaptive movement penalty method for the Newton optimal power flow’, IEEE Trans. Power Syst., 1992, 7, (1), pp. 334–342 6 Burchett, R.C., Happ, H.H., and Veirath, D.R.: ‘Quadratically convergent optimal power flow’, IEEE Trans. Power Appar. Syst., 1984, 103, (11), pp. 3267–3275 7 Glavitsch, H., and Spoerry, M.: ‘Quadratic loss formula for reactive dispatch’, IEEE Trans. Power Appar. Syst., 1983, 102, (12), pp. 3850– 3858 8 Karmarkar, N.: ‘A new polynomial time algorithm for linear programming’, Combinatorica, 1984, 4, pp. 373–395 9 Lu, N., and Unum, M.R.: ‘Network constrained security control using an interior point algorithm’, IEEE Trans. Power Syst., 1993, 8, (3), pp. 1068–1076 10 Vargas, L.S., Quintana, V.H., and Vannelli, A.: ‘A tutorial description of an interior point method and its applications to security-constrained economic dispatch’, IEEE Trans. Power Syst., 1993, 8, (3), pp. 1315– 1323 11 Zhang, X.-P., and Chen, Z.: ‘Security-constrained economic dispatch through interior point methods’, Autom. Electr. Power Syst., 1997, 21, (6), pp. 27–29 12 Momoh, J.A., Guo, S.X., Ogbuobiri, E.C., and Adapa, R.: ‘The quadratic interior point method solving power system optimisation problems’, IEEE Trans. Power Syst., 1994, 9, (3), pp. 1327–1336 13 Granville, S.: ‘Optimal reactive power dispatch through interior point methods’, IEEE Trans. Power Syst., 1994, 9, (1), pp. 136–146 14 Wu, Y.C., Debs, A.S., and Marsten, R.E.: ‘A direct nonlinear predictor-corrector primal-dual interior point algorithm for optimal power flows’, IEEE Trans. Power Syst., 1994, 9, (2), pp. 876–883 15 Irisarri, G.D., Wang, X., Tong, J., and Mokhtari, S.: ‘Maximum loadability of power systems using interior point nonlinear optimisation method’, IEEE Trans. Power Syst., 1997, 12, (1), pp. 167–172 16 Wei, H., Sasaki, H., and Yokoyama, R.: ‘An interior point nonlinear programming for optimal power flow problems within a novel data structure’, IEEE Trans. Power Syst., 1998, 13, (3), pp. 870–877
10 2 101 complementary gap
6
number of iterations
number of iterations
10 0 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 power mismatch method
10 − 6
current mismatch method
10 − 7 0
2
4
6
8
10
12
14
16
18
20
22
number of iterations
Fig. 5 Complementary gap as a function of the number of iterations for the IEEE 300-bus system
current mismatch formulation is comparable in terms of computational performance with that of the power mismatch formulation. Preliminary numerical results indicate that the computational complexity of the current mismatch OPF per iteration is less than that of power mismatch OPF. 6
4
Acknowledgment
Partial financial support of this work from EPSRC under contract GR/R60959 is gratefully acknowledged. IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
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17 Torres, G.L., and Quintana, V.H.: ‘An interior point method for non-linear optimal power flow using voltage rectangular coordinates’, IEEE Trans. Power Syst., 1998, 13, (4), pp. 1211–1218 18 Zhang, X.-P., and Handschin, E.J.: ‘Advanced implementation of UPFC in a nonlinear interior point OPF’, IEE Proc., Gener., Transm. Distrib., 2001, 148, (5), pp. 489–496 19 Xie, K., and Song, Y.H.: ‘Power market oriented optimal power flow via an interior point method’, IEE Proc., Gener., Transm. Distrib., 2001, 148, (6), pp. 549–555 20 Zhang, X.-P., and Handschin, E.: ‘Transfer capability computation of power systems with comprehensive modelling of FACTS controllers’. Presented at the 14th Power System Computation Conf. (PSCC), Sevilla, Spain, 24–28 June 2002 21 El-Bakry, S., Tapia, R.A., Tsuchiya, T., and Zhang, Y.: ‘On the formulation and theory of the Newton interior-point method for nonlinear programming’, J. Optim. Theory Appl., 1996, 89, (3), pp. 507–541 22 Menengoy Da Costa, V., Martins, N., and Pereira, J.-L.R.: ‘Developments in the Newton-Raphson power flow formulation based on current injections’, IEEE Trans. Power Syst., 1999, 14, (4), pp. 1320–1326 23 http://www.ee.washington.edu/research/pstca/, accessed May 2005
constraint is represented in the current formulation, the contribution of the transmission line constraint to Hij becomes zero.
8.2 Transmission line constraint 8.2.1 Power mismatch formulation: The contribution of the transmission line constraint to the elements of the Hessian matrix H is given by the second derivative of Sij2 : @Sij2 @ei @ei
4Qij bij þ 2ð2ei bij Gij fj Bij ej Þ2 @Sij2
8 Appendix: The first and second derivatives
@ei @fi
Some of the key elements of the formulation of both the current and power mismatch approaches are presented in this Appendix allowing easier comparison between them.
mismatch formulation, the first derivatives of (61) and (62) will contribute to the diagonal elements Jii and the nondiagonal elements Jij (voltage dependent). The second derivatives of (61) and (62) will contribute to the nondiagonal elements Hij of the Hessian matrix H (not zero): X ½ei ðGij ej Bij fj Þ þ fi ðGij fj þ Bij ej Þ ð61Þ Pi ¼ X Qi ¼ ½fi ðGij ej Bij fj Þ ei ðGij fj þ Bij ej Þ ð62Þ The second derivatives of Pi and Qi become constant but not zero. @ 2 Pi ¼ Gij ; @ei @ej @ 2 Pi ¼ Bij ; @ei @fj
@ 2 Pi ¼ Gij ; @fi @fj
@ 2 Qi ¼ Bij @fi @fj
2Bij Qij þ2ð2ei bij Gij fj Bij ej ÞðGij fi Bij ei Þ @Sij2 ¼ 2Bij Pij þ 2ð2gij ei þ Gij ej Bij fj ÞðGij fi Bij ei Þ @ei @fj 2Gij Qij 2ð2ei bij Gij fj Bij ej ÞðGij ei þ Bij fi Þ @Sij2 @fi @fi
¼ 4gij Pij þ 2ð2gij fi þ Gij fj þ Bij ej Þ2
þ 2Gij Qij þ2ð2bij fi þ Gij ej Bij fj ÞðGij fi Bij ei Þ @Sij2 ð63Þ
8.1.2 Current mismatch formulation : The second derivatives of the following real and imaginary current components will contribute to the non-diagonal elements Hij of the Hessian matrix H: X ðGij ej Bij fj Þ ð64Þ Ixi ¼ X ðGij fj Bij ej Þ ð65Þ Iyi ¼ From (64) and (65), it can be found that the first derivatives, which contribute to the diagonal elements Jii and nondiagonal elements Jij, are constant, and the second derivatives, which contribute to the non-diagonal elements Hij of the Hessian matrix are zero. This is because the relation between the real and imaginary current components Ixi and Iyi at bus i are linear functions of voltages ej and fj. Furthermore, when the transmission line capacity 804
¼ 2Gij Pij þ 2ð2gij ei þ Gij ej Bij fj ÞðGij ei þ Bij fi Þ
@Sij2 ¼ 2Bij Pij þ 2ð2gij fi þ Gij fj þ Bij ej ÞðGij ei þ Bij fi Þ @fi @ej
@ 2 Qi ¼ Gij @ei @fj @ 2 Qi ¼ Gij @fi @ej
@ei @ej
4Qij bij þ 2ð2fi bij þ Gij ej Bij fj Þ2
@ 2 Qi ¼ Bij @ei @ej
@ 2 Pi ¼ Bij ; @fi @ej
¼ 2ð2gij ei þ Gij ej Bij fj Þð2gij fi þ Gij fj þ Bij ej Þ þ 2ð2bij ei Gij fj Bij ej Þð2bij fi þGij ej Bij fj Þ
@Sij2
8.1 Non-diagonal Hessian matrix elements 8.1.1 Power mismatch formulation: For the power
¼ 2ð2gij Pij Þ þ 2ð2gij ei þ Gij ej Bij fj Þ2
@fi @fj
¼ 2Gij Pij þ 2ð2gij fi þ Gij fj þ Bij ej ÞðGij fi Bij ei Þ 2Bij Qij 2ð2bij fi þGij ej Bij fj ÞðGij ei þBij fi Þ
@Sij2 ¼ 2ðGij ei þ Bij fi Þ2 þ 2ðGij fi Bij ei Þ2 @ej @ej @ 2 Sij2 ¼ 2ðGij fi Bij ei Þ2 þ 2ðGij ei þ Bij fi Þ2 @fj @fj
ð66Þ
8.2.2 Current
mismatch formulation: The contribution of a transmission line constraint to the elements of the Hessian matrix H is primarily given in (59) by the sum of the second derivatives of the following expressions showing the real and imaginary current components: Ixij ¼ gii ei bii fi þ Gij ej Bij fj
ð67Þ
Iyij ¼ gii fi þ bii ei þ Gij fj þ Bij ej
ð68Þ
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
The second derivatives of Ixij and Iyij become constant and are given by: @ 2 Ixij ¼ 2g2ii @ei @ei
@ 2 Iyij ¼ 2b2ii @ei @ei
@ 2 Ixij ¼ 2gii bii @ei @fi
@ 2 Iyij ¼ 2gii bii @ei @fi
@ 2 Ixij ¼ 2gii Gij @ei @ej
@ 2 Iyij ¼ 2bii Bij @ei @ej
@ 2 Ixij ¼ 2gii Bij @ei @fj
@ 2 Iyij ¼ 2bii Gij @ei @fj
@ 2 Ixij ¼ 2b2ii @fi @fi
@ 2 Iyij ¼ 2g2ii @fi @fi
@ 2 Ixij ¼ 2bii Gij @fi @ej
@ 2 Iyij ¼ 2gii Bij @fi @ej
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
@ 2 Ixij ¼ 2bii Bij @fi @fj
@ 2 Iyij ¼ 2gii Gij @fi @fj
@ 2 Ixij ¼ 2G2ij @ej @ej
@ 2 Iyij ¼ 2B2ij @ej @ej
@ 2 Ixij ¼ 2Gij Bij @ej @fj
@ 2 Iyij ¼ 2Bij Gij @ej @fj
@ 2 Ixij ¼ 2B2ij @fj @fj
@ 2 Iyij ¼ 2G2ij @fj @fj
ð69Þ
It can be seen that the second derivatives in (69) are much simpler than those in (66). The second derivatives of (69) are constant and can be calculated before the iterating loop.
805
