Electr Eng (2007) 89:525–533 DOI 10.1007/s00202-006-0035-1
O R I G I NA L PA P E R
Fault location algorithm for cross-bonded cables using the singularity of the sheath impedance matrix Sang-Won Min · Soon-Ryul Nam · Sang-Hee Kang · Jong-Keun Park
Received: 15 April 2006 / Accepted: 22 May 2006 / Published online: 11 July 2006 © Springer-Verlag 2006
Abstract This paper proposes a fault location algorithm for cross-bonded cables. The algorithm uses nonlinear equations that describe the voltage drops between both ends of core conductors and sheaths in terms of the phase voltages and currents of both the local and remote ends. Arranging these equations about the sheath currents yields a sheath impedance matrix, which is singular independently of the fault distance. Due to this singularity, the equations have no solution or an infinite number of solutions. The proposed algorithm finds the condition for which a solution to the equations exists. Since the equation of the condition is linear and explicit, the fault location can be found easily. Test results indicate that the algorithm accurately estimates the fault distance regardless of fault impedance, fault type, and fault section. Keywords Fault location · Cross-bonded cable · Sheath impedance matrix · Singularity S.-W. Min (B) Electro-Fusion Research Division, Korea Electrotechnology Research Institute, Uiwang 437-808, Korea e-mail:
[email protected] S.-R. Nam · S.-H. Kang Next-Generation Power Technology Center and Department of Electrical Engineering, Myongji University, Yongin 449-728, Korea e-mail:
[email protected] S.-H. Kang e-mail:
[email protected] J.-K. Park School of Electrical Engineering, Seoul National University, Seoul 151-742, Korea e-mail:
[email protected]
1 Introduction Since overhead lines can easily be damaged and there often regarded as unsightly, the installation of underground cables has increased; 82.49% of the transmission lines in Seoul, the capital of Korea, are underground [1]. Although underground cables enhance the reliability of power systems by reducing the possibility of a fault occurrence, once a fault occurs in underground cables, the repair is both more costly and time-consuming than for overhead lines. Therefore, it is important to locate the fault in underground cable systems accurately. Until now, offline fault location methods such as the Murray loop method, pulse radar method, and dc thumper have commonly been used for underground cable systems [2, 3]. For these methods, it is necessary to separate the faulted line from the system at the substations at both ends and connect it to the measuring equipment. Therefore, they are time consuming and may cause additional damage to the healthy parts of the cable system. To cope with these drawbacks, considerable research effort has been devoted to developing traveling-wave-based methods for locating faults in underground cable systems [4–8]. Since the fault-generated traveling wave contains information about the fault location, traveling-wave-based methods are capable of locating most faults. However, when a fault occurs near a busbar, the time difference between the arrival of the incident wave and the arrival of its reflection from the busbar will be so short that the waves are unlikely to be detected separately. Furthermore, to improve the accuracy, sampling rate should be much higher by several mega hertz than general sampling rates of relays. Artificial neural networks (ANNs) can be introduced to locate faults in underground cable systems [2, 9].
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Although the performance of neural-network approaches can be enhanced by increasing the quantity of learning data, it is still possible that the solution is ambiguous. Another approach uses impedance-based methods [10, 13, 12], which use both voltage and current signals available from measuring points. The authors of [11] proposed the distance-relaying algorithm with an error compensation factor for overhead lines combined with underground cables. Although the algorithm is easily applied to modern digital impedance relays, it is not clear how to determine the error-compensation factors for different systems and the fault can be located accurately only if it occurs at a joint between major sections. In [12], a distributed-circuit model is used for impedance analysis of cable systems considering their complicated structures. Although the method performs well for solid-bonded cable systems, it is difficult to apply for cross-bonded cable systems, which are frequently used to reduce both the sheath-induced voltage and circulating current. Due to the cross-bonding of sheaths, different models are needed for each minor section of cross-bonded systems and the impedance seen at the terminals shows discontinuity at the cross-bonded link points [10]. This paper proposes a fault location algorithm for cross-bonded cable systems. The algorithm is based on the singularity of the sheath impedance matrix. Using fault signals generated by the electromagnetic transient program (EMTP), the performance of the proposed algorithm was investigated. 2 Underground power cable 2.1 Structure of underground cable Single core coaxial (SC) cables for extra-high voltage systems can be classified into cross-linked polyethylene (XLPE) insulated cables and oil-filled (OF) cables according to the type of insulation. Figure 1 shows the
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structures of OF and XLPE cables. Both have similar structures, consisting of a core conductor, insulation, metallic sheath, and jacket. As shown in Fig. 2, SC cables have two closed loops; loop1 is formed by the core conductor and the metallic sheath, loop2 by metallic sheath and earth [13]. The voltage drops along the two loops can be expressed as Z11 Z12 I1 V1 = (1) V2 Z21 Z22 I2 where, V1 V2 I1 I2 Z11
Z22
Z12 , Z21
: voltage drop per unit length along loop1 : voltage drop per unit length along loop2 : current along loop1 : current along loop2 : impedance per unit length from the outer core conductor to the inner metallic sheath via the insulation : impedance per unit length from the outer metallic sheath to the earth via the jacket : impedance per unit length between loop1 and loop2 .
The voltage and currents along two loops are related to those of the core conductor and sheath as follows: Vcore − Vsheath V1 = (2) V2 Vsheath I1 Icore = (3) I2 Icore + Isheath Substituting (2) and (3) into (1) yields Vcore Zcc Zcs Icore = Vsheath Zcs Zss Isheath where, Zcc = Z11 + 2Z12 + Z22 Zcs = Z12 + Z22 Zss = Z22
Fig. 1 Structures of OF and XLPE cables
Fig. 2 Electrical loops of SC cable
(4)
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Similarly, for the three-phase cable system shown in Fig. 3, voltage drops of the core conductor and sheath can be expressed in terms of an impedance matrix and current vector as follows: ⎞ ⎛ Zcc Va ⎜ Vb ⎟ ⎜Zab ⎟ ⎜ ⎜ ⎜ Vc ⎟ ⎜Zca ⎟ ⎜ ⎜ ⎜Vsa ⎟=⎜ Zcs ⎟ ⎜ ⎜ ⎝Vsb ⎠ ⎝Zab ⎛
Vsc
Zca
Zab Zcc Zbc Zab Zcs Zbc
Zca Zbc Zcc Zca Zbc Zcs
Zcs Zab Zca Zss Zab Zca
Zab Zcs Zbc Zab Zss Zbc
⎞⎛ ⎞ Ia Zca ⎜ Ib ⎟ Zbc ⎟ ⎟⎜ ⎟ ⎜ ⎟ Zcs ⎟ ⎟⎜ Ic ⎟ (5) ⎜ ⎟ Zca ⎟ ⎟⎜Isa ⎟ Zbc ⎠⎝Isb ⎠ Zss
Isc
where Zab , Zbc , and Zca are mutual impedances between each phase. This paper ignores the distance between the core conductor and sheath because it is much smaller than the distance between phases; the mutual impedance between the a-phase core and b-phase core is the same as the impedance between the a-phase sheath and the b-phase core. In Fig. 2, there is no linkage between the current leakages of loop1 and loop2 . Therefore, current leakages can be described as follows: I1 Y1 V1 0 = (6) I2 V2 0 Y2 Substituting (2) and (3) into (6) yields
Icore Isheath
=
Ycc Ycs
Ycs Yss
Vcore Vsheath
(7)
where, Ycc = Y1 Ycs = −Y1 Yss = Y1 + Y2 Since there is no linkage between the current leakages of each phase, the current leakages of the system shown in Fig. 3 are given by
⎞ ⎛ ⎞⎛ ⎞ Ycc 0 Va Ia 0 Ycs 0 0 ⎜ Ib ⎟ ⎜ 0 Ycc 0 ⎟⎜ Vb ⎟ 0 Y 0 cs ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Ic ⎟ ⎜ 0 0 Ycs ⎟ 0 Ycc 0 ⎜ ⎟=⎜ ⎟⎜ Vc ⎟ ⎜Isa ⎟ ⎜Ycs 0 ⎟ ⎟ 0 Yss 0 0 ⎟⎜ ⎜Vsa ⎟ ⎜ ⎟ ⎜ ⎝Isb ⎠ ⎝ 0 Ycs 0 0 Yss 0 ⎠⎝Vsb ⎠ Isc 0 Yss Vsc 0 0 Ycs 0 ⎛
Single core coaxial (SC) cables are generally used for extra-high voltage systems, there are no metallic jackets except for submarine systems. SC cables include crosslinked polyethylene (XLPE) insulated cables as well as oil-filled (OF) cables on the way of insulation. In Korea, it is essential for installation of XLPE cables instead of OF cables after domestic cable companies developed XLPE cables. Currently, more than half of cables in transmission systems are XLPE cables. Figure 1 shows the structures of both OF cables and XLPE cables.
2.2 Impedance of cross-bonded cable To reduce both the sheath-induced voltage and circulating current, a cross-bonding sheath is commonly used in the Korean transmission system, as shown in Fig. 4. From the viewpoint of analysis, the cross-bonding makes it difficult to locate faults because the impedance seen at a terminal has a discontinuity at the cross-bonded link point. As a result of this discontinuity, each minor section of cross-bonded cables must be analyzed with different impedance matrix. The first minor section of the cross-bonded cable has the same structure as the system in Fig. 3. The second and third minor sections have a change in the structure of the sheaths. Isa flows from the a-phase sheath of the first minor section via the b-phase sheath of the second to the c-phase sheath of the third. Similarly, Isb flows from the b-phase sheath via the c-phase sheath to the a-phase sheath, and Isc from the c-phase sheath via the a-phase sheath to the b-phase sheath. This structure yields a different impedance matrix for each minor section. The voltage drops per unit length along the second minor section can be expressed in terms of an impedance matrix and current vector as follows: ⎞ ⎛ Zcc Va ⎜ Vb ⎟ ⎜Zab ⎟ ⎜ ⎜ ⎜ Vc ⎟ ⎜Zca ⎟ ⎜ ⎜ ⎜Vsa ⎟=⎜Zab ⎟ ⎜ ⎜ ⎝Vsb ⎠ ⎝Zca Vsc Zcs ⎛
Fig. 3 Three phase underground cable system
(8)
Zab Zcc Zbc Zcs Zbc Zab
Zca Zbc Zcc Zbc Zcs Zca
Zab Zcs Zbc Zss Zbc Zab
Zca Zbc Zcs Zbc Zss Zca
⎞⎛ ⎞ Ia Zcs ⎜ Ib ⎟ Zab ⎟ ⎟⎜ ⎟ ⎜ ⎟ Zca ⎟ ⎟⎜ Ic ⎟ (9) ⎜ ⎟ Zab ⎟ ⎟⎜Isa ⎟ ⎠ Zca ⎝Isb ⎠ Zss
Isc
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Fig. 4 Cross-bonded underground cable
Similarly, the voltage drops per unit length along the third minor section are given by ⎞ ⎛ ⎞⎛ ⎞ ⎛ Zcc Zab Zca Zca Zcs Zab Ia Va ⎜ Vb ⎟ ⎜Zab Zcc Zbc Zbc Zab Zcs ⎟⎜ Ib ⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎜ Vc ⎟ ⎜Zca Zbc Zcc Zcs Zca Zbc ⎟⎜ Ic ⎟ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎜Vsa ⎟ ⎜Zca Zbc Zcs Zss Zca Zbc ⎟⎜Isa ⎟ (10) ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎝Vsb ⎠ ⎝ Zcs Zab Zca Zca Zss Zab ⎠⎝Isb ⎠ Vsc
Zab Zcs Zbc Zbc Zab Zss
Isc
Voltage drop from the sending end to the fault point is described as follows: V1 − VF = xV11
(15)
Similarly, the voltage drop from the fault point to the receiving end is expressed as follows: VF − V2 = (l1 − x)V12 + l2 V22 + l3 V32
(16)
In this paper, the impedance matrices in (5), (9), and (10) will be called as Z1 , Z2 , and Z3 , respectively.
Eliminating VF from (15) and (16) yields
3 Fault location algorithm for cross-bonded cables
V1 − V2 = xV11 + (l1 − x)V12 + l2 V22 + l3 V32 = xZ1 I1 + (l1 − x)Z1 I2 + l2 Z2 I2 + l3 Z3 I2
3.1 Cross-bonded cable with one major section
(17)
It is assumed that a fault occurs in the first minor section of the cross-bonded cable, as shown in Fig. 5. The voltage of each core and sheath at the fault point F, VF , depends on the fault conditions, such as fault location, fault resistance, and fault type. If VF can be eliminated using the currents and voltages of both the local and remote ends, the fault equation is derived without using the fault location, fault resistance, and fault type. The cable system shown in Fig. 5 can be divided into four parts: the left and right parts of the first minor section from the fault point, the second minor section, and the third minor section. The voltage drops along each part are given by V11 = Z1 I1
(11)
V12 = Z1 I2
(12)
V22 = Z2 I2
(13)
V32 = Z3 I2
(14)
where, Vij I1 I2
: voltage drop per unit length when Ij flows along the ith minor section : current from the sending end S to the fault point : current from the fault point to the receiving ends R.
In (17), the unknown variables are the sheath currents and fault distance (Isa1 , Isb1 , Isc1 , Isa2 , Isb2 , Isc2 , x). Arranging (17) about the sheath currents forms B(x) = A(x)ISh
(18)
where,
T ISh = Isa1 Isb1 Isc1 Isa2 Isb2 Isb2 ⎛ a11 (x) ⎜a21 (x) ⎜ ⎜a31 (x) A(x) = ⎜ ⎜a41 (x) ⎜ ⎝a51 (x) a61 (x)
a12 (x) a22 (x) a32 (x) a42 (x) a52 (x) a62 (x)
a13 (x) a23 (x) a33 (x) a43 (x) a53 (x) a63 (x)
a14 (x) a24 (x) a34 (x) a44 (x) a45 (x) a46 (x)
a15 (x) a25 (x) a35 (x) a45 (x) a55 (x) a56 (x)
⎞ a16 (x) a26 (x)⎟ ⎟ a36 (x)⎟ ⎟ a46 (x)⎟ ⎟ a56 (x)⎠ a66 (x)
T B(x) = b1 (x) b2 (x) b3 (x) b4 (x) b5 (x) b6 (x) . Each element of A(x) and B(x) is shown in the Appendix. Particularly, the elements in each column of A(x) have the following relation: 6 k=1
ck akj (x) = 0,
j = 1, . . . , 6,
(19)
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Fig. 5 Fault in the first minor section of a cross-bonded cable
where,
where ⎛ ⎛ ⎞ ⎞
Ia2
Va1 − Va2 ⎜ ⎜ ⎟ ⎟ f = l e1 e2 e3 ⎝ Ib2 ⎠ + c1 c2 c3 ⎝ Vb1 − Vb2 ⎠ Ic2 Vc1 − Vc2 ⎛ ⎞
Ia1 − Ia2 ⎜ ⎟ g = e4 e5 e6 ⎝ Ib1 − Ib2 ⎠ Ic1 − Ic2
c1 = k−1 (d11 Zcs + d12 Zab ) c2 = k−1 (d21 Zcs + d22 Zca ) c3 = k−1 (d31 Zcs + d32 Zbc ) c4 = c5 = c6 = −1 3 2 2 2 k = Zcs − Zcs (Zab + Zbc + Zca ) + 2Zab Zbc Zca 2 2 d11 = (Zcs −Zss −Zab )(Zbc +Zca )−Zbc −Zca +Zcs Zss
e1 = Zcs + Zab + Zca − aZcc − bZab − cZca
d12 = Zss (Zbc + Zca − Zab ) + 2Zbc Zca
e2 = Zcs + Zab + Zbc − aZab − bZcc − cZbc
d21 =
2 2 (Zcs −Zss −Zca )(Zab +Zbc )−Zab −Zbc +Zcs Zss
e4 = aZcc + bZab + cZca − Zcs − Zab − Zca
d22 = Zss (Zab + Zbc − Zca ) + 2Zab Zbc d31 =
e3 = Zcs + Zbc + Zca − aZca − bZbc − cZcc
2 2 (Zcs −Zss −Zbc )(Zab +Zca )−Zab −Zca +Zcs Zss
e5 = aZab + bZcc + cZbc − Zcs − Zab − Zbc
d32 = Zss (Zab + Zca − Zbc ) + 2Zab Zca
e6 = aZca + bZbc + cZcc − Zcs − Zbc − Zca
From (19), it is found that the rank of A(x) is 5, independent of x. Therefore, the linear equation in (18) has solutions only if the following condition is satisfied:
When a fault occurs on the second minor section of a cross-bonded cable, as shown in Fig. 6, the explicit solution x can be found in a manner similar to that used to find x.
6
x=
ck bk (x) = 0
(20)
k=1
Some minor calculations give the following expression for the difference between x and x :
The solution of (20) is given by x=
f (Va1 , Vb1 , Vc1 , Va2 , Vb2 , Vc2 , Ia2 , Ib2 , Ic2 ) , g(Ia1 , Ib1 , Ic1 , Ia2 , Ib2 , Ic2 )
f (Va1 , Vb1 , Vc1 , Va2 , Vb2 , Vc2 , Ia1 , Ib1 , Ic1 , Ia2 , Ib2 , Ic2 ) g(Ia1 , Ib1 , Ic1 , Ia2 , Ib2 , Ic2 ) (22)
(21)
x − x =
l1 g f − f = = l1 g g
(23)
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Fig. 6 Fault in the second minor section of a cross-bonded cable
Fig. 7 Fault in the third minor section of a cross-bonded cable
When a fault occurs on the third minor section of a cross-bonded cable, as shown in Fig. 7, the difference between x and x can be written as (23): x − x = = l1 + l2
(24)
The results of (23) and (24) mean that the solution of (21), which is for the first minor section, covers every other section. Due to the singularity of A(x), the length of each section has no affect on ck . Therefore, ck is identical in all the cross-bonded cables. These results can be extended to general cable systems that have two or more major sections. In other words, the fault distance can be found from any solution of any section’s fault, independent of the fault impedance, fault type, and fault section.
⎞ ⎛ ⎞ IaS VaS − VaA ⎜ IbS ⎟ ⎜VbS − VbA ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ VcS − VcA ⎟ ⎟ = (l11 Z1 + l12 Z2 + l13 Z3 )⎜ IcS ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎜ IsaS ⎟ ⎟ ⎜ ⎝ ⎠ ⎝ IsbS ⎠ 0 IscS 0 VCoreS − VCoreA M N I CoreS = 0 O P I SHS ⎛
(27)
(28)
where,
T I CoreS = IaS IbS IcS
T I SHS = IsaS IsbS IscS From (28), the currents of the metallic sheaths can be represented as the currents of the core conductors:
3.2 Cross-bonded cable with multi major sections
I SHS = P−1 (−O)I CoreS
As shown in Fig. 8, the shunt capacitances of the system are considered at grounding points, where the sheath voltages are all zero due to the grounding. Considering the shunt capacitances, the currents from the sending end, S, can be described as follows:
Since each voltage and current of the sending end, S, is known, the each voltage and current at point Acan be written as:
VCoreA = VCoreS − M + NP−1 (−O) I CoreS (30)
I S = IS − l1 YVS .
I A = IA − l2 YVA
(25)
Consequently, the voltage drops from S to A can be written as: VS − VA = l11 Z1 I S + l12 Z2 I S + l13 Z3 I S .
(26)
(29)
(31)
Similarly, the current deviations at each grounding point can be considered using (25), and the voltages at grounding points can be found using (30). As mentioned above, the fault distance can be found from any solution of the fault for any section. In the
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Fig. 8 A model 154-kV underground cable system
case study, it is assumed that a fault occurs in the second minor section of the third major section.
4 Performance evaluation 4.1 Simulation data The performance of the algorithm was evaluated for various faults on a 154-kV 4.491-km underground cable system, as shown in Fig. 8. The system consists of five major sections and each major section has three minor sections with the following lengths: Fig. 9 Model cable arrangement
(312m − 291m − 300m) − (330m − 300m − 291m) −(270m − 312m − 300m) − (312m − 330m − 312m) −(270m − 291m − 270m). The EMTP was used to generate fault signals for aphase-core-to-sheath-to-ground (a − s − g) faults and bphase-core-to-sheath (b − s) faults. The sampling frequency was set to 7,760 Hz, i.e., 96 samples per cycle in 60-Hz systems. The phase angle difference between the sending and receiving ends is set to 10◦ . The EMTP output was pre-conditioned using a second-order Butterworth low-pass filter with a cutoff frequency of 300 Hz in order to reject high frequency components. The Discrete Fourier Transform (DFT) was used to extract the accurate phasors of the voltage and current signals. The underground cable parameters considered were obtained using the EMTP cable constant subroutine with the Korea Electric Power Cooperation (KEPCO)
cable arrangement shown in Fig. 9. Details of the cable materials and related parameters are given in Table 1. 4.2 Performance of the proposed algorithm Figure 10 shows the time response of the proposed algorithm for an a-s-g fault (fault inception angle: 0◦ and 90◦ , fault resistance: 1 , fault distance: 0.9 pu). It was found that the algorithm has a transient time of about 20 ms caused by the low-pass filter and DFT. As a performance index, the estimation error of the fault location was calculated using the following equation: %Error =
Estimated distance−Actual distance ×100, Total length (32)
532 Table 1 Specifications of the OF cable materials for a 154-kV system
Table 2 Estimation errors
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Radius (mm)
r1 = 7, r2 = 28.95, r3 = 42.45, r4 = 45.15, r5 = 49.65
Core conductor Insulation Sheath Jacket
2000 mm2 copper, ρ = 1.7241E − 8m, µ = 1.0 µ = 1.0, ε = 3.4 (kraft) Aluminum, ρ = 2.84E − 8m, µ = 1.0 µ = 1.0, ε = 3.4
Fault distance (m)
408 903 1333 1824 2198 2706 3128 3556 4124
Estimation error (%) a − s − g fault
b − s fault
0◦
0◦
90◦
1
10
100
0
0
0.0025 −0.0244 −0.0484 −0.0754 −0.0963 −0.1244 −0.1483 −0.1718 −0.2038
0.0025 −0.0244 −0.0484 −0.0754 −0.0963 −0.1244 −0.1483 −0.1718 −0.2038
0.0025 −0.0244 −0.0484 −0.0754 −0.0963 −0.1244 −0.1483 −0.1718 −0.2038
0.0025 −0.0244 −0.0484 −0.0754 −0.0963 −0.1244 −0.1483 −0.1718 −0.2038
0.0073 0.0141 0.0194 0.0260 0.0305 0.0372 0.0422 0.0480 0.0551
Fig. 10 Estimated fault distances for an a-s-g fault (fault inception angle: 0◦ and 90◦ , fault resistance: 1 , fault distance: 4124 meters)
where the estimated distance indicates the half-cycle average after two cycles following the fault occurrence. Table 2 summarizes the estimation errors for a − s − g and b − s faults. It was easily found that the algorithm estimates the fault distances accurately, independent of fault impedance, fault type, and fault section; the maximum estimation error is about 0.2038%. It was also found that most estimation errors have negative values because of the asymmetry of the length of each section.
when a communication channel between the local and remote ends is available. The proposed algorithm, which is based on the singularity of the sheath impedance matrix, is capable of locating faults independent of the fault impedance, fault type, and fault section. The performance of the algorithm was evaluated for various faults on a 154-kV cross-bonded cable system. The EMTP was used to generate fault signals for different fault locations and fault resistances. The evaluation results show that the proposed algorithm performs well with a maximum estimation error of 0.2038%. Therefore, the algorithm is considered to be useful for locating faults in cross-bonded cable systems. Acknowledgment Authors would like to thank Korea Ministry of Science and Technology and Korea Science and Engineering Foundation for their support through ERC program.
Appendix a11 (x) = xZcs , a12 (x) = xZab , a13 (x) = xZca , a14 (x) = (l1 − x)Zcs + l2 Zab + l3 Zca , a15 (x) = (l1 − x)Zab + l2 Zca + l3 Zcs , a16 (x) = (l1 − x)Zca + l2 Zcs + l3 Zab , a21 (x) = xZab , a22 (x) = xZcs , a23 (x) = xZbc ,
5 Conclusions A new fault location algorithm for cross-bonded cables was proposed. The phase voltages and currents of both the local and remote ends are used as input for the algorithm. The algorithm is capable of locating faults
a24 (x) = (l1 − x)Zab + l2 Zcs + l3 Zbc , a25 (x) = (l1 − x)Zcs + l2 Zbc + l3 Zab , a26 (x) = (l1 − x)Zbc + l2 Zab + l3 Zcs , a31 (x) = xZca , a32 (x) = xZbc , a33 (x) = xZcs , a34 (x) = (l1 − x)Zca + l2 Zbc + l3 Zcs ,
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a35 (x) = (l1 − x)Zbc + l2 Zcs + l3 Zca , a36 (x) = (l1 − x)Zcs + l2 Zca + l3 Zbc , a41 (x) = xZss , a42 (x) = xZab , a43 (x) = xZca , a44 (x) = (l1 − x)Zss + l2 Zss + l3 Zss ,
533
⎛
xZca Ia1 + xZbc Ib1 + xZcs Ic1
⎞
⎜ ⎟ ⎜ + {(l1 − x)Zca + l2 Zcs + l3 Zab }Ia2 ⎟ ⎜ ⎟. b6 (x) = −⎜ ⎟ ⎝ + {(l1 − x)Zbc + l2 Zbc + l3 Zcs }Ib2 ⎠ + {(l1 − x)Zcs + l2 Zca + l3 Zbc }Ic2
a45 (x) = (l1 − x)Zab + l2 Zbc + l3 Zca , a46 (x) = (l1 − x)Zca + l2 Zab + l3 Zbc , a51 (x) = xZab , a52 (x) = xZss , a53 (x) = xZbc , a54 (x) = (l1 − x)Zab + l2 Zbc + l3 Zca , a55 (x) = (l1 − x)Zss + l2 Zss + l3 Zss , a56 (x) = (l1 − x)Zbc + l2 Zca + l3 Zab , a61 (x) = xZca , a62 (x) = xZbc , a63 (x) = xZss , a64 (x) = (l1 − x)Zca + l2 Zab + l3 Zbc , a65 (x) = (l1 − x)Zbc + l2 Zca + l3 Zab , a66 (x) = (l1 − x)Zss + l2 Zss + l3 Zss , b1 (x) = Va1 − Va2 ⎛ ⎞ xZcc Ia1 + xZab Ib1 + xZca Ic1 ⎜ ⎟ ⎜ + {(l1 − x)Zcc + l2 Zcc + l3 Zcc }Ia2 ⎟ ⎜ ⎟, −⎜ ⎟ ⎝ + {(l1 − x)Zab + l2 Zab + l3 Zab }Ib2 ⎠ + {(l1 − x)Zca + l2 Zca + l3 Zca }Ic2 b2 (x) = Vb1 − Vb2 ⎛ ⎞ xZab Ia1 + xZcc Ib1 + xZbc Ic1 ⎜ ⎟ ⎜ + {(l1 − x)Zab + l2 Zab + l3 Zab }Ia2 ⎟ ⎜ ⎟, −⎜ ⎟ }I {(l + − x)Z + l Z + l Z cc 2 cc 3 cc b2 1 ⎝ ⎠ + {(l1 − x)Zbc + l2 Zbc + l3 Zbc }Ic2 b3 (x) = Vc1 − Vc2 ⎛ ⎞ xZca Ia1 + xZbc Ib1 + xZcc Ic1 ⎜ ⎟ ⎜ + {(l1 − x)Zca + l2 Zca + l3 Zca }Ia2 ⎟ ⎟ −⎜ ⎜ + {(l − x)Z + l Z + l Z }I ⎟, 2 bc 3 bc b2 ⎠ bc 1 ⎝ + {(l1 − x)Zcc + l2 Zcc + l3 Zcc }Ic2 ⎛ ⎞ xZcs Ia1 + xZab Ib1 + xZca Ic1 ⎜ ⎟ ⎜ + {(l1 − x)Zcs + l2 Zab + l3 Zca }Ia2 ⎟ ⎜ ⎟, b4 (x) = −⎜ ⎟ ⎝ + {(l1 − x)Zab + l2 Zab + l3 Zbc }Ib2 ⎠ + {(l1 − x)Zca + l2 Zbc + l3 Zcs }Ic2 ⎛ ⎞ xZab Ia1 + xZcs Ib1 + xZbc Ic1 ⎜ ⎟ ⎜ + {(l1 − x)Zab + l2 Zca + l3 Zcs }Ia2 ⎟ ⎟ b5 (x) = −⎜ ⎜ + {(l − x)Z + l Z + l Z }I ⎟, cs 2 cs 3 ab b2 ⎠ 1 ⎝ + {(l1 − x)Zbc + l2 Zcs + l3 Zca }Ic2
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