New Configuration Constraints to Reduce Unbalance in Hexagonal Double-Circuit Transmission Lines Al-Motasem I. Aldaoudeyeh, F. K. Amoura
Hussein M. Al-Masri, Ahmad Abuelrub
Yarmouk University Irbid, Jordan
[email protected],
[email protected]
Texas A&M University College Station, Texas
[email protected],
[email protected]
Abstract—Mutual impedances of transmission lines are not equal due to unequal spacing between conductors which creates problems like circulating current in double circuit lines. In this paper, the authors introduce novel configurations constraints for hexagonal double circuit arrangement that will achieve equal mutual impedances without transposition. Principal results of such arrangement are uncoupling between positive, negative and zero sequence networks, minimizing negative and zero sequence unbalance values, and increasing line loadability. Impact of adding one and two ground wires on the proposed configuration is also evaluated. Index Terms—Negative sequence, zero sequence, unbalance, mutual impedance, transmission line, loadability.
I. INTRODUCTION
P
OWER systems are designed in most cases as 3-phase networks. In general, they are operated under balanced loading. Even if the loads are balanced, an unbalance will take place in the system when transmission lines are not transposed or there is an unbalanced generation [1]. The assumption of a perfectly balanced power system does not hold true in some cases. For example, non-transposed transmission lines supplying large amount of power [1] [2]. Even if the line is short, an inherent asymmetry in its configuration may be a reason for current unbalance to take place [3]. In other words, balanced loading does not necessarily guarantee a balanced operation.
For induction machines, an unbalanced voltage in the system may cause high current magnitude to be induced in the rotor and, eventually, winding and bearing failures. Additional power losses may also take place leading to a necessary derating of the motor [4]. Other problems may include loss of performance, loss of squirrel cage induction generator useful life and increased 2 torque pulsations inside it, overheating of generators is also possible bad effect of negative sequence currents [5] [6]. Even what is considered to be moderate unbalance in the generator loading may raise the temperature inside some spots in the rotor for unacceptable levels [7]. Other adverse effects on induction machines due to unbalance
include large transient currents and reduction of the available torque. These effects are counter measured by disconnection of the motor (which is undesirable action) [8]. Furthermore, unbalance due to non-transposed lines may cause certain faults to make generators lose synchronism just after several cycles of swinging [9]. In non-transposed multi-circuit lines, an unbalance in voltage may create out-of-phase portions of voltage and causes a relatively high circulating current. On the other hand, unbalanced current in non-transposed double-circuit transmission lines, unlike the situation of short single-circuit ones, is not small due to circulating current component. This circulating current may also threaten the reliability of the system as it may cause circuit breakers to trip falsely if exceeded 10% value [10] [11]. These circulating currents will limit value of the useful (load) current that can be transmitted through the line. Reference [12] presents multiple methods to compensate unbalance for EHV (Extra-High Voltage) non-transposed lines, authors of the latterly mentioned study draw a conclusion that none of them achieves perfect balance. Nevertheless, the authors of [12] assumed equal positive, negative and zero sequence impedances for generators. The operation of FACTS (Flexible AC Transmission Systems) devices may suffer due to unbalance. For example, STATCOM (Static Synchronous Compensator) performance may be affected by voltage unbalance and, in severe cases, it may be shut down to protect it from overcurrent [13]. This review shows that unbalance in non-transposed lines can be problematic. Reference [14] cite some references that further discuss the drastic effects of unbalance. Avoiding transposition in many cases is a must, this is because tapping location to a substation from an existing line cannot be related to transposition towers. Furthermore, some engineers consider transposition towers as weak ones because of bypassing needed to change location of phases. Reference [15] tried to find the optimal arrangement of transmission lines that yield minimum electromagnetic
unbalance. In [16], transposed and non-transposed multicircuits have been compared. It is shown that the unbalance level can be significantly reduced by transposition. The study examines the effect of transposition for right-of-way of six circuits and shows that transposition mitigates different current magnitudes in all the phases of circuits. Reference [17] suggests a new three-phase power flow algorithm to mitigate voltage unbalance. The suggested algorithm is tested on IEEE 34-bus and IEEE 123-bus systems to test their ability to mitigate unbalance. The effect of PV panels to assist in voltage unbalance problems is examined. In [18], thyristor controlled reactive power compensator based on phase-tophase voltage control is proposed to reduce unbalance. Both voltage and current unbalances are shown to improve in the proposed method. Nevertheless, the unbalances in phase-toneutral coordinates are still high.
The assumptions of this paper are: 1.
Mutual inductance is dependent on the line geometry.
2.
Whenever the line feeds a load, it is tapped from both circuits. This assumption is necessary so that balance will not be disturbed.
3.
Earth under the line is considered to be an "average damp". Therefore, earth resistivity is 100 . [14]. Unless otherwise is mentioned, we use this resistivity in our calculations.
Authors define the used notations as follows: Distance of any conductor to any conductor . (e.x. = , = ). If = this value will be the conductor's GMR ( ) (e.x. = ).
This paper presents new dimension constraints in double circuit hexagonal transmission line configuration which achieve equal self and mutual GMDs (Geometric Mean Distances) without any need for transposition. Carson's line model is used. It assumes that there is a fictitious conductor inside the earth that models its effect. The fictitious return conductor is beneath surface of the earth at a distance equal (where is square of distance of the fictitious earth conductor below the earth). The earth is considered to have uniform resistivity and to be of infinite extent [14] [19]. In [20], it is shown that in Carson's line model, the fictitious return conductor can be thought of as a conductor with GMR (Geometric Mean Radius) of unit length and located at unit length below the overhead line (where is the distance of any conductor from the fictitious counter-part). Fig.1 shows a hexagonal double-circuit arrangement of conductors, but without determining any numerical values for dimensions. To get balanced transmission line parameters, we need to: 1. make distances between phase conductors equal, and 2. make GMDs between phases equal. In this paper, we obtain constraints to meet both of these conditions.
Carson's line resistance ( /
,
).
AC resistance of phase "a" conductor (equal for all phase conductors) ( / ). AC resistances of ground wires and practice they are equal) ( / ).
Resistivity of the earth ( . ).
System steady state frequency ( System angular frequency (
(in
).
/ ).
3 × 3 Phase domain matrix of transmission line as seen by the system ( / ).
3 × 3 Sequence domain matrix of transmission line as seen by the system ( / ). Self-impedance if . Where or
= , mutual impedance if is any conductor. ( / ).
Zero sequence unbalance.
Negative sequence unbalance. Operator
=
+
Conductor or is any of , , , , , , Where , , , , , and are phase conductors.
or
.
II. METHODOLOGY Our paper suggests new geometric constraints that will decrease differences in mutual as well as self-impedances. Besides, the effects of one and two ground wires was analyzed and taken into account by Kron's reduction. Fig. 1 A Proposed Hexagonal Double Circuit Transmission Line Configuration
Values of negative and zero sequence unbalance factors
were used to calculate the impact of ground wires on the proposed line. Percent negative and zero sequence unbalance factors, as magnitudes only, were calculated from equations [14]: | |
|% =
100% =
|% =
100% =
100% 100%
(1) (2)
Where is the impedance and reflects the amount of voltage drop that would occur in network due to current that flows in network . When = , like in , it accounts for the voltage drop in the negative sequence network for a current that flows inside it.
Consider a three phase double circuit transmission line as shown in Fig.1, the two circuits are assumed to take their voltage feed from the same source. We set three basic dimensions; namely , , and . To achieve balanced line parameters, diagonal elements must be equal to each other and so do the off diagonal elements. According to that, the following conditions have to be satisfied: Equal Self Impedances
The first condition for this is: =
=
(3)
But symmetry of the line around the vertical axis imposes that = , so equation 3 becomes: =
This yields:
(4)
+ (2 ) = 2
Rearranging the expression:
+
(5)
=
(6)
While the second condition is: =
=
=
=
=
(7)
This condition is valid if all conductors are identical, which is, of course, the case for all transmission lines. 2.
Equal Mutual Impedances
This will happen if: =
=
(9)
From figure 1 we note that = and = (2 ) (2 ), substituting the values of N, R, and S in terms of the three basic dimensions yields: ((
+
)(
+2
+
(8)
But as the line is symmetrical around its vertical center, we can say that (for the arrangement shown in particular):
)) =
+
4(
)
(10)
Substituting T as in (6) and rearranging assuming nonnegative values: +
=
2
Solving the equation for
A. Derivation of Dimensional Constraints
1.
=
| yields
| =
(11) 3
or
=
, if
these two results were substituted (each one singly) in (6), the following constraints can be obtained: =
=
1 2
(12)
3 2
(13)
For hexagonal line configuration and phase arrangement shown in Fig.1, if these two constraints were satisfied then self-impedances are equal, and so are mutual impedances. From Fig.1 we note: =
=2 =
3
(14)
This is a desired result because we do not want equal triangular shape of the conductors. If a 6 × 6 impedance matrix was developed and reduced to yield its 3 × 3 equivalent, it could be confirmed that all mutual impedances are equal to each other and the selfimpedances are equal as well. In practice, however, transmission lines must have one or two ground wires to intercept lightning strokes, these ground wires are made of steel and their effect can be modeled by a modification of . As a result, significant increase on the real part of elements will be noticed, but small decrease for their imaginary parts will occur. The existence of ground wire(s) has major impact on the value of zero sequence impedance but not positive and negative sequence impedances. The explanation is that positive and negative sequence currents add to zero but zero sequence currents, by definition, do not. This is because zero sequence currents are in phase. Equations used to calculate including earth wires can be found in reference [14]. Also, the same reference contains the method to reduce the matrix to its final 3 × 3 form.
B. Case 1: Double Circuit Line with One Ground Wire
One ground wire effect can be easily embedded into the impedance matrix by Kron's reduction since the matrix does
TABLE 1
not need any manipulation to apply this technique. One or two ground wires can be used in transmission towers. In this paper, we introduce ) for single ground wire and and for two (see Fig. 2).
PHASE TO PHASE SPACING FOR TYPICAL VOLTAGES (EPRI STANDARD [5])
If a ground wire is introduced, the mutual and selfimpedances are no longer equal even for the configuration presented in this paper. Values of zero and negative sequence unbalance, however, will stay fairly small.
As in Fig.2, a ground wire denoted ) is introduced at the line vertical center and at distance F above a virtual horizontal line connecting and . The shielding angle is taken to be 26.5°. Under such case, matrix will no longer be balanced. But the novelty of this work comes from the fact that unbalance due to the hexagonal arrangement itself has been eliminated.
Line Voltage (KV)
Phase-Phase Spacing (meters)
138
4 to 5
230
6 to 9
345
6 to 9
500
9 to 11
765
13.7
Transmission line parameters are: 500
Voltage (line)
9.74
Minimum phase-phase spacing
0°
Shielding angle, two ground wires Minimum distance of ground wire(s) from fictitious horizontal line connecting the highest two phase conductors (named )
9.75
, , , , , (equivalent of two conductors bundle, and spacing original GMR is 1.75 is 18 ) ,
100 .
(AC resistance of ground wires)
0.0727 / 4 /
.
Sections A and B show numerical calculations for one ground wire and two ground wires cases.
for
A. Case 1: Double Circuit Line with One Ground Wire In this case, values of diagonal elements of matrix are extremely close, and so are the off-diagonal elements. and matrices are: 0.113553 + .457875 0.076280 + 0.304440 0.077203 + 0.304053
III. RESULTS Table 1 shows typical phase-phase clearances for HV (High Voltage) and EHV lines [19]. It is a trend in power industry to make lines as compact as possible by taking the least allowable spacing between conductors to reduce right of way costs. For this reason, authors choose low values of phasephase spacing.
0.014288
For all of next results, units of all matrices are
C. Case 2: Double Circuit Line with Two Ground Wires Two ground wires can be added instead of one, the shielding angle in our case study is 0° .When two ground wires are introduced, zero sequence unbalance is increased further (assuming the new added conductor is of the same resistance and GMR as the old one). This is logical since two ground wires will modify seen by the system two times. Ground wires are denoted and in Fig.2.
0.14256 60
(AC resistance of all line conductors) ,
.
26.5°
Shielding angle, one ground wire
Fig.2 The Proposed Configuration Including One or Two Ground Wires
.
0.266128 + 1.066752 0.000121 0.000978 0.000787 + 0.000594
0.076280 + 0.304440 0.111752 + 0.458640 0.076280 + 0.304440
0.000787 + 0.000594 0.036365 + 0.153819 0.000005 + 0.000015
0.077203 + 0.304053 0.076280 + 0.304440 0.113553 + 0.457875
0.000121 0.000978 0.000010 0.000012 0.036365 + 0.153819
The percent zero and negative sequence unbalances are: | |
|% = 0.0897% |% = 0.0099%
B. Case 2: Double Circuit Line with Two Ground Wires
3.65
75
0.3299
0.8886
3.7
89.8
Adding second ground wire affects mainly zero sequence impedance. This is clear by comparing and matrices listed below to those in case of one ground wire. Further noticeable increases in real parts of impedances can be seen; this is because two ground wires affect the magnetic and electric field surrounding the two circuits (i.e. stronger shielding).
3.41
70
0.3873
1.0347
4.3
104.5
3.17
65
0.4456
1.1680
5.0
118.0
The matrices are: 0.127896 + 0.444351 = 0.090032 + 0.291296 0.091517 + 0.290533
0.307968 + 1.026939 = 0.000082 0.001659 0.001396 + 0.0009
0.090032 + 0.291296 0.124949 + 0.445864 0.090032 + 0.291296
0.001396 + 0.0009 0.036386 + .153814 1.9 7 + 0.000009
0.091517 + 0.290533 0.090032 + 0.291296 0.127896 + 0.444351
0.000082 0.001659 0.000008 0.000005 0.036386 + .153814
|
COMPARISON OF |
|% = 0.1549%
|% = 0.0062%
It is a trend in power industry to make lines as compact as possible to reduce right-of-way costs. This is done by increasing the circuit density to maximize the use of right-ofways [5]. If line dimensions deviated from the constraints mentioned in equations 12 and 13 in section II, both unbalance factors would increase further. We examined effects of reducing to make the line more compact. This is basically a trade-off where negative and zero sequence unbalance factors increase and bigger difference between mutual impedances happens to save construction costs. Please note that, in this section, the arrangement and geometry of the line are the same as in Figures 1 and 2, the only basic dimension modified is . A. Case 2: Double Circuit Line with One ground wire Results are shown in Table 3. Compared with these in section A, there is a considerable increase in | |% for each corresponding change. For example, when = 3.9, | |% is 0.228% for no ground wire and 0.2736% for one ground wire. But for | |% the change is much smaller than in | |%. TABLE 3
4.87 4.62
%
|% AND | |% FOR VARIOUS VALUES OF CIRCUIT LINE WITH O NE GROUND WIRE
100
|
|%
0.0897
|
|%
95
0.1215
0.1968
0.0099
|
| |% |%
1.0
1.4
FOR DOUBLE
|
| |% |%
1.0
19.9
4.38
90
0.1674
0.3839
1.9
38.8
4.14
85
0.2191
0.5622
2.4
56.8
3.9
80
0.2736
0.7308
3.1
73.8
|% AND | |% FOR VARIOUS VALUES OF CIRCUIT LINE WITH TWO GROUND WIRES |%
| |% |%
| |% |%
100
0.1549
|
4.62
95
0.1848
0.1968
1.2
31.7
4.38
90
0.2259
0.3839
1.5
61.9
4.14
85
0.2734
0.5623
1.8
90.7
3.9
80
0.3250
0.7310
2.1
117.9
3.65
75
0.3791
0.8889
2.4
143.4
3.41
70
0.4350
1.0351
2.8
167.0
3.17
65
0.4924
1.1684
3.2
188.5
%
|%
FOR DOUBLE
|
4.87
IV. EFFECTS OF MODIFYING LINE DIMENSIONS TO MAKE IT "MORE COMPACT"
COMPARISON OF |
Value of | |% increases further for each corresponding case compared to section B, for example | |%for = 3.9 is 0.3250%. It is worth noting that | |% for cases B and C is essentially the same when is changed from its original value, this is because ground wires mainly affect zero sequence impedance. Results of this case are shown in Table 4. TABLE 4
Percent zero and negative sequence unbalances are: |
B. Case 2: Double Circuit Line with Two Ground Wires
0.0062
|
1.0
|
1.0
Readers may note that negative sequence unbalances are almost the same for one ground wire and two ground wires at each corresponding modification. A reason why they stay almost unchanged is that negative sequence unbalance happens due to unequal spacing between phases and basically does not depend on the existence of ground wires. On the contrary, zero sequence unbalance is due to the existence of ground wires and has smaller dependence on phase arrangement. As the line becomes more compact, it is more economically advantageous due to reduced strength requirements of the structure as well as a decrease in the total corridor occupied space. For slight adjustments, there is a fair increase in | |% and noticeable increase in | |%. As for the increase in | |%, it may cause
increased magnetic field around the line. For an increase in | |%, it will result in problems for rotating machines
such as overheating. To avoid such issues, it is possible to modify slightly from its original value. In [21], | 0 |% and | 2 |% are calculated for EHV line to be 0.6473 and 4.7409, respectively. From [22], | 0 |% and | 2 |% can be calculated for a 500 kV line to
be 1.03% and 4.17%, respectively. Comparing the results, the unbalances of the proposed configuration are still more acceptable. Results in this section indicate clearly that negative sequence unbalance is remarkably small regardless of how far dimension is decreased; zero sequence unbalance is rather smaller than these reference values as long as the reduction in dimension is not very large as well (40% or less of the original value). Such observations prove the superiority of the proposed geometric constraints even when trying to make the line more compact.
[5]
[6]
[7]
[8]
[9]
V. CONCLUSION
In this paper, we present a hexagonal transmission line configuration constraints with almost equal mutual impedances as well as almost equal self-impedances. As a result, zero, negative and positive sequence networks were roughly uncoupled, and all disadvantages of nontransposed transmission lines would disappear. Negative sequence unbalance factor is almost solely determined by the line geometry itself. Changes in the geometry that violate the proposed constraints result in noticeable changes in negative sequence unbalance; it increases as deviates from the proposed value which proves that the proposed geometry is optimized. Results in section IV show that negative sequence unbalance stays remarkably small for all cases and modifications considered. As for ground wires, adding them results in very small change on negative sequence unbalance factors.
[10]
[11]
[12]
[13]
[14]
[15]
Zero sequence unbalance is determined by both the line geometry and the presence of ground wires. If the line is desired to be very compact, the line geometry will be considerably more influential on zero sequence unbalance than ground wires.
[16]
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