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Dynamic analysis of substation busbar structures
ELSEVIER
Electric Power Systems Research 42 (1997) 47 53
POWER 8WSTBf8 RBSBflRCH
Dynamic analysis of substation busbar structures Marc D. Budinich *, Russell E. Trahan Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148, USA
Received 14 February 1995
Abstract The usual method used to design substation busbar structures is to analyze the short circuit force loading with an equivalent static loading. However, it is shown here that the relatively slow response modes of the structure require that a dynamic analysis be performed for the higher frequency short circuit forces. Full scale test data is compared to simulation results obtained from a finite element model. © 1997 Published by Elsevier Science S.A. Keywords: Substation busbar structure; Dynamic analysis; High frequency short circuit force
I. Introduction Conventional procedures for the structural design of power substations to withstand the electromagnetic forces induced by short circuit currents are based on static methods that often lead to inaccurate, conservative designs with unnecessarily high costs. One such procedure that has been adopted by many power companies is the ANSI/IEEE Std. 605-1987 [1]. The design problem, however, actually requires dynamic methods to achieve accurate results due to the dynamic short circuit forces imparted to the substation structure for very short durations of time and the opposing inertial forces of the structure to these forces. Accurate prediction of the forces and stresses imparted to a substation structure by the short circuit currents is essential for a less conservative design of the substation structure due to the relatively high costs of the substation components affected. While static methods attempt to transform the dynamic forces during a fault into an equivalent static force by employing scale factors, dynamic methods use a mathematical description of the actual dynamic forces and apply them to a mathematical model of the actual substation structure to achieve a much more accurate result and, hence, less conservative design. The primary reason for the wide acceptance of using static methods over dynamic methods for substation design is that * Corresponding author. Tel.: + 1 504 2866650.
static methods involve only algebraic functions to predict the stresses and moments in the components of a substation and, therefore, are easy to implement by substation designers. Dynamic methods involve the solution of differential equations which require complex numerical techniques that may not be familiar to the substation designer. The substation structure design tool described here is a computer program named P C B u s that performs a dynamic finite element analysis to solve the differential equations and presents a spreadsheet in a Windows user-interface to ease the creation and editing of the substation models.
2. Problem description Electric utility substations are sites that serve as distribution or interconnection points for transmission systems. A typical substation includes a bus structure which consists of three parallel, horizontal, and equally spaced busbars mounted atop porcelain or polymer insulators with busbar fittings. The busbars are typically aluminum or copper conductors welded together in sections to form lengths that may exceed a few hundred feet. The conductors distribute the three phase current in the substation through taps mounted transversely along the bus conductors. These are generally called bays and are used to connect transmission lines or transformers to the substation. The insulators are usually mounted atop insulator supports to raise the
0378-7796/97/$17.00 © 1997 Published by Elsevier Science S.A. All rights reserved. PlI S0378-7796(96)01 177-7
M.D. Budinich, R.E. Trahan /Electric Power Systems Research 42 (1997) 47-53
48
high voltage busbars to required heights to provide a safe level of electric insulation between the conductors and ground. Fig. 1 shows a general arrangement for a substation structure with horizontal insulator supports. Under normal operating conditions the load currents in the parallel busbars of a substation structure induce, through the interaction of the electromagnetic fields, tolerable forces on the neighbouring conductors that act in the plane formed by the busbars. During a short circuit fault the currents abruptly increase to many times the load currents to induce extraordinary dynamic forces on the busbars until the fault is removed from the system by protective relays or other means. This is generally accomplished in 6 to 8 cycles of the power system frequency. Possibilities for failure of the substation structure due to these fault currents include the permanent deformation of the busbars and insulator supports and the fracturing of the high cost insulators [5]. The substation structure must, therefore, be designed to withstand the dynamic forces arising from a short circuit fault to avoid the failure of any substation component without over-designing and unnecessarily increasing the overall cost of the structure.
z
1i
~.~/-
- ,
~**'s
/
', / / tI
.,
/
'a
~ e , ~ us/ '~.,..,I
N
/
J , ~
// ~
,~ ,/
///
k.,,.~~ wv
/ / J
F i g . 2. T y p i c a l t h r e e - d i m e n s i o n a l b e a m e l e m e n t .
large enough number of beam elements to define an accurate model of the substation structure. One of the most important aspects for obtaining accurate results from a finite element analysis of the substation structure is providing the analysis with an accurate mathematical model of the structure and its externally applied forces.
3. Finite element analysis 3.1. Modelling the substation structure.
The finite element method is a computer-aided mathematical technique used for obtaining approximate numerical solutions to a wide variety of engineering problems. As applied to substation design, the method is based on partitioning the substation structure into a finite number of discrete elements to reduce the problem from an infinite number of unknowns to a finite number of unknowns. The finite elements used to model the substation structure in PCBus are three-dimensional beam elements as shown in Fig. 2. The true solution to the problem is approached by partitioning a
IInsul"tOr
<~ <~>L"-----HorizonlalInsulatorSupport F i g . 1. G e n e r a l a r r a n g e m e n t f o r a s u b s t a t i o n s t r u c t u r e w i t h h o r i z o n tal i n s u l a t o r supports.
The substation components that are modeled in PCBus include the insulators, insulator supports, busbar conductors, and busbar fittings. The busbar fittings that can be modeled in PCBus include fixed fittings, slide fittings, and expansion fittings. Cantilever ends can also be modeled in PCBus. Each component of the substation structure is modeled with its equivalent mechanical properties assigned to the beam elements that correspond to the component in the three-dimensional model of the structure. The mechanical properties required to define the beam elements are the moment of inertias about the cross sectional axes, Ix and Iy; polar moment of inertia, J; modulus of elasticity, E; density, p; cross-sectional area, A; and Poisson's ratio, v. The required mechanical properties for the beam elements are derived within PCBus when an analysis is performed from the necessary manufacturer data entered into the PCBus model spreadsheet for the various components. The manufacturer data entered into the spreadsheet to model an insulator include height, li, s; core diameter, dins; maximum permissible cantilever force, Fins; maximum tip deflection, Jms; and weight, W,ns. The manufacturer data entered into the spreadsheet to model the other substation components closely match the mechanical properties required to model a beam element and can be found directly from the data sheets in a steel construction handbook. Furthermore, to ease the creation and editing of a substation struc-
M.D. Budinich, R.E. Trahan / Electric Power Systems Research 42 (1997) 47 53
ture model the P C B u s model spreadsheet is linked to separate component libraries for each substation component for which all the necessary manufacturer data is entered into the model spreadsheet when a specific component is selected. This feature of P C B u s eliminates the need for the substation designer to explicitly enter the manufacturer data into the model spreadsheet to model a substation component. The assumption that an insulator can be accurately modeled with beam elements is valid if the insulator exhibits the behaviour of a true beam. Research conducted at the Langen Research Center in Germany on one hundred insulators concluded that an insulator does exhibit the behaviour of a true beam and can accurately be modeled as a cantilever beam [2]. Similar conclusions were also found from the results of research conducted at Tulane University [31. The failure of an insulator occurs in an analysis if the maximum permissible cantilever force rating is exceeded. The maximum permissible cantilever force rating is determined by the manufacturer by applying a static load to the tip of the insulator until a failure occurs. The applied static load at the tip of the insulator produces a maximum moment, Min~, at the base of the insulator given by Mins = Fins/'ins
(1)
The mechanism for failure is this maximum moment since the moment induces local stresses that ultimately cause an insulator to fail. In a dynamic analysis the maximum moment at the base does not correspond directly with the force at the tip of the insulator due to the included inertial forces of the structure. As a result, a maximum equivalent cantilever force, Fm~x, is derived for comparison with the maximum permissible cantilever force, Fm~, by using the largest moment in the insulator predicted by a dynamic analysis, M . . . . in the following relation Fm,x --
mmax
(2}
In the analysis, the insulator fails if the maximum equivalent cantilever force, Fmax, multiplied by the safety factor is greater than the maximum permissible cantilever force, Fi,~. The failure of a conductor occurs in an analysis if the maximum stress in the conductor predicted by the analysis, a .... exceeds the maximum stress rating for the conductor. Like-wise, the failure of an insulator support occurs in an analysis if the maximum stress in the insulator support predicted by the analysis, Osup, exceeds the maximum stress rating for the insulator support.
49
3.2. Modelling the external forces The forces that are modeled in P C B u s include the fault current forces, wind loading, ice loading, and gravitational forces. The forces are modeled with the equivalent mathematical descriptions applied to appropriate nodes of the three-dimensional finite element model of the substation structure. The instantaneous currents at the onset of a three phase fault are given by il(t) = il(0)e - R~t/XL + i~{e- R~,,t/XL COS(E) -- COS(~Ot+ E)}
(3)
i2(t) = i2(O)e R~o,/XL + i ~ I e - R,~,'XL t.
{" 2rC'~ cost,+y)
i3(t ) = i3(0)e - k~,~t/XL + i~ {e - R°~''XLCOS(¢ -- ~-j~ )
where il(t), i2(t), and i3(t) are the instantaneous fault currents at time, t, after the onset of the fault for the center and outer conductors, respectively [amps]; iL(0), i2(0), and i3(0) are the instantaneous load currents the instant of time just before the fault [amps]; R is the system resistance [ohms]; JfL is the system inductive reactance [ohms]; a) is the system [amps]; and e is the fault current phase angle [rad]. Using the instantaneous currents in (3), (4), and (5), the instantaneous forces per unit length of the conductors are found by Ampere's law and assuming the conductors are infinitely long and separated by a distance that is much greater than the diameters of the conductors. The instantaneous forces on the center conductance for which the current i~(t) flows is given by wl(t ) -
5 . 4 x 10 -7 {il(t)i3(t ) - il(t)i2(t)} D
(6)
The instantaneous forces on the outer conductor for which the current i2(t) flows is given by w2(t) -
5.4 × 10-7 D {i2(t)it(t) + ½i2(t)i3(t)}
(7)
Likewise, the instantaneous forces on the outer conductor for which the current i3(t) flows is given by w3(t) -
5.4 x 10- v D {i3(t)i~(t) + ½i3(t)i2(t)}
(8)
In Eqs. (6)-(8), the forces are distributed along the conductors in lbf ft-~ and D is the phase distance
50
M.D. Budinich, R.E. Trahan/Electric Power Systems Research 42 (1997) 47-53
between adjacent conductors in inches. Since worst case conditions are needed for the design of the substation structure, it was determined that the maximum forces are imparted to the structure on the outer conductor through which the current i2(t) flows when the fault current phase angle is optimized to maximize the RMS forces to give
Table 2 Effects of modifying the stiffness of the connections of the fixed fittings to the insulators on the dynamic response of the substation structure
E = 0.86921 rad
Fixed fitting rigid connection Fixed fitting -- loose connection
(9)
The remaining forces modeled in PCBus are static forces and are determined by using the relations found in ANSI/IEEE Std. 605-1987. It is not required to include any of these static forces in an analysis. 3.3. Dynamic response o f the substation structure
Using the finite element models for the substation components and external forces, an assembly of the element equations yield the dynamic governing system equations for the entire substation structure given by M / / + CLi + Ka = f
(10)
where M is the equivalent mass matrix, C is the damping matrix, K is the nodal stiffness matrix, f is the applied nodal actions vector, and a is the nodal displacements vector. Eq. (10) is solved for the nodal displacements in a dynamic finite element analysis by a numerical time-stepping method. The moments and stresses are then derived in a final operation called postprocessing. Several dynamic finite element analyses were performed on an arbitrary single span substation structure to investigate the changes in the dynamic response for modifications of the substation structure. Interesting results were observed for modifications of the substation structure's span length and stiffness of the fittings' connections to the insulators. Table 1 shows the maximum stresses imparted to the conductors and the maximum equivalent cantilever forces imparted to the insulators from the results of a series of dynamic analyses for increases in span length. The results show that an increase in the span length results in an increase in the maximum conductor stress. The results also show that the insulator equivalent
Substation structure description
Maximum value Croo.(psi)
F~.s (lbf)
928 1616
144.61 109.89
cantilever forces are lower for a 30 ft span than a 20 ft span. This suggests that an optimum span length of over 20 ft may be found that minimizes the maximum insulator equivalent cantilever force. A possible explanation for this phenomena is that as the span length increases to a certain length, the natural frequencies of the structure are lowered to give the structure an increasingly low-pass nature that more than offsets the forces added by the increases in span length. Table 2 shows the maximum stresses imparted to the conductors and the maximum equivalent cantilever forces imparted to the insulators from the results of two dynamic analyses for changes in the connections of the fittings to the insulators from rigid to loose. The results show that the maximum conductor stresses greatly increase for fittings with loose connections to the insulators. This is a result of the loose fittings allowing the conductors to have greater displacements. In contrast, the maximum insulator equivalent cantilever forces reduce for fittings with loose connections to the insulators. This is a result of the extra energy dissipated in the conductors with the greater displacements of the conductors. The results show that if it is not known whether the fittings being modeled have rigid or loose connections to the insulators, two separate analyses with the fittings modeled as having rigid connections and loose connections should be performed to investigate the possible range of resuits. 3.4. Using static analysis as a quick design tool
Table 1 Effects of modifying the length of the busbar conductors on the dynamic response of the substation structure
One disadvantage of performing a dynamic analysis is the large amount of time required to complete an analysis. A static analysis requires less time to complete since it does not require numerical time-stepping techniques and involves only solving the static representation of the system equations given by
Substation structure description
Maximum value
Ka = f
crcon(psi)
Fins (lbf}
928 1964 3085 3652
144.61 123.62 143.69 163.07
where the applied nodal actions vector, f, contains only static forces determined by the maximum short circuit forces calculated in Eqs. (6)-(8). However, as mentioned previously, a static analysis results in less accurate results since it cannot account for the dynamic behaviour of the substation structure and, therefore,
20 ft Span, 4 in Conductor 30 ft Span, 4 in Conductor 40 ft Span, 4 in Conductor 50 ft Span, 4 in Conductor
(11)
51
M.D. Budinich, R.E. Trahan //Electric Power Systems Research 42 (1997) 47-53
Table 3 Comparison of results for the dynamic analyses, static analyses employingscale factors, and static analyses with no scale factors Substation structure description
30 fl span 40 fi span 50 fl span
Dynamic analysis
Static analysis Kdy~and K~o,
Static analysis (no scale factors)
~o, (psi)
Fi,s (lbf)
a¢o, (psi)
Fi,~ (lbf)
a¢o, (psi)
Fins (Ibf)
1964 3085 3652
123.62 143.69 163.07
1560 2830 4483
107.2 142.78 178.78
4210 7660 12 140
194.38 259.17 323.96
yields more conservative designs. In order to obtain static analysis results which are closer to the results of a dynamic analysis, a dynamic force scale factor, Kdyn, and a conductor stress scale factor, Kcon, were introduced into the P C B u s static finite element analysis. To determine the scale factors, a series of static analyses were performed on various configurations of the substation structure for different iterations of the dynamic force scale factor to compare the static results with the corresponding dynamic results. It was possible to find a value for the dynamic force scale factor that approximately matched the maximum insulator equivalent cantilever forces from the static analyses to the results of the dynamic analyses by the following reaction Wstat = K d y n m a x { w ~ ( t ) }
(12)
where wi(t ) is the dynamic force imparted to the ith conductor• However, the maximum conductor stresses calculated from the static analyses were still generally much higher than the corresponding maximum conductor stresses calculated from the dynamic analyses. Therefore, the conductor stress scale factor was introduced to lower the conductor stresses resulting from the static analyses by the following relation
4. Verification of the PCBus results
Verification of the P C B u s program results was made possible by the full scale fault current tests performed by the British Columbia Hydro and Power Authority (BCHydro) during August and September of 1993. The verification of the P C B u s results presented here will be performed by comparing the full scale test results of the single span and two-span substation structure to the corresponding P C B u s results. Fig• 3 shows the plan views for the two substation configurations. The insulator equivalent cantilever forces in the direction of the short circuit forces were measured with calibrated fullbridge strain gauge transducers located at the bases of the insulators. In addition, the displacements of a conductor at the middle of a span for selected tests were measured by a laser pointer and video camera. Since the greatest forces were imparted to the conductor through which the current i2(t) flowed, only the result for this A-phase outer conductor will be used for verification. The BCHydro substation structure was constructed with N G K 8A-67971A 230 kV insulators and A L U M 6061-T6 6.0 SCH 40 conductors with 3-0 damping
O'con{dyn} ~ g c o n O ' c o n { s t a t }
While the dynamic force scale factor is applied to the static short circuit forces before an analysis, the conductor stress scale factor is applied to the conductor stresses after an analysis. The following values for the dynamic force scale factor and conductor stress scale factor were selected
-~,_,.,,,~
.9
eY ,
gdyn = 0.55
5
!
Tuqr i
b r~T
I
Kco, = 0.37 Table 3 shows the results of the dynamic analyses and the scaled static analyses for various span lengths of a single span substation structure. Although the results show that improvements in the static analyses are achieved by the scale factors, it is apparent that the static analyses cannot accurately account for the dynamic response of the substation structure by employing scale factors.
O"x rLtl
C~L ~ v
.fl o
.~ I ~ . .
,s
0-, t
l"
,%2," t - !
IP
i
-=-
i,,,, I
Fig. 3. Plan views of the BCHydro single span and two-span substation structures•
52
M.D. Budinich, R.E. Trahan /"Electric Power Systems Research 42 (1997) 47 53
Table 4 Comparison of the maximum conductor displacements for increasing durations of the 50 kA fault current
600 400 2oo
b.
-200
~.mm.u
o
I
o.2
.
I
'
0.4
I
!
I
o.s
o.a
1
Fig. 4. Comparison of the BCHyrdo and PCBus equivalent cantilever forces for Insulator # 1 of the single span structure for a 50 kA, 0.1 s fault.
cables. The insulators were mounted to horizontal insulator supports with l l . 6 i n W10 × 26 mounting adapters and the conductors were mounted to the insulators with fixed fittings. It was necessary to have current feeder lines attached to the conductors at one end of the substation structure and a shorting bar attached to the other end to perform the short circuit tests. 4.1. Single span structure
The single span substation structure was constructed with a 52.49 ft (16000 mm) span. A 50 kA fault current was applied to the substation structure via the current feeder lines for a duration of 0.1 s. Figs. 4 and 5 show a comparison of the measured equivalent cantilever forces from BCHydro to the predicted equivalent cantilever forces from PCBus imparted to Insulators # 1 and # 4, respectively. As shown in Fig. 3, Insulators # 1 and # 4 were located at the current feeder and shorting bar ends of the A-phase outer conductor, respectively. Note that the signals received from the strain gauges were corrupted with 60 Hz noise during the first 0.1 s due to the electromagnetic interference of 600 ..............................................................
200-o U . .
o
02
0.4
06
0a
1
BCHydro (in)
PCBus (in)
%error
50 100 150 200
5.0 7.3 8.6 10.3
4.9 7.3 8.9 9.7
2.0 0.0 3.5 5.8
1.2
T~me (s)
400
50 kA fault duration (ms)
12
T~me (s)
Fig. 5. Comparison of the BCHydro and PCBus equivalentcantilever forces for Insulator # 4 of the single span structure for a 50 kA, 0.1 s fault.
the short circuit currents picked up by the instrumentation cables. The comparisons show a high correlation of the predicted results to the measured results. Note that the measured insulator equivalent cantilever forces from the BCHydro test are different for each insulator since the end conditions were different at each end of the conductor due to the current feeder lines near Insulator # 1 and the shorting bar near Insulator # 4. The end conditions impart large forces to increase the maximum equivalent cantilever forces imparted to the insulators located at the ends of the substation structure. The PCBus results are exactly the same for each insulator since the end conditions were not modeled and, therefore, the model for the single span substation structure was symmetric about the midpoint of the conductor. The PCBus model spread-sheet is not equipped to model the end conditions caused by current feeder lines or shorting bars since these do not occur in practice and are only included in the BCHydro substation structure as part of the test structure setup. Table 4 shows the maximum measured conductor displacements for the A-phase outer conductor of the single span substation structure subjected to increasing durations of a 50 kA fault current. The damping cables were removed from the conductors during these tests. As shown, the PCBus results have very small errors compared to the measured results. 4.2. Two-span structure
The two-span substation structure was constructed with a 52.49 ft (16000 mm) span and a 19.69 ft (6000 mm) span. A 50 kA fault current was again applied to the substation structure via the current feeder lines for a duration of 0.1 s. Figs. 6 - 8 show a comparison of the measured equivalent cantilever forces from BCHydro to the predicted equivalent cantilever forces from PCBus imparted to Insulators # 1, # 4 , and # 7 , respectively. As shown in Fig. 3, Insulator # 7 was located at the far end of the second span near the shorting bar. Again, the comparisons show a high correlation of the predicted results to the measured results. This is especially true for the comparison of the measured and predicted equivalent cantilever forces imparted to Insulator # 4. Verification of the PCBus
M.D. Budinich, R.E. Trahan// Electric Power Systems Research 42 (1997) 47-53
53
800
6OO
600
400
4OO
200
~2,00 azl
~" -200
,~,..20CI
-400
.400
-6OO -800
0
0.2
0.4
0.6
0.8
0
Time (s)
800 60O T ........................................................................ .......................... .. . . . . . . . . . . . . .
o .2oo -40(3 -6OO -800 o
0.2
04
0.4
0.6
0.8
1
"nine (S)
Fig. 6. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 1 of the two-span structure for a 50 kA, 0.1 s fault.
400
0.2
o.6
o,s
Tqme (s)
Fig. 7. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 4 of the two-span structure for a 50 kA, 0.1 s fault.
results for the equivalent cantilever forces imparted to this insulator is the most important verification since the largest forces of the two substation configurations were imparted to this insulator and the contributions of the shorting bar and current feeder lines were at a minimum. The results show that the predicted maximum equivalent cantilever force imparted to Insulator # 4 is very close to the measured maximum equivalent cantilever force.
5. Conclusions
Verification of the PCBus results shows conclusively that utilizing a dynamic finite element analysis to predict the dynamic response of a substation structure subjected to short circuit forces yields accurate results. PCBus is a powerful and convenient tool for substation design since accurate results are obtained from the dynamic finite element analyses and the involved process of generating accurate finite element models for the
Fig. 8. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 7 of the two span structure for a 50 kA, 0.1 s fault.
substation structure and external forces is accomplished within PCBus from the data selected in the PCBus model spreadsheet. In addition, the design process is made more efficient by introducing scale factors to perform quick static analyses to obtain approximate results for a particular substation structure before performing the final dynamic analyses to obtain more accurate results. Research is currently being performed to improve the functionality of PCBus by adding the capability to include bus taps and bay conductors in an analysis.
References [1] IEEE Guide ./'or Design of Substation Rigid-Bus Structures, IEEE/ANS1 Std 605-1987, The Institute of Electrical and Electronics Engineers, New York, NY, 1987. [2] E. Bauer, E. Brandt, H. Dannheim, W. Lehmann, W. Meyer, K. Pietsch and N. Stein, Dynamic behaviour and Strength of High
Voltage Substation Post Insulators Under Short-Circuit Loads,
[3]
[4]
[5]
[6]
[7] [8]
LehrStul ftir Elektriscbe Energieversorgung Universit/it Erlangen-Nfirnberg, 1984. J.E. Borhaug and H.A. Thompson, Mechanical design of electric bus systems to withstand the magnetic forces produced by short circuit currents, Final Report, Tulane University, April 1970. M.D. Budinich, The application of a dynamic finite element analysis to busbar structures subjected to short circuit forces, Graduate Thesis, University of New Orleans, December 1994. B.A. Develle, Finite element analysis applied to substation busbar design, Graduate Thesis, University of New Orleans, May 1992. B.A. DeveUe, J.W. Schilleci and R.E. Trahan, A personal computer based program for the dynamic analysis of busbar structures under short circuit faults, Proc. IEEE Southeastcon '93, April 4-7, 1993, Charlotte, North Carolina, pp. 564-566. A. Opsetmoen, Short circuit test of 230 kV bus, General Analysis, BCHydro Stations Engineering Division, June 1994. T. Stefanski, Tests on 230 kV bus structures for BCHydro and the University of New Orleans, Test Report, Powertech Labs Inc., October 1993.
Electric Power Systems Research 42 (1997) 47 53
POWER 8WSTBf8 RBSBflRCH
Dynamic analysis of substation busbar structures Marc D. Budinich *, Russell E. Trahan Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148, USA
Received 14 February 1995
Abstract The usual method used to design substation busbar structures is to analyze the short circuit force loading with an equivalent static loading. However, it is shown here that the relatively slow response modes of the structure require that a dynamic analysis be performed for the higher frequency short circuit forces. Full scale test data is compared to simulation results obtained from a finite element model. © 1997 Published by Elsevier Science S.A. Keywords: Substation busbar structure; Dynamic analysis; High frequency short circuit force
I. Introduction Conventional procedures for the structural design of power substations to withstand the electromagnetic forces induced by short circuit currents are based on static methods that often lead to inaccurate, conservative designs with unnecessarily high costs. One such procedure that has been adopted by many power companies is the ANSI/IEEE Std. 605-1987 [1]. The design problem, however, actually requires dynamic methods to achieve accurate results due to the dynamic short circuit forces imparted to the substation structure for very short durations of time and the opposing inertial forces of the structure to these forces. Accurate prediction of the forces and stresses imparted to a substation structure by the short circuit currents is essential for a less conservative design of the substation structure due to the relatively high costs of the substation components affected. While static methods attempt to transform the dynamic forces during a fault into an equivalent static force by employing scale factors, dynamic methods use a mathematical description of the actual dynamic forces and apply them to a mathematical model of the actual substation structure to achieve a much more accurate result and, hence, less conservative design. The primary reason for the wide acceptance of using static methods over dynamic methods for substation design is that * Corresponding author. Tel.: + 1 504 2866650.
static methods involve only algebraic functions to predict the stresses and moments in the components of a substation and, therefore, are easy to implement by substation designers. Dynamic methods involve the solution of differential equations which require complex numerical techniques that may not be familiar to the substation designer. The substation structure design tool described here is a computer program named P C B u s that performs a dynamic finite element analysis to solve the differential equations and presents a spreadsheet in a Windows user-interface to ease the creation and editing of the substation models.
2. Problem description Electric utility substations are sites that serve as distribution or interconnection points for transmission systems. A typical substation includes a bus structure which consists of three parallel, horizontal, and equally spaced busbars mounted atop porcelain or polymer insulators with busbar fittings. The busbars are typically aluminum or copper conductors welded together in sections to form lengths that may exceed a few hundred feet. The conductors distribute the three phase current in the substation through taps mounted transversely along the bus conductors. These are generally called bays and are used to connect transmission lines or transformers to the substation. The insulators are usually mounted atop insulator supports to raise the
0378-7796/97/$17.00 © 1997 Published by Elsevier Science S.A. All rights reserved. PlI S0378-7796(96)01 177-7
M.D. Budinich, R.E. Trahan /Electric Power Systems Research 42 (1997) 47-53
48
high voltage busbars to required heights to provide a safe level of electric insulation between the conductors and ground. Fig. 1 shows a general arrangement for a substation structure with horizontal insulator supports. Under normal operating conditions the load currents in the parallel busbars of a substation structure induce, through the interaction of the electromagnetic fields, tolerable forces on the neighbouring conductors that act in the plane formed by the busbars. During a short circuit fault the currents abruptly increase to many times the load currents to induce extraordinary dynamic forces on the busbars until the fault is removed from the system by protective relays or other means. This is generally accomplished in 6 to 8 cycles of the power system frequency. Possibilities for failure of the substation structure due to these fault currents include the permanent deformation of the busbars and insulator supports and the fracturing of the high cost insulators [5]. The substation structure must, therefore, be designed to withstand the dynamic forces arising from a short circuit fault to avoid the failure of any substation component without over-designing and unnecessarily increasing the overall cost of the structure.
z
1i
~.~/-
- ,
~**'s
/
', / / tI
.,
/
'a
~ e , ~ us/ '~.,..,I
N
/
J , ~
// ~
,~ ,/
///
k.,,.~~ wv
/ / J
F i g . 2. T y p i c a l t h r e e - d i m e n s i o n a l b e a m e l e m e n t .
large enough number of beam elements to define an accurate model of the substation structure. One of the most important aspects for obtaining accurate results from a finite element analysis of the substation structure is providing the analysis with an accurate mathematical model of the structure and its externally applied forces.
3. Finite element analysis 3.1. Modelling the substation structure.
The finite element method is a computer-aided mathematical technique used for obtaining approximate numerical solutions to a wide variety of engineering problems. As applied to substation design, the method is based on partitioning the substation structure into a finite number of discrete elements to reduce the problem from an infinite number of unknowns to a finite number of unknowns. The finite elements used to model the substation structure in PCBus are three-dimensional beam elements as shown in Fig. 2. The true solution to the problem is approached by partitioning a
IInsul"tOr
<~ <~>L"-----HorizonlalInsulatorSupport F i g . 1. G e n e r a l a r r a n g e m e n t f o r a s u b s t a t i o n s t r u c t u r e w i t h h o r i z o n tal i n s u l a t o r supports.
The substation components that are modeled in PCBus include the insulators, insulator supports, busbar conductors, and busbar fittings. The busbar fittings that can be modeled in PCBus include fixed fittings, slide fittings, and expansion fittings. Cantilever ends can also be modeled in PCBus. Each component of the substation structure is modeled with its equivalent mechanical properties assigned to the beam elements that correspond to the component in the three-dimensional model of the structure. The mechanical properties required to define the beam elements are the moment of inertias about the cross sectional axes, Ix and Iy; polar moment of inertia, J; modulus of elasticity, E; density, p; cross-sectional area, A; and Poisson's ratio, v. The required mechanical properties for the beam elements are derived within PCBus when an analysis is performed from the necessary manufacturer data entered into the PCBus model spreadsheet for the various components. The manufacturer data entered into the spreadsheet to model an insulator include height, li, s; core diameter, dins; maximum permissible cantilever force, Fins; maximum tip deflection, Jms; and weight, W,ns. The manufacturer data entered into the spreadsheet to model the other substation components closely match the mechanical properties required to model a beam element and can be found directly from the data sheets in a steel construction handbook. Furthermore, to ease the creation and editing of a substation struc-
M.D. Budinich, R.E. Trahan / Electric Power Systems Research 42 (1997) 47 53
ture model the P C B u s model spreadsheet is linked to separate component libraries for each substation component for which all the necessary manufacturer data is entered into the model spreadsheet when a specific component is selected. This feature of P C B u s eliminates the need for the substation designer to explicitly enter the manufacturer data into the model spreadsheet to model a substation component. The assumption that an insulator can be accurately modeled with beam elements is valid if the insulator exhibits the behaviour of a true beam. Research conducted at the Langen Research Center in Germany on one hundred insulators concluded that an insulator does exhibit the behaviour of a true beam and can accurately be modeled as a cantilever beam [2]. Similar conclusions were also found from the results of research conducted at Tulane University [31. The failure of an insulator occurs in an analysis if the maximum permissible cantilever force rating is exceeded. The maximum permissible cantilever force rating is determined by the manufacturer by applying a static load to the tip of the insulator until a failure occurs. The applied static load at the tip of the insulator produces a maximum moment, Min~, at the base of the insulator given by Mins = Fins/'ins
(1)
The mechanism for failure is this maximum moment since the moment induces local stresses that ultimately cause an insulator to fail. In a dynamic analysis the maximum moment at the base does not correspond directly with the force at the tip of the insulator due to the included inertial forces of the structure. As a result, a maximum equivalent cantilever force, Fm~x, is derived for comparison with the maximum permissible cantilever force, Fm~, by using the largest moment in the insulator predicted by a dynamic analysis, M . . . . in the following relation Fm,x --
mmax
(2}
In the analysis, the insulator fails if the maximum equivalent cantilever force, Fmax, multiplied by the safety factor is greater than the maximum permissible cantilever force, Fi,~. The failure of a conductor occurs in an analysis if the maximum stress in the conductor predicted by the analysis, a .... exceeds the maximum stress rating for the conductor. Like-wise, the failure of an insulator support occurs in an analysis if the maximum stress in the insulator support predicted by the analysis, Osup, exceeds the maximum stress rating for the insulator support.
49
3.2. Modelling the external forces The forces that are modeled in P C B u s include the fault current forces, wind loading, ice loading, and gravitational forces. The forces are modeled with the equivalent mathematical descriptions applied to appropriate nodes of the three-dimensional finite element model of the substation structure. The instantaneous currents at the onset of a three phase fault are given by il(t) = il(0)e - R~t/XL + i~{e- R~,,t/XL COS(E) -- COS(~Ot+ E)}
(3)
i2(t) = i2(O)e R~o,/XL + i ~ I e - R,~,'XL t.
{" 2rC'~ cost,+y)
i3(t ) = i3(0)e - k~,~t/XL + i~ {e - R°~''XLCOS(¢ -- ~-j~ )
where il(t), i2(t), and i3(t) are the instantaneous fault currents at time, t, after the onset of the fault for the center and outer conductors, respectively [amps]; iL(0), i2(0), and i3(0) are the instantaneous load currents the instant of time just before the fault [amps]; R is the system resistance [ohms]; JfL is the system inductive reactance [ohms]; a) is the system [amps]; and e is the fault current phase angle [rad]. Using the instantaneous currents in (3), (4), and (5), the instantaneous forces per unit length of the conductors are found by Ampere's law and assuming the conductors are infinitely long and separated by a distance that is much greater than the diameters of the conductors. The instantaneous forces on the center conductance for which the current i~(t) flows is given by wl(t ) -
5 . 4 x 10 -7 {il(t)i3(t ) - il(t)i2(t)} D
(6)
The instantaneous forces on the outer conductor for which the current i2(t) flows is given by w2(t) -
5.4 × 10-7 D {i2(t)it(t) + ½i2(t)i3(t)}
(7)
Likewise, the instantaneous forces on the outer conductor for which the current i3(t) flows is given by w3(t) -
5.4 x 10- v D {i3(t)i~(t) + ½i3(t)i2(t)}
(8)
In Eqs. (6)-(8), the forces are distributed along the conductors in lbf ft-~ and D is the phase distance
50
M.D. Budinich, R.E. Trahan/Electric Power Systems Research 42 (1997) 47-53
between adjacent conductors in inches. Since worst case conditions are needed for the design of the substation structure, it was determined that the maximum forces are imparted to the structure on the outer conductor through which the current i2(t) flows when the fault current phase angle is optimized to maximize the RMS forces to give
Table 2 Effects of modifying the stiffness of the connections of the fixed fittings to the insulators on the dynamic response of the substation structure
E = 0.86921 rad
Fixed fitting rigid connection Fixed fitting -- loose connection
(9)
The remaining forces modeled in PCBus are static forces and are determined by using the relations found in ANSI/IEEE Std. 605-1987. It is not required to include any of these static forces in an analysis. 3.3. Dynamic response o f the substation structure
Using the finite element models for the substation components and external forces, an assembly of the element equations yield the dynamic governing system equations for the entire substation structure given by M / / + CLi + Ka = f
(10)
where M is the equivalent mass matrix, C is the damping matrix, K is the nodal stiffness matrix, f is the applied nodal actions vector, and a is the nodal displacements vector. Eq. (10) is solved for the nodal displacements in a dynamic finite element analysis by a numerical time-stepping method. The moments and stresses are then derived in a final operation called postprocessing. Several dynamic finite element analyses were performed on an arbitrary single span substation structure to investigate the changes in the dynamic response for modifications of the substation structure. Interesting results were observed for modifications of the substation structure's span length and stiffness of the fittings' connections to the insulators. Table 1 shows the maximum stresses imparted to the conductors and the maximum equivalent cantilever forces imparted to the insulators from the results of a series of dynamic analyses for increases in span length. The results show that an increase in the span length results in an increase in the maximum conductor stress. The results also show that the insulator equivalent
Substation structure description
Maximum value Croo.(psi)
F~.s (lbf)
928 1616
144.61 109.89
cantilever forces are lower for a 30 ft span than a 20 ft span. This suggests that an optimum span length of over 20 ft may be found that minimizes the maximum insulator equivalent cantilever force. A possible explanation for this phenomena is that as the span length increases to a certain length, the natural frequencies of the structure are lowered to give the structure an increasingly low-pass nature that more than offsets the forces added by the increases in span length. Table 2 shows the maximum stresses imparted to the conductors and the maximum equivalent cantilever forces imparted to the insulators from the results of two dynamic analyses for changes in the connections of the fittings to the insulators from rigid to loose. The results show that the maximum conductor stresses greatly increase for fittings with loose connections to the insulators. This is a result of the loose fittings allowing the conductors to have greater displacements. In contrast, the maximum insulator equivalent cantilever forces reduce for fittings with loose connections to the insulators. This is a result of the extra energy dissipated in the conductors with the greater displacements of the conductors. The results show that if it is not known whether the fittings being modeled have rigid or loose connections to the insulators, two separate analyses with the fittings modeled as having rigid connections and loose connections should be performed to investigate the possible range of resuits. 3.4. Using static analysis as a quick design tool
Table 1 Effects of modifying the length of the busbar conductors on the dynamic response of the substation structure
One disadvantage of performing a dynamic analysis is the large amount of time required to complete an analysis. A static analysis requires less time to complete since it does not require numerical time-stepping techniques and involves only solving the static representation of the system equations given by
Substation structure description
Maximum value
Ka = f
crcon(psi)
Fins (lbf}
928 1964 3085 3652
144.61 123.62 143.69 163.07
where the applied nodal actions vector, f, contains only static forces determined by the maximum short circuit forces calculated in Eqs. (6)-(8). However, as mentioned previously, a static analysis results in less accurate results since it cannot account for the dynamic behaviour of the substation structure and, therefore,
20 ft Span, 4 in Conductor 30 ft Span, 4 in Conductor 40 ft Span, 4 in Conductor 50 ft Span, 4 in Conductor
(11)
51
M.D. Budinich, R.E. Trahan //Electric Power Systems Research 42 (1997) 47-53
Table 3 Comparison of results for the dynamic analyses, static analyses employingscale factors, and static analyses with no scale factors Substation structure description
30 fl span 40 fi span 50 fl span
Dynamic analysis
Static analysis Kdy~and K~o,
Static analysis (no scale factors)
~o, (psi)
Fi,s (lbf)
a¢o, (psi)
Fi,~ (lbf)
a¢o, (psi)
Fins (Ibf)
1964 3085 3652
123.62 143.69 163.07
1560 2830 4483
107.2 142.78 178.78
4210 7660 12 140
194.38 259.17 323.96
yields more conservative designs. In order to obtain static analysis results which are closer to the results of a dynamic analysis, a dynamic force scale factor, Kdyn, and a conductor stress scale factor, Kcon, were introduced into the P C B u s static finite element analysis. To determine the scale factors, a series of static analyses were performed on various configurations of the substation structure for different iterations of the dynamic force scale factor to compare the static results with the corresponding dynamic results. It was possible to find a value for the dynamic force scale factor that approximately matched the maximum insulator equivalent cantilever forces from the static analyses to the results of the dynamic analyses by the following reaction Wstat = K d y n m a x { w ~ ( t ) }
(12)
where wi(t ) is the dynamic force imparted to the ith conductor• However, the maximum conductor stresses calculated from the static analyses were still generally much higher than the corresponding maximum conductor stresses calculated from the dynamic analyses. Therefore, the conductor stress scale factor was introduced to lower the conductor stresses resulting from the static analyses by the following relation
4. Verification of the PCBus results
Verification of the P C B u s program results was made possible by the full scale fault current tests performed by the British Columbia Hydro and Power Authority (BCHydro) during August and September of 1993. The verification of the P C B u s results presented here will be performed by comparing the full scale test results of the single span and two-span substation structure to the corresponding P C B u s results. Fig• 3 shows the plan views for the two substation configurations. The insulator equivalent cantilever forces in the direction of the short circuit forces were measured with calibrated fullbridge strain gauge transducers located at the bases of the insulators. In addition, the displacements of a conductor at the middle of a span for selected tests were measured by a laser pointer and video camera. Since the greatest forces were imparted to the conductor through which the current i2(t) flowed, only the result for this A-phase outer conductor will be used for verification. The BCHydro substation structure was constructed with N G K 8A-67971A 230 kV insulators and A L U M 6061-T6 6.0 SCH 40 conductors with 3-0 damping
O'con{dyn} ~ g c o n O ' c o n { s t a t }
While the dynamic force scale factor is applied to the static short circuit forces before an analysis, the conductor stress scale factor is applied to the conductor stresses after an analysis. The following values for the dynamic force scale factor and conductor stress scale factor were selected
-~,_,.,,,~
.9
eY ,
gdyn = 0.55
5
!
Tuqr i
b r~T
I
Kco, = 0.37 Table 3 shows the results of the dynamic analyses and the scaled static analyses for various span lengths of a single span substation structure. Although the results show that improvements in the static analyses are achieved by the scale factors, it is apparent that the static analyses cannot accurately account for the dynamic response of the substation structure by employing scale factors.
O"x rLtl
C~L ~ v
.fl o
.~ I ~ . .
,s
0-, t
l"
,%2," t - !
IP
i
-=-
i,,,, I
Fig. 3. Plan views of the BCHydro single span and two-span substation structures•
52
M.D. Budinich, R.E. Trahan /"Electric Power Systems Research 42 (1997) 47 53
Table 4 Comparison of the maximum conductor displacements for increasing durations of the 50 kA fault current
600 400 2oo
b.
-200
~.mm.u
o
I
o.2
.
I
'
0.4
I
!
I
o.s
o.a
1
Fig. 4. Comparison of the BCHyrdo and PCBus equivalent cantilever forces for Insulator # 1 of the single span structure for a 50 kA, 0.1 s fault.
cables. The insulators were mounted to horizontal insulator supports with l l . 6 i n W10 × 26 mounting adapters and the conductors were mounted to the insulators with fixed fittings. It was necessary to have current feeder lines attached to the conductors at one end of the substation structure and a shorting bar attached to the other end to perform the short circuit tests. 4.1. Single span structure
The single span substation structure was constructed with a 52.49 ft (16000 mm) span. A 50 kA fault current was applied to the substation structure via the current feeder lines for a duration of 0.1 s. Figs. 4 and 5 show a comparison of the measured equivalent cantilever forces from BCHydro to the predicted equivalent cantilever forces from PCBus imparted to Insulators # 1 and # 4, respectively. As shown in Fig. 3, Insulators # 1 and # 4 were located at the current feeder and shorting bar ends of the A-phase outer conductor, respectively. Note that the signals received from the strain gauges were corrupted with 60 Hz noise during the first 0.1 s due to the electromagnetic interference of 600 ..............................................................
200-o U . .
o
02
0.4
06
0a
1
BCHydro (in)
PCBus (in)
%error
50 100 150 200
5.0 7.3 8.6 10.3
4.9 7.3 8.9 9.7
2.0 0.0 3.5 5.8
1.2
T~me (s)
400
50 kA fault duration (ms)
12
T~me (s)
Fig. 5. Comparison of the BCHydro and PCBus equivalentcantilever forces for Insulator # 4 of the single span structure for a 50 kA, 0.1 s fault.
the short circuit currents picked up by the instrumentation cables. The comparisons show a high correlation of the predicted results to the measured results. Note that the measured insulator equivalent cantilever forces from the BCHydro test are different for each insulator since the end conditions were different at each end of the conductor due to the current feeder lines near Insulator # 1 and the shorting bar near Insulator # 4. The end conditions impart large forces to increase the maximum equivalent cantilever forces imparted to the insulators located at the ends of the substation structure. The PCBus results are exactly the same for each insulator since the end conditions were not modeled and, therefore, the model for the single span substation structure was symmetric about the midpoint of the conductor. The PCBus model spread-sheet is not equipped to model the end conditions caused by current feeder lines or shorting bars since these do not occur in practice and are only included in the BCHydro substation structure as part of the test structure setup. Table 4 shows the maximum measured conductor displacements for the A-phase outer conductor of the single span substation structure subjected to increasing durations of a 50 kA fault current. The damping cables were removed from the conductors during these tests. As shown, the PCBus results have very small errors compared to the measured results. 4.2. Two-span structure
The two-span substation structure was constructed with a 52.49 ft (16000 mm) span and a 19.69 ft (6000 mm) span. A 50 kA fault current was again applied to the substation structure via the current feeder lines for a duration of 0.1 s. Figs. 6 - 8 show a comparison of the measured equivalent cantilever forces from BCHydro to the predicted equivalent cantilever forces from PCBus imparted to Insulators # 1, # 4 , and # 7 , respectively. As shown in Fig. 3, Insulator # 7 was located at the far end of the second span near the shorting bar. Again, the comparisons show a high correlation of the predicted results to the measured results. This is especially true for the comparison of the measured and predicted equivalent cantilever forces imparted to Insulator # 4. Verification of the PCBus
M.D. Budinich, R.E. Trahan// Electric Power Systems Research 42 (1997) 47-53
53
800
6OO
600
400
4OO
200
~2,00 azl
~" -200
,~,..20CI
-400
.400
-6OO -800
0
0.2
0.4
0.6
0.8
0
Time (s)
800 60O T ........................................................................ .......................... .. . . . . . . . . . . . . .
o .2oo -40(3 -6OO -800 o
0.2
04
0.4
0.6
0.8
1
"nine (S)
Fig. 6. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 1 of the two-span structure for a 50 kA, 0.1 s fault.
400
0.2
o.6
o,s
Tqme (s)
Fig. 7. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 4 of the two-span structure for a 50 kA, 0.1 s fault.
results for the equivalent cantilever forces imparted to this insulator is the most important verification since the largest forces of the two substation configurations were imparted to this insulator and the contributions of the shorting bar and current feeder lines were at a minimum. The results show that the predicted maximum equivalent cantilever force imparted to Insulator # 4 is very close to the measured maximum equivalent cantilever force.
5. Conclusions
Verification of the PCBus results shows conclusively that utilizing a dynamic finite element analysis to predict the dynamic response of a substation structure subjected to short circuit forces yields accurate results. PCBus is a powerful and convenient tool for substation design since accurate results are obtained from the dynamic finite element analyses and the involved process of generating accurate finite element models for the
Fig. 8. Comparison of the BCHydro and PCBus equivalent cantilever forces for Insulator # 7 of the two span structure for a 50 kA, 0.1 s fault.
substation structure and external forces is accomplished within PCBus from the data selected in the PCBus model spreadsheet. In addition, the design process is made more efficient by introducing scale factors to perform quick static analyses to obtain approximate results for a particular substation structure before performing the final dynamic analyses to obtain more accurate results. Research is currently being performed to improve the functionality of PCBus by adding the capability to include bus taps and bay conductors in an analysis.
References [1] IEEE Guide ./'or Design of Substation Rigid-Bus Structures, IEEE/ANS1 Std 605-1987, The Institute of Electrical and Electronics Engineers, New York, NY, 1987. [2] E. Bauer, E. Brandt, H. Dannheim, W. Lehmann, W. Meyer, K. Pietsch and N. Stein, Dynamic behaviour and Strength of High
Voltage Substation Post Insulators Under Short-Circuit Loads,
[3]
[4]
[5]
[6]
[7] [8]
LehrStul ftir Elektriscbe Energieversorgung Universit/it Erlangen-Nfirnberg, 1984. J.E. Borhaug and H.A. Thompson, Mechanical design of electric bus systems to withstand the magnetic forces produced by short circuit currents, Final Report, Tulane University, April 1970. M.D. Budinich, The application of a dynamic finite element analysis to busbar structures subjected to short circuit forces, Graduate Thesis, University of New Orleans, December 1994. B.A. Develle, Finite element analysis applied to substation busbar design, Graduate Thesis, University of New Orleans, May 1992. B.A. DeveUe, J.W. Schilleci and R.E. Trahan, A personal computer based program for the dynamic analysis of busbar structures under short circuit faults, Proc. IEEE Southeastcon '93, April 4-7, 1993, Charlotte, North Carolina, pp. 564-566. A. Opsetmoen, Short circuit test of 230 kV bus, General Analysis, BCHydro Stations Engineering Division, June 1994. T. Stefanski, Tests on 230 kV bus structures for BCHydro and the University of New Orleans, Test Report, Powertech Labs Inc., October 1993.