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Optimum designs for transmission line towers
Cnmpuun
00457949(94)00597-4
OPTIMUM
DESIGNS
FOR TRANSMISSION
& Strwrures Vol. 57. No. I, pp. 81-92, 1995 Copyright ‘3; 1995 Elsevier Science Ltd Printed in Great Brmin. All nghts reserved 0045.7949/95 $9.50 + o.no
LINE
TOWERS
G. Visweswara Rao Senior
Research
Analyst,
Engineering Mechanics Research India (p) Ltd, 907 Barton 84 M. G. Road, Bangalore-560001, India (Received
Centre,
10 April 1994)
Abstract-A method
for the development of optimized tower designs for extra high-voltage transmission lines is presented in this paper. The optimization is with reference to both tower weight and geometry. It is achieved by the control of a chosen set of key design parameters. Fuzziness in the definition of these control variables is also included in the design process. A derivative-free method of nonlinear optimization is incorporated in the program, specially developed for the configuration, analysis and design of transmission line towers. A few interesting results of both crisp and fuzzy optimization, relevant to the design of a typical double circuit transmission line tower under multiple loading conditions, are presented.
1. INTRODUCTION
number of design variables. Vanderplaats [8] has considered the base width and the panel heights also as the decision variables in his study on optimum configuration of a transmission line tower. Such a choice obviously results in fewer design variables. Fuzziness or vagueness in the description of the design variables and the associated constraints are other complicating factors that one has to consider in optimization problems. Optimization studies relevant to structural systems in such a fuzzy environment are of recent origin [9-141. Structural engineers often encounter situations in which it is not possible to precisely define their aspiration levels and constraint functions in a design problem. For example, in the present case of the transmission line tower if the base width is considered as one of the design variables in the optimization process, fixation of crisp lower and upper bounds for the variables mostly depends on the experience of the designer and is obviously affected by impreciseness. The fuzzy-set theory developed by Zadeh [15] has become a useful tool in modelling the intuition and impreciseness in the definition of design optimization problem. The procedure to arrive at an optimum structural design in a fuzzy environment involves the following steps: (i) Assign a transition interval for the variation of the objective function and the constraints from absolute preciseness to absolute impreciseness. (ii) Construct a membership function mapping each transition interval onto the number line between 0 and I. A fuzzy feasible domain is now characterized by a membership function which is an intersection of the membership functions of all the constraints. Within the general framework of the above procedure, there exist two familiar approaches--cc cut approach [ 161 and 2. formulation [ 171, for obtaining an optimum solution in fuzzy environment. In the
Of late, the investment in transmission facilities for extra high-voltage networks has increased considerably. In this connection, minimizing the cost of transmission line structures is an obvious need. While a uniform code of design practice [I] for these structures is followed, more or less, by many countries, there are no established criteria for obtaining an optimal tower design with regard to its weight and geometry. A high-voltage transmission line structure is a complex structure in that its design is characterized by the special requirements to be met from both electrical and structural points of view. The former decides the general shape of the tower in respect of its height and the length of its cross-arms that carry the electrical conductors. While the manoeuvreability in arriving at an optimal tower design is no doubt reduced by these electrical considerations, there is scope for the weight minimization and optimum geometry shaping of a transmission line tower. This is apart from the optimum sizing of the members by a structural design algorithm based on the fully stressed design concept [2]. Optimization of structures in weight and shape through mathematical programming methods has attracted wide attention in the past [2-71. Member sectional areas are usually treated as design variables for weight optimization. The joint coordinates are included as decision variables in the case of shape optimization. In combined shape and weight optimization problems, the main objective function, viz. the weight of the structure, is a highly nonlinear function of the design variables. This is since the nodal coordinates, and hence the member lengths, vary at each stage of iteration. The member sectional areas which are by themselves design variables are dependent on these member lengths. Thus the complexity of such an optimization process lies in the increased 81
G. Visweswara Rao
82
former approach, different satisfactory design levels u between 0 and 1 of the membership function of the fuzzy feasible domain are specified, resulting in a sequence of non-fuzzy optimization problems. It is to be noted that in this approach the objective function is supposed to be devoid of fuzziness. 1 formulation approach considers a new objective function in terms of a single parameter 1 which is characterized by a membership function defined by the intersection of the membership functions of both the constraints and the objective function. The optimization is then obtained by maximizing i with the help of any of the non-fuzzy optimization techniques. In this paper optimum weight and geometry of a transmission line tower under static loads with multiple loading cases are obtained. Few specific design variables that control the optimum solution are identified. Fuzziness in the specification of the boundaries of these control variables is considered and 1 formulation approach is adopted in arriving at the optimum design. Results are obtained from a dedicated program developed for the purpose and based on finite element truss analysis. Numerical results are presented for a typical transmission line tower illustrating the procedure adopted in the crisp and fuzzy optimization schemes.
2. TRANSMISSION LINE TOWER AND DESIGN VARIABLES
In the design of a high-voltage transmission line tower (Fig. 1) the electrical specification decides the general shape of the tower in respect of its height at G.C level
different levels and conductor and groundwire crossarm dimensions. This no doubt introduces a constraint in the form of not having full freedom in arriving at a tower design from a shape optimization point of view. However, the selection of tower basic dimensions, such as base width and widths at crossarm levels, is still open for a designer to shape a transmission line tower in an optimum way. Some of the main parameters of electrical specification are briefly described in Appendix I. 2.1. Normul design practice A complete overview of the different stages involved in a transmission line tower design is shown in the form of a flow-chart in Fig. 2. The optimization of the tower being the main theme of the paper, only salient features of the developed program are described here for the sake of brevity. The first stage of configuration is the distinguishing feature peculiar to a transmission line tower alone, when compared to the design of other normal steel structures. The main considerations in arriving at the tower configuration are: the specified electrical clearances; tower type; wind pressure; maximum and minimum temperature conditions; possible ice loads on the conductor and groundwire and terrain profile. Table 1 lists the details of some of these parameters typical to a 400 kV double circuit tangent type tower. Configuring a transmission line tower basically requires computation of conductor and groundwire sags and their permissible maximum working tensions under the critical wind pressure and temperature conditions. Proper coordination of these sags, in conjunction with maintainence of stipulated electrical clearances
7 s
T.C level f
h
MC level h
B.C level h I
Ground
level
(4 Fig. I. Transmission line tower. (a) Transverse face, (b) longitudinal face and (c) 3-D view. g-ground clearance. h-insulator length + conductor sag. s-groundwire sag. B.C-bottom cross-arm. M.Cmiddle cross-arm. T.C-top cross-arm. G.C-groundwire cross-arm.
Optimum
-
designs
for transmission
I
of the tower
I Nodes & element generation and formation of global stiffness matrix
I
Computation of loads & identification of load cases
I
3.dimensional analysis of the tower for nodal displacements & member forces
I
Structural design of tower members and arriving at the tower weight
I
Start optimization of tower weight and tower geometry
Fig. 2. Flow chart
Table
1. Main electrical
and structural
Parameter
Item No. 1
line tower design pro-
of transmission gram.
Tower
type Circuit Tower
Angle
of deviation
Ground
clearance
Conductor diameter unit weight area of cross-section breaking capacity coefficient of linear expansion Young’s modulus
9 10
11
83
along the tower height, fixes up the required configuration. Table 2 gives the results of the sag-tension computations of the conductor and groundwire. Figure 3 shows the single line diagram of the tower with main dimensions fixed by the configuration part of the design process. With regard to the next two stages of panelling and node, element generation of the transmission line tower design process, it is normal practice to adopt a K or X (double-warren) type of bracing (Fig. 4). The panel heights are obviously dependent on the choice of the bracing angle 4 used in each of these panels. In the analysis of a structure like the transmission line tower having large number of members, the effect of flexure is not significant [18] and it is generally analyzed as a truss with many members. In the static analysis of the tower, the primary loads that are considered are the wind and the dead-weight loads [l] with appropriate factors of safety. It is highlighted here that a transmission line tower is also typified by
Configuration of the tower -> height, widths & crossarm dimensions Panelling
line towers
design
parameters
of the example
Specification
tower
Description
1
400 kV Double Tangent
Fig.
2”
Fig. A2
8.84 m
Fig. 1
31.77 mm 2.002 kg m-t 5.97 cm* 16280 kg 0.1935 x 10e4 deg-’ 0.686 x IO6 kg cm-’
Groundwire diameter unit weight area of cross-section breaking capacity coefficient of expansion Young’s modulus
0.115 x 10-4deg-’ 0.1933 x IO’kgcm-*
Insulator
Suspension
ll.Omm 0.7363 kg m-’ 0.578 cm’ 6950 kg
type
Fig. A4
Shield angle
20”
Fig. A3
Line span
400 m
Between adjacent
Weight
600 m
Between lowest points of adjacent spans
span
Wind pressure conductor and groundwire tower members
45 kg m-’ 195 kgm-’
Temperature maximum minimum
65’ 0”
two towers
To compute wind loads on the projected area
For sag-tension computations
G. Visweswara
84
Rao
Table 2. 400 kV double circuit tangent type tower. Sag-tension values of conductor groundwire under critical wind pressure and temperature conditions Conductor
Description
Item no.
Groundwire
1
Maximum tension in kg at 0 and 45 kgm-? wind pressure
4405.0
1710.0
2
Minimum tension in kg at 65’ and zero wind pressure
3180.0
1250.0
3
Maximum sag in m at 65’ and zero wind pressure
12.6
9.21
4
Minimum sag in m at Or and 45kgm-’
10.05
1.75
the presence of an additional load case of a broken conductor or groundwire loading condition (Appendix II). Figure 5 shows the multiple load cases along with the computed loads for the example tower of 400 kV line. In Fig. 5 it can be observed that an extra load, normally termed as a longitudinal load, acts in a direction normal to the cross-arm axis in each of the broken conductor-groundwire condition cases. A mention has to be made here about the factors of safety to be maintained regarding the maximum working tensions in the conductor and groundwire. While these are dependent on the respective breaking capacities of the conductor and groundwire, the factors of safety impose an upper limit on these tension values in a crisp optimization process. Lowering the working tension values no doubt brings down the tower loads (refer to Appendix II). On the other hand, it obviously increases the conductor and groundwire sags, thereby resulting in an increase in tower height followed by a possible upward or downward trend in tower weight.
and
2.2. Choice of’ control aariables With a specified transmission line voltage the parameters, like line span, conductor and groundwire dimensions, remain invariant in the design process. As one can perceive, the wind pressure and temperature conditions are also out of the purview of control variables. Of the other different parameters that significantly influence the tower weight and geometry, the following are the chosen set of control variables in the present optimization study: (a) conductor tension t,; (b) groundwire tension t,; (c) tower base width b,; (d) tower width at bottom crossarm (e) tower width at tower top b,; (f) panel bracing angle 4.
The above choice is based on a sensitivity study carried out on the effect of each of these variables on the tower weight and geometry. Table 3 shows the results of such a study with respect to the 400 kV
h3 j ;_i
--
a = 8830 b = 3535 c = 2355 d = 6600 e=6355 f=6110 g = 3240 h, = 26035 h2 = 7665 h3 = 7665 h4 = 2975
/;-,
e .I- I
hz
b/-‘_
/J
T
/ I,
.d .-/
I
Maximum working tensions: conductor: 4405 kg groundwire: 17 10 kg
I) ht
:
Panel angle = 0 = 45”
// i
2. Fig. 3. Tower
geometry.
Results
level b2;
a
‘,
from configuration
part of the program.
Optimum designs for transmission line towers --T-T,, {I ;
;/ ( ,,,’ ,‘+’ ,,a’.’ .’
;*‘,. !.’i, e ,...:i.
,,----Y,
I” ., 1‘: ( ‘,<, ‘CL_: “,
,‘:.-,y.-f--y ,,fI/ ,; 1, ,,I’
:’._ s’.,
‘. :,, ’
(4
i:
,
/ ‘! / ‘i, 1 !a,,, ‘{
Fig. 4. Bracing patterns generally adopted for high voltage transmission line towers. H = panel angle. (a) K type bracing. (b) X type bracing.
1690
I
380 --+’ 7700
I
2565 \ 1060 ‘\; 165 ---7--5775
--380 7700
2640
7700
‘--a 7700 ‘--+
2640
I
2640 --et
I
2640 --+I
285 5775 ;
--5775
1980
5775
‘--+ 5775
1980
5775
I
1980 --+’
I
;
1980 --+’
Ground
---
1980
wire broken condition
(e)
(d) 1270
;
1980 --+’
Normal condition
(c)
1270
1270
1270
1270
I
285 ----
i
1270 -r--h
1980
i
----
----
t
5775
5775 ‘--+
--+’
----
,175 __1i-
1980
t
1980
5775
6605 \ 4335 1980
i 5715
5775 1980 ----
285 5775
5775
1980
of
1270
I
7700
details
In most of the earlier truss optimization studies in literature, joint coordinates and the member sizes form the control variable vector. This obviously leads to too large a size optimization problem to handle,
4
2640
the
3. CRISP OPTIMIZATION
1690
‘--+ 7700
2640 --et
double circuit tangent-type tower, which are listed in Tables 1 and 2.
8 (b)
85
----
t
1980
+
Top conductor
broken condition
Middle conductor broken condition
Bottom
conductor broken condition
Fig. 5. Multiple load cases. 400 kV double circuit tangent type transmission line tower. (a) Normal loading condition, (b) groundwire broken loading condition, (c) top conductor broken loading condition, (d) middle conductor broken loading condition and (e) bottom conductor broken loading condition. (The inclined arrow at each crossarm level in the broken loading conditions indicates the additional longitudinal load).
G. Visweswara
86
Rao
Table 3. 400 kV double circuit tower. Sensitivity analysis results. Effect of control variables on tower geometry and weight (percentage values in parenthesis is the effect of each variable with respect to tower weight of 11400 kg at base vector of control variables) 0.8*Base
I. I *Base value
Serial No.
Base value (initial configuration)
Control variable
Total height in mm
No. of panels
44355
13
Total weight in kg
Total height in mm
No. of
11710 (+2.7%) II335 (-0.6%) 11710 (+2.7%)
44265
I3
45705
I3
44340
13
panels
value Total weight in kg
I
1,
4405 kg
2
‘,
1710 kg
42805
13
3
h,
8830 mm
44330
II
4
b>
3535 mm
44470
II
II620 (+ 1.9%)
44260
I4
10805 (-5.2%)
5
b,
2355 mm
44390
IO
44310
I5
6
4
45
44380
9
12085 (+6.0%) II990 (f5.1%)
44310
I7
10635 (-6.7%) 10965 (-3.8%)
especially with PC based software. In the present study, selection of tower widths at a few nodal points on the tower, and the panel bracing angle as elements of the design variable vector X, has an obvious advantage. The variations affected in these variables during the optimization process indirectly amount to changing the nodal coordinates as well, while substantial reduction in the size of the problem is achieved. Moreover, for a transmission line tower, there is lack of freedom in choosing all nodal coordinates as design variables. This is due to the fact that arbitrary changes in these coordinates are likely to lead to violation of electrical clearance requirements. Thus, for the transmission line tower on hand, the crisp optimization problem can be stated as minimize
W(X),
subjected
to g,(X) > 0.0
is possible, while a sequence of accelerating steps of pattern searches are made along this direction till it continues to be a successful direction.
3.1. Results The optimization problem in eqn (1) is solved for the case of the 400 kV double circuit transmissionline tower, the details of which are given in Table 1. It is designed as a self-supporting tower with steel members of L-shaped angles with a bolted type construction. The configuration of Fig. 3 is utilized to arrive at the starting design. The design of tower members is guided by the following slenderness ratio requirements, for tension members L/r d 300,
for i = 1, .
, m,
variable
and
(1)
for compression where X is the control
11585 (+ 1.6%) II650 (+2.2%) II135 (-2.3%)
members
vector given by
W(X) is the tower weight and g,(X) is the vector of functions. In the present case, g,(X) covers the lower and upper limit constraints on the selected design variables. They also include the stress constraints and the restriction of positivity of all elements of the control variable vector. The constraints include both tension and stress compression types. The nonlinear constrained optimization problem is defined in eqn (1) is solved by the Hookes-Jeeves derivative-free method [19,20]. The method is simple, elegant and is easily adoptable for program development. It is basically a hill-climbing technique in that each control variable is varied in succession by a series of exploratory and pattern searches for achieving the desired objective of an optimized solution. The exploratory search is for finding the successful direction in which optimization
kL/r d 150
if it is a leg member crossarm member
kLlr
if it is a member carrying computed stresses
< 200
m constraint
kL/r d 250
or
if it is a redundant member carrying nominal stresses
(2)
where L is the length of the member, r is the radius of gyration. k is a nondimensional factor accounting for different end fixity conditions. k is taken to be unity in this study. The constraint functions g,(X) are
Optimum
f&(X) =
B:
g6(X)=4’<4<#”
i = member
as in eqn (3). For example, the upper bounds on the conductor and groundwire maximum working tension can as well be relaxed, to a small extent, without violating the specified factor of safety by any appreciable margin. In a similar fashion, the earlier selection of the lower and upper bounds on the tower widths in eqn (3) is governed by more intuition than availability of crisp boundaries. Decision making in such imprecise and vague environments is made convenient by the theory of fuzzy sets [15]. The main task in a fuzzy optimization problem is to treat the fuzzy objective function and the fuzzy constraints as fuzzy sets in the space of decision alternatives and characterize them by their membership functions. The membership functions are descriptors of the impreciseness or vagueness with which the boundaries of the fuzzy sets are known. For example, in the present case the membership functions for the lower and upper boundaries of the conductor maximum tension t, can be defined, respectively as,
and
number
gj, 7(X) = x, 2 0.0, i=1,...,6
(3)
1 1 stands for absolute
value.
T, and T, are taken to be the maximum working tension values corresponding to the initial configuration (Fig. 3). The bounds bf and B:, i = 1, . ,3, on the different tower widths are intuitively taken to be 25% below and above the respective initial configuration values. The bounds on C#Jare chosen to be 40 and 60”, respectively. a, is the compressive or tensile stress injth member. The upper bound tr” is taken to be 2600 kg cm-‘. The weight of the tower at the start of the optimization process is 11.400 tonne while the optimum weight of the tower is obtained as 10.000 tonne. The linear dimensions of the configuration corresponding to the final optimized design are shown in Fig. 6. Figure 7 shows the convergence of the tower weight with the number of iterations under the crisp optimization process.
P,, (W = 0
=(-t,+
if t, 2 I.l*T, I.l*TC)/(O.l*TC) if T,g
&2(X)
t,<
l.l*T,
= 1
if t, ,< T,
=
if t, < 0.9*T,
0
(4)
= (t, - 0.9*T,)/(O. 1*T,)
4. FUZZY OPTIMIZATION
if 0.9*T, < t, < T,
It
is intuitively obvious that in reality the bounds on the control variables may not be as crisply defined
a=9300=B, b = 2400 = Bz c= 1180=B3 d = 6150 e = 5880 f=5615 g = 2935 hl = 26770 h2 = 7670 h3 = 7670 h4 = 2205
= I
if t, 2 T,.
;: { j I ;
h,
\
Maximum working T,:4100kg Ts: 1710 kg
tensions:
Panel angle = 0 = 60”
i I i
Fig. 6. Crisp optimization.
The final optimum
tower design configuration.
(5)
G.
88
r.of i 3
5
Number Fig. 7. Fuzzy
The by
shapes
7
9
fl
13
Rao
15
of iterations
optimization. Convergence with number of iterations.
of the ~~ernbershj~
of tower
functions,
weight
as given
eqns (4) and (S), are shown in Fig, 8. The lower
value of T, = 4405 kg (Table 2) in eqns (4) and (5) is the value of the conductor maximum working tension as worked out from sag-tension computations with a resulting factor of safety of 3.7 with respect to its ultimate breaking capacity. A small variation of 10% above this stipulated value is not substantial and may be acceptable. In other words, the assumed membership function in eqn (4) for g,(X) describes this intentional small violation in a mathematical form. Similarly, the membership functions for the other constraints in eqn (3) are defined as
(b)
(a> Fig. 8. Fuzzy
optimization.
Shape
of membership function chosen (b) Upper constraint range.
for r,. (a) Lower
constraint
range.
Optimum
designs
for transmission
line towers
11.5
g,, W):
P~ldX)= 0
if 6, < 0.9*Bi
r
11.3 E
1*B:)
= (b, - 0.9*B:)/(o.
89
2 if 0.9*B:
= 1
< b, < Bi
$
if b, > B:,
i%,,(X):
+\ +\ +-++++-m
.04 11.11 IO.51
if 4 2 I.l*qS”
r+,(X)=0 = (-4
I
I
I
I
I
I
I
3
5
7
9
II
13
1
I 15
Number of iterations
+ 1.1*q5”)/(o.l*q%“)
Fig. 9. Crisp optimization. Convergence of tower weight with number of iterations.
if l.l*$“>f$2&”
if 4 <4”, function and the fuzzy constraints. Consequently the membership function of the fuzzy domain turns out to be
i%(X): y,m
\
2 10.9 3 tF: 10.7
if qb < 0.9*4’
= 0
PD(X) = min{~,$0
= (c#l- 0.9*+ ‘)/(O. 1*cj 1)
if4
The membership structed as I&(X) = 0 = (-
24’.
function
(6) for W(X) is also con-
if W(X) 2 W”
maximize
2,
subjected
to the constraints
W(X) + W”)/(O. 1* W”)
1
if 0.9*W” < W(X) < W” zr 1
if W(X) < W”.
and
1
is now objective
._.g ..
7
a=8880=B, b = 3220 = BZ c = 1920 = B3 d = 6465 e=6185 f = 5900 g = 4700 h, = 26490 h2 = 7680 h3 = 7680 h4 = 2340
e
I
I
j-
j j
ht
Maximum working T, : 4205 kg T,: 1745 kg
;
Fig. IO. Fuzzy optimization.
i = 1, 12.
(9)
The fuzzy optimization problem as stated in eqn (9) is solved by the Hookes and Jeeves method. It is to be noted here that the control variable vector is now augmented by one more element in the form of 1. The
h4 _.
h2 I
(8)
4.1. Results (7)
The optimization in fuzzy environment viewed as an intersection of the fuzzy
i = 1, 12.
If pD(X) is now denoted by 2, the optimum solution is obtained by maximizing 1. At this stage the fuzzy optimization problem can be stated as an equivalent crisp optimization problem as
if0.9*$1<&
P&X)},
II I I // The final optimum
I I I I
tensions:
Panel angle = 8 = 45”
tower design configuration.
G. Visweswara
90 Table
4. Fuzzy
Rao
optimization. Membership function values L-lower constraint range, U-upper constraint
at optimum range
solution;
h, W
(2,
c&
(U)
hL (L)
(hLi)
2)
0.31
0.76
0.65
0.94
0.58
0.06
0.50
as in Fig. 3 is again utilized to start the optimization process. W” in eqn (7) is taken to be the corresponding tower weight (Table 3). The final optimization solution is 10.700 tonne in the form of tower weight. Figure 9 shows the convergence of the optimum solution in terms of tower weight with number of iterations. The linear dimensions of the tower are shown in Fig. 10. The value of 1 corresponding to the optimum solution is 0.58 and amounts to the maximum degree of membership of the optimum vector X* to the feasible domain D. At the end of the optimization process all the elements of the control variable vector X remain active in either upper or lower constraint ranges. The degree of satisfaction expressed by the respective membership function values at the optimum solution are detailed in Table 4. configuration
5. DISCUSSION
The crisp and fuzzy optimization results are presented for transmission line towers of tangent type (Appendix I). This type of tower being used for straight stretches of a transmission line, constitutes 80-90% of the total number of towers. In this context, weight optimization of the tangent type towers assumes significance. The minimization scheme is applicable to other types of transmission line towers as well. It is based on the optimization of tower shape through the judicious selection of a few control parameters. In particular there are six parameters in all and excepting the two tension parameters t, and t,, the others are directly connected with the tower shape. Variations in r, and 1, indirectly affect the tower shape in the form of varying the inter cross-arm heights and hence the tower weight. The selection of the panel angle 4 as a control variable is significant in that its variation controls the panel height, the key parameter deciding the nodal coordinates. The sensitivity of tower weight and geometry with respect to this parameter is comparable to that of b, , b2, and b,. This can be observed from Table 3. It is to be noted however, that geometry shaping and weight minimization strongly depends on the combined influence of these variables. In a crisp optimization problem the saving in weight of the tower is to the extent of 12% with respect to the tower weight, corresponding to the base vector. The fuzzy optimization has resulted in a tower weight higher than that of crisp optimization. However a saving in tower weight of 6% is achieved by the fuzzy optimization process. The convergence of tower weight, by the Hookes-Jeeves method, is fast in the case of both crisp and fuzzy optimization, as can be observed from
Figs 7 and 9. It is to be noted that the membership functions in eqns (4)-(7). chosen for the transition interval of each parameter, are not unique. However the active participation of the control variables chosen for the optimization process is apparent from their respective membership function values in Table 4.
6. CONCLUSIONS
Optimum design of transmission-line towers has been discussed in this paper. A systematic procedure has been presented for obtaining minimum weight tower design in both crisp and fuzzy environments. The efficiency of the optimization scheme is due to the choice of a few key parameters that influence the tower geometry and thus indirectly the weight, as design control variables. This has also reduced the computational effort considerably. A dedicated program has been developed for developing the optimum design from the initial stage of tower configuration generation. to that of analysis, design and optimization. The results presented for a typical high voltage transmission line tower have shown substantial saving in tower weight. REFERENCES
I. Committee on Electrical Transmision Structures of the Committee on Analysis and Design of Structures of the Structural Division. Loadings for electrical transmission structures. J. .s~ruc(. Dir. ASCE 108(5), 1088~1105 (1982). 2. B. H. V. Topping. Shape optimization of skeletal structures: a review. J. .v/ruct. Engng ASCE 109(g), 1933 -1951 (1983). 3. D. J. Sheppard and A. C. Palmer, Optimal design of transmission towers by dynamic programming. Compul. Srruct. 2, 455 468 (1972). and F. Moses, Automated design of 4. G. N. Vanderplaats trusses for optimum geometry. J. SITUCI.Die. ASCE 98, 671690 (1973). and 0. C. Zienkiewicz, Optimum 5. R. H. Gallagher Structuml Design, Theory and Applications. Wiley. New York (1973). of trusses. /. slruc’t. Div. 6. M. P. Saka, Shape optimization ASCE 106(5), 115551174 (1980). 7. W. H. Greene, Minimum weight sizing of guyed antenna tower. J. struct. Engng ASCE lll(lO), 2121~~2137 (1985). Numerical methods for shape 8. G. N. Vanderplaats. optimization-& assessment of the state of the art. Proc. Int. Swap. O~tmum Steel Design, Tuscan. AZ (1981). 9. D. 1. Blockley. The role of fuzzy sets in civil engineering. Fu-_-_J,Sets .S_v.st. 2, 2677278 (1979). IO. C. B. Brown and J. T. P. Yao, Fuzzy sets and structural engineermg. J. .struct. EngnX ASCE 109, 1211~1225 (1983).
Optimum
designs
for transmission
11. W. G. Yuan and W. W. Quart, Fuzzy optimum design of aseismic structures. Earfhquake Engng snuc/. Dyn. 13, 827-837 (1985). 12. W. G. Yuan and W. W. Quart, Fuzzy optimum design of structures. Engng Optbniz. 8, 291-300 (1985). and J. R. Rao, Fuzzy goal 13. R. N. Tiwari, S. Dharmar programming, an additive model. Fuzzy Sets Syst. 24, 27-34 (1987). T. A. Phelps and K. M. Ragsdell, 14. S. U. Mohandas, Structural optimization by goal programming approach. Comput. Strucr. 37(l), l-8 (1990). 15. L. Zedeh. Fuzzy sets. Information Control 8, 338-353 (1965). B. G. Prakash and C. 16. S. S. Rao, K. Sundararaju, Balakrishna, Multiobjective fuzzy optimization techniques for engineering design. Compuf. Strucr. 42(l), 1744 (1992). optimization of fuzzy struc17. S. S. Rao, Multiobjective tural systems. Inr. J. numer. Meth. Engng 24, 1157-I 111 (1987). Loh, Dynamics of 18. W. Weaver Jr, and C. Lawrence trusses by component mode method. J. struct. Engng ASCE 111(12), 256552575 (1985). Applied Nonlinear Programming. 19. D. M. Himmelblau, McGraw-Hill, New York (1972). 20. S. S. Rao, Optimization: Theory and Apphcalions, 2nd Edn. Wiley, New York (1984).
APPENDIX
The main parameters design of transmission
91
Fig. A2. Angle of conductor
specification
in the
deviation arm.
near a typical cross-
(i) Line voltage. The voltage level generally fixes up the safe limits on the conductor-to-ground clearance (Fig. I), conductor-to-metal clearances (Fig. Al) and required clearances over terrain obstacles such as river-rail crossings, teleline and other transmission lines of different voltage. (ii) Number of circuits. number of cross-arms.
I
of the electrical line tower are:
line towers
The number
of circuits decide the
(iii) Angle of conductor deviation, /l (Fig. A2). #’ broadly classifies transmission line towers into tangent and angle type towers. O-2” belongs to the first category and above 2,’ to the angle type. (iv) Shield angle (Fig. A3). Shield angle is the angle between the line joining the groundwire and conductor attachment points and the vertical. It is meant for electrical protection of the current carrying conductor. (u) Znsulators (Fig. A4). Suspension insulators are used for tangent type towers and tension insulators are used for angle type of towers. These insulators and their associated connecting wires are susceptible to deflections due to wind. Maintenance of electrical clearances and hence the cross-arm lengths and heights are strongly dependent on these insulator swing angles.
Fig. Al. Electrical ance diagram.
clearances. c-specified
Conductor electrical
to metal clearance.
clear-
(vi) Conductor and groundM,ire. ACSR (aluminium conductor steel reinforced) stranded conductors are normally used for high voltage transmission lines. High tensile galvanized steel wires are used as groundwires. The sizes of conductor and groundwire depend on the line voltage.
I,
(---T-
I. -----_-_________ .I_
\_____-.------~---- :_,,
Groundwire pomt
attachment
1
~___------
i ‘I Fig. A3. Shield angle.
-
Insulator
-
Conductor point
attachment
G. Visweswara
92
GPPENDIX II
-----_____ ,.. ---___ _.: y.- --_ _:’
_/-
Rao
-----z ____ --__
__._---
$
z
Insulator
The loads on a transmission line tower are mainly due to wind and self weight. The deviation of the line (Fig. A2) also canses a load arising from the tension of a conductor-groundwire at their respective cross-arm ends. This load acts in the direction of the cross-arm axis. Thus, in normal operating condition (Fig. 5a) the loads are of two types. They are given by items (i) and (ii) below. (i) Vertical load at a cross-arm tip =self weight of conductor-groundwire + insulator and its hardware + weight of linemen and tools. (ii) Transverse load at a cross-arm tip =wind load on conductor-groundwire and insulator + tension component due to line deviation.
.f’ -
Conductor
(b) Fig. A4. Insulators. (a) Suspension type used for tangent type towers and (b) tension type used for angle type towers.
The wind load on tower members also forms part of the transverse load. It is either lumped at the cross-arm tips with appropriate apportioning of the tower body for the purpose or treated as a distributed load in the analysis. In the case of a broken condition (Fig. Se) of either a groundwire or a conductor, the resulting unbalanced tension in the intact span acts in a direction parallel to the conductor and a major load component acts in a direction normal to the cross-arm axis.
00457949(94)00597-4
OPTIMUM
DESIGNS
FOR TRANSMISSION
& Strwrures Vol. 57. No. I, pp. 81-92, 1995 Copyright ‘3; 1995 Elsevier Science Ltd Printed in Great Brmin. All nghts reserved 0045.7949/95 $9.50 + o.no
LINE
TOWERS
G. Visweswara Rao Senior
Research
Analyst,
Engineering Mechanics Research India (p) Ltd, 907 Barton 84 M. G. Road, Bangalore-560001, India (Received
Centre,
10 April 1994)
Abstract-A method
for the development of optimized tower designs for extra high-voltage transmission lines is presented in this paper. The optimization is with reference to both tower weight and geometry. It is achieved by the control of a chosen set of key design parameters. Fuzziness in the definition of these control variables is also included in the design process. A derivative-free method of nonlinear optimization is incorporated in the program, specially developed for the configuration, analysis and design of transmission line towers. A few interesting results of both crisp and fuzzy optimization, relevant to the design of a typical double circuit transmission line tower under multiple loading conditions, are presented.
1. INTRODUCTION
number of design variables. Vanderplaats [8] has considered the base width and the panel heights also as the decision variables in his study on optimum configuration of a transmission line tower. Such a choice obviously results in fewer design variables. Fuzziness or vagueness in the description of the design variables and the associated constraints are other complicating factors that one has to consider in optimization problems. Optimization studies relevant to structural systems in such a fuzzy environment are of recent origin [9-141. Structural engineers often encounter situations in which it is not possible to precisely define their aspiration levels and constraint functions in a design problem. For example, in the present case of the transmission line tower if the base width is considered as one of the design variables in the optimization process, fixation of crisp lower and upper bounds for the variables mostly depends on the experience of the designer and is obviously affected by impreciseness. The fuzzy-set theory developed by Zadeh [15] has become a useful tool in modelling the intuition and impreciseness in the definition of design optimization problem. The procedure to arrive at an optimum structural design in a fuzzy environment involves the following steps: (i) Assign a transition interval for the variation of the objective function and the constraints from absolute preciseness to absolute impreciseness. (ii) Construct a membership function mapping each transition interval onto the number line between 0 and I. A fuzzy feasible domain is now characterized by a membership function which is an intersection of the membership functions of all the constraints. Within the general framework of the above procedure, there exist two familiar approaches--cc cut approach [ 161 and 2. formulation [ 171, for obtaining an optimum solution in fuzzy environment. In the
Of late, the investment in transmission facilities for extra high-voltage networks has increased considerably. In this connection, minimizing the cost of transmission line structures is an obvious need. While a uniform code of design practice [I] for these structures is followed, more or less, by many countries, there are no established criteria for obtaining an optimal tower design with regard to its weight and geometry. A high-voltage transmission line structure is a complex structure in that its design is characterized by the special requirements to be met from both electrical and structural points of view. The former decides the general shape of the tower in respect of its height and the length of its cross-arms that carry the electrical conductors. While the manoeuvreability in arriving at an optimal tower design is no doubt reduced by these electrical considerations, there is scope for the weight minimization and optimum geometry shaping of a transmission line tower. This is apart from the optimum sizing of the members by a structural design algorithm based on the fully stressed design concept [2]. Optimization of structures in weight and shape through mathematical programming methods has attracted wide attention in the past [2-71. Member sectional areas are usually treated as design variables for weight optimization. The joint coordinates are included as decision variables in the case of shape optimization. In combined shape and weight optimization problems, the main objective function, viz. the weight of the structure, is a highly nonlinear function of the design variables. This is since the nodal coordinates, and hence the member lengths, vary at each stage of iteration. The member sectional areas which are by themselves design variables are dependent on these member lengths. Thus the complexity of such an optimization process lies in the increased 81
G. Visweswara Rao
82
former approach, different satisfactory design levels u between 0 and 1 of the membership function of the fuzzy feasible domain are specified, resulting in a sequence of non-fuzzy optimization problems. It is to be noted that in this approach the objective function is supposed to be devoid of fuzziness. 1 formulation approach considers a new objective function in terms of a single parameter 1 which is characterized by a membership function defined by the intersection of the membership functions of both the constraints and the objective function. The optimization is then obtained by maximizing i with the help of any of the non-fuzzy optimization techniques. In this paper optimum weight and geometry of a transmission line tower under static loads with multiple loading cases are obtained. Few specific design variables that control the optimum solution are identified. Fuzziness in the specification of the boundaries of these control variables is considered and 1 formulation approach is adopted in arriving at the optimum design. Results are obtained from a dedicated program developed for the purpose and based on finite element truss analysis. Numerical results are presented for a typical transmission line tower illustrating the procedure adopted in the crisp and fuzzy optimization schemes.
2. TRANSMISSION LINE TOWER AND DESIGN VARIABLES
In the design of a high-voltage transmission line tower (Fig. 1) the electrical specification decides the general shape of the tower in respect of its height at G.C level
different levels and conductor and groundwire crossarm dimensions. This no doubt introduces a constraint in the form of not having full freedom in arriving at a tower design from a shape optimization point of view. However, the selection of tower basic dimensions, such as base width and widths at crossarm levels, is still open for a designer to shape a transmission line tower in an optimum way. Some of the main parameters of electrical specification are briefly described in Appendix I. 2.1. Normul design practice A complete overview of the different stages involved in a transmission line tower design is shown in the form of a flow-chart in Fig. 2. The optimization of the tower being the main theme of the paper, only salient features of the developed program are described here for the sake of brevity. The first stage of configuration is the distinguishing feature peculiar to a transmission line tower alone, when compared to the design of other normal steel structures. The main considerations in arriving at the tower configuration are: the specified electrical clearances; tower type; wind pressure; maximum and minimum temperature conditions; possible ice loads on the conductor and groundwire and terrain profile. Table 1 lists the details of some of these parameters typical to a 400 kV double circuit tangent type tower. Configuring a transmission line tower basically requires computation of conductor and groundwire sags and their permissible maximum working tensions under the critical wind pressure and temperature conditions. Proper coordination of these sags, in conjunction with maintainence of stipulated electrical clearances
7 s
T.C level f
h
MC level h
B.C level h I
Ground
level
(4 Fig. I. Transmission line tower. (a) Transverse face, (b) longitudinal face and (c) 3-D view. g-ground clearance. h-insulator length + conductor sag. s-groundwire sag. B.C-bottom cross-arm. M.Cmiddle cross-arm. T.C-top cross-arm. G.C-groundwire cross-arm.
Optimum
-
designs
for transmission
I
of the tower
I Nodes & element generation and formation of global stiffness matrix
I
Computation of loads & identification of load cases
I
3.dimensional analysis of the tower for nodal displacements & member forces
I
Structural design of tower members and arriving at the tower weight
I
Start optimization of tower weight and tower geometry
Fig. 2. Flow chart
Table
1. Main electrical
and structural
Parameter
Item No. 1
line tower design pro-
of transmission gram.
Tower
type Circuit Tower
Angle
of deviation
Ground
clearance
Conductor diameter unit weight area of cross-section breaking capacity coefficient of linear expansion Young’s modulus
9 10
11
83
along the tower height, fixes up the required configuration. Table 2 gives the results of the sag-tension computations of the conductor and groundwire. Figure 3 shows the single line diagram of the tower with main dimensions fixed by the configuration part of the design process. With regard to the next two stages of panelling and node, element generation of the transmission line tower design process, it is normal practice to adopt a K or X (double-warren) type of bracing (Fig. 4). The panel heights are obviously dependent on the choice of the bracing angle 4 used in each of these panels. In the analysis of a structure like the transmission line tower having large number of members, the effect of flexure is not significant [18] and it is generally analyzed as a truss with many members. In the static analysis of the tower, the primary loads that are considered are the wind and the dead-weight loads [l] with appropriate factors of safety. It is highlighted here that a transmission line tower is also typified by
Configuration of the tower -> height, widths & crossarm dimensions Panelling
line towers
design
parameters
of the example
Specification
tower
Description
1
400 kV Double Tangent
Fig.
2”
Fig. A2
8.84 m
Fig. 1
31.77 mm 2.002 kg m-t 5.97 cm* 16280 kg 0.1935 x 10e4 deg-’ 0.686 x IO6 kg cm-’
Groundwire diameter unit weight area of cross-section breaking capacity coefficient of expansion Young’s modulus
0.115 x 10-4deg-’ 0.1933 x IO’kgcm-*
Insulator
Suspension
ll.Omm 0.7363 kg m-’ 0.578 cm’ 6950 kg
type
Fig. A4
Shield angle
20”
Fig. A3
Line span
400 m
Between adjacent
Weight
600 m
Between lowest points of adjacent spans
span
Wind pressure conductor and groundwire tower members
45 kg m-’ 195 kgm-’
Temperature maximum minimum
65’ 0”
two towers
To compute wind loads on the projected area
For sag-tension computations
G. Visweswara
84
Rao
Table 2. 400 kV double circuit tangent type tower. Sag-tension values of conductor groundwire under critical wind pressure and temperature conditions Conductor
Description
Item no.
Groundwire
1
Maximum tension in kg at 0 and 45 kgm-? wind pressure
4405.0
1710.0
2
Minimum tension in kg at 65’ and zero wind pressure
3180.0
1250.0
3
Maximum sag in m at 65’ and zero wind pressure
12.6
9.21
4
Minimum sag in m at Or and 45kgm-’
10.05
1.75
the presence of an additional load case of a broken conductor or groundwire loading condition (Appendix II). Figure 5 shows the multiple load cases along with the computed loads for the example tower of 400 kV line. In Fig. 5 it can be observed that an extra load, normally termed as a longitudinal load, acts in a direction normal to the cross-arm axis in each of the broken conductor-groundwire condition cases. A mention has to be made here about the factors of safety to be maintained regarding the maximum working tensions in the conductor and groundwire. While these are dependent on the respective breaking capacities of the conductor and groundwire, the factors of safety impose an upper limit on these tension values in a crisp optimization process. Lowering the working tension values no doubt brings down the tower loads (refer to Appendix II). On the other hand, it obviously increases the conductor and groundwire sags, thereby resulting in an increase in tower height followed by a possible upward or downward trend in tower weight.
and
2.2. Choice of’ control aariables With a specified transmission line voltage the parameters, like line span, conductor and groundwire dimensions, remain invariant in the design process. As one can perceive, the wind pressure and temperature conditions are also out of the purview of control variables. Of the other different parameters that significantly influence the tower weight and geometry, the following are the chosen set of control variables in the present optimization study: (a) conductor tension t,; (b) groundwire tension t,; (c) tower base width b,; (d) tower width at bottom crossarm (e) tower width at tower top b,; (f) panel bracing angle 4.
The above choice is based on a sensitivity study carried out on the effect of each of these variables on the tower weight and geometry. Table 3 shows the results of such a study with respect to the 400 kV
h3 j ;_i
--
a = 8830 b = 3535 c = 2355 d = 6600 e=6355 f=6110 g = 3240 h, = 26035 h2 = 7665 h3 = 7665 h4 = 2975
/;-,
e .I- I
hz
b/-‘_
/J
T
/ I,
.d .-/
I
Maximum working tensions: conductor: 4405 kg groundwire: 17 10 kg
I) ht
:
Panel angle = 0 = 45”
// i
2. Fig. 3. Tower
geometry.
Results
level b2;
a
‘,
from configuration
part of the program.
Optimum designs for transmission line towers --T-T,, {I ;
;/ ( ,,,’ ,‘+’ ,,a’.’ .’
;*‘,. !.’i, e ,...:i.
,,----Y,
I” ., 1‘: ( ‘,<, ‘CL_: “,
,‘:.-,y.-f--y ,,fI/ ,; 1, ,,I’
:’._ s’.,
‘. :,, ’
(4
i:
,
/ ‘! / ‘i, 1 !a,,, ‘{
Fig. 4. Bracing patterns generally adopted for high voltage transmission line towers. H = panel angle. (a) K type bracing. (b) X type bracing.
1690
I
380 --+’ 7700
I
2565 \ 1060 ‘\; 165 ---7--5775
--380 7700
2640
7700
‘--a 7700 ‘--+
2640
I
2640 --et
I
2640 --+I
285 5775 ;
--5775
1980
5775
‘--+ 5775
1980
5775
I
1980 --+’
I
;
1980 --+’
Ground
---
1980
wire broken condition
(e)
(d) 1270
;
1980 --+’
Normal condition
(c)
1270
1270
1270
1270
I
285 ----
i
1270 -r--h
1980
i
----
----
t
5775
5775 ‘--+
--+’
----
,175 __1i-
1980
t
1980
5775
6605 \ 4335 1980
i 5715
5775 1980 ----
285 5775
5775
1980
of
1270
I
7700
details
In most of the earlier truss optimization studies in literature, joint coordinates and the member sizes form the control variable vector. This obviously leads to too large a size optimization problem to handle,
4
2640
the
3. CRISP OPTIMIZATION
1690
‘--+ 7700
2640 --et
double circuit tangent-type tower, which are listed in Tables 1 and 2.
8 (b)
85
----
t
1980
+
Top conductor
broken condition
Middle conductor broken condition
Bottom
conductor broken condition
Fig. 5. Multiple load cases. 400 kV double circuit tangent type transmission line tower. (a) Normal loading condition, (b) groundwire broken loading condition, (c) top conductor broken loading condition, (d) middle conductor broken loading condition and (e) bottom conductor broken loading condition. (The inclined arrow at each crossarm level in the broken loading conditions indicates the additional longitudinal load).
G. Visweswara
86
Rao
Table 3. 400 kV double circuit tower. Sensitivity analysis results. Effect of control variables on tower geometry and weight (percentage values in parenthesis is the effect of each variable with respect to tower weight of 11400 kg at base vector of control variables) 0.8*Base
I. I *Base value
Serial No.
Base value (initial configuration)
Control variable
Total height in mm
No. of panels
44355
13
Total weight in kg
Total height in mm
No. of
11710 (+2.7%) II335 (-0.6%) 11710 (+2.7%)
44265
I3
45705
I3
44340
13
panels
value Total weight in kg
I
1,
4405 kg
2
‘,
1710 kg
42805
13
3
h,
8830 mm
44330
II
4
b>
3535 mm
44470
II
II620 (+ 1.9%)
44260
I4
10805 (-5.2%)
5
b,
2355 mm
44390
IO
44310
I5
6
4
45
44380
9
12085 (+6.0%) II990 (f5.1%)
44310
I7
10635 (-6.7%) 10965 (-3.8%)
especially with PC based software. In the present study, selection of tower widths at a few nodal points on the tower, and the panel bracing angle as elements of the design variable vector X, has an obvious advantage. The variations affected in these variables during the optimization process indirectly amount to changing the nodal coordinates as well, while substantial reduction in the size of the problem is achieved. Moreover, for a transmission line tower, there is lack of freedom in choosing all nodal coordinates as design variables. This is due to the fact that arbitrary changes in these coordinates are likely to lead to violation of electrical clearance requirements. Thus, for the transmission line tower on hand, the crisp optimization problem can be stated as minimize
W(X),
subjected
to g,(X) > 0.0
is possible, while a sequence of accelerating steps of pattern searches are made along this direction till it continues to be a successful direction.
3.1. Results The optimization problem in eqn (1) is solved for the case of the 400 kV double circuit transmissionline tower, the details of which are given in Table 1. It is designed as a self-supporting tower with steel members of L-shaped angles with a bolted type construction. The configuration of Fig. 3 is utilized to arrive at the starting design. The design of tower members is guided by the following slenderness ratio requirements, for tension members L/r d 300,
for i = 1, .
, m,
variable
and
(1)
for compression where X is the control
11585 (+ 1.6%) II650 (+2.2%) II135 (-2.3%)
members
vector given by
W(X) is the tower weight and g,(X) is the vector of functions. In the present case, g,(X) covers the lower and upper limit constraints on the selected design variables. They also include the stress constraints and the restriction of positivity of all elements of the control variable vector. The constraints include both tension and stress compression types. The nonlinear constrained optimization problem is defined in eqn (1) is solved by the Hookes-Jeeves derivative-free method [19,20]. The method is simple, elegant and is easily adoptable for program development. It is basically a hill-climbing technique in that each control variable is varied in succession by a series of exploratory and pattern searches for achieving the desired objective of an optimized solution. The exploratory search is for finding the successful direction in which optimization
kL/r d 150
if it is a leg member crossarm member
kLlr
if it is a member carrying computed stresses
< 200
m constraint
kL/r d 250
or
if it is a redundant member carrying nominal stresses
(2)
where L is the length of the member, r is the radius of gyration. k is a nondimensional factor accounting for different end fixity conditions. k is taken to be unity in this study. The constraint functions g,(X) are
Optimum
f&(X) =
B:
g6(X)=4’<4<#”
i = member
as in eqn (3). For example, the upper bounds on the conductor and groundwire maximum working tension can as well be relaxed, to a small extent, without violating the specified factor of safety by any appreciable margin. In a similar fashion, the earlier selection of the lower and upper bounds on the tower widths in eqn (3) is governed by more intuition than availability of crisp boundaries. Decision making in such imprecise and vague environments is made convenient by the theory of fuzzy sets [15]. The main task in a fuzzy optimization problem is to treat the fuzzy objective function and the fuzzy constraints as fuzzy sets in the space of decision alternatives and characterize them by their membership functions. The membership functions are descriptors of the impreciseness or vagueness with which the boundaries of the fuzzy sets are known. For example, in the present case the membership functions for the lower and upper boundaries of the conductor maximum tension t, can be defined, respectively as,
and
number
gj, 7(X) = x, 2 0.0, i=1,...,6
(3)
1 1 stands for absolute
value.
T, and T, are taken to be the maximum working tension values corresponding to the initial configuration (Fig. 3). The bounds bf and B:, i = 1, . ,3, on the different tower widths are intuitively taken to be 25% below and above the respective initial configuration values. The bounds on C#Jare chosen to be 40 and 60”, respectively. a, is the compressive or tensile stress injth member. The upper bound tr” is taken to be 2600 kg cm-‘. The weight of the tower at the start of the optimization process is 11.400 tonne while the optimum weight of the tower is obtained as 10.000 tonne. The linear dimensions of the configuration corresponding to the final optimized design are shown in Fig. 6. Figure 7 shows the convergence of the tower weight with the number of iterations under the crisp optimization process.
P,, (W = 0
=(-t,+
if t, 2 I.l*T, I.l*TC)/(O.l*TC) if T,g
&2(X)
t,<
l.l*T,
= 1
if t, ,< T,
=
if t, < 0.9*T,
0
(4)
= (t, - 0.9*T,)/(O. 1*T,)
4. FUZZY OPTIMIZATION
if 0.9*T, < t, < T,
It
is intuitively obvious that in reality the bounds on the control variables may not be as crisply defined
a=9300=B, b = 2400 = Bz c= 1180=B3 d = 6150 e = 5880 f=5615 g = 2935 hl = 26770 h2 = 7670 h3 = 7670 h4 = 2205
= I
if t, 2 T,.
;: { j I ;
h,
\
Maximum working T,:4100kg Ts: 1710 kg
tensions:
Panel angle = 0 = 60”
i I i
Fig. 6. Crisp optimization.
The final optimum
tower design configuration.
(5)
G.
88
r.of i 3
5
Number Fig. 7. Fuzzy
The by
shapes
7
9
fl
13
Rao
15
of iterations
optimization. Convergence with number of iterations.
of the ~~ernbershj~
of tower
functions,
weight
as given
eqns (4) and (S), are shown in Fig, 8. The lower
value of T, = 4405 kg (Table 2) in eqns (4) and (5) is the value of the conductor maximum working tension as worked out from sag-tension computations with a resulting factor of safety of 3.7 with respect to its ultimate breaking capacity. A small variation of 10% above this stipulated value is not substantial and may be acceptable. In other words, the assumed membership function in eqn (4) for g,(X) describes this intentional small violation in a mathematical form. Similarly, the membership functions for the other constraints in eqn (3) are defined as
(b)
(a> Fig. 8. Fuzzy
optimization.
Shape
of membership function chosen (b) Upper constraint range.
for r,. (a) Lower
constraint
range.
Optimum
designs
for transmission
line towers
11.5
g,, W):
P~ldX)= 0
if 6, < 0.9*Bi
r
11.3 E
1*B:)
= (b, - 0.9*B:)/(o.
89
2 if 0.9*B:
= 1
< b, < Bi
$
if b, > B:,
i%,,(X):
+\ +\ +-++++-m
.04 11.11 IO.51
if 4 2 I.l*qS”
r+,(X)=0 = (-4
I
I
I
I
I
I
I
3
5
7
9
II
13
1
I 15
Number of iterations
+ 1.1*q5”)/(o.l*q%“)
Fig. 9. Crisp optimization. Convergence of tower weight with number of iterations.
if l.l*$“>f$2&”
if 4 <4”, function and the fuzzy constraints. Consequently the membership function of the fuzzy domain turns out to be
i%(X): y,m
\
2 10.9 3 tF: 10.7
if qb < 0.9*4’
= 0
PD(X) = min{~,$0
= (c#l- 0.9*+ ‘)/(O. 1*cj 1)
if4
The membership structed as I&(X) = 0 = (-
24’.
function
(6) for W(X) is also con-
if W(X) 2 W”
maximize
2,
subjected
to the constraints
W(X) + W”)/(O. 1* W”)
1
if 0.9*W” < W(X) < W” zr 1
if W(X) < W”.
and
1
is now objective
._.g ..
7
a=8880=B, b = 3220 = BZ c = 1920 = B3 d = 6465 e=6185 f = 5900 g = 4700 h, = 26490 h2 = 7680 h3 = 7680 h4 = 2340
e
I
I
j-
j j
ht
Maximum working T, : 4205 kg T,: 1745 kg
;
Fig. IO. Fuzzy optimization.
i = 1, 12.
(9)
The fuzzy optimization problem as stated in eqn (9) is solved by the Hookes and Jeeves method. It is to be noted here that the control variable vector is now augmented by one more element in the form of 1. The
h4 _.
h2 I
(8)
4.1. Results (7)
The optimization in fuzzy environment viewed as an intersection of the fuzzy
i = 1, 12.
If pD(X) is now denoted by 2, the optimum solution is obtained by maximizing 1. At this stage the fuzzy optimization problem can be stated as an equivalent crisp optimization problem as
if0.9*$1<&
P&X)},
II I I // The final optimum
I I I I
tensions:
Panel angle = 8 = 45”
tower design configuration.
G. Visweswara
90 Table
4. Fuzzy
Rao
optimization. Membership function values L-lower constraint range, U-upper constraint
at optimum range
solution;
h, W
(2,
c&
(U)
hL (L)
(hLi)
2)
0.31
0.76
0.65
0.94
0.58
0.06
0.50
as in Fig. 3 is again utilized to start the optimization process. W” in eqn (7) is taken to be the corresponding tower weight (Table 3). The final optimization solution is 10.700 tonne in the form of tower weight. Figure 9 shows the convergence of the optimum solution in terms of tower weight with number of iterations. The linear dimensions of the tower are shown in Fig. 10. The value of 1 corresponding to the optimum solution is 0.58 and amounts to the maximum degree of membership of the optimum vector X* to the feasible domain D. At the end of the optimization process all the elements of the control variable vector X remain active in either upper or lower constraint ranges. The degree of satisfaction expressed by the respective membership function values at the optimum solution are detailed in Table 4. configuration
5. DISCUSSION
The crisp and fuzzy optimization results are presented for transmission line towers of tangent type (Appendix I). This type of tower being used for straight stretches of a transmission line, constitutes 80-90% of the total number of towers. In this context, weight optimization of the tangent type towers assumes significance. The minimization scheme is applicable to other types of transmission line towers as well. It is based on the optimization of tower shape through the judicious selection of a few control parameters. In particular there are six parameters in all and excepting the two tension parameters t, and t,, the others are directly connected with the tower shape. Variations in r, and 1, indirectly affect the tower shape in the form of varying the inter cross-arm heights and hence the tower weight. The selection of the panel angle 4 as a control variable is significant in that its variation controls the panel height, the key parameter deciding the nodal coordinates. The sensitivity of tower weight and geometry with respect to this parameter is comparable to that of b, , b2, and b,. This can be observed from Table 3. It is to be noted however, that geometry shaping and weight minimization strongly depends on the combined influence of these variables. In a crisp optimization problem the saving in weight of the tower is to the extent of 12% with respect to the tower weight, corresponding to the base vector. The fuzzy optimization has resulted in a tower weight higher than that of crisp optimization. However a saving in tower weight of 6% is achieved by the fuzzy optimization process. The convergence of tower weight, by the Hookes-Jeeves method, is fast in the case of both crisp and fuzzy optimization, as can be observed from
Figs 7 and 9. It is to be noted that the membership functions in eqns (4)-(7). chosen for the transition interval of each parameter, are not unique. However the active participation of the control variables chosen for the optimization process is apparent from their respective membership function values in Table 4.
6. CONCLUSIONS
Optimum design of transmission-line towers has been discussed in this paper. A systematic procedure has been presented for obtaining minimum weight tower design in both crisp and fuzzy environments. The efficiency of the optimization scheme is due to the choice of a few key parameters that influence the tower geometry and thus indirectly the weight, as design control variables. This has also reduced the computational effort considerably. A dedicated program has been developed for developing the optimum design from the initial stage of tower configuration generation. to that of analysis, design and optimization. The results presented for a typical high voltage transmission line tower have shown substantial saving in tower weight. REFERENCES
I. Committee on Electrical Transmision Structures of the Committee on Analysis and Design of Structures of the Structural Division. Loadings for electrical transmission structures. J. .s~ruc(. Dir. ASCE 108(5), 1088~1105 (1982). 2. B. H. V. Topping. Shape optimization of skeletal structures: a review. J. .v/ruct. Engng ASCE 109(g), 1933 -1951 (1983). 3. D. J. Sheppard and A. C. Palmer, Optimal design of transmission towers by dynamic programming. Compul. Srruct. 2, 455 468 (1972). and F. Moses, Automated design of 4. G. N. Vanderplaats trusses for optimum geometry. J. SITUCI.Die. ASCE 98, 671690 (1973). and 0. C. Zienkiewicz, Optimum 5. R. H. Gallagher Structuml Design, Theory and Applications. Wiley. New York (1973). of trusses. /. slruc’t. Div. 6. M. P. Saka, Shape optimization ASCE 106(5), 115551174 (1980). 7. W. H. Greene, Minimum weight sizing of guyed antenna tower. J. struct. Engng ASCE lll(lO), 2121~~2137 (1985). Numerical methods for shape 8. G. N. Vanderplaats. optimization-& assessment of the state of the art. Proc. Int. Swap. O~tmum Steel Design, Tuscan. AZ (1981). 9. D. 1. Blockley. The role of fuzzy sets in civil engineering. Fu-_-_J,Sets .S_v.st. 2, 2677278 (1979). IO. C. B. Brown and J. T. P. Yao, Fuzzy sets and structural engineermg. J. .struct. EngnX ASCE 109, 1211~1225 (1983).
Optimum
designs
for transmission
11. W. G. Yuan and W. W. Quart, Fuzzy optimum design of aseismic structures. Earfhquake Engng snuc/. Dyn. 13, 827-837 (1985). 12. W. G. Yuan and W. W. Quart, Fuzzy optimum design of structures. Engng Optbniz. 8, 291-300 (1985). and J. R. Rao, Fuzzy goal 13. R. N. Tiwari, S. Dharmar programming, an additive model. Fuzzy Sets Syst. 24, 27-34 (1987). T. A. Phelps and K. M. Ragsdell, 14. S. U. Mohandas, Structural optimization by goal programming approach. Comput. Strucr. 37(l), l-8 (1990). 15. L. Zedeh. Fuzzy sets. Information Control 8, 338-353 (1965). B. G. Prakash and C. 16. S. S. Rao, K. Sundararaju, Balakrishna, Multiobjective fuzzy optimization techniques for engineering design. Compuf. Strucr. 42(l), 1744 (1992). optimization of fuzzy struc17. S. S. Rao, Multiobjective tural systems. Inr. J. numer. Meth. Engng 24, 1157-I 111 (1987). Loh, Dynamics of 18. W. Weaver Jr, and C. Lawrence trusses by component mode method. J. struct. Engng ASCE 111(12), 256552575 (1985). Applied Nonlinear Programming. 19. D. M. Himmelblau, McGraw-Hill, New York (1972). 20. S. S. Rao, Optimization: Theory and Apphcalions, 2nd Edn. Wiley, New York (1984).
APPENDIX
The main parameters design of transmission
91
Fig. A2. Angle of conductor
specification
in the
deviation arm.
near a typical cross-
(i) Line voltage. The voltage level generally fixes up the safe limits on the conductor-to-ground clearance (Fig. I), conductor-to-metal clearances (Fig. Al) and required clearances over terrain obstacles such as river-rail crossings, teleline and other transmission lines of different voltage. (ii) Number of circuits. number of cross-arms.
I
of the electrical line tower are:
line towers
The number
of circuits decide the
(iii) Angle of conductor deviation, /l (Fig. A2). #’ broadly classifies transmission line towers into tangent and angle type towers. O-2” belongs to the first category and above 2,’ to the angle type. (iv) Shield angle (Fig. A3). Shield angle is the angle between the line joining the groundwire and conductor attachment points and the vertical. It is meant for electrical protection of the current carrying conductor. (u) Znsulators (Fig. A4). Suspension insulators are used for tangent type towers and tension insulators are used for angle type of towers. These insulators and their associated connecting wires are susceptible to deflections due to wind. Maintenance of electrical clearances and hence the cross-arm lengths and heights are strongly dependent on these insulator swing angles.
Fig. Al. Electrical ance diagram.
clearances. c-specified
Conductor electrical
to metal clearance.
clear-
(vi) Conductor and groundM,ire. ACSR (aluminium conductor steel reinforced) stranded conductors are normally used for high voltage transmission lines. High tensile galvanized steel wires are used as groundwires. The sizes of conductor and groundwire depend on the line voltage.
I,
(---T-
I. -----_-_________ .I_
\_____-.------~---- :_,,
Groundwire pomt
attachment
1
~___------
i ‘I Fig. A3. Shield angle.
-
Insulator
-
Conductor point
attachment
G. Visweswara
92
GPPENDIX II
-----_____ ,.. ---___ _.: y.- --_ _:’
_/-
Rao
-----z ____ --__
__._---
$
z
Insulator
The loads on a transmission line tower are mainly due to wind and self weight. The deviation of the line (Fig. A2) also canses a load arising from the tension of a conductor-groundwire at their respective cross-arm ends. This load acts in the direction of the cross-arm axis. Thus, in normal operating condition (Fig. 5a) the loads are of two types. They are given by items (i) and (ii) below. (i) Vertical load at a cross-arm tip =self weight of conductor-groundwire + insulator and its hardware + weight of linemen and tools. (ii) Transverse load at a cross-arm tip =wind load on conductor-groundwire and insulator + tension component due to line deviation.
.f’ -
Conductor
(b) Fig. A4. Insulators. (a) Suspension type used for tangent type towers and (b) tension type used for angle type towers.
The wind load on tower members also forms part of the transverse load. It is either lumped at the cross-arm tips with appropriate apportioning of the tower body for the purpose or treated as a distributed load in the analysis. In the case of a broken condition (Fig. Se) of either a groundwire or a conductor, the resulting unbalanced tension in the intact span acts in a direction parallel to the conductor and a major load component acts in a direction normal to the cross-arm axis.