EUROPEAN TRANSACTIONS ON ELECTRICAL POWER Euro. Trans. Electr. Power 2008; 18:437–447 Published online 11 June 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/etep.185
Cost related optimum design method for overhead high voltage transmission lines L. Ekonomou1*,y, G. P. Fotis2 and T. I. Maris3 2
1 Hellenic Public Power Corporation S.A., Distribution Division, 5 Psaron Str., Alimos, 174 56 Athens, Greece School of Electrical and Computer Engineering, High Voltage Laboratory, National Technical University of Athens, 9 Iroon Politechniou Str., Zografou, 157 80 Athens, Greece 3 Department of Electrical Engineering, Technological Educational Institute of Chalkida, 334 40 Psachna Evias, Greece
SUMMARY The design of transmission lines is often the key issue for the existence or absence of failures caused by lightning. Detailed engineering studies are usually performed by electric power utilities for the design of new transmission lines. However, there are also cases where the design is based simply on tradition or on utilities’ standardization policy. The paper presents a cost related method for the optimum design of overhead high voltage transmission lines, which intends to reduce or even eliminate the lightning failures. Lightning failures’ cost is related to design parameters’ cost in order to calculate the optimum and most economic design parameter values. In order to validate the effectiveness of the proposed method, the method is applied on several operating Hellenic transmission lines of 150 and 400 kV, respectively, carefully selected among others, due to their high failure rates during lightning thunderstorms. Special attention has been paid on open loop lines, where a possible failure in them could bring the system out of service causing significant problems. The proposed parameters that occurred by the design method and which reduce the failure rates caused by lightning, are compared with the operating transmission lines’ existing design parameters showing the usefulness of the method, which can be proved a valuable tool for the studies of electric power systems designers. Copyright # 2007 John Wiley & Sons, Ltd. key words: cost; lightning performance; optimum design method; overhead transmission lines
1. INTRODUCTION The correct design composes the most important issue for a transmission line, determining the operation and the performance of the line for the rest of its life. Especially for lines running in geographical regions of high lightning level and/or high soil resistivity, a careful design can prevent failures and unscheduled supply interruptions. Plenty of design methodologies have been proposed [1–6] in the technical literature, in an effort to ‘protect’ the pre-constructed transmission lines, since slight differences in the design parameters values can affect significantly their lightning performance.
*Correspondence to: Dr L. Ekonomou, Hellenic Public Power Corporation S.A., Distribution Division, 5 Psaron Str., Alimos, 174 56 Athens, Greece. y E-mail:
[email protected]
Copyright # 2007 John Wiley & Sons, Ltd.
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In addition, improvements and modifications in an existing transmission line are practically impossible, leading again to the conclusion that the key issue for a transmission line is the correct and careful design. The research efforts of the existing design methodologies have been focused on the determination of a minimum cost design of transmission lines, representing the total cost of the system as a function of the system design variables [1]. They have also explored the sensitivity of the required present worth of revenue, to several design variables, in order to achieve design performance at minimum cost [2]. In Reference [3] a range of line optimization techniques has been introduced, which can be applied to decide whether standard or optimized line designs are appropriate, while optimal design methods, for overhead high voltage transmission lines, with main objective the minimization of the line total annual cost, considering the relevant technical constraints and both fixed and running cost items have been presented [4]. Finally the economical aspects of the overhead distribution line lightning performance, taking into account customer and utility costs of line outages, have been analyzed [5] and alternative design procedures for uncompensated overhead transmission lines based on the derivation of closed-form analytical expressions for both line power and current ratings, in terms of the geometrical data of the line tower and its bundled conductors were introduced [6]. In a new transmission line design, designers are mainly focused either on the tower footing resistance or on the line insulation and they face the transmission line alike along its length [7]. The current work presents in detail a design method, which pays special attention on: (a) the optimum selection of the transmission line insulation level, (b) the tower footing resistance, and (c) the average height of the shielding wires. Furthermore the new transmission line is divided into regions and a separate design is performed for each one region, serving with best way lines, which are running at the same time through plain regions, coastlines, and/or mountainous regions with significant different meteorological and geographical characteristics through their length. Suitable performance indices are defined in order to relate the line insulation level, the tower footing resistance, and the average height of the shielding wires cost values to the lightning failure costs. Using an iterative optimization algorithm, optimum values of these three design parameters are calculated in order to minimize the defined performance indices. The developed method is applied on several operating Hellenic transmission lines (including open loop lines) of 150 and 400 kV, with high lightning failure rates, in order to validate its effectiveness. New values for the transmission line design parameters are proposed, having as a result the reduction of the lightning failures. Useful conclusions are extracted from the comparison between the proposed values and the actual line design data. The proposed method can be particular useful to the studies of transmission lines’ designers engineers contributing effectively in the reduction of the lightning failures.
2. LIGHTNING PERFORMANCE ESTIMATION OF TRANSMISSION LINES The number of lightning failures or the outage rate NTotal, is the arithmetic sum of the shielding failures NSF, where the lightning stroke terminates directly on the phase conductor and the failures because of the backflashover phenomenon NBF, where the lightning stroke terminates on the structure or on a shielding wire, changing the potential of the structure sufficiently to cause a flashover to a healthy, until to that moment, phase conductor, as shown in Equation (1). NTotal ¼ NSF þ NBF Copyright # 2007 John Wiley & Sons, Ltd.
(1) Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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Many approaches have been developed the last decades in order to estimate with accuracy the high voltage transmission lines’ lightning performance [7–14]. Shielding failure flashover rate NSF can be estimated according to the method presented in Reference [7] (where it is associated to a required minimum current Imin to cause a line insulation flashover) as follows: NSF ¼ NL
ZImax
DC f ðIÞ dI
(2)
Imin
where: NL is the number of lightning flashes to a line per 100 km per year given from the equation: NL ¼ 0:004 T 1:35 ðb þ 4h1:09 Þ
(3)
T is the yearly lightning or keraunic level in the vicinity of the line in thunderstorm days per year (i.e., average number of days per year, on which thunder is heard), h is the average height in m of the shielding wires, b is the horizontal spacing in m between the shielding wires, Imax is the maximum lightning current in kA, Imin is the minimum current equal to 2Ua/Zsurge, Ua is the insulation level of the transmission line in kV, f(I) is the probability density function for the current, Dc is the shielding failure exposure distance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zsurge is the conductor line surge impedance equal to: 60 ln 4hdc ln 4hDc ; hc is the average conductor height at the tower in m, d is the equivalent conductor diameter without corona, and D is the equivalent conductor diameter with corona. Backflashover failure rate NBF can be estimated according to the method presented in Reference [14] as:
NBF ¼ NL
ðIpeak Z Þmax
ðZdtdiÞmax PðdÞ dIpeak dðdi=dtÞ
ðIpeak Þmin
(4)
ðdtdiÞmin
where: P(d) is the probability distribution function of the random variable d, which is a function of the two random variables Ipeak and di/dt as shown in the following relation: (5) d Ipeak ; di=dt ¼ RIpeak 2 0:85 Ua þ L di=dt with d greater than zero, when there is a backflashover, R is the tower footing resistance in V, L is the total equivalent inductance of the system (tower and grounding system’s inductance) in mH, calculated according to the simplified method presented in Reference [7], di/dt is a random variable denoting the lightning current derivative in kA/ms, and Ipeak is a random variable denoting the peak lightning current in kA. Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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L. EKONOMOU, G. P. FOTIS AND T. I. MARIS
The above analysis shows clearly that the variations of the transmission line insulation level, of the tower footing resistance and of the height of the shielding wires influence significantly the shielding failure and the backflashover failure rates. Therefore, it could be worthwhile to further investigate the most appropriate selection of these design parameters in order to reduce the lightning failure rates. 3. TECHNO ECONOMICAL OPTIMIZATION ANALYSIS The examined transmission line is divided into N regions, due to the different meteorological conditions and the different average values of tower footing resistance, which exist in each one region of the line. For each region an analysis is conducted and suitable values for the insulation level and the tower footing resistance are computed. The total flashover failure rate is also computed for each region, using Equation (1). A performance index is defined for each region of the examined transmission line, in order to relate the annual cost of total flashover failure rate to the total investment cost of the regional values of the three design parameters i.e., insulation level, tower footing resistance, average height of shielding wires. Ji ¼ ki ðNTi Þ þ gUai ðUai Þ þ gRi ðRi Þ þ ghi ðhi Þ
(6)
where: i ¼ 1, ..., N region number, Ji is the performance cost index of the ith region, ki () is the annual line failure cost, gji () is the equivalent annual investment of the ith region line design characteristic j, NTi is the total flashover failure rate of region i, Uai is the insulation level of region i, Ri is the grounding footing resistance of region i, and hi is the average height of shielding wires of region i. The annual line failure cost is given from the equation [5]: kðÞ ¼ CMEU þ CRE þ CFC
(7)
where: CMEU is the mean annual cost of undelivered energy for the utility, equal to zero for non open loop lines, CRE is the mean cost of one permanent failure repair, and CFC is the equivalent annual line failure consumer cost. The equivalent annual investment is calculated by the total cost of investment using: rðr þ 1Þt gðÞ ¼ þ p GðÞ (8) ðr þ 1Þt 1 where: G() is the total cost of investment of the transmission line’s design parameters, r is the annual interest rate, t is the estimated line exploitation period in years, and p is an empirical coefficient which defines the ratio of the annual maintenance cost to the total cost of investment. Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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The design parameters, i.e. insulation level, tower footing resistance of each region, and average height of the shielding wires, form a column vector x: xfx1 ; x2 ; . . . ; x3N g ¼ fUa1 ; Ua2 ; . . . ; UaN ; R1 ; R2 ; . . . ; RN ; h1 ; h2 ; . . . ; hN g
(9)
Optimal selection of xi ’s values minimize the set of the N performance indices defined in Equation (10), i.e.: min Ji ¼ min ½J1
Vai ;Ri ;hi
Vai ;Ri ;hi
J2
JN
...
(10)
under the operating limits: Uai min Uai Uai max Ri min Ri Ri max hi min hi hi max
(11)
where Uaimin, Uaimax, Rimin, Rimax, himin, himax are limit values defined by electrical utilities. Application of an optimization algorithm will determine the optimum values xi, i.e. the insulation level, the tower footing resistance, and the average height of the shielding wires. For the rest of the parameters the typical values of the horizontal spacing, between the shielding wires b, the total inductance of the system L, the equivalent conductor diameter without corona d, and the equivalent conductor corona diameter D, which represent the usual equipment used from electrical utilities, were considered. It must be mentioned that all of the design limits and the design parameters are generally result from safety, reliability, or economic considerations. They can be based on detailed engineering studies or on simple tradition.
4. OPTIMUM DESIGN METHOD Optimization aims to minimize the objective vector function of several variables. The optimum solution vector x can be found for the J function presented in Equation (10). Necessary but not sufficient condition that a point xopt is an extremum (minimum or maximum) is that rJ(xopt) ¼ 0 and sufficient but not necessary condition for the stationary point x0 is that the Hessian matrix H(x) evaluated at x0 is positive or negative definite (for a minimum or maximum, respectively) where: 3 2 2 @ J @2 J @2 J ::: 2 @x1 @x2 @x1 @x3N 7 6 @x2 1 6 @J @2 J @2 J 7 ::: @x2 @x1 @x2 @x3N 7 @x22 (12) HðxÞ ¼ 6 6 ::: ::: ::: ::: 7 5 4 2 @ J @2 J @2 J @x3N @x1 @x3N @x2 ::: @x3N 3N
The optimum solution vector can be found after the iteration of a formula of the form: xnþ1 ¼ xn þ an pn
(13)
where: an is the step size and pn is the step direction, which are both iteratively calculated. Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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L. EKONOMOU, G. P. FOTIS AND T. I. MARIS
Various algorithms give different formulas for the above steps and converging conditions. Among these algorithms, trust region, the Newton-Raphson with line search or with ridging and the Levenberg–Marquardt algorithms [15,16] use both first and second order derivatives, i.e. the gradient vector, and the Hessian matrix or an approximate equivalent. Quasi-Newton, double-dogleg, conjugate gradient algorithms do not use second order derivatives. Taking into account that derivatives are difficult or sometimes even impossible to calculate the Nelder–Mead simplex algorithm does not use derivatives at all [17]. This method requires only the repeated computation of the J function value. The starting point is used to construct an initial polygon shape of nþ1 vertices, where n is the number of parameters. In each step the remaining n vertices are reflected and expanded through the best vertex v0. The reflection is successfully expanded if all the reflected vertices have lower cost than the previous best vertex. If the reflection is successful the algorithm proceeds to an expansion step. The expansion expands each reflected edge to a new expanded vertex. The following algorithm, based on Nelder–Mead [17] algorithm method, has been implemented and applied: Step 1: Determine Ji ’s function in reference to meteorological and tower structure data, from Equation (6). Step 2: Set initial values for insulation level Ua, tower footing resistance R, and average height of shielding wires h. Step 3: Calculate NTi from Equation (1), Ji from Equation (6), and xnþ1 from Equation (13). Step 4 (Initialization): A simplex is constructed on n vertices. Step 5 (Iteration): Vertices vi’s are updated by previous iteration and are sorted Reflection step n vertices are reflected through the best vertex v0 giving the ri if successful makes expansion else sorting of v0, c1, c2, . . ., cn where ci is the middle of v0 and vi Expansion step Expanding each reflected edge (rjv0) to twice its length to give new expansion vertex ej, where is the reflection of ci if successful sorts vertices v0, e1, ... , en according to their increasing function values else sorting of v0, r1, . . . , r Step 6 (Stopping condition): These rules are repeated and the simplex moves over cost surface until the desired bound is found. Step 7: Display converged values xn.
5. APPLICATION OF THE PROPOSED METHOD 5.1. Transmission lines parameters The method presented in this paper has been applied and tested on several 150 and 400 kV operating transmission lines of the Hellenic interconnected system [18]. These lines, which are presented in Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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COST RELATED OPTIMUM DESIGN METHOD
Table I. Line design characteristics of the examined transmission lines. No. Line 1 2 3 4 5 6 7 8
Phase voltage Length No. of Insulation level Conductor dimensions No. of (kV) (km) towers (kV) (ACSR MCM) circuits
Thessaloniki–Amideo Tavropos–Lamia Pirgos–Megalopoli II Igoumenitsa–Sagiada Argos II–Astros Giannena–Kalpaki Megalopoli–Sparti Bolos II–Lafkos
400 150 150 150 150 150 150 150
90.589 75.384 69.665 17.485 29.427 27.850 64.472 35.206
249 219 179 44 85 86 173 90
1550 750 750 750 750 750 750 750
954 636 336.4 336.4 636 and 336.4 336.4 336.4 336.4
2 2 1 1 2 and 1 1 1 1
Table I, were carefully selected among others, due to: (a) their high failure rates during lightning thunderstorms, (b) their consistent construction for at least 90 per cent of their length and (c) their sufficient length and their sufficient time in service in order to present a reasonable exposure to lightning. It must be mentioned that lines 5, 6, 7, and 8 are open loop lines and they have attracted lots of this analysis attention, since a possible failure in them could bring the system out of service, causing significant problems to the customers and the local societies. According to the proposed optimization method, each one of the above lines is divided into regions, due to the different meteorological conditions and the grounding resistance, which exist in each one of them. The performance cost indices, which have been used in the analysis have calculated based on the economical data supplied from the Hellenic Public Power Corporation S.A. [19]. The regions and the different design parameters in each one of them are shown in Table II [18]. Table II. Analytical line parameters of the examined transmission lines. Line Thessaloniki–Amideo Tavropos–Lamia Pirgos–Megalopoli II Igoumenitsa–Sagiada Argos II–Astros Giannena–Kalpaki Megalopoli–Sparti Bolos II–Lafkos
Region
Towers
R (V)
H (m)
NT (average lightning failures 2000–2005)
T (average lightning level 2000–2005)
I II I II III I II III I II I II III I II I II III I II III
1–209 210–249 1–89 90–168 169–219 1–66 67–124 125–179 1–29 30–44 1–23 24–61 62–85 1–40 41–86 1–45 46–75 76–173 1–17 18–40 41–90
6.0 11.6 11.4 24.9 7.3 6.2 10.8 14.5 57.5 14.9 4.2 8.1 60.4 4.2 25.8 5.1 39.7 11.2 9.8 5.1 10.4
44.85 44.85 32.50 32.50 32.50 23.75 23.75 23.75 23.75 23.75 23.75 23.75 32.50 23.75 23.75 23.75 23.75 23.75 23.75 23.75 23.75
2.3 1.8 1.4 0.8 0.3 0.9 1.2 2.1 2.3 1.6 0.3 0.4 0.7 0.2 0.8 0.5 0.7 1.4 0.4 0.5 1.4
32.6 28.8 38.4 31.2 21.3 32.0 25.7 28.6 38.7 27.8 26.3 22.2 21.6 39.8 34.4 28.6 30.2 33.1 23.3 26.9 34.1
Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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L. EKONOMOU, G. P. FOTIS AND T. I. MARIS
Table III. Proposed optimum parameters for the examined transmission lines. Line Thessaloniki–Amideo Tavropos–Lamia Pirgos–Megalopoli II Igoumenitsa–Sagiada Argos II–Astros Giannena–Kalpaki Megalopoli–Sparti Bolos II–Lafkos
Region
Insulation level (kV)
R (V)
H (m)
NT (no. of lightning failures)
I II I II III I II III I II I II III I II I II III I II III
1810 1630 870 890 770 750 780 810 910 770 750 770 830 750 750 750 870 750 750 770 830
2.8 5.3 6.5 10.4 3.1 2.3 7.5 5.9 12.8 8.4 2.0 5.7 22.2 2.6 13.2 3.7 17.6 7.9 6.3 4.0 5.6
47.60 46.40 36.80 34.90 32.50 24.00 25.00 25.50 24.20 23.75 23.75 24.00 35.90 23.75 25.10 24.00 23.75 23.90 23.75 25.40 26.90
0.9 0.7 0.7 0.3 0.1 0.4 0.5 1.0 0.6 1.1 0.1 0.2 0.2 0.1 0.4 0.2 0.3 0.7 0.2 0.1 0.6
5.2. Results of the optimum design method Table III clearly presents the results obtained from the application of the proposed optimum design method to the Hellenic 150 and 400 kV transmission lines of Table I. Optimum values for the average insulation level, the tower footing resistance of each one region of the transmission lines, and the average height of the shielding wires are calculated. It is obvious that the proposed combined values of these three design parameters reduce the total lightning failures of the examined transmission lines. Although any modifications or improvements to the insulation level, the tower footing resistance, and the height of the shielding wires of these operating lines are practically impossible right now, the usefulness of the proposed method can be easily noticed for cases of new transmission lines. If the examined transmission lines were constructed using parameter values calculated by the optimum design method, they would certainly present lower failure rates. Drawback of the method can be considered the proposal of optimum values for the design parameters, which are difficult to be applied in reality. It is very difficult for the construction engineers to construct an insulation level value exactly equal to 770 kVor tower footing resistance values exactly equal to 2.6 or 12.8 Ohm without any variations.
6. CONCLUSIONS The paper describes in detail a cost related optimum design method for improving the lightning performance of overhead high voltage transmission lines. The method calculates and proposes the most suitable line insulation level, tower footing resistance, and average height of shielding wires, for each Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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one region of the examined lines, in an effort to minimize the total failures caused by lightning. Suitable performance indices are defined in order the line insulation level, the tower footing resistance, and the average height of shielding wires cost values to be related to the lightning failure costs. The developed optimum design method has been applied on several operating Hellenic transmission lines of 150 and 400 kV. The obtained results for each one region of the examined lines, i.e. the new selected design parameters, have significantly reduced the failure rates caused by lightning, something really important in the case of the open loop lines. Although, any modifications or improvements to the insulation level, the tower footing resistance, or the height of the shielding wires of these lines are practically impossible, this method can be valuable to electric power utilities, designing new transmission lines, reducing significantly any future failures caused by lightning.
7. LIST OF SYMBOLS b CMEU CRE CFC d D Dc di/dt f(I) gji () G () H(x) h hc Imax Imin Ipeak Ji ki () L NTotal NSF NBF NL p P(d) R r T t Ua Zsurge
horizontal spacing between the shielding wires mean annual cost of undelivered energy for the utility mean costs of one permanent failure repair equivalent annual line failure consumer cost equivalent conductor diameter without corona equivalent conductor diameter with corona shielding failure exposure distance random variable denoting the lightning current derivative probability density function for the current equivalent annual investment of the ith region line design characteristic j total cost of investment of the transmission line’s design parameters Hessian matrix average height of the shielding wires average conductor height at the tower maximum lightning current minimum lightning current random variable denoting the peak lightning current performance cost index of the ith region annual line failure cost total equivalent inductance of the system lightning failures shielding failure rate backflasover failure rate number of lightning flashes to a line empirical coefficient defining the ratio of annual maintenance cost to total cost of investment probability distribution function of the random variable d tower footing resistance annual interest rate yearly lightning or keraunic level estimated line exploitation period insulation level of the transmission line conductor line surge impedance
Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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L. EKONOMOU, G. P. FOTIS AND T. I. MARIS
ACKNOWLEDGEMENTS
The authors want to express their sincere gratitude to the Hellenic Public Power Corporation S.A. for their kind supply of various technical and economical data and the National Meteorological Authority of Hellas for the supply of meteorological data.
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AUTHORS’ BIOGRAPHIES
Lambros Ekonomou was born on January 9, 1976 in Athens, Greece. He received a Bachelor of Engineering (Hons) in Electrical Engineering and Electronics in 1997 and a Master of Science in Advanced Control in 1998 from University of Manchester Institute of Science and Technology (U.M.I.S.T.) in United Kingdom. In 2006 he graduated with a Ph.D. in High Voltage Engineering from the National Technical University of Athens (N.T.U.A.) in Greece. Currently he is working in Hellenic Public Power Corporation S.A. as an Electrical Engineer while he is a Research Associate in the N.T.U.A.’s High Voltage Laboratory. His research interests concern high voltage engineering, transmission and distribution lines, lightning performance, lightning protection, stability analysis, control design, reliability, electrical drives, and artificial neural networks. Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
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Georgios P. Fotis was born on July 26, 1977 in Athens, Greece. He received his diploma and his Ph.D. in Electrical Engineering from the National Technical University of Athens in 2001 and 2006, respectively. He is currently a Research Associate in the N.T.U.A.’s High Voltage Laboratory. His research interests concern high voltages, electromagnetic compatibility, and electrostatic discharges. He is a member of IEEE and the Technical Chamber of Greece. Dr Fotis is the author of 21 papers in scientific journals and conferences proceedings.
Theodoros I. Maris was born in Arta, Greece in 1961. He received his diploma in Electrical Engineering and his Ph.D. from the School of Engineering of University of Patras in Greece in 1984 and 1994, respectively. He became Teaching Assistant (1996–2001) at the Department of Electrical Engineering of the Technological Educational Institute of Chalkida and thereafter Assistant Professor (2001–2005) and Associate Professor (since 2005). His research interests concern electric energy systems, direct current interconnections, high voltage direct current converters, electrical drives, photovoltaic inverters, transmission and distribution lines, and artificial neural networks.
Copyright # 2007 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2008; 18:437–447 DOI: 10.1002/etep
