Application of Statistical Methods in Insulation Coordination of Overhead Power Lines Mariusz Benesz, Wiesław Nowak, Waldemar Szpyra, Rafał Tarko Department of Electrical Engineering and Power Engineering AGH University of Science and Technology Cracow, Poland
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[email protected] Abstract—The process of insulation coordination of power systems is conditioned by a number of factors, among which the improvement of methods of surge analyses occupy a special position. The use of statistical procedures in coordination of insulation of overhead power lines requires knowing statistical distributions of surges and reliability of their parameters. Establishing the type of distribution and the evaluation of values of its parameters may be obtained with the mathematical modeling, computer simulations and respective statistical analyses. The simulations of statistical distributions of switching overvoltages for 400 kV power line are presented in this paper. Keywords—high voltage overhead lines; statistical procedures; insulation coordination
I. INTRODUCTION Switching overvoltages are the important type of surges, determining the dimensioning and coordination of insulation in the HV and extra HV systems, due to the critical electric strength of large air gaps [1]. This applies to phase-to-ground and phase-to-phase insulation of lines and power stations. As far as air insulation is concerned, the most important processes are switching on the lines and their automatic reclosing. The parameters of overvoltages depend, among others, on: short-circuit capacity in the place the line has been connected, length of the line, character and degree to which it is loaded, and also from the moment the contacts of the breaker are touching. The last element is a random variable, therefore the level of overvoltages at the time of switching on the line is a random variable, too [2, 3, 4, 5]. The statistical methods are used for the analysis of surges of HV power lines caused by their switching on or reclosing processes. The obtained statistical distributions of overvoltages are applicable to the coordination of insulation in power lines, the purpose of which is determining minimum distances which would provide electric strength of the insulation system. II. INSULATION COORDINATION OF OVERHEAD POWER LINES Reliable exploitation of overhead HV lines is conditioned by the reliability of their insulating systems. These are two groups of problems to be solved, i.e. coordination of insulation and dimensioning of inner and external clearances in the power lines [8, 9].
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The first issue is connected with selecting a set of standard withstand voltages of the designed line, in compliance with the standard [9], and also determining minimum gaps (Table I), providing electric strength of the insulating system of lines definite surge conditions. These gaps, in turn, create bases for establishing both inner and external clearances. TABLE I. Clearance:
Between:
Used for: Kind of surge:
CLEARANCE DISTANCE TO PREVENT FLASHOVER [9] Del
Dpp
D50Hz-p-e phase conductors and ground potential objects
phase conductors phase and groundconductors potential objects • inner insulation • external insulation
D50Hz-p-p phase conductors
• inner insulation • temporary overvoltages (in extreme wind conditions)
• slow-front overvoltages, • fast-front overvoltages
Among slow-front overvoltages which are important for overhead lines are earth faults and overvoltages taking place while switching on the line. According to [9], for slow-front overvoltages, representative overvoltages are defined on the basis of Ue2%-sf values for phase-to-ground overvoltages, and Up2%-sf for phase-to-phase overvoltages, which may be exceeded with a 2% probability:
(
)
(1)
(
)
(2)
P U m > U e 2% −sf = 0,02 P U m > U p 2% −sf = 0,02
where: Um – the peak value of a representative overvoltage. Formally, the values of Ue2%-sf and Up2%-sf are quantiles of 0.98 of respective distributions of peak values of slow-front overvoltages. Representative overvoltages Urp are obtained by multiplying statistical overvoltage by statistical coordination coefficient Kcs [9] for phase-to-ground insulation:
U rp = K csU e 2% − sf
(3)
and for phase-to-phase insulation:
III. ANALYZED 400 KV POWER SYSTEM
U rp = K csU p 2% − sf
(4)
The risk of the insulation system failure is related with the statistical coordination coefficient Kcs. For example, Kcs equal to 1.05 corresponds to the risk of fault of 10-3. The minimum required clearance distances Del, Dpp and D50Hz are assumed as gap d values, meeting the equation:
U cw = U rp
The subject of the analysis is a 400 kV power system presented schematically in Fig. 1. Power systems A and B (short-circuit power 20 and 15 GVA, respectively) are connected by a 400 kV single-circuit overhead power line 50 km long. In substations, to which it has been connected, surge arresters are installed.
(5)
The coordination withstand voltage Ucw for slow-front overvoltages, is a quantile of 0.1 in the statistical distribution of electric strength. Assuming that this is a normal distribution with expected value U50% and standard deviation σ, then voltage Ucw can be calculated from the relation [9]:
U cw = U 50% + z 0,1σ = U 50% (1 + z 0,1σ w )
(6)
U 50% = K a K g − sf ⋅ 1080 ln (0,46d + 1)
(7)
where: U50% – 50% flashover voltage, z0,1 – quantile of 0.1 of normalized normal distribution N(0,1), σ – standard deviation, σw – relative standard deviation, Ka – correction coefficient for height above the sea level, Kg-sf – coefficient of spark gap for slow-front overvoltages (Table II), d – gap. TABLE II.
TYPICAL VALUES OF SPARK GAPS COEFFICIENTS FOR POWER LINES [9]
Type of gap in air
Configuration
Kg-sf
external
phase conductor — object
1.30
phase conductor — window tower
1.25
phase conductor — tower
1.45
phase conductor — phase conductor
1.60
inner
Fig. 1. Analyzed 400 kV power system.
Using relations (1) ÷ (7), an equation for slow-front overvoltages is obtained, from which the minimum phase-toearth gap Del is defined:
K a K g − sf ⋅ 1080 ln(0,46 Del + 1)(1 + z 0,1σ w ) = K csU e 2% − sf (8)
Del =
K csU e 2% − sf 1 − 1 exp 0,46 1080 K a K g − sf (1 + z 0,1σ w )
(9)
The minimum phase-to-phase gap Dpp is calculated from (8), assuming Up2%-sf values in the place of Ue2%- sf.
For the sake of performing the statistical analysis of distributions of peak values of overvoltages generated in while switching on the line, a computer model of this system was worked out in program EMTP-ATP, accounting for recommendations listed in publications [6]. The model of the analyzed system consists of the following partial models: —
model of 400 kV single-circuit line;
—
models of power systems A and B (built of ideal voltage sources arranged in series with substitute short-circuit impedances defined in the domain of symmetrical components);
—
models of feeder circuit-breakers in substations A and B, where their properties were represented randomly;
—
models of surge arresters in substations A and B, which represent their dynamic properties [7];
—
model of symmetrical and asymmetrical short-circuits.
One of the basic elements of the model is the statistical breaker, which can be closed and opened randomly. Time in which the contacts are touching or are separated is a random variable from a uniform distribution, normal distribution or linear distribution, of expected value T and standard deviation σ. Moreover, the breaker may operate independently (Independent mode) or dependently on another breaker (Master or Slave mode). In the case of a simulation of switching on the analyzed line an assumption was made, that the breaker in phase A is Master, and the time in which the contacts are touching is a random variable from a uniform distribution in an interval from tCa = 15 ms to tCb = 35 ms (one 50 Hz period wide). The function of density of distribution of time, in which the contacts of the breaker are touching is expressed with the dependence:
0 1 f (tC ) = − t Cb tCa 0
dla tC < tCa dla tCa ≤ tC ≤ tCb
(10)
dla tC > tCb
According to standards [8] there are two methods of determining statistical distributions of peak values of overvoltages, applicable for coordinating insulations: method of phase peak value and method of incidence peak value. The simulations involved latter method obtaining statistical distributions of overvoltages which were significant for the coordination of phase-to-earth and phase-to-phase insulation of the analyzed 400 kV line. IV. RESULTS OF ANALYSES Probability simulations were performed for four cases of switching on the 400 kV line, i.e. switching on the line in the absence or in the presence of surge arresters. In each case the simulations were performed 200 times at random parameters of statistical breakers. As a result of the simulation, four 200element random samples were obtained. They contained six random variables (three phase-to-ground voltages and three phase-to-phase voltages), which were statistically analyzed. The probability analysis can be summarized as in Fig. 2 where is shown the probability of exceeding peak values of phase-to-ground and phase-to-phase overvoltages generated in the analyzed 400 kV line.
The expected value T and standard deviation Dev of random tC described by distribution (1) are:
T=
tCa + tCb 2
Dev =
(11)
tCb − tCa
(12)
12
Assumption was made that breakers in the remaining phases are of Slave type, for which switching on is realized with delay ΔtC as compared to the closing of the Master breaker. Moreover, it was also assumed that the delayed switching off was a random variable in normal distribution, having the density function: Fig. 2. Probability of exceeding peak value of phase-to-ground and phase-tophase overvoltages while switching on the 400 kV line.
f (ΔtC ) =
1 Δt − μ C exp − C 2 σ 2 π σC C
1
2
(13)
where: μC, σC – expected value and standard deviation of variable ΔtC (it was assumed that Slave breakers have μC = 5 ms, σC = 1.6 ms). When simulating re-activation of the line, the same assumptions were made as for the parameters of statistical breakers; the only difference was that tCa = 200 ms and tCb = 220 ms because of the previous operation of closing the line after a short-circuit fault incident. The simulation was realized n times for random parameters of statistical breakers. After each i-th cycle (i = 1, ..., n) the global extremums of the observed courses and corresponding times of their occurrence were recorded.
Fig. 3. Probability of exceeding peak value of phase-to-ground and phase-tophase overvoltages while reclosing the 400 kV line.
Analogous probabilities, though for the reclosing of the line, are presented in Fig. 3. From the formal point of view the curves in figures 2 and 3 supplement to unity (100%) the cumulative distribution function F(um) of random variable Um, i.e. peak overvoltage value:
minimum gap to be used equals to a higher value than for gaps calculated for lightning impulse overvoltages and switching overvoltages. Respective statistical methods can be obviously used also for the overvoltages caused by lightning.
P (U m > um ) = 1 − F (um )
The process of insulation coordination is conditioned by a number of factors, among which the improvement of methods of surge analyses and looking for new algorithms of their numerical identification occupy a special position. Theoretical analytical methods, employing mathematical models of overvoltages and the computer calculation technology, are intensely developed.
V. CONCLUSIONS (14)
The peak value of a representative overvoltage may be determined from statistical distributions as in figures 2 and 3. According to (1) and (2) the statistical overvoltages are listed in Tab. III. The minimum insulation gaps (9) of the analyzed 400 kV line are presented in Tab. IV. TABLE III.
VALUES OF STATISTICAL OVERVOLTAGES Slow-front overvoltages generated at
Insulation
Surge arresters
phase-toground short-circuit
switching on the line
reclosing the line
statistical overvoltages Ue2%-sf, kV phase-toground
no
734
765
1272
yes
645
648
682
REFERENCES
statistical overvoltages Up2%-sf, kV phase-tophase TABLE IV.
no
1115
1235
1903
yes
1018
1169
1265
[1]
[2]
MINIMUM CLEARANCE DISTANCES OF 400 KV POWER LINE FOR SLOW-FRONT OVERVOLTAGES Minimum clearance
Clearance
Configuration
no arresters
with arresters
external
phase conductor — object
3.92 m
1.61 m
phase conductor — window tower
4.18 m
1.69 m
phase conductor — tower
3.31 m
1.40 m
phase conductor — phase conductor
5.44 m
inner
The use of statistical procedures in coordination of insulation of overhead power lines requires knowing statistical distributions of surges and reliability of their parameters. Establishing the type of distribution and the evaluation of values of its parameters may be obtained with the mathematical modeling, computer simulations and respective statistical analyses. The improvement of procedures and numerical methods to be used for statistical evaluation of surges is important for precise designing procedures and exploitation of HV and extra HV lines.
[3]
[4]
[5]
[6] 2.83 m
It should be noted that the value of minimum clearance distances strongly depends on the applied surge arresters. The complete analysis requires considering small insulation gaps also for fast-front overvoltages and temporary overvoltages. A
[7]
[8] [9]
CIGRÉ Working Group 33-7, “Guidelines for the evaluation of the dielectric strength of external insulation,” CIGRÉ Technical Brochure no 72 P. Mestas, M.C. Tavares, “Relevant Parameters in a Statistical Analysis—Application to Transmission-Line Energization,” IEEE Transactions on Power Delivery, Vol. 29, Issue 6, pp. 2605 - 2613, December 2014 J.A. Martinez, D. Goldsworthy, R. Horton, “Switching Overvoltage Measurements and Simulations—Part I: Field Test Overvoltage Measurements,” IEEE Transactions on Power Delivery, Vol. 29, Issue 6, pp. 2502 - 2509, December 2014 P. Bunov, et al., “Transmission line arresters application for control of switching overvoltages on 500-kV transmission line,” T&D Conference and Exposition, 2014 IEEE PES A.H. Hamza, S.M. Ghania, A.M.Emam, A.S. Shafy, “Statistical Analysis of Switching Overvoltages and Insulation Coordination For a 500 kV Transmission Line,” Power Systems Conference (MEPCON), 2016 Eighteenth International Middle East, 27-29 December 2016 CIGRE WG 33-02, “Guidelines for representation of network elements when calculating transients,” CIGRE Technical Brochure no 39 IEEE The Working Group 3.4.11, “Modeling of metal oxide surge arresters,” IEEE Transactions on Power Delivery, Vol. 7, No. 1, pp. 302 - 309, January 1992 PN-EN 60071-1:2006+A1:2010 “Insulation co-ordination. Definitions, principles and rules” PN-EN 50341-1:2013-03E “Overhead electrical lines exceeding AC 1 kV. General requirements. Common specifications”
