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Decision Algorithm of Neutral Grounding Mode Based on Weighted Multi-attribute Interval Grey Target Theory Cao Zhenchong, Peng Jian, Xue Zhifang Guangxi Electric Power Industry Investigation Design and Research Institute Jianzheng Road No.10, Nanning, Guangxi, 530023, China caozhenchong@tsinghua.org.cn Abstract: Neutral grounding mode is important to distribution mode, which is derived from the grey system theory, the networks reliability, which has not been calculated theory of analytical hierarchy process, and the multi-attribute quantitatively. Power system is a grey system whose reliability grey target decision algorithm, which involve many parameters are statistic, so interval theory should be introduced disciplines: system engineering, decision theory, and to analysis the impact of neutral grounding mode to distribution networks reliability by interval function comparing method economic theory. based on relative superiority and extended from interval number The algorithm proposed is composed of steps as follows: (1) to interval function. Weighted multi-attribute decision arithmetic to obtain the fault tripping rate of distribution lines and the is given to choose distribution system neutral point mode, which reliability indexes by interval analysis, all of which are the is based on the analytic hierarchy process theory and more interval grey numbers and is not capable of being used directly reasonable, veracious, flexible and applicable. According to for comparison, synthesis and decision; (2) to construct an calculation results of examples, resonance grounding device is optimal objective index that can reflect the integrated better for the reliability than low value resistor in middle voltage performance of neutral grounding mode with the grey distribution overhead line system, in pure cable system or numbers obtained in former step. Thereby, the major research cable-overhead line hybrid system, the determinant factor is the mutual effect between cables when faults occur. in this paper concentrates on three parts: the analysis on interval number of fault tripping rate, the decision algorithm Keywords: Distribution networks; Neutral grounding mode; based on weighted multi-attribute interval grey target theory, Reliability; Interval analysis; Multi-attribute weighted grey and the simulation cases. target theory II. INTERVAL ANALYSIS ON FAULT TRIPPING RATE OF • I. INTRODUCTION Neutral grounding mode is one of the most key problems DISTRIBUTION LINES that affect the reliability of distribution system greatly, and In research filed of power system reliability, the stochastic many researchers have investigated qualitatively the theory is usually applied to quantitative analysis and characteristics of the distribution system reliability with evaluation, from which and many effective methods have been respect to several neutral grounding modes [1-4]. References developed [7]. In addition, the grey system theory is widely [5,6] analyzed in detail the influences to fault tripping rate of used in interval analysis due to the uncertainty of original power distribution lines by four modes including isolated reliability parameters. neutral, neutral point grounded through low-resistance, neutral Definition 1: assume S is a semi-order set, for a point grounded through Peterson coil, and neutral point grounded through adjustable reactance, and emphasized that ( ) given x, x . If x, x ∈ S and x ≤ x , then an interval number the reliability of entire distribution system can’t be evaluated { } can be defined as: [ x] = [ x, x] = x ∈ S | x ≤ x ≤ x , where x accurately only by fault tripping rate of lines. Actually, the reliability should be analyzed according to the specific grid is called lower extreme point, and x upper extreme point. structure of distribution system in terms of reliability indexes Furthermore, if x = x , [ x ] will regress as a real number [8]. such as CAIDI (Customer Average Interruption Duration As shown in Fig. 1, it is an actual 6kV distribution system Index), ASAI (Average System Availability Index), SAIDI with n lines including k cables and n-k overhead lines, each of (System Average Interruption Duration Index), SAIFI (System which corresponds to one load point. Before calculation, some Average Interruption Frequency Index), and ENS (Energy Not reasonable assumptions must be made: Supplied). However, there is still no systematic theory and algorithm 1) When a single-phase-to-ground fault occurs at the ith that can be employed to compare the advantages and cable line, if the arc extinction unsuccessful or the fault disadvantages of different neutral grounding modes line is not tripped out, the accident spreads and quantitatively, which is subject to the reliability index of endangers the near line, which consequently leads to distribution system. In this paper, the decision algorithm based tripping of this line and the near m cables. on weighted multi-attribute interval grey target theory is proposed to solve the decision problem of neutral grounding 2) When the fault occurs at overhead line and even if the [ g ni ]=[λi ](1+ km / n ) (1) accident can’t be cleared in time, the near lines will not be affected. 2) Similarly, in system with neutral grounding through Peterson coil, assume the probability that arc can be extinguished is [α ] . the fault tripping rate of distribution system is derived below, [ g pi ]=[ λi ](1−[α ][η ][δ ])+[ λi ][η ](1−[α ][δ ]) km / n (2) 3) As for system with low-resistance neutral point, when fault occurs, relay protection maybe fails to trip out the fault line, due to the mismatch between relay protection and low-resistance. This condition is described by probability [ β ] , then the fault tripping rate is, [ g ri ]=[ λi ](1+[ β ][λi ][η ]km / n ) (3) 4) For neutral grounding through adjustable reactance, Fig. 1 Schematic diagram of a distribution system assuming the probability that reactance extinguishes arc 3) For the neutral grounding through Peterson coil, if the is [σ ] , and the probability that the reactance can’t match coil can extinguish the arc of single-phase transient fault, the relay protection to successfully trip out the fault line relay protection only records the action event of is [ζ ] , the fault tripping rate is, extinction coil. If the coil can’t eliminate the fault, it quits and the system degenerates as an isolated neutral [ g ci ]=[ λi ](1−[σ ][η ][δ ]) +[ λi ][η ]{1−[σ ][δ ]+ (1−[δ ][ζ ])}km / n (4) system. Finally, relay protection sends the warning signal and records the fault event. As discussed above, the fault tripping rate of distribution lines is usually related to the influencing degree that the fault 4) For the neutral grounding through low-resistance, relay line affects its near lines, which is described by an influencing protection sends tripping command and records the fault factor γ = km / n . event. III. DECISION ALGORITHM BASED ON WEIGHTED 5) For the neutral grounding through adjustable reactance, if the arc of single phase transient fault can be extinguished, MULTI-ATTRIBUTE INTERVAL GREY TARGET THEORY relay protection only records the action event of A. Interval Number of ReliabilityIndex adjustable reactance. Otherwise, relay protection sends directly the over-current quick-break signal used for Typical reliability indexes of distribution system includes tripping and records the fault event. ASAI, CAIDI, SAIDI, SAIFI, and ENS, which are determined by both neutral grounding mode and many other factors such 6) The faults caused by other elements in distribution as elements (generator, transformer, bus, and breaker), main system are not taken into account. connection and grid structure of system[9-11]. Whereas, the 7) The probability that the fault type is influence of neutral grounding mode is relatively independent single-phase-to-ground fault is [η ] , among which the of that of other factors, and this paper only investigates the probability of transient type is [δ ] . The faults including former. two-phase- to-ground, two-phase or three-phase short Setting the schedule repair rate [λi' ] and combining the fault circuit can be eliminated successfully by relay protection. tripping rate of distribution line as shown in Eqs. (1)~ (4), it is Based on the above assumptions, the influence of neutral ready to obtain the outage rate [Gi ] and annual outage time grounding mode on fault tripping rate is discussed as below: [U i ] of the ith line: 1) In isolated neutral system, the fault rate of the ith line [Gi ]=[ gi ]+[ λi' ] (5) is [λi ] . According to the technical code, the line with ' ' [U ] = [ g ][ r ] + [ λ ][ r ] (6) single-phase-to-ground can operate for 2 hours until i i i i i tripping out. Therefore, the fault tripping rate is [λi ] for where the general representations [Gi ] , [ gi ] and [U i ] overhead line, and [ λi ](1+ m ) for cable. In hybrid can be specialized by adding subscripts n , p , r , and c , distribution system consisting of cable and overhead line, which identify respectively the four neutral grounding modes. the probability of being cable fault is k / n , so the fault According to the interval inversion rule [12], the reliability tripping rate of distribution system initiated by the ith indexes of distribution system can be expressed below, line is calculated by, [CAIDI ]=φ ( ∑ [Gi ] Ni , ∑ [U i ] Ni ) i∈R i∈R (7) [ SAIFI ] = φ ( ∑ [Gi ] Ni , ∑ Ni ) n i∈R i∈R (8) [c j , d j ] = [ kij ] ∑ [k ], w ij j ∈ [c j , d j ] (13) [ SAIDI ]=φ ( ∑ [U i ] Ni , ∑ Ni ) i =1 i∈R i∈R (9) In further, by utilizing the weight shown in Eq. (13), the ' [ ASAI ] = ( ∑ 8760 Ni − ∑ [U i ] Ni ) ∑ 8760 Ni normalized decision matrix R is transformed into the (10) i∈R i∈R i∈R weighted normalized decision matrix R = ([ rij , rij ]) m× n . [ ENS ] = ∑ [ Pai ][U i ] i∈R (11) [ rij , rij ] = [c j , d j ] [ rij , rij ] ' ' (14) where [ Ni ] is the customer number at load point i, [ Pai ] the average load at load point i. For different distribution where ri = ([ ri1 , ri1 ],[ ri 2 , ri 2 ], L ,[ rin , rin ]), i = 1, 2, L , m , is systems, the importance of one reliability index varies when compared with the others. To select neutral grounding mode the ith row vector of R, here called the effect vector of drafted from the point of view of the reliability, the decision algorithm decision scheme Si . based on weighted multi-attribute interval grey target theory is Definition 3: used. Let r = max j o {( r + r ) ij ij } 2 | 1 ≤ i ≤ m , j = 1, 2, L , n , and its B. Decision Algorithm Based on Interval Grey Target Theory corresponding decision value in R is written as [ ri j , ri j ] , o o Definition 2: for m-dimension interval numbers o A = [( a11 , a12 ), K , ( am1 , am 2 )] and B = [(b11 , b12 ), K , (bm1 , bm 2 )] then r is called the optimal effect vector of multi-attribute grey target decision, also named bull’s eye. } = {[r } where ai1 ≤ ai 2 , bi1 ≤ bi 2 , i=1,2,L,m , then L p ( A, B ) is called { ro = rj , rj , L , rj o o o io 1 , ri 1 ],[ ri 2 , ri 2 ], L ,[ ri n , ri n ] . o o o o o the distance from A to B. Especially, L2 ( A, B ) is called When constructing the optimal effect vector, if there are (r )/2 Euclidean distance. 1 two or more decision schemes whose values + ri Lp ( A, B) = [( a11 − b11 ) + ( a12 − b12 ) + L p p io j o j 1/ p 2 (12) with respect to one or more indexes are equal, then the optimal + ( am1 − bm1 ) + ( am 2 − bm 2 ) ] p p 1 p effect can be realized. In this case, to select the upper limit Given that the multi-attribute decision problem has m rij as evaluation standard can solve the above problem. evaluation objects or drafted decision schemes, belonging to (n) Definition 4: R is n-dimensional grey target with bull’s the set S = {S1 , S 2 , L , S m } ; and the index set o eye r and radius R, written as: A = { A1 , A2 , L , An } is composed of n evaluation indexes or { 1 ⎡ (r − r ) 2 (n) = ([ ri 1 , ri 1 ], [ ri 2 , ri 2 ], L , [ rin , rin ]) | R = 2 R attributes. Writing the attribute of scheme Si with respect to 2 1/ 2 ⎣ i1 io 1 (13) Aj as [ xij , xij ] , all of which form a decision sample ( ) ( ) + (r − r ) ⎤ ⎫ 1/ 2 2 2 2 ⎦ ⎬ + ri 1 − ri 1 + L + rin − ri n ⎭ o o in io n matrix X = ([ xij , xij ]) m× n , i = 1, 2, L , m , j = 1, 2, L , n . Because the index set A includes both optimistic and Definition 5: pessimistic indexes, it is necessary to transform A into the (n) Let effect vector ri = ([ ri1 , ri1 ],[ ri 2 , ri 2 ], L ,[ rin , rin ]) ∈ R , [12] : R = ([ rij , rij ]) m× n . ' ' ' normalized decision matrix then call ε i the off-target distance of ri . The weight w j owned by each element in index set A can’t ε i = ri − ro be assigned exactly, being able to be described by the (( i1 i 1 ) ) ⎡ r − r 2 + r − r 2 +L+ r − r ( ) ( ) ⎤ 1/ 2 1 2 following: w j ∈ [c j , d j ] , 0 ≤ c j ≤ d j ≤ 1 , w1 + w2 + L + wn = 1 . 2 = + rin − ri n 2 ⎢ ⎣ i1 i 1 ⎥ ⎦ 1/ 2 o in i n o o o According to the meaning of importance standard degree in . analytical hierarchic theory [13], an importance degree matrix The value of off-target distance reflects the bad or good can be constructed, written as K = ([ kij ]) m× n , which is a property of effect vector: the smaller it is, the better the consistent matrix with the following properties (proof omitted): decision scheme Si is; and vice versa. [ kij ] > 0 , [ kij ] = 1 [ k ji ] , and [ kii ] = 1 .By normalizing every This section illustrates the basic principle of decision column of the importance degree matrix K, the weight of each algorithm based on weighted multi-attribute interval grey target theory, which also can be seen in reference [8] for element in the index set A can finally be obtained: detailed derivation process. Essentially, the algorithm is a applications of the algorithm proposed not only in the decision generalization of decision algorithm multi-attribute interval problem of neutral grounding mode, but also in the optimal grey target theory from real space to interval number space, reconfiguration of power system grid, and the optimal design and can comprehensively consider the weight problem of of power system high-voltage transmission line. every index. Compared with traditional decision algorithms, this algorithm takes more advantages of flexibility, rationality, IV. STUDY CASES AND RESULT ANALYSIS and universality: the flexibility is embodied by that all decision indexes with different meanings, dimensions, and Three study cases were performed to validate the algorithm properties can be evaluated in same decision system, thus proposed. expanding the select range of decision indexes to make it more Case 1: given the electric parameters of a real distribution comprehensive of considering the influencing factors on system [14-15] : [λi ] = [0.05, 0.1] , [λ ' ] = [0.5,1] , i decision-making; the rationality by that the importance degree [ηi ] = [0.8, 0.9] , [δ i ] = [0.7, 0.8] , [α ] = [0.9, 0.95] , of every decision index indicated by its importance weight is reflected in the decision process, as seen in the process of [ β ] = [0.05, 0.1] , [σ ] = [0.9, 0.95] , [γ ] = [0.05, 0.1] , transforming the normalized decision matrix into the weighted [ ri ] = [1.0, 2.0] , [ ri ] = [1.0,1.5] , k = 57, n = 57, m = 1. ' normalized decision matrix; the universality by the generalization of decision algorithm multi-attribute interval Case 2: m = 4, and the rest is same as Case 1. grey target theory from real space to interval number space, Case 3: m = 8, and the rest is same as Case 1. thus making the algorithm can deal with more complicated The results are shown in TABLE I ~III respectively for decision problems. The above feature promotes the wide Case 1~3. TABLE I CALCULATION RESULT OF CASE 1 Grounding Mode CAIDI / h SAIFI / time/a SAIDI / h/(a.c) ASAI ENS / MV.A ε isolated [6.000,27.000] [0.0600,0.1000] [0.5600,1.7000] [0.99981,0.99994] [16.993,64.819] 0.102 Peterson coil [13.063,100.430] [0.0152,0.0415] [0.5152,1.5829] [0.99982,0.99994] [15.635,60.354] 0.034 low-resistance [10.569,51.603] [0.0302,0.0523] [0.5302,1.6045] [0.99982,0.99994] [16.090,61.178] 0.061 adjustable-reactance [18.197,146.680] [0.0104,0.0291] [0.5104,1.5581] [0.99982,0.99994] [15.487,59.410] 0.037 TABLE II CALCULATION RESULT OF CASE 2 Grounding Mode CAIDI / h SAIFI / time/a SAIDI / h/(a.c) ASAI ENS / MV.A ε isolated [3.0000,12.0000] [0.1500,0.2500] [0.6500,2.0000] [0.99977,0.99993] [19.724,76.258] 0.159 Peterson coil [6.4705,48.1250] [0.0325,0.0914] [0.5325,1.6828] [0.99981,0.99994] [16.159,64.163] 0.074 low-resistance [9.4746,50.4500] [0.0310,0.0590] [0.5310,1.6180] [0.99982,0.99994] [16.112,61.692] 0.067 adjustable-reactance [12.9330,117.1000] [0.0130,0.0419] [0.5130,1.5838] [0.99982,0.99994] [15.568,60.388] 0.01 TABLE III CALCULATION RESULT OF CASE 3 Grounding Mode CAIDI / h SAIFI / time/a SAIDI / h/(a.c) ASAI ENS / MV.A ε isolated [2.1111,7.5556] [0.2700,0.4500] [0.7700,2.4000] [0.99973,0.99991] [23.365,91509.0] 0.173 Peterson coil [4.1646,28.9980] [0.0556,0.1580] [0.5556,1.8160] [0.99979,0.99994] [16.858,69.242] 0.098 low-resistance [8.3529,48.9920] [0.0319,0.0680] [0.5319,1.6360] [0.99981,0.99994] [16.141,62.379] 0.061 adjustable-reactance [9.4746,92.4490] [0.01658,0.0590] [0.5166,1.6180] [0.99982,0.99994] [15676.000,61692.0] 0.004 [6] Cao Z C, Yang X C, Wu Z S. A novel adjustable TCR arc suppression Cases 1~3 calculated the reliability indexes of the coil [C]. Proceedings of the XIVth International Symposium on High distribution system as shown in Fig. 1, comparing four neutral Voltage Engineering, Tsinghua University, Beijing, China, August grounding modes by the algorithm proposed. 25-29, 2005, 305. From the results shown in Tables 1~3, the neutral [7] Billinton R, Allan R N. Reliability Evaluation of Power Systems [M]. Boston, MA, USA: Pitman Advanced Publishing Program, 1996. grounding mode imposes an extreme impact on the reliability [8] Cao Z C. Research on Reliability and Decision Arithmetic of indexes of distribution system. The performance of isolated Distribution System Neutral Point Mode and Calculation of neutral point is worst, and all the remained three modes can Overvoltage [D]. Beijing, China: Tsinghua University, 2006 improve the reliability of distribution system to some extent, [9] Zhang P，Guo Y J. Interval method for power station and substation affected considerably by the fault influencing coefficient m. reliability evaluation [J]. Automation of Electric Power Systems, 2004, 28(19): 48-52. (in Chinese) the smaller the influencing coefficient m is, the more [10] Zhang P, Wang S X. Interval analysis based multi-objective network remarkable improvement the low-resistance can improve; reconfiguration for distribution system reliability improvement [J]. otherwise, low-resistance and adjustable reactance can Automation of Electric Power Systems, 2004, 28(21): 22-26. (in improve the reliability remarkably. Because adjustable Chinese) [11] Guo Y J. Power system reliability analysis [M]. Beijing, China: reactance has the ability of extinguishing the arc of Tsinghua University Press, 2001 single-phase transient fault and avoiding unnecessary [12] Dang Y G, Liu S F, Liu B. Study on the multi-attribute decision model tripping-out of line, it can improve the reliability of of grey target based on interval number [J]. China Engineering Science, distribution system more effectively than low-resistance. 2005, 7(8): 31-35. (in Chinese) [13] Saaty T L. The analytical hierarchy process [M]. Pittsburgh PA: RW S As discussed above, the algorithm proposed can unite many Publications, 1996. reliability decision indexes with different meanings, [14] Allan R N, Billinton R, Sjariefi I, et al. A reliability test system for dimensions, and weights into a reasonable decision system, educational purposes- basic distribution system data and results [J]. and the decision result is in the form of off-target distance, IEEE Transactions on Power System, 1991, 6(2): 813-820. [15] Billinton R, Jonnavithula S. A test system for teaching overall power which is in agreement with practical decision process. system reliability assessment [J]. IEEE Transactions on Power System, 1996, 11(4): 1670-1676. V. CONCLUSIONS 1) The neutral grounding mode affects reliability of distribution system considerably, so is necessary to take into account the neutral grounding mode in reliability analysis of distribution system. 2) In the distribution system only consists of overhead lines or in the cable distribution system where the fault cable would not affect the near normal operating cables, Peterson coil grounding can improve the system reliability in the maximal degree. 3) In cable system where the fault cable affects severely the near normal operating cables, low-resistance and adjustable reactance are beneficial to the improvement of system reliability. 4) From the point of view of improving system reliability, the decision algorithm based on weighted multi-attribute interval grey target theory can effectively evaluate the advantages and disadvantages of different neutral grounding modes. REFERENCES [1] Yao H N. Practical experiences of EdF (Electricite de France) in using resonance grounding for neutrals of medium voltage power network [J]. Power System Technology, 1998, 22(4): 50-53. (in Chinese) [2] Yi D F. Research on neutral grounding in 6~10kV power networks [J]. Power System Technology, 1998, 22(7): 27-30. (in Chinese) [3] Wu Q Y, Shen P, Chen L M. Neutral grounding in urban power distribution system [J]. High Voltage Engineering, 2002, 28(6): 50, 58. (in Chinese) [4] Yao H N, Cao M Y. On neutral grounding modes of cable network. Power System Technology, 2003, 27(2): 84-89. (in Chinese) [5] Cao Z C, He J S, Yang X C, Zhi X, etc. Interval analysis about Influence on Reliability of Power Distribution Lines With Various Grounded Neutral [J]. Guangxi Electric Power, 2007, 30(5): 5-7. (in Chinese)

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