2003 PD Fuzzy Distribución Incertidumbre

694 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003

Distribution System Performance EvaluationAccounting for Data Uncertainty

Jovan Nahman and Dragoslav Peric

AbstractThe effects of uncertain input data on the perfor-mance evaluation of a distribution system are analyzed. A crite-rion is introduced for assessing the grade of uncertainty of the re-sults obtained in the calculation of maximum loads, voltage drops,energy losses, and characteristic reliability indices of a network ifsome input parameters are only guesses based on limited experi-ence, measurements, and/or statistical data. Reasonable outputsbounds are determined based upon the shape of the function mea-suring the uncertainty. High uncertainty of a result obtained indi-cates that a re-examination of relevant uncertain input data wouldbe recommendable for a more precise quantification. The methodproposed is applied to a real life example for illustration.

Index TermsDistribution systems, fuzzy mathematics, oper-ating performances, uncertainty.

I. INTRODUCTION

FOR a proper planning of construction and exploitation ofdistribution networks, various network performances haveto be analyzed. The most important among them are the max-imum currents to be carried by the distribution feeders and asso-ciated voltage drops, annual energy losses, and the reliability ofsupplying consumers demands. Unfortunately, many of the in-puts forming the basis for these studies are often assessed withsome uncertainty, for many reasons. The annual load durationdiagrams of consumers load demands can be only roughly pre-dicted. The same is the case with the failure transition rates ofnetwork elements and associated renewal duration which areimportant for reliability evaluation [1]. As these input data sub-stantially affect the results of network analysis, it is important tohave some idea how uncertain are the results obtained if someof the inputs can be only roughly assessed. An adequate tool forincorporating the uncertainties in distribution network studiesand for assessing the grade of acceptability of the results ob-tained can provide the fuzzy algebra. Some applications in re-liability analysis of power systems using fuzzy arithmetic andfuzzy logic have been suggested in the past [2][4]. Fuzzy tech-nique was also successfully used in consumer demand predic-tion [5], [6]. This paper proposes a method for quantifying theuncertainties of the input and output data in a distribution systemperformance analysis. The method models the input and outputquantities in distribution network analysis as fuzzy variables.The uncertainty grade for a fuzzy variable is measured by theinterval encompassing its most credible values. This interval isobtained by weighting the possible intervals of the values of the

Manuscript received November 22, 1999.J. Nahman is with the University of Belgrade, Belgrade 11000, Yugoslavia.D. Peric is with the School of Electrical Engineering, Belgrade 11000, Yu-

goslavia.Digital Object Identifier 10.1109/TPWRD.2003.813868

Fig. 1. Characteristic function of a FN.

variable by the uncertainty levels characterizing these intervals.Such a quantification of the uncertainties makes it possible tojudge the credibility of the results of network analysis basedupon uncertain inputs. If the uncertainty grade of a result ob-tained is too high, some of the most uncertain and most effectiveinputs should be reconsidered for a more precise quantification.

II. MATHEMATICAL MODEL

A. Engineering Interpretation of Fuzzy AlgebraConsider a variable which values are not known with cer-

tainty. This variable may be modeled as a normalized unimodalfuzzy number (FN) as depicted in Fig. 1 [7]. FN models ofguessed quantities are further on denoted by capital letters.

Parameter ( cut) is introduced that may beinterpreted as the level of uncertainty of the guess made at .To each , an interval of possible values is attached withlower bound and upper bound . For increasing , thesebounds become closer to one another tending to a single valueas approaches to 1. This value is the kernel of , denoted .The uncertainty of is the highest of all as the presumptionthat has exactly a specified value must be taken with the leastconfidence. If is modeled by a triangular FN, then this FN iscompletely defined by the triple .

The uncertainty grade of , for , may be determinedas

(1)

0885-8977/03$17.00 2003 IEEE

NAHMAN AND PERIC: DISTRIBUTION SYSTEM PERFORMANCE EVALUATION ACCOUNTING FOR DATA UNCERTAINTY 695

As can be seen, the integral in (1) gives the interval of variablevalues obtained by weighting all possible intervals of these

values by associated uncertainty levels. This interval may beconsidered as a most reasonable prediction of variable valuesconcerning its uncertainty. Interval bounds are given in relativeterms with the kernel of being the base value.

The variable lower bound relative declination of maybe determined as

(2)

The corresponding declination of upper bound equals

(3)

It is clear that

(4)On the basis of the former definitions, the following bounds

for variable values may be taken as most reasonable for engi-neering decisions made under uncertainty

(5)

B. Calculation FlowPresume that is a function of inputs , . If

inputs are modeled as FNs, to encompass their uncertainty, thenis also a FN which may be formally expressed as

(6)To define , its characteristic function has to be con-

structed. In order to determine this function, a series of valuesis generated from the whole interval (0,1). For each , the lowerand upper bounds of are determined as

(7)for

(8)Bounds and for all define .

As can be observed from (7), equals the minimum offunction if values are within intervals (8). These intervalsare determined by the lower and upper bonds of for forwhich the calculation is performed. equals the maximumof function obtained for the same intervals of values.

The calculation of and is trivial if is a mono-tonic increasing or decreasing function with regard to all argu-ments being within the intervals in (8). In the first case,is obtained from for and for ,

. In the latter case, should be inserted toobtain and to obtain . As will be shown here-after, all distribution network quantities of interest are mono-tonic increasing functions of their arguments which makes theanalysis easy and straightforward.

Fig. 2. Sample distribution network feeder.

III. NETWORK MODEL

A. Maximum Feeder Branch CurrentsConsider a branching feeder of a radial distribution network

presented in Fig. 2. It is supposed that the feeder consists ofbranches characterized by their length and the consumer de-

mand load at their receiving end. The branches can be separatedfrom each other by opening the disconnectors at their sendingends that are not marked for simplicity.

To determine the maximum feeder branch currents and max-imum voltage drops along the feeder, the peak rms magnitudesof load demand currents are used as input variables. They arepresumed to be fuzzy variables quantified by triangular fuzzynumbers. The following relationships correlate these currentswith currents flowing through feeder branches [8], [9] (here andfurther on fuzzy variables are designated by capital letters)

(9)

where , are by 1 column vectors of real andimaginary parts of branch currents while , arecorresponding column vectors of load demand peak currents.


The elements of the by Boolean matrix are

if branch supplies branchotherwise. (10)

It is implied that .Equations (9) and (10) simply state that the current flowing

through a feeder branch is equal to the sum of the load demandcurrents supplied by this branch.

For illustration, the first 5 rows of for the feeder in Fig. 2are (see equation at the bottom of the page).

From (9), it follows that the maximum branch currents alsoare fuzzy quantities. The rms of the current in branch equals

(11)where is a 1 by row vector built of the row elementsof .

B. Maximum Voltage DropsLet us denote by , the diagonal by matrix of the lengths

of feeder branches. Then, the voltage drops at load points are

(12)where

(13)It is assumed that the feeder branches have the same impedanceper unit length . From (12), it is clear that the elements of by1 column vector are fuzzy quantities as they dependon .

Bearing in mind that

(14)with and being the real and imaginary parts of , from (12)and (14), it follows:

(15)where

(16)The rms values of voltage drops at feeder load points are

(17)with and being elements of vectors

and , respectively. As known, the voltage

Fig. 3. Branch k load duration diagram.

magnitudes are dominantly determined by valueswhile drops affect the voltage phase shifts that arevery small usually. Therefore

(18)which means, according to (16), that voltage drops are practi-cally monotonic increasing functions of load point demand cur-rents. This facilitates the calculation of their bounds for variousuncertainty levels .

C. Load Demand Duration Diagram and Energy LossesThe load duration diagram can be represented by several real

current load levels of given duration, spread in descending order.The load levels are treated as uncertain data and quantified bycorresponding triangular FNs. In this application, three loadlevels are presumed to be sufficiently representative: maximum,medium, and minimum load level (Fig. 3). The load level dura-tion and the power factor are presumed to be the same for allconsumer loads. Consumer loads have been considered to be ofthe same type and synchronous which means that all consumershave the characteristic load levels approximately at the sametime. However, the load levels and their uncertainty grades forvarious consumers generally differ from consumer to consumer.

Based upon the assumptions made, the annual energy lossesequal

(19)It is clear from (19), that the energy losses are monotonic in-creasing functions of load demand current maximum, medium,and minimum values represented as fuzzy variables. It makes itpossible to easily determine bounds for various .


TABLE ILOAD DEMAND CURRENTS

D. Feeder ReliabilityIf there is no back feed available, as presumed, each failure

of a branch, say , interrupts the supply to all branches andassociated loads fed by this branch during its repair. Hence, theexpected energy not supplied due to branch failures is

(20)It is assumed that the feeder failure rate per unit length , sus-tained interruption duration , and the average real power de-mands of customers are fuzzy variables as all of these quan-tities cannot be predicted with certainty.

Here, with reference to Fig. 3

(21)with being the rated network voltage and year.

The total expected energy not supplied annually equals

(22)

The second term in (22) encompasses the energy not suppliedduring the fault location. is the uncertain duration of thisactivity.

The system average interruption frequency index is definedas the ratio of the total number of customer interruptions andthe total number of customers served

(23)

where is the number of customers served at load point .The system average interruption duration index is defined as

the ratio of the sum of customer interruption durations and thetotal number of customers

(24)

with being by 1 column vector of numbers of customersserved at feeder load points.

As , , and are functions of fuzzy vari-ables, these indices are fuzzy variables too. We can observefrom (20), (21), and (22) that is a monotony increasingfunction of consumer currents, failure transition rate per unitfeeder length, and repair duration. and are pro-

TABLE IIBRANCH LENGTHS

TABLE IIINUMBER OF CUSTOMERS

portional to the failure transition rate per unit feeder length.is also a linearly increasing function of fault repair and

location duration. Thus, lower and upper bounds for all of thesethree performance indices can be easily obtained for each uncer-tainty level from the lower and upper bounds of the associatedaforementioned input data for the same .

IV. APPLICATION EXAMPLE

A. Feeder DataA real life example of a 10-kV overhead feeder is considered,

that is displayed in Fig. 2. The main feeder data are given inTables IIV. The data are acquired from experience and limitedmeasurement. Triangular fuzzy numbers quantifying the inputquantities are given in the general form . isthe guess used in the conventional analysis.

B. Calculation ProcedureThe calculation procedure comprises the following steps.

1) Using (10) and (13), determine matrices and forthe feeder under consideration.


Fig. 4. Characteristic functions of calculated performance indices.

TABLE IVOTHER SAMPLE DATA

2) Determine lower bounds of all fuzzy input data forfrom their membership functions.

3) By inserting the values from the previous step into (11)and (18), determine the lower bounds of branch peakcurrents and voltage drops for .

4) Using (9) and data from step 2, determine the lowerbounds of branch currents , , andfor and insert these values in (19) to calculate thelower bound of annual energy losses for .

5) By inserting the lower bounds of , , andfor in (21), the lower bound of elements

of power demands for are obtained.


TABLE VCALCULATION RESULTS

6) By inserting values from the previous step and lowerbounds of , , and for in (20) and (22), lowerbound of is obtained for .

7) By inserting lower bounds of , , and forin (23) and (24), the corresponding lower bounds of

and are calculated.8) Determine upper bounds of all fuzzy input data for

from their membership functions.9) Using the data from step 8 repeat steps 37 with upper

bounds to determine the upper bounds of all consideredperformance indices for .

10) Increase for, say , and repeat steps 29for this increased value of to determine lower andupper bounds of all considered performance indices forthe new value of .

11) Repeat step 10 by gradually increasing values untilvalue is reached. In such a way, the data for theconstruction of membership functions of all consideredperformance indices are obtained.

12) Using (1) and (5) and the data on membership func-tions obtained in previous steps, determine the uncer-tainty grades and reasonable bounds of considered per-formance indices.

C. Calculation ResultsTable V quotes the results of the analysis. The effects of un-

certain input data upon the uncertainty of the quantities charac-terizing the distribution system performances are examined. Thefirst column lists the analyzed outputs. The next three columnsgive the calculated kernels (these coincide with the values ob-tainable using the conventional, crisp approach), and the asso-ciated certainty weighted percentage bounds of declination ofkernels. The last column indicates which input quantity is takento be uncertain and modeled as the corresponding fuzzy numberaccording to Tables IIV. Characteristic functions calculated forthe analyzed outputs are presented in Fig. 4.

The maximum load current that is flowing through branch 1and the maximum voltage drop occurring at node 43 depend onthe load peak currents only. The uncertainty of both of these

quantities is comparatively low with UG being less than 30%.However, the annual energy losses, which depend on the loadduration diagrams of consumers, characterized by three loadlevels for each load point, are more considerably affected bythe fuzziness of input data. The uncertainty of the ishigh if all input data are fuzzy. This particularly holds for theupper bound of , indicating that might be muchgreater than its kernel value which would be taken as relevantin the conventional approach. For illustration, from (5) and thecorresponding data in Table V, we obtain the following boundsof if all inputs are uncertain

MWh MWh (25)

The fuzziness of is closely related to this of the failuretransition rate and may be taken as comparatively low. How-ever, the fuzziness of is more pronounced as it is addi-tionally affected by fault location and repair times that we usu-ally cannot precisely predict. It is worth noting that upper bounddeclinations are high in absolute terms and much greaterthan for the majority of the analyzed outputs. This meansthat, in reality, the crisp valuesbased analysis might underes-timate the examined performance indices and lead to too opti-mistic conclusions. The uncertainty of the calculated outputs istolerable if only one among the inputs is fuzzy to some extent.The fuzziness of two or more inputs produces a high valuethat is greater than the sum of the declinations generated by theinputs individually.

V. CONCLUSIONS

A method for assessing and incorporating the effects of uncer-tain data upon the performance analysis of distribution systemsis proposed. Criteria for determining the uncertainty of both theinput and output quantities are suggested as well as a reason-able way to determine the bounds of calculated performance in-dices reflecting the uncertainties of input data. The applicationof the suggested approach to a real life example has shown thatthe is the performance index most affected by the un-certainties of input data and should not be taken as a relevantindex for any decision making unless the associated inputs arewell known. Second most input data sensitive parameters are

and annual energy losses but to a considerably lowerdegree. Maximum feeder currents and voltage drops and the

index are moderately fuzzy.

REFERENCES[1] R. E. Brown and J. R. Ochoa, Distribution system reliability: Default

data and model validation, IEEE Trans. Power Syst., vol. 13, pp.704709, May 1998.

[2] V. Miranda, Fuzzy reliability analysis of power systems, in Proc. 12thPower Syst. Comput. Conf., Dresden, Aug. 1923, 1996, pp. 558566.

[3] J. Backes, H. J. Koglin, and L. Klein, A flexible tool for planning trans-mission and distribution networks with special regard to uncertain relia-bility criteria, in Proc. 12th Power Syst. Comput. Conf., Dresden, Aug.1923, 1996, pp. 567573.

[4] J. Nahman, Fuzzy logic based network reliability evaluation, Micro-electron. Reliab., vol. 37, no. 8, pp. 11611164, 1997.

[5] M. Chow, J. Zhu, and H. Tram, Application of fuzzy multi-objectivedecision making in spatial load forecasting, IEEE Trans. Power Syst.,vol. 13, pp. 11851190, Aug. 1998.


[6] Sprinivasan, C. S. Chang, and A. C. Liew, Demand forecasting usingfuzzy neural computation, with special emphasis on weekend and publicholiday forecasting, IEEE Trans. Power Syst., vol. 10, pp. 18971903,Nov. 1995.

[7] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic. Englewood Cliffs,NJ: Prentice-Hall, 1995.

[8] M. Papadopoulos, N. D. Hatziargyriou, and M. E. Papadakis, Graphicaided interactive analysis of radial distribution networks, IEEE Trans.Power Delivery, vol. 2, pp. 12971302, Oct. 1987.

[9] J. Burke, Power Distribution Engineering. New York: Marcel Dekker,1994, p. 96.

Jovan Nahman was born in Belgrade, Yugoslavia. He received the Dipl. Eng.Grade and the TechD degree in electric power engineering from the Faculty ofElectrical Engineering at the University in Belgrade, Yugoslavia, in 1960 and1969, respectively.

Currently, he is a Professor with the Power System Department at the Uni-versity of Belgrade, where he has been since 1960.

Dragoslav Peric was born in Raca, Yugoslavia, in 1958. He received theDipl.Eng., M.Sc., and PF.D. degrees in power engineering from the Facultyof Electrical Engineering at the University in Belgrade, Yugoslavia, in 1983,1989, and 1997, respectively.

Currently, he is a Professor with the School of Electrical Engineering, Bel-grade, Yugoslavia. His main fields of interest are distribution systems operationand planning and computer applications.

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2003 PD Fuzzy Distribución Incertidumbre