Interval Estimation for Conditional Failure Rates of Transmission Lines with Limited Samples

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by  Ming Yang

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Interval Estimation for Conditional Failure Rates of
Transmission Lines with Limited Samples
Ming Yang, Member, IEEE, Jianhui Wang, Senior Member, IEEE, Haoran Diao, Student Member, IEEE,
Junjian Qi, Member, IEEE, and Xueshan Han
Abstract—Theestimationoftheconditionalfailurerate(CFR) inant influential factors of the CFR include the aging of the
6
1 of an overhead transmission line (OTL) is essential for power OTL, loading level and external environmentalconditions.
system operational reliability assessment. It is hard to predict
0 Over the last few decades, substantial work has been done
the CFR precisely, although great efforts have been made to
2 on the conditional failure rate (CFR) estimation with respect
improve theestimation accuracy. Onesignificantdifficultyisthe

v lack of available outage samples, due to which the law of large to various operational conditions. In [8], the time-varying
o numbers is no longer applicable and no convincing statistical transformer failure probability is investigated, and a delayed
N result can be obtained. To address this problem, in this paper semi-Markov process based estimation approach is proposed.
a novel impreciseprobabilisticapproach isproposed to estimate In [10], the weather conditionsare dividedinto three classes,
3 the CFR of an OTL. The imprecise Dirichlet model (IDM) is
applied to establish the imprecise probabilistic relation between i.e., normal, adverse and major adverse, to count the failures
] an operational condition and the OTL failure. Then a credal of different weather conditions. The influences of multiple
P
network is constructed to integrate the IDM estimation results weather regions on the failure rate and repair rate of a single
A
corresponding to various operational conditions and infer the transmission line are modeled in [11]. In [12], the Poisson
t. CFRoftheOTL.Insteadofprovidingasingle-valuedestimation regression and Bayesian network are applied to estimate the
a result,theproposedapproachpredictsthepossibleintervalofthe
conditional failure rates of distribution lines. Weather-related
t CFR in order to explicitly indicate the uncertainty of the esti-
s mation and more objectively represent the available knowledge. fuzzy models of failure rate, repair time and unavailability
[
The proposed approach is illustrated by estimating the CFRs of OTLs are established in [13]. The failure rate under both

2 of two LGJ-300 transmission lines located in the same region, normal and adverse weather conditions is expressed by a
v and it is also compared with the existing approaches by using
fuzzy number in the paper. It is pointed out in [14] that
4 datageneratedfromavirtualOTL.Testresultsindicatethatthe
systems with aged components might experience higher than
4 proposedapproachcanobtainmuchtighterandmorereasonable
3 CFR intervals compared with the contrast approaches. averageincidenceoffailures.Byusingthedatacollectedfrom
6 Electricity of France (EDF), the effects of aging and weather
Index Terms—Credal network, failure rate estimation, impre-
0 conditions on transmission line failure rates are analyzed
ciseDirichletmodel,impreciseprobability,overheadtransmission
1. line, reliability. in [15], where the influences of wind speed, temperature,
0 humidity, and lightning intensity are examined. In [16], the
6 time-varyingfailureratesarediscussedconsideringtheeffects
1 I. INTRODUCTION of adverse weather and componentaging, and data collection
:
v ESTIMATIONofthefailureratesofoverheadtransmission efforts for non-constant failure rate estimation are suggested.
Xi lines (OTLs) is crucial for power system reliability as- Although great efforts have been made to improve the
sessment,maintenanceschedulingandoperationalriskcontrol CFR estimation accuracy, it is still difficult if not impossible
r
a [1]–[6]. The failure rates of OTLs are usually assumed to to get convincing CFR estimation results. The major barrier
be constant and are estimated using the long-term mean is the lack of historical outage observations under relevant
values.However,inrealitythefailureratescanbesignificantly operational conditions. In this situation, the law of large
influencedbybothexternalandinternaloperationalconditions numbers is no longer applicable and a precise estimation of
[7]. The constantfailure rate may work well for the relatively thefailureratecannotbeobtained.Thisdata-deficientproblem
long-termapplications,butcanleadtoerroneousresultswhen might be trivial for the constant failure rate estimation, but is
applied to the operational risk analysis [8], [9]. The predom- crucial for the condition-related failure rate prediction.
Infact,theuncertaintyoffailurerateestimationwithlimited
ThisworkissupportedinpartbytheNationalScienceFoundationofChina samples has already been recognized. In [9], an operational
under Grant 51007047 and Grant 51477091, the National Basic Research risk assessment approach based on the credibility theory is
Program of China (973 Program) under Grant 2013CB228205, and the
proposed. In the approach, the uncertainty of the conditional
Fundamental Research Funds of Shandong University. J. Wang’s work is
supported by the U.S. Department of Energy (DOE)’s Office of Electricity failureprobabilityismodeledbyafuzzymembershipfunction.
Delivery andEnergyReliability. Meanwhile, reference [17] and [18] build a general fuzzy
M.Yang, H.DiaoandX.HanarewithKeyLaboratoryofPowerSystem
model to deal with the uncertainty of probabilities and per-
Intelligent Dispatch and Control and Collaborative Innovation Center of
GlobalEnergyInternet,ShandongUniversity,Jinan,Shandong250061China formance levels in the reliability assessment of multi-state
(e-mail: [email protected];[email protected]; [email protected]). systems. However, the approaches mainly focus on the reli-
J. Wang and J. Qi are with the Energy Systems Division, Argonne
ability assessment of the whole system instead of estimating
NationalLaboratory,Argonne,IL60439USA(e-mail:[email protected];
[email protected]). the imprecise failure rate.
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In [12], the Bayesian network is selected as a more prefer- studies, the CFR interval estimated by the proposed
able approach to model the CFR of overhead distribution approach is much narrower and more reasonable than
lines, and the central limit theorem is adopted to estimate the that obtained by the contrast approaches.
confidenceinterval of the predicted CFR. However,as is well The rest of this paper is organized as follows: In Section
known, the sample size should be large when the theorem II a binomial failure rate estimation model is introduced.
is applicable [19], which may restrict the utilization of the Section III discusses the mathematical foundations of the
approach in practice. The central confidence interval of the proposedapproach.DetailsoftheCFRestimationareprovided
mean of a Poisson distribution can be equivalently expressed in Section IV. Case studies are presented in Section V and
by a Chi-square distribution[20]. Therefore,in [13], the OTL conclusions are drawn in Section VI.
failure is assumed to follow a Poisson distribution, and then
the confidence interval of CFR is estimated according to the
II. MODELOF FAILURE RATEESTIMATION
relationshipbetween the Chi-square distributionand the Pois-
son distribution. The approach avoids using the central limit A. Failure Rate Basics
theorem in the data-insufficiency situation, which increases Failure rate is a widely applied reliability index that rep-
the practicability of the approach.However,in the paper only resents the frequency of componentfailures [24]. Mathemati-
the normal and adverse weather conditions are considered. cally, the failure rate at time t can be expressed as
Meanwhile, as will be illustrated in Section V, the Chi-square
Pr t<T ≤t+∆t|T >t
distribution based approach may obtain unreasonable CFR f f
λ(t)= lim , (1)
interval when the sample size is small. ∆t→0 (cid:8) ∆t (cid:9)
Imprecise Dirichlet model (IDM) is an efficient approach where λ(t) is the failure rate at time t, Pr{A|B} is the
for the interval-valued probability estimation [21], [22]. By conditional probability of event A given event B, T is the
f
using a set of prior probabilities, it can objectively estimate failure time, ∆t is a small time interval, and Pr{t < T ≤
f
the possible probability interval of a random event. In [23], t+∆t|T > t} is the probability that the OTL fails in the
f
basedonIDM,aninterval-valuedreliabilityanalysisapproach time interval (t,t+∆t].
is proposed for multi-state systems. Although this approach As expressed in Eq. (1), the failure rate indicates an in-
illustrates the usefulness of the interval-valued failure rate, it stantaneous probability that a normally operating component
ignores the effects of the operational conditions. It has to be breaks down at time t. In practice, the failure rate is usually
admittedthatthedata-insufficiencyproblemofCFRestimation approximatedbytheaveragefailureprobabilitywithinasmall
has not been well addressed. timeinterval(t,t+∆t].Thisapproximationwillhavesufficient
In this paper, a novel approach based on the imprecise accuracy when ∆t is so small that the failure rate can be
probability theory is proposed to estimate the CFR of an considered as constant within the time interval. In this paper,
OTL. Instead of providing a single-valued CFR estimation hourly OTL failure rates are investigated and thus ∆t is
result, the proposed approach predicts the possible interval correspondingly set to 1 hour. Since the time scale is very
of the CFR with limited historical observations, and thus can smallcomparedwiththewholelifeofOTLs,thehourlyfailure
reflect the available estimation information more objectively. rate of an OTL can be approximately represented by
The IDM is adopted to estimate the imprecise probabilistic
relationbetweentheOTLfailureandeachkindofoperational λh(t)≈Pr t<Tf ≤t+1|Tf >t , (2)
condition. Then a credal network that can perform imprecise (cid:8) (cid:9)
where λ (t) is the failure rate in the tth hour.
probabilistic reasoning is established to integrate the IDM h
estimation results corresponding to different operational con-
B. Binomial Model for the Failure Rate Estimation
ditionsandobtaintheinterval-valuedCFR. Theadvantagesof
The hourly failure rate of an OTL can be predicted by
the proposed approach include the following:
estimating the parameter of a binomialdistribution. There are
1) IDM is an efficient statistical approach for drawing out
twopossibleoutcomesofthebinomialdistribution.Oneisthat
imprecise probabilities from available data. By using
the OTL maintains normal operation and the other is that it
IDM, the uncertainty of the probabilistic dependency
fails in the relevant hour, as shown in Fig. 1, where P and
relation between an operational condition and the OTL 1
P are two parameters that indicate the possibilities of the
failure can be properly reflected by the width of the 2
outcomes. According to the definition of the failure rate and
probability interval.
aforementionedassumptions,P andP areequalto1−λ (t)
2) The credal network is a flexible probabilistic inference 1 2 h
andλ (t), respectively.So, the hourlyfailure rate of the OTL
approach.Variousoperationalconditionsthathaveeither h
can be directly obtained from the estimation result of P .
precise or imprecise probabilistic relations with the 2
OTL failure can be simultaneously considered in the
network. Moreover, the occurrence probabilities of the
operational conditions can also be represented in the
network conveniently.
3) BycombiningIDMandthecredalnetwork,theproposed
approach can explicitly model the uncertainty of the
estimated CFR. Meanwhile, as illustrated by the case Fig.1. Diagramofthebinomial distribution model.
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Thispaperwillshowhowto predictthepossibleintervalof By analyzing the estimation results of the determinis-
P under given operational conditions with respect to limited tic Dirichlet model, it can be found that P will be
2 m
historicalobservations.ThisaimisachievedbyusingIDMand a / M a if no available observations exist. This value
m m=1 m
the credalnetwork, which will be introducedin the following isthePpriorestimationoftheoccurrencepossibilityofthemth
section.Itshouldbeemphasizedthattheproposedapproachis outcome. Here the parameter a is called the prior weight of
m
also suitable for the multi-state OTL reliability model. In that theoutcome,and M a isusuallydenotedbys, whichis
m=1 m
case the multinomialdistribution shouldbe applied instead of called the equivalePntsample size. In the parameter estimation
the binomial distribution used in this paper. process, s implies the influence of the prior on the posterior.
The larger s is, the more observations are needed to tune the
III. MATHEMATICALFOUNDATIONS parameters assigned by the prior distribution.
A. Imprecise Probability Equation (5) provides a feasible approach for the multino-
Imprecise probability theory is a generalization of classical mial parameter estimation. However, in practice the available
probability theory allowing partial probability specifications observations may be scarce (like the OTL outage obser-
when the available information is insufficient. The theory vations). In this case, the prior weights will significantly
bloomed in the 1990s owning to the comprehensive founda- influence the parameter estimation results, which causes the
tions put forward by Walley [21]. results too subjective to be useful.
In an imprecise probability model, the uncertainty of each To avoid the shortcomings of the deterministic Dirichlet
outcome is represented by an interval-valued probability. For model, IDM uses a set of prior density functions instead of a
example, the imprecise probabilities of the two outcomes in single density function [22], which can be described as
Fig. 1 can be expressed by P =[P ,P ] and P =[P ,P ]
1 1 1 2 2 2 M
s1a−tisPfy1in,gan0d≤P2P=11≤−PP11.≤e1, 0 ≤ P2 ≤ P2e≤ 1, P2 = f(P1,··· ,PM)= Mm=1ΓΓ(s()s·rm)mY=1Pms·rm−1,
When there is no estimation information at all, the oc-
Q M
currence possibilities of the outcomes will have maximal
∀r ∈[0,1], r =1, (6)
m m
probability intervals, i.e., P =P =0 and P =P =1. If
1 2 1 2 mX=1
sufficient estimation information is available, the probability
where r , m = 1,2,...,M, is the mth prior weight factor
intervalmay shrink to a single pointand a precise probability m
and s is the equivalent sample size.
will be obtained [25].
InEq.(6),s·r playsthesameroleasa inEq.(3).When
m m
B. Imprecise Dirichlet Model rm varies in the interval [0,1], the set will contain all of the
possible priors given a predetermined s, and the prejudice of
IDM is an extension of the deterministic Dirichlet model
the prior can thus be avoided.
[26]. Consider a multinomial distribution which has M types
Then the corresponding posterior density function set can
of outcomes. To estimate the occurrence probabilities of the
be calculated according to Bayesian rules, as
outcomes, the deterministic Dirichlet model uses a Dirichlet
distribution as the prior distribution, which is M
Γ(s+n)
Γ M a M f(P1,··· ,PM)= M Pms·rm+nm−1,
f(P1,··· ,PM)= (cid:16)MmP=m1=Γ1(amm)(cid:17)mY=1Pmam−1, (3) Qm=1Γ(s·rmM+nm)mY=1
∀r ∈[0,1], r =1, (7)
where P ,P ,··· ,P areQthe probabilities of the outcomes, m m
1 2 M mX=1
a ,a ,··· ,a are the positive parameters of the Dirichlet
1 2 M
distribution and Γ is the gamma function. where n= M n is the number of total observations.
m=1 m
Since the Dirichlet distribution is a conjugate prior of So,theimPpreciseparametersofthemultinomialdistribution
the multinomial distribution, the posterior of P1,P2,··· ,PM can be estimated from the posterior density function set as
with respect to the observations also belongs to a Dirichlet
n n +s
distribution, which can be represented as m m
P =[P ,P ]= , , m=1,2,...,M.
m m m (cid:20)n+s n+s (cid:21)
f(P ,··· ,P )
1 M e (8)
Γ M a + M n M
= (cid:16)PmMm==11Γm(amP+mn=m1) m(cid:17)mY=1Pmam+nm−1, (4) calTcuhleabteoduwnditshorfetshpeecptrotobathbeilibtoyuinndtesrvoaflrsmsh,omwn=in1,E2q,..(.8.),Mare,
and more theoretical details can be found in [21].
where n , mQ=1,2,...,M, is the number of times that the
m The interval estimated by IDM intends to include all the
mth outcome is observed.
possible probabilities corresponding to different priors. How-
Therefore, the parameters of the multinomial distribution
ever, in some applications, the probability interval with a
can be estimated by the expected values of the posterior
quantitative confidence index may be preferred. In that case,
distribution, as
the probabilityintervalcan be estimated by usingthe credible
n +a
P = m m , m=1,2,...,M. (5) intervalcorrespondingto the IDM [27], which is presentedin
m Mm=1nm+ Mm=1am Appendix A.
P P
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In fact, several other approaches can also be applied to H =h can be calculated as
obtain the probability interval, such as the central limit theo-
P(h,g,q)
rembasedapproach[12]andtheChi-squaredistributionbased P(h|g,q)=
P(g,q)
approach[13]. The performanceof IDM comparedwith these
P(h)P(g|h)P(q|g,h)
existing approaches is illustrated in Appendix B. =
P(h)P(g|h)P(q|g,h)+P(hc)P(g|hc)P(q|g,hc)
Moreover, in Eq. (8), the equivalent sample size s is the
0.4×0.3×0.5
only exogenous parameter to be specified. It is set to 1 in = ≈0.45.
0.4×0.3×0.5+0.6×0.2×0.6
our model as suggested by [25]. The effects of s on the IDM
estimation result are discussed in Appendix C. The credal network relaxes the Bayesian network by ac-
cepting imprecise probabilistic representations[30], [31]. The
differences between the credal network and the Bayesian
C. Credal Network network lie in the following three levels:
1) The probability corresponding to each outcome: In a
ThecredalnetworkisdevelopedfromtheBayesiannetwork Bayesiannetwork,eachoutcomeofacategoricalvariablehasa
whichisapopulargraphicalmodelrepresentingtheprobabilis- single-valuedprobability.Onthecontrary,inacredalnetwork,
tic relations among the variables of interest [28], [29]. theoccurrencechanceforeachoutcomecanberepresentedby
In a Bayesian network, each node stands for a categorical an interval-valued probability.
variable, while the arcs indicate the dependency relations 2) The mass function of each categorical variable: A
among the variables. A Bayesian network is composed of a categorical variable described by imprecise probabilities has
set of conditional mass functions P(X |π ), X ∈ X and a set of mass functions instead of just one. The convex hull
i i i
πi ∈ ΩΠi, where X stands for the categorical variables of of the mass function set is defined as a credal set [31], [32],
the network, X is the ith categorical variable, Π stands for which can be expressed as
i i
πthieipsaarennotbvsaerrivabatlieosnooffXΠi,i.ΩΠi is the value space of Πi, and K(Xi)=CHnP(Xi):for each xi ∈ΩXi,
A simple Bayesian network is shown in Fig. 2, in which P(x )∈ P(x ),P(x ) and P(x )=1 , (10)
i i i i
the uppercase characters stand for the two-state categorical (cid:2) (cid:3) XΩXi o
variables while the lowercase characters stand for the states whereK(X )isthecredalsetofX ,CHmeansaconvexhull,
i i
of the variables. All necessary conditional mass functions for {P(X ): Descriptions} is a set of probability mass functions
i
the inference have been provided in the figure. specifiedbythedescriptions,Ω isthevaluespaceofX ,and
Xi i
A Bayesian network is identical with a joint mass function P(x ) = 1 means the sum of the probabilities should
over X which can be factorized as P(x) = Ii=1P(xi|πi) bPeΩeXqiual toi1.
for each x ∈ ΩX, where I is the numberQof categorical Althoughthecredalsetcontainsaninfinitenumberofmass
variables in the network, xi is an observation of Xi, x is an functions, it only has a finite number of extreme mass func-
observationofX,andΩX isthevaluespaceofX.Therefore, tions,whicharedenotedbyEXT[K(Xi)].Theseextrememass
with given evidence XE = xE, the conditional probability functions correspond to the vertexes of the convex hull and
of an outcome of the queried variable Xq can be estimated canbeobtainedbycombiningtheendpointsoftheprobability
according to Bayesian rules as intervals. For instance, assume P(h) and P(hc) in Fig. 2 are
imprecise,i.e.,P(h)∈[0.3,0.5]andP(hc)∈[0.5,0.7].Inthis
P(xq|xE)= PP(x(qx,Ex)E) = PXXMM,XQq Ii=Ii1=P1P(x(ix|πi|πi)i), (9) caevanseder,,PtPh(he(Hcm))assaostnifsluyfynihcntaigsontthwPeo(cHeoxn)tsrcteramaninebtemPaa(nshys)cf+uonmPcbt(iiohnnca)sti,o=wn1ho.ifcHPho(awhre)-
P Q {P(h)=0.3,P(hc)=0.7} and {P(h)=0.5,P(hc)=0.5}.
where XM ≡X\(XE ∪Xq) and X means calculating The inference over a credal network is based on the con-
the total probability with respect to XPMM[29]. ditional credal sets K(Xi|πi), i = 1,2,...,I. And it is
For instance, consider the network shown in Fig. 2. With equivalent to the inference based only on the extreme mass
evidence G = g and Q = q, the conditional probability of functions of the conditional credal sets.
3) The joint mass function of the network: While a
Bayesian network defines a joint mass function, a credal
network defines a joint credal set that contains a collection
of joint mass functions. The convex hull of these joint mass
functions is called a strong extension of the credal network
[33], and can be formulated as
K(X)=CH P(X):for each x∈ΩX,
n
I
P(x)= P(x |π ), P(X |π )∈K(X |π ) , (11)
i i i i i i
Fig.2. Diagram ofasimpleBayesian network. iY=1 o
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 5
where K(X) is the strong extension of the credal network, 4) Find the maximum and minimum probabilities of
{P(X):Descriptions}isasetofjointmassfunctionsspecified P(X = x |X = x ) with respect to the inference
q q E E
by the descriptions, and P(X |π )∈K(X |π ) indicates that results of all the extreme joint mass functions.
i i i i
the conditional mass function P(X |π ) should be selected
i i
from the conditional credal set K(X |π ). IV. DETAILSOF IMPRECISE CFR ESTIMATION
i i
ObservefromEq.(11)thatthejointprobabilityP(x)isim- A. Imprecise CFR Estimation Model
precise since each P(xi|πi), whose mass function P(Xi|πi) The failure rate of an OTL has a close relationship with
can be arbitrarily selected from the credal set K(X |π ),
i i surrounding weather conditions. An OTL is more likely to
is imprecise. For this reason, K(X) contains a set of joint break down in adverse weather conditions, e.g., high air
mass functions. On the other hand, when the mass functions
temperature, strong wind, snow/ice, rain, and lightning. The
P(X |π ), i = 1,2,...,I, are selected, the joint probability
i i loading condition can also influence the failure rate of an
P(x) for each x ∈ ΩX will be determined, and hence one OTL. When carrying heavy load, an OTL may suffer from
deterministic joint mass function P(X) will be obtained. It
a higher probability of a short-circuit fault [34]. Moreover,
is easy to imagine that each joint mass function of a strong
the failure rate of an OTL may change gradually with the
extension corresponds to a Bayesian network.
age of the line. An aged transmission line usually has a
If P(X |π ) can only be selected from the extreme mass
i i relatively higher failure rate under the same weather and
functions of K(X |π ), the resulted joint mass functions are
i i loadingconditions.Takingthese factorsinto account,a credal
called the extreme joint mass functions, and they correspond
network is established for estimating the CFR of an OTL, as
tothevertexesofthestrongextension.Theextremejointmass
shown in Fig. 3.
functions can be obtained by
EXT[K(X)]= P(X):for each x∈ΩX,
n
I
P(x)= P(x |π ),P(X |π )∈EXT[K(X |π )] . (12)
i i i i i i
iY=1 o
Inferenceovera credalnetworkis to computethe probabil-
ity bounds of X = x respecting X = x . According to
q q E E
the association between the credal network and the Bayesian
network, this aim can be achieved by inferring over the
Bayesian networks corresponding to the extreme joint mass
functions, as
P(X =x |X =x ) Fig.3. Credalnetworkfortheimprecise CFRestimation.
q q E E
P(Xq =xq,XE =xE) In the figure, the operational conditions are denoted by
= max ,
P(X)∈EXT[K(X)] P(XE =xE) the categorical variables E1,E2,··· ,E6, and the state of
the OTL is indicated by the binary variable H. Mean-
P(X =x |X =x )
q q E E while, the dependency relations between the OTL state and
= min P(Xq =xq,XE =xE), (13) the operational conditions are reflected by the conditional
P(X)∈EXT[K(X)] P(XE =xE) credalsetK(E1|H),K(E2|H),K(E3|H,E1),K(E4|H,E3),
where P(X) ∈ EXT[K(X)] indicates that P(X) should be K(E5|H) and K(E6|H,E1). It should be emphasized that
selected from the extreme joint mass functions of K(X). thedependencyrelationsamongtheoperationalconditionsare
It can be seen from Eq. (13) that when an extreme joint also reflected in the network.
mass function is selected, Eq. (13) will become a Bayesian The aging of the OTL is not explicitly included in the
network.Instead, the averagefailure rate of similar age OTLs
network inference problem which can be conveniently solved
is integrated into the inference model as a prior failure rate,
by Eq. (9). Therefore,inference over a credal network can be
executed by performing the following four steps: by which the aging effects on the failure rate is respected.
The states of the categorical variables are listed in Table
1) For each categorical variable X within the net-
i
I, where T is the hourly average temperature, W is the
work, calculate its extreme conditional mass function
EXT[K(X |π )] by combining the probability interval hourly average wind speed, and L is the hourly average
i i
endpoint P(x |π ) and P(x |π ) of all x ∈Ω . loadingrate.Theoretically,theconditionvalueshavingsimilar
2) Form the extiremie joint maissifunctionsiEXTX[Ki (X)] influences on the failure rate should be classified into the
same state to reduce the loss of information [12]. However,
with respect to the extreme conditional mass functions
EXT[K(Xi|πi)], i = 1,2,...,I, πi ∈ ΩΠi, according dcauteegtooritzheedraecstcroicrdtiionngotof dthaetac,ltahsesifistcaatteisonlisctreidteriina ToafbulteiliItiaerse’
to Eq. (12).
outage records.
3) Infer on each extreme joint mass function of
EXT[K(X)]withgivenevidenceX =x ,andobtain B. Estimation of the Extreme Mass Functions
E E
the conditional probability P(X = x |X = x ) Estimating the extreme mass functions of the credal
q q E E
according to Eq. (9). sets K(E |H), K(E |H), K(E |H,E ), K(E |H,E ),
1 2 3 1 4 3
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 6
TABLEI
C. Imprecise CFR Inference with the Credal Network
STATESOFTHECATEGORICALVARIABLES
With respect to the credal network mentioned above, the
Variables State1 State2 State3
T ≤4°C 4°C<T ≤26°C T >26°C boundsof the CFR underspecified operationalconditionscan
E1 (e1,1) (e1,2) (e1,3) be calculated by the equations deduced from Eqs. (9), (12)
W ≤12km/h 12km/h<W ≤40km/h T >40km/h and (13), as
E2
(e2,1) (e2,2) (e2,3) 6 P ·P(h )
E3 (Re3a,i1n) N(oe3R,2a)in N/A λ=P(h2|e)=max 2iQ=1j=16j=21,jPi,j ·P2(hi),
Lightning NoLightning
E4 N/A P 6QP ·P(h )
(e4,1) (e4,2) λ=P(h |e)=min j=1 2,j 2 ,
E5 L≤80% L>80% N/A 2 2iQ=1 6j=1Pi,j ·P(hi)
(e5,1) (e5,2) when j =1,2,5, P Q
Snow/Ice NoSnow/Ice
E6 N/A
(e6,1) (e6,2) P2,j =P(ej,kj|h2), Pi,j =P(ej,kj|hi),
NormalOperation Failure
H N/A when j =3,
(h1) (h2)
P =P(e |h ,e ), P =P(e |h ,e ),
2,j 3,k3 2 1,k1 i,j 3,k3 i 1,k1
when j =4,
K(E5|H) and K(E6|H,E1) is the first step for the CFR P2,j =P(e4,k4|h2,e3,k3), Pi,j =P(e4,k4|hi,e3,k3),
inference. To achieve this purpose, the endpoints of the con-
when j =6,
ditional probabilities corresponding to P(E |H), P(E |H),
1 2 P =P(e |h ,e ), P =P(e |h ,e ), (14)
P(E |H,E ), P(E |H,E ), P(E |H) and P(E |H,E ) 2,j 6,k6 2 1,k1 i,j 6,k6 i 1,k1
3 1 4 3 5 6 1
have to be calculated using IDM, as expressed in Eq. (8). where i is the index of OTL’s states, j is the index of the
For instance, to estimate the extreme mass functions of evidence variables, kj is used to specify the state of the jth
the conditional credal set K(E5|h2), which describes the evidence variable, P(h2) is the prior probability of h2 which
uncertaintyoftheloadingconditionwhenthetransmissionline is assigned by using the average failure rate of the OTLs
fails, the endpoints of the probability intervals corresponding within the same age group, P(h1) = 1−P(h2), P(ej,kj|hi)
to P(e5,1|h2) and P(e5,2|h2) have to be calculated, as is the conditional probability of the observation ej,kj given
h with respect to the conditional mass function P(E |h ),
i j i
P(e5,1|h2)= nNL , P(e5,1|h2)= nNL+s, P(e3,k3|hi,e1,k1) is the conditional probabilityof e3,k3 given
nF +s nF +s hiande1,k1 withrespecttoP(E3|hi,e1,k1),P(e4,k4|hi,e3,k3)
P(e5,2|h2)= nHL , P(e5,2|h2)= nHL+s, is the conditional probability of e4,k4 given hi and e3,k3
nF +s nF +s with respect to P(E4|hi,e3,k3), and P(e6,k6|hi,e1,k1) is
the conditional probability of e given h and e
where n and n are the numbers of samples for which 6,k6 i 1,k1
NL HL with respect to P(E |h ,e ). Meanwhile, the conditional
the OTL breaks down with a normal and high loading rate, 6 i 1,k1
massfunctionP(E |h ),P(E |h ,e ),P(E |h ,e )and
respectively,and n =n +n is the number of samples j i 3 i 1,k1 4 i 3,k3
F NL HL P(E |h ,e ) are selected from the extreme mass functions
for which the OTL breaks down in the relevant time interval. 6 i 1,k1
of the credal set K(E |h ), K(E |h ,e ), K(E |h ,e )
j i 3 i 1,k1 4 i 3,k3
Then the conditional credal set K(E |h ) can be obtained
5 2 and K(E |h ,e ) to optimize the objective functions.
6 i 1,k1
from Eq. (10) as
In (14), P(e |h ) is the occurrence probability of the
j,kj 2
conditione givenan OTL failure is observed.Considering
j,kj
K(E5|h2)=CH P(E5|h2): the deficiency of the failure samples, the probability should
n
be estimated using IDM. On the other hand, P(e |h ) is
P(e5,1|h2)∈[P(e5,1|h2),P(e5,1|h2)], j,kj 1
the occurrence probability of the condition given no OTL
P(e |h )∈[P(e |h ),P(e |h )],
5,2 2 5,2 2 5,2 2 failure happens. In this case, abundant samples are avail-
P(e |h )+P(e |h )=1 . able since no failure is a large probability event. And this
5,1 2 5,2 2
o probabilitycan be approximatedusing theaverageoccurrence
probability of the condition to simplify the calculation, e.g.,
Also, the extreme mass functions of the credal set can
P(e |h ) ≈ P(e ) can be approximated by the statistical
be obtained by combining the endpoints of the probability 1,3 1 1,3
probability of high temperature in the relevant region. The
intervals as
conditional probability P(e |h ,e ), P(e |h ,e )
3,k3 i 1,k1 4,k4 i 3,k3
andP(e |h ,e )canbeobtainedinthesame way.Since
EXT[K(E |h )]= 6,k6 i 1,k1
5 2
the precise probability is a special case of the imprecise
P(e5,1|h2)=P(e5,1|h2),P(e5,2|h2)=P(e5,2|h2) probability, the CFR can still be estimated by using Eq. (14)
n o
under this circumstance.
and P(e |h )=P(e |h ),P(e |h )=P(e |h ) .
5,1 2 5,1 2 5,2 2 5,2 2
n o V. CASE STUDIES
Other conditional credal sets and corresponding extreme The proposedapproachis tested by estimating the CFRs of
mass functions can be calculated in the same way. two LGJ-300 transmission lines located in the same region.
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 7
TABLEIV
One of the transmission lines, which is denoted as TL , has
1 CONDITIONALMASSFUNCTIONSOFHOURLYAVERAGETEMPERATURE
been in operation for 5 years. Its recent three-year average
hi P(e1,1|hi) P(e1,2|hi) P(e1,3|hi)
failure rate is 0.00027 times per hour. As a result, the pa-
h1 0.28 0.49 0.23
rameter P(h ) and P(h ) of TL are set to 0.99973 and
1 2 1 h2 0.30 0.17 0.53
0.00027,respectively. The other transmission line, denoted as
TL , has been in operation for 10 years, and its recent three-
2
yearaveragefailurerate is0.00042timesper hour.Therefore, Itcanbeobservedfromthetablethatwhenthetransmission
theparameterP(h1)andP(h2)ofTL2 aresetto0.99958and lineTL1 orTL2 experiencesafailure,theambienttemperature
0.00042, respectively. has a large probability of being higher than 26°C.
A total of 40 failure samples are collected from the two The conditional mass functions of other operational condi-
transmission lines during their operation. The failure times tions can be obtained in the same way, and the results are
under various operational conditions are listed in Table II. listed in Table V, VI, VII, VIII and IX.
TABLEII TABLEV
FAILURETIMESUNDERVARIOUSOPERATIONALCONDITIONS CONDITIONALMASSFUNCTIONSOFHOURLYAVERAGEWINDSPEED
Condition Times Condition Times hi P(e2,1|hi) P(e2,2|hi) P(e2,3|hi)
T ≤4°C 12 Lightning,Rain 16 h1 0.40 0.42 0.18
4°C<T ≤26°C 7 Lightning,NoRain 2 h2 0.20 0.10 0.70
T >26°C 21 NoLightning,Rain 10
W ≤12km/h 8 NoLightning,NoRain 12
12km/h<W ≤40km/h 4 L≤80% 22 TABLEVI
CONDITIONALMASSFUNCTIONSOFRAIN
T >40km/h 28 L>80% 18
Rain,T ≤4°C 5 Snow/Ice,T ≤4°C 8 hi e1,k1 P(e3,1|hi,e1,k1) P(e3,2|hi,e1,k1)
Rain,4°C<T ≤26°C 5 Snow/Ice,4°C<T ≤26°C 0 h1 e1,1 0.04 0.96
Rain,T >26°C 16 Snow/Ice,T >26°C 0 h1 e1,2 0.12 0.88
NoRain,T ≤4°C 7 NoSnow/Ice,T ≤4°C 4 h1 e1,3 0.21 0.79
NoRain,4°C<T ≤26°C 2 NoSnow/Ice,4°C<T ≤26°C 7 h2 e1,1 0.40 0.60
NoRain,T >26°C 5 NoSnow/Ice,T >26°C 21 h2 e1,2 0.69 0.31
h2 e1,3 0.75 0.25
A. CFREstimationbyDirichletModelandBayesianNetwork
TABLEVII
Deterministic CFR estimation can be performed by using CONDITIONALMASSFUNCTIONSOFLIGHTNING
the classical Bayesian network expressed by Eq. (9). In the hi e3,k3 P(e4,1|hi,e3,k3) P(e4,2|hi,e3,k3)
equation,theconditionalmassfunctionsareestimatedaccord- h1 e3,1 0.33 0.67
ing to Eq. (5) with data listed in Table II. The selected prior h2 e3,1 0.63 0.37
weights for Eq. (5) are listed in Table III. h1 e3,2 0.05 0.95
h2 e3,2 0.11 0.89
TABLEIII
PRIORWEIGHTSFORCONDITIONALMASSFUNCTIONESTIMATION
TABLEVIII
Condition PW Condition PW CONDITIONALMASSFUNCTIONSOFHOURLYAVERAGELOADINGRATE
T ≤4°C 0.3 Lightning,Rain 0.7
hi P(e5,1|hi) P(e5,2|hi)
4°C<T ≤26°C 0.1 Lightning,NoRain 0.1
h1 0.67 0.33
T >26°C 0.6 NoLightning,Rain 0.3 h2 0.55 0.45
W ≤12km/h 0.2 NoLightning,NoRain 0.9
12km/h<W ≤40km/h 0.1 L≤80% 0.4
T >40km/h 0.7 L>80% 0.6 TABLEIX
Rain,T ≤4°C 0.5 Snow/Ice,T ≤4°C 0.7 CONDITIONALMASSFUNCTIONSOFSNOW/ICE
Rain,4°C<T ≤26°C 0.7 Snow/Ice,4°C<T ≤26°C 0
hi e1,k1 P(e6,1|hi,e1,k1) P(e6,2|hi,e1,k1)
Rain,T >26°C 0.7 Snow/Ice,T >26°C 0
h1 e1,1 0.14 0.86
NoRain,T ≤4°C 0.5 NoSnow/Ice,T ≤4°C 0.3 h1 e1,2 0.01 0.99
NoRain,4°C<T ≤26°C 0.3 NoSnow/Ice,4°C<T ≤26°C 1 h1 e1,3 0 1
NoRain,T >26°C 0.3 NoSnow/Ice,T >26°C 1 h2 e1,1 0.65 0.35
h2 e1,2 0 1
OnthebasisoftheinformationprovidedbyTableIIandIII, h2 e1,3 0 1
the conditional mass function of hourly average temperature
given an OTL failure is estimated, and the result is shown in With respect to the aforementioned statistical results, the
Table IV. In the table, the conditionalmass functionunderno deterministic CFRs of the transmission lines can be estimated
failure situation is approximated by the long-term statistical according to Eq. (9). To illustrate this estimation approach,
distribution of local temperature. threescenarioswithdifferentoperationalconditionsaretested.
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 8
ScenarioI:ThetransmissionlineTL isoperatingnormally B. CFR Estimation by IDM and Credal Network
1
now.Itsaverageloadingrate will be 95%in the nexthour.At
Using the historical observations shown in Table II, the
the same time, according to the short-term weather forecast,
imprecise conditional mass functions given OTL failures can
a thunderstorm will come in the next hour. During the storm,
beobtainedbyusingIDMaccordingtoEq.(8),andtheresults
the average air temperature will be 30°C and the wind speed
are listed in Table XIII, XIV, XV, XVI, XVII and XVIII.
will be 48 km/h. Find the CFR of TL for the coming hour.
1
Scenario II: The transmission line TL is operating nor- TABLEXIII
1
mally now. In the next hour the temperature will be 15°C, IMPRECISECONDITIONALMASSFUNCTIONOFTEMPERATURE
the wind speed will be 10 km/h and the loading rate of the hi P(e1,1|hi) P(e1,2|hi) P(e1,3|hi)
transmissionline will be less than 45%.Find the CFR ofTL h2 [0.29,0.32] [0.17,0.20] [0.51,0.54]
1
for the coming hour.
Scenario III: The transmission line TL is operating nor-
2 TABLEXIV
mally now. All the operational conditions are the same as
IMPRECISECONDITIONALMASSFUNCTIONOFWINDSPEED
Scenario I. Estimate the CFR of TL under this scenario.
2 hi P(e2,1|hi) P(e2,2|hi) P(e2,3|hi)
The CFRs estimated by the Bayesian networkare shown in
h2 [0.19,0.22] [0.10,0.13] [0.68,0.71]
TableX.Itcanbeobservedfromtheestimationresultsthatthe
CFR of TL under Scenario I is much higher than that under
1
Scenario II. This is because all the operational conditions of TABLEXV
Scenario I are adverse to power transmission compared with IMPRECISECONDITIONALMASSFUNCTIONOFRAIN
that of Scenario II. On the other hand, it can be found from hi e1,k1 P(e3,1|hi,e1,k1) P(e3,2|hi,e1,k1)
the resultsofScenarioI andScenarioIII thatthe CFR ofTL2 h2 e1,1 [0.38,0.46] [0.54,0.62]
is obviously higher than that of TL1 under the same weather h2 e1,2 [0.63,0.75] [0.25,0.37]
and loading conditions. TL has a higher CFR because it is h2 e1,3 [0.73,0.77] [0.23,0.27]
2
much older than TL1.
TABLEX TABLEXVI
ESTIMATIONRESULTSOFTHEBAYESIANNETWORK IMPRECISECONDITIONALMASSFUNCTIONOFLIGHTNING
ScenarioI Scenario II ScenarioIII hi e3,k3 P(e4,1|hi,e3,k3) P(e4,2|hi,e3,k3)
2.19E−2 1.16E−5 3.37E−2 h2 e3,1 [0.59,0.63] [0.37,0.41]
h2 e3,2 [0.08,0.15] [0.85,0.92]
The estimation results of the deterministic approach may
be influenced by the prior weights dramatically. To illustrate
TABLEXVII
this, a group of new prior weights are applied for the esti-
IMPRECISECONDITIONALMASSFUNCTIONOFHOURLYAVERAGE
mation, which are listed in Table XI. The correspondingCFR LOADINGRATE
estimation results are shown in Table XII. hi P(e5,1|hi) P(e5,2|hi)
h1 [0.54,0.56] [0.44,0.46]
TABLEXI
ALTERNATIVEPRIORWEIGHTSFORTHEMASSFUNCTIONESTIMATION
Condition PW Condition PW
TABLEXVIII
T ≤4°C 0.4 Lightning,Rain 0.5
IMPRECISECONDITIONALMASSFUNCTIONOFSNOW/ICE
4°C<T ≤26°C 0.3 Lightning,NoRain 0.3
T >26°C 0.3 NoLightning,Rain 0.5 hi e6,k1 P(e6,1|hi,e1,k1) P(e6,2|hi,e1,k1)
W ≤12km/h 0.2 NoLightning,NoRain 0.7 h2 e1,1 [0.62,0.69] [0.31,0.38]
12km/h<W ≤40km/h 0.4 L≤80% 0.8 h2 e1,2 [0,0.12] [0.88,1]
T >40km/h 0.4 L>80% 0.2 h2 e1,3 [0,0.04] [0.96,1]
Rain,T ≤4°C 0.2 Snow/Ice,T ≤4°C 0.5
Rain,4°C<T ≤26°C 0.5 Snow/Ice,4°C<T ≤26°C 0
Rain,T >26°C 0.5 Snow/Ice,T >26°C 0 It is observed from the tables that the conditional proba-
NoRain,T ≤4°C 0.8 NoSnow/Ice,T ≤4°C 0.5 bilities estimated by the deterministic Dirichlet model are all
NoRain,4°C<T ≤26°C 0.5 NoSnow/Ice,4°C<T ≤26°C 1
included in the probability intervals estimated by the IDM,
NoRain,T >26°C 0.5 NoSnow/Ice,T >26°C 1
which illustrates the rationalityof the IDM estimation results.
The widths of the probability intervals estimated by the IDM
TABLEXII quantitatively reflect the uncertainty existed in the probability
ESTIMATIONRESULTSOFTHEBAYESIANNETWORK estimation results.
ScenarioI Scenario II ScenarioIII Respecting the estimation results of IDM, the imprecise
2.05E−2 1.30E−5 3.15E−2 CFRs corresponding to the three scenarios can be calculated
accordingto Eq. (14), and the results are listed in Table XIX.
By comparing the estimation results shown in Table X and It can be found that the estimation results of the Bayesian
XII, it can be found that the estimated CFRs are different network with different prior weights are all included in the
with different prior weights, which illustrates the significant probability intervals estimated by the credal network. In fact,
influence of the prior weights on the CFR estimation results. thisphenomenonisuniversal,whichverifiesthattheproposed
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 9
TABLEXIX
ESTIMATIONRESULTSOFTHECREDALNETWORK
ScenarioI Scenario II ScenarioIII
[1.85E−2,2.38E−2] [8.88E−6,1.94E−5] [2.85E−2,3.66E−2]
approach can eliminate the subjectivity caused by the prior
weightassignment.Furthermore,itcanbefoundbycomparing
theestimationresultscorrespondingtodifferentscenariosthat
the effects of the weather, loading and aging conditions on
the CFRs can also be reflected in the estimation results of the
credal network.
Sometimes, the occurrence of the operational conditions is
uncertain. For instance, it is difficult to exactly predict the air
temperature in the coming hour. The proposed approach can
handlesuchuncertaintyconvenientlybyusingthelawoftotal
probability, as illustrated by the following example.
Scenario IV: The transmission line TL is operating nor-
2
mally now. All the operational conditions are the same as in
Scenario I except that the wind speed has a 70% probability
Fig.4. CFRintervals undertheadverseoperational condition.
ofbeingfasterthan40km/handa30%probabilityofranging
from 12 km/h to 40 km/h.
Under this scenario, the CFR is estimated to be within
3) The central limit theorem based approach cannot per-
[2.05×10−2, 2.65×10−2]. Comparing this result with the
form the interval estimation until the occurrence of the
imprecise CFR estimation result of Scenario III, it can be
first failure. Meanwhile, the lower bound of the esti-
foundthattheCFRislowerunderScenarioIV.Thisisbecause
matedintervalmaybelessthan0becausetheconverted
the wind speed has a considerable probability of being slow
normal distribution is unbounded.
inScenarioIV,whichisbeneficialforthepowertransmission.
The observations illustrate that the proposed approach can
obtainmore reasonableand more accurateCFR intervalsthan
C. Comparison with Other Probability Interval Estimation
the existing approaches.
Approaches
A simulation system is set up to compare the performance
oftheproposedapproachandtwoexistingapproaches,i.e.,the VI. CONCLUSION
central limit theorem based approach [12] and the Chi-square
distributionbased approach[13]. To simplify the analysis, the
Because of the deficiency of the OTL failure samples, the
operational conditions are categorized into two groups, i.e.,
CFR estimation result may be unreliable in practice. Under
the normal operational condition and the adverse operational
thiscircumstance,anovelimpreciseprobabilisticapproachfor
condition, and the occurrence probabilities of these two op-
the CFR estimation, based on IDM and the credal network,
erational conditions are 0.7 and 0.3, respectively. Assume the
is proposed. In the approach, IDM is adopted to estimate
failure rates of the interested OTL under normal and adverse
the imprecise probabilistic dependency relations between the
operationalconditionsare0.0001and0.005,respectively.The
OTL failure and the operational conditions, and the credal
simulation system is virtually operated for 10000 hours, and
network is established to integrate the IDM estimation results
the collected failure and operational condition data are used
andinfertheimpreciseCFR. The proposedapproachistested
to estimate the CFR of the OTL.
by estimating the CFRs of two transmission lines located in
The CFR interval under the adverse operational condition
thesameregion.Thetestresultsillustratethat:a)theproposed
is estimated by using the aforementionedapproaches, and the
approachcanquantitativelyevaluatethe uncertaintyof the es-
results are shown in Fig. 4.
timatedCFRbyusingtheintervalprobability,b)theinfluences
From the figure, the following facts can be observed.
of the operational conditions on the CFR can be properly
1) The CFR intervals estimated by all the approaches can reflected in the estimation result, and c) the uncertainty of
converge to the real CFR as data accumulate. Mean- the operational conditions can also be handled by using the
while, the interval of the proposed approach converges proposed approach. Moreover, a simulation system is set up
much faster than those of the existing approaches. to demonstrate the advantages of the proposed approach over
2) When the sample size is small, the CFR interval esti- the existing approaches, i.e., the central limit theorem based
mated by the Chi-square distribution based approach is approachand the Chi-squaredistributionbasedapproach.The
very large. The upper bound of the estimated interval test results indicate that the CFR interval estimated by the
is even greater than 1, mainly because the equivalent proposed approach is much tighter and more reasonable than
Chi-square distribution is unbounded on the right side. those obtained by the existing approaches.
ACCEPTEDBYIEEETRANSACTIONSONSMARTGRID 10
APPENDIX A 2) The upper bound of the probability interval estimated
THE CREDIBLE INTERVAL by the Chi-square distribution based approach may be
greater than 1 (see the confidence intervals correspond-
The credible interval D =[a,b] is the interval that ensures
ing to the first three samples), which is obviously
the posterior lower probability Pr{P ∈D} reaches a speci-
m unreasonable.
fied credibility γ, e.g., γ =0.95, where P is the probability
m 3) The centrallimit theorembased approachcannotobtain
correspondingtothemthoutcomeofthemultinomialvariable.
a meaningful probability interval until all the states of
The bounds of D can be obtained by
thetwo-state variableoccuratleastonce.Moreover,the
a=0, b=G−1 1+γ , n =0, estimated lower bound may be less than 0 because the
2 m
a=H−1 1−γ , (cid:0)b=(cid:1)G−1 1+γ , 0<n <n, (15) converted normal distribution is unbounded.
a=H−1(cid:0)1−2γ(cid:1), b=1, (cid:0) 2 (cid:1) n =mn, The observationsillustrate that IDM and the corresponding
2 m credibleintervalestimationapproachcanprovidemorereason-
 (cid:0) (cid:1)
where H is the cumulative distribution function (CDF) of ableestimationresultscomparedwiththeexistingapproaches.
beta distribution B(n ,s+n−n ), G is the CDF of beta In this paper, IDM is used to estimate the uncertain proba-
m m
distribution B(s+n ,n−n ), n is the sample size, n is bilistic dependencyrelationsbetween the OTL failure and the
m m m
the number of times that the mth outcome is observed, and s operational conditions. Meanwhile, the credible interval esti-
is the equivalent sample size that is set to 1 [27]. mation approach mentioned in Appendix A is recommended
asanalternativeifa quantitativeconfidenceindexis required.
APPENDIX B
COMPARISON OF THEPROBABILITY INTERVALCOUNTING APPENDIX C
APPROACHES EFFECTSOF THEEQUIVALENT SAMPLE SIZE
Probabilityintervalscountedfromavailablesamplesarethe Theequivalentsamplesizescaninfluencetheestimationre-
basis of the proposed approach. The performance of several sultofIDM.Infact,s determineshowquicklythe probability
countingapproaches,i.e.,IDM,thecredibleintervalestimation intervalwill convergeasdata accumulate.Smaller valuesof s
approach, the central limit theorem based approach [12], and producefaster convergenceandstrongerconclusions,whereas
the Chi-square distribution based approach [13], is tested. larger values of s produce more cautious inferences. More
The samples are randomly selected from Poisson distribution specifically, s indicates the number of observations needed
Pois(0.01), which means for each sample the corresponding to reduce the length of the imprecision interval to half of its
two-staterandomvariablehasa1%chancetobe1anda99% initialvalue(accordingtoEq.(8),thelengthoftheimprecision
chance to be 0. The test results are shown in Fig. 5. intervalis∆Pm =Pm−Pm =s/(n+s),whichwilldecrease
from 1 to 1/2 when n increases from 0 to s).
Therefore, the assignment of s reflects the cautiousness
of the estimation. To obtain objective estimation results, as
suggested by Walley in [27], the value of s should be suffi-
cient large to encompass all reasonable probability intervals
estimated by various objective Bayesian alternatives. Up to
now,severalresearcheshaveled to convincingargumentsthat
s should be chosen from [1,2], and the value of s should
be larger when the multinomial variable has a large number
of states [25]. In our model, since the multinomial variables
only have 2 or 3 states (see Table I), the parameter s is set
to 1 as suggested. Moreover, it should be emphasized that
the influence of s weakens quickly as the number of samples
increases,whichcanbeclearlyseenfromFig.6.Itcanalsobe
observed from the figure that all of the probability intervals
estimated by IDM (1 ≤ s ≤ 2) converge faster than that
estimated by the Chi-square distribution based approach.
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