J. Hyd ol. Hyd omech., 63, 2015, 4, 287–294
DOI: 10.1515/johh-2015-0036
287
Analysis o he in luence o inpu da a unce ain ies on de e mining
he eliabili y o ese oi s o age capaci y
Daniel Ma on*, Miloš S a ý, Pa el Menšík
B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ins i u e o Landscape Wa e Managemen , Ve eří 331/95, 616 00, B no,
Czech epublic.
* Co esponding au ho . Tel.: +420 541 147 773. Fax: +420 541 147 771. E-mails: ma on.[email p o ec ed], s a y.m@ ce. u b .cz,
[email p o ec ed]
Abs ac : The pape con ains a sensi i i y analysis o he in luence o unce ain ies in inpu hyd ological, mo phological
and ope a ing da a equi ed o a p oposal o ac i e ese oi conse a ion s o age capaci y and i s achie ed alues. By
in oducing unce ain ies in o he conside ed inpu s o he wa e managemen analysis o a ese oi , he subsequen
analysed ese oi s o age capaci y is also a ec ed wi h unce ain ies. The alues o wa e ou lows om he ese oi
and he hyd ological eliabili ies a e a ec ed wi h unce ain ies as well. A simula ion model o ese oi beha iou has
been compiled wi h his kind o calcula ion as s a ed below. The model allows e alua ion o he solu ion esul s, aking
unce ain ies in o conside a ion, in con ibu ing o a educ ion in he occu ence o ailu e o lack o wa e du ing
ese oi ope a ion in low-wa e and d y pe iods.
Keywo ds: Unce ain ies; Reliabili y; Rese oi s o age capaci y; Mon e Ca lo me hod; Mean mon hly lows; E apo a-
ion; Ele a ion– olume cu e; Ele a ion–a ea cu e.
INTRODUCTION
The cu en knowledge in he ield o clima ology indica es a
g adual change in hyd oclima ic condi ions all o e he wo ld.
Clima e changes a e e lec ed in he changes in he hyd ological
cycle due o he edis ibu ion o p ecipi a ion du ing he yea
and hey con ibu e o mo e equen occu ences o ex emes in
he o m o loods and d y pe iods.
Clea signs o clima ic changes ha e appea ed in he Czech
Republic in ecen yea s. I should be no ed ha , om he hy-
d ological poin o iew, 2011 and 2012 we e conside ed o be
ex emely d y (Zah adníček e al., 2014). The empe a u es in
he win e in 2014 we e conside ably abo e a e age. In ha
pe iod, he wa e s o age in snow co e was he lowes in he
las wen y yea s. The consequences we e ex ao dina y manip-
ula ions a some wa e ese oi s. I is appa en ha he subjec
o ad anced managemen and con ol o su ace wa e esou ces
is becoming mo e and mo e impo an . The manipula ion ules
o la ge open wa e ese oi s we e app o ed in he pe iod o
cons uc ion o hose wa e wo ks and subsequen ly e iewed o
he cu en hyd ological condi ions. I will be necessa y o ca y
ou a ho ough e iew in he u u e in ela ion o hei adap abil-
i y due o clima e change and hyd ological cyclic e olu ion.
The e o e, he asks o wa e managemen analysis o ese oi s
o ac i e ese oi conse a ion s o age capaci y will always be
necessa y and esea ch in his ield is aluable. In pa icula , he
applica ion o new op imiza ion me hods o wa e managemen
analysis o ese oi s, new ese oi pe o mance de ini ions
and, las bu no leas , he in oduc ion o analysis o unce ain-
ies in hese p oblems o combina ions o he abo e men ioned
applica ions and knowledge.
Unde he gi en condi ions, i is necessa y o in oduce inpu
da a unce ain ies in o he p oposal o con ol o ese oi
s o age capaci y. The p ocedu es which will be desc ibed below
e e o “The in luence o unce ain ies in he calcula ion o
mean mon hly discha ges on ese oi s o age” (Ma on e al.,
2011). This pape desc ibes, in de ail, he in oduc ion o unce -
ain ies o measu emen in de e mining below a e age mon hly
lows o e he s age-discha ge cu e in a i e and a numbe o
measu emen s o hou ly eco ds o i e s ages in a hyd ome ic
p o ile. Fo his so o compu a ion, he Mon e Ca lo me hod
was used. One o he esul s was c ea ing he andom ime se-
ies o mean mon hly lows, which was a ec ed by unce ain-
ies o measu emen in he hyd ome ic p o ile. The andom
se ies o mean mon hly lows se ed as inpu da a o he wa e
managemen analysis o he ese oi s o age capaci y, when he
spec um o ese oi s o age olumes o he maximum hyd o-
logical eliabili y o 100% we e de e mined epea edly using
single-pass simula ions o ese oi ope a ion. The inal spec-
um o andom ese oi s o age capaci ies was e alua ed s a-
is ically and he in e al o possible alues o ese oi s o age
olumes was ound. The o he wo k e e ed o in his pape is a
pape by Ma on e al. (2014) desc ibing he calcula ion o
ese oi s o age capaci y in he condi ions o measu emen
unce ain ies and ex ended o using he AR and ARMA models,
gene a o s o a i icial s eam low se ies o mean mon hly dis-
cha ges. Random discha ge se ies, in his case as da a inpu s o
a i icial s eam low se ies gene a o s, we e used, esul ing in
andom samples o a i icial s eam low se ies. These andom
se ies we e e alua ed using a ese oi simula ion model. The
calcula ion esul was a spec um o ese oi s o age capaci ies
o maximum eliabili y o 100 %, which was s a is ically e al-
ua ed. Bo h pape s ha e indica ed ha he cu en wa e ol-
umes in ese oi s can be unde es ima ed and, in d y pe iods,
may esul in an unexpec ed ailu e in su ace wa e supply.
As ega ds cu en knowledge, unce ain y can be de i ed
om se heo y, bu also unce ain y can be de i ed using s a is-
ics. F om se heo y, i is necessa y o men ion he applica ion
o unce ain y based on he heo y o uzzy se s (Zadeh, 1965)
and he heo y o possibili ies (Kli , 2005). Howe e , he i s
de ini ion o unce ain y was by Knigh (1921), nowadays
known as Knigh ian unce ain y. The unce ain y concep is
cu en ly iewed om di e en aspec s, wi h plen y o de ini-
ions and poin s o iew, such as he unce ain y associa ed wi h
he de ini ion o isk, unce ain ies applied in o ecas ing p ob-
lems, and also he unce ain y o measu emen . Unce ain ies
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Daniel Ma on, Miloš S a ý, Pa el Menšík
288
applied in hyd ology we e desc ibed, o example, by Be en
and Binley (1992). They desc ibed in de ail a me hod called
GLUE – gene alized likelihood unce ain y es ima ion. This
was ollowed by nume ous publica ions which deal wi h his
issue, such as Be en (2007). Unce ain ies in measu emen we e
i s o mula ed on he basis o he WECC (1990) ag eemen . A
s a is ical app oach using he concep o unce ain y o meas-
u emen , which clea ly de ined he in oduc ion and calcula ion
o measu emen unce ain ies, was in oduced as he “Guide o
Exp ession o Unce ain y in Measu emen ” (GUM 1993). The
ISO GUIDE 99998 S anda d (2004) deals wi h he dis ibu ion
and p opaga ion o unce ain ies using Mon e Ca lo simula ion.
Hyd ological applica ions, including p opaga ion o unce ain-
ies in o hyd ological inpu s when measu ing p ecipi a ion,
wa e in lows in o ese oi s and e apo a ion in he wa e bal-
ances o ese oi s, we e deal wi h by Win e (1981). LaBaugh
and Win e (1984) examined he in luence o unce ain ies in
he measu emen s o wa e in low in o ese oi s, wa e ou -
lows om ese oi s and e apo a ion, and o he hyd ological
and ope a ing pa ame e s on he olume and chemical analysis
o wa e in ese oi s. The la es publica ions, o example
Campos e al. (2014), examined he isks and unce ain ies in,
and in luence on ese oi s o age using Mon e Ca lo simula-
ions. Ku ia and Vogel (2014) ca ied ou an analysis o unce -
ain ies in ese oi s o age using he wa e supply yield model.
Coxon e al. (2015) publish pape which apply no el amewo k
o discha ge unce ain y o UK gauging s a ions. Ano he e-
sea ch a ea in he applica ion o wa e managemen simula ion
and modelling is unce ain y associa ed wi h non-s a iona y
p ocesses in ime se ies. This issue is oo expensi e o he
pu poses o his wo k and needs ex ensi e independen esea ch.
The aim o he pape is o p esen ano he possible applica-
ion o he Mon e Ca lo me hod o in oducing unce ain ies
in o he hyd ological, mo phological and ope a ing inpu da a
equi ed o ese oi s o age capaci y design, which is c ucial
in low-wa e pe iods. This is also connec ed wi h he calcula ion
o eliabili y o wa e ou low om he ese oi in adap i e
condi ions. Inpu hyd ological, mo phological and ope a ing
da a o he solu ion a e conside ed, especially wa e in low in o
he ese oi , wa e losses om he ese oi by e apo a ion
om he wa e su ace and by dam seepage, he ese oi ele a-
ion– olume cu e and he ese oi ele a ion–a ea cu e. A
ese oi s o age model was c ea ed o his pu pose using a
single-pass simula ion me hod o de e mine eliabili y o wa e
ou lows om he ese oi , bo h conside ing wa e losses om
he ese oi and igno ing such losses. By in oducing unce -
ain y in o he inpu da a, he Mon e Ca lo me hod is used o
de e mine, by epea ed solu ions, he spec um o eliabili y o
ese oi s o age capaci y. This is hen e alua ed and app op i-
a ely in e p e ed. As he unce ain ies in he inpu da a a e un-
known, he wo k ocuses on compiling sensi i i y analysis
be ween he unce ain y in he inpu da a o a solu ion o he
s o age capaci y o he ese oi and he unce ain y o he
achie ed eliabili y o con olled wa e ou lows om he ese -
oi .
METHOD
Mon e Ca lo me hod
The gene al p ocedu e o gene a ing unce ain y a ec ed hy-
d ological, mo phological and ope a ing inpu da a o he ela -
ed wa e managemen analysis o a ese oi o i s s o age
capaci y is as ollows. Unce ain ies o inpu quan i ies a e in o-
duced in o he calcula ions using he Mon e Ca lo me hod. Using
he dis ibu ion cu e F(X), a andom posi ion o alues NXi
wi hin he in e al o a gi en unce ain y a e gene a ed as inpu
alue Xi. Value Xi is conside ed andom and independen o
alues Xi–1 and Xi+1. This p esump ion will allow he in oduc ion
o he no mal p obabili y dis ibu ion N(
μ
(X),
σ
(X)). Then, each
inpu alue Xi is conside ed as a mean alue μ(X) and he
amoun o unce ain y is de ined as he s anda d de ia ion
σ
(X).
Subsequen ly, a cumula i e dis ibu ion unc ion Fi(X) o no mal
s anda dized p obabili y dis ibu ion is gene a ed o each mean
alue
μ
(Xi). The pseudo andom numbe gene a o gene a es a
andom numbe om an in e al in which andom quan i y alue
NXi is gene a ed.
Fig. 1. P inciple o gene a ing unce ain ies in inpu elemen s using he Mon e Ca lo me hod.
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Analysis o he in luence o inpu da a unce ain ies on de e mining he eliabili y o ese oi s o age capaci y
289
Fig. 2. Symbolic in oduc ion o conside ed quan i ies a ec ed wi h unce ain ies.
The basic p inciple o gene a ing andom posi ions o poin s
in he wo-dimensional coo dina e sys em (NXi,NYi) is iden ical
o he heo y desc ibed abo e. The dissimila i y is gi en by
plo ing a poin which equi es plo ing wo Mon e Ca lo gene a-
o s independen o each o he . Each gene a o p oduces a an-
dom posi ion o poin NXi (e.g. wa e le el ele a ion Nhi) and
wi h i a andom alue NYi (wa e olume in ese oi NVi). The
esul on he ese oi ele a ion– olume cu e is hen a andom
poin coo dina e (NVi,Nhi) o he ese oi ele a ion– olume
cu e. See Fig. 1.
As obse ed, he ese oi in low, wa e su ace e apo a ion,
dam seepage and ese oi ele a ion–a ea and ele a ion–
olume cu es a e conside ed o be hyd ological and ope a ing
inpu s. The p inciple o in oducing unce ain ies in o he calcu-
la ion o ese oi s o age capaci y is shown in Fig. 2.
The gene a ed andom cu es o wa e in lows in o he es-
e oi , wa e e apo a ion om he wa e su ace, seepages and
andom a ea and ele a ion– olume cu es se e as inpu alues
o a simula ion model which, using single-pass simula ion,
simula es he beha iou o he ese oi in he condi ions o da a
a ec ed wi h unce ain y.
Rese oi simula ion model and ese oi pe o mance
calcula ion
The basis o a ese oi simula ion model is an adjus ed
equa ion in he cumula i e o m con e ed o he ollowing
inequali ies (1) S a ý (2005).
0≤∑󰇛−󰇜∆+󰇛 −󰇜∆ ≤,

 (1)
whe e Oi is he ese oi ou low, Qi, he ese oi in low o i =
1, …, n, and Δ is he ime s ep o calcula ion (one mon h). Ok+1
is he ou low om he ese oi in he ollowing ime s ep,
when in s ep i+1 he alue Oi+1 is i s eplaced wi h he alue o
he equi ed ou low Op. The ime cou se o he numbe ed sum
simula es he cou se o emp ying he ese oi s o age by ime
s eps i = 1, …, k. Fo i = 0 i is necessa y o en e he s a ing
solu ion condi ion a e he sum alue. Inequali y (1) is limi ed
om bo h he le and he igh . F om he le i is limi ed by
alue 0 ( ull s o age capaci y) and om he igh by alue Vz,max
(emp y s o age capaci y) cha ac e izing he ese oi s o age
capaci y a ailable o he ese oi . By calcula ing he alue o
he exp ession, he cu en emp ying o he s o age olume
,
󰆒 is ob ained and i is hen es ed as o whe he i lies in a
pa icula in e al 〈0, VZ,max〉. I no , i is necessa y o ind alue
Oi+1 ( o he sum o he exp ession o be equal o ze o, idle dis-
cha ge will occu , o i equal o Vz,max – a ailu e will occu ).
The gene al de ini ion o eliabili y was successi ely
desc ibed by (Hashimo o e al., 1982; Klemeš, 1967; K i skiy
and Menkel, 1952). The classi ica ion o a ailu e in he ese oi
s o age capaci y o he ollowing calcula ion o eliabili y is as
ollows (2).
, = , =1, 
≥

, =0, 
<
 (2)
Z ,i = 1 desc ibes he ese oi s o age capaci y in a no ailu e
si ua ion (sa is ac o y s a e). Z ,i = 0 desc ibes he ese oi s o -
age capaci y in a ailu e si ua ion (unsa is ac o y s a e). The
equi ed eliabili y can be u he calcula ed om alues Z ,i.
Gene ally, eliabili y is calcula ed by ime-based eliabili y as
empo al eliabili y and occu ence eliabili y, and olume ic
eliabili y is calcula ed sepa a ely. The pape uses he o mula
o he calcula ion o empo al eliabili y PT (3).
=
∑,

 (3)
whe e k is he numbe o mon hs in he pe iod being sol ed.
PRACTICAL APPLICATION
The model was applied in p ac ice o he exis ing ese oi ,
Ví I, which is si ua ed in he Vysočina Region, Czech Republic.
This is a mul i-use ese oi se ing mainly as lood p o ec ion
and su ace wa e accumula ion o wa e supply and hyd oelec-
ic pu poses. The ese oi is buil in he S a ka Ri e basin
and has been in ope a ion since 1957. The S a ka is he main
in low in o he ese oi . The mean long- e m in low in o he
ese oi Qa is 3.34 m3 s–1. Inpu alues o he calcula ion we e
made up o a ime se ies o mean mon hly lows o e 60 yea s
wi h he measu emen pe iod om 1950 o 2010. The mean
annual e apo a ion om he wa e su ace EANNUAL = 613 mm.
The mon hly e apo a ion alues om wa e su ace we e de-
i ed in a simpli ied manne acco ding o he pe cen age dis i-
bu ion o e apo a ion acco ding o he ČSN 75 2405 S anda d
(2004) and om he mean annual e apo a ion alues o Ví I
ese oi , see Table 1.
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Daniel Ma on, Miloš S a ý, Pa el Menšík
290
Table 1. Mon hly dis ibu ion o e apo a ion amoun du ing he calenda yea .
Mon h Jan Feb Ma Ap May Jun Jul Aug Sep Oc No Dec
Em [%] 6 9 12 14 16 15 11 7 5 2 1 2
Em [mm] 36.78 55.17 73.56 85.82 98.08 91.95 67.43 42.91 30.65 12.26 6.13 12.26
Table 2. Calcula ion wi hou conside ing wa e losses om he ese oi . Measu emen unce ain ies a e applied o wa e in low in o he
ese oi .
Inpu da a unce ain y Ua = 3
σ
±3% ±6% ±9% ±15%
O
p
[m3 s–1]
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
2.2 100.000 0.000 100.000 0.000 100.000 0.000 99.999 0.033
2.3 99.590 0.001 99.591 0.033 99.598 0.095 99.611 0.200
2.5 98.907 0.001 98.906 0.024 98.896 0.109 98.852 0.277
2.7 96.668 0.354 96.688 0.453 96.701 0.540 96.672 0.753
2.9 93.165 0.302 93.230 0.479 93.250 0.621 93.250 0.879
3.0 90.849 0.338 90.923 0.573 90.984 0.749 91.084 0.987
Fig. 3. Rela ion be ween equi ed ou low Op and empo al eliabili y PT wi hou conside ing.
The o al ese oi olume is VTOTAL 56.193 x 106 m3, ac i e
s o age olume VZ,max is 44.056 x 10
6 m3 and lood ese oi
olume VFLOOD is 8.337 x 106 m3. The o al dam heigh is 67.3 m.
The ecological low om he ese oi QECO is 0.53 m3 s–1. The
alue o seepage h ough he dam was de i ed om empi ical
obse a ion and o he g a i y conc e e dam i is 0.15 l s–1 pe
1000 m2.
The calcula ion o empo al eliabili y PT o an inc eased ou -
low om he ese oi was analysed wi h and wi hou conside -
ing wa e losses om he ese oi . When wa e losses we e
conside ed in he calcula ions o empo al eliabili y, he de-
sc ibed p ocedu es o gene a ing unce ain y-a ec ed hyd olog-
ical, mo phological and ope a ing inpu s we e applied. The anal-
ysis was ca ied ou o he alues o inc eased equi ed ou low
Op lying in he in e al Op∈〈2.1; 3.0〉 m3 s–1. The selec ed num-
be o epe i ions using he Mon e Ca lo me hod was 300. Inpu
unce ain ies o he analysis anged in in e als ±3, ±6, ±9, and
±15%. The algo i hm simula ing he beha iou o he ese oi
hen calcula ed andom discha ges NOi o wa e om he ese -
oi and empo al eliabili y o NPT. Then, andom cou ses o
mon hly illing and emp ying o he ese oi s o age capaci y
we e calcula ed. Fo a be e p esen a ion o he esul s, hese
alues we e e alua ed s a is ically. The mean alue
μ
(X) o each
andom se is conside ed o be he esul an alue and he s and-
a d de ia ion
σ
(X) is conside ed o be he s anda d unce ain y
ela ed o a pa icula esul . The o al, ex ended unce ain y, ype
“Ua” co e ing almos 100% o speci ically 99.97% o occu -
ences o he moni o ed quan i y, co esponded o alue
μ
(X)±3
σ
. Sensi i i y analysis was ca ied ou o he calcula ion
wi hou conside ing wa e losses om he ese oi , when only
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Analysis o he in luence o inpu da a unce ain ies on de e mining he eliabili y o ese oi s o age capaci y
291
in low in o he ese oi was bu dened wi h unce ain y, see
Table 2 and Fig. 3. Calcula ions we e also made while conside -
ing wa e losses om he ese oi . Fi s , only he e apo a ion
alues, ele a ion– olume cu e, ele a ion–a ea cu e and seep-
age h ough he dam we e a ec ed wi h unce ain y, see Table 3
and Fig. 4. Then, ese oi in low, e apo a ion, ele a ion–
olume cu e, ele a ion-a ea cu e and seepage h ough he dam
we e a ec ed wi h unce ain y, see Table 4 and Fig. 5.
The shape o he cu es in Figs. 3, 4, and 5 in he ange om
Op = 2.3 m3 s–1 o Op = 2.5 m3 s–1 is caused by a la ge ime s ep
in he calcula ions (1 mon h), and also by a s ep inc ease in he
numbe o ailu e mon hs, which is a small numbe in he gi en
Table 3. Calcula ion wi h conside ing wa e losses om ese oi . Unce ain ies conside ed o e apo a ion, ele a ion- olume (a ea)
cu es and seepage h ough dam body combina ions.
Inpu da a unce ain y Ua = 3
σ
±6% ±9% ±15%
O
p
[m3 s–1]
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
2.2 100.000 0.000 100.000 0.000 100.000 0.000
2.3 99.590 0.001 99.590 0.001 99.590 0.001
2.5 98.906 0.024 98.899 0.092 98.889 0.140
2.7 96.311 0.001 96.311 0.001 96.311 0.001
2.9 92.896 0.001 92.896 0.001 92.896 0.001
3.0 90.516 0.203 90.514 0.204 90.522 0.224
Fig. 4. Rela ion be ween equi ed ou low Op and empo al eliabili y PT, wi h conside ing wa e losses om ese oi o inpu unce ain-
ies ±6, ±9, ±15, ±30% and e apo a ion, ele a ion- olume(a ea) cu es, dam seepage combina ions.
Table 4. Calcula ion wi h conside ing wa e losses om ese oi . Unce ain ies conside ed o all in low, e apo a ion, ele a ion- olume
(a ea) cu es, seepage h ough dam body combina ions.
Inpu da a unce ain y Ua = 3
σ
±3% ±6% ±9% ±15%
O
p
[m3 s–1]
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
μ
(P
T
) U
a(P
T
)
2.2 100.000 0.000 100.000 0.000 99.996 0.070 99.978 0.191
2.3 99.587 0.057 99.566 0.155 99.555 0.179 99.539 0.204
2.5 98.859 0.196 98.843 0.215 98.811 0.293 98.721 0.451
2.7 96.316 0.084 96.348 0.338 96.350 0.510 96.300 0.797
2.9 92.939 0.290 92.957 0.454 92.943 0.584 92.896 0.838
3.0 90.556 0.402 90.575 0.591 90.589 0.734 90.608 1.038
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Daniel Ma on, Miloš S a ý, Pa el Menšík
292
Fig. 5. Rela ion be ween equi ed ou low Op and empo al eliabili y PT, wi h conside ing wa e losses om ese oi o inpu unce ain-
ies ±3, ±6, ±9 a ±15% and an in low, e apo a ion, ele a ion- olume(a ea) cu es and dam seepage combina ion.
Fig. 6. The cou se o illing ese oi s o age capaci y in he condi ions o en e ed inpu da a unce ain ies Ua = ±3% a ±15% o he selec -
ed low wa e pe iod.
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Analysis o he in luence o inpu da a unce ain ies on de e mining he eliabili y o ese oi s o age capaci y
293
ange o equi ed ou low om he ese oi Op. Fo alues Op
= 2.6 m3 s
–1 and highe , he e is an appa en inc ease in he
ailu e mon hs, due o which he cu es a e smoo h. The mo e
signi ican in e al o spacing in he cu es in Fig. 5, unlike
Figs. 3 and 4, in he ield o equi ed ou low om he ese oi
Op = 2.2 m
3 s
–1 is caused by he numbe o ailu e mon hs
occu ing in he e alua ed se . The p ocess con ingency applied
when unce ain ies a e in oduced in o all inpu da a o he
solu ion esul s in a signi ican inc ease in he numbe o ailu e
mon hs compa ed o he solu ion in which only unce ain ies o
ese oi in low o unce ain ies o e apo a ion, ele a ion–
olume (–a ea) cu es and seepage h ough he dam
combina ion a e applied. The analysis also included alues o
illing he ese oi s o age capaci y. Fig. 6 hen shows he
cou se o illing he ese oi o a pa icula numbe o
epe i ions and o he selec ed low wa e pe iod.
SUMMARY
The inal compa ison is om selec ed equi ed ou low om
he ese oi Op = 2.5 and Op = 3.0 m3 s–1, whe e he in luence
is clea ly g aded. In he a ian wi hou applying wa e losses
om he ese oi , he empo al eliabili y o Op = 2.5 m3 s–1 is
in in e al PT∈〈98.906%; 98,908%〉, i.e. PT = 98.907%
±0.001% o inpu unce ain y ±3% and in in e al
PT∈〈98.575%; 99,129%〉 PT = 98.852% ±0.277% o inpu
unce ain y ±15%. The in e al o empo al eliabili y o he
Op = 3.0 m3 s–1 ange in in e al PT∈〈90.511%; 91,187%〉 is PT
= 90.849% ±0.338% o inpu unce ain y 3% and
PT∈〈90.097%; 92,071%〉 PT = 91.084% ±0.987% o inpu
unce ain y 15%. In he a ian wi hou applying all combina-
ions, i.e. conside ing unce ain ies in bo h in low and wa e
losses om he ese oi , he in e al o empo al eliabili y
was in PT∈〈98.663%; 99,055%〉 PT = 98.859% ±0.196% o
inpu unce ain y ±3% and o unce ain y ±15% in in e al
PT∈〈98.27%; 99,172%〉 PT = 98.721% ±0.451%. Fo Op =
3.0 m3 s
–1 he in e al o empo al eliabili y acqui ed alues
PT∈〈90.154%; 90,958%〉 PT = 90.556% ±0.402% o inpu
unce ain y 3% and PT∈〈89.57%; 91,646%〉 PT = 90.608%
±1.038% o inpu unce ain y 15%. The abo e men ioned
esul s show a logical conclusion ha wi h inc easing inpu da a
unce ain y, he unce ain y in empo al eliabili y also inc eas-
es. Con e ed o he numbe o ailu e mon hs, inc eased dis-
cha ge Op = 2.5 m3 s–1 in he solu ion wi hou conside ing un-
ce ain ies and wi h conside ing ese oi wa e losses, co e-
sponds o eigh ailu e mon hs, and o Op=3.0 m3 s
–1 o 70
mon hs. When conside ing inpu unce ain ies ±3%, he numbe
o ailu e mon hs is om 8 o 9 mon hs o Op = 2.5 m3 s–1 and
66 o 72 mon hs o Op = 3.0 m3 s–1. Fo he inpu da a unce -
ain y o ±15%, he numbe o possible ailu e mon hs is 8 o 12
o Op = 2.5 m3 s–1 and 60 o 76 mon hs o Op = 3.0 m3 s–1. The
p esen ed esul s show how unce ain ies can in luence he
inc ease in ailu e mon hs and which in e als he empo al
eliabili y can hen acqui e.
CONCLUSIONS
In he manipula ion ules o he Ví I ese oi , he s a ed
empo al eliabili y o he hyd ological pe iod 1931 o 1991 is
PT = 99.59% o a equi ed ese oi ou low Op = 2.5 m3 s–1.
This means ha he cu en s a e is unde es ima ed by app oxi-
ma ely 1% compa ed o he calcula ions which we e unde aken
o ha ese oi . Unde es ima ion can be explained by he
leng h o he inpu s eam low se ies in oduced o he calcula-
ions, which we e no upda ed un il 2010. In addi ion, he e
we e ew low wa e yea s in he i s hal o he 1990s. Fo
example, in he p esen ed analysis, he e is he appa en e ec
o unde es ima ion o empo al eliabili y o he alue o he
equi ed ou low om he ese oi Op = 2.3 m3 s–1, which is,
wi h he amoun o empo al eliabili y, nea es o empo al
eliabili y acco ding o he manipula ion ules o he ese oi
o he solu ion including conside ing inpu da a unce ain ies,
and hus he ese oi is classi ied inapp op ia ely in he signi -
ican class (A – PT ≥ 99.5%, B – PT ≥ 98.5%, C – PT ≥ 97.5%,
D – PT ≥ 95%) acco ding o ČSN 75 2405. Fo he unce ain y
o ±6%, he mean alue o empo al eliabili y was PT =
99.566%, which co esponds o he signi icance o he ese oi
A – PT ≥ 99.5%. In conside ing unce ain y, he lowe in e al
o empo al eliabili y co esponded o alue PT = 99.411% and
hus also he ese oi signi icance would all o a lowe signi -
ican class co esponding o class B – PT ≥ 98.5%. F om his
poin o iew, he e is space o u u e e iews o he manipula-
ion ules o ese oi s and possible amendmen o he ČSN 75
2405 s anda d which should ake inpu da a unce ain ies in o
conside a ion. In ha case, i will be necessa y o ake he alue
o empo al eliabili y PT as he lowe limi o he esul an
in e al and hus o incline mo e o he sa e side in he solu ion.
Cu en ly, he esul s canno be gene alized, bu he compu a-
ional algo i hm is w i en in gene al e ms and i can be applied
also o o he ese oi s. When ca ying ou he sensi i i y anal-
ysis, he same alue o unce ain ies was always coun ed o all
inpu da a. Unde hese condi ions i was shown ha wa e
in low was he mos signi ican sou ce o unce ain ies. How-
e e , o he inpu measu emen unce ain ies also ha e an in lu-
ence on he esul , which mus be aken in o accoun . A he
p esen ime, he au ho s a e no awa e o which alues he
ele a ion– olume (–a ea) cu es can acqui e, when hei s a ed
ac ual cou se is a ec ed by sedimen a ion o he ese oi and
o he e ec s. He e i is possible o assume ha highe unce -
ain ies o ele a ion– olume (–a ea) cu es can a ec he e-
sul s mo e. F om his poin o iew, he esul s may be di e en
and hus also he in e als desc ibing he occu ence o he
calcula ed empo al eliabili y PT may be di e en .
Finally, i mus be s a ed ha he p esen ed sensi i i y analy-
sis was only ca ied ou o one ese oi and he esul s canno
be gene alized. I can be assumed ha di e en esul s will be
ob ained o o he ese oi s and all ese oi sys ems wi h
di e en sizes o ese oi s o age capaci y, a ious sou ces o
in lows and lood al i ude. In his espec , he e is space o
u he esea ch, o example using so wa e based on he pape
by Menšík e al. (2015), as well as coope a ion wi h wa e wo ks
adminis a o s.
Acknowledgemen . This pape was suppo ed by he B no Uni-
e si y o Technology’s p ojec “CZ.1.07/2.3.00/30.0039 Ex-
cellen young esea che s a B no Uni e si y o Technology”
and he speci ic esea ch p ojec FAST-S-15-2694 “Unce ain y
p opaga ion in he hyd ological and wa e managemen applica-
ions o mi iga ion o d ough on he open wa e ese oi .”
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