16 h In e na ional Symposium on P ac ical Design o Ships and O he Floa ing S uc u es PRADS 2025
Ann A bo , MI, USA, Oc obe 19 h-23 d 2025
Towa ds Clima e Resilien Inland Wa e way Vessel Design: Concep o
Dis ibu ed Th us o Shallow Wa e Condi ions
Richmond Anku1,*, Je oen P uyn 1,2and Co nel Thill1
1Del Uni e si y o Technology, Del , he Ne he lands
2CoE HRTech, Ma i ime Inno a ion, Ro e dam Uni e si y o Applied Science, Ro e dam, he Ne he lands
Abs ac . Inland wa e way essels a e c i ical o he hin e land anspo a ion ne wo k, o e ing an en-
i onmen ally iendly al e na i e o oad and ail anspo . Howe e , clima e change poses signi ican
challenges, such as luc ua ing wa e le els and ex eme shallow wa e condi ions ha lead o inc eased
esis ance and educed p opulsi e e iciency. These condi ions necessi a e inno a i e design and ope a-
ional s a egies o ensu e he e iciency and sus ainabili y o p opulsion sys ems. Gi en he inc ease in
esis ance and isk o p opelle eme gence in shallow wa e condi ions, his s udy explo es he de elop-
men o clima e- esilien inland essels, by implemen ing he dis ibu ed h us concep , whe e mul iple
smalle p opelle s eplace con en ional single ela i ely la ge uni s, o e ing supe io maneu e abili y,
p opelle load dis ibu ion, and adap abili y o a ying wa e dep hs and condi ions. U ilising s a e-
o - he-a esis ance app oxima ion and a obus op imisa ion me hod, his esea ch p oposes a no el
shallow-wa e model ha enables op imal con igu a ion o p opelle size, numbe , and placemen , con-
side ing key pe o mance me ics such as h us e iciency and en ila ion mi iga ion, con ibu ing o
sus ainable inland wa e way anspo a ion. Resul s om a case s udy demons a e ha he dis ibu ed
p opulsion sys em can e ec i ely shi he ope a ional h eshold o p opulsion, ex ending he na iga-
ional capabili ies and pe o mance in wa e dep hs whe e con en ional design would ace limi a ions.
The indings highligh he po en ial o in eg a ing dis ibu ed p opulsion wi h ad anced op imisa ion
echniques o add ess clima e-induced challenges while ensu ing ope a ional eliabili y.
Key wo ds: shallow wa e , inland wa e way essel, p opulsion, clima e-
esilience, obus op imisa ion
*Co espondence o: [email p o ec ed]
1
1. In oduc ion
Wa e bo ne anspo plays a c ucial ole in hin e land anspo a ion, o e ing a mo e en i onmen ally
sus ainable and cos -e ec i e al e na i e compa ed o o he anspo modali ies. Howe e , his compe i i e
ad an age is inc easingly h ea ened by ex ended pe iods o d ough s and loods ha esul in bo h low and
high wa e le els [1]. Inland essels ope a e in en i onmen s al eady cons ained by wa e dep h, channel
wid h, and he p esence o b idges and locks, making na iga ion a key ac o in de e mining he anspo
e iciency o inland wa e way anspo .
Clima e change exace ba es hese challenges, unde mining he eliabili y and compe i i eness o inland
na iga ion. This also hampe s he achie emen o objec i es ou lined in he EU’s NAIADES III ac ion plan
[2], which seeks o shi a signi ican sha e o eigh anspo om oad o mo e sus ainable modes, such
as inland wa e ways and ail anspo . Nume ous s udies ha e examined he impac o clima e change on
inland wa e anspo , wi h a pa icula ocus on he e ec s o educed wa e dep h in na igable ai ways.
Gi en he expec a ion ha p olonged d ough s will occu mo e equen ly in he u u e, hese condi ions
a e an icipa ed o ha e mo e se e e consequences on anspo e iciency, sa e y, and eliabili y compa ed
o he challenges posed by high wa e le els o discha ge luc ua ions [3].
In a ecen e iew o he li e a u e [4], wo p ima y challenges acing inland wa e way anspo in
achie ing sus ainabili y and eliabili y we e iden i ied. The i s conce ns he ad e se impac o ex eme
shallow-wa e condi ions on bo h hyd odynamic e iciency and ca go-ca ying capaci y, which impedes
e o s o inc ease he modal sha e o inland wa e way anspo , al eady unde u ilised. The second ela es
o he challenges o in eg a ing sui able g een ene gy solu ions onboa d.
This s udy ocuses on add essing he i s challenge by imp o ing he p opulsion pe o mance o inland
essels o ensu e su icien h us unde ex emely low wa e condi ions. By ex ending he echnical ip-
ping poin o ope a ional limi s, he esea ch aims o enhance he echnical esilience and e ec i eness o
inland wa e way anspo in ad e se en i onmen al condi ions. Achie ing his equi es unde s anding he
c i ical in e ac ion be ween he essel hull, p opelle (s), and wa e way, which in luences he hyd odynamic
cha ac e is ics and e iciency in ex eme wa e way condi ions.
This s udy ad ances beyond con en ional hyd odynamic analysis by applying compu a ionally e icien
op imisa ion echniques ha accoun o he pe o mance o he wo s -case scena io, enabling enhanced
ope a ional esilience o hese challenging condi ions.
The es o his pape is s uc u ed as ollows; sec ion 2., p esen s he heo e ical amewo k and backg ound
li e a u e ele an o his s udy, sec ion 3. desc ibes he esea ch me hodology, key indings a e analysed in
sec ion 4. and sec ion 5. concludes wi h con ibu ions and u u e di ec ions.
2
2. Theo i ical backg ound
2.1. Con ined wa e way e ec
Vessel na iga ing in wa e ways wi h es ic ed wid h and dep h, he low a ound i is subs an ially in-
luenced by he con ined en i onmen . In such condi ions, he bounda y laye a ound he la bo om pla e
becomes hinne , esul ing in highe ic ion esis ance. The low a ound he hull is accele a ed, leading o a
p essu e educ ion acco ding o Be noulli’s p inciple. This p essu e dec ease con ibu es o he squa e ec
o he ship [5]. The inc ease in squa also accoun s o inc eased esis ance and loss o speed [6].
[7] summa ised he h eshold o he e ec s o wa e way c oss-sec ion ela i e o he ship dimension
on he hyd odynamic e ec in able 1.. The main pa ame e s a e he blockage ac o (Ac/As), he dep h o
d a a io (h/T) and he channel wid h o beam a io (W/B).
Table 1.: Con inemen E ec Pa ame e s [7]
Pa ame e s S a o con inemen e -
ec
Impo an con inemen Highly con ined
Ac/As50 7–8 4
h/T 15 3–4 1.5
W/B 50–200 10–15 4
In addi ion o he con inemen e ec , he low egime plays a c i ical ole in in luencing he inc eased
esis ance expe ienced by ships. Speci ically, he low egime signi ican ly a ec s he wa e-making esis-
ance componen o inland essels [8]. Flow egimes a e cha ac e ized by he essel’s speed ela i e o
he wa e dep h, exp essed as he dep h F oude numbe (Fnh =V
√gh ). As highligh ed by [9], he shallow
wa e e ec becomes mo e p onounced as he essel app oaches he c i ical low egime (Fnh ≈1). How-
e e , his condi ion is ypically no so common o inland essels, as hey a e low-speed essels ope a ing
p edominan ly wi hin he subc i ical low egime.
In he design o he p opulsion sys em o he ship, he in e ac ions o he hull-p opelle (s)-wa e ways
play an impo an ole in accu a ely de e mining he p opulsi e coe icien s ( h us and wake w ac o s).
Whe e he wake ac ion coe icien desc ibes ac ual in low o he p opelle , and he h us deduc ion ac o
desc ibes he addi ional esis ance due o he suc ion p essu e by he p opelle (s) in ac ion and is used o
de e mine he h us equi ed o o e come he esis ance. These ac o s ha e been heo e ically in es iga ed
by [10], whe e i was obse ed ha , educ ion in wa e dep h weakens he wake (w) and in ensi ies he
p opelle suc ion ( ). In con en ional ship p opulsion sys ems, essels ypically ely on single o win-sc ew
la ge-diame e p opelle s. Howe e , his con igu a ion has no able limi a ions, pa icula ly in shallow-wa e
ope a ions, whe e he isk o p opelle en ila ion becomes a signi ican conce n. The loss o p opulsion
h us and ine iciencies caused by high p opelle loading, esul ing om inc eased esis ance, highligh he
need o a pa adigm shi in he design o p opulsion sys ems o inland wa e way essels.
Mo eo e , when ope a ing in shallow wa e condi ions, he p opelle is also highly loaded due o he
inc ease in h us equi ed o mee he inc eased esis ance. When his happens and, he e o e, he p opelle
is highly loaded, he ope a ing o a ional speed will ha e o be inc eased o main ain he equi ed h us . The
high o a ional speed c ea es condi ions ha lowe he local p essu e on he suc ion side o he blades, which
is lowe han he apo p essu e, leading o p opelle ca i a ion. The a ia ion in open-wa e e iciency is
highly dependen on he change in p opelle loading, which in essence is ela ed o he h us deduc ion
ac o [11]. Ad ancemen s ha e been made in imp o ing he p opulsion e iciency o inland essels o
shallow-wa e condi ions. Ro e eel in [12] in es iga ed he di e en s e n shapes, including unnels, o
imp o e he in low o he p opelle in shallow wa e .
In ecen yea s, nume ical me hods using CFD simula ion ools ha e been used o p edic and op imise
inland essel p opulsion pe o mance. Howe e , hese a e compu a ionally expensi e o ea ly design con-
cep explo a ion. Hence, models ha es ima e esis ance and p opulsion pe o mance based on he essel’s
and he wa e way’s speci ic cha ac e is ics a e essen ial du ing he ea ly design s ages.
To add ess hese challenges and mee he demands o cu en and u u e ope a ions, inno a i e ap-
3
p oaches o p opulsion sys em design mus be conside ed. To bene i om he a ailable beam o he essel
in he s e n egion, a p opulsion concep called he dis ibu ed h us has no ably been in es iga ed by [13],
[14]. As highligh ed by Hages eijn [13], one o he key ad an ages o dis ibu ed h us using mul iple
smalle p opelle s is he edundancy i p o ides. In he e en o ailu e o one o mo e p opelle s, he
sys em can main ain unc ionali y, he eby minimizing down ime. Fu he mo e, dis ibu ed h us allows
o he use o smalle , mo e modula powe d i e ains ins ead o la ge , cen alized sys ems. Fo inland
essels, his app oach enables he use o easily se iceable and eplaceable uck engines as an al e na i e
o specialized ma ine engines [13], signi ican ly simpli ying main enance and educing ope a ional cos s.
2.2. Robus op imisa ion
In oday’s complex echnological landscape, in eg a ing di e se disciplines has become essen ial o
add essing enginee ing challenges. Ope a ions Resea ch (OR), a ield o applied ma hema ics, employs
analy ical me hods o suppo in o med decision-making. A key app oach wi hin OR is obus op imiza-
ion, which speci ically ocuses on decision-making in unce ain en i onmen s by accoun ing o he wo s
possible scena ios.
The e o s o implemen obus op imisa ion o help in decision making in shipping a e no new, ecen
ad ancemen s in ma i ime anspo esea ch ha e le e aged obus op imiza ion in a ious applica ions,
such as a obus op imiza ion model o ship a ic scheduling p esen ed by [15] and essel maneu e ing
con ol, as explo ed by [16]. [17] applied obus op imiza ion o lee deploymen and ac ical ship as-
signmen s, as well as capaci y alloca ion, o accoun o he ola ili y o shipping ma ke s and seasonal
a ia ions, ul ima ely aiming o maximize e enue.
In ship design, obus op imiza ion has made signi ican ad ancemen s by enhancing design me hod-
ologies while minimizing he compu a ional demands associa ed wi h high- ideli y, esou ce-in ensi e sim-
ula ions, such as hose in compu a ional luid dynamics. Addi ionally, he ship design p ocess is becoming
inc easingly complex, d i en by nume ous con lic ing design objec i es ha a e mu ually in e dependen ,
coupled wi h inc easing numbe o design o ope a ional cons ain s and decision a iables. Accoun ing o
his inc easing complexi y poses a challenge commonly known as he so-called ”cu se o dimensionali y”.
As such op imisa ion models ha e become in eg al pa o ship design wo k low. Fo example, [18] ap-
plied obus design op imiza ion o de e mine op imal hull geome ies and loa ing sys em con igu a ions,
ensu ing adap abili y o unce ain design equi emen s.
Conside ing he inhe en unce ain ies in he ope a ional condi ions o inland essels, obus op imi-
sa ion in he au ho s’ iew, p o ides a powe ul amewo k o designing essels ha a e esilien o he
wo s possible condi ions hey may encoun e h oughou hei ope a ional li e ime. As demons a ed in
his s udy, applying obus op imisa ion in o med by physics o ex eme shallow-wa e scena ios acili a es
he de elopmen o p opulsion sys ems ha ensu e bo h e iciency and eliabili y, e en unde a iable wa e
dep hs and challenging cons ain s.
This app oach allows o explo a ion o esilien essels ha can ope a e e ec i ely ac oss a ange o
unp edic able en i onmen al condi ions.
4
3. App oach
3.1. Op imisa ion modelling - ad e sa ial obus op imisa ion
A well-known app oach in obus op imisa ion is he ad e sa ial app oach. The main idea o he ad e -
sa ial app oach is o explici ly model he unce ain y by conside ing he de e minis ic ”wo s -case” scena io
[19]. This me hod seeks o op imise he sys em’s pe o mance while ensu ing i emains obus and e ec-
i e unde he mos un a ou able condi ions he sys em migh encoun e , he e-in an inland wa e way essel.
The op imiza ion p ocess i e a i ely adjus s he design o decision a iables o minimize he impac o hese
ad e sa ial unce ain ies. This ensu es he solu ion is no only op imal unde nominal condi ions bu also
esilien unde ex eme a ia ions.
3.1.1. esis ance
•unce ain y se s
The unce ain ies a e modelled as a se Uo possible wa e way condi ions, and in his case, ela ed
o he con inemen e ec s o he wa e way.
U=u=h
T,W
B, db
h
T∈[1.1,10],W
B∈[2.0,10], db∈[0.1·W, 1.0·W](1)
whe e:
h/T :(wa e dep h- o-d a a io),
W/B :(channel wid h- o-beam a io),
db:(p oximi y o essel o sides o channel wall).
•wo s case o essel esis ance
The wo s possible scena io would esul in inc eased esis ance as gi en below;
RT= (Vs, u)|u∈U (2)
• esis ance es ima ion
Es ima ing he esis ance o he essel, he s a e-o - he-a ic ion co ec ion o shallow wa e used
in his wo k is adop ed om [5]. This is a co ec ion o he ITTC ic ion line, accoun ing o
shallow-wa e e ec ;
C =0.08468
(log Re −1.631)2· 1 + 1.168
log Re −0.5238 ·h
T−1.472!(3)
The addi ional esis ance due o he squa e ec is calcula ed om [20] empi ical ship squa o mula,
which is co ec ed o wa e way condi ions such as he wid h o he channel, he slope o he channel,
and he p oximi y o he essel o he sides.
z= 0.0065 ·e5.2·Fnh + (0.95 ·F6
nh −0.065) (4)
z inal =z·(αW·αM·αW′)(5)
whe e Fnh =V
√gh is he dep h oude numbe and (αw, αm, α′
w) a e co ec ion ac o s o he wid h
o he wa e way, slope o bank, and he p oximi y o he essel o sides.
5
3.1.2. p opulsion
Th us loss in case o p opelle eme gence has been assumed o be p opo ional o he ou -o -wa e a ea
o he p opelle disc. Ven ila ion is a phenomenon ha occu s when he p opelle ope a es close o he
dynamic ee su ace and d awing ai in o he p opelle s eam o when he blades pie ce he ee su ace,
leading o sudden loss in p opelle h us and o que. This becomes mo e p onounced when he p opelle
loading is high and he subme gence is limi ed. Load can be dis ibu ed among mul iple p opelle s o educe
loading on one o wo p opelle s, which is he ocus o his esea ch. To imp o e subme gence below he
su ace, a smalle -diame e p opelle can be used. Hence, o mi iga e he isk o p opelle en ila ion. The
h us loss ac o βis gi en empi ically as a unc ion o he sha subme gence a io h
Rgi en by eq. (6). This
was p oposed by [21], accoun ing o he combined e ec s o loss o disk a ea, wa e, and wagene e ec .
β=(1,i h
R≥1.3
1−0.675 ·(1 −0.769 ·h
R)1.258,o he wise (6)
The sha imme sion a io, which is modelled as a unc ion o he p opelle diame e accoun ing o
hull-p opelle blade clea ance, is gi en by eq. (7).
h
R= 2 Te
D−0.65(7)
whe e Teis he essel d a and Dis p opelle diame e .
•wo se case o p opulsion
Wo se possible scena io o p opulsion would be he po en ial b eakdown in h us gi en below.
TF=β·T(8)
whe e TFand Ta e he e ec i e p opelle h us and openwa e p opelle h us , espec i ely.
In his s udy, Wageningen B-se ies p opelle , a con en ional p opelle se ies widely used in he design
and op imisa ion o p opulsion sys em, is used. O iginally de eloped h ough sys ema ic model es ing
by MARIN [22], his p opelle se ies p o ides comp ehensi e open-wa e pe o mance cha a e is ics o
con en ional ixed pi ch p opelle ac oss pa ame ic a ia ions in pi ch-diame e a io, expanded a ea a io,
numbe o blades, and ad ance numbe .
Recen ad ances ha e ep esen ed he open-wa e cha ac e is ic as high-o de mul i a ia e polynomials as
a unc ions o hese a iables, gi en in eq. (9) and eq. (10).
KT= 1(J, PD, AER, Z) = XCT
s, ,u, (J)s·(PD) ·(AER)u·(Z) (9)
KQ= 2(J, PD, AER, Z) = XCQ
s, ,u, (J)s·(PD) ·(AER)u·(Z) (10)
J=Va
n·D(11)
The coe icien s o he high-o de B-se ies polynomial unc ions a e published in [23]
3.2. MINLP o mula ion
The p oblem, as o mula ed, is a mixed-in ege non-linea (MINLP) op imisa ion p oblem. This is a non-
linea p oblem because he objec i e unc ion, which is he open wa e e iciency, is non-linea , and some
cons ain s, such as he (K −Kq), a e polynomial unc ions o he pi ch-diame e a io, expanded a ea
a io, and numbe o blades. The h us loss ac o (β) is an empi ical hyd odynamic unc ion, as desc ibed
in e e ence [3], and ac s as a penal y ac o ha educes he h us gene a ed by he p opelle . This h us
loss ac o inc eases wi h he p opelle size as discussed in he p e ious subsec ion. Consequen ly, he
op imisa ion p ocess aims o minimise his penal y ac o by dis ibu ing he h us ac oss mul iple smalle -
diame e p opelle s, he eby educing he o e all impac o he h us loss ac o . Hence, o maximise he
6
bene i om he en i e beam (Bmax) o a essel in he s e n egion, he cons ain illus a ed in ig. 1. and
de ined in eq. (12) mus be conside ed.
N
X
np=1
(D+ (np−1) ·(0.2·D)) ≤Bmax (12)
The o al h us p oduced is modelled o accoun o he wo s -case scena io, whe e he o al p opelle (s)
h us (np·T) should o e come he esis ance in he wo s -case (shallow wa e condi ion) RT.
np·(T·β)≥RT,(13)
The h us loss ac o nega i ely impac s he o al h us , p ima ily a unc ion o he subme gence o he
p opelle . This subme gence, in u n, depends on he p opelle ’s size (diame e ).
The o mula ion o he op imisa ion p oblem is summa ised below;
maximise:η=J
2π·KT
KQ
,
subjec o:np·(T·β)≥RT,
N
X
np=1
(D+ (np−1) ·(0.2·D)) ≤Bmax,
KT= 1(J, PD, AER, Z),
KQ= 2(J, PD, AER, Z),
T=KT·n2·D4,
Q=KQ·n2·D5,
decision a iables:
D∈[0.5·Te,1·Te],
PD∈[0.7,1.4],
AER ∈[0.5,1.0],
J∈[0.1,1.6],
Z∈[3,6]
np∈Z+
(a) p opelle spacing (b) p opelle subme -
gence
Figu e 1.: P opelle Con igu a ion Cons ain s
7
This me hod es ima es he diame e , pi ch, numbe o p opelle (s), and e iciency equi ed o achie e
su icien o al h us o inland wa e way essels ope a ing in shallow wa e s. This p og am was imple-
men ed in Py hon, wi h he op imisa ion model o mula ed using Pyomo (Py hon Op imisa ion Modeling
Objec s), an open-sou ce Py hon lib a y designed o modeling a wide ange o op imisa ion p oblems. Py-
omo p o ides a lexible amewo k ha suppo s linea , nonlinea , and mixed-in ege op imisa ion, allowing
use s o in e ace wi h a ious sol e s depending on he p oblem’s equi emen s. Fo his speci ic p oblem,
which is classi ied as a Mixed-In ege Nonlinea P og amming (MINLP) p oblem, he open sou ce sol e
BONMIN (Basic Open Sou ce Nonlinea Mixed IN ege P og amming) was employed. BONMIN is de-
signed o e icien ly handle MINLP p oblems by in eg a ing nonlinea and in ege p og amming echniques.
8
4. Resul s and Discussions
4.1. Resis ance and p opulsion
Implemen ing his me hod on a use case wi h cha ac e is ics in he able able 2., his is he CEMT
Class VI, a ypical essel o Rhine class. Fo his essel, he esis ance i may encoun e is decomposed
Table 2.: Vessel Cha ac e is ics
Pa ame e s alue
Cb[-] 0.86
L[m] 110
B[m] 11.4
Tl[m] 3.5
Te[m] 1.8
Sw[m2] 2974.96
∇[m3] 3795.2
lcb [m] 55.28
lcg [m] 56.27
in o he h ee main componen s: he iscous esis ance and he wa e making esis ance and ship squa
e ec s, as illus a ed in he 1s ,2nd and 3 d ames o ig. 3.. The iscous esis ance is made up o he
esis ance due o ic ion (C ) and he esis ance due o he o m o he essel (1 + k∗). The wa e-making
esis ance exhibi s an inc easing end wi h wa e dep h, as illus a ed in he second ame o ig. 3.. I s
dominance emains ela i ely limi ed. This is because wa e-making esis ance is p ima ily go e ned by
he low egime, cha ac e ized by he dep h F oude numbe (Fnd =V
√gh ). No ably, signi ican wa e-
making esis ance occu s when Fnd app oaches he c i ical low egime Fnd ≈1, Howe e , his condi ion
is gene ally no encoun e ed in inland essels, as hey p edominan ly ope a e wi hin he subc i ical low
egime (Fnd <0.6).
Figu e 2.: Ship squa
Howe e , esis ance due o squa mus also be conside ed, as i a ises om he addi ional ship d a o
sinkage when he essel mo es in shallow wa e . This e ec occu s due o a p essu e d op benea h he keel,
9
[17] X. Lai e al. “Robus Ship Flee Deploymen wi h Shipping Re enue Managemen ”. In: T anspo a-
ion Resea ch Pa B: Me hodological 161 (2022), pp. 169–196. DOI:10.1016/j. b.2022.
05.005.
[18] P. Gallaghe . “No el App oach o Robus Concep De elopmen o O sho e Floa ing Sys ems”. In:
In e na ional Jou nal o Ma i ime Enginee ing 164.A4 (2023). DOI:10.5750/ijme. 164iA4.
1206.
[19] J. Th¨
u au . “Adjus able Robus Nonlinea Ne wo k Design unde Demand Unce ain ies”. In: (2024).
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