Ci a ion: López-Queija, J.; Robles, E.;
Llo en e, J.I.; Touzon, I.;
López-Mendia, J. A Simpli ied
Modeling App oach o Floa ing
O sho e Wind Tu bines o Dynamic
Simula ions. Ene gies 2022,15, 2228.
h ps://doi.o g/10.3390/en15062228
Academic Edi o s: Da ide As ol i
and Uwe Ri schel
Recei ed: 21 Feb ua y 2022
Accep ed: 15 Ma ch 2022
Published: 18 Ma ch 2022
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ene gies
A icle
A Simpli ied Modeling App oach o Floa ing O sho e Wind
Tu bines o Dynamic Simula ions
Ja ie López-Queija 1,2,* , Eide Robles 1,3, Jose Ignacio Llo en e 2, Imanol Touzon 1
and Joseba López-Mendia 1
1TECNALIA, Basque Resea ch and Technology Alliance (BRTA), Pa que Tecnológico de Bizkaia As ondo
Bidea, Edi icio 700 E, 48160 De io, Spain; eide [email p o ec ed] (E.R.); [email p o ec ed] (I.T.);
[email p o ec ed] (J.L.-M.)
2Mechanical Enginee ing Depa men , Uni e si y o he Basque Coun y UPV/EHU, 48013 Bilbao, Spain;
[email p o ec ed]
3Au oma ics and Sys em Enginee ing Depa men , Uni e si y o he Basque Coun y UPV/EHU,
48013 Bilbao, Spain
*Co espondence: ja ie [email p o ec ed]
Abs ac :
Cu en ly, loa ing o sho e wind is expe iencing apid de elopmen owa ds a comme cial
scale. Howe e , he esea ch o design new con ol s a egies equi es nume ical models o low
compu a ional cos accoun ing o he mos ele an dynamics. In his pape , a educed linea
ime-domain model is p esen ed and alida ed. The model ep esen s he main loa ing o sho e
wind u bine dynamics wi h ou plana deg ees o eedom: su ge, hea e, pi ch, i s owe o e-a
de lec ion, and o o speed o accoun o o o dynamics. The model elies on mul ibody and modal
heo ies o de elop he equa ion o mo ion. Ae odynamic loads a e calcula ed using he wind u bine
powe pe o mance cu es ob ained in a p ep ocessing s ep. Hyd odynamic loads a e p ecompu ed
using a panel code sol e and he moo ing o ces a e ob ained using a look-up able o di e en
sys em displacemen s. Wi hou any adjus men , he model accu a ely p edic s he sys em mo ions o
coupled s ochas ic wind–wa e condi ions when i is compa ed agains OpenFAST, wi h e o s below
10% o all he conside ed load cases. The la ges e o s occu due o he ansien e ec s du ing he
simula ion un ime. The model aims o be used in he ea ly design s ages as a dynamic simula ion
ool in ime and equency domains o alida e p elimina y designs. Mo eo e , i could also be used
as a con ol design model due o i s simplici y and low modeling o de .
Keywo ds:
loa ing o sho e wind u bine; simpli ied model; FOWT dynamics; ae odynamics;
hyd odynamics; s uc u al dynamics
1. In oduc ion
In ecen decades, wind ene gy p oduc ion g ow h has pushed enewable ene gies o
di ec ly compe e wi h uel-based ene gies [
1
]. In o de o con inue ising, he wind ene gy
ma ke le e ages he ad an ages o o sho e loca ions whe e he wind esou ce is p e e able,
wi h highe and s eadie winds. In 2019, o sho e wind ene gy capaci y ins alla ion eached
a peak [
2
], which is expec ed o be su passed sho ly, suppo ed by loa ing wind ene gy
de elopmen . Di e en comme cial p ojec s ha e been deployed in his di ec ion such as
Hywind A lan ic and Kinca dine p ojec s [
3
]. Despi e he inc easing amoun o in es men
in loa ing solu ions, he capi al cos s o ins alling a loa ing o sho e wind u bine (FOWT)
make he p oduc ion cos s pe uni gene a ed s ill high [
1
]. Howe e , a iable le elized
cos o ene gy (LCOE) educ ion pa h could be h ough he implemen a ion and es ing o
app op ia e con ol s a egies.
A FOWT is a highly complex nonlinea sys em whose ep esen a ion equi es consid-
e ing hyd odynamics, ae odynamics, s uc u al dynamics, moo ing dynamics, con olle
dynamics, and hei couplings. The FOWT sys em modeling equi emen s a e condi ioned
Ene gies 2022,15, 2228. h ps://doi.o g/10.3390/en15062228 h ps://www.mdpi.com/jou nal/ene gies
Ene gies 2022,15, 2228 2 o 16
by he design s age. The de ail and accu acy needed o he model will di e om he
concep ual design s age o he p e-indus ial s age.
De ailed ma hema ical models o FOWT, including complex ae odynamics, hyd o-
dynamics, moo ing dynamics, and sys em s uc u al dynamics, ha e been de eloped,
allowing complex ae o-hyd odynamic load ep esen a ion and i s in luence on he sys em
h ough high- ideli y ime-domain simula ions [
4
,
5
]. The e a e di e en so wa e packages
a ailable o ep esen in de ail hese complex sys ems [
6
] ha se e as analysis ools o
sys em dynamics, u bine loads, a igue damage, and cos assessmen in he inal design
s ages. In [
7
], a ecen e iew o he FOWT dynamics and modeling app oaches is p esen ed.
Howe e , he mos ex ended simula ion ool o wind u bine design, bo h onsho e and
o sho e, is he so wa e named OpenFAST [8].
Al hough eliable FOWT modeling ools a e a ailable, which a e gene ally based on
e y de ailed and complex ma hema ical desc ip ions, i is o in e es o de elop simple
models ha accu a ely ep esen he main FOWT dynamics. These simpli ied models can
be used o: p o ide a clea unde s anding o he main sys em dynamics, easily modi y
he model pa ame e s o check di e en sys em con igu a ions, design con olle s, and
quickly es bo h sys em designs and con olle s in ea ly design s ages. In he same way,
hese models can be help ul o scale p o o ype es ac i i ies. In [
9
,
10
], educed-o de
models a e p esen ed o ime-domain simula ions. A simila model is desc ibed in [
11
] o
dynamic pe o mance e alua ion, bu simula ed in he equency domain. This low-o de
modeling app oach is also being de eloped o scale p o o ype ac i i ies such as in [
12
],
whe e he model is used o a hyb id ha dwa e-in- he-loop es . A simila model de i a ion
is p oposed in [13], whe e he au ho s alida e he model agains a scaled p o o ype.
In gene al, he FOWT con ol esea ch ield ends owa ds designing con olle s able
o lead wi h mo e han one objec i e a he same ime, which is basic o hese sys ems whe e
pla o m s abiliza ion and powe p oduc ion a e compe ing o wind u bine ope a ion
abo e a ed wind [
14
]. A e iew o he cu en con ol s a egies applied o di e en FOWT
concep s can be ound in [
15
]. The model ypology o mul i a iable con ol design equi es
ep esen ing he sys em dynamics wi h he minimum numbe o a iables. The con ol
design is usually pe o med in a inal s age h ough a sequen ial design app oach [
16
].
Howe e , he complexi y le el needed o assess con ol in luence on global loads and
mo ions is low, allowing modeling o he sys em wi h he dominan sys em dynamics
conside ing only a ew deg ees o eedom [
17
], enabling he use o simpli ied models.
In [
18
], a simpli ied dynamic model o con ol de elopmen is p esen ed and hen u ilized
o i s pu pose in, o example, [
19
]. A simila con ol-o ien ed model is p oposed in [
20
]
which is used o design a obus con olle a e being alida ed wi h OpenFAST.
In his pape , a simpli ied ma hema ical model is p oposed. The de eloped model is
simple and ep esen s he sys em dynamics well enough o be used o ad anced con olle
design. The model’s a chi ec u e is hough o allow changing be ween FOWT concep s by
changing he inpu s o he sys em. The ma hema ical exp ession o he model is ob ained
conside ing he FOWT sys em as wo igid solids: o o –nacelle assembly (RNA) and
subs uc u e linked by a lexible beam ep esen ing he owe . Using a 5 MW wind u bine
a op a spa subs uc u e, ae odynamic and hyd odynamic loads ac ing on he sys em
a e explained. A compa ison o he p esen ed model agains OpenFAST is pe o med o
alida e he sys em dynamic esponse bo h in ime and equency domains.
The pape is s uc u ed as ollows. In Sec ion 2, he FOWT case s udy de ini ion
is gi en, de ining hose p ope ies needed o build he model. Th ough Sec ion 3, he
modeling app oach is in oduced, explaining he ae odynamic, hyd odynamic, and s uc-
u al dynamic ep esen a ion used in he model. Sec ion 4summa izes he load cases and
he compa ison o he p oposed model pe o mance agains he s a e-o - he-a model
OpenFAST. Finally, in Sec ion 5, some conclusions a e d awn conside ing he esul s om
he p e ious sec ion, and possible u u e esea ch di ec ions a e also men ioned in his
las sec ion.
Ene gies 2022,15, 2228 3 o 16
2. Case S udy
The p esen ed model is alida ed using he OC3 Hywind Spa buoy shown in Figu e 1
as a case s udy. The loa ing sys em’s main p ope ies including he pla o m, he wind
u bine, and he moo ing sys em a e summa ized based on he desc ip ion p o ided in [
21
].
Ene gies 2022, 15, x FOR PEER REVIEW 3 o 17
2. Case S udy
The p esen ed model is alida ed using he OC3 Hywind Spa buoy shown in Figu e
1 as a case s udy. The loa ing sys em’s main p ope ies including he pla o m, he wind
u bine, and he moo ing sys em a e summa ized based on he desc ip ion p o ided in
[21].
Figu e 1. OC3 Hywind Spa buoy illus a ion [21].
The pla o m hull eaches 10 m abo e sea wa e le el (SWL) and has a d a o 120 m.
The s abili y o his loa ing pla o m concep is achie ed by a es o ing momen gene a ed
by he di e ence in heigh be ween he mass and buoyancy cen e s. In Table 1, he main
p ope ies used o he de elopmen o he educed model a e summa ized.
Table 1. OC3 Hywind pla o m p ope ies [21].
Pa ame e
Value
Uni s
Dep h o pla o m base below SWL (D a )
120
m
Ele a ion o pla o m op abo e SWL
10
m
Tape op dep h below SWL
4
m
Tape bo om dep h below SWL
12
m
Pla o m diame e abo e he ape
6.5
m
Pla o m diame e below he ape
9.4
m
Pla o m mass
7,466,330.0
kg
Pla o m cen e mass (CM) below SWL
89.9155
m
Pla o m pi ch ine ia abou CM
4,229,230,000
kg m2
Addi ional linea damping (𝐵11
𝑙𝑖𝑛𝑒𝑎𝑟)
100,000
N s m−1
Addi ional linea damping (𝐵33
𝑙𝑖𝑛𝑒𝑎𝑟)
130,000
N s m−1
The wind u bine modelled is he NREL5MW [22], which is a a iable-speed a ia-
ble-pi ch (VSVP) con olled u bine o en used as a e e ence u bine o esea ch pu -
poses. The hub heigh o he onsho e u bine is 90 m abo e g ound, so he owe heigh o
he model is sho ened o ma ch he same hub heigh o he onsho e u bine due o he
pla o m hull heigh abo e wa e le el. Towe op and bo om diame e s a e unchanged.
The wind u bine p ope ies a e shown in Table 2.
Figu e 1. OC3 Hywind Spa buoy illus a ion [21].
The pla o m hull eaches 10 m abo e sea wa e le el (SWL) and has a d a o 120 m.
The s abili y o his loa ing pla o m concep is achie ed by a es o ing momen gene a ed
by he di e ence in heigh be ween he mass and buoyancy cen e s. In Table 1, he main
p ope ies used o he de elopmen o he educed model a e summa ized.
Table 1. OC3 Hywind pla o m p ope ies [21].
Pa ame e Value Uni s
Dep h o pla o m base below SWL (D a ) 120 m
Ele a ion o pla o m op abo e SWL 10 m
Tape op dep h below SWL 4 m
Tape bo om dep h below SWL 12 m
Pla o m diame e abo e he ape 6.5 m
Pla o m diame e below he ape 9.4 m
Pla o m mass 7,466,330.0 kg
Pla o m cen e mass (CM) below SWL 89.9155 m
Pla o m pi ch ine ia abou CM 4,229,230,000 kg m2
Addi ional linea damping (Blinea
11 )100,000 Nsm−1
Addi ional linea damping (Blinea
33 )130,000 Nsm−1
The wind u bine modelled is he NREL5MW [
22
], which is a a iable-speed a iable-
pi ch (VSVP) con olled u bine o en used as a e e ence u bine o esea ch pu poses.
The hub heigh o he onsho e u bine is 90 m abo e g ound, so he owe heigh o he
model is sho ened o ma ch he same hub heigh o he onsho e u bine due o he pla o m
hull heigh abo e wa e le el. Towe op and bo om diame e s a e unchanged. The wind
u bine p ope ies a e shown in Table 2.
Ene gies 2022,15, 2228 4 o 16
Table 2. OC3 Hywind wind u bine p ope ies [22].
Pa ame e Value Uni s
Ra ed powe 5 MW
Ro o diame e 126 m
Hub heigh 90 m
Ro o mass 110,000 kg
Ro o ine ia 35,444,067 kg m2
Gene a o ine ia 534.116 kg m2
Gene a o ic ion 16.5489 N s/m
Gea box a io (high-speed o low-speed) 97 -
Nacelle mass 240,000 kg
Nacelle CM abo e owe op 1.96 m
Towe mass 347,460 kg
The moo ing sys em p ope ies a e desc ibed in [
21
] and summa ized in Table 3. I
consis s o h ee ca ena y lines a ached o he pla o m ia del a connec ion. In he model,
his con igu a ion is simpli ied o educe he sys em complexi y as was done in he OC3
p ojec . The ai leads a e loca ed a 70 m below SWL and symme ically sp ead a a 5.2 m
adius om he pla o m cen e line, moun ing a moo ing sys em con igu a ion whe e lines
a e 120◦sepa a ed. Ancho s a e ixed a a adius o 853.87 m om he pla o m cen e line
and a a dep h o 320 m.
Table 3. OC3 Hywind moo ing sys em p ope ies [21].
Pa ame e Value Uni s
Uns e ched moo ing line leng h 902.2 m
Moo ing line diame e 0.09 m
Equi alen moo ing line mass densi y 77.7066 kg/m
Equi alen moo ing line weigh in wa e 698.094 N/m
Equi alen moo ing line ex ensional s i ness (EA) 384,243,000 N
3. Modeling App oach
As i is explained in [
23
], a con ol-o ien ed wind u bine model is usually de i ed
using he Mul ibody Sys em app oach (MBS). Wi h his echnique, educed low-o de
models can be ob ained allowing he conside a ion o only hose deg ees o eedom ha
a e di ec ly coupled o he con olle ac ions. In a a iable-speed a iable-pi ch (VSVP)
wind u bine ope a ion, he speed con ol in e ac s wi h he modes in he o a ion ame
such as d i e ain o sion mode and blade edgewise bending modes. Howe e , hese
modes’ na u al equencies all beyond he con olle bandwid h equency [
23
], allowing
he simpli ica ion. The pi ch con ol no only a ec s he powe p oduc ion h ough he
ae odynamic o que bu also changes he h us exci a ion o ce. In consequence, owe
bending mode and loa ing sys em igid solid modes, a ec ed by he h us exci a ion o ce,
should also be conside ed.
In he p esen wo k, a plana MBS is de i ed, p esen ing he FOWT as wo lumped
masses, pla o m, and o o nacelle assembly (RNA), connec ed by a lexible owe . The
model desc ibes he FOWT dynamics a ending o he in e ac ion be ween he o o and
he along-wind modes. Su ge, hea e, and pi ch igid-body modes a e conside ed in he
model since hey a e iden i ied as he mos c i ical modes o FOWT con ol design [
24
],
wi h he pi ch mode a limi a ion o adi ional con ol s a egies [
25
,
26
]. The i s o e–a
owe modal de lec ion is also included because i is signi ican ly exci ed by ae odynamic
o ces due o i s low na u al equency. These loads can be educed using he blade pi ch
con ol o wind speeds abo e he a ed alue [
27
] and, consequen ly, a ligh e owe and
ounda ion could be designed. Finally, he o o dynamics a e also ep esen ed by a single
deg ee o eedom equa ion o mo ion.
Ene gies 2022,15, 2228 5 o 16
The ime-domain equa ion o mo ion is based on New on’s second law:
[M]..
x( )+[C].
x( )+[K]{x( )}={FEx ( )}(1)
whe e he mo ion ec o
{x( )}
comp ises he mo ions in each o he conside ed deg ees
o eedom (DOF).
[M]
,
[C]
, and
[K]
a e, espec i ely, he mass, damping, and s i ness
ma ices o he sys em. All he p e iously in oduced ma ices a e 5
×
5 acco ding o he
ep esen ed sys em mo ions. The sys em o al mass sums he con ibu ion o he s uc u al
mass and he hyd odynamic added mass om he ine ial componen o he adia ion o ce.
The damping ma ix is moun ed om he con ibu ion o h ee damping e ms. Fi s , he
linea hyd odynamic damping is p esen ed be o e in he case s udy sec ion. The second is
he s uc u al damping o he sys em. Thi dly, he damping con ibu ion om he adia ion
o ce is added o he model h ough di ec in eg a ion o he con olu ion o he p oduc
be ween adia ion impulse esponse and he s a e eloci y. Finally, he con ibu ion om
he hyd os a ic es o ing and s uc u al s i ness a e included in he sys em s i ness ma ix.
The las e m,
{FEx ( )}
, ep esen s he ex e nal o ces ac ing on each o he sys em DOFs.
In he p esen model, his load ec o encompasses wind
{Fa}
, wa e
{Fh}
, exci a ion o ces,
and moo ing loads
{Fmoo }
. In Figu e 2, a block diag am o he p oposed FOWT model
is p esen ed:
Ene gies 2022, 15, x FOR PEER REVIEW 5 o 17
ounda ion could be designed. Finally, he o o dynamics a e also ep esen ed by a single
deg ee o eedom equa ion o mo ion.
The ime-domain equa ion o mo ion is based on New on’s second law:
[𝑀]{𝑥(𝑡)}+[𝐶]{𝑥(𝑡)}+[𝐾]{𝑥(𝑡)}={𝐹𝐸𝑥𝑡(𝑡)}
(1)
whe e he mo ion ec o {𝑥(𝑡)} comp ises he mo ions in each o he conside ed deg ees
o eedom (DOF). [𝑀], [𝐶], and [𝐾] a e, espec i ely, he mass, damping, and s i ness
ma ices o he sys em. All he p e iously in oduced ma ices a e 5 × 5 acco ding o he
ep esen ed sys em mo ions. The sys em o al mass sums he con ibu ion o he s uc u al
mass and he hyd odynamic added mass om he ine ial componen o he adia ion
o ce. The damping ma ix is moun ed om he con ibu ion o h ee damping e ms.
Fi s , he linea hyd odynamic damping is p esen ed be o e in he case s udy sec ion. The
second is he s uc u al damping o he sys em. Thi dly, he damping con ibu ion om
he adia ion o ce is added o he model h ough di ec in eg a ion o he con olu ion o
he p oduc be ween adia ion impulse esponse and he s a e eloci y. Finally, he con-
ibu ion om he hyd os a ic es o ing and s uc u al s i ness a e included in he sys em
s i ness ma ix. The las e m, {𝐹𝐸𝑥𝑡(𝑡)}, ep esen s he ex e nal o ces ac ing on each o
he sys em DOFs. In he p esen model, his load ec o encompasses wind {𝐹𝑎}, wa e
{𝐹ℎ}, exci a ion o ces, and moo ing loads {𝐹𝑚𝑜𝑜𝑟}. In Figu e 2, a block diag am o he p o-
posed FOWT model is p esen ed:
Figu e 2. Block diag am o he p oposed model conside ing a con olle ac ion.
In he ollowing subsec ions, he di e en e ms o he p e iously p esen ed dynamic
equa ion a e ob ained o he case s udy model.
3.1. Ae odynamics
The in e ac ion be ween he wind and he u bine is de ined by he ae odynamic
model. Commonly, he Blade Elemen Momen um (BEM) heo y is applied o de ine he
wind u bine ae odynamic loads [28]. This heo y is a complex compu a ional me hod ha
equi es i e a ions o ob ain he axial and angen ial induc ion ac o s needed o calcula e
he li and d ag o ces in each o he blade sec ions. The use o his heo y is a oided
du ing simula ion un ime due o he nume ical e o equi ed o induc ion coe icien
de e mina ion. Howe e , i is applied in a p e-p ocessing s age whe e he wind u bine
ae odynamic p ope ies a e ob ained using Ae odyn [29] o di e en o o speeds and
pi ch angles assuming nacelle mo ions a e small so he ae odynamic p ope ies emain
Figu e 2. Block diag am o he p oposed model conside ing a con olle ac ion.
In he ollowing subsec ions, he di e en e ms o he p e iously p esen ed dynamic
equa ion a e ob ained o he case s udy model.
3.1. Ae odynamics
The in e ac ion be ween he wind and he u bine is de ined by he ae odynamic
model. Commonly, he Blade Elemen Momen um (BEM) heo y is applied o de ine he
wind u bine ae odynamic loads [
28
]. This heo y is a complex compu a ional me hod ha
equi es i e a ions o ob ain he axial and angen ial induc ion ac o s needed o calcula e
he li and d ag o ces in each o he blade sec ions. The use o his heo y is a oided
du ing simula ion un ime due o he nume ical e o equi ed o induc ion coe icien
de e mina ion. Howe e , i is applied in a p e-p ocessing s age whe e he wind u bine
ae odynamic p ope ies a e ob ained using Ae odyn [
29
] o di e en o o speeds and
pi ch angles assuming nacelle mo ions a e small so he ae odynamic p ope ies emain
cons an . These p ope ies, powe coe icien ,
CP(λ,β)
, and h us coe icien ,
CT(λ,β)
,
Ene gies 2022,15, 2228 6 o 16
shown in Figu e 3, a e compu ed as unc ions o he blade pi ch angle and he ip speed a io
(TSR o
λ
), which is he ela ion be ween he blade ip linea speed and he inciden wind:
λ=ΩR
w(2)
Ene gies 2022, 15, x FOR PEER REVIEW 6 o 17
cons an . These p ope ies, powe coe icien , 𝐶𝑃(𝜆,𝛽), and h us coe icien , 𝐶𝑇(𝜆,𝛽),
shown in Figu e 3, a e compu ed as unc ions o he blade pi ch angle and he ip speed
a io (TSR o 𝜆), which is he ela ion be ween he blade ip linea speed and he inciden
wind:
𝜆=Ω𝑅
𝑣𝑤
(2)
Figu e 3. Powe coe icien (le ) and Th us coe icien ( igh ).
In he p esen model, he wind u bine ae odynamics a e w i en based on he non-
dimensional powe and h us coe icien s ep esen ed as:
𝐹𝑎=12· 𝜌𝑎𝑖𝑟·𝐴𝑅𝑜𝑡𝑜𝑟·𝐶𝑇(𝜆,𝛽)·𝑣𝑅𝑒𝑙
2
(3)
𝑀𝑎=12· 𝜌𝑎𝑖𝑟·𝐴𝑅𝑜𝑡𝑜𝑟·𝐶𝑃(𝜆,𝛽)
Ω·𝑣𝑅𝑒𝑙
3
(4)
whe e 𝜌𝑎𝑖𝑟 is he ai densi y, 𝐴𝑅𝑜𝑡𝑜𝑟 is he o o a ea, Ω is he o o speed, and 𝑣𝑅𝑒𝑙 is
he ela i e wind speed a hub heigh , calcula ed as he wind speed (𝑣𝑤) educed by he
hub eloci y (𝑣ℎ𝑢𝑏):
𝑣𝑅𝑒𝑙=𝑣𝑤−𝑣ℎ𝑢𝑏
(5)
3.2. S uc u al Dynamics
Floa ing wind u bine mo ions and s uc u al esponses a e consequence o igid-
body mo ions a he han elas ic de o ma ions [30]. Hence, igid-body heo y is accu a e
enough o ep esen he FOWT dynamics o con ol design. Howe e , he p oposed
model is augmen ed o a mul i-body elas ic de o ma ion model o conside he owe as a
lexible elemen due o he in luence o he wind exci a ion o ce in o he owe dynamics
and con olle design. The owe is a con inuous s uc u e ha can be disc e ized in a i-
ous ways. I shea de o ma ion and la e al ine ia e ec s a e neglec ed, he owe de o -
ma ion can be modelled wi h a gene alized displacemen in combina ion wi h he p inci-
ple o i ual displacemen s, as is p oposed in [31] and la e used o model a FOWT in
[11,32]. In he p oposed model, his heo y is ollowed o desc ibe he owe esponse. The
mode is desc ibed by a shape unc ion; hus, he accu acy o he modelled esponse de-
pends on how well he de o ma ion is cap u ed by he shape unc ion. These shape unc-
ions a e commonly chosen o be he mos ele an eigenmodes o he sys em. In his case,
i is he owe ’s i s o e-a bending mode. In he p oposed model, he owe bending
mode shape unc ion is ob ained using BModes [33].
Figu e 3. Powe coe icien (le ) and Th us coe icien ( igh ).
In he p esen model, he wind u bine ae odynamics a e w i en based on he non-
dimensional powe and h us coe icien s ep esen ed as:
Fa=1
2·ρai ·ARo o ·CT(λ,β)· 2
Rel (3)
Ma=1
2·ρai ·ARo o ·CP(λ,β)
Ω· 3
Rel (4)
whe e
ρai
is he ai densi y,
ARo o
is he o o a ea,
Ω
is he o o speed, and
Rel
is he
ela i e wind speed a hub heigh , calcula ed as he wind speed (
w
) educed by he hub
eloci y ( hub):
Rel = w− hub (5)
3.2. S uc u al Dynamics
Floa ing wind u bine mo ions and s uc u al esponses a e consequence o igid-body
mo ions a he han elas ic de o ma ions [
30
]. Hence, igid-body heo y is accu a e enough
o ep esen he FOWT dynamics o con ol design. Howe e , he p oposed model is
augmen ed o a mul i-body elas ic de o ma ion model o conside he owe as a lexible
elemen due o he in luence o he wind exci a ion o ce in o he owe dynamics and
con olle design. The owe is a con inuous s uc u e ha can be disc e ized in a ious
ways. I shea de o ma ion and la e al ine ia e ec s a e neglec ed, he owe de o ma ion
can be modelled wi h a gene alized displacemen in combina ion wi h he p inciple o
i ual displacemen s, as is p oposed in [
31
] and la e used o model a FOWT in [
11
,
32
]. In
he p oposed model, his heo y is ollowed o desc ibe he owe esponse. The mode is
desc ibed by a shape unc ion; hus, he accu acy o he modelled esponse depends on
how well he de o ma ion is cap u ed by he shape unc ion. These shape unc ions a e
commonly chosen o be he mos ele an eigenmodes o he sys em. In his case, i is he
owe ’s i s o e-a bending mode. In he p oposed model, he owe bending mode shape
unc ion is ob ained using BModes [33].
Th ee igid-body modes o mo ion and one lexible mode a e conside ed in he model,
namely, su ge, hea e, pi ch, and he i s owe o e-a bending mode, espec i ely. Mo ions
a e e e enced o he cen e o g a i y o he whole sys em. The owe mode in oduces
o -diagonal e ms in he sys em ma ixes o couple he bending mode wi h su ge and pi ch
modes. The ac ual bending mode shape needed o ep esen he owe o e-a mode is no
Ene gies 2022,15, 2228 7 o 16
known and is, he e o e, ob ained om an eigen alue solu ion. Consequen ly, i is assumed
ha he shape o he o e-a mode is kep in he coupled model.
An addi ional a iable is added o ep esen he o o dynamics. Assuming a igid
d i e ain, a i s -o de dynamic equa ion is applied o conside he o a ional speed o he
o o in he p oposed model:
IR− T·Ig·.
Ω=Ma− T·Mg(6)
whe e he o e all d i e ain ine ia is compu ed by he o o ine ia (
IR
) and he gene a o
ine ia (
Ig
) exp essed in he low-speed sha by he gea box ela ion (
T
). The ine ia is
balanced by he ae odynamic o que (
Ma
) and he gene a o o que (
Mg
), which is kep
cons an o ob ain he equi ed o o speed (Ω).
3.3. Hyd odynamics
In a spa - ype ounda ion, one o he mos widely used me hods is ela ed o he
Mo ison equa ion and s ip heo y [
34
]. The heo y is applied o a ce ain en i onmen al
condi ion whe e wa es a e la ge, and he loa ing s uc u e is conside ed slende . Since he
p esen ed model aims o be used o con ol s a egy design, which is assessed wi hin he
wind u bine ope a ional ange, he en i onmen al sea condi ions a e smoo he , enabling
solu ion o he adia ion–di ac ion p oblem wi h he linea po en ial low heo y [
35
,
36
].
The equency-dependen added mass and adia ion damping a e p ecompu ed in panel
code so wa e, such as he p og ams AQWA [
37
] o WAMIT [
38
], o he speci ic OC3
pla o m shape. The hyd os a ic es o ing ma ix including he con ibu ion om he
buoyancy cen e (CB) and wa e plane a ea oge he wi h he wa e exci a ion o ce a e
ob ained om he panel code sol e based on he p e iously men ioned linea po en ial
low heo y.
To compu e he ime-domain hyd odynamic adia ion o ce, he sum o added mass
and adia ion damping, he so-called ee-su ace memo y e ec , is conside ed by he
con olu ion in eg al o he e a da ion unc ion. In [
39
,
40
], he app oach based on he
po en ial heo y esul s is de eloped and alida ed. The ime-domain alues o he added
mass and adia ion damping a e compu ed as ollows:
A∞=a(ω)+Z∞
0B(τ)·sin(ωτ)d (7)
B(τ)=2
π·Z∞
0b(ω)·cos(ωτ)dω(8)
whe e
a(ω)
is he added mass and
b(ω)
is he adia ion damping, bo h unc ion o equency.
B(τ)
is he e a da ion unc ion ob ained om he cosine ans o ma ion o he impulse
adia ion unc ion. Addi ionally, he hyd odynamic damping model is augmen ed wi h
addi ional linea damping, which is added o ma ch he ee-decay es , as is de ailed in [
21
]
and summa ized in Table 1. Second-o de e ec s would be o signi ican in luence in su ge
mo ions and need o be accoun ed o o conclude he bene i s o speci ic con ol s a egies.
Howe e , his aspec alls beyond he scope o his pape and has no been in oduced.
Slowly a ying second-o de d i o ces will be conside ed in subsequen e sions o he
p esen ed model, enabling i s applicabili y o assess wind u bine con ol s a egies.
3.4. Moo ing Dynamics
The loa ing sys em is ancho ed o he seabed by h ee ca ena y moo ing lines, wi h
a del a connec ion o inc ease he s i ness in yaw. Howe e , he simpli ica ions assumed
in [
21
] a e also conside ed. Each o he moo ing lines is supposed as a con inuous cable
wi h homogeneous p ope ies. I he o ces om ine ia, iscous d ag, in e nal damping,
and bending and o sion modes a e neglec ed, a quasi-s a ic analysis app oach can be
applied o ob ain he non-linea ca ena y s i ness as a unc ion o he pla o m mo emen ,
as is jus i ied in [
41
]. In a p ep ocessing s ep, he moo ing s i ness o ces a e ob ained o
Ene gies 2022,15, 2228 8 o 16
di e en pla o m displacemen s ha a e hen used as a look-up able o compu e he o ces
in each simula ion ime s ep.
3.5. Modeling o Wind and Wa e Resou ce
Wind and wa e modeling is simula ed ollowing he guidelines p o ided in [
42
]. The
Kaimal spec um model is selec ed o include he u bulen componen o he wind and
he JONSWAP spec um is used o ep esen di e en sea s a es. The wind speed Kaimal
spec um is de ined by he ollowing exp ession:
Sk( )=σ2
U
6.868 LU
U10
1+10.32 LU
U10 5/3 (9)
In his exp ession,
deno es he equency,
σU
is he wind s anda d de ia ion,
U10
is he 10 min mean wind speed a 10 m heigh abo e he s ill wa e le el, and
LU
is he
in eg al leng h scale o he wind speed p ocess. This las pa ame e can be ob ained as:
LU=300 z
3000.46+0.074 ln z0(10)
whe e zis he heigh abo e sea wa e le el and z0is he e ain oughness calcula ed by:
z0=Ac
g
kaU(z)
lnz
z0
2
(11)
whe e
ka
is he on Ka man’s cons an (
ka=
0.4),
g
is he g a i y accele a ion, and
Ac
is
he Cha nock’s cons an , which is dependen on he wa e eloci y and he a ailable wa e
e ch. The de ini ion o he implemen ed me hod o
Ac
is gi en in [
43
]. The simula ions
ca ied ou in Sec ion 4a e compu ed using he u bulen wind speed ime se ies de eloped
wi h he p oposed model; see Figu e 4.
Ene gies 2022, 15, x FOR PEER REVIEW 9 o 17
The wa e ele a ion used in Sec ion 4 simula ions is ob ained using his spec um. In Fig-
u e 4, he i egula wa e ele a ion ime se ies can be seen.
Figu e 4. Wind and wa e ime se ies.
4. Model Valida ion
The model is implemen ed in Py hon and i s accu acy is es ed agains OpenFAST
[8]. OpenFAST is a mul i-physics, mul i- ideli y ool o simula ing he coupled dynamic
esponse o wind u bines, bo h ixed and loa ing. The OC3 p ojec [44] collec s esul s
om di e en modeling ools including OpenFAST. To alida e he p oposed low-o de
model, some o he load cases simula ed in ha p ojec a e used. In Table 4, a summa y o
he di e en load cases is shown:
Table 4. Load cases o model compa ison.
Load Case
Models
Wind
Wa es
Analysis
1
OpenFAST
Reduced Model
None
None
Eigenanalysis
Decay Tes s
2
OpenFAST
Reduced Model
None
Regula
RAOs
3
OpenFAST
Reduced Model
None
I egula
JONSWAP spec um
𝐻𝑠 = 6 𝑚
𝑇𝑝=10 𝑠
Time se ies
PSDs
4
OpenFAST
Reduced Model
Tu bulen :
Kaimal spec um
𝑉𝑤 = 8 𝑚/𝑠
None
Time se ies
S a is ics
5
OpenFAST
Reduced Model
Tu bulen :
Kaimal spec um
𝑉𝑤 = 18 𝑚/𝑠
I egula :
JONSWAP spec um
𝐻𝑠 = 6 𝑚
𝑇𝑝=10 𝑠
Time se ies
PSDs
S a is ics
In he ollowing, he compa ison o he esul s be ween he p oposed model and
OpenFAST is p esen ed. All he simula ions ha e a simula ion leng h ime o 1800 s, and
o a oid ini ial ansien e ec s he i s 800 s a e neglec ed. The con olle dynamics a e
simula ed o a speci ic ope a ion poin , which means a cons an o o speed and a con-
s an blade pi ch angle o he comple e simula ion. The alues o he wind u bine s eady
Figu e 4. Wind and wa e ime se ies.
The wa es a e modelled ollowing he Ai y wa e heo y, which conside s ha he
luid laye has a uni o m dep h and i s low is in iscid, incomp essible, and i o a ional [
39
].
The JONSWAP wa e spec um is u ilized due o i s alidi y o ep esen a sea s a e in a
e ch limi ed si ua ion, and i is o mula ed by:
Sj(ω)=AγSPM(ω)γexp (−0.5(ω−ωp
σ ωp)) (12)
Ene gies 2022,15, 2228 9 o 16
whe e he Pie son–Moskowi z spec um (SPM) is de ined as:
SPM(ω)=5
16 H2
Sω4
pω−5exp −5
4ω
ωp−4!(13)
The pa ame e
HS
is he signi ican wa e heigh ,
ωp
is he angula spec al peak
equency ob ained om he peak pe iod alue (
Tp
), and
ω
is he angula equency.
Aγ
is a no malizing ac o unc ion o he non-dimensional peak shape pa ame e ,
γ
, and
σ
is a spec al wid h pa ame e . The alues used o he pa ame e s a e hose p oposed
in [
42
]. The wa e ele a ion used in Sec ion 4simula ions is ob ained using his spec um.
In Figu e 4, he i egula wa e ele a ion ime se ies can be seen.
4. Model Valida ion
The model is implemen ed in Py hon and i s accu acy is es ed agains OpenFAST [
8
].
OpenFAST is a mul i-physics, mul i- ideli y ool o simula ing he coupled dynamic
esponse o wind u bines, bo h ixed and loa ing. The OC3 p ojec [
44
] collec s esul s
om di e en modeling ools including OpenFAST. To alida e he p oposed low-o de
model, some o he load cases simula ed in ha p ojec a e used. In Table 4, a summa y o
he di e en load cases is shown:
Table 4. Load cases o model compa ison.
Load Case Models Wind Wa es Analysis
1OpenFAST
Reduced Model None None Eigenanalysis
Decay Tes s
2OpenFAST
Reduced Model None Regula RAOs
3OpenFAST
Reduced Model None
I egula
JONSWAP spec um
Hs=6 m
Tp=10 s
Time se ies
PSDs
4OpenFAST
Reduced Model
Tu bulen :
Kaimal spec um
Vw=8 m/s
None Time se ies
S a is ics
5OpenFAST
Reduced Model
Tu bulen :
Kaimal spec um
Vw=18 m/s
I egula :
JONSWAP spec um
Hs=6 m
Tp=10 s
Time se ies
PSDs
S a is ics
In he ollowing, he compa ison o he esul s be ween he p oposed model and
OpenFAST is p esen ed. All he simula ions ha e a simula ion leng h ime o 1800 s, and
o a oid ini ial ansien e ec s he i s 800 s a e neglec ed. The con olle dynamics a e
simula ed o a speci ic ope a ion poin , which means a cons an o o speed and a cons an
blade pi ch angle o he comple e simula ion. The alues o he wind u bine s eady
ope a ion a e ob ained om [
22
]. This is done o educe he numbe o a iables ac ing on
he sys em’s dynamic esponse.
4.1. Load Case 1
The sys em na u al equencies a e calcula ed wi h he p oposed model by sol ing he
eigen alue p oblem ma hema ically desc ibed as:
−nω2
no[M]+[K]{ˆ
x(ω)}={0}(14)
To ob ain he na u al equencies in OpenFAST, a PSD o he decay es e ealed
he sys em’s damped na u al equency. A compa ison o na u al equencies is gi en in
Ene gies 2022,15, 2228 16 o 16
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