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Resea ch pape
Elec ic powe op imiza ion in sola ough plan s wi h deep lea ning-based
model p edic i e con ol
Sa a Ruiz-Mo eno ∗, An onio J. Gallego, José Ramón D. F ejo , Edua do F. Camacho
Dep . de Ingenie ía de Sis emas y Au omá ica, Uni e si y o Se ille, Camino de los Descub imien os, no numbe , E-41092, Se ille, Spain
A R T I C L E I N F O
Keywo ds:
Sola ene gy
Pa abolic- ough collec o s
A i icial neu al ne wo ks
Deep lea ning
Powe op imiza ion
A B S T R A C T
On-si e compu a ional capabili ies limi he op imiza ion o elec ici y p oduc ion in comme cial plan s. This
wo k add esses he implemen a ion o a i ual ope a o o elec ic powe maximiza ion in pa abolic- ough
collec o plan s by manipula ing he low a e ci cula ing h ough he pipes wi h a neu al ne wo k-based
p ocedu e. A model p edic i e con ol (MPC) s a egy is p oposed using nonlinea models o p edic he
sys em’s esponse. The e ec o including di e se pa s o he plan in he p edic ion model and using di e en
objec i e unc ions ( he model o he pump and he pipes and he di e ence be ween op imizing he mal and
elec ic powe ) is analyzed. Fi s , wo con ol laye s a e implemen ed: one o ob aining he empe a u e
se poin s and one o compu ing he low a e.
Since he nonlinea MPC canno be compu ed in eal- ime o medium and la ge plan s, an a i icial neu al
ne wo k is ained o lea n he op imal solu ion and lowe he compu a ional bu den, educing a 2-laye MPC
s a egy in o only one neu al con olle ha in e nally decides he ope a ing poin . The simula ion esul s
ob ain a mean ime educ ion o 99.99995%, while he elec ici y p oduc ion o he s udied cases is only
educed by a ound 1%, making he con olle implemen able in eal- ime in ac ual plan s.
1. In oduc ion
The educ ion o emissions o he a mosphe e is one o he main
challenges oday. Fo his pu pose, esea ch in enewable ene gy sou ces
is on he end o educe he impac o ossil uels (Blanco and Mille ,
2017). Renewable ene gy is expe iencing apid global g ow h ha will
help educe ca bon emissions while supplying he expanding elec ici y
demands o he p esen -day (Du e al., 2019). Acco ding o he Wo ld
Ene gy Council, he sha e o sola and wind powe has inc eased o
12% o he global ene gy gene a ion, and he sha e o enewable ene gy
sou ces in he global powe ou pu will be a leas 50% by 2050 (Anon,
2024).
One o he mos ele an ene gy sou ces, i no he mos , is he Sun.
Sola ene gy is he cleanes enewable ene gy sou ce, which makes
i a good al e na i e o ossil uels (Ajba e al., 2022), al hough one
o he challenges oday is s ill o make i e icien and compe i i e.
Due o i s impo ance and abundance, sola ene gy has a ac ed mo e
and mo e in e es , and he U.S. Na ional Academy o Enginee ing and
he Eu opean Commission ha e iden i ied he objec i e o inc easing
i s economic bene i and compe i i eness as one o he signi ican
challenges o he cen u y (Anoun, 2019).
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (S. Ruiz-Mo eno), [email p o ec ed] (A.J. Gallego), [email p o ec ed] (J.R.D. F ejo), [email p o ec ed] (E.F. Camacho).
Sola ene gy echnologies a e di ided in o wo ypes: pho o ol aics
(PV), which di ec ly ans o ms he sunligh in o elec ici y and con-
cen a ing sola powe (CSP), which p oduces s eam o d i e a u bine
gene a o . Wi hin CSP sys ems, his wo k ocuses on pa abolic- ough
collec o s (PTC), which use pa abolic mi o s o concen a e sola ays
in o a luid con ained in a pipe (Gallego e al., 2019). Sola he mal
ene gy can be s o ed in he mal ene gy s o age (TES) anks, which
allows hem o p oduce ene gy du ing he nigh and makes hem
compe i i e. Recen ly, many s udies ha e been conduc ed o imp o e
hei pe o mance, con ibu ing o i s g ow h and making PTCs he
mos ma u e o all CSP echnologies (Ajba e al., 2022). As epo ed
by Tagle-Salaza e al. (2020), PTCs a e used in nume ous indus ial
applica ions, such as elec ici y gene a ion, seawa e desalina ion, o
wa e decon amina ion, and he e a e nume ous plan s unde con-
s uc ion a ound he wo ld. In gene al, he challenges aced by he
CSP indus y a e hei use in coun ies whe e hey ha e ha dly been
buil ye , ma ke esilience, compe i ion om na u al gas, and low
in es men in some coun ies (Anon, 2022a).
T adi ionally, he con ol objec i e in PTC plan s has been o
main ain he ou le empe a u e a he desi ed le el by manipula ing
he low a e Kannaiyan and Bokde (2022). This allows he plan o be
h ps://doi.o g/10.1016/j.engappai.2025.110832
Recei ed 14 No embe 2023; Recei ed in e ised o m 1 Janua y 2025; Accep ed 7 Ap il 2025
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
A ailable online 3 May 2025
0952-1976/© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
S. Ruiz-Mo eno e al.
Nomencla u e
Pa ame e s and a iables
𝛿sDeclina ion (◦)
𝜖E ec i e oughness (m)
𝜂E iciency (−)
𝜇Dynamic iscosi y (mPa s)
𝜔s( ) Hou ly angle (◦)
𝜌(𝑇)Densi y (kg/m3)
𝐴Pipe c oss-sec ional A ea (m2)
𝐶(𝑇)Speci ic hea capaci y (J/(kg ◦C))
𝐺Collec o ape u e (m)
𝑔G a i y o Ea h (m/s2)
𝐻l(𝑇)The mal loss coe icien (W/(m2◦C))
𝐻 (𝑇)Con ec i e hea ans e coe icien (W/
(m2◦C))
𝐼(𝑡)Di ec sola i adiance (W/m2)
𝐾op Op ical e iciency (−)
𝐿Tube pe ime e (m)
𝐿loop Loop leng h (m)
𝐿pPipe leng h (m)
𝑛o(𝑡)Geome ic e iciency (−)
𝑞(𝑡)Volume ic low a e (m3∕s)
𝑆To al a ea o he ield (m2)
𝑡Time (s)
𝑇(𝑡)Tempe a u e (◦C)
𝑋(𝑡)Neu al ne wo k’s inpu ec o
𝜙Weigh ing ac o (−)
𝑎Neu on’s ou pu (−)
𝑏Neu on’s bias uni (−)
𝐸cSola cons an (−)
𝑔Ac i a ion unc ion (−)
𝐻pP edic ion ho izon (−)
𝐻uCon ol ho izon (−)
𝐽dJulian day (−)
𝐽Cos unc ionW
𝑘Times ep (−)
𝑛Numbe o neu ons (−)
𝑡(𝑡)T ansmi ance (W/ m2K)
𝑤Neu on’s weigh ec o (−)
𝑌(𝑡)Neu al ne wo k’s ou pu ec o
Subsc ip s
a Ambien
el Elec ic
Fluid
in Inpu
ini Ini ial
loop Mean o he en i e loop
m Me al
mean Mean be ween inpu and ou pu
ou Ou pu
p Pump
p ed P edic ed
e Re e ence
sg S eam gene a o
h The mal
s abilized be o e changing he se poin o ul ill he di e en equi e-
men s o he ope a o s. On he con a y, some e e ences add ess he
maximiza ion o he powe ob ained. Fo example, he wo k by F ejo
and Camacho (2020) maximizes he he mal powe o he plan and
s a es ha , in e ms o he mal powe , i is p e e able o ope a e he
plan a low empe a u es. Ano he in e es ing app oach is o ocus on
elec ici y p oduc ion, as in he wo k by Camacho and Gallego (2013).
PTC plan s a e highly nonlinea , ime- a ian , and domina ed by
complex dynamics, making i di icul o adi ional con ol echniques
o egula e he low a e. Model p edic i e con ol (MPC) can cope wi h
he dis u bances o sola he mal plan s while aking in o accoun some
cons ain s and ul illing di e en c i e ia.
Di e en linea MPC s a egies ha e been used o add ess he con-
ol speci ica ions. Gallego e al. (2018a) implemen a gain scheduling
MPC con olle ha combines linea MPC wi h a eed o wa d con olle
o linea ize he sys em esponse. Wang e al. (2022) p opose a mul i-
model adap i e p edic i e scheduling con olle o he ou le s eam
empe a u e o a PTC plan . Ma ín e al. (2019) use an MPC s a egy
based on ea ly swi ch-o o maximize he he mal ene gy and imp o e
he e iciency in a swimming pool ha wo ks wi h a sola collec o and
biomass.
Nonlinea MPC con olle s can deal wi h he dynamics o sola
he mal plan s go e ned by high nonlinea i ies. The d awback o his
echnique is he compu a ional cos o sol ing a complex op imiza-
ion p oblem a each sample ime, especially in he case o la ge
plan s. Di e en app oaches ha e been p oposed in he li e a u e o
add ess his p oblem, and i is he subjec o cu en esea ch. Fo
ins ance, Mase o e al. (2022) p opose a coali ional MPC app oach
based on ma ke supply and demand o he mal powe maximiza ion.
A nonlinea uzzy MPC con olle o educing he p edic ion ime is
p oposed by Escaño e al. (2021) o egula ing he ou le empe a u e
a ound gi en se poin s. Pa a o e al. (2021) p opose he use o a mixed-
logical dynamic model, which is linea in he s a e space, o con olling
hyb id sola he mal plan s conside ing a ia ions in load demand. A
quasi-linea pa ame e a ying MPC con olle is p oposed by Mo a o
e al. (2021) o empe a u e egula ion o a sola collec o sol ing a
min–max p oblem. F ejo and Camacho (2020) p opose a dis ibu ed
algo i hm o op imize he plan loops.
The ul ima e goal o comme cial plan s is o ob ain he highes ben-
e i , which is achie ed by he maximiza ion o elec ici y p oduc ion.
This equi es using complex models and sol ing highly ime-demanding
op imiza ion p oblems. Howe e , he compu a ional capabili ies o
on-si e compu e s a e usually minimal, and sola he mal plan s a e
con olled wi h sampling imes o less han one minu e (Camacho e al.,
2012). Ano he issue wi h he implemen a ion o con ol echniques in
comme cial plan s is ha hey a e usually ins alled on a cen alized
se e (Gallego e al., 2022). This u he unde sco es he need o
educe he compu a ional bu den and simpli y he con olle as much
as possible, posing cons ain s on compu a ion imes and equi ing
con olle s ha a e easy o implemen .
A way o app oach his issue is using an a i icial neu al ne wo k
(ANN). The applica ion o ANNs in he li e a u e is ecei ing g owing
a en ion o hei abili y o ep oduce nonlinea unc ions and hei
as implemen a ion. In indus ial applica ions, hei use is gene ally
mo e ocused on sys em modeling o educe p edic ion imes. Fo
example, he wo k by Lee e al. (2022) analyzes he use o an ANN in
he p edic ion model o an MPC con olle o he mal ene gy s o age
wi h a me aheu is ic algo i hm o he sol e , and Hassanpou e al.
(2022) use ecu en neu al ne wo ks o modeling a chemical eac o
in an MPC scheme. The wo k by Zaaoumi e al. (2021) compa es h ee
models o es ima ing he hou ly elec ic p oduc ion in a PTC plan and
concludes ha he bes pe o mance is ob ained wi h a neu al ne wo k.
Rega ding he applica ion o ANNs di ec ly o con ol he sys em,
al hough i is much less equen , some examples can also be ound in
he li e a u e. Fo ins ance, No ouzi e al. (2022) apply ecu en neu al
ne wo ks bo h o model a combus ion engine and o subs i u e an MPC
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
2
S. Ruiz-Mo eno e al.
con olle ha uses i o p edic ions. The wo k o Chan eu e al.
(2021) uses a eed o wa d ANN o ini ialize he Lag ange mul iplie s in
a dis ibu ed MPC based on dual decomposi ion o a 16- ank sys em.
Sola plan s s ill do no ha e many applica ions. Amma e al. (2013)
use an ANN o ob ain he ope a ing poin in pho o ol aic- he mal
panels, Sun e al. (2017) p opose he use o ANNs o con ol esiden ial
pho o ol aic sys ems based on app oxima e dynamic p og amming,
and Ce an es-Bobadilla e al. (2021) use an in e se a i icial neu al
ne wo k wi h pa icle swa m op imiza ion o ob ain he low a e o a
PTC gi en he desi ed empe a u e.
Al hough ANNs ha e been p oposed in he li e a u e o model he
plan s used by he con olle s o educe compu a ion imes, his wo k
aims o di ec ly eplace he comple e con olle , d as ically augmen -
ing he educ ion. A p e ious wo k add essed he compu a ion imes
by applying a neu al ne wo k o ep oduce he beha io o an MPC
con olle (Ruiz-Mo eno e al., 2021), bu he con ol objec i e was o
maximize he he mal powe ob ained ins ead o he elec ic powe ,
which is he ac ual p oduc o in e es in a comme cial plan . Op imiz-
ing he mal powe ins ead o elec ic powe is mo e s aigh o wa d o
sol e since he p oblem can easily be assimila ed in o a empe a u e
minimiza ion p oblem. Fo gi en design pa ame e s, maximizing he -
mal powe implies minimizing he mal losses o he en i onmen . This
is p oduced a lowe wo king empe a u es. When op imizing elec ic
powe , he op imal ope a ing poin a ies h oughou he day (depend-
ing on he empe a u e, posi ion, and adia ion o he Sun). This is
added o he ac ha he plan ope a es in ansien mode, making
he op imiza ion p oblem non i ial and hinde ing he applica ion o a
neu al ne wo k.
The con ol p oblem in PTC plan s is highly nonlinea , and he mo e
accu a e he models a e, he mo e complica ed hei con ol and he
mo e compu a ional ime i equi es, especially in plan s wi h many
loops such as Moja e (282) (Anon, 2022b) o Solana (808) (Anon,
2022c), whe e he loops mus be con olled by sec o s, and no en i ely
dis ibu ed con ol s a egies can be applied.
Gi en he challenge o con olling he plan , his wo k aims o
imp o e he con olle de eloped in Ruiz-Mo eno e al. (2021) wi h a
wo-laye MPC s a egy: one ex e nal laye o compu ing he empe -
a u e se poin s and one in e nal laye o egula ing he plan a ound
hose se poin s by manipula ing he low a e. This allows he sys em
o each op imal empe a u es and acili a es he con ol p ocess. In
his pape , a compa a i e analysis is p esen ed, demons a ing how
esul s imp o e when addi ional elemen s a e inco po a ed in o he
objec i e unc ion. Nex , a neu al ne wo k is applied o lea n om he
wo laye s o MPC con olle s and educe he compu a ional bu den.
This wo k in es iga es he use o one neu al ne wo k pe laye and one
neu al ne wo k o subs i u e he whole con ol s a egy. The esul s a e
ob ained by simula ion o he ACUREX plan . The main ad an ages o
his app oach wi h espec o he p e ious wo k a e as ollows:
•A mo e accu a e model o he ac ual plan . This is achie ed by
including pump consump ion and pipes.
•An easie implemen abili y. The adia ion p o iles a e no as-
sumed o be known by he con olle .
•Elec ic powe op imiza ion, ins ead o he mal powe , o imp o e
he pe o mance and each he neu al ne wo k.
To he bes o he au ho ’s knowledge, his is he i s ime ha
his app oach has been applied o elec ical powe maximiza ion in
sola he mal plan s since neu al ne wo ks had been p e iously used in
his ype o plan o asks such as aul de ec ion (Ruiz-Mo eno e al.,
2022), modeling (Cox e al., 2019), es ima ion (Gh i lah e and P asad,
2018) and he mal powe maximiza ion (Mase o e al., 2023), which
complexi y is di e en om ha o an elec ical powe maximiza ion
p oblem. He eby, he main con ibu ions o his wo k a e he ollowing:
•The use o ANNs o maximize elec ic powe in PTC sys ems wi h
sa is ac o y esul s and negligible compu a ion imes.
Fig. 1. ACUREX collec o loops.
•The educ ion o a wo-laye me hodology in o a single con ol
s ep.
•An analysis o he e ec o conside ing he pipes and he pump
in he con ol sys em, usually neglec ed in he li e a u e.
•A de ailed compa ison o ou le empe a u e maximiza ion and
minimiza ion, he mal powe maximiza ion, and elec ic powe
maximiza ion.
The emainde o his pape is as ollows. Sec ion 2 p o ides a
desc ip ion o he PTC plan and i s models. Sec ion 3 desc ibes he
implemen a ion o he 2-laye MPC con olle and he a i icial neu al
ne wo ks. Then, Sec ion 4 shows some aining and simula ion esul s.
Sec ion 5 gi es some discussion, and inally, Sec ion 6 d aws some
conclusions and ema ks on some lines o u u e wo k.
2. Sys em desc ip ion
This sec ion p esen s he plan used o e alua e he me hodology. In
his wo k, he simula ions a e ca ied ou o one loop o ACUREX (Ca-
macho e al., 2012), a PTC plan o 1 MW ha was loca ed a he
Pla a o ma Sola de Alme ía. I is composed o 10 loops o single-axis
Eas -Wes aligned collec o s a anged in 12 modules o 4 collec o s. The
loops a e 172 m long and include an ac i e pa o 142 m ha ecei es
sola adia ion and a passi e pa o 30 m isola ed om sola adia ion.
The hea ans e luid is The minol 55 he mal oil, wi h he densi y 𝜌
and speci ic hea capaci y 𝐶 o Eqs. (1) and (2). The collec o loop
is shown in Fig. 1. The ACUREX plan is p o ided wi h a sun- acking
sys em ha makes he mi o s o a e a ound an axis pa allel o he pipe
axis. I is based on he ol age di e en ial be ween wo pho odiodes
si ua ed on he collec o axis (Camacho e al., 2012).
𝜌 = 903 − 0.672𝑇 (1)
𝐶 = 1820 + 3.478𝑇 (2)
Fig. 2 shows a scheme o one collec o loop wi h he pipes a he
inpu and ou pu o he ield and he s eam gene a o . The collec o s
and he pipes a e modeled wi h a dis ibu ed pa ame e model, and he
s eam gene a o is modeled as a empe a u e d op o 80 oC. Assuming
he ield is pe ec ly balanced and all loops ha e he same e iciency,
he ield is modeled as an equi alen loop.
2.1. Concen a ed pa ame e model
The concen a ed o lumped pa ame e model is used o implemen
a o wa d con olle (Camacho e al., 2012). I p o ides a simpli ied
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
3
S. Ruiz-Mo eno e al.
Fig. 2. Scheme o one collec o loop wi h pipes and s eam gene a o .
desc ip ion o he plan wi h he in e nal ene gy a ia ion o he luid
and is gi en by Eq. (3), wi h he no a ion o he nomencla u e sec ion.
The es o he coe icien s ha make up he equa ion a e ob ained o
he loop using he mean alues as 𝐶loop =𝐿loop𝜌m𝐶m𝐴 and 𝑃cp =
𝜌m𝐶m.
𝐶loop
𝑑𝑇ou
𝑑𝑡 =𝑛o𝐾op 𝐼𝑆 +𝑞𝑃cp(𝑇in −𝑇ou ) + 𝐻l𝐴(𝑇a−𝑇mean)(3)
2.2. Dis ibu ed pa ame e model
To simula e he sys em, he dis ibu ed pa ame e model gi en by
he pa ial di e en ial Eqs. (4) and (5) p o ides a desc ip ion o he
ene gy balances in he me al and he luid wi h spa ially dis ibu ed
a iables (Camacho e al., 2012). A uni o m local concen a ion a io is
assumed since he dimensions o he e lec o and ecei e a e assumed
o be equal along he loop, excep o he passi e pa s. The empe a u e
o he me al is assumed o be adially cons an . The loop is disc e ized
longi udinally in o 172 segmen s o 1 m and is compu ed wi h an
in eg a ion ime o 0.25 s o acili a e p oblem esolu ion.
𝜌m𝐶m𝐴m
𝜕𝑇m
𝜕𝑡 =𝑛o𝐺𝐾op 𝐼+𝐻l𝐺(𝑇a−𝑇m) + 𝐿𝐻 (𝑇 −𝑇m)(4)
𝜌 𝐶 𝐴
𝜕𝑇
𝜕𝑡 +𝑞𝜌 𝐶
𝜕𝑇
𝜕𝑥 = −𝐿𝐻 (𝑇 −𝑇m)(5)
The pa ame e s o he me al in he sys em a e 𝜌m= 7800 kg/m3,
𝐶m= 550 J/Kg◦ C, 𝐴m= 2.4806 ⋅10−4 m2, 𝐺= 1.82 m, 𝐿= 7.98 ⋅10−2 m
and 𝐴 = 5.0671⋅10−4 m2. The he mal loss coe icien 𝐻l is gi en by Eq.
(6) (i includes he adia ion and ex e nal con ec ion esis ances), and
he coe icien o con ec i e hea ans e o he inne ube 𝐻 is
calcula ed by Eq. (7) (Camacho e al., 2012).
𝐻l= 0.00249 (𝑇 −𝑇a)− 0.06133 (6)
𝐻 =𝑞0.8(2.17 ⋅106− 5.01 ⋅104𝑇 + 4.53 ⋅102𝑇2
− 1.64𝑇3
+ 2.1⋅10−3𝑇4
)(7)
The geome ic e iciency is some imes e e ed o as 𝑐𝑜𝑠(𝜃) and de-
pends on he collec o dimensions, declina ion, la i ude, hou ly angle,
sola hou , and Julian day (Gallego e al., 2018b). I is ob ained wi h
he ela ion be ween he no mal ec o o he mi o and he adia ion
beam ec o . Fo a plan like ACUREX, which is o ien ed Eas -Wes ,
he Sun is acked on ele a ion, and he geome ic e iciency is ob ained
om Eq. (8) (Camacho e al., 2012), whe e 𝛿s and 𝜔s a e he declina ion
and he hou ly angle.
𝑛o=(1 − cos2(𝛿s) sin2(𝜔s))1
2(8)
2.3. Pipes
The pipes a e modeled using he dis ibu ed pa ame e model om
Eqs. (4) and (5), bu in his case, he pipes a e insula ed and he
pa ame e s a e di e en (Camacho and Gallego, 2013). The esul ing
model is desc ibed by Eqs. (9) and (10), wi h 𝐴 = 6.4⋅10−3 m2,
𝐴m= 1.5⋅10−3 m2, 𝐿= 0.09𝜋 m, and 𝑞 ield = 10𝑞. This las conside a ion
is due o he ac ha his pipe collec s he ou pu om all loops. The
ubes a e di ided in o 140 segmen s o 1 m.
𝜌m𝐶m𝐴m
𝜕𝑇m
𝜕𝑡 =𝐿𝐻 (𝑇 −𝑇m)(9)
𝜌 𝐶 𝐴
𝜕𝑇
𝜕𝑡 +𝑞 ield𝜌 𝐶
𝜕𝑇
𝜕𝑥 = −𝐿𝐻 (𝑇 −𝑇m)(10)
2.4. Pump
The consump ion o he pump (𝑃p) (Camacho and Gallego, 2013;
Kundu e al., 2015) a ec s he ne powe ob ained. I is usually ne-
glec ed since i s alue is low compa ed o he he mal and elec ic
powe s. Howe e , i can be conside ed in he op imiza ion p oblem’s
cos unc ion. This wo k s udies he di e ence in he con olled sys em
when i is included.
A s eady s a e is assumed, and he losses o he es o he loops
a e neglec ed since he con ol sys em only conside s one loop. I is
modeled as a sum o ic ion and ine ia and in Eq. (11).
𝑃p=𝑞𝜌 𝑔ℎ
𝜂pump
=𝑞
𝜂pump (8𝑓𝐿p𝑞2
𝑔𝜋2𝐿5+16𝑞2
𝑔𝜋2𝑑4)(11)
whe e 𝜂pump = 0.9, 𝑔= 9.81 m/s2, 𝐿= 2.54⋅10−2. The alue o he pump
e iciency is assumed, as i was no a ailable om he old ACUREX
plan , and i does no o e ide he me hodology and he conclusions
d awn om his s udy.
The pump consump ion depends on he Ba ’s ic ion coe icien
𝑓 and he Reynolds numbe 𝑅𝑒, gi en by Eqs. (12) and (13) (Punmia
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S. Ruiz-Mo eno e al.
e al., 1995), whe e he dynamic iscosi y is compu ed by Eq. (14),
ob ained by leas squa es and wi h manu ac u e da a.
1
√𝑓
= −2 log10 (𝜖
3.7+5.1286
𝑅𝑒0.89 )(12)
whe e 𝜖 =𝜖∕𝐿= 7.874 ⋅10−4 m.
𝑅𝑒 =𝜌 𝑞
𝐴 𝜇(13)
𝜇= 3.558 − 0.0435𝑇 + 2.2860 ⋅10−4𝑇2
− 5.5037 ⋅10−7𝑇3
+ 4.9425 ⋅10−10𝑇4
(14)
2.5. The mal and elec ic powe
The he mal powe (𝑃 h) is compu ed using Eq. (15) e alua ed in he
s eam gene a o , using he mean empe a u e be ween i s inpu 𝑇sg,in
(which is he ou pu o he pipe loca ed a e he collec o s) and i s
ou pu 𝑇sg,ou .
𝑃 h =𝑞𝜌 𝐶 (𝑇sg,ou −𝑇sg,in)(15)
The powe cycle is simpli ied by a Rankine cycle. The powe con-
e sion sys em is modeled as a ela ion be ween oil empe a u e and
cycle e iciency (Goswami e al., 2000). 𝑇sg,ou and 𝑇sg,in co espond o
he hold and cold ocus empe a u es in he Rankine cycle. The elec ic
powe (𝑃el) is he p oduc o he he mal powe and he Rankine
e iciency 𝜂 ank, as in Eq. (16). The ho and cold ocus empe a u es
a e assumed o be he inle empe a u e o he s eam gene a o and he
ambien empe a u e. The emaining pa ame e is 𝐾= 0.772, a cons an
ha models he e iciency loss wi h espec o he Ca no cycle and is
es ima ed wi h he equa ions o he powe con e sion sys em desc ibed
by Camacho e al. (2012). As desc ibed in Camacho and Gallego (2013),
𝐾 is ob ained by adjus ing a cu e wi h h ee poin s om he ac ual
Rankine cycle o app oxima e he modeled Rankine cycle o he eal
one.
𝑃el =𝜂 ank𝑃 h =𝐾(1 − 𝑇a
𝑇sg,in )𝑃 h (16)
3. Me hodology p oposed
The con olle s p oposed in his pape a e desc ibed in his sec ion.
Fi s , wo MPC con olle s a e applied o compu e he op imal con ol
signals along he p edic ion ho izon, and hen a neu al ne wo k is
ained o app oxima e he alues o he low a e wi h much lowe
imes han he MPC con olle .
The MPC con olle s a e applied in wo s ages: one o ob aining
he op imal empe a u es and ano he o calcula ing he low a e
equi ed. The compu a ion o he wo laye s sepa a ely allows he em-
pe a u es o s abilize be o e changing he ope a ing poin , acili a ing
he esolu ion o he op imiza ion p oblems. As ema ked by F ejo and
Camacho (2020), he he mal powe is maximized by inc easing he
low a e (as he he mal powe dependency on he low a e is s onge
han on he luid p ope ies and he he mal d op in he s eam gene a o
is nea ly cons an ), which leads o a minimiza ion o he empe a u e.
Wi h he addi ion o he Rankine e iciency o compu e he elec ic
powe , as shown by Eq. (16), he p oblem loses con exi y and he
op imal empe a u es do no co espond o he minimum because o he
a io 𝑇a∕𝑇in
sg . Figs. 3and 4 show he con ou lines o he he mal and
elec ic powe s ob ained by simula ing he plan wi h cons an alues
o i adiance and low a e, and 𝑇a= 25 oC, 𝑛o= 1. The analysis
highligh s he need o ad anced con ol echniques and models ha
adequa ely desc ibe he sys em. No e ha his he mal powe does no
co espond o ha which would be p o ided by he collec o loop i i
we e sepa a ed om he es o he plan , since he low loop is closed.
Fig. 3. Con ou lines o he he mal powe ecei ed by he s eam gene a o om he
collec o , ob ained om di e en simula ions wi h cons an inpu s. I can be unde s ood
as he main componen o he cos unc ion ha would be sol ed by con olling he
sys em in one s age. The ed lines ep esen he poin s whe e he empe a u e eaches
he limi s o 200 oC and 300 oC.
Fig. 4. Con ou lines o he elec ic powe ecei ed by he s eam gene a o om he
collec o , ob ained om di e en simula ions wi h cons an inpu s. I can be unde s ood
as he main componen o he cos unc ion ha would be sol ed by con olling he
sys em in one s age. The ed lines ep esen he poin s whe e he empe a u e eaches
he limi s o 200 oC and 300 oC.
Fig. 5. Scheme o he p oposed con ol s a egy. A i s MPC con olle p o ides he
empe a u e se -poin and a second one p o ides he low a e.
3.1. Powe op imiza ion
The p oposed con olle is de eloped in wo laye s, as shown in
Fig. 5. The i s laye compu es he op imal empe a u e e e ence o
maximizing he ob ained powe a a highe le el, and he second one
p o ides he low a e equi ed o each hose e e ences wi h a lowe
s ep ime. Bo h con olle s ha e he s uc u e o MPC.
3.1.1. Fi s laye
The i s con ol laye p o ides he op imal empe a u e se poin s
as in Camacho and Gallego (2013), bu using he dis ibu ed pa ame e
model, wi h an MPC con olle e e y 15 min, which is app oxima ely
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S. Ruiz-Mo eno e al.
he anspo delay in his plan (Camacho e al., 2012). The p edic ion
model employed is he dis ibu ed pa ame e model, and he p edic-
ions include a eed o wa d con olle o compu e he u u e alues o
he sys em’s low a e. This allows one o p edic no only he esponse
o he sys em bu also he low a es ha will be necessa y o achie e
each empe a u e, enabling p edic ions o se e al u u e ins an s.
The equa ion o he eed o wa d de i es om he concen a ed
pa ame e model o Eq. (3) in s eady-s a e, and i has a sample ime
o 30 s. The low a e is compu ed by Eq. (17) and is only used o
p edic ion pu poses. The e e ence empe a u e is sa u a ed so ha i
is always 10 deg ees o e he ou le empe a u e o a oid nega i e low
a es. The p edic ion ho izon 𝐻p1 is 6.5 h.
𝑞=𝑛o𝐾op 𝑆𝐼 −𝐻l𝐴(𝑇mean −𝑇a)
𝑃cp(𝑇 e −𝑇in)(17)
The inal aim is o ain a neu al ne wo k using he da a ob ained
wi h his con olle . Al hough u u e i adiance alues a e no a ailable
in he inal applica ion, hey can be used o ob ain he aining da a.
A seconda y ea u e o he neu al ne wo ks is an in e nal i adiance
p edic ion. Fo his pu pose, u u e i adiances a e assumed o be
p e iously known o he MPC con olle along he p edic ion ho izon.
This p o ides p edic ion abili ies o he neu al ne wo k ha will be
used o eal- ime ope a ion. In his way, he ANN sa es signi ican
compu a ional cos s in compu ing he dis ibu ed pa ame e model and
making p edic ions. Ne e heless, an exac model o o ecas ing sola
adia ion could be used, inc easing he compu a ional bu den.
The cos unc ion is gi en by Eq. (18), whe e P can ei he be he
he mal powe (Eq. (15)), he he mal powe wi h he pump consump-
ion, he elec ic powe (Eq. (16)) and he elec ic powe wi h he pump
consump ion. The esul ing op imiza ion p oblem is gi en by Eq. (19).
𝐽1= −𝑃(𝐻p1)(18)
𝑇∗
e (𝑘) = a g min
𝑇 e
𝐽1(𝑘)
𝑠.𝑡. { sys em dynamics
𝑇ini − 15 oC< 𝑇 e < 𝑇ini + 15 oC
(19)
To a oid local minima, he p oblem is sol ed a ound an ini ial poin
𝑇ini(𝑘) in a ange selec ed by ial and e o . This poin is ob ained om
a simple op imiza ion using a s a ic model, whe e all a iables a e con-
side ed cons an , he empe a u e is assumed o ollow he e e ence,
and he low a e is app oxima ed by Eq. (17). The cons ain s o his
p e-op imiza ion p oblem a e a minimum o 205
oC and a maximum o
295 oC in he ou le empe a u e.
3.1.2. Second laye
The MPC con olle has a sample ime o 1 min o alle ia e com-
pu ing ime (F ejo and Camacho, 2020), and p edic ion and con ol
ho izons o 12 and 10 min (𝐻p2= 12 and 𝐻u2= 10). As in he i s
laye , he p edic ions a e made using he dis ibu ed pa ame e model
in he collec o s and he pipes. The cos unc ion is gi en by Eq. (20)
wi h he weigh ing ac o 𝜙= 6 ⋅104 o a oid sudden changes in he
con ol signal ha could damage he ac ua o s. The maximum p essu e
d op in each loop imposes he ha d cons ain s o a minimum low a e
o 0.2 l/s and a maximum o 1.2 l/s. 𝑘 is each simula ion ins an , as
shown by Eq. (21).
𝐽2=
𝐻p2
∑
𝑘=1
(𝑇 e −𝑇ou )2+
𝐻u2
∑
𝑘=1
𝜙(𝑞(𝑘− 1) − 𝑞(𝑘))2(20)
𝑞∗(𝑘) = a g min
𝑞𝐽2(𝑘)
𝑠.𝑡. { sys em dynamics
0.2 l/s < 𝑞 < 1.2 l/s
(21)
3.2. A i icial neu al ne wo ks
A i icial neu al ne wo ks (Abiodun e al., 2018) a e unc ion ap-
p oxima o s o med by di e en nodes ha compu e a linea eg ession
p oblem, as in Eq. (22), whe e 𝑛(𝑙) is he numbe o neu ons in each
laye 𝑙, 𝑤(𝑙−1)
𝑗𝑖 is he ke nel be ween neu ons 𝑗 and 𝑖 o laye 𝑙− 1
and 𝑏(𝑙−1)
𝑖 is he bias uni . The ac i a ion unc ion 𝑔(𝑙)
𝑖 de ines he s a e
o each neu on – ac i e o nonac i e – and acili a es he aining
p ocess by adding limi s o i s ou pu and simpli ying he g adien s. The
combina ion o nodes and ac i a ion unc ions esul s in he esolu ion
o a nonlinea p oblem. The ans e unc ion is known as he one
loca ed a he las laye and con ains he ac i a ion unc ion o he las
laye wi h a ans o ma ion o he da a o he ou pu , al hough in he
li e a u e, i is commonly conside ed ano he e m o he same concep
(ac i a ion unc ion).
𝑎(𝑙)
𝑖=𝑔(𝑙)
𝑖⎛⎜⎜⎝
𝑛(𝑙−1)
∑
𝑗=1
𝑤(𝑙−1)
𝑗𝑖 𝑎(𝑙−1)
𝑗+𝑏(𝑙−1)
𝑖⎞⎟⎟⎠
(22)
In his wo k, he neu al ne wo ks used a e mul ilaye eed- o wa d
back p opaga ion (FFBP) wi h sigmoid angen ac i a ion unc ions in
he hidden laye s and linea unc ions in he ou pu s. The sigmoid
angen unc ion ansla es he neu on’s ou pu o an ac i e-non-ac i e
s a e wi h a smoo h g adien , allowing he in e media e da a o emain
in he [−1,1] ange. The ou pu laye uses a linea unc ion o p oduce
a noncon e ed ou pu . The hidden laye s a e in e media e laye s ha
ans o m he in o ma ion. The da a a e scaled in he ange [−1,1]
because o he use o angen unc ions and di ided in o aining (70%),
alida ion (15%), and es (15%) subse s. The neu al ne wo ks a e
ained wi h Le enbe g–Ma qua d backp opaga ion (Lillic ap e al.,
2020), which is an e icien me hod when he numbe o pa ame e s
in he neu al ne wo k is no excessi e. I op imizes using he sum o
squa ed e o s as a loss unc ion.
Two e sions o neu al ne wo ks we e es ed. The i s is a unique
neu al ne wo k ha subs i u es bo h con ol laye s (ANN 1). The
inpu s o his ANN a e he a iables o he concen a ed pa ame-
e model: 𝑋1(𝑘) = [𝑞(𝑘− 1), 𝑇in, 𝑇ou , 𝑇1,𝑇2, 𝑇3, 𝑇4, 𝑇a, 𝐼(𝑘),𝐼p ed(𝑘+
2|𝑘),𝐼p ed(𝑘+ 4|𝑘),…, 𝐼p ed(𝐻p|𝑘), 𝑛o]𝑇, o med by he p e ious low
a e, he inle and ou le empe a u es, he empe a u es a he cen e
o each collec o , he ambien empe a u e, he i adiance o each wo
p edic ion ins an s, and he geome ic e iciency. The ou pu is he low
a e: 𝑌1(𝑘) = 𝑞(𝑘). The numbe o inpu s and p edic ion ins an s is based
on he p e ious wo k de eloped by he au ho s (Ruiz-Mo eno e al.,
2021).
The second app oach uses one neu al ne wo k pe laye . The i s -
laye ANN (ANN 2.1) ecei es he same inpu s as he unique ANN,
excep o he i adiance, which in his case is aken e e y 15 min
o 1 hou : 𝑋21(𝑘) = [𝑞(𝑘− 1), 𝑇in, 𝑇ou , 𝑇1,𝑇2, 𝑇3, 𝑇4, 𝑇a, 𝐼(𝑘),𝐼p ed(𝑘+
15|𝑘),𝐼p ed(𝑘+ 30|𝑘),…, 𝐼p ed(60|𝑘), 𝑛o]𝑇, 𝑌21(𝑘) = 𝑇 e . The inpu s o
he second-laye ANN (ANN 2.2) a e he same as o he unique ANN
wi h he addi ion o he e e ence empe a u e: 𝑋22(𝑘)=[𝑞(𝑘− 1), 𝑇in,
𝑇ou , 𝑇1,𝑇2, 𝑇3, 𝑇4, 𝑇a, 𝐼(𝑘),𝐼p ed(𝑘+ 2|𝑘),𝐼p ed(𝑘+ 4|𝑘),…, 𝐼p ed(𝐻p|𝑘), 𝑛o,
𝑇 e ]𝑇, 𝑌22(𝑘) = 𝑞(𝑘). Du ing he aining p ocess, he e e ence is
ob ained om he con olle , bu in simula ion, i is eplaced by he one
ob ained om he i s -laye ANN. The ou pu s o he neu al ne wo ks
a e hose om each con olle : e e ence empe a u e o he i s laye
and low a e o he second one.
3.2.1. I adiance p edic ion
In a eal- ime ope a ion, he alues o u u e i adiance a e no
a ailable. Al hough he neu al ne wo ks can lea n om he con olle
and indi ec ly make p edic ions in e nally, hey ecei e simple p edic-
ions o help he p ocess. As s a ed in he p e ious wo k (Ruiz-Mo eno
e al., 2021), he neu al ne wo ks do no need he whole i adiance
in o ma ion. They can pe o m well wi hou knowing all u u e i a-
diances, bu hei pe o mance dec eases when hey a e only ed he
cu en alue.
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S. Ruiz-Mo eno e al.
A simple i adiance p edic ion ob ained om a clea -day model
is ed o he ANNs. No e ha he MPC con olle does no use hese
p edic ions. As i is used o ob aining he da ase and no o eal-
ime ope a ion, he MPC con olle conside s he ac ual u u e alues
o i adiance. Eq. (23) p o ides he e ol en o he i adiance, as
desc ibed by Camacho e al. (2012), which app oxima es he u u e
i adiances gi en he cu en alue a one ins an . The model p o ided
by Ho el (1976) gi es he a mosphe e’s ansmi ance.
𝐼p ed(𝑘+ 1|𝑘) = 𝐼(𝑘) + (𝑡(𝑘+ 1) − 𝑡(𝑘))𝐸c(1+0.033 cos (2𝜋𝐽d
365 )) (23)
whe e 𝐼p ed(𝑘+ 1|𝑘) is he p edic ed alue o i adiance o ins an
𝑘+ 1 calcula ed a ins an 𝑘, 𝑡(𝑘) is he ansmi ance, 𝐸c= 1367 W/m2
is he sola cons an and 𝐽d is he Julian day.
3.2.2. E alua ion me ics
Du ing he neu al ne wo k selec ion p ocess, Pea son’s co ela ion
coe icien (R) is used in each o he subse s o selec he bes neu al
ne wo k and o compa e he a ia ion in pe o mance among di e en
da a. This coe icien is ob ained wi h Eq. (24), whe e 𝑥𝑖 and 𝑦𝑖 a e
wo di e en da a, and 𝑥 and 𝑦 a e he mean alues o a iables 𝑥 and
𝑦. Fo simplici y, in his documen 𝑅 e e s o he Pea son co ela ion
coe icien be ween he da ase and he simula ion da a.
𝑅(𝑥, 𝑦) = ∑𝑁
𝑖=1(𝑥𝑖−𝑥)(𝑦𝑖−𝑦)
√∑𝑁
𝑖=1(𝑥𝑖−𝑥)2∑𝑁
𝑖=1(𝑦𝑖−𝑦)2
(24)
The mean squa ed cons ain iola ion (MSCV) is used o ake in o
accoun he de ia ion o a a iable (in his case, he ou le empe a u e)
om ce ain limi s. I is compu ed wi h Eq. (25).
MSCV = 1
ns
𝑘=𝑘2
∑
𝑘=𝑘1(max (𝑦min −𝑦(𝑘), 𝑦(𝑘) − 𝑦max,0)2)(25)
In addi ion, he esul s ob ained wi h he con olle s a e e alua ed
wi h he mean ime equi ed o compu ing he con ol signals and he
mean powe s ob ained: he mal powe 𝑃 h, he mal powe conside ing
he pump consump ion 𝑃 h −𝑃p, elec ic powe 𝑃el and elec ic powe
conside ing he pump consump ion 𝑃el −𝑃p.
4. Simula ion esul s
This sec ion p esen s he esul s ob ained wi h he di e en con-
olle s p oposed. The simula ions a e ca ied ou using he dis ibu ed
pa ame e model and including he pump and pipes. Fi s , wo con-
olle s we e applied o he sys em o ollow a ixed empe a u e
e e ence. In his case, only he second laye was implemen ed. The i s
con olle uses a low empe a u e, and he second one uses a high one.
These con olle s allow us o compa e he e ec s o using adi ional
con ol objec i es.
4.0.1. Powe op imiza ion
The powe maximiza ion con olle s (con olle s 3–6) we e applied
using he same i adiance p o ile. Fi s , wo con olle s we e imple-
men ed conside ing he mal powe maximiza ion in he cos unc ion.
The ou h con olle includes he pump consump ion in he cos unc-
ion and inco po a es he pipes in he p edic ion model, whe eas he
hi d one only akes he he mal powe . The hi d con olle models
he inle empe a u e wi h a i s -o de il e o he ou le empe a u e
wi h a ime cons an o 10 min. Likewise, wo o he con olle s we e
implemen ed o elec ic powe op imiza ion wi h he same conside a-
ions as he ou h and i h con olle s. The ollowing lis ga he s he
cha ac e is ics o he MPC con olle s used:
1. 220 oC acking, wi hou pipes o pump.
2. 300 oC acking, wi hou pipes o pump.
3. The mal powe maximiza ion, wi hou pipes o pump.
Fig. 6. I adiance and e ec i e i adiance used o es ing he con olle s.
4. The mal powe maximiza ion, wi h pipes and pump.
5. Elec ic powe maximiza ion, wi hou pipes o pump.
6. Elec ic powe maximiza ion, wi h pipes and pump.
A one-day simula ion was pe o med wi h all he con olle s using
he syn he ic i adiance p o ile o Fig. 6. The e ec i e i adiance is
he p oduc o he incoming i adiance ( he one ob ained om he
syn he ic i adiance p o ile) and he geome ic and op ical e iciencies
o he collec o s. The esul s a e ga he ed in Table 1, wi h all me ics
calcula ed be ween hou s 10:30 and 17:30. The MSCV indexes a e
compu ed using 200 oC and 300 oC as limi s, and he mean powe is
compu ed o he 10 loops o he ield, conside ing ha all o hem a e
equally dis ibu ed, i.e., he o al powe o he ield is assumed o be
en imes he powe o one loop. As migh be expec ed, he con olle s
ha p o ide mo e he mal powe a e he hi d and ou h (wi h almos
no di e ence be ween hem because hey bo h wo k minimizing he
ou le empe a u e), and he con olle s ha p o ide mo e elec ic
powe a e he i h and six h. Following he c i e ia o elec ic powe
maximiza ion, since ha is he objec i e in a comme cial plan , he
six h con olle is he bes one.
Fig. 7 shows he esul s ob ained wi h he abo emen ioned MPC
con olle s. No e ha , as well as wi h Table 1, he sys em beha es
simila ly wi h con olle s 3 and 4. This is because when maximizing he
he mal powe , he con olle ends o minimize he ou le empe a u e,
ega dless o he elemen s included in he model. Al hough con olle
h ee neglec s empe a u e and gene a ion d ops due o he pumps and
pipes, he con ol objec i e is s ill o b ing he empe a u e o i s lowe
limi , which is easily achie ed by augmen ing he low a e.
A de ail o he esul s ob ained wi h con olle 6 (maximiza ion o
elec ical powe conside ing pipes and pump) is shown in Fig. 8, which
ep esen s he ou le empe a u e and he e e ences gi en by he i s
con ol laye and he elec ic powe . As his con olle p o ides he
highes mean elec ic powe , i is used o ain he neu al ne wo ks
o ep oduce hei beha io .
4.0.2. A i icial neu al ne wo ks
A e selec ing he con olle ha will be used o ain neu al
ne wo ks, 20 simula ions o one day we e pe o med wi h he di e en
andom syn he ic clouds in Fig. 9. Fo c ea ing he da ase , ins an s
wi h ou le empe a u es below 180 oC we e emo ed (as in hose
si ua ions he plan is shu down), ob aining 8036 ins ances. Then,
he da ase was andomly di ided in o aining, alida ion, and es
subse s o 5626, 1205, and 1205 ins ances, espec i ely. No ice ha
he es subse and he es i adiance p o ile ha e no ela ion o each
o he , as he es se con ains andom ins ances o he esul s wi h
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
7
S. Ruiz-Mo eno e al.
Table 1
Resul s o he MPC con olle s applied wi h di e en cos unc ions.
Con olle 𝑃 h (kW) 𝑃 h −𝑃p (kW) 𝑃el (kW) 𝑃el −𝑃p (kW) MSCV Max ime (s)
1 949.282 936.003 287.197 273.917 0 313.379
2 741.534 735.571 271.452 265.486 0.154 343.471
3 996.151 980.749 276.577 261.175 24.358 392.545
4 996.151 980.749 276.577 261.575 24.358 392.327
5 879.950 870.310 291.280 281.640 0.079 619.376
6 857.887 849.212 290.786 282.110 0 1.274⋅104
Fig. 7. E olu ion o ou le empe a u e, low a e, and p oduc ion wi h he MPC con olle s.
Fig. 8. De ail o he e olu ion o ou le empe a u e, i s e e ence and he p oduc ion wi h he 6◦ MPC con olle (elec ic powe maximiza ion wi h pipes and pump in he cos
unc ion).
he aining i adiance p o iles, and he es i adiance p o ile is used
o pe o m simula ions a e selec ing he bes neu al ne wo ks. The
neu al ne wo ks we e ained, and he simula ions we e pe o med in
Ma lab® R2020b wi h In el® Co e™ i7-9700F CPU a 3.00 GHz and
16 GB RAM. Th ee ypes o neu al ne wo ks we e ained wi h di e en
numbe s o neu ons and laye s in an i e a i e p ocess. Table 2 shows
he aining pa ame e s o he neu al ne wo ks.
The i s expe imen was he aining o neu al ne wo ks o di ec ly
ob ain he low a e in one s ep (ANN 1). Table 3 shows he numbe
o neu ons in he hidden laye s, he R sco es o he ained neu al
ne wo ks in each subse and he comple e da ase , he MSCV wi h he
ain p o iles, and he mean alue o 𝑃el −𝑃p ( he elec ic powe wi h
pump consump ion) be ween 10:30 and 17:30 wi h he aining and
es i adiance p o ile. The R sco es a e ob ained in an open loop,
while he elec ic p oduc ion is compu ed by simula ing he sys em
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
8
S. Ruiz-Mo eno e al.
Table 2
T aining hype pa ame e s o he neu al ne wo ks.
T aining
algo i hm
Range Ac i a ion
unc ions
𝜇0𝜇 inc ease
a io
𝜇 dec ease
a io
Max
𝜇
Max
epochs
Min
g adien
Max
alida ion checks
Le enbe g-
Ma qua d
[-1,1] ansig
pu elin
10−3 10−1 10 1010 10310−9 6
Table 3
Resul s o he neu al ne wo ks ha compu e di ec ly he low a es (ANN 1).
Neu ons R ( ) R ( al) R ( es ) R ( o al) MSCV ( ) 𝑃𝑒𝑙 −𝑃p ( ) 𝑃𝑒𝑙 −𝑃p ( es )
12 99.875 99.828 99.722 99.845 738.644 235.534 kW 280.356 kW
36 99.904 99.850 99.749 99.873 606.069 133.369 kW 280.317 kW
48 99.943 99.827 99.701 99.891 3856.887 205.672 kW 285.249 kW
60 99.943 99.868 99.861 99.920 3047.920 203.257 kW 279.831 kW
72 99.841 99.677 99.770 99.806 2107.935 246.021 kW 276.629 kW
12–4 99.935 99.899 99.743 99.900 1103.012 228.637 kW 277.188 kW
12–6 99.890 99.867 99.808 99.874 744.402 238.693 kW 280.238 kW
12-12 99.803 99.703 99.775 99.784 883.385 237.375 kW 280.054 kW
24–12 99.930 99.840 99.863 99.907 2144.251 226.110 kW 280.389 kW
36–24 99.925 99.884 99.851 99.907 595.590 238.922 kW 279.696 kW
48–24 99.910 99.859 99.847 99.893 1850.920 224.765 kW 280.426 kW
36-24-6 99.927 99.839 99.774 99.890 798.054 235.741 kW 276.634 kW
36-24-12 99.915 99.751 99.798 99.872 1851.284 226.084 kW 261.036 kW
48-24-12 99.977 99.901 99.915 99.956 1168.809 233.489 kW 273.360 kW
Fig. 9. T aining i adiance p o iles.
in a closed loop. This second me ic is a mo e ealis ic indica ion o
he ANN pe o mance, as a e y accu a e ANN in an open loop may
no necessa ily gene alize well and may be o e ly dependen on sligh
de ia ions in he low a e. The selec ed one was a neu al ne wo k
wi h wo hidden laye s, wi h 12 neu ons in he i s hidden laye and
12 in he las one. This ANN was he one ha combined high powe
and low MSCV wi h he aining p o iles and high powe wi h he es
p o iles, and he Pea son co ela ion coe icien was simila in he h ee
subse s, showing a good abili y o gene alize wi hou o e i ing, he
phenomenon whe eby a lea ning algo i hm is o e ained and lea ns o
i only he aining da a. The aining ime o his ANN was 0.896 s.
The second expe imen consis ed o aining wo di e en neu al
ne wo ks: one o emula ing each con ol laye . Table 4 shows he ain-
ing me ics o he i s -laye neu al ne wo ks. The elec ic p oduc ion
alues we e ob ained using a eed o wa d con olle in he second laye
o ack he empe a u es, only o compa ison pu poses. The selec ed
one was a neu al ne wo k wi h 36, 24 and 6 neu ons in he hidden
laye s. The aining ime o his ANN was 13.166 s. The esul s wi h he
second-laye ANN a e shown in Table 5, whe e he absolu e acking
e o o he ou le empe a u e is included. In his case, he e e ences
a e ob ained om he i s -laye MPC con olle . The ANN wi h 36, 24
and 12 neu ons was selec ed because o i s high powe and low loss o
R alue. The aining ime o his ANN was 12.405 s.
Table 6 compa es he simula ion esul s wi h con olle six and
he a i icial neu al ne wo ks in e ms o he mal and elec ic powe s,
cons ain s iola ions, and compu a ion ime. The e m ANN 2 ep-
esen s he combina ion o ANN 2.1 and ANN 2.2. As well as in he
p e ious expe imen s, all me ics a e ob ained be ween hou s 10:30
and 17:30. The e olu ions o ou le empe a u e, low a e, and elec ic
p oduc ion a e ep esen ed in Fig. 10. Du ing ansien s, he e is a
highe disc epancy be ween he esul s ob ained wi h ANNs and hose
wi h MPC. This is because e en mino di e ences be ween he ANN and
MPC esponses can lead o sligh ly di e en dynamic sys em beha io s.
A he nex sampling ime, hese di e ences cause he ANN and MPC
o ecei e sligh ly di e en inpu s again, compounding he e ec . This
phenomenon is common o all app oxima ions o con olle s. How-
e e , he elec ic powe gene a ed emains e y simila . The maximum
compu a ion imes o he neu al ne wo ks a e negligible wi h espec
o he MPC con olle , and he cons ain iola ions a e nea 0. Bo h
ANN app oaches app oxima e success ully he beha io o he MPC
con olle .
Fig. 11 shows he imes ha we e equi ed o ob ain con ol signals
wi h he elec ic powe maximiza ion MPC and wi h he selec ed neu al
ne wo ks. The MPC imes a e ep esen ed on a scale o seconds, while
he ANN imes a e ep esen ed in decimilliseconds. This g aph shows
ha he main objec i e o his wo k is me , which was o con ol he
plan wi h pe o mance simila o ha o he MPC con olle s bu wi h
much lowe compu a ion imes.
One ex a simula ion was pe o med o each con olle o alida e
he esul s using he i adiance p o ile om Fig. 12. The esul s o he
h ee selec ed con olle s a e shown in Table 7. Bo h neu al ne wo k
app oaches a e able o con ol he sys em. The loss o pe o mance in
ANN 2 is highe o his es , al hough i is s ill lowe han 2% (see
Fig. 13).
4.0.3. Robus ness analysis
One limi a ion o he ANNs is ha hey a e ained wi h speci ic
plan pa ame e s. De ia ions be ween he sys em used o aining and
he ac ual plan can lead o a ia ions in he esul s. Di e en simu-
la ions we e pe o med, changing some pa ame e s in he ollowing
condi ions: i) 6 h MPC con olle wi h modi ica ion o he p edic ion
model (as a baseline), ii) MPC con olle wi h a misma ch be ween
Enginee ing Applica ions o A icial In elligence 154 (2025) 110832
9
