LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE
Resea ch A icle
MODELING OF ASYNCHRONOUS UNITS OF A
POWERFUL PUMPING STATION IN THE
PRESSURE STABILIZATION MODE
Vladysla LYSIAK1,∗
1Depa men o Elec ic Powe Enginee ing and Con ol Sys ems, Ins i u e o Powe Enginee ing and Con ol
Sys ems, L i Poly echnic Na ional Uni e si y, 12 Bande a S ee , L i , 79013, Uk aine
ladysla [email protected]
*Co esponding au ho : Vladysla Lysiak; ladysla [email protected]
DOI: 10.15598/aeee. 23i2.240807
A icle his o y: Recei ed Aug 15, 2024; Re ised Dec 19, 2024; Accep ed Ma 15, 2025; Published Jun 30, 2025.
This is an open access a icle unde he BY-CC license.
Abs ac . The pape is de o ed o de eloping a ma he-
ma ical model o a powe ul con olled pumping s a ion
wi h au oma ic p essu e s abiliza ion. The wo k aims
o inc ease he e iciency o he liquid anspo a ion
p ocess h ough main and la ge dis ibu ion pipelines.
The p oposed model wi h a compa able le el o de ail
desc ibes he elec omechanical and hyd aulic subsys-
ems as a single en i y. The equa ions o he hyd aulic
subsys em a e o med based on he p inciple o elec o-
hyd odynamic analogy and e lec he physical p ocesses
in i s componen s. The pa ame e s o he cen i ugal
pump model we e calcula ed based on he geome y and
dimensions o i s in e nal elemen s, aking in o accoun
he in luence o he physical p ope ies o he wo king
luid. I makes i possible o conduc a comp ehensi e
s udy o such objec s wi hou physical impac on hem,
aking in o accoun he mu ual dependence o bo h sub-
sys ems and changes in he pa ame e s o he elemen s
o he hyd aulic subsys em. The pape o e s di ec ions
o use and ways o imp o e he unc ionali y o he
de eloped model.
Keywo ds
Cen i ugal, con ol, induc ion, model, mo o ,
pipeline, pump, s a ion, sys em.
1. In oduc ion
Powe ul pumping s a ions (PS) o main and la ge dis-
ibu ion pipelines (PL) a e s a egically impo an .
They ensu e he mo emen o la ge olumes o liq-
uid [1] and consis o inex icably linked elec ome-
chanical and hyd aulic subsys ems. Powe o e spends
due o non-op imal modes o indi idual powe ul uni s,
subop imal numbe s o less powe ul uni s ope a ing si-
mul aneously, and ansien p ocesses a e qui e signi i-
can . Acco ding o [2], hey can each 14% o he o al
ene gy consump ion o he PS. Acco ding o he same
da a, up o 16 pumping uni s (PU) can occu swi ch-
ing a he main oil s a ion in jus one day. O e spends
o elec ici y a he subs a ion also lead o signi ican
o e spends in he elemen s o elec ical ne wo ks. The
high cos o elec ici y and equipmen and he inadmis-
sibili y o wo k in e up ions complica e and usually
make i impossible o ca y ou physical expe imen s
on ope a ing powe ul PS. The e o e, he de elopmen
o non-con ac means o compu e p edic ion o ope a -
ing modes, aul inding, and au oma ic con ol sys ems
o ope a ing such objec s is ac ual.
Mul i-uni elec ic-d i en PS consis s o inex ica-
bly linked elec omechanical and hyd aulic subsys ems
ha in luence each o he . In u n, he hyd aulic sub-
sys em is cha ac e ized by he mu ual in luences o hy-
d aulically connec ed indi idual CPs and a common
PL. The elec omechanical subsys em is also cha ac-
e ized by he mu ual in luences o he elec ic d i es
o hese CPs elec ically connec ed o a common powe
supply cen e. Thus, he ope a ing modes o o he
uni s o bo h he hyd aulic and elec omechanical sub-
sys ems depend on he ope a ing mode o an indi id-
ual uni . The e o e, o a comp ehensi e analysis o he
ope a ing modes o he PS, i mus be conside ed an in-
eg al elec o echnological complex wi h a compa able
de ailing o he ep esen a ion o i s componen s.
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One o he asks o con olling PS modes is o main-
ain cons an p essu e in a gi en node o he hyd aulic
subsys em a di e en le els o luid consump ion. The
mos common me hods o such con ol a e swi ching
uni s and changing he ope a ing condi ions o cen i u-
gal pumps (CP). The la e can be ca ied ou , in pa -
icula , by changing he cha ac e is ics o he hyd aulic
ne wo k ( h o ling o bypassing o CP), as well as by
equency con ol o he engines o he PU. Th o ling
o bypassing leads o a dec ease in he ene gy e iciency
o pumping. The maximum ene gy e iciency o he PS
can be achie ed by changing se e al simul aneously op-
e a ing PUs and he equency con ol o one o hem.
Such a con ol sys em and he algo i hm o i s ope a-
ion a e p esen ed in [3]. Howe e , i s unc ionali y is
limi ed only o s eady-s a e modes. In addi ion, only
he olume low a e o he wo king luid is measu ed o
s abilize he p essu e, which is ine ec i e o s abilizing
he p essu e in dynamic modes. The speci ied limi a-
ions a e absen in he model [4]. Howe e , an app ox-
ima e polynomial ep esen s he hyd aulic subsys em.
This makes i impossible o s udy he p ocesses in he
hyd aulic subsys em. Mos o he o he wo ks de o ed
o PS ( o example [5–7], and [8]) a e also cha ac e -
ized by ex emely simpli ied ep esen a ions o one o
bo h subsys ems. In many wo ks, he b aking an me-
chanical o que desc ibes he hyd aulic subsys em wi h
CP. In [9] i is shown ha his app oach is co ec only
unde he condi ion o cons an pa ame e s o he hy-
d aulic subsys em. In he gene al case, he pa ame e s
o CP depend on he physical p ope ies o he wo k-
ing luid, he spa ial s uc u e o he elemen s o he
hyd aulic subsys em, and he pumping mode.
Among he la ge numbe o wo ks de o ed o CP, in
ou opinion, i is ad isable o single ou hose in which
he ma hema ical model o CP is based on i s spa ial
s uc u e, which canno e lec ed by scala models. In
pa icula , he au ho s o [10] showed ha he p ojec-
ions o he o ce ac ing on he liquid a he ou le o
he o a ing impelle on o ixed axes a e ha monic ime
unc ions. To desc ibe he mo ion o he luid in he
low pa o he CP impelle , he modi ied Eule equa-
ion was used, and in he spi al pa , he Na ie -S okes
di e en ial equa ions. The la e allowed he au ho s
o w i e he equa ions o he complex CP model in o -
hogonal d-qcoo dina es igidly connec ed o he im-
pelle . This made i possible o ope a e wi h CP egime
pa ame e s ha a e cons an in ime and o de e mine
he dissipa i e and ine ial hyd aulic esis ances o CP
h ough he design pa ame e s and physical cha ac e -
is ics o he wo king luid. I should be no ed ha he
a io o dissipa i e and ine ial hyd aulic esis ances
CP is one o he o ms o he Reynolds numbe , which
de e mines he luid mo ion egime. Conside ing he
physical con en , adap abili y o he applica ion o he
heo y o ci cles, and he o m o w i ing he model
equa ions common in adi ional elec ic powe engi-
nee ing [10], we ook i as a basis. The imp o ed model
was success ully applied in he publica ion de o ed o
s eady-s a e egimes [3] and du ing he modelling o
dynamic egimes [11]. These allow he conside a ion
o he impac o ope a ional and eme gency changes in
CP in e nal pa ame e s on he PS’s ope a ing modes.
The model [11] makes i possible o s udy he mu ual
in luence o elec omechanical and hyd aulic p ocesses
and pa ame e s. Howe e , i does no ha e any uni
con ol sys em. The analysis o he abo e and o he
wo ks gi es g ounds o conclude ha he e is no ma he-
ma ical model o a powe ul PS wi h a closed equency
p essu e con ol sys em and a compa able le el o de ail
o he ma hema ical desc ip ion o he elec omechani-
cal and hyd aulic subsys ems. The pape is de o ed o
de eloping a ma hema ical model o a powe ul egu-
la ed pumping s a ion wi h au oma ic p essu e s abi-
liza ion.
2. Ma hema ical Model
Figu e 1 shows he s uc u al diag am o he PS wi h
a closed equency p essu e con ol sys em. Two ag-
g ega es a e hyd aulically connec ed in se ies: suppo
(PU1) and main (PU2).
The main uni p o ides he necessa y p essu e in he
gi en ange o low o he wo king luid. I s powe is
5-10 imes g ea e han ha o a suppo , and i has
a much highe cos . CP blades a e e y sensi i e o
hyd odynamic ca i a ion, which des oys hem. Bo h
expe imen al [12] and heo e ical [13] s udies a e de-
o ed o he ea ly de ec ion o ca i a ion on CP blades.
To a oid ca i a ion, a suppo uni is used. I c ea es
a small cons an p essu e a he en ance o he main
CP and p o ides a olume ic low a e no less han he
main uni p o ides. To s abilize he p essu e PS wi hin
he equi ed alue H e , he cu en alue o he p es-
su e a he en ance o he PL is used. The senso (SP)
measu es his p essu e. F om he ou pu o he PI con-
olle wi h pa ame e s , and he signal is sen o he
con ol sys em (CS). I p o ides equency and ol -
age con e e (FVC) o ma ion o he equency and
ol age o he powe supply ol age o he IM main
PU. The main uni ’s supply ol age is o med based
on he law o p opo ional equency con olling IM.
s2/ s2=cons The IM o he suppo PUs is uncon-
ollable. Fo simplici y, we assume ha bo h PUs a e
powe ed by h ee-phase symme ical ol age sou ces
ha do no con ain highe ha monics. The ol age
and equency o he supply ol age IM o he suppo
PU a e cons an .
The model equa ions a e w i en in a pe -uni sys-
em. The ollowing sys em o basis alues was used o
w i e he IM equa ions:
©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 99
LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE
Fig. 1: The s uc u al diag am o he PS.
ωIM.b =ωsyn;VIM.b =p2/3VIM.nom;
IIM.b =√2PIM.nom√3VIM.nomηIM.nomcosϕIM.nom;
SIM.b = (3/2) VIM.b/IIM.b;TIM.b =SIM.b/ωIM.b;
ZIM.b =VIM.b/IIM.b, whe e ωsyn,VIM.nom,PIM.nom,
ηIM.nom,cosϕIM.nom a e nominal passpo pa ame e s
o IM. The IM equa ions a e o med based on he gen-
e alized equa ions o he elec omechanical con e e ,
aking in o accoun he ollowing assump ions: he
empe a u e egime is s able, he winding pa ame e s
a e concen a ed; he webe -ampe e cha ac e is ic
o he magne ic sys em is non-linea ; he in luence
o he shape o he co e on he dis ibu ion o he
magne ic ield is no aken in o accoun . The ollowing
sys em o basis alues was used o w i e he CP
equa ions: ωCP.b =ωCP.nom;HCP.b =HCP.nom;
QCP.b =QCP.nom;SCP.b =ρgHCP.bQCP p.b;
TCP.b =SCP.b/ωCP.b;ZCP.b =ρgHCP.b/QCP p.b,
whe e ωCP.nom,HCP.nom,QCP.nom a e he nominal
passpo pa ame e s o CP, ρ,ga e he densi y o
he wo king luid and ee- all accele a ion. The CP
equa ions a e o med based on an elec ohyd ody-
namic analogy acco ding o he complex subs i u ion
scheme [10], aking in o accoun he ollowing assump-
ions: he empe a u e egime is s able, he wo king
luid is homogeneous wi h a cons an densi y, he
image ec o s o he olume ic low a e and p essu e
o he wo king luid a he ou le o he impelle a e
collinea .
We conside he IM o o , he CP impelle , and he
common sha o be igid. Fo he basic alue o he
equency common o bo h subsys ems, we ake he ba-
sic alue o he IM equency: ωb=ωIM.b. PU equa-
ions a e o med in o hogonal d-qcoo dina es igidly
connec ed o he common sha . Complex a iables in-
cluded in he equa ions a e ma ked wi h a do abo e.
Coo dina e dco esponds o he eal componen , and
coo dina e qco esponds o he imagina y componen .
The di e en ial equa ions ha e been sol ed o he i s
de i a i es, which may be con enien o use in com-
pu e ma hema ics sys ems.
˙
h0=H0.nomωb
ωCP.b 2
ω2
ejωbω ,(3)
hCP dq33q−hCP qq33d= 0,(4)
dω
d =1
JΣωIM.b ·(TIM.b (ψδdisq−
−ψδqisd)−TCP.bH0.nom ωIM.b
ωCP.b ω ×
×q(q11d+q44d)2+ (q11q+q44q)2,
(5)
q2
33d1+q2
33q1=QCP b2
QCP b12
q2
33d2+q2
33q2,(6)
qP L =qq2
33d1+q2
33q1,(7)
hP L =qh2
CP d1+h2
CP q1+HCP b2
HCP b1qh2
CP d2+h2
CP q2,
(8)
dqP L
d =− P L
LP L
qP L +1
LP L
hP L −1
LP L
hs ,(9)
dωs2
d =1
2ωs2
(I(H e −hP L)−P y),(10)
dhP L
d =y, (11)
sd2=ωs2,(12)
whe e is ime in (s); ωs,ω a e synch onous syci -
cula equency o he s a o winding ol age IM and
he ci cula speed o o a ion o he common sha ; ˙ s,
˙
is, and ˙
i a e s a o ol age and cu en , as well as
o o cu en IM educed o he s a o winding; ψδ
is educed o he winding o he s a o lux coupling
om he magne ic lux o he ai gap IM; Rs,Lσs,
R ,Lσ ,Raa e IM pa ame e s; ˙
h0is ic i ious p es-
su e o he idealized CP; ˙q11,˙q22,˙q33,˙q44 a e ic i ious
cos s o CP; hP L,qP L a e p essu e and olume low
a e o liquid a PL inle ; hs is s a ic coun e p es-
su e PL; mech (qP L)is equi alen nonlinea hyd aulic
esis ance, which akes in o accoun dissipa i e losses
o mechanical ene gy in CP depending on i s pump-
ing mode [14]; 11, 12, 13, 21, 22, 23, 31, 32, 33,
L′
11,L′
12,L′
13,L′
21,L′
22,L′
23,L′
31,L′
32,L′
33 a e dissipa-
i e hyd aulic esis ances and ine ial hyd aulic induc-
ances CP, which a e calcula ed by he size o i s in e -
nal elemen s and physical cha ac e is ics o he wo k-
ing luid acco ding o he me hod [10], and [11]; P L,
LP L a e dissipa i e hyd aulic esis ance and ine ial
hyd aulic induc ance PL; JΣis ine ia PU in (kgm2).
The magne ic eluc ance o he IM is p esen ed in
he o m o an app oxima ion polynomial Rm(ψδ) =
Imn a0+a2ψ2
δd +ψ2
δq+a4ψ2
δd +ψ2
δq4, whe e
a0= 0.82; a2= 0.148; a4= 0.044; Imn = 1/(xσ+xa),
whe e xσ,xaa e ela i e induc ances o dispe sion and
magne iza ion o he IM in he nominal mode. In his
case, we neglec he e ec o equency on a Rm, since
©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 100
LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE

d˙
is
d
d˙
i
d
d˙
ψδ
d
0

=







−Rs
Lσs −jωs0−1
Lσs −jωs
Lσs
0−R
Lσ −j(ωs−ω )ωs
Lσ jωs−ω
Lσ
0 0 1 1
0 1 1 −Rm(ψδ)−jωs
Ra








×
˙
is
i
˙es
˙
ψδ

+
˙ s
0
0
0

,(1)

d˙q11
d
d˙q22
d
d˙q33
d
d˙q44
d

=








11
L′
11 − 21
L′
12
22
L′
12 − 32
L′
13
33
L′
13 − 23
L′
12 0
11
L′
21 − 21
L′
22
22
L′
22 − 32
L′
23
33
L′
23 − 23
L′
22 0
11
L′
31 − 21
L′
32
22
L′
32 − 32
L′
33
33
L′
33 − 23
L′
32 0
0 0 0 − mech(qP L)
Lmech









×
˙q11
˙q22
˙q33
˙q44

+
1
L11
1
L21
1
L31
1
Lmech

×˙
h0,(2)
i becomes signi ican a equencies abo e 100 Hz [15].
The de ice’s maximal ope a ing equency is he syn-
ch onous equency o he elec ical powe supply sys-
em (in his case 50 Hz).
Equa ion (1) desc ibes he elec omechanical subsys-
em. Equa ions (2), (3), and (4) desc ibe he hyd aulic
PU subsys em. In pa icula , equa ion (4) speci ies he
collinea i y o he image ec o s o he eal head and
he eal low a he ou le o he CP. Equa ion (5) o
he PU common sha mo ion combines he elec ome-
chanical and hyd aulic coo dina es o he IM and CP
modes. Equa ions (1), (2), (3), (4) and (5) a e w i en
sepa a ely o each PU. Equa ions (6), (7), (8), and
(9) desc ibe he hyd aulic connec ions o PU and PL.
Equa ions (10), (11), and (12) de e mine he s abiliza-
ion o he PS head. The basis is aken o ensu e a
cons an a io o he nominal ol age o he nominal
equency o he ol age o he s a o winding o he
elec ic mo o . Taking his in o accoun , in his pa ic-
ula case, in he pe -uni sys em, he law o p opo ion-
ali y o he con ol equency o he main uni ’s induc-
ion mo o is de e mined by he exp ession s2/ s2= 1.
In he ma hema ical model, his is p esen ed as equa-
ion (12). In gene al, equa ions (10), (11), and (12)
can be eplaced by o he app op ia e equa ions ha
desc ibe ano he speci ic con ol sys em. I should be
no ed ha he dissipa i e hyd aulic esis ance o he
CP depends no only on i s in e nal dimensions bu
also on he kinema ic iscosi y o he liquid. In u n,
he kinema ic iscosi y o he liquid depends on he
empe a u e. Thus, i is possible o ake in o accoun
he in luence o empe a u e condi ions on he PS’s op-
e a ing modes.
3. Tes Calcula ions
To e i y he model, es simula ions o he ope a ing
modes o he uni s we e ca ied ou o 200 seconds.
The nominal powe o he mo o is selec ed based on
he nominal powe o he pump. The pump passpo
indica es he nominal use ul hyd aulic powe o he
pump a he ou le o he impelle . The equi ed me-
Tab. 1: Pa ame e s o IM 4AN355M6U3 o he suppo ing PU.
Pnom ,ηnom Vs.nom ,nnom ,cos φnom
kW V pm
250 0.935 380 985 0.9
p0Tmax∗Tmin∗Ts∗Is∗
3 2.2 0.9 1.4 7
Tab. 2: Pa ame e s o IM 4AZMV-1600/6000U2 o he main
PU.
Pnom ,ηnom Vs.nom ,nnom ,cos φnom
kW V pm
1600 0.961 6300 2979 0.9
p0Tmax∗Tmin∗Ts∗Is∗
1 2.6 0.7 1.9 6
Tab. 3: Pa ame e s o CP 14NDs-N o he suppo ing PU.
Hnom ,Qnom ,ηnom nnom ,Phyd .nom ,
m m3·h−1 pm kW
45 1260 0.809 980 154
H0.nom∗R∆Q∗L∆Q∗R∆H ∗L∆H ∗
1.302 29.47 9.49 6.627·0.4144
10−4
L ∗LµH∗LµQ∗Rm∗Lmech∗
0.00876 0.0352 0.2375 7.180 0.02287
Tab. 4: Pa ame e s o CP QG300-2-100b o he main PU.
Hnom ,Qnom ,ηnom nnom ,Phyd .nom ,
m m3·h−1 pm kW
428 800 0.745 2980 932
H0.nom∗R∆Q∗L∆Q∗R∆H ∗L∆H ∗
2.641 43.89 15.12 5.897·0.4675
10−5
L ∗LµH∗LµQ∗Rm∗Lmech∗
1.03311 0.3122 2.3111 20.377 0.00436
©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 101
LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE
chanical powe o he d i es o he pumps gi en in he
manusc ip is signi ican ly highe since he e iciency o
he speci ied pumps is qui e low (0.809 o he suppo
pump and 0.745 o he main one). The nominal powe
o he elec ic mo o s used in his case is o e es ima ed.
To e i y he modelling, we used he nominal pa ame-
e s o he ins alled equipmen o one o he ope a ing
powe ul oil pumping s a ions. Fo e i ying he ope -
abili y and adequacy o he model, he simples scala
au oma ic p essu e s abiliza ion sys em based on a p o-
po ional equency con ol law was chosen. The ime
diag ams shown in Fig. 2 display he main esul s o
he es simula ion. In pa icula , he elec omechan-
ical pa ame e s o he mode a e shown in Fig. 2(a),
2(b), 2(c), and 2(d), and he hyd aulic pa ame e s a e
shown in Fig. 2(e), and 2( ). Figu e 2(d) shows he
elec omagne ic momen s IM. Figu e 2(e) shows he
p essu es o indi idual CPs (on di e en scales). Fig-
u e 2( ) shows he p essu e and olume ic low a e
o he PS wo king luid ( he CP low a e and he PS
low a e a e equal). The g aphs also show he ca a-
logue nominal pa ame e s o IM and CP.
The se alue o he PS p essu e, which was subjec
o s abiliza ion, was 473 m. I was equal o he sum o
he nominal p essu es o he suppo ing CP (45 m) and
he main CP (428 m). Du ing he simula ion, he low
a e o he wo king luid was changed i e imes. By he
ime o 3 s, he pipeline was comple ely blocked. A e
ha , he luid low a e was se a 105 dm3/s (47% o
he nominal low a e o he main uni ). A he ime
poin s o 40 s and 70 s, he consump ion inc eased by
43% o he p e ious alue each ime – o 150 dm3/s
and o 215 dm3/s, espec i ely, and a he ime poin s
o 100 s and 130 s – i dec eased o 150 dm3/s and 105
dm3/s espec i ely. A he momen 179 s he e was
an in e up ion o he powe supply o bo h uni s and
hei subsequen s op. To check he ope abili y o he
model, all changes in pipeline pa ame e s ook place
wi hin 0.1 s, co esponding o he speed o de elop-
men o eme gency modes. This led o co esponding
luc ua ions in p essu e and liquid low. A he same
ime, he PS p essu e was success ully main ained a
he le el o he se alue. I should be no ed ha unde
no mal ope a ing condi ions, he pipeline pa ame e s
change smoo hly and slowly, which does no lead o
such in ense luc ua ions. The e is only simula ion e -
i ica ion in he wo k. The lack o open access o bo h
expe imen al da a and powe equipmen o powe ul
pumping s a ions due o hei s a egic impo ance and
he inadmissibili y o in e up ing hei wo k explain
his. Fo he model’s es ing, we ga e as pe u ba-
ions o he olume ic low a e wi h la ge ampli ude.
The esul s o he ma hema ical expe imen , shown in
Fig. 2, showed he s abili y o he con ol and he
eliabili y o he model. Each dis u bance caused hy-
d aulic and elec omechanical ansien s, which quickly
and s eadily ended in no mal exploi a ion s eady-s a e
egimes. The adequacy o he modelling is e idenced
by he analysis o he ansien modes o equipmen , as
well as he coincidence o he indica o s o he s eady-
s a e modes, which comple e he ansien modes, and
he co esponding ca alogue pa ame e s o he equip-
men . The de elopmen and compa ison o he e i-
ciency o speci ic con igu a ions o he p essu e s abi-
liza ion sys ems we e no he pu pose o his wo k and
a e planned in u he esea ch.
4. Conclusion
A ma hema ical model o asynch onous PUs o a pow-
e ul PS in he p essu e s abiliza ion mode was de-
eloped and e i ied. The equa ions o he hyd aulic
subsys em a e o med based on he p inciple o elec-
ohyd odynamic analogy and e lec he physical p o-
cesses in i s in e nal elemen s. This made i possible
o use he heo y o elec ic ci cui s o w i e he equa-
ions o no only elec omechanical bu also hyd aulic
subsys ems. The composi ion o models o elec ome-
chanical and hyd aulic subsys ems in o a single whole
wi h compa able de ailing o hei elemen s allows si-
mul aneous esea ch o p ocesses in hese subsys ems,
di ec ly calcula ing and analysing he ene gy, elec o-
magne ic and hyd aulic coo dina es o he mode, in-
es iga ing hei mu ual in luence, e alua ing bo h he
s a e o indi idual elemen s o he uni s and he s a e
o he uni s as a whole. The calcula ion o he pa am-
e e s o he CP model is based on he geome ic di-
mensions o he in e nal elemen s o he CP, as well as
he physical p ope ies o he wo king luid. The e o e,
du ing simula ion, i is possible o in es iga e he im-
pac o ope a ional and eme gency de ia ions o hese
ac o s bo h on he egimes o luid anspo a ion sys-
ems and on he egimes o hei elec o echnical com-
plexes. Inc easing he e iciency o exis ing powe ul
PS is possible by op imizing he ope a ing modes. The
de eloped model makes i possible o conduc s udies
o egimes wi hou a physical expe imen on expensi e
ope a ing equipmen . Thanks o his, he e will be
no dis up ion o i s ope a ion and in e up ions in he
anspo a ion o liquid, elimina ing he isks o acci-
den s and inancial losses. The model can be in eg a ed
in o au oma ed design sys ems o powe ul pumping
s a ions and used du ing he sea ch o op imal design
solu ions. The po en ial abili y o ep oduce in he de-
eloped ma hema ical model a ious ypical me hods
o con olling he low and head o CP uni s will al-
low o an e ec i e echnical and economic compa ison
o con ol op ions bo h a he design s age o new and
du ing he mode niza ion o exis ing pumping s a ions.
A p omising di ec ion o imp o emen o he de eloped
ma hema ical model may be o ans o m i in o a hy-
d ide model by supplemen ing i wi h a physical sys em
o au oma ic con ol, which will signi ican ly inc ease
©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 102

LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE
(a) S a o cu en o IM. (b) S a o ol age o IM.
(c) F equency o s a o ol age o he IM. (d) Elec omagne ic o que o IM.
(e) P essu e o CP. ( ) Volume ic low a e and p essu e o he PS.
Fig. 2: Main esul s o he es simula ion
i s unc ionali y. The composi ion o he c ea ed model
wi h a model o a hyd aulic hea ne wo k can also be-
come an e ec i e ool o a comp ehensi e s udy o he
mu ual in luence o he mal, hyd aulic, and elec ome-
chanical p ocesses ha occu in cen alized hea supply
sys ems.
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