Modeling of Asynchronous Units of a Powerful Pumping Station in the Pressure Stabilization Mode

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. ©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 98 LYSIAK, V. VOLUME: 23 |NUMBER: 2 |2025 |JUNE 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. Re e ences [1] PULIDO-CALVO, I., J. C. GUTIERREZ- ESTRADA. Selec ion and Ope a ion o Pumping S a ions o Wa e Dis ibu- ion Sys ems. En i onmen al Resea ch Jou nal. 2011, ol. 5, iss. 3, pp. 1-20. h p://www.uhu.es/juanc.gu ie ez/a iculoPDF/ ERJ2011.pd . [2] BOHDANOV, R. Me hodic o elec ic ene gy consump ion pa e n calcula ion in oil ans- po a ion ia pipelines. Elec onic scien i ic jou - nal "Oil and Gas Business". 2012, iss. 1, pp. 58-68. h ps://elib a y. u/i em.asp?id=20709396. h p://ogbus. u/ iles/ogbus/au ho s/Bogdano / Bogdano _3.pd . [3] PARANCHUK, Y., V. LYSIAK. Ene gy e icien powe supply sys em and au oma ic con ol o modes o he “powe supply – pumping s a ion” ©2025 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 103 LYSIAK, V. 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