Model Crpno Akumulacijske Hirdroelektrane

8/2/2019 Model Crpno Akumulacijske Hirdroelektrane

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38Klepo, M., Mikulii, V., imi, Z., Model crpno-akumulacijske hidroelektrane , Energija, god. 57(2008), br, 1., str. 3863Klepo, M., Mikulii, V., imi, Z., The Pumped-Storage Hydro Power Plant Model , Energija, vol. 57(2008), No. 1., pp. 3863

MODEL CRPNO-AKUMULACIJSK(REVERZIBILNE) HIDROELEK-

TRANE U MODELU POUZDANOSRASPOLOIVOSTI ELEKTROENE-GETSKOG SUSTAVA

THE PUMPED-STORAGE HYDRPOWER PLANT MODEL WITHIN THE POWER SYSTEM RELIABILIAND AVAILABILITY MODE

Mio Klepo - Vladimir Mikulii - Zdenko imi, Zagreb, Hrvatska

U ovom radu izlae se model kojim se u proraune pouzdanosti i raspoloivosti elektroen-ergetskog sustava ukljuuje crpno-akumulacijska hidroelektrana, a zatim i model utjecaja

rizika nedostatka dotoka i zaliha vode na planiranje rada takvih postrojenja u okviru pro-gramskih sustava za operativna planiranja za vremenska razdoblja do razine godinu dana

unaprijed. Utjecaj neizvjesnosti pojava dotoka vee se za karakteristike vodotoka na kojimasu izgraeni akumulacijski bazeni i hidroelektrana, ali i za radne cikluse i mogui nain radau sustavu.

This paper deals with a model that comprises the pumped-storage hydro power plants inthe power system reliability and availability calculations, as well as a model of the impact

of in ow de ciency and water storage risks on the operational planning of such plantswithin the framework of operational planning systems for periods of up to one year. The

impact of in ow uncertainties is related to the characteristics of watercourses where thewater storage reservoirs and hydro power plants are built, but also to the operation cycles

and possible operation modes in the system.Kljune rijei: model crpno-akumulacijske hidroelektrane, model pouzdanosti i

raspoloivosti sustava, rizik nedostatka dotokaKeywords: in ow de ciency risk, power system reliability and availability model,pumped-storage hydro power plant model


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Klepo, M., Mikulii, V., imi, Z., Model crpno-akumulacijske hidroelektrane , Energija, god. 57(2008), br, 1., str. 3863Klepo, M., Mikulii, V., imi, Z., The Pumped-Storage Hydro Power Plant Model , Energija, vol. 57(2008), No. 1., pp. 3863 40

1 UVOD

Kada je rije o problemu ukljuivanja crpno-aku-mulacijske hidroelektrane u model pouzdanostii raspoloivosti elektroenergetskog sustava, unjemu je nuno razlikovati dvije osnovne kompo-nente ili dva dijela. S jedne strane, radi se o os-novnom modelu crpno-akumulacijskog pogonakoji je mogue modelirati na vie naina, bilosloenim modelima koji ukljuuju vie moguihrazliitih stanja i njihovih veza, bilo viestrukimkombiniranjem jednostavnih modela za pojedinefunkcije [1], [2] i [5], a s druge strane je uvjeto-vanost pogona takvih postrojenja njihovom ul-ogom u sustavu i neizvjesnou pojave dotoka istanja gornje i donje akumulacije [3], [4], [6], [8],[9] i [10].

Kod osnovnih modela kod kojih se generatorskii crpni pogon modeliraju odvojeno, osnovni ne-dostatak pristupa proizlazi iz nemogunosti dase usklade razdiobe vjerojatnosti pojedinih grupastanja koje se raunaju na razliitim osnovama.Nadalje, nije mogue jednostavno modelirati brziprelazak iz crpnog u generatorski pogon i even-tualno obrnuto, a tu je i nemogunost da se toizvede zbog odgovarajueg kvara. U ovom raduodabran je pristup kojim se crpno-akumulacijskipogon modelira jedinstvenim modelom s osamstanja u kojem se mogu identi cirati stanja gen-eratorskog i crpnog pogona, ali i sluajevi kvaro-va jedinice pri startu i tijekom pogona, odnosnoprelasci iz jednog pogona u drugi uz pripadnu vje-rojatnost kvara koja ima utjecaja na raspoloivostsustava [7], [10] i [12].

Rad svakog proizvodnog objekta u elektroener-getskom sustavu prati rizik da e prije ili tijekomulaska u pogon, odnosno tijekom pogona, ostatibez primarne pogonske energije, u ovom sluajuvode. Taj rizik vezan je za neizvjesnost pojave do-toka i stanja gornje i donje akumulacije. U pravi-lu, kod hidroelektrana rizik zbog neizvjesnostidotoka i stanja akumulacije puno je izraeniji odrizika pojave kvara. Kod crpno-akumulacijskehidroelektrane pojavljuje se i problem sloenogradnog ciklusa i ovisnosti o prilikama u sustavui nainu rada ostalih proizvodnih postrojenja [1],[10], [11] i [12].

2 OSNOVNI MODEL JEDINICEZA CRPNO-AKUMULACIJSKIPOGON

Kako je u uvodu istaknuto, crpno-akumulacijskipogon mogue je modelirati na vie naina,dakle modelima koji ukljuuju vie moguih

1 INTRODUCTION

Regarding the problem of including the pumped-storage hydro power plants in the power systemreliability and availability model, it is necessaryto distinguish two different basic components ortwo parts. On one hand, there is the basic modelof pumped-storage drive that can be modelled inseveral different ways, either with complex mod-els involving several different states and theirconnections or through multiple combination ofsimple models for speci c functions [1], [2] and[5]. On the other hand, there is the dependence ofthe drive of such plants on their role in the sys-tem and the uncertainty of in ows and the statusof the upper and lower reservoir [3], [4], [6], [8],[9] and [10].

With the basic models, where the generator andpump drive are modelled separately, the funda-mental shortcoming comes from the lack of pos-sibility to balance the probability distribution ofspeci c groups of states that are calculated ondifferent bases. Further, it is not easy to modela quick transition from the pumped to the gen-erator drive and possibly vice versa, and thereis also the impossibility to accomplish this dueto a defect. This work has opted for an approachwhereby the pumped-storage drive is modelledwith a single 8-state model in which the states ofgenerator and pumped drive can be identi ed, aswell as cases of the units failure at start-up andduring the operation, i.e. the alterations from onedrive to another along with the pertaining failureprobability having an impact on the system avail-ability [7], [10] and [12].

The operation of any generating plant in thepower system is exposed to the risk of runningshort of the primary drive energy before or inthe start-up process or during operation, in thiscase water. That risk is linked to the uncertaintyof in ow and the upper and lower reservoir lev-els. As a rule, with hydro power plants the riskassociated with uncertain in ows and reservoirlevels is much higher than the failure risk. Withthe pumped-storage hydro power plants thereis also the problem of a complex operation cycleand dependence on the conditions prevailing inthe system and the operation mode of other gen-erating plants [1], [10], [11] and [12].

2 THE BASIC MODEL OF APUMPED-STORAGE DRIVEUNIT

As emphasized in the Introduction, the pumped-storage drive can be modelled in several ways, i.e.


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Klepo, M., Mikulii, V., imi, Z., Model crpno-akumulacijske hidroelektrane , Energija, god. 57(2008), br, 1., str. 3863Klepo, M., Mikulii, V., imi, Z., The Pumped-Storage Hydro Power Plant Model , Energija, vol. 57(2008), No. 1., pp. 386341

razliitih stanja i njihovih veza, ali i kombiniran- jem odgovarajuih jednostavnih modela jedinica.U ovom radu izloeni su elementi modela kojimse pogon crpno-akumulacijske hidroelektranemodelira jedinstvenim modelom s osam stanja ukojem se mogu identi cirati stanja generatorskogi crpnog pogona, ali i sluajevi kvarova jedinice

pri startu i tijekom pogona, odnosno prelasci iz jednog pogona u drugi uz pripadnu vjerojatnostkvara koja ima utjecaja na raspoloivost sustava(slika 1).

Oznake na slici 1:

0 stanje rezervnog iskljuenja jedinice.1 stanje generatorskog pogona ,2 stanje kvara nastalog tijekom proizvodnje,3 stanje popravka nakon kvara nastalog

tijekom proizvodnje, a kad se pogon netrai,

4 stanje kvara pri ulasku generatora u pogon,5 stanje crpnog pogona,6 stanje kvara nastalog tijekom crpljenja,7 stanje popravka kad se pogon ne trai, a

nakon kvara nastalog tijekom crpljenja,+ / uestalost pojave/prestanka potrebe za

generatorskim pogonom,+ / uestalost pojave/prestanka potrebe za

crpljenjem vode u gornji bazen, GG / uestalost kvara/popravka jedinice u

svezi s generatorskim pogonom,G uestalost popravka jedinice nakon

kvara pri ulasku u proizvodni pogon,PP / uestalost kvara/popravka jedinice u

svezi s crpnim pogonom,

with models that include several possible differ-ent states and their links but also by combiningthe units corresponding simple models. This pa-per presents the elements of the unique 8-statemodel where the states of generator and pumpeddrive can be identi ed as well as cases of start-up and operation incurred failures in the unit, i.e.,

the alternations from one drive to another withthe pertaining failure probability having an im-pact on the systems availability (Figure 1).

Legend (Figure 1):

0 ready for service (reserve) unit state,1 generator drive state,2 generation incurred failure state,3 repair after generation incurred failure

state while operation is not required,4 generator start operation incurred

failure state,5 pump drive state,6 pump drive operation incurred failure

state,7 repair after pump drive operation

incurred failure state while operation isnot required,

+ / transition rate of on/off generator drive,+ / transition rate of on / off water pump

ing into upper reservoir, GG / transition rate of unit failure / repair

related to generator drive,G transition rate of unit repair of start

operation incurred failure,PP / transition rate of unit failure / repairrelated to pump drive,

Slika 1 Model jedinice za crpno-akumulacijski pogonFigure 1 State-space diagram for a pumped-storage unit


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+ z z / uestalost pojave/prestanka potrebe zaprelaskom iz crpnog u proizvodni pogon,

G P vjerojatnost kvara pri startu generatora, PG P vjerojatnost kvara jedinice pri prijelazu

iz crpnog u proizvodni pogon.

Broj i tip kvarova jedinice izrazito su ovisni o rad-

nom ciklusu jedinice, pogotovo to se kod gen-eratorskog naina rada radi o vrnom pogonukod kojeg se kvarovi dogaaju pri samom startu,tijekom pogona ili pri prijelazu iz crpnog pogona ugeneratorski, a traju tijekom potrebe za pogonomili i nakon to ta potreba prestane. Radni ciklusiutjeu na uestalosti prijelaza jedinice iz jednogstanja u drugo, tako da je nuno eksplicitno razlik-ovanje uestalosti nastanka ili prestanka potrebeza odreenim pogonskim stanjem, promjenamapogonskih stanja, kvarova pri startu, kvarova tijekom pogona ili pri prijelazu iz jednog pogon-skog stanja u drugo, te uestalosti popravakanakon kvarova iz razliitih stanja.Sustav linearnih diferencijalnih jednadbi Mark-ovljeva procesa prema slici 1 ima oblik:

Poetni uvjeti jesu: The start-up conditions are:

A stationary solution is sought, i.e., one where:Trai se stacionarno rjeenje, tj. rjeenje kada je:

+ z z / transition rate of alternation frompump to generation drive,

G P failure probability in generator starting-up, PG P failure probability in alternation from

pump to generation drive.

The number and type of unit failures are mark-

edly dependent on the unit operation cycle, es-pecially in the generator operation mode wherefailures occur during the start-up in conditions ofpeak load, operation and drive-to-drive alterna-tion incurred failures and last as long as thereis a need for operation and occasionally after theneed ends. The operation cycles in uence thetransition rate of alternation from one drive toanother, so that it is necessary to clearly distin-guish between the transition rate of the occur-rence or termination of the need for a speci cdrive operation, alternation of drives, failures atstart-up position, failures during operation orduring drive-to-drive alternations, as well as therate of repairs after failures in different states.

The system of linear differential equations of theMarkov process according to Figure 1 has the fol-lowing form:

3G23

32G1G2

5PG4G2G1G0G1

7P53G100

)()(

)()(

)1()()1()(

)()(

P P t P

P P P t P

P z P P P P z P P t P

P P P P P t P

+=

++=

+++++=

+++++=

+

+

++

++

6P5P105

5PG4G0G4

)()(

)(

P P z P z P t P

P z P P P P t P

++++=

+=

++

++

7P67

76P5P6

)()(

)()(

P P t P

P P P t P

+=

++=

+

+

.0)0(,0)0(,0)0(,0)0(

,0)0(,0)0(,0)0(,1)0(

7654

3210

========

P P P P

P P P P

.7,6,5,4,3,2,1,0,0)( == nt P n

(1)

(2)

.


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Uz uvjete (2), sustav (1) poprima novi oblik: With conditions (2), the system (1) assumes a newform:

Uz jednadbu identiteta: With the identity equation:

stacionarno rjeenje, tj. stacionarne vjerojatnostistanja jesu:

the stationary solution, i.e., stationary probabili-ties of the state are:

.)(0

)(0

)(0

0

)(0

)(0

)1()()1(0

)(0

7P6

76P5P

6P5P10

5PG4G0G

3G2

32G1G

5PG4G2G1G0G

7P53G10

P P

P P P

P P z P z P

P z P P P P

P P

P P P

P z P P P P z P P

P P P P P

+=

++=++++=

+=

+=++=

+++++=

+++++=

+

+

++

++

+

+

++

++

176543210 =+++++++ P P P P P P P P

[ ] [ ]{ }

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ].

)(

)()(

)()(

)()(

)()(

)()()(

)()()(

)()()(

PG

7

PPG

6

PPG

5

PG

4

PG

3

GPG

2

GPG

1

PG

0

++

=

+++=

++++

=

++

=

+++

=

++++

=

+++++=

++++++

=

+

+

+

+

+

DI HF L KEI A P

DI HF L KEI A P

DI HF L KEI A P

GH DJ L KJ HM E A P

GK DM I KJ HM F A P

GK DM I KJ HM F A P

GK DM I KJ HM F A P

DI HF L KEI C GK DM I KJ HM F B P

[ ][ ]{ }

[ ][ ]{ })()()(

)()(

)()()(

)(

GGGGG

p

PPPPP

G

A B A

GK DM I KJ HM F

AC A

DI HF L KEI

++++++++++

++++++++=

++

++

gdje je: where:

(4)

(5)

(6)


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Znatno pojednostavljenje rjeenja mogue jeuvae li se ranije iznesene pretpostavke kojevrijede openito za objekte elektroenergetskogsustava, a to su da su vremena ostajanja u stanjima 2, 3, 4, 6 i 7 znatno kraa od vremena os-tajanja u stanjima pogonske spremnosti 0, 1 i5, odnosno da su uestalosti prijelaza u stanjapogonske spremnosti znatno vee od uestalostiulazaka u stanja pogonske nespremnosti. Uz

zanemarivu pogreku koja se pritom ini, postu-pak raunanja vjerojatnosti pojedinih stanjapostaje znatno laki i jednostavniji. Dakle:

It is possible to greatly simplify the solution by takinginto account the previous assumptions which gen-erally apply to power system facilities and accordingto which the periods of remaining in states 2, 3, 4, 6and 7 are much shorter than the periods of remain-ing in the operational stand-by conditions 0, 1 and 5,meaning that the transition rates of alternation intothe stand-by operation condition are signi cantlyhigher than the transition rate of entering into non-

operation conditions. With a marginal error appear-ing in the process, the calculation routine of speci cstate probabilities becomes much easier. Hence:

)].()([

)(

)(

)()1(

)]()([

)1(

)(

)(

PPP pG

G pG

pG

p pPG

pG

pG

p pGPG

pGG

GGG pG

pGPG

p p pG

GG pG

z M

z L

K

z P J

I

P H

z P G

F

z z E

P D

C

B

++++++=

++==

++==

=++=

=

++++++==

+++=+++=

+++

+

+

++

+

++

++

+

+

+

)()()(1

)()()()(1

)()()()(1

])([

)()()()(1

)1()()()(1

)1()()()()(1

)1()()()()(1

)]()1()([

))((1

GPGG*7

GPPGG*6

PGPGG*5

PGG

PGGG*4

GP pGG*3

GPG pGG*2

GPG*1

G

PG pGG*0

++++

+++++

+++++

++++

++++

+++++

++++++

++++++

+++

++

++++

=

+++++

=

++++++

=

++

++++++

=

++++

=

+++++

=

++++++

=

+++

++++

=

z P

z P

z P

P z P z

z P

P z P

P z P

P z P

z P z

P

[ ])(

)()(

pG

GP

A

GH DJ L KJ HM E A

+=+++

++

+

(7)

(8)


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ujednaen, ili se s velikom sigurnou moepredviati, a jednako tako i poeci vodnih valova.S druge strane, postoje vodotoci ije su osci-lacije protoka vrlo visoke i neizvjesne, to nji-hovo predvianje ini vrlo nesigurnim kao i radhidroelektrana s kojima su u vezi.

Kod odreivanja rizika nedostatka dotoka polazise od veliine srednjeg dnevnog protokadQ (m3/s)na mjernom mjestu ili pro lu, za koju je kaostohastiku veliinu vezana neizvjesnost pojave.Time je i za proizvodnju hidroelektrana vezananeizvjesnost pojave budui da su one ovisne i okoliinama vode, ali i o vremenskom rasporedudotoka. Obje te komponente ovisne su dalje o nizuutjecaja koji su ukljueni i u neizvjesnosti pojavasrednjih dnevnih dotoka, koji se dobiju kao rezul-tat svakodnevnih promatranja i mjerenja. Srednjidnevni dotoci poredani kronolokim redom dajutjedne, mjesene i godinje dijagrame protoka

na mjernim mjestima ili pro lima. Na temelju tihdijagrama odreuju se krivulje trajanja protoka,odnosno krivulje vjerojatnosti protoka tjedana,mjeseci ili sezona, te godine.

Meutim, za tonije odreivanje vjerojat-nosti pojave protoka i neizvjesnosti proizvod-nje hidroelektrana radi njihova ukljuivanja uproraun pokazatelja pouzdanosti i raspoloivostielektroenergetskog sustava nuno je odreditipripadajue krivulje vjerojatnosti protoka, i to pokraim vremenskim razdobljima od jedne godine.Prema tako dobivenim krivuljama zatim se mjere

i valoriziraju hidroloki uvjeti, a pripadajue vjerojatnosti pojave odreuju se znatno tonije. Pritomse uvijek mora uzeti u obzir da rad s kraim vre-menskim razdobljima moe izazvati iskrivljenjai pogreke kod obrade malih i velikih protoka, ada su veliine iskrivljenja i pogreaka ovisne oveliinama i uestalostima promjena protoka.

U ovom radu nee se izlagati sam postupakmatematiko-stohastikom obradom, nego e sepretpostaviti da su za mjerna mjesta protoka vena raspolaganju odgovarajue funkcije gustoevjerojatnosti ( )Q f ili frekvencije pojave protoka,odnosno funkcija distribucije (razdiobe) protoka

)(Q F i njen komplement )(1)(* Q F Q F = .Funkcijom distribucije protoka )(Q F odreena jevjerojatnost pojave protoka koji je manji ili jednakprotoku Q, a njezinim komplementom, tj. funkcijom distribucije )(1)(* Q F Q F = vjerojatnostpojave protoka veeg odQ (slika 2).

streams with a rather even ow over the year orpredictable with a fair amount of certainty, thereare also the beginnings of water waves. On theother hand, there are streams with high and un-predictable in ow variations which make theirforecasts extremely insecure, and thereby theoperational planning of hydro power plants as-

sociated with them.In establishing the in ow de ciency risk thestarting point is the average daily in ow value

/s)(m 3Q at the measuring point or pro le, whichas a stochastic value involves the uncertaintyof event. That is why the output of hydro powerplants involves the uncertainty of event, sincehydro power plants depend on both the quantitiesof water in ows and the seasonal in ow pattern.Both of these components are further linked toa series of impacts involved in the uncertaintiesof average daily in ow, obtained as a result of

day-to-day observations and measurements. Av-erage daily in ows lined up chronologically pro-vide weekly, monthly and annual in ow diagramson the measuring points or pro les. The in owduration curves are determined on the basis ofthese diagrams, and so are the weekly, monthlyor annual in ow probability curves. However, formore accurate determination of the in ow prob-ability and the uncertainty of hydro power plantgeneration for their inclusion in the calculationof power system reliability and availability indica-tors it is necessary to de ne the pertaining in owprobability curves for periods shorter than a year.

The curves thus obtained serve then as a basis forcalculating and evaluating the hydrological condi-tions, whereas the appurtenant probabilities arede ned with much greater precision. What mustalways be taken into account in this procedure isthat work with shorter periods of time may leadto distortions and errors in calculations with lowand high in ows, whereas the values of these dis-tortions and errors depend on the in ow variationtransition values and rates.

This work will not describe the mathematical-stochastic procedure itself, it will be assumedinstead that appropriate probability density func-tions ( )Q f or in ow occurrence frequencies orin ow distribution function )(Q F and its comple-ment )(1)(* Q F Q F = are already available forin ow measuring points and pro les. The in owdistribution function )(Q F de nes the in owprobability which is lower than or equal to in-ow Q, whereas its complement, i.e., distributionfunction )(1)(* Q F Q F = de nes in ow prob-ability higher thanQ (Figure 2).


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Slika 2 Funkcije ( )Q f , )(Q F i )(* Q F Figure 2 Functions ( )Q f , )(Q F and )(* Q F

Funkcija distribucije protoka )(Q F ima sljedeioblik:

gdje je:

minQ minimalna vrijednost protoka na mjernommjestu.

Vjerojatnost u krivulji vjerojatnosti protoka moese shvatiti kao relativno trajanje protoka, gdje je ukupno trajanje normirano na jedinicu, imese dobiva i krivulja trajanja protoka. Njezina jetonost tim vea to je dulji niz godina za koji seraspolae podacima i to je postupak aproksi-macije toniji. Prethodno naznaeni postupakprovodi se za vremenske intervale krae od go-dine (mjesec ili tjedan), da bi se dobile krivuljevjerojatnosti koje odraavaju periodike varijacijeprotoka i njima pridruenih vjerojatnosti tijekomgodine.

Ovisno o duljini vremenskog razdoblja, kod elek-troenergetskog bilanciranja mogue proizvodnjehidroelektrana odreuju se bilo izborom referent-ne hidrologije, kada se radi o planiranju za duljarazdoblja (mjesec, vie mjeseci, godina), bilopredvianjem protoka, kada se radi o planiranjuza kraa razdoblja (dan, tjedan ili par tjedana).Bez obzira na trajanje, za svaki dan tijekom razdo-blja planiranja u proraun se ulazi sa zadanimprotokom i njemu pridruenom vjerojatnouostvarenja, ili s protokom koji odgovara unaprijedodabranoj vjerojatnosti pojave hidrolokih uvjeta

where:

minQ is a minimum in ow value at the measuringpoint.

The probability in the in ow probability curve canbe seen as a relative duration of in ow wherethe overall duration is standardised to value one,whereby the in ow duration curve is also ob-tained. Its accuracy rises with the rising numberof years for which data are available, and with therising accuracy of the approximation procedure.The said procedure is carried out for time inter-vals shorter than a year (a month or a week), soas to obtain probability curves which re ect sea-sonal in ow variations and related probabilitiesover the year.

Depending on the length of the period concerned,in the process of power balancing possible out-puts of hydro power plants are established eitherthrough the choice of reference hydrology, when itcomes to longer-term planning (a month, severalmonths, a year), or by in ow forecasting when itcomes to shorter-term planning (a day, a weekor a couple of weeks). Regardless of the duration,for every day of the planning period calculation isstarted with a given in ow and its related ma-terialisation probability or with the in ow cor-responding to a pre-selected probability of the

The in ow distribution function )(Q F has the fol-lowing shape:

=

Q

Q

Q

QQ f Qq f Q F min

d)(d)()( (10),


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prema krivulji vjerojatnosti ili trajanja protoka.Pritom se moe postupiti na dva naina.

Prvi je, to je vie primjereno planiranjima za dul- ja vremenska razdoblja, kada se protoci, a time imogue proizvodnje i optereenja, odreuju pre-ma odabranoj hidrologiji koja u pravilu ima veu

vjerojatnost ostvarenja. Odabrana hidrologijapredstavlja odrednicu za simulaciju rada sustava, jer se kod prorauna pouzdanosti i raspoloivostisustava proizvodnja i optereenje hidroelektraneprema odabranom modelu ukljuuju u dijagramoptereenja, a hoe li izvriti namijenjenu ulogu,ovisi o odnosu prema sluajno odabranom brojukoji je uniformno distribuiran na intervalu (0,1).

Hidroelektrana e izvriti namijenjenu ulogu usvim onim sluajevima kada sluajno odabranibroj u krivulji vjerojatnosti protoka pada u podrujevjerojatnosti protoka veih od protoka prema un-

aprijed odreenoj ili zadanoj hidrologiji. I obrnuto,hidroelektrana nee moi izvriti namijenjenuulogu ako generirani broj u krivulji vjerojatnostiprotoka pada u podruje vjerojatnosti protokamanjih od protoka prema unaprijed odreenojili zadanoj hidrologiji. U oba prethodna sluajau osnovi se promatra odgovarajui komplementfunkcije distribucije. Rizik nedostatka vode kaomjera neizvjesnosti proizvodnje i optereenjahidroelektrane simulacijom se ukljuuje uproraune pouzdanosti sustava.

Drugi je nain da se prije elektroenergetskog

bilanciranja simulacijom, metodom generiranjasluajnih brojeva prema krivulji vjerojatnostiprotoka odredi niz protoka, a to znai i njimapridruene mogue proizvodnje i optereenjahidroelektrane. U tom sluaju rizik nedostat-ka vode kao mjera neizvjesnosti proizvod-nje hidroelektrane ukljuuje se ve na samompoetku prorauna pokazatelja pouzdanosti iraspoloivosti. Pritom se podrazumijeva da etonost postupka rasti s brojem simulacija, to jeu sluaju primjene znaajka Monte Carlo metodesimulacije [5], [9] i [10].

Kod hidroelektrana koje vode koriste kako dotjeui koje zbog ogranienog kapaciteta kompenzaci- jskih bazena nisu u mogunosti akumulirativee koliine vode, rizik nedostatka zaliha vodemoe se zanemariti budui da nema gotovonikakvu vanost. Pogotovo to vrijedi u usporedbis akumulacijskim hidroelektranama. Relativnavanost rizika nedostatka dotoka raste s podi-zanjem tehnike spremnosti postrojenja da tobolje iskoristi raspoloive koliine vode budui dau tom sluaju mogue proizvodnje i optereenjaovise samo o raspoloivim koliinama vode iz do-toka.

occurrence of hydrological events according tothe probability curve or in ow duration. The mat-ter can be approached in two ways.

The rst approach, which is more appropriatefor longer-term planning, is applied where thein ows and thereby possible outputs and loads

are de ned according to the selected hydrologywhich as a rule is more likely to materialise. Theselected hydrology is a guideline for system oper-ation simulation, because in the system reliabilityand availability calculation the output and load ofa hydro power plant are included in the load dia-gram according to the selected model, whereaswhether or not its performance will meet expec-tations depends on a randomly selected numberwhich is evenly distributed on the interval (0,1).

The hydro power plant will play its assigned rolein all cases when the randomly selected number

in the in ow probability curve falls into the areaof in ow probabilities higher than in ows in thepredetermined or pre-given hydrology. Vice ver-sa, the hydro power plant will be unable to play itsassigned role if the number generated in the in-ow probability curve falls into the area of in owprobability lower than the in ows in the prede-termined or pre-given hydrology. In both of thesecases, what is basically observed is the corre-sponding distribution function complement. Thewater de ciency risk as a measure of the plantsoutput and load uncertainty is included throughsimulation in the system reliability calculation.

The second approach is that before the powerbalance simulation the sequence of in ows isdetermined by the method of generating randomnumbers based on the in ow probability curve,and thereby the plants possible outputs and loadsassociated with them. In that case the water de -ciency risk, as a measure of output uncertainty, isincluded at the very beginning of calculating thereliability and availability indicators. In this regardit is understood that that the accuracy of the pro-cedure will grow with the number of simulations,which is a feature of the Monte Carlo simulationmethod [5], [9] and [10].With hydro power plants using in ows as theycome and those which cannot accumulate largeramounts of water due to limited capacities ofcompensation reservoirs, the water de ciencyrisk can be disregarded as it is of no importance.This particularly applies in respect of storageplants. The relative importance of the in ow de-ciency risk grows with the improving capabilityof the plant to make optimum use of availablewater, because in that case potential outputs andloads solely depend on available water in ows.


12/26


4 UTJECAJ RIZIKA NE-DOSTATKA DOTOKA NA RADCRPNO-AKUMULACIJSKEHIDROELEKTRANERad crpno-akumulacijske hidroelektrane u

sluaju da je prirodni dotok u gornji akumulacijskibazen jednak ili vei od potrebnog za rad u razdo-blju vrnih optereenja ili za pokrivanje potrebaza brzim startom ne razlikuje se od rada bilo kojedruge akumulacijske hidroelektrane. Meutim, usluaju da prirodnog dotoka u gornji akumulacijski bazen uope nema, ili je pak nedovoljan zapretpostavljeni nain rada, pojavljuje se potrebaza crpljenjem vode iz donjeg akumulacijskog ba-zena u gornji.

Daljnja razmatranja znatno se olakavaju ako seuvedu sljedee tri pretpostavke. Prva je da u don- jem bazenu, tj. na usisnoj strani u svakom trenut-ku stoje na raspolaganju dovoljne koliine vode zacrpljenje. U protivnom, neizvjesnost pojave dotokai stanja donjeg akumulacijskog bazena mora seukljuiti u ukupnu neizvjesnost ili rizik nedostat-ka vode. Druga je pretpostavka da se voda crpiu razdobljima malih optereenja sustava, imese eli izbjei iskljuivanje iz pogona agregata utermoelektranama tijekom noi ili preko vikenda,ili kada je na bilo koji drugi nain na raspolaganju jeftina energija za crpljenje. Trea pretpostavkavrijedi za sluaj pojave potrebe za brzim startom,a podrazumijeva sposobnost vrlo brzog prelaskaiz stanja crpljenja punim optereenjem u turbins-ki pogon punim optereenjem. Kada je u pogonu,crpno-akumulacijska hidroelektrana treba proiz-voditi uz maksimalno moguu snagu. Rad crpno-akumulacijske hidroelektrane planira se u cik-lusima, najee tjednim, a nain ukljuivanja udijagram optereenja prikazan je na slici 3.

Potrebu za crpljenjem odreuju prirodni dotoci ugornji akumulacijski bazen i potrebna proizvod-nja. Ako prirodnog dotoka nema, odnos energijecrpljenja h p, E i proizvedene energije hg, E odreen je izrazom:

4 THE IMPACT OF INFLOW DE-FICIENCY RISK ON THE OPER-ATION OF PUMPED-STORAGEHYDRO POWER PLANTSThe operation of a pumped-storage hydro power

plant in cases where the natural in ow into theupper storage reservoir equals or exceeds theoperational requirements in peak load periods orthe quick-start requirements does not differ fromthe operation of any other storage hydro powerplant. However, in case of complete absence ofnatural in ow into the upper reservoir or if it doesnot suf ce for the required operation regime, aneed appears to pump water from the lower tothe upper reservoir.

Further considerations will be much easier if thefollowing three conditions are introduced. Therst is that in the lower reservoir, i.e., on the suc-tion side a suf cient amount of water for pump-ing is available at any time. Otherwise the uncer-tainty of in ow and the level of the lower reservoirmust be included in the overall uncertainty or thewater de ciency risk. The second condition isthat water is pumped during low load periods,in order to avoid a shut-down of generator unitsin thermal power plants during the night or overweekends, or in situations when cheap pumpingpower is available from any other source. Thethird condition applies if there is a need for aninstantaneous start and it implies a possibilityof very rapid transition from full-load pumpingoperation to full-load turbine drive. When in op-eration, the pumped-storage hydro power plantshould generate at maximum capacity. The op-eration of a pumped-storage hydro power plant isplanned in cycles, mostly on a weekly basis. Theway of its inclusion in the load diagram is shownin Figure 3.

Pumping requirements are determined by in-ows into the upper reservoir and by the requiredoutput. In absence of the in ow, the relation ofpumping power h p, E and generated energy hg, E isde ned by the following expression:

gdje je:

= hg,h p,h stupanj korisnog djelovanja cilusa, h p, stupanj korisnog djelovanja crpljenja, hg, stupanj korisnog djelovanja proizvodnje

elektrine energije.

where:

= hg,h p,h cycle ef ciency rate, h p, pumping ef ciency rate, hg, power generation rate.

h p,hhg, E E = (11),


13/26


Slika 4 Smjetaj crpno-akumulacijske hidroelektrane u dijagramu optereenjaFigure 4 Location of the pumped-storage hydro power plant in the load diagram

Slika 3 Ukljuivanje crpno-akumulacijske hidroelektrane u dijagram optereenjaFigure 3 Inclusion of pumped-storage hydro power plant in the load diagram

Energiju za crpljenje osiguravaju termoelektranesustava niih speci nih trokova goriva. Proizve-dena snaga crpno-akumulacijske hidroelektranesmjeta se u sam vrh dijagrama optereenja, da bise proizvedenom energijom smanjila proizvodnjatermoelektrana s viim speci nim trokovimaza gorivo (slika 4).

The pumping power is provided by thermal pow-er plants with lower speci c fuel costs. The gen-erated power of a pumped-storage hydro powerplant is placed on the very top of the load curve,so as to reduce the output of thermal powerplants with higher speci c fuel costs (Figure 4).


14/26


Rizik nedostatka vode javlja se zbog mogunostipojave nedovoljnog prirodnog dotoka, kao i ne-dostatka kapaciteta, tj. snage i energije za cr-pljenje. Tu treba ukljuiti i sluaj potrebe da seprekine crpljenje radi brzog pokrivanja iznenad-nih manjkova snage u sustavu.

Osnovni model za ukljuivanje crpno-akumulacijske hidroelektrane u model pouzdanosti elek-troenergetskog sustava jest model vrne jed-inice kojoj se u uestalosti kvarova pri startu, kojiznae nemogunost preuzimanja optereenja,i uestalosti kvarova tijekom pogona, ukljuujurizici nedostatka vode. Zbog vrlo male uestalostikvarova takvih postrojenja, mogua je i potpunazamjena uestalosti kvarova uestalostima pojave nedostatka vode. Pritom se u razmatranjamoraju ukljuiti kriteriji rada koji mogu biti samoekonomski, samo sigurnosni, ili kombinacijaobaju.

isto ekonomski pristup mogu je kada u sus-tavu postoje raspoloivi i drugi kapaciteti koji mogupreuzeti optereenje prema kriteriju ekonominosti.Kriteriji ekonominosti navedeni su ranije.

Ekonomian rad ostvaruje se ako je ispunjen uvjet:

odnosno:

gdje je:

pc - speci ni trokovi goriva termoelektrana kojeproizvode energiju za crpljenje,

gc - specifini trokovi goriva termoelektrana ija seproizvodnja smanjuje proizvodnjom crpnoaku-mulacijske hidroelektrane.

Dakle, ekonomian rad postie se kada je poveanatednja promjenjivih pogonskih trokova, dobivenaneoptereivanjem termoelektrana u vrnom dijelukrivulje trajanja optereenja, jednaka ili vea odpoveanih promjenjivih pogonskih trokova cr-pljenja, uzimajui pritom u obzir korisnost ciklusa.Time se uvjet (13) moe napisati i u obliku:

where:

pc - speci c fuel costs in thermal power plantswhich provide power for pumping;

gc - speci c fuel costs in thermal power plantswhose output is diminishing through the out-put of pumped-storage hydro power plants.

Therefore, economical operation is achievedwhen the increased savings in variable fuelcosts, obtained through reduced capacity ofthermal power plants in the peak section ofthe load curve, equals or exceed the increasedvariable pumping costs, taking into account thecycle ef ciency. Thus the condition (13) can alsobe expressed as follows:

There is a water de ciency risk due to the possibil-ity of in ow de ciency, as well as due to the lack ofcapacity, i.e., capacity and power required for pump-ing. Another contingency to be taken into account isa sudden need to discontinue the pumping opera-tion in order to make up quickly for sudden powershortages in the system.

The basic model for the inclusion of pumped-stor-age hydro power plants in the power system reli-ability and availability model is the peak unit modelto which water de ciency risks are added in thetransition rate of start operation incurred failures,resulting in the inability to take over the load, andin the transition rate of drive operation incurred fail-ures. Owing to a very low failure transition rate insuch plants, it is possible to wholly replace the fail-ure transition rate by the water de ciency transitionrate. In such considerations operation criteria mustbe included which may be exclusively economic orexclusively safety-related, or a combination of both.

A purely economic approach is possible when thereare other capacities available in the system that cantake over the load in accordance with the criteria

Economical operation is achieved if the followingcondition is met:

or:

(12)

(13)

(14)

hg,gh p, p E c E c

cc g p

g pcc

,

,

.


15/26


isto sigurnosni pristup najee je vezan zasluajeve kada je sveukupni raspoloivi kapac-itet sustava nizak, te se, bez obzira na trokove,trai cjelokupni kapacitet crpno-akumulacijskehidroelektrane. Razlog je za to potreba da sepokrije to vei iznos nedostajueg optereenja.Crpno-akumulacijska hidroelektrana zauz-

ima posljednje mjesto u listi prioriteta kakobi se voda drala raspoloivom koliko god jeto mogue. S druge strane, voda se koristi dopotpunog pranjenja gornjeg bazena, kada semanjak snage u sustavu poveava za cjelok-upnu snagu hidroelektrane. Ponovno punjenjegornjeg bazena vri se optereenjem prveraspoloive termoelektrane bez obzira na njezintip ili speci ne trokove proizvodnje. Time seizazivaju visoki dodatni trokovi u sustavu.

Trei sluaj kombinacija je prethodna dva, a u os-novi predstavlja nastojanje da se smanje trokovi

u sustavu i da se izbjegne potpuni gubitak ka-paciteta crpno-akumulacijske hidroelektrane.To se postie planiranjem rada prilagoenogzahtjevima svakog pojedinog pogonskog sluaja.Za isti ukupni iznos neisporuene energijepreraspodjelom nepokrivenog optereenja tijekom sati vrnog optereenja dobije se nizstanja kada potrebe sustava nisu u potpunostipokrivene, ali su njihove amplitude i trajanjapuno nia, to je i prihvatljivije za sustav. Pokrajtoga, smanjuje se i proizvodnja termoelektranas najviim proizvodnim trokovima, a isto tako ibroj potpunih pranjenja gornjeg bazena. Pot-

punim pranjenjem gornjeg bazena gubi semogunost pokrivanja dinamikih obveza kaoto su rotirajua ili pripravna rezerva, regu-lacija snage i frekvencije. Takve obveze utjeuna nastanak rizika nedostatka vode, ali na bitnorazliit nain. Naime, kod rotirajue rezerve ilibrzog starta troenje vode znatno je smanjenoili traje vrlo kratko, za razliku od regulacije kojatrai kontinuiran pogon tijekom dueg razdobljai znatniji utroak vode. U tom drugom sluaju,da bi se smanjio rizik nedostatka vode koji sepojavljuje zbog znatnijih pranjenja, prelazi sena pristup koji znai vee trokove u sustavu.

Kada se odredi osnovni pristup planiranju, tj.pogonu crpno-akumulacijske hidroelektrane,ostaje jo odrediti osnovne parametre mod-ela za simulacijski postupak prorauna pouz-danosti. Ti su parametri u prvom redu ovisni opostojanju prirodnog dotoka, a to znai o nje-govom stohastikom karakteru. Kada u gornjibazen dotjee dovoljno vode iz prirodnog do-toka, pogon se tretira kao pogon akumulacijske hidroelektrane. Meutim, kada tog dotokanema ili nije dovoljan, ukupna proizvodnja crp-no-akumulacijske hidroelektrane neovisna je opromjenjivim koliinama vode iz dotoka, to nevrijedi i za energiju crpljenja vode u gornji bazen

of economic feasibility. The economic criteria werementioned earlier.

A purely safety-motivated approach is mostly as-sociated with the cases where the overall availablecapacity of the system is low, so that, regardless ofcosts, the total capacity of a pumped-storage hydro

power plant is sought. The reason for it is the needto cover the greatest possible amount of the miss-ing load. Pumped-storage hydro power plants areplaced at the bottom of the priority list in order tokeep water available as long as possible. On the oth-er hand, water is used up to the total discharge of theupper reservoir when power shortage in the systemis increased by the hydro plants total power. Re ll-ing of the upper reservoir is done by loading the rstthermal power plant available regardless of its typeor speci c generation costs. This involves high extracosts in the system.

The third case is a combination of the above men-tioned two cases and is basically an attempt to re-duce system costs and to avoid a complete loss ofcapacity of the pumped-storage hydro power plant.That can be achieved with operation planning adapt-ed to the requirements of each particular operationcase. For the same total amount of undelivered en-ergy, through redistribution of uncovered load duringthe peak load hours a series of states are obtainedwhere the system needs are not fully met, but theiramplitudes and duration are much lower, which ismore acceptable for the system. In addition, the out-put of thermal power plants with the highest gen-

eration costs is decreased, and so is the number ofupper reservoir depletions. Upon complete depletionof the upper reservoir the possibility is lost to coverdynamic obligations such as the spinning reserve orstand-by reserve, capacity and frequency regulation.Such obligations have an impact on the occurrenceof water de ciency risk, but in an entirely differentway. Namely, in using the spinning reserve or the in-stantaneous start water consumption is much low-er or takes a very short time, unlike the regulationwhich requires continuous operation over a longerperiod of time and more water consumption. In thelatter case, in order to lower the water de ciency riskresulting from signi cant discharges the approach istaken which involves higher system costs.

Once the basic approach to planning, i.e., to thepumped-storage hydro power plant operation hasbeen de ned, what still remains to be de ned arethe basic parameters of the reliability calculationsimulation model. These parameters depend in therst place on the natural in ow, in other words, on itsstochastic character. When enough water ows intothe upper reservoir from the natural in ow, the driveis treated as a drive of the storage hydro power plant.However, in absence of that in ow, or if it is insuf-cient, the total output of the pumped-storage hydropower plant is independent of the variable quantities


16/26


koju trebaju osigurati ostale elektrane u sus-tavu. Za tu energiju vrijedi razdioba vjerojatnostiuvjetovana prirodnim dotokom.

Maksimalna snaga crpno-akumulacijskehidroelektrane u generatorskom pogonu izno-si:

of in ow water, which does not apply to the energy ofwater pumping into the upper reservoir to be provid-ed by other power plants in the system. What appliesto this energy is probability distribution determinedby the natural in ow.

Maximum capacity of a pumped-storage hydro pow-

er plant in the generator operation mode is:

(15)

(17)

(16)

[ ],MW1081,9 3tnAmaxg, = H Q P

gdje je:

n H neto pad [m], b p H H = ,

b H bruto pad [m], omjer neto i bruto pada,

t stupanj korisnog djelovanja u turbinskompogonu.

Snaga crpljenja potrebna za maksimalni protokcrpljenja max p,Q iznosi:

gdje je:

max p,Q maksimalni protok crpki [m3/s], p stupanj korisnog djelovanja u crpnompogonu.

Budui da se rad crpno-akumulacijske hidroelek-trane u osnovi planira u tjednim ciklusima, tjedniiznos energije koja se bez prirodnog dotoka moeproizvesti maksimalnom snagom u turbinskompogonu iznosi:

gdje je:

r broj radnih dana u tjednu, vtt maksimalno mogue trajanje koritenja mak

simalne snage u turbinskom pogonu, bez obzirana eventualnu dodatnu proizvodnju iz prirodnogdotoka.

Energiju za crpljenje osiguravaju ostale elektranesustava, i to u razdoblju malih optereenja ijetrajanje odreuje sljedei izraz:

where:

n H net fall [m], b p H H = ,

b H gross fall [m], net-gross fall ratio,

t

ef ciency rate in turbine drive.Pumping capacity required for maximum pumpow max p,Q is:

where:

max p,Q maximum pump ow [m3/s], p ef ciency rate in pump operation mode.

As the operation of a pumped-storage hydro pow-er plant is planned in weekly cycles, the weeklyamount of power that can be generated withoutnatural in ow at maximum capacity in turbinedrive is:

where:

r working days in a week, vtt maximum possible duration of using maxi-

mum capacity in turbine drive regardless ofpossible additional generation from naturalin ow

Power for pumping is provided from other plantsin the system in low-load periods, the duration ofwhich is de ned by the following expression:

[ ],MW10181,9 3 p

bmax p,max p,= H Q P

[ ]GWh10 3vtmaxg,mavt, = t r P W x ,


17/26


(18)

(19)

(20)

(21)

(22)

( ) vvm 24 t r nr t r T T +==

gdje je:

T trajanje ciklusa (168 sati za tjedan), n broj neradnih dana u ciklusu, vt trajanje vrnih optereenja u radnom danu.

U sluaju da nema prirodnog dotoka, koliinavode koja se trai u turbinskom pogonu mora biti jednaka koliini vode koja je crpljenjem dovedenau gornji bazen, tako da je maksimalni protok cr-pki odreen izrazom:

gdje je:

AQ maksimalni protok turbina (veliina izgradnje)[m3 /s],

To znai da su veliine max p,Q i max p, P ovisne o oblikudijagrama optereenja, odnosno o nainu radahidroelektrane. Tjedna koliina energije potrebnaza crpljenje vode u gornji bazen iznosi:

Pomou izraza (16) dobije se:

Ako postoji prirodni dotok u gornji bazen, on ese nastojati iskoristiti uz maksimalnu snaguhidroelektrane, a trajanje takvog rada odreujese iz uvjeta jednakosti koliine vode koja dotjeetijekom ciklusa (tjedna) i koliine vode kojom seproizvodi energija uz maksimalni protok. Dakle:

where:

T cycle duration (168 hours per week), n number of non-working days in a cycle, vt peak load duration on a working day

In absence of natural in ow, the amount of wa-ter required in turbine drive must be equal to theamount of water brought by pumping to the upperreservoir, so that maximum pump ow is de nedby the expression:

where:

AQ maximum turbine ow (designed size) [m3 /s],

This means that the values max p,Q and max p, P aredetermined by the shape of the load diagram,i.e., the hydro power plants operation mode.The weekly amount of power required for waterpumping into the upper reservoir is:

With expression (16) the following is obtained:

If there is a natural in ow into the upper reservoir,an attempt will be made to use it at maximumcapacity of the hydro power plant, and the dura-tion of such operation will be determined basedon the equal amount of water coming in duringa cycle (week) and the amount of water used forgenerating electricity at maximum ow. Hence:

v

vtAmax p, )(24 t r nr

t r QQ

+=

[ ].GWh10181,910 6v bA3mmax p,max p, == t r H QT P W p

[ ] .GWh10

pt

maxvt,

pt

3vtmaxg,

max p,

W t r P W ==

Ava

)(24Qr

Qnr t

+=

,

,

.


18/26


(23)

(24)

(25)

(26)

(27)

gdje su :

21 , C C parametri ovisnosti snage crpki

pumpno-akumulacijske hidroelektraneo prirodnom dotoku.

Tijekom vtt sati proizvodi se energija dijelom izprirodnog dotoka, a dijelom crpljenjem vode ugornji akumulacijski bazen. Kada je vtva t t < , cr -pljenje je potrebno u trajanju vavt t t sati, i to zaproizvodnju maksimalnom snagom. Iz prirodnogdotoka proizvodi se energija:

a iz vode koja je crpljena:

Protok crpki je:

Za taj protok potrebna je snaga crpki:

Iz prethodnog slijedi ovisnost snage crpki oprirodnom dotoku:

Za pretpostavljene konstantne bruto padove istupnjeve korisnog djelovanja ta ovisnost je lin-earna i moe se napisati u obliku:

where:

21 , C C parameters of pump capacity depen-

dence on natural in ow.

During vtt hours electricity is generated partlyfrom the natural in ow and partly by pumpingwater into the upper reservoir. When vtva t t < , therequired duration of pumping is vavt t t hours, atfull capacity. Electricity generated from naturalin ow amounts to:

and electricity from pumped water:

The pump ow is:

This ow requires the following pump capacity:

Dependence of pump capacity on the natural in-ow can be derived from the foregoing:

For the assumed constant gross falls and ef-ciency rates the said dependence is linear andcan be expressed as follows:

[ ],GWh10 3vamaxg,va = t r P W

[ ].GWh10)( vamaxvt,3vavtmaxg, pa W W t t r P W ==

v

vavtA p )(24

)(t r nr

t t r QQ

+=

[ ].MW10181,9 3 p

b p p= H Q P

[ ] [ ]Q

T Q

nr P

T

t r P

Qt r nr

nr H

t r nr

t r Q H P

m ptA

maxg,

m pt

vtmaxg,

v p

3 b

v p

3vtA b

p

)(24

)(24

10)(2481,9

)(24

1081,9

+=

++

+=

[MW].

(28)[ ],MW21 p

QC C P =

.


19/26


Funkcija ( )Q f P = p prikazana je na slici 5. Zavie crpki radi se o familiji krivulja. Konstantnidio krivulje nastaje kao posljedica toga to je zasluaj ni < i crpki u pogonu, te protok ( )iQQ t naraspolaganju samo snaga i P Mp, kojom se dodatnocrpi voda da bi se postigla zahtijevana proizvodnja

maxvt,W . Potrebna energija za crpljenje iznosi:

Iz razdiobe vjerojatnosti prirodnog dotoka i ovis-nosti (30) formira se funkcija gustoe vjerojatnosti

( )vaW f , tj. funkcije gustoe vjerojatnosti proizvod-nje crpno-akumulacijske hidroelektrane iskljuivokoristei prirodni dotok. Vjerojatnost da se taproizvodnja nalazi u intervalu [ va'va'va , W W W + ] jednaka je vjerojatnosti pojave dotoka iz intervala[ QQQ +'' , ], dakle:

Ako je prirodni dotok u gornji akumulacijski ba-zen takav da omoguava koritenje maksimalne

snage i due od vremenavtt

, vtvat t >

, crpljenjenije potrebno.

Nakon to je odreena ukupna potrebna proiz-vodnja crpno-akumulacijske hidroelektrane

maxvt,W , neizvjesnost pojave dotoka u gornji bazenzapravo ima utjecaja samo na potrebnu energijucrpljenja koju osiguravaju ostale elektrane sus-tava. Za tu energiju moe se uzeti da ima prirod-nim dotokom uvjetovanu razdiobu vjerojatnosti.Energiju za crpljenje proizvode ostale elektranesustava, i to tijekom razdoblja minimalnihoptereenja, zbog ega dijagram optereenja

sustava raste za iznos ispadima uvjetovanesnage crpljenja. Time se modi cira dijagramoptereenja, to znai da model optereenjatreba prilagoditi novonastaloj situaciji.

Razdioba vjerojatnosti prirodnog dotoka ranije jedetaljno obraena. Mogua proizvodnja raunase prema izrazu:

Function ( )Q f P = p is shown in Figure 5. Morepumps make a family of curves. The constant sec-tion of the curve is a result of the fact that for thecase ni < pumps in operation, and the ow ( )iQQ t only i P Mp, capacity is available used for additionalwater pumping so as to achieve the required output

maxvt,W . Power required for pumping is:

From the natural in ow probability distributionand dependency (30) the probability density func-tion ( )vaW f is formed, i.e., the probability densityfunction of the output of the pumped-storage hydropower plant by using the natural in ow only. Theprobability that such generation is situated in theinterval [ va'va'va , W W W + ] equals the probability ofin ow from the interval [ QQQ +'' , ], hence:

If the natural in ow into the upper reservoir suf cesto allow the use of maximum capacity for periods

longer than vtt

, vtvat t >

, no pumping will be neces-sary.

After determining the total required output of thepumped-storage hydro power plant, maxvt,W , the un-certainty of in ow into the upper reservoir has animpact only on power required for pumping providedfrom other plants within the system. It can be as-sumed that power has a probability distribution de-termined by the natural in ow. It can be assumed thatthis power has a probability distribution determinedby natural in ows. Power for pumping is generatedby other plants in the system during the periods of

minimum load. For that reason the system load dia-gram is rising by the amount of outage-determinedpumping capacity. The load diagram is thereby modi-ed, which means the load model should be adjustedto the newly arisen situation.

The natural in ow probability distribution has al-ready been described above in detail. Possible outputis calculated by means of the following expression:

(29)[GWh].)(

110)(

10)(1

81,910

vamaxvt,

pt pt

3vavtmaxg,

6vavt

p bA

3

m p pm

W W t t r P

t t r H QT P W

=

=

==

(30)

(31)

[ ].GWhA

maxvt,va QQ

W W =

[ ][ ]vavavavavava )()()((

W W f W W W W P QQ f QQQQ P

+


20/26


(36)

(37)

(32)

(33)

(34)

(35)

Funkcija gustoe proizvodnje hidroelektrane izprirodnog dotoka ima oblik:

Dotok tQ za proizvodnju energije maxvt,W tijekomsati vtt rauna se pomou izraza:

Funkcija gustoe vjerojatnosti proizvodnje crp-no-akumulacijske hidroelektrane iz vode koja jecrpljenjem dovedena u gornji bazen odreuje seiz jednakosti vjerojatnosti energija koje povezujuizrazi (23) i (24). Ta jednakost ima oblik:

Funkcija gustoe vjerojatnosti energije dobivenecrpljenjem vode u gornji bazen sada glasi:

Potrebno je odrediti i odgovarajue funkcijepotrebne energije crpljenja )( pmW f . Ovis-nost potrebne energije za crpljenje o energijiiz prirodnog dotoka odreuje izraz (29). Prematom izrazu, razlika u odnosu na proizvedenu en-ergiju koritenjem crpljene vode samo je u kon-stantnom faktoru, odnosno koe cijentu smjerarazliitom od jedinice, tako da vrijede slinarazmatranja. Dakle, prethodni izrazi za vjerojatnost i funkciju gustoe vjerojatnosti mogu sepisati i u obliku:

The density function of pumped-storage hydropower plant generation from the natural in ow isshaped as follows:

In ow tQ for output maxvt,W over vtt hours is calcu-lated by means of the following expression:

The probability density function of pumped-stor-age hydro power plant generation from waterpumped into the upper reservoir is determinedfrom the probability equality of powers linked byexpressions (23) and (24). That equality has thefollowing form:

The probability density function of power obtainedby pumping water into the upper reservoir now

reads as follows:

It is also necessary to determine the correspond-ing functions of required pumping energy )( pmW f .The dependence of the required pumping power onnatural in ow energy is de ned by expression (29).According to it, the difference in relation to powergenerated by using the pumped water lies only inthe constant factor, in other words, in a directioncoef cient other than one, so that similar consid-erations apply. Therefore, the foregoing expres-sions for probability and for the probability densityfunction can also be expressed as follows:

vavava

)()()(

W QQQQ P

W QQ f

W f

+


21/26


Potrebna snaga crpljenja crpno-akumulacijske hidroelektrane kad postoji prirodni dotoku gornji akumulacijski bazen ovisi o koliinivode koja dotjee, ali i o ispadima uvjetovanojraspoloivoj snazi crpki. Podloge za razmatranjaispadima su uvjetovana raspoloiva snaga za cr-pljenje odreena na temelju vjerojatnosti ispada

crpki prema osnovnom modelu jedinica za crp-no-akumulacijski pogon i razdioba vjerojatnostidotoka. Budui da se cijela hidroelektrana tre-tira kao komponenta sustava, a da se u pravilusastoji od vie agregata iste snage0 P , diskretnafunkcija gustoe )( p p P f bit e viestupanjska.Raspoloiva snaga hidroelektrane za crpljenje,uvjetovana ispadima agregata distribuirana jepo binomnoj razdiobi.

Raspoloiva snaga crpljenja ima diskretnufunkciju gustoe vjerojatnosti, koja primjenomDiracove funkcije poprima oblik:

gdje je :

i indeks stupnja snage crpljenja ilibroj crpki spremnih za pogon,

)( MP, i P P vjerojatnost stupnja snage.

Svakom opsegu snage od ukupnon

crpkihidroelektrane pridodaje sem stupnjeva snage.Uz pretpostavku konstantnosti bruto padova istupnjeva korisnog djelovanja, mogua su dvasluaja. Prvi je da su broj crpki spremnih zapogon i njihova snaga dovoljni za crpljenje onekoliine vode u gornji akumulacijski bazen skojim e se uz vodu iz prirodnog dotoka moiproizvesti potrebna energija iz crpno-akumulacijske hidroelektrane maxvt,W . Na krivulji )( p Q f P = (slika 5) za rastue dotoke to znai nalaenje nanjezinom padajuem dijelu.

The required pump capacity of a pumped-storagehydro power plant, given a natural in ow into theupper reservoir, depends on the amount of in-coming water but also on the available outage-determined pump capacity. Considerations arebased on the outage-determined available pumpcapacity determined on the basis of pump outage

probability according to the basic pumped-storageunit model and the in ow probability distribution.Given the fact that the whole hydro power plant istreated as a component of the system and that, asa rule, it consists of a number of generating units ofthe same capacity 0 P , the discrete density function

)( p p P f will be a multi-stage function. The hydropower plant capacity available for pumping, de-termined by generator unit outages, is binomiallydistributed.

The available pumping capacity has a discreteprobability density function which by applying the

Dirac function assumes the following shape:

where:

i index of pumping capacity rate or thenumber of pumps ready for operation,

)( MP, i P P capacity rate probability.

To each capacity volume of n pumps of the hydropower plant m capacity rate is added. Assumingthat gross falls and ef ciency rates are constant,two scenarios are possible. The rst is that thenumber of pumps ready for operation and their ca-pacities are suf cient for pumping as much wateras necessary together with the natural in ow wa-ter to generate required electricity maxvt,W from thepumped-storage hydro power plant. On the curve

)( p Q f P = (Figure 5) for growing in ows it meansthe position in its falling section.

(38)ni P P P P P f iii ...,,2,1,0);()()( MP,MPMP,MP, p ==

Slika 5 Familija krivulja )( p Q f P =Figure 5 Family of curves )( p Q f P =

,


22/26


Treba, dakle, odrediti vjerojatnost da snaga cr-pljenja uvjetovana ispadima i dotokom lei unu-tar pojedinih opsega snage i, tj. u intervalu sgraninim vrijednostima ji P , p, i 1, p, + ji P . Ta vjerojat-nost jednaka je vjerojatnosti da je ispadima uvje-tovana snaga crpljenjaMP P najmanje tolika koliko je potrebno za crpljenje koliine vode pQ , ovisne

o dotokuQ

koji se nalazi u intervalu s granicama jiQ , i 1, + jiQ . Tim graninim vrijednostima dotokaodgovaraju stupnjevi snage crpljenja ji P , p, i 1, p, + ji P , a ovisnost je zadana linearnim dijelom krivuljew

)( p Q f P = . Budui da su dogaaji meusobnoneovisni, vjerojatnost sloenog dogaaja jest:

Funkcija razdiobe snage crpljenja glasi:

Vrijednost funkcije razdiobe na mjestu ( 1, + ji )iznosi:

U drugom sluaju, od ukupnon crpki za pogon je

spremnoi crpki, ali njihovi protoci nisu dovoljni dabi se uz prirodni dotok osigurala dovoljna koliinavode potrebna da bi hidroelektrana proizvelakoliinu energije maxvt,W . Koliine vode koje dotjeuprirodnim dotokom manje su ili jednake odkoliina koje odgovaraju snagama crpki u pogonu

i P MP, , to znai da vrijedi konstantni dio funkcije)( p Q f P = na slici 5. Za svaki stupanj snage i

vrijedi posebna karakteristika .konstMP, p == i P P U tom sluaju raspoloive crpke rade punimoptereenjem i P MP, . Stupnjevi snage crpljenja uv- jetovani ispadima i dotokom su:

Therefore, what should be determined is theprobability that the pump capacity determinedby in ows and outages lies within certain capac-ity volumes and in the interval with the marginalvalues ji P , p, and 1, p, + ji P . Such a probability is equalto the probability that the outage-determinedpump capacity MP P is at least as high as required

for pumping the amount of water pQ

, dependenton in owQ placed in the interval within margins jiQ , and 1, + jiQ . These marginal in ow values are

matched by pump capacity rates ji P , p, and 1, p, + ji P ,whereas dependence is given by the linear sectionof the curve )( p Q f P = . As the events are interde-pendent, the probability of a complex event is:

The pump capacity distribution function reads:

The value of the distribution function at the posi-tion ( 1, + ji ) is:

In the second scenario, out ofn pumpsi pumps

are ready for operation, but their ows are notsuf cient to provide enough water together withnatural in ows required for the hydro power plantto generate electricity in the amount ofmaxvt,W . Theamounts of water from natural in ow are belowor equal to the amounts corresponding to pumpcapacities in operation, i P MP, , meaning that theconstant section of function )( p Q f P = (Figure 5)applies. To each capacity degree a speci c char-acteristic .konstMP, p == i P P , applies. In that casethe available pumps operate at full load i P MP, . Theoutage-determined pump capacity rates and in-ows are:

(39)

(40)

(41)

m j

ni

QQQ P P P P P P P P ji jii ji ji

...,,2,1

...,,2,1,0

)()()( 1,,MP,MP1, p, p, p,

=

=


23/26


ni P i P P ii ...,,2,1,0,0MP, p, ===

Zbog neovisnosti dogaaja, vjerojatnost nastu-panja tih diskretnih stupnjeva snage jednaka jeproduktu vjerojatnosti pojedinanih dogaaja,dakle:

Funkcija gustoe i funkcija razdiobe vjerojatnostiza taj sluaj na mjestui jest:

)( p1 P f Q = inverzna je funkcija funkcije prema

izrazu (28).

Dva prethodno navedena sluaja pojave snagecrpljenja crpno-akumulacijske hidroelektraneuvjetovane dotokom i ispadima meusobno suiskljuiva, tako da je vjerojatnost da se snagacrpljenja nalazi unutar intervala s graninim vrijednostima ji P , p, i 1, p, + ji P jednaka sumi vjerojatnostiprema izrazima (39) i (42). Funkcija razdiobe vje-rojatnosti snage crpljenja jest:

Iz funkcije razdiobe slijedi funkcija gustoe vjerojatnosti:

Na mjestima i P p, pojavljuju se diskretni skokoviija je visina odreena izrazom (43) (slika 6).

Due to the independence of events, the probabilityof the occurrence of these discrete capacity ratesequals the product of probability of single events,hence:

The density function and the probability distribu-tion function for this case at positioni is:

)( p1 P f Q = is an inverse function of the function

according to expression (28).

The above two scenarios of the in ow- and out-age-determined pumped-storage hydro plantspumping power are mutually exclusive, so thatthe probability that the pump capacity is situatedwithin the interval within marginal values ji P , p, and

1, p, + ji P equals the sum of probabilities according to(39) and (42). The probability distribution functionof pumping capacity is:

The probability density function follows from thedistribution function:

On locations i P p, discrete leaps appear whoseheight is de ned by expression (43) (Figure 6).

[ ]{ } = = = = P P P P P P Q Q f P i n p i MP MP i t i MP i( ) ( ) ( ) ; , , ,...,, , ( ) ,1 0 1 2 (42)

(43)

(44)

[ ] 1...,,2,1,0,)()()( MP,1)(tMP, p, === ni P f Q F P f P f iiii

1...,,2,1,0;,)()( 1 p, p p,0

p, p =


24/26


(47)

Slika 6 Funkcija gustoe vjerojatnosti ispadima i dotokom uvjetovane snage crpljenjaFigure 6 Probability density function of outage and in ow determined pumping capacity

Funkcija gustoe vjerojatnosti mora zadovoljavatiuvjet:

Kod ukljuivanja crpno-akumulacijske hidroelek-trane u model za proraun pouzdanosti potreb-no je razmatrati i sluajeve koji mogu izazvatipromjene u nainu rada tih postrojenja, a dolazeiskljuivo od strane sustava. Prvi je takav sluajnemogunost sustava da osigura dovoljno snagei energije za crpljenje vode u gornji akumulacijskibazen. Time se neposredno poveava rizik ne-dostatka vode. Drugi se pojavljuje kao posljedicaiznenadne pojave manjka snage u sustavu kojiizaziva potrebu da hidroelektrana kao jedinica smogunou brzog starta kratkotrajno prekinecrpljenje vode u gornji akumulacijski bazen iprijee u turbinski pogon, tj. preuzme dio ispa-log optereenja ostalih elektrana sustava. U tomdrugom sluaju smatra se da hidroelektranamoe vrlo brzo promijeniti nain rada, te da eu turbinskom pogonu ostati sve dok se na druginain ne pokrije ispalo optereenje. Uz nedo-voljno crpljenje javlja se i dodatni neplaniranipotroak vode iz gornjeg akumulacijskog bazena.U tom sluaju rizik nedostatka vode poveava seako se ne moe relativno brzo, tj. ve tijekom is-tog ciklusa nadoknaditi taj dodatni neplaniranipotroak vode. Isti uinak ima i nemogunostda se osigura snaga i energija za crpljenje zbogznatnog poveanja trokova. Rad u regulaciji nor-malno je pogonsko stanje koje znai i dugotrajnijipotroak vode.

The probability density function must meet thefollowing requirement:

When pumped-storage hydro power plants areincluded in the reliability calculation model it isalso necessary to consider the cases which maybring about changes in the operation mode ofsuch plants and which come exclusively fromthe system itself. The rst such case is the sys-tems inability to provide enough power for pump-ing water into the upper reservoir. This directlyheightens the water de ciency risk. The secondcase occurs as a result of a sudden capacityshortage in the system creating a need that thehydro power plant as a unit with instantaneousstart capacity discontinues water pumping intothe upper reservoir for a short while and switchesover to turbine drive, i.e., takes over a part of out-age load of other plants in the system. In that sec-ond case the hydro power plant is deemed able tochange the operation mode very quickly and stayin turbine drive as long as the load shortage iscovered from other sources. In addition to insuf-cient pumping there is the additional unplannedwater consumption from the upper reservoir.In that case the water de ciency risk increasesif it is not possible to make up relatively quicklyfor such additional unplanned water consump-tion, i.e., already during the same cycle period.The same effect also comes from the inabilityto secure capacity and power for pumping dueto signi cantly increased costs. The regulationoperation is a normal condition, and that meansprolonged water consumption.

1d)(d)(max p,

0 p p p ==

P

p P P f P P f .


25/26


5 ZAKLJUAKRazvijen je model za proraun pouzdanosti iraspoloivosti crpno-akumulacijske hidroelek-trane. Razvijeni su kriteriji, mjerila i pokazateljivrednovanja i iskazivanja razine pouzdanosti iraspoloivosti crpno-akumulacijske hidroelek-trane ukljuujui i odgovarajue rizike nedostatkadotoka i stanja zaliha vode u gornjem i donjemakumulacijskom bazenu, te njihova ukljuivanjau programske sustave za operativna planiranjarada elektroenergetskog sustava. Razvijeni mod-eli uzimaju u obzir sloenost radnog ciklusa i jaku uvjetovanost ulaska u pogon i naina radacrpno-akumulacijske hidroelektrane o prilikamau elektroenergetskom sustavu, te nainu rada ispeci nim trokovima proizvodnje ostalih proiz-vodnih postrojenja, prvenstveno termoelektrana.Time je uspostavljen sustav za cjeloviti obuhvat ivrednovanje crpno-akumulacijske hidroelektranepri operativnom planiranju rada elektroener-getskog sustava.

5 CONCLUSIONA model has been developed for calculating thereliability and availability of pumped-storagehydro power plants. The criteria and indicatorshave also been developed for the evaluation anddetermination of the reliability and availabilitylevels of pumped-storage hydro power plants,including the risks of in ow de ciency and wa-ter storage in the upper and lower reservoirsand their inclusion in the programme systemsof planning the power system operation. The de-veloped models take into account the complexityof the operation cycle and strong dependence ofthe operation mode of the pumped-storage hydropower plants on the conditions prevailing in thepower system and on the operation mode andspeci c generation costs of other power plants,primarily thermoelectric power plants. A systemhas thus been put in place for integrated coverageand evaluation of pumped-storage hydro powerplants in planning the power system operation.


26/26

LITERATURA / REFERENCES[1] MIKULII, V., Matematiki modeli pouzdanosti i raspoloivosti u elektroenergetskom sustavu,

Doktorska disertacija, Sveuilite u Zagrebu, Elektrotehniki fakultet, Zagreb, 1981.[2] MIKULII, V., Matematiki model pouzdanosti komponente, Elektrotehnika EKTTBV24(1981)1,

1981.[3] JZSA, L., Primjena metode pouzdanosti u izgradnji proizvodnih kapaciteta u sustavu hidro i te-

moelektrana, Elektrotehnika EKTTBV24, 1981.[4] Studie Systemzuverlssigkeit, Institut fr Elektrische Anlagen und Energiewirtschaft, R.W.T.H.

ACHEN,1982[5] BILLINTON, R., ALLAN, R. N., Reliability Evaluation of Power Systems, New York, 1984[6] JZSA, L., Analitiki model pouzdanosti akumulacijskih hidroelektrana, I i II dio, Elektrotehnika

ELTHB2 28, 1985.[7] BILLINTON, R., ALLAN, R. N., Reliability Assessment of Large Electric Power Systems, Kluwer

Academic Publishers, Boston, 1988[8] INVERNIZZI, A., MANZONI, G., RIVOIRO, A., Probabilistic Simulation of Generating System Opera-

tion Including Seasonal Hydro Reservoirs and Pumped-Storage Plants, Electric Power & EnergySystems, Vol. 10, No. 1, 1988

[9] BILLINTON, R., LI, W.,Reliability Assessment of Electric Power Systems Using Monte Carlo Meth-ods, New York, 1994

[10] KLEPO, M., Pouzdanost i raspoloivost elektroenergetskog sustava pri operativnim planiranjimarada, Doktorska disertacija, Sveuilite u Zagrebu, Fakultet elektrotehnike i raunarstva, Zagreb,1996.

[11] KLEPO, M., Model neizvjesnosti pojave optereenja u modelu pouzdanosti i raspoloivosti elektro-energetskog sustava, Energija, god.46(1997), br. 3.

[12] KLEPO, M., Modeli proizvodnih jedinica u modelu pouzdanosti i raspoloivosti elektroenergetskogsustava model bazne jedinice, Energija, god.46(2997), br. 4.

Urednitvo primilo rukopis:2007-12-08

Prihvaeno:2008-02-14

Manuscript received on:2007-12-08

Accepted on:2008-02-14

Adrese autora:

Dr. sc.Mio Klepo,[email protected]

Hrvatska regulatorna agencija (HERA)Koturaka 5110000 Zagreb

HrvatskaProf. dr. sc.Vladimir Mikulii

[email protected] u Zagrebu,

Fakultet elektrotehnike i raunarstvaUnska 3

10000 ZagrebHrvatska

doc. dr. sc. Zdenko [email protected]

Sveuilite u Zagrebu,Fakultet elektrotehnike i raunarstva

Unska 310000 Zagreb

Hrvatska

Authors Adresses:

Mio Klepo, [email protected] Energy Regulatory AgencyKoturaka 5110000 ZagrebCroatiaProfVladimir Mikulii, [email protected] of Zagreb, Faculty of ElectricalEngineering and Information TechnologyUnska 310000 ZagrebCroatiaAssistant ProfZdenko imi, [email protected] of Zagreb, Faculty of ElectricalEngineering and Information TechnologyUnska 310000 ZagrebCroatia

Model Crpno Akumulacijske Hirdroelektrane

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