ენერგეტიკული სისტემების მდგრადობის ანალიზი სინერგეტიკის მეთოდებით ვ სესაძე, გ დალაქიშვილი, ვ კეკენაძ by georgianpolitics

saqarTvelos teqnikuri universiteti

v. sesaZe, g. dalaqiSvili, v. kekenaZe, g. WikaZe

energosistemebis mdgradobis gamokvleva sinergetikis meTodebis gamoyenebiT

damtkicebulia stu-s saredaqcio-sagamomcemlo sabWos mier

Tbilisi 2009

Tanamedrove energetikuli sistemebi ganekuTvnebian rTuli sistemebis kategorias da miekuTvnebian Tavisi bunebiT arawrfivi sistemebis klass. monografia eZRvneba energosistemebis Semadgeneli nawilebis mdgradobis analizs sinergetikis meTodebis gamoyenebiT. monografiaSi Seswavlilia energosistemaSi mimdinare gardamavali procesebi, gaanalizebulia energosiste-mis ZiriTadi elementebis maTematikuri modelebi, hopfis bifurkaciis Teoriis gamoyenebiT gaanalizebulia erTgeneratoriani, orgeneratoriani da samgeneratoriani energosistemebis mdgradoba, dadgenilia energosistemebis mdgradobis areebi diskretuli dinamikuri sistemebis marTvis parametris mixedviT, Seswavlilia energosistemis ori an ramdenime procesis qcevis Tavsebadoba droSi, anu sinqronizacia. gamokvleulia energosistemis qaosuri moZraoba sxvadasxva raodenobis generatorisTvis. SerCeuli koeficientebisa da parametrebis safuZvelze miRebuli Sedegebis mixedviT ganisazRvra energosistemis qceva. monografia gankuTvnilia bakalavrebisTvis, magistrebisaT-vis, doqtorantebisTvis da am dargSi momuSave specialistebisa da farTo profilis mecnierebisTvis. monografias vuZRvniT Cvens megobars da maswavlebels baton almasxan guguSvils. recenzentebi: saqarTvelos teqnikuri universitetis sruli profesori konstantine kamkamiZe saqarTvelos teqnikuri universitetis asocirebuli profesori irma kucia

© sagamomcemlo saxli ,,teqnikuri universiteti’’, 2009 ISBN 978-9941-14-373-1 http://www.gtu.ge/publishinghouse/ yvela ufleba daculia. am wignis arc erTi nawili (iqneba es teqsti, foto, ilustracia Tu sxva) aranairi formiT da saSualebiT (iqneba es eleqtronuli Tu meqanikuri), ar SeiZleba gamoyenebul iqnas gamomcemlis werilobiTi nebarTvis gareSe. saavtoro uflebebis darRveva isjeba kanoniT.

Sinaarsi Sesavali................................................................................................................. 1. gardamavali procesebi energosistemaSi ................... 1.1.

energosistemebis

ZiriTadi

6 11

elementebis

maTematikuri modelebi .................................................

1.1.1

sinqronuli manqanis gantolebebi........................ 11

1.1.2

asinqronuli manqanis gantolebebi.....................

1.1.3

eleqtruli qselis gantolebebi............................ 22

1.1.4

mTliani energosistemis maTematikuri modeli ..............................................................................................

1.1.5

mTliani energosistemis mTliani maTematikuri modelis Taviseburebebi................................

1.2

mravalmanqaniani sistemebis gamartivebuli gantolebebi.................................................................................. 29

1.2.1

Cqari eleqtromagnituri procesebis kvlevis maTematikuri modeli................................................ 29

1.2.2

Jdanovis modeli...................................................................

1.2.3

poziciuri modeli................................................................. 32

1.2.4 eleqtrul

qselSi

gasaSualebuli

pro-

cesebis modeli .................................................................

1.3

energosistemis statikuri mdgradoba.............. 39

1.4

ormanqanuri energosistemis dinamika................. 41

1.5

liapunovis veqtoruli funqciis meTodi energosistemis mdgradobisTvis ........................... 45

2. bifurkacia energosistemebSi .................................... 50 2.1

gaormagebis cikli da bifurkacia energosistemebSi......................................................................................

2.2

hopfis bifurkacia energosistemaSi................... 56

2.3

hopfis bifurkacia erTgeneratoriani sistemisTvis.......................................................................................... 59

2.4

hopfis bifurkacia orgeneratoriani energosistemisTvis.......................................................................... 64

2.5

hopfis bifurkacia samgeneratoriani sistemisTvis........................................................................................

3. qaosi energosistemebSi................................................................. 78 3.1

qaosis gamokvlevis meTodebi .................................. 78

3.2

qaosi erTgeneratorian energosistemaSi ....

3.3

qaosi orgeneratorian energosistemaSi ....... 90

3.4

qaosi samgeneratorian energosistemaSi ....... 94

3.5

dinamikuri qaosis kriteriumebi energosistemaSi da liapunovis maxasiaTebeli ..........

101

4. sinqronizacia energosistemebSi............................................. 109 4.1

sinqronizaciis problemebi ....................................... 109

4.2

sinqronuli generatorebis paralelur muSaobaSi avtomaturi CarTva ......................................

4.3

121

avariuli situaciebis analizi .............................. 126

4.3.1. asinqronuli reJimebis likvidacia, resinqronizacia................................................................................. 4.3.2. asinqronuli reJimebis niSnebi ............................. 4

126 129

4.4

sinqronizirebadi marTva ............................................. 131

danarTi..................................................................................................................

144

gamoyenebuli literatura................................................................. 162

Sesavali Tanamedrove ekonomikuri zrdis tempis daCqarebis erT-erTi ZiriTadi safuZveli aris energetika. samecniero-teqnikur progress, warmoebis intensifikacias, teqnikuri donis da Sromis nayofierebis amaRlebas Sedegad moaqvs adamianebis sayofacxovrebo pirobebis gaumjobeseba da warmoebis Tanamedrove aRWurvilobiTa da sayofacxovrebo teqnikiT uzrunvelyofa, raSic ganmsazRvreli

roli

ukavia

Tanamedrove

energo-

sistemas. Tanamedrove

energetikuli

sistemebi

ganekuTvne-

bian rTuli sistemebis kategorias. energosistemebis swraf ganviTarebasTan da avtomatizaciis donis amaRlebasTan

erTad

maTi

sirTule

ufro

izrdeba. rTuli sistemebis qveS SeiZleba vigulisxmoT iseTi sistemebi, romelTac aqvT sakmaod Rrma Siga kavSirebi da Sedgebian didi raodenobis urTierTdakavSirebuli

energetikuli

urTierTqmedi sistema

elementebisgan.

warmoadgens

rTul

amgvarad, sistemas,

romlis ZiriTadi Semadgeneli nawilebia: 1.

sinqronuli da asinqronuli manqanebi (genera-

torebi);

eleqtruli

torebidan

wredi,

sxvadasxva

romelSic

saxis

xdeba

eleqtruli

genera-

signalebis

gadacema; 3. datvirTva. winamdebare

naSromSi

ganvixilavT

energosiste-

mebSi mimdinare gardamaval procesebs, imisaTvis rom warmodgena Segveqmnas energosistemis muSaobis Sesaxeb. Tavdapirvelad vagebT da viyenebT mis maTematikur modelebs. dResdReobiT, rTuli energosistemis funqcionirebisa da marTvis analizisaTvis ZiriTad formirebul saSualebad iTvleba energosistemebis maTematikuri modelireba Tanamedrove gamoTvliTi teqnikis gamoyenebiT. energosistemis Semadgeneli nawilebis maTematikuri modelebis urTierTkavSiri, romlebic adeqvaturad asaxaven sxvadasxva ierarqiul doneze gamosakvlev procesebs, warmoadgenen specifikidan gamomdinare, Sesaswavli procesebis xasiaTis gaTvaliswinebiT, gadasWreli amocanebis Taviseburebebs. cnobilia, rom mdgrad sistemebs aqvT ori mdgeneli: 1. gardamavali procesi; 2. damyarebuli mdgomareoba. damyarebul mdgomareobaSi, roca sistemebSi dasrulebulia yvela gardamavali procesi, sistema warmoadgens an konstantas, an raime periferiul amo7

naxsns

[1].

rogorc

cnobilia,

sistemas

ewodeba

dinamikuri, Tu masSi procesebi mimdinareobs droSi. dinamikur

sistemebSi

damyarebuli

mdgomareoba

niSnavs asimptotur mdgomareobas, roca t

â&#x2C6;&#x17E;, damya-

rebuli reJimi aucileblad unda xasiaTdebodes Sesabamisi funqciis SezRuduli mniSvnelobebiT. Diferencialuri gantolebebis sistemis amonaxsnebsa da mis damyarebul mdgomareobas Soris sxvaobas gardamavali procesi ewodeba. cnobilia, rom sistemis arawrfivobisas arsebobs parametrebis

farTo

diapazoni,

romelSic

sistemis

moZraoba damyarebul reJimSi aris SezRuduli, magram ara perioduli. rxevebi iReben SemTxveviT xasiaTs da xasiaTdebian ara diskretuli, rogorc periodulis dros, aramed farTo, uwyveti speqtriT. amis garda, sistemis

moZraobis

xasiaTi

imdenad

mgrZnobiarea

sawyisi pirobebis variaciaze damokidebulebiT, rom SeuZlebelia zusti amonaxsnebis grZelvadiani prognozireba. aseTi maRali mgrZnobiaroba sawyisi pirobebis variaciaze damokidebulebiT damaxasiaTebelia qaosuri sistemisaTvis, romelic warmoadgens determinirebul sistemas da mis moqmedebas aqvs SemTxveviTi saxe [2,3]. energosistemis gamokvlevaSi mniSvnelovani roli aqvs bifurkaciis movlenas. 8

bifurkacia aris gadasvlebi, romlebic SeiZleba moxdes dinamikuri qcevidan sxvadasxva saxeebs Soris parametrebis cvlilebisas. bifurkaciuli movlenis aRwera Seicavs cnebebs gansakuTrebul Sesaxeb.

amonaxsnebisa

bifurkaciis

kargi

maTi

mdgradobis

magaliTia

SemTxveva,

rodesac sistemis gansakuTrebuli amonaxsnebis wertili iyofa jerad gansakuTrebul amonaxsnebad. Ddakvirvebadi movlenebi â&#x20AC;&#x201C; es ar aris ubralod mdgradobis Secvla, es ufro bifurkaciis magaliTia. unda aRiniSnos Semdegi: gansakuTrebuli wertilis mdgradobis qveS gvesmis mdgradoba liapunovis mixedviT. naSromSi ganvixilavT struqturul mdgradobas, romelic bifurkaciuli analizis kvlevis sagania. gansakuTrebuli

mniSvneloba

energosistemebis

mdgradobis gamokvlevisas eniWeba qaosis arsebobas. qaosi aris arawrfivi movlena, romelic gvxvdeba yvela mecnierul disciplinaSi. misi gaanalizeba intuiciurad Zalze rTulia, magram mainc realurad arsebuli movlenaa. ZvelTaganve mkvlevarebi xvdebodnen qaoss, magram mas Rebulobdnen fizikur xmaurad da gverds uvlidnen. axla ukve cnobili gaxda, rom qaosi Cndeba ara marto xelovnur sistemebSi, aramed sistemebSic, romelTac axasiaTebT arawrfivoba. uxeSad rom vTqvad, qaosi warmoadgens sakmaod araCveulebriv qcevas romeliRaca sistemisa damyare9

bul reJimSi. termini `qaosuriâ&#x20AC;? determinirebul fizikur

maTematikur

sistemebSi

gamoiyeneba

iseTi

moZraobebis mimarT, romelTa traeqtoriac Zlier aris damokidebuli sawyis pirobebze. qaosuri rxevebi Cndeba Zlieri arawrfivobis dros. cnobilia, rom arawrfivobis dros arsebobs

pa-

rametrebis farTo diapazoni, romelTa drosac sistemis moZraoba damyarebul reJimSi aris SezRuduli, magram ara perioduli. rxevebi SemTxveviTia da xasiaTdebian ara diskretuli, rogorc periodulis dros, aramed farTo uwyveti speqtriT. amas garda sistemis moZraobis xasiaTi imdenad mgrZnobiarea sawyis pirobebze damokidebulebiT, rom grZelvadiani prognozireba zusti amonaxsnisa SeuZlebelia. aseTi maRali mgrZnobiaroba sawyisi pirobebis variaciaze damokidebulebiT qaosuri sistemis erT-erTi damaxasiaTebeli Tvisebaa. qaosuri sistema determinirebuli sistemaa, romlis moqmedebasac SemTxveviTi saxe aqvs. energosistemebSi didi yuradReba aris gamaxvilebuli sinqronizaciis movlenaze, romelic Tavis umravles saerTo interpretaciaSi niSnavs ori an ramodenime procesis qcevis Tavsebadobas droSi.

`sinqronizireba~ _ niS-

navs SevaTanxmoT droSi, mivyveT an operireba gavukeToT sistemas mxolod erT da imave siCqareze, gamomdinare erTi da imave droidan. 10

1. gardamavali procesebi energosistemebSi 1.1. energisistemebis ZiriTadi elementebis maTemetikuri modelebi

1.1.1. sinqronuli manqanis gantolebebi. energosistemebSi mimdinare procesebis maTematikuri modelirebis sakiTxebi Seiswavles iseTma gamoCenilma mecnierebma, rogorebic arian: s.a. lebedevi, p.s. Jdanovi, a.a. gorevi, romelTa naSromebic mniSvnelovania energosistemebis mdgradobisa da gardamavali procesebis TeoriaSi. energosistemebis proeqtirebisa da eqsploataciisas aucilebelia eleqtruli amocanebis kompleqsis amoxsna,

romlebic

statikuri

moicaven

dinamikuri

eleqtromeqanikuri

damyarebuli

mdgomareobis,

gardamavali

reJimebis, xangrZlivi

procesebis,

mokled

SerTvis denebis da sxv. amocanebis amoxsnas [4]. sinqronuli konturiT

manqana

warmoadgens

rotorze sistemas,

ori

dempferuli

romelic

Sedgeba

eqvsi magniturad dakavSirebuli konturisgan. am konturebidan sami manqanis fazis sivrceSi umoZrao sami konturia, xolo danarCeni sami konturi â&#x20AC;&#x201C; aRgznebis gragnili da ori dempferuli konturia.

sinqronuli manqanis gardamavali eleqtromagnituri procesebis gantoleba fazur koordinatebSi SeiZleba Semdegnairad Caiweros [1,2,3,4,5]: Uca= rcica+d Ψ ca/dt; Ucb = rcicb+d Ψ cb/dt; Ucc=rcicc+d Ψ cc/dt;

(1.1)

Uf = rfif+d Ψ f/dt; O = rgig+d Ψ g/dt; O = rhih+d Ψ h/dt. sadac UUca, Ucb, Ucc, ica, icb, icc, _ Sesabamisad manqanis eleqtrul wredTan mierTebis fazuri Zabvebi da statoris fazuri denebia; Uf _ aRgznebis gragnilis Zabvaa; rc, rf, rg, rh _ aqtiuri winaRobebia Sesabamisad statoris fazis, aRgznebis gragnilis, horizontaluri da vertikaluri dempferuli konturebisaTvis; if, ig, ih _ denebia aRgznebis gragnilSi, da horizontalur da vertikalur dempferul konturebSi; Ψh

Ψ ca, Ψ cb, Ψ cc, Ψ f, Ψ g,

_ nakadSerTvebi Sesabamisad fazis konturebSi,

aRgznebis gragnilSi, horizontalur da vertikalur dempferul

konturebSi,

romlebic

arian

denebis

funqciebi: Ψ ca=

LaIca + MabIcb+MacIcc+MafIf+MagIg+MahIh;

Ψ cb=

MabIca + LbIcb+MbcIcc+MbfIf+MbgIg+MbhIh;

Ψ cc=MacIca+McbIcb+LcIcc+McfIf+McgIg+MchIh; Ψ f=

MafIca + MbfIcb+MafIcc+LfIf+MfgIg; 12

(1.2)

Ψ g=

MagIca + MbgIcb+McgIcc+MfgIf+LgIg;

Ψ h=

MahIca + MbhIcb+MchIcc+LhIh

sinqromanqanis procesebis Teoriis Camosayalibeblad unda davuSvaT: 1. TviTinduqciisa da urTierTinduqciis koeficientebis damoukidebloba. fazaTa konturebis TviTinduqciis da urTierTinduqciis koeficientebi La, Lb, Lc, Mab, Mac, Mbc, warmoadgenen rotoris horizontaluri RerZisa da a( θ ) fazis magnitur RerZs Soris kuTxis periodul

funqciebs

periodiT П, xolo

urTierT-

induqciis koeficientebi Maf, Mbf, Mcf, Mag, Mbg, Mcg, Mah, Mbh, Mch arian θ kuTxis perioduli funqciebi periodiT 2 Π . 2. mravalwevris umaRlesi wevrebis koeficientebi furies

mwkrivad

daSlisas

arian

mcireni

axdenen mniSvnelovan zegavlenas manqanaSi mimdinare procesebze [1,5,6]. am pirobebis gaTvaliswinebiT TviTinduqciis da urTierTinduqciis

koeficientebi

SeiZleba

Caiweros

Semdegi saxiT: La=Lcp+Lm cos2θ Lb=Lcp+Lm cos(2θ+1200); Lc=Lcp+Lmcos(2θ-1200); Mab= -Mcp+Mo cos(2θ-1200); Mcc= -Mcp + cos(2θ+1200); Mbc = -Mcp + Mo cos2θ. 13

(1.3)

sadac

L cp =

Lmax + Lmin ; 2

M cp =

Lm =

Lmax − Lmin ; 2

M max + M min M − M min ; M o = max . 2 2

Lmax, Lmin, Mmax, Mmin - statoris fazaTa konturis TviTinduqciisa da urTierTinduqciis koeficentebis maqsimaluri da minimaluri mniSvnelobebia. Maf=Mf cosθ; Mbf=Mf cos(θ-1200); Mcf=Mf cos(θ+1200); Mag=Mg cosθ; Mbg=Mgcos(θ-1200);

(1.4)

Mcg=Mg cos(θ-1200); Mah= -Mh sinθ; Mbh= -Mhsin(θ-1200); Mch= -Mhsin(θ+1200); t

θ= ∫ ωdt + θ0. 0

aq Mt, Mg, Mh _ statoris fazis konturis urTierTinduqciis koeficientebis maqsimaluri mniSvnelobebia aRgznebis konturTan, horizontalurad da vertikalur dempferul konturebTan mimarTebaSi, ris brunvis kuTxuri siCqarea.

_ roto-

sinqronuli manqanis moZraobis gantolebas, romelic Cawerilia dalamberis principiT, aqvs Semdegi saxe [1,5]: J

dw =TT_Te dt

(1.5)

sadac TT _ warmoadgens rotoris RerZis Siga Zalebis ganlagebis momentis, anu pirveladi Zravis momentis, J _ rotoris inerciisa da masTan dakavSirebuli

moZravi

nawilebis

momentia;

eleqtro-

magnituri momentia, romelic moqmedebs rotorze da gamoisaxeba damokidebulebiT: Te=

ϑw ϑθ

(1.6)

magnituri velebis energia gamoiTvleba Semdegi formuliT: w= 1

∑ jkΨ k k

(1.7)

am gamosaxulebaSi ajamva xdeba manqanis yvela konturis mixedviT. (1.1) ÷ (1.7) gantolebebi warmoadgenen sinqronuli manqanis gantolebebs fazur koordinatebSi. Tu (1.1) ÷ (1.7) periodulkoeficientebian gantolebebs gardavqmniT mudmivkoeficientebian gantolebebad, SeiZleba mivaRwioT sinqronuli manqanis gardamavali procesebis analizur gamartivebas [5]. gadasvla fazuri koordinatebidan mudmivkoeficientebian gantolebaTa sistemaze xorcieldeba park15

gorevis gardaqmnebis daxmarebiT. denebis, Zabvebis da nakadSerTvebis myisieri fazuri sidideebis nacvlad Semoitaneba

axali

saangariSo

sidideebi

id,

iq,

io, Ψ d, Ψ q, Ψ o, Ud, Uq, Uo, romlebic dakavSirebulni arian fazur sidideebTan Semdegi TanafardobebiT [7]:

⎡Uca ⎤ ⎡Ucd ⎤ ⎢Ucq ⎥ =Ac ⎢Ucb ⎥ ; ⎢ ⎥ ⎢ ⎥ ⎢⎣Uco ⎥⎦ ⎣⎢Ucc ⎥⎦

(1.8)

⎡ Icd ⎤ ⎡ Ica⎤ ⎢ Icq ⎥ =A ⎢ ⎥ ⎢ ⎥ c ⎢ Icb⎥ ; ⎢⎣ Ico ⎥⎦ ⎣⎢ Icc ⎥⎦

(1.9)

⎡Ψcd ⎤ ⎡Ψca ⎤ ⎢ Ψcq ⎥ =A ⎢ ⎥ C Ψcb ; ⎢ ⎥ ⎢ ⎥ ⎢⎣ Ψco ⎥⎦ ⎢⎣ Ψcc ⎥⎦

(1.10)

sadac Ac _ park-gorevis gardasaxvaTa matricaa, romelic Caiwereba Semdegi saxiT: ⎡ ⎢ cos θ 2⎢ Ac = ⎢ − sin θ 3⎢ 1 ⎢ ⎣ 2

cos(θ − 120 ) − sin(θ − 120 0 ) 1 2 0

⎤ cos(θ + 120 ) ⎥ ⎥ − sin(θ − 120 0 ) ⎥ ⎥ 1 ⎥ 2 ⎦ 0

(1.11)

axali saangariSo sidideebi iwodebian A, q, O mdgenelebad, xolo TviT gardasaxva warmoadgens sinqronuli manqanis gantolebis mimarTebas sakoordinato RerZebisadmi,

romlebic

uZravad

arian.

koordinatebidan fazur koordinatebze gadasvla xdeba Semdegi matricis daxmarebiT: cos θ − sin θ 1⎤ ⎡ Αc−1 = ⎢⎢ cos(θ − 1200 ) − sin(θ − 1200 ) 1⎥⎥ ⎢⎣ cos(θ + 1200 ) − sin(θ + 1200 ) 1⎥⎦

(1.12)

axal koordinatul RerZebSi sinqronuli manqanis gardamavali procesebis gantoleba matricul formaSi SeiZleba Caiweros Semdegi saxiT: UI=RIII+(PE+NI) Ψ I; Ψi =LIII;

(1.13)

3 dω = T T - (ψ cd i cq -ψ cq i cd ) 2 dt

sadac UI=[Uf, Ucd, O, Ucq, O, Uco]T II=[if, icd, ig, icq, ih, ico]T Ψ I=[ Ψ f, Ψ cd, Ψ g, Ψ cq, Ψ h, Ψ co]

RI=diag[rf, rc, rg, rc, rb, rc]T

LLI=

3 Mf 2

Mfg

3 Mg Lg 2

3 Mh 2

Lh Lco

-Ď&#x2030; NI= Ď&#x2030;

E _ erTeulovani diagonaluri matricaa. sidideebi Ld=Lcp+Mcp+ 3 Mo;

(1.14)

Lq=Lcp+Mcp_ 3 Mo; 2

warmoadgenen statoris fazis TviTinduqciis koeficientebs horizontaluri da vertikaluri denebisaTvis; Lco=Lcp_2Mcp

(1.15)

warmoadgens statoris fazis TviTinduqciis koeficients nulovani mimdevrobis denebisaTvis. (1.13)

gardamqmnel

gantolebas

uwodeben

park-

gorevis gantolebas, romelic Seicavs periodul koeficientebs imdenad, ramdenadac koordinatul RerZebSi induqciis koeficientebis mniSvnelobebi warmoadgenen mudmiv sidideebs. parkis da gorevis naSromebi erT-erTi pirveli sainJinro namuSevrebia, romlebic efeqturad gamoiyeneba periodulkoeficientebiani gan-

tolebaTa

sistemis

mudmivkoeficientebian

gantole-

baTa sistemad gardaqmnisTvis [8,9].

1.1.2. asinqronuli manqanis gantolebebi asinqronuli

manqanebis

gardamavali

procesebis

gantolebebi fazur koordinatebSi Seicavs periodul koeficientebs, romelTa raodenobac ganpirobebulia im faqtiT, rom statoris fazaTa konturebis urTierTinduqciis koeficientebi arian perioduli funqciebi, romlebic damokidebuli arian rotoris mdgomareobaze. Tu miviRebT, rom stators da rotors aqvT samfaza

simetriuli

gragnilebi,

SeiZleba

asinqro-

nuli manqanis gardamavali el.magnituri procesebis gantolebebi fazur koordinatebSi Caiweros Semdegi saxiT[1,5,7,8,9,10]: Uca=rcica+ dΨca ; dt

Ucb=rcicb+ dΨcb ; dt

Ucc=rcicc+ dΨcc ; dt

Upa=rpipa+ Upb=rpipb+ Upc=rpipc+

dΨ pa dt dΨ pb dt dΨ pc dt

; ; ;

(1.16)

sadac Uca, Ucb, Ucc, ica, icb, icc, Upa, Upb, Upc, ipa, ipb, ipc, _ Sesabamisad fazuri Zabvebi da denebia statoris da rotoris gragnilebisTvis; rc, rp _ statorisa da rotoris

fazaTa

gragnilebis

aqtiuri

winaRobebia;

Ψca , Ψcb , Ψcc , Ψ pa , Ψpb , Ψpc _ statorisa da rotoris fazebis konturebis nakadSerTvebia. asinqronuli manqanis rotoris maZraobis gantolebas aqvs Semdegi saxe:

d 2θ J M 2 =_Tc+Te, dt

(1.17)

sadac Tc _ amZravi meqanizmis winaRobis momentia; Te _ eleqtromeqanikuri momentia, romelic ganisazRvreba (1.6), (1.7) gantolebebiT. d, q, o fazuri sidideebidan moZrav kordinatul sistemaSi ω K Tavisufali siCqariT gadasvla xorcieldeba sinqronuli manqanebis gantolebebis analogiurad. asinqronuli

manqanis

gantoleba

matriculi

formiT Caiwereba Semdegnairad:

U II =RIIIII+(PE+MII) Ψ ; IJ

ΨII =LIJIIJ;

(1.18)

JM dω = 3 Lad(ipdicq_ipqicd)_Tc, dt

sadac: UII=[ Ucd, Ucq, Upd, Upq, Uco, Upo,]T III=[ icd, icq, ipd, ipq, ico, ipo,]T; 20

ΨII = [Ψcd , Ψcq , Ψ pd , Ψ pq , Ψco , Ψ po ,] ; T

RII=diag[ rc, rc, rp, rp, rc, rp]; Lc1

Lad Lc1

LLLI

Lad

Lp1 Lad

Lp1 Lco Lpo

− ωk

ωk ω − ωk

NNII ωk − ω

Lc1= Lc- Mc _

statoris fazebis induqciis koeficientia

pirdapiri mimdevrobis denebisaTvis; Lco=Lc+2Mc _ igive pirdapiri nulovani mimdevrobis denebisaTvis;

Lad= 3 M; 2

Lp1=Lp-Mp _ rotoris fazebis TviTinduqciis koeficienti pirdapiri mimdevrobis denebisaTvis; Lp0=Lp-2Mp _ igive nulovani mimdevrobis denebisaTvis. 21

(1.18)

gantoleba

Seicavs

periodul

koefi-

cientebs, amasTan perioduli koeficientebi (sinqronuli manqanisgan gansxvavebiT) gamoiricxebian asinqronuli manqanis gantolebebidan sakoordinato RerZebis brunvis siCqaris ω k nebismieri SerCevisas, rac ganpirobebulia asinqronuli manqanis simetriulobiT. Tu ganvixilavT

asinqronuli

manqanis

moZraobas,

roca

ω = cos t , maSin misi gardasaxuli gantolebebi Rebuloben wrfivi gantolebebis saxes mudmivi koeficientebiT.

1.1.3. eleqtruli qselis gantolebebi fazur koordinatebSi ra, La, ca wredis gantoleba, boloebSi

U1 ,

ZabvebiT,

SeiZleba

Caiweros

Semdegnairad:

U 2 a = U 1a − r∧ i∧ a − L∧

d i∧a d d 1 − M ∧ i ∧b − M ∧ i ∧c − ∫ i∧ a dt ; dt dt dt C ∧

U 2 b = U 1b − r∧ i ∧ b − M ∧

d i∧a d d 1 − L∧ i ∧ b − M ∧ i ∧ c − ∫ i∧ b dt ; dt dt dt C ∧

U 2 c = U 1c − r∧ i∧ c − M ∧

d i∧a d d 1 − M ∧ i ∧ b − L∧ i ∧ c − ∫ i∧c dt ; dt dt dt C ∧

sadac

(1.19)

r∧ , L∧ , M ∧ _ Sesabamisad fazis aqtiuri wi-

naRobaa, fazis konturis TviTinduqciis koeficienti

ori

fazis

konturis

urTierTinduqciis

koefi-

cientebia; C ∧ _ wredis gamWoli tevadobaa. (1.19) gantolebebi warmoadgenen mudmivkoeficientebian wrfiv diferencialur gantolebebs. maTi gardasaxviT mbrunav koordinatul sistemaSi (U 1 _RerZTan, mbrunavi

ω1 siCqariT;

U 2 _RerZTan

_ ω2

siCqariT;

i∧

RerZTan_ ωk siCqariT)miviRebT[6]: U Id cos(θ k − θ 1 ) + U 1q sin(θ k − θ 1 ) − U 2 d cos(θ k − θ 2 ) − U 2 q sin(θ k − θ 2 ) − rπ I πd − di dθ 1 − Lπ 1 πd + Lπ 1iπ q k = ∫ iπd dt + ∫ [− U 1d sin(θ k − θ 1 ) + U 1q cos(θ k − θ 1 ) + dt dt Cπ

+U2d sin(θk −θ2) −U2q cos(θk −θ2) −rπiπ q − Lπ1

diπq dt

− Lπ1iπ d

dθk ⎤ dθk dt; ⎥ dt ⎦ dt

−U1d sin(θk −θ1)+U1q cos( θk −θ1) +U2d sin(θk −θ2) −U2q cos(θk −θ2)−rπiπq − −Lπ1

diπq dt

−Lπ1iπd

dθk 1 = ∫iπqdt+∫ −U1d cos( θk −θ1)−U1q sin( θk −θ1)+U2d cos( θk −θ2)+ dt Cπ

[

+ U 2 q sin(θ k − θ 2 ) + rπ iπd + Lπ 1 U10 − U 20 − rπ iπ 0 − Lπ 0

sadac,

Lπ 1 = Lπ − M π ;

koeficientebia

diπd dθ ⎤ dθ − Lπ 1iπ q k ⎥ k dt; dt dt ⎦ dt

diπ 0 1 = iπ dt , dt Cπ ∫ 0

Lπ 0 = Lπ + 2M π

pirdapiri

(1.20)

_ wredis TviTinduqciis da

nulovani

mimdevrobis

denebisaTvis. (1.19) da (1.20) gantolebebi Sedgenilia eleqtruli qselis yvela parametrisaTvis da erTiandebian eleqtruli qselis kvanZebisaTvis kirxofis pirveli ka23

nonis gantolebebTan. am damokidebulebebs

j kvanZi-

saTvis SeiZleba hqondeT Semdegi saxe:

∑ i j

= 0

(1.21)

sadac j , j kvanZTan mierTebis Stoebis raodenobaa. (1.19) da (1.20) gantolebebis SerwymiT kirxovis II kanonTan eleqtruli qselis konturebisTvis miviRebT [11]:

∑ U l

sadac

(1.22)

= 0

_ aris elementebis raodenoba, romlebic Se-

m -ur

dian

konturSi.

1.1.4 mTliani energosistemis maTematikuri modeli imisaTvis, rom CavweroT energosistemis gardamavali procesis mTliani maTematikuri modelebis gantoleba, energosistemis calkeuli elementebisaTvis unda miviRoT koordinatTa sistema da denebis mimarTuleba. denebis dadebiT mimarTulebad miviRoT denis mimarTuleba qselidan manqanaSi, q iyos vertikaluri (ordinatTa) RerZi, d – horizontaluri (abscisaTa) [1,5]. mTliani

energosistemis

maTematikuri

modelis

gantolebebis optimalurad CawerisTvis sinqronuli manqanebis

toloba

unda

Caiweros

mbrunav

koordi-

natebSi, romlebic myarad arian dakavSirebuli erTma24

neTTan manqanis rotoriT. asinqronuli manqanebisa da statikuri elementebis gantolebebi Caiwerebian mbrunav

koordinatebSi,

romlebic

xistad

arian

dakav-

Sirebuli erT-erTi sinqronuli manqanis rotorTan an fazur koordinatebTan [1,11]. eleqtruli qselis gantoleba CavweroT kvanZuri Zabvebis meTodiT, roca ar arseboben denis nulovani mdgenelebi.

eleqtruli

qselis

asinqronuli

manqanebis gantolebebi SevufardoT RerZebs, romlebic xistad arian dakavSirebuli erT-erTi sinqronuli manqanis rotorTan. vTqvaT es aris manqana n kvanZSi. am dros manqanis horizontaluri _ namdvili RerZebisa da warmosaxviTi _ vertikaluri RerZebis Tanxvedrisas eleqtruli qselis mdgomareobis gantoleba j kvanZisTvis, (1.20) gantolebis gaTvaliswinebiT, kompleqsur formaSi Caiwereba Semdegnairad [5]:

U jY sadac kvanZSi

+ ∑ (U j − U q )Y

(1.23)

I jc , I ja _ denebia, romlebic ganpirobebulia j sinqronuli

Y jq _

Stos

arsebobiT;

asinqronuli gamtarianoba,

manqanebis romelic

akmayofilebs gamosaxulebas: Y

r + ( p + jw k ) L ∧ 1 +

(1.24) 1 ( p + jw k ) C ∧

Y jq axdens j kvanZSi statikuri datvirTvis modelirebas.

statikuri

datvirTva

warmodgenilia

rH , LH wredis saxiT, maSin

Y jj = Caketili

1 rH + ( p + jwk ) LH

sistemebis

misaRebad

(1.25) saWiroa

sinqro-

nuli manqanebis gantolebebsa da eleqtruli qselis mdgomareobis gantolebas daematos sxvadasxva koordinatTa

sistemis

kavSiris

gantolebebi,

romlebic

SeiZleba CavweroT Semdegi saxiT [5,11,12]:

U 'j = U cdj cos δ jn − U cqj sin δ jn U 'j' = U cdj sin δ jn + U cqj cos δ jn

(1.26)

I 'jc = icdj cos δ jn − icqj sin δ jn I 'jc' = icdj sin δ jn + icqj conδ jn talRuri procesebis gaTvaliswinebiT grZeli xazis gantoleba CavweroT α , β , o koordinatebSi da davumatoT fazuri koordinatebis kavSirebis gantolebebi α , β , o koordinatebSi. d,q,o

koordinatebSi

gardasaxvis

matricas

aqvs

Semdegi saxe:

A d α = A c S II− 1

⎡ cos θ = ⎢ − sin θ ⎢ 0 ⎣⎢ 26

sin θ cos θ 0

0⎤ 0⎥ ⎥ 1 ⎥⎦

(1.27)

1.1.5 energosistemis mTliani maTematikuri modelis Taviseburebebi ganvixiloT umartivesi energosistemebis gardamavali procesebis gantolebebi, romlebic Sedgeba ori sinqronuli manqanisgan da muSaoben statikur datvirTvaze, romelsac warmoadgens parametrebi rH , LH . simartivisaTvis gamoyenebulia araregularuli sinqronuli manqanebi

udemfero

konturebiT

simetriul

reJimSi.

gamoyenebulia mTliani maTematikuri modelis gantolebebi

mbrunav

koordinatTa

RerZebSi.

sinqronuli

manqanebis gantolebebi ukavSirdebian maT rotorebs (1.13) gantolebis safuZvelze da Caiwerebian Semdegnairad [1,5,11,13]:

U cdj = rcj i cdj + U cqj = rcj icqj +

U J

d (Lqj icqj ) + (Ldj icdj + M fj i fj ) ⋅ ω j ; dt

= r fj i fj +

dω

d (L dj icdj + M fj i fj ) − L qj icqj ω j ; dt

= TT j −

d ⎛ 3 M ⎜ dt ⎝ 2

3 ⎡M 2 ⎣

(1.28)

⎞ + i L i fj c d j fj fj ⎟ ; ⎠

i fj i c q j + ( L d j − L q j ) i c d j i c q j ⎤⎦ ,

sadac j=1,2 datvirTvis wredis gantolebebi SeZleba Caiweros Semdegi saxiT:

U cd 1 = rH i Hd + L H

U cq 1 = rH i Hq

di Hd − L H i Hq ω 1 dt

(1.29)

di Hq + LH + L H i Hd ω 1 dt

kirxofis I kanoni datvirTvis kvanZisTvis da I da II sinqronuli manqanebisTvis SeiZleba Caiweros Semdegi saxiT [11]:

icd 1 + icd 2 cos δ + icq 2 sin δ + iHd = 0; icq1 + icq 2 cos δ − icd 2 sin δ + iHq = 0;

(1.30)

U cd 1 = U cd 2 cos δ + U cq 2 sin δ ; U cq1 = −U cd 2 sin δ + U cq 2 cosδ . t

δ = ∫ ( ω 1 − ω 2 ) dt + δ 0

(1.31)

(1.28) ÷ (1.31) gantolebebi aRweren ganxiluli sistemis gardamaval procesebs. maTi erToblivi amoxsna saSualebas

iZleva

aRvweroT

eleqtromagnituri

eleqtromeqanikuri procesebi. (1.28) ÷ (1.31)-dan Cans, rom mbrunav koordnatebSi mTliani maTematikuri modelis Cawerisas sinqronuli manqanebis

gantolebebidan

gamoiricxeba

perioduli

koeficientebi, amasTan es koeficientebi (sin δ , cos δ ) gardaiqmnebian

manqanis

rotorebs

Soris

kuTxis

periodul funqciebad, romelTa cvalebadoba droSi aRiwereba (1.31) gantolebiT. 28

1.2 mravalmanqaniani sistemebis gamartivebuli variantebi

1.2.1 Cqari eleqtromagnituri procesebis kvlevis maTematikuri modeli es modeli aris mTliani modelis kerZo SemTxveva da miiReba misgan Cqari moZraobis gamoyofiT, sinqronuli da asinqronuli manqanebis rotorebis brunvis kuTxuri siCqareebis Semagrebis gziT sinqronuli manqanebis rotorebis brunvis kuTxuri siCqareebis erTnairi mniSvnelobebisas. i-uri sinqronuli manqanebisTvis Cqari eleqtromagnituri procesebis gantoleba (1.13)-is safuZvelze Caiwereba Semdegi saxiT:

U Ii = RIi I Ii + ( PE + N Iio )ΨIi ;

(1.33)

Ψ Ii = L Ii I Ii gantolebebs

wredisTvis

r∧ , L∧ , C ∧

parametrebiT

(1.20)-is safuZvelze eqneba saxe:

U 1d − U 2 d − r∧ i∧ d − L∧1

d i∧d 1 + L∧1i∧ q ω0 = ∫ i∧ d dt + dt C∧

di∧q ⎡ ⎤ U U r i L L i − − − − ω ∧ ∧ ∧ ∧ ∧ 1 q 2 q q 1 1 d o ⎥ωodt ; ∫ ⎢⎣ dt ⎦

di∧q

1 ∫ i∧q dt + dt C∧ dj∧d ⎡ ⎤ − − − − − ω ωodt ; U U r i L L i 1 d 2 d d 1 1 q o ∧ ∧ ∧ ∧ ∧ ∫ ⎢⎣ ⎥ dt ⎦

U1q − U 2 q − r∧ i∧q − L∧1

+ L∧1 j∧d w0 =

U 10 − U 20 − r∧ i∧ 0 − L∧ 0

di ∧ 0 1 = ∫ i ∧ 0 dt . dt C∧

(1.34)

mocemuli gantolebebi miRebulia energosistemis yvela elementisTvis da warmoadgenen Cqari eleqtromagnituri procesebis models, romelic arawrfivia Tavisi mudmivi koeficientebiT simetriuli reJimebisTvis [1,5,13,14].

1.2.2. Jdanovis modeli gasaSualebis pirveli meTodis gamoyeneba iwvevs Jdanovis maTematikuri modelis gamartivebas. energosistemebisTvis

modelis

gantolebebs,

romlebic Sedgebian n-sinqronuli da ℓ-asinqronuli manqanebisgan, kompleqsur formaSi eleqtromagnituri procesebis gauTvaliswineblad, eqnebaT Semdegi saxe: Ji

dωi = TTi (ωi ) − Tei (δ1n ,..., δ ( n−1) n , ω1 ,..., ωn , ωM 1 ,..., ωml ); dt

JMv

dωmv = −Tcv (ωmv ) +Tev (δ1n ,...,δ(n−1)n ,ω1,...,ωn ,ωm1,...,ωml ); dt

dδ in = ωi − ωn . dt 30

(1.35)

sadac i=1,2,...,n; ν = 1, 2,..., l;

δ 1n _

i-uri da n-uri sinqronuli

manqanebis rotorebs Soris urTierTZvris kuTxea;

ωi − i -

uri sinqronuli manqanis rotoris brunvis kuTxuri siCqarea;

ωmν −ν -uri

asinqronuli manqanis rotoris

brunvis kuTxuri siCqarea. (1.35)-Si Semavali funqciebis cxadi analitikuri gamosaxulebebi SeiZleba miRebul iqnes eleqtruli qselis mxolod ubralo wrfivi modelebis SemTxvevaSi. Jdanovis models orgeneratoriani sistemisTvis, romelic mTliani modelidan miiReba, aqvs Semdegi saxe: Eq1Eq 2 ⎤ 1 3 ⎡ Eq12 dω1 sin α11 + sin (δ12 − α12 ) ⎥ ; = TT 1 − J1 ⎢ ω1 2 ⎣ Z11 dt Z12 ⎦ Eq1Eq 2 ⎤ dω2 1 3 ⎡ Eq22 J2 sin α sin (δ 21 − α 21 ) ⎥ ; = TT 2 − + 22 ⎢ dt Z21 ω2 2 ⎣ Z22 ⎦

(1.36)

d δ12 = ω1 − ω2 . dt sadac Eq1 = ω1 M f 1i f 1 , Eq 2 = ω2 M f 2i f 2 - eleqtromamoZravebeli Zalebia; z11, z22, z12, z21,α 11 ,α 22 ,α 12 ,α 21 _ Sesabamisad sakuTrivi

urTierTwinaRobebis

argumentebisTvis

modulebi da damatebiTi kuTxeebia [1,5,15,16,17,18,19,20].

1.2.3 poziciuri modeli poziciuri modelis gantolebebis aracxadi saxiT Cawera statikuri maxasiaTeblebis asinqronuli datvirTvis gaTvaliswinebiT (1.35)-dan gamomdinare, xdeba Semdegnairad:

d 2δin 1 1 ⎡⎣TTn − Ten (δ1n ,..., δ( n−1) n )⎤⎦ = − − [ T T ( δ ,..., δ )] Ti ei 1n ( n−1) n 2 dt Ji Jn (i =1,2,...,n−1)

(1.37)

poziciuri modeli gamoiyeneba sinqronuli manqanebis fardobiTi moZraobis analizisTvis da misi gamoyeneba SeiZleba maSin, roca unda gamovikvlioT procesebi,

damokidebuli

sinqronuli

manqanebis

roto-

rebis brunvis kuTxur siCqareze. arseboben

principze

agebuli

sxva

gamarti-

vebuli maTematikuri modelebi. ganvixiloT maT Soris yvelaze xSirad gamoyenebuli n-kvanZSi ω0 sixSiriani usasrulo simZlavris saltis gamartivebuli maTematikuri modeli [5,19]. sinqronuli generatoris rotoris moZraoba (1.13)is gaTvaliswinebiT ase aRiwereba:

d 2δ rin 3 T ji = PTi − ⎡⎣ Eqi2 Yii sin α ii + EqiU iYEUi sin(δ EVi − α Evi ) ⎤⎦ 2 dt 2 sadac

(1.38)

δ rin − i -uri sinqronuli manqanis rotoris da

eqvivalentur

usasrulo

simZlavris 32

generatoris

saltes Soris kuTxea; Eqi _ i -uri sinqronuli manqanis fiqtiuri emZ; Ui _ i-uri sinqronuli manqanis eleqtrul qselTan

Sexebis

kvanZSi

Zabvis

modulia;

Υ ii , α ii , Υ EUi , α EUi _ sakuTari da saerTo gamtarobis argumentebis modulebi da damatebiTi kuTxeebia EQi da Ui kvanZebisTvis; PTI − i -uri sinqronuli manqanis pirveladi Zravis simZlavrea;

TJi = J iω0 .

i-uri sinqronuli manqanis aRgznebis gragnilSi eleqtromagnituri gardamavali procesebis gantoleba (1.13)-is safuZvelze gardaiqmneba Semdegnairad:

Td oi E 'qi − Χ 'd

sadac

E qei − emZ-is

= E qei − E qi

gardamavali

stacionaluri

proporciulia

T doi −

dE qi'

aRgznebis

(1.39)

winaRobis

mniSvnelobaa,

emZ-a;

romelic

gragnilis

Zabvisa;

aRgznebis gragnilis drois mudmivaa.

gamartivebuli

modelisTvis

i-uri

sinqronuli

manqanis rotoris moZraobis gantolebas aqvs saxe: n ⎤ d2δin 3⎡ 2 Tji 2 = PTi − ⎢EQiYii∋k sinαii∋k +∑EQi EQj sin(δij −αij∋k )⎥ dt 2⎣ j=1 ⎦

i-uri

sinqronuli

manqanisTvis

E 'qi

cvladebis kavSiris gantolebas aqvs saxe:

(1.40) da

EQi

⎛ 1 ⎞ n = EQi ⎜ − Yii∋k cosαii∋k ⎟ + ∑EQiYij∋k cos(δij −αij∋k ) ' ' ⎜ ⎟ j =1 xqi − xdi ⎝ xqi − xdi ⎠ j≠i Eqi'

(1.41)

ufro gamartivebuli modeli miiReba im SemTxvevaSi, roca aRgznebis regulirebis moqmedebiT SeiZleba uzrunvelyofil iqnes zogierTi Ei ∋ k emZ-is mudmivoba eqvivalenturi reaqtiuli winaRobiT xi, romelic damokidebulia aRgznebis regulatoris maxasiaTebelze. am modelis moZraobis gantoleba Caiwereba Semdegnairad: h d 2δ in 2 TJi = PTi − Ei Yii∋k sin α ii∋ k − ∑ Ei E jYij ∋k sin(δ ij − α ij ∋k ) (1.42) 2 j =1 dt

sadac E

3 E 2

i∋ k

; E

3 E 2

i∋ k

;

xi _ gaTvaliswinebulia sakuTriv da urTierTgamtarobaSi

(i=1,2,...,n-1).

modelis

Semdgom

gamartivebad

iTvleba konservatiul modelebze gadasvla, romelTaTvisac damatebiTi kuTxe [5,19,20].

α ij ∋ k

xdeba nulis toli

1.2.4. eleqtrul qselSi gasaSualebuli procesebis modeli eleqtrul qselSi mravalsixSiruli denebis da Zabvebis Secvlas erTsixSiruliT mivyavarT modelis gamartivebamde,

romelsac

SeiZleba

vuwodoT

eleq-

truli qselis gasaSualebuli procesebis maTematikuri modeli. aseTi tipis modelis realizaciisas unda ganvixiloT sinqronuli manqanebis muSaoba SekavSirebis kvanZze, rogorc miwodebuli Zabvis momWerebze da gavaerTianoT sxvadasxva sixSiris wredebi energetikuli balansis gantolebebiT [1,5,19,21,22]. gasaSualebis erT-erT saSualebas, romelic akmayofilebs mocemul pirobebs, warmoadgens el.simZlavreebis an momentis saSualo mniSvnelobis gansazRvra [1,5]: ωe =

∑ω P ∑ P i

∋∧i

anu

∋∧i

(1.43)

∑ ω J ∑ J i

(1.44)

CavweroT

gasaSualebis

energositemisTvis,

romelsac

maTematikuri aqvs

modelebi

n-sinqronuli,

ℓ-

asinqronuli manqana da m-eleqtruli qselis kvanZi. juri sinqronuli manqanis moZraobis gantoleba Caiwereba Semdegi saxiT: 35

U Ij = RIj I Ij + ( PE + N Ij )ΨIj ;

Ψ Ij = LIj I Ij ; Jj

(1.45)

3 dwj = TTj − ( Ψcdj icqj − Ψcqj icqj ) , dt 2

sadac j=1,2,...,n.

ν -uri asinqronuli manqanis moZraobis gantoleba Caiwereba Semdegi saxiT:

U IIν = RIIν I IIν + ( pE + N IIν )Ψ IIν ;

Ψ IIν = LIIν I IIν ; J Mν

(1.46)

d ων 3 = Ladν (i pdν icqν − i pqν icdν ) − Tcν , 2 dt

sadac ν = 1, 2,..., l. yoveli j-uri eleqtruli qselisaTvis (maT Soris sinqronuli

kvanZebisaTvis) mdgomareobis

asinqronuli eleqtruli

gantolebebi

manqanebis qselis

aqtiuri

SeerTebis

stacionaluri da

reaqtiuli

simZlavreebis balansis gantolebebis saxiT Caiwereba Semdegnairad [1,5,22,23]: U 2j Υ jj sinα jjγ y + ∑U jUqYjq sin (δ jn − δqn −α jq ) + PjH (U j ,ωc ) + Pjc + Pja = 0; q

U Υ jj cosα jj − ∑U jUqYjq cos (δ jn − δqn −α jq ) + Qγ H (Uγ ,ωc ) + Qjc + Qja = 0.

(1.47)

2 j

sadac j=1, 2,...., m. gantolebis ricxvis ucnobTa ricxvTan SesabamisobaSi mosayvanad aucilebelia sinqronuli, asinqronuli

manqanebis

eleqtruli 36

qselis

cvladebis

kavSiris damatebiTi gantolebaTa sistemis Sedgena. sinqronuli

manqanisaTvis,

romelic

SeerTebulia

kvanZTan, es gantolebebi Caiwereba Semdegi saxiT [5]:

Pjc = U cdj icdj + U cqj icqj ;

Q jc = U cqj icdj − U cdj icqj ;

(1.48)

2 2 U 2j = U cdj + U cqj ;

δ jn = δ rjn + arctg

dδ rjn dt sadac

δ rjn _kuTxea

U cqj U cdj

;

= ω j − ωn .

sinqronuli

manqanis

j rotoris

horizontalur RerZsa da generatoris horizontalur RerZs Soris. asinqronuli manqanisTvis, romelic mierTebulia

kvanZze,

kavSiris

gantoleba

SeiZleba

Caiweros

Semdegnairad:

Pνa = U cdν icdν + U cdν icqν ; Qν a = U cdν icdν − U cdν icqν ;

U v2 = U c2 dν + U c2 dν ; δν n = arctg ormanqanur

U cqv U cdv

energosistemebSi

(1.49)

sinqronuli

manqa-

nebisTvis mocemul gantolebebs aqvT Semdegi saxe: 37

U cdj = rcj icdj +

U cqj = rcj icqj +

d ( Ldj jcdj + M fji fj ) − Lqj jcqjω j ; dt

d ( Ldjicdj ) − ( Ldj jcdj + M fj j fj ) ω j ; dt

U fj = rfj i fj +

(150)

d ⎛3 ⎞ ⎜ M fj icdj + L fj i fj ⎟ ; dt ⎝ 2 ⎠

dw j

3 = TTj − ⎡⎣ M fj i fj icqj + ( Ldj − Lqj ) icdj icqj ⎤⎦ . dt 2

sadac j=1,2. energetikuli

balansis

gantoleba

datvirTvis

kvanZSi parametrebiT rH, LH da cvladebis kavSiris gantoleba Caiwereba Semdegi saxiT [1,5,24,25,26,27]:

U H2 rH ( ucdj icdj + ucqj icqj ) + r 2 + L2 ω 2 = 0; ∑ j =1 H H c 2

U H2 wc2 LH ( ucqj icdj + ucdj icqj ) + r 2 + L2 ω 2 = 0; ∑ j =1 H H c 2

U H2 = ucd2 1 + ucq2 1;

(1.51)

U H2 = ucd2 2 + ucq2 2 ; δ r12 = arctg

ucq 2 ucd 2

− arctg

dδ r12 = ω1 − ω21 ; dt ωc =

ω1 J1 + ω2 J 2

J1 + J 2

ucq1 ucd 1

;

1.3 energosistemis statikuri mdgradoba amJamad energosistemasTan dakavSirebul sakiTxTa gamartivebam farTo gavrceleba hpova. wrfivi modelebis SemTxvevaSi es sakiTxi SeiZleba gadawyvetili iqnas maxasiaTebeli gantolebis analizis safuZvelze sxvadasxva meTodebiT. am

analizis

mTavari

mizania,

gagebuli

iqnas

Tavisebureba, romelic Tan sdevs energosistemis mTel models, imisaTvis, rom ufro safuZvlianad mivudgeT mis gamartivebas saerTo SemTxvevaSi. naSromSi ganixileba sinqronuli manqana, romelic muSaobs ucvleli sinusoiduri Zabvis wyaroze. ikvleva SemTxveva, romelic miaxloebiT asaxavs erTi sinqronuli manqanis muSaobas Zalze mZlavr wredSi. energosistemis gantoleba CavweroT koordinatTa sistemaSi, romelic emTxveva sinqronuli manqanis rotoris RerZebs. mosaxerxebelia maTi warmodgena Semdegi saxiT (fardobiT erTeulebSi) [1,5,26,27]:

dΨd + Rid − (1 + S )Ψ q = u sin δ ; dt

dΨq dt

dΨ p dt

+ Riq + (1 + S ) Ψ q = u cos δ ;

+ Rpip = u p ; 39

(1.52)

Tj sadac

dS + Ψqiq − Ψd iq − M T (ω ) = 0 , dt

Ψ = {Ψd , Ψq , Ψp };

i = {id , iq , i p };

Ψ p , i p - sinqronuli

manqanis rotoris xviaTa nakadSerTvebisa da denebis veqtorebia;

U p = {U f ,0...,0};

ω0

Zabvis

statoris

xviaze dadebuli sixSirea; δ - kuTxea rotoris ganiv RerZsa da dadebuli Zabvis gamosaxviT veqtors Soris, romelic aiTvleba Zabvis veqtoridan saaTis isris sawinaaRmdego mimarTulebiT. Tj = j

2 ω баз

М баз

meqanikuri momen-

tia; simartivisTvis ugulebulyofelia amZravi meqanizmebis gardamavali procesebi. maxasiaTebel gantolebas aqvs me-5 rigi. a. gorevis naSromSi

mcire

parametrebis

meTodis

daxmarebiT

miRweul iqna me-5 rigis sawyisi gantoleba daSliliyo dabali rigis sam gantolebad:

p 2 + ( ρ a + ρ d ' ) p + (1 + ρ q ρ a ' ) = 0 ; a 4 p + a5 ( 3)

(5)

= 0;

(1.53)

2 (3) (5) a2(1) p 3 + a3(2) э p + q4 p + q 4 = 0 ;

p 5 + a1 p 4 + a 2 p 3 + a3 p 2 + a 4 p + a5 = 0 .

1.4 ormanqanuri energosistemis dinamika energosistemebis maTematikur modelebs Soris poziciuri modeli rCeba energosistemis erT-erT umTavres idializaciad, romelsac klasikuri ewodeba. aseTi sistemis dinamikis sakiTxebs, upirveles yovlisa mdgradobas, eZRvneba uamravi literatura, sadac ganixileba gamokvlevis sxvadasxva aspeqtebi cifruli, analitikuri da xarisxobrivi meTodebis gamoyenebiT. mdgradobis gamokvlevisa ganxilul meTodebs Soris aris liapunovis funqciis meTodi. miuxedavad imisa, rom liapunovis funqciis meTodi ukve didi xania gamoiyeneba energetikaSi, vxvdebiT srulebiT umniSvnelo raodenobis amocanebs, romlebic amoxsnili arian am meTodis daxmarebiT. ganvixiloT ormanqanuri energosistema poziciur idealizaciaSi amoixsneba amocana:

misi

wrfivi

dempfirebiT,

dinamikis

wonasworobis

ramodenime

mdgomareobis

romlisTvisac mniSvnelovani separatrisuli

zedapirebis umravlesobis aRwera, romlebic zRudaven energosistemis wonasworobis mdgomareobas asimptoturi mdgradobiT da wonasworobis mdgomareobis umravlesobis globaluri mizidulobis pirobebis miReba. CamoTvlili amocanebi amoixsneba liapunovis I da II meTodebiT da Sedarebis organzomilebiani sistemis meTodiT.

wrfivad

dempfirebuli 41

organzomilebiani

energosistemis poziciuri modelis fardobiTi moZraoba gamoisaxeba Semdegi saxiT [1,5,26,27]: .

δ 12 = ω12 ; −1 −1 −1 −1 ω12 =−λω 1 12 +(λ2 −λ1)ω2 +μ1 P 12 cos(δ12 −θ12)−μ2 P 12 cos(δ12 −θ12); m1 −μ1 A m2 +μ2 A .

ω 2 = −λ2ω2 + μ 2−1 Pm 2 − μ 2−1 A12 cos(δ12 − θ12 ), sadac

(1.54)

δ 12 , ω12 , ω 2 , _ sistemis fazuri cvladebia

( δ 12 = δ 1 − δ 2 ,

ω12 = ω1 − ω 2 ,

δi ,

ωi = δ i ,

i = 1,2,

Sesabamisad rotoris absoluturi kuTxe da kuTxuri siCqarea

θ12 ,

i -uri

manqanisTvis); λi , μ i , Pmi , i = 1,2,

A12 ,

- mudmivebia, μ i > 0, λi > 0, i = 1,2, A12 > 0. . (1.54)-Si

davuSvaT

δ 12 = 0,

ω12 = 0,

ω 2 = 0,

maSin

vRebulobT wonasworobis gantolebas:

Diω 2 + ϕ (δ 12 ) = Pmi ( i = 1,2, ) romlebSic Di = λi μ i

(1.55)

( i = 1,2, )

ϕ (δ12 ) = A12 cos(δ12 − θ12 ) . Tu

(1.54)

–

sistemis

parametrebi

(1.56) akmayofileben

pirobas:

λ1μ1 ≠ λ2 μ2 ; (λ1 μ1 Pmi − λ 2 μ 2 Pm1) ( A12 (λ1 μ1 − λ 2 μ 2 ) ≤ 1

(1.57)

maSin misi wonasworobis mdgomareoba ganisazRvreba TanafardobebiT:

ω120 = 0;

ω20 = ( Pm − Pm ) (λ1μ1 − λ2 μ2 ); 1

cos(δ 20 − θ12 ) = (λ1μ1 Pm2 − λ2 μ2 Pm1 ) ( A12 (λ1μ1 − λ2 μ2 ))

(1.58)

(1.58)-dan gamomdinareobs, rom (1.57) pirobisTvis (1.54)

sistemas

aqvs

wonasworobis

izolirebuli

mdgomareobis Tvladi simravle. Tu misgan gamovyobT wonasworobis

nebismier

mdgomareobas,

aRniSnuli

( ω12 , ω 2 , δ 12 , )-iT da Tu SevitanT gadaxrebs: z1 = ω12 − ω12 , *

z 2 = ω 2 − ω 2* ,

zσ = δ 12 − δ 12* , vRebulobT (1.54) an (1.55)-is

gaTvaliswinebiT Semdeg gantolebebs: .

σ = z1 ; .

z1 = −λ1 z1 + (λ2 − λ1 ) z2 + b1Ψ (σ );

(1.59)

z 2 = −λ2 z2 + b2 Ψ (σ ).

sadac b1 = A12 ( μ1 − μ 2 ), −1

−1

b2 = A12 μ 2−1 > 0 .

Ψ (σ ) = Sin(σ + α ) − Sin(α );

α = δ12* − θ12 − π 2.

(1.60)

CavTvaloT, rom: λ2 ≥ 1, α ∈ (0, π 2)

(1.61)

amasTan, pirveli gantoleba yovelTvis Sesrulebadia. wonasworobis mdgomareobebs z1 ,

z 2 gadaxrebi-

sTvis eqneba Semdegi saxe:

z i = 0, i = 1,2, .. Ψ (σ ) = Sin(σ + α ) − Sinα = 0 .

(1.62)

am Tanafardobas mivyavarT oradTvlad Tanamimdevrobisken

wonasworobis

obisa: 43

izolirebuli

mdgomare-

θ k ' = zi ' = 0 ; i = 1,2 ; σ k '= 2kπ ; k

= 0,±1,±2...

(1.63)

θ k " = zi " = 0 , i = 1,2 σ k " = kπ − 2α , k = ±1,±3...

(1.64)

(1.59), (1.60) sistemis wrfiv miaxloebas θ k ' wertilis axlos aqvs Semdegi saxe: .

σ = z1 ;

z 1 = −λ1 z1 + (λ2 − λ1 ) z2 + b1 cos α (σ − σ k ');

z 2 = −λ2 z2 + b2 cos α (σ − σ k ').

(1.65)

xolo misi maxasiaTebeli gantolebebi miiReben Semdeg saxes:

Δ' (μ) = μ3 + μ2 (λ1 + λ2 ) + μ(λ2λ1 −b1 cosα) − cosα[b1λ2 + b2 (λ2 − λ1 )] = 0

(1.66)

amasTan, sistemis wrfivi nawilis maxasiaTebeli gantoleba, wonasworobis mdgomareobis farglebSi

θ k''

Caiwereba ase: Δ"(μ ) − μ 3 + μ 2 (λ1 + λ2 ) + μ (λ1λ2 + b1 cosα ) + cosα[b1λ2 + b2 (λ2 − λ1 )] = 0

vTqvaT yofilebs

(1.59)

(1.60)

pirobas:

sistemebis

α ∈ (0, π 2) ,

b1λ2 + b2 (λ2 − λ1 ) < 0 , maSin

mas

izolirebuli mdgomareobebi zRvreba

(1.63)

(1.64)

parametrebi

λ2 > λ1 > 0,

gaaCnia

(1.67) akma-

b2 > 0 ,

wonasworobis

θ k' , θ k'' , romlebic ganisa-

formulebiT,

amasTanave

θ k'

wonasworobis mdgomareobebi asimptoturad mdgradebia ' da es faqti ganisazRvreba imiT, rom θ k wertilebSi

(1.59) da (1.60) sistemebis wrfivi nawilis maxasiaTebel

gantolebebs aqvs fesvebi kompleqsuri sibrtyis fesvebis marcxena Ria naxevarsibrtyeSi [26,27]. 1.5 liapunovis veqtoruli funqciis meTodi energosistemis mdgradobisTvis arawrfivi da arastandartuli elementebis mqone maRali Tanrigis rTul sistemebSi liapunovis erTiani funqciis ageba mdgomareobis yvela cvladisTvis dakavSirebulia

praqtikulad

gadaulaxav

siZneleeb-

Tan. aseTi amocanebis amoxsnis erT-erTi gzaa liapunovis veqtoruli funqciis Simotana. liapunovis veqtorfunqcia

warmoadgens

liapunovis

ufro

martivi

funqciis Î¸i ( xi ) nakrebs, romelTaganac TiToeuli Sedgenili iqneba sistemis (qvesistemis) nawilisaTvis da damokidebulia cvladis umcires ricxvze. amave dros, mTeli

sistemis

mdgradobis

bunebrivia, rom qvesistemebze da

uzrunvelsayofad, Î¸i

funqciebze emateba

damatebiTi moTxovnebi. amJamad ufro metad damuSavebulia is meTodebi, romlebSic es damatebiTi moTxovnebi dayvanilia yoveli S i qvesistemis eqsponencialuri mdgradobis moTxovnamde [28]. energosistemebis

dinamikuri

mdgradobis

gamoT-

vlebis erT-erTi etapi dakavSirebulia gamosakvlevi sistemis fazuri cvladebis avariis Semdgomi damya45

rebuli reJimisgan dasaSvebi gadaxris agebis aucileblobasTan. rig SemTxvevaSi es amocana daiyvaneba gamosakvlevi sistemis modelis avariisSimdgomi mdgomareobis miaxloebiT Sefasebamde. SemoTavazeulia lvf-is (liapunovis veqtoruli funqcia) agebis ori saSualeba energosistemebisTvis: 1) pirveli saSualeba ganekuTvneba lvf-is agebas luries funqciis saxiT (kvadratul funqcias plius arawrfivobis

integrali).

amasTan

sistemis

sawyisi

modeli iSleba rig qvesistemebad, romlebic moicaven erTi

igive

nawils

(mravalmanqanuri

sistemis

gantoleba). 2)

meore

sistemis

modeli

fazuri

ganekuTvneba

koordinatebis

lvf-is wrfivi

agebas formis

modulebis saxis komponentebiT da gankuTvnilia iseTi sistemebisTvis,

romlebic

Seicaven

erTmaneTTan

sustad dakavSirebul qvesistemebs. lvf-is

mixedviT

diferencialuri

gantolebebis

CawerisaTvis gaTvaliswinebulia energosistemaSi arsebuli siCqaris regulatoris muSaoba da mravalmanqanuri sistemebis daxasiaTebisTvis Semoitaneba Semdegi daSvebebi: 1)

sinqronuli manqana icvleba mudmivi emZ-iT, ro-

melsac romelic

aqvs

gardamavali

damokidebulia

reaqtiuli manqanis

muSaobis reJimze; 46

winaRoba

tipze

da misi

pirveladi Zravebis momentebi iTvlebian mud-

mivad; 3)

datvirTvaSi aris mudmivi winaRobebi;

dempfireba gaiTvaliswineba mudmivi koefici-

entiT, romelic proporciulia srialisa; 5)

sistemis sqemis sruli winaRobebi da pasiuri

elementebis gamtarobebi iTvleba wrfivad da ar arian damokidebuli elementebis muSaobis reJimze. Semdegi daSvebebis safuZvelze arawrfivi mravalmanqanuri energosistemis moZraobis gantoleba SeiZleba Caiweros Semdegi saxiT [28,29]:

T jiδ i '+ Diδ i '+ Pei − Pmi = 0 , i = 1, m, sadac

dempfirebis

koeficientia,

(1.68)

Pmi -

manqanis

RerZze meqanikuri simZlavrea, T ji - inerciuli mudmiva,

δ i ' -i-uri manqanis rotoris mobrunebis absoluturi kuTxe;

Pei - aqtiuri simZlavreebia, romlebic ganisa-

zRvrebian Semdegi Tanafardobidan: m

Pei = E Y sin θ ii + ∑ Ei E jYij sin(δ i '− δ j '+ θ ij ) , 2 i ii

sadac

i =1 j ≠1

(1.69)

Ei - aris i-uri siqronuli manqanis emZ; Yij m

manqanis TviTgamtarobebi Yii = Yi + ∑ Yij ; Yij - urTierTj =1 j ≠1

gamtarobebi;

θ ii ,

θ ij - kuTxeebi, romlebic avseben

-iT

TviT da urTierTgamtarobebis kuTxeebs. vTqvaT δ i

- aris (1.68) gantolebis mdgomareobis

cnobili koordinatebi (avariisSemdgomi damyarebuli reJimi).

vuSvebT,

δ i ' = δ i + δ i0 , δ i − δ i ' = yi , sadac

rom

δ i − i − uri manqanis rotoris kuTxis gadaxra damyarebuli mniSvnelobidan

δ i0 da vumatebT ra siCqaris

regulatoris gantolebas, miviRebT arCeuli energosistemis modelis aRgznebuli miZraobis gantolebas:

δ i = yi ; y i = d i z i − d i yi − Ψi ;

(1.70)

z i = −li yi − g i z i − α i yi , sadac ai , d i , li ,

i = 1, m.

gi ≠ 0, α − const , y i

i-uri manqanis

rotoris brunvis absoluturi kuTxuri siCqarea, zi regulatoris

koordinata;

arawrfivi

funqcia

Ψi

gamoisaxeba gantolebiT:

ψ i = ∑ aik ⎡⎣sin(δ i − δ k + δ i0 − δ k0 − θ ik ) − sin(δ i0 − δ k0 − θ ik ) ⎤⎦ k =1 k ≠i

(1.71)

sadac θik = const . advili

SesamCnevia,

rom

(1.70)

gantolebas

aqvs

wonasworobis bevri mdgomareoba, rac gamowveulia am sistemis marjvena mxaris nawilis 48

damokidebulebiT

kuTxeebis

δi −δk .

sxvaobaze

amitom

dinamikuri

mdgradobis gamokvlevisas ganixileba e.w. manqanebis urTierTmoZraobis

gantoleba.

mocemul

SemTxvevaSi

ganixileba (1.70) gantolebidan miRebuli Semdegi gantolebaTa sistema:

u i = y i − y m ; y i = ai z i − d i y i − Ψi ' ; y m = a m z m − d m y m + Ψm ;

(1.72)

z i = −(α i ai + g i ) z i − (li − α i d i ) yi + α i Ψi ' ; z m = −(α m a m + g m ) z m − (l m − α m d m ) y m + α m Ψm ' . sadac U m = δ i − δ m , U i = δ i − δ m , 0

m −1

Ψ i ' = ∑ aik [sin(U i −U k + U i0 + U k0 − θik ) − sin(U i0 − U k0 − θik )] + k =1 k ≠i

(1.73)

+ aim [sin(U i + U − θim ) − sin(U − θim )]; 0 i

0 i

m −1

Ψ m ' = ∑ amk [sin(U k + U k0 + θ mk ) − sin(U k0 + θ mk )] ( i = 1, m − 1 ).. k =1

2. bifurkacia energosistemebSi 2.1. gaormagebis cikli da bifurkacia energosistemebSi rogorc aRvniSneT bifurkacia – es aris gadasvlebi, romlebic SeiZleba moxdes sxvadasxva saxeebs Soris dinamikuri qcevidan parametrebis cvlilebisas [34,35]. bifurkaciis

movleba

zogadi

cnebaa,

romelsac

Seesabameba SesaZlo topologiuri cvlilebebis qvesimravle

axasiaTebs

SemTxvevaSi gansaxilvel

sistemis X

amonaxsns.

zogad

amonaxsnTa sivrce SeiZ-

leba daxasiaTdes garkveuli topologiiT. kerZod, romeliRac metrikuli sivrce SeiZleba SevusabamoT n ganzomilebis Ε evklidur sivrces. Teoriul aspeqtSi n

SeiZleba ganvixiloT Ε -ze ufro zogadi sivrce [30]. n

bifurkaciul TamaSoben

analizSi

gansakuTrebuli

gadamwyvet amonaxsnebi,

rols romlebic

SeiZleba davajgufoT Semdegi saxiT: 1. x0 =|| X | f : x0 → x0 || - stacionaluri amonaxsnebi. 2. X =|| X | f : x(0) → x(0 + T ) || - perioduli amonaxsnebi. 3.

X a =|| X | f : X a → X || -

pirobiT-perioduli

amonax-

snebi. dinamikuri

sistemebisaTvis,

romelic

Semdegi diferencialuri gantolebiT: 50

aRiwereba

dx = f ( x, p ) , dt

(2.1)

amonaxsnTa simravlis pirveli SemTxveva Seesabameba gantolebaTa

sistemis

dx =0 dt

amonaxsns,

romelic

warmoadgens f '( x, p ) = 0 gantolebaTa simrales. meore

SemTxvevaSi

amonaxsni

warmoadgens

martiv, an sakmaod rTul zRvrul ciklebs. zemoT

CamoTvlil

amonaxsnTan

SedarebiT

ufro rTuli xasiaTi gaaCnia an pirobiT-periodul amonaxsnebs an atraqtorebs, romelTa sazRvrebs gareT traeqtoriebi ar gadian. e. i. X a ⊂ X . ukve naxsenebi gansakuTrebuli amonaxsnebi (1,2,3) xasiaTdebian im TvisebebiT, rom maTken miiswrafian yvela regularuli amonaxsnebi, roca

t → +∞

t → −∞ .

mocemuli amonaxsnebi warmoadgenen struqturul mdgradobas, Tu mezobeli regularuli amonaxsnebi parametrebis

cvlilebisas

iqneba

erTmaneTTan

sakma-

risad axlos [31,32,33,34]. ganvixiloT diskretuli dinamikuri sistema, romelic mocemulia QSemdegi gantolebiT

X n +1 = VX n (1 − X n )

(2.2)

V koeficienti marTvis parametria. Tu intervali 0-dan 4-mdea, maSin nebismieri mniSvneloba

0 ≤ X ≤1

aisaxeba amave intervalis romelime wertilSi. V -s 51

sawyisi mniSvnelobis micemis Semdeg advilad ganisazRvrebian X 1 , X 2 ... cnobili formulebiT [34,35].

V -ze { X n }-is

damokidebulebiT qcevis

cvlilebas;

bijebis

gazrda

iwvevs

V <3, maSin { X n }

roca

mimdevroba dadis uZrav wertilze, roca 3<V1 <V2 , maSin { X n } mimdevroba xasiaTdeba T = 2 periodiani wanacvlebiT. atraqtorebi xdebian ucnaurni, roca

V -s

mniSvneloba uaxlovdeba 3,5699471860-s. aq iwyeba qaosi. Tavidan mwyobrad ganSladi xazebi uecrad iqcevian qaosur zigzagebad (nax. 1). amasTan ucnauria isic, rom roca V -s mniSvneloba uaxlovdeba 4-s, qaosi wydeba. aseTi wesiT miRebuli gamosaxuleba aris bifurkaciuli diagrama. Tu mas SevxedavT gverdidan, gamoCndeba

X qaosuri speqtri, romlis dekoratiuli Taviseburebanic warmoadgens gamocanas, romlis amoxsna SeiZleba mxolod Teoriuli saSualebiT. hopfis

Teoriis

safuZvelze

Semogvaqvs

hopfis

bifurkaciisa da zRvruli ciklebis cneba. zRvruli

ciklebis

koncefcia

diferencialuri

gamtolebebis kvlevidan Camoayaliba a. puankarem. aseT sistemebSi SeiZleba miviRoT rxevebi, romlebic ubrundebian sawyis mdgomareobas mcire SeSfoTebis Semdeg rxevebis nebismieri fazis dros. puankarem aseT rxevebs mdgradi zRvruli ciklebi uwoda. sistemebs ki puankares oscilatorebi ewodebaT. 52

nax. 1 bifurkaciuli diagrama

Tu ciklis

CavTvliT, tipis

rom

rxevebs,

sistema maSin

iZleva

SeiZleba

zRvruli gavakeToT

winaswari daskvnebi aseTi sistemebis mdgradobis Sesaxeb, miuxedavad imisa, viciT Tu ara sistemis aRmweri gantolebebi. imisaTvis, rom vimsjeloT sistemis mdgradobis an aramdgradobis Sesaxeb zRvruli ciklis mdgradobis mixedviT, unda davadginoT erTgeneratorian, orgeneratorian da samgeneratorian sistemebSi zRvruli ciklebis

arsebobis

faqti.

mcire

SeSfoTebisas

zRvruli cikli ubrundeba sawyis mdgomareobas, maSin sistema mdgradia, Tu ar ubrundeba â&#x20AC;&#x201C; aramdgradia. Tu dinamikuri sistema damokidebulia parametrze, es parametri SeiZleba iyos temperatura, wneva, Zabva da sxva fizikuri sidideebi, maSin maTi cvlilebisas sistemis

dinamika

icvleba.

dinamikuri

sistemebis

parametrebis cvlilebisas SeiZleba Seicvalos mdgradobis wertilebis ricxvi da maTi mdgradoba. aseTi cvlilebebi

arawrfiv sistemebSi, romlebic damokide-

bulia sistemis parametris cvlilebaze, warmoadgens bifurkaciis Teoriis sagans [34].

2.2 hopfis bifurkacia energosistemaSi arawrfiv dinamikaSi erTi parametris variaciaze damokidebuli aramdgradoba warmoadgens mdgrad bifurkacias, vibraciuli aramdgradoba ki aris hopfis bifurkacia. ganvixiloT organzomilebiani sistema, roca sistemis dinamika aRiwereba Semdegi saxis gantolebebiT.

X 1 = f 1 ( X 1 , X 2 ,η ) ,

X 1 (0) = X 10 ,

X 2 = f 2 ( X 1 , X 2 ,η ) ,

X 2 (0) = X 20 .

(2.3)

η parametris cvlilebisas icvleba sistemis xarisxi. am mniSvnelobas ewodeba bifurkaciuli mniSvneloba. am wertilebis maxloblobaSi xdeba gadasvlebi stacionalur mdgomareobebs Soris, stacionalur mdgomareobebsa da zRvrul ciklebs Soris, an zRvrul ciklebsa da qaoss Soris. bifurkaciis zRvruli

magaliTia

ciklebis

gaCena.

fizikur am

dros

sistemebSi

S 1 , S 2 = ± iω + y

fesvibi ar aris nulis toli da kveTs warmosaxviT RerZs ise, rom y ≥ 0 . am dros Tavidan xdeba mdgradi fokusis bifurkacia zRvrul ciklSi. koeficientis cvlilebam

SeiZleba

gamoiwvios

aramdgradoba (nax.2).

zRvruli

ciklis

nax.2 zRvruli ciklis aramdgradoba

nax.3 bifurkacia spiralidan zRvrul ciklze

Tu erTi namdvili maCvenebeli y xdeba dadebiTi, maSin Zveli zRvruli cikli SeiZleba daiSalos ramodenime zRvrul ciklad. xolo imisaTvis, rom igi Semdgom daubrundes sawyis mdgomareobas, saWiroa 2jer meti dro, anu xdeba periodis gaormageba, e.i. subharmonikis generacia.

nax.3-ze

mocemulia

bifurkacia

spiralidan

zRvrul ciklebze. dasmuli amocana gadavwyvitoT sistemis aRmweri gantolebebis amoxsniT erTi parametris cvlilebis miRebuli

dros. Sedegebi

iZleva

ganzogaddes

saSualebas, da

rom

xelmisawvdomi

gaxdes sxvadasxva msgavsi sistemebis kvlevisaTvis [34]. hopfis bifurkaciuli Teorema organzomilebiani sistemisaTvis ( R ) Camoyalibdeba Semdegnairad: 2

vTqvaT, funqcia cirebadi

f1 da f 2 4-jer maincaa diferen-

TiToeuli

argumentis

mixedviT

f 1 (0.0.4) = f 2 (0.0.4) = 0 yvela namdvili η -Tvis. CavTvaloT, rom 1)

d [Re λ (η )]<0 > 0 , maSin: dη

arsebobs 2-jer diferencirebadi η , ( −ε , ε ) → R . is,

rom sawyisi wertili ( X ,0,η ( X 1 )) Zevs 2π X (η ) peri0

odian Caketil traeqtoriaze, romlis radiusic izrdeba ise, rogorc 2)

η , roca X 10 = 0 , η (0) = 0 .

arsebobs (0,0,0) wertilis V are R -Si, ise, rom 3

nebismieri Caketili traeqtoria U areSi aris zemoT miTiTebuli saxis erT-erTi Caketili traeqtoria. 3)

Tu `0~ wertili atraqtoria, roca η = 0 . maSin

η ( X 10 ) > 0 yvela X 10 ≠ 0 -Tvis da Caketili traeqtoriebi miizidebian. R sistemaSi Caketil traeqtoriad iTvle2

ba nebismieri X , akmayofilebs X (t ) = X (t + T ) = X pi*

robas nebismieri T > 0 droisaTvis. wonasworobis wertilebi Caketili traeqtoriebia. hopfis TeoriiT η -s 0-dan daSorebisas traeqtoriebma SeiZleba Seicvalon qcevebi,

rac

damokidebulia

iakobis

matricis

sakuTari

koordinatTa mniSvnelobis

saTaveSi namdvili

nawilis sidideze. pirveli piroba niSnavs koordinatTa saTaveSi bifurkacias gansazRvrul radiusian Caketil traeqtoriaSi

(~ η ) . rodesac η = 0 , maSin wertili iqneba

mimzidveli. xolo sawyisi traeqtoriebi, warmoSobili fiqsirebuli wertilebidan, Tavad miizidaven.

2.3 hopfis bifurkacia erTgeneratoriani sistemisTvis erTgeneratoriani sistema warmovidginoT Semdegi saxiT:

dS = K (sin δ * − sin(δ * + Δδ * )) , dt

(2.4)

dΔS = S (t ) . dt wonasworoba ganisazRvreba Semdegnairad:

S (t ) = 0 ; K (sin δ * − sin(δ * + Δδ * )) = 0 ,

(2.5)

(0,0 + 2πk ) wertili wonasworobis wertilia. gamovikvlioT (0,0) wertili. gardaqmnebis Sedegad miviRebT [26,34]:

dS ΔS =S = K cos δ * Δδ * , dt dt

(2.6)

nax. 4 gadagvarebuli kvanZi anu aramdgradi wonasworoba

nax. 5 mdgradi kvanZi

nax. 6 aramdgradi mdgomareoba

nax. 7 neitralurad mdgradi mdgomareoba

anu matriculi saxiT ( S , Δδ ) = t

d ⎡ S ⎤ ⎡0 − K cos δ * ⎤ ⎡ S ⎤ = ⎥ ⎢Δδ ⎥ . 0 dt ⎢⎣ Δδ ⎥⎦ ⎣⎢1 ⎦⎣ ⎦ λ -s

(2.7)

mimarT maxasiaTebel gantolebas aqvs saxe:

det

−λ 1

− K cos δ * =0 −λ

(2.8)

anu λ − k cos δ = 0 ; 2

aqedan λ1 =

k cos δ * , λ2 = k cos δ *

radiusi iqneba

λ1 = λ2 .

k . maSin gveqneba oTxi SemTxveva:

1. k > 0 ; λ1 = λ2 > 0 - gadagvarebuli kvanZi – aramdgradi wonasworoba. amonaxsns aqvs saxe (nax. 4): 62

S (t ) = c2 e − λt , Δδ (t ) = ce − λt . 2. k > 0 ; λ1 = λ2 < 0 - mdgradi kvanZi – amonaxsns aqvs saxe:

S (t ) = C1e − λt , Δδ (t ) = −C 2 e − λt da isini akmayofileben (2.5) gantolebas (nax. 5). 3. k = 0 ; λ1 = λ2 = 0 - aramdgradi mdgomareoba. (nax. 6). zogad amonaxsns aqvs saxe:

Δδ = C1 + C 2 t ; S = C 2 ; 4.

K < 0;

λ1 = −λ2

kompleqsuri

fesvebia.

miviRebT

cikls – neitralurad mdgradi mdgomareoba (nax. 7). e.i,

roca

k > 0,

gvaqvs

wonasworobis

mdgradi

mdgomareoba (nax. 5). k = 0 -is dros _ aramdgradi (nax. 6). roca k < 0 _ sistema gadadis rxevis reJimSi. is ar aris mdgradobis dakargva, anu hopfis bifurkacia. urTierTdamikidebuleba S-sa da Δδ -s Soris yvela SemTxvevisaTvis

gamosaxulia

(danarTi).

egm-ze

modelirebiT

2.4 hopfis bifurkacia orgeneratoriani energosistemisaTvis orgeneratoriani sistema aRiwereba Semdegi gantolebaTa sistemiT: dS1 E1 E2 I12 = [sin(δ *1 − δ *2 ) − sin(δ *1 + Δδ *2 − δ *2 )] + dt I1 E UI + 1 1U [sin δ *1 − sin(δ *1 + Δδ1 )] − T1S1 I1

(2.9)

dS 2 E1 E2 I12 = [sin(δ *1 + Δδ *1 − δ*2 − Δδ*2 ) − sin(δ*1 − δ*2 )] + dt I2 E1UI1u [sin δ*2 − sin(δ*2 + Δδ*2 )] − T2 S2 . I2

(2.10)

SemovitanoT aRniSvnebi:

E UI E1 E2 I 12 = K 1 ; 1 1U = K 2 ; I2 I1

(2.11)

E UI E1 E 2 I 12 = K 3 ; 1 1U = K 4 . I2 I1 Tu (2.11) aRniSvnebs SevitanT (2.9)-Si, miviRebT:

dS1 = K1 [sin(δ*1 − δ*2 ) − sin(δ*1 + Δδ*1 − δ*2 + Δδ*2 )] + dt K2 [sinδ*1 −sin(δ*2 +Δδ*1)] −TS 1 1;

(2.12) (2.12)

dS2 = K3 [(δ *1 + Δδ1 − δ *2 − δ 2 ) − sin(δ *1 - δ *2 )] + dt K4 [sinδ*2 − sin(δ*2 + Δδ*2 )] − T2 S2 . aRvniSnoT

dΔδ 1 d Δδ 2 = S2 = S1 dt dt 64

(2.13)

wonasworebis wertili iqneba:

( S1 , Δδ 1 , S 2 , Δδ 2 ) = (0,0,0,0)

(2.14)

da am wertilis areSi sistemas eqneba saxe:

dS1 = − K 1 ⋅ cos(δ *1 − δ *2 )Δδ 1 − K 2 ⋅ cos(δ *1 )Δδ 1 − T1 S1 + dt K 1 cos(δ *1 − δ *2 )Δδ 2 ;

(2.15)

dS 2 = − K 3 cos(δ *1 − δ *2 )Δδ 2 − K 4 ⋅ cos(δ *2 )Δδ 2 − T2 S 2 − dt K 3 cos(δ *1 − δ *2 )Δδ 2 . veqtoruli

saxiT

( S1 , Δδ 1 , S 2 , Δδ 2 ) -Tan

mimarTebaSi

sistema Caiwereba Semdegi saxiT:

⎡ S1 ⎤ ⎡−T −K(δ −δ )+K cos(δ ) 1 *1 *2 2 *1 ⎢Δδ ⎥ ⎢ 1 0 ⎢ 1⎥ ⎢ 1 ⎢ S2 ⎥ =⎢−T K3(δ*1 −δ*2) ⎢ ⎥ ⎢ 2 ⎢Δδ ⎥ ⎣ 0 1 ⎣ 2⎦

K3(δ*1 −δ*2) ⎤ ⎡ S1 ⎤ ⎥ ⎢Δδ ⎥ 0 ⎥×⎢ 1⎥ 0 −K3(δ*1 −δ*2)+K4 cos(δ*2)⎥ ⎢ S2 ⎥ (2.16) ⎥ ⎢ ⎥ 0 0 ⎦ ⎢⎣Δδ2⎥⎦ 0 0

aqedan, maxasiaTebel gantolebas eqneba saxe:

K1 (δ*1 −δ*2 ) ⎡−T1 − λ − K1 (δ*1 −δ*2 ) + K2 cos(δ*1 ) 0 ⎤ ⎢ 1 ⎥ 0 0 0 ⎢ ⎥= − λ − K3 cos(δ*1 −δ*2 ) + K4 cos(δ*2 )⎥ K3 (δ*1 −δ*2 ) ⎢−T2 − λ ⎢ ⎥ −λ 1 1 ⎣ 0 ⎦

⎡− T1 − λ ⎢ 1 = −λ × ⎢ ⎢ − T2 ⎢ ⎣

− K 1 (δ *1 − δ *2 ) + K 2 cos(δ *1 ) 0 −λ 0 −λ K 3 (δ *1 − δ *2 )

⎤ ⎥ ⎥− ⎥ ⎥ ⎦

−T1 −λ −K1 cos( δ*1 −δ*2 ) +K2 cos(δ*1) − 1 −T2 3× 3

−λ K3 cos( δ*1 −δ*2 )

δ*1 −δ*2 ) K1 cos( 0

(2.17)

−K3 cos( δ*1 −δ*2 ) −K4 cos(δ*2 )

determinantis amoxsniT da gamartivebiT miviRebT:

λ4 + T1λ3 + λ2 ( K 1 cos(δ *1 − δ *2 )) + K 2 cos(δ *1 ) + K 3 cos(δ *1 − δ *2 ) − − K 4 cos(δ *2 ) + λ (T1 + K 3 cos(δ *1 − δ *2 ) + T2 cos(δ *1 − δ *2 )) − − K 3 cos(δ *1 − δ *2 ) ⋅ K 1 cos(δ *1 − δ *2 ) + ( K 1 cos(δ *1 − δ *2 )) + + K2 cos(δ *1 ) ⋅ (K3 cos(δ *1 − δ *2 ) + K4 cos(δ *2 )) = 0

(2.18)

Tu miRebuli maxasiaTebeli gantolebis amoxsnis Sedegad K 1 , K 2 , K 3 , K 4 , mniSvnelobebs aRmoaCndaT wyvili SeuRlebuli kompleqsuri fesvebi, sistemaSi adgili eqneba hopfis bifurkacias, e.i. egm-ze modelirebiT fazur sibrtyeze miviRebT zRvrul ciklebs, romelTac fazur sivrceSi Seesabameba tori (nax. 8). upirvelesad ganvixiloT mTlianad es sistema analizurad. gavarkvioT, parametrebis mocemuli mniSvnelobebis dros maxasiaTebeli gantoleba iZleva Tu ara kompleqsur fesvebs. Semdgom is amovxsnaT egm-is saSualebiT da miviRoT fazuri portreti:

dS 2 = − XΔδ 1 + X 2 Δδ X 2 − T1 S1 + X 1 Δδ 2 ; dt

d (Δδ 1 ) = S1 ; (2.19) dt

d ( Δδ 2 ) dS 2 = S2 . = − X 3 Δδ 1 − X 4 ΔδX 2 − T2 S 2 + X 1 Δδ 2 ; dt dt 66

sadac

X 1 = K 1 cos(δ *1 − δ *2 ); X1 = K 2 cos(δ*1 ); X 3 = K 3 cos(δ *1 − δ *2 ); X 4 = K 4 cos(δ *2 );

⎡ S1 ⎤ ⎡− T ⎢ Δδ ⎥ ⎢ 1 ⎢ 1⎥ ⎢ 1 ⎢S2 ⎥ = ⎢ 0 ⎢Δδ 2 ⎥ ⎢ ⎢ ⎥ ⎣ 0 ⎣ ⎦ ⎡ −T − λ ⎢ 1 ⎢ ⎢ 0 ⎢ ⎣ 0

− X1 − X 2 0 X3 0

−( X 1 − X 2 )

−λ X3

0 −T2 − X

S X1 0 ⎤ ⎡ 1 ⎤ ⎥ ⎢ Δδ1 ⎥ O( −T2 ) 0 ⎥ × ⎢S2 ⎥ ⎥ − T2 − X3 − X4⎥ ⎢ ⎥ ⎢Δδ 2 ⎥ 1 0 ⎥ ⎦ ⎢⎣ ⎦ ⎤ ⎥ 0 ⎥ = 0, ⎥ ( X X ) − 3− 4 ⎥ −λ ⎦ X1

0 ⎡− (T1 + λ ) − ( X 1 + X 2 ) ⎢ −λ 1 0 ⎢ −λ − (T2 − λ ) X3 ⎢ 0 ⎢ ⎣ ⎡T1 + λ ⎢ 1 −⎢ ⎢ 0 ⎢ ⎣

(2.20)

− (X1 + X 2 ) X1 −λ 0 X3 − (X 3 + X 4 )

⎤ ⎥ ⎥− ⎥ ⎥ ⎦ ⎤ ⎥ ⎥=0 ⎥ ⎥ ⎦

λ [λ (T1 + λ )(T2 + λ ) + ( X 1 + X 2 )(T1 + λ )] + + λ (T1 + λ )( X 3 + X 4 ) − X 1 X 3 + ( X 1 + X 2 )( X 3 + X 4 )

λ 4 + bλ 3 + C λ 2 + dλ + e = 0 b = T1 + T2

(2.21)

c = T1 + T2 + X 1 + X 2 + X 3 + X 4 67

d = ( X 1 + X 2 )T2 + ( X 3 + X 4 )T1

e = X1X 4 + X 2 (X 3 + X 4 )

nax. 8 orgeneratoriani sistemis fazuri tori

mivaniWoT parametrebs mniSvnelobebi, CaTvaloT, rom

T1 = 10 da T2 = 5 da amovxsnaT miRebuli maxasiaTebeli gantoleba:

X 1 = K 1 cos(δ *1 − δ *2 ) = 0,3 ⋅ 0,9801 = 0,2940

X 2 = K 2 cos(δ *1 ) = 0,44 ⋅ 0,6216 = 0,2735 X 3 = K 3 cos(δ *1 − δ *2 ) = 0,4 ⋅ 0,9801 = 0,3920 X 4 = K 4 cos(δ *2 ) = 0,66 ⋅ 0,7648 = 0,5048 b=15; C=50+0,2940+0,2735+0,3920+0,5048=1,4643

d = (0,2940 + 0,2735) ⋅ 5 + (0,3920 + 0,5048) ⋅ 10 = 11,8055 e = 0,2940 ⋅ 0,5048 + 0,2735 ⋅ (0,3920 + 0,5048) ⋅ 0,3937 (2.9) gantolebis fesvebi emTxveva 68

bY − d λ λ2 + (b + a ) + Y1 + =0 2

(2.22)

kvadratuli gantolebis fesvebs, sadac

a = ± 8Y1 + b 2 − 4c

(2.23)

xolo Y1 akmayofilebs gantolebas:

8Y13 − 4Cy 2 + (2bd − 8e)Y1 + e(4C − b 2 ) − d 2 = 0 da es gantoleba kuburi gantolebis namdvili fesvia. (2.22) ase gadavweroT:

(2.24) romelime

a0 X 3 + b0 X 2 + C 0 X + d 0 = 0

sadac

X = Y1 ;

a0 = 8 ;

(2.25)

b0 = −4C = −205,86 ;

C 0 = 2bd − 8e = 351,015 ; d 0 = e(4C − b 2 ) − d 2 = −146,9052 . Tu

(2.25)-s

gavyofT

a0 -ze

SemovitanT

axal

cvladebs Semdegi saxiT:

Y=X+

b0 = Y1 − 8,5775 , 3a0

(2.26)

miviRebT Semdegi saxis kubur gantolebas:

Y 3 + 3PY + 2q = 0

(2.27)

sadac

C0 b02 P= − 2 = −58 = 9485 3a 0 9d 0 b03 b0 C 0 d 0 q= − + = 451,9653 27 a 03 6a03 2a0 (2.27)

gantolebis

diskriminanti

D = q + p = −1075 <0.

radgan D<0, amitom mocemul gantolebas sami namdvili fesvi aqvs. erT-erTi maTgani Semdegi saxisaa: 69

Y = 2r cos , 3 sadac r = −

cos ϕ = cos

ϕ 3

(2.28)

p = − 58,9485 − 7,6778;

q − 451,9653 = = 0,9977; r 3 − 452,9902

= cos10 20 ' = 0,9977;

ϕ = arccos 0,9977 = 4 0 ,

Y = 2 ⋅ 7,6778 ⋅ 0,9977 = 15,351

(2.26)-is Tanaxmad

Y1 = Y + 8,5775 = 15,351 + 8,5775 = 23,9285 (2.25)-is mixedviT

a = ± 191,428 + 225 − 205,86 = 14,511;

b + a1 = 14,7555 , 2

b + a2 = 14,7555 2

(2.29)

Y1 +

br − d = 47,8498 , a1

Y1 +

br − d = 0,037 a2

(2.20)-sa da (2.25)-is Tanaxmad gvaqvs ori gantoleba:

λ2 + 14,7555λ + 47,8498 = 0

(2.30)

λ2 + 0,2445λ + 0,0370 = 0

(2.31)

(2.30)-Tvis miviRebT:

λ1, 2 = −7,3778 ± 2,5655 xolo (2.31)-Tvis

λ3, 4 = −0,1223 ± i × 0,1483 amgvarad miviRebT Semdeg mniSvnelobebs: 70

λ1 = −4,8123; λ2 = −9,9433; λ3 = −0,1223 + 0,1483i; λ4 = −0,1223 − 0,1483i; zogadi amonaxsni romelime cvladi sididisaTvis Caiwereba Semdegnairad: ξ = ξ1eλ t + ξ1e λ t + ξ1e λ t + ξ1eλ t = 1

= ξ1e 0 , 0012 t ⋅ ξ1e −50 , 0044 t ⋅ ξ1e −0 , 0034 t ⋅ ξ1e −0 , 0034 t ⋅ ξ1e −1, 064 it = = ξ1e −0 , 0012t + ξ1e −50 , 0044t + ξ1e −0 , 003425t ⋅ (cos1,064t + i sin 1,0641t ) +

+ ξ 1 e −0 , 0034 t (cos1,0641 − i sin 1,064t ) t=0

(2.32)

momentSi miviRebT Semdeg gantolebebs:

ξ = X1;

dξ = X 2; dt

d 2ξ = X 3; dt

dξ = X 4. dt 3

(2.33)

amrigad, roca sistemis maxasiaTebel gantolebas namdvil fesvebTan erTad aqvs kompleqsuric, maSin masSi mosalodnelia harmoniuli rxevebis arsebobac, romlebsac fazur sivrceSi zRvruli zedapiri anu ciklebi Seesabameba. analiზუri gamokvlevebidan Cans, rom am dros orgeneratorian sistemaSi miiReba hopfis bifurkacia.

2.5 hopfis bifurkacia samgeneratoriani sistemisaTvis imisaTvis,

rom

bifurkaciis

vaCvenoT

Teorema

sixSiruli

mZlavri

areebis

hopfis

eleqtrosistemebisaTvis,

gamoviyenoT igi samgeneratoriani sistemisaTvis. gantolebebi gadavweroT Semdegi saxiT:

1 + 2D12 sin(θ1 ) + D23 sin(θ1 −θ2 ) + D13 sin(θ2 ) + Q

(1)

C23 cos(θ1 −θ2 ) −C13 cos(θ2 ) = ΔP1

(2.34)

1 + 2D13 sin(θ2 ) + D23 sin(θ2 −θ1 ) + D12 sin(θ1 ) + Q

()

C23 cos(θ 2 − θ1 ) − C12 cos(θ1 ) = ΔP2 sadac

(2.35)

θ1 = δ 2 − δ 1 ;θ 2 = δ 3 − δ 1 ; ΔP1 = P2 − P1 ; ΔP2 = P3 − P1 ; Sesabamisad

aRniSnavs

manqanur

kuTxes

δ1 da

simZlavres. SeiZleba

ganisazRvros

parametrebis

mniSvnelo-

bebis jgufi, romlebic Seesabameba bifurkaciis wertilebs konservatiul sistemaSi da miiReba hopfis bifurkaciis indikatori. bifurkaciis am wertilisaTvis parametrebis mniSvnelobebi mocemulia cxrilSi: Dcxrili C12 C13 C

D12 D23

D13

ΔP1

ΔP2

θ1

θ2

0,577 0,577 4,042 2,887 -1,155 1,047 0,523

dempfirebis damatebis Semdeg (2.34), (2.35) gantolebebi daiyvaneba I rigis gantolebaze, sadac 72

I ⎤ ⎡θ ⎤ ⎡0 ⎡0 ⎤ X = ⎢ ⎥; A = ⎢ ; ; C = [I B = ⎥ ⎢ ⎥ ⎣θ ⎦ ⎣0 − λI ⎦ ⎣I ⎦

0],

⎡P −η − 2D12 sin(θ1 ) − D23 sin(θ1 −θ2 ) − D13 sin(θ2 ) − C23 cos(θ1 −θ2 ) + C13 sin(θ2 )⎤ g(x,η) = ⎢ 2 ⎥ ⎣P3 −η − 2D13 sin(θ2 ) − D23 sin(θ2 −θ1 ) − D23 sin(θ1 ) − C23 cos(θ2 −θ1 ) + C13 sin(θ1 )⎦

η = P1

(2.36) (2.34), (2.35) unda gardavqmnaT ukugadacemis eqvivalentur sistemad:

X = AX + BRY + B[( g ,η ) − RY ]; Y=CX

(2.37)

sadac

L(e) = G ( S ,η ) ⋅ L(U ) G ( S ,η ) = C [SI − ( A + BRC )] β −1

(2.38)

u = ƒ (e,η ) = g (−e,η ) − RY Tu R=1, raTa ar iyos naikvistis konturze erTgvarovneba, maSin miviRebT:

1 ⎡ ⎤ 0 ⎢ 2 ⎥ G ( S ) = ⎢ S + YS − 1 ⎥ 1 0 ⎢ ⎥ ⎣ S 2 + YS − 1⎦

f (e,η ) = g (−e,η ) + e iakobiani J=Detƒ iqneba:

⎡γ + z Detf = ⎢ ⎣β − Y

β +Y⎤ γ − z ⎥⎦

γ = D12 cos(e1 ) + D13 cos(e2 ) + D23 cos(e1 − e2 ) + 1; 73

(2.39)

Z = D12 cos(e1 ) + D13 cos(e2 ) + D23 sin(e1 − e2 ),

(2.40)

β = − D23 cos(e1 − e2 ) + [ D12 cos(e1 ) + D12 cos(e2 )] 2 + [C12 sin(e1 ) + C13 (e2 )] 2; Y = [ D13 cos(e2 ) − D12 cos(e1 )] 2 + C13 sin(e1 − e2 ) + [ −C12 sin(e1 ) + C13 (e2 )] 2. (2.41)

amgvarad,

⎡α + z 1 G(S ) ⋅ J = 2 S + YS − 1 ⎣⎢ β − Y G ( jϖ ) ⋅ J

β +Y⎤ α − z ⎥⎦

(2.42)

funqciis maxasiaTebeli gantoleba iqneba:

(ϖ 2 + 1) + JϖY λ ( jϖ ) = − α ± z + β − y 4 ϖ (2 + Y )ϖ 2 + 1

[

]

(2.43)

naTelia, rom naxevarnormaluri polusi gansazRvruli smit-makmilanis [S] formiT, aris gaxsnil maryuJze gadaadgilebis funqcia. G ( S ) ⋅ J matrica iqneba [34]:

P ( S ) = ( S 2 + TS − 1)

(2.44)

Tu CavsvavT parametrebis mniSvnelobebs cxrilidan, roca Y=0,05, es sakmarisi iqneba saangariSod da egm-ze modelirebisaTvis, radgan bifurkaciis parametri P1 icvleba misi kritikuli mniSvnelobisas. roca dempfirebis

mniSvneloba

Y=0,05 da P1-is

kritikuli

mniSvneloba daZrulia – 1,1551-dan daaxloebiT – 1,146mde.

P1<-1,146 –

Tvis

maxasiaTebeli 74

Semowers

(-1,50)

wertils

orjer

aRwevs

mdgrad

mdgomareobas.

Tavidan ki grafiki miiswrafis am wertilisaken, e.i. sistema

konvergirebs

wonasworobisaken,

rxevebis

sixSiriT 1,2 rad/wm. (nax.9)

nax.9 samgeneratoriani sistemis moZraoba wonasworobisken

P1 gadis

kritikul

mniSvnelobas,

traeqtoria

gaivlis (-1,50) wertilze, magram ar emTxveva mas, e.i. aRwers aramdgrad sistemas. es aris is SemTxveva, roca

nax.10 gardamavali procesis grafiki samgeneratoriani sistemisTvis

nax.11 samgeneratoriani sistemis saxe fazur sibrtyeze

fesvebi kveTen warmosaxviT RerZebs. sistemaSi adgili aqvs hopfis bifurkacias.

nax.12 n â&#x20AC;&#x201C; ganzomilebiani tori

roca

P1=-1,144,

maSin

sistema

miiswrafis

wonas-

worobisaken. Tu P1=-1,145, maSin sistema aRwevs wonasworobas da sabolood, roca P1=-1,146, sistemaSi aqac adgili aqvs hopfis bifurkacias. am eqsperimentis gardamavali procesis grafiki da fazur

sibrtyeze

misi

(nax.11), xolo (nax.12)-ze

saxe

mocemulia

(nax.10)

ki - eqsperimentis saboloo

Sedegebi: n-ganzomilebiani tori. 77

3. qaosi energosistemebSi 3.1 saerTo cnobebi qaosis Sesaxeb qaosi dinamikur sistemebSi â&#x20AC;&#x201C; es aris erT-erTi ZiriTadi problema rxevebis arawrfivi Teoriisa. kvlevebi am mimarTulebiT mimdinareobs rogorc ricxviTi, aseve naturaluri eqsperimentebiT. rxeviT sistemaSi moZraobis xasiaTis Sesaswavlad da

atraqtoris

arsebobis

dasadgenad

gamoiyeneba

Semdegi meTodebi; 1) gamosakvlevi diferencialuri gantolebis ricxviTi integrireba gardamavali procesis uaryofiT.

nax. 13. atraqtoris reJimSi momuSave arawrfivi generatoris gardamavali procesi

inerciuli

Sedegad vRebulobT erT an ramdenime X(t) fazuri cvladis droSi cvlilebis suraTs. (nax. 13.)-ze mocemulia atraqtoris reJimSi momuSave arawrfivi inerciuli generatorisaTvis zemoT aRniSnuli droiTi realizacia [34]: 2) furies gamokvleva. xorcieldeba miRebuli rxevebis speqtrebiT, an naturalur eqsperimentebSi speqtr-analizatoris gamoyenebiT, an kidev gantolebis integrirebiT egm-ze specialuri programiT da furies swrafi gardaqmniT. sistemaSi

rxevebis

evoluciisas

periodulidan

qaosurisaken, erTi an ramodenime mmarTveli parametris cvlilebisas, speqtrSi SeiZleba davakvirdeT harmonikebis doneebis zrdas, Semdeg ki subharmonikebis warmoSobas (rxevebis periodis bifurkacia) da qaosis dros Cndeba sruli speqtri, romelSic Cans sistemis maxasiaTeblebi. 3) qaosuri rxevebis korelaciuri funqciis kvleva. viner-xinCinis arsebobisas,

Teoremis

nebismieri

ZaliT

sruli

cvladisaTvis

speqtris miiReba

korelaciis funqciis daRmavali xasiaTi [34]. 4) dadebiTi entropia dinamikur sestemebSi. igi

iTvleba

qaosis

erT-erT

kriteriumad,

romelic axasiaTebs fazuri traeqtoriebis saSualo siCqares atraqtorSi: 79

K = lim lim ε →0 t →∞

M (ε , t ) , t

(3.1)

sadac, ε - gazomvis cdomilebaa, t – dakvirvebis dro, M – traeqtoriebis maqsimaluri ricxvi, romelTa Soris manZilic metia K-ze. e.i. Tu traeqtoriebi mdgradia liapunovis mixedviT, anu axlo mdebare sawyisi gamomsaxvelebi ar Sordebian erTmaneTs, maSin K=0. 5) dispersiis D = ( X − X ) 2

gansazRvra da histogra-

mebis ageba. es xorcieldeba, maSin, roca atraqtori ikveTeba sibrtyesTan, romelsac puankares sibrtye ewodeba. am dros

mmarTveli

fazuri rxevebis

parametris

sivrcis

zomebis

evoluciis

cvlilebisas

1-iT

suraTis

Semcireba miReba

SeiZleba da

masze

periodulidan

qaosuramde. am dros SeiZleba sivrculi da droiTi diskretizaciis ganxorcieleba [1,5,26,34]. winamdebare samuSaoSi gamoyenebuli maqvs droiTi diskretizacia, radgan is sivrculi diskretizaciisagan

gansxvavebiT

amartivebs

dasmuli

problemis

gadaWras. ganvixiloT

droiTi diskretizacia, rodesac

xdeba gadasvla uwyveti droiTi dinamikuri sistemidan: X=F(X), F : R → R , F ∈ C ( R ) d

gardasaxvaze

(3.2)

f ∈: R d . sainteresoa, Tu ramdenad inar-

Cunebs mocemuli (3.2) sistema Tavis ZiriTad maxasi80

aTeblebs sakmaod mcire diskretizaciis dros. kargadaa cnobili diferencialuri gantolebebidan asaxvaze

gadasvlis

zusti

meTodebi:

esaa

droiT

traeqtoriaze Zvris operatori da puankares asaxva raRac zedapiris mimarT, romelic transversaluria sistemis traeqtoriebTan. praqtikaSi gamoyenebulia droiTi diskretizaciis Semdegi miaxloebiTi meTodebi: eileris, centraluri sxvaobebis, Stermeris meTodebi da a. S. sivrculi diskretizaciisagan gansxvavebiT, sadac tipiur efeqts warmoadgens aramdgradi ciklebis stabilizacia-stoqastikurobis daSla, droiTi diskretizaciisas SesaZlebelia qaosurobis gaCena. davuSvaT, (3.2) sistemas aqvs ori gadaugvarebeli uZravi wertili, romelTa Sorisac erT-erTi aramdgradia. maSin eileris meTodiT diskretizaciisas anu

fε ( X ) = X + εF ( X ) iseTi

ε0 > 0 ,

rom

asaxvaze

gadasvlisas,

ε > ε0

yvela

moiZebneba

diskretizirebuli

sistema qaosuria li-iorkes mixedveT. erT-erT warmogvidgeba zaciis ε

ZiriTad is,

rom

naklovanebad

qaosuroba

Cndeba

SemTxvevaSi diskreti-

parametris mxolod sakmaod didi mniSvne-

lobisas, romlis drosac gansakuTrebuli wertilebis tipi SeiZleba gansxvavdebodes sawyisi sistemisagan. am

naklovanebis

nawilobrivi

acileba

SeiZleba

Semdegnairad: vTqvaT

d=1,

F(X)=X(1-X),

maSin

centraluri

sxvaobebiT diskretizaciis dros, e.i. organzomilebian asaxvaze f (ε (U , V )) = (V + 2εF (U ), U ) gadasvlisas, nebismieri ε > 0 -Tvis, diskretizirebuli sistema qaosuria li-iorkes mixedviT. geometriuli TvalsazrisiT, am dros nebismieri

ε > 0 -Tvis (1,1) wertilis mdgradi da aramdgradi mravalsaxeobebi transversalurad ikveTebian Sesabamisad aramdgrad

mdgrad

(0,0)

wertilis

mravalsaxeo-

bebTan, rac iwvevs qaoss. magram am dros diskretul droiTi sistema mniSvnelovnad gansxvavdeba sawyisi sistemisagan sevrcis

ZiriTadi

ganzomileba,

maxasiaTeblebiT gansakuTrebuli

(fazuri

wertilebis

tipi a.S.) egm-ze modelirebiT SeiZleba gamovikvlioT qaosis Casaxva energosistemebSi. ricxviT analizs safuZvlad udevs sxvadasxva sqemebi, amitom es gvaiZulebs diferencialuri gantolebebidan gadavideT sxvaobiTi saxis

gantolebebze.

sistemis

moZraobis

diskretuli

gantoleba iZens stoqastikurobas nebismieri diskretizaciisas, roca ki sawyisi gantoleba zustad integrirdeba. es niSnavs, rom diskretul gantolebebze

gadasvla

eqvivalenturia

gareSe

perioduli

Zalis

damatebisa. ganvixiloT energosistemebSi qaosis Casaxva da ganviTarebis procesi. qaosuri moZraoba ganvixiloT, rogorc

fazuri

traeqtoriebi

sibrtyeze,

romlis

RerZebad arCeulia: dadgenili reJimebidan gadaxris kuTxe

Sesabamisi

sriali.

traeqtoriebi

miRebulia samganzomilebiani sivrcis kveTiT, romlis mesame RerZad miRebulia integrirebis

bijis sidide.

es ukanaskneli asrulebs gareSe perioduli Zalis da mmarTveli parametris rols. n-generatorebisaTvis

kvlevis

obieqti

Caiwereba

Semdegi saxiT: n dS i Ii = −∑ Ei E j Yij cos(δ *i − δ * j ) ⋅ sin( Δδ i − Δδ j ) + j =1 dt j ≠1 n

+ ∑ Ei E j Yij sin(δ *i − δ * j ) × [1 − cos(Δδ i − Δδ j )], j =1 j ≠1

dΔδ i = Si dt

(3.3)

sadac I – eleqtruli manqanis inerciis mudvivaa; δ * mocemul

reJimSi

kvanZis

sididea;

Δδ

mocemuli

mniSvnelobebidan gadaxris kuTxis sididea; E – generatorebis emZ; Y – ganStoebebis urTierTgamtaroba; S – sriali.

ganvixiloT usasrulo

erTgeneratoriani

simZlavris

salte”,

“generatori-

orgeneratoriani

samgeneratoriani sistemebi.

3.2 qaosi erTgeneratorian energosistemaSi erTgeneratoriani sistemebisaTvis moZraobis gantolebas sistemaSi `generatori-usasrulo simZlavris salte” konservatiuli idealizaciisaTvis eqneba saxe (nax. 14):

dS = − EUY cos δ * sin Δδ + EUY sin δ * (1 − cos Δδ ). dt

(3.4)

nax. 14 `generatori-usasrulo simZlavris salte~ -s bloksqema mocemuli modelisaTvis liapunovis funqcias eqneba saxe [34]:

1 V = IS 2 + EUY cosδ * (1 − cos Δδ ) − EUY sinδ * (1 − cos Δδ ) 2 84

(3.5)

sistemaSi `generatori-usasrulo simZlavris salte” SemovitanoT aRniSvna: EUY=A SeSfoTebaze

miRebuli

(3.6)

SezRudvis

gaTvaliswinebiT

miviRoT, rom S=0, saidanac,

V = A cos δ* (1 − cos Δδ ) − A sin δ* (Δδ − cos Δδ ) qaosis

Sesaswavlad

erTgeneratorian

sistemaSi,

gardavqmnaT (3.4), miviRebT:

dS EUY = (sin δ * − sin(δ * + Δδ )) . dt I (3.7)-is debiT,

amonaxsni

droisa

ganvixiloT fazuri

(3.7)

grafikuli

parametrebis

meTo-

damokide-

bulebaSi. am dros unda dadgindes mdgradobis sazRvrebi

kuTxis

maCvenebliT,

zRvruli

ciklebisa

romelic

atraqtorebis

dgindeba

bifurkaciis

ierarqiiT. (3.7) warmovadginoT Semdegi saxiT: dS EUY EUY = (− cos δ * sin Δδ + sin δ * − sin δ * sin Δδ ) = (sin δ * − sin(δ * + Δδ )); dt I I

δ * =DD;

EUY =K; Δt =T; I

Δδ =D;

Sn +1 − Sn EUY = (sin δ* − sin(δ* + Δδ n )); Δt I

S n +1 = S n + Δt

EUY (sin δ * − sin(δ * + Δδ n )); I

S n +1 = S n + TK

EUY (sin δ * − sin(δ * + Δδ n )); I 85

(3.8)

Δδ n +1 + Δδ n = Sn +1. Δt

Δδ n +1 = Δδ n + TSn +1 ; S n +1 = S n + TK (sin δ * − sin(δ * + Δδ n ));

(3.9)

gantoleba

mixedviTac

avage

warmoadgens fazuri

algoriTms,

portreti

(3.9) romlis

amocanis

dis-

kretizaciiT. programa uzrunvelyofs fazuri portretebis agebas sawyisi pirobebis cvlilebisas. sawyisi pirobebi programis tanSi icvlebian ciklSi (nax. 15). fazuri portretebidan Cans, rom sistemis moZraobis diskretuli analogi iZens stoxasturobis ubans T(s) diskretizaciis nebismier bijze. am dros sawyisi gantoleba zustad integrirdeba. es niSnavs, rom gadasvla moZraobis diskretul gantolebebze (3.9), eqvivalenturia gare perioduli Zalis damatebisa, romelic ganpirobebulia diskretizaciiT. nax. 15-18 naCvenebia, Tu sad Cndeba qaosi sistemaSi `erTi generatori-usasrulo energiis salte” sxvadasxva parametrebis dros. aRwerili

dinamikuri

sistemis

konservatuloba

aRniSnavs, rom winaswari iteraciuli cikli Sualeduri mniSvnelobebis gamosaricxad, gardamaval reJimSi SeiZleba gamoitovos. yoveli gamoTvlili sistema warmoadgens Tavis sakuTar atraqtors. 86

EUY I

parametris nebismieri mniSvnelobis cvlile-

bebis Sedegad sistemas aqvs bevri orbita da konservatulobis Tvisebis Tanaxmad nebismieri sawyisi mniSvnelobebis wyvili S da

Δδ

iqneba wertilis warmom-

dgeni, romelicmdebareobs erT-erT orbitaze.

nax.15 qaosi sistemaSi - `erTi generatori-usasrulo energiis salte” (pirveli cikli)

nax.16 qaosi sistemaSi - `erTi generatori-usasrulo energiis salteâ&#x20AC;? (meore cikli)

nax.17 qaosi sistemaSi - `erTi generatori-usasrulo energiis salteâ&#x20AC;? (mesame cikli)

nax.18 qaosi sistemaSi - `erTi generatori-usasrulo energiis salteâ&#x20AC;? (meoTxe cikli)

aqedan

gamomdinare,

miziduloba

icvleba

myisi-

erad. parametrebis yoveli axali mniSvneloba iZleva axal sistemas. naxazebze 15 Ăˇ 18 naCvenebia orbitebis mimdevroba, romelic agebulia (3.7) asaxvis gantolebiT, romlebsac aqvT Semdegi mniSvnelobebi: d=0,5; centridan

TandaTan

EUY I

=1; d*=0,5; T=0,8.

daSorebisas

Cans,

rogor

qmnian orbitebi budeebs Cakitili traeqtoriebiT. Semdeg uecrad Cndebian patara `kunZulebi~ _ calke or89

bitebi, romlebic arian did orbitebs Soris. centridan kidev ufro daSorebisas vnaxavT Tu rogor monacvleoben aseTi orbitis mqone ubnebi.

3.3. qaosi orgeneratorian energosistemaSi

ganvixiloT

orgeneratoriani

sistema

Jdanovis

modelis magaliTze. diferencialur gantolebas aseTi sistemisaTvis aqvs Semdegi saxe: 2 ⎤ Eq1 Eq2 dω1 1 3 ⎡ Eq1 J1 sin(δ12 − α12 )⎥ ; = TT1 − ⎢ sin α11 + dt Z12 ω1 2 ⎢⎣ Z11 ⎥⎦

2 ⎤ Eq1 Eq2 1 3 ⎡ Eq2 dω2 = TT 2 − + − sin α sin( δ α ) J2 ⎢ 22 12 12 ⎥ ; ω2 2 ⎢⎣ Z 22 dt Z12 ⎥⎦

() (3.10)

dδ12 = ω1 − ω2 . dt am modelis algoriTms eqneba Semdegi saxe: 2 ⎤⎤ Eq1 Eq2 T ⎡ 1 3 ⎡ Eq1 + − ω1i +1 = ω1i + ⎢TT 1 − α δ α sin sin( ) ⎢ 11 12 i 12 ⎥ ⎥ ; ω1i 2 ⎣⎢ Z11 J1 ⎢⎣ Z12 ⎦⎥ ⎥⎦

2 ⎤⎤ Eq1 Eq2 1 3 ⎡ Eq2 T ⎡ sin α22 + sin(δ 21i − α21 )⎥ ⎥ . ω2i +1 = ω2i + ⎢TT 2 − ⎢ J 2 ⎢⎣ Z21 ω2i 2 ⎢⎣ Z22 ⎥⎦ ⎥⎦

(3.11)

egm-is gamoyenebiT movaxdinoT (3.11)-is modelireba. am modelSi cvladebad aviRebT

ω , δ 12 , δ 21 . modelirebis

rezultatebidan Cans, rom fazuri portreti (nax. 19) gansxvavdeba liapunovis modelisagan. orgeneratorian sistemas miekuTvneba agreTve modeli, romelic

iyenebs

erTsixSirian

denebs

Zabvas

manqanis kvanZebsa da qselSi. diskretul - diferencialur gantolebas am modelisaTvis

aqvs

Semdegi

n d 2δ1n 2 T ji = PTi − Ei YijЭκ sin α ijЭκ − ∑ Ei E jYijЭκ sin(δ ij − α ijЭκ ) . dt 2 j =1

saxe: (3.12)

am gantolebas ori generatorisaTvis eqneba saxe:

d 2δ1n 2 T j1 P E = − 1 Y11Эκ sin α11Эk − E1 E2Y12 Эκ sin(δ12 − α12 Эk ) ; Ti 2 dt d 2δ1n Tj 2 = PT2 − E22Y22 Эκ sin α 22 Эk − E1 E2Y21Эκ sin(δ 21 − α 21Эk ) . (3.13) 2 dt am modelis modelirebis Sedegi mocemulia (nax. 20) – ze.

b) nax. 19 orgeneratoriani energosistemis qaosis fazuri

portreti 92

b) nax. 20 orgeneratoriani energosistemis qaosuri moZraobis moderirebis Sedegi kvanZebisa da qselisTvis

3.4. qaosi samgeneratorian energosistemaSi samgeneratorian sistemas agreTve uwodeben `ori generatori-usasrulo energiis salteebi” (nax. 21). gardamavali procesebi rTul eleqtromeqanikur sistemebSi mniSvnelovnad gansxvavdebian moZraobisagan erTgeneratorian sistemaSi. amitom zemoT miRebuli Sedegebi

SevamowmoT

garTulebul

samgeneratorian

sistemaze.

nax. 21 `ori generatori-usasrulo energiis salte” –is blok-sqema es sistema SemTxveviT ar agvirCevia. energosistemebis analizis praqtikaSi es sistema ikavebs gansakuTrebul adgils. bevri mkvlevari Tavis ideebs cdis swored am modelze. es gamowveulia imiT, rom sammanqaniani sistema Seicavs bevr iseT Tvisebas, romelic axasiaTebs realur rTul sistemebs. amitom kvle94

vebi generatorebis nebismieri raodenobiani sistemebisTvis SeiZleba Catardes am sistemaze [34]. samgeneratorian sistemaSi moZraobis aRweris diferencialur gantodebas aqvs saxe [26]:

dS1 = − E1 E2Y12 cos(δ *1 − δ *2 ) sin(Δδ1 − Δδ 2 ) − dt − E1UY1U cos δ *1 sin Δδ1 + E1 E2Y12 sin(δ *1 − δ *2 ) ×

(3.14)

× [1 − cos(Δδ1 − Δδ 2 ) ] + E1UY1U sin δ * (1 − cos Δδ1 ) − T1

d Δδ 1 ; dt

dS 2 = E1 E2Y12 cos(δ *1 − δ *2 ) sin(Δδ1 − Δδ 2 ) − dt − E2UY2U cos δ *2 sin Δδ 2 − E1 E2Y12 sin(δ *1 − δ *2 ) × I2

× [1 − cos(Δδ1 − Δδ 2 ) ] + E1UY2 sin δ *2 (1 − cos Δδ 2 ) − T2

d Δδ 2 ; dt

dΔδ 2 dΔδ 1 = S1 = S2 ; dt dt (3.14)-is Sesabamis liapunovis funqcias aqvs saxe [26]:

V = E1UY1U [cosδ *1 (1 − cos Δδ 1 ) − sin δ *1 (Δδ 1 − sin Δδ 1 )] + + E2UY2U [cosδ *2 (1 − cos Δδ 2 ) − sin δ *2 (Δδ 2 − sin Δδ 2 )] +

+ E1 E2Y12 [cos(δ *1 − δ *2 )[1 − [cos(Δδ 1 − Δδ 2 )] − sin(δ *1 − δ *2 ) ×

(3.15)

× [Δδ 1 − Δδ 2 − sin(Δδ 1 − Δδ 2 )]] samgeneratorian

sistemaSi

qaosis

Sesaswavlad

visargebloT (3.15) moZraobis gantolebiT. gardavqmnaT is Semdeg saxeSi:

dS1 = E1E2Y12 sin(δ*1 − δ*2 ) − sin(δ*1 + Δδ1 − δ*2 − Δδ 2 ) + dt + E1UY1U (sin δ*1 − sin(δ*1 − Δδ1 ) − T1S1; I1

dS 2 = E1 E2Y12 sin(δ *1 + Δδ1 − δ *2 − Δδ 2 ) − (sin(δ *1 − δ *2 )) + dt + E2UY2U (sin δ *2 − sin(δ*2 + Δδ 2 )) − T2 S2 ; I2

dΔδ 2 dΔδ 1 = S1 , = S2 . dt dt uwyveti (3.16) gantolebidan gadavideT diskretulze: S1i +1 = S11 +

Δt [ E1 E2Y12 sin(δ *1 − δ *2 ) − sin(δ *1 + Δδ12 − δ *2 − Δδ 21 )] + I1

+ E1UY1U (sin δ *1 − sin(δ *1 − Δδ1i )) − T1S 21 ;

S 2i +1 = S 21 +

Δt [ E1 E2Y12 sin(δ *1 + Δδ12 − δ *2 − Δδ 21 ) − sin(δ *1 − δ *2 )] + I2

+ E2UY2U (sin δ *2 − sin(δ *2 + Δδ 21 )) − T2 S 21 ;

Δδ 1i +1 = Δδ 1i + ΔtS11 , Δδ 21+1 = Δδ 21 + ΔtS 21 .

(3.17)

(3.17) gantolebebi warmoadgens qaosis Seswavlis algoriTms. fazuri portretis agebis programa iZleva saSualebas

aigos

fazuri

traeqtoriebi

sawyisi

pirobebis cvlilebisas. nax.22 ÷ nax24-ze mocemulia fazuri portretebi, saidanac Cans, rom sistemis moZraobis (3.17) diskretuli analogi iZens stoqasturobis ubnebs diskretizaciis nebismier T bijze, am dros (3.15) sawyisi gantoleba zustad integrirdeba. mocemul naxazze naCvenebia Tu sad Cndeba qaosi. am

paragrafSi,

rxevebis

Teoriis

safuZvelze,

gamokvleulia qaosis aRZvra arawrfiv sistemebSi.

b) nax. 22 fazuri portreti (pirveli biji)

b) nax. 23 fazuri portreti (meore biji)

b) nax. 24 fazuri portreti (mesame biji)

ganvixileT erTgeneratoriani, orgeneratoriani da samgeneratoriani sistemebi. gadaiWra energosistemebSi qaosis

warmoSobis

amocana,

mmarTveli

parametris

cvlilebaze damokidebulebebiT. energosistemebis modelebad warmovadgineT Jdanovis modeli, liapunovis modeli da modeli, romelic iyenebs erTtaqta denebs da daZabulobas generatorebis kvanZebsa da qselSi. energosistemebis modelebis dayvana algoriTmze movaxdine liapunovis funqciis saSualebiT. erTgeneratorian, orgeneratorian da samgeneratorian sistemebSi qaosis warmoSoba da ganviTareba modelirebuli maqvs personalur kompiuterze. kvlevis Sedegebma aCvena, rom qaosi warmoiSoba integrirebis bijis cvlilebisas, roca uwyveti sistemebi gadadian diskretul sistemebSi da amocanis gadasaWrelad gamoyenebuli maqvs droiTi diskretizacia.

100

3.5. dinamikuri qaosis kriteriumebi energosistemaSi da liapunovis maxasiaTebeli dinamikuri qaosis kriteriumebad miRebulia: 1) liapunovis maCveneblebi anu maxasiaTeblebi; 2) korelaciuri funqciis Tvisebebi; 3) speqtraluri simkvrivis Tvisebebi; 4) puankares gardasaxvebi; 5) fraqtaluri ganzomilebani. maT Soris yvelaze xSirad qaosis kriteriumebad gamoiyenebian liapunovis maCveneblebi da fraqtaluri ganzomilebani. zogadad liapunovis dadebiTi maCvenebeli migviTiTebs qaosur dinamikaze, aseve fazur sivrceSi orbitis fraqtaluri struqtura migvaniSnebs ucnauri atraqtorebis arsebobaze. ucnauri atraqtorebis TiTqmis yvela fizikuri magaliTi aRmoCnda qaosuri, e.i. ucnauri atraqtori qaosis kriteriumia. Cemi amocanaa, ganvixilo liapunovis maCveneblebi, rogorc energosistemebSi dinamikuri qaosis Casaxvisa da ganviTarebis kriteriumi. mgrZnobiaroba sawyis aris

pirobebSi gadamwyveti

mcire

fluqtuaciebisadmi,

arsebuli

raRac

dinamikuri

sistemis

anu

ganuzRvreloba, dasaxasiaTeb-

lad. am dros sawyisi ganuzRvreloba aris eqsponencialurad mzardi drois mixedviT manam, sanam sistemis Semdegi mdgomareoba ar gaxdeba ganuzRvreli. infor101

macia sawyisi mdgomareobis Sesaxeb ikargeba sasruli drois

Semdeg.

swored

drois

gavlis

Semdeg

sistemis mdgomareobis winaswarmetyveleba SeuZlebeli xdeba. sawyisi pirobebisadmi mgrZnobiaroba warmoadgens qaosis erT-erT ZiriTad maxasiaTebels. parametri, riTac SeiZleba vimsjeloT saSualo lokalur mdgradobaze an siCqareze, romliTac Cndeba axali informacia nakadis saSualebiT an piriqiT, gvaZlevs informacias sawyis pirobebze da warmoadgens liapunovis maxasiaTebels -

位 -s. igi ar aris xaris-

xobrivi xasiaTis, romelsac damkvirveblis didi gamocdileba sWirdeba, aramed is aris raodenobrivi maxasiaTebeli

misi

gamoyeneba

didi

warmatebiT

SeiZleba. liapunovis

maxasiaTeblis

gamoyeneba

SeiZleba

rogorc disipatiur, aseve aradisipatiur (konservatiul) sistemebSi, fraqtaluri ganzomilebani ki mxolod disipatiurSi gamoiyeneba. liapunovis

maxasiaTeblis

saSualebiT

mowmdeba

sistemis mgrZnobiaroba sawyisi monacemebis variaciis dros. idea mdgomareobs SemdegSi: warmosaxviT fazur sivrceSi gamoiyos mcire sfero, romelSic Tavmoyrili iqneba traeqtoriis sawyisi wertilebi da yuradReba mivaqcioT, Tu rogor moxdeba misi deformacia elifsoidSi sistemis dinamikuri evoluciis procesSi. Tu d aris elifsoidis maqsimaluri zoma, d0_sa102

wyisi pirobebis sferos sawyisi zomaa, maSin liapunovis maxasiaTebeli

λ gamoiTvleba formulidan: d = d 0 2 λ ( t −t ) . 0

(3.18)

amasTan, erTjeradi gazomvebi arasakmarisia, amitom gazomvebis

Sedegi

unda

avirCioT

fazuri

sivrcis

sxvadasxva monakveTze da Semdeg gavasaSualoT: λ = lim N →∞

d 1 N 1 log 2 i . ∑ N i =1 (t i − t 0 i ) d 0i

(3.19)

(3.19)-dan erTjeradi gazomvebiT miviRebT:

λ = log 2 Tu

sistema

aRiwereba

d 1 . d 0 (t − t 0 )

sxvaobiTi

(3.20) gantolebebiT

asaxviT - d n = d 0 2 ΔL , maSin unda aRiniSnos, rom algoriTmis maCvenebeli SeiZleba avirCioT nebismierad. qaosuri traeqtoriebis eqsponencialuri ganSladoba anda iyos lokaluri. Tu sistema SezRudulia, (umetesad

asea),

maSin

d(T)

usasrulod

ver

gaizrdeba. amitom, imisaTvis, rom avirCioT traeqtoriebis ganSladobis zoma, unda movaxdinoT eqsponencialuri

zrdis

gasaSualeba

traeqtoriis

gaswvriv

yvela wertilze (nax. 25 da 26) [34]. liapunovis maxasiaTeblis gamoTvla iwyeba sayrdeni traeqtoriis arCeviT (nax. 25), mezobel traeqtoriaze wertilis arCeviT da d (t ) d 0

sididis gamoT-

vliT. roca d(t) manZili izrdeba Zalian, unda movZeb103

noT sxva traeqtoria da viangariSoT axali sawyisi d0(t) manZili. liapunovis maxasiaTebeli SeiZleba asec Caiweros:

1 λ= t N − t0

∑ log K =1

d (t K ) d 0 − (t K −1 ) .

(3.21)

sabolood davadgineT, rom liapunovis maxasiaTebeli aris sistemis axlo traeqtoriebis gaSlis siCqaris zoma. F(x) – funqciiT mocemuli asaxvisas, erTi iteraciis dros sistemis Sesaxeb informaciis cvlileba dF dx warmoebuliT warmogvidgeba formuliT:

dF . dx

I = log 2

(3.22)

Tu | dF dx |< 1, maSin mocemul wertilSi gardasaxva

moqmedebs,

rogorc

informaciis

mimRebi.

| dF dx |> 1 _ rogorc informaciis wyaro. Tu P(x) mocemuli sawyisi parametris dros traeqtoriebis albaTobebis asimptoturi ganawilebaa, maSin < I > informaciis raodenobis saSualo cvlilebaa liapunovis maxasiaTeblis maCvenebeli mmarTveli parametris mocemuli mniSvnelobisas:

104

b) nax. 25 qaosuri traeqtoriis eqsponencialuri ganSladoba

105

< I >= λ (a) = ∫ P( x) log 2 0

dF dx ; dx

(3.23)

Tu atraqtoris SigniT traeqtoriis qceva ergodikulia,

maSin

arsebobs

liapunovis

maxasiaTeblis

alternatiuli forma: 1 λ (a ) = lim N →∞ N

∑ log 2 i =1

dF ( x, a ) ( dx) X i , dx

(3.24)

sadac N iteraciebis ricxvia, romelsac traeqtoria ganicdis gardasaxvisas. maSasadame, liapunovis maxasiaTeblis zemoT moyvanili gansazRvrebis Tanaxmad Cans, rom rodesac

λ (a ) < 0 ,

traeqtoria mdgradi da araperiodulia. gvaqvs regularuli moZraoba da qaoss adgili ara aqvs. (iteraciuli procesi moqmedebs rogorc sistemis Sesaxeb informaciis Semkrebi).

λ ( a ) = 0 traeqtoria neitralurad mdgradia, e.i. informacia arc ikribeba da arc ixarjeba (am dros xdeba periodis gaormagebis `SexebiTi” bifurkaciebi anu

traeqtoriam

unda

gaiaros

neitralur

mdgrad

atraqtorze). roca λ ( a ) > 0 traeqtoria lokalurad aramdgradia

sistema

qaosuria

(iteraciuli

procesi

moqmedebs, rogorc informaciis wyaro). erTganzomilebian sistemaSi, romelSic atraqtorebi SeiZleba iyos mdgradi gansakuTrebuli werti106

lebi, arsebobs mxolod erTi liapunovis maxasiaTebeli (-). organzomilebiani SeiZleba iyos ori saxis atraqtori: esaa mdgradi saxis stacionaluri wertilebi da zRvruli ciklebi. Tu orive maxasiaTebeli uaryofiTia ( λ1 < 0 ,

λ2 < 0 ), e.i. (-,-), maSin cxdia, rom aseTi

sistema mdgradia. Tu (λ1 , λ2 ) = (−,0) , maSin atraqtori aris zRvrul cikli. samganzomilebian sivrveSi stacionarul mdgomareobas aqvs liapunovis sami maxasiaTebeli da samive uaryofiTi (-,-,-). zRvrul ciklsac sami maxasiaTebeli, maTgan ori – uaryofiTi da erTi (nakadis gaswvriv) nulia anu (-,,0). `ucnauri”

atraqtoric

sammaCvenebeliania

(+,0,_),

anu erTi dadebiTi λ+ maCvenebeli, erTi nulovani

λ0

liapunovis maCvenebliT da erTi uaryofiTi λ− , e.i. `ucnauri” atraqtori erTi mimarTulebiT qmnis sistemis Sesaxeb informacias, meore mimarTulebiT _ kumSavs. nakadis mimarTulebiT _ ki l.m. nulis tolia. (+,0,_) warmoadgens signaturas, rac niSnavs Tu rogori

namdvili

nawili

aqvs

liapunovis

maxasiaTebels: dadebiTi, nulovani Tu uaryofiTi. samganzomilebiani

sevrcisaTvis

(invariantuli

tori) sabolood 4 SemTxveva ase Camoyalibdeba. 107

( λ1 , λ2 , λ3 )=(-,-,-)

–

stacionaluri

mdgomareoba

(mdgradi fokusi an kvanZi); ( λ1 , λ2 , λ3 )=(-,-,0) – mdgradobis zRvruli cikli; ( λ1 , λ2 , λ3 )=(-,0,0) – mdgradi tori; ( λ1 , λ2 , λ3 )=(-,0,+) – `ucnauri” atraqtori. unda aRiniSnos, rom roca

λ > 0 , e.i. aris qaosi,

mezobeli traeqtoriebi ganiSlebian, roca ar aris qaosi – isini ikvrebian.

108

λ < 0 , e.i.

4. sinqronizacia energosistemebSi 4.1 sinqronizaciis problemebi marTvadi sinqronizaciis problemebi imaSi mdgomareobs, rom or an met (xSirad periodul) yj(t) dinamikur process unda hqondes erTnairi ganviTarebis struqtura. kontrolirebadi sistemis sxvadasxva nawilebis qcevis msgavsebis saWiro damTxveva analitikurad gamoxatulia sinqronizaciis pirobebiT, romlebic

SeiZleba

mocemuli

iyos

cxadi

diferen-

cialuri saWiro _ y dj (t ) perioduli amonaxsnebis aR-

y1 = y2 = ... = ym procesiis ganviTarebis identu-

weriT,

robiT an ufro zogadi wrfivi da arawrfivi ho-

Ď&#x2022; ( y1 , y2 ,..., ym ) = 0

lonomiuri duli

orbitebis

miCneulia

kavSirebiT,

analitikuri

aratrevialuri

aseve

aRweriT,

sistemis

perio-

romlebic

atraqtorebad

[34,35,36,37,38,39,40]. marTvadi

sistemis

SemoTavazebuli rxevebis

modelze

sxvadasxva

analizisaTvis.

dayrdnobiT

midgomebi.

isini

Seicaven

iyo

sinqronuli liapunovis

stabilizaciis klasikur meTodebs, gadamkveTi ukukavSirebis arawrfivi

sinqronizirebis meTodebis

koncefcias,

SeTanxmebul

wrfivi

kontrols,

orbitebis da atraqtorebis stabilizacias [41,42,43,44]. 109

sinqronzacias, orbitaluri moZraobis momzadebis saSualebiT, gaaCnia gamokveTilad geometriuli xasiaTi.

dakavSirebulia

meqanikis

ZiriTad

proble-

mebTan. materialuri wertilis moZraoba gluv mrudebze (niutonis, bertrandis da suslovis klasikuri problemebi) aseve Tanamedrove mimarTulebaa arawrfiv marTvaSi [45,46,47]. aq Cven vixilavT lagranJis sistemis oscilaturi qcevis analizs, sinqronizaciis mTavari problema warmodgenilia sistemis gamosavalis cvladebis holonomialuri damikidebulebiT. Sesaferisi diagramebis koordinatebis gamoyeneba yvelaze metad SeTavsebadia sinqronizaciis sistemis mtkicebasTan. moZravi CarCos Tvisebebi, romlebic dakavSirebulia atraqtorTan, iseve, rogorc nawilobrivi

wrfivi

tasTan

sistemis

axlos,

modelis

saSualebas

aproqsimacia

gvaZlevs

orbi-

gamovavlinoT

problemis mTavari geometriuli da dinamikuri specifikuri Tvisebebi da SemovTavazoT arawrfivi marTvis Sesaferisi kanonebi. Cven SeviswavleT gansakuTrebuli SemTxvevebi, dakavSirebuli orkavSiriani lagranJis sistemis organzomilebian rxevebTan da Segegebi warmodgenilia midgomis Tvisebebis gamosavlenad.

110

Cven

vixilavT

sistemas,

dakavSurebuls

Ria

sivrceSi, romelic aRwerilia lagranJis gantolebiT [40,48,49,50]:

A ( y ) y + b ( y , y ) = u sadac

u = {u j }, y = { y j }

marTvis

(4.1) (Sesavali)

gamosavalis cvladebis m-ganzomilebiani veqtorebia,

A = {a jk } invertirebadi matricaa da a jk , b j funqciebi gluvia. (4.1) modeli SeiZleba warmodgenili iqnas, rogorc m _ arawrfivi interaqtiuli SISO qvesistemebis erToblioba. sinqronizaciis martivi problemaa, ganvsazRvroT (4.1) sistemis gaSualebuli moZraobis cvladebi:

1 ∑ yj m

(4.2)

da moviyvanoT m-1 sinqronizaciis pirobebi:

y1 = y2 = ... = ym

(4.3)

sistemuri rxevebis modeli mocemulia gaSualebuli moZraobis modelis saSualebiT:

y + 2ξωy + ω 2 y = 0 sadac

rxevebis

sixSirea

(4.4) ξ

demfirebis

damokidebulebaa -1< ξ <1. maSin, (4.3)-is gaTvaliswinebiT

y j = y (t ) da sistemis yvela cvladi viTardeba (4.4) gantolebis perioduli amoxsnis Sesabamisad.

111

rom

ganvsazRvroT

sinqronuli

reJimisagan

gadaxra, Cven vixilavT Secdomebs:

ε j = y j − y j −1

(4.5)

romlebic unda SevamciroT sistemuri sinqronizaciis procesSi.

maSin

sinqronizaciis

martivi

problema

Semdegnairad Camoyalibdeba: unda moiZebnos iseTi marTva u = U ( y ) , rom yoveli

y ∈ Y -Tvis pirobebis

uzrunvelyofil mxardaWera

(an

iqnas

sinqronizaciis

Secdomebis

asimptoturi

ganuleba) da mocemuli gaSualebuli rxevebi y (t ) (4.4) gantolebidan. mivaqcioT gantolebebi

yuradReba

imas,

ganisazRvreba

rom

m-1

(4.3)

erTganzomilebian

qvesivrceSi S (wrfe) aRWurvili y Sida koordinatiT. problema iRebs aSkara geometriul formas, dakavSirebuls invariantulobis da atraqtivobis aratrivialuri sivrculi obieqtebis koncefciasTan. sinqronizaciis ganzogadebul problemebSi SeiZleba vigulisxmoT is, rom SegviZlia ganvsazRvroT sinqronuli

rxevebis

(1-sixSiriT)

saerTo

reJimis

ufro meti raodenoba

y j cvladebis wrfivi damoki-

debulebis

romlebic

amorCeviT,

iqnebian

eqvivale-

nturi calkeuli rxevebis amplitudebis ranJirebisa.

112

y gaSualebuli moZraobis arawrfivi damokidebuleba

ufro

rTuli

gansazRvra

saSualebas

mogvcems, ganvsazRvroT moTxovnebi arawrfivi rxevebis Tavisufali formebisadmi. Y Ria

simravleSi

ganxilvisas

Cven

(4.1)

sistemis

ganvsazRvravT

moZraobis gaSualebuli

moZraobis cvlads [51,52]:

y = ϕ ( y)

(4.6)

da CavweroT m-1 sinqronizaciis pirobebi veqtorul formaSi:

ϕ sadac

(y ) =

ϕ ε j , j ∈ [1, m − 1 ]

(4.7) _

mudmivi

gluvi

funqciebia. bolo gantoleba Semoxazavs gluv rkals

S ⊂ Rm,

parametruli

Y unda

iyos

Secdomebis

y ∈ Y ⊂ R1

dakavSirebuli veqtori

cvladebiT, Ria

sadac

simravlesTan.

ε = {ε j }( j ∈ [1, m − 1]) miiRebs

Semdeg saxes:

ε = ϕε ( y)

(4.8)

(4.8) dakavSirebulia sisitemis ganiv moZraobasTan. axla Cven SegviZlia ganvixiloT col ( y , ε 1 ,...., ε m −1 ) veqtori, veqtori.

rogorc

(4.1)

Sesaferis

sistemis

axali

koordinatul

gamosavali gardaqmnas,

warmodgenils (4.6), (4.8) gantolebebiT, gaaCnia iakobis matrica: 113

Φ = daSveba

nebismieri

Φ Φε

∂ϕ / ∂y

∂ϕ ε / ∂y

y ∈ S -saTvis, roca

y ∈Y ,

matrica invertirebadia da metic det Φ > δ > 0 . aqedan gamomdinereobs, rom gaxsnili mrudi S erTganzomilebiani gluvi koleqtoria, y ∈ Y lokaluri koordinatiT mrudis garSemo, e.i misi mezoblobiT:

Yε = { y : dist ( y , S ) < ε 0 y ∈ Y } sadac ε 0 > 0

mcire ricxvia. iq arsebobs

y = y( y, ε )

inversiis gluvi asaxva da saTanado iakobis matrica:

J = Φ −1 = J J ε . maSin

marTvadi

sinqronizaciis

saerTo

problema

Semdegnairad JRers: problema1. vipovoT iseTi marTva

u = U ( y) ,

rom nebismieri

y = Y ε -saTvis uzrunvelyofil iqnas (4.7) pirobis sinqronizacia stabiluroba)

ε = 0 amonaxsnis

(an da

mocemuli

asimptoturi

gaSualebuli

rxevebi

y (t ) ∈ Y , gaTvaliswinebuli (4.4)-Si. orbitaluri sinqronizaciis problemebi mdgomareobs SemdegSi: sinqronuli procesebis farTo klasi Seesabameba

moZraobas

periodul

xiloT gluvi rkali Y ∈ R

orbitebze.

ganvi-

Ria dakavSirebul siv-

rceSi, rogorc erTcvladiani koleqtori:

S : φε ( y ) = 0 114

(4.9)

romelic

{S i , ϕ εi }, i = 1,2,....,

aRWurvilia

diagramebiT,

sadac

ϕ ε ,ϕ

εj ,

gluvi

saerTo

regularuli

funqciebia da ∪ S = S . rom warmovadginoT S, rogorc i

Sekruli orbita, Cven virCevT ϕ ε = ϕ ε + (i − 1) s , sadac 0

ϕ ε0 _ gluvia, s 0 _ orbitis periodia, maSin SegviZlia ganvsazRvroT

koleqtoris

diagramis

koordinatebi

s i ∈ S *i ⊂ R 1 ,

sadac

adgilobrivi

∪ S *i = S * _

Ria

sivrcea, dakavSirebuli Semdegi gantolebiT:

s i = ϕεi = ϕε0 + (i − 1) s 0

(4.10)

S rkalis mimarT sistemis qcevis gaanalizebiT, Cven gavarCevT

gaswvriv

moZraobas,

romeluc

aRwerilia

s (t ) = s i (t ) cvladebiT da mkveTi moZraobiT, romelic dakavSirebulia Secdomis

(4.8)

veqtorTan.

gantolebiT (4.9)

piroba

gansazRvrul da

ε (t )

ε (t ) = 0 iden-

turoba ayeneben pirobebs y j (t ) calkeuli perioduli amonaxsnebis koordinirebisaTvis da Seesabamebian y (t ) sistemur traeqtoriebs, romlebic mTlianad ganlagebulia S rkalze. aseTi sistemuri ganviTarebis stabilizacia warmoadgens gaTvaliswinebul marTvis mTavar amocanas proporciuli an cvladi kursiT moZraoba rkalze, romelic xasiaTdeba s cvladiT, adgens procesebis drois masStabs. es SeiZleba gaTvaliswinebuli iyos Sesabamisad s = const , an s = v d (t ) pirobebiT. 115

(4.8) da (4.10) gantolebebi warmoadgenen arawrfiv koordinatul gardaqmnas Φ ( y) = romelic,

rogorc

daSvebas.

maSin

irgvliv

Φs Φε

igulisxmeba,

adgilobriv

romelic,

iakobis matriciT, akmayofilebs

masStabebSi

ganisazRvreba, arsebobs

gluvi

amis

garda,

rkalis rogorc

Yε = { y : dist ( y , S ) < ε 0 , s ∈ S * } , ukuasaxva,

(4.1)

y = y ( s, ε )

rkalebisTvis

SeiZleba moiZebnos S-is adgilobrivi orTogonaluri warmodgena, iseTi, rom Semdeg daSvebas eqneba adgili. daSveba 2. yvela naluria, iakobis

e.i:

y ∈ S -saTvis,

ΦΦ T = I .

matrica

matrica orTogo-

warmovidginoT

nominaluri

Φ * ( s ) = {Φ *jT ( s )} = Φ ( y ( s , 0 )) .

daSvebis gaTvaliswinebiT, Φ j

(4.2)

svetebi gansazRvraven

frenetis moZrav CarCos, dakavSirebuls y = y ( s,0) ∈ S wertilebTan.

Φ*

matricis

dinamika

eqvemdebareba

frenetis magvar gantolebas:

Φ * = s H ( s ) ⋅ Φ * sadac

rkalis

geometriuli

(4.11) invariantebis

damaxinjebuli simetriuli matricaa. amis garda, aseTi gardaqmna (4.8), (4.10) inarCunebs metrikul sivrces da

ε j koordinatebi gansazRvraven S-sagan gadaxras, roca s adgilobrivi koordinata dakavSirebulia S rkalis

116

sigrZesTan. Yε -Si Φ ( y ) matrica gamovTvaleT wrfivi aproqsimaciis gamoyenebiT:

Φ = Φ * ( s ) + ∑ Φ εj ( s )ε

(4.12)

sadac Φ εj = ∂Φ / ∂ε , roca ε = 0 . orbitaluri sinqronizaciis adgilobrivi problema Semdegnairad JRers: problema 2. unda moiZebnos iseTi marTva u = U ( y ) , rom mebismieri

y ∈ Yε -saTvis

uzrunvelyofili

Yε -is

(an

atraqtuloba

axlos

ε =0

S-is

iqnas

amonaxsnis

asimtoturi stabiluroba) CamovayaliboT `sakontrolo~ versia. ganvixiloT K-dinamikuri sistema:

∑ = {T ,U , X , Y , Φ , h } , i = 1,...., K i

(4.13)

sadac T aris drois mudmivebis saerTo jgufi; Ui, Xi, Yi _

Sesasvlelis,

jgufi;

mdgomareobis

Φ i : TxX i xU i → X i

gamosasvlelis

gardamavali

reJimebi,

hi : TxX i xU i → yi _ gamosasvlelebis reJimebi. (aq Cven viyenebT dinamikuri sistemebis erT-erT standartul gansazRvrebas). Tavdapirvelad ganvixiloT SemTxveva, roca yvela Ui aris ubralod erTeulovani, e.i ar gvaqvs

Sesasvlelebi

gamotovebul

Yi _aris

yvela

formirebisas

SeiZleba

mocemulia

i _

g j : Y1 xY2 x...xYk xT → IR 1 , j = 1,..., i .

iqnan.

funqcionalebi:

davuSvaT,

funqciis

rom

jgufi 117

T -dan

Yi -Si,

e.i

Yi = { y : T → Yi } . Semdeg viRebT T-s, rogorc droTa jgufs

aseve

T = IR ≥ 0

(gaxangrZlivebuli

dro)

T = Z ≥ 0 (diskretuli dro). nebismieri romelime J ∈ T sTvis Cven ganvsazRvravT σ τ − 1, rogorc wanacvlebis operators,

σ τ : Yi → Yi

e.i

mocemulia,

rogorc

(σ τ y )(t ) = y (t + Γ) yvela y ∈ Yi -sTvis da yoveli t ∈ T sTvis. Tu Cven Tavs vuyriT yvela amonaxns x1 (⋅),..., x k (⋅) sistemisa ∑ 1 ,..., ∑ k

sawyisi mniSvnelobiT x1 (0),..., x k (0) ,

romelebic sinqronizirebulni arian g 1 ,...g i funqcionalebTan TanafardobiT, maSin

g j (σ Γ1Y1 (⋅),..., σ Γk Yk (⋅), t ) ≡ 0, j = 1,..., i (4.14) gantolebas aqvs mniSvnelobebi yvela t ∈ T -sTvis da J1 ,..., J k ∈ T -sTvis,

zogierTi sistemis

gamosasvlel

Yi (⋅)

sadac

funqcias

aRniSnavs

∑ : y (t ) = h( x (t ), t ), i

t ∈ T , i = 1,..., K . Cven vambobT, rom amonaxsni x1 (⋅);..., x k (⋅) ∑

,..., ∑ k

x1 (0),..., xk (0)

sistemisa daaxloebis

sawyisi

mniSvnelobebiT

sinqronizebulia

g 1 ,..., g k

funqcionalebTan mimarTulebaSi, Tu gvaqvs ε > 0 da

Γ1 ...Γk ∈ T , ase rom

g j (σ Γ1 y1 (⋅),..., σ Tk y k (⋅), t ) ≤ ε , j = 1,..., i yvela

t ∈ T -sTvis. amonaxsnebi

∑ ,...., ∑ 1

sawyisi

x1 (⋅),...., x k (⋅) sistemisa

mniSvnelobebiT 118

(4.15)

asimptoturad

g 1 ,..., g j

sinqronizebul

funqcionalebis

mimarT,

zogierTi Γ1 ,...,Γ k ∈ T -sTvis gvaqvs:

(g j (σ Γ1 y1 (⋅),...,σ Γk y k (⋅), t )) = 0 , j = 1,.., i lim t →∞

(4.16)

Tu sinqronizaciis movlena miRweulia yvela sawyisi mniSvnelobebisaTvis x1 (0),..., x k (0) , maSin SesaZlebelia vTqvaT,

rom

(mocemul

sistemebi

∑ ,...., ∑ k

sinqronizebadia

funqcionalebTan

mimarTebaSi

Sesabamisi

gagebiT). asimptoturi sinqronizaciis SemTxvevaSi SesaZlebelia

ganisazRvros

iseTi

sawyisi

mniSvnelobebi,

romelsac mivyavarT sinqronizaciamde. SemdgomSi Cven ganvixilavT mxolod SemTxvevas, sadac sinqronizacia miRweulia mxolod yvela sawyisi mniSvnelobisaTvis. miuxedavad imisa, rom es gansazRvreba zogadia, is SeiZleba ganzogadebul iqnas SemdgomSi. magaliTad, bevr

praqtikul

problemebSi

droiTi

Zvrebi

Γi , i = 1,..., k ar aris mudmivi, magram miiswrafian mudmivi mniSvnelobebisken, e.w. `asimptoturi fazebisaken~. am SemTxvevaSi Zvris operatoris magier yvela gamosasvleli

funqciisTvis

ganvixiloT

droSi

y 2 (⋅) , daSladi

ufro Zvris

xelsayrelia operatori,

romelic gansazRvrulia Semdegi saxiT:

(σ ) y(t ) = y(t (t )) '

Γi

119

(4.17)

t i' : T → T , i = 1,..., k

sadac

(xangrZlivi

funqciebi,

aris

romelTac

homomorfizmebi aqvT

xangrZlivi

inversiebi), ase, rom

lim t→ ∞

(t (t ) − '

)= ε

(4.18)

SevniSnoT, rom funqcionalebis jgufis nacvlad yovelTvis SesaZlebelia aviRoT erTi funqcionali, romelic asaxavs sinqronizaciis aseTive gamovlinebas. magaliTad, SeiZleba avirCioT funqcionali G, saidanac: G ( y 1 (⋅ ),..., y k (⋅ ), t ) = ∑ g l

j =1

sinqronizaciis moTxovna,

rom

2 j

( y (⋅ ),...,

gamoyenebisas

(4.14)

y k (⋅ ), t )

(4.16)

(4.19)

aucilebelia

pirobebi

iqnas

darRveuli (an darRveuli iqnas umniSvnelod), roca sistemis

zogierTi parametri

gansxvavdeba

zogierT

sferoSi. sxva sityvebiT (4.14) _ (4.16) Tviseba unda iyos myari, magram am SemTxvevaSi fazuri Zvrebi SeiZleba iyos

mudmivi

arc

miiswrafodes

mudmivi

mniSvnelobisaken, Tumca, Semdegi piroba: lim

t → ∞

t i' ( t ) − t

≤ r

(4.20)

SeiZleba warmodgenili iyos (4.18)-is nacvlad. bevr SemTxvevaSi U i , X i , Yi

_ jgufebi _ sivrcis

veqtorebia da ∑ i sistemebi SeiZleba aRwerili iqnas ubralo diferencialuri gantolebebiT.

120

4.2 sinqronuli generatorebis paralelur muSaobaSi avtomaturi CarTva generatoris aucilebelia, CarTvis

qselSi

rom

warmatebiT

mawonasworebeli

momentSi

CarTvisaTvis denis

aRematebodes

biZgi

dasaSveb

mniSvnelobas, xolo CasarTavi generatoris rotori CaerTos sinqronulad xangrZlivi rxevis gareSe. am pirobebis

Sesasruleblad

aucilebelia

winaswar

davareguliroT generatoris brunvis sixSire ise, rom igi

axlos

iyos

sinqronulTan,

gamosasvlelebze

(Tu

xolo

generatori

Zabva

mis

aRgznebulia)

mivuaxlovoT an gavutoloT energosistemis Zabvas da SevarCioT CarTva-gamorTvis brZanebis micemis momenti. brunvis sixSirisa da Zabvis gaTanabrebis am procesis da

generatoris

ewodeba

qselSi

sinqronizacia.

CarTvis

momentis

eqspluataciaSi

SerCevas

gamoiyeneba

energosistemasTan generatoris paraleluri muSaobaSi CarTvis ori ZiriTadi meTodi: zusti sinqronizacia da TviTsinqronizacia. zusti generatori

sinqronizaciiT brundeba

CarTvis

meTodisas

sinqronulTan

miaxloebul

sixSiremde da aRegzneba. Semdeg xeliT an avtomatikis saSualebiT Tanabrdeba sinqronizebuli generatorisa da qselis sixSireebi da Zabva. amis Semdeg miecema generatoris qselSi CarTvis brZaneba. imisaTvis, rom 121

CarTvis

momentis

aRematebodes

gawonasworebuli

dasaSveb

sidides,

denis

xolo

biZgi

generatoris

rotoris rxevebi swrafad Caqres, aucilebelia Zalian zustad

iqnas

qselis

gawonasworebuli

sixSireebi

Zabva.

generatorisa

aucilebelia

SeirCes

generatoris CarTvisa da gamorTvis Sesabamisi momenti. TviTsinqronizaciisas

generatori

brundeba

sin-

qronulTan miaxloebul sixSiremde da qselSi irTveba aRugzneblad. aRgznebis deni miewodeba rotoris xviebs CamrTvelis CarTvisTanave. e.m.Z.-isa

denis

zrda,

Semdeg xdeba rotoris genertori

irTveba

sinqronizmSi. sinqronizaciis ZiriTad Rirsebad iTvleba sinqronizaciis pricesis daCqareba da misi SedarebiT ubraloeba,

ris

gamoc

avtomatizirebuli. gansakuTrebiT

igi

SeiZleba

advilad

TviTsinqronizaciis

mniSvnelovania

iqnes

upiratesoba

avariul

pirobebSi,

energosistemebSi sixSirisa da Zabvis mniSvnelovani meryeobis dros. TviTsinqronizaciis meTodis uaryofiT Tvisebad SeiZleba CaiTvalos CarTvis momentSi denis

SedarebiT

didi

biZgebi,

ris

gamoc

iwveba

CamrTvelis kontaqtebi da saWiro xdeba generatoris da transformatorebis mwyobrSi moyvanisTvis garkveuli damatebiTi Zalisxmeva. zusti

sinqronizaciis

upiratesoba

mdgomareobs

imaSi, rom generatoris CarTvas, rogorc wesi Tan ar 122

axlavs denis didi biZgebi da xangrZlivi rxevebi. amasTanave zusti sinqronizaciisadmi wayenebuli mkacri moTxovnebi mas xdian SedarebiT rTul da xangrZliv operaciad. gansakuTrebiT es exeba avariul pirobas,

roca

soxSirisa

Zabvis

sxvadasxva

meryeobisas praqtikulad SeuZlebeli xdeba sinqronizebuli

generatorisa

qselis

sixSireebisa

Zabvis gawonasworeba. generatorebis

sinqronizacia

Zalian

mniSvnelo-

vani operaciaa, romelic morige persinalisagan moiTxovs muSaobis did gamocdilebas. am operaciis avtomatizacia amsubuqebs operatiuli personalis SromiT pirobebs da saSualebas iZleva, raTa swrafad iqnes CarTuli generatori qselSi, rac avariul pirobebSi Zalze mniSvnelovania. ansxvaveben xelis, avtomatiur da naxevrad avtomatur sinqronizacias. Sesabamisad amisa avtomatikis mowyobilobebi iyofian avtomatikur da naxevrad avtomaturebad. avtomaturi sinqronizacias generatoris qselSi CarTvis mTeli procesi sruldeba avtomaturad, morige personalis Carevis gareSe. ase magaliTad: avtomaturi zusti sinqronizatori axorcielebs sinqronizebuli generatoris brunvis sixSirisa da Zabvis regulirebas, akontrolebs CarTvisaTvis sixSirisa da Zabvis dasaSveb sxvaobas, iZleva impulss CarTvaze, im 123

momentSi

roca

sruldeba

zusti

sinqronizaciis

piroba. naxevradavtomatur

sinqronizaciisas

avtomaturi

mowyobilobebi asruleben damxmare rols, isini exmarebian morige Si.

ase

personals generatoris sinqronizacia-

magaliTad:

TviTsinqronizaciis

naxevradav-

tomaturi mowyobiloba akontrolebs sixSireTa sxvaobas da iZleva CarTvis impulss, roca igi gaxdeba dasaSvebi CarTvisaTvis. sinqronizebuli generatoris brunvis sixSiris regulireba am dros ekisreba morige personals.

rig

SemTxvevaSi

zusti

sinqronizaciis

naxevradavtomaturi mowyobilobebi gamoiyeneba blokirebis saxiT, sinqronizaciisaTvis saWiro pirobebSi generatoris CarTvis nebarTviT. eqspluataciaSi

gamoiyeneba

ori

tipis

sinqro-

nizatorebi: waswrebis mudmivi kuTxiT da waswrebis mudmivi droiT. Tu miviRebT, rom CamrTvelis CarTvisas mudmivi rCeba ara arakuTxuri sixSiris curva, aramed kuTxuri aCqareba,

maSin

qronizatoris

winswrebis amoqmedeba

kuTxis xdeba

SerCeva

momentSi,

sinroca

sruldeba Semdegi toloba:

dδ d 2δ t 2 δ + δ on = δ + t b + 2 = 2π dt 2 dt sadac

(4.21)

t B - winswrebis droa da tolia CamrTvelis

CarTvis droisa. 124

Uδ

gamomdinare

Soris

wrfivi

sinqronizatoris

damikidebulebidan

amoqmedeba

Rebulobs

Semdeg saxes:

dU δ d 2U δ t B2 Uδ + tB + = U 2π , dt dt 2 2

(4.22)

sadac U 2π aris U δ -is mniSvneloba 2π kuTxisas. daCqarebis kuTxuri

as =

gaTvaliswinebiT

sixSire

CamrTvelis

dws dt

kontaqtebis

curvis mokle

CarTvis momentSi ω s ,b = ω s ,c + a s t B , sadac ω s ,b _ curvis kuTxuri

sixSirea

momentSi.

Sesabamisad

sinqronizatoris

amoqmedebis

winswrebis

gamoisaxeba

kuTxe

Semdegnairad:

t B2 δ on = ω s ,c t B + a s 2 sinqronizatorSi

kontroldeba

(4.23) maqsimalurad

dasaSvebi curva ω s ,max,D CamrTvelis kontraqtebis mokle CarTvis momentSi. roca ω s ,b > ω s ,max, D , maSin moqmedebis akrZalvisTvis, curvis mniSvnelobis makontrolebeli elementisaTvis daismis piroba:

δ oπ ,max,D = ωs ,max,Dt B ,

(4.24)

romelSic sinqronizators avtomaturad Seaqvs Sesworeba da am SemTxvevaSi winswrebis dasaSvebi kuTxe gamoisaxeba Semdegnairad: 125

δ oπ ,max,D

t b2 t b2 = (ω s ,c + a s t B )max,D t B − a s = ω s ,max,d t B − a s 2 2

0 SerCeva δ on > 120 avtomaturad ikrZaleba.

4.3 avariuli situaciebis analizi

4.3.1 asinqronuli reJimebis likvidacia. resinqronizacia energosistemaSi avariuli situaciebis analizi da avariasawinaaRmdego RonisZiebebis SerCeva (zemoqmedebis meTodi) muSaobis normaluri reJimis aRsadgenad, win uswrebs teqnikur saSualebaTa damuSavebas. aseTi analizis safuZvels Seadgens energosistemis paraleluri muSaobis mdgomareobis gaangariSeba, romelic sruldeba

gardamaval

procesebze

zemoqmedebis

gaTvaliswinebiT. energosistemis moZraobis gantoleba, Tu ugulebelvyofT xazebSi danakargebs da TviTregulirebis Taviseburebebs (generatoris asinqronuli momentebiT da

datvirTvis

regulirebis

Caiweros Semegi saxiT:

126

efeqtiT),

SeiZleba

T j ,∋k d 2δ 1, 2 + P1, 2 S inδ 1, 2 = ΔP Whom dt 2

sadac

δ1,2 _ e.m.Z-ebis,

(4.25)

E1 -is da E2 -is, generato-

rebis fazebs Soris Zvris kuTxea, ΔP _ warmoqmnili deficiti

simZlavris

danakargia,

T j ,∋ k _

energo-

sistemaSi mbrunavi masebis inerciis drois eqvivalenturi mudmivaa.

T j ,∋k = T j ,1T j , 2 / T j ,1 + T j , 2 aq

T j ,1 , T j , 2

eqvivalenturi

(4.26)

operatorebis

inerciis

droiTi mudmivebia.

ΔP -s

mniSvneloba

SeiZleba

ganvsazRvroT

Semdegnairad:

ΔP = sadac

PT ,1 − PH ,1 P − PH , 2 Tj,2 − T ,2 T j ,1 T j ,1 + T j , 2 T j ,1 + T j , 2

PT ,1 , PT , 2 _

pirveli

meore

(4.27) eleqtro-

sadguris turbinebis sinZlavreebia; PH ,1 , PH , 2 _ eleqtrosadgurebis eleqtruli simZlavris datvirTvebia. (4.25) da

(4.27)

nalizoT

gantolebebis ufro

metad

daxmarebiT

SeiZleba

damaxasiaTebeli

gavaa-

avariuli

situaciebi: gadamcem nawilebSi simZlavris danakargi, mimReb nawilSi simZlavris deficiti da gadacemis Sesusteba.

127

energosistemebis pirvel

etapze

ganviTarebasa

asinqronuli

gaerTianebis

reJimebis

likvidacia

xorcieldeboda operatiuli personalis moqmedebebiT. SeiniSneboda SemTxvevebi

sinqronizmis dinamiuri

TviTneburi

mdgradobis

aRdgenis

darRvevis

dros,

magram ar arsebobda asinqronuli reJimebis daSvebis kriteriumebi da es reJimebi ganixileboda, rogorc mavneni,

ris

gamoc

eqvemdebarebodnen

swraf

likvi-

dacias. Semdgom daiwyo avariis likvidaciasTan dakavSirebuli sareJimo moTxovnebis gadasinjva, rac gamowveuli iyo mdgradebis darRveviT. formulirebuli iqna

asinqronuli

qronuli zRvroT

reJimebis

CarTvis

dasaSvebi

resinqronizaciis

mokle

drois

kriteriumebi.

piroba

sin-

ganvsa-

asinqronuli

reJimis procesis dros sinqronizmis aRdgenas ewodeba resinqronizacia.

resinqronizaciis

pirobas

aqvs

Semdegi saxe:

â&#x2030;¤ S

cp , kp

sadac _ S cp _ damyarebuli saSualo curvaa

S cp ,kp _ damyarebuli saSualo curvis kritikuli mniSvnelobaa, romlis drosac resinqronizacia mimdinareobs sinqronuli momentis gavlenis qveS, rac iwvevs

mrvaljerad

curvis

ryevas

lobasTan axlos.

128

saSualo

mniSvne-

S cp ,kp ricxobrivi SefasebisaTvis Zlier da susti kavSirebisas Sesabamisad miviRoT: P12=1 da P12=0,05

P12

Scp ,kp = 0, 0565

τ j∋k

(4.28)

τ j∋k

τ j1τ j 2 = τ j1 + τ j 2

τ j1 ,τ j 2 _ meqanikuri inerciis mudmivebia; τ

j∋k

_ or-

manqanuri sistemis meqanikuri inerciis eqvivalenturi mumivaa;

P12

asinqronulad

momuSave

nawilebis

urTierT simZlavre fardobiT erTeulebSi, romelic gansazRvravs

simZlavris

ryevis

maqsimalur

ampli-

tudas asinqronuli reJimis xazze.

4.3.2 asinqronuli reJimis niSnebi paralelur

muSaobaze

CarTuli

generatorebi

normalur reJimSi muSaoben sinqronulad. sinqronuli reJimisaTvis generators

damaxasiaTebelia gaaCnia

erTnairi

is,

rom

yvela

aqedan

sixSire

gamomdinare, maTi veqtorebi brunaven erTnairi kuTxuri

siCqariT.

asinqronuli

reJimi

Tavs

iCens

dairRveva generatorebis an energosistemis paraleluri muSaobis mdgradoba. amis garda, asinqronuli 129

reJimi SeiZleba warmoiqmnas im xazebis arasinqronuli CarTvis dros, romlebic aerTeben eleqtrosadgurs energosistemasTan _ asinqronuli reJims Tan sdevs qvemoT warmodgenili movlenebis da niSnebis ganxilva: 1.

asinqronul

e.m.Z.-ebs

Soris

kuTxis

perioduli

cvalebadoba nulidan 3600-mde sixSiris curviT (curva ewodeba brunvis sixSireTa sxvaobas an eleqtruli sixSireTa sxvaobas)

ωs = ωr − ωc , fs = fr − fc , sadac

ωs

sixSirea; ωr

curvis

(4.29) kuTxuri

siCqare

f r _ generatoris an eleqtrosad-

guris kuTxuri siCqare da sixSirea;

energosistemis

kuTxuri

siCqare

sixSirea. 2.

eleqtrogadacemis

yvela

wertilSi

perioduli

sixSiruli curva Zabvis cvlilebiT. 3.

sixSiruli

(rxeva)

yvela

curvis

denis

elementSi,

perioduli

cvlileba

romelic

akavSirebs

asinqronul manqanebs. zemoT

ganxilulidan

Cans,

rom

asinqronuli

reJimebs Tan axlavs Zabvis didi Semcireba, rxevis didi denebis gadineba da aqtiuri simZlavris ryevebi. yvelaferi es warmoadgens normalur reJimSi muSaobas 130

seriozuli darRvevas, saxifaToa mowyobilobebisaTvis da eleqtroenergiis momxmarebelTaTvis. amitom asinqronuli reJimi unda iyos SezRuduli 2-3 ciklamde. asinqronuli reJimis xangrZlivobis dasaSvebi zRvari warmoadgens 15-30 wm-s. am dros ganmavlobaSi unda iqnes miRebuli zomebi sinqronizmis aRsadgenad. arsebobs asinqronuli reJimis likvidaciis ori meTodi: resinqronizacia da energosistemis dayofa. es operaciebi rogorc wesi xorcieldeba, avtomaturad, asinqronuli reJimis likvidaciis avariasawinaaRmdego mowyobilobebis daxmarebiT, xolo iSviaTad (mowyobilobis mwyobridan gamosvlisas) _ operaciuli personalis saSualebiT.

4.4 sinqronizirebadi marTva (4.1) Jis

problemis gadawyvetisaTvis, Sekruli maryu-

marTva

unda

amovirCioT

ise,

rom

mivaRwioT

sasurveli sistemuri sivrcis ganviTrebas, romelic aRwerilia (4.4), (4.7) gantolebebiT. procedura iwyeba (4.1)

modelis

gantolebebis

Cveulebrivi

garadqmniT. (4.6) da (4.8)

diferencirebiT

asaxvebs: 131

Cven

viRebT

Semdeg

y = Φy ε

(4.30)

y y = Φ = Γ y + Γε ε ε −1

axal modeli

koordinatul advilad

(4.31)

diagramaSi

Sedgeba

(4.13)-is

maTematikuri diferencirebiT

drois mimarT da Secvlis Sedegad (4.1), (4.31) Semdeg formas miiRebs:

y y = ΦΓ + ΦA −1 (u − b) ε ε

(4.32)

samuSao proedura Seicavs `sivrcis gawrfivebas~:

u = b + Au~

(4.33)

sadac u~ = {u j }, j ∈ [1, m] _ axali Sesasvleli veqtoria, romelic moiZebneba Semdeg formaSi:

u~ = u~ ( yε )

(4.34)

maSin (4.32) Semdegnairad gadaiwereba:

y y − ΦΓ = Φu~ ε ε ganvixiloT

mdgradi

sinqronizaciis

(4.35) moZraoba,

romelic iRebs ε ≡ 0 da u~0 ( y ) = u~ ( y ,0) mniSvnelobebs. (4.35) Semdegnairad gadaiwereba:

(

)

u~0 = Γ − 2ξωΓ y − ω 2 Γ y

132

(4.36)

rom

gavaanalizoT

Cven

vixilavT

sinqronizaciis

mkveTi

moZraobis

procesebi

models,

Yε -Si,

miRebuls

(4.35) gantolebidan:

Tε Γε ε = Φ Tε Γ y + Φ Tε uˆ ε − Φ

(4.37)

amovirCioT

u~ = u~0 ( y ) − Γε ( K 1ε + K 2 ε )

(4.38)

sadac K1, K2 _ ukukavSiris gadidebis matricebia. problemis sistemaSi

gadawyvetisaTvis,

axali

mdgomareobis

moZraobis ( s, s ) da mkveTi

(ε , ε )

SemovitanoT

cvladebi,

(4.1)

gaswvrivi

cvladebi, romlebic

Semdegnairad gamoiTvlebian:

ϕ si ( y ) s = , = Φy ε ϕε ( y) s s

daSveba

(4.2)-is

gaTvaliswinebiT,

(4.39)

y = y ( s , e)

ukuasaxvis diferenciali iqneba:

y = Φ T

s s

(4.40)

sistemuri modeli axal koordinatul diagaramaSi Seqmnilia (4.39)-is diferencirebiT, Sesabamisi cvlilebebiT

(4.11),

(4.12)

gantolebebis

gamoyenebiT.

Yε

mcire midamoSi, is miiRebs Semdeg formas:

− s H

s − a1ε − a 2 ε = ΦA −1 (u − b) ε

133

(4.41)

Seqmnis

procedura

Seicavs

mdgomareobis

sircis

gawrfivebas (4.33) da amocaniT orientirebuli gancalkevebisaTvis

sistemur

models,

u~j

Sesasvlelebis

gaTvaliswinebiT, wrfivi marTvis cvladis gardaqmna miiReba Semdegnairad:

u = ΦT

sadac

gaswvrivi

ganzomilebis

mkveTi

us uε

(4.42)

marTvaa

marTvis

veqtoria.

uε

m-1

maSin

(4.41)

gantoleba miiRebs Semdeg saxes:

u s s − s H − a1ε − a 2 ε = ( I + Δε ) s uε ε ε romelic

gansazRvravs

gaswvrivi

mkveTi

sustad dinamikis

(4.43) urTierTmoqmedi

modelebs.

isini

SeiZleba Semdegnairad gadavweroT:

T − a11 )ε − a12ε = us + (Δ1ε )uε s − ( sH s − ( s H + a 21 )ε − a 22 ε = u ε − s 2 h( s ) + ( Δ 2 ε )u s

sadac

−h

= H _ mkveTi marTvaa.

uε = s 2 h( s ) − K 1ε − K 2 ε sadac

K1 , K2

gvaZlevs

(4.44)

ukukavSiris

miviRoT

TiTqmis

matricebi avtonomiuri

(4.45) saSualebas Secdomis

modeli:

− a21 )ε + ( K 2 − a22 )ε = 0 ε + ( K1 − sH 134

(4.46)

am modelis matricebis ukukavSiris aucilebeli SezRudvebis amorCeva gvaZlevs asimptoturi stabilurobis

garantias

orbitaluri

amis

stabiluroba,

Sedegad (4.9)

(4.1)

sistemis

atraqtoris

mimarT

gamoisaxeba Semdegnairad:

u s = v d + k s (vd − s ) sadac

(4.47)

k s _ ukukavSiris saTanado amorCeva uzrun-

velyofs vd − s normis Secdomis asimptotur ganulebas da sasurveli rxevebis gadidebas. Sedegad marTvis sistema Seicavs (4.33) gawrfivebis algoriTms,

(4.42)

marTvis

cvladis

gardaqmnas

nawilobrivad (4.45) da (4.47) marTvis kanonebs. SemoTavazebuli

midgomis

warmosadgenad

ganvi-

xiloT orkavSiriani rxeviTi sistemis sinqronizacia:

y + Wy = u

(4.48)

sadac dim y = 2 , W _ diagonaluri matricaa da (4.33) gantoleba miiRebs Semdegi saxis formas:

u = Wy + u~

(4.49)

ganvixiloT zemoT ganxiluli modelebis kerZo SemTxvevebi: 1.

wrfivi

sinqronizacia.

sinqronizaciis

piroba

mdgomareobs SemdegSi:

y1 − 2 ⋅ y 2 = 0 (4.50)

emTxveva

wrfis

gantolebas

cvladi amorCeulia Semdegi formiT: 135

(4.50) da

gaSualebuli

b) nax. 26 wrfivi sinqronizacia

136

y = y1 + y 2

(4.51)

gaSualebuli moZraobis dinamika mocemulia (4.4) gantolebiT, sadac ξ = 0 , ω = 2 . maryuJis wrfivi sinqronizaciis gamoyenebisas iyo gamomuSavebuli mniSvnelovani marTvis mowyobiloba. wrfis

gaswvriv

sistemis

moZraobis

traeqtoriebi,

dawyebuli sxvadasxva Sida wertilebidan naCvenebia nax. 26-ze. 3. arawrfivi sinqronizacia. sinqronizaciis piroba mocemulia

kuburi

y 2 − 0.1 ⋅ y13 = 0 warmodgenilia,

parabolis

gantolebiT

gaSualebuli

rogorc

y = y1 .

rxevebi milevadia, roca ξ = −0.39 .

a) 137

cvladi sasurveli

g) nax. 27 arawrfivi sinqronizacia

138

rogorc pirvel SemTxvevaSi, (4.36), (4.38) arawrfivi sinqronizirebadi gvaZlevs

maryuJebis

CamovayaliboT

gamoyeneba,

saSualebas

mniSvnelovani

marTvis

kanonebi. sxvadasxva

sawyisi

wertilebidan

dawyebuli,

parabolis gaswvriv sistemis moZraobis traeqtoriebi da

sistemis

gamosasvlelebis

droSi

amoqmedeba

mocemulia nax. 27-ze. 3.

sinqronizaci

sinqronizaciis

aelifsuri piroba

orbitis

mimarT.

mocemulia

elifsis

gantolebiT.

(0.87 y

+ 0.5 y 2 ) (0.5 y1 − 0.87 y 2 ) 1 + −1 = 0 16 4 2

(4.52)

gaswvrivi cvladi s warmodgenilia (4.10) gantolebiT, sadac s = 17.89 da ϕ 0

_ gamoTvlilia y1 / y 2 da y 2 / y1

invertuli trigonometriuli funqciebis saSualebiT. proporciuli

gaswvrivi

moZraoba

miRebulia

vd = 1

normirebuli sixSiris amorCeviT. marTvis kanoni iyo gamomuSavebuli (4.33), (4.42), (4.45),

(4.47)

orbitaze sistemis

gantolebebis sistemis

saSualebiT.

moZraobis

gamosasvlelebis

naCvenebia nax. 28-ze.

139

elifsur

traeqtoriebi

droze

damokidebuleba

140

g) nax. 28 siqronizacia elifsuri orbitis mimarT

sinqronizacia

spiralis

mimarT.

sinqronizaciis

piroba mocemulia gantolebiT:

exp s − y12 − y 22 = 0

(4.53)

gaswvrivi cvladi s warmodgenilia (4.10) gantolebiT. sasurveli faza da procesis drois skala mocemulia

vd = −1

normirebuli

sixSiriT.

sistemis

moZraobis traeqtoriebi naCvenebia nax. 29-ze.

141

spiralze

142

g) nax. 29 sinqronizacia spiralis mimarT

modelireba aCvenebs sistemis traeqtoriebis krebadobas sasurvel atraqtorebSi, gaswvrivi moZraobis mocemuli reJimebisaTvis, amitom sxvadasxva sinqronizaciis amocanebs savsebiT asrulebs.

143

danarTi zRvruli cikli Private Sub Command1_Click( ) Y = InputBox("ENTER R=") Scale (0, 0)-(310, 200) Line (0, 0)-(309, 189), , B Print "1"; Print "0"; Print USING; "#. ##"; Y; Print "4"; For J = 0 To 307 X = 0.3: K = Y + (4 - Y) * J / 307 For I = 1 To 200 X = K * X * (1 - X) Next For I = 1 To 300 X = K * X * (1 - X) PSet (J, 187 * X), vbBlue Next Next End Sub

144

mdgradi kvanZi Private Sub Command1_Click( ) Dim S1(1 To 20), DD1(1 To 20), S2(1 To 20), DD2(1 To 20) Scale (0, 0)-(600, 400) Line (0, 0)-(640, 350), vbBlack, BF Line (120, 175)-(520, 175), vbRed Line (320, 50)-(320, 300), vbRed E1 = 200: E2 = 300 F = 0.01: G = 2000: DZ = 0.009 C1 = 1: C2 = 1 T1 = 0: T2 = 2: DT = 1 k = (E1 * E2 * F) / G L = Abs(Sqr(k * Cos(DZ))) L1 = -L: L2 = -L For i = 1 To 20 For C1 = -5 To 5 For C2 = -5 To 5 S1(i) = C2 * (1 / Exp(L1 * T1)) DD1(i) = C1 * (1 / Exp(L2 * T1)) + C2 * T1 * (1 / Exp(L2 * T1)) x = DD1(i) * 10 y = S1(i) * 10 PSet (320 + x, 175 - y), vbYellow For t = T1 + DT To T2 Step DT S1(i) = C2 * (1 / Exp(L1 * t)) DD1(i) = C1 * (1 / Exp(L2 * t)) + C2 * t * (1 / Exp(L2 * t)) x = DD1(i) * 10 y = S1(i) * 10 Line -(320 + x, 175 - y), vbYellow Next t Next C2 Next C1 Next i End Sub

145

aramdgradi mdgomareoba Private Sub Command1_Click( ) Dim S1(1 To 20), DD1(1 To 20), S2(1 To 20), DD2(1 To 20) Scale (0, 0)-(800, 500) Line (0, 0)-(640, 350), vbWhite, BF Line (120, 175)-(520, 175), vbBlack Line (320, 50)-(320, 300), vbBlack T1 = 0: T2 = 2: DT = 1 L = Abs(Sqr(k * Cos(DZ))) For i = 1 To 20 For C1 = -5 To 5 For C2 = -5 To 5 S1(i) = C2 DD1(i) = C1 + C2 * T1 x = DD1(i) * 10 y = S1(i) * 10 PSet (320 + x, 175 - y), vbBlue For t = T1 + DT To T2 Step DT S1(t) = C2 DD1(t) = C1 + C2 * t x = DD1(t) * 10 y = S1(t) * 10 Line -(320 + x, 175 - y), vbRed Next t Next C2 Next C1 Next i End Sub

146

neitralurad mdgradi mdgomareoba Private Sub Command1_Click() Dim S1(1 To 20), DD1(1 To 20), S2(1 To 20), DD2(1 To 20) Scale (0, 0)-(600, 400) Line (0, 0)-(640, 350), vbBlack, BF Line (120, 175)-(520, 175), vbRed Line (320, 50)-(320, 300), vbRed E1 = 200: E2 = 300 F = 0.01: G = 2000: DZ = 0.009 C1 = 1: C2 = 1 T1 = 0: T2 = 20: DT = 1 k = (E1 * E2 * F) / G L = Abs(Sqr(k * Cos(DZ))) PI = 3.14 T1 = -2 * PI: T2 = 2 * PI: DT = 1 Line (0, 0)-(640, 350), vbBlack, BF Line (80, 175)-(560, 175), vbWhite Line (320, 50)-(320, 300), vbWhite L1 = L: L2 = -L For i = 1 To 20 For C1 = -5 To 5 For C2 = -5 To 5 S1(i) = C2 * Cos(L1 * T1) DD1(i) = -C2 * Sin(L1 * T1) x = DD1(i) * 10 y = S1(i) * 10 PSet (320 + x, 175 - y), vbRed For t = T1 + DT To T2 Step DT S1(i) = C2 * Cos(L1 * t) DD1(i) = -C2 * Sin(L1 * t) x = DD1(i) * 10 y = S1(i) * 10 Line -(320 + x, 175 - y), vbRed Next t Next C2 Next C1 Next i End Sub 147

gadagvarebuli kvanZi anu aramdgradi wonasworoba Private Sub Command1_Click( ) Dim S1(1 To 20), DD1(1 To 20), S2(1 To 20), DD2(1 To 20) Scale (0, 0)-(600, 400) Line (0, 0)-(640, 350), vbBlack, BF Line (120, 175)-(520, 175), vbYellow Line (320, 50)-(320, 300), vbYellow E1 = 200: E2 = 300 F = 0.01: G = 2000: DZ = 0.009 C1 = 1: C2 = 1 T1 = 0: T2 = 20: DT = 1 k = (E1 * E2 * F) / G L = Abs(Sqr(k * Cos(DZ))) L1 = L: L2 = L For i = 1 To 20 For C1 = -20 To 55 Step 10 For C2 = -40 To 40 Step 10 S1(i) = C2 * (1 / Exp(L1 * T1)) DD1(i) = C1 * (1 / Exp(L2 * T1)) + C2 * T1 * (1 / Exp(L2 * T1)) x = DD1(i) y = S1(i) PSet (320 + x, 175 - y), vbRed For t = T1 + DT To T2 Step DT S1(i) = C2 * (1 / Exp(L1 * t)) DD1(i) = C1 * (1 / Exp(L2 * t)) + C2 * t * (1 / Exp(L2 * t)) x = DD1(i) y = S1(i) Line -(320 + x, 175 - y), vbRed Next t Next C2 Next C1 Next i End Sub

148

qaosuri atraqtorebi function qazl(action) if nargin<l, action='start'; end; global inertia_field global freq_field global eds_field global volt_field global conductivity_field global eds global volt global conductivity global freq global inertia if strcmp(action,'start'), elf reset; set(gcf,'Units','normalized','backingstore','off); %popup=uicontrol('Style','PopupVString','cosine|square|sawtooth|l 111|2222|3333|4444|',... % 'Position',[.03 .02 .2 .08],'Units','normalized',... % 'CallBack','qazl("redraw")'); inertia_text=uicontrol('Style','text7Position',[.03 .02 .13 .04],... 'Units', 'normalized','BackgroundColor','black',... 'ForegroundColor', 'white',' String', 'Inertia:',... 'HorizontalAlignment','right'); inertia_field=uicontrol('StyleVedit','Position',[.16 .02 .07 .04],... 'Units','normalized','BackgroundColor','blue','ForegroundColor','white','Str ing','1500',. 'CallBack','qazl("setinertia")'); version_text=uicontrol('StyleVtext','Position',[ 0 .96 .16 .04],... 'Units','normalized','BackgroundColor','black',... 'ForegroundColor','white','String','Version 1.0',... 'HorizontalAlignment','left'); freq_text=uicontrol('Style7text7Positiori,[.27 .02 .28 .04],... 'Units','normalized','BackgroundColor','black',.... 'ForegroundColorVwhite','StringVAngle(Gr):',... 'HorizontalAlignment','right'); freq_field=uicontrol('StyleVedit','Position',[.57 .02 .07 .04],... 'Units', 'normalized','B ackgroundColor','blue', 'ForegroundColor', 'white',' String',' 1.1',... 'CallBack','qazl("setfreq")'); eds_text=uicontrol('Style','text','Position',[.03 .88 .10 .04],... 'Units','normalized','BackgroundColor','black',... 'ForegroundColor', 'white','String','EMF(E):',... 'Horizontal Al ignment', 'right'); eds_field=uicontrol('Style','edit','Position',[.13 .88 .07 .04],... 'Units','normalized','BackgroundColor','blue','ForegroundColor','white','Str ing','300',... 'CallBack','qazl("seteds")'); 149

volt_text=uicontrol('Style','text','Position',[.22 .88 .28 .04],... 'Units','normalized','BackgroundColor','black',... 'ForegroundColor', 'white', 'String', 'Tension(Volt):',... 'HorizontalAlignmentVright'); volt_field=uicontrol('Style','edit','Position',[.51 .88 .07 .04],... , 'Units', normalized','BackgroundColor','blue','ForegroundColor','white','Str ing','220',... 'CallBack','qazl("setvolt")'); conductivity_text=uicontrol('Style';'text','Position',[.65 .88 .20 .04],... 'Units','normalized','BackgroundColor','black',... 'ForegroundColor','white','String','Conductivity(R):',... 'HorizontalAlignmentVright'); conductivity_field=uicontrol('Style','edit','Position',[.86 .88 .07 .04],... 'Units','normalized','BackgroundColor','blue','ForegroundColor','white','Stri ng','0.05',... 'CallBack','qazl("setconductivity")'); donejDutton=uicontrol('Style','Pushbutton','Position',[.85 .02... . 12 .08],'Units','normalized','Callback',... 'qazl("done")','StringVClose'); grid_button=uicontrol('StyleVPushbutton','Position',[.3 .02... .08 .06],'Units','norrnalized','Callback',... 'grid','String','Grid'); orbitjMton=uicontrol('Style','Pushbutton','Position',[.75 .02... .07 .05],'Units','normalized','Callback',... 'qazl("rorbit")','String','lef'); rorbit_button=uicontrol('Style','Pushbutton','Position',[.75 .08 ... .07.05]/Units','normalized','Callback',... 'qazl("rorbit")','String','right'); zorbit_button=uicontrol('Style','Pushbutton','Position',[.67 .02 ... .07.05],'Units','normalized','Callback,)... 'qazl(''zorbit'') ','String','up'); rzorbit_button=uicontrol('Style','Pushbutton','Position',[.67 .08 ... .07.05],'Units','normalized','Callback'... 'qazl(''rzorbit'')','String','down'); eds=300; volt=220; conductivity=0.05; freq=l.l; inertia=1500; az=0; el=90; axes(''Position''),[.16 .24 .7 .6]); elseif strcmp(action,'orbit'), [az el] = view; view([az+5 el]) drawnow elseif strcmp(action,'rorbit'), [az el] = view; view([az-5 el]) drawnow elseif strcmp(action'zorbit'), [az el] = view; view([az el+5]) drawnow 150

elseif strcmp(action,'rzorbit'), [az el] = view; view([az el-5]) drawnow elseif strcmp(action,'setfreq'), , freq=str2num(get(freq_fleld,'string )) qazl ('redraw'); elseif strcmp(action,'seteds'), eds=str2num(get(eds_field,'string')) qazl ('redraw'); elseif strcmp(action,'setvolt'), volt=str2num(get(volt_field,'string')) qazl ('redraw'); elseif strcmp(action,'setconductivity'), conductivity=str2num(get(conductivity_field,'string')) qazl ('redraw'); elseif strcmp(action,'setinertia'), inertia=str2num(get(inertia_field,'string')) qazl ('redraw'); elseif strcmp(action,'done'), clf reset; clear global inertia_field clear global freq_field clear global eds_field clear global volt_field clear global conductivity_field clear global eds clear global volt clear global conductivity clear global freq clear global inertia close; elseif strcmp(action,'redraw'), %t0 = 0; %tfmal =10; %x0 = [0 0.1]'; %hold off, %for i=0 : 0.1 : 0.79 % [t,x] = ode23(,vdpol',t0,tfinal,x0+i); % plot(x(:,l),x(:,2)); % hold on, % end [az el] = view; d = freq; k =(eds*volt*conductivity*240)/inertia; t =.06; hold off forj=.2:.2: .8 xl=j; 151

s=0; for i=l:150 plot3(xl,s,j); hold on s=s+t*k*(sin(d)-sin(d+xl)); xl=xl+t*s; end end %grid on axis([-l 1 -10 10 ]); view([az el]) %view([0 90]); xlabel ('d'); ylabel('S'); end function x = str2num(s) if ~isstr(s) | ndims(s)>2 error('Requires string or character array input.') end if isempty(s), x = []; return, end [m,n] = size(s); if m=l, x = protected_conversion(s); % Test flight if isempty(x), x = protected_conversion(['[' s ']']); % Try adding brackets end else semi =';'; space = "; if ~any(any(s == '[' | s == ']')), % String does not contain brackets o = ones(m-l,l); s = [['[';space(o)] s [semi(o) space(o);' ]']]'; elseif ~any(any(s(l:m-l,:) == semi)), % No ;'s in non-last rows s = [s,[semi(ones(m-l,l));space]]'; else, % Put ;'s where appropriate spost = space(ones(m,l)); for i= l:m-l, last = find(fliplr(s(i,:)) ~= space); if s(i,n-last(l)+l) ~= semi, spost(i) = semi; end end s = [s,spost]'; end 152

x = protected_conversion(s); end ifisstr(x) x=[]; end function STR2NUM_VaR = protected_conversion(STR2NUM_StR) STR2NUM_LaSTeRR = lasterr; STR2NUM_VaR = eval(STR2NUM_StR,'[]'); lasterr(STR2NUM_LaSTeRR) function [retl, ret2] = view(argl, arg2) fig = gcf; ax = gca; err = 0; if(nargin==0) if(nargout < 2) retl = get(ax,'xform'); else v = get(ax,'View'); retl = v(l); ret2 = v(2); end elseif (nargin == 1) [r,c] = size(argl); if(r=l)&(c = 2) set(ax,'View',argl); elseif((r==l)&(c==l)) ifargl==2 view(0,90); elseif argl == 3 view(-37.5,30); else error('Single scalar argument must be 2 or 3'); end elseif((r==4)&(c = 4)) set(ax,'xform',argl); elseif r*c == 3 % it's a direction vector unit = argl/norm(argl); az = atan2(unit(2),unit(l))* 180/pi; el = atan2(unit(3),sqrt(unit(l)A2+unit(2)A2))*l 80/pi; set(ax,'View',[az el]); else error('Argument must be scalar, two-vector, or 4-by-4 matrix'); end else [r,c] = size(argl); [r2,c2] = size(arg2); 153

if ((r == 1) & (c == 1) & (r2 = 1) & (c2 = 1)) set(ax,'View',[argl arg2]); else error('Arguments must be scalars'); end function c = strcmp(sl,s2) c = 0; if all(size(sl) == size(s2)) ifall(all(sl=s2)) c = l; end end function udvoenie() y=3.5; forj=0:.5:70, x=.3 ; k = y+(4-y)*j/70; for i=l : 30 x=k*x*(l-x); end for i=l :30, x=k*x*(l-x); plot(j,187*x); hold on end end

154

orgeneratoriani sistemisaTvis > j1=800;j2=700;eq1=1;eq2=5;z11=2.6;z22=7;a11=1.2; a12=0.9;a22=0.6;tt1=100;tt2=120; j1 = 800 j2 = 700 eq1 = 1 eq2 = 5 z11 = 2.6 z22 = 7 a11 = 1.2 a12 = 0.9 a22 = 0.6 tt1 = 100 tt2 = 120

> with(DEtools): > dif1:=diff(w1(t),t)=1/j1*(tt13/(2*w1(t))*(eq1^2/z11*sin(a11)+eq1*eq2/z12*sin( dt1(t)-a12))); eq1 2 sin( a11 ) eq1 eq2 sin( −dt1 ( t ) + a12 ) − z11 z12 3 tt1 − w1( t ) 2 d dif1 := w1( t ) = j1 dt

> dif2:=diff(w2(t),t)=1/j2*(tt23/(2*w2(t))*(eq2^2/z22*sin(a22)+eq1*eq2/z12*sin( dt1(t)-a12))); eq2 2 sin( a22 ) eq1 eq2 sin( −dt1 ( t ) + a12 ) − z22 z12 3 tt2 − w2( t ) 2 d dif2 := w2( t ) = j2 dt

> dif3:=diff(dt1(t),t)=w1(t)-w2(t); dif3 :=

d dt1( t ) = w1( t ) − w2( t ) dt

155

> autonomous({dif1,dif2,dif3},[w1(t),w2(t),dt1(t)] ,t); > true

> phaseportrait({dif1,dif2,dif3},[w1(t),w2(t),dt1( t)],t=4..4,[[w1(1)=1,w2[1]=1,dt1(1)=1],[w1(1)=2,w2[1]= 2,dt1(1)=2],[w1(1)=3,w2[1]=3,dt1(1)=3]],w1(t)=20..20,w2(t)=-20..20,dt1(t)=20..20,linecolor=black,scene=[w1(t),w2(t),dt1(t) ],color=black,stepsize=0.1); > samgeneratoriani sistema > 11=2000;i2=1500;e1=200;e2=300;u=220;y12=0.01;y1u =0.02;y2u=0.15;d1=0.9;d2=0.7;t1=10;t2=5; 11 = 2000 i2 = 1500 e1 = 200 e2 = 300 u = 220 y12 = 0.01 y1u = 0.02 y2u = 0.15 d1 = 0.9 d2 = 0.7

t1=10 t2=5 > with(DEtools): 156

> dif1:=diff(s1(t),t)=1/i1*(-e1*e2*y12*cos(d1d2)*sin(dd1(t)-dd2(t))e1*u*y1u*cos(d1)*sin(dd1(t))+e1*e2*y12*sin(d1d2)*(1-cos(dd1(t)-dd2(t)))+e1*u*y12*sin(d1)*(1cos(dd1(t)))); d s1( t ) = ( −e1 e2 y12 cos( d1 − d2 ) sin( dd1( t ) − dd2( t ) ) dt − e1 u y1u cos( d1 ) sin( dd1( t ) ) + e1 e2 y12 sin( d1 − d2 ) ( 1 − cos( dd1( t ) − dd2( t ) ) ) + e1 u y12 sin( d1 ) ( 1 − cos( dd1( t ) ) ) )/i1

dif1 :=

> dif2:=diff(s2(t),t)=1/i2*(e1*e2*y12*cos(d1d2)*sin(dd1(t)-dd2(t))e1*u*y2u*cos(d2)*sin(dd2(t))-e1*e2*y12*sin(d1d2)*(1-cos(dd1(t)-dd2(t)))+e1*u*y2*sin(d2)*(1cos(dd2(t)))); d s2( t ) = ( e1 e2 y12 cos( d1 − d2 ) sin( dd1( t ) − dd2( t ) ) dt − e1 u y2u cos( d2 ) sin( dd2( t ) ) − e1 e2 y12 sin( d1 − d2 ) ( 1 − cos( dd1( t ) − dd2( t ) ) ) + e1 u y2 sin( d2 ) ( 1 − cos( dd2( t ) ) ) )/i2

dif2 :=

> dif3:=diff(dd1(t),t)=s1(t); dif3 :=

d dd1( t ) = s1( t ) dt

> dif4:=diff(dd2(t),t)=s2(t); > autonomous({dif1,dif2,dif3,dif4},[dd1(t),dd2(t), s11(t),s2(t)],t); 11=2000;i2=1500;e1=200;e2=300;u=220;y12=0.01;y1u =0.02;y2u=0.15;d1=0.9;d2=0.7;t1=20;t2=15; 11 = 2000 i2 = 1500 e1 = 200 e2 = 300 u = 220 y12 = 0.01 y1u = 0.02

157

y2u = 0.15 d1 = 0.9 d2 = 0.7

t1=20 t2=15 > with(DEtools): > dif1:=diff(s1(t),t)=1/i1*(-e1*e2*y12*cos(d1d2)*sin(dd1(t)-dd2(t))e1*u*y1u*cos(d1)*sin(dd1(t))+e1*e2*y12*sin(d1d2)*(1-cos(dd1(t)-dd2(t)))+e1*u*y12*sin(d1)*(1cos(dd1(t)))); d s1( t ) = ( −e1 e2 y12 cos( d1 − d2 ) sin( dd1( t ) − dd2( t ) ) dt − e1 u y1u cos( d1 ) sin( dd1( t ) ) + e1 e2 y12 sin( d1 − d2 ) ( 1 − cos( dd1( t ) − dd2( t ) ) ) + e1 u y12 sin( d1 ) ( 1 − cos( dd1( t ) ) ) )/i1

dif1 :=

dif2 :=

> dif3:=diff(dd1(t),t)=s1(t); dif3 :=

d dd1( t ) = s1( t ) dt

> dif4:=diff(dd2(t),t)=s2(t); > autonomous({dif1,dif2,dif3,dif4},[dd1(t),dd2(t), s11(t),s2(t)],t);

158

11=2000;i2=1500;e1=200;e2=300;u=220;y12=0.01;y1u =0.02;y2u=0.15;d1=0.9;d2=0.7;t1=50;t2=40; 11 = 2000 i2 = 1500 e1 = 200 e2 = 300 u = 220 y12 = 0.01 y1u = 0.02 y2u = 0.15 d1 = 0.9 d2 = 0.7

t1=50 t2=40 > with(DEtools): > dif1:=diff(s1(t),t)=1/i1*(-e1*e2*y12*cos(d1d2)*sin(dd1(t)-dd2(t))e1*u*y1u*cos(d1)*sin(dd1(t))+e1*e2*y12*sin(d1d2)*(1-cos(dd1(t)-dd2(t)))+e1*u*y12*sin(d1)*(1cos(dd1(t)))); d s1( t ) = ( −e1 e2 y12 cos( d1 − d2 ) sin( dd1( t ) − dd2( t ) ) dt − e1 u y1u cos( d1 ) sin( dd1( t ) ) + e1 e2 y12 sin( d1 − d2 ) ( 1 − cos( dd1( t ) − dd2( t ) ) ) + e1 u y12 sin( d1 ) ( 1 − cos( dd1( t ) ) ) )/i1

dif1 :=

> dif2:=diff(s2(t),t)=1/i2*(e1*e2*y12*cos(d1d2)*sin(dd1(t)-dd2(t))e1*u*y2u*cos(d2)*sin(dd2(t))-e1*e2*y12*sin(d1d2)*(1-cos(dd1(t)-dd2(t)))+e1*u*y2*sin(d2)*(1cos(dd2(t))));

159

d s2( t ) = ( e1 e2 y12 cos( d1 − d2 ) sin( dd1( t ) − dd2( t ) ) dt − e1 u y2u cos( d2 ) sin( dd2( t ) ) − e1 e2 y12 sin( d1 − d2 ) ( 1 − cos( dd1( t ) − dd2( t ) ) ) + e1 u y2 sin( d2 ) ( 1 − cos( dd2( t ) ) ) )/i2

dif2 :=

> dif3:=diff(dd1(t),t)=s1(t); dif3 :=

d dd1( t ) = s1( t ) dt

> dif4:=diff(dd2(t),t)=s2(t); > autonomous({dif1,dif2,dif3,dif4},[dd1(t),dd2(t), s11(t),s2(t)],t);

wrfivi sinqronizacia > with(DEtools): > phaseportrait(y1(t)2*y2(t)=0,[y1(t),y2(t)],t=1..7, linecolor=blue,method=classical[foreuler],stepsi ze=.2);

arawrfivi sinqronizacia > with(DEtools): > phaseportrait(y2(t)0.1*y1(t)^3=0,[y1(t),y2(t)],t=1..7, linecolor=blue,method=classical[foreuler],stepsi ze=.2);

sinqronizacia elifsuri orbitis mimarT > with(DEtools): > phaseportrait((0.87*y1(t)+0.5*y2(t))^2/16+(0.5*y 1(t)-0.87*y2(t))^2/4-1=0,[y1(t),y2(t)],t=1..7, 160

linecolor=blue,method=classical[foreuler],stepsi ze=.2); sinqronizacia spiralis mimarT > with(DEtools): > phaseportrait((exp(s)-y1(t)^2y2(t)^2=0,[y1(t),y2(t)],t=1..7, linecolor=blue,method=classical[foreuler],stepsi ze=.2);

161

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