vaxtang bancaZe
,,arasimetriuli damyarebuli reJimebis analizi”
წარმოდგენილია დოქტორის აკადემიური ხარისხის
მოსაპოვებლად
საქართველოს ტექნიკური უნივერსიტეტი
თბილისი, 0175, საქართველო
2010 წელი
საავტორო უფლება © 2010 ,,vaxtang bancaZe” 2010 weli
2
საქართველოს ტექნიკური უნივერსიტეტი
,,energetikisa da telekomunikaciis fakulteti”
ჩვენ, ქვემორე ხელისმომწერნი ვადასტურებთ, რომ გავეცანით
,,vaxtang bancaZis” მიერ შესრულებულ სადისერტაციო ნაშრომს
დასახელებით: ,,arasimetriuli damyarebuli reJimebis analizi” და
ვაძლევთ
რეკომენდაციას
საქარველოს
ტექნიკური
უნივერსიტეტის
,,energetikisa da telekomunikaciis fakultetis” სადისერტაციო
საბჭოში მის განხილვას დოქტორის აკადემიური ხარისხის მოსაპოვებლად.
ხელმძღვანელი:
/nina Turqia/
რეცენზენტი:
რეცენზენტი:
რეცენზენტი:
3
საქართველოს ტექნიკური უნივერსიტეტი
2010 წელი
ავტორი:
,,vaxtang bancaZe”
დასახელება: ,,arasimetriuli damyarebuli reJimebis analizi”
ფაკულტეტი: ,,energetikisa da telekomunikaciis fakulteti”
ხარისხი:
დოქტორი
სხდომა ჩატარდა: თარიღი :
ინდივიდუალური
პიროვნებების
ან
ინსტიტუტების
მიერ
ზემომოყვანილი დასახელების დისერტაციის გაცნობის მიზნით მოთხოვნის
შემთხვევაში მისი არაკომერციული მიზნებით კოპირებისა და გავრცელების
უფლება მინიჭებული აქვს საქართველოს ტექნიკურ უნივერსიტეტს.
ავტორის ხელმოწერა
ავტორი ინარჩუნებს დანარჩენ საგამომცემლო უფლებებს და არც
მთლიანი ნაშრომის და არც მისი ცალკეული კომპონენტების გადაბეჭდვა ან
სხვა რაიმე მეთოდით რეპროდუქცია დაუშვებელია ავტორის წერილობითი
ნებართვის გარეშე.
ავტორი ირწმუნება, რომ ნაშრომში გამოყენებული საავტორო
უფლებებით დაცული მასალებზე მიღებულია შესაბამისი ნებართვა (გარდა
ის მცირე ზომის ციტატებისა, რომლებიც მოითხოვენ მხოლოდ სპეციფიურ
მიმართებას ლიტერატურის ციტირებაში, როგორც ეს მიღებულია
სამეცნიერო ნაშრომების შესრულებისას) და ყველა მათგანზე იღებს
პასუხისმგებლობას.
4
რეზიუმე
vaxtang bancaZis disertacia ,,arasimetriuli damyarebuli
reJimebis analizi” exeba aqtualur problemas. naSromi Sedgeba
141 gverdisagan da doqtoris akademiuri xarisxis mosapoveblad
wardgenili disertaciis gaformebis instruqciis mixedviT
moicavs: titulis gverds, xelmowerebis gverds, saavtoro
uflebebis gverds, reziumes or enaze (qarTul-inglisuri),
Sinaarss (sarCevs), naxazebis nusxas. ZiriTadi teqsti Sedgeba
Sesavlis, literaturis mimoxilvis, eqvsi Tavis, daskvnis da
gamoyenebuli literaturis siisagan.
SesavalSi
ganxilulia
kvlevis
aqtualoba,
problemis
Seswavlis mdgomareoba, kvlevis mizani da amocanebi, kvlevis
sagani, Teoriuli da meTedologiuri safuZvlebi, kvlevis
mecnieruli siaxle, naSromis praqtikuli mniSvneloba, naSromis
aprobacia, misi moculoba da struqtura.
pirvel
TavSi
ganxilulia
aRniSnul
problemasTan
dakavSirebuli Teoriuli safuZvlebi. ganxilulia eleqtruli
qselebis
analizis
topologiuri
meTodebi.
sistemis
mdgomareobis amsaxveli gantolebebi (sxvadasxva safexuris mqone
Zabvis qselebisaTvis), aRwerilia reJimis aqtiuri da pasiuri
parametrebi, Canacvlebis sqemis saxeebi. ganxilulia aRniSnuli
sakiTxTan dakavSirebuli literatura.
meore TavSi mocemulia maTematikuri modelirebis sakiTxebi.
damuSavebuli
da
aRwerilia
ganzogadebuli
parametrebis
gaangariSebis meTodika, romelic iZleva garkveul upiratesobas
arsebulTan SedarebiT da agreTve iZleva kompiuteruli
resursebis dazogvis saSualebas.
mesame TavSi mocemulia normaluri reJimis parametrebis
gaangariSebis algoriTmi. Seicavs Zabvis regulirebis imitacias.
iTvaliswinebs datvirTvis damokidebulebas Zabvaze, asaxuls
statikuri
maxasiaTebliT.
programaSi
gaTvaliswinebulia
informaciis dagroveba monacemTa bazaSi cxrilebis saxiT,
romelic warmoadgens SemdgomSi sawyis informacias sxvadasxva
tipis (mokle SerTvebis reJimebis gaangariSebis SemTxvevaSi)
gaangariSebebSi.
meoTxe TavSi mocemulia kvlevebis ZiriTadi sakiTxebi
dakavSirebuli simetriuli da arasimetriuli mokle SerTvebis
angariSebTan. ganxilulia rTuli simetriuli mokle SerTvebis
angariSi,
simetriul
mdgenelTa
meTodis
gamoyeneba
arasimetriuli
reJimebis
gaangariSebisas.
mexuTe
TavSi
damuSavebulia meTodika, romelic exeba erTdrouli simetriuli
da arasimetriuli mokle SerTvebis gaangariSebas.
arasimetriuli
avariuli
reJimebis
parametrebis
gaangariSebis erTiani meTodikis Camoyalibeba saSualebas iZleva
Catardes Sesabamisi angariSebi nebismieri saxisa da kombinaciis
avariis dros.
rogorc cnobilia arasimetriuli dazianebebis analizi
eyrdnoba
simetriul
mdgenelTa
meTods,
romlis
arsi
5
mdgomareobs arasimetriuli reJimis daSlaSi sam simetriul
reJimad,
raTa
Semdgom
TiToeuli
mimdevrobis
sqemaSi
gamoyenebuli iqnes yvela is meTodi, romelic samarTliania
simetriuli sistemebisaTvis. amis uflebas iZleva is debuleba,
rom calkeuli mimdevrobis sqemaSi cirkulirebs mxolod
Sesabamisi mimdevrobis denebi da urTierTkavSiri Zabvebsa da
denebs Soris aRiwereba cnobili kanonebiT, xolo kavSiri
mimdevrobebs Soris myardeba avariis adgilas im sasazRvro
pirobebis mixedviT, romelic axasiaTebs ama Tu im dazianebas.
amgvarad winaswar unda dadgindes pirdapiri, uku da nulovani
mimdevrobis sqemebis pasiuri parametrebi da gaiTvalos maTi
Sesabamisi kvanZuri gamtareblobisa da kvanZuri winaRobebis
matricebi ganivi asimetriis SemTxvevaSi.
nebismieri saxis arasimetriuli reJimis warmodgena erTfaza
dazianebebis Sesabamisi eleqtruli reJimebis superpoziciiT
saSualebas iZleva sxvadasxva saxis arasimetriuli reJimi, da
maT Soris mokle SerTvebic, aRiweros wrfivi gantolebebiT, rac
Tavis mxriv SeuzRudavs gaxdis SesaZlo avariebis analizs.
gantolebebis koeficientebi, anu kvanZebis sakuTari da
urTierTwinaRobebi ganivi asimetriis dros gaTvlili unda iyos
winaswar (sawyisi qselis Sesabamisad) pirdapiri, uku da
nulovani
mimdevrobis
sqemebisaTvis.
kvanZuri
winaRobebis
matricis elementebi ucvleli rCebian avariebis raodenobisa da
kombinaciebis miuxedavad.
dazianebis unificirebis mizniT sxvadasxva saxis mokle
SerTvebi warmodginda, rogorc erTfaza mokle SerTvebis
kombinacia. raTa eleqtruli reJimis aRmweri gantolebebi Sedges
erTiani wesiT garkveuli (mokle SerTvis saxeobidan gamomdinare)
SemzRudavi (sasazRvro) pirobebis gaTvaliswinebiT. mag. orfaza
mokle SerTva miwaze - rogorc ori erTfaza mokle SerTva,
samfaza mokle SerTva rogorc sami erTfaza mokle SerTva. am
dros aucileblad gaTvaliswinebuli unda iyos kvanZebis
Serwymis piroba (im kvanZebis sadac ganixileba erTfaza mokle
SerTvebi),
rac
gulisxmobs
am
kvanZebis
sakuTari
da
urTierTwinaRobebis tolobas.
erTiani midgomis Camosayalibeblad fazaTaSoriso mokle
SerTvac warmodginda erTfaza mokle SerTvebis kombinaciiT.
daeyrdno orfaza mokle SerTvis (miwaze) SemTxvevisaTvis
miRebul gantolebas da gaiTvaliswina fazaTaSorisi mokle
SerTvis pirobebi, romlebic daedo am gantolebaTa sistemas.
bolos mocemulia daskvnebi da gamoyenebuli literaturis
CamonaTvali.
6
Abstract
Vakhtang Bantsadze’s dissertation “Analysis of asymmetric established regimen”
concerns actual problems. The work consists of 141 pages and according the guidance
for thesis drawing up, presented for receiving academic
degree of Doctor, contains:
pages of a title, a signature, author’s rights, abstract in two languages (Georgian-
English), content, the list of drawings. Main text consists of introduction, discussion
of the literature, six chapters, conclusion and the list of used literature.
Actuality of the research, conditions of problem learning, objects and goals, subject
of the research, theoretical and methodological basis, scientific news of the research,
approbation of the work, its size and structure are reviewed in the introduction.
Theoretical principles associated with mentioned problem are discussed in the first
chapter. Topological methods of electrical network analysis are discussed. Equations
expressing the condition of the system (for networks of various level voltage), active
and passive parameters of regimen, replacement schemes are discussed. Literature
associated to this issue is reviewed.
The issues of mathematical modeling are given in the second chapter. Plural
methods of calculation of generalized parameters are worked out and discussed, which
give certain preference in comparison with existed ones and it lets to preserve
computer resources.
The algorithm of calculation of normal regimen parameters is given in the third
chapter. It contains imitation of voltage regulation. It considers relation of loading and
voltage, expressed with statistical characters. Accumulation of information in database
as tables is considered in the program, which consequently represents an initial
information in various (in the case of calculation of short circuits) calculations.
Main issues associated with symmetric and asymmetric short circuits are given in
the fourth chapter. Calculation of complex symmetric short circuits, using symmetric
part during the calculation of asymmetric regimens are discussed. In the fifth chapter
the plural methods, which concern calculation of simultaneous symmetric and
asymmetric short circuits, are discussed.
Formation of the united plural methods for calculation of parameters of asymmetric
faulty regimen lets conduct corresponding calculations at any type and combination
faults.
As it is known the analysis of asymmetric faults is based on the method of
symmetric parts, the main point of which is division of existing asymmetric regimen
into three symmetric ones to use consequently in the scheme of each sequence all
those methods, which are valid for symmetric systems. This is allowed by the
regulation, which says that in the scheme of separate sequence only currents of
corresponding sequence circulate and interface between voltages and currents are
described by known laws, and a connection between sequences is established at the
area of faults according those boundary conditions which is characteristic for various
faults. Thus passive parameters of direct, back or zero sequences schemes must be
determined and in the case of asymmetry lateral matrixes of corresponding nodular
conductivity and resistance must be calculated preliminarily.
Presenting any type asymmetric regimen with super-position of electrical regimens
corresponding to single-phase faults lets describe various type asymmetric regimens,
including short circuits, with linear equations, which makes the analysis of probable
faults unlimited.
7
Coefficients of equations i.e. own and inter-resistance of nodes at lateral
asymmetry must be calculated preliminarily (corresponding initial network) for the
schemes of direct, back and zero sequences.
In the purpose of faults unification we have represented different types of short
circuits as combination of single-phase short-circuits. To make equations describing
electrical regimens according united rule (coming out of circuit type) considering
restrictive (boundary) conditions. For example: double-phase short-circuit on the
ground – as two single-phase short circuit, three-phase short circuit as three single-
phase short circuit. In this case circumstances of nodes (those nodes which are
reviewed as single-phase short circuits) confluence must be considered, which means
equality of theses nodes own and inter-resistance.
For establishing united attitude we have represented inter-phase short circuits too with
combination of single-phase short circuits. We came out of the equation received for
double-phase short circuits (on the ground) and considered conditions of inter-phase
short circuits which were added to these system of equations.
At the end conclusions and the list of used literature is given.
8
Sinaarsi
Sesavali............................................................................................................................................................. 11
Tavi I
literaturis mimoxilva................................................................................................................... 14
1.1 eleqtruli sistemebis Canacvlebis sqemebi da maTi
elementebi..........................................................................................................................................
16
1.2 sistemis mdgomareobis amsaxveli ZiriTadi gantolebebi...............
21
1.3 kvanZuri Zabvebis gantolebebi.....................................................................................
21
1.4 incidenciis I da II matrica............................................................................................
27
1.5 kirxhofis
I
da
II
kanonebiT asaxuli sistemis mdgomareobis
gantolebebi......................................................................................................................................
32
1.6 kvanZuri Zabvebis gantolebis Sedgena transformatoruli
kavSirebis gaTvaliswinebiT............................................................................................
35
1.7 konturuli gantolebebi....................................................................................................
42
Tavi 2
maTematikuri modelirebis zogierTi sakiTxebi.....................................................
44
2.1 kvanZebis sakuTari da urTierTwinaRoba...........................................................
44
Sedegebi da maTi gansja
2.2 kvanZebis sakuTari da urTierTwinaRobebisa da Stoebis
sakuTari da urTierTgamtarobebis matricebis gaangariSebis
meTodebi...............................................................................................................................................
48
2.3 matricis Sebrunebis operaciis fizikuri arsi......................................... 57
2.4 gausis meTodis Sesabamisoba sqemis gardasaxvebTan............................. 62
Tavi 3
normaluri reJimis parametrebis gaangariSebis algoriTmi...................... 64
Tavi 4
damyarebuli reJimebis angariSi simetriuli da
arasimetriuli mokle SerTvebis dros............................................................................ 69
4.1 rTuli simetriuli avariuli reJimebis analizi.................................... 69
4.2 simetriul mdgenelTa gamoyeneba arasimetriuli reJimebis
angariSisas........................................................................................................................................ 82
4.3 orfaza mokle SerTva........................................................................................................... 86
4.4 erTfaza mokle SerTva......................................................................................................... 88
4.5 orfaza mokle SerTva miwaze........................................................................................ 90
Tavi 5
5.1 droSi Tanxvedrili erTfaza mokle SerTvebi........................................... 93
5.2 sxvadasxva saxis mokle SerTvis warmodgena erTfaza mokle
SerTvebiT............................................................................................................................................ 100
5.3 samfaza mokle SerTva........................................................................................................... 101
5.4 orfaza mokle SerTva miwaze........................................................................................ 103
5.5 110kv-iani qselisaTvis Catarebuli angariSebis Sedegebis
analizi.................................................................................................................................................. 107
Tavi 6
fazaTaSorisi mokle SerTvebi.................................................................................................. 120
daskvna................................................................................................................................................................. 133
danarTebi......................................................................................................................................................... 138
literatura................................................................................................................................................... 141
9
ნახაზების ნუსხა
ნახაზი 1
martivi Sekruli qselis Canacvlebis sqema......................................
21
ნახაზი 2
martivi qselis Canacvlebis sqema incidenciis I
matricis misaRebad..............................................................................
29
ნახაზი 3
martivi qselis Canacvlebis sqema incidenciis II
matricis misaRebad..............................................................................
31
ნახაზი 4
martivi qselis Canacvlebis sqema transformatoruli
kavSirebis gaTvaliswinebiT.............................................................
34
ნახაზი 5
rTuli qselis Canacvlebis sqema konturuli
gantolebebis misaRebad.....................................................................
43
ნახაზი 6
zogadi sqema kvanZebis n raodenobiT.........................................
44
ნახაზი 7
,,T”_sebri sqema..................................................................................................
45
ნახაზი 8
martivi qselis Canacvlebis sqema kvanZuri winaRobebis
misaღebad...................................................................................................
46
ნახაზი 9
martivi qselis Canacvlebis sqemebi kvanZebis sakuTari
da urTierTwinaRobebis matricis misaRebad...........................
49
ნახაზი 10 martivi qselis Canacvlebis sqemebi Stoebis sakuTari
da urTierTgamtarobebis matricis misaRebad........................
55
ნახაზი 11 StoebSi CarTuli Zabvis wyaroebis Secvla denis
wyaroebiT..................................................................................................
58
ნახაზი 12 Zabvisa da denis wyaros moqmedebiT miRebuli Sekruli
da gaxsnili oTxpolusebi................................................................
59
ნახაზი 13 rTuli Sekruli qseli.......................................................................
62
ნახაზი 14 mokle SerTvis dros aqtiur da pasiur qselSi Zabvis
wyaroebis CarTva...................................................................................
70
ნახაზი15
fizikur modelSi Zabvis wyaroebis Secvla denis
wyaroebiT..................................................................................................
72
ნახაზი 16 normaluri da avariuli reJimebis zeddeba............................
74
ნახაზი 17
arasimetriul veqtorTa sistema....................................................
82
ნახაზი 18 arasimetriuli veqtorebis daSla simetriul
mdgenelebad.............................................................................................
83
ნახაზი 19 orfaza mokle SerTva.........................................................................
86
ნახაზი 20 Zabvebisa da denebis veqtoruli diagrama orfaza
mokle SerTvis dros...........................................................................
88
ნახაზი 21 erTfaza mokle SerTva.......................................................................
88
ნახაზი 22 Zabvebisa da denebis veqtorul diagrama erTfaza
mokle SerTvis dros...........................................................................
90
ნახაზი 23
orfaza mokle SerTva miwaze..........................................................
90
ნახაზი 24 Zabvebisa da denebis veqtoruli diagrama orfaza
miwaze mokle SerTvis dros............................................................
92
ნახაზი 25 pirdapiri mimdevrobis sqema simetriuli samfaza mokle
SerTvis dros...................................................................................................................
96
10
ნახაზი 26 uku mimdevrobis sqema simetriuli samfaza mokle
SerTvis dros...................................................................................................................
96
ნახაზი 27 nulovani mimdevrobis sqema simetriuli samfaza
mokle SerTvis dros.................................................................................................
96
ნახაზი 28 pirdapiri mimdevrobis sqemaSi mokle SerTvis
modelireba denis wyaroebiT...........................................................................
97
ნახაზი 29 uku mimdevrobis sqemaSi mokle SerTvis modelireba
denis wyaroebiT.............................................................................................................
97
ნახაზი 30 nulovani mimdevrobis sqemaSi mokle SerTvis
modelireba denis wyaroebiT...........................................................................
97
ნახაზი 31 110kv Zabvis eleqtruli qseli........................................................................ 109
ნახაზი 32 pirdapiri mimdevrobis Canacvlebis sqema denebisa da
Zabvebis angariSiT....................................................................................................... 110
ნახაზი 33 uku mimdevrobis Canacvlebis sqema............................................................ 110
ნახაზი 34 nulovani mimdevrobis Canacvlebis sqema............................................. 110
ნახაზი 35 110kv Zabvis eleqtruli qseli me-2 kvanZSi samfaza
mokle SerTvis Semdeg............................................................................................. 113
ნახაზი 36 pirdapiri mimdevrobis Canacvlebis sqema Secvlili
sqemisTvis.............................................................................................................................. 113
ნახაზი 37 uku mimdevrobis Canacvlebis sqema Secvlili sqemisTvis. 113
ნახაზი 38 nulovani mimdevrobis Canacvlebis sqema Secvlili
sqemisTvis.............................................................................................................................. 113
ნახაზი 39 Secvlili sqemisTvis pirdapiri mimdevrobis Canacvle-
bis sqemaSi denebisa da Zabvebis ganawileba.................................... 114
ნახაზი 40 denebis diagramebi a) erTfaza mokle SerTvis dros
BfazaSi (i kvanZSi); b) erTfaza mokle SerTvis dros
C fazaSi (j kvanZSi)..................................................................................................... 121
ნახაზი 41 denebis diagramebi: a) i kvanZis pirdapiri da uku
mimdevrobis denebis diagrama. b) j kvanZis pirdapiri
da uku mimdevrobis denebis diagrama
0
120
-iT Semobrune-
bis Semdeg............................................................................................................................. 122
ნახაზი 42
pirdapiri mimdevrobis sqemaSi i(A faza) da j (B,C faza)
kvanZebSi mokle SerTvis denebis modelireba denis
wyaroebiT.............................................................................................................................. 124
ნახაზი 43 uku mimdevrobis sqemaSi i (A faza) da j(B,C faza)kvanZebSi
m. S-is denebis modelireba denis wyaroebiT................................... 124
ნახაზი 44 pirdapiri mimdevrobis sqemaSi i(B faza) da j (B,C faza)
kvanZebSi m.S-is denis modelireba denis wyaroebiT................ 126
ნახაზი 45 uku mimdevrobis sqemaSi i(B f) da j (B,C f) kvanZebSi
m.S-is denebis modelireba denis wyaroebiT..................................... 126
ნახაზი 46 pirdapiri mimdevrobis sqemaSi i(C faza) da j (B,C faza)
kvanZebSi m.S-is denis modelireba denis wyaroebiT................ 128
ნახაზი 47 uku mimdevrobis sqemaSi i (C faza) da j (B,C faza)
kvanZebSi m.S-is denebis modelireba denis wyaroebiT
128
11
შესავალი
eleqtruli energiis gadacema da ganawileba unda xdebodes
qselis muSaobis normalur reJimSi, anu uzrunveyofili unda
iyos muSaobis saimedooba da energiis xarisxi. xSirad
normaluri
reJimis
darRveva
xdeba
sxvadasxva
avariuli
situaciebisas. magaliTad mokle SerTvebiT, xazis an fazis
gawyvetebiT, an maTi kombinaciiT (sxvadasxva saxis grZivi da
ganivi dazianebebiT). qselis elementebi daculi unda iyos
avariuli denebis mier dazianebisagan.
energetikuli sistemis mdgradi, uavario da ekonomiuri
muSaoba
SeuZlebelia
sareleo
dacvisa
da
avtomatikis
mowyobilobebis
gareSe.
dazianebuli
ubnis
swrafi
da
seleqtiuri gamorTva xdeba Tanamedrove sareleo dacvis
mowyobilobebiT. maTi saimedo muSaobisaTvis gaTvlili unda
iyos SesaZlo avariuli parametrebi. am parametrebis gaTvla
gaangariSebas eZRveneba
mravali
naSromi da jer kidev
mimdinareobs am mimarTulebiT kvleva Zieba. Cven SevecadeT Cveni
wvlili Segvetana am saqmeSi da davamuSaveT meTodika rTuli
avariuli reJimebis gasaangariSeblad.
arsebuli
meTodebi
saSualebas
iZlevian
gaviangariSoT
erTjeradi avariis dros reJimis parametrebi. dazianebebi
SeiZleba iyos simetriuli da arasimetriuli mokle SerTvebi,
xazisa da fazis gawyvetebi. es meTodebi aseve iZlevian
saSualebas gaangariSebuli iqnas denebi da Zabvebi erTroulad
ori dazianebis SemTxvevaSi, rogoricaa magaliTad erT ubanSi
fazis gawyveta da meore ubanSi simetriuli an arasimetriuli
mokle SerTva.
Cvens
mier
damuSavda
meTodika,
romelic
avariuli
parametrebis gaangariSebis saSualebas iZleva nebismieri saxisa
da raodenobis ganivi dazianebis (anu mokle SerTvis nebismieri
kombinaciis) dros. amis SesaZleblobas arsebuli meTodebi ar
iZleodnen.
12
warmodgenili meTodikis safuZvelze SevqmeniT maTematikuri
modeli. am meTodis gamoyenebiT konkretuli qselisaTvis
CavatareT angariSebi erTdrouli arasimetriuli dazianebebis
SemTxvevaSi. angariSebs vawarmoebdiT programa MATLAB-is
saSualebiT. am modelis safuZvelze unda damuSavdes programa
monacemTa bazis marTvis sistemaSi Tanamedrove dizainiTa da
grafikiT.
naSromSi warmodgenilia arasimetriuli avariuli reJimebis
parametrebis gaangariSebis algoriTmebi, dafuZvnebuli sistemis
mdgomareobis amsaxvel kvanZur gantolebebze.
arasimetriuli mokle SerTvis SemTxvevaSi, simetriuli
sistemebisaTvis Camoyalibebuli Teoria gadaitaneba avariuli
reJimebis simetriul mdgenelebSi. raki avariuli denebis
mdgenelebi
cirkulireben
mxolod
Sesabamisi
mimdevrobis
sqemaSi, yvela is kanoni da debuleba, rac samarTliania
simetriuli
sistemebisaTvis,
samarTliania
TiToeuli
mimdevrobis sqemaSia rsebuli eleqtruli reJimebisaTvis.
e.i TiToeuli mimdevrobis sqemaSi damokidebuleba denebsa da
Zabvebs Soris aRiwereba kvanZuri gantolebebiT, winaswar,
TiToeuli
mimdevrobis
sqemisaTvis
gaTvlili
kvanZuri
winaRobebis matricis saSualebiT. am gantolebaTa gaangariSeba
xdeba erTfaza mokle SerTvisaTvis dadgenili sasazRvro
pirobebis gaTvaliswinebiT, rasac ucnobebis ricxvi dahyavs
dazianebaTa ricxvamde.
amgvarad erTfaza mokle SerTvis sasazRvro pirobebi
mosaxerxebelia unificirebuli gantolebebis misaRebad. amitom
yvela saxis mokle SerTvas ganvixilavT rogorc erTfaza mokle
SerTvis reJimebis zeddebas. ase magaliTad, Tu sami erTfaza
mokle SerTvis (A, B da C fazebSi) aRmwer gantolebebs davadebT
kvanZebis Serwymis pirobas, rac iTvaliswinebs kvanZebis sakuTari
da
urTierTwinaRobebis
tolobas
samive
mimdevrobis
sqemebisaTvis, miviRebT dazianebis adgilas denebis fazur
13
mniSvnelobebs. igive iTqmis miwaze orfaza mokle SerTvis
SemTxvevaSi.
aRniSnuli gantolebebi nebismieri saxis da raodenobis
erTdrouli arasimetriuli mokle SerTvis (miwaze) gaangariSebis
saSualebas iZlevian.
ganxiluli meTodikis Tavisebureba mdgomareobs imaSi, rom
nebismieri saxis dazianeba warmoidgineba rogorc erTfaza mokle
SerTvebis reJimebis zeddeba. am meTodiT movaxdineT aseve
fazaTaSoriso mokle SerTvebis gaangariSebac. fazaTaSorisi
mokle SerTvebic warmovadgineT aRniSnuli gantolebebiT,
romelTac daedoT damatebiTi piroba, ganpirobebuli erTi fazis
meore fazasTan uSualo SeerTebiT. am damatebiT pirobas
Seesabameba erT-erTi dazianebuli fazis denebis (pirdapiri da
uku mimdevrobis sqemebSi) Semobruneba 120
0
-iT. amgvarad Cven
saSualeba mogveca, raki yvela dazianeba warmoadgens erTfaza
mokle
SerTvebis
kombinacias,
gaviangariSoT
nebismieri
simetriuli da arasimetriuli avariuli reJimebi.
14
Tavi I
naSromSi warmodgenil problemasTan dakavSirebiT
arsebuli literaturis mimoxilva
energosistemis proeqtireba da eqspluatacia moiTxovs didi
raodenobiT kvlevebs da gaangariSebis Catarebas normaluri da
avariuli reJimebis parametrebis gansazRvrasTan dakavSirebiT.
am kvlevebs miekuTvnebian damyarebuli normaluri reJimebis
gaangariSebebi, statikuri da dinamikuri mdgradobis analizi,
aqtiuri da reaqtiuli simZlavreebis ganawilebis optimizacia,
mokle
SerTvis
denebis
gansazRvra,
maregulirebeli
mowyobilobebisa da avariis sawinaaRmdego avtomatikis danayenTa
parametrTa ganszRvra da sxva. energosistemis ganviTareba iwvevs
am tipis samuSaoebis garTulebas, rac Tavis mxriv moiTxovs
Tanamedrove kompiuteruli teqnologiebis maRal dones.
damyarebuli
reJimis
gaangariSebas
gansakuTrebuli
mniSvneloba eniWeba im garemoebebis gamo, rom am angariSis
Sedegebi (nakadganawileba, kvanZuri Zabvebi, Zabvisa da simZlavris
danakargebi)
aucilebelia
ara
marto
eqspluataciis
procesisaTvis, aramed aucilebelia sxvadasxva amocanebis
amoxsnisas. rogorebicaa reJimebis optimizacia, gardamavali
procesebis angariSi mokle SerTvebis angariSi da sxva.
eleqtroenergetikuli sistemis reJimebis gaangariSebasTan
dakavSirebiT gamoqveynebulia didi raodenobiT literatura. [1]
am SromebSi ganxilulia am problemis, kerZod reJimis
parametrebis gaangariSebis sxvadasxva meTodebi. mocemulia
topologiuri analizis safuZvlebi da am problemis gadawyvetis
Tanamedrove midgoma kompiuteruli teqnologiebis gamoyenebiT
[2]
detalurad aris ganxiluli sistemis mdgomareobis
ZiriTadi gantolebebi da am gantolebebis klasifikacia [3]
naSromi eZRvneba eleqtruli sistemis damyarebuli reJimebis
gaangariSebis sxvadasxva meTodebs (kvanZurs, konturuls). garda
15
amisa ganxilulia am amocanis amoxsnis sxvadasxva meTodebi
rogorc pirdapiri aseve iteraciuli. gamokveTilia am meTodebis
gansxvaveba da upiratesoba konkretuli amocanebis gadawyvetis
dros. gansakuTrebuli adgili ukavia Tanamedrove algoriTmebs,
romlebic didi ganzomilebis amocanebis gadawyvetis saSualebas
iZleva da uzrunvelyofs kompiuteruli resursebis efeqturad
gamoyenebas.
rogorc ukve aRvniSneT normaluri reJimis parametrebi
gamoiyenebian avariuli reJimebis gasaangariSebladac. TviTon
avariuli reJimis parametrebis gansazRvra aseve aucilebel
amocanas warmoadgens, vinaidan am avariuli parametrebis
winaswari
gansazRvris
gareSe
warmoudgenelia
sistemis
usafrTxo
muSaobis
uzrunvelyofa.
eleqtruli
reJimebis
angariSis Casatareblad aucilebelia eleqtruli sistemis
principialuri sqemis warmodgena misi Canacvlebis sqemiT
nebismieri eleqtruli reJimi rogorc normaluri aseve
avariuli aRiwereba erTi da igive gantolebebiT. amitom
gansakuTrebuli yuradReba eqceva sistemis mdgomareobis aRmwer
gantolebebs. warmodgenil literaturaSi [4] ganxilulia
sistemis mdgomareobis amsaxveli kvanZuri da konturuli
gantolebebi. gaanalizebulia am meTodebis Taviseburebani da
gamokveTilia kvanZuri gantolebebis upiratesoba konturulTan
SedarebiT. ganxilulia agreTve miRebuli gantolebebis amoxsnis
pirdapiri da iteraciuli meTodebi.
mocemulia topolologuri analizis safuZvlebi, sistemis
mdgomareobis amsaxveli gantolebebis warmodgena incidenciis
matricebis saSualebiT, gantolebaTa amoxsnis rogorc pirdapiri
(matricis Sebruneba, gausis meTodi da sxva), aseve iteraciuli
meTodebi (martivi iteracia, zeidelisa da niutonis meTodi).
gansakuTrebuli yuradReba aqvs daTmobili ganzogadoebuli
parametrebiani matriculi gantolebebiT sistemis mdgomareobis
aRweras. am gantolebebis gamoyenebas rogorc normaluri aseve
avariuli reJimebis parametrebis gaangariSebis dros.
16
avariuli reJimebis gaangariSebis Sedegad miRebuli aqtiuri
parametrebi warmoadgenen erT-erT safuZvels sareleo dacvis
samsaxuris
gamarTuli
muSaobisaTvis.
amJamad
arsebuli
meTodebiT
[2,9]
warmoebs
erTjeradi
grZivi
da
ganivi
dazianebebis gaangariSeba. samfaza mokle SerTva, erTfaza mokle
SerTva, fazaTaSorisi mokle SerTva, orfaza mokle SerTva
miwaze, aseve xazisa da fazis gawyvetebi. am monacemebis
safuZvelze xdeba sareleo dacvis danayenis SerCeva - Semowmeba.
bolo
dros
warmodgenili
kompiuteruli
programebiT
gaangariSebas axdenen erTdroulad ori dazianebis, simetriuli
da arasimetriuli, SemTxvevebisTvis. Cven CavatareT samuSaoebi da
kvleva-Zieba raTa mogvexerxebina yvela tipis ganivi dazianebis
gaangariSeba.
arasimetriuli reJimis angariSi ganxilulia simetriul
mdgenelTa
meTodis
bazaze
[2].
am
meTodis
Tanaxmad
arasimetriuli reJimi warmoadgens sam simetriuli reJimis
zeddebas, romlebic Seesabamebian pirdapiri, uku da nulovani
mimdevrobis
sqemebs.
TiToeuli
am
mimdevrobis
reJimis
angariSisas samarTliania yvela is meTodika, rac samarTliania
simetriuli sistemebisaTvis.
$1.1 eleqtruli sistemebis Canacvlebis sqemebi da maTi
elementebi
imisaTvis rom viangariSoT eleqtruli sistemis muSaobis
reJimi saWiroa, rom Tavdapirvelad SevadginoT misi Canacvlebis
sqema. Canacvlebis sqemas (an ubralod sqemas) uwodeben
eleqtruli wredis grafikul gamosaxulebas romelic gviCvenebs
misi monakveTebis Tanmimdevrul SeerTebas da amasTanave asaxavs
gansaxilveli eleqtruli wredis Tvisebebs. Eeleqtruli wredi
da Sesabamisad misi sqema Seicaven Stoebs, kvanZebs da konturebs.
17
Sto_es aris eleqtruli wredis nawili romelic Sedgeba
mimdevrobiT
SeerTebuli
elementebisagan
(maTSi
unda
gaedinebodes erTi da igive sididis deni).
kvanZi_es aris ori an meti Stos SeerTebis adgili.
konturi_es aris nebismieri Caketili gza romelic gadis
ramodenime Stoze. Tu eleqtruli wredis sqemas ar gaaCnia
konturebi, maSin mas ewodeba gaxsnili. Ggaxsnil sqemebSi
TiToeuli kvanZis kveba xorcieldeba mxolod erTi mxridan.
TiToeuli kvanZi iRebs kvebas araumetes erTi Stodan. romelime
Stos gamorTva gamoiwvevs yvela im kvanZis kvebis Sewyvetas
romlebic kvebas iRebdnen am Stos gavliT. sqema, romelic
Seicavs Tundac erT konturs ewodeba Caketili sqema. Caketil
sqemaSi arsebobs erTi kvanZi mainc, romelic kvebas iRebs ori an
meti Stodan. romelime Stos gamorTva ar gamoiwvevs kvebis
Sewyvetas.
eleqtruli sqemis elementebi iyofian aqtiur da pasiur
elementebad. pasiuri elementebi qmnian eleqtruli denebis
gavlis gzebs.
eleqtruli sistemis pasiuri elementebi iyofian grZiv da
ganiv elementebad. ganivi pasiuri elementebi – es aris Stoebi,
romlebic CarTulni arian el. sqemis kvanZebsa da neitrals
Soris, anu Zabvis mqone kvanZsa daNnulovani potencialis mqone
kvanZs Soris. grZivi pasiuri elementebi es aris Stoebi,
romlebic aerTeben yvela kvanZs garda kvanZisa romelsacAaqvs
nulis toli Zabva anu grZivi Stoebi ar arian SeerTebulni
neitralTan. grZivi elementebi Seicaven eleqtrogadamcemi
xazebis aqtiur da induqtiur winaRobebs, transformatorebis
gragnilebis
aqtiur
da
induqtiur
winaRobebs,
grZivi
sakompensacio
mowyobilobebis
tevadur
winaRobebs.
ganivi
pasiuri elementebi Seesabamebian xazebis gamtareblobebs miwis
mimarT, reaqtorebs da kondensatorebs, romlebic CarTulni
arian
miwasTan
(kondensatorTa
batareebi,
maSuntebeli
reaqtorebi). Aaseve transformatoris foladSi simZlavris
18
danakargebi, Canacvlebis sqemaze warmodgeba rogorc ganivi
gamtareblobebi. Canacvlebis sqemebis aqtiuri elementebi arian
Zabvisa da denis wyaroebi, romlebic ZiriTadad gamoiyenebian
mokledSerTvis denebis angariSis dros.
denis
wyaroebs
Seesabamebian
eleqtro
sadgurebis
generatorebi da momxmarebelTa datvirTvebi. denis wyaroebi
arian eqtroenergiis wyaroebi romelTac datvirTvis winaRobis
cvlilebisas denis mniSvneloba ar ecvlebaT. aseTi reJimi
SesaZlebelia maSin roca Sida winaRoba gacilebiT aRemateba
datvirTvis
winaRobas.
marTlac
generatoris
winaRoba
damyarebul reJimSi, 3-4-jer metia gare datvirTvis winaRobaze.
amitom nominaluri parametrebiT muSaobis dros, gare winaRobis
mcire cvlilebisas, generatoris denis mniSvneloba SegviZlia
ucvlelad CavTvaloT.
rac
Seexeba
Zabvis
wyaroebs_isini
aseve
warmoadge-
nenLelqtroenergiis wyaroebs, romelTa gamomyvanebze Zabva ar
icvlis mniSvnelobas datvirTvis cvlilebisas sakmaod did
zRvrebSi. Ees niSnavs imas, rom Zabvis wyaros Sida winaRoba
imdenad mcirea, rom Zabvis vardna masze Zalzed pataraa. Aamis
gamo Zabvis wyaros SeuZlia SeinarCunos ucvleli sididis Zabva
momWerebze gare denis cvlilebisas sakmaod didfarglebSi.
zemoT CamoTvlili pasiuri elementebis ZiriTadi parametrebi
arian aqtiuri winaRoba r, iduqtivoba L , da tevadoba C . wredis
parametrebi ama Tu im xarisxiT damokidebulni arian denze da
Zabvaze. magaliTad winaRoba r damokidebulia denze, radganac
denis gazrdisas izrdeba gamtaris temperatura. kondensatoris
tevadoba damokidebulia Zabvaze, xolo koWas induqtivoba ki
denze. Ees damokidebuleba umravles SemTxvevaSi Zalzed sustia
da amitom SegviZlia misi ugulebelyofa. Ee. i. Cven SegviZlia
wredis pasiuri elementebis maxasiaTeblebi warmovidginoT swori
xazebiT. wredis aseT elementebs ewodebaT wrfivi. wrfiv
elementebSi winaRoba _ r, tevadoba _ C, induqtivoba _ L mudmivi
19
sidideebia anuAar arian elementSi gamaval denze da modebul
Zabvaze damokidebulni.
eleqtruli wredis damyarebuli reJimi, mudmivi denisa da
Zabvis wyaroebis arsebobisas, aris wredis iseTi mdgomareoba,
romlis drosac deni nebismier StoSi da Zabva nebismier kvanZSi
aris ucvleli xangrZlivi drois ganmavlobaSi. damyarebul
reJimSi gvaqvs wrfivi pasiuri elementebi da moduliTa da faziT
ucvleli denis wyaroebi. aseT SemTxvevaSi damyarebuli reJimi
aRiwereba wrfivi algebruli gantolebebiT. aseT wredebs
ewodebaT wrfivi eleqtruli wredebi. es SemTxveva Seesabameba
eleqtruli sistemis damyarebuli reJimis angariSs, rodesac
datvirTvebisa
da
generatorebis
denebs,
eleqtrosistemis
nebismier kvanZSi gaaCniaT ucvleli sididis modulida faza.
Tu eleqtruli wredis pasiuri elementebis parametrebi
damokidebulni arian gamaval denze da modebul Zabvaze, maSin
aseTi
elementebis
maxasiaTeblebic
arawrfivia,
da
aseT
elementebs ewodebaT arawrfivi. viciT, rom iseT eleqtrul
wreds romelic Seicavs Tundac erT arawrfiv elements ewodeba
arawrfivi. E
eleqtruli sistemebis damyarebuli reJimebis angariSisas,
pasiuri elementebis arawrfivoba, rogorc wesi ar aris
gaTvaliswinebuli. am mxriv Canacvlebis sqemis grZivi nawili,
yovelTvis wrfivia. amave dros, rogorc wesi, damyarebuli
reJimebis angariSisas iTvaliswineben denis wyaroebis arawrfiv
maxasiaTeblebs.
denis
wyaroebis
arawrfivoba
Seesabameba
kvanZebSi
generatorebisa da momxmareblebis tvirTebis mocemas mudmivi
simZlavriT,Aaseve
datvirTvebis
mocemas
maTi
statikuri
maxasiaTeblebiT, romlebic warmoadgenen momxmareblis tvirTis
damokidebulebas Zabvaze. aseTi arawrfivi denis wyaroebis
arsebobisas eleqtruli sistemis damyarebuli reJimebi aRiwereba
arawrfivi algebruli gantolebebiT anu damyarebuli reJimis
arawrfivi gantolebebiT.
20
sabolood Cven SegviZlia davaskvnaT: Tu energosistemis
Canacvlebis sqema, romelic warmodgenilia wrfivi pasiuri
elementebiTr, L, Cda denis wyaroebi mocemulia mocemulia mudmivi
sididis deniTa daFfaziT, maSin aseTi reJimi aRiwereba wrfivi
algebruli gantolebebiT. xolo Tu energosistemis Canacvlebis
sqema (romelic warmodgenilia arawrfivi pasiuri elementebiT r,
L, C)
da denis wyaroebi mocemulia simZlavriT: I
kv
=S
kv
*
/√3U
k
*
(1.1.1),
maSin es reJimi aRiwereba arawrfivi algebruli gantolebebiT.
aseTi saxiT warmodgenil denis wyaroze amboben, rom igi
arawrfivia,
radganac
is
damyarebuli
reJimis
amsaxvel
gantolebaSi jdeba ara rogorc mudmivi sidide (rogorc R,X
L),
Aaramed rogorc Zabvis funqcia. es Cans (1.1.1) gantolebidanac,
sadac S
kv
*
=const – K -
uri kvanZis, simZlavris SeuRlebuli
mniSvnelobaa. U
kv
*
- K -
uri kvanZis Zabvis SeuRlebuli komleqsuri
mniSvnelobaa. I
kv
(U) –
arawrfivi denia, romelic damokidebulia
Zabvaze.
21
$ 1.2 sistemis mdgomareobis amsaxveli ZiriTadi gantolebebi
avariuli reJimebis maTematikuri modelebi aRiwerebian
sistemis mdgomareobis amsaxveli igive gantolebebiT rogoric
miRebulia normaluri reJimebis analizis dros.
sistemis mdgomareobis amsaxveli yvela gantoleba emyareba
kirxhofis kanonebs. am gantolebebis pirdapiri amoxsna iZleva
kvanZur Zabvebsa da denebs StoebSi da maTi ricxvi udris
damoukidebeli konturebisa da kvanZebis jams. amitom ufro
xSiri gamoyeneba aqvs kirxhofis gantolebebis gardasaxviT
miRebul
kvanZur
da
konturul
gantolebebs,
romelTa
ganzomilebac mniSvnelovnad naklebia uSualod kirxhofis
kanonebis mixedviT Sedgenil gantolebebTan SedarebiT [1] [3] [11].
$1.3 kvanZuri Zabvebis gantolebebi
ganvixiloT kvanZuri Zabvebis gantolebis miRebis konkretuli
magaliTi:
1
I
I
2
y
1
y
2
II
I
3
y
4
y
3
5
y
4
2
I
3
2
I
4
3
I
1
4
I
nax 1. martivi Sekruli qselis Canacvlebis sqema
me_4 kvanZi aviRoT mabalansirebel kvanZad. amave kvanZis Zabva
iyos bazisuri Zabva
4
U
. movaxdinoT Cveni mosazrebiT denebis
miaxloebiTi
ganawileba
StoebSi
(StoebSi
denebis
mimarTulebebis dadgena). Tu amoxsnis Semdeg denis sidides
miviRebT uaryofiTi niSniT, maSin am denis realuri mimarTuleba
22
ar emTxveva Cvens mier aRebul nakadganawilebas, maSin dens
naxazze Seecvleba mimarTuleba da misi sidide aiReba “+” niSniT.
nax 1_ze mocemuli wredisaTvis 1, 2, 3 kvanZebisaTvis davweroT
kirxofis I kanonis gantolebebi:
0
34
23
2
12
23
24
1
12
41
I
I
I
I
I
I
I
I
I
#3_
kvanZi
#2_
kvanZi
#1_
kvanZi
(1.3.1)
(4) kvanZisTvis gantolebis daweras, rogorc aRvniSneT azri
ar aqvs.
StoSi gamavali denebi gamovsaxoT am StoSi Zabvis vardnebiT,
gveqneba:
0
5
5
4
4
2
3
3
4
4
2
2
1
2
2
1
1
U
Y
U
Y
I
U
Y
U
Y
U
Y
I
U
Y
U
Y
(1.3.2)
StoebSi Zabvis vardnebi gamovsaxoT kvanZuri Zabvebis
meSveobiT:
4
3
5
3
2
4
4
2
3
2
1
2
1
4
1
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
(1.3.3)
Tu (1.3.3) _s SevitanT (1.3.2) _Si gveqneba:
0
)
(
)
(
)
(
)
(
)
(
)
(
)
(
4
3
5
3
2
4
2
4
2
3
3
2
4
2
1
2
1
2
1
2
1
4
1
U
U
Y
U
U
Y
I
U
U
Y
U
U
Y
U
U
Y
I
U
U
Y
U
U
Y
aqedan
4
5
3
5
4
2
4
4
3
2
3
4
2
4
3
2
1
2
4
1
1
1
2
1
2
1
0
)
(
)
(
)
(
U
Y
U
Y
Y
U
Y
U
Y
I
U
Y
U
Y
Y
Y
U
Y
U
Y
I
U
Y
U
Y
Y
(1.3.4)
Y
1
+Y
2
; Y
2
+Y
3
+Y
4
; Y
4
+Y
5
=Y
33
_warmoadgenen kvanZebis sakuTar
gamtareblobebs. isini sididiT tolia kvanZSi SerTuli Stoebis
gamtareblobebis jamis. igulisxmeba is Stoc romelic aerTebs
mabalansirebel kvanZsa da mocemul kvanZs. danarCeni sidideebi
23
Y
2
;
Y
4
warmoadgenen
kvanZebis
urTierT
gamtareblobebs
konkretuli sqemisaTvis (nax.1).
Tu davuSvebT, rom Y
1
+Y
2
=Y
11
; Y
2
+Y
3
+Y
4
=Y
22
; Y
4
+Y
5
=Y
33
;
da
Y
12
=Y
21
=Y
2
; Y
32
=Y
23
=Y
4
;
(1.5.4) gantolebaTa sistema miiRebs Semdeg
saxes:
4
5
3
33
2
32
4
3
2
3
23
2
22
1
21
4
1
1
2
12
1
11
0
U
Y
U
Y
U
Y
U
Y
I
U
Y
U
Y
U
Y
U
Y
I
U
Y
U
Y
(1.3.5)
Tu eleqtruli wredi Sedgeban+1raodenobis kvanZisagan,
maSinnraodenobis damoukidebeli kvanZisaTvis daweril kvanZuri
Zabvebis gantolebebs zogadad eqnebaT Semdegi saxe:
n
n
nn
n
n
n
n
n
n
n
I
U
Y
U
Y
U
Y
U
Y
I
U
Y
U
Y
U
Y
U
Y
I
U
Y
U
Y
U
Y
U
Y
3
3
2
2
1
1
2
2
3
23
2
22
1
21
1
1
3
13
2
12
1
11
(1.3.6)
(1.3.
6) gantolebaTa sistemaSi kvanZebSi denis wyaroebis niSani
ar aris miTiTebuli. Mmisi niSani damokidebulia imaze Tu ra
saxis kvanZs warmoadgens mocemuli kvanZi_datvirTvis Tu
generaciis. Tu or kvanZs Soris, eleqtrul wredSi ar aris
SemaerTebeli Sto, maSin (1.3.5) gantolebaTa sistemaSi Sesabamisi
gamtarebloba aiReba 0-is toli.
Tu wredis StoebSi CarTulia e.m.Z-ebi, maSin K-uri kvanZis
deni I
K
tolia jamisa, am kvanZTan SerTuli Stoebis e.m.Z-ebis
namravli
Sesabamis
gamtareblobebze_
I
k
=
∑E
ki
Y
ki.
Tu
e.m.Z.
mimarTulia kvanZisaken jamSi aiReba “+” niSniT, xolo Tu ar
aris mimarTuli kvanZisaken aiReba “-“ niSniT.
(1.3.6) gantolebaSi Stoebis sakuTari da urTierTgamtareb-
lobebi ganvalagoT matriculad:
33
32
23
22
21
12
11
0
0
Y
Y
Y
Y
Y
Y
Y
aseve kvanZuri denebi da kvanZuri Zabvebi davalagoT veqtor-
svetebis saxiT:
24
0
2
1
I
I
I
;
3
2
1
U
U
U
U
zemoTTqmulis safuZvelze kvanZuri Zabvebis gantoleba
miiRebs saxes:
5
4
3
4
1
4
2
1
3
2
1
33
32
23
22
21
12
11
0
0
0
Y
U
Y
U
Y
U
I
I
U
U
U
Y
Y
Y
Y
Y
Y
Y
anu kvanZuri Zabvebis gantoleba Caiwereba Semdegnairad:
b
b
kv
kv
kv
U
I
U
(1.3.7)
(1.3.5) gantoleba SegviZlia gadavweroT Semdegi saxiTac:
0
)
(
)
(
)
(
)
(
)
(
)
(
)
(
4
3
33
4
2
32
2
4
3
23
4
2
22
4
1
21
1
4
2
12
4
1
11
U
U
Y
U
U
Y
I
U
U
Y
U
U
Y
U
U
Y
I
U
U
Y
U
U
Y
rogorc
ukve
aRvniSneT
bolo
gantolebaTa
sistema
analogiuria (1.3.5) gantolebaTa sistemis. gantolebaTa sistema
warmovadginoT matriculi saxiT:
0
0
0
2
1
4
3
4
2
4
1
33
32
23
22
21
12
11
I
I
U
U
U
U
U
U
Y
Y
Y
Y
Y
Y
Y
Tu U
b
=0,
anu Cvens SemTxvevaSi U
4
=0,
maSin Y
kv
U
kv
=I
kv
;
yuradReba
unda mivaqcioT imas, rom TuU Cven vatarebT wrfivi eleqtruli
wredis angariSs, maSin kvanZebSi Zabvebi da StoebSi denebi ar
arian damokidebuli imaze TuU romel kvanZs aviRebT Cven
mabalansireblad denis mixedviT, Tu yvela (n+1) kvanZis denebis
jami aris 0-is toli. amitom wrfiv eleqtrul sistemebSi,
mabalansirebeli kvanZis arCeva, aseve misi Zabvis arCeva (U
b
=0,
U
b
≠0) ar axdens zegavlenas angariSis rezultatze.
Y
kv
(U-U
b
)=I
kv
gantolebis
amoxsna
gvaZlevs
(U-U
b
)=∆U
veqtorsvets. igi warmoadgens ZabvaTa vardnis sidiedebs
mabalansirebel kvanZsa da mocemul kvanZs Soris _ ∆U
1
; ∆U
2
; ∆U
3
;
……. ∆U
n
.
amis Semdeg Cven SegviZlia ganvsazRvroT ZabvaTa
25
realuri sidideebi kvanZebSi U-U
b
=∆U;U=∆U+U
b
.
da Sesabamisad
U
1
=∆U
1
+U
b
;
U
2
=∆U
2
+U
b
;
U
3
=∆U
3
+U
b
..............
U
n
=∆U
n
+U
b
;
umetes
SemTxvevaSi angariSis mizans warmoadgens denebis gnsazRvra
StoebSi I
ij
= (U
i
-U
j
)Y
ij
; U
i
=U
b
+∆U
i
; U
j
=U
b
+∆U
j
.
amis gaTvaliswinebiT:
I
ij
=(U
b
+∆U
i
-U
b
-∆U
j
)Y
ij
=(∆U
i
-∆U
i
)Y
ij
.
arawrfivi denis wyaroebis SemTxvevaSi kvanZuri Zabvebis
gantolebebs aqvT Semdegi saxe: Y
kv
(U-U
b
)=S
*
/U
*
anuY
kv
∆U= S
*
/U
*
3 faza cvladi denis wredisaTvis, sadac denebi kvanZebSi,
kvanZuri Zabvebi da kvanZuri gamtareblobebi kompleqsuri
sidideebia. am SemTxvevaSiMmatricul gantolebas eqneba Semdegi
saxe: Y
kv
U
kv
=S/U
kv
+ U
b
Y
b
kvanZuri Zabvebis gantolebis amoxsna SesaZlebelia im
SemTxvevaSic, rodesac saqme gvaqvs kompleqsur ricxvebTan.
amisaTvis
mimarTavenn-uri
rigis
kompleqr
wevrebiani
gantolebaTa
sistemis
gardaqmnas
2nrigis
eqvivalentur,
namdvilwevrebian gantolebaTa sistemad. am saxiT Cawerili
gantolebaTa sistema advilad ixsneba e.g.m-is saSualebiT.
am gardaqmnis arsi mdebareobs SemdegSi: Cvenn-ucnobian
kompleqsurwevrebian
gantolebaTa
sistemas,
warmovadgenT
2nucnobian, namdvilwevrebian gantolebaTa sistemad, sadac
pirvelin raodenobis ucnobi aris saZiebeli sidideebis (Zabvebis)
namdvili wevrebi, meoren raodenobis ucnobi aris saZiebeli
sidideebis
warmosaxviTi
nawilebi.
sabolood
gveqneba
2nraodenobis gantoleba 2nucnobiT. vnaxoT, Tu rogor xdeba,
zogadad kvanZuri Zabvebis gantolebaTa sistemis gardaqmna
kompleqsuridan namdvil saxeSi. kompleqsur formaSi kvanZuri
gamtareb-lobebis matrica, Caiwereba Semdegi saxiT:
Y
kv
=G
kv
–jB
kv
(1.3.8).
kvanZuri Zabvebisa da kvanZuri denebis veqtor-svetebi
Caiwereba Semdegi saxiT:
U=U’+jU"
daI=I’+jI" (1.3.9)
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