1579:
1270:
1835:
1573:
945:
956:
1595:
1386:
56:
of the buses in a power system. In realistic systems which contain thousands of buses, the admittance matrix is quite sparse. Each bus in a real power system is usually connected to only a few other buses through the
787:
1265:{\displaystyle I_{k}=\sum _{i=1,2,\ldots ,N \atop i\neq k}{(V_{k}-V_{i})y_{ki}}+V_{k}y_{k}=V_{k}\left(y_{k}+\sum _{i=1,2,\ldots ,N \atop i\neq k}y_{ki}\right)-\sum _{i=1,2,\ldots ,N \atop i\neq k}V_{i}y_{ki}}
622:
1906:
1830:{\displaystyle Y={\begin{pmatrix}y_{1}+y_{12}+y_{13}&-y_{12}&-y_{13}\\-y_{12}&y_{2}+y_{12}+y_{23}&-y_{23}\\-y_{13}&-y_{23}&y_{3}+y_{13}+y_{23}\\\end{pmatrix}}}
1589:
As an example, consider the admittance diagram of a fully connected three bus network of figure 1. The admittance matrix derived from the three bus network in the figure is:
1984:
1319:
261:
164:
1568:{\displaystyle Y_{kj}={\begin{cases}y_{k}+\sum _{i=1,2,\ldots ,N \atop i\neq k}{y_{ki}},&{\mbox{if}}\quad k=j\\-y_{kj},&{\mbox{if}}\quad k\neq j.\end{cases}}}
721:
751:
655:
378:
519:
465:
288:
191:
1351:
2004:
1934:
1374:
1293:
771:
695:
675:
539:
485:
438:
418:
398:
348:
328:
308:
235:
211:
138:
114:
940:{\displaystyle I_{ki}={\begin{cases}V_{k}{y_{k}},&{\mbox{if}}\quad i=k\\(V_{k}-V_{i})y_{ki},&{\mbox{if}}\quad i\neq k.\end{cases}}}
550:
2142:
2095:
2070:
19:
This article is about the admittance matrix in electrical engineering. For the graph theoretic admittance matrix, see
2176:
2161:
1275:
This relation can be written succinctly in matrix form using the admittance matrix. The nodal admittance matrix
1843:
542:
78:
58:
2112:
1411:
812:
1943:
2181:
49:
1298:
240:
143:
2031:
700:
726:
630:
353:
77:
of the nodal admittance diagram of the power system, which is derived by the application of
497:
443:
266:
169:
42:
1327:
8:
1989:
1919:
1359:
1278:
756:
680:
660:
524:
470:
423:
403:
383:
333:
313:
293:
220:
196:
123:
99:
2138:
2091:
2066:
27:
2019:
2007:
1937:
82:
74:
62:
20:
1916:
of the nodes corresponding to the subscripts of the entry. The admittance matrix
2036:
2006:
asymmetrical. For instance, modeling phase-shifting transformers, results in a
2170:
310:
by all loads and sources connected to the bus. The admittance between buses
774:
53:
89:
replacing each line in the diagram with its equivalent admittance, and
1321:
matrix such that bus voltage and current injection satisfy Ohm's law
467:, and is the sum of the admittance of all the loads connected to bus
2162:
A C/C++ Program and Source Code for
Computing Ybus and Zbus Matrices
2018:
The admittance matrix is most often used in the formulation of the
380:, and is the sum of the admittance of all lines connecting busses
92:
converting all voltage sources to their equivalent current source.
81:
to the admittance diagram of the power system. Starting from the
61:. The nodal admittance matrix is used in the formulation of the
1578:
85:
of a power system, the nodal admittance diagram is derived by:
2041:
1561:
933:
73:
The nodal admittance matrix of a power system is a form of
781:
and the line and load currents are linked by the relation
68:
16:
N x N matrix describing a linear power system with N buses
1912:
of the network nodes. The non-diagonal entries are the
2086:
Saadat, Hadi (1999). "6.7 Tap changing transformers".
1610:
1539:
1497:
911:
843:
1992:
1946:
1922:
1846:
1598:
1389:
1362:
1330:
1301:
1281:
959:
790:
759:
729:
703:
683:
663:
633:
553:
527:
500:
473:
446:
426:
406:
386:
356:
336:
316:
296:
269:
243:
223:
199:
172:
146:
126:
102:
617:{\displaystyle I_{k}=\sum _{i=1,2,\ldots ,N}I_{ki}}
1998:
1978:
1928:
1900:
1829:
1567:
1368:
1345:
1313:
1287:
1264:
939:
765:
745:
715:
689:
669:
649:
616:
533:
513:
479:
459:
432:
412:
392:
372:
342:
322:
302:
282:
255:
229:
205:
185:
158:
132:
108:
1986:. However, extensions of the line model may make
2168:
1585:: The admittance diagram of a three bus network.
1376:are then determined by the equations for the
290:is the cumulative current injected at bus
773:to ground through the bus load. Applying
2065:. McGraw-Hill Science/Engineering/Math.
2060:
1901:{\displaystyle Y_{11},Y_{22},...,Y_{nn}}
1577:
69:Construction from a single line diagram
2169:
2132:
2085:
45:describing a linear power system with
2110:
2137:. United Kingdom: WCB/McGraw-Hill.
2090:. United Kingdom: WCB/McGraw-Hill.
777:law to the admittance diagram, the
13:
1432:
1195:
1125:
978:
96:Consider an admittance graph with
14:
2193:
2155:
1356:in vector format. The entries of
420:. The admittance between the bus
215:vector of bus current injections
2013:
1545:
1503:
917:
849:
2126:
2104:
2079:
2054:
1049:
1023:
889:
863:
1:
2047:
1979:{\displaystyle Y_{ki}=Y_{ik}}
7:
2025:
52:. It represents the nodal
10:
2198:
2113:"The Power Flow Equations"
18:
1380:into buses, resulting in
1314:{\displaystyle N\times N}
256:{\displaystyle N\times 1}
159:{\displaystyle N\times 1}
753:is the current from bus
657:is the current from bus
2061:Grainger, John (1994).
716:{\displaystyle k\neq i}
543:Kirchhoff's current law
32:nodal admittance matrix
2177:Electrical engineering
2120:Iowa State Engineering
2000:
1980:
1930:
1902:
1831:
1586:
1569:
1370:
1347:
1315:
1289:
1266:
941:
767:
747:
746:{\displaystyle I_{kk}}
717:
691:
671:
651:
650:{\displaystyle I_{ki}}
618:
535:
515:
481:
461:
434:
414:
394:
374:
373:{\displaystyle y_{ki}}
344:
324:
304:
284:
257:
231:
207:
193:is the voltage of bus
187:
160:
134:
118:vector of bus voltages
110:
2135:Power System Analysis
2133:Saadat, Hadi (1999).
2088:Power System Analysis
2063:Power System Analysis
2032:Admittance parameters
2001:
1981:
1931:
1903:
1840:The diagonal entries
1832:
1581:
1570:
1371:
1348:
1316:
1290:
1267:
942:
768:
748:
718:
692:
672:
652:
619:
536:
516:
514:{\displaystyle I_{k}}
482:
462:
460:{\displaystyle y_{k}}
435:
415:
395:
375:
345:
325:
305:
285:
283:{\displaystyle I_{k}}
258:
232:
208:
188:
186:{\displaystyle V_{k}}
161:
135:
111:
1990:
1944:
1920:
1844:
1596:
1387:
1360:
1346:{\displaystyle YV=I}
1328:
1299:
1279:
957:
788:
757:
727:
701:
681:
661:
631:
551:
525:
498:
471:
444:
424:
404:
384:
354:
350:is a complex number
334:
314:
294:
267:
241:
221:
197:
170:
144:
124:
100:
2010:admittance matrix.
83:single line diagram
2020:power flow problem
1996:
1976:
1926:
1914:mutual admittances
1898:
1827:
1821:
1587:
1565:
1560:
1543:
1501:
1475:
1378:current injections
1366:
1343:
1311:
1285:
1262:
1238:
1168:
1021:
937:
932:
915:
847:
763:
743:
713:
687:
667:
647:
614:
600:
531:
511:
477:
457:
430:
410:
390:
370:
340:
320:
300:
280:
253:
227:
203:
183:
156:
130:
106:
63:power flow problem
59:transmission lines
2111:McCalley, James.
1999:{\displaystyle Y}
1929:{\displaystyle Y}
1542:
1500:
1473:
1427:
1369:{\displaystyle Y}
1288:{\displaystyle Y}
1236:
1190:
1166:
1120:
1019:
973:
914:
846:
766:{\displaystyle k}
690:{\displaystyle i}
670:{\displaystyle k}
567:
534:{\displaystyle k}
492:current injection
480:{\displaystyle k}
433:{\displaystyle i}
413:{\displaystyle i}
393:{\displaystyle k}
343:{\displaystyle i}
323:{\displaystyle k}
303:{\displaystyle k}
230:{\displaystyle I}
206:{\displaystyle k}
133:{\displaystyle V}
109:{\displaystyle N}
36:admittance matrix
28:power engineering
2189:
2149:
2148:
2130:
2124:
2123:
2117:
2108:
2102:
2101:
2083:
2077:
2076:
2058:
2005:
2003:
2002:
1997:
1985:
1983:
1982:
1977:
1975:
1974:
1959:
1958:
1938:symmetric matrix
1935:
1933:
1932:
1927:
1910:self-admittances
1907:
1905:
1904:
1899:
1897:
1896:
1869:
1868:
1856:
1855:
1836:
1834:
1833:
1828:
1826:
1825:
1818:
1817:
1805:
1804:
1792:
1791:
1780:
1779:
1765:
1764:
1748:
1747:
1733:
1732:
1720:
1719:
1707:
1706:
1695:
1694:
1678:
1677:
1663:
1662:
1648:
1647:
1635:
1634:
1622:
1621:
1574:
1572:
1571:
1566:
1564:
1563:
1544:
1540:
1532:
1531:
1502:
1498:
1490:
1489:
1488:
1474:
1472:
1461:
1423:
1422:
1402:
1401:
1375:
1373:
1372:
1367:
1352:
1350:
1349:
1344:
1320:
1318:
1317:
1312:
1294:
1292:
1291:
1286:
1271:
1269:
1268:
1263:
1261:
1260:
1248:
1247:
1237:
1235:
1224:
1186:
1182:
1181:
1180:
1167:
1165:
1154:
1116:
1115:
1101:
1100:
1088:
1087:
1078:
1077:
1065:
1064:
1063:
1048:
1047:
1035:
1034:
1020:
1018:
1007:
969:
968:
946:
944:
943:
938:
936:
935:
916:
912:
904:
903:
888:
887:
875:
874:
848:
844:
836:
835:
834:
824:
823:
803:
802:
772:
770:
769:
764:
752:
750:
749:
744:
742:
741:
722:
720:
719:
714:
696:
694:
693:
688:
676:
674:
673:
668:
656:
654:
653:
648:
646:
645:
623:
621:
620:
615:
613:
612:
599:
563:
562:
540:
538:
537:
532:
520:
518:
517:
512:
510:
509:
486:
484:
483:
478:
466:
464:
463:
458:
456:
455:
439:
437:
436:
431:
419:
417:
416:
411:
399:
397:
396:
391:
379:
377:
376:
371:
369:
368:
349:
347:
346:
341:
329:
327:
326:
321:
309:
307:
306:
301:
289:
287:
286:
281:
279:
278:
262:
260:
259:
254:
236:
234:
233:
228:
212:
210:
209:
204:
192:
190:
189:
184:
182:
181:
165:
163:
162:
157:
139:
137:
136:
131:
115:
113:
112:
107:
79:Kirchhoff's laws
75:Laplacian matrix
21:Laplacian matrix
2197:
2196:
2192:
2191:
2190:
2188:
2187:
2186:
2167:
2166:
2158:
2153:
2152:
2145:
2131:
2127:
2115:
2109:
2105:
2098:
2084:
2080:
2073:
2059:
2055:
2050:
2028:
2016:
1991:
1988:
1987:
1967:
1963:
1951:
1947:
1945:
1942:
1941:
1936:is typically a
1921:
1918:
1917:
1908:are called the
1889:
1885:
1864:
1860:
1851:
1847:
1845:
1842:
1841:
1820:
1819:
1813:
1809:
1800:
1796:
1787:
1783:
1781:
1775:
1771:
1766:
1760:
1756:
1750:
1749:
1743:
1739:
1734:
1728:
1724:
1715:
1711:
1702:
1698:
1696:
1690:
1686:
1680:
1679:
1673:
1669:
1664:
1658:
1654:
1649:
1643:
1639:
1630:
1626:
1617:
1613:
1606:
1605:
1597:
1594:
1593:
1559:
1558:
1538:
1536:
1524:
1520:
1514:
1513:
1496:
1494:
1481:
1477:
1476:
1462:
1433:
1431:
1418:
1414:
1407:
1406:
1394:
1390:
1388:
1385:
1384:
1361:
1358:
1357:
1329:
1326:
1325:
1300:
1297:
1296:
1280:
1277:
1276:
1253:
1249:
1243:
1239:
1225:
1196:
1194:
1173:
1169:
1155:
1126:
1124:
1111:
1107:
1106:
1102:
1096:
1092:
1083:
1079:
1073:
1069:
1056:
1052:
1043:
1039:
1030:
1026:
1022:
1008:
979:
977:
964:
960:
958:
955:
954:
931:
930:
910:
908:
896:
892:
883:
879:
870:
866:
860:
859:
842:
840:
830:
826:
825:
819:
815:
808:
807:
795:
791:
789:
786:
785:
758:
755:
754:
734:
730:
728:
725:
724:
702:
699:
698:
682:
679:
678:
662:
659:
658:
638:
634:
632:
629:
628:
605:
601:
571:
558:
554:
552:
549:
548:
526:
523:
522:
505:
501:
499:
496:
495:
472:
469:
468:
451:
447:
445:
442:
441:
425:
422:
421:
405:
402:
401:
385:
382:
381:
361:
357:
355:
352:
351:
335:
332:
331:
315:
312:
311:
295:
292:
291:
274:
270:
268:
265:
264:
242:
239:
238:
222:
219:
218:
198:
195:
194:
177:
173:
171:
168:
167:
145:
142:
141:
125:
122:
121:
101:
98:
97:
71:
24:
17:
12:
11:
5:
2195:
2185:
2184:
2182:Electric power
2179:
2165:
2164:
2157:
2156:External links
2154:
2151:
2150:
2144:978-0075616344
2143:
2125:
2103:
2097:978-0075616344
2096:
2078:
2072:978-0070612938
2071:
2052:
2051:
2049:
2046:
2045:
2044:
2039:
2037:Nodal analysis
2034:
2027:
2024:
2015:
2012:
1995:
1973:
1970:
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1962:
1957:
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1950:
1925:
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1365:
1354:
1353:
1342:
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1333:
1310:
1307:
1304:
1284:
1273:
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1259:
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1246:
1242:
1234:
1231:
1228:
1223:
1220:
1217:
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1202:
1199:
1193:
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1185:
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1135:
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1129:
1123:
1119:
1114:
1110:
1105:
1099:
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1082:
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1072:
1068:
1062:
1059:
1055:
1051:
1046:
1042:
1038:
1033:
1029:
1025:
1017:
1014:
1011:
1006:
1003:
1000:
997:
994:
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988:
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841:
839:
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829:
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762:
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737:
733:
712:
709:
706:
686:
666:
644:
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637:
625:
624:
611:
608:
604:
598:
595:
592:
589:
586:
583:
580:
577:
574:
570:
566:
561:
557:
530:
508:
504:
476:
454:
450:
440:and ground is
429:
409:
389:
367:
364:
360:
339:
319:
299:
277:
273:
252:
249:
246:
226:
202:
180:
176:
155:
152:
149:
129:
105:
94:
93:
90:
70:
67:
15:
9:
6:
4:
3:
2:
2194:
2183:
2180:
2178:
2175:
2174:
2172:
2163:
2160:
2159:
2146:
2140:
2136:
2129:
2121:
2114:
2107:
2099:
2093:
2089:
2082:
2074:
2068:
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2057:
2053:
2043:
2040:
2038:
2035:
2033:
2030:
2029:
2023:
2021:
2011:
2009:
1993:
1971:
1968:
1964:
1960:
1955:
1952:
1948:
1939:
1923:
1915:
1911:
1893:
1890:
1886:
1882:
1879:
1876:
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1870:
1865:
1861:
1857:
1852:
1848:
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490:Consider the
488:
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271:
263:vector where
250:
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2014:Applications
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779:bus voltages
778:
626:
491:
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950:Therefore,
541:. Applying
521:, into bus
116:buses. The
2171:Categories
2048:References
54:admittance
2008:Hermitian
1769:−
1754:−
1737:−
1684:−
1667:−
1652:−
1550:≠
1518:−
1467:≠
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1429:∑
1306:×
1230:≠
1216:…
1192:∑
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1160:≠
1146:…
1122:∑
1037:−
1013:≠
999:…
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922:≠
877:−
708:≠
591:…
569:∑
248:×
151:×
34:(or just
2026:See also
1583:Figure 1
237:, is an
140:, is an
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2141:
2094:
2069:
627:where
213:, and
43:matrix
2116:(PDF)
1295:is a
775:Ohm's
50:buses
40:N x N
2139:ISBN
2092:ISBN
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723:and
697:for
400:and
330:and
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2173::
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318:k
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276:k
272:I
251:1
245:N
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201:k
179:k
175:V
154:1
148:N
128:V
104:N
47:N
23:.
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