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A Unified Approach for Load-Flow and Short-Circuit Analysis of Large-Scale Underground Distribution Systems
A UNIFIED APPROACH FOR LOAD-FLOW AND SHORT-CIRCUIT ANALYSIS OF LARGE-SCALE UNDERGROUND DISTRIBUTION SYSTEMS S. Deckmann and A. Monticelli Departamento de Engenharia Eletrica, UNICAMP Campinas,' S.P., BRAZIL Abstract. Large underground distribution networks have' special features which can be used in developing efficient simplified methods for load-£low ·and short-circuit calculations. The paper presents a non-iterative method for load-flow and short-circuit analysis, equivalencing the AC system by two equivalent DC networks. When compared with the conventional Newton-Raphson method, the savings in storage requirements' are' around 75%. In addition, the same matrices used in load-flow calculations can be used for contingency analysis (primary feeder outage) and for symmetrical short-circuit calculations. The new method has been verified by simulation of five systems (9, 22, 63, 191 and 307 nodes). The last three form part of the Sao Paulo City underground distribution system. The results have been compared with the solutions obtained using the Newton-Raphson load-flow program and the impe dance method for short-circuit calculations. No significant discrepancies have been observed. The method pre.sented. can be used, when necessary', in conjunction with a two-level approach to solve the complete circuit (primaryand secondary). Sparse techniques make it possible to handle large networks, both for load-flow and short-circuit analysis. The sparse inverse matrix method used in the short-circuit calculations, when compared with the impedance matrix methods ordinarily' used, requires less computer storage and is. faster. Primary feeder outages have been simulated by compensation Keywords. Distribution networks;
load-flow;
INTRODUCTION The most important objectives of load-flow and short-circuit analysis of underground distribution networks are: Calculations of the voltage magnitudes, current flows and transformer loadings for present arid future conditions. . - Simulations of primary-feeders outages. - Simulations of three-phase shortcircuit at any point of the secondary network in order to verify the burn-clear-fault conditions.
short-circuit;
equivalencing
ondary network. Simplifications have been made which introduce a minimum of inaccuracy in the models. These approximations are acceptable since the loads are rarely known very accurately (10% erro~s are typical). SECONDARY NETWORK SYSTEMS
Usually the first two problems are solved using conventional load~flow algorithms (e.g., Newton-Raphson method l ) and short-circuit analysis .is performed through the imped~nce matrix methods 2 • There are several reasons to believe that these methods, originally developed for high-voltage transmission systems, are not the most adequate for underground distribution network analysis.
Secondary network systems are characterized by the following features: ~ Voltage drops are usually less than 5% of the nominal values. - Nodal voltages are nearly in phase (maximum phase difference between any two nodes is less than. 3 degrees). . - Load power factor is nearly uniform (a typical value is .85). - The three phase system is loaded in a balanced manner, even under short-circuit conditions. - R/X ratios of the cables are near ly uniform • and the charging effect is negligible.
This paper presents a unified approach for load-flow and short-circuit analysis of secondary network systems. The classical AC model is repl~ced by two equivalent DC networks. These equivalent circuits are obtained considering the special characteristics of the sec-
This kind of network 3 serves directly indivi dual or lumped consumers. Specially designed distribution transformers ate connected alternately to distinct radial primary feeder~ and supply the loads through a secondary main grid. The primary feeders are radially 278
connected to the same bulk station avoiding unwanted power flows between stations. Fig.l shows a'typical underground distribution net work.
z
.--
BASIC EQUATIONS The method presented in the paper is ~ased on the solution of two linear systems of nod'al equations:
1 '--
3 If
':i
\I~
6 JI
'"
'y"",
11'
. . ':i~ iI
,- ,
'-)'~
where
:
~'I
,I
li~
11'
'~
11 )'~
,/
5
/ ,t.
-
(1)
[I 1
(2)
[rD] and [r] are the nodal admittarce matric~s of the two 'equivalent DC networks; - (I~ and [I] are the equivalent load current v~ctors;
'y~
Fig.l - Underground distribution network 1 2 3 4 5 6
b4
- [DJ and [L] are the nodal volt, age drop and nodal voltage difference vectors. The components of the vectors are given b'y:
Bulk substation Primary feeders (15-20 kv) Manual primary switch Three-phase transformer Secondary mains (110-220 v) Automatic network protector
(3)
(4)
DEFINITIONS AND NOTATIONS
v - magnitude of the line-to-neutral voltao
ge behind the reactance of the network transformers (reference voltage)
V - voltage magnitude at load-bus k k Ok - voltage angle at load-bus k (0 0
•
o
0 )
Ek=Vkexp(jok) - complex voltage at load-bus k Re{Ek}=VkcosO Dk=Vk-V
- real part of the complex voltage E k - nodal voltage drop (bus k)
~l+j~l
- impedance of the branch k-l
k
o ~+j~ - impedance of the transformer k
Zk - impedance magnitude of the transformer k Zkl - impedance magnitude of the branch k-l I
kl
r
- equivalent resistance of the branch k-l kl Sk - apparent power-kva (bus k) k
i-Given the load gowers Sk' the load currents I are calculated from (3).' k ii - The voltage drop vector (D] is obtained from (1). iii - The exact load currents I are k calculated from (4).' iv - The voltage difference vector [L) is obtained from (2). v - The current flows I are obkl ained from (5):
- magnitude of the current flow from bus k to bus 1 - magnitude of the vector difference of the complex voltages (IEk~El I) at the terminals of the-orinch k-l
I
The solution of the problem is obtained using a five-step procedure:
- magnitude of the load current at bus k
Equation (1) gives the voltage drops with, good accuracy, but these voltages cannot be used to calculate the current flows. On the other hand, equation (2) gives the voltage difference vector, which has no physical meaning by itself, but can be used to compute the current flows very accurately.
- power factor of the load k
cos~k
n - number of load buses lE~
(5)
- means that bus 1 is connected to bus k, excluded the case k=l
If - short-circuit currents - vectors and matrices are denoted by brackets
JT
. d'~cates - ~n
THE MODEL FOR VOLTAGE DROP CALCULATIONS A branch k-l of the' secondary network is represented in Fig. 2.(a). The diagram of Fig. 2.(b) shows the relationship among the voltage at terminal nodes, voltages drops and current flow.
, .
transpos~t~on
279
, [k
Rkl
Xkl
El
®'~0
~
Jkl'
.( a1
I~
~ig.
The the Fig. 2 - Voltage drops In the secondary network distribution systems the nodal voltages are nearly in phase. It follows that the nodal voltage magnitude'! an be approximated by:
3 - Load representation t· vo tage dr p ,dlculations
c~nductances (Sk/v~).are ~?Lroduced in of the matnx I r Dj (Eq. (1» •
d~agonal
For example, cons1der the svstem partially repr~sented in Fig. 4(a,b). The equivalent DC network used for voltage drop calculations is represented in Fig. 4.(c).
(6)
For example, consider the critical and unusual case where dk--3°. The error introduced by approxi.tion (6} is only .14% of the nominal voltage Vo. It can be seen from Fig. 2.(b) that for smaller angles the errcrsintro duced in (6) are negligible. It follows from !q. (6) that the voltage drop across the branch k-l is given by D
kl
CD CD~-----:;~---~
- D '-D ;: Re{E } - Re{E }k
1
k
. 1
(7)
(~lcos~ + ~lsen~) I
Fig~
4(a) - Secondary network
kl Eq. (7) is the basic relation for derivation of the first equivalent DC network~. The equ! valent DC resistance is defined as (8)
where ~ is taken as the of the system.
~verage
power factor
The magnitude of the load current is (9)
Ik -
Fig. 4. (b) - AC representation
As a rule ~«v , and thus, Eq. (9) can be approximated as o follows 5 Dk -1 (l + -V-) .0
From Eq. (10) the load current I can be k represented as shown m Fig. 3.
(10)
I Fig, 4.(c) - Equivalent DC network for voltage drop calculaticns 280
THE MODEL FOR CURRENT FLOW
CALCULATIONS The equivalent DC network used to calculate the current flows is represented in Fig. 5. and has the following main characteristics: - The loads are modelled by constant current injections obtained from (4). - The lines and the transformers are represented by the magnitude of the complex impedances (Zkl and Zk).
Let the vectorlHi] be null except for -1 in position ki, where ki is the low voltage ter mina 1 of the transformer i connected to the primary feeder k. The algorithm for multiple network-transform er outage simulation is: i-Compute the n x 1 vector [xi] for i=l ••••• nk. by solving: [rD] [xi]
[Hi)
(11)
ii - Construct the nk x nk matrix defined as: Zip
c xt
Zpi
for i, pc!. •• nk Zii
Xki - Zkl
(12)
iii - Calculate the nk x nk matrixlYJ:
The two DC models are used.to calculate the voltage drops at load buses. current flows in the main grid and the loading of the t~ formers: The following data are assumed known: .
[Y]
PRIMARY FEEDER OUTAGES The triangular factors of the nodal admittan· ce matrices are assumed known from the base= case studies. Since several network-transformers are connec ted to the same primary feeder. a single prY mary feeder outage yields a multiple network transformer outage. as shown in Fig. 6. These changes in the circuit are simulated by the compensation method • Thus, the admit tance-matrix reconstruction and retriangulation are not required.
and network
=
[zt l
(13)
iv - Construct the ~ x 1 vector IT]: IT]
-I z]-l [~Dl
(14).
where the ~ x 1 vector [~Dl is defined as:
- The apparent load power (kva). - Load power factor. - Parameter and topology of the main. grid. - Network-transformer location. rating and impedance. - Primary feeder arrangement.
Fig. 6. Primary-feeder transformers
• i ; k
where Zki is the impedance magnitude of the transformer connec ted to the node ki, and Xki is the ki-th element of the vector[XP] • .
Fig. 5. Equivalent DC network for current flow calculations BASE-CASE LOAD FLOW
[zl
~i
= ~i
' for i-I ••• nk
(15)
(~.1 is the base-case . voltage
drop on bus ki)
v -
Calculate the post-outage volta . ge drop vector [D 1 given by: -
[0)=[00 ]
+[X)[T]
(16)
• where [DO) is the base-case vol tage drop vector, and the n x nk matrix [ X1 is [X] = [xl: x 2 l .. lxnk ] (17)
Note that the same procedure presented here can be used to calculate the current flow&. There is an alternative approximate approach which can be used with computational advanta ge. This method is based on the observationthat if one network-transformer is removed. only the voltages of the adjacent buses are significantly affected. It follows that the multiple network-transformers outage can be simulated as a sequence of single outages , each one considered as an independent event. The . final result is obtained by the superposition of the individual effects •
281
Therefore, it is not necessary to store the n ~ ~ matrix (X.J,. since the n x 1 vectors [X1J are calculated one at a time.
nodes with the branch removed. This situation is show in Fig. 8.
(d)
SHORT CIRCUIT CALCULATIONS Short circpit on load-buses
'CD
The conductors of the three-phase main circuit are usually twisted together to improve voltage regulation. In this situation any fault will affect all phases during the fault clearing process. Therefore, only three-phase short-circuits are considered. The. objective of the short-circuit calculations is to verify the burn-clear-fault condi tions -- the short circuit current is inter-rupted when the conductor is burned; at some critical points of the circuit limiters are used. At this stage only the current flows are needed. The following assumptions are made: - All transformers are operating with voltage V b~hind their internal o reactances. - All loads' are disconnected The sparse bus impedance matrix methodS, 9 is the most efficient when only the short-circuit currents one bus away are needed,as is always. the case in secondary network~.Thus,it is suf ficient to know only those terms of' the imped ance matrix which correspond to the non-zeroelements of. the admittanc7 ~trix [fLI. The sparse bus 1mpedance matr1x 1S constructed from the sparse triangular factors of the admittance matrix. Consider the situation in Fig. 7, representing a short circuit on bus k.
...
" I 'k
-
The fault location is defined by a (O.
P'
I
(19)
11
where Pkk , P , P1k and Pil are elements of kl the sparse impedance matr1x corresponding to the network without the branch k-l. These elements are calculated from the base-case sparse impedance matrix, using the Woodbury formula.
Pkk
Pkk -
2 (~k-~l) -Zkl+(Pkk+Pll-2Pkl) . 2 ) lk -Zkl+(Pkk+Pll-2ekl) (P
P1l
P
11
-
11
-P
(20) Pkl
= Pkl -
P1k
= Pkl
(Pkk-Pkl ) (Plk-P ll ) -Zkl+(Pkk+P1l- 2Pkl )
I~) (21)
L ~ -(V + (I-a) Z If) l o kl 1
-v0
---P
kk °kk-Pkl Pkk
(b)
Fig. 8 - A branch intermediate fault
L :: -(V + aZkl o k
The short circuit currents are given by:
f Ilk
It
Eq. (19) is solved for currents I~ and I~, considering the additional relations:
Fig. 7 - Short-circuit currents
If k
[
Z'd
)f~----"'VVV\N'
./
(t •o.)Zkl - ..
(18) ,
V
0
~
The above approximation is based on the R/X uniformity of the ci rcui t • Substituting Eq. (21) into Eq. (19) we get:
where P and Pkl ~re elements of the sparse bus imp~ance lIUItr1X.
-v0
Branch short-circuit
-v0
The most frequent faults occur at intermedi~ te points in a branch.A solid short-circuit at any point m along branch k-l can be represented by two non-solid ~aults at the extreme
_r'k"'Zkl P1k
Pkl P1l+(l-a)Zkl
[I; If 1 (22)
The line flows to the faulted point m, given by (22), are function of a. So it is easy to verify the burn-clear-fault condition.
282
ACKNOWLEDGEMENTS
NUMERICAL TESTS Table 1 shows voltage drops errors in percent of nominal. branch and transformer loading errors in percent of ratings. These results refer to three of the systems tested: System A - 9 buses. 3 transformers and 12 branches; originally presented in Re£. 6. System B - 63 buses.15 transformers 87 branches; it forms part of the Sao Paulo City underground network. System C - 307 buses.58 transfor mers and 593 branches; it forms pet of the Sao Paulo City underground network. TABLE 1 Results of the base-case tests
ERRORS IN VOLTAGE DROP % OF Vo ERRORS IN BRANCH UW>INGS % OF RATING ERRORS IN 'lRANS FORMER LOAII~ % OF RATINGS
A
B
C
AVERAGE
.18
.02
.05
MAXIMUM
.2
.1
.4
AVERAGE
.8
.8
1.0
MAXIMUM
2.0
3.4
4.4
AVERAGE
.7
.2
.8
MAXIMUM
1.0
.4
3.2
It can be seen that for practical ~urposes the inaccuracies introduced by the simpli.fied models are acceptable. CONCLUSIONS The two DC models have simpler algorithms. less storage_and shorter solution time than the conventional Newton-Raphson method when applied to distribution networks. The same is true for short-circuit analysis. The unified approach presented in the paper is a useful system planning tool. Using the same computer program it is possible to perform any of the following simulations: load-growing studies; primary-feeder. single-transfor mer and branch outages; bus and branch thr~ phase short circuits.
The ·autho~ wish acknowledge their collegues F.Sato and M. Jino for the useful contribution. REFERENCES l-Tinney. W.F. and Hart. C.E. (1967). Power Flow Solutions by Newton's Method. Trans. IEEE.Vol.PAS 86. 1447-1456. ------2-Brown. H.E.(1975).Solution of Large Net works by Matrix Methods. John"Wiley. New York. 3-Westinghouse Electric Corporation (1959). Electric Utility Engineering Reference Book-Distribution Systems. Vol.3. East Pittsburgh 4-Heinhold.L. (1970). Power Cables and their Applications. Siemens Aktiengesellschaft • Berlin. S-Stott. B. (1971). Effective Starting Process for NewtoR-Raphson Load-Flows.Proc. lEE. Vo1. 118, 983-987. -6-Cook. R.F •• and Powers. J.N. (1960). A Digi tal Computer for Secondary Network Analys~ Trans. AIEE. Vol.PAS 79. 941-948. 7~Tinney.·W.F. (1971). Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization. Trans. IEEE. Vol. PAS 91. 123-127. 8-Takahashi. K•• ·Fagan. J •• and Chen. M (1973). Formation of a Sparse Bus Impedance Matrix and its Application to Short Circuit Studies. 8th PICA Conference.Minneapolis. 9-Zo1lenkopf. K. (1975). Sparse Nodal Impedance Matrix Generated by the Bi-factorization Method and Applied to Short Circuit Studies. PSCC Proc., 1-13 (Sec. 3.1/3) • London. .
283
load-flow;
INTRODUCTION The most important objectives of load-flow and short-circuit analysis of underground distribution networks are: Calculations of the voltage magnitudes, current flows and transformer loadings for present arid future conditions. . - Simulations of primary-feeders outages. - Simulations of three-phase shortcircuit at any point of the secondary network in order to verify the burn-clear-fault conditions.
short-circuit;
equivalencing
ondary network. Simplifications have been made which introduce a minimum of inaccuracy in the models. These approximations are acceptable since the loads are rarely known very accurately (10% erro~s are typical). SECONDARY NETWORK SYSTEMS
Usually the first two problems are solved using conventional load~flow algorithms (e.g., Newton-Raphson method l ) and short-circuit analysis .is performed through the imped~nce matrix methods 2 • There are several reasons to believe that these methods, originally developed for high-voltage transmission systems, are not the most adequate for underground distribution network analysis.
Secondary network systems are characterized by the following features: ~ Voltage drops are usually less than 5% of the nominal values. - Nodal voltages are nearly in phase (maximum phase difference between any two nodes is less than. 3 degrees). . - Load power factor is nearly uniform (a typical value is .85). - The three phase system is loaded in a balanced manner, even under short-circuit conditions. - R/X ratios of the cables are near ly uniform • and the charging effect is negligible.
This paper presents a unified approach for load-flow and short-circuit analysis of secondary network systems. The classical AC model is repl~ced by two equivalent DC networks. These equivalent circuits are obtained considering the special characteristics of the sec-
This kind of network 3 serves directly indivi dual or lumped consumers. Specially designed distribution transformers ate connected alternately to distinct radial primary feeder~ and supply the loads through a secondary main grid. The primary feeders are radially 278
connected to the same bulk station avoiding unwanted power flows between stations. Fig.l shows a'typical underground distribution net work.
z
.--
BASIC EQUATIONS The method presented in the paper is ~ased on the solution of two linear systems of nod'al equations:
1 '--
3 If
':i
\I~
6 JI
'"
'y"",
11'
. . ':i~ iI
,- ,
'-)'~
where
:
~'I
,I
li~
11'
'~
11 )'~
,/
5
/ ,t.
-
(1)
[I 1
(2)
[rD] and [r] are the nodal admittarce matric~s of the two 'equivalent DC networks; - (I~ and [I] are the equivalent load current v~ctors;
'y~
Fig.l - Underground distribution network 1 2 3 4 5 6
b4
- [DJ and [L] are the nodal volt, age drop and nodal voltage difference vectors. The components of the vectors are given b'y:
Bulk substation Primary feeders (15-20 kv) Manual primary switch Three-phase transformer Secondary mains (110-220 v) Automatic network protector
(3)
(4)
DEFINITIONS AND NOTATIONS
v - magnitude of the line-to-neutral voltao
ge behind the reactance of the network transformers (reference voltage)
V - voltage magnitude at load-bus k k Ok - voltage angle at load-bus k (0 0
•
o
0 )
Ek=Vkexp(jok) - complex voltage at load-bus k Re{Ek}=VkcosO Dk=Vk-V
- real part of the complex voltage E k - nodal voltage drop (bus k)
~l+j~l
- impedance of the branch k-l
k
o ~+j~ - impedance of the transformer k
Zk - impedance magnitude of the transformer k Zkl - impedance magnitude of the branch k-l I
kl
r
- equivalent resistance of the branch k-l kl Sk - apparent power-kva (bus k) k
i-Given the load gowers Sk' the load currents I are calculated from (3).' k ii - The voltage drop vector (D] is obtained from (1). iii - The exact load currents I are k calculated from (4).' iv - The voltage difference vector [L) is obtained from (2). v - The current flows I are obkl ained from (5):
- magnitude of the current flow from bus k to bus 1 - magnitude of the vector difference of the complex voltages (IEk~El I) at the terminals of the-orinch k-l
I
The solution of the problem is obtained using a five-step procedure:
- magnitude of the load current at bus k
Equation (1) gives the voltage drops with, good accuracy, but these voltages cannot be used to calculate the current flows. On the other hand, equation (2) gives the voltage difference vector, which has no physical meaning by itself, but can be used to compute the current flows very accurately.
- power factor of the load k
cos~k
n - number of load buses lE~
(5)
- means that bus 1 is connected to bus k, excluded the case k=l
If - short-circuit currents - vectors and matrices are denoted by brackets
JT
. d'~cates - ~n
THE MODEL FOR VOLTAGE DROP CALCULATIONS A branch k-l of the' secondary network is represented in Fig. 2.(a). The diagram of Fig. 2.(b) shows the relationship among the voltage at terminal nodes, voltages drops and current flow.
, .
transpos~t~on
279
, [k
Rkl
Xkl
El
®'~0
~
Jkl'
.( a1
I~
~ig.
The the Fig. 2 - Voltage drops In the secondary network distribution systems the nodal voltages are nearly in phase. It follows that the nodal voltage magnitude'! an be approximated by:
3 - Load representation t· vo tage dr p ,dlculations
c~nductances (Sk/v~).are ~?Lroduced in of the matnx I r Dj (Eq. (1» •
d~agonal
For example, cons1der the svstem partially repr~sented in Fig. 4(a,b). The equivalent DC network used for voltage drop calculations is represented in Fig. 4.(c).
(6)
For example, consider the critical and unusual case where dk--3°. The error introduced by approxi.tion (6} is only .14% of the nominal voltage Vo. It can be seen from Fig. 2.(b) that for smaller angles the errcrsintro duced in (6) are negligible. It follows from !q. (6) that the voltage drop across the branch k-l is given by D
kl
CD CD~-----:;~---~
- D '-D ;: Re{E } - Re{E }k
1
k
. 1
(7)
(~lcos~ + ~lsen~) I
Fig~
4(a) - Secondary network
kl Eq. (7) is the basic relation for derivation of the first equivalent DC network~. The equ! valent DC resistance is defined as (8)
where ~ is taken as the of the system.
~verage
power factor
The magnitude of the load current is (9)
Ik -
Fig. 4. (b) - AC representation
As a rule ~«v , and thus, Eq. (9) can be approximated as o follows 5 Dk -1 (l + -V-) .0
From Eq. (10) the load current I can be k represented as shown m Fig. 3.
(10)
I Fig, 4.(c) - Equivalent DC network for voltage drop calculaticns 280
THE MODEL FOR CURRENT FLOW
CALCULATIONS The equivalent DC network used to calculate the current flows is represented in Fig. 5. and has the following main characteristics: - The loads are modelled by constant current injections obtained from (4). - The lines and the transformers are represented by the magnitude of the complex impedances (Zkl and Zk).
Let the vectorlHi] be null except for -1 in position ki, where ki is the low voltage ter mina 1 of the transformer i connected to the primary feeder k. The algorithm for multiple network-transform er outage simulation is: i-Compute the n x 1 vector [xi] for i=l ••••• nk. by solving: [rD] [xi]
[Hi)
(11)
ii - Construct the nk x nk matrix defined as: Zip
c xt
Zpi
for i, pc!. •• nk Zii
Xki - Zkl
(12)
iii - Calculate the nk x nk matrixlYJ:
The two DC models are used.to calculate the voltage drops at load buses. current flows in the main grid and the loading of the t~ formers: The following data are assumed known: .
[Y]
PRIMARY FEEDER OUTAGES The triangular factors of the nodal admittan· ce matrices are assumed known from the base= case studies. Since several network-transformers are connec ted to the same primary feeder. a single prY mary feeder outage yields a multiple network transformer outage. as shown in Fig. 6. These changes in the circuit are simulated by the compensation method • Thus, the admit tance-matrix reconstruction and retriangulation are not required.
and network
=
[zt l
(13)
iv - Construct the ~ x 1 vector IT]: IT]
-I z]-l [~Dl
(14).
where the ~ x 1 vector [~Dl is defined as:
- The apparent load power (kva). - Load power factor. - Parameter and topology of the main. grid. - Network-transformer location. rating and impedance. - Primary feeder arrangement.
Fig. 6. Primary-feeder transformers
• i ; k
where Zki is the impedance magnitude of the transformer connec ted to the node ki, and Xki is the ki-th element of the vector[XP] • .
Fig. 5. Equivalent DC network for current flow calculations BASE-CASE LOAD FLOW
[zl
~i
= ~i
' for i-I ••• nk
(15)
(~.1 is the base-case . voltage
drop on bus ki)
v -
Calculate the post-outage volta . ge drop vector [D 1 given by: -
[0)=[00 ]
+[X)[T]
(16)
• where [DO) is the base-case vol tage drop vector, and the n x nk matrix [ X1 is [X] = [xl: x 2 l .. lxnk ] (17)
Note that the same procedure presented here can be used to calculate the current flow&. There is an alternative approximate approach which can be used with computational advanta ge. This method is based on the observationthat if one network-transformer is removed. only the voltages of the adjacent buses are significantly affected. It follows that the multiple network-transformers outage can be simulated as a sequence of single outages , each one considered as an independent event. The . final result is obtained by the superposition of the individual effects •
281
Therefore, it is not necessary to store the n ~ ~ matrix (X.J,. since the n x 1 vectors [X1J are calculated one at a time.
nodes with the branch removed. This situation is show in Fig. 8.
(d)
SHORT CIRCUIT CALCULATIONS Short circpit on load-buses
'CD
The conductors of the three-phase main circuit are usually twisted together to improve voltage regulation. In this situation any fault will affect all phases during the fault clearing process. Therefore, only three-phase short-circuits are considered. The. objective of the short-circuit calculations is to verify the burn-clear-fault condi tions -- the short circuit current is inter-rupted when the conductor is burned; at some critical points of the circuit limiters are used. At this stage only the current flows are needed. The following assumptions are made: - All transformers are operating with voltage V b~hind their internal o reactances. - All loads' are disconnected The sparse bus impedance matrix methodS, 9 is the most efficient when only the short-circuit currents one bus away are needed,as is always. the case in secondary network~.Thus,it is suf ficient to know only those terms of' the imped ance matrix which correspond to the non-zeroelements of. the admittanc7 ~trix [fLI. The sparse bus 1mpedance matr1x 1S constructed from the sparse triangular factors of the admittance matrix. Consider the situation in Fig. 7, representing a short circuit on bus k.
...
" I 'k
-
The fault location is defined by a (O.
P'
I
(19)
11
where Pkk , P , P1k and Pil are elements of kl the sparse impedance matr1x corresponding to the network without the branch k-l. These elements are calculated from the base-case sparse impedance matrix, using the Woodbury formula.
Pkk
Pkk -
2 (~k-~l) -Zkl+(Pkk+Pll-2Pkl) . 2 ) lk -Zkl+(Pkk+Pll-2ekl) (P
P1l
P
11
-
11
-P
(20) Pkl
= Pkl -
P1k
= Pkl
(Pkk-Pkl ) (Plk-P ll ) -Zkl+(Pkk+P1l- 2Pkl )
I~) (21)
L ~ -(V + (I-a) Z If) l o kl 1
-v0
---P
kk °kk-Pkl Pkk
(b)
Fig. 8 - A branch intermediate fault
L :: -(V + aZkl o k
The short circuit currents are given by:
f Ilk
It
Eq. (19) is solved for currents I~ and I~, considering the additional relations:
Fig. 7 - Short-circuit currents
If k
[
Z'd
)f~----"'VVV\N'
./
(t •o.)Zkl - ..
(18) ,
V
0
~
The above approximation is based on the R/X uniformity of the ci rcui t • Substituting Eq. (21) into Eq. (19) we get:
where P and Pkl ~re elements of the sparse bus imp~ance lIUItr1X.
-v0
Branch short-circuit
-v0
The most frequent faults occur at intermedi~ te points in a branch.A solid short-circuit at any point m along branch k-l can be represented by two non-solid ~aults at the extreme
_r'k"'Zkl P1k
Pkl P1l+(l-a)Zkl
[I; If 1 (22)
The line flows to the faulted point m, given by (22), are function of a. So it is easy to verify the burn-clear-fault condition.
282
ACKNOWLEDGEMENTS
NUMERICAL TESTS Table 1 shows voltage drops errors in percent of nominal. branch and transformer loading errors in percent of ratings. These results refer to three of the systems tested: System A - 9 buses. 3 transformers and 12 branches; originally presented in Re£. 6. System B - 63 buses.15 transformers 87 branches; it forms part of the Sao Paulo City underground network. System C - 307 buses.58 transfor mers and 593 branches; it forms pet of the Sao Paulo City underground network. TABLE 1 Results of the base-case tests
ERRORS IN VOLTAGE DROP % OF Vo ERRORS IN BRANCH UW>INGS % OF RATING ERRORS IN 'lRANS FORMER LOAII~ % OF RATINGS
A
B
C
AVERAGE
.18
.02
.05
MAXIMUM
.2
.1
.4
AVERAGE
.8
.8
1.0
MAXIMUM
2.0
3.4
4.4
AVERAGE
.7
.2
.8
MAXIMUM
1.0
.4
3.2
It can be seen that for practical ~urposes the inaccuracies introduced by the simpli.fied models are acceptable. CONCLUSIONS The two DC models have simpler algorithms. less storage_and shorter solution time than the conventional Newton-Raphson method when applied to distribution networks. The same is true for short-circuit analysis. The unified approach presented in the paper is a useful system planning tool. Using the same computer program it is possible to perform any of the following simulations: load-growing studies; primary-feeder. single-transfor mer and branch outages; bus and branch thr~ phase short circuits.
The ·autho~ wish acknowledge their collegues F.Sato and M. Jino for the useful contribution. REFERENCES l-Tinney. W.F. and Hart. C.E. (1967). Power Flow Solutions by Newton's Method. Trans. IEEE.Vol.PAS 86. 1447-1456. ------2-Brown. H.E.(1975).Solution of Large Net works by Matrix Methods. John"Wiley. New York. 3-Westinghouse Electric Corporation (1959). Electric Utility Engineering Reference Book-Distribution Systems. Vol.3. East Pittsburgh 4-Heinhold.L. (1970). Power Cables and their Applications. Siemens Aktiengesellschaft • Berlin. S-Stott. B. (1971). Effective Starting Process for NewtoR-Raphson Load-Flows.Proc. lEE. Vo1. 118, 983-987. -6-Cook. R.F •• and Powers. J.N. (1960). A Digi tal Computer for Secondary Network Analys~ Trans. AIEE. Vol.PAS 79. 941-948. 7~Tinney.·W.F. (1971). Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization. Trans. IEEE. Vol. PAS 91. 123-127. 8-Takahashi. K•• ·Fagan. J •• and Chen. M (1973). Formation of a Sparse Bus Impedance Matrix and its Application to Short Circuit Studies. 8th PICA Conference.Minneapolis. 9-Zo1lenkopf. K. (1975). Sparse Nodal Impedance Matrix Generated by the Bi-factorization Method and Applied to Short Circuit Studies. PSCC Proc., 1-13 (Sec. 3.1/3) • London. .
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