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Identification of feasible operating machine combinations using efficient combinational techniques
Electric Power Systems Research, 7 (1984) 225 - 229
225
Feasible Operating Machine Combinational Techniques Identification
of
Combinations
Using
Efficient
E. P. M. BROWN, B. J. BROWN and K. R. JOHNSON
New Zealand Electricity, Ministry of Energy, Wellington (New Zealand) (Received October 20, 1983)
SUMMARY
A combinational technique which efficiently identifies all machine combinations whose short-circuit fault level exceeds the circuit breaker rupturing capacity o f the power system is presented. The method recognizes identical machine groupings, identifies only machine combinations which lie within a feasible upper bound and uses partitioning techniques. Only 196 machine combinations require evaluation when the presence o f 13 machines is varied.
of machines which generate short-circuit fault levels in excess of the circuit breaker rupturing capacity of the p o w e r system must be identified and avoided in 'day to day' system operation. In this paper a very efficient 'combinational' technique which identifies all combinations of machines whose short-circuit fault levels exceed the circuit breaker rupturing capacity of the p o w e r system is presented and tested.
2. R A P I D I D E N T I F I C A T I O N O F P E R M I S S I B L E OPERATING MACHINE COMBINATIONS 1. INTRODUCTION
Short-circuit studies are used in the daily operation of p o w e r systems to predict fault MVA levels throughout the system [ 1 - 3]. The concentration of generation in certain areas has resulted in short-circuit fault levels exceeding the circuit breaker rupturing capacity of the system, requiring system splitting and re-arrangement (to lower fault levels) and/or restrictions on the system operation to ensure that the circuit breaker rupturing capacities are n o t exceeded [4]. The New Zealand Electricity Operations Division's policy is to place restrictions on the machine combinations to ensure that the short-circuit fault levels are less than the circuit breaker rupturing capacity of the system. Tables of 'limiting' machine combinations are found b y assessing the fault level associated with the different machine combinations (using a short-circuit analysis program), and excluding any combination whose fault level exceeds the circuit breaker rupturing capacity. Probabilistic short-circuit analysis [ 5 - 7 ] cannot be applied in the 'operational' sense because all combinations 0378-7796/84/$3.00
The m e t h o d for identifying permissible machine combinations assumes that all transmissions lines are in service and only the n u m b e r of machines in service is varied. As a result entries within the permissible machine combination table are conservative. Let A denote the set containing all possible combinations of machines. In a p o w e r system, A is very large. It is impractical to evaluate the short-circuit levels in the p o w e r system for each combination in set A. Machines which are electrically remote from the buses of 'interest' (whose fault MVA levels are close to circuit breaker rupturing capacities) generally only contribute small fault in-feeds. These 'remote' machines can be assumed to be always in service. Set B contains all possible combinations of machines which are electrically close to the buses of interest. Set B is a subset of A, BC A
(I)
The fact that some groups of machines at stations are identical, having the same subtransient reactance and thus the same fault © Elsevier Sequoia/Printed in The Netherlands
226
in-feed, dramatically reduces the number of combinations which are different (see Table 1): CC B
(2)
where C is the set of electrically 'close' machine combinations, and identical machine groupings are evaluated only once. TABLE 1 N u m b e r of d i f f e r e n t c o m b i n a t i o n s w h e n identical m a c h i n e s are p r e s e n t No. o f i d e n t i c a l machines
No. o f d i f f e r e n t combinations
1 2 3 4
2 3 4 5
out out out out
of of of of
2 4 8 16
2.1. Determination o f a permissible machinecombination upper bound Further reductions in the membership of set C can be made by identifying only those permissible combinations of machines whose short-circuit fault MVA level is less than the circuit breaker rupturing capacity b y an a m o u n t which does n o t exceed the fault contribution, ~, of the largest machine in the study, i.e. only the different machine combinations are identified which lie within the 'upper b o u n d ' band (set D), where
D-
(XEC:~--~<
FL(X)< ~)
(3)
where FL(X) is the resulting fault MVA level, at the n busbars of interest, when the machine combination X is present: EL(X) -= {FL1(X), FL2(X), ..., FL,(X)}
(4)
and q' is the set defining the circuit breaker rupturing capacities at the n busbars of interest:
= { ~ , , ~2, . . . , ~ , }
(5)
In more detail (3) becomes D-
(8)
where eIa is the subset of el containing all machines in service, and elb is the subset of el containing all machines out of service. (3) If the resulting fault MVA level satisfies - - ~ ~< EL(X) < @
(9)
then the machine combination is a m e m b e r of set D and should be left in the table. (4) If the fault MVA level satisfies EL(X) < ~ --/3
(10)
then the combination, (ela, elb}, should be removed from the table and all other combinations in the table which have additional machines out of service, satisfying (eia , {3ib} ~---{Cla - - 6, •lb + 67}
(11)
should also be removed, where 6 is some additional group of machines and ~ia C ~la
(12)
EL(X) ~> ~ (6)
Note also that D C C
elb )
(5) If the fault MVA level satisfies
FL2(X) < ~2]
n ... n [ ~ , - - ~ < F L , ( X ) < ~ , ] )
e 1 ~---{ e l a ,
~lb C fib
{[~i--34 ELI(X)< ~i] n [~2--~<
Identifying all machine combinations which lie within the upper b o u n d (6) is the same as identifying all machine combinations whose fault MVA is less than the circuit breaker rupturing capacity, because the removal of any number of machines from any machine combination in the upper feasible b o u n d {6) will still be a feasible operating condition {with reduced fault in-feeds at the busbars of interest). The following simple procedure identifies all machine combinations which lie within the upper b o u n d band, given by (6). (1) Build up a combination table showing the machines in service and those o u t of service, corresponding to each element of set C. (2) Use the short~ircuit analysis program to evaluate the fault MVA levels for the first combination in the table, el, which is the set of machines in and o u t of service corresponding to the first entry in the table:
(7)
(13)
then the combination {ela, elb) should be removed from the table and all other combinations which have additional machines in service, satisfying
227 {eja, eyb- ) = (el a + ~, el b - - ~)
(14)
F -= ( X E E :
should also be removed, where ela C eja
(15) ejb C
elb
(6) Repeat steps (2)-(5) for the next remaining entry in the modified table. (7) After all entries in the modified combination table have been evaluated, the remaining entries correspond to all the members of the upper bound, (6).
2.2. Partitioning the machine combinational problem Further reductions in the number of shortcircuit analysis program re-evaluations required to identify all feasible machine combinations within the upper bound band (given by (6)) can be achieved by partitioning the combinational problem. Let m denote the set of machines which axe to be varied in order to identify the feasible upper bound: m - ( m l , m 2 , . . . , mi . . . . , mk)
(16)
Partition the set of machines m - (ma, mb}
(17)
where ma = ( m l , m2, ..., m i )
(18) m b : (mi+ 1. . . . , mk)
and
maCm (19) mbCm
First evaluate all the m~ combinations with all the mb machines in service. The total number of combinations of the m~ machines is given by set E: E C C
axe the members of set F axe noted, where
(20)
The final number of evaluations will be reduced if the partitioning of m is similar to the partitioning of set A (to give set B), i.e. subset ma should contain those machines with the greatest fault contribution of the buses of interest. All machine combinations which exceed the lower limit of the upper band and which
FL(X)~> ~ - - ~ }
(21)
All members of F axe found simply by applying steps ( 1 ) - ( 4 ) , ( 6 ) - ( 7 ) outlined in the previous section. To 'efficiently' identify the feasible machine combination upper b o u n d band (6): (i) Select the first element of set F and fix the machines belonging to set ma to the first element combination. Vary the machines in set mb and use the procedure ( 1 ) - ( 7 ) outlined in the previous section to identify members of the feasible upper b o u n d band (corresponding to the first element of set ma). (ii) Repeat step (i) for all remaining elements of set F. (iii) The feasible machine combination upper bound, (6), has now been identified by the 'partitioning' process.
3. RESULTS The New Zealand Electricity North Island p o w e r system (which has more than 200 busbars, over 200 transmission lines, and 84 generators) was used in the study. When all machines axe in service, fault MVA levels exceed the circuit breaker rupturing capacities of Otahuhu 110 kV and Penrose 110 kV busbaxs in the northern part of the North Island. The circuit breaker rupturing capacities of these busbaxs axe 5000 and 4000 MVA respectively. The generators which axe electrically 'close' to these busbaxs axe Otahuhu generators Otahuhu generators Meremere generators Marsden generators Huntly generator
G1, G2
G3, G4, Gs, G6 G1, G2, G3, G4 G1, G2
G1
Only the presence of these machines is varied. All other commissioned machines, transmission lines and transformers (which have not been switched out) remain in service. Varying the presence of all 13 machines generates 2 ~3 or 9124 different combinations, i.e. set B has 9124 members. Note that the generators grouped above axe identical within each grouping. The number of different combinations which need evaluating is therefore, from Table 1,
228 TABLE 2 Comparison of methods for evaluating short-circuit studies No. of generators varying
No. of Time to combinations evaluate all combinations
(s)
Time to evaluate upper boundary
Efficiency : No. of evaluations Total No. of combinations
(s)
No. of combinations lying within upper boundary
Identification of machine combinations in the feasible bound (§ 2.1)
6 (2, 2, 2)
27
11
--
6 (2, 4)
15
7
--
7 (2, 4, 1)
30
12
--
8 (2, 4, 2)
45
20
--
7 3 2 13
m
Machine combinational 'partitioning' (§ 2.2) 13 (2, 4, 4, 2, 1) Case A
450
-
67
196/450
69
13 ( 2 , 4 , 4 , 2, 1) Case B
450
--
26
68/450
23
13 (2, 4, 4, 2, 1) Case C
450
--
42
97/450
57
Case A: Otahuhu 110 kV bus solid; case B: Otahuhu 110 kV bus split; case C: Otahuhu 110 kV bus (alternative split).
( 2 - 4 - 4 - 2 - I)=(3)(5)(5)(3)(2) =450set C, i.e. the short-circuit analysis program has to be re-run 450 times. This program uses the sparse nodal impedance matrix formulation of Takahashi et al. [1] and takes 0.42 s on an IBM 3033 to calculate all three-phase and single-phase fault in-feeds throughout the whole North Island. Re-running the program 450 times would take 189 s of CPU time on an IBM 3033. Instead, only the members of the feasible machine combination upper bound are identified. The fault contribution of the largest generating unit, ~, is 350 MVA. Therefore the feasible upper bound becomes: ([4650 ~< FLoTA,,0(X) < 5000] n [3650 ~< FLpEN,,0(X) < 4000] } In practice, it proves easier to identify a broader feasible upper bound band where only the upper boundary is enforced at Otahuhu:
T h e 1 3 - m a c h i n e p r o b l e m is p a r t i t i o n e d into t w o s u b - p r o b l e m s and initially o n l y the six O t a h u h u g e n e r a t o r s are varied. F o r the case w h e r e the O t a h u h u 110 kV bus is solid (and the fault levels c o r r e s p o n d i n g l y high), variat i o n o f the six O t a h u h u g e n e r a t o r s identifies eight m a c h i n e c o m b i n a t i o n s w h i c h are members o f set F. F o r each m a c h i n e c o m b i n a t i o n w h i c h is a m e m b e r o f set F, t h e seven rem a i n i n g g e n e r a t o r s at Meremere, Marsden and H u n t l y are t h e n varied. I d e n t i f i c a t i o n o f all m a c h i n e c o m b i n a t i o n s w h i c h are m e m b e r s o f t h e feasible u p p e r b o u n d (6), using the p a r t i t i o n i n g m e t h o d , o n l y t o o k 67 s o n t h e IBM 3 0 3 3 (a r e d u c t i o n in t h e CPU t i m e o f m o r e t h a n 60%). Test results are s h o w n in Table 2. W h e n O t a h u h u 110 kV bus is split, t h e f a u l t M V A levels are c o r r e s p o n d i n g l y lower; the m e m b e r s h i p o f set F decreases and t h e t i m e r e q u i r e d t o evaluate the 13m a c h i n e c o m b i n a t i o n a l p r o b l e m is even less.
4. FURTHER IMPROVEMENTS
( [ 4 6 5 0 ~< FLOTA,,0 (X) < 5 0 0 0 ]
n [I • FLpEN,,0(X) < 4000]}
The sparse n o d a l i m p e d a n c e m a t r i x m e t h o d o f T a k a h a s h i et al. [1] was used in the s t u d y
229 to evaluate single- and three-phase fault levels throughbut the entire North Island p o w e r system. If the fault level is critically close to the circuit breaker rupturing capacity at only a few buses and no other short-circuit violations are likely to occur, Reichert and Hager's m e t h o d [3] (which involves only computing a restricted 'column-wise' inversion of the nodal admittance matrix) can be used in conjunction with the efficient multi-machine combinational technique outlined in §2. The computing time required to evaluate the m e t h o d of Takahashi et al. is proportional to nw2 (where n is the order of the nodal admittance matrix and w is the average number o f elements), while the computing time required to evaluate Reichert and Hager's m e t h o d is only proportional to w 2. 5. CONCLUSION A technique which rapidly identifies machine combinations whose short-circuit fault level exceeds the circuit breaker rupturing capacity of the p o w e r system has been presented and tested. The proposed technique, which recognizes identical machine groupings at stations (machines having the same sub-transient reactances), only identifies all machine combinations which lie within a feasible upper b o u n d band and employs partitioning o f the combinational problem to reduce the n u m b e r of combinations of a 13-machine problem from 9124 to 196, which can be evaluated in 67 s on an IBM 3033 computer. The resulting table of permissible 'operating' machine combinations (i.e. the feasible upper b o u n d band) is then forwarded to a system control centre for use by the operators. Advantages of the table are: (1) Operators have a better 'overview' when the tables are used because t h e y k n o w in advance all combinations o f machines which will cause fault levels in excess of the circuit breaker rupturing capacity of the p o w e r system. (2) The short-circuit 'permissible' machine combination tables are unaffected by computer downtime. (3) Use of the tables requires no extra skills on the part of the operators. (4) Use of the tables allows the operator to react faster to unplanned loss of genera-
tion. The operator only needs to check the table to see if the plant a b o u t to be ordered into service to cover the contingency is a permissible combination; whereas the operator who only has recourse to an on-line shortcircuit analysis program at the control centre must set up the proposed changes in generation, solve and repeat the process until the plant being considered for service to cover the contingency is f o u n d to generate fault levels which are less than the circuit breaker rupturing capacity of the p o w e r system. The permissible machine combination 'operating' tables produced using the techniques outlined in this paper have been in use at the North Island System Control Centre for nearly a year.
ACKNOWLEDGEMENTS
The authors are grateful to K. D. McCool, General Manager of New Zealand Electricity, for permission to publish this paper and to W. A. Lowrie, Principal System Control Engineer, for his useful comments.
REFERENCES
1 K. Takahashi, J. Fagan and C. Mo-shing, Formation of a sparse bus impedance matrix and its application to short circuit study, Proe. PICA, IEEE, N e w York, 1973, pp. 63 - 69. 2 K. Zollenkopf, Sparse nodal impedance matrix generated by the bifactorization method and applied to short circuitstudies,Proc. Fifth Power
Systems Computation Conf., Dept. of Electrical
and Electronic Engineering, Queen Mary College, Univ. of London, U.K., 1975, paper No. 3.1/3. 3 K. Reichert and H. Hager, Fast evaluation of short-circuit levels in power systems, Proc. Sixth Power Systems Computation Conf., IPC, Guildford, U.K., 1978, pp. 666 - 677. 4 J. A. Dembecki, E. A. Woodley and R. R. Booth, Trends in on-line security analysis in South East Australia, CIGRE SC 31 and 32 Combined Meeting, Australia, March 1973, Paper No. B10, 14 pp. 5 J. Hoffman, On the stochastic calculation of short circuit currents in electric energy systems, Elektrie, 34 (1980) 11 - 13. 6 M. A. EI-Kady, Probabilistic short circuitanalysis by Monte Carlo simulations, IEEE PES S u m m e r Meeting, San Francisco, CA, 1982, Paper No.
82 SM 431-5. 7 M. A. EI-Kady and G. L. Ford, An advanced probabilistic short circuit program, IEEE PES Summer Meeting, San Francisco, CA, 1982, Paper No. 82 SM 303-6.
225
Feasible Operating Machine Combinational Techniques Identification
of
Combinations
Using
Efficient
E. P. M. BROWN, B. J. BROWN and K. R. JOHNSON
New Zealand Electricity, Ministry of Energy, Wellington (New Zealand) (Received October 20, 1983)
SUMMARY
A combinational technique which efficiently identifies all machine combinations whose short-circuit fault level exceeds the circuit breaker rupturing capacity o f the power system is presented. The method recognizes identical machine groupings, identifies only machine combinations which lie within a feasible upper bound and uses partitioning techniques. Only 196 machine combinations require evaluation when the presence o f 13 machines is varied.
of machines which generate short-circuit fault levels in excess of the circuit breaker rupturing capacity of the p o w e r system must be identified and avoided in 'day to day' system operation. In this paper a very efficient 'combinational' technique which identifies all combinations of machines whose short-circuit fault levels exceed the circuit breaker rupturing capacity of the p o w e r system is presented and tested.
2. R A P I D I D E N T I F I C A T I O N O F P E R M I S S I B L E OPERATING MACHINE COMBINATIONS 1. INTRODUCTION
Short-circuit studies are used in the daily operation of p o w e r systems to predict fault MVA levels throughout the system [ 1 - 3]. The concentration of generation in certain areas has resulted in short-circuit fault levels exceeding the circuit breaker rupturing capacity of the system, requiring system splitting and re-arrangement (to lower fault levels) and/or restrictions on the system operation to ensure that the circuit breaker rupturing capacities are n o t exceeded [4]. The New Zealand Electricity Operations Division's policy is to place restrictions on the machine combinations to ensure that the short-circuit fault levels are less than the circuit breaker rupturing capacity of the system. Tables of 'limiting' machine combinations are found b y assessing the fault level associated with the different machine combinations (using a short-circuit analysis program), and excluding any combination whose fault level exceeds the circuit breaker rupturing capacity. Probabilistic short-circuit analysis [ 5 - 7 ] cannot be applied in the 'operational' sense because all combinations 0378-7796/84/$3.00
The m e t h o d for identifying permissible machine combinations assumes that all transmissions lines are in service and only the n u m b e r of machines in service is varied. As a result entries within the permissible machine combination table are conservative. Let A denote the set containing all possible combinations of machines. In a p o w e r system, A is very large. It is impractical to evaluate the short-circuit levels in the p o w e r system for each combination in set A. Machines which are electrically remote from the buses of 'interest' (whose fault MVA levels are close to circuit breaker rupturing capacities) generally only contribute small fault in-feeds. These 'remote' machines can be assumed to be always in service. Set B contains all possible combinations of machines which are electrically close to the buses of interest. Set B is a subset of A, BC A
(I)
The fact that some groups of machines at stations are identical, having the same subtransient reactance and thus the same fault © Elsevier Sequoia/Printed in The Netherlands
226
in-feed, dramatically reduces the number of combinations which are different (see Table 1): CC B
(2)
where C is the set of electrically 'close' machine combinations, and identical machine groupings are evaluated only once. TABLE 1 N u m b e r of d i f f e r e n t c o m b i n a t i o n s w h e n identical m a c h i n e s are p r e s e n t No. o f i d e n t i c a l machines
No. o f d i f f e r e n t combinations
1 2 3 4
2 3 4 5
out out out out
of of of of
2 4 8 16
2.1. Determination o f a permissible machinecombination upper bound Further reductions in the membership of set C can be made by identifying only those permissible combinations of machines whose short-circuit fault MVA level is less than the circuit breaker rupturing capacity b y an a m o u n t which does n o t exceed the fault contribution, ~, of the largest machine in the study, i.e. only the different machine combinations are identified which lie within the 'upper b o u n d ' band (set D), where
D-
(XEC:~--~<
FL(X)< ~)
(3)
where FL(X) is the resulting fault MVA level, at the n busbars of interest, when the machine combination X is present: EL(X) -= {FL1(X), FL2(X), ..., FL,(X)}
(4)
and q' is the set defining the circuit breaker rupturing capacities at the n busbars of interest:
= { ~ , , ~2, . . . , ~ , }
(5)
In more detail (3) becomes D-
(8)
where eIa is the subset of el containing all machines in service, and elb is the subset of el containing all machines out of service. (3) If the resulting fault MVA level satisfies - - ~ ~< EL(X) < @
(9)
then the machine combination is a m e m b e r of set D and should be left in the table. (4) If the fault MVA level satisfies EL(X) < ~ --/3
(10)
then the combination, (ela, elb}, should be removed from the table and all other combinations in the table which have additional machines out of service, satisfying (eia , {3ib} ~---{Cla - - 6, •lb + 67}
(11)
should also be removed, where 6 is some additional group of machines and ~ia C ~la
(12)
EL(X) ~> ~ (6)
Note also that D C C
elb )
(5) If the fault MVA level satisfies
FL2(X) < ~2]
n ... n [ ~ , - - ~ < F L , ( X ) < ~ , ] )
e 1 ~---{ e l a ,
~lb C fib
{[~i--34 ELI(X)< ~i] n [~2--~<
Identifying all machine combinations which lie within the upper b o u n d (6) is the same as identifying all machine combinations whose fault MVA is less than the circuit breaker rupturing capacity, because the removal of any number of machines from any machine combination in the upper feasible b o u n d {6) will still be a feasible operating condition {with reduced fault in-feeds at the busbars of interest). The following simple procedure identifies all machine combinations which lie within the upper b o u n d band, given by (6). (1) Build up a combination table showing the machines in service and those o u t of service, corresponding to each element of set C. (2) Use the short~ircuit analysis program to evaluate the fault MVA levels for the first combination in the table, el, which is the set of machines in and o u t of service corresponding to the first entry in the table:
(7)
(13)
then the combination {ela, elb) should be removed from the table and all other combinations which have additional machines in service, satisfying
227 {eja, eyb- ) = (el a + ~, el b - - ~)
(14)
F -= ( X E E :
should also be removed, where ela C eja
(15) ejb C
elb
(6) Repeat steps (2)-(5) for the next remaining entry in the modified table. (7) After all entries in the modified combination table have been evaluated, the remaining entries correspond to all the members of the upper bound, (6).
2.2. Partitioning the machine combinational problem Further reductions in the number of shortcircuit analysis program re-evaluations required to identify all feasible machine combinations within the upper bound band (given by (6)) can be achieved by partitioning the combinational problem. Let m denote the set of machines which axe to be varied in order to identify the feasible upper bound: m - ( m l , m 2 , . . . , mi . . . . , mk)
(16)
Partition the set of machines m - (ma, mb}
(17)
where ma = ( m l , m2, ..., m i )
(18) m b : (mi+ 1. . . . , mk)
and
maCm (19) mbCm
First evaluate all the m~ combinations with all the mb machines in service. The total number of combinations of the m~ machines is given by set E: E C C
axe the members of set F axe noted, where
(20)
The final number of evaluations will be reduced if the partitioning of m is similar to the partitioning of set A (to give set B), i.e. subset ma should contain those machines with the greatest fault contribution of the buses of interest. All machine combinations which exceed the lower limit of the upper band and which
FL(X)~> ~ - - ~ }
(21)
All members of F axe found simply by applying steps ( 1 ) - ( 4 ) , ( 6 ) - ( 7 ) outlined in the previous section. To 'efficiently' identify the feasible machine combination upper b o u n d band (6): (i) Select the first element of set F and fix the machines belonging to set ma to the first element combination. Vary the machines in set mb and use the procedure ( 1 ) - ( 7 ) outlined in the previous section to identify members of the feasible upper b o u n d band (corresponding to the first element of set ma). (ii) Repeat step (i) for all remaining elements of set F. (iii) The feasible machine combination upper bound, (6), has now been identified by the 'partitioning' process.
3. RESULTS The New Zealand Electricity North Island p o w e r system (which has more than 200 busbars, over 200 transmission lines, and 84 generators) was used in the study. When all machines axe in service, fault MVA levels exceed the circuit breaker rupturing capacities of Otahuhu 110 kV and Penrose 110 kV busbaxs in the northern part of the North Island. The circuit breaker rupturing capacities of these busbaxs axe 5000 and 4000 MVA respectively. The generators which axe electrically 'close' to these busbaxs axe Otahuhu generators Otahuhu generators Meremere generators Marsden generators Huntly generator
G1, G2
G3, G4, Gs, G6 G1, G2, G3, G4 G1, G2
G1
Only the presence of these machines is varied. All other commissioned machines, transmission lines and transformers (which have not been switched out) remain in service. Varying the presence of all 13 machines generates 2 ~3 or 9124 different combinations, i.e. set B has 9124 members. Note that the generators grouped above axe identical within each grouping. The number of different combinations which need evaluating is therefore, from Table 1,
228 TABLE 2 Comparison of methods for evaluating short-circuit studies No. of generators varying
No. of Time to combinations evaluate all combinations
(s)
Time to evaluate upper boundary
Efficiency : No. of evaluations Total No. of combinations
(s)
No. of combinations lying within upper boundary
Identification of machine combinations in the feasible bound (§ 2.1)
6 (2, 2, 2)
27
11
--
6 (2, 4)
15
7
--
7 (2, 4, 1)
30
12
--
8 (2, 4, 2)
45
20
--
7 3 2 13
m
Machine combinational 'partitioning' (§ 2.2) 13 (2, 4, 4, 2, 1) Case A
450
-
67
196/450
69
13 ( 2 , 4 , 4 , 2, 1) Case B
450
--
26
68/450
23
13 (2, 4, 4, 2, 1) Case C
450
--
42
97/450
57
Case A: Otahuhu 110 kV bus solid; case B: Otahuhu 110 kV bus split; case C: Otahuhu 110 kV bus (alternative split).
( 2 - 4 - 4 - 2 - I)=(3)(5)(5)(3)(2) =450set C, i.e. the short-circuit analysis program has to be re-run 450 times. This program uses the sparse nodal impedance matrix formulation of Takahashi et al. [1] and takes 0.42 s on an IBM 3033 to calculate all three-phase and single-phase fault in-feeds throughout the whole North Island. Re-running the program 450 times would take 189 s of CPU time on an IBM 3033. Instead, only the members of the feasible machine combination upper bound are identified. The fault contribution of the largest generating unit, ~, is 350 MVA. Therefore the feasible upper bound becomes: ([4650 ~< FLoTA,,0(X) < 5000] n [3650 ~< FLpEN,,0(X) < 4000] } In practice, it proves easier to identify a broader feasible upper bound band where only the upper boundary is enforced at Otahuhu:
T h e 1 3 - m a c h i n e p r o b l e m is p a r t i t i o n e d into t w o s u b - p r o b l e m s and initially o n l y the six O t a h u h u g e n e r a t o r s are varied. F o r the case w h e r e the O t a h u h u 110 kV bus is solid (and the fault levels c o r r e s p o n d i n g l y high), variat i o n o f the six O t a h u h u g e n e r a t o r s identifies eight m a c h i n e c o m b i n a t i o n s w h i c h are members o f set F. F o r each m a c h i n e c o m b i n a t i o n w h i c h is a m e m b e r o f set F, t h e seven rem a i n i n g g e n e r a t o r s at Meremere, Marsden and H u n t l y are t h e n varied. I d e n t i f i c a t i o n o f all m a c h i n e c o m b i n a t i o n s w h i c h are m e m b e r s o f t h e feasible u p p e r b o u n d (6), using the p a r t i t i o n i n g m e t h o d , o n l y t o o k 67 s o n t h e IBM 3 0 3 3 (a r e d u c t i o n in t h e CPU t i m e o f m o r e t h a n 60%). Test results are s h o w n in Table 2. W h e n O t a h u h u 110 kV bus is split, t h e f a u l t M V A levels are c o r r e s p o n d i n g l y lower; the m e m b e r s h i p o f set F decreases and t h e t i m e r e q u i r e d t o evaluate the 13m a c h i n e c o m b i n a t i o n a l p r o b l e m is even less.
4. FURTHER IMPROVEMENTS
( [ 4 6 5 0 ~< FLOTA,,0 (X) < 5 0 0 0 ]
n [I • FLpEN,,0(X) < 4000]}
The sparse n o d a l i m p e d a n c e m a t r i x m e t h o d o f T a k a h a s h i et al. [1] was used in the s t u d y
229 to evaluate single- and three-phase fault levels throughbut the entire North Island p o w e r system. If the fault level is critically close to the circuit breaker rupturing capacity at only a few buses and no other short-circuit violations are likely to occur, Reichert and Hager's m e t h o d [3] (which involves only computing a restricted 'column-wise' inversion of the nodal admittance matrix) can be used in conjunction with the efficient multi-machine combinational technique outlined in §2. The computing time required to evaluate the m e t h o d of Takahashi et al. is proportional to nw2 (where n is the order of the nodal admittance matrix and w is the average number o f elements), while the computing time required to evaluate Reichert and Hager's m e t h o d is only proportional to w 2. 5. CONCLUSION A technique which rapidly identifies machine combinations whose short-circuit fault level exceeds the circuit breaker rupturing capacity of the p o w e r system has been presented and tested. The proposed technique, which recognizes identical machine groupings at stations (machines having the same sub-transient reactances), only identifies all machine combinations which lie within a feasible upper b o u n d band and employs partitioning o f the combinational problem to reduce the n u m b e r of combinations of a 13-machine problem from 9124 to 196, which can be evaluated in 67 s on an IBM 3033 computer. The resulting table of permissible 'operating' machine combinations (i.e. the feasible upper b o u n d band) is then forwarded to a system control centre for use by the operators. Advantages of the table are: (1) Operators have a better 'overview' when the tables are used because t h e y k n o w in advance all combinations o f machines which will cause fault levels in excess of the circuit breaker rupturing capacity of the p o w e r system. (2) The short-circuit 'permissible' machine combination tables are unaffected by computer downtime. (3) Use of the tables requires no extra skills on the part of the operators. (4) Use of the tables allows the operator to react faster to unplanned loss of genera-
tion. The operator only needs to check the table to see if the plant a b o u t to be ordered into service to cover the contingency is a permissible combination; whereas the operator who only has recourse to an on-line shortcircuit analysis program at the control centre must set up the proposed changes in generation, solve and repeat the process until the plant being considered for service to cover the contingency is f o u n d to generate fault levels which are less than the circuit breaker rupturing capacity of the p o w e r system. The permissible machine combination 'operating' tables produced using the techniques outlined in this paper have been in use at the North Island System Control Centre for nearly a year.
ACKNOWLEDGEMENTS
The authors are grateful to K. D. McCool, General Manager of New Zealand Electricity, for permission to publish this paper and to W. A. Lowrie, Principal System Control Engineer, for his useful comments.
REFERENCES
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