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Optimal operation planning of distribution systems considering security constraints
Optimal operation planning of distribution systems considering secur=ty constra,nts N A Mijuskovic Faculty of Electrical Engineering, Stojana Protica 34, 11 000 Belgrade, Yugoslavia
The paper concerns the development of new approaches to the optimal operation planning of radial distribution networks. In particular the model described attempts to configure the medium voltage networks radially for low losses, while meeting constraints on security (reliability), currents and voltages. The model appears to be a very useful tool for planning the normal operating configuration of the distribution system and .for reconfiguring radial feeder systems during either emergency or planned outages. Keywords: distribution systems, reliability
operation
planning,
I. I n t r o d u c t i o n This paper concerns the development of new approaches to the planning of radial primary distribution systems. In particular, the research has addressed two problem areas. (1) The development of optimization models for arranging or rearranging the radial structure of a system of feeders - for planning the normal operating configuration of the system. (2) The development of optimization models for reconfiguration (made-possible by manual or automatic switching) of a radial system of feeders during either emergencies or planned outages. The models can be used by electric utilities for a variety of distribtion expansion planning functions, including: (1) transferring electric load from one feeder to another or otherwise redistributing load as it changes in magnitude and geographic distribution over a multiyear planning horizon;
Received January 1992; revised April 1992
Vol 1 4 No 6 December 1 992
(2) lowering losses in radial distribution circuits through better feeder configuration; (3) redesigning the arrangement of the radial feeder system in conjunction with the upgrading of the distribution systems as when converting to higher primary voltages. Both the problem of optimizing the normal configuration of a radial distribution, and the problem of optimizing an emergency reconfiguration require manipulation of radial circuit arrangements. A paper by Garret 1 approached the problem of expansion planning of radial subtransmission systems by generating all the spanning trees of the circuit's graph and then checking them for feasibility and cost. A program, called ACORN 2, was developed by SCI and SE in 1974 for expansion planning of radial subtransmission or distribution systems; it used heuristics to add feeder sections to the system, one by one, until all loads were supplied thereby constructing a radial configuration. Several people at Electricite de France (EDF) have developed methods that address this problem. The SECOURS model 3 was developed about fifteen years ago by EDF; it used a heuristic method to attain a radial configuration that minimized the unsupplied load, taking into account feeder capacity constraints. Several years ago EDF extended SECOURS to obtain a new model called 'CORALI 4. CORALI retains the same objective but includes voltage constraints; again, it uses heuristics with some gains in computational efficiency. Another basic technique developed by EDF, called MINIPERT 5, searches for the lowest-losses radial configuration. Backlund and Bubenko 6 describe a model for expansion planning that uses heuristics and that also uses an iterative tree generation algorithm to search for configuration improvements Aoki et al. ~1, developed a method which incorporates both the merits of the mixed integer programming model and the branch-exchange model. But the problem of fast optimal evaluating of a radial network configuration still exists.
0142-0615/92/60383-04 © 1992 Butterworth-Heinemann Ltd
383
II. C h a r a c t e r i s t i c s of t h e model
where
Early in the research phase it was decided that the model would be a static, rather than dynamic, optimization model. That is, it would determine, for each year (or stage) of an expansion plan, an optimum configuration for that year. The planner would then have to combine these static analyses of separate years into an overall expansion plan. This choice of a static approach was made for the sake of keeping the model computationally tractable. A further simplification was that the planner was assumed to have in mind a limited number of reinforcement candidates and/or new locations of ties between pairs of feeders. In each ~run' of the model, the planner would pre-specify these candidates and the output would show which candidates were actually used in obtaining the best configuration. In short, the model would not itself perform the economic selection of alternative expansion candidates; rather, it would identify what configuration met all operational constraints (e.g., on voltages, current, and security) while minimizing some operating cost objective in the year of analysis. It is assumed that annual peak load value is used for calculating the current values and voltage drop. The daily, weekly or seasonal load pattern could be taken into account using an annual load factor. With the above considerations, we formulated the problem as follows. The objective of the model is to find a radial configuration from among all available feeder branches that supplies all the demands while minimizing the operating cost of the system, subject to operating constraints and security constraints, The operating cost of the system is assumed to be adequately represented by the system circuit losses (active power). The operating constraints are to satisfy:
I is the (phase) current in the feeder section, R is the resistance of the feeder section, X is the reactance of the feeder section, cos ~b is the power factor of the distribution system.
(1) all the demands; (2) the current capacity of each feeder branch: (3) the node voltage limits.
(1) The three phases are balanced, so that a single phase equivalent can be used. (2) The distribution system operates at a constant power factor. (3) The shunt capacitance of each feeder section is negligible. The first assumption is valid to the extent that the utility strives to balance phases and taking into account the fact that the method concerns medium voltage networks (10kV, 20 kV, 35 kV). The second assumption is generally valid; economics usually dictate that a power factor of about 0.95 be maintained. The second assumption is very good except for feeders that use shunt capacitance for voltage control; but even under such conditions, the simplified load flow model tends to err on the conservative side - tending to yield lower voltages than are actually present. With these assumptions, the (phase-to-phase) voltage drop along a feeder section between any two load-points k and m becomes: Vk - V , , , = R l c o s ~ + X l s i n ~
384
z = R cos q5+ X sin ~b
(3)
Vt -- Vm=Zl
Elsewhere in this paper, all 'impedances' mentioned will thus be interpreted as equivalent impedances, per equation (2).
III. The n e t w o r k model The approach is based on a linearized network power flow model. The nodal voltage problem for a linear network is that of finding the nodal voltage vector when the nodal current injection vector J is known and is related to V by the admittance matrix Y as: (4)
J= YV=MTYb m
where n number of nodes, Yb the diagonal matrix of the branch admittances, M the branch-node incidences matrix, M T the transpose of M, h(i) the set of branches connected to bus i.
~ii= 2 (-- Yiij) jeh(i)
Yij-~- --1/zij
The equation (4) can be partitioned into the next matrix relation (5):
(5)
where the vector of currents in the feeders between the substations HV/MV and first substation MV/LV ('getaways') Jt the vector of currents in the loading nodes (substation MV/LV) Vg the voltage vector of the buses in the substations HV/MV V~ the voltage vector of the buses in the substations MV/LV Y, the submatrix which has diagonal elements:
Jg
Yii = Yij + S . / V . 2
(6)
For the elements of the matrix Yn defined by equation (6) the elements of the current injection vector El are zero. The elements of the current injection vector J . are defined by equation (7):
where Jc nj
ng (1)
(2)
one obtains the following relation:
~_~ ~JLV~_I
A full AC load flow model was deemed too complex for the analysis. Accordingly, a simplified load flow model is used. This model arises from the following simplifying assumptions.
= l [ R c o s ~ + X sin ~b]
Defining the equivalent impedance, z, of a feeder section as :
the current capacity of each 'getaway' branch the number of the loading nodes the number of the feeders between substations HV/MV and first substation MV/LV ('getaways')
Electrical Power & Energy Systems
Jc often has the same value for all getaways in the distribution network of the same voltage. In that c a s e Jg has the same values for all getaways. The idea is that the optimum operating point is achieved when the getaways are equally loaded in the case when the current capacity of each getaway branch are the same or relatively equally loaded when those values are different. Considering that the elements of the v e c t o r Jg are defined by equation (7) and the elements of the vector J1 are equal to zero, the elements of the vector V~ are given by: V~= (-- y, gyggl Ygl- Yll)-i y~gygg 1j,
(8)
V, = Y**(J,- Yg,V~)
(9)
The equation (8) can be easily reduced to the equation (10). V~= Ab
(10)
The matrix A is the sparse symmetric positive definitive matrix. It can be transformed into L D L T form by the Cholesky method. The algorithm of the heuristic optimization process consists of the following conceptual steps. (1) Form the bus admittance m a t r i x - Y. (2) Perform the L D L T decomposition of the bus admittance matrix. (3) Calculate the elements of the Jg current vector by equation (7). (4) Calculate the elements of the Vg voltage vector by equation (8) and (9). (5) Calculate the currents flow through the branches. (6) If the radialisation is finished ( R L = n b - n 0 go to step 11. R L is the number of branches which should be opened to achieve radial configuration; nb is the number of the branches. (7) Find the unmarked branch with the minimal current flow. If there are no more unmarked branches a feasible solution does not exist. Got to the end of the algorithm. (8) Calculate the number d (see the equations in the next paragraph - the network compensation algorithm) in the case when this branch is eliminated. (9) If the d is less than 0.00015 the part of the network is isolated. Mark this line, and go to the step 7. If the calculated current of any feeder is over the line capacity mark candidate line and go to the step 7. If the calculated voltages of any node is unacceptable mark the candidate line and go to the step 7. (10) Perform the partial refactorization ~ of the bus admittance matrix. Go to the step 4. (11) Calculate the power losses for the obtained radial configuration.
IV. The network compensation algorithm The basic compensation process (see References 7 or 8 for details) is that an existing network solution (the base case) V = Y - 1 I = Z I can be related to the solution, V', after the network configuration is modified, by V' = V - Z C d - 1C T V
where d=(Y1 + C T Z C ) C T - transpose of C
Vol 14 No 6 December 1 9 9 2
(11)
Here C is the branch-node incidence matrix of the added branches, and Y1 is the m by m admittance matrix of the added branches. That is, if m branches are added and there are n nodes, then C is an n by m matrix with non-zero i,j entry only if the added branchj is incident with node i, and of value 1 or - 1 according to whether the branch is directed towards node i or is directed away from node i. Note that the admittance of an existing branch, j say, can b e modified by this process by making column j of C (and row j of C) equal to the nodal incidence vector of that branch, and inserting the change in admittance into positionj,j of Y. This includes the elimination of a branch (open-circuit) by carrying out the process described in the previous sentence but with admittance change of - y , where y is the admittance of the eliminated branch. Note that when the only network change involves a single branch pq with impedance change z, then d is a single number given in terms of the existing values zij of Z (i.e. y - l ) by: d=z+zpp+zqq-2zpq,
Y1 = 1/z
(12)
and the new values, z'ij are z'ij = zij-- (zip -- ziq)(zpj -- zqj)d
The computational simplicity of the compensation process in equation (11) results from the small size of the inverted matrices Y1 and d, which are q by q where q is the number of simultaneous changes, and are in fact single numbers if only one change at a time is made. It is assumed, however, that the existing matrix Z is known. Explicit determination of all the elements of the Z matrix is time-consuming and wasteful. Hence, only the elements of the Z matrix which are needed are computed as follows 1o: Z = (LT) - 1D- 1 + Z ( I - L )
(13)
Using the sparsity of L this equation provides a means of computing particular entries of Z from previously computed ones. A near-singular matrix d can occur in two ways. First the matrix d, if calculated exactly, may be singular but approximations made in computation leave it as only near-singular. Secondly d, if calculated exactly, may be nearly singular, but not singular. In the second case, however, very small changes in circuit parameters can result in large changes to the solution - conversion to the singular case for example. Hence, both cases of near-singularity should be considered as indicating an undesirable situation. Detection of near-singularity of d is not, in general, very easy. If d is a single number then 'near-singularity' is equivalent to d being small, but 'how small' depends on the network. Reasonably satisfactory rough rules can be given such as d is considered near-zero if d is less than x/re, where e is the minimum difference between numbers which the computer can discriminate. It should be noted that for a single number d, the maximum error in calculating 1/d will be roughly of the order e/2d 2 so that if e = 10-lO say, and d is taken by the computer as l0 -5, then the calculation error can be as large as 0.5 in 1/d (in what is admittedly a comparatively large value, i.e. 1/d= 105). For VAX 785 machine we used the limit value of d equal to 0.00015 with very good results.
385
V.
The
case
study
distribution
systems
This section summarizes the characteristics of the small distribution subsystem which is the part of the large distribution system that was used in our case studies. It was based on an actual system of a power distribution utility located in Belgrade. The subsystem predominantly consists of the overhead lines with the number of the load nodes n,=77, the number of getaways ng = 11 and with the peak demand of this area L , , , , = 2 6 (MVA). The current capacity of the getaways is 240 (amps) for normal operating conditions and 300 (amps) for the emergency. Table 1 shows the results obtained for two different optimization criteria: (1) the most equal loading of the getaways; (2) minimal losses. The power losses calculated according to criteria 1 and criteria 2 are 5.66% and 5.1% of the peak demand, respectively. Hence, the power losses of the network configuration determined by criteria 2 are for 11% less than the power losses of the network configuration determined by criteria 1. However, in the network configuration determined by criteria 2 the current value of the getaway number 7 (237.1 amps) is very near to its current capacity (240 amps), so even the small changes in the demand can overload this getaway and cause the customer power interruption. The conclusion is that the solution obtained according to criteria 1 is much more secure than that obtained according to criteria 2, which more than compensates
Table 1. Loading of the getaways Loading of the getaways (amps) Number of the getaways
Criteria l
Criteria2
1 2 3 4 5 6 7 8 9 10 11
147.7 87 111.2 101.5 150.4 60.7 165.9 185.5 140.6 211.5 140.8
173 53.7 63.8 148.8 87.1 60.7 237.1 165.8 162.5 211.5 140.8
386
-
--
-
the costs due to the extra power losses in the network configuration determined by criteria 1. For this size of the distribution system (ng = 11, nl = 77, n b = 92) and for VAX 785 machine the program execution wall clock time is about 2 s. VI.
Conclusions
The model described attempts to configure the distribution network radially for low losses, while meeting constraints on security (reliability), currents and voltages. The heuristic of the method allow examination of a large number of possible configurations of the distribution system and for reconfiguring radial feeder systems during either emergency or planned outages.
VII. References 1 Garret, G P, Fukutome, A and Chen, M 'Expansion planning of radial subtransmission systems', IEEE PES Winter Meeting, New York (Jan. 30-Feb. 4) (IEEE Paper No. F 77 223-1 ) 2 ACORN."A" computer program for the automatic computation for radial networks, System Control, Inc., Palo Alto, California (May 1974) 3 Ludot, J P and Rubinstein, M C 'Methodes pour la plantification a court terme des resaux de distribution', PSCC, Grenoble, Paper 1.1/12 (1972) 4 Hertz, A and Merlin, A 'Survey of methods used by EDF to search for tree operating configurations in power distribution networks', Proceedings of the Sixth Power Systems Computation Conference (PSCC), Darmstadt, Vol 1 (Aug 21-25 1978) pp 122-129 5 Merlin, A and Back, H 'Search for a minimal-loss operating spanning tree configuration for an urban power distribution system', Proceedings of the Fifth Power Systems Computation Conference (PSCC), Cambridge, paper 1.2/6 (1975) 6 Backlund, Y and Bubenko, J A "Computer-aided distribution system planning; Part 2 Primary and secondary circuits modelling', Proceedings of the Sixth Power Systems Computation Conference (PSCC), Darmstadt (Aug. 21-25, 1978) 7 Chan, S i and Brandwajn, V 'Partial matrix refactorization', IEEE Transactions on PWRS, Vol 1, No 1 (Feb. 1986) pp 193-200 8 Pereira, J L R, Brameller, A and Aitchison, P W 'Compensated network solution extensions for fast accurate solutions even when singularities are present', PSCC, Lisbon (1987) pp323- 329 9 Alsac, O, Stott, B and Tinney, W F 'Sparsity-orientated compensation methods for modified network solutions', IEEE Transactions on PAS (May 1983) pp1050-1060 10 Duff, I S, Erisman, A M and Reid, J K Direct methods for space matrices Clarendon Press, Oxford (1986) 11 Aoki, K, Nara, K, Satoh, T, Kitagawa, M and Yamanaka, K 'New approximate optimization method for distribution system planning', IEEE Transactions on PWRS, Vol 5, No 1 (Feb. 1990) pp126-132
Electrical P o w e r & E n e r g y S y s t e m s
The paper concerns the development of new approaches to the optimal operation planning of radial distribution networks. In particular the model described attempts to configure the medium voltage networks radially for low losses, while meeting constraints on security (reliability), currents and voltages. The model appears to be a very useful tool for planning the normal operating configuration of the distribution system and .for reconfiguring radial feeder systems during either emergency or planned outages. Keywords: distribution systems, reliability
operation
planning,
I. I n t r o d u c t i o n This paper concerns the development of new approaches to the planning of radial primary distribution systems. In particular, the research has addressed two problem areas. (1) The development of optimization models for arranging or rearranging the radial structure of a system of feeders - for planning the normal operating configuration of the system. (2) The development of optimization models for reconfiguration (made-possible by manual or automatic switching) of a radial system of feeders during either emergencies or planned outages. The models can be used by electric utilities for a variety of distribtion expansion planning functions, including: (1) transferring electric load from one feeder to another or otherwise redistributing load as it changes in magnitude and geographic distribution over a multiyear planning horizon;
Received January 1992; revised April 1992
Vol 1 4 No 6 December 1 992
(2) lowering losses in radial distribution circuits through better feeder configuration; (3) redesigning the arrangement of the radial feeder system in conjunction with the upgrading of the distribution systems as when converting to higher primary voltages. Both the problem of optimizing the normal configuration of a radial distribution, and the problem of optimizing an emergency reconfiguration require manipulation of radial circuit arrangements. A paper by Garret 1 approached the problem of expansion planning of radial subtransmission systems by generating all the spanning trees of the circuit's graph and then checking them for feasibility and cost. A program, called ACORN 2, was developed by SCI and SE in 1974 for expansion planning of radial subtransmission or distribution systems; it used heuristics to add feeder sections to the system, one by one, until all loads were supplied thereby constructing a radial configuration. Several people at Electricite de France (EDF) have developed methods that address this problem. The SECOURS model 3 was developed about fifteen years ago by EDF; it used a heuristic method to attain a radial configuration that minimized the unsupplied load, taking into account feeder capacity constraints. Several years ago EDF extended SECOURS to obtain a new model called 'CORALI 4. CORALI retains the same objective but includes voltage constraints; again, it uses heuristics with some gains in computational efficiency. Another basic technique developed by EDF, called MINIPERT 5, searches for the lowest-losses radial configuration. Backlund and Bubenko 6 describe a model for expansion planning that uses heuristics and that also uses an iterative tree generation algorithm to search for configuration improvements Aoki et al. ~1, developed a method which incorporates both the merits of the mixed integer programming model and the branch-exchange model. But the problem of fast optimal evaluating of a radial network configuration still exists.
0142-0615/92/60383-04 © 1992 Butterworth-Heinemann Ltd
383
II. C h a r a c t e r i s t i c s of t h e model
where
Early in the research phase it was decided that the model would be a static, rather than dynamic, optimization model. That is, it would determine, for each year (or stage) of an expansion plan, an optimum configuration for that year. The planner would then have to combine these static analyses of separate years into an overall expansion plan. This choice of a static approach was made for the sake of keeping the model computationally tractable. A further simplification was that the planner was assumed to have in mind a limited number of reinforcement candidates and/or new locations of ties between pairs of feeders. In each ~run' of the model, the planner would pre-specify these candidates and the output would show which candidates were actually used in obtaining the best configuration. In short, the model would not itself perform the economic selection of alternative expansion candidates; rather, it would identify what configuration met all operational constraints (e.g., on voltages, current, and security) while minimizing some operating cost objective in the year of analysis. It is assumed that annual peak load value is used for calculating the current values and voltage drop. The daily, weekly or seasonal load pattern could be taken into account using an annual load factor. With the above considerations, we formulated the problem as follows. The objective of the model is to find a radial configuration from among all available feeder branches that supplies all the demands while minimizing the operating cost of the system, subject to operating constraints and security constraints, The operating cost of the system is assumed to be adequately represented by the system circuit losses (active power). The operating constraints are to satisfy:
I is the (phase) current in the feeder section, R is the resistance of the feeder section, X is the reactance of the feeder section, cos ~b is the power factor of the distribution system.
(1) all the demands; (2) the current capacity of each feeder branch: (3) the node voltage limits.
(1) The three phases are balanced, so that a single phase equivalent can be used. (2) The distribution system operates at a constant power factor. (3) The shunt capacitance of each feeder section is negligible. The first assumption is valid to the extent that the utility strives to balance phases and taking into account the fact that the method concerns medium voltage networks (10kV, 20 kV, 35 kV). The second assumption is generally valid; economics usually dictate that a power factor of about 0.95 be maintained. The second assumption is very good except for feeders that use shunt capacitance for voltage control; but even under such conditions, the simplified load flow model tends to err on the conservative side - tending to yield lower voltages than are actually present. With these assumptions, the (phase-to-phase) voltage drop along a feeder section between any two load-points k and m becomes: Vk - V , , , = R l c o s ~ + X l s i n ~
384
z = R cos q5+ X sin ~b
(3)
Vt -- Vm=Zl
Elsewhere in this paper, all 'impedances' mentioned will thus be interpreted as equivalent impedances, per equation (2).
III. The n e t w o r k model The approach is based on a linearized network power flow model. The nodal voltage problem for a linear network is that of finding the nodal voltage vector when the nodal current injection vector J is known and is related to V by the admittance matrix Y as: (4)
J= YV=MTYb m
where n number of nodes, Yb the diagonal matrix of the branch admittances, M the branch-node incidences matrix, M T the transpose of M, h(i) the set of branches connected to bus i.
~ii= 2 (-- Yiij) jeh(i)
Yij-~- --1/zij
The equation (4) can be partitioned into the next matrix relation (5):
(5)
where the vector of currents in the feeders between the substations HV/MV and first substation MV/LV ('getaways') Jt the vector of currents in the loading nodes (substation MV/LV) Vg the voltage vector of the buses in the substations HV/MV V~ the voltage vector of the buses in the substations MV/LV Y, the submatrix which has diagonal elements:
Jg
Yii = Yij + S . / V . 2
(6)
For the elements of the matrix Yn defined by equation (6) the elements of the current injection vector El are zero. The elements of the current injection vector J . are defined by equation (7):
where Jc nj
ng (1)
(2)
one obtains the following relation:
~_~ ~JLV~_I
A full AC load flow model was deemed too complex for the analysis. Accordingly, a simplified load flow model is used. This model arises from the following simplifying assumptions.
= l [ R c o s ~ + X sin ~b]
Defining the equivalent impedance, z, of a feeder section as :
the current capacity of each 'getaway' branch the number of the loading nodes the number of the feeders between substations HV/MV and first substation MV/LV ('getaways')
Electrical Power & Energy Systems
Jc often has the same value for all getaways in the distribution network of the same voltage. In that c a s e Jg has the same values for all getaways. The idea is that the optimum operating point is achieved when the getaways are equally loaded in the case when the current capacity of each getaway branch are the same or relatively equally loaded when those values are different. Considering that the elements of the v e c t o r Jg are defined by equation (7) and the elements of the vector J1 are equal to zero, the elements of the vector V~ are given by: V~= (-- y, gyggl Ygl- Yll)-i y~gygg 1j,
(8)
V, = Y**(J,- Yg,V~)
(9)
The equation (8) can be easily reduced to the equation (10). V~= Ab
(10)
The matrix A is the sparse symmetric positive definitive matrix. It can be transformed into L D L T form by the Cholesky method. The algorithm of the heuristic optimization process consists of the following conceptual steps. (1) Form the bus admittance m a t r i x - Y. (2) Perform the L D L T decomposition of the bus admittance matrix. (3) Calculate the elements of the Jg current vector by equation (7). (4) Calculate the elements of the Vg voltage vector by equation (8) and (9). (5) Calculate the currents flow through the branches. (6) If the radialisation is finished ( R L = n b - n 0 go to step 11. R L is the number of branches which should be opened to achieve radial configuration; nb is the number of the branches. (7) Find the unmarked branch with the minimal current flow. If there are no more unmarked branches a feasible solution does not exist. Got to the end of the algorithm. (8) Calculate the number d (see the equations in the next paragraph - the network compensation algorithm) in the case when this branch is eliminated. (9) If the d is less than 0.00015 the part of the network is isolated. Mark this line, and go to the step 7. If the calculated current of any feeder is over the line capacity mark candidate line and go to the step 7. If the calculated voltages of any node is unacceptable mark the candidate line and go to the step 7. (10) Perform the partial refactorization ~ of the bus admittance matrix. Go to the step 4. (11) Calculate the power losses for the obtained radial configuration.
IV. The network compensation algorithm The basic compensation process (see References 7 or 8 for details) is that an existing network solution (the base case) V = Y - 1 I = Z I can be related to the solution, V', after the network configuration is modified, by V' = V - Z C d - 1C T V
where d=(Y1 + C T Z C ) C T - transpose of C
Vol 14 No 6 December 1 9 9 2
(11)
Here C is the branch-node incidence matrix of the added branches, and Y1 is the m by m admittance matrix of the added branches. That is, if m branches are added and there are n nodes, then C is an n by m matrix with non-zero i,j entry only if the added branchj is incident with node i, and of value 1 or - 1 according to whether the branch is directed towards node i or is directed away from node i. Note that the admittance of an existing branch, j say, can b e modified by this process by making column j of C (and row j of C) equal to the nodal incidence vector of that branch, and inserting the change in admittance into positionj,j of Y. This includes the elimination of a branch (open-circuit) by carrying out the process described in the previous sentence but with admittance change of - y , where y is the admittance of the eliminated branch. Note that when the only network change involves a single branch pq with impedance change z, then d is a single number given in terms of the existing values zij of Z (i.e. y - l ) by: d=z+zpp+zqq-2zpq,
Y1 = 1/z
(12)
and the new values, z'ij are z'ij = zij-- (zip -- ziq)(zpj -- zqj)d
The computational simplicity of the compensation process in equation (11) results from the small size of the inverted matrices Y1 and d, which are q by q where q is the number of simultaneous changes, and are in fact single numbers if only one change at a time is made. It is assumed, however, that the existing matrix Z is known. Explicit determination of all the elements of the Z matrix is time-consuming and wasteful. Hence, only the elements of the Z matrix which are needed are computed as follows 1o: Z = (LT) - 1D- 1 + Z ( I - L )
(13)
Using the sparsity of L this equation provides a means of computing particular entries of Z from previously computed ones. A near-singular matrix d can occur in two ways. First the matrix d, if calculated exactly, may be singular but approximations made in computation leave it as only near-singular. Secondly d, if calculated exactly, may be nearly singular, but not singular. In the second case, however, very small changes in circuit parameters can result in large changes to the solution - conversion to the singular case for example. Hence, both cases of near-singularity should be considered as indicating an undesirable situation. Detection of near-singularity of d is not, in general, very easy. If d is a single number then 'near-singularity' is equivalent to d being small, but 'how small' depends on the network. Reasonably satisfactory rough rules can be given such as d is considered near-zero if d is less than x/re, where e is the minimum difference between numbers which the computer can discriminate. It should be noted that for a single number d, the maximum error in calculating 1/d will be roughly of the order e/2d 2 so that if e = 10-lO say, and d is taken by the computer as l0 -5, then the calculation error can be as large as 0.5 in 1/d (in what is admittedly a comparatively large value, i.e. 1/d= 105). For VAX 785 machine we used the limit value of d equal to 0.00015 with very good results.
385
V.
The
case
study
distribution
systems
This section summarizes the characteristics of the small distribution subsystem which is the part of the large distribution system that was used in our case studies. It was based on an actual system of a power distribution utility located in Belgrade. The subsystem predominantly consists of the overhead lines with the number of the load nodes n,=77, the number of getaways ng = 11 and with the peak demand of this area L , , , , = 2 6 (MVA). The current capacity of the getaways is 240 (amps) for normal operating conditions and 300 (amps) for the emergency. Table 1 shows the results obtained for two different optimization criteria: (1) the most equal loading of the getaways; (2) minimal losses. The power losses calculated according to criteria 1 and criteria 2 are 5.66% and 5.1% of the peak demand, respectively. Hence, the power losses of the network configuration determined by criteria 2 are for 11% less than the power losses of the network configuration determined by criteria 1. However, in the network configuration determined by criteria 2 the current value of the getaway number 7 (237.1 amps) is very near to its current capacity (240 amps), so even the small changes in the demand can overload this getaway and cause the customer power interruption. The conclusion is that the solution obtained according to criteria 1 is much more secure than that obtained according to criteria 2, which more than compensates
Table 1. Loading of the getaways Loading of the getaways (amps) Number of the getaways
Criteria l
Criteria2
1 2 3 4 5 6 7 8 9 10 11
147.7 87 111.2 101.5 150.4 60.7 165.9 185.5 140.6 211.5 140.8
173 53.7 63.8 148.8 87.1 60.7 237.1 165.8 162.5 211.5 140.8
386
-
--
-
the costs due to the extra power losses in the network configuration determined by criteria 1. For this size of the distribution system (ng = 11, nl = 77, n b = 92) and for VAX 785 machine the program execution wall clock time is about 2 s. VI.
Conclusions
The model described attempts to configure the distribution network radially for low losses, while meeting constraints on security (reliability), currents and voltages. The heuristic of the method allow examination of a large number of possible configurations of the distribution system and for reconfiguring radial feeder systems during either emergency or planned outages.
VII. References 1 Garret, G P, Fukutome, A and Chen, M 'Expansion planning of radial subtransmission systems', IEEE PES Winter Meeting, New York (Jan. 30-Feb. 4) (IEEE Paper No. F 77 223-1 ) 2 ACORN."A" computer program for the automatic computation for radial networks, System Control, Inc., Palo Alto, California (May 1974) 3 Ludot, J P and Rubinstein, M C 'Methodes pour la plantification a court terme des resaux de distribution', PSCC, Grenoble, Paper 1.1/12 (1972) 4 Hertz, A and Merlin, A 'Survey of methods used by EDF to search for tree operating configurations in power distribution networks', Proceedings of the Sixth Power Systems Computation Conference (PSCC), Darmstadt, Vol 1 (Aug 21-25 1978) pp 122-129 5 Merlin, A and Back, H 'Search for a minimal-loss operating spanning tree configuration for an urban power distribution system', Proceedings of the Fifth Power Systems Computation Conference (PSCC), Cambridge, paper 1.2/6 (1975) 6 Backlund, Y and Bubenko, J A "Computer-aided distribution system planning; Part 2 Primary and secondary circuits modelling', Proceedings of the Sixth Power Systems Computation Conference (PSCC), Darmstadt (Aug. 21-25, 1978) 7 Chan, S i and Brandwajn, V 'Partial matrix refactorization', IEEE Transactions on PWRS, Vol 1, No 1 (Feb. 1986) pp 193-200 8 Pereira, J L R, Brameller, A and Aitchison, P W 'Compensated network solution extensions for fast accurate solutions even when singularities are present', PSCC, Lisbon (1987) pp323- 329 9 Alsac, O, Stott, B and Tinney, W F 'Sparsity-orientated compensation methods for modified network solutions', IEEE Transactions on PAS (May 1983) pp1050-1060 10 Duff, I S, Erisman, A M and Reid, J K Direct methods for space matrices Clarendon Press, Oxford (1986) 11 Aoki, K, Nara, K, Satoh, T, Kitagawa, M and Yamanaka, K 'New approximate optimization method for distribution system planning', IEEE Transactions on PWRS, Vol 5, No 1 (Feb. 1990) pp126-132
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