![It is essential to have voltage magnitudes of all the
buses and in particular the voltage controlled buses
close to 1pu to achieve load flow convergence. The
equivalent did not make and provisions to avoid
this low voltage condition at the newly created
buses. Even if all the generators are grouped
together a low voltage condition may exist at the
REI buses. So care should be taken in building the
equivalent. It should be avoided to have
incorporating both negative and passive values of
the net injected power in the same equivalent. This
is avoided if two REI equivalents are formed one
for PV buses and one for PQ buses for each
equivalized area.
COMPUTER SIMULATION OF REI
EQUIVALENT STEP BY STEP ALGORITHM
Step 1: Read bus data of a power system and also
data for lines transformers, shunts, all the bus
voltages and powers.
Step 2: From the line, transformer and shunt data Y
bus .
Step 3: Base power flow is carried out to obtain
base case solution before the system is reduced.
Step 4: From the original Y bus only those rows
and columns are selected that belong to external
and boundary buses. Then, the resulting admittance
matrix is of the form.
BBEB
EBEE
YY
YY
Step 5: To this admittance matrix node ‘R’ and
node ‘G’ are added by inserting rows and columns
calculating values corresponding to these nodes
using the formula given above. Then, we get Y bus
of the form
RRRBRGRE
BRBBBGBE
GRGVGGGE
EREBEGEE
YYYY
YYYY
YYYY
YYYY
Y
Step 6: From the above admittance matrix all
external nodes and node ‘G’ are eliminated using
gauss elimination, then it results new connections
between node R and boundary buses.
BBRB
BRBB
eq
YY
YY
Y
Step 7: Now the reduced system [YR] contains
these connections and original lines that are in
between internal system buses and that are in
between internal and boundary buses.
Step 8: If all the eliminated buses are of PQ type,
then bus ‘R’ is being considered as PQ type. If all
are of PV type, then bus ‘R’ is considered of PV
type.
Step 9: With all these values we can formulate REI
equivalent that are from REI 1 to REI 7.
IV WARD EQUIVALENT
One of the most widely used equivalence method is
the one due to ward. This ward technique was
proposed by ward himself in 1949. This equivalent
can be applied to load flow analysis and
contingency analysis. This method suffers from
poor accuracy for security related studies due to
problems in the designation of boundary bus type.
Off line equivalent overcomes this deficiency by
using buffer zones but these can’t be used on line.
The major problem with the ward equivalent is that
it does not allow for reactive power support in the
equivalized area. This can be explained by this. In
any power system the real power is always
specified for every bus except at the slack bus. The
reactive power is specified for few buses and may
vary. If the equivalent assumes that the reactive
power at the regulated buses will remain at its base
case value even under outage, the results are
unacceptable. So for the ward equivalent once the
equivalent has been formed, there is no way of
producing more or less reactive power in the
equivalize area if the out aged case so requires.
Therefore the behaviour of an equivalent network
with reactive power support capabilities would be
closer to the behaviour of the unreduced network.
Fig 4
COMPUTER SIMULATION OF WARD
EQUIVCALENT
Algorithm
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Iaetsd static network equivalents for large power systems
- 1. STATIC NETWORK EQUIVALENTS FOR LARGE POWER SYSTEMS S. Mouli M.E, PSA rajmouli.245@gmail.com Andhra University college of Engineering Dr. V. Bapiraju Professor drbapibv@yahoo.com Andhra University college of Engineering ABSTRACT: This paper describes static network equivalents for power flow studies. Steady state equivalents are of important for the study of the static characteristics of a large power system when the solution time must be decreased. An effort is made in this paper to understand and implement few types of equivalents such as REI equivalents, ward and extended ward. These are tested on several IEEE test systems for power flow solutions under normal and outage conditions. Several equivalents are examined and compared. Among all the equivalent techniques available ward equivalents are widely used in the industries. Test load flow results will be presented in order to assess the accuracy of the examined equivalents. The major conclusion of the paper is that as long as there exists some reactive power support from the equivalent it will perform satisfactorily. I INTRODUCTION Steady state equivalents are of great important when there is an interest in studying a small part of a large system. In such a case it would be desirable to equivalize the system around the area of interest and experiment with the reduced system instead of the large system. Considerable interest is currently being shown in load-flow equivalence especially for control-centre applications. In any country the overall economic development depends on the availability of electric power. The ever-increasing demand for electric power due to rapid industrialization and urbanization the electric power networks are growing in size and complexity years after year. These power systems have several inter connections. Due to this interconnectivity the modern electric power systems are characterised by their large size and complexity with enormous number of generating units transmission lines transformers and other related compensating devices etc., distributed over long distance over several hundreds of kilometres. Several types of equivalence techniques have been considered to date. Since all equivalents are calculated from a solved base case load flow they are exact only for the base case. one of the most widely used equivalence methods is the due to ward. The major problem with the ward equivalent is that it does not allow for reactive power support in the equivalize area. This can be explained by noticing that although the real power is always specified for every bus the slack bus the reactive power is not specified for every bus and may vary. If the equivalent assumes that the reactive power at the regulated buses will remain at its base case value even under out ages the results are really unacceptable. For the ward equivalent once the equivalent has been formed there is no way of producing more or less reactive power in the equivalize area if the out aged case so requires. Therefore it is expected that the behaviour of an equivalent network with the reactive power support capabilities would be closer to the behaviour of the unreduced network. Several equivalence techniques such as REI ward equivalent with buffer and extended ward equivalent satisfy the above requirement. Experiments were conducted with the REI equivalent and the ward equivalent with a buffer zone and although the REI type equivalent performed satisfactorily the final conclusion was that the ward equivalent with a carefully selected buffer zone is a very good equivalence method. This equivalent is the easiest to implement and understand and thus it has been recommended for future usage. II BASIC ISSUES OF EQUIVALENCING Fig illustrates the general problem being deal with. Load flow type studies are to be performed on an interconnected system. An internal system is to be modelled in detail. The remaining external system is to be represented by some equivalent attached at the boundary buses. The solved load flow model of the entire interconnected system is available. The aim of equivalence is simply computational economy, through reduction of the system size. The general problem can be formally stated as follows. Given a solved load flow model of an interconnected power system PS and an area of Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 57
- 2. interest AI with in PS find a new equivalent load flow model PE that has a smaller number of buses and branches than PS. PE should be such that for changes of the operating conditions within AI the results of PE are close to the results of PS. If AE= {PS}-{AI} then schematically the steady state equivalent problem is as shown. Fig 1 If PS were a linear system then the problem posted would be easily and exactly solved. This can be seen from the following matrix manipulations. Since it was assumed that PS can be described by a set of linear equations and without loss of generality these could be the nodal Equations then E I E I EEEI IEII I I V V YY YY AE AI (1) Solving for V1 V1=Y11 -1* Y12*V2+Y11 -1 *I1 (2) Substituting in to (1) Y22EQ*V2=I2EQ (3) Where Y22EQ = Y22 -Y21 * Y11 -1 * Y12 I2EQ= I2 - Y21 * Y11 -1 * I1 According to the previous definition of PE the set of equations (3)is a description of the equalised system and it contains the same number of buses as AI. If any elements of AI are altered the solution of PE is exactly the same with the solution of PS because AE Does not change whenever AI changes. III REI EQUIVALENT The equivalent that has been suggested is the REI (radial equivalent independent) equivalence technique. In this case AE is transformed in to passive network by aggregating all the generation of AE in one bus and all the load of AE in another new bus so that the rest of AE becomes passive. Schematically the REI equivalent inserts a lossless network NREI between the eliminated buses and their aggregated injections as shown where NL is the number of load buses and NG is the number of generator buses. The REI equivalent (NREI) is formed as a function of the injected power of the buses included in the equivalent and the base voltage of these buses. Fig 2 Fig 3 From fig letting VG=0, then SR= i iS IR = i I i VR=SR/*IR YR=*SR/VR 2 YI=- Si*/Vi 2 Notice that the network NREI is a lossless network for the base case conditions. In fact by construction the power injected into R is SR= Si and the power injection in to the ith bus of the REI equivalent is equal to Si. Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 58
- 3. It is essential to have voltage magnitudes of all the buses and in particular the voltage controlled buses close to 1pu to achieve load flow convergence. The equivalent did not make and provisions to avoid this low voltage condition at the newly created buses. Even if all the generators are grouped together a low voltage condition may exist at the REI buses. So care should be taken in building the equivalent. It should be avoided to have incorporating both negative and passive values of the net injected power in the same equivalent. This is avoided if two REI equivalents are formed one for PV buses and one for PQ buses for each equivalized area. COMPUTER SIMULATION OF REI EQUIVALENT STEP BY STEP ALGORITHM Step 1: Read bus data of a power system and also data for lines transformers, shunts, all the bus voltages and powers. Step 2: From the line, transformer and shunt data Y bus . Step 3: Base power flow is carried out to obtain base case solution before the system is reduced. Step 4: From the original Y bus only those rows and columns are selected that belong to external and boundary buses. Then, the resulting admittance matrix is of the form. BBEB EBEE YY YY Step 5: To this admittance matrix node ‘R’ and node ‘G’ are added by inserting rows and columns calculating values corresponding to these nodes using the formula given above. Then, we get Y bus of the form RRRBRGRE BRBBBGBE GRGVGGGE EREBEGEE YYYY YYYY YYYY YYYY Y Step 6: From the above admittance matrix all external nodes and node ‘G’ are eliminated using gauss elimination, then it results new connections between node R and boundary buses. BBRB BRBB eq YY YY Y Step 7: Now the reduced system [YR] contains these connections and original lines that are in between internal system buses and that are in between internal and boundary buses. Step 8: If all the eliminated buses are of PQ type, then bus ‘R’ is being considered as PQ type. If all are of PV type, then bus ‘R’ is considered of PV type. Step 9: With all these values we can formulate REI equivalent that are from REI 1 to REI 7. IV WARD EQUIVALENT One of the most widely used equivalence method is the one due to ward. This ward technique was proposed by ward himself in 1949. This equivalent can be applied to load flow analysis and contingency analysis. This method suffers from poor accuracy for security related studies due to problems in the designation of boundary bus type. Off line equivalent overcomes this deficiency by using buffer zones but these can’t be used on line. The major problem with the ward equivalent is that it does not allow for reactive power support in the equivalized area. This can be explained by this. In any power system the real power is always specified for every bus except at the slack bus. The reactive power is specified for few buses and may vary. If the equivalent assumes that the reactive power at the regulated buses will remain at its base case value even under outage, the results are unacceptable. So for the ward equivalent once the equivalent has been formed, there is no way of producing more or less reactive power in the equivalize area if the out aged case so requires. Therefore the behaviour of an equivalent network with reactive power support capabilities would be closer to the behaviour of the unreduced network. Fig 4 COMPUTER SIMULATION OF WARD EQUIVCALENT Algorithm Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 59
- 4. Step 1: Read system data, bus data of a power system and also data for lines transformers and shunts. Step 2: From the line transformer and shunt data, Y bus is constructed. Step 3: Base case power flow is carried out to obtain base case solution before the system is reduced. Step 4: From the original Y bus only those rows and columns are selected that belong to external or boundary buses. Step 5: From this Y bus by performing gauss elimination on the columns of external buses eliminate external buses. Then one gets Yeq of equivalent network and new connections between boundary buses. Step 6: The reduced system Y bus contains these new connections and original lines that are in between internal system buses and between internal and boundary buses. RESULTS FOR 30BUS SYSTEM FOR DIFFERENT EQUIVALENTS Step 7: With the reduced system Y bus and base case voltage and angles compute the power injections at the boundary buses. These are the total power injections and are kept constant as long as the eliminated network remains same. Step 8: With this injected power starting with initial values of the reduced system, power flow is carried out. V EXTENED WARD In this equivalence technique a different way of providing reactive power support is considered. It involves the creation of a number of new generators equal to the number of boundary load buses. That is one new fictitious generator is connected to each boundary load bus in such a way as to not affect the base case solution of the reduced system. The new generators do not produce any real power but are capable of producing reactive power whenever the internal system so requires. For multisystem planning applications the number of additional generator buses would in general be large. B.NO TYPE GENERATION LOAD DEMAND BUSVOLTAGES REAL REACT REAL REACT VOLTAG ANG(deg) 1 3 2.3873 -0.1765 0 0 1.06 0 2 1 0.4 0.5204 0.217 0.127 1.045 -5.0361 3 1 0.2 0.1917 0 0.3 1.01 -9.8046 4 1 0 0.2271 0.3 0 1.082 -18.3703 5 1 0 0.3574 0.942 0.19 1.01 -13.5302 6 1 0 0.1931 0 0 1.071 -15.0062 7 2 0 0 0 0 1.0399 -15.1913 8 2 0 0 0.058 0.02 1.0244 -16.1105 9 2 0 0 0.112 0.075 1.0458 -15.0062 10 2 0 0 0 0 1.0792 -15.5341 11 2 0 0 0.076 0.016 1.0151 -8.7828 12 2 0 0 0.228 0.109 1.0026 -12.0551 13 2 0 0 0 0 1.0105 -10.1343 14 2 0 0 0.062 0.016 1.0311 -15.9467 15 2 0 0 0.082 0.025 1.0267 -16.0779 16 2 0 0 0.035 0.018 1.0293 -15.7442 17 2 0 0 0.09 0.058 1.0204 -16.2135 18 2 0 0 0.032 0.009 1.0135 -16.8029 19 2 0 0 0.095 0.034 1.0089 -17.0388 20 2 0 0 0.022 0.007 1.0119 -16.8652 21 2 0 0 0.175 0.112 1.0154 -16.5169 22 2 0 0 0 0 1.0171 -16.4866 23 2 0 0 0.032 0.016 1.0206 -16.5169 24 2 0 0 0.087 0.067 1.021 -16.7487 25 2 0 0 0 0 1.0518 -16.1605 26 2 0 0 0.035 0.023 1.0348 -16.5526 27 2 0 0 0.024 0 1.0239 -7.2915 28 2 0 0 0 0 1.0121 -10.7532 Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 60
- 5. 29 2 0 0 0.024 0.009 1.0605 -16.6375 30 2 0 0 0.106 0.019 1.0497 -17.427 Table 1:30 bus system for base case Total generation =2.9873+i1.3132 Total load=2.8340+i1.2500 Real loss=0.1533 Reactive loss=0.0632 BUS TYPE GENERATION LOAD DEMAND BUS VOLTAGES REAL REACTIVE REAL REACTIVE VOLTAGE ANGLE 1 3 2.3873 -0.1765 0 0 1.06 0 2 1 0.4 0.5204 0.217 0.127 1.045 -5.0361 3 1 0.2 0.1917 0 0.3 1.01 -9.8046 4 1 0 0.2271 0.3 0 1.082 -18.3703 5 1 0 0.3574 0.942 0.19 1.01 -13.5302 6 1 0 0.1931 0 0 1.071 -15.0062 7 2 0 0 0 0 1.0399 -15.1913 8 2 0 0 0.058 0.02 1.0244 -16.1105 9 2 0 0 0.112 0.075 1.0458 -15.0062 10 2 0 0 0 0 1.0792 -15.5341 11 2 0 0 0.076 0.016 1.0151 -8.7828 12 2 0 0 0.228 0.109 1.0026 -12.0551 13 2 0 0 0 0 1.0105 -10.1343 14 2 0 0 0.062 0.016 1.0311 -15.9467 15 2 0 0 0.082 0.025 1.0267 -16.0779 16 2 0 0 0.035 0.018 1.0293 -15.7442 17 2 0 0 0.09 0.058 1.0204 -16.2135 20 2 0 0 0.022 0.007 1.0119 -16.8652 21 2 0 0 0.175 0.112 1.0154 -16.5169 22 2 0 0 0 0 1.0171 -16.4866 27 2 0 0 0.024 0 1.0239 -7.2915 28 2 0 0 0 0 1.0121 -10.7532 31 2 0 0 0.411 0.177 1.0269 -16.9978 Table 2: 30 bus system for REI equivalence Total generation=2.9873+i1.3132 Total load=2.8340+i1.2500 Real loss=0.1533 Reactive loss=0.0632 BUS TYPE GENERATION LOAD DEMAND BUS VOLTAGES REAL REACTIVE REAL REACTIVE VOLTAGE ANGLES 1 3 2.3873 -0.1765 0 0 1.06 0 2 1 0.4 0.5204 0.217 0.127 1.045 -5.0361 3 1 0.2 0.1917 0 0.3 1.01 -9.8046 4 1 0 0.2271 0.3 0 1.082 -18.3703 5 1 0 0.3574 0.942 0.19 1.01 -13.5302 Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 61
- 6. 6 1 0 0.1931 0 0 1.071 -15.0062 7 2 0 0 0 0 1.0399 -15.1913 8 2 0 0 0.058 0.02 1.0244 -16.1105 9 2 0 0 0.112 0.075 1.0458 -15.0062 10 2 0 0 0.1777 0.0667 1.0792 -15.5341 11 2 0 0 0.076 0.016 1.0151 -8.7828 12 2 0 0 0.228 0.109 1.0026 -12.0551 13 2 0 0 0 0 1.0105 -10.1343 14 2 0 0 0.062 0.016 1.0311 -15.9467 15 2 0 0 0.1563 0.0664 1.0267 -16.0779 16 2 0 0 0.035 0.018 1.0293 -15.7442 17 2 0 0 0.09 0.058 1.0204 -16.2135 20 2 0 0 0.1184 0.0403 1.0119 -16.8652 21 2 0 0 0.175 0.112 1.0154 -16.5169 22 2 0 0 0.0668 0.0449 1.0171 -16.4866 27 2 0 0 0.024 0 1.0239 -7.2915 28 2 0 0 0 0 1.0121 -10.7532 Table3:30bus system for ward equivalence Total generation=2.9873+i1.3132 Total load=2.8381+i1.2594 Real loss=0.1492 Reactive loss=0.0539 VI CONCLUSION In this thesis an attempt is made to study and analyze the static network equivalents of the power system. The static equivalent reduces the size of the system. The reduced dimensionality of the network would pave the way for real time applications. Ideally the power flow Solution of the unreduced network and the power flow solution of the reduced Network should be same at the identifiable buses. However, in reality due to various reasons, the solutions would slightly differ from each other and also the overall system losses would differ. Some of the issues have been addressed in this thesis. There are several equivalents such as ward; extended ward and REI are in vogue. These equivalents have their merits and demerits. There is no one single type of equivalent which is applicable for all kinds of situations. Most of the times the problems are associated with the reactive power support. This very issue has resulted in different kinds of equivalents. Some of these issues are being tried to understand in this work. Interestingly all these equivalents are designed interconnected systems where in distinct classifications such as study network/systems external systems, which are needed to be equivalence and the interconnectivity of these two systems, called as boundary buses. An attempt is made in this thesis to apply definitions to a single area system itself so that three groups of buses could be formulated. The first group of buses forms a set surrounding the disturbed condition, second group of buses from another set, which is being equivalence and located at the boundary buses. A simple and convenient top down tree model is developed starting from the slack bus growing down as roots of a tree based on the connectivity. Now the grouping of buses can easily be incorporated dividing the network in to the categories of buses. For equivalence the loads are represented as constant admittances using either flat start voltage or an already known base case voltage and conventional gauss elimination technique are used to obtain reduced equivalent. At this stage it is worth mentioning the developed algorithm will segregate all the PQ buses in to the external area without losing the generality for convenience of elimination. However, in this process of segregation if a PV bus falls in to the external area the algorithm takes care about the situation by creating islands of PV buses surrounded by PQ buses. Now each island is again subjected the same rigor of demarcations as if it is an independent network. This situation tantamount to the external network would be reduced to number of PV buses falling in to that group which in turn would be interface to boundary buses. This algorithm would immensely help in contingency analysis for large size networks where in the network size can easily be reduced to the smallest Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 62
- 7. possible dimension and power flow solutions can speedily be obtained using conventional already available power flow methods. The accuracy of the power flow solution greatly depends on the quality of equivalence. In this regard a simple effort is made in representing the loads by constant admittance with appropriate node voltages and reduced networks are tried for several test systems such as IEEE 14, IEEE 30 and IEEE 118 bus test systems. For all these test systems the power flow solutions obtained for reduced networks are in close agreement with those of the unreduced systems. VII References 1. A. S. Debs and A. R. Benson, "Security assessment of power systems", ERDA and EPRI Conf. Publication CONF-750867, Henniker, Aug. 1975. 2. L.P. Hadju and R. Podmore, "Security enhancement for power systems", ibid. 3. J. B. Ward, "Equivalent circuits for power-flow studies", AIEE Trans., vol. 68, pp. 373-382, 1949. 4. T.G. Deville and F.C. Schweppe, "Online identification of interconnected network equivalents from operating data", C 72 464-6 presented at 1972 IEEE PAS Summer Meeting, San Francisco, July 1972. 5. A.S. Debs, "Estimation of external network equivalents from internal system data", IEEE Trans. Power App. Syst., vol. PAS-93, pp. 272- 279, March/April 1975. 6. J.F. Dopazo, M.H. Dwarakanath, J.J. Li and A.M. Sasson, "An external system equivalent model using real-time measurements for system security evaluation", IEEE Trans. Power App. Syst., vol. PAS -96, March/April 1977. 7. G. Contaxis and A.S. Debs, "Identification of external equivalents for steady state security assessment", F 77 526-7 presented at 1977 IEEE PAS Summer Meeting, Mexico City, July 1977. 8. T.E. Dy Liacco, K.A. Ramarao and S. C. Savulescu, "An on-line topological equivalent of a power system", F 77 523-4, ibid. 9. W.F. Tinney and W.L. Powell, "The REI approach to power network equivalents", Proc. IEEE PICA Conf., Toronto, pp. 314-320, May 1977. 10. F.L. Alvarado and E.H. Elkonyaly, "Reduction in power systems", A 77 507-7 presented at 1977 IEEE PAS Suumer Meeting, Mexico City, July 1977. 11. H. Duran and N. Arvanitidis, "Simplification for area security analysis: a new look at equivalence" IEEE Trans. Power App. Syst., vol. PAS-91, pp. 670-679, March/April 1972. 12. E.K. Paulsson, "Network equivalents for on- line systems", C 74 373-7 presented at IEEE PAS Summer Meeting, Annaheim, July 1974. 13. D. Denzel, R. Graf and J. Verstege, "Practical use of equivalents for unobservable networks in online security monitoring", Power Systems Computation Conf., Cambridge, Spt. 1975. 14. L.W. Emark, "Computational implementation of online load flow", F 77 575-4 presented at 1977 IEEE PAS Summer Meeting, Mexico City, July 1977. 15. B. Stott and 0. Alsac, "Fast decoupled load flow", IEEE Trans. Power App. Syst., vol. PAS-93, pp. 859-869, May/June 1974. 16. F. L. Alvarado and E. H. Elkonyaly, "Reduction in Power Systems", A77 5-7-7, IEEE PES Summer Meeting, Mexico City, July 1977. 17. E. H. Elkonyaly and F. L. Alvarado, "External System Static Equivalent for On-Line Implementation", A 78 060-0, IEEE PES Winter Meeting, New York, Jan. 1978. 18. Monticelli A., Deckman S., Garcia A., and Stott B., "Real- Time External Equivalents for Static Security Analysis," IEEE. TPAS, Vol. PAS-98, 498-508. 19. Dimo P., Groza L., lonescu S., Ungureanu B., and Petcu I., "Research Concerning the Generalized Utilization of a Single Equivalent to the Power System State Analysis, “Computer Methods of Power System Analysis and Control, Joint Research Seminar between Romania and USA, Bucharest, Romania, June 1974. 20. Dimo P., Nodal Analysis of Power Systems, Abacus Press, England, 1975. Proceedings of International Conference on Advances in Engineering and Technology www.iaetsd.in ISBN : 978 - 1505606395 International Association of Engineering and Technology for Skill Development 63