ML12088A497
| ML12088A497 | |
| Person / Time | |
|---|---|
| Site: | Indian Point  |
| Issue date: | 03/28/2012 |
| From: | Degeneff R, Mauricio Gutierrez, Mckenny P ABB Power T & D Co, Rensselaer Polytechnic Institute |
| To: | Atomic Safety and Licensing Board Panel |
| SECY RAS | |
| References | |
| RAS 22105, ASLBP 07-858-03-LR-BD01, 50-247-LR, 50-286-LR | |
| Download: ML12088A497 (7) | |
Text
ENT000109 Submitted: March 28, 2012 A METHOD FOR CONSTRUCTING REDUCED ORDER TRANSFORMER MODELS FOR SYSTEM STUDIES FROM DETAILED LUMPED PARAMETER MODELS R.C. Degeneff, Senior Member and P.J. McKenny, Senior Member M. R. Gutierrez, Student Member ABB Power T&D Company, Inc.
Rensselaer Polytechnic Institute Muncie, Indiana 47302 USA Troy, New York 12180 USA the transformer is required to perform realistic system Abstract - All power transformer manufacturers maintain studies.
computer programs that compute the internal transient Failure rates of higher level EHV transformers voltage distribution when the transformer is subjected (2.3% per ye.ar/phase for 765 Kv GSU' s) compared to lower to transient voltages. This information is used to level units (0.7% per year for 345 Kv GSU's) is cur-design the insulation structure for the transformer. rently a significant concern to the utility industry and Utility engineers also need to represent the power considerable effort has been spent investigating the transformer in some detail for their system studies cause of these failures [1]. Preliminary results since significantly higher failure rates on large EHV suggest that the transient voltages at the terminals of units suggest the transient voltage that the system a transformer in the field are, in some measure, a places on the terminals of the transformer are in some function of the impedance characteristic of the trans-measure a function of the impedance characteristic of former itself. As indicated above, however, the the transformer. Currently, no method exists to transformer model used in utility system studies is not conveniently link the detailed lumped parameter model of sufficiently detailed to accurately capture the trans-the transformer designer to that required by the utility former's required frequency characteristic.
engineer. This paper presents a reduction technique A method which starts with the manufacturer's which uses the detailed lumped parameter transformer detailed model and systematically reduces it to a size model as a starting point and allows its reductions to suitable for use in utility system studies is presented.
any size specified by the user. The method is straight- The method requires only that the retained nodes and forward mathematically and while retaining the physical desired size of the model be specified. The result configuration of the transformer does not require any obtained is a group of inductive and capacitive lumped proprietary information from the transformer suppliers. elements suitable for use in utility transient programs The paper presents the necessary mathematics and such as EMTP.
illustrates the method by example using a 500 MVA auto-transformer constructed by ABB for AEP. DETAILED LINEAR TRANSFORMER MODEL INTRODUCTION The following paragraphs describe the basic modules required in a typical program used in computing the To provide a reliable and efficient insulation internal transient voltage distribution in transformers.
structure the transformer design engineer must have a As indicated previously, all transformer manufacturer's knowledge of the transformer voltage distribution under possess similar programs. The detailed lumped parameter impulse and switching waveforms. Transformer manufac- inductance and capacitance models constructed will turing companies generally have some form of computer provide the starting point for the reduction process program which allows the design engineer to calculate developed in this paper. Only a detailed linear the required distribution for various types of waveform. transformer model will be considered in this work.
These programs usually construct an inductance, capaci- An input initialization module will provide a link tance, and resistance model from the transformer between the transformer designers concepts in inches of geometry and then solve for the voltage distribution copper, paper, oil, and steel to the formulas and using either time or frequency domain techniques. The conventions necessary to compute the needed R, L, and validi ty of both these techniques have been verified C' s.
with laboratory and field data from manufacturing and The Topology module will convert users instructions utility companies. specifying desired model detail and descritization into To perform system studies, utility engineers also an orderly system of node numbers associated with the need to represent the power transformer in some detail. winding geometry. In addition, this module would order It is impractical to use the detailed model in system the node numbering sequence to make the node number studies because of its size and the resultant com- sequence as straight-forward as possible. For example, putational burden, therefore, reduced order models are the ground node would be assigned the largest node used in system studies. Reduced order models for number, followed by impulsed nodes, switched nodes, utility system studies are either constructed from nodes connected to non-linear resistors (2nO) , and terminal characteristics or by greatly simplifying the finally the detailed linear model. The resulting detailed model obtained from the transformer design network is shown pictorially in References 5 and 7.
study [2,3,9]. Both these models have impedance vs It is worth noting that the ability of the detailed frequency characteristics which are in considerable model to faithfully reproduce the transient characteris-error above the first resonance. If the transient tic of the transformer depends upon two independent waveform the system places on the transformer terminals model characteristics. First, the model must accurately is a function of both the system and the transformer provide R, L, and C's which are appropriate for the impedance characteristic, then a model which more transformer geometry. (This fact is well appreciated faithfully represents the impedance characteristic of and has been the topic of numerous papers in the literature.) The second requirement is that the model must possess sufficient detail to adequately represent the frequency response for the applied waveshape of interes t. This often overlooked point can produce results which appear to be mathematically valid, but which have little physical basis [4]. The second con-dition is also required to meet the necessary assump-tion for a lumped parameter model - the system must be quasi- static. In other words, the greatest frequency of interest must have a period at least ten times larger than the travel time of the largest element in the model. Hence, for a model to accurately predict the 0-7803-0219-2/91/0009-0532$01.00©1991IEEE
switching surge response it must be valid to at least 10 techniques apply over a much wider frequency range than kHz, 50 kHz for a full wave, and 250 kHz for a chopped is, in fact, the case.
wave [5), since 95% of the energy in the applied wave is The following reduction technique is based on an contained below the frequencies listed. extension of Kron ' s work in the Laplace transform A capacitance module computes the series capaci- domain. The results will be compared with the transient tance along the winding and the shunt capacitance to voltage response and frequency characteristics of the different winding segments, other windings and ground. detailed model which is used as a reference.
The module therefore provides a nodal capacitance matrix for the entire winding structure. In the past, the KRON's REDUCTION METHOD FOR LINEAR TIME-inductance module has been constructed by either INVARIANT LOSSLESS TRANSFORMER MODELS building a self and mutual inductance matrix, inverting the matrix, and then using a loop to nodal transfor- The mathematics behind Kron's reduction method is mation [6,7], or by constructing the inverse inductance straigHtforward [12). Equation (1) shows the relation-matrix directly [8). ship, in frequency domain, between nodal currents and The influence of losses on the reduced model will not voltages for an RLC network representing a linear-time be considered in this paper but will be addressed in invariant model of a power transformer.
subsequent work. Historically, the influence of losses on the transient response has been ignored or handled empirically. Only recently have methods been suggested [I(s) 1 = [.!J.]
I2
=( YYl1/
Z1 Y12 ]
Y 22 (E11 E2
= [Y(s)] [E(s)] (1) that represent losses based upon the geometry of the windings [6]. Ignoring losses will not change the frequency characteristics of the model appreciably and where Y(s) - lis (r] + (G] + sIC]
the transient response will, in general, be more pes-simistic. with [r] - inverse nodal inductance matrix A solution routine applies preselected voltage [G) - nodal conductance matrix and/or current waveforms to the lumped parameter model [C) - nodal capacitance matrix and computes the transient voltages. The solution s - Laplace operator module is normally one of two general types: time domain and frequency domain. Matrix partitioning in equation (1) groups all input, Time domain method solve the problem using some output and nodes of interest under subscript 1. All form of numerical integration. The most common form of remaining nodes in the detailed model are grouped under time domain solution is that proposed Dommel (101. subscript 2. Therefore, 12 = 0 and E2 can be expressed Dommel's approach is based on the trapezoidal rule of as a function of El yielding the reduced matrix equation integration and provides a very robust solution method (2) .
with great flexibility. Nonlinear core characteristics and time dependent switches (used in failure analysis) (2) can both be addressed with this technique. Its prin-cipal disadvantage is that it is difficult to model Equation (1) represents the detailed linear-time frequency dependent parameters such as eddy current invariant transformer model corresponding to a RLC losses. network of n nodes plus ground. On the other hand, A frequency domain approach to solving the problem equation (2) represents the reduced linear-time in-has gained wide acceptance. In this method, a solution variant transformer model corresponding to a RLC network is obtained in the frequency domain and convolution used of m nodes, plus ground, where m < n. The reduction to obtain a time domain solution (11]. With this method outlined yields an exact reduced model valid for approach, frequency dependent parameters can be modeled one frequency. Kron's reduction method allows a with relative ease, whereas, nonlinear characteristics detailed (even turn-to-turn) transformer model to be of the transformer (core and ZnO) tend to be much more reduced to any number of nodes, 0 < m < n. This reduced difficult. For a detailed linear transformer model, model can be applied for"steady-state studies or normal however, both methods provide the same solution. operation of power systems where one single frequency (60 Hz) is normally considered.
REDUCTION GOAL For electrical transient studies in power systems, where a spectruIil of frequencies is involved, Kron's The steady state and transient behavior of a reduction method must be modified in order to obtain a circuit, for any applied voltage, is established by the reduced model which approximates the response of poles and zeros of the circuit impedance in the complex detailed model. This is a trade-off between simplicity frequency plane. By definition, the zeros of the and accuracy.
terminal impedance function coincide with the natural Equation (2) can be rewritten as:
frequencies of the circuit. McNutt [5] has defined terminal resonance as a terminal current maximum and a terminal impedance minimum. (This corresponds to what is also referred to as series resonance.) Terminal where Yr is the reducing nodal admittance and is shown anti-resonance is defined as a terminal current minimum in equation (3):
and impedance maximum. (This is also referred to as (3) parallel resonance.) McNutt also defines internal where:
resonance as an internal voltage maximum and internal anti-resonance as an internal voltage minimum. These (4) relationships are all grouped under the heading of forced oscillations and form the basis of the transient response for a detailed lumped parameter transformer model. If a reduced order model of a transformer is to reprOduce the transient voltage characteristics of the detailed model, the reduced model must essentially Y;; (scn + G22 + P;2 r (5) contain the same poles and zeros in its terminal im- P21 pedance and transfer functions as the complete model Y21 SC21 + G21 +
S (6) over the frequency range of interest. In the past, this relationship has not been emphasized and, if not ap-substituting (4) and (6) into (3):
preciated, will lead to a conclusion that reduction 533
(7)
Obtaining the inverse of Yzz may be cumbersome, however, In equation 11, we have two unknown matrices, Cr1
.this inversion can be simplified if it is reduced one and r r1 and three equations. Taking two equations only, node at a time, by partitioning matrices in equation (1) there are three possibilities:
such that Y Z1 is a row matrix, Y 1Z is a vector and Y zz is a scalar which represents the nodal admittance asso- I Equations (lIb) and (l1c) give the low ciated to the node which is neglected. By reducing one frequency model node at a time, the inverse of the hodal admittance of the neglected node (eq. 5) becarit~ a scalar fraction. II Equations (l1a) and (1113) give the high Substituting equation (5) into equation (7) yields: frequency model s4A4 '+ s3AJ + SZAz + SAL + Ao III Equations (l1a) and (lIe) ,give the disin-Yr = Yr1 =
S(S2C22 + SG22 + r 2Z ) tegrated Kron's reduction model.
(8) The characterization of high or low frequency model where: is given in Appendix 1 for models I and II and the neglected equations for each type may be used for error A4 C12Cl~ evaluation.
It can be shown that model III, called disin-A3 C12Gl~ + G1ZC~~ tegrated Kron's reduction model, can be obtained by applying Kron's reduction method to the nodal capaci-A2 c12rfz + rlZC1~ + G1ZG1~ tance matrix and to the inverse nodal inductance matrix independently. This model can be obtai~ea by neglecting one or more nodes at a time. It can in fact be con-Al G1zrfz + r12Gl~
structed with the application of equation (2) to the L and C matrix independently.
In Table 1 the equivalent reducing nodal capaci-tance and inverse nodal inductance matrices are shown The post subscript t means transpose and Yr1 represents for Models I, II, and III. Those relationships have the reducing admittance by reducing one node at a time. been obtained from equation 11. In Appendix 1 the range Additionally, Y r1 , A4 , Aa, A2 , A1 , Ao, are' n-l x n-l of application for Models I and II is determined by the matrices, with n equal to the number of nodes in the nodal frequency of the node to be deleted. Model I is model not counting ground. applicable for W < wn
- Model II is applicable for Examining equation (8), we see that the associated W > Wn where the nodal frequency is defined by equation RLC parameters of the Y r are frequency dependent, (A4). Model III is a hybrid model, it combines the high however, it is possible to approximate the reducing frequency reducing nodal capacitance with the low nodal admittance by an equivalent that is not frequency frequency reducing inverse nodal inductance.
dependent. Using this approximation maintains the The minimum retained nodes in the reduced model are linearity and time invariant property of , the reduced input, output, and nodes of interest. Additionally, model as indicated in equation (9) nodes may also be retained to i~prove the accuracy of the reduced model. These additional nodes can be
- rrl selected automatically by the program, which optimizes Yr1 = sCrl + Grl + S (9) the reduced model by neglecting .the highest or lowest nodal frequency nodes at each retluction step for the low (model I) and high (model II) frequency model, respec-Equating equations (8) and (9) yields an over-tively. For Model III, (disintegrated Kron' s reduction) determined set of five equations where only three optimization is not necessary because the reduced model matrices are unknowns: Cr1 , Gr1 , and r r1' They must is independent of the order in which nodes are reduced.
satisfy as closely as possible the following equations: The number of nodes contained in the reduced model is a trade-off between simplicity and accuracy. In A4 - CZ2 Cr1 (lOa) general, increasing the'number of retained nodes in the reduced model diminishes its simplicity but improves its Aa = GZ2 Cr1 + C2Z Gr1 (lOb) accuracy.
Az r zz Cr1 + G2Z Gr1 + Czz rrl (lOc) ILLUSTRATIVE EXAMPLE Al = r Z2 Gr1 + G2Z rrl (lOd) The following example will illustrate the computed transient voltage response for standard ful~ and Ao = r 22 rr1 (lOe) switching surge waveforms applied to the HV terminal of a single phase, core-form, 500 MVA, 765/345/34.5 kV YYD Since this analysis is restricted to the lossless case, autotransformer. Voltages presented are either line-to-we can set conductance matrices to zero, therefore ground at the common terminal, or a voltage difference equation (10) becomes between two layers ends in the series winding.
Results from four different models of this trans-(l1a) former are presented. The first is considered to be the base case and is generated from a model of the trans-Az (lIb) former which divided each layer into 5 segment. This model was developed from the transient analysis program (l1c) used to design the insulation structure for the trans-where: former.
534
TABLE 1 Reducing Admittance EIGENVALUES COMPARISON (kHz)
Models Full Model Reduced Models Reducing Parameters I II III 4.53 4.53 4.53 4.53 5.82 5.82 5.83 5.94 12.46 A2 C22 A O A,
~ 12.41 12.41 12.42 r 22 ~2 C22 C22 17.78 17.78 17.98 19.37 19.37 19.66 20.14 Ao ~ 1'22 A , Ao 21.82 21.82 21.96 21.32 r 22 C22 C;2 r 22 31.90 31.91 32.29 23.50 34.56 34.59 36.03 36.34 39.43 39.46 39.53 39.56 40.27 40.78 40.85 41.87 44.01 The transformer contains 1294.5 total turns and 26 47.50 47.58 43.09 48.78 layers. The layer with the greatest number of turns 53.92 53.99 49.80 (69.5) was divided into 5 segments which placed 14 turns 59.93 60.03 55.47 in each segment. It is this segment which determines 60.69 60.98 the highest frequency the detailed model is valid for. 62.49 62.69 62.06 For an average mean turn length of 18 fee't, and 64.26 64.40 69.94 68.49 assuming a velocity of propagation in the winding of 500 69.59 69.77 70.35 feet per microsecond, the highest frequency this model 72.95 73.32 74.31 is valid for is on the order of 200 kHz. The lumped 78.16 78.42 78.70 parameter model for this condition contains 134 nodes. 83.20 83.37 86.12 The following analysis compares the response of the 85.92 87.01 base case to reduced models developed using the techni- 87.96 88.37 90.04 que presented. The three reduced models contain 80, 26, 92.08 92.67 and 10 nodes, respectively. Eighty nodes corresponds to 96.49 96.96 95.29 the number of nodes a lumped parameter model would have 99.95 100.55 if each layer were divided into 3 segments. The 26 node model corresponds to a model constructed with only one segment or division per layer. The results presented correspond to the response of Model I (low-frequency reduced model) only. In general, the response of Model ERROR EVALUATION I I agrees with the reference model for the first few
REFERENCE:
FULL MODEL 134 NODES miaroseconds, and the response of Model III (disin- BASE: FULL MODEL MAXIMUM tegrated Kron's reduction) is comparable to Model I.
However, Model III has shown lower accuracy than Model COMMON TERMINAL I. The applied waves are a standard 1.2/50 p.s full wave ERROR (%)
and a 75/800 p.s switching surge. For this example the common winding is left floating and the tertiary is Reduced Full Wave Switching short circuited. Model RMS MAX RMS MAX Table 2 contains a comparison of the eigenvalues of 80 N 1.19 3.32 1.25 3.05 the base case and three reduced models below 100 kHz. 26 N 4.18 9.83 0.90 2.63 Each model contains the same number of eigenvalues 10 N 4.64 12.22 2.70 7.90 (poles) or natural frequencies (zeros) as their are nodes in the model. Table 2 contains only the lowest VOLTAGE DIFFERENCES BETWEEN LAYER ENDS values. The base case would be expected to contain accurate natural frequencies through approximately 200 ERROR <%)
kHz with those above 200 kHz of questionable accuracy.
It is against these lower frequency eigenvalues the Reduced Full Wave Switching reduced models are compared. Model RMS MAX RMS MAX From Table 2 it is noted that the 80 node reduced 80 N 4.36 12.74 0.77 2.28 model agrees well through 100 kHz or the 25th eigen- 26 N 8.34 22.35 2.19 6.36 value. The 26 node model agrees well through 34 kHz or 10 N 16.20 42.20 3.23 8.89 the 8th eigenvalue and the 10 node model has good agreement up to 12 kHz or the 3rd eigenvalue. From this it would be expected that all the models would give very value of the base case.
good agreement in the switching surge range. For a full Figure 3a, 3b, and 3c compare the base case switch-wave the agreement would be good for the base case, the ing surge response of the common terminal to ground with 80 and 26 node model but the 10 node model would be less that of the 80, 26, and 10 node reduced model, respec-accurate. Subsequent examination of the voltage wave tively; As was anticipated from the natural frequency forms verify this condition. characteristics of the models, the agreement is quite Figures la, lb and lc compare the base case full good for all models. Figures 4a, 4b, and 4c present the wave response of the common terminal to ground with the comparison of the layer to layer voltage differences 80, 26, and 10 node reduced model respectively. Figures when the transformer is subjected to a switching surge.
2a, 2b, and 2c compare the full wave response between All reduced models produce good results.
adj acent layers for the 80, 26, and 10 node reduced Table 3 presents an error analysis for the wave model, respectively. As was anticipated from the forms presented in Figures 1 to 4. The base case analysis, the 80 node reduced model follows the base detailed model response is used as reference and the case response very closely, the 26 node model reasonably error has been computed by the RMS norm and the infinity well, and the 10 node model's response is almost a mean norm.
535
t' (b)
\X0. 1"\.
~-.~
~.~OO--~=----20.~OO----"OO~--'-O.O-O-- ~~.oo--~~---,,~a.JO~~'~~OO---'~"~,"-- ~L,"---------,~-"----,,,-,,,---,-,,-C-,,
~L.oo--~~---,,~oo----,,~co---.-O.O~O- TIME (MICROSECONDS]
TIME [MICROSECONDS] TIME [MICROSECONOSI TIME (MICROSECONDS)
.,~jr
.' II (c) Fig. 1. Reduced Model vs (c) Fig. 3. Reduced Model vs Base Case - Common Terminal Base Case - Common Terminal to Ground Voltage for to Ground Voltage for Full Switching Surge Wave g~
~ a) Base vs 80 node
~
a) Base vs 80 node b) Base vs 26 node b) Base vs 26 node c) Base vs 10 node c) Base vs 10 node Base case is the solid
..' Base case is the solid line.
line.
~~.co~~~--~~~,,~~~~oo--~.a.a~o-- ~'.~OO--~~--~'~~.OO~-=6JO~"~-'~"~."-
TIME (MICROSECONDS) TIME (MICROSECONDS)
/ 1\
~\'
(a) (b)
- \ ~ r( I ~
" ~.
'. I r [
gc> '.! :~ I
- i
. ~ ~ J~\
~Loo----~--_"~"----"O~O---'-ooo~- ~L.oo---------"-.oo----,,.-,,----.O.oo---- ~'~.OO--~~--~'20~.J~O--~"~OOO~--"O~,"-- ~~"--~~--~"~OO~O--~5:0~.~O~-'c..O~"-
TIME [MICROSECONDS) TIME (MICROSECONDS)
TIME [MICROSECONDS) TIME (I1ICROSEC(}NDS)
~~/"~/"'\ (c) Fig. 2. Reduced Model vs Fig. 4. Reduced Model vs Base Case - Layer-to-Layer Base Case - Layer-to-Layer
, '. . Voltage Difference for Full Voltage Differences for
......../ ',,-,,' \. Wave Switching Surge
\ a) Base vs 80 node a) Base vs 80 node
" b) Base vs 26 node b) Base vs 26 node c) Base vs 10 node J~
c) Base vs 10 node Base case is the solid Base case is the solid line. line.
~,~.--~~--~~7.,,----,,~oo---.-oo~o- ~~"~~~---"-O.OO~~6J~OOO~--"O~.CO~
TIME (MICROSECONDS) TIME mICROSECONDS]
536
DISCUSSION The examples have shown that the reduction method formers," paper A78 539-9, presented at IEEE PES presented is valid over a predicted frequency range. Summer Meeting, Los Angeles, California, July 16-Agreement between the -reduced and detailed model's 21, 1978.
transient response improves as the number of eigenvalues that match in each model increase. Agreement between 3. F. de Leon, A. Semlyen, "Reduced Order Model for the voltage versus time profile for a reduced model Transformer Transients," paper 91 WM 126-3 PWRD, compared to the base should not be construed as validity IEEE/PES 1991 Winter Power Meeting, New York, New of the reduced model for all frequencies. The number of York, February 3-7, 1991.
nodes to be retained in the model for the reduction method presented is a user decision and a function of 4. H.A. Haus and J.R. Melcher, Elecrromagneric the task at hand. If the analysis requires the inves- Fields and Energy, Chpt. 3, 1989, Prentice Hall, tigation of the switching surge response for the example Englewood Cliffs, New Jersey.
given, a 10 node model would be adequate. However, if the analysis were for a chopped wave the model would 5. W.J. McNutt, T.J. Blalock and R.A. Hinton, have to retain at least the detail of the base model. "Response of Transformer Windings to System This method does not require the transformer manu- Voltages," IEEE PAS-93, Mar./Apr. 1974, pp. 457-facturer to provide any proprietary information about 467.
transformers dimensions. Finally, it should be pointed out that the accuracy of the reduced model, at best, 6. D.J. Wilcox, W.G. Hurley and M. Conlon, "Calcu-will be that of the detailed model so the detailed model lation of Self and Mutual Impedances Between must be constructed so that it is accurate in the fre- Sections of Transformer Windings, "IEEE Proc.,
quency range of interest. This seems to be a point that Vol. 136, Pt. C, No.5, September 1989.
is overlooked in recent efforts to create reduced order models. 7. R.C. Degeneff, "A General Method for Determining Results presented in this paper have assumed zero Resonances in Transformer Windings," IEEE PAS-96, initial conditions. However, since this model is pp. 423-430, 1977.
completely linear, if the solution routine used to solve the reduced network will allow initial conditions, the 8. F. de Leon and A. Semlyen, "Parameters of Trans-model will reproduce situations with non-zero initial formers," paper 91WM 002-6 PWRD, IEEE/PES 1991 condi tions . Clearly, for the non-linear case this model Winter Power Meeting, New York, NY, February 3-7, is not adequate and a subsequent paper will address this 1991.
concern.
The application of this method requires that a 9. H.W. Dommel, 1.1. Dommel and V. Brandwajn, detailed R,L, and C model of the transformer be availa- "Matrix Representation of Three-Phase N-Winding b1e. Most suppliers of power transformers have this Transformers for Steady State and Transient capability since the design of the transformers in- Studies," IEEE Trans. on Power Appl. and Sys.,
sulation structure depends very greatly on the ability Vol. PAS-I, No.6, June 1982.
to predict the transient voltages prior to construction and this requires the R, L, and C data. There are other 10. H. W. Domme1 and W. S. Meyer, "Computation of techniques available to construct a reduced model if for Electromagnetic Transient," Proc. of IEEE, Vol.
any reason the R, L, and C matrices are not available 62, No.7, July 1974, pp. 983-993.
[2,3].
The presented method is limited in detail only to 11. O. Einarsson, "The Program STATE, High Frequency the model created by the transient program. In the case Transformer Modeling by Modal Analysis," KZEB, presented this is a full multi-winding, three-phase 86-004 (1986).
linear iron core representation.
- 12. G. Kron, Tensor Analysis of Nerworks, John Wiley CONCLUSION and Sons, New York, NY, 1939.
This paper has presented a method of reducing the APPENDIX 1 size of the detailed lumped parameter model normally used for transformer insulation design to a size The reducing admittance matrix generated by acceptable to a utility engineer performing systems eliminating one node at a time is shown in equation (8),
studies. A significant advantage of this is that it can and is rewritten as follows:
retain the frequency response characteristic:;s of the transformer in a range of interest. The reduction technique retains the physical characteristics of the transformer model and the model is suitable to be used in transient analysis programs such as EMTP. The (A1) reduction technique is mathematically straight-forward and requires only the detailed model and user guidance on which nodes to retain. The method does not require that proprietary information about the transformer construction be used in the reduction process. Equation (A1) is obtained from equation (8), by long division where R.n represents the remainder which is REFERENCES given in equation (A2):
- 1. L.B. Wagenar, J.M. Schneider, J.H. Provanzana, D.A. Yannucci, and W.N. Kennedy, "Rationale and Implementation of a New 765 kV Generator Step-up (A2)
Transformer Specification," CIGRE paper 12 - 202, August 1990. where:
- 2. R.C. Degeneff, "A Method for Constructing Ter-minal Models for Single-Phase N-Windings Trans-537
Robert C. Degeneff (M'67,SM'84) was born in Ann Arbor, Michigan on July 9, 1943. He received his B.M.E. from General Motors Institute in 1966. He continued his education at Rensselaer Polytechnic Institute and recei-ved an MS in Electric Power Again, for the lossless case, the conductances in Engineering in 1967. Entering equation (AI) and (A2) can be set to zero which yields: the USAF in 1968, he served as a technical intelligence officer until he was discharged as a Yr.1 = ~s + (~ - r22A4)..!. + R (A3) Captain in 1971. He returned to C22 C22 C;z S m RPI and received his D. Eng. in Electric Power En-where: gineering in 1974.
Upon graduation, he joined General Electric's Large Power Transformer Department as a Senior Development Engineer. After numerous disasters he was promoted in 1978 to Manager of the Advanced Analytic Development Unit and in 1980 to Manager of the Advanced Electrical Development Unit. In 1981 he joined the Electrical Utility Systems Engineering Department as Manager of the HVDC Engineering Subsection and in 1985 assumed the position as Manager of the Software Services Section. In 1989 he joined the faculty of RPI as a Professor of Defining the nodal frequency associated with the node to Electric Power Engineering. Professor Degeneff's work be neglected as: has been in three areas of the utility industry:
transformers, HVDC systems, and planning. He has worked extensively in the modeling of transformers when they (A4) are subjected to system transient voltages. This provided the natural transition to the study of HVDC systems and their interaction with ac systems. The third Then, the remainder is given by: area is the software and procedures used by utilities in their planning efforts for demand side management or in-
- Rm2 tegrated planning.
(AS) He has published over two dozen papers, several chapters in books, and holds three patents. He is a Professional Engineer and a Senior Member of IEEE.
The remainder can be expanded by partial fractions as follows: Moises R. Gutierrez (S'88) was born in Managua, Nicaragua on November 13, 1946. He received (A6) the BS degree in Electromechani-cal Engineering from the Centro-american University of Nicaragua in 1969, and his ME degree from setting s - jw and substituting(A4) in (A6), the Rensselaer Polytechnic Institute remainder can be approximated as: in 1988.
He was an instructor of Elec-trical and Mechanical En-gineering at Centroamerican for W < wn (Al)
University, Head of the En-gineeringDepartment and General Director of Engineering Projects of the Nicaraguan Institute of Energy. Cur-R", - 0 for W > W rently, he is working towards his D. Eng. degree at Rensselaer Polytechnic Institute in the Electric Power Therefore, for the lower frequencies (w < wn ) the Engineering Department.
remainder can be approximated by an equivalent His research interests include transformer model inverse nodal inductance plus an equivalent nodal reduction and electromagnetic transients in electric capacitance matrices. For higher frequencies the power engineering.
remainder can be neglected.
Peter J. McKenny (S'90-M'84) received the B.Sc. in Substituting (A7) in (A3): Electrical and Electronic Engineering from the Univer-sity of Newcastle Upon Tyne, England, UK, and an MS and Ph.D. in Electrical Engineering from Clarkson Univer-sity, Potsdam, NY in 1977, 1980 and 1984, respectively.
Between 1984 and 1990 he was a faculty member in the Electrical Engineering Department of the University of (A8) Maine. In 1990 he joined ABB Power T&D Company as a Senior Development Engineer, with responsibilities in the areas of transformer transient analysis and electri-cal insulation.
Dr. McKenny is a member of both the Electrical In-sulation Society and the Power Engineering Society. He has served as a member of the Board of Directors of CEIDP (1987-89) and has held various executive positions in the IEEE Maine Section.
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