3 Ph Transformer and Generator Models

8/3/2019 3 Ph Transformer and Generator Models

http://slidepdf.com/reader/full/3-ph-transformer-and-generator-models 1/11

IEEE Transactions on Power Delivery, Vol. 6,No. 4, October 1991 1671

THREE-PHASE COGENERATOR AND TRANSFORMER MODELSFOR DISTRIBUTION SYSTEM ANALYSIS

Tsai-Hsiang Chen Mo-Shing Chen Toshio Inoue Paul Kotas Elie A. ChebliMember, IEEE Fellow, IEEE Member, IEEE Member, IEEE

Energy Systems Research CenterThe University of Texas at Arlington

Arlington, Texas 76019

A B S T R A C T - This paper presents detailed three-phasecogenerator and transformer models for analyzing a large-scaledistribution system. The cogenerator model presented here issignificant in that it will represent the inherent generator phaseimbalance due to distribution system imbalance. The cogeneratorscan be synchronous or induction and can be on either primary orsecondary systems. The transformer models consider the copperand core losses, the winding connection, the phase-shifting betweenprimary and secondary windings, and the off-nominal tapping. Anindividual phase, as opposed to a balanced three-phase,representation was employed. This approach is oriented towardapplications in distribution system operational analysis rather thanplanning analysis. This difference should be properly emphasized.The misuse of the planning oriented method to analyze theoperational behavior of the system will distort the explanation of thecalculated results and lead to incorrect conclusions.

Keywords: Cogenerator model, Transformer model, Power flow,Short circuit, System loss, Contingency, Distribution system,Injected current, System imbalance, Subtransient reactance.

JNTRODUCTION

It has becoming increasingly clear that utilities need demand-side management (DSM). The least-cost planning as well asdemand-side strategies are being widely adopted and mandated(e.g.in New York). Moreover, if current legislative proposals becomelaw(e.g. Tim Worth's Energy Efficiency Act of 1989), mandatesmay apply more universally. Both the DSM and least-cost planningand many other distribution analyses, for example energyconservation and switching recommendation studies, need rigorousoperational-type analysis rather than rough planning-type analysis.The difference between these two types of analyses should be

properly emphasized. The misuse of the planning type method toanalyze the operational behavior of the system will distort theexplanation of the calculated results and lead to incorrectconclusions.

The distribution system is basically unbalanced. Many factorscause system imbalance such as untransposed feeders, conductorbundles, single-phase loads, unequal three-phase loads, single anddouble-phase "radial spurs" on primary feeders. Furthermore, evenif the network is balanced, asymmetrical faults introduce imbalance.To avoid significant error arising from inherent system imbalance,rigorous distribution system analysis using detailed componentmodels is required.

91 WM 0 8 3 - 6 PWRD A paper recommended and approved

by the IEEE Transmission and Distribution Committee

of the IEEE Power Engineering Society for presentation

at the IEEE/PES 1991 Winter Meeting, New York, New

York, February 3, 1991. Manuscript submitted

June 26 , 1990 ; made available for printingDecember 11, 1990.

Consolidated Edison Companyof New York, Inc.

4 rving Place, New York, NY 10003

The purpose of this paper is to provide mathematicalcomponent models for this kind of rigorous distribution systemanalysis. The components are modeled by their equivalent circuitsin terms of inductance, capacitance, resistance, and injected current.The form of the models is dependent on the type of study to beperformed. For example, in power flow and short circuit analyses,the same transformer and conductor models are used, but thecogenerator and load models are different.

To develop such a distribution analysis system, the followingcomponent models were determined to be necessary: conductors,cogenerators, transformers, demands or loads, and capacitors.Other components such as network protectors, fuses, automaticswitches, etc., although necessary in contingency analysis, are notimportant in power flow and short circuit studies, and therefore, arenot presented.

Of these required models, conductors, capacitors, and

demands have previously developed standard implementations forthe Gauss solution approach[l-31. This was not the case forthe transformer and cogenerator models, which required anadditional research effort to design and implement new models forthe Z B ~ ~nd other solution methods. Hence, in this paper only thecogenerator and transformer models are introduced. For detaileddiscussion of feeder models for both coupled and uncoupled feederssee references[l] and [2]. For a discussion of the load model used,see reference [3]. These component models may require little, if anymodification to apply them to other solution techniques.

COGENERATOR MO DELS

Cogeneration is an effective means of increasing energyefficiency and reducing energy costs. The cogeneration processputs wasted heat to work. It saves energy by using the reject heat ofone process as an energy input to a subsequent process, effectivelyusing the same fuel. The favorable economics of cogeneration

enable it to play a substantial role in energy development in variousparts of the world[4-61.One utility seeing the effects of these favorable economics is

Consolidated Edison Company of New York, Inc. Con Edisonanticipates the addition of more cogenerators to their system in thefuture, and is developing analysis tools to facilitate the study of theirimpact.

The impact of cogenerators upon the existing system must bestudied because their contribution to the power flow and short-circuit current of the system is significant, When a request is madeto generate electricity by using cogenerators in parallel with a utilitysystem, electric distribution engineers need to study the impact thatthe proposed cogenerators will have upon existing facilities. Thereis currently no capability for studying the effects of cogeneratorsupon distribution systems, especially upon unbalanced distributionsystems.

This paper introduces new cogenerator models for three-phasedistribution power flow and short-circuit studies. The cogeneratormodel used in power flow studies can also be used for system lossand contingency analyses. It is significant because it represents the

generator phase imbalance due to inherent distribution systemimbalance. A program called "Generalized Distribution AnalysisSystem" (GDAS) has been developed to simulate a large scaleunbalanced distribution system. This program can be used toevaluate the performance of a distribution system containing parallelcogenerators[7-81.

0885-8977/91$01.00099 IEEE



1672

ab c

1Z G

Cogenerator M o u

A preliminary investigation of typical voltage control systemsfor synchronous cogenerators was done. According to theinvestigation results, synchronous cogenerators are not controlled tomaintain constant voltage, they are controlled to maintain constantpower and constant power factor. Furthermore, utilities may requirethat cogeneration operators provide such controls. For example,Con Edison, under some conditions, may require a power factor

controller to maintain a constant power factor on the synchronousgenerator by controlling the voltage regulator. The power factorcontroller must be capable of maintaining a power factor within plusor minus one percent at any set point. As a result, the synchronouscogenerators can be represented approximately as constant complexpower devices in the power flow study, i.e. cogenerators can berepresented as P-Q specified devices in the power flow calculation.

As for induction cogenerators, their reactive power will varywith the terminal voltage change. Thus, the reactive powerconsumption of the induction cogenerators is not exactly constant.For simplification, the induction cogenerators can be treated as P-Qspecified devices because the bus voltages are near 1.0 p.u. insteady state cases. As a result, both the synchronous and inductiongenerators can be represented as P-Q specified devices, i.e. constantcomplex power devices.

The cogenerators are modeled as an internal voltage "EI"behind the proper reactance; the generator subtransient reactanceXd"[9,10] as shown in the Figure 1. This is the Theveninequivalent circuit of a cogenerator. This model is different from thetraditional power flow generator bus (PV Bus) which is represented

by specified injected powers at a specified voltage. Using theFigure 1model, injected power of each phase, under an unbalancedterminal voltage condition, can be calculated in detail.

3; is tr ibution 1S y s t e m ,4

I

I, ,

.o n ',4

-Figure 2 Norton Equivalent Cic uit of Cogenerator

Sequence models in symmetrical component coordinates areused to derive the phase model of cogenerator shown in Figure 2.The sequence models shown in Figure 3 are widely used in powersystem analysis.

Z er o P o s i t i v e N e g a ti v eSequence Sequence Sequence

Figure 3 Sequence Models of CogeneratorWhere

20 :Zero-sequence impedanceZ, : Impedance between neutral and groundZ1 :Positive-sequence mpedance (Subtransient reactance &'I)

Z, :Negative-sequence impedanceEl :Generated voltage

The subtransient reactance is used not only in short circuitstudies but also in power flow studies. Rotor effects should beconsidered even in the steady state condition of power flow analysisbecause of phase imbalance.

Figure 4 is the Norton equivalent circuit of Figure 3, where acurrent source 11is used instead of the voltage source El.

Based on the assumptions discussed above, we have

~ o t a leal power

TO^ reactive power

= Pr = pa+pb+pc= Constant

= Qr = qa+ qb+qc=Constant

The internal voltage ebcs a balanced three-phase voltage inboth magnitude and angle, assuming a balanced design of thegenerator windings. In the GDAS program, a Norton equivalentcircuit of Figure 1 is used to represent the cogenerator model shownin Figure 2. The current vector in Figure 2 shouldbe balanced and

y$ = (z$c)-l. TI& = y;JC Efbc

Z e ro P o s i t v e N e g a t i v e

Sequence Sequence Sequence

Figure 4 Norton Equivalent Circuit for CogeneratorWhere

yo =1zo+ 32,

1Y1 =z;

( 1 )

. -

The phase model can be obtained by transforming equation (1)

to phase coordinates using the transformation matrix Ts.

Ts [?]=Ts[ ]-Ts[i;y;]TITs[;] (2)

Where



1673

1 1 1

Thus the phase model is given by equation (4),esulting in thephysical representa$on,shown in Figure (2).

where

(4)

Y$=TsYg12?;1 =Ts (5 )

The current source is balanced, thereby, by constraining the

Multiplying equation(4) n both-side by Vak*t

specified4. andQr in power flow studies.

Then the constraint s represented by

(Pa + ~b + PC) - + qb+ qc)

= 1 1 1 + + av;) - v&*~G ~" h (7)v-3

Pr jQr = @a +pb + PC) -j(qa + qb+ qc)

and

(8)Thus, specifying total PT nd totalQr is sufficient to guarantee a

balanced current source.The cogenerator model shown in Figure 2 can be applied to both

power flow and short circuit analysis. In the power flow study, the

injection current 11 will be updated using equation (4) based on the

updated voltage at each iteration. In the short circuit study, the 11 isconstant, and is calculated using the converged power flow voltagesat the bus where the cogenerator is connected to the system. In

power flow studies the admittance matrix Gbc otal real power PT

and total reactive power Qr are held constant. In short circuit

studies the admittance matrix ydcbc and the internal balanced current

source 11 are held constant. Although some variables are heldconstant during the iterative process, the voltages that are dependenton the system conditionwill change at each iteration. Therefore, inboth power flow and short circuit studies, the injected currents will

change due to the voltage change at the buses where the cogeneratoris connected.

Figure 2 shows the cogenerator model for power flow analysis,where

(9 )

Figure 5 shows the procedure for calculating I,, Ib and IC, thecogenerator injected source currents, for each iteration in the powerflow study. For a more detailed discussion of the distributionpower flow study, see references [111.

From the above discussion, a cogenerator is represented by three

injected currents. This representation is suitable for the Z B " ~rfactorized Y B ~ethods, however, for Newton-Raphson, Gauss-Seidel method or other solution methods, the representation mayrequire modifcation. The basic algorithm s unchanged, however.

Estimated valuesobtainedfrom Specifiedk-th iteration value

$ $vc, P T + j Q r

1

Estimated values

(k+l)-th iterationI; I; tobeusedfor

Figure 5 Procedure to CalculateIa, Ib andfor Power flow Study

. .rator Model for Short-circuit &&

The cogenerator model for short-circuit study is the same as thatof the power flow study except for the calculation of the internal

current source 11 In the short-circuit study, the initial value of the

internal current source 11,obtained from the power flow calculation,is maintained constant since the internal voltage E1of the cogeneratoris assumed constant at the instant of the fault. The constant 11 iscalculated using the converged power flow voltages at the bus wherethe cogenerator is connected to the system. The calculationprocedure of the cogenerator model for short-circuit study is shownin Figure 6.

.--- - - - --Estimatedvalues { 11 is obtained \

obtained from from converged;k-th iteration -4 load flow result II andmaintained

v", $J

Ibtobeusedfor

g g g (k+l)-thiteration

Figure 6 Procedure to CalculateI,, Ib and for Fault StudyFhe impact of the numerous transformers in a distribution

system is significant. Transformers affect system loss, zerosequence current, grounding method, and protection strategy.Although the transformer is one of the most important components



1674

Wye-G Delta

Wye Wye-G

Wve Wve

of modem electric power systems, highly developed transformermodels are not employed in system studies. It is the intention of thispaper to introduce a transformer model and its implementationmethod so that large-scale unbalanced distribution system problemssuch as power flow, short circuit, system loss, and contingencystudies, can be solved.

Recognizing the fact that the system is unbalanced, theconventional transformer.models, based on a balanced three phaseassumption, can no longer be considered suitable. This is done withjustifiable reason. For example in the widely used, delta-grounded

wye connection of distribution step-down transformers, the positiveand negative sequence voltages are shifted in opposite directions,this phase shift must be included in the model to properly simulatethe effects of the system imbalance.

Recent interest in unbalanced system phenomena has alsoproduced a transformer model adaptable to the unbalanced problemwhich is well outlined in [12]. Further information about this modelmay also be found in [2,13,14]. The model developed thus far canbe applied directly to distribution power flow and short-circuitanalyses. However, it is still not accurate for system loss analysisbecause the transformer core loss contribution to total system loss issignificant[l5,16]. To calculate total system loss, the core loss ofthe transformer must be included in the model. The completetransformer model combines the unbalanced and loss models from[12] and [15] in order to integrate system loss analysis in powerflow or short-circuit studies.

It is important to note that the unbalanced transformer modelderived by Dillon in reference [12] cannot be applied directly toeither the factorized Y B ~ ~r direct inverse YB,,~ ethod because ofnumerical considerations. For some connections such as grounded

wye-delta, delta- grounded wye, these models make the system Y B , , ~singular. Therefore, the application of the factorized or directinverse methods becomes impossible. To solve this problem,this paper introduces an implementation method in which artificialinjection currents are used to make the system Y B ~ ~onsingular. Atwelve-bus sample system is used to demonstrate the models and theeffects of various transformer connections on a distribution system.

YI Yn y m y'my n yn -Yn -YE

YTT Yn -Yn -Yn

Derivation of Transformer Mode S

As used in Reference [15], a three-phase transformer isrepresented by two blocks shown in Figure 7. One block represents

the per unit leakage admittance matrix Y+bc, and the other block

models the core loss as a function of voltage on the secondary sideof the transformer.

Wye Delta

Delta Wye-G

AdmittancePrimarv M atnx

y n Yn ym y'm

y n YI YDI

Secondary

a b c

Q b a&

Figure7 Overall Proposed Transformer Model

The presence of the admittance matrix block is the majordistinction between the proposed model and the model used in[15,16]. In the proposed model, Dillon's model is integrated withthe admittance matrix part. As a result, the copper loss, core loss,system imbalance, and phase shift characteristics are taken intoaccount. The implementation method is introduced in the followingsections.

Core LQSS

The core loss of a transformer is approximated by shunt coreloss functions on each phase of the secondary terminal of thetransformer. These core loss approximation functions are based onthe results of EPRI load modeling research [3] which state that realand reactive power losses in the transformer core can be expressed

as functions of the terminal voltage of the transformer. Transformercore loss functions represented in per unit at the system power baseare[15,16]:

(A [VI + B eclvl') (12)P (P.U.) = KVA Rating

System Base

(p'u.) = System Base (13)VA Rating (D VI 2 + E e FIVI2 )

Where, typically,

A=0.00267 B=0.734~10-~ C=13.5

D=0.00167 E=0.268x 10-1 F=22.7

is the voltage magnitude in per unit.It must be noted that the coefficients; A, B, C, D, E, and F; are

machine dependent constants. For the current research project, corelosses are represented by the functions and typical constants shownabove.

Admittance MatriK

The admittance matrix part of the proposed three-phasetransformer models follows the methodology derived by Dillon[l2],but a novel implementation is introduced herein.

For simplification, a single three-phase transformer isapproximated by three identical single-phase ransformers connectedappropriately. This assumption is not essential, however, itsimplifies the ensuing derivation and explanation. Based upon thisassumption, the characteristic submatrices used in forming the three-phase transformer admittance matrices can be developed. The

matrices for the nine common connections of three-phasetransformers are given in Table 1.

Table 1 Characteristic Submatrices used in Forming the

Transformer connection

W e-G W e-G

elf Admittance Mutual Admittance

W e-G W e y n y n -Yn -Yn

Delta Wye I y n Yn I yfn y m

Delta Delta I Yn Yn I -Yn -YnWhere. ~ ~ .

yt 0 0 2Yt -Yt -Yt

(14)

and Y t is the per-unit leakage admittance.

If the transformer has an off-nominal tap ratio a:p between the

primary and secondary windings, where CL and 0 are tappings on theprimary and secondary sides, respectively, then the submatrices aremodified as follows:

a) Divide the self admittance matrix of the primary by a2

b) Divide the self admittance matrix of the secondary by p2

c) Divide the mutual admittance matrices by apThe electrical models (equivalent circuits) and programmingmodels (the modified equivalent circuit models for power flow andfault analysis) of the two most common connections, groundedwye-grounded wye and grounded wye-delta, are presented in thispaper. The programming models of other connections can be

derived through a similar procedure. The derivation of the

I



1675

characteristic submatrices of a three-phase transformer can be foundin more detail in [121.

1. Grounded W ye-Grounded Wye Transformer

grounded wye transformer is shown in Figure 8. A network connection diagram for three-phase grounded wye-

Figure 8 Network Connection Diagram for Three-phaseGrounded Wye-Grounded Wye Transformer

From Table 1,the admittance matrix for the three-phase

0 Yt 0 0 -Yt 0

0 -Y t 0 0 Yt 0

0 0 -Yt 0 0 Yt

If the tappingsa nd P are not equal to 1 i.e. an off-nominal tapratio, then the admittance matrix must be modified using Equation16: the general form of admittance matrix for nominal and off-nominal grounded wye-grounded wye transformer. The equivalentcircuit of equation 16is shown in Figure 9.

Figure 9 Equivalent Circuit for Three-phaseGrounded Wye-Grounded Wye Transformer

Where

equiva lent circui t disappear. For the off-nominal tap ratio

transformer, either a or P or both a and P are not equal to unity,therefore, the shunt elements reflect the effects of the off-nominaltap ratio.

The programming model of a three-phase grounded wye-grounded wye transformer used in the GDAS program is shown inFigure 10.

P r i m a r y SecondaryY3

C O " s o c

Figure 10 Programming Model for Three-phaseGrounded Wye-Grounded Wye Transformer

Where

y3=YtaP

1;- Yt a -P a -P(-)% I:=-y'(-)v:aP P aP P

. .or in vector forms

The artificial current sources reflect the effect of the off-nominal

tap ratio. If a s equal to P then

Ip Is

and there is no injected current on either end of the transformer.Figure 10 is redrawn as Figure 11.

- a k = a h = 0

0rimary Secondary

a a

b b

63

C C

--Figure 11 Another Representation of Programming Model forThree-phase Grounded Wye-Grounded Wye TransformerWhere

For a nominal tap grounded wye-grounded wye transformer,

both a and P are equal to unity so the shunt elements in the



I

1676

2. Grounded Wye-Delta Transformer

wye-delta transformer is shown in Figure 12. The network connection diagram for the three-phase grounded

Secondary

-a

o b

Figure 12 Network Connection Diagram for Three-phaseGrounded Wye-Delta Transformer

' S

The admittance matrix for the three-phase grounded wye-deltatransformer is r Y t O - ~ tf l s O 1

For an off-nominal mounded wve-delta transformer. theadmitta

yp =

The equivalent circuit of equation 24 is shown in Figure 13.

primary Secondary

A

I I

Figure 13 Equivalent Circuit for Three-phaseGrounded Wye-Delta Transformer

where

24)

In Figure 13, the equivalent circuit of grounded wye-deltatransformer, there are three elements ( Y l ) connected to ground.These three elements must be represented by current injectionsbecause the Z B ~ ~ethod (i.e. factorized Y B ~ ~ethod) is used to

solve the system equations for both power flow and short circuitstudies. After replacing these three admittances by current

injections, the system admittance matrix ( Y B ~ )ecomes singular, ifthe grounded wye side is the end bus or swing bus. To avoid thisdifficulty three additional admittances (-Y3) are replaced by currentinjections.

In the GDAS program, a two-step superposition method is usedto implement the ZB,,~method. In the fiist step, there are no loadsapplied to the system and hence no current flows, therefore, thethree Y 2 elements in the delta side are represented by currentinjections. As a result, only the equivalent lines between theprimary and secondary sides in the same phase are used to form thesystem admittance or impedance matrices (YB"~r Z B ~ ~ ) .ll otherequivalent admittances are represented by their equivalent injectedcurrents. The programming model of a three-phase grounded wye-delta transformer used in the GDAS program is shown in Figure 14.

19 8

Figure 14Programming Model for Three-phaseGrounded Wye-Delta Transformer

where

y , = L

flapI1 =yrva 12=y'V"p I , = y t V p

1s=JL v: - VS )

a2 a2

4 = L ( V : - * )P3s2 3B2

I6 =JL v: - v:)3B2

Another representation of programming mddel for three-phase

Butgrounded wye-delta transformer is the same as Figure 11 .

y3=Yt



1677

CO"

Rigid cogenerator and transformer models were introduced inthis paper. Based upon these detailed models, a program, entitled"Generalized Distribution Analysis System (GDAS)," wassuccessfully developed and tested.

The effect of a 400 KVA cogenerator on a 12-bus sampledistribution system[7,11] and a large, 5 MW + j 2 MVARcogenerator on a large, 5000 individual phase bus, distributionsystem was studied. The results show that the cogenerator has asignificant impact on the system. For example, the direction of thepower flow of many branches, especially those near thecogenerator,are changed. This change will affect the protection andcoordination strategy. Furthermore, the voltage profile, systemloss, and power factor are totally different. Hence, to analyze thedistribution system in detail, the three-phase cogenerator model, aswell as the three-phase system representation, are important.

In the future, perhaps very large cogenerators will be used indistribution systems. The cogenerators may be can maintain theterminal voltage constant. In those cases, a big cogenerator bus canbe represented as specified P-V buses. However, for small capacitycogenerators the models presented here can still be applied.

The transformer model has also been successfully tested, fromwhich an important conclusion can be drawn: the effects of thetransformer connections on the distribution system are significantexcept for the balanced power flow study. This means that for thetransmission system, if the system is or very nearly balanced, thetransformers can be modeled as a simple Grounded Wye-GroundedWye connection. Otherwise, for the unbalanced transmission andmost distribution systems, the transformers must be modeled in

greater detail. Core loss, system imbalance, and phase shiftproblems should be taken into account to improve the reliability ofthe results.

REFERENCES

[ l ] N. Vemptati, R.R. Shoults, M.S. Chen and L. Schwobel,"Simplified Feeder Modeling for Load Flow Calculations,"IEEE Trans. on Power Sysrems,Vol. PWRS-2, No.1, pp.168-174, February 1987.

[2] M. S. Chen and W. E. Dillon, "Power System Modeling,"Proc. IEEE, Vo1.62, No.7, PP.901-915, July 1974.

[3] Energy Systems Research Center, The University of Texasat Arlington, "Determining Load Characteristics forTransient Performance," Final Report of EPRI Project RP849-3. Vol. I. II.HI. December 1978.

, I

[4] S . D.'Hu, Cogeneration, Reston, Virginia : RestonPublishing, 1985.

[5] G. Polimeros, Enerev Copeneration Handbook, New York :

Industrial Press, 1981.[6] S . A. Spiewak, Coyeneration and Small Power Production

Manual, Lilburn, Georgia :The Fairmont Press, 1987.[7] Energy Systems Research Center, The University of Texas

at Arlington,"Generalized Distribution Analysis System,Phase I : Feasibility Study," Final Report of Con EdisonR&D Project 92087, June 1989.

[8] Energy Systems Research Center, The University of Texasat Arlington,"GDAS User's manual," Arlington, Texas,1990.

[9] M. S. Sarma, Svnchronous Machines, New York : Gordonand Breach Science Publishers, 1979.

[lo] D. B . Brown and E. P. Hamilton 111, ElectromechanicdEnergv Conversion, New York : Macmillan Publishing,1984.

[1 11 T. H. Chen, M. S . Chen, et al. "Distribution System PowerFlow Analysis - A Rigid Approach," Submitted to IEEEWinter Power Meeting(1990).

1121 W. E. Dillon, "Modeling and Analysis of an ElectricallyPropelled Transportation System," Ph. D. Dissertation, TheUniversi ty of Texas at Arlington, May 1972.

[13] W. E. Dillon and M. S . Chen, "Transformer Modeling inUnbalanced Three-phase Networks," IEEE Summer PowerMeeting, Vancouver, Canada, July 1972.

[14] J. Amllaga and C. P. Arnold, Computer Modelling ofElectrical Power Systems, New Zealand: John Wiley and

Sons, 1983.[151 D. I. Sun, "Distribution System Loss Analysis and Optimal

Planning," Ph. D. Dissertation, The University of Texas atArlington, May 1980.

[16] D. I. Sun, S . Abe, R. R. Shoults, M. S . Chen, P.Eichenberger, and D. Fanis , "Calculation of Energy Lossesin a Distribution System"' IEEE Transactions on PowerApparatus and Systems, Vol. PAS-99, No.4, July/Aug1980.

'Jsai-Hsia was born in Taiwan, R.O.C. on March 15,1953. H e e d i s B.S. and M.S. degrees in elec tricalengineering from the National Taiwan Institute of Technology,Taipei, Taiwan, R.O.C. in 1980 and 1982 respectively. Since1982, he has joined the National Taiwan Institute of Technology asa lecturer. With the support of the Ministry of Education of R.O.C.,he worked for his Ph.D. at the Energy Systems Research Center atthe University of Texas at Arlington and received the Ph.D. degreein May, 1990. He is currently Associate Professor of ElectricalEngineering at the National Taiwan Institute of Technology. He is amember of Tau Beta Pi, Phi Beta Delta, and IEEE.

Mo-Shine Chen received the B.S. degree in electrical engineeringfrom the National Taiwan University, Taipei, Taiwan in 1954 andthe M.S. and Ph.D. degrees from the University of Texas at Austinin 1958 and 1962, respectively. He is currently Professor ofElectrical Engineering and Director of the Energy Systems ResearchCenter at the University of Texas at Arlington, Arlington, Texas.He has served as a consultant to numerous companies andgovernment agencies. Dr. Chen is a member of Eta Kappa Nu, Tau

Beta Pi, Sigma Xi, IEEE, ASEE, Texas Society of ProfessionalEngineers and the New York Academy of Science. In 1976 he wasthe first recipient of the Edison Electric Institute's PowerEngineering Educator Award. He is a Registered ProfessionalEngineer in the state of Texas. Dr. Chen is an IEEE Fellow and haspublished more than 60papers in the Referee Journal.

Toshio Inoue was born in Tokyo, Japan on February 15, 1958.He received his M.S. degree in Electrical Engineering from WasedaUniversity in 1982. In 1982, he joined Central Research Institute ofElectric Power Industry (CREPI ) as a Engineer. Between August,1988 and July 1989, he worked in Energy Systems Research Centerat the University of Texas at Arlington as a Visiting AssistantProfessor. Mr. Inoue is a Senior Research Engineer in PowerSystem Department at CRIEPI and a member of IEEE.

Paul .Kota is a Computer Applications Engineer in the ComputerApplications Engineering Department at Con Edison. He joinedCon Edison in 1983. Mr. Kotas received a BS in EngineeringScience from S.U.N.Y. at Buffalo in 1982 and an MS in

Mathematics from the Courant Institute of Mathematical Sciences atNew York University in 1987. Mr. Kotas is a member of theSociety for Industrial and Applied Mathematics (SIAM).

A. Chebli was born in Beirut Lebanon on April 17, 1936.He received his BEE(Major Electrical) degree from the AmericanUniversity of Beirut, School of Engineering in June 1960. In 1977,he completed the Power Systems Engineering Course given by theGeneral Electric Co. Between 1961 and 1966, he was employed bythe Electricite du Liban. In 1966, he joined Ohio Power in CantonOhio as a Distribution Engineer and was promoted to SeniorEngineer prior to his transfer to AEP in New York in 1973.

At the time he resigned from AEP in July 1980, he wasAssistant Section Head Distribution. From July 1980 to February1984, he worked for the consulting f i i C. T. Main Internationaland its subsidiaries. In February 1984, he joined Con Ed as aPrincipal Engineer and in October 1986 he was transferred tobecome Distribution Design Engineer. Mr. Chebli is a member ofIEEE and is a registered professional engineer in the state of Ohio.



1678DISCUSSION

W.H. KERSTING, New Mexico State University,

Las Cruces, NM 88003. This paper is a very

interesting paper to engineers involved with

distribution system operations and analysis.

The authors model the three-phase

cogenerator as a negative constant P,Q load.

That is not a new idea and is used by many

three-phase power-flow programs.

My major concern with the paper concerns

the models of the transformers. I am

disappointed that the authors do not show in

any of the figures the currents that will be

used in the current vectors for the various

matrix equations. For the wye connected

transformers I assume they are directed into

the wye. For a delta connected transformer,

are the currents the line currents or the

actual transformer currents (delta currents)

For the grounded wye - grounded wye bank I was

able to use Equation 15 and assuming primary

and secondary voltages and transformer

impedances, I was able to compute the primary

and secondary line currents correctly. When I

tried to apply Equation 23 again with assumed

primary and secondary voltages, I was not

successful in computing realistic primary and

secondary currents. In applying Equation 23

it is not clear if the secondary voltages

should be the line-to-line or line-to-neutral

voltges. It is also not clear what base

voltage to use for the line-to-line voltagesshown in Figure 12. It is impossible to

follow the equations under Figure 14. It

appears that the authors have somehow come up

with line-to-neutral voltages on the delta

secondary. If I am given the line-to-line

secondary voltages, how do I get the necessary

line-to-neutral voltages to use in the

equations listed under Figure 14?

It would be of great help if the authors

would include in their discussion an example

on how to apply the method for a wye-delta

transformer bank. Specify the voltages and

currents on the primary wye connected side and

demonstrate how the method can be used to

determine the secondary voltages and currents.

If that can’t be done, at least assume primary

and secondary voltages and compute the primary

and secondary currents using Equation 23.

I have to believe that the method and

equations are correct, it is unfortunate that

it is not possible to read the paper and learn

how to apply the method to even a simple

problem.

Manuscript received February 19, 1991.

I



1679

Closure

T. H. Chen, M. S. Chen, T. Inoue, P. Kotas and E. A. Chebl iThe authors would like to thank the discussor for his interest inthe paper. The questions raised by the discussor will be

answered in turn.

(1) Question about the cogenerator model:

We would like to emphasize that the three-phasecogenerator modeled as a constant p,q device based upon the

investigation results that were done by Consolidated Edison

Company of New York, Inc. is only one of many importantfeatures of the cogenerator model presented in this paper. The

more important one is that it will represent the inherent

generator phase imbalance due to distribution system imbalance.The paper also introduced a cogenerator model for short circuit

study as well as for the load flow study. The cogenerator modelpresented here is much more than only a negative constant p,q

load.

To our knowledge, cogenerator represented as shown inFigure 2 is a new idea. This is the first time cogenerator

modeled for unbalanced distribution load flow study.

(2) Questions about the transformer model:

In this paper, the tedious derivation of the transformermodels was omitted, only the results and its implementation

method were shown. For explanation, a simplified derivation of

the admittance matrix of a grounded wye-delta transformer isshown below:

First, consider the arcuit in Figure D1.

Rl M R 2

a A

I/ / / / / / / / / / / / / /

Figure D1 Coupled Circuit Model of a Grounded Wye-Delta

Transformer

In the figure the six branches are oriented and numbered by

For simplification, branch selfhe labeled branch currents.

impedances are defined as:

Z l = Rl + joLl

Z2= R 2 + joL2

z3=zs=z,z ,= z6= 2

The mutual impedances between branches is now definedas :

and

z12=z34=z56= o M

Z, I oM

The primitive impedance matrix relationship between

branch voltages and currents is :

Z1 Zm

ZmZ20 0

0 0

0 0

0 0

0 0

0 0

z1zmZ m 2 2

0 0

0 0

0

0

I O

0

Z1

Zm~

The inverse relationship is

zz-z, 0 0 0

- zm z l0 0 0

0 0 Z2-Zm 0

0 0 - z m z l 0

o o o o z ~

0 0 0 0 -Zm

Now the branch voltages are related to the node voltages

by the node connection matrix, N,

1 0 0 0 0 0

x 0 0 0 1 - 1a' : I [ ]0 0 1 0 0 0

0 0 0 - 1 0 1 vc~

Where the voltages denoted by letter subscripts are the

node voltages with reference to ground in Figure D1. The node

admittance matrix for these six nodes is defined as

Ynode= Nt YprimNIf the matrix multiplication indicated by the above equation

is carried out, the node admittance matrix becomes:

a b c A B C

o z , - z m - z 1 - z 1 2 z ~ J ~

Now, the node admittance matrix for the transformer can be

written in terms of the primary side short circuit admittance, yt ,defined as:

z2Yt*-

z lZ 2 - z;

Ynode =

Yt O

O Y t

0 0

nl

n l n l

n l

- E Y t 0

E Y t - E Y t

O EYt

If the above admittances are already in the per unit system,

then the effective turns ratio, n)(2, must be :

11=Ln2 6Then equation becomes:



1680

Ynode =

This is equation 23 . In the equation 23 , the Y F as used

to replace ynode.urthermore, for the off-nominal tap ratio

transformer, the equation 23 should be modified to equations 24 .Based on the results, the equivalent circuit for grounded

wye-delta transformer was drawn in Figure 13. From Figure 13 to Figure 14 is very straightforward. The equations under

Figure 14 can be obtained by using simple circuit theory, and canbe proved by inspection. It is clear that the voltages used in

these equations ar e the bus(node) voltages, and the currentsare the bus currents. The relationship of the bus currents and

voltages are shown below:

where

V;bc= IVa V; V, are the three-phase primary voltages;31V:bc= vp vz are the three-phase secondary

voltages;

[a are the three-phase primary currents;

[I: I f are the three-phase secondary currents;

and I is a 6x6 matrix. The I$'] f grounded wye-grounded

wye transformer is shown in Equation 15 and 16 and grounded

wye-delta transformer in Equation 23 and 24 . This pe r unit

leakage admittance matrix Y, is one of the two blocks shownin Figure 7.

Because the bus frame of reference was adopted for the

models and the solution techniques, a set of updated bus

voltages will be obtained each iteration, so it is no problem to

apply the equations under the Figure 14 . Furthermore, becausethe injected currents shown in Figure 14 depend on the bus

voltages, if the voltages change, the currents should be modified.

It means that for each iteration, the injected current should be

updated based on the updated bus voltages or the result will be

incorrect.A two-bus system is given in Figure D 2 as an example.

The system can be represented by its equivalent circuit shown in

Figure D3 and modeled as Figure D 4. After simplification,

Figure D 4 can be represented by Figure D 5. The injected

currents shown in Figure D 5 are the combination of the

contributions of load, and the core loss and some of the

admittance part of the transformer. Al l of the injected currents

are functions of the bus voltages. Therefore the injected currentsshould be updated each iteration until the convergence results

are obtained. For simplification, the transformer in the sample

system is assumed to be nominal, therefore, the a, nd are

equal to 1.0 . Based upon that, the voltages of the swing bus,

abc

bus "p", are assumed to be 1. 0 P.u., three-phase balanced., the

magnitudes and angles of voltages of bus no. "s" each iteration,

and the convergence results are shown in Table D 1. If the

connection of the transformer is changed to delta-grounded wye,and the load is unbalanced, 50% load on phase A, 30% load on

phase B,and 20% load on phase C, the results are shown in

Table D2.

Three-phase

Transformer

lOOOkVAZ = 6 %x / R = 5

13.8kV-208V

Figure D2 Two-bus Sample System

(same as

fFigure D3 Equivalent Circuit of Two-bus Sample System

Swing Bus

1 Transformer - 1I Model II I

a

b

CLoad

(represented

' by injected

cuneats)

Icurrents)

- - - - -Figure D4 Three-phase Model of Two-bus Sample System

a

SwingBus b

C

-Figure D5 Alternative Model of Two-bus Sample System



1681

Iter.

No.

0

hase A hase B hase C

Voltage of Bus "S"

phase A phase B phase C

Ivl e IVl 0 \VI e1.00000 0.00 1.00000 -120.00 1.00000 120.00

0 1.00000 0.00 1.00000

1 0.99668 -32.84 0.9%71

2 0.97590 -31.05 0.975913 0.97685 -31.18 0.97684

4 0.97688 -31.18 0.97685

5 0.97680 -31.18 0.97680

Comments: 1.All voltage magni

2. teration No. "0"

in degree.

tl

II

udes are in p.u. and angles

neans initial guess.

Comments: 1.All voltage magnitudes are in p.u. and angles

in degree.

2. teration No."0" eans initial guess.

Any device models should be discussed with their solution

technologies. This paper presented a near actual transformer

model for distribution system analysis. From some case

studies, we found that by using this model, the distribution

system can be simulated in more detail, and some

characteristics that can not be simulated by the conventional

method can also be obtained. Unfortunately, only the optimallyordered triangular factorization method was used as a solution

method, and not enough cases were studied, therefore, someapplication statuses are still unclear, for example, the

convergence problem. In our research experience, the

convergence characteristic will be affected by many factors, such

as the system configuration, the acceleration factor, the

connection of the transformer, the degree of the systemunbalance, the location and size of cogeneration, the system

grounding type, etc. In some cases, the acceleration factor

should be reduced or the convergence results will be hard toobtain. To model a transformer, the model presented in this

paper is not the only one and may not the best one , but it is

significant that the paper introduced a way to model it in detailthat was not done before. The convergence problem in the

distribution system, especially when the system is modeled indetail, will be more complex than that of a transmission system.We are trying to make it clear and finding a better way to model

the system devices, especially the transformer, and the solutiontechnologies.

Finally, we would like to mention that the transformer

models presented in this paper cannot be applied to thetransformer with mid-tap right now, and we are continuing this

kind of research. It is expected that the transformer with mid-

tap can be modeled in the near future.

Manuscr ipt recei ved June 2 3 , 1991 .

3 Ph Transformer and Generator Models

Documents

Transcript of 3 Ph Transformer and Generator Models