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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006 153

A Unied Three-Phase Transformer Modelfor Distribution Load Flow Calculations

Peng Xiao , Student Member, IEEE , David C. Yu , Member, IEEE , and Wei Yan

Abstract This paper provides a unied method to modelthree-phase transformers for distribution system load ow cal-culations, especially when the matrix singularity caused by thetransformer conguration arises. This paper shows that the singu-larity appears only in certain transformer admittance submatricesand only in certain transformer congurations. The uniedmethod presented in this paper can solve the voltage/current equa-tions in the forward/backward sweep algorithm for various typesof transformer congurations, whether or not the correspondingadmittance submatrices are singular. Comprehensive compar-isons have been made between the proposed approach and othermethods. Test results demonstrate the validity and effectiveness of

the proposed method. Index Terms Admittance matrix, load ow analysis, power

distribution, power transformers.

I. INTRODUCTION

A S the power distribution networks become more and morecomplex, there is a higher demand for efcient and reli-able system operation. Consequently, two of the most importantsystem analysis tools, powerow andshort-circuit studies, musthave the capability to handle various system congurations withadequate accuracy and speed. Of the several dedicated distribu-

tion system load ow methods used in the power industry, theforward/backward sweep algorithm [ 1], with its low memoryand computation requirements and robust convergence charac-teristic, hasgained themost popularity in recentyears.Thealgo-rithm is also known to be the branch current-based feeder anal-ysis. Based on the ladder theory for linear circuit analysis, theforward/backward sweep algorithm can fully utilize the radialstructure of most distribution networks. With minor modica-tions, several methods [ 2], [3] have been proposed to extend itsapplication in weakly meshed distribution systems.

To take into account the existence of multiphase load and un-symmetrical feeders in distribution systems, three-phase systemrepresentation is generally used. The modeling of three-phasetransformers is a vital step in distribution system analysis. Dueto unbalanced system operations, a complete andaccurate three-phase model is desirable for distribution and inline transformersof various core and winding congurations.

Manuscript receivedAugust18,2004; revisedJanuary8, 2005. Thiswork wassupported in part by the Visiting Scholar Foundation of the Key Laboratory of High Voltage Engineering and Electrical New Technology, Education Ministry,China. Paper no. TPWRS-00450-2004.

P. Xiao and D. C. Yu are with the Electrical Engineering Department, Uni-versity of Wisconsin-Milwaukee, Milwaukee, WI 53211 USA.

W. Yan is with the Electrical Power Department, Chongqing University,Chongqing 400044, China.

Digital Object Identier 10.1109/TPWRS.2005.857847

Fig. 1. General three-phase transformer model.

A two-block three-phase transformer model was presentedin [4]. As shown in Fig. 1, a series block represents windingconnections and leakage impedance, and a shunt block modelsreal and reactive power losses in the transformer core. A similarmodel was proposed in [ 5], where the shunt block is connectedto the primary side.

The core loss of a transformer is approximated by shunt coreloss functions on each phase of the secondary terminal of thetransformer. Generally, the functionsare nonlinear and the coef-cients should be determinedby experiments. Since transformerwinding connections have little effect on core loss, this paper fo-cuses mainly on the series block, while the core loss block canbe treated as a three-phase load on either side of the transformer.

To include such a transformer model into the forward/back-ward sweep algorithm, specic voltage/current relationshipsshould be derived. Several approaches have been developed inthe past several years. In [ 4], ctitious injection current sourceswere used to resolve the coupling between the primary and sec-ondary sides, which greatly simplied the admittance matrix.However, this technique faces slowconvergenceproblems whenemployed in the forward/backward sweep algorithm. In [ 5],voltageandcurrent updateequations were developed for three of themost commonly used transformer connections based on their

equivalentcircuits.In[ 9],voltage/currentequationswerederivedin matrix form for transformers of the ungrounded wye-deltaconnection. However, these methods are mainlybased on circuitanalysis with Kirchhoffsvoltage andcurrent laws. Different setof equations are needed to handle each connection type. Trans-formers of various winding connections can only be analyzedon a case-by-case basis, which is not convenient for efcientimplementation and incorporation of new connection types.

In this paper, a unied method is proposed to model thediverse distribution transformers into the forward/backwardsweep load ow algorithm. A brief description of the for-ward/backward sweep algorithm is presented in Section II. InSection III, the proposed modeling procedure is explained in

0885-8950/$20.00 2005 IEEE

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154 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006

Fig. 2. One section of a distribution network.

detail. Extensive computation and comparisons have been doneto verify the method, and the results are presented in Section IV.

II. FORWARD /BACKWARD SWEEP ALGORITHM

Although the forward/backward sweep algorithm can be ex-tended to solve systems with loops and distributed generationbuses, a radial network with only one voltage source is used hereto depict the principles of the algorithm. Such a system can bemodeled as a tree, in which the root is the voltage source, andthe branches can be a segment of feeder, a transformer, or othercomponents between two buses.

With the given voltage magnitude and phase angle at the rootand known system load information, the power ow algorithmneeds to determine the voltages at all other buses and currents ineach branch. The forward/backward sweep algorithm employsan iterative method to update bus voltages and branch currents.During each iteration, a backward sweep is performed to updatebranch currents, and a forward sweep is performed to update busvoltages. The algorithm terminates when the voltages converge.

A brief description of the main steps of the forward/backwardsweep algorithm is shown as follows.

A. InitializationBefore the rst iteration, the voltage at each leaf node, i.e., the

buses that do not havechild buses, is given an initial guess value.The guess values should be as close to the true values as possibleto reduce the number of iterations and to avoid divergence. Typ-ically, the voltage magnitude is set to one per unit, while thevoltage phase angle can be chosen by considering transformerphase shift between the root and the leaf nodes.

B. Backward Sweep

With voltages at all the leaf nodes given, the backward sweepprocedure determines the currents in the branches that connectthese nodes and their parent nodes and the voltages at the parentnodes. When the voltages at the parent nodes are calculated,these nodes can then be treated as leaf nodes and the calcula-tion continues until the root is reached.

Fig. 2 shows one section of the system, in which bus is theparent node and bus is the leaf node. Each step of the backwardsweep is to determine the currents owing through the branch

and the voltages at bus . Therefore, in the backwardsweep, the branch components need to be modeled in a way that

and can be obtained when and are known.

C. Forward Sweep

After backward sweep, the currents in all branches are up-dated. However, since calculations of these currents are based

on estimated bus voltages, incorrect voltage values will resultin incorrect branch currents. In forward sweep, the root voltageinformation, together with the branchcurrents obtained from thelast backward sweep, are used to update the leaf node voltages.

Starting from the root node, the voltages at all the child nodesare calculated based on the parent bus voltages and the currents

in the branches. The procedure continues until all the leaf nodevoltages are updated.As indicated in Fig. 2, the forward sweep requires that the

branch be modeled in a way that can be obtained whenand are known.

The proposed method for modeling distribution transformersatis es the requirements of the forward/backward sweep algo-rithm and can be easily integrated into the algorithm, regardlessof the types of transformer con gurations.

III. T RANSFORMER MODELING PROCEDURE

A. Construction of the Primitive Admittance Matrix

The starting point for the proposed modeling approach is theprimitive admittance matrix of the transformer, whichcan be determined according to the transformer winding con-nections. Depending on the system representation, both per-unitvalues or actual unit values can be used to form the matrix. Adiscussion of the advantages and disadvantages of both unit sys-tems was presented in [ 6].

B. Conversion to Nodal Admittance Matrix

The conversion from the primitive admittance matrixto the nodal admittance matrix was discussed in detail in[8]. Basically, the procedure involves using the node-to-branch

incidence matrix with(1)

The matrix can then be reduced to 6 6 by eliminatingthe neutral point nodes with Kron reduction. The resulted matrixis the nodal admittance matrix .

In [10], a systematic approach utilizing symbolic mathemat-ical tools was proposed to establish the nodal admittance ma-trices for various transformer connection types.Minor modi ca-tions are necessary to take into account the off-nominal tap ratiobetween theprimary andsecondary sides of the transformer. The

matrices for the most common transformer con gurations

were proposed in [ 4].Following the same transformer modeling procedures, it

is straightforward to build the nodal admittance matrices forthe more particular transformer con gurations such as OpenWye/Open Delta. Once these matrices are determined, theycan be used in the forward/backward algorithm. However, thispaper does not discuss these more specialized transformercon gurations due to space limitations.

C. Characteristics of the Submatrices

With the nodal admittance matrix , the transformervoltage-current relationship can be expressed as

(2)

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XIAO et al. : UNIFIED THREE-PHASE TRANSFORMER MODEL FOR DISTRIBUTION LOAD FLOW 155

TABLE I SUBMATRICES FOR COMMON STEP -DOWN TRANSFORMER CONNECTIONS

where the matrix is divided into four 3 3 submatrices:, and . Vectors , and are the

three-phase line-to-neutral bus voltages and injection currents attheprimary and secondary sides of the transformer, respectively.

As described in the last section, in backward sweep proce-dure, and are known, while and are to be calcu-lated. From (2), the following can be derived:

(3)

(4)

In forward sweep, and are known, and needs to becalculated. Similarly

(5)

According to (3) (5), the implementation of the for-ward/backward sweep algorithm requires the inversion of submatrices and . However, a close examinationof the matrices for common transformer con gurationsshows that these submatrices are often singular. Table I showsthe submatrices of for the nine most common step-downtransformer connection types, and Table II shows the matricesfor step-up transformers, where

(6)

(7)

(8)

and is the per-unit transformer leakage admittance. For sim-plication, the leakage admittances of each phase are assumedto be identical. For transformers with unbalanced admittances,their nodal admittance matrices are more complex and do nottake the forms shown in Tables I or II. However, it is provedthat the singularity of the transformer submatrices remains thesame.

TABLE II SUBMATRICES FOR COMMON STEP -UP TRANSFORMER CONNECTIONS

From (7) and (8), it is obvious that both and aresingular. Hence, is invertible only for connection,

and is invertible only for and connections.For connection, (3) (5) can be directly used for forwardand backward sweep calculations. For connection, only(5) can be used in forward sweep. For all other connection types,since the matrices are singular, there is no unique solution to theabove equations. In essence, the singularity in those transformercon gurations arises due to the lack of voltage reference pointon one or both sides of the transformer.

D. Solving the Singularity Problem

To circumvent the singularity issue, it is noted that althoughthe three-phase line-to-neutral voltages and cannot be

obtained by solving (3) and (5), the nonzero-sequence compo-nents of the voltages can be uniquely determined. To illustratethis point in the backward sweep, rewrite (3) as

(9)

Let represent the nonzero-sequence components of ,i.e.,

(10)

where vector is the zero-sequence voltage on the primaryside; thus

(11)

The product of and is always zero for transformercon gurations other than . This is because is rep-resented by or in all other transformer con gurationsexcept . From (7) and (8), it can be seen that

(12)

so (11) can be reduced to

(13)

Equation (13) indicates that the zero-sequence component of does not affect the backward sweep calculation for trans-

formers with a singular matrix. The above analysis shows

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156 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006

that (3) can be used to calculate both and its nonzero-se-quence components . However, if is singular, (13) stillcannot uniquely determine , and a modi cation is needed.Since does not contain zero-sequence component, it satis es

(14)

Equations (13) and (14) can be combined as

(15)

where is obtained by replacing the last row of with, while and are the same as and , except

that elements in their last row are set to 0 so that (14) is satis ed.Now that is not singular, the nonzero-sequence compo-

nents of the voltages on the primary side can be determined by

(16)

Similar results can be obtained for forward sweep calculation

(17)

where is the nonzero-sequence component of is the same as , except that the last row is replaced with

, and are obtained by setting the elements inthe last row of and to 0, respectively.

Once the nonzero-sequence components of or are cal-culated, zero-sequence components are added to them to formthe line-to-neutral voltages so that the forward/backward sweepprocedure can continue.

As an example, consider the backward sweep for an un-grounded transformer. According to Table I

(18)

and

(19)

Thus, (13) becomes

(20)

The singularity of matrix is evident. However, the matrixcan be changed to nonsingular if one of its rows is replaced with

, i.e.,

(21)

Fig. 3. Four-bus example system.

Equation (20) can then be transformed into

(22)

E. Modied Forward/Backward Sweep Algorithm

With the above transformation, the equations for transformervoltage calculation are no longer singular. However, the resultedtransformer voltages and only contain the positive andnegative sequence components. Thus, zero-sequence voltagesmust be added to them to form line-to-neutral voltages. Theprimary-side zero-sequence voltage can be initialized to 0 andis updated during the forward sweep. Since voltages are alsocalculated when line currents are updated, the secondary-sidezero-sequence voltage can be obtained directly from the back-ward sweep.

To illustrate the modi ed procedure, a four-bus exampleshown in Fig. 3 is used.

1) Initialization: The forward/backward sweep algorithmbegins with all the load information and only the source voltage

known. A guess value is given to the voltages on bus 4.2) Backward Sweep: With given, the load currents can

be calculated. If the transformer core loss is modeled on the sec-ondary side, core loss functions can be used to determine theabsorbed power and current. Thus, the three-phase currentsthat ows through feeder 3 4 can be obtained. Assume that linecharging is neglected, then the currents owing through the sec-ondary side of the transformer are equal to . The voltageson the secondary side of the transformer are

where is the line impedance matrix for feeder segment34.

For the -connected transformer, its matrix is notinvertible. According to (16)

Since only contains the positive-and negative-sequencecomponents of , an initial value of zero-sequence voltageis needed to get the transformer primary side voltage

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Next the currents at the ungrounded Wye side can be deter-mined with

The backward sweep calculation continues until the source

bus is reached. If the difference between the computed sourcevoltage and the actual source voltage is not within therequired error limit, a forward sweep is performed.

3) ForwardSweep: The forward sweep beginsby settingto its actual value . Since the currents in feeder sections 1 2have been calculated in the backward sweep, the voltage at bus2 can be obtained by

where is the line impedance matrix for feeder segment12. Note that by updating , the zero-sequence component

is also updated and will be used in the next backward sweepprocedure.

The next step is to determine based on the knowledgeof and . Still, due to matrix singularity, only positive-andnegative-sequence components can be determined

The voltage can then be calculated by adding zero-se-quence voltage , which is obtained from the last value of in backward sweep. The forward sweep then continues until bus4 is reached.

4) Zero-Sequence Voltage Update: As illustrated above, for

transformers with singular or submatrices, zero-se-quence voltage update calculation cannot be performed fromone side to the other. This is due to the fact that zero-se-quence equivalent circuit is interrupted at a transformer withungrounded Y and/or windings. Hence, the zero-sequencevoltage on one side of the transformer cannot be determinedbased on line-to-neutral voltages on the other side, even whenline current information on both sides is available. In thesecases, the zero-sequence voltages on the primary side can beupdated by the source voltage during the next forward sweep.However, the zero-sequence voltages on the secondary sidewith a or ungrounded Y winding will not be updated due to

the lack of voltage reference point on the secondary side, whichmakes determining the real line-to-neutral voltage impossible.To avoid such dif culties, the method in [ 9] used the line-

to-line voltage on the or ungrounded Y side of the trans-former and the line-to-neutral voltage in the remaining part of the system. The method in [ 5] introduced an arbitrary refer-ence neutral point to convert the line-to-line voltages to theline-to-neutral voltages.The proposed method, usingan initiallyguessed zero-sequence voltage on the primary side, does not re-quire any special treatment in handling the or ungroundedY transformers. With this initial zero-sequence voltage, line-to-neutral voltages can be used throughout the system.

The primary-side zero-sequence voltages are updated by thesource voltage. The zero-sequence voltage on the secondaryside with a or ungrounded Y winding are determined by

TABLE III DURING ITERATIONS

grounded elements that are connected to the same side of thetransformer, such as grounded load, line charging, or othergrounded transformers. However, due to the limitations of for-ward/backward sweep algorithm [ 11 ], zero-sequence voltagecannot be updated. Therefore, the secondary line-to-neu-

tral voltages in these conditions are the assumed value. Thecommon practice in the utility industry will seldom see a singlephase-to-ground load connected to an ungrounded three-phasesource or transformer, so the zero-sequence current is generallyvery small compared with load currents. Thus, even if thecalculated line-to-neutral voltage is not real, the correspondingline-to-line voltage is still accurate.

IV. T EST EXAMPLES

To verify the proposed method, different transformer con gu-rations were included in a three-phase distribution load owpro-gram with a forward/backward sweep algorithm. Two differentsized examples were implemented, and results were comparedwith other existing methods.

A. Model Validation

The IEEE four-node test feeder [ 12] was used to provide asimple system for the testing of various three-phase transformerconnections. The one-line diagram of the example is shown inFig. 3.

The two distribution feeder segments have unequal mutualcoupling between the phases. The load is unbalanced with dif-ferent kilovoltampere and power factor in each phase.

For an ungrounded transformer connection, the pro-gram terminates after ve iterations, and the difference betweencomputed source voltage and speci ed source voltageare within 0.0001 p.u. The -phase line-to-line voltages at theload are listed in Table III.

The results match very well with those listed in [ 9], where adetailed step-by-step example was given. Tests have been con-ducted for all other transformer connections, and the resultsmatch those listed in [ 12].

However, the method mentioned in [ 9] and [ 12] requires dif-ferent sets of equations for different transformer con gurations,which is not convenient for ef cient implementation and incor-poration of new connection types. The uni ed method proposedin this paper does not require any special equations to handle thetransformers with the or ungrounded Y windings.

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TABLE IVZERO -SEQUENCE VOLTAGES AT BUS 2

TABLE VZERO -SEQUENCE VOLTAGES AT BUS 4

B. Zero-Sequence Voltage Update

Different transformer con gurations were implemented inthe four-bus example to further test the updating behavior of the zero-sequence voltage. Two different initial values (0 and0.1 p.u.) were chosen for each con guration. If different initialvalues always converge to the same nal value, then it meansthe zero-sequence voltage can be correctly updated during theiterations for the given type of transformer con guration.

Table IV shows the nal zero-sequence voltages on the pri-mary side of the transformer for different connection types. Theresults show that, for each transformer con guration, the pri-mary side zero-sequence voltage always converges to a similarvalue, regardless of its initial value. These results indicate thatthe primary side zero-sequence voltage can be updated. This up-date takes place during the forward sweep, and it is due to theY-grounded source voltage.

Table V shows the nal zero-sequence voltages on the sec-ondary side (bus 4) of the transformer. The results indicate thatfor some transformer con gurations, different initial values willproduce different nal values. This is true for every transformercon guration except and . The results furthershow that forward/backward sweep algorithm cannot produceunique secondary line-to-neutral voltages in these cases. Itshould be noted that the initial zero-sequence voltage isobtained from the initial leaf-node voltages .

For and transformers, zero-sequence voltageon one side can be uniquely determined from voltages on the

TABLE VILINE -TO-L INE VOLTAGES AT BUS 4

Fig. 4. IEEE 123-bus example system.

other side together with current information. For other connec-tions, a close examination reveals that the nal values are in thevicinity of the initial value. In other words, zero-sequence volt-ages are not updated, even though the algorithm converges. Thisis due to the fact that without additional grounding devices onthe secondary side, the subnetwork is isolated, and the zero-se-quence voltage will not be affected by other part of the system.

Table VI shows the corresponding line-to-line voltages atbus 4 for different transformer con gurations with differentinitial zero-sequence voltages. The results indicate that eventhough different initial values may produce different nalzero-sequence voltages, the corresponding line-to-line voltagesare correct when the algorithm converges, which meets therequirement for most load ow analysis.

C. Large System Tests

The IEEE 123-bus example shown in Fig. 4 is used to demon-strate the transformer models in large systems. The load owanalysis calculation was performed using per-unit values on abasis of 115 kV/4.16 kV/480 V and 10 MVA. There are twotransformers in this system. One is located between nodes 150and 149, and the other is between nodes 61 and 610.

Extensive tests have been performed on the system to verifythe validity of the proposed method. Comparisons have beenmade against transformer modeling approaches developed in [ 4]

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