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THREE-PHASE TRANSFORMER MODELING FOR FORWARD/BACKWARD

SWEEP-BASED DISTRIBUTION SYSTEMS POWER FLOW ALGORITHMS

U. Eminoglu M. H. Hocaoglu

Gebze Institute of Technology, Department of Electronics Engineering, Kocaeli 41400, TURKEY

ABSTRACT

This paper provides a new approach for modelling of distribution transformer in the forward/backward sweep-based

algorithms for unbalanced distribution systems. For the transformer modelling symmetrical components theory is used and

zero sequence-voltage and -current updating for the sweep-based methods is shown. The validity and effectiveness of the

proposed method is demonstrated by a simple two-bus balanced and unbalanced system for grounded wye-delta and delta

grounded wye transformer connections. Results are in agreements with the literature and show that the proposed model isvalid and reliable.

Keywords: Distribution system, distribution transformer, forward/backward sweep-based methods, three-phase load flow

analysis, symmetrical components.

1 INTRODUCTION

Load flow analysis is an important task for power system

planning and operational studies [1]. Certain applications,

particularly in distribution automation and optimization of

a power system, require repeated load flow solution and in

these applications, it is very important to solve the load

flow problem as efficiently as possible. The distributionsystem has some characteristic features such as radial

structure, high R/X ratio, untransposed lines and

unbalanced loads along with single-phase and two-phase

laterals. And also due to the unbalance, distribution

network matrices are ill conditioned. Because of these

features of distribution networks, the conventional load

flow methods such as; Newton-Raphson, and Gauss-Seidel

methods generally fail to converge in solving such

networks. Recently many researchers have paid attention

to obtain the load flow solution of distribution networks

and various methods are available in the literature to carry

out the load flow analysis of three-phase radial distribution

systems [2-15, 24]. These methods can be categorized asthe Zbus based methods [2-3, 13-14], the Newton Raphson-

based methods [5, 7, 9, 10, 15], and the Forward/

Backward sweep-based methods [4, 6, 8, 11, 12, 24].

Due to its low memory requirements, computation

efficiency and robust convergence characteristic, Forward/

Backward sweep-based methods have gained the most

popularity for distribution load flow analysis in recent

years. In these methods the problem of how to handle

distribution transformers of various winding connections is

the key to the unbalanced systems of multivoltage levels.

In the past decades, several methods for the modelling of

three-phase transformers have been proposed [17-28] as

an alternative to the classical modelling approach which is

based on proper modification of the nodal admittance

matrices of the transformers [16].

The implementation of nodal admittance matrices based

modelling of transformers [16-18, 20, 22, 25-28] to the

forward/backward sweep based methods is proven to bedifficult to employ and claimed to be unsatisfactory due to

relatively slow convergence problems [27]. It may also

cause singularity in the connection matrix which appears

on some special connection types such as; delta-grounded

wye or grounded wye-delta transformers. The authors of

[19, 21, 23, 24] employ different approaches to model

distribution transformers in a branch current based feeder

analysis. In study [19], voltage and current equations were

developed for three of the most commonly used

transformer connections based on their equivalent circuits.

In studies [21, 23, 24], voltage/current equations were

derived in the matrix form for transformers of the

ungrounded wye-delta connections. However, thesemethods are mainly based on circuit analysis with

Kirchhoff’s voltage and current laws. They are in need to

derive the individual formulae for different winding

connections from scratch.

In this paper, symmetrical components model of

distribution transformers is used for modeling and

incorporated into the forward/ backward sweep-based

power flow methods. General information about the

symmetrical components model of three-phase

transformers are presented in Section 2. In section 3 and 4

a brief description of the sweep algorithms and the

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proposed modeling procedure is explained in detail.

Extensive computation and comparisons have been done

to verify the approach, and the results are presented in

Section 5.

2

SYMMETRICAL COMPONENTS MODEL OF

THREE-PHASE TRANSFORMERS

The method of symmetrical components, first applied to

power system by C. L. Fortescue [29] in 1918, is a

powerful technique for analyzing unbalanced three-phase

systems. Fortescue defined a linear transformation from

phase components to a new set of components. The

advantage of this transformation is that for balanced three-

phase networks the equivalent circuits obtained for the

symmetrical components, called sequence networks, are

separated into three uncoupled networks. As a result,

sequence networks for many cases of unbalanced three-

phase systems are relatively easy to analyze. The

transformation between the phases and sequence

components are defined by a transformation matrix and the

transformation is applied to both voltages and currents of

phase-components (U and I , respectively) as given in eqs.

(1,2) in Appendix. Normally, the three-phase transformer

is modeled in terms of its symmetrical components under

the assumption that the power system is sufficiently

balanced to warrant this. The typical symmetrical

component models of the transformers for the most

common three-phase connection were given in [30].

3 FORWARD/BACKWARD SUBSTITUTION

METHOD

Although the forward/backward substitution algorithm can

be extended to solve systems with loops and distributed

generation buses, a radial network with only one voltage

source is used here to depict the principles of the

algorithm. Such a system can be modeled as a tree, in

which the root is the voltage source and the branches can

be a segment of feeder, a transformer, a shunt capacitor, or

other components between two buses. With the given

voltage magnitude and phase angle at the root and known

system load information, the power flow algorithm needs

to determine the voltages at all other buses and currents in

each branch. The forward/backward substitution algorithm

employs an iterative method to update bus voltages and

branch currents [6, 8].

Several common connections of three-phase transformers

are modeled using the nodal admittance matrices or

different approaches employed in a branch current based

feeder analysis for distribution system load flow

calculation. The type of grounded Wye-grounded Wye

(GY-GY), grounded Wye-Delta (Gy-D), and Delta-

grounded Wye (D-GY) connections for transformers are

most commonly used in the distribution systems. There is

no need to use the nodal admittance matrices and another

approach in forward/backward substitution method, when

the GY-GY connection is used for distribution

transformers. The phase impedance matrices of

transformer can be used directly in the algorithm. The

others connection types of transformers are need to be

modeled and adapted to the forward-backward substitution

algorithm. In this section, symmetrical components for

distribution transformers of Gy-D and D-Gy winding

configurations are implemented into forward/backward

substitution power flow algorithm and the flow chart of the

proposed approach is shown in fig.1.

4 LADDER NETWORK THEORY

The ladder network theory given in [24] is very much

similar to the forward-backward substitution method.

Though the basic principle of both of the methods is the

same, there are differences in the steps of implementation.

In the ladder network theory, the optimal ordering of

nodes is done first. In the backward substitution, the node

voltages are assumed to be equal to some initial value in

the first iteration. The currents in each branch are

computed by Kirchhoff’s Current Law (KCL). In addition

to the branch currents, the node voltages are also

computed by using eq. (9) in Appendix. Thus, the value of

the swing bus voltage is compared with its specified value.

If the error is within the limit, the load flow converges;

otherwise the forward substitution is performed as

explained in the case of forward/backward substitution

method.

The implementation of distribution transformer into the

ladder network theory is similar to the implementation of

transformer into the forward-backward substitution

method given in Section 3. Because of the computation of

node voltages in the backward sweep, the transformer

primer voltage and currents must be calculated. In this

case the distribution transformer is modeled with its

sequence-networks as modeled as in the forward/backward

substitution method. When the node voltages of the system

are calculated in the backward substitution, the sequence-

voltages of transformer primary side can be calculated for

grounded wye-delta and delta-grounded wye as given in

eq. (10) in Appendix. The implementation of distribution

transformers of Gy-D and D-Gy winding configurations

into the ladder network theory is given in flow chart in fig.

2.

5 TEST EXAMPLES

To verify the proposed approach for the transformer

modeling, two transformer configurations were included in

three-phase distribution system power flow program with

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forward/backward substitution method and ladder network

theory, and results were compared with other existing

method [17]. A two-bus three-phase standard test system,

given in fig. 3, were used, and for simplification, the

transformer in the sample system is assumed to be on

nominal, therefore, the taps on the primary and secondary

sides are equal to 1.0. In addition, the voltages of the

swing bus (bus p) is assumed to be 1.0 pu., and load is

balanced. The magnitudes and phase angles of bus ‘s’ are

given in Table 1 and Table 2 for each iteration. Secondly

the connection of 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 given in Table 3 and 4. From results, it can

be said that results match very well with those listed in the

study [17]. Accordingly, from the results it is seen that the

proposed algorithms can reach the tolerance of 0.00001 at

fourth iteration. On the other hand, in the results of the

study [17] the voltages do not reach this tolerance value

for these two connections types of transformer.

Figure 3 Two-bus sample system

Table 1: Voltage magnitudes of two-bus sample systemVoltage of Bus ‘s’

Phase A Phase B Phase CIter.

No F. B.

Sweep

Chen

[17]

F. B.

SweepChen [17]

F. B.

SweepChen [17]

0 1.000000 1.00000 1.000000 1.00000 1.000000 1.000000

1 0.977786 0.99668 0.977786 0.99671 0.977786 0.99672

2 0.976860 0.97590 0.976860 0.97591 0.976860 0.97596

3 0.976838 0.97685 0.976838 0.97684 0.976838 0.97691

4 0.976838 0.97688 0.976838 0.97685 0.976838 0.97680

5 0.976838 0.97680 0.976838 0.97680 0.976838 0.97680

Table 2: Phase angles of two-bus sample systemVoltage of Bus ‘s’


No F. B.

Sweep

Chen

[17]

F. B.

Sweep

Chen

[17]

F. B.

Sweep

Chen

[17]

0 0.000 0.000 -120.000 -120 120.000 120.01 -31.176 -32.84 -151.176 -152.84 88.824 87.16

2 -31.176 -31.05 -151.176 -151.04 88.824 88.96

3 -31.177 -31.18 -151.177 -151.19 88.823 88.82

4 -31.177 -31.18 -151.177 -151.17 88.823 88.83

5 -31.177 -31.18 -151.177 -151.18 88.823 88.82

Table 3: Voltage magnitudes of two-bus sample systemVoltage of Bus ‘s’


No F. B.

SweepChen [17]

F. B.

SweepChen [17]

F. B.

Sweep

Chen

[17]

0 1.000000 1.00000 1.000000 1.00000 1.000000 1.00000

1 0.966838 0.95950 0.979988 0.97780 0.986621 0.98697

2 0.964732 0.96612 0.979240 0.97990 0.986291 0.98655

3 0.964656 0.96460 0.979225 0.97916 0.986287 0.98621

4 0.964651 0.96473 0.979224 0.97917 0.986286 0.98626

5 0.964651 0.96473 0.979224 0.97917 0.986286 0.98602

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

2. Iteration No. ‘0’ means initial guess.

Table 4: Phase angles of two-bus sample systemVoltage of Bus ‘s’


No F. B.

Sweep

Chen

[17]

F. B.

Sweep

Chen

[17]

F. B.

Sweep

Chen

[17]

0 0.0000 0.000 -120.00 -120.00 120.000 120.00

1 28.217 30.60 -91.056 -89.19 149.301 150.91

2 28.217 28.03 -91.056 -91.17 149.301 149.21

3 28.213 28.24 -91.056 -91.05 149.301 149.30

4 28.213 28.21 -91.056 -91.06 149.301 149.30

5 28.213 28.21 -91.056 -91.06 149.301 149.30

Comments: 1. All angles in degree.

2. Iteration No. ‘0’ means initial guess.

6 CONCLUSION

The symmetrical components model of distribution

transformers are incorporated into the sweep-based power

flow algorithm. A simple and practical method to include

three-phase transformers into the forward/backward

sweep-based distribution load flow is presented. Grounded

wye-delta and delta-grounded wye are modeled by using

its sequence-components and adapted to

forward/backward power flow algorithm. The results are

compared with other existing method and it is concluded

that the proposed technique is valid, reliable and

effective, most importantly it is easy to implement.

ACKNOWLEDGEMENTS

This work is supported by the Research Fund of

TUBITAK under the project number of 106E132, 2006.

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Grounded Wye-Delta

P S

Three-Phase transformer

13.8kV-208V, 1000kVA, Z=6%,

Source busLoad

400+j300

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AUTHOR'S ADDRESS

Gebze Institute of Technology

Department of Electronics Engineering,

Kocaeli 41400, TURKEY

E-mail: [email protected]

APPENDIX

The transformation between the phases and sequence

components are defined by;

=

2

2

aa1

aa1

111

A (1)

' AU U = and ' AI I = (2)

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Where ]3

2 jexp[ A

π = , '

U and ' I denotes sequence-

voltages and currents, respectively.

Load injected current can be calculated as follow; *

i

ii

V

S I

= (3)

For the forward sweep, voltages of the receiving end line

segment n are calculated by using Kirchhoff’s Voltage

Law as given in eq. (4).

−

=

c

n

b

n

a

n

cc

n

cb

n

ca

n

bc

n

bb

n

ba

n

ac

n

ab

n

aa

n

c

s

b

s

a

s

c

r

b

r

a

r

J

J

J

Z Z Z

Z Z Z

Z Z Z

V

V

V

V

V

V

(4)

Where,Vs and Vr stand for the sending end, receiving end

voltages of the line segment n, respectively;

Z is the line impedance matrix;

J is the line currents.

Voltage mismatches can be calculated at each bus as

follow;

)1k ()k ()k (V V V

−

−=∆ (5)

Zero-sequence current ( o

p I ) flowing through the primary

side of transformer is defined by;

o

o

po

p Z V I = (6)

Where, o Z denotes the zero-sequence impedance of

transformer. The new sequence-voltages of transformer

secondary and primary bus voltages can be calculated by

using Kirchhoff’s Voltage Law as given in eqs. (7 and 8)

as follow, respectively.

+

−

=

−

+

−

+

−

+

0

0

V

I

I

I

Z 00

0 Z 0

000

V

V

V

V

V

V o

s

s

s

o

s

2

1

p

p

o

p

s

s

o

s

(7)

−

−

=

−

+

−

+

−

+

0

0

Z I

I

I

I

Z 00

0 Z 0

000

V

V

V

V

V

V oo

s

s

0

s

2

1

p

p

0

p

s

s

0

s

(8)

For the backward sweep, voltages of the sending end line

segment n are calculated by using Kirchhoff’s Voltage

Law as follow;

+

=

c

n

b

n

a

n

cc

n

cb

n

ca

n

bc

n

bb

n

ba

n

ac

n

ab

n

aa

n

c

r

b

r

a

r

c

s

b

s

a

s

J

J

J

Z Z Z

Z Z Z

Z Z Z

V

V

V

V

V

V

(9)

The sequence-voltages of transformer primary side can be

calculated for grounded wye-delta and delta-grounded wye

as follow;

+

+

=

−

+

−

+

−

+

0

0

V

I

I

I

Z 00

0 Z 0

000

V

V

V

V

V

V o

p

p

p

o

p

2

1

s

s

o

s

p

p

o

p

(10)

Where,−+0

sV and−+0

pV show sequence voltages of

transformer secondary and primary side, respectively.−+

0 p I shows the sequence current of transformer primary

side and0

pV shows the zero-sequence voltage of

transformer primary side in the previous steps.

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