64
CHAPTER – 3

Three Phase Distribution Transformer Modeling for
Power Flow Calculation
3.1 Introduction
Choosing a proper feeder model for analysis of a distribution
system is a challenging task; one often has to make a compromise
between the detail of system representation and the amount of data
available for analysis. It is usually the amount of load data that limits the
level of detail of system representation at distribution level. The approach
to such problem is limited to the primary feeder analysis. Secondaries
and customer loads are lumped as loads as seen from the primaries of
the distribution transformers (DTs).
The impact of the numerous transformers in a distribution system
is significant. Transformers affect system loss, zero sequence current,
grounding method, and protection strategy. Although transformer is one
of the most important components of modem electric power systems,
highly developed transformer models are not employed in system studies.
Hence it is intended to introduce a transformer model and its
implementation method so that large-scale unbalanced distribution
system problems such as power flow, short circuit, system loss, and
contingency studies, can be solved.
65
Recognizing
the
fact
that
the
system
is
unbalanced,
the
conventional transformer models, based on a balanced three phase
assumption, can no longer be considered suitable. This is done with
justifiable reason. For example in the widely used, delta-grounded wye
connection of distribution step-down transformers, the positive and
negative sequence voltages are shifted in opposite directions, this phase
shift must be included in the model to properly simulate the effects of the
system imbalance.
The purpose of this chapter is to demonstrate how the exact
models for three-phase transformer connections can be developed for use
in power-flow studies. Too many times, approximations are made in the
modelings that result in erroneous results. The exact model of a threephase connection must satisfy Kirchhoff’s voltage and current laws and
the ideal relationship between the voltages and currents on the two sides
of the transformer windings. When this approach is followed, the correct
phase shift, if any, will come out naturally.
This transformer connection is employed in small- to mediumsized commercial loads that have three-phase motors as well as singlephase lighting and appliance load. It is an economical way to provide
both 3-phase and single-phase service with one transformer bank.
The authors of [19, 21, 23, 24] employed different approaches to
model distribution transformers in a branch current based feeder
analysis. In study [19], voltage and current equations were developed for
66
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, these methods are mainly based on circuit
analysis with Kirchhoff’s voltage and current laws. They are in need of
deriving the individual formulae for different winding connections from
scratch.
In this chapter, symmetrical components modeling of 3 phase
distribution
transformers
unbalanced
power
symmetrical
flow
components
is
used
method.
model
and
is
General
of
incorporated
information
three-phase
into
the
about
the
transformers
is
presented. A detailed description of the power flow algorithms used and
the proposed modeling procedure is explained in detail. Extensive
computation and comparisons have been made to verify the approach,
and the results obtained are presented.
3.2 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 sequence
components. The advantage of this transformation is that for balanced
67
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
is defined by;
1 1 1 


A  1 a 2 a 
1 a a 2 


(3.1)
U  AU  And I  AI 
(3.2)
 2 
Where A  exp  j
, U  and I  denotes sequence voltages and currents,
 3 
respectively.
Load injected current can be calculated as follows:
S
I i   i
 Vi




(3.3)
The voltages of the receiving end line segment are calculated by using
Kirchhoff’s voltage law as given in eqn. (3.4)
ab
ac
V qa  V pa  z aa
 I apq 
pq z pq z pq
 b   b   ba bb bc   b 
V q   V p   z pq z pq z pq  I pq 
 c   c   ca cb cc   c 
V q  V p  z pq z pq z pq  I pq 
(3.4)
Where,
V p And Vq stand for the sending end and receiving end voltages of the
line segment pq respectively;
68
Z is the line impedance matrix
I is the line current
Voltage mismatches can be calculated at each bus as
V (k )  V (k )  V (k 1)
(3.5)
 
Zero sequence current I P0 flowing through the primary side of
transformer is defined by
I
0
P
V P0
 0
Z
(3.6)
Where, Z 0 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 eqns. (3.7)
and (3.8) respectively.
VS0  V P0  0 0 0  I S0  VS0 
    
   

VS   V P   0 Z 0  I S    0 
V   V   0 0 Z   I    0 
 S  
 S  P 
(3.7)
VS0  V P0  0 0 0  I S0  I 0 Z 0 
    
  


VS   V P   0 Z 0  I S    0 
V   V   0 0 Z   I    0 
 S 

 S  P 
(3.8)
The voltages of the sending end line segment pq are calculated by using
Kirchhoff’s Voltage Law as follows:
ab
ac
V pa  V qa  z aa
 I apq 
pq z pq z pq
 b   b   ba bb bc   b 
V p   V q   z pq z pq z pq  I pq 
 c   c   ca cb cc   c 
V p  V q  z pq z pq z pq  I pq 
(3.9)
69
The sequence voltages of transformer primary side can be calculated for
delta-grounded wye and grounded wye-delta as follows:
V P0  VS0  0 0 0  I P0  V P0 
    
   

V P   VS   0 Z 0  I P    0 
V   V   0 0 Z   I    0 
 P  
 P  S 
(3.10)
Where
VS0   and V P0   show sequence voltages of transformer secondary and
primary side, respectively.
I P0   Shows the sequence current of transformer primary side and
V P0 Shows the zero-sequence voltage of transformer primary side.
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 eqns. (3.1) and (3.2). Normally, the three-phase transformer is
modeled in terms of its symmetrical components under the assumption
that the power system is sufficiently balanced. The typical symmetrical
component models of the transformers for the most common three-phase
connections were given in [30].
3.3 Three Phase Power Flow
Although the proposed 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.
70
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 a 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 proposed algorithm
employs an iterative method to update bus voltages and branch currents.
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 grounded Wye-grounded Wye (GY-GY), grounded WyeDelta
(GY-D),
and
Delta-grounded
Wye
(D-GY)
connection
type
transformers are most commonly used in the distribution systems. In the
proposed method there is no need to use the nodal admittance matrices
when the GY-GY connection is used for distribution transformers. The
phase impedance matrices of transformer can be used directly in the
algorithm. The other type of transformer connections needs to be
modeled and adapted to the power flow algorithm. In this section,
symmetrical components modeling for distribution transformers of GY-D
and D-GY winding configurations are implemented into power flow
algorithm. The flow chart of the proposed method is shown in Appendix F
as Fig. F.1.
71
3.3.1
Algorithm
for
3-Phase
Power
Flow
with
Transformer
Symmetrical Component Modelling
Step 1: Read the line data and identify the nodes beyond a particular
node of the system.
Step 2: Read load data and Initialize the bus voltages.
Step 3: Calculate each bus current using eqn. (3.3).
Step 4: Calculate each branch current starting from the far end branch
and moving towards transformer secondary side.
Step 5: Calculate the sequence currents (Is’) of transformer secondary
current (Is) using eqn. (3.2).
Step 6: If Gy-D connection
(a) Apply the phase shift Is’ = Is’ *ejπ/6 and set Is0= 0
(b) Calculate the sequence voltages (Vp’) of transformer primary
bus using eq. (2), and zero-sequence current (Ip0) using eqn.
(3.6)
(c) Calculate phase current Ip using Is+ and Is- instead of Ip+ and Iprespectively.
(d) Continue to calculate branch current calculation moving
towards a certain swing bus.
(e) Calculate each line receiving end voltages starting from swing
bus and moving
using eqn. (3.4).
towards
the
transformer
primary
bus
72
(f) Calculate the sequence voltages (Vs’) of transformer secondary
bus using eqn. (3.2).
(g) Calculate the sequence voltages (Vp’) of transformer primary
bus using eqn. (3.2)
and set Vp0=0.
(h) Calculate the new sequence voltages of transformer secondary
bus using eqn. (3.7).
(i) Apply the phase shift Vs’ =Vs’ *ejπ/6 and calculate the phase
voltages of transformer
secondary bus Vs using eqn. (3.2).
(j) Continue the bus voltages calculation moving towards the far
end using eqn. (3.4).
(k) Go to step 8
Step 7: If D-Gy connection
(a) Save the zero-sequence current (Is0= I0 ) and apply the phase
shift Is’ = Is’ *ejπ/6 and
set Is0= 0.
(b) Calculate phase-current Ip using Is+ and Is- instead of Ip+ and
Ip- respectively.
(c) Continue to calculate branch current calculation moving
towards a swing bus.
(d) Calculate each line receiving end voltage starting from swing
bus and moving
towards
the
transformer
primary
bus
using eqn. (3.4).
(e) Calculate the sequence-voltages (Vp’) of transformer primary
bus using eqn. (3.2)
and set Vp0=0.
73
(f) Calculate
the
new
sequence
voltages
of
transformer
secondary bus using eq. (3.8).
(g) Apply phase shift Vs’=Vs’*ejπ/6 and calculate the phase
voltages of transformer secondary bus Vs using eq. (3.2).
(h) Continue the busses voltage calculation moving towards the
far end using eq. (3.4).
Step 8: Calculate voltage mismatches ∆V(k)=||V(k)|-|V(k-1)||
Step 9: Test for convergence, if no, go to step 3
Step 10: Compute branch losses, total losses, quantity of unbalance etc.
Step 11: Stop.
3.4 Simulation Results and Analysis
3.4.1 Case Study 1: 2-bus URDS
To verify the proposed approach for the transformer modeling, two
transformer configurations were included in both the proposed threephase
distribution
system
power
flow
program
and
the
forward/backward substitution power flow method [28] and the results
obtained were compared. A two-bus three-phase standard test system,
given in fig. 3.1, is used, and for simplification, the transformer in the
sample system is assumed to be at nominal rating, therefore, the taps on
the primary and secondary sides are equal to 1.0. In addition, the voltage
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 3.1 for
74
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 3.2. It is observed that results obtained match very well with those
listed in the study [28] and the proposed algorithm can reach the
tolerance of 0.00001 at fourth iteration. On the other hand, in the results
of the study [28], the voltages do not reach this tolerance value for these
two connection types.
p
Source bus
q
Three-Phase transformer
13.8 kV – 208 V, 1000 kVA, Z = 6%
Load
400+j300 kVA
Fig. 3.1 Two Bus Sample System
75
Table 3.1 Voltage magnitudes and Phase angles for Grounded wye-delta of 2-Bus Sample System
Three phase Power Flow Method proposed
Phase a
Phase b
Phase c
|Va|
|Vb|
|Vc|
Va
Vb
Vc
p.u.
p.u.
p.u.
deg.
deg.
deg.
0
1.0000
0.00 1.0000 -120.00 1.0000 120.00
1
0.9778 -31.18 0.9778 -151.18 0.9778
88.82
2
0.9769 -31.18 0.9769 -151.18 0.9769
88.82
3
0.9768 -31.18 0.9768 -151.18 0.9768
88.82
4
0.9768 -31.18 0.9768 -151.18 0.9768
88.82
5
0.9768 -13.18 0.9768 -151.18 0.9768
88.82
Comments: 1. Iteration No. ‘0’ means initial guess
Iter.
No
Forward/Backward Power Flow
Phase a
Phase b
|Va|
|Vb|
Va
Vb
p.u.
p.u.
deg.
deg.
1.0000
0.00 1.0000 -120.00
0.9967 -32.84 0.9967 -152.84
0.9759 -32.05 0.9759 -151.04
0.9769 -31.18 0.9768 -151.19
0.9769 -31.18 0.9769 -151.17
0.9768 -31.18 0.9768 -151.18
Method [28]
Phase c
|Vc|
Vc
p.u.
deg.
1.0000 120.00
0.9967
87.16
0.9760
88.96
0.9769
88.82
0.9768
88.83
0.9768
88.82
Table 3.2 Voltage magnitudes and Phase angles for delta-Grounded wye of 2-Bus Sample System
Three phase Power Flow Method proposed
Phase a
Phase b
Phase c
Iter.
No
|Va|
|Vb|
|Vc|
Va
Vb
Vc
p.u.
p.u.
p.u.
deg.
deg.
deg.
0
1.0000
0.00 1.0000 -120.00 1.0000 120.00
1
0.9668 28.22 0.9800
-91.06 0.9866 146.30
2
0.9647 28.22 0.9792
-91.06 0.9863 146.30
3
0.9647 28.21 0.9792
-91.06 0.9863 146.30
4
0.9647 28.21 0.9792
-91.06 0.9863 146.30
5
0.9647 28.21 0.9792
-91.06 0.9863 146.30
Comments: 1. Iteration No. ‘0’ means initial guess
Forward/Backward Power Flow Method [28]
Phase a
Phase b
Phase c
|Va|
|Vb|
|Vc|
Va
Vb
Vc
p.u.
p.u.
p.u.
deg.
deg.
deg.
1.0000
0.00 1.0000 -120.00 1.0000 120.00
0.9595 30.60 0.9778
-89.19 0.9870 150.91
0.9661 28.03 0.9800
-91.17 0.9866 149.21
0.9646 28.24 0.9792
-91.05 0.9862 149.30
0.9648 28.21 0.9792
-91.06 0.9863 149.30
0.9648 28.21 0.9792
-91.06 0.9860 149.30
76
3.4.2 Case Study II: 37-bus IEEE URDS
799
724
722
707
712
701
742
713
704
720
705
702
714
706
729
744
727
703
718
725
728
730
732
708
709
731
736
733
710
775
734
740
735
737
738
711
741
Fig. 3.2 Single line diagram of 37-bus IEEE URDS [128]
The proposed algorithm is tested on IEEE 37 node unbalanced
radial distribution system [128] illustrated in Fig. 3.2. This feeder is an
actual feeder located in California. The characteristics of the feeder are,
three-wire delta operating at a nominal voltage of 4.8 kV, all line
segments are underground, Substation voltage regulator consisting of
two single phase units connected in open delta, all loads are “spot” loads
and consist of constant PQ, constant current and constant impedance
and the loading is very unbalanced. The line and load, impedance, shunt
admittance; transformer and regulator data are given in [31] and also
77
given in Appendix B Tables B1, B2, B3, B4 and B5 respectively. For the
load flow, base voltage and base MVA are chosen as 4.8 kV and 30 MVA
respectively.
Table 3.3 Voltage magnitudes and Phase angles for IEEE 37 bus URDS
Node
No.
Phase a
Phase b
Phase c
|Va|
Va
|Vb|
Vb
|Vc|
Vc
p.u.
deg.
p.u.
deg.
p.u.
deg.
799
1.0000
0.00
1.0000
-120.00
1.0000
120.00
Reg
1.0435
0.00
1.0200
-120.00
1.0340
120.90
701
1.0308
-0.08
1.0141
-120.39
1.0180
120.61
702
1.0248
-0.14
1.0088
-120.58
1.0098
120.43
703
1.0176
-0.17
1.0049
-120.70
1.0034
120.20
730
1.0125
-0.12
1.0018
-120.73
0.9979
120.10
709
1.0111
-0.11
1.0012
-120.73
0.9967
120.07
708
1.0087
-0.08
1.0002
-120.73
0.9945
120.02
733
1.0063
-0.05
0.9993
-120.73
0.9925
119.96
734
1.0027
-0.01
0.9978
-120.74
0.9893
119.88
737
0.9996
0.02
0.9969
-120.71
0.9871
119.79
738
0.9985
0.04
0.9965
-120.71
0.9859
119.76
711
0.9982
0.06
0.9963
-120.74
0.9852
119.76
741
0.9979
0.07
0.9962
-120.75
0.9849
119.76
713
1.0234
-0.15
1.0070
-120.60
1.0083
120.44
704
1.0217
-0.17
1.0044
-120.61
1.0065
120.46
720
1.0205
-0.21
1.0008
-120.66
1.0041
120.53
706
1.0204
-0.22
1.0007
-120.66
1.0037
120.54
725
1.0202
-0.23
1.0003
-120.65
1.0037
120.55
705
1.0240
-0.13
1.0072
-120.59
1.0088
120.46
742
1.0236
-0.15
1.0064
-120.59
1.0086
120.48
727
1.0167
-0.16
1.0044
-120.69
1.0025
120.19
744
1.0157
-0.16
1.0038
-120.68
1.0019
120.17
729
1.0155
-0.15
1.0037
-120.67
1.0018
120.17
775
1.0111
-0.11
1.0012
-120.73
0.9967
120.07
731
1.0109
-0.13
1.0004
-120.74
0.9964
120.10
Contd . .
.
78
732
1.0086
-0.07
1.0001
-120.74
0.9941
120.02
710
1.0024
0.01
0.9968
-120.77
0.9878
119.91
735
1.0023
0.03
0.9966
-120.78
0.9873
119.91
740
0.9981
0.07
0.9963
-120.75
0.9851
119.76
714
1.0214
-0.17
1.0043
-120.60
1.0064
120.46
718
1.0199
-0.16
1.0040
-120.57
1.0058
120.42
707
1.0185
-0.30
0.9959
-120.62
1.0025
120.67
722
1.0183
-0.30
0.9952
-120.62
1.0023
120.68
724
1.0184
-0.32
0.9950
-120.61
1.0023
120.69
728
1.0156
-0.15
1.0037
-120.68
1.0015
120.18
736
1.0019
-0.02
0.9949
-120.75
0.9872
119.95
712
1.0238
-0.11
1.0072
-120.61
1.0081
120.46
The obtained voltage profile of IEEE 37 bus URDS is given Table
3.3. From Table 3.3, it is observed that the minimum voltage in phases a,
b, and c are 0.9979, 0.9962, and 0.9849 respectively. Voltage regulator
tap positions for the convergence of the power flow in phases a, b and c
are 8, 0 and 5. Table 3.4 shows the power flows for 37 bus URDS. The
active power loss in phases a, b and c are 29.67 kW, 17.80 kW and 24.09
kW respectively and the total reactive power loss in phases a, b and c
are 21.77 kVAr, 13.95 kVAr and 20.73 kVAr respectively.
Table 3.4 Power flow for the IEEE 37 bus unbalanced radial distribution
system
Bus
Bus
From
To
799
701
701
702
Phase a
Phase b
Phase c
P
Q
P
Q
P
Q
(kW)
(kVAr)
(kW)
(kVAr)
(kW)
(kVAr)
964.09
1129.30
1554.00
1130.80
1198.90
1552.50
702
791.90
645.61
1392.40
895.51
826.79
1216.80
703
521.25
410.87
690.46
531.81
471.17
581.96
Contd . .
.
79
703
730
387.58
349.82
549.54
419.96
382.77
494.64
730
709
378.12
255.53
543.44
421.86
296.22
453.90
709
708
377.84
280.68
436.18
394.45
288.92
365.52
702
705
101.52
85.601
228.71
153.77
99.351
216.64
702
713
166.34
147.75
467.57
204.76
251.04
412.55
703
727
132.08
60.64
138.74
110.48
86.51
85.81
709
731
0
25.21
106.67
27.17
6.91
88.19
709
775
0
0
0
0
0
0
708
733
375.24
323.38
196.06
337.27
257.32
164.43
708
707
2.11
42.73
239.43
57.03
31.14
200.98
733
734
289.73
291.41
180.53
247.69
257.12
164.57
734
737
277.78
216.75
74.89
264.67
127.22
60.19
734
710
8.35
34.11
97.50
12.24
87.54
83.45
737
738
137.26
156.84
55.90
113.16
127.14
60.46
738
711
11.14
106.06
36.43
21.65
127.09
60.68
711
741
2.91
36.38
11.27
6.72
42.00
20.87
711
740
8.19
69.86
25.15
14.72
85.02
39.95
713
704
157.87
63.06
454.87
211.86
165.48
372.5
704
714
102.06
13.31
83.75
91.38
1.60
21.60
704
720
10.66
21.25
312.29
60.19
118.91
285.73
720
707
2.10
42.97
240.42
56.98
31.07
201.93
720
706
0
18.39
57.56
11.54
2.311
43.95
706
725
0
18.08
57.54
11.84
2.30
44.21
705
742
8.00
16.09
102.94
38.76
6.99
88.35
705
712
93.42
101.86
125.41
115.08
92.06
128.36
727
744
129.04
32.24
120.46
121.24
44.46
64.95
744
725
44.99
35.49
75.36
50.24
44.45
65.18
744
729
42.00
2.08
22.95
35.28
0
0.07
710
735
8.32
70.99
24.23
14.21
85.02
39.95
710
736
0.01
36.36
73.20
2.48
2.41
43.93
714
718
85.04
8.86
35.13
77.95
0
0.13
707
724
0.00
34.22
71.29
3.705
2.19
44.18
707
722
2.08
8.45
168.44
53.38
28.35
157.86
80
3.5 Conclusion
The symmetrical components model of distribution transformers
are incorporated into the three phase power flow algorithm. A simple and
efficient method to include three-phase transformers into the three phase
distribution load flow is presented. Grounded wye-delta and deltagrounded wye connections are modeled by using their sequencecomponents and adapted to three phase power flow algorithm. The
proposed method is tested on the IEEE 37 bus unbalanced radial
distribution system. The 2 bus sample system results are compared with
that of other existing method and it is concluded that the proposed
technique is valid, reliable and effective. Most importantly it is easy to
implement.