# Three Phase Distribution Transformer Modeling Biology Essay

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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 conventional approach to this problem has been to limit the analysis of feeders at primary level. 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 the transformer is one of the most important components of modem electric power systems, highly developed transformer models are not employed in system studies. It is the intention 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.

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 three-phase 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 medium-sized commercial loads that have three-phase motors as well as single-phase 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] 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, 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 is used and is incorporated into the unbalanced power flow method. General information about the symmetrical components model of three-phase 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 done to verify the approach, and the results 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 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;

(3.1)

And (3.2)

Where, and denotes sequence voltages and currents, respectively.

Load injected current can be calculated as follows:

(3.3)

The voltages of the receiving end line segment are calculated by using Kirchhoff's voltage law as given in eqn. (3.4)

(3.4)

Where,

And stand for the sending end and receiving end voltages of the line segment pq respectively;

Z is the line impedance matrix

I is the line current

Voltage mismatches can be calculated at each bus as

(3.5)

Zero sequence current flowing through the primary side of transformer is defined by

(3.6)

Where, 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.

(3.7)

(3.8)

The voltages of the sending end line segment pq are calculated by using Kirchhoff's Voltage Law as follows:

(3.9)

The sequence voltages of transformer primary side can be calculated for grounded wye-delta and delta-grounded wye as follows:

(3.10)

Where

and show sequence voltages of transformer secondary and primary side, respectively.

Shows the sequence current of transformer primary side and

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. 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 Wye-Delta (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.

## 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' *ejÏ€/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 Ip- respectively.

(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 towards the transformer primary bus using eqn. (3.4).

(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' *ejÏ€/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' *ejÏ€/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 certain 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.

(f) Calculate the new sequence voltages of transformer secondary bus using eq. (3.8).

(g) Apply phase shift Vs'=Vs'*ejÏ€/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 âˆ†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 three-phase 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 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.

## Load 400+j300 kVA

## Three-Phase transformer

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

## Source bus

## q

## p

Fig. 3.1 Two Bus Sample System

Table 3.1 Voltage magnitudes and Phase angles for Grounded wye-delta of 2-Bus Sample System

## Iter.

## No

## Three phase Power Flow Method proposed

## Forward/Backward Power Flow Method [28]

## Phase a

## Phase b

## Phase c

## Phase a

## Phase b

## Phase c

## |Va|

## p.u.

## ïƒVa

## deg.

## |Vb|

## p.u.

## ïƒVb

## deg.

## |Vc|

## p.u.

## ïƒVc

## deg.

## |Va|

## p.u.

## ïƒVa

## deg.

## |Vb|

## p.u.

## ïƒVb

## deg.

## |Vc|

## p.u.

## ïƒVc

## deg.

0

1.0000

0.00

1.0000

-120.00

1.0000

120.00

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

0.9967

-32.84

0.9967

-152.84

0.9967

87.16

2

0.9769

-31.18

0.9769

-151.18

0.9769

88.82

0.9759

-32.05

0.9759

-151.04

0.9760

88.96

3

0.9768

-31.18

0.9768

-151.18

0.9768

88.82

0.9769

-31.18

0.9768

-151.19

0.9769

88.82

4

0.9768

-31.18

0.9768

-151.18

0.9768

88.82

0.9769

-31.18

0.9769

-151.17

0.9768

88.83

5

0.9768

-13.18

0.9768

-151.18

0.9768

88.82

0.9768

-31.18

0.9768

-151.18

0.9768

88.82

Comments: 1. Iteration No. '0' means initial guess

Table 3.2 Voltage magnitudes and Phase angles for delta-Grounded wye of 2-Bus Sample System

## Iter.

## No

## Three phase Power Flow Method proposed

## Forward/Backward Power Flow Method [28]

## Phase a

## Phase b

## Phase c

## Phase a

## Phase b

## Phase c

## |Va|

## p.u.

## ïƒVa

## deg.

## |Vb|

## p.u.

## ïƒVb

## deg.

## |Vc|

## p.u.

## ïƒVc

## deg.

## |Va|

## p.u.

## ïƒVa

## deg.

## |Vb|

## p.u.

## ïƒVb

## deg.

## |Vc|

## p.u.

## ïƒVc

## deg.

0

1.0000

0.00

1.0000

-120.00

1.0000

120.00

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

0.9595

30.60

0.9778

-89.19

0.9870

150.91

2

0.9647

28.22

0.9792

-91.06

0.9863

146.30

0.9661

28.03

0.9800

-91.17

0.9866

149.21

3

0.9647

28.21

0.9792

-91.06

0.9863

146.30

0.9646

28.24

0.9792

-91.05

0.9862

149.30

4

0.9647

28.21

0.9792

-91.06

0.9863

146.30

0.9648

28.21

0.9792

-91.06

0.9863

149.30

5

0.9647

28.21

0.9792

-91.06

0.9863

146.30

0.9648

28.21

0.9792

-91.06

0.9860

149.30

Comments: 1. Iteration No. '0' means initial guess

## 3.4.2 Case Study II: 37-bus IEEE URDS

Fig. 3.2 Single line diagram of 37-bus IEEE URDS

The proposed algorithm is tested on IEEE 37 node unbalanced radial distribution system shown 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 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|

## p.u.

## ïƒVa

## deg.

## |Vb|

## p.u.

## ïƒVb

## deg.

## |Vc|

## p.u.

## ïƒVc

## 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

Contd . . .120.10

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 phase a, b, and c are 0.9979, 0.9962, and 0.9849 respectively. Voltage regulator tap positions of 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

## From

## Bus To

## Phase a

## Phase b

## Phase c

## P

## (kW)

## Q

## (kVAr)

## P

## (kW)

## Q

## (kVAr)

## P

## (kW)

## Q

## (kVAr)

799

701

964.09

1129.30

1554.00

1130.80

1198.90

1552.50

701

702

791.90

645.61

1392.40

895.51

826.79

Contd . . .1216.80

702

703

521.25

410.87

690.46

531.81

471.17

581.96

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

## 3.5 Conclusion

The symmetrical components model of distribution transformers are incorporated into the three phase power flow algorithm. A simple and practical method to include three-phase transformers into the three phase distribution load flow is presented. Grounded wye-delta and delta-grounded wye connections are modeled by using their sequence-components 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.