transformer model

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Transformer Model

Timothy Vismor

February 23, 2012

Abstract

is document describes a transformer model that is useful for theanalysis of large scale electric power systems. It denes a balanced trans-former representation primarily intended for use with analytical techniquesthat are based on the complex nodal admittance matrix. Particular atten-tion is paid to the integration with “fast decoupled load ow” (FDLF)algorithms.

Copyright © 1990-2012 Timothy Vismor



CONTENTS LIST OF FIGURES

Contents

1 Transformer Equivalent Circuit 3

2 Transformer Parameters 42.1 Leakage Impedance . . . . . . . . . . . . . . . . . . . . . . . 52.2 Voltage Transformation . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Transformation Ratio . . . . . . . . . . . . . . . . . . 62.2.2 Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Voltage Regulation . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Transformer Admittance Model 8

4 Computing Admittances 104.1 Self Admittances . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.1 Primary Admittance . . . . . . . . . . . . . . . . . . . 114.1.2 Secondary Admittance . . . . . . . . . . . . . . . . . 11

4.2 Transfer Admittances . . . . . . . . . . . . . . . . . . . . . . . 114.2.1 Primary to Secondary . . . . . . . . . . . . . . . . . . 124.2.2 Secondary to Primary . . . . . . . . . . . . . . . . . . 124.2.3 Observations on Computational Sequence . . . . . . . 13

5 TCUL Devices 135.1 Computing Tap Adjustments . . . . . . . . . . . . . . . . . . 14

5.2 Error Feedback Formulation . . . . . . . . . . . . . . . . . . . 155.3 Controlled Bus Sensitivity . . . . . . . . . . . . . . . . . . . . 165.4 Physical Constraints on Tap Settings . . . . . . . . . . . . . . 16

6 Transformers in the Nodal Admittance Matrix 17

7 e Nodal Admittance Matrix during TCUL 18

8 Integrating TCUL into FDLF 19

List of Figures

1 Transformer Equivalent Circuit Diagram . . . . . . . . . . . . 4

2



LIST OF TABLES LIST OF TABLES

List of Tables

1 Per Unit to Ohm Conversion Constants . . . . . . . . . . . . 62 Transformer Phase Shifts . . . . . . . . . . . . . . . . . . . . . 73 Common Transformer Tap Parameters . . . . . . . . . . . . . 84 Maintenance Costs . . . . . . . . . . . . . . . . . . . . . 20

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1 TRANSFORMER EQUIVALENT CIRCUIT

1 Transformer Equivalent Circuit

e equivalent circuit of a two winding transformer consists of an ideal trans-former, a leakage impedance , and a magnetizing impedance . An idealtransformer is a lossless entity categorized by a complex voltage ratio , i.e.

= (1)

where

is the transformer’s primary voltage. is the transformer’s secondary voltage.

Since the ideal transformer has no losses, it must be true that

= (2)

or

⋆ = ⋆ (3)

Solving for the secondary current

⋆ = ⋆ (4)

= ⋆

(5)

Substituting Equation 1 reveals the relationship between the primary and sec-ondary current of an ideal transformer

= ⋆ (6)

where

is current into the transformer’s primary. is current out of the transformer’s secondary. is the transformer’s voltage transformation ratio.

A transformer’s magnetizing impedance is a shunt element associated withits excitation current, i.e. the “no load” current in the primary windings. ereal component of the magnetizing impedance reects the core losses of thetransformer. Its reactive component describes the mmf required to overcomethe magnetic reluctance of the core.

4



2 TRANSFORMER PARAMETERS

Figure 1: Transformer Equivalent Circuit Diagram

A transformer’s leakage impedance is a series element that reects imper-fections in the windings. e real component of the leakage impedance is theresistance of the windings. Its reactive component is due to leakage ux, i.e. uxin the magnetic circuit that does not contribute to coupling the windings.

Figure 1 depicts a transformer’s equivalent circuit with impedances referredto the secondary.

2 Transformer Parameters

e objective of this section is review commonly published transformer data andexamine techniques for massaging raw transformer data into the form requiredby the parameters of analysis algorithms that are based on the complex nodaladmittance matrix or .

If magnetizing impedance is neglected, the steady state characteristics of a transformer are captured by its leakage impedance, voltage transformation

properties (i.e. transformation ratio and phase shift). Tap-changing-under-load(TCUL) devices are also characterized by their voltage regulation capabilities.

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2.1 Leakage Impedance 2 TRANSFORMER PARAMETERS

2.1 Leakage Impedance

A transformer’s leakage impedance is determined empirically through a short-circuit test. Leakage impedances provided by equipment manufacturers are usu-ally expressed in percent using the device’s rated power as the base. To convertthe impedance from percent to per unit, just divide it by a hundred.

= %100 (7)

To express the impedance in per unit on the system base rather than thetransformer base

′ = ′

′(8)

where

is the transformer power base. is the transformer voltage base.′ is the system power base.′ is the system voltage base.

When the transformer voltage base coincides with the system voltage base, thisequation simplies to

′=

′

(9)

e transformer’s impedance may also be expressed in ohms

= (10)

where

is the transformer’s rated voltage. e computed impedance is referredto the transformer’s primary or secondary by the appropriate choice of . is the transformer’s power rating. is a constant that keeps the units straight. e value of depends on how

the transformer’s power and voltage ratings are specied. Variations in

over a

wide range of unit systems are found in Table 1.

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2.2 Voltage Transformation 2 TRANSFORMER PARAMETERS

Table 1: Per Unit to Ohm Conversion ConstantsVoltage Unit Power Unit Constant

1 1 3 3 1 11,000 3 31,000 1 13 3 1 1 13,000 3 11,000

11,000

3 3,000 1 1 3 3 1 1,0003 3 1,000 1 13 3 1

2.2 Voltage Transformation

A transformer’s voltage ratio is expressed in polar form as

, where

is the

magnitude of the transformation and is its phase shift.

2.2.1 Transformation Ratio

Each transformer is categorized by a nominal operating point, i.e. its nameplateprimary and secondary voltage. e nominal magnitude of the voltage ratiois determined by examining the real part of Equation 1 at nominal operatingconditions.

= (11)

where

is the nominal primary voltage. is the nominal secondary voltage.

When voltages are expressed in per unit, the nominal value of is always one.

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2.3 Voltage Regulation 2 TRANSFORMER PARAMETERS

2.2.2 Phase Shift

In the general case, an arbitrary angular shift

may be introduced by a mul-

tiphase transformer bank. However, omitting phase shifters (angle regulatingtransformers) from the system model, reduces the angle shifts across balancedtransformer congurations to effects introduced by the connections of the wind-ings. Table 2 describes the phase shift associated with common transformerconnections.

Table 2: Transformer Phase Shifts

Winding

Primary Secondary Phase Shift Wye Wye 0∘ Wye Delta −30∘Delta Wye 30∘Delta Delta 0∘

When phase shifters are ignored, the user does not have to specify explicitly.It may be inferred from Table 2.

2.3 Voltage Regulation

Most power transformers permit you to tap into the windings at a variety of different positions. Some transformers are congured with voltage sensors andmotors which permit them to change tap settings automatically in response tovoltage variations on the power system. ese transformers are referred to astap-changing-under-Ioad (TCUL) devices. A TCUL device that controls volt-age but does not transform large amounts of power is referred to as a regulatingtransformer. Regulating transformers that control voltage magnitude on distri-bution systems are usually called voltage regulators or simply regulators.

TCUL devices that regulate voltage magnitudes are categorized by a taprange and a tap increment. e tap range denes the limits of the device’s reg-ulating ability. e tap increment denes the resolution of the device. e tapchanging mechanism of many transformers is categorized by a regulation rangeand a step count, e.g.

±10 percent, 32 steps. e tap increment is computed

from this data as follows.

= − (12)

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3 TRANSFORMER ADMITTANCE MODEL

where

is the transformer’s tap increment.

is maximum increase in voltage provided by the transformer (some-times called boost). If the transformer can not boost the voltage, is zero. is maximum decrease in voltage provided by the transformer (some-times called buck). If the transformer can not decrease the voltage, is zero. is the number of tap steps.

Table 3 catalogs the tap range and increment of common load tap-changers.

Table 3: Common Transformer Tap Parameters

Regulation Range

±10% 32 10.0 -10.0 0.62500±10% 16 10.0 -10.0 1.25000±10% 8 10.0 -10.0 2.50000+10% 32 10.0 0.0 0.31250+10% 16 10.0 0.0 0.62500+10% 8 10.0 0.0 1.25000±7.5% 32 7.5 -7.5 0.46875±7.5% 16 7.5 -7.5 0.93750±7.5% 8 7.5 -7.5 1.87500+7.5% 32 7.5 0.0 0.23437+7.5% 16 7.5 0.0 0.46875

+7.5% 8 7.5 0.0 0.93750

±5% 32 5.0 -5.0 .31250±5% 16 5.0 -5.0 0.62500±5% 8 5.0 -5.0 1.25000+5% 32 10.0 0.0 0.15625+5% 16 10.0 0.0 0.31250+5% 8 10.0 0.0 0.62500

3 Transformer Admittance Model

For analysis purposes, each transformer is described by the admittances of ageneral two-port network. When the current equations of Figure 1 are writtenas follows, the transformer admittances correspond to the coefficient matrix.

9



3 TRANSFORMER ADMITTANCE MODEL

= + (13)

= + (14)

where

is the driving point admittance of the primary. is the driving point admittance of the secondary. is the transfer admittance from the primary to the secondary. is the transfer admittance from the secondary to the primary.

Noting that

= 1and

= 1It is apparent from inspection of Figure 1 that

⋆ = − + and

= − Converting these equations to the desired form

= ⋆ + − ⋆ (15)

= −⋆ + (16)

Recalling that

⋆ = 2and equating the coefficients in Equations 13 and 14

with their counterparts in Equations 15 and 16, yields the transformer’s admit-tances:

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4 COMPUTING ADMITTANCES

= + 2 (17)

= (18)

= −⋆ (19)

= − (20)

In Equation 17 the real number is the absolute value of the complex voltageratio. When the magnetizing impedance is quite large, approaches zero andEquation 17 simplies to

= 2e following sections examine data and computations required to incorpo-

rate this model into balanced power system analyses.

4 Computing Admittances

is section examines Equation 17 through Equation 20 with an eye for stream-lining their computation. e resulting computational sequence explicitly de-composes the complex equations into real operations.

A transformer’s leakage admittance is

is derived from the leakage impedance

in the usual manner. Expressing the leakage impedance in rectangular form

= + (21)

then

= 1 = 1 + = + (22)

where

= 2 + 2 (23)

= − 2 + 2 (24)

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4.1 Self Admittances 4 COMPUTING ADMITTANCES

For the sake of convenience, dene the tap ratio as

= 1 = − (25)

4.1 Self Admittances

In the context of modeling and analysis of electrical networks, the self admit-tances of a transformer can be thought of as the device’s impact on the diagonalelements of of the nodal admittance matrix (i.e. ).

4.1.1 Primary Admittance

Considering the self admittance of the transformer’s primary, Equation 17 states

= + 2 (26)

Expressing the complex equation in rectangular form and making the sub-stitution dened in Equation 25

= 2( + ) + 2( + ) (27)

When the magnetizing admittance is ignored, Equation 27 reduces to

= 2 + 2 (28)

4.1.2 Secondary Admittance

Equation 18 denes the self admittance of the transformer’s secondary as

= (29)

which is expressed in rectangular coordinates as

= + (30)

4.2 Transfer Admittances

In the context of modeling and analysis of electrical networks, the transfer ad-mittances of a transformer can be thought of as the device’s impact on the off-diagonal elements of of the nodal admittance matrix (i.e. ).

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4.2 Transfer Admittances 4 COMPUTING ADMITTANCES

4.2.1 Primary to Secondary

Equation 19 denes the transfer admittance from the transformer’s primary toits secondary as

= −⋆ (31)

Substituting Equation 25 and resolving the tap ratio into polar form

= −⋆ = − Expressing the tap ratio and leakage admittance in rectangular form

= −( + )( + )Carrying out the multiplications then collecting real and imaginary terms

= −((− + ) + ( + ))erefore,

= −( + ) (32)

where

= − + (33)

= + (34)

4.2.2 Secondary to Primary

Equation 20 denes the transfer admittance from the transformer’s secondary to its primary as

= − (35)

Substituting Equation 25 and resolving the tap ratio into polar form

= −⋆ = − Expressing the tap ratio and leakage admittance in rectangular form

= −((−) + (−))( + )13



5 TCUL DEVICES

Invoking the trigonometric identities (−) = and (−) = − the

equation becomes

= −( − )( + )Carrying out the multiplications then collecting real and imaginary terms

= −(( + ) + (− + ))erefore,

= −( + ) (36)

where

= + (37)

= − + (38)

4.2.3 Observations on Computational Sequence

e preceding results suggest the computational sequence is not too crucial. evalues and appearing in Equation 32 and Equation 36 should only becomputed once. Marginal benets may also accrue from a single computationof the terms , , , and 2.

5 TCUL Devices

A voltage regulating transformer changes its tap ratio to maintain a constantvoltage magnitude at its regulation point. e normalized change in voltage(denoted by Δ) required to bring the regulation point to its designated valueis just

Δ = − (39)

where

is the regulated vertex. is the specied (desired) voltage at vertex , the regulation point. is the actual voltage at vertex .

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5.1 Computing Tap Adjustments 5 TCUL DEVICES

Assuming that the normalized change in voltage at the regulation point is

proportional to the normalized change in voltage at the transformer’s secondary,the secondary voltage must change as follows

= + Δ (40)

where is the constant of proportionality.

5.1 Computing Tap Adjustments

Recalling that the secondary voltage of a transformer is a function of the tapratio

= implies that

= and

= Substituting these values into Equation 40 yields

= + Δ which simplies to

= + Δ (41)

To obtain a computational formula, substitute Equation 39

= + − and simplify as follows

=

+

( − 1) =

+

−

or

= (1 − + ) (42)

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5.2 Error Feedback Formulation 5 TCUL DEVICES

Normally, is assumed to be one and Equation 42 reduces to

= (43)

Computation of , the sensitivity of the regulated voltage to transformer tapchanges, is discussed in Section 5.3.

5.2 Error Feedback Formulation

Stott & Alsac(1974) [1] and Chan & Brandwajn(1986) [2] describe transformertap adjustments as an error feedback formula

− =

(44)

where all quantities are expressed in per unit.e per unit voltage transformation ratio is computed as follows

= =

which simplies to

=

= =

where is the nominal voltage transformation ratio.erefore, Equation 44 is equivalent to

− = − or

− = −

in physical quantities. Solving for the updated transformation ratio yields

= + − (45)

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5.3 Controlled Bus Sensitivity 5 TCUL DEVICES

Computation of , the sensitivity of the regulated voltage to changes in

transformation ratio, is discussed in Section 5.3.

5.3 Controlled Bus Sensitivity

e proportionality constant (referenced in Equation 45 and Equation 42)represents the sensitivity of the controlled voltage to tap changes at the regulat-ing transformer. It commonly assumed that is one (the controlled voltage isperfectly sensitive to tap changes). An alternate assumption is that

= −1 (46)

where

is a row vector with 1 in the

ℎposition.

is a column vector with− inthe ℎ position and in the ℎposition.

Carrying out these matrix operations yields

= − −1 + −1 (47)

where

is an element from the bus susceptance matrix .−1 is an element from the inverse of . is magnitude of the transformer’s tap ratio.

Chan & Brandwajn(1986) [2] suggest that sensitivities computed from Equa-tion 47 are useful for coordinating adjustments to a bus that is controlled by several transformers.

5.4 Physical Constraints on Tap Settings

e preceding sections treat a transformer’s tap setting as a continuous quantity.However, tap settings are discrete. ere are three ways to handle discrete taps:

• Modify equations Equation 42 and Equation 45 so that the tap setting iscomputed discretely.

• Compute tap settings from Equation 42 or Equation 45. Round the tapsetting to the nearest discrete value each time the tap changes.

• Compute tap settings from Equation 42 or Equation 45. Round the tapsettings after the load ow has converged. Fix the transformer taps. Con-tinue until the solution converges with the discrete xed tap transformers.

17



6 TRANSFORMERS IN THE NODAL ADMITTANCE MATRIX

As was discussed in Section 2.3, the tap changing mechanism is limited in

its regulating abilities. Each TCUL device is subject to tap limits, i.e. maximumand minimum physically realizable tap settings. e implementation of Equa-tions 42, 43, or 45 should reect this fact. For example, Equation 43 should beimplemented as

= ⎧⎪⎨⎪⎩ if < if ≤ ≤ if >

(48)

6 Transformers in the Nodal Admittance Matrix

e complex nodal admittance matrix is referred to as . In the currentformulation, the admittances of are stored in rectangular form. e realpart of (the conductance matrix) is called . e imaginary part of (the susceptance matrix) is called . is formed according to two simplerules.

1. e diagonal terms of (i.e. ) are the driving point admittances of the network. ey are the algebraic sum of all admittances incident uponvertex . ese values include both the self admittances of incident edges(with respect to vertex ) and shunts to ground at the vertex itself.

2. e off-diagonal terms of

(i.e.

where

≠ ) are the transfer admit-

tances of the edge connecting vertices and .

Assuming that each transformer is modeled as two vertices (representing itsprimary and secondary terminals) and a connecting edge in the network graph,the procedure for incorporating a transformer into the nodal admittance matrixis as follows.

1. Compute the admittance matrix of the transformer (i.e. , , ,and ) at the nominal tap setting. See Section 4 for details.

2. Include as the transformer’s contribution to the self admittance of the

primary vertex (i.e. add it to

term

).

3. Include as the transformer’s contribution to the self admittance of thesecondary vertex (i.e. add it to term ).

18



8 INTEGRATING TCUL INTO FDLF

• Look up shunt elements (e.g. impedance loads) at the primary vertex.

Shunts must be refreshed from the vertex relation itself (unless the auxil-iary store is expanded to contain them).

• Recompute the self admittances of any transformers adjacent to the TCULdevice (incident upon its primary vertex).

e direct update strategy is examined more carefully. e basic computa-tional complexity stems from backing the old value of out of (the self admittance of vertex ). ere are two ways to proceed:

• Remember so that it can be backed out directly, or

• Recompute , then back it out of .

From Equation 27 or Equation 28, it is seen that the square of

(the

magnitude of the original tap ratio) is needed to recompute . e maintenance algorithm based on a direct update approach is

1. Determine the old , subtract it from .

2. Compute the new using Equation 27 or Equation 28, add it to .

3. Compute the new from Equation 32, set to this value.

4. Compute the new using Equation 36, set to this value.

Examining Equation 30, you should observe that is not a function of

the tap setting. erefore, it does not have to be updated when the tap changes. When does not change during the load ow (i.e. no phase angle regula-

tors), the terms, , , in Equation 32 and Equation 36remain constant throughout the load ow solution. If these terms are precom-puted and stored, a considerable reduction in oating point arithmetic results.Obviously, you pay a price in storage. Table 4 summarizes the FLOP versusstorage trade-offs associated with updating the diagonal of and computingEquation 32 and Equation 36 in the absence of phase angle regulators.

8 Integrating TCUL into FDLF

e rst two procedures guarantee that the system remains in a feasible stateas the load ow proceeds. e nal procedure only guarantees a feasible stateafter the load ow is solved. e literature suggests that the nal procedure ispreferred.

A fast decoupled load ow (FDLF) iteration consists of four basic steps:

20



8 INTEGRATING TCUL INTO FDLF

Table 4:

Maintenance Costs

Storage

Computational Approach FLOPS Required Assumed

DiagonalsRecompute 19 + 4 4Store 19 + 2 2 8

AnglesRecompute 19 4Store 9 4 16

is the number of TCUL devices.

is the size of a oating point number in bytes.

19 FLOPS computes , , and . Updates .9 FLOPS computes , , . Updates w/constant angle.2 FLOPS moves into or out of .

2 FLOPS recomputes from 2 .

1. Compute real power mismatches.

2. Update the voltage angles.

3. Compute reactive power mismatches.

4. Update the voltage magnitudes.

e tap adjustment algorithm is applied after the voltage magnitude update.Chan & Brandwajn(1986) [2] suggest that instability is introduced into the

solution by adjusting TCUL devices before the load ow is moderately con-verged. Operationally, moderate convergence is dened by starting criterionfor tap adjustments. e criterion is either:

• A minimum iteration count (no transformer taps are changed before thedesignated iteration) or

• A maximum bus mismatch (no transformer taps are changed unless themaximum reactive bus mismatch is less than the criterion).

Chan & Brandwajn(1986) [2] also suggest that an auxiliary solution be per-formed after the tap update. e tap adjustment creates an incremental reactivemismatch of − ( − ) at the transformer’s terminals.

21



REFERENCES REFERENCES

References

[1] B. Stott and O. Alsac, “Fast decoupled load ow”, IEEE Transactions onPower Apparatus and Systems , PAS 93, No. 3, pp. 859-869, 1974. 15

[2] S. Chan and V. Brandwajn, “Partial matrix refactorization”, IEEE Transac-tions on Power Systems , Volume 1, No. 1, pp.193-200, 1986. 15, 16, 20

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