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EE532 Power System Dynamics and Transients

EUMP Distance Education Services

- Satish J Ranade
- Synchronous Generator Model
- Lecture 12
- Modified for EE531

Topics

- Modeling of synchronous generators
- Modeling machine in more detail
- Results from more detailed modeling

Approaches

- Fields Approach Coupled Coil Model
- Park/Kron/Blondel
- Transformation
- Two reaction theory Transient Studies
- Phasor Model
- Linearized Model
- Steady State Models Stability Studies

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

pd()/dt

? flux linkage ( Kundur Text)

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Kundur notation Note fd, kd, kq subscripts Note

ia, ib, ic now come out of stator coils

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Ignore Dampers

Will fold negative sign on ia ib ic into R and

L Will suppress time variable. Remember

everything Is instantaneous value here

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Laa, etc. are functions of Rotor position ? ?

??dt so T and the inductance terms are

Functions of time

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Inductance Matrix Stator phase a Laa(?)

LalLgoLaa2 cos (2 ?) Round rotor Laa2 0

Leakage

Field q Axis

?

Field d Axis

a

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Phase a Axis

Field q Axis

Inductance Matrix Mutual Inductance

a Lab(?) -(1/2)Labo-Lab2 cos (2

?-2p/3) Round rotor Lab2 0

?

Field d Axis

a

Leakage

Field q Axis

?

Field d Axis

a

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Phase a Axis

Inductance Matrix Mutual Stator phase a-

Field Lafd(?) Lafd cos ( ?)

Field q Axis

a

Field d Axis

Field q Axis

Field d Axis

a

Maximum

Minimum

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Phase a Axis

Inductance Matrix Field Self Lffd(?) Lffd

Field q Axis

a

Field d Axis

Field q Axis

Field d Axis

a

Maximum

Minimum

Modeling of synchronous generators- Round Rotor

L(?)

Modeling of synchronous generators

L(?)

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Ignore Dampers

Will fold negative sign on ia ib ic into R and

L Will suppress time variable. Remember

everything Is instantaneous value here

Modeling of synchronous generators

dqo transformation

Field current ifd is not transformed Io

(iaibic)/3 is the usual zero sequence current

Modeling of synchronous generators

dqo transformation

Direct axis component of stator currents

Quadrature axis component of stator currents

Modeling of synchronous generators

dqo transformation interpretation of dq currents

? is an arbitrary variable ( reference

frame) If Stator currents are balanced three

phase positive sequence

Note unconventional convention

Im peak value of line current

Modeling of synchronous generators

dqo transformation interpretation of dq currents

d

d

d

a

a

a

q

q

q

t0

t2p/3?

t4p/3?

Three phase statorbal.pos.seq. currentgt Uniform

rotating field

Modeling of synchronous generators

d

d

d

a

a

a

q

q

q

dqo transformation interpretation of dq currents

A rotating magnetic field can be created by a

two-phase stator -- coils placed 90 deg

apart -- currents 90 deg out of phase --

simplifies analysis but also practical

Modeling of synchronous generators

dqo transformation interpretation

Replaces 3 phase stator by a 2 phase stator

zero sequence circuit

d

d

a

a

q

q

Modeling of synchronous generators

dqo transformation interpretation

Transformation to synchronously rotating

reference ??st ?o

d

?s

d

?s

?s

q

q

2 phase stator rotating At ?s dc

Sationary 3 phase stator balanced positive

sequence current at frequency ?s

dc!

Modeling of synchronous generators

dqo transformation

dqo inverse transformation

Same transformations for voltage and flux

Modeling of synchronous generators

dqo transformation transformed machine

equations

Similar Notation for voltages and flux

eabc R iabc p?abc ?abc L iabc

Modeling of synchronous generators

dqo transformation transformed machine

equations

eabc R iabc p?abc ?abc L iabc

T(?)eabc T(?) R T-1(?) T(?) iabc

T(?) pT-1(?) T(?) ?abc edqo T(?) R

T-1(?) idqo T(?) pT-1(?) ?dqo ?dqo T(?)

L T-1(?) idqo

Modeling of synchronous generators

dqo transformation transformed machine

equations

edqo T(?) R T-1(?) idqo T(?)

T-1(?) p?dqo T(?) pT-1(?) ?dqo edqo

R idqo p?dqo T(?) pT-1(?)

?dqo ?dqo T(?) L T-1(?) idqo

R is diagonal transformer voltage derivative

of flux Speed voltage

pT-1(?) ? dT-1(?)/d ?

Modeling of synchronous generators

dqo transformation transformed machine

equations

pT-1(?) ? T(?) L T-1(?) Ldqo ?

Modeling of synchronous generators

dqo transformation transformed machine

equations

T (?)pT-1(?) ?r T (?)d(T-1(?))/d ??

- Effect of rotation (generated voltage) captured

by voltage sources - q axis flux linkage induces speed voltage in d

coil - d axis flux linkage induces speed voltage in q

coil

Modeling of synchronous generators

dqo transformation transformed machine

equations

Modeling of synchronous generators

dqo transformation transformed machine

equations

The dqo transformation simplifies machine

equations by transforming to a rotating reference

frame on the rotor The three stator coils are

replaced by a pair of orthogonal d-q coils and a

zero sequence coil Result Inductances are no

longer time varying Coil- coil coupling

simplified - no coupling between d and q

Modeling of synchronous generators

dqo transformation transformed machine

equations

Next Understanding transformed model Using

model for simple transients Steady state

model Slowly varying phasor model for stability

EE532 Power System Dynamics and Transients

EUMP Distance Education Services

- Satish J Ranade
- Synchronous Generator Model
- Lecture 14

Modeling of synchronous generators

dqo transformation transformed machine

equations

Understanding transformed model Using model for

simple transients Steady state model Slowly

varying phasor model for stability

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Kundur notation Note fd, kd, kq subscripts Note

ia, ib, ic now come out of stator coils

Modeling of synchronous generators

Circuits approach Using these principles

develop a model for the generator

Ignore Dampers

Will fold negative sign on ia ib ic into R and

L Will suppress time variable. Remember

everything Is instantaneous value here

Modeling of synchronous generators

L(?)

Modeling of synchronous generators

Simulation of original model

Modeling of synchronous generators

dqo transformation Simulation of original model

Modeling of synchronous generators

dqo transformation Simulation of original model

Modeling of synchronous generators

dqo transformation Simulation of original model

Modeling of synchronous generators

dqo transformation Simulation of original model

Field Current

Time, S

Ia

Modeling of synchronous generators

dqo transformation

dqo inverse transformation

Same transformations for voltage and flux

Modeling of synchronous generators

dqo transformation transformed machine

equations

Modeling of synchronous generators

dqo transformation transformed machine

equations

The dqo transformation simplifies machine

equations by transforming to a rotating reference

frame on the rotor The three stator coils are

replaced by a pair of orthogonal d-q coils and a

zero sequence coil Result Inductances are no

longer time varying Coil- coil coupling

simplified - no coupling between d and q

Modeling of synchronous generators

dqo transformation transformed machine

equations

The dqo transformation simplifies machine

equations by transforming to a rotating reference

frame on the rotor The three stator coils are

replaced by a pair of orthogonal d-q coils and a

zero sequence coil Result Inductances are no

longer time varying Coil- coil coupling

simplified - no coupling between d and q

Modeling of synchronous generators

dqo model simulation

The dqo transformation simplifies machine

equations by transforming to a rotating reference

frame on the rotor The three stator coils are

replaced by a pair of orthogonal d-q coils and a

zero sequence coil Result Inductances are no

longer time varying Coil- coil coupling

simplified - no coupling between d and q

Modeling of synchronous generators

dqo transformation Simulation

Modeling of synchronous generators

dqo transformation Simulation

Modeling of synchronous generators

dqo transformation Simulation-Compare abc

solution

If

Id

Iq

Modeling of synchronous generators

dqo transformation Simulation

Modeling of synchronous generators

dqo transformation Simulation

Ia

Ib

Ic

EE532 Power System Dynamics and Transients

EUMP Distance Education Services

- Satish J Ranade
- Synchronous Generator Model
- Lecture 16

Modeling of synchronous generators

dqo transformation transformed machine

equations

Understanding transformed model Using model for

simple transients Steady state model Slowly

varying phasor model for stability

Modeling of synchronous generators

dqo transformation transformed machine

equations

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

io 0 ?r ?s gt d, q quantities are dc gt

derivative terms 0

ed -Ra id ?r ?q ?d -Ld id Lad ifd eq

-Ra iq ?r ?d ?q -Lq iq efd Rf if

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

jEqEt Ra (id j iq) -Xq iq j Xd id

Eq/dEt /0 Ra (id j iq) /d-90o jXq iq /d j

Xd id /d-90o

q

Im

id It sin (diF) iq It cos (diF)

Eq Eq/d

-Xq iq Ra id

Id id /d-90o Iq iq /d It Id Iq Id It

sin (diF) /d-90o Iq It sin (diF) /d

jXq Iq

d

Iq

Ra iq Xd id

d

Re

jXd Id

Et

It

Id

Ra It

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Final Model Eq/dEt /0 Ra Ia jXq Iq j Xd Id

q

Im

Eq Eq/d

jXq Iq

Iq

d

d

Id id /d-90o Iq iq /d

Re

jXd Id

f

Et

Ia

Id

Ra Ia

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Final Model Eq/dEt /0 Ra Ia jXq Iq j Xd Id

Example Revisited ( Kundur Ex 3.2) 555 MVA, 24

kV, 0.9pf operating at rated conditions Saturated

values of parameters ( includes leakage) in

pu Xad1.386 Xaq 1.344 Xd 1.536 Xq

1.494 Find all quantities

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Example Revisited ( Kundur Ex 3.2)

To find Eq we need Iq and Id this in turns

requires d

Eq/dEt /0 Ra Ia jXq Iq j Xd Id

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Example Revisited ( Kundur Ex 3.2)

To find Eq we need Iq and Id this in turns

requires d

Eq/dEt /0 Ra Ia jXq Iq j Xd Id

Trick, Define

Eq/d Et /0 Ra Ia jXq Iq j Xq Id Et /0

Ra Ia jXq Ia

q

Im

Eq Eq/d

Eq/d is a vector at angle d!

Eq/d

jXq Iq

Find Eq/d first

Iq

d

d

Re

f

Et

jXd Id jXqId

Ia

Id

Ra Ia

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Example Revisited ( Kundur Ex 3.2)

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Example Revisited ( Kundur Ex 3.2)

Now find Id and Iq

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Power and Torque

S Et It Electrical (edjeq)(id-jiq)

(edid eqiq) j(eqid-ediq) T ?d iq

?q id P (1/w) Ra It2 Mechanical

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Alternate power formula( machine to infinite bus)

It Id Iq Make q axis the referenceRa0 Eq

Etcos(d)IdXd 0 Et sin d-XqIq Id -j

(Eq-Etcos d )/Xd Iq Etsin d /Xq It

Etsin d /Xq -j (Eq-Etcos d )/Xd S Et It

q

Im

Eq Eq/d

-Xq iq Ra id

jXq Iq

d

Iq

Ra iq Xd id

d

Re

jXd Id

Et

It

Id

Ra It

Modeling of synchronous generators

Synchronous Steady state model, balanced operation

Alternate power formula( machine to infinite bus)

It Etsin d /Xq -j (Eq-Etcos d )/Xd S Et

It P (EtEq /Xd)sin d (Et2/2)((1/Xq)-(1

/Xd) sin(2 d)