Topic 8 Generators Loads

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Power Systems Analysis ECNG 3012 8-Jan-May 2010

Synchronous Machines

Magnetic Field

DC field on the rotor.

Salient or round. Excitation.

Induced voltages on loops.

3! armature windings onstator.

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! The rotor is supplied by DC current If thatgenerates a DC flux "f .

! The rotor is driven by a turbine with a

constant speed of ns.

! The rotating field flux induces a voltage inthe stator winding. The frequency of theinduced voltage depends upon the speed.

! The frequency - speed relation is

f = (p / 2) ns = p ns / 2p is the number of poles.

! Typical rotor speeds are 3600 rpm for 2-pole, 1800 rpm for 4-pole and 450 rpmfor 16-poles.

N

S

A+

B+

C+B-

A-

C-

Flux " f ns

2





! The value of the induced voltages are:

where:

kw = 0.85-0.95 is the winding factor.

! At no load condition, the induced voltage is equal to the terminal voltage.

! When the generator is loaded, the induced voltage drives a current Ia through the

load.

! The load current produces a flux "ar that reduces the field flux.

! The armature generated flux has constant amplitude and rotates synchronouslywith the field flux.

3





! The armature flux induces a voltage Es in thestator winding.

! This voltage is subtracted vectorially from thefield induced voltage. The terminal voltage is :

Vt = Ef - Es .

! The Es voltage can be represented by an

equivalent armature reactance times thearmature current Es = Ia j X a

! The reactance is:

Xar = # Lar = # ("ar Na / Ia)

! The armature produced voltage is:

Es = Ia j Xar .

Armature

flux "ar

N

S

A+

B+

C+B-

A-

C-

ns

Flux " f

4





! The terminal voltage is:

V t = E f - E s = E f - I a j X ar

! The equation permits the development of an equivalent circuit, that

consists of a voltage source Ef and a reactance Xar connected in

series.

! The stator winding has resistance and some leakage inductance that isadded to armature reactance.

! The Xa r

+ X leakage is called synchronous reactance X syn.

5




Equivalent Circuit

Rs jXsy

IVt

Ef

Ef

VtI!

$

I R sI Xsy

6




Synchronous Reactance

! The synchronous reactance is given in percent xsyn. The ohm value is

calculated by:

Xsyn = xsyn_pu ( V2 /S)

where: V and S are the rated voltage in kV and MVA of the generator.

! Typically generators have several equivalent reactances. Thesynchronous reactance is used for steady-state analysis.

! Reactances are usually measured by manufacturer under standardtesting of the machine.

! The generator is classified as a synchronous machine because it isonly at synchronous speed that it can develop constant electromagnetictorque.

7





! A four pole, three-phase synchronous generator is rated 250 MVA, its

terminal voltage is 24 kV, the synchronous reactance is: 125%.

! Calculate the synchronous reactance in ohm.

! Calculate the rated current and the line to ground terminal voltage.

! Draw the equivalent circuit.

! Calculate the induced voltage, Ef , at rated load and pf = 0.8 lag.

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Generators and Power System Operation

!

A synchronous machine supplies an electric network with constantvoltage under steady state conditions.

! In a network several hundred synchronous generators operate inparallel.

!

Each generator operates with the same electrical speed.

! The generator speed is constant, at the synchronous speed determined

by the network frequency and the number of poles in the machine.

! The proper interconnection of a rotating machine with the network iscalled synchronization.

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! The terminal voltage in the machine is kept constant by the regulationof the field current. It regulates the reactive power as well.

! The load increase is achieved by increasing the input power (increases

the torque). The speed remains constant. Therefore increases theoutput electrical power of the generator (increases the power angle $).

! The synchronous generator starting torque is zero, the machine has tobe driven by a mechanical device (turbine, reciprocating engine, etc).

Generators and Power System Operation

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Transient Behaviour Generator

If a bolted three-phase fault at terminals:

! A fundamental frequency (which decays slowly).

! A d.c. offset decays exponentially.

e(t)

!

!

i(t)

iac(t)

idc

"t

i(t) = iac(t) + idc(t)

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12





I

"t

Subtransient Xd’’

Transient Xd’

Synchronous

" Rotor flux linkages vary then internal voltage source is not constant.

" Armature magnetic flux is initially forced to flow through high reluctancepaths.

" Complex effect on transient circuits (damper windings).

" Short circuit currents contain harmonics (different reactances).

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Fundamental or basic parameters – No directly determined.

Derived parameters – Observed behaviour as viewed fromterminals under suitable test conditions.

Standard parameters – Time constants and reactances.

Obtained from open-circuit and short-circuit tests on unloadedmachines (IEEE standard 115-1983).

Additional tests and analytical methods (e.g: enhanced sudden short-circuit tests, frequency-response tests, finite element analysis).

Rs

jX

IVt

Ef

Subtransient period (lasting for the firstfew cycles – rapid decay).

Transient period (lasting several cycles –slow decay).

Steady-state period (constant amplitude).

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Standard Parameter Hydro unit Thermal unit

Synchronous reactance Xd (p.u.) 0.6 – 1.5 1.0 – 2.3

Transient reactance X’d (p.u.) 0.2 – 0.5 0.15 – 0.4

Subtransient reactance X’’d (p.u.) 0.15 – 0.35 0.12 – 0.25

Transient time constant T’d0(s) 1.5 – 9.0 3.0 – 10.0

Subtransient time constant T’’d0 (s) 0.01 – 0.05 0.02 - 0.05

Stator leakage reactance Xl (p.u.) 0.1 – 0.2 0.1 – 0.2

Stator resistance Ra (p.u.) 0.002 - 0.02 0.0015 – 0.005

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Synchronous Generators - dq

" To facilitate describing synchronous machinecharacteristics, two axes are defined based on symmetry ofthe machine.! The direct (d) axis, centered magnetically in the center of the north

pole.! The quadrature (q) axis, 90 electrical degrees ahead of the d-axis.

" For purposes of analysis, the induced currents in the solid

rotor and/or damper windings may be assumed to flow intwo sets of closed circuits.! One set whose flux is in line with the d-axis.

! The other set whose flux is along the q-axis.

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Synchronous Generators - dq

Axis of phase a

Axis of phase b

Axis of phase c

Air gap

"

d-axis

q-axis

Stator

Rotor

Armature

winding

Field Winding

S

Rotor!r

N

a

a’

b

c

b’

c’

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Modelling Synchronous Generators

"b

"a

"c

ec

ea

eb

ia

ib

ic

b

a

c Axis of phase a

"

q-axis d-axis

ikq

ikd

Rotor Stator

Rotation

!r elec. rads/sec

18





" Dynamics of a synchronous machine is given by the equations of the

coupled stator and rotor circuits" Stator voltage and flux linkage equations for phase a (similar

equations apply to phase b and phase c)

ea = d!a/dt – Ra ia = p!a – Ra ia

!a = -laaia – labib – lac ic + lafd ifd + lakd ikd + lakq ikq

" The equations are complicated by the fact that the inductances arefunctions of rotor position and hence vary with time.

! The self and mutual inductances of stator circuits vary with rotor positionsince the permeance to flux path vary.

! The mutual inductances between stator and rotor circuits vary due torelative motion between the windings.

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" The dqo transformation, also called Park’s transformation,transforms stator phase quantities from the stationary abcreference frame to the dqo reference frame which rotateswith the rotor.

i

v

[L] i

v

[L]

[ T ]

[ T ]-1

Physical Domain D,Q,0 domain

ia

ibic

iabc=

id

iqi0

idq0=

Park’s Transformation

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Park’s Transformation [ T ]

cos# cos (# – 1200) cos (# – 2400)

sin# sin (# – 1200) sin (# – 2400)

1/! 2 1/! 2 1/! 2

2

3

[P] =

[P]-1 = [P]T

idiqi0

= [P]iaibic

vd

vq

v0

= [P]va

vb

vc

"d

"q

"0

= [P]"a

"b

"c


21





Park’s transformation•Defines sets of currents, voltages and flux linkages for three fictitious coils

•One stationary 0-coil•Two rotating coils: d-coil, q-coil

Rotate synchronism with the Rotor

Flux linkages with field and any otherwindings on the rotor are constant

22





#

R, Ldid

R f , Lff If

f

f’

R, Lq

iq

a-axis

c-axisb-axis

Directaxis

Quadrature Axis d & q windings are rotating

at the same speed as the rotor

•d winding and field winding act as two coupled windings•d & f do not couplemagnetically with q

0 sequence winding is stationary and not coupled.(under balanced conditions doesn’t carry current)

23





" The dqo system may be viewed as a means of referring the statorquantities to the rotor side.

" With the stator quantities expressed in the dqo reference frame. All

inductances are independent of rotor position (except for the effects ofmagnetic saturation).

" The mmf due to id and iq are stationary with respect to the rotor and

hence:

! Act on paths of constant permeance, resulting in constant self inductances(Ld, Lq ) of stator windings.

! Maintain fixed orientation with rotor circuits, resulting in constant mutualflux.

" The above simplify computation and analysis of results.

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P S t L d




Stable operation depends on ability to continuously match the electricaloutput of generating units to the electrical load of the system. Loadscharacteristics are influenced by changes in voltage and frequency.

Modelling of loads is difficult:

! Composition (lights, refrigerators, motors,furnaces, etc.)! Composition changes (time, weather, economy)

! Need to simplify.

! Represent a composite load characteristic as seen from bulk powerdelivery points (Includes loads, substation step-down transformers,subtransmission feeders, distribution transformer and feeders, etc.)

! Specific models for particular loads.

Power System Loads

P + jQ

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Daily Load Curve (Example)

• Load Factor• Diversity Factor

P + jQ

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Power System Loads

Load Models

Static Dynamic

• Described by algebraicfunctions

• Voltage and frequency

dependency represented aschanges in P and Q.

• Described by algebraic anddifferential equations

• Voltage and frequency

dependency represented aschanges in different

parameters.• Mostly used for loadcentered studies.

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Power System Loads - Static Models

Exponents a and b define characteristic

0 - constant power1 - constant current

2 - constant impedance

Voltage dependency modeled with exponential model

Composite loads

• a and b depend on the aggregate characteristics.• Typical values: 0.5<a<1.8, 1.5<b<6.• b non-linear.• Tables and references.• Common practice: a=1, b=2.

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ZIP Model(constant impedance,constant current andconstant power).

Voltage dependency modeled with polynomial model

Power System Loads - Static Models

0<Kpf <3.0; -2.0<Kqf <0

f is not a state variable (derivative of bus voltage angle)

Frequency dependency modeled by multiplying the model (exp. orpoly.) by a factor:

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Topic 8 Generators Loads