EIEN15 Electric Power Systems Lecture 1 Olof · PDF file• Electromagnetic dynamics at...

Synchronous generator dynamics

Olof Samuelsson

1 EIEN15 Electric Power Systems L8

Source: O. Elgerd Source: BPA

Outline – Swing equation

– Transient angle stability

– The Equal Area Criterion

– Small-signal stability


Synchronous Generator until now

• Steady state All generators run synchronously ωm ⇔ ωe ⇔ 50 Hz, Pm=Tmωm (think tandem bike) Pe=Pm ⇔ 0=Pm-Pe and Pe(E,V,Xd,δ) and Qe(E,V,Xd,δ)

• Electromagnetic dynamics at short-circuit Subtransient period during the first ms ⇔ X”d

Transient period during the following s ⇔ X’d

Steady state ⇔ Xd • Today electromechanical dynamics


The swing equation


J dωmdt

= Tm − TeGeneral torque balance for rotor

(Newton’s second law Ma=F)

p magnetic rotor poles ωm mech. rad/s( ) =2p

ωe elec. rad/s( )

Multiply torque balance by ωm

Divide by Sbase to get p.u.

Use ωm≈ωm,s

Use ωe as state ( ) ( )....2

,upPupP

dtdH

eme

se−=

ωω

The inertia constant H


=

12

Jωm2

Sbase

Kinetic energy of rotating masses

Generator MVA rating H=

Unit: Ws/VA=s Unit:

H on different MVA bases

– Machine base

»Steam turbines 4-9 s

»Gas turbines 3-4 s

»Hydro turbines 2-4 s

»Synchronous compensator 1-1.5 s

– Common base

»H ~ generator size (kW-GW)

»Infinite bus has infinite H fixed frequency (and phase)


Narrow range!

“Single Machine Infinite Bus”


X V∠0 E’q∠δ

”Classical model”:

•Swing equation for dynamics

•Fixed E’q behind X’d (Thévenin!)

•Constant Pm

•No damping, no saliency

X H, X’d

”Infinite bus” generator:

•Infinite H

•Fixed voltage V∠0

•Zero Thévenin impedance

Pm

”Classical” dynamic generator model


Synchronous generator connected to infinite bus:

• δ in rad, ωe in rad/s, ωe,s typically 100π rad/s

•E’q and X’d for slow transients in Pe (δ)

•Second order system with poor damping

•Electro-mechanical or “swing” dynamics

( )

−=

−=

see

eme

se

dtd

PPdt

dH

,

,

2

ωωδ

δωω

sin

delta

omegaPe

s1

s1

Pmax

wnom/2/HPm

Deviation from ωs,e

with V and X//X

Simulink model

Constant

To Workspace To Workspace

To Workspace

Integrators with Initial value

Two equilibrium points


• Synchronizing torque dPe/dδ

Reaction to disturbance

dPe/dδ>0 for δ<90° - stable equilibrium

dPe/dδ<0 for δ>90° - unstable equilibrium point (UEP)

δ

Pe

Pm

Two solutions for δ:

( ) δδδ sinsin'

maxPX

VEPP

eq

qem ===

−°

==

0

max0

180

arcsin

δ

δδ PPm

Dynamic response


δ

Pe Temporary short-circuit near generator, Pe zero during fault Response?

1. Second order system

2. No damping

3. Oscillator! δ and ω oscillate

4. δ(t) will lag ω(t)

Small disturbance sinusoids linear model OK

Pm

PW Ex 11.5 tcl=0.05 s

Pe=0 at short-circuit near gen (source feeds just X Q)

Step in Pm-Pe

Mechanical states slow

Start at δ0 and Pe(δ0)

Acceleration during fault

Fault removed at δ=δ1=clearing angle

Overshoot to δ2 and Pe(δ2)

Oscillate around equilibrium δ0 so Pe(δ0)=Pm

Second order response


δ

Pe

Pm

Simulation tcl=0.05, 0.1 PW Example 11.5

δ2

δ1

δ0

Transient or large disturbance angle stability


δ0 must be less than steady state limit 90º

δ2 also has limit – transient angle stability limit

Questions:

How large can δ2 be?

What happens when it becomes too large?

What is the largest disturbance that is OK?

PW

The Equal Area Criterion


δ

Pe

Pm

δ2

δ1

δ0

AA

DA Short-circuit: Pe=zero

Mark areas between Pe(δ) and Pm in interval δ0 to δ2

Accelerating Area: Below Pm

Decelerating Area : Above Pm

For stable system AA=DA

EAC derivation

• Textbook 11.3


( )

( )dtdPP

dtd

dtdH

dtdPP

dtd

dtdH

PPdtdH

emes

emes

emes

δδω

δδδω

δω

−=

−=

−=

2

,

2

2

,

2

2

,

2

2

2

( )

( )

( ) ( )

( ) ( ) DAdPPdPPAA

dPPdPP

dPPdtdH

dPPdtddH

meem

emem

emes

emes

=−=−=

=−+−

−=−=

−=

∫∫

∫∫

∫

∫∫

2

1

1

0

2

1

1

0

2

0

2

0

2

0

2

0

0

002

2

2

,

2

,

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δδ

δδ

δδω

δδω

Trick1: multiply with dδ/dt

Trick2: multiply with dt

Integrate both sides over relevant δ range

LHS: Second order time derivative!

Solve LHS

Split δ range

Make integrals equal

Transient stability limit


δ

Pe

Pm

δUEP δ1 δ0

AA DA

More severe disturbance:

Greater clearing angle δ1 makes AA larger

Greater DA makes δ2 larger

Maximum AA and DA when

δ2=δUEP=180º-δ0

then δ1 has its maximum value δ1 = δccl Critical clearing angle

For larger δ2 it is AA that grows…

PW

δ2

UEP=Unstable Equilibrium Point

Beyond stability limit


• dω/dt never becomes zero

• Rotor accelerates even more

• Machine transiently unstable = loses synchronism

• Must disconnect and resynchronise

Demo

Typical EAC scenarios

• Short-circuit at bus (Pmax2=0), self-extinguishes (Pmax3=Pmax1) • Short-circuit on line near bus (Pmax2=0), line trip (Pmax3<Pmax1) • Short-circuit on line (Pmax2>0), line trip (Pmax3<Pmax1)

• Always: Determine Pmax for each stage • With fault on line (not at/near bus) – use Thévenin equivalent


Pmax3=Pmax1 Pmax1 Pmax1

Pmax3

Pmax2 Pmax2

Pmax3

Pmax2 Pm Pm Pm

Transfer capacity

• Total transferred MW on the lines = Pm

• High line transfer ⇔ efficient use of lines, network and system • But higher Pm gives greater AA and smaller DA • Transient stability thus limits line transfer just like voltage

stability and the steady state limit (max angle separation 90°) 18 EIEN15 Electric Power Systems L8

Pmax1 Pmax1 Pmax1

Pmax3

Pmax2 Pmax2

Pmax3

Pmax2

Pm Pm Pm

a. Pe=0 at fault, no line lost b. Pe=0 at fault, one line lost Worse than a.

c. Pe>0 at fault, one line lost Better than b.

Use of Equal Area Criterion


• Stability check for known disturbance Use EAC for δ2 and check δ2<δUEP

• Max disturbance from stability limit Determine disturbance for δ2=δUEP

• The clearing angle corresponds to the fault clearing time where relay protection delay is central

• But EAC calculations do not include time

• Time simulations! (textbook shows approximation)

Stability analysis tools


Analytical – the Equal Area Criterion

• Simple, can be done by hand, but approximate

• Formulated before 1930 by Ivar Herlitz, KTH (First Swedish PhD in engineering)

Time simulation

• Computer application since the beginning

• Voltages and currents as phasors or waveforms

• Multi-machine model with Differential Algebraic Equations

• Set of Differential equations for each generator

• Power flow for Algebraic network equations

Small-signal angle stability


• Also small disturbance angle stability linear model OK

• Linearize at steady state (δ0, ω0, Pm0)

• State space: dxd/dt=Aodexd+Bu

• Compute eigenvalues λi of Aode

• Compute right and left eigenvectors Φi and Ψi of Aode

• Applies also to multi-machine models

• Popular and powerful application of control theory

EIEN15 Electric Power Systems L8

• 37 generators • ≈590 dynamic states xd

• 202 network nodes • ≈810 algebraic states xa • xd=Aodexd… Aode≈590x590 • =Adae … Adae≈1400x1400

DAE matrix for Icelandic system

22

Blue dot = non-zero element

a

d

xx

0dx

Eigenvalues and eigenvectors


• Eigenvalue λi (stable if in Left Half Plane)

Im(λi)=resonance oscillation frequency (e.g. 0.35 Hz)

Re(λi)=resonance oscillation damping ≤0 for all λi system is small-signal stable >0 for any λi system is small-signal unstable

• Right eigenvector Φi:

Which generators participate in mode (resonance) i

E.g. Generators in Finland against those in NO and DK

• Low >0 for uncontrolled system

• Negative damping from controllers

• Automatic Voltage Regulators

• HVDC controllers

• Damping added by dedicated controls

• Power System Stabilizers (PSS) on generator

• Power Oscillation Damper (POD) on HVDC or FACTS

FACTS=MW size power electronic devices

Small-signal damping


Conclusions • On machine base inertia constant H is typically … to … • Infinite bus means a generator with … • Simplest dynamic model of synch. generator has … dyn. states • Loss of transient angle stability = loss of … • Analytical method to assess transient angle stability is…, where

…area and …area being … means stable system • The areas are defined by different P(δ) and relevant values of … • A nonlinear model is needed for … angle stability, while a

linearized model is used for … angle stability • For angle stability an eigenvalue tells … and the corresponding

eigenvector tells …


Conclusions • On machine base inertia constant H is typically 1 to 9s

• Infinite bus means a generator with infinite H (fixed speed)

• Simplest dynamic model of synch. generator has 2 dyn. states

• Loss of transient angle stability = loss of synchronization

• Analytical method to assess transient angle stability is EAC, where accelerating area and decelerating area being equal means stable system

• The areas are defined by different P(δ) and relevant values of Pmax

• A nonlinear model is needed for large disturbance/transient angle stability, while a linearized model is used for small signal/disturbance angle stability

• For angle stability an eigenvalue tells frequency and damping of oscillation and the corresponding eigenvector tells oscillation phase and amplitude of each generator


About textbook sections 11.4-11.8 • 11.4 delta(t) and omega(t) requires integration of differential

equations, as treated in course on numerical analysis. • 11.5 Simulation of larger systems includes solving both

generator and network equations. This is shown in the Icelandic example.

• 11.6 L7 and pm for lab2 shows equations for salient poles. Here the corresponding dynamic model is shown.

• 11.7 Wind turbines use synchronous or asynchronous generators combined with power electronic converters. The resulting structures are shown here.

• 11.8 Transient stability may limit transfer capacity. But if the affecting factors are known, this can be used to increase transfer capacity as exemplified here.


EIEN15 Electric Power Systems Lecture 1 Olof · PDF file• Electromagnetic dynamics at...

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