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

2 EIEN15 Electric Power Systems L8

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

3 EIEN15 Electric Power Systems L8

The swing equation

4 EIEN15 Electric Power Systems L8

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

5 EIEN15 Electric Power Systems L8

=

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)

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Narrow range!

“Single Machine Infinite Bus”

7 EIEN15 Electric Power Systems L8

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

8 EIEN15 Electric Power Systems L8

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

9 EIEN15 Electric Power Systems L8

• 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

10 EIEN15 Electric Power Systems L8

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

11 EIEN15 Electric Power Systems L8

δ

Pe

Pm

Simulation tcl=0.05, 0.1 PW Example 11.5

δ2

δ1

δ0

Transient or large disturbance angle stability

12 EIEN15 Electric Power Systems L8

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

13 EIEN15 Electric Power Systems L8

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

14 EIEN15 Electric Power Systems L8

( )

( )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

15 EIEN15 Electric Power Systems L8

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

16 EIEN15 Electric Power Systems L8

• 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

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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

19 EIEN15 Electric Power Systems L8

• 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

20 EIEN15 Electric Power Systems L8

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

21 EIEN15 Electric Power Systems L8

• 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

23 EIEN15 Electric Power Systems L8

• 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

24 EIEN15 Electric Power Systems L8

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 …

25 EIEN15 Electric Power Systems L8

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

26 EIEN15 Electric Power Systems L8

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.

27 EIEN15 Electric Power Systems L8