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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
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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
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The swing equation
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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
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=
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”
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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
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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
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• 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
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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
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δ
Pe
Pm
Simulation tcl=0.05, 0.1 PW Example 11.5
δ2
δ1
δ0
Transient or large disturbance angle stability
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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
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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
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( )
( )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
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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
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• 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
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• 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
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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
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• 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
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• 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
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• 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
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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 …
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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
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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.
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