Low Frequency Oscillations in Power Systems #2
Nonlinear System ViewJames D. McCalley
Harpole Professor of Electrical and Computer Engineering
Nonlinear System View
p p g g
Hugo Villegas
Iowa State UniversityConference on Power System Oscillations
XM S.A. E.S.P., Medellin, Columbia
11
Wednesday July 14, 2010
Overview of this seminar1 Th C l bi P S t O ill ti1. The Columbian Power System Oscillation2. Model building:
Governor Actuator Interconnected network Hydro turbine Multiple generators Generator dynamics
3. Nonlinear effects4. Model building: nonlinear effects5. Model building:
Voltage regulators and e citation Voltage regulators and excitation Load modeling
6. Simulation illustrationbl l h7. Possible solution approaches
8. Summary and what’s next…2
Case: The Colombian Power System ObservationObservation
Frequency oscillation in the Colombian Power System seen in August the 12th of 2008. Courtesy of XM Colombia.
3
Case: The Colombian Power System ObservationObservation
First Oscillation Sampled (green): 9*4=32s.
S d O ill ti S l d (T ) 8*4=32 Second Oscillation Sampled (Tan): 8*4=32s.
Third Oscillation Sampled (Light green): 5*4=20s.
In the plot the oscillation period of 20 sec is repeated often In the plot, the oscillation period of 20 sec is repeated often. Yet, there are still excursions having a period of 32s. Thus, the observed oscillation ranges from 0.03Hz to 0.05Hz approximately. g pp yIt is not fixed at 0.05Hz.
The signal is negatively damped at the beginning of the observation. l d d f b l dIt is positively damped after system stabilization process occurred.
The signal in the plot is bounded such that
A b d d ill i i i f li i l
59.1 ( ) 61.1 [ ]f t Hz
A bounded oscillation is suggestive of a limit cycle….
4
Limit Cycles:“ l ll “For a linear time-invariant system to oscillate, it must have a pair of eigenvalues on the imaginary axis, which is a non robust condition In real life stable oscillations a non robust condition…. In real life, stable oscillations must be produced by nonlinear systems. There are nonlinear systems that can go into an oscillation of fixed y gamplitude and frequency, irrespective of the initial state [*]. ” This is called a limit cycle.
Such a phenomena may be occurring in the governor-actuator loop, and if so, its observed effect in the power
h l ld h b ll system (as in the previous plot) could exhibit small changes in frequency and amplitude.
5 [*] Khalil Hassan, “Nonlinear Systems,” Prentice Hall, 1996-2002
Some helpful references for the understanding of low frequency oscillationsq y
1. Anderson P., Fouad, A., “Power System Control and Stability,” IEEE series, 2003.
2. Merritt Herbert E., “Hydraulic Control Systems,” Wiley, 1967.
3. Krause Paul C., “Analysis of Electrical Machinery,” McGraw-Hill, 1986.
4. Khalil Hassan K., “Nonlinear Systems,” Prentince Hall, 2002.
5 Tohru Katayama “Subspace Methods for System Identification ” Springer 20055. Tohru Katayama, Subspace Methods for System Identification, Springer, 2005.
6. Tao Gang, “Adaptative control of Systems with Actuator and Sensor Nonlinearities,” Wiley, 1996.
7. Bevrani Hassan, “Robust Power System Frequency Control,” Springer, 2009.
8. Atsushi Izena, Hidemi Kihara, “Practical Hydraulic Turbine Model”, Toshikazu Shimojo, Kaiichirou Hirayama.
9. Taylor C., Lee K. Y., “Automatic Generation Control Analysis with governor Deadband Effects,” IEEE transactions, Vol. PAS-98, 1979.
h h h “ l l l h h b d d d10. Triparthy S.C., Bhatti T.S. Tha C.S., “Sample Data Automatic Generation Control Analysis with reheat Steam Turbines and Governor Dead-Band Effects,” IEEE Transactions, Vol. PAS-103, 1984.
11. Kamwa I, Grondin R, Trudel G, “IEEE2B Versus PSSS4B: The Limits of Performance of Modern Power System Stabilizers,” IEEE Transactions, VOL. 20, May 2005.
12. Youzhong Miao, Tao Wu, Jiayang Guo, Qunju Li, Weimin Su and Yong Tang, “Mechanism Study of Large Power Oscillation of Inter-area Lines Caused by Local Mode,” 2006 International Conference on Power System Technology.
13. Seok-Yong Oh and Dong-Jo Park, Member, “ Design of New Adaptive Fuzzy Logic Controller for Nonlinear Plants with Unknown or Time-Varying Dead-zones,” IEEE Transactions on Fuzzy Systems, Vol. 6, No. 4, Nov. 1998.
14. Mahmut Erkut Cebeci, , Osman Bülent Tör, , Arif Ertaş, “The Effects of Hydro Power Plant’s Governor Settings on the stability of Turkish Power System Frequency.”
15. P. Vuorenpää, T. Rauhala, P. Järventausta, “ Effect of Torsional Mode Coupling on TCSC Related Subsynchronous Resonance Studies.”
16 Li Zh Y Li Mi h l R I D l T B h St E k d M i L C “B lk P S t L F O ill ti 16. Li Zhang, Y. Liu, Michael R. Ingram, Dale T. Brashaw, Steve Eckroad, Mariesa L. Crow, “Bulk Power System Low Frequency Oscillation Suppression By FACTS/ESS.”
17. Youzhong Miao, Tao Wu, Jiayang Guo, Qunju Li, Weimin Su and Yong Tang, “Mechanism Study of Large Power Oscillation of Inter-area Lines Caused by Local Mode.”
18. S.K.Soonee, Vineeta, Agrawal Suruchi Jain, “Reactive Power and System Frequency Relationship: A Case of Study,” at http://www.nrldc.org/docs/Reactive%20Power%20&%20Freq_relationship_CBIP.pdf.
19. R. Grondin, (U) I. Kamwa, (M) L. Soulieres, (M) J. Potvin, (M) R. Champagne, “An Approach to PSS Design for Transient Stability Improvement through Supplementary Damping of the Common Low-Frequency,”
20. F. Schleif and A. Wilbor, “The coordination of hydraulic turbine governors for power system operation,”
21. IEEE Transactions on Power Apparatus and Systems, Vol. PAS-85, No. 7, July 1966.6
Highlights from previous studies
(8) Model and parameters of a hydraulic system and turbine characteristics strongly influence the power system frequency stability.
(13) Dead-zone characteristics are included (typical characteristic) in many practical plants such as electric servo-motors and hydraulic servo-valves. They are usually unknown and, moreover, may vary with time.
(16) Generators from different dynamic groups swing against each other in the ( ) y g p g gpower system 0.1 2.5Hz.
(17) Mechanism of large power oscillations of inter-area lines can be caused by local mode. One generator oscillating in one area influences the rest of the system.y
(18) Studies show that relation of the reactive power with respect to frequency is negative.
(19) Damping through a supplementary governor control at a large hydro-generator unit. Fast valving - surge pressures. PSSs help the damping of low g g g p p p gfrequencies.
(20) Reports the phenomenon that oscillations can be caused by delay introduced by governor dashpot action (backlash). They indicate that this problem is “most sensitive to servosystem delay” and that it “may be either aggravated or y y y ggsuppressed by adjustment of servo-system gain.”
7
Modeling: The Power Plant Overviewg Primary Loop Control: Speed Governor, actuators, turbine-generator dynamics, local
sensors.
• Secondary Loop Control: Generation Control System (AGC), area frequency sensing, telemetry.
8
Modeling: Swing Equation – Rotor Dynamics
T
-Te, ωePower system dynamics is largely
JTm, ωm concerned with the
rotation of a mass. J
From Newton’s second law for the rotation
2H d T t T t
From Newton s second law for the rotation of a mass with inertia J we can derive:
, ,e m pu e pub
T t T tdt
Torque imbalances produce acceleration and deceleration.
9
We want to control this.
Modeling: Primary Control LoopModeling: Primary Control Loop The objective of the primary control is to maintain the speed of
the rotating mass close to its nominal value.
There are two main components to observe in the primary loop l h ll d hcontrol: the controller and the actuator.
Controller (governor): processes signals from the turbine-generator and sends back commands to maintain the speed within generator and sends back commands to maintain the speed within the desired boundaries. If the speed deviates, the governor repositions the valve or gate that injects steam or water.g
Actuator (power amplifier): Because the governor does not have the necessary power to move large gates or valves, it requires
li f h l i l f h ll coupling to transform the low power signal of the controller to a high power signal. 10
Modeling: The Controller – Speed Droop P i C t lPrimary Control:-The Unit is operating at conditions (ωb,Y1)-With a load increase, the new steady-state operating point is (ω, Y) ω<ωb and Y>Y1.
1YR
1R
p g p ( )
-This is done by the governor action.-The new frequency ω is different than ωb.
Secondary Control:
' 0r r e -Yref is controlled till the error in system frequency becomes zero.-Then the new operating point (ωb Y2) is reached(ωb,Y2) is reached.-This is done by the AGC.- Only when Steady State is reached do we have ωr=ωb.
Note: Y can be Power or Gate Position
11 R is the droop characteristic.
Model building: adding governor
iPID d
kC s k k s
It is common to use PID controllers in the governor loop (CPID(s))
PID p dC s k k ss
Water /Steam
The objective of the controller is to keep e=0. This guarantees that Y is set in the
)(sCPID
12
guarantees that Y is set in the desired position.
Modeling: ActuatorA l ll d fi l l d i ( l ) Actuator couples controller and final control device (gates or valves).
The actuator usually uses hydraulic energy. The most basic actuator is the double effect cylinder operated by a servo valve. y p y .
Basic idea: A small energy “input” displacement v(t) results in a large energy “output” displacement y(t).
Give a small linear displacement v(t) to the right, then the spool moves to the right letting flow at pressure P to be routed through inlet A
hich ca ses mass M to mo e (t) linear nitsspool
which causes mass M to move y(t) linear units. Note: The force needed in v is small, but the force obtained in y is big. More details are in [*].
13[*] Merritt Herbert E., “Hydraulic Control Systems,” Wiley, 1967.
Modeling: Actuator
The p.u. closed-loop representation of this actuator is below, which enables it to position the output as indicated by reference input r(t) enables it to position the output as indicated by reference input r(t).
It is important to point out that the servo system has limits in both speed and displacement. These are nonlinearities.p p
14
Modeling: Actuator in-the-loopg p
Water Spraying
15Note: The process is similar for other turbine types.
Model building: Adding actuator & governorg g g
With the previous considerations, our power plant in a reduced fashion ispower plant in a reduced fashion is modeled as below.
ip d
kk k s p ds
16
R is the speed droop.
Modeling: Net torquesg qTm,1Te Tm,2 Tm,3 …. Tm,nFor a network of N
generators and M
Hequ
gbuses, we conceive of an aggregated
pu puP Tprime mover and generator having
i ti H
Because ∆f is always relatively small, we may approximate that
, ,
2 N Mequ
e mi pu ei pu
H d P t P tdt
an inertia Hequ
, ,1 1
p pi ib dt
2H d
17
, , ,
2 eque M pu L pu tie pu
b
H d P t P t P tdt
Modeling: Net torques & load damping
HequPM Ptie
ωe
2 equH d P t P t P t
HequPM Ptie
PL
, , ,q
e M pu L pu tie pub
P t P t P tdt
A t t f th l d i t i tA great percentage of the load in a power system is motors. Thus, small excursions of frequency changes the electric load. Accounting for this, and then linearizing for small excursions
2 dH P t P t D P t
and per-unitizing the angular frequency, we obtain:
, , , , ,2 equ e pu M pu L pu e pu tie puH P t P t D P tdt
18
Modeling: Net torques & frequency
f • Using the fact that:
2H s f s P s P s D f s P s
, ,e pu e puf • And taking the Laplace transform:
, , , , ,2 equ e pu M pu L pu e pu tie puH s f s P s P s D f s P s
• Solving for Δf(s):1 , , , ,1
2e pu M pu L pu tie puequ
f s P s P s P sH s D
19
Modeling: Frequency control in an interconnected networkinterconnected network
Control A 1
Control A 2Ti Li 12
Active Power Flow from Area 1 to Area
1 2,12 1 2sintie
VVPX
Area 1[f1,θ1,V1]
Area 2[f2,θ2,V2]
Tie Line 12 from Area 1 to Area 2 can be written as:
l f 12XGeneralizing from Area i to Area j :In a multiple Area
Power System: sini jVVP
VV
A1 A2
Linearizing:
, sintie ij i jij
PX
, cosi jtie ij io jo i j
ij
VVP
X A3
A4
A5
Synchronizing Power C ff
)cos( 00 jiji
sijVV
T Coefficient
20
)cos( 00 jiij
sij X
)(, jisijijtie TP
Modeling: Frequency Control in an
We know that: Taking the Laplace transform:
Interconnected Network
ee
ss
s
g p
e eddt
C ti f d/ t H d iti i
)(, jisijijtie TP ))()((, sss
TP ji
sijijtie
Converting from rad/sec to Hz, and per-unitizing:
,, , , ,
2 b sij putie ij pu i pu j pu
f TP s f s f s
s
s
21
Modeling: Frequency control in an i d kinterconnected network
Focusing on area j only, it can be shown by induction that:g j y, y
N
i
N
ipujpusijpuipusij
biputie sfTsfT
sfsP
1 1,,,,,,
2
Representing in block Diagram:
ij ij
22
Model building: Adding network frequency controlfrequency control
This is our generator model. We still need to consider turbine dynamics.
23
Modeling: Hydro-turbine Modelsg y
Impulse Turbines (high-head, low speed, H>260m):
-PeltonTurbines
Reaction Turbines (low head high speed H=6m head, high speed , H=6m –244m):
-Francis
- Kaplan
-Deriaz (Modified Kaplan)phttp://electricalandelectronics.org/2008/09/24/constituents-of-hydro-electric-plant/\http://www.tfd.chalmers.se/~hani/phdproject/francispicture.gif
24
Modeling: Hydro-turbine Models
When water is not in motion the pressure at
Surge TankHo
h’’Dam
motion the pressure at A3 is constant (Ho). However, when water is
Forebay
Penstock=LConduit A2
h
h’
A1in motion the pressure at A3 changes dynamically (h h’ h’’);
Turbine
Conduit A2Qo, A
dynamically (h, h ,h ); pressure surges!
Cross-sectional area of
A3From the conservation of energy for a small element of mass travelling from A1 to A3:
the penstock may vary.1 3A A lossesE E E
21mgh mv 2v gh 2v C gh2
mgh mv 2v gh 2v C gh
C=Nozzle coefficient, typically 0.9825
Modeling: Hydro-turbine ModelsWater flow Q and torque Tm in any hydro turbine are functions of the head (h), valve or gate position (y), and speed (ω). The effect of speed on the flow i dl i lt t bi
, ,m mT T h y
is regardless in pelton turbines. , ,Q Q h y
Water hammer: It is the travelling of pressure waves along the penstock (>0 3Hz) It
Linearizing both equations:Q Q QQ h yh y
along the penstock (>0.3Hz). It causes the head (pressure) to be a function of the rate of change of flow.y
T T TT h yh y
When water hammer effects are considered, we have:
.h h Q
When water hammer effects are considered, we have:
Linearizing the above, and taking the Laplace transform we obtain:
H s WH s Q s
where WH(s) is the linear water hammer effect in the s domain.26
Modeling: Hydro turbine ModelsModeling: Hydro-turbine Models
T
Qh
Ty
Qy
ΔyWater
Hammer ΔHΔQ T
h
ΔT
y WH(s) hSpeed signal comes from swing
equation
Partial derivatives come from QT
Δω
Partial derivatives come from field tests.
T
27
Modeling: Hydro-turbine ModelsY is gate position Speed ω=1
∂Q/∂ω
Speed ω (pu)
This data is used to update the hydro turbine model as the speed and gate position change.
Speed ω (pu)
[*] Atsushi Izena, Hidemi Kihara, Toshikazu Shimojo, Kaiichirou Hirayama, Nobuhiko Furukawa, Takahisa Kageyama,Takashi Goto, and Chosei Okamura, “Practical Hydraulic Turbine Model”
28
Modeling: Hydro turbine ModelsModeling: Hydro-turbine ModelsWhen neglecting speed deviations for small excursions, we have:
Qh
Ty ΔT=Torque small signal
ΔP=Power small signal
Q
Δy
h
Water H
ΔHΔQ Th
ΔT= ΔP in p.u.++
y Hammer h
H s T sElastic and inelastic water hammer is represented by:
H sHere:
Tw= water inertia time
+
2 21 0.1
w
e
H s T sQ s T s
w
H sT s
Q s
Elastic water Hammer Inelastic water Hammer
Te= elastic time(in seconds)
LQT 0LT
Note: Detailed derivation of water hammer effects can be found at: P. Anderson and A. Fouad, “Power System Control and Stability,” IEEE series, 2003.
29
AgHTw
0 cTe
L, A are penstock length & cross-sectional area, respectively. C is nozzle coefficient.
Modeling: Hydro turbine ModelsModeling: Hydro-turbine ModelsFrom the previous slide, we use the inelastic form of water hammer with
1Qy
0.5Qh
1Ty
1.5Th
1T T
which are typical constants as shown in [*]. Then we solve the closed loop transfer function to obtain:
11 0.5
w
w
T s T sY s T s
This is the very well known turbine model representation. However, this is notThis is the very well known turbine model representation. However, this is not valid for large excursions in speed. Thus, the former representation (three slides previous) with dependence on speed variation should be used when considering large speed excursions, which we are in analyzing the Colombian oscillation.
[*] Thorne D., H., Hill E.,F., “Field testing and simulation of hydraulic governor performance,” IEEE transactions on power apparatus and systems, Pgs (93)4, July 1974.
30
Model building: adding hydro turbine
SERVO
31
Model building: Adding multiple generatorsF lti hi t i i l ti h hi ’For a multi-machine system in a single area, representing each machine’s individual controls but a single equivalent rotating mass, we have:
32
Modeling: Remaining issues to consider
Generator dynamics (nonlinear description & transient behavior)
Nonlinearities (discontinuities) in the control systems.
Effects of the voltage regulators and excitation.
Load models
33
Modeling: Generator Dynamics
There are various approaches for the modeling of generators, high pp g g , gfidelity models and reduced models.
Reduced Models. Gives approximate dynamical models; however, some dynamics are lost since these models are based on
i ti (th l i l d l i i thi t )some approximations (the classical model is in this category).
High fidelity models Comprehensively models generator High fidelity models. Comprehensively models generator states, but causes required simulation time to be very large.
34
Modeling: Various IEEE standard models
High fidelity models
35
Modeling: Classical ModelModeling: Classical Model For very basic studies the generator can be modeled as a constant y g
voltage behind the transient reactance. The basis for this is that the internal fluxes of the generator don’t change abruptly (within the first second of a disturbance).
This model is sometimes designated model 0.0.
sin'
te
E VPX d
X d
36
Conceptual understanding of generator action: classical model
How do our generators work in the power system?
action: classical model
Before synchronizing a machine to the system (breaker closure) ωb=ωe, δ=0, Vbus=Vgen. In the transient:
sintE VP
21 1sin sin 2t tE V VP
ωe ωb
• To load the machine, the rotor needs some q-axis network
sin'eP
X d sin sin 2
' 2eq d
PX d X X
Round Rotor Salient Pole Rotor
acceleration. E and Vt are assumed constant in the classical model. Then the q-axis of the generator leads the q-axis of the system and desired load is
q-axis rotor
Δδ
ωe
δ
q ydelivered. We adjust that load by governor action.
• When reaching the desired position (load setting) the q axis of the rotor oscillates
ωb
the q-axis of the rotor oscillates.
• If the system is stable, the desired angle δ is reached until ωe=ωb.37
Conceptual understanding of generator action: classical model From our swing equation we have the rotor dynamics:
action: classical model
, ,2
e m pu e pub
H d T t T tdt
b
sin'
te
E VPX d
e
ddt
, ,e pu e puP T We know that generator obeys:
By combining rotor and generator dynamics:
'X d dt, ,p p
2
,2
2 sin'
tm pu
E VH d dT t Ddt X d dt
b dt X d dt38
Conceptual understanding of generator action: classical model
The last equation shows that the rotor does not spin at a t t d i th t i t (i h i l d)
action: classical model
constant speed in the transient (i.e., a change in load).
The dynamics of the rotor (acceleration – deacceleration) depends on the Tm and Tedepends on the Tm and Te.
The power system is never in steady state; thus, it is dynamically changing and with this the rotors are oscillating. g g g
Wh hi k f l h i l ill i i When we think of electro-mechanical oscillations in power systems we should think of rotors changing their q-axis constantly (it means acceleration and deceleration in the
39
constantly (it means acceleration and deceleration in the rotors) by effects of the external torques Tm and Te!!!!!!!!
Gen modeling: High Fidelity Model (2.2)
2.2 means two windings in the q-axis (field winding and damper winding) and twowinding and damper winding) and two windings in the d-axis (damper windings)
40
Gen modeling: High Fidelity Model (2.2)
The expression for the electrical torque is:
41 , ,
2e m pu e pu
b
H d T t T tdt
Where are we?
We have developed a model of the power t th t i l d h dsystem that includes enhanced
representation of the hydro-turbine, the governor/actuator controls and a high-governor/actuator controls, and a high-fidelity model of the generator.
We now consider nonlinearities in the control loops.
42
Nonlinear effects: Li S V N li SLinear Systems Vs. Nonlinear SystemsThe real power system (the real world) is a nonlinear system p y ( ) ywhich most of the time for analysis purposes is approximated with a linear description since it is simpler. However, the real system is nonlinear
Linear System: Non-Linear System
system is nonlinear.
.x Ax Buy Cx Du
y Non Linear System
., ,x f t x u
y Cx Du , ,y g t x u
Tools:C t i l ti
Tools: Transfer function.Analysis in frequency domain -Computer simulation
-Nonlinear analysis methodsAnalysis in frequency domain.
43 DBAsICsG 1
Nonlinear effects: systems & nonlinearities
Nonlinearities are non-smooth discontinuous characteristics of systems.
They may be caused by imperfections of system components.
Th l h l d They are almost everywhere in control systems and are common in actuators, such as servo-valves, mechanical connections (joints), and electrical servomotors.(j ), .
They can limit static and dynamic performance of feedback control systems.
44
Nonlinear effects: types of nonlinearities
The most common nonlinearities found in systems are:
Dead Zone (Dead Band)
Backlash
Saturation
Viscous and Friction Forces
Static and Coulomb Friction, etc.
45
Nonlinear effects: Dead Zone Dead zones are sometimes designed into components intentionally. But they can bring about undesirable effects in control loops.
M l l This type of nonlinearity is memory-less. Memory-less: only matters where you are now, not where you have been.y
46
Nonlinear effects: physical intuition of dead zone effectsof dead zone effects
Dead-zones in the actuator can cause blinding (no response) of the final ll h h h l i h actuator to small changes that the control system might request.
For example, consider1 Steady-state valve opening is 70% but the controller (PID) requests the 1. Steady state valve opening is 70%, but the controller (PID) requests the
actuator to change it due to a variation in the system load conditions to 71%;
2 Under that request the actuator may not respond to the small change 2. Under that request the actuator may not respond to the small change because of the presence of a dead-zone at that operating point;
3. The error signal in the system increases; 4. The control signal out of the controller increases until it is large enough to
surpass the servo dead-zone effect. 5. At this point, the servo operates, but because the controller signal is now p , p , g
very large, the servo overcompensates to 72%. 6. The controller now sends a correction signal and the process repeats. 47
Nonlinear effects: Observed dead zone phenomenazone phenomena From a commissioning
110120
Prueba G6 - Señales Actuador Hidráulico. (Past Comissioning of unit G6 hydraulic servo system )
test, we see that the red line is the set point of the servo system and the
100110
the servo system and the blue line is the actual position of the servo.
80
90
90
100
Observe the error between the setpoint
d t l iti
7080
Actual Servo PositionSetpoint
and actual servo position
The output lags the input, likely as a result 50
60
60
70
input, likely as a result of dead zone.
48
4 9 14 19 24 29 34
Set Point Posición Posición Servomotor
Potencia Activa
Nonlinear effects: BacklashNonlinear effects: Backlash Backlash is typically found in mechanical joints. yp y j
Its effect can be harmful.
This nonlinearity has memory. y y
49
Nonlinear effects: SaturationNonlinear effects: Saturation Saturation represents limits in controls.
This limits can be in speed, position, control signal level, etc.
In servo systems, the hydraulic flow available is controlled by fl l flow valves.
50
Nonlinear effects: Other nonlinearities
51
Nonlinear effects: Backlash as Result of Dead zoneof Dead zone Dead zones in the forward path of a servo-loop can bring about backlash in the
closed loop responses. To demonstrate this in a simple way, let’s compare the output of the close loop servo system with dead zone (y1), and the output through the backlash (y2), after applying a common input r(t) respectively.
52
Note: Saturation is also added in real systems. This is in practice done to avoid water hammer due to fast action of water flow control valves (speed limit). This is also a nonlinearity and may produce oscillatory effects.
Nonlinear effects: Backlash as result of Dead zoneDead zone
When comparing y1(t) and y2(t), we see that both lags the input r(t). The blue signal y1(t) is similar to the green signal y2(t) and both almost follow the same trajectory By y1(t) is similar to the green signal y2(t) and both almost follow the same trajectory. By simulation it is demonstrated that dead zones in the forward path of a system can cause backlash effects in the close loop behavior.
Setpoint and Output Signals Comparison
d f l0.8
0.9
1
Red=Reference Signal r(t).
Blue=Servo-valve loop Output Signal y1(t).
0.5
0.6
0.7
[blue]
, y2
(t) [g
reen
]
Green=Backlash Output Signal y2(t).
0 2
0.3
0.4
r(t) [re
d], y1
(t)
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
t(s)
53
Nonlinear effects: Backlash as result of Dead zoneof Dead zone
Thus, for large frequency control studies the dead zone , g q ynonlinearity found in the servo systems control loop should be modeled as backlash nonlinearity when using very reduced models.
y
speedlimit
3.33 1 yy
y1(t)r1(t)0.07s+1
Servo-valve1
s
IntegratorDead Zoned=0.02
y
y1(t)r1(t) d=0.02
Governor TurbineRotor
Dynamicsωref ωrΔωref Ylp Yhp
Actuator Reduced Model
54Ylp= Gate low power signal.Yhp= Gate high power signal.
Nonlinear effects: Backlash Effects What are the effects of Backlash?
To describe effects of backlash, let’s see what happens to the output , pp pof the backlash block when we apply a sine signal to the input.
0 8Backlash Behavior
I t O t t B kl h Eff tH/2
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
1
(t)
Input - Output Backlash Effect
u(t)y(t)
H
-0.6
-0.4
-0.2
0y(t
-0.8
-0.6
-0.4
-0.2
0
u(t),
y(H
55
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8
u(t)0 1 2 3 4 5 6 7 8 9 10
-1
t(s)
Nonlinear effects: Backlash as result of Dead zone 0.5
Load Step Change
of Dead zoneLet’s analyze the effect of plugging the nonlinearity in 0.1
0.2
0.3
0.4
Load
(p.
u.)
plugging the nonlinearity in the system below. Only Primary control.
60.1Without Nonlinearity
0 5 10 15 20 25 30 35 40 45 500
Time (s)
The test without the nonlinearity shows no limit cycles. 59.95
60
60.05
requ
ency
(H
z)
y
The one with the nonlinearity shows limit
l f f
0 5 10 15 20 25 30 35 40 45 5059.85
59.9
Time (s)F
r
With Nonlinearitycycles of constant frequency.
59 95
60
60.05
60.1With Nonlinearity
uenc
y (H
z)
System with Primary loop control only
56 0 5 10 15 20 25 30 35 40 45 5059.85
59.9
59.95
Time (s)
Fre
q
Nonlinear effects: Backlash as result of Dead zoneDead zone The addition of a The addition of a
secondary loop control does not
Secondary Control without Nonlinearity
damp the limit cycles contained in the primary loop
60
60.02
60.04
60.06y y
uenc
y (H
z)the primary loop control. The step is the same applied 0 5 10 15 20 25 30 35 40 45 50
59.94
59.96
59.98
Time (s)
Fre
qu
Secondary Control with Nonlinearity
before.
60
60.05
60.1
60.15Secondary Control with Nonlinearity
uenc
y (H
z)
57 0 5 10 15 20 25 30 35 40 45 5059.9
59.95
60
Time (s)
Fre
qu
Model building: Inclusion of nonlinearities
Our control loop now looks interesting.
58
Model building: Voltage regulators & excitation
Voltage regulators play an important role in the system stability. Besides controlling the terminal voltage of the
k i ib h l i l i generators as known, it contributes to the electrical torque in the generators, and it can add or reduce damping to an oscillation. We have seen this in the earlier presentation. And pso we do represent a static excitation system in our model.
59
Model building: Load modelingModel building: Load modelingThe power consumed by loads varies with voltage, frequency, and many other p y g , q y, yconditions.
( , , , , ,...)P P V z f t T
( )Q Q V z f t T ( , , , , ,...)Q Q V z f t T
where: z=load demand, V=voltage, f=frequency, t=time, T=temperature, etc.
We know that the load influences the Te of the synchronous machines; thus, loads play a very important role in the analysis of electrical oscillations. The power system controls are designed to follow the load variations and supply the
T t ilib t th T i th hinecessary Tm to equilibrate the Te in the machines.
Our model accounts for load effects only by adding the damping characteristic D in the swing equation. It may be necessary to model load with greater fidelity.the swing equation. It may be necessary to model load with greater fidelity.
60
So summarize - why the power system oscillatesoscillates The underlying phenomena is the torque imbalance between the Tm and
Te, accompanied with the dynamic response of the system.
The mechanical torque Tm is controlled by the prime movers and their controls (turbine and governor/actuator)controls (turbine and governor/actuator)
The electrical torque Te is controlled by the generator dynamics (controller AVR), and the interconnected power grid (loads). This dynamic torque of the system can cause variation in the oscillation damping (negative or positive).
Nonlinearities produce limit cycles If the controls oscillate the controlled Nonlinearities produce limit cycles. If the controls oscillate, the controlled equipment will, too.
61
Simulation illustration: :descriptionSimulation illustration: :description The simulated system is composed of two hydro units supplying a single load The simulated system is composed of two hydro units supplying a single load
which are interconnected by the transmissions lines respectively.
The two generators are modeled using high fidelity generators models (2.2).
A 250MVA hydro unit is modeled with a practical turbine model from [*].
A 100MVA hydro –turbine unit is modeled with the IEEE standard representation for dynamic studies as in [**].p y [ ]
A nonlinearity type dead zone was added to the servo positioning system of the 250MVA hydro unit which is a function of the input, operating point and time.
A PSS 4B ll b dd d d h ll d b h dd f A PSS 4B will be added to damp out the oscillation mode seen by the addition of the nonlinearity [***].[*] Atsushi Izena, Hidemi Kihara, Toshikazu Shimojo, Kaiichirou Hirayama, Nobuhiko Furukawa, Takahisa Kageyama,Takashi Goto, and ChoseiOkamura, “Practical Hydraulic Turbine Model”y[**] IEEE Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance Studies, "Hydraulic Turbine and Turbine Control Models for Dynamic Studies," IEEE Transactions on Power Systems, Vol.7, No.1, February, 1992, pp. 167-179.[***] Kamwa I., Grondin R., “IEEE PSS2B Versus PSS4B: The Limits of Performance of Modern Power System Stabilizers,” Vol. 20, No. 2, 2005.62
Simulation illustration:
0.05
1.0045 Speed Ref erence
Gate Drop Pos. Ref erence Tubine Output<Stator v oltage v q (pu)>
<Stator v oltage v d (pu)>
250 MVA unit with “high-fidelity” modeling
A
B
C
a
b
c
w
Vf _
m
A
B
Actual Power
Hydro-Turbine and Speed Governing System
A
B
C
A
B
C
A
B
C
A
B
C
<Output activ e power Peo (pu)>
<Rotor speed dev iation dw (pu)>
1.0 T1: 900MVA20 kV-230 kV
C
Synchronous Machine250 MVA 20 kV
Switch
dw Vstab
MB-PSS
v ref
v d
v q
v stab
Vf
ExcitationSystem
A B C
300MW80MVAR
25km Area 1 10 km Area 1
Clock
<Stator v oltage v q (pu)>
<Stator v oltage v d (pu)>
<Rotor speed wm (pu)>
<Output activ e power Peo (pu)>
100 MVA unit with “standard”
1.05
0.5
A a
Pm
m
A
wref
Pref
we
Pe0
dw
Pm
gate
HTGA
B
A
B
<Rotor speed dev iation dw (pu)>
modeling
1.0
B
C
b
c
T2: 900MVA 20 kV/230 kV
Vf _B
C
Synchronous Machine100 MVA 20 kV1
HTG
v ref
v d
v q
v stab
Vf
ExcitationSystem1
B
C
B
C
25km Area 2
63
Simulation illustrationSimulation illustrationInside the Prime mover block of the 250MVA power plant which represents the hydroturbine and speed governing system we have:
This block represents thepractical hydro turbine and
lits controls.
1
Gate Output
ka
Ta.s+1
valve
P
uy
Ti V i D dSaturation
1s
Integrator
2
P
1
Gate Input
, ,u t PThis block represents the valveTime Varying Deadzone
Satu at o teg atoGate Inputpservo system (valvepositioning) and the additionof a nonlinearity in it.
64
Simulation illustration: P i l H d li T bi M d lPractical Hydraulic Turbine Model
The practical hydro turbine d l i d b model is so-named because
the model varies under different load conditions [*].
The variations strongly influence the power system fre uenc stabilitfrequency stability.
Improves design accuracy for governor control constants.g
Note: The demonstration for this transfer function model for an hydraulic turbine is very similar to the one we derived as model for our hydro turbine.
[*] Atsushi Izena, Hidemi Kihara, Toshikazu Shimojo, Kaiichirou Hirayama, Nobuhiko Furukawa, Takahisa Kageyama,Takashi Goto, and ChoseiOkamura, “Practical Hydraulic Turbine Model”
65
model for our hydro turbine.
Simulation illustrationSimulation illustration
The simulation was run three times. Each time the system is disturbed using a 10% load shift from 350 MVA machine to the 100 MVA machine.
-The first one without adding the nonlinearity (deadzone).
-The second one with adding the nonlinearity (deadzone).
-A third one to illustrate the use of PSS (to be discussed later)
[*] Atsushi Izena, Hidemi Kihara, Toshikazu Shimojo, Kaiichirou Hirayama, Nobuhiko Furukawa, Takahisa Kageyama,Takashi Goto, and ChoseiOkamura, “Practical Hydraulic Turbine Model”[**] IEEE Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance Studies, "Hydraulic Turbine and Turbine Control Models for Dynamic Studies," IEEE Transactions on Power Systems, Vol.7, No.1, February, 1992, pp. 167-179.66
Simulation illustrationWith no Nonlinearity in the System
The simulation without the nonlinearity is stable throughout the simulation time. This indicates that the hydraulic turbine does not contribute an oscillatory mode.
System oscillates at beginning but scale of plot does not enable it toplot does not enable it to be well observed here.
67
Simulation illustrationWith N li it i th S tWith Nonlinearity in the System
By adding the nonlinearity the systems oscillates continuously.
68
Simulation illustrationWith Nonlinearity in the System
69
Simulation illustrationWith Nonlinearity in the System
The addition of a dead b d b kl h
O S C IL AC IÓ N D E P O T E N C IA E N E L S IN
band (backlash) to our study lead to continuous oscillation in our model systems P eriod o 1 0 , Ago s to 1 2 d e 200 8
6 0
8 0
10 0
12 0
60 .6
60 .8
61
61 .2
systems. The behavior is similar to that of the Colombian power system.
-4 0
-2 0
0
2 0
4 0
59 .6
59 .8
60
60 .2
60 .4
Pote
ncia
(MW
)
Frec
uenc
ia (H
z)
power system.
-10 0
-8 0
-6 0
9:17
:00
9:17
:32
9:18
:04
9:18
:36
9:19
:08
9:19
:40
9:20
:12
9:20
:44
9:21
:16
9:21
:48
9:22
:20
9:22
:52
9:23
:24
9:23
:56
9:24
:28
9:25
:00
9:25
:32
9:26
:04
9:26
:36
9:27
:08
9:27
:40
9:28
:12
9:28
:44
9:29
:16
9:29
:48
9:30
:20
9:30
:52
9:31
:24
9:31
:56
9:32
:28
9:33
:00
9:33
:32
9:34
:04
9:34
:36
9:35
:08
9:35
:40
9:36
:12
9:36
:44
9:37
:16
9:37
:48
9:38
:20
9:38
:52
9:39
:24
9:39
:56
5 9
59 .2
59 .4
BETAN I/13 .8/Gen3/P/ JAM OND/230/POMASQ1/P/JAM OND/230/POM ASQ2/P/ TEBSA/230/TrfGn011/P/TEBSA/230/TrfGn012/P/ FRECUENC/M EDELLIN //Frequ/70
Simulation illustrationWith Nonlinearity in the System
It is possible to see a variety of frequencies of the
71
It is possible to see a variety of frequencies of the oscillation ranging from 0.03 to 0.09 Hz.
Simulation illustrationExplanation
To validate our simulation model let’s think of the following:
V
, ,2
e m pu e pub
H d T t T tdt
TI
Power SystemExfd
V
sinPm K t
R t
Te ωr
t
Pe Te K
TurbineRotor
DynamicsTm
We know that the output power of the turbine oscillates The precedingWe know that the output power of the turbine oscillates. The preceding work suggests that the origin of the oscillation may be a poor performance of the control systems and nonlinearities in it.72
Simulation illustrationExplanation
Assume that the limit cycle causes prime mover torque to oscillate sinusoidally. Then:
2 sine cb
H d K t Kdt
0
sin2
tb
e c bt dtH
b dt
0
1 cos2
be c b
c
tH
This result express that the rotor of the equivalent machine will spin at a This result express that the rotor of the equivalent machine will spin at a constant component ωb=377rad/s (60Hz) + an oscillatory component, oscillating at ωc. Additionally, if the prime mover torque oscillates with a sine function the frequency + oscillatory component will beoscillates with a sine function, the frequency + oscillatory component will be seen as a ‐cosine function. This validates that oscillations in the prime mover output will lead oscillations in the system frequency and that they will be in counter‐phase.
73
Possible solution approaches
Special controls can cancel out the nonlinearities. This stops the oscillation on the prime mover due to the nonlinearities. If we maintain the Tm of the prime mover smooth and non oscillatory the speed of the rotor (related to frequency) will be oscillatory, the speed of the rotor (related to frequency) will be smooth and non-oscillatory also.
According to what we said before in the case of the dead band g(backlash), we could also find ways to avoid the high frequencies in the controls.
l h h l b h h h Controls which equilibrate the torques in the synchronous machine and add damping. The voltage regulator controls the Te and adds damping. Can we excite the generator with a signal such and adds damping. Can we excite the generator with a signal such that the Te equilibrates Tm and damps any oscillation?
74
Possible solution approaches: Cancelation of nonlinearitiesCancelation of nonlinearities Nonlinearities are present in all systems and can be drawn as the system shown below.
Nonlinearities are a drawback for feedback systems to position the controlled variable y pin the desired value.
It is possible to find a function that can cancel out those nonlinearities.
In the block below v=IN(u) In the block below v=IN(u)
ud=N(v)=N(IN(u))=u → ud=u.
If we can find it, the system below would work as if it did not have the nonlinearity.
System
yIN(.) N(.) G(s)u v ud y
System
G(s)
System
u yNow we have an approximation to an ideal system!!!!!!
75
Possible solution approaches: Cancelation of nonlinearitiesCancelation of nonlinearities
Nonlinearities can be canceled very easily when they are non- Nonlinearities can be canceled very easily when they are nontime varying and known since they remain the same for all time t and all conditions.
The problem becomes complicated if the nonlinearities are time varying and dependant of operating conditions, or diffi lt t difficult to measure.
N(.) G(s)
System
u udr+
-How do I access to measure here?????
76
Possible solution approaches: damping with PSSwith PSS As we have seen, PSS detects power oscillations (rotor and power oscillations) and
gives an extra signal to the voltage regulators to damp the oscillation. Generally speaking, it acts over the excitation control to modify the electrical torque and gives the extra signal to equilibrate the Tm with the Te.
www.meppi.com/Products/Generator%20Excitation%20Products%20Documents/Power%20System%20Stabilizer.pdf77
Possible solution approaches: damping with a multiband PSSwith a multiband PSS
Low band for 0.05 Hz
Medium band for 0.2-1.0 Hz
PSS4B multimode damping. High band for 0.8-4.0 Hz
•We have tuned each stage using the conventional tuningWe have tuned each stage using the conventional tuning approach as described yesterday.•We think it feasible and beneficial to adopt an on-line
d ti t i th d i h t [*]78
adaptive tuning method using phasor measurements [*].[*]R. Grondin, (U) I. Kamwa, (M) L. Soulieres, (M) J. Potvin, (M) R. Champagne, “An Approach to PSS Design for Transient Stability Improvement
through Supplementary Damping of the Common Low-Frequency”
Simulation results with PSS IEEE4BSimulation results with PSS IEEE4B
PSS4B is turned on at 200 [s].The 0.05 Hz oscillation is stabilized.
79
SummarySummary
Power system oscillations can be the result of the introductions of ynonlinearities in the control systems, which under certain operating conditions can bring about limit cycles.
These limit cycles cannot be observed from simulation studies unless the model represents the nonlinearities.
W id th d l h d l d f t b We consider the model we have developed so far to be a “demonstration model,” intended only to provide evidence that the 0.05 Hz oscillation could be caused by these nonlinearities.y
Advanced PSS along with AVRs may be good tools for mitigating oscillations in power units in a selected range of frequency. This
80
effect is seen when they add controlled damping.
What next?A. Maintain “open mind” to all possible explanations, including
1. The linearized model is deficient (excludes an important attribute) or has erroneous dataor has erroneous data.
2. There are one or more units having nonlinear elements which create limit cycles that are manifested in the system as slowly varying (in frequency and amplitude) oscillations.
3. Both #1 and #2 are true.4 The 0 05 Hz mode is caused by interactions between modes “Th 4. The 0.05 Hz mode is caused by interactions between modes, “The
results of the numerical simulations show that low-frequency modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation.” A. Messina, V. Vittal, “Assessment of nonlinear , ,interaction between nonlinearly coupled modes using higher order spectra,” IEEE Transactions on Power Systems, Feb. 2005.
5. There are other explanations of which we have not yet considered.
81
p y
What next?B D l l h l f lB. Develop ways to explore each explanation, for example:
1. The linearized model is deficient (excludes an important attribute) or has erroneous data.or has erroneous data.a. Test each major unit and associated controls with small-signal probes and compare frequency response to that of the models. Adjust model as necessary. This is what Psymetrics is doing.This is what Psymetrics is doing.
b. Improve modeling of the interconnection external to Colombia.
c. Improve load models.
2 Th it h i li l t hi h t 2. There are one or more units having nonlinear elements which create limit cycles that are manifested in the system as slowly varying (in frequency and amplitude) oscillations.a. Place “mechanical PMUs” (sensors with time-stamps) on the servo mechanisms of each major unit to monitor its behavior during oscillations.
b. Obtain the process diagrams of all major power plants, and build a simulation
82
p g j p pmodel of the Colombian power system that contains the nonlinear elements.
Gen modeling: 2.2 Vs. 1.1 This exemplification is to show how we may lose valuable information whenvaluable information when we neglect some transient effects via reduced models.
Dynamic Performance of an hydro turbine generator during a 3-phase fault at the terminals
Krause P., Wasynczuk O., Sudhoff S., “Analysis of Electrical Machines and drive Systems,” Wiley, 2002.83
Questions?Questions?
84
