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TRANSCRIPT
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Introduction to Power System Stability
Mohamed A. El-Sharkawi
Department of Electrical Engineering
University of Washington
Seattle, WA 98195
http://SmartEnergyLab.com
Email: [email protected]
1©Mohamed El-Sharkawi, University of Washington
NERC Standard
1. The System remains within acceptable limits;2. The System performs acceptably after credible
contingencies;3. The System contains (limit) instability and
cascading outages;4. The System’s facilities are protected from severe
damage; and5. The System’s integrity can be restored if it is lost.
The bulk power system will achieve an adequate level of reliability when it is planned and operated such that:
2©Mohamed El-Sharkawi, University of Washington
NERC Standard
• “Reliable Operation means operating the elements of the Bulk-Power System within equipment and electric system thermal, voltage, and stability limits so that instability, uncontrolled separation, or cascading failures of such system will not occur as a result of sudden disturbance, including a CybersecurityIncident, or unanticipated failure of system elements.”
• NERC standard is for dynamic performance requirements– Security– Stability
3©Mohamed El-Sharkawi, University of Washington
System Operating Limit (SOL)
• MW, MVar, Amperes, Frequency or Volts that satisfies the most limiting of the operating criteria for a specified system configuration to ensure operation within acceptable reliability criteria
• This is a security measure
4©Mohamed El-Sharkawi, University of Washington
Interconnection Reliability Operating Limit (IROL)
• IROL is a SOL that, if violated, could lead to instability, uncontrolled separation, or cascading outages that adversely impact the reliability of the bulk power system.
• This is a stability measure
5©Mohamed El-Sharkawi, University of Washington
SOL and IROL
• Following a contingency or other system event that cause the system to operate outside set reliability boundaries, Transmission Operators (TO) are obligated to return its transmission system to within SOL or IROL as soon as possible.
6©Mohamed El-Sharkawi, University of Washington
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What is stability?
• Is the ability of the system to achieve steady state operating condition after a disturbance– All oscillations are damped out
f
60Hz
Time7©Mohamed El-Sharkawi, University of Washington
Stability
• During fault, and right after the fault is cleared, the power system stability is determined by the capabilities of its generators to – maintain connected to the grid– provide the extra reactive power– provide fast ramping down of real power
• Generators are permitted to trip off line only in the case of a permanent fault on a directly connected circuit
8©Mohamed El-Sharkawi, University of Washington
What is security?
• If the system is stable after a disturbance, the system is secure if all its key components are operating within their design limits
9©Mohamed El-Sharkawi, University of Washington
Balance of power in Generators
GPm Pe
MechanicalPower
Controlled atthe power plant
ElectricalPower
Controlled bythe customers
Pm = PeAt steady state
10©Mohamed El-Sharkawi, University of Washington
Turbine Speed and Imbalance of Power
dt
dnPP em ~
GPm
Pe
sem nnPPif ;
sem nnPPif ;
0; dt
dnPPif em
speedrotor constant ;nn s
11©Mohamed El-Sharkawi, University of Washington
Turbine Speed and System Frequency
np
f120
f: The frequency of the terminal voltage of the generatorp: The number of poles of the generatorn: The speed of the generator (turbine)
12©Mohamed El-Sharkawi, University of Washington
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If Pm>Pe
• The surplus energy is stored in the rotating mass of the generator in the form of kinetic energy– The machine speeds up (frequency
increases)– The current inside the machine increases– Over-speed and over-current protections will
eventually trip the machine
GPm
Pe
13©Mohamed El-Sharkawi, University of Washington
If Pm<Pe
• The deficit in energy is drawn from the kinetic energy of the rotating mass of the generator– The machine initially slows down (frequency
decreases)
– If not corrected, the machine could operate as a motor
– Machine is tripped to prevent mechanical damages
GPm
Pe
14©Mohamed El-Sharkawi, University of Washington
Transient Stability Analysis
• Transient stability analysis determines whether the generator, after a disturbance, reaches a new stable operating point. – The input mechanical power of the generator
is equal to the output electric power, and
– The frequency of the generator is the same as the frequency of the system before the disturbance (i.e. the generator speed is the same as the prefault synchronous speed)
15©Mohamed El-Sharkawi, University of Washington
Unstable System
n (f)
ns=60Hz
16©Mohamed El-Sharkawi, University of Washington
Unstable System
Time
n (f)
ns=60Hz
17©Mohamed El-Sharkawi, University of Washington
Stable System
Time
n (f)
ns=60Hz
18©Mohamed El-Sharkawi, University of Washington
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Synchronous Generator
El-Sharkawi@University of Washington 20
Power Generation
• 99+ % of all power are generated by the synchronous generators
• Synchronous machines can operate as generators or motors
El-Sharkawi@University of Washington 21 El-Sharkawi@University of Washington 22
El-Sharkawi@University of Washington 23 El-Sharkawi@University of Washington 24
Small Synchronous Machine
5
El-Sharkawi@University of Washington 25
N
S
StatorRotor
X
X
c
b
a
Xc
b
El-Sharkawi@University of Washington 26
Time
Vaa’Vbb’ Vcc’
X
X
c
b
a
Xc
b
N
S
f
El-Sharkawi@University of Washington 27
f
s
Ef
N
S
If
Vf
td
d~E
ff
Open Stator
fE is directly proportional to the excitation current fI
The frequency of fE is proportional to the synchronous speed s
El-Sharkawi@University of Washington 28
Xs R
Ef Vt
sXRcetansisReArmatureR
cetanacResSynchronouX s
Equivalent Circuit
El-Sharkawi@University of Washington 29
Generator Equivalent Circuit
satf XIVE
Ia
Xs
Ef Vt
El-Sharkawi@University of Washington 30
Generator Equivalent Circuit
satf XIVE
Vt
Ia Xs
Ia
Ef
Vt is Fixed (infinite Bus)Ef is function of If
Magnitude and phase of Ia
are dependant variables
Ia
Xs
EfVt
6
El-Sharkawi@University of Washington 31
Power equations
sinIVQ att 3
cosIVP at3
Ia
Xs
Ef Vt
Ia
Vt
Ef
Ia Xs
Vt and Ef are phase quantities
f t
El-Sharkawi@University of Washington 32
Power equation
sinEcosXI fsa
s
fa X
sinEcosI
sinX
EV3P
s
ft
cosIV3P at
Ia
Vt
Ef
Ia Xs
El-Sharkawi@University of Washington 33
Power Characteristics of Generator
sinX
EV3P
s
ft
P
Pmax
l 90o s
ft
X
EVP
3max
Ia
Vt
Ef
Ia Xs
Ia
Xs
Ef Vt
El-Sharkawi@University of Washington 34
Reactive Power equations
sin3 att IVQ
Ia
Xs
Ef Vt
Vt and Ef are phase quantities
f t
tfsa VEXI cossin
Ia
Vt
Ef
Ia Xs
El-Sharkawi@University of Washington 35
sin3 att IVQ
tfsa VEXI cossin
tfs
tatt VE
X
VIVQ cos
3sin3
If Ef cos > Vt ; Qt is positive and Current is lagging
If Ef cos < Vt ; Qt is negative and Current is leading
If Ef cos = Vt ; Qt is zero and Current is in phase
A Synchronous Generator Connected to Large System
Transmission line
G
Terminal busV and f may vary
Infinite busV and f cannot vary
Vt VEf
36©Mohamed El-Sharkawi, University of Washington
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Power Equation
I
X = Xs+Xline
Ef V
Power Reactive
Power Real
VcosEX
VQ
sinX
VEP
f
f
V
Ef
37©Mohamed El-Sharkawi, University of Washington
Power Characteristics of Generator
sinX
EVP f
38©Mohamed El-Sharkawi, University of Washington
P
Pmax
90o
Pm
Maximum Power
P
Pm
Pmax
l 90o
Phasor Diagram atPmax
V
Ef
39©Mohamed El-Sharkawi, University of Washington
Operating Point
Pm
P
Operating point
sinX
VEP f
40©Mohamed El-Sharkawi, University of Washington
Generation Limit
• The generator must generate less than the Pmax
(called pull-out power).
• The difference between Pmax and the actual power generated is the generation margin.
• The generation margin must be positive and large enough to ensure the dynamic stability of the system.
41©Mohamed El-Sharkawi, University of Washington
Dynamic Stability Assessment (DSA)
42©Mohamed El-Sharkawi, University of Washington
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Objective of DSA
• Dynamic stability analysis determines whether the system, after a disturbance, reaches a new stable operating point. – The input mechanical power of the generator is equal
to the output electric power
– The frequencies of the generator is the same as the frequency of the system (i.e. the generator speed is the synchronous speed)
– All voltages and currents are within the allowable limits
43©Mohamed El-Sharkawi, University of Washington
Phasor diagram
I
X
Ef V
V
I X
I
Ef
44©Mohamed El-Sharkawi, University of Washington
Rotation of Phasor Diagram
gf
Hzorf 5060
VInfinite bus
EfRotor
f is the frequency at the infinite bus
fg is the frequency of the generator
45©Mohamed El-Sharkawi, University of Washington
Frequency/Speed Relationship
• The frequency (fg) of the generator’s voltage is proportional to the speed of the generator’s shaft (n).
np
fg 120
p is the number of magnetic poles of the generator
46©Mohamed El-Sharkawi, University of Washington
Frequency/Speed Relationship
• If the frequency of the generator (fg) is 60 Hz (or 50 Hz), the speed of the generator’s shaft is called synchronous speed (ns).
np
fg 120
system Hz50in 120
50
system Hz60in 120
60
s
s
np
np
47©Mohamed El-Sharkawi, University of Washington
Power Control
dt
dn~PPm G
Pm P
increasesn;PPif m decreasesn;PPif m
0dt
dn;PPif m speedrotor constant ;nn s
48©Mohamed El-Sharkawi, University of Washington
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– The current of the machine increases
Power Control• When more power is
needed, the generator speed increases (n>ns) by increasing the mechanical power into the generator– The frequency of the
generator increases (fg>f )
– The angle increases
V
ffg
f1
Ef
Ef
ffg
2I1 X
I2 X
at t1
at t2
– Hence, the power increases
49©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
50©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
1Before disturbance
Pm1= Pe 0 n = ns 1
51©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
1 to 2After increasing Pm
Pm2 > Pe + (n↑) n > ns ↑
52©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
2 Pm2 = Pe 0 n > ns 2
53©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
After 2 Pm2 < Pe - (n↓) n > ns ↑
54©Mohamed El-Sharkawi, University of Washington
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Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
3 Pm2 < Pe - (n↓) n = ns 3
55©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
3 to 2 Pm2 < Pe - (n↓) n < ns ↓
56©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
2 Pm2 = Pe 0 n < ns 2
57©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
2 to 1 Pm2 > Pe + (n↑) n < ns ↓
58©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
3
3
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
1 Pm2 > Pe + (n↑) n = ns1
59©Mohamed El-Sharkawi, University of Washington
Swings due to Sudden increase in Mechanical Power
P
Pm2
2
2
Pm1
1
Operating point Powers Acceleration
Rotor Speed
Power angle
1 to 2 Pm2 > Pe + (n↑) n > ns ↑
60©Mohamed El-Sharkawi, University of Washington
11
Operating point Powers AccelerationRotorSpeed
Power angle
1Before disturbance
Pm1= Pe 0 n = ns 1
1 to 2After Pm increased Pm2 > Pe + (n↑) n > ns ↑
2 Pm2 = Pe 0 n > ns 2
2 to 3 Pm2 < Pe - (n↓) n > ns ↑
3 Pm2 < Pe - (n↓) n = ns 3
3 to 2 Pm2 < Pe - (n↓) n < ns ↓
2 Pm2 = Pe 0 n < ns 2
2 to 1 Pm2 > Pe + (n↑) n < ns ↓
1 Pm2 > Pe + (n↑) n = ns1
Summary
61©Mohamed El-Sharkawi, University of Washington
Swing Angle
3
2
1
Time
dt
dnPPm ~
62©Mohamed El-Sharkawi, University of Washington
Damped Oscillations (Stable)
3
2
1
Time
nDdt
dnMPPm
63©Mohamed El-Sharkawi, University of Washington
Damped Oscillations (Stable)
• Damping is due to several factors such as– Resistances of the various power system
components
– Controllers installed in the power plants
64©Mohamed El-Sharkawi, University of Washington
Undamped Oscillation (Unstable)
3
2
1
Time
When D is negative nDdt
dnMPPm
65©Mohamed El-Sharkawi, University of Washington
P
P m
2
E f2
E f1
E f2 > E f1
1
Effect of Excitation
sinX
EVP f
66©Mohamed El-Sharkawi, University of Washington
12
Effect of Increasing Excitation
• The maximum power that CAN be delivered increases
• The real power is unchanged (Pm
unchanged)
• The power angle decreases
67©Mohamed El-Sharkawi, University of Washington
Increase transmission Capacity
Pm Xl
G
Terminal bus
Xs
Infinite bus
VtVo
sinX
EVP f
68©Mohamed El-Sharkawi, University of Washington
69©Mohamed El-Sharkawi, University of Washington
Ia
Xs
Ef
Xl2
Vt V
Xl1
Pm Xl
G
Xs Vt V
70©Mohamed El-Sharkawi, University of Washington
P
Pm
2
sin)XX(
EVP
ls
f
2
sin)X.X(
EVP
ls
f
501
1
sinX
EVP f
71©Mohamed El-Sharkawi, University of Washington
Dynamic Stability Assessment (DSA)
• DSA Methods• Time-domain solution• Energy Margin• Equal area• Eigenvalues• Pattern Recognition
72©Mohamed El-Sharkawi, University of Washington
13
Time Domain Solution
• Time domain methods seek to set up and solve a set of differential equations that describe the motion of the machines connected to the system
• Advantage:– Direct numerical integration can provide accurate
information on the stability of the system.• Disadvantage:
– The numerical integration is performed in each time interval; time consuming and slow
73©Mohamed El-Sharkawi, University of Washington
Time Domain Solution
Time
Fre
que
ncy
Time
Fre
que
ncy
Unstable System
74©Mohamed El-Sharkawi, University of Washington
Time Domain Solution
Time
Fre
que
ncy
Stable System
75©Mohamed El-Sharkawi, University of Washington
Equal Area Criterion
• A DSA method that represents the kinetic energy gained or lost due to oscillations.
• If the deceleration energy is equal or larger than the acceleration energy, the system is stable.
76©Mohamed El-Sharkawi, University of Washington
Equal Area Criterion
• Kinetic Energy KE
2n~KE Change in Kinetic Energy KE
2n~KE
77©Mohamed El-Sharkawi, University of Washington
Equal Area Criterion• Assume a sudden increase in the mechanical
energy to the generator– The speed of the machines increases from n1 to n2
78©Mohamed El-Sharkawi, University of Washington
P
3
3
Pm2
2
2
Pm1
1
14
Analysis of Equal Area Criterion
• Acceleration Kinetic Energy KEa
21
2212 nn~KEKEKEa
Decelerated Kinetic Energy KEd
23
2232 nn~KEKEKEd
Condition for stable system
ad KEKE
79©Mohamed El-Sharkawi, University of Washington
P
3
3
Pm2
2
2
Pm1
1
Stability Condition
13 nn
21
22
23
22 nnnn
KEKE ad
snn 1Since
Hencesnn 3
80©Mohamed El-Sharkawi, University of Washington
P
3
3
Pm2
2
2
Pm1
1
Equal Area Criterion
d2
a2
3
2
2
1
KEdtPP
KEdtPP
t
t
m
t
t
m
Since
Then
unbalancepower PPm
81©Mohamed El-Sharkawi, University of Washington
Equal Area Criterion
d2
a2
3
2
2
1
1
1
KEdPPn
KEdPPn
m
m
Since
Then
ndt
dnn s
82©Mohamed El-Sharkawi, University of Washington
Stability Condition
da
mm
AA
dPPdPP
areaon Deceleratiareaon Accelerati
3
2
2
1
22
Since
Then da KEKE
n
constant
83©Mohamed El-Sharkawi, University of Washington
Representation of Aa and Ad
P
3
3
Pm2
2
2
Pm1
1
Ad
Aa
For stable SystemAa = Ad
84©Mohamed El-Sharkawi, University of Washington
15
Unstable System
P
3 max
3-maxPm2
2
2
Pm1
1
Ad(max)
AaIf at 3-max
n3 > ns
ThenAa > Ad(max)
85©Mohamed El-Sharkawi, University of Washington
General Stability Condition
(max)
3
2
2
1
22
da
mm
AA
dPPdPPmax
(maximum)da KEKE
86©Mohamed El-Sharkawi, University of Washington
General Stability Condition
(max)
3
2
2
1
22
da
mm
AA
dPPdPPmax
(maximum)da KEKE
87©Mohamed El-Sharkawi, University of Washington
Example: Opened Breaker
TLG
CB
Breaker
Assume that the CB is opened for a short time
88©Mohamed El-Sharkawi, University of Washington
Analysis of Opened Breaker
P
3
3
cClearing angle
2
Pm
1
AdAa
If n3 = ns
OrAa = Ad,
the system is stable
89©Mohamed El-Sharkawi, University of Washington
Critical Clearing Angle
P
3 max
3-max
crCritical Clearing angle
2
Pm
1
Ad min
Aa max
The critical clearing angle (cr)
is the maximum angle for a stable system, i.e. whenAa max = Ad min
90©Mohamed El-Sharkawi, University of Washington
16
Critical Clearing Angle
1
11
0
crmmmmaxa PdPdPPA
crcr
crmcrmax
maxmmmind
PcoscosP
dsinPPdPPAcrcr
11
11
10.91©Mohamed El-Sharkawi, University of Washington
Critical Clearing Angle
1112 cossincos cr
Then
111 crmcrmaxcrm
mindmaxa
PcoscosPP
AA
For stable system
10.92©Mohamed El-Sharkawi, University of Washington
Energy Margin
• To predict the transient behavior of a power system without having to conduct a complete time domain simulation.
• The values of an energy function is calculated and compared with the critical value to determine the stability.
• Accurate only when the operating point is within the region of energy function.
• Because of the approximation in the energy function, results are often pessimistic
93©Mohamed El-Sharkawi, University of Washington
Energy Margin: Steps
• Calculate the transient energy at the instant the disturbance is cleared.
• Determine the critical energy for the current disturbance.
• Calculate the transient energy margin.
94©Mohamed El-Sharkawi, University of Washington
Eigenvalues Method
• The power system is converted into a set of linear equations.
• The dynamic behavior can be analyzed by any of the linear techniques – Eigenvalues– Root locus plots– Nyquist criteria– Routh-Hurwitz criteria
• Inaccuracies resulting from representing a highly nonlinear system by a set of linear equations.
95©Mohamed El-Sharkawi, University of Washington
Eigenvalues Method
• The system is represented by
dxadt
dx
dxax
Where d is a disturbance
96©Mohamed El-Sharkawi, University of Washington
17
Justification
• The solution of the equation is
j
edx t
where
Root (eigenvalue)Real component
Imaginary component
97©Mohamed El-Sharkawi, University of Washington
If is Negative (Stable System)
n
ns
Time
98©Mohamed El-Sharkawi, University of Washington
If is Positive (Unstable System)
n
ns
Time
99©Mohamed El-Sharkawi, University of Washington
Eigenvalues Method
• A linear analysis of the system
• Requires the detailed model of the system and the knowledge of all its parameters
100©Mohamed El-Sharkawi, University of Washington