NERC Standard System Operating Limit (SOL) · Ia Xs cos Ef sin s f a X E sin ... Rotation of Phasor Diagram fg f 60 or50Hz V Infinite bus Ef Rotor f is the frequency at the infinite

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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:


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


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


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


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.


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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


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


Balance of power in Generators

GPm Pe

MechanicalPower

Controlled atthe power plant

ElectricalPower

Controlled bythe customers

Pm = PeAt steady state


Turbine Speed and Imbalance of Power

dt

dnPP em ~

GPm

Pe

sem nnPPif ;

sem nnPPif ;

0; dt

dnPPif em

speedrotor constant ;nn s


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)


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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


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


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)


Unstable System

n (f)

ns=60Hz


Unstable System

Time

n (f)

ns=60Hz


Stable System

Time

n (f)

ns=60Hz


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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

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N

S

StatorRotor

X

X

c

b

a

Xc

b


Time

Vaa’Vbb’ Vcc’

X

X

c

b

a

Xc

b

N

S

f


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


Xs R

Ef Vt

sXRcetansisReArmatureR

cetanacResSynchronouX s

Equivalent Circuit


Generator Equivalent Circuit

satf XIVE

Ia

Xs

Ef Vt


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

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Power equations

sinIVQ att 3

cosIVP at3

Ia

Xs

Ef Vt

Ia

Vt

Ef

Ia Xs

Vt and Ef are phase quantities

f t


Power equation

sinEcosXI fsa

s

fa X

sinEcosI

sinX

EV3P

s

ft

cosIV3P at

Ia

Vt

Ef

Ia Xs


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


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


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


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Power Equation

I

X = Xs+Xline

Ef V

Power Reactive

Power Real

VcosEX

VQ

sinX

VEP

f

f

V

Ef


Power Characteristics of Generator

sinX

EVP f


P

Pmax

90o

Pm

Maximum Power

P

Pm

Pmax

l 90o

Phasor Diagram atPmax

V

Ef


Operating Point

Pm

P

Operating point

sinX

VEP f


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.


Dynamic Stability Assessment (DSA)


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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


Phasor diagram

I

X

Ef V

V

I X

I

Ef


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


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


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


Power Control

dt

dn~PPm G

Pm P

increasesn;PPif m decreasesn;PPif m

0dt

dn;PPif m speedrotor constant ;nn s


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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


Swings due to Sudden increase in Mechanical Power

P

3

3

Pm2

2

2

Pm1

1



P

Pm1

1

Operating point Powers Acceleration

Rotor Speed

Power angle

1Before disturbance

Pm1= Pe 0 n = ns 1



P

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

1 to 2After increasing Pm

Pm2 > Pe + (n↑) n > ns ↑



P

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

2 Pm2 = Pe 0 n > ns 2



P

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

After 2 Pm2 < Pe - (n↓) n > ns ↑


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P

3

3

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

3 Pm2 < Pe - (n↓) n = ns 3



P

3

3

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

3 to 2 Pm2 < Pe - (n↓) n < ns ↓



P

3

3

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

2 Pm2 = Pe 0 n < ns 2



P

3

3

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

2 to 1 Pm2 > Pe + (n↑) n < ns ↓



P

3

3

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

1 Pm2 > Pe + (n↑) n = ns1



P

Pm2

2

2

Pm1

1


Rotor Speed

Power angle

1 to 2 Pm2 > Pe + (n↑) n > ns ↑


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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


Swing Angle

3

2

1

Time

dt

dnPPm ~


Damped Oscillations (Stable)

3

2

1

Time

nDdt

dnMPPm


Damped Oscillations (Stable)

• Damping is due to several factors such as– Resistances of the various power system

components

– Controllers installed in the power plants


Undamped Oscillation (Unstable)

3

2

1

Time

When D is negative nDdt

dnMPPm


P

P m

2

E f2

E f1

E f2 > E f1

1

Effect of Excitation

sinX

EVP f


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Effect of Increasing Excitation

• The maximum power that CAN be delivered increases

• The real power is unchanged (Pm

unchanged)

• The power angle decreases


Increase transmission Capacity

Pm Xl

G

Terminal bus

Xs

Infinite bus

VtVo

sinX

EVP f



Ia

Xs

Ef

Xl2

Vt V

Xl1

Pm Xl

G

Xs Vt V


P

Pm

2

sin)XX(

EVP

ls

f

2

sin)X.X(

EVP

ls

f

501

1

sinX

EVP f


Dynamic Stability Assessment (DSA)

• DSA Methods• Time-domain solution• Energy Margin• Equal area• Eigenvalues• Pattern Recognition


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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



Time

Fre

que

ncy

Time

Fre

que

ncy

Unstable System



Time

Fre

que

ncy

Stable System


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.



• Kinetic Energy KE

2n~KE Change in Kinetic Energy KE

2n~KE


Equal Area Criterion• Assume a sudden increase in the mechanical

energy to the generator– The speed of the machines increases from n1 to n2


P

3

3

Pm2

2

2

Pm1

1

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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


P

3

3

Pm2

2

2

Pm1

1

Stability Condition

13 nn

21

22

23

22 nnnn

KEKE ad

snn 1Since

Hencesnn 3


P

3

3

Pm2

2

2

Pm1

1


d2

a2

3

2

2

1

KEdtPP

KEdtPP

t

t

m

t

t

m

Since

Then

unbalancepower PPm



d2

a2

3

2

2

1

1

1

KEdPPn

KEdPPn

m

m

Since

Then

ndt

dnn s


Stability Condition

da

mm

AA

dPPdPP

areaon Deceleratiareaon Accelerati

3

2

2

1

22

Since

Then da KEKE

n

constant


Representation of Aa and Ad

P

3

3

Pm2

2

2

Pm1

1

Ad

Aa

For stable SystemAa = Ad


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Unstable System

P

3 max

3-maxPm2

2

2

Pm1

1

Ad(max)

AaIf at 3-max

n3 > ns

ThenAa > Ad(max)


General Stability Condition

(max)

3

2

2

1

22

da

mm

AA

dPPdPPmax

(maximum)da KEKE


General Stability Condition

(max)

3

2

2

1

22

da

mm

AA

dPPdPPmax

(maximum)da KEKE


Example: Opened Breaker

TLG

CB

Breaker

Assume that the CB is opened for a short time


Analysis of Opened Breaker

P

3

3

cClearing angle

2

Pm

1

AdAa

If n3 = ns

OrAa = Ad,

the system is stable


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


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1

11

0

crmmmmaxa PdPdPPA

crcr

crmcrmax

maxmmmind

PcoscosP

dsinPPdPPAcrcr

11

11

10.91©Mohamed El-Sharkawi, University of Washington


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


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.


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.


Eigenvalues Method

• The system is represented by

dxadt

dx

dxax

Where d is a disturbance


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Justification

• The solution of the equation is

j

edx t

where

Root (eigenvalue)Real component

Imaginary component


If is Negative (Stable System)

n

ns

Time


If is Positive (Unstable System)

n

ns

Time


Eigenvalues Method

• A linear analysis of the system

• Requires the detailed model of the system and the knowledge of all its parameters


Documents

NERC Standard System Operating Limit (SOL) · Ia Xs cos Ef sin s f a X E sin ... Rotation of Phasor Diagram fg f 60 or50Hz V Infinite bus Ef Rotor f is the frequency at the infinite