Power System Stability Training...

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Power System StabilityTraining Course

DIgSILENT GmbH

Fundamentals on Power System Stability 2

General Definitions

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• „Stability“ - general definition:

Ability of a system to return to a steady state after a disturbance.

• Small signal effects• Large signal effects (nonlinear dynamics)

• Power System Stability - definition according to CIGRE/IEEE:• Rotor angle stability (oscillatory, transient-stability)• Voltage stability (short-term, long-term, dynamic)• Frequency stability

Power System Stability


Ability of a power system to compensate for a power deficit:1. Inertial reserve (network time constant)

Lost power is compensated by the energy stored in rotating masses of all generators -> Frequency decreasing

2. Primary reserve:Lost power is compensated by an increase in production of primary controlled units. -> Frequency drop partly compensated

3. Secondary reserve:Lost power is compensated by secondary controlled units. Frequency and area exchange flows reestablished

4. Re-Dispatch of Generation

Frequency Stability

3


• Frequency disturbance following to an unbalance in active power

Frequency Deviation according to UCTE design criterion

-0,9

-0,8

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

-10 0 10 20 30 40 50 60 70 80 90

dF in Hz

t in s

Rotor Inertia Dynamic Governor Action Steady State Deviation

Frequency Stability


• Mechanical Equation of each Generator:

• ∆P=ω∆T is power provided to the system be each generating unit.• Assuming synchronism:

• Power shared according to generator inertia

nn

elmelm

PPPTTJωω

ω ∆=

−≈−=&

j

i

j

i

ini

JJ

PP

PJ

=∆∆

∆=ωω &

Inertial Reserve

4


• Steady State Property of Speed Governors:

• Total frequency deviation:

• Multiple Generators:

• Power shared reciprocal to droop settings

( )∑∑ ∆

=∆⇒∆=∆i

totitot K

PffKP

i

j

j

i

jjii

RR

PP

PRPR

=∆∆

∆=∆

PRPK

ffKP iii

ii ∆=∆=∆⇒∆=∆1

Primary Control


Turbine 1

Turbine 2

Turbine 3

Generator 1

Generator 2

Generator 3

Network

Secondary Control

PT PG

PT PG

PT PG

f PA

Set Value

Set Value

Set Value

Contribution

• Bringing Back Frequency• Re-establishing area exchange flows• Active power shared according to participation factors

Secondary Control

5


Frequency drop depends on:• Primary Reserve• Speed of primary control• System inertia

Additionally to consider:• Frequency dependence of load

In case of too severe frequency drops:• Load shedding

Frequency Stability


20.0015.0010.005.000.00 [s]

1.025

1.000

0.975

0.950

0.925

0.900

0.875

G 1: Turbine Power in p.u.G2: Turbine Power in p.u.G3: Turbine Power in p.u.

20.0015.0010.005.000.00 [s]

0.125

0.000

-0.125

-0.250

-0.375

-0.500

-0.625

Bus 7: Deviation of the El. Frequency in Hz

DIgSILENT Nine-bus system MechanicalSudden Load Increase

Date: 11/10/2004

Annex: 3-cycle-f. /3

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Frequency Stability

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• Dynamic Simulations

• Sometimes possible: Inertial/Primary controlled or secondary controlled load flows

Frequency Stability - Analysis


Small signal rotor angle stability (Oscillatory stability)Ability of a power system to maintain synchronism under small

disturbances

– Damping torque– Synchronizing torque

Especially the following oscillatory phenomena are a concern:– Local modes– Inter-area modes– Control modes– Torsional modes

Rotor Angle Stability

7


Small signal rotor angle stability (Oscillatory stability) is a system property

Small disturbance -> analysis using linearization around operating point

Analysis using eigenvalues and eigenvectors



Large signal rotor angle stability (Transient stability)Ability of a power system to maintain synchronism during severe

disturbances

– Critical fault clearing time

Large signal stability depends on system properties and the type of disturbance (not only a system property)

– Analysis using time domain simulations


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3.2342.5871.9401.2940.650.00 [s]

200.00

100.00

0.00

-100.00

-200.00

G1: Rotor angle with reference to reference machine angle in deg

DIgSILENT Transient Stability Subplot/Diagramm

Date: 11/11/2004

Annex: 1 /3

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4.9903.9922.9941.9961.000.00 [s]

25.00

12.50

0.00

-12.50

-25.00

-37.50

G1: Rotor angle with reference to reference machine angle in deg

DIgSILENT Transient Stability Subplot/Diagramm

Date: 11/11/2004

Annex: 1 /3

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Transient Stability


Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance.

• Small disturbance voltage stability (Steady state stability)– Ability to maintain steady voltages when subjected to small

disturbances

• Large signal voltage stability (Dynamic voltage stability)

– Ability to maintain steady voltages after following large disturbances

Voltage Stability

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- Dynamic models (short-term), special importance on dynamic load modeling, stall effects etc.

Short-Term

- P-V-Curves (load flows)of the faulted state.- Long-term dynamic models including tap-changers, var-control, excitation limiters, etc.

- P-V-Curves (load flows)- dv/dQ-Sensitivities- Long-term dynamic models including tap-changers, var-control, excitation limiters, etc.

Long-Term

Large-Signal- System fault- Loss of generation

Small-Signal:- Small disturbance

Voltage Stability - Analysis


151.30138.80126.30113.80101.3088.80

1.10

1.00

0.90

0.80

0.70

0.60

0.50

x-Axis: U_P-Curve: Total Load of selected loads in MWAMBOWS51: Voltage, Magnitude in p.u.ANGONS51: Voltage, Magnitude in p.u.BELLES51: Voltage, Magnitude in p.u.BISSES51: Voltage, Magnitude in p.u.BISSES61: Voltage, Magnitude in p.u.

PV-curves U_P-Curve

Date: 11/11/2004

Annex: 1 /1

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Small-Signal Voltage Stability –PV-Curves

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20.0015.0010.005.000.00 [s]

1.25

1.00

0.75

0.50

0.25

0.00

-0.25

APPLE_20: Voltage, Magnitude in p.u.SUMMERTON_20: Voltage, Magnitude in p.u.LILLI_20: Voltage, Magnitude in p.u.BUFF_330: Voltage, Magnitude in p.u.

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Fault with loss of transmission line

Large-Signal Long-TermVoltage Instability


• Dynamic voltage stability problems are resulting from sudden increase in reactive power demand of induction machine loads.

-> Consequences: Undervoltage trip of one or several machines, dynamic voltage collapse

• Small synchronous generators consume increased amount of reactive power after a heavy disturbance -> voltage recovery problems.

-> Consequences: Slow voltage recovery can lead to undervoltagetrips of own supply -> loss of generation

Dynamic Voltage Stability

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1.201.161.121.081.041.00

3.00

2.00

1.00

0.00

-1.00

x-Axis: GWT: Speed in p.u.GWT: Electrical Torque in p.u.

1.201.161.121.081.041.00

0.00

-2.00

-4.00

-6.00

-8.00

x-Axis: GWT: Speed in p.u.GWT: Reactive Power in Mvar

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Dynamic Voltage Stability –Induction Generator (Motor)


1.041.031.021.011.00

3.00

2.00

1.00

0.00

-1.00

x-Axis: GWT: Speed in p.u.GWT: Electrical Torque in p.u.

Constant Y = 1.000 p.u. 1.008 p.u.

1.041.031.021.011.00

0.00

-1.00

-2.00

-3.00

-4.00

-5.00

-6.00

x-Axis: GWT: Speed in p.u.GWT: Reactive Power in Mvar

Constant X = 1.008 p.u.

-1.044 Mvar

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2.001.501.000.500.00 [s]

1.20

1.00

0.80

0.60

0.40

0.20

0.00

G\HV: Voltage, Magnitude in p.u.MV: Voltage, Magnitude in p.u.

2.001.501.000.500.00 [s]

80.00

40.00

0.00

-40.00

-80.00

-120.00

Cub_0.1\PQ PCC: Active Power in p.u.Cub_0.1\PQ PCC: Reactive Power in p.u.

2.001.501.000.500.00 [s]

1.06

1.04

1.02

1.00

0.98

GWT: Speed

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3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

Cub_0.1\PQ RedSunset: Active Power in p.u.Cub_0.1\PQ RedSunset: Reactive Power in p.u.

3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

Cub_0.2\PQ BlueMountain: Active Power in p.u.Cub_0.2\PQ BlueMountain: Reactive Power in p.u.

3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

-60.00

Cub_1.1\PQ GreenField: Active Power in p.u.Cub_1.1\PQ GreenField: Reactive Power in p.u.

3.002.001.000.00 [s]

1.125

1.000

0.875

0.750

0.625

0.500

0.375

GLE\1: Voltage, Magnitude in p.u.GLZ\2: Voltage, Magnitude in p.u.WDH\1: Voltage, Magnitude in p.u.

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Dynamic Voltage Collapse

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3.002.001.000.00 [s]

1.20

1.00

0.80

0.60

0.40

0.20

0.00

HV: Voltage, Magnitude in p.u.MV: Voltage, Magnitude in p.u.

3.002.001.000.00 [s]

120.00

80.00

40.00

0.00

-40.00

-80.00

-120.00

Cub_1\PCC PQ: Active Power in p.u.Cub_1\PCC PQ: Reactive Power in p.u.

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Dynamic Voltage Stability –Voltage Recovery (Synchronous Generators)


Time Domain Simulation

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Fast Transients/Network Transients:Time frame: 10 mys…..500ms

LighteningSwitching OvervoltagesTransformer Inrush/Ferro ResonanceDecaying DC-Components of short circuit currents

Transients in Power Systems


Medium Term Transients / Electromechanical TransientsTime frame: 400ms….10s

Transient StabilityCritical Fault Clearing TimeAVR and PSSTurbine and governorMotor startingLoad Shedding


15


Long Term Transients / Dynamic PhenomenaTime Frame: 10s….several min

Dynamic StabilityTurbine and governorPower-Frequency ControlSecondary Voltage ControlLong Term Behavior of Power Stations



Stability/EMT

Different Network Models used:

Stability:

EMT:

ILjV ω= VCjI ω=

dtdiLv =

dtdvCi =

16


Short Circuit Current EMT

0.50 0.38 0.25 0.12 0.00 [s]

800.0

600.0

400.0

200.0

0.00

-200.0

4x555 MVA: Phase Current B in kA

Short Circuit Current with complete model (EMT-model) Plots

Date: 4/25/2001

Annex: 1 /1

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Short Circuit Current RMS

0.50 0.38 0.25 0.12 0.00 [s]

300.0

250.0

200.0

150.0

100.0

50.00

0.00

4x555 MVA: Current, Magnitude in kA

Short Circuit Current with reduced model (Stability model) Plots

Date: 4/25/2001

Annex: 1 /1

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(X)X

X0

Dynamic voltage stabilitySelf excitation of ASM

X(X)HVDC dynamicsX0Switching Over Voltages

X0Transformer/Motor inrush(X)XAVR and PSS dynamics

((X))XOscillatory stability

XX

X0

Torsional oscillationsSubsynchronous resonance

(X)X

X0

Dynamic motor startupPeak shaft-torque

(X)XCritical fault clearing time

EMT-SimulationRMS-SimulationPhenomena

RMS-EMT-Simulation



Fundamental Concepts

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One Machine Problem

DIgSILENT

PowerFactory 12.1.178

Example

Power System Stability and Control One Machine Problem

Project: Training Graphic: Grid Date: 4/19/2002 Annex: 1

G ~ G1

Gen

222

0MV

A/2

4kV

(1)

1998

.000

MW

967.

920

Mva

r53

.408

kA

1.16

3 p.

u.-0

.000

p.u

.

Trf500kV/24kV/2220MVA

-199

8.00

MW

-634

.89

Mva

r2.

56 k

A

1998

.00

MW

967.

92 M

var

53.4

1 kA

CCT 2Type CCT186.00 km

-698

.60

MW

30.4

4 M

var

0.90

kA

698.

60 M

W22

1.99

Mva

r0.

90 k

A

CCT1Type CCT100.00 km

-129

9.40

MW

56.6

2 M

var

1.67

kA

1299

.40

MW

412.

90 M

var

1.67

kA

V ~

Infin

ite S

ourc

e

-199

8.00

MW

87.0

7 M

var

2.56

kA

Infin

ite B

us50

0.00

kV

450.

41 k

V0.

90 p

.u.

0.00

deg

HT

500.

00 k

V47

2.15

kV

0.94

p.u

.20

.12

deg

LT24

.00

kV24

.00

kV1.

00 p

.u.

28.3

4 de

g

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One Machine Problem

0E

ePX

'GE

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One Machine Problem

• Power transmission over reactance:

• Mechanical Equations:

0

0

ωωϕωω

ω

−=

−≈

−=

G

emem PPPPJ

&

&

( )

( )( )GGG

e

GG

e

EEXEQ

XEEP

ϕ

ϕ

cos

sin

0'

'

'0

−=

=


One Machine Problem

• Differential Equation of a one-machine infinite bus bar system:

• Eigenvalues (Characteristic Frequency):

• Stable Equilibrium points (SEP) exist for:

GGGm

Gm

G

PPPPPJ ϕϕ

ωϕ

ωωϕ

ωωϕ ∆⎟⎟

⎠

⎞⎜⎜⎝

⎛−−≈−= 0

0

max0

0

max

00

max

0

cossinsin&&

00

max2/1 cos GJ

P ϕω

λ −±=

0cos 0 >Gϕ

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Small Signal Stability

180.0144.0108.072.0036.00 0.00

4000.

3000.

2000.

1000.

0.00

-1000...

x-Axis: Plot Power Curve: Generator Angle in degPlot Power Curve: Power 1 in MWPlot Power Curve: Power 2 in MW

Pini y=1998.000 MW

DIgSILENT Single Machine Problem P-phi

Date: 4/19/2002

Annex: 1 /4

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SEP UEP


Transient Stability

• Energy Function:

• At Maximum Angle:

( ) 0)(21

0

2 =+=−

+ ∫ potkinem

G EEdPPJG

ϕω

ϕϕ

ϕ

&

0max =Gϕ&

0)(max

0

=−

= ∫ ϕω

ϕ

ϕ

dPPEG

empot

( )0=kinE

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Equal Area Criterion

180.0144.0108.072.0036.000.00

4000.

3000.

2000.

1000.

0.00

-1000...

x-Axis: Plot Power Curve: Generator Angle in degPlot Power Curve: Power 1 in MWPlot Power Curve: Power 2 in MW

DIgSILENT Single Machine Problem P-phi Date: 4/19/2002

Annex: 1 /4

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E1

E2

0ϕ cϕ

maxϕ

SEP UEP

critϕPm



21 EE −=

∫=c

dPE m

ϕ

ϕ

ϕω

0

11

( )∫ −=max

)sin(1max2

ϕ

ϕ

ϕϕω

c

dPPE m

Stable operation if:

22



)(101 ϕϕ

ω−= cmPE

)cos(cos)( maxmax

max2 ccm PPE ϕϕ

ωϕϕ

ω−+−=

000 cossin)2(cos ϕϕϕπϕ −−=c

Setting and equating E1 and -E2:0ϕπϕ −=crit


Critical Fault Clearing Time

• During Short Circuit:

• Differential Equation:

• Critical Fault Clearing Time:

02

02ϕ

ωϕ += c

mc t

JP

0=eP

0ωϕ m

GPJ =&&

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Voltage Stability



0E

eQX

'GE

( )

( )( )GGG

e

GG

e

EEXEQ

XEEP

ϕ

ϕ

cos

sin

0'

'

'0

−=

=

Voltage Stability

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1762.641462.641162.64862.64562.64262.64

1.40

1.20

1.00

0.80

0.60

0.40

x-Achse: SC: Blindleistung in MvarSC: Voltage in p.u., P=1400MWSC: Voltage in p.u., P=1600MWSC: Voltage in p.u., P=1800MWSC: Voltage in p.u., P=2000MW

P=2000MW

P=1800MW

P=1600MW

P=1400MW

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const. P, variable Q

Voltage Stability – Q-V-Curves


1350.001100.00850.00600.00350.00100.00

1.00

0.90

0.80

0.70

0.60

0.50

x-Achse: U_P-Curve: Total Load of selected loads in MWKlemmleiste(1): Voltage in p.u., pf=1Klemmleiste(1): Voltage in p.u., pf=0.95Klemmleiste(1): Voltage in p.u., pf=0.9

pf=1

pf=0.95

pf=0.9

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const. Power factor, variable P

Voltage Stability – P-V-Curves

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Dynamic Stability / Eigenvalue Analysis



Small signal analysis

• Linear model automatically generated by linearizing the stability model.

• Calculation of eigenvalues, eigenvectors and participation factors

• Calculation of all modes using QR-algorithm -> limited to systems up to 500..1000 state variables

• Calculation of selected modes using implicitly restarted Arnoldi method -> application to large systems (released in Summer 2001)

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Small signal analysis

• Linear System Representation:

• Transformation:

• Transformed System

• Diagonal System

bAxx +=&

xTx ~=

TbxTATx += − ~~ 1&

TbxDx += ~~&

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