Department of Electrical and Computer Engineer Doctoral ... · PDF fileWind injection at location ... rr g dtT g dt r r a dt dt gg r T gg dt gg g g B Jt tK t N BB tTt BK Jt t t NN

Modeling, Analysis and Control of Oscillations in Wind-integrated Power

Systems

North Carolina State UniversityRaleigh, NC

September 18, 20151

Souvik ChandraDepartment of Electrical and Computer Engineer

Doctoral Oral Examination

Souvik Chandra

Growth of Wind Installation Capacity

Modeling, Analysis and Control of Wind-Integrated Power Systems 2

Average annual growth of 20% globally. Clean renewable source of energy. Component level technology has matured

and become cost-effective.

Souvik Chandra

Growth of Wind Power in US


Average annual growth rate of 29.7% for over a decade. Driven by Government mandates: 20% wind by 2030. Research initiatives funded from public and private resources.

Source: AWEA

Installed capacity67,870 MW

(as of June 2015)

Souvik Chandra

Trends in Wind Integration Technology


Fixed-speed wind turbine (Type 1)

DriveTrain

SquirrelCage IM

Transformer

To grid

Variable-speed wind turbine (Type 3)

DriveTrain

DFIG

Transformer

To grid

Controls

Connected asynchronously to the power system via Squirrel cageinduction machines.

Operates at Low efficiency, no reactive power capability

Ratings can only be in 10s of KWs.

Most widely Connected to the power system via Doubly-fed induction machines.

Can operate at maximum efficiency, with reactive power capability.

Ratings can be up to 5-7 MW.

Souvik Chandra

Wind Integration via Large Wind Power Plants


The synchronous generation in power systems is getting replaced by wind.

Large Wind power plants(WPP) with 100s of individual turbinesare connected to the grid.

Ratings of these WPP can be 500-1000 MW, injecting power at a point of common coupling in the grid.

e.g. Alta Wind Energy Center (1320 MW) at California,Shepherds Flat Wind Farm (845 MW) at Oregon,Roscoe Wind farm (781 MW) at Texas,Flower Ridge Wind Farm (600 MW) Indiana.

Souvik Chandra

Motivation of this thesis


Main Questions we address in this thesis are : Will high penetration of wind alter the characteristics of the dynamic model of the

conventional power system ? If so in what respect and how ?Impact on various types of power system dynamics:

Power system stability dynamics

FrequencyStability

Rotor angle stability

VoltageStability

Large-disturbance

Small-disturbance

Large-disturbance

Small-disturbance

[Slootweg et al. 2003 , Tsourakis et.al. 2009, Vittal et. al. 2010]

Souvik Chandra

Outline of the Presentation


Modeling, Analysis and Control of Oscillations in a Wind‐integrated Power System Using a Continuum Model Approach

Time Scale Modeling of Power Systems with Wind Injections

Equilibria Analysis and Voltage Stability Limit of Wind‐injected Power Systems

PART I

PART II

PART III

8

PART I

Modeling, Analysis and Control of Oscillations in a Wind-integrated Power System Using a Continuum Model Approach

Souvik Chandra

Background: Rotor angle stability


i mi siim P P

Maintenance of synchronism of synchronous generators after being subjected to a disturbance

Equilibrium of mechanical and electrical power input in the rotor of a synchronous generator

It can be of two types : Large-disturbance rotor angle stability

System faults, loss of generation, or circuit contingencies etc. Analyzed through nonlinear response of the power system over a

period of time Small-disturbance rotor angle stability

Incremental changes in system load, control action etc. Analyzed through linearized model of power system about an

operating point

Souvik Chandra

Inter and Intra-area oscillations


Two types of oscillations Intra-area oscillations

between generators operating in the same area

1.0-2.0 Hz Inter-area oscillations

between groups of machines in different geographically separated generation areas

0.1-1.0 Hz

In small-signal rotor angle stability the eigenvalues of the linearized system determine the dynamic response or the oscillatory modes of the power system.

Souvik Chandra

Previous Works


1. Slootweg et. al., “The impact of large scale wind power generation on power system oscillations”, Electric Power Systems Research, 2003.

2. Tsourakis et. al. Effect of wind parks with doubly fed asynchronous generators on small-signal stability, Electric Power Systems Research, 2009.

3. Vittal et. al. "Impact of Increased Penetration of DFIG-Based Wind Turbine Generators on Transient and Small Signal Stability of Power Systems," in Power Systems, IEEE Transactions , 2009.

A number of works have been done on this topic :

However these works are based mainly nonlinear simulations over small power system models.

In the proposed work a general wind-integrated power system model is derived to demonstrate the effect of penetration levels on the eigen values of the small-signal model analytically.

Also no literature have looked on the effect of wind penetration on coherency or time scales of the power system.

Souvik Chandra

Organization


Background -Small-signal rotor angle stability and oscillations in power system

Continuum modelling and Spectral response-Representation of a large radial power system-Spectral response via Fourier analysis

Impact assessment of wind penetration-Wind farm model and frequency response-Impact of wind location

Design Controller to damp oscillations-Control of wind farm power-Coordinated control of wind farm and battery energy system

Implementation and simulation results

Souvik Chandra

Continuum model of a radial power system


Radial power system with a string of generators

1, , 1i i i i iM P P

[Cresap & Hauer 1981]Power output of ith machine:

Swing dynamics of ith machine : G1 G2 G3 Gn

1jx 2jx

1,2P 2,3P1E 2E 3E nE, 1n nP

1njx

0u 1u

1

1 12

1i i

i i i ii i i x x

L L

ML L

Taking the limit as 10, , i iL x x n 2 2

22 2

( , ) ( , ) ( , )u t u t u tt t u

A continuum model:

Souvik Chandra

Wind injection at location α


Forcing function

22 T

fH

Averaged damping density

Wind farm injects power2 2

22 2 ( , )W u t

t t u

G1 G2 G3 Gn

1jx 2jx

1,2P 2,3P1E 2E 3E nE, 1n nP

1njx

0u 1u

L

1

ˆ( , ) ( ) ( )W u t P t u

Wave speed-•Inertia density (HT)• Reactance Density(γ) • Electrical Frequency(f)

Souvik Chandra

Frequency Response


We solve the continuum model using Fourier expansions of ( , ) & ( , ) u t W u t

We assume boundary conditions in the form of power flow at the two ends,

1 (0, ) 1 (1, )(0, ) ( 01, )

P t P tttt t

01

01

1( , ) [ ( ) cos( ) ( )sin( )]21( , ) [ ( ) cos( ) ( )sin( )]2

n n n nn

n n n nn

u t A A t k u B t k u

W u t F F t k u G t k u

1

1( , , ) [ ( , ) sin( )]

n n n

nu A k k u

Dependent on wind[Gayme & Chakrabortty, 2012a]

And compute the frequency response of the rotor angle density,

Souvik Chandra

Spectral response dependent on wind


1 ( , )( , ) u tP u tu

1

1 [ ( ) sin( )]n n nn

A t k k u

Frequency response for ( )nA t

2 22 2 2

2 ( ) cos( ) cos( ) sin( )( , ) g n n n

n

n

P k jA

k

Affected by the wind farm Power output and location

12 2 2where tannnk

The spectrum of power flow in the power system

[Gayme & Chakrabortty, 2012a]

Souvik Chandra

Wind farm model


2

Two mass model

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

dtr r g dt T

g

dt r r a

dt dtg g r T

g g

dtg g g

g

BJ t t K tN

B B t T tB KJ t t tN N

B B t T tN

3( ) ( ( ), ( ))( )

Aerodynamic torque i

(

np

)

ut

2s r p

ar

A v t C t tT t

t

Drive Train &

Gear Box

Torsion1( ) ( ) ( )T r g

g

t t tN

( )rv t

( )ref t

Pitch Control System

( )gT t( )g t( )aT t

( )r t

[Sloth et al. 2010]

Generator

( )gP t

,1 1( ) ( ) ( )g g g refg g

T t T t T t

Souvik Chandra

Linearization and frequency response


Parameter values [Sloth et al, ACC 2010]

3 2

6.786 1.939 004( )0.294 816.2 0.7696w

s eG ss s s

Transfer function model

Neglecting the generator pole due to a faster time constant

Souvik Chandra

Spectral Response of power flow


-Spectrum depends on location of the wind farm -Results are robust to wind farm parameter variations (Jg, Jr)

This spectrum is at a position

0.25u

Magnitude of power flow in

the grid

[Gayme & Chakrabortty, 2012a]

Souvik Chandra

Shaping of the spectral response


11 012

21 11 01

( )wq s qH s

p s p s p

( )gP t

G1 G2 G3

1jx 2jx

1,2P 2,3P1E 2E 3E nE, 1n nP

1njx

Controller parameters can be tuned to shape the oscillation spectrum of power flow damp particular inter-area modes

0.2 0.4 0.6 0.8 150

60

70

80

90

100

Frequency(Hz)

Mag

nitu

de(d

b)

=0.25

=0.5

Damping of the sharp peaks

[Gayme & Chakrabortty, 2012b]

Souvik Chandra

Control Problem Formulation


*

1 2

min ( , , ) , n

pcli

S S

*( , )pS

( , , )pclS

Actually the power flow spectrum is Spcl when wind farm and battery is positioned at while output power is controlled by a controller whose parameters are given by

Solution of an optimization problem provides the controller parameters that satisfies

Assume the spectrum Sp of the system over frequencies is optimal when the wind farm is at location * 1 2: , , n

[Chandra, 2013]

Souvik Chandra

Battery Energy System Model


Small signal linearized model

determined by battery characteristics

Usually large battery banks with power electronic control

of the input voltage

[Lu et al., 1995]

0 00

00( )( )

( )BES dc dc

dcP u uid s F s d

s

BESP

BTR

1BR

1BC

1BV

BSR

BPR BPC BOCVdcu

dciaebece

LLL C

Charging/Discharging circuit

Self-discharging circuit

Converter

Battery

Souvik Chandra

Add a battery energy system to the wind injection


• Adding a co-located BES the controller of which is co-optimized with the wind farm controller

( )gP t

11 012

21 11 01

( )wq s qH s

p s p s p

( )BESP t

12 022

22 12 02

( )wq s qH s

p s p s p

0 1 0 1 2: , , , ,i i i i i iq q p p pcontrollers parameterized by

Souvik Chandra

Optimization Strategy


2

1

2

1 2min log ( , , , ) log ( *, )i

cl pS S

Co-optimize the system to find that gives a controlled system with the desired frequency response

Spectrum of power system with wind farm and BES control( , , ) :clS ( *, ) :pS Spectrum at optimal position (a*) with open loop wind farm

Constraints: Closed loop stability for the individual battery and wind farm

'i s

Souvik Chandra

Improved Controller Performance with the BES


0.1 0.15 0.2 0.2555

60

65

70

75

80

windfarm at *=0.25controlled windfarm at =0.5controlled windfarm and battery at =0.5

0.5 0.55 0.6 0.65 0.750

55

60

65

70

75

windfarm at *=0.25controlled windfarm at =0.5controlled windfarm and battery at =0.5

Magnitude

(dB)

Frequency (Hz) Frequency (Hz)

• Controller works well over specific modes• Gain scheduling may produce desired results over a given

set of modes[Chandra et. Al., 2013]

Souvik Chandra

Application to an Actual Wind Farm


Actual Wind farm: Consists of multiple rows of turbines injecting power to a common wind bus

Operating point : Wind speed different in different rows due to Wake Effect, hence different operating point

How can the control be decentralized so as to include each row of turbines?

[Chandra et. Al., 2014]

Souvik Chandra

Decentralized control design


Centralized design difficult to be implemented online

Distribute the desired Power output among different rows of turbines

Design separate controllers for each row

The controllers for each row operate in parallel

2

1

2

,min log( ( , )) log ( )j

ig j j g

i

pP P d

N

2

1 21

* 211 1 2,

min [log( ( , , ,..., , )) log( ( , ))] , 1cl

iP N PS S d i

Centralized design:

Decentralized design:[Chandra, 2014]

Souvik Chandra

Comparison of the Decentralized and Centralized control


Response in centralized case does a better spectral matching In the decentralized case spectral matching is less accurate,

practically applicable

0.4 0.45 0.5 0.5565

70

75

80

85

90

95

100

105

Frequency(Hz)

Mag

nitu

de(d

b)

Reference trajectory (,$ = 0.25)

Uncontrolled system at, = 0.5Controlled distributed system at, = 0.5

Controlled aggregate system at, = 0.5

Souvik Chandra

Controlled time response at location u=0.25


Souvik Chandra

Conclusions of Part I


Derived a dynamic equivalent model of a large radial power system using a continuum approximation.

Obtained the spectral response of the wind integrated power system using Fourier Analysis.

The inter-area oscillatory spectrum of the power system is affected by the location and amount of wind injection.

Designed codependent controllers for the wind farm and storage elements to damp distinct oscillatory modes of the power system.

But obtaining a continuum model of a general power system of any structure is difficult. ODE based models are more relevant from an analysis point of view which we study in the next chapter.

31

PART II

Time Scale Modeling of Power Systems with Wind Injections

Modeling, Analysis and Control of Wind‐Integrated Power Systems

Souvik Chandra

Organization


1. Dynamic model of wind-integrated power system Synchronous generator model

Wind power plant model

Loads and transmission line model

2. Linearized model of wind-integrated power system

3. Time-scale modeling of wind-integrated power system Conventional coherency concepts

Extension to power systems containing wind

4. Simulation test cases 2-area 8-bus system

5-area 68-bus power system

5. Conclusions

Souvik Chandra

Dynamics of a Synchronous Generator


'

i

i

i

mi

di

si

si

mEP

xV

Machine angle

Machine inertia

Machine voltage

Mechanical input

Transient reactance

Bus voltage magnitude

Bus voltage angle

.

Swing dynamics of gen ,

i i

i i mi si

i

m P P

'

' '

2

sin cos

cos

Power output at bus

sin

Re Im

Re Im

isi si i si i

s

di

d

i isi si i s i

i dii

EP V V

E EQ

x

V V

i N

x x[P. Kundur 1994]

Souvik Chandra

Dynamics of a Wind Generator : Mechanical Section (Turbine Rotor)


2

Two mass model

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

dtr r g dt T

g

dt r r a

dt dtg g r T

g g

dtg g g

g

BJ t t K tN

B B t T tB KJ t t tN N

B B t T tN

3

Aerodynamic torque inp( ) ( )

u

(2 )

t

)(

s r pa

r

A v t C tT t

t

Drive Train &

Gear Box

Torsion1( ) ( ) ( )T r g

g

t t tN

( )rv t

( )ref t

Pitch Control System

( )gT t( )g t

( )aT t

( )r t

[Sloth et al. 2010]

Generator

Souvik Chandra

Dynamics of a Wind Generator : Electrical Section (Doubly Fed Induction Generator)


DC/AC AC/DC

Controller

Drive TrainAerodynamics

( )r t

( )sv t

( )si t ( )ai t

( )av t

( )rv t

( )ri t

( )ref t ( )r t

( )gT t

( )g t( )aT t Bus j wN

sj sjV

( ) ( ) ( )

Stator circuit model

( ) ( )

( ) ( ) ( )

( ) ) (

qs s s qs e s ds

m qr e m dr

ds e s qs s s ds

e m qr m dr

v t R sL i t L i t

sL i t L i t

v t L i t R sL i t

L i t sL i t

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

Rotor circuit model

( ) (

)

qr m qs l e m ds

r r qr l e r dr

dr l e m qs m ds

l e r qr r r dr

v t sL i t s L i t

R sL i t s L i t

v t s L i t sL i t

s L i t R sL i t

( ) ( ) ( ) ( ) ( )2g m qs dr ds qrPT t L i t i t i t i t Mechanical

System

Torque equation

[Ugalde-Loo et al. 2010]

Souvik Chandra

Dynamics of a Wind Generator : Decoupled Active and Reactive Power Control in Type IV Turbines


[R. Datta et al. 1999]

Power output equationse e e e ej qs qs ds ds

e e e e ej ds qs qs ds

P v i v i

Q v i v i

Choice of d-q reference

0*

0

00*

0

Current reference

s ewqr

m n

ns ewdr

m e mn

L PiL V

VL QiL w LV

PI control to track current

reference

Souvik Chandra

Dynamics of a Wind Generator : Complete Model


DC/AC AC/DC

Controller

Drive TrainAerodynamics

( )r t

( )sv t

( )si t ( )ai t

( )av t

( )rv t

( )ri t

( )ref t ( )r t

( )gT t

( )g t( )aT t Bus j wN

sj sjV ,Z f Z U

T

r g T qs ds qr drZ i i i i

jU V

Linearized and combined with power flow equations to obtain the integrated system

[Chandra et al. 2014]

Souvik Chandra

Concept of a Wind Power Plant


Power output equationse e e e ej j qs qs ds ds

e e e e ej j ds qs qs ds

P v i v i

Q v i v i

The power output equation each wind plant:

An algebraic sum of the power output of all wind turbines and hence a function of

The stator of each turbine connected directly to the wind bus, hence a function of

j

ejV

Souvik Chandra

Power Flow Equations


*

22 1

1,Re

Njre jim

ej jre jim kre kim j j jk k j Ljk Ljk

V jVP V jV V jV V G g

R jX

* 22

21,

Im2

Njre jim j Lkj

ej jre jim kre kim j j jk k j Ljk Ljk

V jV V BQ V jV V jV V B g

R jX

Algebraic equations for active and reactive power balance at each bus between generation and load

Souvik Chandra

DifferentialEquations

Conventional Power Systems


11 12 m

IM k k V P

Re

Re

Im

Im

1

1

N

N

V

VV

V

V

Algebraic Equations

Kron Reduction

1 412 11

1

mM K P

K k Ak A

1 40 A A V

14 1V A A

Sync gen

Power flow

[Chow et al. 1985]

Souvik Chandra

Wind-integrated Power Systems


Kron reduction :

111 11 12 4 1

112 12 4 2

121 1 4 1

122 1 4 2

:

:

:

:

M

M

M

M

A k k A A

A k A A

A B A A

A A B A A

where,

11 12 m

IM k k V P

1Z A Z B V

21 40 ZA A A V

Sync gen:

Wind gen:

Power flow:

..

M M mM A B P

ZZ

0MB I


Souvik Chandra

Admittance Matrix in Conventional Power Systems


where 1

E EijI Iij

BB

The weighted admittance matrix can be divided into :• internal connections within different

generation areas• external connections within different

generation areas

4,IA

4 ,EA

4 4 4I EA A A

The number of external connections is typically smaller then the number of

internal connections The admittance of external connections is

typically smaller than the admittance of internal connections

E

.I

EijB

.IijB[Chow et al. 1985]

Souvik Chandra

Fast and Slow Time-scales in Conventional Power System


, 1 1f i i

Slow variable for area α

Fast variable in area α

saa ads

fda df

K KK K

1

: /n

s i ii

m m

Similarity transform :

As, 1, Dynamics has fast and slow subsystems leading to Inter and Intra area oscillations[Chow et al. 1985]

Souvik Chandra

Effect of Wind Penetration on the Admittance Matrix


The wind penetration directly affects the admittance matrix by a parameter

The norms of the matrix and are also dependent on bus voltages and angles which are affected by wind penetration

, ,4 4 4 4

I sg I wg EwA A A A

,( )

| |where : smaxw I I

ij min

iB

w

,4I sgA 4

EA

Next we define a similarity transform to identify the time scales

Souvik Chandra

Time-scales in a Wind-integrated Power System


Slow variable in area α with wind

1 1

1 1: /

r r

wg

n nr r r rs i i i

i im m

1

1:sg

n

s i ii

mm

Slow variable in area α without wind

, 1 1f i i

Fast variable in area α

, ,,

, , ,1 1

a wg s wgs wg

wa sg s sg s sg

mf

w d f

w

M

MA B P

MZ

Z

Similarity transform and time-scale separated form :

As the effect of wind penetration is mainly on the slow time-scale as shown next

Time constants of the fast variables and wind variables are fast enough than the slow variables

1w

Souvik Chandra

Slow or Inter-area Dynamics in a Wind-integrated Power System


, ,,11 12

,21 22, ,

a wg s wgs wg

ws sga sg s sg

M k kk kM

We isolate the inter-area dynamics to identify the effect on wind

For typical 15-20% wind penetration and other power system parameters of the order of 1/10. So within practical limits of wind penetration, typically does not change much.

However norms of the matrices are heavily depend on which affect the inter-area oscillatory dynamics.

,w w

11 12 22 21, , , k k k k

4EA

Souvik Chandra

Case Study I: 4-machine 8-bus Power System


Scenario 1:

Scenario 2:

Wind at bus 5 area 1 while major load is at bus 8 area 2

Both wind at bus 7and major load at bus 8 area 2

Souvik Chandra

Case study I: Eigenvalues and Impulse Response of Aggregate Angle Between Area 1 and Area 2


Penetrationlevel

Sloweigenvalues

0 0% 3341 0.00 108.73 -0.10 ± j 1.90

250 5% 3362 4.79 108.93 -0.10 ± j 2.00

500 10% 3381 9.49 107.85 -0.10 ± j 2.10

750 15% 3411 14.08 109.05 -0.10 ± j 2.13 0 5 10 15 20 25 30-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time(secs)

Ang

le(r

adia

ns)

=0=250=500=750

0 5 10 15 20 25 30-0.1

-0.05

0

0.05

0.1

Time(secs)

Ang

le(r

adia

ns)

=0=250=500=750

Scenario 1:

Scenario 2:

γ ,4I sgA , 0

4I wgA 4

EA

Penetrationlevel

Sloweigenvalues

0 0% 3882 0.00 115.87 -0.10 ± j 2.47

250 5% 3892 4.87 116.24 -0.10 ± j 2.47

500 10% 3901 9.77 116.57 -0.10 ± j 2.46

750 15% 3910 14.72 116.64 -0.10 ± j 2.45

γ ,4I sgA , 0

4I wgA 4

EA

Souvik Chandra

Case study II: 16-machine 68-bus Power System


Scenario 1:

Scenario 2:

Wind at bus 66 in area 1 where synchronous generation is less than the load in the area

Wind at bus 38 in area 2 where synchronousgeneration exceedsthe load in the area

Souvik Chandra

Case study II: Eigenvalues and Impulse Response of Aggregate Angle Between Area 1 and Area 2


0 5 10 15 20-0.1

-0.05

0

0.05

0.1

Time(secs)

Ang

le(r

adia

ns)

=0=250=500

0 5 10 15 20-0.1

-0.05

0

0.05

0.1

Time(secs)

Ang

le(r

adia

ns)

=0=250=500=750

Scenario 1:

Scenario 2: Penetration

level Slow eigenvalues

0 0% 2379 0.00 165.63 -0.07 ± j 3.17, -0.06 ± j 2.58,-0.07 ± j 2.02, -0.07 ± j 1.39

250 2.5% 2355 5.88 166.64 -0.07 ± j 3.17, -0.06 ± j 2.46,-0.07 ± j 2.02, -0.07 ± j 1.25

500 5.0% 2298 12.29 165.51 -0.07 ± j 3.17, -0.06 ± j 2.27,-0.07 ± j 2.00, -0.07 ± j 0.92

750 7.5% 2186 18.68 160.99 -0.06 ± j 2.08, -0.07 ± j 1.94,-0.07 ± j 0.92, {0.89, -1.01}

γ ,4I sgA , 0

4I wgA 4

EA

Penetrationlevel Slow eigenvalues

0 0% 2373 0.00 161.56 -0.09 ± j 3.46, -0.06 ± j 2.92,-0.10 ± j 2.38, -0.08 ± j 1.55

250 2.5% 2380 5.60 163.30 -0.09 ± j 3.46, -0.06 ± j 2.92,-0.10 ± j 2.38, -0.08 ± j 1.55

500 5.0% 2379 12.12 167.08 -0.09 ± j 3.46, -0.06 ± j 2.94,-0.10 ± j 2.38, -0.08 ± j 1.54

750 7.5% 2341 19.06 169.08 --0.09 ± j 3.46, -0.06 ± j 2.90,-0.10 ± j 2.38, -0.08 ± j 1.51

γ ,4I sgA , 0

4I wgA 4

EA

Souvik Chandra

Conclusions for Part II


1. The time-scale separation in a wind-integrated power system depend on the level of wind penetration, the system topology and location of the wind plant.

2. may change with increase in wind penetration causing the inter-area oscillation spectrum to change.

3. The is dependent on the structure and relative locations of the generation areas.

4. If reduces to a value lower than a critical limit, the system may become unstable.

4EA

4EA

4EA

52

PART-III

Equilibria Analysis and Transient Voltage Stability Limit

Souvik Chandra

Organization

53

1. Background on equilibrium analysis and voltage stability Definitions

Power flow solution boundary

2. Review methods for locating power flow solution boundary

3. Introduce Numerical Polynomial Homotopy Method Concept

Extension to power flow analysis

4. Detecting multiple equilibria of a wind-integrated power system Case study on a 7-bus power system

5. Power flow solution boundary under multiple wind injection

scenario

6. Conclusions

Souvik Chandra

Concept of voltage stability


Ability to maintain steady voltages at all buses in the powersystem following a disturbance from a given initial operatingcondition.

It can be of two types :

Large-disturbance voltage stability System faults, loss of generation, or circuit contingencies etc. Determined via nonlinear simulation of the power system with

appropriate load characteristics and interactions of continuous and discrete control actions.

Small-disturbance voltage stability incremental changes in system load, continuous controls etc. System equations are linearized for analysis of the stability

condition

Souvik Chandra

Small-disturbance voltage stability


*

2

1,Re 0

N

ej j k j jk k j Ljk

jVP V V V G

Z

*

2

1,Im 00.5

N

ej j k j j Lkjk k

j

j Ljk

Q V V V BZV

B

In a power system the bus voltages are dependent on power flow,

The steady voltage levels in each bus can be uniquely maintained by change in active and reactive power input if the power flow Jacobian is non-singular.

epf

e

P VJ

Q

Linearized power flow equations

Power flow Jacobian matrix

Single machine 2‐bus power system

Thus, small signal voltage stability is dependent on equilibrium values.

Souvik Chandra

P

V2

P2+Q2-(V22/X)2 = 0

-1 -0.5 0 0.5 10.2

0.4

0.6

0.8

1

1.2

Q=0.4

Q=1.0Q=0.8

Q=0.6

Q=0.2



Single machine 2‐bus power system

Consider power balance on bus 2,

1 2 sinVVPX

22 1 2 cosV VVQ

X X

Linearized form,

1 1 2

2

2 1 1 2

sin cos

2 cos sin

V VVP VX XQ V V VV

X X X

Power flow Jacobian

Jacobian is singular when,

1 2 cosV V

2222VP Q

X

Thus for a given value of Q, P can have two real solutions or zero solutions.

Power flow solution boundary or the loadability boundary

Souvik Chandra

Problem Formulation


In a wind integrated power system, The power system equilibrium is affected by wind power plants

via parameters such as penetration levels, voltage control set points or wind speed.

How does wind penetration affect the power flow solution boundary of the power system?

Define mutual penetration levels for multiple wind power plants which guarantee robust operation.

Souvik Chandra

Finding loadability boundary: existing method


3‐bus power system

Problem: Find power flow solution boundary parameterized by λ1 and λ2, the active power levels of generator 1 and generator 2?

[Hiskens & Davy 2001]

( , ) 0f x

( , , ) ( , ) 0xg x v f x v

( ) 1Th v v v

Solution: solve the equations for given range of λ1 and λ2,

Power flow equations:

Singular Jacobian :

v is eigenvector of the zero eigenvalue:

Souvik Chandra

Finding loadability boundary: 2 steps


Step 1:Fix λ2 and vary λ1 to find one particular point on the power flow solution boundary of the power system?


Step 2:

Once an initial point on the loadabilityboundary is obtained track the boundary by a gradient based technique as shown below,

1. Obtain z0

( ) 0iz 1i i iz z

1

1

i ii

i i

z zz z

1z1pz z

2z

( )

where, ( ) ( ) , ( )

f z xz g z z v

h z

2. Solve for zi

3. Update[Hiskens & Davy 2001]

Souvik Chandra

Problems of iterative methods


Depends on the location of an initial point on the solution boundary. Depends on local approximation of the solution boundary

-if solution boundary is smooth local approximations hold-Constraints like generator over excitation limits make the

make the solution boundary non-smooth Knowledge of the solution space is required to determine all solution

boundaries

Proposed Method Find the solution boundary by noting the change in number

of real solutions. Use Numerical polynomial homotopy continuation (NPHC)

method which guarantees to find all solutions of the power flow equations.


Souvik Chandra

NPHC method


1( ) ( ( ), , ( )) 0TmP x p x p x

Set of algebraic polynomial equations,

We form a homotopy as shown below in t,

( , ) (1 ) ( ) ( ) 0hH x t t Q x t P x Where, • Q(x)=0, is chosen as the start problem whose solution is easy to obtain.• 0<t<1 is a continuous homotopy variable, γh is a generic complex number• All solutions of Q(x)=0 are solutions of H(x,0)=0.• Solutions of H(x,t) are tracked via continuity methods from t=0 to t=1 to obtain

all solutions of P(x)=0.• Choice of γh allows all tracking paths to either converge to a solution or diverge

to infinity.• The upper bound on the number of paths to be tracked to guarantee all the

complex roots of P(x)=0 is the Classical Be’zout Bound• Computations grow exponentially with m

1, order of

m

i i ii

p

Souvik Chandra

NPHC method applied to power systems- Step 1


*

2

1,Re 0

N

ej j k j jk

j

k j Ljk

VP V V V G

jZ

*

2

1,Im 00.5

N

ej j k j j Lkjk k

j

j Ljk

Q V V V B BVjZ

3‐bus power system Preconditioning, Represent the voltage at any bus in real and

imaginary form All power flow equations become essentially

quadratic equations Angle equations considered for PV buses Helps in having a tighter Be’zout Bound

Power flow equations,

j jre jimV V iV

Souvik Chandra

NPHC method applied to power systems- Step 2



Solve a start system P(x, λ* )=0, whereλ* is a generic complex quantity.

The number of complex solutions of P(x, λ* )=0 is upper bounded by 2^m if m is the number of equations via Be’zout Bound.

Form a homotopy and track for all paths from 0<t<1,

*( , , ) (1 ) ( , ) ( , ) 0H x t t P x t P x

Tracking via a predictor corrector method compute all solutions of P(x, λ )=0, computed for each different set of λ

Souvik Chandra

Reduced computations


For a parameteric system of polynomial equations, the maximum number of isolated complex solutions over all parameter-points is same as that for a generic complex parameter-point.

Cheater’s homotopy method

When we are solving Q(x, λ* )=0 the maximum number of complex solutions is 2^m if m is the number of equations via Be’zout Bound.

But actually the number of solutions for Q(x, λ* )=0 is usually much less than 2^m as power systems are usually deficient systems.

e.g. In the 3 bus example there were 8 equations with 2^8 possible solutions. But only 6 isolated solutions exist actually for Q(x, λ* )=0.

Thus in the homotopy stage much less number of paths can be tracked

Paths are parallelizable and thus converges fast.

Application

Souvik Chandra

Results for the 3 bus case



IterativeMethod :

NPHCMethod:

2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

4

1

2

initial: 1=2.4524, 2=2

initial: 1=5.5476, 2=2

initial: 1=2.0789, 2=3

initial: 1=4.0389, 2=3

Souvik Chandra Modeling, Analysis and Control of Wind-Integrated Power Systems 66

A wind integrated power system

A number of wind turbines Battery energy storage system to mitigate the

variability of wind speed and provide smooth power A central power controller tracking a power reference

and the output bus voltage The equilibria of the wind installation is dependent on

two parameters the wind speed and wind bus terminal voltage, penetration level as

, a . nde ew r jjV v

A wind integrated power system

Souvik Chandra

Equilibrium equations


*

1,

*

1,

1Re 1

1Im 1 .

Ne e e

si k Lik k i ik

Ne e esi k Li

k k i ik

P V PZ

Q V QZ

For slack bus For any bus*

1,

*

1,

Re

Im .

eNje e e

j jk Ljk k j jk

eNje e e

j jk Ljk k j jk

VP V P

Z

VQ V Q

Z

3 / 4

( ) ( )

e e e e eg m qs dr ds qr

e e e eqs s qs e s ds e m dr

e e e eds e s qs s ds e m qr

e e e eqr e ge m ds r qr e ge r dr

e e e e e e e etj j qs qs ds ds dr dr qr qr

e e e e etj j ds qs qs ds d

T p L i i i i

v R i L i L i

v L i R i L i

v L i R i L i

P v i v i v i v i

Q v i v i v e e e er qr qr dri v i

For wind generator

For dynamic loads

.

sj

sj

e eLj ss j

e eLj ss j

P P V

Q Q V

For BES and WPP

,

,

e e e e eBES j q q d d

e e e e eBES j d q q d

P v i v i

Q v i v i

, ,

, , .

e e ewj j t j BES j

e e ewj j t j BES j

P P P

Q Q Q

Parameters1. Wind speed2. Wind bus voltage3. Wind penetration

amount4. Load types

Souvik Chandra

A case study: 7 bus power system


Brazilian 5-machine 7 bus system with wind at bus 6

Equilibria analysis with different levels of wind speed and wind bus voltage

Multiple feasible equilibria may exist in a wind integrated power system which depend on various local parameters and control actions.

Souvik Chandra

Case study : Multiple wind injection scenario


A 10 bus power system with wind at bus 9 and bus 10

Penetration level: γ1 γ2

How does the penetration levels γ1 and γ2 affect the power flow solution boundary?

We apply the NPHC method to provide a solution for this problem.

Souvik Chandra



0.2 0.3 0.4 0.5 0.6 0.70.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

1

2

# R

eal S

olut

ions

0

7

14 The penetration levels γ1 and γ2

are varied upto 50% of the total connected load of the system.

In the NPHC method, the number of unknown equations is 25 which provides an upper bound of complex solutions to be 2^25, but we only follow around 2^9 complex solutions via cheaters homotopy.

Structure of the solution space is differs substantially from the 3-bus case study.

Souvik Chandra

Conclusion for Part III


1. Parameters of the different dynamic elements of a complex power system like wind penetration affect the number of equilibria.

2. As a consequence tools like the numerical homotopy can be used offline by the power system operators to determine limits like the power flow solution boundary.

3. Robust operating point may be chosen from the information of the power flow solution boundary.

4. As a practical application multiple wind injection scenario determining relative penetration limits for a robust operating point.

Souvik Chandra

Contributions


1. Coordinated control design in wind farms and battery energy systems for inter-area oscillation damping in power systems via spectral matching.

2. Derivation of the wind-integrated power system model where wind turbines are interfaced to the grid via a DFIG.

3. Time-scale modeling of wind-integrated power systems and analysis of the effect of increasing wind penetration on the inter-area modes.

4. Equilibria analysis of wind-integrated power systems via numerical homotopy method and determination power flow boundary or transient voltage stability limit.

Souvik Chandra

Future Work


1. Extend the work to figure out how eigenvectors of the linearizedpower systems are affected by multiple wind injections.

2. Design controllers on the wind farm to improve damping andmaintain voltage stability of the power system.

3. Determine equilibrium boundary based on wind penetrationlevels which guarantee voltage and small–signal rotor anglestability.

Souvik Chandra

Publications


Journals-Published/ Under review S. Chandra, D. F. Gayme, and A. Chakrabortty, “Coordinating Wind Farms and Battery Management

Systems For Inter-area Oscillation Damping Control: A Frequency Domain Approach”. IEEE Transactionson Power Systems, vol. 29(3), 2014.

S. Chandra, D. F. Gayme, and A. Chakrabortty, “Time-Scale Modeling of Wind-Integrated PowerSystems”, currently under review in IEEE Transactions on Power Systems, 2015.

Under Preparation S. Chandra, D. Mehta, and A. Chakrabortty. “Determining Power Flow Solution Boundary in Multiple

Wind power system: A numerical homotopy approach”, to be submitted in IEEE Transactions on PowerSystems

Conferences- Published S. Chandra, D. Mehta, and A. Chakrabortty. “Equilibria Analysis of Power Systems Using a Numerical

Homotopy Method”. IEEE PES General Meeting, Denver, CO, July 2015. S. Chandra, D. Mehta, and A. Chakrabortty. “Exploring Impact of Wind Penetration on Power System

Equilibrium Using a Numerical Continuation Approach”. American Control Conference, Chicago, IL,2015.

S. Chandra, M. D. Weiss, A. Chakrabortty, and D. F. Gayme. “Impact Analysis of Wind Power Injection onTime-Scale Separation of Power System Oscillations”, IEEE PES General Meeting, Washington DC, July2014.

S. Chandra, D. Gayme, and A. Chakrabortty. “Using Battery Management Systems to Augment Inter-area Oscillation Control in Wind-Integrated Power Systems”, in proceedings of American ControlConference, DC, 2013.

Souvik Chandra

Key References


[1] J. H. Chow, Power System Coherency and Model Reduction. Springer New York, Jan. 2013, ch. Slow Coherency and Aggregation, pp. 39–72.

[2] C. Sloth, T. Esbensen, and J. Stoustrup, “Active and passive fault-tolerant LPV control of wind turbines,” in Proceedings of the American Control Conference, Baltimore, MD, 2010.

[3] C. Ugalde-Loo, J. Ekanayake, and N. Jenkins, “State-space modeling of wind turbine generators for power system studies,” IEEE Trans. On Industry Applications, vol. 49, no. 1, pp. 223–232, 2013.

[4] R. Datta and V. Ranganathan, “Decoupled control of active and reactive power for a grid-connected doubly-fed wound rotor induction machine without position sensors,” in Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE, vol. 4. IEEE, 1999, pp. 2623–2630.

[5] P. Kundur, Power system stability and control. McGraw-hill New York, 1994, vol. 7.

[6] S. Chandra, M. Weiss, A. Chakrabortty, and D. Gayme, “Impact analysis of wind power injection on time-scale separation of power system oscillations,” in Proc. of the Power and Energy Soc. Gen. Meeting, 2014.

Souvik Chandra Modeling, Analysis and Control of Wind-Integrated Power Systems 76

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