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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
Modeling, Analysis and Control of Wind-Integrated Power Systems 3
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 4
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 5
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 6
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 Wind-Integrated Power Systems 7
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 9
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 10
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 11
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 12
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 13
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 14
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 15
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 16
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 17
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 18
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 19
-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
Modeling, Analysis and Control of Wind-Integrated Power Systems 20
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 21
*
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 22
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 23
• 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
Modeling, Analysis and Control of Wind-Integrated Power Systems 24
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 25
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 26
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 27
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 28
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 29
Souvik Chandra
Conclusions of Part I
Modeling, Analysis and Control of Wind-Integrated Power Systems 30
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 32
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 33
'
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)
Modeling, Analysis and Control of Wind-Integrated Power Systems 34
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)
Modeling, Analysis and Control of Wind-Integrated Power Systems 35
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 36
[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
Modeling, Analysis and Control of Wind-Integrated Power Systems 37
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 38
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 39
*
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 40
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 41
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
[Chandra et al. 2014]
Souvik Chandra
Admittance Matrix in Conventional Power Systems
Modeling, Analysis and Control of Wind-Integrated Power Systems 42
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 43
, 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
Modeling, Analysis and Control of Wind-Integrated Power Systems 44
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 45
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 46
, ,,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
Modeling, Analysis and Control of Wind-Integrated Power Systems 47
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 48
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 49
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 50
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 51
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
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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
Modeling, Analysis and Control of Wind-Integrated Power Systems 54
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 55
*
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
Power flow solution boundary
Modeling, Analysis and Control of Wind-Integrated Power Systems 56
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 57
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 58
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 59
Step 1:Fix λ2 and vary λ1 to find one particular point on the power flow solution boundary of the power system?
3‐bus 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
Modeling, Analysis and Control of Wind-Integrated Power Systems 60
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.
[Chandra et al. 2015]
Souvik Chandra
NPHC method
Modeling, Analysis and Control of Wind-Integrated Power Systems 61
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 62
*
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
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NPHC method applied to power systems- Step 2
Modeling, Analysis and Control of Wind-Integrated Power Systems 63
3‐bus power system
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 64
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 65
3‐bus power system
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 67
*
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
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A case study: 7 bus power system
Modeling, Analysis and Control of Wind-Integrated Power Systems 68
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 69
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
Power flow solution boundary
Modeling, Analysis and Control of Wind-Integrated Power Systems 70
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 71
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 72
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 73
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 74
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
Modeling, Analysis and Control of Wind-Integrated Power Systems 75
[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
Thanks, Questions?