ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
ECE-620 Lecture on power system modeling anddynamics including wind turbines
Dr. Hector A. Pulgarhpulgar@utk
Department of Electrical Engineering and Computer ScienceUniversity of Tennessee, Knoxville
November 19, 2014
Dr. Hector A. Pulgar hpulgar@utk 1 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
Outline
1. Time scale decomposition
2. Synchronous generator and DFIG-based wind turbine models
3. Small-signal stability analysis (SSSA)
4. Cases of study
4.1. Load parametrization in SSSA
4.2. Inertia parametrization in SSSA
4.3. Effect of wind turbines location on oscillation damping
4.4. Limit-induced bifurcations in wind farms
Dr. Hector A. Pulgar hpulgar@utk 2 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Dr. Hector A. Pulgar hpulgar@utk 3 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Classification of power system dynamics
10-7 10-5 10-3 10-1 103 10510
Lightning surge
Switching surge
Electro-magnetic transient
Electro-mechanical
transientBoiler
dynamics
Time [s]
Should our model represent completely all dynamics?
No, it is not recommended due to numerical issues such as numericalinstability and excessive simulation time
We will focus on the time frame of electromechanical transients
Dr. Hector A. Pulgar hpulgar@utk 4 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Classification of power system dynamics
10-7 10-5 10-3 10-1 103 10510
Lightning surge
Switching surge
Electro-magnetic transient
Electro-mechanical
transientBoiler
dynamics
Time [s]
Should our model represent completely all dynamics?
No, it is not recommended due to numerical issues such as numericalinstability and excessive simulation time
We will focus on the time frame of electromechanical transients
Dr. Hector A. Pulgar hpulgar@utk 4 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Classification of power system dynamics
10-7 10-5 10-3 10-1 103 10510
Lightning surge
Switching surge
Electro-magnetic transient
Electro-mechanical
transientBoiler
dynamics
Time [s]
Should our model represent completely all dynamics?
No, it is not recommended due to numerical issues such as numericalinstability and excessive simulation time
We will focus on the time frame of electromechanical transients
Dr. Hector A. Pulgar hpulgar@utk 4 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Classification of power system dynamics
10-7 10-5 10-3 10-1 103 10510
Lightning surge
Switching surge
Electro-magnetic transient
Electro-mechanical
transientBoiler
dynamics
Time [s]
Should our model represent completely all dynamics?
No, it is not recommended due to numerical issues such as numericalinstability and excessive simulation time
We will focus on the time frame of electromechanical transients
Dr. Hector A. Pulgar hpulgar@utk 4 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
General concept. Consider the following generic system,
x = f(x, z) x(0) = xo
z = g(x, z) z(0) = zo
where x ∈ Rn×1, and z ∈ Rm×1. Assume that the dynamics of x and zare distinctive.
To decouple these dynamics, we seek for an integral manifold forz = h(x) which satisfies the differential equation of z. Thus,
z =∂z
∂xx =
∂h
∂xf (x, h(x)) = g (x, h(x))
If the initial condition belongs to the manifold, zo = h(xo), we say thatthe integral manifold is an exact solution of z = g(x, z), and thefollowing reduced order model is an exact model:
x = f (x, h(x))
Dr. Hector A. Pulgar hpulgar@utk 5 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A complete linear example. Consider the following second order system(assume that ε is a small parameter)
x = −x+ z x(0) = xo
εz = −x− z z(0) = zo
The system eigenvalues are:
λ1,2 =−ε− 1−+
√ε2 − 6ε+ 1
2ε
In the limit,
λ1 = limε→0
−ε− 1−√ε2 − 6ε+ 1
2ε=−1− 1
2ε=−1
ε= −∞
λ2 = limε→0
−ε− 1 +√ε2 − 6ε+ 1
2ε=
0
0indeterminate!
Applying L’Hopital’s rule we can show that λ2 → −2. As a result, thissystem has a very fast mode e−t/ε and a slow mode e−2t—we have twodistinctive dynamics!
Dr. Hector A. Pulgar hpulgar@utk 6 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A complete linear example. Consider the following second order system(assume that ε is a small parameter)
x = −x+ z x(0) = xo
εz = −x− z z(0) = zo
The system eigenvalues are:
λ1,2 =−ε− 1−+
√ε2 − 6ε+ 1
2ε
In the limit,
λ1 = limε→0
−ε− 1−√ε2 − 6ε+ 1
2ε=−1− 1
2ε=−1
ε= −∞
λ2 = limε→0
−ε− 1 +√ε2 − 6ε+ 1
2ε=
0
0indeterminate!
Applying L’Hopital’s rule we can show that λ2 → −2. As a result, thissystem has a very fast mode e−t/ε and a slow mode e−2t—we have twodistinctive dynamics!
Dr. Hector A. Pulgar hpulgar@utk 6 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
We propose a linear manifold of the form z = hx, where h is a realconstant. This manifold must satisfy the differential equation of z, thus
εz = −x− z ⇒ ε∂z
∂xx = −x− hx
⇒ εh (−x+ hx) = −x− hx⇒(−εh+ εh2
)x = −(1 + h)x
⇒(−εh+ εh2
)= −(1 + h)
⇒ εh2 + (1− ε)h+ 1 = 0
If there exists a solution for h, then the manifold z = hx exists.
The solution for h is given by:
h(ε) =−(1− ε) +−
√(1− ε)2 − 4ε
2ε∈ R⇔ φ(ε) = (1− ε)2 − 4ε ≥ 0
Dr. Hector A. Pulgar hpulgar@utk 7 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
We propose a linear manifold of the form z = hx, where h is a realconstant. This manifold must satisfy the differential equation of z, thus
εz = −x− z ⇒ ε∂z
∂xx = −x− hx
⇒ εh (−x+ hx) = −x− hx⇒(−εh+ εh2
)x = −(1 + h)x
⇒(−εh+ εh2
)= −(1 + h)
⇒ εh2 + (1− ε)h+ 1 = 0
If there exists a solution for h, then the manifold z = hx exists.
The solution for h is given by:
h(ε) =−(1− ε) +−
√(1− ε)2 − 4ε
2ε∈ R⇔ φ(ε) = (1− ε)2 − 4ε ≥ 0
Dr. Hector A. Pulgar hpulgar@utk 7 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Roots of φ(ε) are given by:φ(ε) = (1− ε)2 − 4ε = 0⇒ ε1,2 = 3−+
√8
ε
φ(ε)
0 1 2 3 4 5 6
-8
-7
-6
-5
-4
-3
-2
-1
0
1ε1 = 3−
√8 ≈ 0.1716
ε2 = 3 +√8 ≈ 5.8284
If ε < ε1, then we have two solutions for hgiven by:
h1(ε) =−(1−ε)+
√(1+ε)2−4ε
2ε
h2(ε) =−(1−ε)−
√(1−ε)2−4ε
2ε
4 · 10−2 8 · 10−2 0.12 0.18
−10
−8
−6
−4
−2
εh(ε)
h1(ε)
h2(ε)
In general, manifolds may not exist and maynot be unique
Dr. Hector A. Pulgar hpulgar@utk 8 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Roots of φ(ε) are given by:φ(ε) = (1− ε)2 − 4ε = 0⇒ ε1,2 = 3−+
√8
ε
φ(ε)
0 1 2 3 4 5 6
-8
-7
-6
-5
-4
-3
-2
-1
0
1ε1 = 3−
√8 ≈ 0.1716
ε2 = 3 +√8 ≈ 5.8284
If ε < ε1, then we have two solutions for hgiven by:
h1(ε) =−(1−ε)+
√(1+ε)2−4ε
2ε
h2(ε) =−(1−ε)−
√(1−ε)2−4ε
2ε
4 · 10−2 8 · 10−2 0.12 0.18
−10
−8
−6
−4
−2
εh(ε)
h1(ε)
h2(ε)
In general, manifolds may not exist and maynot be unique
Dr. Hector A. Pulgar hpulgar@utk 8 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
With ε ε1, x is the variable associated with the slow mode, while z isassociated with the fast mode. As we are interested in the slow mode, wechoose the manifold z = h1x.
In the case ε = 0,
limε→0
h1(ε) = limε→0
−(1− ε) +√
(1 + ε)2 − 4ε
2ε= −1
In the case ε 6= 0, we use a Taylor expansion series to represent h around−1. Therefore, the manifold becomes:
z = h(ε)x
where h(ε) = h0 + h1ε+ h2ε2 + ...
By using this manifold in the differential equation of z, we get:
εz = −x− z
⇒ ε∂z
∂xx = −x− h(ε)x = −(1 + h(ε))x
Dr. Hector A. Pulgar hpulgar@utk 9 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
⇒ ε(h0 + h1ε+ h2ε
2 + ...) (−x+
[h0 + h1ε+ h2ε
2 + ...]x)
=
−(1 + h0 + h1ε+ h2ε
2 + ...)x
⇒(h0 + h1ε+ h2ε
2 + ...) (
(−1 + h0)ε+ h1ε2 + h2ε
3 + ...)
=
−(1 + h0 + h1ε+ h2ε
2 + ...)
By equating terms based on the powers of ε, we obtain:
ε0 : 1 + h0 = 0⇒ h0 = −1
ε1 : h0(−1 + h0) = −h1 ⇒ h1 = −2
ε2 : h0h1 + h1(−1 + h0) = −h2 ⇒ h2 = −6 , and so on
To sum up, we obtain the following different approximations:
z ≈ −x Zero-order manifold
z ≈ −(1 + 2ε)x First-order manifold
z ≈ −(1 + 2ε+ 6ε2)x Second-order manifold
Dr. Hector A. Pulgar hpulgar@utk 10 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
The reduced-order model becomes:
x = −x+ zz = hxx(0) = x0, z(0) = z0
⇒ x = −(1− h)x⇒ x(t) = x0e−(1−h)t
Considering ε = 0.1, we have:
Zero-ordermanifold
: x(t) = x0e−2t
First-ordermanifold
: x(t) = x0e−(2+2ε)t = x0e
−2.2t
Second-ordermanifold
: x(t) = x0e−(2+2ε+6ε2)t = x0e
−2.26t
Exact solu-tion
: x(t) =(
7.76.4x0 + z0
6.4
)e−2.3t −
(1.36.4x0 + z0
6.4
)e−8.7t
Dr. Hector A. Pulgar hpulgar@utk 11 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A general linear example. In an n-dimensional linear case we have:
x = Ax+Bz
εz = Cx+Dz
where ε is a small constant; x ∈ Rn×1; z ∈ Rm×1; and A, B, C, and Dare all matrices with proper dimensions.
A zero-order manifold is obtained for z by letting ε be equal to zero andsolving in terms of x as follows:
εz = 0 = Cx+Dz ⇒ z = −D−1Cx
⇒ x =(A−BD−1C
)x
We assume that D is full rank and, therefore, invertible.
Dr. Hector A. Pulgar hpulgar@utk 12 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A graphical nonlinear example. Initial value problem with ε = 0.05
x1 = −x1 + x3
x2 = −x2 − x3
εx3 = tan−1 (1− x1 + x2 − x3)⇒ x3 = tan−1(1−x1+x2−x3)
ε
Note there exists δ, small, such that if |1− x1 + x2 − x3| >> δ, then |x3|gets large.
Let M be the set defined by x1, x2, x3 ∈ R : 1− x1 + x2 − x3 = 0
In state space, we say:
If the distance between the system trajectory and M is much larger thanδ, then x3 will exhibit a fast response
Dr. Hector A. Pulgar hpulgar@utk 13 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A graphical nonlinear example. Initial value problem with ε = 0.05
x1 = −x1 + x3
x2 = −x2 − x3
εx3 = tan−1 (1− x1 + x2 − x3)⇒ x3 = tan−1(1−x1+x2−x3)
ε
Note there exists δ, small, such that if |1− x1 + x2 − x3| >> δ, then |x3|gets large.
Let M be the set defined by x1, x2, x3 ∈ R : 1− x1 + x2 − x3 = 0
In state space, we say:
If the distance between the system trajectory and M is much larger thanδ, then x3 will exhibit a fast response
Dr. Hector A. Pulgar hpulgar@utk 13 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
A graphical nonlinear example. Initial value problem with ε = 0.05
x1 = −x1 + x3
x2 = −x2 − x3
εx3 = tan−1 (1− x1 + x2 − x3)⇒ x3 = tan−1(1−x1+x2−x3)
ε
Note there exists δ, small, such that if |1− x1 + x2 − x3| >> δ, then |x3|gets large.
Let M be the set defined by x1, x2, x3 ∈ R : 1− x1 + x2 − x3 = 0
In state space, we say:
If the distance between the system trajectory and M is much larger thanδ, then x3 will exhibit a fast response
Dr. Hector A. Pulgar hpulgar@utk 13 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Example (cont.). Dynamic simulation with ε = 0.05
State-space trajectories from4 different initial points:
Point 1 x1 = 0.8, x2 = −0.2,x3 = 0.5
Point 2 x1 = 0.0, x2 = 0.0,x3 = 0.0
Point 3 x1 = 0.6, x2 = −0.8,x3 = 0.0
Point 4 x1 = 0.0, x2 = −0.8,x3 = 1.0
Dr. Hector A. Pulgar hpulgar@utk 14 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
1. Time scale decomposition
Example (cont.). Considering a zero-order manifold for z (ε = 0)
M : x1, x2, x3 ∈ R : 1− x1 + x2 − x3 = 0
Simplified model:
x1 = −x1 + x3
x2 = −x2 − x3
0 = 1− x1 + x2 − x3
OutsideM
Dynamic of x3
infinitely fast, i.e.,x3 =∞
InsideM
Dynamic of x3 in thesame time scale ofx1 and x2
Dr. Hector A. Pulgar hpulgar@utk 15 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator andDFIG-based wind turbine models
Dr. Hector A. Pulgar hpulgar@utk 16 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Power system models. We typically use a zero-order manifold for the fastdynamics related to, for example, stator flux linkages of synchronousgenerators. Slow dynamics are fully modeled such as those related to theangular motion of synchronous generators. As a result, power systems aredescribed by a set of differential-algebraic equations (DAEs) of the form:
x = f(x, y, µ)
0 = g(x, y, µ)
x Vector of state variables y Vector of algebraic variablesAngular speed of SGs Armature current of SGsAngular rotor position of SGs Bus voltagesRotor flux linkages of SGsVariables of SG’s exciters µ Vector of parametersVariables of SG’s governors Electrical demandVariables of turbines Wind speed, solar radiation, others
Dr. Hector A. Pulgar hpulgar@utk 17 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Two-axis model of SGs
T ′do
dE′q
dt= −E′
q − (Xd −X′d)Id + Efd
T ′qo
dE′d
dt= −E′
d + (Xq −X′q)Iq
dδ
dt= ω − ωs
2H
ωs
dω
dt= TM − E′
dId − E′qIq − (X′
q −X′d)IdIq
Manifold is associated with the following equivalent circuit:
RsjXd Re jXe’+
~ ~
+
Ve jqVS
jq S e
Synchronous generator Thevenin equivalent of the grid
( Ed+(Xq-Xd)Iq+jEq ) ej (d- )’ ’ 2
p
’ ’
( Id+jIq ) ej (d- )2
p
Color definition:
Blue : State vari-ables
Green : Algebraicvariables
Red : Inputs
Dr. Hector A. Pulgar hpulgar@utk 18 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Generator models
Efd: Exciter’s model (IEEE Type 1)
KA
sTA+1
1
sTE+KE
sKF
sTF+1
VR max
VR min
Vt
Vref+ -
-Amplifier Exciter
Stabilizing transformer
VR
SE(Efd)
+
-
Saturation
Rf - EfdKF
TF
Efd
TEdEfd
dt= −
(KE + SE(Efd)
)Efd + VR
TAVR
dt= −VR ++KARf −
KAKF
TFEfd +KA
(Vref − Vt
)TF
dRf
dt= −Rf +
KF
TFEfd con Vmin
R ≤ VR ≤ VmaxR
Dr. Hector A. Pulgar hpulgar@utk 19 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Tm: Governor’s model (IEESGO)
1
1+sT4
K1(1+sT2)
(1+sT1)(1+sT3)
Pmax
Pmin
w
wref+
- +
-
PC
K2
1+sT5
K3
1+sT6
Tm
1-K2
1-K3
+
++
Controller
Turbine
T4dTm
dt= −Tm + PV
T1dy1
dt= −y1 +
1
RD
(ω
ωs− 1
)T3dy3
dt= −y3 + y1
y2 =
(1−
T2
T3
)y3 +
T2
T3y1
DAEs considering the particular caseof K2 = 0
PV = PC − y2, if Pmin ≤ PC − y2 ≤ Pmax
PV = Pmax, if PC − y2 > Pmax
PV = Pmin, if PC − y2 < Pmin
Dr. Hector A. Pulgar hpulgar@utk 20 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Two-axis model of a DFIG
dE′qD
dt= −
1
T ′0
(E′
qD + (Xs −X′s)Ids
)+ ωs
Xm
XrVdr
− (ωs − ωr)E′dD
dE′dD
dt= −
1
T ′0
(E′
dD − (Xs −X′s)Iqs
)− ωs
Xm
XrVqr
+ (ωs − ωr)E′qD
dωr
dt=
ωs
2HD
[Tm − E′
dDIds − E′qDIqs
]
WIND FARM MODEL FOR POWER SYSTEM STABILITY ANALYSIS
BY
HECTOR ARNALDO PULGAR PAINEMAL
DISSERTATION
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Electrical and Computer Engineering
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2010
Urbana, Illinois
Doctoral Committee:
Professor Peter W. Sauer, ChairProfessor Thomas J. OverbyeAssistant Professor Alejandro Dominguez-GarciaAssistant Professor Vinayak V. Shanbhag
Manifold is associated with the following equivalent circuit:
RsjXs Re jXe’
~EqD-jEdD’ ’
IGC
VD+
~
Iqs-jIds
~
+
Ve jq
Pgen+jQgen
VDjq D
e
Machine reference
e-jq : 1
Ideal
shift
transformer
Thevenin equivalent of the grid
Color definition:
Blue : State vari-ables
Green : Algebraicvariables
Red : Inputs
Dr. Hector A. Pulgar hpulgar@utk 21 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Zero-axis model of a DFIG
dωr
dt=
ωs
2HD[Tm −XmIqsIdr +XmIdsIqr]
s =ωs − ωr
ωs
WIND FARM MODEL FOR POWER SYSTEM STABILITY ANALYSIS
BY
HECTOR ARNALDO PULGAR PAINEMAL
DISSERTATION
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Electrical and Computer Engineering
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2010
Urbana, Illinois
Doctoral Committee:
Professor Peter W. Sauer, ChairProfessor Thomas J. OverbyeAssistant Professor Alejandro Dominguez-GarciaAssistant Professor Vinayak V. Shanbhag
Manifold is associated with the following equivalent circuit:
Rs jX sRr jX r
jXm
Vqr-jVdr
s
s
Iqr-jIdr Iqs-jIds
+
~ ~
IGC
VD
Pgen+jQgen
e-jq : 1D
VDjq D
Machine reference
Ideal
shift
transformer
Re jXe
~
+
Ve jqe
Thevenin equivalent of the grid
Color definition:
Blue : State vari-ables
Green : Algebraicvariables
Red : Inputs
Dr. Hector A. Pulgar hpulgar@utk 22 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Vqr, Vdr, Tm: P-Q controllers and pitch angle controller
Vwind
Pref wr
Iqr
Iqrref
Vqr
-
+
-P
KI1
sKP1 +
KI2
sKP2 +
+
DFIG
P
Q
Iqr
Idr
wr
f
Maximum
power
tracking
QrefIdr
Idrref
Vdr+
-
+
-Q
KI3
sKP3 +
KI4
sKP4 +
Vref
+
- V
Supervisory voltage controller
Tm
G3(s)
Wind turbine
G2(s)
Pitch-angle controller
wr
wref
qref
wref
Vwind
1
Tcs+1
Qmax
Qmin
KIv
sKPv +
1
fN
Dr. Hector A. Pulgar hpulgar@utk 23 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
2. Synchronous generator and wind turbine models
Control schemes to participate in frequency regulation (P1 and P2)
Vwind -
+
f
-
+
sTw
sTw+1Kpf +sKdf
DPref
Pref
Washout Filter
wr
fref
Iqr
Iqrref
Vqr
-
+
-P
KI1
sKP1 +
KI2
sKP2 +
+
DFIG
P
Q
Iqr
Idr
wr
f
Maximum
power
tracking
QrefIdr
Idrref
Vdr+
-
+
-Q
KI3
sKP3 +
KI4
sKP4 +
Vref
+
- V
Supervisory voltage controller
Tm
G3(s)
Wind turbine
G2(s)
Pitch-angle controller
wr
wref
qref
wref
Vwind
++
q0
Bfref
+
-+
+
P1: Inertial response
P2: Power reserve
1
Tcs+1
Qmax
Qmin
KIv
sKPv +
1
fN
Dr. Hector A. Pulgar hpulgar@utk 24 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis(Electromechanical oscillations)
Dr. Hector A. Pulgar hpulgar@utk 25 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Consider the aforementioned generic representation by the DAEs:
x = f(x, y, µ)0 = g(x, y, µ)
(?)
Given µ, assume that the system is in equilibrium at the point (xe,ye):
0 = f(xe, ye, µ)0 = g(xe, ye, µ)
In a neighborhood of (xe,ye), the system modeled by (?) can beapproximately represented by:
x = f(xe, ye, µ) +∇fx(xe, ye, µ)∆x+∇fy(xe, ye, µ)∆y + H.O.T.
0 = g(xe, ye, µ) +∇gx(xe, ye, µ)∆x+∇gy(xe, ye, µ)∆y + H.O.T.
where ∆x = x− xe and ∆y = y − ye. Note also that x = d(x−xe)dt = ∆x
Dr. Hector A. Pulgar hpulgar@utk 26 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
By neglecting H.O.T., we obtain:[∆x0
]=
[∇fx(xe, ye, µ) ∇fy(xe, ye, µ)∇gx(xe, ye, µ) ∇gy(xe, ye, µ)
] [∆x∆y
]To eliminate algebraic variables, we proceed as follows:
0 = ∇gx(xe, ye, µ)∆x+∇gy(xe, ye, µ)∆y
⇒ ∆y = −∇g−1y (xe, ye, µ)∇gx(xe, ye, µ)∆x
Finally,
∆x =(∇fx(xe, ye, µ)−∇fy(xe, ye, µ)∇g−1
y (xe, ye, µ)∇gx(xe, ye, µ))︸ ︷︷ ︸
As(xe,ye,µ)
∆x
This is a linear homogenous system, and the spectrum of matrix As giveus information about the system stability around the equilibrium point∆x = 0 (x = xe), ∆y = 0 (y = ye)
Dr. Hector A. Pulgar hpulgar@utk 27 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
To take into account a small perturbation, the following initial valueproblem is considered:
∆x = As∆x
∆x(t0) = x0 6= 0
The explicit solution of this problem is given by:
∆x(t) = eAs(t−t0)∆x0, ∀t ≥ t0
In general, As is a full matrix, and obtaining the exponential of As is notan easy task. However, a solution can be easily obtained using asimilarity transformation.
Dr. Hector A. Pulgar hpulgar@utk 28 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Assume that As ∈ Rn×n has n distinct eigenvalues λi ∈ C∀ i = 1, 2, ..., n. Then, each λi has a unique right eigenvector,vi ∈ Cn×1, and left eigenvector, wi ∈ Cn×1, defined by:
Asvi = λivi
wTi As = λiwTi
Note that:
Right (left) eigenvectors are linearly independent. DefiningV = [v1, v2, ..., vn] and W = [w1, w2, ..., wn], we say that V and Ware full rank matricesThe set of right and left eigenvectors are orthogonal. To prove this,consider λi 6= λj , then
Asvi = λivi (†)wTj As = λjw
Tj (‡)
Multiplying (‡) from the right by vi, we have:
wTj Asvi = λjwTj vi
⇒ wTj λivi = λjwTj vi
⇒ (λi − λj)wTj vi = 0⇒ wTj vi = 0,∀ i 6= j
Dr. Hector A. Pulgar hpulgar@utk 29 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
As we are free to choose any magnitude for the eigenvectors, we assign amagnitude in such a way that the set of right and left eigenvectors areorthonormal. We define wTj vi = δji (Kronecher delta), or in matrix form:[
w1 w2 . . . wn]T︸ ︷︷ ︸
WT
[v1 v2 . . . vn
]︸ ︷︷ ︸V
= In ⇒WT = V −1
Finally,
Asv1 = λ1v1
Asv2 = λ2v2
...Asvn = λnvn
⇒ As [v1 v2 . . . vn]︸ ︷︷ ︸V
= [v1 v2 . . . vn]︸ ︷︷ ︸V
λ1 0 . . . 00 λ2 . . . 0...
.... . .
...0 0 . . . λn
︸ ︷︷ ︸
Λ
⇒ V −1AsV = Λ
⇒WTAsV = Λ : Diagonalization of As!
Dr. Hector A. Pulgar hpulgar@utk 30 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
We define a transformed vector of state variables as q = WT∆x. Notethat ∆x = W−T q = V q. Then, the transformed state equations become:
∆x = As∆x⇒WT∆x = WTAs∆x
⇒ q = WTAsV q ⇒ q = Λq
whose solution is given by:
q(t) = eΛtWT∆x(0)︸ ︷︷ ︸q(0)
= eΛt
wT1 ∆x(0)...
wTn∆x(0)
,∀ t ≥ 0
As Λ is diagonal, eΛt is also diagonal, and the transformed state variablesare decoupled. Therefore,
∀ i = 1, 2, ..., n, qi(t) = eλitwTi ∆x(0)
The term eλit is known as mode i
Dr. Hector A. Pulgar hpulgar@utk 31 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
By transforming back to the original state variables, we obtain:
∆x(t) = V q = [v1 v2 . . . vn]
q1(t)...
qn(t)
=
n∑i=1
qi(t)vi
∆x(t) =
n∑i=1
eλitwTi ∆x(0)vi
⇒ ∀ k = 1, 2, ..., n, ∀ t ≥ 0, ∆xk(t) =
n∑i=1
eλitwTi ∆x(0)vi(k)
Remarks
a. The appearance of mode i in the state variable ∆xk depends on theperturbation ∆x(0) (excitation of the mode)
b. The relative phase of the mode i in the state variable ∆xk depends onthe phase of vi(k) (mode shape)
c. The intensity of the mode i on the state variable ∆xk depends on themagnitude of wTi ∆x(0)vi(k)
Dr. Hector A. Pulgar hpulgar@utk 32 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
By transforming back to the original state variables, we obtain:
∆x(t) = V q = [v1 v2 . . . vn]
q1(t)...
qn(t)
=
n∑i=1
qi(t)vi
∆x(t) =
n∑i=1
eλitwTi ∆x(0)vi
⇒ ∀ k = 1, 2, ..., n, ∀ t ≥ 0, ∆xk(t) =
n∑i=1
eλitwTi ∆x(0)vi(k)
Remarks
a. The appearance of mode i in the state variable ∆xk depends on theperturbation ∆x(0) (excitation of the mode)
b. The relative phase of the mode i in the state variable ∆xk depends onthe phase of vi(k) (mode shape)
c. The intensity of the mode i on the state variable ∆xk depends on themagnitude of wTi ∆x(0)vi(k)
Dr. Hector A. Pulgar hpulgar@utk 32 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
By transforming back to the original state variables, we obtain:
∆x(t) = V q = [v1 v2 . . . vn]
q1(t)...
qn(t)
=
n∑i=1
qi(t)vi
∆x(t) =
n∑i=1
eλitwTi ∆x(0)vi
⇒ ∀ k = 1, 2, ..., n, ∀ t ≥ 0, ∆xk(t) =
n∑i=1
eλitwTi ∆x(0)vi(k)
Remarks
a. The appearance of mode i in the state variable ∆xk depends on theperturbation ∆x(0) (excitation of the mode)
b. The relative phase of the mode i in the state variable ∆xk depends onthe phase of vi(k) (mode shape)
c. The intensity of the mode i on the state variable ∆xk depends on themagnitude of wTi ∆x(0)vi(k)
Dr. Hector A. Pulgar hpulgar@utk 32 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
By transforming back to the original state variables, we obtain:
∆x(t) = V q = [v1 v2 . . . vn]
q1(t)...
qn(t)
=
n∑i=1
qi(t)vi
∆x(t) =
n∑i=1
eλitwTi ∆x(0)vi
⇒ ∀ k = 1, 2, ..., n, ∀ t ≥ 0, ∆xk(t) =
n∑i=1
eλitwTi ∆x(0)vi(k)
Remarks
a. The appearance of mode i in the state variable ∆xk depends on theperturbation ∆x(0) (excitation of the mode)
b. The relative phase of the mode i in the state variable ∆xk depends onthe phase of vi(k) (mode shape)
c. The intensity of the mode i on the state variable ∆xk depends on themagnitude of wTi ∆x(0)vi(k)
Dr. Hector A. Pulgar hpulgar@utk 32 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Common practice (power engineering community): We define the matrixof participation factors as the element-wise product between W and V :
P = W V =
w1(1)v1(1) w2(1)v2(1) . . . wn(1)vn(1)w1(2)v1(2) w2(2)v2(2) . . . wn(2)vn(2)
......
. . ....
w1(n)v1(n) w2(n)v2(n) . . . wn(n)vn(n)
= [pki]
In the particular case that ∆x(0) = ek = [0 ... 1︸︷︷︸k-term
... 0]T
⇒ wTi ∆x(0) = wTi ek = wi(k)
⇒ ∆xk(t) =
n∑i=1
eλitwi(k)vi(k)
⇒ ∆xk(t) =
n∑i=1
eλit pki︸︷︷︸intensity ofmode i onvariable xk
Dr. Hector A. Pulgar hpulgar@utk 33 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Electromechanical oscillations
This refers to low frequency oscillations that occur in synchronousgenerators (typically between 0.1 to 3 Hz)
The physical phenomenon behind these oscillations is the exchangeof kinetic energy stored at the rotating masses of the generators
In mathematical terms, the electromechanical modes are those thathave high participation factors with respect to the angular speed androtor angle of the generators
Depending on the oscillation frequency, these electromechanicalmodes can be classified as:
Local modes (typically between 0.8 to 1.5 Hz)Intra-plant mode (typically between 2 to 3 Hz)Inter-area mode (typically between 0.1 to 0.7 Hz)
Oscillations must have a damping ratio of at least 10% in normaloperation, and at least 5% after a single contingency
Dr. Hector A. Pulgar hpulgar@utk 34 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Electromechanical oscillations
This refers to low frequency oscillations that occur in synchronousgenerators (typically between 0.1 to 3 Hz)
The physical phenomenon behind these oscillations is the exchangeof kinetic energy stored at the rotating masses of the generators
In mathematical terms, the electromechanical modes are those thathave high participation factors with respect to the angular speed androtor angle of the generators
Depending on the oscillation frequency, these electromechanicalmodes can be classified as:
Local modes (typically between 0.8 to 1.5 Hz)Intra-plant mode (typically between 2 to 3 Hz)Inter-area mode (typically between 0.1 to 0.7 Hz)
Oscillations must have a damping ratio of at least 10% in normaloperation, and at least 5% after a single contingency
Dr. Hector A. Pulgar hpulgar@utk 34 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Electromechanical oscillations
This refers to low frequency oscillations that occur in synchronousgenerators (typically between 0.1 to 3 Hz)
The physical phenomenon behind these oscillations is the exchangeof kinetic energy stored at the rotating masses of the generators
In mathematical terms, the electromechanical modes are those thathave high participation factors with respect to the angular speed androtor angle of the generators
Depending on the oscillation frequency, these electromechanicalmodes can be classified as:
Local modes (typically between 0.8 to 1.5 Hz)Intra-plant mode (typically between 2 to 3 Hz)Inter-area mode (typically between 0.1 to 0.7 Hz)
Oscillations must have a damping ratio of at least 10% in normaloperation, and at least 5% after a single contingency
Dr. Hector A. Pulgar hpulgar@utk 34 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Electromechanical oscillations
This refers to low frequency oscillations that occur in synchronousgenerators (typically between 0.1 to 3 Hz)
The physical phenomenon behind these oscillations is the exchangeof kinetic energy stored at the rotating masses of the generators
In mathematical terms, the electromechanical modes are those thathave high participation factors with respect to the angular speed androtor angle of the generators
Depending on the oscillation frequency, these electromechanicalmodes can be classified as:
Local modes (typically between 0.8 to 1.5 Hz)Intra-plant mode (typically between 2 to 3 Hz)Inter-area mode (typically between 0.1 to 0.7 Hz)
Oscillations must have a damping ratio of at least 10% in normaloperation, and at least 5% after a single contingency
Dr. Hector A. Pulgar hpulgar@utk 34 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Electromechanical oscillations
This refers to low frequency oscillations that occur in synchronousgenerators (typically between 0.1 to 3 Hz)
The physical phenomenon behind these oscillations is the exchangeof kinetic energy stored at the rotating masses of the generators
In mathematical terms, the electromechanical modes are those thathave high participation factors with respect to the angular speed androtor angle of the generators
Depending on the oscillation frequency, these electromechanicalmodes can be classified as:
Local modes (typically between 0.8 to 1.5 Hz)Intra-plant mode (typically between 2 to 3 Hz)Inter-area mode (typically between 0.1 to 0.7 Hz)
Oscillations must have a damping ratio of at least 10% in normaloperation, and at least 5% after a single contingency
Dr. Hector A. Pulgar hpulgar@utk 34 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
3. Small-signal stability analysis
Natural frequency (Ω), damping ratio (σ), oscillation frequency (fo)
Consider the mode eλit with λi = λi,x + jλi,y ∈ C
Ωi ,√λ2i,x + λ2
i,y
σi ,−λi,x|λ| =
−λi,xΩ
= − cos θi
⇒λi,x = −σiΩi⇒λi,y =
√|λi|2 − λ2
i,x =√
1− σ2iΩi
⇒fi,o ,λi,y2π
=
√1− σ2
iΩi2π
5%
θi realaxis
imaginaryaxis
λi,x
λi,y
Ωi = |λi|λi
σ10%
σ
Define Ai , wTi ∆x(0)vi(k) = |Ai|ejαi . We can prove that:
∆xk(t) =
n∑i=1
eλitwTi ∆x(0)vi(k) =
n∑i=1
|Ai|e−σiΩit cos
(√1− σ2
i Ωit+ αi
)Dr. Hector A. Pulgar hpulgar@utk 35 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Cases of study
Dr. Hector A. Pulgar hpulgar@utk 36 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: 15-bus test system
2 SGs: Two-axis model, IEESGO governor, IEEE Type-1 exciter5 WTGs: Supervisory control, participation in frequency regulation
~
~
2 7 8 9 3
5 6
4
15
100+j35
MVA
90+j30 MVA125+j50 MVA
Power base: 100 MVA
Rated frequency: 60 Hz
V15 = 1Ð0° p.u. (slack bus)1
10
11
12
13
Wind farm
14 ~
~
~
~
~
Bus regulated by
supervisory controlSG 2, H2=25 [s]
SG 1
H1=10 [s]
PG2 = 163 MW
|V2| = 1 p.u.
Dr. Hector A. Pulgar hpulgar@utk 37 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Cases of study: Base case
Dominant modes - base case
No participation in frequency regulationEigenvalue σ % Mode-0.165 ± j 6.524 2.53 Electromechanical-0.468 ± j 1.575 28.5 SG1 and SG2’s exciters-0.404 ± j 0.723 48.8 SG1 and SG2’s exciters-0.196 ± j 0.165 76.5 SG1 and SG2’s governors
P1: Frequency regulation through inertial responseEigenvalue σ % Mode-0.188 ± j 6.485 2.90 Electromechanical-0.468 ± j 1.575 28.5 SG1 and SG2’s exciters-0.404 ± j 0.723 48.8 SG1 and SG2’s exciters-0.075 ± j 0.128 50.7 SG1 and SG2’s governors
Dr. Hector A. Pulgar hpulgar@utk 38 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Cases of study: Load parametrization
Power transfer from SG2 to bus 8
−0.3 −0.25 −0.2 −0.15 −0.15.9
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Real axis
Imag
inar
y ax
is
Increasing loading atbus 8 from 100 [MW]to 178 [MW]
Increasing loadingat bus 8 from 178[MW] to 225 [MW]
Sudden change due tosupervisory control at 178 [MW]
Electromechanical mode
The abrupt change ineigenvalues pathwayis due to the hittingof the supervisorycontrol limits.
The supervisorycontrol hits its limitswhen the load at bus8 is 178 [MW]
Black-line: No participation in freq. control / Red-line: Participation through inertial response
Dr. Hector A. Pulgar hpulgar@utk 39 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Cases of study: Inertia parametrization
H1 and H2 as parameters (WF at bus 4)
−0.32 −0.3 −0.28 −0.26 −0.24 −0.22 −0.2 −0.18 −0.164
4.5
5
5.5
6
6.5
7
7.5
8
Real part
Imag
inar
y pa
rt
Electromechanical mode
H1=10 [s]
H2=25 [s]
Increasing H1
Decreasing H2
Increasing H1
Decreasing H2
σ=4.39%F
σ=2.24%C
Dσ=2.90%
Aσ=2.53%
Eσ=5.84%
σ=3.47%B
The point Acorresponds to thebase case whenWTGs do notparticipate infrequency regulation.
The point Dcorresponds to thebase case whenWTGs participate infrequency regulationthrough inertialresponse.
Dashed-line: No participation in freq. control / Solid-line: Participation through inertial response
Dr. Hector A. Pulgar hpulgar@utk 40 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Effect of WF’s location
Now, WF connected at bus 7
~
~
2
7
8 9 3
5 6
4
15
SG 2, H2=25 [s]
SG 1
H1=10 [s]
100+j35
MVA
90+j30 MVA125+j50 MVA
PG2 = 163 MW
|V2| = 1 p.u.
Power base: 100 MVA
Rated frequency: 60 Hz
V15 = 1Ð0° p.u. (slack bus)
1
Wind farm
10 11 12 13 14
~ ~ ~ ~ ~Bus regulated by
supervisory control
Dr. Hector A. Pulgar hpulgar@utk 41 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Inertia parametrization
Inertia of SGs as parameters (two cases: WF at bus 4 and bus 7)
1015
2025
30
1015
2025
300
0.02
0.04
0.06
0.08
0.1
H1 [s]
H2 [s]
σ (d
ampi
ng r
atio
)
WF withP1 and
connectedat bus 7
WF with P1 andconnected at bus 4
B
WF with noparticipation
andconnected at:
bus 4bus 7A
C
DE
H
GF
Dr. Hector A. Pulgar hpulgar@utk 42 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Summary
Remarks
X Two schemes to make WTGs participate in frequency regulation arestudied: P1 (inertial response) and P2 (power reserve)
X From a small-signal stability perspective:
- P2 does not alter the typical relation between system inertia anddamping ratio—the larger H, the larger σ
- P1 does alter this typical relation. It is possible to even increase σwhen H decreases
X Results show that locating WTGs closer to SGs with less inertia willincrease the system damping ratio the most. In large systems withmany electromechanical modes, determining the location of WTGsis still an open question
X Discontinuities due to variable limits require exhaustive evaluation,as these may lead to the sudden loss of system stability. Thispossible outcome is discussed in the following case, where WF isconnected instead at bus 9
Dr. Hector A. Pulgar hpulgar@utk 43 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Summary
Remarks
X Two schemes to make WTGs participate in frequency regulation arestudied: P1 (inertial response) and P2 (power reserve)
X From a small-signal stability perspective:
- P2 does not alter the typical relation between system inertia anddamping ratio—the larger H, the larger σ
- P1 does alter this typical relation. It is possible to even increase σwhen H decreases
X Results show that locating WTGs closer to SGs with less inertia willincrease the system damping ratio the most. In large systems withmany electromechanical modes, determining the location of WTGsis still an open question
X Discontinuities due to variable limits require exhaustive evaluation,as these may lead to the sudden loss of system stability. Thispossible outcome is discussed in the following case, where WF isconnected instead at bus 9
Dr. Hector A. Pulgar hpulgar@utk 43 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Summary
Remarks
X Two schemes to make WTGs participate in frequency regulation arestudied: P1 (inertial response) and P2 (power reserve)
X From a small-signal stability perspective:
- P2 does not alter the typical relation between system inertia anddamping ratio—the larger H, the larger σ
- P1 does alter this typical relation. It is possible to even increase σwhen H decreases
X Results show that locating WTGs closer to SGs with less inertia willincrease the system damping ratio the most. In large systems withmany electromechanical modes, determining the location of WTGsis still an open question
X Discontinuities due to variable limits require exhaustive evaluation,as these may lead to the sudden loss of system stability. Thispossible outcome is discussed in the following case, where WF isconnected instead at bus 9
Dr. Hector A. Pulgar hpulgar@utk 43 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Summary
Remarks
X Two schemes to make WTGs participate in frequency regulation arestudied: P1 (inertial response) and P2 (power reserve)
X From a small-signal stability perspective:
- P2 does not alter the typical relation between system inertia anddamping ratio—the larger H, the larger σ
- P1 does alter this typical relation. It is possible to even increase σwhen H decreases
X Results show that locating WTGs closer to SGs with less inertia willincrease the system damping ratio the most. In large systems withmany electromechanical modes, determining the location of WTGsis still an open question
X Discontinuities due to variable limits require exhaustive evaluation,as these may lead to the sudden loss of system stability. Thispossible outcome is discussed in the following case, where WF isconnected instead at bus 9
Dr. Hector A. Pulgar hpulgar@utk 43 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Summary
Remarks
X Two schemes to make WTGs participate in frequency regulation arestudied: P1 (inertial response) and P2 (power reserve)
X From a small-signal stability perspective:
- P2 does not alter the typical relation between system inertia anddamping ratio—the larger H, the larger σ
- P1 does alter this typical relation. It is possible to even increase σwhen H decreases
X Results show that locating WTGs closer to SGs with less inertia willincrease the system damping ratio the most. In large systems withmany electromechanical modes, determining the location of WTGsis still an open question
X Discontinuities due to variable limits require exhaustive evaluation,as these may lead to the sudden loss of system stability. Thispossible outcome is discussed in the following case, where WF isconnected instead at bus 9
Dr. Hector A. Pulgar hpulgar@utk 43 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Limit-induced bifurcation
~
~
2 7 8
9 3
5 6
4
15
110111213
Wind
turbines
SG 1
SG 2
~~ ~ ~ ~
~SG 3
14
PCC
Supervisory
control
Qref (reactive power reference)
Ideal voltage
transformer
V
Vref
100+j35
MVA
90+j30 MVA125+j50 MVA
V15 = 1Ð0° p.u. (slack bus)
PG2 = 163 MW
|V2| = 1 p.u.
Power base: 100 MVA
Rated frequency: 60 Hz
PG3 = 85 MW
|V3| = 1 p.u.
Dr. Hector A. Pulgar hpulgar@utk 44 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Limit-induced bifurcation
Power transfer from SG1 to load at bus 8. Load at bus 8 increases from200 to 400 MW
−15 −10 −5 0 50
5
10
Imag
inar
y A
xis
−15 −10 −5 0 50
5
10
Real Axis
Imag
inar
y A
xis
Withsupervisorycontrol
Withoutsupervisorycontrol
Voltage mode
Electromechanical mode
Electromechanical mode
Voltage modeLIB point
HB point
Dr. Hector A. Pulgar hpulgar@utk 45 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Limit-induced bifurcation
Zoom in of previous trajectory with supervisory voltage control
1.5 1 0.5 0 0.5 1 1.5 23
4
5
6
7
8
9
10
11
12
13
Real Axis
Imag
inar
y A
xis
This unstable eigenvaluetrajectory abruptly appears
when Qref = Qmax(voltage mode)
Qref = Qmin Qmin < Qref < Qmax Qref = Qmax
Stable eigenvalue trajectories(electromechanical modes)
LIB point
Dr. Hector A. Pulgar hpulgar@utk 46 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Limit-induced bifurcation
Trajectories of fixed voltage and fixed reactive power (Qmax)
V
P
Loading trajectory when V=Vref
HB point
V
P
Loading trajectory when hitting limit does not cause instability
PV trajectory when V=Vref
HB point
Supervisory control reaches its limit
V
P
Loading trajectory when Qref=Qmax
PV trajectory when Q=Qmax
HB pointLIB point
(a) (b) (c)
Stable trajectory Unstable trajectory
Vref Vref
Dr. Hector A. Pulgar hpulgar@utk 47 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
4. Case of study: Limit-induced bifurcation
PV curve and limit-induced bifurcation
200 220 240 260 280 300 320 340 360 380 4000.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
Pload
[MW]
VPC
C p
.u. Supervisory control
lower limit binds
Geometric trajectory when Qref
=Qmin
(classical PV curve)
HB point trajectory if Qref
isconstant such that Q
min < Q
ref < Q
max
Geometric trajectory when Qref
=Qmax
(classical PV curve)
Supervisory controlregulates bus voltage
Supervisory controlupper limit binds
LIB point
Dr. Hector A. Pulgar hpulgar@utk 48 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
References
1. P.W. Sauer, M.A. Pai, Power system dynamics and stability, Upper Saddle River,NJ: Prentice Hall, 1998
2. J. Machowski, J. Bialek, J. Bumby, Power System Dynamics: Stability andControl, Chichester, WS: John Wiley & Sons Ltd, 2nd edition, 2008
3. H.K. Khalil, Nonlinear Systems, 3rd ed, Upper Saddle River, NJ: Prentice Hall,2002
4. H. Pulgar-Painemal, Wind farm model for power system stability analysis, Ph.D.Thesis, University of Illinois at Urbana-Champaign, 2010
5. H.M.A. Hamdan, A.M.A. Hamdan, On the coupling measures between modes andstate variables and subsynchronous resonance, Electric Power Systems Research,Volume 13, Issue 3, December 1987, Pages 165-171
6. G.C. Verghese, I.J. Perez-Arriaga, F.C. Schweppe, Selective Modal Analysis WithApplications to Electric Power Systems—Part II: The Dynamic Stability Problem,IEEE Transactions on Power Apparatus and Systems, vol.PAS-101, no.9,pp.3126,3134, Sept. 1982
7. H.A. Pulgar-Painemal, R.I. Galvez-Cubillos, Wind farms participation in frequencyregulation and its impact on power system damping, 2013 IEEE GrenoblePowerTech (POWERTECH), June 16-20, 2013
8. H.A. Pulgar-Painemal, R.I. Galvez-Cubillos, Limit-induced bifurcation by wind farmvoltage supervisory control, Electric Power Systems Research, Volume 103,October 2013, Pages 122-128
Dr. Hector A. Pulgar hpulgar@utk 49 / 50
ECE-620 Lecture on power system modeling and dynamics including wind turbines Knoxville, TN, November 19 2014
Thanks for your attention
Questions?
Dr. Hector A. Pulgar hpulgar@utk 50 / 50