ece-620 lecture on power system modeling and dynamics

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


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




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




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))




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!




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




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




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




⇒ 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




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




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.




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




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




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



2. Synchronous generator andDFIG-based wind turbine models



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




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



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




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




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


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:



Red : Inputs




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


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:



Red : Inputs




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




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



3. Small-signal stability analysis(Electromechanical oscillations)



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




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)




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.




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




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!




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




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)




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




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




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


4. Cases of study



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.



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



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



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


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



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


V15 = 1Ð0° p.u. (slack bus)

1

Wind farm

10 11 12 13 14

~ ~ ~ ~ ~Bus regulated by

supervisory control



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



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



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


PG3 = 85 MW

|V3| = 1 p.u.




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



Voltage modeLIB point

HB point




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




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




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



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



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ece-620 lecture on power system modeling and dynamics

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