A robust adaptive controller for a DFIG wind turbine with grid voltage and frequency support

Bộ điều khiển thích ứng mạnh mẽ cho tuabin gió DFIG có lưới Hỗ trợ điện áp và tần số. A robust adaptive nonlinear controller is designed for a DoublyFed Induction Generator (DFIG) wind turbine connected to a power grid. The controller main objective is to regulate the generator terminal voltage and rotor speed. A model based control design strategy is adopted. The controller structure and equations are obtained following a backstepping control design method using the DFIG reduced order model. Grid parameters are assumed unknown during the design. Therefore, the controller is provided with an adaptation module that automatically readjusts controller parameters when grid conditions change. Simulations are used to assess the proposed DFIG controller effectiveness.

Abstract—A robust adaptive nonlinear controller is designed

for a Doubly-Fed Induction Generator (DFIG) wind turbine

connected to a power grid The controller main objective is to

regulate the generator terminal voltage and rotor speed A

model based control design strategy is adopted The controller

structure and equations are obtained following a backstepping

control design method using the DFIG reduced order model

Grid parameters are assumed unknown during the design

Therefore, the controller is provided with an adaptation module

that automatically readjusts controller parameters when grid

conditions change Simulations are used to assess the proposed

DFIG controller effectiveness

I INTRODUCTION

HE increasing penetration of renewable energy

generators in general and particularly wind power

generators is forcing power utilities to reconsider the way

these generators interact with the grid The trend is that wind

turbines should remain connected to the grid when severe

faults occur while supporting the grid voltage and frequency

during normal operations Several researchers are still

investigating the best control system for wind generators

with these new operating constraints In [1], a combination

of proportional–integral (PI) and Lyapunov based auxiliary

controller is proposed to stabilize and improve the DFIG

post fault behavior In [2], a feedback linearization based

nonlinear voltage and slip controller is proposed for a DFIG

connected to an infinite bus A direct active and reactive

power controller based on stator flux estimation is discussed

in [3] The paper proposes a basic hysteresis controller In

[4], the well-known vector control is proposed to regulate the

active power and reactive power generated by a DFIG In

[5], a DFIG control strategy that is very similar to the

conventional AVR/PSS for synchronous generator is utilized

to support the power grid voltage and frequency The main

drawback of the aforementioned works is their inability to

cope with large changes in grid parameters

This paper proposes a robust adaptive control strategy that

makes possible for a wind generator to adequately support

the grid voltage and frequency A doubly-fed induction

Manuscript received February 15, 2010 This work was supported in

part by the Academic Research Program (ARP)

F A Okou and S Gauthier are with the Royal Military College of

Canada, Po Box 17000 Station Forces, Kingston, Ontario, Canada (phone:

1-613-5416000 x6630; e-mail: aime.okou@ rmc.ca)

O Akhrif is with Ecole de Technologie Superieure, (e-mail:

ouassima.akhrif@etsmtl.ca)

machine is used to converter the mechanical power into an electric energy A fast response wind generator is proposed

to be able to easily follow changes that occur in the power grid The controller adaptive feature helps to considerably compensate for very large variation into the grid parameters

A model based design method is proposed to find the controller equations A backstepping control design method

is used to derive the controller structure The proposed controller main advantages are that it considerably decreases the wind generator time response The active power and reactive power generated are automatically readjusted to maintain the grid frequency and voltage at their nominal values

The rest of this paper is organized as follows: The doubly-fed induction machine model is presented in the section II The proposed controller is designed in the section III Section IV presents the simulation results The paper ends with a conclusion

II DFIGSTATE SPACE MODEL

A doubly-fed induction machine is represented in d/q reference frame by the following equations [6]:

ds

ds s ds s qs

d

dt

Ψ

dt

d i

R

ds s qs s qs

Ψ + Ψ ω +

dt

d i

R

vdr = r dr−ωslΨqr+ Ψdr (1.c)

dt

d i

R

dr sl qr r qr

Ψ + Ψ ω +

where the variables vds, vqs, vdr, vqr represent the d/q reference frame components for the machine stator and rotor voltages Variables ids, iqs, idr, iqr stand for the machine stator and rotor current in the d/q reference frame The machine fluxes in the d/q reference frame are presented by the variables Ψds, Ψqs, Ψdr, Ψqr Finally, variables ωs

and ωsl are stator and rotor current frequencies in rad/s, respectively Parameters Rs and Rr represent rotor and stator resistances

The fluxes and currents are related by the following algebraic relationships

A Robust Adaptive Controller for a DFIG Wind Turbine with Grid

Voltage and Frequency Support

Francis A Okou, Member, IEEE, Ouassima Akhrif, Member, IEEE, and Sebastien Gauthier

T

dr m ds

s

ds=L i +L i

qr m qs

s

qs=L i +L i

ds m dr

r

dr =Li +L i

qs m qr

r

qr =L i +L i

where parameters Ls, Lr, and Lm represent the stator

inductance, the rotor inductance and the mutual inductance

The electric torque generated by the DFIG has the following

expression in terms of the d/q reference frame stator flux and

current

) i i

(

p

The parameter p is the number of pole pairs Note that the

rotor current frequency can be expressed in term of the stator

current frequency and the rotor speed in rad/s as follows:

r

s

sl=ω −ω

When the d-axis for the park transformation used to find the

DFIG dynamics aligned with rotor flux axis, we have the

follow relationships:

0

qs=

s

V

The parameter Vs stands for the grid voltage As the

consequence, rotor flux components, in the d/q reference

frame, have the following expression in terms of the rotor

current components and the grid voltage

m s

s s

L V

L i L

m s r

1 L (L L )

It is common to neglect the stator resistance and to assume a

constant grid voltage magnitude for the design of the wind

power generator controller These assumptions lead to a

reduced order model We get the following equations at the

stator:

0 dt

d i

R

vds= sds−ωsΨqs+ Ψds ≈ (7.a)

qs

d

dt

Ψ

Next equations which represent the rotor dynamics become

the reduced model for the DFIG Note that the model

depends on the grid voltage Vs and frequency ωs

dt

di L i L i

R

vdr = r dr−ωslσ rqr +σ r dr (8.a)

qr

m s

s s

di

L V

The DFIG electrical torque and terminal voltage could be

written in terms of the reduced order model state variables as

follows:

m

s

L

L

s s s ds s m qr

The rotor speed dynamics are represented by the following equation:

p dtω = − ω + + (10)

Parameters J and B represent the generator inertia and the mechanical friction coefficient, respectively Tm is the mechanical torque supplies by the blades

The paper objective is to propose a control system for this wind power generator to regulate its terminal voltage and the rotor speed The rotor speed is related to the grid frequency The following figure represents the wind turbine connected

to a grid AC/DC – DC/AC converters are used to generate the rotor voltage Indeed, the DFIG controller generates the reference signals for the rotor-side converter That converter generates the rotor voltages from the capacitor DC voltage This DC voltage is generated by the grid side AC/DC converter from the grid voltage This paper is concerned with the design of the rotor side converter The grid-side converter controller which regulates the capacitor DC voltage is not treated for the sake of brevity The DC voltage across the capacitor is assumed constant, therefore

Fig 1 A doubly-fed induction machine based wind power generator III ROBUST ADAPTIVE CONTROLLER DESIGN This section presents how the DFIG controller structure and equations are obtained A model based design method is proposed and it is based on the DFIG model presented in the previous section The controller is an adaptive voltage and speed regulator which guarantees that the active power and reactive power delivered by the generator are automatically adjusted to support the grid when a change occurs A systematic method to find the controller equations is now presented

A Rotor Speed Regulator Equations (10) and (8.b) are considered for the design of this rotor speed regulator For the sake of clarity, these equations are repeated below

d

i

dtω = −θ ω + θ − θɶ (11.a)

d

dt = −θ − θ − θ + θ (11.b)

Parameters that appear in this model have the following

expressions:

J

J

s L

J L

θ = , θ = ω5 sl

sl m s

4

r s s

L V

L L

ω

θ =

6 r

R L

θ =

σ , 7

r

1 L

θ =

σ

Note that the difference between the variable ωr to be

regulated and its reference ref

r

ω is denoted by ref

r

r

r

~ =ω −ω

ω It is assumed during the design that the grid

voltage Vs and frequency ωs are unknown or they could

change As a consequence, parameters θ2, θ3 , θ4 and θ5 are

assumed unknown However, θ1, θ6 and θ7 are assumed

perfectly known and they are equal to their nominal values,

θ1N, θ6N and θ7N respectively

The design objective is to find Vqr expression in such a way

that the system described by equations (11.a, 11.b) is stable

and the variable ωɶr is maintained at zero Since, Vqr doesn’t

influence ωɶr directly but does it via the variable iqr, the

value of iqr that will maintain ωɶr at zero is find first using

equation (11.a) It has the following expression

qr ˆ3 1 k 1 r

4

= α + ωɶ (12.a)

2

4

= ω + θ − θ ω +ɶ + ω ωɶ (12.b)

where ˆα3 is an estimation of α = θ3 1 3 The way that

estimation is obtained will be discussed later on The terms

2

1 r

k 4v ωɶ and k (1 )2r r

4 + ω ωɶ appear in equation (12.a) and (12.b) respectively to compensate for the likely difference

between α3, θ1 and θ2; and their corresponding estimation

or nominal values ˆα3, θ1N and ˆθ2 The term ˆθ − θ ω is 2 1N r

included to cancel the corresponding term in equation (11.a)

The parameter k1 is a positive gain selected by the designer

to stabilize the system The parameter kis also a gain Its

role will be explained later on

Next, the error between the signal iqr and its reference i*qr

is defined and it is equal to

*

qr

qr

The control signal Vqr expression will be found in such a way

that its drives this error signal to zero That expression is

obtained from the error dynamics that have the following

form:

6

i 1

d

dtɶ = ϕ + ϕ θ + θ∑= (14)

where the nonlinear functions that appear in the above

equation have the following expressions:

0 ˆ3 1v (ˆ3 kv )1 r ˆ2

2

ϕ = −αɺ − α + ω θɺɶ

ϕ = −ϕω ; ϕ = ϕ2 ; ϕ = −ϕ3 iqr; φ4 =−1; ϕ = −5 idr;

6 iqr

ϕ = −

The control input equation that will stabilize the system while maintaining the error signal at zero has the following form:

2

qr 7N 2 k 2 qr

4

= α − ɶ (15.a)

5

i 2

i r qr

i 1

4

=

=

= −ϕ − ϕ θ − ϕ θ − ϕ θ∑ − + θ ω

− ∑ϕ + ω

ɶ ɶ

(15.b)

The parameter k2 is a positive gain that needs to be selected

by the designer to stabilize the system The term ˆθ ωɶ is 1 r added for stability reason too The terms in bracket are added to cancel their corresponding terms in equation (14) The last terms of equation (15.b) are incorporated to compensate for the possible difference between the estimated parameters or the nominal values and their corresponding parameters

The system closed loop dynamics have the following form:

2

r r

k (1 ) 4

α

ɶ

ɶ

(16.a)

= − + θ ω + ϕ θ −∑ ∑ϕ + ω

α

ɶ ɶ

(16.b)

These equations are necessary to study the system stability The tilde parameters represent the difference between any estimated or nominal parameter and its actual value The speed regulator part of the controller is illustrated at Figure

2 This regulator consists indeed of a ωr-controller which generates the reference for the iqr-controller

Next section discusses the adaptation laws used to update the estimated parameters that appear in the controller equations The adaptation laws are derived from a stability study that involves the closed loop system describes by equations (16.a, 16.b) and the adaptation module The proof of stability guarantees that the two dynamics could coexist without any instability problem

The following candidate Lyapunov function is used for this purpose

2 2 5

α θ

= ω +ɶ + ∑ βɶ + βɶ (17)

ref

r

qr

v

qr

iɶ

r

ωɶ

qr

i

r

ω

qr

i controller

r controller

ω

dr

v

dr

iɶ

v

Ωɶ

+

dr

i

v

Ω

dr

i controller

v controller

Ω

ref t

V

speed controller

Voltage controller

+

Fig 2 The controller interne structure

The function V is positive definite since coefficients β and i

'

3

β are positive real numbers The derivative of the candidate

Lyapunov function has the expression given at the end of this

paragraph (equation 18.a) The adaptation laws are selected

in such a way that the Lyapunov function derivative is

negative semi-definite if αɶ7, θɶ1, and θɶ6 vanish The

resulting adaptation laws have the following expression

θ = βɺ  ϕ ɶ + ω θɶ ,

θ = βɺ  ϕ ɶ − ωɶ ɶ θ ,

( )

ˆ Pr oj i ,ˆ 

θ = βɺ  ϕɶ θ, i=4, 5,

'

The projection function involves in the adaptation laws

expressions is defined as follows:

[ ]

0}

y and

ˆ {

or

0}

y and ˆ

{

or ˆ

{ if

,

y

ˆ

y,

Proj

i iM

im i iM i im i

≤ θ

≤ θ

≥ θ

≤ θ θ

≤ θ

≤ θ

=

θ

) (

ˆ 1

[

y

ˆ

y,

i im

2 im

2 i

2 im

ρ

− θ

− θ

θ

− θ

−

=

θ

) (

ˆ 1

[

y

ˆ

y,

iM

2 i iM

2 iM

2 i

θ

− ρ + θ

θ

− θ

−

=

θ

It will guarantee that estimated parameters remain and

converge inside the predefined domain

[θ + ρ θ −ρim i iM i] Predefined lower and upper bounds of

the estimated parameter ˆθ are represented by i θ and im θiM The positive real number ρ is selected by the designer i The system stability is now discussed The Lyapunov function derivative has the following form:

2

7 2 qr 7

k

4 k

4

θ

β

α

ɺɶ

ɶ

k

Substituting the adaptive laws in equation (18.a) and considering the fact that the projection function has the following properties

Proj(y, ) y , Proj(y, )θ ≤ θɶ θ ≥ θɶy (18.b)

On can show that, the Lyapunov function derivative satisfies the following inequalities,

2

1 r 2 qr 1 6

7 2

1 2

1

k 2min(k ,k )V

k

α

α γ

ɶ

(18.c)

considering the fact that the following inequality is true

2 2

The parameter γ depends on the errors αɶi and θɶj defined before Integrating (18.c), yields;

t 2 2min(k ,k )t 2min(k ,k )(t )

0

k

As consequence, when the time t goes to infinity, the following inequality is true:

2 2

r

1 2

1

k min(k ,k )γ

ω ≤ɶ (18.f) The variable ωɶr can therefore be made arbitrary small independently to the presence of the disturbance γ 2

increasing the gain k The system is said to be ultimately bounded It is therefore stable

B Output Voltage Regulator The output voltage regulator design follows the same scheme presented previously for the rotor speed regulator The design uses equations (8.a), (9.b) and equation (19) given below

ref

0(V V )d

Ω =∫ − τ (19)

ref

s

V is the terminal voltage reference An integrator is

added to the system to be controlled to guarantee a zero

steady state error despite the presence of perturbations that

haven’t been modeled The augmented system state space

representation has the following form, therefore:

ref

d

dtΩ = θ + θ − (20.a)

dr 10 dr 11 qr 12 dr

d

dt = −θ + θ + θ (20.b)

where the parameters appearing in the model have the

following expressions:

8 s mL

θ = ω , θ = ω9 s s dsL i , θ10= ωsl, 11 r

r

R L

θ =

σ ;

12

r

1

L

σ

Since the grid parameters are assumed unknown, parameters

θ9, and θ10 are unknown The design scheme gives the

structure illustrated at Figure 2 inside the box labeled

Voltage controller That regulator consists of two

sub-controller named Ωv-controller and the idr-controller The

first controller is used to synthesize the second

sub-controller reference The errors signals involve in the

diagram given at Figure 2 are defined as follows:

ref

Ω = Ω − Ω = Ωɶ (21.a)

*

dr

dr

i~ = − (21.b)

The d-axis rotor current reference i*dr is obtained using

equation (20.a) It is such that it drives the error Ωɶv to zero

It has the following expression:

4

4

= − Ω +ɶ − θ − Ωɶ (22)

Parameters k3 and k are positive gains selected by the

designer to stabilize the system and α = θ8 1 8 The control

input expression is selected in such a way that it steers the

error signal iɶdr to zero It has the following expression:

2

k

4

= α − ɶ (23.a)

10

i 9

i 8

ˆ

4

=

=

= −ϕ − ϕ θ − ϕ θ − ϕ θ∑ − − θ Ω

ɶ

ɶ (23.b)

The parameter α12= θ1 12 The nonlinear functions

appearing in equation (23.b) have the following expressions

2

ϕ = + α − Ω +ɶ ,ϕ = ϕ8 idr;ϕ = ϕ9

ref

0 ( 8N kv3 V)ˆ9 Vs

2

ϕ = α − Ω θ − ϕɶ ɺ , ϕ = −10 idr; ϕ = −11 iqr

The same stability analysis is performed to obtain the reactive power regulator adaptation laws The following candidate Lyapunov function is used for this purpose

2 10

v dr

i 9 i

θ

= Ω +ɶ + ∑ βɶ (25) Parameters β are positive gains selected by the designer i

The adaptation laws have the following expressions:

θ = βɺ ϕ ɶ θ , i=10,

θ = βɺ ϕ ɶ + Ωɶ θ  Next figure illustrates the system in closed loop and the module involved in the implementation A tachometer is needed to obtain the rotor angular position θr from its speed A phase lock loop (PLL) module is required to obtained the stator flux angular position θs. These angular position are used by the abc/dq module that gives the rotor d-axis and q-d-axis currents The rotor currents along with the active power and reactive power are variables needed to implement the control and adaptation laws

PLL

u

∫

abc/ dq

u

∫

V- controllerω

AC/DC & DC/AC converters Tachometer

r

Ω r ω

r θ

s

θ

2 π

1

θ

2

θ

s,abc

V

s

ω

t r

V ,ω

dr qr

i ,i r,abc

i

+ +

−

−

Fig 3 The DFIG control system The proposed controller is tested in the next section Preliminary results are also shown Simulation results for two common power system contingencies are presented

IV SIMULATION AND RESULTS The power system illustrated at the following figure is used to assess the proposed controller performances The system consists of a 13 Kw wind power generator supplying

a 340 V, 60Hz power grid The controller objective is to maintain the generator terminal voltage and rotor speed at

their reference values (340 V, 96.34 rad/s) The machine

parameters are: Rs=0.05Ω, Rf =0.38Ω, Lm =47.3mH,

s

L =50mH, Lr =50mH

Controller gains are set to 10 except the gain k which is

equal to 1 The controller capability to stabilize the DFIG

signals, when the demanded active power changes or when a

short circuit occurs at the generator terminal, is tested At the

time instant t equal 0.5 second, the generator local load

decreases to about 30% Before that change, the wind power

generator was generating 6150 W Fig.5 shows the generator

active power waveform One can note that the generator

decreases its active power to about 10% to be able to

maintain the same rotor speed

Fig 4 The single DFIG – Infinite bus system

5000

5200

5400

5600

5800

6000

6200

tim e s

Fig 5 The output active power during the first contingency

0.4 0.5 0.6 0.7 0.8 0.9 1 96.25

96.3

96.35

96.4

tim e s

Fig 6 The rotor speed during the first contingency

0.95 1 1.05 1.1

0

100

200

300

400

tim e s

Fig 7 The terminal voltage during the second contingency

1.3 1.32 1.34 1.36 1.38 1.4 1.42x 10

6

tim e s

Fig 8 The output reactive power during the second contingency The rotor speed waveform is illustrated at Fig 6 It takes about 0.3 second to the controller to stabilize the DFIG signals to their steady values The wind power generator is therefore capable to support the power grid frequency when the demanded power changes The second contingency occurs at the time instant equal 1 second It is a 50ms grounded short-circuit at the generator terminals (the stator side) Fig 7 shows the stator voltage waveform and Fig 8 illustrates the generator reactive power profile One can see that these signals return to their pre-fault values quickly after the fault is cleared This fast response should increase the DFIG fault ride-through capability

V CONCLUSION

A nonlinear adaptive controller is proposed to increase a wind power generator grid-fault ride-through capability A control system that considerably improves the generator time response is designed The simulation results show that the generator is capable to readjust quickly the generated active power and reactive power to maintain its terminal voltage and the rotor speed constant when a change occur inside the power grid The controller adaptive feature help to reduce the system sensitivity to the power network condition variations Future works will evaluate the proposed controller in a large scale power system environment

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2008

[3] X Lie, P Cartwright, “Direct active and reactive power control of DFIG for wind energy generation,” IEEE Trans Energy Conversion, vol 21, pp 750–758, Sept 2006

[4] S Muller, M Deicke, and R W De Doncker, “Doubly fed induction generator systems for wind turbines,” IEEE Indus Applications Magazine, vol 8, pp 26–33, June 2002

[5] F M Hughes, O Anaya-Lara, N Jenkins, and G Strbac, “Control of DFIG-based wind generation for power network support,” IEEE Trans Power Systems, vol 4, pp 1958–1966, Nov 2005

[6] P C Krause, O Wasynczuk and S D Sudhoff, Analysis of Electric Machinery and Drive Systems, second edition, USA, Wiley-Interscience, 2002, ch 4