Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter

7/31/2019 Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter

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Hi ngh ton quc viu khin v Tng ho - VCCA-2011

VCCA-2011

Performance of robust controller for DFIM when the rotor angular speed istreated as a time-varying parameter

Nguyen Tien Hung1, Ngo Duc Minh

2

1Thainguyen University of Technology, email: [email protected]

2Thainguyen University of Technology, email: [email protected]

Abstract: This paper describes the design of arobust current controller for doubly-fed induction

machines (DFIM), in which the rotor angular speed

is considered as an uncertain parameter. The

robust controller is then synthesized to guarantee that

the -norm of the closed-loop system is smaller

than some given number for different frozen values of

. Next, the robust performance of the robust

controller with respect to other rotor angular speeds is

investigated for both constant and fast parametervariations. Some simulation results are given to

demonstrate the performance and robustness of thecontrol algorithm.

1. IntroductionIn the literature, the control structure of DFIM

including PI current controllers is described in [1],[2], [3], [4]. In some cases, the cross coupling term in

the rotor equations that includes the mechanical

angular speed is eliminated by adding a feed-forward

term to the output of the q-axis controller [2], [5]. The

rotor mechanical angular speed is treated as an

scheduling parameter that is used for thesecompensators. In these situations the difficulties of

the nonlinear dynamics of the doubly-fed inductionmachine are not taken into account, i.e., the model of

the machine is linearized and it is assumed that the

machine parameters required by the control algorithm

are precisely known. Clearly, such controller designsmight result in a closed-loop behavior that is highly

sensitive to a change in operating conditions and/or

parameters.

In this work, a mixed loop shaping -design for the

rotor current control loop at fixed frozen values of the

rotor angular speed is presented first. Then the

performance of the closed-loop system with

controller designed for different frozen values of

for other rotor angular speeds is investigated. The

performance analysis is also extended for the case

with the face of the stator voltage action. As a further

investigation, the designed controller for a frozen

values of is tested for a fast variation of the rotor

speed along the whole parameter interval.

2. Preliminaries2.1 NotationsLet denote the space of square-integrable signals

defined on the interval . A matrix is calledsymmetric if it is real and satisfies . The set

of all symmetric matrices will be denoted by

. A transfer function with a state-space

realization will be denoted by

2.2 Linear matrix inequalitiesA linear matrix inequality (LMI) has the form

(1)

where denotes the vector of decision

variables and .

2.3 The -norm

Consider a linear input-output system that is

described by

(2)

and whose transfer matrix is given by

If is stable and if we choose the initial condition

to be zero, defines a linear map on

with a finite energy gain defined as

It is well-known that the energy-gain of coincides

with the -norm of the corresponding transfer

matrix given by

where stands for the largest singular value of

the complex matrix matrix .

2.4 The bounded real lemmaIt is not possible to explicitly compute in terms

of the realization matrices. Instead, one can

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characterize stability of and the validity of the

inequality

(3)

as an LMI in some auxiliary matrix and , which is

one version of the celebrated bounded real lemma.Indeed, it can be shown [6] that is stable and that

(3) holds if and only if

and there exits some such that the

Riccati inequality

(4)

is satisfied. By the Schur lemma (4), these conditionsare equivalent to following system of LMIs [7]

(5)

This result is referred to as the bounded real lemma.

Yet another application of the Schur lemma allows to

rewrite these inequalities with into the

following form [8], [9]:

(6)

Note that (6) are LMI constraints on and . This

allows to determine the infimal for which (3) is

true, and hence in turn the value , by

minimizing over the constraint (6) which is a

standard LMI problem. Let us now show how this

procedure of analysis can be successfully generalized

to synthesizing controllers.

2.5 performance

A standard setup for control is presented inFigure 1, where represents the generalized

disturbances, the controlled variable, the control

input and the measurement output, while is a

linear time-invariant system described as

w z

u yP

K

Figure 1.The interconnection of the system

The goal in control is to find a stabilizing linear

time-invariant (LTI) controller that minimizes the

norm of the closed-loop system

(7)

where is lower linear fractional

transformation of and , which is nothing but the

closed-loop transfer function in Figure 1.

2.6 Sub-optimal control

Let us now consider a generalized plant whereweights are incorporated already as follows

(8)

If the linear time-invariant controller is expressed

as

(9)

the closed-loop system admits the followingstate-space description:

(10)

where

(11)

In practice, the control problem is rather

concerned with finding an LTI controller whichrenders stable and such that

(12)

holds true [11], where is a given number that

specifies the performance level. This is the so-called

sub-optimal problem.

2.7 controller synthesis

Using the bounded real lemma for (12), the matrix

is stable and (12) is satisfied if and only if the LMI

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(13)

holds for some . Unfortunately this inequality

is not affine in and in the controller parameterswhich are appearing in the description of , , , .

However, a by now standard procedure [8], [9], [12],

[13] allows to eliminate the controller parameters

from these conditions, which in turn leads to convex

constraints in the matrices and that appear in the

partitioning of

(14)

According to that of in (11) one then arrives at the

following synthesis LMIs for -design [14]:

(15)

(16)

(17)

where and are basis matrices for the

subspaces

(18)

respectively.

Note that these inequalities are defined by open-loopsystem parameters only, and that they depend affinely

on the design variables and . Hence we can

directly minimize over these LMIs in order to

compute the best possible level with (12) that can

be achieved by a stabilizing controller.

After having obtained and that satisfy (15)-(17)

for some level , the controller parameters can be

reconstructed by using the projection lemma [8]. Thisprocedure for -synthesis is implemented in the

robust control toolbox [15].

2.8 Mixed sensitivity approach

Figure 2a shows a simple feedback control system.

This interconnection can be recast into a standard

setup for control as depicted in Figure 2b. For

this control configuration, engineers are usuallyinterested in some specific transfer functions. In

particular, is the sensitivity function

which describes the influence of the external

disturbance to the tracking error .

is the complementary

sensitivity function which describes the influence of

the reference signal to the system output . Finally,

is the transfer function from to the control

input that indicates control activity [16].

++

w y u zK G

(a)

++

w z

G

uP

y

K

(b)

Figure 2.General feedback control configuration

In general, performance of the closed-loop system

that is specified by norm of the channel in

(7) can be formulated as a multi-objective problem

(see Figure 3). This leads to the minimization of

(19)

The multi-variable loop shaping with various

specifications (19) is the so-called the mixed

sensitivity design approach.

++

w yK

uG

1z

2z

3z

z

Figure 3.Mixed sensitivity control

It is well-known in the literature that the transfer

functions , , and need to be small in

magnitude in order to achieve good commandtracking and robust stability. However, the well-

known constraint reveals that these

requirements can not be achieved simultaneously over

the whole frequency range. However, the use of

frequency filters or weighting functions opens up the

possibility to minimize the magnitudes of , , and

over different frequency ranges [17]. Hence, in

practice, instead of minimizing (19) one rather

determines a stabilizing LTI controller that

minimizes the cost function

(20)

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where , , are suitably chosen weighting

functions (Figure 4).

++

w y uK G

SW

PW

TW

z

Figure 4.Weighting functions.

3. The system representationIn this paper, the mechanical angular speed is

considered as a time-varying parameter. This

particular choice is motivated by the fact that ,

which causes the system to be nonlinear, can bemeasured online. The value of the rotor angular speed

varies by around the synchronous speed

, i.e.

(21)

where is the nominal speed, is the

variation of the rotor angular speed around its

nominal value, and is the normalizing factor that

maps the uncertain element into a normalized

uncertain element such that .

In the normal operation of the DFIM, the nominalspeed is close to the synchronous speed .

Hence, if we denote the ratio of the nominal speed

and the synchronous speed by , i.e. we

can write

and

(22)

Here, is a scaling factor that allows to present the

variation of the rotor speed around the

synchronous speed . From that point, the deviation

of the rotor speed by from the nominal speed

can be expressed as

(23)

The representation of in (23) provides a flexible

choice of the rotor speed range in the controller

design for the DFIM. As a result, the system matrices

presented in [18] can now be rewritten as

follows:

where

and

(24)

in which

(25)

(26)

The DFIM model [18] reads as

(27)

where

(28)

(29)

In (28) and (29), and represent the input and

output signals of the disturbance channel

corresponding to the time-varying parameter .

Equations (27), (28), and (29) in combination with the

output equation in [18] can now be expressed as

(30)

(31)

where is an unity matrix, is an zero

matrix, . is also called the

perturbation block.

Since the two last rows of the matrices and

are zero, let , , and

we can write

(32)

where , and are two vectors, is a

matrix.

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Equations (30) and (31) can be easily simplified as

(33)

where

Let be the transfer function with the state-space

realization (33), i.e.

(34)

The system can then be generally described by

(35)

where is the transfer function mapping ,

and is the transfer function

from to .

The linear fractional transformation (LFT)

representation of the system is depicted as shown in

Figure 5.

w

z

sv

rv ryrc

G

Figure 5.LFT representation of the system

4. control of the systemIn this section we start with -synthesis for

mentioned frozen values of the rotor speed . Then

the performance of the LTI controller designed for a

fixed value of is evaluated with other constant

values of as well as with a fast variation of the

rotor speed along the parameter range.

4.1 The control configurationWith the LFT representation of the plant as shown in

Figure 5 we can now derive a standard control

structure for the synthesis of an -controller as

depicted in Figure 6. Here, is the LTI part of the

plant as given in (34), is the uncertainty block as

given in (32), is the controller that is to bedesigned.

+

w

z

sv

rvre

ref

rirc

K

rcG

ry

Figure 6.Structure of the closed-loop system in

design

In this configuration, is the

reference input, is the controller

output, is the controlled output, and

is

the controller input which is equal to the tracking

error. In this case, the transfer function from the

reference input to the tracking error will be

. The transfer function from

reference inputs to controlled outputs is denoted by

, i.e.

(36)

4.2 loop shaping design

The interconnection of the system used for the

controller synthesis is shown in Figure 7. The external

control input consists of stator voltages and

reference rotor currents

. The controller

output is . The controller input or

tracking error is

. The

controlled variable is . Note

that the components of the external

control inputs are considered as disturbances and their

influences on the controlled outputs must be reduced

as much as possible.

The weighting function is

used to shape the function which is corresponding

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to the transfer function from the reference input

to the tracking error . is kept large over the low

frequency range for tracking. The weighting function

is used to shape the transfer

function from the external control input to the

controlled output . The selection of the weightingfunction is not only intended to keep the closed

loop bandwidth at a desired value, but also to reject

the effects of the components and on the

controlled outputs as discussed above. Note that alarge bandwidth corresponds to a faster rise time but

the system is more sensitive to noise and to parameter

variations [16].

+

+

ref

rdi

ref

rqi

rcw

sdv

sqv

rnG

rdv

rqv rcK

rdi

rqi

rcde

rcqe

rtdW rtd

z

rtqW

rtqz

rsdWrsd

z

rsqW

rsqz

rcz

Figure 7.The interconnection of the system

The standard control problem is to find a

stabilizing LTI controller at fixed frozen values

of such that the -norm of the channel

is smaller than a given number .

at fixed frozen values of .

The set of 620kW DFIM parameters is applied for the

controller synthesis. During the controller design

stage, a trial-and-error-repetition technique is used in

order to achieve the desired performance

specifications by adjusting the weighting functions.

The design steps are repeated until we are able to

meet the required performance specifications. Finally,

the following weighting functions were obtained:

(37)

(38)

For the chosen frozen value of (at under-

synchronous speed), the controlled system with

current controller with the above given weighting

functions achieves a norm of 0.36.

4.3 Simulation results with the current

controllerFigure 8 shows the frequency responses of the

controlled system with the current controller and

the inverse of the weighting functions , and

. Figure 8a,b show the relevant magnitude plots

of the complementary sensitivity and sensitivityfunctions of the closed-loop system with the

performance requirements specified by and .

The blue-thick curve shows the response of the output

with respect to the reference inputs . This

curve corresponds to the transfer function (see

equation (36)). Similarly, the red-thick curve shows

the response of the output with respect to the

reference inputs and it corresponds to the transfer

function . Meanwhile the black-solid curve

shows the influence of the reference input on the

output corresponding to the transfer functions

, and the green-solid curve shows the influence

of the reference input on the output

corresponding to the transfer functions . The

inverse of the weighting functions , and(see Figure 7) are depicted by dotted lines A, and B in

Figure 8a, while the inverse of the weighting

functions , and are depicted by dotted lines

C, and D in Figure 8b, respectively. The influences of

the stator voltage on the controlled outputs and

controller inputs are show in Figure 8c,d with the

same color and line styles.

100

101

102

103

104

105

106

-120

-100

-80

-60

-40

-20

0

20

40

Ma

gnitude(dB)

Closed-loop performance of reference inputs to outputs

Frequency (rad/sec)

A

B

(a)

100

101

102

103

104

105

106

-150

-100

-50

0

50

100

Magnitude(dB)

Closed-loop performance of reference inputs to control errors

Frequency (rad/sec)

C

D

(b)

100

101

102

103

104

105

106

-140

-120

-100

-80

-60

-40

-20

0

Magnitude(dB)

The effects of stator voltages to outputs

Frequency (rad/sec)

(c)

100

101

102

103

104

105

106

-140

-120

-100

-80

-60

-40

-20

0

Magnitude(dB)

The effects of stator voltages to control errors

Frequency (rad/sec)

(d)

Figure 8.Performance of the controlled system with

current controller in the frequency domain for

.

It is clear in Figure 8 that the sensitivity andcomplementary sensitivity functions are below the

inverse of the performance weighting functions. The

bandwidths corresponding to the channels

and are about rad/s. The gains of

the frequency responses of the stator voltages to

controlled outputs and controller inputs are all smallerthan -10db. This indicates that the controlled system

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has good disturbance rejection with respect to the

stator voltage . The overshoots of the channels

and are about 15%. Moreover,

the gains corresponding to the frequency responses of

the channels and are smaller

than -22db. This means that the cross-couplinginteraction between and remains quite small, or

in other words, the rotor current components can beconsidered to be no influence on one another. As a

result, the characteristics of electrical torque and

power factor responses are not deteriorated.

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

1

1.2Closed-loop performance of reference inputs to outputs

time (s)

irdref i

rd

irdref i

rq

irqref i

rq

irqref i

rd

(a)

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

1


time (s)

irdref

ercd

irdref

ercq

irqref e

rcq

irqref e

rcd

(b)

0 0.002 0.004 0.006 0.008 0.01

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05The effects of stator voltages to outputs

time (s)

vsdref i

rd

vsdref irq

vsqref i

rq

vsqref i

rd

(c)

0 0.002 0.004 0.006 0.008 0.01-0.05

0

0.05

0.1

0.15

0.2

0.25


time (s)

(d)


current controller in the time domain for

.

Figure 9 shows the time responses of the controlled

system for a step input. The solid line in Figure 9a

shows the response of the output with respect to

the reference input and it corresponds to the

transfer function in equation (36). The dashed

line shows the response of the output with respect

to the reference input and it corresponds to the

transfer function in equation (36). The dotted

curve shows the influence of the reference input

on the output corresponding to the transferfunction , and the dash-dotted curve shows the

influence of the reference input on the output

corresponding to the transfer function . The solid

line in Figure 9b shows the response of the control

error with respect to the reference input . The

dashed line shows the response of the control error

with respect to the reference input . The

dotted curve shows the influence of the reference

input on the control error , and the dash-

dotted curve shows the influence of the reference

input on the control error . The influences of

the stator voltages on the controlled outputs and

control errors are also show in Figure 9c,d with the

same line styles.

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)


ird

ref ird

(m

= 0.63 s)

ird

ref irq

(m

= 0.63 s)

irq

ref irq

(m

= 0.63 s)

irqref

ird (m = 0.63 s)

ird

ref ird

(m

= 0.9 s)

ird

ref irq

(m

= 0.9 s)

irq

ref irq

(m

= 0.9 s)

irq

ref ird

(m

= 0.9 s)

(a)

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

time (s)


(b)

0 0.002 0.004 0.006 0.008 0.01

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

time (s)


(c)

0 0.002 0.004 0.006 0.008 0.010.05

0

0.05

0.1

0.15

0.2

0.25

time (s)


vsd

ref ercd

(m

= 0.63 s)

vsd

ref e

rcq(

m= 0.63

s)

vsq

ref e

rcq(

m= 0.63

s)

vsq

ref ercd

(m

= 0.63 s)

vsd

ref e

rcd(

m= 0.9

s)

vsd

ref e

rcq(

m= 0.9

s)

vsq

ref e

rcq(

m= 0.9

s)

vsq

ref ercd

(m

= 0.9 s)

(d)


current controller for frozen value

for .

Note that the current controller is designed with a

fixed frozen value of the rotor angular speed .

Hence, the obtained performance is not guaranteed for

the whole region of variation of . However, we can

further investigate the performance of the closed-loop

system with the controller designed for the frozenvalue for other angular rotor speeds. In

the following investigation, we consider the

performance of the controlled system with the rotor

speed variations by from the rotor nominal

speed ( ), i.e.

. In order to do so we

synthesized two controllers for the frozen

parameter values

using the same weighting functions as in (37) and(38). Then we plot the time responses of the closed-

loop system with the controller designed for the

frozen value applying for the case where

and , respectively. The

time responses of the closed-loop system with the

local controllers designed for the frozen values

and are also plotted on

each figure for the purpose of comparison of the

achieved performance among these controllers.

Figure 10 shows the performance of the closed-loop

system at with two controllers

designed for the frozen value and

, respectively. The thick-solid lines are

related to the designed controller for the frozen

value . The thin-lines are related to the

local controller designed for . Ascan be seen from Figure 10a, the time responses of the

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outputs and with respect to the step change of

the reference inputs and , respectively, of the

closed-loop system with the controller designed

for the frozen value are maintained for

the frozen values if compared with that

in Figure 9. In addition, these curves are almost the

same with that of the local controller designed for

. The same conclusion can also be

drawn for the curves related to the time responses of

the control errors , with respect to the step

change of the reference inputs , (Figure 10b),

and the influences of the stator voltages , on

the controlled outputs , (Figure 10c) and control

errors , (Figure 10d), respectively. However,

the remarkable difference in the performance among

these controllers is indicated by their cross-

coupling interactions. The effects of the stator

voltages and to the outputs , (Figure10c) and to the control errors , (Figure 10d),

respectively, in the case of the controller

designed for the frozen value are larger

than that of the local controller designed for

.

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)


ird

ref ird

(m

= 1.17 s)

ird

ref irq

(m

= 1.17 s)

irq

ref irq

(m

= 1.17 s)

irq

ref ird

(m

= 1.17 s)

ird

ref ird

(m

= 0.9 s)

ird

ref irq

(m

= 0.9 s)

irqref i

rq(

m= 0.9

s)

irq

ref ird

(m

= 0.9 s)

(a)

0 1 2 3 4 5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

time (s)


ird

ref e

rcd(

m= 1.17

s)

ird

ref ercq

(m

= 1.17 s)

irq

ref e

rcq(

m= 1.17

s)

irq

ref e

rcd(

m= 1.17

s)

ird

ref ercd

(m

= 0.9 s)

ird

ref ercq

(m

= 0.9 s)

irq

ref e

rcq(

m= 0.9

s)

irq

ref e

rcd(

m= 0.9

s)

(b)

0 0.002 0.004 0.006 0.008 0.01

0.25

-0.2

0.15

-0.1

0.05

0

0.05

time (s)


vsdref i

rd(

m= 1.17

s)

vsd

ref irq

(m

= 1.17 s)

vsq

ref irq

(m

= 1.17 s)

vsq

ref ird

(m

= 1.17 s)

vsd

ref ird

(m

= 0.9 s)

vsdref i

rq(

m= 0.9

s)

vsq

ref irq

(m

= 0.9 s)

vsqref i

rd(

m= 0.9

s)

(c)

0 0.002 0.004 0.006 0.008 0.01

-0.05

0

0.05

0.1

0.15

0.2

0.25

time (s)


vsd

ref e

rcd(

m= 1.17

s)

vsd

ref ercq

(m

= 1.17 s)

vsq

ref ercq

(m

= 1.17 s)

vsq

ref e

rcd(

m= 1.17

s)

vsd

ref e

rcd(

m= 0.9

s)

vsd

ref ercq

(m

= 0.9 s)

vsq

ref ercq

(m

= 0.9 s)

vsq

ref e

rcd(

m= 0.9

s)

(d)

Hnh 11. Performance of the controlled system with

current controller for frozen value

for

The performance of the closed-loop system at

with two controllers designed for

the frozen value and ,

respectively, is shown in Figure 11. The thick-solid

lines are related to the designed controller for the

frozen value . The thin-lines are related to

the local controller designed for .

Similarly to the previous simulation, the time

responses of the closed-loop system with the

controller designed for the frozen value

and the local controller designed for

corresponding to the step change of the

reference inputs , , and to the effect of stator

voltages , are almost the same, except their

cross-coupling interactions. In the case of the

controller designed for the frozen value ,

the influences of the stator voltages and to the

outputs , (Figure 11c) and to the control errors

, (Figure 11d) are larger than that of the local

controller designed for .

Obviously, the performance of the the controller

designed for the frozen value for other

rotor angular speeds are not maintained because of the

cross-coupling interactions between the stator

voltages , and the outputs , as well as the

stator voltages , and the control errors ,

. This may cause a large tracking error for the

controlled system since the stator voltages andare the input disturbances.

0 1 2 3 4 5

x 10-3

690

700

710

720

730

740

750

760d component of the rotor currents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(a)

0 1 2 3 4 5

x 10-3

0

50

100

150

200

250

300

350

400q component of the rotor currents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(b)

0 1 2 3 4 5

x 10-3

690

700

710

720

730

740

750


time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(c)

0 1 2 3 4 5

x 10-3

0

50

100

150

200

250

300

350

q component of the rotor currents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(d)

0 1 2 3 4 5

x 10-3

690

700

710

720

730

740

750

760


time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(e)

0 1 2 3 4 5

x 10-3

0

50

100

150

200

250

300

350q component of the rotor currents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(f)


current controllers designed for (a,

b), for (c, d), and for (e, f)

at different constant values of .

In order to evaluate the performance of the closed-

loop system with controller designed for different

frozen values of for other rotor angular speeds in

the face of the stator voltage action, we performed the

simulations with the set value of and the

set value of as shown in Figure 12. The

time responses of the (Figure 12a) and (Figure

12b) components of the rotor currents achieved by the

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designed controller for the frozen value

are plotted by the solid curves. While

the dashed and and the dash-dotted curves show the

performance of this controller for the value

, and , respectively. These

figures reveal that the tracking errors of the and

components of the rotor currents achieved by thiscontroller are increased for and

. This is because of the cross-coupling

interactions between the stator voltages , and

the outputs , as well as the stator voltages ,

and the control errors , become larger for

bigger rotor angular speeds as presented in the

previous simulation. Figures. 12c and 12d show the

time responses of the and components of the rotor

currents achieved by the designed controller for

the frozen value (solid curves) for the

value (dashed curves), and

(dash-dotted curves), respectively.Figures. 12e and 12f show the time responses of the

and components of the rotor currents achieved by

the designed controller for the frozen value

(solid curves) for the value

(dashed curves), and

(dash-dotted curves), respectively. These figures

reveal that performance of the controller

designed for a frozen value of is not guaranteed

for these other values of .

0 0.002 0.004 0.006 0.008 0.010

100

200

300

400

500

600

700


time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(a)

0 0.002 0.004 0.006 0.008 0.010

100

200

300

400

500

600

700

800d component of the rotor c urrents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(b)

0 0.002 0.004 0.006 0.008 0.010

50

100

150

200

250

300

350


time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(c)

0 0.002 0.004 0.006 0.008 0.010

50

100

150

200

250

300

350

400d component of the rotor c urrents

time (s)

Ampere

m

= 0.63s

m

= 0.9s

m

= 1.17s

(d)

0 0.002 0.004 0.006 0.008 0.01

200

250

300

350

m

time (s)

rad/s

(e)

0 0.002 0.004 0.006 0.008 0.01

200

250

300

350

m

time (s)

rad/s

(f)


three current controllers designed for

, , and ,respectively, with fast variations of the rotor speed.

For further investigation, a simulation with the

controller designed for a frozen values of for a

fast variation of the rotor speed along the whole

parameter interval is carried out. We consider three

local controllers designed for the frozen values

, , and as

above. The parameter trajectory is given by the stepresponse of the rotor speed. Figures. 13a,c,e show the

behaviors of the and components of the rotor

currents when the rotor angular speed increases from70% to 130% of the nominal speed of the rotor

, where (rad/s),

. Conversely, the behaviors of the and

components of the rotor currents when the rotorangular speed decreases from 130% down to 70% of

the nominal speed of the rotor are shown in Figures.

13b,d,f. These figures reveal that the controllers

do not guarantee tracking during the fast parameter

transition. The control error increases along the

parameter trajectory and reaches the largest value at

the end of it.

5. ConclusionsThis paper briefly recapitulated the theory of the

mixed loop shaping -design for the rotor current

controller for DFIMs at some fixed frozen values of

the rotor angular speed. The performance of these

current controllers has been investigated for different

values of the mechanical angular speed varied by

% from the rotor nominal speed. The simulation

results showed that the performance of the

controller designed for a frozen value of was notcompletely guaranteed for other rotor angular speeds.

An important point that is needed to be emphasized in

this particular case is that the performance of the

controller is considerably changed for fast parameter

variations. In order to get better performance level for

the controlled system, the designed controller has toadapt to changing of the rotor angular speed. In that

sense, the rotor angular speed can be adopted as a

gain-scheduling parameter.

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[4] Andreas Petersson. Analysis, Modeling andControl of Doubly-Fed Induction Generators for

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[9] P. Gahinet and P. Apkarian. A linear matrixinequality approach to control. Int. J.Robust and Nonlinear Contr., 4:421448, 1994.

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varying and linear parametrically-varying

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Dr. Ngo Duc Minh wasborn in Lang son, Vietnam,

in 1960. He received the

B.S. degree from

Thainguyen University of

Technology in 1982 in

Electrical Engineering,M.S. degree from Hanoi

University of Technology

in 1997 in ElectricalEngineering and in

Industrial Information Technology, and Ph.D degree

from Hanoi University of Technology in 2010 in

Atutomation Technology. He is currently a vice-chairof the Education department of Thainguyen

University of Technology. Dr. Minhs interests are in

the areas of high voltage technology, hydrolic power

plant, power supply, control of electric power

systems, FACTS, BESS, AF, PSS equipments, new

and renewable energy technologies, distributionpower systems.

Nguyen Tien Hung wasborn in Thainguyen,

Vietnam. He received the

B.S. degree fromThainguyen University of

Technology in 1991 and

M.S. degree from Hanoi

University of Technology in1997, both in Electrical

Engineering. He is currently

a Ph.D candidate at Delft Center for Systems andControl (DCSC), Delft University of Technology, the

Netherlands. His main research interests include

topics in robust control, linear parameter varying

control of nonlinear systems, gain-scheduling design,

and their applications in electrical systems.

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Documents

Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter