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

Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter 1 Thainguyen University of Technology, email: h.nguyentien@tnut.edu.vn 2

Performance of robust controller for DFIM when the rotor angular speed is

treated as a time-varying parameter

1

Thainguyen University of Technology, email: h.nguyentien@tnut.edu.vn

2

Thainguyen University of Technology, email: ngoducminh@tnut.edu.vn

robust 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

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 parameter

variations Some simulation results are given to

demonstrate the performance and robustness of the

control algorithm

1 Introduction

In 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

scheduling parameter that is used for these

compensators In these situations the difficulties of

the nonlinear dynamics of the doubly-fed induction

machine 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 designs

might result in a closed-loop behavior that is highly

sensitive to a change in operating conditions and/or

parameters

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

speed along the whole parameter interval

2 Preliminaries

2.1 Notations

2.2 Linear matrix inequalities

A linear matrix inequality (LMI) has the form

2.3 The -norm

described by

and whose transfer matrix is given by

If is stable and if we choose the initial condition

with a finite energy gain defined as

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

matrix given by

2.4 The bounded real lemma

of the realization matrices Instead, one can

characterize stability of and the validity of the

inequality

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

Riccati inequality

(4)

is satisfied By the Schur lemma (4), these conditions

are 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

following form [8], [9]:

(6)

allows to determine the infimal for which (3) is

standard LMI problem Let us now show how this

procedure of analysis can be successfully generalized

to synthesizing controllers

2.5 performance

disturbances, the controlled variable, the control

linear time-invariant system described as

K

Figure 1 The interconnection of the system

norm of the closed-loop system

2.6 Sub-optimal control

weights are incorporated already as follows

as

state-space description:

where

(11)

concerned with finding an LTI controller which renders stable and such that

specifies the performance level This is the so-called

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

(13)

which 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

partitioning of

(14)

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

(15)

(16)

(17)

subspaces

(18) respectively

Note that these inequalities are defined by open-loop

system parameters only, and that they depend affinely

compute the best possible level with (12) that can

be achieved by a stabilizing controller

for some level , the controller parameters can be

reconstructed by using the projection lemma [8] This

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

this control configuration, engineers are usually

interested in some specific transfer functions In

which describes the influence of the external

sensitivity function which describes the influence of

the reference signal to the system output Finally,

input that indicates control activity [16]

(a)

+

¡¡

G

K

(b)

Figure 2 General feedback control configuration

In general, performance of the closed-loop system

(7) can be formulated as a multi-objective problem (see Figure 3) This leads to the minimization of

The multi-variable loop shaping with various specifications (19) is the so-called the mixed

1

z

2

z

3

z z

Figure 3 Mixed sensitivity control

It is well-known in the literature that the transfer

magnitude in order to achieve good command tracking and robust stability However, the

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

minimizes the cost function

where , , are suitably chosen weighting

functions (Figure 4)

S

W

P

W

T

W

z

Figure 4 Weighting functions

3 The system representation

considered as a time-varying parameter This

which causes the system to be nonlinear, can be

measured online The value of the rotor angular speed

, i.e

maps the uncertain element into a normalized

In the normal operation of the DFIM, the nominal

Hence, if we denote the ratio of the nominal speed

can write

and

Here, is a scaling factor that allows to present the

can be expressed as

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

in which

The DFIM model [18] reads as

(27)

where

(29)

Equations (27), (28), and (29) in combination with the output equation in [18] can now be expressed as

(30)

perturbation block

we can write

matrix

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

where

realization (33), i.e

The system can then be generally described by

(35)

representation of the system is depicted as shown in

Figure 5





s v

r

Figure 5 LFT representation of the system

the performance of the LTI controller designed for a

rotor speed along the parameter range

4.1 The control configuration

With the LFT representation of the plant as shown in Figure 5 we can now derive a standard control

designed

+ ¡





s

v

r

v

r

e

ref r

i

rc

K

rc

G

r

y

Figure 6 Structure of the closed-loop system in

design

is the controller input which is equal to the tracking error In this case, the transfer function from the

The transfer function from reference inputs to controlled outputs is denoted by , i.e

4.2 loop shaping design

The interconnection of the system used for the controller synthesis is shown in Figure 7 The external

The controller

control inputs are considered as disturbances and their influences on the controlled outputs must be reduced

as much as possible

to the transfer function from the reference input

frequency range for tracking The weighting function

is used to shape the transfer

loop bandwidth at a desired value, but also to reject

controlled outputs as discussed above Note that a

large bandwidth corresponds to a faster rise time but

the system is more sensitive to noise and to parameter

variations [16]

+

¡¡

+

¡¡

ref

rd

i

ref

rq

i

rc

w

sd

v

sq

v

rn

G

rd

v

rq

v K rc

rd

i

rq

i

rcd

e

rcq

e

rtd

W z rtd

rtq

W z rtq

rsd

W z rsd

rsq

W z rsq

rc

z

Figure 7 The interconnection of the system

is smaller than a given number

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

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:

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

controller

Figure 8 shows the frequency responses of the

Figure 8a,b show the relevant magnitude plots

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

The blue-thick curve shows the response of the output

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

, and the green-solid curve shows the influence

(see Figure 7) are depicted by dotted lines A, and B in Figure 8a, while the inverse of the weighting

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

controller inputs are show in Figure 8c,d with the same color and line styles

10 0

10 1

10 2

10 3

10 4

10 5

10 6

-120 -100 -80 -60 -40 -20 0 20 40

Closed-loop performance of reference inputs to outputs

Frequency (rad/sec)

A B

(a)

10 0

10 1

10 2

10 3

10 4

10 5

10 6

-150 -100 -50 0 50 100

Closed-loop performance of reference inputs to control errors

Frequency (rad/sec)

C D

(b)

10 0

10 1

10 2

10 3

10 4

10 5

10 6

-140 -120 -100 -80 -60 -40 -20 0

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

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 and complementary sensitivity functions are below the inverse of the performance weighting functions The bandwidths corresponding to the channels

the frequency responses of the stator voltages to controlled outputs and controller inputs are all smaller than -10db This indicates that the controlled system

has good disturbance rejection with respect to the

the gains corresponding to the frequency responses of

than -22db This means that the cross-coupling

in other words, the rotor current components can be

considered to be no influence on one another As a

result, the characteristics of electrical torque and

power factor responses are not deteriorated

x 10-3 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

Closed-loop performance of reference inputs to outputs

time (s)

irdref  ird

irdref  irq

irqref  irq

irqref  ird

(a)

x 10-3 -0.2

0 0.2 0.4 0.6 0.8 1 Closed-loop performance of reference inputs to control errors

time (s)

irdref  ercd

irdref  ercq

irqref  ercq

irqref  ercd

(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

The effects of stator voltages to outputs

time (s)

vsdref  ird v sd ref  i rq v sq ref  i rq

vsqref  ird

(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 The effects of stator voltages to control errors

time (s)

(d)

Figure 9 Performance of the controlled system with

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

curve shows the influence of the reference input

line in Figure 9b shows the response of the control

dashed line shows the response of the control error

dotted curve shows the influence of the reference

dash-dotted curve shows the influence of the reference

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

x 10 -3

-0.2 0 0.2 0.4 0.6 0.8 1

time (s)

Closed-loop performance of reference inputs to outputs

irdref  ird ( m = 0.63 s)

irdref  irq ( m = 0.63 s)

irqref  irq ( m = 0.63 s)

irqref  ird ( m = 0.63 s)

irdref  ird ( m = 0.9 s)

irdref  irq ( m = 0.9 s)

irqref  irq ( m = 0.9 s)

irqref  ird ( m = 0.9 s)

(a)

x 10-3 -0.2

0 0.2 0.4 0.6 0.8

time (s) Closed-loop performance of reference inputs to control errors

(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) The effects of stator voltages to outputs

(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)

The effects of stator voltages to control errors

vsdref  ercd ( 

m = 0.63 

s )

vsdref  ercq ( m = 0.63 s)

v sqref e rcq ( m = 0.63 s)

vsqref  ercd ( m = 0.63 s)

vsdref  ercd ( m = 0.9 s)

vsdref  ercq ( m = 0.9 s)

vsqref  ercq ( m = 0.9 s)

vsqref  ercd ( m = 0.9 s)

(d)

Figure 10 Performance of the controlled system with

current controller for frozen value

Hence, the obtained performance is not guaranteed for

further investigate the performance of the closed-loop

the following investigation, we consider the performance of the controlled system with the rotor

In order to do so we

parameter values using the same weighting functions as in (37) and (38) Then we plot the time responses of the

time responses of the closed-loop system with the

each figure for the purpose of comparison of the

Figure 10 shows the performance of the closed-loop

, respectively The thick-solid lines are

can be seen from Figure 10a, the time responses of the

outputs and with respect to the step change of

in Figure 9 In addition, these curves are almost the

The same conclusion can also be

drawn for the curves related to the time responses of

the remarkable difference in the performance among

cross-coupling interactions The effects of the stator

x 10 -3

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)

Closed-loop performance of reference inputs to outputs

irdref  ird ( m = 1.17 s)

irdref  irq ( m = 1.17 s)

irqref  irq ( m = 1.17 s)

irqref  ird ( m = 1.17 s)

irdref  ird ( m = 0.9 s)

irdref  irq ( m = 0.9 s)

irqref  irq ( m = 0.9 s)

irqref  ird ( m = 0.9 s)

(a)

x 10-3 -0.2

0 0.2 0.4 0.6 0.8

time (s)

Closed-loop performance of reference inputs to control errors

i

rd ref  e rcd ( m = 1.17 s) i

rd ref  ercq ( 

m = 1.17 

s ) i

rq ref  ercq ( m = 1.17 s) i

rq ref  e rcd ( m = 1.17 s) i

rd ref  ercd ( m = 0.9 s) i

rd ref  ercq ( m = 0.9 s) i

rq ref  ercq ( 

m = 0.9 

s ) i

rq ref  ercd ( 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)

The effects of stator voltages to outputs

v sd

ref  i rd ( m = 1.17 s)

v sd

ref  i rq ( m = 1.17 s)

vsqref  irq ( m = 1.17 s)

vsqref  ird ( m = 1.17 s)

vsdref  ird ( m = 0.9 s)

vsdref  irq ( m = 0.9 s)

vsqref  irq ( m = 0.9 s)

vsqref  ird ( 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)

The effects of stator voltages to control errors

vsdref  ercd ( m = 1.17 s)

vsdref  ercq ( m = 1.17 s)

vsqref  ercq ( m = 1.17 s)

vsqref  ercd ( m = 1.17 s)

vsdref  ercd ( m = 0.9 s)

vsdref  ercq ( m = 0.9 s)

vsqref  ercq ( m = 0.9 s)

vsqref  ercd ( m = 0.9 s)

(d)

Hình 11 Performance of the controlled system with

current controller for frozen value

for

The performance of the closed-loop system at

respectively, is shown in Figure 11 The thick-solid

Similarly to the previous simulation, the time

responses of the closed-loop system with the

controller designed for the frozen value

corresponding to the step change of the

cross-coupling interactions In the case of the

rotor angular speeds are not maintained because of the cross-coupling interactions between the stator

This may cause a large tracking error for the

are the input disturbances

x 10-3 690

700 710 720 730 740 750 760

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(a)

x 10-3 0

50 100 150 200 250 300 350 400

q component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(b)

x 10-3 690

700 710 720 730 740 750 760

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(c)

x 10 -3

0 50 100 150 200 250 300 350

q component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(d)

x 10 -3

690 700 710 720 730 740 750 760 770

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(e)

x 10 -3

0 50 100 150 200 250 300 350

q component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(f)

Hình 12 Performance of the controlled system with

at different constant values of

In order to evaluate the performance of the

the face of the stator voltage action, we performed the

time responses of the (Figure 12a) and (Figure 12b) components of the rotor currents achieved by the

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

figures reveal that the tracking errors of the and

components of the rotor currents achieved by this

This is because of the cross-coupling

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

(dash-dotted curves), respectively

Figures 12e and 12f show the time responses of the

and components of the rotor currents achieved by

(dashed curves), and

(dash-dotted curves), respectively These figures

0 0.002 0.004 0.006 0.008 0.01

0

100

200

300

400

500

600

700

800

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(a)

0 0.002 0.004 0.006 0.008 0.01 0

100 200 300 400 500 600 700 800

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(b)

0 0.002 0.004 0.006 0.008 0.01

0

50

100

150

200

250

300

350

400

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(c)

0 0.002 0.004 0.006 0.008 0.01 0

50 100 150 200 250 300 350 400

d component of the rotor currents

time (s)

m = 0.63 s

m = 0.9 s

m = 1.17 s

(d)

0 0.002 0.004 0.006 0.008 0.01

200

250

300

350

m

time (s)

(e)

0 0.002 0.004 0.006 0.008 0.01 200

250 300 350

m

time (s)

(f)

Hình 13 Performance of the controlled system with

respectively, with fast variations of the rotor speed

For further investigation, a simulation with the

fast variation of the rotor speed along the whole parameter interval is carried out We consider three

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

currents when the rotor angular speed increases from 70% to 130% of the nominal speed of the rotor

Conversely, the behaviors of the and components of the rotor currents when the rotor angular speed decreases from 130% down to 70% of the nominal speed of the rotor are shown in Figures

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 Conclusions

This paper briefly recapitulated the theory of the

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

completely 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 to adapt 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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Dr Ngo Duc Minh was

born in Lang son, Vietnam,

in 1960 He received the

Thainguyen University of Technology in 1982 in

M.S degree from Hanoi University of Technology

Industrial Information Technology, and Ph.D degree from Hanoi University of Technology in 2010 in Atutomation Technology He is currently a vice-chair

of the Education department of Thainguyen University of Technology Dr Minh’s 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, distribution power systems

Nguyen Tien Hung was

Vietnam He received the

Thainguyen University of Technology in 1991 and M.S degree from Hanoi University of Technology in

1997, both in Electrical Engineering He is currently

a Ph.D candidate at Delft Center for Systems and Control (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