investigation of the robust controller for statcom applied

Investigation of the Robust Controller for STATCOM Applied to Large Induction Motor against Load and

Sinusoidal Disturbance Using Graphical Loop Shaping Method

Mahdi Hedayati

Department of Electrical & Electronic Engineering,

Faculty of Mechtronics, Islamic Azad University, Karaj Branch

E-mail:[email protected]

Norman Mariun Department of Electrical & Electronic Engineering, Faculty of Engineering, University Putra Malaysia,

E-mail: [email protected]

Abstract-Between the different robust H-Infinity methods to design the controller for FACTS, loop shaping is known to be one of the effective and feasible methods. This research presents an investigation on H-Infinity loop shaping procedure via Graphical Loop Shaping (GLS)for STATCOM installed at terminals of a large induction motor. The dynamic behavior of induction motor is analyzed while the load perturbation and sinusoidal perturbation are considered and will be compared with conventional PI controller. Simulation results prove that GLS method has better dynamic response than PI controller against load disturbance and external sinusoidal disturbance, consequently GLS method are more robust than PI controller to remove these perturbations with lower fluctuations. Keywords: Induction Motor; STATCOM; Robust Control; Graphical Loop Shaping; Uncertainty Modeling

I. INTRODUCTION

Conventional PI (or PID controller) is one of the best known as industrial process controller and despite a many research and the great number of different solutions proposed, most industrial control systems are based on this conventional regulator. In this method, the system linearized around nominal point. Although PI controller has some disadvantages such as: the high starting overshoot, long adjusting time, sensitivity to controller gains and lethargic response due to sudden disturbance[1, 2, 3, 4], the mentioned controller has adequate response at this point. But FACTS devices and the other power system elements have nonlinearity nature and acquiring a precise nonlinear model that exactly is compatible with the plant at all working points is a difficult to attain. So, most of advanced controllers are generally designed utilizing a linear model of the process based on fixed information of the plant that is incomplete and defective; consequently, control quality may decline when working circumstances vary [5]. Besides, there exists the uncertainty in the system that makes it hard to design a satisfied controller for the FACTS that assures fast and stable regulation under all operating conditions. The main source of trouble is that the (open loop plant) may be incorrect or may vary with time. In particular, incorrectness in may create problems because the plant is part of the feedback loop. To engage in

a problem, instead of using a single model, a model with should be considered, where is the uncertainty or perturbation [6]. In view of these problems, robust control design methods is appeared to be convenient, since they present linear controllers with acceptable stability margins [7]. The principle of robust control is to model the uncertainties themselves and to include them in the design method of the control system with the target of covering stability and performance criteria at all working points. The change of operating conditions is considered as a source of unstructured uncertainty in the linearized system model. Another important matter is the ability of controller to remove the effects of external disturbances on system execution [8]. Among all the presented robust techniques to design the controller of shunt FACTS, the loop shaping procedure has been chosen, because this method have some benefits when compared with other design techniques. The main advantage of loop shaping method is that it does not require the so called γ – iteration to obtain the optimal controller. Also, there are obtainable comparatively simple formulas to propose the controller and it is proportionately easy to perform based on classical loop shaping ideas [5, 6]. Graphical Loop Shaping (GLS) method is investigated to propose loop shaping theory. The rest of this paper is organized as follows. First, the GLS method is presented. Next, the design of the controller utilizing this procedure for STATCOM connected to a large induction motor is specified and simulation result for step response of motor’s rotor speed based on this method is shown. Finally the load perturbation and sinusoidal perturbation are considered with this suggested controller and will be compared with conventional PI controller.

II. GRAPHICAL LOOP SHAPING METHOD

Loop shaping is a graphical method to design suitable controller fulfilling robust stability and performance [9]. In this section, a concise theory of the uncertainty modeling, robust stability and robust performance criteria will be presented to acquire a robust control design algorithm.

NO –E-13-ELM-2092

A. Uncertainty Modeling Assume that the linearized plant having a nominal transfer function consists a set of transfer functions 𝜬. Consider that the perturbed transfer function, consequence from the variations in operating circumstances, may be stated as:

( ) (1)

Hence, is a fixed stable transfer function that has been defined before as uncertainty weight, also called the weighting function and is a variable transfer function satisfying‖ ‖

Definition: ‖ ‖

| ( )| is the infinity norm

of a function and is the largest value of gain on a bode magnitude plot. For asymptotically stable transfer function, the special form of the infinity norm, called H infinity norm

( norm) is applied [7, 10]. The linearized state model for small perturbation around a nominal operating circumstance is expressed for time invarrying system as:

(2)

(3)

Taking the Laplace transform of Equations 2 and 3, we get:

( ) ( ) ( ) (4)

( ) ( ) ( ) ( ) ( ) (5)

Therefore, the plant transfer function will be calculated as:

( )

( ) ( ) (6)

With considering‖ ‖ , the multiplicative uncertainty model (Equation 1) is rewritten as:

| ( )

( ) |≤ ( ) ω (7)

So, ( ) gives the uncertainty profile, and in the frequency plane is the upper boundary of all the normalized plant transfer functions away from 1 [11, 12].

B. Robust Stability and Robust Performance

Considering a feedback system with reference trajectory and plant output as specified in Figure1. The tracking error signal is defined as thus forming the negative feedback loop. The sensitivity function is written as

(8)

Fig.1. Unity feedback with controller

Where symbolize the plant transfer function, and is the controller. The controller will provide stability if it gives internal stability for every plant in the uncertainty set 𝜬. If represents the open loop transfer function , then the sensitivity transfer function will be written as:

(9)

The complimentary transfer function or the output-input transfer function is:

T=1-

=

(10)

Performance requirements of a feedback controller, utilizing the nominal plant model can be shaped in terms of the Nyquist plot. The requirement for adequate performance with plant uncertainty is a combination of the two requirements. The fist one is a condition for nominal performance in the presence of multiplicative uncertainty parameterized with ( ) that expressed as:

| ( ) ( )| (11)

The second one is necessary and sufficient condition for robust stability in the presence of multiplicative uncertainty parameterized with ( ) that determined as follows:

| ( ) ( ) | (12)

The requirement for adequate performance with plant uncertainty is a combination of the above two requirements. Hence it is clear that with multiplicative uncertainty, nominal performance plus robust stability almost guarantee robust performance [7]. Graphically, as shown in the Figure 2, the disk at the critical point with radius| ( )| should not intersect the disk of radius | ( ) ( ) ( )|, centered on the nominal locus ( ) ( )[11].This can be written as:

P C

-

+ e y r

Investigation of the Performance of Robust Controller for STATCOM Applied to Large Induction Motor against Load and Sinusoidal Disturbance Using Graphical Loop Shaping Method

28th Power System Conference - 2013 Tehran, Iran

2 2

Fig.2.Nyquist plane construction for robust performance under multiplicative uncertainty

| ( )| | ( ) ( ) ( )| | ( ) ( )| (13)

Dividing by | ( ) ( )|gives:

| ( )|

| ( ) ( )|

| ( ) ( ) ( )|

| ( ) ( )| (14)

Or

| ( ) ( )| | ( ) ( ) | (15)

The Equation 15 is necessary and sufficient condition for robust stability and performance in the presence of multiplicative uncertainty.

C. GLS Technique

Loop shaping is a graphical method to design a suitable controller fulfilling robust stability and performance criterion specified in sect 2.2. The basic idea of this method is explained and applied in [11-13] to obtain an approximate solution of this problem, which can be readily extended to different weighting transfer functions. The first step is to construct the loop transfer function to satisfy the robust performance criterion and then to acquire the controller

from the relationship

. Internal stability of the plants

and properness of represent the constraints of the method. Condition on L is such that should not have any pole –zero cancellation. An essential condition for robustness is

that either or both ( ) and ( ) must be less than 1 at any frequency. The robustness condition given in Equation 15 can be obtained as the following equation:

At low frequency, ( ω) is small (S→0 and →1), because ( ) is large, therefore | ( ) ( )| 1 will conclude that ( )<1. At high frequency, ( ω) is small (S→1 and →0), because ( ) is small, therefore | ( ) ( )| 1 will conclude that ( )<1. Sufficient conditions to fulfill the robust performance condition in these regions are [3, 9, 11]:

At low frequency :| | | |

| |or

| | | |

| |

(17)

At high frequency :| | | |

| |or

| | | |

| |or | |

| |

(18)

III. SYSTEM MODELING

The system under study is the machine, a network and infinite bus where STATCOM is installed on the terminal of the induction motor. Figure 3 shows the system modeling with a STATCOM which consists of DC capacitor, a three phase GTO based VSC and a step down transformer.

RL

jXLIM

I LO

VM

0V cVDC

VDC

XSDT

Fig.3. System modeling with a STATCOM

IV. The ALGORITHEM of GLS PROCEDURE BASED on HEORY

The algorithm to produce a control transfer function based on GLS method so that robust stability and robust performance requirements are satisfied, specified by the flowchart of Figure 4 [14].

| ( )( ( ))

| | ( ) ( )( ( ))

| (16)

( )

( ) ( )

( ) ( ) ( )

( ) ( )

3



Fig.4. Algorithm flowchart of GLS method

V. LINEARIZED MODEL of SYSTEM

with STATCOM

For the construction of electromechanical mode damping controllers, the linearized supplementary model around a nominal operation point is commonly utilized. By linearizing the system equations at a given operating state that specified as follows: ( )=2300 V, Power (rated) = 2250 hp, =60 Hz,

=0.0716, =0.0716, =4.13, =4.2,

=4.2, D=0.6,

=0.0091, =0.697,

=0.0023,

=1.0364,

=0.00

26,

=1.0187, =377 rad/s,

=1, =4,

=63.87 , =0.04, =0, =1, =0,

= 0.015, =1,

=1 rad, =1, =1, =2.70,

=9.82,

=0.88, =0.3

=0.9

7, the linearized model for the system with STATCOM can be obtained. The lineraized model for system with a STATCOM can be written as [14, 15]:

[

]

[ ]

[

]

[ ] [

] (19)

[

]

[ ]

[

]

(20)

Matrices[ ],[ ] [ ]are

identified as follows

[ ]

[

]

(21)

Is| ( )

( ) |≤ ( )?

Select

| | | |

| |?and

| |

| |

Is| | | |

and| | ?

Produce

Consider preselected disturbance or input

The stability and

performance of closed loop

system are provided?

Acquire sufficient and

STOP

Yes

Yes

Yes

Yes

No

No

No

No

Build

Acquire the dB magnitude of and different

Choose

START

Find the nominal transfer plant function

4



That

And The quantities of network, induction motor and STATCOM are specified before. Generally, the linearized state model for Equations 19 and 20 are expressed as:

That

Therefore, the plant transfer function will be calculated as:

VI. SIMULATION RESULTS The algorithm flowchart of Figure 4 is considered to design the robust controller for the quoted system. The system parameters at nominal operating point are specified before. The nominal operating point for the design is computed for an infinite bus voltage of 1 at unity power factor (PF), so , and two important

STATCOM parameters, and . A.GLS Method The nominal plant transfer function for the selected operating point is computed as:

Where

Off nominal infinite bus voltage between the ranges of 1-0.1 and power factor between the ranges of 1-0 lagging

[ ]

[

]

(22)

[ ] [ ] (23)

=

(

) =

, = , = , =

= ,

, =

=(

),

, ,

=0, (

), (

)

(

), (

), =0

(24)

(

)

=

=

=

(25)

(34)

(

)

(26)

(27)

(28)

[ ] [ ]

[ ]

(29)

( ) ( ) ( ) (30)

( ) ( ) ( ) ( )

Taking the Laplas transformer of Equations 27 and28 ,we get:

(31)

( )

( ) ( ) (32)

(33)

5



which give steady state stable situation, are considered in the robust design. The dB magnitude versus frequency plots for the nominal plant ( ) and perturbed

plant ( )should be determined. The quantity | ( )

( ) |

for each perturbed plant is constructed and the upper envelope in the frequency plane is fitted to the following function:

So (| ( )

( ) |≤ ( ).A butterwort filter, which satisfies

the properties of ( )is selected as [7]:

(36)

Values of =0.01 and =0.5 are observed to be fulfill the requirements on the open loop transfer function that is

specified in Equations 17 and 18. For and selected above and for choice of the open loop transfer function as:

( )( ) ( )

( )( )( )( )

(37)

The controller transfer function obtained through the

relation

is:

(38)

(39)

The plots for the nominal and robust performance criteria are shown in Figure 5. It can be observed that the nominal

performance measure ( ) is very small relative to 0 dB and well satisfied. The combined robust stability and

performance measure ( + ) has a small peak at the corner frequency that is related to parameters of system considered in design and comparable to [9].

Fig.5. The robust and nominal performance with consideration to step response of motor’s rotor speed

Once the robust stability and performance criteria are plotted in Figure 5 are met, the stability and performance of the closed loop system have to be checked by direct simulation of the system dynamic equations. Figure 6 shows

the step response of motor’s rotor speed ( ) during starting with controller at mentioned nominal operating point.

Figure 6: Step response of motor’s rotor speed with GLS

method Fine Tuning on the controller parameters are performed from monitoring of the transient response of the system. The distortion of the speed response is very small with peak value 8% and has a satisfactory dynamic performance. B. PI Controller

Figure 7 shows the step response of motor’s rotor speed

with GLS method that has been obtained before and

compares it with PI controller. It is obviously indicated that

GLS method has better dynamic attitude than PI controller

so the conventional controller has high fluctuations to reach

the steady state position. The comparison between the

specifications of GLS method and PI controller proves that

quantity of overshoot in PI controller is higher than GLS

method, also the amount of undershoot in GLS method is

obviously lower than PI controller. Although the peak time

in PI controller is lower than GLS method. The PI controller

10-3

10-2

10-1

100

101

102

103

104

-300

-250

-200

-150

-100

-50

0

50

Ma

gn

itu

de

(d

B)

Frequency (rad/sec)

W1S+W2T

W1S

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

T i m e ( s )

A m

p l

i t

u d

e (

p.u

)

(35)

6



designed is checked for the number of other operating points

that has been defined before when the step response of

motor’s rotor speed of was considered.

Fig.7. Comparison between the step Response of motor’s rotor speed with PI controller and GLS method

As mentioned before, the controller can be considered in the system to give good robust performance against load disturbance, besides; the controller has enough robustness to reject or decrease the effect of external sinusoidal disturbance in the system. Figure 8 shows the block diagram of system with controller that is used in Matlab/ Simulink. In this block diagram, the state models of system and controller are specified. Load disturbance and sinusoidal disturbance are connected to output separately via a manual switch. A load disturbance is considered as a step load torque with magnitude 0.5 p.u that is applied at t=2 sec after running up and removed from the motor shaft at t=3 sec.

Fig.8. Block diagram of the system and controller

Figure9 represents the step response of the motor's rotor speed with PI controller and considering this disturbance uncertainty.

Fig.9. Step response of motor’s rotor speed with PI controller when a load disturbance uncertainty is considered

Figure 10 shows the step response of the motor’s rotor speed with robust control theory using GLS method. It is obviously specified that GLS method is more robust than PI Controller to reject the disturbance and have better dynamic responses and lower fluctuations, so the amount of peak value in PI controller is 0.69 p.u (138% of load disturbance) that is high. This magnitude is 0.12 p.u(24%) for GLS method that is lower than PI controller value.

Fig.10. Step response of motor’s rotor speed with GLS method when a load disturbance uncertainty is considered

Another disturbance is sinusoidal disturbance. Since the sinusoidal disturbance frequencies are unfamiliar and the amounts of these frequency bands and their magnitudes depend on the special design, in this research a sinusoidal disturbance with a frequency f =620 Hz and amplitude (peak to peak) of 0.1 p is considered [12]. The step response of motor’s rotor speed with PI controller and GLS method are specified in Figures 11 and 12. It can be seen again that GLS method is considerably better than PI controller to reject sinusoidal disturbances. The amplitude (peak to peak) of sinusoidal disturbance in PI controller is 5× p.u (50%) and is 0.2× p.uin GLS method that is lower than PI controller value, so the amount of existing sinusoidal noise in output of system with GLS method is only 2% of applied disturbance.

XX-dot

AA

To Workspace4

y 2

To Workspace3

y

To Workspace2

states

To Workspace1

y 1

To Workspace

t

Step2

Step1

Step State-Space

x' = Ax+Bu

y = Cx+Du

Sine Wave

Scope8

Scope7

Scope6

Scope5

Scope4Scope3

Scope2

Scope1

Manual Switch

1s

Fcn1

f (u)

Fcn

f (u)

DD

K*uvec

Clock

CC

K*uvec

BB

K*uvec

K*uvec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

T i m e ( s )

A m

p l

i t

u d

e (

p . u

)

Applying a Load Disturbance and Removing after 1 secNormal Working

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

T i m e ( s )

A m

p l

i t

u d

e (

p .u

)

N o r m a l W o r k i n g A p p l y i n g a L o a d D i s t u r b a n c e a n d R e m o v i n g A f t e r 1 S e c

7



Fig. 11. Step response of motor’s rotor speed with PI controller when a

sinusoidal disturbance is considered

Fig.12. Step response of motor’s rotor speed with GLS method when a sinusoidal

disturbance is considered

VII.CONCLUSION

In this research, a method of designing robust loop shaping damping controller for a STATCOM in a power system consist of a large induction motor with STATCOM installed on its terminal, a transmission line and infinite bus through control of motor's rotor speed and motor's terminal voltage is proposed. The design is performed utilizing both robust stability and robust performance considerations. The fixed parameter controller transfer function which provides robust damping characteristics is obtained through the GLS technique. Also, robust controllers and conventional PI controller are tested for a load disturbance such as a step load torque with magnitude 0.5 p.u and a sinusoidal disturbance like a sine wave with a frequency f =620 Hz and amplitude (peak to peak) of 0.1 p.u on system performance. The results of simulations determine that both controllers are able to damp these disturbances, but GLS method is more robust than PI controllers to remove these perturbations with lower oscillations. It is clearly indicated that GLS method has better dynamic execution than PI controller to reject the disturbances and stronger than PI controller against load disturbance and sinusoidal disturbance uncertainties. So

dynamic attitude of the system to return the steady state position after a load disturbance employment in GLS method is faster than PI controller and the output performance has a peak value 0.12 p.u ( 24% of load disturbance), so the amount of peak value in PI controller is 0.69 p.u (138% of load disturbance) that is high. Also the amount (peak to peak) of existing noise in the system response with a sinusoidal noise employment is (2%) while GLS method is considered that is very negligible comparing with PI controller (5× p.u (50%)).

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Description, Arabian Journal for Science and Engineering, 2013.Vol. 38

(10): p. 2713-2723.

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

T i m e ( s )

A m

p li

t u

d e

( p

. u

)

0 0.5 1 1.5 2 2.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

T i m e ( s )

A m

p l

i t

u d

e (

p . u

)

2 2.02 2.04 2.06 2.08 2.10.94

0.96

0.98

1

1.02

1.04

2 2.02 2.04 2.06 2.08 2.1

0.94

0.96

0.98

1

1.02

1.04

8



investigation of the robust controller for statcom applied

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