Nonlinear sliding mode control for three phase UPS

In this paper, a nonlinear sliding mode control with variable damping ratio for uninterrupted power system (UPS) is proposed. In this type of sliding mode controller, the sliding surface is designed with the continuous changed parameters. By this way, the system can have the fast responses without overshoot by giving a low damping ratio at initial time and a high damping ratio as the output reaches the set point.

NONLINEAR SLIDING MODE CONTROL

FOR THREE-PHASE UPS

Vu Thi Thuy Nga*

Abstract: In this paper, a nonlinear sliding mode control with variable damping

ratio for uninterrupted power system (UPS) is proposed In this type of sliding mode controller, the sliding surface is designed with the continuous changed parameters

By this way, the system can have the fast responses without overshoot by giving a low damping ratio at initial time and a high damping ratio as the output reaches the set point The stability of the system is guaranteed by Lyapunov criteria Finally, a three-phase UPS control system is setup in several modes via Matlab/Simulink software Simulation results indicate that the proposed control algorithm have the good effectiveness under the conditions of balanced load, unbalanced load, suddenly changed load, and nonlinear load as well as the system uncertainties

Keywords: Nonlinear surface, Nonlinear sliding mode controller, UPS

1 INTRODUCTION

Nowadays, with the development of technology, the uninterrupted power system (UPS) is become more important day by day to supply continuous power

to critical loads such as computer systems, medical equipment, etc A good UPS not only supplies continuous AC power to loads but it also should guarantee that the AC power is disturbance free, load independent, low THD, etc In order to satisfy that strict requirements, besides the development of the hardware, the software (i.e algorithm control) also has a countable contribution In recent years, there are some advanced control methods have been developed instead of

PI controller which is not easy to accomplish good performances for UPS under some conditions In the dead-beat [1, 2] control the good performances with high quantity output voltage and fast dynamic response are achieved, however, this controller require exactly information about system parameters In [3], a model predictive control (MPC) with load observer is proposed This controller guarantees a small steady-state error, reduces the number of current sensors But, this scheme stills giving a high THD level in the output voltage Next, a repetitive controller is presented in [4] With this algorithm, the voltage distortion

is low but the dynamic response is slow In [5], a feedback linearization control technique is presented, this controller also gives the fast dynamic and low THD But, the drawback of this scheme is that the uncertainties of the system parameters are not considered

In this paper, a nonlinear sliding mode controller is proposed for output voltage control of three-phase UPS with a variable damping ratio This controller ensures the good performances of the output voltage such as fast transient response, small steady-state error, and low THD under various conditions (unbalanced load, nonlinear load, system uncertainties) The effectiveness of the proposed controller

is verified by simulation results

2 CONTENTS 2.1 State-space model of three-phase UPS

Fig 1 shows the block diagram of a three-phase UPS with Y-connection of filter and load

Fig 1 Block diagram of the three-phase UPS

It is easily to get the following results in the abc reference frame by using Kirchhoff’s voltage law and Kirchhoff’s current law for circuit in Fig.1:

























i f L i f

i i

L f i f L

L L

dt d

C C dt d

V V

T I

T

I I V

1 1

1 1

(1)

where





























1 0 1

1 1 0

0 1 1

i

T (2)

Transferring the differential equation in (1) in to dq reference frame as

following:

















































iq id Lq id iq

iq id Ld iq id

Lq iq Ld Lq

Ld id Lq Ld

V k V k V k I I

V k V k V k I I

I k I k V V

I k I k V V

3 4 2

4 3 2

1 1

1 1

















(3)

where ω is the angular frequency ( = 2f), f is the fundamental frequency of output

voltage or current, and

f f

f

C

k

3 2

1 ,

2

1 , 1 ,

1

4 3

2

The reference values (I idr and I iqr ) of the inverter currents (I id and I iq ) in dq axis

are defined as:

Ldr Lq

iqr Lqr Ld

k I I V k I

1 1

1 ,

1







Denote four state variables as the following:

1 Ld Ldr, 2 Lq Lqr, 3 id idr, 4 iq iqr

With this definition, the system model (3) can be rewritten as















































iqr iq id Lq id

idr iq id Ld iq

I V k V k V k I x

I V k V k V k I x

x k x x

x k x x













3 4 2 4

4 3 2 3

4 1 1 2

3 1 2 1









(5)

The state-space form of the model (5) is obtained as:

( )

x AxB ud (6)

T

x x x x x , u  V id V iq T,

1 1

0 0

,

0 0 0 0

0 0 0 0

k k A





0 0

0 0 ,

B

 

 

 

 

  

 

 

 

 

2

2

iq Ld idr

id Lq iqr

d









In (6), the disturbance d is the model nonlinear part, however, d also can be

input noises or uncertainties

2.2 Nonlinear sliding surface design

The nonlinear sliding surface in this paper is a continuous surface with a nonlinear continuous function whose parameters can change by tuning the damping ratio of the system from low value at initial time to the high value as the system reaches the set point By this way, not only the chattering phenomenon is reduced but also the transient response of the system can be fast without overshoot The dynamic model (5) is written in the following form:

1 12 2 13 3

2 21 1 24 4

id iq

id iq

  



  





   











(7)

or it is can be rewriten as:

0 0 ( )

         





where

            

 

Let us define the sliding surface as:

12

( ) T z

z







   

     

          (9)

where F is calculated from the given settling time t s and initial damping ratio ζ1,

and P is a positive definite matrix that is chosen based on the desired final damping

ratio ζ2 and satisfies the following condition:

(10)

where Q is a positive definite weighting matrix The function Φ(y) (in which y is

the output of the system) is selected so that as the output changes from the starting

point to the origin, the value of the function Φ(y) decreases from 0 or a very small

value to –β in which β is a positive scalar Also, the function Φ(y) should have the

first degree differential equation in y

In this study, the function Φ(y) is selected based on [6] with the following form:

2

) (y   eky

  (11)

where k is a positive scalar that should be large enough to ensure a small initial

value of Φ(y)

The stability of the sliding surface (9) is proven by the following theorem [9]

and a positive definite matrix P that is chosen based on the desired final damping

ratio ζ2 Also, it satisfies the condition in (10) with the function Φ(y) defined by

(11), then the nonlinear sliding surface defined by (9) is stable

Proof: Refer to the standard sliding mode control methods presented in [7, 8]

and note that during the sliding mode      0 From (9), it is easy to see that

2 [ ( ) 12T ] 1

z   F y A P z (12)

By using (12), the system dynamics (8) can be expressed by the following

first-order equation

1 [ 11 12 ( ( ) 12T )] 1

(13) For the dynamics (13), the Lyapunov function can be chosen as

1 1

( ) T

V z z Pz (14)

The derivative of V(z) along with (13) is obtained as the following:

( ) T( )T T ( ) 2 ( ) T T

(15) The condition represented by (10) implies that

( ) T( 2 ( ) T )

V z z   Q y PA A P z (16)

In this case, the properties of the function Φ(y) guarantee that it is a negative

function And it can be concluded that the sliding mode dynamics (13) or the sliding surface (9) is asymptotically stable

2.3 Nonlinear sliding mode output voltage control

The above sliding surface guarantee that as the system trajectory reaches the surface then it will slide along the surface to the origin The next section is to design

a controller that satisfies the reaching condition By this way, the trajectory of the system will attain the sliding surface then goes to the origin along the surface from any initial state Such a controller is introduced in the following theorem

Theorem 2: If the uncertain term d is bounded by a constant and there exist

some positive constant γ such that γ ≥||SB|| ||d|| then the following controller will

guarantee that the state of the uncertain system (8) converges to zero

1

u SB  SAz K sign Sz

where K is a positive scalar and σ is a nonlinear sliding surface given by (9)

Proof: It is well known that the sliding motion approaches zero in finite time if

the sliding condition T   0 is guaranteed for all σ(t) ≠ 0 [7,8]

From(8), (9), and (17), the reaching condition can be derived as

2

2

(| | | |) | | | SB | | | | SB | (| | | |) | | || |||| || | | || ||

T

T

T





2

2

|| ||

(| | | |) || || (| | | |)

d

K





(18)

According to (18), it is obviously shown that a stable sliding motion is induced

in finite time Then it completes the proof

3 SIMULATION RESULTS AND DISCUSSION

In this paper, a prototype 450VA DG unit is considered to implement the proposed control algorithm The nominal system parameters are as follows:

DGS rated power (P rated ) : 450VA; dc-link voltage (V dc): 280V; Load output

voltage (V L,rms ): 110V; Output frequency (f): 60Hz; Switching and sampling frequency: 5kHz; LC output filter: L f = 10mH, C f = 6F; Resistive load/unbalanced load: R L = 80; Nonlinear load: L dc = 5mH, R dc = 200Ω

Based on the nominal parameters, the system model (2) can be rewritten as















































iq id

Lq id

iq

iq id

Ld iq

id

Lq iq

Ld Lq

Ld id

Lq Ld

V V

V I

I

V V

V I

I

I I

V V

I I

V V

4 83 1 48 7 166 377

1 48 4 83 7

166 377

7 166666 7

166666 377

7 166666 7

166666 377









(19)

It can be rewritten in the form of state-space model as:

( )

x AxB ud (20) where

, 0 0

0 0

0 0

0 0

7 166666 0

0 377

0 7 166666 377

0





























A





























4 83 11 48

11 48 4 83

0 0

0 0

B

In order to calculate the sliding surface parameters, it is assumed that the initial

settling time t s1 is 0.1 s and the initial damping ratio ζ1 is 0.4 Also, the matrix F

can be calculated using the pole placement technique which gives

1 2

0.0002 0.0028 0.0028 0.0002

F F

F

 

 

      (21)

Solving the Lyapunov function (15) with Q = eye(2), the resulting value of P is

obtained as:

0.0128 0

0 0.0128

   (22)

The nonlinear sliding surface is now built by using the above matrices F and P

with β = 300 and k = 50 as

2

2

50

50

300 ( ) 1 0

300 ( ) 0 1

y T

y T







(23) where ( 12T ) , (1 12T )2

A P

Fig 2 System response with the change

of load (from no load to full load)

Fig 3 System response with

unbanlanced load (two-phase load)

Fig 4 System response with nonlinear load

Figs 2 to 4 show the simulation results under the conditions of no load to full load, unbanlanced load (two-phase load), and nonlinear load It is noted that all

of cases are simulated with the assumption that the system parameters are not known exactly

In Fig 2, the system start up with no load, the load current IL = 0 At the time t

= 0.075s the system operates with full load It can be seen that the load voltage oscillates trivially In this case, the output voltage THD is 0.04 Fig 3 shows the results with one phase opened The load current is unbalanced but the load voltage

is still balanced and THD is 0.04 In the case of nonlinear load (Fig.4), the load current is nonsinusoidal; however the load voltage is sinusoidal and THD is 0.3

4 CONCLUSIONS

In this paper, a nonlinear sliding mode controller for three phase UPS has been proposed The controller is not only robust with system uncertainties and disturbances as linear sliding surface but it also has fast response due to the change

of damping ratio The stability of the presented controller is proven mathematically To support the validity of the proposed controller, the simulation has been carried out through out Matlab/Simulink The simulation results demonstrated that the proposed controller give the good performances for output voltage such as fast dynamic, small steady state error, low THD under various conditions such as change of load, unbanlanced load, nonlinear load with the presence of system uncertainties

REFERENCES

[1] M Kojima, K Hirabayashi, and Y Kawabata, “Novel vector control system using deadbeat-controlled PWM inverter with output LC filter,” IEEE Trans

Ind Appl vol 40, pp 162–169, (2004)

[2] P Mattaveli, “An improved deadbeat control for UPS using disturbance

observers,” IEEE Trans Ind Electr vol 50, pp 206–212, (2005)

[3] Cortes P., Ortiz G., Yuz I J., Rodriguez J., Vazquez S., Franquelo L G

(2009): Model predictive control of an inverter with output LC filter for UPS applications IEEE Trans Ind Electr 56(6), pp 1875–1883

[4] S Jiang, D Cao, Y Li, J Liu, F Z Peng, “Low-THD, fasttransient, and cost-effective synchronous-frame repetitive controller for three-phase UPS

inverters,” IEEE Trans Power Electr vol 27, pp 2994–3004, (2012)

[5] D E Kim, D C Lee, “Feedback linearization control of threephase UPS

inverter systems,” IEEE Trans Ind Electr vol 57, pp 963–968, (2010)

[6] B M Chen, T H Lee, K Peng, and V.Venkataramanan, “Composite nonlinear feedback control for linear systems with input saturation: theory and

applications,” IEEE Trans Automat Control, vol 48, pp 427- 439, (2003)

[7] R A De Carlo, S H Zak, and G P.Matthews “Variable structure control of

nonlinear multivariable systems: A tutorial,” IEEE Proc., vol 76, pp 212

-232, (1998)

[8] V I Utkin, “Variable structure systems with sliding modes,” IEEE Trans

Automat Contr., vol 22, pp 212 -222, (1997)

[9] J W Jung, D Q Dang, N T.- T Vu, J J Justo, T D Do, H H Choi, and T

H Kim, “Nonlinear sliding mode control for IPMSM drives with an adaptive

gain tuning rule,” Jour Power Electr , vol 15, no 3, pp 753-762, 2015

TÓM TẮT

ĐIỀU KHIỂN TRƯỢT PHI TUYẾN CHO BỘ NGUỒN UPS BA PHA

Bài báo đề xuất một bộ điều khiển trượt với mặt trượt phi tuyến có hệ số tắt dần thay đổi cho bộ nguồn UPS Ở bộ điều khiển trượt kiểu này, mặt trượt được thiết kết với các tham số thay đổi liên tục Với cách này, hệ thống

sẽ có đáp ứng nhanh mà không bị quá điều chỉnh bằng cách cho hệ số tắt dần lúc bắt đầu là bé, khi đáp ứng đầu ra gần đạt đến giá trị đặt thì hệ số tắt dần tăng lên Tính ổn định của hệ thống được đảm bảo theo tiêu chuẩn Lyapunov Cuối cùng, việc mô phỏng bằng Matlab/Simulink được tiến hành

ở nhiều chế độ khác nhau Các kết quả mô phỏng chứng tỏ rằng, bộ điều khiển được đề xuất cho đáp ứng tốt ở nhiều chế độ khác nhau của tải như tải cân bằng, tải mất cân bằng, tải thay đổi đột ngột, tải phi tuyến cũng như trường hợp các tham số cua hệ thống là bất định

Từ khóa: Mặt trượt phi tuyến, Điều khiển trượt phi tuyến, UPS

Received date, 11 th Sep., 2017 Revised manuscript, 07 th Oct., 2017

Published, 01 st Nov., 2017

Address:

Ha Noi University of Science and Technology

*

Email: nga.vuthithuy@hust.edu.vn.