Fractional Order IMC-PID Controller Design for the ...

3rd International Conference on Electrical, Electronics, Engineering Trends, Communication, Optimization and Sciences (EEECOS)-2016

Fractional Order IMC-PID Controller Design for the Pressurized Heavy Water Reactor Sumeet Sagar, Sukhman Kaur1and Swati Sondhi Electrical and Instrumentation Engineering Department, Thapar University, Patiala, India (e-mail: [email protected]) Keywords: Internal Model Control, Fractional Order Systems, Fractional Order PID Controller, PHWR.

Abstract A pressurized heavy water reactor (PHWR) is a nuclear power reactor, that commonly uses unenriched natural uranium as its fuel and heavy water as its coolant and moderator. The PHWR has become an important component of the present day power generation industry. Working of a PHWR may often require reduction of nuclear power within a small finite time interval in accordance to the varying load conditions. This phenomenon is referred to as step back condition. Therefore, for the efficient working of the PHWR it is very important to have a control algorithm that has the capability to handle the step back condition very precisely. The fractional order control strategy has emerged as one of the very versatile and accurate control strategies in the recent times. The fractional order PID (FOPID) controllers are known to give very robust performance both in nominal as well as varying operating conditions. However, most of the design techniques available for the FOPID controllers are mathematically very lengthy and complex. Therefore, in this paper an FOPID controller design technique has been suggested using the concept of internal model control (IMCFOPID) for the PHWR under step back conditions. Further, the performance of the proposed controller has been compared with the conventional FOPID controller using the various integral error criterions. The proposed IMC-FOPID controller is found to give an improved performance as compared to the conventional FOPID controller.

I. INTRODUCTION In the rapidly modernizing world of today, automation and control has become a very vital aspect in the area of industrial technology. The pressurized heavy water reactors (PHWR) are an extremely important part of the modern day power generation industry. PHWR is a nuclear power reactor that uses heavy water as its coolant and moderator. The heavy water coolant is kept under pressure that allows it to be heated to higher temperatures without boiling. Although heavy water is significantly more expensive than the ordinary light water, however, it is still preferred mainly because it yields greatly enhanced neutron economy. This allows the reactor to operate without fuel enrichment facilities and generally enhances the ability of the nuclear reactor to efficiently make use of alternate fuel cycles. When

the PHWR works under varying load conditions or under some abnormal conditions, the bulk power of the nuclear reactor is required to be reduced. This is done mainly with the help of control rods. When the power of the reactor needs to be reduced the control rods are inserted into the reactor up to the required level. This condition is called reactor step back condition. The conventional methods used for the insertion of control rods often give rise to undershoot in the output. However, due to the safety constraints much oscillation is not permissible in the reactor output. Therefore, it is very important to have a robust and precise control strategy that can make the PHWR plant work efficiently under the step back conditions. A lot of researchers have been working in the direction of formulating efficient control strategies for this task. The conventional ProportionalIntegral-Derivative (PID) controllers along with many other popular techniques have been very widely used due to the simplicity of their implementation and design [1]-[9]. In order to design an efficient controller for a system it is essential to have an appropriate mathematical model of that system. According to the recent literature, it has been observed that for the case of highly complicated systems such as gas turbine plants, nuclear reactors, non-linear thermal systems, complex biological systems etc. fractional order modeling gives a closer representation of the plant behavior as compared to the integer order modeling [10]. The dynamic behavior of the system represented by the fractional order mathematical model is found to be much closer to the real life dynamics for such systems. Therefore, a lot of research is being carried out these days in order to make the use of fractional order mathematical models easy and widespread [11]-[19]. The fractional order dynamical model for the PHWR plant for different operating conditions is derived in [9]. In recent times fractional order control systems have gained a significant popularity among the research community. The advancements in the field of fractional calculus paved way for the control engineers to explore the area of fractional order control systems [20]. The various conventional controller design schemes like artificial intelligence, adaptive control, optimal control etc., have been suggested for the design of fractional order controllers for various applications [21]-[29]. However, most of the techniques available in the literature are mathematically very complex and lengthy. But at the same time, these fractional order controllers are known to give improved performance, better disturbance rejection and better stability than the conventional controllers [33]. Another control algorithm that is known to give a very robust performance is the internal model control (IMC). This type of controller is mathematically very easy to design and

72


is robust towards the disturbances [32]. Although, IMC based controllers are very popular for the integer order systems, very limited work has been done for designing the IMC controllers for the fractional order systems[35]-[40]. Therefore, in this paper a simplified IMC based fractional order PID controller is proposed for the PHWR which gives the dual advantage of computational simplicity as well as improved performance. Using this technique, the advantages of both internal model control as well as fractional order control can be embedded into a single controller. Hence, through the proposed technique an attempt is made to improve the performance of the PHWR under step back condition, by utilizing the advantageous properties of the two most desirable and versatile control strategies (i.e. internal model control and fractional order control) simultaneously.

II.

  GM ( s )  GM ( s )G M (s)



 Q ( s )  GM ( s)

where, F ( s) 



1

(3)

F ( s)

(4)

1

(5)

(1   s )m

In (5),  is the tuning parameter, known as IMC filter constant or closed-loop time constant. This parameter is responsible for the speed and robustness of the controller response. While m is a value chosen such that Q( s ) becomes physically realizable.

OBJECTIVE

The main objective of this work is to formulate an Internal Model Control based FOPID controller for a PHWR under step back condition that is mathematically easy to design, gives improved response and better disturbance rejection. The work presented here shows that the IMC based fractional order controller when applied to the fractional order systems gives better performance than the conventional fractional PID controller.

Fig.1. IMC strategy- dotted line indicates calculations performed by the model based controller

III. MATHEMATICAL PRELIMINARIES Internal model control (IMC) is a model based controller design technique in which the process model is embedded into the controller. It is a very simple but highly efficient control technique which is known to give better disturbance rejection [34]. The basic IMC structure can be easily modified to form the conventional feedback control system structure. The basic IMC design strategy is shown in Fig.1. The controller Q( s ) and the process model GM ( s ) are combined into one (shown in Fig.1 using dotted lines) to form the IMC controller in the classical feedback form. The modified structure is shown below in Fig.2 where, C( s) Q( s )  1  GM ( s)C( s)

(1)

Q( s ) 1  GM (s)Q(s)

(2)

C(s) 

IMC design procedure Q(s)  GM 1(s) and G ( s )  G M ( s ) is considered. However, if the controller is considered as the inverse of the plant model, it may give rise to instability or physical unrealizability. In order to overcome this problem, the mathematical model of the plant is factored  into the minimum phase GM ( s ) and non-minimum phase In

 GM

Fig. 2 IMC strategy converted into classical feedback form Table 1. Fractional order models of the PHWR plant at different operating points Identified Reduced Fractional order Models Model 30 P100

30 P90 30 P80

30 P70

1522.8947 s

2.0971

 8.1944 s

1.0036

s

 8.1906 s

1.0036

( s)

components and only the inverse of the minimum phase (stable or invertible part) part is considered. This eliminates the instability from the controller to be designed. Further in order to eliminate any physical unrealizability from the controller a suitable filter F ( s) is attached to the inverse plant model as shown in (3), (4) and (5).

50 P70

 7.7075

1027.3027

1074.396 s 2.0961  8.2663s1.0037  7.8641 s

 7.1111s

1.0002

s

 8.871s

1.0321

 9.0873

 11.2993

337.846 s 2.2038  6.7453s1.0132  7.4275 s

 7.1459 s

1.0113

s

6 e 6.510 s

7 e 1.847910 s

325.2142 2.1969

10

5 e 1.604910 s

604.2541 2.2986

s

5 e 2.534610 s

e 3.143110

529.1365 2.1002

12

9 e 1.596810 s

s 2.0163  6.7859 s 0.99388  6.5268

50 P90 50 P80

 7.7684

1359.2345 2.0972

the

50 P100

e 2.004310

 8.3167

7 e 2.34310 s

73


IV. PROPOSED IMC DESIGN PROCEDURE FOR PHWR For designing a model based controller it is very important to have a mathematical model of the system. In this paper, various combinations of the different initial powers of the PHWR and the rod drop levels are considered. The mathematical models for the PHWR under these different operating conditions are illustrated in Table 1. The subscripts of the variable P denote the initial power level while the superscript denotes the level up to which the rod is 30 indicates that the initial power dropped. For example, P100 of the plant is 100% and the control rod is dropped to a level of 30% of its height. For designing the IMC based fractional order PID controller the highest gain mathematical model is taken into 30 in Table 1. consideration. This condition is denoted by P100 Therefore, in this case, the mathematical model of the system is as given below [9]: GM (s) 

12

1522.8947 s

2.0971

1.0036

 8.1944s

 7.7684

e2.004310

From (2), (11) and (12) the IMC based feedback controller is obtained as given in eq.(13), C (s) 

s 2.0971  8.1944s1.0036  7.7684 1522.8947 s

.

(13)

The above controller can be written in the form: C ( s) 

s 2.0971 

1522.8947 s



8.1944s1.0036 1522.8947 s



7.7684 1522.8947 s

.

(14)

By selecting a suitable value of  , the controller in eq.(14) can be transformed into the conventional FOPID form given in eq.(6). Here   1.0036 and   0.5 is chosen. On substituting these values in eq.(14), the controller takes the form given in eq.(15), C(s)  0.0013s1.0935  0.0108 

0.00512 s0.0102

.

(15)

On comparing the controller obtained in eq.(15) with the conventional FOPID form given in (16) below [41]: .

k CF  s   k p  i  k d s  s

(6)

(16)

Following the conventional IMC approach described in the previous section, the plant model GM ( s) is factorized into invertible and non-invertible parts represented as:

We get, k p  0.0108 , ki  0.0102 ,   1.0036 , kd  0.0013 and

GM (s)  GM  (s)GM  (s) .

Therefore, the final FOIMC-PID controller is obtained as given below in eq.(17):

(7)

Where, GM  (s) and GM  (s) represent the non- invertible and invertible parts of the plant, respectively. Here the invertible part is obtained as given in eq.(8), GM  (s) 

1522.8947 s2.0971  8.1944s1.0036  7.7684

.

(8)

The IMC controller Q(s) is defined as: 1 Q(s)  GM  (s) F (s) ,

(9)

1 where GM  (s) is the inverse of the invertible part of the

plant model and F(s) is a suitable filter. In this case the filter is chosen of the form as given in eq.(10) [40], F (s) 

1

 s  1

,

(10)

where  is the filter constant and  is any real number that can be arbitrarily chosen. Therefore from (8), (9) and (10), 1 Q(s)  GM  (s) F (s) 

s 2.0971  8.1944s1.0036  7.7684





1522.8947  s  1

.

(11)

Further, from (6) and (11), GM ( s )Q( s ) 

1 

s 1

.

(12)

  1.0935 .

CFOIMC  PID (s)  0.0108 

0.0102 s1.0036

 0.0013s1.0935 .

(17)

V. PERFORMANCE ANALYSIS And RESULTS In this section, the comparison of the performances of the proposed fractional order IMC-PID (FOIMC-PID) controller and the conventional FOPID controller has been done on the basis of various integral error criterions i.e. Integral Square Error (ISE), Integral Absolute Error (IAE) and Integral Time Absolute Error (ITAE). It is illustrated in Table 2. From the data of Table 2 it is clear that the proposed FOIMC-PID controller gives better results as compared to the FOPID suggested in [9]. The response of the FOPID and FOIMCPID controllers for unity step back under various operating conditions is illustrated in Fig.3. It can be observed from Fig.3 that when a negative step input of unity magnitude is supplied to the system, the FOIMCPID controller designed using the proposed approach gives a faster set point tracking response than the FOPID controllers available in the literature [9]. This indicates that the proposed controller possesses excellent capability to work efficiently even under the varying operating environment. Further, it may be noted that the IMC-FOPID controller proposed here is designed by taking into consideration the 30 highest gain model i.e. P100 and then subjected to variable

74


operating conditions. From the responses illustrated in Fig. 3, it can be seen that the proposed controller gives improved response for all the different operating conditions. Hence it can be concluded that the proposed controller is robust enough to perform well under all possible variable operating conditions.

VI. CONCLUSION In this paper, an IMC based fractional order PID controller is designed for the pressurized heavy water reactor plant. In this work, an attempt is made to improve the performance of the PHWR plant under step back condition for different operating conditions by combining the advantageous properties of internal model control and fractional order control strategies. This is useful because the proposed technique is mathematically very simple and gives better performance than the conventional FOPID controller given in the literature. The design simplicity of the proposed technique can prove to be very beneficial for engineering practitioners working with real time systems. The implementation and testing of the proposed controller in real time can be considered as the future scope of this work.

Thus, the method proposed in this paper helps in formulating an efficient fractional order IMC based PID controller for the fractional order models of the PHWR. It is seen that the proposed controller performs better than the conventional fractional order PID controller available in the literature [9]. Hence, the objective of formulating a fractional order IMCPID controller which can give a better response and is more robust in comparison to the existing controllers is fulfilled successfully. 0.2 P30-100 with FOPID P30-100 with FOIMCPID P30-90 with FOPID P30-90 with FOIMCPID P30-80 with FOPID P30-80 with FOIMCPID P50-100 with FOPID P50-100 with FOIMCPID P50-90 with FOPID P50-90 with FOIMCPID P50-80 with FOPID P50-80 with FOIMCPID P50-70 with FOPID P50-70 with FOIMCPID P30-70 with FOPID P30-70 with FOIMCPID

0

Amplitude

-0.2 -0.4 -0.6 -0.8 -1 -1.2 0

2

4

6

8

10 Time

12

14

16

18

20

Fig.3. Unit step back response of FOPID and FOIMCPID controllers for the PHWR plant for various operating conditions.

Table 2. Performance indices for the FOPID and FOIMCPID controllers. Model

Controller

ISE

IAE

ITAE

30 P100

FOPID [9]

0.5254

1.044

2.439

Proposed FOIMCPID

0.2723

0.4963

0.7563

P9030

FOPID [9]

0.5826

1.153

2.786

Proposed FOIMCPID

0.3008

0.5518

0.8672

P8030

FOPID [9]

0.6489

1.282

3.231

Proposed FOIMCPID

0.3336

0.6203

1.029

P7030

FOPID [9]

0.739

1.456

3.844

Proposed FOIMCPID

0.3764

0.7115

1.235

FOPID [9]

1.583

3.185

13.01

Proposed FOIMCPID

0.7655

1.659

4.788

FOPID [9]

1.711

3.449

14.99

Proposed FOIMCPID

0.7655

1.659

4.788

FOPID [9]

2.08

4.049

19.29

Proposed FOIMCPID

1.031

2.12

6.845

50 P100

P9050 P8050

75


P7050

FOPID [9]

2.397

4.673

24.86

Proposed FOIMCPID

1.182

2.465

8.894

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

R. N. Banavar, and U. V. Deshpande, “Robust Controller Design for a Nuclear Power Plant Using Optimization”, IEEE Trans. Nucl. Sci., vol. 45, no. 2, april 1998.pp129-140. A . P. Tiwari and B. Bandyopadhyay, “An Approach to the Design of Fault-Tolerant Spatial Control System for a Large PHWR”, Proc. IEEE International Conference on Industrial Technology, Jan. 2000, 747 – 752, vol.2, R. Kumar, A. J. Gaikwad, S. F. Vhora, G. Chakraborty, and V. Venkat Raj, “Logic Modification to Avoid ECCS Lineup in the Primary Heat Transport System During Thermal Shrinkage Transients of 220 MWe PHWR Nuclear Power Plant”, IEEE Trans. Nucl. Sci., vol. 50, no. 4, 2003, pp.1229-1237. G. L. Sharma, B. Bandyopadhyay, and A. P. Tiwari, “Spatial Control of a Large Pressurized Heavy Water Reactor by Fast Output Sampling Technique”, IEEE Trans. Nucl. Sci., vol. 50, no. 5, 2003, pp.1740-1751. B. Talange, B. Bandyopadhyay, and A. P. Tiwari, “Spatial Control of a Large PHWR by Decentralized Periodic Output Feedback and Model Reduction Techniques”, IEEE Trans. Nucl. Sci., vol. 53, no. 4, 2006, pp. 2308-2317. D. Reddy, B. Bandyopadhyay, and A. P. Tiwari, “Multirate Output Feedback Based Sliding Mode Spatial Control for a Large PHWR”, IEEE Trans. Nucl. Sci., vol. 54, no. 6, 2007, pp.2677-2686. S. Saha, S. Das, R. Ghosh, B. Goswami, R. Balasubramanian, A. K. Chandra, S. Das, and A. Gupta, “Design of a Fractional Order Phase Shaper for Iso-Damped Control of a PHWR Under Step-Back Condition”, IEEE Trans. Nucl. Sci., vol. 57, no. 3, 2010, pp.1602-1612. S. Banerjee, K. Halder, S. Dasgupta, S. Mukhopadhyay, K. Ghosh, and A. Gupta, “An Interval Approach for Robust Control of a Large PHWR with PID Controllers”, IEEE Trans. Nucl. Sci., vol. 62, no. 1, 2015, pp.281-292. S. Das, A. Gupta, "Fractional Order Modeling of a PHWR Under Step-Back Condition and Control of Its Global Power With a Robust PI  D  Controller," IEEE Trans. Nucl. Sci., vol. 58, no.5, pp.2431-2441, 2011 A. Beaulieu, et.al., "Measurement of Fractional Order Model Parameters of Respiratory Mechanical Impedance in Total Liquid Ventilation," IEEE Trans. Biomed. Eng., vol. 59, no.2, pp.323-331, 2012. S.Victor, et.al., "Parameter and differentiation order estimation in fractional models", Automatica, vol.49, pp.926–935, 2013. J.D. Gabano, T. Poinot, H.Kanoun, "Identification of a thermal system using continuous linear parameter-

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20] [21]

[22]

[23]

[24]

[25]

[26]

varying fractional modelling," IET Control Theory Appl., vol. 5, no.7, pp. 889–899, 2011. R.K.H. Galvão, et.al., "Fractional Order Modeling of Large Three-Dimensional RC Networks," IEEE Trans. Circuits Syst.1, Reg. Papers, vol.60, no.3, pp.624-637, 2013. A.G.Radwan, "Resonance and Quality Factor of the RL C Fractional Circuit," IEEE J. Emerg. Sel. Topic Circuits Syst, vol.3, no.3, pp.377-385, 2013 C.C. Hua, D.Liu, X. P. Guan, "Necessary and Sufficient Stability Criteria for a Class of FractionalOrder Delayed Systems," IEEE Trans. Circuits Syst.II Exp. Briefs, vol. 61, no. 1, pp.59-63, 2014. H. Nasiri, M. Haeri, "How BIBO stability of LTI fractional-order time delayed systems relates to their approximated integer-order counterparts," IET Control Theory Appl., vol. 8, no.8, pp. 598–605, 2014. D. Idiou, A. Charef, A. Djouambi, "Linear fractional order system identification using adjustable fractional order differentiator," IET Signal Process., vol. 8, no. 4, pp. 398–409, 2014. D. G. MEYER "Fractional Balanced Reduction: Model Reduction Via Fractional Representation," IEEE Trans. Automat. Control., vol 35, no.12, pp. 1341-1345, 1990. K. A. Moornani and M. Haeri, "Necessary and Sufficient Conditions for BIBO-Stability of Some Fractional Delay Systems of Neutral Type, " IEEE Trans. Automat. Control, vol. 56, no. 1, pp.125-128, 2011. S. Das, Functoinal Fractional Calculus, 2nd Ed., Springer Berlin Heidelburg, 2011. V. Badri, M. S. Tavazoei, "On tuning fractional order [proportional–derivative] controllers for a class of fractional order systems," Automatica vol.49, pp.2297– 2301, 2013. N. A. Camacho, M. A.D. Mermoud, "Fractional adaptive control for an automatic voltage regulator," ISA Trans. vol.52, pp.807–815, 2013. T. N. L.Vu, M. Lee, "Analytical design of fractionalorder proportional-integral controllers for time-delay processes," ISA Trans. vol.52, pp.583–591, 2013. S.E. Hamamci, "An Algorithm for Stabilization of Fractional-Order Time Delay Systems Using Fractional-Order PID Controllers," IEEE Trans. Automat. Control, vol. 52, no.10, pp.1964-1969, 2007. K. Erenturk, "Fractional-Order PIλDμ and Active Disturbance Rejection Control of Nonlinear Two-Mass Drive System," IEEE Trans. Indus. Electr., vol. 60, no. 9, 3806-3813, 2013. A. Dumlu, K. Erenturk, "Trajectory Tracking Control for a 3-DOF Parallel Manipulator Using FractionalOrder PIλDμ Control," IEEE Trans. Indus. Electr., vol. 61, no. 7, pp.3417-3426, 2014.

76


[27]

[28]

[29]

[30]

[31]

[32]

[33]

Y. Luo, et.al., "Fractional-Order Proportional Derivative Controller Synthesis and Implementation for Hard-Disk-Drive Servo System," IEEE Trans. Control. Syst. Technol., vol. 22, no. 1, pp.281-289, 2014. S.Sondhi, Y.V.Hote, "Fractional order PID controller for load frequency control," Energy Convers. Managmt., vol.85, pp.343-353, 2014. S.Sondhi, Y.V.Hote, “Fractional IMC design for fractional order gas turbine model", 9th IEEE Int. Conf. Industrial Information Systems, (ICIIS) 2014. Gwalior, S. Sondhi and Y. V. Hote, "Stability testing and IMC based fractional order PID controller design for heating furnace system", 11th Annual IEEE India Conference (INDICON) 2014, Pune S. Sondhi and Y. V. Hote, "Fractional Order PI Controller with Specific Gain-Phase Margin for MABP Control", IETE Journal of Research, Taylor & Francis. vol. 61, issue 2, pp. 142-153. 2015 M. S.Tavazoei, M.T.Kakhki, "Compensation by fractional-order phase-lead/lag compensators," IET Control Theory Appl., vol. 8, no. 5, pp. 319–329, 2014. S. Sondhi, Y. V. Hote, "Fractional order controller and its applications: a review," in Proc. of AsiaMIC, Phuket, Thailand, 2012.

[34] [35]

[36]

[37]

[38]

[39]

[40]

[41]

B.W.Bequette, Process Control Modeling Design and Simulation, Prentice Hall, New Jersy, 2003. B.Maâmar, M. Rachid, "IMC-PID-fractional-orderfilter controllers design for integer order systems," ISA Trans., 2014. M. Bettayeba, R. Mansouric, "Fractional IMC-PIDfilter controllers design for non integer order systems," J. Process Control, vol. 24, pp.261–271, 2014. T. Vinopraba, et.al., "Design of internal model control based fractional order PID controller," J Control Theory Appl., vol.10, no.3, pp. 297–302, 2012. M. T. Kakhki, M. Haeri, "Fractional order model reduction approach based on retention of the dominant dynamics: Application in IMC based tuning of FOPI and FOPID controllers," ISA Trans., vol.50, pp. 432– 442, 2011. L. Dazi, et.al., "An IMC- PIλDμ Controller Design for Fractional Calculus System," Proc. 29th Chinese Control Conf., China, 2010. T.Vinopraba, N.Sivakumaran, S.Narayanan, "IMC Based Fractional order PID Controller," Proc. IEEE Int. Conf. on Industrial Technology, 2011. I.Petras, Fractional Order Nonlinear Systems Modeling, Analysis and Simulation, Springer, Higher Education Press, Beijing, 2011.

77