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D04954148

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This document compares the performance of PID and FOPID controllers for automatic voltage regulation (AVR) systems. It presents models for the components of an AVR system and describes integer and fractional order PID controllers. Classical tuning methods like Ziegler-Nichols and Cohen-Coon are discussed for PID tuning. Ziegler-Nichols type rules are presented for FOPID tuning. Simulation results show that a FOPID controller tuned by Ziegler-Nichols methods provides better performance than PID controllers tuned by Ziegler-Nichols or Cohen-Coon, with less overshoot, faster settling time, and reduced rise time.

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IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 04, Issue 09 (September. 2014), ||V5|| PP 41-48 International organization of Scientific Research 41 | P a g e Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation 1Tushar Verma, 2Akhilesh Kumar Mishra 1M.Tech. Scholar, Department of Electrical Engineering, United College of Engineering and Research, Allahabad, Uttar Pradesh, INDIA. 2 Assistant Professor, Electrical Department, United College of Engineering and Research, Allahabad, Uttar Pradesh, INDIA. Abstract: - The aim of this work is to design an automatic voltage regulation system by selection of FOPID parameters. In this work the advantage of FOPID controller over Conventional PID controller is discussed using MATLAB/Simulink. PID (proportional-integral-derivative) control is one of the earlier control strategies. Its early implementation was in pneumatic devices, followed by vacuum and solid state analog electronics, before arriving at today‟s digital implementation of microprocessors. But in the last decade, fractional-order dynamic systems and controllers has been studying widely in many areas of engineering and science. The concept of the fractional-order PID controllers was proposed by Podlubny in 1997[1]. He also demonstrated the better response of this type of controllers, in comparison with the classical PID controllers, when used for the control of fractional-order systems. A method is presented based on idea of the Ziegler–Nichols and Cohen Coon for the tuning of PID controller. Motivated from the fact that the optimization techniques depend on initial estimates, Valerio and Costa have introduced some Ziegler-Nichols-type tuning rules for FOPIDs. Keywords: FOPID controller, PID Controller, Ziegler Nichols and Cohen – Coon tuning. I. INTRODUCTION The Automatic Voltage Regulator (AVR) is widely used in industrial application to obtain the stability and good regulation of different electrical apparatus. A voltage regulator is an electrical regulator designed to automatically maintain a constant voltage level. It may use an electromechanical mechanism, or passive or active electronic components. Depending on the design, it may be used to regulate one or more AC or DC voltages. If the output voltage is too low, the regulation element is commanded up to a point to produce a higher output voltage by dropping less of the input voltage; if the output voltage is too high, the regulation element will normally be commanded to produce a lower voltage [2]. Proportional-Integral-Derivative controllers are widely being used in industries for process control applications. The merit of using PID controllers lie in its simplicity of design and good performance including low percentage overshoot and small settling time for slow industrial processes. The performance of PID controllers can be further improved by appropriate settings of fractional-I and fractional-D actions. This paper attempts to study the behaviour of fractional PID controllers over integer order PID controllers. In a fractional PID controller, the I- and D-actions being fractional have wider scope of design. Naturally, besides setting the proportional, derivative and integral constants Kp, Td and Ti respectively, we have two more parameters: the power of „s‟ in integral and derivative actions- λ and μ respectively. Finding [Kp, Td, Ti, λ, μ] as an optimal solution to a given process thus calls for optimization on the five-dimensional space. The performance of the optimal fractional PID controller is better than its integer counterpart. Thus the proposed design will find extensive applications in real industrial processes. This paper shows the comparative study of PID and FOPID controller response for automatic voltage regulation (AVR) system. The best parameters of for the response of Fractional-Order PID controller consist of proportional gain Kp, integral gain Ki, fractional-order of integrator λ, derivative gain kd and fractional-order of differentiator μ can be determinate for AVR system so that the controlled AVR system has a better control performance than other methods. II. LINEARIZED MODEL OF AN AVR SYSTEM The role of an AVR is to hold the terminal voltage magnitude of a synchronous generator at a specified level. A simple AVR system comprises four main components, namely amplifier, exciter, generator, and sensor [3].
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 42 | P a g e Fig.1 Linearized model of an AVR system Component Transfer Function Parameter Range Amplifier Ka/(1+ Ʈa s) 10 < Ka < 400, 0.02 < Ʈa < 0.1s Exciter Ke/(1+ Ʈe s) 1 < Ke < 400, 0.4 < Ʈe < 1s Generator Kg/(1+ Ʈg s) 0.7 < Kg < 1,1 < Ʈg < 2s from full load to no load Sensor Ks/(1+ Ʈs s) 0.001 < Ʈs < 0.06s Table.1 Parameters value for AVR (a) Amplifier Model: The transfer function of amplifier model is where KA is a gain and ƮA is a time constant. (b) Exciter Model: The transfer function of exciter model is where KE is a gain and ƮE is a time constant. (c) Generator Model: The transfer function of generator model is where KG is a gain and ƮG is a time constant. (d) Sensor Model: The transfer function of sensor model is where KR is a gain and ƮR is a time constant. III. THE INTEGER AND FRACTIONAL ORDER PID CONTROLLERS The PID controller is the most common general purpose controller in the today‟s industries. It can be used as a single unit or it can be a part of a distributed computer control system. After implementing the PID controller, now we have to tune the controller; and there are different approaches to tune the PID parameters like P, I and D. The Proportional (P) part is responsible for following the desired set-point while the Integral (I) and Derivative (D) part account for the accumulation of past errors and the rate of change of error in the process or plant, respectively. PID controller consists of three types of control i.e. Proportional, Integral and Derivative control
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 43 | P a g e Fig.2 Schematic of PID controller The system transfer function in continuous s-domain are given as For P = KP, I = KI/s and D = Kds Gc(s) = P + I + D = Kp + Ki / s + Kd s Gc(s) = Kp { 1+ 1/Ti s + Td s } Where p K is the proportional gain, Ki is the integration coefficient and Kd is the derivative coefficient. Ti is known as the integral action time or reset time and Td is the derivative action time or rate time. In the last decade, fractional-order dynamic systems and controllers has been studying widely in many areas of engineering and science [4-8]. The concept of the fractional-order PID controllers was proposed by Podlubny in 1997. He also demonstrated the better response of this type of controllers, in comparison with the classical PID controllers, when used for the control of fractional order systems. Fractional order control systems are described by fractional order differential equations. The FOPID controller is the expansion of the conventional PID controller based on fractional calculus. A fractional PID controller therefore has the transfer function: Gc(s) = Kp + Tis-λ+ Tdsδ The orders of integration and differentiation are respectively λ and δ (both positive real numbers, not necessarily integers). Taking λ=1 and δ=1, we will have an integer order PID controller. So we see that the integer order PID controller has three parameters, while the fractional order PID controller has five. IV. CLASSICAL TUNING METHODS FOR CONTROLLER There are various tuning strategies based on an open-loop step response. While they all follow the same basic idea, they differ in slightly in how they extract the model parameters from the recorded response, and also differ slightly as to relate appropriate tuning constants to the model parameters. Naturally if the response is not sigmoidal or „S‟ shaped and exhibits overshoot, or an integrator, then this tuning method is not applicable. (a) PID TUNING Ziegler-Nichols Tuning Method The PID tuning parameters as a function of the open loop model parameters K, T and  from the Process reaction curve derived by Ziegler-Nichols [9]. They often form the basis for tuning procedures used by controller manufacturers and process industry. The methods are based on determination of some features of process dynamics. The controller parameters are then expressed in terms of the features by simple formulas. The method presented by Ziegler and Nichols is based on a registration of the open-loop step response of the system, which is characterized by two parameters. First determined, and the tangent at this point is drawn. The intersections between the tangent and the coordinate axes give the parameters T and. A model of the process to be controlled was derived from these parameters. This corresponds to modeling a process by an integrator and a time delay. Ziegler and Nichols have given PID parameters directly as functions of T and. The behavior of the controller is as can be expected. The decay ratio for the step response is close to one quarter. It is smaller for the load disturbance. The overshoot in the set point response is too large.
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 44 | P a g e Controller Kp Ti Td Ziegler- Nichols Method (Open Loop) P T/Kθ - - PI 0.9T/Kθ θ/0.3 - PID 1.2T/Kθ 2θ 0.5θ Table.2 Ziegler Nichols open loop method Cohen-Coon Tuning Method Cohen and Coon based the controller settings on the three parameters, T and K of the open loop step response. The main design criterion is rejection of load disturbances. The method attempts to position closed loop poles such that a quarter decay ration is achieved. Controller Kp Ti Td Cohen- Coon Method (Open Loop) P T/Kϴ.(1 + ϴ/3T) PI T/Kϴ.(0.9 + ϴ/12T) ϴ.{(30+3ϴ/T)/(9+20ϴ/T)} PID T/Kϴ.(4/3 + ϴ/4T) ϴ.{(32+6ϴ/T)/(13+8ϴ/T)} ϴ.{4/(11+2ϴ/T)} Table.3 Cohen Coon open loop method (b) FOPID TUNING Fractional-order calculus is an area of mathematics that deals with derivatives and integrals from non- integer orders. In other words, it is a generalization of the traditional calculus that leads to similar concepts and tools, but with a much wider applicability. The numerical simulation of a fractional differential equation is not simple as that of an ordinary differential equation. Since most of the fractional-order differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Ziegler-Nichols type tuning rules In practice, most solutions found with this optimization method are good enough, but they strongly depend on initial estimates of the parameters provided. Some may be discarded, because they are unfeasible or lead to unstable loops, but in many cases it is possible to find more than one acceptable FOPID [10]. In others, only well-chosen initial estimates of the parameters allow finding a solution. Valerio and Costa have introduced some Ziegler-Nichols-type tuning rules for FOPIDs. These tuning rules are applicable only for systems that have S-shaped step response. (a) First Set of Tuning Rules A first set of rules is given in Tables 4 and 5. These are to be read as P = -0.0048 + 0.2664L + 0.4982T + 0.0232L2 - 0.0720T2 - 0.0348TL and so on. They may be used if 0.1 < T < 50, L < 2 Kp Ki λ Kd μ 1 -0.0048 0.3254 1.5766 0.0662 0.8736 L 0.2664 0.2478 -0.2098 -0.2528 0.2746 T 0.4982 0.1429 -0.1313 0.1081 0.1489 L2 0.0232 -0.133 0.0713 0.0702 -.1557 T2 -0.072 0.0258 0.0016 0.0328 -0.025 LT -0.0348 -0.0171 0.0114 0.2202 -.0323 Table.4 Parameters for the first set of tuning rules when 0.1< T<5
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 45 | P a g e Kp Ki λ Kd μ 1 2.1187 -0.5201 1.0645 1.1421 1.2902 L -3.5207 2.6643 -0.3268 -1.3707 -.5371 T -0.1563 0.3453 -0.0229 0.0357 -.0381 L2 1.5827 -1.0944 0.2018 0.5552 0.2208 T2 0.0025 0.0002 0.0003 -0.0002 0.0007 LT 0.1824 -0.1054 0.0028 0.263 -.0014 Table.5 Parameters for the first set of tuning rules when 5 < T < 50 (b) Second set of tuning rules A second set of rules is given in Table 6. These may be applied for 0.1 < T < 50 and L < 0.5 . Kp Ki λ Kd μ 1 -1.0574 0.6014 1.1851 0.8793 0.2778 L 24.542 0.4025 -0.3464 -15.084 -2.1522 T 0.3544 0.7921 -0.0492 -0.0771 0.0675 L2 -46.732 -0.4508 1.7317 28.0388 2.4387 T2 -0.0021 0.0018 0.0006 0 -0.0013 LT -0.3106 -1.205 0.038 1.6711 0.0021 Table.6 Parameter for the second set of tuning rules for 0.1 < T < 50 and L < 0.5 V. SIMULINK MODEL OF AVR The Simulink model of AVR is shown in Fig 3. This model consist the mathematical function of amplifier, exciter and generator. Fig.3 Simulink model of AVR The Simulink model of various tuning method for AVR using PID tuned by Ziegler Nichols and Cohen Coon and FOPID controller tuned by Ziegler Nichols is shown in Fig 4.
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 46 | P a g e Fig.4 Simulink model of various tuning methods VI. RESULT AND DISCUSSION The Simulink model in Fig. 3 & 4 was simulated and the plots for various tuning method were observed. Fig 6 shows the Step response of AVR using PID tuned by Ziegler Nichols, Fig 7 shows the step response of AVR using PID tuned by Cohen Coon. and Fig 7 shows the Step response of AVR using FOPID controller. Fig.5 Response of PID tuned with Zeigler Nichols & Cohen Coon and FOPID tuned with Zeigler Nichols for AVR Fig.6 Step response of AVR using PID tuned by Ziegler Nichols
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 47 | P a g e Fig.7 Step response of AVR using PID tuned by Cohen Coon Fig.8 Step response of AVR using FOPID It can be seen from the comparison table that while using the FOPID controller tuned by Ziegler Nichols the overshoots obtained is less as compared to the case when the PID Controller was tuned via conventional methods i.e. Ziegler Nichols and Cohen Coon method. The settling time is also lesser in case of the FOPID controller, also the rise time is reduced. The FOPID controller tuned by Z-N tuning rule tends to faster response of AVR. It can be observed from Fig 6 and 7 that the PID controller tuned by Ziegler Nichols and Cohen Coon have larger overshoot than FOPID controller attain a steady state with larger settling time. PID FOPID Parameters Ziegler Nichols Cohen Coon Ziegler Nichols Kp 2.7719 3.3221 0.5543 Ki 3.009 3.4323 0.5841 Kd 0.6384 0.5159 0.2247 λ - - -1.0814 μ - - 1.3628 Tr(sec) 0.149 0.139 0.471 Ts(sec) 5.01 5.55 4.15 Mp( %) 71 75.6 40.5 Table.7 Comparative values of different parameters obtained by PID and FOPID controllers VII. CONCLUSION Performance comparisons of FOPID and PID controllers have been reviewed and it is found that response of FOPID controller is better than PID controller. The settling time and rise time of the response acquired by FOPID controller is found to be less as comparison to response acquired by PID controller.
Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 48 | P a g e VIII. ACKNOWLEDGMENT This work was supported by Mr. Akhilesh Kumar Mishra. Special Thanks to all faculty members of Electrical Engineering Department of United College of Engineering & Research, Allahabad India, Specially Head of Department Mr. Abdul Zeeshan for their co-operation. REFERENCES [1] Podlubny, I.: Fractional-order systems and PI_D_ controllers, IEEE Trans. On Automatic Control, vol.44, no.1, pp. 208 213, 1999. [2] M. Rabiul Alam, Rajib Baran Roy, S.M. Jahangir Alam, Dewan Juel Rahman International Journal of Electrical & Computer Sciences IJECS-IJENS Vol: 11 No: 05 [3] H. Yoshida, K. Kawata, and Y. Fukuyama, “A particle swarm optimization for reactive power and voltage control considering voltage security assessment,” IEEE Trans. Power Syst., vol. 15, pp. 1232– 1239, Nov. 2000. [4] K. B. Oldhamand and J. Spanier, “The Fractional Calculus”, Academic Press, New York, (1974). [5] C. H. Lubich, “Discretized fractional calculus”, SIAM Journal on Mathematical Analysis, (1986), pp. 704-719. [6] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley, New York, (1993). [7] A. Oustaloup, “Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulators”, IEEE Transactions on Circuits and Systems, (1981), pp.1007–1009. [8] M. Chengbin and Y. Hori, “The application of fractional order PID controller for robust two-inertia speed control”, Proceedings of the 4th International Power Electronics and Motion Control Conference, Xi‟an, (2004) August. [9] F. Merrikh-Bayat, M. Karimi-Ghartemani, “Method for designing FOPID stabilisers for minimum-phase fractional-order systems”, IET Control Theory Appl., vol. 4, Issue 1, (2010), pp. 61–70. [10] Padula, F., Visioli, A., (2010a). Tuning rules for optimal PID and fractional-order PID controllers, Journal of Process Control, doi:10.1016/j.jprocont.2010.10.006

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D04954148

  • 1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 04, Issue 09 (September. 2014), ||V5|| PP 41-48 International organization of Scientific Research 41 | P a g e Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation 1Tushar Verma, 2Akhilesh Kumar Mishra 1M.Tech. Scholar, Department of Electrical Engineering, United College of Engineering and Research, Allahabad, Uttar Pradesh, INDIA. 2 Assistant Professor, Electrical Department, United College of Engineering and Research, Allahabad, Uttar Pradesh, INDIA. Abstract: - The aim of this work is to design an automatic voltage regulation system by selection of FOPID parameters. In this work the advantage of FOPID controller over Conventional PID controller is discussed using MATLAB/Simulink. PID (proportional-integral-derivative) control is one of the earlier control strategies. Its early implementation was in pneumatic devices, followed by vacuum and solid state analog electronics, before arriving at today‟s digital implementation of microprocessors. But in the last decade, fractional-order dynamic systems and controllers has been studying widely in many areas of engineering and science. The concept of the fractional-order PID controllers was proposed by Podlubny in 1997[1]. He also demonstrated the better response of this type of controllers, in comparison with the classical PID controllers, when used for the control of fractional-order systems. A method is presented based on idea of the Ziegler–Nichols and Cohen Coon for the tuning of PID controller. Motivated from the fact that the optimization techniques depend on initial estimates, Valerio and Costa have introduced some Ziegler-Nichols-type tuning rules for FOPIDs. Keywords: FOPID controller, PID Controller, Ziegler Nichols and Cohen – Coon tuning. I. INTRODUCTION The Automatic Voltage Regulator (AVR) is widely used in industrial application to obtain the stability and good regulation of different electrical apparatus. A voltage regulator is an electrical regulator designed to automatically maintain a constant voltage level. It may use an electromechanical mechanism, or passive or active electronic components. Depending on the design, it may be used to regulate one or more AC or DC voltages. If the output voltage is too low, the regulation element is commanded up to a point to produce a higher output voltage by dropping less of the input voltage; if the output voltage is too high, the regulation element will normally be commanded to produce a lower voltage [2]. Proportional-Integral-Derivative controllers are widely being used in industries for process control applications. The merit of using PID controllers lie in its simplicity of design and good performance including low percentage overshoot and small settling time for slow industrial processes. The performance of PID controllers can be further improved by appropriate settings of fractional-I and fractional-D actions. This paper attempts to study the behaviour of fractional PID controllers over integer order PID controllers. In a fractional PID controller, the I- and D-actions being fractional have wider scope of design. Naturally, besides setting the proportional, derivative and integral constants Kp, Td and Ti respectively, we have two more parameters: the power of „s‟ in integral and derivative actions- λ and μ respectively. Finding [Kp, Td, Ti, λ, μ] as an optimal solution to a given process thus calls for optimization on the five-dimensional space. The performance of the optimal fractional PID controller is better than its integer counterpart. Thus the proposed design will find extensive applications in real industrial processes. This paper shows the comparative study of PID and FOPID controller response for automatic voltage regulation (AVR) system. The best parameters of for the response of Fractional-Order PID controller consist of proportional gain Kp, integral gain Ki, fractional-order of integrator λ, derivative gain kd and fractional-order of differentiator μ can be determinate for AVR system so that the controlled AVR system has a better control performance than other methods. II. LINEARIZED MODEL OF AN AVR SYSTEM The role of an AVR is to hold the terminal voltage magnitude of a synchronous generator at a specified level. A simple AVR system comprises four main components, namely amplifier, exciter, generator, and sensor [3].
  • 2. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 42 | P a g e Fig.1 Linearized model of an AVR system Component Transfer Function Parameter Range Amplifier Ka/(1+ Ʈa s) 10 < Ka < 400, 0.02 < Ʈa < 0.1s Exciter Ke/(1+ Ʈe s) 1 < Ke < 400, 0.4 < Ʈe < 1s Generator Kg/(1+ Ʈg s) 0.7 < Kg < 1,1 < Ʈg < 2s from full load to no load Sensor Ks/(1+ Ʈs s) 0.001 < Ʈs < 0.06s Table.1 Parameters value for AVR (a) Amplifier Model: The transfer function of amplifier model is where KA is a gain and ƮA is a time constant. (b) Exciter Model: The transfer function of exciter model is where KE is a gain and ƮE is a time constant. (c) Generator Model: The transfer function of generator model is where KG is a gain and ƮG is a time constant. (d) Sensor Model: The transfer function of sensor model is where KR is a gain and ƮR is a time constant. III. THE INTEGER AND FRACTIONAL ORDER PID CONTROLLERS The PID controller is the most common general purpose controller in the today‟s industries. It can be used as a single unit or it can be a part of a distributed computer control system. After implementing the PID controller, now we have to tune the controller; and there are different approaches to tune the PID parameters like P, I and D. The Proportional (P) part is responsible for following the desired set-point while the Integral (I) and Derivative (D) part account for the accumulation of past errors and the rate of change of error in the process or plant, respectively. PID controller consists of three types of control i.e. Proportional, Integral and Derivative control
  • 3. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 43 | P a g e Fig.2 Schematic of PID controller The system transfer function in continuous s-domain are given as For P = KP, I = KI/s and D = Kds Gc(s) = P + I + D = Kp + Ki / s + Kd s Gc(s) = Kp { 1+ 1/Ti s + Td s } Where p K is the proportional gain, Ki is the integration coefficient and Kd is the derivative coefficient. Ti is known as the integral action time or reset time and Td is the derivative action time or rate time. In the last decade, fractional-order dynamic systems and controllers has been studying widely in many areas of engineering and science [4-8]. The concept of the fractional-order PID controllers was proposed by Podlubny in 1997. He also demonstrated the better response of this type of controllers, in comparison with the classical PID controllers, when used for the control of fractional order systems. Fractional order control systems are described by fractional order differential equations. The FOPID controller is the expansion of the conventional PID controller based on fractional calculus. A fractional PID controller therefore has the transfer function: Gc(s) = Kp + Tis-λ+ Tdsδ The orders of integration and differentiation are respectively λ and δ (both positive real numbers, not necessarily integers). Taking λ=1 and δ=1, we will have an integer order PID controller. So we see that the integer order PID controller has three parameters, while the fractional order PID controller has five. IV. CLASSICAL TUNING METHODS FOR CONTROLLER There are various tuning strategies based on an open-loop step response. While they all follow the same basic idea, they differ in slightly in how they extract the model parameters from the recorded response, and also differ slightly as to relate appropriate tuning constants to the model parameters. Naturally if the response is not sigmoidal or „S‟ shaped and exhibits overshoot, or an integrator, then this tuning method is not applicable. (a) PID TUNING Ziegler-Nichols Tuning Method The PID tuning parameters as a function of the open loop model parameters K, T and  from the Process reaction curve derived by Ziegler-Nichols [9]. They often form the basis for tuning procedures used by controller manufacturers and process industry. The methods are based on determination of some features of process dynamics. The controller parameters are then expressed in terms of the features by simple formulas. The method presented by Ziegler and Nichols is based on a registration of the open-loop step response of the system, which is characterized by two parameters. First determined, and the tangent at this point is drawn. The intersections between the tangent and the coordinate axes give the parameters T and. A model of the process to be controlled was derived from these parameters. This corresponds to modeling a process by an integrator and a time delay. Ziegler and Nichols have given PID parameters directly as functions of T and. The behavior of the controller is as can be expected. The decay ratio for the step response is close to one quarter. It is smaller for the load disturbance. The overshoot in the set point response is too large.
  • 4. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 44 | P a g e Controller Kp Ti Td Ziegler- Nichols Method (Open Loop) P T/Kθ - - PI 0.9T/Kθ θ/0.3 - PID 1.2T/Kθ 2θ 0.5θ Table.2 Ziegler Nichols open loop method Cohen-Coon Tuning Method Cohen and Coon based the controller settings on the three parameters, T and K of the open loop step response. The main design criterion is rejection of load disturbances. The method attempts to position closed loop poles such that a quarter decay ration is achieved. Controller Kp Ti Td Cohen- Coon Method (Open Loop) P T/Kϴ.(1 + ϴ/3T) PI T/Kϴ.(0.9 + ϴ/12T) ϴ.{(30+3ϴ/T)/(9+20ϴ/T)} PID T/Kϴ.(4/3 + ϴ/4T) ϴ.{(32+6ϴ/T)/(13+8ϴ/T)} ϴ.{4/(11+2ϴ/T)} Table.3 Cohen Coon open loop method (b) FOPID TUNING Fractional-order calculus is an area of mathematics that deals with derivatives and integrals from non- integer orders. In other words, it is a generalization of the traditional calculus that leads to similar concepts and tools, but with a much wider applicability. The numerical simulation of a fractional differential equation is not simple as that of an ordinary differential equation. Since most of the fractional-order differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Ziegler-Nichols type tuning rules In practice, most solutions found with this optimization method are good enough, but they strongly depend on initial estimates of the parameters provided. Some may be discarded, because they are unfeasible or lead to unstable loops, but in many cases it is possible to find more than one acceptable FOPID [10]. In others, only well-chosen initial estimates of the parameters allow finding a solution. Valerio and Costa have introduced some Ziegler-Nichols-type tuning rules for FOPIDs. These tuning rules are applicable only for systems that have S-shaped step response. (a) First Set of Tuning Rules A first set of rules is given in Tables 4 and 5. These are to be read as P = -0.0048 + 0.2664L + 0.4982T + 0.0232L2 - 0.0720T2 - 0.0348TL and so on. They may be used if 0.1 < T < 50, L < 2 Kp Ki λ Kd μ 1 -0.0048 0.3254 1.5766 0.0662 0.8736 L 0.2664 0.2478 -0.2098 -0.2528 0.2746 T 0.4982 0.1429 -0.1313 0.1081 0.1489 L2 0.0232 -0.133 0.0713 0.0702 -.1557 T2 -0.072 0.0258 0.0016 0.0328 -0.025 LT -0.0348 -0.0171 0.0114 0.2202 -.0323 Table.4 Parameters for the first set of tuning rules when 0.1< T<5
  • 5. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 45 | P a g e Kp Ki λ Kd μ 1 2.1187 -0.5201 1.0645 1.1421 1.2902 L -3.5207 2.6643 -0.3268 -1.3707 -.5371 T -0.1563 0.3453 -0.0229 0.0357 -.0381 L2 1.5827 -1.0944 0.2018 0.5552 0.2208 T2 0.0025 0.0002 0.0003 -0.0002 0.0007 LT 0.1824 -0.1054 0.0028 0.263 -.0014 Table.5 Parameters for the first set of tuning rules when 5 < T < 50 (b) Second set of tuning rules A second set of rules is given in Table 6. These may be applied for 0.1 < T < 50 and L < 0.5 . Kp Ki λ Kd μ 1 -1.0574 0.6014 1.1851 0.8793 0.2778 L 24.542 0.4025 -0.3464 -15.084 -2.1522 T 0.3544 0.7921 -0.0492 -0.0771 0.0675 L2 -46.732 -0.4508 1.7317 28.0388 2.4387 T2 -0.0021 0.0018 0.0006 0 -0.0013 LT -0.3106 -1.205 0.038 1.6711 0.0021 Table.6 Parameter for the second set of tuning rules for 0.1 < T < 50 and L < 0.5 V. SIMULINK MODEL OF AVR The Simulink model of AVR is shown in Fig 3. This model consist the mathematical function of amplifier, exciter and generator. Fig.3 Simulink model of AVR The Simulink model of various tuning method for AVR using PID tuned by Ziegler Nichols and Cohen Coon and FOPID controller tuned by Ziegler Nichols is shown in Fig 4.
  • 6. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 46 | P a g e Fig.4 Simulink model of various tuning methods VI. RESULT AND DISCUSSION The Simulink model in Fig. 3 & 4 was simulated and the plots for various tuning method were observed. Fig 6 shows the Step response of AVR using PID tuned by Ziegler Nichols, Fig 7 shows the step response of AVR using PID tuned by Cohen Coon. and Fig 7 shows the Step response of AVR using FOPID controller. Fig.5 Response of PID tuned with Zeigler Nichols & Cohen Coon and FOPID tuned with Zeigler Nichols for AVR Fig.6 Step response of AVR using PID tuned by Ziegler Nichols
  • 7. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 47 | P a g e Fig.7 Step response of AVR using PID tuned by Cohen Coon Fig.8 Step response of AVR using FOPID It can be seen from the comparison table that while using the FOPID controller tuned by Ziegler Nichols the overshoots obtained is less as compared to the case when the PID Controller was tuned via conventional methods i.e. Ziegler Nichols and Cohen Coon method. The settling time is also lesser in case of the FOPID controller, also the rise time is reduced. The FOPID controller tuned by Z-N tuning rule tends to faster response of AVR. It can be observed from Fig 6 and 7 that the PID controller tuned by Ziegler Nichols and Cohen Coon have larger overshoot than FOPID controller attain a steady state with larger settling time. PID FOPID Parameters Ziegler Nichols Cohen Coon Ziegler Nichols Kp 2.7719 3.3221 0.5543 Ki 3.009 3.4323 0.5841 Kd 0.6384 0.5159 0.2247 λ - - -1.0814 μ - - 1.3628 Tr(sec) 0.149 0.139 0.471 Ts(sec) 5.01 5.55 4.15 Mp( %) 71 75.6 40.5 Table.7 Comparative values of different parameters obtained by PID and FOPID controllers VII. CONCLUSION Performance comparisons of FOPID and PID controllers have been reviewed and it is found that response of FOPID controller is better than PID controller. The settling time and rise time of the response acquired by FOPID controller is found to be less as comparison to response acquired by PID controller.
  • 8. Comparative Study of PID and FOPID Controller Response for Automatic Voltage Regulation International organization of Scientific Research 48 | P a g e VIII. ACKNOWLEDGMENT This work was supported by Mr. Akhilesh Kumar Mishra. Special Thanks to all faculty members of Electrical Engineering Department of United College of Engineering & Research, Allahabad India, Specially Head of Department Mr. Abdul Zeeshan for their co-operation. REFERENCES [1] Podlubny, I.: Fractional-order systems and PI_D_ controllers, IEEE Trans. On Automatic Control, vol.44, no.1, pp. 208 213, 1999. [2] M. Rabiul Alam, Rajib Baran Roy, S.M. Jahangir Alam, Dewan Juel Rahman International Journal of Electrical & Computer Sciences IJECS-IJENS Vol: 11 No: 05 [3] H. Yoshida, K. Kawata, and Y. Fukuyama, “A particle swarm optimization for reactive power and voltage control considering voltage security assessment,” IEEE Trans. Power Syst., vol. 15, pp. 1232– 1239, Nov. 2000. [4] K. B. Oldhamand and J. Spanier, “The Fractional Calculus”, Academic Press, New York, (1974). [5] C. H. Lubich, “Discretized fractional calculus”, SIAM Journal on Mathematical Analysis, (1986), pp. 704-719. [6] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley, New York, (1993). [7] A. Oustaloup, “Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulators”, IEEE Transactions on Circuits and Systems, (1981), pp.1007–1009. [8] M. Chengbin and Y. Hori, “The application of fractional order PID controller for robust two-inertia speed control”, Proceedings of the 4th International Power Electronics and Motion Control Conference, Xi‟an, (2004) August. [9] F. Merrikh-Bayat, M. Karimi-Ghartemani, “Method for designing FOPID stabilisers for minimum-phase fractional-order systems”, IET Control Theory Appl., vol. 4, Issue 1, (2010), pp. 61–70. [10] Padula, F., Visioli, A., (2010a). Tuning rules for optimal PID and fractional-order PID controllers, Journal of Process Control, doi:10.1016/j.jprocont.2010.10.006