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Implementation of Fractional Order PID Controller for an AVR System Using GA and ACO Optimization Techniques
4th International Conference on Advances in Control and 4th 4th International International Conference Conference on on Advances Advances in in Control Control and and Available onlineand at www.sciencedirect.com Optimization of Dynamical Systems 4th International Conference on Advances in Control Optimization of Dynamical Systems 4th International Conference on Advances in Control and Optimization Dynamical Systems February 1-5, of 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India Optimization of Dynamical Systems February February 1-5, 1-5, 2016. 2016. NIT NIT Tiruchirappalli, Tiruchirappalli, India India
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IFAC-PapersOnLine 49-1 (2016) 456–461
Implementation of PID Controller for an AVR System Using Implementation of Fractional Fractional Order Order PID Controller for an AVR System Using Implementation Order PID Controller for an AVR System Using Implementation of of Fractional Fractional Order PID Controller for an AVR System Using GA and ACO Optimization Techniques GA and ACO Optimization Techniques GA and ACO Optimization Techniques GA and ACO Optimization Techniques
First A. G.Suri babu*. Second B.T.Chiranjeevi First A. G.Suri babu*. Second B.T.Chiranjeevi First A. G.Suri babu*. Second B.T.Chiranjeevi *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of Technology,Bhimavaram *Assistant Professor ,Electrical Engineering Department,Vishnu Institute First A. G.Suri babu*. Second B.T.Chiranjeevi *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of of Technology,Bhimavaram Technology,Bhimavaram India(Tel:91-9441901312;email:[email protected]) *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of *Assistant Professor India(Tel:91-9441901312;email:[email protected]) ,Electrical Engineering Department,Vishnu Institute of Technology,Bhimavaram Technology,Bhimavaram India(Tel:91-9441901312;email:[email protected]) India(Tel:91-9441901312;email:[email protected]) **Research Scholar, NIT Silichar,India(email:[email protected]) India(Tel:91-9441901312;email:[email protected]) **Research Scholar, NIT Silichar,India(email:[email protected]) **Research Scholar, Silichar,India(email:[email protected]) **Research Scholar, NIT NIT Abstract: In recent years, the the research research andSilichar,India(email:[email protected]) studies on fractional fractional order (FO) (FO) modelling of of dynamic systems systems Abstract: In recent years, and Abstract: In recent years, the research and studies studies on on fractional order order (FO) modelling modelling of dynamic dynamic systems Abstract: In recent years, the research and studies on fractional order (FO) modelling of dynamic systems and controllers are quite advancing. To improve the stability and dynamic response of automatic voltage Abstract: In recent the researchToand studiesthe onstability fractional order (FO) modelling dynamic systems and controllers are years, quite advancing. improve and dynamic response ofofautomatic voltage regulator (AVR) system, we presented fractional Order PID (FOPID) controller by employing Genetic and controllers are quite advancing. To improve the stability and dynamic response of automatic voltage regulator (AVR) system, we presented fractional Order PID (FOPID) controller by employing Genetic and controllers quite advancing. To fractional improve the stability dynamic responsebyofemploying automatic Genetic voltage regulator (AVR)aresystem, we presented Order PID and (FOPID) controller Algorithm(AVR) (GA) and and Ant Colony Colony Optimization (ACO) techniques. We designed designed masked fractional order regulator system, we fractional Order PID (FOPID) controller by employing Genetic Algorithm (GA) Ant Optimization (ACO) techniques. We aaa masked fractional order regulator (AVR) system, we presented presented fractional Order PID (FOPID) controller by employing Genetic Algorithm (GA) and Ant Colony Optimization (ACO) techniques. We designed masked fractional (ACO) techniques. We designed a masked fractional order Algorithm (GA) and Ant Colony Optimization order PI D controller for any type of system in simulink. Comparisons PID block in Matlab to implement the D (ACO) techniques. We designed a masked fractional order Algorithm (GA) and to Antimplement Colony Optimization for any type of system in simulink. Comparisons PID block in Matlab the PI PI D controller for type system in Comparisons PID block in to implement the are aa PID controller from of response robustness characteristics. It PI D controller controller for any any type of ofand system in simulink. simulink. Comparisons PID blockwith in Matlab Matlab implement thestandpoints are made made with PID to controller from standpoints of transient transient response and robustness characteristics. It is is are made with a PID controller from standpoints of transient response and robustness characteristics. It is shown that the proposed FOPID controller can improve the performance of the AVR in all the aspects. are made PID controller standpoints of transient response and robustness shown thatwith the aproposed FOPID from controller can improve the performance of the AVR in characteristics. all the aspects. It is shown the proposed FOPID controller can improve the of in the aspects. © 2016,that IFAC Federation of Automatic Control) Hostingoptimization. by Elsevier Ltd. All rights shown that the(International proposed FOPID controller can improve the performance performance of the the AVR AVR in all all the reserved. aspects. Keywords: Fractional order PID Controller, Oustaloop, Ant colony Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. FRACTIONAL MODELING FRACTIONAL MODELING I. INTRODUCTION MODELING A. Order A. Integer IntegerFRACTIONAL Order Approximations Approximations I. INTRODUCTION FRACTIONAL MODELING A. Integer Order Approximations I. INTRODUCTION A. Integer Order Approximations Online real-time, fractional-order differentiation may I. INTRODUCTION Even so much research was developed in soft computing A. IntegerOnline Order real-time, Approximations fractional-order differentiation may Even so much research was developed in soft computing Online real-time, fractional-order be required in control systems. Using is best Even so much research was developed in soft computing techniques for designing controllers, still these techniques techniques for controllers, still techniques Online real-time, fractional-order differentiation may be required in control systems. Using filters filtersdifferentiation is one one of of the the may best Even so much research was developed soft computing techniques for designing designing controllers, stillinthese these techniques be required in control systems. Using filters is one of the ways to solve the problems. By using filters, the fractional techniques for designing controllers, still these techniques fails in particular areas. Neuro-controllers works according to be required in control systems.By Using filters is one offractional the best best techniques for designing controllers, stillworks theseaccording techniques ways to solve the problems. using filters, the fails in particular areas. Neuro-controllers to ways to solvefunction the problems. By usingto fractional order is integer order. order transfer transfer function is approximated approximated tofilters, integerthe order. fails in areas. Neuro-controllers works to the given to them and their performance depends on ways to solvefunction the problems. By usingto filters, the fractional failstraining in particular particular areas. Neuro-controllers works according according to order transfer is approximated integer order. the training given to them and their performance depends on order transfer is to training given tovast them and their performance depends on the availability of data of training inputs and target B. Oustaloup’s Recursive Filter the availability of data set of inputs and order transfer function function is approximated approximated to integer integer order. order. B. Oustaloup’s Recursive Filter the given them andset performance on availability of tovast vast data settheir of training training inputsdepends and target target B. Oustaloup’s Recursive Filter the training values of the process. Certainly there is a probability of B. Oustaloup’s Recursive Filter the availability of vast data set of training inputs and target values of the process. Certainly there is a probability of Some continuous filters have been summarized in the availability of vast data set of training inputs and target values of the process. Certainly there is a probability of B. Oustaloup’s Recursive Filter Some filters have in Some continuous continuous have been been summarized summarized in values of process. Certainly there aa controller probability of [13]. the the Oustaloup missed in training set, case [13]. Among Among the filters, filters, filters the well-established well-established Oustaloup values data of the the process. Certainly there isthe probabilityfails of missed data in the the training set, in in such such caseis the controller fails Some continuous filters have been summarized in [13]. Among the filters, the well-established Oustaloup missed data in the training set, in such case the controller fails to generate an accurate output. Also the performance of the [13]. Among the filters, the well-established Oustaloup recursive filter has a very good fitting to the fractional-order missed data an in the training set, inAlso such the caseperformance the controlleroffails to generate accurate output. the [13]. Among the filters, the well-established Oustaloup recursive filter has a very good fitting to the fractional-order to generate an output. Also performance of fuzzy controllers is purely based on thethe rule base, selection of fuzzy controllers is based on rule base, of recursive has fitting the fractional-order differentiators [14,15,16]. Assume thatto the expected fitting to generate an accurate accurate output. performance of the the fuzzy controllers is purely purely based Also on the thethe rule base, selection selection of recursive filter filter [14,15,16]. has aa very very good good fitting tothe the expected fractional-order differentiators Assume that fitting differentiators [14,15,16]. Assume that the expected fitting (𝜔𝜔 ). fuzzy controllers is purely based on the rule base, selection of range is membership functions and their range and it was still , 𝜔𝜔 (𝜔𝜔 ). range is 𝑏𝑏 ℎ , 𝜔𝜔 fuzzy controllers is purely based on the rule base, selection of 𝑏𝑏 ℎ membership functions and their range and it was still differentiators [14,15,16]. Assume that the expected fitting ). (𝜔𝜔 range is 𝑏𝑏 ℎ 𝑏𝑏 ,, 𝜔𝜔 ℎ ). membership functions and their range and it was still challenging to decide the fuzzy parameters. Still today most (𝜔𝜔 range is 𝜔𝜔 𝑏𝑏 , 𝜔𝜔 ℎ 𝑏𝑏 ℎ membership their range and was most still challenging tofunctions decide theand fuzzy parameters. Stillittoday The can range is (𝜔𝜔 𝑏𝑏 be The filter filter can beℎ ).written written as as challenging to the parameters. Still most of the control problems are smoothly solved by PID challenging to decide decide the fuzzy fuzzy parameters. Still today today most of the control problems are smoothly solved by PID The filter can be written as ′′ The filter can be written as 𝑠𝑠+𝜔𝜔 of the control problems are smoothly solved by PID controllers due to simplicity in their design and easy 𝑠𝑠+𝜔𝜔𝑘𝑘 𝑁𝑁 𝑘𝑘 ′′ controllers due simplicity in 𝑁𝑁 of the control problems are smoothly solved and by easy PID 𝑠𝑠+𝜔𝜔 controllers due to to simplicity in their their design design and easy (𝑠𝑠) ∏ 𝑠𝑠+𝜔𝜔 = 𝐾𝐾 (1) 𝐺𝐺 𝑁𝑁 𝑘𝑘 𝑓𝑓 𝑘𝑘=−𝑁𝑁 ′′ 𝑁𝑁 𝑘𝑘 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 𝑠𝑠+𝜔𝜔𝑘𝑘 ∏ = 𝐾𝐾 (1) 𝐺𝐺𝑓𝑓𝑓𝑓𝑓𝑓 (𝑠𝑠) ′ controllers due simplicity in their and 𝑁𝑁 implementation. been two extra of 𝑘𝑘=−𝑁𝑁 𝑁𝑁 𝑘𝑘 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 controllers due It tohas simplicity in that their design and easy easy (𝑠𝑠) ∏ = 𝐾𝐾 𝐺𝐺 (1) implementation. Itto has been shown shown that twodesign extra degrees degrees of 𝑁𝑁 𝑘𝑘 𝑠𝑠+𝜔𝜔 𝑓𝑓 𝑘𝑘=−𝑁𝑁 𝑘𝑘 𝑓𝑓 𝑘𝑘=−𝑁𝑁 (𝑠𝑠) ∏ = 𝐾𝐾 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 𝐺𝐺𝑓𝑓 (1) 𝑠𝑠+𝜔𝜔𝑘𝑘 𝑘𝑘 implementation. It has been shown that two extra degrees of freedom from the use of a fractional-order integrator and 𝑠𝑠+𝜔𝜔 implementation. It has been shown that two extra degreesand of freedom from the use of a fractional-order integrator 𝑘𝑘 can be evaluated where the poles, zeros, and gain of the filter where the poles, zeros, and gain of the filter can be evaluated freedom from the use of a fractional-order integrator and differentiator make it possible to further improve the differentiator make it possible to further improve the freedom from make the useit ofpossible a fractional-order and differentiator to further integrator improve the where the zeros, and from such that where the poles, poles, zeros, and gain gain of of the the filter filter 11can can be be evaluated evaluated from such that differentiator it to improve the performance ofmake traditional PID controllers. Details of past and 1 performance traditional PID Details of 1(1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ from such that differentiator it possible possible to further further improve the (1−𝛾𝛾) (1+𝛾𝛾) performance of ofmake traditional PID controllers. controllers. Details of past past and and 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 1 1 2 2 1 1 2 2 from such 𝜔𝜔that𝑘𝑘+𝑁𝑁+ (1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ (1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 1 1 performance of traditional PID controllers. Details of past and present progress in the analysis of dynamic systems modeled 1 1 𝜔𝜔 2 2 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 2(1−𝛾𝛾) 2(1+𝛾𝛾) h 𝑘𝑘+𝑁𝑁+ h 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 2𝑁𝑁+1 2𝑁𝑁+1 performance of traditional PID controllers. of modeled past and h 1 1 present progress in the analysis of dynamicDetails systems 2(1−𝛾𝛾) 2(1+𝛾𝛾) 𝜔𝜔h 𝜔𝜔 𝜔𝜔𝑘𝑘𝑘𝑘′′′′ = = 𝜔𝜔𝑏𝑏𝑏𝑏 ((𝜔𝜔 )𝑘𝑘+𝑁𝑁+ 𝜔𝜔𝑘𝑘𝑘𝑘 = = 𝜔𝜔 𝜔𝜔𝑏𝑏𝑏𝑏 ((𝜔𝜔 ,𝐾𝐾 = = 𝜔𝜔 𝜔𝜔ℎℎ𝛾𝛾𝛾𝛾𝛾𝛾 2𝑁𝑁+1 2𝑁𝑁+1 2 2 (1−𝛾𝛾) ,, 𝜔𝜔 (1+𝛾𝛾) ,𝐾𝐾 𝑘𝑘+𝑁𝑁+ 𝜔𝜔h 2𝑁𝑁+1 2𝑁𝑁+1 𝜔𝜔 ))𝑘𝑘+𝑁𝑁+ h) 2 2 present progress in analysis dynamic systems by Fractional order differential can be 𝜔𝜔h 𝜔𝜔h 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 𝜔𝜔𝑘𝑘𝑘𝑘′′ = 𝜔𝜔 𝜔𝜔𝑏𝑏𝑏𝑏 (𝜔𝜔 , 𝜔𝜔 = 𝜔𝜔 ( ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔 𝜔𝜔 h) h) 𝑏𝑏 𝑏𝑏 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 𝑘𝑘 𝑏𝑏 present progress in the the analysis of ofequations dynamic (FODEs) systems modeled modeled h h 𝑏𝑏 𝑏𝑏 ℎ by Fractional order differential equations (FODEs) can be 𝑘𝑘 𝑏𝑏 𝜔𝜔 𝜔𝜔 ℎ 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 ′ = 𝜔𝜔𝑏𝑏 𝜔𝜔 ,, 𝜔𝜔 𝜔𝜔 ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔h 𝜔𝜔h 𝑏𝑏 𝑏𝑏 𝑏𝑏 ( 𝑘𝑘 = 𝑏𝑏 ( 𝑘𝑘 ℎ by Fractional order differential equations (FODEs) can be 𝑏𝑏 ) 𝑏𝑏 ) found in [1–14]. Literature survey gives the fractional 𝑘𝑘 𝑏𝑏 𝑘𝑘 ℎ 𝜔𝜔 = 𝜔𝜔 ( ) 𝜔𝜔 = 𝜔𝜔 ( ) ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔 𝜔𝜔 found in [1–14]. Literature survey gives the fractional (2) 𝜔𝜔 𝜔𝜔 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑘𝑘 𝑏𝑏 𝑘𝑘 ℎ by Fractional order Literature differential survey equations (FODEs) can be found in [1–14]. gives the fractional 𝜔𝜔𝑏𝑏 𝜔𝜔𝑏𝑏 (2) 𝑏𝑏 𝑏𝑏 controller which was developed by Crone in [3], while [4, (2) found in [1–14]. Literature survey gives the fractional controller was developed by [3], while found in which [1–14]. survey givesin controller which was Literature developed by Crone Crone in the [3], fractional while [4, [4, (2) 𝜇𝜇 𝜇𝜇 controller which was developed by Crone in [3], while [4, where 𝛾𝛾 is the order of the differentiation and N is the order 12, presented the 𝑃𝑃𝑃𝑃 and [3,14] proposed 𝜇𝜇 where 𝛾𝛾 𝛾𝛾 is is the the order order of of the the differentiation differentiation and and N N is is the the order order 𝜇𝜇 controller controller which was developed by Crone in [3], while the [4, 12,λλ 13] 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the where 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜇𝜇 where 𝛾𝛾 is the order of the differentiation and N is the order 𝜇𝜇 12, 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the 𝐷𝐷 controller. 𝑃𝑃𝐼𝐼 of approximation. λ 𝜇𝜇 𝜇𝜇 controller. 12,λλλ 𝐷𝐷 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the where 𝛾𝛾 is the order of the differentiation and N is the order 𝑃𝑃𝐼𝐼 of approximation. 𝜇𝜇 𝜇𝜇 of approximation. 𝑃𝑃𝐼𝐼 C. A Refined Oustaloup Filter 𝐷𝐷𝜇𝜇 controller. controller. 𝑃𝑃𝐼𝐼λ 𝐷𝐷 of approximation. C. A Refined Oustaloup Filter λ 𝜇𝜇 𝜇𝜇 C. A A Refined Refined Oustaloup Oustaloup Filter Filter λ We extended the benefits of 𝑃𝑃𝐼𝐼 for λ 𝜇𝜇 C. Here we introduce new approximate realization λ 𝐷𝐷 𝜇𝜇 controller HereOustaloup we introduce introduce new approximate approximate realization realization We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 for 𝜇𝜇 C. A Refined Filteraaa new 𝜇𝜇 Here we λ𝜇𝜇 𝜇𝜇 controller λλ 𝐷𝐷λ We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 controller for controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 𝜇𝜇 Here we introduce a new approximate realization λλ 𝐷𝐷λ𝜇𝜇 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 method for the fractional-order derivative in the frequency 𝜇𝜇 We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 controller for AVR system. In Particular a masked 𝑃𝑃𝐼𝐼λλ 𝐷𝐷𝜇𝜇𝜇𝜇 controller block Here we introduce a new approximate realization method for the fractional-order derivative in the frequency is developed in matlab simulink to optimize the parameters λ 𝐷𝐷𝜇𝜇 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 ].Our method for the fractional-order derivative in the frequency is developed in matlab simulink to optimize the parameters range of interest[𝜔𝜔 proposed method here gives aa , 𝜔𝜔 ].Our 𝑏𝑏 ℎ 𝐷𝐷 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 𝑏𝑏 , 𝜔𝜔ℎ ].Our proposed is developed in matlab simulink to optimize the parameters methodoffor the fractional-order derivative in the frequency range interest[𝜔𝜔 method here gives 𝑏𝑏 ℎ based on the theinrecently recently developed optimization techniques 𝑏𝑏 , than ℎ ].Our is developed matlab to optimize range of interest[𝜔𝜔 proposed method here gives 𝜔𝜔 better approximation Oustaloup’s method with respect to based on developed techniques 𝑏𝑏 ℎ better approximation than Oustaloup’s method with respect to is developed matlab simulink simulink to optimization optimize the the parameters parameters 𝑏𝑏 ℎ based on thein recently developed optimization techniques ].Our range approximation of interest[𝜔𝜔𝑏𝑏 , than proposedmethod methodwith here gives toaa 𝜔𝜔ℎ Oustaloup’s better respect based on the recently developed optimization techniques instead of using toolbox which was restrained for using better approximation than Oustaloup’s method with respect to both low frequency and high frequency. Assume that based onof the recently developed optimization techniques better approximation method with respect to instead using toolbox which was restrained for using both low frequency than and Oustaloup’s high frequency. Assume that the the instead of using toolbox which was restrained for using advanced optimization techniques. Genetic Algorithm (GA) advanced optimization techniques. Genetic Algorithm (GA) both low frequency and high frequency. Assume that the (𝜔𝜔 ). frequency range to be fit is defined as Within , 𝜔𝜔 𝑏𝑏 ℎ instead ofoptimization using toolbox which was restrained for using 𝑏𝑏 ,Assume ℎ ). Within both low frequency and high frequency. that the advanced techniques. Genetic Algorithm (GA) (𝜔𝜔 frequency range to be fit is defined as 𝜔𝜔 the advanced optimization techniques. Algorithm (GA) (𝜔𝜔𝑏𝑏𝑏𝑏 ℎℎℎℎ ). and Ant Colony optimization (ACOGenetic ) techniques has already frequency range to is as the pre-specified frequency the fractional-order operator advanced optimization techniques. Genetic Algorithm (GA) frequency range to be be fit fitrange, is defined defined as (𝜔𝜔𝑏𝑏𝑏𝑏𝑏𝑏 ,, 𝜔𝜔 Within the 𝜔𝜔ℎ ). Within and Ant Colony optimization (ACO )) techniques has already pre-specified frequency range, the fractional-order operator 𝛼𝛼 𝛼𝛼 can be approximated and Ant Colony optimization (ACO techniques has already been used to determine optimal solution to several power pre-specified frequency range, the fractional-order operator 𝑠𝑠 by the transfer 𝛼𝛼 can be approximated 𝑠𝑠pre-specified by the fractional-order transfer and Ant Colony optimization (ACO ) techniques has already been used to determine optimal solution to several power 𝛼𝛼 frequency range, the fractional-order operator 𝑠𝑠 can be approximated by the transfer 𝛼𝛼 fractional-order been determine solution to power engineering problems andoptimal we employed these algorithms to function 𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼 can be transfer function as been used used to toproblems determine optimal solutionthese to several several power can as be approximated approximated by by the the fractional-order fractional-order transfer engineering and we employed algorithms to function as engineering problems and we employed these algorithms to design controller for Automatic voltage regulator ′′ ⁄𝜔𝜔 design an an FOPID FOPID controller for Automatic voltage regulator 1+𝑠𝑠 𝜔𝜔 ⁄ function as 1+𝑠𝑠 𝑁𝑁 𝑘𝑘 engineering problems and we employed these algorithms to ′′ FOPID controller for Automatic voltage regulator 𝑁𝑁 𝑘𝑘 function as design an ⁄ (𝑠𝑠) ∏ 1+𝑠𝑠 𝜔𝜔 = lim 𝐾𝐾(𝑠𝑠) = lim 𝐾𝐾 (3) ⁄𝜔𝜔𝑘𝑘 1+𝑠𝑠⁄ 𝑁𝑁 𝑁𝑁→∞ 𝐾𝐾𝑁𝑁 𝑁𝑁 (𝑠𝑠) lim 𝐾𝐾(𝑠𝑠) (3) 𝑘𝑘=−𝑁𝑁 ′′ 𝑁𝑁 design FOPID controller for Automatic regulator (AVR) problem. proposed simulated within 1+𝑠𝑠⁄𝜔𝜔 𝜔𝜔𝑘𝑘 (𝑠𝑠) = 𝑁𝑁→∞ ∏𝑘𝑘=−𝑁𝑁 lim ∏ 𝐾𝐾(𝑠𝑠) = = lim lim𝑁𝑁→∞ (3) 𝑁𝑁→∞ ′ 𝑘𝑘 𝑁𝑁 design an FOPIDThe controller forcontroller Automaticis voltage regulator 𝑁𝑁→∞ 𝑁𝑁 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠 𝑁𝑁 𝑘𝑘 (AVR) an problem. The proposed controller is voltage simulated within 𝑁𝑁→∞ 𝐾𝐾 𝑁𝑁 (𝑠𝑠) = ⁄ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠 𝜔𝜔 ∏ 𝑁𝑁 ⁄ 𝑁𝑁→∞ 𝑘𝑘 = lim 𝐾𝐾(𝑠𝑠) = lim 𝐾𝐾 (3) 1+𝑠𝑠 𝜔𝜔 𝑁𝑁→∞ 𝐾𝐾𝑁𝑁 𝑁𝑁 (𝑠𝑠) = 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠⁄𝜔𝜔𝑘𝑘 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 (AVR) problem. The proposed controller is simulated within various scenarios and its performance is compared with those ∏ lim 𝐾𝐾(𝑠𝑠) = lim (3) According to the recursive distribution of real zeros and 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠⁄𝜔𝜔𝑘𝑘 𝑘𝑘 𝑁𝑁→∞ 𝑁𝑁 𝑁𝑁→∞ (AVR) problem. The proposed controller is simulated within various scenarios and its performance is compared with those ⁄ 𝜔𝜔𝑘𝑘 zeros and According to the recursive distribution of1+𝑠𝑠real 𝑁𝑁→∞ various scenarios and its performance is compared with those of an optimally-designed PID controller. Transient response of an optimally-designed PID controller. Transient response According to the recursive distribution of real zeros and poles, the zero and pole of rank 𝑘𝑘 can be written as various scenarios and its performance is compared those of an optimally-designed PID controller. Transientwith response According to and the pole recursive of real poles, the zero of rankdistribution 𝑘𝑘 can be written as zeros and of anperformance optimally-designed PID controller. response and robustness characteristics of both controllers and robustness characteristics of poles, 𝑘𝑘 can be written as of optimally-designed PID controller. Transient Transient response andanperformance performance robustness characteristics of both both controllers controllers poles, the the zero zero and and pole pole of of rank rank 𝑘𝑘 can be written as 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 and performance robustness characteristics of both are and the in 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 and studied performance robustness of characteristics of controller both controllers controllers 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 are studied and superiority superiority of the proposed proposed controller in all all 𝑑𝑑𝜔𝜔𝑏𝑏 𝑏𝑏𝜔𝜔ℎ 𝑏𝑏 2𝑁𝑁+1 ℎ 2𝑁𝑁+1 2𝑁𝑁+1 2𝑁𝑁+1 ′′ 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝛼𝛼−2𝑘𝑘 , 𝜔𝜔𝑘𝑘 = (𝑏𝑏𝜔𝜔 𝛼𝛼+2𝑘𝑘 𝑑𝑑𝜔𝜔 2𝑁𝑁+1 2𝑁𝑁+1 are studied and superiority of the proposed controller in all two respects is illustrated. 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝑏𝑏 ℎ 𝜔𝜔 = ( ) ) (4) ′ 2𝑁𝑁+1 2𝑁𝑁+1 𝑘𝑘 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝑏𝑏 ℎ ′ = (𝑑𝑑𝜔𝜔 𝑘𝑘 = (𝑏𝑏𝜔𝜔 𝑘𝑘 are studied and superiority of the proposed controller in all 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝑏𝑏 𝑑𝑑 two respects is illustrated. 2𝑁𝑁+1 2𝑁𝑁+1 ) , 𝜔𝜔 ) (4) 𝜔𝜔 𝑏𝑏 ℎ 𝑏𝑏 𝑑𝑑 ′𝑘𝑘 2𝑁𝑁+1 2𝑁𝑁+1 𝑏𝑏 )2𝑁𝑁+1 , 𝜔𝜔𝑘𝑘 ℎ )2𝑁𝑁+1 ′ = (𝑑𝑑𝜔𝜔 𝑘𝑘 𝑘𝑘 𝑏𝑏𝜔𝜔 two respects is illustrated. 𝜔𝜔 = ( (4) 𝑏𝑏 𝑑𝑑 𝑏𝑏 ℎ ′ 𝑏𝑏 𝑑𝑑 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 two respects is illustrated. 𝜔𝜔𝑘𝑘 = ( 𝑏𝑏𝑏𝑏 ) , 𝜔𝜔𝑘𝑘 = ( 𝑑𝑑𝑑𝑑 ) (4)
2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All𝑏𝑏 rights reserved. Copyright 2016 responsibility IFAC 456Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 456 10.1016/j.ifacol.2016.03.096 Copyright © 2016 IFAC 456 Copyright © 2016 IFAC 456
𝑑𝑑
IFAC ACODS 2016 First A. G.Suri babu et al. / IFAC-PapersOnLine 49-1 (2016) 456–461 February 1-5, 2016. NIT Tiruchirappalli, India
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1.013𝑒𝑒 −5 𝑠𝑠 + 𝑠𝑠 3 + 0.2164𝑠𝑠 2 + 0.001564𝑠𝑠 + 4.032𝑒𝑒 −7 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 4 𝑠𝑠 + 0.2172𝑠𝑠 3 + 0.001577𝑠𝑠 2 + 4.234𝑒𝑒 −7 𝑠𝑠 + 4.084𝑒𝑒 −12
Through confirmation by experimentation and theoretical analysis, the synthesis approximation can obtain the good effect when 𝑏𝑏 = 10 and 𝑑𝑑 = 9.
(8)
𝜇𝜇 = 𝑠𝑠 0.9712 is approximated to integer order by using oustaloup filter as follows
II. FRACTIONALORDER PID CONTROLLER The block diagram of Fractional Order PID Controller is shown in Fig.1
7.178𝑒𝑒 4 𝑠𝑠 7 + 2.914𝑒𝑒 8 𝑠𝑠 6 + 4.251𝑒𝑒10 𝑠𝑠 5 + 2.309𝑒𝑒11 𝑠𝑠 4 + 4.674𝑒𝑒10 𝑆𝑆 3 + 3.527𝑒𝑒 8 𝑆𝑆 2 + 9.907𝑒𝑒 4 𝑆𝑆 + 1 𝐺𝐺 = (9) 7 𝑠𝑠 + 9.907𝑒𝑒 4 𝑠𝑠 6 + 3.527𝑒𝑒 8 𝑠𝑠 5 + 4.674𝑒𝑒10 𝑠𝑠 4 + 2.309𝑒𝑒11 𝑠𝑠 3 + 4.251𝑒𝑒10 𝑠𝑠 2 + 2.914𝑒𝑒 8 𝑆𝑆𝑆𝑆 + 7.178𝑒𝑒 4
The above transfer function is reduced to 4th order by using Hankel Minimum Degree Approximation(MDA) as follows 7.178𝑒𝑒 4 𝑠𝑠 4 + 2.913𝑒𝑒 8 𝑠𝑠 3 + 4.218𝑒𝑒10 𝑠𝑠 2 + 2.047𝑒𝑒11 𝑆𝑆 + 3.307𝑒𝑒 9 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑠𝑠 4 + 9.906𝑒𝑒 4 𝑠𝑠 3 + 3.525𝑒𝑒 8 𝑠𝑠 2 + 4.634𝑒𝑒10 𝑠𝑠 + 2.024𝑒𝑒 11
(10)
Fig.1. Block Diagram of Fractional Order PID controller
The differential equation of a fractional order PI λ Dμ controller is described by: 𝜇𝜇
𝒰𝒰(𝑡𝑡) = 𝐾𝐾𝑃𝑃 𝑒𝑒(𝑡𝑡) + 𝐾𝐾𝐼𝐼 𝐷𝐷𝑡𝑡−𝜆𝜆 𝑒𝑒(𝑡𝑡) + 𝐾𝐾𝐷𝐷 𝐷𝐷𝑡𝑡 𝑒𝑒(𝑡𝑡)
(5)
The continuous transfer function of FOPID is obtained through Laplace transform and is given by: 𝐺𝐺𝐶𝐶 (𝑆𝑆) = 𝐾𝐾𝑃𝑃 + 𝐾𝐾𝐼𝐼 𝑠𝑠 −𝜆𝜆 + 𝐾𝐾𝐷𝐷 𝑠𝑠 𝜇𝜇
(6)
We designed masked fractional order PID block in Matlab simulink as shown in Fig.2.The dialog box of fractional PID block is as shown in Fig.3
Fig.2. Masked fractional order PID block
From Fig.3, 𝐾𝐾𝑃𝑃 , 𝐾𝐾𝐼𝐼 and 𝐾𝐾𝐷𝐷 are proportional, integrator and derivative gains respectively. Gamma and delta are orders of the integral and derivative parts, 𝜆𝜆 and 𝜇𝜇 respectively. 𝜔𝜔𝑏𝑏 and 𝜔𝜔ℎ are lower and upper frequencies respectively, for which the fractional order inregral and derivative are approximated to integer order. Here, the fractional order integral and derivative are approximated to integer order by using a oustaloup filter and N is the order of approximation. 𝜆𝜆 = 𝑠𝑠 −0.9989 is approximated to integer order by using oustaloup filter as follows 1.013𝑒𝑒 −5 𝑠𝑠 7 + 1.05𝑠𝑠 6 + 3913𝑠𝑠 5 + 5.427𝑒𝑒 5 𝑠𝑠 4 + 2.806𝑒𝑒 6 𝑆𝑆 3 + 5.407𝑒𝑒 5 𝑆𝑆 2 + 3879𝑆𝑆 + 1 𝐺𝐺 = 𝑠𝑠 7 + 3879𝑠𝑠 6 + 5.407𝑒𝑒 5 𝑠𝑠 5 + 2.806𝑒𝑒 6 𝑠𝑠 4 + 5.427𝑒𝑒 5 𝑠𝑠 3 + 3913𝑠𝑠 2 + 1.05𝑠𝑠 + 1.013𝑒𝑒 −5
Fig.3. Dialog box of fractional PID block
III. AVR DESIGN USING FOPID CONTROLLER A. Linearized Model of Excitation System The role of an Automatic Voltage Regulator (AVR) is to hold the terminal voltage of a synchronous generator at a specified level. We consider an alternator supplied controlled rectifier excitation system [18] for simulation. Five main components, namely amplifier, exciter, excitation voltage limiters, generator, measurement and filtering includes in the mathematical model of AVR system by ignoring the saturation and other non-linearities [19]. The transfer function of these components are represented as follows.
(7)
In-order to make designing process easier the order of the approximated integer order of transfer function is reduced to 4th order by using Hankel Minimum Degree Approximation(MDA) [17].
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Amplifier model. The amplifier model is given by 𝑉𝑉𝑅𝑅 (𝑠𝑠)
𝑉𝑉𝐶𝐶 (𝑠𝑠)
=
𝑘𝑘𝐴𝐴
1+𝜏𝜏𝐴𝐴 𝑠𝑠
integral of time multiplied by absolute value of the error (ITAE),integral of time multiplied by the squared error (ITSE). The ITAE measure, which will be implemented in this paper, is given by the following equation
(11)
Typical values of 𝑘𝑘𝐴𝐴 are in the range of 10 to 400. The amplifier time-constant often ranges from 0.02 to 0.1 s.
(15) ITAE= ∫0 𝑡𝑡|𝑒𝑒(𝑡𝑡)|𝑑𝑑𝑑𝑑 Therefore, the proposed performance criterion 𝐽𝐽(𝑘𝑘) is defined as 𝐽𝐽(𝑘𝑘) =(ITAE) (16) ∞
Exciter model. Exciter model and parameters greatly depend on its type. A simplified transfer function of a modern exciter is 𝑉𝑉𝐹𝐹 (𝑠𝑠) 𝑘𝑘 = 𝐸𝐸 (12) (𝑠𝑠)
where k is [𝐾𝐾𝑃𝑃 , 𝐾𝐾𝐼𝐼 , 𝐾𝐾𝐷𝐷 , 𝜆𝜆, 𝜇𝜇]. The Genetic Algorithm (GA) and Ant Colony Optimization (ACO) are utilized to design these five controller parameters such that the controlled system exhibits desired response and robust stability as evaluated by the proposed performance criterion.
1+𝜏𝜏𝐸𝐸 𝑠𝑠
𝑉𝑉𝑅𝑅
Typical values of 𝑘𝑘𝐸𝐸 are in the range of 0.8 to 1 and the timeconstant 𝜏𝜏𝐸𝐸 for an AC exciter in the range of 0.5 to1.0 s. Generator model. The transfer function relating the generator terminal voltage to its field voltage can be simplified to 𝑉𝑉𝑡𝑡 (𝑠𝑠)
𝑉𝑉𝐹𝐹 (𝑠𝑠)
=
𝑘𝑘𝐺𝐺
1+𝜏𝜏𝐺𝐺 𝑠𝑠
IV. OPTIMIZATION ALGORITHMS In this paper, we preferred two optimization algorithms for tuning the FOPID parameters. They are
(13)
The constants are load dependent, 𝑘𝑘𝐺𝐺 may vary between 0.7 and 1.0, and 𝜏𝜏𝐺𝐺 between 1.0 and 2.0 s from full load to no load.
A) Genetic Algorithm B) Ant Colony Optimization A) Genetic Algorithm
Measurement model. The voltage measurement block, including PT, rectifier and filter, is often modeled with a single time constant.
Genetic algorithm (GA) is the process of searching the most suitable one of the chromosomes that built the population in the potential solutions space. A search like this, tries to balance two opposite objectives: searching the best solutions (exploit) and expanding the search space. Thus, GA has become a robust optimization tool for solving the problems related to different field of the technical and social sciences [21, 22]. B) Ant Colony Optimization
𝑉𝑉𝑠𝑠 (𝑠𝑠)
𝑉𝑉𝑡𝑡 (𝑠𝑠)
=
𝑘𝑘𝑅𝑅
1+𝜏𝜏𝑅𝑅 𝑠𝑠
(14)
𝜏𝜏𝑅𝑅 ranges over 0.001 to 0.06 s. Excitation voltage limiters. AVR and exciter output voltages are limited by windup and non-windup limiters [20]. Also, dedicated over excitation and under excitation limiters are employed to assure safe operation of the generator. Block diagram of the AVR compensated with an FOPID controller is shown in Fig.4. In this figure, the combined effects of these limiters are represented by the upper and lower limits set to three times of the nominal value of the field voltage. B. Performance Criterion In control system design and analysis or for optimal control purposes, performance indices are calculated to be used as quantitative measures to evaluate a system’s performance, where a control system is judged as an optimum system when the system parameters are adjusted so that the index used in the design reaches its minimum value, while constraints of the controlled system are respected. The commonly used indices are integral of the square of the error (ISE), integral of the absolute value of the error (IAE),
The ant colony optimization algorithm (ACO) is an evolutionary meta-heuristic algorithm based on a graph representation that has been applied successfully to solve various hard combinatorial optimization problems. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Artificial ants walk through this graph, looking for good paths. Each ant has a rather simple behavior so that it will typically only find rather poorquality paths on its own. Better paths are found as the emergent result of the global cooperation among ants in the colony [23]. Each ant updates the pheromones deposited to the paths it followed after completing one tour and updates rules as follows 0.01𝜃𝜃 (17) 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 = 𝜏𝜏(𝑘𝑘 − 1)𝑖𝑖𝑖𝑖 + 𝐽𝐽
where 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 is pheromone value between nest i and j at kth iteration, θ is the general pheromone updating coefficient and 𝐽𝐽 is the cost function for the tour travelled by the ant.
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Fig.4. Block diagram of an AVR employing an FOPID controller
𝑘𝑘𝐴𝐴 = 10 , 𝜏𝜏𝐴𝐴 = 0.1 , 𝑘𝑘𝐸𝐸 = 1 , 𝜏𝜏𝐸𝐸 = 0.5 , 𝑉𝑉𝑓𝑓0 = 1 , 𝑘𝑘𝐺𝐺 = 1 , 𝜏𝜏𝐺𝐺 = 1 , 𝑘𝑘𝑅𝑅 = 1 and 𝜏𝜏𝑅𝑅 = 0.06 . Figure 6 shows the original terminal voltage step response of the system with unity gain controller instead of FOPID controller.
Pheromones of the path belonging to the best tour and worst tour of the ant colony are updated as follows [24]: 0.3𝜃𝜃 𝐽𝐽𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝜃𝜃
𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 = 𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 − 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖
(18)
= 𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖
(19)
𝐽𝐽𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
B. Details of FOPID Design Using GA for the AVR The following parameters are used for carrying out the FOPID design using GA:
Where 𝜏𝜏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and 𝜏𝜏 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 are the pheromones of the paths followed by the ant in the tour with the lowest cost value 𝐽𝐽𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and with the highest cost value 𝐽𝐽𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 in one iteration respectively. After pheromone evaporation the ant algorithm forget its history and directed to search towards new direction without being trapped in some local minima as 𝜆𝜆 + [𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ] 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 = τ(𝑘𝑘)𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 where 𝝀𝝀 is the evaporation constant.
Population size =90 Generations=100 Crossover fraction=0.75 Population initial range=[0;0.1] The order of approximation is set to N=3
C. Details of FOPID Design Using ACO for the AVR The following parameters are used for carrying out the FOPID design using ACO:
(20)
Number of ants =100 Pheromone=0.09 Evaporation Parameter =0.65 Positive Pheromone=0.2 Negative Pheromone=0.1 Maxtour=200 Minvalue=0 Maxvalue=1
D. First Test: Basic Performance The designed FOPID controller is approximated by oustaloup filter. The fractional derivative and integrals of FOPID controller have been approximated by N=3 and filter frequency [ 𝜔𝜔𝑏𝑏 , 𝜔𝜔ℎ ]=[1e-5,1e5]. The Bode diagrams of original FOPID and Oustaloup approximated FOPID controllers are shown in Fig.7, depicting very close behaviors in the frequency range of interest. Fig.8 shows the step response of the AVR system with FOPID tuned by both GA and ACO is better than AVR system with normal PID. Also the bode diagrams of the AVR system with FOPID is shown in the Fig.8.Both Figs.7 and 8 and Table I attest that the FOPID type AVR has better performance and higher robust stability than the PID type AVR.
Fig.5. Graphical Representation of ACO for FOPID Tuning Process
V. NUMERICAL RESULTS A. Study System Parameters A practical high-order AVR is used to verify the efficiency of the proposed FOPID controller. The system parameters are:
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E. Second Test: Robustness To examine numerically the robustness of the FOPID controller with respect to parameter uncertainties, the following simulation is performed. Assume that 𝑘𝑘𝐺𝐺 , 𝜏𝜏𝐺𝐺 ,𝑘𝑘𝐸𝐸 and 𝜏𝜏𝐸𝐸 are 20% increases due to the change in loading condition, the actual generator transfer function is given by 𝑉𝑉𝑡𝑡 (𝑠𝑠)
𝑉𝑉𝐹𝐹 (𝑠𝑠)
=
1
(21)
𝑠𝑠+0.769
Also the exciter transfer function is given by 𝑉𝑉𝐹𝐹 (𝑠𝑠)
𝑉𝑉𝑅𝑅 (𝑠𝑠)
=
2
(22)
𝑠𝑠+1.667
The step responses with both generator and exciter uncertainties are shown in Fig.10 and the system is proved to be robust.
Fig.6.Step Response of the original AVR
TABLE I COMPARISION OF THE EVALUATION VALUES BETWEEN BOTH PID AND FOPID CONTROLLERS Type of controller
𝑲𝑲𝑷𝑷
𝑲𝑲𝑰𝑰
𝑲𝑲𝑫𝑫
𝝀𝝀
𝝁𝝁
𝑴𝑴𝑷𝑷 (%)
𝒕𝒕𝒓𝒓
𝒕𝒕𝒔𝒔
𝑬𝑬𝒔𝒔𝒔𝒔
PID
0.2442
0.1423
0.0427
1
1
8.871
0.838
2.641
0
FOPID (ACO)
0.5
0.33
0.1667
0.9897
1.015
5.008
0.377
1.102
0
FOPID (GA)
0.2953
0.2003
0.0939
0.9989
0.9712
0.558
0.693
1.08
0
Fig.7. Bode diagrams of the original and approximated FOPID
Fig.8.Step response of the AVR controlled by PID and FOPID
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[3] [4] [5] [6]
[7] [8]
[9] Fig.9. Bode diagrams of the AVR controlled by PID and FOPID
[10] [11] [12]
[13]
[14] [15]
[16] Fig.10. Step response of the AVR controlled by PID and FOPID in the presence of generator and exciter uncertainties.
[17]
VI. CONCLUSION This paper presents a design method for determining the FOPID controller parameters using the Genetic Algorithm(GA) and Ant Colony Optimization (ACO) techniques. The proposed algorithms performed efficient search for the optimal FOPID controller parameters to the practical AVR system. Furthermore, it can be concluded from the above simulations that the proposed FOPID controller has more robust stability and performance characteristics than the PID controller applied to the AVR system.
[18] [19] [20] [21] [22]
VII. REFERENCES [1] Torvik P. J., Bagley R. L, “On the appearance of the fractional
[23]
derivative in the behavior of real materials”, Transactions of the ASME, 1984, 51(4):294–298. [2] Axtell M., Bise E. M,“Fractional calculus applications in control systems”,in Proceeding of the IEEE 1990 Natational Aerospace and Electronics Conference. New York, 1990, 563– 566.
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C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, “Fractional-Order Systems and Controls: Fundamentals and Applications”.New York: Springer-Verlag, 2010. Matignon D, “Stability result on fractional differential equations with applications to control processing”, in IMACSSMC Proceedings. Lille, France, 1996, 963–968. Oldham K. B., Spanier J, “The Fractional Calculus”, NewYork: Academic Press, 1974. Podlubny I, “The Laplace transform method for linear differential equations of the fractional order”, In Proceedings of the 9th International BERG Conference. Kosice, Slovak Republic, 1997, 119–119 (in Slovak). Podlubny I. “Fractional Differential Equations”, San Diego: Academic Press, 1999. Woon S. C, “Analytic continuation of operators — operators acting complex s-times. Applications: from number theory and group theory to quantum field and string theories”, reviews in Mathematical Physics, 1999, 11(4):463–501. Zavada P, “Operator of fractional derivative in the complex plane”, Communications in Mathematical Physics, 1998, 192(2):261–285. OustaloupA.“LaDérivation non Entière”,Paris:HERMES, 1995. Petras I., Dorcak L., Kostial I, “Control quality enhancement by fractional order controllers”, Acta Montanistica Slovaca, 1998, 2:143–148. Podlubny I, “Fractional-Order Systems and Fractional-Order Controllers”, Technical Report UEF-03-94, The Academy of Sciences Institute of Experimental Physics, Kosice, Slovak Republic, 1994. Podlubny I, “Fractional-order systems and PIλ Dμ – controllers”, IEEE Transactions on Automatic Control,1999, 44(1):208–214. Petras I., Podlubny I., O’Leary P, “Analogue Realization of Fractional Order Controllers”,Fakulta BERG, TU Košice, 2002 Oustaloup A., Levron F., Mathiew B., Nanot F, “Frequency band complex noninteger differentiator: Characterization and synthesis”, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2000, 47(1):25–39. Xue D., Zhao C. N., ChenY. Q, “A modified approximation method of fractional order system”, In Proceedings of the IEEE Conference on Mechatronics and Automation. Luoyang, China, 2006, 1043–1048. M. G. Safonov, R. Y. Chiang and D. J. N. Limebeer, “Optimal hankel model reduction for nonminimalsystems,” IEEE Trans. on Automat. Contr., vol. 35, no. 4, pp. 496-502, April 1990. IEEE Standard Definition for Excitation Systems for Synchronous Machines (ANSI), IEEE Std 421.1-1986. P.Kundur,Power System Stability and Control,McGraw-Hill, 1994. R. A. Krohling and J. P. Rey, “Design of optimal disturbance rejection PID controllers using genetic algorithm,” IEEE Trans. Evol. Comput., Vol. 5, pp. 78–82, Feb. 2001. Coley, D.A.:“An introduction to genetic algorithms for scientists and engineers”,World Scientific Pub.Co. Inc. (1999). Quagliarella, D., Periaux, J., Poloni, C., Winter, G.: “Genetic Algorithms and Evolution Strategy in Engineering Computer Science”, John Wiley, NY (1998). Duan, H.-B., Wang, D.-B., Yu, X.-F.: “Novel Approach to Nonlinear PID Parameter Optimization Using Ant Colony Optimization Algorithm”, Journal of Bionic Engineering 3, 73–78 (2006). Muhammet Ünal, Ayça Ak, Vedat Topuz, and Hasan Erdal.: “Optimization of PID Controllers Using Ant Colony and Genetic Algorithms” Springer 2013.
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IFAC-PapersOnLine 49-1 (2016) 456–461
Implementation of PID Controller for an AVR System Using Implementation of Fractional Fractional Order Order PID Controller for an AVR System Using Implementation Order PID Controller for an AVR System Using Implementation of of Fractional Fractional Order PID Controller for an AVR System Using GA and ACO Optimization Techniques GA and ACO Optimization Techniques GA and ACO Optimization Techniques GA and ACO Optimization Techniques
First A. G.Suri babu*. Second B.T.Chiranjeevi First A. G.Suri babu*. Second B.T.Chiranjeevi First A. G.Suri babu*. Second B.T.Chiranjeevi *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of Technology,Bhimavaram *Assistant Professor ,Electrical Engineering Department,Vishnu Institute First A. G.Suri babu*. Second B.T.Chiranjeevi *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of of Technology,Bhimavaram Technology,Bhimavaram India(Tel:91-9441901312;email:[email protected]) *Assistant Professor ,Electrical Engineering Department,Vishnu Institute of *Assistant Professor India(Tel:91-9441901312;email:[email protected]) ,Electrical Engineering Department,Vishnu Institute of Technology,Bhimavaram Technology,Bhimavaram India(Tel:91-9441901312;email:[email protected]) India(Tel:91-9441901312;email:[email protected]) **Research Scholar, NIT Silichar,India(email:[email protected]) India(Tel:91-9441901312;email:[email protected]) **Research Scholar, NIT Silichar,India(email:[email protected]) **Research Scholar, Silichar,India(email:[email protected]) **Research Scholar, NIT NIT Abstract: In recent years, the the research research andSilichar,India(email:[email protected]) studies on fractional fractional order (FO) (FO) modelling of of dynamic systems systems Abstract: In recent years, and Abstract: In recent years, the research and studies studies on on fractional order order (FO) modelling modelling of dynamic dynamic systems Abstract: In recent years, the research and studies on fractional order (FO) modelling of dynamic systems and controllers are quite advancing. To improve the stability and dynamic response of automatic voltage Abstract: In recent the researchToand studiesthe onstability fractional order (FO) modelling dynamic systems and controllers are years, quite advancing. improve and dynamic response ofofautomatic voltage regulator (AVR) system, we presented fractional Order PID (FOPID) controller by employing Genetic and controllers are quite advancing. To improve the stability and dynamic response of automatic voltage regulator (AVR) system, we presented fractional Order PID (FOPID) controller by employing Genetic and controllers quite advancing. To fractional improve the stability dynamic responsebyofemploying automatic Genetic voltage regulator (AVR)aresystem, we presented Order PID and (FOPID) controller Algorithm(AVR) (GA) and and Ant Colony Colony Optimization (ACO) techniques. We designed designed masked fractional order regulator system, we fractional Order PID (FOPID) controller by employing Genetic Algorithm (GA) Ant Optimization (ACO) techniques. We aaa masked fractional order regulator (AVR) system, we presented presented fractional Order PID (FOPID) controller by employing Genetic Algorithm (GA) and Ant Colony Optimization (ACO) techniques. We designed masked fractional (ACO) techniques. We designed a masked fractional order Algorithm (GA) and Ant Colony Optimization order PI D controller for any type of system in simulink. Comparisons PID block in Matlab to implement the D (ACO) techniques. We designed a masked fractional order Algorithm (GA) and to Antimplement Colony Optimization for any type of system in simulink. Comparisons PID block in Matlab the PI PI D controller for type system in Comparisons PID block in to implement the are aa PID controller from of response robustness characteristics. It PI D controller controller for any any type of ofand system in simulink. simulink. Comparisons PID blockwith in Matlab Matlab implement thestandpoints are made made with PID to controller from standpoints of transient transient response and robustness characteristics. It is is are made with a PID controller from standpoints of transient response and robustness characteristics. It is shown that the proposed FOPID controller can improve the performance of the AVR in all the aspects. are made PID controller standpoints of transient response and robustness shown thatwith the aproposed FOPID from controller can improve the performance of the AVR in characteristics. all the aspects. It is shown the proposed FOPID controller can improve the of in the aspects. © 2016,that IFAC Federation of Automatic Control) Hostingoptimization. by Elsevier Ltd. All rights shown that the(International proposed FOPID controller can improve the performance performance of the the AVR AVR in all all the reserved. aspects. Keywords: Fractional order PID Controller, Oustaloop, Ant colony Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. Keywords: Fractional order PID Controller, Oustaloop, Ant colony optimization. FRACTIONAL MODELING FRACTIONAL MODELING I. INTRODUCTION MODELING A. Order A. Integer IntegerFRACTIONAL Order Approximations Approximations I. INTRODUCTION FRACTIONAL MODELING A. Integer Order Approximations I. INTRODUCTION A. Integer Order Approximations Online real-time, fractional-order differentiation may I. INTRODUCTION Even so much research was developed in soft computing A. IntegerOnline Order real-time, Approximations fractional-order differentiation may Even so much research was developed in soft computing Online real-time, fractional-order be required in control systems. Using is best Even so much research was developed in soft computing techniques for designing controllers, still these techniques techniques for controllers, still techniques Online real-time, fractional-order differentiation may be required in control systems. Using filters filtersdifferentiation is one one of of the the may best Even so much research was developed soft computing techniques for designing designing controllers, stillinthese these techniques be required in control systems. Using filters is one of the ways to solve the problems. By using filters, the fractional techniques for designing controllers, still these techniques fails in particular areas. Neuro-controllers works according to be required in control systems.By Using filters is one offractional the best best techniques for designing controllers, stillworks theseaccording techniques ways to solve the problems. using filters, the fails in particular areas. Neuro-controllers to ways to solvefunction the problems. By usingto fractional order is integer order. order transfer transfer function is approximated approximated tofilters, integerthe order. fails in areas. Neuro-controllers works to the given to them and their performance depends on ways to solvefunction the problems. By usingto filters, the fractional failstraining in particular particular areas. Neuro-controllers works according according to order transfer is approximated integer order. the training given to them and their performance depends on order transfer is to training given tovast them and their performance depends on the availability of data of training inputs and target B. Oustaloup’s Recursive Filter the availability of data set of inputs and order transfer function function is approximated approximated to integer integer order. order. B. Oustaloup’s Recursive Filter the given them andset performance on availability of tovast vast data settheir of training training inputsdepends and target target B. Oustaloup’s Recursive Filter the training values of the process. Certainly there is a probability of B. Oustaloup’s Recursive Filter the availability of vast data set of training inputs and target values of the process. Certainly there is a probability of Some continuous filters have been summarized in the availability of vast data set of training inputs and target values of the process. Certainly there is a probability of B. Oustaloup’s Recursive Filter Some filters have in Some continuous continuous have been been summarized summarized in values of process. Certainly there aa controller probability of [13]. the the Oustaloup missed in training set, case [13]. Among Among the filters, filters, filters the well-established well-established Oustaloup values data of the the process. Certainly there isthe probabilityfails of missed data in the the training set, in in such such caseis the controller fails Some continuous filters have been summarized in [13]. Among the filters, the well-established Oustaloup missed data in the training set, in such case the controller fails to generate an accurate output. Also the performance of the [13]. Among the filters, the well-established Oustaloup recursive filter has a very good fitting to the fractional-order missed data an in the training set, inAlso such the caseperformance the controlleroffails to generate accurate output. the [13]. Among the filters, the well-established Oustaloup recursive filter has a very good fitting to the fractional-order to generate an output. Also performance of fuzzy controllers is purely based on thethe rule base, selection of fuzzy controllers is based on rule base, of recursive has fitting the fractional-order differentiators [14,15,16]. Assume thatto the expected fitting to generate an accurate accurate output. performance of the the fuzzy controllers is purely purely based Also on the thethe rule base, selection selection of recursive filter filter [14,15,16]. has aa very very good good fitting tothe the expected fractional-order differentiators Assume that fitting differentiators [14,15,16]. Assume that the expected fitting (𝜔𝜔 ). fuzzy controllers is purely based on the rule base, selection of range is membership functions and their range and it was still , 𝜔𝜔 (𝜔𝜔 ). range is 𝑏𝑏 ℎ , 𝜔𝜔 fuzzy controllers is purely based on the rule base, selection of 𝑏𝑏 ℎ membership functions and their range and it was still differentiators [14,15,16]. Assume that the expected fitting ). (𝜔𝜔 range is 𝑏𝑏 ℎ 𝑏𝑏 ,, 𝜔𝜔 ℎ ). membership functions and their range and it was still challenging to decide the fuzzy parameters. Still today most (𝜔𝜔 range is 𝜔𝜔 𝑏𝑏 , 𝜔𝜔 ℎ 𝑏𝑏 ℎ membership their range and was most still challenging tofunctions decide theand fuzzy parameters. Stillittoday The can range is (𝜔𝜔 𝑏𝑏 be The filter filter can beℎ ).written written as as challenging to the parameters. Still most of the control problems are smoothly solved by PID challenging to decide decide the fuzzy fuzzy parameters. Still today today most of the control problems are smoothly solved by PID The filter can be written as ′′ The filter can be written as 𝑠𝑠+𝜔𝜔 of the control problems are smoothly solved by PID controllers due to simplicity in their design and easy 𝑠𝑠+𝜔𝜔𝑘𝑘 𝑁𝑁 𝑘𝑘 ′′ controllers due simplicity in 𝑁𝑁 of the control problems are smoothly solved and by easy PID 𝑠𝑠+𝜔𝜔 controllers due to to simplicity in their their design design and easy (𝑠𝑠) ∏ 𝑠𝑠+𝜔𝜔 = 𝐾𝐾 (1) 𝐺𝐺 𝑁𝑁 𝑘𝑘 𝑓𝑓 𝑘𝑘=−𝑁𝑁 ′′ 𝑁𝑁 𝑘𝑘 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 𝑠𝑠+𝜔𝜔𝑘𝑘 ∏ = 𝐾𝐾 (1) 𝐺𝐺𝑓𝑓𝑓𝑓𝑓𝑓 (𝑠𝑠) ′ controllers due simplicity in their and 𝑁𝑁 implementation. been two extra of 𝑘𝑘=−𝑁𝑁 𝑁𝑁 𝑘𝑘 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 controllers due It tohas simplicity in that their design and easy easy (𝑠𝑠) ∏ = 𝐾𝐾 𝐺𝐺 (1) implementation. Itto has been shown shown that twodesign extra degrees degrees of 𝑁𝑁 𝑘𝑘 𝑠𝑠+𝜔𝜔 𝑓𝑓 𝑘𝑘=−𝑁𝑁 𝑘𝑘 𝑓𝑓 𝑘𝑘=−𝑁𝑁 (𝑠𝑠) ∏ = 𝐾𝐾 𝑘𝑘=−𝑁𝑁 𝑠𝑠+𝜔𝜔 𝐺𝐺𝑓𝑓 (1) 𝑠𝑠+𝜔𝜔𝑘𝑘 𝑘𝑘 implementation. It has been shown that two extra degrees of freedom from the use of a fractional-order integrator and 𝑠𝑠+𝜔𝜔 implementation. It has been shown that two extra degreesand of freedom from the use of a fractional-order integrator 𝑘𝑘 can be evaluated where the poles, zeros, and gain of the filter where the poles, zeros, and gain of the filter can be evaluated freedom from the use of a fractional-order integrator and differentiator make it possible to further improve the differentiator make it possible to further improve the freedom from make the useit ofpossible a fractional-order and differentiator to further integrator improve the where the zeros, and from such that where the poles, poles, zeros, and gain gain of of the the filter filter 11can can be be evaluated evaluated from such that differentiator it to improve the performance ofmake traditional PID controllers. Details of past and 1 performance traditional PID Details of 1(1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ from such that differentiator it possible possible to further further improve the (1−𝛾𝛾) (1+𝛾𝛾) performance of ofmake traditional PID controllers. controllers. Details of past past and and 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 1 1 2 2 1 1 2 2 from such 𝜔𝜔that𝑘𝑘+𝑁𝑁+ (1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ (1−𝛾𝛾) (1+𝛾𝛾) 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 1 1 performance of traditional PID controllers. Details of past and present progress in the analysis of dynamic systems modeled 1 1 𝜔𝜔 2 2 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 2(1−𝛾𝛾) 2(1+𝛾𝛾) h 𝑘𝑘+𝑁𝑁+ h 𝑘𝑘+𝑁𝑁+ 𝑘𝑘+𝑁𝑁+ 2𝑁𝑁+1 2𝑁𝑁+1 performance of traditional PID controllers. of modeled past and h 1 1 present progress in the analysis of dynamicDetails systems 2(1−𝛾𝛾) 2(1+𝛾𝛾) 𝜔𝜔h 𝜔𝜔 𝜔𝜔𝑘𝑘𝑘𝑘′′′′ = = 𝜔𝜔𝑏𝑏𝑏𝑏 ((𝜔𝜔 )𝑘𝑘+𝑁𝑁+ 𝜔𝜔𝑘𝑘𝑘𝑘 = = 𝜔𝜔 𝜔𝜔𝑏𝑏𝑏𝑏 ((𝜔𝜔 ,𝐾𝐾 = = 𝜔𝜔 𝜔𝜔ℎℎ𝛾𝛾𝛾𝛾𝛾𝛾 2𝑁𝑁+1 2𝑁𝑁+1 2 2 (1−𝛾𝛾) ,, 𝜔𝜔 (1+𝛾𝛾) ,𝐾𝐾 𝑘𝑘+𝑁𝑁+ 𝜔𝜔h 2𝑁𝑁+1 2𝑁𝑁+1 𝜔𝜔 ))𝑘𝑘+𝑁𝑁+ h) 2 2 present progress in analysis dynamic systems by Fractional order differential can be 𝜔𝜔h 𝜔𝜔h 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 𝜔𝜔𝑘𝑘𝑘𝑘′′ = 𝜔𝜔 𝜔𝜔𝑏𝑏𝑏𝑏 (𝜔𝜔 , 𝜔𝜔 = 𝜔𝜔 ( ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔 𝜔𝜔 h) h) 𝑏𝑏 𝑏𝑏 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 𝑘𝑘 𝑏𝑏 present progress in the the analysis of ofequations dynamic (FODEs) systems modeled modeled h h 𝑏𝑏 𝑏𝑏 ℎ by Fractional order differential equations (FODEs) can be 𝑘𝑘 𝑏𝑏 𝜔𝜔 𝜔𝜔 ℎ 2𝑁𝑁+1 2𝑁𝑁+1 𝛾𝛾 ′ = 𝜔𝜔𝑏𝑏 𝜔𝜔 ,, 𝜔𝜔 𝜔𝜔 ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔h 𝜔𝜔h 𝑏𝑏 𝑏𝑏 𝑏𝑏 ( 𝑘𝑘 = 𝑏𝑏 ( 𝑘𝑘 ℎ by Fractional order differential equations (FODEs) can be 𝑏𝑏 ) 𝑏𝑏 ) found in [1–14]. Literature survey gives the fractional 𝑘𝑘 𝑏𝑏 𝑘𝑘 ℎ 𝜔𝜔 = 𝜔𝜔 ( ) 𝜔𝜔 = 𝜔𝜔 ( ) ,𝐾𝐾 = 𝜔𝜔 𝜔𝜔 𝜔𝜔 found in [1–14]. Literature survey gives the fractional (2) 𝜔𝜔 𝜔𝜔 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑘𝑘 𝑏𝑏 𝑘𝑘 ℎ by Fractional order Literature differential survey equations (FODEs) can be found in [1–14]. gives the fractional 𝜔𝜔𝑏𝑏 𝜔𝜔𝑏𝑏 (2) 𝑏𝑏 𝑏𝑏 controller which was developed by Crone in [3], while [4, (2) found in [1–14]. Literature survey gives the fractional controller was developed by [3], while found in which [1–14]. survey givesin controller which was Literature developed by Crone Crone in the [3], fractional while [4, [4, (2) 𝜇𝜇 𝜇𝜇 controller which was developed by Crone in [3], while [4, where 𝛾𝛾 is the order of the differentiation and N is the order 12, presented the 𝑃𝑃𝑃𝑃 and [3,14] proposed 𝜇𝜇 where 𝛾𝛾 𝛾𝛾 is is the the order order of of the the differentiation differentiation and and N N is is the the order order 𝜇𝜇 controller controller which was developed by Crone in [3], while the [4, 12,λλ 13] 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the where 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜇𝜇 where 𝛾𝛾 is the order of the differentiation and N is the order 𝜇𝜇 12, 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the 𝐷𝐷 controller. 𝑃𝑃𝐼𝐼 of approximation. λ 𝜇𝜇 𝜇𝜇 controller. 12,λλλ 𝐷𝐷 13] presented the 𝑃𝑃𝑃𝑃 controller and [3,14] proposed the where 𝛾𝛾 is the order of the differentiation and N is the order 𝑃𝑃𝐼𝐼 of approximation. 𝜇𝜇 𝜇𝜇 of approximation. 𝑃𝑃𝐼𝐼 C. A Refined Oustaloup Filter 𝐷𝐷𝜇𝜇 controller. controller. 𝑃𝑃𝐼𝐼λ 𝐷𝐷 of approximation. C. A Refined Oustaloup Filter λ 𝜇𝜇 𝜇𝜇 C. A A Refined Refined Oustaloup Oustaloup Filter Filter λ We extended the benefits of 𝑃𝑃𝐼𝐼 for λ 𝜇𝜇 C. Here we introduce new approximate realization λ 𝐷𝐷 𝜇𝜇 controller HereOustaloup we introduce introduce new approximate approximate realization realization We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 for 𝜇𝜇 C. A Refined Filteraaa new 𝜇𝜇 Here we λ𝜇𝜇 𝜇𝜇 controller λλ 𝐷𝐷λ We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 controller for controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 𝜇𝜇 Here we introduce a new approximate realization λλ 𝐷𝐷λ𝜇𝜇 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 method for the fractional-order derivative in the frequency 𝜇𝜇 We extended the benefits of 𝑃𝑃𝐼𝐼 𝐷𝐷 controller for AVR system. In Particular a masked 𝑃𝑃𝐼𝐼λλ 𝐷𝐷𝜇𝜇𝜇𝜇 controller block Here we introduce a new approximate realization method for the fractional-order derivative in the frequency is developed in matlab simulink to optimize the parameters λ 𝐷𝐷𝜇𝜇 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 ].Our method for the fractional-order derivative in the frequency is developed in matlab simulink to optimize the parameters range of interest[𝜔𝜔 proposed method here gives aa , 𝜔𝜔 ].Our 𝑏𝑏 ℎ 𝐷𝐷 controller block AVR system. In Particular a masked 𝑃𝑃𝐼𝐼 𝑏𝑏 , 𝜔𝜔ℎ ].Our proposed is developed in matlab simulink to optimize the parameters methodoffor the fractional-order derivative in the frequency range interest[𝜔𝜔 method here gives 𝑏𝑏 ℎ based on the theinrecently recently developed optimization techniques 𝑏𝑏 , than ℎ ].Our is developed matlab to optimize range of interest[𝜔𝜔 proposed method here gives 𝜔𝜔 better approximation Oustaloup’s method with respect to based on developed techniques 𝑏𝑏 ℎ better approximation than Oustaloup’s method with respect to is developed matlab simulink simulink to optimization optimize the the parameters parameters 𝑏𝑏 ℎ based on thein recently developed optimization techniques ].Our range approximation of interest[𝜔𝜔𝑏𝑏 , than proposedmethod methodwith here gives toaa 𝜔𝜔ℎ Oustaloup’s better respect based on the recently developed optimization techniques instead of using toolbox which was restrained for using better approximation than Oustaloup’s method with respect to both low frequency and high frequency. Assume that based onof the recently developed optimization techniques better approximation method with respect to instead using toolbox which was restrained for using both low frequency than and Oustaloup’s high frequency. Assume that the the instead of using toolbox which was restrained for using advanced optimization techniques. Genetic Algorithm (GA) advanced optimization techniques. Genetic Algorithm (GA) both low frequency and high frequency. Assume that the (𝜔𝜔 ). frequency range to be fit is defined as Within , 𝜔𝜔 𝑏𝑏 ℎ instead ofoptimization using toolbox which was restrained for using 𝑏𝑏 ,Assume ℎ ). Within both low frequency and high frequency. that the advanced techniques. Genetic Algorithm (GA) (𝜔𝜔 frequency range to be fit is defined as 𝜔𝜔 the advanced optimization techniques. Algorithm (GA) (𝜔𝜔𝑏𝑏𝑏𝑏 ℎℎℎℎ ). and Ant Colony optimization (ACOGenetic ) techniques has already frequency range to is as the pre-specified frequency the fractional-order operator advanced optimization techniques. Genetic Algorithm (GA) frequency range to be be fit fitrange, is defined defined as (𝜔𝜔𝑏𝑏𝑏𝑏𝑏𝑏 ,, 𝜔𝜔 Within the 𝜔𝜔ℎ ). Within and Ant Colony optimization (ACO )) techniques has already pre-specified frequency range, the fractional-order operator 𝛼𝛼 𝛼𝛼 can be approximated and Ant Colony optimization (ACO techniques has already been used to determine optimal solution to several power pre-specified frequency range, the fractional-order operator 𝑠𝑠 by the transfer 𝛼𝛼 can be approximated 𝑠𝑠pre-specified by the fractional-order transfer and Ant Colony optimization (ACO ) techniques has already been used to determine optimal solution to several power 𝛼𝛼 frequency range, the fractional-order operator 𝑠𝑠 can be approximated by the transfer 𝛼𝛼 fractional-order been determine solution to power engineering problems andoptimal we employed these algorithms to function 𝑠𝑠𝑠𝑠 𝛼𝛼𝛼𝛼 can be transfer function as been used used to toproblems determine optimal solutionthese to several several power can as be approximated approximated by by the the fractional-order fractional-order transfer engineering and we employed algorithms to function as engineering problems and we employed these algorithms to design controller for Automatic voltage regulator ′′ ⁄𝜔𝜔 design an an FOPID FOPID controller for Automatic voltage regulator 1+𝑠𝑠 𝜔𝜔 ⁄ function as 1+𝑠𝑠 𝑁𝑁 𝑘𝑘 engineering problems and we employed these algorithms to ′′ FOPID controller for Automatic voltage regulator 𝑁𝑁 𝑘𝑘 function as design an ⁄ (𝑠𝑠) ∏ 1+𝑠𝑠 𝜔𝜔 = lim 𝐾𝐾(𝑠𝑠) = lim 𝐾𝐾 (3) ⁄𝜔𝜔𝑘𝑘 1+𝑠𝑠⁄ 𝑁𝑁 𝑁𝑁→∞ 𝐾𝐾𝑁𝑁 𝑁𝑁 (𝑠𝑠) lim 𝐾𝐾(𝑠𝑠) (3) 𝑘𝑘=−𝑁𝑁 ′′ 𝑁𝑁 design FOPID controller for Automatic regulator (AVR) problem. proposed simulated within 1+𝑠𝑠⁄𝜔𝜔 𝜔𝜔𝑘𝑘 (𝑠𝑠) = 𝑁𝑁→∞ ∏𝑘𝑘=−𝑁𝑁 lim ∏ 𝐾𝐾(𝑠𝑠) = = lim lim𝑁𝑁→∞ (3) 𝑁𝑁→∞ ′ 𝑘𝑘 𝑁𝑁 design an FOPIDThe controller forcontroller Automaticis voltage regulator 𝑁𝑁→∞ 𝑁𝑁 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠 𝑁𝑁 𝑘𝑘 (AVR) an problem. The proposed controller is voltage simulated within 𝑁𝑁→∞ 𝐾𝐾 𝑁𝑁 (𝑠𝑠) = ⁄ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠 𝜔𝜔 ∏ 𝑁𝑁 ⁄ 𝑁𝑁→∞ 𝑘𝑘 = lim 𝐾𝐾(𝑠𝑠) = lim 𝐾𝐾 (3) 1+𝑠𝑠 𝜔𝜔 𝑁𝑁→∞ 𝐾𝐾𝑁𝑁 𝑁𝑁 (𝑠𝑠) = 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠⁄𝜔𝜔𝑘𝑘 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 (AVR) problem. The proposed controller is simulated within various scenarios and its performance is compared with those ∏ lim 𝐾𝐾(𝑠𝑠) = lim (3) According to the recursive distribution of real zeros and 𝑁𝑁→∞ 𝑘𝑘=−𝑁𝑁 1+𝑠𝑠⁄𝜔𝜔𝑘𝑘 𝑘𝑘 𝑁𝑁→∞ 𝑁𝑁 𝑁𝑁→∞ (AVR) problem. The proposed controller is simulated within various scenarios and its performance is compared with those ⁄ 𝜔𝜔𝑘𝑘 zeros and According to the recursive distribution of1+𝑠𝑠real 𝑁𝑁→∞ various scenarios and its performance is compared with those of an optimally-designed PID controller. Transient response of an optimally-designed PID controller. Transient response According to the recursive distribution of real zeros and poles, the zero and pole of rank 𝑘𝑘 can be written as various scenarios and its performance is compared those of an optimally-designed PID controller. Transientwith response According to and the pole recursive of real poles, the zero of rankdistribution 𝑘𝑘 can be written as zeros and of anperformance optimally-designed PID controller. response and robustness characteristics of both controllers and robustness characteristics of poles, 𝑘𝑘 can be written as of optimally-designed PID controller. Transient Transient response andanperformance performance robustness characteristics of both both controllers controllers poles, the the zero zero and and pole pole of of rank rank 𝑘𝑘 can be written as 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 and performance robustness characteristics of both are and the in 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 and studied performance robustness of characteristics of controller both controllers controllers 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 are studied and superiority superiority of the proposed proposed controller in all all 𝑑𝑑𝜔𝜔𝑏𝑏 𝑏𝑏𝜔𝜔ℎ 𝑏𝑏 2𝑁𝑁+1 ℎ 2𝑁𝑁+1 2𝑁𝑁+1 2𝑁𝑁+1 ′′ 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝛼𝛼−2𝑘𝑘 , 𝜔𝜔𝑘𝑘 = (𝑏𝑏𝜔𝜔 𝛼𝛼+2𝑘𝑘 𝑑𝑑𝜔𝜔 2𝑁𝑁+1 2𝑁𝑁+1 are studied and superiority of the proposed controller in all two respects is illustrated. 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝑏𝑏 ℎ 𝜔𝜔 = ( ) ) (4) ′ 2𝑁𝑁+1 2𝑁𝑁+1 𝑘𝑘 𝛼𝛼−2𝑘𝑘 𝛼𝛼+2𝑘𝑘 𝑏𝑏 ℎ ′ = (𝑑𝑑𝜔𝜔 𝑘𝑘 = (𝑏𝑏𝜔𝜔 𝑘𝑘 are studied and superiority of the proposed controller in all 𝑑𝑑𝜔𝜔 𝑏𝑏𝜔𝜔 𝑏𝑏 𝑑𝑑 two respects is illustrated. 2𝑁𝑁+1 2𝑁𝑁+1 ) , 𝜔𝜔 ) (4) 𝜔𝜔 𝑏𝑏 ℎ 𝑏𝑏 𝑑𝑑 ′𝑘𝑘 2𝑁𝑁+1 2𝑁𝑁+1 𝑏𝑏 )2𝑁𝑁+1 , 𝜔𝜔𝑘𝑘 ℎ )2𝑁𝑁+1 ′ = (𝑑𝑑𝜔𝜔 𝑘𝑘 𝑘𝑘 𝑏𝑏𝜔𝜔 two respects is illustrated. 𝜔𝜔 = ( (4) 𝑏𝑏 𝑑𝑑 𝑏𝑏 ℎ ′ 𝑏𝑏 𝑑𝑑 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 two respects is illustrated. 𝜔𝜔𝑘𝑘 = ( 𝑏𝑏𝑏𝑏 ) , 𝜔𝜔𝑘𝑘 = ( 𝑑𝑑𝑑𝑑 ) (4)
2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All𝑏𝑏 rights reserved. Copyright 2016 responsibility IFAC 456Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 456 10.1016/j.ifacol.2016.03.096 Copyright © 2016 IFAC 456 Copyright © 2016 IFAC 456
𝑑𝑑
IFAC ACODS 2016 First A. G.Suri babu et al. / IFAC-PapersOnLine 49-1 (2016) 456–461 February 1-5, 2016. NIT Tiruchirappalli, India
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1.013𝑒𝑒 −5 𝑠𝑠 + 𝑠𝑠 3 + 0.2164𝑠𝑠 2 + 0.001564𝑠𝑠 + 4.032𝑒𝑒 −7 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 4 𝑠𝑠 + 0.2172𝑠𝑠 3 + 0.001577𝑠𝑠 2 + 4.234𝑒𝑒 −7 𝑠𝑠 + 4.084𝑒𝑒 −12
Through confirmation by experimentation and theoretical analysis, the synthesis approximation can obtain the good effect when 𝑏𝑏 = 10 and 𝑑𝑑 = 9.
(8)
𝜇𝜇 = 𝑠𝑠 0.9712 is approximated to integer order by using oustaloup filter as follows
II. FRACTIONALORDER PID CONTROLLER The block diagram of Fractional Order PID Controller is shown in Fig.1
7.178𝑒𝑒 4 𝑠𝑠 7 + 2.914𝑒𝑒 8 𝑠𝑠 6 + 4.251𝑒𝑒10 𝑠𝑠 5 + 2.309𝑒𝑒11 𝑠𝑠 4 + 4.674𝑒𝑒10 𝑆𝑆 3 + 3.527𝑒𝑒 8 𝑆𝑆 2 + 9.907𝑒𝑒 4 𝑆𝑆 + 1 𝐺𝐺 = (9) 7 𝑠𝑠 + 9.907𝑒𝑒 4 𝑠𝑠 6 + 3.527𝑒𝑒 8 𝑠𝑠 5 + 4.674𝑒𝑒10 𝑠𝑠 4 + 2.309𝑒𝑒11 𝑠𝑠 3 + 4.251𝑒𝑒10 𝑠𝑠 2 + 2.914𝑒𝑒 8 𝑆𝑆𝑆𝑆 + 7.178𝑒𝑒 4
The above transfer function is reduced to 4th order by using Hankel Minimum Degree Approximation(MDA) as follows 7.178𝑒𝑒 4 𝑠𝑠 4 + 2.913𝑒𝑒 8 𝑠𝑠 3 + 4.218𝑒𝑒10 𝑠𝑠 2 + 2.047𝑒𝑒11 𝑆𝑆 + 3.307𝑒𝑒 9 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑠𝑠 4 + 9.906𝑒𝑒 4 𝑠𝑠 3 + 3.525𝑒𝑒 8 𝑠𝑠 2 + 4.634𝑒𝑒10 𝑠𝑠 + 2.024𝑒𝑒 11
(10)
Fig.1. Block Diagram of Fractional Order PID controller
The differential equation of a fractional order PI λ Dμ controller is described by: 𝜇𝜇
𝒰𝒰(𝑡𝑡) = 𝐾𝐾𝑃𝑃 𝑒𝑒(𝑡𝑡) + 𝐾𝐾𝐼𝐼 𝐷𝐷𝑡𝑡−𝜆𝜆 𝑒𝑒(𝑡𝑡) + 𝐾𝐾𝐷𝐷 𝐷𝐷𝑡𝑡 𝑒𝑒(𝑡𝑡)
(5)
The continuous transfer function of FOPID is obtained through Laplace transform and is given by: 𝐺𝐺𝐶𝐶 (𝑆𝑆) = 𝐾𝐾𝑃𝑃 + 𝐾𝐾𝐼𝐼 𝑠𝑠 −𝜆𝜆 + 𝐾𝐾𝐷𝐷 𝑠𝑠 𝜇𝜇
(6)
We designed masked fractional order PID block in Matlab simulink as shown in Fig.2.The dialog box of fractional PID block is as shown in Fig.3
Fig.2. Masked fractional order PID block
From Fig.3, 𝐾𝐾𝑃𝑃 , 𝐾𝐾𝐼𝐼 and 𝐾𝐾𝐷𝐷 are proportional, integrator and derivative gains respectively. Gamma and delta are orders of the integral and derivative parts, 𝜆𝜆 and 𝜇𝜇 respectively. 𝜔𝜔𝑏𝑏 and 𝜔𝜔ℎ are lower and upper frequencies respectively, for which the fractional order inregral and derivative are approximated to integer order. Here, the fractional order integral and derivative are approximated to integer order by using a oustaloup filter and N is the order of approximation. 𝜆𝜆 = 𝑠𝑠 −0.9989 is approximated to integer order by using oustaloup filter as follows 1.013𝑒𝑒 −5 𝑠𝑠 7 + 1.05𝑠𝑠 6 + 3913𝑠𝑠 5 + 5.427𝑒𝑒 5 𝑠𝑠 4 + 2.806𝑒𝑒 6 𝑆𝑆 3 + 5.407𝑒𝑒 5 𝑆𝑆 2 + 3879𝑆𝑆 + 1 𝐺𝐺 = 𝑠𝑠 7 + 3879𝑠𝑠 6 + 5.407𝑒𝑒 5 𝑠𝑠 5 + 2.806𝑒𝑒 6 𝑠𝑠 4 + 5.427𝑒𝑒 5 𝑠𝑠 3 + 3913𝑠𝑠 2 + 1.05𝑠𝑠 + 1.013𝑒𝑒 −5
Fig.3. Dialog box of fractional PID block
III. AVR DESIGN USING FOPID CONTROLLER A. Linearized Model of Excitation System The role of an Automatic Voltage Regulator (AVR) is to hold the terminal voltage of a synchronous generator at a specified level. We consider an alternator supplied controlled rectifier excitation system [18] for simulation. Five main components, namely amplifier, exciter, excitation voltage limiters, generator, measurement and filtering includes in the mathematical model of AVR system by ignoring the saturation and other non-linearities [19]. The transfer function of these components are represented as follows.
(7)
In-order to make designing process easier the order of the approximated integer order of transfer function is reduced to 4th order by using Hankel Minimum Degree Approximation(MDA) [17].
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Amplifier model. The amplifier model is given by 𝑉𝑉𝑅𝑅 (𝑠𝑠)
𝑉𝑉𝐶𝐶 (𝑠𝑠)
=
𝑘𝑘𝐴𝐴
1+𝜏𝜏𝐴𝐴 𝑠𝑠
integral of time multiplied by absolute value of the error (ITAE),integral of time multiplied by the squared error (ITSE). The ITAE measure, which will be implemented in this paper, is given by the following equation
(11)
Typical values of 𝑘𝑘𝐴𝐴 are in the range of 10 to 400. The amplifier time-constant often ranges from 0.02 to 0.1 s.
(15) ITAE= ∫0 𝑡𝑡|𝑒𝑒(𝑡𝑡)|𝑑𝑑𝑑𝑑 Therefore, the proposed performance criterion 𝐽𝐽(𝑘𝑘) is defined as 𝐽𝐽(𝑘𝑘) =(ITAE) (16) ∞
Exciter model. Exciter model and parameters greatly depend on its type. A simplified transfer function of a modern exciter is 𝑉𝑉𝐹𝐹 (𝑠𝑠) 𝑘𝑘 = 𝐸𝐸 (12) (𝑠𝑠)
where k is [𝐾𝐾𝑃𝑃 , 𝐾𝐾𝐼𝐼 , 𝐾𝐾𝐷𝐷 , 𝜆𝜆, 𝜇𝜇]. The Genetic Algorithm (GA) and Ant Colony Optimization (ACO) are utilized to design these five controller parameters such that the controlled system exhibits desired response and robust stability as evaluated by the proposed performance criterion.
1+𝜏𝜏𝐸𝐸 𝑠𝑠
𝑉𝑉𝑅𝑅
Typical values of 𝑘𝑘𝐸𝐸 are in the range of 0.8 to 1 and the timeconstant 𝜏𝜏𝐸𝐸 for an AC exciter in the range of 0.5 to1.0 s. Generator model. The transfer function relating the generator terminal voltage to its field voltage can be simplified to 𝑉𝑉𝑡𝑡 (𝑠𝑠)
𝑉𝑉𝐹𝐹 (𝑠𝑠)
=
𝑘𝑘𝐺𝐺
1+𝜏𝜏𝐺𝐺 𝑠𝑠
IV. OPTIMIZATION ALGORITHMS In this paper, we preferred two optimization algorithms for tuning the FOPID parameters. They are
(13)
The constants are load dependent, 𝑘𝑘𝐺𝐺 may vary between 0.7 and 1.0, and 𝜏𝜏𝐺𝐺 between 1.0 and 2.0 s from full load to no load.
A) Genetic Algorithm B) Ant Colony Optimization A) Genetic Algorithm
Measurement model. The voltage measurement block, including PT, rectifier and filter, is often modeled with a single time constant.
Genetic algorithm (GA) is the process of searching the most suitable one of the chromosomes that built the population in the potential solutions space. A search like this, tries to balance two opposite objectives: searching the best solutions (exploit) and expanding the search space. Thus, GA has become a robust optimization tool for solving the problems related to different field of the technical and social sciences [21, 22]. B) Ant Colony Optimization
𝑉𝑉𝑠𝑠 (𝑠𝑠)
𝑉𝑉𝑡𝑡 (𝑠𝑠)
=
𝑘𝑘𝑅𝑅
1+𝜏𝜏𝑅𝑅 𝑠𝑠
(14)
𝜏𝜏𝑅𝑅 ranges over 0.001 to 0.06 s. Excitation voltage limiters. AVR and exciter output voltages are limited by windup and non-windup limiters [20]. Also, dedicated over excitation and under excitation limiters are employed to assure safe operation of the generator. Block diagram of the AVR compensated with an FOPID controller is shown in Fig.4. In this figure, the combined effects of these limiters are represented by the upper and lower limits set to three times of the nominal value of the field voltage. B. Performance Criterion In control system design and analysis or for optimal control purposes, performance indices are calculated to be used as quantitative measures to evaluate a system’s performance, where a control system is judged as an optimum system when the system parameters are adjusted so that the index used in the design reaches its minimum value, while constraints of the controlled system are respected. The commonly used indices are integral of the square of the error (ISE), integral of the absolute value of the error (IAE),
The ant colony optimization algorithm (ACO) is an evolutionary meta-heuristic algorithm based on a graph representation that has been applied successfully to solve various hard combinatorial optimization problems. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Artificial ants walk through this graph, looking for good paths. Each ant has a rather simple behavior so that it will typically only find rather poorquality paths on its own. Better paths are found as the emergent result of the global cooperation among ants in the colony [23]. Each ant updates the pheromones deposited to the paths it followed after completing one tour and updates rules as follows 0.01𝜃𝜃 (17) 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 = 𝜏𝜏(𝑘𝑘 − 1)𝑖𝑖𝑖𝑖 + 𝐽𝐽
where 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 is pheromone value between nest i and j at kth iteration, θ is the general pheromone updating coefficient and 𝐽𝐽 is the cost function for the tour travelled by the ant.
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459
Fig.4. Block diagram of an AVR employing an FOPID controller
𝑘𝑘𝐴𝐴 = 10 , 𝜏𝜏𝐴𝐴 = 0.1 , 𝑘𝑘𝐸𝐸 = 1 , 𝜏𝜏𝐸𝐸 = 0.5 , 𝑉𝑉𝑓𝑓0 = 1 , 𝑘𝑘𝐺𝐺 = 1 , 𝜏𝜏𝐺𝐺 = 1 , 𝑘𝑘𝑅𝑅 = 1 and 𝜏𝜏𝑅𝑅 = 0.06 . Figure 6 shows the original terminal voltage step response of the system with unity gain controller instead of FOPID controller.
Pheromones of the path belonging to the best tour and worst tour of the ant colony are updated as follows [24]: 0.3𝜃𝜃 𝐽𝐽𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝜃𝜃
𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 = 𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 − 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖
(18)
= 𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖
(19)
𝐽𝐽𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
B. Details of FOPID Design Using GA for the AVR The following parameters are used for carrying out the FOPID design using GA:
Where 𝜏𝜏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and 𝜏𝜏 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 are the pheromones of the paths followed by the ant in the tour with the lowest cost value 𝐽𝐽𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and with the highest cost value 𝐽𝐽𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 in one iteration respectively. After pheromone evaporation the ant algorithm forget its history and directed to search towards new direction without being trapped in some local minima as 𝜆𝜆 + [𝜏𝜏(𝑘𝑘)𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝜏𝜏(𝑘𝑘)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ] 𝜏𝜏(𝑘𝑘)𝑖𝑖𝑖𝑖 = τ(𝑘𝑘)𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 where 𝝀𝝀 is the evaporation constant.
Population size =90 Generations=100 Crossover fraction=0.75 Population initial range=[0;0.1] The order of approximation is set to N=3
C. Details of FOPID Design Using ACO for the AVR The following parameters are used for carrying out the FOPID design using ACO:
(20)
Number of ants =100 Pheromone=0.09 Evaporation Parameter =0.65 Positive Pheromone=0.2 Negative Pheromone=0.1 Maxtour=200 Minvalue=0 Maxvalue=1
D. First Test: Basic Performance The designed FOPID controller is approximated by oustaloup filter. The fractional derivative and integrals of FOPID controller have been approximated by N=3 and filter frequency [ 𝜔𝜔𝑏𝑏 , 𝜔𝜔ℎ ]=[1e-5,1e5]. The Bode diagrams of original FOPID and Oustaloup approximated FOPID controllers are shown in Fig.7, depicting very close behaviors in the frequency range of interest. Fig.8 shows the step response of the AVR system with FOPID tuned by both GA and ACO is better than AVR system with normal PID. Also the bode diagrams of the AVR system with FOPID is shown in the Fig.8.Both Figs.7 and 8 and Table I attest that the FOPID type AVR has better performance and higher robust stability than the PID type AVR.
Fig.5. Graphical Representation of ACO for FOPID Tuning Process
V. NUMERICAL RESULTS A. Study System Parameters A practical high-order AVR is used to verify the efficiency of the proposed FOPID controller. The system parameters are:
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E. Second Test: Robustness To examine numerically the robustness of the FOPID controller with respect to parameter uncertainties, the following simulation is performed. Assume that 𝑘𝑘𝐺𝐺 , 𝜏𝜏𝐺𝐺 ,𝑘𝑘𝐸𝐸 and 𝜏𝜏𝐸𝐸 are 20% increases due to the change in loading condition, the actual generator transfer function is given by 𝑉𝑉𝑡𝑡 (𝑠𝑠)
𝑉𝑉𝐹𝐹 (𝑠𝑠)
=
1
(21)
𝑠𝑠+0.769
Also the exciter transfer function is given by 𝑉𝑉𝐹𝐹 (𝑠𝑠)
𝑉𝑉𝑅𝑅 (𝑠𝑠)
=
2
(22)
𝑠𝑠+1.667
The step responses with both generator and exciter uncertainties are shown in Fig.10 and the system is proved to be robust.
Fig.6.Step Response of the original AVR
TABLE I COMPARISION OF THE EVALUATION VALUES BETWEEN BOTH PID AND FOPID CONTROLLERS Type of controller
𝑲𝑲𝑷𝑷
𝑲𝑲𝑰𝑰
𝑲𝑲𝑫𝑫
𝝀𝝀
𝝁𝝁
𝑴𝑴𝑷𝑷 (%)
𝒕𝒕𝒓𝒓
𝒕𝒕𝒔𝒔
𝑬𝑬𝒔𝒔𝒔𝒔
PID
0.2442
0.1423
0.0427
1
1
8.871
0.838
2.641
0
FOPID (ACO)
0.5
0.33
0.1667
0.9897
1.015
5.008
0.377
1.102
0
FOPID (GA)
0.2953
0.2003
0.0939
0.9989
0.9712
0.558
0.693
1.08
0
Fig.7. Bode diagrams of the original and approximated FOPID
Fig.8.Step response of the AVR controlled by PID and FOPID
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[3] [4] [5] [6]
[7] [8]
[9] Fig.9. Bode diagrams of the AVR controlled by PID and FOPID
[10] [11] [12]
[13]
[14] [15]
[16] Fig.10. Step response of the AVR controlled by PID and FOPID in the presence of generator and exciter uncertainties.
[17]
VI. CONCLUSION This paper presents a design method for determining the FOPID controller parameters using the Genetic Algorithm(GA) and Ant Colony Optimization (ACO) techniques. The proposed algorithms performed efficient search for the optimal FOPID controller parameters to the practical AVR system. Furthermore, it can be concluded from the above simulations that the proposed FOPID controller has more robust stability and performance characteristics than the PID controller applied to the AVR system.
[18] [19] [20] [21] [22]
VII. REFERENCES [1] Torvik P. J., Bagley R. L, “On the appearance of the fractional
[23]
derivative in the behavior of real materials”, Transactions of the ASME, 1984, 51(4):294–298. [2] Axtell M., Bise E. M,“Fractional calculus applications in control systems”,in Proceeding of the IEEE 1990 Natational Aerospace and Electronics Conference. New York, 1990, 563– 566.
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