multiobjective optimization of a fractional-order pid

Research ArticleMultiobjective Optimization of a Fractional-OrderPID Controller for Pumped Turbine Governing SystemUsing an Improved NSGA-III Algorithm underMultiworking Conditions

Chu Zhang12 Tian Peng 1 Chaoshun Li 2 Wenlong Fu 3

Xin Xia1 and Xiaoming Xue1

1College of Automation Huaiyin Institute of Technology Huaian 223003 China2School of Hydropower and Information Engineering Huazhong University of Science and Technology Wuhan 430074 China3College of Electrical Engineering amp New Energy China Three Gorges University Yichang 443002 China

Correspondence should be addressed to Tian Peng husthydropt126com and Chaoshun Li cslihusteducn

Received 1 November 2018 Revised 19 December 2018 Accepted 6 February 2019 Published 27 February 2019

Guest Editor Izaskun Garrido

Copyright copy 2019 Chu Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to make the pump turbine governing system (PTGS) adaptable to the change of working conditions and suppress thefrequency oscillation caused by the ldquoSrdquo characteristic area running at middle or low working water heads the traditional single-objective optimization for fractional-order PID (FOPID) controller under single working conditions is extended to amultiobjectiveframework in this study To establish themultiobjective FOPID controller optimization (MO-FOPID) problemundermultiworkingconditions the integral of the time multiplied absolute error (ITAE) index of PTGS running at low and high working waterheads is adopted as objective functions An improved nondominated sorting genetic algorithm III based on Latin hypercubesampling and chaos theory (LCNSGA-III) is proposed to solve the optimization problem The Latin hypercube sampling isadopted to generate well-distributed initial population and take full of the feasible domain while the chaos theory is introducedto enhance the global search and local exploration ability of the NSGA-III algorithm The experimental results on eight testfunctions and a real-world PTGS have shown that the proposed multiobjective framework can improve the Pumped storage unitsrsquoadaptability to changeable working conditions and the proposed LCNSGA-III algorithm is able to solve the MO-FOPID problemeffectively

1 Introduction

With the continuous expansion of the power system scalethe power gridrsquos requirements for power quality safety andintelligence have been constantly improving Pumped storageunits (PSUs) have played an important role in maintainingthe balance of power supply and demand because of theirfast start-up and shutdown speed flexible working conditionconversion excellent peak-load regulation and frequencyregulation ability [1ndash3] Pump turbine governing system isthe core control system of the pumped storage power stationwhich is responsible for stabilizing the unit frequency andregulating the unit power [4 5] Due to the huge flow inertia

of the long-distance water pipeline and the existence of theunstable ldquoSrdquo characteristic area the optimal control of PTGSis highly complex [6]Therefore it is of great theoretical valueand practical significance to explore optimization methodsfor PTGS and research new control laws The control qualityof PSU and the dynamic response performance of PTGS canthen be improved

The classical Proportional-Integral-Derivative (PID) con-troller is widely used in the optimal control of PTGS becauseof its simple and reliable structure and easy adjustment ofcontrol parameters [7ndash10] However due to the strong non-linear characteristics of different parts of PSUand the change-able working conditions the traditional PID controller often

HindawiComplexityVolume 2019 Article ID 5826873 18 pageshttpsdoiorg10115520195826873

2 Complexity

fails to realize the global optimization of PTGS The con-trol problems of phase modulation instability and no-loadfrequency fluctuation are becoming increasingly prominentAs an extension of the classical PID controller the fractional-order PID controller (FOPID) has attracted the attention ofmany scholars for its better adaptability and flexibility andgreater potential to obtain better control performance [11ndash15] Li et al [16] proposed an improved gravitational searchalgorithmusing theCauchy andGaussianmutations to adjustthe parameters of the FOPID controller automatically Anumber of tests have shown that the FOPID controller canimprove the dynamic characteristics and stability of regula-tion frequency of PSUs In Xu et al [17] a robust nonfragileFOPID controller was proposed for PTGS The parametersof the FOPID controller were selected using the bacterialforaging algorithm and multiscenario analysis functionsThe FOPID controller turned out to have obtained higherrobustness and stability compared with the traditional PIDcontroller And it can fully track the nonlinear characteristicsof PTGS in the ldquoSrdquo area Xu et al [18] proposed an adaptivelyfast fuzzy fractional-order PID (AFFFOPID) control methodfor PSUs by combining a fuzzy fractional-order PD controllerwith a fuzzy fractional-order PI controller Experiments ofPTGS at variouswater heads under unload running conditionhave shown that the controller can effectively improve theperformance and control quality of PSUs during the transientprocess

Apart from the advantages of the FOPID compared withthe PID controller one of the most important and challengeissues of the employment of FOPID controller in PTGS is theoptimal optimization of its parameters With the continuousdevelopment of optimization algorithms and control theoryin recent years scholars have combined the FOPID controllerwith intelligent algorithms to achieve the optimal tuningof control parameters and improvement of control laws forPTGS [19] The related works on optimal optimization forPID controller or hydroturbine governing system (HTGS)can also provide meaningful reference for the research ofFOPID controller for PTGS Fang et al [20] developedan improved particle swarm optimization (PSO) algorithmfor optimal tuning of PID control parameters for waterturbine governor Simulation results have demonstrated thestable convergence characteristic and good computationalability of the developed optimization stagey Kou et al [21]proposed a novel BFO-PSO algorithm by introducing PSOinto the framework of the bacterial foraging optimization(BFO) algorithm to improve the control performance of thePID controller for HTGS The advantages of the proposedalgorithm to the BFO and PSO algorithms have been demon-strated through experiments at real working conditionsWang et al [22] proposed a three-stage start-up strategy forPSUs by opening guide vanes to a large opening degree firstlyand then reducing the opening degree and finally switchingto the PID controller The switching time and PID controlparameters are optimized synchronously using an integratedoptimization scheme based on artificial sheep algorithm(ASA) Simulation results under various water heads haveshown that the control strategy can shorten the start-up timeand reduce the speed oscillation

Some researchers have paid attention to multiobjectivedesigning to consider multiobjectives that reflect the specificcharacteristics of the control system [23 24] Zamani et al[25] developed a multiobjective cuckoo search approach tooptimize the parameters of a FOPID controller Sanchez etal [26] proposed a multiobjective optimization strategy foridentifying the optimal solution for a robust FOPID con-troller Zhao et al [27] proposed a parameter tuning schemebased on two lbests multiobjective PSO (2LB-MOPSO) tominimize the integral squared error and balanced robustperformance criteria of a robust PID controller simultane-ously Chen et al [28 29] put forward a parameter tuningscheme to optimize the integral of the squared error (ISE)and the integral of the time multiplied squared error (ITSE)performance indices of PTGS simultaneously The adaptivegrid PSO (AGPSO) and the chaotic nondominated sortinggenetic algorithm II (NSGA II) were adopted to realizethe optimal control of the PID and the FOPID controllersrespectively The proposed multiobjective PID and FOPIDcontroller turned out to have achieved better control effectsthan the compared methods

It is noticed that most of the research works on optimalcontrol of the PID or FOPID controllers are designed under asingleworking conditionThe traditionalmultiobjective opti-mization control usually adopts a set of objective functionsto obtain the Pareto optimal solutions for a certain conditionand then select the compromise optimal solution The tra-ditional multiobjective optimization control is essentially anoptimization scheme for singleworking conditionsHoweverthe characteristics of the controlled object of PTGS are notonly related to the nonlinear characteristics of the pumpturbine the penstock system the generator and the otherparts of PTGS but also vary with the changes of the workingconditions The optimal control of a single working conditionis often at the expense of the deterioration of some otherworking conditions The optimal control of PTGS undermultiworking conditions should be a process of overall trade-off and cannot be limited to the pursuit of the optimal controlunder a certain working condition In order to enhance theadaptability of PTGS to the change of working conditionsa multiobjective optimization framework which takes intoaccount the integral of the time multiplied absolute error(ITAE) index of PTGS running at multiworking conditionsis constructed The nondominated sorting genetic algorithmIII based on the Latin hypercube sampling and chaos theory(LCNSGA-III) is proposed to optimize the control parame-ters of the FOPID controller for PTGS under multiworkingconditions

The rest of this paper is arranged as follows Section 2gives a brief introduction to the multiobjective optimiza-tion problem Section 3 builds the mathematical model ofMO-FOPID under multiworking conditions Section 4 pro-poses the LCNSGA-III algorithm based on Latin hyper-cube sampling and the chaos theory to solve the MO-FOPID problem Section 5 employs eight benchmark func-tions and a real-world PTGS to verify the effectiveness ofthe LCNSGA-III algorithm and the developed multiobjec-tive optimization framework Section 6 gives the conclu-sions

Complexity 3

2 Multiobjective OptimizationProblem (MOOP)

The purpose of the single-objective (SO) optimization prob-lem is to obtain the optimal solution by searching for theminimum or maximum value of one single objective func-tion However optimization problems in scientific researchor engineering application usually contain not only oneobjective These objectives are sometimes in concordancewith each other but sometimes there are conflicts betweenthem The objective of the multiobjective optimization prob-lem (MOOP) is to search for the optimal solutions of allobjectives which is also known as the Pareto optimal solution[30] A typical MOOP with D decision variables N objectivefunctions andm+k constraints can be described as follows

min 119865 (119883) = (min1198911 (119909) min1198912 (119909) min119891119894 (119909) min119891119873 (119909))

119904119905 119892119894 (119883) ge 0 119894 = 1 2 119898ℎ119895 (119883) = 0 119894 = 1 2 119896

119883 = [1199091 1199092 119909119889 119909119863 ]119909119889 min le 119909119889 le 119909119889 max 119889 = 1 2 119863

(1)

where 119883 is the decision variable in 119863 dimensions 119865(119883) isthe objective function in 119873 dimensions 119892119894(119883) representsthe 119894th inequality equation ℎ119895(119883) represents the 119894th equalityequation 119909119889 min and 119909119889 min represent the upper and lowerbounds of the 119889th decision variable respectively

To define the Pareto optimal solution four definitions aregiven as follows

Pareto Dominance Relationship A vector 119883lowast = [119909lowast1 119909lowast2 119909lowast119889 119909lowast119863 ] is said to dominate119883 = [1199091 1199092 119909119889 119909119863 ](known as119883lowast ≺ 119883) if and only if the following two conditionsare satisfiedforall119899 119891119899 (119883lowast) le 119891119899 (119883) 119899 = 1 2 N

exist1198990 1198911198990 (119883lowast) lt 1198911198990 (119883) 1 le 1198990 le N (2)

ParetoOptimal Solution Pareto optimal solution is a solutionthat cannot be dominated by any solution in the feasibleregion 119878 which means that if and only if notexist119883 isin 119878 119883 ≺ 119883lowast119883lowast is the Pareto optimal solution

Pareto Optimal Set For a given MOOP the Pareto optimalset 119875lowast can be defined as

119875lowast = 119883lowast isin 119878 | notexist119883 isin 119878 119883 ≺ 119883lowast (3)

Pareto Optimal Front For a given MOOP the Pareto optimalfront 119875119891lowast can be described as

119875119891lowast = 119906 = 119865 (119909) = (1198911 (119909) 1198912 (119909) 119891119873 (119909))119879 | 119883isin 119875lowast (4)

3 Problem Formulation of the MO-FOPIDProblem under Multiworking Conditions

To formulate the multiobjective FOPID controller optimiza-tion problem (MO-FOPID) under multiworking conditionsthe fractional calculus theory and fractional-order PID(FOPID) controller are first introduced

31 Introduction to Fractional Calculus Theory and FOPIDController Fractional calculus theory is an extension of thetraditional calculus theory to the fractional systems Thefractional calculus theory can describe complex systems in aneasy way with a clear physical meaning Among the variousfractional calculus operators the Laplacian transformationdefined by Caputo is the most usual mathematical expressionfor calculating the fractional-order time derivatives and hasbeen widely used in fractional-order PID controller [2831] Given a continuous derivable function 119891(119905) Caputorsquosfractional derivative of order 119886 can be defined as

0119863120572119905 119891 (119905) = 1Γ (119899 minus 120572) int119905

0

119891119899 (119905)(119905 minus 120591)120572+1minus119899119889120591 (5)

where 0119863120572119905 denotes the fractional calculus operator and Γ(sdot)

denotes the Euler Gamma functionFractional calculus equations usually need to be trans-

formed into algebraic equations The Laplacian transforma-tion of (5) under zero initial condition can be expressed as

intinfin

0119890minus119904119905119863120572119891 (119905) 119889119905 = 119904120572119865 (119904) (6)

In recent years with the further research and devel-opment of fractional calculus theory the combination offractional calculus theory and modern control theory isbecoming more and more popular Controllers based onfractional calculus have been implemented and applied inmany research fields [12 25] The Oustaloup recursive filterand its improved version have been widely adopted to realizethe discretization approximation of the fractional calculusoperator The expression of the Oustaloup filter is as follows

119904120572 asymp 119870 119873prod119896=minus119873

((119904 + 1205961015840119896)(119904 + 120596119896) ) (7)

where 120596119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1+120572)2)(2119873+1) 1205961015840119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1minus120572)2)(2119873+1) 119870 = 120596120572

ℎ 120572 denotes the order of thefractional calculus (2N+1) is the order of filter (120596119887 120596ℎ)denotes the expected fitting range 119873 120596119887 and 120596ℎ aredetermined according to the accuracy requirement of thenumerical approximation

The FOPID controller proposed by Professor Podlubny[32] is an extension of the classical PID controller Comparedwith the classical PID controller the range of control rate ofthe FOPID controller is much widerThe transfer function ofthe FOPID controller is as follows

119880 (s)119864 (s) = 119870119901 + 119870119894119904120582 + 119870119889119904119906 (8)

4 Complexity

Kp

bp

+

++

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

FOPID

minus

s

s

Figure 1 The structure of the FOPID controller for pumped turbine governing system

where 119864 denotes the control deviation 119880 denotes the con-troller output 119870119901 119870119894 and 119870119889 represent the gain parameterthe integral order and the differential order respectivelyThe traditional PID controller is a special case of theFOPID controller when 120582 = 1 and 119906 = 1 Because theintegral and differential orders are variational the FOPIDcontroller has better adaptability and flexibility and biggerpotential to obtain better control performance [33] Thestructure of the FOPID controller for PTGS is shown inFigure 1 In Figure 1 119909119888 denotes the given unit speed 119909denotes the unit speed 119887119901 denotes the permanent slipcoefficient 119879119889 denotes the differential time constant 119870119901 119870119894and 119870119889 represent the proportional integral and differentialgain coefficients of the controller respectively 120582 and 120583denote the integral and differential orders of the controllerrespectively

32 Description of the PTGS System PTGS is a complexnonlinear time-varying system of hydraulic mechanicaland electrical connections PTGS mainly contains five partsnamely a controller an electrohydraulic servomechanismsystem a pump turbine a generator and a penstock systemwhere the controller and the servomechanism system consistof the speed governor of PTGS [4] A PID controller hasalways been employed as part of the speed governor of PTGSIn this study the FOPID controller is designed for PTGSTheFOPID controller has been described in Section 31 In whatfollows transfer functions of the other four connectors areilluminated

(1) Servomechanism System Servomechanism is the actuatorof the governor of PSU It is made up of an auxiliaryservomotor and a main servomotor of which the transferfunctions are as follows

119866 (119904) = 119870V1 + 1198791199101119904119866V (119904) = 1119879119910119904

(9)

where 1198791199101 and 119870V are the time constant and scale factor ofthe auxiliary servomotor respectively and 119879119910 represents themain servomotor open-loop time constant

(2) Pump Turbine To reflect the complex nonlinear char-acteristics among water machine and electricity during theoperation process accurately the pump turbine model basedon characteristic curves is constructed The model of pumpturbine is as follows

11987211 = 119891119872 (11988611987311)11987611 = 119891119876 (11988611987311) (10)

where11987211 11987611 and11987311 represent the unit torque unit flowand unit speed respectively 119886 is the guide vane opening119891119872 and 119891119876 denote the functions of the moment and flowcharacteristic curves respectively Because of the strongnonlinear characteristics of the flow and moment charac-teristic curves the improved Suter transform is introducedto transfer the flow and moment characteristic curves intoWH and WM characteristic curves respectively [4] TheWH and WM characteristic curves of a pump turbine in apumped storage power station in China using the improvedSuter transform have been illustrated in Zhou et al [4] Inthis study the WH and WM characteristic curves in twodimensions in [4] have been changed to three dimensionsThe three-dimensional WH and WM characteristic curvesare illustrated as in Figure 2

(3) Generator The common first-order model [6 16 18] isadopted in this study to balance the pump turbine torque andthe generator torque The transfer function of the first-ordermodel is as follows

119866119892 (119904) = 1119879119886119904 + 119890119899 (11)

where 119879119886 and 119890119899 are the inertia time constant and self-adjusting factor of the generator respectively

(4) Penstock System Because of the fluid and tube wallelastic effects on the penstock the second-order elastic waterhammer model is exploited in this study by applying thesecond-order Taylor expansion on the nonlinear hyperbolictangent function The transfer function of the second-orderelastic water hammer model is as follows

119866ℎ (119904) = 119867 (119904)119876 (119904) = minus1198791199081199041 + 05119891119879119903119904 + 012511987911990321199042 (12)

Complexity 5

01

02

04

4

WH

06

3guide vane opening y

08

05

flow angle x

1

21

0 0

(a) WH characteristic curve

01

2

4

4

WM

6


8

05

flow angle x

10

21

0 0

(b) WM characteristic curve

Figure 2 Three-dimensional surface of the characteristic curves of pump turbine

where 119879119908 denotes water flow inertia time constant 119891 repre-sents the water head loss coefficient and 119879119903 is the reflectiontime of water hammer wave

33 Multiobjective Optimization of FOPID Controller forPTGS For single-optimization of the FOPID controllerunder a certain working condition the ITAE [34 35] isusually employed as the objective function to obtain satisfac-tory transient dynamic performance of the systemThe ITAEindex is defined as follows

ITAE = int119879

0119905 |119890 (119905)| 119889119905 (13)

where 119890(119905) denotes the relative deviation of the rotationalspeed of PTGSThe ITAE index considers the stable time andovershoot of the dynamic response of PTGS simultaneouslyIt can evaluate the speed and stability of the system at thesame time The smaller the ITAE the better the speed andstability of the system

When operating at the working condition of low waterhead the PSU is easy to fall into the ldquoSrdquo characteristicregion resulting in oscillation of the unit speed near therated frequency The working condition of medium or highwater head in the other way is the most common workingcondition in the operation process of PSU The optimalcontrol of PTGS under different working conditions shouldbe considered and researched to make the PTGS systembetter adapt to the changeable working environment How-ever the ITAE for PTGS at working condition of low head(referred as ITAE1) and that of high head (referred as ITAE2)usually influence and restrict each other In this study thesingle-objective optimization of the FOPID controller forPTGS is expanded to multiobjective theoretical frameworkto find a compromise solution and ensure that the PTGSsystem can achieve relatively better control performanceunder changeable working conditions To formulate theMO-FOPID problem the ITAE1 and ITAE2 are adoptedas objective functions The five parameters of the FOPID

controller for PTGS including the proportional coefficient119870119901 the integral coefficient 119870119894 the differential coefficient 119870119889the integral order 120582 and the differential order 120583 are takenas decision variables The MO-FOPID problem can then beformulated as

Min

1198911 = ITAE1 = 1198911 (119870119901 119870119894 119870119889 120582 120583)1198912 = ITAE2 = 1198912 (119870119901 119870119894 119870119889 120582 120583) (14)

subject to

119870119901min le 119870119901 le 119870119901max119870119894min le 119870119894 le 119870119894max119870119889min le 119870119889 le 119870119889max120582min le 120582 le 120582max120583min le 120583 le 120583max

(15)

where 1198911(sdot) and 1198912(sdot) are functions of 119870119901 119870119894 119870119889 120582 120583 atworking condition of low water head and high water headrespectively 119883min and 119883max are the lower and upper boundsof 119870119901 119870119894 119870119889 120582 120583 of the FOPID controller respectively

4 Nondominated Sorting GeneticAlgorithm-III Based on Latin-HypercubeSampling and Chaos Theory (LCNSGA-III)

An improved version of NSGA-III based on Latin-hypercubesampling and chaos theory is developed to solve the proposedMO-FOPID problem under multiworking conditions

41 Brief Introduction to NSGA-III TheNSGA-III algorithm[36] first introduced by Deb and Jain in 2014 is a novelreference-point-based nondominated sorting genetic algo-rithm following the NSGA-II framework Unlike the crowd-ing distance operator exploited in NSGA-II [37] NSGA-III employed a reference point-based mechanism to makethe Pareto optimal front well-distributed The step-to-step

6 Complexity

procedures of the NSGA-III algorithm can be expressed asfollows

Step 1 Calculate the number of reference points (119867) to beplaced on the hyper-plane

119867 = (119862 + 119892 minus 1119892 ) (16)

where 119862 represents the number of objective functions and 119892denotes the number of divisions (For 119862 = 3 and 119892 = 4 119867 iscalculated as 15)

Step 2 Generate NP individuals in the feasible regionrandomly to form the initial population of the NSGA-IIIalgorithm and record it as 119875119896 Set the iteration number as119896 = 1Step 3 Generate the offspring population 119876119896 using thesimulation binary crossover (SBX) operator and polynomialmutation operator

Step 4 Let119877119896 = 119876119896cup119875119896 and calculate the fitness value of eachindividual in 119877119896

Step 5 Identify the nondominated level 1198651 1198652 119865119905 for eachindividual in 119877119896 using the nondominated sorting operator

Step 6 Normalize the objectives and associate the solutionsin 119877119896 with the reference points Delete the useless referencepoints and preserve solutions with higher rankings accordingto the niche preservation strategy to construct the nextgeneration of population 119875119896+1Step 7 If 119896 lt 119866max skip to Step 3 else stop iteration andoutput the Pareto optimal set

The crossover and mutation operators in Step 3 and thenondominated sorting operator in Step 5 of the NSGA-IIIalgorithm are the same as those of the NSGA-II algorithm InStep 6 the NSGA-III algorithm generates the next populationusing the reference point-based selection mechanism otherthan the crowding distance operator of NSGA-II Readerscan refer to [36] for more details about NSGA-III Theimprovements of the NSGA-III algorithms including theLatin hypercube sampling-based initialization technique andthe chaotic crossover and mutation operators are introducedin the following subsections

42 Latin Hypercube Sampling Based Initialization TechniqueNSGA-III generates the initial values of the decision variablesin the feasible region randomly to form the initial populationof the algorithm However a large number of individuals mayassemble into a local area of the feasible region because ofthe random initiation leading to premature convergence inthe iterative process In order to make the individuals of theinitial population well-distributed in the feasible region theLatin hypercube sampling based initialization technique [38]is introduced toNSGA-III to improve its search performance

The brief steps to generate the initial population of NSGA-IIIusing Latin hypercube sampling are as follows

Step 1 Suppose the size of the population is 119873 and eachindividual in the population contains119871discrete elementsTherange of the discrete element 119909119897 119897 = 1 2 119871 of individual119909 can be divided into 119873 equal mini zones

119909119897 119898119894119899 = 1199090119897 lt 1199091119897 lt sdot sdot sdot lt 119909119895119897lt sdot sdot sdot lt 119909119873119897 = 119909119897 119898119886119909 (17)

where 119875(119909119895119897lt 119909 lt 119909119895+1

119897) = 1119873 and the value space of 119909 can

be divided into 119873119871 small hypercubes finally

Step 2 Generate a matrix 119872 of which the dimension is119873 times 119871 and every column of 119872 is the full permutation of1 2 119873Step 3 Generate an individual in each row of 119872 randomlyan initial population with 119873 individuals is then generated

43 Chaotic Crossover and Mutation Operators The NSGA-III algorithm generates a random number 119903119888 to determinewhether to apply the crossover operator or not The SBXoperator is implemented when 119903119888 lt 120578119888 where 120578119888 representsthe crossover distribution index For two parent individuals1199091199011 = 11990911199011 1199091198941199011 1199091198991199011 and 1199091199012 = 11990911199012 1199091198941199012 1199091198991199012 the NSGA-III algorithm generates two offspring indi-viduals 1199091198881 = 11990911198881 1199091198941198881 1199091198991198881 and 1199091198882 = 11990911198882 1199091198941198882 1199091198991198882 according to the following

1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)

where 120573 represents the crossover coefficient 120573 can be calcu-lated according to the following

120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)

where 119906 = rand (sdot) is a random number uniformly generatedin [0 1]

The NSGA-III algorithm generates a random number119903119898 to determine whether to apply the mutation operator ornotThe polynomial mutation operator is implemented when119903119898 lt 120578119898 where 120578119898 represents the mutation distributionindex For the feasible solution 119909119904 the mutation individualis generated using the polynomial mutation operator

119909lowast119904 = 119909119904 + (119909119906119904 minus 119909119897119904) times 120575119904 (20)

where 119909lowast119904 represents the mutation individual 119909119906119904 and 119909119897119904represent the upper and lower bounds of 119909119904 respectively 120575119904is the mutation coefficient 120575119904 can be calculated as follows

120575119904 = (2119906119904)1(120578119898+1) minus 1 119906119904 lt 051 minus (2 times (1 minus 119906119904))1(120578119898+1) 119890119897119904119890 (21)

Complexity 7

where 119906119904 = rand(sdot) is a random number uniformly generatedin [0 1]

The standard NSGA-III algorithm has excellent com-putational efficiency and stability However the NSGA-IIIalgorithm sometimes may fall into the local optimal solu-tion because of the insufficient exploration of the feasibleregion Because of the ergodicity and stochasticity of chaoticsequences [39] the chaotic map which can generate chaoticsequences is introduced into the crossover and mutationoperators of NSGA-III to enhance its global search and localexploration ability [40] In this study the tent map [41] isemployed to improve the crossover and mutation operatorsof NSGA-III

The tent chaotic map with uniform distribution probabil-ity density can be expressed as

119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)

where 119909(119896) isin (0 1) denotes the chaotic variable generated inthe 119896th iteration

According to (19) and (21) the crossover and mutationoperators of the standard NSGA-III algorithm need to gen-erate two random numbers 119906 and 119906119904 in [0 1] respectivelyThe two numbers 119906 and 119906119904 are generated using the tentchaotic map in the LCNSGA-III algorithm The two randomnumbers 119906 and 119906119904 are generated according to the following

119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)

44 The Flowchart of LCNSGA-III for MOOPs Based onthe above introduction of LCNSGA-III the Latin hypercubesampling and the chaotic crossover and mutation operatorsthe flowchart of LCNSGA-III for MOOPs is shown inFigure 3

45 Implementation of LCNSGA-III for Solving the MO-FOPID Problem under Multiworking Conditions The aboveLCNSGA-III algorithm is used to optimize the FOPIDcontroller for PTGS under multiworking conditions Theschematic diagram of the MO-FOPID problem optimizedby LCNSGA-III under multiworking conditions is shown inFigure 4 In Figure 4 119909119888 denotes the rotational speed (orfrequency) and 119910119888 denotes the given guide vane openingSince the frequency of the rotational speed is 50 Hz and thefrequency disturbance is 2 Hz the frequency perturbation isset as 4 of the rated frequency

5 Numerical Experiments and Analysis

51 Experiments for Benchmark Functions

511 Benchmark Functions and Performance Metrics Inorder to validity the effectiveness of the newly developedLCNSGA-III algorithm based on Latin-hypercube sampling

Generate the initial population Pk using Latin-hypercube sampling

Generate the offspring population Qk based on chaotic crossover and mutation operators

Let Rk = Pk Qk and calculate the fitness value of each individual in Rk

Start

Initialize the parameters and set the ranges for LCNSGA-III

Apply the non-dominated sorting algorithm to each individual in Rk and identify its

non-dominated level F1 F2 hellipFt

Select NP best individuals as the next population based on the reference point-based mechanism

Reach the maximum iteration

Yes

No

End

cup

Figure 3 Flowchart of LCNSGA-III for MOOPs

and chaotic map a total number of eight test functionsincluding the ZDT1-4 6 [42] and the DTLZ1-2 5 [43] (shownin Table 1) are employed to test its performance Amongthe eight test functions for MOOPs the ZDT benchmarkfunctions are two-objective MOOPs The dimension of thedecision variables and the number of the test functions of theDTLZbenchmark functions can be adjusted In this study thenumber of the objective functions of the DTLZ test functionsis selected as three to display the Pareto front of three-objective MOOPs Four other typical multiobjective algo-rithms including NSGA-II NSGA-III MOEAD and PESA-II [30 44] are adopted as control group The performancesof the multiobjective evolutionary algorithms (MOEAs) areevaluated and compared using the widely used evaluationmetrics of generational distance (GD) and Spread GD isadopted to measure the mean value of the distance betweenthe Pareto solution set and the real Pareto front Spreadis employed to describe the distribution uniformity of thePareto optimal set [40] GD can be expressed as follows

GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y

LCNSGA-III ITAE1 ITAE2

Nonlinear PTGS Generator

q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)

Figure 4 Schematic diagram of the MO-FOPID problem optimized by LCNSGA-III under multiworking conditions

where119863119894 denotes the Euclideandistance between the ith non-dominated solution and the nearest nondominated solutionon the real Pareto front and 119873 denotes the size of the Paretooptimal setThe smaller theGD the closer the Pareto solutionset to the real Pareto front and the better the convergenceaccuracy

Spread can be expressed as follows

Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)

where 119889119894 denotes the distance between ith solution with itsneighboring solution 119889 is the mean of 119889119894 119889119890119895 denotes theEuclidean distance between the extreme solution of the realPareto front and the boundary solution of the obtained Paretooptimal set M denotes the number of objective functionsThe smaller the Spread the better the distribution of thesolutions

512 Results Analysis The parameters of the five algorithmsfor MOOPs with two objectives are set as follows thepopulation size is set as 100 while the number of iterations isset as 300 The parameters of the five algorithms for MOOPswith three objectives are set as follows the population size isset as 150 while the number of iterations is 500The crossoverand mutation probability of the five algorithms are set as 07and 03 respectively The neighborhood size of MOEAD isset as 20 [45 46] All the experiments are implemented inMatlab environment Because of the random initialization oftheMOOPs all the experiments have been repeated ten timesindependently to eliminate the effectiveness of randomnessThe average GD and Spread of the five algorithms for theeight benchmark functions are given in Tables 2 and 3respectively Numbers in bold represent the optimal GD andSpread among the five algorithms for the eight benchmarkfunctions

As can be seen fromTable 2 the GD of the Pareto optimalsolution obtained by the newly developed LCNSGA-III is the

best out of the total eight benchmark functions Althoughthe GD of LCNSGA-III algorithm is slightly worse than thatof NSGA-II and NSGA-III to optimize ZDT4 and DTLZ1the Spread of LCNSGA-III is prominent The Spread ofLCNSGA-III performs best to optimize the ZDT1 ZDT4 andDTLZ5 benchmark problems And the Spread for LCNSGA-III does not differ much from those for the other five MOOPsexcept DTLZ1 which indicates that the LCNSGA-III algo-rithm can obtain good convergence performance for the eightbenchmark functions The Pareto optimal front obtained byLCNSGA-III to optimize the eight benchmark functions isshown in Figure 5 (ZDT1-4) and Figure 6 (ZDT6 DTLZ1-25) (Grey points denote the true Pareto front while blue onesdenote the obtained Pareto front) respectively As depictedin Figures 5 and 6 the Pareto front obtained by LCNSGA-IIIto optimize the eight benchmark MOOPs can approximatethe real Pareto front perfectly and the distributions of thePareto optimal sets are uniform As a result the LCNSGA-III algorithm has prominent convergence performance andoptimization ability compared with the other algorithms

52 Experiments for Nonlinear PTGS

521 Experiments Design and Results In order to fullyverify the performance and effectiveness of the LCNSGA-III algorithm in solving the MO-FOPID problem undermultiworking conditions the nonlinear model of PTGS withdifferent controllers (PID and FOPID) and water heads(198m 205m and 210m) under no-load conditions is simu-lated onMATLAB environment The frequency perturbationis set as 4 of the rated frequency and the simulation time isset as 50s A total number of ten schemes have been designedto obtain the optimal parameters The ten schemes can bedivided into four categories as follows

(1) The backtracking search algorithm (BSA) [47] isexploited to optimize the parameters of PID controller undera single working condition of different water heads (198m205m and 210m) The ITAE of PTGS under a certainworking condition is selected as the objective function for

Complexity 9

Table 1 Eight test functions for MOOPs

Name Functions Dimension ofdecision variable Feasible region Type of Pareto Front

ZDT1

1198911(119883) = 119909130 [0 1] High dimension convex1198912(119883) = 119892 sdot (1 minus radic1198911119892 )

119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2

1198911(119883) = 119909130 [0 1] High dimension convex1198912(119883) = 119892 sdot (1 minus (1198912119892 )2)

119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3

1198911(119883) = 119909130 [0 1] Discontinuous convex1198912 (119883) = 119892 sdot (1 minus radic1198911119892 ) minus (1198911119892 ) sin (101205871198911))

119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110

1199091 isin [0 1]Multi-modal convex1198912(119883) = 119892 sdot (1 minus radic1198911119892 ) 119909119894 isin [minus5 5]

119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6

1198911 (119883) = 1 minus exp (minus41199091) sin6 (61205871199091)10 [0 1] Inhomogeneous1198912 (119883) = 119892 sdot (1 minus (1198911119892 )2)

119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1

1198911(119883) = 1211990911199092 sdot sdot sdot 119909119872minus1 (1 + 119892 (119909119872))

7 [0 1] Linear multimodal

1198912(119883) = 1211990911199092 sdot sdot sdot (1 minus 119909119872minus1) (1 + 119892 (119909119872))119891119872minus1 (119883) = 121199091 (1 minus 1199092) (1 + 119892 (119909119872))

119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872

lowast (119909 minus 05)2 + cos (20120587 (119909 minus 05)))

DTLZ2

1198911 (119883) = (1 + 119892 (119909119872)) cos(11990921205872 ) sdot sdot sdot cos (119909119872minus11205872 )

12 [0 1] Complex nonconvex1198912(119883) = (1 + 119892 (119909119872)) cos (11990911205872 ) sdot sdot sdot cos(119909119872minus11205872 )

119891119872(119883) = (1 + 119892 (119909119872)) sin(11990911205872 )119892(119883119872) = sum lowast (119909 minus 05)2)

DTLZ5

replace the 119909119894 in DTLZ2 with 12057911989412 [0 1] Space arc120579119894 = 1205874 (1 + 119902 (119903)) (1 + 2119892 (119903) 119909119894)

119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity

Table 2 Comparison of GD for eight benchmark functions

Benchmark functions Multi-objective optimization algorithmsPESA-II MOEAD NSGA-II NSGA-III LCNSGA-III

ZDT1 772E-05 366E-04 577E-05 795E-05 380E-05ZDT2 337E-04 827E-04 311E-05 677E-05 284E-05ZDT3 614E-05 190E-03 423E-05 748E-05 356E-05ZDT4 141E-02 205E-03 200E-04 252E-04 315E-04ZDT6 109E-02 738E-04 430E-05 784E-05 363E-05DTLZ1 197E-02 301E-04 475E-04 225E-04 433E-04DTLZ2 977E-04 418E-04 930E-04 414E-04 378E-04DTLZ5 150E-04 788E-05 132E-04 155E-04 534E-05

Table 3 Comparison of Spread for eight benchmark functions


ZDT1 967E-01 453E-01 425E-01 347E-01 324E-01ZDT2 983E-01 673E-01 468E-01 215E-01 299E-01ZDT3 101E+00 679E-01 587E-01 752E-01 665E-01ZDT4 102E+00 551E-01 431E-01 393E-01 330E-01ZDT6 104E+00 155E-01 415E-01 113E-01 120E-01DTLZ1 802E-01 311E-02 495E-01 340E-02 351E-01DTLZ2 388E-01 173E-01 521E-01 174E-01 301E-01DTLZ5 920E-01 203E+00 661E-01 978E-01 458E-01

ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

minus06minus04minus02

002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2

Figure 5 Pareto optimal solutions obtained by LCNSGA-III for ZDT1-4

Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3

Figure 6 Pareto optimal solutions obtained by LCNSGA-III for ZDT6 DTLZ1-2 5

single-objective optimization The experiments are simpli-fied as S-198-PID S-205-PID and S-210-PID respectivelyThe optimized control parameters are applied on the othertwo working conditions to test the adaptability of differentschemes in tracking the dynamic responses of PTGS (eg forscheme S-198-PID the working condition running at 198mwater head is used for training while those at 205m and 210mare used for testing)

(2)TheBSAalgorithm is exploited to optimize the param-eters of FOPID controller under a single working conditionof different water heads (198m 205m and 210m) The exper-iments are simplified as S-198-FOPID S-205-FOPID and S-210-FOPID respectively The optimized control parametersare applied on the other two working conditions to test theadaptability of different schemes in tracking the dynamicresponses of PTGS

(3) The NSGA-III algorithm is adopted to optimize theparameters of PID or FOPID controller under two workingconditions (198m and 210m) The ITAE of PTGS at workingconditions of 198m and 210m water head is selected as theobjective functions for multiobjective optimization and thecompromise optimal solution among the Pareto optimal setis selected The experiments are simplified as NSGA-III-PIDand NSGA-III-FOPID for the PID and FOPID controllerrespectively The compromise optimal control parameters areapplied on the working condition of 205m water head to testthe adaptability of different schemes in tracking the dynamicresponses of PTGS

(4)TheLCNSGA-III algorithm is adopted to optimize theparameters of PID or FOPID controller under two workingconditions (198m and 210m) The experiments are simplified

as LCNSGA-III-PID and LCNSGA-III-FOPID for the PIDand FOPID controller respectively The compromise optimalcontrol parameters are applied on the working condition of205m water head to test the adaptability of different schemesin tracking the dynamic responses of PTGS

The parameters of PTGS are set as follows the rangesof 119870119901 119870119894 and 119870119889 of the PID and FOPID controllers areall set as [0 15] and the ranges of 120582 and 119906 are all set as[0 2]The parameters of the BSA algorithm are set as followsthe population size is set as 50 the number of iterations is200 and the control parameter 119865 is set as the default valueThe parameters of the NSGA-III and LCNSGA-III algorithmsare as follows the population size is set as 50 the iterationnumber is 300 the crossover probability is 07 the mutationprobability is 03

Apart from ITAE the integral of the ITSE [29] the stabletime (ST) and the overshoot (OSO) is adopted to evaluate theperformance of different schemes The ITSE index is definedas follows

ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)

In this study the best compromise solution is selectedaccording to the subjective weighting method based onexpertsrsquo preferences of weights The optimal control param-eters of the ten schemes for PTGS are shown in Table 4 Theperformance indices including ITAE ITSE ST and OSO ofthe ten schemes under different working water heads areshown inTable 5 InTable 5 for scheme S-198-PID the resultsfor 198m water head are training results while those for 205mand 210mheads are testing results For scheme S-205-PID the

12 Complexity

Table 4 Optimal control parameters for nonlinear PTGS using different schemes

Schemes Optimal parameters119870119901 119870119894 119870119889 120582 120583S-198-PID 505 072 208 S-205-PID 800 100 291 S-210-PID 697 087 500 S-198-FOPID 900 052 082 049 097S-205-FOPID 136 058 174 064 098S-210-FOPID 891 056 198 077 099NSGA-III-PID 992 104 162 NSGA-III-FOPID 890 052 102 057 098LCNSGA-III-PID 1472 097 081 LCNSGA-III-FOPID 1062 058 112 056 098

Table 5 Performance indices for nonlinear PTGS using different schemes at different water heads

Experiments 198m 205m 210mITAE ITSE ST (s) OSO () ITAE ITSE ST (s) OSO () ITAE ITSE ST (s) OSO ()

S-198-PID 568 004 499 177 410 002 356 150 316 002 355 124S-205-PID 3728 017 254 269 002 293 229 193 002 201 205S-210-PID 5429 035 228 3788 021 117 134 002 186 93S-198-FOPID 210 002 276 36 150 002 261 05 187 002 271 09S-205-FOPID 2329 010 105 111 002 250 05 148 001 282 07S-210-FOPID 2011 008 79 181 002 257 33 096 002 170 03NSGA-III-PID 588 004 371 282 003 232 345 204 002 237 324NSGA-III-FOPID 218 002 274 56 152 002 241 30 141 002 266 09LCNSGA-III-PID 543 004 333 276 003 184 304 190 002 184 276LCNSGA-III-FOPID 190 002 276 55 137 002 233 23 142 001 259 11

results for 205m water head are training results while thosefor 198m and 210m heads are testing results For scheme S-210-PID the results for 210m water head are training resultswhile those for 198m and 205m heads are testing resultsThe single-objective schemes are designed to compare withthe multiobjective schemes to highlight the effectiveness ofmultiobjective schemes in optimizing PTGS ldquordquo means thatthe system is unstable and when the fluctuation of frequencyis smaller than 0003 it is considered to be stable

522 Comparison of PID and FOPID Controllers underDifferent Working Conditions From Table 5 it is known thatthe FOPID controller generally achieves better performancethan the corresponding PID controller in terms of ITAEITSE ST and OSO for 198m 205m and 210m working waterheadsThe ITAE ITSE ST and OSO for the FOPID controllerare either smaller or in coincidence with those of the PIDcontroller For example the improved percentages of schemeS-198-FOPID compared with scheme S-198-PID are 630500 447 and 797 in terms of ITAE ITSE ST andOSO respectively in the training stage (198m)The improvedpercentages in the testing stage (205m) are 634 267 and967 in terms of ITAE ST and OSO respectively The ITSEfor S-198-FOPID and scheme S-198-PID in the testing stage(205m) is the same To further compare the effects of differentcontrollers in capturing the dynamic performances of PTGS

the test unit frequency at 198m 205m and 210m workingwater heads using different controllers is illustrated in Figures7ndash9 respectively As can be seen fromFigures 7ndash9 the FOPIDcontroller for PTGS can obtain smaller overshoot ITAEITSE ST and OSO in most cases For working condition of198mwater headwhich is easy to fall into the ldquoSrdquo area the unitfrequency oscillates a lot using PID controller The FOPIDcontroller in the other way can effectively restrain the strongnonlinear characteristics of PTGS and significantly improvethe control quality

523 Analysis of Controllers Optimized under a Single Work-ing Condition It is noticed from Table 5 that for controllersoptimized at a single working condition the performanceis only good for the training working condition but thetest working conditions It can also be seen from Table 5and Figures 7ndash9 that the S-210-FOPID scheme can obtaingood control performance when PTGS is running at 210mwater head but not 198m For schemes S-198-PID and S-198-FOPID PTGS can obtain good control performance at 205mand 210mworking water heads which conforms to the actualphysical phenomenon that the control parameters suitablefor low water head may also suitable for middle or highwater heads However due to the lack of comprehensive con-sideration for complex working conditions the adaptabilityand robustness of the PID and FOPID controllers optimized

Complexity 13

S-198-PIDITAE 568 ITSE 004Stable time (s) 499Overshoot () 177

S-205-PIDITAE 3728 ITSE017Stable time (s) Overshoot () 254

S-210-PIDITAE 5429 ITSE 035Stable time (s) Overshoot () 228

LCNSGA-III-PIDITAE 543 ITSE 004Stable time (s) Overshoot () 333

S-198-PIDS-205-PID

S-210-PIDLCNSGA-III-PID

5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller

S-198-FOPIDITAE 210 ITSE 002Stable time (s) 276Overshoot () 36

S-205-FOPIDITAE 2329 ITSE 010Stable time (s) Overshoot () 105


LCNSGA-III-FOPIDITAE 190 ITSE 002Stable time (s) 276Overshoot () 55

S-198-FOPIDS-205-FOPID

S-210-FOPIDLCNSGA-III-FOPID

minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)

(b) FOPID controller

Figure 7 Unit frequency obtained at 198m working water head

at complex operating conditions simultaneously using mul-tiobjective optimization algorithms should be studied andinvestigated

524 Analysis of Controllers Optimized under MultiworkingConditions The Pareto fronts obtained by LCNSGA-III-PID and LCNSGA-III-FOPID are shown in Figure 10 FromFigure 10 it can be noticed that the Pareto front of LCNSGA-III-FOPID can dominate that of LCNSGA-III-PID whichfurther demonstrate the superiority of the FOPID controllerIt can also be noticed from Figure 10 that the ITAE1 for

low water head (198m) and the ITAE2 for high water head(210m) are two conflicting indices The optimal solution forworking condition at low water head is not the best onefor working condition at high water head The employmentof multiobjective optimization algorithms to optimize thetwo objectives simultaneously can help researchers find thecompromise optimal solutions Compare the ITAE indicesfor S-198-PID and S-210-PID in Table 5 with the Paretofront obtained by LCNSGA-III-PID in Figure 10 it can befound that the ITAE indices for the two schemes are thenearest to the two edge solutions of the Pareto front of

14 Complexity




LCNSGA-III-PIDITAE 137 ITSE 002Stable time (s) 233Overshoot () 23

0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)





LCNSGA-III-PID Similar phenomenon also exists betweenthe solutions obtained by S-198-FOPID and S-210-FOPIDand the Pareto optimal solutions obtained by LCNSGA-III-FOPID which demonstrates that the pursing for a singleobjective (high or low water head) is at the cost of theother objective (corresponding low or high water head)What is more compared with the schemes for single workingconditions the adaptability and robustness of the controllerscan be greatly improved using multiobjective optimizationschemes A set of Pareto optimal solutions are obtained usingmultiobjective optimization schemes it is convenient for the

operator to select the most appropriate control parameterswhen the working condition changes ormuchmore attentionshould be paid to the extremely low water head or high waterhead working conditions

The effectiveness of the developed LCNSGA-III algo-rithm has been verified using the eight test functions whichhave been described in Section 51 In what follows the supe-riority of the developed LCNSGA-III algorithm is furtherdemonstrated by applying it to PTGS The Pareto fronts ofPID and FOPID controllers optimized by the two algorithmshave been illustrated in Figures 11(a) and 11(b) respectively

Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy



It can be observed from Figure 11 that the Pareto optimalsolutions obtained by LCNSGA-III can dominate almost allof those obtained byNSGA-III And the Pareto front obtainedby LCNSGA-III distributes more uniformly and extensivelyIt can be found in Table 5 that the performances indicesfor the compromise Pareto optimal solution of LCNSGA-IIIare all smaller than NSGA-III which further demonstratesthe superiority of LCNSGA-III in optimizing the MO-FOPID problem For example the improved percentagesof scheme LCNSGA-III-PID to scheme NSGA-III-PID are629 500 and 849 in terms of ITAE ITSE and OSOrespectively in the training stage (198m) The improved

percentages in the testing stage (205m) are 461 333 and913 in terms of ITAE ITSE and OSO respectively

6 Conclusions

In order to make PSUs adaptable to the changes of workingenvironment and improve their control quality and sta-bility this study constructs a multiobjective optimizationframework to optimize the FOPID controller for PTGSunder multiworking conditions An LCNSGA-III algorithmbased on Latin-hypercube sampling and chaos theory is pro-posed to solve the MO-FOPID problem under multiworking

16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)

LCNSGA-III-PIDLCNSGA-III-FOPID

09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1

Figure 10 Pareto fronts obtained by LCNSGA-III-PID LCNSGA-III-FOPID and their comparison with single-objective solutions

1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID

Figure 11 Comparison of Pareto fronts obtained by LCNSGA-III and NSGA-III for PID and FOPID controllers

conditions Implementation of the MO-FOPID controlleroptimized by LCNSGA-III relies on the simultaneous opti-mization of two complementary features the ITAE indexunder low and high water heads The experiment resultsindicate the following

(1) The classical NSGA-III algorithm is improved usingthe Latin hypercube sampling- based initialization techniqueand the chaotic crossover and mutation operators Exper-iments of the eight test functions show that the improve-ment strategies can effectively improve the convergenceand diversity of the Pareto optimal front of the NSGA-IIIalgorithm

(2) Compared with the traditional PID controller theFOPID can effectively suppress the frequency oscillation ofPSUs in the ldquoSrdquo characteristic area running at middle or lowworking water heads and can enhance the dynamic responseperformance of PTGS

(3) This study extends the single-objective optimiza-tion under single working conditions to the multiobjectiveoptimization framework under multiworking conditionsThe multiobjective framework has provided better dynamicperformances than the single-objective optimization meth-ods The multiobjective implementation of FOPID makesit much more convenient for a operator to select the most

Complexity 17

appropriate control parameters for a certain working condi-tion

This work sets a basis for research on multiobjective opti-mization of a FOPID controller of PTGS under multiworkingconditions The extension of the single-objective optimiza-tion under a single working-condition to the multiobjectiveframework under complex working conditions can providenew control law and optimization algorithms for the optimalcontrol of PTGS What is more the proposed LCNSGA-IIIalgorithm can be easily extended to optimization problems inother fields of scientific research and industrial application

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (No 51741907) the National NaturalScience Foundation of China (NSFC) (No 51709121) andthe National Natural Science Foundation of China (No51709122)

References

[1] R N Allan R Li and M M Elkateb ldquoModelling of pumped-storage generation in sequential Monte Carlo productionsimulationrdquo Generation Transmission and Distribution IEEProceedings vol 145 no 5 pp 611ndash615 1998

[2] W Wang C Li X Liao and H Qin ldquoStudy on unit com-mitment problem considering pumped storage and renewableenergy via a novel binary artificial sheep algorithmrdquo AppliedEnergy vol 187 pp 612ndash626 2017

[3] J Zhou Z Zhao C Zhang C Li and Y Xu ldquoA real-timeaccurate model and its predictive fuzzy PID controller forpumped storage unit via error compensationrdquo Energies vol 11no 1 2018

[4] J Zhou C Zhang T Peng and Y Xu ldquoParameter identificationof pump turbine governing system using an improved back-tracking search algorithmrdquo Energies vol 11 no 7 p 1668 2018

[5] C Zhang C Li T Peng et al ldquoModeling and synchronous opti-mization of pump turbine governing system using sparse robustleast squares support vector machine and hybrid backtrackingsearch algorithmrdquo Energies vol 11 no 11 p 3108 2018

[6] Y Xu Y Zheng Y Du W Yang X Peng and C Li ldquoAdaptivecondition predictive-fuzzy PID optimal control of start-upprocess for pumped storage unit at low head areardquo EnergyConversion and Management vol 177 pp 592ndash604 2018

[7] K Natarajan ldquoRobust PID controller design for hydroturbinesrdquoIEEE Transactions on Energy Conversion vol 20 no 3 pp 661ndash667 2005

[8] R-E Precup A-D Balint M-B Radac and E M PetriuldquoBacktracking search optimization algorithm-based approach

to PID controller tuning for torque motor systemsrdquo in Proceed-ings of the 2015 Annual IEEE Systems Conference pp 127ndash132IEEE Vancouver BC Canada 2015

[9] R-E Precup A-D Balint E M Petriu M-B Radac and E-IVoisan ldquoPI and PID controller tuning for an automotive appli-cation using backtracking search optimization algorithmsrdquo inProceedings of the IEEE Jubilee International Symposium onApplied Computational Intelligence and Informatics pp 161ndash166Timisoara Romania May 2015

[10] A Khodabakhshian and R Hooshmand ldquoA new PID con-troller design for automatic generation control of hydro powersystemsrdquo International Journal of Electrical Power amp EnergySystems vol 32 no 5 pp 375ndash382 2010

[11] R Melıcio V M F Mendes and J P S Catalao ldquoFractional-order control and simulation of wind energy systems withPMSGfull-power converter topologyrdquo Energy Conversion andManagement vol 51 no 6 pp 1250ndash1258 2010

[12] V Haji Haji and C A Monje ldquoFractional order fuzzy-PIDcontrol of a combined cycle power plant using Particle SwarmOptimization algorithm with an improved dynamic parametersselectionrdquo Applied Soft Computing vol 58 pp 256ndash264 2017

[13] S Zhang and L Liu ldquoNormalized robust FOPID controllerregulation based on small gain theoremrdquo Complexity vol 2018Article ID 5690630 10 pages 2018

[14] L Liu and S Zhang ldquoRobust fractional-order PID controllertuning based on bodersquos optimal loop shapingrdquo Complexity vol2018 Article ID 6570560 p 14 2018

[15] J-WPerng G-Y Chen andY-WHsu ldquoFOPID controller opti-mization based on SIWPSO-RBFNN algorithm for fractional-order time delay systemsrdquo Soft Computing vol 21 no 14 pp4005ndash4018 2017

[16] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[17] Y Xu J Zhou Y Zhang W Fu Y Zheng and X ZhangldquoParameter optimization of robust non-fragile fractional orderpid controller for pump turbine governing systemrdquo in Proceed-ings of the in Sixth International Conference on InstrumentationMeasurement Computer pp 15ndash18 2016

[18] Y Xu J Zhou X Xue W FuW Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[19] C Jiang Y Ma and C Wang ldquoPID controller parametersoptimization of hydro-turbine governing systems usingdeterministic-chaotic-mutation evolutionary programming(DCMEP)rdquo Energy Conversion and Management vol 47 no9-10 pp 1222ndash1230 2006

[20] H Fang L Chen and Z Shen ldquoApplication of an improvedPSO algorithm to optimal tuning of PID gains for water turbinegovernorrdquo Energy Conversion and Management vol 52 no 4pp 1763ndash1770 2011

[21] P G Kou J Z Zhou H E Yao-Yao X Q Xiang and L IChao-Shun ldquoOptimal PID governor tuning of hydraulic turbinegenerators with bacterial foraging particle swarm optimizationalgorithmrdquo Proceedings of the Csee vol 29 no 26 pp 101ndash1062009

[22] Z Wang C Li X Lai N Zhang Y Xu and J Hou ldquoAnintegrated start-up method for pumped storage units based on

18 Complexity

a novel artificial sheep algorithmrdquo Energies vol 11 no 1 p 1512018

[23] M J Mahmoodabadi and H Jahanshahi ldquoMulti-objectiveoptimized fuzzy-PID controllers for fourth order nonlinearsystemsrdquo Engineering Science and Technology an InternationalJournal vol 19 no 2 pp 1084ndash1098 2016

[24] S Panda ldquoMulti-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting geneticalgorithm-IIrdquo International Journal of Electrical Power amp EnergySystems vol 33 no 7 pp 1296ndash1308 2011

[25] A-A Zamani S Tavakoli and S Etedali ldquoFractional orderPID control design for semi-active control of smart base-isolated structures a multi-objective cuckoo search approachrdquoISA Transactions vol 67 p 222 2017

[26] H S Sanchez F Padula A Visioli and R Vilanova ldquoTuningrules for robust FOPID controllers based on multi-objectiveoptimization with FOPDT modelsrdquo ISA Transactions vol 66pp 344ndash361 2017

[27] S-Z ZhaoMW Iruthayarajan S Baskar and P N SuganthanldquoMulti-objective robust PID controller tuning using two lbestsmulti-objective particle swarm optimizationrdquo Information Sci-ences vol 181 no 16 pp 3323ndash3335 2011

[28] Z Chen X Yuan B Ji P Wang and H Tian ldquoDesign of afractional order PID controller for hydraulic turbine regulatingsystem using chaotic non-dominated sorting genetic algorithmIIrdquo Energy Conversion and Management vol 84 pp 390ndash4042014

[29] Z Chen Y Yuan X Yuan Y Huang X Li and W LildquoApplication of multi-objective controller to optimal tuningof PID gains for a hydraulic turbine regulating system usingadaptive grid particle swam optimizationrdquo ISA Transactionsvol 56 pp 173ndash187 2015

[30] T Peng J Zhou C Zhang and N Sun ldquoModeling and com-bined application of orthogonal chaoticNSGA-II and ImprovedTOPSIS to optimize a conceptual hydrological modelrdquo WaterResources Management pp 1ndash19 2018

[31] D Xue C Zhao and Y Chen ldquoA modified approximationmethod of fractional order systemrdquo in Proceedings of the IEEEInternational Conference on Mechatronics and Automation pp1043ndash1048 2006

[32] I Podlubny ldquoFractional-order systems and PI120582D120583-controllersrdquoin Proceedings of the IEEE Trans on Automatic Control vol 441999

[33] M ZamaniMKarimi-Ghartemani N Sadati andM ParnianildquoDesign of a fractional order PID controller for an AVR usingparticle swarm optimizationrdquo Control Engineering Practice vol17 no 12 pp 1380ndash1387 2009

[34] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[35] Y Xu C Li Z Wang N Zhang and B Peng ldquoLoad frequencycontrol of a novel renewable energy integrated micro-gridcontaining pumped hydropower energy storagerdquo IEEE Accessvol 6 pp 29067ndash29077 2018

[36] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[37] K Deb S Agrawal A Pratap and T Meyarivan ldquoA fast elitistnon-dominated sorting genetic algorithm for multi-objectiveoptimization NSGA-IIrdquo in Proceedings of the InternationalConference on Parallel Problem Solving From Nature vol 1917pp 849ndash858

[38] S Poles Y Fu and E RigoniThe effect of initial population sam-pling on the convergence of multi-objective genetic algorithmsSpringer BerlinGerman 2009

[39] W Fu K Wang C Li X Li Y Li and H Zhong ldquoVibrationtrend measurement for a hydropower generator based on opti-mal variational mode decomposition and an LSSVM improvedwith chaotic sine cosine algorithm optimizationrdquoMeasurementScience and Technology vol 30 no 1 p 015012 2019

[40] H Lu R Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquoApplied Soft Computing vol 13 no 5pp 2790ndash2802 2013

[41] T Habutsu Y Nishio I Sasase and S Mori ldquoA secret key cryp-tosystemby iterating a chaoticmaprdquo in International ConferenceonTheory and Application of Cryptographic Techniques pp 127ndash140 1991

[42] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[43] K Deb L Thiele M Laumanns and E Zitzler ldquoScalable multi-objective optimization test problemsrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 825ndash830May 2002

[44] X Lai C Li N Zhang and J Zhou ldquoAmulti-objective artificialsheep algorithmrdquo Neural Computing and Applications 2018

[45] K Deb and H Jain ldquoAn evolutionary many-objective optimiza-tion algorithm using reference-point- based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[46] K Deb Multiobjective Optimization Using Evolutionary Algo-rithms New York NY USA Wiley 2001

[47] C Zhang J Zhou C Li W Fu and T Peng ldquoA compoundstructure of ELM based on feature selection and parameteroptimization using hybrid backtracking search algorithm forwind speed forecastingrdquo Energy Conversion and Managementvol 143 pp 360ndash376 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

2 Complexity

fails to realize the global optimization of PTGS The con-trol problems of phase modulation instability and no-loadfrequency fluctuation are becoming increasingly prominentAs an extension of the classical PID controller the fractional-order PID controller (FOPID) has attracted the attention ofmany scholars for its better adaptability and flexibility andgreater potential to obtain better control performance [11ndash15] Li et al [16] proposed an improved gravitational searchalgorithmusing theCauchy andGaussianmutations to adjustthe parameters of the FOPID controller automatically Anumber of tests have shown that the FOPID controller canimprove the dynamic characteristics and stability of regula-tion frequency of PSUs In Xu et al [17] a robust nonfragileFOPID controller was proposed for PTGS The parametersof the FOPID controller were selected using the bacterialforaging algorithm and multiscenario analysis functionsThe FOPID controller turned out to have obtained higherrobustness and stability compared with the traditional PIDcontroller And it can fully track the nonlinear characteristicsof PTGS in the ldquoSrdquo area Xu et al [18] proposed an adaptivelyfast fuzzy fractional-order PID (AFFFOPID) control methodfor PSUs by combining a fuzzy fractional-order PD controllerwith a fuzzy fractional-order PI controller Experiments ofPTGS at variouswater heads under unload running conditionhave shown that the controller can effectively improve theperformance and control quality of PSUs during the transientprocess

Apart from the advantages of the FOPID compared withthe PID controller one of the most important and challengeissues of the employment of FOPID controller in PTGS is theoptimal optimization of its parameters With the continuousdevelopment of optimization algorithms and control theoryin recent years scholars have combined the FOPID controllerwith intelligent algorithms to achieve the optimal tuningof control parameters and improvement of control laws forPTGS [19] The related works on optimal optimization forPID controller or hydroturbine governing system (HTGS)can also provide meaningful reference for the research ofFOPID controller for PTGS Fang et al [20] developedan improved particle swarm optimization (PSO) algorithmfor optimal tuning of PID control parameters for waterturbine governor Simulation results have demonstrated thestable convergence characteristic and good computationalability of the developed optimization stagey Kou et al [21]proposed a novel BFO-PSO algorithm by introducing PSOinto the framework of the bacterial foraging optimization(BFO) algorithm to improve the control performance of thePID controller for HTGS The advantages of the proposedalgorithm to the BFO and PSO algorithms have been demon-strated through experiments at real working conditionsWang et al [22] proposed a three-stage start-up strategy forPSUs by opening guide vanes to a large opening degree firstlyand then reducing the opening degree and finally switchingto the PID controller The switching time and PID controlparameters are optimized synchronously using an integratedoptimization scheme based on artificial sheep algorithm(ASA) Simulation results under various water heads haveshown that the control strategy can shorten the start-up timeand reduce the speed oscillation

Some researchers have paid attention to multiobjectivedesigning to consider multiobjectives that reflect the specificcharacteristics of the control system [23 24] Zamani et al[25] developed a multiobjective cuckoo search approach tooptimize the parameters of a FOPID controller Sanchez etal [26] proposed a multiobjective optimization strategy foridentifying the optimal solution for a robust FOPID con-troller Zhao et al [27] proposed a parameter tuning schemebased on two lbests multiobjective PSO (2LB-MOPSO) tominimize the integral squared error and balanced robustperformance criteria of a robust PID controller simultane-ously Chen et al [28 29] put forward a parameter tuningscheme to optimize the integral of the squared error (ISE)and the integral of the time multiplied squared error (ITSE)performance indices of PTGS simultaneously The adaptivegrid PSO (AGPSO) and the chaotic nondominated sortinggenetic algorithm II (NSGA II) were adopted to realizethe optimal control of the PID and the FOPID controllersrespectively The proposed multiobjective PID and FOPIDcontroller turned out to have achieved better control effectsthan the compared methods

It is noticed that most of the research works on optimalcontrol of the PID or FOPID controllers are designed under asingleworking conditionThe traditionalmultiobjective opti-mization control usually adopts a set of objective functionsto obtain the Pareto optimal solutions for a certain conditionand then select the compromise optimal solution The tra-ditional multiobjective optimization control is essentially anoptimization scheme for singleworking conditionsHoweverthe characteristics of the controlled object of PTGS are notonly related to the nonlinear characteristics of the pumpturbine the penstock system the generator and the otherparts of PTGS but also vary with the changes of the workingconditions The optimal control of a single working conditionis often at the expense of the deterioration of some otherworking conditions The optimal control of PTGS undermultiworking conditions should be a process of overall trade-off and cannot be limited to the pursuit of the optimal controlunder a certain working condition In order to enhance theadaptability of PTGS to the change of working conditionsa multiobjective optimization framework which takes intoaccount the integral of the time multiplied absolute error(ITAE) index of PTGS running at multiworking conditionsis constructed The nondominated sorting genetic algorithmIII based on the Latin hypercube sampling and chaos theory(LCNSGA-III) is proposed to optimize the control parame-ters of the FOPID controller for PTGS under multiworkingconditions

The rest of this paper is arranged as follows Section 2gives a brief introduction to the multiobjective optimiza-tion problem Section 3 builds the mathematical model ofMO-FOPID under multiworking conditions Section 4 pro-poses the LCNSGA-III algorithm based on Latin hyper-cube sampling and the chaos theory to solve the MO-FOPID problem Section 5 employs eight benchmark func-tions and a real-world PTGS to verify the effectiveness ofthe LCNSGA-III algorithm and the developed multiobjec-tive optimization framework Section 6 gives the conclu-sions

Complexity 3




119904119905 119892119894 (119883) ge 0 119894 = 1 2 119898ℎ119895 (119883) = 0 119894 = 1 2 119896

119883 = [1199091 1199092 119909119889 119909119863 ]119909119889 min le 119909119889 le 119909119889 max 119889 = 1 2 119863

(1)













0119863120572119905 119891 (119905) = 1Γ (119899 minus 120572) int119905

0

119891119899 (119905)(119905 minus 120591)120572+1minus119899119889120591 (5)




intinfin

0119890minus119904119905119863120572119891 (119905) 119889119905 = 119904120572119865 (119904) (6)


119904120572 asymp 119870 119873prod119896=minus119873

((119904 + 1205961015840119896)(119904 + 120596119896) ) (7)

where 120596119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1+120572)2)(2119873+1) 1205961015840119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1minus120572)2)(2119873+1) 119870 = 120596120572



119880 (s)119864 (s) = 119870119901 + 119870119894119904120582 + 119870119889119904119906 (8)

4 Complexity

Kp

bp

+

++

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

FOPID

minus

s

s





119866 (119904) = 119870V1 + 1198791199101119904119866V (119904) = 1119879119910119904

(9)



11987211 = 119891119872 (11988611987311)11987611 = 119891119876 (11988611987311) (10)



119866119892 (119904) = 1119879119886119904 + 119890119899 (11)



119866ℎ (119904) = 119867 (119904)119876 (119904) = minus1198791199081199041 + 05119891119879119903119904 + 012511987911990321199042 (12)

Complexity 5

01

02

04

4

WH

06


08

05

flow angle x

1

21

0 0


01

2

4

4

WM

6


8

05

flow angle x

10

21

0 0





ITAE = int119879

0119905 |119890 (119905)| 119889119905 (13)




Min

1198911 = ITAE1 = 1198911 (119870119901 119870119894 119870119889 120582 120583)1198912 = ITAE2 = 1198912 (119870119901 119870119894 119870119889 120582 120583) (14)

subject to


(15)





6 Complexity



119867 = (119862 + 119892 minus 1119892 ) (16)











where 119875(119909119895119897lt 119909 lt 119909119895+1





1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)


120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)






Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3




119904119905 119892119894 (119883) ge 0 119894 = 1 2 119898ℎ119895 (119883) = 0 119894 = 1 2 119896

119883 = [1199091 1199092 119909119889 119909119863 ]119909119889 min le 119909119889 le 119909119889 max 119889 = 1 2 119863

(1)













0119863120572119905 119891 (119905) = 1Γ (119899 minus 120572) int119905

0

119891119899 (119905)(119905 minus 120591)120572+1minus119899119889120591 (5)




intinfin

0119890minus119904119905119863120572119891 (119905) 119889119905 = 119904120572119865 (119904) (6)


119904120572 asymp 119870 119873prod119896=minus119873

((119904 + 1205961015840119896)(119904 + 120596119896) ) (7)

where 120596119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1+120572)2)(2119873+1) 1205961015840119896 = 120596119887(120596ℎ120596119887)(119896+119873+(1minus120572)2)(2119873+1) 119870 = 120596120572



119880 (s)119864 (s) = 119870119901 + 119870119894119904120582 + 119870119889119904119906 (8)

4 Complexity

Kp

bp

+

++

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

FOPID

minus

s

s





119866 (119904) = 119870V1 + 1198791199101119904119866V (119904) = 1119879119910119904

(9)



11987211 = 119891119872 (11988611987311)11987611 = 119891119876 (11988611987311) (10)



119866119892 (119904) = 1119879119886119904 + 119890119899 (11)



119866ℎ (119904) = 119867 (119904)119876 (119904) = minus1198791199081199041 + 05119891119879119903119904 + 012511987911990321199042 (12)

Complexity 5

01

02

04

4

WH

06


08

05

flow angle x

1

21

0 0


01

2

4

4

WM

6


8

05

flow angle x

10

21

0 0





ITAE = int119879

0119905 |119890 (119905)| 119889119905 (13)




Min

1198911 = ITAE1 = 1198911 (119870119901 119870119894 119870119889 120582 120583)1198912 = ITAE2 = 1198912 (119870119901 119870119894 119870119889 120582 120583) (14)

subject to


(15)





6 Complexity



119867 = (119862 + 119892 minus 1119892 ) (16)











where 119875(119909119895119897lt 119909 lt 119909119895+1





1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)


120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)






Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

Kp

bp

+

++

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

FOPID

minus

s

s





119866 (119904) = 119870V1 + 1198791199101119904119866V (119904) = 1119879119910119904

(9)



11987211 = 119891119872 (11988611987311)11987611 = 119891119876 (11988611987311) (10)



119866119892 (119904) = 1119879119886119904 + 119890119899 (11)



119866ℎ (119904) = 119867 (119904)119876 (119904) = minus1198791199081199041 + 05119891119879119903119904 + 012511987911990321199042 (12)

Complexity 5

01

02

04

4

WH

06


08

05

flow angle x

1

21

0 0


01

2

4

4

WM

6


8

05

flow angle x

10

21

0 0





ITAE = int119879

0119905 |119890 (119905)| 119889119905 (13)




Min

1198911 = ITAE1 = 1198911 (119870119901 119870119894 119870119889 120582 120583)1198912 = ITAE2 = 1198912 (119870119901 119870119894 119870119889 120582 120583) (14)

subject to


(15)





6 Complexity



119867 = (119862 + 119892 minus 1119892 ) (16)











where 119875(119909119895119897lt 119909 lt 119909119895+1





1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)


120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)






Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

01

02

04

4

WH

06


08

05

flow angle x

1

21

0 0


01

2

4

4

WM

6


8

05

flow angle x

10

21

0 0





ITAE = int119879

0119905 |119890 (119905)| 119889119905 (13)




Min

1198911 = ITAE1 = 1198911 (119870119901 119870119894 119870119889 120582 120583)1198912 = ITAE2 = 1198912 (119870119901 119870119894 119870119889 120582 120583) (14)

subject to


(15)





6 Complexity



119867 = (119862 + 119892 minus 1119892 ) (16)











where 119875(119909119895119897lt 119909 lt 119909119895+1





1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)


120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)






Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity



119867 = (119862 + 119892 minus 1119892 ) (16)











where 119875(119909119895119897lt 119909 lt 119909119895+1





1199091198941198881 = 12 [(1 minus 120573) 1199091198941199011 + (1 + 120573) 1199091198941199012]1199091198941198882 = 12 [(1 + 120573) 1199091198941199011 + (1 minus 120573) 1199091198941199012]

(18)


120573 = (2119906)1(120578119888+1) 119906 le 05( 12 (1 minus 119906))

1(120578119888+1) 119890119897119904119890 (19)






Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7




119888119909(119896+1) = 119888119909(119896)05 119888119909(119896) lt 052 sdot (1 minus 119888119909(119896)) 119890119897119904119890 (22)



119906 = 119888119909(119896+1) (23)

119906119904 = 119888119909(119896+1) (24)









Start






Yes

No

End

cup



GD = 1119873radic 119873sum119894=1

1198631198942 (25)

8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

-

y



q h

mt+

-x

0mgKp

bp

++

+

minus

minus

xc

x

ycminus

1s

sKd

Td+

Kis

1 1

FOPID controller

KpKi Kd u

minus

s

s

FOPID1 + Ty1s Tys

1

Tas + en

minusTws

1 +1

2fTrs +

1

8T2r s

2

M11 = fM(aN11)

Q11 = fQ(aN11)




Spread = sum119872119895=1 119889119890119895 + sum119873

119894=1

10038161003816100381610038161003816119889119894 minus 11988910038161003816100381610038161003816sum119872119895=1 119889119890119895 + 119873 sdot 119889 (26)








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9



ZDT1


119892 (119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT2


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT3


119892(119883) = 1 + 9 sdot 119899sum119894=2

119909119894(119899 minus 1)ZDT4

1198911(119883) = 119909110


119892(119883) = 1 + 10(119899 minus 1) + 119899sum119894=2

[1199092119894 minus 10 cos(4120587119909119894)] 119894 = 2 3 119899

ZDT6


119892 (119883) = 1 + 9 sdot [ 119899sum119894=2

119909119894(119899 minus 1)]025

DTLZ1




119891119872(119883) = 12 (1 minus 1199091) (1 + 119892 (119909119872))119892 (119883119872) =

100(119909119872 + sum119909119894isin119909119872


DTLZ2




DTLZ5


119892 (119909119872) = Σ119909119894isin11990911987211990901119894

10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity







ZDT1 ZDT2

ZDT3 ZDT4

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1


002040608

1

F2

01 02 03 04 05 06 07 080F1

0010203040506070809

1

F2

01 02 03 04 05 06 07 08 09 10F1

01 02 03 04 05 06 07 08 09 10F1

0010203040506070809

1

F2


Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

0 0005005 0101 015015

DTLZ1

0202F1F2

025025 0303 035035 0404 045045 0505

0 01 010202 0303

DTLZ2

0404F1F2

0505 0606 0707 0808 0909 11

ZDT6

DTLZ5

0 001 010202 03 03F2 F1

04 0405 0506 060707

0010203040506070809

F2

04 05 06 07 08 09 103F1

0005

01015

02025

03035

04045

05

F30

010203040506070809

1

F3

0010203040506070809

1

F3









ITSE = int119879

0119905 (119890 (119905))2 119889119905 (27)


12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity










Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13





S-198-PIDS-205-PID


5 10 15 20 25 30 35 40 45 500time (s)

minus001

000

001

002

003

004

005

006U

nit f

requ

ency

(a) PID controller







minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

5 10 15 20 25 30 35 40 45 500time (s)






14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006U

nit f

requ

ency

time (s)

S-198-PIDS-205-PID


(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

Uni

t fre

quen

cy

time (s)








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 15





0 5 10 15 20 25 30 35 40 45 50minus001

000

002

003

004

005

006

time (s)

S-198-PIDS-205-PID


001

Uni

t fre

quen

cy

(a) PID controller





0 5 10 15 20 25 30 35 40 45 50minus001

000

001

002

003

004

005

006

time (s)



Uni

t fre

quen

cy





6 Conclusions


16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








16 Complexity

S-210-FOPID(2011096)

S-198-FOPID(21187) S-210-PID

(5429134)

S-198-PID(568316)


09

12

15

18

21

24

27

30

33

36

ITA

E2

6 12 18 24 30 36 42 48 54 600ITAE1


1890 27 36 4510

15

20

25

30

NSGA-IIILCNSGA-III

ITAE1

ITA

E2

(a) PID

17 21 25 29

NSGA-IIILCNSGA-III

ITAE1

10

12

14

16

18

ITA

E2

(b) FOPID






Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 17



Data Availability




Acknowledgments


References
























18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018








18 Complexity


































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








multiobjective optimization of a fractional-order pid

Documents