j ourna l h o mepage: www.elsev ier .com/ locate /asoc
nterval type-2 fuzzy PID load frequency controller using Bigang–Big Crunch optimization
ngin Yesil ∗
stanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department, Maslak, TR-34469 Istanbul, Turkey
r t i c l e i n f o
rticle history:eceived 6 September 2012eceived in revised form 16 February 2013ccepted 30 October 2013vailable online xxx
eywords:nterval type-2 fuzzy PID controllersoad frequency controlig Bang–Big Crunch optimization
a b s t r a c t
This paper proposes an optimization based design methodology of interval type-2 fuzzy PID (IT2FPID)controllers for the load frequency control (LFC) problem. Hitherto, numerous fuzzy logic control struc-tures are proposed as a solution of LFC. However, almost all of these solutions use type-1 fuzzy sets thathave a crisp grade of membership. Power systems are large scale complex systems with many differentuncertainties. In order to handle these uncertainties, in this study, type-2 fuzzy sets, which have a gradeof membership that is fuzzy, have been used. Interval type-2 fuzzy sets are used in the design of a loadfrequency controller for a four area interconnected power system, which represents a large power sys-tem. The Big Bang–Big Crunch (BB–BC) algorithm is applied to tune the scaling factors and the footprint ofuncertainty (FOU) membership functions of interval type-2 fuzzy PID (IT2FPID) controllers to minimizefrequency deviations of the system against load disturbances. BB–BC is a global optimization algorithmand has a low computational cost, a high convergence speed, and is therefore very efficient when thenumber of optimization parameters is high as presented in this study. In order to show the benefits of
IT2FPID controllers, a comparison to conventional type-1 fuzzy PID (T1FPID) controllers and conventionalPID controllers is given for the four-area interconnected power system. The gains of conventional PIDand T1FPID controllers are also optimized using the BB–BC algorithm. Simulation results explicitly showthat the performance of the proposed optimum IT2FPID load frequency controller is superior comparedto the conventional T1FPID and PID controller in terms of overshoot, settling time and robustness againstdifferent load disturbances.
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. Introduction
The load frequency control (LFC) is a very critical component inower system operation and control for generation and distribu-ion of electric supply with good reliability and quality [1,2]. Largecale power systems are normally composed of control areas oregions representing coherent groups of generators. Various areasre interconnected through tie lines. The tie lines are utilized forontractual energy exchange between areas and provide inter-areaupport in case of abnormal conditions. Area load changes andbnormal conditions lead to mismatches in frequency and sched-led power interchanges between areas. These mismatches haveo be corrected by load frequency control (LFC), which is defineds the regulation of the power output of generators within a pre-cribed area [3]. The regulation is performed so as to maintain the
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.
cheduled system frequency and/or established interchange withther areas within predetermined limits in response to changes in
system frequency and tie line loading. The key assumptions in theclassical LFC problem are [4,5]:
i. The steady state frequency error following a step load changeshould vanish. The transient frequency and time errors shouldbe small.
ii. The static change in the tie power following a step load in anyarea should be zero, provided each area can accommodate itsown load change.
iii. Any area in need of power during an emergency should beassisted from other areas.
The conventional control strategy for the LFC problem is to takethe integral of the control error as the control signal. An integralcontroller provides zero steady state frequency deviation but itexhibits poor dynamic performance [5,6]. To improve the transientresponse, various control techniques have been proposed. In [7],an adaptive regulator which uses the a priori known information
ad frequency controller using Big Bang–Big Crunch optimization,031
and satisfies the multi-objective character of the control is pro-posed for the Hungarian power system. A new systematic approachto design of sequential decentralized load frequency controllersfor four-area power systems based on � synthesis and analysis
s proposed in [8]. The design strategy includes enough flexibilityo set the desired level of stability and performance and considerhe practical constraints by introducing appropriate uncertainties.n [9] two robust decentralized proportional integral (PI) controlesigns are proposed for load frequency control (LFC) with com-unication delays; and in both methodologies, the PI based LFC
roblem is reduced to a static output feedback (SOF) control syn-hesis for a multiple delay system. An intelligent wavelet neuraletwork (WNN) approach for the load frequency controller (LFC)o damp the frequency oscillations of two area power systems dueo load disturbances is presented in [10]. The comparison betweenhe proposed WNN controller responses with those of the fixed gainI controller indicated the effectiveness of the proposed controllern damping the frequency oscillations and tie line power very fast
ith less undershoot and overshoots. In [11] PID tuning method,hich is based on a two-degree-of-freedom internal model con-
rol (IMC) design method, of load frequency controllers for powerystems is discussed. The resulting PID controller is related to twouning parameters thus it is showed that detuning is easy whenecessary. In addition, some of the researchers used super mag-etic energy storage (SMES) units to improve the transient response12–17]. Artificial neural networks have been successfully appliedo the LFC problem with rather promising results [18–22].
As mentioned above, a multi-area power system comprisesreas that are interconnected by high voltage transmission linesr tie-lines, and the trend of frequency measured in each con-rol area is an indicator of the trend of the mismatch power inhe interconnection and not in the control area alone [8]. In thevent of a decentralized power system, area frequency and tie-lineower interchange vary as power load demand varies randomly;ccordingly, the objectives of LFC are to minimize the transienteviations of these variables and to guarantee their steady staterrors to be zero. When dealing with the LFC problem of powerystems, unexpected external disturbances, parameter uncertainies and the model uncertain ties pose big challenges for controlleresign. Because of the complexity and multi-variable conditions ofhe power system, conventional control methods may not give sat-sfactory solutions for multi-area power system. Fuzzy logic controls an excellent alternative to the conventional control methodology
hen the processes are too complex for analysis by conventionalathematical techniques. Correspondingly, the robustness and the
eliability property of fuzzy controllers make them useful for solv-ng a wide range of control problems in the power systems. Theuzzy control techniques for the LFC problem are applied, and amaz-ng improvements are obtained [2,6,23–26], and a big collection ofll the current LFC techniques can be found in Ibraheem [27] andn Shayeghi et al. [28].
To date, all of the fuzzy logic applications are focused on conven-ional type-1 fuzzy logic system (T1FLS). However, the type-1 fuzzyogic controller (FLC), whose membership functions are type-1uzzy sets, is sometimes not sufficient to directly handle uncer-ainties. To deal with this problem, the concept of type-2 fuzzy setsT2FLS) was introduced by Zadeh [29] as an extension of T1FLS.ompared to T1FLS, T2FLS handle the vagueness inherent in lin-uistic words. The MFs of type-2 fuzzy sets are three dimensionalnd include a footprint of uncertainty (FOU), which is the newhird-dimension of type-2 fuzzy sets. Accordingly, the FOU pro-ides additional degrees of freedom that can make it possible toirectly model and handle the uncertainties in a better way com-are to type-1 fuzzy sets. As a result, FLCs that use type-2 fuzzyets to represent the inputs and outputs of the FLC can handle thehort and long term uncertainties to produce a good performance
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10
30–32].A fuzzy logic system (FLS) that is described using at least one
ype-2 fuzzy set is called a type-2 FLS. The operations of type-2uzzy systems are typically more computationally involved than
PRESSing xxx (2013) xxx–xxx
type-1 systems. This has urged researchers to search for ways toalleviate this high computational burden if type-2 FLCs are to findtheir way to real-world applications; therefore, interval type-2fuzzy sets were introduced [33,34]. It has been acknowledged thatusing interval type-2 fuzzy sets (IT2FLS) to represent the inputsand/or outputs of FLC has many advantages when compared tothe type-1 fuzzy sets (T1FLS) [35]. To optimize the parameters ofT2FS and IT2FS researchers have proposed many different methods[36–43]. Lately, Castillo and Melin published an invaluable reviewon design and optimization of type-2 fuzzy controllers [44].
Some control applications of type-2 fuzzy logic are liquid-levelprocess control [45]; bioreactor control [46]; autonomous mobilerobots [41]; pH neutralization process [47,48] and system identifi-cation [49]. However, the applications of T2FLS in power systemsare limited. Lately, Sudha and Santhi [50] proposed a type-2 fuzzycontroller for two area power system and showed that type-2 fuzzycontroller shows superior performance compare to type-1 fuzzycontroller. Then, the same authors developed their controller struc-ture and included SMES to type-2 fuzzy controller [51]. In [50] and[51], no methodology is shown to tune the type-2 fuzzy controllersoptimally. The authors only show that the performance of type-2fuzzy controllers in power systems are better than the conven-tional type-1 fuzzy controllers since type-2 fuzzy sets representthe uncertainties better than type-1 fuzzy sets.
In this study, interval type-2 fuzzy PID (IT2FPID) controller isproposed as a load frequency controller since a power system is alarge and complex system with many uncertainties. Consequently,for optimizing the parameters of IT2FPID controller Big Bang–BigCrunch (BB–BC) optimization [52] is used for the first time in lit-erature as a method to tune the FOU. It will be illustrated thatthe extra degree of freedom of the antecedent IT2-FSs gives theBB–BC method an opportunity to find a more optimal solutionthan those obtained for the optimized conventional PID and type-1fuzzy PID controllers with respect to the Integral of Time AbsoluteError (ITAE) performance index. Consequently, for the two inputsof IT2FID, which are the error and the change of the error, the totalnumbers to be optimized for the IT2FPID design is 8, and it is obvi-ous that the IT2FPID has these parameters as the extra degrees offreedom.
A multi-area power system includes many areas and conse-quently each area must have a controller, which is in this studyan IT2FPID controller. Since the number of optimization param-eters of type-2 fuzzy controller is relatively big for a multi-areapower system, the BB–BC optimization method [52] which hasa low computational cost and a high convergence speed is pre-ferred. In order to make a fair comparison, the rule base andscaling factors are kept fixed for the fuzzy controller structureswhile only the antecedent parameters are optimized off-line forpossible power load demands. The Big Bang–Big Crunch (BB–BC)optimization method is a natural evolutionary algorithm similarto the genetic algorithms, the ant colony optimization, the particleswarm optimizer and the harmony search [53]. The algorithm gen-erates random points in the Big Bang phase and shrinks those pointsto a single representative point via a center of mass or minimalcost approach in the Big Crunch phase. It is reported to be capableof quick convergence even in long, narrow parabolic shaped flatvalleys or in the existence of several local minima. Though it is anew algorithm, it has been applied to many areas including non-linear controller design [54], design of space trusses [55,56], inversecontroller design [57] and airport gate assignment problem [59].
The optimization results demonstrated that the IT2FPID, whichhas more design parameters, outperforms the optimized type-1
ad frequency controller using Big Bang–Big Crunch optimization,.031
fuzzy and conventional PID controller structures with respect toITAE performance index. Since the FOU provides the antecedenttype-2 fuzzy sets with extra degrees of freedom, it can be con-cluded that an IT2FPID is more capable of providing lower ITAE
erformance measure compared to a type-1 counterpart that hashe same rules and scaling factors. In other words, this extra degreef freedom gives the BB–BC global search method an opportunity tond a more optimal solution than those obtained for the optimizedID and type-1 fuzzy PID (T1FPID). Furthermore, the proposedT2FPID controller decreases the settling time and oscillations inhe frequency deviations.
The rest of the paper is divided into five sections. In Section, a four-area interconnected power system model, which repre-ents a large and complex power system, is presented. The type-2uzzy controllers are detailed in Section 3. The structure of the pro-osed IT2FPID controller is discussed and the BB–BC optimizationlgorithm used for tuning the type-2 membership functions of FLCs presented. In Section 4, simulations are presented to illustratehe effectiveness of the proposed unified method. The proposedT2FPID controllers performance are compared with type-1 fuzzyID (T1FPID) controllers and conventional PID controllers. Possi-le extensions and future work on this issue with conclusions arenally given in Section 5.
. A four-area interconnected power system model
An interconnected four-area [2,60–65] power system has beensed to confirm the effectiveness of the proposed approach in aore realistic power system scenario. The interconnected power
ystem consists of four single areas connected through power linesalled the tie line. Area 1, Area 2 and Area 3 are denoted as therea with reheat unit and they are interconnected with each other,nd Area 4 is denoted as the area with hydro unit, as well Area 4s only connected with Area 1. A simplified representation for annterconnected system in a general form, including both ring andus interconnections, is shown in Fig. 1.
Each area feeds its user pool, and the tie line allows electricower to flow between areas. Since the areas are tied together, a
oad perturbation in one area affects the output frequencies of thereas as well as the power flow on the tie line. The control sys-em of each area needs information about the transient situation ofther areas in order to bring the local frequency back to its steadytate value. Information about the other areas is found in the outputrequency fluctuation of that area and in the tie line power fluctu-tion. Therefore, the tie line power is sensed, and the resulting tieine power signal is fed back into corresponding areas. A block dia-ram representing the four area interconnected power system is
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.
iven in Fig. 2.In the multi-area power system, �fi is the frequency deviation of
rea #i (Hz), Bi is the frequency bias setting of Area #i (p.u. MW/Hz),ij is the synchronizing coefficients between Area #i and Area #j
PRESSing xxx (2013) xxx–xxx 3
(p.u. MW/Hz), �Pi is the load disturbance in Area #i (p.u. MW),�Ptiei is the tie-line power between Area #i and other areas (p.u.MW), and final Ri represents the speed regulation of Area #i (Hz/p.u.MW).
In a conventional system, turbine reference power of each areais tried to be set to its nominal value by an integral controller andthe input of the controller of each area is Bi�fi + �Ptiei (i = 1, 2, 3, 4)called Area Control Error (ACE) of the same area.
3. Type-2 fuzzy PID controllers
Fuzzy logic is credited with being an adequate methodology fordesigning robust systems that are able to deliver a satisfactory per-formance in the face of uncertainty and imprecision. Hence, thefuzzy logic system (FLS) has become established as an adequatetechnique for a variety of applications [35]. As the type-1 fuzzy set(T1FS), the concept of type-2 fuzzy set (T2FS) was introduced byZadeh [29] as an extension of the concept of an ordinary fuzzy set.While type-1 fuzzy logic has been the most popular form of fuzzylogic, recent years have shown a significant increase in researchtoward more complex forms of fuzzy logic, in particular, type-2fuzzy logic. A fuzzy logic system (FLS) that is described using atleast one type-2 fuzzy set is called a type-2 FLS. Type-1 FLSs areunable to directly handle rule uncertainties, because they use type-1 fuzzy sets that are certain. However, type-2 FLSs are very useful incircumstances where it is difficult to determine exact rules [34,59].
Type-2 fuzzy sets (T2FS) are generalized forms of those of type-1 fuzzy sets (T1FS), and characterizing a type-2 fuzzy set is not aseasy as characterizing a type-1 fuzzy set. Mathematically, a type-2fuzzy set, denoted as A, is characterized by a type-2 membershipfunction �A(x, u), where x ∈ X and u ∈ Jx ⊆ [0, 1], i.e.:
A ={
((x, u), �A(x, u))∣∣∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]
}(1)
in which 0 ≤ �A(x, u) ≤ 1. For a continuous universe of discourse,A can be expressed as
A =∫
x ∈ X
∫u ∈ Jx
�A(x, u)(x, u)
Jx ⊆ [0, 1] (2)
where∫ ∫
denotes union over all admissible x and u. Jx is referredto as the primary membership of x, and �A(x, u) is a type-1 fuzzyset known as the secondary set. The uncertainty in the primarymembership of a type-2 fuzzy set A is represented by a regioncalled footprint of uncertainty (FOU) which is also the union ofall primary memberships. When �A(x, u) = 1 for ∀u ∈ Jx ⊆ [0, 1], aninterval type-2 fuzzy set (IT2FS) is obtained. The uniform shadingfor the FOU represents the entire interval type-2 fuzzy set and itcan be described in terms of an upper membership function �A(x)and a lower membership function �
A(x) [34].
An example of a triangular type-2 fuzzy set is given in Fig. 3. Theprimary membership Jx is given in Fig. 3a, and its associated possiblesecondary memberships, that are triangular and interval, are pre-sented in Fig. 3b and c, respectively. When the interval secondarymembership function that is illustrated in Fig. 3c is taken, an IT2FLSis obtained. The term FOU, the shaded region in Fig. 3a, is very use-ful because it not only focuses the attention on the uncertaintiesinherent in a specific type-2 membership function, whose shape isa direct consequence of the nature of these uncertainties, but it alsoprovides a very convenient verbal description of the entire domainof support for all the secondary grades of a type-2 membershipfunction [34].
A FLC described using at least one type-2 fuzzy set is called a
ad frequency controller using Big Bang–Big Crunch optimization,031
type-2 fuzzy logic controller (T2FLC). A block diagram of a T2FLC,that is a special fuzzy logic system, is illustrated in Fig. 4. Similarto a type-1 FLC, a type-2 FLC includes type-2 fuzzifier, rule-base,inference engine, and substitutes the defuzzifier by the output
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID load frequency controller using Big Bang–Big Crunch optimization,Appl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.031
ARTICLE IN PRESSG ModelASOC 2113 1–13
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Fig. 3. Illustration of (a) type-2 fuzzy membership function, (b) triangular secondary membership function and (c) interval secondary membership function.
diagra
ptfitfiowlTo
sgsttatst[
TC
tions (N: Negative, Z: Zero, P: Positive) are chosen and alternativelyfor the output singleton membership functions (NL: Negative Large,
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Fig. 4. Block
rocessor. The type-2 FLC works like a type-1 FLC, of course inhis case the fuzzifier take crisp inputs and this inputs are fuzzi-ed into, in general, input type-2 fuzzy sets, which then activatehe inference engine and the rule base to produce output type-2uzzy sets. A type-2 FLS is again characterized by IF-THEN rules, butts antecedent and/or consequent fuzzy sets are now type-2. Theseutput type-2 fuzzy sets are then processed by the type-reducerhich combines the output sets and then perform a centroid calcu-
ation, which leads to type-1 fuzzy sets called the type-reduced set.he defuzzifier can then defuzzify the type-reduced type-1 fuzzyutputs to produce crisp outputs [34].
According to the rule-base formation, there are three possibletructures for building such FLCs as classified in Table 1. The mosteneral case is called Model I and this structure has type-2 fuzzyets in antecedents, and type-1 fuzzy sets in consequents. There arewo special cases of Model I: Model II with crisp numbers (single-ons) in the consequents; Model III type-1 fuzzy sets in antecedentsnd in consequents. Some may argue that Model III is a Mamdaniype-1 FLC, where the output is a crisp number, but here the con-
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.
equent is type-1 fuzzy set [66]. A special application of Model IIIype-2 fuzzy systems in fault detection is presented by Oblak et al.67].
able 1lassification of Takagi-Sugeno type-2 FLCs.
Consequent MF Antecedent MF
Type-1 fuzzy set Type-2 fuzzy set
Type-0 fuzzy set (crisp) Type-1 FLC Model IIType-1 fuzzy set (interval) Model III Model I
m of T2FLC.
3.1. Type-1 fuzzy PID controllers
A type-1 fuzzy PID (T1FPID) controller can be designed usingone, two or three inputs [68]. From the computational and per-formance point of view, in this study, a T1FPID controller with twoinputs is used as a load frequency controller. To obtain proportional,integral and derivative control action all together, it is appropriateand usual way to combine PI and PD actions together to form afuzzy PID controller. The formulation of fuzzy PID controller can beachieved either by combining fuzzy PI and fuzzy PD controllers withtwo distinct rule-bases or one fuzzy PD controller with an integra-tor at the output and then adding the output of FLC to the outputof the integrator as it is given in Fig. 5 [69].
Here, the ACE and the �ACE are the inputs of the load frequencyfuzzy PID controller and the universe of discourse is chosen to be[−1,1] for the membership functions of input and output variablesas given in Fig. 6. For the inputs, three triangular membership func-
ad frequency controller using Big Bang–Big Crunch optimization,031
NM: Negative Medium, Z: Zero, PM: Positive Medium, PL: Posi-tive Large) are prepared. The T1FPID controller is designed using
Fig. 6. Membership functions of the type-1 fuzzy PID controller.
product–sum inference method with center of gravity defuzzifi-ation method.
The ith fuzzy rule used for T1FPID controller has the followingorm:
Rule Ri : IF ACE is A1i AND �ACE is A2i
THEN u is ci, i = 1, . . ., 9
here A1i and A2i are type-1 fuzzy sets, ci is a singleton. The com-lete rule base of the T1FPID controller with 9 rules is given inable 2.
The input and output parameters are scaled to fit this range viacaling factors as illustrated in Fig. 5. The input scaling factors (Ke
nd Kd) normalize the real world inputs to a range in which mem-ership functions are defined. The output scaling factors ( and ˇ)re used to change the normalized control effort to its practical
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10
alue. The values of the scaling factor play an important role in theerformance of the fuzzy PID controllers [70,71].
able 2ule base of the fuzzy controller.
�ACE ACE
N Z P
N NL NM ZZ NM Z PMP Z PM PL
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PRESSing xxx (2013) xxx–xxx
3.2. Interval type-2 fuzzy PID controllers
The operations of type-2 fuzzy systems are typically morecomputationally involved than type-1 systems. This has urgedresearchers to search for ways to alleviate this high computationalburden if type-2 FLCs are to find their way to real-world applica-tions. For this purpose, interval type-2 fuzzy sets were introduced[33,34]. This type of fuzzy sets provides a simplified and efficientalternative to easily compute the input and antecedent operationsfor FLSs and offers a balanced tradeoff between performance andcomplexity [72].
The distinction between type-1 and type-2 rules is associatedwith the nature of the membership functions; the structure ofthe rules remains exactly the same in the type-2 case [46]. Theantecedent part of the IT2FLC used in this study uses interval type-2 fuzzy sets, and the consequent part is of the singletons that arethe zero-order fuzzy set. The ith fuzzy rule used for the intervaltype-2 load frequency fuzzy PID controller has the following form:
Rule Ri : IF ACE is A1i AND �ACE is A2i
THEN u is ci, i = 1, . . ., 9
where A1i and A2i are interval type-2 fuzzy sets, ci is a singleton.In the interval type-2 FLC, the inference engine combines the
fired rules and gives a mapping from input type-2 fuzzy sets tooutput type-2 fuzzy sets. The type-2 fuzzy outputs of the inferenceengine are then processed by the type-reducer, which combinesthe output sets and performs a centroid calculation that leads totype-1 fuzzy sets called the type reduced sets. The type reducedfuzzy set for an interval type-2 fuzzy model can then be expressedas YTR = [yl, yr] where YTR is the type reduced interval type-1 fuzzyset determined by its two end points and yl and yr. There are manyways to perform type-reduction in type-2 fuzzy systems [34,73].The iterative Karnik and Mendel method (KM) is used most com-monly for type reduction [74]. Wu [74] pointed out two features ofthe Karnik–Mendel type reduction method; namely, novelty andadaptiveness. Novelty means that the upper and lower member-ship functions of the same type-2 fuzzy set may or may not beused simultaneously in computing the type-reduced set. The adap-tiveness has the meaning that the bounds of the type-reducedinterval set change as the inputs change. It has been stated thatthe alternative approximations to the KM algorithms cannot simul-taneously capture novelty and adaptiveness features. Therefore,the Karnik–Mendel type reduction method is preferred. It shouldbe emphasized that the consequent part of the IT2FPID controllerhas singleton membership functions instead of the interval type-2fuzzy sets; therefore Model II IT2FLC is obtained. Then, the defuzzi-fier computes the system output variable y by performing thedefuzzification operation on the interval set [yl, yr] from the typereducer by taking the average of the type-reduced to obtain crispoutputs that are sent to the actuators [35,38].
There are numerous types and shapes of membership functionscan be chosen for inputs of the IT2FPID. Three possible triangularIT2FS that can be used in controller design are shown in Fig. 7.In this study, for each of the IT2FPID controller, triangular IT2FSwith uncertain end-points given in Fig. 8 are used. The membershipfunctions that are in the antecedent part of the fuzzy rules have8 free parameters of FOU to be determined by the designer. Theoutput membership functions of the fuzzy controller are kept assingletons as shown in Fig. 6c.
ad frequency controller using Big Bang–Big Crunch optimization,.031
3.3. A brief overview of Big Bang–Big Crunch optimization
In this study, first the gains of the conventional PID controller,then the scaling factors of the T1FPID and then finally IT2FPID
E. Yesil / Applied Soft Computing xxx (2013) xxx–xxx 7
F(
cirl
oQ2itpf
era
Table 3BB–BC optimization algorithm.
Step 1 (Big Bang Phase)An initial generation of N candidates is generated randomly in the search
space.Step 2The cost function values of all the candidate solutions are computed.Step 3 (Big Crunch Phase)The center of mass is calculated. Either the best fit individual or the center
of mass is chosen as the point of Big Bang Phase.Step 4New candidates are calculated around the new point calculated in Step 3
by adding or subtracting a random number whose value decreases as theiterations elapse.
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ig. 7. Triangular interval type-2 fuzzy sets (IT2FS): (a) with uncertain end-point;b) with uncertain mid point; (c) uncertain width.
ontroller parameters that are the parameters of the IT2FS shownn Fig. 8 are optimized using the Big Bang–Big Crunch (BB–BC) algo-ithm. This global optimization method is prepared since it has aow computational cost and a high convergence speed.
The Big Bang–Big Crunch (BB–BC) optimization method devel-ped by Erol and Eksin [53] consists of two main steps: The first steps the Big Bang phase where candidate solutions are randomly dis-ributed over the search space and the next step is the Big Crunchhase where a contraction procedure calculates a center of massor the population.
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.
The initial Big Bang population is randomly generated over thentire search space just like any other evolutionary search algo-ithms. All subsequent Big Bang phases are randomly distributedbout the center of mass or the best fit individual in a similar
Fig. 8. Interval type-2 fuzzy sets used fo
447
Step 5Return to Step 2 until stopping criteria has been met.
fashion. In [53], the working principle of the Big Bang phase isexplained as energy dissipation or the transformation from anordered state (a convergent solution) to a disordered or chaoticstate (new set of candidate solutions).
After the Big Bang phase, a contraction procedure is applied dur-ing the Big Crunch. In this phase, the contraction operator takes thecurrent positions of each candidate solution in the population andits associated cost function value. In this study, the best fit individ-ual is chosen as the starting point in the Big Bang phase. Instead ofthe best fit individual, as stated in [53] the center of mass, can becomputed as
xc =∑N
i=1(1/f i)xi∑Ni=11/f i
(3)
where xc is the position vector of the center of mass, xi is the positionvector of the ith candidate, fi is the cost function value of the ithcandidate, and N is the population size.
The new generation for the next iteration Big Bang phase isnormally distributed around xc as follows:
xnew = xc + r˛(xmax − xmin)k
(4)
where r is a normal random number, is a parameter limiting thesize of the search space, xmax and xmin are the upper and lowerlimits, and k is the iteration step. Since normally distributed num-bers can be exceeding ±1, it is necessary to limit the populationto the prescribed search space boundaries. This narrowing down
ad frequency controller using Big Bang–Big Crunch optimization,031
restricts the candidate solutions into the search space boundaries.The procedure and the flowchart of the BB–BC optimization is givenin Table 3 and in Fig. 9, respectively.
8 E. Yesil / Applied Soft Computing xxx (2013) xxx–xxx
4
tvmtd
n(ceC
J
Icoi
TP
Table 5Gains of the conventional PID controllers.
Area KP Ki Kd
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Fig. 9. The flowchart of BB–BC optimization algorithm.
. Simulations
The parameters of the four-area interconnected power systemhat has been used in this study are given in Table 4 [2]. As pre-iously stated, the aim of the load frequency control task is toinimize the system frequency deviations in each area and the
ie-line power flow �Ptie between the related areas under the loadisturbances �PD1, �PD2, �PD3 and �PD4 in the power system.
In this study, the performance of the three different controllers,amely conventional PID controller, type-1 fuzzy PID controllerT1FPID), and interval type-2 fuzzy PID (IT2FPID) controller areompared. In order to make a fair comparison, the controller param-ters of each controller are optimized using the Big Bang–Bigrunch optimization method.
In different studies, different cost functions such as ISE, IAE,
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10
TSE, ITAE, are proposed for the optimization of the load–frequencyontrollers. As clearly mentioned in [20], ISE penalize excessivelyverdamped and excessively underdamped responses, ITSE penal-ze large initial and final errors. Nevertheless, in this study, the
expectation from the IT2FPID controller is to reduce long-durationtransients; therefore ITAE cost function given in Eq. (5) is preferredsince ITAE penalize long duration transients and errors occurringlater in the response [6].
For Area 1, 2 and 3 the same controller with the same con-trol parameters is used; on the other hand, for the fourth areawith hydro turbine other controller parameters are determined.First the simulations are performed to determine the gains of theconventional PID, scaling factors of T1FPID, and the membershipparameters of IT2FPID controller. Therefore, for these simulations,always step disturbances of �PD1 = −0.01 pu, �PD2 = −0.01 pu,�PD3 = −0.01 pu and �PD4 = −0.0005 pu are applied to the cor-responding areas. During all the simulations in the parameteroptimization stage, 40 individuals and 50 iteration steps are usedfor BB–BC algorithm.
As a first step, the gains of the conventional parameters aresearched using the cost function given in Eq. (5). As a result ofthe optimization process, the cost function is obtained as 1.135and the obtained gains of conventional PID controllers after BB–BCoptimization are given in Table 5.
As a second step, the scaling factors of the T1FPID controller aresearched using the same method discussed above. The optimumparameters of the F1FPID controller Ke, Kd, ˛, which are all pos-itive constants are listed in Table 6. The optimum cost functionvalue obtained for these scaling factors is 0.848 which is around25% better than the one obtained for conventional PID controllers.
For the third and the last controller structure which is IT2F PIDcontroller the scaling factors kept as the same that are obtainedfor T1FPID controller. The aim of this approach is only to show theeffect of interval type-2 fuzzy sets in the fuzzy controller. On theother hand, it is quite possible for one to make one more searchusing the BB–BC algorithm to set the scaling factors of the IT2FPIDcontrollers. For the IT2FPID controller used in Areas 1, 2, 3 and theother IT2FPID controller designed for Area 4, there exist 8 parame-ters which is the extra degree of freedom. As a result of the BB–BCoptimization search, the parameters of the interval type-2 fuzzysets in the fuzzy rule base are obtained with a cost function valueof 0.411. It is 51.5% better than the one obtained for T1FPID con-troller. The optimized parameters of the interval type-2 fuzzy setsare given in Table 7 and the type-2 fuzzy sets illustrated in Fig. 10.
The simulation result obtained after the BB–BC optimizationprocess is illustrated in Fig. 11. As it can be easily seen from theresults, the conventional PID controller performs more smoothresponse when it is compared to the T1FPID controller but it has abig overshoot and large settling time. When the performance of theIT2FPID controller is compared with the other two controllers, it caneasily be concluded that the transient response is much smoother,the overshoots are less and the settling time is much smaller.
As previously mentioned the power load demand varies ran-
ad frequency controller using Big Bang–Big Crunch optimization,.031
domly, so to demonstrate the benefits of the proposed IT2FPIDcontroller, also other simulation results are given here. This time,there is no load disturbance in the first area are but the disturbances
Table 6Input and output scaling factors of the fuzzy PID controllers.
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID load frequency controller using Big Bang–Big Crunch optimization,Appl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.031
ARTICLE IN PRESSG ModelASOC 2113 1–13
E. Yesil / Applied Soft Computing xxx (2013) xxx–xxx 9
Fig. 10. The interval type-2 fuzzy sets obtained after BB–BC optimization.
Fig. 11. Frequency deviation of the four area power system: (a) �f1, (b) �f2, (c) �f3, (d) �f4 (solid: proposed IT2FPID, dotted: T1FPID, dashed: PID).
Table 9Obtained cost functions for each controller.
JITAE
(trainingphase)
JITAE
(testingphase)
524
525
526
527
528
529
530
531
532
533
534
535
#1, #2, #3 0.0941 0.2593 0
#4 0.3919 0.1846 0.3796
o second and third thermal power plants are �PD2 = 0.01 pu,PD3 = −0.03 pu, and the load disturbances to hydro power plant isPD4 = −0.001 pu. When the simulations are done using the same
ontroller parameters obtained previously by BB–BC optimization,he results given in Fig. 12 are obtained. As a result of the sim-lations, the values of cost functions for PID controller, T1FPIDontroller, and IT2FPID controller are obtained as 1.888, 1.868 and.589, respectively. Once more, it can be attained from the fre-uency deviations of each area that IT2FPID highly improves the
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10
erformance compared to the other controller structures especiallyith regard to overshoots. The obtained results are tabulated in
ables 8 and 9. In addition, the responses of the tie lines, whichrive to zero deviation, are illustrated in Fig. 13.
Fig. 12. Frequency deviation of the four area power system: (a) �f1, (b) �f2,
ad frequency controller using Big Bang–Big Crunch optimization,.031
Fig. 13. Tie-line power deviation of the four area power system: (a) �Ptie1, (b)
. Conclusions
In this study, for the first time in literature, type-2 fuzzy PIDontrollers are suggested for the load frequency control problem.dditionally, for the first time in literature, the BB–BC optimizationlgorithm is successfully used to optimize the scaling factors andembership functions of the fuzzy PID controllers. In the begin-
ing, the definitions of type-2 fuzzy sets and interval type-2 fuzzyets are introduced. Among the discussed fuzzy sets, interval type-2uzzy sets are preferred because of their simplicity and applica-ility. The present study is focused on a four-area interconnectedower system that includes three reheat thermal plants and oneydro power plant. For each of the three reheat thermal turbinenits the same controllers are designed and used; whereas, forhe hydro power plant another controller is designed. First, T1FPIDontrollers were chosen with 9 fuzzy rules and triangular member-hip functions for antecedent parts and singletons for the outputs.B–BC was used to optimize the scaling factors of the controllers.
Please cite this article in press as: E. Yesil, Interval type-2 fuzzy PID loAppl. Soft Comput. J. (2013), http://dx.doi.org/10.1016/j.asoc.2013.10.
Subsequently, T2FPID controllers were designed using the samecaling factors obtained for the T1FPID controllers. The input mem-ership functions changed into IT2FSs and the outputs membership
functions were kept as singletons. Eight design parameters weredefined for each controller for the reheat and hydro turbine plants.These total 8 parameters of interval type-2 sets were optimizedusing BB–BC. Due to the aim of the present study, is mainly to showthe effect of type-2 fuzzy controllers in the load frequency con-trol problem and not the effect of scaling factors, the scaling factoras tuning parameters was not selected to optimize 24 parametersfinding slightly better results. Finally, conventional PID controllerswere designed for the reheat thermal plants and hydro power plantusing the BB–BC optimization. For the optimization process, thecost function was chosen as the integral of time absolute error(ITAE) index since it penalizes long-duration transients. The resultsin terms of the cost function show that T1FPID controllers per-formed almost 25% better than conventional PID controllers; inaddition, IT2FPID controllers operated 51.5% better then T1FPIDcontrollers even with the same scaling factors optimized for T1FPID.Furthermore, more simulation results were presented for differentload disturbances than the ones used for the optimization stage.
ad frequency controller using Big Bang–Big Crunch optimization,031
The IT2FPID controller noticeably showed superior performancescompared to the other two controllers in terms of overshoot andsettling time with fewer oscillations.
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