A FUZZY CULTURAL IMMUNE SYSTEM FOR ECONOMIC LOAD DISPATCH WITH NON-SMOOTH COST FUNCTIONS

Richard Aderbal Gonçalves

UNICENTRO e UTFPR Rua Presidente Zacarias, 875–Guarapuava/Pr e Av. Sete de Setembro, 3165– Curitiba/PR

richardgoncalves@cpgei.cefetpr.br

Josiel Neumann Kuk UNICENTRO e UTFPR

Rua Presidente Zacarias, 875–Guarapuava/Pr e Av. Sete de Setembro, 3165– Curitiba/PR josielkuk@gmail.com

Carolina Paula de Almeida

UNICENTRO e UTFPR Rua Presidente Zacarias, 875–Guarapuava/Pr e Av. Sete de Setembro, 3165– Curitiba/PR

carolpa@cpgei.cefetpr.br

Nátalli Macedo Rodrigues UNICENTRO

Rua Presidente Zacarias, 875–Guarapuava/Pr natalli_macedo@yahoo.com.br

Myriam Regattieri Delgado

UTFPR Av. Sete de Setembro, 3165– Curitiba/PR

myriamdelg@utfpr.edu.br

Abstract This paper presents a novel and efficient method for solving economic load dispatch problems with non-smooth cost functions, by combining an Artificial Immune Systems with Cultural Algorithms and Fuzzy Inference Systems. The proposed method, called Fuzzy Cultural Immune System, uses a real coded AIS that is derived from the clonal selection principle with a pure aging operator and hypermutation operators based on Gaussian and Cauchy mutations that are guided by four knowledge sources stored in the belief space of a Cultural Algorithm with a fuzzy acceptance function. Fuzzy Cultural Immune System also has a local search stage that is based on a quasi-simplex technique and several points of self-adaptation. Three test systems with thermal units whose fuel cost function takes into account valve-point loading effects are used to validate the proposed method. These test systems constitute complex constrained optimization problems. Fuzzy Cultural Immune System is compared with state-of-the-art algorithms. KEYWORDS: Economic Load Dispatch, Cultural Algorithms, Artificial Immune Systems, Fuzzy Inference Systems. Main Areas: Eletrical Engineering and Methaheuristics.

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1. Introduction In the last few years, the use of hybrid methods inspired by Natural Computing (De Castro, 2006) has attracted the attention of many researchers, specially the systems in which two or more methodologies are joined to enhance the final model. Some of such hybrid approaches seems to be robust enough to tackle practical hard optimization problems, such as the economic load dispatch (ELD). The primary objective of the economic load dispatch problem is to determine the optimal quantity of energy that must be generated by each unit to meet the required load demand at minimum operating cost while satisfying system equality and inequality constraints (Park, Jeong and Lee, 2006). Traditionally, the cost of each unit is approximated by a quadratic function which can be exactly solved using mathematical programming based on the optimization techniques such as lambda-iteration method, gradient method, and dynamic programming. These methods require incremental fuel cost curves, which should be monotonically increasing, to find a global optimal solution. Unfortunately, the input-output characteristics of generating units are inherently nonlinear and highly non-smooth because of valve-point loadings. Thus, the practical ELD problem with valve-point effects is addressed as a hard optimization problem with equality and inequality constraints that make the problem of finding the global optimum very difficult. Thus, it normally renders classical optimization methods ineffective. In the absence of exact methods, the stochastic ones (specially the naturally inspired methods such as Artificial Immune Systems (Liu et al., 2006), Genetic Algorithms (Ling et al., 2003; Zhang et al., 2006), Particle Swarm (Sinha and Purkayastha, 2004; Park, Jeong and Lee, 2006; Hou et al., 2005), (Selvakumar and Thanushkodi, 2007) and Differential Evolution (Coelho and Mariani, 2006; Balamurugan and Subramanian, 2007) have grown in popularity for this practical engineering problem.

In this paper we combine an Artificial Immune Systems (AIS) with Cultural Algorithms (CA) and a Fuzzy Inference System (FIS) in order to improve the already good results obtained by AIS and CA for the Economic Load Dispatch (Gonçalves et al., 2007). This paper is organized as described below. In Section 2 the Economic Load Dispatch Problem is briefly described. Section 3, 4 and 5 introduces Cultural Algorithms, Artificial Immune Systems and Fuzzy Inference Systems, respectively. The proposed algorithms are described in Section 6 while Section 7 shows the experiments performed and the results achieved. Finally, Section 8 presents some conclusions. 2. Economic Load Dispatch Economic load dispatch is one of the most important problems to be solved in the operation and planning of a power system (Park, Jeong and Lee, 2006). The objective of the economic dispatch problem is to minimize the total fuel cost of thermal power plants subjected to the operating constraints of a power system. The objective function can be formulated as:

∑ ==

n

j jj PFF1

)(

subject to ∑ =

=n

j jPPD1

and (1) maxminjjj PPP ≤≤

where Fj (Pj) is the fuel cost function of the jth generator (in $/hr), Pj is the power output of the jth unit, n is the number of generating units in the system, PD is the total power demand, Pj

min and Pj

max are, respectively, the minimum and maximum power outputs of the jth unit. The equality constraint of Equation 1 is called power balance constraint while the inequality constraints are called operational constraints.

32

The fuel cost function, considering valve-point effects, of each generator is given by a mix of a quadratic approximation and a sinusoidal function (Park, Jeong and Lee, 2006):

))(*sin(***)( min2jjjjjjjjjjj PPfecPbPaPF −+++= (2)

where Pj is the power output of the jth unit, aj , bj , cj , ej and fj are the fuel cost coefficients of the jth unit with valve point effects. 3. Cultural Algorithms Cultural algorithms (CA) are techniques that add domain knowledge to evolutionary computation methods and are derived from the cultural evolutionary process (Reynolds and Peng, 2005). They assume that the knowledge related to the problem being solved can be extracted from individuals of the population during the evolutionary process and subsequently used to guide the search. Cultural algorithms have two main components: the population space, and the belief space (Saleem, 2001). The population space consists of a set of possible solutions to the problem, and can be modeled using any population based technique, such as Evolutionary Programming (Saleem, 2001), Particle Swarm (Iacoban, Reynolds and Brewster, 2003) and Differential Evolution (Becerra and Coello, 2005; Becerra and Coello, 2006). The belief space is the information repository in which individuals can store their experiences that can be indirectly learned by other individuals (Becerra and Coello, 2006). In cultural algorithms, the information acquired by an individual can be shared with the entire population.

The population space and the belief space are linked through a communication protocol, which states the rules about the individuals that can contribute to the belief space with its experiences (the acceptance function), and the way the belief space can influence the generation of new individuals (the influence function) (Becerra and Coelho, 2005).

The knowledge stored into the belief space can, generally, be divided into five knowledge sources (Reynolds and Peng, 2006; Becerra, 2006): Situational Knowledge, Normative Knowledge, Historical Knowledge, Topographical Knowledge and Domain Knowledge. It is important to note that these knowledge sources were derived from works in cognitive science and semiotics that describe the basic knowledge used by human decision-makers (Reynolds and Peng, 2005). 4. Artificial Immune Systems Artificial Immune Systems can be defined as computational paradigms inspired by theoretical immunology and observed immune functions, principles and models, which are applied to problem solving (De Castro and Timmis, 2002). The natural immune system is highly distributed, highly adaptive, self-organizing in nature, besides it maintains a memory of past encounters and has the ability to continually learn about new encounters (Timmis et al., 2004). Artificial Immune Systems inherit these features, so they can be considered useful, flexible, and powerful computational tools. The most used biological metaphors are the clonal selection principle, the negative selection, the positive selection, and the immune networks (De Castro and Timmis, 2002). As the Artificial Immune System used in this paper is based on the clonal selection principle (particularly in the Clonalg algorithm), it will be detailed below. 4.1. Clonal Selection The basic features of the adaptive immune response to an antigenic stimulus can be explained by clonal selection principle. It establishes the idea that only those cells that recognize the antigens are selected to proliferate. The selected cells are subject to an affinity maturation process, which

33

improves their affinity to some specific antigens (De Castro and Timmis, 2002). The most important algorithm based on this principle is Clonalg (De Castro and Von Zuben, 2002). The Clonalg algorithm works with a population of candidate solutions (B cells or antibody), composed of a subset of memory cells (generally the best ones) and a subset of other good individuals. At each generation the n best individuals of the population are selected, based on their affinity measures. The selected individuals are cloned, giving rise to a temporary population of clones. The clones are submitted to a hypermutation operator, whose rate is proportional (or inversely proportional) to the affinity between the antibody (the solution) and the antigen (the problem to be solved). From this process a maturated antibody population is generated. Some of the individuals of this temporary population are selected to be memory cells or be part of the next population. This whole process is repeated until a termination condition is achieved (De Castro and Von Zuben, 2002). 5. Fuzzy Inference Systems The fuzzy sets theory was introduced by Zadeh (1965). Fuzzy sets appeared as powerful math tools to deal with imprecise or incomplete information. The basic structure of a fuzzy inference system has three conceptual components (Jang, Sun and Mizutani, 1997):

• Rules Base: contains the description of the fuzzy rules of the system; • Data Base: defines the membership functions used in the fuzzy rules; • Reasoning Mechanism: realizes the inference process to obtain the output or conclusion

based on the known rules and the facts representing the inputs. Fuzzy systems design depends on the specification of various elements (including the

number and type of the fuzzy rules, the membership functions parameters, semantic of the rules that participate in the approximated reasoning and the operators of the inference mechanism) to obtain the output from the input data (Pedricz and Gomide, 1998). There are two different ways to determine these parameters: based on the expert knowledge or by means of automatic procedures such as the genetic fuzzy systems (Delgado, Von Zuben and Gomide, 2004). Here, the first one will be considered. FIS have the ability to cope with imprecise information mainly because they do not consider crisp border objects. Instead they handle objects with diffuse frontiers, resulting in smoother surfaces regarding input-output relationships. This can enhance the decision process, since next points might generate soft transitions rather than abrupt changes in decision making. In this paper, a Fuzzy Inference System is designed by an expert to implement the acceptance function of a Cultural Algorithm (see Section 6.1.1. for details). 6. Proposed Artificial Immune System In the Immune System proposed in this paper to solve the Economic Load Dispatch each antibody is a valid combination of power outputs (encoded as real numbers) for the generator units. The affinity of an antibody with the antigen is given by:

)/()( gengenigeni MinCostMaxCostCostMaxCostaff −−= (3) where Costi is the total fuel cost of a particular configuration (represented by the ith antibody), MinCostgen and MaxCostgen are the maximum and minimum fuel cost of an antibody in generation gen, respectively.

The steps of the implemented Immune System are the same as those presented in Pseudo-Code 1 with the following modifications: steps 3, 4.8, 4.9 and 4.10 are eliminated of the algorithm and step 4.2 is substituted by “Apply a Hypermutation to the Clones”.

34

Pseudo-Code 1 – Description of the proposed algorithms.

is satisfied. The term

proportional to its affinity) (Cutello et al., 2007

aptive Cauchy Hypermutation (ACH). These hypermutation operators are given by Equation 4.

−+= (4)

GH or ACH). In both cases the distributions have mean equal to zero and variance

Each antibody is composed of n components (power generators) that are initialized at the

first generation as anti,j = P_imin + U(0,1) * (Pi

max - P_imin), where anti,j is jth component of the ith

antibody (i.e., the power output of the jth unit generator), U(0,1) is a random variable sampled from an uniform distribution in the interval [0,1], P_i

min and P_imax are, respectively, the minimum

and maximum power outputs of the jth unit. This initialization procedure guarantees that no antibody violates the operational constraints of the generator units. After the evaluation of the initial generation, the Immune System enters its main loop (which represents the affinity maturation stage of the algorithm). This loop is repeated until a termination criterion

ination criterion adopted in this work is a maximum number of generations. The affinity maturation process begins by cloning the antibodies of the past generation.

The most common cloning operators are: static cloning operator (where the number of clones of each antibody is dup, independently of the affinity of the antibody) and the proportional cloning operator (where the number of clones of each antibody is

). In this work the static cloning operator is used. The hypermutation operators are subsequently applied. The implemented algorithm

makes use of two hypermutation operators: an Adaptive Gaussian Hypermutation (AGH) and an Ad

)(*)1,0(*)/(* minmax,, jjgenijiji PPRMinCostCostmultantant

where anti,j is the jth component of the ith antibody, mult is an adaptive multiplier given by mult = e-gen/φ (where φ is a parameter that dictates the decrease speed of the multiplier), Pj

max and Pjmin

are the maximum and minimum limits of the jth generator unit, respectively, R(0,1) is a random number sampled according to a Gaussian or a Cauchy distribution depending on the hypermutation type (A

equal to one.

35

The term Costi/MinCostgen makes the mutation more intensive in antibodies with a high fuel cost (low affinity) and smooth in antibodies with low fuel cost (high affinity). Cauchy random numbers allow relatively coarse-grained steps, while Gaussian random numbers produce fine-grained steps, which in theory is a good balance. The multiplier tries to make the search in the begi ion. nning of the evolution intense and smooth at the end of the evolut

The number of mutations applied to each antibody is given by neiMi

*)(ρ

= , where aff

M(i) is t

o instead of penalizing infeasib

population. A clone inherits the age of its parent and is assigned an age equ

ompose the next population. The scheme used is a (μ+λ)-Selection operator (Cutello et l., 2007) that is applied to parents and hypermutated clones that survived after the aging

ence function). Two version of the Cultural Algorithm are plemented: one with a dynamic acceptance function and the other with a Fuzzy Inference

the first version (called Cultural Immune System – CIS) a dynamic acceptance function is used. This acceptance function is given by:

he number of mutations applied to the ith antibody, affi is the affinity of the ith antibody, ρ is a parameter that regulates the number of mutations and n is the number of generators.

After hypermutation a quasi-Simplex method is applied as a local search procedure to the best antibodies of the hypermutated clones. At this point the constraints can be violated, which could cause infeasible antibodies. To avoid such violation a repair process is applied to each clone in order to guarantee that the generated antibodies are feasible. S

le antibodies we repair them. After the application of the repair procedure the affinity given by Equation 3 can be used to evaluate the goodness of an antibody.

The next step in the implemented algorithm is the application of the aging operator. In this work the static pure aging operator (Cutello et al., 2007) is used. This aging operator eliminates old antibodies in order to maintain the diversity of the population and to avoid the premature convergence. In this operator an antibody is allowed to survive for at most tauB generations, after this period it is assumed that this antibody corresponds to local optima and must be eliminated from the

al to zero when it is successfully hypermutated (i.e., when the hypermutation improves the affinity of the antibody).

Finally, the last step of the affinity maturation process is the selection of the antibodies that will caoperator. 6.1. The Proposed Fuzzy Cultural Immune System Our second proposed approach uses the previously described Immune System-based algorithm as the population space of a Cultural Algorithm.Pseudo-Code 1 summarizes the general steps of the proposed algorithm. The Cultural Immune System can be considered as an extension of the previous algorithm where the Belief Space and the communication protocols are added to improve the performance of the original Immune System. The belief space is used to extract information from the antibodies' population and uses this knowledge as a guide to generate new antibodies during the hypermutation operators through the influence functions, i.e. the hypermutation operators are replaced by the influence functions. The communication protocols dictates which antibodies will be considered during the update of the belief space (acceptance function) and the probability of a knowledge stored in the belief space to influence a hypermutation operator (main influimSystem as the acceptance function. 6.1.1. Communication Protocols: Acceptance and Influence Functions In

)(*genperc

acceptacceptpopSize perc+= (5) nAccepted

36

where nAccepted are the number of antibodies of the population that will be used to update the knowledge sources, popSize is the number of antibodies in the population, acceptperc is a parameter that determines the percentage of accepted antibodies and gen is the current generation. This acceptance function allows more antibodies to contribute to the update during

ulation (low, medium

sources have the same probability to be applied (0.25). During the evolutionary process, the probability of the ith knowledge source influence a hypermutation operator is given by Equation 6.

the beginning of the evolution (when there is little accumulated knowledge) and less antibodies at the end of the evolution (when most of the knowledge have already been acquired). The second version (called Fuzzy Cultural Immune System – FCIS) uses a Fuzzy Inference System to determine the percentage of the population that will be accepted. The selection is based on the evolutionary stage of the algorithm (beginning, middle and ending), diversity of the population (low, medium and high) and average affinity of the pop

and high). The FIS used was optimized by a specialist to favor high diversity and high affinity and to accept more individuals at the beginning of the evolutionary process.

The main influence function is responsible for choosing the knowledge source that will influence the hypermutation operators. At the beginning, all the knowledge

⎟⎟⎠

⎜⎝ popSizek

where probKS

⎞⎜⎛

+=ant

kgbKS*6.01.0 (6)

ication of nowledge sources that are capable of maintaining their generated antibodies in the population

wledge source has at least a 10% chance of being applied.

stituted by the term bestk,j, where bestk,j is the jth

l, and the extreme values are associat to note that the limits of the normative knowledge can not violate the operational constraints.

The influence of the Normative Knowledge is as follows:

probKS

k is the probability of selecting the kth knowledge source to influence a hypermutation operator, antgbKSk is number of antibodies that were generated by an influence of the kth knowledge and popSize is the size of the population. Equation 6 favors the applkand guarantees that each kno 6.1.2. Knowledge Sources

The following knowledge sources are utilized in this work: Situational Knowledge: The Situational Knowledge stores the pbest antibodies found

during the evolutionary process (Becerra and Coello, 2006). These antibodies are used as leaders to influence the hypermutation operators. This influence is similar to the hypermutation operation (Equation 4) but the term anti,j on the right side is sub

component of the kth best antibody stored in the Situational Knowledge, and k is an index randomly selected among the best antibodies.

Normative Knowledge: The Normative Knowledge contains the intervals for the power outputs of the generator units where good solutions have been found, and is used to move the outputs of the new solutions towards those intervals. The intervals of the Normative Knowledge are initialized with the lower and upper bounds of the output of the generator units. The update of the normative knowledge can reduce or expand the intervals stored on it. An expansion takes place when the accepted individuals do not fit in the current interval, while a reduction occurs when all the accepted individuals lie inside the current interva

ed with individuals with better fuel cost. It is important

37

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−+

>

−−

<

−+

=

otherwise

NLoNUpRMinCostCostmultant

NUpifant

NLoNUpRAbsMinCostCostmultant

NLoifant

NLoNUpRAbsMinCostCostmultant

ant

jjgeniji

jji

jjgeni

jji

jjgeniji

jiji

),(*)1,0(*/*

),*))1,0((*/*

),(*))1,0((*/*

,

,

,

,

,, (7)

where NUpj and NLoj are, respectively, the upper and lower bounds of the normative interval associated with the jth component, Abs(x) is a function that returns the absolute value of x and the other terms are as defined in Equation 4. This influence function is adaptive: it is intensive when the normative interval is large (the good interval is very uncertain) and it is smooth when the normative interval is small (the good interval is known).

Historical Knowledge: This knowledge source was introduced in Cultural Algorithms as a mean to adapt to environmental changes (Reynolds and Peng, 2006). It stores a list of the best antibodies found before the last HistorySize environmental changes. It also stores the average direction and the average distance (size) of the changes for each component between the environmental changes. In this work, since there is no environmental change, this knowledge is adapted and it happens when the algorithm is trapped at a local optimum (there is no change in the best antibody found during the last p generations). The influence function of the Historical Knowledge used in this work is:

⎪⎪

⎩

⎪⎪

⎨

⎧

<

−

≥

+

=

0

),(*))1,0((*/*

0),(*))1,0((*/*

,

,

,

j

jgeni

j

jgeniji

ji

ifAvDir

AvDistRAbsMinCostCostmultant

ifAvDirAvDistRAbsMinCostCostmultant

antji

(8)

where AvDistj is the average distance change in the jth component, AvDirj is the average direction of the change in the jth component (both are given by Equation 9) and the other terms are as previously defined.

))((

)1/())((1

1 ,,1

1

1 ,,1

∑∑

−

= +

−

= +

−=

−−=eHistorySiz

i jijij

eHistorySiz

i jijij

tHistoryBestHistoryBesSignAvDir

eHistorySiztHistoryBestHistoryBesAbsAvDist (9)

where HistorySize is the number of antibodies stored in historical knowledge, HistoryBesti,j is the jth component of the ith best antibody and Sign(x) is a function that returns +1 if x is positive, -1 if x is negative and 0 otherwise.

This influence tries to increment the jth component of the antibody submitted to hypermutation if in average the jth component of new best antibody was greater than or equal to the jth component of the previous best antibody and tries to decrement this component otherwise. In either case the hypermutation is proportional to the average distance observed between changes (so if the new best antibodies are found far away from the previous ones the hypermutation will be intense and it will be smooth if they are found near the previous ones).

Topographical Knowledge: The Topographical Knowledge is used to create a map of the fitness landscape of the problem during the evolutionary process. It consists of a set of regions and the best individual found on each region. It also stores a sorted list of the r best regions (which is sorted according to its best antibody). A region is represented as a node in a

38

binary tree and stores the upper and lower bounds for each component and the best antibody found so far in that region. The binary tree is initialized with a root node that represents the entire feasible space and has the best antibody of the first generation. If during the update of the Topographical Knowledge an antibody with a better fuel cost than the best antibody represented in a region is accepted and this antibody belongs to this region, the node is split (the region is divided in two). The component where the division is made is the component where there is the highest difference between the previous best antibody of this region and the new one. The division is made in half the distance between the value of the split's component of the new and the old best antibodies of the region.

⎪⎪

⎩

⎪⎪

⎨

⎧

−+

≥

−+

=

otherwiseOldNewOld

NewifOldNewOldNew

SplitPojjj

jj

jjj

,2/)(

,2/)(

int (10)

where SplitPoint is the value of the jth component where the region will be divided, Oldj is the jth component of the previous best antibody of the cell being divided and Newj is the jth component of the new best antibody found in this cell. It is important to note that only leaf nodes can be stored in the ordered list or split. The influence function of the Topographical Knowledge is described by Equation 11.

)(*)1,0(*/* ,,,,lower

jkupper

jkgenijiji RRRMinCostCostmultantant −+= (11) where Rk

upper and Rklower are the upper and lower bounds for the jth component in the kth region

where is randomly selected according to the affinity of the best antibody of each region if a uniform random number in the interval [0,1] is less than pElite (probability of the best regions be chosen more often) and randomly selected independently of the affinity otherwise. The other terms of the equation are as previously defined. This influence function tends to explore good regions of the search space. 7. Experiments and Results Three test cases were used to validate the proposed Immune Systems: 13 generators with a load demand of 1800 MW (Coelho and Mariani, 2006), 13 generators with a load demand of 2520 (Khamswang and Boonseng, 2004) MW and 40 generators with a load demand of 10500 MW (Coelho and Mariani, 2006).

The algorithms were executed 50 independent times with the following set of parameters (when applicable): popSize = 50, termination criterion = 3000 generations, dup = 4, number of elite regions = 40, tauB = 100, pbest = 10, HistorySize = 10, p = 200, acceptB perc = 0.2. Despite the great number of parameters, the proposed Immune Systems are little influenced by them. This fact was observed during the experiments realized to find this particular good set of parameters (the adjustment tests were done using only the first test case).

In order to validate our methodologies, we compared the proposed immune-based systems with other state-of-the-art approaches. Although it would be better to compare the algorithm with the same set of algorithms for all test cases, it was impossible due to the fact that the other works were applied to only one or two of the test cases. Table 1, 2 and 3 summarize the results obtained for the three test cases.

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Table 1 - Result for 13 generators and load demand of 1800 MW. Case with 13 Generators and Load Demand of 1800 MW

Method Minimum Cost

Mean Cost

Maximum Cost σ Time(min)

IS (Gonçalves et al., 2007) 17960.91 17983.03 18074.41 24.96 0.56 CIS (Gonçalves et al., 2007) 17960.73 17970.35 17977.90 1.87 0.66

FCIS 17964.27 17974.19 17981.91 3.03 0.5 DEC-SQP (Coelho and Mariani,

2006) 17963.94 17973.13 17984.81 1.97 0.34

IGA (Ling et al., 2003) 18063.58 18096.40 18293.47 45.79 - MPSO (Hou et al., 2005) 17973.34 - - - -

Table 2 - Result for 13 generators and load demand of 2520 MW.

Case with 13 Generators and Load Demand of 2520 MW

Method Minimum Cost

Mean Cost

Maximum Cost σ Time(min)

IS (Gonçalves et al., 2007) 24164.80 24223.23 24427.29 52.74 0.48 CIS (Gonçalves et al., 2007) 24164.79 24181.67 24255.84 30.16 0.60

FCIS 24170.45 24185.68 24281.89 28.67 0.47 SDE (Balamurugan and Subramanian,

2007) 24164.05 24168.28 24200.05 - -

DTSA (Khamsawang, 2004) 24169.05 - - - -

Table 3 - Result for 40 generators and load demand of 10500 MW. Case with 40 Generators and Load Demand of 10500 MW

Method Minimum Cost

Mean Cost

Maximum Cost σ Time(min)

IS (Gonçalves et al., 2007) 121529.26 121880.13 122283.55 237.09 1.44 CIS (Gonçalves et al., 2007) 121500.43 121735.26 122142.74 149.02 1.5

FCIS 121483.02 121621.39 122157.12 158.36 1.15 DEC-SQP (Coelho and Mariani,

2006) 121741.97 122295.12 122839.29 386.18 10.80

NPSO-LRS (Selvakumar and Thanushkodi, 2007) 121664.43 122209.31 122981.59 - -

CEP-PSO (Sinha and Purkayastha, 2004) 123670.00 124145.60 124900.00 - -

For the test case of 13 generators with a load demand of 1800 MW Table 1 shows that

the best results were obtained with the Cultural Immune System (CIS). The Immune System algorithm (IS) found the second best result concerning the minimum cost, outperforming the non-immune algorithms. The Fuzzy Cultural Immune System (FCIS) obtained results better than IGA (Ling et al., 2003) and MPSO (Hou et al., 2005), losing for the other Immune Systems for DEC-SQP (Coelho and Mariani, 2006). Table 2 presents the comparison of the results obtained by our Immune Systems and other methods reported in the literature for the test case with 13 generators and a load demand of 2520 MW. For this test case the SDE algorithm achieved better results than our Immune Systems. Despite of this, the minimum fuel cost obtained by CIS and IS are less than one dollar more than the minimum fuel cost of the SDE and when compared with DTSA it is more than five dollars lower. FCIS obtained the worst results among all algorithms. The comparison for the test case with 40 generators and load demand of 10500 MW shows that the proposed FCIS was able to attain better values of minimum and mean cost. For this test case IS and CIS were able to outperform all non-immune approaches reported in the literature in all statistics, losing only for FCIS. The computational time of the immune algorithms are very

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similar. For the first test case DEC-SQP is faster than the proposed algorithms, but for the third test case it’s slower (although it’s important to note that DEC-SQP was executed in a different machine and was written in another programming language). It was not possible to compare the computational time of other algorithms because it isn’t reported in the literature. Based on a ranksum test with 95% degree of confiability it is possible to conclude that the quality of the results obtained by CIS are better than those obtained by IS, attesting the superiority of the cultural method. The p-values obtained were 0.013, 7*10-8 and 0.0041 for the first, second and third test case, respectively. When comparing both Cultural Algorithms using the ranksum test it is possible to infer that CIS is better than FCIS for the first and second test cases (p-values of 0.0066 and 0.0094, respectively) while FCIS is better than CIS for the last test case (p-value of 2.85*10-7). Due to the lack of data given by the other compared methods, we were not able to realize statistical tests comparing these methods and the proposed Cultural Immune System. 8. Conclusions and Future Researches In this paper we implemented three Immune Systems based on the Clonal Selection Principle. All algorithms applied static cloning operator, Gaussian and Cauchy Hypermutations, static pure aging, (μ+λ)-selection operator and inversely proportional mutation potential. This set of operators proved to be very effective in finding good solutions to the economic load dispatch problem with valve-point effect, which is an important and complex practical engineering problem. Although the adjustment of the parameters was done on the first test case, the same set of parameters was able to tackle the other problem instances successfully, what corroborates to the robustness of the method. Our proposed novel approach, the Fuzzy Cultural Immune System (CIS), was capable of improving the performance of state-of-the-art algorithm for the test case with 40 generators and a load demand of 10500 MW; improving the best reported minimum and mean cost. Although FCIS obtained good results for this test case, it was unable to obtain good results for the test cases with 13 generators. Further investigation will be performed to assert under which circumstances FCIS is better than CIS, and vice-versa. Based on the results obtained it is possible to argument that Immune Systems are good algorithm to solve the Economic Load Dispatch, especially when hybridized with Cultural Algorithms. In future works we intend to analyze the behavior of the proposed algorithms in other engineering problems. We also intend to substitute the adaptive multiplier used on the hypermutation operators by a chaotic multiplier and to incorporate new knowledge sources. Acknowledgements: Carolina would like to thank the financial support provided by CNPq-Brazil. Josiel would like to thank Capes-Brazil for the financial support.

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