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Ants have always fascinated human beings. What particularly strikes the occasional observer as well as the scientist is the high degree of societal organization that these insects can achieve in spite of very limited individual capabilities (Dorigo et al., 2000). Ants have inspired also a number of optimization algorithms. These algorithms are increasingly successful among researches in computer science and operational research (Blum, 2005; Cordón et al., 2002; Dorigo & Stützle, 2004).

A particular successful metaheuristic—Ant Colony Optimization (ACO)—as a common framework for the existing applications and algorithmic variants of a variety of ant algorithms has been proposed in the early nineties by Marco Dorigo (Dorigo, 1992). ACO takes inspiration from the foraging behavior of some ant species. These ants deposit pheromone on the ground in order to mark some favorable path that should be followed by other members of the colony. ACO exploits a similar mechanism for solving combinatorial optimization problems.

In recent years ACO algorithms have been applied to more challenging and complex problem domains. One such domain is continuous optimization. However, a direct application of the ACO for solving continuous optimization problem is difficult.

The first algorithm designed for continuous function optimization was continuous ant colony optimization (Bilchev & Parmee, 1995) which comprises two levels: global and local; it uses the ant colony framework to perform local searches, whereas global search is handled by a genetic algorithm. Up to now, there are few other adaptations of ACO algorithm to continuous optimization problems: continuous interacting ant colony (Dréo & Siarry, 2002), ACO for continuous and mixed-variable (Socha, 2004), aggregation pheromone system (Tsutsui, 2006), and multilevel ant-stigmergy algorithm (Korošec & Šilc, 2008).

In this chapter we will present so-called Differential Ant-Stigmergy Algorithm (DASA), a new approach to the continuous optimization problem (Korošec, 2006). We start with the DASA description followed by three case studies which show real-world application of the proposed optimization approach. Finally, we conclude with discussion of the obtained results.

2. A differential ant-stigmergy approach

2.1. Continuous optimization

The general continuous optimization problem is to find a set of parameter values,p=(p1p2…pD)that minimizes a function,fcost(p)of D real variables, i.e.,

Find:

p*|fcost(p*)≤fcost(p)∀p∈ℜDE1

To solve this problem, we created a fine-grained discrete form of continuous domain. With it we were able to represent this problem as a graph. This enabled us to use ant-based approach for solving numerical optimization problems.

2.2. The fine-grained discrete form of continuous domain

Letp'ibe the current value of the i-th parameter. During the searching for the optimal parameter value, the new value,piis assigned to the i-th parameter as follows:

pi=p'i+δiE2

Here,δiis the so-called parameter difference and is chosen from the set

Δi=Δi−∪0∪Δi+E3

where

Δi−={δik−|δik−=−bk+Li−1k=1,2,…di}E4

and

Δi+={δik+|δik+=+bk+Li−1k=1,2,…di}E5

Here

di=Ui−Li+1E6

Therefore, for each parameterpithe parameter difference,δihas a range frombLitobUiwhere b is the so-called discrete base,

Li=⌊lgb(εi)E7

and

Ui=⌊lgb(max(pi)−min(pi))E8

With the parameterεithe maximum precision of the parameterpiis set. The precision is limited by the computer's floating-point arithmetic. To enable a more flexible movement over the search space, the weightωis added to Eq. 2:

pi=p'i+ωδiE9

whereω=RandomInteger(1,b−1)

2.3. Graph representation

From all the setsΔi1≤i≤Dwhere D represents the number of parameters, a so-called search graphG=(VE)with a set of vertices, V, and a set of edges, E, between the vertices is constructed (see Fig. 2). Each setΔiis represented by the set of vertices,

Each vertex of the setViis connected to all the vertices that belong to the setVi+1Therefore, this is a directed graph, where each pathνfrom start vertex to any of the ending vertices is of equal length and can be defined withvias:

ν=(v1v2…vD)E13

wherevi∈Vi1≤i≤D

The optimization task is to find a pathνsuch thatfcost(p)≤fcost(p')wherep'is currently the best solution, andp=p'+Δ(ν)(using Eq. 9). Additionally, if the objective functionfcost(p)is smaller thanfcost(p')then thep'values are replaced withpvalues.

2.4. The differential ant stigmergy algorithm

The optimization consists of an iterative improvement of the currently best solution,p'by constructing an appropriate pathνthat uses Eq. 9 and returns a new best solution. This is done as follows (see Fig. 3):

Step 1. A solutionp'is manually set or randomly chosen.

Step 2. A search graph is created and an initial amount of pheromone,τVi0is deposited on all the vertices from the setVi⊂V1≤i≤Daccording to the Cauchy probability density function

C(x)=1sπ+πs(x−l)2E14

wherelis the location offset ands=sglobal−slocalis the scale factor. For an initial pheromone distribution the standard Cauchy distribution (l=0,sglobal=1,andslocal=0) is used and each parameter vertices are equidistantly arranged betweenz=[−4,4]

Step 3. There aremants in a colony, all of which begin simultaneously from the start vertex. Ants use a probability rule to determine which vertex will be chosen next. More specifically, antαin stepimoves from a vertex in setVito vertexvij∈{vi,1vi,2…vi,2di+1 }with a probability, prob, given by:

probj(αi)=τ(vij)∑1≤k≤2di+1τ(vik)E15

whereτ(vik)is the amount of pheromone on vertexvik

The ants repeat this action until they reach the ending vertex. For each ant, pathνis constructed. If for some predetermined number of tries we getν=0the search process is reset by randomly choosing newptemporary_bestand pheromone re-initialization. New solutionpis constructed (see Eq. 9) and evaluated with a calculation offcost(p)

The current best solution,pcurrent_bestout ofmsolutions is compared to the temporary best solutionptemporary_bestIfpcurrent_bestis better thanptemporary_bestthenptemporary_bestvalues are replaced withpcurrent_bestvalues. In this casesglobalis increased (in our case for 1 %) and pheromone amount is redistributed according to the associated pathνbestFurthermore, if newptemporary_bestis better thenpbestthenpbestvalues are replaced withptemporary_bestvalues. So, global best solution is stored. If no better solution is foundslocalis decreased (in our case for 3 %).

Step 4. Pheromone dispersion is defined by some predetermined percentageχThe probability density functionC(x)is changed in the following way:

l←(1−χ)lE16

slocal←(1−χ)slocalE17

Pheromone dispersion has a similar effect as pheromone evaporation in classical ACO algorithm.

Step 5. The whole procedure is then repeated until some ending condition is met.

It is a well known that ant-based algorithms have problems with convergence. This happens when on each step of the walk there is a large number of possible vertices from which ant can choose from. But this is not the case with the DASA where Cauchy distribution of pheromone over each parameter was used. Namely, such distribution reduces the width of the graph to only few dominant parameter values (i.e., vertices). On the other hand, with proper selection of the discrete base, b, we can also improve the algorithm's convergence (larger b reduces the search graph size).

2.5. Software implementation and parameter settings

The DASA was implemented in the Borland® Delphi™ programming language (see Fig. 4). The computer platform used to perform the optimizations was based on AMD Opteron™ 2.6-GHz processors, 2 GB of RAM, and the Microsoft® Windows® XP 32-bit operating system.

The DASA parameters were set as follows: the number of ants,mwas set to 10, the pheromone evaporation factor,χwas set to 0.2, and the maximum parameter precision,εdependent on the discrete step of each parameter.

3. Case studies

Case studies presented in this section are related to the development of a new dry vacuum cleaner (Korošec et al., 2007; Tušar et al., 2007). In particular, the efficiency of the turbo-compressor unit (see Fig. 5) was improved.

3.1. An electric motor power losses minimization

Home appliances, such as vacuum cleaners and mixers, are generally powered by a universal electric motor (UM). These appliances need as low as energy consumption, that is, input power, as possible, while still satisfying the needs of the user by providing sufficient output power. The optimization task is to find the geometrical parameter values that will generate the rotor and the stator geometries resulting in the minimum power losses.

There are several invariable and variable parameters that define the rotor and stator geometries. The invariable parameters are fixed; they cannot be altered, either for technical reasons (e.g., the air gap) or because of the physical constraints on the motor (e.g., the radius of the rotor's shaft). The variable parameters do not have predefined optimum values. Some variable parameters are mutually independent and without any constraints. Others are dependent, either on some invariable parameters or on mutually independent ones.

In our case, 11 mutually independent variable parameters defining the rotor and stator geometries are subject to optimization (see Fig. 6): rotor yoke thickness,p1rotor external radius,p2rotor pole width,p3stator yoke horizontal thickness,p4stator yoke vertical thickness,p5stator middle-part length,p6stator internal edge radius,p7stator teeth radius,p8stator slot radius,p9stator width,p10and stator height,p11We optimized only 10 parameters (the ratio betweenp10andp11was set constant) of the UM rotor and stator geometries.

Power losses,Plossesare the main factor affecting the efficiency of a UM. The efficiency of a UM,ηis defined as the ratio of the output power,Poutto the input power,Pinp:

η=PoutPinp=PoutPout+PlossesE18

and it is very dependent on various power losses (Sen, 1997):

Plosses=PCu+PFe+Pbrush+Pvent+PfrictE19

The overall copper losses,PCuoccurring in the rotor and the stator slots are as follows:

PCu=∑i(J2Aρlturn)iE20

whereistands for each slot,Jis the current density,Ais the slot area,ρis the copper's specific resistance andlturnis the length of the winding turn. The calculation of the iron losses,PFeis less exact because iron has non-linear magnetic characteristics. The iron losses are therefore separated into two components: the hysteresis losses and the eddy-current losses. This means the motor's iron losses can be expressed with the formula

PFe=ceB2fr2mr+ceB2fs2ms+chB2fsmsE21

whereceis the eddy-current material constant at 50Hz,chis the hysteresis material constant at 50Hz,Bis the maximum magnetic flux density,fris the frequency of the magnetic field density in the rotor,fsis the frequency of the magnetic field density in the stator,mris the mass of the rotor, andmsis the mass of the stator. Three additional types of losses occurring in the UM, i.e., the brush losses,Pbrush;the ventilation losses,Pvent;and the friction losses,Pfric;depend mainly on the speed of the motor. When optimizing the geometries of the rotor and the stator, the motor's speed is assumed to be fixed; hencePbrushPventandPfrichave no impact on the motor's efficiency, and so these losses are not significantly affected by the geometries of the rotor and the stator. Therefore, in our case, the overall copper and iron losses can be used to estimate the cost function:

fc(p)=PCu+PFeE22

Our goal is to find such parameter-value settings, wherefc(p)is minimized.

The DASA starts its search with the existent solution. The stopping criterion was set to 1,400 calculations. The calculation of a single solution via the ANSYS Multiphysics simulation takes approximately two minutes, which means that the execution of 1,400 calculations needs about two days. The optimization method was run 20 times. The obtained results in terms of the UM power losses are presented statistically in Table 1.

Existing solution

Optimized solutions

Worst

Mean

Best

177.9

136.7

129.5

113.8

Table 1.

Optimized UM's power losses in Watts after 1,400 calculations.

The engineering rotor and stator design (the existent solution) results in power losses of 177.9 W, and can be seen in Fig. 7 (left). The figure shows the magnetic flux density in the laminations (higher magnetic flux density,Bcauses higher power losses). Figure 7 (right) present a typical example of a feasible rotor and stator geometry, with power losses of 129.1 W. This solution has very low iron losses in the rotor due to its small size and in spite of its high magnetic saturation. The small rotor and its saturation are compensated by large stator poles that ensure large enough a magnetic flux. This design is completely feasible from the technical and production points of view.

3.2. An electric motor casing stiffness maximization

The casing is a part of the dry vacuum cleaner motor which is built into vacuum cleaners. The casing is basically an axisymmetric shell structure which is built of steel suitable for forming. For this procedure which consists of eleven different phases it is important that the radii are growing or do not change while the height is growing. That is an important rule which must not be broken during the geometry optimization. The goal of optimization is to preserve the stiffness while using a thinner shell structure and consequently save material and reduce costs.

The computational model of the casing comprises 26 different parameters which generate the geometry variations. The classical parameters are: radius on the top,p1radius on the side,p2radius at the groove for brushes,p3deviation of the hole for diffuser fixation,p4radius of roundness at the beginning of the bearing groove,p5radius on the top of the bearing groove,p6radius on the top of air culverts,p7radius at the bottom of air culverts,p8roundness of air culverts,p9angle of slope,p22angle of culverts span,p23slope of bearing groove,p24height of bearing groove,p25and slope of the brushes groove,p26

Besides the above listed 14 classical parameters there are 12 more (p_{10} to p_{21}), with which the ribs on areas A, B and C are defined (Fig. 8). These are essential to improve the stiffness so we will search for those which maximize the cost function. With these parameters we can generate a lot of combinations of different rib shapes, depending on the number of given paths and accuracy of intervals for points deviations, positions and number of ribs. In our case there were several billion combinations for ribs. It is obvious that instead of a defined path at position j, we could define displacements for individual points but this would result in many more parameters and time consuming calculation procedure. The question still remains what would be the benefit of that. From previous designs there exist some reasonable shapes for ribs which have been defined intuitively and these then change their position, magnitude and number, which are probably sufficient for a good result, as there has been over ten different rib shapes defined earlier.

Besides the parameters for the creation of ribs there are other classical parameters which also have an influence on their shape. Namely, if the radiip1orp2are varying then this is reflected in the shape of ribs of model A or if the height of bearing groove,p25and slope of groove,p24are modified then this will be reflected in the ribs of model C, etc.

The cost function is relatively simply defined when loadings are deterministic. Usually we optimize such cases with the minimization of displacements or stresses, maximization of elasticity, etc. (Dražumerič & Kosel, 2005). It is difficult to determine the cost function for a dynamically loaded assembly where the loads are stochastic. So the question is how to define the stiffness of a shell structure if we do not know the lateral loads. Let us consider a certain arbitrarily shaped plane shell. We can write its potential energy,Wpas:

Wp=12∫σijεijdVE23

If we consider that if the Poissons shear modulus is neglected, the stress tensor,σijand strain tensor,εijare the following:

σij=−z{∂2w(xy)∂x2∂2w(xy)∂y2∂2w(xy)∂x∂y}TE24

and

εij=−zE{∂2w(xy)∂x2∂2w(xy)∂y2∂2w(xy)∂x∂y}TE25

Herew(xy)is displacement inzdirection andEis elasticity modulus. The potential energy can then be written as:

Wp=12E∫V(σxx2+σyy2+2τxy2)dVE26

We can see that in this case the potential energy is an integral of squares of stresses over the volume. We define also the kinetic energy,Wkfor plate vibrations:

Wk=ρω22∫Vzw2(xy)dVE27

whereρrepresents the density of material. Here the second derivative of displacement is approximated with respect to time with the product of squared displacement and eigen frequency,ω

For conservative systems the maximal potential energy is equal to maximal kinetic energy and from this follows the expression for cost function which can be in our case the stiffness itself:

fc(p)=∫V(σxx2+σyy2+2τxy2)dV∫Vzw2(xy)dVE28

The DASA was run 20 times and each run consisted of 2,000 calculations. The cost function was calculated by the ANSYS Multiphysics simulation tool. The obtained results are presented statistically in Table 2.

Existing solution

Optimized solutions

Worst

Mean

Best

5.87·10-3

6.13·10-3

6.81·10-3

7.34·10-3

Table 2.

Optimized casing's stiffness after 2,000 CFD calculations.

The results of optimization were quite surprising (Fig. 9 bottom row). It was expected that the ribs would form on all the given surface but they did not. The ribs were more distinct where the vertical part of the surface was turning into the horizontal one (ribs A). There were no ribs at the surface where the stator is in a tight fit, probably because of pre-stressing of the casing through the stator. Also in the groove for brushes (ribs B) there were no distinct ribs, probably because the groove itself is a kind of rib. Instead of this there is a given slope for a groove which was not there before. The radii of roundings were in most cases bigger than before as we can clearly see at the air culverts (Fig. 9 middle bottom).

3.3. A turbo-compressor aerodynamic power maximization

Radial air impellers are the basic components of many turbo-machines. In the following we will concentrate on relatively small impellers and subsonic speeds used in a dry vacuum cleaner. Our main aim was to find an impeller shape that has a higher efficiency, i.e., greater aerodynamic power, than the one currently used in production.

An impeller is constructed from blades, an upper and a lower side. The sides enclose the blades and keep them together. The blades, which are all the same, were the main part of the optimization. The geometry of a blade is shown in Fig. 10, where the gray color represents the blade. The method of modeling is as follows: we construct the points at specific locations, draw the splines through them and spread the area on the splines. Once a blade is made an air channel must be constructed in a similar way.

In Fig. 10a the point 1 has two parameters: the radiusp1and the anglep2Similarly, the points 2, 5 and 6 have parameter pairsp3andp4p5andp6p7andp8The points 3 and 4 are fixed on thexaxis. This is because the impeller must have a constant outer radiusp9and the outer side of the blade must be parallel to thezaxis. On the other hand, the outer angle of the bladep10and the angle of the spline at points 3 and 4, can be varied. Analogously, the anglesp11andp12are the inner-blade angles for the upper and lower edges of the blade at the input, respectively.

In Fig. 10c the points 1, 2, and 3 form the upper spline, and the points 4, 5, and 6, the lower spline. Between the points 1 and 6 is the point 7, which defines the spline of the input shape of the blade. In this figure, the points 1, 2, 5, and 6 have the parametersp13p14p15andp16respectively, describing their heights.

Point 3 stays on thexaxis and point 4 has a constant heightp17In other words, the designer of the impeller must know at least the outer diameterp9and the heightp17The parametersp18andp19describe the input angles of the lower and upper parts of the blade with respect to ther–zplane. Similarly, the parametersp20andp21describe the outer blade angle with respect to the same plane.

In Fig. 10b the meaning of point 7 is explained more precisely. The parametersp22p23andp24define the radius, height, and angle, respectively. The radius and angle dictate where the point should appear with respect to thex–yplane and the height with respect to ther–zplane. Similarly, the anglesp25p26p27andp28are needed to define the starting and ending angles of the spline constructed between the points 1, 7, and 6.

If we look closely at Fig. 10c then we can see the contour surrounding the blade. This is the air channel with the following parameters: the inner radiusp29(see Fig. 10a), which is needed for the hexahedral mesh (explained later), the air intake radiusp30the air outflow radiusp31the bolt radiusp32the bolt heightp33and the impeller heightp34

In this way we have successfully modeled the impeller geometry with 34 parameters. For each parameter we have a predefined search interval with a given discrete step. Therefore, the size of the search space can be obtained as the product of the number of possible settings over all the parameters. It turns out that there are approximately 3 10^{34} possible solutions.

The mesh of the air channel between the two blades is constructed with more than 6,000 hexahedral elements. The boundary conditions are zero velocity at all the solid regions and symmetry boundary conditions at the fluid regions. At the influx and outflux the intake velocity,vinand reference pressure,prefare defined, respectively. The intake velocity is parabolically distributed, because we expect that the intake flow is laminar and so:

vin=v(Φ(t))6rrup(rrup−1)E29

Here,v(Φ(t))is a velocity dependent on the stream, which further depends on time, as we shall see later,rupis the upper radius, defined before, andris the radius within the limits fromrstorupThe reference pressure,prefis set to zero.

With respect to the maximum time,tmax, the flux is:

Φ(t)=vAinttmaxE30

whereAinis the influx area andtis the current time.

The distribution of the relative pressure can be used to estimate the cost function. The average pressure,pinis chosen from the air-intake area. Finally, the aerodynamic power, which represents the cost function, is as follows:

fc(p)=Pair=(pout−pin)Φ(topt)E31

wherepoutis the output pressure at the radiusroutandΦ(topt)=40l/s is the flux near the desired optimum performance of the impeller. Our goal is to find such parameter-value settings, wherePairis maximized.

The DASA was run 10 times and each run consisted of 2,000 CFD calculations. For the CFD calculations we used the ANSYS Multiphysics package. A single CFD calculation takes approximately seven minutes. The obtained results, in terms of aerodynamic power, are presented statistically in Table 3.

Existing solution

Optimized solutions

Worst

Mean

Best

411.00

432.00

472.00

483.00

Table 3.

Optimized impeller's aerodynamic power in watts after 2,000 CFD calculations.

Results show that we were able to increase aerodynamic power for approximately 20 %. Figure 11 shows a 3D view of the existent and two optimized impellers (best and worst of 10 runs).

4. Discussion

In this chapter the Differential Ant-Stigmergy Algorithm (DASA) was presented, where the main goal was an evaluation of the DASA on some real-world engineering applications. Case studies were selected from a R&D project where more efficient turbo-compressor for dry vacuum cleaner was developed. Here the DASA was used to improve the efficiency of an electric motor, increase casing stiffness, and increase impeller's aerodynamic power. In all these cases the improvement was evident.

In (Korošec & Šilc, 2009a) we also compared the DASA to the eleven state-of-the-art stochastic algorithms for continuous optimization which were presented in the CEC'2005 Special Session on Real Parameter Optimization. The algorithms are:

Obtained results confirmed that the DASA is comparable to above algorithms and therefore generally applicable to global optimization problems.

The experimental results have shown that the DASA ranks among the best algorithms for real-life-like applications. Its main advantage is in consistent problem solving which is indicated by 19 rankings in top third, 4 ranking in middle third and only 2 in bottom third.

Statistical analysis following the Bonferroni-Dunn post-hoc test withα=0.10showed that the DASA is significantly better than 8 out of 11 compared algorithms.

The algorithm was also applied to dynamic optimization problems with continuous variables proposed for CEC’2009 Special Session on Evolutionary Computation in Dynamic and Uncertain Environments (Korošec & Šilc, 2009b). If we compare the DASA to:

the results show that the DASA can find reasonable solutions for all of the problems. It can be seen that the DASA performs not many worse than a self-adaptive differential evolution and much better than the other three algorithms. One obvious advantage is that was no need any changes to the original algorithm. So, it can be used as such for both cases of numerical optimization, static and dynamic. Furthermore, the algorithm is unsusceptible to different types of changes and can be used with very limited knowledge about problem, only maximal dimension and input problem parameters.

Peter Korosec and Jurij Silc (October 1st 2009). Applications of the Differential Ant-Stigmergy Algorithm on Real-World Continuous Optimization Problems, Evolutionary Computation, Wellington Pinheiro dos Santos, IntechOpen, DOI: 10.5772/9604. Available from:

By Mojtaba Ahmadieh Khanesar, Hassan Tavakoli, Mohammad Teshnehlab and Mahdi Aliyari Shoorehdeli

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