In this paper, PSSE is considered in the rectangular and polar coordinate systems. Up to now, such the investigations have not been considered in existing papers.

In this situation, there were carried out: (i) original calculations of the considered indices for representative operational states of the test PS when SE is performed in the rectangular and polar coordinate systems and in PS there is and there is no PhS, (ii) original statistical analyses of the calculated indices, (iii) original discussion on the causes of observed regularities.

Similarly, in Table II, there are relative changes of parameters of the considered indices when SE is realized in the polar coordinate system. The mentioned changes of the parameters of the considered indices characterizing properties of PSSE are calculated using formula: [DELTA][p.sub.X] = 100([p.sub.X,S] - [p.sub.X,S-])/ [p.sub.X,S-], where [p.sub.X,S], [p.sub.X,S-] are parameters of index X, when SE is performed for PS with or without PhS, respectively.

In the rectangular coordinate system, one utilizes equations in which there are quadratic functions, in the polar coordinate system, one utilizes equations in which there are trigonometric functions (that can be considered as infinite series).

As based on the analysis of the key reasons for the inefficiency of search and poor convergence, this research proposes a novel approach that transforms the coordinate system for design variables and constraint space to local polar coordinate systems. Thereby, ACO algorithm is improved by incorporating the novel approach, in order to optimize the problem of inefficiency in convergence and search strategy in the ACO.

In addition, if a mathematical model is formulated based on independent position coordinates of machines, that is, design variables, the reliability and efficiency of the search process will be poor still, and, in order to improve the efficiency and search reliability, a technique of controlling design variables under local polar coordinate systems is proposed; consequently, a correlation between design variables and objective function is effectively demonstrated.

The Attached Design Variables and Mathematical Model under Local Polar Coordinate Systems

Firstly, a transformation of the rectangular coordinate geometry into a series of local polar coordinate systems is devised as depicted in Figure 2, in which the first local polar coordinate system is set as in Figure 1; thus, every geometric center of machines is set as a local polar origin, and their local polar axes are directed as x-axis.

From Figure 2, the search strategy of the ant colony under local polar coordinate systems can be explained as follows: first, let us assume the machines are in any numbered permutation and say if the location of first machine is determined then the search will automatically go to the next machine that follows, in which search is determine by the azimuth and relative distance of the machines.

An Improved ACO Algorithm Based on Attached Design Variables under Local Polar Coordinate Systems

Caption: Figure 4: Ant perceptive unit model based on attached design variables under local polar coordinate systems.

The positive feedback mechanism is adopted to perceive pheromone concentration, whereby machines are considered as nodes; each time the ant passes a node, the pheromone of that node is distributed under the local polar coordinate system and, hence, is used as a probability condition to choose a relative location of a next node.