**4. Conclusion**

Briefly, Our proposed method as a local solution method has less displacement and by taking advantages of MFA algorithm in comparison to SA algorithm and localizing problem (that reduces number of engaged modules) and therefore by having less variables, is faster. Also having less number of movable modules causes more similarity *if the solution is feasible*.

 Selection of modules for relocation is based on the range that includes enough free space around the extra module so the runtimes of our proposed algorithm almost do not depend on the size of benchmark circuit in compare to the SA-based method, actually size of local relocation range and numbers of movable modules of each problem are the main parameters. Applying ability of rotation of modules inside a fixed distance controller energy function as permissible distances preservation energy and three phases cooling process are main properties of our employed MFA algorithm. Results show our method is almost independent of size and complexity of model placement.

Although the use of SA provides for escaping from the local minima, it results in an excessive computation time requirement that has hindered experimentation with the Boltzmann machine. In order to overcome this major limitation of the Boltzmann machine, a mean field approximation may be used. In mean field network, the binary state stochastic neurons of the Boltzmann machine are replaced by deterministic analogue neurons. A simple formulation of the Traveling Salesman Problems energy function is described which, in combination with a normalized Hopfield-Tank neural network, eliminates the difficulty in finding valid tours[1]. This technique, as the one of the bases of MFA algorithm, is applicable to many other optimization problems involving n-way decisions (such as VLSI layout and resource allocation) and is easily implemented in a VLSI neural network. The solution quality is shown to be dependent on the formation of elements of the problem configuration which are influenced by the constraint penalties and the temperature as what is borrowed from SA technique. The applied algorithm for local relocation problem is modified form of which is applied for cell placement problem. The cooling schedule has three stages that the final stage is very fast cooling with decreasing factor 0.65 that may be what you mean *quenching*. Otherwise other two stages with decreasing factors 0.95 and 0.8 are not so fast and have *annealing* essence. For more information about this topic, one can refer to [1].

### **Author details**

20 Simulated Annealing – Single and Multiple Objective Problems

We implemented the proposed algorithm on a 2.4GHz Intel Pentium IV with 512MB memory using MATLAB 7.2.0.232 (R2006a) in WINDOWS operating system. We applied the proposed algorithm to the relocation of n300a, n200a, and n100a, which are distributed in

For every benchmark five different problems were resolved using our proposed algorithm and maximum and average runtime of 10 runs of them are presented in Table 1. Results show that our MFA based algorithm is faster than SA-based proposed method in SA-based relocation method in [18] because the number of displacements is limited to the number of movable modules of problem and the problem is local relocation. Actually relocation range reflects on number of displacements and also similarity of resultant placement with model

Results show runtimes of our proposed algorithm almost do not depend on the size of benchmark circuit in compare to the method represented in SA-based proposed method, actually size of local relocation range and numbers of movable modules of each problem are the main parameters here. Also feasibility of local relocation solution, to guarantee the similarity of resultant placement with model placement depends on the existence of enough dead space near additional module so that the relocation rage becomes limited and small.

MFA Local Relocation SA

Briefly, Our proposed method as a local solution method has less displacement and by taking advantages of MFA algorithm in comparison to SA algorithm and localizing problem (that reduces number of engaged modules) and therefore by having less variables, is faster. Also having less number of movable modules causes more similarity *if the solution is feasible*. Selection of modules for relocation is based on the range that includes enough free space around the extra module so the runtimes of our proposed algorithm almost do not depend on the size of benchmark circuit in compare to the SA-based method, actually size of local relocation range and numbers of movable modules of each problem are the main parameters. Applying ability of rotation of modules inside a fixed distance controller energy function as permissible distances preservation energy and three phases cooling process are main properties of our employed MFA algorithm. Results show our method is almost

Average runtime (Sec.)

n100 2.0 2.37 2.52 1.0 n200 3.2 3.62 3.72 9.0 n300 3.7 3.92 3.96 60.8

Min. runtime (Sec.)

Max. runtime(Sec.)

*3.2.8. Experimental results* 

GSRC benchmarks in [17].

Min. runtime (Sec.) Benchmark

**4. Conclusion** 

**Table 1.** MFA Local Relocation results for GSRC benchmarks

independent of size and complexity of model placement.

placement.

Gholam Reza Karimi and Ahmad Azizi Verki *Electrical Engineering Department, Engineering Faculty- Razi University, Kermanshah, Iran* 

### **5. References**

	- [8] Hokkinen, J.; Lagerholm, M.; Peterson, C. & Soderberg, B. (1998).A Potts neuron approach to communication routing, *Neural Computation,* vol. 10, pp. 1587–1599.

**Chapter 2** 

© 2012 Miègeville and Guérin, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Miègeville and Guérin, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Optimal Sizing of Harmonic Filters** 

Laurence Miègeville and Patrick Guérin

http://dx.doi.org/10.5772/51661

**1. Introduction** 

process.

Additional information is available at the end of the chapter

**in Electrical Systems: Application of** 

**a Double Simulated Annealing Process** 

Many fields including management science, computer science, electrical and industrial engineering bring into play a number of combinatorial optimisation problems that consist in finding the global minimum of a cost function that may possess several local minima over a finite or infinite set of solutions. In practice, excellent results have been obtained by using local search algorithms for a wide variety of issues, leading thus to a growing interest in theoretical results. However, many problems are still open as a challenge. In the current chapter, the authors present their experience in using a double simulated annealing (SA) optimisation process applied to the search for the optimal sizing of harmonic filters placed in a distribution electrical system. The effectiveness of the SA algorithm will be herein argued and illustrated on a distribution system so as to characterise the suitable placement of filtering devices that leads to a minimum required power. One of its main benefits compared with a popular genetic algorithm for example is to supervise the configuration space at every moment and to control the convergence

Nowadays, harmonic filters are widely installed in distribution systems for harmonic current filtering to achieve harmonic distortion reduction. The extent of this benefit depends greatly on the filtering system placement. Then, the problem focuses on several formulations about filters locations, their types and sizes. In the past, many efforts were put into the capacitor placement for reactive power compensation [1] even with a distorted substation voltage [2]. Many optimisation methods have found a practical application in this problem and the fuzzy logic [3], the simulated annealing [4], as well as the genetic algorithm [5] have been tested on it. A less attention has been paid for harmonic filters. A formulation

