**1. Introduction**

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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 process.

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

© 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.

has been proposed in [6] with analytical expressions, which have been solved by a graphic method. The placement and sizing of a single filter have been also studied in [7] by a graphic approach. More recently, an equivalent resistance approach [8] has been applied to the location of a single-tuned passive filter.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 25

The popularity of the SA theory comes from its ability to solve complex combinatorial optimisation problems which purpose is to develop an efficient technique for finding minimum or maximum values of a function with many degrees of freedom and many local minima. Based on principles of physics, a combinatorial problem can be viewed as a thermodynamic system where all the equilibrium properties can be resolved by standard statistical mechanical methods [13]. Then, states in thermodynamic usage are identified with solutions in a combinatorial optimisation problem. Energy in thermodynamics is the cost function to be minimised in a SA process. The solution space of the optimisation problem is explored by a probabilistic hill climbing search which step size is controlled by a parameter *T* that plays the same role of the temperature in the physical system. Therefore, the abstract system can be described as if it was a thermal physical system which aim is to locate the ground state (i.e. optimal solution) while the temperature

In a typical SA process, the initial temperature is set sufficiently high. A new state *Xj* is generated incrementally from the current state *Xi* by randomly selecting and proposing a

Let the energy of the current state be *f(Xi)* and the energy of the new one be *f(Xj)*. The probability that a proposed move is accepted or rejected in the SA theory is determined by

It can be then appreciated that if the energy is decreased, the so-called Boltzmann probability *PBoltz* is greater than the unity. In that condition, the change is arbitrarily assigned

Conversely, if the energy is increased, the new state is accepted using the acceptance

When the proposed move is accepted, the new state becomes the current state; when it is rejected, the current state remains unchanged. Therefore, by controlling the temperature *T*,

the probability of accepting a hill climbing move which results in a positive

*fij)* equal to one, which means that the system always moves to this state.

p P ij Boltz < (2)

(1)

Δ

*fij* is also

( ) min 1 P exp *Boltz ij ij <sup>Δ</sup><sup>f</sup> <sup>P</sup> <sup>Δ</sup>f , () kT* ì ü ï ï = =- í ý ï ï î þ

*k* is a constant known as Boltzmann's constant relating temperature to energy.

declines.

where

where

to a probability *P(*

Δ

move from a set of predefined ones.

*fij = f(Xj) - f(Xi)* is the proposed energy change,

*pij* stands for a uniform random number between 0 and 1.

controlled and the exploration of the state space too.

Δ

distribution *pij*, as stated in (2):

the Metropolis criterion (1):

In the present chapter, the problem is formulated to minimise the filtering power with respect to the bus voltage constraints by limiting the harmonic currents passing through the filtering system. Corresponding to the harmonic currents either drawn by a passive filter or injected by an active filter, the filtering currents are calculated not to cancel the resulting harmonic voltages as proposed in [9], but to bring their magnitude within the limits recommended by the standards [10]. The optimisation of the filter size is then applied separately for each harmonic order by means of a double SA process ruled by two distinctive objective functions and the results are known in terms of filtering power to install with the resultant harmonic voltages expected on the distribution network.

As the number of busbars able to receive harmonic filters is usually limited in industrial plants or on board power systems, all the possible configurations can be individually considered and analysed thanks to a fast harmonic simulation schedule using the analytical models of static converters. Consequently, the search for the suited number of filters, their best location and their respective dimensioning power can be properly conducted.

The present chapter will clearly stress on the promising results provided by the SA theory when dealing with practical optimisation issues, like the placement and the sizing of harmonic filters that it deals with. The real power system of an electric propulsion ship will be then considered for illustration purposes.
