**2. Simulated annealing overview**

The SA process is motivated by an analogy to annealing in solids. The idea comes first from a paper published by Metropolis et al. in 1953 [11]. An algorithm was then proposed to simulate the cooling of a material in a heat bath. This is a process known as annealing. The structural properties of a material that is heated past melting point and cooled afterwards depend on the rate of cooling. In consequence, if the liquid is cooled slowly enough, thermal mobility is lost and large crystals well ordered are formed, which is the state of minimum energy for the system. Conversely, if the liquid is cooled quickly, the crystals contain imperfections and the process becomes a simulated quenching that cannot ensure the achievement of a low energy state.

Metropolis's algorithm simulates the cooling process by gradually lowering the temperature of the system until it converges to a steady frozen state. In 1982, Kirkpatrick et al. [12] took the idea of the Metropolis's algorithm and applied it to optimisation problem. The SA process is then used to search for feasible solutions and converge to an optimal solution.

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 declines.

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 move from a set of predefined ones.

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 the Metropolis criterion (1):

$$P\left(\Delta f \vec{\eta}\right) = \min\left\{1, \ P\_{\text{Rydz}} = \exp(-\frac{\Delta f \vec{\eta}}{kT})\right\} \tag{1}$$

where

24 Simulated Annealing – Single and Multiple Objective Problems

the location of a single-tuned passive filter.

be then considered for illustration purposes.

**2. Simulated annealing overview** 

achievement of a low energy state.

optimal solution.

conducted.

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

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

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

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

The SA process is motivated by an analogy to annealing in solids. The idea comes first from a paper published by Metropolis et al. in 1953 [11]. An algorithm was then proposed to simulate the cooling of a material in a heat bath. This is a process known as annealing. The structural properties of a material that is heated past melting point and cooled afterwards depend on the rate of cooling. In consequence, if the liquid is cooled slowly enough, thermal mobility is lost and large crystals well ordered are formed, which is the state of minimum energy for the system. Conversely, if the liquid is cooled quickly, the crystals contain imperfections and the process becomes a simulated quenching that cannot ensure the

Metropolis's algorithm simulates the cooling process by gradually lowering the temperature of the system until it converges to a steady frozen state. In 1982, Kirkpatrick et al. [12] took the idea of the Metropolis's algorithm and applied it to optimisation problem. The SA process is then used to search for feasible solutions and converge to an

with the resultant harmonic voltages expected on the distribution network.

Δ*fij = f(Xj) - f(Xi)* is the proposed energy change, *k* is a constant known as Boltzmann's constant relating temperature to energy.

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 to a probability *P(*Δ*fij)* equal to one, which means that the system always moves to this state. Conversely, if the energy is increased, the new state is accepted using the acceptance distribution *pij*, as stated in (2):

$$\mathbf{p}\_{\text{ij}} \preccurlyeq \mathbf{P}\_{\text{Relue}} \tag{2}$$

where

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

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 Δ*fij* is also controlled and the exploration of the state space too.

The driving mechanism of the SA process is described in Figure 1. The process of selecting and proposing a move is repeated until the system is considered in thermal equilibrium. Then, the temperature is reduced according to a temperature schedule and the system is allowed to reach thermal equilibrium again. Then, as the temperature of the system declines, the probability of accepting a worse move is decreased. This is the same as gradually moving to a frozen state in physical annealing. The process is finally stopped when no significant improvement is expected by further lowering temperature. At this point, the current state of the system is the solution to the optimisation problem.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 27

Most of the practical applications of the SA theory are in complex problem domains, where algorithms either did not exist or performed poorly. Among the considerations which have led to the choice of a SA process for solving the current issue are its ease of implementation and its flexibility in applications to complex optimisation problems, especially ones where a desired global extremum can be hidden among many poorer local extrema. Besides, compared to some other popular methods like genetic algorithms, a SA process offers a far well mastered solution space to explore. Indeed, it can be appreciated that the neighbourhood becomes smaller as the global optimal solution is closer, which makes the search process easier and speeds up the convergence. It is also important to ensure that this condition is met when thinking about one's problem. For all those reasons, a SA process applied to the search for the optimal placement and power sizing of harmonic filters on

large-scale electrical systems can be considered as a suitable optimisation method.

The present harmonic filtering study is based on the usual assumptions about the symmetry of the electrical network, the balance of the harmonic currents generated by the non-linear loads, and the independence between the harmonic orders. As a result, a single phase network is considered and the analysis is carried out for each individual harmonic order. Then, the relationship between nodal harmonic currents and voltages is defined through the

The nodal admittance matrix is obtained from the impedance of every network component and the nodal currents are given from the harmonic currents generated by the non-linear loads.

In industrial distribution systems, loads are supplied through transformers by different voltage levels according to their rated power. Due to the voltage supplies, the filtering current is modified by the transformer ratio when the filter is connected to the primary or to the secondary side of the transformer. In consequence, the comparison between the set of

Let *Vn* be a vector composed of the nominal nodal voltages. The admittance matrix formulation (3) can be then rewritten in an equivalent system where the relationship (5) between nodal harmonic currents and voltages becomes independent from the nominal

**I YV** *h hh* = ⋅ (3)

**3. Problem statement** 

**3.1. Network representation** 

admittance matrix [14], as follows:

*Yh* : the harmonic admittance matrix,

*h* : the harmonic order.

nodal voltages.

*Ih* : the vector of the nodal harmonic currents,

*Vh* : the vector of the nodal harmonic voltages,

solutions requires considering a per-unit system.

where

**Figure 1.** SA algorithm

The major advantage of a SA process over other methods is its ability to avoid becoming trapped in local minima. The algorithm employs a random search which not only accepts changes that decrease the objective function, but also some changes that can increase it. In consequence, a worse move can be accepted temporarily by leading subsequently to an improving solution. Then, the system sometimes goes uphill as well as downhill. However, the lower the temperature, the less likely is any significant uphill excursion.

Most of the practical applications of the SA theory are in complex problem domains, where algorithms either did not exist or performed poorly. Among the considerations which have led to the choice of a SA process for solving the current issue are its ease of implementation and its flexibility in applications to complex optimisation problems, especially ones where a desired global extremum can be hidden among many poorer local extrema. Besides, compared to some other popular methods like genetic algorithms, a SA process offers a far well mastered solution space to explore. Indeed, it can be appreciated that the neighbourhood becomes smaller as the global optimal solution is closer, which makes the search process easier and speeds up the convergence. It is also important to ensure that this condition is met when thinking about one's problem. For all those reasons, a SA process applied to the search for the optimal placement and power sizing of harmonic filters on large-scale electrical systems can be considered as a suitable optimisation method.
