**5.** *Candida albicans*

*Candida albicans*, a type of diploid yeast in the Division *Ascomycota*, is the most commonly encountered fungal pathogen in humans. The human infection candidiasis, resulting from overgrowth of *C. albicans*, often occurs in immunocompromised patients [32]. *C. albicans* can be induced to undergo a parasexual cycle that involves mating of diploids (syngamy) to form a tetraploid that subsequently appears to undergo a form of meiosis, followed by chromosome loss leading to approximately diploid cells with high levels of aneuploidy and homozygosity [33]. The *C. albicans* genome contains many genes that are homologous to genes in other species that function in meiosis [34]. One such gene, *Dmc1*, encodes a protein that has a central role in homologous recombination and is only known to express during meiosis [35]. Under appropriate conditions, *C. albicans* is capable of same sex mating and can undergo extensive genetic recombination between homologous chromosomes [36]. The two successive cell divisions that ordinarily occur subsequent to meiotic recombination in other organisms appear to be absent in *C. albicans* meiosis. Instead a reduction from four copies of the genome (tetraploidy) to two copies (diploidy) occurs by random chromosome loss during the mitotic cell division subsequent to meiotic recombination [37]. Although *C. albicans* populations are largely clonal, parasexual recombination can facilitate the evolution of resistance to the antifungal agent fluconazole upon exposure to the agent over successive generations [38].

The parasexual cycle appears to occur with greater frequency under environmental stress condition [33]. Glucose starvation and oxidative stress are environmental stresses that are commonly encountered by pathogenic *C. albicans,* and these stresses efficiently induce same-sex mating between cells from a single progenitor [39]. As suggested by Guan and collaborators [39], same sex mating in *C. albicans* may be an important mode of sexual reproduction that occurs often in nature. Oxidative stress, associated with an increase in reactive oxygen species, causes DNA damage and thus the induction of mating may reflect an adaptive DNA repair response.

 It appears that *C. albicans* has retained meiotic homologous recombination, a principal feature of sexual reproduction that provides the adaptive benefit of DNA repair, while losing the ability to undergo successive cell divisions in an organized fashion to reduce ploidy.

#### **6.** *Paramecium tetraurelia*

*P. tetraurelia* is a unicellular eukaryotic ciliate in the Phylum *Ciliophora* (**Figure 2**). It has two diploid micronuclei and a polyploid macronucleus. The

**141**

repaired micronuclear DNA.

**Figure 2.**

Paramecium tetraurelia *[41].*

tion involves mating with another individual.

micronuclear chromosomal DNA contains the genetic information that is inherited from one generation to the next, whereas the macronucleus contains many chromosomal DNA copies that express cellular functions. *P. tetraurelia* is able to undergo both asexual and sexual reproduction. Asexual reproduction occurs by binary fission in which the micronuclei divide by mitosis and the macronucleus divides by amitotic division [40]. Sexual reproduction involves a meiotic process, either automixis or conjugation. Automixis is a kind of self-fertilization, whereas conjuga-

As *P. tetraurelia* undergoes asexual reproduction by binary fission over many successive generations, the vitality of the lineage declines (clonal aging) until the lineage reaches the end of its clonal lifespan at about 200 fissions [42]. However, if *P. tetraurelia* undergoes automixis or conjugation during clonal aging, vitality is restored. Several laboratories found that clonal aging is associated with a dramatic increase in DNA damage [42–44]. When clonally aged *P. tetraurelia* undergo automixis or conjugation, the micronuclei go through meiosis followed by pairwise nuclear fusion (syngamy) of haploid meiotic products either from the same individual (automixis) or from different individuals (conjugation) to form a new diploid micronucleus. Subsequent to the formation of a new diploid micronucleus, the old macronucleus disintegrates and the new micronucleus replicates to form a new macronucleus. The paramecia that undergo this process have their clonal lifespan restored and are rejuvenated. Clonal aging thus appears to be caused to a large extent by progressive accumulation of DNA damage and rejuvenation likely depends on the repair of such damages in the micronuclear DNA during meiosis followed by the reestablishment of macronuclear DNA by replication of the newly

*Sexual Processes in Microbial Eukaryotes DOI: http://dx.doi.org/10.5772/intechopen.88469* *Parasitology and Microbiology Research*

**5.** *Candida albicans*

successive generations [38].

**6.** *Paramecium tetraurelia*

repair response.

Rad51), a Rec2 protein (more distantly related to mammalian Rad51), and the Brh2 protein [that is related to the mammalian Breast Cancer 2 (Brca2) protein)] [31]. When any of these proteins is inactive, *U. maydis* becomes more sensitive to DNA damaging agents, mitotic recombination is reduced, and there is failure to complete meiosis [31]. Recombinational repair occurring during meiosis as teliospores are formed by the pathogen likely contributes to the maintenance of its genome integ-

*Candida albicans*, a type of diploid yeast in the Division *Ascomycota*, is the most commonly encountered fungal pathogen in humans. The human infection candidiasis, resulting from overgrowth of *C. albicans*, often occurs in immunocompromised patients [32]. *C. albicans* can be induced to undergo a parasexual cycle that involves mating of diploids (syngamy) to form a tetraploid that subsequently appears to undergo a form of meiosis, followed by chromosome loss leading to approximately diploid cells with high levels of aneuploidy and homozygosity [33]. The *C. albicans* genome contains many genes that are homologous to genes in other species that function in meiosis [34]. One such gene, *Dmc1*, encodes a protein that has a central role in homologous recombination and is only known to express during meiosis [35]. Under appropriate conditions, *C. albicans* is capable of same sex mating and can undergo extensive genetic recombination between homologous chromosomes [36]. The two successive cell divisions that ordinarily occur subsequent to meiotic recombination in other organisms appear to be absent in *C. albicans* meiosis. Instead a reduction from four copies of the genome (tetraploidy) to two copies (diploidy) occurs by random chromosome loss during the mitotic cell division subsequent to meiotic recombination [37]. Although *C. albicans* populations are largely clonal, parasexual recombination can facilitate the evolution of resistance to the antifungal agent fluconazole upon exposure to the agent over

The parasexual cycle appears to occur with greater frequency under environmental stress condition [33]. Glucose starvation and oxidative stress are environmental stresses that are commonly encountered by pathogenic *C. albicans,* and these stresses efficiently induce same-sex mating between cells from a single progenitor [39]. As suggested by Guan and collaborators [39], same sex mating in *C. albicans* may be an important mode of sexual reproduction that occurs often in nature. Oxidative stress, associated with an increase in reactive oxygen species, causes DNA damage and thus the induction of mating may reflect an adaptive DNA

Unlike *C. albicans,* several other *Candida* clade species have a sexual cycle that includes ordinary meiosis and formation of sexual spores [37]. It appears that *C. albicans* has retained meiotic homologous recombination, a principal feature of sexual reproduction that provides the adaptive benefit of DNA repair, while losing the ability to undergo successive cell divisions in an organized fashion to reduce

*P. tetraurelia* is a unicellular eukaryotic ciliate in the Phylum *Ciliophora* (**Figure 2**). It has two diploid micronuclei and a polyploid macronucleus. The

rity by removing DNA damages incurred during infection.

**140**

ploidy.

**Figure 2.** Paramecium tetraurelia *[41].*

micronuclear chromosomal DNA contains the genetic information that is inherited from one generation to the next, whereas the macronucleus contains many chromosomal DNA copies that express cellular functions. *P. tetraurelia* is able to undergo both asexual and sexual reproduction. Asexual reproduction occurs by binary fission in which the micronuclei divide by mitosis and the macronucleus divides by amitotic division [40]. Sexual reproduction involves a meiotic process, either automixis or conjugation. Automixis is a kind of self-fertilization, whereas conjugation involves mating with another individual.

As *P. tetraurelia* undergoes asexual reproduction by binary fission over many successive generations, the vitality of the lineage declines (clonal aging) until the lineage reaches the end of its clonal lifespan at about 200 fissions [42]. However, if *P. tetraurelia* undergoes automixis or conjugation during clonal aging, vitality is restored. Several laboratories found that clonal aging is associated with a dramatic increase in DNA damage [42–44]. When clonally aged *P. tetraurelia* undergo automixis or conjugation, the micronuclei go through meiosis followed by pairwise nuclear fusion (syngamy) of haploid meiotic products either from the same individual (automixis) or from different individuals (conjugation) to form a new diploid micronucleus. Subsequent to the formation of a new diploid micronucleus, the old macronucleus disintegrates and the new micronucleus replicates to form a new macronucleus. The paramecia that undergo this process have their clonal lifespan restored and are rejuvenated. Clonal aging thus appears to be caused to a large extent by progressive accumulation of DNA damage and rejuvenation likely depends on the repair of such damages in the micronuclear DNA during meiosis followed by the reestablishment of macronuclear DNA by replication of the newly repaired micronuclear DNA.


**2.2 Calculation of embedment depth**

The factors responsible for the change in fracture width and conductivity after hydraulic fracturing in shale reservoirs include proppant embedment and proppant deformation. The embedment of the proppant involves the penetration of the proppant inside the fracture surface, while proppant deformation is directly related to the strength of the proppant [33]. The deformation of the proppant can be described by Eq. (7), as the proppant can be assumed as an elastic body while the penetration of proppant into a body can be solved using contact mechanics [34]. The contact problem can be formulated as a constrained minimization problem, where the objective function to be minimized is the total potential energy (*Π*) of the bodies in contact. The energy for this system can be written as

$$H(\mathbf{u}) = \frac{1}{2}\mathbf{k}\mathbf{u}^2 - \mathbf{f}\mathbf{u} \tag{7}$$

where *k* is the stiffness matrix, *u* is the displacement field, and *f* is the external force. Several constrained minimization algorithms can be used to solve the problem of the equation, such as the penalty method, the Lagrange multipliers method, and the augmented Lagrangian method. The results presented in this paper are based on the augmented Lagrangian method according to the ANSYS implementation. Augmented Lagrangian methods are a certain class of algorithms for solving c onstrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem with a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term, designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with the meth od of Lagrange multipliers. A general discussion of these techniques can be found in the literature on contact mechanics [35–37]. In this study, a numerical model is developed based on the experimental investigation carried out on the embedment of 20/40 mesh proppant (size between 20 and 40 μm) on shale samples from Sungai Perlis beds, Terengganu, Malaysia. The core samples were subjected to uni-axial stress of 20 MPa to find the proppant embedment in the formation, as shown in **Figure 4**. Such a high compression force was applied to investigate the embedment under reservoir conditions.

The embedment cell consists of a transparent cylindrical tube where a shale sample is placed. A metal loading ram is used to load the shale-proppant stack and deformation is measured as the axial load is increased. The deformation in shale, assuming elastic behavior, is quantified using Young's modulus and the applied load. **Figure 5** shows the universal testing machine (UTM) used for measuring

**Figure 6.** *Illustration of embedment of the surface due to external load.*

compression and tensile strengths of materials and samples used in the proppant embedment tests.

In this study, the contact behavior of proppant and rock was carried out using structural mechanics. Experimental boundary conditions are implemented to find the impact of load on proppant penetration in the rock surface. The resulting properties from experiments are introduced in a finite element model to find the effect of fracture surface on the embedment of proppant, as shown in **Figure 6**. A finite element study with ANSYS Workbench has been performed for the computational contact analysis [38]. In a subsurface reservoir, proppants experiences compression from both sides in the formation; therefore, biaxial test should be carried for precise estimation of embedment. Proppants inside the fracture also contact each other due to subsurface stresses. Due to external force, the stressdisplacement relationship is as follows:

$$
\rho\_n \frac{\partial^2 u\_n}{\partial t^2} = \nabla. \sigma\_n + F\_{\rm vn} \tag{8}
$$

where *ρ* is the density, kg/m3 ; u is the displacement, mm; *σ* is the stress tensor; *Fv* is the external force per unit volume; *n* = 1, indicating the proppant, *n* = 2, indicating the rock matrix. The stress–strain relationship and strain–displacement relationship are shown in Eqs. (9) and (10):

$$
\sigma = \mathbb{C}.\varepsilon \tag{9}
$$

$$\varepsilon = \frac{1}{2} \left[ (\nabla u)^T + \nabla u \right] \tag{10}$$

where *ε* is the strain tensor; *E* is Young's modulus, GPa; *υ* is the Poisson's ratio. As we assume that Young's modulus and Poisson's ratio of the reservoir rock would change from the outer surface of the fracture toward the inside as shown in **Figure 7** and the specific correlation is expressed as follows:

$$E = f(l) \tag{11}$$

$$
v = f(l) \tag{12}$$

where *l* is the distance from the surface of the fracture toward the inside as shown in **Figure 7**. The closure pressure causes the proppant to embed into the fracture surface and the porosity of the propped fracture to change. The embedment of the proppant (*h*em) can be expressed as follows:

$$h\_{\rm em} = u\_m - u\_p \tag{13}$$

where *um* is the displacement of the no-contact part between the fracture and the proppant under closure pressure; *up* is the displacement of the contact part between the fracture and the proppant.

**Figure 7.** *Change in Young's modulus due to fracturing fluid interaction with the shale.*

As the focus is proppant embedment into the shale formation, therefore, proppant is assumed as a rigid body. Thus, no deformation occurs in the proppant. However, the deformation of the shale surface is simulated with the penetration of proppant under uni-axial and bi-axial loads.


**1. Introduction**

Elevated seawater temperatures resulting from climate change are causing widespread coral bleaching events across tropical regions of the world [1–3]. Over the past 30 years the frequency and severity of these bleaching events have increased and they are now occurring on an unprecedented scale [1,4]. In 2015–2016, a severe El Niño event led to widespread coral bleaching across the Pacific and Indian Oceans [1,2,4–8], with some regions experiencing repeated heat stress events throughout this period (e.g., Chagos Archipelago [9], Great Barrier Reef (GBR) [2,10]). During the El Niño, some areas of the Pacific experienced extreme heat stress, reaching over 20 degree heating weeks (DHWs) [5], a metric representing accumulated heat stress over a 90-day window in comparison to historical maximum monthly means (MMMs) [10]. Although not quite as extreme, DHWs from the Western and Eastern Indian Ocean during the same period still greatly exceeded the bleaching threshold, with records ranging from four to 11 [11–13].

Coral bleaching occurs when corals expel their photosynthetic endosymbiotic algae (Symbiodiniaceae), often as a result of temperature stress, leaving a white skeleton visible through translucent tissue [14]. Corals may recover over time or coral mortality may occur

**Citation:** Bessell-Browne, P.; Epstein, H.E.; Hall, N.; Buerger, P.; Berry, K. Severe Heat Stress Resulted in High Coral Mortality on Maldivian Reefs following the 2015–2016 El Niño Event. *Oceans* **2021**, *2*, 233–245. https://doi.org/ 10.3390/oceans2010014

Academic Editor: Rupert Ormond

Received: 28 October 2020 Accepted: 22 February 2021 Published: 3 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

if the algae cells are not regained and temperature stress conditions persist. Susceptibility to bleaching and mortality can vary among coral species due to a variety of factors, which include coral morphology [15,16], history of stress or temperature exposure [17,18], associated algal symbiont type [19–21], and the coral microbiome [22]. This variability in susceptibility can cause not only a loss in overall coral cover, but also a reduction in both the biodiversity and functional diversity of coral assemblages [23]. In severe cases bleaching and subsequent mass mortality events result in decreased larval supply and connectivity, which can result in phase shifts that are often from coral to macroalgal dominated reefs [24]. Other alternate states include sponge- or urchin-dominated reefs [25,26]. It becomes difficult for corals to re-populate reefs and reverse such phase shifts (reviewed in [27]), resulting in obvious changes to benthic and reef-associated pelagic assemblages across the reef ecosystem.

Mass coral bleaching events impact not only the health of the coral reef ecosystem, but also the human populations that may rely on artisanal fisheries, coastal protection, and ecotourism. Coral mortality can be particularly concerning for small island nations because reefs represent a lifeline that provides food and livelihood. Loss of coral in the ecosystem can decrease catches of artisanal reef fisheries and discourage tourism, ultimately impacting the food security and economy of local communities [28,29].

 The Maldives is made up of over 1000 islets situated within 26 atolls. It is the largest group of coral reefs in the Indian Ocean, hosting approximately 1100 reef-associated fish species and 250 species of coral [30]. The economy of the Maldives benefits from marine resources through tourism and ecosystem services such as coastal protection, food, and construction materials [31]. Thus, changes to the state of Maldivian reef health is of concern to the nation. During the 2015–2016 El Niño event, reefs in the Maldives experienced between four and 11 DHWs, suggesting considerable heat stress [32]. This resulted in widespread bleaching across the region [7,33–35], with bleaching observed in 73% of corals from 0–13 m depth [12]. Here we document the effects of the 2015–2016 global bleaching event on the survival and mortality of coral communities in the northern atolls of the Maldives.


**3. Problem statement**

## **3.1. Network representation**

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 admittance matrix [14], as follows:

$$\mathbf{I}\mathbf{\hat{}} = \mathbf{Y}\mathbf{\hat{}} \cdot \mathbf{V}\mathbf{\hat{}}\tag{3}$$

where

26 Simulated Annealing – Single and Multiple Objective Problems

**Figure 1.** SA algorithm

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

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.

current state of the system is the solution to the optimisation problem.

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

*Yh* : the harmonic admittance matrix,

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

*h* : the harmonic order.

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 solutions requires considering a per-unit system.

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 nodal voltages.

$$\mathbf{V}\_n \cdot \mathbf{I}\_h = \mathbf{V}\_n^2 \cdot \mathbf{Y}\_h \cdot \frac{\mathbf{V}\_h}{\mathbf{V}\_n} \tag{4}$$

$$\dot{\mathbf{i}}\_h = \mathbf{y}\_h \cdot \mathbf{v}\_h \tag{5}$$

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 29

**Figure 2.** General flow chart of the optimisation procedure

specific application. It consists herein of the following input data:

*3.2.1. Initialisation procedure*

The implementation of the algorithm is detailed further below and the focus is on the major parts of the procedure that can influence the behaviour of the SA approach to optimisation.

When thinking about any optimisation problem, one of the first considerations is to start with a suitable set of initial parameters in order to ensure a good algorithm performance, which means the quality of the solution returned and a reasonable computation time [15]. The initialisation procedure deals first with the definition of the problem's data linked to the

where

**i VI** *h nh* = ⋅ : the vector of the equivalent injected currents in kVA, *h nh* = ⋅ ² **y VY** : the equivalent admittance matrix in kVA, *h h n* = **V v V** : the vector of the equivalent voltages in %.

Besides, an active filter connected to the node *k* can be considered as a current source which injects a current *jhk* on the network. A passive filter can be also modelled by a current source which magnitude would represent the harmonic current that must be drawn by the filter. In this last case, the phase angle depends on the harmonic voltage at the coupling node. All the additional currents produced by the set of filters are intended to reduce the harmonic voltages at each busbar. The location of these filters is defined in a node list called *ListFilter*. Assuming a given number of filters, the harmonic voltages are deduced from (6):

$$\mathbf{v}\_h = \mathbf{z}\_h \cdot (\mathbf{i}\_h \quad + \mathbf{j}\_h) \tag{6}$$

where

*jh* is the vector of the filtering current and *zh = yh-1* the equivalent impedance matrix.

#### **3.2. Optimisation process**

The aim of the current optimisation problem is to minimise the total filtering power to connect to the grid while the voltage standards are met for each nodal harmonic voltages. Prior to the search for the optimal current to be injected, it is necessary to answer a first question about the possibility to obtain a current vector *jh* able to reduce the voltages within the limiting levels *vlimit* at each node. For this reason, a double SA process is applied, as illustrated by the complete flow chart of figure 2. The purpose of the first one is to minimise the harmonic voltages, while checking if a filtering solution is existing or not.

In cases where no filtering solution can be found, the simulated voltage annealing is fully processed and returns the total filtering current *jtotal* able to check a minimum gap between the nodal harmonic voltages and their expected limits. Instead, when the voltage constraints are properly respected, the voltage annealing is partially processed. Once a local solution is achieved, the procedure is switched to a simulated current annealing which purpose is to minimise the total filtering current *jtotal* that meets the voltage requirements. The search space is actually the same regardless of the nature of the SA process; what is however different is the objective function, as will be argued further in the subsection 3.2.2.

**Figure 2.** General flow chart of the optimisation procedure

The implementation of the algorithm is detailed further below and the focus is on the major parts of the procedure that can influence the behaviour of the SA approach to optimisation.

## *3.2.1. Initialisation procedure*

28 Simulated Annealing – Single and Multiple Objective Problems

**i VI** *h nh* = ⋅ : the vector of the equivalent injected currents in kVA,

*h nh* = ⋅ ² **y VY** : the equivalent admittance matrix in kVA,

**V** : the vector of the equivalent voltages in %.

where

*h*

where

**3.2. Optimisation process**

*h*

*n* = **V v**

2 *h*

*n*

**V** ⋅= ⋅ ⋅ (4)

**i yv** *h hh* = ⋅ (5)

( ) **v zi j** *h hh h* =⋅ + (6)

*nh n h*

**V VI V Y**

Besides, an active filter connected to the node *k* can be considered as a current source which injects a current *jhk* on the network. A passive filter can be also modelled by a current source which magnitude would represent the harmonic current that must be drawn by the filter. In this last case, the phase angle depends on the harmonic voltage at the coupling node. All the additional currents produced by the set of filters are intended to reduce the harmonic voltages at each busbar. The location of these filters is defined in a node list called *ListFilter*.

Assuming a given number of filters, the harmonic voltages are deduced from (6):

*jh* is the vector of the filtering current and *zh = yh-1* the equivalent impedance matrix.

the harmonic voltages, while checking if a filtering solution is existing or not.

different is the objective function, as will be argued further in the subsection 3.2.2.

The aim of the current optimisation problem is to minimise the total filtering power to connect to the grid while the voltage standards are met for each nodal harmonic voltages. Prior to the search for the optimal current to be injected, it is necessary to answer a first question about the possibility to obtain a current vector *jh* able to reduce the voltages within the limiting levels *vlimit* at each node. For this reason, a double SA process is applied, as illustrated by the complete flow chart of figure 2. The purpose of the first one is to minimise

In cases where no filtering solution can be found, the simulated voltage annealing is fully processed and returns the total filtering current *jtotal* able to check a minimum gap between the nodal harmonic voltages and their expected limits. Instead, when the voltage constraints are properly respected, the voltage annealing is partially processed. Once a local solution is achieved, the procedure is switched to a simulated current annealing which purpose is to minimise the total filtering current *jtotal* that meets the voltage requirements. The search space is actually the same regardless of the nature of the SA process; what is however

When thinking about any optimisation problem, one of the first considerations is to start with a suitable set of initial parameters in order to ensure a good algorithm performance, which means the quality of the solution returned and a reasonable computation time [15]. The initialisation procedure deals first with the definition of the problem's data linked to the specific application. It consists herein of the following input data:

The initialisation procedure also starts with the parameters settings for the SA process, i.e. the cooling schedule that consists of the initial temperature and the rules for lowering it as the search for the optimal solution progresses. Each of these specific features will be introduced and explained, regarding the current issue about the optimal placement and sizing of harmonic filters in electrical systems.

• Starting temperature

In a typical SA process, the initial temperature is set sufficiently high to allow a move to almost any neighbourhood state. However, if the temperature starts at a too high value, the search may move to any neighbour and thus transform it into a random search, at least in the early stages. At the moment, there is no known method for finding a suitable starting temperature for a whole range of problems. An idea is to use the information on the cost function difference between one neighbour and another one to calculate a correct initial temperature. Another method suggested in [13] is to rapidly heat the system until a certain proportion of worse solutions are accepted and then slow cooling can start.

In the current application where a double SA process is considered, two initial temperatures for the voltage and the current annealing respectively are set as defined in (7) and (8). The both relationships mean that a 60% tolerance is allowed as regards the acceptance of an unfavourable harmonic voltage equal to twice the required limits and a jump of 30% in the amplitude of the average current is accepted with a same 60% tolerance.

$$Tv = -\frac{2 \times \max(\mathbf{v}\_{\text{limit}})}{\ln(0.6)}\tag{7}$$

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 31

The temperature decreases during the search according to a function known as the cooling (or annealing) schedule. The way in which the temperature declines is critical to the success of the algorithm. Theory states that the number of iterations to execute at each temperature

In the literature, several theoretical and empirical control schemes are suggested [16-19] and can be categorised into classes such as monotonic schedules, geometric schedules, quadratic schedules and adaptive cooling schedules. Actually, many attempts have been made to derive or suggest good annealing schedules. Several comparative studies on the large

In a conventional SA process, the way to decrement the temperature is a simple linear method. By declining the temperature constantly, it provides the search with a higher transition probability in the beginning of the search and lower probability towards the end

The proposed annealing procedure tested on the minimum power sizing scheduling problem involves the above cooling schedule known as an exponential schedule (9), with a

The final decision to make is the number of iterations to consider at each temperature. A constant number seems to be an obvious scheme. An alternative is to dynamically change this number as the algorithm progresses. At lower temperature, it is advisable to set a large value so that the local optimum can be fully explored. At higher temperatures, the number

In the current issue, the number of steps *Nsteps* was set in connection to the number of

where *n* is a suitably large integer (*n* ≅ 200), defined to achieve the thermal equilibrium prior

equal to 0.95 in order to compromise between computation time and

should lie between [0.8 – 0.99], with better results found in the

α

( ) 0 *t Tt T* = ⋅*a* (9)

*N nN steps* =⋅ + ( 2) *filters* (10)

α (0 < α

, the longer it will take to

< 1). Then,

should be large enough to reach the thermal equilibrium.

variety of proposed cooling strategies are discussed in [20-22].

higher end of the range. However, the higher the value of

α

decrement the temperature to the stopping criterion.

of the search. An alternative is a geometric decrease by a constant factor

where *t* (for 'time') is the step count and *T0* is the starting temperature.

possible harmonic filters *Nfilters* to connect to the grid, as given in (10):

• Cooling schedule

experience shows that

α

of iterations can be less.

to a next temperature change.

optimisation performance.

• Iterations at each temperature

coefficient

$$T\_i = -\frac{0.3 \times \sum\_{q \in ListFilter} \left| \dot{h}^{hq} \right|}{\ln(0.6)}\tag{8}$$

• Final temperature

It is usual to let the temperature decrease until it reaches zero. However, this can make the algorithm run for too long, which should be a major drawback. In practice, the stopping criterion is a suitably low temperature, since the chances of accepting a worse move are almost the same when the temperature is null. In other words, the stopping criterion is met when the system is frozen at the current temperature; that is, no better or worse moves are being accepted.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 31

• Cooling schedule

30 Simulated Annealing – Single and Multiple Objective Problems

each node where a filter is connected,

sizing of harmonic filters in electrical systems.

The initialisation procedure also starts with the parameters settings for the SA process, i.e. the cooling schedule that consists of the initial temperature and the rules for lowering it as the search for the optimal solution progresses. Each of these specific features will be introduced and explained, regarding the current issue about the optimal placement and

In a typical SA process, the initial temperature is set sufficiently high to allow a move to almost any neighbourhood state. However, if the temperature starts at a too high value, the search may move to any neighbour and thus transform it into a random search, at least in the early stages. At the moment, there is no known method for finding a suitable starting temperature for a whole range of problems. An idea is to use the information on the cost function difference between one neighbour and another one to calculate a correct initial temperature. Another method suggested in [13] is to rapidly heat the system until a certain

In the current application where a double SA process is considered, two initial temperatures for the voltage and the current annealing respectively are set as defined in (7) and (8). The both relationships mean that a 60% tolerance is allowed as regards the acceptance of an unfavourable harmonic voltage equal to twice the required limits and a jump of 30% in the

> 2 max( ) limit ln(0.6)

0 0.3

*hq*

*j*

ln(0.6)

It is usual to let the temperature decrease until it reaches zero. However, this can make the algorithm run for too long, which should be a major drawback. In practice, the stopping criterion is a suitably low temperature, since the chances of accepting a worse move are almost the same when the temperature is null. In other words, the stopping criterion is met when the system is frozen at the current temperature; that is, no better or worse moves are

*q ListFilter*

å

*Tv* ´ = - **v** (7)

(8)

proportion of worse solutions are accepted and then slow cooling can start.

amplitude of the average current is accepted with a same 60% tolerance.

*i*

*T* Î

= -

´

• Starting temperature

• Final temperature

being accepted.

The temperature decreases during the search according to a function known as the cooling (or annealing) schedule. The way in which the temperature declines is critical to the success of the algorithm. Theory states that the number of iterations to execute at each temperature should be large enough to reach the thermal equilibrium.

In the literature, several theoretical and empirical control schemes are suggested [16-19] and can be categorised into classes such as monotonic schedules, geometric schedules, quadratic schedules and adaptive cooling schedules. Actually, many attempts have been made to derive or suggest good annealing schedules. Several comparative studies on the large variety of proposed cooling strategies are discussed in [20-22].

In a conventional SA process, the way to decrement the temperature is a simple linear method. By declining the temperature constantly, it provides the search with a higher transition probability in the beginning of the search and lower probability towards the end of the search. An alternative is a geometric decrease by a constant factor α (0 < α < 1). Then, experience shows that α should lie between [0.8 – 0.99], with better results found in the higher end of the range. However, the higher the value of α, the longer it will take to decrement the temperature to the stopping criterion.

The proposed annealing procedure tested on the minimum power sizing scheduling problem involves the above cooling schedule known as an exponential schedule (9), with a coefficient α equal to 0.95 in order to compromise between computation time and optimisation performance.

$$T(t) = T \circ \alpha^t \tag{9}$$

where *t* (for 'time') is the step count and *T0* is the starting temperature.

• Iterations at each temperature

The final decision to make is the number of iterations to consider at each temperature. A constant number seems to be an obvious scheme. An alternative is to dynamically change this number as the algorithm progresses. At lower temperature, it is advisable to set a large value so that the local optimum can be fully explored. At higher temperatures, the number of iterations can be less.

In the current issue, the number of steps *Nsteps* was set in connection to the number of possible harmonic filters *Nfilters* to connect to the grid, as given in (10):

$$N\_{\rm str\*} = n \cdot (N\_{\rm filters} + \mathcal{Z}) \tag{10}$$

where *n* is a suitably large integer (*n* ≅ 200), defined to achieve the thermal equilibrium prior to a next temperature change.