ABSTRACT
Thermoelectric coolers (TECs) are applied in a wide range of thermal energy systems� This is due to their superior features where no refrigerant and dynamic parts are needed� TECs generate no electrical or acoustical noise and are environmentally friendly� Over the past decades, many research studies were conducted to improve the efficiency of TECs by enhancing the material parameters and design parameters� The material parameters are the most significant, but they are restricted by currently available materials and module fabricating technologies� Therefore, the main objective of TEC design is to determine a set of design parameters for the purpose of enhancing its performance� Two elements that play an important role when considering the suitability of TEC in applications are: cooling rate and coefficient of performance (COP)� In this chapter, the technical issues of single-stage TECs (STECs) and two-stage TECs (TTECs) are discussed� Consequently, two methods for optimizing the dimensions of the TEC (simulated annealing [SA] and differential evolution [DE]) are introduced and applied to search for the optimal design parameters of both TEC types�
Measurement while drilling (MWD) is a system developed to perform drillingrelated measurements downhole and transmit information to the surface while drilling a well (Scherbatskoy, 1982)� A typical structure of the high-temperature MWD tool is shown in Figure 1�1�
An MWD system can measure several variables such as natural gamma ray, directional survey, tool face, borehole pressure, temperature, vibration, shock, and torque� Maintaining optimal payload temperatures in a typical downhole environment of 230°C requires the MWD cooling system to be capable of pumping a significant load� This requires a low thermal resistance path on the heat rejection side (or hot side)� The application in an extreme environment of high temperature, high pressure, mechanical shock, and vibration requires the use of high temperature TEC materials and assemblies� Cooling of electronic components inside the MWD housing is crucial for maintaining optimal operating conditions� It has been identified that this can be accomplished using thin-film TEC devices�
TEC is a solid-state cooling device that uses the Peltier effect through p-type and n-type semiconductor elements (unlike vapor-cycle-based refrigerators) (Goldsmid, 2009)� TEC is used to convert electrical energy into a temperature gradient� TEC uses
no refrigerant and has no dynamic parts, which make these devices highly reliable and require low maintenance� TECs generate no electrical or acoustical noise and are ecologically clean� TECs are compact in terms of size, are lightweight, and have high precision in temperature control� However, for this application, the most attractive feature of the TECs is that they have the capacity for cooling instruments such as MWD under extreme physical conditions� TECs can be single stage or multistages� Figure 1�2 shows STEC and TTEC�
The commercially available STECs, which can be seen in Figure 1�2, can produce a maximum temperature difference of about 60-70 K when the hot side remains at room temperature (Goldsmid, 2009)� Nevertheless, when a large temperature difference is required in some special applications, the STEC would not be qualified for implementation� To enlarge the maximum temperature difference of TECs, another type of TEC which is a TTEC (Figure 1�2b) or a multistage TEC can be used (Karimi, Culham, & Kazerouni, 2011)� Thermoelectric modules generally work with two heat sinks attached to their hot and cold sides in order to enhance heat transfer and system performance� The application of TECs has been partitioned by their relatively low energy conversion efficiency and ability to dissipate only a limited amount of heat flux� The two parameters that play a crucial role in characterization of TECs are the maximum cooling rate and the maximum COP� TECs operate at about 5-10% of Carnot cycle COP, whereas compressor-based refrigerators normally operate at more than 30%�
Metaheuristics are intelligent techniques that can be used for engineering design optimizations (Deb, 2001)� Until now, Genetic Algorithm (GA) (Davis, 1991) and the extended version of GA, which is Non-dominated Sorting Genetic Algorithm (NSGA-II) (Deb, Pratap, Agarwal, & Meyarivan, 2002; Zhang & Hansen, 2009), are two metaheuristic optimization techniques which have been applied for optimizing
the performance of both types of TECs� However, their behavior and search capability have not been analyzed deeply�
The need for TECs to be utilized as a cooling mechanism for instruments in extreme environments especially in thermal energy and gas drilling operations could be overemphasized� However, setbacks such as the relatively low energy conversion efficiency and the ability to dissipate only a limited amount of heat flux may seriously injure the performance and efficiency of these devices�
The performance evaluation of the applied metaheuristic such as GA and NSGA-II in TECs are limited� Hence, such issues need to be investigated and analyzed in more detail based on some defined criteria such as measuring the stability, reliability, robustness, and so on�
Similar sophisticated techniques in metaheuristics such as SA (Van Laarhoven & Aarts, 1987; Rutenbar, 1989; Wah & Chen, 2000) and other evolutionary algorithms like GA (Davis, 1991) and DE (Storn & Price, 1995; Mezura-Montes, Velázquez-Reyes, & Coello Coello, 2006; Ali, Pant, & Abraham, 2009) are used in their pure and hybridized form to enhance the effectiveness of the TEC model� Besides taking into account the large number of variables, these techniques optimize the model while considering the complex physical and geometrical properties of the TEC�
Although DE is in the same family of algorithms as GA, it has been claimed to have better performance as compared to GA in many applications� SA has the capability to escape from the local optimas to obtain global optimality� Therefore, in this chapter, metaheuristic techniques SA and DE are applied together with the STEC and TTEC models on a MATLAB® platform� In that system, the objectives are the cooling rate and COP� They are optimized individually or simultaneously using single-objective optimization (SOO) or multiobjective optimization (MOO) methods, respectively� The design variables, their constraints, and other fixed parameters need to be identified� The performance of each technique is strategically evaluated based on the simulation results�
The main drawbacks of TECs are poor COP value and cooling rate� These issues could be improved individually or simultaneously� The parameters in the formulation of cooling rate and COP can be classified into three categories: specifications, material properties, and design parameters (Rowe, 1995)� The specifications are: the operating temperatures (Tc and Th), the required output voltage (V), current (I), and power output (P)� The specifications are usually provided by customers depending on the requirement of a particular application�
The material parameters are restricted by current materials and module fabricating technologies� Due to the effectiveness of material properties on the performance of TECs, many research efforts were directed during the past 10 years toward finding a new material/structure for usage in highly efficient cooling applications and energy conversion systems� Bismuth telluride (Bi2Te3) is one of the best thermoelectric materials with the highest value figure of merit (Yamashita & Sugihara, 2005)� Much effort has been made to raise the ZT value (which is the measure of a
material’s capability to efficiently generate thermoelectric power) of bulk materials based on Bi2Te3 by doping or alloying other elements in various fabricating processes� However, in these efforts, it was found that the ZT value was not more than 1� Thus, the ZT value was not sufficient to improve the cooling efficiency significantly� The reason is the difficulty in increasing the electrical conductivity or Seebeck coefficient without increasing the thermal conductivity (Rodgers, 2008)� Recent advancements in improving ZT values include the work of Poudel et al� (2008), who achieved a peak ZT of 1�4 at 100°C from a bismuth antimony telluride (BiSbTe) p-type nanocrystalline bulk alloy� This material is an alloy of Bi2Te3 and is made by hot pressing nanopowders which are ball-milled from crystalline ingots� The ZT value of this alloy is about 1�2 at room temperature and peaks at about 1�4 at 100°C, which makes these materials very useful for microprocessor cooling applications� Consequently, the main objective of the TEC design is to determine a set of design parameters which meet the required specifications or create the best performance at a minimum cost� Table 1�1 shows the optimization techniques which have been applied in the optimization of TECs�
TABLE 1.1 Optimization Techniques Implemented for TEC Improvement
Using SOO, Cheng and Lin (2005) were the first authors implementing metaheuristics (GA) to the STEC model for finding optimal geometric designs� The geometric properties of semiconductor elements are: the total number of units, the length of the unit, and the unit cross-sectional area� These variables were optimized simultaneously to reach the maximum cooling rate under few constraint conditions� Three case studies were performed under various applied currents (I) and various temperature differences (ΔT) to evaluate the effectiveness of the search for generating better design data as compared to the available design on the market� The GA algorithm was described and applied in the TEC model� However, there were many parameters in GA (such as total number of generations, crossover, or mutation coefficient) that needed to be defined� In another study (Cheng & Shih, 2006), these authors also used GA for maximizing separately the cooling rate and the COP but for the TTEC model� In this research, two different types of TTEC which had different types of applied current were considered� Unlike the geometric design of STEC, different types of design parameters of TTEC were used as variables� They supplied current to the hot stage and cold stage of all the TTEC units� By considering the effect of thermal resistance, the mathematical modeling of TTEC was more complicated than that of STEC (Xuan, Ng, Yap, & Chua, 2002)� GA was then applied successfully to solve the defined problems of TTEC� Similar to Cheng and Shih (2006), the parameter selection of GA was not described� The authors claimed that GA had a robust behavior and effective search ability� However, the robustness of an algorithm is based on the optimal capability of handling diverse problems via the changing of parameter settings� Therefore, optimizing the design variables with respect to the maximum cooling rate (and comparing against analytical results) is not sufficient to obtain a clear picture on the robustness of the algorithm�
Huang et al� (2013) developed an optimization approach which integrates a complete multiphysics STEC model and a simplified conjugate-gradient method (nonmetaheuristic)� Similar to Cheng and Shih (2006), the geometric properties of the STEC were optimized to reach the maximum cooling rate under a wide range of operating conditions (temperature differences and applied current variations)� The effects of applied current variations and temperature differences on the optimal geometry are discussed� The author came up with a new better design than the initial one by using the conjugate-gradient method� However, the performance analysis of the conjugategradient technique applied to the STEC model was not complied� And the combined optimization was proven effective only for the miniature TEC with the base area of 9 mm2, which is smaller than the design of Cheng and Lin (2005) with 100 mm2�
By using MOO, Nain et al� (2010a) used another version of GA, which is the NSGA-II for solving the STEC design problem� The values of the geometric properties of the STEC were optimized to achieve Pareto-optimal solutions of cooling rate and COP values at different thermal resistances� The benefits of obtaining the Pareto-frontier is that the balance between cooling rate and COP may be achieved� This benefits the designer during the selection of suitable STEC design in practical applications� However, the obtained results are not reliable since there exists a significant difference in the range of optimal design values between the results obtained using SOO and MOO� This is caused by the instability of the algorithm� In that work, the parameters of NSGA-II algorithm were chosen heuristically�
Venkata Rao and Patel (2013) used teaching-learning-based optimization (TLBO) for optimizing their design parameters of two types of TTEC� TLBO is based on the effects of the influence of a teacher on the output of learners in a classroom setting� In Venkata Rao and Patel (2013), TLBO was modified to increase the exploration and exploitation capacity� The modified TLBO was then applied effectively to maximize the cooling rate and COP simultaneously� The determination of the total number of semiconductor elements as well as the supply current to the hot stage and the cold stage were considered as search variables� This research has improved the ability of the TLBO and the modified TLBO as compared to the GA and PSO approaches by evaluating the convergence rate and the computational time� In addition, the obtained results were analyzed using the Wilcoxon signed-rank test� However, the presentation of parameter selection for TLBO and its modified version was not conducted� The main important parameters such as number of generations, population, and teaching factor need to be indicated clearly in the selection of the values used in optimizing the TTEC�
Operation of TEC is based on the Peltier effect� TEC acts like a solid-state cooling device that can pump heat from one junction to the other junction when a direct current (DC) is applied (Cheng & Lin, 2005)� The energy balance equations at the hot junction and the cold junction for TEC can be described as in Equations 1�1 and 1�2�
( ) = α − ρ + −
κ −
Q N IT I
L A
r
A A T T
L 1 2
(1�1)
( ) = α + ρ +
−
κ −
Q N IT I
L A
r
A A T T
L 1 2
(1�2)
where Qc is the cooling rate that will be absorbed from the cold side of TEC and Qh is the heat rejection that will be released to the environment� Equations 1�1 and 1�2 show the completion between the Seebeck coefficient term, which is responsible for TEC cooling� Besides the parasitic effect of heating, back-heat conduction from the electrical resistance and thermal conductance terms are also represented, respectively� The heat flow αITh and αITc caused by the Peltier effect are absorbed at the cold junction and released from the hot junction� Joule heating ρ + I
L A
r
A 1 2
flow of electrical current through the material is generated on both sides of the TEC semiconductor couples and at the contact surfaces between the TEC semiconductor couples and the two substrates (Cheng & Lin, 2005)� The TEC is operated between temperatures Tc and Th such that the heat conduction ( )κ −A T Th c occurs through the TEC semiconductor couples� The input electrical power P and COP are calculated using Equations 1�3 and 1�4�
= −P Q Qh c (1�3)
=
−
Q Q QCOP
where α, ρr, and κ are the material properties of a thermoelectric couple� Every thermoelectric couple includes one n-type and one p-type semiconductor element which have their own material properties (αn, ρn, κn or αn, ρn, κn)� They represent the thermoelectric material properties� A, L, and N are the cross-sectional area, length, and total number of semiconductor elements, respectively� They represent the geometric properties of the TEC model� COP is a common metric used to quantify the effectiveness of a heat engine� It is also important to quantify the amount of heat that a TEC can transfer and the maximum differential across the TEC� For a STEC, the COP is between 0�3 and 0�7�
As can be seen in Equations 1�1 through 1�4, the important values of the TEC are impacted by three types of parameters� They are operating condition parameters such as temperature difference (ΔT), applied current (I), geometric properties (A, L, N), and finally the material properties (α, ρr, κ)� Based on the work of Huang et al� (2013), the authors claimed that with a fixed ΔT, cooling rate and COP are first increased and then decreased as the supplied current is increased� Unfortunately, with the same supplied current, a maximum cooling rate and a maximum COP value cannot be always reached simultaneously� Similarly, the optimal cooling rate and COP could not be consistently achieved by varying the TEC geometry�
From Equations 1�1 and 1�2, the maximum cooling rate increases with the decrement of the semiconductor length until it reaches a maximum� Then it decreases with a further reduction in the thermo-element length (Huang et al�, 2013)� The COP increases with an increase in thermo-element length� As the COP increases with the semiconductor area, the cooling rate may decrease because the total available volume is limited� As the semiconductor area is reduced, the cooling rate generally increases� A smaller semiconductor area and a greater number of them yield greater cooling capacity� When the semiconductor length is below than this lower bound, the cooling capacity declines enormously (Rowe & Min, 1996)� Other elements do affect the performance of the TEC (e�g�, contact resistance) but since it is very small in most calculations, it can be neglected�
The performance of TECs strongly depends on thermoelectric materials (Huang et al�, 2013)� A good thermoelectric material should have a large Seebeck coefficient to get the greatest possible temperature difference per given amount of electrical potential (voltage)� In addition, low electrical resistance to minimize the Joule heating (Yamashita & Tomiyoshi, 2004) and low thermal conductivity to reduce the conduction from the hot side and back to the cold side are also crucial� Pure metal has a low Seebeck coefficient which leads to low thermal conductivity, whereas in insulators electrical resistivity is low which leads to higher Joule heating� The
performance evaluation index of thermoelectric materials is the figure of merit (Z) or the dimensionless figure of merit (ZT = α2T/ρκ) which combines the aforementioned properties� The increase in Z or ZT leads directly to the improvement in the cooling efficiencies of Peltier modules� The material properties are considered to be dependent on the average temperature of the cold side and hot side of each stage� Their values can be calculated using Equations 1�5 and 1�6 (Cheng & Lin, 2005):
( )α = −α = − + − × α = α − α
−T T263, 38 2.78 0.00406 10
;
( )ρ = ρ = − + × ρ = ρ + ρ
−T T22, 39 0.13 0.00030625 10
;
For TTEC, there are three types which have three typical approaches to supply the electric current to each stage: in series, in parallel, and separated (Cheng & Shih, 2006)� When a STEC is not able to work within the required temperature difference for a specific application, it is possible to use multistage configurations to extend the temperature difference (Enescu & Virjoghe, 2014)� In the research of Enescu and Virjoghe (2014), mathematical modeling of TTEC was presented as follows:
( )=
+ α − − −
Q
N r
I T I R K T T 1
1 2c,c
(1�7)
( )=
+ α + − −
Q
N r
I T I R K T T 1
1 2c,h
(1�8)
( )=
+ α − − −
Q
N r r
I T I R K T T 1
1 2h,c
(1�9)
( )=
+ α + − −
Q
N r r
I T I R K T T 1
1 2h,h
(1�10)
where Qc,c is the cooling capacity at the cold side of the colder stage� Qc,h is the released heat rate at the hot side of the colder stage� Qh,c is the cooling capacity at the cold side of the hotter stage, and Qh,h is the release heat rate at the hot side of the hotter stage� Nt is the total number of semiconductor elements which were put inside TTEC� r is the ratio between the number of semiconductor elements of the hot stage and the number of semiconductor elements of the cold stage� Ic and Ih are the applied current to the colder stage and hotter stage, respectively� Tc,c and Tc,h represent the cold-and hot-side temperatures of the colder stage� Th,c and Th,h represent the coldand hot-side temperatures of the hotter stage while αh, Rh, and Kh are the Seebeck
coefficient, the electrical resistance, and the thermal conductance of the colder stage, respectively� αc, Rc, and Kc are the Seebeck coefficient, the electrical resistance, and the thermal conductance of the colder stage, respectively� The properties of the thermoelectric material of each stage i can be determined as follows:
( )α = α − αi i i T,p ,n i ,ave (1�11)
( ) =
ρ + ρ R
(1�12)
( )= κ + κK Gi i i T,p ,n i ,ave (1�13) The subscripts p and n indicate the properties of the p-type and n-type semiconductors� ρ and κ are the electrical resistivity and thermal conductivity of the material, respectively� The parameter G is the structure parameter of thermocouples which indicated the ratio of cross-sectional area to the length of thermoelectric modules� The material properties are considered to depend on the average temperature Ti,ave of the cold-side temperatures of each stage and their values can be calculated using the following equations (i = c and h) (Venkata Rao & Patel, 2013):
( )α = −α = + − × −T T22224.0 930.6 0.9905 10i i i i,p ,n ,ave ,ave2 9 (1�14) ( )ρ = ρ = + + × −T T5112.0 163.4 0.6279 10i i i i,p ,n ,ave ,ave2 10 (1�15) ( )= = − + × −k k T T62605.0 277.7 0.4131 10i i i i,p ,n ,ave ,ave2 4 (1�16) Equation 1�17 presents the formulation of the COP of TTEC� Similar to STEC, COP is the ratio between the cooling capacity of the cold side and the electrical power consumption, P� It is also important to quantify the amount of heat that a TEC can transfer and the maximum differential across the TEC�
=
−
=
Q Q Q
Q P
(1�17)
Constriction and spreading resistances exist whenever heat flows from one region to another of different cross-sectional area� The term “constriction” is used to describe the situation where heat flows into a narrower region while “spreading resistance” describes the case where heat flows out of a narrow region into a larger cross-sectional area (Song, Au, & Moran, 1995)� The total thermal resistance RSt existing between the interfaces of TTEC is taken into consideration with this result and is given in Equation 1�18, where RScont and RSsprd are contact thermal resistance and spreading thermal resistance between the interfaces of two single stages of TTEC, respectively�
Based on the work of Song et al� (1995) and Venkata Rao and Patel (2013), the RScont and RSsprd are calculated in Equations 1�19 and 1�20:
= +RS RS RSt cont sprd (1�18)
=
+
RS RS aN
r
2 1
(1�19)
=
ψ × pi
RS k radsprd
In Equation 1�19, RSj is the joint resistance at the interface of two single stages of TTEC� The factor 2a represents the linear relationship between the cross-sectional area of the substrate and the thermo-element modules� In Equation 1�20, kh,s is the thermal conductivity of the substrate of the hot stage and radc,s is the equilibrium radius of the substrates of the cold stage� Detailed explanations related to radc,s are available in Song et al� (1995)� From there, the calculation can be expressed as follows:
=
(1�21)
The dimensionless value ψmax of Equation 1�20 is given in Equation 1�22� ε, τ and ϕ are the dimensionless parameters and are calculated in Equations 1�23 through 1�25� Sh,s is the substrate thickness of the hot stage and radh,s is the equilibrium radius of the substrate of the hot stage, respectively� Bi is the Biot number and its value is infinity (Bi = ∞) for the isothermal cold side of the hot stage� The dimensionless parameters λ and radh,s of Equations 1�26 and 1�27 are given by Cheng and Shih (2006)�
ψ = ε × τ
pi +
pi − ε φ1 (1 )max
(1�22)
ε =
r
(1�23)
τ =
S rad
h,s (1�24)
φ = λ × τ + λ
+ λ λ × τ
= λ × τ tanh( )
Bi 1
Bi tanh( )
tanh( )
(1�25)
=
pi +rad
(1�26)
λ = pi +
ε pi
(1�27)
The hot side of the cold stage and cold side of the hot stage are at the interface; so Qh,c = Qc,h, but due to the thermal resistance at the interface, the temperatures of both sides are not the same� The relationship between both these temperatures is given as follows (Cheng & Shih, 2006):
= + ×T T RS Qh,c c,h t c,h (1�28) Since Qh,c = Qc,h, we can obtain Equation 1�29 as follows:
( ) ( )
+ α + − −
= + α + − −
N r
I T I R K T T N r
I T I R K T T 1
1 2 1
1 2
(1�29)
By replacing Th,c via plugging Equation 1�28 into Equation 1�29, the hot-side temperature of the colder stage can be found as in Equation 1�30:
( ) ( )
=
+ α + + + −
− + α − − α
+ −
+ − α +
T I R K T r I RS N
r rK RS N
r r I R K T
I K r I RS N r
rK RS N r
r I K
1 2 1 1
1 1 2
1 1 1
(1�30)
The main objective of the TEC design is to determine a set of design parameters which yields the maximum cooling rate and/or maximum COP while meeting the required specifications at minimal cost� Based on previous works, the main parameters are classified into four groups for the optimization of the design of STEC or TTEC:
1� Group 1-Objective functions: The objective function is the maximum cooling rate and/or COP� For STEC, the formulations of cooling rate and COP are presented in Equations 1�1 and 1�4, respectively� For TTEC, the formulation of the cooling rate and COP are presented in Equations 1�7 and 1�17� These objectives can be optimized individually or simultaneously based on SOO or MOO strategies�
2� Group 2-Variables: For STEC, the design parameters are: a� Length of semiconductor element (L) b� Cross-sectional area of semiconductor element (A) c� Number of semiconductor elements (N)
For TTEC, the design parameters are: a� Supplied current to hot stage (Ih) and cold stage (Ic) b� Ratio of number of semiconductor elements (r) between hot stage and
cold stage 3� Group 3-Fixed parameters: For STEC, some fixed parameters need to be
determined as follows: a� Total volume in which STEC can be placed which is determined by
total cross-sectional area (S) and its height (H) b� Operating conditions such as applied current I and required tempera-
ture at hot side Th and cold side Tc of STEC� Then, the material properties are calculated based on Equations 1�5 and 1�6�
For TTEC, the fixed parameters are as follows: a� Total number of semiconductor elements (Nt) in both stages b� Operating conditions of the system, such as the required temperature at
hot side of the hot stage Th,h and cold side of the cold stage Tc,c 4� Group 4-Constraints: For both types of TEC, the constraints are: a� Boundary constraints on the design variables b� The requirement of satisfying a required value of COP (COP > COPmin)
and a limited value of the manufactured cost (cost < cost max)�
For ease of formulation and implementation, the penalty function approach needs to be used to convert the constrained problem into an unconstrained one to adjust the infeasible search space (Bryan & Shibberu, 2005)� Hence, for the SOO problem, the formulation of the objective function is presented in Equation 1�31:
∑⋅= +β =
F fmaximize Cooling rate
COP i
(1�31)
where F is an objective function, which consists of two parts: first is the objective function cooling rate of COP and the second is the penalty function (which contains the constraint violation function fi,constraint) and the coefficient of violation, β · β is set by the user and is normally chosen as a large value (e�g�, 1015)� During the search, fi,constraint will be aequal to 1 if the search variables do not satisfy one of the constraints, making the function F to blow up to infinity� If all the constraints are satisfied where all fi,constraint is equal to 0 and maximize F = maximize (cooling rate or COP), the search variables are within feasible ranges�
MOO has been defined as finding a vector of the decision variables while maximizing or minimizing several objectives simultaneously within some constraint conditions (Kim & De Weck, 2006)� In MOO, the definition of a performance is more complex than an SOO problem because the optimization goal itself consists of multiple target objectives� Therefore, a single desirable best solution with respect to all the objectives does not exist� However, a series of good solutions which are equally good and known as Pareto optimal solutions could be attained (Babu & Angira, 2003)� Multiobjective programming models are difficult to solve because of
the incoordination of the target vectors and the existence of constraints� There must be a dimensionless or a dimensionally unified treatment for solving MOO problems whose dimensions are not uniform� In this chapter, two main objective functions of the TEC model are the cooling rate (in Watts) and the COP (dimensionless unit)� They need to be optimized simultaneously by combining them into one dimensionless objective function� The weighted sum method or scalarization method is commonly used to solve MOO problems by combining its multiple objectives into one single-objective scalar function (Deb, 2014)� As can be seen in Equations 1�32 and 1�33, the weighted sum method is conducted by multiplying each objective function with a weighting factor and summing up all these weighted objective functions�
∑⋅= + +β
F w w fMaximize cooling rate(cooling rate) COP
1 (1�32)
+ =w w 11 2 (1�33)
w1 and w2 are the weighting factors for each objective function� The Pareto front is obtained by changing the w1 and w2 systemically to create respectively different optimal solutions� In MOO, the image of all optimal solutions is called the Pareto front� The shape of the Pareto front indicates the nature of the trade-off between the different objective functions (Deb, 2014)� The Pareto front contains the Pareto optimal set of solutions such that when going from any point to another in the set, at least one objective function improves and at least one other worsens (Babu & Angira, 2003)� The Pareto front divides the objective function space into two parts: one part contains nonoptimal solutions and another part contains the infeasible solutions� For nonlinear MOO, determining the entire continuous Pareto-optimal surface is practically impossible� However, finding a discrete set of Pareto-optimal points which approximates the true Pareto front is a realistic expectation (Boussaïd, Lepagnot, & Siarry, 2013)�
DE is one of the popular evolutionary metaheuristic algorithms like GA and PSO (Storn & Price, 1995)� DE is divided into ten different strategies� A strategy that works out to be the best for a given problem may not work well when applied for a different problem (Babu & Angira, 2003)� DE has been used widely to solve problems which are non-differentiable, noncontinuous, nonlinear, noisy, flat, multidimensional, and have many local minima� The optimization search of DE proceeds over three operators: mutation, crossover, and selection (Price, Storn, & Lampinen, 2006)� At the first stage, a population of candidate solutions for the optimization task to be solved is randomly initialized (Boussaïd et al�, 2013)� One target vector Xi iPr is randomly selected in the population� For every generation of the evolution process, new individuals are created by applying reproduction mechanisms which are the crossover and mutation operators� During mutation, the mutant vector Vi is generated
by combining three randomly selected vectors X X X, ,1aux 2aux 3aux from the population vector excluding target vector Xi iPr :
= + −V X F X X( )i 1aux 2aux 3aux (1�34)
where F is the mutation amplification factor which is a positive real number that controls the rate at which the population evolves� In the crossover process, DE performs a uniform crossover between the target vector Xi iPr and the mutant vector Vi to create a trial vector Xichild (see Equation 1�35)� The crossover probability, CR, which is used to control the fraction parameter values that are copied from the mutant vector, must be specified within the range [0,1]� To determine which source contributes a given vector, uniform crossover compares CR to the output of a uniform random number generator, rand(0,1)� If the random number is less than or equal to CR, the trial vector is inherited from the mutant vector� Otherwise, the vector is copied from the target vector:
=
≤
>
X V
X
if (rand(0,1) CR)
if (rand(0,1) CR) i
(1�35)
Finally, the fitness of the resulting solutions is evaluated in the selection process and the target vector of the population competes against a trial vector to determine which one will be retained into the next generation (shown in Equation 1�36)� The executional procedures of DE are the same as with GA except for the differences in the mutation type and reproduction mechanisms:
( ) ( ) =
<
+X X f X f X
X
if
otherwise i
(1�36)
Parameter selection of DE is shown in Table 1�2 using deterministic rules� DE is highly sensitive to the choice of scaling factor F� The bigger the value of F,
the higher the exploration capability of DE (Guo et al�, 2014)� A good initial guess is to choose F within the range of [0�5, 1]; for example, F = 0�85 would be a good
TABLE 1.2 Parameter Settings of the DE Algorithm
initial choice (Storn & Price, 1997)� It is said that the value of F smaller than 0�4 and greater than 1 is occasionally effective� To choose the suitable value for crossover probability CR, a bigger CR can increase the convergence speed of the algorithm but a smaller CR could increase the exploitation capability (Guo et al�, 2014)� The value of CR is chosen within the range of [0,1] to help maintain the diversity of the population� However, for most cases, it should be close to 1 (e�g�, CR > 0�9) (Storn & Price, 1997)� When CR is equal to 1, the number of trial solutions will be reduced dramatically� This may lead to search stagnation� Only separable problems do better with CR close to 0 as [0, 0�2] (Price et al�, 2006)� Choosing values for the number of population members, P, is not very critical� An initial guess (10D) is a good choice to obtain global optimum (Guo et al�, 2014), where D stands for a number of variables� Depending on the difficulty of the problem, the number of the population, P, can be lower than (10D) or higher than it to achieve convergence such as 5D to 10D, 3D to 8D, or 2D to 40D (Storn & Price, 1997)� In the stopping condition, the algorithm will stop if number of function evaluations exceeds its maximum value (e�g�, imax = 300). The DE algorithm (Ganesan, Elamvazuthi, Shaari, & Vasant, 2014) is described as follows while the flowchart of the DE algorithm is shown in Figure 1�3�
• Step 1: Initial parameter setting for STEC model and DE algorithm� For DE, set required parameters for the algorithm as in Table 1�2� For a TEC device, set required parameters such as fixed parameters and boundary constraints of the design variables; set all the constraints and apply them into the penalty function�
• Step 2: Randomly initialize the population vectors X0 = [A, L, N] for STEC or X0 = [Ih,Ic,r] for TTEC within the range of boundary constraints by using generated random numbers method�
• Step 3: Randomly select one target vector that can be called as principal parent Xi iPr and start the counter i = i + 1� • Step 4: Randomly select another three vectors that can be called as auxil-
iary parents X X X, , and1aux 2aux 3aux from the population vectors� • Step 5: Perform differential mutation and generate mutated vector Vi� The
mutation vector Vi is defined as ( )= + −V X F X Xi iaux 2aux 3aux � • Step 6: Perform crossover operation by recombining Vi with Xi iPr to generate
a trial vector Xichild� • Step 7: Perform selection by comparing the fitness value to choose the suit-
able vector among the trial vector Xichild and target vector Xi iPr � • Step 8: If the fitness criterion is satisfied and the number of iterations i ≥ imax,
stop the algorithm and print optimal solution, else continue to step 3�
SA is a method employed for solving both unconstrained and constrained optimization problems (Blum & Roli, 2008)� The method is based on the models related to the metallurgical process of heating a metal/material and then slowly lowering the temperature to decrease undesired properties� During this process, the system’s energy is minimized� The objective function of the problem similar to the energy of a material is then minimized by introducing a fictitious temperature which is a simple controllable parameter in the algorithm�
At each iteration of the SA algorithm, a new point xk+1 is randomly generated within the boundary constraints based on the current point xk� The distance of the new point from the current point, or the extent of the search is based on a probability distribution with a scale proportional to the temperature� The algorithm accepts all new points that lower the objective with a certain probability� Points that raise the objective values are evaluated based on the Metropolis calculation (Equation 1�38)�
( ) ( )≤+f x f xk k1
(1�37)
( ) −
−
>
+f x x k T
exp rand(0,1) k k
(1�38)
where Tn is the current annealing temperature and kB is Boltzmann annealing� By accepting points that raise the objective, the algorithm avoids being trapped in the
local minima and is thus able to explore globally for more possible solutions� An annealing schedule is selected to systematically decrease the temperature as the algorithm proceeds� As the temperature decreases, the algorithm reduces the extent of its search to converge to a minimum�
A programmed SA code was used and its parameters were adjusted so that it could be utilized for finding the optimal TEC design� Choosing good algorithm parameters is very important because it greatly affects the whole optimization process� Parameter settings of SA are listed in Table 1�3�
The initial temperature, T0 = 100, should be high enough such that in the first iteration of the algorithm, the probability of accepting a worse solution, is at least 80%� The temperature is the controlled parameter in SA and it is decreased gradually as the algorithm proceeds (Vasant & Barsoum, 2009)� Temperature reduction value α = 0�95 and temperature decrease function is:
= α −T Tn n 1 (1�39)
The numerical experimentation was done with different α values: 0�70, 0�75, 0�85, 0�90, and 0�95 (Abbasi, Niaki, Khalife, & Faize, 2011)� Boltzmann annealing factor, kB, is used in the Metropolis algorithm to calculate the acceptance probability of the points� Maximum number of runs, runmax = 250, determines the length of each temperature level T · accmax = 125 determines the maximum number of acceptance of a new solution point and rejmax = 125 determines the maximum number of rejection of a new solution point (runmax = accmax + rejmax) (Abbasi et al�, 2011)� The stopping criteria determine when the algorithm reaches the desired energy level� The desired or final stopping temperature is set as Tfinal = 10−10� The SA algorithm is described in the following section and the flowchart of SA algorithm is shown in Figure 1�4�
• Step 1: Set the initial parameters and create initial point of the design variables� For SA algorithm, determine required parameters for the algorithm as in Table 1�3� For TEC device, set required parameters such as fixed parameters and boundary constraints of the design variables, and set all the constraints and apply them into penalty function�
TABLE 1.3 Parameter Settings of SA Algorithm
• Step 2: X0 = [A0, L0, N0] for STEC or [Ih0, Ic0, r0] for TTEC-Initial randomly based point of design parameters within the boundary constraint by computer-generated random numbers method� Then, consider its fitness value as the best fitness so far�
• Step 3: Choose a random transition Δx and run = run + 1� • Step 4: Calculate the function value before transition Qc(x) = f (x)� • Step 5: Make the transition as x = x + Δx within the range of boundary
constraints� • Step 6: Calculate the function value after transition Qc(x+Δx) =f(x+Δx)� • Step 7: If Δf = f (x + Δx) − f(x) > 0 then accept the state x = x + Δx. • Step 8: Else if Δf = f (x + Δx) − f (x) ≤ 0, then generate a random number
rand in range (0, 1)� If e[ f(x + Δx) − f (x)]/kB·T > rand(0,1), then accept the state x = x + Δx and acc = acc + 1� Else return to the previous state x = x − Δx and rej = rej + 1�
• Step 9: If acc ≥ accmax or run ≥ runmax then continue to step 9� If not, return to step 2�
• Step 10: If the process meets the stopping conditions, stop running the SA algorithm, get the optimal value xbest and f (xbest)� Otherwise, update T based on temperature reduction function Tn = α·Tn−1 and return to step 2�
In this chapter, the SA and DE algorithms were presented and their application in two types of TEC models (single stage and two stage) was discussed� In the singlestage TEC, the design factors are the geometric parameters of the semiconductor element which are the dimensions of the semiconductor column and the total number of semiconductor units� In the two-stage TEC, the design parameters are the supplied current in the hot and cold stages along with the number of total semiconductor units� The cooling rate and the COP are the two important criteria employed to evaluate the performance of the STEC� They are considered as objective functions and are optimized separately or simultaneously by using a weighted method� Description of the DE and SA algorithms are presented together with the combination of TEC mathematical modelling� Parameter settings of the algorithms are referred from previous studies in combination with the author’s experience when doing programming in MATLAB� For future research, further simulation work could be conducted in MATLAB to get the results using metaheuristics� Different test cases could be executed in a systematic way as described in this chapter to evaluate the performance of every algorithm� The comparison between the practical experiments and simulation works would be necessary to analyze and justify the research claims� Developing a prototype for the optimized TEC design based on the optimization procedures highlighted in this chapter would be the ultimate goal of these research efforts�
