Optimization Genetic Algorithm for geometric constraint solving

Genetic Algorithm for Geometric Optimization of Thermo-electric Coolers and A novel geometric center design method for genetic algorithm optimization
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Published Date:26-08-2017
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Collaborative Simulated Annealing Genetic Algorithm for Geometric Optimization of Thermo-electric Coolers Doan V.K. Khanh, Pandian M. Vasant, Irraivan Elamvazuthi and Vo N. Dieu Abstract Thermo-electricCoolers(TECs)nowadaysareappliedinawiderangeof thermalenergysystems.Thisisduetoitssuperiorfeatureswherenorefrigerantand dynamic parts are needed. TECs generate no electrical or acoustical noise and are environment friendly. Over the past decades, many researches were employed 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 bycurrentlyavailablematerialsandmodulefabricatingtechnologies.Therefore,the main objective of TECs design is to determine a set of design parameters such as leg area, leg length, and the number of legs. Two elements that play an important role when considering the suitability of TECs in applications are rated of refrig- eration (ROR) and coefficient of performance (COP). In this chapter, the technical issues of TECs were discussed. After that, a new method of optimizing the dimension of TECs using collaborative simulated annealing genetic algorithm (CSAGA) to maximize the rate of refrigeration (ROR) was proposed. Equality constraint and inequality constraint were taken into consideration. The results of optimization obtained by using CSAGA were validated by comparing with those obtained by using stand-alone genetic algorithm and simulated annealing optimi- D.V.K. Khanh (&) Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Perak, Malaysia e-mail: kimkhanh2906gmail.com P.M. Vasant Department of Fundamental and Applied Sciences, UTP, Perak, Malaysia e-mail: pvasantgmail.com I. Elamvazuthi Department of Electrical & Electronic Engineering, UTP, Perak, Malaysia e-mail: irraivan_elamvazuthipetronas.com.my V.N. Dieu Department of Power Systems, HCMC University of Technology, Ho Chi Minh City, Vietnam e-mail: vndieugmail.com © Springer India 2016 155 S. Bhattacharyya et al. (eds.), Hybrid Soft Computing Approaches, Studies in Computational Intelligence 611, DOI 10.1007/978-81-322-2544-7_5156 D.V.K. Khanh et al. zation technique. This work revealed that CSAGA was more robust and more reliable than stand-alone genetic algorithm and simulated annealing. Keywords Thermo-electrics coolers Thermal energy system Rate of refriger-   ation Coefficient of performance Collaborative simulated annealing genetic   algorithm Geometric properties Material properties Genetic algorithm     Simulated annealing 1 Introduction Mud-Pulse High-Temperature (MWD) is a system developed to perform drilling-related measurements down-hole and transmit information to the surface while drilling a well 1. MWD systems can take several measurements like a natural gamma ray, directional survey, tool face, borehole pressure, temperature, vibration, shock, and torque.Maintaining optimal payload temperatures in atypical down-hole environment of 230 °C requires that the MWD cooling system is capableof pumping a significant load and requires alow thermal resistance path on the heat rejection (hot side). The application in the extreme environment of high temperature, high pressure, mechanical shock, and vibration requires the use of high-temperature TEC materials and assemblies. A typical High-Temperature MWD tool is shown in Fig. 1. Cooling of electronic components inside MWD housing is crucial for maintaining optimal operating conditions in the MWD. It has beenidentifiedthatthiscanbeaccomplishedusingthin-filmthermo-electriccooling devices. TECs are solid-state cooling devices that use the Peltier effect through p-type and n-type semiconductor elements (unlike vapor-cycle-based refrigerators) 2. These types of coolers are used to convert electrical energy into a temperature gradient. Thermo-electric coolers use no refrigerant and have no dynamic parts which make these devices highly reliable and require low maintenance. These coolers generate no electrical or acoustical noise and are ecologically clean. These coolers are compact in terms of size, light weight, and have high precision in temperaturecontrol.However,forthisapplication,themostattractivefeatureofthe Fig. 1 High-temperature MWD toolCollaborative Simulated Annealing Genetic Algorithm … 157 Fig. 2 Single-stage thermoelectric coolers (STECs) (Ferrotec) Fig. 3 Two-stage thermo-electric coolers (TTECs) (Ferrotec) TECs is that they have the capacity for cooling instruments such as MWDs under extreme physical conditions. TECs can be a single-stage or multi-stages type (Figs. 2 and 3). The commer- cially available single-stage TECs (STECs) can produce a maximum temperature difference of about 60–70 K when hot side remains at room temperature 2. Nevertheless, when a large temperature difference is required for some special applications, STECs will not be qualified. To enlarge the maximum temperature difference of TECs, we use two-stage TECs (TTECs) or multi-stages TEC. Thermo-electric modulegenerally works with two heat sinks attached to its hot and cold sides in order to enhance heat transfer and system performance. As mentioned previously, the application of TECs has been partitioned by their relatively low energy conversion efficiency and ability to dissipate only a limited amountofheatflux.TwoparametersplayacrucialroleincharacterizationofTECs are the maximum rate of refrigeration (ROR) and the maximum coefficient of performance (COP). Thermo-electric coolers operate at about 5–10 % of Carnot cycle COP whereas compressor-based refrigerators normally operates at more than 30 %. Several intelligent techniques that can be used for engineering design optimi- zations are discussed in 3. However, one of the most effective and non-traditional methods used as an optimization technique for TECs is the Non-dominated Sorting Genetic Algorithm (NSGA-II) 4. Similar sophisticated techniques in artificial intelligence, such as Simulated Annealing (SA) 5, other evolutionary algorithms (GA, Differential Evolution (DE) 6, and Particle Swarm Optimization (PSO) Poli158 D.V.K. Khanh et al. et al. 7, can be used in their pure and hybrid form to enhance the effectiveness of the optimization of TECs. To take into account the large number of variables (physical, as well as geometrical properties) and create more performance in the problem, hybridized techniques such as hybrid simulated annealing genetic algo- rithm can be used. This study focuses on optimizing the design parameters of single-stage TECs using collaborative simulated annealing genetic algorithm (CSAGA) optimization technique to create the maximum ROR under some defined constraints. The leg area, number of legs, and leg length of the thermo-electric module were optimized. 2 Optimization Matters in Designing TECs The main drawback of thermo-electric coolers is the poor coefficient of perfor- manceandlowROR.Theycanbeimprovedpersonallyorsimultaneously,fromthe parameters of the equation of TECs performance, we can group them into three categories which are specifications, material properties, and design parameter 8. ThespecificationistheoperatingtemperatureT and T ,therequiredoutputvoltage c h V, current I, and power output P. The specifications are usually provided by cus- tomers depending on the requirement of a particular application. The material parameters are restricted by currently materials and module fabricating technolo- gies. Consequently, the main objective of the TEC design was to determine a set of design parameters which meet the required specifications or create the best per- formance at minimum cost. 3 Geometric Optimization Table 1 lists some research in optimizing the geometric properties of TECs. In single-objective optimization problem, 8 combined a TEC model and a genetic algorithm to optimize the geometry and performance of the STECs. The geometric properties of STECs were considered as the search variables and were optimized Table 1 Previous optimization techniques applied in optimizing performance of TECs Type of optimizations Type of TECs Technique used Author/year SOP STECs GA Cheng/2005 SOP STECs Conjugate-gradient method Huang/2013 SOP STECs GA Nain/2010 SOP TTECs GA Cheng/2006 MOP STECs NSGA-II Nain/2010 MOP TTECs TLBO-II Rao/2013Collaborative Simulated Annealing Genetic Algorithm … 159 simultaneously to reach the maximum ROR under the requirement of minimum COP, the confined volume of STECs, and the restriction on the maximum cost of the material. The optimal search used GA converged so rapidly (around 20 iterations). Huangetal.9developedanoptimizationapproachwhichintegratesacomplete multi-physics TEC model and a simplified conjugate-gradient method. Under a wide range of operating conditions of temperature difference and applied current, the geometric properties of STECs as search variables were optimized to reach the maximum ROR. The effects of applied current and temperature difference in the optimal geometry were also discussed. For TTECs, 10 used GA for maximizing separately the ROR and COP. The author had considered the effect of thermal resistance and determined the optimum value of input current and number of legs for two different design configurations of TEC. The optimal search in this GA converges so rapidly with over 30 runs. These results were not different with those obtained from Xuan’s work 11 and showed that GA had a robust behavior and effective search ability. For multi-objective optimization problems (MOP), STECs will have a better designifwecanfindtheoptimalpointofRORandCOPsimultaneously.Nainetal. 12 used NSGA-II for multi-objective optimization of STECs. The value of geo- metric properties of STECs was optimized to achieve Pareto-optimal solutions at different values of thermal resistance. The authors point out the adverse effects of thermal resistance in obtaining the optimum value of cooling rate or COP. For TTECs, 13 used modified teaching–learning-based optimization (TLBO) inoptimizingthedimensionalstructureofTTECs.TLBOwasbasedontheeffectof theinfluenceofateacherontheoutputlearnersinaclass.Thealgorithmmimicsthe teaching–learning ability of teacher and learners in a classroom; the teacher and learners are the two vital components of the algorithm. TLBO was modified and applied successfully to the multi-objective optimization of TTECs with a better performance than GA. The determination of the number of TE module in hot stage and cold stage as well as the supply currentto the hotstage and the cold stage were considered as search variables. Two different configurations of TTECs, electrically separated and electrically connected in series, were investigated for the optimization. 4 Material Properties Optimization Matters As shown in the above part, a good thermoelectric material should have high Seebeck coefficient, high electrical conductivity, and low thermal conductivity. However,sincethesethreeparametersareinterrelated,followingtheWiedenmann– Franz law, researches have to optimize these conflicting parameters to get the maximize ZT. With the effectiveness of material properties on the performance of TEC, there have been conducted many research during the past ten years in finding a new160 D.V.K. Khanh et al. material and structure for use in green, highly efficient cooling, and energy con- version system. Bismuth–Telluride (Bi Te ) is one of the best thermo-electric 2 3 materials with the highest value figure of merit 14. Much effort has been made to raise ZT of bulk materials based on Bi Te by doping or alloying other elements in 2 3 various fabricating processes. However, ZT was not much more than one and are not sufficient to improve dramatically the cooling efficiency. The reason is due to the difficulty to increase the electrical conductivity or Seeback coefficient without increasing the thermal conductivity 15. Recent advancements in improving ZT values include the work of Poudel et al., who achieved a peak ZT by 1.4 at 100 °C from a bismuth antimony Telluride (BiSbTe) p-type Nano crystalline bulk alloy 16. This material is an alloy of Bi Te and is made by hot pressing Nano powders that are ball-milled from 2 3 crystalline ingots. ZT is about 1.2 at room temperature and peaks at about 1.4 at 100 °C, which makes these materials useful for microprocessor cooling applications. 5 Mathematical Modeling of Thermo-electric Coolers 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 DC current is applied. The energy balance equations at the hot junction and the cold junction for TEC can be described as in Eqs. 1, 2. ROR is the net rate of heat transfer in Watts. These equations show the completion between the Seebeck coefficient term, which is responsible for TEC cooling, and the parasitic effect of Joule heating and back heat conduction from the electrical resistance and thermal conductanceterms,respectively.Theheatflows αIT and αIT causedbythePeltier h c effect are absorbed at the cold junction and released from the hot junction, 2 respectively. Joule heating1/2I (ρL/A+2r /A)due to theflowof electrical current r c through the material is generated both inside the TEC legs and at the contact surfacesbetween theTEClegsand thetwo substrates8.TEC isoperated between temperatures T and T , so heat conduction κA(T − T ) occurs through the TEC c h h c legs.   1 L 2r kAðT T Þ c h c 2 ROR¼ N aIT  I q þ  ð1Þ c r 2 A A L   1 L 2r kAðT T Þ c h c 2 Q ¼ N aIT þ I q þ  : ð2Þ h h r 2 A A L The input electrical power and coefficient of performance (COP) can be calcu- lated using following relations (Eqs. 3–4):Collaborative Simulated Annealing Genetic Algorithm … 161 P¼ Q ROR ð3Þ h ROR COP ¼ : ð4Þ Q  ROR h α, ρ, k are Seebeck coefficient, electrical resistivity, and thermal conductivity of r TEelements,respectively.Theyrepresentforthermo-electricmaterialproperties. A, L, N are geometric properties of TEC model. COPisacommonmetricusedtoquantifytheeffectivenessofaheatengine.Itis also important to quantify the amount of heat that a TEC can transfer and the maximum differential across the TEC. For an STEC, basically COP is between 0.3 and0.7.TheCOPcanbegreaterthan1.0onlywhenthemoduleispumpingagainst the positive temperature gradient. 6 Relation Between COP and ROR For a TEC with a specific geometry, ROR and COP are all dependent on its operating conditions which are the temperature difference ΔT and applied current. With a fixed ΔT, ROR and COP are first increased and then decreased as I is increased 9. Unfortunately, with the same applied current, maximum ROR and maximum COP always cannot reach simultaneously. Similarly, with the same operating conditions, as the TEC geometry is varied, ROR and COP are all varied, but maybe cannot reach the maximums simultaneously 17. 7 Affections of Material Properties and Geometric Properties on TEC Performance The performance of TEC (COP and ROR) strongly depends on thermo-electric materials. A good thermo-electric material should have a large Seebeck coefficient to get the greatest possible temperature difference per given amount of electrical potential (voltage), low electrical resistance to minimize the Joule heating 18, and lowthermalconductivitytoreducetheconductionfromthehotsideandbacktothe cold side. Pure metal has a low Seebeck coefficient, which leads to low thermal conductivity, whereas in insulators electrical resistivity is low which lead to higher Joule heating. The performance evaluation index of thermo-electric materials is the figure of 2 merit Z or dimensionless figure of merit (ZT = α T/ρK), which combines the above properties. The increase in Z or ZT leads directly to the improvement in the cooling efficiency of Peltier modules.162 D.V.K. Khanh et al. The material properties are considered to be dependent on the average temper- ature of the cold side and hot side temperatures of each stage. Their values can be calculated from the following equations 8: 2 6 a ¼a ¼ð263;38þ2:78T 0:00406T Þ10 p n ave ave ð5Þ a¼a a ; p n 2 6 q ¼q ¼ð22;390:13T þ0:00030625T Þ10 ave p n ave ð6Þ q ¼q þq ; r p n 2 j ¼j ¼ 3:950:014T þ0:00001875T p p ave ave ð7Þ j¼j þj : n p From the Eqs. 1, 2, the geometric structure has remarkable effect on the TEC. The maximum ROR increases with the decrease of leg length until it reaches a maximum and then decreases with a further reduction in the thermo-element length 9. The COP increases with an increase in thermo-element length. As the COP increases with the leg area, the ROR may decrease because the total available volume is limited. As the leg area is reduced, the ROR generally increases. A smaller leg area and a greater number of legs yield greater cooling capacity. When the leg length is below than this lower bound, the cooling capacity declines enormously19.OtherelementshaveaffectionontheperformanceofTEClikethe contact resistance, but it is very small in some calculation it can be neglected. 8 Meta-heuristic Optimization Algorithm In the thermal energy sector, meta-heuristics have been used recently, to solve industrial problems as well as enhance current processes, equipment, and field operations. Table 2 lists some applications of meta-heuristic in thermal energy systems such as GA, SA, Particle Swarm Optimization, and Ant Colony Optimization. Meta-heuristics are widely recognized as efficient approaches for many hard optimization problems. A meta-heuristic is an algorithm designed to solve approximately a wide range of hard optimization problems without having to deeply adapt to each problem. Almost all meta-heuristics share the following characteristic: they are nature-inspired (based on some principles from physic, biology or ethnology) 20. A meta-heuristic will be successful on a given opti- mization problem if it can provide a balance between the exploration and exploi- tation 21. Roughly speaking, the basic single-solution-based meta-heuristics are more exploitation oriented, whereas basic population-based meta-heuristics are more exploration oriented. Exploration (diversification) is needed to identify the parts of the search space with high-quality solutions. Exploitation (intensification)Collaborative Simulated Annealing Genetic Algorithm … 163 Table 2 Recent application of intelligent strategies in thermal energy system Author/year Application Technique Xu and Wang 35 Optimal thermal models of building envelope GA based on frequency domain Gozde and Automatic generation control application in a PSO Taplamacioglu 36 thermal power system Kou et al. 37 Optimum thermal design of micro-channel heat SA sinks Pezzini et al. 38 Optimize energy efficiency ACO, PSO, GA, ES, EP Sharma 39 Optimization of thermal performance of a smooth PSO flat solar air heater Eynard et al. 40 Forecasting temperature and thermal power Wavelet and consumption ANN is important to intensify the search in some promising areas of the accumulated search experience 21. The main differences between existing meta-heuristics concern the particular way in which they try to achieve this balance. Categories of meta-heuristic are introduced in Fig. 4. SA is a point-based meta-heuristics which is normally started single initial solution and move away from it. GA is a population-based meta-heuristics which can deal with a set of solutionsratherthanwithasinglesolution.ThisresearchmainlyfocusesonSAand GA meta-heuristic techniques. 9 Genetic Algorithm Optimization Technique Genetic Algorithm (GA) is a method for solving both constrained and uncon- strained optimization problems that are a random searching based on the mecha- nism of natural selection and survival of the fittest 3. GA has repeatedly modified META-HEURISTICS Population Based Collaborative Meth- Point-based methods Methods ods Genetic Algorithm, Collaborative GA and Simulated Annealing, Particle Swarm Opti- SA, Collaborative Tabu Search, etc. mization, etc. PSO and SA, etc. Fig. 4 Categories of meta-heuristic164 D.V.K. Khanh et al. a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over succeeding generations, the population“evolves” toward an optimal solution. The three most important phases involved in GA are selected, crossover, and mutation: � Selection: Select the individuals, called parents that contribute to the population in the next generation. � Crossover: Combine two parents to form children for the next generation. � Mutation: Apply random changes to individual parents to form children. GAcanbeusedtosolveaconstrainedoptimizationproblemandcanfindagood local optimum solution 22. GA is simple and quick to execute 23. GA can be effectively applied in highly nonlinear problems 24 and solve a variety of opti- mizationproblemsbysearchingalargersolutionspace20.However,GArequires determination of optimum controlling parameters such as crossover rate and mutationrate. Moreover, GA has apoor global search capability. Flow chart of GA algorithm is shown in (Fig. 5). Fig. 5 Genetic algorithm Start flow chart Generate initial population randomly and initialize crossover and mutation probability Calculate the fitness value of each individual Selection Crossover No Mutation New generation Meet the stopping condition? Yes StartCollaborative Simulated Annealing Genetic Algorithm … 165 10 Simulated Annealing Optimization Technique Simulated Annealing (SA) is a method for solving unconstrained and bound-constrained optimization problems 25. The method models the physical process of heating a material and then slowly lowering the temperature to decrease defects, thus minimizing the system energy. 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 of the algorithm. At each iteration of the SA algorithm, a new point is randomly generated. The distanceofthenewpointfromthecurrentpoint,ortheextentofthesearch,isbasedon a probability distribution with a scale proportional to the temperature. The algorithm not only accepts all new points that lower the objective, but also, with a certain probability, accepts points that raise the objective. By accepting points that raise the objective, the algorithm avoids being trapped in local minima, and is able to explore globally for more possible solutions. An annealing schedule is selected to systemat- ically decrease the temperature as the algorithm proceeds. As the temperature decreases, the algorithm reduces the extent of its search to converge to a minimum. In contrary to GA, SA has the ability to escape from local optima 26, flex- ibility, and ability to approach global optimality. SA can be applied to large-scale problems regardless of the conditions of differentiability, continuity, and convexity those are normally required in conventional optimization methods 27. SA is easy to code even for complex systems and can deal with highly nonlinear models with many constraints. However, SA still suffers some disadvantages such as the diffi- culty in defining a good cooling schedule. Figure 6 presents the flow chart of SA. Fig. 6 Simulated annealing Start flow chart Choose an initial value for T and a random base point x Update T Max 1, expf (x) – f(x+s)/T great- er than random (0, 1) No No Yes Replace x with (x + s) Termination condition meet? Yes Stop166 D.V.K. Khanh et al. 11 Introduction of Collaborative Simulated Annealing Genetic Algorithm Traditional SA not only has strong global search ability in solving NP-hard prob- lem, but also has defects such as premature and weak local search ability. GA has stronglocalsearchabilityandnoprematureproblem.Therefore,thecombinationof GA and SA can overcome the defects of each of the two methods, bring into play their respective advantages, and improve the solving efficiency. Flow chart of CSAGA is shown in Fig. 7. The algorithmic statement of CSAGA has been combined with STECs mathematical modeling and is described as follows: � Step1:SettheinitialparametersofSTECsmodel,CSAGAalgorithmandcreate the initial point of design variables: – For STEC model,determinerequired initial parameters (T , T and I), setthe h c boundary constraint of the design variables A , L , N and A , min min min max L , N , nonlinear equality constraints as well. Consequently, the max max material properties of STEC are then calculated. Refer to “Parameter selection of STEC and CSAGA” for more details about choosing these parameters. – For SA, determine required parameters for the algorithm such as Initial annealing temperature T , Boltzmann annealing k , temperature reduction α, o B maximum number of iterations, stopping condition criteria: tolerance func- tion value and final stopping temperature. Refer to “Parameter selection of STEC and CSAGA” for more details about choosing these parameters. – Toimplementthealgorithm,first randomlyinitializing basedpointofdesign parameters x=A, L, N within the boundary constraint A , L , N min min min and A , L , N . max max max � Step 2: Choose a random transition Δx and count the number of iterations. � Step 3: Calculated the function value before transition Q = f(x). c(x) � Step 4: Make the transition as x = x + Δx within the range of boundary constraint. � Step 5: Calculate the function value after transition Q = f(x + Δx). c(x+Δx) � Step 6: If Δf = f(x + Δx) − f(x) 0 then accept the state x = x + Δx. � Step 7: Else If Δf = f(x) − f(x + Δx) ≤ 0 then generate a random number (0,1). ½fðxþDxÞfðxÞ=k :T B – If e is greater than random (0,1) then accept the state x = x + Δx. – Else then return to the previous state x = x – Δx. � Step 8: Check number of integration with maximum number of iterations. If the number of iterations meets, return to step 2. � Step 9: If the process meets the stopping condition, stop running the SA algorithm, get the optimal value and start to run GA. Otherwise, update T based on temperature reduction function T = α · T and return to step 2. n n-1Collaborative Simulated Annealing Genetic Algorithm … 167 Fig. 7 Flow chart of collaborative simulated annealing genetic algorithm � Step 10: GA optimization tool box in MAtlab is used. Set initial parameters of GA and create initial population points: – Population size, number of generations, crossover probability, mutation probability, and stopping condition values. Refer to“Parameter selection of STEC and CSAGA” for more details about choosing these parameters. – The initial population points A, L, N of STEC are created randomly at each iteration within the range of boundary constraint (upper bound and lower bound) by using Matlab random number generator. Number of initial168 D.V.K. Khanh et al. population points is limited by population size. Optimal value obtained from SA algorithm is put into initial population of GA. After this, the algorithm uses the individuals in the current generation to create the next population. � Step 11: Calculate the fitness value of each individual that includes optimal design variables obtained from SA algorithm. � Step 12: Perform genetic operators: – Selection: specify how the GA chooses parents for the next generation. – Crossover: specifies how the GA combines two individuals to form a crossover child for the next generation. – Mutation: specifies how the GA makes small random changes in the indi- viduals in the population to create mutation children. � Step 13: Get new generation with best offspring and fitness function. � Step 14: The GA will stop when one of the stopping criteria is met as follows: – Maximum number of generations – Average change in the fitness value less than defined value. Otherwise, return to step 11. 12 Optimization of STECs TECs can be single-stage of multi-stages. This work used single-stage TECs (STECs) and found good geometric properties which were the optimal leg length (L), the leg area (A), and the number of legs (N). Maximizing ROR or maximizing COP is an main important criterion to evaluate the performance of TECs. In this work, one objective function, namely maximizing ROR of STECs isconsidered for single-objectiveoptimization.Thedesignvariables(A, L, N)ofSTECsareputinan inequality constraint which is bound by upper and lower limits of the design variables and the total area S of STECs. Additionally, STECs are put in some requirements which are the confined volume of STECs (S), the minimum requirement of the COP, and the maximum cost of material 8. Because optimi- zationofTECsgeometrymaycausethereduction intheCOP,theCOPisusedasa constraint condition during the optimization in order to guarantee that the TECs with the optimal geometry have a relatively high COP 28. Referring to Eqs. 1–4, the parameters T , T , and I are defined in the beginning of the calculation. The c h unknown term is material properties of TECs, which will be determined based on the Eqs. 5–7 with the values of T and T . h cCollaborative Simulated Annealing Genetic Algorithm … 169 13 Test System Details To verify the effectiveness of meta-heuristic optimization techniques, simulations tests were carried out on STECs model to find the optimal value of geometric properties. The purpose of the tests should be related to the ultimate goal of meta-heuristic methods: fast, high-quality solutions to important problems. Single-objectiveoptimizationisusedunderconstraints.ParametersettingofSTECs system, the proposed techniques CSAGA, and stand-alone optimization technique GA, SA are chosen. After testing the performance of the proposed technique CSAGA, better design parameters of STECs have been explored by running the system under various operating conditions such as various input currents, various cold side temperatures with constraint condition of COP. 14 Parameters Selection of STECs and CSAGA Parameters of STECs are referred from Cheng’s work 8. Table 3 lists the parameters of STECs. STEC is placed in a confined volume with total area 2 100 mm and a height of 1 mm. The objective function is to maximize the ROR. The temperatures of the cold side stage and the hot side stage are both fixed to −8 2 323 K. The effect of electrical resistance r is considered with the value 10 Ωm . c Parameter selection of CSAGA is shown in Table 4. Initial temperature (T = 100), the temperature is the control parameter in simulated annealing and it is o decreased gradually as the algorithm proceeds 29. Temperature reduction (α = 0.95), temperature decrease is T = α · T . Experimentation is done with n n−1 different alpha values: 0.70, 0.75, 0.85, 0.90, and 0.95. Boltzmann annealing (k = 1), k will be used in the Metropolis algorithm to calculate the acceptance B B −6 probability of the points. Stopping criteria, the function tolerance is set as 10 and −10 final stopping temperature is set as 10 . This value can be obtained as a function of minimum possible deterioration the objective function. Table 3 Parameters setting Group Parameters setting Specific values of STECs 1 Objective function Maximize ROR 2 Variables 0.03 mm L1mm 2 2 0.09 mm A 100 mm 1 N 1000 2 3 Fixed parameters S = 100 mm T = T = 323 K h c −8 2 r =10 Ωm c 2 4 Constraints A.N 100 mm availability Maximum cost 385 Required COP = 0.75; 1170 D.V.K. Khanh et al. Table 4 Parameters setting of collaborative simulated annealing genetic algorithm (CSAGA) No. Parameters setting Specific values 1 Initial temperature T = 100 o 2 Temperature reduction α = 0.95 3 Boltzmann annealing k =1 B −10 4 Stopping criteria of SA Final stopping temperature 10 −6 Function tolerance 10 5 Population size 100 6 Fitness scaling function Fitness scaling rank 7 Selection function Selection tournament, 4 8 Crossover function Crossover arithmetic 9 Crossover fraction 0.6 10 Mutation function Mutation Adaptive Feasible −6 11 Stopping condition of GA Function tolerance 10 Maximum number of generations 10,000 For GA, the population size, the maximum number of generations must be determined. Generation specifies the maximum number of iterations the genetic algorithm performs. Population specifies how many individuals are in each gen- eration. Increasing the population size will increase the accuracy of GA 30in finding a global optimum but cause the algorithm to run slowly. A population size of 100 for 10,000 generations is run. Furthermore, the selection operator, crossover operator,andthemutationoperatorareappliedwithprobabilities,soastoconstruct a new set of solutions randomly. In selection process, option is selected by using selection function which has some options such as stochastic uniform, uniform or roulette, tournament. Tournament selection chooses each parent by choosing Tournament size four players at random and then choosing the best individual out of that set to be a parent. In crossover process, crossover arithmetic typewithfraction0.6ischosentocreatechildrenthatareweightarithmeticmeanof two parents 31 so that children are always feasible with respect linear constraints andbounds.Inmutationprocess,mutationadaptivefeasibleoptionischosenwhich satisfies the constraints. Finally, for stopping criteria, the function tolerance was set −6 at 10 or the algorithm will stop after getting over the maximum number of iterations 10,000. The optimization process terminated until one of these criteria is satisfied. 15 Robustness Test Robustness of the meta-heuristic method is evaluated by measuring the less sen- sitive capability to different types of applied system as TEC’s operating conditions (Barr et al. 1995), because the meta-heuristic method may not converge to an exactCollaborative Simulated Annealing Genetic Algorithm … 171 same solution at each random run. Therefore, their performances could not be judged by the results of a single run. Many trials should be done with some case studies of STEC’s system to reach a useful conclusion about the robustness of the algorithm 32. Based on Matlab 2013a platform, CSAGA was programmed and then run in 30 ® trialsonacomputer(CPU:Intel Core™i5-3470CPU3.2GHz3.2GHz; RAM: 4 GB DDR; OS: Windows 7) with three case studies as follows: � Casestudy1:BasedonthesettingofSTECsmodelinTable3,CSAGAistested under one nonlinear inequality constraint which is confined volume of STECs 2 S = 100 mm (A.N S). � Casestudy2:BasedonthesettingofSTECsmodelinTable3,CSAGAistested under one nonlinear equality constraint which is a limitation of COP = 1. � Casestudy3:BasedonthesettingofSTECsmodelinTable3,CSAGAistested under two above constraints which are nonlinear inequality constraint (A. N S) and nonlinear equality constraint COP = 0.75. To evaluate and compare the performance of the proposed technique, the stand-aloneoptimizationtechniqueswhichareGAandSAwerealsorunonthistest in the same operating condition of STECs model. The best value, average value, lowest value, and standard deviation of 30 trials of each technique were collected. After testing and collecting the data, CSAGA and GA can find the optimal dimension and satisfy the constraint in all three case studies; SA can solve the problemincasestudy1butgetstuckinsolvingtheproblemwithnonlinearequality constraint (case study 2). The comparison between maximum ROR obtained from theoptimizationtechniquewithanalyticalresultsrevealsthatCSAGA,GA,andSA perform well and give exactly the same value as analysis results. Figures 8, 9 and 10 show the graphs of best fitness values which are maximum ROR and maximum COPafter30trials.Table5showsthecollecteddataofthreecasestudies.Fromthe figures, the line created by 30 trial runs of CSAGA is more stable than by SA and GA. As shown in Table 5, the range between maximum and minimum values of Fig. 8 Case study 1—run STECs system using CSAGA, GA, and SA under nonlinear inequality 2 constraint A.N 100 mm172 D.V.K. Khanh et al. Fig. 9 Case study 2—run STECs system using CSAGA, GA, and SA under nonlinear equality constraint COP = 1 Fig.10 Casestudy3—runSTECssystemusingCSAGA,GA,andSAundernonlinearinequality 2 constraint A.N 100 mm and nonlinear equality constraint COP = 0.75 bestfitnessobtainedbyCSAGAapproachiscloserthanbySAandGAapproaches. In case study 1, the average value of maximum ROR obtained after 30 trial runs using CSAGA is 9.7874 W, 24.43 % higher than the obtained value by using GA (7.8654 W). In case study 2, the average value of maximum ROR obtained using CSAGA is 40.7097 W, increasing 38.83 % as compared to the GA approach (29.3253 W). In terms of standard deviation, CSAGA performs 0.0269 W for case study 1 and 0.5409 W for case study 2. The ranges of maximum value of ROR of CSAGA for case study 1 and case study 2 are (9.7874 ± 0.0269) and (7.916 ± 0.5409) W, respectively. As compared with other techniques, SA shows larger range (9.5893 ± 0.3681) W than CSAGA for case study 1; GA shows larger ranges (7.8654 ± 1.4389) and (7.8491 ± 1.5734) W than CSAGA for case study 1 and case study 2, respectively. A low standard deviation indicates that data points tend to be very close to the mean. These data demonstrate that the performance of CSAGA is more stable and more reliable when yields smaller range of maximum ROR than GA and SA. CSAGA has better robustness in solving optimization problem under constraints. CSAGA is more helpful for the designers to save their timeinfindingtheoptimaldesignparameters.ByrunningthealgorithmofCSAGA in only 1 time, the optimal design parameters of STECs can exactly better than other algorithms.Collaborative Simulated Annealing Genetic Algorithm … 173 Table 5 Test results after running system for 30 trials Technique used CSAGA SA GA Case study 1 Max ROR (W) Max ROR (W) Max ROR (W) Standard deviation 0.0269 0.3681 1.4389 Average value of best fitness 9.7874 9.5893 7.8654 Minimum value of best fitness 9.6486 8.663 4.932 Maximum value of best fitness 9.7956 9.795 9.7798 Technique used CSAGA SA GA Case study 2 Max ROR Max ROR (W) Max ROR (W) (W) Standard deviation 13.3064 SA cannot find optimal 12.6172 value Average value of best 40.7097 29.3253 fitness Minimum value of best 0.6868 2.7407 fitness Maximum value of best 45.4319 45.1371 fitness Technique used CSAGA SA GA Case study 3 Max ROR Max ROR (W) Max ROR (W) (W) Standard deviation 0.5409 SA cannot find optimal 1.5734 value Average value of best 7.916 7.8491 fitness Minimum value of best 6.907 0.0409 fitness Maximum value of best 9.1576 9.26 fitness 16 Computational Efficiency Computational efficiency of all methods is compared based on the average CPU time taken to converge the solution. The CPU time taken by each solution is given in Table 6.SA can perform well in case study 1 with smallest time-consuming. GA takes longer computational time, especially in case study 3, about 44.97 s which is double as compared with CSAGA approach. CSAGA solves the problem for three cases with the same time-consuming. It demonstrates the stability of CSAGA in term of computational efficiency. Table 6 Comparison of Technique CPU time (s) average execution times Case study 1 Case study 2 Case study 3 CSAGA 27.24 28.78 20.35 GA 6.012 30.64 44.97 SA 14.06 ––174 D.V.K. Khanh et al. 17 Results AfterevaluatingandcomparingtheperformanceofSTECs,resultsfromoptimizing the design of STECs were produced. Some case studies which were taken from previous works of 8 are as follows: � Case study 1: STECs model is run with CSAGA under the two constraints 2 which are the constraint of total area A.N 100 mm which is nonlinear inequality constraint and the maximum cost of material 385. The input current is varied from 0.1 to 8 A; hot side and cold side temperatures are set as 323 K (T = T = 323 K); the requirement of COP was neglected. Because COP is not h c taken into account, stand-alone SA can perform well and is run in the same condition, and results of Cheng’s work using GA are taken for comparison. � Case study 2: Same condition with case study 1 but cold side temperature is varied from 283 to 323 K; input current is set as 1 A. � Case study 3: STECs model is run with CSAGA under various input currents, with three constraints which is the requirement of confined volume of STECs 2 (S = 100 mm ), maximum cost of material 385, and the requirement of COP = 0.75 which is a non-linear equality constraint. In this case, SA cannot solve the problem with non-linear equality constraint; stand-alone GA is run in the same condition with CSAGA for comparison. For case study 1, Table 7 presents the optimal design parameters of STECs model for various input currents from 0.1 to 8 A and the comparison is shown in Fig. 11. In Table 7, when maximum ROR is increased, leg area increases and numberoflegsdecreases;leglength doesnotchangeand reachthelower boundby thelimit(0.3mm).AsshowninFig.11,whentheinputcurrentislargerthan0.5A, the maximum values of ROR obtained by CSAGA and SA are approximately 7.42 W which seem unchanged. The table demonstrates that STECs using CSAGA Table7 Case study1—collected dataafter runningCSAGAandSAundervarious inputcurrents I (A) CSAGA SA 2 2 MaxROR N (unit) A (mm ) L (mm) MaxROR N (unit) A (mm ) L (mm) (W) (W) 0.1 4.63 841.75 0.09 0.3 4.63 841.75 0.09 0.3 0.2 7.04 841.75 0.09 0.3 7.04 841.75 0.09 0.3 0.5 7.42 437.18 0.17 0.3 7.42 437.14 0.17 0.3 1 7.42 216.63 0.35 0.3 7.42 221.38 0.34 0.3 2 7.42 107.64 0.7 0.3 7.42 107.79 0.7 0.3 4 7.42 54.67 1.39 0.3 7.42 54.66 1.39 0.3 6 7.42 36.34 2.09 0.3 7.41 36.34 2.08 0.3 8 7.42 26.91 2.82 0.3 7.42 26.91 2.82 0.3 T = T = 323 K, maximum cost of the material was 385 and STECs were put in a confined h c 2 volume 100 mm . The requirement of COP was ignored

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