Four-Objective Optimizations of a Single Resonance Energy Selective Electron Refrigerator

According to the established model of a single resonance energy selective electron refrigerator with heat leakage in the previous literature, this paper performs multi-objective optimization with finite-time thermodynamic theory and NSGA-II algorithm. Cooling load (R¯), coefficient of performance (ε), ecological function (ECO¯), and figure of merit (χ¯) of the ESER are taken as objective functions. Energy boundary (E′/kB) and resonance width (ΔE/kB) are regarded as optimization variables and their optimal intervals are obtained. The optimal solutions of quadru-, tri-, bi-, and single-objective optimizations are obtained by selecting the minimum deviation indices with three approaches of TOPSIS, LINMAP, and Shannon Entropy; the smaller the value of deviation index, the better the result. The results show that values of E′/kB and ΔE/kB are closely related to the values of the four optimization objectives; selecting the appropriate values of the system can design the system for optimal performance. The deviation indices are 0.0812 with LINMAP and TOPSIS approaches for four-objective optimization (ECO¯−R¯−ε−χ¯), while the deviation indices are 0.1085, 0.8455, 0.1865, and 0.1780 for four single-objective optimizations of maximum ECO¯, R¯, ε, and χ¯, respectively. Compared with single-objective optimization, four-objective optimization can better take different optimization objectives into account by choosing appropriate decision-making approaches. The optimal values of E′/kB and ΔE/kB range mainly from 12 to 13, and 1.5 to 2.5, respectively, for the four-objective optimization.


Introduction
With the development of micro-nano technology, the micro energy conversion systems have gradually entered the field of view of modern thermodynamic researchers. As with motors in the macro world, there are also micro "motors" at the micro level, including quantum thermal cycles [1,2], Brownian motors [3,4], and electron engine systems [5][6][7], etc. They also function as motors in a mechanical sense; that is, they obtain mechanical energy by converting other forms of energy. In the studies on micro energy conversion systems, thermodynamic performance of the system has always been one of the core issues that people pay attention to. Due to their advantages of high efficiency, controllability, and easy integration, the optimal performance of these micro energy conversion systems has become a very active focus of modern thermodynamic research.
Since its establishment, the theory of finite time thermodynamics (FTT) [8][9][10][11][12][13][14][15][16][17][18][19] has been applied to research the optimal configurations  and the optimal performances  of thermal devices. The ideas of FTT considering performance optimization with heat leakage loss and finite rate heat transfer are also applicable to analyses and optimizations of micro energy conversion systems. Applying FTT to optimize micro energy conversion Figure 1 gives a model of ESER with heat leakage [89]. It includes two electron reservoirs and an energy filter, which only allows electrons in a certain energy range to pass through. In the cold-and hot-electron reservoirs, the electrochemical potentials are µ C and µ H , and the temperatures are T C and T H , respectively. ∆E and E are important structural parameters of the energy filter, and they are the energy boundary and resonance width, respectively. According to Ref. [78], to make the ESE system work as an ESER, the value of the energy E of electrons involved in energy transfer must meet µ C < E < E 0 , where E 0 = (µ C T H − µ H T C )/(T H − T C ). During the working of the ESER, the heat transfer of electrons is irreversible due to the heat leakage loss caused by phonon propagation.

Model Description and Performance Indicators
where h is Planck constant, and C f and H f are Fermi distributions of electrons: where B k is Boltzmann constant.  According to Refs. [78,85], in a small energy range, the heat transferred into hotreservoir and from cold-reservoir are: where h is Planck constant, and f C and f H are Fermi distributions of electrons: where k B is Boltzmann constant. From Equations (1)-(4), the total heat transferred into hot-reservoir ( . Q HE ) and from cold-reservoir ( (5) .
The heat leakage loss rate ( . Q L ) caused by phonon propagation is where k L is heat-leakage coefficient. Combining Equations (5)- (7), the actual heats transferred into hot-reservoir ( . Q H ) and from cold -reservoir ( .

Multi-Objective Optimizations
In the MOO process, the solution set is not the only one, but a series of feasible alternatives; that is, the optimal solutions set in the multivariate space, which are called the "Pareto frontiers". They are obtained by NSGA-II (see Figure 2 for flow chart). The steps of NSGA-II are: (1) the initialization population with size N is performed by non-dominated sorting; then, the first offspring population is obtained by selection, crossover, and variation through genetic algorithm; (2) in the second generation, merging the parent population and first offspring population for rapid non-dominated sorting, the crowding distance of individuals is calculated to select suitable individuals to form a new parent population; and (3) a new population of offspring is obtained with the genetic algorithm; when it meets the program requirements, the genetic algorithm will end and a Pareto optimal frontier will be obtained. hierarchical sorting and elite strategies. In order to obtain an ideal design scheme by applying NSGA-II, three strategies including TOPSIS, LINMAP, and SE are usually adopted and their pros and cons are compared by deviation indices ( D s) obtained from them. The positive and negative ideal points are the solutions at which all performance indicators of Pareto frontiers reach the maximum and minimum values, respectively. TOPSIS strategy makes the solutions farthest from the negative ideal point, LINMAP strategy makes the solutions closest to positive ideal points, and SE strategy attains the solutions when the last performance indicator reaches its maximum. The D represents the average distance between the solution and positive ideal point; the smaller the value, the better the result. The definition of D is  According to Ref. [89], 14 1.5 10 are set to 10~15 and 0~5 , respectively. Table 1 lists the setting parameters of NSGA-II. Table 2 gives the outcomes of various single-objective and multi-objective optimizations. From Table 2  NSGA-II is an improvement for the shortcomings of NSGA, which can improve the convergence speed of the algorithm and maintain the diversity of results by adopting fast hierarchical sorting and elite strategies. In order to obtain an ideal design scheme by applying NSGA-II, three strategies including TOPSIS, LINMAP, and SE are usually adopted and their pros and cons are compared by deviation indices (Ds) obtained from them. The positive and negative ideal points are the solutions at which all performance indicators of Pareto frontiers reach the maximum and minimum values, respectively. TOPSIS strategy makes the solutions farthest from the negative ideal point, LINMAP strategy makes the solutions closest to positive ideal points, and SE strategy attains the solutions when the last performance indicator reaches its maximum.
The D represents the average distance between the solution and positive ideal point; the smaller the value, the better the result. The definition of D is According to Ref. [89], k L = 1.5 × 10 −14 W/K, T 0 = 1.6 K, T C = 1.2 K, µ C /k B = 12 K, T H = 2.2 K and µ H /k B = 10 K are set. The value ranges of energy boundary E /k B and resonance width ∆E/k B are set to 10 ∼ 15 and 0 ∼ 5, respectively. Table 1 lists the setting parameters of NSGA-II. Table 2 gives the outcomes of various single-objective and multi-objective optimizations. From Table 2, the Ds are 0.0812 with LINMAP and TOPSIS approaches when the MOO is performed on four-objective optimization (ECO − R − ε − χ), while the Ds are 0.1085, 0.8455, 0.1865, and 0.1780 for four single-objective optimizations of maximum ECO, R, ε, and χ, respectively. It indicates that compared with single-objective optimization, MOO can better take into account different optimization objectives by choosing appropriate decision-making methods. For the MOO of ECO − R, the D obtained with the TOPSIS approach is 0.0809, which is the smallest and the most perfect scheme; and the corresponding values of the E /k B and ∆E/k B are 12.5887 and 1.8050, respectively.

Optimization
Variables Objective Functions Deviation Index  Figure 3 gives the outcomes of different (six) bi-objective optimization combinations. It can be concluded from Figure 3 that two different performance indicators cannot achieve the best at the same time, and the improvement of one performance indicator will often lead to the deterioration of the other. From Figure 3 and Table 2, SE solutions represent the points when the performance indicators on the ordinate reach the maximum, LINMAP and TOPSIS solutions represent the points closest to the positive ideal point (1.0000, 1.0000, 0.4916, 1.0000) and the points farthest from the negative ideal point (−2.0171, 0.4739, 0.2943, 0.6532), respectively. As ECO grows, R, ε, and χ all decline. As R grows, both ε and χ decline. As ε grows, χ declines. From Table 2, for the MOOs of ECO − ε and ECO − χ, the Ds (0.1229, 0.0884) obtained with the LINMAP approach are the same as those obtained with the TOPSIS approach, and are better than those obtained with the SE approach. For the MOOs of ECO − R, R − ε, and ε − χ, the Ds (0.0809, 0.3786, 0.0814) obtained with the TOPSIS approach are better than those obtained with LINMAP and SE approaches. For the MOO of R − χ, the D (0.1780) obtained with SE is more perfect than that obtained with LINMAP and TOPSIS approaches.  Figure 3 gives the outcomes of different (six) bi-objective optimization combinations. It can be concluded from Figure 3 that two different performance indicators cannot achieve the best at the same time, and the improvement of one performance indicator will often lead to the deterioration of the other. From Figure 3 and Table 2, SE solutions represent the points when the performance indicators on the ordinate reach the maximum, LINMAP and TOPSIS solutions represent the points closest to the positive ideal point (1.0000, 1.0000, 0.4916, 1.0000) and the points farthest from the negative ideal point (−2.0171, 0.4739, 0.2943, 0.6532), respectively. As ECO grows, R ,  , and  all decline. As R grows, both  and  decline. As  grows,  declines. From   Figure 4 gives the outcomes of different (four) tri-objective optimization combinations. As ECO grows, R declines,  grows, and  first grows and then declines. As R grows,  declines, and  first grows and then declines. From   Figure 4 gives the outcomes of different (four) tri-objective optimization combinations. As ECO grows, R declines, ε grows, and χ first grows and then declines. As R grows, ε declines, and χ first grows and then declines. From Table 2, for the MOO of ECO − R − ε, the D (0.0814) obtained with the LINMAP approach is the same as that obtained with the TOPSIS approach, and is better than that obtained with the SE approach. For the MOO of ECO − R − χ, the D (0.0816) obtained with the TOPSIS approach is better than that obtained with LINMAP and SE approaches. For the MOO of ECO − ε − χ and R − ε − χ, the Ds (0.0888, 0.1692) obtained with the LINMAP approach are better than those obtained with TOPSIS and SE approaches.    The numerical changes of R and  are represented by three axes, and the numerical change of  is represented by color change. The positive and negative points lie outside the Pareto frontier, which means that the four optimization objectives ECO , R ,  , and  cannot be optimal or worst at the same time. As ECO grows, R declines,  grows, and  first grows and then declines. From Table 2, for the MOO of ECO R      , the D s (0.0812, 0.0812) obtained with LINMAP and TOPSIS approaches are equal, which are more reasonable than those obtained with SE.   Figure 5 gives the Pareto frontier for four-objective (ECO − R − ε − χ) optimization. The numerical changes of R and ε are represented by three axes, and the numerical change of χ is represented by color change. The positive and negative points lie outside the Pareto frontier, which means that the four optimization objectives ECO, R, ε, and χ cannot be optimal or worst at the same time. As ECO grows, R declines, ε grows, and χ first grows and then declines. From Table 2, for the MOO of ECO − R − ε − χ, the Ds (0.0812, 0.0812) obtained with LINMAP and TOPSIS approaches are equal, which are more reasonable than those obtained with SE.   The numerical changes of R and  are represented by three axes, and the numerical change of  is represented by color change. The positive and negative points lie outside the Pareto frontier, which means that the four optimization objectives ECO , R ,  , and  cannot be optimal or worst at the same time. As ECO grows, R declines,  grows, and  first grows and then declines. From Table 2, for the MOO of ECO R      , the D s (0.0812, 0.0812) obtained with LINMAP and TOPSIS approaches are equal, which are more reasonable than those obtained with SE.   Figure 6 gives the distributions of (E /k B ) opt and (∆E/k B ) opt obtained with fourobjective optimization. In Figure 6a, the value of (E /k B ) opt ranges mainly from 12 to 13; as (E /k B ) opt grows, R continues to decline, ε continues to grow, ECO and χ grow first and then decline. From Figure 6b, the value of (∆E/k B ) opt ranges mainly from 1.5 to 2.5; as (∆E/k B ) opt grows, R continues to grow, ε continues to decline, ECO and χ grow first and then decline. From Figure 6, the values of E and ∆E are closely related to the values of the four optimization objectives (ECO, R, ε and χ), and the selection of the parameters of the energy filter is very important to improve the performance of ESERs. Figure 6 gives the distributions of ( / )  Figure 6a, the value of ( / ) grows, R continues to grow,  continues to decline, ECO and  grow first and then decline. From Figure 6, the values of E and E  are closely related to the values of the four optimization objectives ( ECO , R ,  and  ), and the selection of the parameters of the energy filter is very important to improve the performance of ESERs.
B opt E k  Figure 6. Distributions of design variables in Pareto frontier. Figure 6. Distributions of design variables in Pareto frontier. Figures 7 and 8 give the average distance and average spread in relation to generations obtained from two different MOOs. From the two figures, the genetic algorithm will end when it reaches convergence, which occurs at the 302th and 455th generations for four-objective (ECO − R − ε − χ) and two-objective (ECO − R) optimizations, respectively. Figures 7 and 8 give the average distance and average spread in relation to generations obtained from two different MOOs. From the two figures, the genetic algorithm will end when it reaches convergence, which occurs at the 302th and 455th generations for four-objective ( ECO R      ) and two-objective ( ECO R  ) optimizations, respectively.

Conclusions
In this paper, according to the model established in Ref. [89], the NSGA-II is applied to perform MOO for a single resonance energy selective electron refrigerator with heat leakage. Four objective functions are introduced, including cooling load, coefficient of performance, figure of merit, and ecological function. The E and E  are regarded as op-  Figures 7 and 8 give the average distance and average spread in relation to generations obtained from two different MOOs. From the two figures, the genetic algorithm will end when it reaches convergence, which occurs at the 302th and 455th generations for four-objective ( ECO R      ) and two-objective ( ECO R  ) optimizations, respectively.

Conclusions
In this paper, according to the model established in Ref. [89], the NSGA-II is applied to perform MOO for a single resonance energy selective electron refrigerator with heat leakage. Four objective functions are introduced, including cooling load, coefficient of performance, figure of merit, and ecological function. The E and E  are regarded as op-

Conclusions
In this paper, according to the model established in Ref. [89], the NSGA-II is applied to perform MOO for a single resonance energy selective electron refrigerator with heat leakage. Four objective functions are introduced, including cooling load, coefficient of performance, figure of merit, and ecological function. The E and ∆E are regarded as optimization variables, their optimal intervals are obtained, and their effects on four objective functions are analyzed when the MOO is performed for ECO − R − ε − χ. The results show that: 1.
The For the MOO of ECO − R − ε − χ, the value of (E /k B ) opt ranges mainly from 12 to 13; as (E /k B ) opt grows, R continues to decline, ε continues to grow, ECO and χ grow first and then decline. The value of (∆E/k B ) opt ranges mainly from 1.5 to 2.5; as (∆E/k B ) opt grows, R continues to grow, ε continues to decline, ECO and χ grow first and then decline. It indicates that the values of E and ∆E are closely related to values of the four optimization objectives (ECO, R, ε, and χ), and the selection of the parameters of the energy filter is very important to improve the performance of energy selective electron refrigerators.

4.
For the MOO of ECO − R − ε − χ and ECO − R, the average distances range mainly from 0 to 0.5 and change slightly; the average spreads range mainly from 0 to 0.2, vary significantly before the 100th generations, and then remain stable. 5.
NSGA-II and FTT theory are effective tools to guide the designs of energy selective electron refrigerators. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Acknowledgments:
The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.