Multi-Objective Optimization of an Irreversible Single Resonance Energy-Selective Electron Heat Engine

Abstract

Based on the model of irreversible single resonance energy-selective electron heat engine established in the previous literature, this paper applies finite-time thermodynamic theory and NSGA-II algorithm to perform multi-objective optimization. Single-, bi-, tri- and quadru-objective optimizations are performed when the energy boundary and the resonance width are taken as the optimization variables, and the power output, thermal efficiency, efficient power and ecological function are taken as the optimization objectives. The deviation indexes of different optimization objective combinations are obtained by using LINMAP, TOPSIS and Shannon entropy approaches. The results show that the values of energy boundary and resonance width can be reasonably selected according to the design requirements of the system. When power output and efficiency are optimized, the minimal deviation index is obtained by TOPSIS approach and the value is 0.0748, which is the most ideal design scheme.

1. Introduction

Since the establishment of finite time thermodynamics (FTT) theory, many scholars have introduced it into various cycle studies and have made great progress [1,2,3,4,5,6], including optimal performances [7,8,9,10,11,12,13,14,15] and optimal configurations [16,17,18,19,20,21,22,23,24,25]. The early FTT research objects involved mainly traditional macro thermal devices, such as macro heat engines, refrigerators and heat pumps. The idea of FTT considering various irreversible loss factors to analyze and optimize thermodynamic cycle and process is also of great significance to micro energy conversion devices. With the developments of FTT, the research objects involve quantum cycles [26,27], Brownian motor systems [28,29], electron engine systems [30], and other micro energy conversion devices related to time and rate.

In 2002, Humphrey et al. [31] found that between two electron reservoirs with different electrochemical potentials and temperatures, electrons can be exchanged reversibly by choosing an appropriate energy filter, and this novel electronic device was named as energy selective electron (ESE) heat engine [32]. When not accounting for heat leakage losses, Humphrey [32] applied the FTT theory to explain the operation mechanism of the single resonance ESE heat engine at the earliest and analyzed the efficiency at maximum power (EMP) characteristics of the system. Ding et al. [33] introduced the ecological function [34] into an endoreversible ESE heat engine. Luo et al. [35] further analyzed effect of resonance width. When accounting for heat leakage losses, Ding et al. [36,37,38] studied the power output (POW) and thermal efficiency (TEF) of irreversible single resonance ESE heat engines, and Zhou et al. [39] studied the ecological function performance.

There are also other performance indicators for evaluating thermodynamic devices, besides POW, TEF and ecological function. For example, Yan [40] first took the product of POW and TEF as a new objective function. Later, Yilmaz [41,42] named this new objective function as efficient power (EP). Patodi and Maheshwari [43] studied the maximum EP performance of the macro Atkinson cycle. Singh and Johal [44] and Nilavarasi and Ponmurugan [45] studied the maximum EP performances of low-dissipation heat engines. There is no work concerning the maximum EP performances of an ESE heat engine in the available literature.

The above work focused on single-objective optimization of cycle performance, and there may be contradictions between various objective functions. In order to coordinate the relationship between the objective functions, some scholars have carried out multi-objective optimization (MOO) for macro cycle performances by using NSGA-II algorithm [46,47,48,49,50,51,52,53,54,55,56,57,58], including the thermoelectric generator [46], macro [47] and micro [48] Stirling cycles, Atkinson cycle [49], Diesel cycle [50], Brayton cycle [51], ORC [52], dual cycle [53], porous-medium engines [54,55], MHD plant [56] and chemical reactors [57,58], with objectives of POW, TEF, ecological function and power density.

There is no work concerning the MOO for an ESE heat engine in the available literature. Based on the model of an irreversible single-resonance ESE heat engine established in References [36,39], this paper will perform MOO for the model by applying NSGA-II algorithm. The boundary energy and resonance width will be used as the optimization variables, and the POW, TEF, EP and ecological function will be used as the optimization objectives. The three decision-making methods of LINMAP, TOPSIS and Shannon Entropy will be used to select the optimal scheme with the smallest deviation index (D) under different optimization objective combinations.

2. Model Description

Figure 1 shows the irreversible single-resonance ESE heat engine model [36,39], which consists of two electron sources connected by an energy filter.

There is no heat exchange between two electron reservoirs, and electrons are exchanged only by filter, which allows for electrons in a specific energy range to be freely transported, and there is also heat leakage loss due to phonon propagation during the transmission process. There are different temperatures, $T_H$ and $T_C$, and two different electrochemical potentials, ${\mu }_H$ and ${\mu }_C$, in two electron reservoirs. $E^{\prime }$ is the energy boundary, $\mathrm{\Delta }E$ is the resonance width and $e{V}_0$ is the bias voltage. During the operation of the ESE system, electrons are transferred from the hot electronic reservoir through the energy filter to the cold electronic reservoir due to factors such as electrochemical potential difference and temperature difference. Since the electrons carry energy, the transfer process is accompanied by energy changes, thereby realizing an energy conversion.

The detailed description of the model is shown in Appendix A. From Equations (A8) and (A9) in Appendix A, the POW ($P$) and TEF ($\eta$) can be expressed as [39]

$$\begin{array}{ll}P& ={\dot{Q}}_H-{\dot{Q}}_C\\ & =\frac{2}{h}{k}_B({\mu }_C-{\mu }_H)[T_H\ln \frac{1+{\mathrm{e}}^{(r_H)}}{1+{\mathrm{e}}^{(R_H)}}-T_C\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]\end{array}$$

(1)

$$\begin{array}{ll}\eta & =\frac{P}{{\dot{Q}}_H}\\ & =({\mu }_C-{\mu }_H)[T_H\ln \frac{1+{\mathrm{e}}^{(r_H)}}{1+{\mathrm{e}}^{(R_H)}}-T_C\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]/\{k_B{T}_H{}^2[\frac{1}{2}{R}_H{}^2-g({\mathrm{e}}^{(R_H)})\\ & -R_H\ln (1+{\mathrm{e}}^{(R_H)})-\frac{1}{2}{r}_H{}^2+r_H\ln (1+{\mathrm{e}}^{(r_H)})+g({\mathrm{e}}^{(r_H)})]-k_B{T}_C{}^2[\frac{1}{2}{R}_C{}^2-\\ & R_C\ln (1+{\mathrm{e}}^{(R_C)})-g({\mathrm{e}}^{(R_C)})-\frac{1}{2}{r}_C{}^2+r_C\ln (1+{\mathrm{e}}^{(r_C)})+g({\mathrm{e}}^{(r_C)})]-T_C\times \\ & ({\mu }_C-{\mu }_H)[R_C-r_C+\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]+(T_H-T_C)\frac{k_{L}h}{2k_B}\}\end{array}$$

(2)

where $R_H=\frac{E^{\prime }+\mathrm{\Delta }E-{\mu }_H}{k_B{T}_H}$, $r_H=\frac{E^{\prime }-{\mu }_H}{k_B{T}_H}$, $R_C=\frac{E^{\prime }+\mathrm{\Delta }E-{\mu }_C}{k_B{T}_C}$ and $r_C=\frac{E^{\prime }-{\mu }_C}{k_B{T}_C}$.

Combining Equations (1) and (2), according to the definition of EP ($E_P$) [40,41,42], one has

$$\begin{array}{ll}{E}_P& =P\eta \\ & =\frac{2}{h}{({\mu }_C-{\mu }_H)}^2{[T_H\ln \frac{1+{\mathrm{e}}^{(r_H)}}{1+{\mathrm{e}}^{(R_H)}}-T_C\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]}^2/\{T_H{}^2[\frac{1}{2}{R}_H{}^2-g({\mathrm{e}}^{(R_H)})\\ & -R_H\ln (1+{\mathrm{e}}^{(R_H)})-\frac{1}{2}{r}_H{}^2+r_H\ln (1+{\mathrm{e}}^{(r_H)})+g({\mathrm{e}}^{(r_H)})]-T_C{}^2[\frac{1}{2}{R}_C{}^2-\\ & R_C\ln (1+{\mathrm{e}}^{(R_C)})-g({\mathrm{e}}^{(R_C)})-\frac{1}{2}{r}_C{}^2+r_C\ln (1+{\mathrm{e}}^{(r_C)})+g({\mathrm{e}}^{(r_C)})]-\frac{T_C}{k_b}\times \\ & ({\mu }_C-{\mu }_H)[R_C-r_C+\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]+(T_H-T_C)\frac{k_{L}h}{2k_B{}^2}\}\end{array}$$

(3)

From Equations (A8) and (A9) in Appendix A, for an ESE heat engine system, the entropy generation rate ($\sigma$) is

$$\begin{array}{ll}\sigma & ={\dot{Q}}_C/T_C-{\dot{Q}}_H/T_H\\ & =\frac{2}{h}{k}_B{}^2\frac{T_H(T_H-T_C)}{T_C}[\frac{1}{2}{R}_H{}^2-R_H\ln (1+{\mathrm{e}}^{(R_H)})-g({\mathrm{e}}^{(R_H)})-\frac{1}{2}{r}_H{}^2+g({\mathrm{e}}^{(r_H)})\\ & +r_H\ln (1+{\mathrm{e}}^{(r_H)})]-\frac{2}{h}{k}_B{}^2\frac{T_C(T_H-T_C)}{T_H}[\frac{1}{2}{R}_C{}^2-R_C\ln (1+{\mathrm{e}}^{(R_C)})-g({\mathrm{e}}^{(R_C)})\\ & -\frac{1}{2}{r}_C{}^2+r_C\ln (1+{\mathrm{e}}^{(r_C)})+g({\mathrm{e}}^{(r_C)})]-\frac{2}{h}{k}_B({\mu }_C-{\mu }_H)[(R_C-r_C)(1-\frac{T_C}{T_H})+\frac{T_H}{T_C}\times \\ & \ln \frac{1+{\mathrm{e}}^{(r_H)}}{1+{\mathrm{e}}^{(R_H)}}-\frac{T_C}{T_H}\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}]+{(T_H-T_C)}^2\frac{k_L}{T_H{T}_C}\end{array}$$

(4)

From Equations (1) and (4), according to the definition formula of ecological function ($E$) [34], one has

$$\begin{array}{ll}E& =P-T_0\sigma \\ & =P(1+\frac{T_0}{T_C})-T_0(\frac{1}{T_C}-\frac{1}{T_H}){\dot{Q}}_H\\ & =\frac{2}{h}{k}_B({\mu }_C-{\mu }_H)[(1+\frac{T_0}{T_C})T_H\ln \frac{1+{\mathrm{e}}^{(r_H)}}{1+{\mathrm{e}}^{(R_H)}}-(1+\frac{T_0}{T_H})T_C\ln \frac{1+{\mathrm{e}}^{(r_C)}}{1+{\mathrm{e}}^{(R_C)}}+T_0(1-\frac{T_C}{T_H})\times \\ & (R_C-r_C)]-T_0(\frac{1}{T_C}-\frac{1}{T_H})\{\frac{2}{h}{(k_B{T}_H)}^2[\frac{1}{2}{R}_H{}^2-R_H\ln (1+{\mathrm{e}}^{(R_H)})-g({\mathrm{e}}^{(R_H)})-\frac{1}{2}{r}_H{}^2\\ & +r_H\ln (1+{\mathrm{e}}^{(r_H)})+g({\mathrm{e}}^{(r_H)})]-\frac{2}{h}{(k_B{T}_C)}^2[\frac{1}{2}{R}_C{}^2-R_C\ln (1+{\mathrm{e}}^{(R_C)})-g({\mathrm{e}}^{(R_C)})\\ & -\frac{1}{2}{r}_C{}^2+r_C\ln (1+{\mathrm{e}}^{(r_C)})+g({\mathrm{e}}^{(r_C)})]+k_L(T_H-T_C)\}\end{array}$$

(5)

The dimensionless POW ($\overline{P}$), EP (${\overline{E}}_P$) and ecological function ($\overline{E}$) are defined as:

$$\overline{P}=P/P_{\max }$$

(6)

$${\overline{E}}_P=E_P/{(E_P)}_{\max }$$

(7)

$$\overline{E}=E/E_{\max }$$

(8)

3. Multi-Objective Optimizations

For the MOO problem, if the solution cannot be improved for any objective without deteriorating at least one objective, it is a “Pareto optimal solution”, and the corresponding set of objective function values are “Pareto frontiers”. The Pareto frontier contains multiple feasible solutions, and the commonly used three decision-making approaches are LINMAP, TOPSIS and Shannon Entropy. The deviation indices ($D$ s) are used to compare the pros and cons of three decision-making approaches to select the optimal design. The positive and the negative ideal points represent the optimal and worst virtual points for all objective functions, respectively.

The LINMAP decision approach obtains the solution with the smallest distance from the positive ideal point. For the LINMAP approach:

$$B_{ij}=F_{ij}/\sqrt{{\displaystyle {\sum }_{i=1}^m{F}_{ij}{}^2}}$$

(9)

$$G_{ij}=w_j^{\mathrm{LINMAP}}\cdot B_{ij}$$

(10)

$$i_{\mathrm{opt}}^{\mathrm{LINMAP}}\in \min \left\{E{D}_i^{+}\right\}$$

(11)

where the value of $B_{ij}$ is $F_{ij}$ normalized, $w_j^{\mathrm{LINMAP}}$ is the proportion of the j-th optimization goal and the value of $G_{ij}$ is $B_{ij}$ weighted. $G_j^{\mathrm{positive}}$ is the normalized and weighted value of the j-th goal of the positive ideal point, $E{D}_i^{+}$ is the Euclidean distance between the i-th feasible solution and the positive ideal point and $i_{\mathrm{opt}}^{\mathrm{LINMAP}}$ is the best feasible solution.

The TOPSIS decision approach not only requires the solution with the smallest distance from the positive ideal point, but also with the farthest distance from the positive negative point. For the TOPSIS approach:

$$B_{ij}=F_{ij}/\sqrt{{\displaystyle {\sum }_{i=1}^m{F}_{ij}{}^2}}$$

(12)

$$G_{ij}=w_j^{\mathrm{TOPSIS}}\cdot B_{ij}$$

(13)

$$E{D}_i^{-}=\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_{ij}-G_j^{\mathrm{negative}})}^2}}$$

(14)

$$i_{\mathrm{opt}}^{\mathrm{TOPSIS}}\in \max \{\frac{E{D}_i^{-}}{E{D}_i^{+}+E{D}_i^{-}}\}$$

(15)

where $w_j^{\mathrm{TOPSIS}}$ is the proportion of the j-th optimization goal, $G_j^{\mathrm{negative}}$ is the normalized and weighted value of the j-th goal of the negative ideal point, $E{D}_i^{-}$ is the Euclidean distance between the i-th feasible solution and the negative ideal point and $i_{\mathrm{opt}}^{\mathrm{TOPSIS}}$ is the best feasible solution.

The Shannon Entropy approach obtains the point when the last objective function of the optimization reaches the maximum. For the Shannon Entropy approach:

$$P_{ij}=F_{ij}/{\displaystyle \sum _{i=1}^n{F}_{ij}}$$

(16)

$$S{E}_j=-\frac{1}{\ln n}{\displaystyle \sum _{i=1}^n{P}_{ij}}\ln P_{ij}$$

(17)

$$w_j^{\mathrm{Shannon} \mathrm{Entropy}}=(1-S{E}_j)/{\displaystyle \sum _{j=1}^n(1-S{E}_j)}$$

(18)

$$i_{\mathrm{opt}}^{\mathrm{Shannon} \mathrm{Entropy}}\in \min \{P_{ij}\cdot w_j\}$$

(19)

where the value of $P_{ij}$ is $F_{ij}$ normalized, $S{E}_j$ is the Shannon Entropy of the j-th optimization objective, $w_j^{\mathrm{Shannon} \mathrm{Entropy}}$ is the proportion of the j-th optimization goal and $i_{\mathrm{opt}}^{\mathrm{Shannon} \mathrm{Entropy}}$ is the best feasible solution.

The $D$ is defined as:

$$D=\frac{\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_{i_{opt}j}-G_j^{\mathrm{positive}})}^2}}}{\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_{i_{opt}j}-G_j^{\mathrm{positive}})}^2}}+\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_{i_{opt}j}-G_j^{\mathrm{negative}})}^2}}}$$

(20)

The $D$ can calculate the average distance between the values obtained by different decision-making methods and the positive and negative ideal points. The smaller $D$ is obtained under a certain decision-making approach, which means the design scheme is better. Figure 2 illustrates NSGA-II algorithm flow chart.

This paper takes $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ as optimization objectives; takes the boundary energy ($E^{\prime }$) and resonance width ($\mathrm{\Delta }E$) as optimization variables; and applies NSGA-II algorithm to optimize the ESE heat engine with single-, bi-, tri- and quadru-objective optimizations.

According to Reference [39], $T_0=1K$, $T_C=1.2K$, $T_H=2.2K$,${\mu }_H/k_B=10K$, ${\mu }_C/k_B=12K$ and $k_L=1.5\times {10}^{-14}W/K$ are set. Ranges of optimization variables $E^{\prime }/k_B$ and $\mathrm{\Delta }E/k_B$ are $13.5~18$ and $0~15$, respectively.

Appendix B lists the results of quadru-, tri-, bi- and single-objective optimizations under three decision-making approaches, respectively. When $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ reach their maximum values, it can be seen that $D$ s are 0.1007, 0.9331, 0.0882 and 0.1254, respectively. When the $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$ is optimized, the $D$ s obtained by the TOPSIS and LINMAP decision approaches are both 0.0755, indicating that the result of the optimization of four objectives is more perfect than that of a single objective. The $D$ of $\overline{P}$ and $\eta$ obtained by the TOPSIS solution is 0.0748, which is the smallest and the most ideal.

Figure 3 shows the Pareto frontier of quadru-objective ($\overline{P}-\eta -{\overline{E}}_P-\overline{E}$) optimization. In Figure 3, the change in color represents the change in the value of $\overline{E}$. The positive and the negative ideal points are not on the pareto frontier, indicating that the variables $E^{\prime }$ and $\mathrm{\Delta }E$ cannot make $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ simultaneously optimal or worse. As $\overline{P}$ increases, $\eta$ continues to decrease, and $\overline{E}$ and ${\overline{E}}_P$ first reach their maximum values and then decrease. When $\overline{P}$ reaches the maximum value, $\eta$ reaches the minimum, and ${\overline{E}}_P$ and $\overline{E}$ can take into account both $\overline{P}$ and $\eta$. From Appendix B, for the MOO of $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$, the $D$ s obtained by TOPSIS and LINMAP are the same, which is smaller than that obtained by Shannon Entropy, and the result is more ideal.

Figure 4 shows the distributions of ${(E^{\prime }/k_B)}_{opt}$ and ${(\mathrm{\Delta }E/k_B)}_{opt}$ within the value ranges in the Pareto front obtained by taking quadru-objective ($\overline{P}-\eta -{\overline{E}}_P-\overline{E}$) optimization. From Figure 4a, ${(E^{\prime }/k_B)}_{opt}$ is distributed between 13.95 and 14.10, but mostly concentrated between 14.00 and 14.05, with the increase of ${(E^{\prime }/k_B)}_{opt}$, the change trends of $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ are not obvious; therefore, the value of $E^{\prime }/k_B$ can be designed between 14 and 14.05. From Figure 4b, ${(\mathrm{\Delta }E/k_B)}_{opt}$ is distributed between 0 and 15, since energy filter allows for electrons in a specific energy range to be freely transported; as the resonance width increases, the number of electrons no longer increases with the increase in resonance width when all electrons in a specific energy range pass through. So as ${(\mathrm{\Delta }E/k_B)}_{opt}$ increases, $\overline{P}$ increases, $\eta$ decreases, ${\overline{E}}_P$ and $\overline{E}$ increase first and then decrease and the value of $\mathrm{\Delta }E$ can be reasonably selected according to the actual needs.

Figure 5 show the Pareto frontiers obtained by different (six) bi-objective optimization combinations. As $\overline{P}$ increases, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ both decrease. As $\eta$ increases, both ${\overline{E}}_P$ and $\overline{E}$ decrease. As ${\overline{E}}_P$ increases, $\overline{E}$ decreases. From Appendix B, for the MOO of $\overline{P}-\overline{E}$ and ${\overline{E}}_P-\overline{E}$, the $D$ with the LINMAP approach is better. For the MOO of $\overline{P}-{\overline{E}}_P$, $\eta -{\overline{E}}_P$ and $\eta -\overline{E}$, the $D$ with the Shannon Entropy approach is better. For the MOO of $\overline{P}-\eta$, the $D$ with TOPSIS approach is better.

Figure 6 show the Pareto frontiers obtained by different (four) tri-objective optimizations. As $\overline{P}$ increases, $\eta$ decreases, and $\overline{E}$ and ${\overline{E}}_P$ reach their maximum values and then decrease. As $\eta$ increases, ${\overline{E}}_P$ decreases, and $\overline{E}$ first increases and then decreases.

From Appendix B, for the MOO of $\overline{P}-\eta -\overline{E}$ and $\overline{P}-{\overline{E}}_P-\overline{E}$, the $D$ with the LINMAP approach is better. For the MOO of $\overline{P}-\eta -{\overline{E}}_P$, the $D$ with the TOPSIS approach is better. For the MOO of $\eta -{\overline{E}}_P-\overline{E}$, the $D$ s with the Shannon Entropy and TOPSIS approaches are the same, which is less than that with the LINMAP method.

Figure 7 and Figure 8 show the average distance versus generations and average spread versus generations obtained by taking quadru-objectives ($\overline{P}-\eta -{\overline{E}}_P-\overline{E}$) and bi-objectives ($\overline{P}-\eta$) as the optimization objectives, respectively. From two figures, if genetic algorithm converges, it will stop. The MOO of $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$ and $\overline{P}-\eta$ converges at 307 and 1107 generations, respectively.

4. Conclusions

Based on the established model of irreversible single ESE heat engine, this paper takes the energy boundary and the resonance width as the optimization variables and the $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ as the optimization objectives to perform MOO by applying the NSGA-II algorithm and FTT theory. The EP performance indicator is introduced to the optimization. The effects of the values of the two variables ($E^{\prime }$ and $\mathrm{\Delta }E$) on four optimization objectives are analyzed with the MOO of $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$. The results show that:

• When $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ reach their maximum values, it can be seen that $D$ values are 0.1007, 0.9331, 0.0882 and 0.1254, respectively. For the MOO of $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$, the $D$ s obtained by the TOPSIS and LINMAP decision approaches are both 0.0755. The $D$ obtained by MOO is smaller and better compared with single-objective optimization, which means that the MOO results are better.
• For the MOO of $\overline{P}-\eta -{\overline{E}}_P-\overline{E}$, ${(E^{\prime }/k_B)}_{opt}$ is distributed between 13.95 and 14.10, and ${(\mathrm{\Delta }E/k_B)}_{opt}$ is distributed between 0 and 15. As ${(E^{\prime }/k_B)}_{opt}$ increases, the change trends of $\overline{P}$, $\eta$, ${\overline{E}}_P$ and $\overline{E}$ are not obvious; as ${(\mathrm{\Delta }E/k_B)}_{opt}$ increases, $\overline{P}$ increases, $\eta$ decreases and ${\overline{E}}_P$ and $\overline{E}$ first increase then decrease. In the design of the ESE heat engine, it is very important to choose appropriate values of $E^{\prime }$ and $\mathrm{\Delta }E$.
• For the MOO of $\overline{P}-\eta$, the $D$ is the minimum obtained by the TOPSIS approach and the value is 0.0748; at this time, the $E^{\prime }/k_B$ and $\mathrm{\Delta }E/k_B$ are 14.0091 and 8.2266, respectively, which is the most ideal design scheme. For the MOO of other combinations, the appropriate decision-making approach can be selected according to the actual requirements and needs.
• It is meaningful to introduce the MOO to the performance optimization ESE heat engines.