## 2 Materials and Methods
### Multi-Objective Optimization Model  
In this paper, multi-objective grey wolf optimizer (MOGWO) algorithm is employed for optimizing the ORC system. Some standard unconstrained, constrained, and engineering design problems have been used to validate MOGWO [26]. The results are compared with other multi-objective algorithms including Multi-objective Particle Swarm Optimization (MOPSO), Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D), non-dominated sorting genetic algorithm II (NSGA-II), and Strength Pareto Evolutionary Algorithm 2 (SPEA2) to evaluate the efficiency and effectiveness of MOGWO [26; 27; 28; 29; 30]. The comparison results show that the MOGWO can provide very competitive optimization results. And in terms of convergence, spread and coverage MOGWO presents excellent performance. MOGWO is developed on the basis of grey wolf optimizer (GWO). Two components including an archive which is used to store the non-dominated Pareto  
Figure 5: Validation of the turbine efficiency prediction model.  
optimization solutions and a leader selection mechanism that is used to select \(\alpha\), \(\beta\), and \(\delta\) wolves from the archive are integrated with GWO to perform multi-objective optimization, which is called MOGWO. The grey wolf optimizer (GWO) is inspired by the grey wolves, including their social leadership and hunting mechanism. The flowchart of MOGWO is shown in Figure 6. First, initialize the grey wolves population including the number and position of the grey wolves, maximum iteration and the archive size. Calculate the fitness of each grey wolf according to their position, sort the non-dominated solutions and initialize the Prato archive through saving the non-dominated solutions to it. Select the fittest, second and third fittest solutions (the wolves that gain the best, second and third best positions) as \(\alpha\), \(\beta\), and \(\delta\) wolves by leader selection mechanism. Update the positions of the current wolves according to the positions of \(\alpha\), \(\beta\), and \(\delta\) wolves and the distance between the current wolves and \(\alpha\), \(\beta\), and \(\delta\) wolves to generate new wolves (solutions). Calculate the fitness of the updated grey wolves and find new non-dominated solutions. Save new non-dominated solutions in the Prato archive and remove the dominated solutions simultaneously. If the Prato archive is full, remove some of the archive members based on crowding distance to meet the archive size. Then conduct non-dominated sorting for the Prato archive members and update the leaders (\(\alpha\), \(\beta\), and \(\delta\) wolves). If the termination condition (maximum iteration number) is satisfied, report the optimal Pareto solutions and the MOGWO ends.  
To compare the ORCCTE system and ORCDTE system, the thermal efficiency and specific investment cost are selected as the thermodynamic and economic objectives respectively. Two parameters including evaporation temperature and condensation temperature are selected as independent variables for optimization of the ORC system. The mathematical model of the multi-objective optimization for the ORC system can be described as follows:  
Figure 6: Flowchart of the multi-objective grey wolf optimizer (MOGWO) algorithm.  
\[\left\{\begin{array}{l}\max(\eta_{\text{the}})=f_{1}(T_{0},T_{4})\\ \min(SIC)=f_{2}(T_{0},T_{4})\\ \end{array}\right.. \tag{24}\]  
The logical bounds of the decision variables can be described as follows:  
\[\left\{\begin{array}{c}T_{0}<T_{\text{g1}}\\ T_{0}<T_{\text{critical}}\\ \Delta T_{\text{p,eva}}>5\\ \Delta T_{\text{p,con}}>5\\ 303.15<T_{4}<323.15\\ \end{array}\right.. \tag{25}\]