As node-wise non-structural superstructure model is applied to heat exchanger network optimization, heat exchanger crossover may appear. It is found that crossover structure causes an increase of total heat transfer area for heat exchange unit with same heat loads. At the same time, it reduces obviously computational efficiency of the algorithm. A vertical node-wise non-structural superstructure model is established. Compared with node-wise non-structural superstructure, the vertical node-wise non-structural superstructure model reduces computational complexity in the optimization process and eliminates adverse effects of crossover structure, which improves efficiency and accuracy of optimization. Finally, optimization performance of RWCE algorithm based on vertical non-structural superstructure model is analyzed in two examples. The results are superior to those in literature.

As heat exchange loads are treated as continuous variables to be optimized in heat exchanger network (HEN) problems, temperature cross could be generated in stream matches. Infeasible structures with temperature cross are penalized by penalty function method in literature. However, in random walk algorithm with compulsive evolution (RWCE) algorithm to HEN optimization, structures with temperature cross may be accepted as imperfect solutions due to its mutation operation of accepting imperfect solutions, which may negatively affect optimization process and decrease algorithm efficiency. Hence, temperature cross in HEN is firstly explained. Then negative effects of infeasible structures on optimization process of RWCE is described. A treatment method is proposed: Individuals trapped in infeasible regions are compulsively returned to their original positions in feasible regions to accept re-optimization. It was demonstrated that the proposed method improves algorithm efficiency and achieves better results than reported ones in literature.

Individual gathering could cause decline of search ability as heuristic methods are applied to heat exchanger network (HEN) optimization. An evaluation methodology for HEN structure diversity was designed to measure degree of individual structure gathering, and guided algorithm improvement. Firstly, group division of population was performed, individuals of a certain scale with a common structure were classified as a group to get individual structure distribution. Then dispersal search strategy was proposed to give perturbation to heat exchangers randomly selected from common structure of individuals in each group except the best one, which aimed at dispersing individual structures in groups. Concentration search strategy was then proposed to enhance exploitation for excellent structure by making other individuals accept a common structure of the optimal group. Finally, two cases involving nine and fifteen streams proved that dispersal search strategy strengthened global search ability and concentration search strategy strengthened local search ability. It obtained results decreased by 7 008 $·a^-1 and 17 973 $·a^-1, respectively, compared to those obtained by original algorithm. They are superior to results in literature.

To avoid the problem of being disturbed by stochastic acceptance of imperfect solution for evolution process of individual optimal solution existing in optimization of heat exchanger network by random walk algorithm with compulsive evolution, an improved RWCE based on trilevel protection strategy is proposed. Individuals in population are divided into three levels. The lower-level is optimized by basic RWCE to protect global search ability of individuals. The middle-level reads historical optimal solution of the lower-level's individuals, and optimized by RWCE with fine tuning to protect evolution process of each individual's optimal solution from disruption. All individuals in the upper-level are initialized by solution of the best individuals in the middle-level, and optimized by RWCE with automatic fine search to ensure that the best individuals are fully searched. Finally, result of the upper-level is passed to corresponding individual at the lower-level. Two cases are optimized by using the algorithm,and results are better than those in literature. Evolution process of individual optimal solution is protected while accepting imperfect solution, therefore, both global search ability and local search ability are realized.

Aiming at popularing diversity is disappared or other reasons in the later stage of optimiging heat exchanger network synthesis problem, it is difficult to find direction of evolution which makes total annual cost further reduce in optimization. A coupling evolution strategy of heat exchanger was proposed. In later stage of heuristic algorithm optimization, heat exchangers with no heat load are taken into coupled evolution by a certain probability distribution, to find coupling match to reduce cost. It shows that the strategy is effective. The strategy was combined with RWCE algorithm to form a hybrid algorithm. Firstly, RWCE algorithm was used to explore solution domain to find potential solutions by its strong global searching ability. Secondly, the coupled evolution strategy is applied to further optimize for these explored solutions. Thirdly, the further optimized solutions are fed back to RWCE algorithm after mutated. The hybrid algorithm is applied to 10SP2 and 15SP, and better optimization results are obtained.

Optimization in random walking algorithm with compulsive evolution (RWCE) for heat exchanger network synthesis (HENS) turns to slow down in late evolution as heat load of all heat exchangers are optimized simultaneously. A random walk algorithm with compulsive evolution combined with restrictive-evolution strategy for heat unit (RS-RWCE) is proposed, in which number of evolved heat units in HEN is restricted in each random walk evolution to keep fast convergence of total annual cost in early evolution process and fine search in late evolution process. Specific examples are applied to verify high computational efficiency and accuracy of the strategy. It gives consideration to integer variable and continuous variable. The results are encouraging.

Due to non-convex and non-continuous complexity of MINLP problems, heuristic methods are subject to numerous local optima. A novel random walking algorithm with compulsive evolution (RWCE) is an efficient global optimization method for HENS, where probability δ of accepting imperfect network is able to keep population diversity and jump out of local optima effectively during evolution process. In this paper, strong ability of parameter δ to detect promising structure is proved. Structure of evolution process is compared and analyzed under different parameters. Furthermore, a strategy for accepting imperfect network oriented with number of heat exchangers is proposed to improve computational time and evolution ability of optimization. The strategy was applied to several cases taken from literature. It shows better performance of global optimization.

Particle swarm optimization (PSO) algorithm has strong ability to explore global optimal region for heat exchanger networks synthesis. However, particles may trap into local optima and converge prematurely in late evolution. Therefore, an improved particle swarm optimization algorithm based on diversity feedback and real-time updating strategy is proposed. Firstly, index of population health degree is established to evaluate population diversity during evolution. Secondly, a random perturbation strategy and a centrifugal strategy are combined respectively with PSO algorithm to enrich population diversity and enhance global search ability. Furthermore, gradient search strategy is applied to search efficiently local optima and improve computational efficiency of PSO algorithm. Finally, a feedback mechanism of population health degree is proposed to real-time monitor health status of population and further to adopt different update strategies for keeping particles healthy during evolution. The method was applied to several cases taken from literature and results are encouraging. They are better than those of other improvements for PSO.

By analyzing mechanism of premature phenomenon in particle swarm optimization (PSO), we found nature of premature convergence and proposed a "leap" strategy to jump out of local minimum, making halted particles "renewed" when they are trapped into a local optimum. The strategy is applied to nonlinear programming and results are encouraging. The improved PSO solves efficiently premature convergence of the algorithm applying in nonlinear optimizations and improves global search ability of PSO.