Heuristics with performance guarantees for the minimum number of matches problem in heat recovery network design

Abstract

Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an N P-hard mixedinteger linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods.This manuscript is dedicated, with deepest respect, to the memory of Professor C. A. Floudas.Professor Floudas showed that, given many provably-strong solutions to the minimum number of matches problem, he could design effective heat recovery networks. So the diverse solutions generated by this manuscript directly improve Professor Floudas' method for automatically generating heat exchanger network configurations. 1988, Furman and Sahinidis 2002, Escobar and Trierweiler 2013). Floudas et al. (2012) review the critical role of heat integration for energy systems producing liquid transportation * fuels (Niziolek et al. 2015). Other important applications of HENS include: refrigeration systems (Shelton and Grossmann 1986), batch semi-continuous processes (Zhao et al. 1998, Castro et al. 2015 and water utilization systems (Bagajewicz et al. 2002).Heat exchanger network design is a mixed-integer nonlinear optimization (MINLP) problem (Yee and Grossmann 1990, Ciric and Floudas 1991, Papalexandri and Pistikopoulos 1994, Hasan et al. 2010. Mistry and Misener (2016) recently showed that expressions incorporating logarithmic mean temperature difference, i.e. the nonlinear nature of heat exchange, may be reformulated to decrease the number of nonconvex nonlinear terms in the optimization problem. But HENS remains a difficult MINLP with many nonconvex nonlinearities. One way to generate good HENS solutions is to use the so-called sequential method (Furman and Sahinidis 2002). The sequential method decomposes the original HENS MINLP into three tasks: (i) minimizing utility cost, (ii) minimizing the number of matches, and (iii) minimizing the investment cost. The method optimizes the three mathematical models sequentially with: (i) a linear program (LP) (Cerda et al. 1983, Papoulias andGrossmann 1983), (ii) a mixed-integer linear program (MILP) (Cerda and Westerberg 1983, Papoulias and Grossmann 1983), and (iii) a nonlinear program (NLP) (Floudas et al. 1986). The sequential method may not return the global solution of the origi...

Globally optimal synthesis of heat exchanger networks. Part I: Minimal networks

“…Such definition almost always leads to a total number of units that is larger than N min (as defined above), and consequently these structures are rarely MSTR structures. Similarly, structures with nonisothermal mixing and with bypasses that cannot be represented by the staged superstructure have been studied and rendered values given by N min that are virtually impossible using other limited superstructures.Remark There have been several studies in the literature regarding the a‐priori calculation of the minimum number of units, sometimes referred to as minimum number of matches . Letsios et al presented a proof that the minimum number is N min = NH + NC + NHU + NCU‐L , where NHU and NCU are the number of hot and cold utilities types respectively and L ∈ [1, Min { NH + NHU , NC + NCU }].…”

“…The issue is also complicated by the fact that the heat transfer is limited by the second law of thermodynamics, that is, temperature differences are limited to be higher than a certain minimum, an issue that complicates the proofs. Letsios et al make such a distinction when they introduce multiple temperature intervals. Even after this limitation is introduced, they cannot obtain a solution directly, and have to rely on “approximation” algorithms.Remark We believe that the aforementioned works identify the problem as NP‐hard correctly, but only because of their incomplete formulation when multiple intervals are introduced.…”

“…Letsios et al make such a distinction when they introduce multiple temperature intervals. Even after this limitation is introduced, they cannot obtain a solution directly, and have to rely on “approximation” algorithms.Remark We believe that the aforementioned works identify the problem as NP‐hard correctly, but only because of their incomplete formulation when multiple intervals are introduced. Bagajewicz and Valtinson showed that linear constraints can be added to solve the problem rigorously and globally using small computational time.…”

“…Remark 5 There have been several studies in the literature regarding the a-priori calculation of the minimum number of units, sometimes referred to as minimum number of matches. 31,32 Letsios et al 32 presented a proof that the minimum number is Min{NH + NHU, NC + NCU}]. For these authors, 31 the big issue is to obtain the maximum value of L. They state the problem is NP-hard, and provide a series of theorems, lemmas, and approximation procedures that intend to obtain bounds of the solution.…”

Evaluation of the Complexity, Controllability and Observability of Heat Exchanger Networks Based on Structural Analysis of Network Representations

“…The optimisation of HENs can be formalised in several ways. Algorithms can focus on the minimum number of matches [13], the Maximum Energy Recovery (MER) [14], the Minimum Energy-Capital cost [15], the minimum Total Annualised Cost (TAC) [16], the minimum number of exchangers [17], or, in the case of retrofit design, the minimum number of additional exchangers and the additional area of the exchangers or piping costs [18]. The goals listed above can be achieved by different algorithms, such as the Pinch methodology [19], dual-temperature approach method [20], pseudo-pinch [21], Supertargeting [14], State-Space approach [16], branch-and bound-based algorithms [22], or the application of Genetic Algorithm and Simulated Annealing (GA and SA, respectively) [23].…”