980a5f28111405e62506cc1f073c2aa9.ppt

- Количество слайдов: 29

ESCAPE-16 & PSE 2006 Garmisch-Partenkirchen, Germany Developments in the Sequential Framework for Heat Exchanger Network Synthesis of industrial size problems Rahul Anantharaman and Truls Gundersen Dept of Energy and Process Engineering Norwegian University of Science and Technology Trondheim, Norway Anantharaman & Gundersen, PSE/ESCAPE ’ 06 1

Overview 1. Introducing the Sequential Framework 1. 2. 3. Motivation Our Goal Our Engine 1. 2. Challenges 1. Combinatorial Explosion – MILP 1. 2. Automated starting values Modal trimming method Examples 1. 2. 4. Temperature Intervals EMAT as an area variable Non-convexities - NLP 1. 2. 3. Subproblems Loops 7 stream problem 15 stream problem Concluding remarks Anantharaman & Gundersen, PSE/ESCAPE ’ 06 2

Motivation for the Sequential Framework q Pinch Methods for Network Design q Improper trade-off handling q Time consuming q Several topological traps q MINLP Methods for Network Design q Severe numerical problems q Difficult user interaction q Fail to solve large scale problems q Stochastic Optimization Methods for Network Design q Non-rigorous algorithms q Quality of solution depends on time spent on search Anantharaman & Gundersen, PSE/ESCAPE ’ 06 3

Motivation for the Sequential Framework q HENS techniques decompose the main problem q Pinch Design Method is sequential and evolutionary q Simultaneous MINLP methods let math considerations define the decomposition q The Sequential Framework decomposes the problem into subproblems based on knowledge of the HENS problem § Engineer acts as optimizer at the top level § Quantitative and qualitative considerations Anantharaman & Gundersen, PSE/ESCAPE ’ 06 4

Our Ultimate Goal q Solve Industrial Size Problems q Defined to involve 30 or more streams q Include Industrial Realism q Multiple Utilities q Constraints in Heat Utilization (Forbidden matches) q Heat exchanger models beyond pure countercurrent q Avoid Heuristics and Simplifications q No global or fixed ΔTmin q No Pinch Decomposition q Develop Semi-Automatic Design Tool q A tool Seq. HENS is under development § EXCEL/VBA (preprocessing and front end) § MATLAB (mathematical processing) § GAMS (core optimization engine) q Allow significant user interaction and control q Identify near optimal and practical networks Anantharaman & Gundersen, PSE/ESCAPE ’ 06 5

Our Engine – A Sequential Framework Adjust EMAT New HLD Preoptim. HRAT LP QH MILP QC (EMAT=0) U EMAT Vertical MILP HLD 1 Final NLP Adjust Units Adjust HRAT 2 Network 3 4 Compromise between Pinch Design and MINLP Methods Anantharaman & Gundersen, PSE/ESCAPE ’ 06 6

Challenges q Combinatorial explosion (binary variables) q Problem proved to be NP -complete in the strong sense § Any algorithm may take exponential number of steps to reach optimality q Use physical/engineering insights based on understanding of the problem § Will not remove the problem but help mitigate it q MILP and VMILP are currently the bottlenecks w. r. t. time (and thus size) q Local optima (non-convexities in the NLP model) q Convex estimators developed for MINLP models are computationally intensive § Only very small problems have been solved q Explore other options q Time to solve the NLP is not a problem q Relatively easier to solve than MINLP formulations Anantharaman & Gundersen, PSE/ESCAPE ’ 06 7

Anantharaman & Gundersen, PSE/ESCAPE ’ 06 8

Temperature Intervals (TIs) in the Vert. MILP model q Objective function is minimizing pseudo area H 1 q Vert. MILP model works best when the pseudo area accurately reflects the actual HX area q This happens when the number of TIs approaches infinity m-1 n-1 C 1 i m n j HH m+1 n+1 CC § Size of the Vert. MILP model increases exponentially with the number of temperature intervals § The transportation model has a polynomial time algorithm → Keep number of TIs to a minimum while ensuring the model achieves its objective Anantharaman & Gundersen, PSE/ESCAPE ’ 06 9

Temperature Intervals (TIs) in the Vert. MILP model q Original philosophy of the Vert. MILP model q Minimum area is achieved by vertical heat transfer q Temperature intervals must facilitate vertical heat transfer q Use Enthalpy Intervals to develop the vertical TIs q The Normal and Enthalpy based (vertical) TIs are the basis for the Vert. MILP model EMAT q Elaborate testing show that the Vert. MILP model achieves its objective with this set of TIs q Size of the model is reduced, on an average, by 10% (more for larger models) Anantharaman & Gundersen, PSE/ESCAPE ’ 06 10

EMAT as an Area Variable q Choosing EMAT is not straightforward q EMAT set too low (close to zero) § non-vertical heat transfer (m=n) will have very small ΔTLM, mn and very large penalties in the objective function q EMAT set too high (close to HRAT) § Potentially good HLDs will be excluded from the feasible set of solutions q HLDs are affected by the choice of EMAT q EMAT comes into play only when there is an extra degree of freedom in the system : U > Umin Anantharaman & Gundersen, PSE/ESCAPE ’ 06 11

Automated Starting Values and Bounds for the NLP subproblem q Multiple starting values for the NLP subproblem q Ensure a feasible solution q Explore different local optima q Use physical insight to ensure `good´ local optima q Heat Capacity Flowrates (m. Cps) identified to be the decision variables q Lower Bounds for Area were found to be crucial in getting a feasible solution q Information from the Vert. MILP subproblem is utilized q 4 different strategies for starting values were explored q Ref. : Hilmersen S. E. and Stokke A. , M. Sc Thesis , NTNU 2006 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 12

Serial/Parallel m. Cp Generator q Simple & flexible method q Little physical insight needed q Parallel arrangement gives feasible solution to most problems (90%) Anantharaman & Gundersen, PSE/ESCAPE ’ 06 13

Clever Serial m. Cp Generator q Serial configuration assumed for all streams q Assigns demanding exchangers at the supply end q Only stream temperatures are considered q Heat exchanger duties & stream m. Cp values are not considered q Assumed sequence of heat exchangers § Hot supply end matched with ranked set of cold targets & vice versa q Similar to the Ponton/Donaldson heuristic synthesis approach q Only serial configuration is limiting in many cases q Feasible solution in 50% of cases tested Anantharaman & Gundersen, PSE/ESCAPE ’ 06 14

Combinatorial m. Cp Generator q Utilizes heat loads, temperatures and overall m. Cp values to assign stream flows q Uses physical insight to determine flows q Provides a feasible solution to the NLP subproblem in all cases tested Anantharaman & Gundersen, PSE/ESCAPE ’ 06 15

Modal Trimming Method for Global Optimization of NLP subproblem Anantharaman & Gundersen, PSE/ESCAPE ’ 06 16

Modal Trimming Method for Global Optimization of NLP subproblem q Search for feasible solutions is the most important step Testing showed the Modal Trimming method to be inefficient and computationally expensive for solving the NLP model Anantharaman & Gundersen, PSE/ESCAPE ’ 06 17

Illustrating Example 1 Stream Tin (K) Tout (K) m. Cp (k. W/K) ΔH (k. W) h (k. W/m 2 K) H 1 626 586 9. 802 392. 08 1. 25 H 2 620 519 2. 931 296. 03 0. 05 H 3 528 353 6. 161 1078. 18 3. 20 C 1 497 613 7. 179 832. 76 0. 65 C 2 389 576 0. 641 119. 87 0. 25 C 3 326 386 7. 627 457. 62 0. 33 C 4 313 566 1. 690 427. 57 3. 20 ST 650 - - 3. 50 CW 293 308 - - 3. 50 Exchanger cost ($) = 8, 600 + 670 A 0. 83 (A is in m 2) References: Example 3 in Colberg, R. D. and Morari M. , Area and Capital Cost Targets for Heat Exchanger Network Synthesis with Constrained Matches and Unequal Heat Transfer Coefficients, Computers chem. Engng. Vol. 14, No. 1, 1990 Example 4 in Yee, T. F. and Grossmann I. E. , Simulataneous Optimization Models for Heat Integration - II. Heat Exchanger Network Synthesis, Computers chem. Engng. Vol. 14, No. 10, 1990 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 18

Example 1 – Initial Values Adjust EMAT 2 New HLD HRAT LP QH MILP QC (EMAT=0) U EMAT Vertical MILP HLD Adjust Units HRAT fixed at 20 K QH = 244. 1 k. W QC = 172. 6 k. W 1 Final NLP Network 3 Absolute Minimum Number of Units = 8 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 19

Example 1 – Looping to Solution Adjust EMAT 2 New HLD HRAT LP QH MILP QC (EMAT=0) U EMAT Vertical MILP 1 Final HLD NLP Adjust Units Network 3 Soln. No U EMAT (K) HLD# INVESTMENT COST ($) 1 8 2. 5 A 199, 914 2 8 2. 5 B Not feasible 3 9 2. 5 A 147, 861 4 9 2. 5 B 151, 477 5 9 5. 0 A 147, 867 6 9 5. 0 B 151, 508 7 10 2. 5 A 164, 381 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 20

Example 1 – `Best´ Solution HRAT = 20, EMAT = 2. 5, ΔTsmall= 3 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 21

Example 1 – Solution Comparisons No of Units Area (m 2) Cost Remarks Colberg & Morari (1990) 22 173. 6 - Colberg & Morari (1990) 12 188. 9 $177, 385 Synthesized network by evolution Yee and Grossmann (1990) 9 217. 8 $150, 998 Optimized w. r. t. cost $147, 861 MILP optimized w. r. t ”area” NLP optimized w. r. t cost Our work 9 189. 7 Optimized w. r. t area Spaghetti design Anantharaman & Gundersen, PSE/ESCAPE ’ 06 22

Illustrating Example 2 Stream Tin (°C) Tout (°C) m. Cp (k. W/°C) ΔH (k. W) h (k. W/m 2 °C) H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 C 1 C 2 C 3 C 4 C 5 C 6 C 7 ST CW 180 280 140 220 180 200 120 40 100 40 50 50 90 160 325 25 75 120 75 40 120 55 60 40 230 220 290 250 190 250 325 40 30 60 30 30 50 35 30 100 20 60 35 30 60 50 60 3150 9600 3150 3000 5000 4375 4200 8000 3800 7200 8750 7200 12000 5400 2 1 1 2 0. 4 0. 5 1 1 2 2 2 1 3 1 2 Exchanger cost ($) = 8, 000 + 500 A 0. 75 (A is in m 2) Reference: Björk K. M and Nordman R. , Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods, Chemical Engineering and Processing. Vol. 44, 2005 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 23

Example 2 – Initial Values Adjust EMAT 2 New HLD HRAT LP QH MILP QC (EMAT=0) U EMAT Vertical MILP HLD 1 Network 3 Adjust Units HRAT fixed at 20. 35 C QH = 11539. 25 k. W QC = 9164. 25 k. W Final NLP Absolute Minimum Number of Units = 14 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 24

Example 2 – Looping to Solution Adjust EMAT 2 New HLD HRAT LP QH MILP QC (EMAT=0) U EMAT Vertical MILP 1 Final HLD NLP Adjust Units Network 3 Soln. No U EMAT (K) HLD# TAC ($) 1 14 2. 5 A 1, 545, 375 2 15 2. 5 A 1, 532, 148 3 15 2. 5 B 1, 536, 900 4 15 5. 0 A 1, 529, 968 5 15 5. 0 B 1, 533, 261 6 16 2. 5 A 1, 547, 353 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 25

Example 2 – `Best´ Solution HRAT = 20. 35 EMAT = 5 ΔTsmall= 4. 9 Anantharaman & Gundersen, PSE/ESCAPE ’ 06 26

Example – Solution Comparison q The solution given here with a TAC of $1, 529, 968, about the same cost as the solution presented in the original paper (TAC $1, 530, 063) q When only one match was allowed between a pair of streams the TAC is reported as $1, 568, 745 - Björk & Nordman (2005) q The Sequential Framework allows only 1 match between a pair of streams q Solution at Iteration 2 (TAC $ 1, 532, 148) provides a slightly more expensive but slightly less compless network q Unable to compare the solutions apart from cost as the paper did not present the networks in their work Anantharaman & Gundersen, PSE/ESCAPE ’ 06 27

Global vs Local Optimum q Global optima in each of the subproblems does not, by itself, ensure overall global optimum for the HENS problem q Inherent feature of any problem decomposition q The emphasis has been on utilizing knowledge of the problem and engineering insight to achieve a network close to global optimum Anantharaman & Gundersen, PSE/ESCAPE ’ 06 28

Concluding Remarks q Sequential Framework has many advantages q Automates the design process q Allows significant User interaction q Numerically much easier than MINLPs q Progress q q EMAT identified as an optimizing `area variable´ Improved HLDs from Vert. MILP subproblem Algorithm for generating optimal TIs for the Vert. MILP Significantly better and automated starting values for NLP subproblem q Limiting elements q NLP model for Network Generation and Optimization § Enhanced convex estimators are required to ensure global optimum q Vert. MILP Transportation Model for promising HLDs § Significant improvements required to fight combinatorial explosion q MILP Transhipment model for minimum number of units § Similar combinatorial problems as the Transportation model Anantharaman & Gundersen, PSE/ESCAPE ’ 06 29