Abstract
In this paper, we consider the design of water usage and treatment systems in industrial plants. In such a system, the demand of water using units as well as environmental regulations for wastewater have to be met. To this end, water treatment units have to be installed and operated to remove contaminants from the water. The objective of the design problem is to simultaneously optimize the network structure and water allocation of the system at minimum total cost. Due to many bilinear mass balance constraints, this water allocation problem is a nonconvex mixed integer nonlinear program (MINLP) where nonlinear solvers have difficulties to find feasible solutions for real world instances. Therefore, we present a problem specific algorithm to iteratively solve this MINLP. In each iteration, this algorithm deals with an interplay of a mixed integer linear program (MILP) and a quadratically constrained program (QCP). First, an MILP approximates the original problem via discretization and provides a suitable network structure. Then, by fixing this structure, the original MINLP turns into a QCP which yields feasible solutions to the original problem. To improve the accuracy of the generated structure, the discretization of the MILP is adapted after each iteration based on the previous MILP solution. In many cases where nonlinear solvers fail, this approach leads to feasible solutions with good solution quality in short running time.
Similar content being viewed by others
References
Bagajewicz M (2000) A review of recent design procedures for water networks in refineries and process plants. Comput Chem Eng 24(9):2093–2113
Castro PM (2015) Tightening piecewise McCormick relaxations for bilinear problems. Comput Chem Eng 72:300–311
Chaturvedi ND, Bandyopadhyay S (2014) Optimization of multiple freshwater resources in a flexible-schedule batch water network. Ind Eng Chem Res 53(14):5996–6005
Faria DC, Bagajewicz M (2008) A new approach for the design of multicomponent water/wastewater networks. Comput Aided Chem Eng 25:43–48
Galan B, Grossmann IE (1998) Optimal design of distributed wastewater treatment networks. Ind Eng Chem Res 37(10):4036–4048
Gamrath G, Fischer T, Gally T, Gleixner AM, Hendel G, Koch T, Maher SJ, Miltenberger M, Müller B, Pfetsch ME, Puchert C, Rehfeldt D, Schenker S, Schwarz R, Serrano F, Shinano Y, Vigerske S, Weninger D, Winkler M, Witt JT, Witzig J (2016) The SCIP Optimization Suite 3.2. Technical Report 15-60, ZIB, Takustr. 7, 14195 Berlin
GAMS Development Corporation (2016) General algebraic modeling system (GAMS) release 24.7.3. http://www.gams.com. Accessed 9 April 2018
Goderbauer S, Bahl B, Voll P, Lübbecke ME, Bardow A, Koster AMCA (2016) An adaptive discretization MINLP algorithm for optimal synthesis of decentralized energy supply systems. Comput Chem Eng 95:38–48
Gouws JF, Majozi T, Foo DCY, Chen CL, Lee JY (2010) Water minimization techniques for batch processes. Ind Eng Chem Res 49(19):8877–8893
Gunaratnam M, Alva-Argáez A, Kokossis A, Kim JK, Smith R (2005) Automated design of total water systems. Ind Eng Chem Res 44(3):588–599
IBM (2015) ILOG CPLEX optimization studio release 12.6.3. http://www.ibm.com/software/integration/optimization/cplex-optimizer/. Accessed 9 April 2018
Jezowski J (2010) Review of water network design methods with literature annotations. Ind Eng Chem Res 49(10):4475–4516
Karuppiah R, Grossmann IE (2006) Global optimization for the synthesis of integrated water systems in chemical processes. Comput Chem Eng 30(4):650–673
Kolodziej S, Castro PM, Grossmann IE (2013) Global optimization of bilinear programs with a multiparametric disaggregation technique. J Global Optim 57(4):1039–1063
McLaughlin LA, McLaughlin HS, Groff KA (1992) Develop an effective wastewater treatment strategy. Chem Eng Progress 88(9):34–42
MINO Initial Training Network (2016) MINO industrial challenge. http://www.mino-itn.unibo.it/challenge-2016/. Accessed 9 April 2018
Misener R, Thompson JP, Floudas CA (2011) APOGEE: global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput Chem Eng 35(5):876–892
Nagarajan H, Lu M, Yamangil E, Bent R (2016) Tightening McCormick relaxations for nonlinear programs via dynamic multivariate partitioning. In: International conference on principles and practice of constraint programming, Springer, pp 369–387
Ng DKS, Foo DCY, Rabie A, El-Halwagi MM (2008) Simultaneous synthesis of property-based water reuse/recycle and interception networks for batch processes. AIChE J 54(10):2624–2632
PassMark Software Pty Ltd (2017) CPU benchmarks. https://www.cpubenchmark.net/cpu_list.php. Accessed 9 April 2018
Takama N, Kuriyama T, Shiroko K, Umeda T (1980) Optimal water allocation in a petroleum refinery. Comput Chem Eng 4(4):251–258
Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103:225–249
Teles JP, Castro PM, Matos HA (2012) Global optimization of water networks design using multiparametric disaggregation. Comput Chem Eng 40:132–147
Ullmer C, Kunde N, Lassahn A, Gruhn G, Schulz K (2005) WADO\(^{{ \text{TM}}}\): water design optimization-methodology and software for the synthesis of process water systems. J Clean Prod 13(5):485–494
Wang YP, Smith R (1994a) Design of distributed effluent treatment systems. Chem Eng Sci 49(18):3127–3145
Wang YP, Smith R (1994b) Wastewater minimisation. Chem Eng Sci 49(7):981–1006
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Log information
For the MILPs and QCPs of ADISCM-P with \(m = 5\) and for the MINLPs, Table 4 shows the number of constraints and number of binary and continuous variables. Furthermore, the number of performed iterations of ADISCM-P is stated. For these eight instances, ADISCM-P was able to solve 56% of the QCPs within the desired 1% gap. Another 20% of the QCPs has not been solved within the 1% gap but terminated with an average gap of 17%. For 24% of the QCPs, no feasible solution has been found within the time limit.
Appendix 2: Detailed computational results
Tables 5 and 6 show detailed results for all 99 test instances of the computational study from Sect. 5.5. In Table 6, the objective values of each instance are divided by the best solution among ADISCM-P and MINLP.
Rights and permissions
About this article
Cite this article
Koster, A.M.C.A., Kuhnke, S. An adaptive discretization algorithm for the design of water usage and treatment networks. Optim Eng 20, 497–542 (2019). https://doi.org/10.1007/s11081-018-9413-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-018-9413-6