LP-based solution strategies for the optimal design of industrial water networks with multiple contaminants
Introduction
Freshwater supply around the world is under pressure due to a growing population and a significant boost in agricultural and industrial wide range of products and applications. Growing freshwater demand is raising new concerns about water quality and generating greater awareness of both public and local control and management. Water is probably the most broadly used raw material in the process industries and it has been intensively used in abundant quantities by the chemical, petrochemical, petroleum refinery, food and drink, pulp and paper industries and many other for multiple purposes: materials separating agent, reactant, heating and cooling utility, cleaning agent for equipment, etc. However, few companies or investors have thought strategically about the risks to business posed by growing water scarcity. Corporations and investors are largely unaware of both freshwater-related risks and of management approaches and tools available to reduce them. Yet water scarcity already poses serious challenges to many companies and will dramatically affect many more if current trends continue. As a consequence, reducing wastewater has become one of the greatest challenges facing the process industries. As a first step, companies need to understand and measure their current water use and wastewater discharges associated with their own operations and production, or in any supply chain in which they are involved. This would provide the baseline data for assessing risks, prioritizing efforts and measuring progress.
Increasing water costs, restrictions on water use and increased environmental awareness have driven designers towards more efficient water systems, targeting for minimum water consumption through the identification of re-use and recycle opportunities. Typically, two approaches have been used to obtain good designs of these systems: pinch technology and mathematical programming. A comprehensive review of the various graphical and mathematical programming techniques to design and retrofit water networks can be found in Bagajewicz (2000). Recently, design methods based on genetic algorithms have also appeared (Prakotpol and Srinophakun, 2004, Lavric et al., 2005).
Pinch technology has proved very useful for targeting minimum freshwater consumption (Wang and Smith, 1994). After the targeting step, a few set of rules is used to design networks that verify those targets. Since then, the design procedure has become more systematic, and the design process can itself ensure that the target is not exceeded (Castro et al., 1999, Relvas et al., 2007). A similar design procedure has been used by Feng and Seider (2001) to simplify the piping network in large plants. For multiple contaminant systems, Wang et al. (2003) presented a related design methodology that achieves water networks that are easy to design, operate and control. Their method relies on the use of a single internal water main and can sharply reduce freshwater consumption, although it cannot guarantee optimal solutions. Ng and Foo (2006), recently introduced the new concept of water path analysis as a non-iterative means to reduce the number of interconnections within the network. By shifting the amount of water flowrate along a water path from a sink–source counterpart of lower concentration to one of higher concentration, an equal amount of freshwater can be added as a penalty to replace the shifted water load.
While the previous methods focus on mass transfer based water-using operations, important work has appeared in the literature dealing with fixed flowrate operations. A targeting method for single contaminant systems that is based on composite curves and on the concept of water surplus is given in Hallale (2002). A design method that meets the targets on freshwater use and wastewater generation is also proposed. Manan et al. (2004) described such graphical procedure as tedious and presented the water cascade analysis as the numerical equivalent that is able to generate the exact utility targets and pinch location more quickly. Similar work has been performed by Aly et al. (2005), who use the load problem table. More recently, Prakash and Shenoy (2005) have proposed a targeting method that deals with problems with both fixed contaminant load and fixed flowrate operations. Their approach is non-iterative, is completely analogous to the original composite curves used in heat integration and nearly identical to that of El-Halwagi et al. (2003). Network design is achieved through an algorithm that uses the principle of nearest neighbours, i.e., the water source streams that are chosen to satisfy a particular water demand are the nearest available neighbours in terms of contaminant concentration.
The drawback of pinch technology is that for a large number of operations, the piping network becomes very complicated and hard to design and it only provides optimal solutions for single contaminant systems. Although it can be adapted to multi-contaminant systems, the targeting and design step have proved to be very cumbersome and unreliable.
Mathematical programming approaches are more general and powerful. Takama et al. (1980) were the first to address the simultaneous design of water usage and treatment networks with a nonlinear (nonlinear program, NLP) formulation. Although few additional features have been introduced since then (Huang et al., 1999), the non-convex NLP model is very difficult to initialize. This is particularly important, since the solution returned from local optimization solvers (a local optimum) is very dependent on the starting point. More recently, Gunaratnam et al. (2005) proposed an automated design methodology based on the optimization of a superstructure that gives rise to a mixed integer nonlinear (MINLP) formulation. The binary variables are used to enforce certain network connections and/or to eliminate some substructures from consideration. In the first stage, a decomposition strategy divides the problem into MILP and LP problems. These are then solved in an iterative manner to provide an initial starting point that is then refined in the second stage through the solution of the general MINLP. The basis of the most important design stage comes from physical insights gained through conceptual approaches and aspects of mathematical programming. Overall, their approach is able to explore the synergies between the two subsystems, is capable of solving both systems separately and provides a robust technique, although it does not necessarily lead to the global optimum. Building on this work, Alva-Argaéz et al. (2006) proposed a novel decomposition approach that simplifies the challenges of the optimization problem, making systematic use of water-pinch insights to define successive projections in the solution space. Their generic methodology is applied for total water system design in petroleum refineries and has the potential to address retrofit problems. Karuppiah and Grossman (2006) solved the simultaneous design problem to global optimality through a new spatial branch and contract algorithm. Piecewise linear under- and over-estimators were used to approximate the non-convex terms in the original NLP/MINLP to obtain a MILP relaxation whose solution provides a tight lower bound at every node of the tree. Their algorithm is computationally faster than the global optimization solver BARON, but it is still computationally expensive for medium sized problems.
This paper focus on the design of the water-using network without considering the treatment units, neither in an integrated way nor as a part of a system that is located downstream, that are required to lower the concentration of the network wastewater stream's down to the discharge limits. The design problem can be formulated as a non-convex NLP, that can be initialized with the procedure of Doyle and Smith (1997). The starting point is generated with an approximated version of the NLP, where bilinear terms are removed by fixing the outlet concentrations in all operations to their maximum values (for all contaminants), thus leading to an LP. An alternative and also simple initialization procedure that is explored in this paper is to eliminate from the superstructure the connecting streams between the fixed contaminant load operations, which allows for the design problem to be formulated as an LP. However, the resulting network, where the units are arranged in parallel, lacks the possibility of water reuse, which is the responsible factor for most of the savings in freshwater consumption and wastewater generation.
Simple initialization procedures require the solution of a single LP before tackling the NLP, resulting in very fast solution strategies due to the smaller size of the mathematical problems. However, a sole starting point may lead to the local optimization solver to be trapped in a suboptimal solution and miss the global optimum. To overcome this, two new initialization procedures are proposed that provide multiple starting points, which may prove beneficial in finding better solutions. Both methods approximate the general NLP by a sequence of structurally different subproblems, one for each possible sequence of operations, in terms of the path followed by fresh/wastewater streams. A somewhat related technique was successfully used by Castro et al. (2007) in the design of wastewater treatment systems, where each subproblem involved the solution of a single LP. One of the new methods solves the LP resulting from the approximated method of Doyle and Smith (1997) for all operation sequences and achieves a trade-off between total computational effort and quality of the solution. The other, more original, tackles each operation unit, one at a time, with the purpose of previously determining the concentrations of all water streams that can be used to meet the demand of the operation unit under consideration. Such concentrations can be considered as parameters and hence bilinear terms are avoided, thus leading to a succession of LP problems. The new initialization procedure has the advantage of providing starting points that correspond to feasible networks where a few of them may even be global optimal solutions to the problem, but it is computationally more demanding. Due to the significant number of structurally different starting points that are generated by the four different solution procedures tested in this paper, it is highly likely that the best found solution is the global optimum and so it will be referenced as such, although unless the global optimization solver is able to solve the problem to zero optimality, there is no firm theoretical guarantee that it is so.
The remainder of the paper is structured as follows. The next section defines the water-using network design problem. The general superstructure is given in Section 3 while Section 4 presents the nonlinear programming formulation taken from the literature. The central part of the paper can be found in Section 5 were four alternative initialization procedures for the NLP are given. The computational results are given in Section 6 together with a detailed analysis in terms of a few performance measures. The optimal water-using networks for the majority of problems solved is shown in Section 7, while the conclusions are left for Section 8.
Section snippets
Problem statement
Industrial water-using networks comprise operations (set O) that may be classified into two broad categories (Prakash and Shenoy, 2005). Fixed contaminant load operations (set are quality controlled and may be modelled as mass transfer units (e.g. washing, scrubbing, extraction). Their data are often expressed by a limiting flowrate together with maximum inlet and outlet concentrations, which can be related to the mass exchange through Eq. (1). It
System superstructure
In order to find the optimal network, we must first consider all possibilities for combining the inlet freshwater streams and interconnecting the several water-using units. These are included in a general superstructure of the network, identical to the one proposed by Wang and Smith (1994) and used by Doyle and Smith (1997), among others. The superstructure shown in Fig. 1 includes the full set of freshwater streams (i.e. more than one quality of freshwater can be handled) and water-using
General NLP formulation
The optimization problem defined in the previous section can be formulated as a nonlinear programming model (Doyle and Smith, 1997). It uses total flows and concentrations as the model variables, which are represented in capital letters, contrary to the model parameters, to make it easier to identify bilinear terms. The required model variables are the following: , represents the flowrate of fresh water source needed to satisfy operation unit o; and are, respectively,
Efficient solution methods
The nonlinear model presented in the previous section features constraints containing bilinear terms in the mass balances. Such terms impose significant difficulties for the commercial NLP solvers, which may be unable to find a feasible solution to the problem or end up with a suboptimal local solution (an overview of NLP solution methods can be found in Biegler et al., 1997). Hence, initialization procedures that rely on a single starting point, such as the one by Doyle and Smith (1997) next
Computational results
The performance of the alternative initialization procedures is illustrated through the solution of nine example problems, for which the data were generated randomly. The computational studies were performed in a Pentium-4 3.4 GHz processor, with 2 GB of memory RAM, running Windows XP Professional. All mathematical formulations and algorithms were implemented in GAMS (Brooke et al., 2005) 22.2. The resulting LPs were solved with CPLEX while the NLPs were solved by MINOS, CONOPT and BARON to
Optimal water-using networks
In this section we present most of the global optimal water-using networks together with a description of their most relevant features. The network representation is achieved through an automated procedure that is supported on the superstructure given in Fig. 1 and features the problem data as well as the optimal values of all the model variables.
Conclusions
This paper has presented a new method for the optimal design of water using networks with multiple contaminants and a few freshwater sources. It relies on a standard superstructure that includes all possibilities for freshwater use and water reuse, and uses a standard non-convex NLP formulation as the second and final phase of the solution process. The novelty consists in the initialization procedure, the first phase of the solution process, which uses multiple starting points resulting from
Acknowledgements
A shorter version of this paper was presented in the 9th Conference on Process Integration, Modelling and Optimisation for Energy Saving and Pollution Reduction, 27-31 August 2006, Prague - Czech Republic.
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