EIP problem statement and model

Water integration in EIP is modeled as an industrial water network (IWN) allocation problem, according to numerous previous works^2, (M. A. Ramos et al. 2014)^,19,20 and based on the same model employed by Ramos et al. (M. A. Ramos et al. 2016). Indeed, the way to model a IWN allocation problem is based on the concept of superstructure (Yeomans & Grossmann, 1999;

Biegler, Grossmann & Westerberg, 1997). Note that the present work focuses on an EIP without water-regeneration units. From a given number and processes, all possible connections between them may exist, except recycling to the same process. For each water-intensive using process, input water may be freshwater and/or output water from other processes. In the same way, output water from a process may be directly discharged and/or distributed to another process. For the sake of simplicity and generalization, the problem is built as a set of black boxes. In this kind of approach, physical or chemical phenomena occurring inside each process is not taken into account. In addition, each process has a contaminant load over the input flowrate of water. As aforementioned, only one contaminant is considered in the presented EIP. A general view of the superstructure is given in Figure 4.

Figure 2. General view of the superstructure for IWN allocation problem (M. A. Ramos et al. 2016).

- 180 -

Mathematically speaking, let np denote the given number of processes per plant, {1,2,..., }

P np denote the index set of processes, and let nep denote the given number of plants in the EIP, EP{1,2,...,nep} denote the index set of plants. Each process p P of each plant ep EP has a given contaminant load, denoted by _{Mep p,} , a given maximum concentration of contaminant allowed either in the inlet as in the outlet, denoted by Cmax_{ep p}ⁱⁿ_, , Cmax_{ep p}^out_, respectively. It is important to highlight that contaminant partial flows are neglected, since their magnitude is considerably lower in comparison to water flows. Therefore, it is assumed that the total flow between processes is equivalent to water flowrate. Moreover, it is assumed that processes will only consume the exact amount of water needed to satisfy concentration constraints.

Consequently, processes water outlet will have a concentration equivalent to Cmax_{ep p}^out_, (cf.

Bagajewicz and Faria (Bagajewicz & Faria, 2009) for detailed explanation). In terms of variables, each process of each plant p P ep EP ,  sends water to process p P' of plant

' ,{ ', '} { , }

ep EP ep p  ep p , taken into account by variableFpart_{ep p ep p, , ', '}, receives water, denoted by variable _{Fpartep p ep p', ', ,} and has an inlet flow of freshwater, denoted by _{Fwep p,} . In addition, each process may send polluted water to directly to the discharge, denoted by _{Fdisep p,} . Finally, it is to be noted that the original model (e.g. Bagajewicz and Faria (Bagajewicz &

Faria, 2009), Boix et al. (Boix, Montastruc, Pibouleau, et al., 2012) and Ramos et al. (M. A. Ramos et al. 2014)) was formulated as a mixed-integer linear program (MILP), since it takes into account minimum allowable flowrate between processes and/or regeneration units (namely, the minimum allowed water flowrate was fixed at 2 T/h in Boix et al.⁸). Nevertheless, in a MLSFG formulation discrete variables are rather impossible to handle in a deterministic way if discrete decisions are part of the follower problem, since no continuous optimality conditions exist for these kind of problems. Note that the solution methodology of MLSFG models includes the reformulation of the follower problem via his KKT conditions. On the other hand, note that MLSFG are solved in this work by formulating strong-stationarity conditions on all leaders’ problems, leading to a MOPEC formulation. Strong-stationarity conditions for non-continuous functions is rather a subject under research. In consequence, in the present article minimum flowrate minf is handled by an elimination algorithm which is explained afterwards.

Given the aforementioned notation, the model without regeneration units presented by Ramos et al. (M. A. Ramos et al. 2016) for the IWN without regeneration units in EIP is presented

- 181 -

below. Note that the model is presented in its final form after manipulation (cf. Ramos et al. (M. A.

Ramos et al. 2016) for details):

- Mass balance around a process unit:

, ', ' ', ', , , ', ',

M Cmax Fpart Cmax Fw Fpart

ep EP p P

-Inlet/outlet concentration constraints for a process unit:

', ' ', ', , , ', ',

-Freshwater positivity for a process unit:

, 0, ,

Fwep p   ep EP p P _{Eq. 4}

-Flow between processes positivity

, , ', ' 0, , ' , , ' ,{ , } { ', '}

ep p ep p

Fpart  ep ep EP p p P ep p   ep p _{Eq. 5} The respective objective function of each plant ep EP aims to minimize his annualized operating cost, defined by:

- 182 -

where _fw^cost stands for the purchase price of freshwater, dis for the cost associated ^cost to polluted water discharge and ex for the cost of pumping polluted water from one process to ^cost another. Indeed, each plant pays the cost of pumping water both to a process and from a process between processes of the same plant. Then, according to the payment policy adopted in the EIP, i.e. pol pol pol , shared payment, who sends water pays and who receives water pays ¹, ², ³ respectively, plants account for the cost of in-coming and out-coming water to other plants. On the other hand, the EIP authority aims to minimize total freshwater consumption in the EIP.

In the MLSFG problem, plants act as leaders and the EIP authority as the common follower.

In order to maintain the same notation as above, we define:

   

Formally, each plant ep EP bi-level optimization problem is the following:

- 183 -

As it can be seen from Prob. 2, each plant controls the flows from each one of its processes to all other processes (included those to other plants), while his problem is parameterized by the same respective variables of other plants and the freshwater flow to its processes, controlled by the follower.

2.2.3. All equilibrium MPEC reformulation

Assuming that a follower k problem (PFk) is convex, i.e. _z and m are respectively convex functions and concave functions in w , then for any solution ( , )w v of the following Karush-Kuhn-k k

Tucker (KKT) optimality conditions, w is a global optimal solution of (PF_k ^k). Note that KKT conditions are equivalent to the parametric nonlinear complementarity problem (NCP) (Leyffer &

Munson, 2010) ; (Kulkarni & Shanbhag, 2014):

( , , ) ( , , ) 0 0

By substituting follower’s problem in each leader problem, the all-equilibrium bilevel MLSFG described in Prob. is transformed into the following MPEC for each leader (Prob. 4):

- 184 -

The reader is encouraged to refer to Ramos et al. (M. A. Ramos et al. 2016) for detailed explanations regarding the transformation of the bi-level problem into a MPEC.

In Prob. 4 it can be seen that each variable of the follower is duplicated for each leader (even multipliers), in a way consistent with the bilevel ^ae formulation. Then, each leader is now constrained by the KKT conditions of the follower regarding both his own conjecture as other leaders’ conjectures. In other words, leaders now control both their own variables, and their own conjectures about follower’s response (multipliers included), while they are parameterized by other leaders’ variables and their conjectures about follower’s response.

Note that Prob. 4   constitutes a so-called MOPEC (multiple optimization problems i L with equilibrium constraints). The MLSFG in this form is indeed in a more convenient form in order to solve it.