1. Introduction  

Pollutant trading in a watershed is a market based strategy to economically achieve environmental waste management. Trading programs allow facilities facing higher pollution control costs to meet their regulatory obligations by purchasing environmentally equivalent (or superior) pollution reductions from another source at lower cost [1]. Since the relative success of trading for air pollutants, its application in dealing with water pollutants needs to be assessed. It is believed that successful implementation of credit trading policy can substantially lower the compliance costs for water pollutants. In the wake of the added flexibility due to trading, decision making for polluters such as process industries becomes complicated. This is because trading decisions, such as if and how much pollutant to trade, can have implications on the technology selection and design decisions for the industries. In order to utilize the full potential of trading in reducing compliance costs, simultaneous decision making related to trading and technology implementation is advantageous. Optimization theory provides the necessary tools to achieve this task. Its application in this area of simultaneous trading and technology selection decision making has been illustrated in [2]. This work extends that formulation by simultaneously optimizing the design decisions of the selected technology.

In this approach, it is important to account for two important aspects: nonlinearity and uncertainty. The performance and cost relationships for the technologies are often governed by complex nonlinear relationships. In addition, there are various sources of uncertainties in this setup. The efficiency of a treatment technology and its operating costs, even for an established technology, are often not known deterministically. Moreover, new treatment technologies are coming up, for which data availability is scarce. This further complicates the technology selection and design decisions.

Incorporation of nonlinearity and uncertainty necessitates the formulation of a stochastic nonlinear programming problem for the mentioned task. Such problems are often computationally difficult to solve, particularly in the presence of nonlinear relationships. This work proposes to use the L-shaped BONUS algorithm that has been recently proposed for the solution of such problems [3]. The new formulation will be implemented for decision making in the Savannah River watershed, with mercury as the pollutant, taking advantage of data availability and preliminary results from the previous work [2].  

2. Approach

2.1. Problem Formulation  

The formulation considers that TMDL (Total Maximum Daily Load) regulation has already been developed, translating into specific load allocations for each point source. Consider a set of point sources (PS_i), i=1,…N, disposing pollutant containing waste water to a common water body or watershed. Here D_i, and red_i are the volumetric discharge quantity and desired pollutant quantity reduction for PS_i. P_i is the treatment cost incurred by PS_i to reduce pollution when trading is not possible. Let j = 1,…,M  be the set of reduction technologies available to the point sources for implementation. Trading is possible between all point sources and a single trading policy exists between all possible pairs of point sources. Let r be the trading ratio and F be the trading transaction cost in $/Mass. The objective of the model is to achieve the desired TMDL goal at minimum overall cost. Let b_ij be the binary variables representing the point source-technology correlation. The variable is 1 when PS_i implements technology j, and 0 otherwise. Let t_ik (mass/year) be the amount of pollutant traded by PS_i with PS_k.           

Let U_ij represent the design variables for PS_i if it implements technology j. The performance (reduction capabilities) and costs for these technologies are governed by nonlinear relationships, and some parameters in these nonlinear models are uncertain. The quality and quantity of the waste being treated can vary, leading to uncertainty about the performance of the technology. The operating costs for the technology are subject to constant fluctuations depending on the market variables and material costs. These factors can lead to performance uncertainty even for well established processes. For new technologies, non availability of data can lead to uncertainties.

The objective in the problem is to reduce the overall compliance cost by optimizing the binary decision variables b_ij along with the continuous decision variables t_ik and U_ij for the point sources. The constraints ensure that all the targeted reductions red_i are achieved and no industry spends more while trading as compared to when not trading. The cost and performance of the technologies are governed by technology models. The presence of nonlinearity and uncertainty in the technology models leads to a mixed integer stochastic nonlinear programming problem (SMINLP).  

2.2. Two Stage Problem Formulation

            To make the problem solution computationally efficient, the SMINLP problem is decomposed into a two stage programming problem with recourse [4]. The idea behind the decomposition strategy is to take certain decisions in the first stage without the complete realization of uncertainty, while the uncertain recourse part is computed exactly only in the second stage. Sampling based L-shaped method is a well known decomposition based method in stochastic programming literature.

            For the presented problem, the decomposition strategy is used to separate the technology design decisions from the trading and technology selection decisions. The binary (technology selection) decisions b_ij and continuous decisions t_ik are made in the first stage, where the technology costs are linearly approximated. The nonlinear uncertain models for the technologies are then used in the second stage. Here, using the first stage decision variables, the technology design decisions U_ij are made and the expected value of the technology costs is computed.  

2.3. Solution Methodology

            The two stage formulation mentioned before can still be computationally demanding if the technology models are nonlinear and/or high dimensional. In such cases, high model simulation requirements in sampling based algorithm can seriously impede the solution speed. To circumvent this problem, L-Shaped BONUS algorithm has been recently proposed, which combines the sampling based L-shaped method with BONUS algorithm [3]. It uses reweighting scheme to bypass repeated model simulations in the second stage, which results in significant computational savings. The computational properties are further improved by using the efficient Hammersley Sequence Sampling technique.  

2.4 Savannah River Basin  

TMDL of 32.8 Kg/year of total mercury has been established for five contiguous segments of the Savannah River in Georgia, U.S. which corresponds to a water quality standard (WQS) of 2.8 ng/liter [5]. Based on the current volumetric discharge of each of the point sources, waste load allocation is carried out. In all, there are 29 significant point sources discharging mercury in the Savannah River watershed. The combined targeted reduction for point sources is taken to be 40%.

Three treatment technologies, coagulation and filtration, activated carbon adsorption and ion exchange process are considered. The cost data for the technologies is reported in [6, 7]. The trading ratio r is 1.1 and trading transaction fee, based on the average treatment costs of the processes, is around 4x10⁸ $/Kg.

3. Summary  

With the option of pollutant trading, simultaneous decision making related to trading and technology selection and design is advantageous. Consideration of nonlinearity and uncertainty in the technology performance and cost models is essential for a realistic analysis. The proposed work formulates a two stage stochastic mixed integer nonlinear programming problem and proposes to solve the problem using a recently proposed L-shaped BONUS algorithm. The methodology is to be validated using the Savannah River basin case study for mercury waste management. The results are expected to illustrate the potential of simultaneous decision making under uncertainty for performance optimization.  

References  

[1] USEPA. Draft framework for watershed-based trading. Technical report, EPA 800-R-96-001. Washington, DC: United States Environmental Protection Agency, Office of Water, 1996.  

[2] Shastri, Y., Diwekar, U. & Mehrotra, S. (2006). Water waste management and mercury trading: An optimization approach, Journal of Environmental Management, under review.  

[3] Shastri, Y. & Diwekar, U. (2006). An efficient algorithm for large scale stochastic nonlinear programming problems, Computers and Chemical Engineering, 30, 864-877.  

[4] U.M. Diwekar. Introduction to Applied Optimization. Kluwer Academic Publishers, Dordrecht, 2003.  

[5] USEPA. Total maximum daily load (TMDL) for total mercury in fish tissue residue in the middle and lower savannah River watershed. Report, United States Environmental Protection Agency, Region 4, 2001.  

[6] USDOI. Total plant costs: For contaminant fact sheets. Technical report, U.S.

Department of Interior, Bureau of Reclamation, Water treatment engineering and research group, Denver CO 80225, 2001.  

[7] USDOI. Water treatment estimation routine (WATER) user manual. Technical report, U.S. Department of Interior, Bureau of Reclamation, Denver CO 80225, 1999.