(422e) Mercury Trading in Water: Application of Stochastic Programming for Decision Making

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, andred_i are the
volumetric discharge quantity and desired pollutant quantity reduction for PS_i.
P_iis the treatment cost incurred by PS_ito
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_ijbe the binary variables
representing the point source-technology correlation. The variable is 1 when PS_iimplements technology j, and 0 otherwise. Let t_ik(mass/year)
be the amount of pollutant traded by PS_iwith PS_k.           

Let U_ijrepresent the design variables for PS_iif 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.