# An Indirect Simulation-Optimization Model for Determining Optimal TMDL Allocation under Uncertainty

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. BRRT-EILP Model

#### 2.2. Application to the Swift Creek Reservoir Watershed

^{2}; Figure 1). Its major environmental concerns are algal blooms and hypolimnetic hypoxia in summer, the majority caused by two stressors: the loadings of inorganic nitrogen (NO

_{3}

^{−}and NH

_{4}

^{+}) and inorganic phosphorus (PO

_{4}

^{3−}) from eight sub-watersheds [28]. A hydrodynamic and water-quality model, CE-QUAL-W2 v3.1, was developed to represent such stressor-response relationships from 1998–2000 [28]. Previously, a nutrient TMDL allocation for the SCR watershed was developed through a stochastic simulation method by stochastically changing the pollutant removal rates among nonpoint sources from the eight sub-watersheds [15,18,28]. In our study, we applied the proposed BRRT-EILP model to determine optimal nutrient TMDL allocation under uncertainty for the SCR watershed. The goal of such TMDL allocation was to identify the critical load reductions from the eight sub-watersheds for the maintenance of summer (June–September) maxima of vertically averaged chlorophyll-a (Chl-a) concentrations at the reservoir outlet (defined as ${y}^{*\pm}$) below Chl-a criteria. According to Equations (1) and (2), the minimum total load reduction model for inorganic phosphorus and nitrogen for the SCR watershed could be formulated as follows:

_{jl}) at river mouths divided by baseline loadings, that is ${x}_{jl}^{\pm}=({c}_{jl}^{\pm}-{A}_{jl}^{\pm})/{c}_{jl}^{\pm}$, where ${c}_{jl}^{\pm}$ is the baseline load of pollutant l in sub-watershed j, computed using Hydrological Simulation Program—FORTRAN (HSPF) v11 for the period from January–December 1998 (Table 1) [18]. ${\mathsf{\beta}}_{jl}^{ik}$ (including ${\mathsf{\beta}}_{0}^{ik}$) and ${\mathsf{\epsilon}}_{\mathsf{\alpha}}^{ik}$ are the regression coefficients and predictive error of the i-th interval linear regression equation at compliance point k, respectively. To derive the interval linear regression equations, the CE-QUAL-W2 v3.1 generated the simulated Chl-a (defined as ${y}^{ik\pm}$) with randomly-generated pollutant removal rates from 8 sub-watersheds, resulting in 2000 stressor-response datasets. We then calibrated these coefficients and determined the associated confidence intervals using BRRT v2. Since no Chl-a criteria have been published by either the United States Environmental Protection Agency or the State of Virginia, ${y}^{*\pm}$ was set at 12−15 μg/L in this study based on North Carolina water-quality standards and other references [28,29]. Three Chl-a criteria scenarios had been set as [12, 15], [13, 15] and [14, 15] μg/L; ${x}_{l}^{max}$ is the maximum removal level of the BMPs’ performance for pollutant l, which was set at 80% according to the American Society of Civil Engineers’ (ASCE) national BMP database; ${x}_{l}^{min}$ was set as zero, indicating that no nutrient load was removed; ${d}_{jl}^{i-}\in {D}^{i-}$, ${d}_{jl}^{i+}\in {D}^{i+}$.

**Table 1.**Baseline loadings (inorganic phosphorus and inorganic nitrogen) of Swift Creek Reservoir watershed.

Sub-Watershed j | Area, km^{2} | c_{i}_{1}, kg | c_{i}_{2}, kg |
---|---|---|---|

1. Swift Creek | 59.47 | 4451.7 | 157.4 |

2. Horsepen-Otterdale-Blackman Creek | 40.47 | 5169.3 | 388.4 |

3. Tomahawk Creek | 25.30 | 2677.2 | 126.8 |

4. West Branch | 7.50 | 286.7 | 87.9 |

5. Dry-Ashbrook Creek | 14.52 | 516.5 | 161.1 |

6. Direct runoff (1) | 3.73 | 1916.3 | 114.2 |

7. Direct runoff (2) | 6.13 | 1000.7 | 90.2 |

8. Direct runoff (3) | 3.31 | 471.7 | 146.1 |

Total | 160.4 | 16,490 | 1272 |

#### 2.3. Algorithmic Processes

_{j};

**Figure 2.**Algorithmic process for solving Bayesian recursive regression tree (BRRT)-enhanced-interval linear programming (EILP) model for risk-based optimal total maximum daily load (TMDL) allocation.

## 3. Results

^{2}= 0.73; Figure S1c).

**Table 2.**Optimal pollution removal levels of inorganic phosphorus and inorganic nitrogen for the Swift Creek Reservoir (SCR) watershed for three scenarios.

Sub-watershed j | Minimum Total Load Reduction of Inorganic Nitrogen | Minimum Total Load Reduction of Inorganic Phosphorus | ||||||||||

Scenario 1 (Chl-a: [14, 15] μg/L) | Scenario 2 (Chl-a: [13, 15] μg/L) | Scenario 3 (Chl-a: [12, 15] μg/L) | Scenario 1 | Scenario 2 | Scenario 3 | |||||||

${x}_{j1}^{\pm}$ * | ${c}_{j1}^{\pm}{x}_{j1}^{\pm}$ *, kg | ${x}_{j1}^{\pm}$ | ${c}_{j1}^{\pm}{x}_{j1}^{\pm}$, kg | ${x}_{j1}^{\pm}$ | ${c}_{j1}^{\pm}{x}_{j1}^{\pm}$, kg | ${x}_{j2}^{\pm}$ | ${c}_{j2}^{\pm}{x}_{j2}^{\pm}$, kg | ${x}_{j2}^{\pm}$ | ${c}_{j2}^{\pm}{x}_{j2}^{\pm}$, kg | ${x}_{j2}^{\pm}$ | ${c}_{j2}^{\pm}{x}_{j2}^{\pm}$, kg | |

1 | [0.42, 0.42] | [1870, 1870] | [0.42, 0.42] | [1870, 1870] | [0.42, 0.42] | [1870, 1870] | [0.62, 0.74] | [98.2, 116.2] | [0.62, 0.8] | [98.2, 125.6] | [0.62, 0.8] | [98.2, 125.6] |

2 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0.79, 0.79] | [308, 308] | [0.79, 0.79] | [308, 308] | [0.79, 0.8] | [308, 310.7] |

3 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0.51] | [0, 64.8] |

4 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] |

5 | [0, 0.23] | [0, 120] | [0, 0.71] | [0, 360] | [0, 0.8] | [0, 413] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] |

6 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0.12] | [0, 238] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] |

7 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0.8] | [0, 72.2] |

8 | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0] | [0, 0.8] | [0, 116.9] |

Total load reduction | [1870, 1989] | [1870, 2235] | [1870, 2521] | [406.2, 424.2] | [406.2, 433.6] | [406.2, 690.2] | ||||||

$E[{Z}_{opt}^{\pm}]$ | 1929.5 | 2052.5 | 2195.5 | 415.2 | 419.9 | 548.2 |

#### Minimum Total Load Reductions of Inorganic Nitrogen

## 4. Discussion

^{®}Core™ 2 Duo 2.4-GHz computer (CPU T8300) to complete one CE-QUAL-W2 model run. Other than that, the calibration and verification of the BRRT model takes only about 5 min, and finding the optimal solution of the BRRT-EILP model for three scenarios with 16 decision variables takes less than 1 min. Taken together, the total computational time taken to find out the approximate optimal solutions was only about 33 h. In contrast, running a traditional direct SOM framework using a GA approach might require several days (e.g., if a population size of 50 is used and the model is iterated for 300 generations, it takes over 10 days to obtain a solution) [11]. More time would be required to increase the chances of obtaining globally-approximate optimal solutions, because multiple GA runs would need to be executed for the same problem [21,22]. Moreover, the computation efficiency of the proposed BRRT-EILP model can be significantly higher than that of the traditional direct SOM approach when Equation (3) is used to evaluate solutions for different objective functions, water-quality criteria or system constraints. For example, assume that decision makers are interested in the optimal TMDL allocation in cases where the total cost, rather than the total reduction level, is to be minimized, or when water quality targets for dissolved oxygen are also considered, or when they are interested in achieving compliance at more than one location. In such cases, traditional direct SOM would involve re-running the stochastic search algorithm, which is very inefficient and can be computationally prohibitive if many scenarios need to be analyzed. In contrast, the BRRT-EILP model would require less than 1 min to compute such a new scenario, since it is not necessary to repeat the time-consuming data generation process required for BRRT calibration/verification in the scenario analysis stage.

Method | Calibration | New Application |
---|---|---|

BRRT-EILP model | 33 h | 1 min |

Direct SOM framework | ~10 days | ~10 days |

^{3}tonnes were identified as feasible inorganic nitrogen load reductions to meet Chl-a maximum criteria of 12, 13, 14 and 15 μg/L, respectively. Additionally, alternatives of 0.74, 0.50, 0.44 and 0.43 × 10

^{3}tonnes were identified as feasible load reductions for inorganic phosphorus. Second, a TAE analysis of inorganic nitrogen or phosphorus load reduction was conducted by iteratively reducing loads using different reduction ratios for different sub-watersheds. In this case, we first implemented the load reductions for the three sub-watersheds (Direct Runoff (1)–(3)) closest to the outlet until a certain pre-specified maximum achievable reduction ratio was reached and then reduced loads from the five upstream sub-watersheds until compliance with Chl-a criteria was achieved in the three scenarios. The minimum load reductions identified by the TAE method (4.3–5.7 × 10

^{3}tonnes for inorganic nitrogen, 0.62–0.91 × 10

^{3}tonnes for phosphorus inorganic) are much larger than those of the BRRT-EILP method. Figure 3, Figure 4 and Figure 5 show the minimum load reductions and pollution removal levels of the BRRT-EILP model compared to those of the two kinds of feasible alternatives generated by stochastic simulation and TAE methods. Specifically, the minimum inorganic phosphorus load reductions derived by the BRRT-EILP model were 7.0%, 14%, 4.8% and 5.9% less than those of the best stochastic simulation (SS) alternatives when the Chl-a criteria were levels below 12, 13, 14 and 15 μg/L, respectively (Figure 3a). The SS method required reducing inorganic phosphorus loads in all eight of the sub-watersheds, while the BRRT-EILP model required reductions in no more than five sub-watersheds (Figure 4). The reductions of inorganic nitrogen loads required by the BRRT-EILP solutions were 31%, 32%, 23% and 23% less than those of the best SS alternatives (Figure 3b). Additionally, the BRRT-EILP solutions only required inorganic nitrogen pollution removal in Swift Creek, Dry-Ashbrook Creek and Direct Runoff (1), while the removal rates of best SS alternatives required reductions greater than 0.3 in Sub-watersheds 1, 4, 5 and 8 (Figure 5). The plausible explanation for such sub-watershed selection by BRRT-EILP model is that the summer maximum vertically-averaged Chl-a concentration is more sensitive to load reductions in the selected sub-watersheds than the others. Another possible explanation is that the SS method uses random search, rather than the heuristic search applied by the BRRT-EILP model. In addition, the BRRT-EILP solutions were superior to those of the TAE method, since the latter ignores the difference in the sensitivity of Chl-a concentration to a unit of incremental load reduction across different sub-watersheds, while the BRRT-EILP model could rapidly identify critical sub-watersheds, resulting in the approximately most effective and reliable TMDL allocation. Therefore, the BRRT-EILP model may provide a new framework for updating the current TMDL allocation paradigm.

**Figure 3.**Minimum load reductions of the BRRT-EILP model, best stochastic simulation (SS) alternatives and trial-and-error (TAE) solutions that meet Chl-a criteria below 12, 13, 14 and 15 μg/L. (

**a**) Inorganic phosphorus; (

**b**) inorganic nitrogen. The percentage in the panels is calculated as the difference in minimum load reductions between the BRRT-EILP model and the SS method divided by that of the SS method.

**Figure 4.**Pollution removal levels of inorganic phosphorus solved by the BRRT-EILP model, best stochastic simulation (SS) alternatives and the TAE method that meet diverse Chl-a criteria. (

**a**) Chl-a = 12 μg/L; (

**b**) Chl-a = 13 μg/L; (

**c**) Chl-a = 14 μg/L and (

**d**) Chl-a = 15 μg/L. The red, blue and green solid lines indicate the solutions of the BRRT-EILP model, the best stochastic simulation (SS) alternatives and the TAE method, while the other lines (in gray) represent the other alternatives with the SS method.

**Figure 5.**Pollution removal levels of inorganic nitrogen solved by the BRRT-EILP model and the TAE simulation method that meet diverse Chl-a criteria. (

**a**) Chl-a = 12 μg/L; (

**b**) Chl-a = 13 μg/L; (

**c**) Chl-a = 14 μg/L and (

**d**) Chl-a = 15 μg/L. The meaning of the colors is the same as in Figure 4.

## 5. Conclusions

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- USEPA. Allocating Loads and Wasteloads, Technical Support Documents, Total Maximum Daily Load (TMDL); US EPA Press: Washington, DC, USA, 2003.
- Jia, Y.B.; Culver, T.B. Robust optimization for total maximum daily load allocations. Water Resour. Res.
**2006**, 42. [Google Scholar] [CrossRef] - Cerco, C.F.; Noel, M.R.; Kim, S.C. Three-dimensional management model for lake washington, part ii: Eutrophication modeling and skill assessment. Lake Reserv. Manag.
**2006**, 22, 115–131. [Google Scholar] [CrossRef] - Cole, T.; Wells, S.A. CE-QUAL-W2: A Two-Dimensional, Laterally Averaged, Hydrodynamic and Water Quality Model, Version 3.1; US Army Engineering and Research Development Center: Vicksburg, MS, USA, 2003. [Google Scholar]
- Boesch, D.F. The gulf of Mexico’s dead zone. Science
**2004**, 306, 977–978. [Google Scholar] [CrossRef] - Dhar, A.; Datta, B. Optimal operation of reservoirs for downstream water quality control using linked simulation optimization. Hydrol. Process.
**2008**, 22, 842–853. [Google Scholar] [CrossRef] - He, L.; Huang, G.H.; Lu, H.W. Health-risk-based groundwater remediation system optimization through clusterwise linear regression. Environ. Sci. Technol.
**2008**, 42, 9237–9243. [Google Scholar] [CrossRef] [PubMed] - He, L.; Huang, G.H.; Zeng, G.M.; Lu, H.W. An integrated simulation, inference, and optimization method for identifying groundwater remediation strategies at petroleum-contaminated aquifers in western canada. Water Res.
**2008**, 42, 2629–2639. [Google Scholar] [CrossRef] - Huang, G.H.; Huang, Y.F.; Wang, G.Q.; Xiao, H.N. Development of a forecasting system for supporting remediation design and process control based on NAPL-biodegradation simulation and stepwise-cluster analysis. Water Resour. Res.
**2006**, 42. [Google Scholar] [CrossRef] - Rejani, R.; Jha, M.K.; Panda, S.N. Simulation-optimization modelling for sustainable groundwater management in a coastal basin of orissa, india. Water Resour. Manag.
**2009**, 23, 235–263. [Google Scholar] [CrossRef] - Saadatpour, M.; Afshar, A. Waste load allocation modeling with fuzzy goals; simulation-optimization approach. Water Resour. Manag.
**2007**, 21, 1207–1224. [Google Scholar] [CrossRef] - Zhou, F.; Chen, G.X.; Huang, Y.F.; Yang, J.Z.; Feng, H. An adaptive moving finite volume scheme for modeling flood inundation over dry and complex topography. Water Resour. Res.
**2013**, 49, 1914–1928. [Google Scholar] [CrossRef] - Zhou, F.; Chen, G.X.; Noelle, S.; Guo, H.C. A well-balanced stable generalized riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes. Int. J. Numer. Methods Fluids
**2013**, 73, 266–283. [Google Scholar] [CrossRef] - Qin, X.S.; Huang, G.H.; Chakma, A. A stepwise-inference-based optimization system for supporting remediation of petroleum-contaminated sites. Water Air Soil Pollut.
**2007**, 185, 349–368. [Google Scholar] [CrossRef] - Guo, H.C.; Gao, W.; Zhou, F. Three-level trade-off analysis for decision making in environmental engineering under interval uncertainty. Eng. Optim.
**2014**, 46, 377–392. [Google Scholar] [CrossRef] - Huang, G.H. A stepwise cluster-analysis method for predicting air-quality in an urban-environment. Atmos. Environ. B Urban Atmos.
**1992**, 26, 349–357. [Google Scholar] [CrossRef] - Sahinidis, N.V. Optimization under uncertainty: State-of-the-art and opportunities. Comput. Chem. Eng.
**2004**, 28, 971–983. [Google Scholar] [CrossRef] - Wang, Z.; Gao, W.; Cai, Y.L.; Guo, H.C.; Zhou, F. Joint optimization of population pattern and end-of-pipe control under uncertainty for lake dianchi water-quality management. Fresenius Environ. Bull.
**2012**, 21, 3693–3704. [Google Scholar] - Zhou, F.; Huang, G.H.; Chen, G.X.; Guo, H.C. Enhanced-interval linear programming. Eur. J. Oper. Res.
**2009**, 199, 323–333. [Google Scholar] [CrossRef] - Birge, J.R.; Louveaux, F.V. Introduction to Stochastic Programming; Springer: New York, NY, USA, 1997. [Google Scholar]
- Zou, R.; Liu, Y.; Riverson, J.; Parker, A.; Carter, S. A nonlinearity interval mapping scheme for efficient waste load allocation simulation-optimization analysis. Water Resour. Res.
**2010**, 46. [Google Scholar] [CrossRef] - Zou, R.; Lung, W.S.; Wu, J. An adaptive neural network embedded genetic algorithm approach for inverse water quality modeling. Water Resour. Res.
**2007**, 43. [Google Scholar] [CrossRef] - Zhou, F.; Guo, H.C.; Chen, G.X.; Huang, G.H. The interval linear programming: A revisit. J. Environ. Inform.
**2008**, 11, 1–10. [Google Scholar] [CrossRef] - Iorgulescu, I.; Beven, K.J. Nonparametric direct mapping of rainfall-runoff relationships: An alternative approach to data analysis and modeling? Water Resour. Res.
**2004**, 40. [Google Scholar] [CrossRef] - Breiman, L.; Friedman, J.H.; Olshen, R.A.; Stone, C.G. Classification and Regression Trees; Wadsworth International Group: Belmont, CA, USA, 1984. [Google Scholar]
- Zhou, F.; Shang, Z.Y.; Zeng, Z.Z.; Piao, S.L.; Ciais, P.; Raymond, P.; Wang, X.H.; Wang, R.; Chen, M.P.; Yang, C.L.; et al. New model for capturing the variations of fertilizer-induced emission factors of n2o. Glob. Biogeochem. Cycles
**2015**, 29. [Google Scholar] [CrossRef] - Harmel, R.D.; Cooper, R.J.; Slade, R.M.; Haney, R.L.; Arnold, J.G. Cumulative uncertainty in measured streamflow and water quality data for small watersheds. Trans. ASABE
**2006**, 49, 689–701. [Google Scholar] [CrossRef] - Wu, J.; Zou, R.; Yu, S.L. Uncertainty analysis for coupled watershed and water quality modeling systems. J. Water Resour. Plann. Manag.
**2006**, 132, 351–361. [Google Scholar] [CrossRef] - Wetzel, R.G. Limnology—Lake and River Ecosystems, 3rd ed.; Academic Press: New York, NY, USA, 2001. [Google Scholar]
- U.S. Environmental Protection Agency (USEPA). Protocol for Developing Nutrient TMDLs; EPA 841-B-99-007; Office of Water (4503F), United States Environmental Protection Agency: Washington, DC, USA, 1999.
- Conley, D.J.; Paerl, H.W.; Howarth, R.W.; Boesch, D.F.; Seitzinger, S.P.; Havens, K.E.; Lancelot, C.; Likens, G.E. Ecology controlling eutrophication: Nitrogen and phosphorus. Science
**2009**, 323, 1014–1015. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Zhou, F.; Dong, Y.; Wu, J.; Zheng, J.; Zhao, Y.
An Indirect Simulation-Optimization Model for Determining Optimal TMDL Allocation under Uncertainty. *Water* **2015**, *7*, 6634-6650.
https://doi.org/10.3390/w7116634

**AMA Style**

Zhou F, Dong Y, Wu J, Zheng J, Zhao Y.
An Indirect Simulation-Optimization Model for Determining Optimal TMDL Allocation under Uncertainty. *Water*. 2015; 7(11):6634-6650.
https://doi.org/10.3390/w7116634

**Chicago/Turabian Style**

Zhou, Feng, Yanjun Dong, Jing Wu, Jiangli Zheng, and Yue Zhao.
2015. "An Indirect Simulation-Optimization Model for Determining Optimal TMDL Allocation under Uncertainty" *Water* 7, no. 11: 6634-6650.
https://doi.org/10.3390/w7116634