Application of High Dimensional Model Representations (hdmr) and Interpolation to Optimize Nutrient Removal under Uncertainty

In order to circumvent difficulty caused by model uncertainty, high dimensional model representation (HDMR) and interpolation were employed to approximate model outputs with different combinations of manipulated variables and model parameters. The results showed that HDMR and interpolation could be successfully applied to optimize nutrient removal under uncertainty. INTRODUCTION Uncertainty is inevitable when using mathematical model to simulate real process that subject to both anthropentic and natural disturbance. If uncertainty associated with a model is neglected, an optimal solution induced from this model may be far from optimal when applied to reality. Some control strategies have been proposed to take model uncertainty into account. The most popular method is to perform Mante Carlo simulations. However, with so many parameters in ASM, the computation cost is prohibitively high. On the other hand, high dimensional model representation (HDMR) is a fast algorithm that can circumvent the apparent exponential difficulty of highdimensional mapping problem. It has been successfully applied in atmospheric chemistry modeling (Li et al, 2000). Interpolation is an algorithm used to estimate function values between data points. Here we adopted HDMR and interpolation to approximate model outputs under different combinations of manipulated variables and model parameters. PROCESS AND MODEL DESCRIPTION The basic process investigated here was an activated sludge process (AS) to which an anaerobic tank and anoxic tank were added to enhance nutrient removal (Figure 1). The dimensions of the units were listed in Table 1. Table 1. Main dimensions of units Construction Item Dimension Bioreactor Anaerobic tank Volume 884 m Anoxic tank Volume 1768 m Aerobic tank Volume 6375 m Clarifier Area 500 m Height 4m Table 2. Flux-based average influent characterization Parameters Unit Dimension Total COD mgCOD/l 404 Ortho-P mgP/l 1.94 Total phosphorus mgP/l 5.43 NH4-N mgN/l 12.72 TSS mgSS/l 261 INFLUENT LOAD The influent data were collected in Athens No. 2 wastewater treatment plant (WWTP) of Georgia in 1998 (Liu, 2000; Liu and Beck, 2000). Generally, the influent quality can be classified as medium. Its main characteristics were listed in Table 2. The influent COD was fractioned into its components as in ASM No. 2d (Henze et al, 1999). The range of components listed in Table 3 was induced from the results of previous model calibration. MODEL AND SIMULATION DESCRIPTION ASM 2d was selected as it includes both nitrogen and phosphorus removal. The model was calibrated with the data collected in Athens No. 2 WWTP in 1998. Totally, there were 41 parameters in this model. We selected 16 parameters according to their sensitivity, and included the fractionation of influent COD in the framework. Thus, totally 24 parameters were adjusted in each simulation. All simulations were performed on WEST simulation platform (Hemmis nv, Kortrijk, Belgium). The implementation of the process was shown in Figure 2. Figure 1. Flowchart of the process Figure 2. Implementation of process in WEST Table 3. Characterization of influent COD (Ratio of total COD) Component Definition Range (%) SI Inert soluble organic material 4.33~5.75 SA Fermentation products 4.59~6.31 SF Fermentable, readily biodegradable organic substrates 8.08~11.81 XI Inert particulate organic material 25.4~28.7 XS Slowly biodegradable substrates 36.5~51.8 XH Heterotrophic organisms 5.13~9.81 XAUT Nitrifying organisms 0.39~0.61 XPAO Phosphate-accumulating organisms 0.20~0.34 XPP Poly-phosphate 0 XPHA A cell internal storage product of PAO 0.08~0.15 CONTROL STRATEGY DESCRIPTION The control strategy used here was essentially stochastic optimization of manipulated variables under model uncertainty. The values of manipulated variables were chosen such that the expected objective function assumed a minimum (Infanger, 1993): z = min E f(x, ω) s/t x∈ C = ∩ω∈Ω Cω Where x – manipulated variables; ω model parameters; Ω set of possible realizations of ω; f(x, ω) – objective function; C – set of feasible solutions. The optimal solution represented the realistic solution of the stochastic optimization problem. X* ∈ arg min {Ef(x, ω) | x∈∩ω∈Ω C } Table 4. Range of model parameters Parame ter Definition Unit Range μPAO Maximum growth rate of PAO d 1.62~1.79 qPP Rate constant for storage of XPP gXppg XPAO d 1.98~2.16 YPHA PHA requirement for PP storage gCOD gP 0.06~0.16 KPS Saturation coefficient for phosphorus in storage of PP gPm 0.08~0.12 ηNO3_HE T Reduction factor for denitrification 0.33~0.49 KO2_AUT Saturation/inhibition coefficient for oxygen for XAUT gO2m 0.29~0.45 μAUT Maximum growth rate of XAUT d 0.87~1.04 bH Rate constant for lysis and decay d 0.33~0.45 bPP Rate for lysis of XPP d 0.0019~0.10 KNH4_AU T Saturation coefficient for ammonia for XAUT gNm 0.6~1.03 KA Saturation coefficient for acetate gCOD m 3~5 KF Saturation coefficient for growth on SF gCOD m 3~5 μH Maximum growth rate on substrate d 4.93~5.96 fXI Fraction of inert COD generated in biomass lysis 0.10~0.15 KX Saturation coefficient for particulate COD gCOD m 0.11~0.15 V0 Settling velocity of sludge m/d 617~916 For this research, 6 manipulated variables were employed, i.e. allocation of influent and return sludge among the three tanks, dissolved oxygen in aerobic tank, internal recirculation, outer recycle, and waste sludge. For each manipulated variable, we selected 5 values. Thus, totally we had 5 = 15,625 different combinations of manipulated variable values. Obviously, the simulations of all these combinations cannot be finished in a reasonable period of time. Alternatively, we used HDMR (Li et al, 2000) to approach model outputs as the following