Efficient Economic Model Predictive Control of Water Treatment Process with Learning-based Koopman Operator

TL;DR

The paper tackles the challenge of economically operating wastewater treatment plants with nonlinear, high-dimensional dynamics by proposing a Deep Input-Output Koopman Operator (DIOKO) that learns a linear latent model and cost predictor from available measurements. This enables a convex economic model predictive control (EMPC) that solves as a quadratic program, significantly reducing computation time while improving both effluent quality and operating cost. Compared with baselines including first-principles EMPC, nonlinear MPC, and SAC, the DIOKO-EMPC yields lower cost and better EQ under dry, rainy, and stormy conditions, with computation times on the order of 3–4 ms per step (thousands of times faster than nonconvex EMPC). The approach supports partial-state sensing and robust performance, and shows strong generalization to unseen weather scenarios; extensions to probabilistic and hybrid Koopman models are noted as future directions.

Abstract

Used water treatment plays a pivotal role in advancing environmental sustainability. Economic model predictive control holds the promise of enhancing the overall operational performance of the water treatment facilities. In this study, we propose a data-driven economic predictive control approach within the Koopman modeling framework. First, we propose a deep learning-enabled input-output Koopman modeling approach, which predicts the overall economic operational cost of the wastewater treatment process based on input data and available output measurements that are directly linked to the operational costs. Subsequently, by leveraging this learned input-output Koopman model, a convex economic predictive control scheme is developed. The resulting predictive control problem can be efficiently solved by leveraging quadratic programming solvers, and complex non-convex optimization problems are bypassed. The proposed method is applied to a benchmark wastewater treatment process. The proposed method significantly improves the overall economic operational performance of the water treatment process. Additionally, the computational efficiency of the proposed method is significantly enhanced as compared to benchmark control solutions.

Figures (8)

• Figure 1: A schematic of the biological wastewater treatment plant alex2008benchmark.
• Figure 2: The pipeline of the proposed DIOKO modeling approach. The neural network encodes the measurement vector $y_k$ and the known disturbance $d_k$ into the observable vector $\psi_{k|k}$. Then, the learned Koopman matrix $A$ and matrix $B$ propagate $\psi_{k|k}$ and $u_k$ to the next step, and produce observable vector $\psi_{k+1|k}$. Finally, the predicted observable vector $\psi_{k+1|k}$ is transformed by the output module through equation \ref{['eq:output function']} to predict the next step stage cost $\hat{c}_{k+1|k}$.
• Figure 3: Cumulative prediction error of the proposed DIOKO modeling approach on the validation data set and the test data set. The Y-axis indicates the cumulative mean-squared prediction error in log space over 16 instants, and the X-axis indicates the training epochs. The shaded region shows the confidence interval (one standard deviation) over 10 random initializations.
• Figure 4: Control performance comparison between the proposed DIOKO-based EMPC and baselines under the dry weather conditions. The trajectories of the overall economic stage cost ($c_k$), EQ, and OCI for each of the methods are presented.
• Figure 5: Control performance comparison between the proposed DIOKO-based EMPC and baselines under the rainy weather condition. The trajectories of the overall economic stage cost ($c_k$), EQ, and OCI for each of the methods are presented.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5