Simulating cyclical behavior in water and wastewater treatment processes

Membrane processes in water and wastewater treatment (W/WWT) often produce cyclical data. The ability to simulate functions that resemble what is produced by these processes allows researchers to compare system parameters, train additional models, or test fault detection methods. However, changes in cyclical behavior, distinctive water quality, and unique treatment processes make this challenging. In this paper, we develop an approach for simulating these functions that is decomposed into four parts, accounting for (i) a global trend that allows for cycles to evolve in magnitude; (ii) a local trend that captures the individual cycle shape; (iii) the cross-cycle variance that may change over time; and (iv) the within cycle variance that may differ based on the position in the cycle. Not every process requires each of these steps, so we discuss when and how to implement each of them, as well as adjustments for specific processes. The flexibility of this method is demonstrated using datasets from ultrafiltration, membrane bioreactors, and closed-circuit reverse osmosis systems. We show that important features of the simulated data match those of the real data and describe ways in which these simulations can be used to support additional study of these processes.