**Abstract**

There are a series of sensitivity analysis performed around activated sludge models for wastewater treatment. Comparison is presented both for local and global approaches, and the most used methods are reported. It is observed that sensitivities depend on the modeling objectives. Furthermore, local methods are applicable only for linear models, thus, the global ones are often preferred. Due to the current wastewater resource recovery trend, more sensitivity analysis regarding phosphorus removal and model refinement will be required. Finally, knowledge gaps are identified in association with uncertainty in the influent fractions, and variancebased methods for factor interaction. The sensitivity analyses are quality assurance tools that, if applied properly, it is expected to improve complex phenomena understanding as well as decision making.

**Keywords:** activated sludge models (ASM), benchmark simulation model (BSM), membrane bioreactor (MBR), uncertainty, sensitivity analysis, local sensitivity analysis (LSA), global sensitivity analysis (GSA)

### **1. Introduction**

Disposal of urban wastewater (WW) with adequate treatment is a major concern in developing countries. In most of them, a considerable amount of WW is discharged into the environment (rivers, lakes, and oceans) as raw WW or poorly treated WW. Consequently, surface water and groundwater get polluted, affecting human health, aquatic ecosystems, food production, and drinking water availability [1]. Thereupon, it is vital to treat wastewater to mitigate the environmental impact.

Wastewater treatment plants (WWTP) are infrastructure dedicated to water sanitation. The most commonly applied process is activated sludge, a biological treatment consisting of a bioreactor coupled with a secondary settler. Within the bioreactor, biomass (heterotrophic, autotrophic, and/or phosphorus accumulating) is synthesized for biodegradation of the pollutants as well as for the removal of nutrients suchlike Nitrogen and Phosphorus [2]. Then, the secondary settler concentrates the biomass for its removal and further solid treatment. Finally, a clarified effluent result after the treatment.

However, in developing countries, most WWTPs only aim for primary (physical) and secondary (biological) treatment, without tertiary treatment nor sludge treatment (anaerobic digestion) [3]. Hence, the lack of advanced treatment techniques as well as inefficient operation/control of the WWTPs results in increased water pollution.

Lately, it has surged a trend to conceive wastewater treatment plants as water resource recovery facilities (WRRF). This is because it is possible to recover organic matter, nutrient-rich by-products, energy, and water itself, representing an economic revenue for the WRRF [4]. Consequently, there is a need for designing new infrastructure and for process optimization to meet stringent water quality standards together with resource recovery.

Either for design or diagnosis, it is vital to consider the processes influencing the WWTP performance. In the AS process, it is governed by the interaction of raw wastewater fluctuations (quality and quantity), the biokinetics, the mixing conditions, the aeration system, together with the operational conditions [5]. Due to process complexity, mathematical models have arisen as an ideal tool for assessment of the AS performance, allowing to provide continuous feedback in an understandable, faster, and cheaper manner.

Over the past few decades, process models have been established for designing, upgrading, and optimizing wastewater treatment plants [4]. In the wastewater industry, specifically for the biological processes area, the activated sludge models (ASM) were introduced for the latter, given its capability to proximately simulate the process kinetics taking place in the bioreactor in a simpler fashion. Mind that the ASMs, are deemed as core models, i.e., that can be modified according to modelers' needs. While benchmarking frameworks-also known as BSMs-have been proposed to assess environmental and economic aspects in an activated sludge plant-wide context [6]. Moreover, recently membrane bioreactor (MBR) models-activated sludge process plus membrane filtration have surged as an alternative for meeting stringent water regulations and for resource recovery of water given its high-quality effluent post-membrane treatment [7, 8]. Mind that the above-mentioned activated sludge modeling frameworks have been developed by various modeling task groups of the International Water Association (IWA). Thus, the group of them will be referred to as the IWA models.

According to Saltelli et al. [9], building any kind of model requires specifying model archetype, parameters, resolutions, and calibration data including its acceptance criteria, and so on. Nevertheless, sometimes information and data are missing or are not well-known, resulting in uncertainty in each of the previous requirements. Hence, model implementation highly depends on the understanding of the AS process, as lack of it results in augmentation of model uncertainty.

Mind that model applicability relies on how proximate model inputs and outputs are to the real-plant data. Therefore, appropriate modeling practices based on highquality data collection and model calibration are essential. According to Rieger et al. [10], a good modeling practice (GMP) of the activated sludge process consists of 5 phases: project definition, data collection and reconciliation, plant model set-up, calibration, and validation, as well as scenario simulation. Inherently, uncertainty is present in the input data required in each phase (e.g., influent flow rate, pollutant fluxes, seasonal conditions, model parameters), that if not heed and reduced, will spoil model applicability.

Consequently, Belia et al. [11] stated the importance of identifying the sources of uncertainty in WWTP modeling for project risk reduction and model validation. Thereby, identification and classification of the sources of uncertainty as input data (influent, operational settings, etc.), model data (e.g., structure and process interaction), model parameters (hydraulic, biokinetic, settling), and technical aspects

(solver setting and computational thresholds) within each of the GMP phases is strongly recommended [11]. After sources identification, an uncertainty analysis is to be conducted. It consists of propagating the model input-also called input factoruncertainty in the desired model output(s) via Monte Carlo simulations or by probabilistic methods [9, 12]. Hence, determining probabilistic distributions of the model output given uncertain input factors.

Nevertheless, the IWA models (including model refinements) are often over parameterized. Hence, a vast number of input factors may be uncertain, troubling calibration principally of complex models. Therefore, after an uncertainty analysis, a sensitivity analysis is conducted for quantifying how much uncertainty is related to an induvial input factor or a group of them [13]. So, a sensibility analysis (SA) is a method used to characterize and prioritize uncertainty. According to Al et al. [14], and SA can be used as a quality assurance technique for modelers as it improves promotes a better understanding of the activated sludge model behavior.

There are a series of sensitivity analysis performed around activated sludge models for wastewater treatment. These can be classified according to their nature, is to say, local or global. Local approaches-also called OAT approaches or LSA-assess parameters sensitivity in the function of partial derivatives of the outputs given small perturbations of an input factor for control/identification of problems [15]. Nevertheless, local approaches are fiercely criticized due to method limitations such as linearity, normality assumptions, and local variations of the input space [16]. Still, there is a significant amount of local sensitivity analysis (LSA) in the activated sludge modeling field.

To overcome the limitations of LSAs, global approaches have been established for the assessment of the entire domain of the input space of parameters variation. Global sensitivity analysis (GSAs) methods can be deemed as an analysis of variance (ANOVA), thus, it fractions variance among the uncertain input factors to elucidate its influence in model output [9, 15]. Therefore, unlike local approaches, the GSAs allow studying mathematical models as a whole, even, some methods account for the effect of factor interaction [14, 16]. Fortunately, over the past years, most activated sludge modelers have taken previous considerations and conducted several GSA methods to reduce model uncertainty in predicting system performance.

However, either for local or global approaches, in the activated sludge modeling area, there has been published a wide range of sensibility analysis methods, along with a different focus, i.e., for SA method introduction or application of it. The latter under different modeling goals and their respective scenarios demonstrating the applicability of the sensibility analysis in the field. Consequently, due to the advantages of the SAs, together with the need for complex models for improving process understanding, it is expected that more AS stakeholders will rely on sensitivity analysis results for process design, control, and upgrade.

Yet, up to the author's knowledge, there is not a systematic review concerning the sensibility analysis around IWA models. Hence, according to the statements above mentioned, the objective of this review is to (I) compare the sensitivity analysis performed in the IWA models, distinguishing them from local and global approaches, (II) report the used method, (III) look for similarities and misinterpretations found in the reviewed papers, (IV) determine if the purpose was developing a methodology or the sensitivity analysis were used for an application, (V) catalog the papers according to the aim of the papers (e.g., control, operation, etc.), and (VI) demonstrate lacunae in knowledge concerning the sensitivity analysis in the IWA models.

According to these objectives, this paper presents the collective effort of the authors to collect the up-to-date most relevant works in the activated sludge modeling area. We summarize what we consider the most relevant features that current and future AS modeling practitioners must heed. For improving readers' understanding the paper is divided into six major sections. First, an overview of the IWA models is presented for the readers to become au fait of the activated sludge models, the benchmark simulations models, as well as the membrane bioreactor models. Then, some of the most applied sensitivity analysis methods in the area are presented, principally distinguishing them as local or global approaches. In the third section, we outline the systematic selection of the papers across the activated sludge modeling area. The results of the systematic review are presented in section four. Finally, the results discussion and our main conclusions are reported in sections five and six, respectively.
