# Modelling and simulation of non-Gaussian stochastic processes for optimization of water-systems under uncertainty

I. Tsoukalas, Modelling and simulation of non-Gaussian stochastic processes for optimization of water-systems under uncertainty, PhD thesis, 339 pages, Department of Water Resources and Environmental Engineering – National Technical University of Athens, December 2018.

[doc_id=1933]

[English]

Hydrometeorological inputs are a key ingredient and simultaneously one of the main sources of uncertainty of every hydrological study. This type of uncertainty is referred to as hydrometeorological uncertainty and is of utmost importance in risk-based engineering works, due the high variability and randomness that is naturally embedded in physical processes. Considering hydrometeorological time series as realizations of stochastic processes allow their analysis, modeling, simulation and forecasting. Embracing the existence of randomness and unpredictability in such processes is a first step towards their understanding and the development of uncertainty-aware methodologies for water-systems optimization.

In this vein, due to the typical size of historical data, which is not (neither will ever be) sufficient to extract safe conclusions about the long-term performance of a system, the common procedure entails driving the typically deterministic water-system models (conceptual or physical-based) using stochastic inputs (that in a statistical sense resemble the parent information; typically, but not exclusively derived from the historical time series). This essentially enables the establishment of Monte Carlo experiments where the intrinsic uncertainty of the inputs (i.e., hydrometeorological processes) is propagated through a deterministic filter (i.e., a water-system simulation model) in order to derive, or assess, the probabilistic behavior of the output of interest (e.g., water supply coverage). Further to this, when the objective is the optimization of the deterministic model’s control variables (i.e., model’s parameters) with respect to some quantity or metric (i.e., objective), this procedure can (and should) be embed within an iterative scheme driven by an optimization algorithm (i.e., establishing uncertainty-aware simulation-optimization frameworks). An important step of this procedure is the realistic simulation of hydrometeorological processes, since they are the main drivers of the whole procedure, and eventually determine its accuracy, as well as the probabilistic behavior of the output of interest. This in turn, poses an intriguing challenge that arises from a series of unique peculiarities that characterize such processes, namely, non-Gaussianity, intermittency, auto-dependence (short- or long-range), cross-dependence and periodicity. Despite the significant amount of research during last decades, these challenges remain partially unresolved. To a large extent, this is due to the standard hypothesis of most simulation schemes that does not lie in the reproduction of a specific distribution, but on the reproduction of low-order statistics (e.g., mean, variance, skewness) and correlations in time and space. This is a problem because, a) for a given set of low-order statistics multiple distributions may be represented, thus making the simulation problem only partially defined, and b) as shown herein, this practice may lead to bounded, and thus unrealistic dependence forms among consecutive time steps and/or processes.

Further to this, driving water-system simulation models with long stochastically generated sequences, thus accounting for input (hydrometeorological) uncertainty, inevitably increases the required computational effort, especially within the context of simulation-optimization frameworks. This in turn, poses the challenge of addressing and ensuring the practical implementation of water-system optimization problems under uncertainty.

Thereby, the main research objectives and contributions of this Thesis are related to:

1. The development of novel non-Gaussian stochastic simulation models, able to account also for the other peculiarities typically encountered in hydrometeorological processes, such as, intermittency, auto- and cross- dependence, periodicity, as well as their scale-varying probabilistic and stochastic behavior.
2. The development of surrogate-based optimization methodologies and algorithms that can efficiently and effectively confront water-system simulation-optimization problems under uncertainty, i.e., when using stochastic inputs to drive the simulation-optimization procedure.

Specifically, herein a by building upon copula concepts, probability laws and the theory of stochastic processes, a theoretically justified family of univariate and multivariate non-Gaussian stationary and cyclostationary models is defined and thoroughly investigated. This type of models have been unknown to the hydrological community, and this Thesis is the first attempt to align them with hydrological stochastics. The developed models are shown to be able to account for all the typical characteristics of hydrometeorological processes and simultaneously exhibit a simple and parsimonious character. Furthermore, these models are then coupled, using a disaggregation approach, thus eventually enabling the development of a modular stochastic simulation framework that allows the simultaneous reproduction of the probabilistic and stochastic behavior (including non-Gaussian distributions) of hydrometeorological processes at multiple time scales (from annual to daily; as well as finer time scales). The advantages of this class of stochastic processes and models, as well as of the modular stochastic simulation framework for multi-scale simulations, are demonstrated and verified through numerous hypothetical and real-world simulation studies.

Finally, in order to ensure the effective exploitation and practical implementation of these new developments in the stochastic simulation of hydrometeorological processes within the uncertainty-aware, engineering design and management of water-systems (i.e., driven by stochastic inputs), this Thesis develops appropriate surrogate-based computationally-efficient methodologies and algorithms, that effectively handle water-system simulation-optimization problems under hydrometeorological uncertainty, thus alleviating the associated computational barrier.

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Our works that reference this work:

 1 I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Building a puzzle to solve a riddle: A multi-scale disaggregation approach for multivariate stochastic processes with any marginal distribution and correlation structure, Journal of Hydrology, 575, 354–380, doi:10.1016/j.jhydrol.2019.05.017, 2019. 2 D. Nikolopoulos, G. Moraitis, D. Bouziotas, A. Lykou, G. Karavokiros, and C. Makropoulos, Cyber-physical stress-testing platform for water distribution networks, Journal of Environmental Engineering, 146 (7), 04020061, doi:10.1061/(ASCE)EE.1943-7870.0001722, 2020. 3 I. Tsoukalas, P. Kossieris, and C. Makropoulos, Simulation of non-Gaussian correlated random variables, stochastic processes and random fields: Introducing the anySim R-Package for environmental applications and beyond, Water, 12 (6), 1645, doi:10.3390/w12061645, 2020.