Grey‐box modelling of flow in sewer systems with state‐dependent diffusion

Abstract

Generating flow forecasts with uncertainty limits from rain gauge inputs in sewer systems require simple models with identifiable parameters that can adequately describe the stochastic phenomena of the system. In this paper, a simple grey-box model is proposed that is attractive for both forecasting and control purposes. The grey-box model is based on stochastic differential equations and a key feature is the separation of the total noise into process and measurement noise. The grey-box approach is properly introduced and hypothesis regarding the noise terms are formulated. Three different hypotheses for the diffusion term are investigated and compared: one that assumes additive diffusion; one that assumes state proportional diffusion; and one that assumes state exponentiated diffusion. To implement the state dependent diffusion terms Itô's formula and the Lamperti transform are applied. It is shown that an additive diffusion noise term description leads to a violation of the physical constraints of the system, whereas a state dependent diffusion noise avoids this problem and should be favoured. It is also shown that a logarithmic transformation of the flow measurements secures positive lower flow prediction limits, because the observation noise is proportionally scaled with the modelled output. Finally it is concluded that a state proportional diffusion term best and adequately describes the one-step flow prediction uncertainty, and a proper description of the system noise is important for ascertaining the physical parameters in question.

A generalised Dynamic Overflow Risk Assessment (DORA) for Real Time Control of urban drainage systems

“…Current research is now focusing on the application of stochastic models (such as those presented by Breinholt et al, 2011 and for runoff estimation . This will provide a more accurate and dynamic estimation of the uncertainty affecting runoff volume, thus affecting the estimation of overflow risk.…”

A formal statistical approach to representing uncertainty in rainfall–runoff modelling with focus on residual analysis and probabilistic output evaluation – Distinguishing simulation and prediction

“…These steps were also described in Breinholt et al (2011) and are only summarised here to provide some methodology background. Three different approaches to quantify the inherent uncertainties of the system are incorporated into three models denoted M1, M2 and M3.…”

“…This is an important but often neglected analysis by authors who apply formal statistical approaches for model conditioning (Beven et al, 2011). The modelling philosophy adopted here is that of the grey box modelling principle that unites prior physical knowledge with information from data, and uses statistical tools for parameter significance testing and estimation and thus adhere to the principle of parsimony (Kristensen et al, 2004a;Kristensen et al, 2004b;Breinholt et al, 2011).…”

Flow Forecasting using Deterministic Updating of Water Levels in Distributed Hydrodynamic Urban Drainage Models

“…For simple, linear models this can be achieved using a version of the Kalman Filter [5]; recent examples, using a lumped conceptual urban runoff model and an Extended Kalman filter, are provided by [6,7]. For large or very non-linear models ensemble-based data assimilation methods, such as the Ensemble Kalman Filter [8,9] or the Particle Filter [10,11], would usually be required to obtain satisfactory results.…”