## Grey Box Modelling of Flow in Sewer Systems with State Dependent Diffusion

Publication: Research - peer-review › Journal article – Annual report year: 2011

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**Grey Box Modelling of Flow in Sewer Systems with State Dependent Diffusion.** / Breinholt, Anders; Thordarson, Fannar Örn; Møller, Jan Kloppenborg; Grum, Morten; Mikkelsen, Peter Steen; Madsen, Henrik.

Publication: Research - peer-review › Journal article – Annual report year: 2011

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*Environmetrics*, vol 22, no. 8, pp. 946-961. DOI: 10.1002/env.1135

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*Environmetrics*,

*22*(8), 946-961. DOI: 10.1002/env.1135

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*Environmetrics*. 2011, 22(8). 946-961. Available: 10.1002/env.1135

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TY - JOUR

T1 - Grey Box Modelling of Flow in Sewer Systems with State Dependent Diffusion

AU - Breinholt,Anders

AU - Thordarson,Fannar Örn

AU - Møller,Jan Kloppenborg

AU - Grum,Morten

AU - Mikkelsen,Peter Steen

AU - Madsen,Henrik

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.1002/env.1135

DO - 10.1002/env.1135

M3 - Journal article

VL - 22

SP - 946

EP - 961

JO - Environmetrics

T2 - Environmetrics

JF - Environmetrics

SN - 1180-4009

IS - 8

ER -