Incorporating concepts from physical theory into stochastic modelling of urban runoff pollution

tS)

Pergamon

Wal. ScL Tech. Vol. 37, No. I, pp. 179-18~ ,1998. Q 1998 1AwQ. Published by Elsevier Science Ltd Printed in Great Britain .

PIT: S0273-1223(97)00768-3

0273-1223/98 S19'00 + 0-00

INCORPORATING CONCEPTS FROM PHYSICAL THEORY INTO STOCHASTIC MODELLING OF URBAN RUNOFF POLLUTION Morten Grum Department of Water Quality Management and Aquatic Ecology. P. O. Box 8080, 6700 DD Wageningen, Wageningen Agricultural University. The Netherlands

ABSTRACT On evaluating the present or future state of integrated urban water systems, sewer drainage models, with rainfall as primary input, are often used to calculate the expected return periods of given detrimental acute pollution events and the uncertainty thereof. The model studied in the present paper incorporates notions of physical theory in a stochastic model of water level and particulate chemical oxygen demand (COD) at the overflow point of a Dutch combined sewer system. A stochastic model based on physical mechan isms has been formulated in cont inuous time. The extended Kalman filter has been used in conjunction with a maximum likelihood criteria and a non-linear state space formulation to decompose the error term into system noise terms and measurement errors. The bias generally obtained in determin istic modelling, by invariably and often inappropriately assuming all error to result from measurement inaccuracies, is thus avoided . Continuous time stochastic modelling incorporating physical , chemical and biological theory presents a possible modelling alternative. These preliminary results suggest that further work is needed in order to fully appreciate the method's potential and limitations in the field of urban runoff pollution modelling. @ 19981AWQ. Published by Elsevier Science Ltd

KEYWORDS Urban storm drainage;runoff pollution;eso; sedimenttransport; stochastic modelling. INTRODUCTION Since the construction of seweragetreatment plants, the effects of sewer overflows during wet weather have become an increasingly important limiting factor to the physical, chemical and ecological state of the receiving waters. Evaluating these effects under present conditions and possible future scenarios is often done with the use of models. Traditional approaches to mathematical modelling of urban drainage systems have involveda compilation of mechanistic theories from several independent laboratory studies performed under ideal conditions. MOdels are often over parameterised, have a large amount of uncertain input parameters and calibration is invariably and often inappropriately based on the assumption that all deviation betweenthe modelled output and the observed output resultsfrom measurement error, Based on the available data, the present study had, as an objective, the use of stochastic differential equations to create an urban runoff pollution model with a few, and as far as possible interpretable, 179

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M.GRUM

parameters. At the same time as tackling implementation problems, the aim was thus to evaluate the importance of suitable assumptions with respect to sources of error. Grum and Aalderink (1997) present the results of a study carried out on the same data using the equivalent deterministic model formulation and thus assuming that all deviation between modelled and observed results from measurement error. DATA The data used in the present study originate from a Dutch combined sewer catchment with identified characteristics which are summarised in Table I. The data set was collected from 1981 to 1986 as part of a study carried out by the National Working Party on Sewerage and Water Quality (NWRW/STORA). The data have previously been used and presented in studies by van Walraven et al. (1985), Van der Heijden et al, (1986), Bakker et al. (1988), Benoist and Lijklema (1989) and by van Sluis et aI. (199\). The Loenen sewer is a looped system with a single overflow structure at which the water level had been monitored continuously. Water is pumped to the treatment plant from the lowest point in the sewer. During actual overflow, samples for chemical analysis were taken at approximately volume proportional intervals. Only the water level and suspended COD concentration at the overflow structure have been modelled. The rainfall events that resulted in a combined sewer overflow (eSO) have been used in the study. Table 2 contains a summary of the data. Table I. Summary of catchment characteristics (Loenen, The Netherlands) Total catchment area Impervious area Mean Pipe gradient Volume below weir Depth below weir

15.8 ha 28 % 3.3 %0 895 m' 5.7 mm

Table 2. Summary of the data studied Mean Minimum

Maximum No. events No. observations

Level at Overflow 17.5 m 16.3m 18.5 m 22 623

Suspended COD cone. 150.0 rnz/l OmWl 997 mgtl 19 253

METHODS ModelJjn~ usin~

stochastjc differential eQuations

Modelling using stochastic differential equations is a good tool for combining information from physical, chemical and biological theory with information from data and is therefore often also called grey box modelling (Madsen and Holst, 1996). In this section is a brief outline of a few essential elements of modelling using stochastic differential equations. Refer to Madsen and Holst (1996) and Madsen and Melgaard (1991) for further details on the mathematics behind modelling with stochastic differential equations. In the field of water resources research these methods have been applied to practical problems by

Stochastic modelling of urban runoffpollution

181

several works including Carstensen and Harremoes (1997), Jacobsen et a!. (1996) and Carstensen and Harremoes (1995). Consider, as an example, the usual (determ inistic) continuity equation for a reservoir which could be written as:

dV di"= Q.-Q.•,

(1)

v= the volume stored in the reservoir, =

Qjn and Qou' the flow into and out of the reservoirrespectively.

The flows would often through some linear or non-linear storage function be connected to the reservoir volume and input flows. In accepting that the above differential equation does not cover 'the whole truth', a stochasticterm is added to (I) to give: dV

dro

di" = QI. - Q,.,., +di"

Cl)

(2)

= a stochasticprocess assumed to have independent increments (a Weinerprocess).

Consider the case in which the water level in the reservoirhas been monitored. Assumingthe reservoir to be such that the level can be calculatedas a function of the volume,j{), the following discrete time observation equationcan be formulated:

h, =!(V(t»+E,

(3)

h, =the monitoredwater level, V(t) =the modelled volume in the reservoir, £, =measurement error (independent of the stochastic process). The system and observation errors can not be found directly. However, given their variances the Extended Kalmanfilter can, on the presenceof a new observation, be used to calculatethe best estimatesof the system variables' present values.Thus, at every availableobservation, the Kalmanfilter makes a weightingbetween 'what we calculate' and 'what we see'. Prediction then proceeds from these new estimates of the system variables. The variances of the system and observation errors, and the other constant model parameters are estimated by an off-line optimisation of a maximum likelihood criteria. In the present study a modified version of the program CfLSM, due to Melgaard and Madsen (1993), was used. This program, which is specifically aimed at parameter estimation in non-linear stochastic differential equations, uses a modified quasi-Newton optimisation. Water Quantity modelJjn~ (Jeye)) Water quantity has been modelled as three linear reservoirs in series followed by a final reservoir representing the static storage volume in the pipe system immediately before the overflowweir. In Table 3 is a combined representation of the reservoirs as sketches with the corresponding storage equations and stochastic differential equations. The flow out of the first three reservoirs is assumed proportional to the volume of water stored in each reservoir. The flow out of the final reservoir is that pumped to the treatment plant (here assumed equal to the pump capacity) and the flow out of the system through the CSO structure which is calculated as a function of the water level at this structure. The level is in tum calculated as a function of the volume stored in this final reservoirbased on the pipe dimensions of the sewer system itself. If the water level is above the assumed known 'pump on level' then the flow to the treatment plant is set

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M.GRUM

equal to the pump capacity and otherwise to zero. In order to maintain a low number of parameters during these preliminary studies the rainfall is multiplied by a runoff coefficient and there are therefore assumed to be no initial losses. Five model parameters : runoff coefficient, reservoir storage constant, overflow weir coefficient, variance of reservoir noise and the variance of the measurement error are estimated. Water guaHly mQdel!jn~ (suspended com The suspended COD model is based on the premise that the main source Qf particulate pollution in the overflow is as a result of resuspended sewer sediments. The input to the first reservoir is thus assumed to contain no suspended material. It is further assumed that each reservo ir has an infinite amount of available deposited sediment (see Results and Discussion for comments on this assumption). The water quantity model provides no flow velocities to which resuspension rate could be related. The resuspension rate is therefore calculated as a function, g( ), of the flow rates in and out of the reservoirs. Above an estimated threshold value, resuspension is assumed to be proportional to the flow rate. The stochastic differential equations for suspended mass of COD are shown in the far left column of Table 3. In this suspended COD model, three parameters are estimated: settling rate, threshold flow rate and a proportionality constant. RESULTS AND DISCUSSION Water quantjty (Jeye)) The water quantity model presented here contains 5 estimated parameters. These are listed in Table 4 where their estimated values and confidence limits for the final model are also given. All parameters except the observation error variance were found to be significantly different from zero. This suggests that there, in practical terms, is no real observation error when compared to the incompleteness of the models description of system inputs and behaviour. The values listed in the far right column are those reported in some of the literature listed earlier in the data section . It is apparent from the values in Table 5 that the parameter correlation were generally rather high . This is particularly the case for correlation between the observation error variances and both the runoff coefficient and the system noise variance. A number of parameters including pump on level, pump capacity and overflow weir coefficient were fixed to their 'known' values. Future efforts should aim at having also these parameters estimated from the data as has been done in the equivalent deterministic model in Grum and Aalderink (1997). Attempts were also made to identify a separate noise term for each of the four reservoirs and also to ident ify a common variance for the first three linear reservoirs with a fourth variance on the final reservoir. In both cases it was not possible to obtain convergence to a single or reproducible likelihood optimum. This could relate both to structural aspects of the model but also to the quality of data, in terms of both excitation and sampling frequency. The identifiability of stochastic terms in reservoir modelling (linear and non-linear) is clearly an area in need of much more experience. Water guaHty (suspended com Reproducible results proved hard to obtain in the case of the water quality model. This was probably for a large part due to the little observed data available, namely only an average of only 13 observations per event for 19 rainfall-runoff events. In table 6 and 7 are the results of the parameter estimation only the threshold flow rate of the resuspension equation is not significantly different from zero and could therefore be excluded in order to obtain an improved model.

Table 3. System model overview Reservoir Storage Relationship

System sketch

Stochastic Differential Equation:WATER QUAUIY (Suspended COD concentration)

Q",•• =~A~_-. r(t)

+Q"""

~

V. =kQ.

dV. = Qro•• _ Q. dt

dM. =

~

V2=kQ2

dV1 = Q. -Q2 dt

dM2 =

v] =kQ)

dV I =Q2 -Q] dt

~c. ~C2

~ rl V).M)

Q).

Q",nJI-

f~--.

.r

Stochastic Differential Equation: WATER QUAN1ITY (Level at overflow structure)

Q,...

dt

~

dM) = ~

3

Q""~11- = w-, ·(h- h.... )2 and

h=f(V.)

am,

Q2' C2 + g((Q. + Q2)/2) - /3•. M 2+ dtiJ2 ~

Q2 ' C2 - Q3' C) + g((Q2 + Q))/2) - /31 ' M 3+ dtiJ1 ~

hobm vrd ••

=f(V.(t)) + E•••

i

c.

n

3

S·

0<>

So

1

+ g((Q3 + Q.- + f4.",,)/2) - /3,' M. + dtiJ dt

ft C - M.(t)/ +E J obsm-rd •• /V.(t) 2~ ,

Where • The stochastic terms drn./dt and drnz/dt are the derivatives of additive random processes (Wiener processes) for the quantity and quality equations respectively. • The discrete time observation error terms £1 and £2 are assumed to be normally distributed random variables with mean zero and variance _.... is the impervious area (fixed). 4> is a runoff coefficient. r(t) is the rainfall series and Q.... is the flow into the first reservoir. • V" Q.. M, and C. are the volume, flow out. suspendedCOD mass and concentrations respectively(of the j'th reservoir). • k is the storage constant for the first three linear reservoirs, hwea- the weir level (fixed) and Wooer the weir coefficient. • g() is resuspensionof sewer sediments given as a function of the flows in the reservoir: 0 for Q:S;Q_d g(Q) = { /3 2'(Q-Q_d) for Q> Q_d and /3 I is the settling rate of the suspended COD. • f{) is a function giving the water level at the overflow point as a function of the static storage in the sewer system. This relationship is based on the pipe dimensionsof

the actual sewer system

en

8g,

(

! Observation Equations:

~ . C1 -

dt

+dt

Ii,

dt

su, -=Q3 , C3- (O-.n..+~)· C.

dV. = Q] _ Q-rfl- _ Qp•.." dt

- Q, . C. + g{(Qro•• + ~)/2) - /31' MI + dtiJ1

c

a~

2

'o::::"

~c

c. o

'"

00 IN

184

M.GRUM

Table 4. Results of the parameter estimation in the final water quantity model No.

Parameter name

Parameter symbol

Estimated value

95% confidence limns lower unoer

Reported value

I

Runoff coefficient [none]

Q

0.71

0.62

0.80

2

Proportionahty constant [ lIs1

k

5.6

5.1

6.0

Overtlow weir coef. [m1lsj System noise variance [(m 1/s)1 I

-

3 4

Wl~r

3.6 0.221

5

Observation error var. [(m)1 I

0.097 1

-0.0038

5.3 0.271 0.151

2.79

cr,

1.9 0.151

cr",

0.7-1.7

-

-

Table 5. Parameter correlation matrix (for the final water quantity model)

Runoff coefficient

I

I

2

3

4

5

1.00

-0.65

-0.5 1

0.87

-0.96

2

Proportionality constant

-0.65

1.00

-0.03

-0.65

0.70

3 4

Overflow weir coefticient System noise variance

-0.5 1 0.87

-0.03 -0.65

1.00

-0.08

-0.08

5

Observation error variance

-0.96

0.70

0.26

1.00 -0.92

0.26 -0.92 1.00

The variance of the system noise term is again seen to be significantly different from zero as is the variance of the observation error. These results would suggest that both system noise and observation error should be modelled . The absolute value of 1.58.10-5 mgll for the observation error standard deviation would however appear suspiciously small. Suspiciouslly small were also the correlation coefficients between the observation error variance and the other model parameters. It should be noted that though the results were reproducible from different parameter starting values not all starting values converged to this optimum point. Other resuspension models were examined. These included the estimation of a maximum amount of available deposited sediment and other functions relating the resuspension rate to the flow rate. None of these gave better results than the presented model. Table 6. Results of the parameter estimation in the final water quality model (suspended COD) Parameter name

No.

I

Aow rate threshold [m 1/s1

2

Settling rate [lIs1

3 4

System noise variance l(l!IS)ll

5

Observation error var. [imi!/I)lj

Proportionality constant rwmll

Parameter symbol

Estimated value

95% confidence limits upper lower

Q'hmhold

8.7·10" "

-2.0·10''''

2.0·10'·

B1

0.00294

0.00293

0.00295

B1

1749 13781 iI.58 .10·?

1705 13641 ( 1.57·\0,')1

14001 ( 1.68·10'5)1

cr",

cr,

1791

Table 7. Parameter correlation matrix (for the final water quality model, suspended COD)

I

Flow rate threshold

2

Settlmg rate

3

Proportionality constant

4

System noise variance

5

Observat ion error variance

I

2

3

4

5

1.00

0.35

0.62

-0.85

0.00

0.35

1.00

0.11

-0.07

0.00

0.62

0.11

1.00

-0.87

0.00

-0.85

-0.07

-0.87

1.00

-0.00

0.00

0.00

0.00

-0.00

1.00

CONCLUSIONS It has been possible to identify and estimate the parameters of a sewer system water quantity model. The results of the water quantity model suggest the generally accepted assumption that all error results from measurement error (the basis of deterministic model calibration) is not valid. On the contrary these results

Stochastic modelling of urban runoff pollution

185

suggest that most of the deviation between model prediction and observation are a result of sewer system behaviour which the model does not explain. It is difficult to conclude anything from the results of the water quality model. They suggest that more research efforts should be put into applying existing experimental design and sampling frequency theory to water quality modelling. The results from both the models and from the study as a whole suggest that more efforts should be put into exploring and gaining experience in stochastic modelling with physically interpretable parameters. This is particularly important with respect to the identifiability of both the mechanistically interpretable model parameters and of the system noise components. ACKNOWLEDGEMENTS The present study was carried out under the EU's HCM Network MATECH. My supervisor Hans Aalderink and the MATECH project partners are thanked for valuable discussion. RIONED is thanked for permission to use the data. Peter Thyregod and Judith Jacobsen are thank for sharing experience on reservoir modelling with CfLSM and Rasko Jovin for his assistance in getting the original CTLSM UNIX version to run on a PC. REFERENCES Bakker, K., Ten Hove, D., De Ruiter, M. A. and Van Walraven. J. H. A. (1988). Stormwater overflows from sewer systems: reduction of pollution emission by storage sedimentation tanks. H20 21(19), 568-572. Benoist, A. P. and LijkJema, L. (1989). A methodology for the assessment of frequency distributions of combined sewer overflow volumes. Wal. Sci. Tech.• 23(4). I Cartstensen, J. and Harrernoes, P. (1995). A grey-box modelling approach for the description of a storage basin in a sewer system . Submittedfor ASCE Journal ofHydraulic Engineering . Cartstensen , J. and Harremoes , P. (1997). Time Series Modelling of Overflow Structures. Wal. Sci. Tech .• 36(8-9). 45-50 . Grurn, M. and Aalderink, R. H. (1997). A Statistical Approach To Urban Runoff Pollution Modelling. Wal. Sci. Tech., 36(5). 117124. Jacobsen, J. L., Madsen, H. and Harremoes, P. (1996). Modelling the Transient Impact of Rain Events on the Oxygen Content of a Small Creek . War. Sci. Tech., 33(2).177-185. Madsen. H. and Holst, 1. (1996). Modelling Non-Linear and Non-Stationary Time Series. Lecture notes for the PhD. cource DTU0417 Advanced Time Series Analysis . IMM-Institute of Mathematical Modelling , building 321. Technical University of Denmark , OK 2800 Lyngby. Madsen, H. and Melgaard. H (1991). The Mathematical and Numerical Methods used in CTLSM - a program for ML-estimation in. continuous time dynamic models. IMM-Institute of Mathematical Modelling, building 321. Technical University of Denmark. DK 2800 Lyngby. Melgaard, H. and Madsen . H. (1993). CTLSM version 2.6 - a program for ML-estimation in stochastic differential equations . Tech . rep. No. 1/1993, IMM-Institute of Mathematical Modelling, building 321. Technical University of Denmark. OK 2800 Lyngby . Van der Heijden, R. T. J. M., LijkJema, L. and Aalderink, R. H. (1986). A statistical methodology for the assessment of water quality effect of storm water discharges. In: Urban Runoff Pollution, NATO ASI series G; Ecol. Sci. Vol. 10. Springer . Van Sluis, J. M., Ten Hove, D. and de Boer, B. (1991). Final Report of the 1982-1989 NWRW Research Program: Conclusions and Recommendations. Min. of Housing, Phys. Planning and Env., SOU Publs., P.O. Box 20014. 2500EA 's-Gravenhage. Van Walraven, J. H. A., Bakker, K. and Wensveen, L. D. M. (1985). The STORA results on discharge from sewer systems into surface waters.