384 Hydraulic Engineering Software - WIT Press · 2014-05-17 · and nonlinear dynamic systems to...

Simulating transformation inflow-outflow

phenomena in technical hydrology by linear

time-invariant models

O. Giustolisi

Dipartimento di Ingegneria delle Acque,II Facolta di Ingegneria di Taranto, Politecnico di BanEmail: [email protected]

Abstract

The author is studying, Giustolisi et al. , the possibility which offers the SystemIdentification Theory, Ljung , to build models to simulate the transformation inflow-outflow phenomena in technical hydrology. The aim is to resolve the complex physi-cal system represented by an urban basin in which hydrological and hydraulic proc-esses influence how the rain, summarized in a global vision in a single pluviogram,produces a variable discharge for example in the closing section of the pipe drainagenetwork. In fact, the author is investigating different possibilities offered by linearand nonlinear dynamic systems to resolve the problem of predicting the outflowduring a meteoric event or giving an hypothetical inflow. It has been reached en-couraging results, Giustolisi et al?, in simulating an urban basin by means of Box-Jenkins dynamic system using all the fifty-nine pairs of pluviogram-hydrogram ofLuzzi experimental basin (Italy) measured during three years, Calomino et al AThis paper is a further step of the research which aims to generalize the previous re-sults; the author simulates, using linear time-invariant dynamic systems, the trans-formation inflow-outflow phenomena of other six experimental urban basins, Calo-mino et alA The need to extend the research towards the study of the capabilities tosimulate by linear time-invariant dynamic systems other hydrological systems, is re-lated to the complexity of the phenomenon analyzed and also to the variableamplness of the knowledge about physical system. In fact the seven urban basins, theLuzzi basin plus the six new ones, are quite different regarding to their morphology,extension, climatic characteristics, ratio between permeable and impermeable areas,pipe drainage net and so on and this situation is helpful in testing the possibility tosolve nonlinear physical phenomena by means of linear models utilizing SystemIdentification Theory, Ljung \

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

384 Hydraulic Engineering Software

1 Introduction

The present paper has identified and estimated the linear time-invariantcausal models, by means of the dynamic system identification theory, seeLjung \ to simulate seven experimental urban basins, see Calomino et alAThe aim is to test linear model to simulate hydrological systems which arequite different as for known physical parameters like morphology, exten-sion, climatic characteristics, geographical location, destination of thearea, land characteristics, pipe drainage net and having a variable numberof rainfall-runoff events experimentally measured. Table 1 reports someglobal parameters of the basins which show the differences among the

physical systems.

Table 1. Summary of the global characteristics of the urban basins

Basins

Bolpgna

Cagliari

Con*)

Luzzi

PalermoPaviaPotenza

A

4121

13.31

2229

1.89

14.35

11.35

8.10

Sr

25

17

24

63

18

22

0

Si

50

39

18

28

52

43

85

Sp

25

44

58

9

30

35

15

Cm

0238

0.300

0.376

0.623

0240

0.384

0.451

Cmax

0.397

0.360

0.648

0.836

0.343

0.734

0.694

Cmin

0.132

0244

0.160

0.309

0.168

0.171

0.309

T

528

127

351

54

230260

97

Im

034

0,48

0,11

0,45

0,52

0,75

0,57

Imax

1,10,7

1,7

1,8

1,72,8

1,6

Imin

0,10,3

0,1

0,10,2

0,10,2

Ev

9

6

7

59

14

31

23A=> Total basin area in ha (10.000 nr).Sr+Si+Sp = 100 % => roof area (%) + impervious area (%) + pervious area (%)Cm => mean outflow coefficient among rainfall-runoff events.Cmax => maximum outflow coefficient among rainfall-runoff events.Cmin => minimum outflow coefficient among rainfall-runoff events.T => mean time length of the meteoric events in minutes.Im => mean of the maximum intensity of the meteoric events in mm/minutes.Imax => maximum intensity measured during the meteoric events in mm/minutes.Imin => minimum intensity measured during the meteoric events in mm/minutes.Ev => number of rainfall-runoff events experimentally measured in the basin.

2 Dynamic Systems

The starting point to set up a model simulating rainfall-runoff transforma-tion in the seven hydrological systems of table 1 was the dynamic systemstheory, in particular the linear, time-invariant, causal systems theory. Ta-ble 1 shows that the physical systems are not time-invariant because theglobal outflow coefficient C varies, in every basin, for each inflow-outflow


Hydraulic Engineering Software 385

event and throughout each single event as it is known. However, it is in-

teresting to solve, in first approximation, the problem of modelling suchcomplex physical systems by means of linear time-invariant system, seeLjung\ Giustolisi*, Natale? et al., and then to study, non-linear and time-variant behaviour using other tools, also not conventional ones, Giustolisi*.

2.1 Dynamic Systems with a Stochastic Process

In order to model the rainfall-runoff phenomena in the seven urban basins

a dynamic linear time-invariant causal system having two components: the

deterministic and the stochastic, has been chosen:

(" - *> + S «(*> ' *(» - *>*=0

v(n) = v(n) - Ev(ri) = £ g(k) - s(n - k)

In the first equation of eqns (1), where usually g(0)=l and h(n)=0, thedischarge sequence q(n) is calculated as sum of three values:* the first addendum represents the deterministic framework and, for this

reason, is the convolution between h(n), the impulse response of thedynamic linear time-invariant system, andp(n), the rainfall sequence;

# the second addendum represents the probabilistic framework, useful in

prediction, and is a stochastic process written as the convolution be-

tween g(n), its impulse response, and s(n), a sequence of independentrandom variables distributed according a gaussian probability densityfunction with zero mean and variance A (white noise);

• the third addendum is the mean of the disturbance v(n), which globallyexpresses the fraction of the actual runoff that the deterministic frame-work, also due to its hypothesis of linear time-invariant behaviour, is

not able to model.It is important to underline that, due to the complex phenomena in hydro-logical basins, the stochastic process aims at modelling the disturbancev(n) fraction generated by the errors in the experimental measures or un-

controllable inputs, which influence discharge, see Giustolisi *.

2.2 Nonparametric and Parametric System Identification

The estimation of the rainfall-runoff model from eqns (1) has beenachieved in two ways. Without parametrizing (Nonparametric Identifica-tion) the dynamic system expressed in eqns (1) it is possible to estimate



the impulse response h(n) of the deterministic framework by means of thecrosscorrelation function Rpq between the rain sequence and the dischargeone and by means of the autocorrelation Rpp of the rain sequence

(Correlation Analysis) thanks to

R,f("-&) (2)

Eqn (2) can be written assuming that the disturbance v(n) and the rain se-quence p(n) are not correlated (open loop) and the rain-discharge se-quences are limited, see Ljung*, Giustolisi et al.''*, Oppenheim et al/. Eqn

(2) has been solved by the prewhitening technique, see Box et al.*°, that isfiltering the sequence/? , Ep(n) is its mean value, by means of a 10th-

order AutoRegressive filter in order to obtain

e\n) = e(Q,X) + Ep(n) = AR(IQ) • p(n) (3)

The filter AR(10), calculated by eqn (3), has been applied to the sequenceq(n) and, being the correlation function ofe(n), white noise, equal to zeroexcept in n=0 where it is equal to the variance A, we can calculate h(n) by

where q '(n) is the filtered sequence q(n) by means of AR(10).The q(n) and p(n) sequences, coherently with the hypothesis of a dynamicsystem from eqns (1) (linear and time-invariant behaviour), have beenconstructed considering the total number of rainfall-runoff events, table 1,coming one after the other separated by zeros, see Giustolisi et al.\ Theevents were divided into two sets: one of about 85 % used to identify andto estimate the model and the remaining 15 % to validate it. Four se-quences, two pairs of input-output for the dynamic system, were obtained.Moreover, it is possible to estimate also the power spectrum of v(n) bySpectral Analysis, see Ljung , Giustolisi et al. . The previous way to cal-culate h(n), is useful to evaluate the dynamic characteristics of the deter-ministic framework in eqns (1) such as the order of the dynamic system,the impulse response, the response time delay and so on, but it does notprovide an efficient way to simulate the physical system. The ParametricIdentification, based on the results of the Nonparametric one, is the methodto obtain a simulating and predicting model from eqns (1).The starting point of the parametric identification is in representing thedynamic system, eqns (1), in the z-transform domain, see Ljung\ Gius-tolisi*. The following eqns (5) are the general form of different model sets



with two components, the deterministic one and the stochastic process. The

different model structures, table 2, are function of the dynamic relationship

between the probabilistic and deterministic schemes (note that z-transformis consistency with the q operator in Box et al.'°). The rational functionsH(z) and G(z), in eqns (5), represent the z-transform of h(n) and g(n)which are usually chosen stable, see Ljung* (nk is the delay).

- Ev(n)q(n) = H(z)-p(n) + G(z)-s(n)nbSL -i-nk+\b,-z

H(z) = B(z)na "f

G(z) =

•E^-z"

nd

A(z)-F(z)

C(z)

A(z)-D(z)

(5)

na=0=00#00#0=0=0*0#0

nb#0=0=0=0*0#0#0#0#0#0

nc=0*0=0#0=00=0#0=0*Q

nd=0=0=0=0=0=0=0#0#0#0

nf=0=0=0=0=0=0*0*0=0=0

ModelFIRMAAR

ARMAARX o IIRARM AX

Output ErrorBox-JenkinsARARX

ARARMAX

2.3 Simulation and Prediction by Dynamic Systems

The system described by means of eqns (1) or (5) has the goal to simulateand to predict the physical system behaviour through two components: thedeterministic and the stochastic ones. The deterministic framework is ableto simulate the rigorous repetitive linear behaviour of the hydrologicalsystem in steady condition but, for this reason, it is not able to forecast thefraction ofq(nj, which derives from non-linear time-variant phenomena orstatistical, but non deterministic, steady behaviour which are involved in

disturbance v(n). The stochastic process has the goal to predict the frac-



tion involved in v(n) due to the statistical quasi-stationary behaviour of the

physical system. The prediction can be performed by the knowledge ofv(n) forecasting the k-step-ahead value v(n+k) as in eqns (6),

"* (6)

/=o

It is important to underline that the knowledge of the discharge measuredsequence qm(n) until n and the rain sequence p(n) until n+k-1, due to thetime delay between input-output of the physical system, gives v(n), as

The eqns (6) and the first of eqns (5) gives the k-step-ahead prediction as

/C*-') ^ (8)

The eqn (8) is the prediction ofqfn) at the line horizon k performed by theselection of an observer of the system through the stochastic process trans-fer function G(z), see Ljung , Giustolisi *, as follow

%; (z)) ' 9(" + X:) _.

In other words, forecasting q(n+k\n) in eqns (8) and (9) can be obtainedby the disturbance, calculated until %, which allow to compute the path ofthe stochastic process until /?, s(n), thanks to the filter G(z) which, for thisreason, must be invertible or inversely stable. The stochastic process path,then, can be extrapolated until n+k to perform prediction.

2.4 Parameter Estimation and ARX Structure

The parametric model estimation started from the ARX structure.It is the simplest structure, table 2, which can be viewed as derived fromthe AR model, autoregressive structure, adding an extra input, B(z)-p(n).The choice of the ARX structure has been made because it is possible toestimate the parameters of the model by a linear regression as follows.The eqns (5) for the ARX structure, with nk=l, give



q(ri) + ai • q(n - 1)+.. .+<,„, • q(n -na) =

= bfp(n-l)+...+b^-p(n-nb) + s(n)

In the eqn (10) we can write the unknown parameters in a column vector

0 = [0, 2•••«„„ *i *2---bnbf

A(z) = 1 + a, • *-' +a, -z-*+...+a • z'"" (11)

B(z) = b> -z-' + 63 -z-*+.,.+brt 'Z~*

To calculate the one step-ahead prediction from eqns (9) we need to write

A(z) (12)

) = X(z)

The eqns (9) together with eqns (11) give ARX one step-ahead prediction

) (13)

In the eqn (13), written after changing the variable n to n-1, the predictedvalue q(n) is function of the known row vector

cp(n) = [-q(n - 1). . .-q(n - na) p(n - 1). . . p(n - nb)] ( 1 4)

The vector <p(n), which varies with the values of q(n) to be predicted,must be taken from the set of events selected to estimate the ARX model.The eqn (13) transformed seeing eqns (11), (13) and (14) gives

which is a linear regression with # column vector of the unknown values.A way, see Ljung\ to estimate 6 of the ARX model can be performed byminimizing, according the Last Squares Criterion, the error function

?=,(„) (16)

2.5.1 ARX Structure: Selection of the Best Order

The LS criterion gives a way to estimate parameters of a single ARX(na,nb, nk=l) model, (during the estimation nk, system response delay, is notimportant because it is only useful to shift the sequences p(n)). Increasingthe order of A(z) and B(z) the fitting between the model and the data al-ways improves because the degree of freedom in eqn (15) decreases. For



this reason, it is important to validate the model with the validation set notused to estimate the parameter 0, cross-validation.There are also other complementary ways, see Ljung\ to validate a model,here is performed the computation at different line horizon k to test thestochastic component and the model quality. The Nonparametric identifi-cation, eqns (2), (3) and (4), has been advantageous to estimate the systemresponse delay and to validate the model impulse response.However, during the validation of the model, it is helpful to improve thechoice of the orders in 9 making use of a criterion which can be able tocompensate decreasing of the loss function in (16) due to an higher orderna+nb. The selection of the best ARX model has given by the Akaike's In-formation Theoretic Criterion (AIC) or by the Rissanen's Minimum De-scription Length MDL, see Ljung\ formed as

(17)

where d represents the number of the estimated parameters of 9 (na+nb),N is the length of the estimation sequence, see eqn (14), and N/ is thelength of the data record used to calculate e(n, 9). In fact, the criteria in(17) aim at simulating the cross-validation when applied by means of theestimation set, but they can be also performed using the validation set.The comparison between the results of the dynamic model, tested with thevalidation set of the rainfall-runoff events in the seven hydrological basins,has been performed by the percent of not explained outflow variance,which is an objectively measure able to evaluate how linear time-invariantdynamic system structures predict discharge in different hydrological sys-tems. The formulation of the not explained outflow variance is

100 (18)

where qm(n) and q^ean are the measured discharge and its mean value ofthe validation set respectively, q(n) is the computed discharge by means ofdynamic model. The eqns (17) has been utilized also to evaluate the ca-pability of the model to predict (see follow) at different line horizon



(k=/m, 20, 75, 10, 8, 6, 4, 2J). This is useful to test the stochastic compo-

nent as the simulation (k=oo) to test the deterministic one.

The selection of the ARX structure, to model the seven hydrological sys-tems, has been made because it is the simplest one to estimate the parame-ters, see eqn (15). In fact, here the goal is not to improve the model qualitythrough the finest tuning of the probabilistic component.Moreover, the author, see Giustolisi et al/, built a BJ model for the Luzzibasin but the model improvement (in prediction) was not significant as thatobtained selecting the ARX one. The author believe that modelling theprobabilistic component, with the hypothesis it has the same dynamics of

the deterministic one through A(z), perhaps, is satisfactory to model quasi-stationary behaviour of the hydrological systems.

2.5.1.1 Results

The models for every basin are selected, after identification and estimationphase, by means of the AIC and MDL criteria. The data record used tocalculate the loss function were from the estimation set but also from thevalidation set. Table 3 reports the results of the model orders and delays.

pplimb

nb

nk

cr

A

1

5

M

Table 3. Orders and delays of the calculated models

ologna

3

1

5

A

V

2

10

1

M

6

10

1A

E

Cagliari

3

3

9

M

3

3

9

A

V

2

10

3

M

6

10

3

A

E

Como

l

1

1

M

1

1

1

A

V

9

6

1

M

9

6

1

A

E

Luzzi

"7

5

1

M

5

5

1

A

V

5

10

1

M

5

10

1

A

E

Palermo

4

4

3

M

4

5

2

A

V

6

9

4M

10

10

3

A

E

Pavia

3

1

5

M

3

5

1

A

V

10

10

3

M

10

10

3

A

E

Potenza

1

3

6

M

1

3

6

A

V

9

5

2

M

9

9

1

A

Ecr: criterion - M:MDL criterion used - A:AIC criterion used - qioutflow se-quence for criterion - V: Validation sequence used - E [Estimation sequence used

Table 3 shows that MDL criterion gives less, sometime equal, orders thanAIC one because they both penalize model data overfitting, but the firstpenalizes complex structures more than the second. The models obtainedby means of criteria using validation set have less orders, table 3, than theones selected by means of criteria using estimation set, because the loga-rithmic shape of eqns (17) penalize higher values of the loss function.Moreover the models, obtained by AIC and MDL criteria calculating lossfunction using estimation set, do not improve the cross-validation bymeans of eqn (18). For this reason, the best model is selected from table 3,for each basin, given by MDL or AIC criterion using validation set.

Table 4 reports the parameter values of this models for the seven basins.



As for the comparison among the seven basins of the ARX model selectedcapability to simulate, table 4, the results, tested by means of eqn (18),show that it is possible to divide basins in two sets as reported in figs 1and 2.

Table 4. Best models selected byMDL or AIC criterion

•IjLBO 2.41

'W

CO

LZ

PA

PV

PO

0.12

5.30

0.03

0.03

0.06

0.48

at-1.269

-1.115

•0.882

-1282

-1.745

-1.787

0.897

a%

0.379

0.091

0.509

0.714

0.886

&i

0.081

0.042

0.126

0.078

a*

0.117

•0.071

&5

0.030

b,0.036

0.010

0.033

•0.014

0.0017

0.003

0.056

bz

•0.015

0.007

0.0015

•0.002

•0.008

ba

0.018

0.026

0.0018

0.007

•0.005

b4

0.065

0.0021

•0.009

bs

0.034

0.008

(q(n) andpfn) are in mVmin and n is in minutes)

Bologna, Luzzi, Palermo, Potenza Basins

K=2 K=4 K=6 K=8

line-horizon K

- Bologna —»— Palermo -A— Potenza —*— Luzzi

Figure 1. Less pervious basins

Figure 1 represents the model simulation (deterministic component) andprediction (stochastic component) at various line-horizon in less perviousbasins. These basins are the best simulated, by linear, time-invariant modelbecause the time-variant behaviour of the pervious areas are smaller thanthe dynamics of the impervious areas. The model simulating capabilityseems strictly inversely correlated to percent of the pervious areas, in fact,it decreases from Luzzi, the less pervious, to Como, the less impervious,and the Potenza basin is a single exception, figs 1 and 2.

Figure 2 shows the results of the model simulation and prediction in more



pervious basins. As already said, the simulation is not good because the

non-linear, time-variant behaviour of the great fraction of the pervious ar-

eas. The stochastic component, seems to give results at line-horizon with kless than 6 -*- 4, as figs 1 and 2 show. It is useful to underline that themodel delay is important to evaluate the possibility to use prediction,thanks to real time acquisition of data, for the system control. For this rea-

son, the rainfall measure station should be located to make large delaywithout decreasing model quality.

Finally, the utility of the stochastic process seems to be correlated to theextension areas, perhaps because the approximation of uniform rainfall,through a single input to the model, is greater.

Cagliari, Co mo, Pa via Basins

K=2 K=8 K=10

Hue-horizon K

- Cagliari —a— Corno —A— Pavia

Figure 2. More pervious basins

3 Conclusions

The paper aim is to test a linear time-variant model simulating and predict-ing hydrological urban basin behaviour. It has been reached thanks to thedata records from seven experimental basins, see Calomino et alA

After the model identification and estimation, it has been performed thevalidation by means of the MDL and AIC criteria which allow to selectmodel structures penalizing data overfitting and model complexity.The seven ARX models chosen for every basin, have been used to com-pare, thanks to NEXV%, the ability to simulate the physical system behav-iour through the deterministic framework. The results show that this pos-sibility is strictly inversely correlated to the fraction of pervious areas.



The stochastic process capability to improve the model forecasting hasbeen tested and compared among the seven basins. The utility of the prob-abilistic framework seems to be more important for greater basins and it isefficacious at line-horizon less than 6.Starting from the present results, the author aims at improving modelsimulation and prediction by means of non-linear models or non conven-

tional tools as Artificial Neural Networks.

References

[1] Giustolisi 0, Masini P., Mastrorilli M, Identificazione non Para-

metrica e Parametrica di Sistemi Dinamici applicata all 'Idrologia,Giornate di Studio in Onore del Prof. E Orabona, Bari, Italy, 13-14ottobre 1997.

[2] Giustolisi O, Porcaro F., Taratura di un Si sterna Dinamico per un

Bacino Urbano, Giornate di Studio in Onore del Prof. E. Orabona,Bari, Italy 13-14 ottobre 1997.

[3] Ljung L, System Identification: Theory for the User, Prentice-Hall

Inc., Englewood Cliffs, New Jersey, U.S.A 1987.[4] Calomino F, Caputo V., Galasso L, Piro P., // Bacino Sperimentale

Urbano di Luzzi (CS) - Osservazioni sperimentali nelperiodo 1987-1990, Editoriale Bios s.a.s., Cosenza, Italy 1993.

[5] Calomino F, Maksimovic C, Molino B, URBAN DRAINAGE, Experi-mental Catchments in Italy, Editoriale Bios s.a.s., Cosenza, Italy1995.

[6] Giustolisi O., Simulazione e Predizione Stocastica nei SistemiIdrologici, XXVI Convegno di Idraulica e Costruzioni Idrauliche,Catania, Italy 10-12 settembre 1998.

[7] Natale L., Todini E. A Constrained Parameter Estimation Techniquefor Linear Models in Hydrology, Matematical Models for SurfaceWater Hydrology, John Wiley & Sons, London, U.K. 1977.

[8] Giustolisi O , Mastrorilli M., Porcaro F, Application of a NeuralNetwork to Transformation Inflow-Outflow Phenomena, XII AIENGConf., Wessex Institute of Technology - Capri, Italy 7-9 July 1997.

[9] Oppenheim A.V., Schafer R.W., Elaborazione Numerica dei Seg-nali, editrice FrancoAngeli, Milano, Italia 1990.

[10] Box G.E.P., Jenkins G.M., Time Series Analysis, Forecasting andControl Holden-Day, San Francisco, U.S.A. 1989.


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