REAL-TIME FLOW FORECASTING IN THE PARANA RIVER: A COMPARISON BETWEEN ARMAX AND ANN MODELS

REAL-TIME FLOW FORECASTING IN THE PARANA RIVER: A COMPARISON BETWEEN ARMAX AND ANN MODELS Pujol Reig, Lucas (1); Ortiz, Enrique (1); Cifres, Enrique (2) and García-Bartual, Rafael (2) (1)

HidroGaia S.L. Avda. Juan de la Cierva 27, 46980 - Paterna - Valencia - Spain phone: +34 96136 6072; fax: +34 96136 6073; emails: lucaspujol@yahoo.com.ar; eortiz@hidrogaia.com (2) Universidad Politécnica de Valencia (Valencia-Spain) phone: +34 963879891 ; fax: +34 963877618; e-mail: enrique@cifres.com ; rgarciab@hma.upv.es

ABSTRACT Linear and nonlinear approaches are compared herein, in terms of their performance in 5 and 10 days ahead flow forecasts for a given section of the Paraná River, located 34 km upstream the city of Santa Fe (Argentina). The models used are an autoregressive moving average with exogenous variable (ARMAX) and a multilayer feedforward artificial neural network (ANN). The variables used as inputs to the models are daily flows at the mentioned section and another cross section located 571 km upstream, together with daily rainfall measured in an intermediate rain gauge station, all during the period 1994-1998.

Different configurations of the models have been tested, varying the number of inputs and parameters, but in general the resulting forecast quality with either method is similar and quite satisfactory.Nash Sutcliffe coefficient of efficiency is in all cases over 0.94 for the 10days ahead forecasts in the validation sample. Among the compared models, best results are obtained with the non linear approach ANN. Some uncertainty considerations are also pointed out, after the analysis of the empirical error distributions under different modelling strategies. It is shown that the apparently centred normal distribution expected for errors really departs significantly from zero-average when different parts of the hydrograph are taken into account.

Keywords: ARMAX, Neural Network, flow forecast, Real-time 1 INTRODUCTION Forecasting flows and corresponding water levels at different sections of the Paraná River (average flow around 18,000 m3/s), is crucial to minimize the effects of floods affecting properties and human activities. On the other hand, low flows can affect navigation conditions through the river. The National Institute of Water and Hydrologic Alert System for the Plata Basin - INA SiyAH – provides forecasts with time leads of 5 and 10 days, at several river harbours located along the middle and lower Paraná. This anticipation is enough to alert population in different agricultural and urban areas and take the appropriate actions to reduce damages and losses. Such predicted flows estimations are obtained with the hydraulic model Ezeiza-V [Jaime and Menéndez, 1997].

The research reported herein is developed under the same practical framework, applying alternative modelling schemes for an improved flow forecasts in the Paraná River, using exactly the same time horizons.

The application of artificial neural networks (ANNs) to various aspects of hydrological modelling has yielded to interesting and promising results during recent years. In particular, rainfall-runoff models and real time forecasting models based on ANN’s schemes have received special attention [Kang et al., 1993; Karunanithi et al., 1994; Smith and Eli, 1995; Minns and Hall, 1996; Shamseldin, 1997; Fernando and Jayawardena, 1998; Sajikumar and Thandaveswara, 1999; Tokar and Markus, 2000; Kumar and Minocha, 2001; García-Bartual, 2002; Jain and Prasad, 2003; Abrahart et al., 2004; Cigizoglu, 2005; Sahoo et al., 2006]. Their ability to incorporate in a systematic approach non linear relationships between variables

represents an attractive aspect in favour of ANNs modelling.

On the other hand, the classical and well known ARMAX linear modelling approach (autoregressive moving average with exogenous inputs) has proved to be an efficient forecasting tool in many practical instances when rainfall and flow registers are available at different points or river sections, while other information about the catchment itself required by other more complex models (i.e., physically based distributed models) is not available.

[Burn and McBean, 1985; Awwad and Valdés, 1992; Hipel and McLeod, 1994; Papamichail and Georgiou, 2001].

Both, ARMAX and ANN approaches have their clear advantages for real-time modelling purposes because once calibrated and implemented, they are computationally very fast and adapt perfectly well to the various restrictions and limitations imposed by real-time systems under operation. Others papers also reported comparison of this two type of models showing similar results [Abrahart and See, 2000].

Paper contents are organized in the following way: The case study and data set is presented in section 2. Section 3 presents briefly the general formulation of the two modelling approaches, while section 4 reports the application with its results and finally discussion and some relevant conclusions are presented in section 5.

2 CASE STUDY AND DATA SET The Plata Basin has a drainage area of 3,005,000 km2, being the fifth largest of the world.

It covers part of 5 countries: Argentina, Bolivia, Brazil, Paraguay and Uruguay.

The area of study is located on the lower Paraná as shown in Figure 1. The basin area between Chapetón and Corrientes stations is about 350,000 km2, and the main river length between them is 571 km. This reach is characterized by its low slope (5.7cm/km) and a diversity of cross section shapes depending on water levels.

The maximum mean daily flow in Chapetón’s station registered in the past is 31,930 m3/s (year 1982) with an annual mean flow of 18,248 m3/s.

Figure 1: Plata basin and study area.

The available data for this research was taken from the “Ministerio de Planificación Federal. Inversion Pública y Servicios.

Secretaría de Obras Públicas. Subsecretaría de Recursos Hídricos Argentina, 2004”, and include mean daily flows at Chapetón and Corrientes gauge stations, and also daily precipitation series at Batel- Paso Cerrito rain gauge, situated between both river sections (see Figure 1). Continuous data are available during period 1994-98.

Figure 2 shows values of mean monthly flows at Chapetón and Corrientes. It can be seen that upstream flows are significantly lower during the first part of the hydrologic year, while the situation is inverted from January to April, with higher flows in Corrientes than those measured downstream (Chapetón). The average travel time of a given peak flow from Corrientes to Chapetón has been estimated in 15 days. The estimation was carried out by means of a least-squares objective function derived from comparison of delayed Corrientes series and Chapetón series. The system shows a long memory, with significant autocorrelations (higher than 0.8), lasting over 22 lagged daily intervals.

22,000

For the modelling purposes, the data series have been conveniently splitted into two subsets. The first two years (1994-95) are reserved for validation (testing) purposes, while the remaining three (199698) are to be used for calibration (training).

21,000 20,000

Flows (m3/s)

19,000 18,000 17,000 16,000 15,000 14,000 13,000 12,000 SET

OCT

NOV

DIC

JAN

FEB

Corrientes 1903/2003

MAR

APR

MAY

JUN

JUL

AGO

Chapetón 1975/2003

Figure 2: Mean monthly flows for Chapetón (red) and Corrientes (blue).

3 MODELS DESCRIPTION 3.1

ARTIFICIAL NEURAL NETWORK An ANN is a massively parallel-distributed information processing system that has certain performance characteristics resembling biological neural networks of the human brain [Haykin, 1994]. Each net is characterized by its architecture, which is represented in terms of an organized number of hidden layers and nodes, input nodes, activation functions and the direction of the information’s flux through the network (see Figure 3).

The vast majority of the networks are trained with the popular back-propagation learning algorithm BPNN [Sahoo et al., 2006], which is also used herein, with the MSE (mean squared error function) as the objective function. The number of hidden layers is fixed to one, which is sufficient to approximate any complex nonlinear function with the desired Figure 3: Typical 3-layered feed forward accuracy [Cybenko, 1989; Hornik et al., 1989]. network with a single hidden layer NN(11-3-2).

Following a typical and well tested configuration in the literature, the activation function of the input and output layer has been set to a linear function, while the hidden layer incorporates the non linear sigmoid function (eq. [3.1]), providing the ability of the network to capture possible complex non linear relationships between inputs and target outputs.

1 f ( y) = (0;1) [3.1] 1 + e−y Figure 3 shows an example of one of the artificial neural networks topologies used in this research. In such scheme, input layer have passive nodes allowing the presentation to the network of several selected variables (i.e. past flows in the gauge stations and cumulative rainfall), while the output layer nodes will be producing the forecasted flows values for 5 and 10 days ahead at Chapetón station. It should be noted that simultaneous forecasts for the two chosen time horizons are produced by the same network.

The input to each of the nodes contained in the hidden and output layer consist of a linear combination (eq. [3.2]), usually known as PSP (Post-Synaptic-Potential). n

y j = ∑ wi , j xi − b j i =1

[3.2]

where; wi,j are the weights of node j coming from a node i of the previous layer; bj is the bias of node j and xi is the input to the node j from each i node of the previous layer..

Further details about the back propagation training algorithm and operation of this typical network configuration used here can be found in Wasserman, 1989 and Fausett, 1994.

3.2

AUTOREGRESIVE MOVING AVERAGE WITH EXOGENOUS VARIABLE MODEL Alternatively to the ANN, classical modelling approach for flow forecasting has been used, by means of an ARMAX stochastic model. Further details about ARMAX modelling can be found in [Box and Jenkins, 1976].

In our case, two different ARMAX models are formulated, one for the 5-day forecast, and another for the 10-day forecast. The analytical form of the model ARMAX (p, r1, r2, q) is as follows: p

r1

r2

q

i =0

i =0

i =1

i =1

Qtpred + k = ∑ φ i Qt − i + ∑ γ i SPt − Li + ∑ λi MFt − Ji + ∑ θ i ε t + k −i + ε t + k

Where, Qt : Flow at Chapetón station.

SPt : Precipitation at Batel-Paso Cerrito rain gauge station MFt : Flow at Corrientes station. εi : Error of the model with N(0; σε); εi = Qtpred + k − Qt + k k: Time horizon for forecasts p, r1 , r2 and q: Orders of the model φ , γ , λ and θ : Parameters of the model.

L and J: Factors for SP and MF to consider different time delays.

The moving average term has been set to zero (q = 0) for this application.

4 APPLICATION 4.1

DATA PREPROCESSING AND VARIABLES SELECTION For the models under consideration, basically data-driven type, it is fundamental to do a careful exploration of data series previous to the modelling stage itself. Cross-correlation, autocorrelation, and partial autocorrelation analysis with the three series (Flows at Corrientes and Chapetón and rainfall at Batel-Paso Cerrito) was carried out. This analysis provides good hints for an appropriate selection of predictors containing maximum information for the forecasts purposes. Similarly, a number of time aggregation intervals have been explored concerning rainfall data. After the analysis, an interval of 10 days cumulative rainfall is selected, appearing to be a relevant time period, more than daily data, from the perspective of flow forecasts at Chapetón. In the same manner, this previous exploratory analysis of variables showed that a smoothed Corrientes flow series yields also to higher correlations with those of Chapetón. Moving averages with 11 daily flows were finally used as inputs for all tested models. Table 1 comprises the set of prediction variables used.

For an improved numerical efficiency, both in the ARMAX and ANN models, previous data transformation is applied, reducing skewness and providing homogeneous variation ranges for the different variables. A number of criteria are found in the literature for this purpose. A simple straight-forward pre-processing function is the one use in [García-Bartual, α

 x  2002], which uses two parameters: x'i =  i  where x is the variable and M is a reference Mx  value (close to the maximum). The parameters used here are:

Precipitation at Batel-Paso Cerrito [α = 0.3; M = 635 mm], Corrientes’ flows [α = 0.6; M = 50856 m3/s] and Chaperon’s flows [α = 0.6; M = 33817 m3/s] Rainfall at Batel-Paso Cerrito rain gauge station [mm] SPt = SumP( t; t - 9) SPt-10 = SumP(t - 10; t - 19) SPt-20 = SumP(t - 20; t - 29)

Flow at Corrientes station [m3/s]

Flow at Chapetón station [m3/s]

MFt-5 = MeanF(t; t - 10) MFt-10 = MeanF(t - 5; t - 15) MFt-15 = MeanF(t - 10; t – 20) MFt-20 = MeanF(t - 15; t - 25) MFt-25 = MeanF(t - 20; t - 30)

Qt Qt – 1 Qt – 2

Table 1: Prediction variables used.

4.2

MODELS The models chosen for the analysis are described in Table 2.

Models

5 days ahead forecast 10 days ahead forecast 3 previous intervals of 10 days cumulative rainfall at Batel-Paso Cerrito 5 previous moving averages with 11 daily flows at Corrientes station 3 consecutive previous flows at Chapetón station

NN (R) X X

NN (NR) X X

X

ARMAX (R5) X

ARMAX (NR5) X

ARMAX (R10)

ARMAX (NR10)

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Table 2: Models analyzed.

4.3

RESULTS AND PERFORMANCE EVALUATION Evaluation of models performance was done through the following indexes: n

•

Nash and Sutcliffe (1970) coefficient (E)

E = 1−

∑ (Q

− Qi ) 2

pre

i

i =1 n

∑ (Q

i

i =1

− Q )2

Where, Qi is the ith value of the observed series; Qipre is the ith value of the predicted series and n is the total quantity of elements. E = 1 would indicate a perfect forecast, while E = 0 is the E coefficient for a forecast value equal to the mean of the series.

∑ (Q n

•

Root Mean Squared Error (RMSE)

RMSE =

i =1

i

− Qipre n

)

2

Qipre − Qi • Absolute Relative Error (ARE) AREi = Qi The RMSE error statistic is the most common error indicator used in practice. But it indicates only the model’s ability to predict a value away from mean [Hsu et al., 1995].

Alternatively, it has been computed the PFC (Peak Flow Criterion) [Ribeiro et al., 1998] gives a good idea of the quality for the predictions of peak flows.

1/ 4

 np 2   ∑ (Qip − Qipre ) 2 Qip    • Peak Flow Criterion (PFC) PFC =  i =1 np  p2   Q ∑ i i =1   th p Where, Qi is the i value of the peak flows of the observed series at time t; Qipre is the ith value that corresponds to the predicted flow at time t; np is the total number of peak flows selected. Low values of PFC indicate good flow peak predictions, and a perfect prediction yields to PFC = 0.

In order to assess forecasts quality, all given coefficients have been also calculated for the naïve – persistence model, that is Qtpred + k = Qt .

Calibration Models

t+5

t + 10

ARMAX(R5) ARMAX(NR5) NN(NR) NN(R) Naïve ARMAX(R10) ARMAX(NR10) NN(NR) NN(R) Naïve

Validation

E (Nash and Sutcliffe) 0.9919 0.9916 0.9912 0.9905 0.9650 0.9666 0.9645 0.9679 0.9724 0.8756

0.9859 0.9856 0.9863 0.9854 0.9556 0.9416 0.9428 0.9428 0.9468 0.8542

Table 3: Nash and Sutcliffe coefficient for the validation and calibration data.

Errors distribution plays a central role for empirical assessment in forecasts uncertainty [Jain, 2004]. Figure 6 shows the empirical errors distribution after the application of best ANN model (10-days forecast). Figure 7 shows the same empirical distributions derived only for error predictions produced during the rising limb of the hydrograph (in blue), and the corresponding one obtained only during recession curves of the observed hydrographs at Chapetón.

Calibration

Models

Validation

Calibration

RMSE

t+5

t + 10

316.94 323.14 329.94 343.56 655.74 640.89 660.98 628.18 582.85 1218.08

ARMAX(R5) ARMAX(NR5) NN(NR) NN(R) Naïve ARMAX(R10) ARMAX(NR10) NN(NR) NN(R) Naïve

Validation

PFC 286.81 289.36 282.12 291.46 511.30 584.99 578.96 578.96 558.63 938.58

15.54 15.05 17.50 20.40 26.76 29.84 29.64 27.44 27.98 41.15

15.03 14.94 15.62 17.06 20.16 23.32 23.92 23.92 20.34 29.85

Table 4: RMSE and PFC coefficients for the validation and calibration data.

5 DAYS AHEAD FORECAST (validation period) ARMAX(R5) NN(NR) 5 days ahead forecast with NN(NR) model - Validation data 1994/1995 25000

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5 days ahead forecast with ARMAX(R5) model - Validation data 1994/1995

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Forecasted Flows

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[m /s]

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26000

0 12000 17/12/1993 16/04/1994 14/08/1994 12/12/1994 11/04/1995 09/08/1995 07/12/1995 05/04/1996 Forecasted Flows

Observed Flows

ARE

Figure 4: 5 days ahead forecast of the best models of ARMAX and NN for the validation period at Chapetón station. Red lines in the top figures represent errors equal to ± 10% of the observed flows.

10 DAYS AHEAD FORECAST (validation period) ARMAX(NR10) NN(R) 10 days ahead forecast with NN(R) model - Validation data 1994/1995 25000

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[m /s]

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ARE

Forecasted Flows

Observed Flows

ARE

Figure 5: 10 days ahead forecast of the best models of ARMAX and NN for the validation period at Chapet贸n station. Red lines in the top figures represent errors equal to 卤 10% of the observed flows.

-4

-3

x 10

x 10

9

1.2

Error data Normal

8

Rising limb Falling Limb Rising Limb Falling Limb

1 7 0.8

5

Density

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6

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2 0.2 1 0

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0 Errors

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Figure 6: Distribution probability plot for the Figure 7: Distribution probability plot for the errors in the validation period obtained with a 10 errors grouped. Red, errors produced during the days ahead forecast using NN(R) model. falling limbs of the hygrograph, and blue during the rising limbs.

5 DISCUSSION AND CONCLUSIONS In this case study there is a high lineal correlation between flows in Corrientes and Chapet贸n. The linear model ARMAX produces very good quality forecasts, particularly for the t + 5 time horizon. An efficiency coefficient of E = 0.986 (see Table 3) for the validation period was obtained, reflecting the good performance of the model. The model also shows good ability to forecasts peaks (see Table 4), as shown in Figure 4 and Figure 5.

ANN model produces, with one single network, Qt+5 and Qt+10 forecasts, showing also very good performance, even superior to the former at Qt+10. It should be pointed out that rainfall data consideration in either model does not improve Qt+5 forecasts, as can be seen in Table 3 and Table 4 where the best models (cells painted) are no rain models. On the longer Qt+10 forecasts, though, rainfall inclusion helps to improve performance. Best results are obtained for NN(R) (E = 0.947) for the Qt+10 forecasts and NN(NR) for the Qt+5.

Concerning empirical error distribution, Figure 6 shows that it can be reasonably well described by a Gaussian distribution with µ = 150 m3/s and σ = 539 m3/s , which gives approximate 90% confidence intervals of ± 840 m3/s. Though, this first approximation to quantify overall uncertainty requires further aspects to be investigated. First, variance of the error should in principle be conditioned to the flow values themselves, which in turn, can be done in practice as ε = f(Q|Qpred), and not as f(Q|Qobs), as Qt+10 values are not know in present time.

On the other hand, error distributions described in Figure 6 and Figure 7 show significant differences between errors at the rising limb of the hydrograph when compared to those during the falling limb. In the first case, a general flow underestimation occurs, while the second has a positive mean value, that is, an overestimation tendency. This question clearly points out interesting possibilities of future modelling refinements using a combination of models, depending on the hydrograph evolution by means of some automatic selection procedures of the active model that could be done by means of classification artificial networks.

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