A neuro-fuzzy model for inflow forecasting of the Nile river at Aswan high dam

Water Resour ManageDOI 10.1007/s11269-006-9027-1

A neuro-fuzzy model for inflow forecasting of the Nileriver at Aswan high dam

Ahmed El-Shafie · Mahmoud Reda Taha ·Aboelmagd Noureldin

Received: 3 February 2005 / Accepted: 7 March 2006C© Springer Science + Business Media B.V. 2006

Abstract River flow forecasting is an essential procedure that is necessary for proper reser-voir operation. Accurate forecasting results in good control of water availability, refinedoperation of reservoirs and improved hydropower generation. Therefore, it becomes crucialto develop forecasting models for river inflow. Several approaches have been proposed overthe past few years based on stochastic modeling or artificial intelligence (AI) techniques.

In this article, an adaptive neuro-fuzzy inference system (ANFIS) model is proposed toforecast the inflow for the Nile River at Aswan High Dam (AHD) on monthly basis. Amajor advantage of the fuzzy system is its ability to deal with imprecision and vagueness ininflow database. The ANFIS model divides the input space into fuzzy sub-spaces and mapsthe output using a set of linear functions. A historical database of monthly inflows at AHDrecorded over the past 130 years is used to train the ANFIS model and test its performance.The performance of the ANFIS model is compared to a recently developed artificial neuralnetworks (ANN) model. The results show that the ANFIS model was capable of providinghigher inflow forecasting accuracy specially at extreme inflow events compared with thatof the ANN model. It is concluded that the ANFIS model can be quite beneficial in watermanagement of Lake Nasser reservoir at AHD.

Keywords Inflow forecasting . Reservoir management . Fuzzy systems . Neuro-fuzzysystems . Nile river

1. Introduction

River inflow forecasting has a significant role in mitigation of impacts of deficit(surplus)on water resources systems (Wood, 1980) and therefore is essential for reservoir systemcontrol and management. Conventionally, statistical models have been utilized in the lastfour decades for inflow forecasting based on time series analysis (Box and Jenkins, 1970;

A. El-Shafie · M. Reda Taha (�) · A. NoureldinDepartment of Civil Engineering, University of New Mexico, Albuquerque, NM 87131, USAe-mail: [email protected]

Springer

Water Resour Manage

Salas et al., 1988). Regression and autoregressive moving average (ARMA) models arestandard forecasting models for statistical time series analysis. One source of complexity inforecasting river inflow is the non-stationary nature of inflow series (Grino, 1992). Whilemost existing models can be classified as linear models with limited non-stationary modellingabilities the inflow pattern is highly nonlinear and non-stationary (Hsu et al., 1995). Despitethe inadequacies of ARMA, inflow forecasting has been commonly modeled using ARMAmodels during the past several decades (Salem and Dorrah, 1982; Karunithi et al., 1994).As Artificial intelligence (AI) paradigms have been successfully deployed in modellingnon-linear systems in different engineering fields (e.g. Tsai and Hsu, 2002; Seker et al.,2003; Colulibaly et al., 1998; Colulibaly and Anctil, 1999), it became necessary to examinethe ability of these paradigms in modelling the nonlinear and stochastic inflow. Recentinvestigations using some of these models for inflow forecasting showed promising resultsas discussed below.

1.1. Hydrological modeling using AI paradigms

1.1.1. Hydrological modeling utilizing neural network

Motivated by its nonlinear modeling capabilities, artificial neural networks (ANN) wereutilized in several inflow forecasting models (Olason et al., 1997). The ANN approachhas proven to be effective in modelling the river flow process in situations where explicitknowledge of the internal hydrologic sub-process is not available (Subramanian et al., 1999).Literature survey showed that ANN models were successfully applied to problems involvingriver watershed and weather prediction. French et al. (1992) demonstrated that ANN iscapable of forecasting the complex temporal and spatial distribution of rainfall generatedby a rainfall simulation model. Chang and Tsang (1992) used ANN to model snow waterequivalent from multi-channel brightness temperatures and obtained results better than amultiple regression model.

Moreover, several types and architectures of ANN were investigated in the field of waterresources management specially for river flow forecasting. Kang et al. (1993) developed ANNfor daily and hourly inflow forecasting using one of four pre-specified network structures.Gautam (1999) developed an auto regressive neural network (ARNN) model to predict waterlevel of the lake IJsselmeer at the North-Holland on the basis of incoming river discharge,water level at the sea-side of the sluices and wind event. Seno et al. (2003) proposed a methodfor inflow forecasting of the Karogawa Dam by using ANN. The methodology was appliedusing the rain data outside and inside the dam basin. It was shown that a model consideringthe rain data inside and outside the dam basin improved the forecasting accuracy by about30% over other models that considered the rain data inside the dam basin only. Coluibalyet al. (2000, 2001a,b) reported that recurrent neural network (RNN) could be successfullyused for inflow forecasting while taking into consideration the precipitation, snowmelt andtemperature. Computational time and complexity was a major obstacle to achieve the desiredaccuracy.

1.1.2. Hydrological modeling using fuzzy systems

Fuzzy set theory has also been examined in hydrological modelling such as rainfall forecasting(Yu and Chen, 2000), inflow prediction (Chang and Chen, 2001) and reservoir operation(Russell and Campbell, 1996). These models included fuzzy optimization techniques, fuzzyrule based systems, and hybrid fuzzy systems. In these models, fuzzy rules were created

Springer

Water Resour Manage

using expert knowledge or observed data. Several reservoir operation techniques based onfuzzy systems can be found in the work of Fontane et al. (1997), Huang (1996), and Saadet al. (1996). Fuzzy control systems for reservoir operations are also presented by Russelland Campbell (1996) and Shrestra et al. (1996). Ballini et al. (2001) developed a neuro-fuzzynetwork for inflow forecasting that was used in Brazilian hydroelectric plants. The modelwas established to achieve a root mean square error (RMSE) of 0.001 m3/sec.

Apparently, neural networks and fuzzy models are suitable alternatives for modelling ofthe nonlinear and non-stationery inflow problem when compared to conventional modellingapproaches especially in situations that do not require modeling of the environmental andgeometrical parameters affecting the river flow.

2. Aswan high dam (AHD) – case study

Aswan’s High Dam (AHD) is the major irrigation structure in Egypt and is located on theNile River near the city of Aswan. With the completion of AHD in 1970, a heuristic operationwas established and adopted for the management and operation of the AHD reservoir. Itsreservoir “Lake Nasser” is considered as the largest man made lake worldwide. Lake Nasseris supplied by Nile river flow with an average annual inflow of 84 Billion cubic meter (BCM).The reservoir supplies water for irrigation, municipal and industrial and energy productionas well as navigation purposes.

Developing optimal release policies of a multi-purpose reservoir like Lake Nasser is acomplex process, especially for reservoirs where explicit stochastic environments exist (i.e.with high uncertainty in future inflows). The development of management tools that targetadopting optimal operating policies for reservoirs spans over four decades of research (AbuZeid and Abdel Daym, 1990; Sadek et al., 1997; Fahmy, 2001). The significance of thisresearch is attributed to the fact that the consequences of reservoir release decision cannotbe realized until future unknown (inflow) events occur.

Operation policy of the reservoir is based on dividing Lake Nasser storage into six zones,as illustrated in Figure 1. The dead storage zone, that is allocated to receive sediments comingwith the river flow during the flood period, has a top elevation of 147 m with total volumeof about 31 BCM. The operation rule for this zone is to release no flow regardless of the

Fig. 1 Inflow as the main component for different operation zones of AHD reservoir

Springer

Water Resour Manage

Fig. 2 The natural inflow at Aswan for period 1871–2000 for month August

downstream requirements. The live storage zone, which amounts 90 BCM includes the bufferzone and the conservation zone. The buffer zone lies between elevation 147 m and elevation150 m while the conservation zone lies between elevation 150 and 175. Within the live storagezone, the dam operators make their releases to meet the downstream requirements. The totalannual release should not exceed Egypt’s quota (55.5 BCM). The remaining storage betweenelevations 175 m and 183 m is divided into the flood buffer zone and flood control zone.Although the emergency spill zone is designed to have a crest level of 178 m, it is decided tocontrol the reservoir releases so that the water elevation does not exceed 175 m at the end ofthe hydrologic year (July 31st). As shown in Figure 1, the level of 178 m is separating theflood buffer zone from the flood control zone at which any accumulated volume has to bespilled (Abu Zeid and Abdel Daym, 1990; Sadek et al., 1997; Fahmy, 2001).

A database of historical natural monthly inflow series at AHD for the period between1871 and 2000 has been made available (National Water Research Center, 2005). The naturalinflows in Aswan for months August and March are demonstrated in the Figures 2 and 3.Table 1 presents the maximum and minimum values of inflow that have been occurred duringthe period 1870 to 2000. It is noteworthy, that the water year starts in month of August thatmarks the start of the flooding season in Egypt. It is obvious from the inflow values thatthe period between August and January represents a wet season, while the rest of the yearrepresents a dry season.

2.1. Problem statement

In fact, the nature of the Nile river inflow can be described as a multivariate process. Physi-cally, the flow at a given station depends on the past and present flow rates at the upstreamstations. However, accurate information about the flow rates at the upstream stations is veryscarce. Therefore, the modelling process shall be based on historical inflow data at the AHD.Fortunately, accurate historical inflow data at the AHD over the past 130 years are available.Information about the flow rates at the upstream stations as well as the climatic conditionaffecting the inflow are all embedded in these data.

Most existing models used univariate autoregressive moving average representation ofthe natural inflow at AHD and thus tend to either overestimate the inflow for low flood

Springer

Water Resour Manage

Table 1 Maximum andminimum inflow values at AHDfor the period of 1871–2000

Maximum inflow Minimum inflowMonth (BCM) (BCM)

August 29.10 6.50September 32.79 7.31October 27.40 5.97November 14.40 4.12December 11.00 2.83January 7.70 1.72February 6.04 1.15March 5.81 1.07April 5.26 0.95May 4.72 0.80June 5.16 0.90July 11.03 1.74

Fig. 3 The natural inflow at Aswan for period 1871–2000 for month March

time or underestimate for high flood time. Given the size of Lake Nasser and the high damthis drawback represents a major challenge to ensure acceptable storage limits, namely thedead storage and the reservoir capacity. Recently, inflow forecasting model for the Nileriver based on multilayer perceptron neural network (MLP-NN) has been developed by El-Shafie et al. (2004). The MLP-NN model showed reasonable accuracies for average inflowforecasting overcoming most of the drawbacks of ARMA models. However, the MLP-NNmodel still has a major limitation for inflow forecasting at extreme inflow events as the MLP-NN has a relatively high average error of 9% (El-Shafie et al., 2004). This limitation doesnot provide acceptable margin of errors for managing Lake Nasser reservoir. Therefore, anapproach that can provide accurate inflow forecast of the Nile river at average and extremeinflows events is highly necessitated for efficient decision making at the AHD and LakeNasser.

2.2. Objectives

The objective of this paper is to investigate the potential use of adaptive neuro-fuzzy infer-ence system (ANFIS) to improve the forecasting accuracy for the inflow at AHD especially at

Springer

Water Resour Manage

extreme inflow events. Successful development of ANFIS will warrant the planning commit-tee at Lake Naser an efficient tool that can forecast the natural inflow at average and extremeinflow events at AHD without the need to explicitly represent the internal hydrologic orclimatic parameters which is necessary for efficient decision making.

3. Methods

3.1. Fuzzy modeling

One of the major advantages in using fuzzy systems for modelling is accounting for uncer-tainties in the modelling variables. This is accomplished either by considering input variablesassociated with preferences to address modeling ambiguity or by considering input variablesin the form of interval data rather than crisp point data to address modeling fuzziness (Klirand Yuan, 1995). Ambiguity in modelling represents the difficulty in characterizing inflowparameters as it remains unclear which of several alternatives shall be used to represent inflowmeasurements. Ambiguity thus represents a modelling uncertainty that is met when a lackof certain distinctions or existing of conflicting distinctions that characterize the subject ofinterest (here river inflow) occurs. On the other hand, fuzziness represents the lack of definiteor sharp distinction. Therefore, fuzziness does not address the uncertainty of the measurebut the uncertainty in characterizing this measure. We argue that both types of uncertaintiesexist in inflow modelling. While we introduce here a model that addresses the fuzziness kindof uncertainty in inflow variables we ascertain that this modelling paradigm can include ameasure of ambiguity using fuzzy set theory or possibility theory.

We consider fuzziness in historical inflow input data variables by modelling inflow intervaldata through the use of membership functions to address our uncertainty in classifyingobservations (Laviolette et al., 1995). In other words, considering an inflow observation “Q”,the membership function μA(Q) that lies between 0 and 1 symbolizes the assessor’s viewof the extent to which Q belongs to the fuzzy set A. The fact that μA(Q) ∈ [0, 1] shall notbe observed within the context of probability theory but simply used for scaling handiness(Singpurwalla and Booker, 2004) recognizing that the sum of membership functions mightnot be 1.0. The capability of fuzzy systems thus lies in being subjective (thus includesuncertainty) as being dependent on the assessor’s view (represented mathematically by theshape and overlap of the membership functions). However, in doing so we assume that forany inflow measurement Q we shall be able to assign a membership value μAi (Q) for all1 ≤ i ≤ N where N represents the number of all fuzzy subsets that can be recognized withinthe inflow measurement space (Laviolette et al., 1995 and Singpurwall and Booker, 2004).

Moreover, fuzzy systems have been widely recognized as universal approximators forcomplex systems behaviour when systematic modelling of the system does not exist due toour ignorance of significant system variables or due to the vagueness or imprecision of thesystem inputs (Kreinovich et al., 2000). Therefore, fuzzy systems are used here to modelthe Nile river inflow due to our lacking of systemized set of functions that can correctlyforecast the Nile river flow without forcing pre-defined mathematical abstraction whichcan be misleading. Several fuzzy based models for input-output mapping have been usedsuccessfully in modelling engineering systems with different sources of uncertainties. Suchsystems include batch least squares, gradient least squares and fuzzy learning from exampleswith and without recursive learning (Passino and Yurkovich, 1998) as well as adaptive-neuro fuzzy system (ANFIS) (Jang et al., 1997). In some of these methods the membershipfunctions of the input and output variables are specified prior to modelling (e.g. batch least

Springer

Water Resour Manage

squares) while in other models the membership functions are developed using the input dataset used for training the model (e.g. recursive learning from examples and ANFIS). Anotherimportant difference between these methods includes the method by which nonlinearity ismapped by the system. This is accomplished by either changing the number of fuzzy rulesor by tuning the number and shape of membership functions or by changing both such thatthe fuzzy model achieves a pre-specified accuracy with the training dataset (Ross, 2004).

3.2. Adaptive-neuro fuzzy inference system (ANFIS)

Adaptive neuro fuzzy inference system (ANFIS) is a fuzzy mapping algorithm that is based onTagaki-Sugeno-Kang (TSK) fuzzy inference system (Jang et al., 1997; Loukas, 2001). ANFIShas been successfully used for mapping input-output relationship based on available datatuples (Gallo et al., 1999). The system acquires its adaptability by utilizing a hybrid learningmethod that combines backpropagation and least mean square optimization algorithms. Whena considerably large number of data tuples exist, learning can be achieved by tuning themembership functions using gradient descent method to determine the premise parameters,along with applying the least mean square method to modify the consequent parameters sothat the model output matches the system output with a minimum root mean square error.

In this context, ANFIS utilizes the TSK rules. Thus, ANFIS builds “R” rules to map lvariables (x1, x2, . . . , xl ) to a single output y. Such rules will include

if x1 is A1 and x2 is B1 . . . and x1 is K1 then f1 = m1x1 + n1x2 + · · · + w1x1 + q1 (1)

if x1 is A2 and x2 is B2 . . . and x1 is K2 then f2 = m2x1 + n2x2 + · · · + w2x1 + q2

(2)

if x1 is A f1 and x2 is B f2 . . . and x1 is K f 1 then fR = m R x1 + nR x2 + · · · + wR x1 + qR

(3)

where fi is the number of membership functions for each variable and R is the number ofrules that can be used to describe the relationship. A regression parameter of the ith fuzzyrule ξi can be defined as

ξi =

∏lj=1

1

1+(

x j −ckj

σkj

)2bkj

∑Ri=1

∏lj=1

1

1 +(

x j −ckj

σkj

)2bkj

for i = 1 . . . R (4)

The fuzzy operator (�) represents the T-norm (minimum-multiplication) operation (Guptaand Qi, 1991) that is activated at the second layer of the ANFIS network. The number of fuzzyrules “R” is a function of the number of input variables (l) and the number of membershipfunctions for each variable (fi ). It is assumed in Equation (4) that all the universe of discoursefor all input variables can be defined using bell-shape membership functions. The final outputis computed using the regression parameters for R rules as:

y =R∑

i=1

ξi fi . (5)

Springer

Water Resour Manage

The premise parameters bkj and ck

j (Equation (4)) define the kth bell shape membershipfunction for the jth input variable where (1 ≤ k ≤ f) and (1 ≤ j ≤ l) while the consequenceparameters are represented by the consequence vectors mi , ni . . . wi and qi where (1 ≤ i ≤R). The training process aims at tuning the premise and consequence parameters to achievethe desired learning. This is performed through a hybrid learning algorithm the consequenceparameters are tuned using least mean squared optimization in a forward path while thepremise parameters are tuned using backpropagation in a backward path. The number andshape of the membership functions for each variable shall be predefined. However, the originalspread and overlap of the membership functions is defined using fuzzy clustering algorithm(Jang et al., 1997; Ross, 2004).

3.3. ANFIS model for forecasting inflow

Comprehensive analysis of data including autocorrelation and cross-correlation sequences(C.C.S.) of the historical inflow data suggested a three-input, single-output network struc-ture for all models (El-Shafie et al., 2004). Figure 4 shows a sample cross-correlation se-quences between the inflow during the month of August and the previous three months(July–June–May). Similar cross correlation sequences for other months were also investi-gated. Such cross correlation pattern indicated the existence of strong correlation betweenthe examined variables (Brown and Hwang, 1997). It thus becomes obvious from Figure 4that there is a strong correlation between the inflows in consecutive months which suggeststhat forecasting the inflow at certain month shall be based on the monitored inflow at previousmonths.

A major constraint in the modelling process is the fact that a single month forecastingis not satisfactory for managing the water flow at AHD. Therefore, it was decided to es-tablish the forecast model network that targeted forecasting the inflow at month t, t + 1and t + 2 based on inflow measurements at month t − 1, t − 2 and t − 3. Schematicrepresentation of one of these proposed networks is shown in Figure 5. The �/N opera-tor represents the combined T-norm and normalization processes of the membership val-ues as expressed by Equation (4). The proposed network targets forecasting the inflow atmonth t denoted Qt using the measured inflow at months t − 1, t − 2 and t − 3 de-noted Qt −1, Qt −2 and Qt −3. The other two networks will have a similar architecture andthe same input variables with the only difference being the output variables as Qt +1 and Qt +2

respectively.Analysis of the proposed ANFIS models showed that such networks did not yield satis-

factory results specially in forecasting the inflow at time t + 1 and t + 2. This is because ofthe fact that such structure increases the time distance between the input and output inflowswhere the relation between the temporal patterns between the input and output inflows be-come weak. Therefore, it was decided to develop the model in sequence that can make useof the non-linear capabilities of ANFIS while performing a one-step-ahead forecasting at atime using a combination of monitored and forecasted inflow at three previous months. Thisarchitecture will provide the ANFIS model with a multi-lead forecasting capability. Archi-tecture of the proposed inflow forecasting ANFIS-based model is available in this manuscriptin Figure 6.

Using the proposed architecture, the inflow Q f (t) forecasted at month t based on the inflowmonitored Qm at the previous three months can be expressed as:

Q f (t) = f (Qm(t − 1), Qm(t − 2), Qm(t − 3)) (6)

Springer

Water Resour Manage

Fig. 4 The cross-correlationsequence for the inflow formonths of (a) August–July, (b)August–June and (c)August–May

Springer

Water Resour Manage

Fig. 5 Schematic representation of ANFIS used to forecast inflow (Qt ) at time t from measured inflow attime t−1, t−2 and t−3 respectively

Fig. 6 Schematic representation of the proposed ANFIS model for forecasting inflow of the Nile river atAHD

Springer

Water Resour Manage

Consequently, the inflow for month t+1 can be forecasted as follows:

Q f (t + 1) = f (Q f (t), Qm(t − 1), Qm(t − 2)) (7)

Similarly, the inflow for month t + 2 can be forecasted using the following equation:

Q f (t + 2) = f (Q f (t + 1), Q f (t), Qm(t − 1)) (8)

In fact, Q f (t) in all of the above equations represents forecasted inflow while Qm is a monitoredinflow. The multi-lead capability of the ANFIS model adds to the ANFIS ability to modelthe inflow and thus shall be capable of providing a high accuracy for inflow forecasting forthree months ahead.

3.4. Training the ANFIS model

The ANFIS model was trained using historical inflow data measured from 1870 to 1930, andwas tested with two historical inflow datasets measured from 1931 to 1960 and from 1961to 2000. The inflow forecasting model was formed for each month resulting into 12 ANFISmodels. Each training dataset was divided into a training dataset and a checking dataset.This method is recommended to avoid over-fitting of the system to the training dataset (Janget al., 1997). The ability of the ANFIS model to achieve the performance goal depends onthe predefined internal ANFIS parameters such as the number and shape of membershipfunctions and the step size. The type and number of the membership functions define thecontribution of each input parameter to the regression parameter ξ and thus to the desiredoutput. Whereas, the step size controls the adaptive changes that ANFIS introduces to themembership functions during the training procedure. Optimization of the internal ANFISparameters is an important process for adequate mapping. This optimization was performedvia an extensive trial and error process where the ANFIS parameters were tuned to obtainthe optimal values of internal ANFIS learning parameters that were capable of mimickingthe sequences of the inflow patterns.

In fact, one of the challenges in modeling with ANFIS is determining its optimal learn-ing parameters (number of membership function and initial value of step size) prior totraining such that optimal training is accomplished. Two approaches have been recom-mended by many researchers for determining the optimal learning parameters of learn-ing environments such as ANFIS: optimization algorithms (Hassanain et al., 2004) or bytrial-and-error (Kim et al., 2002). While finding the optimal learning parameters might beguaranteed by optimization techniques (derivative free or derivative based optimization),the optimization alternative has the drawback of being computationally expensive. On theother hand, trial-and-error approach has been proven efficient if the target root mean squareerror can be met. The trial-and-error approach has also the advantage of yielding a knowl-edge rule-base that has a lower probability of over-fitting the training dataset comparedto that of the optimization approach. We, therefore, excluded the optimization alterna-tive and determined the optimal learning parameters of ANFIS using the trial-and errorapproach.

Training of the ANFIS model was performed to minimize the error between the ANFISoutput (Q f (training)) and the desired response (Qm) defined as the root mean square error

Springer

Water Resour Manage

Fig. 7 The ANFIS performance curve for the training and checking data while training for month of August

(RMSE) (Equation (9)). The targeted RMSE for inflow training was set to 10−2. The choiceof the best ANFIS model for each month was based on achieving a minimum RMSE for boththe training and checking datasets.

RMSE =(

1

N

N∑1

(Q f (training) − Qm)2)

)0.5

(9)

where Qm and Q f (training) represent the monitored (desired) and forecasted inflows in thetraining and checking datasets respectively. In fact, RMSEs less than 10−2 were achieved forall months after different number of training epochs ranging between 100 and 200. Figure 7presents an example of the training process showing the change of the RMSE for both thetraining and checking datasets for the different epochs. It should be noted that the ANFISmodel parameters that correspond to the minimum checking data error are utilized afterwardsfor forecasting the inflow.

During the training process, ANFIS adjusts the initial value of step size to minimizethe RMSE in order to reach the performance goal. This step size is incremented (by acertain predefined ratio) if the RMSE continuously decreases while training in order tospeed up system convergence. On the other hand, if an increase in the RMSE is noticedduring the training procedure, the step size is decremented (by a certain predefined ra-tio). Figure 8 demonstrates the adjustment of the step size parameter during the train-ing process to achieve the target RMSE. Table 2 summarizes the internal ANFIS param-eters that were used in the modelling process. All the development made on this studywas implemented with the aide of Fuzzy logic toolbox in MATLAB (Beale and Demuth,2001).

Springer

Water Resour Manage

Ta

ble

2In

tern

alA

NFI

Spa

ram

eter

sas

soci

ated

with

each

mon

th

Cro

ss-c

orre

latio

nva

lues

Num

ber

ofM

Fs

Mon

thM

onth

Mon

thM

onth

Mon

thM

onth

Step

step

decr

emen

tM

onth

(t−

1)(t

−2)

(t−

3)(t

−1)

(t−

2)(t

−3)

Step

size

incr

emen

trat

io(%

)ra

tio(%

)

Aug

ust

0.64

0.47

0.24

22

50.

6512

080

Sept

embe

r0.

690.

530.

153

34

0.3

150

75O

ctob

er0.

830.

490.

122

45

0.5

120

80N

ovem

ber

0.89

0.67

0.38

23

60.

2515

060

Dec

embe

r0.

870.

640.

62

44

0.20

120

80Ja

nuar

y0.

430.

350.

313

33

0.30

110

40Fe

brua

ry0.

930.

770.

292

24

0.45

180

20M

arch

0.92

0.84

0.76

23

30.

7512

080

Apr

il0.

870.

790.

683

45

0.65

110

10M

ay0.

800.

720.

583

34

0.4

150

30Ju

ne0.

550.

220.

122

45

0.35

120

80Ju

ly0.

640.

340.

172

35

0.25

150

75

Springer

Water Resour Manage

Fig. 8 The adjustment of the step size during ANFIS training for month of August

Fig. 9 Architecture of the MLP-NN developed for forecasting inflow of the Nile River at AHD for August

3.5. Testing the ANFIS model

The ANFIS model was tested using two historical inflow datasets monitored during thetime period from 1931 to 1960 and from 1961 to 2000. To examine the relative accuracyof forecasting of the ANFIS model it was decided to compare the ANFIS model accuracyand forecast with the recently developed MLP-NN by El-Shafie et al. (2004). The MLP-NNmodel architecture is shown in Figure 9. It is the authors’ knowledge that no other model forforecasting the Nile river inflow at AHD with similar forecasting accuracy to the MLP-NNexists. Details about MLP-NN can be found elsewhere (El-Shafie et al., 2004).

Springer

Water Resour Manage

Three statistical measures were used to examine the goodness to fit of the ANFIS model tothe testing data. These measures include the forecasting error (FE) and the maximum relativeerror (REmax%) to examine the relative accuracy of both models for the extreme inflow eventsas represented by Equations (10) and (11).

FE =(

1

N

N∑1

(Q f (testing) − Qm)2

)0.5

(10)

REmax(%) = max

( |Q f (testing) − Qm |Qm

)% (11)

Another important performance evaluation measure in forecasting is the correlation coeffi-cient; as follow:

ρ =∑N

i=1 (Q f − Q f )(Qm − Qm)√∑Ni=1 (Q f − Q f )2

∑Ni=1 (Qm − Qm)2

(12)

Correlation coefficient measures how well the inflows predicted correlate with the inflowsobserved. Clearly, correlation coefficient value closer to unity means better forecasting. InEquation (12), Q f and Qm are the mean of forecasted and observed inflows, respectivelywhile N is the number of datasets (years).

4. Results and Discussions

The proposed inflow forecasting ANFIS-based model is developed according to the procedurediscussed in the previous section. Table 3 summarizes the forecasting error evaluated forboth the ANFIS model (suggested in this study) and the recently developed MLP-NN modelfrom 1931 to 1960. Obviously, the inflow forecasting accuracy is significantly improvedwhen using the ANFIS model compared with the MLP-NN model. It can be depicted thatthe accuracy improvement over the 12 months ranged from 3% to 83% with the majoraccuracy enhancement occurring in the months of January, May and June (over 80% accuracy

Table 3 Forecasting error (FE) in BCM associated with ANFIS and MLP-NN (El-Shafieet al., 2004) models for each month during the period between 1931 and 1960

Month ANFIS MLP-NN (El-Shafie et al., 2004) Percentage improvement (%)

August 0.25 0.40 37September 0.28 0.56 50October 0.34 0.66 49November 0.13 0.13 3December 0.10 0.27 61January 0.05 0.22 78February 0.07 0.22 70March 0.04 0.09 57April 0.07 0.14 47May 0.04 0.17 77June 0.05 0.27 83July 0.16 0.39 58

Springer

Water Resour Manage

Fig. 10 The performance of both ANFIS and MLP-NN models for month of November for the period 1931–1960 (a) Natural inflow at AHD showing the low and peak inflow events (b) The percentage of forecastingerror distribution

enhancement). The small improvement of 3% at the month of November is due to the factthat the Nile river inflow is usually experiencing a significant transition from high inflow(during the months from August to October) to low inflow (starting the month of December)at this month as can be depicted from Table 1.

Moreover, It was also observed from Table 2 that the number of membership functionsassociated with each input Q(t−1), Q(t−2) and Q(t−3) is inversely related to the cross-correlation coefficient between the forecasted inflow Q(t) and each monthly monitored inputinflow. Table 2 shows the cross-correlation values between model output (Q(t)) and eachof all corresponding model input (Q(t−1), Q(t−2) and Q(t−3)). It can be noticed that thehigher the cross-correlation coefficient value, the lower the number of membership functionsneeded to represent the input parameter so that the performance goal is achieved duringtraining. This observation can be attributed to the fact that high cross-correlation coefficientswould usually indicate small changes in the dynamic patterns between the both input andforecasted inflow within the considered time period. It is evident that one or two membershipfunctions can adequately map the inflow parameter when significantly low level of dynamicsoccurs (insignificant fluctuation in the monitored inflow). On the other hand, when highlevel dynamics (significant fluctuation in the monitored inflow) takes place, there is a high

Springer

Water Resour Manage

Fig. 11 The performance of both ANFIS and MLP-NN models for month of May for the period 1931–1960(a) Natural inflow at AHD showing the low and peak inflow events (b) The percentage of forecasting errordistribution

likelihood of having a low cross-correlation coefficient between monitored and forecastedinflows. High number of membership functions would typically be needed to describe inflowwith high level dynamics.

It is of particular importance for the reservoir operation to provide accurate forecastingfor the peak and low inflows. Figures 10 to 13 presents the natural inflow between 1931and 1960 as well as the percentage inflow forecasting error provided by both ANFIS andMLP-NN (El-Shafie et al., 2004) model for the months of November, May, March andSeptember, respectively. The choice of these specific four months (November, May, Marchand September) was done to consider two distinct patterns of inflow events at AHD. The firsttwo months (November and May) represent the inflow pattern at transition period from highto low and low to high inflows respectively. The second two months (March and September)represent the seasonal low and high inflows respectively. The inflow values between 1931 and1960 as depicted from Figures 10 to 13 enlighten the fact that such inflow patterns containingextreme inflow events.

Considering the inflow forecasting error from Figure 10b along with the natural inflowgiven in Figure 10a, it can be determined that although similar accuracy levels can be provided

Springer

Water Resour Manage

Fig. 12 The performance of both ANFIS and MLP-NN models for month of March for the period 1931–1960(a) Natural inflow at AHD showing the low and peak inflow events (b) The percentage of forecasting errordistribution

by both ANFIS and MLP-NN models between the year 1931 and 1950, the accuracy issignificantly improved by the ANFIS during the period from 1951 to 1960 where peakinflow events were experienced. This demonstrates the capabilities of the ANFIS model indetecting the extreme inflow events. Similar performances can be depicted for the month ofMay in Figure 11. Between the year 1940 and 1950, extreme inflow events were forecastedby the ANFIS model with forecasting errors less than 2%, while the forecasting error ofthe MLP-NN model reached 12% for the same time period. Moreover, the results for themonths of March (Figure 12) and September (Figure 13) show the reliability of the ANFISmodel over the MLP-NN model to accurately forecast not only the extreme events but alsothe average inflow events.

The ability of the ANFIS model to accurately forecast extreme inflow events might beattributed to the method by which the ANFIS model recognizes the input variables. Theuse of fuzzy membership functions to represent the different classes of inflow data allowsconsidering the input data as a range rather than as a crisp input. This method provides theextreme inflow events with specific membership values that permit recognizing occurrenceof extreme events. This recognition does not take place in MLP-NN where the weights of

Springer

Water Resour Manage

Table 4 FE, REmax% and correlation coefficient ρ associated with ANFIS and MLP-NN (El-Shafieet al., 2004) during the period between 1961 and 2000

MLP-NN PercentageANFIS (El-Shafie et al., 2004) improprement

Month FE REmax(%) ρ FE REmax(%) ρ FE (%) REmax(%)

August 0.26 3.9 0.978 0.45 4.6 0.968 43 15September 0.19 2.6 0.986 0.63 3.9 0.966 70 32October 0.21 2.5 0.972 0.77 4.9 0.943 73 49November 0.13 4.1 0.975 0.18 5.0 0.936 20 18December 0.11 2.2 0.991 0.31 2.8 0.943 64 22January 0.05 4.3 0.984 0.26 8.5 0.926 80 50February 0.06 4.2 0.985 0.26 8.0 0.918 74 47March 0.04 5.3 0.986 0.10 9.9 0.910 52 46April 0.07 6.7 0.964 0.15 12.4 0.881 52 46May 0.04 4.7 0.955 0.20 8.3 0.864 80 43June 0.04 4.7 0.993 0.29 8.5 0.889 84 45July 0.18 6.5 0.882 0.44 11.6 0. 832 64 44

the network are not determined in accordance to the degree by which the input parameterbelongs to a fuzzy set but are mainly governed by the backpropagation algorithm that targetsreducing the RMSE.

To further assess and validate the ANFIS model, the model was tested using historicalinflow data between the years 1961 and 2000. The significance of this testing period isattributed to the fact that this time period is historically distant from the training time periodof the ANFIS model (Training period between 1870 and 1930) and that there are noticeabledynamics in the inflow pattern during this period which experienced different cycles ofhigh flood and drought normal than that normally observed at AHD. Table 4 compares theperformances of the ANFIS to the MLP-NN models over the period between 1961 and 2000using the three statistical forecasting measures as discussed in Section (3.5). It is obviousthat the ANFIS model performed more adequately than the MLP-NN model with significantimprovement in the FE ranging from 20% for month of November to 84% for the month ofJune. Moreover, ANFIS model provides relative improvement in the REmax% ranging from15% for the month of August to 50% for the month of January. Table 4 also demonstratesthat the ANFIS model provides higher correlation coefficient between the forecasted andmonitored inflows compared to MLP-NN. Figure 14 demonstrates the natural inflow patternof the month of January over the period from 1961 to 2000. It can be observed from thisfigure that events of significant extreme inflows existed between 1985 and 2000 (low in1987 and high in 1995). Figure 15 shows the error distribution for the month of Januaryutilizing the ANFIS and MLP-NN models with one-step-ahead inflow forecasting (Figure15a), two-step-ahead inflow forecasting (Figure 15b) and three-step-ahead inflow forecasting(Figure 15c).

It can also be observed that the ANFIS model outperformed the MLP-NN model and wasable to provide significant improvement in forecasting accuracy over the whole period. Thisis represented by reducing the forecasting error for one-step-ahead inflow forecasting forlow and high extreme inflow events at the years 1987 and 1995 by 83% and 57%, respec-tively. Results from Figures 15b and 13c show that the ANFIS model provides remarkableimprovement over the MLP-NN model for forecasting the inflow for two or three months

Springer

Water Resour Manage

Fig. 13 The performance of both ANFIS and MLP-NN models for month of September for the period 1931–1960 (a) Natural inflow at AHD showing the low and peak inflow events (b) The percentage of forecastingerror distribution

ahead. These results show that the ANFIS model was not only capable of improving theaccuracy of one-step-ahead inflow forecasting but the model also was capable of capturingthe temporal patterns of the inflow which allowed it to provide significant enhancement inmulti-step ahead inflow forecasting. In addition, if Figures 15a, 15b and 15c are carefullyexamined, it can be noticed that the ANFIS model was capable of achieving consistent accu-racies especially at the extreme inflow events (at 1985 and 1997) while the accuracy of theMLP-NN model was considerably reduced at the same events. This result is of major sig-nificance for managing very large reservoirs such as Lake Nasser where accurate multi-leadforecasting of natural inflow at AHD can provide efficient planning.

While the ANFIS model was computationally inexpensive compared to MLP-NN, theonly drawback of ANFIS might be its inability to model more than one output. This mightbe attributed to the defuzzification method used for aggregating the knowledge rule-base toinfer the output (Equations 4 and 5) (Hellendoorn and Thomas, 1993). While this did notaffect the modeling problem discussed here, it can represent a limitation for using ANFISin water resources management. Recent research showed that such modeling problem can

Springer

Water Resour Manage

Table 5 Forecasting error (FE) for one, two and three months ahead in BCM associated with ANFIS andMLP-NN (El-Shafie et al., 2004) for the period (1961–2000)

ANFIS MLP-NN

Month Month (t) Month (t + 1) Month (t + 2) Month (t) Month (t + 1) Month (t + 2)

August 0.26 0.25 0.40 0.45 0.70 0.90September 0.24 0.31 0.22 0.64 0.83 0.86October 0.29 0.18 0.16 0.78 0.14 0.37November 0.13 0.15 0.08 0.14 0.34 0.31December 0.11 0.06 0.07 0.31 0.28 0.30January 0.05 0.07 0.06 0.26 0.27 0.12February 0.07 0.06 0.11 0.26 0.11 0.18March 0.04 0.09 0.11 0.10 0.17 0.22April 0.07 0.06 0.05 0.15 0.21 0.35May 0.04 0.05 0.19 0.20 0.32 0.50June 0.05 0.18 0.28 0.29 0.46 0.62July 0.16 0. 26 0.27 0.44 0.52 0.81

Fig. 14 The natural inflow at AHD for month January showing low and peak inflow events for the period1961–2000

be handled by using parallel ANFIS architectures known as coactive neuro-fuzzy models(CANFIS) (Jang et al., 1997; Craven et al., 1999).

To substantiate the functionality of the ANFIS model, Table 5 shows the FE value ofthe inflow error over these 40 years utilizing ANFIS and MLP-NN models for each monthbetween 1961 and 2000. The second column shows the FE forecasting error when only themonitored inflows from the previous months are utilized at the input of the model. The thirdcolumn corresponds to the case when one forecasted inflow and two monitored inflows areused at the network inputs while the fourth column corresponds to the case of two forecastedinflows and one monitored inflow are utilized. Similarly and with the same sequence, thefifth, sixth and seventh columns are for the MLP-NN model. Apparently, Table 5 shows that

Springer

Water Resour Manage

Fig. 15 The forecasting error distribution for month January utilizing ANFIS and MLP-NN models for theperiod 1961–2000 (a) one-month-ahead (b) two-month-ahead (c) three-month-ahead

the multi-lead inflow forecasting using the ANFIS model outperforms that of the MLP-NNmodel.

5. Conclusions

A model for inflow forecasting of the Nile River at AHD using adaptive neuro fuzzy infer-ence system (ANFIS) is developed. The ANFIS model provides accurate multi-lead inflowforecasting. The ANFIS model accuracy outperformed a recently developed MLP-NN toforecast inflow at AHD (El-Shafie et al., 2004). While the ANFIS model showed high andconsistent accuracy in forecasting average inflow events, it showed significantly higher accu-racy in forecasting extreme inflow events compared to MLP-NN model. The ANFIS modelalso showed robustness and reliable performance in forecasting inflow data with differentinflow patterns than that used in training the model.

Acknowledgements This research was based on the Nile river inflow database made available by the NationalWater Research Center, Cairo, Egypt. The authors greatefully acknowledge this collaboration. This work was

Springer

Water Resour Manage

developed while the first author was funded by The Canadian International Development Agency (CIDA) andby a research grant from the Natural Science and Engineering Research Council (NSERC) of Canada to thethird author. The authors gratefully acknowledge these supports. The financial support by the Defense ThreatReduction Agency (DTRA)-University of New Mexico Strategic Partnership to the second author is greatlyacknowledged.

References

Abu Zeid M, Abdel Daym S (1990) The Nile, the Aswan high dam and the 1979–1988 drought, Proceedingsof International Commission on Irrigation and Drainage, Volume 1-c, Q. 43, 14th Congress on Irrigationand Drainage, Rio De Janeiro, Brazil

Ballini R, Soares S, Andrade MG (2001) Multi-step-ahead monthly streamflow forecasting by a neurofuzzynetwork model. Proceedings of Annual Conference of the North American Fuzzy Information ProcessingSociety 2:992–997

Beale H, Demuth HB (2001) Fuzzy Systems Toolbox for Use with MATLAB. 1st ed., International ThomsonPublishing, MA, USA

Box GH, Jenkins G (1970) Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, USABrown RG, Hwang PYC (1997) Introduction to Random Signals and Applied Kalman Filtering. John Wiley

and Sons, New York, USAChang FJ, Chen YC (2001) A counter propagation fuzzy-neural network modeling approach to real time

streamflow prediction. J Hydrol 245:153–164Chang ATC, Tsang L (1992) A neural network approach to inversion of snow water equivalent from passive

microwave measurements. Nordic Hydrol 23:173–182Coulibaly P, Anctil F, Bobee B (1998) Real time neural network-based forecasting system for hydropower

reservoirs. Proceedings of First International Conference On New Information Technologies For DecisionMaking In Civil Engineering, Montreal (October 11–13), Vol. II:1001–1011

Coulibaly P, Anctil F (1999) Real-time short-term natural water inflows forecasting using recurrent neuralnetworks. International Joint Conference On Neural Networks, Washington Dc, Proceedings Ijcnn’996:3802–3805

Coulibaly P, Anctil F, Bobee B (2000) Daily reservoir inflow forecasting using artificial neural networks withstopped Training Approach. J Hydrol 230:244–257

Coulibaly P, Anctil F, Bobee B (2001a) Multivariate reservoir inflow forecasting using temporal neural net-works. ASCE J Hydrol Eng 6(5):367–376

Coulibaly P, Bobee B, Anctil F (2001b) Improving extreme hydrologic events forecasting using a new criterionfor ANN selection. Hydrol Process 15(8):1533–1536

Craven PJ, Sutton R, Burns RS (1999) Multivariable Neurofuzzy Control of an Autonomous UnderwaterVehicle. Integr. Compt-Aid Eng 6(4):275–288

El-Shafie A, Noureldin A, Abdin A (2004) Neural network model for Nile river inflow forecasting basedon correlation analysis of historical inflow data, Technical Report # 223(2004, National Water ResearchCenter, Egypt

Fahmy H (2001) Modification and re-calibration of the simulation model of Lake Nasser. Water Int. – off. JInt Water Res Assoc (IWRA) 26(1):129–135

Fontane DG, Gates TK, Moncada E (1997) Planning reservoir operations with imprecise objectives. J WaterRes Plng Mgmt 123(3):154–162

French MN, Krajewski WF, Cuykendall RR (1992) Rainfall forecasting in space and time using a neuralnetwork. J Hydrol 137:1–31

Gallo S, Murino T, Santilllo LC (1999) Time manufacturing prediction: preprocess model in neuro fuzzyexpert system. CD Proceedings of European Symposium on Intelligent Techniques, Greece, 11 p

Gautam DK (1999) Classification of wind events using Kohonen networks. Proceedings of Joint workshop onNeural networks in Civil Engineering, Delft, The Netherlands

Gupta MM, Qi J (1991) Theory of T-norms and fuzzy inference methods. Fuzzy Sets Syst 40(3):431–450Grino R (1992) Neural networks for univariate time series forecasting and their application to water demand

prediction. Neural Netw World 5:437–450Hassanain MA, Reda Taha MM, Noureldin A, El-Sheimy N (2004) Automization of INS(GPS Integration

System Using Genetic Optimization. Proceedings of the 5th International Symposium on Soft Computingfor Industry, Seville, Spain, 6p

Hellendoorn H, Thomas C (1993) Defuzzification in fuzzy controllers. J Intell Fuzzy Syst 1:109–123Hsu K, Gupta HV, Sorooshian S (1995) Artificial neural network modeling of the rainfall-runoff process.

Water Resour Res 31(10):2517–2530

Springer

Water Resour Manage

Huang WC (1996) Decision support system for reservoir operation. Water Resour Bull 32(6):1221–1232Jang JSR, Sun CT, Mizutani E (1997) Neuro-Fuzzy and Soft Computing, A Computational Approach to

Learning and Machine Intelligence. Prentice Hall, NJ, USAKang KW, Park CY, Kim JH (1993) Neural network and its application to rainfall-runoff forecasting. Korean

J Hydrosci 4:1–9Karunithi N, Grenney WJ, Whitley D, Bovee K (1994) Neural networks for river flow prediction. J Comput

In Civil Eng 8(2):201–220Kim B, Park JH, Kim B-S (2002) Fuzzy logic model of Langmuir probe discharge data. Comput Chem

26(6):573–581Klir GJ, Yuan B (1995) Fuzzy Sets and Fuzzy Logic, Theory and Applications. Prentice Hall, NJ, USAKreinovich V, Nguyen HT, Yam, Y (2000) Fuzzy systems are universal approximators for a smooth function

and its derivatives. Int. J. Intell. Syst 15(6):565–574Laviolette M, Seaman JW, Barrett JD, Woodall WH (1995) A probabilistic and statistical view of fuzzy

methods. Technometrics 37:249–261Loukas YL (2001) Adaptive neuro-fuzzy inference system: an instant and architecture-free predictor for

improved QSAR studies. J Med Chem 44(17):2772–2783National Water Research Center, Cairo, Egypt, 2005Olason T, Huysentruyt J, Hurdowar-Castro D, Kirshen P (1997) Inflow forecasting for short term hydro

scheduling. Proceedings of International Conference on Hydropower, Atlanta, GA, USAPassino K, Yurkovich S (1998) Fuzzy Control. Addison Wesley Longman, Menlo Park, CA, USARoss TJ (2004) Fuzzy Logic with Engineering Applications. 2nd ed., Chichester, John Wiley & Sons, UKRussell SO, Campbell PF (1996) Reservoir operating rules with fuzzy programming. J. Water Resour. Plng.

Mgmt 122(3):165–170Saad M, Bigras P, Turgeon A, Duquette R (1996) Fuzzy learning decomposition for the scheduling of hydro-

electric power systems. Water Resour Res 32(1):179–186Sadek MF, Shahin MM, Stigter CJ (1997) Evaporation from the Reservoir of the High Aswan Dam, Egypt: A

New Comparison of Relevant Methods with Limited Data. J Theor Appl Climatol l56:57–66Salas JD, Delleur JW, Yevjevich V, Lane WL (1988) Applied Modeling Hydrological Time Series. Water

Resources Publications, P.O. Box 2841, Littleton, Colorado, 80161, USASalem MH, Dorrah HT (1982) Stochastic Generation and Forecasting Models for the River Nile. Proceedings

of International Workshop on Water Resources Planning, Alexandria, EgyptSeker H, Odetayo MO, Petrovic D, Naguib RNG (2003) A fuzzy logic based method for prognostic decision

making in breast and prostate cancers. IEEE Trans Inf Technol Biomed 7(2):114–122Seno E, Murakami K, Izumida M, Matsumoto S (2003) Inflow forecasting of the dam by the neural network.

Proceedings of Artificial Neural Networks in Engineering Conference, St. Louis, USAShrestra BP, Duckstein L, Stakhiv EZ (1996) Fuzzy rule-based modeling of reservoir operation, operation. J

Water Resour Plan Mgmt 122(4):262–269Singpurwalla ND, Booker JM (2004) Membership functions and probability measures of fuzzy sets. J Am

Stat Assoc 99(467):867–877Subramanian C, Manry MT, Naccarino J (1999) Reservoir inflow forecasting using neural network, Proceed-

ings of American Power Conference, Chicago, Illinois, USATsai CH, Hsu DS (2002) Diagnosis of Reinforced Concrete Structural Damage Based on Displacement Time

History Using the Back-propagation Neural Network Technique. ASCE J Comput Civil Eng 16(1):49–58Wood EF (1980) Real Time Forecasting Control of Water Resource Systems. Workshop Report, Pergamon

Press, New York, USAYu PS, Chen CJ (2000) Application of gray and fuzzy methods for rainfall forecasting. J Hydrol Eng 4(5):339–

345

Springer