standard equations for predicting the ...scientiairanica.sharif.edu/article_4198_b1660aa8ec...1...

Scientia Iranica A (2018) 25(3), 1057{1069

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Standard equations for predicting the dischargecoe�cient of a modi�ed high-performance side weir

A.H. Zajia;�, H. Bonakdaria, and Sh. Shamshirbandb

a. Department of Civil Engineering, Razi University, Kermanshah, Iran.b. Department of Computer System, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala

Lumpur, Malaysia.

Received 4 March 2016; received in revised form 15 December 2016; accepted 31 July 2017

KEYWORDSDischarge coe�cient;Modi�ed triangularside weir;Nonlinear regression;Particle swarmoptimization;Standard equation.

Abstract. Side weirs are hydraulic structures that are used as discharge adjustments todivert the surplus water owing from the main channel. Predicting the discharge coe�cientis one of the most important parameters in the side weir design process. In practicalsituations, it is preferred to predict the discharge coe�cient with simple equations. The goalof this study is to develop accurate standard equations for use in predicting the dischargecoe�cient of a high-performance, modi�ed triangular side weir. The Particle SwarmOptimization (PSO) algorithm was used to optimize the parameters of the equations.Four di�erent forms of the equations and two non-dimensional input combinations wereused to develop the most appropriate model. The results obtained by our simple standardequations optimized by the PSO algorithm were compared with those of complex nonlinearregression equations, and our equations were more accurate in modeling the dischargecoe�cient. Our method reduced the error in the results by as much as 43% compared tothe regression methods, and its simplicity makes it useful in solving practical problems.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

In some rivers, water and wastewater, irrigation, anddrainage channel zones, the ow rate may exceed thetolerance capacity of a river or channel. In such acase, discharge control structures, such as side weirs,can be used to control the ow and prevent over ows.Side weirs are also used to keep oods away fromdam reservoirs and the diversion of ow for the sakeof protection. The study of side weirs dates back tothe early 20th century when De Marchi [1] laid thegroundwork for other studies with his mathematical

*. Corresponding author.E-mail addresses: [email protected] (A.H. Zaji);[email protected] (H. Bonakdari);[email protected] (Sh. Shamshirband)

doi: 10.24200/sci.2017.4198

model of side weirs. The equations he developed todescribe side weirs were based on the assumption thatthe speci�c energy was the same before and after theweirs.

In this paper, a model with a decreasing dischargeequation is presented for a rectangular channel witha horizontal bed and a spatially-varied ow. Theequation is:

dydx

=Q

gb2y3

��dQdx

�1� Q2

gb2y2

=Qy��dQ

dx

�gb2y3 �Q2 ; (1)

where dy=dx is the depth change along the channel, Q isthe discharge of the ow, y is the depth of the ow, b isthe width of the main channel, dQ=dx is the dischargechange along the channel, and g is the acceleration ofgravity. The discharge per unit length over the sideweir equals:

1058 A.H. Zaji et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 1057{1069

�dQdx

=23CM

p2g(y � w)1=5; (2)

in which CM is the discharge coe�cient, also knownas the De Marchi coe�cient. By assuming that thespeci�c energy is constant, the discharge in the channelwould be:Q = by

p2g(E � y): (3)

By inserting Eqs. (2) and (3) into Eq. (1), we obtain:

dydx

=43CMb

p(E � y)(y � w)

3y � 2E: (4)

Integrating the above equations and assuming CMindependent of x direction, we will have Eq. (5):

x =3b

2CM�(y;E;w) + c; (5)

where � is obtained from Eq. (6). Therefore, CMobtained in the laboratory is calculated by substituting� through Eq. (5) as follows:

�(y;E;w)=2E�3wE�w

sE�yy�w�3 sin�1

rE�yE�w: (6)

Moreover, the length of the side weir is obtained fromEq. (7):

L = x2 � x1 =32

bCM

(�2 � �1); (7)

in which �1 and �2 are obtained from Eq. (6) forupstream and downstream of the weirs, respectively.Assuming that Q1 and Q2 are the upstream and down-stream discharges, the amount of side weir discharge isobtained from Eq. (3).

Following De Marchi [1], many researchers havetried to calculate the side weir coe�cients (CM ), andmost of them have considered CM as a function of theupstream Froude number (Fr1) [2-6].

A signi�cant number of researches have beenconducted to increase the e�ciency of the side weirby changing its geometrical situation, i.e. changing itsrectangular crest to triangular, circular, or elliptical,causing an increase in both the length of the weir andthe discharge coe�cient. Borghei and Parvaneh [7]designed a modi�ed triangular side weir whose dis-charge coe�cient that was 2.33 times higher thana conventional side weir's in a rectangular channel.The authors used 200 test measurements and, �nally,using the nonlinear regression method, determined anequation for the modi�ed triangular side weir dischargecoe�cient, i.e. Eq. (8):

CM =��0:18

�Fr1

sin �0�0:71

� 0:15(Fr1)0:44 +�wY1

�0:7��2:37 + 2:58

�w sin �0Y1

��0:36�:

(8)

Soft computing methods are used as a new tool forcalculating the side weir coe�cient, and many re-searchers have considered their application in predict-ing scour depth [8-13], determining ow characteristicsin di�erent open channels [14,15], modeling rainfalland predicting stream ow [16-22], modeling coastalalgal blooms [23], evapotranspiration [24,25], combinedopen channel ow [26,27], sediment transport [28,29],predicting ground water levels [30], and forecasting thedemand for water [31-33]. In estimating the side weircoe�cient, Bilhan et al. [34] studied di�erent Arti�cialNeural Network (ANN) methods, and found that theyhave much higher accuracy than multiple nonlinearand linear regression models do. Using Radial BasisNeural Network (RBNN), Regression Neural Network(RNN), and Genetic Expression Programming (GEP),Kisi et al. [35] predicted the triangular side weirdischarge coe�cient. The results were compared with2500 laboratory measurements, and it was found thatANN and GEP gave better results than regressionmethods. To determine side weir discharge coe�cients,Emin Emiroglu et al. [36] used an adaptive neuro-fuzzytechnique (ANFIS), the results of which indicated thehigh accuracy of the ANFIS method for modeling theside weir discharge coe�cient. Some other studies havebeen published on modeling the triangular labyrinthside weir discharge using soft computing, all of whichwere proved better than the regression methods [37-42].

The objective of this study is to provide a prac-tical, simple equation for calculating the dischargecoe�cient for a modi�ed, high-performance triangu-lar side weir. The Particle Swarm Optimization(PSO) algorithm was used to optimize each of theequations considered. Four di�erent equation formswere developed using the input parameters of Fr1(upstream Froude number), w=L (weir height/weirlength), Fr1=sine(�0) (upstream Froude number/sine(weir included angle/2)), w=Y1 (weir height/upstream ow depth), and w � sine(�0)=Y1 (weir height �sine(weir included angle/2)/upstream ow depth). Con-sidering dimension L (length) for w, L, and Y1, it isobvious that all of the used input parameters are non-dimensional. Finally, the accuracy of each equationwas examined and compared.

2. Experimental setup

In this study, the laboratory model of Borghei andParvaneh [7] was used, including a rectangular umethat was 11 m long and 0.4 m wide. The glass wallsof the channel were 0.66 m high. The outlet dischargesfrom the main and tributary channels (Q2 and Qw)were measured by standard V notches. The accura-cies of the water head and discharge measurementswere �1 mm and �0:0001 m3/s, respectively. Theauthors performed 200 tests for di�erent geometrical

A.H. Zaji et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 1057{1069 1059

Table 1. Interval variations in experimental tests of Borghei and Parvaneh [7].

� (�) L (m) w (mm) w=Y1 Q1 (m3/s) Fr1 Number of runs

30 0.3 50, 75, 100, 150 0.46-0.83 0.019-0.030 0.19-0.96 400.4 50, 75, 100, 150

450.3 50, 75, 100, 150

0.46-0.83 0.019-0.030 0.19-0.96 550.4 50, 75, 100, 1500.6 50, 100, 150

600.3 50, 75, 100, 150

0.46-0.83 0.019-0.030 0.19-0.96 500.4 50, 100, 1500.6 50, 100, 150

700.3 50, 75, 100, 150

0.46-0.83 0.019-0.030 0.19-0.96 550.4 50, 75, 100, 1500.6 50, 100, 150

Figure 1. Plan view of modi�ed triangular side weir and the parameters used.

conditions, opening lengths (L), weir heights (w), weiroblique angles (�0), and input discharges (Q1). All ofthe tests were conducted under certain conditions inwhich the upstream Froude number (Fr1) was at theinterval of 0.19-0.96. Table 1 provides the parameterswhose variation intervals are observed in the laboratorytests. Figure 1 shows the schema of the modi�edtriangular side weir and the parameters used.

3. Methods and materials

In this study, the Particle Swarm Optimization (PSO)method was used to determine an accurate equationto predict the discharge coe�cient. The design of thismethod was inspired by ocks of birds or schools of�sh, because when they stay together, the likelihood ofbeing caught reduces. This is called swarm intelligence.When �sh make up a school, they follow only a fewsimple, yet systematic, rules; however, their behaviorappears very complicated. In addition to the PSOalgorithm, Ant Colony Optimization (ACO), BeesAlgorithm (BA), and Arti�cial Fish Swarm (AFS) areall considered swarm intelligence methods.

3.1. Particle Swarm Optimization (PSO)Kennedy and Eberhart [43] introduced the PSO.Initially, the authors intended to create a sort ofsocial-based computational intelligence, independentof individual capabilities, i.e. individuals with normalintelligence quotients could make up a very intelligentcommunity. Therefore, their study resulted in a reliablealgorithm for optimization called Particle Swarm Op-timization (PSO), in which there are some particles inspace, and each particle has a potential solution to theproblem. Each particle follows a general and �xed ruleto move, called self-organization. Based on this rule,each particle remembers the present and past situationsas well as the best-experienced situation through boththe single particle and all of the particles. The newposition of the particle is calculated from Eq. (9) asfollows:

vi[t+ 1] =wvi[t] + c1r1(xpbest[t]� xi[t])+ c2r2(xgbest[t]� xi[t]): (9)

In this equation, the �rst term on the right side of theequation represents a particle's movement toward its


last movement; the second term indicates the velocityof a particle moving toward the best-known formerposition, and the third term shows the velocity ofa particle moving toward the best previous positionexperienced by all particles. Herein, xpbest is thebest personal memory, and xgbest is the best collectivememory. The inertia coe�cient (w) is an arbitraryfactor that has values less than 1; r1 and r2 are randomnumbers with a uniform distribution, while c1 and c2are the individual and collective learning coe�cients,respectively, ranging between 0-2 [43]. The three termsare the resultant vector of the amount and direction ofand movement towards a new position. Thus, fromEq. (10), we will get a new position of the particle asfollows:

xi[t+ 1] = xi[t] + vi[t+ 1]: (10)

Figure 2 demonstrates the steps of the PSO algorithm.Initially, the primary population is formed; then,the �tness function of each particle is determined.Subsequently, pbest and gbest are determined and,then, memorized in the particle's memory. Eq. (9)is used to determine the particle's velocity, namelythe direction and amount of the movement of theparticle; subsequently, the velocity is added to the

Figure 2. Particle swarm optimization ow chart.

present particle's position to determine the particle'snew position. Finally, the stopping criteria of thealgorithm are checked. If the criteria have beenmet, the algorithm stops the process; otherwise, theloop is repeated. In all models here, the Number ofFunction Evaluation (NFE) is considered as a criterionfor stopping the iterations at NFE = 100,000.

If the inertia parameter (w) reaches its maximum(w = 1), the algorithm's exploration property goesup; however, if w is low, the algorithm's exploitationproperty increases. It is generally expected that theexploration property be high at the beginning, whileit increase at the end; therefore, in the algorithm usedhere, the damping coe�cient (wdamp = 0:98) was used[44,45]. At the end of each repetition, the inertiaparameter (w) decreases gradually, and as the resultsget closer to the end of the solution, w decreases,leading to an increase in the exploitation of the model(Eq. (11)):

w = w � wdamp: (11)

3.2. Objective function and model performanceIn this study which uses the PSO algorithm, someequations are given for modeling the discharge coef-�cient of a modi�ed triangular side weir. The primarygoal of PSO is to reduce the error that results fromthe proposed equation application, compared to thelaboratory model. Therefore, according to Eq. (12),we have the following:

ei = fi � f̂i; (12)

where fi and f̂i are the results of the laboratorymodel and the proposed equation, respectively. Thealgorithm uses Mean Square Error (MSE) to determinethe iteration error, as shown in Eq. (13). In addition,in order to investigate the models' performance, RootMean Squared Error (RMSE), Mean Average Error(MAE), and Mean Absolute Percentage Error (MAPE)are used (Eqs. (14)-(16)):

MSE =1n

nXi=1

e2i ; (13)

RMSE =

vuut 1n

nXi=1

e2i ; (14)

MAE =1n

nXi=1

jeij; (15)

MAPE =Pni=1 jeijPni=1 fi

� 100n; (16)

where n is the quantity of the input dataset.


4. Results

To estimate CM , four non-dimensional parameters(Var1 to Var4) are considered as inputs in each ofthe following equations. These parameters are acombination of the upstream Froude number (Fr1), sideweir oblique angle (�0), weir height (w), and upstreamdepth (Y1). As Eqs. (17) and (18) show, there aretwo possible input sets, i.e. INPUT1 and INPUT2,that can be considered for predicting the dischargecoe�cient (CM ). The di�erence between the inputsis that INPUT2 considers the four parameters (Y1, w,�0, Fr1) as well as the length e�ect of the side weir(L):

INPUT1:

CM = '�

Var1 = Fr1; Var2 =Fr1

sin �0 ;

Var3 =wY1; Var4 =

w sin �0Y1

�; (17)

INPUT2:

CM = '�

Var1 =wL; Var2 =

Fr1

sin �0 ;

Var3 =wY1; Var4 =

w sin �0Y1

�: (18)

In order to study the e�ects of the considered inputvariables in Eqs. (17) and (18) on CM , the sensitivityanalysis of partial di�erential equations is used [46,47].In this method, the ratio of the change in CM (as the

output) to the change in the input variables consideredin Eqs. (17) and (18) is examined. For the practicaluse of this method, the partial di�erential equation canbe approximated as a �nite di�erence [48] and outputvalues calculated for the small changes in the input pa-rameters [49]. The sensitivities of Fr1; Fr1

sin �0 ;wY1; w sin �

Y1,

and wL variables are {0.66, {0.22, �0:89, {3.28, and

1.27, respectively. Thus, w sin �0Y1

and wL are the most

important input parameters where w sin �0Y1

has a neg-ative relation with CM and w

L has a positive relationwith CM . In what follows, some standard equations areoptimized to estimate the discharge coe�cient (CM ) bythe PSO algorithm; then, the error of each equation iscalculated.

4.1. Optimized equation of Borghei andParvaneh [7]

Eq. (19) represents the structure of Eq. (8) used byBorghei and Parvaneh [7]. Table 2 shows the coe�-cients, i.e. a1 to a8, proposed in this study, estimatedthrough nonlinear regression. In Eq. (19), INPUT1 isconsidered as the input variable. MSE of the equation,related to the laboratory results, is equal to 0.0037(Table 2). Figure 3(a) compares the calculated andlaboratory results. The 45� line is the exact solution tothe problem, and the closeness of the scatter to this lineshows the accuracy of the model. Moreover, the �t lineequation (assuming that the equation is y = c1x+ c2)in Figure 3(a) shows the accuracy of the model and thedeviation of the results from the exact line. Therefore,the closer c1 is to one and c2 is to zero, the greaterthe accuracy of the model will be. In Figure 3(b), thelaboratory results are compared with those of Eq. (8)

Table 2. Characteristics and statistical errors of Eq. (8).

INPUT1Eq. (8)

a1 a2 a3 a4 a5 a6 a7 a8

{0.18 0.71 {0.15 0.44 0.70 {2.37 2.58 {0.36

MSE RMSE MAE MAPE

0.0037 0.016 0.0482 0.035

Figure 3. (a) Scatter plot of comparing the observed CM and the estimated CM by Eq. (8). (b) Comparison of theobserved CM and the estimated CM by Eq. (8).


to estimate CM . The horizontal axis represents thenumber of datasets (200 tests), and the vertical axisshows the discharge coe�cient (CM ). Figure 3(a)shows that, in some parts, such as experiment numbers20 to 40 and 100 to 120, there is a greater error inEq. (8) than in the laboratory results. However, forintervals with a minor sudden change in the dischargecoe�cient CM , such as experiment numbers 120 to 160,the results are closer together:

CM = [a1(Var2)a2 + a3(Var1)a4 + (Var 3)a5 ]

� [a6 + a7(Var4)a8 ]: (19)

To �nd an accurate and e�cient equation for calcu-lating CM , coe�cients a1 to a8 were calculated by the

PSO algorithm, instead of nonlinear regression. Table 3shows Eq. (19.1) calculated based on INPUT1 as theinput variable. The table shows that MSE is equal to0.0024, which is much lower than that of Eq. (8). In thenext step, the input non-dimensional parameters werechanged, and INPUT2 was used as an input variable forEq. (19). Table 3 shows the related results obtained byEq. (19.2). Since MSE decreased to 0.0021 while MSEof Eq. (19.1) was 0.0024, it is apparent that higheraccuracy was attained using non-dimensional INPUT2.In Figure 4(a), the results from Eqs. (19.1) and (19.2)are given on the vertical axis; the laboratory test resultsare given on the horizontal axis. In both cases, the �tline is very close to the exact line. Eq. (19.2), calculatedbased on INPUT2, has much higher e�ciency than Eq.

Table 3. Characteristics and statistical errors of Eqs. (19.1) and (19.2).

INPUT1Eq. (19.1)

a1 a2 a3 a4 a5 a6 a7 a8

0.34 {0.37 {0.18 3.07 2.34 {0.53 0.87 {0.63MSE RMSE MAE MAPE0.0024 0.049 0.0386 0.029

INPUT2Eq. (19.2)

a1 a2 a3 a4 a5 a6 a7 a8

0.37 {0.43 {0.30 0.20 2.24 {3.00 3.26 {0.30MSE RMSE MAE MAPE0.0021 0.046 0.0354 0.026

Figure 4. (a) Scatter plot of comparing the observed CM and the estimated CM by Eqs. (19.1) and (19.2). (b)Comparison of the observed CM and the estimated CM by Eq. (19.2).


(19.1). In Figure 4(b), the results of laboratory testsare compared with those obtained from Eq. (19.2).It is evident that the accuracy problems in Eq. (8)(Figure 3(b)) in test numbers 20 to 40 and 100 to 120are almost solved; the results are agreeable.

4.2. Optimized 2nd-order equationThe main objective herein is to determine equationsof high accuracy featuring simple standard forms.Therefore, the accuracy of the 2nd-order equations wasinvestigated. Thus, CM equation resembles Eq. (20).In this equation, a1 is the equation's constant value,and a2 to a9 are the coe�cients that make the 2nd-order equation. In Eq. (20), Var1 to Var4 are used asinputs from INPUT1 and INPUT2. Table 4 presents a1to a9 coe�cients as calculated from the PSO algorithm

for Eqs. (20.1) and (20.2) with INPUT1 and INPUT2as input variables, respectively. It is apparent thatMSE in Eq. (20.2) using INPUT2 is less than thatof Eq. (20.1) using INPUT1, indicating that INPUT2combination gives more accurate results. Note thatthe very simple Eq. (20) gives much more accurateresults than the complex Eq. (8) does. Higher MSEsin Eqs. (20.1) and (20.2) than those in Eqs. (19.1)and (19.2) may demonstrate that Eq. (19) has anoptimized shape and is more e�cient for modelingthe discharge coe�cient of the modi�ed triangular sideweir. However, as stated above, the main advantageof Eq. (20) is its simplicity and its standard shape.In Figure 5(a), the estimated values of Eqs. (20.1)and (20.2) for INPUT1 and INPUT2 are plotted onthe ordinate; the laboratory values are plotted on the


INPUT1Eq. (20.1)

a1 a2 a3 a4 a5 a6 a7 a8 a9

0.25 {2.58 1.16 1.42 {0.47 0.31 0.03 2.14 {2.44MSE RMSE MAE MAPE0.0032 0.057 0.044 0.033

INPUT2Eq. (20.2)

a1 a2 a3 a4 a5 a6 a7 a8 a9

1.41 {1.64 1.98 {0.53 0.020 2.67 {0.77 {4.43 2.39MSE RMSE MAE MAPE0.0029 0.054 0.0413 0.031




INPUT1Eq. (21.1)

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13

0.42 3.11 {0.13 {1.20 { 2.37 { 5.91 { 0.46 { 2.95 {MSE RMSE MAE MAPE0.0031 0.056 0.0439 0.033

INPUT2Eq. (21.2)

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13

{0.20 0.37 {3.18 4.45 { 0.82 { 3.19 0 { 0.36 { 2.07MSE RMSE MAE MAPE0.0029 0.054 0.0405 0.030

abscissa. Note the excessive closeness of the �t andexact lines of this equation to the extent that theyoverlap in Eq. (20.2). The closeness between thesetwo lines means that errors in the data are scatteredequally around the exact line. It could be inferredthat this symmetry is due to the greater symmetryof Eq. (20) as compared to Eq. (19). Notice thatthis symmetry bears no relation to the accuracy ofthe model; for example, in Eq. (19.2), MSE (0.0021)is less than MSE of Eq. (20.2) (0.0029); however, inEq. (20.2), c1 and c2 coe�cients are closer to 1 and0, respectively, making the error results completelysymmetrical. In Figure 5(b), the suggested resultsderived from Eq. (20.2) are presented with bold fontson the ordinate, and the laboratory test results areplotted with empty circles. The number of tests isplotted on the abscissa. The comparison made betweenFigures 5(b) and 4(b) shows that the results in areaswhere CM does not experience dramatic uctuations(e.g., experiment numbers 0 to 30) are more accuratethan those of Eq. (19.2) are. The disadvantage of usingthis equation in CM modeling is clear in areas where thedischarge coe�cient undergoes a variation (experimentnumbers 100 to 120). The results indicate that thissimple equation is valid and can have practical usesdue to the non-dimensionality of the data:

CM = a1 +�a2(Var1) + a3(Var1)2�

+�a4(Var2) + a5(Var2)2�

+�a6(Var3) + a7(Var3)2�

+�a8(Var4) + a9(Var4)2� : (20)

4.3. Optimized 3rd-order equationAfter studying the equation with the 2nd-order struc-ture and having favorite equations to calculate dis-charge coe�cient CM of the modi�ed triangular sideweir (Eqs. (20.1) and (20.2)), the standard structure ofthe polynomial of a 3rd-order equation was investigatedalong with the general structure, as shown in Eq. (21).Herein, a1 is a constant, and a2 to a13 are the equation's

coe�cients. The PSO algorithm was used to solve a 13-dimensional problem and calculate the proper valuesfor a1 to a13. Table 5 shows the calculated valuesof INPUT1 (Eq. (21.1)) and INPUT2 (Eq. (21.2)) inEq. (21). From this table, as introduced in otherequations, the equation's error in which INPUT2 isused is less than that when INPUT1 is used. Figure6(a) shows CM results obtained from Eqs. (21.1) and(21.2) and compares them with the laboratory values.The comparison between Eq. (20.2) and Eqs. (21.1)and (21.2) indicates that if the power of the multi-termequation is changed from 2 to 3, the accuracy of themodel does not increase noticeably. As Figure 6(a)shows, the �t and exact lines have less adaptationthan Figure 5(a) do. A comparison of Figure 5(b)with Figure 6(b) shows that Eq. (20.2) provides moreaccurate results in both areas, i.e. where the changes inCM are uniform and where there are sudden changes inCM . Since the objective of this study is to determinean applicable equation to calculate CM of a modi�edtriangular side weir and considering simplicity andshortness as the major criteria of applicability, it canbe inferred that increasing the polynomial's power from2 to 3 does not provide a more e�cient equation forcalculating CM :

CM = a1 +�a2(Var1) + a3(Var1)2 + a4(Var1)3�

+�a5(Var2) + a6(Var2)2 + a7(Var2)3�

+�a8(Var3) + a9(Var3)2 + a10(Var3)3�

+�a11(Var4) + a12(Var4)2 + a13(Var4)3� :(21)

4.4. Optimized nonlinear equationA comparison of MSEs resulting from Eqs. (19.2) and(20.2), i.e. 0.0021 and 0.0029, respectively, shows thatthe lower error in Eq. (19.2) is more likely due tothe constant values considered as powers of the inputvariables. If we intend to reduce MSE, the idea ofusing a standard, nonlinear equation, such as Eq. (22),may play an instrumental role. In Eq. (22), for eachinput variable, one coe�cient (a8, a6, a4, a2) and one




INPUT1Eq. (22.1)

a1 a2 a3 a4 a5 a6 a7 a8 a9

0.93 {0.08 0.95 {0.20 1.19 1.06 1.44 {1.48 1.11MSE RMSE MAE MAPE0.0022 0.047 0.0373 0.028

INPUT2Eq. (22.2)

a1 a2 a3 a4 a5 a6 a7 a8 a9

0.28 {0.22 9.99 {0.27 1.12 1.84 0.61 {1.66 0.91MSE RMSE MAE MAPE0.0020 0.045 0.035 0.026

power (a9, a7, a5, a3) were considered. As was thecase with previous equations, one constant value (a1)was considered for all of the equations. Then, a1 toa9 were calculated with the PSO algorithm, and theoptimization results and MSE values for INPUT1 andINPUT2 are given in Table 6 as Eqs. (22.1) and (22.2).As with other previous equations, the error value waslower in INPUT2. The interesting point here is that theerror value was at its minimum compared with otherequations. Since the shape of Eq. (22) is simple andstandard, it can model the discharge coe�cient (CM )of a modi�ed triangular side weir coe�cient with aninsigni�cant error. Figure 7(a) shows that this equationreduces the error to a minimum; however, the �t and

exact lines do not overlap as they do in Eq. (20.2).A comparison of Figure 5(b) with Figure 7(b) showsthat Eq. (20.2) has more e�cient performance thanEq. (22.2) for experiment numbers 0 to 40, whileEq. (22.2) has higher accuracy with respect to the otherexperiment numbers. A comparison of Figure 4(b) withFigure 7(b) shows that both models are too weak tomodel experiment numbers 0 to 40; however, Eq. (19.2)functions better in areas where CM experiences suddenchanges (experiment numbers 100 to 120):

CM =a1 + a2(Var1)a3 + a4(Var2)a5 + a6(Var3)a7

+ a8(Var4)a9 : (22)


Table 7. AIC amounts for the considered models.

Eq. Nos.(8) (19.1) (19.2) (20.1) (20.2) (21.1) (21.2) (22.1) (22.2)

SSE 0.732 0.472 0.426 0.642 0.575 0.624 0.573 0.443 0.425k 9 9 9 10 10 14 14 10 10

AIC {1104 {1192 {1212 {1128 {1150 {1126 {1143 {1202 {1211

Figure 7. (a) Scatter plot of comparing observed CM and the estimated CM by Eqs. (22.1) and (22.2). (b) Comparisonof the observed CM and the estimated CM by Eq. (22.2).

4.5. Akaike Information Criterion (AIC)By adding parameters to the model constantly, itmay work somewhat better; however, over�tting andinformation loss concerning the real pattern may occur.Therefore, AIC [50-52] represents a trade-o� betweenthe number of parameters involved in the model andthe increase or decrease of error. AIC is used to selecta model by considering not only the error, but also thenumber of parameters used to develop the model. Byassuming that the errors are normally distributed inthe model, AIC equation is obtained as follow:

AIC = n log�SSEn

�+ 2k; (23)

where n is the number of observations, k is the numberof used parameters used in the model plus one, andSSE is the Sum of Squared Errors of the model.

The parameters used in this equation for the modelsdeveloped in the present study are shown in Table 7.

According to AIC formulation, the model enjoy-ing a better �t has lower AIC value. Based on Tables 3and 6, Eq. (22.2) with RMSE of 0.045 performedsomewhat better than the Eq. (19.2) with RMSE of0.049 did. However, Table 7 shows that the numberof involved parameters of Eq. (22.2) is more than thatof Eq. (19.2)'s parameters. Therefore, Eq. (19.2) withAIC of {1212 has the best-�t score according to thiscriterion.

5. Conclusions

In this research, a PSO model was developed to deter-mine the discharge coe�cients of modi�ed triangularside weirs. Two hundred experimental datasets wereused for the PSO model. Four di�erent equation forms


were used to determine the most accurate form; in eachform, two non-dimensional parameters were used asinputs to the PSO model. As shown, the second inputseries provided more accurate results and includedw=L (weir height/weir length), Fr1=sine(�0) (upstreamFroude number/sine (weir included angle/2)), w=Y1(weir height/upstream ow depth), and w�sine(�0)=Y1((weir included angle/2)/upstream ow depth). Theequations based on the PSO algorithm were comparedwith a nonlinear regression model, and it was foundthat they were more accurate than the regression mod-els. After comparing four forms of the equations usedin this study, it was determined that the performanceof the power equation is better than those of the otherforms are. It also was determined that a combinationof the PSO algorithm with a power equation could beused successfully to compute the discharge coe�cientsof modi�ed triangular side weirs.

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Biographies

Amir Hossein Zaji is now a PhD Student in Hy-draulic Structures (Civil Engineering), Department ofCivil Engineering, University of Razi, Kermanshah,Iran. He has 20 published papers in ISI journals.He works in the �eld of soft computing methods inengineering applications. He has been rated as adistinguished researcher in Razi University, 2015.

Hossein Bonakdari is a Professor at the Departmentof Civil Engineering of University of Razi Univer-sity; he earned his PhD in Civil Engineering at theUniversity of Caen-France. After obtaining PhD,Dr. Bonakdari joined University of Razi as a facultymember in 2006; presently, he is a Professor at theDepartment of Civil Engineering. He has supervised 5PhD and 30 MS theses. He enjoys brilliant teachingexperience of more than 16 years in the �eld of CivilEngineering. Furthermore, From 2013 till 2015, hewas the Director General of Training, Research and

Technology Development at Ministry of Energy, Iranand also the Deputy of Planning & Development inNational Water and Wastewater Engineering Com-pany, Iran, from 2011-2013. His �elds of special-ization and interest include practical application ofsoft computing in engineering, modeling of wastewaterurban drainage systems, sediment transport, com-putational uid dynamic and hydraulics, design ofhydraulic structures and uid mechanics. From 2010till 2011, he has been a researcher at Laboratory ofCivil and Environmental Engineering, INSA of Lyon,France. Results obtained from his researches have beenpublished in more than 100 papers in internationaljournals (h-index=13). He has also more than 150presentations in national and international conferences.He published two books. He has been rated as adistinguished researcher in Razi University, 2014, 2015and 2016.

Shahaboddin Shamshirband is now a Senior Lec-turer at the Department of Computer System andTechnology, University of Malaya. He received his MScdegree in Computer Science from Islamic Azad Univer-sity of Mashhad (IAUM), Iran in 2006. He joined theFaculty of Computer Science, Islamic Azad University,Iran for seven years. Currently, he is pursuing hisPhD from the University of Malaya, Malaysia. He isworking in a multi-disciplinary environment involvingapplications of computational intelligence and neuro-computing in Civil Engineering. He has published morethan one hundred papers in ISI journals.

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