A new integrated model of the group method of data handling and the firefly algorithm (GMDH-FA): application to aeration modelling on spillways

Due to the high flow velocity over dam spillways and outlets, severe cavitation damage might occur to the structures. Aeration (introducing air into the passing flow) is a useful remedy for preventing or decreasing cavitation, however, proper estimation of aerators air demand is a complex problem. On that account, the standard GMDH model, integrated GMDH-HS (with the harmony search algorithm) model and a novel integrated GMDH-FA model (with the firefly algorithm), were developed and applied to estimate air demand on spillway aerators in dams. Input parameters including flow rate (Q_w), flow depth (d₀), relative pressure under the jet (h_s), ramp angle (α), step height (s), and spillway slope (θ) were applied as the effective factors for estimating the amount of air flow of the aerators (Q_a). General results based on several statistical measures (NRMSE, PCC, NMAE, NSE) and the test of ANOVA for models’ residuals, showed that the standard GMDH improved the accuracy of estimating air flow in comparison to empirical equations (an average enhanced efficiency of 59.86% in terms of NRMSE) and multiple linear regression method (an enhanced efficiency of 37.15% in terms of NRMSE). Moreover, findings of the research revealed that the FA and HS algorithms improved the performance of the standard GMDH equal to 17% and 13%, respectively.

α :

Ramp angle

β :

Attractiveness of firefly

β :

Ratio of air flow rate to water flow rate

β ₀ :

Attractiveness at r equal to 0 in FA

ε _i :

A random number in FA

γ :

Light absorption coefficient in FA

ΔP :

Pressure difference between the atmosphere and the aerator nappe

η :

Randomization coefficient between 0 and 1 in FA

λ :

Mutation coefficient

ρ :

Fluid density

σ:

Cavitation index

θ :

Spillway slope

ω :

Selection pressure parameter (is between 0 and 1) in GMDH

A _d :

Effective air duct area per unit width of chute

bw :

A bandwidth used in pitch adjusting in HS

C :

(c₀, c₁, c₂, c₃, c₄, c₅) is solution candidate of FA and HS

c₀, c_i, c_js, c_jsd, …:

Weights op polynomials in GMDH

C _max :

Maximum of search space in HS

C _min :

Minimum of search space in HS

D :

Dimension of problem in FA and HS

d :

Depth of non-aerated flow

d ₀ :

Flow depth

e _c :

Selection pressure criteria in GMDH

f :

Friction factor of ramp surface

Fr _c :

Froude number in the aerator location

FW :

Fret width damp ratio in HS

g :

Acceleration due to gravity

HMCR :

Harmony memory consideration rate in HS

HMS :

Harmony memory size in HS

h _s :

Pressure difference under the flow jet

K₁, K₂, K₃ and n :

Empirical coefficients

L _X :

Horizontal distance from the beginning of the free jet over the aerator to the impact point

m :

Number of inputs in GMDH

MCDR :

Mutation coefficient damping ratio

Nd :

Number of data

NMAE :

Normalized mean absolute error

N _n :

Number of next layer neurons in GMDH

N _np :

Number of neuron of present layer in GMDH

NRMSE :

Normalized root-mean-square error

NSE :

Nash–Sutcliffe coefficient

p _abs :

Absolute pressure

PAR :

Pitch adjustment rate in HS

p _v :

Fluid vapor pressure

q _a :

Unit air discharge rate

Q _air :

Required air flow of the spillway

Q _w :

Flow rate

r :

Correlation coefficient

r _ij :

Distance between a pair of fireflies

RMSE _best :

RMSE of best neuron in GMDH

RMSE _worst :

RMSE of worst neuron in GMDH

s :

Step height

SR :

Search range

SR :

Search range in FA and HS

U :

Flow velocity

V :

Mean velocity of flow

\( x_{k,t}^{i,j} \) :

Neuron number j of layer ith layer which is connected to the variables/neurons number k and t of the previous layer

\( \bar{y} \) :

Average of measured air flow

y ^o :

Measured air flow

y ^s :

Predicted air flow

Authors and Affiliations

1. Department of Water Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

   Amin Mahdavi-Meymand & Mohammad Zounemat-Kermani

Corresponding author

Correspondence to Mohammad Zounemat-Kermani.

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