MODEL FORMULATION FOR LARGE WATERSHED WITH DIFFERENT OWNERSHIP

OPTIMAL ECOLOGICAL MANAGEMENT PRACTICES (EMPs)

6.4. MODEL FORMULATION

6.4.2. MODEL FORMULATION FOR LARGE WATERSHED WITH DIFFERENT OWNERSHIP

C_i = cover factor for the i^th EMP in the plot (dimensionless) A_j = Area of the j^th land cover in the plot (m²)

C_j= Cover factor for the j^th land cover in the plot (dimensionless) Q_max = Maximum allowable peak rate of runoff from the plot (cumec)

Q_min = Minimum allowable peak rate of runoff at downstream from theplot, (cumec) RC_i = Runoff coefficient for the i^th EMP in the plot (dimensionless)

RC_j= Runoff coefficient for the j^th landcover in the plot (dimensionless) Rc = Runoff coefficient of the coverage area (dimensionless)

Ac= Coverage area of the plot (m²)

I = Maximum intensity of rainfall for the time of concentration of the selected design storm for the plot (mm/hr)

(as)_i= Suitable area available for i^th EMP in the plot (m²) (amin)i = Minimum area required for i^th EMP (m²) AL= Total available EMP area in the plot (m²) C_max = Maximum coverage allowed in the plot (m²)

(a_min)_i is the minimum area kept for i^th EMP in the plot according to owner’s choice (m²) (a_max)_i is the maximum area kept for i^th EMP in the plot according to owner’s choice (m²)

6.4.2. MODEL FORMULATION FOR LARGE WATERSHED WITH

(a) OPTEMP-LM Model Mathematical formulation

Objective function: Minimization of construction and maintenance cost for the EMPs in p number of plots

(6.8)

Where,

i= 1,2,……n, where, n is number of possible EMPs that can be applied k=1,2,3….p, p is the number of plots in the watershed

Cc_ik = Capital cost of the i^th EMP in the k^th plot (`) Cm<sub>ik</sub> = Maintenance cost of the i<sup>th</sup> EMP in the k<sup>th</sup> plot (`) Constraints:

Sediment yield constraints:

Stmax k

n

1

= i

m

1

= j

k k k

j ik

ik k K K p

1

= k

min ∑ ∑ ∑n +Cc Ac }P ]≤

1

= i

)cik a ik A -

+ ( a { c

) LS ( K

∑[R

≤

St

(6.9) Peak discharge constraints:

( )

}÷A ]IA≤Q_k

∑RkC a +(∑ A -∑a )Rc + RckAc

≤ [{

Qk

k max n

1

= i

m

1

=

j jk k

k n

1

= i k i k j

k i i

min

(6.10) Available EMP area constraints:

∑a ≤AL_k =A_k -A_kCk_max

n

1

= i

i_k (6.11)

Suitable EMP area constraints:

ak ≤ask

i

i (6.12)

( )

_a

∑ Cc +Cm

∑

=

MinimizeZ _ik

n

1

= i

ik ik p

1

= k

Owner’s Choice constraints:

( ) ( )

a min k

≥ ak

≥ a maxk

i i

i (6.13) Non negativity constraints:

ak ≥0

i (6.14)

where,

j= 1,2,……m, where m is the number of different landcover in the plot except the EMPs and coverage area

(Cc)k_i = Capital cost of the i^th EMP in the k^th plot (`) (Cm )k<sub>i</sub> = Maintenance cost of the i<sup>th</sup> EMP in the k<sup>th</sup> plot (`) a_ki = Area of the i^th EMP in the k^th plot (m²)

St_min = Minimum sediment yield required from the watershed (tonnes/yr) St_{max =} Maximum sediment yield allowed from the watershed (tonnes/yr) A_k=Area of the k^th plot in the watershed (m²)

R_k = Rainfall and runoff erosivity index of the k^th plot (100 ft·tonf·in/acre/hr/yr) K _k = Soil erodibility factor of the k^th plot (tonnes/acre per unit of R),

(LS) _k = LS factor of the k^th plot (dimensionless)

C_{i k} = cover factor for the i^th EMP in the k^th plot (dimensionless) A_{j k} = Area of the j^th land cover in the k^th plot (m²)

C_{j k} = Cover factor for the j^th land cover in the k^th plot (dimensionless) Qk_max = Maximum allowable peak rate of runoff from the k^th plot (cumec)

Qkmin = Minimum allowable peak rate of runoff at downstream from thek^th plot, (cumec) RC_ki = Runoff coefficient for the i^th EMP in the k^th plot (dimensionless)

RC_kj= Runoff coefficient for the j^th landcover in the k^th plot (dimensionless) Rc_k = Runoff coefficient of the coverage area in the k^th plot (dimensionless)

Ac_k= Coverage area of the plot (m²) in the k^th plot

Ik = Maximum intensity of rainfall for the time of concentration of the selected design storm for the k^th plot (mm/hr)

(as) _ki= Suitable area available for i^th EMP in the k^th plot (m²) (a_min) _ki = Minimum area required for i^th EMP in the k^th plot (m²) AL_k= Available EMP area in the k^th plot (m²)

C_kmax = Maximum coverage allowed in the k^th plot (%)

(a_min) k_i is the minimum area kept for i^th EMP in the plot according to owner’s choice in the k^th plot (m²)

(a_max) k_i is the maximum area kept for i^th EMP in the plot according to owner’s choice in the k^th plot (m²)

(b) OPTEMP-LDM Model

In the OPTEMP-LM model, numbers of variable representing EMPs increase with the increase in number of plot. Therefore, with large number of EMPs and with large number of plots, the numbers of variables in the model becomes considerably high, which may sometimes become problem from computational point of view. Therefore, a model with DP approach is developed (OPTEMP-LDM), where number of variable remains same as number of EMPs considered for the study. Here number of stages increases with increase in number of plots, which is computationally advantageous.

In the OPTEMP-LDM model, the watershed is considered as combination of various plots belongs to different land owners. Plots from upstream to downstream in sequence were considered as stages for the DP model. The OPTEMP-LS model provides optimal combination of EMPs for the different plots of the watershed for various desired sediment yield and peak discharge from the plots. The optimal values obtained for each of these plots for different sediment and peak discharge values are then used in the DP model

to obtain optimal combination of EMPs for the entire watershed for desired sediment yield and peak discharge at the outlet of the watershed.

The sequence of plots considered in the DP model is as shown in the Figure 6.1.

Figure 6.1: Arrangement of the plots

Objective function:To determine the best combination of EMPs for the entire watershed so that the otherwise allowable coverage area of the watershed can be developed completely at minimum cost maintaining sediment yield and peak discharge from the watershed within permissible limits.

Recursive equation:

f= cost of EMP combination

f^p (S_p) = Min [Z(X_p,Q_p)+ f^p-1 (S_p-X_p)] (6.15) where, S_pis the state variable representing amount of sediment yield from the p^th stage, and X_p is the trial discrete sediment yield from the current plot in p^th stage and f^p-1(S_p-X_p) is the optimal cost of EMPs combination up to the previous stage for allowing (S_p-X_p) sediment yield from (p-1) stage. Z(X_p,Q_p) is the cost of the EMPs for sediment yield X_p and is it peak discharge of Q_p