Problem Formulation
5.3 Objective Functions and Constraints
5.3.1 Objective Functions
The purpose of optimization is to choose the best of many acceptable designs or policies available. Thus a criterion has to be chosen for comparing the different acceptable
designs or policies and for selecting the best one. The criterion with respect to which a design or policy is optimized, when expressed as a function of the design variable, is known as the objective function (Rao, 1996). The choice of the objective function is governed by the nature of the problem.
Considering the target power demand maximization of the power profit is taken as the objective function in this study with release as decision variable. This project is planned for the peaking hour power demands; therefore maximization of the power benefit is more advantageous form the management point of view. However considering the fact that the reservoir induced diurnal variations will have some adverse impact on the downstream and will cause some losses, the objective function in this study has been designed to have maximum net benefit, i.e. benefit obtained after deducting losses that occurs downstream from the power benefit. The major losses that affect the livelihood of the downstream community directly are considered for assessing the losses. The losses taken into account are;
loss of agriculture and loss of fish production due to diurnal variations. Details of these losses are given in chapter 4. Mathematical expression of the objective function is given below:
Maximize
( )
1
( )
=
=
∑
T bit− fat + fpt tf P L L 5.1 where,
Pbit = profit from power production during time period t;
Lfat =loss of agriculture at downstream due to water scarcity at time period t;
Lfpt= loss of fish production at downstream due to water scarcity at time period t;
T = 3600; 36 stages per year for 100 years.
The expanded form of equation 5.1 is written in equations 5.2 to 5.6
Maximize
( )
00
0.0006 250
1 ( ) 28987
2 250
nt
36
Q nt
dt n nn p r t fft f
t=1
f = ηgQ H +H R H − − +w × −Q ×L ×R
∑
5.2 106
3600
t dt
r
Q R H
= ×
× 5.3
( )
( )
106
240 3600
t t t t El
nt
r
S I R E K
Q H
+ − − − ×
= − × 5.4
n t tail f
H = El - El - h 5.5
nn nt tail f
H = El - El - h 5.6
t = index of time period (10 days);
Hr= Duration of turbine operation (hour);
η = the combined efficiency of turbine and generator in percent in %;
g = gravitational acceleration (9.81) m/s2;
Qdt = discharge passing through turbines m3/s for time t;
Qnt =discharge in non-operating hours m3/s for time t;
Lfft = fish production (kg) for time t;
wt= weightage given to fish production;
Rf = cost of fish per kg (Rs);
Hn= Net hydraulic head (difference of reservoir elevation at time t and normal tailrace level) at beginning of time t (m);
Hnn= Net hydraulic head (difference of reservoir elevation at time t and normal tailrace level) at end of time t (m);
St = reservoir storage (a state variable) at the beginning of time period t (Mm3);
It = inflow at time t (Mm3);
Et = evaporation at time t (Mm3);
Rt = release at time t (Mm3);
KEl = Storage capacity of reservoir at time t (Mm3);
φt = discrete set of characteristic storage volumes considered at the beginning of time
period t;
n = total number of time periods remaining including the current period before;
Elt =elevation of reservoir at the beginning of time t (m);
Elnt =elevation of reservoir at the end of time t (m);
Eltail =elevation of normal tail race water (m);
hf = head loss due to friction (m) corresponding to Elt and Eln;
Rp= cost of power (Rs.) per unit (kWh).
5.3.2 Constraints
a) Continuity Constraint
t+1 t t t t m
S = S + I - R - E - R 5.7
where,
St+1 = storage in Mm3 at the end of the time period t or beginning of time period t+1;
Rm = minimum downstream release = 6 ×10×24×3600/106 which the project proposes to release primarily to meet the water requirement of river reach between dam and the tail race confluence.
b) Reservoir Storage Constraint
Reservoir storage cannot be lowered below the dead storage volume of 720 Mm3. The maximum storage of the reservoir at beginning or the end of ten days cannot be more than storage capacity of the reservoir which is given in table no 3.1 in chapter 3 in order to
accommodate for coming flood in the reservoir. But in any case it should not be greater than full reservoir level (FRL) volume of the 1365 Mm3.
Thus the reservoir storage constraint is given as;
d t+1 max
S <S <S ` 5.8
where,
Sd = Storage capacity of the reservoir at MDDL = dead storage volume = 720 Mm3;
Smax = storage capacity of reservoir at FRL =1365 Mm3. c) Release constraint
Release from the reservoir should be such that at the end of time t it the reservoir storage level does not go above FRL and does not go below MDDL
Rtmin <Rt<Rtmax 5.9 where,
Rtmin =max[0, (St+It−Et−KEl−Rm)] 5.10
Rtmax =St+It−Et−Sd −R m 5.11
In above equations
Rtmin = minimum release in time t based on reservoir storage constraint;
Rtmax = maximum release in time t;
KEl = reservoir storage capacity considering flood cushioning at time t;
CKd = downstream channel capacity;
d) Downstream Environmental Constraint
In the lean period, inflow to the reservoir reduces and LSHE project being run-off- the-river scheme, the release made from the reservoir in lean period also gets reduced. In such situation it becomes very important to maintain the minimum release to meet the
downstream water quality standards which is essential for the survival of the aquatic life and human being residing at the downstream of the river. Thus another release constraint is.
Rt ≥Rtdm 5.12
where, Rtdm is the minimum mandatory flow that policy maker may decide to provide at downstream to maintain near natural flow. 250 m3/s is considered for the present study to conduct the case study.