Tamtam Proceedings - lamsin

Global Optimization of Water Resources 107where K j is the price for the KW h consumed during the period ∆t j .The objective of our formulation of the objective function is to determine, at the givenmoment, the optimal combination of the drillings which have to work simultaneously tosatisfy the drinking water requirements of the population. Or (α i , Q i ) debit supplies bythe drilling F iIf one finds as result of our model of optimization: α i = 1, it means that drilling F i hasto work. If one finds α i = 0, then drilling F i does not have to work.Our objective function will be so the sum of costs functions of every drilling F i . Thatis [7]: Z = ∑ i,j C i,j, and because electric fixing of a price scale imposes the followingthree slices:Objective function will be so: Z = ∑ 3 ∑ nj=1 i=1 C i,j, where n is the number of drillings.By replacing function cost C i,j by its expression, one will have:Z = (3∑j=1( ∑nK j .∆t j ).i=1ρg.(α i .Q i ).(H i )).η giIt is so necessary to determine the theoretical capacity of a reservoir. The calculationof this capacity makes on the basis of the variation of the advanced debit schedule Q ph :V Reservoir = f(Q pj , Q ph ), with Q pj is a daily advanced debit see Ref. [2].Notice that the reservoir has to store the surplus of water during the hours of weak consumptionand fill the deficit if water V 1 during the rush hours.This condition is translated in practice, by the fact that the sum of the initial volume V 0of the reservoir, at the given moment, and the difference between entering volume andoutput of the reservoir has to be superior to the volume V 1 . What means mathematically:V 1 ≤ V 0 + (V put − V output ) (1)Now, the volume of water pumped in the reservoir does not have to exceed its maximalcapacity V max . So disparity (1) becomes: V 1 ≤ V 0 + (V put − V output ) ≤ V max .Besides, the volume of water pumped in the reservoir during the duration ∆t expressesitself as follows: V put = ∆t. ∑ ni=1 (α i.Q i ). Also for the outgoing volume of water:V output = ∆t.Q d (Q d is wanted debit).Mathematical formulation to minimize the cost of the electrical energy of the pumpingspells so:⎧⎪⎨ Minimize f(α) = ( ∑ 3j=1 K j.∆t j ).( ∑ n ρg.(α i .Q i ).(H gi + 1.1.∆H iL )i=1)η gi⎪⎩ subject to V 1 ≤ V 0 + ∆t.( ∑ ni=1 (α (2)i.Q i ) − Q d ) ≤ V maxα i ∈ {0, 1}2.1. Example in KenitraThe application of our model of optimization requires the data collection concerningthe parameters of definition of the objective function and its constraints. So, one is goingTAMTAM –Tunis– 2005