# Design of Optimal Water Distribution Systems

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5 ALPEROVITS AND SHAMIR: DI-SIGN 01 WATER DISTRIBUTION' SYSTEMS 889 TABLE 3. Structure of the Initial Linear Program: Strings of Pipes for Pressure or Loop Constraints 1 he constraint data include 39 variables, six pressure equations, two loop equations, no equations between sources, and a coefficient matrix of 16 rows and 55 columns. linear program which was set up initially. There are 39 variables (4-5 candidate diameters for each of the eight pipes), six pressure equations for each node except I. at which the head is fixed, and two loop equations. Because only H min is specified at each node (the maximum is unrestricted in this example), there are six head constraints (equation (4)). Their structure is listed in Table 3 by showing the 'strings* of pipes in each constraint. For example, for node 7 the constraint is formulated by going along pipes 1, 3, 5, and 6. The constraint is thus where, for example. (14) (15) There are two loop equations, whose strings of pipes are listed also in Table 3. A negative sign means that the flow initially assumed is in a direction opposite to that taken in formulating the hydraulic head lin e continuity constraint (equation (7)). For example, the equation for the upper loop starts and ends at node 2 and goes along pipes 3, 4, 7. and 2. This loop equation is Fig. 3. (a) initial flow distribution and i t s linear programing design at a cost of (b) Final flow distribution and optimal design at a cost of Table 4 summarizes the intermediate results of the computa- TABLE 4. Intermediate Results of the Computations for the Network in Figure 2

8 892 ALPEROVITS AND SHAMIR: DESIGN OF WATER DISTRIBUTION SYSTEMS Fig. 6. Schematic diagram of a real network. reflect the total length of time that the specific loading condition is assumed to prevail throughout the design horizon (say, 25 yr) and a coefficient that converts streams of annual expenditures to present value. Recall now that the LPG procedure is to fix the discharges throughout the network, optimize, then change the flows. From (20), if the efficiency is assumed to be constant, the power is linearly proportional to the head added by the pump, for each linear program. The operating cost of the pump is therefore linearly proportional to the decision variable, which is XP. The capital cost, however, is not linear, as seen from Figure 4. This is where an iterative procedure is developed. (I) Assume values for the cost per horsepower, from the data represented in Figure 4, for each pump. (2) Solve the linear program with these values as the coefficients of the XP in the objective function. (3) For the resulting XP after the linear program has been solved, compute the cost per horsepower. I f all values are close enough to those assumed, this step is complete, and one proceeds to a flow iteration by the gradient method. Otherwise one takes the new costs and solves the linear program again. This procedure has been found to work very well, owing probably to the relatively mild and regular slope of the cost curve. No more than 2-5 repetitions of the linear program at any gradient move were required to converge to within reasonable accuracy, with up to three pumps in the system. An alternative would have been to use separable programing, but due to the success of the relatively simple procedure outlined above, this was deemed unnecessary. Each pump designed by this procedure may represent a pumping station. One now takes the values of the flow and TABLE 6b. Basic Pump Data: Cost Function of Pumps TABLE 6a. Basic Design Data for The Network in Figure 5 With Two Loops, One Pump, One Source, One Reservoir, and Two Loads Other data include the following. Pump 1 was connected to pipe 1. The assumed initial cost for pump 1 was The additional storage elevation cost {per unit of elevation) was TABLE 6c. Basic Cost Data for Pipes The computation stopped when max DQ{!) equaled 1.0 or after 50 major iterations. Results will be printed out every 10 major itcralions or whenever best total cost is further improved. Flow change in loop / will be executed only if DQ(l)/DQ max is greater than Number of minor iterations allowed after a feasible solution has been reached for a flow distribution was 20, Local solution is considered to be reached if last improvement per iteration equaled 0.0%.

13 ALPEROVITS AND SHAMIR; DESIGN OF WATER DISTRIBUTION SYSTEMS 897 TABLH 9c. Section Data supply from external sources, and a balancing reservoir. (The computer output shows 52 nodes, because the reservoir is given two numbers, one for each loading. It shows 34 loop equations: 15 for loops and two for equations between the sources and the reservoir, one equation for each loading.) This network was designed several years ago to serve one out of

14 898 ALPBROVITS AND SHAMIR: DESIGN OF WATER DISTRIBUTION SYSTEMS TABLB 9/ Structure of the Initial Linear Program: Strings of Pipes for Pressure or Loop Constraints The constraint data include 340 variables, 22 pressure equations, four source equations, 30 loop equations, and a coefficient matrix of 121 rows and 461 columns.

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