Design of optimal water distribution systems

Surendra Chandniha

Design of optimal water distribution systems

Surendra Chandniha

1977, Water Resources Research

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A method called linear programing gradient (LPG) is presented, by which the optimal design of a water distribution system can be obtained. The system is a pipeline network, which delivers known demands from sources to consumers and may contain pumps, valves, and reservoirs. Operation of the system under each of a set of demand loadings is considered explicitly in the optimization. The decision variables thus include design parameters, i.e.. pipe diameters, pump capacities and reservoir elevations, and operational parameters, i.e., the pumps to be operated and the valve settings for each of the loading conditions. The objective function, to be minimized, reflects the overall cost capital plus present value of operating costs. The constraints are that demands are to be met and pressures at selected nodes in the network are to be within specilied limits. The solution is obtained via a hierarchial decomposition of the optimization problem. The primary variables are the flows in the network. For each flow distribution the other decision variables arc optimized by linear programing. Postoptimality analysis of the linear program provides the information necessary to compute the gradient of the total cost with respect to changes in the How distribution. The gradient is used to change the flows so that a (local) optimum is approached. The method was implemented in a computer program. Solved examples are presented.

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Fig. 1. Overview of the LPG method. The first step in developing the LPG method is to consider optimization of the design when the distribution of flows'in the network is assumed to be known. We adopt the formulation given by equations (1), (2), (4), and (5), in which the lengths of

Fig. 2. A two-loop network supplied by gravity.

TABLE l<j. Basic Design Data for the Network in Figure 2 With Two Loops, One Source, and a Single Load The computation stopped when max DQ(J) equaled 1.0 or after 50 major iterations; Results will be printed out every five major iterations or whenever best total cost is further improved. Flow change in loop / will be executed only if DQ{I)//DQ max is greater than 0.20. The number of minor iterations allowed after a feasible solution has been reached for a flow distribution was 20. The local solution is con- sidered to be reached if last improvement per iteration equals 0.0%.

The first identity on the left holds because both 8(AQp) and 8(Qp) are incremental changes in flow in the same path. Since

TABLE 2b. Section Data No simple way was found to do this, and the heuristic ap- proach was adopted. A step size, given in terms of a change in flow, is fixed at the start of the program (given by the user as input data). The flow component which has the largest (abso- lute) value of the gradient component is given a flow change of the specified step size, and the other flows are changed by quantities reduced by the ratio of the appropriate gradient components to the largest gradient component. The step is thus in the direction of the gradient, its maximum component being determined by a user-supplied value. The program also contains a routine for increasing or reducing the step size from one iteration to the next, based on the success or failure of previous steps. The overall iterative procedure stops when no improvement is achieved with the minimum step size allowed (a user-supplied parameter), or after a prescribed number of iterations has been exceeded.

the constant 1.852 appears in all components of the gradient, and we are interested only in their relative magnitude, we can leave the constant out and write the component of the gradient vector G as

TABLE 4. Intermediate Results of the Computations for the Network in Figure 2 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 line 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

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.

Table 4 summarizes the intermediate results of the computa- Fig. 3. (a) initial flow distribution and its linear programing design at a cost of 493.779. (b) Final flow distribution and optimal design at a cost of 479.525.

TABLE 5a. Final Results for the Design of the Network in Figure 2: Section Data and Optimal Diameters for Iteration 18 The solution was reached after six minor iterations. The total network cost (there were no penalty costs) was 479,525.

TABLE 5b. Final Results for the Design of the Network in Figure 2: Node Data for Iteration 18

Fig. 4. Cos! per installed horsepower (schematic).

Fig. 5. Network with a pump, a balancing reservoir, loads, and initial and final flow distributions. where K, is a constant. In equation (19), y is computed to Pumps. When there are to be pumps in the system, the problem is one of the design-operation type; i.e., one has to select the capacities of the pumps as well as to decide which pumps should operate for each of the loading conditions. The locations at which pumps may be installed are selected by the designer, but since the program can set certain pump capaci- ties to zero, if that is the optimal solution, the program ac-

TABLE 6a. Basic Design Data for The Network in Figure 5 With Two Loops, One Pump, One Source, One Reservoir, and Two Loads 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(1)/DQ max is greater than 0.15. 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%.

Fig. 6. Schematic diagram of a real network.

Other data include the following. Pump | was connected to pipe 1. The assumed initial cost for pump | was 1000. The additional storage elevation cost {per unit of elevation) was 2000.

TABLE Jc, Structure of the Initial Linear Program: Strings of Pipes for Pressure or Loop Constraints The constraint data include 51 variables, iive pressure equations, two source equations, four loop equations and a coefficient matrix of 20 rows and 71 columns.

“ABLE 8«. Optimal Solution for the Network in Figure 5: Section Data and Optimal Diameters for Iteration 22 The solution was reached after one minor iteration. The total network cost (there were no penalty costs) was 299,851; the total pipeline cost, 267,113; the additional storage elevation cost, 4330; and the total pumping cost, 28,408. The total lengths of pipes Tor class 1 were ihe following: for diameters of 4, 6, 8, 10, 12, 14, and 16 in., the lengths were, respectively, 70.4, 2120.2, 983.6, 2859.5, 966.1, 1000, and 100 m.

TABLF- 8/). Optimal Solution for the Network in Figure 5: Node Data for Iteration 22 Additions to an existing system. When parts of the network already exist and only new parts are to be designed, the exist- ing components are specified as being fixed, and the program solves for the rest. For each existing pipe, its diameter is specified as being the only one on its candidate list. When

TABLE 9a. Basic Design Data for a Real Network With Two Loads

The method was applied to the system shown in Figure 6, which has 51 nodes, 65 pipes. 15 loops, two pumps which

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 ALPEROVITS AND SHAMIR; DESIGN OF WATER DISTRIBUTION SYSTEMS

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.

As an example, the information in Table | of this comment can be used to calculate the correct gradient for iteration 18 of the example problem. That iteration is described in Figure 3b and Tables 4, 5a, and 56 of the original paper. Loop A in- cludes links 3 and 4 in the direction of flow and links 2 and 7 in the opposite direction. With the exception of the path to node 2, every path in Table 1 includes at least one of the links in loop A and must therefore be included in the calculation of the gradient, O(cost)/O(AQ,). The equation for the gradient component would be Similarly, the gradient component for loop B can be ob- tained as follows:

Since each gradient is positive, both AQ, and AQ; should be negative. Alperovits and Shamir's computer program was modified to incorporate the new gradient calculation for the small ex- ample problem. The programing change is specific to that

TABLE 1. Link and Path Data for Interation 18 of Example Problem

*Using the same rules as Alperovits and Shamir, these changes are not made because AQ, < 0.2AQ ma, TABLE 2. Example Problem With Corrected Gradient

The loop constraints from the linear program can be writ- ten in the direction shown in Figure 1,

We wish to determine the flow changes necessary to cancel y and 6; the total change in head loss around the loops due to the changes in flows should be -y and — 6, respectively. Thus the total increment in the head loss around each loop should be zero.

Apurba Gupta

Environmental Software, 1993

An optimization algorithm with recourse to a nonlinear programming technique based on the interior penalty function with Davidon-Fletcher-Powell method, incorporating the graph theoretic approach for analysis of networks, has been developed for design of functional and least-cost water distribution systems with multiple reservoirs. The model has the capability to design a new system as well as to reorganize an existing system. Reorganization of a system becomes necessary due to increased water demand resulting from an increase in population density and enhanced service levels or expansion to the new areas. Further, the hydraulic simulator based on graph theory enables design of all types of systems; viz. branched. looped or a combination thereof including nodes with specified fixed energy grades. The software package has been developed for use on an IBM-PC compatible micrrDcomputer. Tim efficacy of the algorithm has been demonstrated through an example on design of a typical water distribution system.

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