An improved genetic algorithm (GA) formulation for pipe network optimization has been developed. The new GA uses variable power scaling of the fitness function. The exponent introduced into the fitness function is increased in magnitude as the GA computer run proceeds. In addition to the more commonly used bitwise mutation operator, an adjacency or creeping mutation operator is introduced. Finally, Gray codes rather than binary codes are used to represent the set of decision variables which make up i the pipe network design. Results are presented comparing the performance of the traditional or simple GA formulation and the improved GA formulation for the New York City tunnels problem. The case study results indicate the improved GA performs significantly better than the simple GA. In addition, the improved GA performs better than previously used traditional optimization methods such as linear, dynamic, and nonlinear programming methods and an enumerative search method. The improved GA found a solution for the New York tunriels problem which is the lowest-cost feasible discrete size solution yet presented in the literature.

https://digital.library.adelaide.edu.au/dspace/bitstream/2440/1073/1/hdl_1073.pdf

An Improved Genetic Algorithm for Pipe Network Optimization

The bandwidth problem for a graph G is to label its n vertices vi with distinct integers f(vi) so that the quantity max{| f(vi) − f(vi)| : (vi vj) ∈ E(G)} is minimized. The corresponding problem for a real symmetric matrix M is to find a symmetric permutation M' of M so that the quantity max{| i − j| : m'ij ≠ 0} is minimized. This survey describes all the results known to the authors as of approximately August 1981. These results include the effect on bandwidth of local operations such as refinement and contraction of graphs, bounds on bandwidth in terms of other graph invariants, the bandwidth of special classes of graphs, and approximate bandwidth algorithms for graphs and matrices. The survey concludes with a brief discussion of some problems related to bandwidth.

The bandwidth problem for graphs and matrices—a survey

This note suggests a technique for including pressure dependent demand and leakage terms in simulation models for water distribution systems. Empirical functions are used to relate consumer outflows and leakage losses to the network pressures and the inclusion of these functions in the mathematical formulation of the network analysis problem is described. An application to an existing distribution system is presented where it is shown that the extended network model proposed leads to more realistic simulation results when the network pressures are too low to provide specified consumer demands, or high enough to cause significant leakage losses. It is also found that the computational requirements of network analysis and simulation are not significantly affected by the inclusion of the additional terms.

A technical note on the inclusion of pressure dependent demand and leakage terms in water supply network models

The requirements and basic components of a typical optimal control environment for water‐supply pumping systems are presented and discussed Examined model components include hydraulic network models, demand forecast models, and optimal control models A review of existing optimal control methodologies for water‐supply pumping systems is also provided Examined methodologies are classified on the basis of the type of system to which the methodology can be applied (single source‐single tank or multiple source‐multiple tank), the type of hydraulic model used (mass balance, regression, or hydraulic simulation), the type of demand model used (distributed or proportional), the type of optimization method used (linear programming, dynamic programming, or nonlinear programming), and the nature of the resulting control policy (implicit or explicit) Advantages and disadvantages of each approach are presented, along with recommendations for future work The applicability of current technology to an existing water‐

Optimal Control of Water Supply Pumping Systems

For more than forty years, approximate solutions for the classical pipe network analysis problem have been obtained by direct solution of the nonlinear stationary point conditions. We propose a revolutionary new approach involving optimization techniques for solving this well-known engineering problem. It is shown that the pipe network analysis problem may be described mathematically in terms of a nonlinear convex cost network flow problem. Three mathematical programming algorithms for solving this problem have been coded and are computationally compared with a code using the traditional Newton-Raphson technique. The computational experience demonstrates that this new approach provides an attractive alternative for solving this important problem.

Solving the Pipe Network Analysis Problem Using Optimization Techniques

A complete algorithm for the computer solution of steady-state fluid flows in networks is given. Particular stress is placed on fast solution, minimal storage requirements and simplicity of the input data. Although the Hardy Cross method is the classical method of solution of this type of problem, convergence is slow for large networks. To overcome this problem, the whole network is considered simultaneous, and this produces a large system of non-linear equations. Newton's method is applied, which results in an iterative solution of a system of linear equations. In order to reduce computer storage requirements and to simplify the data input, a number of algorithms from graph theory are involved. The resulting matrix of coefficients associated with the system of linear equations is banded and symmetric for which efficient (in time and memory requirements) methods of solution exist.

Closure to “Efficient Code for Steady-State Flows in Networks”

Efficient Code for Steady-State Flows in Networks

Estimating modeling parameters based on a prescribed optimization target requires to solve an inverse problem, which is commonly ill-posed. Consequently, either infinitely many or no solutions may exist, depending on whether the system is under- or overdetermined, and whether it is consistent or inconsistent. This paper focuses on scenarios where the solution is ambiguous and infinitely many combinations of possible parameter values can accurately achieve the optimization target. Selecting the most suitable solution requires incorporating additional constraints into the model, which is achieved by regularizing the inverse problem. However, common regularization approaches require the specification of a priori unknown regularization hyperparameters that are difficult and tedious to obtain, and can have a large impact on the result. Here, a novel strategy to reduce the ambiguity of such inverse problems is presented, ensuring that the primary optimization target is always reached accurately. To further reduce the solution space, additional constraints are included, until the optimal modeling parameters are found. Importantly, the required regularization parameters have a direct physical meaning and can be derived sequentially, starting from an initial guess that can be obtained conveniently by solving the system without regularization. By considering several illustrative examples, the applicability of the method is demonstrated, and its potential for various comparable inverse problems is highlighted. 

Robert Epp

Papers

Closure to “Efficient Code for Steady-State Flows in Networks”

Efficient Code for Steady-State Flows in Networks

Hierarchical regularization of solution ambiguity in underdetermined inverse and optimization problems

Fast convergence strategy for ambiguous inverse problems based on hierarchical regularization