  Optimal design of water distribution networks, which are governed by a series
of linear and nonlinear equations, has been extensively studied in the past
decades. Due to their NP-hardness, methods to solve the optimization problem
have changed from traditional mathematical programming to modern intelligent
optimization techniques. In this study, with respect to the model formulation,
we have demonstrated that the network system can be reduced to the
dimensionality of the number of closed simple loops or required independent
paths, and the reduced nonlinear system can be solved efficiently by the
Newton-Raphson method. Regarding the optimization technique, a discrete state
transition algorithm (STA) is introduced to solve several cases of water
distribution networks. In discrete STA, there exist four basic intelligent
operators, namely, swap, shift, symmetry and substitute as well as the "risk
and restore in probability" strategy. Firstly, we focus on a parametric study
of the restore probability $p_1$ and risk probability $p_2$. To effectively
deal with the head pressure constraints, we then investigate the effect of
penalty coefficient and search enforcement on the performance of the algorithm.
Based on the experience gained from the training of the Two-Loop network
problem, the discrete STA has successfully achieved the best known solutions
for the Hanoi and New York problems. A detailed comparison of our results with
those gained by other algorithms is also presented.
