Increasing concerns on the security and quality of water distribution systems (WDS), call for computational tools with performance guarantees. To this end, this work revisits the physical laws governing water flow (WF) and provides a hierarchy of solvers of complementary value. Given the water injection or pressure at each WDS node, finding the WFs within pipes and pumps along with the pressures at all WDS nodes, constitutes the WF problem. The latter entails solving a set of (non)-linear equations. We extend uniqueness claims on the solution to the WF equations in setups with multiple fixed-pressure nodes and detailed pump models. For networks without pumps, the WF solution is already known to be the minimizer of a convex function. The latter approach is extended to networks with pumps but not in cycles, through a stitching algorithm. For networks with nonoverlapping cycles, a provably exact convex relaxation of the pressure drop equations yields a mixed-integer quadratically constrained quadratic program (MI-QCQP) solver. A hybrid scheme combining the MI-QCQP with the stitching algorithm can handle a WDS with overlapping cycles, but without pumps on them. Each solver is guaranteed to converge regardless of initialization, as numerically validated on a benchmark WDS.

中文翻译：

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配水网络中的水流问题：唯一性和求解器

对水分配系统（WDS）的安全性和质量的关注日益增加，因此需要具有性能保证的计算工具。为此，这项工作重新审视了控制水流量（WF）的物理定律，并提供了具有互补价值的求解器层次结构。给定每个WDS节点处的注水量或压力，在管道和泵内找到WF以及所有WDS节点处的压力都会构成WF问题。后者需要求解一组（非线性）方程。在具有多个固定压力节点和详细的泵模型的设置中，我们将解决方案的唯一性要求扩展到WF方程。对于没有泵的网络，WF解决方案是凸函数的最小化方法。通过拼接算法，后一种方法扩展到具有泵的网络，但不能周期性地扩展。对于具有非重叠周期的网络，压降方程的可证明的精确凸松弛会产生混合整数二次约束二次规划（MI-QCQP）求解器。将MI-QCQP与拼接算法结合在一起的混合方案可以处理具有重叠循环的WDS，但循环上没有泵。在基准WDS上进行了数值验证，保证了每个求解器都可以收敛，而无需初始化。