Non Uniqueness of Power-Law Flows

Plane potential flows simplify fluid motion by assuming the fluid to be irrotational and incompressible. These characteristics allow these flows to be described by a velocity potential function, ϕ, representing the flow speed in a given direction, and a stream function, ψ, that visualizes the flow path, both governed by Laplace's equation. These parameters help in estimating flow patterns, velocity distributions, and pressure fields around various hydraulic structures.
Uniform...

Any fluid in a horizontal tube can flow due to pressure differences—fluid flows from high to low pressure. The flow rate (Q) is the ratio of pressure difference and resistance through a horizontal tube. The greater the pressure difference, the higher the flow rate. The flow resistance is expressed as:

Fluid flows are categorized by dimensionality and behavior, with one-dimensional flow being the simplest form, where properties like velocity and pressure change only along a single axis. Water moving through straight pipes exemplifies this flow type, as variations in other directions are minimal. One-dimensional analysis helps simplify understanding such flows, focusing solely on changes along the pipe's length.
Two-dimensional flow involves changes in both length and height, as seen in...

Power flow problem analysis is fundamental for determining real and reactive power flows in network components, such as transmission lines, transformers, and loads. The power system's single-line diagram provides data on the bus, transmission line, and transformer. Each bus k in the system is characterized by four key variables: voltage magnitude Vk​, phase angle δk​, real power Pk​, and reactive power Qk​. Two of these four variables are inputs, while the...

The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the...