Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Approximate Linear Power Flow Using Logarithmic Transform of Voltage Magnitudes With Reactive Power and Transmission Loss Consideration

Abstract: Alternating current (ac) power flow (PF) presents difficulties for power system analysis and optimization due to its nonlinearity. Progress has been made to approximately linearize ac PF in recent decades. However, few studies have reported the simultaneous accurate approximation of reactive power and transmission losses. To bridge this gap, this paper investigates the linear approximation of ac PF considering the accuracy of the reactive load flows and transmission losses. Using the logarithmic transform of voltage magnitudes, a linear PF (LPF) model involving tap changers and phase shifters is derived from the approximation analysis of general branch flows. Transmission power loss and loss-concerned complex branch flow are also formulated. Cold-start and warm-start LPF calculation methods associated with injection compensation are also developed. Numerical simulations are performed to compare the proposed models and several state-of-the-art LPF models using 25 practical-scale test systems. The simulation results demonstrate the advantages of the proposed model over the other models for approximating voltage magnitudes, branch flows, and power losses. The effectiveness of using proper compensation injection in improving the solution accuracy is also verified.

Higher order linear approximations to nonlinear control systems

Abstract: One traditional approach in the analysis and design of nonlinear control systems is a first order approximation by a linear system. A new approach is to use nonlinear change of coordinates and feedback to construct linear approximations that are accurate to second and higher orders. However, the algebraic calculations required to obtain these aproximations are somewhat lengthy. In this paper, the theoretical framework for finding such change of coordinates for a nonlinear system are described. A software package that symbolically solves these transformations is currently being prepared.

Piecewise-Linear Approximations of Multidimensional Functions

Abstract: We develop explicit, piecewise-linear formulations of functions f(x):ℝ n ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.