[Paper Review] DC power grids with constant-power loads -- Part I: A full characterization of power flow feasibility, long-term voltage stability and their correspondence

Mark Jeeninga, Claudio De Persis+1 more|arXiv (Cornell University)|Oct 2, 2020

Optimal Power Flow Distribution4 citations

TL;DR

This paper presents a comprehensive characterization of power flow feasibility and long-term voltage stability in DC power grids with constant-power loads. By establishing a one-to-one correspondence between feasible power demands and long-term voltage semi-stable operating points, it proves convexity of the feasible power demand set and provides an LMI-based necessary and sufficient condition for feasibility under small perturbations, using a parametrized solution to an initial value problem to trace feasible operating points across the entire feasible region.

ABSTRACT

In this two-part paper we develop a unifying framework for the analysis of the feasibility of the power flow equations for DC power grids with constant-power loads. In Part I of this paper we present a detailed introduction to the problem of power flow feasibility of such power grids, and the associated problem of selecting a desirable operating point which satisfies the power flow equations. We introduce and identify all long-term voltage semi-stable operating points, and show that there exists a one-to-one correspondence between such operating points and the constant power demands for which the power flow equations are feasible. Such operating points can be found by solving an initial value problem, and a parametrization of these operating points is also obtained. In addition, we give a full characterization of the set of all feasible power demands, and give a novel proof for the convexity of this set. Moreover, we present a necessary and sufficient LMI condition for the feasibility of a vector of power demands under small perturbation, which extends a necessary condition in the literature.

Motivation & Objective

• To develop a unifying framework for analyzing power flow feasibility in DC power grids with constant-power loads.
• To identify and characterize all long-term voltage semi-stable operating points in such systems.
• To establish a one-to-one correspondence between feasible power demands and these stable operating points.
• To prove the convexity of the set of feasible power demands using a novel approach.
• To derive a necessary and sufficient LMI condition for feasibility under small perturbations.

Proposed method

• Formulates the power flow equations for DC grids with constant-power loads and defines the feasible set of power demands.
• Introduces a parametrized initial value problem to trace operating points from a known stable solution to the boundary of the feasible region.
• Uses the inverse of the Jacobian of the power function to define a vector field that guides the path of feasible solutions.
• Applies results from ordinary differential equations theory to prove existence and uniqueness of the solution path within the domain of feasible voltages.
• Employs convex analysis and generalized quadratic forms to prove the convexity of the feasible power demand set.
• Derives a necessary and sufficient linear matrix inequality (LMI) condition for feasibility under small perturbations, extending prior results.

Experimental results

Research questions

• RQ1 What is the complete set of feasible power demands for a DC power grid with constant-power loads?
• RQ2 How are long-term voltage semi-stable operating points related to feasible power demands?
• RQ3 Is the set of feasible power demands convex, and if so, how can this be proven?
• RQ4 What conditions ensure that a given power demand vector is feasible under small perturbations?
• RQ5 Can all feasible operating points be systematically traced from a known stable solution?

Key findings

• There exists a one-to-one correspondence between feasible power demand vectors and long-term voltage semi-stable operating points in DC power grids with constant-power loads.
• The set of all feasible power demands is convex, and this is proven via properties of the non-homogeneous numerical range of a generalized quadratic form.
• A novel necessary and sufficient LMI condition is derived for the feasibility of a power demand vector under small perturbations, extending previous necessary conditions.
• All feasible operating points can be uniquely traced via a solution to an initial value problem starting from a known stable voltage profile.
• The solution path remains within the domain of feasible voltages for the entire interval [0,1], ensuring that any feasible demand vector lies within the image of the power function over the feasible voltage set.
• The method guarantees that the entire feasible region is spanned by a continuous, unique path, enabling systematic exploration of all stable operating points.