Tianran Chen
Department of Mathematics
Auburn University at Montgomery
Oct. 4, 2020
AMS Fall Eastern Virtual Sectional Meeting
Joint work with Robert Davis (Colgate University)
Power networks carry alternating current (AC) electric power and make modern life possible.
Power-flow equations describes the intricate balancing conditions on "buses" of a AC power network derived from Kirchhoff's circuit laws.
Their solutions, i.e.
"power-flow solutions",
corresponds power network's state of operation
(including physically impossible states).
We can model an AC power network as graph $G$ whose vertices represent "buses" and edges represent junctions.
Three basic types of buses
PV and PQ buses result in very different formulations.
Here we only focus on a much simplified formulation in which all non-reference buses are modeled as PQ buses, and no electric energy is lost in the network.
Derived from Kirchhoff's circuit laws, the PQ-type power-flow equations for power network $G$ consisting of $N$ buses is a system of $N-1$ equations
\[ S_i = \sum_{j \in N_G(i)} \overline{Y}_{ij} v_i \overline{v}_j \quad\text{for } i = 2,\dots,N \]
in the $N-1$ complex variables $v_2, \dots, v_N$ with $v_1 = 1$ , where $\overline{v}_j$ denotes the complex conjugate of $v_j$
\[ S_i = \sum_{j \in N_G(i)} \overline{Y}_{ij} v_i \overline{v}_j \quad\text{for } i = 2,\dots,N \]
One of the central object in "power-flow study" in the analysis of power networks.
Similar systems also appear in the study of wind turbine farm operations and other applications.
Closely related to Kuramoto equations, reactive power balancing equations for solar farms, and bilinear TL-type consensus network equations.
\[ S_i = \sum_{j \in N_G(i)} \overline{Y}_{ij} v_i \overline{v}_j \quad\text{for } i = 2,\dots,N \]
Despite the simplicity, these questions are not trivial.
Algebraic geometry approaches to PQ-type power-flow:
\[ S_i = \sum_{j \in N_G(i)} \overline{Y}_{ij} v_i \overline{v}_j \quad\text{for } i = 2,\dots,N \]
How many nonzero complex solutions can there be?
J. Baillieul and C. Byrnes (1982)
Geometric Critical Point Analysis of Lossless Power System Models
T.-Y. Li, T. Sauer, and J. Yorke (1987)
Numerical Solution of a Class of Deficient Polynomial Systems
PQ-type power-flow equations: Upper bound on the number of complex solutions is $\binom{2(N-1)}{N-1}$
For sparse networks the upper bound for the number of power-flow solutions depends on network topology.
We focused on an algebraic/convex/tropical geometry approach to certain aspects of this problem.
Theorem.
(R. Davis and Chen, 2020)
arXiv:2007.11051
Given a network $G$,
the maximum number of power-flow solutions on this network
is bounded above by the number of
$D(G)$-draconian sequences
derived from $G$.
Definition. For a network/graph $G$ with vertices $1,\dots,N$. We call $(a_1,\dots,a_N) \in \mathbb{Z}_{\geq 0}^N$ a $D(G)$-draconian sequence if $\sum a_i = N-1$ and, for any $1 \leq i_1 < \cdots < i_k \leq N$, \[ a_{i_1} + \cdots + a_{i_k} < \left|\{i_1,\dots,i_k\} \cup \left(\bigcup_{j=1}^k \mathcal{N}_G(i_j)\right)\right| \]
What's the point?
Another direction of our project is to study power-flow equations from a new "tropical" point of view.
Tropical geometry studies polynomials by turning them into piece-wise linear functions.
\[ \operatorname{Trop}( a x^3 + b x^4 y^5 ) = \min \{ A + 3x \,,\, B+4x+5y \, \} \] where $A$ and $B$ are the "valuation" of $a$ and $b$.
Definition. The tropical hypersurface defined by a tropical polynomial $\operatorname{Trop}(f)(x_1,\dots,x_n)$ is the set of all points in $\mathbb{R}^n$ such that the minimum in $\operatorname{Trop}(f)(x_1,\dots,x_n)$ is attained at least twice.
This is the widely used definition, but it is not sufficient in our study. Motivated by the PQ-type power-flow equations, we define the concept of generalized tropical hypersurface .
Definition. The tropical hypersurface of type $t$ defined by a tropical polynomial $\operatorname{Trop}(f)(x_1,\dots,x_n)$ is the set of all points in $\mathbb{R}^n$ such that the minimum in $\operatorname{Trop}(f)(x_1,\dots,x_n)$ is attained at least $t+1$ times.
The original definition of a tropical hypersurface is a special case of this definition with type $t=1$.
Generalized intersection of $n$ tropical hypersurfaces $(H_1,\dots,H_N)$ of type $(t_1,\dots,t_N)$ is their set-theoretic intersection in $\mathbb{R}^N$.
Theorem. (T. Chen & R. Davis) Counting multiplicity, the power-flow solutions are in one-to-one correspondence with a subset of the generalized intersections of tropical hypersurfaces defined by $L_i$ of all possible types $(t_1,\dots,t_N) \in \mathbb{Z}_0^N$ with $\sum_{i=1}^N t_i = N$, where \[ L_{i}(z_1,\dots,z_N) = z_i + \sum_{j \in \mathcal{N}_(i)} c_{ij} \, z_j. \]
We focused on the PQ-type power-flow equations \[ S_i = \sum_{j \in N_G(i)} \overline{Y}_{ij} v_i \overline{v}_j \quad\text{for } i = 2,\dots,N \]
My collaborators on this and related projects: Rob Davis, Evgeniia Korchevskaia, Wuwei Li, Jakub Marecek, Dhagash Mehta, Dan Molzahn, Matthew Niemerg.
My advisor Tien-Yien Li (1945-2020).
This research is supported, in part, by NSF and AUM grant-in-aid program
ti@nranchen.org
http://www.tianranchen.org/