[Paper Review] Synchronization and Aggregation of Nonlinear Power Systems with Consideration of Bus Network Structures

Petar Mlinarić, Takayuki Ishizaki+4 more|arXiv (Cornell University)|Mar 1, 2018

Nonlinear Dynamics and Pattern Formation9 references3 citations

TL;DR

This paper introduces a synchronization framework for nonlinear power systems by modeling generators and their associated buses as joint dynamical units, linking synchronization to graph symmetry and equitable partitions. It proposes an exact, structure-preserving model order reduction via aggregation when synchronization conditions are met, yielding a reduced power system that preserves physical dynamics and network structure.

ABSTRACT

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power system.

Motivation & Objective

• To develop a synchronization concept for generator-bus pairs in nonlinear power systems using graph-theoretic tools.
• To extend synchronization to partitions of generators and relate it to equitable partitions and graph symmetry.
• To enable exact model order reduction through aggregation of synchronized generators and their buses.
• To ensure the reduced model remains a valid power system with preserved structure and dynamics.
• To provide a framework for simulating or controlling specific grid components or phenomena via reduced-order models.

Proposed method

• Introduces a joint state representation combining generator voltage angles and amplitudes with their associated bus voltages.
• Defines synchronization between generator-bus pairs using symmetry in the Kron-reduced system's Laplacian matrix.
• Extends synchronization to partitions using equitable partition theory, linking it to symmetry in the system's matrix structure.
• Derives exact aggregation rules by exploiting invariance under the partition, ensuring the reduced system matches the original dynamics.
• Uses matrix decomposition and projection techniques (e.g., Schur complement) to derive the reduced system's dynamics.
• Validates the method via simulation on a 5-bus system with different partitions and initial conditions.

Experimental results

Research questions

• RQ1 How can synchronization be defined for a pair of generators and their associated buses in a nonlinear power system?
• RQ2 What graph-theoretic conditions (symmetry or equitable partition) ensure synchronization across a partition of generators?
• RQ3 Under what conditions can the full power system be exactly reduced via aggregation of synchronized generators and buses?
• RQ4 How does the reduced system preserve the physical structure and dynamics of the original power system?
• RQ5 Can the aggregation framework be applied to arbitrary partitions and initial conditions, even when weak synchronization does not hold?

Key findings

• Synchronization of generator-bus pairs is equivalent to symmetry in the Kron-reduced Laplacian matrix of the network.
• Two notions of partition-based synchronization are equivalent to the system's Laplacian being symmetric or forming an equitable partition.
• When the system is synchronized with respect to a partition, exact model order reduction is possible via aggregation of generators and their buses.
• The reduced system is itself a valid power system, preserving the structure and dynamics of the original network.
• Simulations show the reduced model matches the steady-state behavior and approximates transient dynamics even when weak synchronization does not hold.
• The aggregation framework is applicable to arbitrary partitions and initial conditions, though error bounds for transient approximation remain an open problem.