Stability And Uncertainty Propagation In Power Networks: A Lyapunov-based Approach With Applications To Renewable Resources Allocation

TL;DR

This work addresses stability and uncertainty propagation in power networks with intermittent renewable injections by formulating a model-based NL-DAE representation that retains both differential and algebraic dynamics. It develops a Lyapunov-spectrum stability framework that computes the full set of Lyapunov exponents via a discrete-time, QR-based method, enabling a parameterized deformation matrix to quantify how renewables perturb one node and propagate uncertainty across the network. A log-determinant stability measure ties the deformation to the sum of Lyapunov exponents, leading to a submodular, greedy optimization (P1) that identifies stable nodes and optimally locates renewable injections to minimize global instability. The framework is validated on 4th- and 9th-order networks, including PV integration scenarios, showing faster transient damping when renewables are allocated to stable nodes and establishing the practical utility for stability-guided renewable planning in real power systems.

Abstract

The rapid increase in the integration of intermittent and stochastic renewable energy resources (RER) introduces challenging issues related to power system stability. Interestingly, identifying grid nodes that can best support stochastic loads from RER, has gained recent interest. Methods based on Lyapunov stability are commonly exploited to assess the stability of power networks. These strategies approach quantifying system stability while considering: (i) simplified reduced order power system models that do not model power flow constraints, or (ii) data-driven methods that are prone to measurement noise and hence can inaccurately depict stochastic loads as system instability. In this paper, while considering a nonlinear differential algebraic equation (NL-DAE) model, we introduce a new method for assessing the impact of uncertain renewable power injections on the stability of power system nodes/buses. The identification of stable nodes informs the operator/utility on how renewables injections affect the stability of the grid. The proposed method is based on optimizing metrics equivalent to the Lyapunov spectrum of exponents; its underlying properties result in a computationally efficient and scalable stable node identification algorithm for renewable energy resources allocation. The developed framework is studied on various standard power networks.

Figures (7)

• Figure 1: Trajectory of nearby orbits starting from initial state $\boldsymbol{x}_0$ and perturbed initial state $\boldsymbol{x}_{0}+\boldsymbol{\delta}\boldsymbol{x}_{0}$: $(a)$ converging and $(b)$ diverging trajectories.
• Figure 2: Lyapunov spectrum of exponents for (a.1 and a.2) $\mathrm{case}$-$\mathrm{9}$ (9-buses) and (b.1 and b.2) $\mathrm{case}$-$\mathrm{200}$ (200-buses). Column one depicts the spectrum of LEs computed for all the system states. Column two depicts the LEs for each bus within the network, i.e., the stability index.
• Figure 3: Nodes from most to least stable obtained by solving $\mathrm{\textbf{P1}}$ for networks (a) $\mathrm{case}$-$\mathrm{9}$, (b) $\mathrm{case}$-$\mathrm{39}$ and (c) $\mathrm{case}$-$\mathrm{200}$. The square nodes represent generator buses while the circles represent loads/renewables buses.
• Figure 4: Clearing time for frequency transients induced by allocating an uncertain renewable load injection at a random bus (a, c, e) and at the most stable bus (b, d, f), computed by solving $\mathrm{\textbf{P1}}$ for each case system.
• Figure 5: One-line diagram of the modified WSCC power system Roy2023 ($\mathrm{case}$-$\mathrm{9PV}$). It includes a motor load at Bus 8, a synchronous generator at Bus 1, and two PV plants $\mathrm{S} 1$ and $\mathrm{S} 2$ at Buses 2 and 3.

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Remark 3
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 4