⚡ electrical engineering

Analytical Framework for Power System Strength

This paper proposes a general analytical framework for evaluating power system strength by introducing twelve indicators across three dynamical orders, utilizing a novel Delta operator and complex frequency concept to quantify voltage phasor resistance to current injection changes across various system devices.

Original authors: Ignacio Ponce, Federico Milano

Published 2026-05-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ignacio Ponce, Federico Milano

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the electrical grid not as a complex web of wires and machines, but as a giant, invisible trampoline. In this analogy, voltage is the height of the trampoline's surface, and current is the weight of a person jumping on it.

The paper you shared is about a new way to measure how "strong" this trampoline is. In the real world, engineers need to know: If a heavy weight (a sudden change in power demand or a fault) hits the grid, how much will the voltage bounce? Will it wobble wildly, or will it stay steady?

Here is a breakdown of the paper's ideas using simple analogies:

1. The Problem: We Need a Better "Strength" Test

Traditionally, engineers have tried to measure grid strength using two separate rulers:

Voltage Strength: How much does the height of the trampoline dip when someone jumps?
Frequency Strength: How much does the speed of the trampoline's bounce change?

The authors argue that these two rulers are actually measuring the same object from different angles. They propose a single, unified "super-ruler" that measures both at the same time. They call this System Strength.

2. The New Tool: The "Delta Operator" (The Jump Detector)

To measure strength, the authors invented a new mathematical tool they call the Delta Operator.

Think of a video camera filming a trampoline.

Normal cameras (standard math) look at the smooth motion between frames.
The Delta Operator is like a camera that can freeze time exactly at the moment a heavy weight hits the trampoline. It measures the instantaneous "jump" or "shock" that happens in a split second before the trampoline starts to settle down.

This is crucial because modern power grids (with solar panels and wind turbines) can react instantly to shocks, unlike old-school power plants (synchronous machines) which react more slowly. The new tool captures these instant "jumps" that older math missed.

3. The Three Levels of Strength

The paper suggests that to truly understand how strong the grid is, you need to look at three different "orders" of reaction, like looking at a car crash from three different perspectives:

Level 1: The Immediate Bump (Zero-Order)
- Analogy: How far does the trampoline surface sink the exact moment the weight lands?
- What it measures: The immediate drop in voltage. This is similar to the old "Short-Circuit Ratio" but more precise.
Level 2: The First Wobble (First-Order)
- Analogy: How fast does the trampoline start to move immediately after the impact? Does it jerk up or down instantly?
- What it measures: The sudden change in the speed of the voltage (frequency). The paper found that in grids with many solar/wind inverters, this "jerk" can be much sharper than in traditional grids.
Level 3: The Acceleration of the Wobble (Second-Order)
- Analogy: How quickly is that "jerk" itself speeding up or slowing down?
- What it measures: The rate of change of the frequency. This helps predict if the grid will stabilize quickly or start shaking violently.

4. The "Complex Frequency" Concept

The authors use a concept called Complex Frequency.

Analogy: Imagine a spinning top.
- Real Frequency: How fast it spins.
- Imaginary Frequency: How fast it is wobbling or changing size.
The paper combines these two into one single number (a "complex" number) to describe the voltage's behavior. This allows them to measure the "strength" of the grid in a way that captures both the spin and the wobble simultaneously.

5. Testing the Theory

The authors tested their new "super-ruler" on a standard model of a power grid (the IEEE 39-bus system) with three different setups:

Old School: All traditional spinning generators (like big steam turbines).
Mixed: Some traditional generators replaced by solar/wind inverters.
Future: 100% solar/wind inverters.

The Results:

In the Old School grid, the voltage didn't "jump" instantly when disturbed; it moved smoothly. The "First-Order" strength was effectively infinite (no sudden jumps).
In the Future grid (100% inverters), the voltage and frequency did jump instantly. The new tool successfully measured these sharp "jumps," proving that the grid is "weaker" (more sensitive to sudden shocks) in this scenario.

Summary

This paper doesn't just say "the grid is weak." It provides a mathematical toolkit to measure exactly how and where the grid is weak.

It unifies voltage and frequency into one concept.
It introduces a new math trick (the Delta Operator) to catch instant "shocks."
It breaks strength down into three layers: the immediate dip, the initial jerk, and the acceleration of that jerk.

The authors emphasize that this is a measurement tool, not a stability guarantee. Knowing a trampoline is "weak" (sensitive to jumps) doesn't mean it will break, but it tells you exactly how it will react when someone jumps on it, allowing engineers to design better safety nets.

Technical Summary: Analytical Framework for Power System Strength

Problem Statement
The concept of "system strength" refers to a power system's resistance to disturbances, typically evaluated through voltage magnitude and frequency stability. However, quantifying this ability remains an open question. Existing metrics often rely on strong approximations, empirical equations, or separate analyses for voltage magnitude (e.g., Short-Circuit Ratio, SCR) and frequency (e.g., system inertia). These approaches frequently fail to capture the spatial distribution of inertia, the effects of neighboring converters, or the unified nature of voltage as a three-phase AC entity comprising both magnitude and phase. Furthermore, current methods often lack a rigorous analytical derivation that accounts for the "jumps" in algebraic variables during discrete events, such as faults or sudden current injections.

Methodology
The paper proposes a general, purely analytical framework to evaluate power system strength based on the response of bus voltage phasors to sudden changes in current injection. The methodology is built upon three core mathematical foundations:

Complex Frequency (CF): The framework utilizes the concept of complex frequency ( $\bar{\eta}$ ), which unifies the rate of change of voltage magnitude ( $\rho$ ) and phase angle ( $\omega$ ). It extends this to a second-order complex frequency ( $\bar{\eta}''$ ) to capture the acceleration of these dynamics.
Delta Operator ( $\Delta$ ): A novel finite-difference operator is introduced to quantify the "jumps" in algebraic variables (discontinuities) at the instant of a disturbance ( $t^+$ vs. $t^-$ ). This operator allows for the exact handling of discontinuities in differential-algebraic equations (DAEs) without resorting to small-signal linearization.
Generalized Formulation: The system is modeled as a set of DAEs. The framework derives sensitivity matrices ( $S$ $S$ , $S'$ $S^{'}$ , $S''$ $S^{''}$ ) that relate changes in current injection ( $\Delta \mathbf{i}_{pq}$ $Δ i_{pq}$ ) to changes in:
- Zero-order: The voltage vector itself ( $\Delta \mathbf{v}_{v\theta}$ ).
- First-order: The first-order complex frequency ( $\Delta \bar{\eta}'$ ).
- Second-order: The second-order complex frequency ( $\Delta \bar{\eta}''$ ).

The derivation involves transforming the network equations and device models (synchronous machines, converters, loads) into a form where the sensitivity of the voltage to current steps can be calculated analytically. This results in twelve specific indicators per bus, grouped into three dynamical orders.

Key Contributions
The paper makes three primary contributions:

A General Analytical Framework: It provides a unified formulation to evaluate system strength in both steady-state and dynamic conditions, applicable to any system device model.
Systematic Device Modeling: It offers a methodology to derive specific strength expressions for diverse devices, including synchronous machines (SMs), grid-following (GFL) converters, grid-forming (GFM) converters, and constant impedance loads.
The Delta Operator: It defines a new mathematical operator and its properties (linearity, multiplication, division, chain rule) specifically designed to handle discontinuities in power system variables, ensuring the formulation remains exact for the given system model.

Results
The framework was validated using the IEEE 39-bus system under three scenarios:

Conventional Synchronous Generators: The results confirmed that in a system dominated by SMs, the first-order complex frequency is continuous (no jump), while zero-order strength is driven by network topology and internal impedance, and second-order strength is driven by local inertia.
Grid-Following (GFL) Converters: Replacing SMs with GFLs introduced discontinuities in the first-order complex frequency (jumps in $\rho$ and $\omega$ ), which the framework accurately predicted.
100% Inverter-Based Resources (IBRs): A system with a mix of GFL and GFM converters showed significantly higher jumps and rates of change in the complex frequency, driven by the control dynamics (e.g., power-frequency droop) of the converters.

In all cases, the analytical predictions of the initial jumps and rates of change (Predicted Rate of Change - PRoC) matched time-domain simulation (TDS) results with high precision, validating the exactness of the formulation.

Significance and Claims
The authors claim that this framework provides a "unifying" approach that bridges the gap between voltage magnitude and frequency strength assessment. By avoiding approximations and small-signal assumptions, the formulation is "exact" provided the underlying system model is exact.

Crucially, the paper maintains a modest stance regarding the relationship between strength and stability. The authors explicitly state that system strength and system stability are "fundamentally decoupled" concepts; a weak system is not necessarily unstable, and vice versa. Therefore, the framework is presented as a tool for quantifying resistance to perturbations, not as a direct surrogate for stability analysis. The paper concludes that future work is required to establish the specific links between these new metrics and the occurrence of instabilities, as well as to define security thresholds based on statistical analysis of real-world systems.