⚡ electrical engineering

Localization and Reshaping of Non-Minimum-Phase Zeros in Multi-Converter Systems

This paper proposes a Jacobian-based framework to quantify non-minimum-phase zeros in multi-converter systems using only grid admittance and power injection data, and introduces a zero reshaping strategy that optimally deploys voltage droop to steer these zeros away from the origin, thereby enhancing stability margins without requiring internal converter models.

Original authors: Ailixier Yaermaimaiti, Jiaxin Wang, Yunjie Gu, Huanhai Xin

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ailixier Yaermaimaiti, Jiaxin Wang, Yunjie Gu, Huanhai Xin

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 a modern power grid not as a giant machine, but as a symphony orchestra. In the past, the "musicians" were big, heavy, predictable power plants (like steam turbines). Today, we are replacing them with hundreds of smaller, faster, and more flexible musicians: solar panels, wind turbines, and battery storage systems. These are all connected to the grid through electronic devices called converters.

The problem? When you have too many of these fast, electronic musicians playing together in a weak room (a "weak grid"), they start to trip over their own feet. They create a hidden trap called a Non-Minimum-Phase (NMP) Zero.

Here is a simple breakdown of what this paper does, using everyday analogies.

1. The Problem: The "Ghost in the Machine"

In a normal system, if you push a button (change the power), the machine responds immediately. But in these modern converter systems, there is a weird glitch: when you push, the system sometimes pulls back first.

The Analogy: Imagine trying to steer a car that has a broken steering wheel. You turn the wheel to the right to go right, but for a split second, the car jerks violently to the left before correcting itself.
The Consequence: This "wrong-way" reaction creates a bandwidth ceiling. It's like a speed limit sign that says, "You cannot drive faster than 10 mph, or you will crash." No matter how good your driver (the controller) is, they cannot go faster than this limit without the car spinning out of control.

2. The Mystery: Why Can't We See the Problem?

Usually, to fix a car, you need the engine manual. But these power converters are "black boxes." The companies that make them keep their internal software secrets. Power grid operators can see the outside (voltage and current), but they can't see the engine.

The Analogy: Imagine trying to tune a piano, but you aren't allowed to look inside the piano or touch the strings. You can only listen to the sound.
The Paper's Breakthrough: The authors realized they don't need to see inside the piano. They found that the "ghost" (the NMP zero) is actually determined entirely by the room the piano is in (the grid's wiring and how much power is flowing), not by the piano itself.

3. The Solution: The "X-Ray Vision"

The authors developed a new mathematical tool (a "Jacobian-based framework") that acts like X-ray vision.

How it works: Instead of needing the secret engine manuals, they use a simple formula based on two things the grid operators already know:
1. The layout of the wires (Grid Admittance).
2. How much power is currently flowing (Steady-state injections).
The Result: They can calculate exactly where the "speed limit" is and how close the system is to crashing, without ever opening a single converter. They proved these "ghosts" are always real numbers, making them predictable.

4. The Danger Zone: The "Exponential Cliff"

The paper explains a scary relationship: The closer the "ghost" gets to the center of the room (zero), the more dangerous the system becomes.

The Analogy: Imagine a tightrope walker. If the wind (the NMP zero) is far away, they are fine. But as the wind gets closer to their feet, the chance of falling doesn't just get a little worse; it gets exponentially worse. A tiny shift in the wind can cause a massive crash.
The Metric: They created a formula showing that as this "ghost" moves closer to zero, the system's tolerance for errors drops to near zero. The system becomes incredibly fragile.

5. The Fix: "Reshaping" the Ghost

Since we can't change the secret software inside the converters, how do we move the "ghost" away from the danger zone?

The Strategy: The authors propose adding a specific type of control called Voltage Droop. Think of this as giving the musicians a new rule: "If the room gets too loud, turn your volume down slightly."
The Magic Trick: You can't just add this rule to any musician. If you add it to the wrong one, nothing happens. If you add it to the right one, the "ghost" moves far away, and the speed limit goes up.
The "Participation Factor" Ranking: The paper provides a simple ranking list. It calculates which specific converter is most "connected" to the problem.
- The Analogy: Imagine a group of people trying to push a heavy boulder. If you push the boulder from the back, it moves. If you push the guy standing next to the boulder, nothing happens. The paper tells you exactly who is standing on the boulder so you know who to push.
- No Guessing: You don't need to try every converter one by one. You just look at the list, pick the top-ranked one, apply the fix, and the system becomes stable again.

Summary

This paper solves a major headache for the future of green energy:

The Problem: Modern power grids have hidden "speed limits" that cause instability, and we can't see them because the equipment is a black box.
The Discovery: These limits are actually caused by the grid itself, not the equipment. We can calculate them using simple data we already have.
The Fix: By adding a specific control to the one converter that matters most (identified by a simple ranking), we can push the danger away and make the grid safe and fast again.

It turns a complex, black-box engineering nightmare into a simple, data-driven checklist for grid operators.

1. Problem Statement

The increasing penetration of Voltage Source Converters (VSCs) in modern power systems introduces significant small-signal stability challenges, particularly in weak grids. A critical issue is the emergence of Non-Minimum-Phase (NMP) zeros in the open-loop transfer functions of grid-tied converters.

Impact: NMP zeros impose a hard "bandwidth ceiling" on feedback control. According to Bode integral theory, pushing the control bandwidth closer to the magnitude of an NMP zero causes the peak magnitude of the complementary sensitivity function ( $M_T$ ) to grow exponentially. This degrades the system's tolerance to modeling uncertainties and reduces stability margins.
Challenges in Multi-Converter Systems:
1. Global Dependence: Unlike single-converter systems where zeros are local, NMP zeros in multi-converter systems depend globally on the network admittance matrix, nodal voltages, and power injections. A change in one converter affects zeros at distant buses.
2. Black-Box Barrier: Commercial converters are proprietary. System operators lack access to internal state-space models, making traditional system-level zero analysis impossible.
3. Lack of Reshaping Strategy: There is no systematic method to determine which converter node requires control modification to effectively shift critical NMP zeros away from the origin (rightward shift) to restore stability.

2. Methodology

The authors propose a Jacobian-based framework that decouples NMP zeros from individual converter dynamics, allowing for analysis using only network data.

A. System Modeling

The system is modeled as a Multiple-Input-Multiple-Output (MIMO) feedback interconnection between a Network Jacobian ( $J_{NET}(s)$ ) and Device Jacobians ( $J_{VSC}(s)$ ).
The network model is derived using Kron reduction to eliminate interior nodes, resulting in a reduced Laplacian matrix ( $B_r$ ) and a complex grid admittance matrix ( $Y$ ).
The open-loop transfer function is defined as $L(s) = J_{NET}(s)J_{VSC}^{-1}(s)$ .

B. Theoretical Derivation

Decoupling: The paper proves that NMP zeros originate exclusively from the network Jacobian ( $J_{NET}(s)$ ), not the device controllers. This is because the device Jacobian is assumed stable, and its inverse does not introduce NMP zeros. Thus, $\det(J_{NET}(z)) = 0$ determines the zeros.
Closed-Form Expression: Using a similarity transformation and Schur complement decomposition, the authors derive an explicit formula for the NMP zeros ( $z_i$ ):
$z_i = \omega_0 \sqrt{\sigma_i^2(B_r^{1/2} D B_r^{1/2}) - 1}$
Where:
- $\sigma_i$ is the $i$ -th singular value.
- $B_r$ is the reduced nodal susceptance matrix.
- $D$ is a diagonal matrix encoding the steady-state operating point (voltage magnitudes, angles, and power injections).
- Key Finding: The NMP zeros are proven to be strictly real.
Stability Bound: An exponential lower bound for the peak complementary sensitivity function ( $M_T$ ) is derived as a function of the dominant (smallest magnitude) NMP zero ( $z_0$ ):
$M_T \geq \exp\left(\frac{\pi \omega_c}{2 z_0}\right)$
This establishes that as $z_0 \to 0$ , $M_T \to \infty$ , inevitably degrading stability.

C. Reshaping Strategy

To move $z_0$ away from the origin, the authors propose deploying voltage droop control ( $\Delta Q_i = -k_i \Delta U_i$ ) at specific nodes.

Sensitivity Analysis: Using the Implicit Function Theorem, the sensitivity of the zero to the droop gain ( $dz_0/dk_i$ ) is derived.
Participation Factor Ranking: The sensitivity is proven to be proportional to the real part of the nodal participation factor ( $Re(p_i)$ ).
Optimal Placement: The optimal node for control deployment is simply the one with the maximum real participation factor, eliminating the need for iterative search or full eigenvalue recomputation.

3. Key Contributions

Black-Box Elimination: Proved that NMP zeros in multi-converter systems are determined solely by grid admittance and steady-state power injections, bypassing the need for proprietary converter models.
Analytical Quantification: Derived a closed-form expression for NMP zeros based on singular values of a matrix constructed from network data, proving they are strictly real.
Stability Metric: Established an exponential lower bound linking the dominant NMP zero to the peak sensitivity magnitude, quantifying the unavoidable stability margin degradation.
Optimal Control Strategy: Developed a non-iterative zero reshaping strategy that identifies the optimal control location based on a scalar ranking of nodal participation factors.

4. Results (Case Studies)

The framework was validated using a modified IEEE 9-bus system with three VSCs in MATLAB/Simulink.

Operating Point Impact: As active power injection increased (Case 1 to Case 3), the dominant NMP zero migrated leftward (from 640 rad/s to 341 rad/s).
- Consequence: The peak sensitivity ( $M_T$ ) increased from 26 dB to 43 dB.
- Stability: The system transitioned from stable (Case 1) to marginally stable (Case 2) and finally unstable (Case 3), confirmed by time-domain oscillations and Generalized Nyquist plots.
Reshaping Effectiveness: In the unstable Case 3 ( $z_0 = 341$ $z_{0} = 341$ rad/s):
- Participation Analysis: Node 3 had the highest real participation factor ( $Re(p_3) = 0.497$ ), while Nodes 1 and 2 were negligible.
- Deployment: Applying voltage droop at Node 3 shifted $z_0$ to 863 rad/s (a 153% increase), restoring stability and damping oscillations.
- Ineffective Deployment: Applying the same control at Nodes 1 or 2 resulted in negligible zero shifts and failed to stabilize the system.

5. Significance

This paper provides a paradigm shift in analyzing converter-dominated grids:

Data-Driven Stability: It enables system operators to assess stability margins using only publicly available network data (admittance and power flow) without needing access to confidential converter controller models.
Predictive Control: The derived exponential bound offers a clear warning signal: if the dominant NMP zero approaches the origin, stability is mathematically guaranteed to degrade regardless of controller tuning.
Targeted Mitigation: The participation factor-based reshaping strategy offers a computationally efficient, non-iterative method to restore stability by targeting the specific nodes that physically influence the system's NMP dynamics. This is crucial for the reliable integration of renewable energy in weak grids.