[Paper Review] A simple method for shifting local dq impedance models to a global reference frame for stability analysis

Atle Rygg, Marta Molinas+3 more|arXiv (Cornell University)|Jun 26, 2017

Microgrid Control and OptimizationEngineering9 references19 citations

TL;DR

This paper presents a simple rotation matrix method to transform local dq and modified sequence domain impedance models to a global reference frame, enabling standard series/parallel circuit rules in stability analysis. The key contribution is frame-invariant analysis with minimal computational overhead, validated via MATLAB Simulink frequency sweeps showing exact match between analytical and simulated impedance responses.

ABSTRACT

Impedance-based stability analysis in the dq-domain is a widely applied method for power electronic dominated systems. An inconvenient property with this method is that impedance models are normally referred to their own local reference frame, and need to be recalculated when referring to a global reference frame in a given network. This letter presents a simple method for translating impedance sub-models within a complex network, from their own reference frames to any given point in the network. What distinguishes this method is that by using a simple rotational matrix, it only needs impedance models in their own local reference frames, to be translated to a global reference in the network. By way of this method, standard circuit analysis rules for series and parallel connection are applicable, as proven in the letter. The method is defined and validated for impedances in the dq and modified sequence domains, and it is shown that the dependency on reference frame is marginal in the latter. An additional finding from the application of this method is that components or subsystems with a certain symmetry property called Mirror Frequency Decoupling are invariant to the choice of reference frame.

Motivation & Objective

• To address the challenge of inconsistent reference frames in impedance-based stability analysis of power electronic systems.
• To enable consistent system-level stability analysis by transforming local impedance models to a common global reference frame.
• To simplify the application of standard series and parallel impedance connection rules across multiple subsystems with different local reference frames.
• To validate the method through analytical and simulation-based comparison of impedance responses in both dq and modified sequence domains.

Proposed method

• Define a global reference frame at a selected network node, typically the source or load interface.
• Perform a power flow calculation to determine fundamental voltage angles at all nodes, with the global reference angle set to zero.
• Obtain local impedance models (Z^l) for each component, referenced to their own local terminals and operating points.
• Apply a rotation matrix T to transform each local impedance model to the global reference frame using Z^g = T Z^l T^(-1).
• Use the dq-domain rotation matrix T_dq = [[cosθ_i, sinθ_i], [-sinθ_i, cosθ_1]] and the modified sequence domain matrix T_pn = [[e^{jθ_i}, 0], [0, e^{-jθ_i}]] for transformation.
• After transformation, apply standard circuit analysis rules (series/parallel connections) to compute aggregate impedances for stability analysis.

Experimental results

Research questions

• RQ1 How can local impedance models in different reference frames be consistently aligned to a global reference frame for system-level stability analysis?
• RQ2 What is the minimal computational overhead required to transform local impedance models to a global frame without modifying the original models?
• RQ3 How do the dq-domain and modified sequence domain impedance models behave under rotation, and what are the differences in their transformation properties?
• RQ4 Can standard series and parallel connection rules be reliably applied after frame transformation, and under what conditions?
• RQ5 Which system properties remain invariant under reference frame rotation, and how does this affect system analysis?

Key findings

• The proposed rotation matrix method enables exact transformation of local impedance models to a global reference frame with minimal computational complexity.
• The method allows the direct application of standard series and parallel impedance connection rules after transformation, significantly simplifying system-level analysis.
• In the dq-domain, all four elements of the impedance matrix are affected by rotation, while in the modified sequence domain, only the off-diagonal elements' angles are altered.
• Components or subsystems that satisfy the Mirror Frequency Decoupling (MFD) condition are invariant to reference frame rotation and can be directly connected without transformation.
• Validation via MATLAB Simulink frequency sweep shows perfect agreement between analytical impedance responses and simulated results in both dq and modified sequence domains.
• The method is robust even in no-load conditions, where all voltage angles are equal and no rotation is required.