---
title: Extended Impedance Modal Analysis (EMAI)
url: https://www.emergentmind.com/topics/extended-impedance-modal-analysis-emai
type: topic
---

# Extended Impedance Modal Analysis (EMAI)

Searching arXiv for the specified paper and closely related impedance-based modal analysis work to ground the article in the cited literature.
Extended Impedance Modal Analysis (EMAI) is an impedance-based modal analysis method for revealing the internal dynamic mechanisms of grid-following inverters (GFLs) in inverter-rich power systems. It was introduced to overcome a specific limitation of modal analysis based on the impedance model (MAI): MAI can quantify how strongly a device participates in a mode, but it treats each GFL as a single integrated entity and therefore does not directly expose which internal control loop drives poor damping or instability. EMAI extends the impedance framework by decomposing the equivalent dynamics of a GFL into synchronous and electromagnetic components, calculating participation factors and participation ratios for those components, and introducing parameter participation factors to identify the key control parameters of the dominant loop, all without requiring a full state-space model [2503.00797].

## 1. Position within impedance-based modal analysis

The immediate background to EMAI is the contrast between modal analysis based on the state-space model (MASS) and modal analysis based on the impedance model (MAI). MASS identifies critical factors affecting system stability at the state-variable level and quantifies state contributions via participation factors, but it requires full and detailed state-space models of the system. MAI, by contrast, leverages frequency-domain impedance or admittance models and is therefore compatible with black-box or gray-box components, yet its standard formulation remains device-level rather than control-loop-level [2406.00421].

The broader MAI literature provides the theoretical setting in which EMAI operates. One recent study reports the theoretical equivalency between MASS and MAI in transfer functions, eigenvalues, and sensitivities, and also introduces a revised element participation index, a transformer ratio-based admittance sensitivity adjustment, and an impedance splitting-based sensitivity analysis considering parameter variations [2406.00421]. Within that setting, EMAI can be understood as a refinement of MAI’s diagnostic granularity: it preserves the impedance-model workflow while extending participation analysis from the element level to the internal-dynamics level.

The motivating problem is the increasing penetration of GFLs in power systems and the corresponding transformation of system dynamic characteristics. In that regime, the internal multi-timescale dynamics of GFLs and their interaction with the grid become central to oscillatory stability. EMAI is designed specifically for that setting, where full controller disclosure may be impractical but impedance or admittance models are available or measurable.

## 2. Decomposition of GFL internal dynamics

The central innovation of EMAI is the decomposition of the GFL admittance into internal dynamic components. The method separates the equivalent dynamics of a GFL into:

- **Synchronous dynamics (SD)**, dominated by the phase-locked loop (PLL)
- **Electromagnetic dynamics (ED)**, dominated by the current control loop (CCL)

This decomposition is expressed as
$$
Y_m^{DQ} = Y_{me}^{DQ} + Y_{me,s}^{DQ} + Y_{ms}^{DQ},
$$
where \(Y_{me}^{DQ}\) denotes the ED component, \(Y_{ms}^{DQ}\) denotes the SD component, and \(Y_{me,s}^{DQ}\) denotes cross terms reflecting PLL-CCL coupling [2503.00797].

The paper states that this separation is achieved using matrix operations and the matrix inversion lemma to analytically isolate the synchronous and electromagnetic parts in the original admittance formula. The significance of this step is methodological rather than merely notational. Conventional MAI retains only the port-level input-output behavior of the GFL, whereas EMAI attributes modal influence to dynamic substructures within the converter. This enables the analysis of the dynamic characteristics of each control loop in the GFL based on the impedance model, rather than only the integrated converter behavior.

At the system level, after aligning inverter and grid impedance or admittance models in the global DQ coordinate frame, EMAI forms the whole-system dynamic impedance matrix as
$$
Z = (E + Z_N Y_N) Z_N.
$$
This is the same network-level setting in which modal information is extracted, but the subsequent sensitivity analysis is applied to decomposed internal admittances rather than to an undifferentiated device model [2503.00797].

## 3. Participation factors, participation ratios, and parameter participation factors

EMAI extends the participation analysis of MAI in three layers: participation factors (PF), participation ratios (PR), and parameter participation factors (PPF). For the \(i\)-th eigenvalue \(\lambda_i\), MAI writes the mode perturbation associated with element \(m\) as
$$
\Delta \lambda_i = - \operatorname{Res}_{\lambda_i}(Z_{wm}, \Delta Y_m^{DQ}(\lambda_i)),
$$
where \(Z_{wm}\) is the local system impedance seen at element \(m\), and \(Y_m^{DQ}\) is the element’s DQ-frame admittance [2503.00797].

After decomposition, EMAI attributes modal influence to the ED and SD parts of a given GFL. The paper gives
$$
\Delta \lambda_i
= -\operatorname{Res}_{\lambda_i}(Z_{wm}, \Delta Y_m^{DQ}(\lambda_i))
= \epsilon_{me} PF_{me,1} + \epsilon_{ms} PF_{ms,1},
$$
where \(\epsilon_{me}\) and \(\epsilon_{ms}\) represent perturbation degrees in the ED and SD portions. In this formulation, \(PF_{me,1}\) and \(PF_{ms,1}\) measure the contribution of the electromagnetic and synchronous components of element \(m\) to mode \(i\) [2503.00797].

To enable normalized comparison, EMAI defines participation ratios. For the ED part of GFL \(m\),
$$
PR_{me,1} = \frac{|PF_{me,1}|}{|PF_{me,1}| + |PF_{ms,1}|}.
$$
The same idea is applied to the SD part and, according to the paper, can also be normalized over all GFLs for system-level comparison. PR therefore converts raw modal participation into a relative measure of dominance between internal dynamic components.

The third layer is the parameter participation factor,
$$
\operatorname{PPF}_{m,\rho}
= -\operatorname{Res}_{\lambda_i}\!\left(Z_{wm}, \frac{\partial Y_m^{DQ}(\lambda_i)}{\partial \rho}\right),
$$
where \(\rho\) is a control parameter such as a CCL PI gain or a PLL gain. The sign and magnitude of the PPF express how much the eigenvalue \(\lambda_i\) would change if \(\rho\) is perturbed, i.e., its leverage on the mode’s damping and frequency [2503.00797].

The paper further reports explicit closed-form expressions for \(\partial Y_m^{DQ}/\partial \rho\), derived by leveraging the decomposed admittance structure and symbolic differentiation, including expressions for CCL and PLL parameter sensitivities. This is the step that turns internal-loop attribution into parameter-level diagnosis.

## 4. Relation to adjacent extended impedance and admittance frameworks

EMAI belongs to a broader family of methods that attempt to make control-loop dynamics visible within frequency-domain network models. A closely related development is an extended admittance modeling method for converters-interlinked systems, which proposes a four-port Extended Impedance Model (EIM) with one virtual synchronization node and constructs an Extended Impedance Network (EIN) so that frequency-domain modal analysis can be applied directly to virtual sync nodes and branches [2410.00619].

The emphasis of that four-port formulation is different. It explicitly characterizes synchronization dynamics in GFL/GFM interlinked systems by augmenting the electrical network with a virtual sync port, thereby making sync-loop participation intuitive in the network representation. EMAI, by contrast, decomposes the GFL’s DQ-frame admittance into synchronous and electromagnetic contributions and then evaluates PF, PR, and PPF for those contributions. The former centers on explicit virtual sync nodes in an EIN; the latter centers on internal-dynamics decomposition inside the inverter admittance [2410.00619].

This suggests that the two approaches are complementary rather than identical. The virtual-sync-node framework is oriented toward intuitive evaluation of sync dynamics in converters-interlinked systems containing both GFL and GFM converters, whereas EMAI is oriented toward root-cause tracing inside GFLs, especially the distinction between PLL-dominated synchronous dynamics and CCL-dominated electromagnetic dynamics. Both approaches are part of a larger movement in impedance-based stability analysis away from purely aggregated port descriptions and toward physically interpretable internal mechanisms.

## 5. Validation on benchmark systems

The reported validation of EMAI uses simulations on modified IEEE 14-bus and 68-bus systems. In the modified IEEE 14-bus system, the setup includes several GFLs; cases with and without DC voltage loops (DVLs) are considered; parameters vary among inverters; and forced oscillations are applied to test the dynamic response [2503.00797].

For that benchmark, the paper reports that EMAI can identify, for each mode, which GFL contributes most at the device level, which internal loop is dominant, and which control parameter is critical for damping the mode. The examples given in the provided material include a case in which GFL8 is dominated by ED and another in which GFL6 is dominated by SD. The comparison with other methods is central to the interpretation of these results: MAI can assess which GFL most affects a mode, but not which internal loop or parameter; MASS can identify dominant loops and parameters, but requires full state-space information; EMAI is reported to achieve nearly identical insight using only locally available or black-box impedance models [2503.00797].

The modified IEEE 68-bus system extends the validation to a large-scale network with many synchronous machines, infinite buses, and numerous GFLs. In that setting, the paper reports that system modes were analyzed and that control parameters together with PR and PPF were evaluated. Subtle instabilities and improvement opportunities were identified purely from admittance models. The reported large-scale tests are presented as evidence that EMAI remains computationally tractable and effective for large-scale networks, in contrast to the dimensional burden associated with MASS [2503.00797].

Taken together, the two case studies are used to support three claims: first, that EMAI preserves the black-box compatibility of impedance-based methods; second, that it adds internal-loop interpretability beyond standard MAI; and third, that it remains applicable at a scale where full state-space modeling becomes difficult.

## 6. Diagnostic use, scope, and interpretive issues

The practical role of EMAI is diagnostic. The method is presented as enabling the tracing of a problematic oscillatory mode to a specific GFL device, the responsible internal control loop, and the exact control parameter whose adjustment will most efficiently improve damping or eliminate the mode [2503.00797]. In that sense, EMAI is not merely a modal screening tool; it is a structured sensitivity framework for controller-oriented stability improvement.

Several interpretive issues arise in comparing EMAI with earlier techniques. One is whether impedance-based analysis can provide insight comparable to state-space analysis. The MAI literature cited above argues that MAI and MASS are theoretically equivalent in transfer functions, eigenvalues, and sensitivities [2406.00421]. EMAI builds on that foundation and pushes the impedance-based viewpoint to internal-loop diagnosis. Another issue is whether impedance methods necessarily remain confined to electrical ports. The related extended admittance modeling literature answers that question by making synchronization loops explicit through virtual sync nodes [2410.00619], whereas EMAI answers it by decomposing the converter admittance itself [2503.00797].

The principal scope of EMAI, as presented, is the analysis of GFL internal dynamics based on the impedance model, with particular emphasis on PLL-dominated synchronous dynamics and CCL-dominated electromagnetic dynamics. A plausible implication is that the method is especially suited to systems in which internal converter detail is partially unavailable but port admittance data can be measured, identified, or supplied in gray-box form. The paper also presents controller tuning, stability margin assessment, and scalability to large systems as practical implications of the framework [2503.00797].

In summary, EMAI occupies a specific place in modern impedance-based stability analysis: it extends MAI beyond element-level participation, preserves compatibility with black-box and gray-box modeling, and provides an explicit route from oscillatory mode to internal loop and parameter. Within converter-dominated power systems, that shift from aggregate device attribution to internal dynamic attribution is its defining contribution.

Source: https://www.emergentmind.com/topics/extended-impedance-modal-analysis-emai