#### Identifying Critical Clusters in Microgrids

Ph.D. Student: Andrey Gorbunov | Advisors: Jimmy C. -H. Peng | Collaborator: Petr Vorobev, Skolkovo Institute of Science and Technology, Russia | Project Duration: 2018-2019

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Grid-forming inverters via droop control are thought to be the core technology for inverter-based microgrids, allowing them to operate in stand-alone modes without being connected to the main power grid. Ideally, it is best to have all inverters (those that are dispatchable) operating in such grid-forming mode as this will maximize the reliability of the microgrid to provide uninterrupted services to consumers. Droop-controlled inverters, working in parallel, are prone to instabilities and the allowed region for values of droop coefficients can be quite restricted. It has been shown in literature, that the fast electromagnetic dynamics of the network can not be neglected when analysing the small-signal stability for power controllers, thus making the dynamic model of the system very complex, since all the line currents have to be modeled as dynamic states.

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The term ”critical clusters” refers to a group of adjacent inverters that are tightly connected and make the dominant contribution to the unstable mode. It is thus essential, to identify these critical clusters since it is the parameters of this group that need to be modified to restore the system’s stability or enhance its stability margin. Identification of critical clusters can be challenging (even if the full-scale direct numerical stability analysis is performed) since their proximity to the instability onset depends on both the network parameters and the inverter control settings. In particular, it is not the most tightly connected cluster that is critical, but rather the one with the unfortunate combination of line parameters and droop values.

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Methodology

We develop a method for generalization of the critical clusters concept: we find an equivalent representation for a microgrid as a set of clusters, that are ranked according to their ”criticality.” Each of the clusters appears to be equivalent to an inverter-infinite bus system with some effective parameters, that makes the stability analysis straightforward as shown in Figure 1.

Such a representation becomes possible by analysing the system susceptance matrix. Every eigenvalue of a ”weighted” susceptance matrix corresponds to one cluster, and clusters can be naturally arranged in the order of their ”criticality” according to the corresponding eigenvalues. After the clusters are identified, we can immediately determine whether the system is stable and specify the parameters that need to be changed to stabilize the system or enhance its stability. Namely, the procedure for critical cluster identification is depicted in Figure 2

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Figure 1. Clustering of inverters for a two-bus system.

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Figure 2. Flow chart of critical cluster identification.

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Line Parameter Variations

First, we illustrate the system stabilization by changing the line impedance between inverters 3 and 4 (referring to Figure 1). Figure 3 shows the dependence of all μ values on the length of the line 3−4, or, equivalently, on its impedance value. It is observed that for the values of the line length smaller than about 6 km, only the μ2 value -- the critical one is affected. The system gains stability starting from the length of this line around 3.3 km, which is slightly bigger than the starting value for this line. The system also remains stable when line 3-4 bigger than this value.

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On the other hand, Figure 4, that the variation of the line 2−3 length, even in a much higher range -- up to 50 km could not stabilize the system as μ3 stays above the critical value. This is the consequence of the fact that the cluster 3 remains unstable even for an infinite length of the line 2−3 when the system splits into two separate areas.

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Figure 3. Variation μ with respect to line 3-4 length.

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Figure 4. Variation μ with respect to line 2-3 length.

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Droop Gain Variations

The system can also be stabilized by changing the droop gains of inverters 3 and/or 4. Figure 5 shows the dependence of the eigenvalues μ on the M1 -- frequency droop of inverter 1. We see that across all the region of the values of M1, the value of μ3 stays above μcr. Therefore, it is impossible to stabilize the system by any variations of the droop gains of inverter 1. This is in agreement with the fact that the cluster, responsible for instability, is composed of inverters 3 and 4. Moreover, with the increase of M1 above a certain threshold (∼3%), another eigenvalue, namely, μ1 crosses the critical value, so that the system now has two critical clusters, making it unstable.

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The situation is different, with the variation of the inverter 3 droop gain M3. Figure 6 demonstrates, that the value of μ2 is greatly affected by this variation, so the system can be efficiently stabilized by adjusting (decreasing) the M3 coefficient.

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Figure 5. Variation μ with respect to the first frequency droop gain, M1.

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Figure 6. Variation μ with respect to the second frequency droop gain, M3.

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Remarks

We have developed a method for stability assessment of inverter-based microgrids by means of representing it as a set of 2-bus equivalent clusters, which can be arranged in order of their proximity to instability. The proposed method is based on the analysis of the spectrum of a special weighted admittance matrix of the network and determining the eigenvalues, that lie above a certain threshold. Our findings are consistent with the number of previous results on account of the fact that groups of tightly connected neighboring inverters typically cause instabilities in inverter-based microgrids. This, so-called, critical clusters, are identified by the eigenvectors of the weighted admittancen matrix. Therefore, our method allows us to assess stability and determine the most critical parts of the system in a single step.

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We have validated and illustrated our method on a particular system of 4 inverters and demonstrated, that variation only of the very specific parameters can stabilize the system or enhance its stability margin. The developed method has an excellent practical perspective as a method for ”weak spots” identification in nearly built microgrids, or microgrids with planned reconfiguration. Further research will focus on deriv- ing closed-form expressions for stability enhancement rules, considering the possible worst-case scenarios in R/X ratios and/or frequency/voltage droop ratios, that can potentially allow formulating robust stability assessment methods.

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