Grid-forming inverters: the nexus between inertia and small-signal stability


Research Fellows: Gurupraanesh Raman and Gururaghav Raman (equally contributing authors) | Advisor: Jimmy C.-H. Peng | Project Duration: 2021 - 2022

Cite this work as:

G. Raman, G. Raman, and J. C.-H. Peng, "Coupled power generators require stability buffers in addition to inertia", Scientific Reports, vol. 12 no. 13714, Aug. 2022. DOI:

Grid-forming inverters are being recognized as key enablers in achieving large-scale integration of renewables as conventional fossil-fuel-based power plants are being phased out. Utilities are now starting to take steps to incorporate them into their own grids.

While there exists much research separately on the need for inertia in renewable-dominated grids​, and the behavior of droop-controlled inverters, the two problems have rarely been considered simultaneously. For instance, while a primary goal of including grid-forming inverters is to provide inertia, much of the research on droop-controlled inverters assumes fixed coefficients for the first-order power filter in the droop loop (typically based on the timescale separation between the droop controller and inner voltage/current loops), that directly affects the contributed inertia.

In this work, we attempt to bridge this key gap. We take the Greater London distribution system as a case study, where the grid operator National Grid is looking to contract Virtual Synchronous Generators (VSGs) to provide grid-forming services. We focus on the following questions: Can high inertia guarantee the small-signal stability of a practical distribution grid? How many Virtual Synchronous Generators are needed for a given inertia requirement, i.e., what should their individual capacities be? Is there a scalable integration philosophy that takes the power grid up to 100% non-synchronous operation?

Varying the non-synchronous penetration


We built the distribution system network model for Greater London as a set of spanning trees based on the assumption that distribution lines are usually laid alongside roads. We then studied the small-signal stability of the system with VSGs of various capacities and number for a variety of projected inertia scenarios from National Grid's decarbonization pathways (see Fig. 1). The heatmaps we obtained demonstrate a limit on the number of VSGs that can stably operate for each value of system non-synchronous penetration (SNSP).


Figure 1. Small-signal instability in the Greater London power grid for various energy futures considering virtual synchronous generators. (a) Projections of the system inertia (H) and system non-synchronous penetration (SNSP) for four future energy scenarios. (b) Distributions of the damping ratio for the projected grid in 2024, 2027, and 2030. (c) Impact on the grid stability as the number of VSGs and SNSP change.

Interestingly, these results also show the need to phase out generators of small capacities, as these are detrimental to stability by virtue of their large equivalent droop gains. This is a key result in charting the transition towards 100% SNSP.

The need for decentralized interventions

Small-signal instability amongst droop-controlled sources results from their primary control gains (the droop gains), their inertia, and the network impedances. Here, the addition of new droop-controlled sources cannot improve the stability of a poorly-stable system (see Fig. 2). The implication for future grids is that utilities cannot intervene externally to maintain the stability of a set of grid-forming inverters. Moreover, centralized supervisory control of all these (independent) prosumer-owned inverters to influence the global stability, is not scalable. Therefore, we argue that they must necessarily incentivize these inverters themselves to act to rectify instabilities as they arise. Not only would such a framework be scalable, it is arguably the only scalable one.


Figure 2. Demonstrating how external interventions may not guarantee stability of the future grid. (a) The figure presents three scenarios where 10 VSGs share the load with the existing rotating synchronous generators. In each scenario, a new VSG, marked in green, is added to provide infinite inertia in an exaggerated attempt to stabilize the grid. (b) Damping ratios of the scenarios in (a) before and after the external intervention, showing no improvement in stability.

Stability storage: the analogue to energy storage

Here, we propose the concept of stability storage as a decentralized framework for maintaining stability, whereby the same inverters that lead to instability act to improve the stability. We envisioned this as the participating inverters (acting individually and locally) providing additional output impedance when needed, to increase the system's damping ratio to the required value. This is shown in Fig. 3. Note that we chose output impedance control as the stabilizing, as it is one of the commonly deployed control strategies in the literature.


Figure 3. How VSGs can stabilize the power grid by acting as stability stores, which contribute output impedance and get rewarded for the change in their respective stability contributions. 

The stability storage framework is analogous to how energy storage works: if there is an energy deficit, contribute energy locally, until balance is achieved. In our framework, the global stability is analogous to energy deficit, and the local output impedance contributions are analogous to the energy contributed by energy storage. Finally, to quantify the contributed stability so that it can be suitably compensated, we also developed a metric called Distributed Stability Metric (DSM), that measures the stability of an individual inverter's interaction with the rest of the grid. The individual inverters calculate their DSM, and contribute output impedance until the DSM reduces to a certain threshold. This is illustrated in Fig. 4.


Figure 4. Stability storage in action: (a)-(c) illustrating the use of stability storage to safely integrate a new inverter (marked Inv-1) which, without stability storage, would cause instability. (d) Stabilizing the three unstable scenarios from Fig. 2(b), where the stability contributions with and without stability storage contribution are drawn.

Overall, our results show that the stability storage framework can ensure the stability of the Greater London power grid for a variety of SNSP and inertia scenarios and number of VSGs (see Fig. 5).


Figure 5. Effectiveness of stability storage in the Greater London grid for various energy futures. (a) Same as Fig. 1(b) but with stability storage. (b) Same as Fig. 1(c) but with stability storage.


This study provides context of inertia to research on small-signal stability of grid-following inverters. Particularly, we have shown that there is an upper limit in the number of grid-forming inverters or VSGs that can be present in the grid while maintaining its small-signal stability. We also showed that recruiting VSGs with smaller capacities jeopardizes the stability of the grid. Given these constraints, operators can contract prosumers of suitable capacities to participate in forming the grid and providing inertia through the stability market, so that the number VSGs remains below the limit.

Despite long-term planning by the utilities, the number of VSGs operating can change in the short-term depending on, e.g., generation stochasticity, changing prosumer needs, faults, or other failures. Then, the stability storage service proposed in this paper can enable the robust operation, while adequately compensating the prosumers for this additional service.

The prosumers continue to provide stability storage as a stop-gap measure until grid-level coordination obviates its need by calling for an increase in the overall inertia from prosumers, bringing online generators of higher capacity, removing generators with smaller capacities, or effecting updates on primary controller parameters through secondary control. Meanwhile, the secondary control structure can also be utilized to tackle the side-effects arising from implementing the stability storage functionality such as voltage sag and changing reactive power share due to the changes in the VSGs' output impedance.

Overall, our study helps energy policymakers and power utilities from diverse power systems to develop a viable roadmap towards achieving higher renewable penetrations.