Adaptive Virtual Inertia Provision for AC and MT HVDC Grids Based on Converters' Capabilities

IET Generation, Transmission & Distribution ORIGINAL RESEARCH Adaptive Virtual Inertia Provision for AC and MT HVDC Grids Based on Converters’ Capabilities Amir Arsalan Astereki1Mehdi Monadi1Seyed Ghodratolah Seifossadat1Alireza Saffarian1 Kumars Rouzbehi2 1Department of Electrical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran 2Department of System Engineering and Automatic Control, University of Seville, Seville, Spain Correspondence: Mehdi Monadi ([email protected]) Received: 26 May 2025 Revised: 6 August 2025 Accepted: 2 September 2025 Funding: The authors received no specific funding for this work. ABSTRACT This paper presents a novel perspective on providing adaptive virtual inertia (AVI), aimed at improving DC voltage stability in Multi-Terminal High Voltage DC (MT-HVDC) grids while simultaneously enhancing frequency response in AC grids. The proposed approach introduces an innovative Virtual Synchronous Generator (VSG) that supplies AVI for the AC systems. Additionally, a new control strategy for the Power Electronics Converters (PECs) that supply the MT-HVDC grid is presented, referred to as dcVSG, to provide AVI for this grid. Utilising both controllers concurrently enables adaptive and simultaneous virtual inertia provision on both DC and AC grids, while effectively leveraging the operational capabilities of the PECs. In this regard, the DC voltage and the AC grid frequency are considered as control parameters. The AVI is dynamically adjusted according to the PEC operating point. Specifically, the calculated maximum AVI is sensitive to the increase and reduction of the control parameter, demonstrating appropriate distinct values in response. This behaviour aims to utilise the PEC’s maximum power capacity. The small-signal stability of the proposed system is analysed by focusing on the influence of virtual inertia on overall stability. Also, to assess the stability of the proposed controllers, Lyapunov stability theory, alongside a series of detailed simulation analyses, is conducted utilising the Cigre-DCS3 test grid. The simulation outcomes indicate that the proposed coordinated strategy yields a 20% reduction in DC voltage deviation while also enhancing frequency nadir. Additionally, it achieves over a 60% decrease in the rate of change of voltage (RoCoV) on the DC side and a 68% reduction in the rate of change of frequency (RoCoF), specifically when compared to methods that rely solely on the headroom power of the PEC to deliver maximum virtual inertia. 1 Introduction The shift from traditional power systems to advanced configurations that predominantly utilise renewable energy sources, DC transmission systems, PEC-based loads, and grid compensators has introduced a range of technical challenges, such as lowfrequency oscillation. However, one of the critical challenges is the diminishing inertia across these entire modern power grids [1–5]. In a hybrid power system incorporating both AC and MT-HVDC networks, the inertia in the AC segments plays a vital role in dampening frequency fluctuations. The inertia not only enhances the frequency nadir but also reduces the RoCoF. Concurrently, on the DC side, inertia functions to stabilise variations in DC voltage, thereby improving the DC voltage nadir and decreasing the RoCoV [6–9]. To address the challenges associated with insufficient inertia in power systems, several strategies have been employed to effectively provide AVI through the control of PECs. One methodology is discussed in [10]. This approach leverages the RoCoF and This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). IET Generation, Transmission & Distribution published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology. IET Generation, Transmission & Distribution,2025;19:e70154 https://doi.org/10.1049/gtd2.70154 1of15 FIGURE 1 Provision of virtual inertia based on grid conditions. (a) For AC grids, large virtual inertia in frequency deviation periods. (b) For DC grids, large virtual inertia in DC voltage deviation periods. frequency deviations, reformulating the swing equation into a quadratic format. It eliminates the derivative term, which is sensitive to the noise, utilising the Vieta Theorem. It should be highlighted that virtual inertia can also be realised through the concept of virtual inductance, as detailed in [11]. In this framework, virtual inductance is derived by taking into account both the effects of virtual damping and virtual inertia dynamics. Furthermore, the AVI can be generated based on the system’s RoCoF or by taking into account both RoCoF and frequency deviations as outlined in [12, 13]. Additionally, in [14–16]AVIis provided based on the sign of frequency variation and RoCoF, as illustrated in Figure 1a. This dynamic adjustment increases virtual inertia during periods of frequency deviation while reducing it as the frequency stabilises towards its reference value. Further exploration in [17] introduces an arctangent function to modulate virtual inertia in reaction to power fluctuations and frequency deviations. One conventional method for providing inertia in AC systems is the VSG control technique. This method effectively emulates the damping characteristics of synchronous generators (SGs) and provides virtual inertia for AC systems. However, the integration of VSGs in MT-HVDC systems presents certain challenges. These challenges arise from design constraints associated with the VSG’s outer DC voltage control loop, which can adversely impact the VSG dynamics when it tracks the active power reference. This issue is explored in [18], where a two-degree-of-freedom VSG is proposed. This design decouples changes in active power references from the outer DC voltage control loop, thereby providing virtual inertia while controlling the DC voltage. The management of DC voltage is further explored in [19], where the authors propose AVI and adaptive damping strategies implemented through an optimisation algorithm. This approach aims to establish a trade-off among frequency stability, power stability, and the constraints related to DC-side voltage stability. Moreover, a method rooted in model predictive control that offers AVI based on specific system contingencies is highlighted in [20]. Also, in [21] a linear quadratic regulator is outlined aimed at adaptively tuning the virtual inertia and damping parameters of virtual synchronous machines embedded within PECs. Incorporating adaptive damping with AVI is also explored in [22]. In addition, [23] proposes a dual-adaptivity inertia control methodology to enhance both frequency and power regulation through the dynamic adjustment of virtual inertia. While efforts to provide AVI in AC systems are ongoing, various methods have been proposed to enhance the inertia of DC systems. Indeed, the analysis presented in [24] indicates that a comparable oscillation mechanism exists between DC microgrids and traditional AC power systems. Therefore, the authors propose a virtual synchronous control aimed at enhancing the DC voltage nadir and RoCoV. Another approach is presented in [25] that employs a sigmoid-like function for the adaptive modulation of a virtual capacitor, aimed at achieving enhanced virtual inertia within the MTDC grid. Another strategy is presented in [26], where AVI is incorporated into a droop control loop to stabilise DC grids based on the RoCoV and deviations in DC voltage. Additionally, in [27, 28]avirtual capacitor is designed that offers AVI for the DC system in response to RoCoV. Moreover, [29] presents an adaptive virtual capacitance droop control mechanism designed for hybrid energy storage systems within DC microgrids. This approach aims to enhance power management and mitigate DC voltage fluctuations. Similarly, [30] employs fuzzy logic to optimise the virtual capacitor droop parameters for these hybrid storage systems. Also, [31] proposes a cascaded control strategy with a negative feedback effect that preserves the original control architecture of the PEC to enhance the DC voltage nadir by providing virtual inertia and damping for the DC microgrid. In addition, there are also approaches targeting the simultaneous provision of virtual inertia across both DC and AC systems. For instance, an approach is outlined in [32], which involves the integration of a dual-port grid-forming modular multilevel converter (MMC) operating in parallel with energy storage systems to enhance both AC and DC stability. Furthermore, a methodology described in [33] leverages coordinated converters via synchronverter and virtual DC machine controllers to facilitate AVI provision for both AC/DC systems. Furthermore, works highlighted in [34, 35] explore the use of bidirectional interfacing converters to create AVI that serves both AC and DC networks, enhancing overall system stability. However, to the best of the author’s knowledge, no method has yet been proposed that provides AVI by taking into account the converter’s capability to adjust its active power in response to load change conditions. The maximum value of AVI has typically been calculated solely based on the converter’s reserve power. This paper introduces a novel approach wherein the AVI is adjusted according to the network’s conditions and its behaviour during disturbances, allowing for different maximum AVI selections. This leads to optimal utilisation of the converter’s capacity and enhances network stability. Specifically, in scenarios of increasing load, it is the reserve power of the converter that primarily influences the calculation of maximum AVI. Conversely, during load reductions, the maximum AVI can be determined based on the converter’s operational power. Additionally, in this paper, a novel AVI method is proposed to enhance the performance of PECs during grid contingencies, leveraging a hyperbolic tangent function. This technique is applied to both PECs injecting power into AC and MT-HVDC 2of15 IET Generation, Transmission & Distribution,2025 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License TABLE 1 Comparison between the proposed method and the existing methods. Ref. Method Virtual inertia in the AC grid Virtual inertia in the DC grid AVI Online tracking of PEC’s equilibrium point for AVI [11] Virtual inductance ✓✗✗ ✗ [18] Two-degree-of-freedom VSG ✓✗✗ ✗ [19]ImprovedVSG✓✗✓ ✗ [24] Virtual synchronous control ✗✓✗ ✗ [25] Virtual Capacitor ✗✓✓ ✗ [31] Cascaded virtual inertia ✗✓✗ ✗ [32] Dual-port grid-forming ✓✓✗ ✗ [33] Synchronverter, virtual DC machine ✓✓✓ ✗ [34, 35] Bidirectional PECs ✓✓✓ ✗ This paper VSG, dcVSG ✓✓✓ ✓ grids, aiming to enhance DC voltage and AC frequency nadirs and their rate of change. Table 1provides a comparison between the proposed method and the recently presented methods in the field of virtual inertia provision. To elaborate on the proposed methods for DC and AC grids, Section 2begins by outlining the overall structure of the presented strategy for simultaneously providing virtual inertia in both AC and MT-HVDC grids. Section 3analyses the impact of virtual inertia within a hybrid AC/MT-HVDC grid, using small-signal stability analysis. Then, Section 4provides a comprehensive explanation of the proposed approach for establishing AVI in AC grids, accompanied by Liyapanov’s stability proof, while Section 5 examines the approach for employing AVI in MT-HVDC grids. Section 6details the simulation outcomes, demonstrating the efficacy of the proposed methods. Subsequently, Section 7offers a comprehensive summary of the key findings. 2Method Description Unlike traditional SGs, the virtual inertia provided by PECs offers notable flexibility, as it can be dynamically adjusted in accordance with grid conditions. This adaptability allows for improved resilience and stability in modern power systems. Indeed, a greater amount of virtual inertia leads to a slower deviation in frequency or DC voltage, resulting in a lower RoCoF or RoCoV, respectively. However, under small disturbances, the converter’s response may be sluggish, and during the recovery phase, excessive virtual inertia can negatively affect the restoration of frequency or DC voltage. Therefore, as illustrated in Figure 1, the provision of AVI for AC and MT-HVDC grids can be tailored according to the behaviour of AC frequency and DC voltage. Specifically, substantial AC virtual inertia can be provided when the frequency and its rate of change align in sign. A similar strategy applies to an MT-HVDC grid; as the signs of voltage and its rate of change match, the level of AVI increases. Conversely, as shown in Figure 1b, when the system FIGURE 2 Active power control loop of the VSG. returns to its reference point (where the signs are opposite), the amount of AVI is minimised. So, in this paper, for concurrently providing AVI in the AC and MTDC grids, the VSG is tasked to enhance RoCoF and frequency nadir, while dcVSGs are employed to improve RoCoV and DC voltage nadir. 2.1 Providing Virtual Inertia for AC Grids According to the swing equation presented by the following equation, the inertia constant acts as the resistive parameter in response to power fluctuations: 2𝐻𝑣 𝑑𝜔 𝑑𝑡 =𝑃∗∗ 𝑒−𝑃𝑚𝑒𝑎𝑠 −𝐷𝑣𝑠𝑔(𝜔 −𝜔∗), 𝑃𝑖𝑛𝑟 =2𝐻𝑣𝑠𝑔 𝑑𝜔 𝑑𝑡 (1) in which, 𝑃meas and 𝑃∗∗ 𝑒are measured electrical power and its reference in per-unit (p.u.), respectively, 𝐷vsg is the damping coefficient, (𝜔 −𝜔∗)is the frequency deviation from its reference in p.u. Also, 𝐻𝑣is the virtual inertia time constant, and 𝑃inr is the active power injected or absorbed by the converter for inertia emulation. The conventional VSG controller is depicted in Figure 2,where𝐻𝑣is the inertia time constant, 𝑃∗is the initial reference value for active power, 𝐾𝜔and 𝐾𝑣are the frequency and DC voltage droop coefficients, respectively. 3of15 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 3 Active power control loop of the dcVSG. To enable AVI, it’s essential to adjust 𝐻𝑣in response to the variation of the grid’s conditions. This adaptability provides increased operational flexibility for the VSG. It should be noted that the active power reference (𝑃∗∗) is changed by the droops asshowninFigure2, in which Δ𝑃𝑣and Δ𝑃𝜔are the output signals from the V-P and 𝜔-P droops, respectively. To prevent the converter from overloading, the droop coefficients have to be designed to satisfy 𝑃∗+Δ𝑃𝑣max ++Δ𝑃𝜔max ≤𝑃max in which 𝑃max is the maximum active power of the converter. 2.2 Providing Virtual Inertia for MT-HVDC Grid According to the analogy between AC and DC grids, the following equation can be established that represents the relationship between the virtual inertia provided by the virtual capacitor (𝐶𝑣) in response to the changes in active power and DC voltage, 𝑃∗∗ 𝑒−𝑃𝑚𝑒𝑎𝑠 −𝐷𝑣(𝑉𝑣−𝑉∗ 𝑑𝑐)=𝐶𝑣𝑉𝑣 𝑑𝑉𝑣 𝑑𝑡 ≈𝐶 𝑣𝑉𝑑𝑐𝑛 𝑑𝑉𝑣 𝑑𝑡 (2) where 𝑉𝑣is the output signal of the active power loop to generate 𝐼∗ 𝑑.Also,𝑃∗ 𝑒and 𝐷𝑣are the active power reference and DC voltage droop coefficient, respectively. The active control loop associated with the presented controller is illustrated in Figure 3. The AVI is managed through the dcVSG by adjusting 𝐶𝑣to meet grid conditions. The objective is to dynamically adjust the virtual inertia in response to RoCoV and DC voltage variations. In this paper, a safe operational range for DC voltage variations has been determined, specifically between 0.95 and 1.05 p.u., while considering a maximum RoCoV of ±0.1 p.u.∕s. Therefore, utilising both controllers concurrently can enhance the overall inertia within the AC/MTDC grid. To illustrate this, a 5-terminal configuration that incorporates VSG, dcVSG, and droop control mechanisms is examined. The case under consideration is illustrated in Figure 4, in which the PECs for C2 and D1 AC grids are integrated with the dcVSG controller. Notably, PEC station A1, connected to AC grid A0, has been determined to be the prime candidate for the VSG method. It’s important to clarify that PEC station B1 operates bidirectionally and is not appropriate for either the VSG or dcVSG methodologies. Additionally, VSGs are not ideally suited for parallel operation with stiff grids. Consequently, PEC station B2, which is connected to the stiff grid represented by AC grid B0, is not appropriate for consideration in this study. FIGURE 4 Modified Cigre DCS-3 equipped with the VSG and dcVSG methods. 3 Effect of Virtual Inertia on System Stability In this section, a linearised model of the presented system is developed that allows for a detailed assessment of the influence of virtual inertia in each grid on the system’s stability. 3.1 Linearised Model of MT-HVDC Grid Presuming that the equivalent πmodelisusedforeveryDCcable in the bipolar MT-HVDC grid, the dynamics of each node voltage and each DC line current can be calculated by: ⎧ ⎪ ⎨ ⎪ ⎩ Δ 𝑉𝑖=1 𝐶𝑒𝑞𝑖∑ 𝑗∈𝑁𝑖 Δ𝐼𝑙𝑖,𝑗 −Δ𝐼𝑜𝑢𝑡𝑖,Δ𝐼 𝑜𝑢𝑡𝑖=Δ𝑃𝑖 𝑉𝑖0 −𝑃𝑖0 𝑉2 𝑖0 Δ𝑉𝑖, Δ 𝐼𝑙𝑖,𝑗 =1 𝐿𝑖,𝑗 [Δ𝑉𝑗−Δ𝑉𝑖−𝑅𝑖,𝑗Δ𝐼𝑖,𝑗] (3) in which 𝐶𝑒𝑞𝑖denotes the equivalent DC capacitance at node i, while Δ𝐼out𝑖represents the output current flowing from the MT-HVDC grid to each interconnected AC grid. The term 𝑁𝑖 indicates the count of neighbouring terminals associated with node i. Furthermore, Δ𝑃𝑖reflects the fluctuations in active power injected into the AC grid. Also, 𝑉𝑖0,𝑃𝑖0 are the DC voltage and the active power delivered by the converter station iat their equilibrium points, respectively. The terms 𝐿and 𝑅denote the total inductance and resistance, equivalently defined as (2/3) 𝐿𝑎𝑟𝑚 and (2/3) 𝑅arm, respectively, where 𝐿arm and 𝑅arm correspond to the inductance and resistance of an individual arm of the converter. 3.2 Linearised Model of the AC Grids The AC grids can be modelled as a second-order generator with a first-order governor and a non-reheated turbine, assuming a 4of15 IET Generation, Transmission & Distribution,2025 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License negligible synchronising torque as: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Δ 𝜔𝑔𝑖=1 2𝐻𝑔𝑖 [Δ𝑃𝑚𝑒𝑐ℎ𝑖−Δ𝑃𝑔𝑒𝑛𝑖−𝐷𝑖(Δ𝜔𝑔𝑖)], Δ 𝑃𝑚𝑒𝑐ℎ𝑖=− 1 𝑇𝑡𝑢𝑟𝑖[𝐾𝑖 𝑇𝑔𝑖 Δ𝜔𝑔𝑖+Δ𝑃𝑚𝑒𝑐ℎ𝑖], Δ𝑃𝑔𝑒𝑛𝑖=∑Δ𝑃𝑙𝑜𝑎𝑑𝑖−∑Δ𝑃𝑖 (4) in which the subscripts irefer to the AC grid iparameters. Also, 𝑇𝑡𝑢𝑟𝑖and 𝑇𝑔𝑖are the turbine and governor time constants, respectively, which are small values. The term ∑Δ𝑃load𝑖signifies the total variations in connected loads, whereas ∑Δ𝑃𝑖represents the active power fluctuations occurring within the converter stations associated with the AC grid i. 3.3 Linearised Model of the dcVSGs In a system where a fast phase-locked loop (PLL) is employed, the d-axis can align closely with the AC terminal voltage, rendering variations in the AC voltage effectively negligible. Consequently, variations in active power can be directly assessed through changes in the d-axis current. Additionally, the current control loop operates at a speed sufficient to be considered as a lowpass filter characterised by a time constant 𝑇Inn [36]. Thus, the dynamics of the dcVSG can be represented by: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Δ𝑃𝑖=−𝐸0𝑖Δ𝐼𝑑𝑖,Δ 𝐼𝑑𝑖=1 𝑇𝑖𝑛𝑛𝑖 [Δ𝐼𝑟𝑒𝑓𝑖−Δ𝐼𝑑𝑖], Δ𝐼𝑟𝑒𝑓𝑖=𝐾𝑝𝑖(Δ𝑉𝑣𝑖−Δ𝑉𝑖)+𝐾𝐼𝑖(ΔΦ𝑖), Δ Φ𝑖=Δ𝑉𝑣𝑖−Δ𝑉𝑖,Δ𝑃 𝑟𝑒𝑓𝑑𝑖 =Δ𝑃𝑟𝑒𝑓0𝑑𝑖 +𝑘𝜔𝑖(Δ𝜔𝑔𝑖), Δ 𝑉𝑣𝑖=1 𝐶𝑣𝑖𝑉𝑛𝑖 [Δ𝑃𝑟𝑒𝑓𝑑𝑖 −Δ𝑃𝑔𝑒𝑛𝑖−𝐷𝑉𝑖(Δ𝑉𝑣𝑖−Δ𝑉𝑟𝑒𝑓0𝑖)] (5) where Δ𝐼𝑑𝑖is the d-axis current change of the ith AC grid, and 𝐸0𝑖 denotes its AC voltage amplitude. Also, 𝐾𝑃𝑖and 𝐾𝐼𝑖correspond to the proportional and integral gains within the DC voltage control loop, respectively. Moreover, ΦIndicates the state of the integral part of the DC voltage control loop. Additionally, 𝑘𝜔𝑖denotes the frequency droop coefficient of the supplementary controller in dcVSGi. 3.4 Linearised Model of the Droop-Controlled Stations Droop-controlled PECs interfacing with the AC grid B0 can be modelled as grid-following current sources when analysed from the perspective of the AC output. Therefore, the linearised droop station’s dynamics can be represented as: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Δ𝑃𝑖=𝐸0𝑖Δ𝐼𝑑𝑖,Δ 𝐼𝑑𝑖=1 𝑇𝑖𝑛𝑛𝑖 [Δ𝐼𝑟𝑒𝑓𝑖−Δ𝐼𝑑𝑖], Δ𝐼𝑟𝑒𝑓𝑖=𝐾𝑝𝑖(Δ𝑃𝑟𝑒𝑓𝑖−Δ𝑃𝑖)+𝐾𝐼𝑖(ΔΦ𝑖), Δ Φ𝑖=Δ𝑃𝑟𝑒𝑓𝑖−Δ𝑃𝑖,Δ𝑃 𝑟𝑒𝑓𝑖=Δ𝑃𝑟𝑒𝑓0𝑖+𝐾𝐷𝑖(Δ𝑉𝑖) (6) in which 𝐾𝐷𝑖is the P-V droop coefficient of the ith droop station. 3.5 Linearised Model of the VSG In a grid-connected mode, the reactive power output of the VSG can be configured to zero. This indicates that any variations in power on the DC side directly correspond to variations in active power on the AC side of the VSG. Furthermore, the active power adjustments on the AC side of the VSG are contingent upon the phase angle discrepancy between the VSG (𝛿VSG)and AC grid (𝜃𝑖), specifically represented as (𝛿VSG −𝜃𝑤), consequently, the linearised model of the VSG can be formulated as: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Δ 𝜃𝑖=Δ𝜔𝑔𝑖,Δ 𝛿𝑉𝑆𝐺 =Δ𝜔𝑣 Δ 𝜔𝑣=1 2𝐻𝑣 [Δ𝑃𝑟𝑒𝑓𝑣−Δ𝑃𝑖−𝐷𝑣𝑠𝑔(Δ𝜔𝑣)], Δ𝑃𝑖=𝐸𝑈 𝑋(Δ𝛿𝑉𝑆𝐺 −Δ𝜃𝑖), Δ𝑃𝑟𝑒𝑓𝑣=Δ𝑃𝑟𝑒𝑓0𝑣 +𝐾𝑣(Δ𝑉𝑖)−𝐾𝜔(Δ𝜔𝑣), (7) where Eand Uare the amplitudes of the internal voltage of the VSG and the amplitude of the PCC voltage, respectively. Also, 𝑋 signifies the total equivalent reactance of the VSG’s transformer, phase, filter, and virtual reactances. 3.6 State Space Model of the AC/MT-HVDC Grids The state variables are characterised according to the dynamics outlined in Equations (3)through(7). So, the state space model of the system is made by: { 𝑋=𝐴𝑋 +𝐵𝑢𝑈+𝐵𝑑𝐷 𝑌=𝐶𝑋 (8) in which 𝑈is the input vector that is defined as the variations in the active power setpoints of the converter stations and the fluctuations in the DC voltage references of the dcVSGs, that is 𝑈=[Δ𝑃ref01,Δ𝑃 ref02,Δ𝑃 ref0𝑑1 ,Δ𝑃 𝑟𝑒𝑓0𝑑2 ,Δ𝑃 ref0𝑣,Δ𝑉 ref01,Δ𝑉 ref02]. Also, 𝐷denotes the disturbance vector which is selected as the load changes at AC grids that is 𝐷=[Δ𝑃load1,Δ𝑃 load2,Δ𝑃 load3]. Furthermore, the output vector is 𝑌which is selected as the angular frequency of the AC grid A0 and the DC voltage of the MT-HVDC terminals. Moreover, Arepresents the state matrix, 𝐵𝑢denotes the input matrix, and 𝐵𝑑is the disturbance matrix. Also, 𝐶and Xsignify the output matrix and state variable matrix, respectively. The presented linear AC/MT-HVDC grid is implemented in MATLAB code that enables a thorough small-signal stability assessment. The parameters of the system used for this analysis are presented in Table A1 in the Appendix A. Additionally, in this analysis, the control parameters for the VSG are set as follows: 𝐷𝑣𝑠𝑔 =50 p.u.,𝐾𝜔=20 p.u.,and𝐾𝑣=1p.u., while for both dcVSGs, the control parameters are: 𝐷𝑣=6p.u.,𝑘𝜔=1p.u.,and, 𝐶𝑣=500 µF. In the first step, the virtual inertia time constant in VSG is adjusted from 0.5 s to 8 s. Figure 5illustrates the trajectories of the sensitive modes of the system in response to 𝐻𝑣adjustments. Alongside the colour scale representing the damping ratio of the system’s modes, the trajectories of the lowdamped modes are illustrated with distinct arrows, such as those used for modes 𝜆3,𝜆 4. According to the figure, increasing 𝐻𝑣 5of15 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 5 Sensitive mode trajectories in response to the changes in 𝐻𝑣from 0.5 s to 8 s. within the AC system linked to the MT-HVDC grid results in an enhancement of the damping ratios for oscillatory modes 𝜆23,𝜆 24,and𝜆1,𝜆 2. Conversely, this increment leads to a reduction in damping ratios for the well-damped modes 𝜆26,𝜆 27 and for a range of oscillatory modes, that is 𝜆3,𝜆 4,𝜆9,𝜆 10,and𝜆11,𝜆 12. Therefore, while augmenting virtual inertia in the AC grid can bolster system stability, excessively high values may introduce dynamic complications. To illustrate the impact of varying virtual inertia within the dcVSGs on the AC/MT-HVDC system, the capacitance 𝐶𝑣in both dcVSG is increased from 500 μF to 6mFwhile 𝐻𝑣is setat0.5s.Figure6presents the trajectories of the sensitive modes of the system in response to these adjustments. The figure illustrates that an increase in 𝐶𝑣enhances the damping ratios of the oscillatory modes 𝜆3,𝜆 4,and𝜆7,𝜆 8,aswellasthedamped modes 𝜆21,𝜆 22. However, this increase simultaneously detracts from the damping ratios of other oscillatory modes, specifically 𝜆5,𝜆 6,𝜆9,𝜆 10,𝜆13,𝜆 14 and 𝜆15,𝜆 15 as well as the damped modes 𝜆26,𝜆 27. It is evident that a majority of the critical modes are migrating towards the origin, suggesting that while elevated virtual inertia in DC grids is beneficial, excessive 𝐶𝑣may induce instability. Consequently, the virtual inertia in both AC and MT-HVDC grids should be adaptively changed based on the system dynamics. In response to small disturbances, the PEC should maintain low virtual inertia; conversely, this parameter should be augmented in the presence of significant voltage or frequency deviations. Such adaptability can enhance the reliability of the control system and improve system responsiveness to contingencies. Thus, the subsequent sections will propose AVIs tailored for both AC and FIGURE 6 Sensitive modes trajectories in response to the changes in 𝐶𝑣from 500 μF to 6mFin both dcVSGs. MT-HVDC grids, focusing on optimising system dynamics while accounting for the PEC’s capability in responses to different disturbances. 4 Proposed AVI Provision in AC Grid One of the primary considerations in designing the VSG is determining the maximum inertia time constant (𝐻max).It’s essential to understand that the dynamic of virtual inertia emulation within the active power control loop is faster than those of supplementary droop controllers. This is primarily because, during the initial moments of contingencies, the variations in frequency and DC voltage are minimal, whereas the RoCoF is considerably large. As a result, supplementary droops can be disregarded when calculating 𝐻max. It is important to note that the maximum RoCoF and frequency deviation must be considered to protect the converter from exceeding the active power limits during the 𝐻max calculation. According to ENTSO-E guidelines, frequency should be maintained within a range of 49.0 Hz to 51.0 Hz, and RoCoF should not exceed ±1.0 Hz/s. Consequently, 𝑑𝜔∕𝑑𝑡max =±0.02 p.u.∕s and Δ𝜔max =0.02 p.u. are considered for the calculation of 𝐻max. The converter’s capacity to modulate active power varies across each steady-state equilibrium point, as shown in Figure 7.Atthe initial equilibrium point (eq1), the VSG reserve power is |𝑃res|, while the active power output is |𝑃of |. These parameters are dynamic and can adjust in response to various contingencies. For instance, if the frequency deviates further from the reference (path 2), an increase in the output active power alongside a decrease in reserved active power is observed. Therefore, to determine the maximum virtual inertia time constant, it is essential to evaluate the VSG’s active power change capability at each steady-state equilibrium position. Additionally, the converter can 6of15 IET Generation, Transmission & Distribution,2025 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 7 Active power and frequency variations in different contingencies. provide varying virtual inertia time constants based on different contingency scenarios, which is related to the disparity between |Pres|and |Pof |. It means that in over-frequency conditions, the converter can reduce its active power output by about |𝑃of |, whereas it can inject |𝑃res|during under-frequency events. This dynamic capability enhances system reliability and flexibility during large disturbances. Thus, in this paper, the methodology to calculate the virtual inertia time constant at each equilibrium point is proposed as follows: 𝑃𝑖𝑛𝑟max =2𝐻𝑠𝑠max 𝑑𝜔 𝑑𝑡 max =Δ𝑃𝑐𝑎𝑝 −𝐷𝑣𝑠𝑔(Δ𝜔𝑠𝑠max ), 𝐻𝑠𝑠max = Δ𝑃𝑐𝑎𝑝 −𝐷𝑣𝑠𝑔(Δ𝜔𝑠𝑠max ) 2𝑑𝜔 𝑑𝑡 max (9) in which Δ𝑃cap is the active power change capability of the VSG and Δ𝜔𝑠𝑠max is the maximum allowable frequency deviation according to each steady-state equilibrium point. Specifically, for steady-state equilibrium point 1 (𝑒𝑞1) in Figure 7, along path 1, ΔP cap =𝑃of and Δ𝜔 𝑠𝑠max =𝜔of , while for path 2, Δ𝑃 cap =𝑃𝑟𝑒s and Δ𝜔 𝑠𝑠max =𝜔𝑢𝑓. At each steady-state equilibrium point, the trajectory of frequency changes, whether increasing or decreasing, is governed by the sign of 𝑑𝜔∕𝑑𝑡. Consequently, the maximum inertia in either direction can be determined as follows: 𝐻𝑢𝑓max = 𝑃𝑟𝑒𝑠 −𝐷𝑣𝑠𝑔(𝜔𝑢𝑓) 2𝑑𝜔 𝑑𝑡 max ,𝐻 𝑜𝑓max = 𝑃𝑜𝑓 −𝐷𝑣𝑠𝑔(𝜔𝑜𝑓) 2𝑑𝜔 𝑑𝑡 max 𝐻𝑠𝑠max ={𝐻𝑢𝑓max ,𝑑𝜔∕𝑑𝑡<0 𝐻𝑜𝑓max 𝑑𝜔∕𝑑𝑡 >0,(10) Furthermore, identifying the new equilibrium point is crucial. From the control theory perspective, the frequency serves as a state variable within the AC system. The system is in an equilibrium point when 𝑑𝜔∕𝑑𝑡 =0. However, in practical applications, RoCoF can momentarily drop to zero during transient conditions and may exhibit small fluctuations under steadystate scenarios. Indeed, the duration of transients can extend over several hundred milliseconds [36, 37]. As such, it’s essential to incorporate a time period to guarantee that the system has stabilised when the condition |𝑑𝜔∕𝑑𝑡|<𝛼is satisfied, in which 𝛼is a pre-defined threshold. This threshold adjusts the sensitivity of the controller in response to RoCoF. It should be noted that the calculated values (𝐻uf 𝑎𝑛𝑑 𝐻of ) will remain constant during transients. FIGURE 8 Proposed AVI variation in response to the RoCoF. Therefore, based on the derived description in this section, the AVI is proposed as: 𝐻𝑣=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝐻0+(𝐻𝑢𝑓max −𝐻0)tanh𝑘(−𝑑𝜔 𝑑𝑡 ),𝑑𝜔 𝑑𝑡 <0 𝐻0+(𝐻𝑜𝑓max −𝐻0)tanh𝑘(𝑑𝜔 𝑑𝑡 ),𝑑𝜔 𝑑𝑡 >0 (11) in which 𝑘is an adjusting parameter that adjusts the sensitivity of the adaptive inertia to the RoCoF. Indeed, as indicated in (11), an increase in the value of 𝑘enhances the responsiveness of the hyperbolic tangent function to changes in RoCoF. The variation of the adaptive inertia in response to the RoCoF is depicted in Figure 8.Inaddition,asdemonstratedinFigure1, the objective is to implement large virtual inertia in response to frequency deviations from the reference value, while a smooth decrement facilitates the convergence phase. Thus, Equation (11)canbe revised as follows: 𝐻𝑣= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐻0+(𝐻𝑢𝑓max −𝐻0)tanh𝑘(−𝑑𝜔 𝑑𝑡 ), 𝑑𝜔 𝑑𝑡 <0 &Δ𝜔 <0 𝐻0+(𝐻𝑜𝑓max −𝐻0)tanh𝑘(𝑑𝜔 𝑑𝑡 ), 𝑑𝜔 𝑑𝑡 >0 &Δ𝜔 >0 𝐻0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (12) The flowchart of the AVI provision for the VSG is shown in Figure 9. In this flowchart, 𝐻ufmax,prv and 𝐻ofmax,prv denote the previously calculated values of virtual inertia. When the PEC is in a stable equilibrium state, the parameters 𝐻ufmax and 𝐻ofmax are updated, accordingly. Conversely, during transient events, these parameters retain their last calculated values. 4.1 Stability Proof of the Proposed Controller This section employs Lyapunov theory to evaluate the stability of the proposed controller. In this context, (11) can be reformulated as follows: (𝐻0+𝐻𝑎𝑑𝑑 tanh (𝑘|||| 𝑑𝜔 𝑑𝑡 ||||)𝑑𝜔 𝑑𝑡 =Δ𝑃 −𝐷𝑣𝑠𝑔Δ𝜔 (13) where 𝐻add is equal to either (𝐻ofmax −𝐻0)or(𝐻ufmax −𝐻0), both of which are positive. Let 𝑥=Δ𝜔,so𝑑𝜔 𝑑𝑡 =𝑑𝑥 𝑑𝑡 . Therefore, the 7of15 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 9 Flowchart of adjusting virtual inertia based on the VSG’s equilibrium point. equation becomes: (𝐻0+𝐻𝑎𝑑𝑑 tanh (𝑘|||| 𝑑𝑥 𝑑𝑡 ||||))𝑑𝑥 𝑑𝑡 =Δ𝑃 −𝐷𝑣𝑠𝑔𝑥(14) here, |𝑑𝑥∕𝑑𝑡|≥0,sotanh |𝑑𝑥∕𝑑𝑡|≥0. To find the equilibrium point, 𝑑𝑥 𝑑𝑡 is set to zero. Therefore, 𝑥=Δ𝑃 𝐷vsg =𝑥𝑒,where𝑥𝑒is the equilibrium point with 𝑑𝑥 𝑑𝑡 =0. By shifting the equilibrium point to the origin, and defining 𝑧=𝑥−𝑥𝑒,𝑥=𝑧+Δ𝑃 𝐷vsg is obtained. Also, 𝑑𝑥 𝑑𝑡 =𝑑𝑧 𝑑𝑡 .So,(14) is rewritten as: (𝐻0+𝐻𝑎𝑑𝑑 tanh (𝑘|||| 𝑑𝑧 𝑑𝑡 ||||))𝑑𝑧 𝑑𝑡 =Δ𝑃 −𝐷𝑣𝑠𝑔(𝑧+Δ𝑃 𝐷𝑣𝑠𝑔 ) =Δ𝑃 −𝐷𝑣𝑠𝑔𝑧−Δ𝑃 =−𝐷𝑣𝑠𝑔𝑧(15) now, the stability of 𝑧=0is analysed. by constructing a Lyapunov function as: 𝑉=1 2𝐷vsg𝑧2,theterm𝑉(𝑧) ≥0and 𝑉(𝑧)=0only at 𝑧=0are satisfied, so it is positive definite. The derivative term of the candidate Lyapunov is calculated as:  𝑉(𝑧) =𝑑𝑉 𝑑𝑧 .𝑑𝑧 𝑑𝑡 =(𝐷𝑣𝑠𝑔𝑧).𝑑𝑧 𝑑𝑡 (16) and by considering 𝑚=𝑑𝑧 𝑑𝑡 ,(15) become: 𝑧=− (𝐻0+𝐻𝑎𝑑𝑑 tanh(𝑘 |𝑚|))𝑚 𝐷𝑣𝑠𝑔 (17) therefore, by substituting (17) into (16):  𝑉(𝑧) =(𝐷𝑣𝑠𝑔𝑧).𝑚 =𝐷𝑣𝑠𝑔 (−(𝐻0+𝐻𝑎𝑑𝑑 tanh(𝑘 |𝑚|))𝑚 𝐷𝑣𝑠𝑔 ).𝑚 =− (𝐻0+𝐻𝑎𝑑𝑑 tanh(𝑘 |𝑚|))𝑚2(18) since 𝐻0+𝐻𝑎𝑑𝑑 tanh 𝑘|𝑚|>0,(18) is a non-positive term. consequently, ∙ 𝑉(𝑧) ≤0. ∙ 𝑉(𝑧) =0if and only if 𝑚=𝑑𝑧 𝑑𝑡 =0. At the point where 𝑚=𝑑𝑧 𝑑𝑡 =0, the equilibrium point is calculated as: (𝐻0+𝐻add tanh 𝑘|0|).0 =−𝐷vsg𝑧,gives𝑧=0.So,  𝑉=0 only at the equilibrium point, where 𝑧=0. Therefore, based on LaSalle’s Invariance Principle, the equilibrium point, where 𝑧= 0,thatis𝑥=𝑥𝑒is globally asymptotically stable, and the system converges to its equilibrium point as 𝑡→∞. 5Proposed AVI Provision for MT-HVDC According to (2), to safeguard the PEC from overload conditions, the maximum voltage deviation and RoCoV must be integrated into the calculations for the maximum virtual capacitor (𝐶ssmax ), 𝐶𝑠𝑠max 𝑉𝑑𝑐𝑛 𝑑𝑉𝑣 𝑑𝑡 max =Δ𝑃𝑐𝑎𝑝 −𝐷𝑣(Δ𝑉𝑠𝑠max ) 𝐶𝑠𝑠max = Δ𝑃𝑐𝑎𝑝 −𝐷𝑣(Δ𝑉𝑠𝑠max ) 𝑉𝑑𝑐𝑛 𝑑𝑉 𝑑𝑡 max (19) where, Δ𝑉ssmax denotes the maximum allowable variation in DC voltage from the steady-state equilibrium point and ΔPcap refers to the active power change capability of the power converter in response to contingencies. The primary source connected to the dcVSG plays a crucial role in determining the value of the 𝐶ssmax . When considering a stiff grid with rapid dynamics as the primary source for the dcVSG, the ability to change active power is a crucial parameter that influences the calculation of 𝐶ssmax . In this context, the active power of the dcVSG can exhibit variability, spanning from 0 to 1 p.u. Therefore, Δ𝑃cap can have different values depending on the sign of Δ𝑉dc asshowninFigure10.Based on the pathways illustrated, 𝐶ssmax is determined for each path using (20), named 𝐶uvmax and 𝐶ovmax . Following the system’s path, 8of15 IET Generation, Transmission & Distribution,2025 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License FIGURE 10 Active power and DC voltage variations in different contingencies. the corresponding virtual capacitor is applied accordingly, 𝐶𝑜𝑣max =𝑃𝑜𝑣 −𝐷𝑣(Δ𝑉𝑜𝑣) 𝑉𝑑𝑐𝑛 𝑑𝑉𝑣 𝑑𝑡 max ,𝐶 𝑢𝑣max =𝑃𝑟𝑒𝑠 −𝐷𝑣(Δ𝑉𝑢𝑣) 𝑉𝑑𝑐𝑛 𝑑𝑉𝑣 𝑑𝑡 max , 𝐶𝑠𝑠max ={𝐶𝑜𝑣max ,𝑑𝑣∕𝑑𝑡>0 𝐶𝑢𝑣max ,𝑑𝑣∕𝑑𝑡<0(20) where 𝐶uvmax and 𝐶ovmax represent the maximum virtual capacitor values corresponding to undervoltage and overvoltage conditions, respectively. Also, Δ𝑉uvmax denotes the peak voltage reduction for undervoltage scenarios, while Δ𝑉ovmax indicates the highest voltage increase in overvoltage conditions. Consequently, based on the discussions outlined in the preceding section, the AVI for the dcVSG is considered as follows: 𝐶𝑣= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐶0+(𝐶𝑢𝑣max −𝐶0)tanh𝑘(−𝑑𝑉 𝑑𝑡 ), 𝑑𝑉 𝑑𝑡 <0 &Δ𝑉 <0 𝐶0+(𝐶𝑜𝑣max −𝐶0)tanh𝑘(𝑑𝑉 𝑑𝑡 ), 𝑑𝑉 𝑑𝑡 >0 &Δ𝑉 >0 𝐶0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (21) It is essential to highlight that the cable capacitances and DC link capacitors play a significant role in stabilising fluctuations in DC voltage. However, Equation (21), derived from (19), relies on the capabilities of the PEC by considering its reserved power and the output power at its operating point as well as the DC voltage level. This design strategy allows the dcVSG to operate independently of the grid’s structural configuration, including cable capacitors, enhancing its adaptability to various grid architectures with differing parameters. The approach focuses on analysing the dynamics of DC voltage fluctuation and its rate of change, facilitating improved performance across diverse operational conditions. Also, as outlined in (20), the provided AVI by the dcVSG is contingent upon the direction of DC voltage deviations and the equilibrium point parameters. Therefore, it is essential to detect the equilibrium point. Considering the DC voltage as a state FIGURE 11 Flowchart of adjusting virtual inertia based on the dcVSG’s equilibrium point. variable within the MT-HVDC system allows the detection of the system’s equilibrium point, when 𝑑𝑣∕𝑑𝑡 =0. Still, in practical scenarios, RoCoV may temporarily decrease to zero during transient events and can display minor oscillations in steadystate conditions. Consequently, the system is deemed stable if the condition |𝑑𝑣∕𝑑𝑡|<𝛽is met for a specified duration, in which 𝛽is a pre-defined threshold. By modifying this threshold, the sensitivity of the controller to the RoCoF can be adjusted. Furthermore, the flowchart illustrating AVI provision in dcVSG can also be presented in Figure 11. Additionally, the same Lyapunov function can be employed to demonstrate the stability of the presented controller for dcVSG. 6 Simulation Results To assess the effectiveness of the proposed methodologies for delivering AVI in both AC and MT-HVDC grids, simulation results are presented, derived from the test grid depicted in Figure 4. The simulation results are conducted using MATLAB Simulink. The grid parameters, model, and additional details regarding this simulation are presented in Table A1 in the Appendix A. This analysis aims to demonstrate the effects of the 9of15 17518695, 2025, 1, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/gtd2.70154 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [23/10/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License