Virtual Inertia Scheduling (VIS)

• Virtual Inertia Scheduling (VIS) is the coordinated allocation, tuning, and procurement of synthetic inertia and damping from converter-based resources to preserve grid stability.
• VIS formulations leverage network-reduced swing equations, H2 norm minimization, and dynamic control to optimize performance and economic cost under frequency constraints.
• Practical VIS implementations demonstrate reduced RoCoF, improved frequency nadir, and decreased renewable energy curtailment, thereby enhancing grid resilience and market efficiency.

Searching arXiv for recent and foundational papers on virtual inertia scheduling and related allocation/market/control formulations. Virtual Inertia Scheduling (VIS) is the coordinated allocation, tuning, and, in some formulations, procurement of virtual inertia and virtual damping from inverter-based resources, energy storage systems, and converter-interfaced devices so that frequency quality and transient stability are preserved as synchronous-machine inertia declines. Across the literature, VIS appears as a network-constrained control-design problem, a real-time dispatch problem, a stochastic look-ahead scheduling problem, a dynamic optimal control problem, and a market-clearing problem. Common objectives are to reduce rate-of-change-of-frequency (RoCoF), improve frequency nadir, damp inter-area oscillations, reduce primary control effort, and satisfy device power and energy limits at minimum cost or minimum social cost (She et al., 2022, Poolla et al., 2017, Tuo et al., 2021).

1. Definition, scope, and physical role

VIS is rooted in the distinction between physical rotational inertia and converter-emulated inertia. Physical inertia is the kinetic energy stored in rotating masses of synchronous generators. Virtual inertia reproduces the frequency–power behavior of rotating machines using inverter control, including “synthetic inertia,” inertia emulation, virtual synchronous machine, synchronverter, and grid-forming virtual inertia implementations. In the formulations surveyed here, VIS determines where virtual inertia should be placed, how much should be allocated, how it should vary over time, and under what constraints or market rules it should be deployed (Poolla et al., 2017, Li et al., 2021).

At the device level, a recurring control law is an active-power injection proportional to frequency deviation and RoCoF. Representative forms include

$$P_{\mathrm{VI},k}^\star=\tilde{d}_k\,\hat{\omega}_k + \tilde{m}_k\,\dot{\hat{\omega}}_k,$$

for grid-following virtual inertia with PLL-based measurements, and

$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$

for grid-forming virtual inertia modeled as a voltage source with droop and inertia (Poolla et al., 2018). Other works use equivalent swing-like formulations such as

$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$

or

$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$

making the scheduling variables explicit as inertia and damping gains (Tuo et al., 2021, Ademola-Idowu et al., 2018).

A central theme is that virtual inertia is not a scalar system-wide commodity in the same sense as conventional reserve. Its value is strongly location dependent because it enters bus frequency dynamics through network coupling, disturbance location, and heterogeneous damping. This is why several VIS formulations are explicitly locational, and why one market design concludes that no single global clearing price emerges (Poolla et al., 2017).

2. Network models and performance criteria

Most VIS formulations start from network-reduced swing equations. A standard linearized model is

$$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$

with $\omega_i=\dot{\theta}_i$, bus inertia $m_i$, damping or droop $d_i$, and susceptance-based Laplacian coupling. With virtual inertia procurement or allocation,

$m_i = m_i^0 + \sum_{k\in\mathcal{A}_i}\mu_k,$

or, in grid-forming formulations,

$$M_{\mathrm{cl}}=M+M_{vi},\qquad D_{\mathrm{cl}}=D+D_{vi}.$$

This state-space structure underlies market-based VIS, H$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$0-norm allocation, and reduced-model methods (Poolla et al., 2017, Tuo et al., 2021).

A widely used performance language is based on H$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$1 metrics. For output $$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$2, the squared H$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$3 norm is

$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$4

where $$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$5 solves a Lyapunov equation. Different choices of $$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$6 encode different operational priorities. When the output is chosen as primary control effort,

$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$7

the resulting metric becomes

$$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$8

which is convex in inertia and linear in disturbance strengths (Poolla et al., 2017).

Other works encode frequency quality and coherency more directly. One reduced-model formulation penalizes angle differences and frequency excursions weighted by the Fiedler mode of the reduced Laplacian: $$\dot{\theta}_{\mathrm{VI},k}=\omega_{\mathrm{VI},k},\qquad \tilde{m}_k\,\dot{\omega}_{\mathrm{VI},k}=-\tilde{d}_k\,\omega_{\mathrm{VI},k}-P_{\mathrm{VI},k},$$9 so that the objective is the H$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$0 norm of the Fiedler Mode Weighted Coherency Index (Tuo et al., 2021). Another uses

$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$1

to trade frequency deviation, RoCoF, governor effort, and converter effort within a single H$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$2-norm design (Poolla et al., 2018).

Closed-form frequency-security formulas are also used when VIS is embedded in dispatch. In a center-of-inertia approximation for software-defined microgrids,

$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$3

with explicit expressions for RoCoF, nadir time, nadir magnitude, and steady-state deviation. This allows frequency limits to become second-order-cone or linear constraints in the scheduler (Chu et al., 2022). In stochastic look-ahead dispatch, aggregated inertia and damping enter formulas such as

$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$4

while the nadir constraint is approximated by a convex hull to preserve tractability (Shen et al., 2022).

3. Core scheduling formulations

VIS spans several mathematically distinct formulations. The following summary organizes representative approaches without implying a single canonical model.

Formulation family	Decision variables	Representative papers
Network H$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$5 allocation	$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$6, $$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$7, or $$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$8 by bus	(Poolla et al., 2015, Tuo et al., 2021, Ademola-Idowu et al., 2018)
Real-time dispatch with frequency constraints	$$P_{vi,i}=m_{vi,i}\,\dot{\omega}_i + d_{vi,i}\,\omega_i$$9, reserves, $$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$0, $$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$1	(She et al., 2022, Shen et al., 2022, Chu et al., 2022)
Dynamic optimal control	$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$2 trajectories	(Yan, 2019, Yan, 2019)
Market-based procurement	$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$3, bids $$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$4, VCG payments	(Poolla et al., 2017)
Learning-based coordination	$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$5, $$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$6 setpoints	(Stock et al., 2024)
Transient-stability-aware redispatch	$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$7	(Masoumi et al., 17 Jul 2025)

In centralized benchmark formulations, the operator minimizes performance cost plus procurement or operating cost. One market-oriented benchmark is

$$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$8

subject to $$\Delta P_{vsm,i}(t)=M_{v,i}\,\frac{d\Delta\omega_i}{dt}+D_{v,i}\,\Delta\omega_i,$$9. For the primary-effort metric, the robust performance target becomes the linear “valley-filling” condition

$$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$0

so buses below this threshold require procurement (Poolla et al., 2017).

In H$$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$1-based placement and design, VIS is often posed as constrained optimization over inertia and damping coefficients. A representative formulation is

$$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$2

subject to bounds on $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$3 and $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$4, together with Lyapunov equations for the observability and controllability Gramians. The regularization parameter $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$5 explicitly trades frequency nadir against damping or settling. Negative $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$6 encourages larger inertia and lower RoCoF or nadir, while positive $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$7 discourages inertia and favors faster settling (Ademola-Idowu et al., 2018).

Real-time economic dispatch formulations make inertia and damping part of the dispatch vector. VIS-based RTED schedules generator setpoints, reserves, IBR control modes, and virtual inertia or damping parameters every 5 minutes under RoCoF and nadir constraints. In one such model, the initial RoCoF constraint is

$$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$8

and the peak inertial power requirement of each IBR is computed analytically and enforced as reserve headroom (She et al., 2022).

Dynamic optimal-control VIS instead schedules a trajectory $$m_i\,\ddot{\theta}_i+d_i\,\dot{\theta}_i = p_{\mathrm{in},i}-\sum_{\{i,j\}\in\mathcal{E}} b_{ij}(\theta_i-\theta_j),$$9 over a disturbance window. In a structure-preserving network model, the storage bus obeys

$\omega_i=\dot{\theta}_i$0

and the optimal-control problem penalizes frequency deviations, angle deviations, and deviations from desired inertia while enforcing bus frequency bounds and storage power and energy constraints (Yan, 2019).

4. Spatial allocation, placement logic, and time variation

A major result across the literature is that VIS is intrinsically spatial. Uniform allocation is generally suboptimal. In a three-region, 12-bus case study, optimal inertia placement materially outperformed heuristic or uniform allocations, and under localized disturbance the optimal schedule concentrated strongly near the affected bus (Poolla et al., 2015). In the 12-bus market case study, buses needing additional inertia were 2, 4, 8, and 12, and cheapest co-located agents were selected first, but not always to full capacity because location and performance were coupled (Poolla et al., 2017).

Several works provide explicit spatial heuristics. One robust H$\omega_i=\dot{\theta}_i$1-allocation result shows that, when the objective is primary control effort, the optimal unconstrained allocation is proportional to the square root of disturbance strength,

$\omega_i=\dot{\theta}_i$2

whereas a robust worst-case target leads to a valley-filling structure that raises low-inertia buses toward a common threshold (Poolla et al., 2015). In the Fiedler-mode approach, the largest virtual inertia is assigned to buses with the largest $\omega_i=\dot{\theta}_i$3. In the IEEE 24-bus case, bus 7 had the largest $\omega_i=\dot{\theta}_i$4, experienced the largest oscillation under a contingency on bus 18, and received the largest virtual inertia allocation, while bus 23, with small $\omega_i=\dot{\theta}_i$5, received much less (Tuo et al., 2021).

Time variation in VIS arises in two distinct ways. First, schedulers update static setpoints in repeated operational intervals, such as hourly windows in software-defined microgrids or 5-minute real-time dispatch in transmission systems (Chu et al., 2022, She et al., 2022). Second, some controllers deliberately make inertia itself state dependent during the disturbance. An adaptive inertia law is

$\omega_i=\dot{\theta}_i$6

so inertia increases rapidly when RoCoF is large, then decays back to a baseline to avoid sustaining inter-area oscillations (Fritzsch et al., 2023). That work reports that, on the RTS-96 and PanTaGruEl systems, adaptive inertia improved resynchronization time and RoCoF norm in most scenarios, with resynchronization improved in 91% of cases and RoCoF norm improved in 92% of cases for the PanTaGruEl study (Fritzsch et al., 2023).

A related but distinct dynamic-grid-forming formulation uses an auxiliary signal $\omega_i=\dot{\theta}_i$7 driven by power-limit violations to schedule both virtual inertia and virtual damping: $\omega_i=\dot{\theta}_i$8 together with phase and frequency shift functions. In RTDS case studies, this improved robustness under faults, islanding, and large power-balance oscillations relative to traditional synchronous generation of comparable size (Khamisov et al., 2024).

5. Procurement, dispatch, and market implementation

A distinctive line of VIS research treats virtual inertia as a procured ancillary service. In the market mechanism of (Poolla et al., 2017), each provider submits a convex, non-decreasing bid $\omega_i=\dot{\theta}_i$9, a location, and a capacity $m_i$0. The system operator clears the market by solving

$m_i$1

and compensates each provider via a Vickrey–Clarke–Groves payment equal to the externality that provider removes from total system cost (Poolla et al., 2017).

The formal properties are strong: truthful bidding is a dominant strategy, individual rationality holds, and the resulting allocation matches the centralized social-welfare optimum under truthful bids. In the three-region, 12-bus case study, market-based allocation achieved the same performance target as a regulatory approach but at much lower total procurement cost: market-based or centralized cost was approximately $m_i$2 arbitrary units, whereas the regulatory allocation cost was approximately $m_i$3 (Poolla et al., 2017).

Dispatch-oriented VIS embeds inertia directly in security-constrained economic dispatch. In VIS-RTED, the scheduled variables include synchronous-generator power and regulation reserves, IBR power setpoints, IBR headroom, control mode, and virtual inertia and damping parameters. On the modified IEEE 39-bus system, four methods were compared. Ordinary RTED without inertia support had four RoCoF violations; RTED with fixed VSG mode but without inertia-support reserve had four IBR capacity violations; VIS-RTED with fixed $m_i$4 was secure but more expensive; complete VIS-RTED with scheduled $m_i$5 and reserves achieved zero violations with lower reserve requirement than the fixed-parameter case (She et al., 2022).

Stochastic look-ahead dispatch generalizes this further by making RES and ESS inertia and droop coefficients rolling-horizon decision variables under chance constraints derived from Gaussian mixture models. On the modified IEEE 24-bus system, online VIS reduced RES curtailment from $m_i$6 to $m_i$7; on a provincial power system in China, it reduced RES curtailment from $m_i$8 to $m_i$9, while keeping RoCoF, nadir, and steady-state metrics within limits (Shen et al., 2022).

A recent reformulation treats inertia itself as a bidirectional power-oriented service. Under this view, the inertial power contribution is

$d_i$0

and the committed inertia service is

$d_i$1

This makes synchronous inertia and virtual inertia commensurable in SCED and yields explicit inertia prices from dual variables. In a modified IEEE 30-bus system, the low-synchronous-inertia case showed that bidirectional scheduling increased inertia price from $d_i$2, reflecting scarcity of nadir capability (Park et al., 8 Jan 2026).

6. Learning-based, distributed, and transient-stability-aware VIS; limitations

Beyond model-based optimization, VIS has been approached through reinforcement learning and distributed optimization. A physics-informed actor–critic method coordinates virtual inertia and damping from inverter-based resources in distribution systems when network models are uncertain. The decision variables are plant-level $d_i$3 and $d_i$4, the reward penalizes economic cost, voltage deviation, and budget violations, and the critic is regularized by a swing-equation surrogate. In the IEEE 37-bus reference scenario, the reported final reward was $d_i$5 in 365 iterations for PI-AC, versus $d_i$6 in 686 iterations for a data-driven actor–critic and $d_i$7 in 928 iterations for a genetic algorithm (Stock et al., 2024).

Distributed VIS also appears in MPC-based control of storage devices that schedule both direct power and virtual inertia. In that framework, storage power $d_i$8 and virtual inertia $d_i$9 are co-optimized under system dynamics, frequency constraints, and storage energy limits. A key qualitative result is that direct power scheduling dominates early steady-state correction, whereas time-varying inertia becomes more important once power or energy limits bind (Yan, 2019).

Transient stability introduces a further layer beyond nadir and RoCoF. In an IEEE 39-bus, 70% IBR system, a deep-learning predictor of post-fault dynamics is coupled with Information Gap Decision Theory to produce risk-averse redispatch and VIS. Under a severe three-phase fault on Line 33 near Bus 26, conventional VIS failed to restore COI stability and triggered UFLS, whereas the proposed method restored COI stability within approximately 12 s. The operating cost increased from $m_i = m_i^0 + \sum_{k\in\mathcal{A}_i}\mu_k,$0, described as approximately a 5% increase relative to conventional VIS alone, but avoided the $m_i = m_i^0 + \sum_{k\in\mathcal{A}_i}\mu_k,$1 UFLS cost and prevented collapse (Masoumi et al., 17 Jul 2025).

Several recurring limitations define the current boundaries of VIS. Most formulations rely on linearized or reduced-order models, small-angle or small-signal approximations, fixed voltage magnitudes, and simplified disturbance sets. Detailed converter dynamics, measurement noise, current saturation, response delays, and energy-buffer degradation are often excluded or only indirectly represented (Poolla et al., 2017, Poolla et al., 2018). Some solution methods are nonconvex and depend on gradient heuristics or relaxations; when VCG-type market properties or distributed-optimization guarantees require global optima, these assumptions become significant (Poolla et al., 2017, Han et al., 2021).

A recurrent misconception is that “more inertia is always better.” Multiple sources contradict this. Regularized H$m_i = m_i^0 + \sum_{k\in\mathcal{A}_i}\mu_k,$2 design explicitly treats inertia as a trade-off against settling (Ademola-Idowu et al., 2018). Adaptive-inertia studies show that high inertia is desirable immediately after a fault but can sustain coherent inter-area oscillations if not reduced later (Fritzsch et al., 2023). In DC grids with constant-power loads, there is even an analytical maximum virtual inertia beyond which stability deteriorates; the optimal and maximum inertia are derived in closed form from converter droop, line inductance, and CPL parameters (Tu et al., 2022). This suggests that VIS is best viewed not as monotonic inertia maximization but as constrained spatiotemporal shaping of effective electromechanical response.

Taken together, the literature defines VIS as a family of technically connected but operationally distinct problems: locational inertia allocation, dynamic parameter scheduling, stochastic economic dispatch, market procurement, distributed coordination, and transient-stability-aware redispatch. The unifying principle is that virtual inertia has to be scheduled jointly with damping, reserves, power headroom, and network location rather than treated as a fixed add-on parameter (She et al., 2022, Poolla et al., 2017, Masoumi et al., 17 Jul 2025).