Regional Inertia Security Region (R-ISR)

• Regional Inertia Security Region (R-ISR) is a framework that defines secure sets of regional inertia, frequency-response, and network variables to ensure stability after disturbances.
• It integrates detailed electromechanical dynamics with computational models that account for nonuniform inertia distribution and oscillatory behavior across interconnected regions.
• The approach supports robust operational scheduling and planning by informing ancillary service procurement and system expansion through regional, rather than uniform, inertia assessments.

Regional Inertia Security Region (R-ISR) denotes the secure set of regional inertia allocations—and, in several formulations, associated regional frequency-response, demand, disturbance, and network variables—for which regional frequency trajectories satisfy prescribed security limits after credible contingencies. The term is stated explicitly for RoCoF-constrained regional inertia security in multi-region systems (Liu et al., 21 Jul 2025). Closely related constructions appear in regional frequency-secured scheduling and planning, where the secure set is expressed through regional RoCoF, nadir, quasi-steady-state, or local oscillatory constraints even when the exact term is not used (Badesa et al., 2020, Wang et al., 24 Jul 2025), and in topology-aware effective regional inertia assessment (Pinheiro et al., 4 Nov 2025).

1. Electromechanical rationale

The central premise of R-ISR is that a single Centre-of-Inertia (COI) or single-bus frequency is not sufficient in systems with high RES penetration, non-uniform inertia distribution, and geographically distinct areas. In the regional-frequency formulation of power-system planning, “the unbalanced allocation of RESs at different areas creates a non-uniform distribution of inertia, leading to spatial gradients and distinct regional frequencies,” and “even though the frequency requirement under COI is satisfactory, the RoCoF or nadir of some regions may still go beyond the limits” (Wang et al., 24 Jul 2025). In the Great Britain two-region scheduling model, frequency security therefore has to hold in each electromechanical region, not only for a single COI frequency (Badesa et al., 2020).

Within this perspective, a “region” is dynamical rather than merely administrative. Regions are defined from electromechanical coupling to consider intra-area oscillations (Wang et al., 24 Jul 2025), or as coherent areas in which machines show coherent frequency behaviour (Badesa et al., 2020). The review literature generalizes this point by arguing that zonal inertia estimates are more flexible than system inertia estimates, because aggregating zonal inertia contributions gives the total system inertia, while zonal or individual contributions cannot be derived from the system estimate (Heylen et al., 2020).

Regional security is then expressed through region-specific trajectories $\Delta f_n(t)$, region-specific RoCoF $\frac{d\Delta f_n(t)}{dt}$, region-specific nadir, and a system-wide quasi-steady-state condition. A standard regional-frequency statement is

$$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$

for every region $n$ and relevant time instant, together with a quasi-steady-state reserve condition (Wang et al., 24 Jul 2025). This makes R-ISR a regional-frequency security construct rather than a purely inertial inventory.

2. Formal definitions and state variables

In the explicit RoCoF-constrained formulation, fix an operating condition $\boldsymbol{x}$, a disturbance $\Delta \boldsymbol{P}^{D}$, and a RoCoF limit $RoCoF^{LIM}$. The RoCoF-secure inertia set is

$$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$

with inertia bounds

$$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$

and enclosed Regional Inertia Security Region

$$\boldsymbol{\tilde{\Omega}} = \boldsymbol{\Omega} \cap \boldsymbol{\Upsilon}.$$

Inertia security is therefore equivalent to $\frac{d\Delta f_n(t)}{dt}$0 (Liu et al., 21 Jul 2025). In this formulation, $\frac{d\Delta f_n(t)}{dt}$1 is the vector of regional inertias, and the boundary is the set where the maximum regional RoCoF reaches the limit.

Related formulations enlarge the state space beyond $\frac{d\Delta f_n(t)}{dt}$2. In the regional frequency-constrained planning model, the ICNN input vector $\frac{d\Delta f_n(t)}{dt}$3 contains “inertia, PFR, EFR, demand, fault information (i.e., largest loss and fault location), etc.”, and the feasible regional security region is an admissible set

$\frac{d\Delta f_n(t)}{dt}$4

in the feature space used by the learned surrogate (Wang et al., 24 Jul 2025). In the Great Britain regional unit-commitment formulation, the admissible tuples are $\frac{d\Delta f_n(t)}{dt}$5, and the regional security region becomes the intersection of region-wise linear inequalities enforcing RoCoF, nadir, and possibly quasi-steady-state constraints (Badesa et al., 2020).

This suggests two nested interpretations. A narrow interpretation treats R-ISR as a region in regional inertia space $\frac{d\Delta f_n(t)}{dt}$6. A broader interpretation treats it as a secure region in the joint space of regional inertia and fast frequency services, with disturbance location, damping, and network coupling either fixed or parameterized.

3. Dynamics, boundary geometry, and tractable representations

The nontriviality of R-ISR arises from multi-region inertial dynamics. In the RoCoF-constrained regional model, the regional COI frequencies $\frac{d\Delta f_n(t)}{dt}$7 and regional COI angles $\frac{d\Delta f_n(t)}{dt}$8 satisfy a linearized multi-region state-space model whose coefficients depend on the diagonal regional inertia matrix and the inter-regional synchronizing matrix $\frac{d\Delta f_n(t)}{dt}$9 (Liu et al., 21 Jul 2025). After modal decomposition, the regional RoCoF in region $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$0 under a disturbance in region $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$1 can be written as a superposition of one decaying exponential and multiple decaying sinusoids,

$$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$2

Because the maximum RoCoF occurs after several oscillatory swings, and because the maximum-swing number can change with inertia, the R-ISR boundary is nonlinear and non-convex (Liu et al., 21 Jul 2025).

A closely related analytical form appears in the Great Britain two-region model, where the regional RoCoF condition is

$$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$3

with the first term corresponding to the COI RoCoF and the second to the inter-area oscillation contribution (Badesa et al., 2020). In the planning formulation based on coupled regional swing equations, inter-area power transfer is explicitly represented through $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$4 and electrical stiffness $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$5, so spatial differences in frequency are generated by the interaction among regional inertia, damping, and tie-line coupling (Wang et al., 24 Jul 2025).

Three computational representations have emerged. The explicit R-ISR paper computes candidate local maxima of regional RoCoF, decomposes the global non-convex boundary into multiple “simple” boundaries $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$6, and then convexifies the enclosed secure region by convex decomposition and big-$$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$7 disjunctive constraints (Liu et al., 21 Jul 2025). The Great Britain formulation replaces the analytical regional conditions by conservative linear inequalities obtained from constrained least squares fitted on boundary samples, producing a convex polytope in $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$8 space (Badesa et al., 2020). The regional planning formulation learns a convex surrogate with an enhanced ICNN and embeds it through MILP ReLU constraints with $$\left|\frac{d\Delta f_n(t)}{dt}\right| \le \overline{RoCoF}, \qquad |\Delta f_n(t)| \le \Delta \overline{f},$$9, using principled weight initialization to address gradient-vanishing issues of non-negative ICNN weights (Wang et al., 24 Jul 2025).

A plausible implication is that R-ISR is not a single mathematical object but a family of secure-set representations: exact but non-convex, conservative linearized, or learned convex.

4. Effective regional inertia and estimation of regional inertia states

R-ISR depends on how regional inertia is defined. A topology-aware formulation begins with nodal inertia. For bus $n$0,

$n$1

so $n$2 is the effective inertia seen at that bus. On a coherent region $n$3, the effective regional inertia is defined as

$n$4

the mean of topology-dependent nodal inertias in that region (Pinheiro et al., 4 Nov 2025). This differs from the conventional regional inertia obtained by summing device inertia constants, because $n$5 depends on topology, operating point, and electrical coupling. The same work extends slow coherency to all buses through a quadratic eigenvalue problem $n$6, then applies eigengap selection and clustering to produce coherent regions (Pinheiro et al., 4 Nov 2025).

A fully data-driven approach estimates regional inertia through the regional COI proxy. The pilot-bus is selected by typicality-based data analysis using filtered frequency and active-power deviations over a $n$7 s inertial window. Tie-line power deviations are then used as input to an ARMAX model, reduced to a first-order transfer function $n$8, from which the equivalent regional inertia is obtained as

$n$9

On the IEEE 68-bus system, all regional inertia estimation errors are under about $\boldsymbol{x}$0 without motors; with motors they remain mostly below $\boldsymbol{x}$1, with one region at about $\boldsymbol{x}$2 (Lugnani et al., 2022). This framework is explicitly designed to include load inertial contribution and COI displacement.

A measurement-based regional approach estimates effective regional inertia directly from disturbance data. In the CAISO study, regional frequency is the average of CAISO FDR measurements, a two-point mean filter is applied, disturbance onset is detected by maximizing the difference between pre- and post-event RoCoFs, and the representative regional RoCoF is the peak slope in a $\boldsymbol{x}$3 s moving window over the first $\boldsymbol{x}$4 s after onset. Regional inertia is then inferred as

$\boldsymbol{x}$5

Across seven major CAISO events, interconnection RoCoF ranges from $\boldsymbol{x}$6 to $\boldsymbol{x}$7 mHz/s, regional RoCoF from $\boldsymbol{x}$8 to $\boldsymbol{x}$9 mHz/s, and local RoCoF from $\Delta \boldsymbol{P}^{D}$0 to $\Delta \boldsymbol{P}^{D}$1 mHz/s; regional-to-system inertia ratio ranges from $\Delta \boldsymbol{P}^{D}$2 to $\Delta \boldsymbol{P}^{D}$3; inertial support arrival time is typically about $\Delta \boldsymbol{P}^{D}$4 s (Dulal et al., 26 Nov 2025).

The broader estimation literature classifies inertia methods by time horizon—offline post mortem, online discrete, online continuous, and forecast—and by scope—system, zone, synchronous generation, embedded generation, clustered generators or node, and demand (Heylen et al., 2020). In R-ISR terms, these methods determine whether the secure-set boundary is evaluated with nameplate-based, event-inferred, ambient-estimated, or forecasted regional inertia.

5. Embedding R-ISR in operation, scheduling, and planning

R-ISR becomes operational when it is inserted into optimization models. In regional-frequency stochastic unit commitment for Great Britain, the model simultaneously optimizes energy production and ancillary services while enforcing regional RoCoF and nadir constraints at every scenario-tree node. Regional inertia $\Delta \boldsymbol{P}^{D}$5 and regional PFR $\Delta \boldsymbol{P}^{D}$6 are linear functions of commitment and dispatch variables, the scenario tree spans a 24-hour horizon with 5 quantiles of net-demand forecast error, and a rolling planning approach resolves the problem hourly (Badesa et al., 2020).

In the RoCoF-constrained regional inertia adjustment model, R-ISR is embedded through convexified polyhedral constraints. Regional inertia aggregation is

$\Delta \boldsymbol{P}^{D}$7

and the optimization minimizes generation costs, start-up and shut-down costs, and virtual inertia costs subject to power balance and disjunctive R-ISR membership constraints $\Delta \boldsymbol{P}^{D}$8 with $\Delta \boldsymbol{P}^{D}$9 (Liu et al., 21 Jul 2025).

In expansion planning, regional frequency security is enforced through a learned convex surrogate. The planning model first extracts regional frequency constraints via an enhanced ICNN and then embeds them into the original optimization. An adaptive genetic algorithm with sparsity calculation and local search separates the planning model into two stages and solves it iteratively. The input space contains inertia, PFR, EFR, demand, and fault information; the output is a regional frequency security indicator constrained by a threshold $RoCoF^{LIM}$0 (Wang et al., 24 Jul 2025).

The planning and scheduling implications are explicit across the literature. Regional constraints allow the procurement of ancillary services in each region (Badesa et al., 2020). They also provide a basis for regional reserve or inertia products, security rules conditional on loss location, transmission reinforcement valuation, and the location of synchronous condensers, fast resources, or virtual inertia devices (Badesa et al., 2020, Pinheiro et al., 4 Nov 2025). Forecasting frameworks further suggest using regional inertia estimates and forecasts in the procurement of very fast frequency response services and enhanced and coordinated frequency control methods (Heylen et al., 2020).

6. Empirical findings, misconceptions, and limitations

A recurring misconception is that system-wide inertia or uniform-frequency security is sufficient. The Great Britain studies reject this directly. For a $RoCoF^{LIM}$1 GW loss in England, the regional-frequency-secured schedule requires additional $RoCoF^{LIM}$2 inertia and PFR relative to the uniform-frequency model, specifically about $RoCoF^{LIM}$3 GW·s of inertia and $RoCoF^{LIM}$4 MW of PFR, with carbon intensity increases of $RoCoF^{LIM}$5 gCO$RoCoF^{LIM}$6/kWh versus no frequency security and $RoCoF^{LIM}$7 gCO$RoCoF^{LIM}$8/kWh versus uniform-frequency security (Badesa et al., 2020). For a $RoCoF^{LIM}$9 GW loss in Scotland, a minimum inertia of about $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$0 GW·s is required in Scotland; under the uniform model, the SUC can schedule zero inertia in Scotland, which the regional model identifies as insecure (Badesa et al., 2020).

A second misconception is that increasing inertia in one region must monotonically improve that region’s RoCoF. The explicit R-ISR study reports the opposite as a counter-intuitive result: increasing $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$1 can first reduce, then increase, and then reduce maximum RoCoF because the timing and constructive addition of oscillatory components change as inertia varies (Liu et al., 21 Jul 2025). In that same three-region case, the proposed R-ISR boundary approximation has $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$2 error relative to simulation, compared with $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$3 for a COI-based approximation and $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$4 for a conservative approximation; over the 4-hour inertia-adjustment case, the proposed formulation yields a total cost of $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$5k for the conservative method, whereas the $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$6k COI-based solution is insecure (Liu et al., 21 Jul 2025).

A third misconception is that any added device inertia is necessarily beneficial. The topology-aware effective regional inertia framework shows that adding inertia does not uniformly improve frequency response and can even worsen it. In the IEEE 39-bus case, a GFM at bus $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$7 has a derived minimum inertia requirement $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$8 s: $$\boldsymbol{\Omega} = \left\{ \boldsymbol{H} \;\middle|\; \mathcal{F}\!\left[\boldsymbol{H}, \boldsymbol{x}, \Delta \boldsymbol{P}^{D}\right] \le RoCoF^{LIM} \right\},$$9 s worsens nodal and regional inertia, $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$0 s is approximately neutral, and $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$1 s improves both inertia and RoCoF (Pinheiro et al., 4 Nov 2025). In the IEEE 68-bus case, replacing a low-inertia synchronous generator at bus $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$2 with a GFL improves effective regional inertia and frequency behaviour, while adding four $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$3 s GFMs at already strong buses reduces effective regional inertia and worsens frequency behaviour (Pinheiro et al., 4 Nov 2025).

Measurement-based evidence also undermines system-average interpretations. In CAISO, regional RoCoF is consistently about $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$4–$$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$5 interconnection RoCoF, and local RoCoF can be about $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$6–$$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$7 the regional average, with the worst local value reaching $$\boldsymbol{\Upsilon} = \left\{ \boldsymbol{H} \;\middle|\; \boldsymbol{H}^{LO} \le \boldsymbol{H} \le \boldsymbol{H}^{UP} \right\},$$8 mHz/s (Dulal et al., 26 Nov 2025). This is consistent with the broader review claim that lower inertia levels increase the risk for angle swings and oscillations between areas, and motivates zonal or spatial inertia profiles rather than a single system scalar (Heylen et al., 2020).

The main limitations are methodological rather than conceptual. The effective regional inertia framework is based on linearized swing equations and a lossless DC network, is operating-point dependent, and focuses on the inertial timescale rather than longer-term controls (Pinheiro et al., 4 Nov 2025). The fully data-driven pilot-bus and ARMAX method is disturbance-based, uses a short inertial window, assumes coherent regions, and depends on sufficient PMU coverage (Lugnani et al., 2022). The forecasting and estimation review emphasizes that uncertainty quantification is still limited and argues for probabilistic inertia forecasting to assess the risk of RoCoF and underfrequency relay tripping (Heylen et al., 2020). A plausible implication is that practical R-ISR deployment requires margins, multi-point validation, and continuous recalibration as topology, resource mix, and controller settings change.