Virtual Capacity Curves in Energy Systems

• Virtual capacity curves are mathematical representations that map latent flexibility and hidden system characteristics into operationally useful boundaries.
• They are applied across diverse domains—including battery diagnostics, virtual power plants, flexible demand, and biometric provisioning—to derive counterfactual or aggregated performance measures.
• By reducing complex behaviors into simplified curves or envelopes, these methods facilitate reliability assessment, cost optimization, and real-time decision making.

Searching arXiv for papers that use or closely match “virtual capacity curves” across domains. Virtual capacity curves are constructed representations of capability, separability, or diagnostic structure that are not read directly from a single physical asset’s native characteristic. Recent arXiv literature uses the notion in several technically distinct ways: as counterfactual diagnostic curves for batteries, as reliability-adjusted reserve supply curves for virtual power plants, as charge-only virtual storage envelopes for large flexible loads, as interface capability and adequacy-allocation frontiers in power systems, and as capacity relations for provisioning non-colliding virtual biometric identities (Zhou et al., 26 Feb 2025, Zapparoli et al., 6 Oct 2025, Chaudhary et al., 6 Jul 2026, Singhal et al., 2018, Tindemans et al., 2018, Ji et al., 18 May 2026). This suggests that the unifying theme is not a single canonical curve, but a family of domain-specific constructions that translate latent or distributed flexibility into an operationally meaningful boundary.

1. Conceptual scope

Across the cited literature, “virtual” does not denote mere simulation. In battery diagnostics, it denotes a counterfactual constant-current curve reconstructed from non-constant-current data. In power-system applications, it often denotes an aggregated or market-facing capability that no individual device possesses alone. In adequacy studies, it denotes an equivalent firm-capacity contribution. In biometric provisioning, it denotes the achievable count of synthetic identities that can be placed in unclaimed regions of an embedding manifold without colliding with enrolled real identities.

Domain	Curve object	Operational meaning
Battery diagnostics	Virtual incremental capacity / virtual differential voltage	Counterfactual CC ICA/DVA signatures from general charging
Virtual power plants	Reserve supply curve / capability envelope	Reliability-guaranteed reserve quantity and cost
Flexible demand	Charge-only virtual storage envelope	Storage-equivalent curtailment capability
Power-system interfaces	Aggregated VAR capability / capacity allocation curve	Feeder-level reactive support or interconnector adequacy trade-off
Biometric identities	Provisioning-capacity relation	Achievable non-colliding virtual identity count

A common structural pattern is visible. First, a physically distributed or nonstandard resource is mapped into a simpler external representation. Second, the mapping is constrained by hidden state, uncertainty, or network physics. Third, the resulting curve is operationally useful precisely because it is not a raw device characteristic. This suggests that virtual capacity curves are best understood as reduced-order boundary objects rather than as universal constitutive laws.

2. Counterfactual diagnostic curves in battery health estimation

In battery diagnostics, the closest precise usage is the introduction of virtual incremental capacity (VIC) and virtual differential voltage (VDV) for degradation monitoring under general charging profiles (Zhou et al., 26 Feb 2025). Conventional incremental capacity analysis and differential voltage analysis are defined under constant-current charging as

$$IC_{\mathrm{CC}}=\frac{dQ_{\mathrm{CC}}}{dV_{\mathrm{CC}}}, \qquad DV_{\mathrm{CC}}=\frac{dV_{\mathrm{CC}}}{dQ_{\mathrm{CC}}},$$

and their peaks, shifts, heights, and areas are diagnostically informative. The central limitation is that under general charging current profiles, direct differentiation of measured $Q$–$V$ data does not produce ICA/DVA curves that are meaningful in the standard diagnostic sense. The virtual curves are therefore defined as the curves the same cells or modules, in the identical state of degradation, would exhibit under CC charging at the same reference C-rate.

The battery paper is explicit that this is not a generic virtual $Q(V)$ construction. Rather, it reconstructs $\widehat Q_{\mathrm{CC}}(k)$, $\widehat V_{\mathrm{CC}}(k)$, and especially $\widehat{IC}_{\mathrm{CC}}(k)$ from general charging inputs, then obtains

$$\widehat{DV}_{\mathrm{CC}}(k)=\frac{1}{\widehat{IC}_{\mathrm{CC}}(k)}.$$

Measured data are the current, voltage, and Coulomb-counted transferred charge,

$$\{I(t)\},\quad \{V(t)\},\quad \{\Delta Q(t)\},\qquad \Delta Q(t)=\int_0^t I(\tau)\,d\tau,$$

which are reparameterized as $\{I(\Delta Q)\}$ and $Q$0. Notably, SOC is not used as an online model input. Instead, admissible charging snippets are restricted by minimum SOC-window conditions,

$Q$1

The reconstruction pipeline standardizes variable-length charging segments by downsampling in the $Q$2 domain with

$Q$3

then applying one-sided symmetric padding and standardization

$Q$4

A 1D U-Net or Mobile U-Net maps the two-channel input $Q$5 to standardized CC targets; VIC/VDV are then used either directly for diagnosis or indirectly through feature extraction. For SOH regression, Conv-Net and Mobile-Net reuse the encoder through transfer learning and output $Q$6.

The architecture details matter because they show that the “virtual curve” is a learned surrogate for a prescribed counterfactual experiment. U-Net uses 1D convolutions, max pooling, transposed convolutions, skip connections, PReLU, batch normalization, He normal initialization, and large kernel size 11 in designated layers to help filter noise and generate smoother reconstructed profiles. Mobile U-Net replaces standard convolutions with depthwise separable convolutions and replaces transposed convolution with simple upsampling by repetition. The reported parameter counts are 95,503 for U-Net, 32,425 for Mobile U-Net, 73,361 for Conv-Net, and 27,118 for Mobile-Net.

Training uses a proprietary dataset of 96 NMC622 battery modules with nominal module capacity 208 Ah, module configuration of 3 cells in parallel, SOH range 100% to 86%, three fast-charging protocols, 40,512 input-output pairs, and 405,120 pairs after random truncation. The low-C-rate reference profile contains CC regimes at 86 A and 64.5 A followed by constant voltage; ground-truth IC/DV targets are taken from the first CC regime. The train/validation/test split is 60%/20%/20%, with $Q$7, MSE loss, ADAM, and early stopping.

The principal result is that most virtual IC/DV curves are close to their corresponding actual IC/DV curves and preserve diagnostically meaningful peaks and feature regions. In a feature-based SOH case study, actual CC features yield 0.53% SOH RMSE, U-Net virtual features under fast charge yield 0.60%, and Mobile U-Net virtual features yield 0.61%. Performance remains similar for 15%–90%, 30%–80%, and 45%–90% SOC windows without retraining, though shorter segments degrade slightly because they contain less information. A frequent misconception is therefore incorrect: these virtual capacity-related curves are not arbitrary learned embeddings, but explicit surrogates for standard CC diagnostic signatures.

3. Reserve, cost, and frequency-support envelopes in virtual power plants

In virtual power plants, the closest analogue to a virtual capacity curve is a reserve-capacity supply curve or a capability envelope, not a static device rating. One formulation computes the maximum feasible reserve under uncertainty and then attaches marginal cost, thereby producing a reserve supply curve that maps reserve quantity to risk-adjusted price (Zapparoli et al., 6 Oct 2025). For uncertain inputs $Q$8, the technical layer solves a MILP

$Q$9

subject to product-defining constraints, network constraints, and DER constraints. The offered reserve is then a quantile,

$V$0

with $V$1. Cost is obtained by solving a quantity-constrained cost-minimization model and reading the dual variable of the reserve-quantity constraint,

$V$2

then taking a cost quantile

$V$3

This construction is explicitly product-specific. Reserve quantity must be deliverable across the full delivery interval and all required activation states, not merely at an instant. Reliability therefore reshapes the curve horizontally, while explicit and opportunity costs shape it vertically. In the Swiss LV case study, the representative VPP uses a 97-bus radial network with 150 kW peak non-dispatchable load, about 150 kWp rooftop PV, 85 kW heat pumps, 75 kWh BESS, and 40 EV charging events. For a day-ahead, symmetrical, 4 h duration product with 8:00–12:00 delivery, 5 min ramp time, and 99.9% reliability, the VPP can reliably offer

$V$4

while direct Monte Carlo gives 56.2 kW. Subset simulation reaches 1% estimator COV with 2800 function evaluations instead of 9000, a 69% reduction, and the resulting average reserve costs span roughly 0.34 to 0.40 CHF/kW. The paper emphasizes that opportunity costs drive pricing.

A related but distinct formulation appears for dynamic virtual power plants providing fast frequency regulation (Zhu et al., 2024). There, the virtual capacity object is an implicit capability region in the $V$5 plane rather than a reserve-price curve. The aggregated DVPP is represented through virtual inertia and virtual damping,

$V$6

with security constraints on RoCoF, nadir, and quasi-steady-state deviation. The RoCoF relation is

$V$7

which directly yields a minimum required inertia. The nadir boundary is convexified as

$V$8

and the QSS condition is expressed through

$V$9

The resulting feasible region is therefore a control-dependent capacity envelope.

This envelope is not monotone in any naive sense. The case study reports that, with $Q(V)$0, increasing $Q(V)$1 from 22.25 to 60.00 raises the overshoot metric $Q(V)$2 from 0.6287 to 0.9161. A common simplification—that more virtual inertia is always better—is therefore not supported by the reported dynamics. The paper’s own classification is careful: it provides an implicit analytical relationship and feasible operating envelope, not an explicit named virtual capacity curve.

4. Charge-only virtual storage envelopes for flexible demand

A different usage appears in the representation of large flexible loads as virtual storage (Chaudhary et al., 6 Jul 2026). The load model is defined by per-interval power bounds and a horizon-wide energy window,

$Q(V)$3

By defining curtailment relative to rated draw as

$Q(V)$4

the feasible set becomes

$Q(V)$5

with

$Q(V)$6

Cumulative curtailment energy,

$Q(V)$7

then acts as a virtual storage state.

The paper’s equivalence result is exact within the base model: every feasible large-load trajectory under power-bound and energy-window constraints is a valid charge trajectory of a charge-only virtual storage device operating at unity accounting efficiency in the grid power balance. This is a strong statement of curve equivalence. The corresponding virtual storage feasible set enforces

$Q(V)$8

with no discharge analogue. The operational consequence is that the appropriate virtual capacity object is not a symmetric battery chart but a finite-horizon cumulative-energy envelope. A direct implication of the stated model is the pair of reachable bounds

$Q(V)$9

which define a time-indexed band within which any feasible nondecreasing virtual-storage trajectory must remain. This is a plausible interpretation of a virtual capacity curve in the strict geometric sense.

For a portfolio co-located with BESS, aggregation reduces constraint count from $\widehat Q_{\mathrm{CC}}(k)$0 to $\widehat Q_{\mathrm{CC}}(k)$1 and yields a co-dispatch price for both resources. In the IEEE RTS-GMLC validation with a 200 MW PEM electrolyzer at bus 118, a 250 MW data center at bus 218, a 300 MW aluminum potline at bus 318, and a 200 MW / 400 MWh BESS, the aggregate virtual-storage parameters are $\widehat Q_{\mathrm{CC}}(k)$2 MW, $\widehat Q_{\mathrm{CC}}(k)$3 MWh, and $\widehat Q_{\mathrm{CC}}(k)$4 MWh. On the July 5 reference day, baseline procurement cost is $\widehat Q_{\mathrm{CC}}(k)$5275,473 for savings of $\widehat Q_{\mathrm{CC}}(k)$$611,398 (3.03$$112,575 (29.89%). Over 14 days, mean daily co-dispatch savings are 33.9%.

The paper is also explicit about scope. The equivalence is exact for single-horizon window models. If richer process physics such as rolling thermal windows are added, a single cumulative envelope is no longer sufficient. Likewise, aggregate exactness requires the proportionality condition

$\widehat Q_{\mathrm{CC}}(k)$7

otherwise the aggregate set is an outer approximation. The virtual capacity curve is therefore exact under a sharply specified abstraction, not universally.

5. Capacity-credit curves for virtual energy storage

Virtual energy storage in adequacy studies is yet another construction. Here the object is not a power-balance envelope but a reliability-equivalent capacity relation between VES characteristics and adequacy contribution (Qi et al., 2023). The paper proposes a two-stage coordinated dispatch that combines day-ahead self-energy management with real-time emergency response and recovery, while modeling decision-independent uncertainties (DIUs) and decision-dependent uncertainties (DDUs). It distinguishes theoretical and practical reliability indices,

$\widehat Q_{\mathrm{CC}}(k)$8

$\widehat Q_{\mathrm{CC}}(k)$9

so that practical adequacy explicitly includes unavailable response under uncertainty.

The most relevant capacity metric is equivalent physical storage capacity (EPSC), defined by reliability equivalence between a VES system and a benchmark system with physical storage:

$\widehat V_{\mathrm{CC}}(k)$0

More generally, the paper treats EFC, ECC, ELCC, EGCS, and EPSC as mappings from storage characteristics to equivalent adequacy contribution. In this sense, a virtual capacity curve is a curve such as VES power versus EGCS, VES duration versus normalized CC, or VES size versus EPSC.

The contribution of behavioral realism is central. VES availability is modeled through operating-state uncertainty,

$\widehat V_{\mathrm{CC}}(k)$1

stochastic baseline self-consumption,

$\widehat V_{\mathrm{CC}}(k)$2

and DDU-adjusted SoC bounds,

$\widehat V_{\mathrm{CC}}(k)$3

$\widehat V_{\mathrm{CC}}(k)$4

This gives the virtual capacity curve a market- and behavior-dependent form rather than a purely technical one.

The case studies show that overlooking DIUs and DDUs causes about 10%–70% overestimated CC. For VES, the uncertainty scenarios are U1 without uncertainty modeling, U2 with DIUs, and U3 with DIUs plus DDUs. At peak load time, DIUs reduce available SoC or energy capacity by about 40%, DIUs plus DDUs reduce it by about 50%, and VES power actions fall by about 50% and 60%, respectively. In normalized capacity-value terms, U1 theoretical values are 11.1%–15.6% for 1 h VES, 12.8%–22.1% for 1.5 h VES, and 12.9%–22.9% for 2 h VES; U1 practical values fall to 3.2%–5.4%, 3.0%–6.6%, and 3.1%–6.0%. For U2 practical values, the ranges are 2.7%–5.4%, 4.4%–7.2%, and 6.5%–9.0%. The paper states that U3 theoretical and practical values are approximately the same as U2 practical values because chance constraints already control unavailability.

EPSC makes the virtual-to-physical comparison explicit. For 2 h VES, the paper reports that without uncertainty the VES is equivalent to 60%–95% of ES, whereas with practical DIUs and DDUs this falls to about 25%–60% EPSC. The value of both ES and VES also declines with increasing renewable penetration: VES normalized capacity value is 5.4%–6.0%, 4.4%–13.1%, and 6.5%–16.1% in low-RES systems for 1 h, 1.5 h, and 2 h duration, but many high-RES cases are near zero. These are properly understood as context-conditioned virtual capacity curves rather than universal derating factors.

6. Interface capability and adequacy-allocation curves in networked power systems

At the transmission–distribution and cross-border interface, virtual capacity curves become aggregated boundary objects. One example is the feeder-level aggregated net VAR capability curve for DER-rich distribution systems (Singhal et al., 2018). The feeder is represented to the TSO as a single virtual reactive resource, analogous to a conventional bulk generator, with net reactive capability

$\widehat V_{\mathrm{CC}}(k)$5

where $\widehat V_{\mathrm{CC}}(k)$6 is aggregated DER headroom. The curve is derived by repeatedly solving an OPF based on unbalanced LinDist3Flow, inverter apparent-power limits,

$\widehat V_{\mathrm{CC}}(k)$7

voltage constraints, and OLTC coupling

$\widehat V_{\mathrm{CC}}(k)$8

The lower branch $\widehat V_{\mathrm{CC}}(k)$9 corresponds to capacitive support and the upper branch $\widehat{IC}_{\mathrm{CC}}(k)$0 to inductive support. Because the DS capability is contingent on operating condition, network constraints, unbalance, and grid-side voltage, the curve is explicitly virtual rather than nameplate.

The numerical results show why aggregation matters. On the IEEE 37-bus feeder, with base substation reactive demand $\widehat{IC}_{\mathrm{CC}}(k)$1 Mvar, the reactive power flexibility region at peak load is $\widehat{IC}_{\mathrm{CC}}(k)$2 Mvar for 10% headroom, $\widehat{IC}_{\mathrm{CC}}(k)$3 Mvar for 45%, $\widehat{IC}_{\mathrm{CC}}(k)$4 Mvar for 65%, and $\widehat{IC}_{\mathrm{CC}}(k)$5 Mvar for 85%. Increasing headroom expands capacitive support, but under peak loading the inductive branch eventually contracts because lower-voltage constraints become binding. The paper also reports infeasibility for $\widehat{IC}_{\mathrm{CC}}(k)$6 pu and for $\widehat{IC}_{\mathrm{CC}}(k)$7 pu, underscoring that the communicated capability is valid only over a bounded upstream-voltage range.

A second interface-level construction is the capacity allocation curve of interconnection between two systems (Tindemans et al., 2018). Here the virtual capacity object is the efficient frontier of paired load additions

$\widehat{IC}_{\mathrm{CC}}(k)$8

that can be sustained while preserving baseline adequacy in both areas. The acceptable set is

$\widehat{IC}_{\mathrm{CC}}(k)$9

and the capacity allocation curve is

$$\widehat{DV}_{\mathrm{CC}}(k)=\frac{1}{\widehat{IC}_{\mathrm{CC}}(k)}.$$0

Because adding constant load is mathematically equivalent to removing firm capacity, each point on $$\widehat{DV}_{\mathrm{CC}}(k)=\frac{1}{\widehat{IC}_{\mathrm{CC}}(k)}.$$1 is an allocation of equivalent firm-capacity value created by the interconnector.

This formulation is fundamentally policy-dependent. The paper evaluates four flow policies—Veto, Share, Assist A, and Assist B—using a convolution-based adequacy method. Share is identified as LOLE-maximizing and therefore conservative under LOLE, while Assist A and Assist B provide upper bounds for each area’s allocation. In the GB/continental-Europe-inspired case study, both systems are calibrated to LOLE = 3 hours/year, with four interconnectors each having 95% availability. For 3 GW, 5 GW, and 10 GW of installed interconnection under Share, the total capacity value rises while the geometry changes from a sharp-cornered curve to a nearly flat trade-off frontier. A key misconception is therefore ruled out: interconnection capacity value is not a single scalar property of the asset, but an allocation problem over a two-area adequacy frontier.

7. Capacity relations for non-colliding virtual identities

In biometric identity provisioning, the nearest analogue to a virtual capacity curve is the relation between the number of non-colliding virtual identities, the enrolled real gallery, and the matching threshold (Ji et al., 18 May 2026). The paper introduces Biometric Identity Provisioning (BIP) for digital entities and argues that real face identities occupy a low-dimensional subspace of the embedding hypersphere, leaving no residual subspace for virtual identities. The consequence is geometric: virtual identities must be allocated as unclaimed gaps within the real face manifold itself. The provisioning problem is therefore a constrained packing problem rather than a simple sampling problem in unused ambient dimensions.

The paper’s flagship empirical point is that repulsion-based allocation is not bounded by any fixed provisioning count and demonstrates 10M non-colliding virtual identity embeddings against a gallery of 360K real identities. Because realizability matters, it also introduces GapGen, a gap-aware generator trained with a curriculum that progressively extends synthesis into non-colliding regions, and reports validation at 1M photorealistic virtual face images. It further constructs v-LFW, with protocols for virtual face verification, cross-reality matching, real-vs-virtual detection, and unified recognition and detection.

In this setting, a virtual capacity curve is plausibly interpreted as the relation between achievable provisioning count and system parameters such as enrolled real identity count, threshold, margin, packing strategy, and image-realizability constraints. The paper’s stated geometric thesis—that there is no residual subspace—makes the capacity object sharply different from classical spherical packing. Capacity is not determined by the full ambient sphere, but by the availability of sparse, image-realizable gaps within the occupied manifold. This shifts the meaning of “virtual capacity” from an engineering output boundary to a packing limit under semantic and verification constraints.

Across these domains, virtual capacity curves are best treated as rigorously defined reductions from latent capability to externally actionable boundaries. Their technical value lies in converting inaccessible internal structure—counterfactual battery signatures, aggregated DER flexibility, adequacy-equivalent storage, interconnector sharing benefit, or manifold gaps in biometric embedding space—into curves or frontiers that can be optimized against product definitions, reliability targets, or verification thresholds. The term is therefore coherent only at the level of method: each field uses a virtual curve to expose a capability that exists operationally or counterfactually, but is not directly observable as a native physical characteristic.