Virtual Capacity Curve (VCC)

• Virtual Capacity Curve (VCC) is a quantitative construct that defines the available resource envelope across temporal, operational, or market dimensions for engineered systems.
• It supports optimization and risk-aware admission control in cloud datacenters, virtual power plants, and aggregated DER management.
• VCCs integrate stochastic modeling and operational constraints to balance cost, carbon reduction, reliability, and grid support under real-world conditions.

A Virtual Capacity Curve (VCC) is a quantitative construct delineating the available resource envelope—across temporal, operational, or market dimensions—for flexible workload, reserve, or grid-support capabilities in large-scale engineered systems. VCCs are used as an optimization and admission-control artifact in contexts such as cloud datacenters (Radovanovic et al., 2021), virtual power plant (VPP) reserve capacity provisioning (Zapparoli et al., 6 Oct 2025), and aggregated distributed energy resource (DER) reactive-power management at the TSO–DSO interface (Singhal et al., 2018). VCCs encode risk-aware, constraint-satisfying boundaries—which may be reservation limits, supply curves, or capability boundaries—facilitating co-optimization of system objectives (carbon, cost, reliability, grid support) under operational and market realism.

1. Formal Definitions and Application Contexts

VCCs are instantiated according to the resource, system, and context:

• Datacenter Batch Capacity Envelope: A VCC is a 24-point sequence $\bigl(c_{1}, c_{2}, \dots, c_{24}\bigr)$ where each $c_t$ is an hourly, day-ahead reservation limit of CPU-hours (or GCU-hours) dedicated to temporally flexible tasks. The VCC guarantees that total flexible workload scheduled over the day meets risk-inflated demand, with hourly allocations shaped to minimize carbon cost and infrastructure peak (Radovanovic et al., 2021).
• VPP Market Reserve Supply Curve: The VCC is a mapping $q \mapsto$ price, such that, for each reserve offer quantity $q$, the curve reports the lowest price at which the VPP is willing to deliver $q$ as reserve capability, accounting for technical limits and reliability constraints (Zapparoli et al., 6 Oct 2025).
• DER Q–P Capability Envelope: At the TSO–DSO interface, the VCC is the aggregated Q–P capability curve, i.e., $\bigl(P_\mathrm{hr}, Q_\mathrm{substation}\bigr)$ pairs encoding the feasible net VAR injected or absorbed for a given real-power headroom, derived via OPF under prevailing system conditions (Singhal et al., 2018).

2. Mathematical Formulations

The structure and derivation of VCCs depend on system objectives and constraints.

Datacenter VCC (OPT–VCC LP):

For each cluster and day, the core risk-aware optimization is

$$\begin{aligned} \min_{x_{1:24},\,c_{1:24},\,y}\quad & \lambda_{e}\sum_{t=1}^{24} e_{t}P\bigl(\hat U_{IF}(t)+x_{t}\bigr) + \lambda_{p}y \ \text{subject to}\quad & \sum_{t=1}^{24} x_{t} = \tau_F \ & 0 \le x_{t} \le c_{t}-\hat U_{IF}(t), \quad c_{t} \le C_{\max}, \ & y \ge P\bigl(\hat U_{IF}(t)+x_{t}\bigr), \ & c_{t} \ge \hat U_{IF}(t)+x_{t} \qquad t=1,\dots,24 \end{aligned} \tag{OPT­VCC}$$

Key stochastic model: $\tau_F = \alpha \cdot \widehat{T_F}$ with $\alpha$ as the empirical 97th quantile of 90-day rolling forecast errors

VPP VCC Under Reliability Constraint:

Given uncertain inputs $\boldsymbol{z}$, maximum feasible reserve: $$g(\boldsymbol{z}) = \max_{\boldsymbol{x},\boldsymbol{y}}\,q^{\mathrm{p}}(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z})$$ Reliability-adjusted reserve: $$q^{\mathrm{p,max}} = \inf\left\{\tilde q : \mathbb{P}\{g(\boldsymbol{z}) \le \tilde q\} = \alpha^{\mathrm p}\right\}$$ Supply curve: For an increment $q_n^{\epsilon}$, risk-adjusted marginal price $c^{\mathrm{p},\alpha_{\mathrm c}}(q_n^{\epsilon})$ is computed by cost quantile over uncertainty draws (Zapparoli et al., 6 Oct 2025).

DER Var Capability VCC (OPF Sweep):

For a set of DERs indexed by $j$ and aggregated headroom $p_{sub}^{g,hr}$,

$$\min_{p_j^{g,hr},\,q_j^g,\,y_j,\,v_0}\quad q_{sub}^{net}(q_j^g, y_j)$$

subject to LinDist3Flow relations, inverter constraints, load limits, and IEEE 1547 compliance (Singhal et al., 2018). Envelope traced by sweep over $p_{sub}^{g,hr}$ produces $(Q_{min}(P_{hr}), Q_{max}(P_{hr}))$ for the VCC.

3. Data Sources, Analytical Pipelines, and Model Assumptions

The derivation of a VCC is predicated on high-fidelity data ingest and model calibration.

• Carbon-Aware Pipelines (Radovanovic et al., 2021):
  • Carbon Forecasts: 48-h ahead, hourly values from Tomorrow.io; median absolute percentage error (MAPE) 0.4–26%
  • Power Modeling: Piecewise-linear regression on 5-min data (daily retrain, 95% PDU MAPE < 5%)
  • Load Forecasting: EWMA time series on historical inflexible, flexible usage (correction for yesterday’s residuals, risk inflation via 97% quantile)
  • Reservation Translation: Regression maps usage envelope to actual reservations
• VPP Reserve Assessment (Zapparoli et al., 6 Oct 2025):
  • Uncertainty Sampling: Joint PDFs, subset simulation (SS) for extreme quantiles, rare-event probability structure
  • Cost Attribution: Explicit DER operating costs (fuel, wear, thermal penalties); opportunity cost of day-ahead energy market returns; dual-variable based price-quantity mapping
• DER Q–P Capability (Singhal et al., 2018):
  • LinDist3Flow: Linearized unbalanced three-phase power flow model governs voltage–power relationships
  • Headroom Computation: Aggregated and per-node curtailment constraints
  • IEEE 1547: Headroom or oversizing mandate (≥44% for Q modulation)

4. Operational Procedures, Updating, and Compliance Enforcement

Effective deployment of VCCs requires operational rigor and feedback mechanisms:

• Datacenter (Radovanovic et al., 2021):
  • Daily pipeline: input ingestion → optimization → enforcement in real-time scheduler
  • SLO violation auto-suspension: if measured flexible work allocation misses risk-inflated budget in two consecutive days, pause shaping for model retrain
  • Envelope fallback: Infeasibility (flexible load > max physical capacity or missing models) triggers unconstrained ($c_{t}=C_{\max}$) scheduling
• VPP (Zapparoli et al., 6 Oct 2025):
  • Fast SS enables feasible reserve quantile estimation with minimal samples (e.g., 69% reduction in computation vs. direct MCMC)
  • Reliability threshold and product specs direct offering quantity
• DER Q–P Capability (Singhal et al., 2018):
  • Updated and published to TSO at 10–15 min intervals with associated decoupling range $D = [v_0^*/\overline{r}, v_0^*/\underline{r}]$
  • TSO uses VCC and $D$ as constraints in OPF/UC models
  • Parametric voltage studies for worst-case bounds when $v_{tm}$ outside $D$
  • Nonlinear validation: LinDist3Flow produces base-case voltage error $<$0.07%, VCC Q error $<$1.5%

5. Quantitative Results and Performance Metrics

Empirical studies have evaluated VCC impacts on cost, reliability, carbon reduction, reserve provisioning, and grid support.

• Datacenter (Radovanovic et al., 2021):
  • A/B test (Feb–Mar 2021): VCC shaping in 50% of clusters
  • Carbon reduction: 5%–25% per-cluster, dependent on grid ramp and flexible-load mix
  • Peak power reduction: 2%–10% (daily 95th percentile)
  • Missed clusters: ≈10% per day (insufficient flexible demand or forecast uncertainty)
• VPP (Zapparoli et al., 6 Oct 2025):
  • 99.9% reliability: $q^{\mathrm{p,max}} \approx 56$kW (2800 SS samples vs. 9000 DMC)
  • VCC shape: stepwise fan chart, marginal prices 0.34–0.40 CHF/kW; steps reflect hourly energy price variation
  • Opportunity costs dominate explicit costs for reserve blocks compelling expensive hour dispatch
  • Reliability loosening: Increase from 95% to 99.9% reduces offerable reserve by ≈20%
• DERs at TSO–DSO Interface (Singhal et al., 2018):
  • IEEE 37-bus: 90% PV penetration, headroom VT sweep expands capacitive support from ≈1.8 MVar to ≈4 MVar
  • Voltage constraints, unbalanced loading shrink VCC envelope
  • Nonlinear power-flow validation: LinDist3Flow vs. GridLAB-D, voltage error across operating points $<$0.5%

Context	VCC Role	Key Metric/Result
Datacenter	Envelope for batch CPU allocations	Carbon reduction 5–25%; 97% SLO
VPP	Reserve bid supply curve	99.9% reliable $\sim$56 kW, stepwise price
TSO–DSO DER	Aggregated Q–P capability curve	Capacitive support up to 4 MVar

6. Limitations, Insights, and Future Directions

VCC techniques are constrained by modeling and operational realities:

• Model Limitations:
  • Linear approximations (e.g., LinDist3Flow) may not capture extreme distribution system nonlinearities; higher-order models advised for low X/R feeders (Singhal et al., 2018).
  • Day-ahead forecast uncertainty and insufficient flexible load limit shaping efficacy in some clusters (Radovanovic et al., 2021).
  • DER headroom constraints assume consent or contractual flexibility; economic frameworks required (Singhal et al., 2018).
• Operational Insights:
  • VCCs enhance system-wide transparency; aggregation enables competitive bidding and dispatch (Zapparoli et al., 6 Oct 2025).
  • Product technical requirements (ramp rates, duration, reliability) dictate aggregation efficacy and price formation.
  • DER portfolio diversity (PV, BESS, EVs, HPs) is essential for round-the-clock and high-reliability reserve blocks.
• Extensions:
  • Mixed-integer OPF for discrete device modeling in DER VCC (Singhal et al., 2018).
  • Robust uncertainty-aware VCCs, meshed topologies, and extended T–D co-optimization frameworks.

A plausible implication is that with further integration of uncertainty modeling and market mechanisms, VCCs will become central artifacts linking resource adequacy, cost minimization, grid stability, and carbon objectives at system and interface boundaries.