Will it work on the wires? Voltage, reactive power, inertia, ramps and hosting capacity

Check a plan that balances every hour against the wires: a line’s heating, a feeder’s voltage, rooftop solar’s limit, a ramp and the first second after a trip

A17 of Energy Systems, Step by Step: take a plan that balances every hour and check it against the wires: a line heated by reactive power, a feeder’s voltage sag and its rise under rooftop solar (hosting capacity), a ramp an hourly plan assumed, and the first second after a plant trips.
  • Analyst · L2
  • 30 min guided · about 20 min practice
  • After F6 and lesson 3

Five checks a plan that balances every hour can still fail (illustrative network)

Limit In this lesson
A line's heating M1's Ridge–Town line carries 110 MW at 12:00, its MW limit. Rated 120 MVA at 138 kV (illustrative), it passes that rating above 48 Mvar of reactive power (at 50 Mvar: 120.8 MVA, 101%).
Voltage drop A 12.47 kV feeder, 8 km long (illustrative): a first estimate puts its far end below 0.95 per unit (114 V of 120 V) above 4.1 MW of evening load. An AC check puts that limit near 3.8 MW.
Hosting capacity At noon (1.0 MW of load), a first estimate puts the far end over 1.05 per unit (126 V) above 3.1 MW of rooftop solar; inverters absorbing reactive power (power factor 0.95) raise that to 6.5 MW.
A ramp Without M1's ramp rule the gas plant is asked to change by −55 MW into 08:00, +50 MW into 14:00, +50 MW into 18:00; it can change by 40 MW an hour (illustrative).
Inertia After a 100 MW trip on lesson 3's Plains Grid, frequency starts falling at 0.10 Hz per second with 30 GW·s of stored rotational energy, and at 0.50 with 6 GW·s.
Inertia: inverters' help A battery assumed to give 50 MW at once (grid-forming) halves that starting fall; one assumed to wait 0.5 s (grid-following) helps only after that. Real responses depend on controls, spare power and current limits.

With JavaScript on, you work through each check yourself: a slider for each limit, a gauge that shows when it binds, then a sort of which check would catch it.

Illustrative: the line's voltage and rating, the feeder's length, ohms, set point and loads, the plant's ramp limit, the inverters' assumed responses and the Plains Grid are invented.
From earlier lessons: M1's 12:00 plan and its day, M1 (3 September 2025, Southwest Power Pool shapes from EIA-930, accessed 2 November 2026), and lesson 3's Plains Grid, lesson 3.
Real: the voltage band, ANSI C84.1 Range A as tabulated by NREL (2021), Table 1, p. 4 (PDF).
First estimates: closed-form formulas, not a power-flow solution; within 0.01 per unit of a voltage limit, the gauges leave the verdict to an AC check (Go deeper).

About 30 guided minutes. Practice, about 20 more: the practice set under Check.

Need a refresher? F6 follows power through each voltage step; lesson 3 keeps frequency at 60 Hz.

What you just did, as diagrams

Heating
A line's thermal rating caps its current; at a given voltage that is a limit on apparent power (MVA), not MW alone.
Voltage sag
Power drawn along a feeder lowers the voltage toward its far end; the normal band is 0.95 to 1.05 per unit.
Hosting capacity
Rooftop solar that sends power back up a feeder raises its voltage; the most it can take within the limits is its hosting capacity.
Ramps
A plant can change its output only so fast; an hourly plan without a ramp rule can ask for more.
Inertia
Stored rotational energy sets how fast frequency starts to fall after a trip. Inverters can help too; how much and how fast depends on their controls, spare power and current limits.
What catches them
An AC check of voltage and reactive power; ramp limits checked at the plan's time step, with finer steps for changes inside the hour; and a check of the first seconds.
Go deeper · AnalystThe five first estimates, with this lesson's numbers
Check First estimate In this lesson
Apparent power \(S = \sqrt{P^2 + Q^2}\) \(\sqrt{110^2 + 50^2}\) = 120.8 MVA
Current in each wire \(I_\text{A} = 1000\,S_\text{MVA} \div (\sqrt{3}\,V_\text{kV})\) 1000 × 120 ÷ (√3 × 138) = 502 A
Voltage at the far end \(V_\text{end} = V_\text{sub} - (R P + X Q) \div V_\text{nom}^2\) 1.03 − (2.0 × 4.0 + 3.2 × 1.31) ÷ 155.5 = 0.951 per unit
Hosting capacity \(P_\text{pv} = (R P_L + X Q_L + (1.05 - V_\text{sub}) V_\text{nom}^2) \div (R - X q)\) (2.0 × 1.0 + 3.2 × 0.33 + 0.02 × 155.5) ÷ 2.0 = 3.1 MW
Starting rate of change of frequency \(f_0\,\Delta P \div (2E)\) 60 × 100 ÷ (2 × 16,000) = 0.19 Hz per second
  • Power and wires. \(P\) in MW, \(Q\) in Mvar, \(S\) in MVA; \(R\) and \(X\) in ohms (Ω); the factor 1,000 turns kA into A.
  • Voltages. \(V_\text{nom}\) is the nominal line-to-line voltage in kV (12.47² = 155.5); \(V_\text{sub}\) and \(V_\text{end}\) are in per unit of it.
  • Noon symbols. \(P_L\) and \(Q_L\) are the noon load’s MW and Mvar; \(q\) is the Mvar absorbed per MW of solar: 0 at a power factor of 1, 0.33 at 0.95.
  • Frequency symbols. \(f_0\) = 60 Hz; \(\Delta P\) is the MW lost in the trip; \(E\) is the stored rotational energy in MW·s (16 GW·s = 16,000 MW·s).
  • How close is the estimate? An AC calculation of this feeder, exact for one load at its far end, gives 0.945 per unit at 4.0 MW, not 0.951: the evening limit is near 3.8 MW, not 4.1.
  • At noon it errs the other way: the AC check finds a little more room than 3.1 MW of solar, and with absorbing inverters no voltage limit up to 8 MW.
  • The starting slope holds only for a moment: lesson 3 adds load relief and governors.
Go deeper · ModelerWhat M1's DC model drops, and what replaces it
  • DC power flow (M1) holds every voltage at 1.0 per unit, assumes small angle differences, and leaves out losses and reactive power.
  • An AC power flow keeps them: non-linear equations for every voltage’s size and angle, solved by iteration, for example with Newton’s method (PyPSA’s documentation, accessed 5 November 2026).
  • The workflow. Plan with a fast linear model, then check the plan’s hours with an AC power flow and its largest trips with a dynamics study.
  • Out of this course’s scope: AC optimal power flow and dynamic simulation.
Go deeperWhat this lesson leaves out
  • The drop through the last transformer and the service wires: the far end of the feeder stands in for a home's meter.
  • Load and solar spread along the feeder, regulators that change taps through the day, and the feeder's own heating and protection, which can bind first.
  • Moves inside the hour: real-time markets re-dispatch every five minutes, as PJM's does.
  • Load relief, governors and reserves after the first second (lesson 3); inverters' current limits and control settings.
  • N−1 checks of the network after any one line fails (F6), and stability problems such as oscillations.

Check yourself

Myth or fact? “If every line is within its MW limit in the model, no line will overheat.”

  • Myth
  • Fact
Answer

Myth. Heating follows current, and reactive power adds current: 110 MW with 50 Mvar is 120.8 MVA, over a 120 MVA rating.

Myth or fact? “Rooftop solar can only help a feeder's voltage.”

  • Myth
  • Fact
Answer

Myth. At noon, rooftop solar above about 3 MW at this feeder's far end lifts its voltage over 1.05 per unit.

Myth or fact? “Batteries cannot help in the first second after a trip, because nothing in them spins.”

  • Myth
  • Fact
Answer

Myth. A grid-forming inverter can respond from the first instant. In the sketch, a battery assumed to give its 50 MW at once halves the starting fall after a 100 MW trip.

Myth or fact? “A plan that balances every hour can always be run.”

  • Myth
  • Fact
Answer

Myth. Without a ramp rule, M1's day asks the gas plant to climb 50 MW into 14:00, though it can change by only 40 MW an hour.

Which check would catch a feeder's far end sagging below 0.95 per unit?

  • An AC check of voltage and reactive power
  • A ramp rule
  • A check of the first seconds
Answer

An AC check. M1's model omits feeders, and its DC approximation does not calculate voltage-magnitude changes or reactive power.

Double the grid's stored rotational energy. What happens to the starting rate of change of frequency after the same trip?

  • It doubles
  • It halves
  • It stays the same
Answer

It halves: the starting fall is f₀ × ΔP ÷ (2E), so twice the energy E gives half the fall.

Practice · about 20 minutes on your own

  1. Heating. A 230 kV line rated 320 MVA carries 300 MW and 100 Mvar. Is it within its rating, and what current flows in each wire?

    Answer

    Within, barely: √(300² + 100²) = 316.2 MVA, 99% of its rating; 316.2 MVA ÷ (√3 × 230 kV) = 794 A.

  2. Power factor. What is the Ridge–Town line's power factor at 110 MW and 50 Mvar?

    Answer

    110 ÷ 120.8 = 0.91.

  3. A shorter feeder. 12.47 kV, 1.5 Ω and 2.4 Ω (resistance, reactance); far end 3 MW at a power factor of 0.95; substation 1.02 per unit. Is the far end in the band, by a first estimate?

    Answer

    Yes: Q = 0.99 Mvar, and 1.02 − (1.5 × 3 + 2.4 × 0.99) ÷ 155.5 = 0.976 per unit.

  4. A longer feeder. Double the lesson's feeder to 16 km (4.0 Ω and 6.4 Ω), same noon load and set point. What is its hosting capacity at a power factor of 1?

    Answer

    About 2.3 MW: (4.0 × 1.0 + 6.4 × 0.33 + 0.02 × 155.5) ÷ 4.0, less than the 3.1 MW of the 8 km feeder.

  5. A bigger trip. A 200 MW plant trips on a grid with 10 GW·s of stored rotational energy. How fast does frequency start to fall?

    Answer

    60 × 200 ÷ (2 × 10,000) = 0.6 Hz per second.

  6. A ramp. Within an hour, the gas plants must go from 100 to 210 MW. The combined-cycle plant is at 100 MW (150 at most, 40 MW an hour); the 100 MW peaker is off but fast. Can they follow?

    Answer

    Yes, if the peaker starts: the combined-cycle plant climbs 40 MW, to 140 MW, and the peaker gives the other 70 MW.

  7. Read a claim critically. A report says: “Our plan is feasible: it meets demand in every hour of the year.” What four checks would you ask about?

    Answer

    Lines with reactive power counted; feeder voltages where rooftop solar grows; ramps at the plan's time step and below it; the first seconds after the largest trip.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
0.95–1.05 per unit The normal range of service voltage, ANSI C84.1 Range A: 114 to 126 V on a 120 V service. NREL (2021), Table 1, p. 4 (PDF) · September 2021
1,500 MW Data-center load that dropped off at once, across 60 load points, after a normally cleared 230 kV fault in Loudoun County, Virginia, in July 2024. NERC slides, posted by FERC (PDF) · 17 April 2025
100 GW·s The critical inertia level for the Texas grid that ERCOT's 2024 ancillary-services study identified. ERCOT, Ancillary Services Study (PDF) · 7 October 2024

Keep going

Back to the course map

In the real world

Loads can test the wires too. In July 2024, after a fault on a 230 kV line in Loudoun County, Virginia, was cleared normally, about 1,500 MW of data-center load dropped off at once (NERC).

An hourly energy plan cannot see that.

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.