HVDC, FACTS, and Power Quality Conditioners

Because a cable's capacitance is roughly sixteen to twenty times an overhead line's, and that capacitance draws current continuously whether or not any power flows.

Why the cable is so much more capacitive. An overhead line's conductors are metres apart with air between them; a cable's conductor and its earthed screen are separated by a few millimetres of polymer with a relative permittivity around 2.3. Capacitance scales with area over distance, so the geometry alone accounts for the factor.

Put numbers on both at 400 kV, 50 Hz:

• Cable, \( C = 0.2\ \mu \)F/km: \( I_c = 14.5 \) A/km. Over 200 km that is 2900 A — several times the conductor's rating, before any power is transmitted.
• Overhead line, \( C = 0.012\ \mu \)F/km: \( I_c = 0.87 \) A/km. Over 200 km, 174 A — a modest fraction of rating, easily absorbed by shunt reactors.

So the cable case is not an efficiency question at all. The conductor is physically full of reactive current and there is no room left for real power. No increase in cross-section helps, because a thicker cable has more capacitance too.

The partial remedies and why they run out: shunt reactors at each end compensate at their own location only — the middle of the cable still carries the charging current. Mid-point compensation solves it and is used on land, but it is not available under the sea. Lower frequency reduces \( \omega C \), which is why low-frequency AC is occasionally proposed for offshore wind.

On DC the term vanishes. The capacitance charges once at energisation and then draws nothing. This is why every long submarine link in the world is DC — not because DC is cheaper there, but because AC does not work at all.

Because a thyristor is turned off by the external circuit, so an LCC needs an AC voltage that already exists. A VSC's devices turn off on command, so it can create one.

The mechanism. In an LCC, current transfers from one valve to the next because the AC system's line-to-line voltage drives it — line-commutated. With no AC voltage there is no commutating force, and the converter cannot operate at all. It is not weak in that condition; it is inoperable.

A VSC is a controlled voltage source. It synthesises its own AC waveform from the DC link, so it can define voltage and frequency for a network that has none — which is exactly what black starting means.

Now the offshore consequence, which is decisive. An offshore wind farm has:

• No short-circuit capacity of its own. The turbines are converter-interfaced and current-limited; they contribute almost nothing to fault current. An LCC needs a short-circuit ratio above about 2–3 and would see something near zero.
• No voltage until something creates one. The turbines are grid-following (Chapter 28) — they need a grid to synchronise to before they can produce anything. Something must energise the array first.

So the offshore converter must be the grid-forming source. It energises the array cables, establishes voltage and frequency, and the turbines then synchronise to it and begin exporting. An LCC cannot perform the first step, so it cannot be used — irrespective of cost or efficiency.

Two further VSC advantages that reinforce the choice: the platform is roughly half the size, because there are no large filter and reactive-compensation banks, and offshore platform steel is extremely expensive; and independent reactive control lets the converter support the onshore grid even when the wind is not blowing.

And the honest counterpoint, so the comparison stays fair: for bulk transfer between two strong systems over 2000 km, an LCC at 12 GW and 0.7% station losses remains unbeaten, and such links are still being built. The two technologies serve different problems rather than competing for the same one.

Because the harmonics are removed by amplitude resolution rather than by switching speed. The two are independent, and a two-level converter conflates them.

The two-level case. A two-level bridge has only \( \pm V_{dc}/2 \) available, so it must approximate a sine by dwelling at the wrong voltage and correcting rapidly. Its only lever is time, so it needs a high switching frequency — and pays for it in loss and in filters.

The MMC case. With \( N = 400 \) cells per arm the converter has 401 distinct output levels, each \( V_{dc}/N \) apart:

• The staircase never departs from the target sine by more than half a step — about 0.25% of peak.
• The error is therefore tiny at every instant, and no fast correction is needed.
• The waveform meets grid harmonic limits before any filtering.

Now the arithmetic that resolves the apparent paradox. The output changes level frequently — thousands of times a second. But each change involves inserting or bypassing one cell out of 400, and the control rotates which cell that is, both to share the duty and to balance the capacitor voltages. So:

An effective output rate of 40–60 kHz divided among 400 cells leaves each device switching at 100–150 Hz.

Three consequences follow, and together they are why the MMC displaced everything else:

1. Switching loss collapses, since it is proportional to device switching frequency. Station losses fall towards 1%.
2. Each device switches only \( V_C = 1.6 \) kV, not the full 640 kV. The \( dv/dt \) is modest, so EMI is far lower and the microsecond voltage-sharing problem of series-connected valves disappears.
3. Redundancy becomes trivial. Fit 5% extra cells; a failed cell bypasses itself and the converter runs on.

The general principle is worth keeping: harmonic content is set by how closely the output approximates the target, and there are two independent ways to improve that — finer steps in amplitude, or faster correction in time. Multilevel converters buy quality with amplitude resolution, which is far cheaper in loss than buying it with speed.

Not by any electrical rating, but by a mechanical resonance in generator shafts up to hundreds of kilometres away.

The electrical part. The series capacitor and the line inductance form a resonant circuit at

which is always below the line frequency — hence subsynchronous. At 50% compensation on 50 Hz, \( f_{er} = 35 \) Hz.

The mechanical part. A large turbine-generator is not a rigid body. High-pressure, intermediate-pressure and low-pressure turbines plus the generator and exciter are coupled by a long shaft, giving torsional natural modes typically between 10 and 45 Hz.

The coupling, which is the dangerous bit. A rotor oscillating at \( f_m \) produces stator currents at \( f_0 \pm f_m \). If \( f_0 - f_m \) coincides with \( f_{er} \), the electrical circuit responds strongly, and the resulting torque reinforces the original oscillation:

1. A small torsional oscillation induces current at the electrical resonance.
2. That current produces a torque at the torsional frequency, in phase with the motion.
3. Net damping is negative, so the amplitude grows — potentially to shaft failure.

This is documented, not hypothetical: two shafts at the Mohave plant in Nevada were damaged in 1970 and 1971 before the mechanism was understood, and torsional interaction studies have been mandatory for series-compensated schemes ever since.

What is done about it:

• Keep \( k \) low enough that \( f_0 - f_{er} \) avoids the shaft modes. The usual answer, and the source of the 50–60% rule of thumb.
• Use a TCSC. The thyristor-controlled reactor makes the branch appear inductive at subsynchronous frequencies while remaining capacitive at 50 Hz — so it compensates without creating a subsynchronous resonance. This is the main technical argument for the TCSC over a fixed bank.
• Blocking filters tuned to the complementary frequency, supplementary excitation damping, or torsional relays that trip on detecting shaft oscillation.

And the lesson that generalises past this device: a power-electronic component acts on a system with mechanical and control dynamics it was never designed against. The binding constraint here is not the capacitor's voltage rating or the thyristor's current — it is a resonance in a steam turbine somebody else owns.

Because cost and simplicity still win where the load is large, well-characterised and unchanging — and a substantial share of industrial loads are exactly that.

What the active filter genuinely does better:

• All harmonics at once, up to its bandwidth, instead of one per trap.
• No detuning. A trap drifts as capacitors age and as the supply impedance changes with network switching; an active filter measures what is actually present every cycle.
• Cannot resonate with the supply. A trap bank forms a parallel resonance with the source inductance and can amplify a harmonic it was not tuned for. This has closed plants.
• Corrects reactive power and imbalance too, potentially removing a separate capacitor bank.
• Adapts when the plant adds equipment. A passive design becomes wrong; an active one does not.

What the passive filter still wins on:

1. Cost — roughly a quarter to a third for a comparable job on a well-defined spectrum.
2. Losses — a trap is a capacitor and a reactor, with essentially no switching loss. An active filter's converter dissipates 1–3% of its rating continuously.
3. Reliability and life. No semiconductors, no control system, no firmware, no cooling fans. Thirty years is unremarkable.
4. It supplies reactive power as a by-product, which the site may need anyway — so a detuned capacitor bank does two jobs at once.
5. No bandwidth limit. An active filter cannot correct harmonics beyond its switching frequency; a trap tuned to the 25th works regardless.

Where each belongs, stated as a rule:

• Passive: one large dominant load with a known, stable spectrum — a big six-pulse drive, a smelter, an arc furnace's characteristic harmonics — where reactive compensation is wanted anyway.
• Active: many varied loads, a changing plant, imbalance to correct, or a site where a resonance has already caused trouble.
• Hybrid: in practice, very common — a passive trap on the 5th and 7th carrying the bulk of the energy cheaply, with a small active filter cleaning up the rest. This is often the economic optimum, and it is worth proposing before either extreme.

A switch that is either fully on or fully off dissipates nothing — so power can be converted at high efficiency, and everything else in this book is a consequence of working around what that costs.

A linear regulator dropping 400 V at 100 A wastes 40 kW. A switching converter doing the same job wastes a few hundred watts. That factor of a hundred is why the field exists.

Everything after Chapter 1 is a consequence of one of four things:

1. Real switches are not ideal. They take time to turn on and off, and they dissipate during the transition. Hence gate drives (Ch. 5), snubbers (Ch. 24), soft switching (Ch. 20), and the whole efficiency-versus-frequency trade that recurs in every chapter.
2. Switching produces the wrong waveform. A switch gives a square wave, and almost nothing wants one. Hence PWM (Ch. 18), space vector modulation (Ch. 19), multilevel converters (Ch. 20, 30), filters (Ch. 24) and active filters (Ch. 30).
3. Energy must be stored between switching instants. Hence inductors and capacitors everywhere, the magnetics chapter (Ch. 23), and the fact that most topological invention in this book is a different arrangement of the same few storage elements.
4. The converter must be told what to do. Hence feedback (Ch. 15), MPPT (Ch. 28), field-oriented control (Ch. 26–27) and the control loops that decide almost every real design's behaviour.

Three patterns recurred often enough to be worth naming explicitly:

• Volt-second balance. An inductor in steady state has zero average voltage; a capacitor has zero average current. Nearly every conversion ratio in this book, from the buck to the DAB, falls out of applying that to a repeating waveform.
• Orthogonality makes torque. The commutator does it mechanically (Ch. 25), field orientation computationally (Ch. 26), and a PM drive by placing current against a measured rotor position (Ch. 27). One equation, three engineering answers.
• Measure and synthesise, rather than fitting a bigger passive component. Active filters instead of traps, active front ends instead of capacitor banks, STATCOMs instead of switched reactors, vector control instead of V/f. This shift — from passive components to measurement and computation — is close to a one-line description of what the field has done to electrical engineering.

And a fourth pattern that is not technical at all, but which decided several chapters' outcomes: the best converter is rarely the most efficient one. Grid codes chose the full-converter turbine over the DFIG; safety standards size Y-capacitors and forced the active short circuit; warranties and settlement rules, not hardware, are what hold back V2G. Knowing when the converter has stopped being the problem is part of the engineering.

Thirty chapters, one premise. Everything else was working out what it costs, and what it makes possible.