HVDC, FACTS and Power Quality

The pole current:

Three kiloamps per pole, with the two poles at +500 and −500 kV so that normal operation needs no earth return. Losing one pole halves the capacity rather than losing it entirely — the reason bipolar arrangements dominate.

The converter. Line-commutated HVDC uses the six-pulse bridge of Set 14, doubled:

The 30° phase shift between the secondaries cancels the 5th and 7th harmonics on the AC side, exactly as Set 11 showed for a 12-pulse rectifier. Several such units in series reach the full pole voltage, and each is a valve group that can be bypassed for maintenance.

Why DC wins over distance. Three separate effects:

The last row is why distance matters. DC has a large fixed cost at each end and a lower cost per kilometre, so the two curves cross — and where they cross depends entirely on whether the route is overhead or submarine.

The breakeven distances:

The third row is the case where distance is irrelevant. Two grids at different frequencies, or at the same nominal frequency but not synchronised, cannot be joined by AC at any length — so back-to-back HVDC stations exist with no transmission line between them, purely to make the connection.

What line commutation costs. Set 31 established the mechanism and its price:

The second row is decisive for offshore wind. An LCC needs an existing AC voltage to commutate against, and an offshore platform surrounded by converter-connected turbines has no such source — which is exactly why VSC-HVDC exists, and Problem 2.

What LCC retains:

Nothing else moves that much power. Line-commutated thyristors switch at line frequency with no switching loss and carry thousands of amps per device, so the largest links in the world — bulk hydro transmission across China and Brazil — remain LCC and will for the foreseeable future.

The submodule count:

Nearly two thousand independent half-bridge cells, each with its own capacitor, gate drive, local controller and communication link — and each is essentially the cascaded cell of Set 29 scaled up. The MMC is the cascaded H-bridge idea taken to its conclusion.

The output levels:

Three hundred and twenty-one voltage levels per phase. Each step is 2 kV out of 640 — 0.3% — so the output is sinusoidal to well within any harmonic limit.

What that buys:

The fifth row is what made VSC-HVDC practical. A two-level converter at 640 kV needs hundreds of IGBTs in series switching within nanoseconds of each other — the string problem of Set 5, at a scale where a single mismatch destroys the valve. The MMC sidesteps it: each cell manages its own 2 kV independently, and they need not switch together at all.

What VSC does that LCC cannot:

The first row is exactly the offshore wind requirement from Set 38. An offshore platform surrounded by converter-connected turbines has no synchronous source to commutate against, so the converter must create the AC voltage itself — then the turbines synchronise to it. Only a VSC can do this.

The MMC's own problems:

The first row is the MMC's defining control task and it has an elegant solution: at each switching decision, sort the arm's cells by capacitor voltage and insert whichever ones the current direction will move towards balance. The redundancy that Set 28's zero states provided appears here as a choice of which cells to use.

The DC fault problem, which is genuinely hard:

On a DC-side short circuit, the AC system feeds the fault through those diodes as an uncontrolled rectifier — and the converter cannot stop it. Full-bridge cells can block DC fault current but cost twice the devices and more loss; the alternative is a DC circuit breaker, which must interrupt thousands of amps with no natural current zero. Both are active areas of development, and it is the main obstacle to true multi-terminal DC grids.

The base quantities:

The mechanism. A STATCOM is a voltage source behind a reactance, exactly like a synchronous condenser:

Fifteen per cent above the bus voltage delivers 50 MVAr into it. Making \(E\) lower than \(V\) by the same amount absorbs 50 MVAr — symmetric operation with no change of hardware, which is one of the STATCOM's advantages over a switched capacitor bank.

The current:

Which is also the converter's rating, and here is the key point: that current is set by the converter's own capability, not by the bus voltage.

The comparison that matters. Consider what each device does as the bus voltage falls — the situation in which reactive support is most needed:

At half voltage the STATCOM delivers twice what an SVC can. An SVC is a capacitor, and a capacitor's current falls with the voltage across it — so precisely when the network is collapsing, an SVC withdraws its support. A STATCOM holds its rated current down to very low voltage and keeps pushing.

The full comparison:

The sixth row hints at the modern direction: add a battery to a STATCOM's DC link and it becomes a device that supports voltage and frequency — combining this problem with Set 39.

The FACTS family, for orientation:

All five are converters placed on a transmission system to control quantities that were historically fixed by its geometry. The unified power flow controller of the last row is the most general — a shunt and a series converter sharing a DC link, able to control all three parameters independently.

The power transfer relation. For a line between two buses:

Inserting a series capacitor reduces the effective reactance:

The line's capability doubles — without a single new tower, conductor or right of way. Series compensation is the cheapest capacity increase available on a transmission system, which is exactly why it is used despite what follows.

The additional benefits:

The last row is elegant: the capacitor's voltage rises with the line current, so its compensating effect automatically increases exactly when the line is most heavily loaded and most needs it.

Now the hazard. A series capacitor and the line's inductance form a series resonant circuit:

Below the power frequency — hence sub-synchronous. A disturbance excites a current at 35.4 Hz in the electrical network, and that current interacts with the generator rotor.

The complement is what reaches the shaft:

A 35.4 Hz stator current produces a torque component at \(50-35.4 = 14.6\) Hz on the rotor. A turbine-generator shaft is not a rigid body — it is a chain of masses on a torsional spring, with natural modes typically between 10 and 50 Hz. If one of them lies near 14.6 Hz, the electrical and mechanical systems exchange energy at that frequency.

And the exchange can grow. Three distinct phenomena:

The consequences are mechanical and permanent. The Mohave incidents of the 1970s cracked two generator shafts before the mechanism was understood, and SSR studies have been mandatory for series-compensated systems ever since.

The TCSC solution. Placing a thyristor-controlled reactor in parallel with the series capacitor makes the reactance adjustable:

The third row is the important one and is not obvious: the thyristor-controlled branch makes the assembly behave as a capacitor at 50 Hz — where compensation is wanted — and not as a capacitor at 35 Hz, where it would resonate. One device, two different characters at two frequencies.

The load's harmonic content:

The load is a capacitor-input rectifier of the kind Set 10 analysed — the harmonics are not incidental, they are what such a load draws by construction.

The active filter's principle. Measure the load current, extract its harmonic content, and inject the negative of it:

The supply then sees a purely sinusoidal current in phase with the voltage, while the harmonics circulate between the load and the filter. It is an inverter — the same six switches as Set 26 — controlled to produce a specific distorted current rather than a sinusoid.

The rating:

Under a third — because the filter handles only the harmonic current, never the fundamental. That is the same partial-rating argument that made the DFIG attractive in Set 38, and it is what makes active filtering economically viable.

The bandwidth requirement. To cancel a harmonic, the filter must be able to produce it:

A factor of ten between the switching frequency and the highest harmonic to be cancelled is the practical rule — the current loop needs that much margin to track. Cancelling to the 50th would require 25 kHz of bandwidth and 50 kHz of switching, which at this power is demanding.

Against the passive filter of Set 32:

Rows two and three are the passive filter's fundamental weaknesses, and they were the subject of Set 32, Problem 4: a tuned branch drifts upward in frequency as it ages and can become an amplifying parallel resonance. An active filter has no resonance to drift and no tuning to maintain.

The hybrid answer, which is what large installations actually use:

The passive branches carry the bulk of the harmonic current cheaply, and a much smaller active filter handles the residual, the higher orders, and — importantly — damps the passive branch's resonance. The active unit can then be rated at perhaps 10% of the load rather than 29%.

What the standards require:

Note that IEEE 519 applies at the point of common coupling — the shared connection — not at each item of equipment. It is a limit on what a customer injects into the network, which is why the responsibility falls on the installation as a whole and why site-level filtering is the usual answer.

What a synchronous machine provides for free:

None of the first three is designed in — they are consequences of the machine's physics, and the entire power system was built assuming they would always be present.

What a conventional converter provides instead:

The third row is the deepest problem. A grid-following converter uses a phase-locked loop to synchronise to an existing voltage — and if every source is grid-following, there is nothing for any of them to follow. The system cannot start, and at high converter penetration it becomes unstable well before that point.

The consequences, quantified:

The rate of change of frequency after a generation loss is inversely proportional to system inertia \(H\). As synchronous plant retires, \(H\) falls, and the same loss produces a steeper frequency fall — giving protection and response services less time to act, and risking cascading disconnection of generation on RoCoF protection.

Grid-forming control reverses the converter's role:

A grid-forming converter emulates a synchronous machine's behaviour in software — an internal angle that responds to power imbalance with a chosen inertia constant. It still needs energy behind it to deliver that inertia, which is where Set 39's storage comes in.

The converter-dominated toolkit that replaces what was lost:

Every entry in the middle column is a power electronic converter. The transition is not from one generation technology to another — it is from a grid whose properties emerged from spinning iron to one whose properties are chosen, in software, by the engineers who write the control laws.

Which is where this book has been going. Trace it back:

The MMC of Problem 2 is 1920 switching cells, each obeying the volt-second balance of Set 1, arranged by the multilevel argument of Set 29, modulated by the vector reasoning of Set 28, protected by the thermal analysis of Set 8 and filtered by the principles of Set 32. Nothing in this last set required an idea that was not already built.