Wind Energy Conversion Systems

The swept area:

Two-thirds of a hectare of moving air, intercepted by three blades. Power scales with \(D^2\), which is why turbines have grown relentlessly — doubling the diameter quadruples the output from the same wind and the same tower footprint.

The power:

The \(v^3\) is the dominant term and the source of every control difficulty in this set. A 10% increase in wind speed is a 33% increase in power, and a gust from 12 to 15 m/s nearly doubles it.

Against the Betz limit:

The Betz limit is not an engineering constraint but a conservation argument: extracting all the kinetic energy would leave the air stationary behind the rotor, and stationary air cannot get out of the way of the air behind it. The optimum slows the wind to a third of its upstream speed, and 16/27 is what falls out.

The rotor speed. The tip speed ratio is the dimensionless variable that determines \(C_p\):

Nineteen revolutions per minute — and the blade tips are moving at \((2.0)(45) = 90\) m/s, seven and a half times the wind speed. That tip speed is what caps rotor size: beyond about 90 m/s the blades erode from rain and the aerodynamic noise becomes unacceptable, so larger rotors must turn proportionally slower.

Why \(\lambda\) has an optimum rather than simply increasing:

The optimum depends on blade count — a two-blade rotor peaks nearer \(\lambda = 10\) and a multi-blade water pumper near 1. Three blades at \(\lambda \approx 7.5\) is the compromise between \(C_p\), tip speed, noise and structural cost that the industry converged on.

The immediate consequence for the drive train:

One and a half meganewton-metres — six hundred times the mill drive of Set 31. Either a gearbox steps the speed up by a factor of about a hundred so a conventional generator can be used, or the generator is built to take that torque directly. Problems 3 and 4 take those two paths.

The four regions:

Two different actuators for two different regions. Below rated the converter controls the torque and the pitch stays fixed; above rated the pitch spills the excess and the converter simply holds rated power. The handover between them is one of the trickier parts of turbine control.

The optimal tracking law. In region II, hold \(\lambda = \lambda_{opt}\) so that \(C_p = C_{p,max}\). Substituting \(v = \omega R/\lambda_{opt}\):

Every wind-speed term has vanished. The controller does not need to measure the wind at all — it commands a torque based purely on the measured rotor speed, and the aerodynamics do the rest.

Evaluate the constant:

Equivalently, as a torque command:

Which is all the region II controller needs: measure the speed, square it, multiply by a constant, command that torque. No wind measurement, no search, no hill climbing — a striking contrast with the PV tracker of Set 37, and it works because the turbine's characteristic is known in advance where a panel's is not.

The power at lower wind speeds:

Halving the wind speed leaves an eighth of the power. That is why turbine siting is dominated by mean wind speed rather than by anything else — a site with 20% more wind yields 73% more energy, which dwarfs any conceivable improvement in the machine.

Why variable speed is worth the converter. A fixed-speed turbine holds \(\omega\) constant, so \(\lambda\) varies inversely with wind speed:

The second row is worth as much as the first. A gust hitting a fixed-speed turbine becomes a torque spike through the gearbox; hitting a variable-speed one, it accelerates the rotor instead and the energy is recovered later. The rotor's inertia becomes a buffer, and gearbox life improves markedly.

The arrangement. The stator connects directly to the grid; the rotor windings are brought out through slip rings to a back-to-back converter:

Feeding the rotor at slip frequency lets the machine run at any speed while the stator stays synchronised at 50 Hz — hence "doubly fed". It is a wound-rotor induction machine used as Set 34's equations describe, with the slip power recovered rather than dissipated.

The power split. From the induction machine relations of Set 34:

The converter handles only \(sP_{airgap}\) — and above synchronous speed the rotor delivers that power to the grid rather than absorbing it, which is why the converter must be bidirectional.

The rating at the worst case:

Under a quarter. That is the DFIG's whole reason for existing: a 3 MW turbine with a 0.7 MW converter, saving three quarters of the semiconductors, the filters, the cooling and the losses.

The complete picture:

The third row is what turned the industry away. Because the stator is bolted to the grid, a voltage dip appears directly across the machine, inducing large rotor currents that would destroy the converter — so a crowbar must short the rotor, at which point all control is lost precisely when the grid code demands controlled reactive support.

Why maintenance dominates the economics. The turbine is at the top of a tower, often offshore:

A routine job at ground level becomes a vessel, a weather window and a crew offshore. The same argument that removed brushes from industrial drives in Set 34 applies here with the costs multiplied — which is why offshore turbines have moved almost entirely to the full-converter arrangement of Problem 4.

Where it still makes sense:

The torque, with no gearbox:

Machine size scales with torque, not power — so a direct-drive generator for 3 MW is enormous, typically 5–7 m in diameter and weighing 50–80 tonnes. That mass sits at the top of the tower and drives the whole structural design.

The pole count:

Ninety-six poles — forty-eight pole pairs, against the four of the industrial machines in Part 6. High pole count is how a slow rotor produces a usable electrical frequency, and it is also why the machine must be large in diameter: the poles have to fit around the circumference.

What the full converter buys:

Rows one and three are the decisive pair. With a full converter the machine is completely isolated from grid events — a voltage dip is handled entirely by the grid-side converter, which keeps controlling current throughout and can inject the reactive support that grid codes demand. The DFIG's crowbar has no counterpart here.

What it costs:

Two extra points of loss on every megawatt-hour, forever. Against that, the gearbox removed was itself worth 1–2% and was the single largest source of unplanned downtime — so the lifetime energy comparison is much closer than the instantaneous efficiency suggests.

The three architectures compared:

The last two rows show what decided it. As grid codes tightened to demand ride-through and reactive support — the requirements of Problem 5 — the full converter's advantages stopped being optional refinements and became conditions of connection.

The problem. During a deep voltage dip the grid-side converter's export capability collapses:

Meanwhile the wind keeps blowing and the generator keeps producing. Power arrives at the DC link at nearly the full rate and cannot depart, so the link capacitor charges — and its voltage rises until something gives.

The energy involved:

Nearly half a megajoule in a sixth of a second. For comparison, the entire link capacitor of the drive in Set 26 stored a few hundred joules.

Could a capacitor absorb it? Allowing the link to rise from 1100 to 1200 V:

Four farads at 1200 V. That is not a capacitor bank; it is a room full of them, costing more than the converter and occupying more space than the nacelle allows. The approach is not marginal — it is off by orders of magnitude.

So the energy must be dissipated. A braking chopper — the same circuit that Set 33 used for a non-regenerative DC drive:

A three-megawatt resistor sounds extravagant until the duty is considered: it operates for a fraction of a second, a handful of times a year. Like the soft starter of Set 30, it is rated for a transient rather than a continuous load, so it is far smaller than its peak power suggests.

The alternative uses the rotor. A 3 MW rotor stores an enormous amount of kinetic energy:

Instead of dissipating the power, stop taking it — the wind then accelerates the rotor, banking the energy as speed, to be recovered once the grid returns. The pitch system simultaneously begins to feather. This works because 150 ms is short compared with the rotor's mechanical time constant, so the overspeed is only a few per cent.

What the grid code actually demands goes beyond merely staying connected:

The second row is the one that requires a full converter. Injecting reactive current during a dip — helping to support the very voltage that has collapsed — needs a converter in full control of its output, which a directly-connected stator with a crowbar across its rotor cannot provide.

Synthetic inertia is the same rotor energy used for a different purpose:

A converter-connected turbine is invisible to grid frequency — its rotor speed has nothing to do with the grid's — so as conventional synchronous plant retires, system inertia falls and frequency excursions become sharper. Synthetic inertia deliberately couples the two: on a frequency drop, the converter briefly takes more power than the wind is providing, slowing the rotor and releasing its stored energy. The rotor's inertia was always there; the control makes it available.

The turbine terminals. Each machine produces low voltage and is stepped up immediately:

The transformer sits in the tower base or the nacelle. Stepping up at the turbine is essential: 3 MW at 690 V is 2500 A, which cannot be carried more than a few tens of metres economically.

The collection network:

The last row matters. In a conventional distribution network fault current flows one way from the source; in a wind farm every turbine is a source, so protection must be directional and coordinated with converters whose fault contribution is limited to about 1.1 pu rather than the several per unit a synchronous machine delivers.

Farm-level control is what the network operator actually interacts with:

The fourth row is a genuinely modern idea: deliberately misaligning an upwind turbine costs it a little power but moves its wake off the machine behind, and the farm total improves by a few per cent. It is only possible because every turbine's converter allows independent control.

Wake losses are the farm-scale version of Set 37's shading:

And because power goes as \(v^3\), a 10% velocity deficit costs 27% of that turbine's output. Spacing is therefore a direct trade against land or seabed area — typically 5–7 rotor diameters crosswind and 8–10 downwind.

Offshore at 80 km changes the transmission question entirely. AC cable has capacitance:

Beyond about 80 km the charging current fills the cable and there is no room left for real power — so the connection becomes a VSC-HVDC link with an offshore converter platform. That is Set 40, and it is the largest single cost item in a far-offshore project.

The complete offshore specification:

Every choice traces to the same fact: access costs a boat. The technology that wins offshore is whichever needs the fewest visits, which is why the full-converter direct-drive machine — more expensive and slightly less efficient than a DFIG — dominates there completely.