Solar PV and Wind Energy Conversion Systems

The hot one — by roughly 20%, because a silicon module loses about 0.4% of its power per kelvin of cell temperature.

So the hot array needs about \( 1/0.808 = 1.24 \) times the rated capacity to match — call it 24% more modules.

Where the loss physically comes from. Temperature raises the diode saturation current \( I_0 \) steeply, and \( V_{oc} = n_sAV_T\ln(I_{ph}/I_0+1) \) falls as a result. The current barely changes — it even rises very slightly, by about 0.05% per kelvin, because a narrower band gap absorbs a little more of the spectrum. Temperature is a voltage effect, essentially not a current one.

Three consequences that follow directly, and are worth knowing:

• Cold and bright is the best condition, not hot and bright. High-altitude and high-latitude sites can outperform their irradiance figures.
• Mounting matters as much as the module. A roof-integrated array runs 20–30 K hotter than a rack-mounted one with airflow behind it — worth 8–12% of annual yield for no change in equipment.
• Standard Test Conditions are not real conditions. STC specifies a 25 °C cell temperature at 1000 W/m², which essentially never happens in the field. This is why NOCT and PVUSA ratings exist, and why installed arrays routinely produce 10–20% below their nameplate at noon.

And the design trap in the same fact: the hot site needs more modules for the same energy, but the cold extreme still sets the maximum string voltage. The two ends of the calculation are governed by opposite conditions, and both must be checked.

Because it cannot distinguish "power rose because I moved the right way" from "power rose because the sun came out".

Trace the failure step by step:

1. The tracker increases the operating voltage by one step, moving away from the true MPP.
2. Between that sample and the next, a cloud edge passes and the irradiance rises 30%.
3. Power is measured higher than before — because of the sun, not the step.
4. P&O reads a power increase as confirmation and steps further in the same wrong direction.
5. The irradiance keeps rising, so the reasoning repeats, and the tracker walks steadily away from the peak for as long as the ramp lasts.

The root cause is that P&O has one equation and two unknowns. It observes \( \Delta P \), which contains both \( (\partial P/\partial V)\Delta V \) and \( (\partial P/\partial G)\Delta G \), and it attributes all of it to the first.

The defences, and what each costs:

1. Sample faster. If the tracker runs at 10–100 Hz, it outruns most cloud dynamics. Cheap, and the usual first answer.
2. Three-point weight comparison. Take three measurements before deciding, so a monotonic irradiance ramp produces a recognisable signature. Slower to respond, more robust.
3. Measure irradiance directly and freeze the tracker while it changes quickly. Needs a sensor, which is an extra part to fail.
4. Incremental conductance. Testing \( dI/dV \) against \( -I/V \) uses the shape of the curve rather than a change in power, so a uniform change in irradiance does not fool it in the same way. More computation, and it needs a clean current measurement.

Keep the significance proportionate. The permanent small oscillation costs perhaps 0.5% of annual energy; the wrong-way tracking costs a few percent on genuinely broken-cloud days and nothing on clear ones. P&O is used everywhere because 98–99% for three lines of code is an excellent bargain — but the 1–2% is real, and on a 100 MW plant it is worth a great deal of money.

Because it turns one smooth power hill into several, and a conventional tracker climbs the nearest one.

What the diode does. Cells in series carry the same current, so one shaded cell throttles the whole string and is driven into reverse by the others, dissipating rather than generating — a hot spot that can reach 150 °C. A bypass diode across each group of about twenty cells conducts when that group cannot carry the string current, routing the current around it.

What that does to the curve. Each combination of bypassed and active groups is a different circuit, with its own I–V characteristic and its own local power peak. An array with several partially shaded modules can present three or four local maxima.

• P&O and incremental conductance are both local algorithms — they follow the gradient uphill and stop where it is zero.
• Started on the wrong hill, they converge correctly to a maximum that is not the best one. Nothing has malfunctioned.
• Documented losses on shaded arrays that are tracking a local peak run to 10–30%.

The remedies, and where each makes sense:

1. Periodic global scan. Sweep the whole voltage range every few minutes, find the global peak, return and track locally. Costs a little energy each sweep; standard in modern string inverters.
2. Power optimisers — a DC–DC converter per module, feeding a shared inverter. Each module reaches its own MPP.
3. Microinverters — a complete grid-tied inverter per module. Same benefit, plus per-module monitoring and no high-voltage DC on the roof.
4. Layout. String modules that shade together on the same input, so shading takes a whole string rather than fragmenting several. Free, and frequently the largest single improvement available.

And the economics, which decide it in practice: module-level electronics cost 10–20% more and put active components on a hot roof for twenty-five years. On a shaded domestic roof they recover 10–25% of annual yield and are clearly worth it. On an unshaded utility plant they recover almost nothing, and are not used.

Because the dangerous case is the one where nothing changes at the moment the grid disconnects.

The mechanism. Before the breaker opens, the inverter supplies some power and the grid makes up the difference to the local load. If the local load happens to consume exactly what the inverter produces — in both real and reactive power — then the grid is contributing nothing, and removing it changes nothing measurable. Voltage stays put, frequency stays put, and every passive detector sees a perfectly normal grid.

That region is the non-detection zone, and it is defined in the \( \Delta P, \Delta Q \) plane. It shrinks as the mismatch grows, which is why islanding is easy to detect in almost every case and impossible in the one that matters.

Why a resonant load is the standard test. Standards specify a parallel RLC load resonant at the nominal frequency with a quality factor of about 1.0, tuned so the inverter's output exactly matches it:

• Matching real power means the voltage will not move when the grid goes.
• Matching reactive power at resonance means the frequency will not move either — and a high-\( Q \) load actively resists being pushed off frequency, defeating the active methods that work by nudging it.

It is a deliberately unrepresentative worst case. Real loads are rarely resonant and rarely matched. The point of testing there is that the consequence of failure is a lethal hazard, so the method must survive the hardest case rather than the typical one.

How the active methods get around it:

• Slip-mode frequency shift makes the inverter's current phase a function of frequency, giving positive feedback that runs away once the grid is not there to hold it.
• Active frequency drift injects a small dead time into the current zero crossings, continuously pushing the frequency.
• Impedance measurement injects a small non-fundamental signal and measures the response — a stiff grid responds with a low impedance, an island with a high one.

All of them trade a small deliberate degradation of power quality for detection inside the non-detection zone, and all of them can interact when many inverters share a feeder — which is itself part of the test regime.

And the tension that defines current practice: grid codes now also require inverters to ride through voltage dips rather than disconnecting, because with high inverter penetration a mass disconnection is itself a grid emergency. Riding through and refusing to island pull against each other, and reconciling them is why grid-forming inverter control is an active field rather than a settled one.

Because extracting all the kinetic energy would require the air to stop dead behind the rotor — and stationary air blocks the air arriving behind it.

The argument in four steps, using only mass and energy conservation:

1. Air approaches at \( v_1 \) and leaves at \( v_2 \). By continuity the speed at the disc is the average, \( v = (v_1+v_2)/2 \).
2. The mass flow is \( \dot m = \rho A(v_1+v_2)/2 \), and the power extracted is the rate of loss of kinetic energy, \( P = \tfrac12\dot m(v_1^2-v_2^2) \).
3. Normalising by \( \tfrac12\rho Av_1^3 \) and writing \( b = v_2/v_1 \) gives \( C_p = \tfrac12(1+b)(1-b^2) \).
4. Maximising gives \( b = 1/3 \) and \( C_p = 16/27 = 0.593 \).

The physical tension is visible in the expression. Slowing the air more extracts more energy per kilogram — the \( (1-b^2) \) factor — but slows the flow through the disc, so fewer kilograms arrive — the \( (1+b) \) factor. The product peaks when the air leaves at one third of its arrival speed.

Why it applies to every design. Nothing in the derivation mentions blades, aerofoils, the number of blades, or even that the device is a turbine. It is a control-volume argument about a disc that extracts momentum from a fluid stream. Any device that works by slowing an unducted airstream obeys it — three-blade, vertical-axis, multi-blade, or something not yet invented.

The honest caveats, since claims of "exceeding Betz" recur:

• A ducted or shrouded turbine draws in air from a larger area than its own disc. It can exceed 0.593 measured against its rotor area — but not against the area of the duct's inlet, which is the fair comparison.
• The classical derivation neglects wake rotation. Including it (the Glauert analysis) gives a slightly lower limit at low tip-speed ratios, not a higher one.
• Real rotors reach 0.45–0.50 — about 80% of Betz — after tip losses, drag and finite blade count. That is the number worth remembering for engineering, and a design claiming much more deserves a careful look at what area it is being compared against.

In wind, the optimum is a known function of a measurable state. In solar, it is not.

The wind case. The power coefficient depends on the tip-speed ratio, \( \lambda = \omega R/v \), and \( C_p(\lambda) \) has a single peak at a \( \lambda_{opt} \) that is a fixed property of the rotor's geometry. Holding \( \lambda = \lambda_{opt} \) means \( \omega \propto v \), and eliminating \( v \) gives

Measure the generator speed, demand a torque proportional to its square, and the rotor settles at the optimum by itself. No anemometer, no perturbation, no oscillation — and it is stable, because if the turbine is running slow the applied torque is too small and it accelerates.

The solar case. The MPP voltage depends on irradiance, temperature, module age, soiling and shading pattern. A model could be written, but it would need parameters that drift over a twenty-five-year life and a temperature the system does not measure. So the tracker searches instead — which is why it oscillates, and why it can be fooled by a change in the curve it is standing on.

The general principle, which is worth carrying beyond these two: if the optimum can be expressed as a function of measurable states with stable parameters, use a control law. If it cannot, search — and accept the cost of searching, which is permanent perturbation and vulnerability to changes in the thing being searched.

Two honest qualifications:

• Wind's advantage is not free. \( K_{opt} \) depends on \( \rho \), which varies with air density, and on \( C_{p,max} \) and \( \lambda_{opt} \), which change as blades erode and soil. Large turbines therefore add slow adaptation on top of the law — a search, but a much easier one.
• Solar can move partway towards a law: the fractional-\( V_{oc} \) method exploits the observation that \( V_{mp} \approx 0.78\,V_{oc} \) across a wide range. It is only approximate, but it is a model rather than a search, and it is used to seed a search — a useful hybrid, and a good illustration that the two approaches are ends of a spectrum rather than alternatives.