Gradient Noise in Extremum Seeking for Wind Turbines

How turbulence sets the power that LP-ESC and LP-PIESC lose, and the tuning that loses least

Aykut C. Satici

Robot Control Lab · Systems Engineering · UT Dallas

The result in brief

41% less energy lost by the proposed LP-ESC than by the best published design, the LP-ESC of Ciri, Leonardi & Rotea 2019 (same one-day tracking time): 0.385% → 0.227%, about 27 MWh per turbine-year at 8 m/s

12× less by the proposed LP-PIESC than by the LP-PIESC of Kumar & Rotea 2022, same estimator: 3.19% → 0.275%, about 500 MWh per turbine-year

46% less by the proposed LP-ESC than by the LP-ESC tuning of Kumar & Rotea 2022, both tracking the optimum in 740 s: 4.51% → 2.42%, about 360 MWh per turbine-year

Six-day simulations of the nonlinear NREL 5 MW turbine at 8 m/s, turbulence intensity 0.10, four wind records per spectrum; every controller runs in the same wind as a turbine held at the optimum. Each published design is paired with the proposed design of the same type, tracking at least as fast. Bars: mean over records, lines: range. Solid: Kaimal; light: von Kármán.

The headline. The proposed design, an ordinary demodulation loop tuned from the turbulence spectrum, loses 0.23 percent of the energy. The best published design, also demodulation and at the same one-day tracking time, loses 0.39. Each comparison is like for like: same controller type, same or slower tracking. The 2022 demodulation tuning tracks in 740 seconds and loses 4.5 percent; the proposed design for 740 seconds loses 2.4. The 2022 least-squares LP-PIESC tracks in 5.1 days and loses 3.2 percent; the same estimator with the proposed tuning tracks in one day and loses 0.275. The rest of the talk is where these numbers come from, and why the prediction and the simulation agree.

One gain, unknown and drifting

Below rated wind speed the turbine should run at the tip-speed ratio \(\lambda = R\Omega/V\) that maximizes the power coefficient \(C_P(\lambda)\). The standard controller sets the generator torque to \(u\,\Omega^2\); the gain \(u\) alone fixes where the rotor settles, at every wind speed (Unit 1).

The best gain \(u^\star\) depends on the blades, which erode, ice and foul. It must be found online, and found again when it moves.

At a fixed tip-speed ratio, both torques scale as V squared, so the wind cancels and the gain alone sets lambda. The optimum is one number, but it drifts with the blades.

Extremum seeking finds it, by two routes

1. Dither: add \(a\sin\omega t\) to the gain, period \(T = 2\pi/\omega\).
2. Estimate: read from log-power \(y = \ln P\) how power changes with the gain.
3. Step: move the gain uphill, once per period (step size \(\kappa\)) or continuously.

LP-ESC: demodulation Correlate \(y\) with the dither over each period: the in-phase channel \(\hat g\) [Rotea 2017; Ciri, Leonardi & Rotea 2019].

LP-PIESC: least squares Fit \(\dot y \approx \theta_0 + \theta_1(\nu - \hat\nu)\) by recursive least squares; proportional-integral update [Kumar & Rotea 2022]. Here \(\nu = \nu^\star u/u^\star\) is the normalized gain (\(\nu^\star = 0.061\)) and \(\hat\nu\) the integrator state, about which the dither swings.

Both are log-power schemes: the log removes the mean wind speed from the loop gain. They differ in how they turn the response into a gradient.

LP-PIESC: a least-squares fit in place of the correlation

Published tuning [Kumar & Rotea 2022]: \(T = 314\) s, \(a = 0.082\,\nu^\star\), estimator memory \(1/k_T = 0.05\) s, ridge \(q = 10^{-6}\), \(k_p = 5\times10^{-6}\) s, \(\tau_I = 2.8\). Reported to reach the optimum almost at once, with a quarter of the classical dither.

Keep in mind two numbers: a memory of a twentieth of a second, and a period of five minutes. Their ratio is what decides what this estimator reads.

Turbulence makes the gain scatter

Scatter \(\sigma(\tilde u)\): the standard deviation of the gain error \(\tilde u = u - u^\star\) once the loop has settled, a spread over time quoted as a fraction of \(u^\star\). It concerns the center of the dither only; the dither’s own cost is counted separately. Every excursion costs power. With \(J(u)\) the log-power at rest at gain \(u\) (apart from a constant set by the wind), which peaks at \(u^\star\), the loss is \(\tfrac12|J''|\,\sigma(\tilde u)^2\) plus the dither’s cost.

Left: the 2022 LP-PIESC in four wind records over six days, started at the optimum. It random-walks. Right: the same LP-PIESC estimator with the proposed design (60 s period, amplitude 0.111, one-day tracking time), same winds. The scatter is the size of this motion.

What this work answers

Given the wind’s spectral density and the tuning of each controller:

1. How noisy is one gradient estimate at the optimum, and what does LP-PIESC’s estimate measure?
2. How large is the scatter of the gain in steady state?
3. How much power is lost, and which tuning loses least for a required tracking time?
4. How do LP-ESC and LP-PIESC compare, and do the predictions survive simulation, including the published runs?

The answer in one line. Write the wind speed as \(V = \bar V(1+\varepsilon)\), with mean \(\bar V\) and relative fluctuation \(\varepsilon\). At the optimum the wind enters log-power as \(3\varepsilon\), so each estimate is the wind seen through a known window; a slow loop (one whose tracking time spans many dither periods) sees only the wind’s density at the dither frequency; the scatter, the loss and the best tuning then follow in closed form from three numbers: that density, the curvature of \(C_P\) at its peak, and the rotor time constant.

Everything that follows is this one sentence, unpacked and checked.

The argument in one line

The wind \(\Phi(f_1)\): the turbulence density at the dither frequency \(f_1 = 1/T\) (IEC Kaimal or von Kármán).

The rotor \(\tau_r = 6.88\) s: the time constant with which the rotor follows the gain (at 8 m/s).

The power curve \(\chi = 6.3\): how sharply \(C_P\) peaks; \(|J''(u^\star)|\,u^{\star2} = \chi/9 = 0.70\).

Five links, each checked in simulation. Then the design, and the evidence on energy.

Link 1: at the optimum, the wind enters log-power as \(3\varepsilon\)

With \(V = \bar V(1 + \varepsilon)\), power is \(\propto V^3 C_P(\lambda)\), so \[y = \ln P = \text{const} + 3\ln(1+\varepsilon) + \ln C_P(\lambda).\]

At \(u^\star\) the power curve is flat: \(C_P'(\lambda^\star) = 0\), so the wind’s effect through \(\lambda\) drops out at first order, and \[y - \mathbb{E}[y] = 3\,\varepsilon + O(\sigma_\varepsilon^2),\] where \(\sigma_\varepsilon\), the standard deviation of \(\varepsilon\), is the turbulence intensity (0.10).

Simulation: slope 3.086 (Kaimal), 3.090 (von Kármán); correlation 0.991.

A gust changes power only through the cube of the wind speed, because at the peak the power curve is flat. That single term makes everything else linear.

Link 2: one estimate sees the spectrum through a window

The demodulated estimate correlates \(y\) with \(\sin\omega t\), so its noise is \(6Z_n/(aT)\) with \(Z_n = \int_0^T \varepsilon(nT+t)\sin\omega t\,dt\), and \[\operatorname{Var}(Z) = \int \Phi(f)\,|\hat w(f)|^2\,df ,\] where \(\hat w(f) = \int_0^T \sin(\omega t)\,e^{-i2\pi ft}\,dt\), the dither’s Fourier transform over one period, is the estimate’s window on the wind.

Both windows pass \(f_1\) equally; the sine window leans to low frequencies, where the wind has its energy, so the quadrature (cosine) channel is the quieter one per estimate (Units 12–14).

A linear filter of a stationary wind: its variance is the spectrum integrated against the filter’s squared gain.

Checked over a tenfold range of dither periods

Predicted and simulated standard deviation of one estimate agree within 0.5 – 2.2 % for \(T\) from 60 to 600 s under both spectra, with no fitted parameter; the small excess is the size and sign of the terms the prediction leaves out.

240 periods per point; sampling uncertainty about 4.6 percent per point.

The two IEC turbulence spectra

• Same flat level \(4\sigma_\varepsilon^2 L/\bar V\), so \(L\) is the integral length scale: 340 m (Kaimal), 147 m (von Kármán) at the 90 m hub.
• Both fall as \(f^{-5/3}\); they differ mainly near their corners.
• At the published period (314 s) the densities agree within 3 %; at 60, 150, 600 s their ratio is 0.79, 0.81, 1.35.
• Effect on the predicted scatter: at most 11 % over 60–314 s. No conclusion changes.

Everything is worked out twice, on equal footing. The spectrum enters only through its value at the dither frequency.

Link 3: the published LP-PIESC reads the quadrature channel

The regressor is \([1, d]\), with \(d = \nu - \hat\nu = a\sin\omega t\) the dither (proportional term neglected). Over a 0.05 s memory it is frozen, so the ridge fit, with effective ridge weight \(\gamma = 4\times10^{-4}\), returns the minimum-norm product \[\hat\theta_1 = \frac{d\,\dot y}{1 + d^2 + \gamma},\] whose period average is the cosine (quadrature) coefficient of \(y\). The rotor returns the dither in phase with gain \(G_1 = 1/(1+(\omega\tau_r)^2)\), which demodulation reads, and a quarter period late with gain \(G_q = \omega\tau_r G_1\), which this estimate reads: 0.135 against 0.981 at 314 s.

Simulation: \(\hat\theta_1\) follows \(d\,\dot y\) with correlation 0.997; in-phase / quadrature 7.24, predicted \(1/(\omega\tau_r) = 7.27\).

The estimator sees the gradient only through the rotor’s lag. A rotor without lag would give it no gradient signal at all.

Link 4: a slow loop sees one frequency

Each period a loop corrects a fraction \(K = T/\tau_c\) of the gain error; it is slow when \({K \ll 1}\) (one day at 60 s: \({K = 7\times10^{-4}}\)). Averaging \(N\) estimates is one estimate over \(N\) periods; its window narrows onto \(f_1\) (Fejér). So the noise a slow loop feels is the long-run variance \[\sigma_\infty^2 = \frac{2\Psi}{a^2T},\qquad \Psi = 9\,\Phi(f_1),\] and the scatter at tracking time \(\tau_c\) is \[\sigma(\tilde u)^2 = \frac{\Psi}{a^2 g^2\,\tau_c\,J''^2},\quad g = G_1 \text{ or } G_q .\]

Simulated loops: within 4 % of the long-run prediction; one period’s variance overstates by up to 18 %.

The period cancels from the scatter once you fix the tracking time; the wind enters through its density at one frequency.

Link 5: loss = scatter + dither; the best amplitude balances them

\[\ell(a) = \underbrace{\frac{\Psi}{2a^2g^2\tau_c|J''|}}_{\text{scatter}} + \underbrace{\frac{|J''|G_1a^2}{4\vphantom{2a^2g^2\tau_c|J''|}}}_{\text{dither}}\]

Least when the two are equal: \[a_{\mathrm{opt}} = \Big(\frac{2\Psi}{g^2G_1\tau_cJ''^2}\Big)^{1/4},\qquad \ell_{\mathrm{min}} = \sqrt{\frac{\Psi G_1}{2\tau_c g^2}}.\]

The step size or integral time then follows from the tracking time: \[\begin{aligned}\kappa &= \frac{T}{\tau_cG_1|J''|} &&\text{(LP-ESC)},\\[2pt] \tau_I &= \frac{\tau_c\pi a^2G_q|J''|}{T} &&\text{(LP-PIESC)}.\end{aligned}\]

A larger dither lets a loop of fixed tracking time take smaller steps and scatter less, but the dither itself costs power. The least loss balances the two.

Choosing the period

Shorter periods move \(f_1\) up the falling spectrum, so the least loss falls, until the rotor’s lag takes over. Recommendation: \(T = 60\) s (about nine rotor time constants): LP-ESC 0.21 %, LP-PIESC 0.29 % (Kaimal, one day), against 0.41 % and 2.98 % at 314 s. The quadrature penalty \(1/(\omega\tau_r)\) falls from 7.3 to 1.4.

Below about 45 seconds the static power curve and the first-order rotor are least reliable, so we stop at 60.

The published LP-PIESC tracks in 5.1 days

Its integral path is a slow loop on the quadrature channel with loop gain \(7.13\times10^{-4}\) per period: tracking time 5.1 days, stationary scatter 0.33 \(u^\star\), loss 3.9 % (Kaimal). The published fast convergence appears only when the integrator starts at the optimum, which the closed-form optimum makes possible.

LP-ESC with its published tuning converges in about 1000 seconds, as reported. LP-PIESC barely moves from any other start in 3000 seconds.

Energy over six days, every controller in the same wind

controller (\(T\), \(a/u^\star\))	Kaimal, days 2–6	predicted	von Kármán, days 2–6	predicted
LP-ESC, Kumar & Rotea 2022 (314 s, 0.33)	4.51 %		4.48 %
LP-ESC, proposed design at 740 s (60 s, 0.31)	2.42 %	2.22 %	2.70 %	2.50 %
LP-PIESC, Kumar & Rotea 2022 (314 s, 0.082)	3.19 % (0.9–7.1)	3.9 %a	3.09 %
LP-ESC, Ciri, Leonardi & Rotea 2019 (150 s, 0.136)	0.385 %	0.38 %	0.406 %	0.39 %
LP-PIESC, proposed design (60 s, 0.111)	0.275 %	0.29 %	0.308 %	0.32 %
LP-ESC, proposed design (60 s, 0.094)	0.227 %	0.21 %	0.258 %	0.23 %

Energy lost against a turbine held at \(u^\star\) in the same wind; mean of four six-day records. ^a stationary prediction for a 5.1-day tracking time, which six days do not reach. Tracking times: 740 s for the 2022 LP-ESC and the design beside it; 5.1 days for the 2022 LP-PIESC; one day for every other row.

The designs lose what the formulas predict. The 2022 LP-PIESC depends on the wind record (0.9 to 7.1 %). At the 2022 LP-ESC’s 740 s tracking time the proposed design loses 2.42 %, 9 % above the slow-loop prediction: that loop is no longer slow.

Every number here is in the claim checker. The prediction for the designs comes from three numbers and was made before the simulation.

After the optimum moves

Gain started at 0.8 \(u^\star\). Energy lost on the first day: proposed LP-ESC 0.67 % against LP-ESC 2019 0.89 %; proposed LP-ESC at 740 s 2.29 % against LP-ESC 2022 4.25 %; proposed LP-PIESC 0.68 % against LP-PIESC 2022 3.2 % (0.6 to 9 % with the wind record). The 2022 LP-PIESC’s gain wanders instead of converging (0.78–1.45 \(u^\star\) after six days).

One wind record shown. The designs return within about a day, as their tracking time says.

Gains over the published designs

−41 % proposed LP-ESC vs. the best published design (LP-ESC, Ciri, Leonardi & Rotea 2019, same tracking time): 0.16 points, 27 MWh per turbine-year at 8 m/s. Von Kármán: −36 %.

÷12 proposed LP-PIESC vs. LP-PIESC, Kumar & Rotea 2022, same estimator: 3.19 % → 0.275 %, 2.92 points, 499 MWh per turbine-year. Von Kármán: ÷10.

−46 % proposed LP-ESC vs. LP-ESC, Kumar & Rotea 2022, same tracking time (740 s): 4.51 % → 2.42 %, 2.09 points, 358 MWh per turbine-year. Von Kármán: −40 %.

Remark. Measured in our simulator: NREL 5 MW rotor with a static power curve and a first-order rotor, aerodynamic power, 8 m/s, turbulence intensity 0.10, synthesized turbulence. The published designs were tuned in other simulators; the comparison holds turbine, wind and tracking time fixed for both LP-ESC comparisons, so the design is the only difference; the proposed LP-PIESC tracks five times faster than the published one. Reference turbine: 1.95 MW mean, 17.1 GWh per year at this wind.

The gain over the best published design comes entirely from the period and the amplitude: shorter period, amplitude set where scatter and dither cost the same. Against the 2022 LP-ESC the amplitude is almost the same (0.31 against 0.33); the gain is the period. At 60 seconds the rotor passes less of the dither, so the dither costs 1.1 percent rather than 1.9, and the wind’s density at the dither frequency is lower. A 740-second tracking time is expensive: at one day the same rules lose 0.23 percent.

When is LP-PIESC the right choice?

The gap is a tuning choice The penalty comes from the 0.05 s memory plus fitting \(\dot y\). Fit \(y\) itself over a full period and least squares reads the in-phase channel, with demodulation’s loss. At 60 s the published estimator costs only 20 % more (0.275 vs 0.227 %).

Least squares gives more than an estimate An explicit model, an information matrix that can serve as an error bar, and updates in continuous time: useful for a loop that acts on its own uncertainty.

The quadrature channel’s one merit A slow wind drift within a period reads as gradient in the sine window but cancels in the cosine window. It loses more signal (\(\times\omega\tau_r\)) than it saves in noise.

Speed was the published motivation This work measures steady-state loss of a slow loop and neglects the proportional path. Whether a retuned least-squares loop recovers faster after the optimum moves is open.

For steady-state energy capture, choose demodulation or a full-period least-squares fit. Choose LP-PIESC’s structure where its model, its error bar or a faster transient is needed, and tune it by these rules.

The design procedure

Three numbers

1. \(\tau_r = \dfrac{2I\lambda^{\star2}}{3\rho\pi R^4\bar V C_P^{\mathrm{max}}}\) from the rotor data.
2. \(\chi\) from the design power curve: \({|J''(u^\star)| = \chi/(9u^{\star2})}\).
3. \({\Psi = 9\,\Phi(1/T)}\) from the site’s spectrum (or the IEC model).

One choice: the tracking time \(\tau_c\) (blades change over weeks; recover from start-up within a day: \({\tau_c = 1}\) day).

Then

4. \(T \approx 9\,\tau_r\) (60 s here).
5. \(a = a_{\mathrm{opt}}\).
6. \({\kappa = T/(\tau_cG_1|J''|)}\) or \({\tau_I = \tau_c\pi a^2G_q|J''|/T}\).

A design curvature \({J''_\mathrm{d} = rJ''}\) leaves the loop unbiased, scales the tracking time by \(r\), and costs \({(\sqrt r + 1/\sqrt r)/2}\) of the least loss for that tracking time: 6 % at \(r = 2\) (B15).

Everything needed before any simulation is run.

Limits, and the next step

• The design needs \(J''\). It comes from the design power curve, and the true curvature drifts with the blades, as \(u^\star\) does. A design curvature \({J''_\mathrm{d} = rJ''}\) leaves the loop unbiased, scales its tracking time by \(r\), and costs \({(\sqrt r + 1/\sqrt r)/2}\) of the least loss for that tracking time: measured +5 % at \(r = \tfrac12\) and +8 % at \(r = 2\) (B15). The design also needs \(\Phi(f_1)\) at the site, and \(\tau_r\), which varies as \(1/\bar V\).
• The wind model: stationary, with fixed mean speed and turbulence intensity, while a day mixes wind states; point spectra, while the rotor averages the wind over its disk.
• The turbine model: static power curve and first-order rotor; aerodynamic, not electrical, power (the term \(I\Omega\dot\Omega\) matters when \(\omega\tau_r\) is not small: 0.72 at 60 s).
• The simulations use the same spectra as the analysis: they test the derivation, not the wind models.
• LP-PIESC: its proportional path \(k_p\hat\theta_1\), the term that adds the current estimate to the gain, moves the gain \({4.5\times10^{-8}}\) as much as the integral path per period at the published \(k_p\). A \(k_p\) raised for speed, and the projection, are not analyzed.

Next: measure \(J''\) online; test on measured turbine data and electrical power; measure recovery after the optimum moves, where LP-PIESC’s proportional path was meant to act.

Summary

1. At the optimum the wind enters log-power as \(3\varepsilon\), so each gradient estimate is the wind seen through a known window.
2. A slow loop sees only the wind’s density at the dither frequency; the scatter is \(\sqrt{\Psi/(a^2g^2\tau_cJ''^2)}\).
3. The published LP-PIESC reads the quadrature channel, \(\omega\tau_r\) of the signal, and tracks in 5.1 days.
4. Loss = scatter + dither; the minimum-loss design follows from three numbers, at \(T \approx 9\tau_r\).
5. Measured over six days: 0.23 % lost, 41 % less than the best published design; 46 % less than the 2022 LP-ESC tuning at its 740 s tracking time; the same rules cut the 2022 LP-PIESC’s loss 12× while tracking five times faster.

Technical report and workbook: rotea repository, papers/esc-spectral-noise/.

Backup

Derivations and checks behind the main slides. Each cites the units of the technical report that prove it.

• B1 The rest point, and why the wind cancels
• B2 The rotor time constant, and the inertia
• B3 The rotor seen from the gain
• B4 The estimate in calm wind
• B5 The wind as a random process
• B6 The least-squares estimator, unpacked
• B7 Correlated noise and the long-run variance
• B8 Short records under-measure the scatter
• B9 Away from the optimum
• B10 The synthesized wind
• B11 Where the optimum sits in turbulence
• B12 Why the 2022 LP-ESC tuning loses 4.5 %
• B13 Simulation setup
• B14 Running example
• B15 A wrong design curvature

B1: The rest point, and why the wind cancels

At a fixed tip-speed ratio \(\lambda = R\Omega/V\), both torques on the rotor scale as \(V^2\):

Aerodynamic torque \[\tfrac12\rho\pi R^3V^2\,\frac{C_P(\lambda)}{\lambda}\]

Generator torque \[u\,\Omega^2 = u\,\frac{\lambda^2V^2}{R^2}\]

so their balance is free of the wind:

\[\frac{C_P(\lambda)}{\lambda^3} = c_u\,u,\qquad c_u = \frac{2}{\rho\pi R^5}.\]

One stable rest point For every gain below \(1.86\,u^\star\) there is exactly one, \(\lambda_{\mathrm{eq}}(u)\), on the branch through \(\lambda^\star\); near it the rotor is first order with time constant \(\tau_r\) (Units 1–2).

The objective Log-power at rest is a wind constant plus \(J(u) = \ln C_P(\lambda_{\mathrm{eq}}(u))\), which peaks at \(u^\star = 2.23\times10^6\) N m s² with \(J''(u^\star) = -\chi/(9u^{\star2})\) (Unit 3).

B2: The rotor time constant, and the inertia

\[\tau_r = \frac{2I\lambda^{\star2}}{3\rho\pi R^4\bar V C_P^{\mathrm{max}}} = 6.88\ \text{s at } 8\ \text{m/s},\qquad \tau_r\propto 1/\bar V.\]

The rotor’s inertia \(I_{\mathrm{rotor}}\) follows from the NREL 5 MW blade and hub data, moving each blade to the shaft by the parallel-axis theorem: \[I_{\mathrm{rotor}} = 3\cos^2\beta_p\,(I_b + 2r_hS_b + m_br_h^2) + 115\,926\ \text{kg m}^2,\]

with \(\beta_p\) the precone angle, \(r_h\) the hub radius, \(m_b\), \(S_b\), \(I_b\) the blade’s mass and its first and second mass moments about its root, and 115 926 kg m² the hub.

3.88 × 10⁷ kg m² the rotor, \(I_{\mathrm{rotor}}\)

+ 97² × 534 kg m² the generator, seen through the 97 : 1 gearbox

4.38 × 10⁷ kg m² the total, \(I\)

Remark. [Kumar & Rotea 2022] use \(3.54\times10^7\) kg m², which omits the hub-radius offset \(r_h = 1.5\) m.

B3: The rotor seen from the gain

From the gain to log-power, the rotor is a first-order lag with unit static gain and time constant \(\tau_r\), followed by the slope \(J'\). A dither \(a\sin\omega t\) therefore returns as

\[\delta y = J'a\,(G_1\sin\omega t - G_q\cos\omega t),\qquad G_1 = \frac{1}{1+(\omega\tau_r)^2},\quad G_q = \omega\tau_rG_1 .\]

0.981 / 0.135 \(G_1\) / \(G_q\) at the published period, \(T\) = 314 s

0.66 / 0.48 \(G_1\) / \(G_q\) at the recommended period, \(T\) = 60 s

43 s \(T = 2\pi\tau_r\): the two cross, and \(G_q = 1/2\) is largest (Unit 4)

Demodulation reads the in-phase part \(G_1\); the published LP-PIESC estimator reads the quadrature part \(G_q\) (Link 3).

B4: The estimate in calm wind

A Fourier coefficient \[\hat g_n = B_1/a,\qquad \hat g^c_n = A_1/a,\] with \(B_1\) and \(A_1\) the first sine and cosine coefficients of \(y\) over period \(n\). It ignores the mean and every higher harmonic, but a drift within the period reads as gradient (Unit 5).

Its bias \[\hat g_n = J'(u_n) + \tfrac{a^2}{8}J'''(u_n) + O(a^4),\] with \(u_n\) the center of the dither in period \(n\) (Unit 6).

At \(a = 0.136\,u^\star\):

0.37 % above \(u^\star\): where the bias makes the loop settle

5 × 10⁻⁶ of the power: what that offset costs

0.32 % of the power: what the dither itself costs

The bias is negligible. The error that matters is random.

B5: The wind as a random process

The model \[V = \bar V(1+\varepsilon),\qquad \Gamma(\tau) = \mathbb{E}[\varepsilon(t)\varepsilon(t+\tau)],\] with \(\varepsilon\) stationary and \(|\Gamma| \le \Gamma(0) = \sigma_\varepsilon^2 = \mathrm{TI}^2\).

Its spectrum \[\Phi(f) = \int\Gamma(\tau)e^{-i2\pi f\tau}d\tau,\qquad \sigma_\varepsilon^2 = \int_0^\infty S\,df,\] with \(S = 2\Phi\) the one-sided density (Units 7–8).

Orders of smallness Remainders are ranked by RMS in powers of \(\sigma_\varepsilon\). With sixth moments finite, products of \(k \le 3\) filtered winds are \(O(\sigma_\varepsilon^k)\) (Unit 9).

What the tip-speed ratio sees The high-passed wind, with corner \(1/(2\pi\tau_r) = 0.023\) Hz; the mean of \(\lambda\) stays at \(\lambda^\star\) to second order (Unit 10).

72 % of the Kaimal variance lies below that corner, where the rotor follows the wind

B6: The least-squares estimator, unpacked

1 · A weighted ridge fit At every instant the recursion returns the minimizer of an exponentially weighted ridge cost with memory \(1/k_T = 0.05\) s (Unit 15).

2 · A fit of filtered \(\dot y\) With \(\tilde y\) the prediction error and \(c\) the filtered regressor, the states combine into \[z = \tilde y + c^\top\hat\theta,\quad \dot z = -k_fz + \dot y.\] So the estimator regresses \(\dot y\), low-passed at rate \(k_f\), on the equally filtered regressor (Unit 16).

3 · A frozen regressor Over the memory the regressor barely moves, so the fit is the minimum-norm product, whose period average is the quadrature channel (Unit 17).

γ = 4 × 10⁻⁴ effective ridge weight, \(qk_f^2\)

𝓜 = 0.0025 memory number \(\omega^2/(k_T^2\gamma)\): small means frozen

A fit of \(y\) itself on \([1, d]\) over a full period returns the in-phase channel instead.

B7: Correlated noise and the long-run variance

A first-order recursion \(x_{n+1} = \alpha x_n + \kappa v_n\), driven by noise \(v_n\) of variance \(\sigma_v^2\) and correlation \(\rho_k\) between values \(k\) steps apart, has variance

\[\operatorname{Var}(x) = \frac{\kappa^2\sigma_v^2}{1-\alpha^2}\Big[1 + 2\sum_{k\ge1}\alpha^k\rho_k\Big].\]

A slow loop, \(\alpha \to 1\) The bracket tends to \(1 + 2\sum_{k\ge1}\rho_k\), and \(\sigma_\infty^2 = \sigma_v^2\big(1 + 2\sum_{k\ge1}\rho_k\big)\) is the long-run variance of the noise (Units 20–21).

In the frequency domain Summed with Fejér’s kernel, \(\sigma_\infty^2 = 2\Psi/(a^2T)\), for both channels (Unit 22).

1.29 × one period’s variance against the long-run value, in-phase channel (Kaimal, \(T\) = 150 s)

0.87 × the same ratio for the quadrature channel

B8: Short records under-measure the scatter

The sample variance \(s^2\) of a record falls short of the true variance \(\sigma^2\) by the variance of the record’s own mean \(\bar x\) (Unit 21):

\[\mathbb{E}[s^2] = \sigma^2 - \operatorname{Var}(\bar x).\]

2 days record length

6 h tracking time of the loop: the record spans eight tracking times

0.88 sample standard deviation over true: a 12 % shortfall

The nonlinear comparison of LP-PIESC against demodulation applies this factor to its predictions.

B9: Away from the optimum

Away from \(u^\star\) the rotor adds a high-passed copy of the wind to log-power, with weight \(\eta\), the slope of \(\ln C_P\) against \(\ln\lambda\) at the mean tip-speed ratio. The standard deviation of the noise is multiplied by \(|1 + \eta H(f_1)/3|\), where

\[\eta \approx 2.1\,\frac{\tilde u}{u^\star},\qquad H(f) = \frac{-i2\pi f\tau_r}{1+i2\pi f\tau_r}\ \ \text{(the rotor's high-pass)}.\]

0.1 \(u^\star\) scatter of a converged loop

≈ 2 % change in the noise at that scatter (Unit 26)

The noise model derived at \(u^\star\) holds over the gains a converged loop visits.

B10: The synthesized wind

A sum of cosines with independent uniform phases \(\Theta_k\):

\[\varepsilon(t) = \sum_{k=1}^{M} A_k\cos(2\pi f_kt + \Theta_k),\qquad f_k = \frac{k}{D},\quad A_k = \sqrt{2S(f_k)/D},\]

with \(D\) the record length and \(f_M = 1/(2\Delta t)\) the Nyquist frequency of the sampling interval \(\Delta t\). The sum is strictly stationary, with the target spectrum on its frequency grid (Unit 27).

2.7 % of the variance missing from a ten-hour record at \(\Delta t\) = 0.5 s, mostly above the Nyquist frequency

9.86 % so its turbulence intensity (Kaimal; target 10 %)

−0.002 excess kurtosis: Gaussian

Runs under the two spectra share their random phases.

B11: Where the optimum sits in turbulence

Fixed gains, no dither: the energy peaks at 0.99 \(u^\star\); mean log-power, which the loops climb, at 1.013 \(u^\star\) (1.014 von Kármán). The difference costs about 0.02 % of the energy. The dither alone, center held at \(u^\star\), costs exactly the calm-wind \(\tfrac14|J''|G_1a^2\): 0.102 % at 60 s and 0.094 \(u^\star\), 0.116 % for the published LP-PIESC dither.

B12: Why the 2022 LP-ESC tuning loses 4.5 %

Its dither \(a = 0.33\,u^\star\) at \(T\) = 314 s costs 1.94 % on its own (measured; the quadratic formula gives 1.88 %).

Its integrator Fast: it reaches the optimum’s neighborhood within the first hour, but swings the gain by 0.17 \(u^\star\) root-mean-square, about a mean 10 % above \(u^\star\).

4.5 % lost in all, steady, independent of the start

740 s its tracking time, from the linearized loop

2.42 % lost by the proposed LP-ESC for the same 740 s (\(T\) = 60 s, \(a\) = 0.31 \(u^\star\)); predicted 2.22 %

Linearized loop of the 2022 tuning: high-pass 0.018 s⁻¹, demodulation phase 0.36 rad, integrator gain 0.001.

Same amplitude, shorter period: the proposed design’s dither costs 1.11 % instead of 1.88 %.

B13: Simulation setup

turbine	NREL 5 MW rotor, Heier power curve rescaled to \((7.48, 0.4895)\), \(I = 4.38\times10^7\) kg m²
wind	8 m/s, TI 0.10, Kaimal (\(L\) = 340 m) and von Kármán (147 m), synthesized, 0.5 s
integration	adaptive Runge–Kutta (open-loop tests of one estimate); explicit Euler, 5 ms (every closed loop, both controllers)
energy runs	six controllers + reference at \(u^\star\), same wind; 6 days × 4 records × 2 spectra × 2 starts
left out	pitch, power limit, generator and converter dynamics, drivetrain flexibility; one wind speed acts on the whole rotor
constants	\(u^\star\) by Newton on \(J'\), \(J''\) by automatic differentiation: \(|J''|u^{\star2} = 0.7017\) (designs use \(0.7022 = 6.32/9\)); \(\tau_r\) = 6.878 s
checks	629 stated numbers recomputed from formulas and stored results by the claim checker

Every run is specified in full (equations as implemented, initial states, seeds, record lengths, statistics) in Appendix D of the technical report.

B14: Running example

quantity	value
rotor radius, air density	63 m, 1.225 kg/m³
optimum	\(\lambda^\star = 7.48\), \(C_P^{\mathrm{max}} = 0.4895\)
optimal gain	\({u^\star = 2.23\times10^6}\) N m s², normalized \({\nu^\star = 0.061}\)
curvature	\(\chi = 6.3\), \(|J''(u^\star)|u^{\star2} = 0.70\)
rotor time constant	6.88 s at 8 m/s
dither periods	60 s (recommended), 150 s (2019 design), 314 s (2022 tunings)
tracking time	one day (designs); 740 s for the design matched to the 2022 LP-ESC

B15: A wrong design curvature

Tuned with \({J''_\mathrm{d} = rJ''}\), the loop tracks in \({r\tau_c}\), stays unbiased, and has amplitude \({a_\mathrm{opt}r^{-1/2}}\). The loss of Link 5 becomes

\[\ell = \frac{\ell_\mathrm{min}}{2}\Big(1 + \frac1r\Big) = \frac{\sqrt r + 1/\sqrt r}{2}\cdot\frac{\ell_\mathrm{min}}{\sqrt r},\]

where \(\ell_\mathrm{min}/\sqrt r\) is the least loss at the tracking time \(r\tau_c\) the loop actually has.

LP-ESC, \(T\) = 60 s (Kaimal)	tracking time	\(a/u^\star\)	lost, days 2–6	predicted
tuned with \(J''/2\) (\(r = \tfrac12\))	0.5 day	0.133	0.328 %	0.31 %
correct design for 0.5 day	0.5 day	0.112	0.311 %	0.29 %
proposed design (\(r = 1\))	1 day	0.094	0.227 %	0.21 %
correct design for 2 days	2 days	0.079	0.160 %	0.15 %
tuned with \(2J''\) (\(r = 2\))	2 days	0.067	0.172 %	0.15 %

Against the correct design for the same tracking time: +5 % (\(r = \tfrac12\)) and +8 % (\(r = 2\)), predicted +6 %; von Kármán gives the same. The error shows in the tracking time, not in the energy. Mean of four six-day records; same winds as the energy table.

References

• [R17] M. A. Rotea, “Logarithmic power feedback for extremum seeking control of wind turbines,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 4504–4509, 2017.
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• [GD17] M. Guay, D. Dochain, “A proportional-integral extremum-seeking controller design technique,” Automatica, vol. 77, pp. 61–67, 2017.
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• [IEC] IEC 61400-1, Wind turbines – Part 1: Design requirements.