Extremum Seeking Control of Wind Turbines

The Region-2 problem, the extremum-seeking machinery built for it, and five open questions

Aykut C. Satici

Robot Control Lab · Systems Engineering · UT Dallas

Two regimes, one optimization problem

Region 3: above rated

There is more wind than the machine can use. Throw the excess away and hold rated power. A setpoint, an error signal and a PI controller.

Region 2: below rated

Take everything the wind offers. There is no setpoint, only an efficiency to maximize. This is the only regime containing an extremum to seek.

Everything in this deck is Region 2. The machine spends most of its operating hours there.

The hill

With the blades held at their design pitch, the rotor’s efficiency depends on one number: the tip-speed ratio \(\lambda = R\Omega/V\), blade tip speed divided by wind speed. Hold \(\lambda\) at \(\lambda^\star\) and you capture the most power available.

Two features of this curve set up everything that follows. The peak is flat: being wrong by half a unit in lambda costs about one percent of the power, so power is a weak signal about where you are. The curve also moves. Blades erode, insects accumulate, and the factory value of lambda-star stops being right.

Worth being precise about the qualifier, because it matters later. C_P is really a surface in tip-speed ratio and blade pitch, and it also depends on yaw misalignment and on the condition of the blades. Region 2 pins the pitch at its design value and the yaw at zero, which is what collapses the surface to this curve.

Under real inflow even that is an idealization. Blade sections see different angles of attack at different azimuths, so what you measure is an average over that variation rather than a point on a curve. In the large-eddy simulations the realized C_P falls from 0.49 to 0.44 for exactly that reason, which is a ten per cent violation of the picture on this slide.

You cannot measure either axis

Both coordinates of the hill are written in terms of \(V\), the wind speed actually reaching the rotor, which is exactly what a turbine cannot measure. So this is the route extremum seeking declines to take.

Concretely: an anemometer on the nacelle sits behind the rotor, in air the rotor has already slowed, and how much it has slowed depends on the thrust, which depends on the control parameter. The measurement responds to the thing you are trying to tune.

Be careful how this slide is used. It is not saying that extremum seeking estimates the wind badly. Extremum seeking never estimates the wind at all. This slide rules out the obvious alternative, which is to reconstruct the curve from a wind estimate and find its peak, and it is worth ruling out because the error there is systematic rather than random: it rescales the curve, so no amount of averaging removes it.

Where wind accuracy does still matter, later: reporting the answer as a tip-speed ratio rather than a torque gain, and the dither-free identification in Q3. Neither is the main loop.

Sitting on a hill you cannot see

How the controller finds the peak without measuring where it is.

The \(k\Omega^2\) law

The rotor is one rigid body, so what governs it is

\[I\dot\Omega \;=\; \tau_{\text{aero}} - \tau_g, \qquad \tau_{\text{aero}} = \tfrac12\rho\pi R^3 V^2\,C_Q(\lambda), \qquad \tau_g = k\Omega^2\]

Put \(\dot\Omega = 0\), substitute \(\Omega = \lambda V/R\), and divide through by \(V^2\). Both \(V\) and \(\Omega\) disappear, leaving a condition on \(\lambda\) alone:

\[\boxed{\;\frac{C_P(\lambda)}{\lambda^3} \;=\; \frac{2k}{\rho\pi R^5}\;}\]

So \(\dot\Omega = 0\) does not leave the speed free to have drifted anywhere. The left side is a function of \(\lambda\) whose shape is set by the blade geometry; the right side is a genuine constant, the one you pick with \(k\). The rotor can only rest where the two meet.

This is the point to make slowly, because it is the one that is easy to misread. Setting Omega-dot to zero looks like it only says the speed stopped changing, which would leave the speed itself unconstrained. It does not, because after the substitution neither Omega nor V survives. What is left is an equation in the ratio alone, and its solutions are isolated rather than a continuum.

Second, why the wind cancels at all. It goes in the step where you divide by V squared, and that works only because the generator torque is quadratic in speed. Try k times Omega to the n and you are left with something proportional to V to the n minus two. Only n equal to two removes it, so the quadratic law is the unique exponent that makes the resting tip-speed ratio independent of the wind.

Third, on a real machine V never holds still, so Omega-dot is never exactly zero. The rotor tracks a slowly moving root with a lag, which is what the next slide is about.

Picking \(k\) picks a resting point

Raise \(k\) and the line rises, so the crossing slides left to a lower \(\lambda\). Nothing here needed the wind speed, and nothing needed a measurement.

This is the picture behind the boxed equation, and it answers the obvious objection to it. The left-hand side is not a constant, and of course it is not: lambda moves with rotor speed, so the value of C_P over lambda cubed moves too. What is fixed is the shape of the curve, which is a property of the blades.

So the equation is not saying a quantity equals a number. It is saying: here is a curve, here is a horizontal line at a height you chose, and the rotor rests where they intersect. Reading it that way also makes the next slide obvious, because whether a crossing attracts depends on which way the curve is going through it.

Why the rotor goes to that root

Resting there is not the same as being drawn there. Write the net torque as a function of the state, \(I\dot\Omega = F(\Omega)\) with

\[F(\Omega) \;=\; \tfrac12\rho\pi R^3 V^2\,C_Q\!\left(\frac{R\Omega}{V}\right) - k\Omega^2\]

A small perturbation \(\delta = \Omega - \Omega_{eq}\) obeys \(I\dot\delta = F'(\Omega_{eq})\,\delta\), so it decays exactly when \(F'(\Omega_{eq}) < 0\). Substituting the equilibrium relation collapses that to a statement about the curve alone:

\[F'(\Omega_{eq}) < 0 \qquad\Longleftrightarrow\qquad \frac{d}{d\lambda}\!\left(\frac{C_P}{\lambda^3}\right) < 0 \qquad\Longleftrightarrow\qquad \frac{d \ln C_P}{d \ln \lambda} < 3\]

At the peak \(C_P' = 0\), so the middle expression is \(-3C_P^{\max}/\lambda^4\), negative for any turbine. The design point is always attracting, whatever the blades look like.

The log-slope form is the one worth remembering: the rotor is drawn to a resting point wherever C_P grows more slowly than lambda cubed, and pushed away from it wherever C_P grows faster.

A caveat I should flag rather than gloss. Real power-coefficient curves have a cut-in tip-speed ratio with a steep rise just above it, and in that region the log slope can exceed three, which would give a second root that repels. That is the usual textbook reason a turbine needs help getting started rather than being left to spin up from rest. But the analytic surface these figures are drawn from does not reproduce it: its C_P stays positive down to very small lambda, so the condition never flips and there is exactly one root. So treat the second equilibrium as physically expected and not demonstrated here.

There is no rotor speed to regulate to

The crossing gives \(\lambda_{\text{eq}}(k)\), with no \(V\) in it. Convert back through \(\lambda = R\Omega/V\):

\[\Omega_{eq}(V) \;=\; \frac{\lambda_{\text{eq}}(k)}{R}\,V\]

So the crossing is this ray, in rotor-speed coordinates: one point there, one line here, every point on it the same \(\lambda\). The controller never rejects the wind. It makes that line an attracting manifold and lets the wind slide the machine along it.

This inverts the usual picture. In ordinary regulation you form an error and drive it to zero. Here there is no error signal: the closed loop was designed so that its natural resting place is the set you want. The wind is a parameter that moves you along that set rather than a disturbance pushing you off it.

Region 3 is the opposite and matches the textbook picture, with a real speed setpoint tracked by blade pitch.

Two things worth adding if anyone asks. The stability argument carries over for free: at fixed V, Omega is just lambda rescaled by V over R, so the condition on the slope of the curve is the same condition either way.

And the ray is not quite as fixed as the picture suggests. Air density survives in the equilibrium relation, so the slope drifts slowly with temperature and altitude. That is why the expression for the optimal torque gain carries a rho. It is small and seasonal, and extremum seeking re-tunes straight through it.

So the problem reduces to one number

What we have

A single scalar knob \(k\), and a monotone map from \(k\) to the tip-speed ratio the machine settles at.

What we want

The \(k\) that lands us on the peak. The factory value is wrong at commissioning and gets worse as the blades age.

Find the maximum of a function you cannot evaluate, by turning a knob and watching a noisy meter. That is extremum seeking control, and it is what the last decade of work at UTD has been about.

Extremum seeking

Estimate a derivative you cannot measure by wiggling the input.

The loop

Add a slow sinusoid \(a\sin\omega t\) to the knob. If the output wiggles in phase, you are on the uphill side; out of phase, the downhill side. Multiply and average, and what survives is proportional to the slope.

The multiply-and-average step is a lock-in amplifier, the same device used to pull tiny optical signals out of noise. Its whole purpose is to answer one question: how much of the tone I injected came back out, and with what sign?

The one identity that made it practical

Power is proportional to the cube of the wind speed, so the gradient of power is too, and the wind varies by 3:1 across Region 2. A loop tuned at 8 m/s is 27 times too slow at 4 m/s and 27 times too aggressive at 12 m/s.

Rotea’s 2017 observation is one line. Since \(\ln P = \ln(\tfrac12\rho A) + 3\ln V + \ln C_P(u)\), and the wind term does not depend on \(u\):

\[\frac{\partial \ln P}{\partial u} \;=\; \frac{1}{C_P}\frac{\partial C_P}{\partial u}\]

Feed the algorithm \(\ln P\) instead of \(P\) and the loop bandwidth becomes \(\nu_c = (\kappa/C_P^{\max})\,|\partial^2 C_P/\partial u^2|\), free of both the wind speed and the air density. Tune it once; it works everywhere.

What the logarithm fixes

Note also a small free bonus. The 1/C_P factor is a state-dependent gain: far from the peak, C_P is small, so the step is large. The algorithm accelerates itself during the transient and settles down near the optimum, with nothing to tune.

It holds up in high-fidelity simulation

Ciri, Leonardi & Rotea 2019

Large-eddy simulation of the full flow around an NREL 5 MW rotor, with shear and 10% turbulence. No analytic gradient anywhere.

The result

Log-power holds 8 minutes across a 3:1 wind range. Plain ESC is four times slower at the bottom and goes unstable at the top.

Making it fast: LP-PIESC

Replace demodulation with recursive least squares on \(\dot y = \theta_0 + \theta_1(u-\hat u)\) and add a proportional term. The gradient \(\theta_1\) now comes with a covariance matrix \(\Sigma\) that the algorithm already propagates, and that none of these papers reports.

On the energy number: LP-ESC, measured against a controller that already knows the optimal gain, loses 14.2% of the energy. That is the cost of searching. LP-PIESC brings it to 0.3%, which is what makes the approach deployable.

The wind tunnel result is a twelve-turbine array steering wakes by yaw, and the algorithms converge in seconds at model scale.

LP-PIESC in full

The whole demodulation chain is gone: no multiplier, no high-pass, no low-pass, no phase angle. The gradient is \(\hat\theta_1\), read directly out of a least-squares fit to \(\dot y = \theta_0 + \theta_1(u - \hat u)\).

Walk the loop once. The control law at top left is Equation 5 of Kumar & Rotea [KR22] paper: a proportional term in the gradient estimate, an integrated term, and the dither. The turbine responds, power is normalized by rated and logged, and that signal y is the only measurement the estimator sees.

Inside the estimator, the prediction error e drives the parameter update, c is a filtered version of the regressor, and Sigma is a covariance that is propagated in inverse form. The projection operator keeps the estimates inside a set you specify in advance.

Then theta-1-hat comes back out and does two jobs: it feeds the proportional path directly, and it feeds the integrator that produces u-hat.

What the estimator is fitting

The local model is affine in the distance from the current estimate:

\[\dot y \;=\; \theta_0 + \theta_1\,(u - \hat u) \;=\; \phi^{T}\theta, \qquad \phi = \begin{bmatrix} 1 \\ u - \hat u\end{bmatrix}, \qquad \theta = \begin{bmatrix}\theta_0 \\ \theta_1\end{bmatrix}\]

so \(\phi\) and \(\theta\) are two-vectors, and \(\theta_1\) is the gradient we are after. Everything else in the box has a shape forced by these:

vectors in \(\mathbb{R}^2\) \(\phi\), \(\theta\), \(\hat\theta\), and \(c\)

scalars \(y\), \(\hat y\), \(e\), and the constants \(K\), \(k_T\), \(\sigma\)

\(2\times2\) matrices \(\Sigma\) and \(\Sigma^{-1}\)

The obstacle: the model is written in \(\dot y\), and \(\dot y\) is not measured. Differentiating a turbulent power signal to get it would be hopeless.

Why there is a filtered regressor

Predict \(y\) with \(\dot{\hat y} = \phi^{T}\hat\theta + Ke + c^{T}\dot{\hat\theta}\) and let \(\tilde\theta = \theta - \hat\theta\). Subtracting from \(\dot y = \phi^T\theta\),

\[\dot e \;=\; \phi^{T}\tilde\theta - Ke - c^{T}\dot{\hat\theta}\]

Now define \(\eta = e - c^{T}\tilde\theta\) and differentiate, using \(\dot{\tilde\theta} = -\dot{\hat\theta}\) and \(\dot c^{T} = -Kc^{T} + \phi^{T}\):

\[\dot\eta = \underbrace{\phi^{T}\tilde\theta - Ke - c^{T}\dot{\hat\theta}}_{\dot e} \;-\;\underbrace{(-Kc^{T} + \phi^{T})\tilde\theta}_{\dot c^{T}\tilde\theta} \;+\;\underbrace{c^{T}\dot{\hat\theta}}_{-c^{T}\dot{\tilde\theta}} \;=\; -K\big(e - c^{T}\tilde\theta\big) \;=\; -K\eta\]

Everything cancels but \(-K\eta\). So \(\eta \to 0\) exponentially, which means \(e \to c^{T}\tilde\theta\): the measurable error becomes the regressor \(c\) dotted with the parameter error. The odd \(c^{T}\dot{\hat\theta}\) term in the predictor is there for exactly this, since it is what cancels \(c^{T}\dot{\tilde\theta}\).

This is the whole trick, and it is worth saying plainly. You cannot regress on y-dot because you cannot measure it. So you build a predictor whose error obeys a regression you can run, and the filtered regressor c is what appears in it.

The obvious question is where the filter came from, and the answer is that it was reverse-engineered. Run the derivation backwards: differentiate eta, substitute, and you are left with eta-dot equal to phi transpose minus c-dot transpose, all times theta-tilde, minus K e. Demand that this equal minus K eta and the requirement collapses to c-dot transpose equals minus K c transpose plus phi transpose. The filter is forced. There is no freedom in it once the goal is chosen, and the same is true of the c-transpose theta-hat-dot term in the predictor, which exists only to cancel its counterpart.

This is the augmented error idea from adaptive control, going back to Monopoli in the mid-1970s for model-reference adaptive systems. Worth naming, because it makes the box look like standard machinery rather than something invented for this paper.

It also explains the value of K. It is the rate at which the regression becomes exact, and Kumar and Rotea set it to 20 radians per second, a fifty millisecond time constant against a dither period of 314 seconds. So eta is zero on every timescale that matters and the regression holds exactly. That is not a tuning compromise, it is fast enough to be instantaneous.

Which makes it a ridge regression

Rearranging \(e = c^{T}\theta - c^{T}\hat\theta\) and defining \(z := e + c^{T}\hat\theta\), we get \(z = c^{T}\theta\): known regressor, computable observation, unknown parameter. Fit it with forgetting and a penalty on the size of \(\hat\theta\):

\[J(t,\hat\theta) = \int_0^t e^{-k_T(t-s)}\Big[\,\big|z - c^{T}\hat\theta\big|^2 \;+\; \sigma\big|\hat\theta\big|^2\,\Big]ds\]

Setting \(\partial J/\partial\hat\theta = 0\) gives the normal equations \(\Sigma\hat\theta = b\), where, with \(w = e^{-k_T(t-s)}\),

\[\Sigma = \underbrace{\int_0^t w\,cc^{T}ds}_{A} \;+\; \sigma\underbrace{\int_0^t w\,ds}_{m}I, \qquad b = \int_0^t w\,c\,z\;ds\]

\(\Sigma\) is the information matrix of a ridge-regularized least squares, and it is the \(\Sigma\) of the block diagram. The next slide shows that both places \(\sigma\) appears in the algorithm come from the one penalty written here.

One cost, two equations

The \(\Sigma\) recursion. Since \(\dot A = cc^{T} - k_TA\) and \(\dot m = 1 - k_Tm\),

\[\dot\Sigma = \big(cc^{T} - k_TA\big) + \sigma\big(1 - k_Tm\big)I = cc^{T} - k_T\underbrace{\big(A + \sigma mI\big)}_{\Sigma} + \sigma I\]

The parameter update. Differentiate \(\hat\theta = \Sigma^{-1}b\), substitute \(\dot\Sigma\) and \(\dot b = cz - k_Tb\), and the \(k_T\) terms cancel:

\[\dot{\hat\theta} = -\Sigma^{-1}\dot\Sigma\,\hat\theta + \Sigma^{-1}\dot b = \Sigma^{-1}c\big(z - c^{T}\hat\theta\big) - \sigma\Sigma^{-1}\hat\theta = \Sigma^{-1}\big(ce - \sigma\hat\theta\big)\]

That is why \(c\) multiplies \(e\), and why the same \(\sigma\) appears twice. It is not a robustness patch bolted on in two places: it is one ridge penalty, showing up once through \(\Sigma\) and once through the shrinkage of \(\hat\theta\).

What sigma is actually doing is shrinkage, not slowing. In Sigma it puts a floor of sigma m under the information matrix, so the gain cannot diverge. In the update it pulls theta-hat toward zero at rate sigma times Sigma inverse. Those are the same statement from either side of the inverse.

The floor matters more than it looks. c is a two-vector, so c c-transpose is rank one at every instant and the instantaneous problem is always underdetermined. Only accumulation over time, or sigma, makes it invertible, and when excitation fades the accumulation stops and sigma is all that is left.

Kumar and Rotea use sigma equal to ten to the minus six, and m tends to one over k-T, which is 0.05, so the floor is about five times ten to the minus eight. Small enough that the ridge bias is negligible and it is doing pure conditioning work.

The projection is the one genuinely separate piece. It confines theta-hat to a compact set for the boundedness argument, and it needs a priori bounds that the cost does not supply.

Intuition for the update itself: e is a scalar, so it says how wrong the prediction is but not which parameter is at fault. c is the sensitivity of the error to each parameter, so the product spreads one scalar across the parameter vector, and Sigma inverse weights it by what is already known in each direction.

So why is demodulation still the default?

What LP-PIESC assumes

That \(\dot y\) is affine in \((u - \hat u)\), that the regressor stays persistently exciting, and that you can bound the parameters in advance for the projection.

What demodulation assumes

That the plant is roughly static at one frequency and the peak is locally unimodal. No model structure at all.

And the count goes the wrong way. LP-ESC has six parameters: \(a\), \(\omega\), \(\theta\), two cutoffs, \(k\). LP-PIESC has seven: \(a\), \(\omega\), \(k_T\), \(K\), \(\sigma\), \(k_p\), \(\tau_I\), and Rotea’s 2024 multi-row paper says all but two were set by trial and error.

Which is exactly the complaint we were given. The fast algorithm is the one that “takes too long to tune its parameters.”

Four more things on the other side of the ledger, worth having ready.

Persistent excitation is a real requirement, not a formality. The 2024 multi-row paper computes it and reports rho equal to about 0.011 for one turbine. If excitation dies, as it tends to near convergence with a small dither, the covariance recursion misbehaves, which is what the forgetting factor and the sigma regularization are there to prevent.

The projection operator needs a priori bounds on the parameters. That is model knowledge the demodulation route never asks for.

Frequency selectivity. A lock-in listens in one narrow bin, which you place where turbulence is weak. Least squares on y-dot is broadband, so it is exposed to the whole spectrum including the energy-containing range.

And for the curvature question, demodulation hands you the second harmonic for free, whereas a regressor that is linear in the parameter gives the gradient and nothing else.

Set against all of that: in this application LP-PIESC simply wins. Energy loss of 0.3 percent against 14.2, with a dither a quarter the size. That is why the UTD group moved to it, and why the question is not whether to use it but how to tune it without trial and error.

Where things stand

Solved

Consistency across wind speed. The logarithm removes the \(V^3\) dependence exactly, in theory, in LES, and in the tunnel.

Solved

Speed. LP-PIESC converges fast enough that the search costs 0.3% of energy rather than 14%.

Not solved

Everything to do with noise. “The enemy is process noise”, and none of these papers reports a variance.

Extremum seeking works well when the process noise is negligible. Otherwise you must average, and averaging costs convergence time. That trade-off has not been quantified for this problem.

Five questions, audited

Two of them closed under inspection. Q1 turned out not to be a control problem and Q2 was fixed in 2017. What is left is Q3, Q4 and Q5.

Q1 · Where does the torque gain converge?

offset from \(\lambda^\star = 7.5\)
[CLR19] Table 4, shear \(+10\%\) TI	\(+0.22\) to \(+0.35\)
[CLR19] Table 3, uniform inflow	\(+0.10\) to \(+0.25\)
finite-difference bias, computed	\(-0.013\)
turbulent peak shift, computed	\(+0.005\) to \(+0.037\)

Most of the gap is already there in inflow with no turbulence, where neither computed candidate can act, and the scatter is bounded at \(\sigma_\lambda \lesssim 0.14\). What is left is the reference: design \(\lambda^\star\) is a blade-element number, the LES rotor is actuator-line. Closed as a control question, open as a modeling one. Worked through in the Q1 deck.

Two things to say out loud. First, extremum seeking never adjusts, estimates or holds lambda. It adapts k, and lambda is the state that follows. The anomaly is in k: 7.7 percent low, which is 0.19 in lambda.

Second, why this matters beyond bookkeeping. Blade erosion moves lambda-star by about 0.2. If the reference itself is uncertain at 0.1 to 0.25, the detection problem this work is aimed at is harder than the published framing suggests.

The deck also carries two results worth more than the anomaly: a constant-gain loop never averages down, so N is a tuning rather than a run length; and predicting the gradient-estimate scatter from the turbulence overshoots the published spread by 24 to 40 times, which is unresolved.

Q2 · What sets the stability limit?

[R17] Eq. 8 reduces the loop to \(\dot{\tilde u} = -\omega_c\tilde u\) with \(\omega_c \propto \rho V^3\). One pole, negative real, for any gain: it cannot go unstable. [CLR19] report plain ESC unstable at 12 m/s, in uniform inflow and again in turbulence.

[R17] p. 4506 names the effect and cannot model it: “an increase in crossover frequency implies a decrease of the stability margins.” A first-order lag has no margin to lose. The missing element is that the loop is sampled: one update per 150 s dither period, so \(\tilde u_{n+1} = (1-K)\tilde u_n\) with \(K = \kappa_{\text{eff}}|J''|\), giving \(0 < K < 2\) where the continuous model gives no bound at all. Since \(K \propto V^3\), high wind walks across it.

Closed, and closed by the existing fix. The logarithm removes the \(V^3\) exactly, so LP-ESC holds one \(K\) at every wind speed. Nor is \(K<2\) a speed ceiling: \(K=1\) is deadbeat in one update. What survives is that \(K = \kappa_{\text{eff}}|J''|\) still needs a curvature nobody measures, which is Q4.

Why the rotor is not the answer: linearizing the rotor ODE gives a time constant of 11 seconds at 4 meters per second, 5.6 at 8 and 3.7 at 12, against a 150-second dither period. It is fully settled between updates.

The number worth quoting is that nothing has to be fitted. Their own settling time places K at 8 meters per second between 0.44 and 1.57. Cube that band for 12 and 86 percent of it exceeds 2. The boundary lands between 8.7 and 13.3, and any further phase in the loop, rotor lag or the blanked 27 percent of each period, moves it down.

The sharp test costs nothing: above K of one the pole is negative, so the torque gain should flip sign on consecutive updates, a five-minute square oscillation in the existing 12 meter per second series.

But say the verdict plainly. This diagnoses a published anomaly and the remedy already exists, so it is not a research direction. The residual is the curvature in K, and that hands over to Q4.

And this is what the logarithm really bought. Its effect on the loop gain is to remove the V-cubed, so LP-ESC holds one K at every wind speed and never approaches the boundary. That is a stability story, not a convergence-rate story, and it is not how the benefit is usually described.

Q3 · Does the turbine need to probe at all?

Two things are true and encouraging. \(\tau_{\text{aero}}\) is observable from the encoder alone, no wind sensor and no \(C_P\): augment it as a state on \(I_{\text{eff}}\dot\Omega = \tau_{\text{aero}} - \tau_g\) and \([C;CA]\) has rank 2. And turbulence already moves \(\lambda\) about as far as the probe does, \(\sigma_\lambda = 0.41\) against \(0.34\), verified in simulation.

Neither helps. Given a candidate curve, each instant supplies one equation and one unknown \(V(t)\), so the curve is unconstrained pointwise. The degeneracy is not a direction, it is the whole function space.

posited curve	solvable	implied mean \(V\)	implied TI	corr. with true \(V\)
true	\(100\%\)	\(8.00\)	\(9.3\%\)	\(1.0000\)
peak flattened \(30\%\)	\(100\%\)	\(7.99\)	\(9.3\%\)	\(0.9999\)

A curve that is simply wrong reproduces the record exactly, with an implied wind indistinguishable from the real one. This is why extremum seeking exists. A known excitation is not a convenience; it is the only thing that breaks the degeneracy. The answer to Q3 is yes, and structurally so.

Worth stating as a positive result rather than a failure. The published record does not say why probing is necessary, it just probes. This says why: an unknown wind absorbs any curve you propose, at any noise level, so no identification scheme on operating data alone can work. That is a cleaner justification for the whole method than anything currently written down.

The near-miss is instructive too. The one invariance I first found, scaling the wind by c and reshaping the curve to match, leaves the optimal torque gain exactly fixed, so had the degeneracy really been one-dimensional the scheme would have worked without any absolute wind reference. It is not one-dimensional.

Q4 · What the curvature is doing

Look again at the result of Rotea (2017):

\[\nu_c \;=\; \frac{\kappa}{C_P^{\max}}\left|\frac{\partial^2 C_P}{\partial u^2}(u_o)\right|\]

Choosing the gain \(\kappa\) to hit a target settling time requires the curvature at the peak, currently taken from a design curve that the LES says is wrong by 10% in \(C_P\). The gain calibration is not self-consistent, and “it takes too long to tune” is partly a symptom of that.

The curvature also sets the terminal scatter, \(\sigma_\lambda \approx \sigma_{\hat g}/|H|\), and the energy lost to mistuning is quadratic in \(\sigma_\lambda\). The same number sets the gain, the achievable accuracy and the cost of missing.

Q4 · Two routes to the same covariance

Meyn does not estimate the Hessian in the ESC papers. It appears in the analysis, as \(A = \nabla^2\Gamma(\theta^\star)\) in the asymptotic covariance. Where he does estimate it, in Zap SA, it is a two-timescale Jacobian estimate used as a matrix gain.

Polyak-Ruppert averaging attains the same optimal asymptotic covariance without estimating any curvature. You average the iterates. Meyn’s own recommendation is that this is the first thing to try.

So if the goal is optimal terminal variance, the Hessian is not needed. A matrix gain buys transient behavior and stability under weak assumptions, which are worth having but are not what the variance question asks for.

Which raises a question to settle early: how much of what looks open in wind is already answered in stochastic approximation and simply has not been imported?

Q5 · How good could any algorithm be?

Given a turbulence spectrum, an observation time and a fatigue budget, how well can anyone locate \(\lambda^\star\)? Nobody has written the bound down.

Without it there is no way to know whether LP-PIESC is within twenty percent of optimal, in which case this is engineering rather than research, or a factor of twenty off, in which case there is a great deal of work here.

The structure suits a Cramer-Rao argument: the measurement equation is known, the noise spectrum is measurable, and the nuisance parameter is the wind. I would want this before committing to anything else, since it is the only item that says whether the other four are worth doing.

Why turbulence is not ordinary noise

In classical stochastic approximation

Noise is pure nuisance. It corrupts the measurement and you average it away.

Here

Turbulence is simultaneously the noise and the excitation. It is what corrupts the gradient estimate and what sweeps the rotor across the curve for free.

Both sides are now measured, and they are the same size:

	\(\sigma_\lambda\) contributed
turbulence, through the rotor high-pass at \(0.029\) Hz	\(0.41\)
the probe, at the published \(a = 13.6\%\) of \(k_{\text{opt}}\)	\(0.34\)

The excitation the wind supplies for free is comparable to the one the controller pays for. That is what made Q3 worth asking. It still fails, but on identifiability rather than on signal strength: an unknown wind absorbs any curve you propose, so excitation of unknown magnitude cannot substitute for excitation you chose.

The experiment I would run first

Open the loop. Hold the torque gain at a series of fixed values, run each long enough to average the turbulence, and measure the simulator’s own \(C_P(\lambda)\). Then ask where its peak is.

Why this one

Every published offset is scored against a blade-element \(\lambda^\star = 7.5\). Q1 showed the offset survives in inflow with no turbulence, so the reference is the only candidate left. This measures it directly.

What it costs

No new machinery, no closed loop, a few days of compute. And the same runs give \(\sigma_\lambda\) at each gain, which is the \(K_v\) nobody can currently predict.

Note the earlier obvious experiment, sweeping the dither amplitude to look for an \(a^2\) scaling in the offset, is not worth running: the predicted effect is \(0.013\) against a seed spread larger than that.

One methodological prerequisite. The LES of Ciri et al. recycles a twenty-minute precursor wind field, so a sixty-minute run contains three copies of the same turbulence. No variance estimate survives that, so non-repeating inflow has to come first whatever else is run.

Where this leaves things

The algorithms work. What is missing is the statistics of the quantity they compute.

What exists

A model-free loop, consistent across wind speed, with results from LES, a wind tunnel and a field test.

What does not

A variance for the gradient estimate, a reference the simulations can honestly be scored against, and a performance bound.

	verdict
Q1 estimand	closed; the offset is in the reference, not the algorithm. A measurement, not a program
Q2 stability limit	closed; sampling explains it and the \(\ln\) of 2017 already fixes it
Q3 is the dither needed	closed; an unknown wind absorbs any curve, so a known probe is the only thing that identifies one
Q4 matrix gain	open, and simulated. The curvature does leave the loop gain
Q5 performance bound	untouched

References

[R17]  M. A. Rotea, Logarithmic power feedback for extremum seeking control of wind turbines, IFAC-PapersOnLine 50(1) (2017) 4504–4509. doi:10.1016/j.ifacol.2017.08.381

[CLR19]  U. Ciri, S. Leonardi & M. A. Rotea, Evaluation of log-of-power extremum seeking control for wind turbines using large eddy simulations, Wind Energy 22 (2019) 992–1002. doi:10.1002/we.2336

[KR22]  D. Kumar & M. A. Rotea, Wind turbine power maximization using log-power proportional-integral extremum seeking, Energies 15 (2022) 1004. doi:10.3390/en15031004

[KR24]  D. Kumar & M. A. Rotea, Optimal tip-speed ratio for degraded blades, Wind Energy Science 9 (2024) 2133–2146. doi:10.5194/wes-9-2133-2024

[RKAJ24]  M. A. Rotea, D. Kumar, E. J. Aju & Y. Jin, Multi-row extremum seeking for wind farm power maximization, J. Phys. Conf. Ser. 2767 (2024) 032043. doi:10.1088/1742-6596/2767/3/032043

[MGR24]  S. P. Mulders, N. Gallo & M. A. Rotea, Analysis of extremum seeking control for wind turbine torque controller optimization by aerodynamic and generator power objectives, ACC (2024). arXiv:2407.08059

[GKN12]  A. Ghaffari, M. Krstić & D. Nešić, Multivariable Newton-based extremum seeking, Automatica 48 (2012) 1759–1767. doi:10.1016/j.automatica.2012.05.059

[S00]  J. C. Spall, Adaptive stochastic approximation by the simultaneous perturbation method, IEEE Trans. Automat. Contr. 45 (2000) 1839–1853. PDF

[LM23]  C. Lauand & S. Meyn, Quasi-stochastic approximation: design principles with applications to extremum seeking control, IEEE Control Systems Magazine 43 (2023). doi:10.1109/MCS.2023.3291884

[A22]  N. J. Abbas et al., A reference open-source controller for fixed and floating offshore wind turbines, Wind Energy Science 7 (2022) 53–73. doi:10.5194/wes-7-53-2022

[J09]  J. Jonkman, S. Butterfield, W. Musial & G. Scott, Definition of a 5-MW reference wind turbine, NREL/TP-500-38060 (2009). PDF  ·  [IEC]  IEC 61400-1 ed. 3, normal turbulence model.