Climbing a Hill You Cannot See

Gradient estimation for wind turbine control, and five things nobody has settled

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

The rotor’s efficiency depends on one number, the tip-speed ratio \(\lambda = R\Omega/V\), which is 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 per cent 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.

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.

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. That circularity is why this line of work is model-free.

Sitting on a hill you cannot see

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

The \(k\Omega^2\) law

The generator is commanded to resist the rotor with a torque quadratic in speed,

\[\tau_{\text{gen}} = k\,\Omega^2 .\]

Balance it against the aerodynamic torque \(T_{\text{aero}} = \tfrac12\rho\pi R^3V^2\,C_Q(\lambda)\), with \(C_Q = C_P/\lambda\), and the equilibrium condition is

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

The wind speed cancels. Try \(\tau_{\text{gen}} = k\Omega^n\) and you get \(C_Q(\lambda)/\lambda^{n} \propto V^{\,n-2}\). Only \(n = 2\) removes \(V\), so it is the unique exponent for which the equilibrium tip-speed ratio does not depend on the wind.

Read the boxed equation carefully: the left side is a pure function of lambda, the right side is a constant you choose. So picking k picks an equilibrium lambda, permanently, without anyone measuring anything.

There is no rotor speed to regulate to

The equilibrium of the rotor equation is a ray, not a point. Every wind speed gets its own \(\Omega\); all of them share the same \(\lambda\). The controller never rejects the wind. It makes the desired behaviour 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.

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.

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 things nobody has settled

Each one is a well-posed question with an answer somebody could get.

Q1 · What is the estimator estimating?

In the 2019 LES, both algorithms converge to \(\bar\lambda \approx 7.85\) against a design value of \(7.5\), at every wind speed, in every condition. The paper calls this “close to the ideal optimum” and moves on.

Three candidate targets, and they are not the same number:

1. the argmax of the steady C_P curve, which is what every paper scores against;
2. the argmax of the expected log-power under turbulence, which differs by a Jensen gap plus rotor inertial lag;
3. the algorithm’s actual fixed point, which adds a finite-difference bias.

The gap between (a) and (c) in the published data is 0.35 in lambda. The effect everybody wants to detect, blade erosion, is 0.2. You cannot do estimation theory when the ambiguity in the definition of the target exceeds the signal.

Q1 · Why it is a variance problem

\(3\ln V\) is independent of the knob, so it vanishes from the gradient exactly. It does not vanish from any finite-time estimate of that gradient, and it is fifteen times larger than the signal.

Writing the estimator out: one dither period produces one number,

g-hat = (1/aT) times the integral of ln P against the dither.

Substituting the decomposition, the constant drops out by symmetry and two terms survive: the projection of the turbulence onto the dither, which has nothing to do with the turbine, and the projection of the C_P response, which carries the signal. A back-of-the-envelope for the first gives

sigma(g-hat) is about 3 · TI · sqrt(tau_c/T) / a

which falls as one over root T, the averaging Rotea mentions, and as one over a.

Q1 · The trade that was never computed

A central difference estimates \(J'(q) + \tfrac{a^2}{6}J'''(q)\), so its zero sits off the true optimum by \(-\tfrac{a^2}{6}\,J'''/J''\). Bigger dither, less noise, more bias. There is an optimal \(a\), and both published values were chosen by trial and error.

This is Kiefer-Wolfowitz structure, and it has not been written down for wind turbines.

It also suggests the experiment that settles Q1. Run the existing setup at two or three dither amplitudes with several independent turbulence seeds. If the offset from 7.5 scales as a-squared, it is estimator bias. If it is flat in a, the turbulent optimum has moved, and the reference value these papers score against is wrong. Either answer is publishable, and the seed spread gives you the error bars none of them report.

Q2 · What sets the stability limit?

The 2017 model is first order: \(\dot{\tilde u} = -\omega_c\tilde u\). It cannot go unstable for any positive gain. The LES did, twice: in uniform flow and in turbulence.

So the failure mode that actually kills the algorithm in practice has no theory behind it in this line of work.

The likely mechanism is sampled data. One parameter update per 150-second dither period, against a rotor whose own time constant is about six seconds, with the loop gain tripling between 8 and 12 m/s. That is a discrete-time stability boundary, and a continuous first-order model is blind to it.

This boundary, rather than the noise, is the ceiling on how fast any of these algorithms can be made to converge, which is the quantity Rotea says he cares about.

Q3 · Does the turbine need to probe at all?

The rotor equation is \(I\dot\Omega = T_{\text{aero}} - \tau_{\text{gen}}\). Every term on the right is known: \(\tau_{\text{gen}}\) is commanded, \(\Omega\) is measured, and \(I\) is a design constant. So

\[T_{\text{aero}}(t) \;=\; I\dot\Omega(t) + \tau_{\text{gen}}(t)\]

is exactly reconstructible without any anemometer. And turbulence already swings \(\lambda\) between 6 and 8.5 all day long, for free.

The dither exists because the excitation is unknown, not because excitation is lacking. So: are \(V(t)\) and the curve \(C_Q(\cdot)\) jointly identifiable from the record of \((\Omega,\,T_{\text{aero}})\) alone?

This is the question with the largest payoff, because a yes would dissolve the probing-versus-loads trade-off rather than optimize it. Without a dither there is no fatigue cost to probing and no amplitude to choose.

Q3 · Exactly one blind direction

Substituting \(\tilde V = cV\) into \(T_{\text{aero}} = \tfrac12\rho\pi R^3V^2C_Q(R\Omega/V)\):

\[\tilde C_Q(x) = \frac{C_Q(cx)}{c^2} \quad\Longrightarrow\quad \tilde C_P(x) = \frac{C_P(cx)}{c^3} \quad\Longrightarrow\quad \tilde\lambda^\star = \frac{\lambda^\star}{c}\]

There is one unidentifiable direction, an overall scaling of the wind, and it maps straight onto a scaling of \(\lambda^\star\). Everything else about the curve is determined. Air density errors, by contrast, scale \(C_P\) uniformly and do not move the peak.

If this holds, you do not need a fast rotor-effective wind estimate. You need one slowly updated absolute wind scale to pin the constant c: a met mast, a lidar, or a long-run SCADA average. The fast fluctuations come free from the rotor itself.

A slow reference is also harder to contaminate with the control parameter, because the induction response you would worry about is fast. That sidesteps the circularity that makes nacelle wind estimates unusable.

Caveat: this sits close to the wind-speed-estimator literature, which assumes C_P known and solves for V. The joint version is what would be new, and it needs checking against that literature before anyone builds on it.

Q4 · What the curvature is doing

Look again at the 2017 result:

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

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 the matrix \(A = \nabla^2\Gamma(\theta^\star)\) governing the asymptotic covariance. Where he does estimate it, in Zap SA, it is a two-timescale recursive Jacobian estimate used as a matrix gain: stochastic Newton-Raphson.

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 behaviour 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 per cent 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.

That duality is what the stochastic-approximation literature does not cover, and it is what makes Q3, estimating the curve without probing it, plausible.

The experiment I would run first

Take the existing simulation setup. Run it at two or three dither amplitudes, with several independent, non-repeating turbulence seeds. Look at where the torque gain lands.

If the offset scales as \(a^2\)

It is estimator bias. The algorithm is systematically missing the peak, and the fix is a known one.

If it is flat in \(a\)

The turbulent optimum has moved, and the published results have been scored against a reference that is off.

Either way the seed spread yields \(\sigma(\hat g)\) and \(\sigma_\lambda\), which none of these papers reports. Two outcomes, both publishable, a few days of compute.

One methodological point: the 2019 LES 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 the inflow has to be fixed first.

Climbing a hill you cannot see

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

What exists

A model-free loop, consistent across wind speed and fast enough to deploy, with results from large-eddy simulation, a wind tunnel and a field test.

What does not

A definition of the estimand, a variance for it, an account of the stability limit, a performance bound, and an answer on whether the dither is needed.

Rotea, IFAC 2017 · Ciri, Leonardi & Rotea, Wind Energy 2019 · Kumar & Rotea, Energies 2022 · Rotea et al., TORQUE 2024 · Lauand & Meyn, IEEE CSM 2023

Prepared for a collaboration with M. A. Rotea, UTD Wind Energy Center.