Analysis of extremum seeking control for wind turbine torque controller optimization by aerodynamic and generator power objectives

TL;DR

The paper analyzes the feasibility of optimizing wind-turbine torque gain $K$ using extremum seeking control with either aerodynamic power $P_r$ or generator power $P_g$ as the objective. Through dynamic and frequency-domain analysis, it reveals that $P_g$ introduces a zero in its transfer function, causing minimum-/nonminimum-phase behavior near the optimum $K^*$ and hindering high-frequency ESC convergence unless the demodulation phase $\psi$ is carefully tuned. To overcome this, the authors propose reconstructing an aerodynamic-power objective by augmenting generator power with rotor-acceleration dynamics, i.e., $\hat P_r(\dot{\hat{\omega}}_r)= I \omega_r \dot{\hat{\omega}}_r + P_g$, and implement it via both perfect acceleration and filtered numerical differentiation. Simulation results show that this approach enables faster, more robust ESC convergence, allowing higher dither frequencies and reducing phase-sensitivity, with practical applicability to online torque-controller calibration in large wind turbines.

Abstract

Wind turbines degrade over time, resulting in varying structural, aeroelastic, and aerodynamic properties. In contrast, the turbine controller calibrations generally remain constant, leading to suboptimal performance and potential stability issues. The calibration of wind turbine controller parameters is therefore of high interest. To this end, several adaptive control schemes based on extremum seeking control (ESC) have been proposed in the literature. These schemes have been successfully employed to maximize turbine power capture by optimization of the $Kω^2$-type torque controller. In practice, ESC is performed using electrical generator power, which is easily obtained. This paper analyses the feasibility of torque gain optimization using aerodynamic and generator powers. It is shown that, unlike aerodynamic power, using the generator power objective limits the dither frequency to lower values, reducing the convergence rate unless the phase of the demodulation ESC signal is properly adjusted. A frequency-domain analysis of both systems shows distinct phase behavior, impacting ESC performance. A solution is proposed by constructing an objective measure based on an estimate of the aerodynamic power.

Figures (8)

• Figure 1: Dither-demodulation extremum seeking control scheme. The reader is referred to the main text for interpretation of this schematic.
• Figure 2: ESC for torque gain optimization for a $K\omega^2$ type of turbine torque controller. Two objectives, aerodynamic power and generator power, are considered for analysis.
• Figure 3: Torque gain optimization on aerodynamic power ($J=\mathcal{P}_\mathrm{r}$) and generator power ($J=\mathcal{P}_\mathrm{g}$) objectives, without demodulation phase compensation ($\psi=0$). ESC optimization based on aerodynamic power shows convergence to the actual optimum, whereas generator power-based optimization shows convergence to a nonoptimal value.
• Figure 4: Torque gain optimization on aerodynamic power ($J=\mathcal{P}_\mathrm{r}$) and generator power ($J=\mathcal{P}_\mathrm{g}$) objectives, with demodulation phase compensation. Plots (a) and (b) respectively show the results for dither frequencies $\omega\textunderscore{d}=0.1$ rad/s and $\omega\textunderscore{d}=0.01$ rad/s. The final convergence value of ESC based on the generator power objective is more sensitive to the chosen demodulation phase when the dither frequency is chosen as $0.1$ rad/sec. An arbitrary phase offset is included for aerodynamic power (--) to show that the phase offset only influences the convergence rate.
• Figure 5: Bode plots of the transfer functions $\mathcal{P}\textunderscore{r}$ from torque gain input $K$ to aerodynamic power (a), and transfer functions from the same input to generator power (b). The frequency responses represent the dynamics around the optimal torque gain value $K^*$. The dynamics of both systems differ by introducing a zero for the generator power objective. This zero crosses the imaginary axis when surpassing $K^*$, transitioning the system from minimum- to nonminimum-phase. Whereas $\mathcal{P}\textunderscore{r}$ shows a consistent $180$ deg phase difference around $K^*$, this phase difference is only present at lower frequencies of $\mathcal{P}\textunderscore{g}$. This impacts ESC based on generator power and makes the final convergence value dependent on the demodulation phase compensation angle $\psi$ when the dither frequency is chosen too high.

Theorems & Definitions (1)

• Remark 1