On the Analysis and Synthesis of Wind Turbine Side-Side Tower Load Control via Demodulation

Abstract

As wind turbine power capacities continue to rise, taller and more flexible tower designs are needed for support. These designs often have the tower's natural frequency in the turbine's operating regime, increasing the risk of resonance excitation and fatigue damage. Advanced load-reducing control methods are needed to enable flexible tower designs that consider the complex dynamics of flexible turbine towers during partial-load operation. This paper proposes a novel modulation-demodulation control (MDC) strategy for side-side tower load reduction driven by the varying speed of the turbine. The MDC method demodulates the periodic content at the once-per-revolution (1P) frequency in the tower motion measurements into two orthogonal channels. The proposed scheme extends the conventional tower controller by augmentation of the MDC contribution to the generator torque signal. A linear analysis framework into the multivariable system in the demodulated domain reveals varying degrees of coupling at different rotational speeds and a gain sign flip. As a solution, a decoupling strategy has been developed, which simplifies the controller design process and allows for a straightforward (but highly effective) diagonal linear time-invariant controller design. The high-fidelity OpenFAST wind turbine software evaluates the proposed controller scheme, demonstrating effective reduction of the 1P periodic loading and the tower's natural frequency excitation in the side-side tower motion.

Figures (14)

• Figure 1: A wind turbine is excited at the side-side direction by a periodic load due to the rotor imbalance at the 1P frequency ${F_\text{sd}(t) = a\textunderscore{sd} \cos{(\psi(t) + \phi\textunderscore{sd})}}$, with the azimuth $\psi(t)=\omega\textunderscore{r}(t)t$. The tangential speed of the periodic load is indicated by $v\textunderscore{t}(t)$, and $x(t)$ denotes tower top displacement in the horizontal direction.
• Figure 2: The modulated-demodulated control scheme for the cancellation of a side-side periodic load $F\textunderscore{sd} = a\textunderscore{sd} \cos{(\omega\textunderscore{r}t+\phi\textunderscore{sd})}$ affecting wind turbine tower $\mathrm{G}(s)$. The demodulation operation is driven by the disturbance frequency $\omega\textunderscore{r}$ and creates a separation of the output signal $\dot{x}$ into quadrature and in-phase signals, $\dot{x}\textunderscore{c}$ and $\dot{x}\textunderscore{s}$, respectively. On these channels, two identical SISO controllers $\mathrm{C}(s) \mathbf{I}_{2\times2}$ are designed, generating the control inputs $\Delta T\textunderscore{g,c}$ and $\Delta T\textunderscore{g,s}$. Finally, modulation at $\omega\textunderscore{r}$ combines these two control inputs into a single signal $\Delta T\textunderscore{g}$ that is fed into $\mathrm{G}(s)$ to alleviate the periodic loading. The phase offset $\psi\textunderscore{off}$ can be added to the demodulator to influence the system's behavior, such as channel decoupling. Note that the negative sign preceding $\mathrm{C}(s)$ indicates the negative feedback convention used in the framework.
• Figure 3: (a) SISO modulated controller $\mathrm{C}\textunderscore{m}(s,\omega\textunderscore{r})$ and (b) MIMO demodulated plant $\mathbf{H}(s,\omega\textunderscore{r})$ in the MDC scheme.
• Figure 4: Bode magnitude plots of the demodulated controller $\mathrm{C}_{n}(s)$ (top) and the resulting SISO modulated controller $\mathrm{C}_{\mathrm{m},n}(s, \omega\textunderscore{r})$ (bottom), $n = \{1,2,3\}$. The proportional controller $\mathrm{C}_{1}$ is mapped into $\mathrm{C}_{\mathrm{m},1}$, being the same proportional controller but with an additional factor of 2. The integral controller $\mathrm{C}_{2}(s)$ is rendered into an undamped inverted notch filter $\mathrm{C}_{\mathrm{m},2}(s, \omega\textunderscore{r})$ with infinite gain at $\omega\textunderscore{r} = 0.5$ rad/s. The low-pass filter $\mathrm{C}_{3}(s)$ with a cut-off frequency of $\omega\textunderscore{LPF} = 0.01$ rad/s results into a damped inverted notch filter $\mathrm{C}_{\mathrm{m},3}(s, \omega\textunderscore{r})$.
• Figure 5: Bode magnitude and phase plots of the nominal (a) and the demodulated (b) wind turbine models. In (a), vertical dashed lines indicate the operating range of a soft-soft wind turbine $\mathrm{G}(s)$, where a resonance peak at about $\omega = \omega\textunderscore{n} = 0.7071$ rad/s is apparent in the magnitude plot. A $180^\circ$ phase shift occurs due to the presence of this resonance, as shown in the corresponding phase plot. The points indicated by labeled arrows $i=\{1,2,3\}$ represent three sample points $\omega\textunderscore{r}^{(i)} = \{\omega\textunderscore{r,min}, \omega\textunderscore{n}, \omega\textunderscore{r,rated}\}$, with ${\omega\textunderscore{r,min} = 0.5}$ rad/s and ${\omega\textunderscore{r,rated} = 1.2}$ rad/s, to evaluate the mapping from $\mathrm{G}(s)$ into the steady-state components of $\mathbf{G}_{2}(s,\omega\textunderscore{r})$, as shown in (b), before, during, and after the resonance. Note the $40$ dB/dec slope in the magnitude plot of $\mathrm{G}_{2,12}(s,\omega\textunderscore{r}^{(2)})$ at low frequencies, which indicates the presence of two zeros at the origin.

Theorems & Definitions (2)

• Remark 1
• Remark 2