Kalman-based interacting multiple-model wind speed estimator for wind turbines (2104.10063v1)

Abstract: The use of state estimation technique offers a means of inferring the rotor-effective wind speed based upon solely standard measurements of the turbine. For the ease of design and computational concerns, such estimators are typically built based upon simplified turbine models that characterise the turbine with rigid blades. Large model mismatch, particularly in the power coefficient, could lead to degradation in estimation performance. Therefore, in order to effectively reduce the adverse impact of parameter uncertainties in the estimator model, this paper develops a wind sped estimator based on the concept of interacting multiple-model adaptive estimation. The proposed estimator is composed of a bank of extended Kalman filters and each filter model is developed based on different power coefficient mapping to match the operating turbine parameter. Subsequently, the algorithm combines the wind speed estimates provided by each filter based on their statistical properties. In addition, the proposed estimator not only can infer the rotor-effective wind speed, but also the uncertain system parameters, namely, the power coefficient. Simulation results demonstrate the proposed estimator achieved better improvement in estimating the rotor-effective wind speed and power coefficient compared to the standard Kalman filter approach.

• The paper proposes a Kalman-based IMM estimator that utilizes multiple EKFs with varied Cₚ models to enhance wind speed estimation accuracy.
• The methodology iteratively mixes filter estimates using mode probabilities and measurement residuals to adjust for dynamic aerodynamic variations.
• Simulations on the DTU10MW turbine demonstrate reduced errors (e.g., -0.42% vs 3.23% wind speed error) compared to a standard EKF.

This paper addresses the challenge of accurately estimating the rotor-effective wind speed in wind turbines using standard sensor measurements like rotor speed and generator torque. Traditional state estimation techniques, often based on simplified turbine models, struggle with significant model mismatches, particularly concerning the power coefficient ($C_\mathrm{p}$). The $C_\mathrm{p}$ mapping, which relates aerodynamic efficiency to tip-speed ratio and pitch angle, is often based on static simulations but varies dynamically in real turbulent wind conditions, especially in the above-rated operating region. This variation leads to degraded estimation performance for standard methods like the Extended Kalman Filter (EKF).

To overcome this, the paper proposes a Kalman-based Interacting Multiple-Model (IMM) estimator. The core idea is to run a bank of parallel estimators (in this case, EKFs) where each estimator is based on a different model assumption regarding the uncertain parameter – the power coefficient $C_\mathrm{p}$. The estimates from these individual filters are then combined based on how well each filter's model matches the current system behavior, as inferred from the measurement residuals.

The IMM structure implemented in the paper consists of the following key steps performed iteratively at each time step:

1. Interaction (Mixing): Based on the mode probabilities (confidence in each model) from the previous step and a defined Markov transition matrix ($\Pi$) representing the probability of switching between models, initial conditions (state estimate and covariance) for each filter in the current step are computed as weighted averages of the previous step's estimates from all filters. This ensures filters less likely to be correct are re-initialized with better estimates.

   $$\tilde{x}^{(j)+}_k = \sum_i \mu^{(i|j)-}_k \hat{x}_k^{(i)+}$$

   $$\tilde{P}_k^{(j)+} = \sum_i \mu^{(i|j)-}_k \left[ P_k^{(i)+} + ( \tilde{x}^{(j)+}_k - \hat{x}^{(i)+}_k)) ( \tilde{x}^{(j)+}_k - \hat{x}^{(i)+}_k)^T \right]$$

   where $\mu^{(i|j)-}_k$ are mixing weights derived from the mode probabilities and transition matrix.

2. Filtering: Each of the parallel filters (EKFs) processes the measurement using its specific model variant. The paper uses a simplified nonlinear turbine model: Drive-train dynamics: $$J\dot{\omega} = \tau_\mathrm{a}(\lambda,\theta) -\tau_\mathrm{g}$$, where $$\tau_\mathrm{a}(\lambda,\theta) = \frac{1}{2} \rho \pi r^2 C_\mathrm{p}(\lambda,\theta ) v^3 \omega^{-1}$$ Wind speed dynamics: ${v}_{k+1} = {v}_k+n_k$ (random walk) The state vector is $x = [\omega, v]^T$. Measurements used are typically generator speed and controller inputs (pitch angle, generator torque command). The paper employs three EKFs, each using a different $C_\mathrm{p}$ mapping:
   • Filter 1: Nominal $C_\mathrm{p,0}(\lambda,\theta)$
   • Filter 2: $$C_\mathrm{p,0}(\lambda,\theta) + \Delta C_\mathrm{p}$$ (nominal plus offset)
   • Filter 3: $$C_\mathrm{p,0}(\lambda,\theta) - \Delta C_\mathrm{p}$$ (nominal minus offset) Each EKF performs prediction and measurement update steps based on its specific $C_\mathrm{p}$ model, yielding state estimates $\hat{x}^{(j)+}_k$ and covariances $P^{(j)+}_k$ for filter $j$.
3. Mode Probability Update: For each filter $j$, the likelihood $\Lambda^{(j)}_k$ of its model given the current measurement $y_k$ is calculated using the measurement residual $z^{(j)}_k = y_k - \hat{y}^{(j)}_k$ and its covariance $S^{(j)}_k$. Assuming Gaussian errors, $\Lambda^{(j)}_k$ is calculated. The mode probability $\mu^{(j)+}_k$ for filter $j$ is then updated proportionally to its prior probability and the likelihood.

   $$\mu^{(j)+}_k = \frac{\mu^{(j)-}_k \Lambda^{(j)}_k}{\sum_j \mu^{(j)-}_k \Lambda^{(j)}_k}$$

4. Combination: The final IMM state estimate $\hat{x}^{+}_k$ and its covariance $P^{+}_k$ are computed as the weighted average of the individual filter estimates and covariances, using the updated mode probabilities $\mu^{(j)+}_k$ as weights.

   $$\hat{x}^{+}_k = \sum_j \mu^{(j)+}_k \hat{x}^{(j)+}_k$$

   $$P^{+}_k = \sum_j \mu^{(j)+}_k \left[ P^{(j)+}_k + ( \hat{x}^{+}_k - \hat{x}^{(j)+}_k ) (\hat{x}^{+}_k - \hat{x}^{(j)+}_k )^T \right]$$

The practical implementation involves setting up the discrete-time nonlinear state-space model for the wind turbine, linearizing it for the EKF Jacobian matrices ($F_k$, $H_k$), defining the process and measurement noise covariances ($Q_k$, $R_k$), selecting the number of filters and their corresponding $C_\mathrm{p}$ variations ($\Delta C_\mathrm{p}$), and tuning the Markov transition matrix ($\Pi$). The paper empirically chose $\Pi$ with high diagonal values (0.99) and low off-diagonal values (0.005), suggesting a high probability of staying in the current mode but allowing for switches. The choice of $\Delta C_\mathrm{p}$ would typically be informed by historical data or expected variations.

Simulation results using the DTU10MW reference turbine in the HAWC2 aeroelastic code demonstrated the benefits. Comparing the IMM estimator (with three $C_\mathrm{p}$ models) against a standard single EKF (with nominal $C_\mathrm{p,0}$), the IMM showed significantly reduced estimation errors for both rotor-effective wind speed and estimated $C_\mathrm{p}$, particularly in the above-rated 15 m/s turbulent wind case where $C_\mathrm{p}$ variations are more pronounced. For example, the mean error in wind speed estimation for the 15 m/s case was -0.42% for IMM vs 3.23% for standard KF. For $C_\mathrm{p}$ estimation (compared to a low-pass filtered true $C_\mathrm{p}$), the mean error was 4.93% for IMM vs 8.07% for standard KF at 15 m/s. This reduced bias in estimation is critical for applications like advanced control strategies, health monitoring, and power reserve estimation.

Key implementation considerations include the increased computational burden of running multiple filters in parallel compared to a single EKF. Tuning the Markov transition matrix and the magnitude of the $C_\mathrm{p}$ offsets ($\Delta C_\mathrm{p}$) are important steps influencing performance. The IMM framework provides a practical way to handle known uncertainties in specific model parameters by explicitly modeling alternative scenarios.

The research suggests that extending the IMM approach to incorporate more diverse $C_\mathrm{p}$ mappings or using a time-varying transition matrix could further improve performance, highlighting areas for practical refinement and application.