Wind turbine performance monitoring is a complex task because of the non-stationary operation conditions and because the power has a multivariate dependence on the ambient conditions and working parameters. This motivates the research about the use of SCADA data for constructing reliable models applicable in wind turbine performance monitoring. The present work is devoted to multivariate wind turbine power curves, which can be conceived of as multiple input, single output models. The output is the power of the target wind turbine, and the input variables are the wind speed and additional covariates, which in this work are the blade pitch and rotor speed. The objective of this study is to contribute to the formulation of multivariate wind turbine power curve models, which conjugate precision and simplicity and are therefore appropriate for industrial applications. The non-linearity of the relation between the input variables and the output was taken into account through the simplification of a polynomial LASSO regression: the advantages of this are that the input variables selection is performed automatically. The k-means algorithm was employed for automatic multi-dimensional data clustering, and a separate sub-model was formulated for each cluster, whose total number was selected by analyzing the silhouette score. The proposed method was tested on the SCADA data of an industrial Vestas V52 wind turbine. It resulted that the most appropriate number of clusters was three, which fairly resembles the main features of the wind turbine control. As expected, the importance of the different input variables varied with the cluster. The achieved model validation error metrics are the following: the mean absolute percentage error was in the order of 7.2%, and the average difference of mean percentage errors on random subsets of the target data set was of the order of 0.001%. This indicates that the proposed model, despite its simplicity, can be reliably employed for wind turbine power monitoring and for evaluating accumulated performance changes due to aging and/or optimization.