**5. Conclusions**

As the accuracy and applicability of many JPDF modeling methods for multiple wind farms need to be improved, it is of significance to promote the accuracy and applicability of the modeling method of JPDF. This is exactly the purpose of this study.

The specific steps of AMNKDE are as follows. First, the model of AMNKDE was constructed using the optimal bandwidth. Second, an optimal model of bandwidth based on Euclidean distance and maximum distance was constructed and the comprehensive minimum of these distances was used as a measure of optimal bandwidth. Finally, the OO algorithm was used to solve this model.

The specific conclusions of this paper are as follows:

(1) The adaptive bandwidth improvement strategy proposed in this paper replaces the traditional fixed bandwidth of MNKDE with the adaptive bandwidth. It effectively facilitates the overall modeling accuracy of MNKDE by adjusting the bandwidth of local sample interval adaptively. Simulation results in this paper indicate that the overall fitness error of AMNKDE was 8.81% lower than that of traditional MNKDE.

(2) Based on the above strategy, an AMNKDE approach was proposed and utilized to build the JPDF model for multiple wind farms. The simulation results in this paper indicated that the overall fitness error of AMNKDE was 11.6% lower than that of Copula-based PE method. After replacing the modeling object the overall fitness error of the comprehensive Copula method increased by as much as 1.94 times that of AMNKDE. Consequently, the accuracy and applicability of the AMNKDE approach were better than that of the traditional Copula PE method.

In summary, the proposed AMNKDE approach clearly performed better than MNKDE and Copula hybrid models and was suitable for building a multi-wind farm joint probability density model.

This study can be further extended. The proposed approach in this paper can be further applied to many other fields of uncertain modeling, such as the JPDF modeling problems of photovoltaic power systems and wind-solar combined power systems.

**Author Contributions:** Conceptualization, N.Y. and Y.H.; Data curation, Y.H., D.H., S.L. and D.Y.; Formal analysis, Y.H. and B.D.; Project administration, B.D.; Software, Y.H., D.H. and D.Y.; Supervision, S.L.; Visualization, D.Y.; Writing – original draft, N.Y. and Y.H.; Writing – review & editing, N.Y. and Y.F.

**Funding:** This research was funded by the National Natural Science Foundation of China No. 51607104.

**Conflicts of Interest:** The authors declare no conflict of interest.
