Taking the temporal order of the data into account, one can consider (seasonal or) monthly wind speeds. Consequently, for every month of a year the Weibull PDF is estimated with the maximum likelihood method. Then, we consider a convex combination of these 12 Weibull PDFs, i.e. we formulate a PDF \(\bar{f}_{\bar{\lambda },\bar{k},\bar{c}}\) for the whole year with
$$\begin{aligned} {\bar{f}_{\bar{\lambda },\bar{k},\bar{c}}(v)}\\& {}:=\, \sum \limits _{j=1}^{12} \lambda _j f_{k_j, c_j}(v)\\& {}= \left\{ \begin{array}{ll} 0 &{}\quad \text {if } v < 0\\ \sum \limits _{j=1}^{12} \lambda _j \frac{k_j}{c_j} \left( \frac{v}{c_j}\right) ^{k_j1}e^{(\frac{v}{c_j})^{k_j}} &{}\quad\text {if } v \ge 0. \end{array}\right. \end{aligned}$$
Here we have \(\bar{\lambda } := (\lambda _1, \dots , \lambda _{12})\), \(\bar{k} := (k_1, \dots , k_{12})\), \(\bar{c} := (c_1, \dots , c_{12})\) with \(k_j, c_j >0\), \(\lambda _j \in [0,1]\) for all \(j\in \{ 1,\ldots ,12\}\) and \(\sum\nolimits_{j=1}^{12} \lambda _j =1\). The coefficients \(\lambda _1,\ldots ,\lambda _{12}\) can be chosen as quotient of the number of days per considered month and the number of days per year. Since wind speeds with 0 m/s are possible, we consider an exponential PDF as a special Weibull PDF \(f_{1, c_0}\) with \(k_0=1\), \(c_0 >0\) and
$$\begin{aligned} f_{1, c_0}(v) = \left\{ \begin{array}{ll} 0 & \quad \text {if } v < 0\\ \frac{1}{c_0}e^{\frac{v}{c_0}} & \quad \text {if } v \ge 0. \end{array}\right. \end{aligned}$$
This special Weibull PDF is then added to the convex combination of the 12 PDFs so that we investigate the new convex combination \(\tilde{f}_{ \tilde{\lambda },\bar{k},\tilde{c} }\) with \(\tilde{\lambda } := (\lambda _0, \bar{\lambda })\), \(\tilde{c} := (c_0, \bar{c})\) where \(c_0>0\), \(\lambda _j \in [0,1]\) for all \(j\in \{0,\ldots ,12\}\), and \(\sum\nolimits_{j=0}^{12} \lambda _j =1\). This new convex combination is then given by
$$\begin{aligned} {\tilde{f}_{\tilde{\lambda },\bar{k},\tilde{c}}(v)}\nonumber \\:=\, & {} \sum \limits _{j=0}^{12} \lambda _j f_{k_j, c_j}(v)\nonumber \\=\, & {} \left\{ \begin{array}{ll} 0 &{}\quad\text {if } v < 0\\ \sum \limits _{j=0}^{12} \lambda _j \frac{k_j}{c_j}\left( \frac{v}{c_j}\right) ^{k_j1}e^{(\frac{v}{c_j})^{k_j}} &{}\quad \text {if } v \ge 0. \end{array}\right. \end{aligned}$$
(3)
An example of such a convex combination is illustrated in Fig. 3.
If we apply the maximum likelihood method to the convex combination (3) of Weibull PDFs with the logarithmic simplification to Eq. (2), we get the following nonlinear optimization problem
$$\begin{aligned} \begin{array}{l} \displaystyle \max \quad \sum _{i=1}^{8760} \ln \sum _{j=0}^{12} \lambda _j \frac{k_j}{c_j}\Bigl ( \frac{v_i}{c_j}\Bigr )^{k_j1} e^{(\frac{v_i}{c_j})^{k_j}} \\ \text {subject to the constraints}\\ \lambda _j \ge 0 \ \forall \ j\in \{ 0,\ldots ,12\}\\ k_j \ge \varepsilon \ \forall \ j\in \{ 1,\ldots ,12\}\\ c_j \ge \delta \ \forall \ j\in \{ 0,\ldots ,12\}\\ \displaystyle \sum _{j=0}^{12} \lambda _j =1\\ \lambda _0 = c_0h_0\\ (\tilde{\lambda },\bar{k}, \tilde{c})\in \mathbb {R}^{13}\times \mathbb {R}^{12}\times \mathbb {R}^{13}, \end{array} \end{aligned}$$
(4)
where \(\epsilon , \delta >0\) are given lower bounds, \(k_{0}:=1\) and \(h_0\) equals the relative frequency of the wind speeds with 0 m/s. In problem (4), the following adaptations are already modelled:

1.
The original objective function appears in a logarithmic form.

2.
All observed wind speeds are taken into account including wind speeds with 0 m/s.

3.
The positivity of the parameters \(k_j\) (\(j\in \{ 1,\ldots , 12\}\)) and \(c_j\) (\(j\in \{ 0,\ldots ,12\}\)) is ensured by the lower bounds \(\varepsilon \) and \(\delta \).

4.
The last constraint ensures the right PDF value at 0 m/s.
The optimization problem (4) is a constrained problem with a highly nonlinear objective function. In general, methods of continuous optimization determine at most local optima. Figure 4 illustrates the graph of the logarithmic objective function (2) for the classical maximum likelihood method using wind speeds at Jamaica. This figure already highlights the complexity of this problem.