## 2 Mathematical Model of OED of Combined Heat and Power-Thermal-Wind-Photovoltaic Systems
### Renewable Energy  
#### 2.1.1 Relationship between Wind Speed and Wind Power Output  
The power output of a WT unit is mainly determined by the probability distribution parameter of wind speed. In general, the wind speed probability distribution can be described by different models, including Weibull distribution, Rayleigh distribution, log-normal distribution, and normal distribution. Since the two-parameter Weibull model can adjust the parameters to adapt to the periodic variation of wind speed, it is the most frequently used model for wind speed probability distribution, in which the cumulative distribution function can be written as follows [15; 16]:  
\[f_{V}(v)=\frac{k}{c}(\frac{v}{c})^{(k-1)}\exp[-(\frac{v}{c})^{k}] \tag{1}\]  
\[F_{V}(v)=\int_{0}^{v}f_{V}(\xi)d\xi=1-\exp[-(\frac{v}{c})^{k}] \tag{2}\]  
where \(V\) represents the speed random variable, \(v\) is the wind speed, \(k\) is the shape factor of the wind speed probability distribution function, which determines the overall shape of the wind speed probability distribution density curve, and \(c\) is the scale factor of the wind speed probability distribution function, which reflects the average wind speed of the wind field and can enlarge or reduce the curve, but does not affect the shape of the distribution density curve.  
Assuming that the output characteristic of a WT is a simplified piecewise linear function, its power output can be calculated according to the given wind speed [17]:  
\[p_{wt}=\left\{\begin{array}{ll}0&v<v_{in},v>v_{out}\\ p_{wt}^{r}\frac{v-v_{in}}{v_{in}-v_{in}}&v_{in}\leq v\leq v_{r}\\ p_{wt}^{r}&v_{r}<v\leq v_{out}\end{array}\right. \tag{3}\]  
where \(p_{wt}\) is the current maximum power points of the WT unit, \(p_{wt}^{r}\) is the rated power of the WT unit, \(v_{r}\) is the rated wind speed, \(v\) is the current wind speed, and \(v_{in}\) and \(v_{out}\) are the cut-in and cut-out wind speeds, respectively.  
The probability distribution of wind speed is converted into the probability distribution of wind power by a linear transformation [18]:  
\[P_{WT}=T(V)=eV+f \tag{4}\]  
Figure 1: Optimal energy dispatch (OED) framework of combined heat and power-thermal-wind-photovoltaic systems.  
\[f_{P_{WT}}(p_{wt})=f_{V}[T^{-1}(p_{wt})][\frac{dT^{-1}(p_{wt})}{dp_{wt}}]=f_{V}( \frac{p_{wt}-f}{e})|\frac{1}{e}| \tag{5}\]  
where \(V\) and \(P_{WT}\) are the wind speed random variable and the wind power random variable, respectively, and \(T\) represents a transformation.  
For the Weibull function, the discrete part of the wind power output can be written as follows:  
\[F_{P_{WT}}(p_{wt}=0) =F_{V}(v_{in})+(1-F_{V}(v_{out})) \tag{6}\] \[=1-\exp(-(\frac{v_{in}}{c})^{k})+\exp(-(\frac{v_{out}}{c})^{k})\]  
\[F_{P_{WT}}(p_{wt}=p_{wt}^{r}) =F_{V}(v_{out})-F_{V}(v_{r}) \tag{7}\] \[=\exp(-(\frac{v_{r}}{c})^{k})-\exp(-(\frac{v_{out}}{c})^{k})\]  
If the output power of the fan is in a continuously changing range, the probability density function of the Weibull model can be described as follows:  
\[f_{P_{WT}}(p_{wt})=(\frac{k\varphi v_{in}}{cp_{wt}^{r}})[\frac{(1+\varphi p_{ wt}/p_{wt}^{r})v_{in}}{c}]\exp\{-[\frac{(1+\varphi p_{wt}/p_{wt}^{r})v_{in}}{c}] ^{k}\} \tag{8}\]  
\[\varphi=\frac{v_{r}-v_{in}}{v_{in}} \tag{9}\]