# A Maximum Power Point Tracking Control Method Based on Rotor Speed PDF Shape for Wind Turbines
## 3 MPPT Control Based on PDF Shape
### PDF Control Based on FPK Equation  
For one-dimensional controlled nonlinear stochastic system, according to [PERSON]'s lemma, its dynamic equation can be expressed as:  
\[\dot{x}(t)=\varphi(x)+G(x)\varepsilon(t),\ (x(t_{0})=x_{0},\ t\in[t_{0},T]) \tag{10}\]  
where \(x(t)\) is the system status variable and has an initial state \(x(t_{0})\), \(\varphi(x)\) is the nonlinear function of system state, \(G(x)\) is the diffusion coefficient and \(\varepsilon(t)\) is the stochastic input vector or interference vector of the system. The correlation coefficient matrix of \(\varepsilon(t)\) is \(E\{\varepsilon(t)\varepsilon(t^{\prime})\}=2\pi S_{\varepsilon 1}\delta(t-t^{ \prime})\), where \(S_{\varepsilon 1}\) is the power spectral density of \(\varepsilon(t)\) and often take constant value. The polynomial nonlinear system is the most common system type. For this kind of system, \(\varphi(x)\) is calculated by:  
\[\varphi(x)=F(x)+u(x) \tag{11}\]  
where \(u(x)\) is the control function to be obtained and \(F(x)\) is the nonlinear part of the system. Thus, a PDF controller in nonlinear form can be designed:  
\[u(x) =\underbrace{b_{l}x^{l}+b_{l-1}x^{l-1}+\cdots+b_{1}x+b_{0}}_{ \varphi(x)}-\underbrace{\Big{(}a_{r}x^{r}+a_{r-1}x^{r-1}+\cdots+a_{1}x+a_{0} \Big{)}}_{F(x)} \tag{12}\] \[=b_{l}x^{l}+b_{l-1}x^{l-1}+\cdots+(b_{r}-a_{r})x^{r}+(b_{r-1}-a_{ r-1})x^{r-1}+\cdots+(b_{1}-a_{1})+(b_{0}-a_{0})\]  
where \(b_{l},b_{l-1},\cdots,b_{0}\) and \(a_{r},a_{r-1},\cdots,a_{0}\) are the coefficients, and \(r\leq l\). In the system of (10), the stationary FPK equation corresponding to the PDF of the state variable is:  
\[-\frac{d\Big{[}a(x)p_{trf}(x)\Big{]}}{dx}+\frac{1}{2}\frac{d^{2}\Big{[}\gamma (x)p_{trf}(x)\Big{]}}{dx^{2}}=0 \tag{13}\]  
where \(p_{trf}(x)\) is the PDF of system's state variable, \(\alpha(x)=\varphi(x)\) and \(\gamma(x)=2\pi S_{\varepsilon 1}G^{2}(x)\). Since the precise solution of the FPK equation is difficult to obtain, the optimal performance index can be constructed according to the distribution shape of the goal function (\(p_{trf}(x)\)), and the approximate solution of the FPK equation (\(p_{trf}^{\infty}(x)\)). Control inputs are determined by minimizing the performance index:  
\[min\ e(l)=\int_{-\infty}^{+\infty}\Big{(}p_{trf}(x)-p_{trf}^{\infty}(x)\Big{)} ^{2}dx \tag{14}\]  
where \(e(l)\) is the system tracking error of \(l\)-order controller. For simple calculation, let \(S_{\varepsilon 1}=\frac{1}{\pi}\). Then, the FPK equation corresponding to \(p_{trf}^{\infty}(x)\) is:  
\[-\frac{d\Big{[}\varphi(x)p_{trf}^{\infty}(x)\Big{]}}{dx}+\frac{d^{2}\Big{[}G(x )^{2}p_{trf}^{\infty}(x)\Big{]}}{dx^{2}}=0. \tag{15}\]  
When \(G(x)=1\), the above FPK equation is transformed into the following second-order homogeneous differential equation:  
\[p_{trf}^{\infty}(x)-\varphi^{\prime}(x)p_{trf}^{\infty}(x)-\varphi(x)p_{trf}^ {\infty}(x)=0. \tag{16}\]  
Assuming that one of the special solutions of \(p_{trf}^{\infty}(x)\) is \(p_{1}(x)=e^{q(x)}\) and considering \(e^{q(x)}\
eq 0\), the following equation is obtained by substituting \(p_{1}(x)\) into Equation (16):  
\[q^{\prime\prime}(x)+\big{(}q^{\prime}(x)\big{)}^{2}-\varphi(x)q^{\prime}(x)- \varphi^{\prime}(x)=0. \tag{17}\]Similarly, when another special solution of \(p_{trf}^{\infty}(x)\) is \(p_{2}(x)=c(x)e^{q(x)}\) and \(c(x)\
eq 0\), there is:  
\[c^{\prime\prime}(x)+\big{[}2q^{\prime}(x)-\varphi(x)\big{]}c^{\prime}(x)=0. \tag{18}\]  
Solving Equation (17) and Equation (18) respectively to obtain the unknown variables in \(p_{1}(x)\) or \(p_{2}(x)\):  
\[\left\{ \begin{array}{l}q(x)=\int\varphi(x)dx\\ c(x)=\int e^{\int\varphi(x)-2q^{\prime}(x))dx}dx.\end{array} \right.