Squirrel-cage induction generator system using hybrid wavelet fuzzy neural network control for wind power applications

Abstract

An intelligent controlled three-phase squirrel-cage induction generator (SCIG) system for grid-connected wind power applications using hybrid wavelet fuzzy neural network (WFNN) is proposed in this study. First, the indirect field-oriented mechanism is implemented for the control of the SCIG system. Then, an AC/DC power converter and a DC/AC power inverter are developed to convert the electric power generated by a three-phase SCIG to power grid. Moreover, the dynamic model of the SCIG system and an ideal computed torque controller are developed for the control of the square of DC-link voltage. Furthermore, an intelligent hybrid WFNN controller and two WFNN controllers, which are computation intensive approaches, are proposed for the AC/DC power converter and the DC/AC power inverter, respectively, to improve the transient and steady-state responses of the SCIG system at different operating conditions. In the intelligent hybrid WFNN controller, to relax the requirement of the lumped uncertainty in the design of the ideal computed torque controller, a WFNN is designed as an uncertainty observer to adapt the lumped uncertainty online. Finally, the feasibility and effectiveness of the SCIG system for grid-connected wind power applications are verified with experimental results.

Appendix

Theorem 1

Considering the SCIG system represented by (16), if the hybrid WFNN control law is designed in (20), in which the computed torque control law is designed in (24), the adaptation law of the WFNN is designed in (43), and the compensated control is designed in (44), then the asymptotical stability of the control system is guaranteed.

$$({\mathbf{w}}^{{\mathbf{5}}} )^{{\prime }} = \eta {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{y}}^{{\mathbf{4}}}$$

(43)

$$U_{{C}} (t) = - B_{{n}}^{ - 1} (t)\delta \text{sgn} ({\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} )$$

(44)

Proof

A Lyapunov function is defined as

$$V(t) = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PE}} + \frac{1}{2\eta }[{\mathbf{w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} ]^{\rm T} [{\mathbf{w}}^{{{\mathbf{5}}*}} - {\mathbf{w}}^{{\mathbf{5}}} ]$$

(45)

where ${\mathbf{P}}$ satisfies the following Lyapunov equation:

$${\varvec{\Lambda}}^{{\mathbf{T}}} {\mathbf{P}} + {\mathbf{P\Lambda }} = - {\mathbf{Q}}$$

(46)

and ${\mathbf{Q}} > 0$ is selected by the designer. Take the derivative of the Lyapunov function and use (31), (32) and (46), then

$$\begin{aligned} V^{{\prime }} (t) & = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PE}}^{{\prime }} + \frac{1}{2}{\mathbf{E}}^{{{\prime }{\mathbf{T}}}} {\mathbf{PE}} - \frac{1}{\eta }[{\mathbf{w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} ]^{{\mathbf{T}}} ({\mathbf{w}}^{{\mathbf{5}}} )^{{\prime }} \\ & {\kern 1pt} = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{P}}\Lambda {\mathbf{E}} + \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\varvec{\Lambda}}^{{\mathbf{T}}} {\mathbf{PE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \{ [{\mathbf{w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} ]^{{\mathbf{T}}} {\mathbf{y}}^{{\mathbf{4}}} + \varepsilon (t) + B_{{n}} (t)U_{{C}} (t)\} \\ & \quad - \frac{1}{\eta }{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{w}}^{{\prime }} \\ & = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} [\varepsilon (t) + B_{{n}} (t)U_{C} (t)] \\ & \quad + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{y}}^{{\mathbf{4}}} - \frac{1}{\eta }{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{(w}}^{{\mathbf{5}}} {\mathbf{)}}^{{\prime }} \\ \end{aligned}$$

(47)

To satisfy $V^{{\prime }} (t) \le 0$, the update law $({\mathbf{w}}^{{\mathbf{5}}} )^{{\prime }}$ and the compensated controller $U_{{C}} (t)$ are designed as (43) and (44). Substitute (43) into (47) and use (44), then

$$\begin{aligned} {\kern 1pt} V^{{\prime }} (t) & = - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \varepsilon (t) + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} B_{{n}} (t)U_{{C}} (t) \\ & \le - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + \left| {{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} } \right|{\kern 1pt} {\kern 1pt} \left| {\varepsilon (t)} \right| + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} B_{{n}} (t)U_{{C}} (t) \\ & \le - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} \le 0 \\ \end{aligned}$$

(48)

Since $V^{{\prime }} (t) \le 0$, $V^{{\prime }} (t)$ is negative semidefinite (i.e. $V(t) \le V(0)$), which implies ${\kern 1pt} {\mathbf{E}}$ and ${\mathbf{[w}}^{{{\mathbf{5}}*}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}}$ are bounded. Define function $\Theta (t) = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} \le - V^{\prime}(t)$, and integrate function $\Theta (t)$ with respect to time

$$\int_{0}^{t} {\Theta (\tau ){\text{d}}\tau } \le V(0) - V(t)$$

(49)

Because $V(0)$ is bounded, and $V(t)$ is nonincreasing and bounded, then

$$\mathop {\lim }\limits_{t \to \infty } \int_{0}^{t} {\Theta (\tau ){\text{d}}\tau } \le \infty$$

(50)

Differentiate with respect to time

$$\Theta^{{\prime }} (t) = {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}}^{{\prime }}$$

(51)

Since all the variables in the right-hand side of (32) are bounded, it implies ${\mathbf{E}}^{{\prime }}$ is also bounded. Then, $\Theta^{{\prime }} (t)$ is uniformly continuous. By using the Barbalat’s lemma, it can be shown that $\mathop {\lim }\limits_{t \to \infty } \Theta (t) = 0$. Thus, ${\kern 1pt} {\mathbf{E}} \to 0$ as $t \to \infty$. As a result, the hybrid WFNN control system is asymptotically stable.

Theorem 2

Considering the SCIG system shown in (16 ), if the hybrid WFNN control law is designed in (20), in which the computed torque control law is designed in (24), the adaptation law of the WFNN is designed in (42), and the compensated control is designed in (44), then the asymptotical stability of the control system is guaranteed.

Proof

When the condition $\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\| \le M_{{b}}$ or $\left( {\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\| = M_{{b}} \,{\text{and}}\,{\kern 1pt} {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{w}}^{{{\mathbf{5}}{\text{T}}}} {\mathbf{y}}^{{\mathbf{4}}} \le 0} \right)$ holds, the stability analysis is the same as Theorem 1. On the other hand, when $\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\| = M_{{b}} {\kern 1pt}$ and ${\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{w}}^{{{\mathbf{5T}}}} {\mathbf{y}}^{{\mathbf{4}}} > 0$, the derivative of the Lyapunov function shown in (47) can be redescribed as follows:

$$\begin{aligned} V^{{\prime }} (t) & = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} [\varepsilon (t) + B_{{n}} (t)U_{{C}} (t)] + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{y}}^{{\mathbf{4}}} - \frac{1}{\eta }{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} ({\mathbf{w}}^{{\mathbf{5}}} )^{{\prime }} \\ & = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} [\varepsilon (t) + B_{{n}} (t)U_{{C}} (t)] + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{y}}^{{\mathbf{4}}} - \frac{1}{\eta }{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} \\ & \quad \times \left( {\eta {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} {\mathbf{y}}^{{\mathbf{4}}} - {\mathbf{\eta E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \frac{{{\mathbf{w}}^{{\mathbf{5}}} {\mathbf{w}}^{{{\mathbf{5T}}}} {\mathbf{y}}^{{\mathbf{4}}} }}{{\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} }}} \right) \\ & = \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} [\varepsilon (t) + B_{{n}} (t)U_{{C}} (t)] + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \frac{{{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{w}}^{{\mathbf{5}}} }}{{\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} }}{\mathbf{w}}^{{{\mathbf{5T}}}} {\mathbf{y}}^{{\mathbf{4}}} \\ \end{aligned}$$

(52)

Using the characteristic ${\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{w}}^{{\mathbf{5}}} = \frac{1}{2}\left( {\left\| {{\mathbf{w}}^{{{\mathbf{5*}}}} } \right\|^{2} - \left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} - \left\| {{\mathbf{w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} } \right) < 0$, which is based on $\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\| = M_{{b}} > {\kern 1pt} {\kern 1pt} \left\| {{\mathbf{w}}^{{{\mathbf{5}}*}} } \right\|$, then

$$\begin{aligned} V^{{\prime }} (t) & \le - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + \left| {{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} } \right|{\kern 1pt} {\kern 1pt} \left| {\varepsilon (t)} \right| + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} B_{{n}} (t)U_{{C}} (t) + {\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \frac{{{\mathbf{[w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} {\mathbf{]}}^{{\mathbf{T}}} {\mathbf{w}}^{{\mathbf{5}}} }}{{\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} }}{\mathbf{w}}^{{{\mathbf{5T}}}} {\mathbf{y}}^{{\mathbf{4}}} \\ & \le - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} + \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{PB}}_{{\mathbf{P}}} \frac{{\left\| {{\mathbf{w}}^{{{\mathbf{5*}}}} } \right\|^{2} - \left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} - \left\| {{\mathbf{w}}^{{{\mathbf{5*}}}} - {\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} }}{{\left\| {{\mathbf{w}}^{{\mathbf{5}}} } \right\|^{2} }}{\mathbf{w}}^{{{\mathbf{5T}}}} {\mathbf{y}}^{{\mathbf{4}}} \\ & \le - \frac{1}{2}{\mathbf{E}}^{{\mathbf{T}}} {\mathbf{QE}} \le 0 \\ \end{aligned}$$

(53)

By using Barbalat’s lemma as shown in Theorem 1, ${\kern 1pt} {\mathbf{E}} \to 0$ as $t \to \infty$. Therefore, the stability property also can be guaranteed.