# Cascade Control of the Ground Station Module of an Airborne Wind Energy System

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Research Gap

#### 1.3. Major Contributions

## 2. Modeling of Ground Station Module

#### 2.1. Mathematical Model of Winch

#### 2.2. Mathematical Model of Induction Machine

#### 2.2.1. $d-q$ Model of an Induction Machine in Synchronously Rotating Reference Frame

#### 2.2.2. $d-q$ Model of an Induction Machine in Stationary Reference Frame

#### 2.3. Coupling of Winch and Induction Machine

## 3. Cascade Control Strategy for Ground Station Module

#### 3.1. Sliding Mode Control Design for Winch System

#### 3.2. Rotor-Based Field-Oriented Control of Induction Machine

#### 3.2.1. Torque and Flux Controller Design

#### 3.2.2. Discrete-time Kalman Filter for Flux Estimation

- The initialization of DKF is carried out as:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\widehat{x}}_{0}^{+}=E\left({x}_{0}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {P}_{0}^{+}=E\left[\left({x}_{0}-{\widehat{x}}_{0}^{+}\right){\left({x}_{0}-{\widehat{x}}_{0}^{+}\right)}^{T}\right],\hfill \end{array}$$
- The main algorithm of DKF comprises the following equations, which are sequentially solved for each time step $k=1,2,\dots $$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {P}_{k}^{-}={F}_{k-1}{P}_{k-1}^{+}{F}_{k-1}^{T}+{Q}_{k-1},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {K}_{k}^{-}={P}_{k}^{-}{C}_{k}^{T}{\left(C{P}_{k}^{-}{C}^{T}+{R}_{k}\right)}^{-1},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\widehat{x}}_{k}^{-}={F}_{k-1}{\widehat{x}}_{k-1}^{+}+G{u}_{k-1},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\widehat{x}}_{k}^{+}={\widehat{x}}_{k}^{-}+{K}_{k}\left({y}_{k}-{C}_{k}{\widehat{x}}_{k}^{-}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {P}_{k}^{+}=\left({I}_{5}-{K}_{k}C\right){P}_{k}^{-}{\left({I}_{5}-{K}_{k}C\right)}^{T}+{K}_{k}{R}_{k}{K}_{k}^{T},\hfill \end{array}$$

#### 3.2.3. Implementation of RFOC on IM

## 4. Results and Discussions

- A variable step MATLAB solver: Ode23t is used to simulate the AWES in Figure 1. The minimum and maximum step sizes used for simulation are 5 $\mathsf{\mu}$s and 50 $\mathsf{\mu}$s, respectively.
- The minimum height of the tether from where the KM is launched into the cross wind flight is ${L}_{min}=50$ m. Whereas, the power production phase is stopped when the tether length reaches its maximum value: ${L}_{max}=150$ m.
- To evaluate the robustness of the control strategy, it is assumed that the wind gusts generate a constant disturbance torque, i.e., $\zeta \left(t\right)=12.4$ Nm in (1). It is important to mention here that we have not used any specific dynamic model of the wind gust. The constant value for the wind gust used was the maximum value of the disturbance considered, corresponding to the simulation of the worst case scenario.
- The desired angular speed of the winch ${\omega}_{{w}_{r}}=10$ r/s for the traction pahse. Positive angular speed refers to counter-clockwise rotation, whereas, negative angular speed indicates clockwise rotation.
- A 5.5 KW IM with rated line-line voltage ${u}_{LL}=400$ V (${u}_{\varphi}=231$ V), operating at 50 Hz is considered for the simulation study. The rated speed, rated torque, and efficiency of the IM are 1461 rev/min (153 r/s), 780 Nm, and $89.6$%, respectively.
- It is assumed that the gain of voltage fed inverter in Figure 2 is unity.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

${\omega}_{r}$, ${\omega}_{m}$ | electrical and mechanical speeds of rotor of the induction |

machine (IM), respectively $\left(\mathrm{r}/\mathrm{s}\right)$ | |

${\theta}_{e}$ | electrical angle required for field oriented control of the IM $\left(\mathrm{r}\right)$ |

${\varphi}_{dr}$, ${\varphi}_{qr}$ | d and q axes rotor fluxes, respectively $\left(\mathrm{Wb}\right)$ |

${\varphi}_{opt}$ | optimal rotor flux $\left(\mathrm{Wb}\right)$ |

${i}_{ds}$, ${i}_{qs}$ | d and q axes stator currents, respectively $\left(\mathrm{A}\right)$ |

${R}_{s}$, ${R}_{r}$ | stator and rotor resistances, respectively $\left(\Omega \right)$ |

${L}_{s}$, ${L}_{r}$ | stator and rotor leakage inductances, respectively $\left(\mathrm{mH}\right)$ |

${L}_{m}$ | magnetizing inductance $\left(\mathrm{mH}\right)$ |

${N}_{P}$ | number of pole pairs $\left(--\right)$ |

f | frequency of $3-\varphi $ stator |

J | inertia of the IM rotor $\left(\mathrm{kg}.{\mathrm{m}}^{2}\right)$ |

b | damping coefficient of the IM rotor $\left(\mathrm{N}.\mathrm{m}.\mathrm{s}/\mathrm{r}\right)$ |

${F}_{T}$ | tether tension $\left(\mathrm{N}\right)$ |

${\omega}_{{w}_{r}}$, ${\omega}_{w}$ | desired and measured speeds of the winch $\left(\mathrm{r}/\mathrm{s}\right)$ |

${T}_{{w}_{c}}$ | torque generated by winch speed controller $\left(\mathrm{Nm}\right)$ |

${T}_{r},{T}_{e}$ | desired and actual induced torques of the IM, respectively $\left(\mathrm{Nm}\right)$ |

${T}_{{w}_{i}}$ | input torque to the winch system $\left(\mathrm{Nm}\right)$ |

${J}_{w}$ | winch inertia $\left(\mathrm{kg}{\mathrm{m}}^{2}\right)$ |

${b}_{w}$ | damping coefficient of winch $\left(\mathrm{Nms}/\mathrm{r}\right)$ |

${r}_{w}$ | winch radius $\left(\mathrm{m}\right)$ |

${N}_{w}$, ${N}_{{}_{IM}}$ | Number of teeth of winch and IM gears, respectively $\left(--\right)$ |

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**Figure 2.**Rotor flux-oriented control of IM. (

**a**) Alignment of rotor flux. (

**b**) Rotor flux angle calculation.

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|

${N}_{p}$ | 2 | ${R}_{s}$ | 0.295 | ${R}_{r}$ | 0.379 | ${L}_{r}$ | 0.0608 |

${L}_{s}$ | 0.0608 | ${L}_{m}$ | 0.059 | J | 0.62 | b | 0.1 |

${b}_{w}$ | 0.01 | ${J}_{w}$ | 0.124 | ${r}_{w}$ | 0.25 | ${N}_{w}/{N}_{IM}$ | 12 |

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**MDPI and ACS Style**

Uppal, A.A.; Fernandes, M.C.R.M.; Vinha, S.; Fontes, F.A.C.C. Cascade Control of the Ground Station Module of an Airborne Wind Energy System. *Energies* **2021**, *14*, 8337.
https://doi.org/10.3390/en14248337

**AMA Style**

Uppal AA, Fernandes MCRM, Vinha S, Fontes FACC. Cascade Control of the Ground Station Module of an Airborne Wind Energy System. *Energies*. 2021; 14(24):8337.
https://doi.org/10.3390/en14248337

**Chicago/Turabian Style**

Uppal, Ali Arshad, Manuel C. R. M. Fernandes, Sérgio Vinha, and Fernando A. C. C. Fontes. 2021. "Cascade Control of the Ground Station Module of an Airborne Wind Energy System" *Energies* 14, no. 24: 8337.
https://doi.org/10.3390/en14248337