# Enhanced Quadratic V/f-Based Induction Motor Control of Solar Water Pumping System

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## Abstract

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## 1. Introduction

- Performance of the two-stage solar PV water pumping system with quadratic V/f control compared to linear V/f control.
- The dynamic response and the system efficiency due to a sudden change in the input power under the proposed quadratic V/f control.
- Replacing the conventional PI control with fuzzy logic and sliding mode controllers in order to improve the system performance.

## 2. SPWPS Layout and Design

#### 2.1. Pump Selection

#### 2.2. PV Array Design

#### 2.3. Design of System Passive Elements

- $m$ is the modulation index.
- ${V}_{L}$ is the motor line to line voltage = 380 V and the step-up transformer turns ratio is 1:2, so that the inverter RMS line voltage output is 190 V.
- ${V}_{DC}$ is the DC link voltage. From (1), it is selected to be 300 V.

- ${V}_{DC}^{*}$ is the set value of the DC link voltage = 300 V.
- ${V}_{DC}$ is the acceptable lower voltage at the DC link during transients.
- α is an overloading factor and $t$ is the duration of transient.
- $V$ and $I$ are the phase RMS voltage and current, respectively.

- ${V}_{in}$ is the input DC voltage = 200 V.
- $D$ is the duty cycle = 0.3333.
- $\Delta {I}_{L}$ is the value of the input current ripple, which is set to 20% of the rated PV current.
- ${f}_{s}$ is the switching frequency, which is set to 10 kHz.

## 3. Two-Stage SPWPS Control

#### 3.1. Quadratic V/f Combined with PI Controller

_{n}and f

_{n}represents the values of the rated voltage and the rated frequency, respectively.

#### 3.2. Quadratic V/f Combined with Other Alternative Controllers

#### 3.2.1. Sliding Mode Control (SMC)

#### 3.2.2. Fuzzy Logic Control (FLC)

## 4. Simulation Results

#### 4.1. Comparsion between Q V/f and L V/f-Based Controllers

#### 4.2. Simulation Results of the Other Alternative Controller Combined with the Q V/f

## 5. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${V}_{in}$ | Input voltage [V] |

${I}_{L}$ | Inductor current [A] |

${V}_{DC}$ | DC link voltage [V] |

${V}_{L}$ | Motor line to line voltage [V] |

P_{m} | Maximum power [W] |

${V}_{OC},{\text{}V}_{m}$ | Open circuit and MPP voltage [V] |

${I}_{SC}$, ${I}_{m}$ | Short circuit and MPP current [A] |

$m$ | Modulation index |

$C$ | DC link capacitor [f] |

$L$ | DC-DC boost inductor [H] |

${f}_{s}$ | Switching frequency [HZ] |

$D$ | Duty Cycle |

${T}_{L}$ | Load Torque [N·m] |

${K}_{pump}$ | Pump constant |

$\omega $ | Motor speed [rad/s] |

$p$ | Motor number of poles |

${P}_{in}$, ${P}_{L}$ | Input and output power [W] |

$K$ | Constant derived from pump constant |

${f}_{ref}$ | Reference frequency [Hz] |

$S$ | Sliding surface |

${K}_{V},{\text{}K}_{I}$ | Sliding surface coefficients |

$\eta $ | Efficiency [%] |

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**Figure 1.**The power circuit for the direct-coupled solar photovoltaic energy (PV) water pumping system.

**Figure 2.**The I-V and P-V characteristic curves of the PV array: (

**a**) at different atmospheric temperature values and conScheme 1000 W/m

^{2}); (

**b**) at different solar irradiation levels at constant atmospheric temperature value (25°).

**Figure 4.**The relation between the stator voltage and frequency in (

**a**) Linear V/f control; (

**b**) Quadratic V/f control.

**Figure 7.**Simulation results of the proposed Q V/f compared to the conventional L V/f: (

**a**) Inductor current; (

**b**) DC link voltage; (

**c**) Motor speed; (

**d**) Motor torque; (

**e**) RMS phase current.

**Figure 8.**Simulation results of the proposed Q V/f PI-FLC compared to Q V/f PI-PI: (

**a**) Inductor current; (

**b**) DC link voltage; (

**c**) Motor speed.

**Figure 10.**Experimental results at 4 A of the inductor current: (

**a**) the proposed Q V/f; (

**b**) the conventional L V/f PI-PI. (Trace 1: Inductor current, Trace 2: Inverter phase current, Trace 3: DC-link voltage).

**Figure 11.**Experimental results at 2 A of the inductor current: (

**a**) the proposed Q V/f; (

**b**) the conventional L V/f PI-PI (Trace 1: Inductor current, Trace 2: Inverter phase current, Trace 3: DC-link voltage).

**Figure 12.**Experimental results at step change from 2A to 4A of the inductor current: (

**a**) the proposed Q V/f; (

**b**) the conventional L V/f PI-PI (Trace 1: Inductor current, Trace 2: Inverter phase current, Trace 3: DC-link voltage).

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

Rated line voltage | 380 V |

Rated phase current | 2.7 A |

Rated frequency | 50 Hz |

Rated speed | 2850 rpm |

Stator resistance | 12.6 Ω |

Stator leakage inductance | 5 mH |

Rotor resistance | 12.1 Ω |

Rotor leakage inductance | 5 mH |

Magnetizing inductance | 250 mH |

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

Module maximum power (P_{m}) | 185 W |

Module open Circuit Voltage (V_{OC}) | 121 V |

Module Short Circuit Current (I_{SC}) | 2.19 A |

Module MPP voltage (V_{m}) | 95 V |

Module MPP current (I_{m}) | 1.95 A |

Array maximum power (P_{m}) | 740 W |

Array open Circuit Voltage (V_{OC}) | 242 V |

Array Short Circuit Current (I_{SC}) | 4.38 A |

Array MPP voltage (V_{m}) | 190 V |

Array MPP current (I_{m}) | 3.9 A |

E/CE | NB | NS | Z | PS | PB |
---|---|---|---|---|---|

NB | NB | NB | NB | NS | Z |

NS | NB | NB | NS | Z | PS |

Z | NB | NS | Z | PS | PB |

PS | NS | Z | PS | PB | PB |

PB | Z | PS | PB | PB | PB |

Time (S) | Type of Control | Inductor Current (A) | DC Link Voltage (V) | Motor Speed (rpm) | Motor Torque (N.m) | RMS Phase Current (A) | Efficiency (%) |
---|---|---|---|---|---|---|---|

0–5 | Quadratic V/f | 3.65 | 300 | 2840 | 1.376 | 2.54 | 56.04 |

Linear V/f | 3.65 | 300 | 2836 | 1.372 | 2.54 | 55.79 | |

5–10 | Quadratic V/f | 2.74 | 300 | 2544 | 1.104 | 2.302 | 53.649 |

Linear V/f | 2.74 | 299.6 | 2379 | 0.9655 | 2.52 | 43.88 | |

10–15 | Quadratic V/f | 1.825 | 300 | 2169 | 0.802 | 1.91 | 49.917 |

Linear V/f | 1.825 | 299.5 | 1757 | 0.5269 | 2.461 | 26.567 |

Time (S) | Type of Control | Inductor Current (A) | DC Link Voltage (V) | Motor Speed (rpm) | Efficiency (%) |
---|---|---|---|---|---|

0–5 | Q V/f PI-PI | 3.65 | 300 | 2840 | 56.04 |

Q V/f PI-FLC | 3.65 | 298.8 | 2840 | 56.08 | |

Q V/f SM-FLC | 3.652 | 299.1 | 2837 | 55.72 | |

5–10 | Q V/f PI-PI | 2.74 | 300 | 2544 | 53.649 |

Q V/f PI-FLC | 2.74 | 298.9 | 2554 | 54.32 | |

Q V/f SM-FLC | 2.743 | 298.9 | 2554 | 54.21 | |

10–15 | Q V/f PI-PI | 1.825 | 300 | 2169 | 49.917 |

Q V/f PI-FLC | 1.825 | 299 | 2173 | 50.24 | |

Q V/f SM-FLC | 1.822 | 299 | 2169 | 50.05 |

Type of Control | Inductor RMS Current (A) | DC Link Voltage (V) | Frequency (HZ) | RMS Phase Current (A) |
---|---|---|---|---|

Quadratic V/f | 2.05 | 298 | 37.2 | 1.91 |

Linear V/f | 2.04 | 298 | 35.07 | 2.25 |

Quadratic V/f | 4.33 | 304 | 46.54 | 2.46 |

Linear V/f | 4.27 | 300 | 42.26 | 2.64 |

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**MDPI and ACS Style**

Yussif, N.; Sabry, O.H.; Abdel-Khalik, A.S.; Ahmed, S.; Mohamed, A.M.
Enhanced Quadratic V/f-Based Induction Motor Control of Solar Water Pumping System. *Energies* **2021**, *14*, 104.
https://doi.org/10.3390/en14010104

**AMA Style**

Yussif N, Sabry OH, Abdel-Khalik AS, Ahmed S, Mohamed AM.
Enhanced Quadratic V/f-Based Induction Motor Control of Solar Water Pumping System. *Energies*. 2021; 14(1):104.
https://doi.org/10.3390/en14010104

**Chicago/Turabian Style**

Yussif, Neama, Omar H. Sabry, Ayman S. Abdel-Khalik, Shehab Ahmed, and Abdelfatah M. Mohamed.
2021. "Enhanced Quadratic V/f-Based Induction Motor Control of Solar Water Pumping System" *Energies* 14, no. 1: 104.
https://doi.org/10.3390/en14010104