# Magnetization-Dependent Core-Loss Model in a Three-Phase Self-Excited Induction Generator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis

_{c}, and the magnetizing reactance, X

_{m}. To determine the values of the circuit parameters, the generator is conventionally tested under DC, locked rotor, and no load [3,4,5,6,7,8]. Values of R

_{s}, R

_{r}, X

_{s}, and X

_{r}are found from the DC and locked rotor tests. The magnetization curve of the machine, which includes the relation of R

_{c}and X

_{m}against air-gap voltage (or magnetization current), is obtained from a no load test (at slip = 0), as shown in Figure 3. As clearly shown, X

_{m}and R

_{c}are variable according to the level of saturation as it is linked with the air-gap voltage. The magnetization curve of the machine in Figure 3 is redeveloped and depicted in Figure 4a, to be used with the circuit shown in Figure 2 to yield the SEIG performance measures. As the saturation level in the generator is variable, X

_{m}is obviously variable and R

_{c}must also be variable. To the best of the authors’ knowledge, this fact has been ignored in all the published research concerning SEIGs [5,6,7,8,9,10,11,12,13].

#### 2.1. Core-Loss Modeling

_{c}with X

_{m}, as shown in Figure 4b. From the experimental results in Figure 4b, the core loss, R

_{c}, varies substantially with X

_{m}, as illustrated by the 4th-degree polynomial fitted curve. Now, any change in load, speed or/and excitation capacitance will change the level of saturation, which, consecutively, will change the value of X

_{m}and, hence, the value of R

_{c}, which results in a variable core loss. For computational purposes, the curve of the air-gap voltage (E

_{g}) versus X

_{m}in Figure 4a is expressed either by a set of piecewise linear approximations [4,5], or by fitting the curve as a polynomial function of a suitable degree, as developed by the authors in [7].

_{m}is also fitted as another polynomial function, as shown in Figure 4b. The fitted curves can be written as:

_{i}and m

_{i}are the polynomial coefficients of the fitted curves that can be determined from experimental results. These two polynomial functions are as given in Appendix A. This approach does not change the characterization given in [5], yet it can solve the three unknown variables simultaneously because R

_{c}is considered as a function of X

_{m}.

#### 2.2. Loop-Impedance Solution

_{s}Z

_{t}= 0

_{t}is the total impedance of the circuit across X

_{m}and R

_{c}branch, as given in Appendix A.

_{s}≠ 0, which indicates that Z

_{t}= 0, or

_{t}) = 0

_{t}) = 0

_{c}), (F and X

_{m}), (F and u), or (F and Z

_{L}).

#### 2.3. Method of Solution

_{c}or X

_{m}, by minimizing the value of the total impedance (i.e., |Z

_{t}| = 0). The performance of the generator described by the circuit of Figure 2 can be derived once the values of the unknowns are obtained utilizing data provided by the magnetization curve.

_{m}when varying the speed of the prime mover. Similar programs were developed to solve for other unknowns such as (F and X

_{c}), (F and u), and (F and Z

_{L}).

## 3. Results and Discussion

_{m}, R

_{c}, F as well as other performance parameters of the generator vary, as these three parameters are varied. Figure 7a,b show the variations of X

_{m}, R

_{c}, and V

_{o}, I

_{s}versus the excitation capacitor, respectively, under different loading conditions. Results confirm the reliability, accuracy, and feasibility of the proposed core modeling. In Figure 7a, X

_{m}decreases to a minimum as C is being increased and then starts increasing. R

_{c}on the other hand increases and decreases independently from X

_{m}. In Figure 7b, V

_{o}changes in a concave manner, whereas I

_{s}increases and then decreases. When X

_{m}is greater than X

_{o}, the machine does not generate voltage. Figure 7b is plotted for a case when the machine is generating voltage (i.e., when X

_{m}is less than or equal to X

_{o}) [5,6,7,8].

_{min}) and F versus power factor (pf) at different loads. In this case, X

_{m}is kept constant at a value equal to X

_{o}, and speed (u) is fixed at 1 p.u. C

_{min}is higher for lower loads and stays nearly constant at lower pfs. When pf increases to a certain value, C

_{min}begins to decrease. F is higher for higher loads, but decreases in very small amounts as the pf increases. Figure 9 shows the variations of X

_{m}and F against pf with C fixed at 40 µF. It can be seen that X

_{m}is larger for smaller loads. In addition, F is decreasing at smaller amounts as pf increases, and it decreases more for smaller loads.

_{o}, and I

_{s}as pf is being varied at a speed of 1 p.u. while C is fixed at 40 µF. At higher loads, V

_{o}is almost constant, and it is obvious that it is higher when X

_{m}is lower by comparing Figure 9 and Figure 10.

_{m}and R

_{c}against speed (Figure 11a) with C fixed at 30 µF for different loads, as well as V

_{o}and I

_{s}against speed (Figure 11b), at the same value of C. As stated above, the machine will not generate voltage for values of X

_{m}above X

_{o}. It is clear from this figure that R

_{c}varies as the speed changes which agrees with the measured results depicted in Figure 3. The assumption in many documented research publications is that it remains constant [16,17,20].

## 4. Experimental Verification

#### 4.1. Setup

#### 4.2. Performance Measurements

_{o}, and stator current, I

_{s}, against excitation capacitor. Figure 14 shows the variations of terminal voltage, frequency, and stator current against generator speed. Figure 14 is repeated in Figure 15 but under different excitation capacitor values. From these figures, V

_{o}and I

_{s}increase as C, or speed, increases. Frequency also increases, as expected, as speed increases. These figures show the superiority and accuracy of the modeling presented, as can be seen from the perfect correlation between computed and experimental results.

## 5. Influence of Core Loss

_{c}.

_{o}) and efficiency (η) are analyzed under different conditions for the generator under study and are shown in Figure 16, Figure 17 and Figure 18. Figure 16 shows the error variation versus excitation capacitance under fixed load and speed, while Figure 17 shows the error variation versus speed under fixed load and excitation capacitance. It can be deduced from Figure 16 and Figure 17 that the error in the value of V

_{o}is relatively high for low C and u values, and then this error rapidly decreases as C, or u increase before it reaches an almost constant low value. On the other hand, the efficiency error variation is relatively high even at high values of C, or u.

_{o}is relatively high at low impedance values and then it rapidly decreases as the load impedance increases before it reaches a nearly constant low value. On the other hand, the efficiency error variation increases with a high percentage as the load increases.

## 6. Conclusions

_{c}or X

_{m}by minimizing the total impedance. Accordingly, the performance curves are computed for the machine as shown in Figure 9, Figure 10 and Figure 11. Experimental verifications were carried out to compare theoretical results with measurements. Perfect agreement between the analytical and the experimental results confirms the feasibility and accuracy as well as the functionality of the modeling presented. It has been found that representing core loss with a fixed resistance causes an error between (2–12)% in computing terminal voltage while it reaches between (15–40)% in the value of the efficiency.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

F, u | p.u frequency and speed, respectively |

C, X_{c} | value of excitation capacitance (µF) and its p.u reactance (at base frequency), respectively |

C_{min} | minimum excitation capacitance (µF) |

R_{s}, R_{r}, R_{L} | p.u stator, rotor, and load resistances, respectively |

X_{s}, X_{r}, X_{L} | p.u stator, rotor leakage, and load reactances (at base frequency), respectively |

X_{m}, X_{o} | p.u saturated and unsaturated magnetizing reactances at base frequency, respectively |

I_{c}, I_{L}, I_{s} | p.u. excitation capacitance, load, and stator currents, respectively |

E_{g}, V_{o} | air-gap and terminal voltages, respectively |

V_{b}, I_{b}, Z_{b} | base voltage, current, and impedance, respectively |

f_{b}, N_{b} | base frequency and speed in Hz and rpm, respectively |

## Appendix A

#### Appendix A.1. Machine Parameters

V_{b} (V) | I_{b} (A) | Z_{b} = V_{b}/I_{b} (Ω) | N_{b} (rpm) | f_{b} (Hz) | R_{s} (p.u.) | R_{r} (p.u.) | X_{s} = X_{r} (p.u.) | X_{o} (p.u.) |
---|---|---|---|---|---|---|---|---|

220 | 2.9 | 75.862 | 1800 | 60 | 0.086 | 0.044 | 0.19 | 1.89 |

#### Appendix A.2. Fitted Curves

_{m}of Figure 3 can be, respectively, fitted by two polynomials of 3rd-degree as follows:

_{0}= 1.1, k

_{1}= −0.636, k

_{2}= 0.727, k

_{3}= −0.321, m

_{0}= 270.67, m

_{1}= −472.71, m

_{2}= 303.76, and m

_{3}= −67.045.

#### Appendix A.3. Total Impedance

_{t}, of Figure 2 is given by:

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**Figure 2.**Per-phase equivalent circuit of the induction generator under the proposed core-loss model.

**Figure 3.**Variation of magnetizing reactance X

_{m}and core loss resistance R

_{c}/F versus air-gap voltage E

_{g}/F in the machine under study.

**Figure 4.**Variation of air-gap voltage and core loss versus magnetizing reactance X

_{m}: (

**a**) Air-gap voltage E

_{g}/F (

**b**) Core loss resistance R

_{c}/(F X

_{m}).

**Figure 6.**Flowchart of the developed optimization program to obtain the performance of the self-excited induction generator (SEIG).

**Figure 7.**Variation versus excitation capacitance C for different loads at fixed speed (u = 1.0 p.u.): (

**a**) Magnetizing reactance X

_{m}and core loss resistance R

_{c}(

**b**) Terminal voltage V

_{o}and Stator current I

_{s}.

**Figure 8.**Variation of minimum excitation capacitance C

_{min}and frequency F versus power factor pf for different loads at fixed speed (u = 1.0 p.u.).

**Figure 9.**Variation of magnetizing reactance X

_{m}and frequency F versus power factor pf for different loads at fixed speed (u = 1.0 p.u.).

**Figure 10.**Terminal voltage V

_{o}and Stator current I

_{s}versus power factor pf for different loads at fixed speed (u = 1.0 p.u.).

**Figure 11.**Variation versus speed for different loads at capacitance C = 30 µF: (

**a**) Magnetizing reactance X

_{m}and core loss resistance R

_{c}(

**b**) Terminal voltage V

_{o}and stator current I

_{s}.

**Figure 13.**Terminal voltage V

_{o}and stator current I

_{s}versus excitation capacitance C under no load when speed (u) = 1 p.u.

**Figure 14.**Variation of terminal voltage V

_{o}, stator current I

_{s}, and frequency F versus speed under no load.

**Figure 16.**Error variation of terminal voltage V

_{o}and efficiency η versus excitation capacitance C.

**Figure 18.**Error variation of terminal voltage V

_{o}and efficiency η versus load impedance |Z

_{L}|.

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**MDPI and ACS Style**

Al-Senaidi, S.H.; Alolah, A.I.; Alkanhal, M.A.
Magnetization-Dependent Core-Loss Model in a Three-Phase Self-Excited Induction Generator. *Energies* **2018**, *11*, 3228.
https://doi.org/10.3390/en11113228

**AMA Style**

Al-Senaidi SH, Alolah AI, Alkanhal MA.
Magnetization-Dependent Core-Loss Model in a Three-Phase Self-Excited Induction Generator. *Energies*. 2018; 11(11):3228.
https://doi.org/10.3390/en11113228

**Chicago/Turabian Style**

Al-Senaidi, Saleh H., Abdulrahman I. Alolah, and Majeed A. Alkanhal.
2018. "Magnetization-Dependent Core-Loss Model in a Three-Phase Self-Excited Induction Generator" *Energies* 11, no. 11: 3228.
https://doi.org/10.3390/en11113228