Conversion of AC wind energy system model to DC model for integration to optimization software with large scales

Abstract

The work presented in this paper deals with an AC model of wind turbine system conversion to a DC model with reduced simulation time, for possible integration to optimization software with larges scales permitting a multi-objective optimization, such as the constrained optimization conjointly of the cost and power losses of the wind turbine energy system. The DC model is based on average calculation of the DC voltage recharging the battery energy accumulator used for recovering the converted wind energy and the electromagnetic torque. Indeed, classical model of wind turbine using generally electric generator associated to a PD3 rectifier to convert the alternative energy on DC energy recoverable on battery energy accumulator with need a large simulation time and thereafter it is non integrable to optimization software for multi-objective optimization problem resolution. The two models are implemented under Matlab-Simulink simulation environment. Simulation results valid entirely the wind turbine system DC model. Finally, as perspective it is interesting to use a booster chopper as an interface between the rectifier and the battery to optimize the recovered energy. An average model of the booster chopper can be integrated into the DC model for performance improvement of the conversion chain.

Keywords

Classical wind turbine DC wind energy system equivalent model permanent magnet synchronous generator mathematical calculations validation multi-objective optimization genetic algorithms

Introduction

The optimization of wind energy system recovered energy is generally achieved by controlling the speed of electric generator (Koutroulis and Kalaitzakis 2006; Nhidi et al., 2015). In particular, the rotor speed should vary in accordance with the wind speed by maintaining the tip speed ratio to the value that maximizes aerodynamic efficiency. For that purpose, many different Maximum Power Point Tracking (MPPT) control strategies have been developed, allowing the regulation of the generator voltage (Koutroulis and Kalaitzakis, 2006). The high efficiency of these generators is counterbalanced with higher costs and greater complexity of the electronic devices (Neji et al., 2006; Nhidi et al., 2015). Elsisi et al. (2021) Presents a robust design of ANFIS-Based Blade Pitch Controller for Wind Energy Conversion Systems against wind speed fluctuations. This study permit to reduce speed fluctuations by using a new robust wind turbine speed control technique. Despite being interesting, this control technique requires a significant study and production cost, given the complexity of the electronic, and software elements necessary to achieve this control technique. Elsisi (2019a) concerns a new design of adaptive model predictive control for energy conversion system with wind torque effect. This study presents several aspects of innovation in the field of speed regulation of wind turbines minimizing speed fluctuations, but it slows down the possibility of approaching optimizations for example of the efficiency of the entire power chain by coupling the proposed model to optimization algorithms of the manufacturing parameters of the power chain components, given that the simulation time of the proposed model is high as well as the divergence of the system during the iterative variation of the optimization parameters. Elsisi (2019b) illustrates a future search algorithm for optimization (FSA). This study is interesting and may be a way of extending the study presented in this paper. Elsisi et al. (2015) and Elsisi (2019c) presents robust optimization methods of PID controller. This study can complement the study presented in this paper. Indeed, after the phase of power chain parameters optimization, a second phase of speed regulator parameters optimizing can be carried out by these techniques.

The added value of the study presented in this paper is the development of a model equivalent to the classic models existing under Simulink simulation environment, with reduced simulation time and highly parameterized, in view of a good compatibility with large dimension stochastic optimization algorithms. This study is based on the coupling of an analytical model of a synchronous generator with permanent magnets to the other models of the components of the power chain of the wind turbine studied.

The system cost has to be drastically minimized for instance by simplifying the structure by the use only of permanent magnet synchronous generator feeding a diode rectifier associated with a battery.

On the other hand, thanks to a suitable choice of the system design variables associated with the wind turbine generator sizing (especially electrical and geometrical parameters) it is possible to improve significantly the global system efficiency (Elsisi 2019a, 2019b, 2019c; Elsisi et al., 2015, 2021; Koutroulis and Kalaitzakis; Neji et al., 2006; Nhidi et al., 2015; Ouyang et al., 2019; Zhang et al., 2020).

The design model of the generator is based on design process of an axial motor. Indeed, the model is achieved by conversion of the motor design model in generator design model (Neji et al., 2006).

The method used for the generator design is a combined Analytic-Finite element method (Ali et al., 2020a, 2020b, 2021; Aly et al., 2017; Beltran et al., 2008; Bhattacharyya et al., 2020; Cheng et al., 2018; Cross et al., 2015; Das et al., 2020; Ebrahim et al., 2012; Ebrahim et al., 2012; Ebrahim et al., 2018; Elsisi et al., 2021; Elsisi, 2020; Geng and Yang, 2014; Hussain and Mishra 2020; Jang, 1993; Kennedy and Eberhart, 1995; Khosravi et al., 2020; Li et al., 2021; Liang et al., 2021; Liu et al., 2020; Ma et al., 2020; Man et al., 1996; Nikiforow et al., 2018; Ouyang et al., 2019; Pan et al., 2020; Przybylek and Gielniak, 2019; Soued et al., 2017; Sugeno and Tanaka, 1991; Yang, 2014; Zervoudakis and Tsafarakis, 2020).

In this paper we propose a very low cost generator structure for remote applications without active control unit and with a minimum number of sensors. A modular axial generator structure with permanent magnet reducing the cost of manufacture is chosen to generate renewable energy. The analytic method is chosen to conceive the permanent magnet generator seen its compatibility to optimization approaches. Indeed, it’s fast and product results quickly and without iterations. The generator mass and the extracted output power are considered as optimization criteria.

Classical models of the wind energy systems need an elevated simulation time, and thereafter are not compatible to constrained multi-objective optimization problem resolution with stochastic optimization software such as Genetic Algorithms.

The paper is organized as follows. In first part of the study, the architecture and the design method of the wind turbine are presented. In Sections 2, the DC mathematical model with reduced simulation time of the wind turbine is presented. Section 4 illustrates the validation process of the wind turbine DC model.

Wind turbine design process

Wind turbine structure

To optimize an energy conversion system, it is necessary to know the different parts of the system, from the source to the use. Our choice is based on a wind conversion chain consisting of a synchronous generator with permanent magnets with axial flux coupled to a diode rectifier, ensuring the storage of the energy recovered in batteries.

The synoptic diagram of our passive wind turbine system is illustrated by Figure 1. This approach aims to simplify the model for optimization.

Figure 1.

Synoptic of the wind turbine system.

Design process

Choice of the electric generator configuration

The studied generator configurations are the following:

Permanents magnets generator with trapezoidal waves-forms.

Permanents magnets generator with sinusoidal waves-forms.

Coiled rotor generator with trapezoidal waves-forms.

Coiled rotor generator with sinusoidal waves-forms.

Hybrid excitation generator with trapezoidal waves-forms.

Hybrid excitation generator with sinusoidal waves-forms.

These generators can be with axial or radial configurations. The axial configurations present the following advantages:

Opened and straight slots easy the manufacturing.

Possibility of increasing the power by adding additional modules on the generator axis.

Higher mass power to weight ratio than radial flux generators.

End coil smaller in size than radial flux generators leading to reduced copper losses.

Axial structures are compact and occupy little space than radial structures.

Permanent magnet generator structures are used for high current and power applications since magnets can significantly reduce magnetic armature reaction. On the other hand, the coiled rotor generators are applied for applications with low current and power since the magnetic armature reaction is large compared to the permanent magnet generators.

The generators having an excitation coil have an additional degree of freedom with respect to the motors only with permanent magnets.

Renewable energy system is an application with high power and current, in the consequence, an axial structure of generator with permanent magnet is chosen for this application.

Design simplifying assumptions

Simplified assumptions are taken into account to develop the dimensioning analytical model of the permanent magnets synchronous generator with axial flux are namely:

Magnitude of the stator currents very lower than the demagnetization currents of the magnets.

Magnetic reaction of the armature negligible, which amounts to saying that the stator currents are perfectly sinusoidal.

Negligible cogging torque.

The magnets are perfectly mounted on the surface.

The phase’s inductances are perfectly constant.

The inductances are considered negligible only for the design model and are considered for the control model.

Permeability of magnets close to that of air.

Permeability of infinite iron.

Invariance of the remanent induction of the magnets as a function of the temperature.

The phase’s resistances of the generator are considered equal, invariant of the temperature, and are calculated in hot steady state. .

Negligible leaks Flux.

Negligible leakage inductance in the coils heads.

Generator structure

The chosen generator is a permanent magnet generator with five pairs of poles and axial flux (Figure 2). This generator is designed by the analytical method. The analytical model is validated by the finite element method using Maxwell-2 D software (Elsisi, 2019a, 2019b; Elsisi et al., 2021).

Figure 2.

Generator structure.

The analytical model has as input the rated speed, battery maximal charging current, and magnitude of the stator phase’s currents.

Generator design

The wind energy power chain is designed by the analytical method validated by finite element simulations. Taking in account of the maximal current recharging batteries as main constraint fixed to its optimal value, the DC bus voltage recharging the batteries is regulated at the value corresponding to this value. At this functioning point, the generator phase’s currents are regulated by acting on the phase between generator phases induced electromotive forces and phases voltages delivered by the generator. The generator sizing current taking also as sizing constraint is fixed by the constructor notebook. The design process is based on study developed on (Ouyang et al., 2019). The analytical calculations are based on the following theorems:

Ampere Theorem.

Flux conservation theorem.

The iron Saturation phenomena take place for a flux density equal to 1.6 T deduced from iron B-H curve.

Theorem of flux superposition.

The generator is drawn according sizing parameters extracted from analytical model by the FEEM 2-D software and analyzed in to dimension to validate the sizing model. The results obtained by analytical model are near to those obtained by finite element method. The generator sizing model is coupled the models of all other power chain components. Finally, a high parameterized model of the wind turbine power system is developed. This model is compatible to stochastic optimization program with large scales.

The chosen design method is analytical one, since it is flexible to provide solutions based on the power requirement. This method is based on general theorems of design of electrical device as on simplifying assumptions justified (Koutroulis and Kalaitzakis, 2006; Neji et al., 2006; Nhidi et al., 2015).

If phase’s equivalent inductance is neglected, the equivalent schema of the assembly generator-rectifier-battery is simplified as show in Figure 3.

Figure 3.

Equivalent schema of the assembly generator-inverter-battery.

Where R is the generator phase’s current, R_b is the battery internal resistance, U_b is the battery internal voltage, U_r is the battery regulated voltage, I_b is the battery current, e_a, e_b, and e_c are respectively the induced electromotive forces of the phases a, b, and c, and i_a, i_b, and i_c are respectively the currents of the phases a, b, and c.

The dimensioning current of the generator (I_bmax) is calculated for recharge at maximal battery current regime.

From equivalent schema (Figure 3), the generator’s dimensioning current (I_bmax) can be expressed by the following relation:

I {\frac{U b_{rmax}}{R_{b}}}_{bmax}

(1)

Where I_bmax is the dimensioning current, U_rmax is the maximal battery recharge voltage.

The maximal value of the induced electromotive forces, considering that the battery recharge voltage (U_rmax) is filtered at its average voltage by a capacitor with high capacity, is expressed by equation (2).

E_{max} = \frac{π}{3} \times \frac{U_{b} + (2 \times R + R_{b}) \times I_{b max}}{\sqrt{3}}

(2)

Equation (3) is deduced from equations (1) and (2).

E_{max} = \frac{π}{3} \times \frac{U_{b} + (2 \times R + R_{b}) \times \frac{U_{r max} - U_{b}}{R_{b}}}{\sqrt{3}}

(3)

The maximal generator phase’s induced electromotive force value is expressed by the relation (4):

E_{max} = K_{e} \times Ω_{max}

(4)

Where K_e is the induced electromotive forces constant:

K_{e} = N_{sph} \times \frac{(D_{e}^{2} - D_{i}^{2})}{4} \times B_{e}

(5)

Where N_sph is number of turn per phase, D_e and D_i are respectively the external and internal diameters of the motor and B_e is the flux density in the Air-gap

The generator’s maximal angular speed (Ω_max) is deduced from equations (3) and (4):

Ω_{max} = \frac{\frac{π}{3} \times \frac{U_{b} + (2 \times R + R_{b}) \times \frac{U_{r max} - U_{b}}{R_{b}}}{\sqrt{3}}}{K_{e}}

(6)

Equation (7) is deduced from equations (5) and (6):

Ω_{max} = \frac{\frac{π}{3} \times \frac{U_{b} + (2 \times R + R_{b}) \times \frac{U_{r} - U_{b}}{R_{b}}}{\sqrt{3}}}{N_{sph} \times \frac{(D_{e}^{2} - D_{i}^{2})}{4} \times B_{e}}

(7)

The height of the teeth is expressed as follows:

H_{d} = \frac{N_{sph} \times I_{bmax}}{K_{f} \times δ \times L_{encr}}

(8)

Where K_f is the filling factor and δ copper admissible current density in coils.

The height of the rotor yoke is calculated by applying the flux conservation theorem for a maximum flux position in the cylinder head:

H_{cr} = \frac{B_{e}}{B_{cr}} \times \frac{s_{d}}{\frac{D_{e} - D_{i}}{2}}

(9)

Where B_cr is the flux density in the rotor yoke, B_e is the flux density in the Air-gap, and s_d is the section of the main tooth.

The height of magnets (H_a) necessary for a magnetic induction in the gap B_e is derived by applying the Ampere theorem on a closed contour at a tooth:

H_{a} = \frac{μ_{r} \times B_{e} \times e}{B_{r} - B_{e}}

(10)

Where B_r is the residual induction and µ_r is the relative permeability of magnets and e is the air-gap thickness.

The height of the stator yoke is calculated by applying the flow conservation theorem for a maximum flux position in the cylinder head:

H_{cs} = \frac{B_{e}}{B_{cs}} \times \frac{s_{d}}{\frac{D_{e} - D_{i}}{2}}

(11)

Where B_cs is the flux density in the stator yoke.

Electrical parameters of the electric generator

The expression of the magnitude of the induced back electromotive forces is as follow:

E = N_{sph} \times \frac{(D_{e}^{2} - D_{i}^{2})}{4} \times B_{e} \times Ω

(12)

Where Ω is the angular speed of the motor, and N_sph is the number of turns per phase.

Equation (13) leads to the general expression of back electromotive force:

E = K_{e} \times Ω

(13)

The inductance of the rotor winding is given by the following relation:

L = 2 \times μ_{0} \times (\frac{\frac{s_{d}}{2}}{2 \times (e + H_{a})} + \frac{(\frac{D_{e} - D_{i}}{2}) \times H_{d}}{2 \times L_{en c_{r}}}) \times {(\frac{N_{sph}}{2})}^{2}

(14)

The mutual inductance of the rotor winding is given by the following relation:

M = 2 \times μ_{0} \times (\frac{\frac{s_{d}}{2}}{2 \times (e + H_{a})}) \times {(\frac{N_{sph}}{2})}^{2}

(15)

The flux density in the air-gap due to the power of the stator winding by the demagnetization current of the magnet (I_d) is given by the following relation:

B_{ei} = \frac{μ_{0} \times \frac{N_{sph}}{2} \times I_{d}}{H_{a} + e}

(16)

The demagnetization of the magnets is provided when:

B_{e} - B_{ei} = B_{c}

(17)

Where B_c is the demagnetization flux density of the magnets.

The stator current does not exceed the demagnetization current to avoid demagnetization of the magnets. The demagnetization current is expressed by the following relation:

I_{d} = (\frac{H_{a} \times B_{r}}{H_{a} + μ_{r} \times e} - B_{c}) \times \frac{(H_{a} + e)}{μ_{0} \times \frac{N_{sph}}{2}}

(18)

Classical modeling of the wind turbine system

The Simulink model of the induced electromotive forces is illustrated in Figure 4.

Figure 4.

Simulink model of the induced electromotive forces.

The Simulink model of the assembly Generator-Rectifier-Battery is illustrated in Figure 5.

Figure 5.

Simulink model of the assembly Generator-Rectifier-Battery.

The Simulink model of motion equation is illustrated in Figure 6.

Figure 6.

Simulink model of the motion equation.

The Simulink model of the global classical wind energy system is illustrated in Figure 7.

Figure 7.

Simulink model of the global classical wind energy system.

For classical wind energy system model, the power generated by the wind and the power recovered by the batteries are illustrated by the Figure 8.

Figure 8.

Generated and recovered powers for classical wind energy model.

Figure 8 demonstrates that the power transferred to the batteries is lower than that developed by the wind, which is explained by the different losses of energy generation chain.

Conversion of the wind turbine AC model to an equivalent DC model

Mathematical model

For inductive load, the vector diagram of synchronous generator operation is shown in Figure 9.

Figure 9.

Vector diagram of synchronous generator.

Where V is the phase’s voltage vector, E is the electromotive force vector, I is the phase’s current vector, L is the phase’s inductance, M is the phase’s mutual inductance, Ω is the generator angular speed and p is the number of poles pairs.

For resistive load, the vector diagram becomes Figure 10:

Figure 10.

The equivalent vector diagram.

The equivalent schema of the assembly generator-inverter-battery is showed in Figure 11.

Figure 11.

Equivalent schema of the assembly generator-inverter-battery.

Figure 12 illustrates le paces of the phase’s currents of the generator.

Figure 12.

Paces of the currents i_a, i_b, i_c, i_a–i_b, i_b–i_c, i_a–i_c, i_b–i_a, i_c–i_b, and i_c–i_a_.

According to Figure 12, the average value of the battery recharging voltage is expressed by the following relation.

U_{dc} = \frac{1}{\frac{π}{3}} \int_{\frac{π}{3}}^{\frac{2 \times π}{3}} (e_{a} - e_{b} - R \times (i_{a} - i_{b}) - (L - M) \times (\frac{d i_{a}}{dt} - \frac{d i_{b}}{dt})) d θ

(19)

Equation (19) can be transformed to equation (20).

U_{dc} = \frac{1}{\frac{π}{6}} \int_{\frac{π}{6}}^{\frac{2 \times π}{6}} (e_{ab} - R \times (i_{a} - i_{b}) - (L - M) \times (\frac{d (i_{a} - i_{b})}{dt})) d θ

(20)

The generator’s phase’s i_a and i_b currents are expressed by relation (21) and (22).

i_{a} = I_{s} \times \sin (ω \times t - ϕ)

(21)

i_{b} = I_{s} \times \sin (ω \times t - \frac{2 \times π}{3} - ϕ)

(22)

The difference between the currents i_a and i_b can be expressed by relation (23).

i_{a} - i_{b} = I_{s} \times (\sin (ω \times t - ϕ) - \sin (ω \times t - \frac{2 \times π}{3} - ϕ))

(23)

Relation (23) can be transformed into relation (24).

i_{a} - i_{b} = \sqrt{3} \times I_{s} \times \sin (ω \times t + \frac{π}{6} - ϕ)

(24)

Considering equation (25).

I_{s} = I_{b}

(25)

the average value of the difference between currents i_a and i_b becomes:

\frac{ω}{\frac{π}{6}} \times \int_{\frac{π}{6 \times ω} + \frac{ϕ}{ω}}^{\frac{2 \times π}{6 \times ω} + \frac{ϕ}{ω}} (i_{a} - i_{b}) dt = \sqrt{3} \times I_{s} \times \frac{3}{π} = \sqrt{3} \times \frac{3}{π} \times I_{b}

(26)

The average value of the derivation of the difference between currents i_a and i_b can be calculated from relation (27):

\frac{ω}{\frac{π}{6}} \times \int_{\frac{π}{6 \times ω} + \frac{ϕ}{ω}}^{\frac{2 \times π}{6 \times ω} + \frac{ϕ}{ω}} \frac{d}{dt} (i_{a} - i_{b}) dt = I_{b} \times \frac{ω}{\frac{π}{6}} {[i_{a} - i_{b}]}_{\frac{π}{6 \times ω}}^{\frac{2 \times π}{6 \times ω}} = \frac{6}{π} \times ω \times I_{b} \times \frac{1}{2} \times \sqrt{3} = \frac{3}{π} \times \sqrt{3} \times ω \times I_{b}

(27)

In the same way as for the currents, the average value of the difference of the induced electromotive forces e_a and e_b (equation 34), is deduced from the relations (28)–(33).

e_{a} = \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times \cos (ω \times t - \frac{π}{2})

(28)

e_{b} = \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times \cos (ω \times t - \frac{2 \times π}{3} - \frac{π}{2})

(29)

e_{a} - e_{b} = \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times (\cos (ω \times t + \frac{π}{2}) - \cos (ω \times t + \frac{π}{2} - \frac{2 \times π}{3}))

(30)

e_{a} - e_{b} = \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times \cos (ω \times t + \frac{π}{2} + \frac{π}{6})

(31)

e_{ab} = e_{a} - e_{b} = - \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times \sin (ω \times t + \frac{π}{6})

(32)

e_{ab} = e_{a} - e_{b} = \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} \times \sin (ω \times t + \frac{π}{6} + π)

(33)

\frac{ω}{\frac{π}{6}} \times \int_{\frac{π}{6 \times ω} + \frac{π}{6 \times ω}}^{\frac{2 \times π}{6 \times ω} + \frac{π}{6 \times ω}} e_{ab} dt = \frac{ω}{\frac{π}{6}} {[e_{ab}]}_{\frac{π}{6 \times ω} - \frac{π}{6 \times ω} + \frac{π}{ω}}^{\frac{2 \times π}{6 \times ω} - \frac{π}{6 \times ω} + \frac{π}{ω}} = \frac{6}{π} \times \frac{1}{2} \times \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} = \frac{3}{π} \times \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p}

(34)

The average value of the battery recharging voltage (equation 36) is deduced from equations (26), (27), (34), and (35).

U_{dc} = \int_{\frac{π}{6 \times ω} + \frac{π}{6 \times ω}}^{\frac{2 \times π}{6 \times ω} + \frac{π}{6 \times ω}} e_{ab} dt - R \times \frac{ω}{\frac{π}{6}} \times \int_{\frac{π}{6 \times ω} + \frac{ϕ}{ω}}^{\frac{2 \times π}{6 \times ω} + \frac{ϕ}{ω}} (i_{a} - i_{b}) dt - (L - M) \times \frac{ω}{\frac{π}{6}} \times \int_{\frac{π}{6 \times ω} + \frac{ϕ}{ω}}^{\frac{2 \times π}{6 \times ω} + \frac{ϕ}{ω}} \frac{d}{dt} (i_{a} - i_{b}) dt

(35)

U_{dc} = \frac{3}{π} \times \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} - R \times \frac{3}{π} \times \sqrt{3} \times I_{b} - (L - M) \times \frac{3}{π} \times \sqrt{3} \times ω \times I_{b}

(36)

Equation (36) can be simplified into equation (37).

U_{dc} = E_{dc} - R_{dc 1} \times I_{b} - R_{dc 2} \times I_{b} = E_{dc} - (R_{dc 1} + R_{dc 2} (ω)) \times I_{b}

(37)

Where:

U_{dc} = \frac{3}{π} \times \sqrt{3} \times \frac{2}{3} \times K_{e} \times \frac{ω}{p} - R \times \frac{3}{π} \times \sqrt{3} \times I_{b} - (L - M) \times \frac{3}{π} \times \sqrt{3} \times ω \times I_{b}

(38)

R_{dc 1} = R \times \frac{3}{π} \times \sqrt{3}

(39)

R_{dc 1} = R \times \frac{3}{π} \times \sqrt{3}

(40)

ω = p \times Ω

(41)

The electromagnetic torque is expressed by relation (42).

T_{em} = K_{e} \times I_{b} \times \cos (ϕ)

(42)

The angle φ is deduced from vector diagram.

ϕ = A \tan (\frac{(L - M) \times p \times Ω \times I_{b}}{(\frac{U_{dc}}{\sqrt{3}} \times \frac{π}{3}) + (R + R_{d}) \times I_{b}})

(43)

DC Model Implementation

The wind turbine DC model is implemented under Matlab–Simulink Simulation environment according to Figures 13 to 15.

Figure 13.

Average model of the induced electromotive force.

Figure 14.

Average model of the generator-rectifier-battery-assembly.

Figure 15.

Global DC average model of the wind turbine.

Formulation of the optimal integrated design problem

The continuous useful power is obtained from the wind power P_eol and all the losses dP of the system according to the power balance represented by Figure 16.

Figure 16.

Power assessment in the passive wind system.

The coupling of power chain losses model and the model of the generator mass to the program dimensioning the generator, pose an optimization problem. This last is solved by the software of optimization based on the Genetic Algorithm method.

We are conducting a design approach by optimization using a multi-criteria genetic algorithm, pursuing as a first objective the increase of the energy efficiency of the wind chain during a wind cycle. It should be noted that maximizing the useful power implies the maximization of the extracted wind power as well as the minimization of all losses (dP) in the system. We also integrate as a second objective the minimization of the mass of the generator (M_g), which on the one hand reduces costs and on the other hand limits the amount of suspended mass.

Thus, the function to optimize is expressed by the following expression:

F_{o} = M_{g} + a \times dp

(44)

Where a is a coefficient fixing the influence degree of dP at the global objective function compared to M_g. Indeed, a brings closer the value of (a × dP) to the value of M_g.

The optimization problem consists on the determination of the generator sizes optimizing F_o while respecting all the constraints of design.

The optimization parameters are found from the analytical modeling of the losses and the generator mass.

The generator’s weight is expressed as follows

M_{g} = M_{cs} + M_{ds} + M_{cr} + M_{a} + M_{cu}

(45)

Where M_cs is the weight of stator yoke M_ds is the weight of the stator tooth M_cu is the weight of the copper M_cr is the weight of the rotor yoke and M_a is the weight of magnets.

The iron losses are expressed by the following relation:

P_{iron} = q \times {(\frac{f}{50})}^{\frac{3}{2}} \times (M_{cs} \times B_{cs}^{2} + M_{ds} \times B_{d}^{2})

(46)

Wherein c is the core loss, f is the motor supplying frequency, M_ds is the teeth weight, M_cs is the stator yoke weight, B_d is the air-gap flux density, and B_cs is the flux density in stator yoke.

The mechanical losses are expressed by the following relation:

T_{mec} = s + k \times Ω + γ \times Ω

(47)

Where s is a dry friction coefficient, k is a viscous friction coefficient and γ is a fluid coefficient of friction, and Ω is the angular speed of the electric generator.

The optimization process is illustrated by Figure 17.

Figure 17.

Optimization process.

The F_o model is coupled to a program of optimization by the method of the genetic algorithm. The progress of the program of optimization of the F_o with constraints is described by the organization diagram (Figure 18).

Figure 18.

Progress of the optimization program.

Simulation results and discussions

Simulations of the power chain model are realized according the data extracted from wind turbine analytical model as shown in Table 1.

Table 1.

Simulation parameters.

Parameters	Values	Units
Battery nominal voltage (U_dc)	$\sqrt{2} \times 220$	Volts
Internal resistance of the battery (R_b).	0.02	Ω
Transient regime battery resistance (R_t).	0.001	Ω
Transient regime battery capacitor (C_t).	0.1	Farad
Diodes internal resistance (R_d)	0.001	Ω
Generator electric constant (K_e).	0.4	Volt/(rad/s)
Generator phase’s resistance (R).	88.2 10⁻³	Ω
Generator phase’s inductance (L).	0.02213	H
Generator phase’s mutual inductance (M).	0.00213	mH
Gear ratio (r_d).	8	/
Battery rated optimal angular speed (Ω_m).	400	rad/s
Battery rated optimal recharging voltage (U_b).	320	Volts
Battery rated optimal recharging current (I_b).	200	A

Figure 19 illustrate the evolutions of the current and voltage of the phase a for classical AC model. Figure 19 demonstrates the shift angle between the current and the voltage of the phase “a” is equal to zero which validates the accuracy of the vector diagram presented on Figure 9.

Figure 19.

Evolutions of the current and voltage of the phase a in versus time for classical AC model.

Figures 20 to 24 demonstrates that the evolution of the generator’s angular speed, electromagnetic torque, turbine resulting torque, battery charging voltage, and battery charging voltage are similar for simulation by classical AC model and developed DC model. In conclusion, simulation result validates the accuracy of the developed DC model.

Figure 20.

Evolutions of the generator’s angular speed in versus time for the two cases.

Figure 21.

Evolutions of the generator’s electromagnetic torque in versus time for the two cases.

Figure 22.

Evolutions of the turbine resulting torque in versus time for the two cases.

Figure 23.

Evolutions of the battery recharging voltage in versus time for the two cases.

Figure 24.

Evolutions of the battery recharging current in versus time for the two cases.

Figure 25 illustrates the evolutions of the shift angle between currents and electromotive forces and power factor in versus time for DC model.

Figure 25.

Evolutions of the shift angle between currents and electromotive forces and power factor in versus time for DC model.

Conclusion

The study presented in this paper illustrates an AC model conversion of wind energy system to a DC model with reduced simulation time, for possible integration to optimization software with larges scales permitting a multi-objective optimization, such as the constrained optimization conjointly of the cost and power losses of the wind turbine energy system. In this context, an analytical design model of the permanent magnets generator is developed according simplifying assumptions. In addition, a DC model based on average calculation of the DC voltage recharging the battery energy accumulator used for recovering the converted wind energy and the electromagnetic torque is developed. This model is validated by comparison to Simulink wind turbine classical model. Indeed, simulation results are superposed, and thereafter valid entirely the wind turbine system DC model

As perspective it is interesting to use a booster chopper as an interface between the rectifier and the battery to optimize the recovered energy. An average model of the booster chopper can be integrated into the DC model for performance improvement of the conversion chain.

Other interesting perspectives are as follow:

Optimization of the structure and parameters of the speed controller.

Industrialization of the global system.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Souhir Tounsi

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