Average model of wind energy system dedicated to optimal design of the global system

Abstract

This paper presents an average model based on justified simplifications dedicated to the design and optimization of wind energy systems. Indeed, the classic models of wind energy systems are complex and their use is not efficient for the optimal design of the components of the power chain given the complexity, the significant time of resolution and the strong correlation of the physical parameters of these models. For these reasons, a model based on the theory of average values with reduced simulation time of a wind turbine structure is developed. This model is validated against the classic model of the wind chain using the SimPowerSystem library of power component models integrated under the Matlab-Simulink simulation environment.

Keywords

Average model wind energy system classical model validation design optimization

Introduction

The optimal design of electrical systems is a complex problem given the complexity of their models. The main constraint for the optimal design of an electrical system is the formulation of the quantities to be optimized according to the geometric and physical quantities of manufacture of their components. Two types of models exist in the literature are namely:

Control models: These models are precise and complex, and are usually intended for integration into electronic control boards of the system under study. The use of these models to design the studied system is not very profitable given the possibility of divergence of the model and the important simulation time.

Analytical or average models: These models offer good precision and can be integrated into high-dimensional design approaches since they have a reduced simulation time. These models are much more cost-effective than control models for the design of electrical systems since they provide a local and global view of the system under study and have a reduced simulation time. They can then be integrated into large-scale stochastic optimization algorithms.

In particular, wind power generation systems are part of complex electrical systems. They have several energy conversion stages.

In Singh and Sundaram (2022), an interesting study showing the performance of systemic design approaches for a better interconnection of components, since these methods allows a vision of the electromechanical behavior of components with the possibility of intervention to improve the performance of the overall system. The article (Singh and Sundaram, 2021) presents a solution to improve the performance of the wind turbine system and at the same time making it a commercially attractive choice. This can be achieved by reducing generator windings and bearings operating temperature, and also reducing its cost and weight at the same time. The proposed solution to reduce overheating is achieved and verified through results. The achieved results also explain how one solution is advantageous over the other. Singh et al. (2021) illustrates the interest of the systemic design in the significant improvement of the overall performance of a system to be studied. Singh et al. (2019a) presents a Reduced Ingress Protection Class & Proper Filter Selection for Open Ventilated (IC3A1) Wind Turbine Generator demonstrating the importance of wind turbines components heating in the significant improvement of their efficiency and lifetime. Singh et al. (2019b) illustrates the impact of imbalanced wind turbine generator cooling on reliability. Studies presented in Singh and Sundaram (2022), Singh and Sundaram (2021), Singh et al. (2019a, 2019b, 2021) are interesting but can’t be used directly to solve the wind turbine global system efficiency under several constraints solved classically by stochastic method.

In Singh et al. (2019c) an interesting study consists in combining an analytical model of a synchronous magnet generator to a simple algorithmic model with reduced simulation time of a three-phase rectifier PD3 shows the interest of this model for the optimal design of the wind energy system. This model is very accurate compared to the models presented in the literature. In Tounsi (2022b), a conversion model of an alternative model of a wind turbine to a continuous equivalent model also with reduced simulation time is presented and also shows the interest of this model for the optimal design of the wind energy system. These models are with comparative accuracy, but they differ in simulation time and in the degree of compatibility with optimization algorithms. In Tounsi (2021a), a behavioral model of a wind turbine is presented. This model is based on the analytical modeling of a permanent magnet synchronous generator. This model is less flexible than the other models.

In this context, this study concerns the development of an average model of a wind turbine based on the average value theory of a wind turbine combined with an analytical model of a permanent magnet generator. This model is precise and has a shorter simulation time than the models mentioned above. The performance of this model is validated against conventional models developed under the Matlab-Simulink simulation environment. The developed model allows a local and global vision of the system and provides component manufacturing data in a reduced time and without iterations, and it can be coupled with multi-objective and high-dimensional optimization algorithms. The analytical model of the synchronous generator is based on a sizing study of this type of generator presented in Tounsi (2015b).

In this context, this paper presents essentially a comparative study of the developed average model and classical model for developed model validation:

Wind turbine structure

This paragraph attempts to describe the architecture adopted in the rest of this work. In order to minimize the system cost and to maximize its reliability, the full passive wind energy conversion system presented in Figure 1 is put forward. It is a very simple solution. For this architecture, the wind kinetic energy is captured by a three-bladed horizontal axis wind turbine. This type of propeller has taken over those with vertical axis because it represents a lower cost, it is less exposed to mechanical stresses, and the three-blade rotor is a compromise between the power coefficient, cost, and the wind sensor rotational speed. For these reasons our choice feels on the three-bladed horizontal axis propeller driving a permanent magnet synchronous generator with axial flow. Such a generator structure allows stacking to increase the power supplied and achieves significant current values as the energetic armature reaction is reduced by the magnetic effect of the magnets. In addition, this structure is with reduced production cost, since it has a right and open slot easy to make, the winding is concentrated which allows their insertion in a single block. The three-phase current of the generator is rectified by a diode bridge rectifier which is coupled directly on a battery accumulator.

Figure 1.

Structure of the wind turbine system.

By eliminating the control part, we greatly reduce the complexity and thus the cost of the system. In addition, several additional devices of the active systems are removed (sensors, regulators). This approach aims to simplify the model for optimization.

Simplifying assumptions of the study

This study is developed based on the following simplifying assumptions:

The temperature in the different parts of the generator is regulated by a cooling system.

The temperature in the diodes is regulated by a cooling system.

Diodes are equivalent in conduction to a resistor in series with the threshold voltage.

The spontaneous switching of the diodes is considered fast (switching time considered zero).

The equivalent resistances of the diodes and the phases of the generator are calculated at the setting temperature of the cooling systems.

The generator operates in linear mode (saturation not reached).

Leakage flux in the generator negligible.

Amplitude of the phase’s currents lower than that of the magnet demagnetization current.

Negligible magnetic field in the iron of the generator yoke.

Magnetization of the magnets perfectly parallel to the axis of the generator (axial magnetization).

The battery is equivalent to a resistance in series with a constant internal voltage.

The electromotive forces of the generator are considered perfectly sinusoidal.

The generator currents are considered perfectly sinusoidal.

Classical model of the wind energy system under Simulink

The three induced electromotive forces e_a, e_b, and e_c are expressed by the following three equations (Tounsi, 2015a, 2022a, 2021b):

e_{a} = \frac{2}{3} \times K_{e} \times Ω_{g} \times \cos (p \times Ω \times t + \frac{π}{2})

(1)

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

(2)

e_{c} = \frac{2}{3} \times K_{e} \times Ω_{g} \times \cos (p \times Ω \times t + \frac{π}{2} - \frac{4 \times π}{3})

(3)

where K_e is the electric constant of the generator, Ω_g is the generator’s angular speed, and p is the pole pairs number.

The Simulink model of the induced electromotive forces is illustrated in Figure 2.

Figure 2.

Simulink model of the induced electromotive forces.

Each phase of the generator is equivalent to a resistor in series with an inductance and an induced electromotive force. The generator three phase’s model is described by the following equations (Tounsi, 2015a, 2022a, 2021b):

e_{a} = v_{a} + R \times i_{a} + (L - M) \times \frac{d i_{a}}{dt}

(4)

e_{b} = v_{b} + R \times i_{b} + (L - M) \times \frac{d i_{b}}{dt}

(5)

e_{c} = v_{c} + R \times i_{c} + (L - M) \times \frac{d i_{c}}{dt}

(6)

where R, L, and M are respectively the resistance, the inductance, and the mutual inductance of the phases a, b, and c, i_a,b,c, e_a,b,c, and v_a,b,c are respectively the currents, the induced electromotive forces, and the voltages of the phase a, b, and c.

The electromagnetic torque is calculated from equation (7).

T_{em} = - \frac{1}{(Ω)} \times (e_{a} \times i_{a} + e_{b} \times i_{b} + e_{c} \times i_{c})

(7)

The Simulink model of the assembly Generator-Rectifier-Battery is illustrated in Figure 3.

Figure 3.

Simulink model of the assembly Generator-Rectifier-Battery.

The Simulink model of motion equation is illustrated in Figure 4.

Figure 4.

Simulink model of motion equation.

The global model of wind energy system connecting subsystems model illustrated in Figure 2 to 4 is illustrated in Figure 5.

Figure 5.

Classical wind energy system model.

Average model of wind energy system

The equivalent schema of the assembly generator-inverter-battery is showed in Figure 6.

Figure 6.

Equivalent schema of the assembly generator-inverter-battery.

For battery load (battery equivalent to an internal resistance in series with a continuous internal voltage), the vector diagram is illustrating in Figure 7:

Figure 7.

The equivalent vector diagram.

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, and M is the phase’s mutual inductance.

Equation (8) is deduced from vector diagram (Figure 7).

{(V + R \times I)}^{2} + {((L - M) \times Ω_{g} \times p \times I)}^{2} = E^{2}

(8)

Equation (8) can be transformed to equation (9).

V = {\sqrt{E^{2} - {((L - M) \times Ω_{g} \times p \times I)}^{2}}}^{2} - R \times I

(9)

The average value of phase 1–2 voltage (u₁₂) is expressed by ration (10).

u_{a} = \frac{1}{T} \times \int_{0}^{T} u_{12} \times dt = \sqrt{3} \times \frac{3}{π} \times V

(10)

From equations (9) and (10) is deduced the expression of the dc bus voltage recharging the battery (U_dc).

U_{d c} = ({\sqrt{E^{2} - {((L - M) \times Ω_{g} \times p \times I)}^{2}}}^{2} - (R + R_{d}) \times I) \times \sqrt{3} \times \frac{3}{π} + 2 \times V_{d}

(11)

where V_d and R_d are respectively the infernal voltage and resistance of diodes.

The generator phase’s currents magnitude is expressed by relation (12).

I = \frac{π}{3} \times I_{b}

(12)

where I_b is the battery recharging current.

From equations (11) and (12) is deduced the relation (13).

U_{d c} = ({\sqrt{E^{2} - {((L - M) \times Ω_{g} \times p \times I_{b} \times \frac{π}{3})}^{2}}}^{2} - (R + R_{d}) \times I_{b} \times \frac{π}{3}) \times \sqrt{3} \times \frac{3}{π} + 2 \times V_{d}

(13)

The electromotive force magnitude is epressed by the relation (14).

E = \frac{2}{3} \times K_{e} \times Ω_{g}

(14)

From equations (13) and (14) is deduced the relation (15).

U_{d c} = ({\sqrt{{(\frac{2}{3} \times K_{e} \times Ω_{g})}^{2} - {((L - M) \times Ω_{g} \times p \times I_{b} \times \frac{π}{3})}^{2}}}^{2} - (R + R_{d}) \times I_{b} \times \frac{π}{3}) \times \sqrt{3} \times \frac{3}{π} + 2 \times V_{d}

(15)

I_{b} = \frac{U_{d c} - E_{b}}{R_{b}}

(16)

Where E_b is the internal voltage of the battery.

The recovered energy (W_r) is estimated by relation (17).

W_{r} = \int E_{b} \times I_{b} \times dt

(17)

The power factor is estimated by relation (18).

φ = t g^{- 1} (\frac{L \times Ω_{g} \times p \times I_{b}}{\frac{(U_{d c} - 2 \times V_{d})}{\sqrt{3} \times \frac{π}{3}} + (R + R_{d}) \times I_{b}})

(18)

The power system average model is illustrated by Figure 8.

Figure 8.

Simulink power system average model.

The global power system average model is illustrated by Figure 9.

Figure 9.

Simulink global power system average model.

Simulation results

Simulation of the power chain model for the two cases of models are realized according the data extracted from wind turbine analytical model as shown in Table 1.

Table1.

Simulation parameters.

Parameters	Values	Units
Battery nominal voltage (U_dc)	40	Volts
Internal resistance of the battery (R_b)	0.02	Ω
Diodes internal resistane (R_d)	0.001	Ω
Generator electric constant (K_e)	24	Volt/(rad/s)
Generator phase’s resistance (R)	8.8200 × 10⁻⁴	Ω
Generator phase’s inductance equivalent (L)	0.0014123	H
Generator phase’s mutual inductance (M)	0.0000123	mH
Gear ratio (r_d)	2	/

Figure 10 illustrates the evolution in versus time of the wind speed. This characteristic presents areas of high acceleration and overspeed to take into account the limits of the turbine speed and the dimensioning current of the power chain.

Figure 10.

Wind speed.

Figure 11 illustrates the evolution in versus time of the power factor for the two cases of models. Figure 11 demonstrates that the two results are close, which fully validates the conducted study.

Figure 11.

Power factor.

Figure 12 illustrates the evolution in versus time of electromagnetic torque for the two cases of models. Figure 12 demonstrates that the two results are close, which fully validates the developed study.

Figure 12.

Electromagnetic torque.

Figure 13 illustrates the evolution in versus time of recharging voltage of the battery. Figure 13 also demonstrates that the two results are close, which fully validates the developed study.

Figure 13.

Recharging voltage of the battery.

Figure 14 illustrates the evolution in versus time of the recharging current of the battery. Figure 14 also demonstrates that the two results are close, which fully validates the developed study.

Figure 14.

Recharging current of the battery.

Figure 15 illustrates the evolution in versus time recovered energy. Figure 15 also demonstrates that the two results are close, which fully validates the developed study.

Figure 15.

Recovered energy.

Conclusion

This paper presents an average model based on justified simplifications dedicated to the design and optimization of wind energy systems for robust and optimal design of wind turbine. Indeed, the classic models of wind energy systems are complex and their use is not efficient for the optimal design of the components of the power chain given the complexity, the significant time of resolution, the strong correlation of the physical parameters of these models. For these reasons, a model based on the theory of average values with reduced simulation time of a wind turbine structure is developed. This model is validated against the classic model of the wind chain using the SimPowerSystem library of power component models integrated under the Matlab-Simulink simulation environment. Simulation results valid the full study.

As a perspective it will be interesting to solve the problem of optimal design of the turbine by the developed average model using stochastic optimization algorithms.

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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