The Behavior of Wind Turbines Equipped with Induction Generators and Stator Converters Under Significant Variations in Wind Speed

Abstract

This study investigates the performance of medium-power wind turbines (within kilowatt range) in response to substantial fluctuations in wind speed. The wind turbines utilize induction generators that have a short-circuited rotor and are controlled by a power converter within the stator circuit. This configuration facilitates the adjustment of the stator frequency, thereby allowing the desired rotational speed to be achieved and guaranteeing that the turbine operates at the maximum power point (MPP). Specific mathematical models for the turbine and generator have been developed using technical data from an operational wind turbine. The study demonstrated that utilizing a power converter within the stator circuit enhances the turbine’s operation at its maximum power point. A crucial aspect of effective MPP operation is the accurate determination of the relationship between wind speed and the corresponding optimal angular mechanical speed. Precise understanding and implementation of the interdependence among the primary generator parameters—namely power, frequency, current, and power factor—in relation to wind speed is essential for maximizing power generation and achieving grid stability for wind turbines operating in variable wind speed.

1. Introduction

To achieve sustainable development goals, it is essential that green energy constitute an increasing share of gross energy consumption. Consequently, in accordance with the latest European directives, it has been established that by 2023, the proportion of renewable sources in gross energy consumption should be at least 42.5%, with a target of reaching 45% [1]. Figure 1 illustrates the variation in the primary sources contributing to electricity production [2]. The analysis reveals that in the year 2000, the energy generated from solid fuels was 800,340 GWh, whereas in 2023 it decreased to 321,498 GWh, representing 40.17% of the initial value. Concurrently, the share of renewable sources has increased significantly, from 406,997 GWh in 2000 to 1,213,388 GWh in 2023, indicating an approximate growth of 33 times compared to the value in 2000 [2].

Regarding the development of installed capacity in wind power plants across Europe, the period from 2013 to 2022 shows a doubling of the installed power (Figure 2) [3].

Regarding the utilization of renewable energy, several challenges have been identified that are relevant to this study, as noted in [4]:

-

The high cost of the initial investment, particularly in cases where there are no government subsidies for the large-scale implementation of these solutions;

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The shortage of qualified human resources capable of installing, operating, and maintaining electricity generation systems;

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The inadequate communication among the parties involved regarding the benefits associated with the implementation of renewable energy generation systems;

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The low reliability of transport networks, which prevents multiple individuals or legal entities from injecting any power into the grid.

Recently, an increasing trend has been identified regarding the integration of wind systems in the production of electricity [5]. This fact is also determined by the consideration that the amortization period of the investment is an acceptable one. Thus, following new studies, it was established that for a wind farm with a power of 105 MW, investment is amortized in 9.1 years, so this sector is in continuous development due to the benefits brought in the long term [6].

Regarding the integration of wind turbines into electricity production systems, it is essential that this process also ensures the stability of the electricity transport/distribution system. To verify the fulfillment of this requirement, prior simulations are necessary, utilizing dedicated programs to determine the impact of renewable sources on the stationery and dynamic stability of the system [7].

It is well known that wind turbines are designed and selected to operate under specific weather conditions and mechanical stress requirements. During operation, these parameters may vary, leading to the conclusion that a continuous correlation between wind speed and optimal angular mechanical speed is essential for delivering maximum power [8,9,10].

To measure or calculate wind speed, various methods are employed, such as

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The use of a small power turbine operating at maximum power in free air [11];

-

Reversing the aerodynamic model of the turbine based on the estimated values of the turbine’s torque and rotor speed [12];

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Estimating the rotor speed of the turbine and its torque, utilizing the inverse aerodynamic model and nonlinear control theory [13].

The operation of wind turbines encounters the issue of unpredictable variability in wind speed, impacting both the output power and the structural stability of the turbine system [14]. To manage the output power and, by extension, the wind turbines at varying wind speeds, different control algorithms and methods are employed, such as

-

The implementation of an advanced torque controller, which relies on an efficient estimation of wind speed, has been shown to enhance the accuracy of wind speed estimation by 2–7% compared to alternative methods [15];

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The application of a PID controller, with results assessed through performance index parameters in the time domain [16];

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The use of a controller that, based on the power curve corresponding to various wind speeds, aims to track the optimal power curve identified by the control system [17];

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The application of modified virtual inertial controllers, which also serve to enhance the damping capacity of power oscillations within the energy system [18];

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The application of vector control oriented by stator flux, achieving a reduction in the generator’s impact on system stability [19].

In certain instances, various dispatch commands for the wind generator are utilized, enabling control of the generator across different wind speed profiles [20], particularly in systems that also allow energy storage [21].

In the case of isolated networks, relying solely on green energy creates challenges regarding stability and frequency control. Following a simulation analysis of a hybrid installation composed of hydropower units, a pumping station, and a wind farm, it was determined that the implementation of a battery storage system would enhance frequency control and ensure the operational security of the electricity generation system [22,23]. Another challenge that must be considered in the operation of wind turbines involves controlling the speed during sudden discharges, which can lead to the rotor accelerating beyond the established limits [24].

There are situations in which wind farms are utilized for secondary frequency regulation [25]. In such cases, it has been observed that the use of energy for frequency regulation generates significantly higher revenue compared to merely supplying electrical energy [26].

This study addresses several significant issues in wind energy, especially concerning variable wind speeds over time: determining the mathematical model for the turbine and generator based on the technical data of the wind system; computing the values of stator frequency, current, and power factor based on wind speed; based on numerical simulations, the analysis highlights notable differences in wind energy capture, particularly at constant and variable generator power under conditions of significant temporal variations in wind speed.

In contrast to the existing research, this paper places particular emphasis on cases where wind speed fluctuates significantly over time, a topic that has been inadequately addressed in the literature, which predominantly focuses on constant wind speeds. The simulations conducted highlight the effects of temporal variations in wind speed.

The primary objectives pursued in this work are

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Presentation of the current state of research concerning the behavior of wind turbines under significant variations in wind speed;

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Determining the potential power output based on the characteristics of the wind turbine and the power curve as a function of angular velocity;

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Determining the power output of the induction generator and constructing the power characteristic as a function of angular velocity, based on its rated values;

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Establishing the optimal angular mechanical speed by utilizing the mathematical model of the turbine based on its technical parameters, specifically the wind speed;

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Determining the maximum value of the power supplied by the wind turbine.

The primary objective is to visualize the essential elements—power, energy, and currents—as they relate to the variable speed of the wind.

2. Properties and Characteristics of the Analyzed Wind Turbine

This paper examines the operation of the wind turbine, WT, which is equipped with a squirrel-cage induction generator, under substantial variations in wind speed [16,22]. The technical characteristics of the turbines are rated power at 10 m/s—55 kW, wind speed—2.5 m/s to 25 m/s, survival wind speed—59.5 m/s, no of blades—3, diameter of rotor—20 m; swept area—314 m, and tower height—18 m [27]. We consider that the wind speed varies within the range of (0 to 120) km/h. It is known that at low wind speeds, specifically below 3 m/s, wind turbines remain stationary, with a rotational speed of zero [8,26]. They become operational at speeds exceeding 3 m/s, and within the range of (3 to 10) km/h, they operate at maximum power points (MPPs) (Figure 3) [28]. In certain studies, an asynchronous generator equipped with a dual AC-DC-AC power converter has been analyzed, which, through its control methodology, enables the maximization of power output across a wide range of wind speeds [29].

The turbine’s power, WT, depends on the wind speed, V, and the MAS, ω, whereas the power of the electric generator, EG, is influenced by the stator voltage, U_S, along with the mechanical angular speed, ω. Additionally, it is essential to consider the altitude at which the turbines are installed, as a lower air density results in a decrease in the power output [30].

For the purpose of analysis, technical data pertaining to a 55 kW wind turbine was examined [31]. The power characteristics of the turbine at two wind speeds, V₁ = 3 m/s and V₂ = 10 m/s, are illustrated in Figure 3.

The fluctuation in wind speed, represented within these parameters, is highlighted in green within the operational zone. The turbine’s maximum power characteristic, P_WT-MAX(ω), intersects the maximum power points, MPP 1, at a wind speed of V₁ = 3 m/s, and MPP 2, at a wind speed of V₂ = 10 m/s.

For the analyzed case, the power characteristic at V₁ = 3 m/s is defined by the relationship

$$P_{WT-1}\left(\omega ,3\right)=19182\left({\displaystyle \frac{3}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.63\cdot {\displaystyle \frac{3}{\omega }}}\cdot 3^3 [\mathrm{W}]$$

(1)

and for V₂ = 10 m/s by the relationship

$$P_{WT-2}\left(\omega ,10\right)=19182\left({\displaystyle \frac{10}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.63\cdot {\displaystyle \frac{10}{\omega }}}\cdot {10}^3 [\mathrm{W}]$$

(2)

The maximum power characteristic developed by the turbine is

$$P_{WT-MAX}\left(\omega \right)=1.6744\cdot {10}^{-3}\cdot {\omega }^3 [\mathrm{W}]$$

(3)

Fixed-blade wind turbines are protected against strong winds (over 25 m/s) by several methods, because they do not have the ability to adjust the pitch of the blades to reduce speed.

When wind speeds exceed 10 m/s, to prevent damage to the structural integrity of the wind system, it is necessary to engage the turbine rotor’s braking mechanism using a furling system. This allows the turbine to automatically turn away from the direction of strong winds, thereby reducing the force exerted on the blades and limiting the generator’s rotational speed. As the turbine blades are no longer perpendicular to the airflow, the rotational speed decreases and the load on the generator diminishes, thereby safeguarding the turbine from overloading. In the absence of a furling system, a mechanical brake will be employed at high wind speeds to keep the rotor’s speed within the nominal operating limits [12,13].

Consequently, the speed of the generator must be adjusted in accordance with the ratio 1/3.3.

To achieve optimal energy performance within this speed range, adjustments are made to the voltage and frequency at the generator by utilizing power converters [32,33] that are connected to the stator of induction generators with a short-circuited rotor or to the rotor of induction generators equipped with slip rings.

Induction generators with a wound rotor can utilize two converters: one for the stator winding and the other for the rotor winding. In this scenario, the costs are high, so, generally, this solution is not justified.

In [34,35], the analysis of induction generators with a wound rotor was conducted in detail, utilizing a power converter in the rotor circuit to achieve the MPP operation of the turbine. In this scenario, the voltage at the rings exceeds twice the nominal stator voltage at a wind speed of 10 m/s, thus making the specially designed rotor more expensive compared to the conventional type.

The variations in wind speed cause the wind turbine to operate in a dynamic regime and at variable speeds [8,26]. Given that the process is dynamic, the equation for kinetic momentum is utilized, in the form [20,21,22].

$$J\cdot {\displaystyle \frac{d\omega }{dt}}=M_{WT}-M_{EG}$$

(4)

where ω is the MAS at the generator shaft, J is the equivalent inertia moment, dω/dt is the MAS time derivative, M_WT is the moment provided by WT, relative to the EG shaft, and M_EG is the electromagnetic moment at the EG shaft.

Multiplying with ω, we obtain the power equation [36,37].

$$J\cdot {\displaystyle \frac{d\omega }{dt}}\cdot \omega =P_{WT}-P_{EG}$$

(5)

where P_WT is the useful power provided by WT, referenced to the electric generator shaft, and P_EG is the electromagnetic power of EG at the shaft.

The WT functions at the convergence of the power characteristics of the EG and that of the turbine, P_WT(ω).

The maximum wind energy is harnessed when the wind turbine (WT) functions at its maximum power points (MPPs) [38]. In this regard, the power characteristic of the electric generator, P_EG(ω), has to correspond to the maximum power curve generated by the turbines (Figure 3) [39].

The electric generator power characteristic, P_EG(ω), is influenced by the power values related from the stator via the power converter located between the generator and the grid.

In the simulations provided, the rotational speeds of wind turbines that are fitted with a gearbox are related to the shaft of the electric generator. The rotational speeds at the turbine shaft are k_T times less; k_T represents the transmission multiplication ratio.

3. Mathematical Models of the Induction Generator and the Three-Blade Horizontal Axis Turbine

The control of wind systems relies on mathematical models of the induction generator and the turbine [40].

3.1. The Mathematical Model of the Induction Generator

An analysis was conducted on wind turbines equipped with induction generators with the following nominal data [1,31,41]:

• The moment of inertia, J = 0.34 kgm²;
• The nominal power P_N = P_EG = 55 kW;
• The rated voltage U_N = 400 V;
• The nominal current I_N = 98 A;
• The number of poles 2p₁ = 2;
• Nominal speed n_N = 2940 rpm;
• The nominal sliding s_N = 0.02.

The parameters of the generator are established based on the nominal data, at the nominal frequency f_N = 50 Hz. The nominal operating slip is

$$s_N={\displaystyle \frac{n_1-n}{n_1}}={\displaystyle \frac{3000-2940}{3000}}=0.02$$

(6)

n₁ represents the rotational speed of the spinning field, while n is the rotor’s rotational speed.

The rotor resistance, denoted as R_R, is determined from the power balance equation

$$s_N\cdot P_N=3\cdot R_R\cdot I_N^2$$

(7)

or

$$0.02\cdot 55000=3\cdot R_R\cdot {98}^2$$

(8)

with the solution R_R = 3.8179 × 10⁻² Ω.

The stator resistance is equal to the rotor resistance: R_S = R_R = 3.8179 × 10⁻² Ω.

The short-circuit reactance, X_sc, is derived from the nominal current, I_N, as follows:

$$I_N={\displaystyle \frac{230}{\sqrt{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{0.02}\right)}^2+{X_{sc}}^2}}}$$

(9)

or

$$98={\displaystyle \frac{230}{\sqrt{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{0.02}\right)}^2+{X_{sc}}^2}}}$$

(10)

with the solution X_sc = 1.31 Ω.

The rotor parameters R_R and X_R are minimized at the stator in the following manner:

$$R_R=k^2\cdot R_{R-REAL};R_R=k^2\cdot X_{R-REAL}$$

(11)

where k = N₁/N_2, N₁ is the number of turns per phase in the stator, and N₂ is the number of turns per phase in the rotor.

The magnetizing reactance, X_M, is regarded as significantly larger in comparison to the short-circuit reactance.

With these parameters of the generator, the stator current I_S and rotor current I_R at the stator voltage U_S are

$$\underset{¯}{I_S}={\displaystyle \frac{\underset{¯}{U_S}}{R_S+\frac{R_R}{s}+j\cdot X_{sc}}}={\displaystyle \frac{\underset{¯}{U_S}}{3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{s}+j\cdot 1.31}}$$

(12)

$$\underset{¯}{I_R}=-\underset{¯}{I_S}$$

(13)

The electromagnetic power of the induction generator at the shaft is

$$P_{EG}=3\cdot {\displaystyle \frac{U_S^2\left(\frac{R_R}{s}\right)}{{\left(R_S+\frac{R_R}{s}\right)}^2+{1.31}^2}}=-{\displaystyle \frac{3\cdot {230}^2\left(\frac{3.8179\cdot {10}^{-2}}{s}\right)}{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{s}\right)}^2+{1.31}^2}}$$

(14)

By substituting the slip s with the mechanical angular velocity ω, the following results are obtained:

$$s={\displaystyle \frac{{\omega }_1-\omega }{{\omega }_1}}=1-{\displaystyle \frac{\omega }{{\omega }_1}}=1-{\displaystyle \frac{\omega }{314}}=\left(1-3.1847\cdot {10}^{-3}\cdot \omega \right)$$

(15)

The power characteristic of the induction generator, P_EG, is obtained:

$$P_{EG}\left(\omega \right)=-{\displaystyle \frac{3\cdot {230}^2\left(\frac{3.8179\cdot {10}^{-2}}{s}\right)}{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{s}\right)}^2+{1.31}^2}}=-{\displaystyle \frac{3\cdot {230}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-3.1847\cdot {10}^{-3}\cdot \omega }\right)}{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-3.1847\cdot {10}^{-3}\cdot \omega }\right)}^2+{1.31}^2}}$$

(16)

at U_N = 230 V, f = 50 Hz. Figure 4 illustrates the generator’s power characteristics.

The coordinates of the generator’s nominal power point are nominal power P_N = P_EG = 55 kW. The nominal slip is

$$s_N={\displaystyle \frac{{\omega }_1-{\omega }_N}{{\omega }_1}}={\displaystyle \frac{314-{\omega }_N}{314}}=-0.02$$

(17)

the nominal mechanical angular velocity is ω_N = 320.28 rad./s.

The converter positioned between the generator and the grid facilitates the operation of the wind system across a broad range of speeds, frequencies, and voltages by maintaining a constant steady-state flow:

$${\displaystyle \frac{U}{f_1}}={\displaystyle \frac{230}{50}}=4.6 \mathrm{V}/\mathrm{Hz}$$

(18)

The ratio of voltage to frequency serves as an indicator of the saturation level of the magnetic core. Utilizing the expression for generator power

$$P_{EG}=3\cdot {\displaystyle \frac{U_S^2\left(\frac{R_R}{s}\right)}{{\left(R_S+\frac{R_R}{s}\right)}^2+{1.31}^2}}$$

(19)

and replacing s with

$$s={\displaystyle \frac{{\omega }_1-\omega }{{\omega }_1}}=1-{\displaystyle \frac{\omega }{2\cdot \pi \cdot f}}=1-{\displaystyle \frac{0.15915}{f}}\cdot \omega$$

(20)

we obtain

$$P_{EG}=3\cdot {\displaystyle \frac{U_S^2\left(\frac{R_R}{s}\right)}{{\left(R_S+\frac{R_R}{s}\right)}^2+{1.31}^2}}=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}$$

(21)

Which is visible in Figure 5.

As will be demonstrated through the mathematical model of the turbine, the wind system operates optimally at the mechanical angular velocity ω_OPTIM = 32.026 V, which is determined by the wind speed, V.

The power of the generator, under constant stator flux, is influenced by the mechanical angular velocity, ω, and the stator frequency [40]:

$$P_{EG}\left(\omega ,f\right)=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}$$

(22)

3.2. The Mathematical Model of the Turbine

The mathematical model of the turbine is established using the technical data from [31]. The power output provided by the WT is calculated using the relationship outlined in [8,16]:

$$P_{WT}\left(\omega ,V,\beta \right)=a\left({\displaystyle \frac{V}{\omega }}-b\right)\cdot e^{-c\left({\displaystyle \frac{V}{\omega }}\right)}\cdot V^3$$

(23)

where the values of the parameters a, b, and c can be determined from the technical data of the turbine as follows:

(1)

The maximum power value is achieved at ω_OPTIM by setting the derivative of power to zero:

$${\displaystyle \frac{d{P}_{WT}}{d\omega }}={\displaystyle \frac{d}{d\omega }}\left[a\left({\displaystyle \frac{V}{\omega }}-b\right)\cdot e^{-c\left({\displaystyle \frac{V}{\omega }}\right)}\cdot V^3\right]=-V^4\cdot {\displaystyle \frac{a}{{\omega }^3}}\cdot e^{-V{\displaystyle \frac{c}{\omega }}}\left(\omega -V\cdot c+b\cdot c\cdot \omega \right)=0$$

(24)

or

$$\omega -V\cdot c+b\cdot c\cdot \omega =0$$

(25)

or

$${\omega }_{OPTIM}={\displaystyle \frac{c}{1+b\cdot c}}\cdot V=k\cdot V$$

(26)

Given that the wind speed V is 10 m/s, the optimal mechanical angular velocity, ω_OPTIM, is equal to the nominal mechanical angular velocity of the generator [37]:

$${\omega }_{OPTIM}={\omega }_N=320.28 [\mathrm{rad}./\mathrm{s}.]$$

(27)

achieving

$${\displaystyle \frac{{\omega }_{OPTIM}}{V}}={\displaystyle \frac{320.28}{10}}=32.028$$

(28)

or

$${\displaystyle \frac{c}{1+b\cdot c}}=32.028$$

(29)

This is the initial equation of the system involving the unknowns a, b, and c.

(2)

The maximum power value of the WT corresponds to the optimal MAS, ω_OPTIM:

$${\displaystyle \frac{V}{{\omega }_{OPTIM}}}={\displaystyle \frac{1+b\cdot c}{c}}$$

(30)

and it is

$$P_{WT-MAX}\left(V\right)=a\left({\displaystyle \frac{1+b\cdot c}{c}}-b\right)\cdot e^{-c\left({\displaystyle \frac{1+b\cdot c}{c}}\right)}\cdot V^3={\displaystyle \frac{a}{c}}\cdot e^{-1-c\cdot b}\cdot V^3$$

(31)

following that

$$k_p={\displaystyle \frac{a}{c}}\cdot e^{-1-c\cdot b}$$

(32)

At a wind speed of V = 10 m/s, the maximum power output WT is P_WT-MAX = 55,000 W, which indicates that the proportionality factor k_p is determined as follows:

$$k_p={\displaystyle \frac{P_{WT-MAX}}{V^3}}={\displaystyle \frac{55000}{{10}^3}}=55$$

(33)

and the second equation is obtained:

$${\displaystyle \frac{a}{c}}\cdot e^{-1-c\cdot b}=55$$

(34)

These two equations are fundamental in establishing the mathematical model of the wind turbine (MM-WT) and in determining the coordinates of the maximum power point (MPP), optimal angular velocity (ω_OPTIM), and maximum wind power (P_WT-MAX).

For a wide range of WT, the report ω_OPTIM/ω_MAXIM, [15,42] has the value ω_OPTIM/ω_MAXIM = 4/9; ω_MAXIM represents the maximum mechanical angular velocity during the no-load operation of the wind turbine.

(3)

The value of the ratio ω_OPTIM/ω_MAXIM leads to Equation (3).

Since the power of the wind turbine is zero when operating under no load

$$P_{WT}=a\left({\displaystyle \frac{V}{\omega }}-b\right)\cdot e^{-c\left({\displaystyle \frac{V}{\omega }}\right)}\cdot V^3=0$$

(35)

at MAS

$$\omega ={\omega }_{MAXIM}={\displaystyle \frac{V}{b}}$$

(36)

By substituting ω_MAXIM into the ratio ω_OPTIM/ω_MAXIM, we obtain

$${\displaystyle \frac{{\omega }_{OPTIM}}{{\omega }_{MAXIM}}}={\displaystyle \frac{\frac{c}{1+b\cdot c}}{\frac{1}{b}}}={\displaystyle \frac{b\cdot c}{1+b\cdot c}}$$

(37)

and it results in

$${\displaystyle \frac{4}{9}}={\displaystyle \frac{b\cdot c}{1+b\cdot c}}$$

(38)

Thus, the system in the unknowns a, b, and c was obtained:

$$\left\{\begin{array}{c}{\displaystyle \frac{c}{1+b\cdot c}}=32.028\\ {\displaystyle \frac{a}{c}}\cdot e^{-1-c\cdot b}=55\\ {\displaystyle \frac{4}{9}}={\displaystyle \frac{b\cdot c}{1+b\cdot c}}\end{array}\right.$$

(39)

with the solution a = 19,182, b = 1.3877 × 10⁻², and c = 57.65.

The mathematical model of WT was obtained in the form

$$P_{WT}\left(\omega ,V\right)=19182\left({\displaystyle \frac{V}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left({\displaystyle \frac{V}{\omega }}\right)}\cdot V^3 [\mathrm{W}]$$

(40)

The maximum power point, MPP, is achieved by setting the derivative of power with respect to ω to zero, resulting in the following:

$${\displaystyle \frac{d{P}_{WT}\left(\omega ,V\right)}{d\omega }}={\displaystyle \frac{d\left[19182\left(\frac{V}{\omega }-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left(\frac{V}{\omega }\right)}\cdot V^3\right]}{d\omega }}=0$$

(41)

or

$$1.1058\cdot {10}^{13}\cdot V-3.4528\cdot {10}^{11}\cdot \omega =0$$

(42)

with the solution ω_OPTIM = 32.026·V.

The maximum power value of the WT corresponds to the optimal MAS and is

$$P_{WT-MAX}\left(V\right)=55\cdot V^3$$

(43)

depending on the wind speed:

$$P_{WT-MAX}\left(\omega \right)=55\cdot {\left({\displaystyle \frac{\omega }{32.026}}\right)}^3=1.6744\cdot {10}^{-3}\cdot {\omega }^3$$

(44)

or the mechanical angular speed.

4. System Operation in the Optimal Zone: Analysis of Various Dependencies

The adjustment of the frequency and the stator voltage is performed using the converter interposed between the generator and the grid (Figure 6).

By adjusting the frequency, f, and the stator voltage, the optimal mechanical angular speed, ω_OPTIM, is achieved based on wind speed.

The generator power at constant stator flux is defined in (22) and in the steady state it is equal to the turbine power defined in (40). It results in

$$19182\left({\displaystyle \frac{V}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left({\displaystyle \frac{V}{\omega }}\right)}\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}$$

(45)

Given that the inertia of the converter situated between the generator and the grid is significantly lower than the mechanical inertia of the turbine, it can be assumed that the stator frequency, f, is achieved without any dead time.

4.1. The Dependencies of Frequency, Steady-State Current, and Power Factor in Relation to Wind Speed

The static frequency, f, is adjusted to enable the operation of the wind turbine at maximum power points (MPPs) under optimal conditions, where ω_OPTIM = 32.026·V, which is contingent upon the wind velocity, V. We consider the stationary regime, where the powers are equal, P_WT(ω, V) = P_EG(ω, f).

The wind speed varies from 3 m/s to 10 m/s, yielding the following results.

• For V = 3 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=15.233 \left[\mathrm{Hz}\right]\end{array}$$

  (46)
• For V = 4 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=20.284 \left[\mathrm{Hz}\right]\end{array}$$

  (47)
• For V = 5 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=25.231 \left[\mathrm{Hz}\right]\end{array}$$

  (48)
• For V = 6 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=30.344 \left[\mathrm{Hz}\right]\end{array}$$

  (49)
• For V = 7 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=35.347 \left[\mathrm{Hz}\right]\end{array}$$

  (50)
• For V = 8 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=40.325 \left[\mathrm{Hz}\right]\end{array}$$

  (51)
• For V = 9 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=45.260 \left[\mathrm{Hz}\right]\end{array}$$

  (52)
• For V = 10 m/s:

  $$\begin{array}{c}55\cdot V^3=-{\displaystyle \frac{3\cdot {4.6}^2\left(\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}\\ \omega =32.026\cdot V\\ f=50.093 \left[\mathrm{Hz}\right]\end{array}$$

  (53)

Table 1. Frequency dependence on wind speed.

V [m/s]	f [Hz]
3	15.233
4	20.493
5	25.321
6	30.344
7	35.347
8	40.325
9	45.26
10	50.877

systematizes the results obtained:

As can be seen from Figure 7, the stator frequency increases linearly with the wind speed value.

The value of the stator current, at constant stator flux, is

$$I_s={\displaystyle \frac{U_s}{\sqrt{{\left(R_s+\frac{R_s}{s}\right)}^2+X_{sc}^2}}}={\displaystyle \frac{4.6}{\sqrt{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{f}\cdot \omega }\right)}^2}{f^2}+{\left(\frac{1.31}{50}\right)}^2}}}$$

(54)

By substituting the mechanical angular speed, ω, with the optimal value ω = ω_OPTIM = 32.026·V and incorporating the relationship between stator frequency and wind speed, f = 5·V, the resulting value is obtained:

$$I_s={\displaystyle \frac{4.6}{\sqrt{\frac{{\left(3.8179\cdot {10}^{-2}+\frac{3.8179\cdot {10}^{-2}}{1-\frac{0.15915}{5\cdot V}\cdot \omega }\right)}^2}{{\left(5\cdot V\right)}^2}+{\left(\frac{1.31}{50}\right)}^2}}}={\displaystyle \frac{4.6}{\sqrt{\frac{1}{V^2}\left(6.8644\cdot {10}^{-4}\cdot V^2+0.14916\right)}}}$$

(55)

The variation of the static current with wind speed is illustrated in Figure 8.

At a wind speed of 10 m/s, the nominal current I_N is determined to be 98 A. A linear variation of the current, expressed as I = 9.8·V, is validated by the relationship between the steady-state current and wind speed illustrated in Figure 9. Consequently, the electrical power delivered to the grid is

$$P_{El}=3\cdot U\cdot I\cdot \cos \phi =3\cdot 4.6\cdot 5\cdot V\cdot 9.8\cdot V\cdot I\cdot \cos \phi =676.2\cdot V^2\cdot \cos \phi$$

(56)

From the equality of the turbine powers, P_WT(V) = 55·V³, with the generator P_WT(V) = P_EG(V), the generator power factor cos φ value is obtained:

$$\cos \phi ={\displaystyle \frac{55\cdot V^3}{676.2\cdot V^2}}=8.1337\cdot {10}^{-2}\cdot V$$

(57)

Figure 9 illustrates that the power factor ranges from 0.24401 to 0.81337, with the minimum value occurring at a wind speed of 3 m/s and the maximum value at a wind speed of 10 m/s.

The speed of the wind significantly affects both the quality of power delivered to the grid, as indicated by the power factor, and the quantity of power supplied, which is dependent on the cube of the wind speed. The higher the wind speed value, the higher the quality and value of power delivered to the grid.

4.2. Wind System Behavior at Time-Varying Wind Speeds by Estimating the Wind Energy Captured by the Turbine

At constant wind speeds over time, the wind turbine is capable of operating at the maximum power point, MPP, by utilizing a suitable power prescription at the EG [8,26]. This can be accomplished through the Perturb and Observe Method, which is thoroughly examined in the literature [28] concerning the operation of wind installations in energetically optimal areas under stable wind speeds.

Given the variable wind speeds over time [11,12,43], the issue becomes complex due to the significant inertia moments. To operate within the optimal energy-efficient range, the mechanical angular speed must correspond with the temporal variations in wind speed.

As will be demonstrated subsequently, employing the mathematical model of WT, if the wind speed varies over time while the power output of the electric generator stays constant, the maximum wind energy that the turbines can capture will not be achieved. It is known that the optimal power value at the generator is contingent upon the mechanical angular speed, which is in turn is influenced by the wind speed.

Owing to the significant inertia moment [12,42], it is not feasible for the operation to consistently maintain the MPP, as the wind turbine’s rotational speed is unable to sufficiently adjust to the swift variations in wind speed over time [16,42], which requires an optimal rotational speed from an energy efficiency standpoint [11].

It is considered that the wind speed changes from V₁ = 8 m/s to V₂ = 10 m/s, having a sinusoidal variation of the form

$$V\left(t\right)=9+\sin 9.43\cdot {10}^{-2}\cdot t$$

(58)

At t = 0, the wind speed has the value V(0) = 9 m/s and the optimal MAS, ω_OPTIM, corresponding to the speed of 9 m/s is

$${\omega }_{OPTIM}=32.026\cdot V=32.026\cdot 9=288.23 \mathrm{rad}./\mathrm{s}$$

(59)

At ω_OPTIM = 288.23 rad/s, the turbine operates at its maximum power point (Figure 10), with a constant generator power output.

$$P_{EG}\left(0\right)=33\cdot 9^3=24057 \left[W\right] \mathrm{rad}./\mathrm{s}$$

(60)

The turbine power characteristics are

• For V₁ = 8 m/s:

  $$P_{WT-1}\left(\omega ,8\right)=19182\left({\displaystyle \frac{8}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left({\displaystyle \frac{8}{\omega }}\right)}\cdot 8^3 \left[W\right]$$

  (61)
• For V₂ = 10 m/s:

  $$P_{WT-2}\left(\omega ,10\right)=19182\left({\displaystyle \frac{10}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left({\displaystyle \frac{10}{\omega }}\right)}\cdot {10}^3 \left[W\right]$$

  (62)
• For V₃ = 9 m/s:

  $$P_{WT-3}\left(\omega ,9\right)=19182\left({\displaystyle \frac{9}{\omega }}-1.3877\cdot {10}^{-2}\right)\cdot e^{-57.65\left({\displaystyle \frac{9}{\omega }}\right)}\cdot 9^3 \left[W\right]$$

  (63)

From the power equation

$$J\cdot {\displaystyle \frac{d\omega }{dt}}\cdot \omega =P_{WT}-P_{EG} \left[W\right]$$

(64)

with the total moment of inertia of the generator and turbine system, J = 0.7 kg m², the obtained equation of motion is

$$0.7\cdot {\displaystyle \frac{d\omega }{dt}}\cdot \omega =P_{WT}-P_{EG} \left[W\right]$$

(65)

or

$$\left\{\begin{array}{c}\begin{array}{l}0.7\cdot {\displaystyle \frac{d\omega }{dt}}\cdot \omega =19182\left[{\displaystyle \frac{\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}{\omega }}-1.3877\cdot {10}^{-2}\right]\cdot e^{-57.65\cdot \left({\displaystyle \frac{\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}{\omega }}\right)}\cdot \\ {\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}^3-33\cdot 9^3\end{array}\\ \begin{array}{l}{\displaystyle \frac{d{E}_W}{dt}}=19182\left[{\displaystyle \frac{\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}{\omega }}-1.3877\cdot {10}^{-2}\right]\cdot e^{-57.65\cdot \left({\displaystyle \frac{\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}{\omega }}\right)}\cdot \\ {\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}^3\end{array}\\ \omega \left(0\right)=288.23\\ E_W\left(0\right)=0\end{array}\right.$$

(66)

By solving the differential equations system, the MAS variation, ω, is obtained as illustrated in Figure 11, under the condition that the power at the generator remains constant at 24,057 W.

The wind energy harnessed by the turbine over a time span of 3500 s, E_W(3500), is derived from the integration of the differential equation

$${\displaystyle \frac{d{E}_W}{dt}}=P_{WT}\left[W\right]$$

(67)

resulting in E_W(3500) = 8.4251 × 10⁷ J.

The maximum wind energy that the turbine can capture in a time interval of 3500 s, E_W-MAX(3500), is obtained by integrating the maximum power P_WT-MAX:

$$E_{W-MAX}\left(3500\right)={\displaystyle \underset{0}{\overset{3500}{\int }}{P}_{WT-MAX}}\cdot dt={\displaystyle \underset{0}{\overset{3500}{\int }}55\cdot V^3}\cdot dt={\displaystyle \underset{0}{\overset{3500}{\int }}55\cdot {\left(9+\sin 9.43\cdot {10}^{-2}\cdot t\right)}^3}\cdot dt=1.4321\cdot {10}^8\left[J\right]$$

(68)

The electric energy supplied to the grid E_E is derived from the integration of electrical power:

$$E_E\left(3500\right)={\displaystyle \underset{0}{\overset{3500}{\int }}{P}_{EG}}\cdot dt={\displaystyle \underset{0}{\overset{3500}{\int }}33\cdot 9^3}\cdot dt=0.841995\cdot {10}^8\left[J\right]$$

(69)

Upon comparing the energy values, it is evident that

(1)

The amount of electric energy supplied to the grid is 41.206% lower than the maximum wind energy that can be harnessed by the turbines:

$${\displaystyle \frac{E_{W-MAX}\left(3500\right)-E_E\left(3500\right)}{E_{W-MAX}\left(3500\right)}}\cdot 100={\displaystyle \frac{1.4321\cdot {10}^8-0.841995\cdot {10}^8}{1.4321\cdot {10}^8}}\cdot 100=41.206\%$$

(70)

(2)

The wind energy harnessed by the turbine at P_EG = 24,057 W is 41.17% than the maximum wind energy that the turbine is capable of capturing:

$${\displaystyle \frac{E_{W-MAX}\left(3500\right)-E_W\left(3500\right)}{E_{W-MAX}\left(3500\right)}}\cdot 100={\displaystyle \frac{1.4321\cdot {10}^8-8.425\cdot {10}^7}{1.4321\cdot {10}^8}}\cdot 100=41.17\%$$

(71)

5. Discussion

This study examines the performance of medium-power wind turbines under significant variations in wind speed, with the turbines being equipped with short-circuited rotor induction generators and regulated through a power converter in the stator circuit. The load at the terminals of the electric generator was assumed to be constant.

5.1. Results

The most significant results are

• Mathematical models for the generator and turbine have been established based on technical data from a 55 kW turbine;
• The generator’s power was determined at a variable frequency while maintaining a constant stator flux;
• The dependencies of the stator current and frequency on wind speed have been determined;
• The wind-turbine-captured wind energy values were compared with the electrical energy supplied to the grid.

5.2. Fundamental Aspects

The results obtained highlight the following significant aspects:

• When the load on the electric generator remains constant, the electrical energy supplied to the grid is more than twice as small as the maximum wind energy that can be harnessed by a wind turbine;
• The operation of the turbine at its maximum power points necessitates the adjustment of the generator’s power output by regulating the stator frequency, thereby achieving optimal rotational speed.

5.3. Discussions

• Utilizing a power converter within the stator circuit enables the turbine to operate at its maximum power point;
• For optimal control, it is essential to accurately understand the dependence of the optimal mechanical angular speed on wind velocity.

6. Conclusions

This study examines the performance of high-power wind turbines equipped with short-circuited rotor induction generators and regulated through the use of a power converter in the stator circuit under significant variations in wind speed.

It has been demonstrated that adjusting the generator’s power according to the wind speed is essential.

Thus, the main contributions of the research can be highlighted as determining the dependences between generator power on wind speed and stator frequency and between generator frequency on wind speed, generator current, and generator power factor on wind speed. Comparative studies highlighted the captured wind energy at constant and variable wind speeds over time.

The primary outcomes achieved in this study are as follows:

-

Deduction of mathematical models for the 55 kW turbine and the induction generator from the technical data of the turbine (power characteristics, power dependence on rotational speed).

-

Analysis of the behavior of the induction generator under variable frequencies and controlled stator flux.

-

Determination of wind speed dependencies for the stator frequency and current as well as the power factor in the MPP operation mode of the turbine.

Also, simulation-based analysis at variable wind speeds with constant and variable power at the generator were conducted and the harvested wind energy at these power levels for the electric generator were provided.

The results obtained are genuinely beneficial for those who use wind energy systems and thus contribute to the existing specialized literature. Understanding and exploiting the relationship between wind speed and optimal MAS can unlock the full potential of variable-speed wind turbines and contribute to a sustainable energy future.