A Simple Model of Turbine Control Under Stochastic Fluctuations of Internal Parameters

Abstract

This article considers a model of a wind power generation system. It is assumed that the wind torque is transmitted to the generator via a gear. At the same time, the gear itself can have backlash with stochastic parameters. This kind of nonlinearity simulates an inevitable aging and wear of the mechanical parts of wind power generation systems over time. The purpose of the study was to identify a control system that would allow for establishing and maintaining the stability of the desired characteristics. The control system is formalized in the form of a second-order linear system. Numerical experiments demonstrated that the suggested control system is robust to stochastic perturbations resulting from both external and internal factors.

1. Introduction

Turbines are currently top priority for the production and energy industries. They transform kinetic energy and/or extract energy from a fluid flow (steam, gas, or water) and convert it into useful work. An electromechanical turbine is usually a turbomachine with a moving part called a rotor assembly. The design and elaboration of turbines, as well as the improvement of their main characteristics, pose an important scientific and technical problem, which can be solved using a combination of methods of continuum mechanics, materials science, mathematical modeling, and control theory.

The growing popularity of alternative energy technologies makes it important to use wind power with maximum effectiveness, which can be done by means of highly efficient wind turbines.

At the moment, a lot of studies focus on the factors that determine the cost of wind power, with the most important factor being the efficiency of wind turbines. The modeling of wind turbines requires an analysis of their flexible parts that move relative to each other and have various mechanical connections. These parts interact with the aerohydrodynamic environment on one side and with various drives (e.g., pitch and yaw drives, the generator, and the turbine brakes) on the other. Various aspects of the design and elaboration of wind turbines are considered in monographs (e.g., [1,2]), and articles [3,4,5,6,7,8,9,10,11,12,13,14,15].

The traditional approach to the development of wind power generation control systems is based on a hierarchical system: supervisory control, operational control, and the control of individual links. High-level control is responsible for the procedures for starting and shutting down the turbine. Operational control is focused on the optimal functioning of the entire generation system in real time, and finally, low-level control is responsible for the optimal functioning of individual links of the generation system. Operational control is the most important from the point of view of the optimization of the wind power generation system. A fairly detailed review of modern turbine control methods in wind power generation systems is given in [16,17]. Obviously, each of the control levels is important; however, since this article is generally devoted to optimizing the functioning of the mechanism for the direct transformation of wind energy into useful energy, we will provide a brief review of methods related to this area.

The operating conditions of wind systems are traditionally divided into three classes by wind speed, corresponding to different control objectives. Control strategies prevent rotation at low speeds (class 1), regulate torque to maximize the amount of transmitted power (class 2), and limit excess power at high wind speeds (class 3). In this regard, we note the work of [18], which analyzes, in detail, various control models, such as pitch control, gain scheduling control, and adaptive control. All of these methods are applicable within the framework of the feedback control paradigm. In this case, the control signal is formed in accordance with the set control objective, depending on external conditions. We also note the work of [19], where the control strategy for generation systems is based on the experience of managing economic systems.

One of the areas ensuring the optimal operation of the wind generation mode is related to the structural stability of the generating systems themselves, as well as the control systems. In this regard, we note the work [20], devoted to modern methods of combating the erosion of single-pile foundations.

The development of wind energy requires the design of increasingly larger turbines, which places increased demands on structural materials due to high dynamic loads. Passive, active, and semi-active control are the main methods for solving these problems. The article [21] considers the characteristics of these control methods, and it also discusses the latest developments in new control tools. To evaluate the effectiveness of control methods in various conditions and environments, a comparative procedure is proposed, within the framework of which the output characteristics and performance of the system are related to various types of input data through modeling the structure and external influences.

Due to the growth of the wind power industry, it is also important to predict the interactions between turbines and electric grids. Fluctuations in wind speed and its direction, as well as irregular variations in the required amount of power, mean that is necessary to perform the high-precision online modeling of complex systems, which include elements described by the laws of aerodynamics, mechanics, and electrodynamics. As noted above, the structural stability of generation systems is a prerequisite for their successful operation. It should be taken into account that, during long-term operation, the mechanical properties of turbines inevitably undergo degradation associated with the wear and fatigue of materials. The consequence of this is the emergence of backlashes in the units of mechanical energy transmission (gear transmissions). A number of recent studies have focused on the imperfections of mechanical subsystems and the specifics of modeling of gear trains and gear mechanisms in particular [22,23,24,25]. They demonstrate that the presence of characteristic gear backlashes is a source of nonlinear dependences described by dynamic equations. Furthermore, such backlashes are usually stochastic and cannot be identified without non-destructive testing. Therefore, it is especially important to consider the nondeterministic nature of the main parameters when modeling electromechanic systems.

Backlashes and stochastic backlashes belong to the class of hysteresis nonlinearities. There is a fairly extensive literature devoted to hysteresis nonlinearity models; each of the known models (Bouc–Wen model, Jiles–Atherton model, Iwan model, Preisach model, etc.) is devoted to hundreds of articles and many reviews. However, hysteresis models with stochastic parameters are found in a relatively small number of works, of which we note [26,27,28]. In this article, the stochastic backlash model is based on the constructions from [27]. Thus, this article is devoted to the issue of the structural stability of wind power generation systems in terms of the system’s response to backlash in gears with unknown parameters of a stochastic nature.

This article considers a simple model of a turbine with its gearbox characterized by hysteretic nonlinearities. The main parts of the turbine are presented at Figure 1.

• Blades $A_1{A}_2$;
• Gearbox T with meshed gears;
• The generator axis $B_1{B}_2$.

Let us denote the turbine and the rotor rotation angles as ${\phi }_1$ and ${\phi }_2$, respectively. They are the main dynamic parameters of the system.

Gearbox T contains gears engaged in different ways, depending on the current rotation angle of the axis. Specifically, when the rotation angle is small, the teeth of the gears practically do not contact, and the transmitted torque is very small. When a certain threshold $({\phi }_1-{\phi }_2)\mathrm{mod}2\pi >\epsilon$ is exceeded, the teeth of the gears contact on the drive side, which results in the torque $N_H({\phi }_1,{\phi }_2)={\displaystyle \frac{I_2{\omega }_2}{I_1{\omega }_1}}$, where $I_{1,2}$ are the moments of inertia of the axes, and ${\omega }_{1,2}$ are their angular velocities of rotation. Provided that $({\phi }_1-{\phi }_2)\mathrm{mod}2\pi <\epsilon$$N_H({\phi }_1,{\phi }_2)=0$.

Therefore, the torque in the gearbox T depends on the difference between the rotation angles ${\phi }_1$ and ${\phi }_2$ according to the hysteresis law:

$$N_H[{\phi }_1-{\phi }_2]=\left\{\begin{array}{c}0,\mathrm{if}0⩽({\phi }_1-{\phi }_2)\mathrm{mod}2\pi ⩽\epsilon ,\hfill \\ {\displaystyle \frac{I_2{\omega }_2}{I_1{\omega }_1}},\mathrm{if}({\phi }_1-{\phi }_2)\mathrm{mod}2\pi >\epsilon .\hfill \end{array}\right.$$

(1)

2. Stochastic Backlash

A physical backlash model can be presented as a system including a cylinder with the length h and a piston, which can both move horizontally (Figure 2). Let the position of the piston be the input coordinate and the position of the cylinder be the output coordinate. Depending on the time on a finite interval $t\in [t_0,T]$, let us denote the input of the system as $x\left(t\right)$ and the output as $u\left(t\right)$.

Assuming that the right-hand and left-hand walls of the cylinder are randomly distributed, the corresponding operator is a backlash with stochastic parameters. Specified below are functions $f_{-}\left(x\right)$ and $f_{+}\left(x\right)$, which are meant as probability densities corresponding to the left-hand and right-hand boundaries of the cylinder, respectively.

Let us suggest that the supports of functions $f_{-}\left(x\right)$ and $f_{+}\left(x\right)$ are located on disjoint sets; specifically, $f_{-}\left(x\right)$ is not zero only on the $(x_{-},-h/2)$ interval, and $f_{+}\left(x\right)$ is not zero on the $(h/2,x_{+})$ interval (Figure 2). The output of the backlash operator with stochastic parameters is then a stochastic process, $u\left(t\right)$, with a distribution function, $P\left\{u\right(t)<u\}$.

Based on a classical scheme [29], we can determine the output on monotonic inputs. The stochastic process $u\left(t\right)$ at any moment t complies with the following relation:

$$u\left(t\right)=\widehat{L}[u_0,x_0;{\Gamma }_l,{\Gamma }_r]x\left(t\right),t_0⩽t⩽T,$$

(2)

where $x_0=x\left(t_0\right)$ and $u_0=u\left(t_0\right)$ are the initial values of the input and output function, respectively. Here, ${\Gamma }_l$, ${\Gamma }_r$ are the defining curves of the backlash: ${\Gamma }_l:u=x+{\xi }_l$ and ${\Gamma }_r:u=x-{\xi }_r$, ${\xi }_l$, and ${\xi }_r$ are random variables with probability densities $f_{-}\left(x\right)$ and $f_{+}\left(x\right)$, respectively (Figure 3).

The distribution function of the stochastic process is defined on monotonically increasing inputs, $x\left(t\right)$, as follows [27]:

$$P\left\{u\right(t)<u\}=$$

(3)

$$=\left\{\begin{array}{c}\theta (u-u_0),\mathrm{if}x\left(t\right)<u_0+h/2;\hfill \\ \left.\left\{\begin{array}{c}\phantom{{\psi }_{+}(u,x\left(t\right))}0,\mathrm{if}u<u_0,\hfill \\ {\psi }_{+}(u,x\left(t\right)),\mathrm{if}{u}_0⩽u⩽x\left(t\right)-h/2,\hfill \\ \phantom{{\psi }_{+}(u,x\left(t\right))}1,\mathrm{if}u>x\left(t\right)-h/2;\hfill \end{array}\right.\right\}\mathrm{if}{u}_0+h/2⩽x\left(t\right)⩽x_{+}+u_0;\hfill \\ \left.\left\{\begin{array}{c}\phantom{{\psi }_{+}\left(\right)}0,\mathrm{if}u<u_0,\hfill \\ {\psi }_{+}(u-(x\left(t\right)-x_{+}-u_0),x_{+}+u_0),\mathrm{if}{u}_0⩽u⩽x\left(t\right)-h/2,\hfill \\ \phantom{{\psi }_{+}\left(\right)}1,\mathrm{if}u>x\left(t\right)-h/2;\hfill \end{array}\right.\right\}\mathrm{if}x\left(t\right)>x_{+}+u_0;\hfill \end{array}\right.$$

where

$${\psi }_{+}(u,x\left(t\right))=\underset{h/2}{\overset{u-u_0+h/2}{\int }}{f}_{+}\left(\tilde{u}\right)\mathrm{d}\tilde{u}+\underset{x\left(t\right)-u_0}{\overset{x_{+}}{\int }}{f}_{+}\left(\tilde{u}\right)\mathrm{d}\tilde{u},$$

(4)

resulting in equations

$$\begin{array}{cc}\hfill & {\psi }_{+}(u,x_{+}+u_0)=\underset{h/2}{\overset{u-u_0+h/2}{\int }}{f}_{+}\left(\tilde{u}\right)\mathrm{d}\tilde{u},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\psi }_{+}(u-(x\left(t\right)-x_{+}-u_0),x_{+}+u_0)=\underset{h/2}{\overset{u-(x\left(t\right)-x_{+})+h/2}{\int }}{f}_{+}\left(\tilde{u}\right)\mathrm{d}\tilde{u},\hfill \end{array}$$

(6)

and the Heaviside step function $\theta \left(x\right)$ is used.

Similarly, using

$${\psi }_{-}(u,x\left(t\right))=\underset{x_{-}}{\overset{x_{-}+x\left(t\right)-u_0}{\int }}{f}_{-}\left(\tilde{u}\right)\mathrm{d}\tilde{u}+\underset{-u+x\left(t\right)}{\overset{-h/2}{\int }}{f}_{-}\left(\tilde{u}\right)\mathrm{d}\tilde{u},$$

(7)

for monotonically decreasing inputs, we obtain

$$P\left\{u\right(t)<u\}=$$

(8)

$$\begin{array}{cc}\hfill & =\left\{\begin{array}{c}\theta (u-u_0),\mathrm{if}x\left(t\right)>u_0-h/2;\hfill \\ \left.\left\{\begin{array}{c}\phantom{{\psi }_{-}\left(\right)}1,\mathrm{if}u>u_0,\hfill \\ {\psi }_{-}(u,x\left(t\right)),\mathrm{if}x\left(t\right)+h/2⩽u⩽u_0,\hfill \\ \phantom{{\psi }_{-}\left(\right)}0,\mathrm{if}u<x\left(t\right)+h/2;\hfill \end{array}\right.\right\}\mathrm{if}{x}_{-}+u_0⩽x\left(t\right)⩽u_0-h/2;\hfill \\ \left.\left\{\begin{array}{c}\phantom{{\psi }_{-}\left(\right)}1,\mathrm{if}u>u_0,\hfill \\ {\psi }_{-}(u-(x\left(t\right)-x_{-}-u_0),x_{-}+u_0),\mathrm{if}x\left(t\right)+h/2⩽u⩽u_0,\hfill \\ \phantom{{\psi }_{-}\left(\right)}0,\mathrm{if}u<x\left(t\right)+h/2;\hfill \end{array}\right.\right\}\mathrm{if}x\left(t\right)<x_{-}+u_0.\hfill \end{array}\right.\hfill \end{array}$$

To determine $\widehat{L}$ on piecewise monotonic inputs, when t is finite, we should partition interval $[0,T]$, presented as $[0,t_1]\cup [t_1,t_2]\cup \dots \cup [t_{n-1},t_n]\cup [t_n,T]$, into intervals of monotonicity and determine the corresponding operator on each interval as the operator on the monotonic input. The initial state is defined as a state at the moment of the transition to a different interval of monotonicity. Then, if, on the interval $t_0⩽t⩽T$, a piecewise sequence of monotonic functions $\left\{x_n\left(t\right)\right\}$, $n=1,2,\dots$ uniformly converges to the function $x_{*}\left(t\right)$:

$$x_n\left(t\right)\rightrightarrows x_{*}\left(t\right)\forall t\in [t_0,T].$$

(9)

The output is a stochastic process, which converges distributionwise to a stochastic process,

$$u_{*}\left(t\right)=\widehat{L}[u_0,x_0]x_{*}\left(t\right).$$

(10)

Thus, the stochastic backlash operator is determined on all the continuous outputs.

3. A Turbine Model

The laws of rigid body dynamics provide us with a system of differential equations with regard to variables ${\phi }_{1,2}$:

$$\left\{\begin{array}{cc}\hfill & I_1{\ddot{\phi }}_1=N\left(t\right)-N_H·N\left(t\right),\hfill \\ \hfill & I_2{\ddot{\phi }}_2=N_H·N\left(t\right)-N_{\mathrm{el}},\hfill \\ \hfill & N_H=N_H[{\phi }_1-{\phi }_2],\hfill \end{array}\right.$$

(11)

where $I_1$, $I_2$ are the moments of inertia of the turbine and the rotor (with respect to the common rotation axis), respectively;

$N\left(t\right)$ is the external torque rotating the turbine;

$N_H$ is the coefficient in the hysteretic gear;

$N_{\mathrm{el}}$ is the torque transmitted to the rotor;

and the two dots above the variables denote the second derivative with respect to time.

Let us denote the angular velocity as ${\omega }_{1,2}={\dot{\phi }}_{1,2}$. Taking into account the laws of thermodynamics, we can present the torque rotating the turbine via the wind speed and other parameters: $N\left(t\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{\rho S\mathcal{C}}{{\omega }_1}}{v}^3$. Here, $\rho$ is the density of air, $\mathcal{C}$ is the power coefficient (a ratio of the rotor power coefficient to the wind power), S is the swept area, and V is the wind speed [30].

For the sake of a numerical experiment, system (11) can be presented as $\dot{\mathbf{z}}=F\left[N_H\right](\mathbf{z},t)$, with the designations $\mathbf{z}={({\omega }_1,{\omega }_2,{\phi }_1,{\phi }_2)}^T$ and

$$F\left[N_H\right](\mathbf{z},t)=\left\{\begin{array}{c}{\displaystyle \frac{1}{I_1}}(N\left(t\right)-N_H·N\left(t\right)),\hfill \\ {\displaystyle \frac{1}{I_2}}(N_H·N\left(t\right)-N_{\mathrm{el}}),\hfill \\ {\omega }_1,\hfill \\ {\omega }_2.\hfill \end{array}\right.$$

(12)

Let the external driving force be a constant with superimposed harmonic vibrations at a frequency of ${\omega }_0$: $N\left(t\right)=B+Acos\left({\omega }_{0}t\right)$. The rotor torque is $N_{\mathrm{el}}=R{\omega }_2$, where $R=\mathrm{const}$.

Figure 4 (left panel) presents the results of the numerical solution of the system (12) for the following parameter values: $I_1=2.0$, $I_2=1.0$, $A=2.0$, ${\omega }_0=5.0$, $B=5.0$, $R=2.0$.

All the parameters in Figure 4 (right panel) are the same except for $R=0.2$. In this case, the angular acceleration of the turbine is greater.

It is known that the mechanical elements of various systems are susceptible to aging and wear. In most cases, these processes are nondeterministic and can only be adequately described by means of the probability theory. In particular, the backlash parameter $\epsilon$ in (1) should be considered as a discrete random variable changing its value at points ${\phi }_1-{\phi }_2=2\pi k$, $k\in \mathbb{Z}$, i.e., after each rotation cycle. In our calculations, we used $\epsilon =A_{\epsilon }\xi$, where $\xi$ is a random variable uniformly distributed on [0,1], and amplitude $A_{\epsilon }$ equals $A_{\epsilon }=2.0$.

Figure 5 (left panel) presents the results of the numerical solution of the system (12) with a random $\epsilon$ for the following parameter values: $I_1=2.0$, $I_2=1.0$, $A=2.0$, ${\omega }_0=5.0$, $B=5.0$, $R=2.0$. In Figure 5 (right panel), $R=0.2$, and the other parameters are the same. As compared to Figure 4, the behavior of the frequencies ${\omega }_1$ and ${\omega }_2$ is less regular, depending on time.

Visually, the graphs shown in Figure 4 and Figure 5 differ slightly, but the parameters for which they were obtained are fundamentally different: the backlash parameter in Figure 4 has a constant value $\epsilon =0.7$, and in Figure 5, the backlash parameter is a random value with a uniform distribution on the segment $[0,0.7]$. The visual similarity of the graphs is due to the structural stability of the system under stochastic disturbances of the backlash parameter.

4. Dissipation Effects

The idealized model of the dynamics of the turbine (12) does not take into account the friction force, which is present in any engineering system. Dissipation effects are usually more obvious in the transmission shafts moving the blades. In order to take into account the impact of the dissipation effects on the system, let us consider the friction force, the torque of which is determined according to the following law: $D\left({\dot{\phi }}_1\right)\sim {\dot{\phi }}_1$. In this case, Equation (12) is presented as follows:

$$F\left[N_H\right](\mathbf{z},t)=\left\{\begin{array}{c}{\displaystyle \frac{1}{I_1}}(N\left(t\right)-N_{H}N\left(t\right))-c{\omega }_1,\hfill \\ {\displaystyle \frac{1}{I_2}}(N_{H}N\left(t\right)-N_{\mathrm{el}}),\hfill \\ {\omega }_1,\hfill \\ {\omega }_2,\hfill \end{array}\right.$$

(13)

where $c>0$.

5. Control of the Electricity Generation

The suggested model can be implemented in real-life systems for the control of the generation of electricity via wind turbines.

Such a control system includes a pitch actuator, which can be simulated using a simple second-order differential equation:

$$\ddot{\beta }\left(t\right)+2\eta {\omega }_n\dot{\beta }\left(t\right)+{\omega }_n^2\beta \left(t\right)={\omega }_n^2{\beta }_r\left(t\right),$$

(14)

where $\beta \left(t\right)$ (the pitch angle) is used to control the system, $\eta$ is a positive parameter with the frequency dimension, ${\beta }_r\left(t\right)$ is the reference angle $\beta \left(t\right)$, and ${\omega }_n$ is the frequency of the pitch actuator.

The reference values ${\beta }_t\left(t\right)$ and the frequency are connected via the following equation:

$${\beta }_r\left(t\right)=k_1({\omega }_2\left(t\right)-{\omega }_n)+k_2{\int }_0^t({\omega }_2\left(\tau \right)-{\omega }_n)d\tau ,$$

(15)

where $k_1>0$, $k_2>0$ are constants.

We should note that the described control mode was used earlier in [31,32,33]. We should also note that, in [31], the control mode was hysteretically reconfigured, which allowed for more flexibility when reacting to changes in the environment.

Let us determine the desired control modes. Assume that they are modes implemented at a constant wind speed and described by (12), where $A=0$. In the stationary mode, the desired angular velocity and the actual velocity of the shaft rotation are the same.

To a first approximation, the torque of the generator shaft complies with the equation $\tau \left(t\right)=C_{\mathrm{el}}{I}_2{\omega }_n\beta \left(t\right)$; i.e., it is proportional to the angle $\beta$, the moment of inertia of the shaft, and the frequency of the pitch actuator. The positive coefficient of proportionality is denoted as $C_{\mathrm{el}}$.

The dynamics of the wind generator with a hysteretic element is then described via a system of differential equations, $\dot{\mathbf{x}}=G(\mathbf{x},t)$, with designations $\mathbf{x}={({\omega }_1,{\omega }_2,{\phi }_1,{\phi }_2,\beta ,v)}^T$ and

$$G(\mathbf{x},t)=\left\{\begin{array}{c}{\displaystyle \frac{1}{I_1}}(N\left(t\right)-N_H·N\left(t\right))-c{\omega }_1,\hfill \\ {\displaystyle \frac{1}{I_2}}(N_H·N\left(t\right)-C_{\mathrm{el}}{I}_2{\omega }_n\beta ),\hfill \\ {\omega }_1,\hfill \\ {\omega }_2,\hfill \\ v,\hfill \\ -2\eta {\omega }_{n}v-{\omega }_n^2\beta +{\omega }_n^2{\beta }_r\left(t\right).\hfill \end{array}\right.$$

(16)

Figure 6 presents a graph of the wind torque $N\left(t\right)$ and the graphs of functions ${\omega }_2\left(t\right)$ and $\beta \left(t\right)$. The numerical calculations were performed for the following control parameters: $C_{\mathrm{el}}=1.0$, $\eta =0.6$, ${\omega }_n=10.0$, $k_1=k_2=1.0$.

Numerical experiments, the results of which are presented in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, demonstrated that the chosen control algorithm allows for the stabilization of the main parameters closely to the desired values. Figure 6 and Figure 7 show that the transition mode takes 100–150 model time units. Then, the main parameters (the angular velocity of the driven shaft and the dump load) are stabilized very close to the desired values. We should stress that the chosen control system is robust to stochastic perturbations, namely random wind speed and stochastic backlash in gears. Figure 8 demonstrates that the controlled variable $\beta \left(t\right)$ varies only insignificantly close to zero. In other words, the chosen control system damps stochastic perturbations resulting from both external and internal factors.

Having compared Figure 9 and Figure 10, we can see that an increase in the desired value of the frequency ${\omega }_N$ results in changes in the dynamics of the control parameter. Namely, when ${\omega }_N=3.0$, stochastic perturbations have a significantly greater impact on the dynamics of $\beta \left(t\right)$ than when ${\omega }_N=0.5$. In the latter case, $\beta \left(t\right)$ behaves normally.

6. Conclusions

At the initial stage of operation of wind power generation systems, their mechanical components operate in an ideal mode. However, during long-term operation, nonlinear effects may occur that are not formally taken into account when systems are designed. Such effects include backlashes, the parameters of which are difficult to identify in the standard operating mode. Moreover, in torque transmission systems (gear transmissions), the backlash parameters can vary significantly. As a result, to ensure the efficient operation of generation systems, it is necessary to configure the control system to neutralize the potentially undesirable manifestation of these nonlinearities. This paper has been devoted to the development of control methods aimed at eliminating stochastic disturbances in the parameters of mechanical components of wind power generation systems.

The algorithm proposed in the paper can be used to ensure the efficient operation of wind power generation systems under conditions of stochastic disturbances in the parameters of power plants. In our study, the damping of stochastic perturbations, which are natural for the studied system, was observed. When modeling gear backlashes, we took into account the fact that backlash parameters can be subjected to stochastic perturbations resulting from the inevitable aging and wear of the mechanical parts. Furthermore, the wind load was also modeled while the stochastic component was taken into account. A numerical analysis of the studied system demonstrated that the chosen control system is robust to stochastic perturbations of various natures. The effect of the stochastic factors differed, depending on the desired parameters of the system. An increase in the desired frequency resulted in a significant level of stochasticity of the control parameters. We should also note that the transition processes (in particular, their duration) also depend on the desired parameter values: greater desired frequencies correspond to quicker transition processes.