Digital Twin Formation Method for Distributed Generation Plants of Cyber–Physical Power Supply Systems ^†

Abstract

The purpose of the study presented in the article was to develop a method for the formation of digital twins for distributed generation plants operating as part of cyber–physical power supply systems. A method of forming a digital twin for a system for automatic regulation of the voltage and rotor speed of a synchronous generator is considered. The structure of a digital twin is presented in the form of a multiply connected model using experimental data. The possibility of using a fuzzy inference system, artificial neural networks, and a genetic algorithm for solving the problem is shown. As a result of the research, neuro-fuzzy models of the elements of the distributed generation plant were obtained, which are an integral part of the digital twin. Based on the simulation results, the following conclusions were drawn: the proposed method for constructing an optimized fuzzy model gives acceptable results when compared with experimental data and shows practical applicability in constructing a digital twin. In the future, in order to simplify the model, it is necessary to solve the problem of optimizing the number of rules in the fuzzy inference system. It is also advisable to direct further research to the formation of a complete hierarchical fuzzy system that connects all elements of the digital twin.

1. Introduction

To introduce Smart Grid technologies [1,2] to contemporary power supply systems (PSS) [3], which broadly use distributed generation (DG) plants [4,5] and energy storage units (ESU) [6,7], algorithms and smart mode control devices must be developed. The cyber–physical energy systems concept [8,9] is one of the prospective lines ensuring such development. This concept involves the construction of modern PSS not only using physical objects (power system elements) but also the deep integration of digital systems [10] that contain a lot of data and perform various calculations, simulation of physical objects, and their control. In cyber–physical PSS, physical and digital systems are combined, overlapping and interacting with each other in all operating modes. For proper functioning of physical and digital systems, there is a need for a developed interface of their interaction that allows collecting experimental information based on numerous sensors to build full digital models [11,12] and control physical objects using various mechanisms in automatic mode or provide this action to the operator. Such an interaction interface can be built based on digital twins (DTs) [13,14], allowing us to obtain all necessary data characterizing the functioning of the physical object and the digital system interrelated with it.

In contemporary understanding, the DT concept is that a physical object is represented as a physical system and an associated virtual system [15]. The physical part should be equipped with sensors to collect data on the object’s state and its operating modes and to refine (update) the mathematical model. Such refinement makes it possible to create an adequate model that accurately describes the object behavior in order to optimize its operating mode and optimal control. This computer model is the digital twin [16,17].

The concept of a “digital shadow” exists in parallel with a “digital twin”. Their main difference is that in using the digital shadow, it is possible to predict behavior similar to what has already been observed, but it is impossible to predict accidents if they have never occurred. At the stage of operation, a smart digital shadow can be generated by obtaining up-to-date information about the functioning of a physical object with the help of industrial Internet technologies. This increases the level of model adequacy, which allows further prediction of the object behavior in all operating modes and leads to DT creation.

Methods of mathematical physics are often used to build the digital models of physical objects. In this case, the problem of modeling a physical object is represented as a set of boundary value problems for ordinary differential equations and (or) partial differential equations, for example, for the Euler–Lagrange and Navier–Stokes equations [18].

Modeling technical objects involves engineering analysis software systems, for example, those based on the finite element methods [19,20]. These methods have some drawbacks:

• There is no accurate information about the differential equations describing the behavior of the object and therefore simplification methods are used;
• The finite element method requires knowledge of the initial and boundary conditions, information which may not be available;
• It is difficult to adapt the model built on the basis of the finite element method when the operating parameters and conditions of a physical object change.

Another method allows searching for a solution in the form of a neural network function, for which the parameters of the mathematical model are among the arguments. Relying on this methodology, a unified procedure for building the digital twins of real-world objects and maintaining them up-to-date has been developed [21]. This method can be used to build models that are not of the neural network type [22].

Artificial neural networks [23] are used to solve a variety of problems in various fields, including forecasting of processes [24,25], modeling of complex systems [26,27], identification [28], and control of complex objects [29,30].

Fuzzy inference systems [31] can also be used to build models and control complex systems. Application of a fuzzy logic inference system in the problem of the model building is based on the proof of the FAT (Fuzzy Approximation Theorem), according to which any mathematical system can be approximated by a system based on fuzzy logic [32]. In other words, using the natural language statements “IF–THEN”, followed by their formalization by means of the theory of fuzzy sets, it is possible to describe any arbitrary input–output relationship of an object as precisely as possible without using complicated calculus apparatus, traditionally used in control and identification.

The application of neural networks when combined with fuzzy inference systems allows one to increase control efficiency [33], improve forecasting results [34], diagnose faults in industrial facilities [35], increase accuracy and speed of convergence when training neural networks [36], and much more.

The following evolutionary algorithms are also used in building a digital twin in the form of neural network models [37,38]:

• ▪ Generalized error clustering algorithm;
• ▪ Generalized Schwartz method;
• ▪ Genetic algorithm for building a set of neural networks [39,40];
• ▪ Method of training a set of expert models, etc.

The development of digital twins can also involve machine-learning technology [41,42] as a class of methods for training neural networks. The distinguishing feature in this case is not the direct solution of the problem, but learning in the process of applying solutions to many similar problems.

The paper [43] proposes the construction of a “right” digital twin using an evolutionary method for parallel modeling of the digital twin based on conventional modeling methods, machine learning algorithms [41], and methods for evaluating digital twin properties, such as fuzzy analytic hierarchy process (AHP) [44], Bayesian networks [45], neural networks [21,22], etc.

In [46], digital twin modeling is performed for photovoltaic panels based on a hybrid neural network and an optimized genetic algorithm [47,48] using data from the Internet of Things (IoT) control platform [49].

The article [50] describes constructing digital twins for energy on the basis of ontological modeling [51,52] that determines the stages of the transition from mathematical models and computer programs to digital twins based on ontologies.

The paper [53] describes the design of a digital twin based on data obtained from sensors and used for the automatic rule construction of fuzzy systems, for which an appropriate algorithm is proposed, allowing the model to learn and reduce the response error in the short term.

In [54], an expert set of fuzzy rules generated from production data is used to create digital twins; a large number of artificial neural network models with different configurations are created, from which the best models are selected and then improved with additional training datasets generated by fuzzy inference expert system modeling.

Although the above methods can provide reliable support for the construction of digital twins, there are still many challenges in creating and validating complex dynamic models, such as distributed generation plants operating in a cyber–physical power supply system.

The paper considers a method of DT generation for DG plants equipped with a digital system for the automatic adjustment of excitation controllers and the rotor rotational frequency of synchronous generators [55]. The aim of the study was to determine the possibility of building neuro-fuzzy digital models based on experimental data by the proposed method and to test this method by building models of individual elements of digital twins of a DG plant. The DG plant DT structure is presented in the form of a multiply connected model using experimental data. The description of the proposed method for the DG plant’s DT building is given, and possibilities of using a fuzzy logic inference system, artificial neural networks, and genetic algorithm for solving this problem are shown. The study resulted in obtaining neuro-fuzzy models of individual elements of the DG plant, which are a part of the digital twin.

The novelty of the study and its contribution to research are that it develops a structure and method for building a digital twin of the distributed generation plant based on hierarchical fuzzy inference systems. To this end, the method uses subtractive data clustering and neural networks to generate fuzzy rules as well as a genetic algorithm to optimize the fuzzy model membership functions. The novelty of the research is also in the results of the experimental studies on the construction of a neuro-fuzzy model of a separate link of the digital twin of a DG plant.

2. Description of the DG Plant Working in Cyber–Physical PSS

For a DG plant working in cyber–physical PSS, we consider a synchronous generator of relatively low power, the rotor of which is driven by a thermal, hydraulic or gas turbine. The DG plant under consideration (Figure 1), for which the DT building method is described, also includes a thyristor exciter, automatic voltage regulator (AVR), and automatic speed regulator (ASR), in which proportional–integral–differential (PID) laws are implemented [38]. In addition, there is a fuzzy controller that corrects the AVR and ASR tuning coefficients when there is a significant change in the DG plant operating mode. The DG plant operating mode is identified by a digital device, which implements an adaptive neuro-fuzzy network, which represents the Sugeno fuzzy inference system in the form of a five-layer neural network of direct signal propagation [56]. The architecture of such a network is isomorphic to a fuzzy knowledge base. The neuro-fuzzy network that differentiates the implementation of triangular norms (multiplication and probabilistic OR) is used, as well as differentiable membership functions. This makes it possible to apply fast learning algorithms based on the method of the error back propagation for tuning the neural-fuzzy networks.

The building of Sugeno type fuzzy models to solve the problems of DG plant working mode identification requires two stages. In the first stage, based on experimental data on voltage U_g, frequency ω_g, active and reactive power P_g, Q_g, and power quality indicators (PQI) at the connection node of the DG plant for different operating modes, fuzzy rules are determined by the subtractive clustering method (generalized method of mining clustering, proposed by R. Jaeger and D. Filev) [57]. The total harmonic distortions k_U and the negative sequence voltage unbalance factors k_2U are taken into account as PQI at the connection point of the DG plant. In the second stage, with the trained neural network employed, the parameters of the fuzzy model, which determines the DG plant operating modes, are tuned.

The fuzzy controller is a fuzzy logic inference system (Figure 2), which determines the AVR and ASR settings in different operating modes of the DG plant. The small inertia constant of the DG rotor requires taking into account the mutual influence of AVR and ASR. Therefore, the fuzzy controller rule base is formed based on the method of AVR and ASR concordant tuning [55,58].

As a footnote, fuzzy models are widely used in distributed generation, including Takagi–Sugeno fuzzy modes, see [59] and references therein.

3. Method for DG Plant Digital Twin Building

In the proposed concept of the DG plant digital twin, the object is represented in the form of a multi-connected structure, of which some input and output parameters and relations are formed on the basis of experimental data. The main elements of the object, such as the regulators, turbine, and excitation system are proposed to be represented in the form of fuzzy models, determining the dependences of input and output variables based on fuzzy logical inference. The division of the object into elements allows us to adequately describe the dependencies of the input and output variables, using a small number of rules. On the whole, the building of the whole set of links of the proposed DT structure, provided in Figure 3, is based on the application of a hierarchical fuzzy logical inference system [60], which allows us to significantly reduce the dimensionality of the fuzzy rules base of the whole system. If it is necessary to add other controlled parameters, the proposed DT structure can be expanded and modernized by adding the necessary elements and links.

The purpose of fuzzy simulation is the description of dependence (approximation of some vector function)

Y = f(X)

(1)

where Y is the vector of output linguistic variables; X is the vector of input linguistic variables; and f maps X into Y, described by a set of fuzzy rules.

Fuzzy models, similar to the one provided in Figure 2, are a generalization of intervallic-assessed models, which, in turn, are a generalization of coherent models. The rules base stores a lot of logical inference rules and the order (hierarchical structure) of their use. The base fuzzy variables contain the names of linguistic terms and parameters of their membership functions. The rules base together with the fuzzy variables base form the knowledge base of the fuzzy inference system.

In accordance with the proposed DT structure (Figure 3), sensors should measure all of these parameters, which are inputs and outputs of individual elements of the DG plant. The measuring system must ensure the identification of power quality indicators in the form of total harmonic distortions and a negative sequence voltage unbalance factor, the violation of which may lead to the need to limit the power generated by the DG plant or to disconnect it by relay protection. For training and updating fuzzy models, the parameters should be measured during the transient process caused by the change in the DG plant operating mode in the course of operation and when the special test signals of low intensity that do not violate the normal operation are applied to the AVR and ASR outputs. When fuzzy models are updated, the parameters of the accessory functions of the term sets and reduction are optimized, and if possible—the number of fuzzy rules. The subtractive clustering method [57] and adaptive neuro-fuzzy network [56] can be used to reduce the size of the rule bases of individual blocks. This allows us to train neural networks based on experimental data obtained from sensors and synthesize and update the fuzzy systems’ knowledge bases.

To build an optimized fuzzy model of a digital twin separate link based on experimental data, we developed an algorithm using fuzzy logic, neural networks, and genetic algorithms (GA) (Figure 4a). To build an optimized fuzzy model, it is necessary to take the experimental characteristics of the input and output signals when some perturbation is introduced and train the neural network using this data. The trained neural network allows one to determine the parameters of the fuzzy inference systems of the Sugeno type, using the subtractive clustering method to synthesize fuzzy “IF–THEN” rules.

Artificial neural networks of direct propagation (sigmoidal, radial), as well as networks with feedbacks, can be used to solve the problem. Networks with feedbacks have a higher approximation ability with a comparable number of neurons. Therefore, the proposed algorithm uses the recurrent Elman neural network (Figure 4b), which consists of N inputs, k hidden layer neurons, covered by feedbacks through delay elements $z^{-1}$, and M output layer neurons.

The description of the knowledge base terms, for example, input linguistic variables of the fuzzy model of the selected element, can be performed using the Gaussian membership function:

$${\mathsf{\mu }}_A(x)=e^{-{\left(\frac{x-c}{\mathsf{\sigma }}\right)}^2}$$

(2)

where c is coordinate of the membership function maximum; σ is the mean square deviation, which determines the function width; x is the variable from the base set.

The procedure of GA-based membership function optimization is to enumerate different parameter values and estimate the difference between the responses of the fuzzy model and the experimental data. The block schematic diagram of the algorithm for optimizing the membership function parameters is presented in Figure 5. The construction of a fuzzy model for different parameter values during optimization can be represented as follows:

$$F={(\mathrm{X}}_n, Y_n, C)$$

(3)

where ${\mathrm{X}}_n$ is the vector of the input linguistic variable values of n dimensionality; $Y_n$ is the vector of the output linguistic variable values; n is the total number of lines in the fuzzy knowledge base; C is the vector of the membership functions’ parameter values that have to be optimized.

Determination of the individuals’ fitness function is carried out by the difference between the responses of the fuzzy model and the experimental data using the following quadratic criterion:

$$J={\displaystyle \sum _{i=1}^n{\left[\mathrm{Def}(Y_n,C)-y_i\right]}^2\to \min }$$

(4)

where Def is the defuzzification operator used, which performs the translation of fuzzy variables of the corresponding parameters of the membership functions into the final values of the model output parameters; $y_i$ represents the values of the experimental data output parameters under the corresponding input influences.

When using GA, the membership function parameters are represented in the chromosome by real numbers or binary sequences using one of the known coding methods. Then, a GA procedure is applied to the obtained population of individuals, of which a block schematic diagram for the problem of optimizing the membership functions of a fuzzy model is provided in Figure 6.

The choice of coding method is an important step in finding the optimal parameters of the fuzzy model membership functions and can affect the performance and accuracy of the result. A real-valued coding method is often used, in which a single gene represents one of the required parameters as a real number, and their set is a chromosome, i.e., a possible set of fuzzy model membership function parameters (Figure 7). From the chain of chromosomes, individuals are randomly formed, and the initial population is determined. Figure 7 shows the chromosome when encoding c parameters and σ the Gaussian membership function (1) of the individual term of the input linguistic variables.

Currently, GA researchers propose various methods of selection, crossover, and mutation [61]. Among the systems that implement GA with modern selection operators, crossover and mutation operators, we can highlight the Flex Tool program of the Genetic Algorithm package, which is the counterpart of the MATLAB system. The GA was employed in [62] in the centralized reconfiguration algorithm for hybrid AC/DC shipboard power systems.

It should be noted that the determination of the initial population, crossover fraction probability (crossover probability), and mutation algorithm for the target function has a great impact on the final result and can lead to ambiguous solutions and higher CPU time consumption. In addition, since GA is stochastic, different results may occur as a result of its application to the same optimization problem. To obtain the best GA results and automatically determine the family of the initial population with the highest suitable probability of crossover and mutation, we propose the use of an adaptive algorithm that provides the optimal GA settings for a particular target function. The essence of this algorithm is in the repeated performance of the GA procedure. In the first stage for the given target function, the search range and initial reference point are formed; in this case it is sufficient to perform 10...100 GA iterations depending on the function complexity. Then, based on the use of the settings obtained in the first settings stage and additional procedures, a global solution is formed. For example, the finite family of individuals determined in the first stage is assumed as the initial population, and the adaptive algorithm (adaptive feasible) implemented in Flex Tool software is used as the mutation function, and the optimal value of the crossover fraction probability determined in the first stage of the algorithm is set.

The proposed adaptive GA uses the tournament selection method and two-point crossover, since these methods possess a sufficiently high efficiency. The efficiency of tournament selection lies in the ease of implementation. This method solves the problem of selecting the least adapted individuals, thereby maintaining the diversity of the population. In the two-point crossover, chromosomes are regarded as cycles formed by joining the ends of a linear chromosome together. Therefore, the two-point crossover method solves the same problem as the one-point crossover method, but more completely. In addition, the Nelder–Meade method is used to reduce the number of GA iterations and to refine the global solution after the GA procedure.

Figure 8 shows the results of GA operation implemented in the MATLAB system with default settings (random initial population formation, crossing probability 0.8, standard mutation algorithm with a fraction of selected chromosome components 0.01) and the proposed adaptive algorithm. As can be seen from Figure 8, when using the adaptive GA, the solution is achieved with fewer iterations, and a more accurate result is obtained (Figure 8b), which shows the effectiveness of the proposed algorithm.

To build and optimize a fuzzy model, a program has been developed that has a Windows–oriented interface in which the algorithms described above are implemented. Further, the specified program is used in the fuzzy model building and optimization.

4. Simulation Results and Discussion

Modeling was performed with respect to the power supply scheme, in two nodes of which a mini-hydro power plant (HPP) with a nominal capacity of 24 MW each are connected, operating on the basis of synchronous generators, and the third node has a connection with a high-capacity electrical energy system. A complete structural diagram of the network under study in MATLAB, which takes into account the above DG plants with AVR, ASR, and fuzzy controllers that change the settings of regulators, is provided in Figure 9. The parameters of the DG plan’ts synchronous generators are shown in

Table 1. Parameters used in the model of the DG plant synchronous generator.

Parameter	Value
Rated generator power, S	24 MVA
Rated voltage, U	6000 V
Frequency, f	50 Hz
Stator active resistance, Rs	0.008979 p.u.
Stator leakage inductance, LI	0.05 p.u.
Stator inductance in the longitudinal axis, Lmd	2.35 p.u.
Stator inductance in the transverse axis, Lmq	1.72 p.u.
Excitation-winding resistance, Rf	0.00206 p.u.
Rotor excitation-winding inductance, LIfd	0.511 p.u.
Damper-winding resistance in the longitudinal axis, Rkd	0.0652 p.u.
Damper-winding inductance in the longitudinal axis, LIkd	0.5134 p.u.
Damper-winding resistance in the transverse axis, Rkq1	0.0287 p.u.
Damper-winding inductance in the transverse axis, LIkq1′	0.2553 p.u.
Inertia coefficient (inertial constant), H	2.8485 s
Friction coefficient, F	0.009238 p.u.
Number of pairs of poles, p	3

.

The thyristor exciter of the synchronous generator (Excitation System block in Figure 9) was modeled by a first-order aperiodic link with coefficient k_e = 1 and time constant T_e = 0.025 s. The description of the used AVR and ASR models is given in work [55]. The structural diagram of the hydraulic turbine model used (Hydraulic Turbine units in Figure 4) is provided in Figure 10. The model consists of the main servomotor described by a transfer function of the form

$$W_{Servo}(s)=\frac{0.1\cdot s+1}{(0.25s+1)\cdot (0.1\cdot s+1)+s}$$

(5)

where s is a Laplace operator and the wicket gate limiter (Limiter unit in Figure 10) and a hydraulic turbine with a transfer function:

$$W_{Hydro}(s)=\frac{1-\mathsf{\alpha }\cdot T_{Hydro}\cdot s}{1+0.5\cdot \mathsf{\alpha }\cdot T_{Hydro}\cdot s}$$

(6)

where $T_{Hydro}$—hydraulic turbine time constant (was assumed equal to 0.344 s when modeling); $\mathsf{\alpha }$—position of the wicket gate opening (changes within the range 0…1).

The model diagram (Figure 9) also shows the measurement points of the experimental data to build a fuzzy model of the individual element of the DG plant under consideration (turbine). Experimental data were measured in the emergency and post-emergency mode of network operation, when a three-phase short-circuit caused disconnection of one of the lines by the relay protection with a time delay of 0.3 s, and the DG plant’s controllers were unloading generators to maintain stability. The osscillograms of the measured input signals (control action on the turbines from ASR) and output signals (mechanical power on the turbine shaft) for the described mode are presented in Figure 11. These signals were used to create a fuzzy model of the turbine using fuzzy logic inference system, subtractive data clustering method, neural network, and genetic algorithm used for optimization. The results of a comparison of experimental data and the responses obtained by different methods of the turbine fuzzy models for the considered operating mode of the DG plant are provided in Figure 12. Simulation results show that the fuzzy turbine model optimized by the proposed method, built using a trained neural network, provides a response with an acceptable root-mean-square (RMS) deviation from the experimental data; in this case, 25 rules are used compared to the fuzzy model built using the subtractive clustering method, which requires seven rules to implement. It should be noted that the turbine fuzzy model was obtained for one of the possible operating modes only with the variation of the input and output parameters within a limited range. The refinement and further optimization of a fuzzy turbine model should be carried out for the other DG plant’s operating modes as well. It should also be noted that an increase in the number of fuzzy rules results in a more accurate model, but leads to its complication. Moreover, apart from the increase in the dimension of the fuzzy model, the requirement for computing resources, in particular, for the amount of memory, increases.

Figure 13 shows the results of comparing the experimental turbine input–output data with the trained optimized fuzzy model built using 144 rules of the IF–THEN type. The RMS deviation of the fuzzy model response from the experimental data used for training was 0.0126. However, the training and optimization of such a fuzzy model requires a powerful computational resource. The fuzzy logic inference diagram of the resulting fuzzy model is provided in Figure 14.

The building and optimization of the fuzzy model were also performed for the physical plant, of which the photo and functional diagram are provided in Figure 15. The physical model of the DG plant included the following elements: synchronous generator; DC machine used as a primary motor—turbine (T); exciter; excitation winding (EW); angular transducer (AT); active-inductive load. The rotor speed was regulated with the ASR model in the form of a PID controller implemented using the Real-Time Windows Target and Simulink libraries units of the MATLAB system. The rotor speed was measured using an angular movement transducer. The AVR model was also performed as a PID controller. The physical model and the computer communicated via the PCI 6025E board (Figure 16). The diagram shown in Figure 17 was assembled to control the physical model in the MATLAB system, where the model’s regulators (ASR and AVR blocks) received signals of effective values of current, voltage, and rotor speed using the Analog Input block. The effective values of current and voltage were determined using standard RMS blocks. The control signals through the Analog Output block were received by the DC machine power supply (to control the rotor rotational frequency) and the synchronous machine exciter (to control the generator voltage).

The experimental data were registered using Scope units at various setpoints and ASR tuning coefficients. Based on the experimental data, a fuzzy model of rotor rotational frequency control was built and optimized. It was tested with random tuning coefficients of the regulator (Figure 18).

The results of the computer simulation show that the proposed method of building an optimized fuzzy model of a single element of the DG plant’s digital twin produces acceptable results when compared with experimental data. At the same time, training a fuzzy model with the aid of a neural network allows one to reduce the mean square deviation of the response of this model from the experimental data by 2.5 times while maintaining the same the number of “If–then” rules, which attests to the efficacy of the joint application of artificial neural networks and fuzzy inference systems when constructing digital twins [54]. At the same time, optimizing the parameters of the membership functions of the fuzzy terms of the model using a genetic algorithm that determines the least-squares difference between the fuzzy model response and the experimental data further improves the model by reducing the root-mean-square deviation of the fuzzy model response. These findings are consistent with the findings of studies focused on the joint use of a hybrid neural network and an improved genetic algorithm to build digital twins of engineering systems [47,48].

Optimizing a fuzzy model and training it with the aid of a neural network provides the best results with an increasing number of fuzzy rules. In the example considered here, increasing the number of rules by a factor of almost 6 reduced the mean square deviation of the fuzzy model response from the experimental data by a factor of 6. The obtained results confirm the possibility of reducing the errors of neuro-fuzzy models during their training [53].

The construction and optimization of fuzzy models of complex transients with a large number of rules (144 rules or more) require a considerable amount of time. Therefore, further attention should be paid to the optimization of the rule base of the resulting fuzzy model.

The results of the study on the physical model of the DG plant confirmed a sufficiently high efficiency of using neural networks, fuzzy logic inference system, and a genetic algorithm for building numerical models based on experimental data. In the example of the study of a simple transient with increasing rotor speed of the generator considered here, the fuzzy model response fits well with the experimental data with a relatively small number of rules (25 rules) of the “If–then” type. These findings allow us to suggest that the proposed method for constructing a fuzzy model is valid and applicable to physical systems.

A comparison of the results of the studies based on a computer model and a physical model for a complex and relatively simple transient, respectively, attests to the need to increase the number of fuzzy system rules to ensure high accuracy of the complex model. This circumstance must be taken into account in future work on optimizing the number of rules of the fuzzy hierarchical system of the digital twin of a distributed generation plant.

Generally, the findings of the study are consistent with the results obtained earlier [12] and confirm the approximating abilities of fuzzy logic inference systems [47] and the possibility of obtaining an accurate neuro-fuzzy model based on experimental input–output data.

Thus, the simulation results allow us to back up the claim that we have achieved the research goal of constructing neuro-fuzzy numerical models based on experimental data using the proposed method of training and optimization of a fuzzy inference system. The presented practical evaluation of the proposed method for building models of individual elements of the digital twin of the distributed generation plant shows acceptable results, which sets the stage for further work to improve all steps of the method and its testing on more complex models of distributed generation plants with a subsequent transition to real-world systems.

5. Conclusions

In the example of a DG plant equipped with a digital regulator tuning system, the method of formation of a digital twin based on experimental data using hierarchical fuzzy systems is considered. The method can be used to build digital twins of cyber–physical power systems’ individual DG plants. The study allowed us to obtain neuro-fuzzy models of individual elements of the DG plant, which are the counterparts of the digital twin.

Based on the calculation results and simulation, the following conclusions can be drawn:

The proposed method of building an optimized fuzzy model of an individual element of the DG plant’s digital twin produces acceptable results when compared with experimental data. The root-mean-square deviation of the fuzzy model response from the experimental data used for training was 0.0126. In the future, in order to simplify the model, the problem of optimizing the number of rules in the fuzzy logical inference system shall be solved.

The results of fuzzy model building and optimization according to the experimental data obtained from the physical plant showed high accuracy (RMS deviation was less than 0.008) and practical suitability of the developed program for building individual links of the DG plant’s digital twin.

Further research should be focused on the building and optimization of fuzzy models of other elements of the DG plant’s digital twin, as well as on the research of the full hierarchical fuzzy system operation.