Volume 2024, Issue 1 5947120
Research Article
Open Access

Phenomenological Model of Electrotechnical Systems Based on a Synchronous Generator With an Excitation System

Volodymyr Moroz

Volodymyr Moroz

Institute of Power Engineering and Control Systems , Lviv Polytechnic National University , 12 S. Bandera Str., Lviv , 79013 , Ukraine , lp.edu.ua

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Yaroslav Paranchuk

Yaroslav Paranchuk

Institute of Power Engineering and Control Systems , Lviv Polytechnic National University , 12 S. Bandera Str., Lviv , 79013 , Ukraine , lp.edu.ua

Department of Electromechanics and Electronics , Hetman Petro Sahaidachnyi National Army Academy , 32 Heroiv Maidanu Str., Lviv , 79026 , Ukraine , asv.gov.ua

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Volodymyr Konoval

Volodymyr Konoval

Institute of Power Engineering and Control Systems , Lviv Polytechnic National University , 12 S. Bandera Str., Lviv , 79013 , Ukraine , lp.edu.ua

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Oleksiy Kuznyetsov

Corresponding Author

Oleksiy Kuznyetsov

Institute of Power Engineering and Control Systems , Lviv Polytechnic National University , 12 S. Bandera Str., Lviv , 79013 , Ukraine , lp.edu.ua

Department of Electromechanics and Electronics , Hetman Petro Sahaidachnyi National Army Academy , 32 Heroiv Maidanu Str., Lviv , 79026 , Ukraine , asv.gov.ua

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First published: 26 July 2024
Academic Editor: Ali Khalfallah

Abstract

We propose a variant of the application of a phenomenological approach using the “black box” concept to the modeling of electrotechnical systems. The method enables obtaining simplified models of a complex electrotechnical system based on a synchronous generator (SG) with an excitation system and an integral estimation of the accuracy of the reproduction of the original step response. The compromise between complexity and accuracy is the basis for obtaining the computer model of different orders utilizing the proposed method. The application of the proposed phenomenological approach is illustrated by the examples.

1. Introduction

Simulation of transients in complex electrotechnical systems, which describe not only tens or hundreds but also thousands of ordinary differential equations (ODE) (e.g., complex electromechanical systems with a power supply [14] and complex electrotechnical systems [5, 6]), is a quite complicated problem even for the modern status of computer technology and methods for solving ODEs [711]. Another problem arises from the fact that, in the case of the application of numerical methods, the behavior of the utilized numerical method also affects the behavior of the studied system. As demonstrated in [12], due to the discretization by numerical integrators of the continuous model of the dynamic system, there are additional zeros and poles of the resulting discrete transfer function in the obtained digital model, and corresponding changes appear in Bode plots as compared to the analog prototype (the initial ODE system). As a result, the complexity of the digital model of the electrotechnical system increases several times compared to the continuous mathematical model described by differential equations. To some extent, this is consistent with the hypothesis of von Neumann, according to which the simplest description of an object that has reached a certain threshold of complexity is the object itself, and any attempt at its strict formal description (in this situation, a digital model) leads to something more complex and confusing [13, 14].

Another problem that needs to be solved is the implementation of transient calculations in power systems in real-time, or tens, hundreds of times faster for the system’s operational control [7, 10, 11, 15].

The traditional approach in modeling electrotechnical systems by solving ODEs using known numerical methods and writing differential equations in the Cauchy form (initial value problem) often encounters problems with increasing calculation time and accumulation of errors, which is observed in the case of complex electrotechnical systems. The use of classical numerical methods in such cases is often ineffective, even for low-order problems [16]. An example of the practical use of classical real-time modeling methods is the Dakar software package, which is designed to analyze the dynamic and quasistatic modes of power systems [15, 17]. However, in many cases, increasing the efficiency of the simulation of the dynamics of electrotechnical systems by only choosing the most efficient numerical method is not always successful.

The way out of this situation is through methods of simplification of complex models with the application of a phenomenological method of modeling (phaenomena, from Latin—“an observable event”).

2. About the Phenomenological Method

The use of a phenomenological method of analyzing the dynamics of electrotechnical systems is based on the fact that the behavior of a real high-order system is often only slightly different from the behavior of a lower order system. In particular, in [18], it is shown by example that a linear model of the fifth order provides almost the same accuracy of reproduction of dynamic behavior as a nonlinear 11th-order model. The practice of the phenomenological method is also facilitated by the fact that, in many cases, there is no need to reproduce the internal phenomena of the system but only to consider it as a “black box” that has an input and output. In particular, this method is effective for the analysis of power system stability [15, 17].

The concept of the “black box,” we recall, was introduced by the “father of cybernetics,” Norbert Wiener, in particular, in [19]. The concept of the “black box” allows one to ignore the internal structure of the object and focus only on its behavior, which should be reproduced in a computer model.

Thus, the advantage of the phenomenological method of computer modeling, which is based on the “black box” concept, is a significant time saving during the computer calculations. This is due to the need to reproduce only the external behavior of the object within the model; it can be achieved in a much simpler way. Note that this is possible in the situation when the researcher is not interested in the internal phenomena in the part of the system (or its element), which allows to consider this element only as an object of the “input-output” type, that is, “black box.”

It is clear that when there is no need to take into account the internal phenomena in the element of the studied system but only to reproduce its response to external impacts, this can be done with a certain accuracy, which is acceptable under research conditions. In many cases, the accuracy at the level of a couple of percent (e.g., up to 5%) is sufficient, which corresponds to the accuracy of the analog elements of the system—resistors and capacitors [20, 21]. Reducing the accuracy requirements of reproducing the behavior of a system element makes it possible to reduce the order of mathematical description (e.g., the transfer function or ODE order), which, accordingly, increases the calculation speed.

Therefore, the adequacy of digital models of power systems in accordance with their description by ODE systems and their higher calculation speed can be achieved by replacing the original model (prototype) with a simulation (phenomenological) model by order reduction. At present, the methods of reducing the model’s order are well known and have become widespread as an effective means of accelerating calculations [22, 23]. Model order reduction is widespread in power system studies, for example, for transient stability studies [15, 24] and for the detection of instability sources [25]. It is also of interest for control-oriented modeling to efficiently design a controller [26] for the purposes of state identification and parameter estimation [27], et cetera.

3. Method of Reducing the Order

Without claiming originality, the current research used a fairly simple method, which is to approximate the transients of the original system (prototype), and a simplified model, which is represented as a transfer function. To find the coefficients of the transfer function of the simplified model, the minimization of the quadratic deviation integral of both transient characteristics is used:
( )
where hp(τ) is the step response of a prototype (an original complicated) system and hs(τ) is the step response of a simplified (a reduced-order) system.
The search for the minimum of the target function is carried out by one of the classical methods, for example, the Nelder–Mead [28] or Levenberg–Marquardt algorithm [29], which are implemented in many mathematical applications [30, 31]. These methods work effectively with no more than 3–4 independent variables, but in our situation that is not a problem—the behavior of a complex object is simulated by the behavior of transfer functions of second and third order:
( )
where K2 and K3 are the gain coefficients, T1 and T2 are the time constants (system’s inertia), and ξ is a damping factor.

In the first case, we have three independent variables, and in the second case, we have four variables. However, their number can be immediately reduced by one because the gain K2 or K3 is directly found from the step response after its setting into a steady state. Thus, to describe the behavior of the system by a phenomenological model of the second order, it is required to find only two coefficients, for a phenomenological model of the third order—three coefficients.

The fourth-order phenomenological model with a transfer function is as follows:
( )
where K4 is a gain coefficient, T1 and T2 are the time constants (system’s inertia), and ξ1 and ξ2 are the damping factors; the number of independent variables is five, similarly reducing to four by finding K4 from the step response.

The block diagrams of the abovementioned phenomenological models in a generalized representation (with an input signal Xin and an output signal Xout) are given in Figure 1.

Details are in the caption following the image
Generalized block-diagram representation of the phenomenological models of the (a) second, (b) third, and (c) fourth order.
Details are in the caption following the image
Generalized block-diagram representation of the phenomenological models of the (a) second, (b) third, and (c) fourth order.
Details are in the caption following the image
Generalized block-diagram representation of the phenomenological models of the (a) second, (b) third, and (c) fourth order.

The efficiency and expediency of using the phenomenological approach to represent complex dynamical electrotechnical objects or their elements by the “black box” low-order models are illustrated by the two examples. These are typical for the problems of mathematical description and dynamic studies in electrical systems. The obtained results allow us to conclude about the efficiency of the phenomenological approach as a compromise between the model’s computational complexity (and the calculation speed) and the accuracy of representing the step response of the complex electrotechnical systems.

In particular, in the fourth section, we demonstrate that decreasing the order of the model from fifth to second and third allows us to obtain satisfactory results by means of maximum deviation and root mean square error. The accuracy of representing the step response by the third-order model is better.

In the fifth section, we present the second example of the higher order model. When the step response for the excitation (11th-order model) is used for obtaining the phenomenological models, the results are similar. But with the 12th-order model representing the step response for restoring the previous status after external loading, we also demonstrate that there is no improvement by using the higher order phenomenological model, and the best results are obtained by using the lower order ones.

4. First Example

The first example of finding phenomenological models for power systems is shown below in the example taken from [32]. Consider a simplified block diagram of a synchronous generator (SG) with an excitation system (Figure 2).

Details are in the caption following the image
Block diagram of a synchronous generator with an excitation system [32].
The excitation system regulates the EMF of the generator and, therefore, not only controls the output voltage Vt but also the power factor and current. The state-space mathematical model of a system and the structural model are described in [32] (Figure 3). Based on them, an ODE system describing the behavior of a SG with an excitation system is compiled and used for further research:
(1)
where the parameters of the system used for the simulations are as follows [32]:
  • o.

    Regulator gain KА = 40 and filter time constant τA = 0.05 s.

  • o.

    Exciter circuit gain KG = 0.8 and time constant (inertia) τG = 1.5 s.

  • o.

    Exciter constant (related to self-excitation) KE = –0.05 and time constant τE = 0.5 s.

  • o.

    Feedback loop parameters: KR = 1; τR = 0.01 s.

  • o.

    Generator stabilizing feedback loop gain KF = 0.04 and time constant τF = 0.715.

Details are in the caption following the image
Structural model of synchronous machine and excitation system [32].
Based on the data described above, a mathematical model of a synchronous machine (SM) with an excitation system in the form of transfer functions is synthesized. After elementary transformations, the inner loop of the structural model in Figure 3 can be reduced to the transfer function:
(2)
where
(3)
(4)
(5)
The next step is to reduce the external loop to one transfer function:
(6)
where
(7)
(8)
For the obtained general transfer function of a closed system, we substitute specific values of the block diagram’s parameters (Figure 3):
(9)

For the obtained transfer function of the model of SM with an excitation system, a step response is found (Figure 4).

Details are in the caption following the image
Step response of a synchronous machine’s model with an excitation system.

Reproduction of the behavior of such a fifth-order system using a phenomenological approach is possible using lower order simulation models that would provide sufficient engineering accuracy to represent the prototype’s transients.

4.1. Second-Order Phenomenological Model

Due to the oscillatory nature of the prototype model’s transients, the minimum order of the simulation phenomenological model must be at least the second, which corresponds to a pair of complex-conjugate poles and the transfer function given as
(10)

In this case, it is possible to find the appropriate approximation coefficients T2 and ξ that would ensure maximum convergence of the step responses of the prototype (model of SM with excitation system) and its simplified (phenomenological) model. The procedure for finding the coefficients can be reduced to minimizing the integral quadratic deviation between the step responses of both models. The result is shown in Figure 5 for such values of model parameters: T2 = 0.163; ξ = 0.2.

Details are in the caption following the image
Approximation of the prototype’s (SM with an excitation system) behavior by a second-order phenomenological model.

The maximum deviation of the step response of the phenomenological model from the step response of the prototype is below 13.2%, and the root mean square error does not exceed 0.7%.

4.2. Third-Order Phenomenological Model

The higher accuracy of step response reproduction of a prototype is provided by the third-order simulation model with a transfer function.
(11)

After the procedure for finding the approximation coefficients, the result is shown in Figure 6 for such values of model parameters: T1 = 0.039; T2 = 0.155; ξ = 0.183.

Details are in the caption following the image
Approximation of the prototype’s (SM with an excitation system) behavior by a third-order phenomenological model.

The maximum deviation of the step responses of the third-order phenomenological model for the step responses of the prototype is below 5.8%, and the root mean square error does not exceed 0.19%.

The results are summarized in Table 1.

Table 1. Numerical comparison of phenomenological models of different orders for the first example.
Original model (prototype) Reduced-order phenomenological model Maximum deviation (%) Root mean square error (%)
5th order 2nd order 13.2 0.7
3rd order 5.8 0.19

5. Second Example

The second example is based on 11th- or 12th-order mathematical models that are used in DAKAR software [15, 17] for a SG with an excitation system and a steam turbine, taking into account the joint to the generator. These models include such submodels (mathematical models in equation form are known (see references below) and not included due to the large size of the mathematical description):
  • o.

    Classical SG model in dq coordinates [7, 3336].

The mathematical model of a SM comprises rotor motion equations, rotor electromagnetic equations, et cetera, together with stator voltage equations and the expressions for electromagnetic powers. Rotor motion equations are as follows:
(12)
Rotor electromagnetic equations are as follows:
(13)
where and .
  • o.

    Typical excitation system includes PSS (power system stabilizers) [7, 37]. Its mathematical model is given by the ODE system (1) and with the structural diagram as given in Figure 3.

  • o.

    Equivalent complex electrical loading as an equivalent electrical circuit (based on [7, 36]).

  • o.

    The shaft of a hydraulic turbine is connected to the SG shaft. The research is performed using the conventional classical transfer function of a hydraulic turbine, as shown in Figure 7 [7].

Details are in the caption following the image
Transfer function of a hydraulic turbine (after [7]).
SG type TBB-320-2/20 kV is used for those simulations for two types of regimes:
  • 1.

    Step response for excitation (output voltage as the output coordinate).

  • 2.

    Step response for restoring the previous status after external loading.

5.1. Step Response for the Excitation

Such a model has the 11th order of mathematical description by the differential equations and reproduces the start regime for the output voltage including the excitation system. This regime is simulated by the step response of the input reference. The original model’s step response of the whole system is denoted on the plots below as the prototype.

5.1.1. Second-Order Phenomenological Model

Due to the complicated oscillatory nature of the prototype model’s transients, the minimum order of the simulation phenomenological model must be at least the second again, which corresponds, as mentioned above, to a pair of complex-conjugate poles and the transfer function as
(14)

In this case, it is possible to find the appropriate approximation coefficients T2 and ξ that would ensure the maximum convergence of the step responses of the prototype (complete model of a SM with an excitation system) and its simplified (phenomenological) model. As mentioned above, the procedure of the search of the coefficients can be reduced to minimize the integral quadratic deviation between the step responses of both models. The result is shown in Figure 8 for such parameters: T2 = 0.346; ξ = 1.80.

Details are in the caption following the image
Approximation of the prototype’s behavior (step response for the excitation) by a second-order phenomenological model.

The maximum deviation of the step response of the second-order phenomenological model from the step response of the prototype is below 6.4%, and the root mean square error does not exceed 0.22%.

5.1.2. Third-Order Phenomenological Model

The higher accuracy of step response reproduction of a prototype is provided by the third-order simulation model with a transfer function.
(15)

After the procedure for finding the approximation coefficients, the result is shown in Figure 9 for the values of model parameters: T1 = 1.203; T2 = 0.151; ξ = 0.207.

Details are in the caption following the image
Approximation of the prototype’s behavior (step response for the excitation) by a third-order phenomenological model.

The maximum deviation of the step responses of the third-order phenomenological model from the step responses of the prototype is below 0.6%, and the root mean square error does not exceed 0.0023%.

5.2. Step Response for Restoring Previous Status After External Loading

Such a model has the 12th order of mathematical description by the ODEs, and it reproduces the regime (system’s mode) for restoring the previous status of output voltage after external loading including the excitation system. This regime is simulated by the step response of the external loading (i.e., increasing the loading current). The original model’s step response to the whole system is denoted as the prototype in the plots below.

5.2.1. Second-Order Phenomenological Model

Due to the complicated oscillatory nature of the prototype model’s transients, the minimum order of the simulation phenomenological model must be at least the second again, which corresponds, as mentioned above, to a pair of complex-conjugate poles and the transfer function as
(16)

In this case, it is possible to find the appropriate approximation coefficients T2 and ξ that would ensure maximum convergence of the step responses of the prototype (a complete model of SM with an excitation system and loading) and its simplified (phenomenological) model. As mentioned above, the procedure for finding the coefficients can be reduced to minimizing the integral quadratic deviation between the step responses of both models. The result is shown in Figure 10 for the values of model parameters: T2 = 0.163; ξ = 2.083.

Details are in the caption following the image
Approximation of the prototype’s behavior (restoring previous status after external loading) by a second-order phenomenological model.

The maximum deviation of the step response of the second-order phenomenological model from the step response of the prototype is below 27%, and the root mean square error does not exceed 4.55%.

5.2.2. Third-Order Phenomenological Model

The higher accuracy of step response reproduction of a prototype is provided by the third-order phenomenological model with a transfer function.
(17)

After the procedure for finding the approximation coefficients, the result is shown in Figure 11 for the values of model parameters: T1 = 0.686; T2 = 0.155; ξ = 0.061.

Details are in the caption following the image
Approximation of the prototype’s behavior (restoring previous status after external loading) by a third-order phenomenological model.

The maximum deviation of the step responses of the third-order phenomenological model for the step responses of the prototype is below 15.9%, and the root mean square error does not exceed 1.66%.

5.2.3. Fourth-Order Phenomenological Model

And now, the actual question arises: “Is it really good to increase the order of the phenomenological model to increase the accuracy of such a model?” The answer is “no,” for some reasons:
  • o.

    First, models with more than third-order complexity are more complicated, and the expectations of an increase in the accuracy of the reproduction of transient behavior are rarely realized (see, e.g., [18]).

  • o.

    Second, additional coefficients of the higher order transfer function result in a more complicated target minimization function. That correspondingly complicates the minimization process using traditional minimization procedures [2831] as far as some local nonglobal minimums appear.

The example below demonstrates the obtained accuracy of the step response reproduction of a prototype by the fourth-order phenomenological model with the transfer function.
(18)

After applying the search procedure for the approximation coefficients, one of the variants is shown in Figure 12 for the values of phenomenological model parameters: T1 = 0.154; T2 = 0.088; ξ1 = 0.063; ξ2 = 3.886.

Details are in the caption following the image
Approximation of the prototype’s behavior (restoring previous status after external loading) by a fourth-order phenomenological model.

Note that the obtained result is only one minimum point from some local minimums of the target minimization function, and results strongly depend on the initial values of variables. The maximum deviation of the step responses of this fourth-order phenomenological model for the step responses of the prototype is below 16.8%, and the root mean square error does not exceed 1.84%.

This example demonstrates that there is no improvement by using the fourth-order phenomenological model, and the best results are obtained by the lower order phenomenological models (see Table 2).

Table 2. Numerical comparison of phenomenological models of different orders for the second example.
Original model (prototype) Reduced-order phenomenological model Maximum deviation (%) Root mean square error (%)
11th order 2nd order 6.4 0.22
3rd order 0.6 0.0023
  
12th order 2nd order 27 4.55
3rd order 15.9 1.66
4th order 16.8 1.84

6. Planned Future Research

Increasing the speed of calculation using phenomenological models is not a single solution for fast computer calculations for the problem of electrical power system stability. From this point of view, the authors see the continuation of the research in two ways:
  • o.

    Using fast computational methods to create high-speed computer phenomenological models. One of the solutions is using recurrent equations based on the z-transform (e.g., old but still actual book [38]).

  • o.

    Using a simple neural network as a phenomenological model instead of transfer functions. It is planned to test the effectiveness and speed of operation of these two methods to reproduce the behavior of complex systems.

7. Conclusion

The use of a phenomenological approach with a model’s description as a “black box” to the stability analysis of power systems provides a simplified computer model of the studied system of sufficient accuracy, which allows high-speed calculations during computer analysis of the dynamics of electrotechnical systems.

The studies have shown that simple phenomenological models of the second and third orders allow us to obtain sufficient accuracy for reproducing the behavior of complex parts of electrical power systems.

Disclosure

The work has been performed as part of the authors’ employment at Lviv Polytechnic National University.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The authors have nothing to report.

    Data Availability Statement

    The data used to support the findings can be obtained using the basic functions available in most mathematical software.

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