1.1. Preliminaries
The energy policy of the European Union forces progressive modernization changes in the structure of generation sources in power systems related to reducing the impact of energy on the environment, especially for technologies related to the energy combustion of fuels. The generation system in Poland is based 70% on coal-fired power plants. Due to the high emissivity of these sources and the high costs of the fluid gas treatment installation, the Polish Energy Policy by 2040 [
1] aims to introduce progressive changes in the energy generation structure. According to the assumptions [
1], the transformation of the generation system in Poland is to be based on the replacement of conventional power plants with renewable energy sources and nuclear energy. The first block of the nuclear power plant with a capacity of 1–1.6 GW is to be commissioned in 2033, for which the location was already selected at the end of 2021. The government company, Polish Nuclear Power Plants, selected the coastal location of Lubiatowo-Kopalino in the Choczewo commune in Pomerania for the first nuclear reactor [
2]. In the following years, it is planned to launch another five such units at intervals of 2–3 years. As for renewable energy sources, it is expected that by 2040 the installed capacity of renewable sources will increase almost twice, from the current 15% to 28.5% [
3]. However, due to rapid climate changes, the transition towards a non-nuclear renewable energy system remains a major political challenge, which can be made even worse by increases in energy supply chain uncertainty given rise by the shift away from fossil fuels [
3]. The described development activities are aimed at replacing the shut down of conventional power plants in a way that conditions the fulfillment of the power balance in the system, as well as a significant reduction in the national emission of greenhouse gases and air pollutants. However, from the point of view of introducing a significant share of renewable sources, it is necessary to take the stochasticity of their work and significant fluctuations in generated power into account, which are associated with the need to equip classic coal or nuclear units with modern control systems capable of responding to dynamic changes in the power generated in the system power industry. Therefore, the optimization and analysis of dynamic responses of power unit control systems is a critical issue from the point of view of the planned development and upgrade in the generation structure of the Polish Power System. The latter has been carried out in the reported application by the use of integral-based performance indices.
There is a number of indices for evaluation of electrical energy quality, which differ in their nature and allow simultaneous, and yet independent from one another, evaluation of various characteristics of the electric power generation process. In the case of control system design, a single performance index is usually needed to clearly mirror the control quality. Introduction of the indices is, however, necessary both for the evaluation of the operation of control systems and the synthesis of controllers.
The aim of the research work described in this paper is to find the index which is related to a set of those used hitherto, as per relation to the same phenomena, and which is at the same time useful for implementation and analysis of control systems.
As the quality of electrical energy is a major factor when the development of modern societies is concerned, as well as there is a growing demand on the generation of electric power in accordance with the speed of economic development of societies, one can clearly identify a growing need to increase power plants’ efficiency and improve electrical energy quality.
In order to develop solutions improving this quality, it is necessary to have appropriate methods for their assessment. Thanks to the suitably selected indices, it is possible to compare solutions and seek the superior one which satisfies all the requirements. Currently, various indices are used, derived from the control theory and energy engineering. To facilitate the synthesis and analysis of control systems, one aggregated index is needed. This article proposes to use the integral of the squared error (ISE) or integral of the squared error multiplied by time (ITSE) indices as candidates to mirror the quality of operation of the turbine-generator control system. Obviously, the performance of the turbine-generator control loop is a function of a suitable design and tuning of its controller.
In addition, as far specific characteristics of a turbine-generator set are concerned, using the proposed control method (by appropriate tuning of performance indices) clearly leads to improvement in the quality of energy generated by the turbine-generator set. It reduces rotor speed fluctuations in dynamic conditions for arbitrary load changes or for changes in a shaft torque, being typical cases in the system with renewable energy sources present. A proper tuning of controllers is followed by the possibility to fix the properties of transient- and steady-state performance in closed-loop systems [
4]. To be able to evaluate the properties of such systems in a qualitative manner, performance measures need to be introduced first. These should be related to the desired properties of specific responses, and thus should be expressed in terms of both transient- and steady-state performance [
5].
The obtained closed-loop system, after tuning, is a result of a compromise between the aforementioned factors, as usually steady-state error rejection is the adverse requirement with respect to the stability margins one, and it is impossible to meet contradictory aims simultaneously [
6]. A shortlist of these might include: good regulation against disturbances, desirable response to commands, maintaining low amplitudes of critical signals, etc. To override this problem, it is advantageous to introduce integral performance indices to conveniently mirror the performance of the control system [
7].
The integral indices assessing the control performance are widely used at the optimization stage in a minimization task of a selected performance-related cost function. To make the results portable across various platforms and applications, asymptotic tracking is usually evaluated for a span of different types of reference signals, such as a unit step, ramp or parabolic inputs, as a basic control task in feedback systems [
8]. In such an application, the presented integral performance-related indices, using the information concerning the tracking error in the closed-loop system, are used as quantitative measures of the control quality. In this case, the attained minimum, coupled with selection of controller gains, mirrors the fact that optimal control policy is obtained [
9].
Typically, quality indices found in the literature focus either on long-term quality assessment in order to analyze the network or on evaluating electric power suppliers. In [
10], the indices are proposed to analyze the quality of electricity every month to give scores to different utility companies, whereas in [
11], the authors analyze the situation in the Italian power system in the long term, looking into the continuity of supply and differences in quality between the north and the south of the country. In [
12], the usage of wavelet packet transforms to measure non-stationary power quality disturbances, and in [
13], wavelets are used to visualize time-varying power quality indices. Some authors use probabilistic methods to assess the power quality using Markov models [
14] or the Monte Carlo method [
15]. The authors of [
16] use neural networks to classify the disturbances occurring in the system offline. Paper [
17] contains an overview of commonly used voltage characteristics such as 10 min voltage and total harmonic distortion factor (THD) averages, 10 s frequency averages or 2 h long-term voltage fluctuation analysis. What is more, the authors of [
18] use machine learning and artificial neural networks to analyze the power quality in a specific site using common parameters counted as 10 min averages as input. The aforementioned approaches focus on a single power quality index (e.g., total harmonic distortion). On the contrary, the global quality indices (GPQI) [
19] proposed in the literature are based on 10/15 min time frames for which quality indices are calculated and then aggregated into a single index showing the global quality.
Despite a very valuable analysis of power quality indices, all of these approaches have, unfortunately, general (site assessment and company assessment) and long-term (10 min, months and years) focus and cannot be used in the case of control system analysis and synthesis where physical processes occur within milliseconds. Therefore, a new approach focused primarily on this particular issue is needed, which constitutes the motivation to undertake the proposed research project. The proposed quality index is expected to be able to properly assess power quality but at the same time must be useful for control systems analysis, synthesis and parameter tuning.
The presented literature review shows some separation of electrical engineering from control engineering. New publications in the field of control engineering analyze new approaches in isolation from the specific case of controlling the turbine-generator set of a nuclear power plant. Publications in the field of electrical engineering, in turn, focus on a detailed analysis of individual energy parameters, or on a global analysis of energy quality but over long periods of time. The presented article connects the achievements of control theory with the achievements of electrical engineering, and thus fills a certain gap between these areas. The conducted research is the starting point for further development of better control systems in the power industry.
The main novelty of the solution presented in the paper is the use of global quality indices with characteristics suitable for use in the synthesis of used control systems. The presented solution takes the analysis of critical parameters from the point of view of the control system of the turbine-generator set and analyzes the data taking very high resolution into account. This is to correctly assess the quality of the control system and to find the optimal controller parameters, which is not possible with the use of widely used individual, classic quality indices.
The main contribution of the work is the confirmation that the integral indices of electric power quality developed in this paper allow to determine the quality of the control system with a single value capturing its viable characteristics. This type of approach significantly facilitates the analysis and synthesis of control systems. The model predictive control technique was applied to control the turbo-generator set, and its synthesis is based on the proposed indices.
The paper is structured as follows:
Section 1.2 describes the electrical power quality definition used within this paper.
Section 2 discusses the problem including the description of the power quality indices that can be found in the literature.
Section 3 describes the turbine-generator set control problem which is the basis for further considerations, while
Section 3.2 presents the proposed integral index. Simulation results of computing indices’ values for several control systems with different parameters are presented in
Section 4. The article ends with a conclusion resulting from the conducted research and lists the directions for further research.
1.2. Electrical Power Quality
As per the selected subject of the presented research, it is to be mentioned that the definition of energy generation quality depends on the loads considered and the way in which energy parameter deviations affect their final values. The definition is very customer-oriented, where a quality-related problem is defined as follows [
20]:
Any power problem manifested in voltage, current, or frequency deviations that results in failure or misoperation of customer equipment.
A few types of such quality problems can be found in the literature, for example, see [
17,
21], among which one can list amplitude related, waveform disruption related, balance related and frequency related (
Figure 1).
As a nearly perfect sine-wave, voltage is generated by the generator [
20], the current- and harmonics-related problems are omitted in this paper, since the turbo-generator set control system has no influence on these quality problems, and hence, including these phenomena in the control-oriented indices would not result in any solution improvement. Therefore, only voltage dips, swells, undervoltage, overvoltage, voltage fluctuations and frequency variations are considered in the paper, (highlighted in bold in the previous paragraph).
The turbo-generator set control system should not introduce such problems as they can influence a large group of energy consumers [
17], namely: CNC machines, adjustable speed drives, personal computers, programmable logic controllers (PLCs), relays, contactors, motor starters, fax machines, metal-halid and high pressure sodium lighting, telcom switching equipment, electronic ballast fluorescent lighting, etc.
When solving voltage and frequency stabilization control problems resulting with no disruptions in the electrical power system (EPS) given rise by the turbine-generator set, all the considerations presented above must be taken into account. To ensure high quality of stabilization, quality indices must be introduced to properly measure the results of the controller’s actions, and mirror the expectations of the designer. From this viewpoint, the power quality index for control purposes is defined in the paper as:
The index that can quantify the behavior of the power system assessing the scale of power problems manifested in voltage, current, or frequency and that can be used for control system synthesis and evaluation.
The next Sections describe the details of the introduced control problem and introduce such control-oriented indices in a form of ISE/ITSE integrals.