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Frequency behaviour of two-source systems
Introduction
The interactions between synchronous machines and power electronics-based sources are not easy to analyse. Seeking to facilitate the understanding of this matter, the key idea of this chapter is to study the interactions between these sources in a very simple system composed of two generators and a load. For this purpose, the analysis relies on simplified models of the sources and other elements of the system, providing analytical interpretations of the interactions between a synchronous machine and a PE-based source either behaving as a voltage or a current source, i.e. VCI or CCI mode.
It has been shown in Chapter 1 that the simplified model of a PE-based source operating in VCI mode presents similarities to that of the synchronous machine. Therefore, considering some adjustments, the classical methodology proposed in the specialised literature [52], [54], [57] can be applied in analysis of interactions between the VCI and the synchronous machine.
In order to recall the classical methodology, the case of a two-synchronous machine system is first analysed. From a highly detailed level representation of the system, it is possible to design a simplified model used to observe the frequency behaviour displayed by each synchronous machine. Considering some simplifying hypotheses, the simplified model can be reduced to observe the main dynamics of the system, which are decomposed into two components: the “dominant behaviour” illustrates the overall frequency dynamics displayed by both synchronous machines, whereas the “oscillating behaviour” illustrates the frequency oscillations between them. The same methodology can be used to analyse the interactions between a VCI and a synchronous machine. However, due to the difference in the magnitude order of both sources, the simulation results show a quite different frequency behaviour.
Regarding the CCI, as also mentioned in Chapter 1, its behaviour is very different to that of the VCI, and the simplification method is not the same. It is still possible to determine the dominant behaviour of a system composed of a CCI and a synchronous machine, however its representativeness is limited since the CCI can easily display stability issues. Therefore, a method based on the linearised representation of the system is proposed in order to determine its stability limits.
In this chapter, multiple scenarios considering the variation of macroscopic parameters of the power system — e.g. electrical distance, inertial effect, power frequency characteristics — are analysed. It is emphasised that the findings from the analyses carried out in this chapter can be considered as the foundations for the study of interconnected power systems further in this work.
The reference system: association of two synchronous machines
The two-synchronous machine system is considered as reference for the present chapter. The objective of this section is to describe the system and to introduce the different repres-entations which can be employed on the analysis of the frequency behaviour of a two-voltage source system.
Description of the studied system: the reference model
Figure 2.1 illustrates the single line diagram of the Reference Model of the two-source system in which both generators are synchronous machines. The control chain of each synchronous machine is composed of an IEESGO governor and a steam turbine, a ST1C excitation system and a PSS1A power system stabiliser. The modelling of the synchronous machines and their controllers are introduced in Chapter 1, and the parameters adopted for the system are given in Tables 1.2, 1.5, 1.7 and 1.9. The design of the grid elements are described as follows, and their adopted parameters are given in Table 1.1.
the transmission line is represented as a series impedance,
the step-up transformers are represented only with their leakage inductance, neglecting the active losses,
the load is modelled as a composite of a shunt resistance, inductance and capacitance in parallel, with constant impedance.
A linearised form of the reference model is proposed to perform the analysis in the frequency domain. Since the dynamics of the reference representation of the system are described with 44 differential equations, the linearised form of the model has 44 differential states:
eight for the grid:
– two for the equivalent T1 + L13,
– two for the transformer T2,
– four for the load.
18 for each generator:
– eight for the synchronous machine: eqs. (1.1) to (1.6) and eq. (1.13),
– five for the IEESGO: Figure 1.8,
– one for the ST1C: Figure 1.10c,
– four for the PSS1A: Figure 1.11.
In order to verify the compliance of the linearised form of the reference model, the dynamics obtained with both representations in time domain simulations are compared. For this purpose, the Transmission Line Length (TLL) of L13 is 25 km, and the initial operating point of the system is given in Table 2.1.
The following analyses are carried out in time and frequency domains. In further fre-quency domain analyses, the reference eigenvalues are extracted from the linearised form of the reference model of the system, and therefore, these eigenvalues are hereinafter referred to as Reference Model. An in-depth analysis regarding the dynamics of the system is realised further in this section.
Reduced order representations of the system
It is nearly impossible to draw out any physical meaning using a 44th order model such as that described in Section 2.2.1. Therefore, several levels of simplifications are proposed in order to understand the main dynamics of the system. As described in Chapter 1, the association of E0X0H models of synchronous machines with their power frequency regulators and the grid is employed as a first level of simplification. From this model, it is possible to deduce another representation, the dominant model, which is much more simplified than the reference model, to highlight the dominant behaviour of the frequency. Furthermore, based on the frequency decoupling due to the high time constant of the steam turbines, it is also possible to neglect the dynamics of the regulators from the simplified model in order to analyse the oscillating frequency behaviour of the system.
The simplified model of the SM vs SM system
Figure 2.3 illustrates the block diagram of the Simplified Model of the SM vs SM system. As introduced in Chapter 1, each synchronous machine is represented with the E0X0H model associated with an IEESGO as power frequency regulator. In this system, differently than the synchronous machine vs infinite bus system, the frequency of the grid is not constant, but it is a composite of the frequencies of both synchronous machines. This composite frequency displays the behaviour of the Centre of Inertia (COI) of the system, and is computed as given in eq. (2.1) [68], [98]–[100].
The dynamics of the simplified representation of the system are described with 14 dif-ferential equations, being, for each generator, two for the synchronous machine and five for the IEESGO. The parameters adopted for this model are the same as those of the reference model of the synchronous machine, excepting the values of Kd;1 and Kd;2 which, as well as in Section 1.2, they are estimated in order to reasonably represent the damping of the system.
The network linking the two synchronous machines and the load is represented with a quasi static model (Kcplg). The modelling and numerical application to design the grid are presented in Appendix D. The size of Kcplg is 2-by-3, and, for the following analysis, its elements are noted as K11 to K23. The principal steps to define Kcplg are summarised as follows:
1. Determination of the admittance matrix of the system (Y);
2. Computation of @Pi=@ i (Jacobian matrix, J);
3. Reduction of the dependent variable ( 3).
The eigenvalues maps of both simplified and reference representations of the system are illustrated in Figure 2.4. The eigenvalues of the simplified representation display a good estimation of the corresponding ones of the reference model, and therefore, it might be possible to determine the expected frequency behaviour of the system using the simplified model. However, this representation still has 14 eigenvalues, and not all of them are related to the frequency dynamics. Therefore, in order to identify the eigenvalues which have an influence on the frequency behaviour, their participation factors in the electromechanical states of the system are illustrated in Figure 2.5. The colour intensities indicate the factors according to the sidebar on the right of the figure. The factors smaller than 5% are neglected and represented in grey.
Between the 14 eigenvalues of the simplified representation of the system, only 3;4, 5;6, 8 and 9 have influence on !1 and !2, and therefore, focus is applied to these eigenvalues, summarising them in Table 2.2 with their corresponding ones of the reference model for numerical comparison. The eigenvalue 1 is overlooked in the analysis, since it corresponds to the null eigenvalue related to the absence of common angle reference [52].
Table of contents :
Abstract
Résumé étendu en Français
Contexte de l’étude
Objectifs de la thèse
Liste des publications issues de cette thèse
Les conclusions de la thèse
Les contributions de la thèse
General Introduction
Background and motivations
Thesis scope & objectives
Outline of chapters
List of publications derived from this work
1 Fundamental behaviour of grid sources
1.1 Introduction
1.2 Synchronous Machines
1.2.1 Reference model of Synchronous Machines
1.2.2 E0X0H SM model
1.2.3 Comparison between SM models
1.2.4 Prime movers and governor
1.2.5 Excitation system
1.2.6 The Power System Stabiliser
1.2.7 Considerations about the E0X0H model
1.3 Power Electronics (PE)-based sources
1.3.1 Voltage Control mode Inverters (VCIs)
1.3.2 Current Control mode Inverters (CCIs)
1.4 Chapter conclusions
2 Frequency behaviour of two-source systems
2.1 Introduction
2.2 The reference system: association of two synchronous machines
2.2.1 Description of the studied system: the reference model
2.2.2 Reduced order representations of the system
2.2.3 Comparison between the models
2.3 Replacement of a synchronous machine with a VCI
2.3.1 Description of the studied system: the reference model
2.3.2 Reduced order representations of the system
2.3.3 Impact of the frequency support of the VCI on the dynamics of the system
2.3.4 Impact of power ratio between the sources
2.3.5 Influence of the VCI control strategy
2.3.6 Summary of the findings
2.4 Two-voltage source system composed of VCIs
2.4.1 Description of the studied system: the reference model
2.4.2 A completely different frequency behaviour
2.4.3 Comparison with the two-synchronous machine system
2.5 The association of a CCI with a synchronous machine
2.5.1 Description of the studied system: the reference model
2.5.2 The dominant model of the CCI vs SM system
2.5.3 Dynamic analysis of a CCI vs SM system
2.6 Chapter conclusions
3 Frequency dynamics assessment of interconnected power systems
3.1 Introduction
3.2 The 2-area 4-generator system: the Kundur system with 100% of synchronous machines
3.2.1 Description of the reference model of the system
3.2.2 Frequency behaviour of the system
3.2.3 Methodology employed for the design of reduced order models
3.2.4 Considerations about the aggregated model of the system
3.3 Introduction of PE-based sources in the Kundur system
3.3.1 Replacement of a SM with a PE-based source: the VCI case
3.3.2 Replacement of a SM with a PE-based source: the CCI case
3.3.3 Replacement of a synchronous area with a PE-based area
3.4 Conclusions
4 Conclusions and future work
4.1 Thesis contributions
4.2 Recommendations for future work
Bibliography
