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## Modelling of transmission systems with a high power electronics penetration

This chapter presents the diﬀerent models that are used in this report. For each model, its diﬀerential and algebraic equations are given, as well as some block diagrams to illustrate them. In all cases, the equations are written in the DQ0 reference frame (the frequency/angle are specified in each case) and in pu (the basis is also specified in each case) to simplify the linear analysis and accelerate the simulation (see Appendix for more explanations). This is possible because all the models are considered as balanced. This hypothesis is made for the sake of simplicity but the developed MOR methods are easily applicable to unbalanced systems, provided that the model describing them is given.

The first section presents the basic elements in a power system: branches, trans-formers, loads, shunt capacitors and synchronous machines. The second section presents the diﬀerent models used for PE converters: grid feeding two-level voltage source con-verter, grid forming converter and Modular Multilevel Converter (MMC). Finally, the third section presents how a complete model is deduced when all these elements are assembled in a transmission system.

### Modelling of the basic elements of a transmis-sion system

In this section, the models that are used for the branches, transformers, loads, shunt capacitors and synchronous machines are presented.

**Branches**

The modelling of the lines has been a particularly important topic in studying transmission systems [67] and it is still today [68]. Several models exist, from the most complex one to the simplest one. In this thesis, two models are used: a pi-line model and an RL-line model. They are presented in the following.

**Pi-line model**

The Pi-line model is presented in Figure 2.1. This model takes into account the capacitive eﬀects of the line.

**Transformers**

As for the lines, the choice of the model for transformers is a very important topic [69]. In this report, a generalized RL-model is used to model them in pu and in the DQ0 reference frame of a frequency ω. A phase shift δ can be taken into account with this model, as well as a change in the transformer ratio r. This model is represented in Figure 2.3.

**Shunt capacitors**

In transmission systems there can be shunt capacitors that are put to help maintain-ing the voltage at a certain level. In this report, they are modelled as simple capacitors, as shown in Figure 2.5.

**Synchronous machines**

In one test case in chapter 4, there is a PE converter but also one SM to study the interactions between the two. This is why a good synchronous machine model is needed. It is presented here and is taken from [11]. In this model, the equations are given in pu and in the reference frame of a frequency ω, the frequency of the SM rotor.

The general structure, detailed hereafter, is given in Figure 2.6. It is made of the machine itself, an RL line modelling a transformer and a control. Each part is detailed in the following.

**Modelling of the machine**

The equations modelling the physical part of the SM are given here. The first six equations model the dynamics of the fluxes in the SM.

**Converters models**

As presented in the introduction, this section focuses on the models of PE converters. The first subsection presents the model of a grid feeding converter, the second one the model of a grid forming converter and the third one the model of an MMC.

#### Grid feeding converter

A grid feeding [70] or grid following converter is a PE converter that just injects power/current into the grid. Unlike the grid forming converter it is considered as a current source and not voltage source because it does not create the voltage. It is the most used type of converter today, mainly to connect renewable sources to the system [71].

**General structure of the grid feeding converter**

The structure of the grid feeding converter studied in this thesis and its control is presented in Figure 2.9.

This figure shows that the grid feeding converter is made of the DC/AC converter itself (here the DC part is modelled as a perfect voltage source), an RLC filter, an RL transformer and the control part is made of an external loop, a Phase-Locked Loop (PLL) to measure the angle and the frequency of the grid, and a current loop. All these parts are presented in the following.

**Table of contents :**

**Introduction **

General context

Presentation of the MIGRATE project

Contributions

Outline of the thesis

Publications

**1 Scientific context **

1.1 Tools and methods for the simulation and analysis of transmission systems

1.1.1 Simulation tools

1.1.2 Analysis tools

1.1.3 Conclusion

1.2 Model order reduction

1.2.1 Modal truncation

1.2.2 Balanced truncation and singular perturbation approximation

1.2.3 Proper orthogonal decomposition

1.2.4 Krylov methods

1.2.5 Methods comparison and applicability on transmission systems with a high PE penetration

1.3 Chapter conclusion

**2 Modelling of transmission systems with a high power electronics penetration**

2.1 Modelling of the basic elements of a transmission system

2.1.1 Branches

2.1.2 Transformers

2.1.3 Loads

2.1.4 Shunt capacitors

2.1.5 Synchronous machines

2.2 Converters models

2.2.1 Grid feeding converter

2.2.2 Grid forming converter

2.2.3 Modular multilevel converter

2.3 Modelling of a complete transmission system

2.4 Chapter conclusion

**3 Development of model order reduction methods**

3.1 Common principles of the developed methods

3.1.1 The state residualization

3.1.2 The modal approach

3.1.3 Conclusion

3.2 Developed strategies to choose the groups of states to residualize/the groups of poles to discard

3.2.1 Strategy 1: discarding the fastest poles

3.2.2 Strategy 2: discarding the poles that depend on the less observable and reachable states in the balanced realization

3.2.3 Strategy 3: discarding some poles to minimize an error criterion

3.2.4 Conclusion

3.3 Chapter conclusion

**4 Application of the methods for the simulation and analysis of transmission systems with a high power electronics penetration**

4.1 Simple test cases

4.1.1 One grid forming converter connected to an infinite grid

4.1.2 Two-converter system

4.1.3 Three-converter system

4.1.4 Conclusion

4.2 Transmission system test case

4.3 Test case with one grid forming converter and one modular multilevel converter

4.4 Test case with one grid feeding converter and one synchronous machine

4.5 Chapter conclusion

**Conclusion and perspectives**

Conclusion

Perspectives

**Bibliography **