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## Hypotheses, computational complexity and solution method

The selected methodological approach has one structural hypothesis, which supposes that the cost functions are convex and the constraint functions are jointly convex with respect to the continuous relaxation. Then, some simplification hypotheses have been made to limit the computational complexity of the optimisation problem, however, raising them is quite straightforward. For instance:

1. Variable generation costs are considered linear, but a piece-wise linear or even a quadratic model could be used. In the latter case the problem becomes a Mixed- Integer Quadratic Programming problem (MIQP) that can be handled by efficient and reliable commercial software (such as GUROBI [110]) at a (significantly) higher computational cost.

2. Single bus model (no network consideration) is used, but DC power flow or quadratic formulation of AC constraints could be included. In the latter case the problem becomes Mixed-Integer Quadratically Constrained Quadratic Program (MI-QCQP).

3. Conventional units are considered to be thermal, but different terms can be added to the objective function to account for other types of power plants.

4. Only units operational constraints, power balance equation and three types of reserve are modelled, but differentiated variables can be used for the secondary and tertiary reserves and additional constraints, such as emission, fuel, must run units, modulation etc., can be added as long as they preserve the MILP problem structure.

### Reduced order system frequency response model

This section analyses different models proposed in literature to characterise the primary frequency response in an electrical power system. Their pros and cons in view of the construction of suitable frequency nadir constraints for the FCUC problem are discussed. The need of modelling some additional features is highlighted and a improved reduced order system frequency response (ROSFR) model is proposed.

In power systems, different dynamic phenomena can be categorised according to their time scale; from wave phenomena (microseconds) to thermodynamic phenomena (hours), passing through electromagnetic and electromechanical phenomena [2, 9]. On the one hand, electromagnetic interactions occur within the generator at a time scale of some milliseconds after a disturbance appears on the system. In this case, the interest is placed in the fault current, internal voltages and fluxes, while the rotor speed is considered to be constant. On the other hand, electromechanical dynamics analysis includes the variation of the rotor speed, and the action of the turbine and generator control systems. In this work, the interest is place in the latter. Then, different hypotheses can be made to achieve a trade off between the complexity of the generator model and the accuracy of the overall system model. A “detailed » or “complete » electromechanical model of a generator is composed by a system of ordinary differential equations (ODE) that may reach several orders (4-8) to include its internal dynamics, as well as the inertial and controllers response.

In transient stability analysis, where the attention is placed on the rotor angular position following a sudden disturbance, reduced order models tend to be used. In fact, following a large power imbalance the angular speed of generating units varies since part of the kinetic energy stored in the rotating masses is injected into the network, making the system frequency evolve. This evolution can be studied with the torque (couple or moment of force) equation of rotating masses, referred in the transient stability terminology as the swing equation, and represented by equation (2.15).

#### Multi-machine ROSFR model

The model presented here was proposed by D.L. Hau in 2006 [128] and tackles some of the aforementioned issues. In this case, the individual response on each unit to the frequency variation is represented in a separate way, which enables the consideration of the heterogeneous characteristics of different units. As a consequence, the contribution (and impact) of individual units to the frequency response can now be examined. Nevertheless, hypothesis 1-5 of the previous model are still required for the MM-ROSFR. In this case, the response of the overall system is given by the addition of different first order subsystems as shown in figure 2.7.

**Proposed ROSFR model**

In this work a test system is built based on existing units in different French Non Interconnected areas (ZNI, by its French acronym), to ensure representativeness of the results. A simplified model of their regulation includes, at least, a proportional controller (the power/frequency droop) and a lead-lag compensator, typically used to improve the frequency response of a feedback loop in a control system. Therefore, the MM-ROSFR is improved for taking into account:

1. The inertial response of surviving synchronous production units.

2. The primary frequency regulation characteristics and limits of the units providing this ancillary service; this is their governor droops, lead-lag compensators, maximal power outputs and the saturation of the regulations.

3. The load damping following frequency drop.

The model of the primary frequency regulation of a unit j is depicted in figure 2.8. In this representation, the outage of unit k has been considered as the disturbance (Pstep = gk).

**Table of contents :**

**1 Introduction **

1.1 Context and challenges

1.1.1 Power system operation and security

1.1.2 Evolution of power systems

1.1.3 Emerging frequency regulation issues

1.2 Motivation and problem statement

1.3 Objectives and scope

1.4 Thesis outline

1.5 Main contributions and originality claim

**2 Formulation of classic models to study the primary frequency response **

2.1 Introduction

2.2 Unit commitment model

2.2.1 Background

2.2.2 Notation

2.2.3 Deterministic UC model

2.2.4 Hypotheses, computational complexity and solution method

2.3 Reduced order system frequency response model

2.3.1 Notation

2.3.2 Equivalent machine ROSFR model

2.3.3 Multi-machine ROSFR model

2.3.4 Proposed ROSFR model

2.4 Numerical analysis

2.4.1 UC solution

2.4.2 Primary frequency response

2.4.3 Relationship between frequency nadirs and generation schedules

2.4.4 Computational details

2.5 Conclusion

**3 Impact of PV generation on the primary frequency response **

3.1 Introduction

L Contents

3.1.1 Background

3.1.2 Methodology

3.2 Understanding generation scheduling changes with PV

3.2.1 Day-ahead demand and PV generation forecast

3.2.2 PV integration scenarios and residual demand

3.2.3 Optimisation results

3.3 Evolution of the primary frequency response

3.4 Case study

3.4.1 Energy mix

3.4.2 Primary frequency response

3.4.3 Periods with an insufficient dynamic response

3.4.4 Relation between the frequency and the PV generation share

3.5 Conclusion

**4 Limiting UFLS risk with high share of non-synchronous generation **

4.1 Introduction

4.1.1 Background

4.1.2 Methodology

4.2 Implemented models

4.2.1 Enhanced security constraints

4.2.2 V-RES dispatch-down

4.2.3 Dynamic support from non-conventional providers

4.3 Case study

4.3.1 Primary reserve volume

4.3.2 Inertia constraints

4.3.3 Relation between the V-RES dispatch-down and the UFLS risk .

4.3.4 Contribution of frequency regulation resources by new providers

4.4 Conclusion

**5 A convex formulation for the FCUC problem **

5.1 Introduction

5.1.1 Generalities on convex optimisation

5.1.2 Interest of convex optimisation for the FCUC problem

5.1.3 Decomposition methods

5.2 Benders’ decomposition approach for the FCUC

5.2.1 FCUC decomposed formulation

5.2.2 Proposed algorithm

5.2.3 Numerical implementation

5.3 Quadratic stabilisation of the Benders’ method

5.4 Conclusion

**6 General conclusions and perspectives **

6.1 Dissertation overview

6.2 General remarks

6.3 Perspectives

**Bibliography**