Stability-Constrained Optimization for Modern Power System Operation and Planning E-Book

Yan Xu

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: IEEE Press Series on Power Engineering
• Sprache: Englisch

Seitenzahl: 649

Veröffentlichungsjahr: 2023

Beschreibung

Stability-Constrained Optimization for Modern Power System Operation and Planning Comprehensive treatment of an aspect of stability constrained operations and planning, including the latest research and engineering practices Stability-Constrained Optimization for Modern Power System Operation and Planning focuses on the subject of power system stability. Unlike other books in this field, which focus mainly on the dynamic modeling, stability analysis, and controller design for power systems, this book is instead dedicated to stability-constrained optimization methodologies for power system stability enhancement, including transient stability-constrained power system dispatch and operational control, and voltage stability-constrained dynamic VAR Resources planning in the power grid. Authored by experts with established track records in both research and industry, Stability-Constrained Optimization for Modern Power System Operation and Planning covers three parts: * Overview of power system stability, including definition, classification, phenomenon, mathematical models and analysis tools for stability assessment, as well as a review of recent large-scale blackouts in the world * Transient stability-constrained optimal power flow (TSC-OPF) and transient stability constrained-unit commitment (TSC-UC) for power system dispatch and operational control, including a series of optimization model formulations, transient stability constraint construction and extraction methods, and efficient solution approaches * Optimal planning of dynamic VAR Resources (such as STATCOM and SVC) in power system for voltage stability enhancement, including a set of voltage stability indices, candidate bus selection methods, multi-objective optimization model formulations, and high-quality solution approaches Stability-Constrained Optimization for Modern Power System Operation and Planning provides the latest research findings to scholars, researchers, and postgraduate students who are seeking optimization methodologies for power system stability enhancement, while also offering key practical methods to power system operators, planners, and optimization algorithm developers in the power industry.

Leseprobe

Table of Contents

Cover

Series Page

Title Page

Copyright Page

About the Authors

Foreword

Preface

Part I: Power System Stability Preliminaries

List of Acronyms

1 Power System Stability: Definition, Classification, and Phenomenon

1.1 Introduction

1.2 Definition

1.3 Classification

1.4 Rotor Angle Stability

1.5 Voltage Stability

1.6 Frequency Stability

1.7 Resonance Stability

1.8 Converter‐Driven Stability

References

2 Mathematical Models and Analysis Methods for Power System Stability

2.1 Introduction

2.2 General Mathematical Model

2.3 Transient Stability Criteria

2.4 Time‐Domain Simulation

2.5 Extended Equal‐Area Criterion (EEAC)

2.6 Trajectory Sensitivity Analysis

Nomenclature

References

3 Recent Large‐Scale Blackouts in the World

3.1 Introduction

3.2 Major Blackouts in the World

References

Part II: Transient Stability-Constrained Dispatch and Operational Control

List of Acronyms

4 Power System Operation and Optimization Models

4.1 Introduction

4.2 Overview and Framework of Power System Operation

4.3 Mathematical Models for Power System Optimal Operation

4.4 Power System Operation Practices

Nomenclature

References

5 Transient Stability‐Constrained Optimal Power Flow (TSC‐OPF): Modeling and Classic Solution Methods

5.1 Mathematical Model

5.2 Discretization‐based Method

5.3 Direct Method

5.4 Evolutionary Algorithm‐based Method

5.5 Discussion and Summary

Nomenclature

References

6 Hybrid Method for Transient Stability‐Constrained Optimal Power Flow

6.1 Introduction

6.2 Proposed Hybrid Method

6.3 Technical Specification

6.4 Case Studies

Nomenclature

References

7 Data‐Driven Method for Transient Stability‐Constrained Optimal Power Flow

7.1 Introduction

7.2 Decision Tree‐based Method

7.3 Pattern Discovery‐based Method

7.4 Case Studies

Nomenclature

References

8 Transient Stability‐Constrained Unit Commitment (TSCUC)

8.1 Introduction

8.2 TSC‐UC model

8.3 Transient Stability Control

8.4 Decomposition‐based Solution Approach

8.5 Case Studies

Nomenclature

References

9 Transient Stability‐Constrained Optimal Power Flow under Uncertainties

9.1 Introduction

9.2 TSC‐OPF Model with Uncertain Dynamic Load Models

9.3 Case Studies for TSC‐OPF Under Uncertain Dynamic Loads

9.4 TSC‐OPF Model with Uncertain Wind Power Generation

9.5 Case Studies for TSC‐OPF Under Uncertain Wind Power

9.6 Discussions and Concluding Remarks

Nomenclature

References

10 Optimal Generation Rescheduling for Preventive Transient Stability Control

10.1 Introduction

10.2 Trajectory Sensitivity Analysis for Transient Stability

10.3 Transient Stability Preventive Control Based on Critical OMIB

10.4 Case Studies of Transient Stability Preventive Control Based on the Critical OMIB

10.5 Transient Stability Preventive Control Based on Stability Margin

10.6 Case Studies of Transient Stability Preventive Control Based on Stability Margin

Nomenclature

References

11 Preventive‐Corrective Coordinated Transient Stability‐Constrained Optimal Power Flow under Uncertain Wind Power

11.1 Introduction

11.2 Framework of the PC–CC Coordinated TSC‐OPF

11.3 PC–CC Coordinated Mathematical Model

11.4 Solution Method for the PC–CC Coordinated Model

11.5 Case Studies

Nomenclature

References

12 Robust Coordination of Preventive Control and Emergency Control for Transient Stability Enhancement under Uncertain Wind Power

12.1 Introduction

12.2 Mathematical Formulation

12.3 Transient Stability Constraint Construction

12.4 Solution Approach

12.5 Case Studies

Nomenclature

References

Part III: Voltage Stability-Constrained Dynamic VAR Resources Planning

List of Acronyms

13 Dynamic VAR Resource Planning for Voltage Stability Enhancement

13.1 Framework of Power System VAR Resource Planning

13.2 Mathematical Models for Optimal VAR Resource Planning

13.3 Power System Planning Practices

References

14 Voltage Stability Indices

14.1 Conventional Voltage Stability Criteria

14.2 Steady‐State and Short‐term Voltage Stability Indices

14.3 Time‐Constrained Short‐term Voltage Stability Index

Nomenclature

References

15 Dynamic VAR Resources

15.1 Fundamentals of Dynamic VAR Resources

15.2 Dynamic Models of Dynamic VAR Resources

References

16 Candidate Bus Selection for Dynamic VAR Resource Allocation

16.1 Introduction

16.2 General Framework of Candidate Bus Selection

16.3 Zoning‐based Candidate Bus Selection Method

16.4 Correlated Candidate Bus Selection Method

16.5 Case Studies

Nomenclature

References

17 Multi‐objective Dynamic VAR Resource Planning

17.1 Introduction

17.2 Multi‐objective Optimization Model

17.3 Decomposition‐based Solution Method

17.4 Case Studies

Nomenclature

References

18 Retirement‐Driven Dynamic VAR Resource Planning

18.1 Introduction

18.2 Equipment Retirement Model

18.3 Retirement‐Driven Dynamic VAR Planning Model

18.4 Solution Method

18.5 Case Studies

Nomenclature

References

19 Multi‐stage Coordinated Dynamic VAR Resource Planning

19.1 Introduction

19.2 Coordinated Planning and Operation Model

19.3 Solution Method

19.4 Case Studies

Nomenclature

References

20 Many‐objective Robust Optimization‐based Dynamic VAR Resource Planning

20.1 Introduction

20.2 Robustness Assessment of Planning Decisions

20.3 Many‐objective Dynamic VAR Planning Model

20.4 Many‐objective Optimization Algorithm

20.5 Case Studies

Nomenclature

References

Index

IEEE Press Series on Power and Energy Systems

End User License Agreement

List of Tables

Chapter 3

Table 3.1 Summary of 9 blackouts in the world.

Chapter 5

Table 5.1 Reported simulation results for WSCC 3‐machine 9‐bus system.

Table 5.2 Reported simulation results for New England 39‐bus system.

Chapter 6

Table 6.1 Initial optimal OP of New England system.

Table 6.2 Contingency list for New England system.

Table 6.3 TSC‐OPF calculation results of New England System.

Table 6.4 Result comparison on New England system.

Chapter 7

Table 7.1 Initial operating point of 10‐machine 39‐bus system.

Table 7.2 New stable OP (under Fault 1).

Table 7.3 New stable OP (under multi‐contingency).

Chapter 8

Table 8.1 SCUC/TSCUC results (New England 10‐machine 39‐bus system).

Table 8.2 CPU time (New England 10‐machine system).

Table 8.3 CPU time (IEEE 50‐machine system).

Table 8.4 Total CPU time for TSCUC computations.

Chapter 9

Table 9.1 Testing scenarios determined of OA

L

8

(2

7

).

Table 9.2 Contingency set.

Table 9.3 Base case‐initial generation dispatch (MW).

Table 9.4 Mean value (

μ

) of the load model parameters.

Table 9.5 Single contingency (C1) solution results (MW).

Table 9.6 Multi‐contingency solution results (MW).

Table 9.7 Robustness degree.

Table 9.8 Testing scenarios OA

L

4

(2

3

).

Table 9.9 Contingency set.

Table 9.10 Initial base dispatch (MW).

Table 9.11 Transient stability margin for initial dispatch.

Table 9.12 C1‐constrained robust stability dispatch results.

Table 9.13 Multi‐contingency robust stability dispatch results.

Table 9.14 Transient stability margin for multi‐contingency dispatch.

Table 9.15 CPU time (s) for computational tasks.

Chapter 10

Table 10.1 Active generation output (MW) at base operating point.

Table 10.2 Trajectory sensitivities at

T

u

= 0.5917 second to generator activ...

Table 10.3 Active generation output (MW) and trajectory sensitivity at

T

u

of...

Table 10.4 Contingency set.

Table 10.5 Active generation output (MW) and wind variation (MW) at the base...

Table 10.6 Active generation output (MW) and wind variation (MW) at the base...

Table 10.7 Sensitivities of 7 synchronous generators for C1 and C2.

Table 10.8 Active generation output (MW) and wind variation (MW) for multi‐c...

Chapter 11

Table 11.1 Contingency set.

Table 11.2 Base case dispatch.

Table 11.3 Stability margin for base case.

Table 11.4 Generation rescheduling and load shedding for three contingencies...

Table 11.5 Robust generation for C2.

Table 11.6 Transient stability margin for three contingencies under four sel...

Table 11.7 Generation rescheduling for multi‐contingency (C1).

Table 11.8 Transient stability margin for multi‐contingency under four selec...

Chapter 12

Table 12.1 Contingency set.

Table 12.2 Initial generator dispatch results (MW).

Table 12.3 Transient stability margin for initial dispatch.

Table 12.4 Load shedding amount with 0.9–1.1 uncertainty budget

Table 12.5 Robustness check with different uncertainty budget pair for C1.

Table 12.6 Robustness check with different uncertainty budget pair for C2.

Table 12.7 Robustness check with different uncertainty budget pair for C3.

Table 12.8 Robustness check with different uncertainty budget pair of Nordic...

Table 12.9 Time consumption for different calculation tasks.

Chapter 15

Table 15.1 Comparison between SVC and STATCOM.

Table 15.2 SVSMO3U1 model parameters.

Chapter 16

Table 16.1 Pair Copula families.

Table 16.2 Parameters for simulation setting.

Table 16.3 Simulation results for case study 1.

Table 16.4 Simulation results for case study 2.

Table 16.5 Capacity impact on the result of other zoning methods.

Table 16.6 Comparison between direct averaging and RMS.

Table 16.7 Parameters for simulation analysis.

Table 16.8 Rankings of different candidate selection methods.

Table 16.9 Optimized capacity of STATCOM (in MVar).

Table 16.10 Results comparison (in million US dollar).

Chapter 17

Table 17.1 Parameters for simulation setting.

Table 17.2 Short‐term voltage stability evaluation results with different lo...

Table 17.3 STATCOM installation decision.

Table 17.4 Short‐term voltage stability results with STATCOMs.

Chapter 18

Table 18.1 Parameters for simulation setting.

Table 18.2 Installation and upgrade scheme.

Table 18.3 Capacitor banks retirement scheme.

Table 18.4 Objective values.

Table 18.5 The comparison of the proposed method and previous research.

Chapter 19

Table 19.1 Decomposition of wind power scenarios.

Table 19.2 Parameters for simulation setting.

Table 19.3 Installation decisions (MVAR, 11 candidate buses).

Table 19.4 Compromise solutions.

Table 19.5 Installation decisions (MVAR, 15 candidate buses).

Table 19.6 Robust design comparison.

Table 19.7 Comparison results (MVAR, MW, and Million).

Table 19.8

RI

RVSI

results comparison.

Table 19.9 Details of resilience evaluation results.

Table 19.10 Computation times for different test systems.

Chapter 20

Table 20.1 Parameters for simulation setting.

Table 20.2 Installation decisions (in MVAR).

Table 20.3 Simulation results.

Table 20.4 Computation costs for different test systems.

List of Illustrations

Chapter 1

Figure 1.1 Classification of power system stability.

Figure 1.2 Simulated rotor angles of a transient stable case.

Figure 1.3 Simulated rotor angles of a transient unstable case.

Figure 1.4 Active power oscillation due to the small‐disturbance instability...

Figure 1.5 Simulated voltage trajectories of the large‐disturbance voltage s...

Figure 1.6 Simulated voltage trajectories of the large‐disturbance voltage u...

Figure 1.7 Simulated power–voltage (PV) curve.

Figure 1.8 Simulated post‐disturbance frequency trajectory of a frequency st...

Figure 1.9 Simulated post‐disturbance frequency trajectory of a frequency un...

Figure 1.10 Phenomenon of resonance stability. (a) Torsional torque in LPB‐G...

Figure 1.11 Phenomenon of converter‐driven stability. (a) Measured waveforms...

Chapter 2

Figure 2.1 Illustration of OMIB

P

 − 

δ

plane: (a) unstable case, (b) uns...

Figure 2.2 OMIB trajectory sensitivities with respect to generator active ou...

Chapter 3

Figure 3.1 Schematic map of UCTE area splitting into three areas.

Figure 3.2 South Australia generation mix.

Chapter 4

Figure 4.1 Conflicting triangle for power system operation.

Figure 4.2 Power system operation framework.

Figure 4.3 Power system operation practice classification.

Chapter 5

Figure 5.1 Direct method for TSC‐OPF computation process.

Figure 5.2 EA‐based method for TSC‐OPF computation process.

Chapter 6

Figure 6.1 Flowchart of the hybrid method.

Figure 6.2 Conceptual expression of the optimization process by the hybrid m...

Figure 6.3 Generator output changes (MW) with respect to the initial OP.

Figure 6.4 Convergence curves for the New England system test.

Figure 6.5 Unstable trajectories of initial OP under C1.

Figure 6.6 Stable trajectories of optimal OP under C1.

Figure 6.7 Unstable trajectories of the dynamic equivalent power system.

Figure 6.8 Stable trajectories of the dynamic equivalent power system.

Chapter 7

Figure 7.1 A typical CT for power system stability assessment.

Figure 7.2 An example of PD by residual analysis and recursive partitioning....

Figure 7.3 Computation flowchart of the proposed preventive control.

Figure 7.4 New England 10‐machine 39‐bus system.

Figure 7.5 Rotor angle swing curves of the initial OP under Fault 1 without ...

Figure 7.6 Weight of each generator feature under Fault 1.

Figure 7.7 Critical and noncritical generator distance spaces.

Figure 7.8 DT for preventive control of Fault 1.

Figure 7.9 Rotor angle swing curves under Fault 1 after preventive control....

Figure 7.10 Rotor angle swing curves of the initial OP under Fault 2 without...

Figure 7.11 Weight of each generator feature under Fault 2.

Figure 7.12 DT for preventive control of Fault 2.

Figure 7.13 Rotor angle swing curves after multi‐contingency preventive cont...

Figure 7.14 Rotor angle swing curves of the initial OP under Fault 3.

Figure 7.15 Histogram of TSI distribution of the generated database (Fault 3...

Figure 7.16

RELIEF

feature estimation result (Fault 3).

Figure 7.17 Noncritical feature space constructed by G32 and G33.

Figure 7.18 Different classes of instances and discovered patterns.

Figure 7.19 Secure/insecure regions and corresponding centroids.

Figure 7.20 Rotor angle swing curves of the 1st new OP under Fault 3.

Figure 7.21 Rotor angle swing curves of the 2nd new OP under Fault 3.

Figure 7.22

RELIEF

feature estimation result (Fault 3 and Fault 4).

Figure 7.23

RELIEF

feature estimation result (Fault 4).

Figure 7.24 Rotor angle swing curves of the new OP for multi‐contingency con...

Chapter 8

Figure 8.1 Decomposition strategy for TSCUC.

Figure 8.2 Computation flowchart.

Figure 8.3 Implementation of the introduced approach.

Figure 8.4 SCUC generation dispatch.

Figure 8.5 Multi‐machine rotor angle trajectories (left window) and the corr...

Figure 8.6 Multi‐machine rotor angle trajectories (left window) and the corr...

Figure 8.7 TSCUC generation dispatch–single contingency (C1).

Figure 8.8 TSCUC generation dispatch–multiple contingencies (C1 and C2).

Figure 8.9 A portion of one‐line diagram of the IEEE 50‐machine system.

Figure 8.10 Multi‐machine rotor angle trajectories (upper window) and the co...

Figure 8.11 Multi‐machine rotor angle trajectories (upper window) and the co...

Chapter 9

Figure 9.1 Structure of the complex load model “CLOD.”

Figure 9.2 Decomposition‐based solution approach.

Figure 9.3 System trajectories for base case with dynamic load model.

Figure 9.4 Transient stability margin with varying load component percentage...

Figure 9.5 System trajectories for the multi‐contingency TSCOPF solution und...

Figure 9.6 Decomposition scheme.

Figure 9.7 Computation flowchart.

Figure 9.8 Rotor angle trajectories for C1 under

l

1

: (a)‐unstable (under ini...

Figure 9.9 Pe‐OMIB angle plane for C1 under

l

1

: (a)‐unstable (under initial ...

Figure 9.10 OMIB angle trajectories for C1 under

l

1

.

Figure 9.11 Trajectory sensitivities of synchronous machine's output to OMIB...

Figure 9.12 Transient stability margin versus key parameters: (a)‐fault clea...

Figure 9.13 Rotor angle trajectories for C2 under

l

2

: (a)‐unstable (before d...

Figure 9.14 Pe‐OMIB angle plane for C2 under

l

2

: (a)‐unstable (before dispat...

Figure 9.15 One‐line diagram of Nordic32 system.

Figure 9.16 Unstable trajectories for Nordic32 system before dispatch: (a)‐m...

Figure 9.17 Extremely unstable margin versus fault clearing time.

Figure 9.18 Illustration of stable trajectories for Nordic32 system after di...

Chapter 10

Figure 10.1 Fault clearing time (CT) versus stability margin

η

.

Figure 10.2 Unstable and critical OMIB trajectories – New England test syste...

Figure 10.3 Computation flowchart of the proposed approach.

Figure 10.4 Unstable multi‐machine system trajectories – New England test sy...

Figure 10.5 Equivalent trajectories of CM, NMs, and OMIB – New England test ...

Figure 10.6 Unstable OMIB power‐angle curve – New England test system.

Figure 10.7 OMIB trajectory sensitivities with respect to generator active o...

Figure 10.8 OMIB trajectories – New England test system.

Figure 10.9 System trajectories after stability control – New England test s...

Figure 10.10 System trajectories after stability control – New England test ...

Figure 10.11 System trajectories after only 160.6 MW generation rescheduling...

Figure 10.12 Unstable multi‐machine system trajectories – the 285‐machine 16...

Figure 10.13 Unstable OMIB power‐angle curve – the 285‐machine 1648‐bus syst...

Figure 10.14 Unstable and critical OMIB trajectories – the 285‐machine 1648‐...

Figure 10.15 Stable OMIB power‐angle curve of the system – the 285‐machine 1...

Figure 10.16 Stable multi‐machine system trajectories – the 285‐machine 1648...

Figure 10.17 Computation flowchart of the proposed approach.

Figure 10.18 The structure of the wind generators.

Figure 10.19 Rotor angle trajectories for C1: (a) unstable (before optimizat...

Figure 10.20 Pe‐OMIB angle plane for C1: (a) unstable (before optimization);...

Figure 10.21 Rotor angle trajectories for C2: (a) unstable (before optimizat...

Figure 10.22 Pe‐OMIB angle plane for C2: (a) unstable (before optimization);...

Figure 10.23 Rotor angle trajectories for multi‐contingency before optimizat...

Figure 10.24 Pe‐OMIB angle plane for C2 with the step size of 5 and 40 MW....

Figure 10.25 The stability margin versus step size for contingency 1.

Chapter 11

Figure 11.1 Proposed coordinated framework of PC and CC.

Figure 11.2 Illustration of the golden section search.

Figure 11.3 Flowchart of the solution process.

Figure 11.4 Rotor angle trajectories for C1 under base case: (a)‐unstable (b...

Figure 11.5 Pe‐OMIB angle plane for C2 under only PC control.

Figure 11.6 Pe‐OMIB angle plane for C3 under both PC and CC control.

Figure 11.7 Cost function with respect to the risk coordination parameters f...

Chapter 12

Figure 12.1 Framework of the proposed method.

Figure 12.2 Flowchart of the proposed solution algorithm.

Figure 12.3 Illustration of the worst case of wind power uncertainty.

Figure 12.4 Rotor angle curve under initial case.

Figure 12.5 Rotor angle curve under worst‐case without ELS.

Figure 12.6 Rotor angle curve under worst‐case with ELS.

Figure 12.7 Rotor angle curve under a non‐worst case with ELS.

Figure 12.8 Rotor angle curve under initial case for Nordic system.

Figure 12.9 Rotor angle curve under worst‐case without ELS for Nordic system...

Figure 12.10 Rotor angle curve under worst‐case with ELS for Nordic system....

Chapter 13

Figure 13.1 Power system VAR source planning framework for voltage stability...

Figure 13.2 Time scales of planning stage, pre‐contingency stage, and post‐c...

Figure 13.3 Power system planning practices of electric utilities.

Chapter 14

Figure 14.1 Illustration of

P

–

V

curve.

Figure 14.2 Illustration of

Q

–

V

curve.

Figure 14.3 The two‐bus representation of a power system.

Figure 14.4 Illustration of Typical Post‐contingency Scenario.

Figure 14.5 Illustration of Typical Scenario A.

Figure 14.6 Illustration of Typical Scenario B.

Figure 14.7 Illustration of Typical Scenario C.

Figure 14.8 Illustration of Typical Scenario D.

Figure 14.9 Illustration of Typical Scenario E.

Figure 14.10 Values of against the varying .

Chapter 15

Figure 15.1 Thyristor‐based SVC and STATCOM.

Figure 15.2 Voltage–current characteristics of SVC and STATCOM.

Figure 15.3 STATCOM reactive power output during a contingency.

Figure 15.4 Block diagram of the SVSMO3 STATCOM model [8].

Chapter 16

Figure 16.1 General framework of the conventional candidate bus selection me...

Figure 16.2 General framework of the zoning‐based candidate bus selection me...

Figure 16.3 Structure of initial matrices group.

Figure 16.4 Computation flowcharts of the proposed candidate bus selection m...

Figure 16.5 Illustration of a 5‐dimensional D‐vine copula.

Figure 16.6 Illustration of two‐dimension candidate bus selection criterion....

Figure 16.7 Evolution of bus groups with different priorities.

Figure 16.8 General computation flowchart.

Figure 16.9 New England 39‐bus test system.

Figure 16.10 Contingency severity analysis results.

Figure 16.11 Evaluation results for each bus.

Figure 16.12 Capacity sensitivity comparison of

VCPI

.

Figure 16.13 Capacity sensitivity comparison of

TVSI

r

.

Figure 16.14 Capacity sensitivity comparison of

TVSI

s

.

Figure 16.15 Capacity sensitivity comparison of

ISGA

.

Figure 16.16 Single line diagram of the modified Nordic 74‐bus test system....

Figure 16.17 The correlation between different uncertainties over a year....

Figure 16.18 Optimized D‐vine copulas.

Figure 16.19 Uncertainties sensitivity analysis results (correlation is not ...

Figure 16.20 Dependent uncertainties sensitivity analysis results (D‐vine co...

Figure 16.21 Dependent uncertainties sensitivity analysis results (Gaussian ...

Figure 16.22 Independent capacity sensitivity analysis results.

Figure 16.23 Two‐dimensional candidate bus selection.

Figure 16.24 Comparison between

TVSI

a

and conventional index.

Chapter 17

Figure 17.1 Illustration of Pareto Front.

Figure 17.2 Single line diagram of the New England 39‐bus test system.

Figure 17.3 Complex load model “CLOD”.

Figure 17.4 Pareto optimal solutions and the Pareto Front.

Chapter 18

Figure 18.1 One‐line diagram of the New England 39‐bus system.

Figure 18.2 Illustration of Pareto front.

Chapter 19

Figure 19.1 Different voltage stability mitigations comparison.

Figure 19.2 Multi‐stage coordinated VAR planning framework.

Figure 19.3 Comparison of characteristics of different measures.

Figure 19.4 Illustration of a general DFIG.

Figure 19.5 Active and reactive power limits of stator (DFIG).

Figure 19.6 Impact of wind power penetration level on the voltage profile....

Figure 19.7 Impact of DR shifting on the peak load after a contingency.

Figure 19.8 Characteristics of load shedding cost functions.

Figure 19.9 Illustration of priorities assigned to different indices.

Figure 19.10 Computation flowchart of NSGA‐II.

Figure 19.11 An individual code structure.

Figure 19.12 Computation flowchart.

Figure 19.13 Single‐line diagram of the New England 39‐bus test system.

Figure 19.14 Candidate bus selection results for STATCOMs.

Figure 19.15 Pareto optimal solutions.

Figure 19.16 Post‐contingency voltage response (fault on line 16–24) at bus ...

Figure 19.17 Post‐contingency voltage response (fault on line 2–3) at bus 24...

Figure 19.18 Post‐contingency voltage response (fault on line 6–7) at bus 24...

Figure 19.19 Post‐contingency (N‐2) voltage response at bus 8.

Chapter 20

Figure 20.1 Impact of variations of uncertainties on the voltage dynamics....

Figure 20.2 Illustration of (a) acceptable sensitivity region (ASR) and (b) ...

Figure 20.3 Impact of wind power penetration level on the voltage dynamics....

Figure 20.4 Comparison between nominal Pareto front and Robust Pareto front....

Figure 20.5 Illustrations of WCSR.

Figure 20.6 Illustrations of different

L

p

‐norms.

Figure 20.7 Illustrations of FWCSR.

Figure 20.8 Illustrations of FWCSR.

Figure 20.9 New England 39‐bus test system.

Figure 20.10 Pareto optimal solutions of the proposed method.

Figure 20.11 Voltage trajectories comparison between method A and method B....

Figure 20.12 Pareto optimal solutions comparison of method A and method B.

Figure 20.13 Illustration of the un‐penalized and penalized voltage deviatio...

Guide

Cover Page

Series Page

Title Page

Copyright

About the Authors

Foreword

Preface

Table of Contents

Begin Reading

Index

IEEE Press Series on Power and Energy Systems

Wiley End User License Agreement

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IEEE Press445 Hoes LanePiscataway, NJ 08854

IEEE Press Editorial BoardSarah Spurgeon, Editor in Chief

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Behzad Razavi  

Jeffrey Reed  

Anjan Bose  

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Diomidis Spinellis  

James Duncan  

Hai Li  

Adam Drobot  

Amin Moeness  

Brian Johnson  

Tom Robertazzi  

Desineni Subbaram Naidu  

Ahmet Murat Tekalp  

Stability‐Constrained Optimization for Modern Power System Operation and Planning

Yan Xu

Nanyang Technological University

Singapore

Yuan Chi

Chongqing University

China

Heling Yuan

Nanyang Technological University

Singapore

Copyright © 2023 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging‐in‐Publication Data:Names: Xu, Yan (Associate professor), author.Title: Stability-constrained optimization for modern power system operation and planning / Yan Xu, Nanyang Technological University, Singapore, Yuan Chi, Chongqing University, China, Heling Yuan, Nanyang Technological University, Singapore.Description: Hoboken, NJ : Wiley-IEEE Press, [2023] | Series: IEEE Press series on power and energy systems | Includes index.Identifiers: LCCN 2023002456 (print) | LCCN 2023002457 (ebook) | ISBN 9781119848868 | ISBN 9781119848875 (adobe pdf) | ISBN 9781119848882 (epub)Subjects: LCSH: Electric power system stability.Classification: LCC TK1010 .X83 2023 (print) | LCC TK1010 (ebook) | DDC 621.319–dc23/eng/20230214LC record available at https://lccn.loc.gov/2023002456LC ebook record available at https://lccn.loc.gov/2023002457

Cover Design: WileyCover Image: © Leo Pakhomov/Shutterstock

About the Authors

Yan Xu received the B.E. and M.E. degrees from South China University of Technology, China, and the Ph.D. degree from University of Newcastle, Australia, in 2008, 2011, and 2013, respectively. He conducted postdoctoral research with the University of Sydney Postdoctoral Fellowship, and then joined Nanyang Technological University (NTU) with the Nanyang Assistant Professorship. He is now an Associate Professor at School of Electrical and Electronic Engineering and a Cluster Director at Energy Research Institute @ NTU (ERI@N). His research interests include power system stability, microgrid, and data analytics for smart grid applications. Dr Xu’s research in Singapore is funded by a range of funding agencies (including Singapore NRF, EMA, MOE, HDB, etc.) and industry partners (including Rolls-Royce Electrical, Singapore Power Group, Singtel, Infineon, EDF Lab, Lite-On, etc.). Many of his research outcomes have been practically applied/licensed to industry partners. Dr Xu has received 10 IEEE/IET paper contest and conference best paper awards, the 2022 IET Premium Award (Best Paper), the 2021 IEEE Transactions on Smart Grid Outstanding Paper Award, and the 2018 Applied Energy Highly Cited Paper Award. His professional service roles include Associate Editor for IEEE Trans. Smart Grid and IEEE Trans. Power Systems, Chairman of the IEEE Power & Energy Society (PES) Singapore Chapter (2021 to 2022) and the General Co-Chair of the 11th IEEE ISGT-Asia Conference, Nov. 2022.

Yuan Chi received the B.E. degree from Southeast University, Nanjing, China, in 2009, and the M.E. degree from Chongqing University, Chongqing, China, in 2012, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2021. From 2012 to 2017, he worked as an Electrical Engineer of Power System Planning consecutively with State Grid Chongqing Electric Power Research Institute and Chongqing Economic and Technological Research Institute. He is currently a Research Associate with Chongqing University. His research interests include planning, resilience, and voltage stability of power systems. Dr Chi’s research in China is funded by a range of funding agencies and industry partners, including Ministry of Finance of PRC, China Postdoctoral Science Foundation, State Grid Corporation of China, China Southern Power Grid, etc.

Heling Yuan received the B.E., M.Sc., and Ph.D. degrees from North China Electric Power University, Beijing, China, the University of Manchester, UK, and Nanyang Technological University, Singapore, in 2016, 2017, and 2022, respectively. She is currently a Research Fellow at Rolls-Royce @ NTU Corporate Lab, Singapore. Her research interests include modeling, optimization, stability analysis and control of power systems. Dr Yuan’s research in Singapore is funded by Singapore NRF, MOE and Rolls-Royce Electrical.

Foreword

The stability of a power system is defined by the IEEE as its ability “for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact”. Recent years have seen significant integration of renewable energy resources such as wind and solar power into power grids globally. Yet, such renewable energy‐based generators can significantly complicate the power system's dynamic behavior and introduce considerable operational uncertainties due to their power‐electronic converter interface and variable power output.

Practically, the stability of a power system can be maintained and enhanced through three general approaches: (i) accurately modeling and analyzing the power system's dynamic characteristics, then designing and deploying real‐time controllers to make the system well behaved under disturbances; (ii) dispatching the power system to a state that can better withstand the disturbances; (iii) reinforcing the power grid with fast‐responding resources such as FACTS (Flexible AC Transmission Systems) to support the power system dynamics to ride through the disturbances. While the first approach involves power system dynamics modeling, stability analysis, and controller design, the latter two require advanced optimization methods to optimally operate the power system and determine the optimal size and site of such resources for maximum cost effectiveness.

While most existing books are focused on the first approach to power system dynamics modeling, stability analysis, and controller design, very few address the latter two approaches that require advanced optimization. This book fills this gap by presenting a series of stability‐constrained optimization methodologies for power system operation and planning. Two major foci of the book are transient stability enhancement through optimal power system dispatch and operational control and voltage stability enhancement through optimally sizing and siting dynamic VAR resources in the power grid, respectively. The book presents a series of corresponding mathematical models and solution methods to achieve the these objectives.

This book is written by a dedicated research team with over 10 years of research experience in this field, and the presented methodologies are based on their original research outcomes and insights into these topics. With a balance between theory and practice, this book serves as a timely reference and guidebook for graduate students, researchers, and power system operation and planning engineers in this field.

David John Hill, PhD

Professor of Power and Energy Systems, Monash University, Australia

Emeritus Professor, University of Sydney, and University of Hong Kong

Fellow of IEEE, IFAC, SIAM,

Australian Academy of Science,

Australian Academy of Technological Sciences and Engineering

Hong Kong Academy of Engineering Sciences

Foreign Member of Royal Swedish Academy of Engineering Science

Preface

The electrical power system is essential to a modern society, and its stability is a fundamental requirement during both online operation and offline planning studies. In general, the stability of a power system refers to its ability to regain a state of operating equilibrium after experiencing a physical disturbance, such as a short‐circuit fault. In practice, the stability of the power system mainly depends on both its inherent dynamic characteristics, i.e. how the system responds to disturbances, and its steady‐state operating conditions, i.e. how the system is dispatched.

In recent years, renewable energy sources such as solar photovoltaic and wind power have rapidly penetrated modern power systems, which have inherently stochastic and intermittent power output and are connected to the grid through power electronic converters. Consequently, both the static and dynamic behaviors of the power system have become much more complex, creating a series of challenges for maintaining system stability. These challenges include long‐distance power transmission from renewable power stations to load centers, reduced synchronous inertia in the power systems, complex dynamics of power electronic interfaced devices, lack of reactive power resources, and fast fluctuation of magnitude and direction of power flows through the transmission network. Recent large‐scale blackout events, e.g. the September 2016 South Australia blackout and the August 2019 UK blackout, have clearly demonstrated the adverse impact of these challenges on power system stability.

This book focuses on two power system stability problems, namely transient stability and voltage stability. Transient stability, also known as large‐disturbance rotor angle stability, is the most stringent requirement for a power system because instability can develop rapidly within several cycles after a disturbance. Voltage stability is becoming increasingly critical since poor dynamic voltage performance of the power system could lead to the failure of wind and solar power generators riding through disturbances. This book presents a series of optimization methodologies that we have originally proposed to (i) optimally dispatch the power system to an operating state that can maintain transient stability in the event of a large disturbance and (ii) optimally allocate dynamic VAR resources, including STATCOM and SVC, in the power grid to reinforce the grid's capability to counteract voltage instability.

The book consists of 20 chapters, which are organized into three parts:

Part I

(

Chapters 1

–

3

) provides an overall introduction to power system stability, including fundamental concepts, definitions, mathematical models, metrics, and analysis methods, as well as presents a review of major large‐scale blackout events in recent years.

Part II

(

Chapters 4

–

12

) focuses on transient stability‐constrained (TSC) power system dispatch and operational control. The problems are generally modeled as TSC‐optimal power flow (TSCOPF) and TSC‐unit commitment (TSCUC), and a series of mathematical formulations are presented in this part. The formulations cover deterministic, stochastic, and robust optimization models with uncertainties arising from dynamic load components and renewable power generation, and two‐stage optimization models for coordinating preventive and corrective control actions. To solve these problems, a set of computationally efficient methods have been correspondingly presented, which are based on quantitative transient stability assessment, trajectory sensitivity analysis, linear transient stability constraint construction, machine learning‐based stability constraint extraction, hybrid computation, and decomposition‐based solution frameworks. The proposed methodologies are numerically tested on IEEE benchmark testing systems, showing their effectiveness with simulation results.

Part III

(

Chapters 13

–

20

) focuses on voltage stability enhancement through dynamic VAR resource allocation in the power system. This part first introduces the general framework and mathematical models for optimal VAR resource planning in the power grid, then presents several quantitative voltage stability indices, followed by an overview of the fundamentals and the mathematical models of the dynamic VAR resources. To reduce the size of the optimization model, two methods for candidate bus selection are presented afterward. After that, a series of different mathematical formulations and solution methods for optimal dynamic VAR resource allocation are presented, including multi‐objective optimization model, retirement‐driven planning model, multi‐stage coordinated planning model, and many‐objective robust optimization model. The proposed methodologies are numerically demonstrated on IEEE benchmark testing systems with simulation results.

The book is targeted at scholars, researchers, and postgraduate students who are seeking optimization methodologies for power system stability enhancement. Additionally, it provides practical solutions to operational dispatch and network reinforcement planning for power system operators, planners, and optimization algorithm developers in the power industry.

We would like to express our sincere gratitude to the funding supports for the research presented in this book, including the Hong Kong Research Grant Council (RGC) GRF Grant, the Australia Research Council (ARC) Linkage Project, the University of Newcastle Faculty Strategic Pilot Grant, the University of Sydney Postdoctoral Fellowship (USYD‐PF), the Nanyang Assistant Professorship (NAP) and PhD scholarships from Nanyang Technological University, and the Singapore Ministry of Education (MOE) Tier‐1 Grants.

Yan Xu

Nanyang Technological University

Singapore

Yuan Chi

Chongqing University

China

Heling Yuan

Nanyang Technological University

Singapore

Part IPower System Stability Preliminaries

List of Acronyms

ACC

 alternative current control

CCT

 critical clearing time

CIG

 converter interfaced generation

CMs

 critical machines

COI

 center of inertia

DAE

 differential algebraic equations

DFIG

 doubly‐fed induction generators

EEA

 energy emergency alert

EEAC

 extended equal‐area criterion

ERCOT

 electric reliability council of texas

FE

 first energy

IGE

 induction generator effect

LPB‐G

 low pressure turbine B to generator

LVRT

 low voltage ride through

MISO

 midwest independent system operator

NMs

 non‐critical machines

PLL

 phase‐locked loop

SA

 South Australia

SIME

 single machine equivalence

SOL

 system operational limits

SSR

 subsynchronous resonance

T&D

 transmission and distribution

TDS

 time domain simulation

TEF

 transient energy function

TSA

 transient stability assessment

TSC

 transient stability control

TSIs

 transient stability indexes

TSO

 transmission system operator

UCTE

 Union for the Co‐ordination of Transmission of Electricity

1Power System Stability: Definition, Classification, and Phenomenon

1.1 Introduction

An electrical power system is a fundamental infrastructure of a society. As a large‐scale time‐varying dynamic system, maintaining its stability is a basic and essential requirement during its operation and planning decision‐making process. In general, the stability of a power system refers to its ability to regain a state of operating equilibrium after being subjected to a physical disturbance (such as a short‐circuit fault) [1]. In practice, the stability of the power system depends on both its dynamic characteristics, i.e. how the system would behave in response to disturbances, and its steady‐state operating conditions, i.e. how the power system is dispatched.

In recent years, modern power systems have started to integrate high shares of renewable energy sources, such as solar photovoltaic and wind power, which are inherently stochastic and intermittent in their power outputs and are interfaced with the power grid through power electronic converters. While these renewable energy‐based converter interfaced generators (CIGs) are environmentally beneficial, they significantly complicate the power grid's static and dynamic characteristics. As a result, the dynamic behaviors of the power system become much more complex, which introduces a series of challenges to the control, operation, and planning for maintaining system stability.

In a nutshell, this chapter gives a brief introduction to the modern power system stability, including its definition, classification, and phenomenon.

1.2 Definition

In this book, the definition of power system stability given in [1] is adopted. Conforming to definitions from system theory, the definition is based in physics, thus easily understood and readily applied by power system engineering practitioners:

Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.

The definition is applied to interconnected power systems at large while also concerning the stability of a single generator or a group of generators. When a power system is subjected to a disturbance, its stability depends on its initial operating point and the nature of the disturbance. In practice, a power system can suffer from various disturbances regardless of small or large. A small disturbance can be a continuous load change that will not alter the system topologies. A large disturbance is one which may result in structural changes due to the isolation of the faults, such as a short circuit on the transmission line or losses of a large generator. After small or large disturbances, the system must be capable of riding through the disturbances and returning to a viable equilibrium.

It should be noted that, at an initial operating point, a power system may be stable for a given large disturbance but become unstable for another. It is in general not practical or economical to ensure stability against all possible disturbances. It is more reasonable to select contingencies that show a high probability of occurrence and criticality based on historical data and system topology information. Large‐disturbance stability is generally validated under a set of specified disturbances. A stable equilibrium has a finite region of attraction; the larger the region, the more robust the system is against large disturbances. However, it should be noted that the region of attraction changes with the operating point of the power system. Though the power system continually fluctuates with small magnitudes, it is usually acceptable to assume that the system is initially in a steady‐state operating point when assessing its stability after being subjected to a specific disturbance.

1.3 Classification

Stability is a condition of equilibrium between opposing forces undergoing continuous imbalance, which results in different forms of instability subject to the network topology, system operating point, and the type of disturbance. The different forms of instabilities need to be properly analyzed. To facilitate the analysis of stability problems which are in nature of high dimension and complexity, it is necessary to firstly classify the stability into appropriate categories.

Figure 1.1 Classification of power system stability.

Figure 1.1 shows the overall picture of the power system stability classification with its categories and subcategories. Traditionally, the power system stability was classified into three categories: rotor angle stability, voltage stability, and frequency stability [1]. The power system stability primarily dealt with fairly slow electromechanical phenomena, which are typically caused by synchronous machines and induction machines. Given the increased penetration of CIGs in modern power systems and their substantial impacts on system dynamic behavior, two new stability categories have been added [2], i.e. resonance stability and converter‐driven stability, to deal with faster dynamics within electromagnetic time scales.

The focus of this book is on the rotor angle stability and voltage stability, which are the two most stringent stability requirements for the power system operation and planning. Rotor angle stability and voltage stability can be approached by optimization‐based operating point dispatches and network reinforcement (dynamic reactive power device deployment).

1.4 Rotor Angle Stability

Rotor angle stability is the ability of synchronous generators to remain in synchronism after being subjected to a disturbance. It depends on the ability of each synchronous generator to maintain equilibrium between electromagnetic torque and mechanical torque. Instability will occur when the rotor angles of some generators increase continuously with regard to other generators. Namely, the generators lose synchronism with others. The loss of synchronism can occur between one machine and the rest of the system, or between groups of machines, with synchronism maintained within each group after separating from each other. The possible outcome of the instability is generator tripping and/or separation of the power systems.

As presented in Figure 1.1, the rotor angle stability can be divided into large‐disturbance rotor angle stability (also called transient stability) and small‐disturbance rotor angle stability, based on the severity of the disturbance.

1.4.1 Large‐Disturbance Rotor Angle Stability

For transient stability, it is always relative to a severe disturbance, such as a short circuit on a transmission line, which will result in large excursions of rotor angles and is involved by the nonlinear power–angle relationship. Transient stability depends on both the initial operating point and the severity of the disturbance. Instability is usually in the form of aperiodic angular separation due to insufficient synchronizing torque, indicating first‐swing instability. Figures 1.2 and 1.3 indicate the simulated post‐disturbance rotor angle trajectories of a real large power grid for a stable case and an unstable case, respectively. From the two figures, the stable case corresponds to keeping the synchronism of all the generators, while the unstable case corresponds to the loss of synchronism of some generators after the disturbance.

Figure 1.2 Simulated rotor angles of a transient stable case.

Source: Xu [3].

Figure 1.3 Simulated rotor angles of a transient unstable case.

Source: Xu [3].

1.4.2 Small‐Disturbance Rotor Angle Stability

Small‐disturbance rotor angle stability refers to the ability of the power system to maintain synchronism under small disturbances. The disturbances are considered to be sufficiently small that linearization of system equations is permissible for purposes of analysis, e.g. the sudden variation of the load demands and/or the change of generation output. Small‐disturbance rotor angle stability usually depends on the initial operating point. Instability is often in two forms: (i) non‐oscillatory increment of rotor angle due to lack of synchronizing torque; or (ii) rotor oscillations of amplitude due to lack of damping torque. For modern power systems, small‐disturbance rotor angle stability is usually associated with insufficient damping torque. The illustration of the small‐disturbance stability problem due to the lack of damping torque is presented in Figure 1.4.

Figure 1.4 Active power oscillation due to the small‐disturbance instability.

Source: Xu [3].

1.5 Voltage Stability

Voltage stability is the ability of the power system to maintain steady voltages at all buses after being subjected to a disturbance. It depends on the ability to maintain equilibrium between load demand and load supply from the power system. Instability will occur in the form of a successive fall or rise of voltage in some buses. A possible outcome of instability is loss of load in an area or transmission lines and other elements tripping by their protective systems, leading to cascading outages. It should be noted that a progressive drop in bus voltage may be associated with rotor angle instability.

The driving force for voltage instability is usually the loads. A run‐down situation causing voltage instability occurs when load dynamics attempt to restore power consumption beyond the capability of the transmission network and the connected generation.

As in the case of rotor angle stability, classifying voltage stability into large‐disturbance voltage stability and small‐disturbance voltage stability is applicable.

1.5.1 Large‐Disturbance Voltage Stability

Large‐disturbance voltage stability refers to the system's ability to maintain steady voltages following large disturbances such as system faults, loss of generation, or circuit contingencies. This ability is determined by the system and load characteristics, and the interactions of both continuous and discrete controls and protections.

The simulated post‐disturbance voltage trajectories of the New England 10‐machine 39‐bus system are indicated in Figures 1.5 and 1.6 for a stable and unstable case, respectively. In the stable case, all voltage trajectories fluctuate and return to the nominal level finally, while in the unstable case, the voltage trajectories fluctuated dramatically without recovery.

Figure 1.5 Simulated voltage trajectories of the large‐disturbance voltage stable case.

Figure 1.6 Simulated voltage trajectories of the large‐disturbance voltage unstable case.

Figure 1.7 Simulated power–voltage (PV) curve.

Source: Xu [3].

1.5.2 Small‐Disturbance Voltage Stability

Small‐disturbance voltage stability refers to the system's ability to maintain steady voltages when subjected to small perturbations, such as incremental changes in system load. Under this case, the system voltage will drop continuously with the load increment until a critical point is reached. After the critical point, the system voltage will drop dramatically and finally collapse.

A simulated power–voltage curve is presented in Figure 1.7, where the loading parameter means the load‐increasing rate. When it increases to 3.7115 times the initial operating point, voltage instability will occur.

1.6 Frequency Stability

Frequency stability refers to the ability of a power system to maintain steady frequency following a severe system upset resulting in a significant imbalance between generation and load. It depends on the ability to maintain/restore equilibrium between system generation and load, with minimum unintentional loss of load. The instability that may result occurs in the form of sustained frequency swings, leading to the tripping of generating units and/or loads.

Figures 1.8 and 1.9 indicate the simulated post‐disturbance frequency trajectory for a stable and unstable case.

Figure 1.8 Simulated post‐disturbance frequency trajectory of a frequency stable case.

Source: Xu [3].

Figure 1.9 Simulated post‐disturbance frequency trajectory of a frequency unstable case.

Source: Xu [3].

1.7 Resonance Stability

Generally, resonance instability may occur when the oscillatory magnitude of voltage/current/torque exceeds specified thresholds due to insufficient dissipation of energy in the flow path. The term resonance stability encompasses subsynchronous resonance (SSR) and can be indicated in two forms: (i) due to a resonance between series compensation and the mechanical torsional frequencies of the turbine‐generator shaft; (ii) due to a resonance between series compensation and the electrical characteristics of the generator. Thus, the resonance stability has been divided into two categories shown in Figure 1.1.

1.7.1 Torsional Resonance

Torsional resonance is the SSR due to torsional interactions between the series compensated line(s) and a turbine‐generator mechanical shaft, particularly as it pertains to conventional synchronous generation.

Figure 1.10a indicates the torsional torque in low‐pressure turbine B to generator 1 (LPB‐G1) shaft section with 70% and 30% compensation levels. The dark blue shade shows the torsional torque with a 70% compensation level, and the gray shade shows the torque with a 30% compensation level. From the figure, the torque is growing slowly for the dark blue shade, which means that the 70% compensation level leads the system unstable. While for the gray shade, the torque is decaying. It represents the system becomes stable with the 30% compensation level. It can be found that the higher the compensation level, the weaker the damping.

1.7.2 Electrical Resonance

As the variable‐speed DFIG generator is an induction generator connected to the grid, the electrical resonance between the generator and series compensation may exist, which would be highly susceptible to IGE‐self‐excitation type SSR. In this case, the self‐excitation type SSR occurs when the series capacitor forms a resonant circuit, at subsynchronous frequencies, with the effective inductance of the induction generator.

The phenomena can be realized by the IGE simulation performed with the torsional system disabled. Figure 1.10b indicates the dynamic responses with various wind speeds and a constant 75% compensation level. From the figure, it can be found that the higher the wind speed, the better the SSR damping for the doubly‐fed induction generator (DFIG) system.

Figure 1.10 Phenomenon of resonance stability. (a) Torsional torque in LPB‐G shaft section.

Source: Adrees and Milanović [4]/(© [2014] IEEE.

(b) Dynamic response of Pe under different wind speeds. Compensation level: 75%.

Source: Fan et al. [5]/© [2010] IEEE.

1.8 Converter‐Driven Stability

Since the timescale referring to the controls of CIGs is wide, the cross‐coupling occurs with both the electromechanical dynamics of machines and the electromagnetic transients of the network, which may lead to unstable power system oscillations within a wide frequency range. Hence, slow‐interaction converter‐driven stability (less than 10 Hz) and fast‐interaction converter‐driven stability (tens to hundreds of Hz to kHz) are categorized.

1.8.1 Fast‐Interaction Converter‐Driven Stability

Fast‐interaction converter‐driven stability is driven by fast dynamic interactions of power electronic‐based control systems, such as CIGs, HVDC, and FACTS, which are fast response elements. Instabilities produced by fast converter interactions may indicate various forms. For example, interactions of the fast inner‐current loops of CIG with passive system components may cause high‐frequency oscillations.

Figure 1.11a shows the harmonic instability phenomenon due to the interactions between the inner alternative current control (ACC) loops of the three paralleled VSCs. The bandwidth of the ACC loop is increased from fs/20 to fs/15 at the time instant of Ti. It is obvious to see that the three paralleled VSCs become unstable with the bandwidth of fs/15.

1.8.2 Slow‐Interaction Converter‐Driven Stability

Slow‐interaction converter‐driven stability is driven by slow dynamic interactions of power electronic‐based devices' control systems, such as the electromechanical dynamics of synchronous generators. Like voltage stability, a weak system can be the primary cause of instability. However, these two types of stability are fundamentally different since voltage instability is driven by loads, while converter‐driven instability is involved by power electronic converters.

Figure 1.11b shows low‐frequency oscillation and subsynchronous frequency oscillation phenomena for four phase‐locked loop (PLL) parameters. For PLL1, there is a 4.5 Hz undamped oscillation, which is defined as low‐frequency oscillation. For PLL2, 30 Hz oscillations are dominant. The low‐frequency mode can also be found in the initial few cycles. For PLL3, both modes can be damped. For PLL4, the 40 Hz oscillation mode is dominant and the system becomes unstable.

This book mainly focuses on transient stability and large‐disturbance voltage stability, as marked in the dashed square in Figure 1.1. Besides, transient stability in this book refers to first‐swing stability. Multi‐swing stability is not considered. Thus, the transient stability in the subsequent chapters all refers to the first‐swing stability.

Figure 1.11 Phenomenon of converter‐driven stability. (a) Measured waveforms for the harmonics instability.

Source: Wang and F. Blaabjerg [6]/(© [2019] IEEE.

(b) Response of VPCC for four sets of PLL parameters, from upper to bottom: PLL1, PLL2, PLL3, and PLL4.

Source: Fan and Miao [7]/© [2018] IEEE.

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